text
sequencelengths 2
2.54k
| id
stringlengths 9
16
|
|---|---|
[
[
"Open loop amplitudes and causality to all orders and powers from the\n loop-tree duality"
],
[
"Abstract Multiloop scattering amplitudes describing the quantum fluctuations at high-energy scattering processes are the main bottleneck in perturbative quantum field theory.",
"The loop-tree duality is a novel method aimed at overcoming this bottleneck by opening the loop amplitudes into trees and combining them at integrand level with the real-emission matrix elements.",
"In this Letter, we generalize the loop-tree duality to all orders in the perturbative expansion by using the complex Lorentz-covariant prescription of the original one-loop formulation.",
"We introduce a series of mutiloop topologies with arbitrary internal configurations and derive very compact and factorizable expressions of their open-to-trees representation in the loop-tree duality formalism.",
"Furthermore, these expressions are entirely independent at integrand level of the initial assignments of momentum flows in the Feynman representation and remarkably free of noncausal singularities.",
"These properties, that we conjecture to hold to other topologies at all orders, provide integrand representations of scattering amplitudes that exhibit manifest causal singular structures and better numerical stability than in other representations."
],
[
"Introduction",
"Precision modeling of fundamental interactions relies mostly on perturbative quantum field theory.",
"Quantum fluctuations in perturbative quantum field theory are encoded by Feynman diagrams with closed loop circuits.",
"These loop diagrams are the main bottleneck to achieve higher perturbative orders and therefore more precise theoretical predictions for high-energy colliders [1], [2].",
"Whereas loop integrals are defined in the Minkowski space of the loop four-momenta, the loop-tree duality (LTD) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] exploits the Cauchy residue theorem to reduce the dimensions of the integration domain by one unit in each loop.",
"In the most general version of LTD the loop momentum component that is integrated out is arbitrary [3], [4].",
"In numerical implementations [7], [8], [10], [11], [13], [14], [16], [21], [22] and asymptotic expansions [12], [17], it is convenient to select the energy component because the remaining integration domain, the loop three-momenta, is Euclidean.",
"LTD opens any loop diagram to a forest (a sum) of nondisjoint trees by introducing as many on-shell conditions on the internal loop propagators as the number of loops, and is realized by modifying the usual infinitesimal complex prescription of the Feynman propagators.",
"The new propagators with modified prescription are called dual propagators.",
"LTD at higher orders proceeds iteratively, or in the words of Feynman [23], [24], by opening the loops in succession.",
"While the position of the poles of Feynman propagators in the complex plane is well defined, i.e., the positive (negative) energy modes feature a negative (positive) imaginary component due to the momentum independent $+\\imath 0$ imaginary prescription, the dual prescription of dual propagators is momentum dependent.",
"Therefore, after applying LTD to the first loop, the position of the poles in the complex plane of the subsequent loop momenta is modified.",
"The solution found in Refs.",
"[4], [5] was to reshuffle the imaginary components of the dual propagators by using a general identity that relates dual with Feynman propagators in such a way that propagators entering the second and successive applications of LTD are Feynman propagators only.",
"This procedure requires to reverse the momentum flow of a few subsets of propagators in order to keep a coherent momentum flow in each LTD round.",
"Figure: Maximal loop topology (left) and the corresponding open dual representation (right).An arbitrary number of external legs is attached to each loop line.All the propagators in the set ii on the rhs are off shell, whilethe dashed line represents the on-shell cut over the other n-1n-1 sets: one on-shell propagator in eachset and an implicit sum over all possible on-shell configurations.Bars indicate a reversal of the momentum flow.Recent papers have proposed alternative dual representations [19], [20], [21], [22].",
"In Refs.",
"[19], [20], an average of all the possible momentum flows is proposed, which requires a detailed calculation of symmetry factors.",
"We show in this Letter that this average is unnecessary.",
"In Refs.",
"[21], [22], the Cauchy residue theorem is applied iteratively by keeping track of the actual position of the poles in the complex plane.",
"The procedure requires to close the Cauchy contours at infinity from either below or above the real axis, in order to cancel the dependence on the position of the poles.",
"In this Letter, we follow a new strategy to generalize LTD to all orders, and with the original Lorentz-covariant prescription [3], [4].",
"As in Refs.",
"[4], [5], [13], [14], we reverse sets of internal momenta whenever it is necessary to keep a coherent momentum flow, and we close the Cauchy contours always in the lower complex half-plane.",
"Causality [6], [15], [25], [26], [27], [28], [29], [30] is also used as a powerful guide to select which kind of dual contributions are endorsed, and then construct suitable Ansätze that are proven by induction.",
"This procedure allows us to obtain explicit and very compact analytic expressions of the LTD representation for a series of loop topologies to all orders and arbitrary internal configurations."
],
[
"Loop-tree duality to all orders and powers",
"The internal propagators of any multiloop integral or scattering amplitude can be classified into different sets or loop lines, each set collecting all the propagators that depend on the same single loop momentum or a linear combination of them.",
"To simplify the notation, $s$ labels the set of all the internal propagators $i_s \\in s$ carrying momenta of the form $q_{i_s}=\\ell _s+k_{i_s}$ , where $\\ell _s$ is the loop momentum identifying this set, and where $k_{i_s}$ is a linear combination of external momenta $\\left\\lbrace p_j\\right\\rbrace _N$ .",
"Note that $\\ell _s$ may be a linear combination of loop momenta, so long as it is the same fixed combination in all the elements in the set $s$ .",
"The usual Feynman propagator of one single internal particle is $G_F(q_{i_s}) = \\frac{1}{q_{i_s,0}^2 - \\left(q_{i_s,0}^{(+)}\\right)^2}~,$ where $q_{i_s,0}^{(+)} = \\sqrt{\\mathbf {q}_{i_s}^2+m_{i_s}^2-\\imath 0}~,$ with $q_{i_s,0}$ and $\\mathbf {q}_{i_s}$ the time and spatial components of the momentum $q_{i_s}$ , respectively, $m_{i_s}$ its mass, and $\\imath 0$ the usual Feynman's infinitesimal imaginary prescription.",
"We extend this definition to encode in a compact way the product of the Feynman propagators of one set or the union of several sets: $G_F(1,\\ldots , n) = \\prod _{i\\in 1\\cup \\cdots \\cup n} \\left( G_F(q_i) \\right)^{a_i}~.$ Here, we contemplate the general case where the Feynman propagators are raised to arbitrary powers.",
"Still, the powers $a_i$ will appear only implicitly in the following.",
"A typical $L$ -loop scattering amplitude is expressed as ${\\cal A}^{(L)}_N(1,\\dots ,n) = \\int _{\\ell _1,\\ldots , \\ell _L} {\\cal N}(\\lbrace \\ell _i \\rbrace _L, \\lbrace p_j\\rbrace _N) \\, G_F(1, \\ldots , n)$ in the Feynman representation, i.e.",
"as an integral in the Minkowski space of the $L$ -loop momenta over the product of Feynman propagators and the numerator ${\\cal N}(\\lbrace \\ell _i \\rbrace _L, \\lbrace p_j\\rbrace _N)$ , which is given by the Feynman rules of the theory.",
"The integration measure reads $\\int _{\\ell _i} = -\\imath \\, \\mu ^{4-d} \\int d^d\\ell _i/(2\\pi )^d$ in dimensional regularization [31], [32], with $d$ the number of space-time dimensions.",
"Beyond one loop, any loop subtopology involves at least two loop lines that depend on the same loop momentum.",
"We define the dual function that accounts for the sum of residues in the complex plane of the common loop momentum as $G_D(s; t) = -2\\pi \\imath \\sum _{i_s \\in s} {\\rm Res}\\left(G_F(s,t), {\\rm Im}(\\eta \\cdot q_{i_s}) < 0\\right)~,$ where $G_F(s,t)$ represents the product of the Feynman propagators that belong to the two sets $s$ and $t$ .",
"Each of the Feynman propagators can be raised to an arbitrary power.",
"Notice that in Eq.",
"(REF ) only the propagators that belong to the set $s$ are set consecutively on shell.",
"The Cauchy contour is always closed from below the real axis, ${\\rm Im}(\\eta \\cdot q_{i_s}) < 0$ .",
"The vector $\\eta $ is futurelike and was introduced in the original formulation of LTD [3] to regularize the dual propagators in a Lorentz-covariant form.",
"For single power propagators, $s=t$ and $\\eta =(1,{\\bf 0})$ , Eq.",
"(REF ) provides the customary dual function at one loop with the energy component integrated out $G_D(s) = - \\sum _{i_s\\in s} \\tilde{\\delta }\\left(q_{i_s}\\right) \\prod _{\\stackrel{j_s\\ne i_s}{j_s\\in s }}\\frac{1}{(q_{i_s,0}^{(+)}+ k_{j_s i_s,0})^2 - (q_{j_s,0}^{(+)})^2}~,$ with $k_{j_s i_s} = q_{j_s}-q_{i_s}$ , and $\\tilde{\\delta }\\left(q_{i_s}\\right)= 2\\pi \\imath \\, \\theta (q_{i_s,0}) \\delta (q_{i_s}^2-m_{i_s}^2)$ selecting the on-shell positive energy mode, $q_{i_s,0}>0$ .",
"If some of the Feynman propagators are raised to multiple powers, then Eq.",
"(REF ) leads to heavier expressions [5] but the location of the poles in the complex plane is the same as in the single power case.",
"Then, we construct the dual function of nested residues involving several sets of momenta $&& G_D(1, \\ldots , r; n) = -2\\pi \\imath \\nonumber \\\\ &&\\times \\sum _{i_r \\in r} {\\rm Res}\\left(G_D(1, \\dots , r-1; r, n), {\\rm Im}(\\eta \\cdot q_{i_r})<0\\right)~.$ In the rhs of Eqs.",
"(REF ) and (REF ), we can introduce numerators or replace the Feynman propagators by the integrand of Eq.",
"(REF ) to define the corresponding unintegrated open dual amplitudes ${\\cal A}^{(L)}_D(1,\\ldots ,r; n)$ .",
"An example of dual amplitudes at two loops was presented in Ref. [13].",
"In the next sections, we will derive the LTD representation of the multiloop scattering amplitude in Eq.",
"(REF ) and we will present explicit expressions for several benchmark topologies to all orders.",
"The notation introduced above allows us to express the LTD representations in a very compact way, since it only requires to label and specify the overall structure of the loop sets, regardless of their internal and specific configuration."
],
[
"Maximal loop topology",
"The maximal loop topology (MLT), see Fig.",
"REF , is defined by $L$ -loop topologies with $n=L+1$ sets of propagators, where the momenta of the propagators belonging to the first $L$ sets depend on one single loop momentum, $q_{i_s} = \\ell _s + k_{i_s}$ with $s\\in \\lbrace 1,\\ldots ,L\\rbrace $ , and the momenta of the extra set, denoted by $n$ , are a linear combination of all the loop momenta, $q_{i_n} = -\\sum _{s=1}^{L} \\ell _s + k_{i_n}$ .",
"The minus sign in front of the sum is imposed by momentum conservation.",
"The momenta $k_{i_s}$ and $k_{i_n}$ are linear combinations of external momenta.",
"At two loops ($n=3$ ), this is the only possible topology, and therefore sufficient to describe any two-loop scattering amplitude.",
"The LTD representation of the multiloop MLT amplitude, starting at two loops, is extremely simple and symmetric $&& {\\cal A}^{(L)}_{\\rm MLT}(1,\\dots ,n) \\nonumber \\\\ &&= \\int _{\\ell _1,\\ldots , \\ell _L}\\sum _{i=1}^{n} {\\cal A}^{(L)}_D(1, \\ldots , i-1, \\overline{i+1}, \\ldots , \\overline{n} ; i)~,$ with ${\\cal A}^{(L)}_D(\\overline{2}, \\ldots , \\overline{n}; 1)$ and ${\\cal A}^{(L)}_D(1, \\ldots , n-1; n)$ as the first and the last elements of the sum, respectively.",
"The bar in $\\overline{s}$ indicates that the momentum flow of the set $s$ is reversed ($q_{i_s}\\rightarrow -q_{i_s}$ ), which is equivalent to selecting the on-shell modes with negative energy of the original momentum flow.",
"The compact expression in Eq.",
"(REF ) was obtained by first evaluating the nested residues, Eq.",
"(REF ), of several representative multiloop integrals.",
"The derived expressions were then used to formulate an Ansatz to all orders that was proven by induction.",
"It is noteworthy that there is no dependence in this expression on the position of the poles in the complex plane.",
"In each term of the sum in the integrand of Eq.",
"(REF ) there is one set $i$ with all its propagators off shell, and there is one on-shell propagator in each of the other $n-1$ sets.",
"This is the necessary condition to open the multiloop amplitude into nondisjoint trees.",
"Note also that there is an implicit sum over all possible on-shell configurations of the $n-1$ sets.",
"The LTD representation in Eq.",
"(REF ) is displayed graphically in Fig.",
"REF , and represents the basic building block entering other topologies.",
"The causal behavior of Eq.",
"(REF ) is also clear and manifest.",
"The dual representation in Eq.",
"(REF ) becomes singular when one or more off-shell propagators eventually become on shell and generate a disjoint tree dual subamplitude.",
"If these propagators belong to a set where there is already one on-shell propagator then the discussion reduces to the one-loop case [6].",
"We do not comment further on this case.",
"The interesting case occurs when the propagator becoming singular belongs to the set with all the propagators off shell [15].",
"For example, the first element of the sum in Eq.",
"(REF ) features all the propagators in the set 1 off shell.",
"One of those propagators might become on shell, and there are two potential singular solutions, one with positive energy and another with negative energy, depending on the magnitude and direction of the external momenta [6], [15].",
"The solution with negative energy represents a singular configuration where there is at least one on-shell propagator in each set.",
"Therefore, the amplitude splits into two disjoint trees, with all the momenta over the causal on-shell cut pointing to the same direction.",
"Abusing notation: ${\\cal A}_D^{(L)} (\\overline{2}, \\ldots , \\overline{n}; 1) \\stackrel{1~\\rm {on-shell}}{\\rightarrow }{\\cal A}_D^{(L)} (\\overline{1}, \\overline{2}, \\ldots , \\overline{n})~.$ The on-shell singular solution with positive energy, however, is locally entangled with the next term in Eq.",
"(REF ) such that the full LTD representation remains nonsingular in this configuration: $&&{\\cal A}_D^{(L)} (\\overline{2}, \\overline{3}, \\ldots , \\overline{n}; 1) + {\\cal A}_D^{(L)} (1, \\overline{3}, \\ldots , \\overline{n}; 2) \\\\&& \\stackrel{(1,2)~\\rm {on-shell}}{\\rightarrow } {\\cal A}_D^{(L)} (1, \\overline{2}, \\overline{3}, \\ldots , \\overline{n}) - {\\cal A}_D^{(L)} (1, \\overline{2}, \\overline{3}, \\ldots , \\overline{n})~.\\nonumber $ These local cancellations also occur with multiple power propagators.",
"They are the known dual cancellations of unphysical or noncausal singularities [6], [13], [14], [15] and their cancellation is essential to support that the remaining causal and anomalous thresholds as well as infrared singularities are restricted to a compact region of the loop three-momenta.",
"Causality determines that the only surviving singularities fall on ellipsoid surfaces in the loop three-momenta space [7], [8], [22], that collapse to finite segments for massless particles leading to infrared singularities.",
"These causal singularities are bounded by the magnitude of the external momenta, thus enabling the simultaneous generation with the tree contributions describing emission of extra radiation through suitable momentum mappings, as defined in four-dimensional unsubstraction (FDU) [9], [10], [11].",
"Another potential causal singularity occurs from the last term in Eq.",
"(REF ) when all the on-shell momenta are aligned in the opposite direction over the causal on-shell cut, ${\\cal A}_D^{(L)} (1, \\ldots , n-1; n) \\stackrel{n~\\rm {on-shell}}{\\rightarrow } {\\cal A}_D^{(L)} (1, \\ldots , n)$ .",
"It is also interesting to note the remarkable structure that the LTD representation exhibits when expressed in terms of dual propagators.",
"For example, the scalar MLT integral with only one single propagator in each set, e.g.",
"the sunrise diagram at two loops, reduces to the extremely compact expression ${\\cal A}^{(L)}_{\\rm MLT}(1,\\ldots ,n) = - \\int _{\\vec{\\ell }_1, \\ldots , \\vec{\\ell }_L} \\frac{1}{2q_{n,0}^{(+)}}\\left( \\frac{1}{\\lambda ^-_{1,n}} + \\frac{1}{\\lambda ^+_{1,n}} \\right)~,$ where $\\lambda ^\\pm _{1,n} = \\sum _{i=1}^{n} q_{i,0}^{(+)} \\pm k_{0,n}$ , with $k_n = \\sum _{i=1}^{n} q_i $ , and $\\int _{\\vec{\\ell }_s} = - \\mu ^{4-d} \\, \\int d^{d-1} \\ell _s / (2\\pi )^{d-1} / (2 q_{s,0}^{(+)})$ .",
"The most notable property of this expression is that it is explicitly free of unphysical singularities, and the causal singularities occur, as expected, when either $\\lambda ^+_{1,n}$ or $\\lambda ^-_{1,n}$ vanishes, depending on the sign of the energy component of $k_n$ , in the loop three-momenta region where the on-shell energies are bounded, $q_{i,0}^{(+)}< |k_{0,n}|$ .",
"This property also holds for powered propagators, nonscalar integrals, and more than one propagator in each set.",
"Furthermore, Eq.",
"(REF ) is independent of the initial momentum flows in the Feynman representation.",
"Figure: Next-to-maximal loop topology (left) and its convoluted dual representation (right).Each MLT subtopology opens according to Eq. ().",
"Only the on-shell cut ofthe last MLT-like subtopology with reversed momentum flow is shown."
],
[
"Next-to-Maximal loop topology",
"The next multiloop topology in complexity, see Fig.",
"REF , contains one extra set of momenta, denoted by 12, that depends on the sum of two loop momenta, $q_{i_{12}} = - \\ell _1 - \\ell _2 + k_{i_{12}}$ .",
"We call it next-to-maximal loop topology (NMLT).",
"This topology appears for the first time at three loops, i.e.",
"$n+1$ sets with $L=n-1$ and $n\\ge 4$ , and its LTD representation is given by the compact and factorized expression $&& {\\cal A}^{(L)}_{\\rm NMLT}(1,\\dots ,n,12)= {\\cal A}^{(2)}_{\\rm MLT}(1, 2, 12) \\otimes {\\cal A}^{(L-2)}_{\\rm MLT}(3, \\ldots , n) \\nonumber \\\\&&\\qquad + {\\cal A}^{(1)}_{\\rm MLT}(1,2)\\otimes {\\cal A}^{(0)}(12)\\otimes {\\cal A}^{(L-1)}_{\\rm MLT}(\\overline{3},\\ldots , \\overline{n})~.$ The first term on the rhs of Eq.",
"(REF ) represents a convolution of the two-loop MLT subtopology involving the sets $(1,2,12)$ with the rest of the amplitude, which is also MLT.",
"Each MLT component of the convolution opens according to Eq.",
"(REF ).",
"In the second term on the rhs of Eq.",
"(REF ), the set 12 remains off shell while there are on-shell propagators in either 1 or $\\overline{2}$ , and all the inverted sets from 3 to $n$ contain on-shell propagators.",
"For example, at three loops ($n=4$ ), these convolutions are interpreted as $&& {\\cal A}^{(2)}_{\\rm MLT}(1, 2, 12) \\otimes {\\cal A}^{(1)}_{\\rm MLT}(3, 4) \\nonumber \\\\&& \\qquad = \\int _{\\ell _1,\\ell _2, \\ell _3} \\left( {\\cal A}^{(3)}_D(\\overline{2}, \\overline{12}, \\overline{4}; 1, 3)+ {\\cal A}^{(3)}_D(1, \\overline{12}, \\overline{4}; 2, 3) \\right.",
"\\nonumber \\\\&& \\qquad \\left.",
"+ {\\cal A}^{(3)}_D(1, 2, \\overline{4}; 12, 3) + (\\overline{4}\\leftrightarrow 3) \\right)~,$ and $&& {\\cal A}^{(1)}_{\\rm MLT}(1,2)\\otimes {\\cal A}^{(0)}(12)\\otimes {\\cal A}^{(2)}_{\\rm MLT}(\\overline{3}, \\overline{4}) \\\\&& \\qquad = \\int _{\\ell _1,\\ell _2, \\ell _3}\\left( {\\cal A}^{(3)}_D(\\overline{2}, \\overline{3}, \\overline{4}; 1, 12) +{\\cal A}^{(3)}_D(1, \\overline{3}, \\overline{4}; 2, 12) \\right)~.",
"\\nonumber $ The two sets after the semicolon remain off shell.",
"In total, the number of terms generated by Eq.",
"(REF ) is $3L-1$ .",
"Figure: Next-to-next-to-maximal loop topology (left) and its convoluted dual representation (right).Opening according to Eq.",
"().Only the on-shell cut of the last MLT-like subtopology with reversed momentum flow is shown.Causal thresholds and infrared singularities are then determined by the singular structure of the ${\\cal A}^{(2)}_{\\rm MLT}(1, 2, 12)$ subtopology, and by the singular configurations that split the NMLT topology into two disjoint trees with all the on-shell momenta aligned over the causal cut.",
"Again, the singular surfaces in the loop three-momenta space are limited by the external momenta, and all the noncausal singular configurations that arise in individual contributions undergo dual cancellations."
],
[
"Next-to-next-to-Maximal loop topology",
"The last multiloop topology that we consider explicitly is the next-to-next-to-maximal loop topology (N$^2$ MLT) shown in Fig.",
"REF .",
"At three loops, it corresponds to the so-called Mercedes-Benz topology.",
"Besides the 12-set, there is another set denoted by 23 with $q_{i_{23}} = - \\ell _2 - \\ell _3 + k_{i_{23}}$ .",
"Its LTD representation is given by the following convolution of factorized subtopologies $&& {\\cal A}^{(L)}_{{\\rm N}^2{\\rm MLT}}(1,\\dots ,n,12,23) \\\\&& \\qquad = {\\cal A}^{(3)}_{\\rm NMLT}(1, 2, 3, 12, 23) \\otimes {\\cal A}^{(L-3)}_{\\rm MLT}(4, \\ldots , n) \\nonumber \\\\&& \\qquad + {\\cal A}^{(2)}_{\\rm MLT}(1\\cup 23, 2, 3\\cup 12) \\otimes {\\cal A}^{(L-2)}_{\\rm MLT}(\\overline{4}, \\ldots , \\overline{n})~.",
"\\nonumber $ The sets $(1,2,3,12,23)$ form a NMLT subtopology.",
"Therefore, the first component of the first term on the rhs of Eq.",
"(REF ) opens iteratively as $&& {\\cal A}^{(3)}_{\\rm NMLT}(1, 2, 3, 12, 23)= {\\cal A}^{(2)}_{\\rm MLT}(1, 2, 12) \\otimes {\\cal A}^{(1)}_{\\rm MLT}(3, 23) \\nonumber \\\\&& + \\int _{\\ell _1, \\ell _2, \\ell _3} \\left( {\\cal A}^{(3)}_D(1, \\overline{3}, \\overline{23}; 2, 12) + {\\cal A}^{(3)}_D(\\overline{12}, 3, 23; 1, 2) \\right)~.",
"\\nonumber \\\\$ The last two terms on the rhs of Eq.",
"(REF ) are fixed by the condition that the sets $(2, 3, 23)$ cannot generate a disjoint subtree.",
"The second term on the rhs of Eq.",
"(REF ) contains a two-loop subtopology made of five sets of momenta, ${\\cal A}^{(2)}_{\\rm MLT}(1\\cup 23, 2, 3\\cup 12)$ , which are grouped into three sets and dualized through Eq.",
"(REF ).",
"For example, propagators in the sets 1 and 23 are not set simultaneously on shell.",
"The number of terms generated by Eq.",
"(REF ) is $8(L-1)$ .",
"As for the NMLT, the causal singularities of the N$^2$ MLT topology are determined by its subtopologies and by the singular configurations that split the open amplitude into disjoint trees with all the on-shell momenta aligned over the causal cut.",
"Any other singular configuration is entangled among dual amplitudes and cancels.",
"We would like to emphasize that Eq.",
"(REF ) accounts properly for the NMLT and MLT topologies as well, if either 23 or both 12 and 23 are taken as empty sets.",
"At three loops, therefore, Eq.",
"(REF ) emerges as the LTD master topology for opening any scattering amplitude from its Feynman representation.",
"Finally, let us comment on more complex topologies at higher orders.",
"For example, let's consider the multiloop topology made of one MLT and two two-loop NMLT subtopologies that appears for the first time at four loops.",
"This topology opens into nondisjoint trees by leaving three loop sets off shell and by introducing on-shell conditions in the others under certain conditions: either one off-shell set in each subtopology or two in one NMLT subtopology and one in the other with on-shell propagators in all the sets of the MLT subtopology.",
"Once the loop amplitude is open into trees, the singular causal structure is determined by the causal singularities of its subtopologies, and all entangled noncausal singularities of the forest cancel."
],
[
"Conclusions",
"We have reformulated the loop-tree duality at higher orders and have obtained very compact open-into-tree analytical representations of selected loop topologies to all orders.",
"These loop-tree dual representations exhibit a factorized cascade form in terms of simpler subtopologies.",
"Since this factorized structure is imposed by the opening into nondisjoint trees and by causality, we conjecture that it holds to all loop orders and topologies.",
"Remarkably, specific multiloop configurations are described by extremely compact dual representations which are, moreover, free of unphysical singularities and independent of the initial momentum flow.",
"This property has been tested with all the topologies and several internal configurations.",
"We also conjecture that analytic dual representations in terms of only causal denominators are always plausible.",
"The explicit expressions presented in this Letter are sufficient to describe any scattering amplitude up to three loops.",
"Other topologies that appear for the first time at four loops and beyond have been anticipated, and will be presented in a forthcoming publication.",
"This reformulation allows for a direct and efficient application to physical scattering processes, and is also advantageous to unveil formal aspects of multiloop scattering amplitudes.",
"Acknowledgements: We thank Stefano Catani for very stimulating discussions.",
"This work is supported by the Spanish Government (Agencia Estatal de Investigación) and ERDF funds from European Commission (Grants No.",
"FPA2017-84445-P and No.",
"SEV-2014-0398), Generalitat Valenciana (Grant No.",
"PROMETEO/2017/053), Consejo Superior de Investigaciones Científicas (Grant No.",
"PIE-201750E021) and the COST Action CA16201 PARTICLEFACE.",
"JP acknowledges support from \"la Caixa\" Foundation (No.",
"100010434, LCF/BQ/IN17/11620037), and the European Union's H2020-MSCA Grant Agreement No.",
"713673; SRU from CONACyT and Universidad Autónoma de Sinaloa; JJAV from Generalitat Valenciana (GRISOLIAP/2018/101); WJT and AERO from the Spanish Government (FJCI-2017-32128, PRE2018-085925); and RJHP from Departament de Física Teòrica, Universitat de València, CONACyT through the project A1-S-33202 (Ciencia Básica) and Sistema Nacional de Investigadores."
]
] | 2001.03564 |
[
[
"Tidal disruptions of main sequence stars -- IV. Relativistic effects and\n dependence on black hole mass"
],
[
"Abstract Using a suite of fully relativistic hydrodynamic simulations applied to main-sequence stars with realistic internal density profiles, we examine full and partial tidal disruptions across a wide range of black hole mass ($10^{5}\\leq M_{\\rm BH}/\\mathrm{M}_{\\odot}\\leq 5\\times 10^{7}$) and stellar mass ($0.3 \\leq M_{\\star} /\\mathrm{M}_{\\odot}\\leq 3$) as larger $M_{\\rm BH}$ leads to stronger relativistic effects.",
"For fixed $M_{\\star}$, as $M_{\\rm BH}$ increases, the ratio of the maximum pericenter distance yielding full disruptions ($\\mathcal{R}_{\\rm t}$) to its Newtonian prediction rises rapidly, becoming triple the Newtonian value for $M_{\\rm BH} = 5\\times10^{7}~{\\rm M}_\\odot$, while the ratio of the energy width of the stellar debris for full disruptions to the Newtonian prediction decreases steeply, resulting in a factor of two correction at $M_{\\rm BH} = 5 \\times 10^7~{\\rm M}_\\odot$.",
"We find that for partial disruptions, the fractional remnant mass for a given ratio of the pericenter to $\\mathcal{R}_{\\rm t}$ is higher for larger $M_{\\rm BH}$.",
"These results have several implications.",
"As $M_{\\rm BH}$ increases above $\\sim 10^7~{\\rm M}_\\odot$, the cross section for complete disruptions is suppressed by competition with direct capture.",
"However, the cross section ratio for partial to complete disruptions depends only weakly on $M_{\\rm BH}$.",
"The relativistic correction to the debris energy width delays the time of peak mass-return rate and diminishes the magnitude of the peak return rate.",
"For $M_{\\rm BH} \\gtrsim 10^7~{\\rm M}_\\odot$, the $M_{\\rm BH}$-dependence of the full disruption cross section and the peak mass-return rate and time is influenced more by relativistic effects than by Newtonian dynamics."
],
[
"Introduction",
"Supermassive black holes (SMBHs) tidally disrupt stars when their separation becomes smaller than the so-called “tidal radius”.",
"Roughly half the mass removed from the star is bound to the black hole and may produce a luminous flare when it returns to the black hole, while the other half is expelled.",
"Tidal disruption events (TDEs) caused by a $10^{6}\\;\\mathrm {M}_{\\odot }$ SMBH have been considered a representative case in many theoretical studies [2], [8], [15], [7].",
"However, in reality, TDEs can occur for a wide range of mass $M_{\\rm BH}$ .",
"It is therefore useful to study how the key properties of tidal disruptions depend on $M_{\\rm BH}$ .",
"The interest of this study is enhanced by the fact that Newtonian order of magnitude estimates suggest that the characteristic tidal radius measured in gravitational units, i.e., $r_{\\rm t}/r_{\\rm g}\\equiv (R_{\\star }/r_{\\rm g}) (M_{\\rm BH}/M_{\\star })^{1/3} \\propto M_{\\rm BH}^{-2/3}$ , where $R_{\\star }$ is the stellar radius, $M_{\\star }$ is the stellar mass and $r_{\\rm g}$ is the gravitational radius, $r_{\\rm g}=G M_{\\rm BH}/c^{2}$ .",
"Given that scaling, these events take place in increasingly relativistic environments as $M_{\\rm BH}$ increases.",
"A study of black hole mass-dependence is therefore a study of how relativistic effects alter the course of these events (see a recent review by [26] for TDEs in relativity).",
"We aim to accomplish this study by performing relativistic hydrodynamic simulations (using Harm3d: [17]) whose initial conditions are realistic main-sequence stellar models taken from the stellar evolution code MESA.",
"In particular, we will examine a small sample of stellar masses ($0.3 \\;\\mathrm {M}_{\\odot }$ , $1.0\\;\\mathrm {M}_{\\odot }$ , and $3.0 \\;\\mathrm {M}_{\\odot })$ being disrupted by black holes of six different masses: $10^{5}\\;\\mathrm {M}_{\\odot }$ , $10^{6}\\;\\mathrm {M}_{\\odot }$ , $5\\times 10^{6}\\;\\mathrm {M}_{\\odot }$ , $10^{7}\\;\\mathrm {M}_{\\odot }$ , $3\\times 10^{7}\\;\\mathrm {M}_{\\odot }$ and $5\\times 10^{7}\\;\\mathrm {M}_{\\odot }$ .",
"In Section , we present results for the physical tidal radius $\\mathcal {R}_{\\rm t}$ (Section REF ), the energy distribution of stellar debris and the resulting fallback rate (Section REF ), and the remnant mass of partial disruptions (Section REF ).",
"In Section , we discuss the TDE event rate (Section REF ).",
"We also reconsider the maximum black hole mass for tidal disruptions (Section REF ).",
"Lastly, we summarize our findings in Section .",
"Throughout the remainder of this paper, all masses will be measured in units of ${\\rm M}_\\odot $ and all stellar radii in units of ${\\rm R}_\\odot $ .",
"Table: Values of r p /r t r_{\\rm p}/r_{\\rm t} considered in these experiments.",
"The units of M ☆ M_{\\star } and M BH M_{\\rm BH} are M ⊙ \\rm {M}_{\\odot }.",
"We also show the range of the “penetration factor\" β\\beta .Table: The physical tidal radii ℛ t \\mathcal {R}_{\\rm t} for different M BH M_{\\rm BH}, in units of r g r_{\\rm g}; the specific angular momentum ℒ t ≡L(ℛ t )\\mathcal {L}_{\\rm t}\\equiv L(\\mathcal {R}_{\\rm t}), in units of r g cr_{\\rm g}c; ℛ t /r t (≡Ψ)\\mathcal {R}_{\\rm t}/r_{\\rm t}(\\equiv \\Psi ) and β d (≡Ψ -1 )\\beta _{\\rm d}(\\equiv \\Psi ^{-1}).",
"The units of M ☆ M_{\\star } and M BH M_{\\rm BH} are M ☆ M_{\\star }."
],
[
"Simulations",
"Our simulations differ from those described in Ryu2+2019 and Ryu3+2019 only by using a wider range of black hole masses: $10^{5}\\;\\mathrm {M}_{\\odot }$ , $10^{6}\\;\\mathrm {M}_{\\odot }$ , $5\\times 10^{6}\\;\\mathrm {M}_{\\odot }$ , $10^{7}\\;\\mathrm {M}_{\\odot }$ , $3\\times 10^{7}\\;\\mathrm {M}_{\\odot }$ and $5\\times 10^{7}\\;\\mathrm {M}_{\\odot }$ .",
"In all cases we use the fully general relativistic hydrodynamics code Harm3d [17] operating in a Schwarzschild spacetime, but in a coordinate frame we call the box frame that follows the star's center-of-mass trajectory.",
"The initial internal structure of each star is taken from a MESA model at an age equal to half its main-sequence lifetime [18].",
"The case with mass $M_{\\star }=0.3$ represents fully convective stars; $M_{\\star }=1$ is our example of a (nearly) fully radiative star; like other high-mass stars, $M_{\\star }=3$ is radiative outside a convective core (see their density profiles in Ryu2+2019).",
"The choice of these three masses was motivated by the fact that for $M_{\\rm BH}=10^{6}$ , $\\mathcal {R}_{\\rm t}$ for $0.15\\le M_{\\star }\\le 3$ is bounded below by its value for $1\\;\\mathrm {M}_{\\odot }$ and bounded above by its value for $3\\;\\mathrm {M}_{\\odot }$ , while $\\mathcal {R}_{\\rm t}$ for $0.3\\;\\mathrm {M}_{\\odot }$ is closest to the average value ($\\mathcal {R}_{\\rm t}\\simeq 27~r_{\\rm g}$ ) within the range of masses $0.15\\le M_{\\star }\\le 3$ (Ryu1+2019).",
"As we showed in Ryu1+2019, relativistic corrections to ${\\cal R}_{\\rm t}$ are almost independent of $M_{\\star }$ .",
"This fact suggests that these three masses should play the same roles (average, lower, and upper bound) for any $M_{\\rm BH}$ .",
"Although the background spacetime is fully relativistic, the star's self-gravity is calculated using a Newtonian Poisson solver in a frame comoving with the star defined by a tetrad system at the star's center-of-mass.",
"In this frame, the metric is exactly Minkowski at the origin, but deviates from Minkowski elsewhere (see Ryu2+2019 for details).",
"The approximation of Newtonian self-gravity is valid when both the self-gravity and, more importantly, the non-Minkowski terms associated with tidal gravity, are small throughout the simulation volume.",
"This criterion is satisfied in the tetrad frame, but not in the box frame.",
"The stellar potential is added to $g_{\\rm tt}$ in the tetrad frame as a well-justified post-Newtonian approximation because in relativistic units it is $\\lesssim 10^{-6}$ .",
"To obtain the metric in the box frame, we then apply an inverse tetrad transformation.",
"Quantitative limits for the applicability of this approximation are presented in Appendix A in Ryu2+2019.",
"As remarked in [21], if stellar self-gravity is added to $g_{\\rm tt}$ in the box frame, where tidal gravity is significant, rather than in the tetrad frame, errors in the gravitational acceleration at the tens of percent level can be created.",
"Although the departure of the background metric from Minkowski grows as the separation to the BH falls, these departures are always small in our simulations.",
"Even along the outer edges of the simulation box, where they are largest, at a distance from the black hole $\\simeq 100r_{\\rm g}$ they are $\\sim 10^{-4}$ and rise to only $\\sim 10^{-2}$ at $\\simeq 5r_{\\rm g}$ .",
"For each stellar mass, we performed a suite of simulations for TDEs with various pericenter distances $r_{\\rm p}/r_{\\rm t}$ separated by increments $r_{\\rm p}/r_{\\rm t}=0.05-0.25$ .",
"We tabulate the values of $r_{\\rm p}/r_{\\rm t}$ considered in these experiments in Table REF .",
"The quantity $r_{\\rm p}/r_{\\rm t}$ is the inverse of the “penetration factor\" $\\beta $ .",
"To distinguish full from partial disruptions, we employ the same criteria introduced in Ryu2+2019, i.e., requiring full disruptions to have: No approximately-spherical bound structure.",
"Monotonic (as a function of time) decrease in the maximum pressure of the stellar debris.",
"Monotonic (as a function of time) decrease in the mass within the computational box.",
"We refer to events satisfying all of those conditions as “full”, others we call “partial\".",
"We estimate the physical tidal radius $\\mathcal {R}_{\\rm t}$ , the maximal radius at which a full tidal disruption takes place, as the mean of the greatest $r_{\\rm p}$ yielding a full disruption and the smallest $r_{\\rm p}$ producing a partial disruption.",
"The uncertainty in $\\mathcal {R}_{\\rm t}$ is due to our discrete sampling of $r_{\\rm p}$ .",
"Figure: The physical tidal radius in units of the nominal tidal radius, ℛ t /r t (≡Ψ)\\mathcal {R}_{\\rm t}/r_{\\rm t}(\\equiv \\Psi ), shown by filled symbols color-coded to indicate mass as shown in the legend.",
"The curves indicate the fitting formula (Equation ), multiplied by Ψ(M ☆ ,M BH =10 6 )\\Psi (M_{\\star },M_{\\rm BH}=10^6).",
"The hollow symbols show Ψ\\Psi for M ☆ =1M_{\\star }=1 from (diamonds), (pentagons) and (crosses)."
],
[
"Physical tidal radius $\\mathcal {R}_{\\rm t}$",
"The physical tidal radius $\\mathcal {R}_{\\rm t}$ is the maximum radius within which a full tidal disruption takes place.",
"The actual values measured in our numerical experiments are tabulated in Table REF .",
"Figure REF illustrates them graphically, showing $\\mathcal {R}_{\\rm t}/r_{\\rm t}(\\equiv \\Psi )$ as a function of $M_{\\rm BH}$ for the three stellar models.",
"For comparison, it also shows the equivalent predictions of two other studies employing relativistic calculations of the tidal stresses.",
"As can be seen easily, both $\\Psi $ and $d\\Psi /dM_{\\rm BH}$ increase with greater $M_{\\rm BH}$ .",
"Tidal forces are more destructive as relativistic effects become more significant, which leads to larger $\\Psi $ .",
"From the Newtonian limit ($M_{\\rm BH}=10^{5}$ ) to the strongly relativistic conditions of $M_{\\rm BH} = 5 \\times 10^7$ , $\\Psi $ grows by a factor $\\sim 3$ .",
"Figure REF also shows the $M_{\\rm BH}$ -dependence of $\\Psi $ has only a weak dependence on $M_{\\star }$ (also see the left panel of Figure 2 in Ryu1+2019).",
"This fact allows us to find an analytic expression for the $M_{\\rm BH}$ -dependence of $\\Psi $ separate from that for the $M_{\\star }$ -dependence.",
"The expression for the $M_{\\rm BH}$ -dependent term, which we call $\\Psi _{\\rm BH}$ in Ryu1+2019, is, $\\Psi _{\\rm BH}(M_{\\rm BH})=0.80 + 0.26~\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{0.5},$ which is depicted in Figure REF using dashed lines.",
"By comparing the logarithmic derivative of $\\Psi _{\\rm BH}$ with respect to $M_{\\rm BH}$ (i.e.",
"$d\\ln \\Psi _{\\rm BH}/d\\ln M_{\\rm BH}>1$ ), we find that for black holes more massive than $\\sim 3 \\times 10^7 \\;\\mathrm {M}_{\\odot }$ , the size of the physical tidal radius is more sensitive to relativistic corrections than to the simple Newtonian comparison of stellar self-gravity to black hole tidal gravity.",
"Several previous efforts have also explored this trend, [10], [6] and [24], which are indicated using hollow symbols in Figure REF .",
"All sought to explore relativistic effects in TDEs, but did so with a variety of approximations.",
"[10] calculated the tidal stress exactly, but described their star as a set of ellipsoidal shells whose initial structure was that of a $M_{\\star }=1$ $\\gamma =5/3$ polytrope (i.e., having the internal density profile of a low-mass star), and whose pressure and self-gravity were computed in a 1$-d$ approximation.",
"[6] employed a “generalized Newtonian potential” [29] that reproduces test-particle motion in a Schwarzschild spacetime very well when the specific energy is unity; it is unclear how well it reproduces relativistic tidal stresses and debris motion.",
"Their stars were supposed to be $\\gamma =5/3$ polytropes with $1\\;\\mathrm {M}_{\\odot }$ , and the stellar self-gravity was computed in an entirely Newtonian fashion.",
"[24] constructed an analytic expression for mapping Newtonian hydrodynamics simulations of $\\gamma = 4/3$ polytropes with $M_{\\star }=1$ to Schwarzschild geodesics by matching the magnitude of the tidal stresses at pericenter.",
"As shown in Figure REFThe data plotted were read from Figure 5 in [10], Figure 3 in [6] and Figure 8 in [24]., the alteration to the tidal radius due solely to relativistic effects found by the first and third efforts [10], [24] is similar to ours, but [6] found a weaker dependence on $M_{\\rm BH}$ .",
"Because relativity enters this part of the problem largely through the tidal stress, this should, perhaps, be unsurprising.",
"Where the results of [10] and [24] differ from ours, as well as each other's, is in the normalization.",
"Compared to our results for $M_{\\star }=1$ , $\\Psi $ from [10] is $50-80\\%$ larger, while the predictions of [24] are closer to ours, $10-30\\%$ larger.",
"The closer agreement with [24] is likely due to the coincidence that $\\gamma =4/3$ , although physically inappropriate, produces a good approximation to the density profile of a realistic main sequence star with $M_{\\star }=1$ .",
"Lastly we note that [28] and [5] used a relativistic hydrodynamics SPH code with Newtonian self-gravity to probe the relativistic regime.",
"Their study employed a $\\gamma =5/3$ polytrope for $M_{\\star }=1$ stars and considered how the encounters depended on $\\beta $ and spin parameter $a/M$ for a single black hole mass, $M_{\\rm BH}= 10^6$ , paying special attention to debris geometry due to black hole spin.",
"In contrast, we have determined how the tidal disruption properties of realistic main sequence stars depend on $M_{\\rm BH}$ over a wide range of masses.",
"Figure: The energy distribution dM/dEdM/dE for full disruptions of M ☆ =0.3M_{\\star }=0.3 (top panel), 1.01.0 (middle panel) and 3.03.0 (bottom panel)."
],
[
"Energy distribution and fallback rate of stellar debris for full disruptions",
"The energy distribution of stellar debris directly determines their orbits.",
"In the conventional description of TDEs [20], the energy distribution $dM/dE$ is approximated as flat within a characteristic energy width $\\pm \\Delta E$ .",
"In relativistic language, the classical specific orbital energy $E \\equiv -u_{\\rm t} - 1$ evaluated in the black hole frame, i.e., it is the conserved relativistic specific orbital energy exclusive of the rest mass energy.",
"This characteristic width is often estimated [12], [25] as $\\Delta \\epsilon =\\frac{GM_{\\rm BH}R_{\\star }}{r_{\\rm t}^{2}}.$ In this section, we focus on how $dM/dE$ varies as a function of $M_{\\rm BH}$ .",
"Figure REF shows $dM/dE$ for all 18 combinations of $M_{\\star }$ and $M_{\\rm BH}$ .",
"For all $M_{\\star }$ , $dM/dE$ becomes narrower and the “shoulders” (local maxima near the outer edges) become more conspicuous for higher $M_{\\rm BH}$ .",
"As a result, the energy width $\\Delta E$ containing 90% of the total mass, when measured in units of $\\Delta \\epsilon $ is smaller for higher $M_{\\rm BH}$ (see also Figure 5 in Ryu1+2019), with small variations ($<5-10\\%$ ) within the range of $r_{\\rm p} < \\mathcal {R}_{\\rm t}$ considered.",
"In Ryu1+2019, we provide an analytic expression for the $M_{\\rm BH}$ -dependence of $\\Delta E/\\Delta \\epsilon (\\equiv \\Xi _{\\rm BH})$ , $\\Xi _{\\rm BH}=1.27 - 0.300\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{0.242}.$ In Ryu1+2019, we also showed that $\\Xi _{\\rm BH}$ could be more crudely, but more simply, approximated by $\\Psi _{\\rm BH}^{-1}$ .",
"It is interesting that any prediction for a spread in energy due solely to the tidal potential would have suggested this dependence would have been $\\propto \\Psi _{\\rm BH}^{-2}$ rather than $\\propto \\Psi _{\\rm BH}^{-1}$ .",
"This is yet another piece of evidence supporting the argument given in Ryu1+2019 that the “frozen-in\" approximation is not a good basis on which to predict the debris energy spread.",
"Unlike $\\Delta E$ , the shape of the outer edge of the energy distribution depends on $M_{\\rm BH}$ in a way that does depend on stellar mass.",
"The distributions $dM/dE$ for $M_{\\star }=1$ and $M_{\\star }=3$ have significant tails for low $M_{\\rm BH}$ , but these become narrower for larger $M_{\\rm BH}$ .",
"In contrast, $dM/dE$ for $M_{\\star }=0.3$ has very sharp edges for the entire range of $M_{\\rm BH}$ .",
"Because Newtonian gravity is scale-free, it would not predict any changes in the shape of $dM/d(E/\\Delta \\epsilon )$ as a function of $M_{\\rm BH}$ ; only in general relativity, for which there is a special spatial scale and ${\\cal R}_{\\rm t}/r_{\\rm g}$ is a function of $M_{\\rm BH}$ , can these trends emerge.",
"Figure: The mass fall back rate M ˙ fb \\dot{M}_{\\rm fb} for M ☆ =0.3M_{\\star }=0.3 (top panel), 1.01.0 (middle panel) and 3.03.0 (bottom panel), using the energy distributions shown in Figure .",
"The time and rate are normalized by P Δϵ P_{\\Delta \\epsilon } and M ˙ 0 =M ☆ /3P Δϵ \\dot{M}_{0}=M_{\\star }/3P_{\\Delta \\epsilon }, respectively.",
"Here, P Δϵ P_{\\Delta \\epsilon } is the orbital period for the specific orbital energy of Δϵ\\Delta \\epsilon .",
"The diagonal line in each panel indicates the t -5/3 t^{-5/3} power-law.",
"[24] have also estimated the change in energy spread due to relativistic effects.",
"Phrased in terms of our language, they assumed that the energy distribution is zero for $|E| > GM_{\\rm BH}R_{\\star }/\\mathcal {R}_{\\rm t}^{2}$ and a constant value for $|E| \\le GM_{\\rm BH}R_{\\star }/\\mathcal {R}_{\\rm t}^{2}$ .",
"However, as we have seen, the character of the energy distribution is more complicated than a simple square wave, and its characteristic width is not $\\propto \\Psi _{\\rm BH}^{-2}$ as this assumption would predict.",
"For reasons like these, and because mass-loss takes place across a wide span of radii at which stellar gravity, hydrodynamic forces, and tidal gravity are all competitive (Ryu2+2019), approximating the energy spread in terms of the potential energy range at a particular location is not a particularly good approximation (Ryu1+2019).",
"Using the expression for the mass fallback rate of stellar debris on ballistic orbits [20], [19], $\\dot{M}_{\\rm fb}&=\\frac{dM}{dE}\\left|\\frac{dE}{dt}\\right|=\\frac{(2G M_{\\rm BH})^{2/3}}{3}\\frac{dM}{dE}t^{-5/3},$ and the energy distributions for the full disruptions in Figure REF , we determine the mass fallback rate as a function of time.",
"The results are depicted in Figure REF , where the rate and time are normalized by $\\dot{M}_{0} \\equiv M_{\\star }/(3P_{\\Delta \\epsilon })$ and $P_{\\Delta \\epsilon } \\equiv \\frac{}{\\sqrt{2}}G M_{\\rm BH}{\\Delta \\epsilon }^{-3/2}$ , respectively.",
"The shapes of the fallback curves are all qualitatively similar, possessing a rapid rise and a decline that is not far from the classical expectation, $\\propto t^{-5/3}$ .",
"However, it is also clear that, as a consequence of the decrease in $\\Delta E$ with increasing $M_{\\rm BH}$ , the time at which the peak is reached increases for larger black holes and the associated fallback rate decreases (because for any given $M_{\\star }$ , the total amount of mass returning is fixed).",
"The largest $t_{\\rm peak}/P_{\\Delta \\epsilon }$ (for $M_{\\rm BH}=5\\times 10^{7}$ ) and shortest one (for $M_{\\rm BH}=10^{5}$ ) differ by a factor of $2-4$ .",
"For $M_{\\star }=1$ , $t_{\\rm peak}/P_{\\Delta \\epsilon }$ rises from 0.50 for $M_{\\rm BH}=10^{5}$ to 0.55 for $M_{\\rm BH}=10^{6}$ , and 1.0 for $M_{\\rm BH}=10^{7}$ .",
"These shifts are superimposed upon those created by the internal structure of the stars.",
"There are also finer-scale features that depend on black hole mass, such as the steepness of the initial rise and the shape of the peak.",
"$\\dot{M}_{\\rm fb}/\\dot{M}_{0}$ for the $0.3\\;\\mathrm {M}_{\\odot }$ star increases very sharply as a result of the sharp edge at the low-energy end of $dM/dE$ , whereas $\\dot{M}_{\\rm fb}/\\dot{M}_{0}$ for the $1\\;\\mathrm {M}_{\\odot }$ and $3\\;\\mathrm {M}_{\\odot }$ stars begins to rise sooner and approaches the peak more gradually due to the wider tails in their energy distributions.",
"In addition, the maximum in $\\dot{M}_{\\rm fb}/\\dot{M}_0$ for $M_{\\star }=1$ is rather flat and broad, particularly for larger $M_{\\rm BH}$ .",
"Figure: The fractional remnant mass M rem /M ☆ M_{\\rm rem}/M_{\\star } as a function of pericenter distance r p r_{\\rm p} normalized by the physical tidal radius ℛ t \\mathcal {R}_{\\rm t} for M ☆ =0.3M_{\\star }=0.3 (top panel), 1 (middle panel) and 3 (bottom panel).",
"The 50% and 90% levels are marked by horizontal dotted lines.",
"The shaded regions demarcate the ranges determined by the uncertainties of ℛ t \\mathcal {R}_{\\rm t}, filled with the same colors as the solid lines.Because of cases like these, we do not define $t_{\\rm peak}$ as the actual time when $\\dot{M}_{\\rm fb}\\dot{M}_0$ reaches its absolute maximum, but rather as the time at which 5% of $M_{\\star }$ has returned to the black hole.",
"This time corresponds to the time of the absolute maximum when the peak is sharp, and the beginning of the maximum when the peak is relatively flat.",
"In addition, it is very close to the orbital period of matter with $E\\simeq -\\Delta E$ , making it consistent with the traditional definition of the characteristic timescale of mass-return even though our $dM/dE$ distributions are not square waves.",
"Several previous efforts have been made to determine how relativistic dynamics alter fallback rates.",
"Using Newtonian and relativistic hydrodynamic simulations, [3] studied the tidal encounters of a $1\\;\\mathrm {M}_{\\odot }$ polytropic star with $\\gamma =5/3$ with BHs of varying masses ($10^{5},~10^{6}$ and $10^{7}$ ).",
"The treatment of the star's self-gravity in their relativistic simulations is quite similar to ours: the self-gravity is calculated using a Newtonian Poisson solver in a frame comoving with the star and defined to be nearly Minkowski.",
"The only difference is that they used Fermi normal coordinates to define this frame [4] rather than a tetrad system as we did.",
"The results from their relativistic simulations show a shift in $t_{\\rm peak} / P_{\\Delta \\epsilon }$ with the same sign as ours, but significantly smaller amplitude: rather than a factor of 2–4 from $M_{\\rm BH}=10^5$ to $M_{\\rm BH}=10^7$ , they found only a factor 1.1.",
"[24] also estimated $\\dot{M}_{\\rm peak}$ and $t_{\\rm peak}$ using relativistic corrections to the energy width for $10^{5}\\le M_{\\rm BH}\\le 10^{7}$ .",
"They found results qualitatively consistent with ours in that $\\dot{M}_{\\rm peak}$ decreases and $t_{\\rm peak}$ increases.",
"However, they found a significantly shallower slope for $t_{\\rm peak}/P_{\\Delta \\epsilon }$ and $\\dot{M}_{\\rm fb}/\\dot{M}_0$ between $M_{\\rm BH}=10^5$ and $M_{\\rm BH} = 10^7$ than we do.",
"[24] predicted that $\\dot{M}_{\\rm peak}$ decreases by only 20% from $M_{\\rm BH}=10^{5}$ to $M_{\\rm BH}=10^{7}$ whereas, over the same $M_{\\rm BH}$ range, our calculations indicate that $\\dot{M}_{\\rm peak}$ decreases by a factor of 2.5."
],
[
"Partial disruption and the remnant mass",
"Stars are partially disrupted when $r_{\\rm p} > \\mathcal {R}_{\\rm t}$ , but less than a few times $\\mathcal {R}_{\\rm t}$ (Ryu3+2019).",
"Figure REF shows the ratio of the mass of the remnant to the initial stellar mass, $M_{\\rm rem}/M_{\\star }$ , as a function of $r_{\\rm p}/\\mathcal {R}_{\\rm t}$ .",
"The mass of a remnant is defined as the mass enclosed in the computational domain when the mass settles to an asymptotic value.",
"The fractional remnant masses for $M_{\\rm BH}=10^{5}$ and $10^{6}$ are similar for a given $r_{\\rm p}/\\mathcal {R}_{\\rm t}$ .",
"However, for larger $M_{\\rm BH}$ , $M_{\\rm rem}/M_{\\star }$ at fixed $r_{\\rm p}/\\mathcal {R}_{\\rm t}$ grows.",
"In other words, for a fixed ratio of the pericenter to the physical tidal radius, stars are better able to hold onto their mass when the event is more realistic.",
"[10], [6] and [24] also found that the remnant mass fraction for $1\\;\\mathrm {M}_{\\odot }$ stars depends on $M_{\\rm BH}$ in a fashion qualitatively similar to what we find, i.e., less mass is lost for higher $M_{\\rm BH}$ .",
"For a more quantitative comparison, we used the curves shown in their papers to determine their expectation for $M_{\\rm rem}/M_{\\star }$ at values of $r_{\\rm p}$ matching those used in our simulations.",
"In Figure REF , we show the average fractional difference between $M_{\\rm rem}/M_{\\star }$ as found by the three studies (for $M_{\\star }=1$ ) and the remnant mass fraction we determined.",
"For almost the entire range of black hole mass considered, the values of $M_{\\rm rem}/M_{\\star }$ from [10] and [24] are higher than ours by 20-60%.",
"These rather small differences from ours are remarkable given the approximate methods used in these calculations.",
"Although the remnant mass fractions from [6] are similar to ours for $M_{\\rm BH}=10^{7}$ , those for $M_{\\rm BH}=10^{6}$ are higher by almost a factor of two.",
"Figure: The average fractional difference between three other estimates of the remnant masses produced by partial disruptions (M rem others M_{\\rm rem}^{\\rm others} as estimated by [red crosses], [green pentagons] and [blue diamonds] ) and our simulations' estimates (M rem Harm3d M_{\\rm rem}^{{\\sc Harm3d}}), i.e., 〈M rem others /M rem Harm3d -1〉\\langle M_{\\rm rem}^{\\rm others}/M_{\\rm rem}^{{\\sc Harm3d}}-1\\rangle .",
"The error bars show the entire range of variation of the fractional differences over the span of r p /ℛ t r_{\\rm p}/\\mathcal {R}_{\\rm t} shown in Figure .",
"These ranges of variation are not standard deviations.",
"For better readability, the symbols for M BH =10 6 M_{\\rm BH}=10^{6} from and for M BH =10 7 M_{\\rm BH}=10^{7} from are shifted horizontally by a small amount."
],
[
"Implications",
"As our results illustrate, relativistic effects create $M_{\\rm BH}$ -dependence for all the principal properties of tidal disruptions: the physical tidal radius, the debris energy distribution, and the relation between orbital pericenter and remnant mass for partial disruptions.",
"These relativistic effects can produce quite noticeable departures from the Newtonian predictions for these physical quantities.",
"Relativistic effects also lead to significant changes in observable quantities.",
"Changes in the range of pericenters producing tidal disruptions translate directly into changes in event rates, particularly for galaxies in which the stellar angular momentum distribution is in the “full loss-cone\" limit.",
"Because the debris energy distribution determines the debris orbital period distribution, these changes alter the predicted fallback rate.",
"In this section we develop the consequences of these relativistic effects.",
"This entire discussion is made simpler by our demonstration that the relativistic corrections to $\\mathcal {R}_{\\rm t}$ and $\\Delta E$ depend only very weakly on $M_{\\star }$ .",
"The relativistic corrections to both $\\mathcal {R}_{\\rm t}$ and $\\Delta E$ can therefore be described by functions of $M_{\\rm BH}$ wholly independent of $M_{\\star }$ .",
"As we did in the previous three papers of this series, we refer to stars with $M_{\\star }\\le 0.5$ as “low-mass” stars and those with $M_{\\star }\\ge 1$ as “high-mass” stars."
],
[
"Physical tidal radii",
"The range in physical radii for main sequence stars of all masses at a single value of the black hole mass $M_{\\rm BH} = 10^6$ is considerably narrower than would be predicted on the basis of $r_{\\rm t}$ (Table 2 in Ryu2+2019 or the right panel of Figure 3 in Ryu1+2019).",
"From $M_{\\star } = 0.15$ to $M_{\\star }= 3$ , the maximum pericenter at which a total disruption occurs has a range of only $\\simeq 1.5$ , whereas the range of $r_{\\rm t}$ is $> 5$ .",
"The reason for this narrowing is that the shape of the internal density profile as a function of $M_{\\star }$ runs counter to the dependence of stellar radius on $M_{\\star }$ .",
"Because the relativistic corrections to $\\mathcal {R}_{\\rm t}$ are nearly independent of $M_{\\star }$ , this range is almost preserved; in fact, the sense in which the relativistic corrections do depend mildly on $M_{\\star }$ is such as to narrow the range even further (see Table REF ): at $M_{\\rm BH} = 10^7$ , it is only a factor of $\\simeq 1.25$ .",
"Thus, for the great majority of main sequence stars, $\\mathcal {R}_{\\rm t}$ is at most weakly dependent on $M_{\\star }$ for any given $M_{\\rm BH}$ , no matter what that black hole mass is.",
"Figure: (Left panel) Ratio Σ 1 \\Sigma _1, relevant to “full loss-cone\" event rates.",
"(Right panel) Ratio Σ 2 \\Sigma _2, relevant to “empty loss-cone\" event rates.",
"The error bars indicate the errors propagated from the uncertainties of ℛ t \\mathcal {R}_{\\rm t}."
],
[
"Relation between physical tidal radii and event rates",
"The rate of TDEs depends on the specific angular momentum $L$ associated with an orbit whose pericenter is $\\mathcal {R}_{\\rm t}$ : $L^2(r_{\\rm p}=\\mathcal {R}_{\\rm t}) \\equiv \\mathcal {L}_{\\rm t}^{2} = \\frac{2 (\\mathcal {R}_{\\rm t}/ r_{\\rm g})^{2}}{\\mathcal {R}_{\\rm t}/r_{\\rm g}- 2}.$ When the per-orbit root-mean-square change in $L$ is larger than $\\mathcal {L}_{\\rm t}$ (the “full loss-cone\" or “pinhole\" regime), the stars' velocities (when far from the black hole) are distributed uniformly across the solid angle of the loss-cone.",
"It is then appropriate to speak of event “cross sections\".",
"Because stars with $L < L_{\\rm dc} (= 4r_{\\rm g}c$ for parabolic orbits in Schwarzschild spacetime) plunge directly into the black hole without first being disrupted, the rate of total tidal disruptions is $\\propto \\mathcal {L}_{\\rm t}^2 - L_{\\rm dc}^2$ [11], [23].",
"On the other hand, when the rate at which a star's angular momentum changes is slow compared to the orbital frequency (the “empty loss-cone\" or “diffusive\" regime), the velocities of stars in the loss-cone are mostly directed very close to its edge.",
"In this situation, the “cross section\" language is inappropriate because the distribution of impact parameters is not uniform.",
"In this regime, the event rate depends logarithmically on $\\mathcal {L}_{\\rm t}$ [14], [16], [1] with a $\\sim 10\\%$ enhancement by occasional stronger encounters [30].",
"Direct capture is almost irrelevant in this regime until $M_{\\rm BH}$ approaches the Hills mass.",
"Progression toward full disruption through the range of angular momenta larger than ${\\cal L}_t$ is also interrupted by partial disruptions, which may lead to changes in the remnant's specific energy as well as its mass Ryu1+2019,Ryu3+2019.",
"For these reasons, we focus here on how our calculations affect estimates of $\\mathcal {L}_{\\rm t}$ , rather than their quantitative impact on actual event rates."
],
[
"Comparison between relativistic and estimated values of $\\mathcal {L}_{\\rm t}$",
"For “full loss-cone\" angular momentum evolution, the rate of an event with $r_{\\rm p} \\le \\mathcal {R}_{\\rm t}$ is $\\propto \\mathcal {L}_{\\rm t}^2$ , a quantity in which relativity alters the relation between $L$ and $\\mathcal {R}_{\\rm t}$ , and $\\mathcal {R}_{\\rm t}$ itself differs from $\\;r_{\\rm t}$ by effects both relativistic and derived from realistic stellar structure.",
"In addition, the actual rate of total disruptions is diminished by the rate at which direct capture, rather than tidal disruption, occurs.",
"On the other hand, in the “empty loss-cone\" regime (when one ignores the effects of partial disruptions), the rate is $\\propto \\ln (\\mathcal {L}_{\\rm t})$ .",
"Consequently, to demonstrate how our predictions alter rates, we examine two ratios: $\\Sigma _{1} &= \\frac{\\mathcal {L}_{\\rm t}^2 - L_{\\rm dc}^2}{L_{\\rm N}(r_{\\rm t})^2-L_{\\rm N,dc}^{2}},\\\\\\Sigma _{2} &= \\frac{\\mathcal {L}_{\\rm t}^2 }{L_{\\rm N}(r_{\\rm t})^2},$ where the subscript N denotes the Newtonian functional relationship.",
"$\\Sigma _1$ is the ratio between our predicted rate and the rate predicted by simple Newtonian estimates of disruption and direct capture; $\\Sigma _2$ is the ratio between $\\mathcal {L}_{\\rm t}^2$ and the square of the Newtonian angular momentum associated with the simple estimate.",
"The contrast between “full loss-cone\" event rates as we predict them and the simple estimate is given by the multiplicative factor $\\Sigma _1$ ; the contrast between our predicted “empty loss-cone\" rates and those given by the traditional estimate is the additive factor $\\ln \\Sigma _2$ .",
"The left panel of Figure REF shows $\\Sigma _{1}$ as a function of $M_{\\rm BH}$ .",
"$\\Sigma _1$ remains constant for $10^{5}<M_{\\rm BH}<10^{6}$ because relativistic corrections remain relatively small for this range of $M_{\\rm BH}$ .",
"The departures from unity in $\\Sigma _1$ in this range of $M_{\\rm BH}$ reflect the corrections to the cross section due to our use of realistic internal stellar density profiles (for the low $M_{\\rm BH}$ limit, $\\Sigma _{1}\\rightarrow \\Psi $ ).",
"Above $M_{\\rm BH} \\approx 10^6$ , $\\Sigma _1$ for low-mass stars increases, while it falls for high-mass stars.",
"This behavior is due to the competition between different relativistic effects, a competition that balances out differently depending on stellar structure.",
"Due to stronger tidal stress, $\\mathcal {R}_{\\rm t}/r_{\\rm t}$ increases with growing $M_{\\rm BH}$ , but the band of angular momentum outside $L_{\\rm dc}$ and inside $\\mathcal {L}_{\\rm t}$ rapidly becomes narrower, approaching zero for $M_{\\rm BH} > 5 \\times 10^7$ .",
"Stronger tidal stress plays the dominant role for $M_{\\star }=0.3$ , whereas the contribution from direct captures becomes more important for $M_{\\star }=1$ and 3.",
"The right panel of Figure REF shows these comparisons for $\\Sigma _2$ , the parameter more relevant to the empty loss-cone limit.",
"Independent of stellar mass, this ratio increases with $M_{\\rm BH}$ at an accelerating rate, reflecting the way in which stronger tidal stresses steeper relationship between $\\mathcal {L}_{\\rm t}^2$ and $r_{\\rm p}$ when the orbit runs deep into the relativistic potential.",
"Unlike $\\Sigma _1$ , $\\Sigma _2$ ignores losses due to direct capture.",
"$\\Sigma _2$ grows by a factor of 3–5 from the Newtonian limit to $M_{\\rm BH} = 5 \\times 10^7$ , depending on the stellar mass."
],
[
"Ratio of tidal disruption and direct capture cross sections in the full-loss cone regime",
"To illustrate how relativistic effects alter the outcome of tidal disruption events taking place in the full loss-cone context, Figure REF shows the ratio of the cross sections for direct capture to those for full tidal disruptions, i.e., $L_{\\rm dc}^{2}/[\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}]$ for the three stellar masses.",
"This ratio increases from being rather small for low $M_{\\rm BH} ($ $\\lesssim 0.1$ for $M_{\\rm BH} = 10^5$ ) to greater than unity for $M_{\\rm BH} \\gtrsim 5 \\times 10^6$ , although the precise value of the ratio depends weakly on $M_{\\star }$ .",
"It becomes $\\gtrsim 10$ for $M_{\\rm BH} \\gtrsim 5 \\times 10^7$ .",
"[11] also estimated this ratio, but in a different framework.",
"His dynamical calculation also used relativistic tidal stresses and orbital dynamics, but he defilned $\\mathcal {L}_{\\rm t}$ by the condition that the Newtonian surface gravity of a star with solar mass and radius match the magnitude of the eigenvalue for tidal stretch at the orbital pericenter; in other words, neither hydrodynamics nor the star's internal density profile played a role.",
"In addition, rather than present the cross section ratio, he presented the ratio of rates corresponding to a particular full loss-cone model.",
"This approach yielded $L_{\\rm dc}^{2}/[\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}]$ at $M_{\\rm BH} = 10^6$ $\\sim 3-4 \\times $ smaller than our value for $M_{\\star }=1$ , and a factor of 2 smaller for $M_{\\rm BH} > 10^7$ .",
"These quantitative contrasts may be due to both the stellar orbital population model used by [11] and the lack of hydrodynamics in his calculations.",
"Figure: The fractional remnant mass M rem /M ☆ M_{\\rm rem}/M_{\\star } as a function of the ratio of the cross-section for full+partial to full disruptions for M ☆ =0.3M_{\\star }=0.3 (top panel), 1 (middle panel) and 3 (bottom panel).",
"The 50% and 90% levels are marked by horizontal dotted lines.",
"The shaded regions delineate the uncertainties of the cross section ratio propagated from the uncertainties of ℛ t \\mathcal {R}_{\\rm t}, filled with the same colors as the solid lines.",
"The black dashed lines in each panel depict the fit given in Equation ."
],
[
"Maximum black hole mass for tidal disruption",
"The replacement of tidal disruption with direct capture places a fundamental limit on the range of black hole masses relevant to TDEs.",
"Indeed, to the degree that we can be confident about this limit, it can be used to constrain the inference of $M_{\\rm BH}$ in observed TDE events [13].",
"However, the concept of “maximum black hole mass\" is necessarily somewhat fuzzy.",
"As shown by [11], when the black hole has non-zero spin, the maximum mass depends on the black hole's spin parameter and the angle between the black hole's angular momentum and the star's orbital angular momentum.",
"More fundamentally, as was noted by [11] and can be seen in our study of the $M_{\\rm BH}$ -dependence of $\\mathcal {L}_{\\rm t}^2 - L_{\\rm dc}^2$ , even for masses a factor of several below the absolute maximum mass, the rate of tidal disruptions (when stellar angular momentum evolves rapidly, the “full loss-cone\" case) can be very strongly suppressed by the competition with direct capture.",
"On the other hand, if the limit of slow stellar angular momentum evolution applies (the “empty loss-cone\" regime), a condition that might apply to spherical stellar distributions around high-mass black holes [27], direct capture is irrelevant until $M_{\\rm BH}$ is large enough that $\\mathcal {L}_{\\rm t}^2$ becomes very close to $L^2_{\\rm dc}$ .",
"In our special case of non-spinning black holes, we define $M_{\\rm BH,max}$ as the value of $M_{\\rm BH}$ for which $L=L_{\\rm dc}$ , the angular momentum at which $\\mathcal {R}_{\\rm t}= 4r_{\\rm g}$ (note that the data presented in [11] indicate that $L_{\\rm dc}$ is very weakly dependent on spin when the orientation of the orbital axis relative to the spin axis is averaged over solid angle).",
"Because the smallest $\\mathcal {R}_{\\rm t}/r_{\\rm g}$ in Table REF is $\\simeq 6-7$ , we can not directly determine $M_{\\rm BH, max}$ from the simulation results, but it is clear that $M_{\\rm BH,max} > 5\\times 10^{7}$ .",
"Note that our lower bound on $M_{\\rm BH, max}$ is larger than some previous estimates, e.g., $M_{\\rm BH,max} \\simeq 2.5\\times 10^{7}$ for a solar-type star suggested by [24].",
"On the other hand, we also find that the rate of direct capture becomes comparable to that of tidal disruption at a mass a factor $\\sim 2$ smaller, so that the range of black hole masses in which the two rates compete is significantly broader than previously estimated.",
"The disagreement can probably be attributed to differences in method: [24] determined $M_{\\rm BH,max}$ by defining $\\mathcal {L}_{\\rm t}$ in terms of a match between the Newtonian self-gravity and an eigenvalue of the relativistic tidal tensor, but adjusted with a parameter derived from the Newtonian calculations of [8] applied to polytropic stars."
],
[
"Ratio of partial to total disruption cross sections",
"Partial disruptions, by definition, involve stars outside the loss-cone.",
"For these stars, the cross section approach is appropriate.",
"It is then convenient to compare the rates for these events to the rates for total disruptions.",
"Just as for total disruptions, the cross section is $\\propto L^2 = 2(r_{\\rm p}/r_{\\rm g})^{2}/(r_{\\rm p}/r_{\\rm g} - 2)$ .",
"We show in Figure REF the remnant mass fraction $M_{\\rm rem}/M_{\\star }$ as a function of the ratio $[L(r_{\\rm p})^{2}-L_{\\rm dc}^{2}]/[\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}]$ .",
"This ratio compares the cross section for all events (full+partial) with pericenter up to $r_{\\rm p}$ with the cross section for full disruptions; in the Newtonian limit, it reduces to $r_{\\rm p}/{\\cal R}_{\\rm t}$ .",
"The curves for different black hole masses coincide significantly more closely than the curves in Figure REF , where the same remnant mass fraction is plotted as a function of $r_{\\rm p}/{\\cal R}_{\\rm t}$ .",
"Due to the near coincidence of the curves plotted in Figure REF , all of them can be described—to the same accuracy as our expression for the $M_{\\rm BH}=10^6$ case—by a single curve, first presented in Ryu1+2019: $\\frac{M_{\\rm rem}}{M_{\\star }} = 1 - \\left[\\frac{L(r_{\\rm p})^{2}-L_{\\rm dc}^{2}}{\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}}\\right]^{-3}.$ The cross section ratio of all partial disruptions to all full disruption events is $[\\widehat{L}_{\\rm t}^{2}-\\mathcal {L}_{\\rm t}^{2}]/[\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}]$ , which is depicted in Figure REF .",
"Here, $\\widehat{R}_{\\rm t}$ is the largest pericenter distance yielding partial disruptions.",
"To use our data in order to measure $\\widehat{R}_{\\rm t}$ , we define it to be $r_{\\rm p}$ for $M_{\\rm rem}/M_{\\star }=0.9$ .",
"We locate this point by linear interpolation between the two data points closest to $M_{\\rm rem}/M_{\\star }=0.9$ .",
"Experimentation with other interpolation methods led to only slight changes in the results.",
"Figure: The ratio of the partial disruption to full disruption cross section [L ^ t 2 -ℒ t 2 ]/[ℒ t 2 -L dc 2 ][\\widehat{L}_{\\rm t}^{2}-\\mathcal {L}_{\\rm t}^{2}]/[\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}], estimated from analytic fits to the remnant mass curves in Figure , as a function of M BH M_{\\rm BH}.As is clear from Figure REF , the ratio of the partial to full disruption cross section depends quite weakly on $M_{\\rm BH}$ , varying by less than a factor of two from the Newtonian limit to the highest black hole masses probed.",
"It does, however, depend somewhat on $M_{\\star }$ : it is $\\approx 0.5$ for $M_{\\star }=0.3$ , $\\approx 2$ for $M_{\\star }=1$ , and $\\approx 1$ for $M_{\\star }=3$ .",
"The weak $M_{\\rm BH}$ -dependence is because as $M_{\\rm BH}$ increases, the full disruption cross section decreases due to direct capture events while the partial disruption cross section also declines owing to the decrease in $\\widehat{R}_{\\rm t}/\\mathcal {R}_{\\rm t}$ (see Figure REF )."
],
[
"Summary",
"This paper is the fourth in a series presenting the results of tidal disruption event simulations that, for the first time, combine general relativistic hydrodynamics, careful calculation of stellar self-gravity in a relativistic spacetime, and realistic main-sequence stellar structures for a wide range of stellar masses.",
"In this paper, we have focused on how properties of TDEs depend on black hole mass for non-spinning black holes; because the characteristic distance scales measured in gravitational units decrease with increasing $M_{\\rm BH}$ , studying TDEs at higher black hole mass means studying them in increasingly relativistic conditions.",
"Although qualitative results have been obtained previously on some of the issues we consider [10], [11], [24], [6], [28], [5], our more powerful methods (see Ryu2+2019 for details) have enabled quantitative characterization—and therefore greater insight—about how TDE properties depend on $M_{\\rm BH}$ : $\\bullet $ The dependence on $M_{\\rm BH}$ of the maximum radius for total disruption $\\mathcal {R}_{\\rm t}$ can be factored out from its weak dependence on $M_{\\star }$ .",
"We find that for a fixed $M_{\\star }$ , the ratio of $\\mathcal {R}_{\\rm t}$ to the classical estimator, $r_{\\rm t}$ , can be well approximated as $ \\Psi _{\\rm BH}(M_{\\rm BH})\\equiv \\mathcal {R}_{\\rm t}/r_{\\rm t} = 0.80 + 0.26~({M_{\\rm BH}}/{10^{6}})^{0.5}$ .",
"This function can and should be used a simple correction factor for the Newtonian estimates.",
"As $M_{\\rm BH}$ increases, this ratio steadily grows, increasing by a factor $\\simeq 3$ from the Newtonian limit, $M_{\\rm BH} = 10^5$ to the relativistic one, $M_{\\rm BH} = 5 \\times 10^7$ .",
"$\\bullet $ A direct corollary of the increase in $\\mathcal {R}_{\\rm t}/r_{\\rm t}$ is that the rate of events with pericenters $\\le \\mathcal {R}_{\\rm t}$ increases, relative to a Newtonian estimate based upon $r_{\\rm t}$ , by a factor $\\simeq 5$ from the Newtonian limit to $M_{\\rm BH} = 5 \\times 10^7$ .",
"However, at the same time, the fraction of direct captures also increases, becoming a majority of these events for $M_{\\rm BH} > 5 \\times 10^6$ .",
"Although our results are all calculated in Schwarzschild spacetime, they would change little in Kerr if averaged over orbital orientation because, as shown by [11], the orientation-averaged angular momentum for direct capture in Kerr almost exactly coincides with Schwarzschild.",
"Our main-sequence structures and hydrodynamics permit us to calculate $\\mathcal {R}_{\\rm t}$ , and therefore the flare fraction.",
"$\\bullet $ The Newtonian estimate $\\Delta \\epsilon $ for the width of the debris energy distribution is $\\propto M_{\\rm BH}^{1/3}$ .",
"However, the energy spread becomes narrower than this for higher SMBH masses: the ratio of the actual energy width $\\Delta E$ to $\\Delta \\epsilon $ falls by a factor $\\simeq 2$ from the Newtonian limit $M_{\\rm BH} = 10^5$ to the relativistic regime, $M_{\\rm BH} = 5 \\times 10^7$ .",
"This lengthens the return time and reduces the return rate of the debris stream.",
"$\\bullet $ Despite all these strong dependences on $M_{\\rm BH}$ , the full loss-cone rates of partial disruptions and total disruptions remain approximately equal for all $M_{\\star } \\lesssim 3$ and across the entire range of $M_{\\rm BH}$ ; the latter effect is due to the increasing fraction of direct captures as $M_{\\rm BH}$ grows.",
"Still more surprisingly, the fraction of the star's incoming mass lost in a partial disruption can be reasonably approximated by a single function that depends only on the angular momentum of the star's orbit and ${\\cal L}_t (M_{\\rm BH})$ , with almost no dependence on $M_{\\star }$ or any separate function of $M_{\\rm BH}$ (Equation REF )."
],
[
"Acknowledgements",
"This work was partially supported by NSF grant AST-1715032, Simons Foundation grant 559794 and an advanced ERC grant TReX.",
"S. C. N. was supported by the grants NSF AST 1515982, NSF OAC 1515969, and NASA 17-TCAN17-0018, and an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center administrated by USRA through a contract with NASA.",
"This research project (or part of this research project) was conducted using computational resources (and/or scientific computing services) at the Maryland Advanced Research Computing Center (MARCC).",
"The authors would like to thank Stony Brook Research Computing and Cyberinfrastructure, and the Institute for Advanced Computational Science at Stony Brook University for access to the high-performance SeaWulf computing system, which was made possible by a $\\$1.4$ M National Science Foundation grant (#1531492).",
"matplotlib [9]; MESA [18]; Harm3d [17]."
]
] | 2001.03504 |
[
[
"Raychaudhuri-based reconstruction of anisotropic Einstein-Maxwell\n equation in 1+3 covariant formalism of $f(R)$-gravity"
],
[
"Abstract Recently, a new strategy to the reconstruction of $f(R)$-gravity models based on the Raychaudhuri equation has been suggested by Choudhury et al.",
"In this paper, utilizing this method, the reconstruction of anisotropic Einstein-Maxwell equation in the $1+3$ covariant formalism of $f(R)$-gravity is investigated in four modes: $i.$ Reconstruction from a negative constant deceleration parameter refereeing to an ever-accelerating universe; $ii.$ Reconstruction from a constant jerk parameter $j=1$ which recovers celebrated $\\Lambda \\text{CDM}$ mode of evolution; $iii.$ Reconstruction from a variable jerk parameter $j=Q(t)$; and $iv.$ Reconstruction from a slowly varying jerk parameter.",
"Furthermore, two suggestions for enhancing the method are proposed."
],
[
"11em [rgb]0.00,0.40,0.80Raychaudhuri-based reconstruction of anisotropic Einstein-Maxwell equation in 1+3 covariant formalism of $f(R)$ -gravity Behzad Tajahmad [email protected] Faculty of Physics, University of Tabriz, Tabriz, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM)-Maragha, Iran, P.O.",
"Box: 55134-441 [breakable,colback=white, colframe=cyan,width=,enlarge left by=-17mm,enlarge right by=-6mm ] Abstract: Recently, a new strategy to the reconstruction of $f(R)$ -gravity models based on the Raychaudhuri equation has been suggested by Choudhury et al.",
".",
"In this paper, utilizing this method, the reconstruction of anisotropic Einstein-Maxwell equation in the $1+3$ covariant formalism of $f(R)$ -gravity is investigated in four modes: i. Reconstruction from a negative constant deceleration parameter refereeing to an ever-accelerating universe; ii.",
"Reconstruction from a constant jerk parameter $j=1$ which recovers celebrated $\\Lambda \\text{CDM}$ mode of evolution; iii.",
"Reconstruction from a variable jerk parameter $j=Q(t)$ ; and iv.",
"Reconstruction from a slowly varying jerk parameter.",
"Furthermore, two suggestions for enhancing the method are proposed.",
"[rgb]0.00,0.00,0.00"
]
] | 2001.03613 |
[
[
"Microscopic theory of OMAR based on kinetic equations for quantum spin\n correlations"
],
[
"Abstract The correlation kinetic equation approach is developed that allows describing spin correlations in a material with hopping transport.",
"The quantum nature of spin is taken into account.",
"The approach is applied to the problem of the bipolaron mechanism of organic magnetoresistance (OMAR) in the limit of large Hubbard energy and small applied electric field.",
"The spin relaxation that is important to magnetoresistance is considered to be due to hyperfine interaction with atomic nuclei.",
"It is shown that the lineshape of magnetoresistance depends on short-range transport properties.",
"Different model systems with identical hyperfine interaction but different statistics of electron hops lead to different lineshapes of magnetoresistance including the two empirical laws $H^2/(H^2 + H_0^2)$ and $H^2/(|H| + H_0)^2$ that are commonly used to fit experimental results."
],
[
"Introduction",
"Hopping conductivity is one of the fundamental types of electron transport in solid-state materials.",
"It exists when the electron wavefunctions are localized.",
"The conductivity is achieved due to the acts of hopping when the electron hops between localized functions (sites) with different energies due to the emission or absorption of phonons.",
"The conventional theory of hopping conductivity is closely related to doped semiconductors with compensation.",
"It is based on the mean-field approximation and Miller-Abrahams resistor network, which follows from this approximation in weak applied electric field [1], [2].",
"The drawback of the mean-field approximation is that it neglects correlations between occupation numbers of different sites.",
"One type of materials with hopping transport that is actively developed right now is the organic semiconductors.",
"They are already widely applied in OLEDs [3] and have other possible applications, for ex.",
"in organic solar cells [4].",
"These materials very often display an intriguing property that is called “organic magnetoresistance” or “OMAR” [5], [6], [7], [8], [9], [10], [11].",
"It is quite a strong magnetoresistance observed in magnetic fields $10-100gs$ both at low and room temperatures.",
"Although the qualitative explanations [12], [13] and semi-qualitative theories [14], [15], [16], [17], [18], [19], [20] of this phenomenon started to appear ten years ago, the detailed microscopic theory of OMAR is not yet developed.",
"One of the reasons for it is the close relation of OMAR to non-equilibrium spin correlations.",
"The magnetoresistance is equal to zero in the mean-field approximation [21] but re-appears when the correlations are included in the theory even in an oversimplified model [22].",
"The physical reason for OMAR is the dependence of the relaxation of spin correlations on the applied magnetic field.",
"This relaxation is often associated with hyperfine interaction with atomic nuclei.",
"With some simplification, it can be described as spin rotation around the so-called “hyperfine fields”.",
"These fields are different at different sites therefore random hopping of electron with rotation around these fields leads to spin relaxation.",
"When the external magnetic field is large compared to hyperfine fields, the spin rotates around approximately the same direction on all the sites and its relaxation is suppressed.",
"The microscopic theory of OMAR requires a theoretical approach that takes into account the non-equilibrium correlations including the spin correlations.",
"Up to very recent times, practically the only theoretical tool to do this was the Monte-Carlo numerical simulation.",
"It was used in one of the pioneering studies of OMAR to show the possibility of its bipolaron mechanism (the mechanism related to double-occupation of a single site with two electrons in the spin-singlet state) [13].",
"However, this method has its drawbacks.",
"It is a numerical method not suited for analytical theory.",
"Also, it is based on the semi-classical nature of hopping transport were all the quantum mechanics is included in the electron hopping rates.",
"It has some problems with spin correlations that actually have quantum nature.",
"In [13] the spin was described semi-classically as the two possibilities for an electron: to have spin up or spin down.",
"This approximation can readily be used in Monte-Carlo simulation, however, it cannot describe actual spin rotation around hyperfine fields.",
"To describe it the spin should be allowed to be directed along any axis, not only “up” and “down”.",
"It will be shown that it requires a more rigorous description of spin correlations based on quantum mechanics.",
"There is an approach to consider pair correlations in close pairs of sites as a modification of Miller-Abrahams resistors [22], [23], [24].",
"However, up to now, electron spin was considered with this approach only in the semi-classical “up” and “down” model.",
"Very recently the approach that allows to include correlations of arbitrary order into the analytical theory of hopping transport was developed [25], [26].",
"The approach operates with correlation kinetic equations (CKE) that relate occupation numbers and their correlations.",
"It is based on Bogolubov chain of equations [27], [28].",
"In [25], [26] this approach is developed only for charge correlations, i.e.",
"it does not consider electron spin in any model.",
"The goal of this study is to develop CKE theory that includes spin correlations and explicitly takes into account their quantum nature and to apply this theory to the problem of organic magnetoresistance.",
"The study is restricted to the bipolaron mechanism of OMAR in low electric fields and large Hubbard energy.",
"The spin relaxation is considered to be provided by hyperfine interaction with atomic nuclei.",
"Although the qualitative theories can give a general understanding of OMAR it is desirable to have an approach that can be used to calculate OMAR explicitly.",
"The progress in general understanding of organic semiconductors and in simulation technics leads to the possibility to calculate the microscopic properties of organic materials: the energy of molecular orbitals and their overlap integrals [29].",
"Some additional study of electron-phonon interaction in organic materials may lead to the possibility of direct calculation of hopping rates.",
"If all these properties will be known the CKE approach can be used to quantitatively calculate the magnetoresistance.",
"In the present paper, the magnetoresistance is calculated in model systems.",
"It is shown that the so-called lineshape of magnetoresistance (the shape of the dependence of resistivity on the applied magnetic field) depends on the properties of short-range electron transport.",
"Different model systems with identical hyperfine interaction but different statistics of hopping rates show different lineshapes of organic magnetoresistance.",
"The obtained lineshapes include (but are not limited to) the two empirical laws that are most often used to fit experimental data [6], [7]: $H^2/(H^2 + H_0^2)$ and $H^2/(|H| + H_0 )^2$ .",
"Here $H$ is the applied magnetic field and $H_0$ is a fitting parameter.",
"It gives hope that the calculations of OMAR can be used to relate microscopic models of hopping transport in organics with experimental results.",
"The paper is organized as follows.",
"In Sec.",
"I discuss the model that is used to describe organic semiconductors.",
"In Sec.",
"the general mathematical definitions of quantum spin and charge correlations are introduced.",
"In Sec.",
"the kinetic equations that relate these correlations to currents are derived.",
"In Sec.",
"the obtained system of CKE is used to describe the possible lineshapes of the bipolaron mechanism of OMAR.",
"In Sec.",
"REF it is done analytically in the model of modified Miller-Abrahams resistors.",
"In Sec.",
"REF it is done in the more general case with the numerical solution of CKE.",
"In Sec.",
"the general discussion of the obtained result is provided.",
"In Sec.",
"the conclusion is given.",
"Some part of quantum mechanical calculations that are made to derive spin CKE is discussed in the appendix ."
],
[
"model",
"The following model is considered in the present work.",
"The material contains a number of hopping sites were electrons (or polarons) are localized.",
"In principle a hopping site can contain two polarons, however, the energy of its double occupation is larger than the energy of its single occupation.",
"If the energy of the single occupation of site $i$ is $\\varepsilon _i$ , the energy of its double occupation is $\\varepsilon _i + U_h$ , were $U_h$ is the Hubbard energy.",
"The spins of electrons on a double-occupied site should form spin-singlet.",
"The possibility of double occupation with electrons in a triplet state is neglected.",
"It is considered that the system has some concentration of electrons and the Fermi level $\\mu $ .",
"The current in small applied electric fields in the linear response regime is discussed.",
"I consider the Hubbard energy to be much larger than temperature and energy differences in the hopping process.",
"In this situation all the sites participating in hopping transport can be divided into the two groups.",
"$A$ -type sites have the energy of single-occupation near Fermi energy $\\varepsilon _i \\sim \\mu $ .",
"They have 0 or 1 electrons but are never double-occupied because $\\varepsilon _i + U_h$ is too large for an $A$ -type site $i$ .",
"$B$ -type sites have the energy of double occupation near chemical potential $\\varepsilon _j + U_h \\sim \\mu $ .",
"They always have at least one electron because $\\varepsilon _j \\ll \\mu $ for a $B$ -type site $j$ .",
"Therefore “unoccupied” $B$ -type site is a $B$ -type site with one electron and has the spin degree of freedom.",
"An occupied $B$ -type site has two electrons in the spin-singlet state.",
"This model is most convenient for the description of a situation when the distribution of site energies $\\varepsilon _i$ is broad not only compared to temperature but also to the Hubbard energy $U_h$ (Fig.",
"REF (A)).",
"In this case, only a small part of hopping sites effectively participate in transport.",
"The consideration of sites without an energy level near $\\mu $ significantly complicates the numeric simulation and has little impact on the result.",
"I adopt the simplified model where there are two independent densities of states for $A$ -type sites and for $B$ -type sites (Fig.",
"REF (B)).",
"Figure: The density of states in the system with a broad distribution of energies (A) and in the adopted simplified model (B).The AA-type and BB-type sites marked with red and blue color correspondingly.In the following part of the text, the energy of double occupation of $B$ -type site $i$ is denoted as $\\varepsilon _i$ because the energy of its single occupation does not appear in the theory.",
"The hopping between sites is controlled by hopping rates $W_{ij}$ .",
"It is assumed that the electron spin is always conserved in the hopping process.",
"When sites $i$ and $j$ have the same type, the site $j$ is occupied and site $i$ is not, the electron hops from $j$ to $i$ with rate $W_{ij}$ .",
"$W_{ij} = W_0 |t_{ij}|^2 \\exp \\left( - \\frac{\\max [(\\varepsilon _i - \\varepsilon _j), 0]}{T} \\right).$ Here $t_{ij}$ is the overlap integral between localized states on sites $i$ and $j$ .",
"$W_0$ describes the strength of electron-phonon interaction.",
"In principle, it can have a power-law dependence of site energies $\\varepsilon _i$ , $\\varepsilon _j$ .",
"However, this dependence is not universal (is material-depend) and is not considered in the present study.",
"Therefore $W_0$ is treated as a constant.",
"When site $j$ is of type $B$ and $i$ is $A$ -type site ($B\\rightarrow A$ hop) the hopping occurs with the rate $2W_{ij}$ because both of the two electrons on site $j$ can hop to $i$ .",
"After the hop, the spins of electrons are in the singlet state.",
"In the situation of $A \\rightarrow B$ hop, when spins are in thermal equilibrium, the hopping occurs with the rate $W_{ij}/2$ .",
"However, this rate increases to $2W_{ij}$ when the spins are in singlet state.",
"It ensures that the detailed balance holds in the thermal equilibrium in a pair of sites of different kinds.",
"The $A \\rightarrow B$ hop is impossible when the spins are in a triplet state because the electron spin is conserved during the hop.",
"The equilibrium occupation number of an $A$ -type site $i$ is equal to $n_i^{(0)} = 1/(1+e^{(\\varepsilon _i-\\mu )/T}/2)$ .",
"The factor 2 in this expression follows from the degeneracy of the occupied state of an $A$ -type site [2].",
"The $A$ -type site $i$ has two occupied states with different spin.",
"The joint probability of the occupied states is $2 e^{-(\\varepsilon _i-\\mu )/T}/Z_i$ where $Z_i$ is the statistical sum of site $i$ : $Z_i = 1 + 2e^{-(\\varepsilon _i-\\mu )/T}$ .",
"It leads to the mentioned expression for the occupation probability $n_i^{(0)}$ .",
"For the $B$ -type site $j$ the free state is degenerate.",
"With similar arguments it leads to slightly different expression for the equilibrium occupation number of site $j$ : $n_j^{(0)} = 1/(1+ 2e^{(\\varepsilon _j-\\mu )/T})$ .",
"In any pair of sites $i-j$ , the equality holds without respect for site types $\\Gamma _{ij} = W_{ij}p_{sp}^{(ij)}(1-n_i^{(0)}) n_j^{(0)} = \\\\W_{ji} p_{sp}^{(ji)} (1-n_j^{(0)}) n_i^{(0)} = \\Gamma _{ji}.$ $\\Gamma _{ij}$ is the number of electrons that hops from site $j$ to site $i$ in unit of time in thermal equilibrium.",
"$p_{sp}^{(ij)}$ is the spin term in the hopping probability.",
"It is equal to $1/2$ when site $j$ has type $A$ and site $i$ has type $B$ .",
"When site $j$ is $B$ -type site and $i$ is $A$ -type site, $p_{sp}^{(ij)} = 2$ .",
"When the types of sites $i$ and $j$ are the same, $p_{sp}^{(ij)} = 1$ .",
"The equation (REF ) shows that detailed balance holds in thermal equilibrium.",
"Figure: Rotation of spins on sites ii and jj with Larmor frequencies Ω i \\mathbf {\\Omega }_i and Ω j \\mathbf {\\Omega }_j.The organic magnetoresistance is closely related to the dynamics of spin correlations.",
"The mathematical description of these correlations is introduced in Sec. .",
"Here I describe the physics that is considered.",
"The main reason for spin dynamics is the hyperfine interaction with atomic nuclei.",
"It is described as effective on-site magnetic fields ${\\bf H}_{\\rm hf}^{(i)}$ that are different on different sites.",
"These fields should be added to the external magnetic field ${\\bf H}$ .",
"The electron spin rotates around the total field with Larmor frequency $\\mathbf {\\Omega }_i = \\mu _b g ({\\bf H} + {\\bf H}_{\\rm hf}^{(i)})$ .",
"This description is valid when the slow dynamics of nuclear spins can be neglected.",
"The rotation in hyperfine fields can modify spin correlations.",
"Consider that at some point of time the spins of electrons on sites $i$ and $j$ were parallel.",
"After some time due to rotation with different vector frequencies $\\mathbf {\\Omega }_i$ and $\\mathbf {\\Omega }_j$ there will be an angle between spin directions.",
"In combination with electron hops, this rotation leads to spin relaxation [30], [31], [32].",
"I also introduce a phenomenological time of spin relaxation $\\tau _s$ .",
"It can describe the time of on-site spin relaxation related to (for example) the spin-phonon interaction.",
"However, the main reason to introduce $\\tau _s$ is the possibility to compare the results with previous theories where spin relaxation was treated in this way [13], [22]."
],
[
"Spin correlations",
"The site occupation numbers are insufficient for the description of OMAR that is controlled by spin degrees of freedom.",
"OMAR appears due to spin correlations.",
"Starting from the first qualitative studies of OMAR [13], [12] the magnetoresistance is attributed to different probabilities of parallel and anti-parallel configurations of two spins.",
"In terms of statistics the description of these probabilities is possible only beyond the mean-field approximation.",
"By definition the mean-field approximation deals with occupation numbers on a single site.",
"It can describe only the situation when one spin direction on some site is more probable than another without respect to other sites.",
"It corresponds to the appearance of averaged on-site spin polarization that is impossible at room temperatures and magnetic fields $\\sim 100 gs$ .",
"OMAR is controlled by the probabilities of the relative directions of two spins.",
"For example let us consider the situation when parallel configuration is more probable than anti-parallel.",
"In this case the probability for both spins to have up direction $\\uparrow \\uparrow $ is equal to the probability of $\\downarrow \\downarrow $ direction but is larger than the probability of $\\uparrow \\downarrow $ and $\\downarrow \\uparrow $ directions.",
"In terms of statistics this situation is described with correlations.",
"When the correlations are neglected in the mean-field approximation, it is impossible to distinguish between parallel configuration of two spins (without averaged polarization) and the equilibrium statistics.",
"Therefore a microscopic theory of OMAR should be developed beyond the mean-field approximation and take at least some correlations into account.",
"The approach adopted in the present paper is to write the equations for arbitrary correlations and neglect the insignificant ones at the last step of the theory when the equations are solved numerically.",
"In this section, the general notations for spin correlations between electrons on different sites are introduced.",
"The special attention is paid to the quantum nature of spin.",
"It allows to describe simultaneously the hopping transport and the spin rotation around local hyperfine fields.",
"These fields are responsible for spin relaxation in different materials [30], [31], [32] including many organic semiconductors.",
"At first, I describe the electron state on a single A-type site $i$ .",
"The electron has the two quantum-mechanical states: with spin up $|\\uparrow \\rangle $ or down $|\\downarrow \\rangle $ .",
"The general description of its state can be given by $2 \\times 2$ density matrix $\\widehat{\\rho }_i$ .",
"Its diagonal terms are real.",
"Their sum is unity because the site $i$ is considered to be occupied.",
"The non-diagonal terms are complex and are conjugate.",
"It leads to three independent parameters describing the density matrix.",
"These parameters can be selected to have clear physical meaning [33]: the averaged spin polarizations of site in the directions $x$ , $y$ and $z$ .",
"The density matrix $\\widehat{\\rho }_i$ can be re-constructed with this averaged polarizations.",
"$\\widehat{\\rho }_i = (1/2)\\left( \\widehat{1} + \\overline{s}_{i}^x \\widehat{\\sigma }_x + \\overline{s}_{i}^y \\widehat{\\sigma }_y + \\overline{s}_{i}^z \\widehat{\\sigma }_z\\right)$ .",
"Here $\\overline{s}_{i}^{x,y,z}$ are the averaged values of operators $\\hat{s}_{i}^{x,y,z}$ of spin polarization along the axes $x$ , $y$ and $z$ .",
"$\\widehat{\\sigma }_{x,y,z}$ are the Pauli matrices.",
"$\\widehat{1}$ is the unit $2\\times 2$ matrix.",
"In this work it also will be denoted as $\\widehat{\\sigma }_0$ Now consider an $A$ -type site $i$ with finite occupation probability $\\overline{n}_i \\equiv \\overline{s}_i^0$ .",
"When the conductivity is due to electron hops the terms of density matrix with uncertain occupation numbers can be neglected between hops (although they should be treated with perturbation theory when the hopping rates are calculated).",
"The density matrix $\\widehat{\\rho }_i$ can be given in block-diagonal form.",
"The free site is described by the block $\\widehat{\\rho }_i^{(0)}$ that actually is a single number equal to $1-\\overline{n}_i$ .",
"The single-occupied site is described by the block $\\widehat{\\rho }_i^{(1)}$ that can be expressed as follows $\\widehat{\\rho }_i^{(1)} = \\frac{\\sum _p \\overline{s_i^p} \\sigma _p}{2} = \\\\\\frac{1}{2}\\left( \\overline{s}_{i}^0 \\widehat{\\sigma }_0 + \\overline{s}_{i}^x \\widehat{\\sigma }_x + \\overline{s}_{i}^y \\widehat{\\sigma }_y + \\overline{s}_{i}^z \\widehat{\\sigma }_z\\right).$ Here index $p$ can have one of the four values $(0,x,y,z)$ .",
"The averaged quantities $\\overline{s}_{i}^0$ , $\\overline{s}_{i}^x$ , $\\overline{s}_{i}^y$ and $\\overline{s}_{i}^z$ can describe any state of the site $i$ .",
"It is possible to describe spin correlations in a similar manner.",
"The density matrix of two sites $i$ and $j$ can be expressed as the four blocks with well-defined occupation numbers.",
"Let $i$ be an $A$ -type site and $j$ be a $B$ -type site.",
"In this case the site $i$ can have 0 or 1 electrons and $j$ can have 1 or 2 ones.",
"The joint density matrix of sites $i$ and $j$ can be expressed in terms of four blocks $\\widehat{\\rho }_{ij} = \\left(\\begin{array}{cccc}\\widehat{\\rho }_{ij}^{(12)} & 0 & 0 & 0 \\\\0& \\widehat{\\rho }_{ij}^{(11)} &0 & 0 \\\\0 & 0 & \\widehat{\\rho }_{ij}^{(02)} & 0 \\\\0 & 0 & 0 & \\widehat{\\rho }_{ij}^{(01)}\\end{array}\\right).$ Here the upper indexes stand for the occupation numbers of the sites.",
"For example, $\\widehat{\\rho }_{ij}^{(12)}$ describes the part of density matrix related to single-occupied site $i$ and double-occupied site $j$ .",
"$\\widehat{\\rho }_{ij}^{(11)}$ is $4 \\times 4$ matrix that describes the single-occupied state of both sites and contain all the spin correlations $\\widehat{\\rho }_{ij}^{(11)} = \\frac{1}{4}\\sum _{pq} \\overline{s^{p}_{i} s^{q}_{j}} \\left(\\widehat{\\sigma }_{p}^{(i)} \\otimes \\widehat{\\sigma }_{q}^{(j)}\\right)$ where $\\overline{s^{p}_{i} s^{q}_{j}}$ is the quantum mechanical average of operator $\\hat{s}^{p}_{i} \\hat{s}^{q}_{j}$ .",
"Here $\\hat{s}^{x,y,z}_{i}$ and $\\hat{s}^{x,y,z}_{j}$ are the operators of spin polarizations of sites $i$ and $j$ correspondingly.",
"$s^{0}_i$ and $s^{0}_j$ are the operators of single occupations of sites $i$ and $j$ .",
"They always have well-defined values between hops.",
"Note that the relation between occupation number and $\\overline{s}^{0}$ is different for different types of sites.",
"For A-type site $i$ , $\\overline{s}^0_{i} = \\overline{n}_i$ .",
"For a B-type site $j$ , $\\overline{s}^0_{j} = 1- \\overline{n}_j$ because $B$ -type sites have one electron in unoccupied state.",
"I keep these double notations because the notation $n_i$ is useful to track the charge conservation law while the notation $\\hat{s}^0$ allows to give the expression for density matrix in terms of correlations in unified form for both A-type and B-type sites.",
"The sign $\\otimes $ in Eq.",
"(REF ) denotes the Cartesian product of matrices.",
"The upper indexes $(i)$ and $(j)$ in $\\widehat{\\sigma }_{p}^{(i)}$ and $\\widehat{\\sigma }_{q}^{(j)}$ have no mathematical meaning but help to track what Pauli matrix is related to what site.",
"The block $\\widehat{\\rho }_{ij}^{(02)}$ is a single number equal to $\\overline{(1-n_i)n_j}$ The blocks $\\widehat{\\rho }_{ij}^{(12)}$ and $\\widehat{\\rho }_{ij}^{(01)}$ are $2\\times 2$ matrices $\\widehat{\\rho }_{ij}^{(12)} = \\sum _p \\overline{ (1 - s^0_{j}) s^{p}_{i}} \\widehat{\\sigma }_p^{(i)}, \\\\\\widehat{\\rho }_{ij}^{(01)} = \\sum _p \\overline{(1 - s^0_{i})s^{p}_{j}} \\widehat{\\sigma }_p^{(j)}.$ The whole matrix $\\rho _{ij}$ can be parameterized with 24 averaged values: $\\overline{s^{p}_{i}}$ , $\\overline{s^{p}_{j}}$ and $\\overline{s^{p}_{i}s^{q}_{j}}$ .",
"These values have clear physical meaning: they describe the occupation probabilities, mean spin polarizations and their correlations.",
"The kinetic equations for these averaged values would allow to describe all the dynamics of the density matrix for a system with hopping transport.",
"This result can be generalized for arbitrary number of sites.",
"Let $I$ be some set of sites.",
"The density matrix $\\widehat{\\rho }_I$ of this set can be described by averaged products of $\\hat{s}_i^{p}$ in all the subsets $I_n$ of the set $I$ .",
"$\\overline{s_{I_n}^{P}} = \\overline{\\prod _{i \\in I_n} \\hat{s}_{i}^{p_i} }, \\quad P = \\lbrace p_1,... p_{n(I_n)} \\rbrace , \\quad I_n \\subset I.$ Here $P$ is the set of upper indexes $p_i$ equal to 0, $x$ , $y$ or $z$ related to the sites $i$ in the set $I_n$ .",
"All the kinetics of a hopping system can be described with $\\overline{s_I^{P}}$ .",
"However, these values cannot be considered as correlations.",
"When two A-type sites $i$ and $j$ are not correlated $\\overline{s_i^{0} s_j^{0}} = \\overline{n}_i \\overline{n}_j \\ne 0$ in thermal equilibrium.",
"Therefore $\\overline{s_I^{P}}$ cannot be neglected even when correlations inside the set $I$ are not important.",
"The idea of correlation kinetic approach [25] is to write equations for correlations themselves and neglect the correlations at large distance and of high order.",
"To follow this idea the correlations $\\overline{c_{I}^{P}}$ are introduced $\\overline{c_{I}^{P}} = \\overline{\\prod _{i \\in I} c_{i}^{p_i} }, \\quad c_{i}^{0} = n_i - n_i^{(0)}, \\quad c_i^{\\alpha } = \\hat{s}_i^{\\alpha }.$ Here the Greek letter $\\alpha $ stands for the spin projection on $x$ , $y$ or $z$ .",
"$n_i^{(0)}$ is the equilibrium filling number.",
"In this notations $\\overline{c}_{i}^{\\alpha } = 0$ when we neglect averaged spin polarization and $\\overline{c}_{i}^{0} = \\overline{n}_i - n_i^{(0)}$ is the perturbation of the occupation number due to applied electric field.",
"When the hopping system is close to equilibrium, eq.",
"(REF ) describes the correlations that can be neglected if occupation numbers and spins in the set $I$ are not considered to be correlated."
],
[
"Kinetic equations for spin and charge correlations",
"In this section the kinetic equations are derived for the correlations defined in Sec. .",
"Consider the correlation $\\overline{c}_I^P$ in some set $I$ of sites.",
"It can be changed due to one of the following processes: the hopping of electrons between sites of the set and outer sites, the hopping of electrons inside set $I$ and due to the internal spin dynamics.",
"$\\frac{d}{dt} \\overline{c}_I^P = \\sum _{i\\in I, k \\notin I} \\left(\\frac{d}{dt}\\right)_{ik} \\overline{c}_I^P + \\\\\\sum _{i,j \\in I} \\left(\\frac{d}{dt}\\right)_{ij} \\overline{c}_I^P+ \\sum _{i\\in I} \\left(\\frac{d}{dt}\\right)_{i} \\overline{c}_I^P$ Here in r.h.s of Eq.",
"(REF ) the symbolical expressions for different terms are used.",
"$(d/dt)_{ij} \\overline{c}_I^P$ means the changing rate of $\\overline{c}_I^P$ due to the hops between sites $i$ and $j$ of the set.",
"$(d/dt)_{ik} \\overline{c}_I^P$ stands for the transitions between site $i$ from the set and outer site $k$ .",
"$(d/dt)_{i} \\overline{c}_I^P$ describes the changing rate of $\\overline{c}_I^P$ due to the internal spin dynamics (spin rotation and phenomenological spin relaxation) related to site $i$ .",
"The term $(d/dt)_{ik}$ describes the transition of correlations between sets of sites.",
"It can be expressed as follows.",
"$\\left( \\frac{d}{dt} \\right)_{ik} \\overline{c_i^p c_{I^{\\prime }}^{P^{\\prime }}} = T_{ik}^{(p)} \\overline{c_k^p c_{I^{\\prime }}^{P^{\\prime }}} - T_{ki}^{(p)}\\overline{c_i^p c_{I^{\\prime }}^{P^{\\prime }}}.$ Here $T_{ik}^{(p)}$ is the rate of charge or spin transition from site $k$ to site $i$ .",
"$c_{I^{\\prime }}^{P^{\\prime }}$ describes the part of correlation that is not related to site $i$ and is conserved during $i \\leftrightarrow k$ hops.",
"$I^{\\prime } = I \\backslash \\lbrace i\\rbrace $ where the notation $\\backslash $ stands for the set difference.",
"$P^{\\prime }$ is the set of indexes from $P$ other than the index $p$ related to the site $i$ .",
"$ T_{ik}^{(0)} = W_{ik}p_{sp}^{(ik)}\\left(1-n_i^{(0)}\\right) + W_{ki} p_{sp}^{(ki)}n_i^{(0)}$ $T_{ik}^{(0)}$ describes the rate of transition of small perturbation of charge density from site $k$ to site $i$ [25].",
"Figure: The spin transfer from site ii to site jj in different pairs of sites.",
"On the left hand side of the figure the site ii is spin-polarized and site jj is not.",
"On the right-hand side of the figure (after the hop) the site jj acquires spin polarization in the direction of the initial polarization of site ii.$T_{ik}^{(\\alpha )}$ describes the process of spin polarization transfer between sites.",
"This process is different for different types of sites, as shown in Fig.",
"REF .",
"In a pair of A-type sites the spin transfer is achieved due to hops of spin-polarized electrons.",
"In a pair of B-type sites the hops of spin-polarized holes are responsible for the spin transfer.",
"In a mixed $AB$ pair the spin transfer occurs because electron with one spin projection can hop from the $A$ -type site to the $B$ -type site while the electron with other spin projection cannot.",
"It leads to the following expressions for spin transfer rates in different pairs of sites (see [21] for details).",
"$T_{ik}^{(\\alpha )} = \\left\\lbrace \\begin{array}{ll}W_{ik}(1-n_i^{(0)}), & AA \\\\W_{ki} n_i^{(0)}, &BB \\\\W_{ki} n_i^{(0)}/2, &AB \\\\W_{ik} (1-n_i^{(0)})/2, &BA\\end{array}\\right.$ The term $(d/dt)_{ij}$ for charge correlations is derived in [25] using the fact that joint occupation of sites $i$ and $j$ cannot be changed due to $i \\leftrightarrow j$ hops, $(d/dt)_{ij} \\overline{n_in_j} = 0$ .",
"This argument should be generalized to include spin and different types of sites.",
"When both the sites $i$ and $j$ have the same type the hops between them are impossible when both of them are single occupied.",
"It leads to the expression $(d/dt)_{ij} \\overline{s_i^p s_j^q s_{I^{\\prime }}^{P^{\\prime }}} = 0$ for $AA$ and $BB$ pairs of sites $i$ , $j$ .",
"When the sites $i$ and $j$ have different types the hop between them is possible when both are single-occupied.",
"Actually it is the process that relates spin correlations and charge transport in bipolaron mechanism of OMAR .",
"Rate equations for this process when $i$ is $A$ -type site and $j$ is $B$ -type site are derived in appendix with quantum mechanic approach: $\\left( \\frac{d}{dt} \\right)_{ij} \\overline{s_i^0s_j^0 s_{I^{\\prime }}^{P^{\\prime }}} = 2W_{ij} \\overline{n_{j\\uparrow } n_{j\\downarrow } (1 - n_{i\\uparrow })(1 - n_{i\\downarrow }) s_{I^{\\prime }}^{P^{\\prime }} } - \\\\\\frac{W_{ji}}{2} \\overline{(s_i^0 s_j^0 - \\sum _{\\alpha } s_i^{\\alpha }s_j^{\\alpha } )s_{I^{\\prime }}^{P^{\\prime }}},$ $ \\left( \\frac{d}{dt} \\right)_{ij} \\overline{s_i^0s_j^\\alpha s_{I^{\\prime }}^{P^{\\prime }}} = \\frac{W_{ji}}{2}\\overline{(s_i^{\\alpha }s_j^0 - s_i^{0}s_j^\\alpha ) s_{I^{\\prime }}^{P^{\\prime }}},$ $\\left( \\frac{d}{dt} \\right)_{ij} \\overline{s_i^\\alpha s_j^0 s_{I^{\\prime }}^{P^{\\prime }}} = \\frac{W_{ji}}{2}\\overline{(s_i^{0}s_j^\\alpha - s_i^{\\alpha }s_j^0) s_{I^{\\prime }}^{P^{\\prime }}},$ $\\left( \\frac{d}{dt} \\right)_{ij} \\overline{s_i^\\alpha s_j^\\beta s_{I^{\\prime }}^{P^{\\prime }}} = \\\\-2W_{ij} \\delta _{\\alpha \\beta } \\overline{n_{j\\uparrow } n_{j\\downarrow } (1 - n_{i\\uparrow })(1 - n_{i\\downarrow }) s_{I^{\\prime }}^{P^{\\prime }} } + \\\\\\frac{W_{ji}}{2} \\delta _{\\alpha \\beta } \\overline{(s_i^0 s_j^0 - \\sum _{\\alpha } s_i^{\\gamma }s_j^{\\gamma } )s_{I^{\\prime }}^{P^{\\prime }}} - \\\\\\frac{W_{ji}}{2} \\overline{(s_i^{\\alpha }s_j^{\\beta } - s_i^{\\beta } s_j^{\\alpha })s_{I^{\\prime }}^{P^{\\prime }}}$ Here Greek indexes $\\alpha $ , $\\beta $ and $\\gamma $ stand for spin polarizations along Cartesian axes.",
"The physical meaning of Eqs.",
"(REF -REF ) is as follows.",
"Eq.",
"(REF ) and the first two terms in r.h.s.",
"of Eq.",
"(REF ) show that the hop from site $i$ to site $j$ is possible only when both sites are single-occupied and electron spins on these sites are in singlet state.",
"It decreases the probability of single occupation of both sites.",
"The backward hop from $j$ to $i$ leads to the single occupation of sites with electrons in singlet state.",
"The third term in Eq.",
"(REF ) leads to relaxation of the antisymmetric combinations $\\overline{s_i^{\\alpha }s_j^{\\beta } - s_i^{\\beta } s_j^{\\alpha }}$ .",
"It can be shown that these combinations are related to a coherent combination of singlet and triplet states of the two electrons.",
"When the electrons are either in the singlet or in a triplet state $\\overline{s_i^{\\alpha }s_j^{\\beta } - s_i^{\\beta } s_j^{\\alpha }} =0$ .",
"However, when their state is a coherent combination of singlet and triplet $\\overline{s_i^{\\alpha }s_j^{\\beta } - s_i^{\\beta } s_j^{\\alpha }} \\ne 0$ .",
"Even if due to some reason the probabilities of singlet and triplet states are conserved, their coherent combinations relax because the hopping $i\\leftrightarrow j$ can occur in the singlet state and cannot occur in a triplet state.",
"Similar term appears in spin dynamic of a double quantum dot [34].",
"Eqs.",
"(REF ,REF ) show that spin transfer process in $AB$ pairs of sites is correlated with occupation numbers.",
"The known relations between $\\overline{c}_I^P$ and $\\overline{s}_I^P$ allow to obtain the expressions for $(d/dt)_{ij}\\overline{c}_I^P$ similar to Eqs.",
"(REF -REF ).",
"These expressions are not provided here because they are quite cumbersome but can be given in a much more compact form when the correlation potentials are introduced.",
"The term $(d/dt)_i \\overline{c}_{I}^P$ is responsible for the internal spin dynamics.",
"It is present only when the index $p$ corresponding to site $i$ is a spin index.",
"$\\left(\\frac{d}{dt}\\right)_i \\overline{c}_{i,I^{\\prime }}^{\\alpha ,P^{\\prime }} = \\epsilon _{\\alpha \\beta \\gamma }\\Omega _{i,\\beta }\\overline{c}_{i,I^{\\prime }}^{\\gamma ,P^{\\prime }} - \\frac{1}{\\tau _s} \\overline{c}_{i,I^{\\prime }}^{\\alpha ,P^{\\prime }}$ Here $\\Omega _{i,\\beta }$ is the projection of spin rotation frequency vector on site $i$ to the axis $\\beta $ .",
"$\\tau _s$ is a phenomenological on-site spin relaxation time.",
"$\\epsilon _{\\alpha \\beta \\gamma }$ is Levi-Civita symbol.",
"In the linear response regime it is useful to introduce effective correlation potentials $\\overline{\\varphi }_{I}^P$ .",
"$ \\varphi _i^0 = c_i^0 / (1-n_i^{(0)})n_i^{(0)}, \\quad \\varphi _i^{\\alpha } = c_i^{\\alpha }/(s_i^0)_{eq}.$ Here $(s_i^0)_{eq}$ is the equilibrium probability of single occupation of site $i$ .",
"It is equal to $n_i^{(0)}$ if site $i$ has type $A$ and to $1-n_i^{(0)}$ if site $i$ has type $B$ .",
"There is no averaging in the definition (REF ).",
"$\\varphi _i^p$ should be ensemble averaged in some combination to have the meaning of potential.",
"For example $\\overline{\\varphi }_{ijk}^{0xy} = \\overline{\\varphi _i^0\\varphi _j^x\\varphi _k^y}$ can be considered as a potential of correlation $\\overline{c}_{ijk}^{0xy}$ .",
"In these notations $(d/dt)_{ik}\\overline{c}_{i,I^{\\prime }}^{p,P^{\\prime }}$ corresponds to a “correlation flow” $J_{ik;I^{\\prime }}^{p;P^{\\prime }}$ between correlations $\\overline{c}_{i,I^{\\prime }}^{p,P^{\\prime }}$ and $\\overline{c}_{k,I^{\\prime }}^{p,P^{\\prime }}$ : $\\left( \\frac{d}{dt} \\right)_{ik} \\overline{c}_{i,I^{\\prime }}^{p,P^{\\prime }} = - \\left( \\frac{d}{dt} \\right)_{ik} \\overline{c}_{k,I^{\\prime }}^{p,P^{\\prime }} =J_{ik;I^{\\prime }}^{p;P^{\\prime }}.$ When $I^{\\prime }$ is the empty set $J_{ik}^0$ is the particle flow from site $k$ to site $i$ .",
"The flow $J_{ik;I^{\\prime }}^{p;P^{\\prime }}$ is expressed as follows $J_{ik;I^{\\prime }}^{p;P^{\\prime }} = \\Gamma _{ik} \\Theta _{I^{\\prime }}^{P^{\\prime }} \\left( \\overline{\\varphi }_{k,I^{\\prime }}^{p,P^{\\prime }} - \\overline{\\varphi }_{i,I^{\\prime }}^{p,P^{\\prime }}+S_{ik,I^{\\prime }}^{p} \\right.+ \\\\ \\left.\\Phi _p^{qq^{\\prime }}(ik) \\overline{\\varphi }_{ik,I^{\\prime }}^{qq^{\\prime },P^{\\prime }} \\right).$ Here $\\Gamma _{ik}$ is the average number of electrons that hops from site $k$ to site $i$ in unit time in equilibrium (REF ).",
"$\\Theta _{I^{\\prime }}^{P^{\\prime }}$ is the coefficient related to sites of $I$ that do not participate in transition $i \\leftrightarrow k$ .",
"$\\Theta _{j,l,m,...}^{p_j,p_l,p_m,...} = \\theta _j^{p_j} \\cdot \\theta _l^{p_l} \\cdot \\theta _m^{p_m}\\cdot ...$ $\\theta _j^0 = n_j^{(0)}\\left(1-n_j^{(0)}\\right), \\quad \\theta _j^{\\alpha } = \\left(s_j^0\\right)_{eq}.$ $S_{ik,I^{\\prime }}^p$ is the source term related to the external electrical field $\\bf E$ .",
"It is not equal to zero only when $I^{\\prime }$ is empty set and $p=0$ .",
"In this case $S_{ik,\\emptyset }^0 = e{\\bf E} {\\bf r}_{ik}$ where $e$ is electron charge, ${\\bf r}_{ik}$ is vector of distance between $i$ and $k$ .",
"The term $\\Phi _p^{qq^{\\prime }}(ik) \\overline{\\varphi }_{ik,I^{\\prime }}^{qq^{\\prime },P^{\\prime }}$ describes the effect of higher-order correlations to the flow of lower-order correlations.",
"For different indexes $p$ , $q$ and $q^{\\prime }$ , the coefficients $\\Phi _p^{qq^{\\prime }}(ik)$ are $\\begin{array}{l}\\Phi _0^{00}(ik) = n_k^{(0)}-n_i^{(0)}, \\\\\\Phi _0^{\\alpha \\alpha }(ik) = \\tau _k - \\tau _i, \\\\\\Phi _\\alpha ^{\\alpha 0}(ik) = (1-\\tau _i)n_k^{(0)} - \\tau _i(1-n_k^{(0)}), \\\\\\Phi _\\alpha ^{0\\alpha }(ik) = \\tau _k(1-n_i^{(0)}) - (1-\\tau _k)n_i^{(0)}.\\end{array}$ Here $\\tau _i=0$ for $A$ -type site $i$ and $\\tau _i=1$ if site $i$ has type $B$ .",
"All the coefficients $\\Phi _p^{qq^{\\prime }}$ not listed in (REF ) are equal to zero.",
"For example when all the three indexes $p$ , $q$ and $q^{\\prime }$ are spin indexes $\\Phi _p^{qq^{\\prime }} = \\Phi _\\alpha ^{\\beta \\gamma } = 0$ .",
"The term $(d/dt)_{ij}\\overline{c}_{ij,I^{\\prime }}^{pq,P^{\\prime }}$ is closely related to the correlation flow between correlations $\\overline{c}_{i,I^{\\prime }}^{q^{\\prime },P^{\\prime }}$ and $\\overline{c}_{k,I^{\\prime }}^{q^{\\prime },P^{\\prime }}$ : $\\left(\\frac{d}{dt}\\right)_{ij}\\overline{c}_{ij,I^{\\prime }}^{pq,P^{\\prime }} = - \\Phi ^{pq}_{q^{\\prime }}(ij) J_{ij;I^{\\prime }}^{q^{\\prime };P^{\\prime }} - \\\\\\Upsilon _{ij}^{pq} \\Theta _{ij,I^{\\prime }}^{pq,P^{\\prime }} \\left( \\overline{\\varphi }_{ij,I^{\\prime }}^{pq,P^{\\prime }} - \\overline{\\varphi }_{ij,I^{\\prime }}^{qp,P^{\\prime }} \\right)$ Note that the same coefficients $\\Phi _p^{qq^{\\prime }}$ enter the equations (REF ) and (REF ).",
"The second term in r.h.s.",
"of Eq.",
"(REF ) is related to the relaxation of coherent singlet-triplet combinations.",
"It contains the coefficient $\\Upsilon _{ij}^{pq}$ that is equal to unity when sites $i$ and $j$ have different types and $p$ and $q$ are spin indexes.",
"Otherwise, $\\Upsilon _{ij}^{pq} = 0$ .",
"The term $(d/dt)_i \\overline{c}_{i,I^{\\prime }}^{\\alpha ,P^{\\prime }}$ related to the spin rotation and relaxation should also be expressed in terms of potentials $\\left(\\frac{d}{dt}\\right)_i \\overline{c}_{i,I^{\\prime }}^{\\alpha ,P^{\\prime }} = \\Theta _{i,I^{\\prime }}^{\\alpha ,P^{\\prime }} \\left( \\epsilon _{\\alpha \\beta \\gamma } \\Omega _{i,\\beta } \\overline{\\varphi }_{i,I^{\\prime }}^{\\gamma ,P^{\\prime }} - \\frac{\\overline{\\varphi }_{i,I^{\\prime }}^{\\alpha ,P^{\\prime }}}{\\tau _s} \\right).$ In a stationary system the derivatives $d \\overline{c}_I^P/dt$ are equal to zero.",
"Therefore the equations (REF ), (REF ),(REF ), (REF ) and (REF ) compose a closed system of linear equations for the correlations potentials.",
"It incudes all the charge and spin correlations.",
"The total number of these correlations is extremely large, $4^N$ where $N$ is the number of sites.",
"However, one can hope that correlations between sites at very large distances and the correlations of very high order are not relevant for the electron transport and can be neglected.",
"Actually, to treat reasonably large systems some of the correlations should be neglected to make the system of equations solvable.",
"The idea of CKE approach is to write the equations in general form relevant for arbitrary correlations and make the cutoff at the “final step” taking into account the structure of considered system (that defines the correlations that are really relevant) and the possibility to numerically solve the system of equations of the desired size.",
"When some correlations are neglected in this way, the potentials $\\overline{\\varphi }_I^P$ of these correlations are considered to be equal to zero in all the equations."
],
[
"Magnetoresistance due to the relaxation of spin correlations",
"In this section, the discussed approach to the theory of hopping transport with spin correlations is applied to the bipolaron mechanism of OMAR.",
"As it was discussed in the previous section, it is necessary to cut the system of kinetic equations at some point.",
"The most simple cutoff is the model when only the correlations in close pairs of sites are considered.",
"In this case, it is possible to reduce the problem to a network of modified Miller-Abrahams resistors.",
"This approach was used in [24], [22] with the semi-classical model of spins up and down.",
"However, in the present study, the quantum nature of spin correlations is taken into account and the expressions for resistors are different from [22].",
"This model is discussed in Sec.",
"REF .",
"It allows the analytical solution in the limiting cases of fast and slow hopping.",
"The long-range and high-order correlations can be taken into account with the numerical solution of kinetic equations.",
"Such solutions are provided in Sec.",
"REF and are compared with analytical results."
],
[
"Modified resistor model",
"When the long-range spin correlations are neglected the rate equation for spin correlations $\\overline{c}_{ij}^{\\alpha \\beta }$ includes only the spin generation due to the electron flow $J_{ij}$ and the internal spin dynamics.",
"The effect of other sites is reduced to spin relaxation.",
"When the correlation $i-j$ is considered in this model the spins on other sites are assumed to be in thermal equilibrium.",
"The transition of correlation $i-j$ to other sites can be formally included into internal spin dynamics as additional source of relaxation.",
"$\\frac{d}{dt} \\overline{c}_{ij}^{\\alpha \\beta } = -R_{\\alpha \\beta ;\\alpha ^{\\prime }\\beta ^{\\prime }}^{(ij)}\\overline{c}_{ij}^{\\alpha ^{\\prime }\\beta ^{\\prime }} +\\delta _{\\alpha \\beta }(\\tau _i - \\tau _j)J_{ij}^{0}.$ Here ${R}_{\\alpha \\beta ;\\alpha ^{\\prime }\\beta ^{\\prime }}^{(ij)}$ is a matrix that describes the dynamics and relaxation of correlations.",
"${ R}_{\\alpha \\beta ;\\alpha ^{\\prime }\\beta ^{\\prime }}^{(ij)} = \\gamma _{ij} \\delta _{\\alpha \\alpha ^{\\prime }}\\delta _{\\beta \\beta ^{\\prime }} - \\epsilon _{\\alpha \\gamma \\alpha ^{\\prime }}\\delta _{\\beta \\beta ^{\\prime }} \\Omega _{i,\\gamma } - \\\\ \\epsilon _{\\beta \\gamma \\beta ^{\\prime }} \\delta _{\\alpha \\alpha ^{\\prime }}\\Omega _{j,\\gamma } + \\frac{\\Gamma _{ij}}{(s_i^0)_{eq} (s_j^0)_{eq}}(\\delta _{\\alpha \\alpha ^{\\prime }}\\delta _{\\beta \\beta ^{\\prime }} - \\delta _{\\alpha \\beta ^{\\prime }}\\delta _{\\beta \\alpha ^{\\prime }}).$ $\\gamma _{ij}$ is the effective rate of relaxation of spin correlations either due to phenomenological on-site spin relaxation mechanism or due to electron transition to or from other sites.",
"$\\gamma _{ij} = \\frac{1}{\\tau _s} + \\sum _{k\\ne i,j}\\left( T_{ki}^{(\\alpha )} + T_{kj}^{(\\alpha )} \\right).$ Note that spin transition rates $T_{ki}^{\\alpha }$ do not depend on value of spin index $\\alpha = x,y,z$ .",
"The index is kept only to show that it is a spin index, not the charge index 0.",
"It is assumed in the present section that $i-j$ is the resistor that controls the resistivity of some mesoscopic part of the sample.",
"It happens when the rate of hopping inside the pair $i-j$ is slow compared to the hopping between this pair and other sites.",
"It allows to assume that $\\Gamma _{ij} \\ll \\gamma _{ij}$ .",
"Therefore $\\Gamma _{ij}$ and the last term in r.h.s.",
"part of Eq.",
"(REF ) is neglected in the present section.",
"It is possible to give a closed expression for $J_{ij}^0$ with account to short-range pair correlations in terms of inverse matrix $({R}^{(ij)})^{-1}$ .",
"$J_{ij}^0 = \\frac{\\varphi _j^0 - \\varphi _i^0}{\\Gamma _{ij}^{-1} + {\\cal F}_s(ij) + {\\cal F}_c(ij)},$ ${\\cal F}_s(ij) = \\frac{ \\delta _{\\alpha \\beta }\\delta _{\\alpha ^{\\prime }\\beta ^{\\prime }} \\left({R}^{(ij)}\\right)^{-1}_{\\alpha \\beta ;\\alpha ^{\\prime }\\beta ^{\\prime }} (\\tau _i -\\tau _j)^2}{(s_{i}^0)_{eq} (s_j^0)_{eq}},$ ${\\cal F}_c (ij) = \\frac{\\left(n_i^{(0)} - n_j^{(0)}\\right)^2 \\left( \\sum _{k \\ne i,j} T_{ki}^{(0)} + T_{kj}^{(0)} \\right)^{-1}}{n_i^{(0)}n_j^{(0)} (1-n_i^{(0)}) (1-n_j^{(0)}) }.$ Here $\\Gamma _{ij}^{-1}$ corresponds to ordinary Miller-Abrahams resistance.",
"${\\cal F}_s$ and ${\\cal F}_c$ describe the additional resistance that appear due to spin and charge correlations correspondingly.",
"The effect of the external magnetic field on the conductivity is incorporated in ${\\cal F}_s$ .",
"As shown in eqs.",
"(REF ,REF ) it depends on the vectors of on-site rotation frequencies $\\mathbf {\\Omega }_i$ and $\\mathbf {\\Omega }_j$ that are proportional to the sum of hyperfine field and applied external magnetic field $\\mathbf {\\Omega }_i = \\mu _b g ({\\bf H} + {\\bf H}_{\\rm hf}^{(i)})/\\hbar $ .",
"However, even if the system is composed from only one resistor the averaging over $\\mathbf {\\Omega }_i$ and $\\mathbf {\\Omega }_j$ is required.",
"The hyperfine fields are slowly changed due to the nuclear spin dynamics.",
"Although this dynamics is considered to be slow compared to electron hops and electron spin rotation, it is usually fast compared to the current measurement procedures.",
"Therefore the final expression for dc resistivity should be averaged over hyperfine fields.",
"I assume that hyperfine fields have normal distribution $p(H_{{\\rm hf},\\alpha }^{(i)}) = \\frac{1}{\\sqrt{2\\pi }H_{\\rm hf}} \\exp \\left[ - \\frac{\\left(H_{{\\rm hf},\\alpha }^{(i)}\\right)^2}{2H_{\\rm hf}^2} \\right].$ Here $H_{\\rm hf}$ is the typical value of hyperfine fields.",
"The different components of the hyperfine field on a given site and the fields on different sites are considered not to be correlated.",
"The spin correlation part of resistance is related to the reverse relaxation function ${\\cal R}(\\gamma _{ij}, H, H_{\\rm hf})$ : ${\\cal F}_s = (\\tau _i-\\tau _j)^2 {\\cal R}/(s^{0}_i)_{eq}(s^{0}_j)_{eq}$ .",
"${\\cal R} = \\left\\langle \\delta _{\\alpha \\beta }\\delta _{\\alpha ^{\\prime }\\beta ^{\\prime }} (R^{(ij)})^{-1}_{\\alpha \\beta ;\\alpha ^{\\prime }\\beta ^{\\prime }} \\right\\rangle _{\\rm hf}.$ Here $\\langle ... \\rangle _{\\rm hf}$ means the averaging over the hyperfine fields.",
"${\\cal R}$ can be thought of as the time of relaxation of probabilities for the two spins to be in singlet or triplet state.",
"In the general case ${\\cal R}(\\gamma _{ij}, H, H_{\\rm hf})$ can be found numerically.",
"However, it is possible to find it analytically in the limiting cases of slow and fast hops.",
"Figure: Reverse relaxation function ℛ(h ext /h hf ){\\cal R}(h_{\\rm ext}/h_{\\rm hf}) corresponding to different relations γ ij /h hf \\gamma _{ij}/h_{\\rm hf}.",
"Solid blue curve in all the figures is numeric calculation of ℛ{\\cal R}.",
"Yellow dash-dot curve in Fig.",
"(A) and (B) is approximation with Eq. ().",
"Red dashed curve in Fig.",
"(B) is the non-Lorentz fit described in text.",
"Red dashed curve on Fig.",
"(C)-(D) is the Lorentzian fit.",
"Yellow dash-dot curve in Fig.",
"(D)-(F) is approximation with Eq. ().",
"Inset in Fig.",
"(C) shows the curves at small magnetic fields.In the limit of fast hops the rate of relaxation $\\gamma _{ij}$ due to electron transition to other sites is fast compared to the typical rate of rotation in hyperfine fields $h_{\\rm hf} = \\mu _b g H_{\\rm hf}/\\hbar $ .",
"The rate of rotation in the external magnetic field $h_{\\rm ext} = \\mu _b g H/\\hbar $ is arbitrary.",
"In this case relaxation matrix $R^{(ij)}$ can be divided into $R^{(ij)}_1$ related to hopping and rotation in the external magnetic field $R^{(ij)}_1 = - \\gamma _{ij} \\delta _{\\alpha \\alpha ^{\\prime }}\\delta _{\\beta \\beta ^{\\prime }} + \\mu _b g H(\\epsilon _{\\alpha z \\alpha ^{\\prime }}\\delta _{\\beta \\beta ^{\\prime }} + \\epsilon _{\\beta z \\beta ^{\\prime }} \\delta _{\\alpha \\alpha ^{\\prime }})/\\hbar $ and $R^{(ij)}_2$ related to the rotation in hyperfine fields.",
"The first of this matrices $R^{(ij)}_1$ can be inverted analytically.",
"The second one can be considered as a small perturbation.",
"The total inverse relaxation matrix averaged over hyperfine fields can be approximately expressed as $\\left\\langle \\left(R^{(ij)}\\right)^{-1} \\right\\rangle _{\\rm hf}= \\left(R^{(ij)}_1\\right)^{-1} + \\\\ \\left\\langle \\left(R^{(ij)}_1\\right)^{-1} R^{(ij)}_2 \\left(R^{(ij)}_1\\right)^{-1} R^{(ij)}_2 \\left(R^{(ij)}_1\\right)^{-1}\\right\\rangle _{\\rm hf}.$ In principle the expression for $(R^{(ij)})^{-1}$ also includes the first order term $(R^{(ij)}_1)^{-1} R^{(ij)}_2 (R^{(ij)}_1)^{-1}$ , however, it becomes equal to zero after the averaging over hyperfine fields.",
"With straightforward calculations, Eq.",
"(REF ) leads to explicit expression for the function ${\\cal R}$ ${\\cal R} = \\frac{1}{\\gamma _{ij}} \\left[3 - 4 \\frac{h_{\\rm hf}^2}{\\gamma _{ij}^2} \\left(1 + \\frac{2}{1 + h_{\\rm ext}^2/\\gamma _{ij}^2}\\right) \\right].$ The dependence of resistance on the magnetic field corresponding to eq.",
"(REF ) is described by Lorentz function.",
"Note that its width is controlled not by the relation of external magnetic and hyperfine fields $h_{\\rm ext}/h_{\\rm hf}$ but by the relation of hopping rate ant the rate of rotation in the external field $h_{\\rm ext}/\\gamma _{ij}$ .",
"The magnetoresistance is relatively weak due to the small prefactor $h_{\\rm hf}^2/\\gamma _{ij}^2$ .",
"The opposite limit is the situation of slow hopping, $\\gamma _{ij} \\ll h_{\\rm hf}$ .",
"In this case it is possible to use the random phase approximation.",
"It is assumed that component of electron spin on site $i$ normal to the on-site effective magnetic field ${\\bf H} + {\\bf H}_{\\rm hf}^{(i)}$ relaxes very fast due to the rotation around this field.",
"However, the component along ${\\bf H} + {\\bf H}_{\\rm hf}^{(i)}$ is conserved until the electron is transferred to some other site.",
"In this case ${\\cal R}$ is proportional to the averaged squared cosinus of the angle between on-site fields ${\\cal R} = \\left\\langle \\cos ^2 \\left({\\bf H}^{(i)}, {\\bf H}^{(j)}\\right) \\right\\rangle _{\\rm hf}/\\gamma _{ij}.$ Here ${\\bf H}^{(i)} = {\\bf H} + {\\bf H}_{\\rm hf}^{(i)}$ .",
"The averaging in (REF ) can be done analytically in terms of special functions ${\\cal R} = \\frac{1}{3\\gamma _{ij}} + \\frac{12}{\\gamma _{ij}H^6} \\times \\\\ \\left[\\frac{\\sqrt{2}}{6}H(H^2-3H^2_{\\rm hf}) + H_{\\rm hf}^3 D_+\\left( \\frac{H}{\\sqrt{2}H_{\\rm hf}}\\right)\\right]^2.$ Here $D_+(x)$ is the Dawson function $D_+(x) = e^{-x^2} \\int _0^x e^{t^2} dt$ .",
"Eq.",
"(REF ) shows that in slow hopping limit the dependance of resistance on magnetic field has non-Lorentz shape.",
"It saturates when applied magnetic field is much larger than hyperfine fields, while $\\gamma _{ij}$ controls its overall strength.",
"In Fig.",
"REF I compare the approximate expressions (REF ) and (REF ) with the function ${\\cal R}$ calculated numerically.",
"It can be seen that Eq.",
"(REF ) can quantitatively describe ${\\cal R}$ only for quite large values of hopping rate $\\gamma _{ij} \\gtrsim 30h_{\\rm hf}$ .",
"However, the dependence of ${\\cal R}$ on the applied magnetic field can be described by the Lorentzian $R = A + B/(h_{\\rm ext}^2 + \\widetilde{\\gamma }^2 )$ for significantly smaller $\\gamma _{ij} \\gtrsim 3h_{\\rm hf}$ .",
"Here $A$ , $B$ and $\\widetilde{\\gamma }$ are fitting parameters.",
"At smaller hopping rates $\\gamma _{ij} \\lesssim 0.3 h_{\\rm hf}$ the function ${\\cal R}$ becomes non-Loretzian.",
"It is most clearly seen when comparing the numeric results for ${\\cal R}$ with its Lorentzian fit at small magnetic fields as shown on the inset in Fig.",
"REF (C).",
"For small hopping rates $\\gamma _{ij} \\lesssim 0.05h_{\\rm hf}$ the non-Lorentzian fit related to Eq.",
"(REF ) becomes relevant.",
"It is the fit ${\\cal R} = A+B\\widetilde{\\cal R}(H,H_{\\rm hf},\\gamma _{ij})$ where $\\widetilde{\\cal R}$ is described with Eq.",
"(REF ).",
"For $\\gamma _{ij} \\lesssim 0.01 h_{\\rm hf}$ Eq.",
"(REF ) can describe the reverse relaxation function quantitatively.",
"It is interesting to compare the results of the present section with results of [22] where the correlations in close pairs of sites were considered with semi-classical model of spin.",
"In [22] OMAR was related to spin relaxation time $\\tau _s(H)$ that had a phenomenological dependence on the applied magnetic field.",
"The approach of the present section can describe quantum spin correlations with the same relaxation time.",
"When the spin relaxation is reduced to a single time $\\tau _s(H)$ the reverse relaxation function can be found analytically and the spin part of resistance is equal to ${\\cal F}_s = 3(\\tau _i-\\tau _j)^2\\tau _s(H)/[(1+\\gamma _{ij}\\tau _s(H))(s_i^0)_{eq} (s_j^0)_{eq} ]$ .",
"It is exactly 3 times larger than the correction to Miller-Abrahams resistor due to spin correlations obtained in [22].",
"Note that ${\\cal F}_c$ quantitatively agree with the correction to Miller-Abrahams resistor due to charge correlations obtained in [22].",
"The difference between the results is related to the quantum nature of spin correlations taken into account in the present paper.",
"It appears that semi-classical model of spin cannot quantitatively describe OMAR even if the spin relaxation is reduced to a single time $\\tau _s(H)$ ."
],
[
"Numerical results",
"This section includes the results of the numerical solution of correlation kinetic equations for several disordered systems.",
"The calculation is made with some of the long-range and high-order correlations taken into account.",
"The results are compared with the model described in Sec.",
"REF to show when the theory that includes only the correlations in close pairs of sites is applicable and when it is not.",
"I start from a single numerical sample consisting of 25 sites.",
"The sample is shown in Fig.",
"REF (A).",
"The sites are placed on a square lattice with the same overlap integrals $t_{ij}$ between neighbor sites.",
"The site energies are selected independently with normal distribution with the standard deviation $\\Delta E$ .",
"The temperature is $T = \\Delta E$ .",
"Some sites are randomly selected to have type B.",
"They are marked with crosses in Fig.",
"REF (A).",
"Other sites have type A.",
"Each site is ascribed with a random hyperfine field.",
"The typical rate of rotation in random fields is equal to the pre-exponential term in the hopping rate between neighbors $h_{\\rm hf} = w_0$ .",
"Here I define the pre-exponential term in the hopping rates as $w_0 = W_0|t_{ij}|^2$ .",
"The periodic boundary conditions are applied.",
"Figure: Results of CKE solution for random 10×1010\\times 10 numerical samples.",
"(A) the structure of a single sample.",
"(B) and (C) calculated magnetoresistance in different scales averaged over 10 random samples.In Fig.",
"REF (B) the calculated conductivity of this sample is presented.",
"The conductivity is calculated in four different approximations.",
"The blue dashed curve corresponds to $(1,2)$ approximation where pair correlations between sites $i$ and $j$ are included in the theory when the distance between sites $i$ and $j$ along the lattice bonds is 1.",
"It is the approximation used in Sec.",
"REF .",
"In general, the notation $(p,q)$ stands for the approximation when correlations up to the order $q$ are considered provided that the distance between sites in the correlations along lattice bonds is no longer than $p$ .",
"The yellow dash-dot curve corresponds to $(3,2)$ approximation when most pair correlations are taken into account.",
"Green solid curve corresponds to joint $(3,2)$ and $(2,4)$ approximation when most of pair correlations and correlations up to the 4-th order in close complexes of sites are considered.",
"The red dots stand for joint $(3,3)$ and $(2,5)$ approximation where most of the correlations of the third order and the correlations up to 5-th order in close complexes are included in the theory.",
"The introduction of new correlations into the theory decreases the calculated conductivity and increases its calculated dependence on the applied magnetic field.",
"However, the difference between $(3,2)+(2,4)$ and $(3,3)+(2,5)$ approximations is rather small and one can hope that $(3,2)+(2,4)$ approximation adequately describes the system.",
"In the following analysis of larger numerical samples, the correlations of order $q>2$ will be considered only for the distance between sites $p \\le 2$ .",
"The pair correlations will be considered at slightly larger distances.",
"Unfortunately, the number of spin correlations grows extremely fast with the correlation order and it was technically impossible to go beyond the $(3,3)+(2,5)$ approximation even for the quite small $5 \\times 5$ sample.",
"Note that the magnetic field dependence of conductivity shown in Fig.",
"REF (B) is not symmetric with respect to the inversion of the sign of the magnetic field.",
"It is related to the mesoscopic nature of the considered numerical sample.",
"There is a finite number of sites and each site is ascribed with the well-defined on-site hyperfine field.",
"It breaks the time-reversal symmetry of the calculated system.",
"In real samples (even in mesoscopic ones) the hyperfine fields slowly change in time.",
"It restores the time-reversal symmetry.",
"Therefore the dependence $\\sigma (H)$ for real samples is symmetric even if they are mesoscopic.",
"In Fig.",
"REF I show results for larger $10 \\times 10$ samples.",
"The results are averaged over 30 disorder configurations.",
"The properties of the numerical samples (except their size) are the same as in Fig.",
"REF .",
"Fig.",
"REF (A) shows the structure of one of these samples.",
"Fig.",
"REF (B) and (C) show the magnetoresistance $(R(H) - R(0))/R(0)$ in different scales.",
"Here $R(H)$ is the sample resistance in the external magnetic field $H$ .",
"The blue points stand for the magnetoresistance calculated with the numeric solution of CKE.",
"The pair correlations at the distance no longer than 4 and 4-th order correlations with the distance 2 were taken into account ($(4,2)+(2,4)$ approximation).",
"Green dashed curve is the fit with Lorentzian.",
"Yellow solid curve is the fit with function ${\\cal R}$ described in Sec.",
"REF .",
"It means that the expression $\\frac{R(H)-R(0)}{R(0)} = A \\times \\frac{{\\cal R}(\\widetilde{\\gamma }, H, H_{\\rm hf}) - {\\cal R}(\\widetilde{\\gamma }, 0, H_{\\rm hf}) }{{\\cal R}(\\widetilde{\\gamma }, 0, H_{\\rm hf})}$ was used for fitting.",
"Here $A$ and $\\widetilde{\\gamma }$ are the fitting parameters.",
"$\\widetilde{\\gamma }$ can be considered as the effective rate for a correlation to leave the pair of sites where it appeared.",
"The results shown in Fig.",
"REF (B) correspond to $\\widetilde{\\gamma } \\approx 2.3 h_{\\rm hf}$ .",
"$A$ is the general amplitude of magnetoresistance, it describes the relative part of sample resistance that is related to spin correlations.",
"The shape of resistance dependence on the magnetic field significantly deviates from Lorentzian.",
"It can be easily seen in Fig.",
"REF (C) where the results for small magnetic fields are shown on a close scale.",
"The fitting (REF ) has better agreement with numerical simulation.",
"Figure: Magnetoresistance in numerical samples with slow w 0 =0.1h hf w_0 = 0.1h_{\\rm hf} and fast w 0 =5h hf w_0 = 5h_{\\rm hf} hopping rates.",
"Blue dots on all subplots correspond to results of numerical solution of CKE.",
"Yellow solid curves correspond to Eq. ().",
"Green dashed curves in Figs (A) and (B) is the fit with Eq.",
"() where function ℛ{\\cal R} is described by Eq.",
"() and fitting hyperfine fields are artificially increased h hf →1.08h hf h_{\\rm hf} \\rightarrow 1.08h_{\\rm hf}.",
"Green dashed curve in Figs (C) and (D) is the fit with Lorentzian.In Fig.",
"REF the similar analysis is provided for similar numeric samples with different rates of hopping between neighbors $w_0 = 0.1h_{\\rm hf}$ and $w_0 = 5h_{\\rm hf}$ .",
"Other characteristics of the samples are the same as in the previous numeric experiment including the averaging over 30 disorder configurations.",
"Fig.",
"REF (A) and (B) shows the results for the samples were the hopping is slow, $w_0 = 0.1 h_{\\rm hf}$ .",
"Blue points correspond to the numeric solution of CKE.",
"Yellow solid curve corresponds to the fit with Eq.",
"(REF ), the fitting parameter $\\widetilde{\\gamma }$ was $\\widetilde{\\gamma } = 1.1h_{\\rm hf}$ .",
"Green dashed curve corresponds to fit with Eq.",
"(REF ), where function ${\\cal R}$ is described by Eq.",
"(REF ) (that is valid in the limit $\\widetilde{\\gamma } \\ll h_{\\rm hf}$ ) but the strength of hyperfine fields was artificially increased $h_{\\rm hf} \\rightarrow 1.08 h_{\\rm hf}$ to achieve better fit with numerical results.",
"Figure: Results for the numeric samples constructed with 3×33\\times 3 blocs.",
"(A) the structure of a numeric sample.",
"(B) magnetoresistance in (1,2)(1,2) approximation.",
"(C) magnetoresistance in (7,2) approximation.Note that the model from Sec.",
"REF does not take into account long-range and high-order correlations.",
"Therefore, the possibility of the description of numerical results with this model could not be taken for granted.",
"However, for the considered numerical samples, this description is possible and the effect of the simplifications made in Sec.",
"REF is reduced to small modification of $h_{\\rm hf}$ and to some changes in the amplitude of magnetoresistance (described with fitting parameter $A$ ).",
"In Fig.",
"REF (C) and (D) the results for numeric samples with fast hopping $w_0 = 5 h_{\\rm hf}$ are shown.",
"The results of the simulation (blue dots) agree with fitting with Eq.",
"(REF ) (yellow solid curve), where the value of fitting parameter $\\widetilde{\\gamma }$ is $8.7 h_{\\rm hf}$ .",
"At this $\\widetilde{\\gamma }$ the function ${\\cal R}$ can be described by Lorentzian, as shown with green dashed curve in Fig.",
"REF (C) and (D).",
"The provided numeric results show that in some cases the theory from Sec.",
"REF can be used as a toy model for the understanding more complex situations when long-range and high-order correlations are required to quantitatively calculate the magnetoresistance.",
"However, not all the lineshapes that appear in numeric experiments can be described with this toy model.",
"Let us consider the numeric sample shown in Fig.",
"REF .",
"It is constructed from $3\\times 3$ blocks, inside a block the sites have the same type and the hopping between them is fast $w_{0} = 3h_{\\rm hf}$ .",
"The blocks are connected with links with slow hopping $w_{0} = 0.3 h_{\\rm hf}$ .",
"The energies of sites are random with normal distribution with the standard deviation $\\Delta E = T$ .",
"The average energy of sites in $A$ -type block is $1.5T$ and in $B$ -type block it is $-1.5T$ .",
"The idea behind this sample is as follows.",
"The conductivity of the sample is controlled by the process of generation and recombination of electron-hole pairs in $AB$ pairs of sites.",
"The structure of the sample ensures that the generated electron and hole will stay near the pair of sites were they are generated for quite a long time.",
"However, they are trapped not on a single site but on a cluster of nine sites, therefore their spins not only rotate around local hyperfine fields but also relax due to hops between sites with different hyperfine fields.",
"The positive average energy of $A$ -type sites and the negative one of $B$ -type sites ensures that $A$ -type clusters contain a small number of electrons and $B$ -type clusters contain a small number of holes.",
"Physically this situation can correspond to small polymer molecules where the hopping between monomers of a single polymer is fast while the hops between different molecules are slow.",
"Note that for the relaxation of spin correlation in the discussed electron-hole pair all the 18 hyperfine fields in two neighbor blocks are relevant.",
"It cannot be captured with the toy model from Sec.",
"REF .",
"The structure of the described sample is shown in Fig.",
"REF (A).",
"Fig.",
"REF (B) shows the magnetoresistance of such samples calculated in the $(1,2)$ approximation that corresponds to the model of modified Miller-Abrahams resistor.",
"In Fig.",
"REF (C) the magnetoresistance calculated in $(7,2)$ approximation is shown.",
"In both of the cases, the magnetoresistance is averaged over 30 random disorder configurations (i.e.",
"the random hyperfine fields and site energies).",
"The magnetoresistance calculated in $(1,2)$ approximation is quite weak $\\sim 2\\%$ and its lineshape is Lorentzian.",
"However, $(1,2)$ approximation is not adequate for the description of the magnetoresistance in these samples because the correlations can easily leave the initial sites but are trapped in the clusters.",
"To take this trapping into account it is required to consider the correlations at the inter-site distance equal to 7.",
"When these correlations are taken into account the estimated magnetoresistance increases $\\sim 4$ times and its lineshape becomes non-Lorentzian.",
"It can be described with the expression $\\frac{R(H)-R(0)}{R(0)} \\propto \\frac{H^2}{(|H| + H_0)^2},$ where $H_0$ is a fitting parameter.",
"The expression (REF ) was used to describe the lineshape of OMAR in a number of experimental works [6], [7], [8], [9], [10].",
"I want to stress that the statistics of hyperfine fields is exactly the same in all the considered numerical samples.",
"However, the obtained OMAR lineshapes are different including the two shapes most commonly obtained in experiments: $H^2/(H^2 + H_{0}^2)$ and $H^2/(|H| + H_0)^2$ .",
"What is different in the numerical samples is the statistics of electron hops.",
"One can conclude that the shape of OMAR contains information about short-range transport in organic materials."
],
[
"discussion",
"Up to the recent time, the most known method for the calculation of hopping transport with the account to correlations of filling numbers was the numeric Monte-Carlo simulation.",
"When only the charge correlations are important it leads to the correct description of transport provided that the time of simulation is enough to achieve the averaging.",
"In [25] the Monte-Carlo simulation was used to prove that CKE approach also leads to correct results when a sufficient number of correlations are taken into account.",
"Therefore the Monte-Carlo simulation was considered to be “an arbiter” for CKE approximations.",
"However, the situation is different when the spin correlations are relevant.",
"In [13] the Monte-Carlo simulation with the semi-classical description of electron spins in terms of “spin up” and “down” was used to show the possibility of bipolaron mechanism of OMAR.",
"However, this simplified description cannot include spin rotation around hyperfine on-site fields.",
"Naturally, “up” and “down” spins cannot rotate.",
"Therefore the spin relaxation in [13] was described by a single relaxation time $\\tau _s$ with phenomenological dependence on the applied magnetic field.",
"Even when the spin relaxation is reduced to a single time $\\tau _s$ the semi-classical description of spin does not lead to the correct qualitative estimate of the spin-correlation part of resistance.",
"In [22] the semi-classical spin correlations were considered in the approximation when only pair correlations in close pairs of sites are taken into account.",
"It leads to correlation corrections of Miller-Abrahams resistors.",
"These corrections are compared with similar corrections due to quantum spin correlations in Sec.",
"REF .",
"The corrections due to charge correlations obtained in [22] and in the present study agree.",
"However, the corrections due to spin correlations obtained in Sec.",
"REF are exactly three times larger than the spin corrections to resistances in [22].",
"I believe that the reason for this difference is related to the quantum nature of spin that was not taken into account in [22].",
"Note that in [22] only the correlations in close pairs are taken into account.",
"This approximation is insufficient for the description of the OMAR shape $H^2/(|H|+H_0)^2$ that appears in the sample on Fig.",
"REF only when long-range pair correlations are included into the theory.",
"The author does not know about any way to make Monte-Carlo simulations where electron spin is allowed to have arbitrary direction (instead of only “up” and “down”) and take into account quantum spin correlations.",
"Any wavefunction of a single spin $1/2$ is the eigenfunction of some operator of spin projection $\\hat{s}_{\\alpha } = c_x \\hat{s}_x + c_y \\hat{s}_y + c_z \\hat{s}_z$ , where $c_x^2 + c_y^2 + c_z^2 = 1$ .",
"Therefore it is tempting to describe the electron spins $1/2$ as classical units vectors.",
"However, this model cannot describe the real quantum statistics of spins.",
"Consider for example the scalar product of two spins averaged over some ensemble $\\langle {\\bf s}_i {\\bf s}_j \\rangle = \\overline{s_{i}^x s_j^x} + \\overline{s_{i}^y s_j^y} + \\overline{s_{i}^z s_j^z}$ .",
"In classical statistics of unit vectors $-1 \\le \\langle {\\bf s}_i {\\bf s}_j \\rangle \\le 1$ .",
"The value 1 describes the vectors that always have the same direction.",
"The value $-1$ corresponds to the vectors that have opposite directions.",
"In quantum mechanics $-3 \\le \\langle {\\bf s}_i {\\bf s}_j \\rangle \\le 1$ .",
"The value $-3$ corresponds to the singlet state of the spins.",
"In the present study operator $\\hat{s}_i^z$ was selected to have eigenvalues 1 and $-1$ , however, it corresponds to the actual spin $1/2$ .",
"In quantum mechanics the eigenvalues of the operator $\\hat{l}^2$ of squared angular momentum are equal to $l(l+1)$ .",
"It the case of the angular momentum $1/2$ the only existing eigenvalue is $3/4$ .",
"Therefore the operator $(\\hat{\\bf s}_i)^2 = \\hat{s}_i^z\\hat{s}_i^z + \\hat{s}_i^x\\hat{s}_i^x + \\hat{s}_i^y\\hat{s}_i^y$ is always equal to 3.",
"When two spins are in the singlet state their total spin is equal to zero $(\\hat{\\bf s}_{i} + \\hat{\\bf s}_j)^2 = 0$ .",
"The product $\\hat{\\bf s}_i \\hat{\\bf s}_j$ in this case is well defined and is equal to ${\\bf s}_i {\\bf s}_j = [(\\hat{\\bf s}_{i} + \\hat{\\bf s}_j)^2 - (\\hat{\\bf s}_i)^2 - (\\hat{\\bf s}_i)^2]/2 = -3$ .",
"When the spins are in a triplet state, the total momentum of the system is equal to $l=1$ and $(\\hat{\\bf s}_{i} + \\hat{\\bf s}_j)^2 = 4l(l+1) = 8$ .",
"It leads to ${\\bf s}_i {\\bf s}_j = 1$ .",
"Therefore the quantum spin statistics is different from the statistics of classical vectors, although it can be described with the classical values: averaged spin polarizations and their correlations.",
"The Monte-Carlo calculations at least with a naive description of electron spins cannot be used to quantitatively calculate the magnetoresistance related to the spin correlations and act as “an arbiter” for CKE approach.",
"Although the properties of transport in organic semiconductors are studied for some time they are not completely understood.",
"The low-field mobility in organic semiconductors is often extremely small $\\sim 10^{-8} - 10^{-6}cm^2/Vs$ [35], [36], [37].",
"However, these small values can be related to the long-range correlations of electrostatic potential produced by the unscreened molecular dipoles [38], [39], [40].",
"There is an evidence that at small length-scales the electron mobility can be much higher [41].",
"The provided results show that some information about short-range mobility can be obtained from the measurements of OMAR.",
"The lineshape of organic magnetoresistance depends on hopping rates.",
"However, it is not related to the time for an electron to cross the macroscopic sample.",
"What is important is the time $\\tau _{sep}$ that is required for two spins to become separated with sufficient distance that prevents their meeting before their spin correlation relaxes.",
"A theoretical estimate of $\\tau _{sep}$ can be possible in the framework of complex numerical modeling of organic semiconductors similar to the one made in [29].",
"However, even the analysis of the provided simplified models can give some hints on $\\tau _{sep}$ and nature of the short-range transport.",
"When $\\tau _{sep}$ is small compared to the period of precession in hyperfine field $\\tau _{sep} h_{hf} \\ll 1$ OMAR is suppressed.",
"When $\\tau _{sep} h_{hf} \\gtrsim 1$ due to overall slow rate of hopping OMAR can be relatively strong $\\sim 10\\%$ as shown on Fig.",
"REF and its lineshape is described by Lorentzian or by Eq.",
"(REF ).",
"When some of the hops are fast but $\\tau _{sep} h_{hf} \\gtrsim 1$ due to bottlenecks with slow hops as in the sample shown in Fig.",
"REF the size of OMAR is still $\\sim 10\\%$ but is shape is close to $H^2/(|H| + H_0)^2$ .",
"The present study deals only with the most simple model with large Hubbard energy and small applied electric field.",
"In principle, it is possible to generalize CKE theory to include other cases.",
"In [25] the far from equilibrium CKE that can be applied for high electric fields are derived for charge correlations only.",
"In [22] the situation with arbitrary Hubbard energy is considered for close-range pair correlations with the semi-classical spin model.",
"It is shown in [22], [25] that the discussed generalizations make the theory much more complex.",
"The present work shows that different lineshapes of OMAR appear even in the simplest model due to the different properties of short-range transport."
],
[
"Conclusions",
"The system of correlation kinetic equations (CKE) is derived for the spin correlations in materials with hopping transport with large Hubbard energy for a small applied electric field.",
"The spins are assumed to be conserved in the hopping process and can rotate around on-site hyperfine fields.",
"The spin degrees of freedom are described with quantum mechanics as averaged products of spin operators.",
"The derived CKE approach allows to describe the bipolaron mechanism of OMAR.",
"It is shown that the shape of the magnetic field dependence of resistivity contains information about short-range electron transport.",
"Different statistics of hopping rates lead to different OMAR lineshapes including the empirical laws $H^2/(H^2 + H_0^2)$ and $H^2/(|H| + H_0 )^2$ that are often used to describe experimental data.",
"The author is grateful for many fruitful discussions to Y.M.",
"Beltukov, A.V.",
"Nenashev, D.S.",
"Smirnov, V.I.",
"Kozub and V.V.",
"Kabanov.",
"The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “Basis”.",
"A.V.S.",
"acknowledges support from Russian Foundation for Basic Research (Grant No.",
"19-02-00184)."
],
[
"Derivation of $(d/dt)_{ij}$ term in kinetic equations",
"In this appendix I derive the term $(d/dt)_{ij} \\overline{s}_I^P$ in the kinetic equation.",
"It is supposed that sites $i$ and $j$ are included into set $I$ .",
"For definiteness the site $i$ is considered to be an $A$ -type site and $j$ to have type $B$ .",
"$\\overline{s}_I^P$ can be expressed as quantum mechanical average of the operator $\\overline{s}_I^P = \\left\\langle \\hat{s}_I^P \\right\\rangle = \\left\\langle \\hat{s}_i^p \\hat{s}_j^q \\hat{s}_{I^{\\prime }}^{P^{\\prime }} \\right\\rangle .$ Here when the index $p$ corresponds to $x$ ,$y$ and $z$ , $\\hat{s}_i^p$ is the spin polarization operator $\\hat{s}_i^{\\alpha } = a_{i,n}^+ (\\sigma _\\alpha )_{nm} a_{i,m}$ .",
"$\\sigma _\\alpha $ is a Pauli matrix.",
"$a_{i,n}^+$ and $a_{i,m}$ are the creation and destruction operators for electron on site $i$ .",
"Indexes $n$ and $m$ correspond to spin states $\\uparrow $ or $\\downarrow $ .",
"The operator $\\hat{s}_i^0$ is the operator of single occupation and can be expressed as follows $\\hat{s}_i^0 = a_{i\\uparrow }^+ a_{i\\uparrow } + a_{i\\downarrow }^+ a_{i\\downarrow } - 2 a_{i\\uparrow }^+ a_{i\\uparrow } a_{i\\downarrow }^+ a_{i\\downarrow }.$ The expression (REF ) is valid for any type of site, therefore $\\hat{s}_{j}^0$ can be expressed in a similar way.",
"The transitions between sites $i$ and $j$ can be described by the operator $\\widehat{H}_{ij}$ (it is the term in Hamiltoinan related to these transitions).",
"$\\widehat{H}_{ij} = t_{ij} \\widehat{\\tau }_{ij} \\Phi _{ij} + t_{ji} \\widehat{\\tau }_{ji}\\Phi _{ji}, \\quad \\tau _{ij} = a_{i\\uparrow }^+a_{j\\uparrow } + a_{i\\downarrow }^+a_{j\\downarrow }.$ The operator $\\Phi _{ij}$ describes the interaction with phonons that appears in the transition term of Hamiltonian $\\widehat{H}_{ij}$ after the polaron transformation [42].",
"With the approximations corresponding to hopping transport the time derivative of operator $\\hat{s}_I^P$ can be expressed as follows $ \\left(\\frac{d}{dt}\\right)_{ij} \\hat{s}_I^P = -\\frac{1}{\\hbar ^2} \\int _{-\\infty }^{t} \\left\\langle \\left[ \\widehat{H}_{ij} (t), \\left[\\widehat{H}_{ij} (t^{\\prime }) \\hat{s}_I^P(t) \\right] \\right] \\right\\rangle _{ph} dt^{\\prime }.$ Here square brackets denote commutator $[\\widehat{A},\\widehat{B}] = \\widehat{A}\\widehat{B} - \\widehat{B}\\widehat{A}$ .",
"$\\left\\langle ...\\right\\rangle _{ph}$ means the averaging over phonon variables with equilibrium distribution of phonons.",
"Note that even with the simplifications made eq.",
"(REF ) contains not only hopping terms but also terms corresponding to quantum mechanical perturbation of electron wavefunctions on sites $i$ and $j$ and to exchange interaction between electrons that is neglected in our study.",
"In further calculations I keep only the terms related to hopping process that are proportional to the hopping rates $W_{ij} = \\frac{1}{\\hbar ^2} |t_{ij}|^2 \\left\\langle \\int _{-\\infty }^0 \\Phi _{ji}(t) \\Phi _{ij}(0) e^{(i/\\hbar )(\\varepsilon _j - \\varepsilon _i)t} +\\Phi _{ji}(0)\\Phi _{ij}(t) e^{(i/\\hbar )(\\varepsilon _i-\\varepsilon _j)t} dt\\right\\rangle _{ph},$ $W_{ji} = \\frac{1}{\\hbar ^2} |t_{ij}|^2 \\left\\langle \\int _{-\\infty }^0 \\Phi _{ij}(t) \\Phi _{ji}(0) e^{(i/\\hbar )(\\varepsilon _i - \\varepsilon _j)t} +\\Phi _{ij}(0)\\Phi _{ji}(t) e^{(i/\\hbar )(\\varepsilon _j-\\varepsilon _i)t} dt\\right\\rangle _{ph}.$ The term in $(d/dt)_{ij} \\widehat{s}_I^P$ proportional to $W_{ji}$ and related to hops $i\\rightarrow j$ is equal to $- W_{ji}(\\widehat{\\tau }_{ij}\\widehat{\\tau }_{ji}\\widehat{s}_I^P + \\overline{s}_I^P\\widehat{\\tau }_{ij}\\widehat{\\tau }_{ji})/2 $ .",
"The term related to $j \\rightarrow i$ hops is $W_{ij} \\widehat{\\tau }_{ji} \\widehat{s}_I^P \\widehat{\\tau }_{ij}$ .",
"Here I took into account that site $i$ cannot be double-occupied and site $j$ cannot have zero electrons.",
"The following computation is quite cumbersome but straightforward operator algebra.",
"There are two useful relations that make it simpler $\\hat{\\tau }_{ij}\\hat{\\tau }_{ji} = \\frac{\\hat{s}_i^{0}\\hat{s}_j^{0} - \\sum _{\\alpha } \\hat{s}_i^{\\alpha }\\hat{s}_j^{\\alpha } }{2},\\quad \\hat{s}_{i}^{\\alpha }\\hat{s}_i^{\\beta } = \\delta _{\\alpha \\beta } \\hat{s}_i^{0} + i\\epsilon _{\\alpha \\beta \\gamma } \\hat{s}_i^{\\gamma },$ where $\\epsilon _{\\alpha \\beta \\gamma }$ is Levi-Civita symbol.",
"The operator algebra yields $\\left( \\frac{d}{dt} \\right)_{ij} \\hat{s}_i^0\\hat{s}_j^0 \\hat{s}_{I^{\\prime }}^{P^{\\prime }} = 2W_{ij} \\hat{n}_{j\\uparrow } \\hat{n}_{j\\downarrow } (1 - \\hat{n}_{i\\uparrow })(1 - \\hat{n}_{i\\downarrow }) \\hat{s}_{I^{\\prime }}^{P^{\\prime }} -\\frac{W_{ji}}{2} \\left(\\hat{s}_i^0 \\hat{s}_j^0 - \\sum _{\\alpha } \\hat{s}_i^{\\alpha }\\hat{s}_j^{\\alpha } \\right)\\hat{s}_{I^{\\prime }}^{P^{\\prime }},$ $ \\left( \\frac{d}{dt} \\right)_{ij} \\hat{s}_i^0\\hat{s}_j^\\alpha \\hat{s}_{I^{\\prime }}^{P^{\\prime }} = \\frac{W_{ji}}{2}\\left(\\hat{s}_i^{\\alpha }\\hat{s}_j^0 - \\hat{s}_i^{0}\\hat{s}_j^\\alpha \\right) \\hat{s}_{I^{\\prime }}^{P^{\\prime }},$ $\\left( \\frac{d}{dt} \\right)_{ij} \\hat{s}_i^\\alpha \\hat{s}_j^0 \\hat{s}_{I^{\\prime }}^{P^{\\prime }} = \\frac{W_{ji}}{2}\\left(\\hat{s}_i^{0}\\hat{s}_j^\\alpha - \\hat{s}_i^{\\alpha }s_j^0\\right) \\hat{s}_{I^{\\prime }}^{P^{\\prime }},$ $\\left( \\frac{d}{dt} \\right)_{ij} \\hat{s}_i^\\alpha \\hat{s}_j^\\beta \\hat{s}_{I^{\\prime }}^{P^{\\prime }} =-2W_{ij} \\delta _{\\alpha \\beta } \\hat{n}_{j\\uparrow } \\hat{n}_{j\\downarrow } (1 - \\hat{n}_{i\\uparrow })(1 - \\hat{n}_{i\\downarrow }) \\hat{s}_{I^{\\prime }}^{P^{\\prime }} +\\frac{W_{ji}}{2} \\delta _{\\alpha \\beta } \\left(\\hat{s}_i^0 \\hat{s}_j^0 - \\sum _{\\alpha } \\hat{s}_i^{\\gamma }\\hat{s}_j^{\\gamma } \\right)\\hat{s}_{I^{\\prime }}^{P^{\\prime }} - \\\\\\frac{W_{ji}}{2} \\left(\\hat{s}_i^{\\alpha }\\hat{s}_j^{\\beta } - \\hat{s}_i^{\\beta } \\hat{s}_j^{\\alpha } \\right)\\hat{s}_{I^{\\prime }}^{P^{\\prime }}$ The quantum mechanical averaging of these equations leads to eqs.",
"(REF -REF ) from the main text."
]
] | 2001.03404 |
[
[
"Accelerated and nonaccelerated stochastic gradient descent with model\n conception"
],
[
"Abstract In this paper, we describe a new way to get convergence rates for optimal methods in smooth (strongly) convex optimization tasks.",
"Our approach is based on results for tasks where gradients have nonrandom small noises.",
"Unlike previous results, we obtain convergence rates with model conception."
],
[
"Введение",
"В данной работе рассматривается задача стохастической оптимизации , , $f(x) = \\mathbb {E}[f(x,\\xi )] \\rightarrow \\min _{x\\in Q \\subseteq \\mathbb {R}^n},$ где множество $Q$ предполагается выпуклым и замкнутым, $\\xi $ — случайная величина, математическое ожидание $\\mathbb {E}[f(x,\\xi )]$ определено и конечно для любого $x \\in Q$ , функция $f(x)$ – $\\mu $ -сильно выпуклая в 2-норме ($\\mu \\ge 0$ ) и имеющая $L$ -Липшицев градиент, т.е.",
"для всех $x,y\\in Q$ $f(x) + \\langle \\nabla f(x), y - x \\rangle + \\frac{\\mu }{2}\\Vert y-x\\Vert _2^2 \\le f(y) \\le f(x) + \\langle \\nabla f(x), y - x \\rangle + \\frac{L}{2}\\Vert y-x\\Vert _2^2.$ Предположим, что есть доступ к $\\nabla f(x,\\xi )$ – стохастическому градиенту $f(x)$ , удовлетворяющему следующим условиямЗаметим, что для задач минимизации функционалов вида суммы условие ограниченности (субгауссовской) дисперсии может не выполняться даже в очень простых (квадратичных) ситуациях.",
"Как следствие, в общем случае приводимые далее результаты не распространяются на задачи минимизации функционалов вида суммы, в которых в качестве стохастического градиента выбирается градиент случайно выбранного слагаемого .",
"(несмещенность и субгауссовость хвостов распределения, с субгауссовской дисперсией $\\sigma ^2$ ) $\\mathbb {E}\\left[\\nabla f(x, \\xi )\\right] \\equiv \\nabla f(x),\\mathbb {E}\\left[\\exp \\left(\\frac{ \\Vert \\nabla f(x, \\xi )-\\mathbb {E}[\\nabla f(x, \\xi )]\\Vert _2^2}{\\sigma ^2}\\right)\\right]\\le \\exp (1),$ для всех $x \\in Q$ .",
"Тогда после $N$ вычислений $\\nabla f(x,\\xi )$ с большой вероятностью имеемЗдесь и далее “с большой вероятностью” – означает с вероятностью $\\ge 1 - \\gamma $ , а $\\tilde{O}(\\cdot )$ означает то же самое, что $O(\\cdot )$ , только числовой множитель зависит от $\\ln \\left( N/\\gamma \\right)$ .",
", , $f(x_N) - f(x_*) = \\tilde{O}\\left(\\min \\left\\lbrace \\frac{LR^2}{N^p} + \\frac{\\sigma R}{\\sqrt{N}},{\\color {black}LR^2}\\exp \\left(-\\left(\\frac{\\mu }{L}\\right)^{\\frac{1}{p}}\\frac{N}{2}\\right) + \\frac{\\sigma ^2}{\\mu N}\\right\\rbrace \\right),$ где $x_*$ – решение задачи (REF ), $R = \\Vert x_0 - x_*\\Vert _2$ , $x_0$ – точка старта, $p=1$ отвечает стохастическому градиентному спуску, а $p=2$ ускоренному стохастическому спуску.",
"С другой стороны известно (см.",
", , , , ), что если для задачи (REF ) доступен неточный градиент $\\nabla _{\\delta } f(x)$ , удовлетворяющий для всех $x,y \\in Q$ ослабленному условию $L$ -Липшицевости градиента $f(x) + \\langle \\nabla _{\\delta } f(x), y-x \\rangle + \\frac{\\mu }{2}\\Vert y-x\\Vert ^2_2 - \\delta _1 \\le f(y) \\le f(x) + \\langle \\nabla _{\\delta } f(x), y-x \\rangle + \\frac{L}{2}\\Vert y-x\\Vert ^2_2 + \\delta _2,$ то после $N$ вычислений $\\nabla _{\\delta } f(x)$ для соответствующих модификаций градиентного и ускоренного градиентного спуска можно получить оценку, аналогичную (REF )Для $p=2$ нужно ввести дополнительные ограничения на $\\delta _1$ , см.",
"ниже.",
"$\\tilde{O}\\left(\\min \\left\\lbrace \\frac{LR^2}{N^p} + \\delta _1 + N^{p-1}\\delta _2,{\\color {black}LR^2}\\exp \\left(-\\left(\\frac{\\mu }{L}\\right)^{\\frac{1}{p}}\\frac{N}{2}\\right) + \\delta _1 + \\left(\\frac{L}{\\mu }\\right)^{\\frac{p-1}{2}}\\delta _2 \\right\\rbrace \\right).$ В данной статье подмечается, что результат (REF ) может быть полученВ сильно выпуклом случае только в смысле сходимости по математическому ожиданию, без оценки вероятностей больших отклонений.",
"из результата (REF ).",
"Более того, сделанное наблюдение, оказывается возможным провести и в модельной общности.",
"Данная работа имеет следующую структуру.",
"В разделе рассматривается концепция неточного градиента функции и для нее приводится соответствующая теорема сходимости, дополнительно для задачи из (REF ) приводится простой способ того, как можно получить оптимальные оценки.",
"В разделе рассматривается концепция неточной модели функции и доказываются все основные результаты.",
"Отметим, что в начале раздела и разделе REF приводятся примеры некоторых классов негладких задач (композитная оптимизация и оптимизация максимума нескольких гладких функций), для которых возможно применять предложенные концепции неточной модели функции.",
"Это позволяет говорить о том, что полученные в работе результаты об оценках сложности, эффективные на классах выпуклых гладких задач, верны и для некоторых типов выпуклых негладких задач."
],
[
"Основные результаты",
"Ограничимся для компактности изложения пояснением перехода от (REF ) к (REF ) для случая $\\mu = 0$ , и с теми же целями переопределим $R = \\max _{x,y \\in Q} \\Vert x-y\\Vert _2$ (в действительности, все приведенные далее в этом разделе результаты верны для $R = \\Vert x_0 - x_*\\Vert _2$ ; показывается аналогично ).",
"Будем далее дополнительно предполагать, что в ослабленном условии $L$ -Липшицевости градиента для неточного градиента $\\nabla _{\\delta } f(x)$ ошибка $\\delta _1$ зависит от $y$ и $x$ : $f(x) + \\langle \\nabla _{\\delta } f(x), y-x \\rangle - \\delta _1(y,x) \\le f(y) \\le f(x) + \\langle \\nabla _{\\delta } f(x), y-x \\rangle + \\frac{L}{2}\\Vert y-x\\Vert ^2_2 + \\delta _2,$ Первое важное наблюдение заключается, в следующем (доказательство более общего утверждения вынесено в Раздел 3).",
"Предположение 1 Пусть даны две последовательности $\\delta _1^k(y,x)$ и $\\delta _2^k$ ($k \\ge 0$ ).",
"Будем предполагать, что $\\mathbb {E}\\left[\\delta _1^k(y,x) |\\delta _{1,2}^{k-1},\\delta _{1,2}^{k-2},... \\right]= 0$ , (условная несмещенность) $\\delta _1^k(y,x)$ имеет $\\left(\\hat{\\delta }_1\\right)^2$ -субгауссовскую условную дисперсию, $\\sqrt{\\delta _2^k}$ имеет $\\hat{\\delta }_2$ -субгауссовский условный второй момент.",
"Предположение 2 Пусть даны две последовательности $\\delta _1^k(x,y)$ и $\\delta _2^k$ ($k \\ge 0$ ).",
"Случайная величина $\\delta _1^k(x, y)$ имеет $\\left(\\hat{\\delta }_1^k(x - y)\\right)^2$ -субгауссовский условный момент ($\\hat{\\delta }_1^k(\\cdot )$ есть неслучайная функция от одного аргумента) такой, что $\\hat{\\delta }_1^k(\\alpha z) \\le \\alpha \\hat{\\delta }_1^k(z)$ для всех $\\alpha \\ge 0$ и $z \\in B(0, R)$ .",
"$\\hat{\\delta }_1 < +\\infty $ , где $\\hat{\\delta }_1 \\ge \\sup _{z \\in B(0, R)} \\hat{\\delta }_1^k(z)$ .",
"Теорема 1 Пусть для последовательностей $\\delta _1^k(y,x)$ и $\\delta _2^k$ ($k \\ge 0$ ) верно предположение REF , тогда После $N$ шагов соответствующей модификации градиентного спуска будет верно следующее неравенство: $\\mathbb {E}[f(x_N)] - f(x_*) \\le O\\left(\\frac{LR^2}{N} + \\hat{\\delta }_2\\right).$ И с большой вероятностью $f(x_N) - f(x_*) = \\tilde{O}\\left(\\frac{LR^2}{N} + \\frac{\\hat{\\delta }_1}{\\sqrt{N}} + \\hat{\\delta }_2\\right).$ После $N$ шагов соответствующей модификации ускоренного градиентного спуска будет верно следующее неравенство: $\\mathbb {E}[f(x_N)] - f(x_*) \\le O\\left(\\frac{LR^2}{N^2} + N \\hat{\\delta }_2\\right).$ Если дополнительно верно предположение REF , то с большой вероятностью $f(x_N) - f(x_*) = \\tilde{O}\\left(\\frac{LR^2}{N^2} + \\frac{\\hat{\\delta }_1}{\\sqrt{N}} + N\\hat{\\delta }_2\\right).$ Данная теорема является следствием теоремы REF и REF из Раздела 3 для модели вида $\\psi _{\\delta }(y,x) = \\langle \\nabla _{\\delta } f(x), y-x \\rangle $ .",
"Если в качестве $\\nabla _{\\delta }f(x)$ взять $\\nabla f(x,\\xi )$ , тогда будет выполнено неравенство (REF ) с $\\delta _1(y,x) = \\langle \\nabla f(x) - \\nabla f(x,\\xi ), y - x \\rangle $ , $\\delta _2 = \\frac{1}{2L}\\Vert \\nabla f(x,\\xi ) - \\nabla f(x)\\Vert _2^2$ , с $L:=2L$ .",
"Чтобы это понять, достаточно заметить (первое важное наблюдение), что $\\langle \\nabla f(x) - \\nabla f(x,\\xi ) , y - x \\rangle \\le \\frac{1}{2L}\\Vert \\nabla f(x,\\xi ) - \\nabla f(x)\\Vert _2^2 + \\frac{L}{2}\\Vert y-x\\Vert _2^2.$ Более того, для $\\delta _1(y,x)$ и $\\delta _2$ верно (см.",
"обозначения предположения REF ), что $\\hat{\\delta }_1 = O(\\sigma R)$ и $\\hat{\\delta }_2 = O(\\sigma ^2/L)$ , и для $\\delta _1(y,x)$ верно предположение REF .",
"Сделанное наблюдение позволяет с помощью теоремы REF получить, например, что с большой вероятностью $f(x_N) - f(x_*) = \\tilde{O}\\left(\\frac{LR^2}{N^p} + \\frac{\\sigma R}{\\sqrt{N}} + N^{p-1}\\frac{\\sigma ^2}{L}\\right),$ причем $\\mathbb {E}[f(x_N)] - f(x_*) = O\\left(\\frac{LR^2}{N^p} + N^{p-1}\\frac{\\sigma ^2}{L}\\right).$ Отметим также возможность при $p=1$ выбора шага $h$ в базовом детерминированном градиентном спуске меньше чем $1/L$ .",
"В этом случае оценка будет иметь вид $\\mathbb {E}[f(x_N)] - f(x_*) = O\\left(\\frac{R^2}{hN} + h\\sigma ^2\\right).$ Минимизируя правую часть по $h$ , получим $h = R/\\left(\\sigma \\sqrt{N}\\right)$ и $\\mathbb {E}[f(x_N)] - f(x_*) = O\\left(\\frac{\\sigma R}{\\sqrt{N}}\\right).$ Аналогичные оценки можно выписать и в категориях больших отклонений.",
"Вторым важным наблюдением является следующая теорема (см., например, ).",
"Теорема 2 (Батчинг) Пусть $\\lbrace \\xi ^l\\rbrace _{l=1}^r$ – независимые одинаково распределенные случайные величины (также как случайная величина $\\xi $ , которая имеет субгауссовскую дисперсию $\\sigma ^2$ ).",
"Тогда для $\\sigma ^2_r$ – субгауссовской дисперсии $\\nabla ^r f\\left(x,\\lbrace \\xi \\rbrace _{l=1}^r\\right)=\\frac{1}{r}\\sum _{l=1}^r \\nabla f(x,\\xi ^l),$ справедлива оценка $\\sigma ^2_r = O\\left(\\sigma ^2/r\\right)$ .",
"Для обоснования перехода от (REF ) к (REF ) положим в (REF ) $\\nabla _{\\delta } f(x) = \\nabla ^r f\\left(x,\\lbrace \\xi \\rbrace _{l=1}^r\\right)$ и подберем должным образом $r$ .",
"Для подбора $r$ потребуем, чтобы правая часть в оценке (REF ) была равна $\\varepsilon $ (желаемой точности решения задачи по функции).",
"Чтобы добиться этого исходя из формулы (REF ) согласно теореме REF нужно выбрать $r$ так, чтобы все слагаемые в (REF ) были порядка $\\varepsilon $ .",
"То есть $\\left(\\frac{LR^2}{N^p}\\right) \\simeq \\varepsilon $ , $\\frac{\\sigma R}{\\sqrt{N}} \\simeq \\varepsilon $ , $N^{p-1}\\frac{\\sigma ^2}{L}\\simeq \\varepsilon $ .",
"Получается переопределенная система уравнений на $N,r$ , которая, тем не менее, оказывается совместной $r = \\tilde{O}\\left(\\frac{\\sigma ^2}{L\\varepsilon }\\left(\\frac{LR^2}{\\varepsilon }\\right)^\\frac{p-1}{p}\\right)$ .",
"При этом, число итераций алгоритма – $N = \\tilde{O}\\left(\\left(\\frac{LR^2}{\\varepsilon }\\right)^{\\frac{1}{p}}\\right)$ , а число вычислений $\\nabla f(x,\\xi )$ – $\\tilde{O}\\left(\\max \\left\\lbrace \\left(\\frac{LR^2}{\\varepsilon }\\right)^{\\frac{1}{p}},\\frac{\\sigma ^2R^2}{\\varepsilon ^2}\\right\\rbrace \\right) = \\tilde{O}\\left(\\left(\\frac{LR^2}{\\varepsilon }\\right)^{\\frac{1}{p}} + \\frac{\\sigma ^2R^2}{\\varepsilon ^2}\\right)$ .",
"Данные оценки в точности соответствуют тому, что можно получить с помощью батчинга из оценки (REF ).",
"Отметим, что при $p=2$ данные оценки оптимальны как по числу итераций, так и по числу параллельно вычисляемых стохастических градиентов на каждой итерации ."
],
[
"Модельная общность",
"Результаты раздела можно воспроизвести и в модельной общности , , .",
"Будем говорить, что функция $\\psi _{\\delta }(y,x)$ является $(\\delta ,L)$ -моделью целевой функции $f(x)$ , если для всех $x,y\\in Q$ функция $\\psi _{\\delta }(y,x)$ – выпукла по $y$ , $\\psi _{\\delta }(x,x)\\equiv 0$ , $f(x) + \\psi _{\\delta }(y, x) + \\frac{\\mu }{2}\\Vert y-x\\Vert ^2_2 - \\delta _1 \\le f(y) \\le f(x) + \\psi _{\\delta }(y, x)+ \\frac{L}{2}\\Vert y-x\\Vert ^2_2 + \\delta _2.$ Для задач композитной оптимизации (см., например, , ), в которых целевая функция имеет вид $F(x) = f(x)+h(x)$ , где $h(x)$ достаточна простая функция, для которой доступен субградиент, а функция $f(x)$ имеет $L$ -Липшицев градиент, и для нее доступен только стохастический градиент $\\nabla f(x,\\xi )$ , в качестве модели можно взять $\\psi _{\\delta } (y,x) = \\langle \\nabla ^r f\\left(x,\\lbrace \\xi \\rbrace _{l=1}^r\\right), y - x \\rangle + h(y) - h(x)$ .",
"Тогда аналогично разделу , получим, что в (REF ) можно положить $L:=2L$ , $\\hat{\\delta }_1 = O(\\sigma R/r)$ , $\\hat{\\delta }_2 = O(\\sigma ^2/(Lr))$ .",
"Это наблюдение позволяет перенести все результаты раздела на задачи стохастической композитной оптимизации.",
"Введем следующее предположение.",
"Предположение 3 Пусть даны две последовательности $\\delta _1^k$ и $\\delta _2^k$ ($k \\ge 0$ ).",
"Будем предполагать, что имеется некоторая константа $\\tilde{\\delta }_1$ такая, что $\\mathbb {E}\\left[\\delta _1^k |\\delta _{1,2}^{k-1},\\delta _{1,2}^{k-2},... \\right] \\le \\tilde{\\delta }_1$ , $\\delta _1^k$ имеет $\\left(\\hat{\\delta }_1\\right)^2$ -субгауссовский условный второй момент, $\\sqrt{\\delta _2^k}$ имеет $\\hat{\\delta }_2$ -субгауссовский условный второй момент.",
"Отметим, что предположение REF является более общим, чем предположение REF .",
"Обозначим $q = 1 - \\frac{\\mu }{L}$ .",
"Представим градиентный и быстрый градинетный метод в модельной общности (Алгоритм 1 и 2).",
"В разделах REF и REF представлены теоремы сходимости и соотвествующие доказательства.",
"Градиентный метод [1] Input: Начальная точка $x_0$ , константа сильной выпуклости $\\mu \\ge 0$ , константа липшевости градиента $L > 0$ .",
"$k \\ge 0$ $\\phi _{k+1}(x) := \\psi _{\\delta _k}(x, x_k)+ \\frac{L}{2} \\Vert x-x_k\\Vert ^2_2,$ $x_{k+1} := \\arg \\min _{x \\in Q} \\phi _{k+1}(x).$ Output: $y_N = \\frac{1}{\\sum _{i = 1}^{N} q^{N-i}}\\sum _{i = 1}^{N} q^{N-i} x_i$ и Быстрый градиентный метод [1] Input: Начальная точка $x_0$ , константа сильноый выпуклости $\\mu \\ge 0$ , константа липшевости градиента $L > 0$ .",
"Set $y_0 := x_0$ , $u_0 := x_0$ , $\\alpha _0 := 0$ , $A_0 := \\alpha _0$ $k \\ge 0$ Константа $\\alpha _{k+1}$ — это наибольший корень уравнения $A_{k+1}{(1 + A_k \\mu )}=L\\alpha ^2_{k+1},\\quad A_{k+1} := A_k + \\alpha _{k+1}.$ $y_{k+1} := \\frac{\\alpha _{k+1}u_k + A_k x_k}{A_{k+1}}.$ $\\phi _{k+1}(x)=\\alpha _{k+1}\\psi _{\\delta _k}(x, y_{k+1}) + \\frac{1 + A_k\\mu }{2} \\Vert x-u_k\\Vert ^2_2 + \\frac{\\alpha _{k+1} \\mu }{2} \\Vert x-y_{k+1}\\Vert ^2_2,$ $u_{k+1} := \\arg \\min _{x \\in Q}\\phi _{k+1}(x).$ $x_{k+1} := \\frac{\\alpha _{k+1}u_{k+1} + A_k x_k}{A_{k+1}}.",
"$ Output: $x_N$ ,"
],
[
"Градиентный метод для оптимизационных\nзадач, допускающих модель функции",
"Лемма 1 Пусть $\\psi (x)$ — выпуклая функция и $y = {\\arg \\min _{x \\in Q}} \\lbrace \\psi (x) + \\tfrac{\\beta }{2}\\left\\Vert x - z\\right\\Vert _2^2 + \\tfrac{\\gamma }{2}\\left\\Vert x - u\\right\\Vert _2^2\\rbrace ,$ где $\\beta \\ge 0$ и $\\gamma \\ge 0$ .",
"Тогда $\\psi (x) &+ \\tfrac{\\beta }{2}\\left\\Vert x - z\\right\\Vert _2^2 + \\tfrac{\\gamma }{2}\\left\\Vert x - u\\right\\Vert _2^2\\\\&\\ge \\psi (y) + \\tfrac{\\beta }{2}\\left\\Vert y - z\\right\\Vert _2^2 + \\tfrac{\\gamma }{2}\\left\\Vert y - u\\right\\Vert _2^2 + \\tfrac{\\beta + \\gamma }{2}\\left\\Vert x - y\\right\\Vert _2^2 ,\\,\\,\\, \\forall x \\in Q.$ Доказательство Из критерия оптимальности следует, что: $\\exists g \\in \\partial \\psi (y), \\,\\,\\, \\langle g + \\tfrac{\\beta }{2} \\nabla _y \\left\\Vert y - z\\right\\Vert _2^2 + \\tfrac{\\gamma }{2} \\nabla _y \\left\\Vert y - u\\right\\Vert _2^2, x - y \\rangle \\ge 0 ,\\,\\,\\, \\forall x \\in Q.$ Из $\\beta + \\gamma $ –сильной выпуклости $\\psi (x) + \\tfrac{\\beta }{2}\\left\\Vert x - z\\right\\Vert _2^2 + \\tfrac{\\gamma }{2}\\left\\Vert x - u\\right\\Vert _2^2$ получаем, что $\\psi (x) + \\tfrac{\\beta }{2}\\left\\Vert x - z\\right\\Vert _2^2 &+ \\tfrac{\\gamma }{2}\\left\\Vert x - u\\right\\Vert _2^2 \\ge \\psi (y) + \\tfrac{\\beta }{2}\\left\\Vert y - z\\right\\Vert _2^2 + \\tfrac{\\gamma }{2}\\left\\Vert y - u\\right\\Vert _2^2 \\\\&+ \\langle g + \\tfrac{\\beta }{2} \\nabla _y \\left\\Vert y - u\\right\\Vert _2^2 + \\tfrac{\\gamma }{2} \\nabla _y \\left\\Vert y - z\\right\\Vert _2^2, x - y\\rangle + \\tfrac{\\beta + \\gamma }{2}\\left\\Vert x - y\\right\\Vert _2^2$ Последние два неравенства доказывают лемму.",
"Теорема 3 После $N$ шагов Алгоритма будет верно следующее неравенство: $f(y_{N}) - f(x_*) \\le \\min \\left\\lbrace \\frac{LR^2}{2N}, \\frac{LR^2}{2}\\exp \\left(-\\frac{\\mu }{L}N\\right)\\right\\rbrace + \\frac{1}{\\sum _{i=1}^{N}q^{N-i}}\\sum _{i=1}^{N}q^{N-i}(\\delta _1^{i-1} + \\delta _2^{i-1}).$ Доказательство Из (REF ) получаем: $f(x_{N})&\\le f(x_{N-1}) + \\psi _{\\delta _{N-1}}(x_{N}, x_{N-1}) + \\frac{L}{2}\\left\\Vert x_{N} - x_{N-1}\\right\\Vert _2^2 + \\delta _2^{N-1}.$ Воспользуемся леммой REF для (REF ): $f(x_{N})&\\le f(x_{N-1}) + \\psi _{\\delta _{N-1}}(x, x_{N-1}) + \\frac{L}{2}\\left\\Vert x - x_{N-1}\\right\\Vert _2^2 - \\frac{L}{2}\\left\\Vert x - x_{N}\\right\\Vert _2^2 + \\delta _2^{N-1}.$ Воспользуемся левым неравенством из (REF ): $f(x_{N}) \\le f(x) + \\frac{L - \\mu }{2}\\left\\Vert x - x_{N-1}\\right\\Vert _2^2 - \\frac{L}{2}\\left\\Vert x - x_{N}\\right\\Vert _2^2 + \\delta _1^{N-1} + \\delta _2^{N-1}.$ Перепишем неравенство для $x = x_*$ : $\\frac{1}{2}\\left\\Vert x_* - x_{N}\\right\\Vert _2^2 \\le \\frac{1}{L}\\left(f(x_*) - f(x_{N}) + \\delta _1^{N-1} + \\delta _2^{N-1}\\right) + \\frac{q}{2}\\left\\Vert x_* - x_{N-1}\\right\\Vert _2^2.$ Рекусрсивно получаем, что $\\frac{1}{2}\\left\\Vert x_* - x_{N}\\right\\Vert _2^2 \\le \\sum _{i=1}^{N}\\left(\\frac{q^{N-i}}{L}(f(x_*) - f(x_{i})+ \\delta _1^{i-1} + \\delta _2^{i-1})\\right) + \\frac{q^{N}}{2}\\left\\Vert x_* - x_0\\right\\Vert _2^2.$ Учитывая, что $\\frac{1}{2}\\left\\Vert x_* - x_{N}\\right\\Vert _2^2 \\ge 0$ и определение $y_{N}$ , мы получим: $\\frac{q^{N}}{2}\\left\\Vert x_* - x_0\\right\\Vert _2^2&\\ge \\sum _{i=1}^{N}\\left(\\frac{q^{N-i}}{L}(f(x_i) - f(x_*) - \\delta _1^{i-1} - \\delta _2^{i-1})\\right)\\\\&\\ge (f(y_{N}) - f(x_*))\\sum _{i=1}^{N}\\frac{q^{N-i}}{L} - \\frac{1}{L}\\sum _{i=1}^{N}q^{N-i}(\\delta _1^{i-1} + \\delta _2^{i-1}).$ Разделим обе части последнего неравенства на $\\sum _{i=1}^{N}\\frac{q^{N-i}}{L}$ : $f(y_{N}) - f(x_*)&\\le \\frac{\\frac{q^{N}}{2}}{\\sum _{i=1}^{N}\\frac{q^{N-i}}{L}} \\left\\Vert x_* - x_0\\right\\Vert _2^2 + \\frac{1}{\\sum _{i=1}^{N}q^{N-i}}\\sum _{i=1}^{N}q^{N-i}(\\delta _1^{i-1} + \\delta _2^{i-1}).$ Используя то, что $\\sum _{i=1}^{N}\\frac{q^{N-i}}{L} \\ge \\frac{1}{L}$ и $q^{N-i} \\ge q^{N}$ для всех $i \\ge 0$ , мы получим неравенство: $f(y_{N}) - f(x_*) \\le \\frac{L}{2}\\min \\left\\lbrace q^{N}, \\frac{1}{N}\\right\\rbrace \\left\\Vert x_* - x_0\\right\\Vert _2^2 + \\frac{1}{\\sum _{i=1}^{N}q^{N-i}}\\sum _{i=1}^{N}q^{N-i}(\\delta _1^{i-1} + \\delta _2^{i-1}).$ Данное неравенство и $q^N \\le \\exp (-\\frac{\\mu }{L}N)$ завершают доказательство теоремы.",
"Рассмотрим следствие теоремы REF .",
"Теорема 4 Пусть для последовательностей $\\delta _1^k$ и $\\delta _2^k$ ($k \\ge 0$ ) верно предположение REF , тогда после $N$ шагов Алгоритма будет верно следующее неравенство: $\\mathbb {E}[f(y_N)] - f(x_*) \\le \\min \\left\\lbrace \\frac{LR^2}{2N}, \\frac{LR^2}{2}\\exp \\left(-\\frac{\\mu }{L}N\\right)\\right\\rbrace + \\tilde{\\delta }_1 + O(\\hat{\\delta }_2).$ Если предположить, что $\\mu = 0$ , то с большой вероятностью $f(y_N) - f(x_*) = \\tilde{O}\\left(\\frac{LR^2}{N} + \\frac{\\hat{\\delta }_1}{\\sqrt{N}} + \\tilde{\\delta }_1 + \\hat{\\delta }_2\\right).$ Доказательство Первое неравенство можно получить используя стандартные неравенства для моментов субгуассовских случайных величин .",
"Для второго неравенства надо заметить, что в случае $\\mu = 0$ выполнено равенство: $\\frac{1}{\\sum _{i=1}^{N}q^{N-i}}\\sum _{i=1}^{N}q^{N-i}(\\delta _1^{i-1} + \\delta _2^{i-1}) = \\frac{1}{N}\\sum _{i=1}^{N}(\\delta _1^{i-1} + \\delta _2^{i-1}).$ Для последнего слагаемого надо воспользоваться неравенствами концентрации для субгауссовских и субэкспоненциальных случайных величин ."
],
[
"Быстрый градиентный метод для оптимизационных\nзадач, допускающих модель функции",
"В случае быстрого градиентного метода нам понадобится изменить определение модели функции.",
"Будем говорить, что функция $\\psi _{\\delta }(y,x)$ является $(\\delta ,L)$ -моделью целевой функции $f(x)$ , если для всех $x,y\\in Q$ функция $\\psi _{\\delta }(y,x)$ – выпукла по $y$ , $\\psi _{\\delta }(x,x)\\equiv 0$ , $f(x) + \\psi _{\\delta }(y, x) + \\frac{\\mu }{2}\\Vert y-x\\Vert ^2_2 - \\delta _1(y, x) \\le f(y) \\le f(x) + \\psi _{\\delta }(y, x)+ \\frac{L}{2}\\Vert y-x\\Vert ^2_2 + \\delta _2.$ Отметим, что теперь мы в общем случае предполагаем, что $\\delta _1$ является функцией от двух аргументов $y, x \\in Q$ .",
"Лемма 2 Для всех $x \\in Q$ выполнено неравенство $&A_{k+1} f(x_{k+1}) - A_{k} f(x_{k}) + \\frac{{1 + A_{k+1} \\mu }}{2}\\left\\Vert x - u_{k+1}\\right\\Vert _2^2 - \\frac{{1 + A_k \\mu }}{2}\\left\\Vert x - u_k\\right\\Vert _2^2\\\\&\\le \\alpha _{k+1}f(x) + A_{k}\\delta _1^k(x_k,y_{k+1}) + \\alpha _{k+1}\\delta _1^k(x,y_{k+1}) + A_{k+1}\\delta _2^k.$ Доказательство Воспользуемся (REF ): $f(x_{k+1}) \\le f(y_{k+1}) + \\psi _{\\delta _k}(x_{k+1},y_{k+1}) + \\frac{L}{2}\\left\\Vert x_{k+1} - y_{k+1}\\right\\Vert _2^2 + \\delta _2^k.$ Из (REF ) и (REF ) для последовательностей $x_{k+1}$ и $y_{k+1}$ мы получим, что $f(x_{k+1})&\\le f(y_{k+1}) + \\psi _{\\delta _k}\\left(\\frac{\\alpha _{k+1}u_{k+1} + A_k x_k}{A_{k+1}},y_{k+1}\\right)\\\\&\\hspace{20.0pt}+ \\frac{L}{2}\\left\\Vert \\frac{\\alpha _{k+1}u_{k+1} + A_k x_k}{A_{k+1}} - y_{k+1}\\right\\Vert _2^2 + \\delta _1^k\\\\&= f(y_{k+1}) + \\psi _{\\delta _k}\\left(\\frac{\\alpha _{k+1}u_{k+1} + A_k x_k}{A_{k+1}},y_{k+1}\\right)+\\frac{L \\alpha ^2_{k+1}}{2 A^2_{k+1}}\\left\\Vert u_{k+1} - u_k\\right\\Vert _2^2 + \\delta _2^k.$ Так как модель $\\psi _{\\delta _k}(\\cdot ,y_{k+1})$ выпуклая, то $f(x_{k+1})&\\le \\frac{A_k}{A_{k+1}}\\left(f(y_{k+1}) + \\psi _{\\delta _k}(x_k, y_{k+1})\\right)+\\frac{\\alpha _{k+1}}{A_{k+1}}\\left(f(y_{k+1}) +\\psi _{\\delta _k}(u_{k+1}, y_{k+1})\\right)\\\\&\\hspace{20.0pt}+ \\frac{L \\alpha ^2_{k+1}}{2 A^2_{k+1}}\\left\\Vert u_{k+1} - u_k\\right\\Vert _2^2 + \\delta _2^k.$ Из (REF ) для последовательности $\\alpha _{k+1}$ будет верно: $\\begin{split}f(x_{k+1})&\\le \\frac{A_k}{A_{k+1}}\\left(f(y_{k+1}) + \\psi _{\\delta _k}(x_k,y_{k+1})\\right) +\\frac{\\alpha _{k+1}}{A_{k+1}}\\Big (f(y_{k+1}) + \\psi _{\\delta _k}(u_{k+1},y_{k+1})\\\\&\\hspace{20.0pt}+ \\frac{{1 + A_k \\mu }}{2 \\alpha _{k+1}}\\left\\Vert u_{k+1} - u_k\\right\\Vert _2^2\\Big ) + \\delta _2^k.\\end{split}$ Из леммы REF для оптимизационной задачи (REF ) будет следовать, что $&\\alpha _{k+1}\\psi _{\\delta _k}(u_{k+1}, y_{k+1}) + \\frac{1 + A_k \\mu }{2}\\left\\Vert u_{k+1} - u_k\\right\\Vert _2^2 + \\frac{\\alpha _{k+1} \\mu }{2}\\left\\Vert u_{k+1} - y_{k+1}\\right\\Vert _2^2 \\\\&\\hspace{20.0pt}+ \\frac{1 + A_{k+1}\\mu }{2}\\left\\Vert x - u_{k+1}\\right\\Vert _2^2 \\\\&\\le \\alpha _{k+1}\\psi _{\\delta _k}(x, y_{k+1}) + \\frac{1 + A_k \\mu }{2}\\left\\Vert x - u_k\\right\\Vert _2^2 + \\frac{\\alpha _{k+1} \\mu }{2}\\left\\Vert x - y_{k+1}\\right\\Vert _2^2.$ Так как $\\frac{1}{2}\\left\\Vert u_{k+1} - y_{k+1}\\right\\Vert _2^2 \\ge 0$ , то $\\begin{split}&\\alpha _{k+1}\\psi _{\\delta _k}(u_{k+1}, y_{k+1}) + \\frac{1 + A_k\\mu }{2}\\left\\Vert u_{k+1} - u_k\\right\\Vert _2^2\\\\&\\le \\alpha _{k+1}\\psi _{\\delta _k}(x, y_{k+1}) + \\frac{1 + A_k\\mu }{2}\\left\\Vert x - u_k\\right\\Vert _2^2\\\\&\\hspace{20.0pt}- \\frac{1 + A_{k+1}\\mu }{2}\\left\\Vert x - u_{k+1}\\right\\Vert _2^2 + \\frac{\\alpha _{k+1} \\mu }{2}\\left\\Vert x - y_{k+1}\\right\\Vert _2^2.\\end{split}$ Обьединим неравенства (REF ) и (REF ), тогда $f(x_{k+1})&\\le \\frac{A_k}{A_{k+1}} \\left(f(y_{k+1}) + \\psi _{\\delta _k}(x_k,y_{k+1})\\right)\\\\&\\hspace{20.0pt}+\\frac{\\alpha _{k+1}}{A_{k+1}}\\Big (f(y_{k+1})+ \\psi _{\\delta _k}(x,y_{k+1}) + { \\frac{\\mu }{2}\\left\\Vert x - y_{k+1}\\right\\Vert _2^2}\\\\&\\hspace{20.0pt}+ \\frac{{1 + A_k \\mu }}{2\\alpha _{k+1}}\\left\\Vert x - u_k\\right\\Vert _2^2 - \\frac{{1 + A_{k+1} \\mu }}{2\\alpha _{k+1}}\\left\\Vert x - u_{k+1}\\right\\Vert _2^2 \\Big ) + \\delta _2^k.$ Воспользуемся левым неравенством из (REF ): $f(x_{k+1})&\\le \\frac{A_k}{A_{k+1}} f(x_k) +\\frac{\\alpha _{k+1}}{A_{k+1}} f(x) \\\\&\\hspace{20.0pt}+ \\frac{{1 + A_k \\mu }}{2A_{k+1}}\\left\\Vert x - u_k\\right\\Vert _2^2 - \\frac{{1 + A_{k+1} \\mu }}{2A_{k+1}}\\left\\Vert x - u_{k+1}\\right\\Vert _2^2 \\\\&\\hspace{20.0pt}+ \\frac{A_{k}}{A_{k+1}}\\delta _1^k(x_k,y_{k+1}) + \\frac{\\alpha _{k+1}}{A_{k+1}}\\delta _1^k(x,y_{k+1}) + \\delta _2^k.$ Данное неравенство завершает доказательство леммы.",
"Теорема 5 После $N$ шагов Алгоритма будет верно следующее неравенство: $f(x_{N}) - f(x_*) &\\le \\frac{R^2}{2A_N} + \\frac{1}{A_N}\\sum _{k=0}^{N-1}A_{k}\\delta _1^k(x_k,y_{k+1}) \\\\&\\hspace{20.0pt}+ \\frac{1}{A_N}\\sum _{k=0}^{N-1}\\alpha _{k+1}\\delta _1^k(x_*,y_{k+1}) + \\frac{1}{A_N}\\sum _{k=0}^{N-1}A_{k+1}\\delta _2^k.$ Доказательство Суммирая неравентсва из леммы REF для $k$ от 0 и $N-1$ и, взяв $x = x_*$ , мы получим, что $A_{N} f(x_{N})&\\le A_{N}f(x_*) + \\frac{1}{2}\\left\\Vert x_* - u_0\\right\\Vert _2^2 - \\frac{1 + A_N\\mu }{2}\\left\\Vert x_* - u_N\\right\\Vert _2^2 + \\sum _{k=0}^{N-1}A_{k}\\delta _1^k(x_k,y_{k+1}) \\\\&\\hspace{20.0pt}+ \\sum _{k=0}^{N-1}\\alpha _{k+1}\\delta _1^k(x_*,y_{k+1}) + \\sum _{k=0}^{N-1}A_{k+1}\\delta _2^k.$ Так как $\\frac{1 + A_N\\mu }{2}\\left\\Vert x_* - u_N\\right\\Vert _2^2 \\ge 0$ , то $A_{N} f(x_{N}) - A_{N}f(x_*) &\\le \\frac{1}{2}\\left\\Vert x_* - u_0\\right\\Vert _2^2 + \\sum _{k=0}^{N-1}A_{k}\\delta _1^k(x_k,y_{k+1}) \\\\&\\hspace{20.0pt}+ \\sum _{k=0}^{N-1}\\alpha _{k+1}\\delta _1^k(x_*,y_{k+1}) + \\sum _{k=0}^{N-1}A_{k+1}\\delta _2^k.$ Последнее неравенство доказывает теорему.",
"Лемма 3 Для всех $N \\ge 1$ , $\\frac{1}{A_N} \\le \\min \\left\\lbrace \\frac{4L}{N^2}, 2L\\exp \\left(-\\frac{N-1}{2}\\sqrt{\\frac{\\mu }{L}}\\right)\\right\\rbrace .$ Результат леммы можно получить по аналогии с , (см.",
"замечание 5.11.).",
"Далее нам будут полезны следующие предположения.",
"Похожее на предположение REF условие на детерминированный шум в градиенте можно встретить в работах , .",
"Предположение 4 Пусть даны две последовательности $\\delta _1^k(x,y)$ и $\\delta _2^k$ ($k \\ge 0$ ).",
"Случайная величина $\\delta _1^k(x,y)$ имеет такое условное математическое ожидание, что $\\mathbb {E}\\left[\\delta _1^k(x,y) |\\delta _{1}^{k-1}(x,y),\\delta _{2}^{k-1},\\delta _{1}^{k-2}(x,y)\\dots \\right] \\le \\tilde{\\delta }_1^k(x - y)$ $ \\forall x, y \\in Q$ , где $\\tilde{\\delta }_1^k(\\cdot )$ есть неслучайная функция от одного аргумента.",
"$\\tilde{\\delta }_1^k(\\alpha z) \\le \\alpha \\tilde{\\delta }_1^k(z)$ для всех $\\alpha \\ge 0$ и $z \\in B(0, R)$ .",
"$\\tilde{\\delta }_1 < +\\infty $ , где $\\tilde{\\delta }_1 \\ge \\sup _{z \\in B(0, R)} \\tilde{\\delta }_1^k(z)$ .",
"Теорема 6 Пусть для последовательностей $\\delta _1^k(x, y)$ и $\\delta _2^k$ ($k \\ge 0$ ) верно предположение REF и REF для любых $x, y \\in Q$ , тогда после $N$ шагов Алгоритма будет верно следующее неравенство: $\\mathbb {E}[f(x_N)] - f(x_*) &\\le \\min \\left\\lbrace \\frac{4LR^2}{N^2}, 2LR^2\\exp \\left(-\\frac{N-1}{2}\\sqrt{\\frac{\\mu }{L}}\\right)\\right\\rbrace + \\tilde{\\delta }_1 \\\\&\\hspace{20.0pt}+ O\\left(\\min \\left\\lbrace N, \\sqrt{\\frac{L}{\\mu }}\\right\\rbrace \\hat{\\delta }_2\\right).$ Предположим дополнительно, что $\\mu = 0$ , и для последовательности $\\delta _1^k(x, y)$ выполнено предположение REF , тогда с большой вероятностью $f(x_N) - f(x_*) = \\tilde{O}\\left(\\frac{LR^2}{N^2} + \\frac{\\hat{\\delta }_1}{\\sqrt{N}} + \\tilde{\\delta }_1 + N\\hat{\\delta }_2\\right).$ Доказательство Первое неравенство получается из тех же соображений, что и в доказательстве теоремы REF с учетом того, что $\\frac{1}{A_N}\\sum _{k=0}^{N-1}A_{k+1} \\le O\\left(\\min \\left\\lbrace N, \\sqrt{\\frac{L}{\\mu }}\\right\\rbrace \\right)$ (см.",
", лемма 5.11) и серии неравенств: $A_{k}\\tilde{\\delta }_1^k(x_k - y_{k+1}) = A_{k}\\tilde{\\delta }_1^k\\left(\\frac{\\alpha _{k+1}}{A_k}(y_{k+1} - u_k)\\right) \\le \\alpha _{k+1} \\tilde{\\delta }_1^k(y_{k+1} - u_k) \\le \\alpha _{k+1} \\tilde{\\delta }_1.$ В первом переходе мы воспользовались (REF ).",
"В предпоследнем и последнем переходе использовали предположение REF .",
"Теперь докажем (REF ).",
"Для доказательства неравенства $\\frac{1}{A_N}\\sum _{k=0}^{N-1}A_{k+1}\\delta _2^k \\le \\tilde{O}(N\\hat{\\delta }_2)$ нужно воспользоваться неравенством концентрации для субэкспоненциальных случайных величин.",
"Чтобы показать неравенство $\\frac{1}{A_N}\\sum _{k=0}^{N-1}\\alpha _{k+1}\\delta _1^k(x_*,y_{k+1}) \\le \\tilde{O}\\left(\\frac{\\hat{\\delta }_1}{\\sqrt{N}}\\right)$ нужно воспользоваться неравенством концентрации для субгауссовских случайных величин.",
"Неравенство $\\frac{1}{A_N}\\sum _{k=0}^{N-1}A_{k}\\delta _1^k(x_k,y_{k+1}) \\le \\tilde{O}\\left(\\frac{\\hat{\\delta }_1}{\\sqrt{N}}\\right)$ доказывается аналогично, как и предыдущее, но с учетом того, что $A_{k}\\hat{\\delta }_1^k(x_k - y_{k+1}) = A_{k}\\hat{\\delta }_1^k\\left(\\frac{\\alpha _{k+1}}{A_k}(y_{k+1} - u_k)\\right) \\le \\alpha _{k+1} \\hat{\\delta }_1^k(y_{k+1} - u_k) \\le \\alpha _{k+1} \\hat{\\delta }_1.$ Во первом равенстве мы воспользовались (REF )."
],
[
"Максимум гладких функций",
"Рассмотрим следующую задачу: $F(x) := \\max _{1 \\le i \\le m} f_i(x) \\rightarrow \\min _{x\\in Q \\subseteq \\mathbb {R}^n}.$ Будем предполагать, что $f_i(x)$ ($i \\in [1, m]$ ) выпуклые и имеют $L$ -Липшицев градиент, т.е.",
"для всех $x,y\\in Q$ $f_i(x) + \\langle \\nabla f_i(x), y - x \\rangle \\le f_i(y) \\le f_i(x) + \\langle \\nabla f_i(x), y - x \\rangle + \\frac{L}{2}\\Vert y-x\\Vert _2^2.$ Выберем в качестве модели функции $F(x)$ : $\\psi _{\\delta }(y, x) = \\max _{1 \\le i \\le m} \\left\\lbrace f_i(x) + \\langle \\nabla f_i(x,\\xi _i), y - x \\rangle \\right\\rbrace - F(x),$ где $\\nabla f_i(x,\\xi _i)$ — независимые стохастические градиенты функций $f_i$ , для которых выполнены условия (REF ).",
"Из (REF ) будет выполнено левое неравенство $F(y) &\\ge F(x) + \\psi _{\\delta }(y, x) + \\max _{1 \\le i \\le m} \\left\\lbrace f_i(x) + \\langle \\nabla f_i(x), y - x \\rangle \\right\\rbrace \\\\&\\hspace{20.0pt}- \\max _{1 \\le i \\le m} \\left\\lbrace f_i(x) + \\langle \\nabla f_i(x,\\xi _i), y - x \\rangle \\right\\rbrace \\\\&\\ge F(x) + \\psi _{\\delta }(y, x) - \\max _{1 \\le i \\le m} \\left\\lbrace \\langle \\nabla f_i(x,\\xi _i) - \\nabla f_i(x), y - x \\rangle \\right\\rbrace $ и правое неравенство $F(y) &\\le F(x) + \\psi _{\\delta }(y, x) + \\max _{1 \\le i \\le m} \\left\\lbrace f_i(x) + \\langle \\nabla f_i(x), y - x \\rangle \\right\\rbrace \\\\&\\hspace{20.0pt}- \\max _{1 \\le i \\le m} \\left\\lbrace f_i(x) + \\langle \\nabla f_i(x,\\xi _i), y - x \\rangle \\right\\rbrace + \\frac{L}{2}\\left\\Vert y - x\\right\\Vert _2^2\\\\&\\le F(x) + \\psi _{\\delta }(y, x) + \\frac{1}{2L}\\max _{1 \\le i \\le m}\\left\\Vert \\nabla f_i(x) - \\nabla f_i(x,\\xi _i)\\right\\Vert _2^2 + L \\left\\Vert y - x\\right\\Vert _2^2.$ из (REF ) с $\\delta _2 = \\frac{1}{2L}\\max _{1 \\le i \\le m}\\left\\Vert \\nabla f_i(x) - \\nabla f_i(x,\\xi _i)\\right\\Vert _2^2,$ $\\delta _1(y,x) = \\max _{1 \\le i \\le m} \\left\\lbrace \\langle \\nabla f_i(x,\\xi _i) - \\nabla f_i(x), y - x \\rangle \\right\\rbrace $ и $L := 2L$ .",
"Из условий (REF ) получаем, что $\\delta _1(y,x)$ является субгауссовской, а $\\delta _2$ — субэкспоненциальной случайной величиной, так как максимум субгауссовских (субэкспоненциальных) случайных величин есть субгауссовская (субэкспоненциальная) случайная величина.",
"Более того, будут выполнены предположения REF , REF и REF .",
"Таким образом для текущей задачи с выбранной моделью применимы теоремы REF и REF ."
]
] | 2001.03443 |
[
[
"Encode, Shuffle, Analyze Privacy Revisited: Formalizations and Empirical\n Evaluation"
],
[
"Abstract Recently, a number of approaches and techniques have been introduced for reporting software statistics with strong privacy guarantees.",
"These range from abstract algorithms to comprehensive systems with varying assumptions and built upon local differential privacy mechanisms and anonymity.",
"Based on the Encode-Shuffle-Analyze (ESA) framework, notable results formally clarified large improvements in privacy guarantees without loss of utility by making reports anonymous.",
"However, these results either comprise of systems with seemingly disparate mechanisms and attack models, or formal statements with little guidance to practitioners.",
"Addressing this, we provide a formal treatment and offer prescriptive guidelines for privacy-preserving reporting with anonymity.",
"We revisit the ESA framework with a simple, abstract model of attackers as well as assumptions covering it and other proposed systems of anonymity.",
"In light of new formal privacy bounds, we examine the limitations of sketch-based encodings and ESA mechanisms such as data-dependent crowds.",
"We also demonstrate how the ESA notion of fragmentation (reporting data aspects in separate, unlinkable messages) improves privacy/utility tradeoffs both in terms of local and central differential-privacy guarantees.",
"Finally, to help practitioners understand the applicability and limitations of privacy-preserving reporting, we report on a large number of empirical experiments.",
"We use real-world datasets with heavy-tailed or near-flat distributions, which pose the greatest difficulty for our techniques; in particular, we focus on data drawn from images that can be easily visualized in a way that highlights reconstruction errors.",
"Showing the promise of the approach, and of independent interest, we also report on experiments using anonymous, privacy-preserving reporting to train high-accuracy deep neural networks on standard tasks---MNIST and CIFAR-10."
],
[
"Introduction",
"To guide their efforts, public health officials must sometimes gather statistics based on sensitive, private information (e.g., to survey the prevalence of vaping among middle-school children).",
"Due to privacy concerns—or simple reluctance to admit the truth—respondents may fail to answer such surveys, or purposefully answer incorrectly, despite the societal benefits of improved public-health measures.",
"To remove such discouragement, and still compute accurate statistics, epidemiologists can turn to randomized response and have respondents not report their true answer, but instead report the results of random coin flips that are just biased by that true answer [6].",
"In computing, such randomized-response mechanisms that guarantee local differential privacy (LDP) have become a widely-deployed, best-practice means of gathering potentially-sensitive information about software and its users in a responsible manner [1], [7].",
"Simultaneously, many systems have been developed for anonymous communication and messaging [8], [9], many of which are designed to gather aggregate statistics with privacy [2], [10], [11], [12].",
"As shown in Figure REF , when combined with anonymity, LDP reports can permit high-accuracy central visibility into distributed, sensitive data (e.g., different users' private attributes) with strong worst-case privacy guarantees that hold for even the most unlucky respondents—even when fate and other parties conspire against them.",
"Thereby, a key dilemma can be resolved: how to usefully learn about a population's data distribution without collecting distinct, identifiable population data into a database whose very existence forms an unbounded privacy risk, especially as it may be abused for surveillance.",
"Unfortunately, in practice, there remains little clarity on how statistical reporting should be implemented and deployed with strong privacy guarantees—especially if LDP reports are to be made anonymous [2], [3], [13], [14].",
"A daunting number of LDP reporting protocols have been recently proposed and formally analyzed, each using slightly different assumptions and techniques, such as strategies for randomization and encoding of binary, categorical, and other types of data [1], [15], [16], [17].",
"However, these protocols may not be suitable to the specifics of any given application domain, due to their different assumptions, (e.g., about adaptivity [3], [14], sketching [1], [18], [15], [16], or succinctness of communication [15], [16], [19]).",
"Thus, these protocols may exhibit lackluster performance on real-world data distributions of limited size, even when accompanied by a formal proof of asymptotically-optimal privacy/utility tradeoffs.",
"In particular, many of these protocols perform dimensionality-reduction using sketches whose added noise may greatly thwart visibility into the tail of distributions (as shown in the experiments of Section ).",
"Finally, the option of simply replicating the details of prominent LDP-reporting deployments is not very attractive, since these have been criticized both for a lack of privacy and a lack of utility [1], [2], [20].",
"Similarly, multiple, disparate approaches have been developed for ensuring anonymity, including some comprehensive systems that have seen wide deployment [8].",
"However, most of these are not well suited to gathering statistics with strong privacy, as they are focused on low-latency communication or point-to-point messaging [8], [9], [21].",
"The few that are well-suited to ensuring the anonymity of long-term, high-latency statistical reporting are somewhat incomparable, due to their different technical mechanisms and their varying assumptions and threat models.",
"Whether they rely on Tor-like mixnets or trusted hardware, some proposed systems output sets of reports unlinkable to their origin [2], [13], while others output only a summary of the reports made anonymous by the use of a commutative, associative aggregation operation [10], [11].",
"Also, these systems' abilities are constrained by the specifics of their construction and mechanisms (e.g., built-in sampling rates and means of multi-party cryptographic computation, as in [11]); some systems are more specific still, and focus only on certain applications, such as the maintenance of (statistical) models [12], [22].",
"Finally, all of these systems have slightly different threat models, e.g., with some assuming an honest-but-curious central coordinator [11] and other assuming a non-colluding, trusted set of parties [2], [10].",
"(Interestingly, these threat models typically exclude the risk of statistical inference, even though limiting this risk is often a primary privacy goal, as it is in this paper.)",
"All of this tends to obscure how these anonymity systems can be best applied to learning statistics with strong privacy guarantees.",
"This lack of clarity is especially concerning because of recent formal results—known colloquially as “privacy amplification by shuffling” [3], [13], [14], [19]—which have fundamentally changed privacy/utility tradeoffs and forced a reconsideration of previous approaches, like those described above.",
"These amplification results prove how central privacy guarantees can be strengthened by orders of magnitude when LDP reports can be made anonymous—i.e., unlinkable to their source—in particular, by having them get “lost in the crowd” through their shuffling or aggregate summarization with a sufficiently-large set of other reports.",
"The source of these privacy amplification results are efforts to formalize how LDP reporting mechanisms benefit from anonymity in the Encode, Shuffle, Analyze (ESA) framework [2].",
"The ESA architecture is rather abstract—placing few restrictions on specifics such as randomization schemes, report encoding, or the means of establishing anonymity—and, not surprisingly, can be a suitable foundation for implementations that aim to benefit from privacy amplification by anonymity."
],
[
"Practical Experiments, Primitives, and Attack Models",
"In this work, we revisit the specifics of the ESA framework and explore statistical reporting with strong privacy guarantees augmented by anonymity, with the goal of providing clear, practical implementation guidelines.",
"At the center of this paper are a set of empirical experiments, modeled on real-world monitoring tasks, that achieve different levels of privacy on a representative set of data distributions.",
"For most of our experiments we use data distributions derived from images, which we choose because they are both representative of certain sensitive data—such as user-location data, as in Figure REF —and their reconstruction accuracy can be easily estimated, visually.",
"Reconstructing images with strong privacy is particularly challenging since images are a naturally high-dimensional dataset with a low maximum amplitude (e.g., the per-pixel distribution of an 8-bit gray-scale image will have a luminescence bound of 255), and which can be either dense or sparse.",
"In addition, following most previous work, we also include experiments that use a real-world, Zipfian dataset with high-amplitude heavy hitters.",
"The overall conclusion of this paper is that high-accuracy statistical reporting with strong, anonymity-amplified privacy guarantees can be implemented using a small set of simple primitives: (i) a new “removal” basis for the analysis of LDP reporting, (ii) one-hot encoding of categorical data, (iii) fragmenting of data and reports into multiple messages, and (iv) anonymous shuffling or aggregate sums.",
"Although novel in combination, most of these individual primitives have been explored in previous work; the exception is our “removal LDP” report definition which can strengthen the local privacy guarantees by a factor of two.",
"For several common statistical reporting tasks, we argue that these four primitives are difficult to improve upon, and we verify this in experiments.",
"Interestingly, we find that some of the more advanced primitives from the related work may offer little benefits and can, in some cases, be detrimental to privacy and utility.",
"These include ESA's Crowd IDs and the heterogeneous privacy levels they induce, by identifying subsets of reports, as well as—most surprisingly—the use of the sketch-based encodings like those popularized by RAPPOR [1], [16].",
"As we shown in experiments, while sketching will always reduce the number of sent reports, sketching may add noise that greatly exceeds that required for privacy, unless the sketch construction is fine-tuned to the target data-distribution specifics.",
"However, we find great benefits in the ESA concept of fragments: breaking up the information to be reported and leveraging anonymity to send multiple unlinkable reports, instead of sending the same information in just one report.",
"As an example, using attribute fragmentation, a respondent with different attributes encoded into a long, sparse Boolean bitvector can send multiple, separately-anonymous reports for the index of each bit set in an LDP version of the bitvector.",
"In particular, we show how privacy/utility tradeoffs can be greatly improved by applying such attribute fragmentation to LDP reports based on one-hot encodings of categorical data.",
"Another useful form is report fragmentation, where respondents send multiple, re-randomized reports based on an LDP backstop (e.g., an underlying, permanent LDP report, like the PRR of [1]); this can allow for a more refined attack model and lower the per-report privacy risk, while maintaining a strict cap on the overall, long-term privacy loss.",
"Finally, we propose a simple, abstract model of threats and assumptions that abstracts away from the how shuffling is performed and assumes only that LDP report anonymization satisfy a few, clear requirements; thereby, we hope to help practitioners reason about and choose from the disparate set of anonymization systems, both current and future.",
"The requirements of our attack model can be met using a variety of mechanisms, such as mixnets, anonymous messaging, or a variety of cryptographic multi-party mechanisms including ESA's “blinded shuffling” [2].",
"Furthermore, while simple, our attack model still allows for refinements—such as efficient in-system aggregation of summaries, and gradual loss of privacy due to partial compromise or collusion—which may be necessary for practical, real-world deployments."
],
[
"Summary of Contributions",
"This paper gives clear guidelines for how practitioners can implement high-accuracy statistical reporting with strong privacy guarantees—even for difficult, high-dimensional data distributions, and as little as a few dozen respondents—and best leverage recent privacy-amplification results based on anonymity.",
"In particular, this paper contributes the following: We explain how the reports in anonymous statistical monitoring are well suited to a “removal LDP” definition of local differential privacy and how this can strengthen respondents' local privacy guarantees by a factor of two, without compromise.",
"We give the results of numerous experiments that are representative of real-world tasks and data distributions and show that strong central privacy guarantees are compatible with high utility—even for low-amplitude and long-tail distributions—but that this requires high-epsilon LDP reports and, correspondingly, great trust in how reports are anonymized.",
"We clarify how—given the strong central privacy guarantees allowed by anonymity—the use of higher-epsilon LDP reports is almost always preferable to mechanisms, like ESA Crowd IDs, which perform data-dependent grouping of reports during anonymization.",
"We outline how privacy and utility can be maximized by having respondents use attribute fragmentation to break up their data (such as the different bits of their reports) and send as separate, unlinkable LDP reports.",
"We formally analyze how—along the lines of RAPPOR's permanent randomized response [1]—report fragmentation can reduce the per-report privacy risk, while strictly bounding the overall, long-term privacy loss.",
"We empirically show the advantages of simple one-hot LDP report encodings and—as a warning to practitioners—empirically highlight the need to fine-tune the parameters of sketch-based encodings.",
"We provide a simple, abstract attack model that makes it easier to reason about the assumptions and specifics of anonymity mechanisms and LDP reporting schemes, and compose them into practical systems.",
"Finally, we demonstrate how anonymous LDP reports can be usefully applied to the training of benchmark deep learning models with high accuracy, with clear central privacy guarantees and minimal empirical loss of privacy."
],
[
"Definitions and Assumptions",
"We first lay a foundation for the remainder of this paper by defining notation, terms, and stating clear assumptions.",
"In particular, we clarify what we mean by LDP reports, their encoding and fragmentation, as well as our model of attackers and anonymization."
],
[
"Local Differential Privacy and Removal vs. Replacement",
"Differential privacy (DP), introduced by Dwork et al.",
"[23], [24], is a privacy definition that captures how randomized algorithms that operate on a dataset can be bounded in their sensitivity to the presence or absence of any particular data item.",
"Differential privacy is measured as the maximum possible divergence between the output distributions of such algorithms when applied to two datasets that differ by any one record.",
"The most common definition of this metric is based on the worst-case replacement of any dataset record: Definition 2.1 (Replacement $(\\varepsilon ,\\delta )$ -DP [25]) A randomized algorithm $\\mathcal {M}\\colon \\mathcal {D}^n \\rightarrow \\mathcal {S}$ satisfies replacement $(\\varepsilon ,\\delta )$ -differential privacy if for all $S \\subset \\mathcal {S}$ and for all $i\\in [n]$ and datasets $D=(x_1,\\ldots ,x_n), D^{\\prime }=(x^{\\prime }_1,\\ldots ,x^{\\prime }_n) \\in \\mathcal {D}^n$ such that $x_j = x^{\\prime }_j$ for all $j \\ne i$ we have: $\\mathop {\\mathbf {Pr}}[\\mathcal {M}(D) \\in S] \\le e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {M}(D^{\\prime }) \\in S] + \\delta .$ Above, as in the rest of this paper, we let $[n]$ denote the set of integers $\\lbrace 1, \\ldots , n\\rbrace $ , $[a,b]$ denote $\\lbrace v \\colon a \\le v \\le b\\rbrace $ , and $(a \\wedge b)$ denote $\\text{max}(a, b)$ .",
"Symbols such as $x$ typically represent scalars, symbols such as $x$ represent vectors of appropriate length.",
"Elements of $x$ are represented by $x_i$ .",
"Respectively, $\\Vert x\\Vert _1$ and $\\Vert x\\Vert _2$ represent $\\sum |x_i|$ and $\\sqrt{\\sum x_i^2}$ .",
"Additionally, all logarithms in this paper are natural logarithms, unless the base is explicitly mentioned.",
"Local differential privacy (LDP) considers a distributed dataset or data collection task where an attacker is assumed to see and control the reports or records for all-but-one respondent, and where the entire transcript of all communication must satisfy differential privacy for each respondent.",
"Commonly, LDP guarantees are achieved by having respondents communicate only randomized reports that result from applying a differentially private algorithm $\\mathcal {R}$ to their data.",
"For any given level of privacy, there are strict limits to the utility of datasets gathered via LDP reporting.",
"The uncertainty in each LDP report creates a “noise floor” below which no signal can be detected.",
"This noise floor grows with the dimensionality of the reported data; therefore, compared to a Boolean question (“Do you vape?”), a high-dimensional question about location (“Where in the world are you?”) can be expected to have dramatically more noise and a correspondingly worse signal.",
"This noise floor also grows in proportion to the square root of the number of reports; therefore, somewhat counter-intuitively, as more data is collected it will become harder to detect any fixed-magnitude signal (e.g., the global distribution of the limited, fixed set of people named Sandiego).",
"The algorithms used to create per-respondent LDP reports—referred to as local randomizers—must satisfy the definition of differential privacy for a dataset of size one; in particular, they may satisfy the following definition based on replacement: Definition 2.2 (Replacement LDP) An algorithm $\\mathcal {R}\\colon \\mathcal {D}\\rightarrow \\mathcal {S}$ is a replacement $(\\varepsilon ,\\delta )$ -differentially private local randomizer if for all $S \\subseteq \\mathcal {S}$ and for all $x,x^{\\prime } \\in \\mathcal {D}$ : $\\mathop {\\mathbf {Pr}}[\\mathcal {R}(x) \\in S] \\le e^{\\varepsilon } \\mathop {\\mathbf {Pr}}[\\mathcal {R}(x) \\in S] + \\delta .$ However, this replacement-based LDP definition is unnecessarily conservative—at least for finding good privacy/utility tradeoffs in statistical reporting—although it has often been used in prior work, because it simplifies certain analyses.",
"Replacement LDP compares the presence of any respondent's report against the counterfactual of being replaced with its worst-case alternative.",
"For distributed monitoring, a more suitable counterfactual is one where the respondent has decided not to send any report, and thereby has removed themselves from the dataset.",
"It is well known that replacement LDP has a differential-privacy $\\varepsilon $ upper bound that for some mechanisms can be twice that of an $\\varepsilon $ based on the removal of a respondent's report.",
"For the $\\varepsilon > 1$ regime that is typical in LDP applications, this factor-of-two change makes a major difference because the probability of $S$ depends exponentially on $\\varepsilon $ .",
"Thus, a removal-based definition is more appropriate for our practical privacy/utility tradeoffs.",
"Unfortunately, a removal-based LDP definition cannot be directly adopted in the local model due to a technicality: removing any report will change the support of the output distribution because the attacker is assumed to observe all communication.",
"To avoid this, we can define removal-based differential privacy generally with respect to algorithms defined only on inputs of fixed length $n$ , and from this define a corresponding local randomizer: Definition 2.3 (Generalized removal $(\\varepsilon ,\\delta )$ -DP) A randomized algorithm $\\mathcal {M}\\colon \\mathcal {D}^n \\rightarrow \\mathcal {S}$ satisfies removal $(\\varepsilon ,\\delta )$ -differential privacy if there exists an algorithm $\\mathcal {M}^{\\prime }\\colon \\mathcal {D}^n \\times 2^{[n]} \\rightarrow \\mathcal {S}$ with the following properties: for all $D\\in \\mathcal {D}^n$ , $\\mathcal {M}^{\\prime }(D,[n])$ is identical to $\\mathcal {M}(D)$ ; for all $D\\in \\mathcal {D}^n$ and $I\\subseteq [n]$ , $\\mathcal {M}^{\\prime }(D,I)$ depends only on the elements of $D$ with indices in $I$ ; for all $S \\subset \\mathcal {S}$ , $D\\in \\mathcal {D}^n$ and $I, I^{\\prime }\\subseteq [n]$ where we have that $|I \\bigtriangleup I^{\\prime }|=1$ : $\\mathop {\\mathbf {Pr}}[\\mathcal {M}^{\\prime }(D,I) \\in S] \\le e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {M}^{\\prime }(D,I^{\\prime }) \\in S] + \\delta .$ (Notably, this definition generalizes the more standard definition of removal-based differential privacy where $\\mathcal {M}$ is defined for datasets of all sizes, by setting $\\mathcal {M}^{\\prime }(D,I):=\\mathcal {M}((x_i)_{i\\in I})$ —i.e., by defining $\\mathcal {M}^{\\prime }(D,I)$ to be $\\mathcal {M}$ applied to the elements of $D$ with indices in $I$ .)",
"In the distributed setting it suffices to define removal-based LDP—as follows—by combining the above definition with the use of a local randomizer whose properties satisfy Definition REF when restricted to datasets of size 1.",
"(For convenience, we state this only for $\\delta =0$ , since extensions to $\\delta >0$ and other notions of DP are straightforward.)",
"Definition 2.4 (Removal LDP) An algorithm $\\mathcal {R}\\colon \\mathcal {D}\\rightarrow \\mathcal {S}$ is a removal $\\varepsilon $ -differentially private local randomizer if there exists a random variable $\\mathcal {R}_0$ such that for all $S \\subseteq \\mathcal {S}$ and for all $x \\in \\mathcal {D}$ : $e^{-\\varepsilon } \\mathop {\\mathbf {Pr}}[\\mathcal {R}_0 \\in S] \\le \\mathop {\\mathbf {Pr}}[\\mathcal {R}(x) \\in S] \\le e^{\\varepsilon } \\mathop {\\mathbf {Pr}}[\\mathcal {R}_0 \\in S].$ Here $\\mathcal {R}_0$ should be thought of as the output of the randomizer when a respondent's data is absent.",
"This definition is equivalent, up to a factor of two, to the replacement version of the definitions.",
"To distinguish between these two notions we will always explicitly state “removal differential privacy\" but often omit “replacement\" to refer to the more common notion."
],
[
"Attributes, Encodings, and Fragments of Reports",
"There are various means by which LDP reports can be crafted from a respondent's data record, $x\\in \\mathcal {D}$ in a domain $\\mathcal {D}$ , using a local randomizer $\\mathcal {R}$ .",
"This paper considers three specific LDP report constructions, that stem from the ESA framework [2]—report encoding, attribute fragmentation, and report fragmentation—each of which provides a lever for controlling different aspects of the utility/privacy tradeoffs.",
"Encodings: Given a data record $x$ , depending on its domain $\\mathcal {D}$ , the type of encoding can have a strong impact on the utility of a differentially private algorithm.",
"Concretely, consider a setting where the domain $\\mathcal {D}$ is a dictionary of elements (e.g., words in a language), and one wants to estimate the frequency of elements in this domain, with each data record $x$ holding an element.",
"One natural way to encode $x$ is via one-hot encoding if the cardinality of $\\mathcal {D}$ is not too large.",
"For large domains, in order to reduce communication/storage one can use a sketching algorithm (e.g., count-mean-sketch [26]) to establish a compact encoding.",
"(For any given dataset and task, and at any given level of privacy, the choice of such an encoding will impact the empirical utility; we explore this empirical tradeoff in the evaluations of Section .)",
"Attribute fragments: Respondents' data records may hold multiple independent or dependent aspects.",
"We can, without restriction, consider the setting where each such data record $x$ is encoded as a binary vector with $k$ or fewer bits set (i.e., no more than $k$ non-zero coordinates).",
"We can refer to each of those $k$ vector coordinates as attributes and write $x=\\sum \\limits _{i=1}^k x_i$ , where each $x_i$ is a one-hot vector.",
"Given any bounded LDP budget, there are two distinct choices for satisfying privacy by randomizing $x$ : either send each $x_i$ independently through the randomizer $\\mathcal {R}$ , splitting the privacy budget accordingly, or sample one of the $x_i$ 's at random and spend all of privacy budget to send it through $\\mathcal {R}$ .",
"As demonstrated empirically in Section , we find that sampling is always better for the privacy/utility tradeoff (thereby, we verify what has been shown analytically [16], [27]).",
"[color=cyan!30,inline]VF: I don't see a discussion in Sec III that supports this.",
"Also not in experiments.",
"[color=blue!30,inline]AT: @Shuang: Can you verify the reference is correct, and hence the comment by Vitaly is addressed?",
"Once a one-hot vector $z$ is sampled from $\\lbrace x_i\\colon i\\in [k]\\rbrace $ , we establish analytically and empirically that for both local and central differential-privacy tradeoffs it is advantageous to send each attribute of $z$ independently to LDP randomizers that produce anonymous reports.",
"(There are other natural variants of attributes based on this encoding scheme e.g., in the context of learning algorithms [28], but these are not considered in this paper.)",
"Report fragments: Given an $\\varepsilon $ LDP budget and an encoded data record $x$ , a sequence of LDP reports may be generated by multiple independent applications of the randomizer $\\mathcal {R}$ to $x$ , while still ensuring an overall $\\varepsilon $ bound on the privacy loss.",
"Each such report is a report fragment, containing less information than the entire LDP report sequence.",
"Anonymous report fragments allow improved privacy guarantees in more refined threat models, as we show in Section .",
"Sketch-based reports: Locally-differentially-private variants of sketching [16], [7], [19] have been used for optimizing communication, computation, and storage tradeoffs w.r.t.",
"privacy/utility in the context of estimating distributions.",
"Given a domain $\\lbrace 0,1\\rbrace ^k$ , the main idea is to reduce the domain to $\\lbrace 0,1\\rbrace ^{\\kappa }$ , with $\\kappa \\ll k$ , via hashing and then use locally private protocols to operate over a domain of size $\\kappa $ .",
"To avoid significant loss of information due to hashing, and in turn boost the accuracy, the above procedure is performed with multiple independent hash functions.",
"Sketching techniques can be used in conjunction with all of the fragmentation schemes explored in this paper, with the benefits of sketching extending seamlessly, as we corroborate in experiments.",
"As a warning to practitioners, we note that sketching must be deployed carefully, and only in conjunction with tuning of its parameters.",
"Sketching will add additional estimation error—on top of the error introduced by differential privacy—and this error can easily exceed the error introduced by differential privacy, unless the sketching parameters are tuned to a specific, known target dataset, We also observe that sketching is not a requirement for practical deployments in regimes with high local-differential privacy, such as those explored in this paper.",
"A primary reason for using sketching is to reduce communication cost, by reducing the domain size from $k$ to $\\kappa \\ll k$ , but for high-epsilon LDP reports only a small number of bit may need to be sent, even without sketching.",
"If the probability of flipping a bit is $p$ for one-hot encodings of a domain size $d$ , then only the indices of $p (d-1) + (1-p)$ bits need be sent—the non-zero bits—and each such index can be sent in $\\log _2 d$ bits or less.",
"For high-epsilon one-hot-encoded LDP reports, which apply small $p$ to domains of modest size $d$ , the resulting communication cost may well be acceptable, in practice.",
"Table REF shows some examples of applying one-hot and sketch-based LDP report encodings to a real-world dataset, with sketching configured as in a practical deployment [7].",
"As the table shows, for a central privacy guarantee of $\\varepsilon _c=1$ , only the indices of one or two bits must be sent in sketch-based LDP reports; on the other hand, five or six bit indices must be sent using one-hot encodings (because the attribute-fragmented LDP reports must have $\\varepsilon _{\\ell ^{\\infty }}=12.99$ , which corresponds to $p=2.28\\times 10^{-6}$ ).",
"However, this sixfold increase in communication cost is coupled with greatly increased utility: the top $10\\mathrm {,}000$ items can be recovered quite accurately using the one-hot encoding, while only the top 100 or so can be recovered using the count sketch.",
"Such a balance of utility/privacy and communication-cost tradeoffs arises naturally in high-epsilon one-hot encodings, while with sketching it can be achieved only by hand-tuning the configuration of sketching parameters to the target data distribution."
],
[
"Anonymity and Attack Models",
"The basis of our attack model are the guarantees of local differential privacy, which are quantified by $\\varepsilon _\\ell $ and place an $e^{\\varepsilon _\\ell }$ worst-case upper bound on the information loss of each respondent that contributes reports to the statistical monitoring.",
"These guarantees are consistent with a particularly simple attack model for any one respondent, because the $\\varepsilon _\\ell $ privacy guarantees hold true even when all other parties (including other respondents) conspire to attack them—as long as that one respondent constructs reports correctly using good randomness.",
"We write $\\varepsilon _{\\ell ^{\\infty }}$ when this guarantee holds even if the respondent invokes the protocol multiple (possibly unbounded) number of times, without changing its private input.",
"Statistical reporting with strong privacy is also quantified by $\\varepsilon _c$ , as its goal is to ensure that a central analyzer can never reduce by more than $e^{\\varepsilon _c}$ its uncertainty about any respondent's data—even in the worst case, for the most vulnerable and unlucky respondent.",
"The analyzer is assumed to be a potential attacker which may adversarially compromise or collude with anyone involved in the statistical reporting; if successful in such attacks, the analyzer may be able to reduce their uncertainty from $e^{\\varepsilon _c}$ to $e^{\\varepsilon _\\ell }$ for at least some respondents.",
"Unless the analyzer is successful in such collusion, our attack model assumes that its $\\varepsilon _c$ privacy guarantee will hold.",
"In addition to the above, as in the ESA [2] architecture, an intermediary termed the shuffler can be used to ensure the anonymity of reports without having visibility into report contents (thanks to cryptography).",
"Our attack model includes such a middleman even though it adds complexity, because anonymization can greatly strengthen the $\\varepsilon _c$ guarantee that guards privacy against the prying eyes of the analyzer, as established in recent amplification results [3], [13], [29].",
"However, our attack model requires that the shuffler can learn nothing about the content of reports unless it colludes with the analyzer (this entails assumptions, e.g., about traffic analysis, which are discussed below).",
"Anonymization Intermediary: In our attack model, the shuffler is assumed to be an intermediary layer that consists of $K$ independent shuffler instances that can transport multiple reporting channels.",
"The shuffler must be a well-authenticated, networked system that can securely receive and collect reports from identifiable respondents—simultaneously, on separate reporting channels, to efficiently use resources—and forward those reports to the analyzer after their anonymization, without ever having visibility into report contents (due to encryption).",
"Each shuffler instance must separately collect reports on each channel into a sufficiently large set, or crowd, from enough distinct respondents, and must output that crowd only to the analyzer destination that is appropriate for the channel, and only in a manner that guarantees anonymity: i.e., that origin, order, and timing of report reception is hidden.",
"In particular, this anonymity can be achieved by outputting the crowd's records in a randomly-shuffled order, stripped of any metadata.",
"Our attack model abstracts away from the specifics of disparate anonymity techniques and is not limited to shuffler instances that output reports in a randomly shuffled order.",
"Depending on the primitives used to encrypt the reports, shuffler instances may output an aggregate summary of the reports by using a commutative, associative operator that can compute such a summary without decryption.",
"Such anonymous summaries are less general than shuffled reports (from which they can be constructed by post-processing), but they can be practically computed using cryptographic means [10], [11], [30] and have seen formal analysis [19], [31].",
"However, if the output is only an aggregate summary, the shuffler instance must provide quantified means of guaranteeing the integrity of that summary; in particular, summaries must be robust in the face of corruption or malicious construction of any single respondent's report, e.g., via techniques like those in [10].",
"By utilizing $K$ separate shuffler instances, each in a different trust domain, our attack model captures the possibility of partial compromise.",
"The $K$ instances should be appropriately isolated to represent a variety of different trust assumptions, e.g., by being resident in separate administrative domains (physical, legislative, etc.",
"); thereby, by choosing to which instance they sent their reports, respondents can limit their potential privacy risk (e.g., by choosing randomly, or in a manner that represents their trust beliefs).",
"Thereby, respondents may retain some privacy guarantees even when certain shuffler instances collude with attackers or are compromised.",
"The effects of any compromise may be further limited, temporally, in realizations that regularly reset to a known good state; when a respondent uses fragmentation techniques to send multiple reports, simultaneously, or over time, we quantify as $\\varepsilon _{\\ell ^{1}}$ the worst-case privacy loss due to attacker capture of a single report, noting that $\\varepsilon _{\\ell ^{1}} \\le \\varepsilon _{\\ell ^{\\infty }}$ will always hold.",
"Our attack model assumes a binary state for each shuffler instance, in which it is either fully compromised, or fully trustworthy and, further, that the compromise of one instance does not affect the others.",
"However, notably, in many realizations—such as those based on Prio [10], mixnets [8], or ESA's blinding [2]—a single shuffler instance can be constructed from $M$ independent entities, such that attackers must compromise all $M$ entities, to be successful.",
"Thereby, by using a large $M$ number of entities, and placing them in different, separately-trusted protection domains, each shuffler instance can be made arbitrarily trustworthy—albeit at the cost of reduced efficiency.",
"Our attack model assumes that an adversary (colluding with the analyzer) is able to monitor the network without breaking cryptography.",
"As a result, attackers must not benefit from learning the identity of shufflers or reporting channels to which respondents are reporting; this may entail that respondents must send more reports, and send to more destinations than strictly necessary, e.g., creating cover traffic using incorrectly-encrypted “chaff” that will be discarded by the analyzer.",
"Our attack model also abstracts away from most other concerns relating to how information may leak due to the manner in which respondents send reports, such as via timing- or traffic-analysis, via mistakes like report encodings that accidentally include an identifier, or include insufficient randomization such that reports can be linked (see the PRR discussion in [1]), or via respondents' participation in multiple reporting systems that convey overlapping information.",
"Much like in [2], our attack model abstracts away from the choice of cryptographic mechanisms or how respondents acquire trusted software or keys, and how those are updated.",
"Finally, our attack model also abstracts away from policy decisions such as which of their attributes respondents should report upon, what privacy guarantees should be considered acceptable, the manner or frequency by which respondents' self-select for reporting, how they sample what attributes to report upon, when or whether they should send empty chaff reports, and what an adequate size of a crowd should be."
],
[
"Central Differential Privacy and Amplification by Shuffling",
"To state the differential privacy guarantees that hold for the view of the analyzer (to which we often refer as central privacy) we rely on privacy amplification properties of shuffling.",
"First results of this type were established by Erlingsson et al.",
"[3] who showed that shuffling amplifies privacy of arbitrary local randomizers and Cheu et al.",
"[13] who gave a tighter analysis for the shuffled binary randomized response.",
"Balle et al.",
"[29] showed tighter bounds for non-interactive local randomizers via an elegant analysis.",
"We state here two results we use in the rest of the paper.",
"The first [29] is for general non-interactive mechanisms, and the second for a binary mechanism [13].",
"Lemma 2.5 For $\\delta \\in [0,1]$ and $\\varepsilon _\\ell \\le \\log (n/\\log (1/\\delta ))/2$ , the output of a shuffler that shuffles $n$ reports that are outputs of a $\\varepsilon _\\ell $ -DP local randomizers satisfy $(\\varepsilon , \\delta )$ -DP where $\\varepsilon = O\\left((e^{\\varepsilon _\\ell }-1)\\sqrt{\\log (1/\\delta )/n}\\right)$ .",
"Lemma 2.6 Let $\\delta \\in [0,1]$ , $n\\in \\mathbb {N}$ , and $\\lambda \\in \\left[14\\log (4/\\delta ),n\\right]$ .",
"Consider a dataset $X=(x_1,\\ldots ,x_n)\\in \\lbrace 0,1\\rbrace ^n$ .",
"For each bit $x_i$ consider the following randomization: $\\hat{x}_i\\leftarrow x_i$ w.p.",
"$\\left(1-\\frac{\\lambda }{2n}\\right)$ , and $1-x_i$ otherwise.",
"The algorithm computing an estimation of the sum $S^{\\sf priv}=\\frac{1}{n-\\lambda }\\left(\\sum \\limits _{i=1}^n \\hat{x}_i-\\frac{\\lambda }{2}\\right)$ satisfies $(\\varepsilon ,\\delta )$ -central differential privacy where $\\varepsilon =\\sqrt{\\frac{32\\log (4/\\delta )}{\\lambda -\\sqrt{2\\lambda \\log (2/\\delta )}}}\\left(1-\\frac{\\lambda -\\sqrt{2\\lambda \\log (2/\\delta )}}{n}\\right).$ We will also use the advanced composition results for differential privacy by Dwork, Rothblum and Vadhan [32] and sharpened by Bun and Steinke [33].",
"Theorem 2.7 (Advanced Composition Theorem [33]) Let $\\mathcal {M}_1, \\ldots , \\mathcal {M}_k\\colon \\mathcal {D}^n \\times \\mathcal {S}\\rightarrow \\mathcal {S}$ be algorithms such that for all $z \\in \\mathcal {S}$ , $i\\in [k]$ , $\\mathcal {M}_i(\\cdot ,z)$ satisfies $(\\varepsilon ,\\delta )$ -DP.",
"The adaptive composition of these algorithms is the algorithm that given $D\\in \\mathcal {D}^n$ and $z_0\\in \\mathcal {S}$ , outputs $(z_1,\\ldots ,z_k)$ , where $z_i$ is the output of $\\mathcal {M}_i(D,z_{i-1})$ for $i\\in [k]$ .",
"Then $\\forall \\,\\delta ^{\\prime }>0$ and $z_0\\in \\mathcal {S}$ , the adaptive composition satisfies $\\left(k\\varepsilon ^2/2 + \\sqrt{k}\\varepsilon \\cdot \\sqrt{2\\log (\\sqrt{k\\pi /2}\\varepsilon /\\delta ^{\\prime })}, \\delta ^{\\prime } + k\\delta \\right)$ -DP.",
"When these amplification and composition results are used to derive central privacy guarantees for collections of LDP reports, the details matter.",
"Depending on how information is encoded and fragmented into the LDP reports that are sent by each respondent, the resulting central privacy guarantee that can be derived may vary greatly.",
"For some types of LDP reports, new amplification results may be required to precisely account for the balance of utility and privacy.",
"Specifically—as described in the next section and further detailed in our experiments—for sketch-based LDP reports, more precise analysis have yet to be developed; as a result, the central privacy guarantees that are known to hold for anonymous, sketch-based reporting are quite unfavorable compared to those known to hold for one-hot-encoded LDP reports.",
"[t] $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ : Attribute fragmented $k$ -RAPPOR.",
"[1] Respondent data $x \\in \\mathcal {D}$ , LDP parameter $\\varepsilon _\\ell $ .",
"Compute $x\\in \\lbrace 0,1\\rbrace ^k$ , a one-hot encoding of $x$ .",
"For each $j \\in [k]$ , define $\\mathcal {R}_j(b, \\varepsilon ) := \\left\\lbrace \\begin{array}{cr}b & \\text{w.p.~}\\; e^{\\varepsilon }/\\left(1+e^{\\varepsilon }\\right) \\\\1-b & \\text{w.p.~}\\; 1/\\left(1+e^{\\varepsilon }\\right)\\end{array}\\right.$ send $\\mathcal {R}_j(x^{(j)}, \\varepsilon _\\ell )$ to shuffler $\\mathcal {S}_j$ for $j \\in [k]$"
],
[
"Histograms via Attribute Fragmenting",
"In this section we revisit and formalize the idea of attribute fragmenting [2].",
"We demonstrate its applicability in estimating high-dimensional histogramsFollowing a tradition in the differential-privacy literature [34], this paper uses the term histogram for a count of the frequency of each distinct element in a multiset drawn from a finite domain of elements.",
"with strong privacy/utility tradeoffs.",
"By applying recent results on privacy amplification by shuffling [3], [13], [14], we show that attribute fragmenting helps achieve nearly optimal privacy/utility tradeoffs both in the central and local differential privacy models w.r.t the $\\ell _\\infty $ -error in the estimated distribution.",
"Through an extensive set of experiments with data sets having long-tail distributions we show that attribute fragmenting help recover much larger fraction of the tail for the same central privacy guarantee (as compared to generically applying privacy amplification by shuffling for locally private algorithms [3], [29]).",
"In the rest of this section, we formally state the idea of attribute fragmenting and provide the theoretical guarantees.",
"We defer the experimental evaluation to Section REF .",
"Consider a local randomizer $\\mathcal {R}$ taking inputs with $k$ attributes, i.e., inputs are of the form $x_i = (x_i^{(1)}, \\ldots , x_i^{(k)})$ .",
"Attribute fragmenting comprises two ideas: First, decompose the local randomizer $\\mathcal {R}$ into $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}) := (\\mathcal {R}_1, \\ldots , \\mathcal {R}_k)$ , a tuple of independent randomizers each acting on a single attribute.",
"Second, have each respondent report $\\mathcal {R}_j(x_i^{(j)})$ to $\\mathcal {S}_j$ , one of $k$ independent shuffler instances $\\mathcal {S}_1, \\ldots , \\mathcal {S}_k$ that separately anonymize all reports of a single attribute.",
"Attribute fragmenting is applicable whenever LDP reports about individual attributes are sufficient for the task at hand, such as when estimating marginals.",
"Attribute fragmenting can also be applied to scenarios where the respondent's data is not naturally in the form of fragmented tuples.",
"Thus, we can consider two broad scenarios when applying attribute fragments: (1) Natural attributes such as when reporting demographic information about age, gender, etc., which constitute the attributes.",
"Another example would be app usage statistics across different apps with disjoint information about load times, screen usage etc.",
"(2) Synthetic fragments where a single piece of respondent data can be cast into a form that comprises several attributes to apply this fragmenting technique.",
"An immediate application of (synthetic) fragments is to the problem of learning histograms over a domain $\\mathcal {D}$ of size $k$ where each input $x_i \\in \\mathcal {D}$ can be represented as a “one-hot vector” in $\\lbrace 0,1\\rbrace ^k$ .",
"Algorithm REF shows how to (naturally) apply attribute fragmenting when the local randomizer $\\mathcal {R}$ is what is referred to as the $k$ -RAPPOR randomizer [35].",
"Theorems REF and REF demonstrate the near optimal utility/privacy tradeoff of this scheme.",
"We remark that Algorithm REF is briefly described and analyzed in [13] (for replacement LDP).",
"To estimate the histogram of reports from $n$ respondents, the server receives and sums up bits from each shuffler instance $\\mathcal {S}_j$ to construct attribute-wise sums.",
"The estimate for element $j \\in \\mathcal {D}$ is computed as: $\\hat{h}_j = \\frac{1}{n} \\cdot \\frac{e^{\\varepsilon _\\ell }+1}{e^{\\varepsilon _\\ell }-1} \\cdot \\underbrace{\\sum _{i=1}^n\\mathcal {R}_j(x_i^{(j)}, \\varepsilon _\\ell )}_{\\text{from shuffler $\\mathcal {S}_j$}} - \\frac{1}{e^{\\varepsilon _\\ell }-1}.$ We show that $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ achieves nearly optimal utility/privacy tradeoffs both for local and central privacy guarantees.",
"Accuracy is defined via the $\\ell _\\infty $ error: $\\alpha := \\max \\limits _{j\\in [k]}\\left|\\hat{h}_j-\\frac{1}{n}\\sum _{i=1}^n x_i^{(j)}\\right|$ .",
"Informally, the following theorems state that in the high-epsilon regime, $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ achieves privacy amplification satisfying $\\left(O(e^{\\varepsilon _\\ell /2}/\\sqrt{n}), \\delta \\right)$ -central DP, and achieves error bounded by $\\Theta \\left(\\sqrt{\\frac{\\log k}{ne^{\\varepsilon _\\ell }}}\\right)$ and $\\Theta \\left(\\frac{\\sqrt{\\log k}}{n\\varepsilon _c}\\right)$ in terms of its local ($\\varepsilon _\\ell $ ) and central ($\\varepsilon _c$ ) privacy respectively.",
"Proofs are deferred to Appendix REF .",
"Standard lower bounds for central differential privacy imply that the dependence of $\\alpha $ on $k$ , $n$ , and $\\varepsilon _c$ are within logarithmic factors of optimal.",
"To the best of our knowledge, the analogous dependence for $\\varepsilon _\\ell $ in the local DP model is the best known.",
"Theorem 3.1 (Privacy guarantee) Algorithm $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ satisfies removal $\\varepsilon _\\ell $ -local differential privacy and for $\\varepsilon _\\ell \\in \\left[1,\\log n-\\log \\left(14\\log \\left(\\frac{4}{\\delta }\\right)\\right)\\right]$ and $\\delta \\ge n^{-\\log n}$ , $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ satisfies removal $(\\varepsilon _c, \\delta )$ -central differential privacy in the Shuffle model where: $\\varepsilon _c= \\sqrt{\\frac{64 \\cdot e^{\\varepsilon _\\ell }\\cdot \\log (4/\\delta )}{n}}.$ Theorem 3.2 (Utility/privacy tradeoff) Algorithm $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ simultaneously satisfies $\\varepsilon _\\ell $ -local differential privacy, $(\\varepsilon _c, \\delta )$ -central differential privacy (in the Shuffle model), and has $\\ell _\\infty $ -error at most $\\alpha $ with probability at least $1-\\beta $ , where $\\alpha =\\Theta \\left(\\sqrt{\\frac{\\log (k/\\beta )}{ne^{\\varepsilon _\\ell /2}}}\\right);\\text{ equiv.\\ } \\alpha =\\Theta \\left(\\frac{\\sqrt{\\log (k/\\beta )\\log (1/\\delta )}}{n\\varepsilon _c}\\right).$ Unlike one-hot-encoded LDP reports, for deployed sketch-based LDP reporting schemes—such as the count sketch of [16], [7]—there are no analyses that are known to derive precise central privacy guarantees, while both leveraging amplification-by-shuffling and being able to account for attribute fragmentation.",
"One known approach to analyzing sketch-based LDP reports is to ignore all fragmentation and apply a generic privacy amplification-by-shuffling result, such as Lemma REF ; since it ignores attribute fragments its $\\varepsilon _\\ell $ dependence is $e^{\\varepsilon _\\ell }$ , instead of $e^{\\varepsilon _\\ell /2}$ , and its central privacy bound suffers compared to that of $k$ -RAPPOR.",
"A second known approach observes that the randomizer for each individual hash function is an instance of $k$ -RAPPOR, for which the lower $e^{\\varepsilon _\\ell /2}$ -type dependence holds.",
"However, for this second analysis, the effective size of the crowd $n$ is reduced by the number of hash functions used—making anonymity less effective in amplifying privacy—and a large number of hash functions is often required to achieve good utility.",
"Thus, for sketch-based LDP reports, the best known privacy/utility tradeoffs may not be favorable, in the eyes of practitioners, compared to those of one-hot-encoded LDP reports.",
"[color=green!30]ÚE: comment on communication cost?",
"In real-world applications—unlike what is proposed above—the number of attributes may be far too large for it to be practical to use a separate shuffler instance for each attribute.",
"For example, this can be seen in the datasets of Table REF , which we use in our experiments.",
"However, in our attack model, efficient realizations of shuffling are possible for high-epsilon LDP reports with attribute fragmenting.",
"For this, there need only be $K$ shuffler instances with each instance having a separate reporting channel for every single attribute, for a number $K$ that is sufficiently large for the dataset and task at hand.",
"For high-epsilon LDP reports, the report encoding can be constructed such that each respondent will send only a few LDP reports for a few attributes—and if this number is small enough, those reports can still be arranged to be sent to independent shuffler instances, e.g., in expectation, by randomly selecting the destination shuffler instance.",
"In particular, for the experiments of Table REF , our assumption of independence will hold as long as the number of $K$ shuffler instances is large enough for each bit to be sent to a separate instance, with high confidence, in expectation."
],
[
"Report Fragmenting",
"While the shuffle model enables respondents to send randomized reports of local data with large local differential privacy values and still enjoy the benefits of privacy amplification, it might be desirable to further reduce the risk to respondents' privacy by reducing the privacy cost of each individual report.",
"As an example, consider randomizing a single bit with the randomizer defined in Section .",
"For $\\varepsilon _\\ell =10$ , the probability of sending a flipped bit is $\\approx e^{-10}$ .",
"Therefore, given a report from a respondent, there is a roughly $99.996\\%$ chance of the report being identical to the respondent's data.",
"This probability drops to $63.21\\%$ with $\\varepsilon _\\ell =1$ .",
"Extending the ideas of fragmenting from Section , one might be tempted to consider the following different way to fragment the reports: given an LDP budget of $\\varepsilon _\\ell $ , send several reports (specifically, $\\varepsilon _\\ell /\\varepsilon _{\\ell ^{f}}$ reports) each with LDP $\\varepsilon _{\\ell ^{f}} \\ll \\varepsilon _\\ell $ .",
"While this certainly reduces the privacy cost of each report, it has an impact on the utility.",
"To replace one report of $\\varepsilon _\\ell =4$ , with several reports of $\\varepsilon _{\\ell ^{f}}=2$ while achieving the same utility one would need roughly $\\exp (\\varepsilon _\\ell /\\varepsilon _{\\ell ^{f}})=\\exp (2)\\sim 7$ reports, which blows up the local privacy loss.",
"Equivalently (see Corollary REF in Appendix REF ), for a given local privacy budget $\\varepsilon _\\ell $ , the $\\ell _\\infty $ error increases by a factor of roughly $\\sqrt{\\exp (\\varepsilon _\\ell /2)/\\varepsilon _\\ell }$ .",
"[t] $\\mathsf {r}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_b, \\mathcal {R}_f, \\varepsilon _{\\ell ^{b}}, \\varepsilon _{\\ell ^{f}}, \\tau )$ : Report fragmenting.",
"[1] Respondent data $x$ , LDP $\\varepsilon _{\\ell ^{b}}$ , fragment LDP $\\varepsilon _{\\ell ^{f}}$ , number of fragments $\\tau $ $x^{\\prime } \\leftarrow \\mathcal {R}_b(x;\\varepsilon _{\\ell ^{b}})$ $i \\in [\\tau ]$ $y_i = \\mathcal {R}_f(x^{\\prime }; \\varepsilon _{\\ell ^{f}})$ send $(i, y_i)$ to shuffler $\\mathcal {S}_i$ for $i \\in [\\tau ]$ Report fragments with privacy backstops: Inspired by the concept of a permanent randomized response [1], we propose a simple fix to the unfavorable tradeoff described above.",
"Instead of working with reports of local privacy $\\varepsilon _{\\ell ^{f}}$ on the original respondent data, the respondent first constructs a randomized response of the original data with a higher epsilon $\\varepsilon _{\\ell ^{b}}$ (for backstop) and only outputs lower epsilon reports on this randomized data.",
"More precisely, given $\\varepsilon $ -DP local randomizers $\\mathcal {R}_b(\\cdot ;\\varepsilon )$ and $\\mathcal {R}_f(\\cdot ;\\varepsilon )$ , on input data $d$ , a backstop randomized report $d^{\\prime } \\leftarrow \\mathcal {R}_b(d;\\varepsilon _{\\ell ^{b}})$ is first computed.",
"Then, we fragment the report into several reports $r_i \\leftarrow \\mathcal {R}_f(d^{\\prime }; \\varepsilon _{\\ell ^{f}})$ for several independent applications of $\\mathcal {R}_f$ .",
"We claim to get the best of both worlds with this construction.",
"With sufficiently many reports, we get utility/privacy results that are essentially what we can achieve with local privacy budget of $\\varepsilon _{\\ell ^{b}}$ while ensuring that each report continues to have small LDP.",
"The backstop ensures that even with sufficiently many reports sent to the same shuffler, the privacy guarantee does not degrade linearly with the number of reports, but stops degrading beyond the backstop $\\varepsilon _{\\ell ^{b}}$ .",
"The only price we pay is in additional communication overhead.",
"The number of fragments is only constrained by the communication costs, though beyond a few fragments there are diminishing returns for utility (at no cost to privacy).",
"The following theorem states the privacy guarantees of report fragmenting.",
"It analyzes the situation in which an adversary has gained access to $t \\le \\tau $ fragments.",
"It demonstrated that the privacy of a respondent degrades gracefully as more fragments are exposed to an adversary.",
"Theorem 4.1 For any $\\varepsilon _{\\ell ^{f}},\\varepsilon _{\\ell ^{b}}>0$ , an $\\varepsilon _{\\ell ^{b}}$ -DP local randomizer $\\mathcal {R}_b$ , an $\\varepsilon _{\\ell ^{f}}$ -DP local randomizer $\\mathcal {R}_f$ , an integer $\\tau $ , and a set of indices $J\\subseteq [\\tau ]$ of size $t$ , consider the algorithm $\\mathcal {M}_J$ that for $(y_1,\\ldots ,y_\\tau ) = \\mathsf {r}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_b, \\mathcal {R}_f, \\varepsilon _{\\ell ^{b}}, \\varepsilon _{\\ell ^{f}}, \\tau )$ outputs $y_J = (y_i)_{i\\in J}$ .",
"Then $\\mathcal {M}_J$ is an $\\varepsilon $ -DP local randomizer for $\\varepsilon = \\ln \\left(\\frac{e^{\\varepsilon _{\\ell ^{b}}+t\\varepsilon _{\\ell ^{f}}}+1}{e^{\\varepsilon _{\\ell ^{b}}}+e^{t\\varepsilon _{\\ell ^{f}}}}\\right) \\le \\min \\lbrace \\varepsilon _{\\ell ^{b}},t\\varepsilon _{\\ell ^{f}}\\rbrace $ .",
"We stated Theorem REF for the standard replacement DP.",
"If $\\mathcal {R}_b$ satisfies only removal $\\varepsilon _{\\ell ^{b}}$ -DP then $\\mathcal {M}_J$ has the same $\\varepsilon _{\\ell ^{b}}$ for removal DP.",
"The proof is based on a general result showing how DP guarantees are amplified when each data element is preprocessed by a local randomizer.",
"(Details in Appendix REF .)",
"Report fragmenting for histograms: Here we instantiate report fragmenting in the context of histograms.",
"Recall, for the histogram computation problem described in Section , each data sample is $x=\\left(x^{(1)},\\cdots ,x^{(k)}\\right)$ is a one-hot vector in $k$ dimensions.",
"In report fragmenting with privacy backstop, we do the following: For each $i\\in [k]$ , we run an instance of Algorithm independently, with $x^{(i)}$ as respondent data.",
"One can view the set of report fragments generated by all the execution of Algorithm as a matrix: $M(x)=[m_{i,j}]_{\\tau \\times k}$ , where $m_{i,j}$ refers to the $i$ -th report generated for the $j$ -th domain element.",
"To be most effective, report fragments should be sent according to respondent's trust in shuffler instances.",
"For the report fragmenting above, we obtain the following accuracy/privacy tradeoff (proof in Appendix REF ).",
"Theorem 4.2 (Utility/privacy tradeoff) For a per-report local privacy budget of $\\varepsilon _{\\ell ^{f}} > 1$ , backstop privacy budget of $\\varepsilon _{\\ell ^{b}}$ , and number of reports $\\tau $ , Algorithm $\\mathsf {r}\\textsf {-}\\mathsf {frag}(\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}}), \\tau )$ satisfies removal $\\ln \\left(\\frac{e^{\\varepsilon _{\\ell ^{b}}+\\tau \\varepsilon _{\\ell ^{f}}}+1}{e^{\\varepsilon _{\\ell ^{b}}}+e^{\\tau \\varepsilon _{\\ell ^{f}}}}\\right)$ -local differential privacy and $(\\varepsilon _c,\\delta )$ -central DP where, for any $\\delta < 1/2$ : $\\varepsilon _c= \\min \\left\\lbrace \\sqrt{\\frac{8\\tau \\varepsilon _{\\ell ^{f}} \\log ^2(\\tau \\varepsilon _{\\ell ^{f}}/\\delta )}{n}}, \\sqrt{\\frac{}{}}{64{e^{\\varepsilon _{\\ell ^{b}}}\\log (4/\\delta )}}{n}\\right\\rbrace ,$ has accuracy $\\alpha $ with probability at least $1-\\beta $ with: $\\alpha =O\\left(\\sqrt{\\frac{\\log (k/\\beta )}{n\\tau \\varepsilon _{\\ell ^{f}}}}+\\sqrt{\\frac{}{}}{\\log (k/\\beta )}{ne^{\\varepsilon _{\\ell ^{b}}}}\\right).$"
],
[
"Crowds and crowd IDs",
"Foundational to this work is the concept of a crowd: a sufficiently large set of LDP reports gathered from a large enough set of distinct respondents, such that each LDP report can become “lost in the crowd” and thereby anonymous.",
"As discussed in Section , the shuffler intermediary must ensure, independently, that a sufficiently large crowd is present on every one of the shuffler's reporting channels.",
"Channels are equivalent to (but more efficient than) a distinct shuffler with its own public identity, and channels are only hosted on a single shuffler for efficiency.",
"As such, the identity of the channel that a report is sent on must be assumed to be public.",
"As an alternative, the ESA architecture described how respondents could send LDP reports annotated by a “Crowd ID” that could be hidden by cryptographic means from both network attackers and the shuffler intermediaries (using blinded shuffling).",
"In ESA, the reports for each Crowd ID were grouped together, shuffled separately, and only output if their cardinality was sufficient; furthermore, this cardinality threshold was randomized for privacy.",
"Revisiting this alternative, we find that annotating LDP reports by IDs can be helpful, in those cases where respondents have an existing reason to publicly self-identify as belonging to a data partition—e.g., because they are unable to hide their use of certain computer hardware or software, or do not want to hide their coarse-grained location, nationality, or language preferences.",
"On the other hand, given the strength of the recent privacy amplification results based on anonymity, we find little to no value remaining in the use of ESA's Crowd IDs as a distinct reporting channel (i.e., reporting some data via an LDP report and some data via that report's ID annotation).",
"We can formally define ESA's Crowd IDs as being the set of indices $\\lbrace 1,\\ldots ,m\\rbrace $ for any partitioning of an underlying dataset of LDP records $D=\\lbrace x_1,\\dots ,x_n\\rbrace \\in \\mathcal {D}^n$ into disjoint subsets $D=D_1\\cup \\dots \\cup D_m$ .",
"For tasks like those in Section , separately analyzing each subset $D_i$ can significantly improve utility whenever reports that carry the same signal are partitioned into the same subset—i.e., if reports about the same values are associated with the same ID.",
"The expected (un-normalized) $\\ell _\\infty $ -norm estimation error for each partition $D_i$ will be $\\sqrt{{|D_i|}/{e^{\\varepsilon _\\ell }}}$ , if the records in the dataset have an $\\varepsilon _\\ell $ privacy guarantee, compared to $\\sqrt{{|D|}/{e^{\\varepsilon _\\ell }}}$ for the whole dataset.",
"Therefore, for equal-size Crowd ID partitions, the utility of monitoring can be improved by a factor of $\\sqrt{m}$ , and, if partition sizes vary a lot, the estimation error may be improved much more for the smaller partitions.",
"However, the utility improvement of Crowd IDs must come at a cost to privacy.",
"After all, Crowd IDs are visible to the analyzer and can be considered as the first component of a report pair, along with their associated LDP report.",
"As such, their total privacy cost can only be bounded by $\\varepsilon _\\ell + \\widehat{\\varepsilon _\\ell }$ : the sum of each LDP report's $\\varepsilon _\\ell $ bound and any bound $\\widehat{\\varepsilon _\\ell }$ that holds for its associated Crowd ID (and this $\\widehat{\\varepsilon _\\ell }$ may be $\\infty $ ).",
"Even without a bound on the Crowd ID privacy loss, respondents may want to send ID-annotated LDP reports.",
"In particular, this may be because partitioning is based on aspects of data that raise few privacy concerns, or are seen as being public already (e.g., the rough geographic origin of a mobile phone's data connection).",
"Alternatively, this may be because respondents see a direct benefit from sending reports in a manner that improves the utility of monitoring.",
"For example, respondents may desire to receive improved services by sending reports whose IDs depend on the version of their software, the type of their hardware device, and their general location (e.g., metropolitan area).",
"Or, to help build better predictive keyboards, respondents may send LDP reports about the words they type annotated by their software's preferred-language settings (e.g., EN-US, EN-IN, CHS, or CHT); such partitioned LDP reporting is realistic and has been deployed in practice [7], [36].",
"For lack of a better term, we can refer to such partitioning as natural Crowd IDs.",
"However, even when Crowd IDs are derived from public data, the cardinality of each partition may be a privacy concern—at least for small partitions—if Crowd IDs are derived without randomization.",
"The shuffler intermediary can address this privacy concern by applying randomized thresholding, as outlined in the original ESA paper [2].",
"For a more complete description, Algorithm shows how the shuffler can drop reports before applying a fixed threshold in order to make each partition's cardinality differentially private; furthermore, formal privacy and utility guarantee is given in Theorem REF and Theorem REF and Appendix REF includes proofs.",
"Randomized Report Deletion.",
"[1] reports partitioned by Crowd ID: $\\lbrace R_i\\rbrace _{[P]}$ , privacy parameters: $\\left(\\varepsilon ^{\\sf cr},\\delta ^{\\sf cr}\\right)$ .",
"$i\\in [P]$ $n_i\\leftarrow |R_i|$ $\\hat{n}_i\\leftarrow \\max \\lbrace n_i +{\\sf Laplace}\\left(\\frac{2}{\\varepsilon ^{\\sf cr}}\\right) - \\frac{2}{\\varepsilon ^{\\sf cr}}\\log \\left(\\frac{2}{\\delta ^{\\sf cr}}\\right),0\\rbrace $ $\\hat{n}_i \\le n_i$ $R^{\\prime }_i \\leftarrow R_i \\backslash \\left\\lbrace (n_i-\\hat{n}_i) \\text{ uniformly chosen records} \\right\\rbrace $ Abort The new partitioning by Crowd ID: $(R^{\\prime }_1, \\ldots , R^{\\prime }_P)$ Theorem 5.1 (Privacy guarantee) Algorithm satisfies $\\left(\\varepsilon ^{\\sf cr},\\delta ^{\\sf cr}\\right)$ -central differential privacy on the counts of records in each crowd.",
"Theorem 5.2 (Utility guarantee) Algorithm ensures that for all crowds $i$ , $\\left|R_i \\setminus R^{\\prime }_i\\right| \\le \\frac{4}{\\varepsilon ^{\\sf cr}}\\log \\left(\\frac{2P}{\\delta ^{\\sf cr}}\\right)$ with prob.",
"$\\ge 1-\\delta ^{\\sf cr}$ .",
"Data-derived Crowds IDs: In addition to natural Crowd IDs, ESA proposed that LDP reports could be partitioned in a purely data-dependent manner—e.g., by deriving Crowd IDs by using deterministic hash functions on the data being reported—and reported on the utility of such partitioning in experiments [2].",
"While such data-derived Crowd IDs can improve utility, their privacy/utility tradoffs cannot compete with those offered by recent privacy amplification results based on anonymity.",
"The following simple example serves to illustrate how amplification-by-shuffling have made data-derived Crowd IDs obsolete.",
"Let's assume LDP records are partitioned by a hash function $h\\colon \\mathcal {D}\\rightarrow [m]$ , for $m=2$ , with the output of $h$ defining a binary data-derived Crowd ID.",
"For worst-case analysis, we must assume a degenerate $h$ that maps any particular $z\\in \\mathcal {D}$ to 0 and all other values in $\\mathcal {D}$ to 1.",
"Therefore, the Crowd ID must be treated as holding the same information as any value $z$ contained in an LDP report with an $\\varepsilon _\\ell $ privacy guarantee; this entails that the Crowd ID must be randomized to establish for it a privacy bound $\\widehat{\\varepsilon _\\ell }$ , if the privacy loss for any value $z$ is to be limited.",
"As a result, ID-annotated LDP reports have a combined privacy bound of $\\varepsilon _\\ell + \\widehat{\\varepsilon _\\ell }$ , and any fixed privacy budget must be split between those two parameters.",
"ESA proposed that data-derived Crowd IDs could be subjected to little randomization (i.e., that $\\widehat{\\varepsilon _\\ell } \\gg \\varepsilon _\\ell $ ).",
"Thereby, ESA implicitly discounted the privacy loss of data-derived Crowd IDs, with the justification that they were only revealed when the cardinality of report subsets was above a randomized, large threshold.",
"In certain special cases—e.g., when $\\varepsilon _\\ell =0$ —such discounting may be appropriate, since randomized aggregate cardinality counts can limit the risk due to circumstances like that of the degenerate hash function $h$ above.",
"However, in general, accurately accounting for the privacy loss bounded by $\\varepsilon _\\ell + \\widehat{\\varepsilon _\\ell }$ reveals that it is best to not utilize data-derived Crowd IDs at all.",
"The best privacy/utility tradeoff is achieved by setting $\\widehat{\\varepsilon _\\ell }=0$ and not splitting the privacy budget at all (cf.",
"Table REF and Table REF ), while amplification-by-shuffling with attribute fragmenting can be used to establish meaningful central privacy guarantees.",
"[th] LDP-SGD; client-side [1] Local privacy parameter: $\\varepsilon _{\\ell e}$ , current model: $\\theta _t\\in \\mathbb {R}^d$ , $\\ell _2$ -clipping norm: $L$ .",
"Compute clipped gradient $x\\leftarrow \\nabla \\ell (\\theta _t;d)\\cdot \\min \\left\\lbrace 1,\\frac{L}{\\Vert \\nabla \\ell (\\theta _t;d)\\Vert _2}\\right\\rbrace .$ $z_i\\leftarrow {\\left\\lbrace \\begin{array}{ll}L\\cdot \\frac{x}{\\Vert x\\Vert _2} & \\text{w.p. }",
"\\frac{1}{2} + \\frac{\\Vert x\\Vert _2}{2L}, \\\\-L\\cdot \\frac{x}{\\Vert x\\Vert _2} & \\text{otherwise}.\\end{array}\\right.",
"}$ Sample $v\\sim _u \\mathbf {S}^d$ , the unit sphere in $d$ dimensions.",
"$\\hat{z}\\leftarrow {\\left\\lbrace \\begin{array}{ll}\\text{sgn}(\\langle z,v\\rangle )\\cdot v& \\text{w.p. }",
"\\frac{e^{\\varepsilon _{\\ell e}}}{1+e^{\\varepsilon _{\\ell e}}}.\\\\-\\text{sgn}(\\langle z,v\\rangle )\\cdot v& \\text{otherwise}.\\\\\\end{array}\\right.",
"}$ $\\hat{z}$ ."
],
[
"Machine Learning in the ESA framework",
"In this section we demonstrate that ESA framework is suitable for training machine learning models with strong local and central differential privacy guarantees.",
"We show both theoretically (for convex models), and empirically (in general) that one can have strong per epoch local differential privacy (denoted by $\\varepsilon _{\\ell e}$ ), and good central differential privacy overall, while achieving nearly state-of-the-art (for differentially private models) accuracy on benchmark data sets (e.g., MNIST and CIFAR-10).",
"Per-epoch local differential privacy refers to the LDP guarantee for a respondent over a single pass over the dataset.",
"Here we assume that each epoch is executed on a separate shuffler, and the adversary can observe the traffic onto only one of those shufflers.",
"However, it is worth mentioning that the central differential privacy guarantee we provide is over the complete execution of the model training algorithm.",
"Formally, we show the following: For convex Empirical Risk Minimization problems (ERMs), with local differential privacy guarantees per report on the data sample, and amplification via shuffling in the ESA framework, we achieve optimal privacy/utility tradeoffs w.r.t.",
"excess empirical risk and the corresponding central differential privacy guarantee.",
"Empirically, we show that one can achieve accuracies of $95\\%$ on MNIST, $70\\%$ on CIFAR-10, and $78\\%$ on Fashion-MNIST, with per epoch $\\varepsilon _{\\ell e}\\approx 1.9$ .",
"[thb] LDP-SGD; server-side [1] Local privacy budget per epoch: $\\varepsilon _{\\ell e}$ , number of epochs: $T$ , parameter set: $\\mathcal {C}$ .",
"$\\theta _0\\leftarrow \\lbrace 0\\rbrace ^d$ .",
"$t\\in [T]$ Send $\\theta _t$ to all clients.",
"Collect shuffled responses $(\\hat{z}_i)_{i\\in [n]}$ .",
"Noisy gradient: $g_t\\leftarrow \\frac{L\\sqrt{\\pi }}{2}\\cdot \\frac{\\Gamma \\left(\\frac{d-1}{2}+1\\right)}{\\Gamma \\left(\\frac{d}{2}+1\\right)}\\cdot \\frac{e^{\\varepsilon _{\\ell e}}+1}{e^{\\varepsilon _{\\ell e}}-1}\\left(\\frac{1}{n}\\sum \\limits _{i\\in [n]} \\hat{z}_i\\right)$.",
"Update: $\\theta _{t+1}\\leftarrow \\Pi _{\\mathcal {C}}\\left(\\theta _t - \\eta _t\\cdot g_t\\right)$ , where $\\Pi _\\mathcal {C}(\\cdot )$ is the $\\ell _2$ -projection onto set $\\mathcal {C}$ , and $\\eta _t=\\frac{\\Vert \\mathcal {C}\\Vert _2\\sqrt{n}}{L\\sqrt{d}}\\cdot \\frac{e^{\\varepsilon _{\\ell e}} - 1}{e^{\\varepsilon _{\\ell e}} + 1}$ .",
"$\\theta _{\\sf priv}\\leftarrow \\theta _T$ .",
"In the rest of this section, we state the algorithm, privacy analysis, and the utility analysis for convex losses.",
"We defer the empirical evaluation to Section REF .",
"Empirical Risk Minimization (ERM): Consider a dataset $D=(x_1,\\ldots ,x_n)\\in \\mathcal {D}^n$ , a set of models $\\mathcal {C}\\subseteq \\mathbb {R}^d$ which is not necessarily convex, and a loss function $\\ell \\colon \\mathcal {C}\\times \\mathcal {D}\\rightarrow \\mathbb {R}$ .",
"The problem of ERM is to estimate a model $\\hat{\\theta }\\in \\mathcal {C}$ such that: $R(\\hat{\\theta }):=\\frac{1}{n}\\sum \\limits _{i=1}^n\\ell (\\hat{\\theta };x_i)-\\min \\limits _{\\theta \\in \\mathcal {C}}\\frac{1}{n}\\sum \\limits _{i=1}^n\\ell (\\theta ;x_i)$ is small.",
"In this work we revisit the locally differentially private SGD algorithm of Duchi et al.",
"[17], denoted LDP-SGD (Algorithms and ), to estimate a $\\theta _{\\sf priv}\\in \\mathcal {C}$ s.t.",
"i) $R(\\theta _{\\sf priv})$ is small, and ii) the computation of $R(\\theta _{\\sf priv})$ satisfies per-epoch local differential privacy of $\\varepsilon _{\\ell e}$ , and overall central differential privacy of $(\\varepsilon _c,\\delta )$ (Theorem REF ).",
"We remark that, by adapting the analysis from [37], one can similarly address the problem of stochastic convex optimization in which the goal is to minimize the expected population loss on a dataset drawn i.i.d.",
"from some distribution.",
"At a high level, LDP-SGD follows the following template of noisy stochastic gradient descent [38], [39], [40].",
"Encode: Given a current state $\\theta _t$ , apply $\\varepsilon _{\\ell e}$ -DP randomizer from [17] to the gradient at $\\theta _t$ on all (or a subset of) the data samples in $D$ .",
"Shuffle: Shuffle all the gradients received.",
"Analyze: Average these gradients, and call it $g_t$ .",
"Update the current model as $\\theta _{t+1}\\leftarrow \\theta _t-\\eta _t\\cdot g_t$ , where $\\eta $ is the learning rate.",
"Perform steps (1)–(3) for $T$ iterations.",
"In Theorem REF , we state the privacy guarantees for LDP-SGD.",
"Furthermore, we show that under central differential guarantee achieved via shuffling, in the case of convex ERM (i.e., when the the loss function $\\ell $ is convex in its first parameter), we are able to recover the optimal privacy/utility tradeoffs (up to logarithmic factors in $n$ ) w.r.t.",
"the central differential privacy stated in [40].",
"(proof in Appendix REF ).",
"Theorem 6.1 (Privacy/utility tradeoff) Let per-epoch local differential privacy budget be $\\varepsilon _{\\ell e}\\le (\\log n)/4$ .",
"Privacy guarantee; applicable generally: Over $T$ iterations, in the shuffle model, LDP-SGD satisfies $\\left(\\varepsilon _c,\\delta \\right)$ -central differential privacy where: $\\varepsilon _c= O\\left(\\frac{e^{\\varepsilon _{\\ell e}}-1}{\\sqrt{n}}\\cdot \\sqrt{T\\log ^2(T/\\delta )}\\right).$ Utility guarantee; applicable with convexity: If we set $T=n/\\log ^2 n$ , and the loss function $\\ell (\\cdot ;\\cdot )$ is convex in its first parameter and $L$ -Lipschitz w.r.t.",
"$\\ell _2$ -norm, the expected excess empirical loss satisfies $ &\\mathop {\\mathbf {E}}\\left[\\frac{1}{n}\\sum \\limits _{i=1}^n\\ell (\\theta _{\\sf priv};x_i)\\right]-\\min \\limits _{\\theta \\in \\mathcal {C}}\\frac{1}{n}\\sum \\limits _{i=1}^n\\ell (\\theta ;x_i)\\\\&=O\\left(\\frac{L\\Vert \\mathcal {C}\\Vert _2\\sqrt{d}\\log ^2 n}{n}\\cdot \\frac{e^{\\varepsilon _{\\ell e}} + 1}{e^{\\varepsilon _{\\ell e}} -1 }\\right).",
"$ Here $\\Vert \\mathcal {C}\\Vert _2$ is the $\\ell _2$ -diameter of the set $\\mathcal {C}$ .",
"Reducing communication cost using PRGs: LDP-SGD is designed to operate in a distributed setting and it is useful to design techniques to minimize the overall communication from devices to a server.",
"Observe that in the client-side algorithm (Algorithm ) the only object that depends on data is the sign of the inner product in the computation of $\\hat{z}$ .",
"By agreeing with the server on a common sampling procedure $\\mathsf {Samp}\\colon \\lbrace 0,1\\rbrace ^\\mathsf {len} \\rightarrow \\mathbf {S}^d$ taking $\\mathsf {len}$ uniform bits and producing a uniform sample in $\\mathbf {S}^d$ , clients can communicate $\\text{sgn}(\\langle z,\\mathsf {Samp}(r)\\rangle )$ and randomness $r$ instead of $\\hat{z}$ .",
"This can be further minimized by replacing randomness $r$ of length $\\mathsf {len}$ with the seed $s$ of length 128 bits and producing $r \\leftarrow \\mathsf {PRG}(s)$ where $\\mathsf {PRG}$ is a pseudorandom generator stretching uniform short seeds to potentially much longer pseudorandom sequences.",
"Thus, communication can be reduced to 129 bits by sending $(\\text{sgn},s)$ and the server reconstructing $\\hat{z}= (\\text{sgn}) \\mathsf {Samp}(\\mathsf {PRG}(s))$ .",
"Note that only the utility of this scheme is affected by the quality of the pseudorandom generator (i.e., the uniform randomness of the PRG).",
"Revealing the PRG seed $s$ is equivalent to publishing $v$ , which is independent of the user's input $z$ ; therefore, reducing communication through the use of a PRG with suitable security properties does not affect the privacy guarantees of the resulting mechanism."
],
[
"Experimental Evaluation",
"This section covers the experimental evaluation of the ideas described in Sections –.",
"We consider three scenarios.",
"In the first set of experiments, we consider a typical power law distribution for discovering heavy hitters [16] that is derived from real data collected on a popular browser platform.",
"The second, inspired by increasing uses of differential privacy for hiding potentially sensitive location data, considers histogram estimation over flat-tailed distributions, where a small number of respondents contribute to a great many number of categories.",
"In order to visualize the privacy/utility tradeoffs, as is natural in these distributions over locations, we select three distributions that correspond to pixel values in three images.",
"The third set of experiments apply ideas in Section to train models to within state-of-the-art guarantees on standard benchmark datasets."
],
[
"A Dataset with a Heavy-Hitter Powerlaw Distribution",
"We consider the “Heavy-hitter” distribution shown in Table REF , as it is representative of on-line behavioral patterns.",
"It comprises 200 million reports collected over a period of one week from a 1.7-million-value domain.",
"The distribution is a mixture of about a hundred heavy hitters and a power law distribution with the probability density function $p(x) \\propto x^{-1.35}$ .",
"Our experiments target different central DP $\\varepsilon _c$ values to demonstrate the utility of the techniques described in previous sections.",
"Specifically, we experiment with a few central DP guarantees.",
"For each given $\\varepsilon _c$ , we consider attribute fragmenting with the corresponding $\\varepsilon _\\ell $ computed using Theorem REF , and report fragmenting with 4, 16 and 256 reports.",
"The fragmenting parameters $\\varepsilon _{\\ell ^{b}}$ and $\\varepsilon _{\\ell ^{f}}$ are selected so that the central DP is $\\varepsilon _c$ and the variance introduced in the report fragmenting step is roughly the same as that of the backstop step.",
"We compare the results with a baseline method—the Gaussian mechanism that guarantees only central DP.",
"We enforce local differential privacy by randomizing the one-hot encoding of the item, as well as using the private count-sketch algorithm [16], [7], which has been demonstrated to work well over distributions with a very large support.",
"When using private count-sketch, as in [16], [7], we use the protocol where each respondent sends one report of their data to one randomly sampled hash function.",
"This setting is different from the original non-private count-sketch algorithm, where each respondent sends their data to all hash functions.",
"This is because we need to take into consideration the noise used to guarantee local differential privacy.",
"In fact, for the count-sketch algorithm we use, it can be shown [16], [27] that under the same local DP budget used in the experiments, the utility is always the best when each respondent sends their data only to one hash function.",
"Table REF shows our experimental results.",
"In each experiment, we report $\\varepsilon _{\\ell ^{\\infty }}$ —the LDP guarantee when the adversary observes all reports from the respondent, corresponding to Theorem REF with $t=\\tau $ , and (when using report fragmenting) $\\varepsilon _{\\ell ^{1}}$ —the LDP guarantee when the adversary observes only one report from the respondent, corresponding to Theorem REF with $t=1$ .",
"For the Gaussian mechanism, we report $\\sigma $ —the standard deviation of the zero mean Gaussian noise used to achieve the desired level of central privacy.",
"To measure the utility of the algorithms, we compare the true and estimated frequencies.",
"We also report the expected communication cost for one-hot encoding and count-sketch, as discussed in Section REF .",
"The specific sketching algorithm we consider is the one described in [7].",
"Our experimental results demonstrate that: With attribute fragmenting and report fragmenting with various number of reports, we achieve close to optimal privacy-utility tradeoffs and recover the top 10,000 frequent items of the total probability mass with good central differential privacy $\\varepsilon _c\\le 1$ .",
"It is harder to bound the central privacy of count-sketch LDP reports; using off-the-shelf parameters [16], [7] results in slightly less communication cost, but this can come at a very high cost to utility.",
"As we discuss in Section , and elsewhere, one-hot encodings may be preferable in in the high-epsilon regime, at least until stronger results exist for sketch-based encodings.",
"Table: Alternativecentral differential-privacy bounds for LDP reports like those in the first three rows of Table ,computed without the use of attribute fragmentingas the minimum of the LDP guarantee and the central bound from ."
],
[
"Datasets with Low-amplitude and Flat-tailed Distributions",
"We consider three datasets described below.",
"Phone Location Dataset: We consider a real-world dataset created by Richard Harris, a graphics editor on The Times's Investigations team showing 235 million points gathered from $1.2$ million smartphones [5].Direct link to image: https://static01.nyt.com/images/2018/12/14/business/10location-insider/10location-promo-superJumbo-v2.jpg.",
"The resulting dataset is constructed by taking $2.5\\times $ the luminosity values (ranging from 0 to 255) of the image to scale up the number of datapoints such that the total number of reports is around 235 million, with each person reporting coordinates in a $1365 \\times 2048$ grid.",
"Horse Image Dataset: As in the phone location dataset, we consider the dataset corresponding to the image of a sketch of a horse with contours highlighted in white.",
"Due to the majority black nature of this image, it serves as a good test-case for the scenario where the tail is flat, but somewhat sparse.",
"Child Image Dataset: We use this drawing of a child originally used by Ledig et al.",
"[41] (converted to a grayscale) to represent a dense distribution with an average luminosity of roughly 140 and no black pixels.",
"A dense, flat tail distribution is one of the more challenging scenarios for accurately estimating differentially private histograms.",
"Table REF shows the distributions and statistics of each of these datasets.",
"As stated before, in our experiments we assume that for every $(x,y)$ with luminosity $L \\in [0,255]$ , there are $L$ respondents (for the phone location dataset, this count is scaled) each holding a message $(x,y)$ .",
"Each $(x,y)$ is converted into a one-hot-encoded LDP report sent using attribute and report fragmenting for improved central privacy.",
"In Tables REF –REF we report for each dataset on the results of experiments similar to those we performed for the heavy-hitters dataset (shown in Table REF ).",
"At various central privacy levels, we show the measured utility of anonymous LDP reporting with attribute and report fragmenting compared to the utility of analysis without any local privacy guarantee (the Gaussian mechanism applied to the original data).",
"To measure utility, we report the Root Mean Square Error (RMSE) of the resulting histogram estimate.",
"The essence of our results can be seen in Table REF , and its companion Table REF .",
"At relatively low LDP report privacy of $\\varepsilon _\\ell =2.0$ , none of the three datasets can be reconstructed, at all, whereas at higher $\\varepsilon _\\ell $ reconstruction becomes feasible; at $\\varepsilon _c=1.0$ , reconstruction is very good, and the number of LDP report messages sent per respondent is very low.",
"As shown in Table REF , such high utility at a strong central privacy is only made feasible by the application of both amplification-by-shuffling and attribute fragmenting.",
"For each of these three datasets, Tables REF –REF give detailed results of further experiments.The reconstructed images missing in these tables are included in ancillary files at https://arxiv.org/abs/XXXX.YYYY.",
"Most of these follow the pattern set by Table REF , while giving more details.",
"The exceptions are Table REF and Table REF ), which empirically demonstrate how each respondent's LDP budget is best spent on sending a single LDP report (while appropriately applying attribute or report fragmentation to that single report).",
"In our experiments we show: attribute fragmenting helps us achieve nearly optimal central privacy/accuracy tradeoff, report fragmenting helps us achieve reasonable central privacy with strong per-report local privacy under various number of reports.",
"Table: Upper bound of privacy loss as ε c \\varepsilon _c, and lower bound from membership inference attack using the averaged TPR--FPR over 10 runs.Attribute fragmenting: Each of Tables REF , REF , and REF demonstrate how attribute fragmenting achieves close to optimal privacy/utility tradeoffs comparable to central DP algorithms.",
"The improvements on reconstructing the histogram as $\\varepsilon _c$ values go up demonstrate that the optimality results hold asymptotically and bounds arguing the guarantees of privacy amplification could be tightened.",
"Report & Attribute fragmenting: Tables REF –REF demonstrate that by combining report and attribute fragmenting, in a variety of scenarios, we can achieve reasonable accuracy while guaranteeing local and central privacy guarantees and never producing highly-identifying individual reports (per-report privacy $\\varepsilon _{\\ell ^{1}}$ 's are small)."
],
[
"Machine Learning in the ESA Framework",
"In this section we provide the empirical evidence of the usefulness of the ESA framework in training machine learning model (using variants of LDP-SGD) with both local and central DP guarantees.",
"In particular, we show that with per-epoch local DP as small as $\\approx 2$ , one can can achieve close to state-of-the-art accuracy on benchmark data sets with reasonable central differential privacy guarantees.",
"We want to emphasize that state-of-the-art results [42], [44], [43] we compare against do not offer any local DP guarantees.",
"We consider three data sets, MNIST, Fashion-MNIST, and CIFAR-10.",
"We first describe the privacy budget accounting for central differential privacy and then we state the empirical results.",
"We train our learning models using LDP-SGD, with the modification that we train with randomly sub-sampled mini-batches, rather than full-batch gradient as described in Algorithm .",
"The privacy accounting is done as follows.",
"i) Fix the $\\varepsilon _{\\ell e}$ per mini-batch gradient computation, ii) Amplify the privacy via privacy amplification by shuffling using [14], and iii) Use advanced composition over all the iterations [33].",
"Because of the LDP randomness added in Algorithm (LDP-SGD; client-side) of Section , Algorithm (LDP-SGD; server-side) typically requires large mini-batches.",
"Due to engineering considerations, we simulate large batches via report fragmenting, as we do not envision the behavior to be significantly different on a real mini-batch of the same size.Note that this is only to overcome engineering constraints; we do not need group privacy accounting as this only simulates a larger implementation.",
"Formally, to simulate a batch size of $m$ with a set of $s$ individual gradients, we report $\\tau =m/s$ i.i.d.",
"LDP reports of the gradient from each respondent, with $\\varepsilon _{\\ell e}$ -local differential privacy/report.",
"(To distinguish it from actual batch size, throughout this section we refer to it as effective batch size.",
"For privacy amplification by shuffling and sampling, we consider batch size to be $m$ .)",
"Table: Architecture for MNIST and Fashion-MNIST.Table: Architecture for CIFAR-10.Implementation framework: To implement LDP-SGD, we modify the DP-SGD algorithm in Tensorflow Privacy [44] to include the new client-side noise generation algorithm (Algorithm ) and the privacy accountant.",
"MNIST and Fashion-MNIST Experiments: We train models whose architecture is described in Table REF .",
"The results on this dataset is summarized in Table REF .",
"The non-private accuracy baselines using this architecture are 99% and 89% for MNIST and Fashion-MNIST respectively.",
"We reiterate that the privacy accounting after Shuffling should be considered to be a loose upper bound.",
"To test how much higher the accuracy might reach without accounting for central DP, we also plot the entire learning curve until it saturates (varying batch sizes) at LDP $\\varepsilon _{\\ell e}=1.9$ per epoch.",
"The accuracy tops out at 95% and 78% respectively.",
"CIFAR-10 Experiments: For the CIFAR-10 dataset, we consider the model architecture in Table REF following recent work [42], [43].",
"Along the lines of work done in these papers, we first train the model without privacy all but the last layer on CIFAR-100 using the same architecture but replacing the softmax layer with one having 100 units (for the 100 classes).",
"Next, we transfer all but the last layer to a new model and only re-train the last layer with differential privacy on CIFAR-10.",
"Our non-private training baseline (of training on all layers) achieves 86% accuracy.",
"The results of this training method are summarized in Table REF .",
"As done with the MNIST experiments, keeping in mind the looseness of the central DP accounting, we also plot in Figure REF the complete learning curve up to saturation at LDP $\\varepsilon _{\\ell e}=1.8$ per epoch.",
"We see that the best achieved accuracy is 70%.",
"Figure: Accuracy vs iterations tradeoff on various data sets, with local differential privacy-per-record-per-iteration ε ℓe =1.8\\varepsilon _{\\ell e}=1.8 for CIFAR-10, and ε ℓe =1.9\\varepsilon _{\\ell e}=1.9 for MNIST and Fashion-MNIST.",
"The plots are over at least ten independent runs.Note on central differential privacy: Since we are translating local differential privacy guarantees to central differential privacy guarantees, our notion of central differential privacy is in the replacement model, i.e., two neighboring data sets of the same size but differ by one record.",
"However, the results in [42], [44], [43] are in the add/removal model, i.e., two neighboring data sets differ in the presence or absence of one data record.",
"As a blackbox, for any algorithm, $\\varepsilon _\\mathrm {Add/Remove} \\le \\varepsilon _\\mathrm {Replace} \\le 2\\varepsilon _\\mathrm {Add/Remove}$ , and for commonly used algorithms, the upper bound is close to tight.",
"Open question: We believe that the current accounting for central differential privacy via advanced composition is potentially loose, and one may get stronger guarantees via Rényi differential privacy accounting (similar to that in [42]).",
"We leave the problem of tightening the overall central differential privacy guarantee for future work.",
"Estimating lower bounds through membership inference attacks: We use the membership inference attack to measure the privacy leakage of a model [45], [46], [47], [48].",
"While these measurements yield loose lower bounds on how much information is leaked by a model, it can serve as an effective comparison across models trained with noise subject to different privacy analyses (with their separate upper bounds on differential privacy).",
"Along the lines of Yeom et al.",
"[46], for each model, we measure the average log-loss between true labels and predicted outputs over a set of samples used in training and not in training.",
"One measure of privacy leakage involves the best binary (threshold) classifier based on these loss values to distinguish between in-training and out-training examples.",
"The resulting ROC curve of the classifier across different thresholds can be used to estimate a lower bound on the privacy parameter.",
"Specifically, it is easy to strengthen the results in Yeom et al.",
"[46] to show that the difference between the true positive rate (TPR) and false positive rate (FPR) at any threshold is bound by $1-e^{-\\varepsilon }$ for a model satisfying $\\varepsilon $ -differential privacy.",
"Thus, the lower bound $\\varepsilon \\ge -\\log \\left(\\text{max}(\\text{TPR}-\\text{FPR})\\right)$ .",
"The results are shown in Table REF for all models trained.",
"As can be seen from the results, even though the $\\varepsilon _c$ upper bound are different for models trained under the ESA framework and those with DPSGD, there is no much difference in the lower bound."
],
[
"Conclusions",
"This paper's overall conclusion that it is feasible to implement high-accuracy statistical reporting with strong central privacy guarantees, as long as respondent's randomized reports are anonymized by a trustworthy intermediary.",
"Sufficient for this are a small set of primitives—applied within a relatively simple, abstract attack model—for both analysis techniques and practical technical mechanisms.",
"Apart from anonymization itself, the most critical of these primitives are those that involve fragmenting of respondents' randomized reports; first explored in the original ESA paper [2], such fragmentation turns out to be critical to achieving strong central privacy guarantees with high utility, in our empirical applications on real-world datasets.",
"As we show here, those primitives are sufficient to achieve high utility for difficult tasks such as iterative training of deep-learning neural networks, while providing both central and local guarantees of differential privacy.",
"In addition, this paper makes it clear that when it comes to practical applications of anonymous, differentially-private reporting, significantly more exploration and empirical evaluation is needed, along with more refined analysis.",
"Specifically, this need is made very clear by the discrepancy this paper finds between the utility and central privacy guarantees of anonymous one-hot-encoded LDP reports and anonymous sketch-based LDP reports, witch sketching parameters drawn from those used in real-world deployments.",
"At the very least, this discrepancy highlights how practitioners must carefully choose the mechanisms they use in sketch-based encodings, and the parameters by which they tune those mechanisms, in order to achieve good tradeoffs for the dataset and task at hand.",
"However, the lack of precise central privacy guarantees for anonymous sketch-based LDP reports also shows the pressing needs for better sketch constructions and analyses that properly account for the anonymity and fragmentation of respondents' reports.",
"While some recent work has started to look at better analysis of sketching (e.g., asymptotically [19]), practitioners should look towards the excellent tradeoffs shown here for one-hot-encoded LDP reports, until further, more practical results are derived in the large alphabet setting."
],
[
"Missing details from Section ",
"[Proof of Theorem REF ] To prove removal LDP we use the reference distribution $\\mathcal {R}_0$ to be randomized response with $\\varepsilon _\\ell $ on the $k$ -dimensional all-zeros vector 0.",
"For any $x\\in \\mathcal {D}$ (represented as one-hot binary vector in $k$ dimensions), $x$ and 0 differ in one position and therefore, by standard properties of randomized response Algorithm $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ computing $\\hat{x}^{(j)} := \\mathcal {R}_j(x^{(j)}, \\varepsilon _\\ell )$ for $j \\in [k]$ satisfies removal $\\varepsilon _\\ell $ -local differential privacy.",
"Furthermore, each $\\hat{x}^{(j)}$ by itself is computed with (replacement) $\\varepsilon _\\ell $ -DP.",
"We obtain the central differential privacy guarantee (through amplification via shuffling) by invoking Lemma REF with $\\lambda =\\frac{2n}{1+e^{\\varepsilon _\\ell }}$ .",
"The lower bound of $14\\log (4/\\delta )$ for $\\lambda $ translates (with some simplification) to an upper bound of $\\varepsilon _\\ell \\le \\log n - \\log (14\\log (4/\\delta ))$ assumed in the Theorem statement.",
"Furthermore, as $\\lambda \\ge 14\\log (2/\\delta ) \\ge 8\\log (2/\\delta )$ , we have that $\\lambda -\\sqrt{2\\lambda \\log (2/\\delta )}$ in Lemma REF is at least $\\lambda /2$ .",
"Simplifying the expression in (REF ), the central privacy guarantee for each individual bit of any $\\hat{x}$ is: $\\varepsilon _c^{\\sf bit}& \\le \\sqrt{\\frac{64\\log (4/\\delta )}{\\lambda }}=\\sqrt{\\frac{64(1+e^{\\varepsilon _\\ell })\\log (4/\\delta )}{2n}}\\nonumber \\\\&\\le \\sqrt{\\frac{64\\cdot e^{\\varepsilon _\\ell }\\log (4/\\delta )}{n}}.$ To prove removal central differential privacy for the entire output we define the algorithm $\\mathcal {M}^{\\prime }\\colon \\mathcal {D}^n \\times 2^{[n]}$ as follows.",
"Given $D=(x_1,\\ldots ,x_n)$ and a set of indices $I$ , $\\mathcal {M}^{\\prime }$ uses the reference distribution $\\mathcal {R}_0$ in place of the local randomizer for each element $x_i$ for which $i\\notin I$ .",
"Changing any $x$ to 0 for the $i$ -th element changes only one input bit.",
"It follows from Eq.",
"(REF ) that the overall $\\varepsilon _c$ for removal central differential privacy guarantee is $\\varepsilon _c=\\sqrt{\\frac{64\\cdot e^{\\varepsilon _\\ell }\\log (4/\\delta )}{n}}$ , which completes the proof.",
"[Proof of Theorem REF ] In $\\mathsf {att}\\textsf {-}\\mathsf {frag}(\\mathcal {R}_{k\\text{-RAPPOR}})$ (Algorithm REF ) consider any $x$ and the corresponding $\\hat{x}$ , the list of randomized responses $\\mathcal {R}_j(x^{(j)}, \\varepsilon _\\ell )$ .",
"For brevity, consider the random variable $\\zeta =\\left(\\frac{e^{\\varepsilon _\\ell }+1}{e^{\\varepsilon _\\ell }-1}\\cdot \\hat{x}-\\frac{1}{e^{\\varepsilon _\\ell }-1}\\right)$ .",
"It follows that $\\mathbb {E}\\left[\\zeta \\right]=x$ and furthermore ${\\sf Var}[\\zeta ]=\\frac{e^{\\varepsilon _\\ell }+1}{e^{\\varepsilon _\\ell }-1}-1=\\Theta \\left(1/e^{\\varepsilon _\\ell }\\right)$ .",
"Using standard sub-Gaussian tail bounds, and taking an union bound over the domain $[k]$ , one can show that w.p.",
"at least $1-\\beta $ , over all $n$ respondents with data $x_i$ , $\\alpha =\\left\\Vert \\hat{h}-\\frac{1}{n}\\sum x_i\\right\\Vert _\\infty =\\Theta \\left(\\sqrt{\\frac{}{}}{\\log (k/\\beta )}{n e^{\\varepsilon _\\ell }}\\right).$ Applying Theorem REF to compute $\\varepsilon _c$ in terms of $\\varepsilon _\\ell $ completes the proof."
],
[
"Missing details from Section ",
"We start by analyzing the privacy of an arbitrary combination of local DP randomizer followed by an arbitrary differentially private algorithm.",
"To simplify this analysis we show that it suffices to restrict our attention to binary domains.",
"Lemma 8.1 Assume that for every replacement $(\\varepsilon _1,\\delta _1)$ -DP local randomizer $\\mathcal {Q}_1 \\colon \\lbrace 0,1\\rbrace \\rightarrow \\lbrace 0,1\\rbrace $ and every replacement $(\\varepsilon _2,\\delta _2)$ -DP local randomizer $\\mathcal {Q}_2 \\colon \\lbrace 0,1\\rbrace \\rightarrow \\lbrace 0,1\\rbrace $ we have that $\\mathcal {Q}_2 \\circ \\mathcal {Q}_1$ is a replacement $(\\varepsilon ,\\delta )$ -DP local randomizer.",
"Then for every replacement $(\\varepsilon _1,\\delta _1)$ -DP local randomizer $\\mathcal {R}_1 \\colon X \\rightarrow Y$ and replacement $(\\varepsilon _2,\\delta _2)$ -DP local randomizer $\\mathcal {R}_2: Y \\rightarrow Z$ we have that $\\mathcal {R}_2 \\circ \\mathcal {R}_1$ is a replacement $(\\varepsilon ,\\delta )$ -DP local randomizer.",
"Let $\\mathcal {R}_1 \\colon X \\rightarrow Y$ be a replacement $(\\varepsilon _1,\\delta _1)$ -local randomizer and $\\mathcal {R}_2\\colon Y \\rightarrow Z$ be a replacement $(\\varepsilon _2,\\delta _2)$ -DP randomizer.",
"Assume for the sake of contradiction that for some $(\\varepsilon ,\\delta )$ there exists an event $S\\subseteq Z$ such that for some $x,x^{\\prime }$ : $\\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(\\mathcal {R}_1(x)) \\in S] > e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(\\mathcal {R}_1(x^{\\prime })) \\in S] + \\delta .$ We will show that then there exist an $(\\varepsilon _1,\\delta _1)$ -DP local randomizer $\\mathcal {Q}_1 \\colon \\lbrace 0,1\\rbrace \\rightarrow \\lbrace 0,1\\rbrace $ and $(\\varepsilon _2,\\delta _2)$ -DP local randomizer $\\mathcal {Q}_2 \\colon \\lbrace 0,1\\rbrace \\rightarrow \\lbrace 0,1\\rbrace $ such that $\\mathop {\\mathbf {Pr}}[\\mathcal {Q}_2(\\mathcal {Q}_1(0)) = 1] > e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {Q}_2(\\mathcal {Q}_1(1)) = 1] + \\delta ,$ contradicting the conditions of the lemma.",
"Let $y_0 := \\operatornamewithlimits{arg\\,min}_{y\\in Y}\\lbrace \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y)\\in S] \\rbrace ,$ $y_1 := \\operatornamewithlimits{arg\\,max}_{y\\in Y}\\lbrace \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y)\\in S] \\rbrace $ and let $P_1 := \\lbrace y \\in Y \\ |\\ \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x)=y] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x^{\\prime })=y] > 0 \\rbrace .$ Using this definition and our assumption we get: $ & \\left(\\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x) \\notin P_1] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x^{\\prime }) \\notin P_1]\\right) \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y_0)\\in S] \\\\& + \\left(\\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x) \\in P_1] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x^{\\prime }) \\in P_1]\\right) \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y_1)\\in S]\\\\&\\ge \\sum _{y\\in Y} \\left(\\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x)=y] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x^{\\prime })=y] \\right) \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y)\\in S] \\\\&> \\delta .",
"$ We now define $\\mathcal {Q}_1(0) := \\mathbb {1}\\left(\\mathcal {R}_1(x) \\in P_1\\right)$ and $\\mathcal {Q}_1(1) := \\mathbb {1}\\left(\\mathcal {R}_1(x^{\\prime }) \\in P_1\\right)$ , where $\\mathbb {1}\\left(\\cdot \\right)$ denotes the indicator function.",
"By this definition $\\mathcal {Q}_1$ is obtained from $\\mathcal {R}_1$ by restricting the set of inputs and postprocessing the output.",
"Thus $\\mathcal {Q}_1$ is a replacement $(\\varepsilon _1,\\delta _1)$ -DP local randomizer.",
"Next define for $b\\in \\lbrace 0,1\\rbrace $ , $\\mathcal {Q}_2(b) := \\mathbb {1}\\left(\\mathcal {R}_2(y_b) \\in S\\right)$ .",
"Again, it is easy to see that $\\mathcal {Q}_2$ is a replacement $(\\varepsilon _2,\\delta _2)$ -DP.",
"We now obtain that $ &\\mathop {\\mathbf {Pr}}[\\mathcal {Q}_2(\\mathcal {Q}_1(0)) = 1] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {Q}_2(\\mathcal {Q}_1(1)) = 1] \\\\&= \\sum _{b\\in \\lbrace 0,1\\rbrace } \\left(\\mathop {\\mathbf {Pr}}[\\mathcal {Q}_1(0)=b] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {Q}_1(1)=b] \\right) \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {Q}_2(b) = 1] \\\\&= \\left(\\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x) \\notin P_1] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x^{\\prime }) \\notin P_1]\\right) \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y_0)\\in S] \\\\& + \\left(\\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x) \\in P_1] - e^\\varepsilon \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(x^{\\prime }) \\in P_1]\\right) \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(y_1)\\in S]\\\\& > \\delta $ as needed for contradiction.",
"As an easy corollary of Lemma REF we obtain a tight upper bound in the pure differential privacy case.",
"Corollary 8.2 For every replacement $\\varepsilon _1$ -DP local randomizer $\\mathcal {R}_1 \\colon X \\rightarrow Y$ and every replacement $\\varepsilon _2$ -DP local randomizer $\\mathcal {R}_2 \\colon Y \\rightarrow Z$ we have that $\\mathcal {R}_2 \\circ \\mathcal {R}_1$ is a replacement $\\varepsilon $ -DP local randomizer for $\\varepsilon = \\ln \\left(\\frac{e^{\\varepsilon _1+\\varepsilon _2}+1}{e^{\\varepsilon _1}+e^{\\varepsilon _2}}\\right)$ .",
"In addition, if $\\mathcal {R}_1$ is removal $\\varepsilon _1$ -DP then $\\mathcal {R}_2 \\circ \\mathcal {R}_1$ is a removal $\\varepsilon $ -DP.",
"By Lemma REF it suffices to consider the case where $X=Y=Z=\\lbrace 0,1\\rbrace $ .",
"Thus it suffices to upper bound the expression: $ \\frac{ \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(0)=0] \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(0)=1] + \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(0)=1] \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(1)=1] }{\\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(1)=0] \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(0)=1] + \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(1)=1] \\cdot \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(1)=1] } .$ Denoting by $p_0 := \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(0)=0]$ , $p_1 := \\mathop {\\mathbf {Pr}}[\\mathcal {R}_1(1)=0]$ and $\\alpha = \\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(0)=1]/\\mathop {\\mathbf {Pr}}[\\mathcal {R}_2(1)=1]$ the expression becomes: $\\frac{1 + (\\alpha -1)p_0}{1 + (\\alpha -1)p_1} .$ The conditions on $\\mathcal {R}_1$ imply that $\\frac{p_0}{p_1}, \\frac{1-p_0}{1-p_1} \\in [e^{-\\varepsilon _1},e^{\\varepsilon _1}]$ and $\\alpha \\in [e^{-\\varepsilon _2},e^{\\varepsilon _2}]$ .",
"Without loss of generality we can assume that $\\alpha \\ge 1$ and thus the expression is maximized when $\\alpha = e^{\\varepsilon _2}$ and $p_0>p_1$ .",
"Maximizing the expression under these constraints we obtain that the maximum is $\\frac{e^{\\varepsilon _1+\\varepsilon _2}+1}{e^{\\varepsilon _1}+e^{\\varepsilon _2}}$ and is achieved when $p_0=1-p_1= e^{\\varepsilon _1}/(1+e^{\\varepsilon _1})$ .",
"In particular, the claimed value of $\\varepsilon $ is achieved by the standard binary randomized response with $\\varepsilon _1$ and $\\varepsilon _2$ .",
"To deal with the case of removal we can simply substitute $\\mathcal {R}_1(x^{\\prime })$ with the reference distribution $\\mathcal {R}_0$ in the analysis to obtain removal DP guarantees for $\\mathcal {R}_2 \\circ \\mathcal {R}_1$ .",
"We remark that it is easy to see that $\\frac{e^{\\varepsilon _1+\\varepsilon _2}+1}{e^{\\varepsilon _1}+e^{\\varepsilon _2}} \\le \\min \\lbrace e^{\\varepsilon _1},e^{\\varepsilon _2}\\rbrace $ .",
"Also in the regime where $\\varepsilon _1,\\varepsilon _2 \\le 1$ we obtain that $\\varepsilon = O(\\varepsilon _1 \\varepsilon _2)$ , namely the privacy is amplified by applying local randomization.",
"[Proof of Theorem REF ] The proof of local differential privacy is immediate based on Theorem REF .",
"To obtain the central differential privacy guarantee, we consider each of the terms in the $\\min $ expression for $\\varepsilon _c$ .",
"From the central differential privacy context, each of the shufflers in the execution of Algorithm can be considered to be a post-processing of the output of a single shuffler, and the privacy guarantee from this single shuffler should prevail.",
"Each of the individual reports are at most $\\varepsilon _b$ -locally differentially private, and hence by using the generic privacy amplification by shuffling result from Lemma REF , the second term in the $\\varepsilon _c$ follows.",
"To obtain the first term, recall the matrix $M(x)$ in Section .",
"Each row of the matrix satisfies $\\varepsilon _0$ -local differential privacy, and there are $\\tau $ rows in this matrix.",
"Hence, first applying privacy amplification theorem from Lemma REF on each of the rows independently, and then using advanced composition from Theorem REF over the $\\tau $ rows, we obtain the first term in $\\varepsilon _c$ , which completes the proof of the central differential privacy guarantee.",
"The utility guarantee follows immediately from the utility proof of Theorem REF ."
],
[
"Missing Details from Section ",
"[Proof of Theorem REF ] The proof follows a similar argument as [49].",
"Consider two neighboring data sets $D$ and $D^{\\prime }$ , there are only two crowd IDs whose counts get affected.",
"Since the randomization for each of the counts are done independently, we can analyze their privacy independently and then perform standard composition [34].",
"Consider a crowd $\\mathcal {D}_i$ , and the corresponding counts $n_i \\ne n^{\\prime }_i$ on data sets $D$ and $D^{\\prime }$ respectively.",
"Notice that the computation of $\\hat{n}_i$ satisfies $\\frac{\\varepsilon _\\ell ^{\\sf cr}}{2}$ -differential privacy by the Laplace mechanism [23].",
"Now, by the tail probability of Laplace noise, with probability at least $1-\\frac{\\delta ^{\\sf cr}}{2}$ , the algorithm does not abort on crowd $\\mathcal {D}_i$ .",
"In that case, the shuffler can ensure $\\hat{n}_i$ records in $\\mathcal {D}_i$ via dropping records.",
"This would ensure $\\left(\\frac{\\varepsilon _\\ell ^{\\sf cr}}{2},\\frac{\\delta ^{\\sf cr}}{2}\\right)$ -differential privacy.",
"Therefore, composing the above over the two crowds that are affected by $D$ and $D^{\\prime }$ , we complete the proof.",
"[Proof of Theorem REF ] The proof of this theorem follows from standard tail probabilities of the Laplace mechanism.",
"With probability at least $1-\\delta ^{\\sf cr}$ , for a given crowd $\\mathcal {D}_i$ , the error in the reported count is at most $2T=\\frac{4}{\\varepsilon _\\ell ^{\\sf cr}}\\log \\left(\\frac{4}{\\delta ^{\\sf cr}}\\right)$ .",
"Taking an union bound over all the $\\xi $ crowds, completes the proof."
],
[
"Missing Details from Section ",
"[Proof of Theorem REF ] We will prove the privacy and utility guarantees separately.",
"Privacy guarantee: We will prove this guarantee in two steps: (i) Amplify the local differential privacy guarantee $\\varepsilon _\\ell $ per epoch via [29] (see Theorem REF ), and (ii) Use advanced composition [49] to account for the privacy budget.",
"Combination of these two immediately implies the theorem.",
"Theorem 8.3 (Corollary 5.1 from [29]) Let $\\mathcal {R}:\\mathbb {X}\\rightarrow \\mathbb {Y}$ be an $\\varepsilon _{\\ell e}$ -local differentially private randomizer, and $\\mathcal {M}$ be the corresponding shuffled mechanism (that shuffles all the locally randomized reports).",
"If $\\varepsilon _{\\ell e}\\le \\log (n)/4$ , then $\\mathcal {M}$ satisfies $\\left(O\\left(\\frac{(e^{\\varepsilon _{\\ell e}}-1)\\sqrt{\\log (1/\\delta )}}{\\sqrt{n}}\\right),\\delta \\right)$ -central differential privacy in the shuffled setting.",
"Utility guarantee: Here we use the a standard bound on the convergence of Stochastic Gradient Descent (SGD) stated in Theorem REF .",
"One can instantiate Theorem REF in the context of this paper as follows: $F(\\theta )=\\frac{1}{n}\\sum \\limits _{i=1}^n\\ell (\\theta ;x_i)$ , and $g_t$ is the randomized gradient computed in Algorithm .",
"Theorem 8.4 (Theorem 2 from [50]) Consider a convex function $F:\\mathcal {C}\\rightarrow \\mathbb {R}$ defined over a convex set $\\mathcal {C}\\subseteq \\mathbb {R}^d$ , and consider the following SGD algorithm: $\\theta _{t+1}\\leftarrow \\Pi _{\\mathcal {C}}\\left(\\theta _t-\\frac{c}{\\sqrt{t}}g_t\\right)$ , where $\\Pi _{\\mathcal {C}}\\left(\\cdot \\right)$ is the $\\ell _2$ -projection operator onto the set $\\mathcal {C}$ , $c>0$ is a constant, and $g_t$ has the following properties.",
"i) [Unbiasedness] $\\mathop {\\mathbf {E}}[g_t]=\\bigtriangledown F(\\theta _t)$ , and ii) [Bounded Variance] $\\mathop {\\mathbf {E}}\\left[\\Vert g_t\\Vert ^2_2\\right]=G^2$ .",
"The following is true for any $T>1$ .",
"$\\mathop {\\mathbf {E}}[F(\\theta _t)]-\\min \\limits _{\\theta \\in \\mathcal {C}}F(\\theta )\\le \\left(\\frac{\\Vert \\mathcal {C}\\Vert ^2_2}{c}+cG^2\\right)\\frac{2+\\log T}{\\sqrt{T}}.$ Following the instantiation above, by the property of the noise distribution, one can easily show that $\\mathop {\\mathbf {E}}[g_t]=\\frac{1}{n}\\sum \\limits _{i=1}^n\\bigtriangledown \\ell (\\theta _t;x_i)$ , and furthermore $\\mathop {\\mathbf {E}}[\\Vert g_t\\Vert _2]=O\\left(\\frac{L\\sqrt{d}}{\\sqrt{n}}\\cdot \\frac{e^{\\varepsilon _{\\ell e}}+1}{e^{\\varepsilon _{\\ell e}}-1}\\right)=G$ .",
"(See [17] for the full derivation.",
"Setting $c=\\frac{\\Vert \\mathcal {C}\\Vert _2}{G}$ , and setting $T=n/\\log ^2 n$ completes the proof."
]
] | 2001.03618 |
[
[
"Vetting the optical transient candidates detected by the GWAC network\n using convolutional neural networks"
],
[
"Abstract The observation of the transient sky through a multitude of astrophysical messengers hasled to several scientific breakthroughs these last two decades thanks to the fast evolution ofthe observational techniques and strategies employed by the astronomers.",
"Now, it requiresto be able to coordinate multi-wavelength and multi-messenger follow-up campaign withinstruments both in space and on ground jointly capable of scanning a large fraction of thesky with a high imaging cadency and duty cycle.",
"In the optical domain, the key challengeof the wide field of view telescopes covering tens to hundreds of square degrees is to dealwith the detection, the identification and the classification of hundreds to thousands of opticaltransient (OT) candidates every night in a reasonable amount of time.",
"In the last decade, newautomated tools based on machine learning approaches have been developed to perform thosetasks with a low computing time and a high classification efficiency.",
"In this paper, we presentan efficient classification method using Convolutional Neural Networks (CNN) to discard anybogus falsely detected in astrophysical images in the optical domain.",
"We designed this toolto improve the performances of the OT detection pipeline of the Ground Wide field AngleCameras (GWAC) telescopes, a network of robotic telescopes aiming at monitoring the opticaltransient sky down to R=16 with a 15 seconds imaging cadency.",
"We applied our trainedCNN classifier on a sample of 1472 GWAC OT candidates detected by the real-time detectionpipeline.",
"It yields a good classification performance with 94% of well classified event and afalse positive rate of 4%."
],
[
"Introduction",
"The time domain astronomy aim at studying transient phenomena having a wide variety of flux and time scales and detected with a very broad range of localization accuracies in the sky depending on the astrophysical messengers emitted (electromagnetic, gravitational waves and high-energy particles).",
"For several centuries, the main observed transient phenomena were the supernovae (SNe) in the optical domain, tracing the violent fate of the most massive stars undergoing a core collapse or the thermonuclear explosion of white dwarfs accreting the matter of a companion star [18].",
"In the last century, the SNe were detected only at a rate of few per yearSee for example http://www.rochesterastronomy.org/snimages/snactive.html, mainly because the observational techniques and strategies were not optimized to frequently detect such rare eventsThe observed local (within 100 Mpc) supernovae rate is about 10$\\mathrm {^{-4}~SNe\\cdot yr^{-1}\\cdot Mpc^{-3}}$ [23]..",
"Therefore, the workload pressure on the detection pipelines and classification procedures of those transients were easily manageable by involving human actions in several steps, especially knowing that SNe can be observed during several days to months after the initial explosion with a 1-meter class telescope.",
"A first major revolution in the transient sky astronomy came with the development of the high-energy x-ray and gamma-ray telescopes and the detection of new classes of transients such as the Gamma-ray bursts [27] or the flaring blazars [11], [39], [22].",
"In addition to the high-energy emission, those transients also produce low energy broadband emission up to the radio wavelengths.",
"Hence, multi-wavelength follow-up observations across the whole electromagnetic spectrum became crucial to get a global picture of the physical processes.",
"The GRBs certainly represent one of the most extreme observational challenge for the follow-up telescopes as the short-living initial gamma-ray signal [29] can be very poorly localized within up to several tens of square degrees depending on the trigger instrument.",
"Then, a race against time is engaged to catch the so-called multi-wavelength afterglow emission that is fading very quickly so that it usually becomes unreachable for a detection 1-2 days after the trigger time by any x-ray or optical facility.",
"This kind of transient event has definitely led to the birth of a new type of astronomy where different type of electromagnetic facilities have to work together in near real-time to complete the scientific data sets.",
"Two decades ago, in the optical domain, several groups started to develop networks of small aperture robotic telescopes (for example ROTSE, TAROT, BOOTES, MASTER) that wereMost of them are still in operation.",
"capable to respond to any alert and scan a large fraction of the night sky continuously with a high cadence [34], [4], [10], [28], [12], [32].",
"The multiplication of the synergies between the space and ground-based telescopes, all broadcasting alerts about a large variety of transient sources, has largely contributed to increase the flow of data to be analyzed on real-time (photometry, spectroscopy and polarimetry).",
"Currently, the increasing pressure on the data processing of the follow-up telescopes studying the transient sky is significantly accelerating with the recent birth of the multi-messenger (MM) astronomy adding the high energy-neutrinos (HEN) and gravitational wave (GW) events in the global alert broadcasting system.",
"With the constant sensitivity improvements of the electromagnetic and the MM facilities, one can now regularly deal with the reception of several valuable transient alerts of any astrophysical type every night.",
"In the next decade, the multiplication of the facilities dedicated to the study of the transient sky and being able to make an all-sky monitoring at even deeper sensitivities will continue to progress, e.g.",
"the Large Synoptic Survey Telescope (LSST) [24], the Square Kilometer Array (SKA) [42], KM3NeT [3], SVOM [46] or the next generation of GW detectors LIGO/Virgo and Kagra [1].",
"Those projects will definitely make the time-domain astronomy enter into the big data era.",
"As an example, the LSST project [25] would produce 20 terabytes of data every night with the possibility of having several hundreds of thousands alerts per night starting from 2021 and running over ten years of operation.",
"It should extend the known SNe catalog with more than three billions of new entries (more than two orders of magnitude in terms of detection rate compared to any current survey).",
"In the optical domain several groups already developed synoptic surveys, like the Catalina Real-time Transient Survey [15], PTF [30], ASAS-SNhttp://www.astronomy.ohio-state.edu/~assassin/index.shtml, PanSTARRS [13], ATLAS [43], ZTF [5], DES [21] or Gaia [17], that explore the transient sky in addition to their participation to the various multi-messenger follow-up campaigns [2].",
"The data flow generated by those surveys are already no longer manageable in a reasonable amount of time by the standard techniques previously used for narrow field of view telescopes as shown for example for ZTF [33].",
"The standard transient detection pipelines were usually based on PSF-matching and the catalog cross-matching methods for the detection of new sources, followed by a human validation of each transient candidate for the classification task.",
"The growing alert rates and data flows now force the astronomers to develop new observational strategies and techniques to quickly detect, identify and classify the numerous uncatalogued sources they catch every night in their extensive searches.",
"New techniques using machine learning algorithm are developed to perform robust automated classifications of hundreds up to thousands of sources every night in real-time.",
"The classification task is usually split into two steps independently performed.",
"First, the goal is to filter out the bogus sources from the real uncatalogued sources of interest [35], [40], [33], [26] immediately after the detection.",
"The second step goes deeper in the classification procedure by associating an astrophysical category to an identified transient based on its temporal and/or spectral properties [36], [38], [37].",
"Among the zoo of machine learning algorithms, convolutional neural networks (CNN) are now massively used for such tasks as they are well-adapted to ingest data containing multiple arrays like images [8], [31].",
"They employ multiple interconnected layers, similar to a neuronal network, to efficiently identify patterns in images and are therefore particularly suitable for the time-domain astronomy [20].",
"In this paper, we investigate the possibility of using CNN for the vetting of the optical transient (OT) candidates that will be detected by the Ground Wide field Angle Cameras network (GWAC).",
"The GWAC system is a synoptic optical survey which is currently able to instantaneously cover 2000 square degrees on the sky with a high imaging cadency of one frame every 15 seconds.",
"In operation since 2017, GWAC is a part of the ground-based follow-up system of the SVOM mission [46], the next generation of space mission dedicated to the study of the multi-wavelength transient sky.",
"It already provides a large data flow that must be smoothly digested by the real-time data processing pipeline as well as a significant amount of OT candidates sometimes well identified as real transients such as dwarf novae outbursts recently discovered in the GWAC survey [45].",
"The GWAC network is a perfect example of the evolution of the optical facilities that emerge nowadays to study the transient phenomena.",
"It brings new observational challenges which have to be solved in order to exploit the full capabilities of the instruments.",
"In the section , we will describe the GWAC system and the transient detection pipeline which is currently running.",
"Then, in the section , we will introduce the deep machine learning classifier we set up for the vetting of the GWAC OT.",
"The classification results and performances will be presented in the section and we finally draw our conclusions and perspectives for this work in the section .",
"Since the end of 2017, the Ground Wide field Angle Cameras telescopes are under development in China at the Xinglong Observatory.",
"Each GWAC telescope mount is equipped with five cameras: four JFoV cameras (4k $\\times $ 4k CCD E2V camera with an aperture of 180 mm) and 1 FFoV camera (3k $\\times $ 3k CCD camera with an aperture of 35 mm) used to monitor the sky seeing and brightness conditions, see Figure REF .",
"The main scientific instruments, the JFoV cameras, cover a field of view of about $12.4^\\circ \\times 12.4^\\circ $ per camera ($\\sim $ 150 square degrees per camera).",
"Taking into account the overlaps between the fields of view of the 4 JFoV cameras, a GWAC mount finally covers 500 square degrees on the sky.",
"Each JFoV camera is designed to reach an unfiltered limiting magnitude of about 16 in a dark night for 10 seconds of exposure.",
"A stacking analysis of the single frames can be performed on real-time to reach a maximum limiting magnitude of R$\\sim $ 18 in clear and dark night as shown in [44].",
"Figure: The GWAC telescope network at the Xinglong observatory in China.",
"Currently, 4 mounts are operational among 10 at completion.",
"Each mount is equipped with four JFoV camera (18 cm) and one FFoV camera (3.5 cm) located at the center of the mount.",
"The total FoV of the current GWAC network is about 2000 sq.deg.Table: Characteristics of the GWAC JFoV cameras."
],
[
"The GWAC optical transient detection pipeline",
"The search for OT in GWAC data is made through several steps from the detection of candidates to their identification as being real variable/transient sources.",
"The raw images are first pre-processed camera per camera to correct them from the Dark and the Bias offsets and to make the WCS (World Coordinates System) calibration.",
"Those calibrated images are then automatically and independently analyzed by two pipelines to search for OT candidates.",
"These two pipelines make use of standard methods comparing the scientific images with reference images taken much earlier such as the catalog cross-matching and the differential image analysis (DIA).",
"Concerning the GWAC system more details can be found in [44], [45] but typically a new source is detected once it fulfills the following criteria: The source has a signal-to-noise (SNR) ratio $\\ge $ 5 and is not detected down a SNR = 5 in the reference images The source is detected in several successive images The point spread function (PSF) of the source shall be stellar-like profile, i.e.",
"a 2D gaussian profile.",
"no any CCD defect is detected in a region of 6 pixels around the source.",
"The uncatalogued sources extracted from those analysis form the preliminary OT candidate list named OT1 candidates.",
"Then, several filters are applied on the source candidate parameters (the Full Width at Half Maximum -FWHM-, the SNR, the optical peak flux, the source position, etc.)",
"on at least 5 successive images.",
"Practically speaking, these filters aim to clean the OT1 candidates from most of the spurious sources like the hot pixels or cosmic ray tracks.",
"If at least 2/5 images pass the selection criteria, the OT candidates is kept otherwise it is rejected.",
"A catalog cross matching filter using deeper catalogs is then applied to the OT1 candidates that passed the first selection criteria, see Figure REF .",
"Catalogs such as Gaia DR2 [17], PanStarrs DR1 [13], 2MASS [41], Galex DR5 [7] or public databases on solar system objects such as the Minor Planet centerhttps://minorplanetcenter.net/iau/mpc.html are used to perform this task.",
"Figure: A schematic view of the current GWAC detection pipeline setup to detect and identify the optical transient sources in both single and stacked images.After passing all of those filters, the remaining candidates are grouped in the OT2 candidates.",
"Sub-images are then cropped from each initial 4k $\\times $ 4k JFoV images and subtracted from the sky background contribution to make 100 $\\times $ 100 pixel-sized finding charts centered at the positions of each selected OT2 candidate.",
"These finding charts are then checked one by one by a human eyed-check analysis.",
"Simultaneously, two 60 cm robotic telescopes (GWAC-F60A and GWAC-F60B) located beside the GWAC telescopes at the Xinglong Observatory automatically perform follow-up observations of any source found by the GWAC system in order to help the GWAC scientist on duty to finally confirm the genuineness of a given OT2 candidate.",
"Once the OT candidates are confirmed as being real transient sources, additional follow-up observations can be triggered with larger telescopes and public alerts can be released.",
"This kind of detection pipeline is commonly used in the time domain astronomy.",
"However, while it is robust enough for telescopes with a very limited field of view (typically few tens of arcminute), it turns to be no longer the optimal solution for telescopes covering hundreds of square degrees in the sky like the GWAC system as explained in the next section."
],
[
"Data flow and false detection rate",
"The GWAC telescope network is operated in a sky survey mode following a pre-defined sky grid pointing strategy searching for bright optical transient events with a minimum of sub-minute time scale.",
"Since the beginning of 2019, four GWAC mounts (16 JFoV cameras) are operational but at completion, the full system will be composed of 10 mounts.",
"This setup implies the collection of a huge amount of data every night with typically between 6000-8000 images taken each night for a single telescope mount.",
"When the observational conditions are optimum, the current network can generate as a whole as many as 24 000 images per night (up to 80k images per night for the complete network).",
"When using the detection pipeline described above and in Figure REF , the difficulties encountered with the GWAC telescopes system mainly come from the data flow and subsequently, the large false detection rate it can produce.",
"The data flow is generated by the image cadency (the exposure time) and the number of operated cameras.",
"The false detection rate is partly due to the data flow itself but it is also strongly dependent on the optical sensitivity of the instruments, their field of views and the strictness of the transient selection criteria.",
"In addition, the large field of view of the single GWAC cameras (150 sq$\\cdot $ deg) combined with the limited size of the CCD detectors produces a large pixel scale of $\\rm {11.7~arcsec\\cdot pix^{-1}}$ and image distortion effects (while corrected in our images).",
"These two factors make the use of the catalog cross matching and the differential image analysis even more complicated.",
"This usually results in the production of additional fake detections populating the OT1 candidates category.",
"While the standard filtering algorithms are able to clean many fake OT1 candidates, there is still a large fraction of them that pass through the filters.",
"Typically, for one GWAC telescope mount, the number of OT1 candidates can be as numerous as several hundreds in a single night depending on the observational conditions.",
"Our standard filtering algorithms then reduces this number to several tens up to a few hundreds.",
"Those ones then must be manually vetted both by humans and further follow-up observations.",
"Multiplying this task to the number of GWAC mounts and one can easily understand that this \"true or false\" classification task becomes no longer manageable both by the GWAC-F60 follow-up telescopes and the GWAC scientists in a reasonable amount of time.",
"Therefore, our GWAC-F60 telescopes can be rapidly unable to ingest the quantity of triggers and additionally they can no longer smoothly follow their own observation plans independently of the GWAC camera activities.",
"Moreover, the increase in our duty scientists workload finally make them no longer being able to focus their efforts on the most promising events.",
"The identification and classification processes of a genuine transient source then undergo a long delay which is not compatible with the scientific purposes of the GWAC system that aim to quickly identify short-lived optical transient sources.",
"Our goal is to improve the current detection pipeline of the GWAC system, especially in easing the OT1 candidates classification and making the human decision-taking process more responsive.",
"As shown previously, there is a crucial need for a classification that distinguishes the astrophysical sources from the GWAC alert stream prior to build the OT2 candidates list.",
"Before going deeper into the details, we start to define few acronyms that we will use all along the paper: ROS: real optical sources in an image.",
"FOS: fake optical sources in an image.",
"ROT: real optical transients.",
"A ROT is actually a ROS present in a series of images and showing a significant flux variation.",
"FOT: fake optical transients.",
"TP: true positives, i.e.",
"the OT candidates well classified as ROT or ROS.",
"TN: true negatives, i.e.",
"the OT candidates well classified as FOT or FOS.",
"FP: false positives, i.e.",
"the OT candidates classified as ROT or ROS while there are actually FOT or FOS.",
"FN: false negatives, i.e.",
"the OT candidates classified as FOT or FOS while there are actually ROT or ROS.",
"One immediately understands that our classifier must minimize the number of FP and FN to limit the contamination of the OT2 candidates sample by any bogus in one hand and to avoid too many losses of ROT because of misclassifications in the other hand.",
"The final goal is to obtain a classification accuracy greater than 90% with a FN classification not as great as 2%.",
"Indeed, we prefer to keep more false positives (FP) instead of losing too many transients falsely classified as bogus (FN) in the classification process.",
"To perform this task, we used a Convolutional Neural Networks algorithm.",
"This choice is firstly motivated by the fact that CNNs are very well adapted for pattern recognition in images [8], [31] and have been already robustly tested with success for many different purposes in astronomy [9], [14], [16], [33].",
"Secondly, the CNNs have demonstrated excellent classification performances compared to other standard and deep machine learning methods with a minimum of implementation [20]."
],
[
"The CNN model architecture and implementation",
"While this kind of \"true or False\" classification game does not require in principle a very deep and complex network structure, a too basic network may also have limited performances even considering such a \"simple\" task as noticed by [20].",
"We therefore built a CNN code using an architecture composed of two convolutional layers , two pooling layers, one ReLu and one softmax hidden layers, see the details in Table REF .",
"The pooling layers were kept to $2 \\times 2 $ bin size due to the small size of some objects projected in the large GWAC pixel scale.",
"The cross-entropy function was used as a loss function to give a high weight for very confident false positives, which we strongly want to avoid.",
"Table: The CNN structure used in this work.The CNN was implemented in Python v3.6, using the Kerashttps://keras.io/.",
"See also [19] for a review of the usages of Keras.",
"package with TensorFlow2https://www.tensorflow.org/ https://github.com/tensorflow/tensorflow .",
"The Keras package has the advantages to provide built-in diagnostic tools and a compact code writing which allow for a relative ease of use.",
"A Keras Adam optimizer was used with a low learning rate (lr=0.0001) after witnessing disappointing convergence properties.",
"As an input, our CNN algorithm uses background-subtracted finding charts (100 $\\times $ 100 pixels) of the OT1 candidates.",
"We then select only the central part of those images (35 $\\times $ 35 pixels) for the classification.",
"This choice is motivated to have a high learning rate as the CNN requires to be trained on an extensive amount of data (typically of the order of a minimum of 10$^5$ images) while keeping enough informations (background and a minimum number of sources) in the sub-images for the pattern recognition.",
"Before being able to give any classification on our OT candidates, the CNN must be trained to recognize patterns in our images.",
"When a CNN layer receives an input, an output is then produced to feed the next layer.",
"As long as the inputs are transformed into outputs, a series of several weights is produced to finally converge and build a final probabilistic rank between 0 and 1.",
"The training phase contains several epochs of test to make the final convergence.",
"The CNN ranking is then compared to the image labeling previously made by our expert scientists which consists in giving either a mark \"1\" to sub-images containing a ROT or \"0\" if they contain FOT.",
"Therefore, this comparison method gives an idea on the level of agreement or disagreement of the CNN decision with the human classification.",
"If a disagreement is frequently observed, it means either the CNN architecture is not optimised for our classification purpose or the human labeling is not correct.",
"In such case, the CNN architecture and/or the labeling has to be revised until a good agreement is found.",
"At the end of the training, we build a Keras python model of our CNN to be used later to classify any OT candidate detected by the GWAC real-time detection pipeline.",
"Before using the CNN model in production, a final test of the classification performances is usually performed on an image sample that has never been used previously.",
"If the CNN model reaches the classification requirements, i.e.",
"if the number of misclassified sources is consistent with our scientific requirements, it can then be used to classify the genuineness of any GWAC optical transient candidates.",
"On the contrary, if the classification is not good enough, a new training with a data set more representative of the GWAC OT candidate images must be performed until the classification requirements are fulfilled.",
"We illustrate, in figure REF , the implementation we set up for both the training and the validation of the CNN algorithm as well as how it should be inserted in the detection pipeline of GWAC during real-time data taking.",
"Figure: Schematic view of the implementation of the classifier tool from the training of the CNN algorithm (left side) to the use of the CNN Keras model to make the vetting of the GWAC OT1 candidates on real-time.",
"We used a large data set of N = 200 000 images to train the CNN."
],
[
"Training data set",
"The classification of the sources into different astrophysical categories can be challenging.",
"Indeed, transient sources are rare events and one might not have collected enough images of transient sources for the training.",
"Some techniques can be used for the augmentation of the training data set such as simulating images of transient sources with a physical or empirical model or adding rotated images of real transient sources which artificially produces a new background and source distribution compared to the initial images as suggested by [20].",
"Typically, several tens of thousands images are needed to obtain a well trained CNN model.",
"Classifying our detected transients into several astrophysical categories based on additional informations such as the spectral and flux time evolution is actually beyond the scope of this work.",
"For our purpose, our bogus/real source classification tasks is independent of the nature of the transient as long as it is supposed to be a point-like source in the images.",
"As a consequence, we avoid the problem of having too few images of real GWAC transients to train the CNN.",
"Instead, we can directly extract point-like sources in GWAC images to build our sample of ROS images.",
"Our training data set is finally composed of 200 000 sub-images (35 $\\times $ 35 pixels) with an equal distribution between FOS and ROS.",
"Among them, 180 000 are directly used to train the algorithm while the 20 000 remaining images are used to validate each training epoch."
],
[
"Details on the ",
"The ROS sub-image sample is built from several 4k $\\times $ 4k GWAC images taken from the same camera during one year of operation.",
"Therefore, we have at our disposal a complete overview of the observational and sky background conditions we can encounter at the GWAC site.",
"The 4k $\\times $ 4k initial images are chosen randomly and background-subtracted to follow the GWAC detection pipeline process previously described in section REF .",
"In each of the selected images, we extracted the position of the point sources detected by the Sextractor software [6] at the 3$\\sigma $ confidence level.",
"From this list of sources, we then randomly cropped 35 $\\times $ 35 pixels sub-images around the Sextractor positions of 100 randomly chosen sources.",
"However, we make a selection cut on the instrumental magnitudes estimated by Sextractor as we want to avoid very bright or \"saturated\" stars that may produce artefacts such as blooming effect.",
"During the source extraction process and the creation of the finding charts, we noticed that the current GWAC detection pipeline can sometimes shift the centroid of the OT candidate from the center of the finding charts from 1 pixel at maximum in any direction.",
"To be as close as possible to the GWAC pipeline output, we also reproduced this trend for each of our ROS sub-image.",
"The centroid of each extracted ROS is therefore shifted in position in all the direction possible by an increment (uniformly) randomly chosen in the range [-1;1] pixel.",
"We reproduced this operation on 1000 different 4k $\\times $ 4k images to obtain a final sample of 100 000 images of ROS.",
"We show, in figure REF , a sub-sample of ROS images we used for the training of our CNN.",
"Figure: Example of some sub-images (35×\\times 35 pixels) centered (±\\pm 2 pixel) at the position of point-like sources (ROTs) extracted from the GWAC 4k ×\\times 4k images.",
"100 000 images similar to these ones are produced to build our ROT training data set.",
"Note that we extract point-like source with no prescription relative to their position in the original image (close to the edge or not, located in a dense star field, etc.",
")."
],
[
"Details on the ",
"While the ROS should all have a similar 2D gaussian profile (in the ideal case with negligible distortion effects) it is no longer the case for FOS.",
"Indeed, a large variety of bogus can lead to false detections such as cosmic-ray tracks, hot pixels, bad pixels, crosstalks artefacts, dusts, irregularities in the sky background contribution, etc.",
"Therefore, our FOS training data set must reproduce as close as possible such bogus shape distribution.",
"As it is actually very complicated to exactly mimic all the types of bogus we may encounter, we finally divided our bogus in several categories that are easily reproducible and correspond to the most frequent type of the bogus we encounter in GWAC images: hot pixels, background noise, bad column of pixels, dark pixels and a sky background with a significant light gradient.",
"To reach the same statistics than the ROS training sample we had to use data augmentation techniques as we did not get enough images of all the categories of bogus.",
"We simulated 100 000 images of bogus (50% of the full training data set) in equal proportions between our five categories defined above.",
"Our bogus simulator starts with the same process than for extracting ROS from the 4k $\\times $ 4k GWAC images.",
"From the background-subtracted initial images, we extract 35 $\\times $ 35 pixel sub-images and add a bogus in the central position similarly as we did for the ROS.",
"Then comes the difference in the process, depending on the bogus to simulate we crop different parts of the 4k $\\times $ 4k images according to the following criteria: For the hot pixel sub-images: no Sextractor sources should have a position (X,Y) in the sub-image consistent within a region of at least 6 pixels around the central pixel (X0,Y0): $(X-X0)^{2} + (Y-Y0)^{2} \\ge 36$ For the noisy sub-images: no Sextractor sources should be present in the sub-images only the background residual noise.",
"For the bad pixel columns sub-images: any sub-image randomly chosen is suitable.",
"For the dark pixels sub-images: any sub-image randomly chosen is suitable.",
"For the non uniform sky background sub-images: we select only the position of brightest stars (estimated by Sextractor), even the saturated stars, that produce a light gradient in the surrounding pixels.",
"The distance from the center of the image to the position of bright star centroid can span from $d \\in [6;15]$ pixels.",
"For the hot pixels, we actually choose to randomly put a single or a group of hot pixels (2 $\\times $ 2 pixels at maximum) at the central position of the sub-images following the pixel intensities we observed from real data.",
"We also add a random increment spanning in [-1;1] pixel to slightly shift the position of the bogus from the center.",
"The bad column of pixels were simulated as an excess of light observed normalised to the pixel intensities we observed for this kind of bogus in real data.",
"Also according to the real data the number of bad columns ranges from 1 to 3 either in the X or Y direction of the image.",
"In figure REF , we compare typical bogus we simulated with the observed ones in the OT1 candidates finding charts.",
"Figure: The five categories of bogus simulated for the FOS training data set (a) compared with (b) the same kind of bogus we indeed observed in the GWAC OT1 finding charts (100 ×\\times 100 pixel-sized)."
],
[
"Analysis and results",
"The analysis of the classification performance of our CNN is made in two steps: the training to build the keras model and the validation step of the classification procedure on a previously unseen image sample.",
"For the training, the ROS images are labeled \"1\" while the FOS are labeled \"0\".",
"We then compared this labeling with the CNN model probabilistic prediction spanning in the range $P_{CNN}\\in [0;1]$ .",
"Therefore, a source in a given image is considered as an FOS if $P_{CNN}<0.5$ and as an ROS if $P_{CNN}\\ge 0.5$ .",
"The mid value 0.5 represents a perfect random guess by the CNN model between the two categories."
],
[
"The training",
"We trained our CNN algorithm on the 200 000 simulated images (50% ROS, 50% FOS) making 10 training epochs to build the final Keras model.",
"Based on the $P_{CNN}$ criteria, we can build the normalized confusion matrix for a quick look of the classification results.",
"The normalized confusion matrices allow to display the fraction of the well classified instances as TN, TP and the fraction of the mis-classified ones in the FN and FP categories, as shown in figure REF .",
"The normalized values of TN, TP, FN, FP obey to the following rule: $\\frac{TN + FN}{N_{FOS}} = 1,~N_{FOS}=10^{5}~~;~~\\frac{TP + FP}{N_{ROS}} = 1,~N_{ROS}=10^{5}$ Figure: The normalized confusion matrix produced after the training of the CNN algorithm on 200 000 simulated images of bogus and real sources.",
"The numbers in each blue square indicate the fraction of the total instances correctly classified as FP (top left) and TP (bottom right) while in the white squares are shown the mis-classified instances as TN (top right) and FN (bottom left).",
"For a perfect classifier, the blue squares would indicate \"1\" while the white squares would indicate \"0\".Based on the training data set, the normalized confusion matrix shows that the CNN algorithm has been well trained to recognize bogus and real sources with classification perfomances close the ideal case where the TP and the FP instances would be maximized up to a normalize value of \"1\" while the FP and the FN would have been minimize to \"0\".",
"To better characterize the classification response of our CNN model, we also computed three diagnosis: The receiver operating characteristic (ROC) curves that display on graph the True Positive Rate (TPR) as function of the False Positive Rate.",
"The Area Under the ROC Curve (AUC) which corresponds to the integral of the ROC curve $\\in [0;1]$ .",
"\"0\" or \"1\" correspond to an ideal case where 100% of the instances are mis- or well classified.",
"The Accuracy coefficient (AC) $\\in [0;1]$ .",
"\"0\" or \"1\" correspond to an ideal case where 100% of the instances are mis- or well classified.",
": $AC = \\frac{TN+TP}{TN+FP+FN+TP}$ The ROC curve of the CNN model applied to the training data set also shows that we obtain a very high TPR (close to the maximum value \"1\") while keeping a extremely low FPR (close to \"0\"), see figure REF .",
"This trend is a very convincing proof that a classifier is behaving well as it falsely classifies a very few number of detected events.",
"Figure: The ROC curve of our CNN model applied to the training data set.",
"The AUC value is also indicate at the bottom.Finally, the corresponding AUC and AC are 0.99 and 0.986, respectively, and also point out a very good classification performance of our CNN model.",
"All these diagnosis confirm that the architecture of our trained CNN model is well adapted to distinguish bogus form real sources.",
"However, while it helps to confirm that the architecture of the CNN is robust enough to perform this kind of classification task, it does not guarantee at all that our CNN model will have the same performance on real GWAC images as our implementation has a limitation.",
"Indeed, we could not simulate all the types of bogus we encounter in the real GWAC images as it would require a too large amount data for the simulation which translates in a higher computational cost and a severe worsening of the simulation complexity.",
"Nevertheless, our FOS simulations and the architecture of our CNN are expected to be generic enough to deal with unseen bogus that may share the same properties than our simulated ones.",
"As an example, we did not simulated any cosmic-ray track in our bogus sample but we simulated some groups of defective pixels that share common properties with those of cosmic-ray tracks (elongated shape with no PSF model or having a very sharp PSF model)."
],
[
"The validation",
"To finally validate the classification performance of our CNN we confront it with a new sample of images representative of the zoo of bogus and real sources the GWAC pipeline generally detects.",
"Each of those images have been previously labeled by our expert scientists following the same labeling rule described in REF .",
"In addition to this labeling, each image has been manually classified into representative categories such as real moving objects, hot pixels, flaring stars, variable stars, bad pixels, dark pixels, incorrectly processed columns of pixels.",
"This categorization fits the different groups of bogus used in the simulated training data set and are the most common bogus encountered in GWAC images.",
"Our validation image sample is finally composed of 7841 images of 1472 objects detected by the GWAC transient search pipeline in 2017 and 2018.",
"The detail of the object distribution into each source category is shown in the table REF .",
"Table: The different categories of the image sample used to validate the classification performances of our CNN.As for the training sample, the probability given by the CNN on each image, P$_{CNN}$ , is compared with the image labeling to make the diagnosis of the classification performance.",
"When we applied our trained CNN model on those images, we finally found that the overall accuracy of the classifier is AC =0.94 with a very low number of FN classification, around 2% of the total sample.",
"Around only 4% of the images containing a bogus are misclassified as ROS (FP) as shown in Figure REF .",
"These classification performances are in good agreement with our scientific requirements mentioned in section and hence, we consider that our generic deep learning classifier is robust enough to be automatized in the transient detection pipeline as a tool to vet the GWAC optical transient candidates.",
"In table REF , we give more details on the classification performance for each source category used in the validation sample.",
"Figure: The confusion matrix and the ROC curve of our trained CNN model applied on a complete unseen data set of 7841 images of bogus and real astrophysical sources.",
"The classification diagnosis AUC is around 0.95 in good agreement with our scientific classification requirements.Table: The results of the CNN classification in the different categories of real/bogus sources tested during the validation step.In addition, we also explored the capabilities of our generic CNN model in classifying bogus images that were not included in the simulation of the training sample.",
"We added to our initial validation samples around 1700 images (a data augmentation of $\\sim 20$ %) of completely new bogus types such as dust obstructions, suspected ROS or low signal-to-noise ratio candidates and a large variety of bad pixels.",
"While the addition of those new bogus make the classification accuracy dropped to AC = 0.91, we found that the performances are still good enough with respect to our scientific requirements.",
"It reinforces the validation test and overall shows how powerful and generic are the CNN algorithms, even with relatively simple layer architectures, in distinguishing any type of bogus from real point-like sources."
],
[
"Analysis of the bogus rejection and false positive detections",
"The analysis presented above only considered the classification of the individual images of bogus and real sources.",
"However, the vetting of the OT candidate must also include the time evolution of the source candidates, i.e.",
"taking into account the image time series.",
"Therefore, the rejection of bogus is made on the basis of the evolution of the CNN score across several images.",
"The mean of the CNN probabilities that tracks the stability of the CNN ranking over the image time series is used as a criterion of rejection.",
"Playing on this criterion allows to determine the final rate of FP and FN the system will tolerate.",
"The current GWAC pipeline is taking a decision on the classification of the candidate after analyzing five consecutive images.",
"For comparison, we used the same numbers of images to take a decision with the CNN.",
"If we have less than five images for a given candidate we computed the mean of P$_{CNN}$ on a minimum of two images.",
"as shown in equation REF , we choose different rejection criteria in order to analyse the evolution of the FP and FN as function of the strictness of our rejection.",
"$R=\\sum _{i=0}^{N=5}\\frac{P_{CNN,i}}{N}\\le 5\\sigma ,~0.997~(3\\sigma ),~0.95~(2\\sigma )~\\rm {and }~0.68~(1\\sigma )$ A candidate is finally classified as a bogus if it satisfies the rejection criterion otherwise it is classified as a real point-like source.",
"We applied these criteria to the full validation sample of candidates (1861 candidates including the bogus not simulated in the training data set) and show, in Table REF , the evolution of FN, FP.",
"Table: The evolution of the false positives (FP) and false negatives (FN) as function of the rejection criterion R≥0.99,0.95,0.90R \\ge 0.99,~0.95,~0.90.The goal is to find the good trade-off in the rejection criterion in order to minimize both FP and FN.",
"A too strict rejection may enhance too much the FN while keeping the FP very low, i.e.we miss some real events but do not get any fake.",
"On the contrary, a too shallow rejection will go into the opposite direction, i.e.",
"we would keep many bogus by ensuring to keep all the real events.",
"We find that a rejection criterion at R=3$\\sigma $ confidence level is finally a good trade-off with less than 2% of ROT loss and about 7% of false positive detections ($\\sim $ 91% of OT candidate well classified as bogus or real sources).",
"We noticed that the FN candidates are actually sources having brightness very close to the detection threshold with a SNR$\\le 3$ which make them hard to be clearly identified by our CNN algorithm trained on securely detected OT.",
"These results translate into the following scenario.",
"In a typical night, where a hundred of candidates would have been detected, only 7 bogus would have been eventually stored in the OT1 candidates list before passing through the series of filter described in section REF and the human/GWAC-F60 telescope vetting step.",
"We find that this number of false positives is now easily manageable on a whole night by the GWAC data processing pipeline, the scientist on duty and in addition it will significantly reduce the workload pressure on the GWAC-F60 telescopes scheduler."
],
[
"Conclusion",
"The fast identification and classification of the transient sources are the major challenge to take up for the current and the near upcoming wide field angle facilities dedicated to the time-domain astronomy.",
"In this paper, we have presented a method to distinguish real astrophysical sources from many types of bogus detected by the GWAC survey telescopes (FoV = 25$^\\circ $ , R$_{lim}$ = 16 in 10s) based on a deep machine learning approach.",
"The machine learning methods are usually easy-to-set-up, cost-effective, time-effective and bring a valuable automated classification procedure to any transient detection pipeline.",
"The first \"True or False\" classification step is now unavoidable to obtain efficient transient search pipeline and quick human reaction to validate the optical transient candidates.",
"To solve the problem of optical transient vetting in the GWAC images, we used a convolutional neural network classifier trained on computationally-enhanced data relying on the GWAC database to generate images of real sources and bogus.",
"The CNN classifier proved to be very efficient in filtering out many types of bogus using a few amount of images for the decision.",
"The final false positive alarm ratio is less than 5% when it is applied to the individual images.",
"When applying the CNN classifier on the image time series of each OT candidate, we end up with about 7% of FP optical transient classifications at the level of the OT1 candidate sample.",
"The great advantage of our classifier is that it keeps the loss of real OT (FN) as low as 2% of the total transient candidate sample.",
"This is a key parameter to maintain a high level of transient detection rate every night.",
"Including such classifier tool in the transient detection pipeline of GWAC will significantly lighten the workload pressure of the pipeline itself and the GWAC duty scientists.",
"These performances are in well agreement with the scientific requirements of the GWAC system that aim at detecting and quickly identifying optical transient sources.",
"Therefore, the output CNN score is a precious information for the scientists who will have to take important decisions and actions with respect to any detected OT candidate.",
"Our classifier is generic enough so that a quick configuration of the CNN parameters can also make it usable for other kind of optical facilities.",
"This work amongst others shows how it is important now for wide field angle telescopes in the time domain astronomy to use such machine learning techniques to deal with huge data flow and big data analysis."
],
[
"Acknowledgements",
"D.Turpin acknowledges the financial support from the Chinese Academy of Sciences (CAS) PIFI post-doctoral fellowship program (program C).",
"D. Turpin is now supported by the CNES Postdoctoral Fellowship at Département d'astrophysique du CEA-Saclay.",
"M. Ganet acknowledges the financial support of LAL and NAOC.",
"S. Antier is supported by the CNES Postdoctoral Fellowship at Laboratoire AstroParticule et Cosmologie.",
"This work is supported by the National Natural Science Foundation of China (Grant No.",
"11533003 and 11973055 ) as well as the Strategic Priority Research Program of the Chinese Academy of SciencesGrant No.XDB23040000 and the Strategic Pionner Program on Space Science, Chinese Academy of Sciences, Grant No.XDA15052600."
]
] | 2001.03424 |
[
[
"Modelling an electricity market oligopoly with a competitive fringe and\n generation investments"
],
[
"Abstract Market power behaviour often occurs in modern wholesale electricity markets.",
"Mixed Complementarity Problems (MCPs) have been typically used for computational modelling of market power when it is characterised by an oligopoly with competitive fringe.",
"However, such models can lead to myopic and contradictory behaviour.",
"Previous works in the literature have suggested using conjectural variations to overcome this modelling issue.",
"We first show however, that an oligopoly with competitive fringe where all firms have investment decisions, will also lead to myopic and contradictory behaviour when modelled using conjectural variations.",
"Consequently, we develop an Equilibrium Problem with Equilibrium Constraints (EPEC) to model such an electricity market structure.",
"The EPEC models two types of players: price-making firms, who have market power, and price-taking firms, who do not.",
"In addition to generation decisions, all firms have endogenous investment decisions for multiple new generating technologies.",
"The results indicate that, when modelling an oligopoly with a competitive fringe and generation investment decisions, an EPEC model can represent a more realistic market structure and overcome the myopic behaviour observed in MCPs.",
"The EPEC considered found multiple equilibria for investment decisions and firms' profits.",
"However, market prices and consumer costs were found to remain relatively constant across the equilibria.",
"In addition, the model shows how it may be optimal for price-making firms to occasionally sell some of their electricity below marginal cost in order to de-incentivize price-taking firms from investing further into the market.",
"Such strategic behaviour would not be captured by MCP or cost-minimisation models."
],
[
"Introduction",
"Electricity market modelling is an area of research that has attracted much attention in the Operations Research literature.",
"Optimisation and equilibrium models in particular have been extensively used to better understand the behaviour of electricity generators.",
"Such tools provide insights from planning, operations and regulatory perspectives.",
"Regulators may use them to monitor market inefficiencies, profit-maximising generators may use them to gain insights on possible trading strategies while policy-makers may use them to test the impact of different proposed policy mechanisms.",
"Since the 1980s, countries have been deregulating their electricity markets with the intention of splitting ownership of market activities [36].",
"Governments' goals are to foster competition, increase market efficiencies and thus reduce consumer costs.",
"As a result, individual market participants, also known as market players, have been behaving selfishly by seeking to independently maximise their profits [13].",
"Deregulation has resulted in many electricity markets showing evidence of market power [28].",
"Market power is present when one (or more) seller(s) in the market can strategically maximise their profits by influencing the selling price through the quantity they supply to the market.",
"When such behaviour is not present in the market, the market is perfectly competitive.",
"Accurately modelling market power in electricity markets is a challenging area of operations research.",
"However, there has been many electricity market models that have incorporated market power.",
"For a comprehensive review of electricity market models that incorporate market power, we refer the reader to [36].",
"More recent examples in the operations research literature include [5] who use a Mixed Complementarity Problem (MCP) to study different consumer led load shedding strategies.",
"MCPs solve multiple constrained optimisation problems simultaneously and in equilibrium and they allow players with market power to be modelled as Cournot players.",
"[14] proposed a Mathematical Program with Equilibrium Constraints (MPEC) for strategic bidding in an electricity market.",
"An MPEC model solves a bi-level optimisation problem which is a mathematical program where one or more optimisation problems are embedded within another optimisation problem.",
"The outer optimisation is the upper-level optimisation while the inner optimisations, which are represented in the outer problem as constraints, are the lower-level optimisation problems.",
"In an electricity market setting, MPEC problems can be used to model markets where a single player has market power.",
"The upper level represents the optimisation problem of the player who has market power while the lower level problems represents the problems of players that do not have market power.",
"[39] present a methodology that combines Lagrangian relaxation and nested Benders decomposition to model a single hydro producer with market power.",
"Similarly, [19] us an optimisation-driven heuristic approach to model a large electricity consumer with market power.",
"In both of these works, only one market participant has a strategic advantage.",
"The papers in the previous paragraphs do not consider strategic behaviour when the overall market was characterised by an by oligopoly with competitive fringe.",
"Such a market structure occurs when more than one generator (the oligopolists) have market power and at least one generator does not (the competitive fringe).",
"Many modern electricity markets are characterised by an oligopoly with competitive fringe [4], [40].",
"However, when it comes to the energy market modelling literature, such markets structures are under-represented.",
"Some exceptions include [22] and [1], who use a Mixed Complemenatrity Problem (MCP) to model an oligopoly with competitive fringe in international oil market contexts.",
"Similarly, [6] develop a MCP model of an oligopoly with competitive fringe to investigate the impact demand response has on market power in an electricity market.",
"[22] highlights how modelling an oligopoly with competitive fringe using a MCP can lead to myopic, counter-intuitive and thus unrealistic optimal decisions from the oligopolists.",
"In a MCP framework, each oligopolist optimises its own position but does not take into account the optimal reaction of the competitive fringe.",
"The oligopolists reduce their generation levels with the intention of increasing the market price and hence increasing their overall profits.",
"However, when the oligopolists do this, the competitive fringe increase their generation and fill the generation gap.",
"This results in market prices not increasing as the oligopolists would anticipate.",
"[22] proposes using conjectural variations to overcome the issue.",
"Conjectural variations make assumptions about how players react to other players' quantity changes and have been widely used in the energy market modelling literature [10], [20], [24], [2], [11].",
"These assumptions allow the oligopolists to somewhat incorporate the reactions of the competitive into their decision making process.",
"However, the resulting decisions from the oligopolists may not be necessarily be optimal.",
"Neither [22], [1] nor [6] consider investment in new generation decisions.",
"Consequently, in this work, we show that when investment decisions are incorporated into a MCP model of an oligopoly with competitive fringe, conjectural variations still lead to contradictory behaviour from the oligopolists.",
"Moreover, to overcome the modelling issue, we develop an Equilibrium Problem with Equilibrium Constraints (EPEC) model of an oligopoly with competitive fringe where both oligopolists and the competitive fringe have investment decisions.",
"Each firm may initially hold, and invest in, multiple generating technologies.",
"An EPEC model solves multiple interconnected MPEC problems in equilibrium [16].",
"In this work, each MPEC represents the optimisation problem of a different electricity generating firm who has market power (also known as a price-making firm).",
"The price-making firms each seek to maximise profits subject to capacity constraints.",
"In addition, the equilibrium conditions representing the optimisation problems of the competitive fringe are embedded into each price-making firm's problem as constraints.",
"Consequently, an EPEC approach can overcome the limiting assumptions associated with conjectural variations.",
"Instead of making assumptions of how players react to other players' quantity changes, an EPEC model allows oligopolists to explicitly account for optimal reactions of the competitive fringe and thus to make optimal decisions by anticipating those reactions.",
"There has been many examples in the literature where EPECs have been used to model electricity markets.",
"Many of the first EPEC models for electricity markets consider electricity generators in the upper level and an Independent System Operator (ISO) in the lower level [21], [38], [34].",
"Building on these works, [41] and [42] develop an EPEC model that incorporates both capacity expansion and generation decisions amongst electricity generators.",
"In the upper level, the generators decide investment decisions whilst accounting for operational decisions in the lower level.",
"[35] and [37] developed a model similar to [41] but add an extra level to the model; an ISO who makes transmission expansion decisions whilst accounting for generators' capacity investment and operational decisions.",
"[25] consider a similar EPEC model as well but, in contrast to [35] and [37], model price-responsive demand and strategic interactions amongst the generators.",
"[26] and [27] use EPEC models where the upper-level problem determines the optimal investment for strategic producers while lower-level problems represent different market clearing scenarios.",
"Similarly, [43] use a EPEC model to investigate the impact consumer led demand shifting has on market power and find that demand response can reduce the negative impacts of market power.",
"The upper level again represents the producers' problems while the lower-level represent the market clearing process, in addition to the consumers decisions.",
"An EPEC model is used in [23] as part of a three-stage equilibrium model between a supra-national planner, zonal planners, and an ISO.",
"[33] develop an EPEC model that considers generators' operational decisions in the lower level and their ramping decisions in the upper level.",
"More recently, [18] introduce another EPEC model where the upper level maximises generators' decisions while the lower level represents an ISO.",
"Interestingly, [18] account for risk-averse decision making by incorporating Conditional Vale at Risk (CVaR) into their model.",
"Despite the rich literature of EPEC models for electricity markets, none of the aforementioned EPEC problems model a market characterised by an oligopoly with competitive fringe, where some generators have market power and other do not.",
"The closest work to the present paper is [44].",
"They propose a three-stage game to model transmission network expansion in an imperfectly competitive market where some generators have market power while do not.",
"They solve the model using backward induction.",
"The third stage represents the problem of the ISO and the competitive fringe.",
"The second stage represents the firms who have market power and thus account for the third stage.",
"In the first stage, social welfare is maximised using network expansion decisions whilst accounting for the second and third stages.",
"Significantly, we advance the work of [44] by including generation expansion/investments for generating firms in our model.",
"We apply the EPEC developed in this work to an electricity market representative of the Irish power system in 2025 using data from [30] and [3].",
"EPEC models can be challenging to solve computationally.",
"We utilise the Gauss Seidel algorithm in to order use diagonalization for solving the EPEC, and we solve each individual MPEC using disjunctive constraints [15].",
"In addition, to improve computational efficiency, we utilise the approach developed in [29] to provide an initial strong stationary point of the EPEC to use as a starting point for our diagonlization algorithm.",
"We solve the model numerically as it is too large to be solved in closed form, which is another contribution of this work.",
"A closed-form solution is possible using standard techniques but we combine two techniques from the literature in order to solve our problem.",
"Our results show that may it be optimal for generating firms with market power to occasionally operate some of their generating units at a loss.",
"The driving factor behind this model outcome is the fact we allow both price-making and price-taking firms to make investment decisions.",
"Price-taking firms' ability to invest further into the market motivates the price-making firms to depress prices in some timepoints.",
"This reduces the revenues price-taking firms could make from new investments and thus prevents them from making such investments.",
"Such strategic behaviour would not be captured by MCP or cost-minimisation models.",
"Consequently, this result highlights the suitability of the EPEC modelling approach and the importance of including investment decisions in models of oligopolies with competitive fringes.",
"The remainder of this paper is structured as follows: firstly, in Section , we describe the model data inputs.",
"Secondly, in Section we demonstrate the naivety of using a MCP to model an oligopoly with a competitive fringe where both price-making and price-taking firms have investment decisions.",
"Thirdly, in Section , we introduce the EPEC model.",
"In Sections and , we provide some discussion and conclusions, respectively.",
"Finally, in , we provide additional material related to the case study."
],
[
"Input data",
"In this section, we introduce the market we consider and describe the data inputs for models we use in Sections and .",
"The electricity market we consider consists of two types of players: price-making firms and price-taking firms.",
"Price-making firms may exert market power by using generation decisions to influence the market price.",
"Price-taking firms do not have such ability.",
"Each firm chooses its forward market generation decision so as to maximise its profits.",
"Each firm may also hold multiple generating units with the technologies considered being baseload, mid merit and peakload.",
"The firms are distinguished by their price-making ability and their initial generation portfolio they may hold.",
"However, each firm may also invest in new generation capacity in any of the technologies.",
"Table: Initial power generation portfolio by firm (CAP f,t CAP_{f,t}).We consider a electricity market that consists of four generating firms; firms $l=1$ and $l=2$ are price-making firms and while firms $f=3$ and $f=4$ are price-taking firms.",
"We consider $|T|=6$ generating technologies; existing baseload, existing mid-merit, existing peaking, new baseload, new mid-merit and new peaking.",
"Each of the four firms hold different initial generating capacities.",
"Firms $l=2$ , $f=3$ and $f=4$ are, initially, specialised baseload, mid-merit and peaking firms, respectively.",
"In contrast, firm $l=1$ is a integrated firm initially holding capacity across each of the existing technologies.",
"Because of their sizes, the integrated firm and the specialised baseload firm are modelled as the price-making firms while the specialised mid-merit and peaking firms are the price-taking firms.",
"Initially, each firm only holds `existing' technologies but, through their respective optimisation problems, may invest in any of the `new' technologies.",
"Given the stylised nature of the model and following [7] and [32] , we do not explicitly model renewable technologies.",
"Wind is incorporated into the model via the (net) demand intercept (see Market Clearing Condition (REF )).",
"We assume wind is not owned by any generation firm and its sole function is to reduce net demand.",
"This is because wind has a marginal cost of zero and furthermore can only be dispatched downwards, and so given an exogenously-determined level of wind capacity, wind generation itself is unlikely to ever be strategically withheld by a generation firm [7].",
"The initial portfolios of each firm are displayed in Table REF .",
"These capacities follow from [30] and [3] and are broadly based on [12].",
"Table: Summary of techno-economic input data of considered supply side technologies.The different characteristics associated with the technologies are displayed in Table REF .",
"Both the marginal generation and investment costs again follow from [30] and [3].",
"The marginal investment costs represent annualised investment costs.",
"We consider $|P|=5$ forward time periods.",
"Table REF displays the demand curve intercept values which correspond to average hourly values for each time period.",
"However, each time period $p$ is assigned a weight $W_{p}=\\frac{8760}{5}$ .",
"Thus, the test case in this work represents one year.",
"Following [31], the five time periods represent summer low demand, summer high demand, winter low demand, winter high demand and winter peak demand.",
"The demand curve slope value chosen is $B=9.091$ .",
"This parameter choice follows from [7] and [8].",
"Table: Demand curve intercept (A p A_{p}) values"
],
[
"Modelling electricity market as a mixed complementarity problem",
"In this section, we motivate our EPEC modelling approach, by applying the above data to a Mixed Complementarity Problem (MCP).",
"This analysis demonstrates the naivety of using a MCP to model an oligopoly with a competitive fringe where both price-making and price-taking firms have investment decisions.",
"A MCP determines an equilibrium of multiple optimisation problems by finding a point that satisfies the KKT conditions of each optimisation simultaneously as a system of non-linear equations [17].",
"MCPs have used to model many energy markets [22], [9].",
"However, in a MCP modelling framework, a price-making firm's optimisation problem does not contain the optimal reactions of price-taking firms as constraints.",
"As this subsection shows, this omission leads to myopic and contradictory outcomes.",
"This is in contrast to the EPEC model described in Section .",
"describes the MCP problem.",
"The MCP consists of the market clearing condition (REF ), the price-taking firms' KKT conditions and the KKT conditions for all price-making firms.",
"The parameter $CV_{l}$ represents the Conjectural Variation (CV) associated with firm $l$ .",
"CVs have been implemented in many cases MCP models [10], [20], [24], [11] as they allow price-making firms to somewhat account for the optimal reactions of competitors.",
"Conjectural variations take a value in the range $[0,1]$ .",
"[22] proposes a methodology to determine CVs that can be used to overcome myopic behaviour in models of an oligopoly with a competitive fringe.",
"We advance the work of [22] by incorporating investment decisions.",
"We solve the MCP eleven times.",
"Each time with a different CV for the two price-making firms; both firms have the same CV in each case.",
"When $CV_{l=1}=CV_{l=2}=0$ , both price-making firms lose their price-making ability and thus the market outcome corresponds to perfect competition.",
"The remaining cases correspond to an oligopoly with competitive fringe modeled through conjectural variations.",
"Figure: Price-making firm l=1l=1's profits using MCP setting framework.Figure: Price-making firms' investment into new mid-merit generation under MCP setting framework.Figure REF describes the profits of the price-making firm $l=1$ for the MCP cases.",
"It shows that firm $l=1$ actually make less profits in the oligopoly with competitive fringe cases compared with perfect competition case.",
"Clearly, if firms have price-making ability then they should be able to to use that ability, at the very least, to make the same profits as they would have in a perfect competition setting.",
"The result can be explained by Figure REF which shows the investment in new mid-merit generation for perfect competition case and $CV_{l=1}=CV_{l=2}=1$ case (similar results are observed for the $0<CV_{l}<1$ cases).",
"In the perfect competition case, all firms invest 713MW into of new mid-merit generation.",
"However, in the oligopoly with competitive fringe case, the two price-making firms do not invest in any new technology while the two price-taking firms each increase their investment in new mid-merit generation to 1736MW.",
"Note: in both cases, there are zero investments in new baseload or new peaking generation.",
"As equations (REF ) and (REF ) show, price-making firms 1 and 2 assume $\\frac{\\partial \\gamma _{p}}{\\partial gen^{\\text{PM}}_{l, t,p}}=-CV_{l}\\times B$ in the oligopoly with competitive fringe case.",
"These means that these two firms assume that if they decrease the amount of electricity they generate by one MW, then the equilibrium market price will increase by $CV_{l}\\times B$ .",
"In seeking to increase profits, price-making firms $l=1$ and $l=2$ decrease their generation in this way and hence do no invest in any new technology, as there is no point if they are not going to fully use that new generation.",
"However, in the MCP with CV setting, price-making firms do not correctly account for the optimal reactions of the competitive fringe.",
"Consequently, when price-making firms decrease their generation, the price-taking firms increase their generation and replace the price-making firms generation.",
"Thus, the market price does not increase as much the price-making firms anticipate, if it increases at all.",
"The expanded opportunity for price-taking firms to generate enables them to invest further into new mid-merit generation, as evidenced by Figure REF .",
"The assumption that $\\frac{\\partial \\gamma _{p}}{\\partial gen^{\\text{PM}}_{l, t,p}}=-CV_{l}\\times B$ in the oligopoly with competitive fringe case is not valid in a MCP setting where investment decisions are also incorporated.",
"However, this assumption is valid in a MCP setting if all firms are price-making firms and hence behave in the same manner.",
"In other words, when one firms seeks to increase the market price by decreases its generation, so does the rest of the firms and no one firm replaces the decreased generation from any other firm.",
"This section demonstrates how the MCP modelling approach is unsuited to modelling an oligopoly with competitive fringe when investment decisions are also accounted for.",
"Moreover, conjectural variations cannot overcome this modelling issue.",
"In the following section, we show how the EPEC modelling approach overcomes the short-sighted/myopic behaviour observed in this section."
],
[
"Equilibrium problem with equilibrium constraints",
"In this section we describe the Equilibrium Problem with Equilibrium Constraints (EPEC) we introduce in this work.",
"As before, it represents an electricity market with two types of players: price-making firms and price-taking firms.",
"Price-making firms may exert market power by using generation decisions to influence the market price.",
"Price-taking firms do not have such ability.",
"Each firm chooses its forward market generation decision so as to maximise its profits.",
"Each firm may also hold multiple generating units with the technologies considered being baseload, mid merit and peakload.",
"The firms are distinguished by their price-making ability and their initial generation portfolio they may hold.",
"However, each firm may also invest in new generation capacity in any of the technologies.",
"The optimisation problems of each price-taking firm are embedded into the optimisation problem of each price-making firm.",
"Thus, each price-making firm's problem is a bi-level optimisation problem and can be described as a Mathematical Program with Equilibrium Constraints (MPEC); the equilibrium constraints are the optimality conditions of the price-taking firms.",
"This problem formulation enables each price-making firm to influence the market price through their decision variables, account for the optimal reactions of price-taking firms and, consequently, maximise profits.",
"The overall EPEC problem is to find an equilibrium among the MPEC problems of each price-making firm which represent Nash Equilibria amongst them.",
"Each MPEC problem can be represented as a Mixed Integer Non-Linear Problem (MINLP) and, thus, finding Nash Equilibria is a challenging task.",
"To do so, we employ the Gauss-Seidel algorithm .",
"Furthermore to obtain an initial starting solution for this algorithm we utilise the approach taken in [29] for solving EPEC problems (henceforth known as the Leyffer-Munson approach).",
"Both the Guass-Seidel algorithm and the Leyffer-Munson approach are described in detail in this section.",
"Throughout this section the following conventions are used: lower-case Roman letters indicate indices or primal variables, upper-case Roman letters represent parameters (i.e., data), while Greek letters indicate prices or dual variables.",
"The variables in parentheses alongside each constraint in this section are the Lagrange multipliers associated with those constraints.",
"Tables REF - REF explain the indices, variables and parameters, respectively, associated with both the price-making and price-taking firms' optimisation problems."
],
[
"Price-taking firm $f$ 's problem",
"Price-taking firm $f$ seeks to maximise its profits (revenue less costs) by choosing investments in new capacity ($inv^{\\text{PT}}_{f,t}$ ) and by choosing the amount of electricity to generate from each technology at each time period ($gen^{\\text{PT}}_{f, t,p}$ ).",
"We assume each time period represents a forward time-period.",
"Thus $gen^{\\text{PT}}_{f, t,p}$ represents forward market generation decisions.",
"Firm $f$ 's costs include the per unit investment cost ($IC^{\\text{GEN}}_{t}$ ) and the marginal cost of generation ($C^{\\text{GEN}}_{t}$ ) while its revenues comes from the forward market price $\\gamma _{p}$ .",
"Price-taking firm $f$ 's optimisation problem is as follows: $\\begin{split}\\max _{\\begin{array}{c} gen^{\\text{PT}}_{f,t,p},inv^{\\text{PT}}_{f,t}\\\\\\end{array}}\\:\\:&\\sum _{t,p} W_{p}\\times gen^{\\text{PT}}_{f, t,p} \\times \\big ( \\gamma _{p} - C^{\\text{GEN}}_{t}\\big )-\\sum _{t} IC^{\\text{GEN}}_{t}\\times inv^{\\text{PT}}_{f,t},\\end{split}$ subject to: $gen^{\\text{PT}}_{f, t,p} \\le CAP^{\\text{PT}}_{f,t}+inv^{\\text{PT}}_{f,t}, \\:\\: \\forall t,p,$ where the parameter $W_{p}$ is the weight associated with timestep $p$ .",
"Constraint (REF ) ensures, for each generating technology in each timestep, firm $f$ cannot generate more than its initial capacity ($CAP^{\\text{PT}}_{f,t}$ ) plus any new investments.",
"The variable alongside constraint (REF ) ($\\lambda ^{\\text{PT},1}_{f,t,p}$ ) is the Lagrange multiplier associated with this constraint.",
"In addition, each of firm $f$ 's generation and investment decisions are constrained to be non-negative.",
"As firm $f$ is a price-taker, it cannot influence the market price with its generation decision.",
"The variable $\\gamma _{p}$ is exogenous to firm $f$ 's problem but a variable of the overall EPEC problem.",
"When determining firm $f$ 's Karush-Kuhn-Tucker (KKT) conditions, it is assumed $\\frac{\\partial \\gamma _{p}}{\\partial gen^{\\text{PT}}_{f, t,p}}=0$ .",
"Moreover, firm $f$ cannot see and hence account for the optimal decisions of other price-taking firms in addition to those of price-making firms.",
"As firm $f$ 's problem is linear, solving its associated KKT conditions ensures its problem is optimised [16]."
],
[
"Market clearing conditions",
"The forward market price for each time period is determined from the following market clearing condition: $\\gamma _{p}=A_{p} -B \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big ), \\:\\: \\forall p,$ where $A_{p}$ represents the demand curve intercept for each time period while $B$ is the time independent demand curve slope.",
"Condition (REF ) represents a linear demand curve and allows the market price to increases as the total market generation decreases and vice-versa."
],
[
"Price-maker $l$ 's MPEC",
"Price-making firm $l$ 's optimisation problem is similar to firm $f$ 's problem in that it too seeks to maximise profits (revenues less cost) by choosing its investment ($inv^{\\text{PM}}_{l,t}$ ) and forward market generation ($gen^{\\text{PM}}_{l, t,p}$ ) decisions.",
"As before, firm $l$ 's revenues come the forward market price while its costs include marginal generation and investment costs.",
"In contrast to Section REF however, price-making firm $l$ can use their generation decisions to influence the market price.",
"Price-making firm $l$ can also account for the optimal reactions of the price-taking firms.",
"Its objective function is $\\begin{split}\\max _{\\begin{array}{c}gen^{\\text{PM}}_{l,t,p}, inv^{\\text{PM}}_{l,t}\\\\gen^{\\text{PT}}_{f,t,p}, inv^{\\text{PT}}_{f,t}\\\\\\gamma _{p}, \\lambda ^{\\text{PT}}_{f, t,p}\\end{array}}\\:\\:&\\sum _{t,p} W_{p}\\times gen^{\\text{PM}}_{l, t,p} \\times \\big ( \\gamma _{p} - C^{\\text{GEN}}_{t}\\big )-\\sum _{t} IC^{\\text{GEN}}_{t}\\times inv^{\\text{PM}}_{l,t}.\\end{split}$ As firm $l$ can influence the market price through its generation decisions, we re-write objective function (REF ) using market clearing condition (REF ) as follows: $\\begin{split}\\max _{\\begin{array}{c}gen^{\\text{PM}}_{l,t,p}, inv^{\\text{PM}}_{l,t}\\\\gen^{\\text{PT}}_{f,t,p}, inv^{\\text{PT}}_{f,t}\\\\\\gamma _{p}, \\lambda ^{\\text{PT}}_{f, t,p}\\end{array}}\\:\\:&\\sum _{t,p} W_{p} \\times \\bigg ( A_{p} -B \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big ) - C^{\\text{GEN}}_{t} \\bigg ) \\times gen^{\\text{PM}}_{l, t,p} -\\sum _{t} IC^{\\text{GEN}}_{t}\\times inv^{\\text{PM}}_{l,t}.\\end{split}$ The constraints of price-making firm $l$ 's problem are $gen^{\\text{PM}}_{l, t,p} &\\le & CAP^{\\text{PM}}_{l,t}+inv^{\\text{PM}}_{l,t}, \\:\\: \\forall t,p, $ where each of firm $l$ 's generation and investment decisions are also constrained to be non-negative.",
"As with the price-taking firms, constraint (REF ) ensures, for each technology at each time period, firm $l$ cannot generate more electricity than its initial initial capacity plus any new investments.",
"In addition to constraint (REF ), firm $l$ 's constraints also include the KKT conditions of the each price-taking firm: $0\\le gen^{\\text{PT}}_{f, t,p} &\\perp & -W_{p} \\times \\big ( \\gamma _{p}-C^{\\text{GEN}}_{t} \\big )+\\lambda ^{\\text{PT}}_{f, t,p} \\ge 0, \\:\\: \\forall f,t,p,\\\\0 \\le inv^{\\text{PT}}_{f, t} &\\perp & IC^{\\text{GEN}}_{t}-\\sum _{p}\\lambda ^{\\text{PT}}_{f, t,p} \\ge 0, \\:\\: \\forall f,t,\\\\0 \\le \\lambda ^{\\text{PT},1}_{f,t,p} &\\perp & -gen^{\\text{PT}}_{f, t,p}+CAP^{\\text{PT}}_{f,t}+inv^{\\text{PT}}_{f,t} \\ge 0, \\:\\: \\forall f,t,p.$ Using market clearing condition (REF ) leads to condition (REF ) being re-written as follows: $0\\le gen^{\\text{PT}}_{f, t,p} \\perp -W_{p} \\times \\bigg (A_{p} -B \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big )-C^{\\text{GEN}}_{t} \\bigg )+\\lambda ^{\\text{PT}}_{f, t,p} \\ge 0, \\:\\: \\forall f,t,p.$ Constraints () - (REF ) represent the optimal reactions, of each price-taking firm.",
"As firm $f$ 's problem (equations (REF ) and (REF )) is a linear optimisation problem, these KKT conditions are both necessary and sufficient for optimality for the price-taking firms [16].",
"Incorporating these conditions as constraints makes firm $l$ 's problem a bi-level optimisation problem and ensures firm $l$ correctly anticipates how each price-taking firm will react to its decisions.",
"Thus, this allows firm $l$ to adjust its decisions accordingly when seeking maximise its profits.",
"Firm $l$ 's optimisation problem is affected by the generation decisions of all other price-making firms; see objective function (REF ) and constraint (REF ).",
"However, when solving firm $l$ 's problem we assume the decisions of all other price-making firms are fixed and exogenous to firm $l$ 's problem.",
"Sections REF – REF describes how the optimisation problems all price-making firms are solved such that solutions represent Nash equilibria.",
"As the KKT conditions () - (REF ) represent the equilibrium constraints (optimal reactions) of the price-taking firms, firm $l$ 's problem is a Mathematical Program with Equilibrium Constraints.",
"We denote this problem as MPEC$_{l}$ , which is a non-linear mathematical program because of the bi-linear terms in objective function (REF ) involving firms $l$ 's generation decisions and the generation decisions of all price-taking firms and because of the complementarity conditions incorporated into constraints () - (REF ).",
"However, following the approach presented in [15], we can remove the latter source of non-linearity using disjunctive constraints and big $M$ notation.",
"Consequently, this leads to constraints () - (REF ) being re-written as follows: $0 &\\le & gen^{\\text{PT}}_{f, t,p} \\le M \\times r^{\\text{1}}_{f, t,p},\\:\\: \\forall f,t,p,\\\\0 &\\le & -W_{p} \\times \\bigg (A_{p} -B \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big )-C^{\\text{GEN}}_{t} \\bigg )+\\lambda ^{\\text{PT}}_{f, t,p} \\le M \\times (1-r^{\\text{1}}_{f, t,p}),\\:\\: \\forall f,t,p,\\\\0 &\\le & inv^{\\text{PT}}_{f, t} \\le M \\times r^{\\text{2}}_{f, t}, \\:\\: \\forall f,t,\\\\0 &\\le & IC^{\\text{GEN}}_{t}-\\sum _{p}\\lambda ^{\\text{PT}}_{f, t,p} \\le M \\times (1-r^{\\text{2}}_{f, t} ), \\:\\: \\forall f,t,\\\\0 &\\le & \\lambda ^{\\text{PT},1}_{f, t,p} \\le M \\times r^{\\text{3}}_{f, t, p}, \\:\\: \\forall f,t,p.\\\\0 &\\le & -gen^{\\text{PT}}_{f, t,p}+CAP^{\\text{PT}}_{f,t}+inv^{\\text{PT}}_{f,t} \\le M \\times (1-r^{\\text{3}}_{f, t,p} ), \\:\\: \\forall f,t,p.$ where $r^{\\text{1}}_{f, t,p}$ , $r^{\\text{2}}_{f, t}$ and $r^{\\text{3}}_{f, t,p}$ all represent binary 0-1 variables.",
"When solving the overall EPEC problem using the Gauss-Seidel algorithm (see Section REF ), MPEC$_{l}$ is characterized by objective function (REF ), subject to constraint (REF ) and constraints (REF ) - ().",
"Consequently price-making firm $l$ 's optimisation problem is a Mixed Integer Non-Linear Problem.",
"In Section , we use the DICOPT solver in GAMS to solve it."
],
[
"Overall EPEC",
"The overall EPEC can be expressed as the problem of finding Nash equilibria among the price-makers $l$ : $\\begin{split}\\text{Find: } & \\bigg \\lbrace inv^{\\text{PM}}_{l=1,t},...,inv^{\\text{PM}}_{l=L,t},\\\\&gen^{\\text{PM}}_{l=1, t,p},...,gen^{\\text{PM}}_{l=L, t,p},\\\\&inv^{\\text{PT}}_{f=1,t},...,inv^{\\text{PT}}_{f=F,t},\\\\&gen^{\\text{PT}}_{f=1, t,p},...,gen^{\\text{PT}}_{f=F, t,p},\\\\&\\lambda ^{\\text{PT}}_{f=1,t,p},...,\\lambda ^{\\text{PT}}_{f=F,t,p},\\\\& \\gamma _{p} \\bigg \\rbrace \\text{ that solve:}\\\\& \\text{MPEC}_{l} \\text{ for each } l=1,...,L.\\end{split}$ To find the such equilbiria, we implement the following Gauss-Seidel [16] algorithm.",
"The algorithm iteratively solves each price-making firm's MPEC problem by fixing every other price-making firms' decisions, until it converges to a point where neither leader has an optimal deviation.",
"[H] $\\sum _{l} | x_{l,k}-x_{l,k-1} | > TOL $ and $k < K$ $l=1,..,L$ Assume price maker$_{-l}$ 's decision variables are fixed Solve MPEC$_{l}$ Gauss-Seidel algorithm where $TOL$ and $K$ represent a pre-defined convergence tolerance and a maximum number of allowable iterations, respectively.",
"The vector $x_{l,t}$ represents the vector of all MPEC$_{l}$ 's primal variables at iteration $k$ ."
],
[
"Obtaining an initial solution",
"To improve computational efficiency, we utilise the approach to obtaining a strong stationary point for EPECs, as described in [29].",
"We then use the statioanry point obtained as a starting point to the Gauss-Seidel algorithm.",
"In this subsection, we describe how the Leyffer-Munson method as applied to the EPEC presented in this work.",
"Firstly, we re-write price-making firm $l$ 's problem, as defined by equations (REF ) - (REF ), using slack variables.",
"We do this by converting firm $l$ 's inequality constraints into equality constraints as follows: $\\begin{split}\\max \\:\\:&\\sum _{t,p} W_{p} \\times \\bigg ( A_{p} -B \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big ) - C^{\\text{GEN}}_{t} \\bigg ) \\times gen^{\\text{PM}}_{l, t,p} -\\sum _{t} IC^{\\text{GEN}}_{t}\\times inv^{\\text{PM}}_{l,t},\\end{split}$ subject to: $CAP^{\\text{PM}}_{l,t}+inv^{\\text{PM}}_{l,t}- gen^{\\text{PM}}_{l, t,p}-s^{\\text{CON\\_LR}}_{l, t,p}&=&0, \\:\\: \\forall t,p, \\:\\: (\\lambda ^{PM}_{l, t,p}), \\\\gen^{\\text{PM}}_{l, t,p} &\\ge & 0, \\:\\: \\forall t,p,\\:\\: (\\chi ^{\\text{GEN}}_{l,t,p}), \\\\inv^{\\text{PM}}_{l, t} &\\ge & 0, \\:\\: \\forall t,\\:\\: (\\chi ^{\\text{INV}}_{l,t}), \\\\s^{\\text{CON\\_LR}}_{l, t,p}&\\ge & 0, \\:\\: \\forall t,p,\\:\\: (\\mu ^{\\text{CON\\_LR}}_{l, t,p}), \\\\-W_{p} \\times \\bigg (A_{p} -B \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big )-C^{\\text{GEN}}_{t} \\bigg )+\\lambda ^{\\text{PT}}_{f, t,p}-s^{\\text{KKT\\_GEN}}_{f, t,p} &=& 0, \\:\\: \\forall f,t,p,\\:\\: (\\alpha ^{\\text{KKT\\_GEN}}_{l, f, t,p}), \\\\gen^{\\text{PT}}_{f, t,p} &\\ge & 0, \\:\\: \\forall f,t,p,\\:\\: (\\mu ^{\\text{KKT\\_GEN}}_{l, f, t,p}), \\\\s^{\\text{KKT\\_GEN}}_{f, t,p} &\\ge & 0, \\:\\: \\forall f,t,p,\\:\\: (\\mu ^{\\text{s\\_KKT\\_GEN}}_{l, f, t,p}), \\\\gen^{\\text{PT}}_{f, t,p} \\times s^{\\text{KKT\\_GEN}}_{f, t,p} &=& 0, \\:\\: \\forall f,t,p,\\:\\: (\\mu ^{\\text{GEN\\_s\\_KKT\\_GEN}}_{l, f, t,p}), \\\\IC^{\\text{GEN}}_{t}-\\sum _{p}\\lambda ^{\\text{PT}}_{f, t,p}-s^{\\text{KKT\\_INV}}_{f, t} &=& 0, \\:\\: \\forall f,t,\\:\\: (\\alpha ^{\\text{KKT\\_INV}}_{l, f, t}), \\\\inv^{\\text{PT}}_{f, t} &\\ge & 0, \\:\\: \\forall f,t,\\:\\: (\\mu ^{\\text{KKT\\_INV}}_{l, f, t}), \\\\s^{\\text{KKT\\_INV}}_{f, t} &\\ge & 0, \\:\\: \\forall f,t,\\:\\: (\\mu ^{\\text{KKT\\_s\\_INV}}_{l, f, t}), \\\\inv^{\\text{PT}}_{f, t} \\times s^{\\text{KKT\\_INV}}_{f, t} &=& 0, \\:\\: \\forall f,t,\\:\\: (\\mu ^{\\text{INV\\_s\\_KKT\\_INV}}_{l, f, t}), \\\\-gen^{\\text{PT}}_{f, t,p}+CAP^{\\text{PT}}_{f,t}+inv^{\\text{PT}}_{f,t} - s^{\\text{CON\\_FR}}_{f, t,p} &=& 0\\:\\: \\forall f,t,p, \\:\\: (\\alpha ^{\\text{CON}}_{f, t, p}), \\\\\\lambda ^{\\text{PT}}_{f, t,p} &\\ge & 0, \\:\\: \\forall f,t,p,\\:\\: (\\mu ^{\\text{CON}}_{l, f, t,p}), \\\\s^{\\text{CON\\_FR}}_{f, t,p} &\\ge & 0, \\:\\: \\forall f,t,p,\\:\\: (\\mu ^{\\text{s\\_CON}}_{l, f, t,p}), \\\\\\lambda ^{\\text{PT}}_{f, t,p} \\times s^{\\text{CON\\_FR}}_{f, t,p} &=& 0, \\:\\: \\forall f,t,p,\\:\\: (\\mu ^{\\text{CON\\_s\\_CON}}_{l, f, t,p}).",
"$ The variables in brackets alongside each of these constraints are the Lagrange multipliers associated with those constraints.",
"Note: each of the multipliers has a subscript $l$ associated with it showing how there are unique multipliers for each of the price-making firms problems.",
"Secondly, we find the stationary KKT conditions of the optimisation problem (REF ) - ().",
"Let $\\mathcal {L}_{l}$ be the Lagrangian associated with that problem.",
"$\\begin{split}&\\\\ \\frac{\\partial \\mathcal {L}_{l}}{\\partial gen^{\\text{PM}}_{l,t,p}}:- W_{p} \\times \\bigg (A_{p} -B \\times \\big ( \\sum _{ll,t} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big )-B \\times gen^{\\text{PM}}_{l,t,p}&\\\\-C^{\\text{GEN}}_{t} \\bigg )+\\lambda ^{\\text{PT}}_{f, t,p}+\\sum _{ff,tt}W_{p} \\times B \\times \\alpha ^{\\text{KKT\\_GEN}}_{l, ff, tt,p} - \\chi ^{\\text{GEN}}_{l,t,p}=0, \\:\\: \\forall t,p,\\end{split}$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial inv^{\\text{PM}}_{l,t}}: IC^{\\text{GEN}}_{t}-\\sum _{p}\\lambda ^{\\text{PM}}_{l, t,p}- \\chi ^{\\text{INV}}_{l,t} =0, \\:\\: \\forall t,\\\\$ $\\begin{split}&\\\\\\frac{\\partial \\mathcal {L}_{l}}{\\partial gen^{\\text{PT}}_{f,t,p}}:\\sum _{tt}W_{p} \\times B \\times gen^{\\text{PM}}_{l,tt,p} + \\sum _{ff,tt}W_{p} \\times B \\times \\alpha ^{\\text{KKT\\_GEN}}_{l, ff, tt,p} - \\mu ^{\\text{KKT\\_GEN}}_{l, f, t,p}&\\\\+ s^{\\text{KKT\\_GEN}}_{f, t,p} \\times \\mu ^{\\text{GEN\\_s\\_KKT\\_GEN}}_{l, f, t,p} -\\alpha ^{\\text{CON}}_{f, t, p} =0, \\:\\: \\forall f,t,p,\\\\\\end{split}$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial inv^{\\text{PT}}_{f,t}}:-\\mu ^{\\text{KKT\\_INV}}_{l, f, t}+s^{\\text{KKT\\_INV}}_{f, t} \\times \\mu ^{\\text{INV\\_s\\_KKT\\_INV}}_{l, f, t}-\\sum _{p}\\alpha ^{\\text{CON}}_{f, t, p}=0, \\:\\: \\forall f,t,\\\\$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial \\lambda ^{\\text{PT}}_{f, t,p}}: -\\alpha ^{\\text{KKT\\_GEN}}_{l, f, t,p}+\\alpha ^{\\text{KKT\\_INV}}_{l, f, t}-\\mu ^{\\text{CON}}_{l, f, t,p}+s^{\\text{CON\\_FR}}_{f, t,p} \\times \\mu ^{\\text{CON\\_s\\_CON}}_{l, f, t,p}=0,\\:\\: \\forall f,t,p,\\\\$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial s^{\\text{CON\\_LR}}_{l, t,p}}: \\lambda ^{PM}_{l, t,p}-\\mu ^{\\text{CON\\_LR}}_{l, t,p}=0,\\:\\: \\forall t,p,\\\\$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial s^{\\text{KKT\\_GEN}}_{f, t,p}}: -\\alpha ^{\\text{KKT\\_GEN}}_{l, f, t,p}-\\mu ^{\\text{s\\_KKT\\_GEN}}_{l, f, t,p}+ gen^{\\text{PT}}_{f, t,p} \\times \\mu ^{\\text{GEN\\_s\\_KKT\\_GEN}}_{l, f, t,p}=0,\\:\\: \\forall f,t,p,\\\\$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial s^{\\text{KKT\\_INV}}_{f, t}}: -\\alpha ^{\\text{KKT\\_INV}}_{l, f, t}-\\mu ^{\\text{KKT\\_s\\_INV}}_{l, f, t}+inv^{\\text{PT}}_{f, t} \\times \\mu ^{\\text{INV\\_s\\_KKT\\_INV}}_{l, f, t}=0,\\:\\: \\forall f,t,\\\\$ $\\frac{\\partial \\mathcal {L}_{l}}{\\partial s^{\\text{CON\\_FR}}_{f, t,p}}: -\\alpha ^{\\text{CON}}_{f, t, p}-\\mu ^{\\text{s\\_CON}}_{l, f, t,p}+\\lambda ^{\\text{PT}}_{f, t,p} \\times \\mu ^{\\text{CON\\_s\\_CON}}_{l, f, t,p}=0,\\:\\: \\forall f,t,p.\\\\$ In addition, each of the Lagrange multipliers associated with inequality constraints in (REF ) - () are constrained to be non-negative.",
"Following this, we find the complementary KKT conditions of the optimisation problem (REF ) - () as follows: $gen^{\\text{PM}}_{l,t,p} \\times \\chi ^{\\text{GEN}}_{l,t,p} &=&0, \\:\\: \\forall t,p,\\\\inv^{\\text{PM}}_{l,t} \\times \\chi ^{\\text{INV}}_{l,t} &=&0, \\:\\: \\forall t,\\\\s^{\\text{CON\\_LR}}_{l, t,p} \\times \\mu ^{\\text{CON\\_LR}}_{l, t,p}&=&0, \\:\\: \\forall t,p,\\\\gen^{\\text{PT}}_{f, t,p} \\times \\mu ^{\\text{KKT\\_GEN}}_{l, f, t,p}&=&0, \\:\\: \\forall f,t,p,\\\\s^{\\text{KKT\\_GEN}}_{f, t,p} \\times \\mu ^{\\text{s\\_KKT\\_GEN}}_{l, f, t,p}&=&0, \\:\\: \\forall f,t,p,\\\\inv^{\\text{PT}}_{f, t} \\times \\mu ^{\\text{KKT\\_INV}}_{l, f, t}&=&0, \\:\\: \\forall f,t,\\\\s^{\\text{KKT\\_INV}}_{f, t} \\times \\mu ^{\\text{KKT\\_s\\_INV}}_{l, f, t}&=&0, \\:\\: \\forall f,t,\\\\\\lambda ^{\\text{PT}}_{f, t,p} \\times \\mu ^{\\text{CON}}_{l, f, t,p}&=&0, \\:\\: \\forall f,t,p,\\\\s^{\\text{CON\\_FR}}_{f, t,p} \\times \\mu ^{\\text{s\\_CON}}_{l, f, t,p}&=&0, \\:\\: \\forall f,t,p.$ The Leyffer-Munson method, as applied to this work, is obtain a solution set that satisfies conditions (REF ) - () of each price-making firm $l$ simultaneously.",
"To do this, each KKT condition with bi-linear terms (equations (), (), () and (REF ) - ()) are removed as constraints and are summed together to create the following objective function: $\\begin{split}\\min & \\sum _{f,t,p} gen^{\\text{PT}}_{f, t,p} \\times s^{\\text{KKT\\_GEN}}_{f, t,p} +\\sum _{f,t}inv^{\\text{PT}}_{f, t} \\times s^{\\text{KKT\\_INV}}_{f, t} +\\sum _{f,t,p} \\lambda ^{\\text{PT}}_{f, t,p} \\times s^{\\text{CON\\_FR}}_{f, t,p} +\\sum _{l,t,p} gen^{\\text{PM}}_{l,t,p} \\times \\chi ^{\\text{GEN}}_{l,t,p}+\\\\& \\sum _{l,t} inv^{\\text{PM}}_{l,t} \\times \\chi ^{\\text{INV}}_{l,t} +\\sum _{l,t,p} s^{\\text{CON\\_LR}}_{l, t,p} \\times \\mu ^{\\text{CON\\_LR}}_{l, t,p}+\\sum _{f,t,p} gen^{\\text{PT}}_{f, t,p} \\times \\mu ^{\\text{KKT\\_GEN}}_{l, f, t,p}+\\\\& \\sum _{f,t,p} s^{\\text{KKT\\_GEN}}_{f, t,p} \\times \\mu ^{\\text{s\\_KKT\\_GEN}}_{l, f, t,p}+\\sum _{f,t} inv^{\\text{PT}}_{f, t} \\times \\mu ^{\\text{KKT\\_INV}}_{l, f, t}+\\sum _{f,t} s^{\\text{KKT\\_INV}}_{f, t} \\times \\mu ^{\\text{KKT\\_s\\_INV}}_{l, f, t}+\\\\& \\sum _{f,t,p} \\lambda ^{\\text{PT}}_{f, t,p} \\times \\mu ^{\\text{CON}}_{l, f, t,p}+\\sum _{f,t,p} s^{\\text{CON\\_FR}}_{f, t,p} \\times \\mu ^{\\text{s\\_CON}}_{l, f, t,p}.\\end{split}$ Thus, the Leyffer-Munson optimisation problem is to minimise equation (REF ) subject to constraints (REF ) - (), () - (), () - () and (REF ) - (REF ).",
"In addition, each of the Lagrange multipliers associated with inequality constraints in (REF ) - () are constrained to be non-negative.",
"The Leyffer-Munson optimization problem is a Non-Linear Program (NLP) and, in Section , we use the CONOPT solver in GAMS to solve it."
],
[
"Overall algorithm",
"Algorithm REF describes the overall algorithm for finding Nash equilibria from the EPEC problem.",
"For iteration $i$ , we firstly provide a random solution set from the search space and use these as initial starting point solutions for the Leyffer-Munson approach.",
"As the Leyffer-Munson approach is a non-linear optimisation problem, the CONOPT solver does not always find a local minimum.",
"If the Leyffer-Munson method does not converge to a locally optimal solution, then iteration $i$ is deemed unsuccessful and the algorithm skips ahead to iteration $i+1$ .",
"If the Leyffer-Munson approach does converge however, the locally optimal solution is then used as starting point solution for the Gauss-Seidel algorithm.",
"If the Gauss-Seidel does (not) converge to a Nash equilibrium solution, then iteration $i$ is (not) deemed successful.",
"This process is repeated for $I$ iterations.",
"[H] $i=1,..,I$ Provide random initial solutions Solve Leyffer-Munson optimisation problem Solution from LM is locally optimal Solve EPEC using Gauss-Seidel algorithm using solutions from LM as starting point Gauss-Seidel algorithm converges Save solution Overall algorithm for finding Nash Equilibria"
],
[
"Results from EPEC model",
"In this section, we present the results when the data presented in Section is applied to the EPEC model described in Section .",
"We focus on the firms' profits, firms' investment decisions, market prices, consumer costs and carbon emissions.",
"To obtain these results we utilise the algorithm described in Section REF for $I=2000$ iterations.",
"For the first 1000 iterations firm $l=1$ 's MPEC problem is solved before firm $l=2$ 's MPEC problem.",
"For the subsequent 1000 iterations the opposite applies and firm $l=2$ 's MPEC problem is solved before firm $l=1$ 's MPEC problem.",
"The algorithm did not always find a Nash Equilibrium (NE) solution.",
"In fact, in the results to follow, only 72 of the 2000 iterations successfully found a NE solution, henceforth know as successful iterations.",
"Of these, 62 iterations occurred when firm $l=1$ 's MPEC problem was solved before firm $l=2$ 's MPEC problem while the remaining 10 successful iterations occurred when firm $l=2$ 's MPEC problem was solved first.",
"For unsuccessful iterations the algorithm failed to find a NE solution for one of two reasons: For the random initial solution provided, the Leyffer Munson was found to be locally infeasible by the CONOPT solver.",
"For the Gauss-Seidel algorithm, the convergence tolerance remained greater than $TOL=10^{-3}$ after $K=100$ iterations.",
"For each of the 72 successful iterations, the Leyffer-Munson approach solved to a locally optimal solution which implied that it is not necessarily a feasible solution to the EPEC.",
"For 43% of these successful iterations, the objective function for the Leyffer-Munson approach (equation (REF )) converged to zero.",
"For the remaining 57% of successful iterations, the objective function converged to a strictly positive objective function value.",
"When the Leyffer-Munson approach gives a non-zero objective function value, the solutions cannot guaranteed to be a feasible point for the overall EPEC.",
"However, the results in this section show that, despite this, such solutions can still provide good starting point solutions to the Gauss-Seidel algorithm.",
"Figure: Profits for price-making firm l=1l=1 for each successful iteration.Figure: Profits for price-making firm l=2l=2 for each successful iteration.Figure: Combined profits for price-making firms for each successful iteration.Figures REF and REF display price-making firms $l=1$ and $l=2$ profits, respectively, for each of the successful iterations.",
"The horizontal lines in each figure represent the profits each firm would make from the perfect competition case in the Section .",
"Figures REF and REF both show that the algorithm found multiple NE solutions.",
"Firm $l=1$ 's profits varies from 0 to 12.98M while firm $l=2$ 's profits ranged from 50,000 to 6.8M.",
"For the majority of equilibria found, both firms made profits greater than they would in a perfect framework.",
"This occurred in 66.7% and 100% of the successful iterations for firms $l=1$ and $l=2$ profits, respectively.",
"Thus showing that there was no NE found where both firms' profits were below their perfect competition equivalent.",
"Figure REF display the combined profits of the two price-making firms.",
"It shows that the combined profits varied across the equilibria suggesting that there was not a zero-sum game between the price-making firms on how profits were split between them.",
"We describe below why there are multiple equilibria and why, in some equilibria, one of the price-making firms makes a profit less than it would in a perfectly competitive market.",
"Figure: Combined investment in new mid-merit for price-making firms for each successful iteration.Figure REF displays the combined investments into new mid-merit generation for the two price-making firms for each successful iteration.",
"In comparison to Section , both of these firms did not invest in any baseload or peaking generation.",
"However, in contrast to Section , there was only one equilibrium point found where the price-taking firms invested in any new generation technology.",
"For the majority of equilibria found (80%), the combined investments were 2941MW.",
"But, for some equilibria, the combined investments were slightly higher with maximum combined investment reaching 2967MW at one equilibrium point while the lowest combined investment was 2831MW.",
"The first 62 successful iterations in Figures REF - REF came when firm $l=1$ 's MPEC problem was solved before firm $l=2$ 's.",
"At these equilibria, firm $l=1$ and $l=2$ 's investments in new mid-merit generation averaged at 2807MW and 496MW, respectively.",
"The final 10 successful iterations occurred when firm $l=2$ 's MPEC problem was solved first.",
"As a result, at these equilibria, firm $l=1$ average investments in new generation decreased to 1467MW while firm $l=2$ 's increased to 2618MW.",
"The results show that firm $l=1$ makes a profit less than it would of in a perfectly competitive market, in some of the equilibria found.",
"This is because we model two price-making firms.",
"When price-making firm $l$ commits to a large amount of forward generation, it can leave firm $\\hat{l} \\ne l$ with a reduced opportunity to generate and hence reduced profits.",
"We explain this in further detail in Section REF .",
"Interestingly, in each of the successful iterations where firm $l=2$ MPEC is solved first, firm $l=1$ 's profits are significantly below their perfect competition result while firm $l=2$ are significantly higher - this result highlights the importance of the order in which MPEC problems of an EPEC are searching, when searching for equilibria.",
"Figure: Equilibrium forward market prices (γ p \\gamma _{p}) for EPEC model versus perfect competition.Despite Figures REF - REF presenting the multiple equilibria, the forward prices rested at one of three price time series.",
"Time series one and two were observed in 81.9% and 16.7% of the equilbira found while the third series was only observed at one of the equilibrium points found.",
"Figure REF displays these three price time series along with the prices from the perfect competition case of Section .",
"Interestingly, for time periods $p=2,3$ , the forward market prices in the oligopoly with competitive fringe case are less than those from the perfect competition case.",
"This is despite half the firms having price-making ability.",
"However, the market prices in the oligopoly with competitive fringe case are higher at later time steps.",
"Note: both the first and second equilibrium price time series were found when both firm $l=1$ and firm $l=2$ MPEC's were solved first while the only instance of the third equilibrium price time series occurred when firm $l=2$ 's MPEC was solved first.",
"Figure: Generation mix for the first successful iteration.Figure: Revenue earned by firm l=1l=1 for the first successful iteration.Figure: Revenue earned by firm l=2l=2 for the first successful iteration.The forward prices in Figure REF can be explained by Figures REF – REF .",
"Figure REF shows the generation mix for the first successful iteration while Figures REF and REF display the revenues earned/lost by firms $l=1$ and $l=2$ , respectively, in each time period for the same iteration.",
"At this equilibrium point, forward prices converged to first time series in Figure REF and firm $l=1$ and firm $l=2$ invested 2765MW and 176MW into new mid-merit generation, respectively.",
"In time periods $p=1$ and $p=2$ , only firm $l=1$ and $l=2$ 's new mid-merit units are generating leading to forward prices of $\\gamma _{p=1}=\\gamma _{p=2}=34$ , the marginal cost of new mid-merit.",
"Consequently, neither price-making firm earns, nor loses, revenue at these time periods.",
"The forward price is the same for $p=3$ but, because the demand curve intercept is higher (see Table REF ), more generation is needed to meet demand.",
"The increased demand is primarily met by firm $l=2$ 's new mid-merit unit.",
"In addition however, firm $l=1$ 's existing mid-merit unit generates 403MW.",
"This is despite existing mid-merit having a marginal cost of 41.1.",
"Thus, as Figure REF outlines, firm $l=1$ losses revenue at this timepoint.",
"Firm $l=2$ does not earn revenue, nor does it lose revenue, at $p=3$ .",
"In time period $p=4$ , the forward price is 41.1 which is the marginal cost of an existing mid-merit unit.",
"Consequently, all mid-merit units, for firm $l=1$ , $l=2$ and $f=3$ are utilised.",
"In addition, firm $l=1$ also utilises is existing baseload despite the marginal cost of exiting baseload being 48.87.",
"The forward price is $\\gamma _{p=5}=65.19$ at timestep $p=5$ .",
"Because this price is higher than the marginal cost of exiting baseload, both price-making firms utilise their existing baseload units and make a profit from doing so.",
"The two price-making firms use their generation to set $\\gamma _{p=5}=65.19$ and hence maximise their respective profits.",
"This forward price allows the two price-making firms to partially recover the investments cost associated with investing in new mid-merit generation.",
"Because they both do not earn any revenue from new mid-merit in timesteps $p=1,2,3$ , the remaining investment costs are recovered in timestep $p=4$ where the market price of $\\gamma _{p=4}=41.1$ allows both firm $l=1$ and $l=2$ to earn enough revenue from their new mid-merit units to break even on their investments.",
"If either price-making firm adjusted their generation so as to set a price higher than 41.1 in $p=4$ or higher than 65.19 in $p=5$ , then the two price-taking firms would invest in new mid-merit generation also, as it would be a profitable decision.",
"The price-making firms prevent this because investment from the price-taking fringe would erode the substantial revenues they earn in timestep $p=5$ .",
"Similarly, it is profit maximising for firm $l=1$ to generate using its existing mid-merit unit, at below marginal cost, in time period $p=3$ .",
"If firm $l=1$ did not do this, the remaining demand would be met by firm $f=3$ 's existing mid-merit unit, which would drive up the market price and thus, make investing in a new mid-merit a profitable option for both price-taking firms.",
"Again, such a market outcome, is not profit-maximising for firm $l=1$ .",
"Instead, it is optimal for firm $l=1$ to take take the small losses in time period $p=3$ so as to prevent the fringe from eroding its large profits in timestep $p=5$ .",
"As Figure REF shows, firm $l=1$ 's revenues from $p=4$ and $p=5$ far exceed its losses from $p=3$ .",
"Additionally, in time periods $p=1,2$ , it is optimal for firm $l=1$ to ensure the market price is $\\gamma _{p=1}=34$ .",
"However, in these timesteps, firm $l=1$ does not need its existing mid-merit unit to maintain the price at this level.",
"These results are in contrast to the prices observed in the perfect competition case; see Figure REF .",
"In the perfect competition setting, firms only utilise a generating unit if the market price is at or above the marginal cost of that unit.",
"Consequently, the market price is set by the marginal cost of the most expensive unit that is generating.",
"Hence, there is no below marginal cost operation of units in time periods $p=3$ and $p=4$ , which leads to higher forward prices compared with the oligopoly with a competitive fringe case.",
"Similarly, in time period $p=5$ , the forward price in the perfect competition case is set by the most expensive unit that is generating; existing baselaod.",
"In contrast, in the oligopoly with a competitive fringe case, it is optimal for the price-making firms to adjust their generation to ensure the forward price is higher than the perfect competition case.",
"Similar qualitative results to those in Figures REF and REF can be seen in the rest of the successful iterations.",
"The exact level of revenue earned or lost in each timestep, for both price-making firms, varies in a similar manner to Figures REF - REF .",
"and equation (REF ) shows that when using a MCP model, it is never optimal for a generator to operate one of its units at below marginal cost.",
"In contrast, when the EPEC approach of this work is utilised, equation (REF ) shows that when $gen^{\\text{PM}}_{l,t,p}>0$ , $\\gamma _{p}$ can be less than $C^{\\text{GEN}}_{t}$ .",
"This is because of the additional $\\alpha ^{\\text{KKT\\_GEN}}_{l, ff, tt,p}$ that is in equation (REF ) but not in equation (REF ).",
"Moreover, this further highlights the benefit of the EPEC approach and the limitations of the MCP approach when modelling an oligopoly with a competitive fringe and investment decisions.",
"The generation levels of price-taking firms $f=3$ and $f=4$ were similar for all equilibria that converged to the first two equilibrium price series.",
"Both price-taking firms utilised their existing mid-merit and peaking units, respectively, to maximum capacity in time period $p=5$ only as this was the only time period where the price was high enough for them to make earn profits.",
"As the equilibrium forward prices converged to one of only three series, the price-taking firms' profits similarly converged to one of three levels.",
"For equilibria that converged on the first price time series in Figure REF , the profits were 17.1M and 0.7M for firms $f=3$ and $f=4$ , respectively, while for equilibria with the second time series, the profits were 22.14M and 3.6M, respectively.",
"At the equilibrium point where the third price time series was observed firm $f=3$ also utilised its existing at time period $p=4$ in addition to $p=5$ .",
"At this equilibrium point, firm $f=4$ did not generate any electricity as the price was never high enough for them to so.",
"Consequently, firm $f=4$ made zero profits while firm $f=3$ made a profit of 17.1M.",
"Figure: Consumer costs as % of perfect competition case.Figure REF displays the consumer costs, as defined by the following equation: $\\sum _{p} \\bigg ( W_{p}\\times \\gamma _{p} \\times \\big ( \\sum _{ll,tt} gen^{\\text{PM}}_{ll,tt,p}+\\sum _{ff,tt} gen^{\\text{PT}}_{ff,tt,p} \\big )\\bigg ).$ As above, because the equilibrium prices landed at one of three price series, consumer costs also converged to one of three levels.",
"This is because the amount of energy consumed has a fixed relationship with market prices; see market clearing condition (REF ).",
"Figure REF shows how the consumer costs increase by 1.7%, 2.1%, 0.9% for equilibria that converged to first, second and third time series of forward prices, respectively.",
"In previous works that use similar data, the presence of price-making behaviour was found to lead to a larger increases in consumer costs [5].",
"However, the ability of the price-taking fringe to invest in new generation motivates the price-making firms to reduce forward market prices in some time periods.",
"While the market prices increase again in subsequent time periods, these consumer cost results shows how the presence of a competitive fringe helps mitigate against the negative effects of market power.",
"Figure: C0 2 _2 emission levelsFigures REF and REF display the carbon dioxide emissions level for equilibria that converged at the first and second set of price time series, respectively.",
"These emissions were calculated as follows: $\\sum _{p,t} \\bigg ( W_{p}\\times E_{t} \\times \\big ( \\sum _{ll} gen^{\\text{PM}}_{ll,t,p}+\\sum _{ff} gen^{\\text{PT}}_{ff,p} \\big )\\bigg ),$ where the parameter $E_{t}$ gives the emissions factor level for technology $t$ , as displayed in Table REF .",
"Figure REF show that, despite equilibrium prices remaining constant across subsets of the equilibria found, the emissions levels varied across the equilibira.",
"This is particularly evident in Figure REF for equilibria with the second price time series.",
"The reasons behind this results will now be explained."
],
[
"Reasons for multiple equilibria",
"Figures REF - REF show that there are multiple equilibria to the EPEC presented in this work.",
"In this subsection, we explore the reasons behind this finding.",
"Firstly, we look at two equilibria where the forward prices were the same, i.e., the first equilibrium time series from Figure REF .",
"The multiple equilibria are driven by the market's indifference to what firm is providing electricity when firms are generating at the same price.",
"For example, Figures REF and REF display the generation mixes for the first two successful iterations, respectively.",
"In the first successful iteration firms $l=1$ and $l=2$ invest 2765MW and 176MW into new mid-merit generation, respectively.",
"In contrast, in the second successful iteration, they invest 2941MW and 0MW into new mid-merit generation, respectively.",
"At time period $p=5$ in the first iteration, firm $l=1$ uses its new and existing mid-merit units to maximum capacity while also generating 21MW from its baseload unit.",
"As the same iteration, firm $l=2$ uses its new mid-merit unit to full capacity and also generates 2MW from its baseload unit.",
"In the second successful iteration, firm $l=1$ decreases its baseload generation at $p=5$ from 21MW to 4MW but increases its generation from new mid-merit from 2765MW to 2941MW.",
"This allows firm $l=1$ to make less profits in Figure REF ; firms break even on their new mid-merit investments but make profits from existing baseload generation.",
"Firm $l=2$ increases its baselaod generation from 2MW to 19MW but decreases generation from new mid-merit from 176MW to 0MW.",
"This allows firm $l=1$ to make more profits in Figure REF .",
"Because the market prices are the same across both equilibria considered, the market is indifferent to whether the electricity comes firm $l=1$ 's baseload or mid-merit or from firm $l=2$ 's baseload or mid-merit units.",
"Once firm $l$ commits to forward generation decisions, firm $\\hat{l} \\ne l$ is not willing to adjust its generation levels so as to either increase or decrease forward market price of $\\gamma _{p=5}=65.19$ .",
"If either price-making firm increased any of the forward prices, then the price-taking firms would invest in new mid-merit generation, as explained in the previous subsection.",
"It is also not profit-maximising for firm $l$ to undercut firm $\\hat{l}$ $\\ne $ $l$ at a price lower than 65.19.",
"To do so, would mean firm $l$ would make a loss on its new mid-merit investment.",
"Furthermore, if firm $l$ adjusted its generation so as to decrease $\\gamma _{p=5}$ by 1, then it would only be able to, at most, increase its generation from existing baseload $\\frac{1}{B}=0.11$ MW (see market clearing condition (REF )).",
"This is because it would continue to be profitable for firm $\\hat{l}$ $\\ne $ $l$ to utilise its existing baseload at the reduced price.",
"The small increase in generation opportunity would not make up for the decreased revenues resulting from the reduced price.",
"This paragraph explains why in some of the equilbria found, firm $l=1$ makes less profits than in the perfect competition case.",
"Similar market in-differences are also observed in time periods $p=2$ -4 and in the other 57 successful iterations that converge to the same price time series, thus explaining the multiple equilibria displayed in Figures REF - REF .",
"In some of other equilibria found, both price-making firms generate significant amounts from their baselaod units in $p=5$ , thus preventing each other from generating and investing in new mid-merit generation.",
"Consequently, both firms do not make as large a profit as they otherwise could.",
"Such equilibria are also evident in Figures REF - REF .",
"This particular result highlights the absence of collusion between the two price-making firms modelled in this work.",
"Figure: Generation mix for the first successful iteration that results in the second time series for forward prices.Figure: Revenue earned by firm l=1l=1 for the first successful iteration that results in the second time series for forward prices.Figure: Revenue earned by firm l=2l=2 for the first successful iteration that results in the second time series for forward prices.We now examine the differences between two equilibria that converged to different forward price time series.",
"Figure REF displays the generation mix for the first successful iteration where the forward prices converged to the second time series in Figure REF while Figures REF and REF show the revenues for firms $l=1$ and $l=2$ , respectively, for the same iteration.",
"Firms $l=1$ and $l=2$ made profits of 4.92M and 3.21M, respectively, at the equilibrium point.",
"In Figure REF , firm $l=2$ generated 17MW from its existing baseload unit at time period $p=4$ .",
"In contrast, in Figure REF , firm $l=2$ increased its baseload generation to 93MW at time period $p=4$ .",
"Following from market clearing condition (REF ), this lead to the forward price decreasing from $\\gamma _{p=4}=41.1$ to $\\gamma _{p=4}=31$ between the two time series.",
"This decrease in forward price meant that firm $l=1$ needed to decrease its overall generation in $p=5$ from 3456MW in Figure REF to 3337MW in Figure REF .",
"This resulted in a higher forward price in $p=5$ for in the second time series and thus allowed both price-making firms to recover its investment capital costs, despite the decreased price in $p=4$ .",
"Firm $l=2$ cannot make a profit from its existing baseload unit in time period $p=4$ at either $\\gamma _{p=4}=41.1$ or $\\gamma _{p=4}=31$ .",
"Consequently, firm $l=2$ prefers a higher forward price in $p=5$ as this allows it to maximise its profits on its existing baselaod unit; see Figure REF compared with Figure REF .",
"In contrast, firm $l=1$ prefers the first forward price time series, i.e., a higher price in $p=4$ and a slightly lower price in $p=5$ .",
"In time period $p=4$ , firm $l=1$ can earn positive revenues from its new mid-merit unit and not make a loss from its existing mid-merit unit in $p=4$ , if the forward price is 41.1.",
"In contrast however, firm $l=2$ does not own an exiting mid-merit unit and, in Figures REF – REF , only invests in 1MW of new mid-merit generation.",
"Consequently, firm $l=2$ prefers a lower forward price in $p=4$ and a higher price in $p=5$ as this allows firm $l=2$ to maximise its profits from its existing baselaod unit.",
"For both forward price time series, the forward price is not high enough for existing baseload units to earn positive revenues in $p=4$ .",
"In general, the equilibria resulting from the second time series represent equilibria where firm $l=2$ has invested in a relatively small amount of new mid-merit generation, if any at all, but where firm $l=2$ has also made forward generation decisions before firm $l=1$ .",
"When firm $l=2$ commits to a large amount of generation in $p=4$ , firm $l=1$ must reduce its generation in $p=5$ in order to allow the forward price increase and hence break even on its and firm $l=2$ 's new mid-merit investments.",
"Interestingly, there is one equilibrium point where firm $l=1$ does not invest in any new technology and consequently, commits to a large amount of generation in $p=4$ .",
"This leads to the second equilibrium price time series from Figure REF .",
"This also forces firm $l=2$ to reduce its generation in $p=5$ and motivates it to not make any investment decisions either.",
"As a result, this is the only equilibrium point where the followers make investment decisions; firm $f=3$ invests 2766MW into a new mid-merit facility while firm $f=4$ invest 86MW into the same technology.",
"Because of the generation commitments of the price-making firms set the equilibrium prices, both price-taking firms break exactly even on these investments.",
"Thus, in the model, the price-making firms are indifferent to whether they do the investment at this equilibrium point or the price-taking firms do.",
"However, this indifference may not reflect reality.",
"In the real-world, price-making firms may fear losing their price-making ability if they allow the competitive fringe to invest.",
"The EPEC model presented in this work does not account for this as the price-making/price-taking characteristics of all firms remain unchanged throughout the model.",
"Finally, as mentioned above, there was one equilibrium point found where the prices converged to third equilibrium time series in Figure REF .",
"In comparison with the second equilibrium time series, firm $l=2$ commits to investing in new mid-merit generation before $l=1$ .",
"However, at this equilibrium point, firm $l=1$ also commits to a large amount of generation in $p=5$ which leads to a reduced price of $\\gamma _{p=5}=47.79$ , from which its existing mid-merit unit profits from.",
"Consequently, in order for firm $l=2$ to break even on its new mid-merit investment, firm $l=2$ is forced to ensure its generation in $p=4$ is low enough to allow $\\gamma _{p=4}=58.59$ ."
],
[
"Discussion",
"The following summarises the five main findings of our research.",
"Firstly, an Equilibrium Problem with Equilibrium Constraints (EPEC) is a prudent model choice when modelling an oligopoly with competitive fringe and investments.",
"As outlined in Section , when investment decisions are included in the model, using a Mixed Complementarity Problem (MCP) can lead to myopic model behaviour and thus contradictory results.",
"Our analysis shows that an EPEC model can overcome this issue and moreover does not require the limiting assumption of conjectural variations.",
"Secondly, our results show that may it be optimal for generating firms with market power to occasionally operate some of their generating units at a loss.",
"The driving factor behind this model outcome is the fact we allow both price-making and price-taking firms to make investment decisions.",
"The ability of price-taking firms to invest further into the market motivates the price-making firms to depress prices in some timepoints.",
"This reduces the revenues price-taking firms could make from new investments and thus prevents them from making such investments.",
"Such behaviour would not be captured by MCP or cost-minimisation unit commitment models.",
"Consequently, this result again highlights the suitability of the EPEC modelling approach and the importance of including investment decisions in models of oligopolies with competitive fringes.",
"Thirdly, the analysis in Section found multiple market equilibria.",
"This led to varied investment decisions, and thus profits, for the price-making firms.",
"These results will be of interest to generating firms, particularly those with market power.",
"Figures REF - REF highlight the benefit of making investment decisions before other competing price-making firms do so.",
"In fact, our results indicate that if firms do not expand their generation portfolios, then they may face profits lower than they would if the market was perfectly competitive.",
"The multiple equilbira result also indicates that that generation from existing baseload generation may be higher in some equilibria compared with others.",
"Such market outcomes will be of interest to energy policymakers who are concerned about carbon emission levels.",
"Older baseload generators tend to be coal-based and thus emit higher levels of carbon.",
"Consequently, while the market may be indifferent to where the electricity comes from, policymakers may seek to put measures in places to encourage the equilibrium outcomes where existing baseload generation is reduced.",
"Fourthly, Figure REF showed that the presence of market power increases consumer costs by 1% – 2%.",
"While this outcome is not surprising, the level is relatively small compared with the literature.",
"For instance, using similar data, [6] estimate market power in an oligopoly with competitive fringe context could double consumer costs compared with a perfectly competitive market.",
"However, [6] do not include investment decisions in their model.",
"Thus, this result again highlights the impact of including investment decisions in models of oligopolies with competitive fringes.",
"It also highlights the importance for policymakers to encourage new entrants into electricity markets and, moreover, the benefits of encouraging smaller generating firms to expand their portfolios, or at least threaten to.",
"Finally, as the literature details [36], solving EPEC problems can be computational challenging.",
"In this work we utilised the method outlined in [29] to obtain an initial starting point solution to our algorithm.",
"Using this approach our algorithm successfully found an equilibrium from 72 of the 200 iterations attempted.",
"When instead we used a random initial starting point solution, we found an equilibrium from only 2 of the 2000 iterations attempted.",
"Critically reflecting on our approach, we wish to acknowledge some limitations.",
"Firstly, because EPEC problems are challenging to solve, we choose the relatively small number of five timesteps.",
"These represented hours in summer low demand, summer high demand, winter low demand, winter high demand and winter peak demand.",
"Thus, the net demand intercept values represent average values for these timesteps.",
"In reality, particularly in systems with a large amount of renewables, these intercept values will fluctuate from hour to hour.",
"As a result, the average values may over- or under-estimate the total profits each generating firm could make in each time period.",
"This would impact investments decisions and consumer costs.",
"Secondly, we did not account for any stochasticity in the model.",
"Due to the intermittent and uncertain nature of wind energy, stochasticity is a feature of many electricity market models.",
"Such stochasticity is typically introduced by making generation capacity scenario-dependent [30].",
"Deterministic capacity values may also over- or under-estimate the profits each firm may make in each timestep.",
"However, while including further timesteps and stochastic capacity values would most likely affect the exact numbers presented in Section , we do not anticipate them changing the qualitative findings discussed above.",
"Finally, we did not consider a capacity market as part of the market modelled in this work.",
"Capacity payments exists when firms get paid for simply owning generation units and making them available to the grid.",
"Capacity payments do not depend on the extent that the unit(s) are utilised.",
"Regulators and policymakers include such payments so as to ensure security of supply [31].",
"The market we considered was an 'energy-only' market, where the generating firms only get paid on the basis if how much they generate.",
"A capacity market can affect the level of investment into new generation.",
"Future research activities will address each of these modelling limitations."
],
[
"Conclusion",
"In this paper, we developed a novel mathematical model of an imperfect electricity market, one that is characterised by an oligopoly with a competitive fringe.",
"We modelled two types of generating firms; price-making firms, who have market power, and price-taking firms who do not.",
"All firms had both investment and forward generation decisions.",
"The model took the form of an Equilibrium Problem with Equilibrium Constraints (EPEC), which finds an equilibrium of multiple bi-level optimisation problems.",
"The bi-level formulation allowed the optimisation problems of the price-taking firms to be embedded into the optimisation problems of the price-making firms.",
"This enabled the price-making firms to correctly anticipate the optimal reactions of the price-taking firms to their decisions.",
"We applied the model to data representative of the Irish power system for 2025.",
"To solve the EPEC problem, we utilised the Gauss-Seidel algorithm.",
"Furthermore, we found the computational efficiency of the algorithm was improved when the algorithm's starting point solution was provided by the approach detailed in [29].",
"Overall, we found that an EPEC problem is a prudent model choice when modelling an oligopoly with competitive fringe.",
"This is because it overcomes modelling issues previously found in the literature and requires a fewer limiting assumptions.",
"The model found multiple equilibria.",
"This was due to the market's indifference to which price-making firm generates electricity.",
"Although consumer costs were found to be relatively constant across the equilibria found, this result is important to energy policymakers who who wish to avoid equilibrium outcomes that higher carbon emission levels.",
"We also observed that it may be strategically optimal for price-making firms to occasionally to generate at a price that is lower than their marginal cost.",
"This is because we incorporated investment decisions into the optimisation problems of both types of generating firms.",
"Consequently, the price-making firms seek to depress prices occasionally so as to discourage the fringe from investing further into the market.",
"Furthermore, we found that consumer costs only decreased by 1% – 2% when market power was removed from the model.",
"In future research, we will study the effects of increasing the number of timesteps in the model.",
"Moreover, we will explore the impact stochasticity, particularly from wind generation, would have.",
"In addition, future research will analyse how the introduction of a capacity market would affect equilibrium outcomes."
],
[
"Acknowledgements",
"M. T. Devine acknowledges funding from Science Foundation Ireland (SFI) under the SFI Strategic Partnership Programme Grant number SFI/15/SPP/E3125.",
"S. Siddiqui acknowledges funding by NSF Grant #1745375 [EAGER: SSDIM: Generating Synthetic Data on Interdependent Food, Energy, and Transportation Networks via Stochastic, Bi-level Optimization].",
"The authors also sincerely thank Dr. M. Lynch and Dr. S. Lyons from the Economic and Social Research Institute (ESRI) in Dublin and who provided invaluable feedback and advice."
],
[
"Alternative Price-making firm $l$ 's problem",
"When the problem is solved as a Mixed Complementarity Problem (MCP), price-making firm $l$ 's optimisation problem takes the following form, where all variables an parameters are as defined previously: $\\begin{split}\\max _{\\begin{array}{c}gen^{\\text{PM}}_{l,t,p}, inv^{\\text{PM}}_{l,t}\\\\gen^{\\text{PT}}_{f,t,p}, inv^{\\text{PT}}_{f,t}\\\\\\gamma _{p}, \\lambda ^{\\text{PT}}_{f, t,p}\\end{array}}\\:\\:&\\sum _{t,p} W_{p}\\times gen^{\\text{PM}}_{l, t,p} \\times \\big ( \\gamma _{p} - C^{\\text{GEN}}_{t}\\big )-\\sum _{t} IC^{\\text{GEN}}_{t}\\times inv^{\\text{PM}}_{l,t}.\\end{split}$ subject to: $gen^{\\text{PM}}_{l, t,p} &\\le & CAP^{\\text{PM}}_{l,t}+inv^{\\text{PM}}_{l,t}, \\:\\: \\forall t,p.",
"$ The KKT conditions associated with this optimisation problem are $0\\le gen^{\\text{PM}}_{l, t,p} &\\perp & -W_{p}\\times \\big (\\gamma _{p} + \\frac{\\partial \\gamma _{p}}{\\partial gen^{\\text{PM}}_{l, t,p}} \\times gen^{\\text{PM}}_{l, t,p}-C^{\\text{GEN}}_{t}\\big )+\\lambda ^{\\text{PM}}_{l, t,p} \\ge 0, \\:\\: \\forall t,p,\\\\0 \\le inv^{\\text{PM}}_{l, t} &\\perp & IC^{\\text{GEN}}_{t}-\\sum _{p}\\lambda ^{\\text{PM}}_{l, t,p} \\ge 0, \\:\\: \\forall t,\\\\0 \\le \\lambda ^{\\text{PM},1}_{l,t,p} &\\perp & -gen^{\\text{PM}}_{l, t,p}+CAP^{\\text{PM}}_{l,t}+inv^{\\text{PM}}_{l,t} \\ge 0, \\:\\: \\forall t,p,$ where $\\frac{\\partial \\gamma _{p}}{\\partial gen^{\\text{PM}}_{l, t,p}}=-CV_{l} \\times B ,\\:\\: \\forall l,t,p,$ is determined via market clearing condition (REF ).",
"Furthermore, the parameter $CV_{l} \\in [0,1]$ represents the Conjectural Variation associated with firm $l$ .",
"When firm $l$ 's problem is described by equations (REF ) and (REF ), it is a convex optimisation problem and hence the KKT conditions (REF ) - () are both necessary and sufficient for optimality [16].",
"When the overall market problem is solved as a MCP, the problem consists of the market clearing condition (REF ), the price-taking firms' KKT conditions (equations (REF ) - ()) and the KKT conditions for all price-making firms (equations (REF ) - ()).",
"It is important to note that when $gen^{\\text{PM}}_{l, t,p}>0$ , then condition (REF ) is only satisfied if $\\gamma _{p} \\ge C^{\\text{GEN}}_{t}$ .",
"Thus, when the above MCP is used in Section , not one generating unit will operate at below marginal cost.",
"This is in contrast to the EPEC analysis in Section and highlights a further limitation of the MCP modelling approach."
]
] | 2001.03526 |
[
[
"microbatchGAN: Stimulating Diversity with Multi-Adversarial\n Discrimination"
],
[
"Abstract We propose to tackle the mode collapse problem in generative adversarial networks (GANs) by using multiple discriminators and assigning a different portion of each minibatch, called microbatch, to each discriminator.",
"We gradually change each discriminator's task from distinguishing between real and fake samples to discriminating samples coming from inside or outside its assigned microbatch by using a diversity parameter $\\alpha$.",
"The generator is then forced to promote variety in each minibatch to make the microbatch discrimination harder to achieve by each discriminator.",
"Thus, all models in our framework benefit from having variety in the generated set to reduce their respective losses.",
"We show evidence that our solution promotes sample diversity since early training stages on multiple datasets."
],
[
"Introduction",
"Generative adversarial networks [12], or GANs, consist of a framework describing the interaction between two different models - one generator (G) and one discriminator (D) - that are trained together.",
"While $G$ tries to learn the real data distribution by generating realistic looking samples that are able to fool $D$ , $D$ tries to do a better job at distinguishing between real and the fake samples produced by $G$ .",
"Although showing very promising results across various domains [11], [14], [41], [40], [8] , GANs have also been continually associated with instability in training, more specifically mode collapse [16], [1], [25], [5], [2].",
"This behavior is observed when $G$ is able to fool $D$ by only generating samples from the same data mode, leading to very similar looking generated samples.",
"This suggests that $G$ did not succeed in learning the full data distribution but, instead, only a small part of it.",
"This is the main problem we are trying to solve with this work.",
"The proposed solution is to use multiple discriminators and assign each $D$ a different portion of the real and fake minibatches, i.e., microbatch.",
"Then, we update each $D$ 's task to discriminate between samples coming from its assigned fake microbatch and samples from the microbatches assigned to the other discriminators, together with the real samples.",
"We call this microbatch discrimination.",
"Throughout training, we gradually change from the originally proposed real and fake discrimination by [12] to the introduced microbatch discrimination by the use of an additional diversity parameter $\\alpha $ that ultimately controls the diversity in the overall minibatch.",
"The main idea of this work is to force $G$ to reduce its loss by inducing variety in the generated set, complicating each $D$ 's task on separating the samples in its microbatch from the rest.",
"Even though only producing very similar images would also complicate the desired discrimination, it would not benefit any of the models.",
"This is due to the attribution of distinct probabilities by each $D$ to samples from and outside its microbatch being required to minimize $G$ and $D$ 's losses.",
"Hence, all models in the proposed framework, called microbatchGAN, benefit directly from diversity in the generated set.",
"Our main contributions can be stated as follows: [label=()] proposal of a novel multi-adversarial GANs framework (Section ) that mitigates the inherent mode collapse problem in GANs; empirical evidence on multiple datasets showing the success of our approach in promoting sample variety since early stages of training (Section ) Competitiveness against other previously proposed methods on multiple datasets and evaluation metrics (Section ).",
"Related Work Previous works have optimized GANs training by changing the overall models' objectives, either by using discrepancy measurements [20], [35] or different divergence functions [31], [36] to approximate the real data distribution.",
"Moreover, [42], [4], [37] proposed to use energy-driven objective functions to encourage sample variety, [28] tried to match the mean and covariance of the real data, and [26] used an unrolled optimization of $D$ to train $G$ .",
"[5], [39], [38], [4] penalized missing modes by using an extra autoenconder in the framework.",
"[33] performed minibatch discrimination by forcing $D$ to condition its output on the similarity between the samples in the minibatch.",
"[34] increased $D$ 's robustness by maximizing the mutual information between inputs and corresponding labels, while [21] forced $D$ to make decisions over multiple samples of the same class, instead of independently.",
"Regarding using multiple discriminators, [29] extended the framework to several discriminators with each focusing in a low-dimensional projection of the data, set a priori.",
"[10] proposed GMAN, consisting of an ensemble of discriminators that could be accessed by the single generator according to different levels of difficulty.",
"[30] introduced D2GAN, introducing a single generator dual discriminator architecture where one discriminator rewards samples coming from the true data distribution whilst the other rewards samples coming from the generator, forcing the generator to continuously change its output.",
"[27] proposed Dropout-GAN, applying adversarial dropout by omitting the feedback of a given $D$ at the end of each batch.",
"Generative Adversarial Networks The original GANs framework [12] consists of two models: a generator ($G$ ) and a discriminator ($D$ ).",
"Both models are assigned different tasks: whilst $G$ tries to capture the real data distribution $p_r$ , $D$ learns how to distinguish real from fake samples.",
"$G$ maps a noise vector $z$ , retrieved from a noise distribution $p_{z}$ , to a realistic looking sample belonging to the data space.",
"$D$ maps a sample to a probability $p$ , representing the likeliness of that given sample coming from $p_{r}$ rather than from $p_g$ .",
"The two models are trained together and play the following minimax game: $\\begin{split}\\min _{G}\\max _{D}V(D,G) =\\\\ \\operatorname{\\mathbb {E}}_{x \\sim p_{r}(x)}[\\log D(x)] + \\operatorname{\\mathbb {E}}_{z \\sim p_{z}(z)}[\\log (1-D(G(z)))],\\end{split}$ where $D$ maximizes the probability of assigning samples to the correct distribution and $G$ minimizes the probability of its samples being considered from the fake data distribution.",
"Alternatively, one can also train $G$ to maximize the probability of its output being considered from the real data distribution, i.e., $\\log D(G(z))$ .",
"Even though this changes the type of the game, by being no longer minimax, it avoids the saturation of the gradient signals at the beginning of training [12], where $G$ only receives continuously negative feedback, making training more stable in practice.",
"However, since we employ multiple discriminators in the proposed framework, it is less likely that $G$ does not receive any positive feedback from the whole adversarial ensemble [10].",
"Therefore, we make use of the original value function in this work.",
"microbatchGAN In this work, we propose a novel generative multi-adversarial framework named microbatchGAN, where we start by splitting each minibatch into several microbatches and assigning a unique one to each $D$ .",
"The key aspect of this work is the usage of microbatch discrimination, where we change the original discrimination task of distinguishing between real and fake samples, as proposed in [12], to each $D$ distinguishing between samples coming or not from its fake microbatch.",
"This change is performed in a gradual fashion, using an additional diversity parameter $\\alpha $ .",
"Thus, each $D$ 's output gradually changes from the probability of a given sample being real to the probability of a given sample not belonging to its fake microbatch.",
"Moreover, since each $D$ is trained with different fake and real samples, we encourage them to focus on different data properties.",
"Figure REF illustrates the proposed framework.",
"Figure: microbatchGAN framework assuming a positive diversity parameter α\\alpha .",
"Each discriminator D k D_k is assigned a different microbatch x G D k x_{G_{D_{k}}}, where it discriminates between samples coming from inside its microbatch and samples coming from the microbatches assigned to the rest of the discriminators (x G ∖x G D k x_G \\setminus x_{G_{D_{k}}}) together the real samples x r D k x_{r_{D_{k}}}.The proposed microbatch-level discrimination task leads to $G$ making such discrimination harder for each $D$ to lower its loss.",
"Hence, $G$ is forced to induce variety on the overall minibatch, making it a substantially harder task for each $D$ to be able to separate its subset of fake samples in the diverse minibatch.",
"Note that producing very similar samples across the whole minibatch would also make such discrimination difficult by making the whole minibatch the same.",
"However, $G$ also benefits from each $D$ assigning distinct probabilities to samples from inside and outside its designed microbatch to lower its loss, making the generation of different samples in the minibatch a necessary requirement to obtain different outputs from $D$ .",
"Hence, all models in our framework benefit directly from sample variety in the generated set.",
"In the microbatchGAN scenario with a positive diversity parameter $\\alpha $ , each $D$ assigns low probabilities to fake samples from its microbatch and high probabilities to fake samples from the rest of the microbatches as well as samples from the real data distribution.",
"Hence, fake samples in the rest of the minibatch, i.e., not coming from its assigned microbatch, shall be given distinct output probabilities by each $D$ .",
"On the other hand, $G$ minimizes the probability given by each $D$ to the samples outside its microbatch and maximizes the probability given to the fake samples assigned to that specific $D$ .",
"The value function of our minimax game is as follows: $\\begin{split}\\min _{G}\\max _{\\big \\lbrace D_k\\big \\rbrace } \\sum _{k=1}^{K} V(D_k,G) =\\sum _{k=1}^{K} \\operatorname{\\mathbb {E}}_{x \\sim p_{r_{D_{k}}}(x)}[\\log D_k(x)] \\\\+ \\operatorname{\\mathbb {E}}_{z \\sim p_{z_{G_{D_{k}}}}(z)}[\\log (1-D_k(G(z)))] \\\\+ \\alpha \\times \\operatorname{\\mathbb {E}}_{z^{\\prime } \\sim p_{z_{G_{D}}\\setminus \\small \\lbrace z_{G_{D_{k}}}\\small \\rbrace }(z^{\\prime })}[\\log D_{k}(G(z^{\\prime }))],\\end{split}$ where K represents the number of total discriminators in the set.",
"$p_{r_{D_{k}}}$ represents real samples from $D_k$ 's real microbatch, $p_{z_{G_{D_{k}}}}$ indicates fake samples from $D_k$ 's fake microbatch, and $p_{z_{G_{D}}\\setminus \\small \\lbrace z_{G_{D_{k}}}\\small \\rbrace }$ relates to the rest of the fake samples in the minibatch but not in $p_{z_{G_{D_{k}}}}$ .",
"$\\alpha $ represents the diversity parameter responsible for penalizing the incorrect discrimination of fake samples coming from $p_{z_{G_{D}}\\setminus \\small \\lbrace z_{G_{D_{k}}}\\small \\rbrace }$ by each $D_k$ .",
"Note that $\\alpha $ = 0 would represent the original GANs objective for each $D$ in the set.",
"The training procedure of microbatchGAN is presented in Algorithm .",
"microbatchGAN.",
"Input: $K$ number of discriminators, $\\alpha $ diversity parameter, $B$ minibatch size Initialize: $m \\leftarrow \\frac{B}{K}$ number of training iterations • Sample minibatch $z_i$ , $i=1 \\ldots B$ , $z_i \\sim p_g(z)$ • Sample minibatch $x_i$ , $i=1 \\ldots B$ , $x_i \\sim p_{r}(x)$ $k = 1$ to $k = K$ • Sample microbatch $z_{k_{j}}$ , $j = 1 \\ldots m$ , $z_{k_{j}} = z_{(k-1) \\times m + 1 : k \\times m}$ • Sample microbatch $x_{k_{j}}$ , $j = 1 \\ldots m$ , $x_{k_{j}} = x_{(k-1) \\times m + 1 : k \\times m}$ • Sample microbatch $z^{\\prime }_{k_{j}}$ , $j = 1 \\ldots m$ , $z^{\\prime }_{k_{j}} \\subset z_i \\setminus \\small \\lbrace z_{k_{j}}\\small \\rbrace $ • Update $D_k$ by ascending its stochastic gradient: $\\nabla _{\\theta _{D_k}} \\frac{1}{m} \\sum _{j=1}^{m} [\\log D_k(x_{k_{j}}) +\\log (1-D_k(G(z_{k_{j}})))$ $+ \\alpha \\times \\log D_{k}(G(z^{\\prime }_{k_{j}}))]$ • Update $G$ by descending its stochastic gradient: $\\nabla _{\\theta _{G}} \\sum _{k=1}^{K} \\big [ \\frac{1}{m} \\sum _{j=1}^{m} [\\log (1-D_k(G(z_{k_{j}})))$ $+ \\alpha \\times \\log D_{k}(G(z^{\\prime }_{k_{j}}))]\\big ]$ Theoretical Discussion To better understand how our approach differs from the original GANs in promoting variety in the generated set, we study a simplified version of the minimax game where we freeze each $D_k$ and train $G$ until convergence.",
"In the most extreme case, we say that we have mode collapse when: $\\text{For all } z^{\\prime } \\sim p_g(z), G(z^{\\prime })=x$ Theorem 1 In original GANs, mode collapse fully minimizes $G$ 's loss when we train $G$ exhaustively without updating $D$ .",
"The optimal $x^{\\ast }$ is the one that maximizes $D$ 's output, where: $x^{\\ast } = \\underset{x}{\\text{argmax}} D(x)$ .",
"Thus, assuming $G$ would eventually learn how to produce $x^{\\ast }$ , mode collapse on $x^{\\ast }$ would fully minimize its loss, making $x^{\\ast }$ independent of $z$ .",
"Theorem 2 In microbatchGAN, assuming $\\alpha > 0$ , $x \\sim p_g$ must be dependent of $z$ for $G$ to fully minimize its loss, mitigating mode collapse when we train $G$ exhaustively without updating any $D_k$ .",
"From Eq.",
"REF , the value function between $G$ and each $D_k$ can be expressed as $\\begin{split}V(D_k,G) = \\operatorname{\\mathbb {E}}_{x \\sim p_r}[\\log D_k(x)] + \\operatorname{\\mathbb {E}}_{x^{\\prime } \\sim p_g}[\\log (1-D_k(x^{\\prime }))] \\\\+ \\alpha \\times \\operatorname{\\mathbb {E}}_{x^{\\prime \\prime } \\sim p_g}[\\log D_{k}(x^{\\prime \\prime })].\\end{split}$ To fully minimize its loss in relation to $D_k$ , $G$ must find $x^{\\prime } = \\underset{x}{\\text{argmax}} D_k(x) \\text{ and } x^{\\prime \\prime } = \\underset{x}{\\text{argmin}} D_k(x),$ which implies $D_k(x^{\\prime }) \\ne D_k(x^{\\prime \\prime }) \\Rightarrow x^{\\prime } \\ne x^{\\prime \\prime }.$ Thus, generating different outputs for different $z$ is a requirement to fully minimize $G$ 's loss regarding each $D_k$ .",
"Since we sum all $V(D_k,G)$ to calculate $G$ 's final loss, this also applies to overall adversarial set, concluding the proof.",
"Diversity Parameter $\\alpha $ We control the weight of the microbatch discrimination in the models' losses by introducing an additional diversity parameter $\\alpha $ .",
"Lower $\\alpha $ values lead to $G$ significantly lowering its loss by generating realistic looking samples on each microbatch without taking much consideration on the variety of the overall minibatch.",
"On the other hand, higher $\\alpha $ values induce a stronger effect on $G$ 's loss if each $D$ is able to discriminate between samples inside and outside its microbatch.",
"However, high values of $\\alpha $ might compromise the realistic properties of the produced samples, since too much weight is given to the last part of Eq.",
"REF , being sufficient to effectively minimize $G$ 's loss.",
"Thus, using $\\alpha > 0$ represents an additional way of ensuring data variety within the minibatch produced by $G$ at each iteration.",
"An overview of different possible $\\alpha $ settings follows below.",
"Static $\\alpha $ .",
"First, we statically set $\\alpha $ to values between 0 and 1 throughout the whole training.",
"For the evaluation of the effects of each $\\alpha $ value, we used a toy experiment of a 2D mixture of 8 Gaussian distributions (representing 8 data modes) firstly presented by [26], and further adopted by [30].",
"We used 8 discriminators for all the experiments.",
"Results are shown in Figure REF .",
"Figure: Toy experiment using static α\\alpha values.",
"Real data is presented in red while generated data is in blue.When setting $\\alpha = 0$ , $G$ mode collapses on a specific mode, showing the importance of using positive $\\alpha $ values to mitigate mode collapse.",
"When setting $0.1 \\le \\alpha \\le 0.5$ , $G$ is able to capture all data modes during training.",
"However, learning problems in the early stages are observed, with $G$ only focusing on promoting variety in the generated samples.",
"For higher $\\alpha $ values ($\\alpha \\ge 0.6$ ), $G$ was unable to produce any realistic looking samples throughout the whole training, focusing solely on sample diversity to lower its loss, suggesting the dominance of the last part of Eq.",
"REF .",
"Hence, a mild, dynamic, manipulation of $\\alpha $ values seems to be necessary for a successful training of $G$ , ultimately meaning both realistic and diverse samples from an early training stage.",
"Self-learned $\\alpha $ .",
"We dynamically set $\\alpha $ over time by adding it as a parameter of $G$ and letting it self-learn its values to lower its loss.",
"However, we observed that $G$ takes advantage of being able to reduce its loss by increasing $\\alpha $ at a large rate, focusing simply on promoting diversity in the generated samples without much realism, similarly to what was observed when using $\\alpha = 0.6$ in the toy experiment (Figure REF ).",
"Hence, we suggest several properties that $\\alpha $ should have so that diversity does not compromise the veracity of the generated samples.",
"First, $\\alpha $ should be upper bounded so that the last part of Eq.",
"REF (responsible for sample diversity) does not overpower the first part (responsible for sample realism), ultimately not compromising the feedback given to $G$ to also be able to generate realistic samples.",
"Second, $\\alpha $ 's growth should saturate over time, meaning that continuously increasing at large rates $\\alpha $ is no longer an option to substantially decrease $G$ 's loss over time.",
"Lastly, to tackle the problem in learning of early to mid stages, we suggest that $\\alpha $ should grow in a controlled fashion, so focus can also be given in the realistic aspect of the samples since the beginning of training.",
"Thus, we propose to make $\\alpha $ a function of $\\beta $ , where $\\alpha (\\beta ) \\in \\small [0,1\\small [$ , and let $G$ regulate $\\beta $ instead of directly learning $\\alpha $ .",
"We evaluated regulating $\\alpha $ over three different functions that have the desired properties: $\\alpha (\\beta ) =\\begin{dcases*}\\alpha _{sigm}(\\beta ) = Sigmoid(\\beta ), \\beta \\ge \\beta _{sigm}\\\\\\alpha _{soft}(\\beta ) = Softsign(\\beta ), \\beta \\ge \\beta _{soft}\\\\\\alpha _{tanh}(\\beta ) = Tanh(\\beta ), \\beta \\ge \\beta _{tanh}\\end{dcases*}$ with $\\beta _{sigm}$ , $\\beta _{soft}$ , and $\\beta _{tanh}$ representing the initial values of $\\beta $ when training begins for the respective functions.",
"For all the experiments of this paper, we set $\\beta _{tanh} = \\beta _{soft} = 0$ , to obtain a positive codomain, and $\\beta _{sigm} = -1.8$ , since we achieved better empirical results by starting $\\beta $ with this value (for further discussion about the effects of using different $\\beta _{sigm}$ on $\\alpha _{sigm}(\\beta )$ 's growth please see the Appendix).",
"Note that learning $\\alpha $ without any constraints can be characterized as using the identity function ($\\alpha (\\beta ) = \\alpha _{ident}(\\beta ) = \\beta $ ).",
"Thus, each used function promotes a different $\\alpha $ growth over time.",
"To ease presentation, we neglect to write $\\beta $ 's dependence for the rest of the manuscript and use only the function names to described each $\\alpha $ setting: $\\alpha _{sigm}$ , $\\alpha _{soft}$ , $\\alpha _{tanh}$ , and $\\alpha _{ident}$ .",
"Figure: α\\alpha evolution.Results on the toy dataset using the different proposed $\\alpha $ functions are shown in Figure REF .",
"The benefits of increasing $\\alpha $ in a milder fashion, as performed when using $\\alpha _{sigm}$ , are observed especially early on training, with $G$ being concerned with the realism of the generated samples.",
"On the other hand, when using $\\alpha _{tanh}$ and $\\alpha _{soft}$ , the network takes longer to focus on the data realism (10K steps) since it is able to reduce its loss significantly by simply promoting variety due to the steeper growth of $\\alpha $ in the earlier stages on both functions.",
"Nevertheless, as the functions gradually saturate, all $\\alpha $ settings manage to eventually capture the real data distribution while still keeping the diversity in the generated samples.",
"In conclusion, one can summarize microbatchGAN's training using these variations of self-learned $\\alpha $ as the following: in the first iterations, $G$ increases $\\alpha $ to reduce its loss, expanding its output.",
"As $\\alpha $ starts to saturate and each $D$ learns how to distinguish between real and fake samples, $G$ is forced to lower its loss by creating both realistic and diverse samples.",
"Experimental Results We validated the effects of using different $\\alpha $ functions on MNIST [19], CIFAR-10 [18], and cropped CelebA [22].",
"To quantitatively evaluate such effects, we used the Fréchet Inception Distance [13], or FID, since it has been shown to be sensitive to image quality as well as mode collapse [23], with the returned distance increasing notably when modes are missing from the generated data.",
"We used several variations of the standard FID for a thorough study of $\\alpha $ 's effects in training, as well as the influence of using a different number of discriminators in our framework.",
"Intra FID To measure the variety of samples of the generated set, we propose to calculate the FID between two subsets of 10K randomly picked fake samples generated at the end of every thousand iterations.",
"We call this metric Intra FID.",
"Important to note that Intra FID only measures the diversity in the generated set, not its realism.",
"Hence, higher values indicate more diversity within the generated samples while lower values might indicate mode collapse in the generated set.",
"The relation between Intra FID and progressive values of $\\alpha $ is shown in Figure REF .",
"Figure: Intra FID as α\\alpha progresses.",
"Higher values represent higher variety in the generated set.We observe a strong correlation between $\\alpha $ 's growth and variety in the set, especially in beginning to mid-training.",
"Later on, as $\\alpha $ saturates, the variety is kept (represented by the stability of the Intra FID).",
"It is further visible that $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ converge to similar Intra FID on all datasets.",
"Important to note, that, to ease the visualization, the graphs only represent $0 \\le \\alpha \\le 1$ , with $\\alpha _{ident}$ 's values naturally surpassing 1 as time progresses.",
"Cumulative Intra FID To analyze the sample variety over time, we summed the Intra FID values obtained from every thousand iterations.",
"Hence, higher values indicate that the model was able to promote more variety in the set across time.",
"Results are shown in Figure REF , where we observe that using more discriminators leads to more variety across all datasets and $\\alpha $ functions.",
"Moreover, using $\\alpha = 0$ leads to lower variety compared to using positive $\\alpha $ values, with $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ obtaining similar values throughout the different datasets.",
"Even though $\\alpha _{ident}$ promotes the highest variety, the generated samples lack realism, as previously witnessed in the toy experiment and further discussed next.",
"Figure: Cumulative Intra FID using a different number of discriminators and α\\alpha functions on the different datasets.",
"Higher values correlate to higher variety in the produced samples across time.",
"Values obtained using standard GANs are represented by the grey plane as a baseline.",
"Mean and Minimum FID To analyze both the realism and variety of the generated samples, we used the standard FID calculated between 10K fake samples and the real training data.",
"Lower values should indicate both diversity and high-quality samples.",
"The Mean FID and Minimum FID across 50K iterations are presented in Table REF for each dataset.",
"We observe that the best values, both in terms of mean and minimum, are obtained when using a higher number of discriminators, i.e., 5 or 10, and $\\alpha _{tanh}$ , $\\alpha _{soft}$ , and $\\alpha _{sigm}$ .",
"Moreover, the high distances obtained when using $\\alpha _{ident}$ confirm the lack of realism of the generated samples, highlighting the importance of constraining $\\alpha $ by the properties previously stated in Section .",
"Table: Mean and Minimum FID over 50K iterations on the different datasets.",
"Generated samples The generated samples on each dataset using 1 and 10 discriminators with different $\\alpha $ are presented in Figure REF .",
"For an objective assessment of the variety by the end of each iteration, the Intra FID is also provided.",
"We observe the superiority of the generated samples, both in terms of realism and variety, when using $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ on all datasets.",
"However, $\\alpha _{tanh}$ seems to show a delayed ability in generating realistic samples, possibly due to the increase of $\\alpha $ at a steeper fashion.",
"The inability of generating realistic samples when using $\\alpha _{ident}$ is also clearly detected on all datasets, as previously discussed.",
"More importantly, the high variety on the generated set, observed by the high Intra FID, is witnessed since very early iterations when using $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ .",
"The observed mitigation of mode collapse is carried out throughout the whole training.",
"Figure: Generated samples from 1K, 2K, 5K and 50K iteration with the respective Intra FID.When using standard GANs, we notice severe mode collapse, especially early on training.",
"When using 10 discriminators and $\\alpha $ set to 0, we notice a slight variation in the generated set, yet, this is only detected after a decent number of iterations, when each $D$ has seen enough samples to guide its judgment to a specific data mode due to the usage of different microbatch for each $D$ , delaying sample variety substantially.",
"Thus, using positive $\\alpha $ values is shown to be a necessary measure to stimulate variety since the beginning and until the end of training.",
"Method Comparison We proceeded to compare different settings of microbatchGAN to other existing methods on 3 different datasets: CIFAR-10, STL-10 [6], and ImageNet [7].",
"We down-sampled the images of the last two datasets down to 32x32 pixels.",
"We used Inception Score [33] or IS (higher is better) as the first quantitative metric.",
"Even though IS has been shown to be less correlated with human judgment than FID, most previous works only report results on this metric, making it a useful measure for model comparisons.",
"Out of fairness to the single discriminator methods that we compare our method against, we used only 2 discriminators in our experiments.",
"The architectures and training settings used for all the experiments can be found in the Appendix.",
"The comparison results are shown in Table REF .",
"We point special attention to the underlined method representing standard GANs, since it was the only method executed with our own implementation and identical training settings as microbatchGAN.",
"Thus, this represents the only method directly comparable to ours.",
"We notice a fair improvement of IS on all the tested datasets, observing an increase up to around 15% for CIFAR-10, 7% for STL-10, and 5% for ImageNet.",
"This indicates the success of our approach on improving the standard GANs framework on multiple datasets with different sizes and challenges.",
"Table: Inception scores.",
"For a fair comparison, only unsupervised methods are compared.On CIFAR-10, microbatchGAN achieves competitive results, significantly outperforming GMAN with 5 discriminators while using a similar architecture.",
"We argue that the use of more powerful architectures in the higher ranked methods plays a big role in their end score, especially for DCGAN.",
"Nonetheless, we acknowledge that using different objectives for each $D$ (as proposed in D2GAN) seems to be beneficial in a multi-discriminator setting, representing a good path to follow in the future.",
"Moreover, we observe that using extra autoencoders (DFM) or classifiers (MGAN) in the framework can help to achieve a better performance in the end.",
"However, we note that MGAN makes use of a 10 generator framework, on top of an extra classifier, to achieve the presented results.",
"Furthermore, the generated samples presented in their paper ([15]) indicate signs of partial mode collapse, which is not reflected in its high IS.",
"Table: Minimum FID comparison.We further compared our best FID with a subset of the reported methods in [23], namely GANs, both with the original and modified objective, LSGAN, and DRAGAN on CIFAR-10.",
"These methods were chosen since they represent interesting variants of standard GANs, as presented in [23].",
"Figure: CIFAR-10, STL-10, and ImageNet results.We extended each method to an ensemble of discriminators, for a fair comparison to our multiple discriminator approach.",
"Furthermore, we compare against additional results with adversarial dropout at a dropout rate of $0.5$ , as proposed in [27].",
"We used the same architecture of the last experiment for all methods.",
"Results are shown in Table REF .",
"We observe that all variants of microbatchGAN outperform the rest of the compared methods under controlled and equal experiments.",
"A subset of the generated samples produced by the different variations of microbatchGAN reported in Table REF are shown in Figure REF , where we observe high variety and realism across all generated sets.",
"Extended results are provided in the Appendix.",
"Conclusions In this work, we present a novel framework, named microbatchGAN, where each $D$ performs microbatch discrimination, differentiating between samples within and outside its fake microbatch.",
"This behavior is enforced by the diversity parameter $\\alpha $ , that is indirectly self-learned by $G$ .",
"In the first iterations, $G$ increases $\\alpha $ to lower its loss, expanding its output.",
"Then, as $\\alpha $ gradually saturates and each $D$ learns how to better distinguish between real and fake samples, $G$ is forced to fool each $D$ by promoting realism in its output, while keeping the diversity in the generated set.",
"We show evidence that our solution produces realistic and diverse samples on multiple datasets of different sizes and nature, ultimately mitigating mode collapse.",
"Training settings The architectural and training settings used in Sections , , and are presented in Tables REF , REF , and REF , respectively.",
"For the FID comparison on CIFAR-10 and CelebA in Section , we used the same architectures as Table REF but with a batch size of 64 on both datasets, and ran for 78K iterations on CIFAR-10 and 125K iterations on CelebA.",
"Table: Training settings for the toy dataset.Table: Training settings for MNIST, CIFAR-10, and CelebA.Table: Training settings for CIFAR-10, STL-10, and ImageNet.",
"Sigmoid initial value In Figure REF , we show and discuss the effects of using different $\\beta _{sigm}$ on $\\alpha _{sigm}$ on the toy dataset, giving more insights regarding the choice of $\\beta _{sigm} = -1.8$ mentioned in Section .",
"Figure: α\\alpha evolution.",
"Toy dataset comparisons Figure REF shows how different methods compare using the above mentioned toy dataset.",
"We compared microbatchGAN's results (K = 8, $\\alpha _{sigm}$ ) to the standard GAN ([12]), UnrollledGAN ([26]), D2GAN ([30]), and MGAN ([15]).",
"We observe bigger sample diversity with our method, while still approximating the real data distribution.",
"Figure: Method comparisons on the toy dataset.",
"Extended Results Additional results for CIFAR-10, STL-10, and ImageNet are presented bellow.",
"Figure: CIFAR-10 extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: CIFAR-10 extended results using K = 2 and α soft \\alpha _{soft}.Figure: CIFAR-10 extended results using K = 2 and α tanh \\alpha _{tanh}.Figure: STL-10 extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: STL-10 extended results using K = 2 and α soft \\alpha _{soft}.Figure: STL-10 extended results using K = 2 and α tanh \\alpha _{tanh}.Figure: ImageNet extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: ImageNet extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: ImageNet extended results using K = 2 and α sigm \\alpha _{sigm}."
],
[
"Generative Adversarial Networks",
"The original GANs framework [12] consists of two models: a generator ($G$ ) and a discriminator ($D$ ).",
"Both models are assigned different tasks: whilst $G$ tries to capture the real data distribution $p_r$ , $D$ learns how to distinguish real from fake samples.",
"$G$ maps a noise vector $z$ , retrieved from a noise distribution $p_{z}$ , to a realistic looking sample belonging to the data space.",
"$D$ maps a sample to a probability $p$ , representing the likeliness of that given sample coming from $p_{r}$ rather than from $p_g$ .",
"The two models are trained together and play the following minimax game: $\\begin{split}\\min _{G}\\max _{D}V(D,G) =\\\\ \\operatorname{\\mathbb {E}}_{x \\sim p_{r}(x)}[\\log D(x)] + \\operatorname{\\mathbb {E}}_{z \\sim p_{z}(z)}[\\log (1-D(G(z)))],\\end{split}$ where $D$ maximizes the probability of assigning samples to the correct distribution and $G$ minimizes the probability of its samples being considered from the fake data distribution.",
"Alternatively, one can also train $G$ to maximize the probability of its output being considered from the real data distribution, i.e., $\\log D(G(z))$ .",
"Even though this changes the type of the game, by being no longer minimax, it avoids the saturation of the gradient signals at the beginning of training [12], where $G$ only receives continuously negative feedback, making training more stable in practice.",
"However, since we employ multiple discriminators in the proposed framework, it is less likely that $G$ does not receive any positive feedback from the whole adversarial ensemble [10].",
"Therefore, we make use of the original value function in this work."
],
[
"microbatchGAN",
"In this work, we propose a novel generative multi-adversarial framework named microbatchGAN, where we start by splitting each minibatch into several microbatches and assigning a unique one to each $D$ .",
"The key aspect of this work is the usage of microbatch discrimination, where we change the original discrimination task of distinguishing between real and fake samples, as proposed in [12], to each $D$ distinguishing between samples coming or not from its fake microbatch.",
"This change is performed in a gradual fashion, using an additional diversity parameter $\\alpha $ .",
"Thus, each $D$ 's output gradually changes from the probability of a given sample being real to the probability of a given sample not belonging to its fake microbatch.",
"Moreover, since each $D$ is trained with different fake and real samples, we encourage them to focus on different data properties.",
"Figure REF illustrates the proposed framework.",
"Figure: microbatchGAN framework assuming a positive diversity parameter α\\alpha .",
"Each discriminator D k D_k is assigned a different microbatch x G D k x_{G_{D_{k}}}, where it discriminates between samples coming from inside its microbatch and samples coming from the microbatches assigned to the rest of the discriminators (x G ∖x G D k x_G \\setminus x_{G_{D_{k}}}) together the real samples x r D k x_{r_{D_{k}}}.The proposed microbatch-level discrimination task leads to $G$ making such discrimination harder for each $D$ to lower its loss.",
"Hence, $G$ is forced to induce variety on the overall minibatch, making it a substantially harder task for each $D$ to be able to separate its subset of fake samples in the diverse minibatch.",
"Note that producing very similar samples across the whole minibatch would also make such discrimination difficult by making the whole minibatch the same.",
"However, $G$ also benefits from each $D$ assigning distinct probabilities to samples from inside and outside its designed microbatch to lower its loss, making the generation of different samples in the minibatch a necessary requirement to obtain different outputs from $D$ .",
"Hence, all models in our framework benefit directly from sample variety in the generated set.",
"In the microbatchGAN scenario with a positive diversity parameter $\\alpha $ , each $D$ assigns low probabilities to fake samples from its microbatch and high probabilities to fake samples from the rest of the microbatches as well as samples from the real data distribution.",
"Hence, fake samples in the rest of the minibatch, i.e., not coming from its assigned microbatch, shall be given distinct output probabilities by each $D$ .",
"On the other hand, $G$ minimizes the probability given by each $D$ to the samples outside its microbatch and maximizes the probability given to the fake samples assigned to that specific $D$ .",
"The value function of our minimax game is as follows: $\\begin{split}\\min _{G}\\max _{\\big \\lbrace D_k\\big \\rbrace } \\sum _{k=1}^{K} V(D_k,G) =\\sum _{k=1}^{K} \\operatorname{\\mathbb {E}}_{x \\sim p_{r_{D_{k}}}(x)}[\\log D_k(x)] \\\\+ \\operatorname{\\mathbb {E}}_{z \\sim p_{z_{G_{D_{k}}}}(z)}[\\log (1-D_k(G(z)))] \\\\+ \\alpha \\times \\operatorname{\\mathbb {E}}_{z^{\\prime } \\sim p_{z_{G_{D}}\\setminus \\small \\lbrace z_{G_{D_{k}}}\\small \\rbrace }(z^{\\prime })}[\\log D_{k}(G(z^{\\prime }))],\\end{split}$ where K represents the number of total discriminators in the set.",
"$p_{r_{D_{k}}}$ represents real samples from $D_k$ 's real microbatch, $p_{z_{G_{D_{k}}}}$ indicates fake samples from $D_k$ 's fake microbatch, and $p_{z_{G_{D}}\\setminus \\small \\lbrace z_{G_{D_{k}}}\\small \\rbrace }$ relates to the rest of the fake samples in the minibatch but not in $p_{z_{G_{D_{k}}}}$ .",
"$\\alpha $ represents the diversity parameter responsible for penalizing the incorrect discrimination of fake samples coming from $p_{z_{G_{D}}\\setminus \\small \\lbrace z_{G_{D_{k}}}\\small \\rbrace }$ by each $D_k$ .",
"Note that $\\alpha $ = 0 would represent the original GANs objective for each $D$ in the set.",
"The training procedure of microbatchGAN is presented in Algorithm .",
"microbatchGAN.",
"Input: $K$ number of discriminators, $\\alpha $ diversity parameter, $B$ minibatch size Initialize: $m \\leftarrow \\frac{B}{K}$ number of training iterations • Sample minibatch $z_i$ , $i=1 \\ldots B$ , $z_i \\sim p_g(z)$ • Sample minibatch $x_i$ , $i=1 \\ldots B$ , $x_i \\sim p_{r}(x)$ $k = 1$ to $k = K$ • Sample microbatch $z_{k_{j}}$ , $j = 1 \\ldots m$ , $z_{k_{j}} = z_{(k-1) \\times m + 1 : k \\times m}$ • Sample microbatch $x_{k_{j}}$ , $j = 1 \\ldots m$ , $x_{k_{j}} = x_{(k-1) \\times m + 1 : k \\times m}$ • Sample microbatch $z^{\\prime }_{k_{j}}$ , $j = 1 \\ldots m$ , $z^{\\prime }_{k_{j}} \\subset z_i \\setminus \\small \\lbrace z_{k_{j}}\\small \\rbrace $ • Update $D_k$ by ascending its stochastic gradient: $\\nabla _{\\theta _{D_k}} \\frac{1}{m} \\sum _{j=1}^{m} [\\log D_k(x_{k_{j}}) +\\log (1-D_k(G(z_{k_{j}})))$ $+ \\alpha \\times \\log D_{k}(G(z^{\\prime }_{k_{j}}))]$ • Update $G$ by descending its stochastic gradient: $\\nabla _{\\theta _{G}} \\sum _{k=1}^{K} \\big [ \\frac{1}{m} \\sum _{j=1}^{m} [\\log (1-D_k(G(z_{k_{j}})))$ $+ \\alpha \\times \\log D_{k}(G(z^{\\prime }_{k_{j}}))]\\big ]$"
],
[
"Theoretical Discussion",
"To better understand how our approach differs from the original GANs in promoting variety in the generated set, we study a simplified version of the minimax game where we freeze each $D_k$ and train $G$ until convergence.",
"In the most extreme case, we say that we have mode collapse when: $\\text{For all } z^{\\prime } \\sim p_g(z), G(z^{\\prime })=x$ Theorem 1 In original GANs, mode collapse fully minimizes $G$ 's loss when we train $G$ exhaustively without updating $D$ .",
"The optimal $x^{\\ast }$ is the one that maximizes $D$ 's output, where: $x^{\\ast } = \\underset{x}{\\text{argmax}} D(x)$ .",
"Thus, assuming $G$ would eventually learn how to produce $x^{\\ast }$ , mode collapse on $x^{\\ast }$ would fully minimize its loss, making $x^{\\ast }$ independent of $z$ .",
"Theorem 2 In microbatchGAN, assuming $\\alpha > 0$ , $x \\sim p_g$ must be dependent of $z$ for $G$ to fully minimize its loss, mitigating mode collapse when we train $G$ exhaustively without updating any $D_k$ .",
"From Eq.",
"REF , the value function between $G$ and each $D_k$ can be expressed as $\\begin{split}V(D_k,G) = \\operatorname{\\mathbb {E}}_{x \\sim p_r}[\\log D_k(x)] + \\operatorname{\\mathbb {E}}_{x^{\\prime } \\sim p_g}[\\log (1-D_k(x^{\\prime }))] \\\\+ \\alpha \\times \\operatorname{\\mathbb {E}}_{x^{\\prime \\prime } \\sim p_g}[\\log D_{k}(x^{\\prime \\prime })].\\end{split}$ To fully minimize its loss in relation to $D_k$ , $G$ must find $x^{\\prime } = \\underset{x}{\\text{argmax}} D_k(x) \\text{ and } x^{\\prime \\prime } = \\underset{x}{\\text{argmin}} D_k(x),$ which implies $D_k(x^{\\prime }) \\ne D_k(x^{\\prime \\prime }) \\Rightarrow x^{\\prime } \\ne x^{\\prime \\prime }.$ Thus, generating different outputs for different $z$ is a requirement to fully minimize $G$ 's loss regarding each $D_k$ .",
"Since we sum all $V(D_k,G)$ to calculate $G$ 's final loss, this also applies to overall adversarial set, concluding the proof."
],
[
"Diversity Parameter $\\alpha $",
"We control the weight of the microbatch discrimination in the models' losses by introducing an additional diversity parameter $\\alpha $ .",
"Lower $\\alpha $ values lead to $G$ significantly lowering its loss by generating realistic looking samples on each microbatch without taking much consideration on the variety of the overall minibatch.",
"On the other hand, higher $\\alpha $ values induce a stronger effect on $G$ 's loss if each $D$ is able to discriminate between samples inside and outside its microbatch.",
"However, high values of $\\alpha $ might compromise the realistic properties of the produced samples, since too much weight is given to the last part of Eq.",
"REF , being sufficient to effectively minimize $G$ 's loss.",
"Thus, using $\\alpha > 0$ represents an additional way of ensuring data variety within the minibatch produced by $G$ at each iteration.",
"An overview of different possible $\\alpha $ settings follows below.",
"Static $\\alpha $ .",
"First, we statically set $\\alpha $ to values between 0 and 1 throughout the whole training.",
"For the evaluation of the effects of each $\\alpha $ value, we used a toy experiment of a 2D mixture of 8 Gaussian distributions (representing 8 data modes) firstly presented by [26], and further adopted by [30].",
"We used 8 discriminators for all the experiments.",
"Results are shown in Figure REF .",
"Figure: Toy experiment using static α\\alpha values.",
"Real data is presented in red while generated data is in blue.When setting $\\alpha = 0$ , $G$ mode collapses on a specific mode, showing the importance of using positive $\\alpha $ values to mitigate mode collapse.",
"When setting $0.1 \\le \\alpha \\le 0.5$ , $G$ is able to capture all data modes during training.",
"However, learning problems in the early stages are observed, with $G$ only focusing on promoting variety in the generated samples.",
"For higher $\\alpha $ values ($\\alpha \\ge 0.6$ ), $G$ was unable to produce any realistic looking samples throughout the whole training, focusing solely on sample diversity to lower its loss, suggesting the dominance of the last part of Eq.",
"REF .",
"Hence, a mild, dynamic, manipulation of $\\alpha $ values seems to be necessary for a successful training of $G$ , ultimately meaning both realistic and diverse samples from an early training stage.",
"Self-learned $\\alpha $ .",
"We dynamically set $\\alpha $ over time by adding it as a parameter of $G$ and letting it self-learn its values to lower its loss.",
"However, we observed that $G$ takes advantage of being able to reduce its loss by increasing $\\alpha $ at a large rate, focusing simply on promoting diversity in the generated samples without much realism, similarly to what was observed when using $\\alpha = 0.6$ in the toy experiment (Figure REF ).",
"Hence, we suggest several properties that $\\alpha $ should have so that diversity does not compromise the veracity of the generated samples.",
"First, $\\alpha $ should be upper bounded so that the last part of Eq.",
"REF (responsible for sample diversity) does not overpower the first part (responsible for sample realism), ultimately not compromising the feedback given to $G$ to also be able to generate realistic samples.",
"Second, $\\alpha $ 's growth should saturate over time, meaning that continuously increasing at large rates $\\alpha $ is no longer an option to substantially decrease $G$ 's loss over time.",
"Lastly, to tackle the problem in learning of early to mid stages, we suggest that $\\alpha $ should grow in a controlled fashion, so focus can also be given in the realistic aspect of the samples since the beginning of training.",
"Thus, we propose to make $\\alpha $ a function of $\\beta $ , where $\\alpha (\\beta ) \\in \\small [0,1\\small [$ , and let $G$ regulate $\\beta $ instead of directly learning $\\alpha $ .",
"We evaluated regulating $\\alpha $ over three different functions that have the desired properties: $\\alpha (\\beta ) =\\begin{dcases*}\\alpha _{sigm}(\\beta ) = Sigmoid(\\beta ), \\beta \\ge \\beta _{sigm}\\\\\\alpha _{soft}(\\beta ) = Softsign(\\beta ), \\beta \\ge \\beta _{soft}\\\\\\alpha _{tanh}(\\beta ) = Tanh(\\beta ), \\beta \\ge \\beta _{tanh}\\end{dcases*}$ with $\\beta _{sigm}$ , $\\beta _{soft}$ , and $\\beta _{tanh}$ representing the initial values of $\\beta $ when training begins for the respective functions.",
"For all the experiments of this paper, we set $\\beta _{tanh} = \\beta _{soft} = 0$ , to obtain a positive codomain, and $\\beta _{sigm} = -1.8$ , since we achieved better empirical results by starting $\\beta $ with this value (for further discussion about the effects of using different $\\beta _{sigm}$ on $\\alpha _{sigm}(\\beta )$ 's growth please see the Appendix).",
"Note that learning $\\alpha $ without any constraints can be characterized as using the identity function ($\\alpha (\\beta ) = \\alpha _{ident}(\\beta ) = \\beta $ ).",
"Thus, each used function promotes a different $\\alpha $ growth over time.",
"To ease presentation, we neglect to write $\\beta $ 's dependence for the rest of the manuscript and use only the function names to described each $\\alpha $ setting: $\\alpha _{sigm}$ , $\\alpha _{soft}$ , $\\alpha _{tanh}$ , and $\\alpha _{ident}$ .",
"Figure: α\\alpha evolution.Results on the toy dataset using the different proposed $\\alpha $ functions are shown in Figure REF .",
"The benefits of increasing $\\alpha $ in a milder fashion, as performed when using $\\alpha _{sigm}$ , are observed especially early on training, with $G$ being concerned with the realism of the generated samples.",
"On the other hand, when using $\\alpha _{tanh}$ and $\\alpha _{soft}$ , the network takes longer to focus on the data realism (10K steps) since it is able to reduce its loss significantly by simply promoting variety due to the steeper growth of $\\alpha $ in the earlier stages on both functions.",
"Nevertheless, as the functions gradually saturate, all $\\alpha $ settings manage to eventually capture the real data distribution while still keeping the diversity in the generated samples.",
"In conclusion, one can summarize microbatchGAN's training using these variations of self-learned $\\alpha $ as the following: in the first iterations, $G$ increases $\\alpha $ to reduce its loss, expanding its output.",
"As $\\alpha $ starts to saturate and each $D$ learns how to distinguish between real and fake samples, $G$ is forced to lower its loss by creating both realistic and diverse samples."
],
[
"Experimental Results",
"We validated the effects of using different $\\alpha $ functions on MNIST [19], CIFAR-10 [18], and cropped CelebA [22].",
"To quantitatively evaluate such effects, we used the Fréchet Inception Distance [13], or FID, since it has been shown to be sensitive to image quality as well as mode collapse [23], with the returned distance increasing notably when modes are missing from the generated data.",
"We used several variations of the standard FID for a thorough study of $\\alpha $ 's effects in training, as well as the influence of using a different number of discriminators in our framework."
],
[
"Intra FID",
"To measure the variety of samples of the generated set, we propose to calculate the FID between two subsets of 10K randomly picked fake samples generated at the end of every thousand iterations.",
"We call this metric Intra FID.",
"Important to note that Intra FID only measures the diversity in the generated set, not its realism.",
"Hence, higher values indicate more diversity within the generated samples while lower values might indicate mode collapse in the generated set.",
"The relation between Intra FID and progressive values of $\\alpha $ is shown in Figure REF .",
"Figure: Intra FID as α\\alpha progresses.",
"Higher values represent higher variety in the generated set.We observe a strong correlation between $\\alpha $ 's growth and variety in the set, especially in beginning to mid-training.",
"Later on, as $\\alpha $ saturates, the variety is kept (represented by the stability of the Intra FID).",
"It is further visible that $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ converge to similar Intra FID on all datasets.",
"Important to note, that, to ease the visualization, the graphs only represent $0 \\le \\alpha \\le 1$ , with $\\alpha _{ident}$ 's values naturally surpassing 1 as time progresses."
],
[
"Cumulative Intra FID",
"To analyze the sample variety over time, we summed the Intra FID values obtained from every thousand iterations.",
"Hence, higher values indicate that the model was able to promote more variety in the set across time.",
"Results are shown in Figure REF , where we observe that using more discriminators leads to more variety across all datasets and $\\alpha $ functions.",
"Moreover, using $\\alpha = 0$ leads to lower variety compared to using positive $\\alpha $ values, with $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ obtaining similar values throughout the different datasets.",
"Even though $\\alpha _{ident}$ promotes the highest variety, the generated samples lack realism, as previously witnessed in the toy experiment and further discussed next.",
"Figure: Cumulative Intra FID using a different number of discriminators and α\\alpha functions on the different datasets.",
"Higher values correlate to higher variety in the produced samples across time.",
"Values obtained using standard GANs are represented by the grey plane as a baseline."
],
[
"Mean and Minimum FID",
"To analyze both the realism and variety of the generated samples, we used the standard FID calculated between 10K fake samples and the real training data.",
"Lower values should indicate both diversity and high-quality samples.",
"The Mean FID and Minimum FID across 50K iterations are presented in Table REF for each dataset.",
"We observe that the best values, both in terms of mean and minimum, are obtained when using a higher number of discriminators, i.e., 5 or 10, and $\\alpha _{tanh}$ , $\\alpha _{soft}$ , and $\\alpha _{sigm}$ .",
"Moreover, the high distances obtained when using $\\alpha _{ident}$ confirm the lack of realism of the generated samples, highlighting the importance of constraining $\\alpha $ by the properties previously stated in Section .",
"Table: Mean and Minimum FID over 50K iterations on the different datasets."
],
[
"Generated samples",
"The generated samples on each dataset using 1 and 10 discriminators with different $\\alpha $ are presented in Figure REF .",
"For an objective assessment of the variety by the end of each iteration, the Intra FID is also provided.",
"We observe the superiority of the generated samples, both in terms of realism and variety, when using $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ on all datasets.",
"However, $\\alpha _{tanh}$ seems to show a delayed ability in generating realistic samples, possibly due to the increase of $\\alpha $ at a steeper fashion.",
"The inability of generating realistic samples when using $\\alpha _{ident}$ is also clearly detected on all datasets, as previously discussed.",
"More importantly, the high variety on the generated set, observed by the high Intra FID, is witnessed since very early iterations when using $\\alpha _{sigm}$ , $\\alpha _{soft}$ , and $\\alpha _{tanh}$ .",
"The observed mitigation of mode collapse is carried out throughout the whole training.",
"Figure: Generated samples from 1K, 2K, 5K and 50K iteration with the respective Intra FID.When using standard GANs, we notice severe mode collapse, especially early on training.",
"When using 10 discriminators and $\\alpha $ set to 0, we notice a slight variation in the generated set, yet, this is only detected after a decent number of iterations, when each $D$ has seen enough samples to guide its judgment to a specific data mode due to the usage of different microbatch for each $D$ , delaying sample variety substantially.",
"Thus, using positive $\\alpha $ values is shown to be a necessary measure to stimulate variety since the beginning and until the end of training."
],
[
"Method Comparison",
"We proceeded to compare different settings of microbatchGAN to other existing methods on 3 different datasets: CIFAR-10, STL-10 [6], and ImageNet [7].",
"We down-sampled the images of the last two datasets down to 32x32 pixels.",
"We used Inception Score [33] or IS (higher is better) as the first quantitative metric.",
"Even though IS has been shown to be less correlated with human judgment than FID, most previous works only report results on this metric, making it a useful measure for model comparisons.",
"Out of fairness to the single discriminator methods that we compare our method against, we used only 2 discriminators in our experiments.",
"The architectures and training settings used for all the experiments can be found in the Appendix.",
"The comparison results are shown in Table REF .",
"We point special attention to the underlined method representing standard GANs, since it was the only method executed with our own implementation and identical training settings as microbatchGAN.",
"Thus, this represents the only method directly comparable to ours.",
"We notice a fair improvement of IS on all the tested datasets, observing an increase up to around 15% for CIFAR-10, 7% for STL-10, and 5% for ImageNet.",
"This indicates the success of our approach on improving the standard GANs framework on multiple datasets with different sizes and challenges.",
"Table: Inception scores.",
"For a fair comparison, only unsupervised methods are compared.On CIFAR-10, microbatchGAN achieves competitive results, significantly outperforming GMAN with 5 discriminators while using a similar architecture.",
"We argue that the use of more powerful architectures in the higher ranked methods plays a big role in their end score, especially for DCGAN.",
"Nonetheless, we acknowledge that using different objectives for each $D$ (as proposed in D2GAN) seems to be beneficial in a multi-discriminator setting, representing a good path to follow in the future.",
"Moreover, we observe that using extra autoencoders (DFM) or classifiers (MGAN) in the framework can help to achieve a better performance in the end.",
"However, we note that MGAN makes use of a 10 generator framework, on top of an extra classifier, to achieve the presented results.",
"Furthermore, the generated samples presented in their paper ([15]) indicate signs of partial mode collapse, which is not reflected in its high IS.",
"Table: Minimum FID comparison.We further compared our best FID with a subset of the reported methods in [23], namely GANs, both with the original and modified objective, LSGAN, and DRAGAN on CIFAR-10.",
"These methods were chosen since they represent interesting variants of standard GANs, as presented in [23].",
"Figure: CIFAR-10, STL-10, and ImageNet results.We extended each method to an ensemble of discriminators, for a fair comparison to our multiple discriminator approach.",
"Furthermore, we compare against additional results with adversarial dropout at a dropout rate of $0.5$ , as proposed in [27].",
"We used the same architecture of the last experiment for all methods.",
"Results are shown in Table REF .",
"We observe that all variants of microbatchGAN outperform the rest of the compared methods under controlled and equal experiments.",
"A subset of the generated samples produced by the different variations of microbatchGAN reported in Table REF are shown in Figure REF , where we observe high variety and realism across all generated sets.",
"Extended results are provided in the Appendix."
],
[
"Conclusions",
"In this work, we present a novel framework, named microbatchGAN, where each $D$ performs microbatch discrimination, differentiating between samples within and outside its fake microbatch.",
"This behavior is enforced by the diversity parameter $\\alpha $ , that is indirectly self-learned by $G$ .",
"In the first iterations, $G$ increases $\\alpha $ to lower its loss, expanding its output.",
"Then, as $\\alpha $ gradually saturates and each $D$ learns how to better distinguish between real and fake samples, $G$ is forced to fool each $D$ by promoting realism in its output, while keeping the diversity in the generated set.",
"We show evidence that our solution produces realistic and diverse samples on multiple datasets of different sizes and nature, ultimately mitigating mode collapse."
],
[
"Training settings",
"The architectural and training settings used in Sections , , and are presented in Tables REF , REF , and REF , respectively.",
"For the FID comparison on CIFAR-10 and CelebA in Section , we used the same architectures as Table REF but with a batch size of 64 on both datasets, and ran for 78K iterations on CIFAR-10 and 125K iterations on CelebA.",
"Table: Training settings for the toy dataset.Table: Training settings for MNIST, CIFAR-10, and CelebA.Table: Training settings for CIFAR-10, STL-10, and ImageNet."
],
[
"Sigmoid initial value",
"In Figure REF , we show and discuss the effects of using different $\\beta _{sigm}$ on $\\alpha _{sigm}$ on the toy dataset, giving more insights regarding the choice of $\\beta _{sigm} = -1.8$ mentioned in Section .",
"Figure: α\\alpha evolution."
],
[
"Toy dataset comparisons",
"Figure REF shows how different methods compare using the above mentioned toy dataset.",
"We compared microbatchGAN's results (K = 8, $\\alpha _{sigm}$ ) to the standard GAN ([12]), UnrollledGAN ([26]), D2GAN ([30]), and MGAN ([15]).",
"We observe bigger sample diversity with our method, while still approximating the real data distribution.",
"Figure: Method comparisons on the toy dataset."
],
[
"Extended Results",
"Additional results for CIFAR-10, STL-10, and ImageNet are presented bellow.",
"Figure: CIFAR-10 extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: CIFAR-10 extended results using K = 2 and α soft \\alpha _{soft}.Figure: CIFAR-10 extended results using K = 2 and α tanh \\alpha _{tanh}.Figure: STL-10 extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: STL-10 extended results using K = 2 and α soft \\alpha _{soft}.Figure: STL-10 extended results using K = 2 and α tanh \\alpha _{tanh}.Figure: ImageNet extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: ImageNet extended results using K = 2 and α sigm \\alpha _{sigm}.Figure: ImageNet extended results using K = 2 and α sigm \\alpha _{sigm}."
]
] | 2001.03376 |
[
[
"Dissociation dynamics of the diamondoid adamantane upon photoionization\n by XUV femtosecond pulses"
],
[
"Abstract This work presents a photodissociation study of the diamondoid adamantane using extreme ultraviolet femtosecond pulses.",
"The fragmentation dynamics of the dication is unraveled by the use of advanced ion and electron spectroscopy giving access to the dissociation channels as well as their energetics.",
"To get insight into the fragmentation dynamics, we use a theoretical approach combining potential energy surface determination, statistical fragmentation methods and molecular dynamics simulations.",
"We demonstrate that the dissociation dynamics of adamantane dications takes place in a two-step process: barrierless cage opening followed by Coulomb repulsion-driven fragmentation."
],
[
"Introduction",
"Diamondoids are a class of carbon nanomaterials based on carbon cages with well-defined structures formed by $\\rm C(sp^3)-C(sp^3)$ -hybridized bonds and fully terminated by hydrogen atoms.",
"All diamondoids are variants of the adamantane molecule, the most stable among all of the isomers with the formula $\\rm C_{10}H_{16}$ , shown in Figure REF .",
"On Earth, diamondoids are naturally found in petroleum deposits and natural gas reservoirs, and their most common applications are for the characterization of petroleum and gas fields, offering possibilities to e.g.",
"trace the source of oil spills [1].",
"Today, diamondoids are attracting increasing interest for use as an applied nanomaterial in e.g.",
"nano- and optoelectronics as well as in biotechnology and medicine due to their high thermal stability and well-defined structure in combination with no known toxicity [2].",
"In space, diamondoids have been found to be the most abundant component of presolar grains [3], and due to their high stability they are thus also expected to be abundant in the interstellar medium [4].",
"Figure: Structure of adamantane.However, when compared to laboratory measurements based on infrared spectroscopy [5], astronomical observations show a deficiency of diamondoids in the interstellar medium which to date is not completely understood [6].",
"The first ionization limit in diamondoids lies around $8-9$ eV with a maximum in the ionization yield between 10 and 11 eV [7], close to the hydrogen Lyman-$\\alpha $ line, and the efficient production of cations followed by dissociation has been suggested as a possible explanation for the apparent lack of diamondoids in the interstellar medium.",
"Steglich et al.",
"investigated the stability of diamondoid cations using ultraviolet irradiation, finding that rapid loss of a neutral hydrogen followed ionization [8].",
"Since then, further studies have suggested that small hydrocarbons are also created as dissociation products.",
"In a recent work at the Swiss Light Source, vacuum ultraviolet radiation (9-12 eV) was used in combination with threshold photoelectron and photoion coincidence detection to determine the appearance energies and branching ratios of the resulting photofragments of the singly charged adamantane cation [9].",
"The study reveals, in addition to the expected hydrogen loss, dissociation via a number of parallel channels which all start with an opening of the carbon cage and hydrogen migration indicating that the low photostability of adamantane could explain its deficiency in astronomical observations.",
"While this study was recently complemented by a first time-resolved study [10], to date no results have been published on the dissociation dynamics of multiply charged adamantane molecules.",
"In this work, we study the fragmentation dynamics of the adamantane dication after ionization by extreme ultraviolet (XUV) femtosecond pulses, the use of which ensures prompt and well-defined ionization.",
"The experimental technique used in this study is based on correlated ion and electron spectroscopy, enabling the characterization of the charged products of interaction (identification and energetics).",
"The support of various theoretical methods such as molecular dynamics simulations, potential energy surface determination and statistical fragmentation models, helps to unravel the fragmentation dynamics of such a complex molecular system.",
"The high-intensity XUV beamline at the Lund Attosecond Science Centre provides trains of attosecond pulses in the XUV spectral region using the high-order harmonic generation (HHG) technique [11], [12].",
"This is achieved by focusing an intense infrared (IR) pulse (high-power Ti:Sapphire chirped pulse amplification laser with a pulse energy of 50 mJ, a central wavelength of 810 nm, a pulse duration of 45 fs and a repetition rate of 10 Hz) into an argon gas medium (6 cm long cell) in a loose focusing geometry ($\\sim 8$ m focal length) [13].",
"The train of XUV attosecond pulses, with a total duration of 20 fs, contains around 15 attosecond pulses with an estimated individual pulse duration of approximately 300 as, spaced by 1.35 fs.",
"The photon energy spectrum of the produced XUV light is a characteristic harmonic comb spanning from $\\sim 20$ to 45 eV (Figure REF ).",
"Then, the XUV light is micro-focused ($\\rm \\sim 5\\times 5 \\, \\mu m^2$ ) on target via a double toroidal mirror [14] and with a pulse train energy on target of around 10 nJ this leads to intensities of the order of $10^{12}$ W$\\cdot $ cm$^{-2}$ .",
"Figure: XUV spectrum and the first three ionization thresholds of adamantane.Adamantane molecules, $\\rm C_{10}H_{16}$ (powder from Aldrich with $>$ 99% purity), are produced in the gas phase by a pulsed Even-Lavie valve [15], [16], heated to 100$^{\\circ }$ C and using He as a carrier gas, in the form of a cold and collimated supersonic jet.",
"The photon-molecule interaction leads to the formation of highly excited singly and doubly charged adamantane molecules (single and double ionization thresholds $\\rm IT_1 \\sim 9.2$ eV and $\\rm IT_2 \\sim 23.9$ eV - see Figure REF ).",
"The trication is only produced in negligible amounts since the triple ionization threshold is $\\rm IT_3 \\sim 43.6$ eV (Figure REF ) and thus is not discussed in the following.",
"The charged products of interaction are detected by a double-sided velocity map imaging (VMI) spectrometer [17] giving access to the kinetic energy distributions of ions and electrons on a shot-to-shot basis.",
"In addition, the ion side of the spectrometer can measure the time-of-flight (TOF) of the ions, providing the mass spectrum.",
"Despite the high count rates (several tens of counts per pulse), the use of the partial covariance technique [18] enables us to disentangle the contributions of the different fragmentation channels.",
"For instance, applying this technique on single-shot ion TOF spectra gives the possibility to produce ion-ion correlation maps that reveal the dissociation dynamics of the doubly charged adamantane molecules.",
"In addition, the use of this technique on single-shot ion TOF spectra and the single-shot ion VMI data gives access to the kinetic energy distribution of specific ionic fragments."
],
[
"Theory",
"Three different theoretical methods have been used: (i) molecular dynamics (MD) simulations in the framework of the density functional theory (DFT) and the density functional tight binding (DFTB) method, (ii) exploration of the potential energy surface (PES) employing the DFT, and (iii) statistical fragmentation using the Microcanonical Metropolis Monte-Carlo (M3C) method."
],
[
"Molecular dynamics simulations",
"The molecular dynamics approach, using the DFTB method[19], [20], has been used to compute the lifetime of doubly-ionized and excited adamantane.",
"To this end, we have considered double ionization of adamantane in a Franck–Condon way; that is, our molecular dynamics simulations start from the optimized geometry of the neutral adamantane molecule after removal of two electrons.",
"We assume that the electronic excitation energy ($\\rm E_{exc}$ ) is rapidly redistributed into the nuclear degrees of freedom and thus we run these simulations in the electronic ground state.",
"We have taken four values of excitation energy corresponding to the relative energy of the highest order of harmonics in the XUV spectrum (Figure REF ) with respect to IT$_2$ ($\\rm E_{exc}$ = 8.46, 11.50, 14.63 and 17.80 eV, which correspond to temperature values 2520, 3450, 4360 and 5300 K, respectively).",
"The used $\\rm E_{exc}$ values can be considered as the upper limits of the remaining electronic excitation energy after ionization with the four highest energy harmonics.",
"This excitation energy is randomly distributed into the nuclear degrees of freedom in each trajectory.",
"Molecular dynamics trajectories are propagated up to a maximum time of $\\rm t_{max}$ = 1, 5, 10, 20 and 100 ps.",
"For each value of excitation energy and propagation time a set of 1000 independent trajectories are considered (that is, the initial conditions are separately established in each set of trajectories).",
"Statistics are then carried out over these trajectories to obtain information on the survival time of the doubly-ionized adamantane with different excitation energies.",
"To ensure adiabaticity in the simulations a time step of $\\Delta $ t=0.1 fs is used.",
"These simulations have been carried out with the deMonNano code [21] and the results show that even with these relatively high excitation energies the dication does not fully fragment until after tens to hundreds of picoseconds making it computationally too heavy to perform full MD simulations at the DFT level (see SI).",
"We have explored the PES using DFT, in particular we have employed the B3LYP functional[22], [23], [24] in combination with the 6-31G(d) basis set.",
"This part of the study provides useful energetic and structural information of the experimentally measured exit channels.",
"In order to identify the most relevant stationary points of the PES we have adopted the following strategy: 1) We have first performed molecular dynamics simulations at the same DFT-B3LYP/6-31G(d) level, to mimic the evolution of the system during the first femtoseconds after the ionization and excitation.",
"This part of the simulations also starts by computing the energy required to doubly ionize adamantane in a Franck–Condon transition from the optimized neutral structure.",
"This is our starting point for the dynamics.",
"160 trajectories were carried out using the ADMP method with a maximum propagation time of 500 fs, and considering a time step of $\\Delta t=$ 0.1 fs, and a fictitious mass of $\\mu =$ 0.1 au.",
"Thus, after propagation of the doubly charged excited adamantane, we have obtained the evolution of the system in the first femtoseconds.",
"2) Then, using the last step in the dynamics as an initial guess, we have optimized the geometry of the produced species.",
"3) Finally, geometry optimization of fragments observed in the experiments has also been computed.",
"To this end, we have considered several structures for each ${\\rm C}_n{\\rm H}_x^{q+}$ fragment, thus obtaining the relative energy of the exit channels observed in the experiment.",
"Harmonic frequencies have been computed after geometry optimization to confirm that the obtained structures are actual minima in the PES (no imaginary frequencies) and to evaluate the Zero-Point-Energy correction.",
"These calculations were carried out with the Gaussian09 package[25].",
"The proposed strategy was used in the past with success to study the fragmentation dynamics of ionized biomolecules (see e.g.",
"[26], [27], [28]).",
"We study statistical fragmentation of doubly ionized adamantane with the recently developed M3C method[29], [30], using the constrained approach presented in [31].",
"The key aspect of this methodology is that it provides a random way to move in the phase space (the so-called Markov chain) until a region of maximum entropy is reached, where the physical observables are computed.",
"This description should be equivalent to an MD simulation in the infinite integration time limit.",
"In this work, we focus on two observables: 1) the probability of each fragmentation channel as a function of internal energy (the so-called breakdown curves), and 2) the distribution of the internal energy of the system in its components.",
"This method was successfully used in the past to describe the fragmentation of carbonaceous species[32], [33], [29], [34], [31].",
"The main ingredients of the M3C simulations, i.e.",
"structures, energies, and vibrational frequencies of the fragments, are those obtained in the PES exploration at the B3LYP/6-31G(d) level.",
"In total, 148 molecules are included in the fragmentation model (see SI for details).",
"Geometries are available as an additional file in the SI and in the M3C-store project database [35].",
"The statistical simulations have been carried out such that the sum of angular momenta from all fragments exactly compensates the orbital momentum resulting in a total angular momentum equal to zero.",
"We set the radius of the system to 30.0 Å.",
"Implementation of larger radii implies similar Coulomb interaction among fragments, but requires increased sampling to achieve convergence; on the other hand, a smaller radius results in an artificial overestimation of the fragments' angular momentum.",
"We have performed a scan of the internal energy from 0 to 10 eV.",
"10000 numerical experiments for each value of internal energy have been carried out in order to estimate the error in the computed observables (and thus to use them as convergence criteria).",
"The numerical experiments each differ from one another in their initial values for vibrational energy, angular momentum, and molecular orientation, which were randomly chosen.",
"All numerical experiments start from the most stable structure of the doubly charged $\\rm C_{10}H_{16}^{2+}$ .",
"The sequence 5*V,T,R,S:0,5*V,T,R,S:-1:1 has been used as a Markov chain, including a total of 2000 events; among them 10% have been used as a burn-in period (see ref.",
"[29] for details).",
"In summary, a complete picture of the fragmentation of excited doubly-charged adamantane is obtained with the theoretical simulations: dynamic, energetic and entropic approaches allow us to infer the main factors governing the experimentally observed processes, also providing complementary information."
],
[
"Results and discussion",
"The experimental total mass spectrum of the charged products of interaction (Figure REF ) is obtained by calibration of the TOF spectrum recorded after 275000 laser shots.",
"The most intense peak (excluding helium) corresponds to the singly charged parent ion at $m/z = 136$ .",
"The loss of one hydrogen atom is observed at a mass-to-charge ratio of $m/z = 135$ with an intensity of 9% of the parent ion.",
"The losses of two and three hydrogen atoms are also observed, however they have an intensity two orders of magnitude lower than that of the single hydrogen loss.",
"As a general feature, the production of a wide distribution of $\\rm C_nH_x^+$ fragments resulting from dissociation of singly and doubly charged adamantane molecules is observed.",
"The most intense peaks of each $\\rm C_n$ group are attributed to $\\rm CH_3^+$ , $\\rm C_2H_5^+$ , $\\rm C_3H_5^+$ , $\\rm C_4H_7^+$ , $\\rm C_5H_7^+$ , $\\rm C_6H_7^+$ , $\\rm C_7H_9^+$ , $\\rm C_8H_{11}^+$ and $\\rm C_9H_{13}^+$ (respectively $m/z = 15$ , 29, 41, 55, 67, 79, 93, 107 and 121) and are rather similar to the ones observed in the case of ionization by electron impact at 70 eV (dashed line in Figure REF ).",
"On the other hand, the main fragments of the $\\rm C_n$ groups $n=3,4$ and 8 are different from the ones found by Candian et al.",
"[9] (photodissociation around first ionization threshold), i.e.",
"$\\rm C_3H_7^+$ ,$\\rm C_4H_8^+$ and $\\rm C_8H_{12}^+$ respectively, demonstrating that the dynamics of fragmentation is sensitive to the ionization/excitation energy.",
"Most of the fragments indicate strong intramolecular rearrangements with multiple hydrogen migrations and/or hydrogen losses.",
"Some of these rearrangements may occur before fragmentation and lead to the cage opening of adamantane cations.",
"The corresponding cage opening of the singly charged cation was already studied by Candian et al.",
"using DFT and RRKM simulations [9].",
"In the case of the doubly charged adamantane, the results of our ab initio molecular dynamics calculations (ADMP using DFT-B3LYP up to 500 fs) followed by PES exploration are summarized in Figure REF .",
"The three lowest energy configurations (Figure REF (a)), appearing $\\approx 4$ eV below the double ionization threshold, have an open-cage geometry and have at least one hydrogen migration ($\\rm CH_3$ termination).",
"Figure: Key energy levels of adamantane dication processes.",
"(a) Double ionization threshold and lowest energy configurations for doubly charged adamantane found in the PES exploration.",
"(b) and (c), final energy levels of the fragmentation channels of the adamantane dication (2- and 3-body breakups) corresponding to the ones in Table .",
"Energy levels in gray are not explicitly labelled but can be found in Table .",
"The energy values are relative to the neutral ground state in units of eV."
],
[
"Dication fragmentation pattern",
"In order to help to understand the complex energetic picture of the dication dynamics, Figure REF shows a schematic dissociative potential energy curve for the dication, including the different energetic quantities that are useful for the discussion.",
"We consider the vertical ionization (Franck-Condon region) from the neutral ground state with a photon of energy $h \\nu $ (purple arrow), such as $h \\nu > \\rm IT_2$ .",
"The excess energy after photoionization is defined as $\\rm {E_{ph}} = \\it {h \\nu } - \\rm IT_2$ .",
"We already know that the relaxation from the double ionization threshold $\\rm IT_2$ to the ground state of the dication has a fixed energy of $\\rm E_{relax} \\approx 4.1$ eV (see Figure REF (a)).",
"The kinetic energy of the electron pair involved in the double ionization is called $\\rm E_{2e^-}$ and the internal energy of the dication ground state is denoted $\\rm E_{int}$ .",
"Under our assumptions, the internal energy is the sum of the rapidly redistributed electronic excitation energy ($\\rm E_{exc}$ ) and the relaxation energy ($E_{relax}$ ), and thus $\\rm E_{int} \\ge E_{relax} \\approx 4.1$ eV.",
"The final ionic products of the dissociation have a total energy equal to the sum of their internal energy ($\\rm E_{int\\,frag}$ ) and the kinetic energy release (KER).",
"This total energy also corresponds to the sum of the initial internal energy of the dication ($\\rm E_{int}$ ) and the energy difference between the dication ground state and the energy levels of the exit channel.",
"Experimentally, the intact doubly charged parent ion is not observed in the mass spectrum (Figure REF ) at the timescale of the detection (a few microseconds).",
"Moreover, no doubly charged fragments are detected.",
"While the total mass spectrum is dominated by the fragments of singly charged adamantane molecules, the use of the partial covariance technique [37], [18] on the ion TOF spectrum enables to case correlate singly charged ions coming from the dissociation of the dication of adamantane using an ion-ion correlation map representation (Figure REF ).",
"Correlation islands in this map give the ion pairs that are summarized in Table REF as well as the branching ratios (BR) of these fragmentation channels.",
"In addition, the PES exploration provides the final energy levels of the dication fragmentation channels referred to the neutral ground state ($\\rm \\Delta E$ ), which are represented in Figure REF and given in Table REF .",
"The energy levels of the 2-body breakup channels appear at lower energies than most of the 3-body breakup ones.",
"It is interesting to notice that all the energy levels are below the double ionization threshold, meaning that the dication of adamantane is metastable and will spontaneously dissociate (if no barrier, i.e.",
"transition state, at higher energy than the double ionization threshold is involved).",
"Figure: Ion-ion correlation map resulting from the fragmentation of adamantane dications.Table: List of correlated singly charged fragments coming from the dissociation of adamantane dication observed experimentally.",
"In the case of n\\rm n-body breakups with n>2\\rm n>2, the neutral losses are given in mass losses such that the chemical formulae have to be seen as chemical element indicators and not necessarily as fragments.",
"BR stands for branching ratio and is given in percent.",
"ΔE\\rm \\Delta E is the calculated final energy level of the dication fragmentation channels (2- and 3-body breakups) referred to the neutral ground state (in eV).Assuming a low internal energy after the ionization, the fragmentation time of doubly charged adamantane is expected to be very long ($\\gtrsim $ 100 ps, according to our DFTB simulation, see SI) and therefore we cannot afford to carry out simulations with ab initio molecular dynamics.",
"Thus, we use the M3C statistical method to obtain complementary information.",
"This method was developed to study the fragmentation of molecular systems based on entropic criteria (see [29], [31], [30] for details).",
"In Figure REF , showing the breakdown curves, the gray areas mark the inaccessible regions of internal energy ($\\rm E_{int} < E_{relax}$ ).",
"The probabilities for the 2-body channels (Figure REF (a)) all peak below $\\rm E_{relax}$ , with tails reaching into the accessible internal energy region.",
"This is consistent with the fact that they appear in the lowest energy region in the PES (Figure REF ).",
"Above $\\rm E_{relax}$ , only the $\\rm C_2H_5^+/C_8H_{11}^+$ has a significant probability, while the other 2-body breakup channels $\\rm C_3H_5^+/C_7H_{11}^+$ , $\\rm C_3H_6^+/C_7H_{10}^+$ and $\\rm C_4H_7^+/C_6H_9^+$ are almost not populated.",
"The 3-body breakups channels (Figure REF (b)) all peak around 6 eV, also consistent with their higher energy levels according to the PES in Figure REF .",
"Comparing the calculated breakdown cuves with the experimentally measured branching ratios in Table REF , suggests that the internal energy of the adamantane dications is close to $\\rm E_{relax}$ under the current conditions, since in this region the $\\rm C_2H_5^+/C_8H_{11}^+$ channel dominates the other 2-body breakup channels, in good agreement with the experiment (BR$>$ 23% vs. BR$<$ 3%).",
"If the internal energy was higher, the 3-body breakup channels would start to dominate, which is not observed in the experiment.",
"Under our assumption that the electronic excitation energy is considered to be rapidly redistributed into the nuclear degrees of freedom, this implies that most of the dications remain in the ground state or low excited states after ionization, and that the internal energy of the system mainly corresponds to the relaxation energy.",
"Figure: Breakdown curves of the channels observed in the experiments for 2-body breakups (a) and 3-body breakups (b).",
"The errors (shaded areas around curves) correspond to the standard deviation.",
"The black dashed lines indicate the minimum of internal energy that we can reach in our case, meaning E relax \\rm E_{relax}, and the gray areas mark the regions of internal energy that are inaccessible in the experiment.We have further obtained additional valuable information by analyzing how the internal energy is distributed after ionization using the statistical simulations with the M3C code.",
"Note that this analysis is based on the ergodic assumption, i.e.",
"at infinite time when the system has reached equilibrium and the maximum entropy region in the phase space is populated.",
"We can decompose the internal energy as $\\rm E_\\text{int}=E_\\text{kin}+E_\\text{pot}+E_\\text{vib}+E_\\text{rot}$ , where $\\rm E_\\text{kin}$ , $\\rm E_\\text{pot}$ , $\\rm E_\\text{vib}$ , and $\\rm E_\\text{rot}$ represent the kinetic or translational, the potential, the vibrational, and the rotational energy components, respectively.",
"The potential energy $\\rm E_\\text{pot}$ is the energy difference between different geometrical configurations, in this case the exit channel and the most stable dication structure (see Figure REF ).",
"While the potential energy is important for the total available energy, we now focus on how the latter is distributed between the remaining degrees of freedom, i.e.",
"between $\\rm E_\\text{kin}$ , $\\rm E_\\text{vib}$ and $\\rm E_\\text{rot}$ .",
"Figure REF shows the average and the standard deviation of these energy components as a function of the internal energy $\\rm E_\\text{int}$ .",
"It is clear that in the considered internal energy range the vibrational contribution is most prominent, with smaller contributions of the rotational and kinetic energy components.",
"As already shown, the fragmentation is the dominant process in the relaxation of the adamantane dication; thus, the available energy of the system is primarily absorbed by the vibrational component, i.e.",
"the produced fragments can store a large amount of energy in nuclear degrees of freedom (mainly vibrations).",
"Assuming an internal energy of $\\sim 4.1$ eV (corresponding to the relaxation energy), $\\sim 70\\%$ of the available energy is stored in vibration and $\\sim 15\\%$ in rotation while the remaining $15\\%$ are shared among the other components.",
"As we have seen, the channel C$_2$ H$_{5}^+$ /C$_{8}$ H$_{11}^+$ is strongly dominating the fragmentation dynamics of the dication with a branching ratio $\\sim 23.3$ % whereas the other channels are at least three times less intense (Figure REF and Table REF ).",
"This can be roughly interpreted by looking at the energy levels of the different fragmentation channels obtained from the exploration of the PES of the adamantane dication (Figure REF and Table REF ).",
"This main channel is energetically favorable since it has the lowest energy level at around $\\sim $ 7 eV below the double ionization threshold.",
"In addition, we have seen from the M3C calculations that the breakdown curve (Figure REF (a)) of this channel was dominant at low accessible internal energy ($\\sim 4-5$ eV), thus being also entropically favorable in this energy region.",
"Figure: VMI images obtained after filtering using the partial covariance method on the TOF peaks correspond to the fragments C 2 _2H 5 + _5^+ (a) and C 8 _8H 11 + _{11}^+ (b) (left part: raw data and right part: inverted data).",
"Artefacts are coming from intense signal (helium and parent ion) that were not filtered out by covariance analysis (see SI for more details).",
"(c) and (d) Ion kinetic energy distributions of the respective fragments obtained by angular integration of the inverted data avoiding the artefacts signal and energy calibrated using ion trajectory simulations (SIMION®).",
"(e) Kinetic energy release distribution (KERd) for the channel C 2 _2H 5 + _5^+ / C 8 _8H 11 + _{11}^+ obtained by convolution of the kinetic energy distributions of the two fragments.In order to have a better insight into the energetics of the main fragmentation channel, we can regard the ion kinematics of the dissociation process, particularly the kinetic energy release distribution (KERd).",
"Performing partial covariance analysis between the ion TOF and the ion VMI data gives the ”mass-selected” velocity map images displayed on the left parts of panels (a) and (b) in Figure REF .",
"The right parts in Figure REF (a) and (b) show the result of Abel inversion using an iterative method [39].",
"The angular integration of the inverted images gives, after energy calibration, the kinetic energy distributions of the fragments C$_2$ H$_5^+$ and C$_8$ H$_{11}^+$ (Figure REF (c) and (d)).",
"The intense signal close to zero corresponds to the contribution from the dissociation of the singly charged adamantane.",
"At higher energies, clear peaks show contributions at $\\sim $ 2.5 eV and $\\sim $ 0.7 eV respectively due to the Coulomb repulsion of the 2-body breakup of the dication and verify the momentum conservation principle.",
"It is possible to obtain the KERd of the channel C$_2$ H$_5^+$ /C$_8$ H$_{11}^+$ by convolution of the two individual kinetic energy distributions (Figure REF (e)).",
"The peak at $\\sim $ 3 eV indicates the main energy contribution of this channel.",
"The small value of the KER reflects complex fragmentation dynamics with strong molecular rearrangement before the charge separation takes place: In a first step, a cage opening leading, most probably, to one of the structures in Figure REF ; then a second molecular reorganization producing both charged fragments; and finally charge repulsion between them.",
"The energy difference between the open structures in Figure REF and the C$_2$ H$_{5}^+$ /C$_{8}$ H$_{11}^+$ exit channel is between $\\sim $ 2.6 and $\\sim $ 2.9 eV, which is consistent within a multiple step fragmentation as the one presented here.",
"Considering the Coulomb repulsion between the two positive charges, the energy released is $C/R$ , with $C=14.4$ eV$\\rm \\cdot \\mathring{A}$ and $R$ the initial inter-charge distance in $\\rm [\\mathring{A}]$ .",
"An energy release of $\\sim $ 3.0 eV, as measured in the experiment, corresponds to an inter-charge distance of 4.8 $\\rm \\mathring{A}$ .",
"However, the maximum distance between two C atoms in the closed-cage adamantane structure is $\\sim ~3.5~\\rm \\mathring{A}$ , discarding the assumption of an instantaneous double ionization followed by prompt fragmentation.",
"Thus, structural rearrangements before the charge separation would produce a considerable extension of the structure, thus increasing the inter-charge distance up to 4.8 $\\rm \\mathring{A}$ (as inferred from the experiment).",
"This further confirms the multi-step processes with cage opening preceding Coulomb repulsion.",
"Figure REF (a) (purple line) shows the total photoelectron energy spectrum obtained by angular integration of the electron VMI data after inversion using an iterative method [39].",
"Electrons coming from the ionization of the helium buffer gas can be seen around the dashed lines and were used for energy calibration.",
"The contribution to the photoelectron spectrum associated with single ionization by the harmonics is found in the higher energy part of the total spectrum.",
"The expected contribution, shown by the gray line Figure REF (a), is calculated using the measured photoelectron spectrum for singly charged adamantane [40], the harmonic spectrum (see Figure REF ), and the photoabsorption cross section of adamantane [8].",
"In this estimated contribution, the calculated photoelectron spectrum from ionization of helium has also been included, and by subtracting this from the total photoelectron spectrum, the expected spectrum of the photoelectrons associated with double ionization is obtained and shown in Figure REF (b).",
"Although the absolute signal in the resulting spectrum is sensitive to the scaling of the calculated spectrum, it is clear that the photoelectrons primarily occupy the low-energy part of the spectrum ($<10$ eV) compared to the photoelectrons from single ionization.",
"From the harmonic spectrum (Figure REF ) the maximum total electron pair energy is $\\rm E_{2e-} \\approx 21$ eV, assuming that the dications remain in the ground state or in low excited states after ionization.",
"Thus, the observed cut-off at $\\sim 10$ eV, suggests that the available energy is shared rather evenly between the two electrons.",
"All-in-all, the shape of the photoelectron energy distribution supports the assumption of low excitation, apart from the unexpectedly strong contribution at energies below 2 eV.",
"The latter feature is a possible indication of excitation of higher-lying electronic states, resulting in low energy electrons.",
"Possible candidates for such states are found through a calculation of the excited states of the adamantane dication (see SI for more details) exhibiting a dense band of excited states between 35 and 40 eV (relative to the ground state of the neutral) that would result in large internal energy values $\\rm E_{int}$ between 15 and 20 eV, and values of $\\rm E_{2e-}$ in the 5-10 eV range following ionization by the cut-off harmonics.",
"Such excitation could not be inferred from the experimental fragmentation pattern, nor is included in the current level of theory, which calls for further investigation."
],
[
"Conclusion",
"We have performed a detailed study of the photodissociation of adamantane, focused on the fragmentation dynamics of the dication.",
"By combining the experimental analysis with multiple theoretical methods we unraveled key processes governing charge and energy distribution after the ultrafast photoionizaton and the subsequent fragmentation dynamics.",
"We found that the most stable structures of the dication of adamantane present an open-cage geometry, appearing at $\\sim 4$ eV below the double ionization threshold, that can be reached in a few tens of femtoseconds after the ionization.",
"However, these structures are metastable and evolve producing several fragments in a Coulomb repulsion process.",
"Much like other carbonaceous species, adamantane dications are quite efficient energy reservoirs, being able to store a large amount of energy in particular in vibrational modes.",
"This occurs when the internal energy generated in the photoionization, together with the energy produced in the Coulomb explosion, is redistributed among the nuclear degrees of freedom of the produced fragments.",
"Among the different fragmentation pathways, the most populated channel $\\rm C_2H_5^+/C_8H_{11}^+$ is the lowest in energy in the PES ($\\sim 2.6$ eV below the most stable structure of the adamantane dication) leading us to conclude that the fragment distribution is mainly governed by energetic criteria.",
"We have shown that the ion KERd of this channel peaks at $\\sim 3$ eV with a width of 1 eV allowing us to experimentally confirm that the cage opening takes place prior to fragmentation.",
"A qualitative comparison between the experimental branching ratios and the results of statistical fragmentation simulations suggests that the internal energy of the dication largely consists of the relaxation energy from the cage opening, with only minor contribution from redistribution of electronic excitation energy.",
"Finally, measurements of the photoelectron kinetic energy spectrum largely confirms these observations, but also indicate the possible existence of electronic excitation which could not be further investigated in the current experiment.",
"The presented results highlight the complexity of, and provides pieces of information on, the fragmentation of multiply charged diamondoids.",
"While this study was able to identify the dominant fragmentation pathways and shine light on the redistribution of energy and charge, the further elucidation of the ultrafast excitation dynamics, and in particular the timescales involved, calls for time-resolved experiments.",
"Such experiments can be envisaged using ultrashort single wavelength XUV pulses, e.g.",
"from free electron lasers, in combination with multicoincidence ion-electron spectroscopy techniques."
],
[
"Acknowledgements",
"A special thanks is given to A. Persson for his support for the high-power laser and to M. Gisselbrecht for critical reading of the manuscript.",
"This research was supported by the Swedish Research Council, the Swedish Foundation for Strategic Research and the Crafoord Foundation.",
"This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No.",
"641789 MEDEA and under grant agreement no 654148 Laserlab-Europe.",
"The research was conducted in the framework of the International Associated Laboratory (LIA) Fragmentation DYNAmics of complex MOlecular systems - DYNAMO, funded by the Centre National de la Recherche Scientifique (CNRS).",
"The authors acknowledge the generous allocation of computer time at the Centro de Computación Científica at the Universidad Autónoma de Madrid (CCC-UAM).",
"This work was partially supported by the project CTQ2016-76061-P of the Spanish Ministerio de Economía y Competitividad (MINECO).",
"Financial support from the MINECO through the “María de Maeztu” Program for Units of Excellence in R&D (MDM-2014-0377) is also acknowledged.",
"P.R, P.J and S.M.",
"designed the experiments, S.M., J.L., J.P., H.W., P.R., P.R., S.I., F.B., H.C.-A., and B.A.H.",
"performed the experiment, S.M.",
"and J.L analysed the results.",
"S.D.-T. performed the potential energy surface and the molecular dynamics calculations, and N.F.A.",
"the M3C simulations.",
"S.M., S.D.-T. and P.J wrote the original draft, S.D.-T. and N.F.A.",
"wrote the theoretical sections and part of the discussion and all authors reviewed the manuscript.",
"The authors declare no competing interests.",
"Supplementary information is available for this paper at link"
]
] | 2001.03373 |
[
[
"Thirty Years of Radio Observations of Type Ia SN 1972E and SN 1895B:\n Constraints on Circumstellar Shells"
],
[
"Abstract We have imaged over 35 years of archival Very Large Array (VLA) observations of the nearby (d$_{\\rm{L}}$ $=$ 3.15 Mpc) Type Ia supernovae SN\\,1972E and SN\\,1895B between 9 and 121 years post-explosion.",
"No radio emission is detected, constraining the 8.5 GHz luminosities of SN\\,1972E and SN\\,1895B to be L$_{\\nu,8.5\\rm{GHz}}$ $<$ 6.0 $\\times$ 10$^{23}$ erg s$^{-1}$ Hz$^{-1}$ 45 years post-explosion and L$_{\\nu,8.5\\rm{GHz}}$ $<$ 8.9 $\\times$ 10$^{23}$ erg s$^{-1}$ Hz$^{-1}$ 121 years post-explosion, respectively.",
"These limits imply a clean circumstellar medium (CSM), with $n$ $<$ 0.9 cm$^{-3}$ out to radii of a few $\\times$ 10$^{18}$ cm, if the SN blastwave is expanding into uniform density material.",
"Due to the extensive time coverage of our observations, we also constrain the presence of CSM shells surrounding the progenitor of SN\\,1972E.",
"We rule out essentially all medium and thick shells with masses of 0.05$-$0.3 M$_\\odot$ at radii between $\\sim$10$^{17}$ and 10$^{18}$ cm, and thin shells at specific radii with masses down to $\\lesssim$0.01 M$_\\odot$.",
"These constraints rule out swaths of parameter space for a range of single and double degenerate progenitor scenarios, including recurrent nova, core-degenerate objects, ultra-prompt explosions and white dwarf (WD) mergers with delays of a few hundred years between the onset of merger and explosion.",
"Allowed progenitors include WD-WD systems with a significant ($>$ 10$^{4}$ years) delay from the last episode of common envelope evolution and single degenerate systems undergoing recurrent nova, provided that the recurrence timescale i short and the system has been in the nova phase for $\\gtrsim$10$^{4}$ yr, such that a large ($>$ 10$^{18}$ cm) cavity has been evacuated.",
"Future multi-epoch observations of additional intermediate-aged Type Ia SNe will provide a comprehensive view of the large-scale CSM environments around these explosions."
],
[
"Introduction",
"Type Ia supernovae (SNe) are caused by the explosion of a carbon-oxygen white dwarf [81].",
"They have become an important cornerstone of cosmological distance calculations as \"standardizable candles\" for measuring the expansion of the universe via their measured luminosity distances as a function of redshift [95], [88].",
"However, despite their importance, debates still remain regarding both the progenitor systems and explosion mechanism of Type Ia SNe [71].",
"There are two broad scenarios in which a carbon-oxygen WD can explode as Type Ia SNe, and both involve binary systems [45], [124].",
"The first is the single degenerate (SD) scenario, in which the WD accretes material from a non-degenerate stellar companion [82], [117], [46].",
"The second is the double degenerate (DD) scenario, where the secondary companion is also a WD [125], [50], [71], [66].",
"The term \"double degenerate\" is broad and currently encompasses multiple combinations of progenitor binary systems and explosion mechanisms, including direct collisions [63], mergers [108], and double detonations due to accretion from a helium WD companion [109], [37].",
"It is also debated whether Type Ia SN can only be produced near the Chandrasekhar Mass (M$_{\\rm {Ch}}$ ), or if sub-M$_{\\rm {Ch}}$ WDs can also produce normal Type Ia SNe while undergoing double detonations or violent mergers [127], [62], [110].",
"Some observations show evidence for a population of sub-M$_{\\rm {Ch}}$ explosions [104].",
"One strategy to shed light on these open questions is to search for circumstellar material (CSM) surrounding Type Ia SNe.",
"The CSM is produced by the pre-explosion evolution of binary system—including winds, outbursts and episodes of mass transfer—and can therefore reflect the nature of the SN progenitor.",
"However, for decades searches for CSM around Type Ia SNe in the X-ray and radio have yielded non-detections [84], [43], [73], [18], [100], [72], [19], implying low-density environments.",
"Most of these observations were taken within a few hundred days of the SN explosion, constraining the density of the CSM at distances $\\lesssim $ 10$^{16}$ cm from the progenitor star.",
"Of these, observations of three nearby events—SN 2011fe, SN 2014J, and SN 2012cg—have constrained the pre-explosion mass-loss rates of the progenitor systems to $\\dot{M} < 10^{-9} M_{\\odot }$ yr$^{-1}$ , ruling out all but the lowest mass SD systems [73], [18], [72], [19].",
"At the same time, larger samples of more distant events systematically rule out winds from more massive or evolved stellar companions [100], [19].",
"In recent years, however, other types of observations have painted a more complex picture of the CSM surrounding Type Ia SNe.",
"First, a new class of SNe (SNe Ia-CSM) spectroscopically resemble SNe Ia but have strong hydrogen emission lines [111].",
"This has been interpreted as the SN shockwave interacting with a significant amount of CSM ($\\sim $ few $M_{\\odot })$ located directly around the explosion site (distributed out to radii of $\\sim $ 10$^{16}$ cm).",
"SNe Ia-CSM are rare, and the most nearby (SN 2012ca; d$_{\\rm {L}}$ $\\sim $ 80 Mpc) is the only Type Ia SN detected in X-rays to date [9].",
"Additionally, blue-shifted Na I D absorbing material has been detected in some normal Type Ia SNe spectra, which is interpreted as CSM surrounding the SNe that has been ionized [86], [8], [115], [70].",
"Modeling has indicated the material is not distributed continuously with radius, but is more likely located in shell-like structures at radii $\\ge 10^{17}$ cm [20].",
"Such absorbing material is estimated to have a total mass of up to $\\sim 1 M_{\\odot }$ , and is thought to be present in $\\ge $ 20% of SNe Ia in spiral galaxies [115].",
"Most recently, [38] reported evidence of CSM interaction surrounding SN 2015cp at $\\sim 730$ days post-explosion, consistent with a CSM shell that contains hydrogen at distances $\\ge $ 10$^{16}$ cm, and [60] reported the detection of H$\\alpha $ in a late-time nebular spectrum of ASASSN-18tb, interpreted as the signature of CSM interaction.",
"Despite these intriguing results, constraints on the CSM surrounding Type Ia SNe at radii $\\gtrsim $ 10$^{17}$ cm have been relatively sparse.",
"These distances can be probed by radio observations obtained between $\\sim $ 5 and 50 years post-explosion.",
"These timescales have typically been neglected because the deepest constraints on the presence of a stellar wind density profile can be made in the first $\\sim $ year post-explosion.",
"However, if an uniform density medium is present, deeper limits on CSM would be possible via radio observations at greater times post-SN, as the shockwave continues to interact with the ambient material [15].",
"Additionally, if multiple observations are taken over the course of several years, the presence of CSM shells at a range of radii can be probed.",
"On even longer time scales ($\\sim $ 100 years) radio observations can yield information on the CSM density and structure as a SN transitions to the SN remnant (SNR) stage.",
"In our own galaxy, young Type Ia SNRs have been observed in radio wavelengths.",
"For example, G1.9+0.3, was first discovered by the Very Large Array (VLA) and is estimated to be between 125 and 140 years old [92].",
"Additionally, Kepler's SNR is radio bright $\\sim $ 400 years after the explosion [29].",
"However, whether this emission is due to interaction with CSM ejected by the progenitor system, or the interstellar medium (ISM), is still debated.",
"In contrast, [103] made deep radio images of the SN 1885A area in the Andromeda Galaxy (M31; 0.785 $\\pm $ 0.025 Mpc distant).",
"The resulting upper limits constrain SN 1885A to be fainter than G1.9+0.3 at a similar timescale of $\\sim $ 120 years post-explosion, placing strict limits on the density of the ambient medium and the transition to the SNR stage.",
"This appears to favor a sub-M$_{\\rm {Ch}}$ model for the explosion.",
"While observations of SNe within our Local Group (e.g.",
"SN 1885A) can provide the deepest individual limits on the CSM density surrounding the progenitors of Type Ia SNe, the number of Type Ia SNe with ages $\\lesssim $ 100 years is limited.",
"Therefore, in order to build up a statistical sample of intermediate-aged SNe, we must look to galaxies farther afield.",
"In this paper, we have compiled over 30 years of radio observations of NGC 5253 for this purpose.",
"NGC 5253 offers an ideal example for such studies because (i) it has hosted two Type Ia SNe in the past $\\sim $ 150 years (SN 1972E and SN 1895B), (ii) it is located at very close proximity (d$=$ 3.15 Mpc; [36]), and (iii) it has been observed with the historic VLA and upgraded Karl G. Jansky VLA multiple times between 1981 and 2016.",
"Such a data set over so many years allows us to probe the density of the CSM out to large radii from the SNe, constrain the presence of CSM shells, and provide insight into various progenitor scenarios for Type Ia SNe.",
"This paper is structured as follows.",
"In Section , we summarize information known on SN 1895B and SN 1972E.",
"In Section , we describe 30 years of archival radio observations of these systems.",
"In Section , we use these observations to place deep limits on the density of a uniform ambient medium and the presence of CSM shells surrounding SN 1972E and SN 1895B at radii between 10$^{17}$ and 10$^{18}$ cm.",
"In Section , we discuss these results in the context of multiple Type Ia SN progenitor scenarios, and the future of SN 1972E and SN 1895B as they transition to the SNR stage."
],
[
"Background: SN 1895B and SN 1972E",
"Two independent Type Ia SNe, SN 1895B and SN 1972E, occurred within a century of each other in the nearby blue compact dwarf galaxy, NGC 5253.",
"NGC 5253 is located within the M83/Centaurus A Group, and throughout this work we adopt the Cepheid distance of 3.15 Mpc from [36]This distance includes a metallicity correction factor.. NGC 5253 is currently undergoing a starburst phase with a compact, young star forming region at its center [77], thought to be triggered by an earlier interaction with M83 [123].",
"SN 1895B (J2000 Coordinates: RA $=$ 13:39:55.9, Dec $=$ $-$ 31:38:31) was discovered by Wilhelmina Fleming on December 12, 1895 from a spectrum plate taken on July 18, 1895 [89].",
"Throughout this manuscript, we adopt the discovery date as the explosion epoch for our analysis, although the explosion likely occurred some days earlier.",
"Three direct image plates and one spectrum plate taken within the first five months of the SN are available.",
"Re-analysis of these plates with a scanning microdensiometer have resulted in a light curve that is consistent with a normal Type Ia SN $\\sim $ 15 days after maximum light [105].",
"From this analysis, it is estimated that SN 1895B peaked at a visual magnitude of $<8.49\\pm 0.03$ mag.",
"Significantly more information is available for SN 1972E, which was the second-brightest SN of the 20th century.",
"Discovered on May 13, 1972 (J2000 coordinates: RA $=$ 13:39:52.7, Dec $=$ $-$ 31:40:09), SN 1972E was identified just prior to maximum light [64], peaked at a visual of 8.5 mag and was observed for 700 days after initial discovery [58], [1], [10].",
"As with SN 1895B, we adopt the discovery date as the explosion date for the analysis belowWe note that differences of $\\sim $ 1 month in adopted explosion epoch will not influence our results, as our observations take place tens to 100 years after the explosion..",
"The exquisite late-time coverage of SN 1972E at optical wavebands played a key role in our understanding of the link between Type Ia SNe and nucleosynthesis [120] as it was shown that the energy deposition during the optical-thin phase was consistent with the radioactive decay of $^{56}$ Ni and $^{56}$ Co [3].",
"SN 1972E is now considered an archetype for Type Ia SN, and was one of the events used to define the spectroscopic features of “Branch normal” events [12].",
"Figure: Top: Radio image of NGC 5253 from a December 2016 VLA observation at 8.35 GHz, with the positions SN 1972E and SN 1895B marked.",
"The bright central radio source in NGC 5253 is a compact star forming region in the galaxy core .",
"The synthesized beam is drawn as an ellipse in the lower left corner.",
"Bottom: Close-ups of the regions surrounding each SN.After the initial optical light faded, neither SN 1895B nor SN 1972E has been detected at any wavelength.",
"Observations of NGC 5253 with the Chandra X-ray observatory yielded non-detections at the locations of both SNe [116].",
"In the radio, two upper limits for the SNe have been published to date, which are listed in Table , below.",
"[23] observed both SN 1895B and SN 1972E with the VLA for 3 hours at 1.45 GHz in April 1981.",
"They report non-detections with upper limits of 0.9 mJy for both SNe.",
"Subsequently, [33], reported upper limits on the radio flux from both SNe of 0.15 mJy based on 9.1 hours of VLA data obtained in November 1984, also at 1.45 GHz.",
"Modeling these limits assuming a CSM with a $\\rho \\propto r^{-2}$ density profile, [33] find upper limits on mass-loss rates of the progenitor systems of SN 1972E and SN 1895B of $<8.60\\times 10^{-6}$ M$_{\\odot }$ yr$^{-1}$ and $<7.2\\times 10^{-5}$ M$_{\\odot }$ yr$^{-1}$ , respectively.",
"These mass-loss rate estimates, which assumed wind speeds of 10 km s$^{-1}$ , are not strongly constraining in the context of Type Ia SN progenitors, ruling out only a few specific Galactic symbiotic systems [107].",
"lccccccr Observation Details for Archival VLA Data 0pt Observation Project Configuration Integration Central Freq.",
"Receiver Bandwidth Referencea Date Code (hr) (GHz) (MHz) Apr.",
"1981b N/A BnA 3.0 1.45 L 12.5 (1) Nov. 1984b AB0305 A 9.1 1.45 L 25 (2) Oct. 13, 1991b AB0626 A 1.36 8.40 X 50 This Work Feb. 18, 1999b AN0081 D 3.60 8.30 X 25 This Work May 5, 2012 12A-184 CnB 1.16 5.85 C 2048 This Work Mar.",
"23, 2016 TDEM0022 C 0.66 9.00 X 4096 This Work Dec. 16, 2016 16B-067 A 0.75 8.35 X 4096 This Work a(1) [23]; (2) [33] bHistorical VLA observations These two SNe are worthy of further study at radio wavelengths for several reasons.",
"First, at 3.15 Mpc, SN 1972E and SN 1895B are among the closest known extragalactic SNe.",
"Second, while radio observations of SNe years after explosion are generally not constraining in the content of a $\\rho \\propto r^{-2}$ wind environments, even comparatively shallow limits can provide useful constraints on the presence of a constant density CSM [103] and low-density CSM shells [44]—physical models that were not considered in the analysis of [33].",
"Third, NGC 5253 has been observed multiple times by the VLA since 1984, and these observations are currently in the VLA archive.",
"This gives us the unique opportunity of being able to set limits at multiple epochs for two SNe, as the shockwave has traversed a wide range of radii—and potentially, CSM environments."
],
[
"VLA Observations",
"For our study, we examined all archival VLA observations of the galaxy NGC 5253.",
"While over 85 observations of NGC 5253 have been obtained since 1979, the location of SN 1972E [54] is too far to be visible in higher frequency images centered on the galaxy core.",
"As a result, we initially restrict ourselves to 24 observations that contain either SN 1895B or SN 1972E within their primary beam, and occurred in C and X bands (4-12 GHz).",
"Subsequently, we further restrict ourselves to observations that can provide constraints on constant density CSM surrounding the SNe, as described by the model outlined in Section REF .",
"In particular, while a higher density CSM will lead to brighter overall radio emission, it will also cause the SN to enter the Sedov-Taylor phase (and therefore fade at radio wavelengths) at an earlier epoch.",
"Thus, in the context of this physical model, there is a maximum radio luminosity that can be achieved at a given time post-explosion.",
"This translates to a minimum image sensitivity that must be achieved for a given intermediate-aged SN.",
"For the cases of SN 1972E and SN 1895B, we find that we require radio images with RMS noise less than 85 mJy/beam.",
"After performing a number tests with historical VLA data of NGC 5253, we find that observations with total on-source integration times less than 20 minutes do not meet this threshold.",
"After applying these cuts, we are left with two historical (pre-2010) VLA observations in addition to the observations published in [23] and [33], and three observations taken with the upgraded Karl G. Jansky VLA (post-2012).",
"The information for each observation including date, project code, exposure time, configuration, frequency, and band are shown in Table .",
"Overall, these observations provide constraints on the radio luminosity from SN 1972E and SN 1895B between 9$-$ 44 years and 86$-$ 121 years post-SN, respectively."
],
[
"Data Reduction and Imaging",
"All VLA data were analyzed with the Common Astronomy Software Applications (CASA; [76]).",
"For the 2012 and 2016 data, taken with the ungraded VLA, CASA tasks were accessed through the python-based pwkit packageavailable at: https://github.com/pkgw/pwkit [126], while historical data was reduced manually.",
"We flagged for RFI using the automatic AOFlagger [83].",
"After calibration, we imaged the total intensity component (Stokes I) of the source visibilities, setting the cell size so there would be 4$-$ 5 pixels across the width of the beam.",
"All data was imaged using the CLEAN algorithm [22], and for post-2010 data we utilize mfsclean [91] with nterms $=$ 2.",
"Due to the large distance of SN 1972E from the galaxy center (and thus image pointing) we also image using the w-projection with wprojplanes $=$ 128.",
"Finally, images were produced setting robust to 0 and for all observations, we used the flux scaling as defined by [87].",
"llccccc Radio Observations of SN 1972E and SN 1895B Supernova Obs.",
"Time Since Central RMS Luminosity Density Date Explosiona Freq.",
"Noise Upper Limit Upper Limitb (UT) (yrs) (GHz) ($\\mu $ Jy/beam) (ergs/s/Hz) (cm$^{-3}$ ) 1895B Apr.",
"1981 85.8 1.45 900 3.2E+25 4.2 Nov. 1984 89.3 1.45 150 5.3E+24 1.0 Oct. 1991 96.3 8.40 820 2.9E+25 16 Feb. 1999 103.6 8.30 33 1.2E+24 1.1 May 2012 116.9 5.85 33 1.2E+24 0.8 Mar.",
"2016 120.7 9.00 77 2.7E+24 2.1 Dec. 2016 121.5 8.35 25 8.9E+23 0.8 1972E Apr.",
"1981 8.9 1.45 900 3.2E+25 14 Nov. 1984 12.5 1.45 150 5.3E+24 2.6 Oct. 1991 19.4 8.40 270 9.6E+24 15 Feb. 1999 27.8 8.30 26 9.2E+23 1.7 May 2012 40.0 5.85 26 9.2E+23 1.0 Mar.",
"2016 43.9 9.00 40 1.4E+24 2.0 Dec. 2016 44.6 8.35 17 6.0E+23 0.9 aAssuming the explosion epochs adopted in Section .",
"bAssuming a constant CSM density, n$_0$ , and the fiducial model described in Section REF .",
"For all observations, the center of the radio image is dominated by the bright central radio source in NGC 5253 located at RA $=$ $13\\rm {h}39\\rm {m}55.96\\rm {s}$ and Dec. $=$ $-31^{\\circ }38^{\\prime }24.5^{\\prime \\prime }$ (J2000; [5]).",
"An example images can be seen in Figure REF , with the positions of SN 1972E and SN 1895 marked for reference."
],
[
"Flux Limits",
"We did not detect radio emission at the location of either SN 1895B or SN 1972E in any of our images.",
"To obtain flux upper limits, we measured the RMS noise at the locations of the SNe using the imtool program within the pwkit package [126].",
"These values are listed in Table REF .",
"Throughout this manuscript, we will assume 3$\\sigma $ upper limits radio flux from SN 1972E and SN 1895B.",
"In general, the upper limits obtained on the flux from SN 1972E were a factor of $\\sim $ 2$-$ 3 deeper than for SN 1895B.",
"This primarily due to that fact that SN 1895B occurred significantly closer to the radio-bright center of the galaxy (see Figure REF ).",
"The deepest individual flux limits for both SNe were provided by the December 2016 observation, with 3$\\sigma $ upper limits of F$_{\\nu }$ $<51\\ \\mu $ Jy/beam and F$_{\\nu }$ $<75\\ \\mu $ Jy/beam for SN 1972E and SN 1895B, respectively."
],
[
"Radio Luminosity Limits: Comparison to Previously Observed SNe and SNRs",
"Upper limits on the radio luminosity to each SNe, computed using a distance of 3.15 Mpc to NGC 5253, are listed in Table REF .",
"We find limits ranging from $\\lesssim $ 3 $\\times 10^{25}$ erg s$^{-1}$ Hz$^{-1}$ in 1981 to $\\lesssim $ 6 $\\times 10^{23}$ erg s$^{-1}$ Hz$^{-1}$ in 2016.",
"These limits are shown in Figure REF , along with observations of previously observed SNe and SNRs for comparison.",
"Each SN or SNR is plotted in a different color, while symbols indicate the frequency of each observation.",
"Upper limits are designated by black arrows.",
"Figure REF demonstrates the unique timescales and luminosities probed by SN 1972E and SN 1895B.",
"In one of the most thorough reviews of radio emission from Type Ia SNe to date [19] provided observations of 85 Type Ia SNe within 1 year post-explosion.",
"The deepest limits cited in [19] correspond to luminosities of $\\sim $ 3$-$ 6 $\\times $ $10^{23}$ erg/s/Hz for SN 2014J between 84 and 146 days post-explosion, and $\\sim $ 4$-$ 6 $\\times $ $10^{24}$ erg/s/Hz for SN 2012cg between 43 and 216 days post-explosion.",
"These are comparable to the limits obtained for SN 1972E and SN 1895B, but at a significantly shorter time post-explosion.",
"In Figure REF , we plot the Type Ia SNe with the deepest luminosity limits obtained between 3 months and 1 year post-explosion [19], [84].",
"Figure: Radio luminosity upper limits for the intermediate-aged Type Ia SN 1972E (blue) and SN 1895B (red) spanning three decades using data from this work (see Table and previous observations.",
", ).",
"Also shown, for comparison, are observed radio luminosities and luminosity upper limits (black arrows) for Galactic SNRs and other extagalactic Type Ia SNe for a range of times post-SN , , , .",
"We have distinguished the different observed frequency bands present in this data set as different symbols: squares correspond to L-band (1-2 GHz), diamonds to C-band (4-8 GHz), and circles to X-band (8-12 GHz) observations.",
"For illustrative purposes, we have included solid lines to represent two potential model radio light curves expected for a SN blast wave expanding into a uniform medium with a density of 1 cm -3 ^{-3} (blue) and 10 cm -3 ^{-3} (orange), assuming our baseline S17 model described in Section .",
"See Table for the precise density limits that can be derived from each point.While observations of SNe and SNRs within the Milky Way and other Local Group galaxies can provide deeper constraints on the radio luminosity from Type Ia SNe, such observations have typically been obtained at longer timescales post-explosion.",
"This is demonstrated in Figure REF , where we also plot a radio upper limit for SN 1885A in M31 and observed radio luminosities for the Galactic SNRs G1.9+0.3, Tycho, and Kepler, all associated with events of thermonuclear origin [28], [35], [92], [99], [93].",
"By co-adding VLA observations in the 4-8 GHz frequency range, [103] produced a deep radio image with RMS noise of 1.3 $\\mu $ Jy/ beam at the location of SN 1885A in M31.",
"Some radio emission with 2.6$\\sigma $ confidence is also present, but the association with SN 1885A for this emission is uncertain due to the large amount of diffuse radio emission in the central regions of M31 where the SN is located.",
"The resulting luminosity upper limit of 8.5$\\times 10^{21}$ erg s$^{-1}$ Hz$^{-1}$ at 127 years post-explosion is approximately two orders of magnitude deeper and at timescales just beyond those probed by SN 1895B.",
"In comparison, the Galactic SNR G1.9+0.3 was detected at 1.4 GHz with a flux of 0.74 $\\pm $ 0.04 Jy in 1993 [21], and 0.935 $\\pm $ 0.047 Jy in 2008 [40], corresponding to ages of $\\sim $ 125$-$ 140 years post-explosion [92], [40].",
"Based on a high absorbing column density in observed X-ray observations, [92] place the distance to G1.9+0.3 to be $\\sim $ 8.5 kpc, with corresponding radio luminosities of $\\sim 10^{23}$ erg/s/Hz.",
"Finally, the Catalog of Galactic Supernova Remnants [39], lists 1 GHz fluxes of 56 Jy and 19 Jy for Tycho's SNR and Kepler's SNR, respectively.",
"At estimated distances of 2.8 kpc [61] and 6.4 kpc [94], respectively, these translate to radio luminosities of $\\sim 5 \\times 10^{23}$ erg/s/Hz.",
"However, we emphasize that these SNe are over 400 years old, and have transitioned to the SNR phase.",
"Given that the observed luminosities of these Galactic intermediate-aged Type Ia SNe/SNRs are below the luminosity upper limits obtained for SN 1972E and SN 1895B, we also calculate the flux densities that they would be observed with at the distance of NGC 5253.",
"We find that the observed flux densities of G1.9+0.3-like, Kepler-like, and Tycho-like SNRs would be $\\sim $ 2 $\\mu $ Jy, $\\sim $ 15 $\\mu $ Jy, and $\\sim $ 26 $\\mu $ Jy in NGC 5253, respectively.",
"These flux levels for Kepler's and Tycho's SNR are within the sensativity limits that can be achieved through dedicated JVLA observations, and the implications for the future evolution of SN 1972E and SN 1895B are discussed in Section , below."
],
[
"Constraints on a Uniform Density CSM",
"The radio emission from a SN expanding into a relatively low density medium is described by a synchrotron spectrum.",
"As the shockwave expands into the CSM, electrons are accelerated to relativistic speeds and interact with shock-amplified magnetic fields [14], [17].",
"Here, we use a quantitative model for the radio luminosity from a SN blast wave expanding into a constant density CSM and our luminosity upper limits to place constraints on the density of the media surrounding the progenitors SN 1972E and SN 1895B."
],
[
"Radio Light Curve Model",
"We adopt the radio luminosity model outlined in [101], based on the radio synchrotron formalism of [15].",
"This model self consistently treats the evolution of the SN from early (ejecta dominated) to late (Sedov-Taylor) phases, and is therefore ideal for the intermediate-aged SNe considered here.",
"While we refer the reader to S17 for a complete model description, we summarize salient features here.",
"The luminosity of the radio emission from a Type Ia SN will depend on the density profiles of the outer SN ejecta and CSM, the ejecta mass (M$_{\\rm {ej}}$ ) and kinetic energy (E$_{\\rm {K}}$ ) of the SN explosion, the power spectrum of the relativistically accelerated electrons, and the fraction of post-shock energy contained in amplified magnetic fields and relativistic electrons ($\\epsilon _{\\rm {B}}$ and $\\epsilon _e$ , respectively).",
"S17 use standard model assumptions in many cases: adopting a power-law density profile with a “core-envelope” structure for the SN ejecta as defined by [121] with $\\rho \\propto v^{-n}$ and $n=10$ in the outer ejecta [75], a constant density CSM, and a distribution of relativistic electrons of the form $N(E) \\propto E^{-p}$ .",
"However, S17 deviate from standard assumptions in their treatment of the magnetic field amplification.",
"In most analytic models of SN radio light curves, $\\epsilon _{\\rm {e}}$ and $\\epsilon _{\\mathrm {B}}$ are free parameters, assumed to be constant.",
"This is generally considered to be one of the most significant uncertainties in converting observed radio luminosities to CSM densities [47], [48].",
"In contrast, S17 develop a new parameterization for $\\epsilon _{\\mathrm {b}}$ , as a scaling of the Alfven Mach number of the shock and the cosmic ray acceleration efficiency, based on the results of numerical simulations of particle acceleration [13].",
"$\\epsilon _{\\mathrm {B}}$ is therefore determined as a function of time and equipartition is not assumed.",
"As a result, the models of S17 contain five free parameters: $p$ , $\\epsilon _e$ , $M_{\\rm {ej}}$ , and $E_{\\rm {K}}$ , and $n_0$ (the density of the CSM).",
"Given their ages and the analytic models for SN blast wave dynamics of [121], SN 1972E and SN 1895B should still be in the free-expansion (ejecta-dominated) phase during the VLA observations described above.",
"During this phase, the radius and velocity of the forward shock can be described by: $R_s = (1.29\\ \\mathrm {pc})\\ t_2^{0.7}\\ E_{\\mathrm {51}}^{0.35}\\ n_0^{-0.1}\\ M_{\\mathrm {ej}}^{-0.25}$ and $v_s = (8797\\ \\mathrm {km/s})\\ t_2^{-0.3}\\ E_{\\mathrm {51}}^{0.35}\\ n_0^{-0.1}\\ M_{\\mathrm {ej}}^{-0.25}$ where $t_2 = t/(100\\ \\rm {yrs})$ , is the time post-explosion, $E_{\\rm {51}} = E/(10^{51}\\ \\rm {ergs})$ is the kinetic energy of explosion, $M_{\\rm {ej}} = M/ (1\\ \\rm {\\rm {M_{\\odot }}})$ is the ejecta mass, and $n_0$ is the ambient medium density in units of 1 cm$^{-3}$ .",
"Using these relations, and equations A1-A11 in S17, we can then calculate the radio luminosity of a Type Ia SN interacting with a uniform density CSM under the assumption that the resulting synchrotron emission is optically thin and the forward shock will dominate the radio luminosityPlease note corrections to these equations provided in the erratum [102]..",
"These assumptions hold for the low density ambient media we consider here.",
"In Figure REF we plot example S17 light curves for two CSM densities (1 and 10 cm$^{-3}$ ), assuming a fiducial “baseline” model with $M_{\\rm ej} =$ 1.4 M$_{\\odot }$ , $E_{\\rm K} = 10^{51}$ erg, $p=3$ , and $\\epsilon _e$ = 0.1.",
"The latter two values are widely adopted in the literature and are motivated by radio observations of SNe and gamma-ray bursts [17].",
"However, we emphasize that both $p$ and $\\epsilon _e$ may vary based on the source population and $\\epsilon _e$ , in particular, is subject to significant uncertainty.",
"Observations of young SNRs, such as Tycho, are consistent with a very small $\\epsilon _e$ ($\\lesssim $ 10$^{-4}$ ; [79], [7], [6]), while the luminosity function of older SNRs in local galaxies requires an intermediate value ($\\epsilon _e$ $\\approx $ 10$^{-3}$ ; S17).",
"Similarly, while young radio SNe are often consistent with $p=3$ [17], the spectral index of young SNRs is usually in the range of $p=2.0-2.4$[32].",
"We have chosen our baseline values for $p$ and $\\epsilon _e$ both because SN 1972E and SN 1895B should still be in the ejecta-dominated phase, and to allow for direct comparison to the observational results of [19] and the hydrodynamic models of SN-CSM shell interaction described in Section REF .",
"Effects of varying these parameters will be examined below.",
"From the baseline S17 models presented in Figure REF it is clear the predicted radio luminosity increases steadily during the free-expansion phase—over a timescale of centuries—thus allowing later observations to place deeper constraints on the density of the ambient medium.",
"This is in sharp contrast to a $\\rho \\propto r^{-2}$ wind environment, where the predicted radio luminosity fades with time as a result of the decreasing density [19].",
"In the uniform CSM scenario, the radio light curve peaks a few hundred years after SN, around the Sedov time, and subsequently the radio luminosity declines throughout the Sedov-Taylor phase (S17)."
],
[
"Limits on Uniform Density CSM",
"We have applied the radio model of S17 to the luminosity upper limits derived for SN 1972E and SN 1895B (Section REF ; Figure REF ) in order to place limits on the density of any uniform medium surrounding the SNe.",
"In Table REF we list the density upper limits that result when assuming our baseline model described above ($M_{\\rm {ej}} =$ 1.4 M$_{\\odot }$ , $E_{\\rm {K}}=$ 10$^{51}$ erg, $p = 3$ , and $\\epsilon _e$ = 0.1).",
"For each point, we run a large grid of S17 models and the quoted density upper limit corresponds to the curve which goes directly through the 3$\\sigma $ luminosity limit plotted in Figure REF ).",
"These density upper limits, which were computed assuming a mean molecular weight of 1.4, range from $\\sim $ 1 to $\\sim $ 15 cm$^{-3}$ , depending on the epoch, frequency, and sensitivity of the observation.",
"In the top panel of Figure REF , we plot example 8 GHz light curves for this baseline model at a range of CSM densities, along with the X-band (8-10 GHz) upper limits for SN 1972E and SN 1895B.",
"For both SNe, our deepest constraints on the density of the ambient medium come from the Dec. 2016 observations, due to a combination of their deeper sensitivity and longer time post-explosion.",
"Assuming our baseline model, these limits correspond to n$_0$ $\\lesssim 0.8$ cm$^{-3}$ for SN 1895B, and n$_0$ $\\lesssim 0.9$ cm$^{-3}$ for SN 1972E.",
"In Figure REF we plot these density limits in comparison to those for SN 1885A, SNR G1.9+0.3, and the $\\sim $ 200 observations of 85 extra-galactic Type Ia SN from [19].",
"For SN 1885A and G1.9+0.3 we have taken the observed luminosities from [103] and computed new density limits assuming our baseline model, as [103] adopted significantly different values of p$=$ 2.2 and $\\epsilon _e = 10^{-4}$ .",
"Given both the small distance to NGC 5253 and the fact that the radio luminosity of a SN expanding into a uniform density CSM will continue to increase over time, we are able to place limits on the CSM density surrounding SN 1972E and SN 1895B that are several orders of magnitude lower than the bulk of the population presented in [19].",
"In Figure REF , we also examine the influence on our derived density upper limits if we deviate from our baseline model described above.",
"In the middle panel we plot the 8 GHz light curves that result if we consider an ejecta mass of $M_{ej} = 0.8\\ M_{\\odot }$ , representative of sub-M$_{ch}$ explosions [112].",
"For these parameters, our best ambient density constraints correspond to n$_0$ $< 0.38$ cm$^{-3}$ (SN 1972E) and n$_0$ $< 0.31$ cm$^{-3}$ (SN 1895B).",
"Overall, assuming a sub-M$_{ch}$ explosion yields upper limits on the CSM density that are a factor of $\\sim $ 2.5 more constraining (assuming $E_{\\rm {K}}=$ is held fixed at 10$^{51}$ erg).",
"Finally, in the lower panel of Figure REF we highlight the influence of varying the adopted value for $\\epsilon _e$ .",
"Lowering the value of $\\epsilon _e$ by a factor of 10 will yield a predicted luminosity for a given density that is a factor of 10 fainter, and a density constraint for a given luminosity upper limit that is a factor of $\\sim $ 7 weaker (for $p = 3$ ).",
"If we adopt $\\epsilon _e$ $=$ 10$^{-4}$ and p $=$ 2.2 as assumed by [103] when modeling SN 1885A (based on values consistent with young SNRs), our best ambient density constraints become $\\sim $ 17 cm$^{-3}$ (SN 1972E) and $\\sim $ 16 cm$^{-3}$ (SN 1895B).",
"In this case, the impact of a lower adopted $p$ value partially cancels the effect of a dramatically lower $\\epsilon _e$ .",
"The uniqueness of a data set spanning two decades also allows us to place constraints on the density of the CSM as a function of radius from the progenitor star.",
"In Figure REF we plot the uniform density CSM limits obtained for each SN 1972E and SN 1895B observation, assuming our baseline S17 model.",
"On the top axis we also provide the radius probed as a function of time, assuming a constant CSM density of 1 cm$^{-3}$ .",
"We note that the exact radius probed by each point will vary depending on the density of the CSM (see Equation REF ).",
"These densities and radii are similar to those observed in several known CSM shells.",
"For illustrative purposes, we have provided a simple density profile for two such examples: the inner ring of SN 1987A, and the planetary nebula Abell 39.",
"These density profiles should be associated with the top axis of Figure REF , which lists the radius from the SN progenitor star.",
"The radius and density of for inner ring of SN 1987A are obtained from [74], who provide both upper and lower limits on the ring density (plotted as dashed and dotted orange lines, respectively).",
"For the planetary nebula Abell 39, the radius and density of the shell were obtained via spectroscopic analysis from [53].",
"We chose Abell 39 because it is the simplest possible planetary nebula: a one-dimensional projected shell that is used as a benchmark for numerical modeling of these structures [53], [25].",
"In the case of Abell 39, the shell has a radius of 0.78 pc, a thickness of 0.10 pc, and a density of 30 cm$^{-3}$ [53].",
"We have plotted a simple step function where the density is 2 cm$^{-3}$ outside of the shell, consistent with the number density observed within the shell [118].",
"This illustrative comparison highlights that even the less sensitive luminosity limits obtained for SN 1972E and SN 1895B are useful in constraining the presence of CSM shells.",
"We consider a more detailed model for the radio emission from a SN interacting with CSM shells below.",
"Figure: The upper limits on density, in cm -3 ^{-3}, obtained for SN 1972E (blue triangles) and SN 1895B (red) assuming our baseline model.",
"The top axis shows the radius probed by each observation, assuming a constant density of 1 cm -3 ^{-3}.",
"For reference, we have also provided a simple density profile for the planetary nebula Abell 39 (green dashed line; ), and the upper and lower limits on the density of the inner ring of SN 1987A (orange dashed and dotted lines, respectively; )."
],
[
"Constraints on the Presence of CSM Shells",
"In addition to placing deep limits on the density of uniform CSM, the multi-epoch nature of our radio observations allow us to investigate the possibility of shells of CSM surrounding the progenitors of SN 1972E and SN 1895B.",
"Here we outline a parameterized radio light curve model for SN ejecta interacting with spherical shells of finite extent, the applicability of these models to the regimes probed by our observations of SN 1972E and SN 1895B, and the types of shells that can be ruled out for these systems."
],
[
"Radio Light Curve Model: Shell Interaction",
"To constrain the presence of CSM shells surrounding the progenitors of SN 1972E and SN 1895B we use the parameterized light curve models of [44] (H16, hereafter).",
"H16 model the interaction of expanding SN ejecta with a CSM shell of constant density using the Lagrangian hydrodynamics code of [96] and compute radio synchrotron light curves based on the gas property outputs of these simulations.",
"While these models can be run for a wide variety of ejecta and CSM configurations, for ease of parameterization, H16 also produced a set of fiducial models for a M$_{\\rm {ej}}$ = M$_{\\rm {Ch}}$ = 1.38 M$_\\odot $ and E$_{\\rm {K}}$ = 10$^{51}$ erg Type Ia SN, with a physical set-up that is based off of the self-similar formalism of [14].",
"Specifically, for this fiducial model set, H16 adopt power law density profiles for both the SN ejecta and CSM, and set the initial conditions of the simulations such that the initial contact discontinuity radius equals the contact discontinuity radius at the time of impact from [14].",
"Following [16] and [56], the SN ejecta is defined by a broken power law with shallow and steep density profiles ($\\rho $ $\\propto $ r$^{-1}$ vs. $\\rho $ $\\propto $ r$^{-10}$ ) for material interior and beyond a transition velocity, $v_{t}$ , respectively.",
"The CSM is defined as a shell with a finite fixed width, $\\Delta $ R, and constant density, n. In constructing radio synchrotron light curves from the outputs of this fiducial set of models H16 assume that $\\epsilon _e = \\epsilon _B = 0.1$ and that that the accelerated electrons possess a power-law structure with respect to their Lorentz factor, $\\gamma $ , of the form $n(\\gamma ) \\propto \\gamma ^{-p}$ with $p=3$ .",
"It is also assumed that the radio emission is dominated by the forward shock and that the resulting emission is optically thin to synchrotron self-absorption, assumptions that were shown to be valid for their model set.",
"With these assumptions, H16 find a “family” of resulting radio synchrotron light curves that can be defined by three key parameters: $r_{1}$ : the inner radius of the CSM shell.",
"$n$ : the density of the CSM shell.",
"$f$ : the fractional width of the CSM shell ($\\Delta $ R/$r_{1}$ ).",
"H16 provide analytic expressions describing radio light curves as a function of these three parameters.",
"In Figure REF we plot the resulting radio light curves (lower panels) for various CSM shells (top panels) as each of $r_1$ , $n$ , and $f$ are varied individually.",
"Also shown, for context, are the luminosity upper limits measured for SN 1972E and the radio light curve for a 0.1 cm$^{-3}$ constant density CSM from S17.",
"Overall, the resulting radio light curves are strongly peaked in time, with a rapid decline occurring once the forward shock reaches the outer radius of the CSM shell.",
"For a constant shell density and fractional width (left panels), adjusting the inner radius of the shell will primarily influence the time of impact and therefore the onset of radio emission.",
"Adjusting the density of the CSM shell (center panels) will primarily influence the peak luminosity of the resulting radio emission—although the onset of radio emission will also be delayed slightly for higher density shells (see below).",
"Finally, increasing the fractional width of the shell (right panels) will increase both the overall timescale and peak luminosity of the resulting radio signature as the interaction continues for a longer time period.",
"Thus, a given observed data point will constrain the presence of a thick shell over a larger range of $r_1$ , compared to thin shells with similar densities."
],
[
"Applicability to SN 1972E and SN 1895B",
"H16 first developed and applied their fiducial models to investigate the case of low-density CSM shells located at radii $\\lesssim $ a few $\\times 10^{16}$ cm, whose presence would manifest in radio light curves within the first $\\sim $ 1 year post-explosion.",
"We now examine whether the assumptions made in H16 are applicable for CSM shells that would manifest at the timescales of the observations of SN 1972E and SN 1895B described above.",
"Figure: Visual representation of the CSM shell inner radii (r 1 _1) and densities (n) probed by observations of SN 1972E and SN 1895B (aqua and red boxes, respectively) and regions where the model assumptions of H16 are valid.",
"The shaded blue region highlights the parameter space where the assumption that the CSM impacts the outer portion of the SN ejecta is violated.",
"Violet lines indicate densities for which the total shell mass equals the total mass in the outer ejecta (∼\\sim 0.3M ⊙ _\\odot ) for fiducial thin, medium, and thick shells.",
"Yellow lines designate the time when the SN ejecta would impact the CSM shell in the models of H16.",
"See text for details.Figure: Grid of H16 CSM shell models tested against observations of SN 1972E.",
"Red squares designate the shell radii and densities ruled out for representative thin (left panel), medium (center panel), and thick (right panel) shells.",
"The blue shaded area designates the region where H16 model assumptions are violated.",
"In each panel dotted, dashed, dot-dashed, and triple-dot-dashed lines designate shells with total masses 0.01, 0.05, 0.1, and 0.3 M ⊙ _\\odot , respectively.",
"For all shell thicknesses, we can rule out shells with masses down to 0.005--0.01 M ⊙ _\\odot for specific radii, and for medium and thick shells our observation exclude the presence of essentially any shell with masses >>0.05 M ⊙ _\\odot at radii between 10 17 ^{17} and 10 18 ^{18} cm.",
"See text for further details.The main assumption that may be violated for the case of shells at the radii probed by the observations of SN 1972E and SN 1895B is that the CSM impacts the outer portion of the SN ejecta, which has a steep density profile.",
"For this to hold true, first the total mass swept up by the SN shock prior to impacting the shell should not approach the mass in the outer SN ejecta.",
"For the broken power-law ejecta profile adopted in H16, $\\sim $ 2/9 of the SN ejecta mass is located in the outer ejecta, corresponding to $\\sim $ 0.3 M$_\\odot $ for a Chandrasekhar mass explosion.",
"H16 assume that the shell occurs essentially in a vacuum.",
"If we instead assume a low density medium interior to the shell of $<$ 0.1 cm$^{-3}$ [4] we find that that mass of the internal material swept up should be $\\lesssim $ 0.002 M$_{\\odot }$ for the shell radii probed by the observations of SN 1972E.",
"Thus, the CSM shell density and radius are the primary determinants of whether the interaction is with the outer SN ejecta.",
"In setting the initial conditions of their simulations, H16 assume that the “impact”, and hence the beginning of the radio light curve, occurs when the ratio of the CSM and SN ejecta density at the contact discontinuity reaches a specific value ($\\rho _{\\rm {CSM}} = 0.33\\, \\rho _{\\rm {ej}}$ ).",
"This requirement is the cause of the shift in radio emission onset time when considering shells of various densities at a fixed radius.",
"For denser shells, the H16 impact will occur when a slightly denser—more slowly moving—portion of the SN ejecta reaches $r_1$ .",
"Thus, at every radius, there is a density that corresponds to 0.33$\\,\\rho _{\\rm {ej,vt}}$ where $\\rho _{\\rm {ej,vt}}$ is the density of the ejecta at the transition velocity, $v_t$ , between the outer and inner density profiles.",
"This is the maximum density of a CSM shell at this radius that does not violate the model assumption that the impact occurs in the outer portion of the SN ejecta.",
"Because the density of the expanding SN ejecta decreases with time, as we consider shells at larger and larger radii, this model assumption will break down for lower and lower densities.",
"Assuming CSM shells with fractional widths between 0.1 and 1.0, we find that the observations of SN 1972E and SN 1895B will probe CSM shells with inner radii ranging between [1$-$ 15] $\\times $ 10$^{17}$ cm and [1.5$-$ 4.0] $\\times $ 10$^{18}$ cm, respectively.",
"In Figure REF we show these ranges in comparison to the model assumption constraints described above.",
"For SN 1972E, we find that there are large swaths of parameter space that can be probed using the parameterized light curves of H16.",
"However, for SN 1895B, we find that only shells with very low densities ($\\lesssim $ 10 cm$^{-3}$ ) will not violate model assumptions.",
"Finally, we note one other requirement based on the assumption that the interaction primarily occurs in the outer SN ejecta: the total mass in the CSM shell should not exceed the total mass in the outer SN ejecta ($\\sim $ 0.3 M$_{\\odot }$ ).",
"Parameter space where this requirement is met and violated are discussed in Section REF , below."
],
[
"CSM Shell Models Excluded",
"Finding that the H16 model assumptions are valid over a portion of the parameter space of CSM shells probed by SN 1972E and SN 1895B, we run large grids of parameterized light curve models for comparison with our observations.",
"For SN 1972E, we run 3,200 models for shell radii spanning $r_1 = [1-15] \\times 10^{17}$ cm and shell densities spanning $n = 1-16,000$ cm$^{-3}$ ($\\sim 2.3 \\times 10^{-24}$ to $3.7 \\times 10^{-20}$ g cm$^{-3}$ ).",
"This grid is chosen to encompass the full range of densities that can be probed without violating the the model assumptions described above.",
"For the highest densities considered these models assumptions are only valid at the smallest radii (see Figures REF and REF ).",
"For SN 1895B we consider 450 models spanning shell radii of $r_1 = [1.6-4] \\times 10^{18}$ cm and shell densities of $n = 1-15$ cm$^{-3}$ .",
"For each event, we run models for three representative shell widths, chosen to span the range of astrophysical shells predicted surrounding some putative Type Ia SN progenitors (see Section ).",
"Specifically, we consider $f$ values of: [leftmargin=1.5cm] $f=0.15$ : A thin shell based on the based on the observed width of the Abell 39 planetary nebula, and in line with widths predicted for some material swept up in nova outbursts [78].",
"$f=0.33$ : A medium thickness shell based on models of “nova super shells” [27].",
"$f=1.00$ : A representative thick shell.",
"For each combination of $f$ , $r_1$ , and $n$ , we compute the resulting radio light curve at the frequencies of all of our observations and determine whether any of the flux upper limits described above rule out a shell with those parameters.",
"Results from this process for SN 1972E are shown in Figure REF .",
"Shells excluded by the data are displayed in red.",
"For reference, we also plot lines that indicate constant shell masses of 0.01, 0.05, 0.1, and 0.3 M$_\\odot $ for each shell thickness.",
"Shells with total masses $>$ 0.3 M$_\\odot $ violate the H16 requirement that the total shell mass be less than the mass in the outer SN ejecta.",
"Regions where the condition that the initial interaction occurs in the outer SN ejecta is violated are also shown in blue.",
"For the medium-thickness shells considered here, theses two conditions are violated at very similar shell densities, while for thick-shells the constraint that the total shell mass be less than 0.3 M$_\\odot $ is the more restrictive requirement (see Figure REF ; right panel).",
"Each observed luminosity limit leads to a diagonal line of excluded CSM shells in the density-inner radius plane.",
"This is due to the interplay between the shell density and the onset time of strong interaction that leads to radio emission (see above).",
"For thin shells (left panel) these individual “tracks” of excluded models are visible, while for thicker shells (center and right panels) they broaden and overlap.",
"Thus, excluding a complete set of thin shell models for an individual SN progenitor would require higher cadence radio observations than those available for SN 1972E.",
"In contrast, for thicker shells, we are primarily limited by the depth of individual observations.",
"Overall, for SN 1972E, we can rule out CSM shells down to masses of $\\sim $ 0.01 M$_\\odot $ at a range of radii, which vary depending on the shell thickness.",
"We can also rule out the presence of all thick shells with masses $\\gtrsim $ 0.05 M$_\\odot $ at radii between 1$\\times $ 10$^{17}$ and 1$\\times $ 10$^{18}$ cm, and most medium-width shells of similar mass at radii between 2$\\times $ 10$^{17}$ and 1.5$\\times $ 10$^{18}$ cm.",
"In terms of raw CSM shell density, our deepest limits come between 1 and 1.5 $\\times $ 10$^{18}$ cm, where we can rule out shells with densities between 1 and 3 cm$^{-3}$ .",
"We emphasize that these radii are larger than those probed by most other observations searching for CSM surrounding Type Ia SNe to date, including time-varying absorption features [86] and late-time optical photometry/spectroscopy [38], which tend to constrain the presence of CSM around $\\sim $ 10$^{16}$ cm.",
"[113] do find a radius of $\\sim $ 3 $\\times 10^{17}$ cm for the material responsible for time-varying Na absorption lines around the Type Ia SN 2007le.",
"However, the density inferred is much higher ($\\sim $ 10$^{7}$ cm$^{-3}$ ) and fractional width much narrower ($f \\approx 3\\times 10^{-4}$ ) than those considered here, possibly suggesting a clumpy or aspherical CSM.",
"Our observations constrain a unique parameter space of CSM shells.",
"For SN 1895B, we find that essentially all of the shell models that would be excluded by the depth and timing of our observations fall in the regime where the H16 assumption that the CSM impacts the outer SN ejecta is violated.",
"However, a few specific exceptions to this exist.",
"For example, we can rule out the presence of an $f=0.33$ medium width shell with a density of 6 cm$^{-3}$ at a radius of $\\sim 2 \\times 10^{18}$ cm (total shell mass $\\sim $ 0.3 M$_\\odot $ ).",
"These borderline cases demonstrate that the observations of SN 1895B are likely useful to constrain the presence of shells at these radii, but updated models that include interaction with the dense inner SN ejecta are required for a quantitative assessment."
],
[
"Discussion",
"The CSM environment surrounding a Type Ia SN is dependent on pre-explosion evolutionary history of the progenitor system.",
"In this section, we will consider different types of CSM that are both allowed and ruled out by our results (Section ), and what they indicate in the context of various Type Ia SN progenitor scenarios.",
"In Section REF , we consider the presence of constant density material, the only material expected in DD scenarios with significant delay times.",
"We next consider the presence of shells (Section REF ), as may be expected for SD progenitors if they contain nova shells or planetary nebula and DD progenitors in the case of a prompt explosion post-common envelope.",
"We also consider the presence of other types of CSM (Section REF ).",
"Finally, in Section REF , we make predictions for the future of both SN 1895B and SN 1972E as the SNe evolve and future observations are taken."
],
[
"Presence of Constant Density CSM or ISM",
"Our deepest luminosity limits constrain the density of a uniform ambient medium surrounding SN 1972E and SN 1895B to be $\\lesssim $ 0.9 cm$^{-3}$ out to radii of $\\sim $ 10$^{17}$ $-$ 10$^{18}$ cm.",
"This implies a clean circumstellar environment out to distances 1$-$ 2 orders of magnitude further than those previously probed by prompt radio and X-ray observations [18], [72].",
"Densities of this level are consistent with the warm phase of the ISM in some galaxies [34], and we examine whether our density constraints for SN 1972E and SN 1895B are consistent with expectations for the ISM in their local environments within the intensely star-forming galaxy NGC 5253.",
"Using the HI observations of [59], [116] estimate the ISM density at the location of SN 1972E, which is $>$ 1.5 kpc from the central star-forming region, to be $\\lesssim $ 1 cm$^{-3}$ —comparable to our radio limits.",
"In contrast, SN 1895B exploded $\\sim $ 100 pc from the nucleus of NGC 5253, in a complex region with multiple large stellar clusters (Section ).",
"Excluding the dense stellar clusters themselves, [77] use IFU spectroscopy with VLT-FLAMES to conclude that the ISM density in this central region is $<$ 100 cm$^{-3}$ , and could potentially be 1$-$ 2 orders of magnitude lower and the explosion site of SN 1895B, depending on the local distribution of material.",
"Thus, despite some uncertainty, we find that our deepest radio limits constrain the density surrounding SN 1972E and SN 1895B to be at levels comparable to, or below, the local ISM at distances of $\\sim $ 10$^{17}$ $-$ 10$^{18}$ cm.",
"Low density media surrounding Type Ia SNe can be achieved through multiple progenitor scenarios.",
"Clean, ISM-like, environments are most commonly evoked for DD models produced by the merger of two WDs.",
"The components of such systems have low intrinsic mass loss rates and current population synthesis models predict that $>$ 90% of WD mergers should occur $>$ 10$^{5}$ years after the last phase of common envelope evolution [98].",
"Thus, the material ejected during this phase should fully disperse into the ISM at radii beyond 10$^{18}$ cm by the time of explosion.",
"While WD mergers may also pollute the CSM via a number of other physical mechanisms including tidal tail ejections [90], outflows during a phase of rapid mass transfer pre-merger [41], [24] and accretion disk winds in systems that fail to detonate promptly [55], this material will be located at radii $<$ a few $\\times $ 10$^{17}$ cm, unless there is a significant ($\\gtrsim $ 100 years) delay between the onset of merger and the subsequent Type Ia explosion.",
"In this case, the small amount of material ejected via these mechanisms ($\\sim $ 10$^{-3}$ $-$ 10$^{-2}$ M$_\\odot $ ) will have either dispersed to densities below our measurements or swept up material into a thin shell [90], whose presence will be assessed below.",
"Thus, we conclude that our low inferred densities surrounding SN 1972E and SN 1895B are be consistent with expectations for a majority of DD explosions due to WD mergers.",
"However, low density ambient media can also be produced by SD and DD Type Ia SN models in which either fast winds or shells of material are ejected from the progenitor system prior to explosion.",
"This high velocity material will subsequently “sweep-up” the surrounding ISM, yielding low density cavities surrounding the stellar system [4].",
"For example, recent hydrodynamical simulations of recurrent nova systems find cavity densities of 10$^{-1}$ $-$ 10$^{-3}$ cm$^{-3}$ , far below the density of the ambient ISM [31], [27].",
"Our radio observations would require a cavity that extends to a few $\\times $ 10$^{18}$ cm.",
"These distances are consistent with the large (r $>$ 10$^{19}$ cm) cavities predicted to be carved by fast accretion wind outflows from the WD surface in some SD models [42], although such cavities may be inconsistent with observed SNR dynamics [4].",
"In the context of recurrent nova systems such large cavities would require a system that had been undergoing outbursts for $\\gtrsim $ 10,000 years [31], [27].",
"In the section below, we discuss constraints on the presence of CSM shells surrounding SN 1972E and SN 1895B, and thus further implications for this class of progenitor model if a cavity is the source of the clean CSM environments observed."
],
[
"Presence of Shells",
"Several putative progenitor systems for Type Ia SNe predict the presence of shells surrounding the system at distances in the range of those probed by our observations ($\\sim 10^{17}-10^{18}$ cm).",
"These include both SD and DD systems, with examples of shell creation mechanisms ranging from a recurrent nova to common envelope ejections.",
"In Section REF , we utilized the models of [44] to explore the basic parameter space of shells that can be constrained and ruled out by our data.",
"Here, we discuss the implications of these results for various progenitor scenarios."
],
[
"Recurrent Nova Progenitors",
"A recurrent nova is a high mass accreting WD system that undergoes repeating thermonuclear outbursts due to unstable hydrogen burning on its surface, ejecting mass from the system every $\\sim 1-100$ yr.",
"The identification of time variable absorption and blue shifted Na I D lines in some Type Ia SNe [86], [8], [115], [70] have raised the question of a connection between recurrent novae and Type Ia SNe, particularly in light of the discovery of blue-shifted Na I D lines in the recurrent nova RS Ophiuchi (RS Oph) during outburst [85], [11].",
"Individual nova eruptions eject a small mass of material (M$_{\\rm {ej}}$ $\\sim $ 10$^{-7}$ $-$ 10$^{-5}$ M$_\\odot $ ) at high velocities ($v_{\\rm {ej}} \\gtrsim 3000$ km/s; [78], [27]).",
"However, this material will rapidly decelerate to velocities on the order of tens of km s$^{-1}$ as it sweeps up material from the ISM, CSM, or collides previously ejected shells.",
"The result is a complex CSM structure consisting of of low-density ($n$ $\\sim $ 10$^{-1}$ $-$ 10$^{-3}$ cm$^{-3}$ ) cavities enclosed by a dense outer shell [80], [4].",
"For a for a $10^4$ year recurring nova phase, such as that seen in RS Oph-like stars, the outer cavity wall predicted to be at radii of $\\gtrsim 3\\times 10^{17}$ cm [11], [31], within the regime probed by our observations.",
"The constraints that our observations can provide on the presence of nova shells surrounding SN 1972E depend primarily on their predicted densities, radii, and thicknesses, which in turn depend on the density of the ambient ISM, the total time the system has been in an active nova phase, and the recurrence timescale between eruptions.",
"Two recent hydrodynamic models for the CSM structure surrounding such systems are presented by [31] and [27].",
"The former models nova eruptions with 25, 100, 200 year recurrence timescales expanding into a CSM shaped by winds from a red giant donor star with $\\dot{M}$ $=$ 10$^{-6}$ M$_\\odot $ and v$_{\\rm {w}}$ $=$ 10 km s $^{-1}$ .",
"The the latter simulated eruptions with both a shorter recurrence timescale (350 days) and a lower density CSM (shaped by a red giant star with $\\dot{M}$ $=$ 2.6 $\\times $ 10$^{-8}$ M$_\\odot $ and v$_{\\rm {w}}$ $=$ 20 km s$^{-1}$ ).",
"This model was specifically designed to reproduce the CSM surrounding the M31 nova system M31N 2008-12a.",
"M31N 2008-12a is particularly interesting system as it is the most frequently recurring nova known, the WD is predicted to surpass the Chandrasekhar limit in $<$ 20,000 years [26], and it is surrounded by an observed cavity-shell system with a total projected size of $\\sim $ $134 \\times 90$ pc [27].",
"[31] find that the density of individual nova ejections expanding into the main cavity depends on the nova recurrence timescale.",
"For longer recurrence times the densities will be higher, as the donor star has additional time to pollute the CSM.",
"For the donor mass-loss rate and recurrence timescales considered by [31] these shells are predicted to have densities $\\gtrsim $ 10$^{2}$ cm$^{-3}$ , while the low density and short recurrence timescale of [27] yield individual shell densities below the detection threshold of our observations ($n$ $\\lesssim $ 0.1 cm$^{-3}$ ).",
"However, while our observations can rule out high-density shells from some individual nova eruptions, they are predicted to be too thin ($f$ $\\sim $ 0.01; [31]) for our sparse observations to conclusively rule our a system of shells predicted for any specific recurrence time.",
"In contrast, the outer cavity wall is expected to be thicker.",
"[27] find that this “nova super-remnant shell” converges a width of $f$ $=$ 0.22 and density approximately 4 times that of the ISM in their simulations ($\\sim $ 4 cm$^{-3}$ ).",
"Our observations can rule out the presence of even these low-density medium-thickness shells at radii between $\\sim $ 5 $\\times $ 10$^{17}$ cm and 2 $\\times $ 10$^{18}$ cm.",
"[27] find that the outer cavity would be located at these radii for nova systems that have been active for between $\\sim $ 10$^{3}$ and $10^4$ years (having undergone $\\sim $ 1000 $-$ 10,000 total eruptions).",
"For higher density CSM and longer recurrence times, [31] find that the cavities will expand more slowly, and thus our observations will rule out older systems."
],
[
"Core Degenerate Scenario",
"In the core degenerate scenario for Type Ia SNe, a WD companion merges with the hot core of an asymptotic giant branch (AGB) star at the end of a common envelope (CE) or planetary nebula (PN) phase [57], [114].",
"The result of this merger is a massive (M $\\gtrsim $ M$_{\\rm {Ch}}$ ), rapidly-rotating, and highly magnetized WD [119], [57], which can subsequently explode as a Type Ia SN.",
"In this scenario, the delay time between the merger and the SN—and hence the location of the CE or PN shell—is primarily set by the spin-down timescale of the merger remnant [51].",
"While originally proposed as a mechanism for prompt explosion after CE ejection (in order to explain Type Ia SN with strong hydrogen emission; [67]), a wide range of spin-down timescales are permitted [65], [129], [51].",
"Based on a number of observational probes, [122] have suggested that $\\sim $ 20% of all Type Ia SN should occur within a PN that ejected within the $\\sim $ 10$^5$ years prior to explosion due to the core-degenerate scenario.",
"Assuming average expansion velocities of tens of km s$^{-1}$ , our observations of SN1972E constrain the presence of PN ejected between a few $\\times $ 10$^{3}$ and a few $\\times $ 10$^{4}$ years prior to explosion.",
"We find we can rule out the presence of roughly Abell 39-like PN (with $n$ $\\sim $ 30 cm$^{-3}$ and $f$ $=$ 0.15 at $r_1$ $\\sim $ 10$^{18}$ cm) for most of this range of delay times.",
"More broadly, observed PN have masses in the range of $\\sim $ 0.1 M$_\\odot $ to 1 M$_\\odot $ .",
"Our observations rule out most shells with masses between 0.05 M$_\\odot $ and 0.3 M$_\\odot $ and thicknesses greater than f$=$ 0.15.",
"Our observations likely also constrain higher mass PN—relevant as the core-degenerate scenario may require massive AGB stars [67], [57]—but updated theoretical models, which include the effects of the inner SN ejecta impacting the CSM, are required for quantitative assessment."
],
[
"Shell Ejections in DD Progenitors",
"There are multiple mechanisms by which DD Type Ia progenitors may also eject shells of material pre-explosion.",
"First, all putative DD progenitor scenarios must undergo at least one episode of CE evolution, in order to yield the requisite tight double WD system [52].",
"For WD merger models, the delay between CE ejection and SN is primarily set by the binary separation post-CE and the gravitational-wave timescale.",
"While current binary population synthesis models predict a majority of WD mergers will occur with a significant delay post-CE, [98] highlight a channel wherein $\\sim $ 3.5% of WD binaries with a massive ($>$ 0.9 M$_\\odot $ ) primary will merge between 10$^3$ and 10$^{4}$ post-CE.",
"As described above, assuming expansion velocities of a few tens to 100 km s$^{-1}$ , our observations of SN 1972E constrain shells ejected on these timescales.",
"While the CE mass ejection process is uncertain, the total envelop ejected for putative Type Ia progenitors ranges from a few tenths to $\\sim $ 1 M$_\\odot $ [69].",
"We can rule out most CE shells with masses between 0.05 M$_\\odot $ and 0.3 M$_\\odot $ , unless they are very thin ($f \\lesssim 0.1$ ).",
"Thus, it is unlikely that SN 1972E underwent and ultra-prompt explosion, although we caution additional theoretical models are required to quantitatively rule out CE shells with masses of $\\sim $ 1 M$_\\odot $ .",
"For DD models that are triggered by the detonation of thin surface layer of helium accreted from a low-mass WD companion (the “double detonation” model; e.g., [128], [68], [109]), the explosion is predicted to occur between 10$^{8}$ and 10$^{9}$ years after CE [97], [109].",
"As such, any CE shell will have long since dispersed into the ISM.",
"However, [109] outline a model whereby such systems can also eject small amounts of hydrogen-rich material (a few $\\times $ 10$^{-5}$ M$_\\odot $ ) at high velocities ($\\sim $ 15000 km s$^{-1}$ ) in the hundreds to thousands of years before the SN.",
"Analogous to classical novae, this material will sweep up the ISM, forming a cavity and outer shell structure whose properties (mass, radius, thickness) depend on both the evolutionary history of the WD and the ambient ISM density.",
"For ISM densities of 1 cm$^{-3}$ , [109] predict shells with $n$ $\\sim $ 5 cm$^{-3}$ and widths of $f$ $\\sim $ 0.25 at radii ranging from $r_1$ $\\sim $ 5 $\\times $ 10$^{17}$ cm (for older WD progenitors) to $r_1$ $\\sim $ 1 $\\times $ 10$^{18}$ cm (for younger WD progenitors).",
"Our deepest limits just rule out the presence of such shells around SN1972E, although some intermediate ages are permitted.",
"For sparser ambient ISM densities, such shells would be below our detection limits."
],
[
"Tidal Tail Ejections",
"In WD-WD merger scenarios, a small amount of material (a few $\\times $ 10$^{-3}$ M$_\\odot $ ) can be ejected in the form of tidal tails, which are stripped from the system just prior to coalescence [90].",
"The ultimate location of this material depends on the delay between the initiation of the merger and the ultimate explosion, and the non-detection of Type Ia SN in prompt (t $\\lesssim $ year) radio and X-ray observations have been used to argue for either very short ($\\lesssim $ 100 s) or long ($>$ 100 years) delays [72], [90].",
"For a delay time of $\\sim $ 100 year, [90] predict that the tidal tails should appears as a wide ($f = 1$ ) shell-like structure with a density of $n$ $\\sim $ 100 cm$^{-3}$ at a radius of $r_1$ $\\sim $ 2 $\\times $ 10$^{17}$ cm.",
"Our observations rule out such a CSM structure for SN 1972E.",
"From this time onward, the tidal material will sweep-up ISM material, decelerating and narrowing in the process.",
"Thus, our observation likely rule out delay times of a few hundred years for this scenario, with the exact range depending on the ISM density and deceleration timescale.",
"[90] predict that by 3000 years post-ejection, the tidal material will be located at a radius of $\\sim $ 8 $\\times $ 10$^{18}$ cm, well beyond those probed by our observations."
],
[
"Other CSM Structures",
"There are several putative Type Ia SN explosion models that predict the presence of CSM, which is neither constant in density nor strictly in the form of shells.",
"Here, we discuss two such cases."
],
[
"Stellar Winds",
"If the CSM surrounding surrounding the Type Ia SN has a stellar wind-like density distribution ($\\rho $ $\\propto $ $r^{-2}$ ), observations from the first $\\sim $ year post-explosion would provide the deepest constraints on the mass-loss rate of the progenitor system.",
"This density distribution is what is typically expected in SD models that undergo quasi-steady mass-loss due to either winds from a giant (symbiotic) donor star [107], optically-thick winds from the WD itself during phases of high-accretion [42], or non-conservative mass-loss through the second Lagrange point during Roche Lobe overflow for some binary configurations [30].",
"In all such cases, emission from the CSM interaction would be strongest in the first days after the SN event when the density of the CSM is highest [19].",
"As described in Section , the deepest limits on the mass-loss rates for SN 1972E and SN 1895B come from the 1984 observations, 12.5 and 8.3 years post-explosion.",
"The constraints of $<8.60\\times 10^{-6}$ M$_{\\odot }$ yr$^{-1}$ and $<7.2\\times 10^{-5}$ M$_{\\odot }$ yr$^{-1}$ (for wind velocities of 10 km s$^{-1}$ ) rule out a number of Galactic symbiotic systems [107], but are otherwise unconstraining.",
"We note that these limits depend linearly on the assumed wind speed, and hence for v$_{\\rm {w}}$ $>$ 10 km s$^{-1}$ the mass-loss constraints would be even weaker."
],
[
"Mass Loss from a Radially Extended Envelope",
"[108] present updated model for the long-term evolution of the remnants of WD mergers, in which the lower mass WD is disrupted and forms a hot radially extended (r $\\sim $ 10$^{13}$ cm) envelope around the central remnant rather than an accretion disk.",
"While the final fate of such remnants are debated, it should persist for $\\gtrsim $ 10$^{4}$ years as a carbon burning shell, ignited off-axis, propagates inward to the core.",
"While they neglect mass loss in their calculations, [108] note that with a typical escape velocities of 60 km s$^{-1}$ , material lost during this phase in the remnant's evolution could reach radii of $\\sim $ 2$\\times $ 10$^{18}$ cm, within the radius range probed by our observations.",
"Subsequently, [106] perform updated models and examine the consequences of different mass loss prescriptions on the evolution of such merger remnants.",
"In particular, they note the similarities between the observed properties of these remnants and AGB stars, raising the possibility that a dusty wind may form during an $\\sim $ 5000 year phase in their evolution.",
"Within this context, we note that our observations rule out mass loss on the level observed in extreme AGB stars ($\\dot{M}$ $\\sim $ 10$^{-4}$ M$_\\odot $ ) out to radii of a few times 10$^{17}$ cm for wind speeds between 10 and 100 km s$^{-1}$ .",
"However, [106] also find that the temperature of the merger remnant will eventually increase, in a process analogous to PN formation in AGB stars.",
"As a consequence of this evolution, any phase of intense dusty mass loss should cease and the increased UV radiation from the central star could yield an ionized nebulae with total mass of $\\sim $ 0.1 M$_\\odot $ at a distance of $\\gtrsim $ 3 $\\times $ 10$^{17}$ cm.",
"Our observations rule out the presence of such shells over a wide range of radii, unless they are very thin ($f$ $<$ 0.1)."
],
[
"The Future: SN to SNR Transition",
"Our upper limits on the radio luminosity from SN 1972E and SN 1895B are consistent with both SN blastwaves expanding into low density CSM environments out to radii of a few $\\times $ 10$^{18}$ cm.",
"Assuming a constant density CSM, the radio emission from both events is predicted to continue rising over time (see Figure REF ), and we can use our baseline S17 model described in Section REF to project their future evolution and thus prospects for subsequent radio detections.",
"If we assume that no CSM shells are present, and that the SNe are expanding into ambient densities of 0.7 cm$^{-3}$ (just below our Dec. 2016 limits; Table REF ), then both SN would peak at a flux level of $\\sim $ 200 $\\mu $ Jy (at 5 GHz) $\\sim $ 300 years post-explosion.",
"In this scenario SN 1972E and SN 1895B would reach maximum observed brightness in 2272 and 2195, respectively.",
"If we assume that targeted VLA observations of each SN could achieve C-band images with RMS noise levels of $\\sim $ 5$-$ 10 $\\mu $ Jy (consistent with the sensitivity limits obtained by [19]), then both SN1972E and SN 1895B would currently be detectable at a greater than 5$\\sigma $ level.",
"However, the ambient density surrounding both SNe may be significantly lower than the upper limits found in Section .",
"In this case the radio light curve would peak at later times and fainter flux levels (S17; See Figure REF ).",
"For example, the youngest SN in our own Galaxy, G1.9+0.3, is detected at radio wavelengths at a level consistent ambient density of 0.02 cm$^{-3}$ ([103]; adjusted for consistency with our baseline S17 model; see Figure REF ).",
"If SN 1972E and SN 1895B are expanding into similar CSM environments, then we project that they would peak at 5 GHz flux levels of $\\sim $ 6 $\\mu $ Jy $\\sim $ 990 years post-explosion.",
"In such a scenario, their current 5 GHz fluxes would be only $\\sim $ 1 $\\mu $ Jy and $\\sim $ 2 $\\mu $ Jy, respectively, and they would never rise above the optimal VLA sensitivity limits described above.",
"This indicates that observations of intermediate-aged Type Ia SNe in nearby galaxies may still be sensitivity limited without significant time investment [103] from current instruments.",
"Limits with future radio telescopes such as the Square Kilometer Array (SKA) and the Next Generation VLA will allow us to study radio emission from Type Ia SNe throughout the local volume, even when they are expanding into low density ($\\sim 0.1$ cm$^{-3}$ ) surroundings.",
"Once additional detections of intermediate aged-SN and young SNRs are made, interpretation of the results will require careful consideration of whether the emission is due to CSM shaped by the progenitor system, or simply the ambient ISM.",
"For example, recent analysis of radio observations of SN 1885A by [103] conclude the density surrounding the system must be approximately a factor 5 lower density that that surrounding G1.9+0.3.",
"However, they argue that the higher density found for G1.9+0.3 may be due to a higher density in the Milky Way's center—as compared to M31's—and does not require CSM from the progenitor.",
"Currently, we cannot distinguish between these scenarios based on our data for SN 1895B and SN 1972E."
],
[
"Summary and Conclusions",
"We have conducted a study of the circumstellar environments of the nearby Type Ia SN 1972E and SN 1895B by analyzing seven epoch of archival VLA observations from obtained between 1981 and 2016.",
"We do not detect emission from the location of either SN in our data set.",
"The most stringent upper limits on the radio luminosity from each event are L$_{\\nu ,8.5\\rm {GHz}}$ $<$ 8.9 $\\times $ 10$^{23}$ erg s$^{-1}$ Hz$^{-1}$ 121 years post-explosion for SN 1895B and L$_{\\nu ,8.5\\rm {GHz}}$ $<$ 6.0 $\\times $ 10$^{23}$ erg s$^{-1}$ Hz$^{-1}$ 45 years post-explosion for SN 1972E.",
"These imply low-density environments with $n$ $<$ 0.9 cm$^{-3}$ out to radii of a few $\\times $ 10$^{18}$ cm — nearly two orders of magnitude further from the progenitor star than those previously probed by prompt (t $\\lesssim $ 1 year) radio and X-ray observations [84], [73], [18], [100], [72], [19].",
"These ambient densities are consistent with progenitor scenarios that produce either ISM-like environments or low-density evacuated cavities out to large distances.",
"Given the multi-epoch nature of our dataset, we also investigate the possibility of shells surrounding the progenitor of SN 19722E.",
"Using the models of H16, we rule out the presence of essentially all medium and thick CSM shells with total masses of 0.05 M$_\\odot $ to 0.3 M$_\\odot $ located at radii between a few $\\times $ $10^{17}$ and a few $\\times $ $10^{18}$ cm.",
"We also exclude specific CSM shells down to masses of $\\lesssim $ 0.01 M$_\\odot $ at a range of radii, which vary depending on the shell thickness (see Figure REF ).",
"Quantitative assessment of the presence of more massive CSM shells will require updated theoretical models that include the effect of the inner SN ejecta impacting the CSM shell.",
"These shell constraints rule out swaths of parameter space for various SD and DD Type Ia SN progenitor models including recurrent nova, core-degenerate objects, ultra-prompt explosions post-CE, shells ejections from CO$+$ He WD systems, and WD mergers with delays of a few hundred years between the onset of merger and explosion.",
"Allowed progenitor systems include DD in which the delay from the last episode of CE is long ($>$ 10$^{4}$ yrs) as well as SD models that exhibit nova eruptions—provided the system has a relatively short recurrence timescale and has been in the nova phase for either a short ($\\lesssim $ 100 yrs) or long ($\\gtrsim $ 10$^{4}$ yrs) time.",
"It is clear that multi-epoch radio observations of nearby intermediate-aged Type Ia SNe explore useful regions of parameter space for distinguishing between the plethora of theoretical progenitor models.",
"In the future, a statistical sample of such events will provide even more robust discriminating power, as different models predict a range of delay times and hence a variety of locations for CSM material."
],
[
"Acknowledgements",
"We would like to thank C. Harris for their generous assistance in our modeling work and answering our questions and B. Shappee for conversations that inspired this work.",
"Support for this work was provided to MRD through Hubble Fellowship grant NSG-HF2-51373, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. for NASA, under contract NAS5-26555.",
"MRD acknowledges support from the Dunlap Institute at the University of Toronto and the Canadian Institute for Advanced Research (CIFAR).",
"LC and SS are grateful for the support of NSF AST-1412980, NSF AST-1412549, and NSF AST-1751874.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.",
"The Dunlap Institute is funded through an endowment established by the David Dunlap family and the University of Toronto.",
"CASA [76], pwkit [126], Astropy [2], Matplotlib [49]"
]
] | 2001.03558 |
[
[
"Computable Lower Bounds for Capacities of Input-Driven Finite-State\n Channels"
],
[
"Abstract This paper studies the capacities of input-driven finite-state channels, i.e., channels whose current state is a time-invariant deterministic function of the previous state and the current input.",
"We lower bound the capacity of such a channel using a dynamic programming formulation of a bound on the maximum reverse directed information rate.",
"We show that the dynamic programming-based bounds can be simplified by solving the corresponding Bellman equation explicitly.",
"In particular, we provide analytical lower bounds on the capacities of $(d, k)$-runlength-limited input-constrained binary symmetric and binary erasure channels.",
"Furthermore, we provide a single-letter lower bound based on a class of input distributions with memory."
],
[
"Introduction",
"Discrete finite-state channels (DFSCs or FSCs) are mathematical models for channels with finite memory that are applied to magnetic recording [1], and wireless communications [2].",
"The channel memory is encapsulated in the channel state, which takes values from a finite set.",
"Although Shannon's single-letter expression exists for the capacity of discrete memoryless channels (DMCs) [3], namely, $C_{DMC} = \\max _{p(x)} I(X;Y)$ , the capacity of FSCs, with the exception of special cases, is characterized only as a multi-letter expression.",
"In this paper, we derive lower bounds on the capacity of input-driven FSCs (Fig.",
"REF ), where the channel state evolves as a time-invariant function of the state at the previous time instant, and the current input.",
"This class of channels includes the collection of DMCs with input constraints, which will be treated in some detail in our work.",
"Existing work on lower bounding the capacity of FSCs includes the simulation-based approach in [4], the generalized Blahut-Arimoto algorithm developed in [5], and the stochastic approximation algorithm proposed in [6], all of which are numerical methods.",
"Analytical lower bounds were derived by Zehavi and Wolf [7] for binary symmetric channels with a $(d,k)$ -runlength-limited (RLL) constraint — see Definition REF — at the input.",
"Later works gave capacity lower bounds for input-constrained binary symmetric and binary erasure channels in the asymptotic (very low or very high noise) regimes [8], [9], [10].",
"Our work applies to a larger class of channels, and provides bounds for all values of the channel parameters.",
"The key idea is the lower bounding of the $N$ -letter mutual information between the channel inputs and the outputs, $I(X^N;Y^N)$ , by the reverse directed information [11], $I(Y^N\\rightarrow X^N)$ .",
"We show that the derived lower bound on the capacity can be formulated as an infinite-horizon average-reward dynamic programming (DP) problem, and is hence computable.",
"Further, we show that the lower bound derived can, in turn, be bounded below by a single-letter expression obtained using input distributions on a directed “${V}$ -graph”.",
"Our treatment here is entirely analogous to the single-letter lower bounding technique introduced in [15], which uses input distributions on a “Q-graph” that is obtained by a recursive quantization of channel outputs on a directed graph.",
"We then apply the DP-based lower bound to the class of input-constrained binary symmetric channels (BSCs) and binary erasure channels (BECs).",
"DP problems are typically handled by solving the corresponding Bellman equations.",
"We consider the $(d,k)$ -RLL input-constrained (RIC) BSC and BEC, and explicitly solve the Bellman equations for the DP-based lower bounds for each of these channels.",
"Interestingly, our techniques recover the lower bounds given in [7], for the $(d,k)$ -RIC BSC, for $k<\\infty $ .",
"For the $(1,\\infty )$ -RIC BSC and BEC, the analytical lower bounds thus found compare favourably with asymptotic lower bounds given in [8], [9], and extend to all values of the channel parameters.",
"Figure: System model of an input-driven FSC.The paper is organized as follows.",
"Section contains the necessary information-theoretic preliminaries, Section states the main results, and Section explains the DP formulation, which is used in Section to derive explicit lower bounds on the capacity of the $(d,k)$ -RIC BEC and BSC.",
"The proofs of our main results are presented in Section .",
"Some concluding remarks are made in Section ."
],
[
"Notation and Preliminaries",
"In this section, we introduce notation, the channel model, the lower bound on mutual information rate, and the definitions of ${V}$ -graphs and $({S},{V})$ -graphs."
],
[
"Notation",
"In what follows, random variables will be denoted by capital letters, and their realizations by lower-case letters, e.g., $X$ and $x$ , respectively.",
"Calligraphic letters, e.g., ${X}$ , denote sets.",
"The notations $X^{N}$ and $x^N$ denote the random vector $(X_1,\\ldots ,X_N)$ and the realization $(x_1,\\ldots ,x_N)$ , respectively.",
"Further, $P(x), P(y)$ and $P(y|x)$ are used to denote the probabilities $P_X(x), P_Y(y)$ and $P_{Y|X}(y|x)$ , respectively.",
"As is usual, the notations $H(X)$ and $I(X;Y)$ stand for the entropy of the random variable $X$ , and the mutual information between the random variables $X$ and $Y$ , respectively, and $h_b(p)$ and $H_{ter}(p,q)$ are the binary and ternary entropy functions, respectively.",
"Finally, for a real number $\\alpha \\in [0,1]$ , we define $\\bar{\\alpha }=1-\\alpha $ ."
],
[
"Channel Model",
"Consider the family of channels with input $x_t\\in {X}$ , output $y_t\\in {Y}$ , and state $s_t\\in {S}$ at time $t$ , with ${X},{Y},{S}$ having finite cardinalities.",
"At each time $t$ , an FSC obeys $P(s_t,y_t|x^t,s^{t-1},y^{t-1})=P(s_t,y_t|x_t,s_{t-1})$ .",
"An input-driven FSC has the additional property that there exists a time-invariant function, $f: {S}\\times {X}\\rightarrow {S}$ , such that $s_t=f(s_{t-1},x_t)$ .",
"We shall assume throughout that the initial state $s_0$ of the FSC is known to the encoder and decoder.",
"While this assumption can be removed through a suitable notion of channel indecomposability [12], we retain the assumption as it is realistic in the context of input-constrained DMCs, which is the main application of interest to us.",
"The following theorem gives an expression for the capacity of such FSCs.",
"Theorem 2.1 ([12], Ch.",
"$4.6$ ) The capacity of an FSC with known initial state $s_0$ is given by $C = \\lim _{N\\rightarrow \\infty } \\max _{P(x^N | s_0)} \\frac{1}{N}I(X^N;Y^N|s_0).$ From here on, we will drop the explicit conditioning on $s_0$ in our notation, including it only when there is need.",
"We also introduce below the definition of a connected FSC, which we shall use in Theorem REF .",
"Definition 2.1 An FSC is connected if, for each $s \\in {S}$ , there is an input distribution $\\lbrace P(x_n|s_{n-1})\\rbrace _{n \\ge 1}$ and an integer $N$ such that $\\sum _{n=1}^N P_{S_n|S_0} (s|s^{\\prime }) > 0$ , for all $s^{\\prime } \\in {S}$ ."
],
[
"Directed Information",
"We recall the definition of directed information, introduced by Massey [13]: Definition 2.2 The (forward) directed information is given by $I(X^N\\rightarrow Y^N)&:=\\sum _{t=1}^{N} I(X^t;Y_t|Y^{t-1})\\\\&=\\mathbb {E}\\left[\\log _2\\left(\\frac{P(Y^N||X^N)}{P(Y^N)}\\right)\\right],$ where $P(y^N||x^N):=\\prod _{t=1}^{N}P(y_t|x^{t},y^{t-1})$ is the causal conditioning distribution.",
"Analogously, we define the reverse directed information by $I(Y^N\\rightarrow X^N):=\\sum _{t=1}^{N} I(Y^t;X_t|X^{t-1}).$ In addition, we make use of the following definition: $I(X^{N-1}\\rightarrow Y^N)&:=\\sum _{t=1}^{N} I(X^{t-1};Y_t|Y^{t-1})\\\\&=\\mathbb {E}\\left[\\log _2\\left(\\frac{P(Y^N||X^{N-1})}{P(Y^N)}\\right)\\right],$ where $P(y^N||x^{N-1}):=\\prod _{t=1}^{N}P(y_t|x^{t-1},y^{t-1})$ .",
"The following conservation law for information is well-known [14]: $I(X^N;Y^N)=I(Y^N\\rightarrow X^N)+I(X^{N-1}\\rightarrow Y^N).$ In particular, mutual information is bounded below by reverse directed information, i.e., $I(X^N;Y^N)\\ge I(Y^N\\rightarrow X^N).$"
],
[
"The ${V}$ -graph and {{formula:2853cd44-3aa2-4cc1-987b-c019bbcc84e0}} -graph",
"Similar to the $Q$ -graph and the $({S},Q)$ -graph of [15], we introduce the following definitions: Definition 2.3 A ${V}$ -graph is a finite irreducible labelled directed graph on a vertex set ${V}$ , with the property that each $v\\in {V}$ has at most $|{X}|$ outgoing edges, each labelled by a unique $x\\in {X}$ .",
"Thus, there exists a function $\\Phi : {V} \\times {X} \\rightarrow {V}$ , such that $\\Phi (v,x)=v^{\\prime }$ iff there is an edge $v \\stackrel{x}{\\longrightarrow } v^{\\prime }$ in the ${V}$ -graph.",
"We arbitrarily label one vertex of the ${V}$ -graph as $v_0$ .",
"For any positive integer $n$ , there is a one-to-one correspondence between sequences in $(x_1,x_2,\\ldots ,x_n) \\in {X}^n$ and directed paths in the ${V}$ -graph starting from $v_0$ : $v_0 \\stackrel{x_1}{\\longrightarrow } v_1 \\stackrel{x_2}{\\longrightarrow }\\cdots \\stackrel{x_n}{\\longrightarrow } v_n$ .",
"Fig.",
"REF depicts an example of a ${V}$ -graph.",
"Figure: A V{V}-graph where each node represents the last channel input, where X={0,1}{X}=\\lbrace 0,1\\rbrace .Definition 2.4 Given an input-driven FSC specified by $\\lbrace P(y|x,s)\\rbrace $ and $s_t=f(s_{t-1},x_t)$ , and a ${V}$ -graph with vertex set ${V}$ , the $({S},{V})$ -graph is defined to be a directed graph on the vertex set ${S}\\times {V}$ , with edges $(s,v) \\xrightarrow{} (s^{\\prime },v^{\\prime })$ if and only if $P(y|x,s)>0$ , $s^{\\prime }=f(s,x)$ , and $v^{\\prime }=\\Phi (v,x)$ .",
"Now, given an input distribution $\\lbrace Q(x|s,v)\\rbrace $ defined for each $(s,v)$ in the $({S},{V})$ -graph, we have a Markov chain on ${S}\\times {V}$ , where the transition probability associated with any edge $(x,y)$ emanating from $(s,v)\\in {S}\\times {V}$ is $P(y|x,s)Q(x|s,v)$ .",
"Let ${G}(\\lbrace Q(x|s,v)\\rbrace )$ be the subgraph remaining after discarding edges of zero probability.",
"We then define ${Q} \\triangleq \\bigl \\lbrace \\lbrace Q(x | s,v)\\rbrace : {G}& (\\lbrace Q(x|s,v)\\rbrace )\\text{ has a single} \\\\& \\text{closed communicating class}\\bigr \\rbrace .$ Given an irreducible ${V}$ -graph, an input distribution $\\lbrace Q(x|s,v)\\rbrace \\in {Q}$ is said to be aperiodic, if the corresponding graph, ${G}(\\lbrace Q(x|s,v)\\rbrace )$ , is aperiodic.",
"For such distributions, the Markov chain on ${S}\\times {V}$ has a unique stationary distribution $\\pi (s,v)$ ."
],
[
"Main Results",
"We shall now restrict attention to input-driven FSCs, defined in Section REF .",
"We assume that the initial channel state, $s_0$ , is chosen deterministically, and is known to both the encoder and the decoder.",
"We present a lower bound on the capacity of indecomposable input-driven FSCs.",
"Theorem 3.1 The capacity of an input-driven FSC with known initial state is bounded below as: $C&\\ge \\lim _{N\\rightarrow \\infty } \\max _{\\lbrace P(x_t|s_{t-1})\\rbrace _{t= 1}^{N}}\\frac{1}{N}\\sum _{t=1}^{N}I(X_t,S_{t-1};Y_t \\mid X^{t-1})\\\\&=\\sup _{\\lbrace P(x_t|s_{t-1})\\rbrace _{t \\ge 1}} \\liminf _{N\\rightarrow \\infty } \\frac{1}{N}\\sum _{t=1}^{N}I(X_t,S_{t-1};Y_t \\mid X^{t-1}),$ where the conditional distribution $P_{X_t,S_{t-1},Y_t|X^{t-1}}(x_t,s_{t-1},y_{t}|x^{t-1})=\\beta _{t-1}(s_{t-1})\\times P(x_t|s_{t-1})P(y_t|x_t,s_{t-1})$ , with $\\beta _{t-1}(s_{t-1})=P(s_{t-1}|x^{t-1})$ .",
"Remark When $Y^N=X^N$ , it can be seen that: $\\sum _{t=1}^{N}I(X_t,S_{t-1};Y_t \\mid X^{t-1})&=\\sum _{t=1}^{N} \\left[H(X_t|X^{t-1})- H(X_t|X_t)\\right]\\\\&=H(X^N).$ Hence, the lower bound is tight, and is equal to the maximum entropy rate of the input process.",
"The proof of Theorem REF is given in section .",
"The computability of the lower bound follows from Theorem REF .",
"Theorem 3.2 The lower bound expression in Theorem REF can be formulated as an infinite-horizon average-reward dynamic program, where the DP state is the probability vector $\\beta _{t-1} = \\bigl (P(s_{t-1}|x^{t-1}): s_{t-1}\\in {S}\\bigr )$ , the action is the stochastic matrix $[P(x_t|s_{t-1})]$ , and the disturbance is the channel input, $x_t$ .",
"We also propose the following alternative single-letter lower bound, when the channel also obeys the connectedness property defined in Section REF .",
"Theorem 3.3 For a connected input-driven FSC, given a ${V}$ -graph on the inputs, $C\\ge I_{{Q}}(X;Y|S,V),$ where $\\lbrace Q(x|s,v)\\rbrace \\in {Q}$ is an aperiodic distribution that induces the stationary distribution $\\pi (s,v)$ on the corresponding $({S},{V})$ -graph, and the random variables $X,Y,S,V$ are jointly distributed as $P_{X,Y,S,V}(x,y,s,v) = \\pi (s,v) Q(x|s,v) P(y|x,s)$ .",
"The proof of Theorem REF is presented in Section .",
"Theorems REF and REF provide powerful techniques to arrive at analytical lower bounds.",
"We then applied Theorem REF to input-constrained memoryless channels, and obtained the following lower bounds: The capacity of the $(d,\\infty )$ -RIC BSC($p$ ) satisfies $C\\ge \\max _{a\\in [0,1]}\\frac{h_b(ap+\\bar{a}\\bar{p})-h_b(p)}{ad+1}.$ This result holds for all $p\\in [0,1]$ , and, for $d=1$ , numerical evaluations indicate that the DP bound is close to the asymptotic bounds of [8] (as $p\\rightarrow 0$ ), and of [10] (as $p\\rightarrow 0.5$ ).",
"The capacity of the $(d,k)$ -RIC BSC($p$ ) obeys $C\\ge \\max _{a_d,\\ldots ,a_{k-1}}\\frac{\\sum \\limits _{i=d}^{k-1}(h_b(a_ip+\\bar{a_i}\\bar{p})-h_b(p))\\prod \\limits _{j=d}^{i-1}(1-a_j)}{d+1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j)},$ where $a_d,\\ldots ,a_{k-1}\\in [0,1]$ .",
"These lower bounds hold for arbitrary $0\\le d<k<\\infty $ .",
"For $0 \\le d < k \\le \\infty $ , the capacity of the $(d,k)$ -RIC BEC($\\epsilon $ ) satisfies $C\\ge C_{d,k} \\cdot \\bar{\\epsilon }$ , where $C_{d,k}$ is the noiseless capacity of the $(d,k)$ -RLL constraint.",
"In particular, when $d=0$ , the bound becomes tight as $k\\rightarrow \\infty $ , and for $d=1$ and $k=\\infty $ , it extends the asymptotic results of [9]."
],
[
"DP Formulation",
"In this section, we shall formulate the lower bound in Theorem REF as a DP problem, thereby showing the validity of Theorem REF .",
"We also introduce the Bellman equation, that provides a sufficient condition for optimality of the reward."
],
[
"DP Problem",
"An infinite-horizon average-reward DP is defined by the tuple $({Z},{U},{W},F,P_Z,P_w,g)$ .",
"We consider a discrete-time dynamic system evolving according to: $z_t=F(z_{t-1},u_t,w_t),\\quad t=1,2,\\ldots $ Each state, $z_t$ , takes values in a Borel space ${Z}$ , each action, $u_t$ , in a compact subset ${U}$ of a Borel space, and each disturbance, $w_t$ , in a measurable space ${W}$ .",
"The initial state, $z_0$ , is drawn from $P_Z$ , and the disturbance, $w_t$ , from $P_w(\\cdot |z_{t-1},u_t)$ .",
"At time $t$ , the action $u_t$ is equal to $\\mu _t(h_t)$ , where $h_t:=(z_0,w^{t-1})$ is the history up to time $t$ .",
"A policy $\\pi $ is defined as $\\pi :=\\lbrace \\mu _1,\\mu _2,\\ldots \\rbrace $ .",
"The aim is to maximize the average reward, given a bounded reward function $g:{Z}\\times {U}\\rightarrow \\mathbb {R}$ .",
"For a policy $\\pi $ , the average reward is: $\\rho _{\\pi }:=\\liminf _{N\\rightarrow \\infty } \\frac{1}{N}\\mathbb {E}_{\\pi }\\left[\\sum _{t=1}^{N}g(Z_{t-1},\\mu _t(h_t))\\right],$ where the subscript $\\pi $ indicates that the actions are generated by $\\pi =(\\mu _1,\\mu _2,\\ldots )$ .",
"The optimal average reward is defined as $\\rho ^{*}:=\\sup _{\\pi } \\rho _{\\pi }$ ."
],
[
"Lower Bound of Theorem ",
"From the DP formulation of the average reward in equation (REF ) and from Theorem REF , the DP state is chosen to be $z_{t-1}\\stackrel{\\Delta }{=}\\beta _{t-1}=\\bigl (P(s_{t-1}|x^{t-1}): s_{t-1}\\in {S}\\bigr )$ .",
"The action, $u_t$ , is the stochastic matrix $[P(x_t|s_{t-1})]$ , and the disturbance, $w_t$ , is the channel input, $x_t$ , which takes values in ${X}$ .",
"Further, we define the reward function, $g(\\beta _{t-1},u_t)=I(X_t,S_{t-1};Y_t|x^{t-1}).$ It is easy to see that the average of the reward function is indeed the lower bound of Theorem REF .",
"We now verify that the formulation above satisfies the properties of a DP problem.",
"Firstly, we note that the conditional distribution $P(x_t,s_{t-1},y_t|x^{t-1})=z_{t-1}(s_{t-1})P(x_t|s_{t-1})P(y_t|x_t,s_{t-1}),$ which depends only on the previous state and the action.",
"Therefore, the reward function, $g(\\cdot )$ at time $t$ , is a function of only $z_{t-1}$ and $u_t$ .",
"Secondly, it is easy to check that the disturbance distribution depends only on $z_{t-1}$ and $u_t$ : $&P(w_t|w^{t-1},z^{t-1},u^t)\\\\&=P(x_t|x^{t-1},\\beta ^{t-1},u^t)\\\\&\\stackrel{(a)}{=}\\sum _{y_t,s_{t-1}} \\beta _{t-1}(s_{t-1})P(x_t|s_{t-1},x^{t-1},\\beta ^{t-1},u^{t})P(y_t|x_t,s_{t-1})\\\\&\\stackrel{(b)}{=}\\sum _{y_t,s_{t-1}}\\beta _{t-1}(s_{t-1})P(x_t|s_{t-1},\\beta _{t-1},u_t)P(y_t|x_t,s_{t-1})\\\\&=P(x_t|\\beta _{t-1},u_t)\\\\&=P(w_t|\\beta _{t-1},u_t),$ where (a) follows from the fact that the value of $P(s_{t-1}|x^{t-1},\\beta ^{t-1},u^{t})$ is determined by $\\beta _{t-1}(s_{t-1})$ , and (b) follows from the fact that the distribution of $x_t$ depends only on the triplet $(s_{t-1},\\beta _{t-1},u_t)$ .",
"Lastly, we need to show that there exists a deterministic function $F:{Z}\\times {U}\\times {W}\\rightarrow {Z}$ , such that $z_t=F(z_{t-1},u_t,w_t)$ .",
"But we know that $z_t(s_t) = P(s_t|x^t) = \\sum _{s_{t-1}} P(s_{t-1}|x^{t-1}){1}\\lbrace f(s_{t-1},x_t)=s_t\\rbrace .$ Clearly, the next DP state is a function of the previous state and disturbance alone, and hence, the formulation above is a DP problem."
],
[
"Bellman Equation",
"The Bellman equation provides a sufficient condition that helps us verify that a given average reward is indeed optimal.",
"Theorem 4.1 ([16], Thm.",
"6.2) If $\\rho \\in \\mathbb {R}$ and a bounded function $h:{Z}\\rightarrow \\mathbb {R}$ satisfies $\\forall z\\in {Z}$ , $\\rho + h(z) = \\sup _{u\\in {U}}\\left[g(z,u) + \\int P_W(dw|z,u)h(F(z,u,w))\\right],$ then $\\rho ^{*}=\\rho $ ."
],
[
"$(d,k)$ -RIC Channels",
"In this section, we study the application of Theorem REF to certain input-constrained DMCs.",
"Specifically, we impose $(d,k)$ -RLL constraints, defined below, on input sequences.",
"Definition 5.1 A binary sequence $\\mathbf {x}=(x_1,x_2,\\ldots )\\in \\lbrace 0,1\\rbrace ^{*}$ is said to obey the $(d,k)$ -RLL constraint, (for $0\\le d<k\\le \\infty $ ) if each run of 0s in $\\mathbf {x}$ has length at most $k$ , and any pair of successive 1s is separated by at least $d$ 0s.",
"It is easily verified that $(d,k)$ -RIC DMCs are input-driven.",
"Indeed, we take the state space ${S}$ to be $\\lbrace 0,1,2,\\ldots ,d\\rbrace $ if $k = \\infty $ , and $\\lbrace 0,1,2,\\ldots ,k\\rbrace $ if $k < \\infty $ .",
"The state transitions are shown in the edge-labelled directed graphs in Figs.",
"REF and REF : an edge $s \\stackrel{x}{\\longrightarrow } s^{\\prime }$ represents the transition $s^{\\prime } = f(s,x)$ .",
"Figure: State transitions for the (d,∞)(d,\\infty )-RLL constraint.Figure: State transitions for the (d,k)(d,k)-RLL constraint, k<∞k < \\infty .Our assumption that the encoder and decoder share knowledge of the initial state $s_0$ is easily realized in this context, as they can a priori agree upon a choice of $s_0$ , e.g., $s_0 = 0$ .",
"Figure: Comparison of the DP lower bound for the (1,∞)(1,\\infty )-RIC BSC(pp) with bounds in , and .Theorem 5.1 The capacity of the $(d,\\infty )$ -RIC binary symmetric channel with cross-over probability $p$ obeys $C_{d,\\infty }^{\\text{BSC}(p)}\\ge \\max _{a\\in [0,1]} \\frac{h_b(ap+\\bar{a}\\bar{p})-h_b(p)}{ad+1}.$ The DP state, $\\mathbf {z}$ , is a probability vector on ${S} = \\lbrace 0,1,2,\\ldots ,d\\rbrace $ , with elements indexed as $z_s,$ $s\\in {S}$ .",
"As the channel is input-driven, we have $z_s \\in \\lbrace 0,1\\rbrace \\, \\forall s\\in {S}$ , and exactly one $z_s$ can be equal to 1.",
"With some abuse of notation, we write the DP state as $z=i$ when $z_i=1$ , for $0\\le i\\le d$ .",
"The disturbance $w$ , is equal to $x$ , and the action, $u$ , is the stochastic matrix $u\\text{ }= \\begin{blockarray}{ccc}& 0 & 1 \\\\\\begin{block}{c[cc]}0 & 1 & 0 \\\\1 & 1 & 0 \\\\\\vdots & \\vdots & \\vdots \\\\d-1 & 1 & 0 \\\\d & 1-a & a \\\\\\end{block}\\end{blockarray}$ where the rows correspond to the channel states, and $a\\in [0,1]$ .",
"The next DP state is given by: $F(z,u,x)=\\psi (z,x),$ where $\\psi (z,x)=z^\\prime $ , if the edge $(z,z^\\prime )$ labelled by $x$ exists in the presentation.",
"From the conditional distribution in (REF ), the reward function can be computed as: $g(z,u)&=H(Y|z,u)-H(Y|X)\\\\&=h_b\\left(z_d(a+p-2ap)+p\\sum \\limits _{i=0}^{d-1}z_i\\right)-h_b(p).$ Solving the Bellman equation of Theorem REF entails identifying a scalar $\\rho _p$ such that for a function $h_p:{Z}\\rightarrow \\mathbb {R}$ , $\\rho _p + h_p(z)=\\max _{a\\in [0,1]} [g(z,a)+(1-az_d)h_p(\\psi (z,0))\\\\+az_dh_p(\\psi (z,1))],$ for each $z\\in {Z}$ .",
"The set of $d+1$ equations in (REF ) can be split as: $\\rho _p + h_p(i) = h_b(p)-h_b(p)+h_p(i+1)=h_p(i+1),$ for $0\\le i\\le d-1$ , since $z_d=0$ , and $\\rho _p = \\max _a \\left[h_b(a+p-2ap)-h_b(p)+a(h_p(0)-h_p(d))\\right].$ From the first set of $d$ equations, we arrive at the fact that $d\\rho _p=h_p(d)-h_p(0)$ .",
"Substituting this in equation (REF ), we note that: $\\max _{a\\in [0,1]}\\left[h_b(ap+\\bar{a}\\bar{p})-h_b(p)-(ad+1)\\rho _p\\right]=0.$ Clearly, the choice $\\rho _p = \\max _{a\\in [0,1]}\\frac{h_b(ap+\\bar{a}\\bar{p})-h_b(p)}{ad+1}$ satisfies (REF ).",
"For $d=1$ , figure REF shows plots of our DP lower bound, alongside the lower bound of Ordentlich [10].",
"Upper bounds on the capacity in the form of the feedback capacity of the $(1,\\infty )$ -RIC BSC($p$ ) [17], and the dual capacity upper bound of Thangaraj [18] are also shown.",
"Numerical evaluations indicate that the DP lower bound is close to the asymptotic bounds in [8] as $p\\rightarrow 0$ , and in [10] as $p\\rightarrow 0.5$ , and extends these results to all values of $p$ .",
"Plots of the DP lower bound for $d=1,2,3$ , are given in figure REF , with the unconstrained ($d=0$ ) capacity also indicated.",
"Figure: DP lower bounds for the (1,∞)(1,\\infty ), (2,∞)(2,\\infty ), (3,∞)(3,\\infty )-RIC BSC(pp).We now consider the $(d,k)$ -RIC BSC($p$ ), with $k < \\infty $ .",
"Theorem 5.2 The capacity of the $(d,k)$ -RIC $(k<\\infty )$ binary symmetric channel with cross-over probability $p$ satisfies $C_{d,k}^{\\text{BSC}(p)}\\ge \\max _{a_d,\\ldots ,a_{k-1}}\\frac{\\sum \\limits _{i=d}^{k-1}\\bigl (h_b(a_ip+\\bar{a}_i\\bar{p})-h_b(p)\\bigr )\\prod \\limits _{j=d}^{i-1}(1-a_j)}{d+1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j)},$ where the maximization is over $a_d,\\ldots ,a_{k-1}\\in [0,1]$ , and an empty product is, by convention, equal to 1.",
"This time, the DP state, $\\mathbf {z}$ , is a probability vector on ${S} = \\lbrace 0,1,2,\\ldots ,k\\rbrace $ , with elements indexed as $z_s,$ $s\\in {S}$ .",
"As in the proof of Theorem REF , we have $z_s \\in \\lbrace 0,1\\rbrace \\, \\forall s\\in {S}$ , and exactly one $z_s$ can be equal to 1, and so again, we write the DP state as $z=i$ when $z_i=1$ , for $0\\le i\\le k$ .",
"The disturbance, $w$ , is equal to $x$ , and the action, $u$ , is the stochastic matrix $u\\text{ }= \\begin{blockarray}{ccc}& 0 & 1 \\\\\\begin{block}{c[cc]}0 & 1 & 0 \\\\1 & 1 & 0 \\\\\\vdots & \\vdots & \\vdots \\\\d-1 & 1 & 0 \\\\d & 1-a_d & a_d \\\\\\vdots & \\vdots & \\vdots \\\\k-1 & 1-a_{k-1} & a_{k-1}\\\\k & 0 & 1\\\\\\end{block}\\end{blockarray}$ where $a_d,\\ldots ,a_{k-1}\\in [0,1]$ .",
"The next DP state is given by $F(z,u,x)=\\psi (z,x)$ , where $\\psi (z,x)=z^\\prime $ , if the edge $(z,z^\\prime )$ labelled by $x$ exists in the presentation.",
"The reward function then is: $g(z,u)=h_b(p\\delta + \\bar{p}\\bar{\\delta })-h_b(p),$ where $\\delta := \\sum \\limits _{i=0}^{d-1}z_i + \\sum \\limits _{i=d}^{k-1}z_i(1-a_i)$ .",
"The Bellman equation in Theorem REF simplifies to $\\rho _p + h_p(i) = h_p(i+1)$ for $0\\le i\\le d-1$ , $\\rho _p + h_p(i)=\\max _{a_i} [h_b(p\\bar{a}_i+\\bar{p}a_i)+(1-a_i)h_p(i+1)\\\\+a_ih_p(0)],$ for $d\\le i\\le k-1$ , and $\\rho _p =h_p(0)-h_p(k),$ for $\\rho _p\\in \\mathbb {R}$ , and $h_p:{Z}\\rightarrow \\mathbb {R}$ .",
"Adding together the set of $d$ equations in (REF ) yields $d\\rho _p = h_p(d)-h_p(0).$ Now, since we have that $\\rho _p + h_p(k-1) = \\max _{a_{k-1}}[h_b(p\\bar{a}_{k-1}+\\bar{p}a_{k-1})-h_b(p)\\\\+(1-a_{k-1})h_p(k)+a_{k-1}h_p(0)],$ we substitute $h_p(k)$ as $h_p(0)-\\rho _p$ from (REF ), giving $h_p(k-1)-h_p(0)=\\max _{a_{k-1}}[h_b(p\\bar{a}_{k-1}+\\bar{p}a_{k-1})-h_b(p)\\\\+(a_{k-1}-2)\\rho _p].$ We now substitute $h_p(k-1)$ from (REF ) in the penultimate equation of (REF ), to get $h_p(k-2)-h_p(0)$ in terms of $\\rho _p$ and $p$ alone.",
"Proceeding similarly, we arrive at: $h_p(d)-h_p(0)=\\max _{a_d,\\ldots ,a_{k-1}}[h_b(p\\bar{a}_d+a_d\\bar{p})-h_b(p)+\\\\\\sum \\limits _{i=d+1}^{k-1}(h_b(a_ip+\\bar{a_i}\\bar{p})-h_b(p))\\prod \\limits _{j=d}^{i-1}(1-a_j)\\\\-\\rho _p(1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j))].$ Using (REF ), it is clear that the choice $\\rho _p = \\max _{a_d,\\ldots ,a_{k-1}}\\frac{\\sum \\limits _{i=d}^{k-1}(h_b(a_ip+\\bar{a_i}\\bar{p})-h_b(p))\\prod \\limits _{j=d}^{i-1}(1-a_j)}{d+1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j)},$ satisfies (REF ), where the empty product is taken to be 1.",
"We note here that the lower bound in the theorem is exactly equal to that presented in Lemma 5 of [7], which evaluates a lower bound on the maximum mutual information rate among stationary Markovian input distributions on the graph in Fig.",
"3.",
"Fig.",
"REF shows plots of the DP lower bound for the $(0,k)$ -RIC BSC($p$ ), for $k=1,2,3$ , alongside the capacity of the unconstrained ($k\\rightarrow \\infty $ ) BSC($p$ ).",
"Figure: DP lower bounds for the (0,1)(0,1), (0,2)(0,2) and (0,3)(0,3)-RIC BSC(pp).We now move on to the input-constrained BEC($\\epsilon $ ).",
"Theorem 5.3 The capacity of the $(d,\\infty )$ -RIC binary erasure channel with erasure probability $\\epsilon $ satisfies $C_{d,\\infty }^{\\text{BEC}(\\epsilon )}\\ge C_{d,\\infty }\\cdot \\bar{\\epsilon },$ where $C_{d,\\infty }=\\underset{a\\in [0,1]}{\\max } \\frac{h_b(a)}{ad+1}$ is the (noiseless) capacity of the $(d,\\infty )$ -RLL constraint.",
"Just as in the proof of Theorem REF , the DP state, $\\mathbf {z}$ , is a probability vector on ${S} = \\lbrace 0,1,\\ldots ,d\\rbrace $ , with elements indexed as $z_s\\in \\lbrace 0,1\\rbrace $ , $s\\in {S}$ , and we write the DP state, $z=i$ , when $z_i=1$ , for $0\\le i\\le d$ .",
"The disturbance, $w$ , is equal to $x$ , and the action, $u$ , is the same stochastic matrix as in the proof of Theorem REF .",
"Also, the next DP state is decided by the deterministic function, $\\psi (\\cdot )$ , as defined in the proof of Theorem REF .",
"Now, the reward function can be computed to be: $g(z,u)&=H(Y|z,u)-H(Y|X)\\\\&= H_{ter}(az_d\\bar{\\epsilon },\\epsilon )-h_b(\\epsilon )\\\\&\\stackrel{(a)}{=} \\bar{\\epsilon }h_b(az_d),$ where (a) follows from the fact that $H_{ter}(c\\bar{d},d)=h_b(d)+\\bar{d}h_b(c)$ .",
"Solving the Bellman equation entails identifying a scalar $\\rho _\\epsilon $ such that for a function $h_\\epsilon :{Z}\\rightarrow \\mathbb {R}$ , $\\rho _\\epsilon + h_\\epsilon (z)=\\max _{a\\in [0,1]} [g(z,a)+(1-az_d)h_\\epsilon (\\psi (z,0))\\\\+az_dh_\\epsilon (\\psi (z,1))],$ for each $z\\in {Z}$ .",
"The set of $d+1$ equations in (REF ) can be split as: $\\rho _\\epsilon + h_\\epsilon (i) = h_\\epsilon (i+1),$ for $0\\le i\\le d-1$ , since $z_d=0$ , and $\\rho _\\epsilon = \\max _a \\left[\\bar{\\epsilon }h_b(a)+a(h_\\epsilon (0)-h_\\epsilon (d))\\right].$ From the first set of $d$ equations, we arrive at $d\\rho _\\epsilon =h_\\epsilon (d)-h_\\epsilon (0)$ .",
"Substituting this in equation (REF ), we get: $\\max _{a\\in [0,1]}[\\bar{\\epsilon }h_b(a)-\\rho _\\epsilon (ad+1)]=0.$ The choice $\\rho _\\epsilon = \\bar{\\epsilon }\\max _{a}\\left\\lbrace \\frac{h_b(a)}{ad+1}\\right\\rbrace $ satisfies (REF ).",
"We note that at $\\epsilon =0$ , $Y^N$ is equal to $X^N$ .",
"Hence, the coefficient of $\\bar{\\epsilon }$ is equal to the noiseless capacity of the $(d,\\infty )$ -RIC, by the remark following Theorem REF .",
"For $d=1$ , a comparison between the DP lower bound and the “memory 1” dual capacity upper bound of Thangaraj [18] are shown in figure REF , along with a plot of the feedback capacity [19].",
"Numerical evaluations indicate that the DP lower bound also closely approximates the asymptotic ($\\epsilon \\rightarrow 0$ ) lower bound of Li et al.",
"[9].",
"We now provide a lower bound on the capacity of the $(d,k)$ -RIC BEC($\\epsilon $ ).",
"Figure: Comparison of the DP lower bound for the (1,∞)(1,\\infty )-RIC BEC(ϵ\\epsilon ) with bounds in and .Theorem 5.4 The capacity of the $(d,k)$ -RIC $(k<\\infty )$ binary erasure channel with erasure probability $\\epsilon $ satisfies $C_{d,k}^{\\text{BEC}(\\epsilon )}\\ge C_{d,k} \\cdot \\bar{\\epsilon },$ where $C_{d,k}=\\underset{a_d,\\ldots ,a_{k-1}}{\\max } \\frac{\\sum \\limits _{i=d}^{k-1}h_b(a_i)\\prod \\limits _{j=d}^{i-1}(1-a_j)}{d+1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j)}$ is the (noiseless) capacity of the $(d,k)$ -RLL constraint.",
"The DP state, $\\mathbf {z}$ , is a probability vector on ${S} = \\lbrace 0,1,\\ldots , k\\rbrace $ , with elements indexed as $z_s\\in \\lbrace 0,1\\rbrace $ , $s\\in {S}$ , and we write the DP state, $z=i$ , when $z_i=1$ , for $0\\le i\\le k$ .",
"The disturbance, $w$ , is equal to $x$ , and the action, $u$ , is the same stochastic matrix as in the proof of Theorem REF .",
"The next DP state is dictated by the deterministic function, $\\psi (\\cdot )$ , defined in the proof of Theorem REF .",
"The reward function is computed to be: $g(z,u)&=H(Y|z,u)-H(Y|X)\\\\&= H_{ter}\\left(\\bar{\\epsilon }\\left(z_k+\\sum \\limits _{i=d}^{k-1}z_ia_i\\right),\\epsilon \\right)-h_b(\\epsilon )\\\\&= \\bar{\\epsilon }h_b\\left(z_k+\\sum \\limits _{i=d}^{k-1}z_ia_i\\right),$ where $H_{ter}(\\cdot ,\\cdot )$ denotes the ternary entropy function, and the last equality follows from the property of the ternary entropy function.",
"The Bellman equations can then be split as: $\\rho _\\epsilon + h_\\epsilon (i)&=\\max _{a_d,\\ldots ,a_{k-1}} [g(z,u)+(1-z_k-\\sum \\limits _{i=d}^{k-1}a_iz_i)h_\\epsilon (\\psi (z,0))]\\\\&=h_\\epsilon (i+1),$ for $0\\le i\\le d-1$ , and $\\rho _\\epsilon + h_\\epsilon (i)=\\max _{a_i} [\\bar{\\epsilon }h_b(a_i)+(1-a_i)h_\\epsilon (i+1)+a_ih_\\epsilon (0)],$ for $d\\le i\\le k-1$ , and $\\rho _\\epsilon =h_\\epsilon (0)-h_\\epsilon (k),$ for $\\rho _\\epsilon \\in \\mathbb {R}$ , and $h_\\epsilon :{Z}\\rightarrow \\mathbb {R}$ .",
"Again, the first set of $d$ equations gives us $d\\rho _\\epsilon = h_\\epsilon (d)-h_\\epsilon (0).$ Now, since we have that $\\rho _\\epsilon + h_\\epsilon (k-1) = \\max _{a_{k-1}}[\\bar{\\epsilon }h_b(a_{k-1})+(1-a_{k-1})h_\\epsilon (k)+\\\\a_{k-1}h_\\epsilon (0)],$ we substitute (REF ), giving $\\rho _\\epsilon + h_\\epsilon (k-1)=\\max _{a_{k-1}}[\\bar{\\epsilon }h_b(a_{k-1})+h_\\epsilon (0)-(1-a_{k-1})\\rho _\\epsilon ],$ and hence, $h_\\epsilon (k-1)-h_\\epsilon (0)=\\max _{a_{k-1}}[\\bar{\\epsilon }h_b(a_{k-1})+(a_{k-1}-2)\\rho _\\epsilon ].$ Proceeding similarly, we arrive at $h_\\epsilon (d)-h_\\epsilon (0)=\\max _{a_d,\\ldots ,a_{k-1}}[\\bar{\\epsilon }h_b(a_d)+\\bar{\\epsilon }\\sum \\limits _{i=d+1}^{k-1}h_b(a_{i})\\prod \\limits _{j=d}^{i-1}(1-a_j)\\\\-\\rho _\\epsilon (1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j))].$ Now, using (REF ), we see that $\\rho _\\epsilon $ obeys $\\rho _\\epsilon = \\bar{\\epsilon }\\cdot \\max _{a_d,\\ldots ,a_{k-1}}\\left(\\frac{\\sum \\limits _{i=d}^{k-1}h_b(a_i)\\prod \\limits _{j=d}^{i-1}(1-a_j)}{d+1+\\sum \\limits _{i=d}^{k-1}\\prod \\limits _{j=d}^{i}(1-a_j)}\\right),$ where the empty product is taken to be 1.",
"Again, the coefficient of $\\bar{\\epsilon }$ is equal to the noiseless capacity of the $(d,k)$ -RIC, by the remark following Theorem REF .",
"Remark In the theorems above, the fact that the coefficients of $\\bar{\\epsilon }$ are indeed equal to the noiseless capacities also follows from the observation that the coefficients are the entropies of the maxentropic Markov chains on the graphs in Figs.",
"REF and REF , and hence, by Theorem 3.23 of [20], are equal to the noiseless capacities."
],
[
"Proofs",
"[Proof of Theorem REF ] By way of (REF ), we have $I(X^N;Y^N \\mid s_0) & \\ge I(Y^N \\rightarrow X^N \\mid s_0) \\\\&\\ge \\sum _{t=1}^N I(X_t;Y_t \\mid X^{t-1},s_0) \\\\&= \\sum _{t=1}^N I(X_t,S_{t-1};Y_t \\mid X^{t-1},s_0), $ the last equality following from the state evolution of input-driven FSCs.",
"Hence, via Theorem REF , we have $C &= \\lim _{N\\rightarrow \\infty } \\max _{\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^{N}} \\frac{1}{N} I(X^N;Y^N \\mid s_0) \\\\&\\ge \\lim _{N\\rightarrow \\infty } \\max _{\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^{N}} \\frac{1}{N} \\sum _{t=1}^N I(X_t,S_{t-1};Y_t \\mid X^{t-1},s_0)\\\\&= \\sup _{\\lbrace P(x_t|x^{t-1})\\rbrace _{t\\ge 1}} \\liminf _{N\\rightarrow \\infty } \\frac{1}{N} \\sum _{t=1}^N I(X_t,S_{t-1};Y_t \\mid X^{t-1}),$ the last equality above following by the arguments in Lemma 4 of [21], the conditioning on $s_0$ being suppressed in the notation.",
"Finally, we can replace the supremum over $\\lbrace P(x_t|x^{t-1})\\rbrace _{t \\ge 1}$ by a supremum over input distributions of the form $\\lbrace P(x_t|s_{t-1})\\rbrace _{t \\ge 1}$ , possibly at the expense of another inequality.",
"[Proof of Theorem REF ] For any fixed $s_0$ , we have $C_N &:= \\max _{\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^{N}} \\frac{1}{N}I(X^{N};Y^{N} \\mid s_0) \\\\&\\stackrel{(a)}{\\ge } \\max _{\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^{N}} \\frac{1}{N}\\sum _{t=1}^{N} I(X_t;Y_t \\mid X^{t-1},s_0) \\\\&\\stackrel{(b)}{=} \\max _{\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^{N}} \\frac{1}{N} \\sum _{t=1}^{N} I(X_t;Y_t \\mid S_{t-1},X^{t-1},s_0), $ where (a) is by (REF ), and (b) is by the fact that for an input-driven channel, $S_{t-1}$ is determined by $X^{t-1}$ and $s_0$ .",
"Now, let $\\lbrace Q(x|s,v)\\rbrace \\in {Q}$ be an aperiodic input distribution, so that ${G}(\\lbrace Q(x|s,v)\\rbrace )$ has a single closed communicating class, ${{G}}_0$ , that is also aperiodic.",
"By the connectedness of the FSC, arguing as in Lemma 1 of [15], there is some vertex $v_0$ of the ${V}$ -graph such that $(s_0,v_0)$ is in ${{G}}_0$ .",
"We will continue the chain of inequalities from (REF ) by specifying an input distribution $\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^N$ in terms of $\\lbrace Q(x|s,v)\\rbrace $ .",
"For all $t \\ge 1$ , set $P(x_t \\mid x^{t-1},s_0) = Q(x_t \\mid s_{t-1},v_{t-1}),$ where $s_{t-1} = f(s_0,x^{t-1})$ is the state at time $t-1$ reached by the FSC starting at $s_0$ and driven by the inputs $x^{t-1}$ , and $v_{t-1} = \\Phi (v_0,x^{t-1})$ is the vertex of the ${V}$ -graph at the end of the path labelled by $x^{t-1}$ starting at $v_0$ .",
"Note that this choice of input distribution induces the following Markov chain: $X^{t-1}\\textbf {—}(S_{t-1},V_{t-1})\\textbf {—}(X_t,Y_t),$ where $S_{t-1} = f(s_0,X^{t-1})$ and $V_{t-1} = \\Phi (v_0,X^{t-1})$ .",
"Thus, carrying on from (REF ), with the input distribution $\\lbrace P(x_t|x^{t-1},s_0)\\rbrace _{t=1}^{N}$ specified as above, we have $C_N &\\ge \\frac{1}{N} \\sum _{t=1}^N I(X_t;Y_t \\mid S_{t-1},X^{t-1},s_0) \\\\&= \\frac{1}{N} \\sum _{t=1}^N I(X_t;Y_t \\mid S_{t-1},V_{t-1}).",
"$ The last equality is due to the fact that $I(X_t;Y_t \\mid S_{t-1},X^{t-1},s_0) = I(X_t;Y_t \\mid S_{t-1},V_{t-1},X^{t-1},s_0)$ , since $V_{t-1}$ is determined by $X^{t-1}$ and the (fixed) vertex $v_0$ , and the latter mutual information equals $I(X_t;Y_t \\mid S_{t-1},V_{t-1})$ by the Markov chain in (REF ).",
"Finally, taking limits as $N \\rightarrow \\infty $ in (REF ), we obtain the desired bound $C \\ge I_Q(X;Y|S,V)$ , since $\\lim _{N \\rightarrow \\infty } \\frac{1}{N} \\sum _{t=1}^N I(X_t;Y_t \\mid S_{t-1},V_{t-1}) = I_Q(X;Y \\mid S,V)$ by the ergodicity of the Markov chain on ${S} \\times {V}$ induced by the aperiodic input distribution $\\lbrace Q(x|s,v)\\rbrace $ ."
],
[
"Conclusions and Future Work",
"In this work, novel lower bounds on the capacities of input-driven FSCs were derived.",
"The main idea was the lower bounding of the mutual information rate by the reverse directed information rate.",
"A DP formulation of the lower bound was given, which was then applied to input-constrained memoryless channels, resulting in simple analytical expressions that are valid for all values of the channel parameters, thereby extending known asymptotic results.",
"Furthermore, an alternative single-letter lower bound on the capacity was derived, using the concept of directed ${V}$ -graphs.",
"This paper focuses on lower bounding the reverse directed information rate only.",
"The lower bounds on capacity that this approach yields can be improved by estimating or bounding the forward directed information rate from the input distribution that optimizes the reverse rate."
],
[
"Acknowledgment",
"This work was supported in part by a MATRICS grant (no.",
"MTR/2017/000368) administered by the Science and Engineering Research Board (SERB), Govt.",
"of India."
]
] | 2001.03423 |
[
[
"Electrodynamics and radiation from rotating neutron star magnetospheres"
],
[
"Abstract Neutron stars are compact objects rotating at high speed, up to a substantial fraction of the speed of light (up to 20\\% for millisecond pulsars) and possessing ultra-strong electromagnetic fields (close to and sometimes above the quantum critical field of \\numprint{4.4e9}~\\SIunits{\\tesla}).",
"Moreover, due to copious $e^\\pm$ pair creation within the magnetosphere, the relativistic plasma surrounding the star is forced into corotation up to the light cylinder where the corotation speed reaches the speed of light.",
"The neutron star electromagnetic activity is powered by its rotation which becomes relativistic in the neighbourhood of this light cylinder.",
"These objects naturally induce relativistic rotation on macroscopic scales about several thousands of kilometers, a crucial ingredient to trigger the central engine as observed on Earth.",
"In this paper, we elucidate some of the salient features of this corotating plasma subject to efficient particle acceleration and radiation, emphasizing several problems and limitations concerning current theories of neutron star magnetospheres.",
"Relativistic rotation in these systems is indirectly probed by the radiation produced within the magnetosphere.",
"Depending on the underlying assumptions about particle motion and radiation mechanisms, different signatures on their light-curves, spectra, pulse profiles and polarisation angles are expected in their broadband electromagnetic emission.",
"We show that these measurements put stringent constraints on the way to describe particle electrodynamics in a rotating neutron star magnetosphere."
],
[
"Introduction",
"Neutron stars are compact objects produced by the explosion of a massive star or by the collapse of an accreting white dwarf reaching the Chandrasekhar limit of about $1.44\\,M_\\odot $ [1], [2] where $M_\\odot $ is the solar mass.",
"They represent the ultimate fate of the stellar evolution of massive stars before the black hole stage.",
"During their birth, their angular speed and magnetic field are amplified by several orders of magnitude.",
"It is not yet clear what mechanisms are able to produce the expected fields in the range of $B \\approx {e8}-{e11}~{}$ , but a dynamo effect and magnetic flux freezing during the collapse are certainly key processes.",
"Neutron star rotation periods span four decades from several milliseconds to tenths of seconds.",
"The stellar magnetic field drags charged particles into corotation with the star.",
"Relativistic corotating speeds are reached at the light cylinder defined by $r_{\\rm L}= \\frac{c}{\\omega } = \\frac{c\\,P}{2\\,\\pi } = 48~{} \\left( \\frac{P}{1~{}} \\right)$ where $c$ is the speed of light, $P=2\\pi /\\omega $ the pulsar period and $\\omega $ its rotation rate.",
"Moreover, their compactness places them closest to the black hole stage because $\\frac{R_{\\rm s}}{R} = 0.345 \\, \\left( \\frac{M}{1.4~M_\\odot } \\right) \\, \\left( \\frac{R}{12~{}} \\right)^{-1}$ where $R_{\\rm s}=2\\,G\\,M/c^2$ is the Schwarzschild radius, $M$ and $R$ the neutron star mass and radius respectively and $G$ the gravitational constant.",
"Neutron stars are therefore places in the universe where general relativity and quantum electrodynamics act together to sustain their electromagnetic activity.",
"Simple but realistic orders of magnitude for neutron star rotation periods and magnetic field strengths are easily derived from conservation of angular momentum and magnetic flux during the collapse of the progenitor.",
"These conservation laws imply $M \\, \\omega \\, R^2 & = M_* \\, \\omega _* \\, R_*^2 \\\\B \\, R^2 & = B_* \\, R_*^2$ where the star subscript $*$ refers to quantities relative to the progenitor.",
"Magnetic field and rotation frequency therefore increase by a factor as large as $R_*^2/R^2 \\approx 10^{10}$ if dissipation processes are neglected and all angular momentum and magnetic flux of the progenitor go to the neutron star.",
"The magnetic flux conservation argument, equation (REF ), was first discussed by [3].",
"The above estimates are however probably largely overestimated because not all of the progenitor is collapsing and because its mass is not conserved during the implosion [4].",
"Only the iron core of a massive star produces a neutron star star whereas the outer shells are expelled.",
"Electrodynamics in a relativistically rotating frame is a crucial ingredient in our understanding and modelling of neutron star magnetospheres.",
"In this paper, we discuss some issues about rotating magnetospheres and their radiative properties.",
"First we briefly remind the rotating coordinate system used and its implication for field transformations and especially for Maxwell equations in section .",
"Next we discuss the electromagnetic field expected inside and outside a neutron star in section .",
"The motion of charged particles in this field is exposed in section .",
"Such trajectories can be computed in solutions found from numerical simulations as explained in section .",
"However, corotation is not compulsory and differentially rotating magetospheres have been found as shown in section .",
"Some clues about the electrodynamics of pulsar magnetospheres can be gained from their radiation as explained in section .",
"Possible extensions to general relativity and multipolar magnetic fields are discussed in Sec. .",
"We conclude with a brief summary in section ."
],
[
"Rotating vs inertial frame",
"Neutron stars are mainly observed through their pulsed emission detected in the radio wavelength [5] but also at very high-energy by gamma-ray photons in the MeV/GeV band [6] or through thermal X-ray emission from hot spots on the surface [7].",
"Such emission is attributed to ultra-relativistic charged particles flowing inside the magnetosphere.",
"The very stable and periodic pulsation is explained by the stellar rotation.",
"It is believed that these particles corotate with the star almost up to the light cylinder and generate a relativistic magnetized outflow outside the light cylinder known as the pulsar wind [8].",
"Description of the flow taking properly into account this corotation is therefore central to our understanding of neutron star magnetospheric emission.",
"Before going into the dynamics of this plasma and its underlying particle trajectories let us briefly review relativistic rotating frames and related electrodynamics problems.",
"An excellent review about relativistically rotating frames can be found in [9]."
],
[
"Metric",
"The space-time geometry of a rotating system is best described in a cylindrical coordinate system labelled by $(t,r,\\varphi ,z)$ .",
"The coordinate transformation from an inertial observer $(t,r,\\varphi ,z)$ to an observer rotating at the stellar angular speed $(t^{\\prime },r^{\\prime },\\varphi ^{\\prime },z^{\\prime })$ is usually written as $t^{\\prime }=t \\qquad ; \\qquad r^{\\prime }=r \\qquad ; \\qquad \\varphi ^{\\prime } = \\varphi - \\omega \\,t \\qquad ; \\qquad z^{\\prime }=z .$ It is essential to realise that this transformation does not lead to a new orthonormal basis in the rotating frame.",
"Therefore physical quantities measured locally by an observer in this coordinate system can not be directly read off from these basis vectors.",
"Indeed, the space-time geometry of an uniformly rotating frame is given in the rotating observer frame by the metric $ds^2 = c^2\\,dt^2 - dr^2 - \\frac{r^2\\,d\\varphi ^2}{1-r^2\\,\\omega ^2/c^2} - dz^2 .$ The coordinates $(t^{\\prime },r^{\\prime },\\varphi ^{\\prime },z^{\\prime })$ are not related to any observer because they do not form an orthonormal basis.",
"In order to relate measurements between the inertial and the rotating observers, it is more convenient to introduce two orthonormal bases attached to each observer.",
"The transformation from the inertial to the rotating frame is simply a Lorentz boost as is always the case when performing reference frame transformations between orthonormal bases.",
"Explicitly, the orthonormal basis vectors of the rotating frame are $\\mathbf {e}_0^{\\prime } & = \\Gamma \\left( \\mathbf {e}_t + \\beta \\, ( - \\sin \\Phi \\, \\mathbf {e}_x + \\cos \\Phi \\, \\mathbf {e}_y ) \\right) = \\Gamma \\, ( 1, - \\beta \\, \\sin \\Phi , \\beta \\, \\cos \\Phi , 0 ) \\\\\\mathbf {e}_1^{\\prime } & = \\cos \\Phi \\, \\mathbf {e}_x + \\sin \\Phi \\, \\mathbf {e}_y = (0, \\cos \\Phi , \\sin \\Phi , 0) \\\\\\mathbf {e}_2^{\\prime } & = \\Gamma \\, \\left( \\beta \\, \\mathbf {e}_t - \\sin \\Phi \\, \\mathbf {e}_x + \\cos \\Phi \\, \\mathbf {e}_y \\right) = \\Gamma \\,( \\beta , - \\sin \\Phi , \\cos \\Phi , 0 ) \\\\\\mathbf {e}_3^{\\prime } & = \\mathbf {e}_z = (0,0,0,1)$ where the phase is $\\Phi =\\omega \\,t$ , the relative speed is $\\beta = \\frac{\\omega \\,r}{c}$ , and the Lorentz factor $\\Gamma = (1-\\beta ^2)^{-1/2}$ .",
"This coordinate transformation is of Lorentz type, going from a Minkowskian metric to another Minkowskian metric.",
"Building on this orthonormal basis, it is easy to deduce the relation between the electromagnetic fields measured by the two observers as we now show."
],
[
"Electrodynamics",
"The electromagnetic field tensor in an inertial frame is derived from the electromagnetic quadri-potential $A_i = (\\phi /c, - \\mathbf {A})$ ($\\phi $ being the scalar potential and $\\mathbf {A}$ the vector potential adopting the metric with signature (+,-,-,-)) by $F_{ik} = \\partial _i A_k - \\partial _k A_i$ .",
"The electromagnetic field tensors in the inertial and rotating frame are related by a special relativistic transformation according to the previous section from the basis transformation eq.",
"(REF ).",
"They are also found from the definition using the observer 4-velocity $\\mathbf {u}$ by projection of the electromagnetic field tensor $F_{ik}$ and its dual ${^*\\!F}_{ik} = \\frac{1}{2} \\, \\epsilon _{ikmn} \\, F^{mn}$ [10] onto the observer world line such that $E_i/c & = F_{ik} \\, u^k \\\\B_i & = {^*\\!F}_{ik} \\, u^k .$ Explicit computations of these transformations show that it is simply the Lorentz transformation of the electromagnetic field between two inertial observers with relative velocity $\\mathbf {V} = r \\, \\omega \\, \\mathbf {e}_\\varphi $ .",
"Introducing the normalized velocity by $\\beta = \\mathbf {V}/c$ , the transformations of the electric and magnetic field vectors are $\\mathbf {E}^{\\prime } & = \\Gamma \\, \\left[ \\mathbf {E} - \\frac{\\Gamma }{\\Gamma +1} \\, (\\beta \\cdot \\mathbf {E}) \\, \\beta + c \\, \\beta \\wedge \\mathbf {B} \\right] \\\\\\mathbf {B}^{\\prime } & = \\Gamma \\, \\left[ \\mathbf {B} - \\frac{\\Gamma }{\\Gamma +1} \\, (\\beta \\cdot \\mathbf {B}) \\, \\beta - \\beta \\wedge \\mathbf {E}/c\\right]$ where primed quantities are defined in the rotating frame.",
"There is nothing special about rotation if transformations are made locally between inertial observers.",
"This holds for aberration and Doppler effects when emission emanates from within the light cylinder ($r<r_{\\rm L}$ ).",
"These results are easily extended to general relativity when gravitation is included.",
"However special relativity applies if transformations are made locally between inertial observers [11]."
],
[
"Doppler effect",
"The Doppler effect is subject to the same transformation as the electromagnetic field.",
"A simple Lorentz transformation between inertial frames holds to relate photon propagation direction $\\mathbf {n}=(n_r,n_\\theta ,n_\\varphi )$ and frequency $\\nu $ in the observer frame and in the instantaneous inertial frame coinciding locally with the corotating frame.",
"It can be checked by a direct derivation from the transformation of coordinates between both frames as given by eq.",
"(REF ).",
"For the sake of completeness, they are given in a spherical coordinate systems $(r,\\vartheta ,\\varphi )$ by $\\nu & = \\gamma _{\\rm obs} \\, \\nu ^{\\prime } \\, ( 1 + \\beta _{\\rm obs} \\, n_\\varphi ^{\\prime } ) \\\\n_r & = \\frac{n_r^{\\prime }}{\\gamma _{\\rm obs} \\, ( 1 + \\beta _{\\rm obs} \\, n_\\varphi ^{\\prime } )} \\\\n_\\vartheta & = \\frac{n_\\vartheta ^{\\prime }}{\\gamma _{\\rm obs} \\, ( 1 + \\beta _{\\rm obs} \\, n_\\varphi ^{\\prime } )} \\\\n_\\varphi & = \\frac{\\beta _{\\rm obs} + n_\\varphi ^{\\prime }}{1 + \\beta _{\\rm obs} \\, n_\\varphi ^{\\prime }} .$ This aberration formula also holds in general relativity if the Lorentz factor $\\gamma _{\\rm obs}$ and velocity $\\beta _{\\rm obs}$ are properly defined as the true physical quantities measured by a corotating observer.",
"Mathematically, this requires to switch from a general curvilinear coordinate system, like the metric of a rotating disk, to an orthonormal coordinate system associated to the Minkowskian metric.",
"Only the latter coordinates have a clear physical interpretation, the former being only appropriate (or not) coordinates to describe the problem.",
"To summarize, in any local Lorentzian frame, using an orthonormal basis, the Doppler factor is written geometrically as $\\mathcal {D} = \\frac{1}{\\Gamma \\, (1 - \\beta \\cdot \\mathbf {n})} .$ It enables to relate photon propagation directions in both frames as $\\mathbf {n} = \\frac{1}{\\mathcal {D}} \\, \\left[ \\mathbf {n}^{\\prime } + \\Gamma \\, \\left( \\frac{\\Gamma }{\\Gamma +1} \\, ( \\beta \\cdot \\mathbf {n}^{\\prime } ) + 1 \\right) \\, \\beta \\right]$ with the usual redshift phenomenon relating the frequency in the rotating frame $\\nu ^{\\prime }$ to the frequency in the observer frame $\\nu $ by $\\nu = \\mathcal {D} \\, \\nu ^{\\prime } .$ The aberration effect can only be computed for a physical realisation of a corotating frame.",
"It would fail right at the light cylinder or outside it.",
"Unfortunately, a smooth transition from the magnetosphere $r\\le r_{\\rm L}$ to the wind $r\\ge r_{\\rm L}$ is required for modelling the pulsar broad band emission.",
"This tells us that the introduction of a corotating frame where the electromagnetic field and plasma motion are both stationary will not help in advancing our understanding of neutron star electrodynamics.",
"It is preferable to keep the physical quantities expressed in an inertial frame even if the coordinate system can be advantageously described in a corotating frame (not related to any comoving observer).",
"We expose such a technique in the next section for the evolution of the electromagnetic field."
],
[
"Maxwell equations in a rotating coordinate system",
"Measuring the electromagnetic field in the corotating system is possible up to the light-cylinder.",
"However, outside this surface, the metric has no physical significance any more.",
"It is impossible to describe the neutron star electrodynamics in whole space with the field locally measured by a corotating observer because such an observer does not exist when $r\\ge r_{\\rm L}$ .",
"Nevertheless, it is relevant and useful to keep the definition of the electromagnetic field as measured in the inertial reference frame but using the rotating coordinate system to localize it with $(t^{\\prime },r^{\\prime },\\varphi ^{\\prime },z^{\\prime })$ .",
"In such a case the time derivative of any vector field $\\mathbf {A}$ is given by [12] $\\frac{\\partial \\mathbf {A}}{\\partial t} = \\frac{\\partial \\mathbf {A}}{\\partial t^{\\prime }} + \\textrm {curl} \\, ( \\mathbf {V}_{\\rm rot} \\wedge \\mathbf {A} ) - \\mathbf {V}_{\\rm rot} \\, \\textrm {div} \\mathbf {A} .$ The solid body corotation velocity, expressed in the inertial frame, is simply $\\mathbf {V}_{\\rm rot} = \\omega \\wedge \\mathbf {r} = r\\,\\omega \\, \\mathbf {e}_\\varphi .$ Note that it is not restricted to remain less than the speed of light.",
"There is no singularity in eq.",
"(REF ) when crossing the light cylinder.",
"With the correspondence established in eq.",
"(REF ), in the rotating coordinate system, Maxwell equations become $\\frac{\\partial \\mathbf {B}}{\\partial t^{\\prime }} & = - \\, \\textrm {curl} \\, ( \\mathbf {E} + \\mathbf {V}_{\\rm rot} \\wedge \\mathbf {B} ) \\\\\\frac{\\partial \\mathbf {E}}{\\partial t^{\\prime }} & = \\textrm {curl} \\, ( c^2 \\, \\mathbf {B} - \\mathbf {V}_{\\rm rot} \\wedge \\mathbf {E} ) - \\frac{\\mathbf {j}}{\\varepsilon _0} + \\mathbf {V}_{\\rm rot} \\, \\textrm {div} \\mathbf {E} .$ Note however the subtleties that $\\mathbf {E}$ and $\\mathbf {B}$ are still defined as observed in the inertial frame.",
"No particular problem arises at the light cylinder when Maxwell equations are written in this way.",
"In section about numerical simulations, we use this mathematical formulation to solve for the force-free (FFE) and radiative magnetosphere for an oblique rotator as a typical example.",
"But first we have to define the electromagnetic field inside and outside the star in vacuum."
],
[
"Electromagnetic field inside and outside the star",
"To first approximation, a neutron star can be assimilated to a very good conductor.",
"We therefore assume that the electric field inside the star as seen by an observer at rest with respect to the star, the so called comoving observer, vanishes.",
"In the inertial frame of a distant observer, the electric field $\\mathbf {E}$ is given by the usual Lorentz transformation $\\mathbf {E} + \\mathbf {V}_{\\rm rot} \\wedge \\mathbf {B} = \\mathbf {0}$ where $\\mathbf {B}$ is the magnetic field in the same observer frame.",
"At this stage, already several implicit assumptions must be done.",
"Should we assume a prescribed magnetic field in the observer frame $\\mathbf {B}$ or in the corotating frame $\\mathbf {B}^{\\prime }$ ?",
"Both situations are obviously not identical.",
"If the magnetic field is fixed in the observer frame, then in the star corotating frame we find $\\mathbf {B}^{\\prime } = \\frac{\\mathbf {B}}{\\Gamma _{\\rm rot}} + \\frac{(\\beta _{\\rm rot} \\cdot \\mathbf {B})}{\\Gamma _{\\rm rot}+1} \\, \\beta _{\\rm rot}$ with $\\Gamma _{\\rm rot} = (1-\\beta _{\\rm rot}^2)^{-1/2}$ .",
"If the magnetic field is fixed in the rest frame of the star, which seems more reasonable, then the observer will measure a magnetic field $\\mathbf {B} = \\Gamma _{\\rm rot} \\, \\left[ \\mathbf {B}^{\\prime } - \\frac{\\Gamma _{\\rm rot}}{\\Gamma _{\\rm rot}+1} \\, (\\beta _{\\rm rot} \\cdot \\mathbf {B}^{\\prime }) \\, \\beta _{\\rm rot} \\right] .$ Note that both assumptions leads to $\\beta _{\\rm rot} \\wedge \\mathbf {B} = \\Gamma _{\\rm rot}\\, \\beta _{\\rm rot} \\wedge \\mathbf {B}^{\\prime }$ so eq.",
"(REF ) remains valid in any case inside the perfectly conducting star.",
"Practically, the corotation speed inside the star is always weakly or mildly relativistic, therefore $\\beta _{\\rm rot} \\ll 1$ reducing both approaches to $\\mathbf {B} \\approx \\Gamma _{\\rm rot} \\, \\mathbf {B}^{\\prime }$ .",
"An exact analytical solution for the radiating electromagnetic field has been given by [13] assuming a dipolar magnetic field in the inertial frame and neglecting relativistic effects.",
"Close to the light-cylinder, relativistic effects have been taken into account as investigated by [14].",
"Moreover, the current induced inside the star by its rotation influences the magnetic field itself as shown by [15].",
"Consequently, the description of the magnetic field inside the star is already biased by some assumptions on its geometry in the rotating or inertial frame.",
"What happens outside, in its magnetosphere?",
"Also, at large distances, outside the light cylinder $r>r_{\\rm L}$ there exist no more physical frame in corotation with the star.",
"The Lorentz transformation of the electromagnetic field does not apply anymore.",
"It is preferable to stay in the observer frame without any reference to a corotating frame.",
"The strong electric field induced by the rotating star pulls out charged particles, filling the magnetosphere with relativistic electron/positron pairs [8].",
"To good accuracy, this magnetosphere is assumed to be in force-free equilibrium, neglecting particle inertia and fluid temperature, as well as gravity because of the electromagnetic field strength producing forces several orders of magnitude stronger than the gravitational attraction.",
"The magnetosphere is therefore set into corotation with the star up to the light cylinder at a radius $r=r_{\\rm L}$ .",
"Outside this cylinder, plasma corotation cannot be maintained.",
"The flow of leptons generates space charges and currents that significantly modify the electromagnetic structure.",
"The charge density $\\rho _{\\rm e}$ produced by the electric field, following Maxwell-Gauss law and the perfect conductor hypothesis eq.",
"(REF ) diverges right at the light-cylinder because $\\rho _{\\rm e} = \\varepsilon _0 \\, \\textrm {div} \\, \\mathbf {E} = - 2\\,\\varepsilon _0\\,\\frac{\\omega \\cdot \\mathbf {B}}{1 - r^2/r_{\\rm L}^2}$ unless $\\omega \\cdot \\mathbf {B}=0$ on this surface.",
"The magnetic field must therefore adjust itself in order to keep the constrain $\\omega \\cdot \\mathbf {B} = 0$ at $r=r_{\\rm L}$ which implies $B_{\\rm z}(r_{\\rm L}) = 0$ .",
"Another possibility would be to break the corotation approximation by introducing some dissipation through an ad-hoc resistivity or through a radiative dissipation mechanism as shown in subsequent sections.",
"Once the electromagnetic field settled down, we need to investigate particle motion in these fields."
],
[
"Particle motion in the magnetosphere",
"In order to predict the radiation emanating from neutron star magnetospheres, a good understanding of particle trajectories in the electromagnetic field produced by these stars is required.",
"Several attempts focused exclusively on motion restricted to within the light-cylinder.",
"This limitation avoids the problem of transformations involving larger than the speed of light relative velocities between inertial frames.",
"Radiation from outside the light cylinder therefore requires another description.",
"This artificial transition between the magnetosphere and the wind is far from satisfactory.",
"Claims have been made about a new technique to compute emission everywhere, for instance in the vacuum Deutsch field [16], but we will show that at least in some cases it fails too to smoothly join inside and outside light cylinder electrodynamics.",
"Moreover, their model based on the assumption that eq.",
"(REF ) is valid within the magnetosphere is inconsistent with the vacuum assumption of a Deutsch field.",
"A more satisfactory solution includes the plasma feedback self-consistently as proposed by [17] in order to avoid superluminal particle speed everywhere.",
"In the following subsections, we summarize the most studied maybe not the most effective models of particle trajectories.",
"Three different approximations for the particle motion have been tried.",
"Indeed a charged particle can be seen as [leftmargin=*,labelsep=5.8mm] following magnetic field lines $\\mathbf {B^{\\prime }}$ in the corotating frame.",
"following magnetic field lines $\\mathbf {B}$ in the inertial frame in addition to a corotation imposed by the stellar rotation.",
"following the ultra-relativistic radiation reaction limit leading to the so-called Aristotelian dynamics.",
"Acutally, this limit can be explained by Newtonian dynamics in a stationary regime balancing electric acceleration and radiation friction.",
"These different views are not equivalent to each other because they assume different electromagnetic field structures, either fixed in the rotating frame or in the inertial frame.",
"Let us discuss these approaches in depth starting with the corotating frame view."
],
[
"Corotating frame",
"Viewed from the corotating frame, particles are assumed to follow magnetic field lines along $\\mathbf {B}^{\\prime }$ .",
"Therefore in this frame the particle velocity is given by $\\mathbf {v}^{\\prime } = v^{\\prime } \\, \\frac{\\mathbf {B^{\\prime }}}{B^{\\prime }} = v^{\\prime } \\, \\mathbf {n}^{\\prime }_{\\rm B}$ where $v^{\\prime }$ is the particle speed along the field line $\\mathbf {B}^{\\prime }$ in the normalized direction $\\mathbf {n}^{\\prime }_{\\rm B} = \\mathbf {B^{\\prime }}/B^{\\prime }$ .",
"How then to choose this field $\\mathbf {B}^{\\prime }$ ?",
"Some authors used in the past the relation $\\mathbf {B}^{\\prime } = \\mathbf {B}$ which is only correct to second order in $\\beta _{\\rm rot}$ , a results derived from the coordinate transformation between inertial frame and rotating coordinate systems.",
"However, this equality was used by several authors to compute pulsar high-energy light-curves at high altitude, up to a substantial fraction of the light-cylinder [18], [19], [20], [21].",
"Light curves and sky maps derived from this model are sensitive to the upper boundary of the radiating zone.",
"However this dependence is undesirable.",
"Moreover, the rotating coordinate system is not an orthonormal basis, therefore $\\mathbf {B}^{\\prime }$ should not be interpreted as the local magnetic field measured by a rotating observer.",
"It must be computed according to the Lorentz transformation [16].",
"Nevertheless, both descriptions agree to good accuracy for non relativistic corotating speeds.",
"Going into the rotating frame synchronous with the neutron star rotation can lead to misinterpretation of the physical electromagnetic field measured by a local observer.",
"Moreover, the corotating frame can not be extended beyond the light cylinder radius $r_{\\rm L}$ .",
"Such description therefore faces severe difficulties to deal with the entire neutron star magnetosphere and is inadequate to efficiently model them from the surface to large distances within the striped wind $r\\gtrsim r_{\\rm L}$ .",
"Much better we think is to perform all calculations in the inertial frame of a distant observer as we now describe in the next two sections."
],
[
"Corotating velocity",
"When staying in the observer frame, without reference to any rotating frame, the velocity is described by a velocity component along the magnetic field lines $\\mathbf {B}$ , now expressed in the inertial frame and denoted by $v_\\parallel $ , and a velocity component due to the dragging by the star denoted by $\\mathbf {V}_{\\rm rot}$ .",
"Therefore we write $\\mathbf {v} = v_\\parallel \\, \\frac{\\mathbf {B}}{B} + \\mathbf {V}_{\\rm rot} .",
"$ The velocity along the field line $v_\\parallel $ must be chosen in order to keep the total speed smaller than the speed of light, $v<c$ .",
"It requires a special configuration of the magnetic field $\\mathbf {B}$ with an increasing toroidal component to compensate for the linear increase in corotation speed as given by $\\mathbf {V}_{\\rm rot}$ .",
"Therefore not all magnetic field configurations are permitted to fulfil this constrain.",
"By assumption, in some models [18], [19], [20], [21], particles follow magnetic field lines in the corotating frame.",
"Their distribution function is isotropic in the rest frame of the fluid.",
"Following the previous prescriptions by [16], we assume that their Lorentz factor $\\Gamma $ is constant in the observer frame such that the velocity, being a combination between propagation along field lines and corotation at speed $\\mathbf {V}_{\\rm rot}$ , is $\\mathbf {v} = v_\\parallel ^{\\rm c} \\, \\mathbf {t} + \\mathbf {V}_{\\rm rot}$ where $\\mathbf {t} = \\pm \\mathbf {B}/B$ is the outward pointing tangent vector to the field line.",
"Solving for the parallel velocity $v_\\parallel ^{\\rm c}$ the only real and positive solution is $v_\\parallel ^{\\rm c} = - \\mathbf {t} \\cdot \\mathbf {V}_{\\rm rot} + \\sqrt{(\\mathbf {t} \\cdot \\mathbf {V}_{\\rm rot})^2 + v^2 - V_{\\rm rot}^2} .$ $V_{\\rm rot}$ exceeds the speed of light outside the light cylinder by definition.",
"The term $v^2 - V_{\\rm rot}^2$ in the square root becomes negative and must be compensated by the term $(\\mathbf {t} \\cdot \\mathbf {V}_{\\rm rot})^2$ meaning that the magnetic field must be strongly bend toward the azimuthal direction $\\mathbf {e}_\\varphi $ .",
"The Deutsch field does not satisfy this requirement and cannot be used to study photon emission within the wind if this view is adopted.",
"Knowing the velocity, we get the Doppler factor for radiation as explained in eq.",
"(REF ).",
"This velocity field assumes that the electric field vanishes in the corotating frame.",
"But this requires a large amount of plasma to screen the electric field, in contradiction with the vacuum assumption made in [16].",
"Therefore, the aberration formula eq.",
"(REF ) can only be an approximation in this case.",
"Moreover, this approximation also fails at sufficiently large distances because the Deutsch field solution [13] possesses a magnetic field structure for which the poloïdal component does not decay fast enough with respect to the toroidal component.",
"Real solutions to eq.",
"(REF ) do not exists at several light-cylinder radii because the square root in eq.",
"(REF ) becomes negative.",
"Indeed, taking an orthogonal rotator, it can be shown that in the equatorial plane the term in the square root of eq.",
"(REF ) tends to $v^2-4\\,c^2<0$ for $r\\rightarrow +\\infty $ on the spiral given by $\\varphi + r/r_{\\rm L}- \\omega \\,t = \\pi /2$ .",
"In the most favourable case for which $v=c$ , it actually becomes negative already at the light-cylinder.",
"Using the corotating frame does not help to go beyond the light-cylinder for vacuum fields.",
"Nevertheless, the description exposed in this section is applicable to force-free magnetospheres that exactly cancel the electric field in the frame comoving with the plasma at the electric drift speed.",
"Only in such FFE models can this prescription be correctly applied in whole space, within the magnetosphere $(r\\le r_{\\rm L}$ ) and within the wind $(r\\ge r_{\\rm L}$ ).",
"Is it possible to find a formulation alleviating the need for special magnetic field configurations?",
"In our opinion, there exist a simple and efficient way to compute particle trajectories in any electromagnetic field when moving at the radiation reaction limit.",
"We detail this last approach in the next section."
],
[
"Aristotelian dynamics",
"Particles in the neutron star magnetosphere are ultra-relativistic.",
"They copiously radiate photons during their motion.",
"This has to be taken into account.",
"The simplest approximation is given by the radiation reaction limit, where the radiation force, acting as a damping working against the motion, a kind of radiative friction, exactly compensates for the electric acceleration.",
"It is sometime called Aristotelian electrodynamics because the velocity is completely and solely determined by the electromagnetic field felt locally by the particles although we believe that it is more appropriate to speak about radiation reaction motion because it can be derived from the Lorentz force with radiative friction and this according to Newtonian dynamics.",
"The expression for the velocity has been derived in [22], but see also [23].",
"Assuming that particles move exactly at the speed of light (which is an excellent approximation in neutron star magnetospheres), depending on the sign of their charge, their velocity reads $\\mathbf {v}_\\pm = \\frac{\\mathbf {E} \\wedge \\mathbf {B} \\pm ( E_0 \\, \\mathbf {E} / c + c \\, B_0 \\, \\mathbf {B})}{E_0^2/c^2+B^2}$ where the plus sign corresponds to positive charges and the minus sign to negative charges.",
"Actually, the velocity is independent of the mass $m$ over charge $q$ ratio $q/m$ , it only depends on the sign of its charge.",
"Moreover, we introduced the electromagnetic field strengths $E_0$ and $B_0$ according to the two electromagnetic invariants $(\\mathcal {I}_1,\\mathcal {I}_2)$ such that $\\mathcal {I}_1 & = \\mathbf {E}^2 - c^2 \\, \\mathbf {B}^2 = E_0^2 - c^2 \\, B_0^2 \\\\\\mathcal {I}_2 & = c\\,\\mathbf {E} \\cdot \\mathbf {B} = c\\,E_0 \\, B_0$ with the subsidiary condition $E_0 \\geqslant 0$ ensuring that the radiation reaction force is always directed oppositely to the velocity direction.",
"As explained in [24] these invariants are related to the electromagnetic field strength in a frame where $\\mathbf {E}$ and $\\mathbf {B}$ are parallel.",
"The lepton motion can be decomposed into an electric drift part along the vector $\\mathbf {E} \\wedge \\mathbf {B}$ , a motion along magnetic field lines $\\mathbf {B}$ and a motion along electric field lines $\\mathbf {E}$ .",
"This last part of the motion is responsible for dissipation because the power of the Lorentz force is $q\\,(\\mathbf {E} + \\mathbf {v}_\\pm \\wedge \\mathbf {B}) \\cdot \\mathbf {v}_\\pm = q \\, \\mathbf {v}_\\pm \\cdot \\mathbf {E} = Z\\,e\\,c\\,E_0 \\ge 0$ where $q=\\pm Z\\,e$ depending on the charge $Z$ of the particle: positrons and electrons have $Z=1$ whereas ions have $Z$ arbitrary.",
"The velocity field (REF ) is regular in whole space, nothing singular happens at the light-cylinder.",
"It can be implemented to compute realistic pulsar light-curves and spectra even in vacuum Deutsch solution as shown by [25].",
"This latest work serves as a starting point to investigate more deeply pulsar magnetospheric radiation by including for instance the plasma feedback onto the Deutsch field as will be shown in section .",
"In the near field zone, i.e.",
"close to the neutron star surface, where $E\\ll c\\,B$ , the particle velocity simplifies into a motion solely along $\\mathbf {B}$ such that $\\mathbf {v}_\\pm = \\pm c \\, \\frac{( \\mathbf {E} \\cdot \\mathbf {B} ) \\, \\mathbf {B}}{E_0 \\, (E_0^2/c^2+B^2)} .$ This expression can be reduced to $\\mathbf {v}_\\pm = \\pm c \\, \\textrm {sign}(B_0) \\, \\frac{\\mathbf {B}}{B}$ by noting that in this weak electric field limit the magnitude of $\\mathbf {B}$ is almost equal to the invariant $B_0$ , namely $B^2 \\approx B_0^2$ .",
"Particles are accelerated mostly by the electric component parallel to the magnetic field.",
"The surfaces $\\mathbf {E} \\cdot \\mathbf {B}=0$ are of particular interest because the velocity changes sign when the particle crosses this region.",
"It is called a force-free surface and represents trapping regions for those particles [26], [27], [28].",
"The concept of magnetic field line in vacuum is misleading and specifying motion along a particular field line is not well defined in the general case.",
"This requires some caution about the interpretation of the corotation speed $\\mathbf {V}_{\\rm rot}$ .",
"The way to follow the particle trajectory replaces this velocity by a special frame in which the electric field is parallel to the magnetic field leading to the Aristotelean dynamics discussed before.",
"Indeed, the velocity $\\beta _\\parallel \\, c$ required by the Lorentz transformation to get this condition is [29] $\\frac{\\beta _\\parallel }{1+\\beta _\\parallel ^2} = \\frac{c \\, \\mathbf {E} \\wedge \\mathbf {B}}{E^2 + c^2\\, B^2}$ neglecting all other curvature, gradient and polarization drifts in the limit of vanishing Larmor radius which is correct in a super strong magnetic field.",
"In that frame, where quantities are denoted by a prime, motion is along the common direction of $\\mathbf {E}^{\\prime }$ and $\\mathbf {B}^{\\prime }$ .",
"To get the useful solution, we write the frame velocity as $\\mathbf {V}_\\parallel = \\frac{\\mathbf {E} \\wedge \\mathbf {B}}{E_0^2/c^2 + B^2} .$ The electric and magnetic fields in the frame moving at speed $\\mathbf {V}_\\parallel $ are found by a special-relativistic Lorentz boost of the electromagnetic field and gives $\\mathbf {E}^{\\prime } & = \\Gamma \\, \\frac{E_0}{E_0^2/c^2 + B^2} \\, \\left[ \\frac{E_0}{c^2} \\, \\mathbf {E} + B_0 \\, \\mathbf {B} \\right] \\\\\\mathbf {B}^{\\prime } & = \\Gamma \\, \\frac{B_0}{E_0^2/c^2 + B^2} \\, \\left[ B_0 \\, \\mathbf {B} + \\frac{E_0}{c^2} \\, \\mathbf {E} \\right] .$ Electric and magnetic fields are indeed collinear because $E_0 \\, \\mathbf {B}^{\\prime } = B_0 \\, \\mathbf {E}^{\\prime }$ .",
"In this frame, particles move along the common direction of $\\mathbf {E}^{\\prime }$ and $\\mathbf {B}^{\\prime }$ .",
"Thus the local tangent vector to the trajectory becomes $\\mathbf {t}^{\\prime }_\\parallel = \\pm \\mathbf {E}^{\\prime }/E^{\\prime } = \\pm \\mathbf {B}^{\\prime }/B^{\\prime }$ , the sign being chosen such that particles flow outwards.",
"Therefore we replace $\\beta $ by $\\mathbf {V}_\\parallel /c$ in eq.",
"(REF ) to get a velocity field that should not be confused or seen as motion along field lines because this concept is usually ill defined for non-ideal plasmas when $\\mathbf {E} \\cdot \\mathbf {B} \\ne 0$ .",
"Our expression for the particle velocity resembles to the Aristotelian expression given by [30].",
"Our velocity prescription is however more general because we do not assume that particles travel exactly at the speed of light.",
"The speed along the common $\\mathbf {E}$ and $\\mathbf {B}$ direction is unconstrained and fixed by the “user” contrary to Aristotelian electrodynamics.",
"If particles exactly move at the speed of light, in the comoving frame this velocity becomes $\\mathbf {v}^{\\prime }= \\pm \\mathbf {E}^{\\prime }/E^{\\prime } = \\pm \\mathbf {B}^{\\prime }/B^{\\prime }$ , the sign depending on the charge.",
"Note also that the electromagnetic field strengthes are $E^{\\prime }=E_0$ and $B^{\\prime }=B_0$ .",
"Doing the Lorentz transformation to the observer frame, noting that $\\mathbf {V}_\\parallel $ and $\\mathbf {v}^{\\prime }$ are orthogonal, this is nothing but Aristotelian electrodynamics.",
"Our treatment is more general because we do not enforce the speed of light in this frame.",
"The prescription for the velocity impacts the high-energy light-curves from pulsars.",
"This has been shown in depth by [24].",
"The parallel velocity $\\mathbf {V}_\\parallel $ in eq.",
"(REF ) generalizes the electric drift approximation to field configurations with an electric field $E$ exceeding the magnetic field $c\\,B$ .",
"There is no need to impose the condition $E<c\\,B$ to respect the force-free condition.",
"However, it reduces to force-free if $E_0=0$ , meaning no radiation reaction and no dissipation meanwhile requiring $\\mathbf {E} \\cdot \\mathbf {B} = 0$ .",
"In order to look for plasma filled magnetospheres, we have to resort to numerical simulations in the force-free regime or in a dissipative regime because of resistivity and/or radiation damping.",
"In the next section, we show some new results for radiative pulsar magnetospheres to be compared with the standard force-free solution for oblique rotators."
],
[
"Numerical simulation of rotating magnetospheres",
"Neutron stars cannot be surrounded by vacuum because particles are expelled from the surface and accelerated in the surrounding strong electromagnetic field.",
"This is indirectly deduced from their broad band electromagnetic spectrum for which the Crab pulsar is an archetypal example [31].",
"Although an exact analytical solution for a rotating dipole in vacuum exists, known as Deutsch solution [13], realistic magnetospheres require the presence of plasma producing charges and currents that retroact to the electromagnetic field.",
"Because the problem is highly non linear, numerical simulations are compulsory.",
"Two dimensional neutron star magnetospheres have been computed in the force-free regime two decades ago starting with the aligned FFE case [32] and followed several years later by the general three dimensional oblique cases by [33].",
"Since then, these results have been retrieved by several other authors using different numerical approaches like finite difference/finite volume methods [34], [35], [36] or pseudo-spectral methods [37], [38], [39].",
"Even a combined spectral/discontinuous Galerkin method has been tried including general-relativistic effects for a monopole [40] or a dipole [41].",
"Some extension to dissipative magnetospheres was undertaken by [42], [39], [43] assuming an ad hoc prescription for the dissipation.",
"Here we show three models of pulsar magnetosphere for an oblique rotator with obliquity (angle between rotation axis and magnetic axis) $\\chi = \\lbrace 0, 30, 60, 90\\rbrace $ , namely the vacuum, the force-free and the radiative cases.",
"Simulations are performed in the observer inertial frame but using the corotating coordinate system leading to Maxwell equations written as eq.",
"(REF ).",
"This particular frame ensures that the solution relaxes to a time independent solution where the current sheet remains at a fixed position in space in order to ease its location for subsequent purposes.",
"In other words, the time derivatives in eq.",
"(REF ) must vanish when the solution becomes stationary.",
"The three models correspond to three prescriptions for the electric current density $\\mathbf {j}$ .",
"In vacuum, for the Deutsch solution it is obviously $\\mathbf {j} = \\mathbf {0}$ .",
"This is our reference solution for checking our algorithm and accuracy of the computed solution.",
"Simulations are performed using our pseudo-spectral Maxwell solver explained in depth in [37].",
"Before discussing our new results, we remind the essential features of our pseudo-spectral code in the following paragraph."
],
[
"Numerical schemes",
"Spectral and pseudo-spectral numerical schemes convert a system of partial differential equations (PDE) into a larger system of ordinary differential equations (ODE) much easier to integrate numerically with standard ODE integration techniques like the explicit Runge-Kutta and Adams-Bashforth schemes.",
"See [44] for a detailed review on these techniques.",
"We emphasize that spectral methods do not approximate the equations of the problem but the solution itself.",
"Therefore the numerical problem exactly reflects the mathematical problem with the same boundary conditions which need to be properly imposed without any under or over-determinacy.",
"Note that finite volume/finite difference codes are prone to large (with respect to spectral codes) diffusion/dissipation and are therefore able to damp boundary conditions that are not exactly identical to the mathematical problem making it analytically an ill-posed problem (mathematically speaking not from a numerical point of view).",
"Spectral methods are primarily dealing with expansion coefficients of the unknown quantities not their value themself at the grid points.",
"This expansion possesses the great advantage of removing singularities of differential operators like the gradient, the divergence and the curl in spherical coordinates along the polar axis.",
"We use this flexibility to solve Maxwell equations in polar spherical coordinates $(r,\\theta ,\\varphi )$ with no special care about the polar axis.",
"Boundary conditions on the stellar surface can thus be properly and exactly imposed as required by the original mathematical problem.",
"Specifically, the components of the electromagnetic field are expanded onto a real Fourier-Legendre-Chebyshev basis.",
"The azimuthal dependence is expanded into a standard Fourier series in $\\cos (m\\,\\varphi )$ and $\\sin (m\\,\\varphi )$ whereas the latitude is expanded into Legendre functions $P_\\ell ^m(\\theta )$ where $\\ell $ and $m$ are integers related to the spherical harmonics $Y_{\\ell ,m}(\\theta , \\varphi )$ [45].",
"The radial part is expanded into Chebyshev polynomials $T_n(x(r))$ where $r \\in [R_1, R_2]$ is mapped into the normalized range $x\\in [-1,1]$ by a linear transformation.",
"The straightforward implementation of this mapping accumulates the discrete grid found from the Chebyshev-Gauss-Lobatto points unevenly near the boundary points where the resolution becomes prohibitively high.",
"The constrain on the time step is therefore to severe.",
"In order to distribute more evenly the grid points, we use the Kozloff/Tal-Ezer mapping [46].",
"See also [47] for a similar implementation of this technique for axisymmetric neutron star magnetospheres.",
"Derivatives are computed in the Fourier-Legendre-Chebyshev space by simple algebraic operations, instead of pure function derivatives, and then transformed back to real space on grid points.",
"The outer boundary conditions are outgoing waves with a sponge layer absorbing spurious reflections.",
"The inner boundary conditions enforce the tangential part of the electric field and the normal component of the magnetic field at the stellar surface.",
"To keep a mathematically well-posed problem, we employ the characteristic compatibility method described in [44].",
"Time integration is performed via a standard third order Runge-Kutta scheme.",
"Spectral methods are known to converge to the exact solution faster than finite difference or finite volume schemes for sufficiently smooth problems without discontinuities.",
"They require less resolution for the same accuracy [48].",
"Because spectral methods rely on Fourier-like series expansions, they are also sensitive to the Gibbs phenomenon [45], spoiling the solution with overshoot possibly leading to unphysical quantities like negative densities or pressures.",
"In our strong electromagnetic field limit, however, no positivity constrain is required for the unknown field.",
"However, in order to stabilize the algorithm, tending to put more and more energy into small scales because of the Gibbs effects, we need to filter the highest frequencies by applying for instance an exponential filter damping the highest order coefficients in the Fourier-Legendre-Chebyshev expansion.",
"Eventually, we check a posteriori that the simulation has converged to the desired solution to good accuracy by performing a resolution analysis, meaning that increasing by a factor two the grid resolution in each direction, the solution does not significantly changes.",
"We found that for the simulations shown below, a resolution $N_r\\times N_\\theta \\times N_\\varphi = 257 \\times 32 \\times 64$ already gave reasonable results.",
"We checked on a few cases that increasing by a factor 2 the resolution in all directions did not change the results (but drastically increased the computational time on a single core).",
"Consequently, we adopted a resolution of $N_r\\times N_\\theta \\times N_\\varphi = 257 \\times 32 \\times 64$ for accurate and converged results.",
"In the special case of a aligned rotator, the Gibbs phenomenon is strongest.",
"We had to resort to higher resolution of $N_r\\times N_\\theta \\times N_\\varphi = 513 \\times 64 \\times 1$ for accurate and converged results."
],
[
"Force-free magnetospheres",
"In the force-free regime where $\\mathbf {E} \\cdot \\mathbf {B}=0$ and $E<c\\,B$ , the electric current density is uniquely defined by [49] $\\mathbf {j} = \\rho _{\\rm e} \\, \\frac{\\mathbf {E}\\wedge \\mathbf {B}}{B^2} + \\frac{\\mathbf {B} \\cdot \\mathbf {\\nabla } \\times \\mathbf {B} / \\mu _0 - \\varepsilon _0 \\, \\mathbf {E} \\cdot \\mathbf {\\nabla } \\times \\mathbf {E}}{B^2} \\, \\mathbf {B} .$ This current is decomposed into an electric drift part, first term on the right hand side, depending only on the total electric charge density $\\rho _{\\rm e}$ , and a part along the magnetic field that is not constrained but deduced a posteriori from the simulation output.",
"Because all particles drift with the same velocity, contribution to the drift part of the electric current arises solely from the non-neutrality of the plasma, meaning $\\rho _{\\rm e}\\ne 0$ .",
"The electric drift speed must remains strictly less than the speed of light.",
"If the condition $E<c\\,B$ is violates, force-free breaks down and the plasma becomes dissipative.",
"In force-free simulations however, we enforce by hand the condition $E<c\\,B$ everywhere in space in order to stay in the sub-relativistic drift speed limit.",
"An example of magnetic field lines in the equatorial plane for an orthogonal rotator with $\\chi =90$ is shown in figure REF with $R/r_{\\rm L}=0.2$ .",
"The force-free regime tries to put the field lines out of the current sheet which becomes singular, but due to numerical resistivity, field lines also tend to close by crossing this singular surface.",
"Figure: Magnetic field lines for orthogonal (χ=90\\chi =90) force-free (FFE) and radiative magnetospheres for different values of pair multiplicity κ\\kappa ."
],
[
"Radiative magnetospheres",
"For the radiative magnetosphere, we introduce a additional free parameter represented by the pair multiplicity factor $\\kappa $ such that the electric current derived from the Aristotelian electrodynamics becomes $\\mathbf {j} = \\rho _{\\rm e} \\, \\frac{\\mathbf {E} \\wedge \\mathbf {B}}{E_0^2/c^2 + B^2} + |\\rho _{\\rm e}| \\, (1+2\\,\\kappa ) \\, \\frac{E_0 \\, \\mathbf {E}/c^2 + B_0 \\, \\mathbf {B}}{E_0^2/c^2 + B^2} .$ It is decomposed into a $\\mathbf {E} \\wedge \\mathbf {B}$ drift similar to force-free but without the additional constraint $E<c\\,B$ and a part along $\\mathbf {E}$ and $\\mathbf {B}$ which reduces in the drift frame to a motion along the common direction of $\\mathbf {E}^{\\prime }$ and $\\mathbf {B}^{\\prime }$ .",
"Fig.",
"REF shows some field lines in the equatorial plane for the orthogonal rotator with $\\chi =90$ in the radiative regime with pair multiplicity $\\kappa =\\lbrace 0,1,2,5\\rbrace $ .",
"In the most dissipative case corresponding to $\\kappa =0$ , field lines cross the current sheet at smaller distances compared to less dissipative cases with $\\kappa =2$ or $\\kappa =5$ .",
"Fig.",
"REF shows the associated radial dependence of the Poynting flux for force-free and radiative cases with $\\kappa =\\lbrace 0,1,2,5\\rbrace $ .",
"The radiative magnetosphere dissipates a small fraction of the Poynting flux into particle acceleration and radiation, most efficiently when $\\kappa =0$ , corresponding to a charge separated plasma.",
"Increasing the pair multiplicity factor $\\kappa $ to higher values shifts the radiative model towards the force-free limit.",
"In the aligned case, the decrease in Poynting flux is abrupt right at the light-cylinder.",
"It is most prominent for $\\kappa =0$ .",
"However, due to the intrinsic dissipation of our algorithm, even in the FFE case there some Poynting flux dissipation is observed.",
"This is due to the infinitely thin current sheet with discontinuous toroidal magnetic field that is smeared by our spectral methods (Gibbs phenomenon).",
"The situation improves for oblique cases as the displacement current take over some fraction of the electric current within the sheet.",
"Figure: The radial dependence of the Poynting flux for an oblique rotator in force-free and radiative regimes.In fig.",
"REF we show the Poynting flux crossing the light-cylinder for force-free and radiative cases with $\\kappa =\\lbrace 0,1,2,5\\rbrace $ and depending on the inclination angle $\\chi $ .",
"All cases can be fitted with a single formal expression summarized as $L = L_\\perp \\, ( a + b \\, \\sin ^2 \\chi )$ with different coefficients depending on the regime considered.",
"The fitted values extracted from the numerical simulations are listed in Table REF .",
"The most dissipative case $\\kappa =0$ slightly decreases the Poynting flux for the aligned rotator already inside the light-cylinder.",
"The decrease is accurately quantified by the fitting parameter $a$ .",
"The FFE normalized Poynting flux is 1.42 whereas for the radiative $\\kappa =0$ case it is 1.36.",
"The fitting parameter $b$ seems less dependent to the regime considered.",
"The aligned rotator also shows the most prominent gradual decrease in the Poynting flux with respect to distance.",
"Dissipation starts at the light cylinder but goes on at several light cylinder radii.",
"For oblique rotators, the slope of this radial decrease slowly diminishes, becoming negligible for the orthogonal rotator.",
"Figure: The Poynting flux crossing the light-cylinder for oblique rotators in force-free and radiative regimes corresponding to fig.",
".Table: Fitting coefficients aa and bb for the spin-down luminosity as fitted in eq.",
"().In all regimes, the electromagnetic fluxes are very similar while inside the light-cylinder.",
"The discrepancies occur outside the light-cylinder, in regions where the electric field is dominant and not fully screened by the plasma because of the too low pair multiplicity.",
"A corotative ideal and dissipationlessness magnetosphere inside the star is therefore a good approximation, whereas outside, efficient dissipation sets in right at the light-cylinder, around the current sheet.",
"Fig.",
"REF shows a summary of the Poynting flux crossing the light-cylinder (larger markers) and crossing a sphere of radius $4\\,r_{\\rm L}$ (smaller markers) for oblique rotators in force-free and radiative regimes.",
"The dissipation going on at large distances is most visible for the aligned rotator with green triangles.",
"Figure: The Poynting flux crossing the light-cylinder (larger markers) and crossing a sphere of radius 4r L 4\\,r_{\\rm L} (smaller markers) for oblique rotators in force-free and radiative regimes.Some fraction of the electromagnetic flux goes into particle acceleration and radiation.",
"Quantitatively, this dissipation of the electromagnetic energy is computed as a work done on the plasma such that $\\mathbf {j} \\cdot \\mathbf {E} = |\\rho _e| \\, ( 1 + 2 \\, \\kappa ) \\, c \\, E_0 \\ge 0 .$ This dissipation rate, for $\\kappa =\\lbrace 0,1,2,5\\rbrace $ , is shown in Fig.",
"REF on a log scale.",
"It shows the location of largest dissipation for an orthogonal rotator according to the dissipation rate controlled by $\\kappa $ .",
"Poynting flux goes into particle acceleration and radiation mainly outside the light-cylinder along the current sheet starting from the Y-point.",
"We expect therefore gamma-rays to be produced along this sheet, emitting pulses at the neutron star rotation frequency.",
"Such models have already been put forward and known as the striped wind.",
"See for instance [50] for the production of high-energy emission and [51] for demonstrating the pulsation.",
"Figure: Dissipation in the equatorial plane of an orthogonal rotator for κ={0,1,2,5}\\kappa =\\lbrace 0,1,2,5\\rbrace (from left to right, top to bottom).We conclude that energy conversion occurs mainly around the current sheet.",
"Within the light-cylinder, the electric field is always less than the magnetic field $E<c\\,B$ .",
"Therefore the force-free condition can be maintained without resorting to artificial damping of $\\mathbf {E}$ .",
"However, dissipation sets in right at the light-cylinder, where magnetic field lines start to cross the light-cylinder.",
"The dissipation region follows a spiral pattern with decreasing amplitude with distance from the star.",
"These new simulations offer for the first time a fully self-consistent description of a dissipative and radiative magnetosphere, where feedback between plasma flow, particle radiation and electromagnetic field is included.",
"Note that emission occurs only along the current sheet outside the light-cylinder.",
"This conclusion supports the idea of the striped wind model introduced by [52] and by [53].",
"It also explains pulsed high-energy emission from gamma-ray pulsars as demonstrated by [51], [54] and [55].",
"Contrary to the vacuum case, by construction, particles cannot move faster than the speed of light, even if the corotating velocity eq.",
"(REF ) is used.",
"This is because the magnetic field is now sufficiently bent to counterbalance the effect of adding the corotation velocity given by eq.",
"(REF ).",
"Note also that radiative magnetospheres presented in our study do not tend to the vacuum solution when the pair multiplicity vanishes $\\kappa =0$ because inside the light-cylinder we enforce force-free conditions by construction.",
"Dissipative losses in the current sheet, also called striped wind, have also been proposed by other authors.",
"For instance [56] found a new standard solution for the aligned rotator, free of separatrix current layer within the light-cylinder.",
"Dissipation occurs only in the equatorial current sheet where acceleration and radiation of particle is allowed.",
"They found an increase of 23% of the spindown with respect to $L_\\perp $ , 40% of which goes into the current sheet dissipation.",
"In our solutions, we found a spindown increase from 36% to 42% depending on the pair multiplicity, see table REF .",
"The crux of the matter is the microphysical description of this current sheet that conditions the whole magnetospheric solution.",
"In order to prescribe the electric current in this sheet, [56] assumed a null-like current everywhere, a prescription which is questionable.",
"Moreover 60% of the magnetic flux crossing the light-cylinder opens up to infinity.",
"It is not clear how this percentage is controlled by the solution.",
"A better solution would get all magnetic flux dissipated sooner or later in the equatorial current sheet.",
"[57] used another approach, performing Particle In Cell (PIC) simulations of pulsar magnetospheres.",
"Here the sensitive parameter is the unconstrained pair injection rate, from the surface or from the whole magnetosphere.",
"The stationary solution crucially depends on this injection mechanisms, going from an electrosphere to an almost force-free magnetosphere.",
"They found that less than 15% of the Poynting flux is dissipated within $2\\,r_{\\rm L}$ .",
"It is not clear how much additional decrease is expected if the solution would have been computed to larger distances.",
"A partial answer is given in [58] where the dissipation is as high as 35% at $5\\,r_{\\rm L}$ .",
"Comparing both models is difficult because they are not performed with the same set up.",
"The most critical variable being the pair multiplicity which is not fixed by the user and not easily controlled.",
"We showed that $\\kappa $ strongly affects the asymptotic large distance dissipation in the axisymmetric case.",
"These different approaches can only be reconciled in light of the pair content within the magnetosphere.",
"To summarize, all results performed with different numerical codes and different assumptions demonstrated that the magnetosphere relaxes automatically to a state where corotation with the star is enforce by the electric current prescription.",
"However, while this picture is simple and easily understood, nothing forbids solutions with differentially rotating plasmas.",
"Such solutions are discussed in the next section."
],
[
"Differentially rotating magnetospheres",
"The neutron star magnetosphere is often described as perfectly corotating with the star, dragged by the electromagnetic field to enforce strict corotation as in the simulations performed in the previous section.",
"However, it is well known from Ferraro isorotation law [59] that to keep corotation, the plasma must be connected magnetically to the star everywhere in the magnetosphere.",
"If some vacuum gaps exist between the surface and the magnetospheric plasma, corotation becomes impossible.",
"The plasma around the equatorial plane will start to rotate differentially, leading to a much complexer variety of physical processes like non neutral plasma instabilities [60] and efficient particle diffusion across magnetic field lines [61], [62].",
"A perfectly corotating magnetospheric plasma, as simple as it could be, does not represent a realistic pulsar magnetosphere.",
"Differential rotation or lagging of particle motion is permitted when vacuum gaps are allowed.",
"Indeed, in the electrospheric solution found by [63] and detailed by [64] for an aligned rotator, the domes are corotating because magnetically connected to the star but the equatorial disc over-rotates with respect to the star.",
"This differential rotation is induced by a charge density given for an aligned rotator by $\\rho = - \\varepsilon _0 \\, (2 \\, \\mathbf {\\Omega } \\cdot \\mathbf {B} + r^2 \\, B^2 \\, \\Omega ^{\\prime }(a))$ where $a$ is the magnetic flux function.",
"The quantity $\\rho _{\\rm gj} = - 2 \\, \\varepsilon _0 \\, \\mathbf {\\Omega } \\cdot \\mathbf {B}$ is usually referred as the Goldreich-Julian charge density [8].",
"The charge density $\\rho $ in absolute value is much higher than the one required for corotation [65] because $\\Omega ^{\\prime }(a)>0$ .",
"Therefore, the location of the light cylinder shifts nearer towards the surface.",
"Actually it is no more a cylinder but an azimuthally symmetric surface.",
"In the general case, it is more suitable to speak about a light-surface rather than about a light-cylinder.",
"This new structure has profound consequence on the secular evolution of the magnetosphere.",
"Indeed, [66] showed that the equatorial disc is unstable with respect to the diocotron instability.",
"A quasilinear theory has been developed by [67] demonstrating the possibility to transport charges across magnetic field lines.",
"This is of paramount importance for the magnetosphere.",
"Later [68] investigated through electrostatic PIC simulations the fully non-linear evolution of the diocotron instability.",
"He found that particles are transported radially outwards in the equatorial plane when the system is feed with fresh electron/positron pairs from the innermost part of the magnetosphere (produced for instance by magnetic photon absorption or photon-photon collisions).",
"However, [69] proved that the relativistic rotation tends to stabilize the diocotron instability.",
"Moreover particle inertia becomes significant close to the light-cylinder, but the instability still survives, switching to the magnetron case [70].",
"The presence of such instabilities destroys the picture of a stationary and corotating magnetosphere.",
"By nature the pulsar electrodynamics is non stationary as can be witnessed from the highly erratic emission feature of single radio pulse profiles [71].",
"This is inherent to the relativistic plasma flow and to the pair creation process occurring close to the stellar surface and/or close to the light-cylinder.",
"This remark leads us to the last topic concerning radiative signatures within the magnetosphere."
],
[
"Radiation from the magnetosphere",
"Pulsars show a broadband emission from radio through optical up to X-rays and gamma-rays.",
"Photons are produced within the magnetosphere and wind.",
"They must indirectly carry information about their production site, therefore showing an imprint of relativistic rotation if produced close to the light-cylinder.",
"The location of high-energy emission from gamma-ray pulsars is poorly constrained by observations.",
"Several competing models interpret the measurements within the magnetosphere or wind, with either curvature, synchrotron or inverse Compton radiation.",
"So far, there is little hope to get a clear insight about the magnetosphere from these observations.",
"Much more interesting in our view is the wealth of data in radio pulse profiles and their associated polarisation feature.",
"Radio emission height in normal radio pulsars with slow periods, larger than 100 ms, is well constrained to lie at altitudes around several hundreds of kilometres [72].",
"This is deduced from the shift in polarization angle traverse with respect to the middle of the pulse profile.",
"This shift $\\Delta \\phi $ is explained in the framework of aberration/retardation effects of photons propagating in the magnetosphere [73] and amounts to $\\Delta \\phi \\approx 4\\,r/r_{\\rm L}.$ This shift cannot be explained by emitting a photon along a field line in the corotating frame and then using aberration formulas because the pulse profile would be subject to the same shift in phase as the polarization angle traverse and therefore cancelling the possible time delay between the middle of the pulse profile and the steepest gradient in the polarization angle.",
"The only way to correctly catch this shift, which is a well defined fact seen in many observations of radio pulsars, is through photons emitted along field lines dragged into corotation as seen in the observer inertial frame.",
"This corotation velocity leads naturally to a time lag between the middle of the pulse profile and the polarization angle inflexion point.",
"Nevertheless, the magnetic field topology has to adjust to compensate for the corotation velocity in order to keep the particle velocity less than the speed of light.",
"This is not always possible outside the light-cylinder as already explained in paragraph REF .",
"The approximate estimate given in eq.",
"(REF ) has been checked for off-centred and rotating dipoles by [74].",
"It is therefore a very robust result, sharply constraining the radio emission heights.",
"Consequently, the view presented in paragraph REF must be rejected because it cannot reproduce aberration/retardation effects in pulsar radio polarization observations.",
"However, the particle motion described in paragraph REF seems more appropriate.",
"But the best choice in our view is represented in paragraph REF where trajectories are computed according to the full electromagnetic field taking into account radiative effects.",
"The latter option gives the simplest plausible scenario where radiation reaction acts efficiently and self-consistently backwards onto the particle motion and onto the electromagnetic field."
],
[
"Discussion",
"Pulsar magnetospheres have been extensively computed for stellar centred dipolar fields in special relativity.",
"However, there are increasing evidences for off-centred or even multipolar components anchored in the neutron star crust.",
"Indeed, joined modelling of pulsed radio emission and thermal X-rays from the polar cap hot spots requires decentred dipoles [75].",
"Moreover detailed investigations of X-ray light curves of millisecond pulsars also favours off-centred and quadrupolar components according to recent observations from NICER [76].",
"Nevertheless, as shown by [77], we do not expect drastic changes in the spin-down luminosity and magnetic field structure outside the light-cylinder for slowly rotating pulsars with period $P>10$ ms.",
"In the case of radiative magnetospheres, we expect a similar trend because radiation occurs outside the light-cylinder where the multipolar components have sufficiently decreased to become negligible.",
"Indeed, a dipole magnetic field decreases like $1/r^3$ whereas a multipole of order $\\ell $ decreases like $1/r^{\\ell +2}$ .",
"Therefore a multipole of strength $B_{\\rm m}$ at the surface contributes only a ratio $(B_{\\rm m} / B_{\\rm d} ) \\, (R/r_{\\rm L})^\\ell $ compared to a dipole of strength $B_{\\rm d}$ at the surface.",
"We can draw the same conclusions for general-relativistic radiative magnetospheres.",
"Indeed, for force-free magnetospheres, [78] already showed that qualitatively the picture does not vary and that the spindown luminosity scales like the magnetic field strength at the light-cylinder.",
"Extrapolating to the radiative case, general-relativistic effects remain very weak outside the light-cylinder for any pulsar, millisecond or second and the overall picture discussed above remains valid."
],
[
"Conclusions",
"Rotation in neutron stars plays a central role to sustain their electromagnetic activity of particle acceleration, pair creation and the subsequent broadband radiation from radio wavelengths to very high energies.",
"A corotating magnetosphere is often used as a good approximation to describe its electrodynamics.",
"However, due to their fast rotation, relativistic speeds are reached already close to the stellar surface, around the light-cylinder.",
"This hypothetical surface separates the inner magnetosphere from the wind.",
"We showed that the transition zone between the quasi-static magnetosphere and the wave zone in the wind is difficult to treat satisfactorily and smoothly because of the absence of a physical frame rotating with the star outside this light cylinder.",
"Several prescriptions where exposed, starting from different assumptions.",
"We also showed that pulsar radio observations give some hint about promising paths to follow particle trajectories and their emission.",
"Clearly, a deeper and better understanding of the neutron star electrodynamics is required to faithfully explain the wealth of data about their radiation.",
"Neutron stars are one of the only macroscopic objects that produce relativistic rotation on a length scale of the order of Earth radius about several hundreds to several thousands of kilometres.",
"Investigating jointly theoretical models and observational facts will reveal relativistic rotation effects in astrophysics in an unusual way, hoping to solve half a century mystery bout their functioning.",
"This research was funded by CEFIPRA grant number IFC/F5904-B/2018.",
"We would like to acknowledge the High Performance Computing center of the University of Strasbourg for supporting this work by providing scientific support and access to computing resources.",
"Part of the computing resources were funded by the Equipex Equip@Meso project (Programme Investissements d'Avenir) and the CPER Alsacalcul/Big Data.",
"We also thank the International Space Science Institute, Berne, Switzerland for their hospitality and for providing financial support during the meeting led by I. Contopoulos & D. Kazanas that helped to improve the present work.",
"References"
]
] | 2001.03422 |
[
[
"Design and Control of a Variable Aerial Cable Towed System"
],
[
"Abstract Aerial Cable Towed Systems (ACTS) are composed of several Unmanned Aerial Vehicles (UAVs) connected to a payload by cables.",
"Compared to towing objects from individual aerial vehicles, an ACTS has significant advantages such as heavier payload capacity, modularity, and full control of the payload pose.",
"They are however generally large with limited ability to meet geometric constraints while avoiding collisions between UAVs.",
"This paper presents the modelling, performance analysis, design, and a proposed controller for a novel ACTS with variable cable lengths, named Variable Aerial Cable Towed System (VACTS).Winches are embedded on the UAVs for actuating the cable lengths similar to a Cable-Driven Parallel Robot to increase the versatility of the ACTS.",
"The general geometric, kinematic and dynamic models of the VACTS are derived, followed by the development of a centralized feedback linearization controller.",
"The design is based on a wrench analysis of the VACTS, without constraining the cables to pass through the UAV center of mass, as in current works.",
"Additionally, the performance of the VACTS and ACTS are compared showing that the added versatility comes at the cost of payload and configuration flexibility.",
"A prototype confirms the feasibility of the system."
],
[
"INTRODUCTION",
"Combining the agility of aerial vehicles with the manipulation capability of manipulators makes aerial manipulation an attractive topic[1].",
"Unmanned Aerial Vehicles (UAVs) are becoming more and more popular during last decades, not only in research fields but also in commercial applications.",
"Amongst UAVs, quadrotors are widely used in photographing, inspection, transportation and (recently) manipulation because of their agility, versatility, and low cost.",
"However, the coupling between their translational and rotational dynamics complicates tasks requiring fine pose control.",
"Moreover, most have limited payload capacity, which makes it difficult to transport large objects by an individual quadrotor.",
"Multi-quadrotor collaboration is studied as a solution to compensate the previous two drawbacks inherent to individual quadrotors.",
"In this paper, we study a payload suspended via cables, which reduce the couplings between the platform and quadrotors attitude dynamics, allow reconfigurability, and reduce aerodynamic interference between the quadrotor downwash and the payload.",
"Moreover, Cable-Driven Parallel Robots (CDPRs) are well-studied, providing a part of theoretical support, such as tension distribution algorithms in [2], [3], [4], [5], controllers [6], [7] and wrench performance evaluations [8].",
"Reconfigurable CDPRs with both continuously moving [6], [9] and discrete [10] cable anchor points are proposed to solve problems such as wrench infeasiblility and self or environmental collisions.",
"For the past ten years, the Aerial Cable Towed System (ACTS) composed of several UAVs, a payload, and cables to cooperatively manipulate objects, has attracted researchers' attention and has been well developed from a controls viewpoint.",
"An ACTS prototype for 6-dimensional manipulation \"FlyCrane\" and a motion planning approach called Transition-based Rapidly-exploring Random Tree (T-RRT) are proposed in [11].",
"In [12] an ACTS prototype for 3-dimensional manipulation and a decentralized linear quadratic control law are proposed and [6] develops a general feedback linearization control scheme for over-actuated ACTS with a 6-DOF payload.",
"An ACTS prototype with three quadrotors and a point mass is implemented in [13] and the performance was evaluated using capacity margin, a wrench-based robustness index adapted from the CDPR in [14].",
"Figure: The sketch structure of a VACTS with 4 quadrotors, 6 winches and cables, and a platformThe ACTS shows promise both in aerial manipulation and sharing heavy burden, however there are still some limitations for current designs.",
"The size of ACTS is usually large.",
"As a consequence, it is not suitable for motion in cluttered environments.",
"Besides, short cables may lead to self collisions, particularly on takeoff where cable tension discontinuities may lead to poor control.",
"Furthermore, the ACTS may have to change the distribution of UAVs for motion in a cluttered environment, which may lead to wrench infeasibility and self collisions.",
"Therefore, a novel Aerial Cable Towed System with actuated cable lengths, the Variable Aerial Cable Towed System (VACTS) is proposed in this paper to make up for these shortcomings.",
"The actuated cable lengths can reshape the size of overall system, which implies the possibility of passing through a constrained environment or limited spacehttps://drive.google.com/file/d/1IO3qvFyWSTnUMLefXeysFpoqPMR585ud/view?usp=sharing.",
"It also allows flexibility in choosing between external load resistance and energy efficiency for some systems, with the possibility of modifying the wrench capabilities of certain ACTS systems such as the Flycrane.",
"Moreover, it is conceivable that combining force control of the quadrotors with velocity control by actuating cable lengths might improve payload positioning precision.",
"This modification to the common ACTS is first proposed in this paper, and a general model is rigorously built with few assumptions.",
"A prototype of this novel system (to our knowledge, the first ACTS with actuated cables) is presented in this paper, showing the system is feasible.",
"Section derives the geometric, kinematic and dynamic models.",
"Section illustrates the architecture of a centralized feedback linearization controller.",
"The design of the VACTS and its performance relative to the ACTS are compared in Sec. .",
"Section discusses the experimental results.",
"Conclusions and future work are presented in Sec.",
"."
],
[
"Modelling",
"The geometric, kinematic and dynamic models of the VACTS are derived in this section.",
"The sketch structure of a VACTS is shown in Fig.",
"REF .",
"It is composed of $n$ quadrotors, $m$ cables and winches ($m\\ge n$ ), and a payload.",
"There are $s_j \\in [1,2]$ winches mounted on the $j^{\\text{th}}$ quadrotor.",
"Unlike existing ACTS models where the cable passes through the quadrotor center of mass (COM), each winch of the VACTS may impose geometric constraints, therefor the model considers the cable attached to an arbitrary point on the quadrotor."
],
[
"Geometric Modelling of the VACTS",
"The geometric parametrization is presented in Fig.",
"REF with symbolic interpretation listed in Table REF .",
"Matrix $^a{\\mathbf {T}}_b$ is the homogeneous transformation matrix from frame $\\mathcal {F}_a$ to $\\mathcal {F}_b$ , and consists of a rotation matrix ${}^a{\\mathbf {R}}_b$ and a translation vector ${}^a{\\mathbf {x}}_b$ .",
"Matrix $^j{\\mathbf {T}}_{wi}$ expresses the transform of the $i^{\\text{th}}$ winch from the $j^{\\text{th}}$ quadrotors frame $\\mathcal {F}_j$ located at it's center of mass (COM).",
"The cable length is $l_i$ , and the unit vector along cable direction is expressed in the payload frame $\\mathcal {F}_p$ as ${}^p{\\mathbf {u}}_i={\\begin{bmatrix}c_{{\\phi }_i}s_{{\\theta }_i} &s_{{\\phi }_i}s_{{\\theta }_i} &c_{{\\theta }_i}\\end{bmatrix}}^T$ via azimuth angle ${\\phi }_i$ and inclination angle ${\\theta }_i$ , as used in [13].",
"The gravity vector is $\\mathbf {g}=[0 \\; 0 \\; -9.81]\\text{~ms}^{-2}$ .",
"Note that if $s_j = 2$ , the two cables share a coupled motion wrt the payload.",
"The $i^{\\text{th}}$ loop closure equation can be derived considering the forward and backward derivation of ${}^0{\\mathbf {x}}_{Ii}$ .",
"${}^0{\\mathbf {x}}_p + {}^0{\\mathbf {R}}_p {}^p{\\mathbf {x}}_{B_i} + l_i {}^0{\\mathbf {R}}_p {}^p{\\mathbf {u}}_i = {}^0{\\mathbf {x}}_j + {}^0{\\mathbf {R}}_j {}^j{\\mathbf {x}}_{Ii}$"
],
[
"Kinematic Modelling of the VACTS",
"After differentiating (REF ) with respect to time, the first order kinematic model for each limb can be derived as (REF ), where the payload has translational velocity ${}^0{\\dot{\\mathbf {x}}}_p$ and angular velocity ${}^0{\\mathbf {\\omega }}_p$ , and the $j^{\\text{th}}$ quadrotor has translational velocity ${}^0{\\dot{\\mathbf {x}}}_j$ and angular velocity ${}^0{\\mathbf {\\omega }}_j$ .",
"$\\footnotesize \\begin{aligned}{}^0{\\dot{\\mathbf {x}}}_{p} + {}^0{\\mathbf {\\omega }}_p \\times ({}^0{\\mathbf {R}}_p {}^p{\\mathbf {x}}_{B_i}) + l_i {}^0{\\mathbf {R}}_p {}^p{\\dot{\\mathbf {u}}}_i + l_i {}^0{\\mathbf {\\omega }}_p \\times ({}^0{\\mathbf {R}}_p {}^p{{\\mathbf {u}}}_i) \\\\= {}^0{\\dot{\\mathbf {x}}}_{j} - {\\dot{l}}_i {}^0{\\mathbf {R}}_p {}^p{\\mathbf {u}}_i + {}^0{\\mathbf {\\omega }}_j \\times ({}^0{\\mathbf {R}}_j {}^j{\\mathbf {x}}_{Ii}) + {}^0{\\mathbf {R}}_j {}^j{\\dot{\\mathbf {x}}}_{Ii}\\end{aligned}\\normalsize $ Figure: Parametrization of VACTS with the quadrotor, winch and platformTable: NomenclatureThe first order kinematic model can be re-expressed as (REF ), where the Jacobian matrix ${\\mathbf {J}}_{(6+3m) \\times 3n}$ can be obtained from the forward Jacobian matrix ${\\mathbf {A}}_{3m \\times (6+3m)}$ and the inverse Jacobian matrix ${\\mathbf {B}}_{3m \\times 3n}$ .",
"${\\mathbf {A}}^{+}$ is the pseudo-inverse of $\\mathbf {A}$ , which minimizes the 2-norm of ${\\dot{\\mathbf {x}}}_{\\mathbf {t}}$ since the system is under-determined.",
"For example, if $s_1=1$ and $s_2=2$ , the matrix $\\mathbf {B}$ will be expressed as (REF ).",
"The task space vector is ${\\mathbf {x_t}}_{(6+3m) \\times 1}$ and the joint space vector is ${\\mathbf {q_a}}_{3n \\times 1}$ .",
"Note that ${\\dot{\\mathbf {x}}}_{\\mathbf {t}}$ is not the time derivative of $\\mathbf {x_t}$ considering that the angular velocity ${}^0{\\mathbf {\\omega }}_p$ is not the time derivative of payload orientation, which can be expressed in the form of Euler angles or quaternions.",
"The form ${[\\mathbf {x}]}_{\\times }$ is the cross product matrix of vector ${\\mathbf {x}}$ and ${\\bf {I}}_3$ is a $3\\times 3$ identity matrix.",
"${\\dot{\\mathbf {x}}}_{\\mathbf {t}} = \\mathbf {J} {\\dot{\\mathbf {q}}}_{\\mathbf {a}} + \\mathbf {a} \\text{, with } \\mathbf {J} = {\\mathbf {A}}^{+} \\mathbf {B}$ $\\small {\\dot{\\mathbf {x}}}_{\\mathbf {t}}={\\begin{bmatrix} {}^0{\\dot{\\mathbf {x}}}_{p}^T & {}^0{\\mathbf {\\omega }}_p^T & {\\dot{\\phi }}_1 & {\\dot{\\theta }}_1 & {\\dot{l}}_1 & \\cdots & {\\dot{\\phi }}_m & {\\dot{\\theta }}_m & {\\dot{l}}_m \\end{bmatrix}}^T \\normalsize $ ${\\dot{\\mathbf {q}}}_{\\mathbf {a}}={\\begin{bmatrix} {}^0{\\dot{\\mathbf {x}}}_{1}^T & \\cdots & {}^0{\\dot{\\mathbf {x}}}_{n}^T \\end{bmatrix}}^T$ $\\footnotesize \\begin{aligned}\\mathbf {A}=&\\begin{bmatrix}{\\mathbb {I}}_3 & {\\mathbf {A}}_{12} & l_1 {}^0{\\mathbf {R}}_p {\\mathbf {C}}_1 & {}^0{\\mathbf {R}}_p {}^p{\\mathbf {u}}_1 \\\\\\vdots & \\vdots &&& \\ddots & \\ddots \\\\{\\mathbb {I}}_3 & {\\mathbf {A}}_{m2} &&&&& l_m {}^0{\\mathbf {R}}_p {\\mathbf {C}}_m & {}^0{\\mathbf {R}}_p {}^p{\\mathbf {u}}_m\\end{bmatrix} \\\\&\\text{with } {\\mathbf {A}}_{i2} = {[{}^0{\\mathbf {R}}_p ({}^p{\\mathbf {x}}_{B_i} + l_i {}^p{\\mathbf {u}}_i)]}_{\\times }^T\\end{aligned}\\normalsize $ $\\mathbf {B}=\\begin{bmatrix}{\\bf {I}}_3 \\\\& {\\bf {I}}_3 \\\\& {\\bf {I}}_3 \\\\&& \\ddots \\\\&&& {\\bf {I}}_3\\end{bmatrix} \\begin{matrix}\\Rightarrow s_1 \\\\ \\\\\\Rightarrow s_2 \\\\\\vdots \\\\\\Rightarrow s_n\\end{matrix}$ ${\\mathbf {C}}_i = \\begin{bmatrix}-s_{{\\phi }_i}s_{{\\theta }_i} & c_{{\\phi }_i}c_{{\\theta }_i} \\\\c_{{\\phi }_i}s_{{\\theta }_i} & s_{{\\phi }_i}c_{{\\theta }_i} \\\\0 & -s_{{\\theta }_i}\\end{bmatrix}$ $\\\\\\mathbf {a} = {\\mathbf {A}}^{+} \\begin{bmatrix}{}^0{\\mathbf {\\omega }}_1 \\times ({}^0{\\mathbf {R}}_1 {}^1{\\mathbf {x}}_{I_1}) + {}^0{\\mathbf {R}}_1 {}^1{\\dot{\\mathbf {x}}}_{I_1} \\\\{}^0{\\mathbf {\\omega }}_2 \\times ({}^0{\\mathbf {R}}_2 {}^2{\\mathbf {x}}_{I_2}) + {}^0{\\mathbf {R}}_2 {}^2{\\dot{\\mathbf {x}}}_{I_2} \\\\\\vdots \\\\{}^0{\\mathbf {\\omega }}_n \\times ({}^0{\\mathbf {R}}_n {}^n{\\mathbf {x}}_{I_m}) + {}^0{\\mathbf {R}}_n {}^n{\\dot{\\mathbf {x}}}_{I_m}\\end{bmatrix}$ Additionally, the second order kinematic model is also derived in matrix form as (REF ), where $\\mathbf {b}$ is the component related to the derivatives of $\\mathbf {A}$ and $\\mathbf {B}$ .",
"${\\mathbf {B}}^{+}$ is the pseudo-inverse of $\\mathbf {B}$ , which will minimize the residue of the equation system if the system is over-determined, i.e.",
"$m>n$ .",
"${\\ddot{\\mathbf {q}}}_{\\mathbf {a}} = {\\mathbf {B}}^{+} \\mathbf {A} {\\ddot{\\mathbf {x}}}_{\\mathbf {t}} + {\\mathbf {B}}^{+} \\mathbf {b} \\text{, with }\\mathbf {b}={\\begin{bmatrix} {\\mathbf {b}}_1^T & \\cdots & {\\mathbf {b}}_m^T \\end{bmatrix}}^T$ $\\small \\begin{aligned}{\\mathbf {b}}_i= & {}^0{\\mathbf {\\omega }}_p \\times ({}^0{\\mathbf {\\omega }}_p \\times {}^0{\\mathbf {R}}_p ({}^p{\\mathbf {x}}_{B_i}+l_i {}^p{{\\mathbf {u}}}_i)) \\\\& + 2{\\dot{l}}_i {}^0{\\mathbf {\\omega }}_p \\times ({}^0{\\mathbf {R}}_p {}^p{{\\mathbf {u}}}_i) + 2{\\dot{l}}_i {}^0{\\mathbf {R}}_p {\\mathbf {C}}_i {\\begin{bmatrix}{\\dot{\\phi }}_i & {\\dot{\\theta }}_i\\end{bmatrix}}^T \\\\& + 2 l_i {}^0{\\mathbf {\\omega }}_p \\times ({}^0{\\mathbf {R}}_p {\\mathbf {C}}_i {\\begin{bmatrix}{\\dot{\\phi }}_i & {\\dot{\\theta }}_i \\end{bmatrix}}^T) + l_i {}^0{\\mathbf {R}}_p {\\dot{\\mathbf {C}}}_i {\\begin{bmatrix}{\\dot{\\phi }}_i & {\\dot{\\theta }}_i\\end{bmatrix}}^T \\\\& - {}^0{\\dot{\\mathbf {\\omega }}}_j \\times ({}^0{\\mathbf {R}}_j {}^j{\\mathbf {x}}_{Ii}) - {}^0{\\mathbf {\\omega }}_j \\times ({}^0{\\mathbf {\\omega }}_j \\times {}^0{\\mathbf {R}}_j {}^j{\\mathbf {x}}_{Ii}) \\\\& -2{}^0{\\mathbf {\\omega }}_j \\times {}^0{\\mathbf {R}}_j {}^j{\\dot{\\mathbf {x}}}_{Ii} - {}^0{\\mathbf {R}}_j {}^j{\\ddot{\\mathbf {x}}}_{Ii}\\end{aligned}\\normalsize $"
],
[
"For the payload",
"By using the Newton-Euler formalism, the following dynamic equation is presented, taking into account of an external wrench ${\\mathbf {w}}_e$ exerted on the payload by the environment, consisting of force ${\\mathbf {f}}_e$ and moment ${\\mathbf {m}}_e$ .",
"Along with ${\\mathbf {w}}_e$ , inertial forces from the mass $m_p$ and inertia matrix ${\\mathbf {I}}_p$ of the payload, and cable tensions $t_i {}^p{\\mathbf {u}}_i$ act on the payload.",
"Eqns.",
"(REF ,REF ) express the dynamics of the platform.",
"${\\mathbf {f}}_e + m_p\\mathbf {g}+{}^0{\\mathbf {R}}_p \\sum _{i=1}^{m} t_i {}^p{\\mathbf {u}}_i = m_p {}^0{\\ddot{\\mathbf {x}}}_p$ $\\begin{aligned}{\\mathbf {m}}_e + ({}^0{\\mathbf {R}}_p {}^p{\\mathbf {x}}_C) \\times m_p\\mathbf {g} + {}^0{\\mathbf {R}}_p \\sum _{i=1}^{m}{}^p{\\mathbf {x}}_{B_i} \\times t_i {}^p{\\mathbf {u}}_i \\\\={\\mathbf {I}}_p {}^0{\\dot{\\mathbf {\\omega }}}_p + {}^0{\\mathbf {\\omega }}_p \\times ({\\mathbf {I}}_p {}^0{\\mathbf {\\omega }}_p)\\end{aligned}$ If these equations are expressed in matrix form, the cable tension vector $\\mathbf {t}={\\begin{bmatrix}t_1 & \\cdots & t_m\\end{bmatrix}}^T$ can be derived as (REF ) considering a wrench matrix $\\mathbf {W}$ and a mass matrix ${\\mathbf {M}}_p$ .",
"${\\mathbf {W}}^{+}$ is the pseudo-inverse of $\\mathbf {W}$ and $\\mathcal {N}(\\mathbf {W})$ represents the null-space of $\\mathbf {W}$ , used in some tension distribution algorithms if the payload is over-actuated [4], [5].",
"$\\mathbf {t} = {\\mathbf {W}}^{+} \\left( {\\mathbf {M}}_p {\\begin{bmatrix}{}^0{\\ddot{\\mathbf {x}}}_p^T & {}^0{\\dot{\\mathbf {\\omega }}}_p^T\\end{bmatrix}}^T + \\mathbf {c} \\right) + \\mathcal {N}(\\mathbf {W})$ ${\\mathbf {M}}_p = \\begin{bmatrix}m_p {\\mathbb {I}}_3 & {\\mathbf {0}} \\\\{\\mathbf {0}} & {\\mathbf {I}}_p\\end{bmatrix}$ $\\mathbf {W}=\\begin{bmatrix}{{}^0{\\mathbf {R}}_p {}^p\\mathbf {u}}_1 &\\cdots & {}^0{\\mathbf {R}}_p {}^p{\\mathbf {u}}_m\\\\{}^0{\\mathbf {R}}_p ({}^p{\\mathbf {x}}_{B_1}\\times {}^p{\\mathbf {u}}_1) &\\cdots & {}^0{\\mathbf {R}}_p ({}^p{\\mathbf {x}}_{B_m}\\times {}^p{\\mathbf {u}}_m)\\end{bmatrix}$ $\\mathbf {c}= \\begin{bmatrix}{\\mathbf {0}}\\\\{}^0{\\mathbf {\\omega }}_p \\times ({\\mathbf {I}}_p {}^0{\\mathbf {\\omega }}_p)\\end{bmatrix} - \\begin{bmatrix}m_p\\mathbf {g} \\\\ ({}^0{\\mathbf {R}}_p {}^p{\\mathbf {x}}_C) \\times m_p\\mathbf {g}\\end{bmatrix} - {\\mathbf {w}}_e$"
],
[
"For the winch",
"All winches are assumed to have the same design, specifically the drum radius of all winches is $r_{d}$ , the mass is $m_w$ , and the inertia matrix is ${\\mathbf {I}}_w$ .",
"The $x$ -axis of the winch frame is along the winch drum axis of rotation.",
"The rotational rate of the $i^{\\text{th}}$ winch is ${\\omega }_{r_i}$ .",
"It relates to the change rate of cable length as (REF ) considering that the radius of cable is small (reasonable, as payloads are generally light).",
"${\\dot{l}}_i= r_d {\\omega }_{r_i}$ By using the Newton-Euler formalism, the following dynamic equation is expressed in $\\mathcal {F}_{wi}$ , where the winch torque is ${\\tau }_i$ .",
"Additionally, we define a constant vector $\\mathbf {k}=\\begin{bmatrix} 1&0&0 \\end{bmatrix}$ to represent the torque generated from cable tension orthogonal to the drum surface.",
"${{\\tau }}_{i} - \\mathbf {k} ({}^w{\\mathbf {x}}_{Ii} \\times t_i{}^w{\\mathbf {u}}_i) = {{I}}_{xx} {\\dot{\\omega }}_{r_i}$"
],
[
"For the quadrotor",
"By using the Newton-Euler formalism, the following dynamic equations can be derived taking into account of the thrust force ${\\mathbf {f}}_j$ and thrust moment ${\\mathbf {m}}_j$ generated by the quadrotor's motors.",
"The mass of the $j^{\\text{th}}$ quadrotor with $s_j$ embedded winches is $m_j= m_q + s_j m_w$ and the inertia matrix ${\\mathbf {I}}_j$ , is from the parallel axis theorem.",
"${\\mathbf {f}}_j + m_j\\mathbf {g} - {}^0{\\mathbf {R}}_p \\sum _{k=1}^{s_j} t_{k} {}^p{\\mathbf {u}}_{k} = m_{j} {}^0{\\ddot{\\mathbf {x}}}_j$ $\\begin{aligned}{\\mathbf {m}}_j + \\underbrace{({}^0{\\mathbf {R}}_j{}^j{\\mathbf {x}}_G) \\times m_j \\mathbf {g}}_{\\text{gravity}} - {}^0{\\mathbf {R}}_j \\sum _{k=1}^{s_j} \\underbrace{{}^j{\\mathbf {x}}_{I_{k}} \\times t_{k} {}^p{\\mathbf {u}}_{k}}_{\\text{cable tension}} \\\\= {\\mathbf {I}}_j {}^0{\\dot{\\mathbf {\\omega }}}_j+{}^0{\\mathbf {\\omega }}_j\\times ({\\mathbf {I}}_j {}^0{\\mathbf {\\omega }}_j)\\end{aligned}$ The dynamic model is expressed in matrix form, where the thrust force of all quadrotors $\\mathbf {f}={\\begin{bmatrix} {\\mathbf {f}}_1^T {\\mathbf {f}}_2^T & \\cdots & {\\mathbf {f}}_n^T \\end{bmatrix}}^T$ are derived as (REF ), with the mass matrix ${\\mathbf {M}}_q$ , and a wrench matrix $\\mathbf {U}={\\begin{bmatrix} {\\mathbf {U}}_1^T & \\cdots & {\\mathbf {U}}_n^T \\end{bmatrix}}^T$ such that ${\\mathbf {U}}_j=\\begin{bmatrix}\\mathbf {0}& {}^0{\\mathbf {R}}_p {}^p{\\mathbf {u}}_i & \\cdots &\\mathbf {0}\\end{bmatrix}$ has $s_j$ non-zero columns.",
"$\\mathbf {f} = {\\mathbf {M}}_q {\\ddot{\\mathbf {q}}}_{\\mathbf {a}} + \\mathbf {U}\\mathbf {t}+\\mathbf {d}$ $\\scriptsize {\\mathbf {M}}_q=\\begin{bmatrix}m_1 {\\bf {I}}_3 \\\\& m_2{\\bf {I}}_3 \\\\&&\\ddots \\\\&&& m_n {\\bf {I}}_3 \\\\\\end{bmatrix} \\text{ and }{\\mathbf {d}}= \\begin{bmatrix}m_1\\mathbf {g} \\\\ m_2\\mathbf {g} \\\\ \\vdots \\\\ m_n\\mathbf {g}\\end{bmatrix}\\normalsize $ The inverse dynamic model can be derived as (REF ) from (REF ,REF ,REF ).",
"${\\mathbf {f}} = {\\mathbf {D}}_q {\\ddot{\\mathbf {x}}}_{\\mathbf {t}} + {\\mathbf {G}}_q$ ${\\mathbf {D}}_q = {\\mathbf {M}}_q {\\mathbf {B}}^{+} \\mathbf {A} + \\mathbf {U} {\\mathbf {W}}^{+} \\begin{bmatrix} {\\mathbf {M}}_p & \\mathbf {0} \\end{bmatrix}$ ${\\mathbf {G}}_q = {\\mathbf {M}}_q {\\mathbf {B}}^{+} \\mathbf {b} + \\mathbf {U}{\\mathbf {W}}^{+} \\mathbf {c} + \\mathbf {d}$ Several types of quadrotors exist in the literature.",
"In this paper, standard quadrotors with 4 co-planar propellers are considered.",
"For each quadrotor, the thrust force and thrust moment can be derived in the following way [15].",
"As shown in Fig.",
"REF , each propeller produces a thrust force $f_{p_k}$ and a drag moment $m_{p_k}$ , for $k=1,2,3,4$ .",
"Considering that the aerodynamic coefficients are almost constant for small propellers, the following expression can be obtained after simplification: $f_{p_k}=k_f{\\Omega _k}^2$ $m_{p_k}=k_m{\\Omega _k}^2$ where $k_f$ , $k_m$ are constants and $\\Omega _k$ is the rotational rate of the $k^{\\text{th}}$ propeller.",
"In our experiments, $k_f={3.55}\\mathrm {e}{-6}$ $N\\cdot s^2/{Rad}^2$ and $k_m={5.4}\\mathrm {e}{-8}$ $N\\cdot s^2/{Rad}^2$ were empirically identified using an free-flight identification method proposed by some of the authors, currently under review.",
"The total thrust force and moments expressed in the $j^{\\text{th}}$ quadrotor frame $\\mathcal {F}_j$ are ${\\begin{bmatrix}0&0&f_z\\end{bmatrix}}^T$ and ${\\begin{bmatrix}m_x&m_y&m_z\\end{bmatrix}}^T$ , respectively, where $r$ is the distance between the propeller and the center of the quadrotor.",
"Importantly, we could indicate that quadrotors have four DOFs corresponding to roll, pitch, yaw and upward motions.",
"$\\begin{bmatrix}f_z\\\\m_x\\\\m_y&\\\\m_z\\end{bmatrix}=\\begin{bmatrix}1&1&1&1\\\\0&r&0&-r \\\\ -r&0&r&0 \\\\ -\\frac{k_m}{k_f}&\\frac{k_m}{k_f}&-\\frac{k_m}{k_f}&\\frac{k_m}{k_f}\\end{bmatrix}\\begin{bmatrix}f_{p_1} \\\\ f_{p_2} \\\\ f_{p_3} \\\\ f_{p_4}\\end{bmatrix}$ Figure: Free Body Diagram of the quadrotor jjThe overview of a centralized feedback linearization VACTS control diagram is shown in Fig.",
"REF , which includes the control loop of quadrotors, as well as winches.",
"Once the thrusts and the cable velocities are decided by the PD control law (REF ) and the P control law (REF ), respectively, the quadrotors can be stabilized by attitude controller and the cable lengths can be actuated by servo motors.",
"The poses of quadrotors and the payload can be tracked by the Motion Capture System (MoCap).",
"Therefrom the task space state vector $\\mathbf {x_t}$ and its derivative are derived based on the geometric and kinematic models."
],
[
"Desired Thrust",
"The desired thrust represents the reference thrust generation of quadrotors for sharing burden.",
"The control law is proposed based on the inverse dynamic model of the VACTS.",
"This control law for a regular parallel robot guarantees asymptotic convergence is the absence of modelling errors [16].",
"As the error behaves as a second-order system, the gains can be determined by a cutoff frequency and a damping ratio.",
"${\\mathbf {f}}^d = {\\mathbf {D}}_q({{\\ddot{\\mathbf {x}}}_{\\mathbf {t}}}^d + k_d({{\\dot{\\mathbf {x}}}_{\\mathbf {t}}}^d-{\\dot{\\mathbf {x}}}_{\\mathbf {t}}) + k_p({\\mathbf {x}}_{\\mathbf {t}}^d-{\\mathbf {x_t}}) ) +{\\mathbf {G}}_q$"
],
[
"Velocity Control Loop",
"The velocity control loop represents the control loop of winches for actuating cable lengths.",
"${\\dot{\\mathbf {l}}}^d$ is obtained from the desired trajectory.",
"Moreover, the function of saturation in the control diagram will be introduced in detail.",
"Let us assume that there is a linear relationship between the maximum speed of winch ${\\omega }_{r_{max}}$ and its torque, which is called the speed-torque characteristics.",
"For a certain torque, the servo motor can tune the winch speed from $-{\\omega }_{r_{max}}$ to $+{\\omega }_{r_{max}}$ .",
"Therefore, the output signal can be determined based on the knowledge of cable tension and desired cable velocity.",
"Furthermore, the output signal is limited within the $70\\%$ of its working range for ensuring safety.",
"$\\dot{\\mathbf {l}}={\\dot{\\mathbf {l}}}^d+k_c({\\mathbf {l}}^d-\\mathbf {l})$"
],
[
"Attitude Control",
"The aim of the attitude controller [17] is to provide the desired thrust force and stabilize the orientation of quadrotor.",
"An estimated attitude ${\\mathbf {q}}_j$ in quaternion representation is used as the feedback signal.",
"The yaw angle is set to be a given value such as 0 in order to fully constrain the problem, however it may be interesting in the future to optimize it as a function of the state to reduce moments applied by the cable to the quadrotor.",
"In [18], a second-order attitude controller with feed-forward moment is proposed, to which a feed forward moment from the desired cable tension could be applied.",
"Figure: The control diagram showing the desired thrust of quadrotors (blue), the control loop of winches (green) and the VACTS plant (red)"
],
[
"Wrench Analysis",
"The available wrench set ${\\mathcal {W}}_a$ is the set of wrenches that the mechanism can generate and depends on the state.",
"${\\mathcal {W}}_a$ can be obtained by three space mappings, which is an extension of that introduced in [14] for a general classical ACTS.",
"This paper details the differences, which is caused 1) by the winch preventing cables from passing through the COM of the quadrotor, and 2) by the winch torque limits.",
"Generally, for an UAV with $n_p$ propellers, its propeller space $\\mathcal {P}$ can be determined by the lower (to prevent the motors from stalling) and upper bounds of thrust force that each propeller can generate.",
"Considering $n$ UAVs, the propeller space is (REF ).",
"$\\mathcal {P}=\\left\\lbrace {\\mathbf {f}}_p\\in {\\rm I\\!R}^{(n\\cdot n_p)}: \\underline{{\\mathbf {f}}_p}\\le {\\mathbf {f}}_p\\le \\bar{{\\mathbf {f}}_p}\\right\\rbrace $ For a certain set of cable tensions, the compensation moment can be obtained according to (REF ), which is risen from non-coincident COM and geometric center of quadrotors, as well as the cables not passing through the geometric center.",
"Then the maximum value of thrust force of the $j^{\\text{th}}$ quadrotor can be found by analyzing an optimization problem as (REF ).",
"$\\bar{f_z}=\\underset{f_{p_k}\\in \\mathcal {P}}{\\arg \\max } f_z \\text{ s.t.",
"(\\ref {eq:quad})}$ This method accounts for individual propeller saturation and may be applied to any (V)ACTS, for any co-planar multi-rotor type UAV."
],
[
"Thrust, tension and wrench spaces",
"The thrust space $\\mathcal {H}$ is obtained by space mapping from the propeller space $\\mathcal {P}$ as (REF ).",
"It represents the set of available thrusts of quadrotors.",
"The tension space $\\mathcal {T}$ shows the available cable tensions, which considers not only the space mapping from the thrust space, but also the torque capacity of winch.",
"The available wrench set ${\\mathcal {W}}_a$ can be obtained by mapping tension space $\\mathcal {T}$ to wrench space $\\mathcal {W}$ [14] by using the convex hull method [8].",
"The capacity margin $\\gamma $ is a robustness index that is used to analyse the degree of feasibility of a configuration [19].",
"It is defined as the shortest signed distance from the task wrench ${\\mathcal {W}}_t$ (here $\\mathcal {W}_t = {m_p{\\bf g}}$ ) to the boundary of the ${\\mathcal {W}}_a$ zonotope.",
"The payload mass, the cable directions and the capability of UAVs and winches all affect the value of capacity margin.",
"For example, these four spaces can be presented visually as Fig.",
"REF for the VACTS with three quadrotors, three winches and cables, and a point-mass however the method is general for all ACTS designs [14].",
"Note that $\\mathcal {P}\\in \\mathbb {R}^{12}$ is shown as three dimensional, although in reality it is a 12 dimensional hypercube (as all motors and propellers are identical).",
"The most notable difference of wrench analysis between ACTS and VACTS is that $\\mathcal {H}$ of the ACTS is constant, while $\\mathcal {P}$ of the VACTS is constant and $\\mathcal {H}$ is variable.",
"The case study parameters for a VACTS with three quadrotors, three cables and winches, and one point-mass are shown in Table REF .",
"Figure REF shows the effect of different parameters related to winch (drum radius, offset, and orientation).",
"From this case study, we conclude that (i) For the same system configuration, the VACTS needs extra energy to compensate the moment generated by cable tension because of the offset of winch.",
"Therefore the winch should be mounted to the quadrotor as close to its centroid as possible; (ii) Additionally, the drum radius should be set less than a certain value according to the simulation results, such as $2cm$ for this case study; (iii) The optimal inclination angle is different after embedding winches; (iv) Last but not least, the servo motor selection is also under constraints.",
"The stall torque of servo motor should be large enough to support the force transmission.",
"Figure: Capacity margin for different mounted ways and drum radii with the offset along drum 2cm2cm of the coincident point of the i th i^{\\text{th}} cable-winch pair in the winch frame w 𝐱 Ii {}^w{\\mathbf {x}}_{Ii}.",
"The \"offset\" in the legend represents the translation vector j 𝐱 w {}^j{\\mathbf {x}}_w between the winch frame and the quadrotor frame.",
"The \"RotZRotZ\" represents the rotation angle around zz axis between the winch frame and the quadrotor frame.The performance of ACTS and VACTS can be evaluated from different points of view: ($i$ ) For the same system configuration for ACTS and VACTS, the VACTS needs extra energy to compensate the moment generated by cable tensions.",
"Therefore the VACTS has a smaller thrust space than the ACTS.",
"So we can come up with the conclusion that the ACTS has a better performance in terms of available wrench set than the VACTS in unconstrained environments; ($ii$ ) In constrained environments, maybe the ACTS can not achieve the optimal system configuration because of its large size.",
"For example, the ACTS can not achieve the optimal inclination angle (50) because of the big width of the overall system which can not be implemented in an environment with limited width.",
"While the VACTS can reshape its size and achieve an optimal configuration.",
"Therefore, the VACTS can behave better than the ACTS in constrained environments and it will reshape the available wrench set; ($iii$ ) The VACTS has a better manipulability than ACTS from the fact that the VACTS has a larger velocity capacity along cable directions than ACTS because of the actuated cable lengths, which can also be proven by simulation results as Fig.",
"REF .",
"The manipulability index [20] $w_s$ as defined in (REF ) is proportional to the volume of the ellipsoid of instantaneous velocities and that it can be calculated as the product of the singular values of the normalized Jacobian matrix ${\\mathbf {J}}_{norm}$ .",
"For example, for the VACTS with three quadrotors, three cables and winches, and one point-mass, ${\\mathbf {J}}_{norm}$ is derived as (REF ) considering the actuated cable lengths in joint space rather than task space; $w_s = \\sqrt{det({\\mathbf {J}}_{norm}{\\mathbf {J}}_{norm}^T)}\\text{, or } w_s = {\\lambda }_1{\\lambda }_2\\cdots {\\lambda }_r$ ${\\mathbf {J}}_{norm} = diag(1,1,1,l_1,l_1,l_2,l_2,\\cdots ,l_m,l_m) \\mathbf {J}$ Figure: The reciprocal of the manipulability index for ACTS and VACTS with three quadrotors and a point-mass.",
"The lower, the better.",
"(iv) On one hand, the actuated cable lengths increase the control complexity from $4n$ to $4n$ +$m$ .",
"On the other hand, they improve the reconfigurability from $3n$ -$m$ to $4n$ -$m$ at the same time, which means the system is more flexible.",
"Specifically, each quadrotor has four control variables, while each actuated cable length has one control variable.",
"Therefore the control complexity is $4n$ for an ACTS with $n$ quadrotors, while it’s $4n$ +$m$ for a VACTS with $n$ quadrotors and $m$ cables.",
"In the meanwhile, a single cable has two reconfigurable variables (azimuth and inclination angles) and a pair of coupled cables has one reconfigurable variable (inclination angle).",
"In an ACTS with $n$ quadrotors and $m$ cables, there are $2n$ -$m$ single cables and $m$ -$n$ pairs of coupled cables.",
"So an ACTS has $3n$ -$m$ reconfigurable variables.",
"If the cable lengths are actuated, extra $n$ independent reconfigurable variables appear, i.e., a VACTS has $4n$ -$m$ reconfigurable variables."
],
[
"Experimental results",
"The VACTS prototype with three quadrotors, winches and cables, and a point-mass $m_p=670g$ is shown in Fig.",
"REF .",
"The Quadrotor body is a Lynxmotion Crazy2fly as Fig.",
"REF because of low cost and mass, while the embedded winch is shown in Fig.",
"REF .",
"The servo motor winch is FS5106R.",
"The poses of quadrotors and the payload are tracked by the MoCap as we mentioned in Sec. .",
"Therefrom the cable lengths and the unit vectors along cable directions are computed.",
"The hardware and software platform and their working frequencies are detailed in [13].",
"The winch servo motor is controlled via ROS messages over Pixhawk AUX outputs.",
"The translation vector ${}^j{\\mathbf {x}}_{Ii}$ is between the coincident point $I_i$ and the origin of the quadrotor frame, as shown in Fig.",
"REF .",
"It is assumed to be a constant value ${[0,0,-6.4]}^T$ cm, thanks to the existence of guide hole in the V-shape bar of winch mount piece as shown in Fig.",
"REF .",
"Figure: The VACTS prototype with 3 quadrotors, winches and a point-mass Figure: Winch prototype and supportThe motion planning for a VACTS can be divided into several parts based on the task space state vector: the motion of payload, the system configuration of cable directions, and the cable lengths: (i) The motion of payload can be designed under the knowledge of environment; (ii) The design methodology for system configuration[13], [21] varies from quasi-static to dynamic case.",
"In short, it is designed under capacity margin and cost optimization; (iii) The cable lengths are determined on the basis of limited volume space.",
"In order to prove that this novel system is feasible and verify the ability of changing overall size, a motion planning is considered as following.",
"This system takes off firstly and then hover for several seconds.",
"Then the cable lengths are changed from initial values to 1.4m, followed by decreasing from 1.4m to 1.0m by using a fifth-order polynomial trajectory planning method [22].",
"During the period of cable displacements, the desired payload position and system configuration are kept as constants.",
"In other words, the quadrotors move along the cable direction while the payload stays at the same position.",
"Figure: The payload tracking of VACTS with 3 quadrotors, 3 winches and a point-mass m p =670m_p=670 gFigure: The cable lengths tracking of VACTS with 3 quadrotors, 3 winches and a point-mass m p =670m_p=670 gThe experimental results of payload tracking and cable lengths tracking are shown in Fig.",
"REF and Fig.",
"REF .",
"Moreover the corresponding video that shows the experimental results step by step can be found in the linkhttps://drive.google.com/open?id=139fZmalOvZGxuETJfh39mbEexZ2P5At.",
"From the experimental results, it's shown that the desired cable lengths are followed well and the system is stable under control.",
"The payload tracking mean errors along $x$ , $y$ and $z$ axis are $1.90cm$ , $9.41cm$ and $10.67cm$ , respectively.",
"The standard deviations are $3.15cm$ , $3.24cm$ and $3.29cm$ , respectively.",
"The cable lengths tracking mean errors for $l_1$ , $l_2$ and $l_3$ are $-2.25cm$ , $0.24cm$ and $0.10cm$ , respectively.",
"The standard deviations are $3.09cm$ , $0.73cm$ and $0.23cm$ , respectively.",
"Even though there was a cable length that did not follow the trajectory well, it is acceptable.",
"Specifically, the cable length velocity is limited by the winch capability.",
"In the experiments, we define the limitation that the maximum speed of servo motor is 70% of its actual maximum speed for security.",
"At some point, the servo motor reaches its safety limit which leads to the small slope of $l_1$ (red line) in Fig.",
"REF .",
"The error of payload tracking along $z$ -axis is mainly due to the low precision of attitude control for hovering or slow motion.",
"The positioning error of the payload can be decreased by using the moment feedback in the attitude controller [18].",
"The main control loop in Sec.",
"runs at 50 Hz and the attitude controller runs at 200 Hz.",
"The observed positioning errors are mainly due to this low frequency and the imperfect controller.",
"Experimental results show that the VACTS is feasible and that the change of cable lengths can reshape the VACTS, which implies the possibility of passing through a constrained environment or limited space.",
"Moreover, the precision of cable lengths control is much higher than the torque control of ACTS, which also implies the (untested) potential for improving precision of the VACTS when the payload position is fine-tuned by changing cable lengths."
],
[
"CONCLUSIONS",
"This paper dealt with a novel aerial cable towed system with actuated cable lengths.",
"Its non-linear models were derived and a centralized controller was developed.",
"The wrench analysis was extended to account for propeller saturation resulting from the offset of the cable connections with the quadrotor and the winch was designed based on the wrench performance.",
"The advantage of this novel system is in reshaping the overall size and wrench space in constrained environments, while it comes at the price of lower peak performance in unconstrained situations.",
"The feasibility of the system is experimentally confirmed, showing the system to be capable of resizing while hovering.",
"Later on, ($i$ ) a decentralized control method will be designed and applied in order to increase the flexibility and precision, with better fault tolerance; ($ii$ ) cameras will be used to obtain the relative pose of the payload instead of motion capture system as in [23] and [24] to develop a system that can be deployed in an external environment; ($iii$ ) some criteria will be investigated to quantify the versatility of VACTS; ($iv$ ) for the experimental demonstrations, a VACTS prototype with a moving-platform instead of point-mass will be developed to perform more complex aerial manipulations in a cluttered environment$^1$ ; ($v$ ) a quadrotor attitude controller with feed forward moments from cable tension measurements or estimations will be implemented to improve performance."
]
] | 2001.03435 |
[
[
"A quantitative study of some sources of uncertainty in opacity\n measurements"
],
[
"Abstract Laboratory (laser and Z-pinch) opacity measurements of well-characterized plasmas provide data to assist inertial confinement fusion, astrophysics and atomic-physics research.",
"In order to test the atomic-physics codes devoted to the calculation of radiative properties of hot plasmas, such experiments must fulfill a number of requirements.",
"In this work, we discuss some sources of uncertainty in absorption-spectroscopy experiments, concerning areal mass, background emission, intensity of the backlighter and self-emission of the plasma.",
"We also study the impact of spatial non-uniformities of the sample."
],
[
"Introduction",
"In the introduction to the article “Opacity calculations: past and future” in 1964 [1], H. Mayer writes “Initial steps for this symposium began a few billion years ago.",
"As soon as the stars were formed, opacities became one of the basic subjects determining the structure of the physical world in which we live”.",
"He is also the author of a famous report [2] written in 1947 and presenting many theoretical methods for opacity calculation.",
"Since Mayer's 1947 report, there have been few comprehensive reviews of methods for opacity calculations, although several reviews of limited scope have appeared, focusing on specific aspects [3].",
"In early papers, the authors were mainly concerned with photo-ionization [4], [5], [6], [7].",
"Many interesting summary articles about atomic opacities from the astrophysicists viewpoint were published [8], [9], [10], [11], [12], [13].",
"Penner and Olfe discussed atomic and molecular opacities as applied to atmospheric reentry phenomena [14] and the opacity of heated air was the subject of work by Armstrong et al.",
"[15], [16] and Avilova et al.",
"[17], [18].",
"Proceedings of three opacity conferences were published by Mayer [1], Huebner et al.",
"[19] and Adelman and Wiese [20].",
"Rickert [21] and Serduke et al.",
"[22] summarized workshops in which opacities were compared.",
"Efforts to calculate opacities and the underlying equations of state (EOS) have been renewed by the Los Alamos group, the Livermore group, and the opacity project at University College London and the University of Illinois.",
"While the first two groups cover the entire range of atomic opacities, the last one concentrates on detailed EOS and opacities of light elements for stellar envelopes in the high temperature 3 10$^3$ $\\le T\\le $ 10$^7$ K and low density regions of the astrophysical plasma domain [23], [24], [25], [26].",
"The low temperature limit avoids the presence of molecules and the high density limit is chosen so that the isolated atom or ion remains a reasonably good approximation.",
"Seventy-one years have passed, and despite the progress made, many researchers are still working on radiative opacity, due to its applications in the fields of inertial confinement fusion, defense and astrophysics.",
"Several experiments were performed during the past three decades, but the recent experiment on the Z machine at Sandia National Laboratory (SNL), dedicated to the measurement of the transmission of iron in conditions close to the ones of the base of the convective zone of the Sun [27], reveals that our computations may be wrong or at least incomplete.",
"The two main facilities used for opacity measurements are lasers and Z-pinches.",
"High-power lasers and Z-pinches can be used to irradiate high-$Z$ targets with intense X-ray fluxes which volumetrically heat materials in local thermodynamic equilibrium (LTE) to substantial temperatures.",
"These X-ray fluxes produce a state of high-energy density matter that can be studied by the technique of absorption spectroscopy.",
"In such experiments, also known as pump-probe experiments, the X-ray source creating the plasma is expected to be Planckian.",
"The radiative quantities which are measured are the reference $I_{0,\\nu }$ (unattenuated radiation intensity) and $I_{\\nu }$ , the radiation attenuated by the plasma.",
"$\\nu $ is the photon frequency.",
"The transmission of the sample is given by $\\mathbb {T}_{\\nu }=\\frac{I_{\\nu }}{I_{0,\\nu }},$ and the knowledge of $\\mathbb {T}_{\\nu }$ and of the areal mass of the sample enables one to deduce the opacity (see section ).",
"In laser experiments, the sample is heated by the X rays resulting from the conversion of the energy of laser beams focused inside a gold Hohlraum, and measurement of absorption coefficients in plasmas may be done for instance using the technique of point-projection spectroscopy, first introduced by Lewis and McGlinchey in 1985 [28].",
"The technique involves a small plasma produced by tightly focusing a laser on a massive or a fiber target to create a point-like X-ray source with a high continuum emission used to probe the heated sample.",
"It was first used to probe expanding plasmas [29], [30] and applied to probe a radiatively heated plasma for the first time in 1988 [31].",
"The point-projection spectroscopy technique can be used to infer the plasma conditions and/or its spectral absorption.",
"Z-pinch experiments mentioned above proceed as follows [32], [33], [34], [35], [36].",
"The process entails accelerating an annular tungsten Z-pinch plasma radially inward onto a cylindrical low density CH$_2$ foam, launching a radiating shock propagating toward the cylinder axis.",
"Radiation trapped by the tungsten plasma forms a Hohlraum and a sample attached on the top diagnostic aperture is heated during a few nanoseconds when the shock is propagating inward and the radiation temperature rises.",
"The radiation at the stagnation is used to probe the sample.",
"For a quantitative X-ray opacity experiment, great care must be taken in the preparation of the plasma, as the latter must be spatially uniform in both temperature and density.",
"Masses and dimensions of the sample must be well-known.",
"In order to compare with LTE opacity codes (see for instance [37], [38], [39], [40]), it is crucial to ensure that the plasma is actually in LTE.",
"Quantitative information on the opacity can be obtained only if the following requirements are satisfied [41], [42], [43]: (i) The instrumental spectral resolution has to be sufficiently high to resolve key line features and measured accurately prior to the experiments.",
"(ii) Backlight radiation and tamper transmission have to be free of a wavelength-dependent structures.",
"(iii) Plasma self-emission has to be minimized.",
"(iv) The tamper-transmission difference has to be minimized.",
"(v) The sample conditions must be uniform, achieving near-local thermodynamic equilibrium.",
"(vi) The sample temperature, density, and drive radiation should be independently measured.",
"(vii) Measurements should be repeated with multiple sample thicknesses.",
"(viii) Both quantities $I_{0,\\nu }$ and $I_{\\nu }$ must be measured during the same experimental “shot” together with the plasma conditions.",
"The lack of simultaneous measurement of plasma conditions and absorption coefficient is a weakness of most absorption measurements.",
"For example, some experiments rely on radiative-hydrodynamics simulations to infer the plasma temperature and density, while other provide measurements of temperature, density and absorption spectrum, but on different shots.",
"However, even if many experimental teams devote lots of efforts to perform simultaneous measurements, the inferred quantities are always known with a limited accuracy.",
"There are many sources of uncertainty: areal mass of the sample, background radiation, intensity of the backlighter, plasma temperature and density, etc.",
"Strictly speaking, some of error quantification depends on platform (such as self-emission).",
"Opacity-measurement uncertainty is challenging because it consists of three sources of errors that are complicated in different ways: (i) transmission error, (ii) areal density error, and (iii) temperature and density errors.",
"In addition, there are multiple sources for each category.",
"When there is a lateral areal-density non-uniformity, effective areal density (and its error) become transmission dependent.",
"Therefore, areal-density non-uniformity must be treated either as transmission error or as a special category.",
"In the present work, we investigate some uncertainty sources in absorption spectroscopy measurements.",
"In section , we show how the relative uncertainties on the transmission and on the areal mass $\\rho L$ are related to the uncertainty on the opacity.",
"The question we want to answer is: if we seek a particular value of the relative precision on opacity $\\Delta \\kappa _{\\nu }/\\kappa _{\\nu }$ , knowing the relative uncertainty on the areal mass $\\Delta (\\rho L)/(\\rho L)$ , which precision $\\Delta \\mathbb {T}_{\\nu }$ on the transmission do we need?",
"As an example, we impose the requirement $\\Delta \\kappa _{\\nu }/\\kappa _{\\nu }$ =10 %.",
"The uncertainties on the background emission and on the self-emission of the sample are discussed in section .",
"We chose to discuss the latter areal-mass non-uniformities in a special category: different sources of uncertainty due to defects in the areal mass are examined in section : wedge shape, bulge (concave distortion), hollow (convex distortion), holes in the sample and oscillations.",
"In section , we address the issue of the temperature and density uncertainties.",
"Section is the conclusion."
],
[
"Required precision on the measured transmission: error and uncertainty propagations",
"For a homogeneous and optically thin (non-emissive) material, the transmission is related to the opacity by the Beer-Lambert-Bouguer law [44], [45], [46]: $\\mathbb {T}_{\\nu }=e^{-\\rho L\\kappa _{\\nu }},$ where $\\rho $ is the density and $L$ the thickness of the material, $\\kappa _{\\nu }$ its spectral opacity and $\\mathbb {T}_{\\nu }$ its transmission.",
"$\\rho L$ is the areal mass.",
"Formula (REF ) is valid for $\\rho L\\kappa _{\\nu }\\lesssim 1$ , i.e.",
"$\\mathbb {T}_{\\nu }\\gtrsim 0.37$ .",
"First we have to specify what we mean by uncertainty.",
"In particular, it is important to separate error and uncertainty.",
"Error can be defined as the difference between measured value and the true value, $\\Delta \\mu =\\mu _{\\mathrm {meas}}-\\mu _{\\mathrm {true}}$ , from a single measurement, which can go either positive or negative.",
"On the contrary, uncertainty can be defined as interval (or width) of likelihood where a measured value could fall in.",
"Usually, an uncertainty $\\sigma $ given by an analysis or measurement represents a width of Gaussian probability distribution where the true value can be found.",
"For example, if a measurement found $\\mu _{\\mathrm {meas}}\\pm \\sigma _{\\mathrm {meas}}$ , the true value $\\mu _{\\mathrm {true}}$ can be any value but its likelihood follows the Gaussian probability distribution defined as $\\frac{1}{\\sqrt{2\\pi }\\sigma _{\\mathrm {meas}}}\\exp \\left[-\\frac{\\left(\\mu -\\mu _{\\mathrm {meas}}\\right)^2}{2\\sigma _{\\mathrm {meas}}^2}\\right].$ This is why it is usually considered that the true value exists within the measured $\\mu \\pm \\sigma $ for 68 % of the time.",
"It is tempting to split the error propagation $\\Delta \\mathbb {T}_{\\nu }\\rightarrow \\Delta \\kappa _{\\nu }$ in two separate steps: $\\Delta \\mathbb {T}_{\\nu }\\rightarrow \\Delta \\tau _{\\nu }$ ($\\tau _{\\nu }=\\rho L\\kappa _{\\nu }$ being the optical depth) and then $\\Delta \\tau _{\\nu }\\rightarrow \\Delta \\kappa _{\\nu }$ .",
"The conversion from $\\Delta \\mathbb {T}_{\\nu }$ to $\\Delta \\tau _{\\nu }$ is complicated by nature due to their non-linear relation and $\\mathbb {T}_{\\nu }$ dependence.",
"There are two main sources of $\\Delta \\mathbb {T}_{\\nu }$ : (i) miscalibration between unattenuated intensity to attenuated intensity and (ii) background subtraction error.",
"Plasma self-emission (i.e., sample and tamper) can be considered as a special case of background (see Sec.",
").",
"The second phase is the conversion from $\\Delta \\tau _{\\nu }$ to $\\Delta \\kappa _{\\nu }$ .",
"This conversion is mathematically much less complicated than the $\\Delta \\mathbb {T}_{\\nu }\\rightarrow \\Delta \\tau _{\\nu }$ one, and only areal-density errors $\\Delta (\\rho L)$ have to be quantified.",
"For example, if $\\rho L$ is perfectly known, the percent errors on $\\tau _{\\nu }$ and $\\kappa _{\\nu }$ are the same.",
"If $\\rho L$ is underestimated by 10 %, opacity $\\kappa _{\\nu }$ ($=\\tau _{\\nu }/(\\rho L)$ ) is (additionally) overestimated by 11 % (i.e., 1/0.9 $\\approx $ 1.11).",
"This is well separable from $\\Delta \\mathbb {T}_{\\nu }$ (or $\\Delta \\tau _{\\nu }$ ).",
"For example, if $\\tau _{\\nu }$ is overestimated by 10 % due to transmission error (whatever the source is), 10 %-underestimated $\\rho L$ ends up in giving 1.1/0.9=1.22, which ends up in 22 % overestimate in opacity.",
"So, the impact of areal-density error can be separately computed from transmission (or optical-depth) error.",
"The quantity of interest in fine is opacity $\\kappa _{\\nu }$ , which is considered to be a function of areal mass $\\rho L$ (measured by specific techniques such as Rutherford back-scattering for instance) and transmission $\\mathbb {T}_{\\nu }$ .",
"Actually, $\\mathbb {T}_{\\nu }$ is not measured directly; the quantities that are measured are transmitted intensity $I_{\\nu }$ , backlighter intensity $I_{0,\\nu }$ , electron density $n_e$ , electron temperature $T$ , etc.).",
"Therefore, the proper way to study propagation error would be to consider $\\kappa _{\\nu }=f(n_e, L, T, I_{\\nu }, I_{0,\\nu }, \\mathrm {etc.",
"})$ .",
"In order to simplify the problem, we gather all these variables into two ones: areal mass $\\rho L$ and transmission $\\mathcal {T}_{\\nu }=I_{\\nu }/I_{0,\\nu }$ .",
"Error propagation must absolutely be performed using $\\kappa _{\\nu }=f(\\rho L, T_{\\nu })$ and not $T_{\\nu }=f(\\rho L, \\kappa _{\\nu })$ (the latter procedure gives unrealistic uncertainties).",
"Carrying out the same measurement operation many times and calculating the standard deviation of the obtained values is one of the most common practices in measurement uncertainty estimation.",
"Either the full measurement or only some parts of it can be repeated.",
"In both cases useful information can be obtained.",
"The obtained standard deviation is then the standard uncertainty estimate.",
"If $q=f(x,y)$ , the propagation of uncertainty reads [47]: $\\sigma _q=\\sqrt{\\left[\\frac{\\partial f}{\\partial x}\\right]^2~\\left(\\sigma _x\\right)^2+\\left[\\frac{\\partial f}{\\partial y}\\right]^2\\left(\\sigma _y\\right)^2},$ i.e.",
"if $q=x+y$ : $\\sigma _q=\\sqrt{\\left(\\sigma _x\\right)^2+\\left(\\sigma _y\\right)^2}.$ In our case, $q=\\kappa _{\\nu }$ , $x=\\rho L$ , $y=\\mathbb {T}_{\\nu }$ , $f(x,y)=-\\ln (y)/x$ and Eq.",
"(REF ) becomes $\\frac{\\sigma _{\\kappa _{\\nu }}}{\\kappa _{\\nu }}=\\sqrt{\\left(\\frac{\\sigma _{\\rho L}}{\\rho L}\\right)^2+\\left(\\frac{1}{\\ln \\mathbb {T}_{\\nu }}\\frac{\\sigma _{\\mathbb {T}_{\\nu }}}{\\mathbb {T}_{\\nu }}\\right)^2}.$ Uncertainty estimates obtained as standard deviations of repeated measurement results are called A-type uncertainty estimates.",
"If uncertainty is estimated using some means other than statistical treatment of repeated measurement results then the obtained estimates are called B-type uncertainty estimates.",
"The latter represent upper bounds on the variable of interest and have no statistical meaning.",
"Therefore, if the quantity $z$ has the value $z_0$ with a relative uncertainty of $\\Delta z/z$ , it means that $z$ lies between $z_0-\\Delta z$ and $z_0+\\Delta z$ , where $\\Delta z$ is the absolute value of errors on any quantity $z$ and thus positive.",
"The other means can be e.g.",
"certificates of reference materials, specifications or manuals of instruments, estimates based on long-term experience, etc.",
"The propagation of error bounds of a quantity $q$ depending on two independent variables $x$ and $y$ is $\\Delta _q=\\left|\\frac{\\partial f}{\\partial x}\\right|\\Delta _x+\\left|\\frac{\\partial f}{\\partial y}\\right|\\Delta _y.$ If $q=x+y$ , then $\\Delta _q=\\Delta _x+\\Delta _y$ .",
"One has therefore $\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}=-\\frac{1}{\\ln \\mathbb {T}_{\\nu }}\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}+\\frac{\\Delta (\\rho L)}{\\rho L}.$ Since Eq.",
"(REF ) is the simple derivative, this is just a propagation of error while Eq.",
"(REF ) is the propagation of uncertainty stricto sensu.",
"To differentiate these two better, let us say we want to find $f$ , which is a known function of $x$ and $y$ , and we measure $x$ and $y$ .",
"Then, there are three cases: If $x$ and $y$ are known perfectly, $f(x,y)$ is perfectly known.",
"If errors in $x$ and $y$ are known perfectly (i.e., $\\Delta x$ and $\\Delta y$ , respectively), you can correct the errors in $f(x,y)$ perfectly using Eq.",
"(REF ).",
"If only uncertainties of $x$ and $y$ are known (i.e., $\\sigma _x$ and $\\sigma _y$ but not actual errors $\\Delta x$ and $\\Delta y$ ), one can only compute the likelihood interval (or the probability distribution) of true $f$ using $\\sigma _f$ found with Eq.",
"(REF ).",
"Sometimes, it is possible to know perfectly the uncertainty of one of the two variables $x$ or $y$ , and to have an upper bound for the other.",
"In such a case, neighther Eq.",
"(REF ) nor Eq.",
"(REF ) are relevant for the propagation of errors or uncertainties.",
"In this work, we therefore choose to always use Eq.",
"(REF ) which is more restrictive than Eq.",
"(REF ).",
"Using Eq.",
"(REF ) one gets, in terms of error bars, the error on spectral transmission $\\Delta \\mathbb {T}_{\\nu }=-\\mathbb {T}_{\\nu }\\ln (\\mathbb {T}_{\\nu })\\left(\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}-\\frac{\\Delta (\\rho L)}{\\rho L}\\right).$ In that case, errors represent upper bounds on the variables.",
"The same result can be obtained using inequalities (i.e.",
"assuming that transmission lies between $\\mathbb {T}_{\\nu }-\\Delta \\mathbb {T}_{\\nu }$ and $\\mathbb {T}_{\\nu }+\\Delta \\mathbb {T}_{\\nu }$ and that areal mass lies between $\\rho L-\\Delta (\\rho L)$ and $\\rho L+\\Delta (\\rho L)$ ).",
"The corresponding calculation is provided in Appendix A, since it might be helpful for didactic reasons (the connection between mathematical differentiation and positive variations $\\Delta $ is hidden and not easy to understand).",
"Eq.",
"(REF ) shows also that the error on the areal mass must not exceed the required precision on the opacity, which can be easily understood.",
"One can also “invert” the formula (REF ) in order to express the transmission that should be sought in order to ensure a given value of $\\Delta \\mathbb {T}_{\\nu }$ : $\\mathbb {T}_{\\nu }=-\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\left(\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}-\\frac{\\Delta (\\rho L)}{\\rho L}\\right)W\\left(-\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\left[\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}-\\frac{\\Delta (\\rho L)}{\\rho L}\\right]}\\right)},$ where $W$ is Lambert's function ($w=W(z)$ is the solution of $we^w=z$ , see Fig.",
"REF ) [48], [49], [50], [51], [52], [53], [54].",
"The Lambert function is named “ProductLog” in the Mathematica$^{ ®}$ software.",
"Table REF contains uncertainties mentioned in several publications about absorption-spectroscopy measurements over the past decades (see Refs.",
"[32], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66]).",
"For the enigmatic iron experiment on Z [27], as well as for the more recent measurements on chromium, iron and nickel [36], the relative uncertainty on the areal mass $\\Delta (\\rho L)/(\\rho L)$ was close to 4 % (estimated from Rutherford back-scattering).",
"In a recent paper devoted to the ongoing National Ignition Facility (NIF) experiment on iron [67], Heeter et al.",
"estimate the relative uncertainty on the areal mass $\\Delta (\\rho L)/(\\rho L)\\approx 7 \\%$ .",
"The definition of Lambert's function (for the principal branch $W_0$ in black in Fig.",
"REF ) implies that: $W\\left(-\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\left[\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}-\\frac{\\Delta (\\rho L)}{\\rho L}\\right]}\\right)\\ge -1$ and $\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\left[\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}-\\frac{\\Delta (\\rho L)}{\\rho L}\\right]}\\le \\frac{1}{e},$ which means that the largest acceptable value of $\\Delta \\mathbb {T}_{\\nu }$ corresponds to $\\mathbb {T}_{\\nu }=1/e\\approx 0.37$ .",
"If we want a precision $\\Delta \\kappa _{\\nu }/\\kappa _{\\nu }$ of 10 % on the opacity, assuming a relative uncertainty of 7 % on the areal mass, formula (REF ) implies that, for a transmission $\\mathbb {T}_{\\nu }=0.4$ , a precision of 0.011 is required on the transmission, which corresponds to $\\approx $ 2.75 % (see table REF and Fig.",
"REF ).",
"Therefore, the value of 0.02 (5 %) given by Heeter et al.",
"was a bit overestimated.",
"If $\\Delta \\rho L$ is perfectly known, in order to achieve $\\Delta \\kappa _{\\nu }/\\kappa _{\\nu }=10 \\%$ , Eq.",
"(REF ) becomes $\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}=-0.1\\times \\ln (\\mathbb {T}_{\\nu }).$ Assuming $\\Delta \\rho L\\ne 0$ ends up in tightening the $\\Delta \\mathbb {T}_{\\nu }/\\mathbb {T}_{\\nu }$ measurement.",
"For $\\Delta (\\rho L)/(\\rho L)=7 \\%$ , Eq.",
"(REF ) becomes $\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}=-0.03\\times \\ln (\\mathbb {T}_{\\nu }).$ Figure REF represents $\\Delta \\mathbb {T}_{\\nu }/\\mathbb {T}_{\\nu }$ as a function of $\\mathbb {T}_{\\nu }$ in both cases (REF ) and (REF ).",
"We note that if we had used Eq.",
"(REF ) for uncertainty propagation, in order to achieve $\\Delta \\kappa _{\\nu }/\\kappa _{\\nu }=10 \\%$ with $\\Delta (\\rho L)/(\\rho L)=7 \\%$ , we would have obtained, for $\\mathbb {T}_{\\nu }=0.4$ , $\\Delta \\mathbb {T}_{\\nu }=0.026$ , which is larger than the estimate made using Eq.",
"(REF ), i.e.",
"0.011."
],
[
"Comparisons with Monte Carlo simulations",
"Since Eq.",
"(REF ) reflects a simple derivative, it is interesting to check the error propagation using Monte Carlo simulations.",
"For given values of the transmission $\\mathbb {T}_{\\nu }$ , as well as of the relative uncertainties $\\Delta \\mathbb {T}_{\\nu }/\\mathbb {T}_{\\nu }$ and $\\Delta (\\rho L)/(\\rho L)$ , we have generated two sets of random numbers uniformly distributed, namely $X_i$ and $Y_i$ such as $1-\\frac{\\Delta (\\rho L)}{\\rho L}\\le X_i=\\frac{\\rho _iL_i}{\\rho L}\\le 1+\\frac{\\Delta (\\rho L)}{\\rho L}$ and $1-\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}\\le Y_i=\\frac{\\mathbb {T}_i}{\\mathbb {T}_{\\nu }}\\le 1+\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}.$ Therefore, setting $\\kappa _i=-\\ln \\mathbb {T}_i/\\left(\\rho _iL_i\\right)$ , one has $\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}=\\max _{\\mathrm {random}~i}\\left(\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}\\right)_i$ with $\\left(\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}\\right)_i=\\frac{\\left|\\kappa _i-\\kappa _{\\nu }\\right|}{\\kappa _{\\nu }}=\\left|1-\\frac{1}{X_i}\\left(\\frac{\\ln Y_i}{\\ln \\mathbb {T}_0}+1\\right)\\right|,$ independent of the choice of $\\rho L$ and $\\kappa _{\\nu }$ .",
"In the present case, we have chosen $\\mathbb {T}_{\\nu }$ =0.4.",
"Fig 2 shows the relative uncertainty on the opacity as a function of the transmission $\\mathbb {T}_{\\nu }$ for $\\Delta (\\rho L)/(\\rho L)$ =7% and various values of $\\Delta \\mathbb {T}_{\\nu }/\\mathbb {T}_{\\nu }$ .",
"As can be seen, in order to get a precision of 10 % on the opacity for a transmission $\\mathbb {T}_{\\nu }$ =0.4, a precision of 2.75 % is required on the transmission.",
"This confirms the value obtained mentioned above."
],
[
"Uncertainty on the background signal",
"Let us consider a wavelength-independent backlight signal $I_{0,\\nu }\\equiv I_0$ (the effect of wavelength-dependent structures was studied by Iglesias [68]).",
"Assuming a background radiation of intensity $b_{\\nu }$ in a restricted spectral region, we have $\\tilde{\\mathbb {T}_{\\nu }}=\\frac{I_{\\nu }-b_{\\nu }}{I_0-b_{\\nu }}.$ The quantity $\\tilde{\\mathbb {T}_{\\nu }}$ represents the true transmission and $\\mathbb {T}_{\\nu }$ is the apparent transmission, i.e.",
"the transmission defined by Eq.",
"(REF ) without substracting the background $\\delta $ .",
"Expanding the latter expression up to second order yields $\\tilde{\\mathbb {T}_{\\nu }}\\approx \\mathbb {T}_{\\nu }\\left(1-\\frac{b_{\\nu }}{\\mathbb {T}_{\\nu }I_0}\\right)\\left(1+\\frac{b_{\\nu }}{I_0}\\right)=\\mathbb {T}_{\\nu }\\left(1+\\frac{b_{\\nu }}{I_0}\\left[1-\\frac{1}{\\mathbb {T}_{\\nu }}\\right]-\\frac{b_{\\nu }^2}{\\mathbb {T}_{\\nu }I_0^2}\\right)+O(\\delta ^3).$ Therefore, at first order, we have $\\tilde{\\mathbb {T}_{\\nu }}\\approx \\mathbb {T}_{\\nu }\\left(1+\\epsilon \\right)-\\epsilon ,$ where $\\epsilon =b_{\\nu }/I_0$ .",
"As can be seen in figures REF , REF and REF , for a transmission of $\\mathbb {T}_{\\nu }\\approx $ 0.4, if background level is as high as 10 % of backlight intensity, transmission inferred without accounting for it (i.e.",
"using Eq.",
"(REF ) instead of Eq.",
"(REF )) would significantly misinfer the sample transmission.",
"This background error is bigger when expected transmission value is low (see Figs.",
"REF and REF ).",
"This is the reason why it is required to repeat many experiments with varied sample thicknesses in order to measure strong lines at sufficiently high transmission.",
"It is important to mention that Eq.",
"(REF ) can also be used to address the question of how the uncertainty $\\Delta b_{\\nu }$ on the background $b_{\\nu }$ would impact the inferred opacity.",
"Indeed, the modified transmission can be rewritten $\\tilde{\\mathbb {T}_{\\nu }}=\\frac{I_{\\nu }-\\left(b_{\\nu }+\\Delta b_{\\nu }\\right)}{I_0-\\left(b_{\\nu }+\\Delta b_{\\nu }\\right)}=\\frac{I_{\\nu }^{\\prime }-\\Delta b_{\\nu }}{I_0^{\\prime }-\\Delta b_{\\nu }}$ with $I_{\\nu }^{\\prime }=I_{\\nu }-b_{\\nu }$ and $I_0^{\\prime }=I_0-b_{\\nu }$ ."
],
[
"Effect of self-emission",
"The radiative-transfer equation for stationary, homogeneous and non-diffusive material reads $\\frac{dI_{\\nu }}{dx}=-\\rho \\kappa _{\\nu }I_{\\nu }+j_{\\nu },$ where $x$ represents the position along the line of sight of the spectrometer, $I_{\\nu }$ is the intensity of the radiation field and $j_{\\nu }$ the emissivity.",
"For a plasma in LTE, using Kirchhoff's law $j_{\\nu }=B_{\\nu }\\kappa _{\\nu }$ , where $B_{\\nu }$ is the Planckian distribution $B_{\\nu }=\\frac{2h\\nu ^3}{c^2}\\frac{1}{e^{\\frac{h\\nu }{k_BT}}-1},$ one gets the solution $\\mathbb {T}_{\\nu }=e^{-\\rho L\\kappa _{\\nu }}+\\frac{B_{\\nu }}{I_0}\\left(1-e^{-\\rho L\\kappa _{\\nu }}\\right).$ The transmission is therefore higher than the one predicted by Beer's law.",
"However, Eq.",
"(REF ) holds for a point-like source with a time-independent emission in one direction [69], which is of course not representative at all of what really happens.",
"The quantities $B_{\\nu }$ and $I_0$ are usually given in erg/s/eV/cm$^2$ /sr.",
"However, the measured self-emission and backlight signals integrate $B_{\\nu }$ and $I_0$ , respectively, over their emitting area (observable from each point on the detector) and duration.",
"Comparing these quantities without the integrations has limited applicability.",
"In reality, the measurements also integrate over small energy range and solid angle as well, but for many platforms, the integrations over these quantities have similar impacts on backlight radiation and self-emission.",
"A point-projection method [75] has a greater chance of self-emission contamination.",
"For example, if a 2 mm$^2$ sample foil is heated over 2.5 ns and backlit by a 200-$\\mu $ m backlight source over 300 ps, the ratio of sample-self-emission-to-backlight is close to 1/3 (in terms of expected photons per mm$^2$ , see Figure 5 of Ref.",
"[70]).",
"In the NIF experiment, there is no dedicated aperture [70], and thus, every point on the detector sees most of the emitting region.",
"As a result, there is a huge difference in emitting area as well as in duration, which signifies the self-emission contamination relative to the backlight radiation.",
"On the other hand, the Z-pinch experiment performed at SNL [27], [36] uses a larger backlight area (800 $\\mu $ m), and the detector's view is limited by an aperture and a slit.",
"As a result, the detector sees similar emitting surface areas for backlight radiation and sample self-emission (see the red rectangle of Fig.",
"16(a) of Ref.",
"[42]).",
"In that way, the measurements are less subject to self-emission issues than the laser experiments.",
"The durations of self-emission and backlight radiation are similar too, because the same source works as heating and backlight radiation.",
"This might be the reason why NIF experiment suffers from self-emission and background, on the contrary to the SNL experiment.",
"However, one has to be cautious; the background can also originate from other sources (such as Hohlraum itself, crystal second- (or higher-) order reflection, some sort of fluorescence or hard X rays, etc.).",
"In order to quantify the impact of self-emission on the opacity measurements, one should consider the differences in the emission areas as well as the integration over their time histories (see Sec.",
"IV-C of Ref.",
"[43]).",
"It is worth mentioning that the point-projection method evoked in the introduction is not the best method at high temperature due to this reason."
],
[
"Modeling spatial non-uniformities of the sample",
"Figure REF represents microscopy views of several copper samples (before the experiment) used during a recent (2017) experimental campaign in “Laboratoire pour l'Utilisation des Lasers Intenses” (LULI) in France [71].",
"As can be seen, the surface of the sample is far from being perfect.",
"Spatial non-uniformities of targets have been widely investigated in the past (see for instance the non-exhaustive list of references [72], [73], [74]).",
"There are various possible modulations.",
"In reality, some modulation that exists in the target fabrication is relaxed during the experiment, while some other modulation might be produced by hydrodynamics.",
"In the present work, we focus on five spatial deformations of the target in two dimensions: wedge, bulge (convex deformation), hollow (concave deformation), holes and oscillations (modulations of the surface), assuming that they are still present at the time of probe.",
"Our goal is to find a simple analytical modeling of such non-uniformities in order to get a realistic idea of their respective impact.",
"The areal mass, written $\\rho L$ , can have defects in both directions $x$ and $y$ (we do not, for simplicity, separate the variations of $\\rho $ and $L$ with respect to $x$ and $y$ but $\\rho L$ is taken as a global quantity) and its average reads $\\langle \\rho L(x,y)\\rangle =\\frac{1}{ab}\\int _0^a\\int _0^b\\rho L(x,y)dxdy,$ where $a$ and $b$ are the dimensions of the sample in directions $x$ and $y$ respectively.",
"Sometimes, the interpretation of an experimental spectrum reveals that the main structures have the right energy and relative intensities which seem consistent with the experiment, but the general level of transmission is not satisfactory.",
"Even if the areal mass of the sample is guaranteed by the manufacturer with a good accuracy, it might happen that some variations of the areal mass occur, during the experiment.",
"There might be a difference between the areal density used in the analysis and the true areal density.",
"In the following, we assume that area-averaged areal density $\\langle \\rho L(x,y)\\rangle $ during the experiment is known and defined as Eq.",
"(REF ) and try to quantify how different types of lateral variations would affect the opacity inferred with $\\langle \\rho L(x,y)\\rangle $ .",
"In the present work, $\\rho L$ is considered as a function of lateral (or transverse) position $x$ only (we still do not separate the dependence of $\\rho $ and $L$ with respect to $x$ and take $\\rho L$ as a global quantity) and in both cases, we preserve the average areal mass: $\\langle \\rho L(x)\\rangle =\\frac{1}{a}\\int _0^a\\rho L(x)dx=\\rho _0L_0,$ $a$ being the transverse dimension of the target.",
"For a perfect sample (no defects), we have $\\tilde{\\mathbb {T}_{\\nu }}=\\mathbb {T}_{\\nu }=e^{-\\kappa _{\\nu }\\rho _0L_0}$ and for a corrugated sample, we have $\\tilde{\\mathbb {T}_{\\nu }}=\\frac{1}{a}\\int _0^ae^{-\\kappa _{\\nu }\\rho L(x)}dx.$ Equation (REF ) is correct only when backlight radiation is uniformly filling the observed sample area.",
"For NIF, it should not be a problem since backlight is only bright over 100 $\\mu $ m (although it has a self-emission issue).",
"The concern is modulation over 100 $\\times $ 100 $\\mu $ m$^2$ backlit region.",
"For SNL, it is a bigger concern since backlighter is bigger (approximately 800 $\\mu $ m).",
"However, it should not be a serious problem because the measurement resolves in one direction and takes lineout only over brightest 300 $\\mu $ m. Since it does not resolve in other direction, the modulation concern for SNL experiment is over 800 (backlight width) $\\times $ 300 (lineout width) $\\mu $ m$^2$ region.",
"The convexity of the exponential function implies $\\langle e^{-\\kappa _{\\nu }\\rho L(x)}\\rangle \\ge e^{-\\kappa _{\\nu }\\langle \\rho L(x)\\rangle }$ and therefore, due to Eq.",
"(REF ), we have $\\tilde{\\mathbb {T}_{\\nu }}\\ge \\mathbb {T}_{\\nu }.$ Assuming constant areal mass, the presence of spatial non-uniformities or distortions of the sample tends to make the foil more transparent.",
"This is correct but may mislead the reader to think effective areal density is always lower.",
"This is not true if the sample is tilted somehow by an angle $\\theta $ , maybe due to misalignment or some weird hydrodynamics.",
"If tilt happens, the apparent areal density along the line of sight is elongated by $1/\\cos (\\theta )$ .",
"We have considered a few distortions: Wedge: in order to model a wedge-shape distortion of the sample (see Fig.",
"REF ), we use the linear form $\\rho L(x)=\\rho _1L_1\\left(1-\\epsilon \\frac{x}{a}\\right),$ where $\\epsilon $ controls the slope of the surface of the target.",
"Bulge: a concave distortion of the sample (see Fig.",
"REF ) is modeled here as $\\rho L(x)=\\rho _1L_1\\left[-\\frac{4\\epsilon }{a^2}x(x-a)+1\\right].$ Hollow: a convex distortion (see Fig.",
"REF ) is modeled here as $\\rho L(x)=\\rho _1L_1\\left[\\frac{4\\epsilon }{a^2}x(x-a)+1\\right].$ Impact of holes: we choose to model the presence of holes (see Fig.",
"REF ) in the sample by the replacement $\\rho _0L_0 \\rightarrow \\rho _1L_1(1-\\epsilon ),$ where $0<\\epsilon <1$ quantifies the amount of holes (the areal mass of the sample must therefore be increased from $\\rho _0L_0$ to $\\rho _1L_1$ ).",
"The different expressions of the modified transmission are summarized in table REF and the derivations are provided in Appendix B.",
"The effect of the holes is stronger than the effect of the thickness modulations (25 % of holes yield the same result as 100 % of modulations).",
"Nevertheless, to have a visible effect, one needs around 75-100 % of spatial modulations of the areal mass; it seems unrealistic to have such a perturbed hydrodynamic evolution and / or such a bad conception of the targets.",
"The impact of the different defects, as $\\epsilon $ varies, is illustrated respectively in Figs.",
"REF , REF and REF (wedge), Figs.",
"REF , REF and REF (bulge), Figs.",
"REF , REF and REF (hollow), Figs.",
"REF , REF and REF (holes) and Figs.",
"REF , REF and REF (modulations).",
"Values of $\\epsilon $ required to obtain an uncertainty of 7 % on the areal mass (as in the NIF experiment [67], [75], [76]) are displayed in table REF .",
"It is difficult for us to clarify what causes each type of non-uniformity (e.g., target fabrication, non 1-D expansion during experiment, instabilities during experiments).",
"Of course, if the temperature is sufficiently high (which is the case in the laser or Z-pinch experiments mentioned above), the sample becomes a plasma, and the inhomogeneities will not be the same as the ones before the experiment (in the solid phase).",
"For instance, one can imagine that the holes will be filled very quickly.",
"In fact, the above considerations imply that we consider the defects of the sample at the instant of the probe."
],
[
"A complex issue",
"The temperature and density errors do not contribute to the opacity measurement itself.",
"They are important only when comparing with models.",
"Even if temperature and density are off by 50 %, it would be fine if the calculations were identical at the misinferred conditions.",
"In fact, temperature and density uncertainties are very different from $\\Delta \\mathbb {T}$ and $\\Delta (\\rho L)$ ones, and cannot be discussed in a general way.",
"The criteria must depend on the level of model-data discrepancies.",
"The relevant question for this uncertainty is: can the observed discrepancy between measurement and modeling be explained by temperature and density uncertainties?"
],
[
"Upper bound on the uncertainty on temperature and density due to the cross-section calculation",
"Uncertainties in opacity calculations stem from the fundamental atomic cross-sections, plasma effects caused by perturbing ions, computational limitations, etc.",
"Measurements of fundamental cross-sections are usually carried out on neutral atoms, rather than on charged ions, due to the difficulty in preparing a sample in a specific ion stage and because of the myriad possibilities of excited levels.",
"The problem is that cross-sections of neutral atoms are more difficult to calculate accurately because of the many-body electron-electron interaction.",
"Thus, comparison of calculations with measured cross-sections for neutral species should provide an upper bound on uncertainties.",
"Huebner and Barfield [3] estimate that: (i) When scattering dominates (high temperature, low density), the uncertainty in the opacity is 5 %.",
"(ii) As the density increases, free-free processes become more important, the uncertainty is less than 10 %.",
"(iii) As the temperature decreases and bound-free processes become important, the uncertainty increases to 15-20 % and as the temperature decreases still further (photo-excitation can contribute), the uncertainty increases to 30 %.",
"The calculated opacity error purely due to electron temperature and density errors is not just cross-section error, but it involves cross-sections (oscillator strengths) combined with error on quantum-state populations as well as on line broadening (in the case of photo-excitation) and edge broadening (in the case of photo-ionization).",
"Since we are mostly dealing with LTE opacities in the present work, the population factor reduces to a Boltzmann factor, and depends on the energies of the configurations (and therefore in particular on the way electron-electron interactions are taken into account).",
"Line shapes depend on many factors such as the atomic-physics basis used in the computation, the microfield distribution, etc.",
"In the present subsection, we try to find upper bounds on temperature and density uncertainties required to ensure a given relative uncertainty on opacity; for that purpose we simplify the problem as much as we can, and restrict ourselves to one process (the simplest one to model approximately), namely inverse Bremsstrahlung.",
"Kramers' formula for inverse Bremsstrahlung reads: $\\kappa _{\\nu }\\approx \\left(\\frac{\\mathcal {N}_A}{A}\\right)^2\\frac{32\\pi ^3}{3\\sqrt{3}}\\left(\\frac{e^2}{4\\pi \\epsilon _0}\\right)^3\\frac{\\hbar ^2}{mc\\sqrt{2\\pi mk_BT}}\\frac{Z^{*3}\\rho }{\\left(h\\nu \\right)^3},$ which can be put in the form $\\kappa _{\\nu }\\approx CT^{-1/2}\\rho /\\nu ^3$ yielding $\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}\\approx \\frac{1}{2}\\frac{\\Delta T}{T}+\\frac{\\Delta \\rho }{\\rho }.$ Of course, this is not realistic; in the experiments, our main goal is to study plasmas for which the photo-excitation is important.",
"In general, it is difficult to find a simple scaling with density and temperature.",
"If we require $\\Delta \\kappa _{\\nu }/\\kappa _{\\nu }\\approx 0.1$ , relation (REF ) implies that $\\frac{\\Delta T}{T}<2\\times 0.1=20~\\%.$ and $\\frac{\\Delta \\rho }{\\rho }<10~\\%.$ Thus, if observed discrepancy is 10 %, $T$ and $n_e$ need to be known better than 20 % and 10 %.",
"This is a crude approximation, but since it is reasonable to assume that $d\\kappa _{\\nu }/dT$ and $d\\kappa _{\\nu }/dn_e$ are similar between models than $\\kappa _{\\nu }$ itself, such estimates should be relevant."
],
[
"A remark on temperature inhomogeneities",
"Recently, Busquet [77] paid attention to the fact that under some circumstances, the transmission of a L- or M-shell weakly inhomogeneous plasma is identical to the transmission of a one-temperature plasma.",
"This is clearly demonstrated in the case of an opacity varying linearly with the temperature.",
"Indeed, if $\\mathbb {T}_{\\nu }=\\exp \\left\\lbrace -\\int _{y_1}^{y_2}\\kappa _{\\nu }\\left[T(y),\\rho (y)\\right]\\rho (y)dy\\right\\rbrace ,$ where $T(y)$ and $\\rho (y)$ are temperature and density of the sample at depth $y$ , $y_1$ and $y_2$ being the limits of the sample (see Fig.",
"REF ).",
"Defining the average temperature $\\bar{T}$ as $\\bar{T}=\\int _{y_1}^{y_2}T(y)\\rho (y) dy/\\int _{y_1}^{y_2}\\rho (y) dy,$ Busquet assumed a uniform density $\\rho $ (but this is not necessary), and a linear dependence of $\\kappa _{\\nu }$ with respect to temperature: $\\kappa _{\\nu }(T,\\rho )=\\kappa _{\\nu }(\\bar{T},\\rho )+(T-\\bar{T})\\times d,$ where $d=\\left.\\frac{d\\kappa _{\\nu }}{dT}\\right|_{T=\\bar{T}}$ and therefore $\\mathbb {T}_{\\nu }=\\exp \\left[-\\int _{y_1}^{y_2}\\kappa _{\\nu }(\\bar{T})\\rho (y)dy\\right],$ which means that $\\mathbb {T}_{\\nu }$ is identical to the transmission of a sample at the average temperature.",
"This implies that, under particular circumstances, one can find a temperature for which the experiment can be interpreted, although the plasma is subject to gradients...",
"This enforces the need for independent diagnostics of the plasma conditions (K-shell spectroscopy of a light element, Thomson scattering, shadowgraphy, etc.",
")."
],
[
"Conclusion",
"It is important to study the effect of relative uncertainties in photo-absorption measurements (areal mass, backlighter, background radiation, self-emission, etc.).",
"In the present work, we discussed, assuming the knowledge of the uncertainty on the areal mass, the required uncertainty on the transmission measurement in order to infer the opacity with a given accuracy.",
"The issue of the uncertainty on the backlighter emission was also briefly investigated.",
"We quantified the impact of several spatial non-uniformities of the areal mass on the transmission, considering fives cases: wedge, bulge, hollow, holes and modulations.",
"The corresponding formulas can provide an insight on the effect of the amount of corrugations, depending in each case on a single parameter $\\epsilon $ (e.g.",
"depth of the hollow relative to the nominal thickness, density of holes, amplitude of modulations, etc.).",
"Non-uniformities of the sample can be detected by analysis techniques (Rutherford back-scattering, scanning electron microscopy, etc.).",
"Such defects always overestimate the transmission, i.e.",
"make the plasma more transparent, due to convexity of the exponential function.",
"They are more important when the spectral transmission is low.",
"Holes, modulations (oscillations) and hollows (convex distortions) are expected to have the strongest impact."
],
[
"Appendix A: Expression of opacity error in terms of areal-mass and transmission relative uncertainties",
"Let us denote $\\kappa $ the opacity, $\\mathbb {T}$ the transmission and $\\mathcal {A}$ the areal mass.",
"One has: $\\mathbb {T}_{\\nu }-\\Delta \\mathbb {T}_{\\nu } \\le \\mathbb {T} \\le \\mathbb {T}_{\\nu }+\\Delta \\mathbb {T}_{\\nu }$ $\\mathbb {\\kappa }_{\\nu }-\\Delta \\kappa _{\\nu } \\le \\kappa \\le \\kappa _{\\nu }+\\Delta \\kappa _{\\nu }$ $\\rho L-\\Delta (\\rho L) \\le \\mathcal {A}\\le \\rho L+\\Delta (\\rho L)$ One has therefore $-\\ln \\left[\\mathbb {T}_{\\nu }-\\Delta \\mathbb {T}_{\\nu }\\right] \\ge -\\ln \\mathbb {T}\\ge -\\ln \\left[\\mathbb {T}_{\\nu }+\\Delta \\mathbb {T}_{\\nu }\\right]$ and $\\frac{1}{\\rho L-\\Delta (\\rho L)} \\ge \\mathcal {A} \\ge \\frac{1}{\\rho L+\\Delta (\\rho L)},$ which implies $\\frac{-\\ln \\left[\\mathbb {T}_{\\nu }-\\Delta \\mathbb {T}_{\\nu }\\right]}{\\rho L-\\Delta (\\rho L)} \\ge \\kappa \\ge \\frac{-\\ln \\left[\\mathbb {T}_{\\nu }+\\Delta \\mathbb {T}_{\\nu }\\right]}{\\rho L+\\Delta (\\rho L)}.$ Setting $\\kappa _{\\mathrm {min}}=\\frac{-\\ln \\left[\\mathbb {T}_{\\nu }+\\Delta \\mathbb {T}_{\\nu }\\right]}{\\rho L+\\Delta (\\rho L)}=\\kappa _{\\nu }-\\Delta \\kappa _{\\nu }$ and $\\kappa _{\\mathrm {max}}=\\frac{-\\ln \\left[\\mathbb {T}_{\\nu }-\\Delta \\mathbb {T}_{\\nu }\\right]}{\\rho L-\\Delta (\\rho L)}=\\kappa _{\\nu }+\\Delta \\kappa _{\\nu }$ where $\\kappa _{\\nu }=-\\ln \\mathbb {T}_{\\nu }/(\\rho L)$ , we have $\\kappa _{\\mathrm {max}}\\approx -\\frac{\\ln \\mathbb {T}_{\\nu }}{\\rho L}\\left(1+\\frac{\\Delta (\\rho L)}{\\rho L}\\right)-\\frac{1}{\\rho L}\\ln \\left[1-\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}\\right]\\left[1+\\frac{\\Delta (\\rho L)}{\\rho L}\\right]$ and taking only the first-order terms in $\\Delta \\mathbb {T}_{\\nu }$ and $\\Delta (\\rho L)$ and neglecting the second-order terms in $\\Delta \\mathbb {T}_{\\nu }\\Delta (\\rho L)$ : $\\kappa _{\\mathrm {max}}=\\kappa _{\\nu }+\\Delta \\kappa _{\\nu }\\approx \\kappa _{\\nu }+\\kappa _{\\nu }\\frac{\\Delta (\\rho L)}{\\rho L}+\\frac{1}{\\rho L}\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }}$ or $\\frac{\\Delta \\kappa _{\\nu }}{\\kappa _{\\nu }}\\approx \\frac{\\Delta (\\rho L)}{\\rho L}-\\frac{1}{\\ln \\mathbb {T}_{\\nu }}\\frac{\\Delta \\mathbb {T}_{\\nu }}{\\mathbb {T}_{\\nu }},$ which is exactly Eq.",
"(REF ).",
"The same result can of course be obtained using $\\kappa _{\\mathrm {min}}$ (see Eq.",
"(REF ))."
],
[
"Appendix A: Impact of areal-mass distortions on the transmission - analytical formulas",
"In this appendix, we provide the main steps of the derivations of expressions givven in table REF ."
],
[
"Wedge",
"In order to model a wedge-shape distortion of the sample (see Fig.",
"REF ), we use the linear form $\\rho L(x)=\\rho _1L_1\\left(1-\\epsilon \\frac{x}{a}\\right),$ where $\\epsilon $ controls the slope of the surface of the target.",
"The preservation of the areal mass reads $\\rho _1L_1\\frac{1}{a}\\int _0^a\\left(1-\\epsilon \\frac{x}{a}\\right)dx=\\rho _0L_0,$ which yields $\\rho _1L_1=\\frac{\\rho _0L_0}{1-\\epsilon /2}$ and we get $\\tilde{\\mathbb {T}_{\\nu }}=\\frac{\\left(1-\\epsilon /2\\right)}{\\epsilon \\ln \\left(\\mathbb {T}_{\\nu }\\right)}\\left[\\mathbb {T}_{\\nu }\\right]^{\\frac{1}{1-\\epsilon /2}}\\left[1-\\mathbb {T}_{\\nu }^{-\\frac{\\epsilon }{1-\\epsilon /2}}\\right].$"
],
[
"Bulge (convex distortion)",
"A bulge-shape (concave) distortion of the sample (see Fig.",
"REF ) is modeled here as $\\rho L(x)=\\rho _1L_1\\left[-\\frac{4\\epsilon }{a^2}x(x-a)+1\\right].$ The quantity $\\rho _1L_1$ , given by the preservation of the areal mass, is $\\rho _1L_1=\\frac{\\rho _0L_0}{1+2\\epsilon /3}.$ We have therefore $\\tilde{\\mathbb {T}}_{\\nu }=\\sqrt{\\frac{1+2\\epsilon /3}{-\\epsilon \\ln \\mathbb {T}_{\\nu }}}~\\left[\\mathbb {T}_{\\nu }\\right]^{\\frac{1}{1+2\\epsilon /3}}~D\\left(\\sqrt{\\frac{-\\epsilon \\ln \\mathbb {T}_{\\nu }}{1+2\\epsilon /3}}\\right),$ where $D(x)$ represents Dawson's function $D(x)=e^{-x^2}\\int _0^x e^{t^2}dt.$"
],
[
"Hollow (convex distortion)",
"In a similar way, a hollow (convex distortion, see Fig.",
"REF ) is modeled here as $\\rho L(x)=\\rho _1L_1\\left[\\frac{4\\epsilon }{a^2}x(x-a)+1\\right].$ The thickness $L_1$ , given by the preservation of the areal mass, is $\\rho _1L_1=\\frac{\\rho _0L_0}{1-2\\epsilon /3}.$ In the case of a hollow, we have $\\tilde{\\mathbb {T}}_{\\nu }=\\frac{\\sqrt{\\pi }}{2}\\left[\\mathbb {T}_{\\nu }\\right]^{\\frac{1-\\epsilon }{1-2\\epsilon /3}}~\\sqrt{\\frac{1-2\\epsilon /3}{-\\epsilon \\ln \\mathbb {T}_{\\nu }}}~\\mathrm {Erf}\\left[\\sqrt{\\frac{-\\epsilon \\ln \\mathbb {T}_{\\nu }}{1-2\\epsilon /3}}\\right],$ where $\\mathrm {Erf}$ is the usual error function $\\mathrm {Erf}(x)=\\frac{2}{\\sqrt{\\pi }}\\int _0^xe^{-t^2}dt.$ Nevertheless, to have a visible effect, at least 25 % of holes are required in the target (this is very important, but since the considered thicknesses are of the order of a few hundreds of Angströms, it might be realistic)."
],
[
"Impact of holes",
"We choose to model the presence of holes (see Fig.",
"REF ) in the sample by the replacement $\\rho _0L_0 \\rightarrow \\rho _1L_1(1-\\epsilon ),$ where $0<\\epsilon <1$ quantifies the amount of holes (the areal mass of the sample must therefore be increased from $\\rho _0L_0$ to $\\rho _1L_1$ ).",
"The preservation of areal mass implies $\\rho _1L_1=\\frac{\\rho _0L_0}{1-\\epsilon }$ and the transmission becomes $\\tilde{\\mathbb {T}_{\\nu }}=(1-\\epsilon )\\left[\\mathbb {T}_{\\nu }\\right]^{\\frac{1}{1-\\epsilon }}+\\epsilon .$"
],
[
"Modulations",
"We consider the following modulations of the sample (see Fig.",
"REF ): $\\rho L(x)=\\rho _0L_0\\left[1+\\epsilon \\cos (x)\\right]$ with $0\\le \\epsilon \\le 1$ .",
"We have, taking $a=2\\pi N$ with $N\\in \\mathbb {N}$ : $\\langle \\rho L(x)\\rangle =\\rho _0L_0\\frac{1}{a}\\int _0^a\\left[1+\\epsilon \\cos (x)\\right]dx=\\rho _0L_0\\left(1+\\frac{\\epsilon }{a}\\sin (a)\\right)=\\rho _0L_0.$ We get $\\tilde{\\mathbb {T}_{\\nu }}=\\mathbb {T}_{\\nu }\\times \\frac{1}{2\\pi N}\\int _0^{2\\pi N}\\left(\\mathbb {T}_{\\nu }\\right)^{\\epsilon \\cos (x)}dx=\\mathbb {T}_{\\nu }\\times \\frac{1}{2\\pi }\\int _0^{2\\pi }e^{\\epsilon \\cos (x)\\ln \\left(\\mathbb {T}_{\\nu }\\right)}dx$ and finally $\\tilde{\\mathbb {T}_{\\nu }}=\\mathbb {T}_{\\nu }\\times I_0\\left(-\\epsilon \\ln \\mathbb {T}_{\\nu }\\right),$ where $I_0$ is the Bessel function of the first kind of order zero.",
"Table: Uncertainties mentioned in several publications about absorption-spectroscopy measurements (non-exhaustive list).",
"T BL T_{\\mathrm {BL}} represents the effective temperature deduced from the backlighter flux (which is proportional to the fourth power of T BL T_{\\mathrm {BL}} according to Stefan's law).Table: Modified transmission 𝕋 ˜ ν \\tilde{\\mathbb {T}}_{\\nu } as a function of the “unperturbed” transmission 𝕋 ν \\mathbb {T}_{\\nu } in the case of the different areal-mass defects considered in the present paper.",
"D(x)=e -x 2 ∫ 0 x e t 2 dtD(x)=e^{-x^2}\\int _0^x e^{t^2}dt represents Dawson's function, Erf (x)=2 π∫ 0 x e -t 2 dt\\mathrm {Erf}(x)=\\frac{2}{\\sqrt{\\pi }}\\int _0^xe^{-t^2}dt is the usual error function, and I 0 (x)I_0(x) is the Bessel function of the first kind of order zero.",
"aa is the areal size of the sample and ϵ\\epsilon quantifies the amplitude of the perturbation.Table: Values of ϵ\\epsilon required to obtain an uncertainty of 7 % on the areal mass (estimated from Rutherford back-scattering), as in the NIF experiment , , ."
]
] | 2001.03559 |
[
[
"Edge-plasmon assisted electro-optical modulator"
],
[
"Abstract An efficient electro-optical modulation has been demonstrated here by using an edge plasmon mode specific for the hybrid plasmonic waveguide.",
"Our approach addresses a major obstacle of the integrated microwave photonics caused by the polarization constraints of both active and passive components.",
"In addition to sub-wavelength confinement, typical for surface plasmon polaritons, the edge plasmon modes enable exact matching of the polarization requirements for silicon based input/output grating couplers, waveguides and electro-optical modulators.",
"A concept of the hybrid waveguide, implemented in a sandwich-like structure, implies a coupling of propagating plasmon modes with a waveguide mode.",
"The vertically arranged sandwich includes a thin layer of epsilon-near-zero material (indium tin oxide) providing an efficient modulation at small length scales.",
"Employed edge plasmons possess a mixed polarization state and can be excited with horizontally polarized waveguide modes.",
"It allows the resulting modulator to work directly with efficient grating couplers and avoid using bulky and lossy polarization converters.",
"A 3D optical model based on Maxwell equations combined with drift-diffusion semiconductor equations is developed.",
"Numerically heavy computations involving the optimization of materials and geometry have been performed.",
"Effective modes, stationary state field distribution, an extinction coefficient, optical losses and charge transport properties are computed and analyzed.",
"In addition to the polarization matching, the advantages of the proposed model include the compact planar geometry of the silicon waveguide, reduced active electric resistance R and a relatively simple design, attractive for experimental realization."
],
[
"Introduction",
"New problems of great practical significance arise as electronic devices like transistors get downscaled to atomic dimensions [1].",
"As we seek the ability to engineer materials and devices on an atomic scale, a prediction for the structural, electrical and mechanical properties of new materials, and the rates of chemical reactions become crucial for the transition from nano to molecular electronics [2], [3].",
"Attempts to use light to interconnect electronic units at small scales face the mismatch between typical sizes in electronics (tens of nanometers) and communication wavelengths (thousands of nanometers) [4].",
"A weak light-matter interaction makes the efficiency of classic opto-electronic devices poor [5].",
"Hybrid quasiparticles appear as an elegant approach, which can be used to overcome the existing issues and bring the electronics to the next level.",
"For example, exotic fundamental properties of exciton-polaritons allow to exceed limitations of classical photonics [6].",
"As it was shown, the exciton-polaritons may form condensates at room temperatures and demonstrate nonlinear coherent behavior typical for quantum fluids [7], [8], [9], [10].",
"Some applications of their properties in polariton simulators and potential exciton-polariton integrated circuits have been discussed recently [11], [12], [13].",
"The technology of subwavelength optics based on surface plasmon polaritons (SPP) is used in numerous applications [14], [15].",
"It is known that SPP modes may combine strong optical confinement and efficient light-matter interaction [4].",
"Both properties might improve the performance of active devices for integrated electro-optical systems.",
"Here we suggest a model of efficient and compact electro-optical modulator based on a hybrid plasmonic waveguide (HPWG) concept employing an epsilon-near-zero (ENZ) effect.",
"While ordinary plasmonic waveguides allow to reach a high degree of optical confinement, they also introduce unavoidable losses, caused by the presence of metallic components.",
"On the other hand, losses in ordinary dielectric waveguides can be made negligible, but the optical mode confinement is obviously much worse.",
"An idea of HPWG is to combine two types of waveguides into a single structure, where a hybrid mode can be formed to compromise losses and confinement [16], [17].",
"The strength of the coupling can be controlled by geometrical parameters, such as the distance between the metal surface and the dielectric waveguide.",
"The ENZ effect, useful for optical switching, appears at infrared frequencies in a certain classes of materials, including transparent conductive oxides (TCO) [18].",
"Applying an external electric field an increased concentration of electrons can be achieved, for instance, at a boundary with a dielectric [19].",
"When the critical concentration is reached, the real part of the dielectric permittivity tends to zero in a good correspondence with the Drude theory [20].",
"The ENZ effect was recently demonstrated experimentally in indium-tin-oxide (ITO) [21].",
"When the voltage is applied and the real part of permittivity turns into zero, field intensity inside ITO accumulation layer grows in a resonant manner.",
"The imaginary part of the dielectric constant also grows significantly in the ENZ regime, causing a strong dissipation of localized electromagnetic energy in the ITO accumulation layer.",
"The described concepts in different manifestations were used in many theoretical and experimental research works.",
"The concept of plasmostor is introduced in the experiments of [22].",
"A silicon based plasmonic waveguide with a thin dielectric layer and silver claddings is used.",
"The applied voltage influences available modes, allowing them to interact constructively or destructively at the output and providing the modulation effect.",
"Doped silicon acts as an active material.",
"Another model based on a plasmonic slot waveguide is presented in the work by [23].",
"A dielectric core with a thin layer of ITO is used.",
"The conducting oxide accumulates electrons under the influence of applied voltage, which changes its permittivity and allows to absorb a propagating plasmonic mode.",
"Charge induced changes of the refractive index are more pronounced in ITO than in Si.",
"A numerical model of ultrasmall fully plasmonic absorption modulator is proposed in [24].",
"It is suggested to use a thin plasmonic nanowire with rectangular cross-section separated from a metallic surface by a 5 nm thick sandwiched ITO/insulator spacer.",
"A compact geometry ($25{\\times }30{\\times }100$ nm$^3$ ) and the implementation of high quality dielectric HfO$_2$ allows to make the switching voltage as low as 1 V. Different gate metals are compared and a better performance of gold is reported.",
"Further development of the plasmonic slot waveguide modulator appears in experiments of [25].",
"In their model ITO fills the space between golden claddings with an additional thin layer of insulator.",
"In such a geometry ENZ regime can be reached at relatively low voltages (${\\sim }2$ V), making the model attractive for applications.",
"Having in mind applications in photonic circuits, where the optical signal is confined in a low loss waveguide, the listed models require conversion of the waveguide mode to the plasmonic slot mode and back, which is accompanied by losses.",
"The classic example of a fully waveguide based absorption modulator is presented in the work of [26].",
"Modulating sandwich, which consists of a thin layers of ITO and an insulator, covered by a golden electrode, is placed on top of a waveguide, forming HPWG.",
"More works developing this idea are available [27], [28].",
"There is a family of modulators where ENZ effect is exploited without the implementation of hybrid waveguides.",
"In calculations of [29] it is suggested to incorporate thin layers of an active material (aluminum zinc oxide) and insulator into the silicon waveguide, without metallic layers and corresponding SPP modes.",
"Similar approach is suggested in theoretical works [30] and [31].",
"ITO based capacitor either surrounds the waveguide or partially penetrates inside it.",
"High quality dielectric HfO$_2$ is used in these works, which allows to reach larger charge concentrations at lower voltages.",
"In general ENZ modulators with no plasmonic waveguides implemented are noticeably larger.",
"On the other hand, fully plasmonic solutions with many metallic elements [32] introduce significant losses.",
"HPWG with accurately chosen geometry may provide a useful compromise between two approaches.",
"Advanced ways to couple waveguide modes with plasmonic modes may also help to fight plasmonic losses and bring HPWG idea to the next level [33].",
"More details about plasmonic modulators are available in corresponding reviews [34], [4], [35] In current work we remove one of the main obstacles of integrated photonic devices based on HPWG - the polarization constraint.",
"Indeed, plasmons can be excited only by light with the polarization perpendicular to the surface of the metal.",
"Thus, a planar device can only operate with the vertically polarized modes [26].",
"It makes the design incompatible with modern grating couplers, which can provide almost 100 percent efficiency [36].",
"To match the polarization requirements for grating couplers and the modulator one should develop and use polarization converters [37], [38], [39], [40], [41].",
"Obviously, a significant part of the light intensity is lost after the double conversion, before and after the modulation.",
"Besides, it results in additional complexity at the manufacturing side.",
"In this paper we propose a model of a vertically assembled plasmonic modulator employing edge plasmons.",
"Our device works with the same polarization as the couplers and does not require the usage of converters.",
"In contrast with SPP, edge modes have a mixed polarization state, which allows them to serve as coupling providers between a horizontally polarized optical waveguide mode and a metallic electrode.",
"Fundamental properties and dispersion relations of edge plasmons at different boundaries are studied in order to deduce an efficient devise structure.",
"Our modulator has a simple geometry designed for the subsequent experimental realization.",
"The proposed planar structure possess many practical benefits discussed in the text.",
"Technically, to achieve the goal, we propose an advanced HPWG that mixes two edge plasmons with a waveguide mode.",
"The corresponding model is essentially 3D and requires numerically demanding computations.",
"The designed structure provides a comfortable waveguide-plasmon-waveguide conversion length (tunable via geometrical parameters) and allows to reach a required modulation depth.",
"The paper is organized as following.",
"In Sec.",
"we give a short review of a theoretical framework used to perform numerical computations.",
"In Sec.",
"edge plasmon properties are investigated.",
"The dispersion relation is computed numerically and compared with the SPP dispersion.",
"A polarization state of edge plasmons and their stability with respect to the edge roughness are discussed.",
"Sec.",
"presents the details of the modulator geometry.",
"3D computations of the electric field distribution inside the device are presented and discussed.",
"Optical losses are evaluated.",
"High frequency charge transport properties of the modulator are discussed in Sec. .",
"The response of the electrons distribution in the active material on the applied voltage is calculated.",
"Optimal parameters for the modulator electrical capacity and related bandwidth are discussed.",
"ENZ wavelength in ITO, as well as the detailed properties of the accumulation layer are investigated.",
"In the last section (Sec. )",
"the on- to off-state transition of the modulator is presented and the extinction coefficient is evaluated.",
"The influence of the modulating sandwich length on the performance of the device is discussed."
],
[
"Theory",
"Optical model of the modulator is based on Maxwell equations in frequency domain [42] $\\nabla \\times \\mathbf {E}(\\mathbf {r}) = +i\\omega \\mu \\mathbf {H}(\\mathbf {r}),$ $\\nabla \\times \\mathbf {H}(\\mathbf {r}) = -i\\omega \\varepsilon \\mathbf {E}(\\mathbf {r}),$ where $\\varepsilon =\\varepsilon _r\\varepsilon _0$ is the total permittivity (defined as a product of relative and vacuum permittivities) and $\\mu $ is a permeability of the medium.",
"The time dependence of electric and magnetic fields is assumed to be harmonic, i.e.",
"$\\mathbf {E}(\\mathbf {r},t) = \\mathbf {E}(\\mathbf {r}) \\exp (-i\\omega {t})$ and $\\mathbf {H}(\\mathbf {r},t) = \\mathbf {H}(\\mathbf {r}) \\exp (-i\\omega {t})$ .",
"For the convenience of the numerical treatment two equations are combined into a single second-order vector wave equation with respect to the electric field $\\nabla \\times \\nabla \\times \\mathbf {E}(\\mathbf {r}) - k_0^2 n^2 \\mathbf {E}(\\mathbf {r}) = 0.$ Here we introduce a vacuum wave vector $k_0=\\omega /c$ , a speed of light $c=1/\\sqrt{\\varepsilon _0\\mu _0}$ and a refractive index $n=\\sqrt{\\varepsilon _r}$ ($\\mu =1$ is assumed in this work).",
"Eq.",
"REF is solved numerically in 3D for a modulator geometry specified in Sec. .",
"This task is numerically expensive and requires HPC (high performance computing) cluster to be performed.",
"To build the numerical model complex refractive indices (or permittivities) should be fixed for all the materials used in the simulation for a specific wavelength ($\\lambda = 1550$ nm is assumed in the paper).",
"Optical parameters for metallic and semiconducting materials are presented in the Tab.",
"REF .",
"Table: Optical model parametersAlong with full 3D numerical computations we use the mode analysis technique.",
"Considering a wave propagating in $x$ -direction and confined in $y$ and $z$ directions, one may write $\\mathbf {E}(\\mathbf {r}) = \\mathbf {E}(y,z)e^{i\\beta {x}},$ where the propagation constant $\\beta \\equiv n_{\\text{eff}}k_0$ , expressed through effective mode indices $n_{\\text{eff}}$ , is introduced.",
"Such a substitution (Eq.",
"REF ) makes Eq.",
"REF effectively 2D and, thus, allows to perform fast calculations and obtain useful results without the necessity to use supercomputers.",
"Effective mode indices $n_{\\text{eff}}$ and corresponding field distributions $\\mathbf {E}(y,z)$ are computed numerically using ordinary PC.",
"In our model (see Sec. )",
"the cross-section of the modulator has a rather complicated structure containing many areas and the permittivity distribution $\\varepsilon =n^2(y,z)$ should be considered as a function of $y$ and $z$ .",
"Therefore, Eq.",
"REF under condition REF in a scalar form prepared for the numerical treatment reads: $\\frac{\\partial ^2E_x}{\\partial {y}^2} + \\frac{\\partial ^2E_x}{\\partial {z}^2} + k_0^2n^2E_x =i\\beta \\frac{\\partial E_y}{\\partial {y}} + i\\beta \\frac{\\partial E_z}{\\partial {z}},$ $\\frac{\\partial ^2E_y}{\\partial {z}^2} + (k_0^2n^2-\\beta ^2)E_y =i\\beta \\frac{\\partial {E_x}}{\\partial {y}} + \\frac{\\partial ^2E_z}{\\partial {y}\\partial {z}},$ $\\frac{\\partial ^2E_z}{\\partial {y}^2} + (k_0^2n^2-\\beta ^2)E_z =i\\beta \\frac{\\partial {E_x}}{\\partial {z}} + \\frac{\\partial ^2E_y}{\\partial {z}\\partial {y}} .$ The propagation constant $\\beta $ should be considered as a discrete eigenvalue here.",
"To investigate charge transport in the modulator, semiconductor drift-diffusion system of equations for the electron density $n=n(\\mathbf {r},t)$ and potential $\\varphi =\\varphi (\\mathbf {r},t)$ in the form [48], [49] $\\nabla \\cdot (\\varepsilon _0\\bar{\\varepsilon }\\nabla \\varphi ) = e(n-N_d),$ $\\frac{{\\partial }n}{{\\partial }t} = \\nabla \\cdot (D_n\\nabla {n}-n\\mu _n\\nabla (\\varphi +\\chi )),$ is solved numerically.",
"Here $\\bar{\\varepsilon }$ is a static permittivity [50], $N_d$ - doping level, $\\mu _n$ - mobility of electrons, $\\chi $ - electron affinity and $D_n$ - diffusion coefficient.",
"Since both ITO and silicon are n-doped, only donor type conductance is considered.",
"Model parameters for silicon and ITO are collected in Tab.",
"REF .",
"In certain regimes, when the Maxwell-Boltzmann statistics is applicable, diffusion coefficient can be expressed simply as $D_n=\\mu _nk_bT/e$ .",
"In our computations the complete Fermi-Dirac statistics is taken into account, and the diffusion coefficient reads [30] $D_n = \\frac{\\mu _nk_bT}{e}\\frac{F_{1/2}(\\eta )}{F_{-1/2}(\\eta )},$ with $\\eta = F_{1/2}^{-1}(n/N_c),$ where $F_{\\pm 1/2}$ and $F^{-1}_{1/2}$ are direct and inverse Fermi-Dirac integrals of the order $\\pm 1/2$ .",
"Effective density of states in the conduction band is expressed as [50] $N_c = 2 \\left( \\frac{m^{\\ast }k_bT}{2\\pi \\hbar ^2} \\right)^{3/2}.$ Reduced masses of electrons $m^{\\ast }$ for ITO and n-Si are shown in Tab.",
"REF .",
"The room temperature $T=300$ K is assumed in computations.",
"The Drude theory links the charge density evolution under the applied voltage with the optical model.",
"Drude based frequency domain permittivity [51], [52] $\\varepsilon (\\omega ) = \\varepsilon _{\\infty } - \\frac{\\omega _p^2}{(\\omega ^2 + \\gamma ^2)}+ i\\frac{\\gamma \\omega _p^2}{\\omega (\\omega ^2 + \\gamma ^2)},$ describes well optical properties of ITO and n-Si in C+L telecom range ($1530 - 1625$ nm).",
"Here $\\omega _p=\\sqrt{n_ce^2/(\\varepsilon _0m^{\\ast })}$ is the plasmonic frequency and $\\gamma = e/(\\mu _nm^{\\ast })$ is the relaxation coefficient.",
"Asymptotic material permittivity at large frequencies is denoted as $\\varepsilon _{\\infty }$ and $n_c$ stands for the charge carriers concentration.",
"Inhomogeneous concentration profile $n(\\mathbf {r})$ obtained from Eqs.",
"REF and REF can be substituted here to obtain the distribution of permittivity.",
"In particular, the structure of the accumulation layer at the boundary between ITO and dielectric can be computed with high precision and used in the optical model, as shown in Sec. .",
"Drude based permittivities and refractive indices, used in optical computations are collected in Tab.",
"REF .",
"They are computed at the frequency $\\omega _0 = 1.215\\cdot 10^{15}$ s$^{-1}$ (corresponding to the wavelength $\\lambda _0=1550$ nm).",
"Refractive indices of dielectrics, namely HfO$_2$ and SiO$_2$ , are assumed to be constant and real in C+L range with $n=2.0$ and $n=1.97$ respectively [53], [54], [55].",
"Table: Drift-diffusion equations parametersNumerical experiments are performed using the commercial software Comsol Multiphysics 5.3a with additional modules WaveOptics and Semiconductors.",
"HPC cluster of Skoltech called Pardus is used for calculations."
],
[
"Edge plasmons",
"Before the introduction of the modulator and its parameters (see the next section) we review some important fundamental properties of edge plasmons.",
"As it is mentioned in the introduction, these modes provide a useful alternative to SPP in the case of plasmonic modulators.",
"In contrast to SPP they have a symmetric mixed polarization state, which allows to interact with both polarizations of the waveguide mode.",
"A field distribution of the edge mode at the ITO/Au boundary is shown in Fig.",
"REF (c).",
"The plasmon propagates in the $x$ -direction (perpendicular to the surface of the picture).",
"Projections $E_y$ and $E_z$ of the field on the transverse directions, revealing the polarization state, are shown separately (Fig.",
"REF (d)-(e)).",
"Dispersion relations of edge plasmons are shown in Fig.",
"REF (a) and (b).",
"Instead of usual dependence of angular frequency on a wave vector $\\omega (k)$ , real and imaginary parts of $n_{\\text{eff}}(\\lambda )$ are plotted (as defined in the previous section).",
"To go back to the usual notations one may use $k=n_{\\text{eff}}\\,\\omega /c$ and $\\omega =2{\\pi }c/\\lambda $ , where $c$ is a speed of light and $\\lambda $ is a wavelength.",
"While the real part of $n_{\\text{eff}}$ is associated with the mode propagation, the imaginary part reflects its decay.",
"A family of curves describing plasmons excited at the golden edge surrounded by SiO$_2$ or ITO (see below) is presented.",
"The solid black line shows the dispersion relation of SPP given by the analytical formula [51] $k = \\frac{\\omega }{c}\\sqrt{\\frac{\\varepsilon _1(\\omega )\\varepsilon _2}{\\varepsilon _1(\\omega )+\\varepsilon _2}},$ where $\\varepsilon _2 = 3.9$ is a permittivity of SiO$_2$ and $\\varepsilon _1(\\omega )$ is associated with gold and expressed by the Drude formula (Eq.",
"REF ).",
"C+L telecom range ($1530 - 1625$ nm), crucial for applications, is shown in Fig.",
"REF (a)-(b) using vertical dashed lines.",
"In our modulator design plasmons are excited at the Au/ITO boundary.",
"The advantages of the selected sandwich structure, materials and their order in the sandwich, are discussed in detail in Sec.",
".The difference between the Au/ITO boundary and frequently used Au/SiO$_2$ boundary [26] can be understood with a help of the dispersion relations in Fig REF (a)-(b) (orange and blue lines respectively).",
"Au/ITO edge plasmons are slightly more lossy than Au/SiO$_2$ plasmons at $\\lambda =1550$ nm (Fig.",
"REF (b)).",
"That is a consequence of the fact that ITO permittivity has a nonzero imaginary part.",
"However, gently doped ITO films remain mostly transparent at near-infrared and electro-magnetic losses are low [56], [57].",
"It is evident from the picture that the edge plasmons in ITO decay fast at larger wavelengths, interacting stronger with the medium.",
"One can see very different asymptotics of the orange line and the blue line.",
"Both curves demonstrate growth at small wavelengths.",
"An interplay of two trends produces a point with minimal losses for Au/ITO plasmons which appears at 1250 nm.",
"Losses in C+L range, which is slightly shifted with respect to the minimum, are still quite low.",
"Additional tuning can be performed by changing the doping level in ITO with limitations discussed in Sec. .",
"Real parts of ITO permittivity and SiO$_2$ permittivity have the values close to each other, which results in a family of similar curves in Fig.",
"REF (a).",
"The SPP case is shown for comparison (black solid curve).",
"In the experimental implementation metallic edges cannot be perfectly sharp.",
"To investigate the stability of edge modes with respect to the edge roughness we perform the comparison between sharp edge solutions and rounded edge solutions (the radius of the rounded corner is 300 nm), which is shown in Fig.REF (a) and (b) with orange and green curves respectively.",
"For long-wavelength plasmons the difference is negligible, since the plasmon characteristic sizes are much larger than the curvature radius.",
"Solutions are also similar around $\\lambda =1550$ nm.",
"At small wavelengths the difference can be significant.",
"When the size of the plasmon becomes smaller than the curvature radius it perceives the edge rather like a surface and its dispersion curve moves closer to SPP (see how the green line approaches the black SPP line at small $\\lambda $ in Fig.",
"REF (a)).",
"It is interesting to note, that, due to this effect, rounded edge plasmon losses in ITO can be even smaller than losses of edge plasmons in SiO$_2$ (compare blue and green curves in Fig.",
"REF (b) below 1000 nm)."
],
[
"Modulator design and field distribution",
"The proposed modulator design is presented in Fig.",
"REF .",
"Single mode silicon waveguide (600 nm x 200 nm) is covered with the modulating sandwich consisting of few layers: high quality dielectric HfO$_2$ (10 nm), ITO (15 nm) and golden contact (155 nm).",
"Additional structural element, which we call the plasmonic rail (or just rail), is placed on top of the ITO layer (see Fig.",
"REF (a)).",
"The resulting geometry contains two Au edges, which support plasmons, as described in the previous section.",
"Such modes can be coupled to the $y$ -polarized waveguide mode.",
"On the contrary, a flat structure without the rail supports only $z$ -polarized SPP modes which do not interact with the waveguide mode.",
"It is relatively easy to fabricate such a geometry which makes the resulting modulator design promising for applications.",
"Certain deviations from the specified geometry of the rail are acceptable.",
"The precision of vertical lines, for example, does not have to be very strict, since edge plasmons are quite stable to such variations.",
"The edges can also be smoothed, as it was shown in the previous section.",
"Since the rail brakes the symmetry in $y$ -direction, 2D computations would not be sufficient for such a device and full 3D modeling is required.",
"The size of the plasmonic rail (80 nm $\\times $ 55 nm in our model) is chosen to maximize the modulator performance, which means at least several things.",
"HPWG should provide an effective conversion of the waveguide mode to the plasmon (which can be efficiently modulated) and back.",
"In the on-state, when the voltage is not applied, conversion losses should be minimal.",
"According to our computations, conversion between modes occurs in a manner of beats.",
"Such a behavior is typically expected when two or more interacting modes exist in the system.",
"Even in a simplest case HPWG contains at least two modes, which appear as a result of hybridization of one waveguide mode and one plasmonic mode [26].",
"The analysis for our edge modulator reveals three modes (two edge plasmons hybridized with one waveguide mode), but just two of them are excited.",
"Strong interaction between modes, caused by small distances and large confinement, results in a strong mixing.",
"In contrast with directional couplers (and similar devices), where the coupled mode theory can be used to describe weak interaction between modes, there is no simple analytical theory describing HPWG [16].",
"For this reason different geometries of the rail, along with other parameters of the modulator, are tested numerically in order to obtain the configuration with suitable insertion losses and beats period.",
"Another factor, which defines the performance, is related to the off-state.",
"When the voltage is applied the signal attenuation has to be maximal.",
"It should be also taken into account that the accumulation layer in ITO, being formed, changes the character of interaction between modes.",
"The final geometry of the modulator is defined by both on- and off-states.",
"The latter one is discussed in Sec. .",
"The absolute value of the electric field $|\\mathbf {E}(x,y,z)|$ inside the modulator is shown in Fig.",
"REF .",
"To visualize a 3D distribution, five different cross-sections are plotted.",
"The position and orientation of the selected cross-sections is denoted with thin redlines in Fig.",
"REF .",
"In Fig.",
"REF (a) 'horizontal' cross-section $|\\mathbf {E}(x,y,z=z_1)|$ is fixed at the center of the gap between the silicon waveguide and the electrode.",
"In Fig.",
"REF (b) the cross-section $|\\mathbf {E}(x,y,z=z_2)|$ shows the field in the center of the waveguide.",
"It is evident from the pictures that the waveguide mode, passing through the sandwich is smoothly transformed into the plasmon and back.",
"The 'side' view $|\\mathbf {E}(x,y=y_1,z)|$ in Fig.",
"REF (c) illustrates the same process.",
"Since the modes inside the modulator are mixed by strong interaction, we recognize a plasmon by the high value of the field intensity near the golden surface.",
"Note the scaling factor, which is different in Fig.",
"REF (a) and (b), and was introduced for the better visualization.",
"Fig.",
"REF (e) and (f) show the transverse cross-section that illustrate the distribution of the field at the entrance of the modulator, and in the center, where the plasmonic state is most populated.",
"One can see that the metal edges act as coupling providers (see Fig.",
"REF (d)) for the $y$ -polarized waveguide mode and allow to concentrate the field in the plasmonic gap (Fig.",
"REF (f)), where it can be efficiently modulated.",
"Geometrical parameters of the plasmonic rail, in particular its width $R_w$ , can be used to tune the interaction between modes, which influences the period of beats.",
"In the proposed model (Fig.",
"REF ) the period is equal to $6.8 \\,\\mu $ m and the length $L$ of the modulating sandwich is chosen accordingly.",
"If these lengths are not synchronised, losses at the output of the modulator, related to the mode conversion (namely, the conversion of the hybrid mode into the waveguide mode), are generally larger.",
"It is suggested that the sandwich length must be equal to an integer number of beats periods.",
"An example of the devise with a double length is presented in the last section.",
"However, multiple conversions of the signal to the plasmon and back are also connected with additional losses.",
"According to our computations, $6.8 \\,\\mu $ m long sandwich, containing just one period, allows to reach an acceptable modulation depth as it is discussed in Sec. .",
"For the selected geometry the transmission coefficient (the ratio of the output and input intensities $I_{\\text{out}}/I_{\\text{in}}$ ) $T=0.747$ corresponds to the optical losses coefficient $\\kappa _{\\text{ol}}=10\\log _{10}{\\frac{I_{\\text{in}}}{I_{\\text{out}}}}=1.27 \\, \\text{dB}.$ The second important geometrical parameter of the plasmonic rail is its height $R_h$ .",
"If it is too large, an additional gap mode appears which interacts with existing modes and makes the picture more complicated.",
"In our geometry (55 nm height of the rail) this mode is attenuated."
],
[
"Switching",
"To understand the state switching in the modulator and design the optimal structure of the sandwich, the detailed drift-diffusion analysis based on Eq.",
"REF and Eq.",
"REF is performed.",
"Main results are summarized in Fig.",
"REF .",
"Charge density evolution in the modulator is symmetric in the $x$ -direction.",
"Despite the absence of the symmetry in the $y$ -direction, caused by the plasmonic rail, one dimensional charge transport models provide a precise description of the accumulation layer at the ITO/HfO$_2$ boundary.",
"For drift-diffusion computations the rail has been ignored and one dimensional models used to obtain the charge concentration profiles.",
"We also perform 2D computations in some regimes for the comparison, where the rail was modeled explicitly (see the inset in Fig.",
"REF (b)).",
"In order to develop an essential understanding of the modulating sandwich one should think about it as a nano-capacitor.",
"In our model the capacitor is produced by the combination of a doped silicon, HfO$_2$ and ITO (see the inset in Fig.",
"REF (a)).",
"The Au contact is attached to ITO.",
"It is not explicitly modelled in the drift-diffusion approach, but enters the equations through the boundary conditions.",
"Silicon is assumed to be grounded.",
"The thin layer of hafnium oxide acts as an insulator.",
"Electron densities in silicon and ITO are $5{\\cdot }10^{17}$ cm$^{-3}$ and $10^{19}$ cm$^{-3}$ respectively, which allows them to conduct current.",
"Note that ITO films can be doped up to the level $10^{21}$ $cm^{-3}$ [44].",
"If the voltage is applied, the charge is accumulated at the capacitor.",
"When the critical charge concentration is reached, ENZ effect is initiated in ITO leading to the strong attenuation of the optical signal.",
"It is easy to evaluate the critical concentration from Eq.",
"REF , assuming the real part is equal to zero: $n_c^{enz} = \\frac{m^{\\ast }}{e^2}\\varepsilon _{\\infty }\\varepsilon _0 (\\omega ^2 + \\gamma ^2) = 6.5\\cdot 10 ^{20} \\text{cm}^{-3}$ ($\\lambda =1550$ nm is assumed).",
"Having in mind the classic formula for a capacitor $Q=CU$ , where $C$ is a capacitance, $U$ - voltage and $Q$ - accumulated charge, one may conclude that since $Q$ is fixed by Eq.",
"REF , and we want $U$ to be small (for practical applications), the capacitance, on the contrary, should be large.",
"To increase $C$ one may decrease the thickness of the insulating layer, which is practically difficult and potentially allows electrons to tunnel through the capacitor as, for example, in the floating gate transistor, producing undesired effects.",
"Another way to increase $C$ is to use high a quality dielectric with a large static permittivity, like HfO$_2$ with $\\bar{\\varepsilon }=25$ [55] [54].",
"In our model, 10 nm thick layer of the hafnium oxide allows to keep the voltage below 10 V. Based on this analysis one should avoid also structures with multiple layers of insulator, since the capacitance of such junction is decreased.",
"Of course, if $C$ is increased, the charging time also becomes larger and makes the modulation process slower.",
"Thereby, there is a certain trade between the modulation speed and power consumption.",
"Having in mind a sandwich with a single insulating layer, an active material (ITO) and a metallic contact, two possible configurations can be considered.",
"Instead of Si/HfO$_2$ /ITO/Au structure used in our model, alternative Si/ITO/HfO$_2$ /Au sandwich can be used.",
"In such a configuration ITO is in a direct contact with silicon, the capacitor is formed between ITO and Au and plasmons are excited at the boundary between Au and HfO$_2$ .",
"Since Si and ITO have different electron affinities, there is a contact potential difference at the Si/ITO boundary and the charge density distribution function has an extra peak there.",
"It produces undesirable refractive index gradients in the modulator.",
"It is also hard to model this peak, since its amplitude depends on the details of the fabrication.",
"On the other hand, the described effect is noticeable only at low voltages and becomes small at high voltages (like the modulator switching voltage, around 8 V).",
"From this point of view both types of sandwiches can be used.",
"In our model we prefer to use Si/HfO$_2$ /ITO/Au structure, minimizing uncertainty in parameters and possible parasitic effects.",
"The charge density distribution in the modulator (along $z$ -axis) at the voltage switched off is plotted in Fig.",
"REF (a).",
"Boundaries of materials are shown with vertical black lines.",
"The structure of the sandwich is demonstrated in the inset.",
"It is clear from the picture, that there is a small charge at the capacitor even at zero voltage.",
"It is a consequence of difference between electron affinities of ITO and Si which results in a small potential difference (around $0.75$ V).",
"Asymptotical values of the charge distribution far from the capacitor correspond to the doping level of materials.",
"To switch the state of the modulator the voltage is applied to the golden contact.",
"Electron density profiles are presented for voltages $-5$ V, $-7$ V and $-10$ V in Fig.",
"REF (b) (note the scale difference with Fig.",
"REF (a)).",
"The horizontal dashed line shows the ENZ value of the concentration (Eq.",
"REF ) that should be reached for the efficient optical modulation.",
"For a given capacitance $C_0$ , the critical value is reached starting from voltage $-5$ V, which is low enough to be practical.",
"Using the data from Fig.",
"REF (b) it is easy to evaluate $C_0$ as $C_0 = \\frac{eS_{mod}}{U} \\int \\limits _{z{\\in }ITO} (n_c^U(z)-n_c^0(z))dz = 0.1 \\,\\text{[pF]},$ where $S_{mod}=6\\cdot 10^6\\,\\text{nm}^2$ is a transverse area of the modulating sandwich and the integral is taken through the ITO lead of the capacitor, to evaluate the accumulated charge in the active material.",
"Further, $n_c^U$ is the charge density profile at the voltage $U$ and $n_c^0$ is the profile for $U=0$ (blue line in Fig.",
"REF ).",
"The classic formula for a parallel-plate capacitor applied to our nano-capacitor gives very close values $C_0 \\approx \\varepsilon _0\\bar{\\varepsilon }\\frac{S_{mod}}{d},$ where $d$ is a thickness of HfO$_2$ .",
"It is clear from this formula that by changing the insulating material from HfO$_2$ to SiO$_2$ the capacitance becomes approximately 6 times smaller and requires 6 times larger voltage to switch the modulator.",
"The lack of capacitance can be compensated by decreasing the thickness of the insulator from 10 nm to $1.7$ nm which is not practical for fabrication.",
"Thus, HfO$_2$ remains the main candidate for the insulating material in the proposed device.",
"One important characteristic number obtained from the drift-diffusion model is the thickness of the accumulation layer in ITO.",
"According to our computations (Fig.",
"REF (b)) this layer is quite thin.",
"The specific number can be defined using the equation $\\frac{1}{t} \\int \\limits _{z_0}^{z_0+t} n_c^U(z) dz = n_c^{enz},$ where $z_0$ is a coordinate of HfO$_2$ /ITO boundary and $t$ is a thickness of the active layer.",
"The numerical evaluation procedure returns the value $t\\sim 1$ nm.",
"Thus, independently on how thick the ITO layer is, the active layer is very thin.",
"The layout of the accumulation layer in 2D computations is also shown in Fig.",
"REF (b) (the inset).",
"It corresponds to the thin red line at the ITO/insulator boundary.",
"Charge density in the insulator is equal to zero (white color), and the charge density in the Au contact is not modeled explicitly, but added as a boundary condition in drift-diffusion calculations.",
"The fact that the accumulation layer is very thin is taken into account in optical calculations that require an advanced resolution in this area of the model (see the next section).",
"Equations REF and REF are solved in both time-dependent and stationary regimes (when $\\partial {n}/\\partial {t}=0$ ).",
"The formation of the accumulation layer, depicted in Fig.",
"REF , is also studied dynamically.",
"According to the computations, the formation time of the layer is $\\Delta {t} \\sim 1$ ps, which corresponds to the modulator bandwidth $\\Delta {f} \\sim 1$ THz.",
"On the other hand, the bandwidth of the modulator is defined by the charging time of the nano capacitor.",
"Classically, this time is of the order of $\\Delta {t}\\sim RC_0$ .",
"To evaluate the active resistance of the leads $R$ one can use the formula $R=l {\\sigma }^{-1} S_{mod}^{-1}$ , where $\\sigma =n_c\\mu _ne$ is the conductivity and $l$ is the conductor length (in $z$ -direction).",
"The active resistance of our modulator is defined by the resistance of the silicon waveguide, mainly because of its transverse size (the resistance of 'Si lead of the capacitor' is roughly one order of magnitude larger, than the resistance of 'ITO lead').",
"Using the value of $R$ for doped silicon to evaluate the charging time gives the value $RC_0 \\sim 1$ ps which well coincides with our numerical result.",
"Consequently, the doping level of silicon directly influences the modulator bandwidth and increasing the concentration of electrons 10 times one may obtain 10 times wider band.",
"On the other hand the larger doping leads to the increased imaginary part of permittivity in silicon, which attenuates the optical signal, propagating in such a material.",
"Thus, a compromise between the waveguide losses and the modulator bandwidth should be reached."
],
[
"Modulator off-state",
"The accumulation layer with special ENZ properties is formed at the ITO/HfO$_2$ boundary as the voltage is applied.",
"The layer does not support its own optical modes due to the small thickness.",
"Despite the fact, that ITO in ENZ regime becomes 'more metallic' in a sense that its charge density becomes larger, it cannot support subwavelength SPP modes at the boundary with a dielectric either.",
"To provide appropriate conditions for plasmon excitation the ITO layer should have a large negative real part of the permittivity, which is not the case here.",
"The interaction of ENZ layer with the existing modes is, thus, dictated by boundary conditions.",
"Since the component of the displacement field vector normal to the surface of ITO should be continuous, the condition $E_2=E_1\\varepsilon _1/(\\varepsilon _2^{\\prime }+i\\varepsilon _2^{\\prime \\prime })$ is fulfilled [31], where $E_1$ and $E_2$ are electric fields outside and inside ENZ layer.",
"If the real part of ITO permittivity $\\varepsilon _1^{\\prime }$ passes through zero, a resonance in the local field intensity takes place, with a width defined by the imaginary part of permittivity $\\varepsilon _2^{\\prime \\prime }$ .",
"At the same time $\\varepsilon _2^{\\prime \\prime }$ defines optical losses in ITO and they grow notably, approximately 65 times, inside the accumulation layer at the ENZ regime.",
"Therefore, an efficient absorption occurs, when the significant part of the field is concentrated inside ITO accumulation layer.",
"An opposite effect appears when we change the polarity of the voltage.",
"In the resulting depletion layer $\\varepsilon _2^{\\prime }$ may be larger than $\\varepsilon _1$ .",
"In this case ITO pushes the field out of the active layer.",
"The distribution of the electric field inside the modulator under the applied voltage is shown in Fig.",
"REF (a)-(c).",
"The output in this case is significantly weaker than the input.",
"The value of the transmission coefficient is $T_{\\text{off}}=0.019$ .",
"Using the previously obtained on-state transmission ($T_{\\text{on}}=0.747$ ) we evaluate the extinction coefficient as $\\kappa _{\\text{ext}}=10\\log _{10}{\\frac{I_{\\text{on}}}{I_{\\text{off}}}}=15.95 \\, \\text{dB}.$ An oscillatory mode behavior, typical for the on-state (Fig.",
"REF ), cannot be observed for the off-state in Fig.",
"REF .",
"The beats do not appear also for the elongated model as it will be shown below.",
"It suggests that the presence of the accumulation layer changes the character of interaction between modes in the modulator and may affect the period of beats.",
"Figure: Dependence of the modulator transmission coefficient on the applied voltage (a) and on the wavelength of light for both on-state and off-state (b).",
"Vertical solid black line corresponds to the wavelength 1550 nm.",
"C+L telecom band is shown using vertical dashed black lines.",
"The optical bandwidth of the modulator, defined in the text, is shown with vertical gray dash-dotted lines.Figure: Electric field distribution in the enlarged model (length LL of the modulating sandwich is 13.713.7 μ\\mu m) of the electro-optical modulator.",
"On-state (a) and off-state (b) are demonstrated.",
"Obtained values of optical losses coefficient and extinction coefficient are 2.342.34 dB and 29.3629.36 dB respectively.A small, about 1 nm, width of the accumulation layer creates a resolution challenge in numerical computations.",
"It is complicated to resolve density and, thus, permittivity variation given in Fig.",
"REF (b) in 3D computations, since the profile varies significantly at a small scale.",
"To build an efficient numerical model we used the concept of an effective accumulation layer with constant ENZ parameters placed in a contact with ordinary unperturbed ITO.",
"For a verification of the selected approach we performed 2D mode analysis, where two cases were compared.",
"First, the hybrid plasmonic mode was computed when the electrons density distribution (and corresponding permittivity distribution in ITO) given in Fig.",
"REF (b) was used explicitly.",
"Second, we computed the same mode, but using an effective layer with a fixed width and constant ENZ parameters instead of a distribution.",
"Computations were repeated with varying effective layer width until the effective index of the mode and the field distribution became very close to the original computation with continuous permittivity distribution.",
"This procedure can be considered as an alternative way to determine the width of the accumulation layer.",
"It gives 1 nm thickness, which coincides with the evaluation in Sec. .",
"The effective layer concept is used then to build a 3D numerical model.",
"The field distribution in the hybrid mode is shown in Fig.",
"REF (e) (note the intensity maximum inside the thin accumulation layer at ITO/HfO$_2$ boundary).",
"Taking into account, that only a small fraction of ITO can be switched to ENZ regime, we obtain the modulator characteristics that are slightly less impressive than those where the whole volume of ITO is considered as active.",
"Nevertheless, our evaluation is realistic and obtained numbers are sufficient for potential applications.",
"Despite the simplification provided by the effective layer model, it is still a challenge to resolve a 1 nm thick material in 3D.",
"To compute the field numerically, strongly inhomogeneous mesh is used (see Fig.",
"REF (d)).",
"First, the planar triangular mesh is built at the boundary between the insulator and ITO with a variable element size of 10 - 20 nm.",
"Then the plane is copied in the directions of HfO$_2$ and ITO ($z$ -direction) with steps 3 nm and $0.3$ nm respectively to reproduce 10 nm thick insulator and 1 nm thick active layer of ITO.",
"After that, a 3D tetrahedral mesh is generated in other areas with variable elements sizes depending on the refractive indices of materials.",
"Resulting mesh contains roughly 6 million elements with the size varied from $0.3$ nm to 155 nm.",
"One numerical experiment takes approximately 200 Gb of RAM and 24 hours of computational time at HPC cluster of Skoltech.",
"Additional characteristics of the modulator are presented in Fig.",
"REF .",
"The switching process is illustrated in details by a smooth dependence of the transmission on the applied voltage in Fig.",
"REF a. Maximal absorption regime (off-state) corresponds to the point where the accumulation layer with ENZ properties is fully formed ($-8.5$ V).",
"Further increase of the voltage does not improve the extinction.",
"Positive voltages allow to form the depletion layer at HfO$_2$ /ITO boundary instead of the accumulation layer.",
"Nevertheless, according to Drude theory (see Eq.",
"REF ), ENZ effect is not possible in this regime and corresponding variation of the refractive index is much less pronounced.",
"For this reason, strong modulation regime does not exist at positive voltages and we do not consider them.",
"Optical bandwidth of the modulator can be defined using the data in Fig.",
"REF b, where the wavelength dependence of the transmission in both on-state and off-state is shown.",
"We define the band, using Eq.",
"REF , as an interval around the central wavelength (1550 nm) where the extinction drop is less that 5 dB.",
"The obtained 421 nm wide band is shown in Fig.",
"REF b using vertical gray dash-dotted lines (at 1385 nm and 1806 nm).",
"It is obviously much larger than the C+L telecom band shown using vertical black dashed lines.",
"The performance and main characteristics of plasmonic electro-optical modulators depend on the length of the modulating sandwich in a nontrivial way.",
"As discussed in Sec.",
", the period of a population exchange between the waveguide mode and the plasmonic mode should be consistent with the length of the modulator.",
"Optical computations for the model with $13.7$ $\\mu $ m long modulating sandwich, i.e.",
"twice the oscillation period, are shown in Fig.",
"REF .",
"Both on- and off-state are demonstrated with corresponding optical losses and the extinction coefficients.",
"The transmission coefficient for the doubled model ($T=0.583$ ) is just $1.3$ times smaller than the corresponding coefficient of $6.8$ $\\mu $ m long model ($T = 0.747$ ), which, probably, means that the system loose approximately $25\\%$ of the signal per each oscillation period plus $10\\%$ for input and output.",
"Optical losses is a typical issue in plasmonic applications and there are different suggestions on how to overcome them in the future [58], [59].",
"Since the attenuation of the signal in the off-state (see Fig.",
"REF (b)) is exponential, the enlarged model leads to the significant increase in the modulation depth (the transmission is almost 30 times smaller than in the $6.8$ $\\mu $ m long model), which can be an advantage for a certain types of applications.",
"Another advantage of the long model is the decreased active resistance.",
"Since the area of the sandwich $S_{\\text{mod}}$ and, consequently, the electric contact area in the $xy$ -plane is twice larger, the resistance $R$ is twice smaller, which decreases the $RC_0$ time of the capacitor and makes the modulator bandwidth larger.",
"It is remarkable that $13.7$ $\\mu $ m long waveguide based electro-optical modulator is still much more compact than many solutions without HPWG.",
"At the same time, low optical losses and predicted THz bandwidth limit of the short model (Fig.",
"REF ) also look promising for applications.",
"If the geometry of the sandwich is modified, the character of the interaction between modes changes as well, which influences the period of beats in HPWG.",
"Therefore, the length of the modulator should be optimized for each geometry of the sandwich (taking into account both on- and off-states).",
"Since the numerical optimization is computationally expensive and time consuming, the development of a reasonable analytic theory of the modes interplay in HPWG would be a nice task for the future."
],
[
"Conclusion",
"The new approach utilizing edge plasmons in optoelectronics is developed.",
"It allows to couple a horizontally polarized waveguide mode to the plasmonic mode via appropriately designed HPWG.",
"The described idea helps to design compact and efficient electro-optical modulators.",
"The following advantages of the proposed design should be emphasized: (a) the possibility to remove the polarization constraint, thus matching the modulator with recently proposed highly efficient grating couplers [36] without the need to use lossy polarization converters; (b) steady planar geometry following from the fact that silicon waveguides for horizontally polarized modes have the width lager than height; (c) consequently, the possibility to keep the active electric resistance $R$ lower; (d) the horizontal polarization makes it possible to put an electrode at the bottom of the waveguide and avoid parasitic plasmon modes excitation.",
"The proposed design implies the usage of a plasmonic rail with two golden ribs at the Au/ITO boundary supporting edge plasmon modes.",
"The design is relatively simple, which is crucial for potential experimental realizations and future applications.",
"To optimize the geometry and structure of the device, numerically heavy 3D optical model based on Maxwell equations is developed.",
"The details of charge density behavior are obtained from the drift-diffusion system of equations.",
"The electron distribution in the accumulation layer at ITO/HfO$_2$ boundary is studied and implemented in the optical computations.",
"The most important characteristics of the device, such as optical losses, the extinction coefficient and the bandwidth are computed.",
"The obtained numbers make the device attractive for potential applications.",
"This work was financially supported by the Ministry of Science and Higher Education of the Russian Federation, project No.RFMEFI58117X0026.",
"I.A.P.",
"thanks Victor Vysotskiy for the support of Pardus cluster and fruitful discussions."
]
] | 2001.03578 |
[
[
"Compressive sensing based privacy for fall detection"
],
[
"Abstract Fall detection holds immense importance in the field of healthcare, where timely detection allows for instant medical assistance.",
"In this context, we propose a 3D ConvNet architecture which consists of 3D Inception modules for fall detection.",
"The proposed architecture is a custom version of Inflated 3D (I3D) architecture, that takes compressed measurements of video sequence as spatio-temporal input, obtained from compressive sensing framework, rather than video sequence as input, as in the case of I3D convolutional neural network.",
"This is adopted since privacy raises a huge concern for patients being monitored through these RGB cameras.",
"The proposed framework for fall detection is flexible enough with respect to a wide variety of measurement matrices.",
"Ten action classes randomly selected from Kinetics-400 with no fall examples, are employed to train our 3D ConvNet post compressive sensing with different types of sensing matrices on the original video clips.",
"Our results show that 3D ConvNet performance remains unchanged with different sensing matrices.",
"Also, the performance obtained with Kinetics pre-trained 3D ConvNet on compressively sensed fall videos from benchmark datasets is better than the state-of-the-art techniques."
],
[
"Introduction",
"As per WHO report [12], India is the second most populous country in the world with more than 75 million people lying in the age group of more than 60 years.",
"Human fall is a serious problem concerning people with this age group and is considered as one of the \"Geriatric Giants\" [12].",
"Therefore, to address this issue, the need for intelligent monitoring system of the elderly people has risen over the past years.",
"The precise objective for these systems is to automatically detect falls while minimizing false negatives and then to intimate the caregivers/family members.",
"Several deep learning based fall detection techniques [17], [8], [3], [19] have been presented and for generalization few depend on large action recognition datasets for pre-training.",
"In [17] authors proposed a scheme for fall detection through ambient camera, where they employed 3D convolutional neural network (3D CNN) to obtain coarse spatio-temporal features, This was followed by Long short-term memory (LSTM) based visual attention mechanism to extract the motion information encoded within the region of interest from coarse spatio-temporal features of the video sequence.",
"The kinetic database Sports-1M which does not have fall data was used for training the 3DCNN.",
"In [3] fall events are detected as a series of sequential change in human pose and these different poses are recognized using CNN.",
"They tried different input image combinations of RGB, Depth, background subtracted RGB to name a few as input to the CNN.",
"Their focus was on human silhouette extracts for recognizing human pose for fall detection.",
"In this paper, we propose 3D ConvNet architecture which consists of 3D Inception modules for the task of fall detection.",
"The architecture takes spatio-temporal input in compressed domain, rather than spatio-temporal input in image domain as done in Inflated 3D (I3D) architecture.",
"The compressive sensing captures the measurements which are then used for performing classification as a fall or other daily activities (labelled as non fall).",
"In visual systems, while training the fall data is usually generated by simulated falls under a variety of circumstances, that makes it difficult to obtain large quantity of training instances and thus trained classifier has high chance of overfitting the training data.",
"Also, since both the fall dataset used for experiments do not have sufficient training samples, we pre-train the architecture on action recognition datasets for learning better representation of the input videos.",
"This significantly improves the generalization of the deep neural network by giving good detection rates [26], [5].",
"The authors adopt compressive sensing step in the recognition framework which render the compressive samples visually imperceptible.",
"This is essential in circumstances where one might prefer a system which doesn't disclose their identity and capturing all personal activities/details via visual systems/cameras used for detecting falls poses a serious threat to one's privacy.",
"Compressive sensing demonstrates that a signal that is K-sparse in one basis called sparsity basis can be recovered or classified from K linear projections onto a second basis.",
"The latter is called measurement basis which is incoherent with the first.",
"While the measurement process is linear, the reconstruction or classification process has to be done through non linear transformations.",
"It is also a well known fact that the compressive samples of images/video frames containing personal information can essentially be used to achieve privacy.",
"This is because CS transformation is viewed as a symmetric cipher resulting in computational secrecy when the secret sensing matrix is unknown to the adversary [20], [21], [10].",
"Although, several privacy based intelligent systems for fall detection have been designed in the past [18].",
"These systems employ action recognition algorithms which run directly on the camera monitoring the person thus enhancing privacy.",
"Their deployment is done in such a manner that only the fall alarms are transmitted but the the video frames are not.",
"Other popular systems [19] are usually based on thermal heat- maps although capable of masking the person’s identity effectively but are an expensive option.",
"The earlier in-house implementation will be problematic to update when new instances are available [18].",
"In contrast to the aforementioned approaches, compressive sensing field suggests that a small group of linear projections of a compressible signal contains enough information for reconstruction, classification and processing [15], [14], [28], [27], [9], [6], [13]."
],
[
"Related works",
"Existing non-deep learning fall detection techniques depends on extracting the person (foreground) first, which is highly influenced by image noise (background), illumination variation and occlusion.",
"In [23] authors presented the fall detection by quantifying human shape deformation.",
"For human shape change analysis, they extract and compare two consecutive silhouettes of a person.",
"The landmarks/edge points extracted from silhouette are then matched through video sequence to quantify the silhouette deformation.",
"They compare the mean matching cost of silhouette landmarks and the full Procrustes distance [7] as body shape deformation measures.",
"Based on these shape deformation measures during the fall followed by a lack of significant movement after the fall are fed to Gaussian Mixture Model (GMM) to classify the different activities as fall or not.",
"In [18] the authors presented a fall detection system that uses silhouette area as a feature.",
"Their approach works irrespective of the direction of the movement of the person with respect to the camera.",
"They present a mathematical analysis to confirm the relation between silhouette area and a fall event.",
"The classification is done separately based on the variations of silhouette area as features for SVM classifier.",
"In [8] authors have proposed a spatial-temporal fall detection method, which can present specific spatial and temporal locations of fall events in complex scenes.",
"In their method, an object detector YOLO v3 [22] is used for person detection, later a deep learning based method for multi-object tracking is used.",
"The features from the tracker are fed to an attention guided LSTM model to detect specific fall events.",
"In [19] the authors presented the use of thermal camera for fall detection which is privacy preserving as it effectively masks the identity of those being monitored.",
"They formulated the fall detection problem as an anomaly detection problem and used Convolutional LSTM Autoencoders to identify unseen falls.",
"In compressive sensing, random Gaussian matrix or random Bernoulli matrix has been widely used to generate linear measurements of natural images, frames of video, etc.",
"[9].",
"In practice there are several problems with GRM such as GRM is non-sparse and complicated, and hence highly computational complex and highly difficult in hardware implementation.",
"The other issue is that the measurements generated by GRM are random, neither are data-driven nor adjacent measurements have enough correlation.",
"In literature other measurement matrices have been proposed to solve the above issues.",
"In [6], the authors proposed structural measurement matrix (SMM) to achieve a better Rate-Distortion performance in CS based image coding, in which the image is sampled by small blocks for better measurement coding while CS recovery can be performed in large blocks for better quality of recovered images.",
"Their method of measurement coding with SMM, helps exploit the spatial correlation in measurement domain, which is represented by directional pixel behaviour (i.e object edges), that improves measurement prediction scheme and reconstructed with large blocks spliced from small correlated blocks improves CS recovery.",
"In [9], the authors proposed a novel local structural measurement matrix (LSMM) for block-based CS coding of natural images by utilizing the local smooth property of images.",
"Their proposed LSMM is a highly sparse matrix and the adjacent measurement elements generated by LSMM have high correlation that has been shown to improve the coding efficiency of spatial information.",
"Outline of the paper is as follows: Section introduces methodology to solve the problem and the proposed architecture.",
"Section presents experimental results to show the effectiveness of the framework and Section concludes the paper."
],
[
"Methodology",
"We use 3D ConvNet which includes submodules designed from Inception-V1 network architecture for fall detection.",
"The submodules present in Inception-V1 architecture are inflated as done in I3D Convolutional neural network [5] to construct 3D ConvNet.",
"The inflated Inception-V1 modules are found to be more effective in action recognition compared to VGG-style 3D CNN [5].",
"There are four inflated Inception submodules in our 3D ConvNet architecture.",
"For fall detection, our 3D ConvNet takes compressed measurements of video sequence as spatio-temporal input, obtained from compressive sensing framework (as shown in Figure REF ), rather than video sequence as input, as in the case of I3D convolutional neural networks.",
"Here, the compressed measurements for RGB frames of given video sequence are stacked together along the color (RGB) channel dimension.",
"Figure REF shows the fall detection architecture.",
"We adopt a compressive sensing step in the recognition framework which render the compressive samples visually imperceptible, a necessity for privacy.",
"When block based compressive sensing is performed over video frame, we get compressed measurements for the corresponding block.",
"If the dimension of block is $N (=B^2)$ and when it is multiplied with a sensing matrix of size $M$ x$N$ , we get $M$ measurements and the compression ratio is defined as $r=\\frac{N}{M}$ .",
"The compressed measurement vectors obtained for corresponding blocks in a frame, are arranged across channel dimension as shown in figure before given as input to fall detection architecture.",
"Hence, when compressive sensing is applied to the frame at block level, the output compressed representation will have spatial dimension depending on the number of blocks in video frame and the channel dimension depending on the compression ratio.",
"Similar rearrangement of images or video frame is also performed in the inverse pixel shuffling operation present in sub-convolutional layer of image or video super-resolution frameworks [24].",
"The difference between their inverse pixel shuffling operation is that it does not involve dimensionality reduction.",
"Moreover, the linear transformation involved in CS of the video frame blocks into compressed measurements makes rearrangement of the measurements back to the input frame difficult compared to pixel shuffling in sub-convolutional layer.",
"Figure: Compression techniqueWe show that our CS based privacy for fall detection architecture can work with different compressive sensing matrices.",
"Random Gaussian matrix or random Bernoulli matrix has been used to generate random linear measurements of the video frame blocks.",
"We have also used structural measurement matrix and local structural measurement matrix which exploits intra-block correlation in spatial domain.",
"Figure: Fall detection architecture"
],
[
"Experimental Results",
"In this section we report performance of our framework over action recognition and fall datasets with a wide variety of sensing matrices.",
"Once our 3D ConvNet is trained on action recognition dataset, we fine-tune the network for fall detection dataset."
],
[
"Fall and Action Datasets",
"In [4], the authors collected a dataset of fall and normal activities from a calibrated Multi-camera system, of eight inexpensive IP cameras with a wide angle to cover the whole room.",
"There are 22 scenarios of fall captured by 8 cameras which include sequences of forward falls or backward falls while walking, falls when inappropriately sitting down, loss of balance etc.",
"and 2 scenarios of normal daily activities such as walking in different directions, housekeeping, activities with characteristics similar to falls (sitting down/standing up, crouching down).",
"The fall sequences in dataset are not trimmed action videos as they involve frames containing walking before fall, recovery phase and walking after fall.",
"The temporal annotations of fall is also provided in the dataset which we use to create fall and non-fall sequences.",
"The fall and non-fall video sequences from the first 17 scenarios along with 23rd scenario, are used as training set while the video sequences from 18th to 22nd along with 24th scenario, are used as test set.",
"In [16], the authors collected dataset containing 70 videos, comprising of 30 fall videos and 40 videos with activities of daily living.",
"Fall and daily activities sequences were recorded with Microsoft Kinect cameras in form of RGB and depth data.",
"Here we create the learning set containing 70 fall and 642 non-fall sequences with temporal strides.",
"Fall sequences from first 24 fall videos and non-fall sequences from first 32 non-fall videos are used as training set and the rest are used as test set.",
"For pretraining our 3D CovNet, we create a learning set by randomly selecting 10 classes$^{*}$ from Kinetics-400 dataset [5].",
"The actions involved in these 10 classes from Kinetics-400 are archery, belly dancing, cheerleading, dodgeball, high jump, playing cello, push up, swimming backstroke, tying tie and washing hair.",
"This subset is composed of around 8K clips of YouTube videos.",
"Each video includes only one actions.",
"The training set, validation set and test set is divided as given in Kinetics-400 dataset.",
"Table: Accuracy on test split of Kinetics dataset with different deep learning architecturesTable: Accuracy on test split of Kinetics-10 * ^{*} with our 3D ConvNet architectureTable REF , shows the accuracy performance on test split of Kinetics dataset with different deep learning architectures.",
"Table REF shows the accuracy results over 10 classes of Kinetics dataset with random Gaussian, random Bernoulli, structural measurement matrix, local structural measurement matrix and Convolutional CS measurement matrix at different compression ratios.",
"We train separately, from scratch, the 3D ConvNet for different compression ratios and different measurement matrices.",
"The performance of 3D ConvNet is more or less similar for the reported measurement matrices.",
"If we train I3D [5] network from scratch over the given classes from Kinetics dataset, the performance comes out to be 79.73% and the performance of our 3D ConvNet comes out to be 78.98%.",
"Since there is small difference in performance between I3D and our 3D ConvNet with compressive sensing, it is safe to say our 3D ConvNet is sufficient to learn actions for the reported action recognition dataset.",
"Table: Performance of various techniques over Multi-camera fall dataset and UR fall datasetIn Table REF , we report the performance on fall detection dataset using pre-trained 3D ConvNet (over reported action recognition dataset) with structural measurement matrix at different compression ratios.",
"Since fall detection is a binary classification problem, we report 100% accuracy with pre-trained 3D ConvNet.",
"We found that our 3D ConvNet architecture performs better than 3D CNN from [17] for fall detection."
],
[
"Implementation Details",
"All action sequences (including fall and non-fall), were resized to 224x320 before compressed using measurement matrix.",
"We train our model using ADAM optimizer with initial learning rate of $10^{-3}$ which is reduced by a factor of 10 when validation loss doesn't decrease for 10 consecutive epochs and training is terminated when validation loss doesn't decrease for 22 consecutive epochs.",
"We implemented all the models in TensorFlow [2] and trained and evaluated them on nvidia-docker [1] for Tensorflow on NVIDIA DGX-1."
],
[
"Conclusion",
"A compressive sensing based fall detection framework has been presented in the paper that also enables privacy preserving since it is a huge concern for patients being monitored through regular cameras.",
"Our deep learning architecture performs similar to I3D network [5], when trained from scratch, in accuracy for reported action recognition dataset, even with wide variety of compressive sensing measurement matrices.",
"Experimental results on Multi-camera fall dataset and UR-Fall dataset were presented to show the effectiveness of the framework at different compression ratios."
],
[
"Acknowledgment",
"The NVIDIA DGX-1 for experiments was provided by CSIR-CEERI, Pilani, India"
]
] | 2001.03463 |
[
[
"Oscillation-like diffusion of two-dimensional liquid dusty plasmas on\n one-dimensional periodic substrates with varied widths"
],
[
"Abstract The long-time diffusion of two-dimensional dusty plasmas on a one-dimensional periodic substrate with varied widths is investigated using Langevin dynamical simulations.",
"When the substrate is narrow and the dust particles form a single row, the diffusion is the smallest in both directions.",
"We find that as the substrate width gradually increases to twice its initial value, the long-time diffusion of the two-dimensional dusty plasmas first increases, then decreases, and finally increases again, giving an oscillation-like diffusion with varied substrate width.",
"When the width increases to a specific value, the dust particles within each potential well arrange themselves in a stable zigzag pattern, greatly reducing the diffusion, and leading to the observed oscillation in the diffusion with the increasing width.",
"In addition, the long-time oscillation-like diffusion is consistent with the number of dust particles that are hopping across the potential wells of the substrate."
],
[
"Introduction",
"Diffusive motion on a substrate is a fundamental transport problem with numerous applications in various fields of science and technology [1], [2], [3], [4].",
"Two-dimensional (2D) diffusion is of great interest [5], [6], [7], [8], and it has been widely investigated in many 2D systems, such as colloidal suspensions [9], strongly correlated electrons on the surface of liquid helium [10] and, in particular, strongly coupled dusty plasmas [11].",
"When a periodic substrate is applied to these systems, the arrangement of particles is distorted as the particles tends to move toward the local energy minima, producing a structure that is determined by the substrate.",
"The diffusive motion of particles on the substrate is also of interest, and this is the focus of the present work.",
"A dusty plasma [12], [13], [14], [15], [16], [17], or a complex plasma, refers to a partially ionized gas containing micron-sized particles of solid matter, called dust particles.",
"Under laboratory conditions, these dust particles are typically charged to $\\approx -10^4e$ , and their mutual repulsion can be described by a Yukawa or Debye-Hückel potential, $\\phi (r)=Q^2\\exp (-r/\\lambda _D)/4\\pi \\epsilon _0r$ , produced by the shielding effects of free electrons and ions in plasmas [18], [19], [20].",
"Here $Q$ is the particle charge and $\\lambda _D$ is the Debye screening length.",
"Due to their extremely low charge-to-mass ratio, the dust particles are in the strongly coupled limit, causing them to exhibit typical liquid- [20], [21] or solid-like [22], [23] properties.",
"In typical laboratory conditions, these charged dust particles can self-organize into a single layer, i.e., a 2D suspension [24], [25] with negligible out-of-plane motion.",
"Dusty plasmas are amenable to direct video imaging and individual particle tracking, allowing the motion of individual dust particles to be investigated at the kinetic level [14], [24], [11].",
"The dusty plasma is recognized as a promising model system in which to investigate many physical processes in solids and liquids, such as heat conduction [26], [27], [28], [29], shear viscosity [30] and diffusion [31], [32], [11], [34], [35], [36], [37], [38], [39].",
"Previous diffusion studies in 2D dusty plasmas (2DDP) [31], [32], [11], [34], [35], [36], [37], [38], [39] focused on systems without a substrate, while the diffusion of 2DDP on a substrate was briefly considered in Ref. [40].",
"To our knowledge, diffusion of 2DDP on a one-dimensional periodic substrate (1DPS) with varied width has not been studied previously.",
"Here, we report our systematic investigation of the diffusive motion of 2DDP on 1DPS with varied widths.",
"We perform Langevin dynamical simulations to mimic a 2D dusty plasma liquid on 1DPS [22], [41], [42], [43].",
"We find that, as the width of 1DPS varies, the long-time diffusion of 2DDP exhibits an oscillation-like behavior, which we attribute to the stable structure of dust particles within the potential wells of 1DPS combined with hopping [44], [45], [46] of dust particles between potential wells.",
"This paper is organized as follows.",
"In Sec.",
"II, we briefly explain the Langevin dynamical simulation method used here.",
"In Sec.",
"III, we present our finding of the oscillation-like diffusion of 2DDP on the 1DPS with varied widths, as well as an analysis of this behavior using structural and dynamical properties.",
"Finally, we provide a brief summary.",
"Figure: (Color online) Mean-squared displacement (MSD) calculated from the motion in the xx direction (XMSD) (a) and in the yy direction (YMSD) (b) for the motion of the simulated 2DDP on the 1DPS of U 0 =0.05E 0 U_0=0.05E_0 with the different widths.",
"The motion in the xx direction is clearly strongly suppressed by the 1DPS.",
"Also, changing the width of the 1DPS substantially modifies the XMSD and YMSD of 2DDP.",
"We find that as the width of the 1DPS gradually increases from 1.0021b1.0021b to 2.0041b2.0041b, the long-time diffusion (at ω pd t=1000\\omega _{pd} t =1000) in the xx direction does not vary monotonically.",
"Similarly, the long-time diffusion in the yy direction also exhibits multiple increases and decreases as the width of the 1DPS changes."
],
[
"Simulation method",
"To study the diffusion mechanism of two-dimensional dusty plasmas on one-dimensional periodic substrates, we perform Langevin dynamical simulations of 2D Yukawa systems [41].",
"We numerically integrate the equation of motion ${ m \\ddot{\\bf r}_i = -\\nabla \\Sigma \\phi _{ij} - \\nu m\\dot{\\bf r}_i + \\xi _i(t)+{\\bf F}^{S}_i,}$ for 1024 dust particles, confined in a rectangular box with dimensions $61.1 a \\times 52.9 a$ , where $a = (n\\pi )^{-1}$ is the Wigner-Seitz radius for an areal number density $n$ for 2D systems.",
"The four forces on the RHS of Eq.",
"(REF ) are fully described in [41].",
"To characterize our 2DDP, we fix the coupling parameter $\\Gamma = Q^2/(4 \\pi \\epsilon _0 a k_B T) = 200$ and the screening parameter $\\kappa = a/\\lambda _{D} = 2$ , where $T$ is the kinetic temperature of the simulated dust particles.",
"In addition to the value of $a$ , we use the lattice constant $b$ to normalize the length, with $b = 1.9046a$ for a 2D defect-free triangular lattice.",
"The force ${\\bf F}^{S}_i$ from the 1DPS is ${ {\\bf F}^{S}_i = - \\frac{\\partial U(x)}{\\partial x} = (2\\pi U_0/w) \\sin (2\\pi x/w) \\hat{\\bf x}, }$ where $U(x) = U_0 \\cos (2\\pi x/w)$ is an array of potential wells parallel to the $y$ axis.",
"Here, $U_0$ is the substrate strength in units of $E_0 = Q^2/4\\pi \\epsilon _0 a $ and $w$ is the width of the substrate in units of $b$ .",
"We specify the substrate strength $U_0 = 0.05E_0$ , and gradually change the substrate width $w$ from $1.0021b$ to $2.0041b$ .",
"Since the simulated size is $61.1a \\approx 32.07b$ in the $x$ direction, to satisfy the periodic boundary conditions, we choose $w/b = 1.0021$ , $1.1057$ , $1.1876$ , $1.2333$ , $1.2826$ , $1.3942$ , $1.5270$ , $1.6033$ , $1.6877$ , $1.7814$ , $1.8862$ , and $2.0041$ , corresponding to $32, 29, 27, 26, 25, 23, 21, 20, 19, 18, 17$ , and 16 full potential wells, respectively.",
"For each simulation run, we begin with a random configuration of dust particles and integrate for $3 \\times 10^5$ simulation time steps to achieve a steady state.",
"We then record the particle positions and velocities during the next $10^7$ simulation steps.",
"The time step is $0.0037 {\\omega }_{pd}^{-1}$ and ${\\omega }_{pd} = (Q^2/2\\pi \\epsilon _0 m a^3)^{1/2}$ is the nominal dusty plasma frequency.",
"Other simulation details are the same as those in [41], [40].",
"In addition, we also perform a few test runs with a much larger system containing 4096 dust particles, and find that there is no substantial difference in the results reported here."
],
[
"Diffusion of 2DDP on 1DPS",
"Here, we study the diffusive motion of 2DDP on 1DPS with different widths.",
"We calculate the time series of the mean-square displacement (MSD) from the motion of dust particles, defined as ${{\\rm MSD} = \\langle |{\\bf r}_{i}(t) - {\\bf r}_{i}(0)|^{2} \\rangle = 4Dt^{\\alpha },}$ where $\\langle ~ \\rangle $ denotes the ensemble average and ${\\bf r}_{i}(t)$ is the position of the $i$ th particle at time $t$ .",
"For the long-time diffusive motion, the exponent $\\alpha $ reflects the diffusion properties.",
"The exponent $\\alpha =1$ indicates normal diffusion, while $\\alpha < 1$ and $\\alpha > 1$ correspond to sub- and super-diffusion, respectively.",
"Figure 1 shows the calculated MSD of 2DDP due to the motion in two directions on 1DPS with different widths.",
"In the presence of the 1DPS, the initial ballistic and final long-time diffusive motion is accompanied by sub-diffusive motion at intermediate timescales, as clearly shown in Fig.",
"REF (b) and first observed in [40].",
"Here, we focus primarily on the long-time diffusive motion.",
"In Fig.",
"REF (a), we shown XMSD, which is the MSD calculated from the motion in only the $x$ direction, perpendicular to the potential wells.",
"Similarly, in Fig.",
"REF (b), YMSD is calculated from the motion in only the $y$ direction, parallel to the potential wells.",
"Figure: (Color online) The long-time MSD at ω pd t=1102\\omega _{pd}t=1102, determined from the motion in two directions for 2DDP on 1DPS with different widths.",
"As the substrate width increases monotonically from w=1.0021bw = 1.0021b to w=2.0041bw = 2.0041b, XMSD and YMSD both initially increase, then decrease, and finally increase again.",
"Although XMSD is much smaller than YMSD due to the suppressed motion of dust particles in the xx direction by the 1DPS, the magnitude of YMSD is exactly synchronized with that of XMSD.",
"We speculate that there is a coupling of the motion in one direction with the motion in the other direction.",
"To further analyze this oscillation-like long-time diffusion as the substrate width varies, we select six typical data points, which are marked with arrows.As the width of the 1DPS gradually increases from $w=1.0021b$ to $2.0041b$ , the long-time XMSD and YMSD (at $\\omega _{pd}t=1000$ ) do not vary monotonically, as shown in Fig.",
"REF .",
"For some specific values of the width of the 1DPS, such as $w = 1.7814b$ , we find a remarkable decrease of the diffusion in both directions.",
"Although there is no constraint from the potential wells along $y$ direction, the long-time YMSD also varies nonmonotonically in the same way as XMSD as the 1DPS width increases.",
"We speculate that this is due to the coupling of the motion of dust particles in one direction with the motion in the other direction.",
"Note that when the width of 1DPS is close to the lattice constant $b$ , i.e., $w=1.0021b$ , the diffusion at long timescales is completely suppressed, suggesting that almost no dust particles can hop [44], [45], [46] out of the potential wells in which they were originally located.",
"To study the variation of the long-time MSD with the increasing width of the 1DPS, we plot the long-time MSD at $\\omega _{pd}t=1102$ for different widths of 1DPS ranging from $w=1.0021b$ to $2.0041b$ in Fig.",
"REF .",
"Clearly, the long-time XMSD and YMSD do not vary monotonically at all.",
"We find that the long-time YMSD is always much larger than the long-time XMSD due to the suppressed motion of dust particles in the $x$ direction by the 1DPS.",
"Figure: (Color online).Snapshots of the simulated dust particle positions (dots) in a 2DDP with Γ=200\\Gamma =200 and κ=2\\kappa =2 on the 1DPS (curves) for the six different widths indicated in Fig.",
": w=w= (a) 1.0021b1.0021b, (b) 1.1876b1.1876b, (c) 1.3942b1.3942b, (d) 1.6033b1.6033b, (e) 1.7814b1.7814b, and (f) 2.0041b2.0041b.",
"The substrate strength is fixed at U 0 =0.05E 0 U_0 = 0.05E_0.",
"For each panel, the inset illustrates the corresponding 2D distribution function g(x,y)g(x,y) calculated from the simulated dust particle positions.",
"When ww is small as in (a) and (b), all of the dust particles are pinned at the bottom of each potential well, forming 1D or quasi-1D chains.",
"When the width is larger as in (c) and (d), the dust particle arrangement is much more disordered, giving a liquid-like signature in g(x,y)g(x,y).",
"When the width increases to w=1.7814bw=1.7814b as in (e), the dust particles form an ordered zigzag lattice arrangement within each potential well, as also clearly indicated by the features in g(x,y)g(x,y).",
"When the width increases further to w=2.0041bw=2.0041b as in (f), the dust particles become disordered again, giving a nearly liquid-like signature in g(x,y)g(x,y).As the major result of this paper, we find the oscillation-like diffusion of dust particles as a function of the width of the 1DPS.",
"As the width of 1DPS increases monotonically from $w=1.0021b$ to $w=2.0041b$ , the XMSD and YMSD both initially increase, then decrease, and finally increase again, as shown in Fig.",
"REF .",
"Changes in the width of the 1DPS modifies XMSD and YMSD of the 2DDP substantially.",
"For some specific values of the width of the 1DPS, such as $w=1.7814b$ , we find a remarkable decrease of the long-time diffusion.",
"Figure: (Color online).Typical trajectories of dust particles from the simulated 2DDP on the 1DPS with six different widths, for the same conditions as in Fig. .",
"When w=1.0021bw = 1.0021b in (a), all of the dust particles are pinned at the bottom of the potential wells, and almost no particles can hop across the potential wells of the 1DPS.",
"As the width gradually increases to w=1.1876bw = 1.1876b in (b) and 1.3942b1.3942b in (c), dust particles are able to move much more freely in the xx direction inside the potential wells, and at the same time, more and more dust particles are able to hop across the potential wells.",
"When w=1.6033bw=1.6033b in (d), the number of hopping dust particles decreases compared to (c).",
"When the width increases further to w=1.7814bw = 1.7814b in (e), the hopping of the dust particles greatly diminishes, and two ordered rows of dust particles can be clearly observed at the bottom of each potential well.",
"When the width further increases to w=2.0041bw = 2.0041b in (f), the number of hopping particles increase again and the system is much more disordered.",
"These panels show that as the width increases from w=1.0021bw = 1.0021b to w=2.0041bw=2.0041b, the number of hopping dust particles across the potential wells first increases, then decreases, and finally increases again.Interestingly, the long-time YMSD is exactly synchronized with the XMSD, as indicated by the fact that XMSD and YMSD have the same oscillation behavior.",
"We speculate that this synchronization is a result of the coupling of the motion of the dust particles in one direction with the motion in the other direction.",
"To further investigate the underlying physics of this oscillation-like diffusion, we select six typical data points (a,b,c,d,e,f), marked with arrows in Fig.",
"REF , at which to analyze the static arrangement and dynamics of the dust particles."
],
[
"Structure of 2DDP on 1DPS",
"To determine the underlying physics of the observed oscillation-like diffusion, we first study the structure or arrangement of the dust particles for the six different widths highlighted in Fig.",
"REF .",
"In Figure REF , we present snapshots of the simulated dust particles of the 2DDP with $\\Gamma =200$ and $\\kappa =2$ on 1DPS with different values of $w$ ranging from $w=1.0021b$ to $w=2.0041b$ .",
"The inset of each panel indicates the corresponding 2D distribution function [47] $g(x,y)$ calculated from the simulated dust particle positions.",
"The 2D distribution function $g(x,y)$ gives the probability density of finding a particle at position ${\\bf r}_2$ , given that a particle is located at ${\\bf r}_1$ .",
"Unlike the pair correlation function $g(r)$ widely used for isotropic systems, the 2D distribution function $g(x,y)$ is the static structural measure employed for anisotropic systems such as ours.",
"Using $g(x,y)$ , we can clearly distinguish whether the structure of the dust particles is ordered or disordered.",
"As the width of 1DPS gradually increases from $w=1.0021b$ to $2.0041b$ , the structure of the 2DDP within the 1DPS first changes from ordered to disordered, then orders again, and finally returns to a disordered state, as shown in Fig.",
"REF .",
"Initially when $w$ is small, as shown in Figs.",
"REF (a) and REF (b), all of the dust particles are pinned at the bottom of each potential well, forming 1D or quasi-1D chains.",
"The constraint from the substrate is larger when the width of the 1DPS is smaller, so the dust particles are in an ordered arrangement for small $w$ and their long-time diffusive motion is greatly suppressed in both the $x$ and $y$ directions.",
"When the width is larger as in Figs.",
"REF (c) and REF (d), the dust particle arrangement is much more disordered as clearly observed from the liquid-like distribution function $g(x,y)$ , and at the same time the long-time diffusive motion of the dust particles is much larger.",
"When the width of 1DPS increases to $w=1.7814b$ as shown in Fig.",
"REF (e), we find that, within each potential well, the dust particles form a stable ordered zig-zag arrangement.",
"From the calculated 2D distribution function $g(x,y)$ in Fig.",
"REF (e), this zigzag structure produces sixfold-symmetric enhanced peaks in probability around the center.",
"The long-time diffusive motion of the dust particles is greatly suppressed by this stable zigzag structure for the 1DPS of width $w=1.7814b$ .",
"When the width of the 1DPS increases further to $w=2.0041b$ as shown in Fig.",
"REF (f), the dust particles become disordered again, forming a nearly liquid-like state as indicated by the ring-like signature in $g(x,y)$ .",
"As a result, the diffusive motion of the dust particles increases again.",
"The changing trend of the structure of the 2DDP in Fig.",
"REF matches well with the oscillation-like diffusion for varied widths of the 1DPS.",
"The arrangement of the dust particle at the specific width of $w=1.7814b$ results in a dramatic decrease in the long-time diffusion due to the stable zigzag structure of the 2DDP.",
"We speculate that the dynamics of the 2DDP, such as the hopping of dust particles across the potential wells of the 1DPS, would also be substantially affected by this stable structure, as we study below."
],
[
"Dynamical behavior of 2DDP on 1DPS",
"To verify our speculation regarding the hopping of dust particles across the potential wells of the 1DPS, we measure typical trajectories over a time duration of $\\omega _{pd}t=1102$ for the dust particles from the simulated 2DDP on 1DPS with six different widths, as shown in Fig.",
"REF , with the same conditions as in Fig.",
"REF .",
"When $w=1.0021b$ in Fig.",
"REF (a), all of the particles are pinned at the bottom of the potential wells, and almost no particles can overcome the potential barrier to hop across the potential wells.",
"As the width gradually increases from $w=1.1876b$ in Fig.",
"REF (b) to $w=1.3942b$ in Fig.",
"REF (c), the dust particles are able to move much more freely in the $x$ direction inside the potential wells.",
"More and more dust particles are able to overcome the potential barrier to hop across the potential wells.",
"As a result, the long-time MSD is much larger for these widths.",
"When $w=1.6033b$ in Fig.",
"REF (d), within each potential well we find that two rows of dust particles are able to form due to the increased width of the 1DPS.",
"The number of hopping dust particles is, however, smaller than in Fig.",
"REF (c), most likely due to the increased repulsion of particles in adjacent columns that results from the smaller spacing between the particles in the buckled configuration.",
"When the width increases to $w=1.7814b$ in Fig.",
"REF (e), two rows of dust particles can be clearly observed at the bottom of each potential well, corresponding to the stable zigzag arrangement, and coinciding with a dramatic reduction of the number of hopping dust particles.",
"When the width increases further to $w=2.0041b$ in Fig.",
"REF (f), the number of hopping dust particles increases again, which is likely facilitated by the the disordered arrangement of the dust particles, as shown in Fig.",
"REF (f).",
"Figure: (Color online).",
"(a) The number NN of dust particles hopping across the potential wells and (b) the long-time MSD due to the motion of dust particles in the xx direction during a time duration of ω pd t=1102\\omega _{pd}t=1102 as functions of the width ww of the 1DPS for a constant strength of U 0 =0.05E 0 U_0=0.05E_0.",
"Clearly, the XMSD in (b) is synchronized with the number of hopping dust particles NN in (a), since the hopping dust particles make a much larger contribution to XMSD.",
"As the width of the 1DPS increases from w=1.0021bw=1.0021b to 1.3942b1.3942b, the number of hopping dust particles NN gradually increases, since more and more dust particles are able to move across the potential wells.",
"When the width increases further from w=1.3942bw=1.3942b to 1.7814b1.7814b, the number of hopping dust particles gradually diminishes.",
"We attribute this to the appearance of a stable zigzag arrangement of dust particles, as also shown in Figs.",
"and .",
"When the width increases further to w=2.0041bw=2.0041b, the number of hopping dust particles NN gradually increases again.Note that, as the width of the 1DPS increases from $1.0021b$ to $2.0041b$ , the spatial region inside each potential well over which the motion of dust particles in the $x$ direction occurs first increases, then diminishes, and finally increases again.",
"At some widths, such as $w=1.7814b$ in Fig.",
"REF (e), two rows of dust particles clearly appear at the bottom of each potential well, corresponding to the stable zigzag arrangement, and there is probably some diffusive motion between these two rows.",
"However, in the MSD measurement, and especially in XMSD, the diffusive motion between rows inside each potential well has a much smaller effect than the hopping diffusion between potential wells, since the displacement due to inter-well hopping is much larger.",
"We next quantify the hopping dust particles by counting the number $N$ of hopping dust particles from our simulation data, as shown in Fig.",
"REF (a).",
"For comparison, we also plot the corresponding XMSD in Fig.",
"REF (b).",
"Clearly, the long-time XMSD in Fig.",
"REF (b) due to the motion of dust particles in the $x$ direction is synchronized with $N$ as a function of the width of 1DPS for the time duration of $\\omega _{pd}t=1102$ .",
"As we speculated above, the long-time XMSD is determined mainly by the number $N$ of dust particles that hop across the potential wells, and $N$ is affected by the structural arrangement of the dust particles inside the potential wells.",
"As the width of the 1DPS increases from $w=1.0021b$ to $2.0041b$ , the number of hopping dust particles $N$ first increases, then decreases, and finally increases again.",
"When the width of 1DPS increases from $w=1.0021b$ to $1.3942b$ , the number of hopping dust particles $N$ gradually increases since more and more dust particles can overcome the potential energy barrier to hop across the potential wells due to the unstable disordered arrangement of dust particles inside the potential wells, as shown in Figs.",
"REF and REF .",
"When the width increases further from $w=1.3942b$ to $1.7814b$ , the number of hopping dust particles $N$ gradually decreases due to the emergence of the stable zigzag arrangement of dust particles inside the potential wells, as also shown in Figs.",
"REF and REF .",
"When the width increases further to $w=2.0041b$ , the number of hopping dust particles $N$ gradually increases again, since now the arrangement of dust particles changes from zigzag to a nearly liquid-like state.",
"Thus, as the width of the 1DPS varies, the oscillation-like diffusion is correlated with the static arrangement of the dust particles which modifies the dynamical dust particle hopping across the potential wells of the 1DPS."
],
[
"Summary",
"In summary, using Langevin dynamical simulations, we study the diffusion of 2DDP on the 1DPS.",
"We find that, as the width of the 1DPS increases, the diffusion of 2DDP exhibits an oscillation-like feature.",
"For small 1DPS widths, the dust particles arrange into a single row at the bottom of each potential well and the diffusion is greatly suppressed.",
"As the width of the 1DPS gradually increases, the diffusion of the dust particles first increases, then decrease, and finally increases again.",
"Based on the static structural measures and trajectories, we attribute the diffusion decrease to the stable zigzag arrangement of dust particles within the potential well that occurs when the width is about $w=1.7814b$ .",
"We also find that the long-time diffusion of 2DDP in one direction is synchronized with the diffusion in the other direction.",
"Future directions include studying the sliding dynamics under a drive or tilt to see whether different types of depinning correlate with the oscillations in the diffusion or whether different types of creep behavior occur [48]."
],
[
"Acknowledgments",
"Work in China was supported by the National Natural Science Foundation of China under Grants No.",
"11875199 and No.",
"11505124, the 1000 Youth Talents Plan, startup funds from Soochow University, and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.",
"Work at LANL was supported by the US Department of Energy through the Los Alamos National Laboratory.",
"Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U. S. Department of Energy (Contract No.",
"892333218NCA000001)."
]
] | 2001.03585 |
[
[
"Tidal disruptions of main sequence stars -- III. Stellar mass dependence\n of the character of partial disruptions"
],
[
"Abstract In this paper, the third in this series, we continue our study of tidal disruption events of main-sequence stars by a non-spinning $10^{6}~\\rm{M}_\\odot$ supermassive black hole.",
"Here we focus on the stellar mass dependence of the outcomes of partial disruptions.",
"As the encounter becomes weaker, the debris mass is increasingly concentrated near the outer edges of the energy distribution.",
"As a result, the mass fallback rate can deviate substantially from a $t^{-5/3}$ power-law, becoming more like a single peak with a tail declining as $t^{-p}$ with $p\\simeq2-5$.",
"Surviving remnants are spun-up in the prograde direction and are hotter than main sequence stars of the same mass.",
"Their specific orbital energy is $\\simeq10^{-3}\\times$ that of the debris, but of either sign with respect to the black hole potential, while their specific angular momentum is close to that of the original star.",
"Even for strong encounters, remnants have speeds at infinity relative to the black hole potential $\\lesssim 300$ km s$^{-1}$, so they are unable to travel far out into the galactic bulge.",
"The remnants most deeply bound to the black hole go through a second tidal disruption event upon their first return to pericenter; if they have not thermally relaxed, they will be completely disrupted."
],
[
"Introduction",
"Supermassive black holes (SMBHs) exert a significant tidal gravity on stars when their separation becomes comparable to or shorter than the “tidal radius”.",
"Only if the star passes inside the physical tidal radius $\\mathcal {R}_{\\rm t}$ is it fully disrupted; otherwise, if its pericenter $r_{\\rm p}\\gtrsim \\mathcal {R}_{\\rm t}$ , it is partially disrupted and loses only a fraction of its mass.",
"In both cases, roughly half of the mass removed from the star is bound to the black hole.",
"When the bound debris returns to the vicinity of the BH, it may produce a luminous flare.",
"This is the third paper in a series of four whose aim is to study quantitatively the key properties of tidal disruption events (TDEs) as a function of stellar mass $M_{\\star }$ and black hole mass $M_{\\rm BH}$ .",
"To do so, we have performed a suite of hydrodynamic simulations employing the intrinsically-conservative grid-based general relativistic hydrodynamics code Harm3d[17].",
"With initial data for the stars created using main-sequence models generated by MESA, we compute the time-dependent stellar self-gravity in relativistically consistent fashion (further methodological details can be found in [22]).",
"This apparatus is then applied to events involving stars of eight different masses, ranging from $0.15\\;\\mathrm {M}_{\\odot }$ to $10\\;\\mathrm {M}_{\\odot }$ , and with multiple pericenter distances $r_{\\rm p}$ for each stellar mass.",
"In this paper, we focus on how the outcomes of partial disruptions (surviving remnants and stellar debris) depend on stellar mass $M_{\\star }$ and orbital pericenter $r_{\\rm p}$ when the black hole has no spin and mass $M_{\\rm BH} = 10^6$ (from this point on, all masses will be given in solar mass units).",
"We provide a short overview of our simulation setup in Section .",
"In Section , we present the distribution of energy and the fallback rate of stellar debris (Section REF ).",
"Then we analyze the properties of the surviving remnants (Section REF ): the mass of surviving remnants for different degrees of partial disruption (Section REF ); the specific orbital energy of the remnants (Section REF ); remnant spin (Section REF ) and remnant internal structure (Section REF ).",
"We discuss the future fate of partially disrupted stars in Section .",
"Finally, we conclude with a summary of our findings in Section .",
"Throughout this paper, symbols with the subscript $\\star $ , such as $R_{\\star }$ (stellar radius) and $M_{\\star }$ (stellar mass), always refer to the properties of the star at the beginning of the tidal encounter.",
"All masses are measured in units of ${\\rm M}_\\odot $ and stellar radii in units of ${\\rm R}_\\odot $ .",
"Figure: dM/dEdM/dE for the stellar debris produced in partial TDEs with M rem /M ☆ ≃40-60%M_{\\rm rem}/M_{\\star }\\simeq 40-60\\% (left panel) and ≳90%\\gtrsim 90\\% (right panel).",
"We normalize the distribution with M ☆ /ΔϵM_{\\star }/\\Delta \\epsilon , where Δϵ=GM BH R ☆ /r t 2 \\Delta \\epsilon =G M_{\\rm BH}R_{\\star }/r_{\\rm t}^{2}.",
"The integrated area under each curve is therefore the fractional mass of the stellar debris (1.0-M rem /M ☆ )(1.0-M_{\\rm rem}/M_{\\star }).",
"M rem /M ☆ M_{\\rm rem}/M_{\\star } is given in Table .",
"The diagonal dotted line in each panel represents dM/dE∝e -k|E|/Δϵ dM/dE\\propto e^{-k |E|/\\Delta \\epsilon } with k=4.0k=4.0 (left panel) and 7.57.5 (right panel)."
],
[
"Simulations",
"We treated stellar masses $M_{\\star } = 0.15$ , 0.3, 0.4, 0.5, 0.7, 1.0, 3.0, 10.0.",
"For each, we ran a set of simulations with pericenters $r_{\\rm p}$ chosen so as to span the range from total disruptions to weakly partial disruptions.",
"These pericenters may be described in terms of the order-of-magnitude estimate for the tidal radius $\\;r_{\\rm t}$ by writing $r_{\\rm p} = \\;r_{\\rm t}/\\beta $ , where $\\beta $ is the so-called “penetration factor”.",
"The largest pericenter studied was chosen so that mass lost from the star was several percent of the star's initial mass.",
"We distinguish full from partial disruptions by three conditions: Lack of any approximately-spherical bound structure.",
"Monotonic (as a function of time) decrease in the maximum pressure of the stellar debris.",
"Monotonic decrease in the mass within the computational box.",
"The mass remaining in the box for complete disruption falls with increasing distance from the BH $\\propto r^{-\\alpha }$ with $\\alpha \\simeq 1.5-2.0$ , whereas for partial disruptions the remaining mass eventually becomes constant, which signifies a persistent self-gravitating object.",
"Events violating any one of these conditions we deem “partial\"; in all cases, if one is violated, all are.",
"We estimate the physical tidal radius $\\mathcal {R}_{\\rm t}$ as the mean of the largest $r_{\\rm p}$ yielding a full disruption and the smallest $r_{\\rm p}$ producing a partial disruption.",
"As shown in Ryu1+2019, for $M_{\\rm BH}=10^{6}$ , $\\mathcal {R}_{\\rm t}/\\;r_{\\rm t}\\simeq 1$ –1.4 for low-mass stars ($0.15\\le M_{\\star }\\le 0.5$ ); falls rapidly between $M_{\\star }\\simeq 0.5$ and $1.0$ ; and is roughly constant at $\\simeq 0.45$ for high-mass stars ($M_{\\star }\\ge 1$ ).",
"As a result, for stars with $0.15 \\le M_{\\star } \\le 3$ , all orbits with $r_{\\rm p}\\gtrsim 27~r_{\\rm g}$ lead to at most partial disruption.",
"Here, $r_{\\rm g}=G M_{\\rm BH}/c^{2}$ refers to the gravitational radius of the BH.",
"Figure REF shows the evolution of the density distribution of a $1\\;\\mathrm {M}_{\\odot }$ star when it is partially disrupted as it traverses an orbit with $r_{\\rm p} = 0.55\\;r_{\\rm t}= 1.16~\\mathcal {R}_{\\rm t}$ .",
"Note how it begins to stretch shortly before reaching pericenter, but continues to lose mass until it swings out to $\\gtrsim 10\\;r_{\\rm t}$ .",
"For the partial disruptions discussed in this paper, we followed the progress of the event until the remnant reached distances from the black hole $\\gtrsim 20\\;r_{\\rm t}$ , equivalent to a time past pericenter $\\gtrsim 30\\times $ the initial star's vibrational time.",
"The precise distance at which we stopped the simulation was determined by the point at which the remnant mass ceased changing."
],
[
"Results",
"Partial tidal disruptions produce two distinct products: a remnant and gaseous debris.",
"The debris resembles that of full disruptions in the sense that roughly half is unbound and half is bound to the black hole.",
"The bound debris can return to the black hole, generating a bright flare.",
"On the other hand, there is a remnant, of course, only in a partial disruption."
],
[
"Stellar debris - Distribution of specific energy and fallback rate",
"The most observationally-significant property of the debris is its energy distribution $dM/dE$ .",
"This quantity determines the fallback rate of bound debris and the ejection speeds of unbound debris.",
"[12] pointed out that there is a characteristic scale for the energy of tidal disruption debris, $\\Delta \\epsilon \\sim \\frac{GM_{\\rm BH} R_{\\star }}{\\;r_{\\rm t}^2},$ and the distribution $dM/dE$ should be roughly symmetric around $E=0$ .",
"We measure $dM/dE$ by continuously adding up the mass and energy of each fluid element leaving the simulation box.",
"For this purpose, we define $E$ as the relativistic specific orbital energy evaluated in the BH frame, minus rest mass energy.",
"It is a well-defined quantity because, for all but the final $\\lesssim 0.5\\%$ of mass-loss, very nearly all the gas leaves the simulation box unbound to the remnant (the bound fraction is $\\lesssim 10^{-4}$ ).",
"Because we employ a simulation box elongated in the direction of debris flow and most of the work done on the gas by the remnant's gravity happens when the gas is relatively close, we capture most of the change in energy due to this effect (see, e.g., [8]).",
"Put another way, the box is long enough that it contains the remnant's Hills radius until roughly the end of the simulation, and by this point the overwhelming majority of mass lost has traveled far outside the Hills radius.",
"The finite size of the box may, however, lead to a small overestimate of the orbital energy of unbound gas and a similarly small underestimate of the energy of bound gas.",
"The fractional error is $\\sim \\langle \\cos \\theta \\rangle (\\Delta \\epsilon /\\Delta E) (R_{\\star }/L_x)^{1/2}(r_{\\rm t}/\\langle r\\rangle )^{1/2}$ , which is $\\simeq 0.05$ for typical parameter values.",
"Here $\\theta $ is the angle between the line connecting a debris fluid element to the remnant and the velocity of the fluid element, $\\Delta E$ is the characteristic scale of the energy distribution, $L_x$ is the size of the box in its long dimension, and $\\langle r\\rangle $ is the mean distance of the star from the black hole when the mass is lost.",
"In the left panel of Figure REF , we show $dM/dE$ for the stellar debris produced by severe partial disruptions.",
"By “severe\", we mean events in which the remnant mass $M_{\\rm rem}/M_{\\star }\\simeq 40-60\\%$ .",
"These events have pericenters not much greater than $\\mathcal {R}_{\\rm t}$ ($r_{\\rm p}/\\mathcal {R}_{\\rm t}\\simeq 1.2$ ).",
"The right panel of Figure REF shows $dM/dE$ for “weak\" partial disruptions, those in which $M_{\\rm rem}/M_{\\star }\\gtrsim 90\\%$ and $r_{\\rm p}/\\mathcal {R}_{\\rm t}\\simeq 1.5-2.0$ .",
"Because our sample was bimodal in terms of mass-loss (only 3 of our 32 cases had fractional mass-loss between 10% and 40%), these two extremes comprise most of the cases we studied.",
"Figure: The fallback rate M ˙ fb \\dot{M}_{\\rm fb} for partial TDEs using the energy distribution in Figure .",
"We normalize the time tt by the orbital period P Δϵ P_{\\Delta \\epsilon } and the fallback rate M ˙ fb \\dot{M}_{\\rm fb} by M ˙ 0 =M ☆ /(3P Δϵ )\\dot{M}_{0}=M_{\\star }/(3P_{\\Delta \\epsilon }).",
"The diagonal solid lines show the power-law t -p t^{-p} with p=8/3p=8/3 (left panel) and p=5p=5 (right panel).",
"The fractional mass of the debris bound to the BH is ≃0.5(1.0-M rem /M ☆ )\\simeq 0.5(1.0-M_{\\rm rem}/M_{\\star }), and M rem /M ☆ M_{\\rm rem}/M_{\\star } is given in Table .As we showed in Ryu2+2019, explicit calculations find that the actual distribution $dM/dE$ in complete disruptions is, indeed, very symmetric as [12] predicted, but the magnitude of the energy is correct only at the order of magnitude level.",
"The characteristic spread in energy $\\Delta E$ , defined as the energy width containing 90% of the total mass, is $\\simeq 0.8 \\Delta \\epsilon $ for low-mass stars ($0.15\\le M_{\\star }\\le 0.5$ ), but jumps to $\\simeq 1.5\\Delta \\epsilon $ for $M_{\\star }\\approx 1$ and rises to almost 2 for higher-mass stars.",
"For all masses, $dM/dE$ has local maxima at $E \\simeq \\pm \\Delta E$ , but drops smoothly toward $E \\approx 0$ , where there is a local minimum whose value is only $\\simeq 2/3$ that found at the maxima.",
"In low-mass stars, $dM/dE$ plummets for $|E| > \\Delta E$ ; in high-mass stars, it falls exponentially toward larger $|E|$ , but on a scale $\\simeq \\Delta \\epsilon /3$ , so that there can be a noticeable amount of mass in the wings.",
"As shown in Figure REF , some of these characteristics are replicated in partial disruptions, but with the notable contrasts that the local minimum near $E=0$ is much deeper, and $\\Delta E$ is a function of $r_{\\rm p}/\\mathcal {R}_{\\rm t}$ as well as of $M_{\\star }$ .",
"Not too surprisingly, in severe partial disruptions $\\Delta E$ is consistently close to its value in full disruptions.",
"However, it drops by a factor $\\simeq 2$ going from severe disruptions to weak ones.",
"Severe disruptions also resemble full disruptions in that $dM/dE$ for high-mass stars, but not low-mass stars, has exponential wings.",
"These differ, however, in that they are somewhat steeper: $dM/dE \\propto e^{-4|E|/\\Delta \\epsilon }$ rather than $\\propto e^{-3 |E|/\\Delta \\epsilon }$ .",
"In weaker partial disruptions, the exponential wings decline more rapidly, on scales a factor $\\sim 2$ shorter than in the severe cases.",
"The greatest contrast between partial disruptions and full disruptions is in the depth of the central minimum.",
"The factor $\\simeq 2/3$ between $dM/dE(E=0)$ and $dM/dE(E=\\Delta E)$ for full disruptions becomes a factor $\\sim 10^{-2}$ for partial disruptions.",
"The very deep central minimum results in nearly all the debris mass being concentrated near $E \\simeq \\pm \\Delta E$ .",
"In Figure REF , we show the fallback rate for the two partial disruption cases, calculated using the energy distributions shown in Figure REF and the expression for the fallback rate [21], [20], $\\dot{M}_{\\rm fb}=\\left(\\frac{M_{\\star }}{3P_{\\Delta \\epsilon }}\\right)\\left( \\frac{dM/M_{\\star }}{d\\epsilon /2\\Delta \\epsilon }\\right)\\left(\\frac{t}{P_{\\Delta \\epsilon }}\\right)^{-5/3},$ where $P_{\\Delta \\epsilon }= (/\\sqrt{2})G M_{\\rm BH}\\Delta \\epsilon ^{-3/2}$ is the orbital period for orbital energy $-\\Delta \\epsilon $ .",
"The most noticeable feature is greater deviations from the $t^{-5/3}$ power-law for weaker tidal encounters.",
"This effect is directly due to the progressively smaller amount of mass with $E \\simeq 0$ as the events weaken.",
"Even for the severe events, however, the decline is noticeably steeper than $t^{-5/3}$ .",
"As shown in the left panel of Figure REF , the slope is $\\simeq -2.7$ for the high-mass stars, and somewhat shallower for low-mass stars (between $\\simeq -2$ and $\\simeq -2.7$ ).",
"For weak events, the fallback rate declines fastest for the low-mass stars ($\\propto t^{-6}$ ) and a bit more gently for the high-mass stars ($\\propto t^{-5}$ ).",
"These power-laws are best-determined for times when ${\\dot{M}}_{\\rm fb}/{\\dot{M}}_0 \\gtrsim 10^{-3}$ ; the total mass returning at later times is so small that it could radiate very little energy.",
"As is true of total disruptions, the peak in the fallback rate for low-mass stars is both sharper than for high-mass stars and delayed by factor $\\simeq 3$ ; these contrasts directly reflect the narrower energy width in the debris from low-mass stars (Figure REF ).",
"These results bear a qualitative resemblance to those of [8], but also disagree in some aspects.",
"Direct comparison is possible only for their $M_{\\star }=1$ polytrope with $\\gamma =4/3$ .",
"In both their calculations and ours, the slope of the decline is greater for weaker events.",
"However, in their case the contrast is substantial only for the first $\\sim 3-5$ $P_{\\Delta \\epsilon }$ , after which the logarithmic slope for the weakest encounters, whose most negative value is $-3.7$ , becomes as shallow as $\\simeq -2.3$ (see their Figure 7).",
"By contrast, our $M_{\\star }=1$ results show a fairly constant power-law slope $\\simeq -2.7$ for severe disruptions up to the point at which ${\\dot{M}}_{\\rm fb}/{\\dot{M}}_0$ falls below $10^{-3}$ (at $\\simeq 10 P_{\\Delta \\epsilon }$ ) and a similarly constant power-law slope $\\simeq -5$ up to the same fallback rate cut-off for a weaker one.",
"Some of these contrasts may be due to our coarser sampling in $\\beta $ ; however, especially for weak partial disruptions, a more important source of contrast may be the differing density profiles in the outer portions of $M_{\\star }=1$ stars predicted by a realistic density profile and a $\\gamma =4/3$ polytrope (see Figure 2 in Ryu2+2019).",
"Our results also conflict with the claim of [3] that the post-peak logarithmic slope $p$ for partial disruptions gradually steepens to an asymptote of $\\simeq 9/4$ independent of $M_{\\rm rem}$ , owing to a continuous gravitational influence of the remnant on the debris marginally bound to the BH.",
"Several methodological contrasts may account for this disagreement.",
"Whereas we use a full $3-$ dimensional hydrodynamic simulation to describe the complex geometry of the tidal streams and remnant, [3] use a $1-$ dimensional analytic model in which both the debris streams and the remnant move exclusively in the radial direction with respect to the black hole.",
"This assumption has the consequences that the gravitational force exerted by the remnant on a gas parcel is purely radial, and its magnitude is determined by the difference between their distances from the black hole.",
"It also implies that the work done by the remnant on the fluid elements does not reflect any obliquity between the direction of motion of the fluid and the direction between it and the remnant.",
"Finally, whereas we compute the self-gravity of both the mass in the stellar remnant and the debris contained within a large box around the remnant ($17~R_{\\star } \\times 9~R_{\\star } \\times 14~R_{\\star }$ ), [3] ignore the self-gravity of the debris.",
"Our approach accurately calculates the work done on the fluid by the remnant while it remains within the simulation box; because the total amount of work is dominated by the portion done while the fluid element is nearest the remnant, our box is large enough to account for the majority of this effect.",
"[6] presented one example of a partial TDE taking place in a star directly comparable to one of ours: a $3\\;\\mathrm {M}_{\\odot }$ star whose structure was computed with MESAand was halfway through its main-sequence lifetime.",
"Using the SPH code PHANTOM, they found a fallback rate exhibiting a late-time slope $\\simeq -9/4$ .",
"The pericenter for this encounter, $r_{\\rm p} = 0.33 \\;r_{\\rm t}$ , was, however, smaller than $\\mathcal {R}_{\\rm t}$ as determined by our simulations ($\\simeq 0.4-0.45\\;r_{\\rm t}$ ).",
"It is possible that they found only a partial disruption, but perhaps a rather strong one, because they employed Newtonian rather than relativistic gravity, even though this pericenter is only $27~r_{\\rm g}$ .",
"[5] also studied the shape of the debris energy distribution and the consequent fallback rate for a $M_{\\star }=1$ star whose initial mass profile was taken from MESA data.",
"Comparing their $\\beta =1.6$ and $\\beta =1.1$ cases with ours having $\\beta = 1.54$ and $\\beta = 1.0$ , we find (comparing to their Figure 5a) good consistency: from the time of peak fallback rate to a time $10\\times $ greater, we both find a mean slope $\\simeq -2.5$ in the former case and $\\simeq -3$ in the latter.",
"Similarly to ours, the $dM/dE$ distribution in their Figure 4 shows the appearance of wings near the outer boundaries, and these wings become steeper for weaker encounters.",
"Given the consistency in $dM/dE$ , it is not surprising to find similar fallback rates as well.",
"Figure: The fractional remnant mass M rem /M ☆ M_{\\rm rem}/M_{\\star } as a function of pericenter distance normalized to physical tidal radius, i.e., r p /ℛ t r_{\\rm p}/\\mathcal {R}_{\\rm t}.",
"The shaded regions around the solid lines demarcate the ranges determined by the uncertainties of ℛ t \\mathcal {R}_{\\rm t}, filled with the same colors as the solid lines.",
"The uncertainty in ℛ t \\mathcal {R}_{\\rm t} is due to our discrete sampling of r p r_{\\rm p} (0.05-0.10.05-0.1 in r p /r t r_{\\rm p}/r_{\\rm t}).",
"The dotted horizontal lines show the 50% and 90% remnant mass-fraction levels.The fitting formula given in Equation is plotted using a thicker black dashed line.",
"The fitting formulae for 1M ⊙ 1\\;\\mathrm {M}_{\\odot } polytropic stars with γ=5/3\\gamma =5/3 and γ=4/3\\gamma =4/3 by (GR-R) are depicted using thinner dot-dashed and dotted curves, respsectively.",
"The circle markers indicate whether each remnant has a positive (unfilled) or negative (filled) orbital energy with the BH potential.Figure REF shows the fractional remnant mass $M_{\\rm rem}/M_{\\star }$ as a function of $r_{\\rm p}$ .",
"When low-mass stars have $r_{\\rm p}\\gtrsim 1.5\\mathcal {~}\\mathcal {R}_{\\rm t}$ , even though they are tidally deformed near the pericenter, they recover their (quasi-) spherical structures without a significant loss of mass ($\\lesssim 10\\%$ ).",
"For high-mass stars, such weak mass-loss occurs for $r_{\\rm p}\\gtrsim 1.8~\\mathcal {R}_{\\rm t}$ .",
"We find that a simple functional form, $\\frac{M_{\\rm rem}}{M_{\\star }} &= 1.0 - \\left(\\frac{r_{\\rm p}}{\\mathcal {R}_{\\rm t}}\\right)^{-3.0},$ captures the key features of the pericenter-dependence of $M_{\\rm rem}/M_{\\star }$ .",
"In fact, by coincidence, it reproduces the curve for $M_{\\star }=3$ almost exactly.",
"[8] also provides fitting formulae for the remnant mass of polytropic stars with $\\gamma =4/3$ and $5/3$ ($1.0-C_{\\gamma }$ in their Appendix), as a function of $\\;r_{\\rm t}/r_{\\rm p}$ .",
"Their formulae for these two values of $\\gamma $ run along the envelope of the remnant mass curves shown in Figure REF : the curve for $\\gamma =5/3$ lies slightly above that for $M_{\\star }=0.15$ , while the curve for $\\gamma =4/3$ is close to that for $M_{\\star }=1$ .",
"In other words, compared to our calculations for fully convective low-mass stars, their remnants retain greater mass, while compared to our calculations for $M_{\\star }=1$ stars, there is reasonable agreement.",
"Although it is remarkable that such a simple expression can well characterize the remnant mass, Equation REF does not attempt to describe $M_{\\rm BH}$ -dependence of $M_{\\rm rem}/M_{\\star }$ .",
"Ryu1+2019 shows that when $r_{\\rm p}/r_{\\rm t}$ is rewritten in terms of the specific orbital angular momentum in units of $r_g c$ (Equation 11 in Ryu1+2019), it becomes valid independent of $M_{\\rm BH}$ .",
"In Ryu2+2019, we introduced a semi-analytic model which predicts the physical tidal radius $\\mathcal {R}_{\\rm t}$ and the maximum radius at which significant mass can be removed in a partial disruption $\\widehat{R}_{\\rm t}$ on the basis of the star's central density and mean density, respectively.",
"By combining the empirical Equation REF and this semi-analytic model, we can obtain a direct relationship between three dimensionless spatial scales, i.e., $\\mathcal {R}_{\\rm t}/r_{\\rm t}(=\\Psi )$ , $r_{\\rm p}/r_{\\rm t}(=\\beta ^{-1}\\ge \\Psi )$ and $R/R_{\\star }$ , the fractional radius within the star containing $M_{\\rm rem}$ .",
"Inserting $M_{\\rm rem}/M_{\\star }$ from Equation REF into the equation defining this model's basic assumption (that lost mass is taken from outside the point at which the tidal gravity matches an empirically-determined multiple of the star's self-gravity), we find: $\\frac{R}{R_{\\star }} \\simeq 0.47 \\left(\\left[\\frac{r_{\\rm p}}{r_{\\rm t}}\\right]^{3}-\\left[\\frac{\\mathcal {R}_{\\rm t}}{r_{\\rm t}}\\right]^{3}\\right)^{1/3}.$ This relation behaves correctly in simple limits: at $\\beta ^{-1} = \\Psi $ , $R=0$ , and at $\\beta ^{-1}=\\widehat{R}_{\\rm t}/r_{\\rm t}$ ($\\widehat{R}_{\\rm t}$ the largest pericenter distance for tidal mass-loss, see Equation 17 in [22]), $R/R_{\\star }\\simeq 1$ with no more than 5% errors.",
"Thus, with a model for the star's initial mass profile and knowledge of $\\mathcal {R}_{\\rm t}$ , the remnant mass can be predicted easily for any pericenter larger than the physical tidal radius.",
"Table: The properties of partial disruption remnants.",
"In the left-hand columns, we list the original mass of our model stars M ☆ [M ⊙ ]M_{\\star }~[\\rm {M}_{\\odot }], r t /r p (≡β)r_{\\rm t}/r_{\\rm p}(\\equiv \\beta ), r p /ℛ t r_{\\rm p}/\\mathcal {R}_{\\rm t}, the remnant mass M rem [M ⊙ ]M_{\\rm rem}~[\\rm {M}_{\\odot }], the mass fraction M rem /M ☆ M_{\\rm rem}/M_{\\star }, the sign of the mass-weighted specific energy E ¯{\\bar{E}} (B:E ¯<0{\\bar{E}}<0 and U:E ¯>0{\\bar{E}}>0) and the magnitude of the average specific energy in units of Δϵ\\Delta \\epsilon .The right-hand four columns give orbital parameters for the remnants: for unbound stars, only the ejection velocity v ejec v_{\\rm ejec}; for bound stars, the eccentricity e ¯\\bar{e}, semimajor axis aa and orbital period PP.",
"The orbital parameters and the remnant mass are measured when those quantities have settled into asymptotic values (at r≃20-30r t r \\simeq 20-30\\;r_{\\rm t}).",
"Note that we do not show v ejec v_{\\rm ejec} for the M ☆ =3M_{\\star }= 3 and 10 cases' most severe disruptions.",
"This is because even at r≃20-30r t r \\simeq 20-30\\;r_{\\rm t}, they had not settled into an approximate steady state; in addition, their mean specific energy was so different from that of the initial star's that the remnant was offset far enough from the center of the simulation box that some of its mass was no longer inside the box.",
"We exclude these two cases from the analysis of the unbound population in the text.Figure: Rotational properties of the remnant from an event in which a star with M ☆ =1M_{\\star }=1 passes through a pericenter r p =0.65r t r_{\\rm p}=0.65\\;r_{\\rm t}.",
"The remnant mass M rem ≃0.48M_{\\rm rem}\\simeq 0.48.",
"(left panel) The mean angular frequency Ω ¯(R)\\bar{\\Omega }(R) at cylindrical radius RR on three horizontal planes; their heights above the orbital plane are: z=0R ⊙ z=0\\;\\mathrm {R}_{\\odot } (solid), 0.4R ⊙ 0.4\\;\\mathrm {R}_{\\odot } (dashed) and 0.8R ⊙ 0.8\\;\\mathrm {R}_{\\odot } (dotted).",
"The radius RR on the x-x-axis is normalized by the radius R ☆,99% R_{\\star ,99\\%} for 99% of the remnant mass.",
"The red dotted line shows the equatorial break-up angular frequency.",
"The vertical magenta solid line at R/R ☆,99% ≃1.16R/R_{\\star ,99\\%}\\simeq 1.16 is placed at the radius R ☆ R_{\\star } of our 1M ⊙ 1\\;\\mathrm {M}_{\\odot } MS star.",
"(right panel) Azimuthal velocity v φ v_\\phi .",
"xx and yy are normalized by R ☆,99% R_{\\star ,99\\%}.",
"The solid magenta circle delineates R ☆ R_{\\star }."
],
[
"Specific energy - bound or unbound",
"In this section, we focus on the specific energies of surviving remnants to see whether or not they are bound to the BH, and to determine their orbital motion in either case (see Table REF for the results).",
"We consider the question of whether they are bound to the galaxy's bulge separately.",
"As a prologue to this topic, it is useful to lay out the hierarchy of orbital energy scales in this problem.",
"The most useful unit for this hierarchy is the specific kinetic energy of stars in the region of the galaxy from which the disrupted stars are drawn, i.e., $(1/2)\\sigma ^{2}$ , where $\\sigma $ is the $3-$ dimensional bulge velocity dispersion.",
"In terms of this unit, the initial orbital energy of stars in our simulations counting only the black hole's contribution to the gravitational potential is very small, $\\sim -10^{-3}(\\sigma ^2/2)$ , which, in relativistic terms, is a specific energy $\\sim -10^{-10}c^2$ for $\\sigma \\sim 100 - 300$ km s$^{-1}$ .",
"In this sense, one might think of our stars as having, prior to the disruption, energy very close to the middle of the bulge stars' energy distribution.",
"On the other hand, the magnitude of the typical remnant's specific energy is relatively large, $\\sim 1-10$ .",
"Because the typical remnant energy changes by an amount greater than the actual energy with which stars begin the event, we can approximate the remnant's final energy as its actual energy with respect to the BH potential.",
"Moreover, because it is also several times larger than the potential associated with the stars of the inner galaxy, it is appropriate to label remnants with positive final energy as “unbound\" with respect to the innermost portion of the galaxy.",
"However, we must also emphasize that “large\" is a relative term.",
"Although the remnants' energies are comparable to or larger than the kinetic energy of bulge stars, they are tiny compared to the magnitude of the debris energy, whether bound or unbound—they are $\\sim 10^{-3}$ on that scale.",
"It is a good approximation to suppose that the BH potential dominates the entire region through which bound remnants travel because all but one of their apocenters ($\\simeq 0.05-1\\;\\mathrm {pc}$ , Table REF ) are smaller than the BH's radius of influence ($ \\sim 1-10$ pc; see Section REF for further discussion of this point).",
"The corresponding periods are between $\\simeq 400$ and $\\simeq 40,000\\;\\mathrm {yr}$ .",
"Their eccentricities are exceedingly close to 1, mostly with $|1-e| \\sim 10^{-5}$ .",
"There is also one case ($M_{\\star }=0.7$ , $r_{\\rm p}/\\;r_{\\rm t}= 0.9$ ) that is intermediate between bound and unbound in the sense that it is bound, but only weakly, having $a\\simeq 1.7\\;\\mathrm {pc}$ and $P\\simeq 0.2\\;\\mathrm {Myr}$ .",
"The comparative rarity of remnants whose net energy is very close to zero is likely due to the small associated phase space.",
"With specific energies similar in magnitude to those of the bound remnants, but opposite sign, the unbound remnants have ejection speeds $v_{\\rm ejec}\\simeq 100-330 \\;\\mathrm {km}\\;\\mathrm {s}^{-1}$ .",
"Figure REF distinguishes bound from unbound remnants by using filled circles for the former and unfilled circles for the latter.",
"For low-mass stars, the unbound remnants are associated with the most severe partial disruptions, whereas relatively weak encounters yield bound remnants.",
"However, for high-mass stars, even some severe partial disruptions yield bound remnants.",
"Because the specific angular momentum of a remnant (either bound or unbound) is essentially identical to the specific angular momentum of the original star, its pericenter (when bound) is very nearly unchanged by the tidal encounter.",
"A similar studies were reported by [14].",
"Using Newtonian hydrodynamics simulations of tidal disruption of polytropic stars with $\\gamma =4/3$ , they determined the orbital energies of remnants at a time $\\simeq 100 \\sqrt{R_{\\star }^{3}/GM_{\\star }}$ ) after pericenter passage.",
"Contrary to what we found, all their surviving remnants were, in our language, unbound, and their ejection speeds were considerably greater than ours.",
"For example, in the case of stars with $M_{\\star }=1$ (for which a $\\gamma =4/3$ polytrope is a reasonable approximation), the ejection speed for their remnants ranged from $\\simeq 100$ km/s (for $r_{\\rm p}/r_{\\rm t} = 1$ ) to $\\simeq 600$ km/s (for $r_{\\rm p}/r_{\\rm t} = 0.55$ ).",
"By contrast, the remnants of our $M_{\\star }=1$ simulations with $0.5 \\le r_{\\rm p}/r_{\\rm t} \\le 1$ were all bound, and the greatest ejection speed we found for any other case was $\\simeq 330$ km/s.",
"It is unclear how to account for these differing results; the difference between relativistic and Newtonian tidal forces might play a part.",
"Figure: The ratio Ω ¯/Ω bk \\bar{\\Omega }/\\Omega _{\\rm bk} as a function of M en /M rem M_{\\rm en}/M_{\\rm rem}, the ratio of the enclosed mass to the remnant mass.",
"In all cases displayed, M rem /M ☆ ≃0.5M_{\\rm rem}/M_{\\star } \\simeq 0.5, or equivalently, r p /ℛ≃1.1-1.3r_{\\rm p}/\\mathcal {R}\\simeq 1.1-1.3."
],
[
"Spin ",
"All surviving remnants are spun-up in the prograde direction as they are tidally torqued near the pericenter [21], [5].",
"As a result, they are approximately oblate spheroids in shape, with the minor axis perpendicular to the orbital plane.",
"In all cases, the angular frequency increases outward.",
"As an example, we present in the left panel of Figure REF the angular frequency $\\bar{\\Omega }(R)$ , an azimuthal average over cells at the same cylindrical radius from an axis through the remnant's center of mass perpendicular to the orbital plane, at three different heights.",
"The star in this simulation began with mass $M_{\\star }=1$ , passed through a pericenter $r_{\\rm p}=0.65\\;r_{\\rm t}$ , and emerges from the event with $M_{\\rm rem}\\simeq 0.48$ .",
"The angular frequency $\\bar{\\Omega }$ at each height increases outwards until it reaches a maximum at $R\\simeq 0.8-1.0$ .",
"The maximum frequency at the equator is around $25-30\\%$ of the equatorial break-up angular frequency $\\Omega _{\\rm bk}$ , defined as $\\Omega _{\\rm bk}(R)=\\sqrt{GM_{\\rm en}(R)/R^{3}}$ .",
"Here $M_{\\rm en}(R)$ is the enclosed mass inside cylindrical radius $R$ on the equatorial plane.",
"The rotational velocity $v_{\\phi }$ (right panel) therefore rises steeply at small radius and then $\\propto R$ for $R \\gtrsim 0.6\\;\\mathrm {R}_{\\odot }$ .",
"Its maximum is $\\simeq 100-120\\;\\mathrm {km}\\;\\mathrm {s}^{-1}$ .",
"We find a general trend that, for fixed fractional mass loss, the more massive the initial star, the closer its remnant comes to break-up rotation.",
"This trend is illustrated in Figure REF , in which we present data for partially disrupted stars with $M_{\\rm rem}/M_{\\star } \\simeq 0.5$ , corresponding to $r_{\\rm p}/\\mathcal {R}_{\\rm t}\\simeq 1.1-1.3$ .",
"That the high-mass stars reach higher fractions of the break-up rotation rate than the lower-mass stars can be explained simply.",
"To zeroth order, when a star passes through pericenter, tidal forces torque it so that its outer layers rotate at roughly the local orbital frequency.",
"But the local orbital frequency is, by definition, about the same as the vibrational frequency when the distance from the black hole is similar to $r_{\\rm t}$ .",
"By the same token, the break-up rotational frequency is similar to the vibrational frequency.",
"Consequently, $\\Omega /\\Omega _{\\rm bk} \\simeq \\Omega (r_{\\rm p})/\\Omega _{\\rm bk} \\propto \\beta ^{3/2}$ .",
"It is also worth noting that if the star spins at near break-up rates before the encounter, tidal dynamics can be quite different [25].",
"Figure: The specific entropy P/ρ Γ P/\\rho ^{\\Gamma } (Γ=5/3\\Gamma =5/3, in cgs units) of the same remnant star shown in Figure (M ☆ =1M_{\\star }=1, M rem ≃0.48M_{\\rm rem}\\simeq 0.48, ψ=0.65\\psi =0.65 and r≃23r t r\\simeq 23\\;r_{\\rm t}).",
"The black curves represent the entropy profile in the equatorial plane ( azimuthally-averaged, solid) and along the z-z-axis (dashed).",
"The blue and red dotted curves indicate the entropy profile for main sequence stars with mass M ☆ M_{\\star } and M rem M_{\\rm rem}, respectively.",
"The radius RR on the x-x-axis is normalized by the radius R ☆,99% R_{\\star ,99\\%} for 99% of the remnant mass.",
"The vertical magenta solid line indicates the radius of the original 1M ⊙ 1\\;\\mathrm {M}_{\\odot } MS star, R ☆ R_{\\star }.Figure: The density ρ\\rho of the same remnant star in Figure (M ☆ =1M_{\\star }=1, M rem ≃0.48M_{\\rm rem}\\simeq 0.48, β=1.54\\beta = 1.54 and r≃23r t r\\simeq 23\\;r_{\\rm t}).The spatial scales (RR, xx, yy) are normalized to R ☆,99% R_{\\star ,99\\%}, the radius containing 99% of the remnant mass.",
"(top panel) The density on the equatorial plane is shown by the black solid curve and along the z-z-axis by the black dashed curve.",
"The red dashed curve depicts the density profile of a MESA-MS analog.",
"The vertical magenta solid line indicates the original R ☆ R_{\\star }.",
"(Middle and bottom panels) 2-2-dimensional density maps of the star in the equatorial plane (x-yx-y) and in the vertical plane (x-zx-z), respectively.",
"The solid (larger) magenta circle delineates R ☆ R_{\\star } and the red (smaller) dashed line the radius for 99% of the mass of the MESA-MS star."
],
[
"Internal structure",
"Figure REF shows the specific entropy as a function of distance from the center of the star portrayed in Figure REF , a partial disruption of a $1 \\;\\mathrm {M}_{\\odot }$ star that leaves a $0.48 \\;\\mathrm {M}_{\\odot }$ remnant.",
"As we have assumed adiabatic behavior and found that the tidally-induced motions are laminar, the range of specific entropy found in the remnant matches the range found in the original star.",
"However, the mean entropy in the remnant is a bit lower than in the initial star because $1\\;\\mathrm {M}_{\\odot }$ main sequence stars have positive radial entropy gradients, and most of the mass lost in the encounter is taken from the star's outer layers.",
"Because the remnant rotates so rapidly, its specific entropy rises more gradually outward in the equatorial plane than along the rotational axis.",
"Although the specific entropy of the remnant is similar to that of the initial star, it is in general greater than in a main sequence star of the remnant mass because higher-mass main sequence stars have higher specific entropy than lower-mass stars.",
"For this reason, remnants of severe partial disruptions are, in general, far from thermal equilibrium.",
"A direct consequence of this departure from thermal equilibrium is shown in Figure REF , where we compare the density distributions of this remnant and its main-sequence counterpart.",
"The top panel of Figure REF shows its density profile both in the equatorial plane and along the $z-$ axis.",
"The density on the equatorial plane is calculated in the same way as the entropy in that plane.",
"The middle and bottom panels depict $2-$ dimensional snapshots of the star's density in the $x-y$ and $x-z$ planes, respectively.",
"The most noticeable feature in the top panel of Figure REF is how much more extended the density distribution is in the remnant than in its main-sequence counterpart.",
"At the center of the star, the density is about a factor of 3 smaller.",
"It is hard to determine an outer radius for this remnant because the density drops so smoothly outward: for radii outside $\\simeq 0.3R_{\\star }$ , in the equatorial plane it is very well described by $\\rho \\propto \\exp [-R/(0.15 R_{\\star })]$ .",
"The photosphere lies well outside the range portrayed: the Thomson optical depth over an exponential scale-length at $R/R_{\\star } \\simeq 1.2$ is $\\sim 10^8$ .",
"On the other hand, the majority of the star's mass is confined much more tightly, and can be found within a distance similar to that of a main sequence star of this mass, $\\simeq 0.5R_{\\star }$ .",
"All these trends are reproduced in our other remnants, but, as might be expected, with the contrast between the remnant and its main sequence partner greater for more severe encounters.",
"In one case, the central density is a factor of 30 smaller than a main sequence star of the same mass.",
"All three panels of Figure REF portray the star's oblate spheroidal shape.",
"The top panel shows how the density drops outward more rapidly along the $z-$ direction than on the equatorial plane.",
"The two lower panels show its shape in the equatorial and poloidal planes.",
"It is clear from them that, although the star is very nearly axisymmetric, it is substantially oblate, and the oblateness increases with distance from the center.",
"Due to slightly asymmetric mass-loss, remnants whose parent stars had very nearly zero energy with respect to the BH have a small, but non-zero, orbital energy per unit mass after their tidal encounters.",
"In real events, the initial stellar orbital energy can also be slightly non-zero, but the magnitude of the surviving remnants' energy is sufficiently larger than the initial energy that the latter can be neglected (Section REF ).",
"The orbits of the remnants can then be conveniently divided into two classes according to the sign of their energy considering only the black hole potential: those with positive energy are unbound, and those with negative energy are bound."
],
[
"Unbound population",
"The ejection velocities of the unbound remnants we simulated range from $90-330\\;\\mathrm {km}\\;\\mathrm {s}^{-1}$ .",
"Extrapolating from the bulge dispersion data of galaxies with central BHs slightly more massive than $10^{6}\\;\\mathrm {M}_{\\odot }$ , we find that the dispersions of galaxy bulges containing BHs with $M_{\\rm BH}\\simeq 10^{6}$ are $\\sim \\sigma =60-90\\;\\mathrm {km}/{\\rm s}^{-1}$ [26], [11], [7].",
"Our unbound remnants can therefore easily escape the radius of influence, $r_{\\rm inf}= GM_{\\rm BH}/\\sigma ^{2}\\simeq 0.5\\;\\mathrm {pc}(M_{\\rm BH}/10^{6})(\\sigma /90\\;\\mathrm {km}\\;\\mathrm {s}^{-1})^{-2}$ , of the central BH.",
"Nonetheless, if the potential beyond the sphere of influence is logarithmic, the remnants are likely to reach a turning point $r_{\\rm max}$ at only a few $r_{\\rm inf}$ , i.e., $r_{\\rm max} \\simeq \\Lambda ~r_{\\rm inf}$ with $\\Lambda = e^{(v_{\\rm ejec}/2\\sigma )^{2}}\\simeq 1-10$ .",
"Such a turning point would be well within the bulge region.",
"As the angular momentum of a remnant is much smaller than the value corresponding to a circular orbit at its semimajor axis, the pericenter distance is determined almost purely by the angular momentum.",
"If it is unchanged during the time spent near apocenter, any such remnant will return to the black hole with the same pericenter as the original stellar orbit, raising the prospect of a second tidal interaction.",
"To estimate how large these perturbations may be, as a crude approximation we compare the travel time for a partial disruption remnant to reach its turning point with the time required for weak stellar encounters to alter the remnant's original specific angular momentum $L_{0}$ by a factor of order unity.",
"The travel time is $t_{\\rm travel}=r_{\\rm max}/v_{\\rm ejec}\\simeq 10^{5} (r_{\\rm max}/10\\;\\mathrm {pc})(v_{\\rm ejec}/90\\;\\mathrm {km}\\;\\mathrm {s}^{-1})^{-1}\\;\\mathrm {yr}$ .",
"On the other hand, the evolution time for remnant angular momentum is $t_{\\rm L}\\simeq (L_{0}/L_{\\rm r})^{2}t_{\\rm r}$ [15] where $L_{\\rm r}$ refers to the specific angular momentum change during a collisional relaxation time $t_{\\rm r}$ ; by definition, $L_{\\rm r}\\simeq \\sigma ~r_{\\rm max}$ .",
"Using the relation $t_{\\rm r}\\simeq 0.1(N/\\ln N)t_{\\rm cross}$ [2], where $N$ is the number of stars within the region the test-particle star travels through, the definitions $t_{\\rm cross}=r_{\\rm max}/\\sigma $ and $r_{\\rm max}=\\Lambda ~ r_{\\rm inf}$ , and the fact $L_{0}\\simeq \\sqrt{2GM_{\\rm BH}r_{\\rm t}}$ , we find that the ratio between the two characteristic times when $r_{\\rm max}>r_{\\rm inf}$ is, $\\frac{t_{\\rm L}}{t_{\\rm travel}}&\\simeq 2 \\left(\\frac{0.1N}{\\ln N}\\right) \\left(\\frac{v_{\\rm ejec}}{\\sigma }\\right)\\left(\\frac{ r_{\\rm inf} r_{\\rm t}}{r_{\\rm max}^{2}}\\right),\\nonumber \\\\&\\simeq 10^{-2}\\left(\\frac{v_{\\rm ejec}}{\\sigma }\\right)\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{4/3}\\left(\\frac{r_{\\rm max}}{5\\;\\mathrm {pc}}\\right)^{-1}.$ For this estimate, we also assumed the mass of background stars is $1\\;\\mathrm {M}_{\\odot }$ , giving $N(<r_{\\rm max})\\simeq 2~M_{\\rm BH}~\\Lambda $ for a logarithmic potential.",
"This estimate implies that gravitational encounters are very likely to result in changes of the unbound remnants' angular momenta large enough to alter their pericenter distances (a situation also called “full loss-cone\" evolution).",
"Because $r_{\\rm p} \\propto L^2$ for these highly-eccentric orbits, the resulting change of $r_{\\rm p}$ should be $\\propto t_{\\rm travel}/t_{\\rm L}$ .",
"Thus, for these unbound remnants, the pericenter upon return is likely to be considerably larger than the value of $\\;r_{\\rm t}$ of the returning remnant.",
"It is also possible for their angular momenta to be affected by other mechanisms, e.g.",
"scattering by giant molecular clouds [19] or torques due to non-spherical galactic pontentials [16].",
"These remnants, although on unclosed orbits, will nonetheless return to the galactic center close to the BH, but their pericenters are likely to be altered enough that the probability of an interesting tidal encounter is small."
],
[
"Bound population",
"Every remnant in our bound sample (except for one that is exceptionally weakly bound) has an eccentricity less than unity by $\\sim 10^{-4}-10^{-5}$ , a semimajor axis $a\\sim 0.03-0.5\\;\\mathrm {pc}$ , and an orbital period $P\\sim 400-40000\\;\\mathrm {yr}$ .",
"Although it is likely that our sample does not span the full range of possibilities, these numbers may be taken as indicative of the typical magnitudes for events with $M_{\\rm BH} \\sim 10^6$ .",
"These bound remnants are also subject to stellar encounters, but within the black hole sphere of influence.",
"For this case, we can not use the same expression for $t_{\\rm L}$ used above as it is derived for remnants whose motions are dominated by the potential from surrounding stars while, within $r_{\\rm inf}$ , the BH potential dominates.",
"The typical velocity of stars at $r_{\\rm max}=2a<r_{\\rm inf}$ is roughly $\\sigma \\simeq \\sqrt{GM_{\\rm BH}/2a}$ .",
"This leads to a relaxation time $t_{\\rm r}=0.1 (M_{\\rm BH}/m)^{2} /[N\\ln (M_{\\rm BH}/m)]~t_{\\rm cross}$ , where $m$ is the mean mass per star.",
"With $r_{\\rm max}=2a$ and $t_{\\rm travel}=P/2$ , we find that $t_{\\rm L}/t_{\\rm travel}$ for our fiducial values is not very different from the value estimated for the unbound population: $\\frac{t_{\\rm L}}{t_{\\rm travel}}&\\simeq \\frac{0.1\\times 2^{3/2}}{}\\frac{(M_{\\rm BH}/m)^{2}}{N \\ln (M_{\\rm BH}/m)}\\left(\\frac{r_{\\rm t}}{a}\\right),\\nonumber \\\\&\\simeq 2\\times 10^{-2} \\left(\\frac{N}{2\\times 10^{6}}\\right)^{-1}\\left(\\frac{\\ln (M_{\\rm BH}/m)}{13.8}\\right)^{-1},\\nonumber \\\\&\\times m^{-2}\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{7/3}\\left(\\frac{a}{0.5\\;\\mathrm {pc}}\\right)^{-1},$ where we have scaled to values appropriate to the one of the longer semi-major axes in our sample.",
"The apocenter distance for such a semi-major axis is comparable to $r_{\\rm inf}$ for $M_{\\rm BH}=10^{6}$ , within which, by definition, $N(<r_{\\rm inf})\\simeq 2\\times 10^{6}$ .",
"However, this timescale ratio is sensitive to the dependence of $N$ on $a$ .",
"If the stellar density $\\rho _{\\star }\\propto r^{-n}$ , $N(<r)\\propto r^{3-n}$ .",
"The ratio $t_{\\rm L}/t_{\\rm travel}$ then scales $\\propto a^{n-4}$ .",
"Therefore, for a density profile near the BH with $n<4$ , $t_{\\rm L}/t_{\\rm travel}$ increases as $a$ decreases, possibly becoming larger than unity at a sufficiently small $a$ (e.g., for $n=7/4$ , the steady-state solution of [1], the ratio becomes larger than unity at $a\\lesssim 0.07-0.08\\;\\mathrm {pc}$ ).",
"This means that for bound remnants with sufficiently small semimajor axes, the pericenter upon return remains almost unchanged from its value during the first passage.",
"Because our sample includes some remnants with semimajor axes as small as $\\simeq 0.03$ pc, a fraction of the bound remnant population will return with pericenters either the same as during their first passage, or enlarged by only a little."
],
[
"A second tidal disruption?",
"Whether a significant tidal disruption event takes place at the next pericenter passage depends on how the (possibly larger) pericenter compares to the star's new tidal radius.",
"If the remnant returns to the main sequence before returning to the vicinity of the black hole, its smaller mass would imply a smaller size and a smaller $r_{\\rm t}$ , whereas its new pericenter is likely to be at least as large as in the original event.",
"Significant disruption would probably not occur.",
"However, return to the main sequence in time for the next return to pericenter may be problematic.",
"Relative to main sequence structure, these remnants are expanded by both extra heat and rapid rotation.",
"In terms of its enclosed mass profile, the example shown in Figure REF resembles a red giant: most of its mass is contained within a relatively small radius, while a low-density envelope extends out to large distances.",
"Employing our semi-analytic model Ryu1+2019, we might then estimate a critical distance for complete disruption $\\simeq 1.5 \\times $ that expected for the same-mass main sequence star, which is $\\simeq 1.8~\\mathcal {R}_{\\rm t}$ for the parent star, but a critical distance for partial disruptions $\\simeq 1.4\\times $ that of the parent star.",
"Both distances are also enlarged by a modest amount because the ratio $(M_{\\rm BH}/M_{\\star })^{1/3}$ is greater by 28%.",
"Thus, if there is too little time for it to cool before the next pericenter passage, a significant tidal encounter might well take place upon its first return to the vicinity of the black hole.",
"Whether thermal relaxation can be completed by the time the remnant returns to periastron depends upon the ratio of the cooling time to the orbital period.",
"The photon diffusion time from the center of a star to its edge is $t_{\\rm th}&\\simeq \\kappa _{\\rm c}\\rho _{\\rm c} R_{\\rm c}^{2} /c,\\nonumber \\\\&\\simeq 2\\times 10^{4}\\;\\mathrm {yr}\\left(\\frac{\\kappa _{\\rm c}}{10\\kappa _{\\rm e}}\\right)\\left(\\frac{\\rho _{\\rm c}}{10^{2}\\;\\mathrm {g}\\;\\mathrm {cm}^{-3}}\\right)\\left(\\frac{R_{\\rm c}}{0.1}\\right)^{2},$ where $\\kappa _{\\rm c}$ is the core opacity, $\\rho _{\\rm c}$ is the core density and $R_{\\rm c}$ is the radial length scale of the core.",
"The Thomson opacity is $\\kappa _{\\rm e}$ .",
"In the conditions of our stellar remnants ($\\rho _{\\rm c}\\sim 1- 10^{2} \\;\\mathrm {g}\\;\\mathrm {cm}^{-3}$ , core temperature $T_{\\rm c}\\sim 10^{6} - 10^{7} \\;\\mathrm {K}$ ), $\\kappa _{\\rm c}/\\kappa _{\\rm e}\\sim 1- 10^{2}$ [9].",
"Comparing this time to the orbital periods shown in Table REF demonstrates that the more tightly bound remnants ($P<t_{\\rm th}$ ) would return back to the BH without significant changes in their internal structures.",
"These are also the remnants likely to suffer the least increase in orbital pericenter due to scattering with background stars.",
"Thus, for both reasons, the more tightly bound remnants have the greatest probability of going through a second TDE.",
"However, we caution that a more careful calculation of the remnant's cooling is necessary to determine what happens when it next passes through pericenter.",
"The evolution of the remnant star's rotation may also influence its fate.",
"Angular momentum may be lost through magnetic braking [4]; it may also be mixed inward from the outer $\\sim 10\\%$ of the star's mass where it initially resides by any of a variety of processes [13].",
"Because only a minority of the remnants' mass rotates rapidly, evolution in the star's rotation may be a next-order correction to the effect of cooling."
],
[
"Summary",
"In this paper, the third in this series, we continue our study of tidal disruption events of main-sequence stars, focusing on the properties of partial disruptions.",
"Our results are based upon a suite of fully general relativistic simulations in which the stars' initial states are described by realistic main-sequence models.",
"We examined tidal disruption events for eight different stellar masses, from $M_{\\star }=0.15$ to $M_{\\star }=10$ with a fixed black hole mass ($10^6\\;\\mathrm {M}_{\\odot }$ ).",
"In Ryu4+2019, we will explore how increasingly strong relativistic effects alter the properties of partial disruptions involving higher-mass black holes.",
"We find that the energy distribution $dM/dE$ of the stellar debris created from partial disruptions is different from the one that arises in full disruptions, with the contrast growing for weaker encounters.",
"For full disruptions, the characteristic energy width $\\Delta E$ of the stellar debris for low-mass stars is $\\simeq 0.8 \\Delta \\epsilon $ , while that for high-mass stars can be as large as $\\simeq 2\\Delta \\epsilon $ , where $\\Delta \\epsilon $ is the traditional order of magnitude estimate for this width.",
"The energy distribution $dM/dE$ for all masses has a local minimum near $E\\simeq 0$ and “shoulders” near the outer boundaries, with a contrast between the two $\\simeq 1.5$ (Ryu2+2019).",
"On the other hand, for partial disruptions, most of the mass of the stellar debris is concentrated near the shoulders, with little mass near $E\\simeq 0$ : the contrast is $\\sim 10$ for strong disruptions, in which a large fraction of the stellar mass is lost, and it increases to $\\sim 100-1000$ for weaker disruptions.",
"Although the outer edges of the distribution are quite sharp for low-mass stars subjected to either partial or full disruption, there can be significant tails for high-mass stars.",
"These become progressively steeper for weaker partial disruptions.",
"Because there is so little mass near $E\\simeq 0$ , late-time fallback is suppressed, and the overall shape of the fallback rate becomes more and more like a single peak as the mass lost in the event diminishes.",
"On the declining side of the peak, the mass-return rate is $\\propto t^{-p}$ with $p\\simeq 2-5$ , very unlike the consistent $p=5/3$ for full disruptions.",
"Another product of partial disruptions is surviving remnants.",
"We have found a simple analytic expression linking the ratio between the stellar orbit's pericenter and the physical tidal radius for that stellar mass to the ratio between the remnant mass and the original stellar mass (see Equation REF ).",
"The remnants retain around $50\\%$ of the original mass at $r_{\\rm p}/\\mathcal {R}_{\\rm t}\\simeq 1.2-1.5$ , while the mass loss becomes less than $10\\%$ at $r_{\\rm p}/\\mathcal {R}_{\\rm t}\\gtrsim 1.5-1.8$ .",
"Because higher-mass main sequence stars have higher entropy than lower-mass stars, surviving remnants are out of thermal equilibrium and tend to be larger in size than a MS star of the same mass.",
"They are also rapidly-rotating, reaching angular frequencies near break-up in the outer layers of the remnants left by events causing substantial mass-loss from initially massive stars.",
"The rapid rotation makes these stars oblate spheroids.",
"The change in specific orbital energy of partially-disrupted stars is quite small compared to the spread in energy of the debris: $\\simeq 10^{-3} \\Delta \\epsilon $ (see Table REF ), but it can be of either sign.",
"Particularly for low-mass stars, weaker encounters lead to remnants that lose orbital energy and therefore remain within the sphere of influence of the black hole, while the strongest encounters can create remnants able to travel some distance out into the galaxy's bulge.",
"For high-mass stars, most partial disruptions lead to bound remnants, except for those that are nearly strong enough to cause total disruption.",
"When a stellar remnant, whether bound to the black hole or able to travel out into the bulge, reaches its orbital apocenter, weak gravitational interactions with buldge stars can alter its angular momentum.",
"The change can be large compared to the remnant's original angular momentum when the remnant goes as far as the stellar bulge, or even the outer portion of the black hole's sphere of influence, but if the remnant's apocenter is smaller than the black hole's sphere of influence, the change can be comparable to the original angular momentum or even less.",
"When the increase in specific angular momentum is relatively small, the remnant may become a victim of another TDE if its cooling time is longer than its orbital period.",
"Because the most tightly-bound remnants have substantially shorter orbital periods than those able to reach the bulge, their prospects for a second tidal event are further enhanced."
],
[
"Acknowledgements",
"We would like to thank an anonymous referee for an insightful question about the specific entropy in partial disruption remnants.",
"This work was partially supported by NSF grant AST-1715032, Simons Foundation grant 559794 and an advanced ERC grant TReX.",
"S. C. N. was supported by the grants NSF AST 1515982, NSF OAC 1515969, and NASA 17-TCAN17-0018, and an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center administrated by USRA through a contract with NASA.",
"This research project (or part of this research project) was conducted using computational resources (and/or scientific computing services) at the Maryland Advanced Research Computing Center (MARCC).",
"The authors would like to thank Stony Brook Research Computing and Cyberinfrastructure, and the Institute for Advanced Computational Science at Stony Brook University for access to the high-performance SeaWulf computing system, which was made possible by a $\\$1.4$ M National Science Foundation grant (#1531492).",
"matplotlib [10]; MESA[18]; Harm3d[17];"
]
] | 2001.03503 |
[
[
"Explaining the Explainer: A First Theoretical Analysis of LIME"
],
[
"Abstract Machine learning is used more and more often for sensitive applications, sometimes replacing humans in critical decision-making processes.",
"As such, interpretability of these algorithms is a pressing need.",
"One popular algorithm to provide interpretability is LIME (Local Interpretable Model-Agnostic Explanation).",
"In this paper, we provide the first theoretical analysis of LIME.",
"We derive closed-form expressions for the coefficients of the interpretable model when the function to explain is linear.",
"The good news is that these coefficients are proportional to the gradient of the function to explain: LIME indeed discovers meaningful features.",
"However, our analysis also reveals that poor choices of parameters can lead LIME to miss important features."
],
[
"Interpretability",
"The recent advance of machine learning methods is partly due to the widespread use of very complicated models, for instance deep neural networks.",
"As an example, the Inception Network [12] depends on approximately 23 million parameters.",
"While these models achieve and sometimes surpass human-level performance on certain tasks (image classification being one of the most famous), they are often perceived as black boxes, with little understanding of how they make individual predictions.",
"This lack of understanding is a problem for several reasons.",
"First, it can be a source of catastrophic errors when these models are deployed in the wild.",
"For instance, for any safety system recognizing cars in images, we want to be absolutely certain that the algorithm is using features related to cars, and not exploiting some artifacts of the images.",
"Second, this opacity prevents these models from being socially accepted.",
"It is important to get a basic understanding of the decision making process to accept it.",
"Model-agnostic explanation techniques aim to solve this interpretability problem by providing qualitative or quantitative help to understand how black-box algorithms make decisions.",
"Since the global complexity of the black-box models is hard to understand, they often rely on a local point of view, and produce an interpretation for a specific instance.",
"In this article, we focus on such an explanation technique: Local Interpretable Model-Agnostic Explanations (LIME, [8]).",
"Figure: LIME explanation for object identification in images.",
"We used Inception as a black-box model.",
"Terrapin, a sort of turtle, is the top label predicted for the image in panel (a).Panel (b) shows the results of LIME, explaining how this prediction was made.The highlighted parts of the image are the superpixels with the top coefficients in the surrogate linear model.",
"We ran the same experiment for the `strawberry' label in panel (c)."
],
[
"Contributions",
"Our main goal in this paper is to provide theoretical guarantees for LIME.",
"On the way, we shed light on some interesting behavior of the algorithm in a simple setting.",
"Our analysis is based on the Euclidean version of LIME, called “tabular LIME.” Our main results are the following: When the model to explain is linear, we compute in closed-form the average coefficients of the surrogate linear model obtained by TabularLIME.",
"In particular, these coefficients are proportional to the partial derivatives of the black-box model at the instance to explain.",
"This implies that TabularLIME indeed highlights important features.",
"On the negative side, using the closed-form expressions we show that it is possible to make some important features disappear in the interpretation, just by changing a parameter of the method.",
"We also compute the local error of the surrogate model, and show that it is bounded away from 0 in general.",
"We explain how TabularLIME works in more details in Section .",
"In Section , we state our main results.",
"They are discussed in Section , and we provide an outline of the proof of our main result in Section .",
"We conclude in Section ."
],
[
"Intuition",
"From now on, we will consider a particular model encoded as a function $f:\\operatorname{\\mathbb {R}}^d\\rightarrow \\operatorname{\\mathbb {R}}$ and a particular instance $\\xi \\in \\operatorname{\\mathbb {R}}^d$ to explain.",
"We make no assumptions on this function, e.g., how it might have been learned.",
"We simply consider $f$ as a black-box model giving us predictions for all points of the input space.",
"Our goal will be to explain the decision $f(\\xi )$ that this model makes for one particular instance $\\xi $ .",
"As soon as $f$ is too complicated, it is hopeless to try and fit an interpretable model globally, since the interpretable model will be too simple to capture all the complexity of $f$ .",
"Thus a reasonable course of action is to consider a local point of view, and to explain $f$ in the neighborhood of some fixed instance $\\xi $ .",
"This is the main idea behind LIME: To explain a decision for some fixed input $\\xi $ , sample other examples around $\\xi $ , use these samples to build a simple interpretable model in the neighborhood of $\\xi $ , and use this surrogate model to explain the decision for $\\xi $ .",
"One additional idea that makes a huge difference with other existing methods is to use discretized features of smaller dimension $d^{\\prime }$ to build the local model.",
"These new categorical features are easier to interpret, since they are categorical.",
"In the case of images, they are built by using a split of the image $\\xi $ into superpixels [7].",
"See Figure REF for an example of LIME output in the case of image classification.",
"In this situation, the surrogate model highlights the superpixels of the image that are the most “active” in predicting a given label.",
"Whereas LIME is most famous for its results on images, it is easier to understand how it operates and to analyze theoretically on tabular data.",
"In the case of tabular data, LIME works essentially in the same way, with a main difference: tabular LIME requires a train set, and each feature is discretized according to the empirical quantiles of this training set.",
"Figure: General setting of TabularLIME along coordinate jj.",
"Given a specific datapoint ξ\\xi (in red), we want to build a local model for ff (in blue), given new samples x 1 ,...,x n x_1,\\ldots ,x_n (in black).Discretizing with respect to the quantiles of the distribution (in green), these new samples are transformed into categorical features z i z_i (in purple).In the construction of the surrogate model, they are weighted with respect to their proximity with ξ\\xi (here exponential weights given by Eq.",
"(), in black).",
"In red, we plotted the tangent line, the best linear approximation one could hope for.We now describe the general operation of LIME on Euclidean data, which we call TabularLIME.",
"We provide synthetic description of TabularLIME in Algorithm REF , and we refer to Figure REF for a depiction of our setting along a given coordinate.",
"Suppose that we want to explain the prediction of the model $f$ at the instance $\\xi $ .",
"TabularLIME has an intricate way to sample points in a local neighborhood of $\\xi $ .",
"First, TabularLIME constructs empirical quantiles of the train set on each dimension, for a given number $p$ of bins.",
"These quantile boxes are then used to construct a discrete representation of the data: if $\\xi _j$ falls between $\\hat{q}_k$ and $\\hat{q}_{k+1}$ , it receives the value $k$ .",
"We now have a discrete version of $\\xi $ , say $(2,3,\\dots )^\\top $ .",
"The next step is to sample discrete examples in $\\lbrace 1,\\ldots ,p\\rbrace ^d$ uniformly at random: for instance, $(1,3,\\ldots )^\\top $ means that TabularLIME sampled an encoding such that the first coordinate falls into the first quantile box, the second coordinate into the third, etc.",
"TabularLIME subsequently un-discretizes these encodings by sampling from a normal distribution truncated to the corresponding quantile boxes, obtaining new examples $x_1,\\ldots ,x_n$ .",
"For example, for sample $(1,3,\\ldots )^\\top $ we now sample the first coordinate from a normal distribution restricted to quantile box $\\#1$ , the second coordinate from quantile box $\\#3$ , etc.",
"This sampling procedure ensures that we have samples in each part of the space.",
"The next step is to convert these sampled points to binary features, indicating for each coordinate if the new example falls into the same quantile box as $\\xi $ .",
"Here, $z_i$ would be $(1,0,\\ldots )^\\top $ .",
"Finally, an interpretable model (say linear) is learned using these binary features.",
"[ht] TabularLIME for regression [1]Model $f$ , $\\#$ of new samples $n$ , instance $\\xi $ , bandwidth $\\nu $ , $\\#$ of bins $p$ , mean $\\mu $ , variance $\\sigma ^2$ $q\\leftarrow $ GetQuantiles($p$ ,$\\mu $ ,$\\sigma $ ) $t \\leftarrow $ Discretize($\\xi $ ,$q$ ) $i=1$ to $n$ $j=1$ to $d$ $y_{i,j}\\leftarrow $ SampleUniform($\\lbrace 1,\\ldots ,p\\rbrace $ ) $(q_\\ell ,q_u)\\leftarrow (q_{j,y_{ij}},q_{j,y_{ij}+1})$ $x_{i,j}\\leftarrow $ SampleTruncGaussian($q_\\ell ,q_u,\\mu ,\\sigma $ ) $z_{i,j}\\leftarrow \\mathbf {1}_{t_j=y_{i,j}}$ $\\pi _i\\leftarrow \\exp \\left(\\frac{-\\left\\Vert x_i-\\xi \\right\\Vert ^2}{2\\nu ^2}\\right)$ $\\widehat{\\beta }\\leftarrow $WeightedLeastSquares($z,f(x),\\pi $ ) $\\widehat{\\beta }$"
],
[
"Implementation choices and notation",
"LIME is a quite general framework and leaves some freedom to the user regarding each brick of the algorithm.",
"We now discuss each step of TabularLIME in more detail, presenting our implementation choices and introducing our notation on the way.",
"Discretization.",
"As said previously, the first step of TabularLIME is to create a partition of the input space using a train set.",
"Intuitively, TabularLIME produces interpretable features by discretizing each dimension.",
"Formally, given a fixed number of bins $p$ , for each feature $j$ , the empirical quantiles $\\hat{q}_{j,0},\\ldots ,\\hat{q}_{j,p}$ are computed.",
"Thus, along each dimension, there is a mapping $\\hat{\\phi }_j:\\operatorname{\\mathbb {R}}\\rightarrow \\lbrace 1,\\ldots ,p\\rbrace $ associating each real number to the index of the quantile box it belongs to.",
"For any point $x\\in \\operatorname{\\mathbb {R}}^d$ , the interpretable features are then defined as a $0-1$ vector corresponding to the discretization of $x$ being the same as the discretization of $\\xi $ .",
"Namely, $z_{j}=\\mathbf {1}_{\\hat{\\phi }_j(x)=\\hat{\\phi }_j(\\xi )}$ for all $1\\le j\\le d$ .",
"Intuitively, these categorical features correspond to the absence or presence of interpretable components.",
"The discretization process makes a huge difference with respect to other methods: we lose the obvious link with the gradient of the function, and it is much more complicated to see how the local properties of $f$ influence the result of the LIME algorithm, even in a simple setting.",
"In all our experiments, we took $p=4$ (quartile discretization, the default setting).",
"Figure: A visualization of the train set in dimension d=2d=2 with μ=(0,0) ⊤ \\mu =(0,0)^\\top , and σ 2 =1\\sigma ^2=1.",
"The empirical quantiles (dashed green lines) are already very close to the theoretical quantiles (green lines) for n train =500n_{\\text{train}}=500.The main difference in the procedure appears if ξ\\xi (red cross) is chosen at the edge of a quantile box, changing the way all the new samples are encoded.But for a train set containing enough observations and a generic ξ\\xi , there is virtually no difference between using the theoretical quantiles and the empirical quantiles.Sampling strategy.",
"Along with $\\hat{\\phi }$ , TabularLIME creates an un-discretization procedure $\\hat{\\psi }:\\lbrace 1,\\ldots ,p\\rbrace \\rightarrow \\operatorname{\\mathbb {R}}$ .",
"Simply put, given a coordinate $j$ and a bin index $k$ , $\\hat{\\psi }_j(k)$ samples a truncated Gaussian on the corresponding bin, with parameters computed from the training set.",
"The TabularLIME sampling strategy for a new example amounts to (i) sample $y_i\\in \\lbrace 1,\\ldots ,p\\rbrace ^d$ a random variable such that the $y_{ij}$ are independent samples of the discrete uniform distribution on $\\lbrace 1,\\ldots ,p\\rbrace $ , and (ii) apply the un-discretization step, that is, return $\\hat{\\psi }(y)$ .",
"We will denote by $x_1,\\ldots ,x_n\\in \\operatorname{\\mathbb {R}}^d$ these new examples, and $z_1,\\ldots ,z_n\\in \\lbrace 0,1\\rbrace ^d$ their discretized counterparts.",
"Note that it is possible to take other bin boxes than those given by the empirical quantiles, the $y_{ij}$ s are then sampled according to the frequency observed in the dataset.",
"The sampling step of TabularLIME helps to explore the values of the function in the neighborhood of the instance to explain.",
"Thus it is not so important to sample according to the distribution of the data, and a Gaussian sampling that mimics it is enough.",
"Assuming that we know the distribution of the train data, it is possible to use the theoretical quantiles instead of the empirical ones.",
"For a large number of examples, they are arbitrary close (see, for instance, Lemma 21.2 in [13]).",
"See Figure REF for an illustration.",
"It is this approach that we will take from now on: we denote the discretization step by $\\phi $ and denote the quantiles by $q_{jk}$ for $1\\le j\\le d$ and $0\\le k\\le p$ to mark this slight difference.",
"Also note that, for every $1\\le j\\le d$ , we set $q_{j\\pm }$ the quantiles bounding $\\xi _j$ , that is, $q_{j-}\\le \\xi _j < q_{j+}$ (see Figure REF )."
],
[
"Train set. ",
"TabularLIME requires a train set, which is left free to the user.",
"In spirit, one should sample according to the distribution of the train set used to fit the model $f$ .",
"Nevertheless, this train set is rarely available, and from now on, we choose to consider draws from a $\\operatorname{\\mathcal {N}}\\left(\\mu ,\\sigma ^2\\mathrm {I}_{d}\\right)$ .",
"The parameters of this Gaussian can be estimated from the training data that was used for $f$ if available.",
"Thus, in our setting, along each dimension $j$ , the $\\left(q_{jk}\\right)_{0\\le k\\le p}$ are the (rescaled) quantiles of the normal distribution.",
"In particular, they are identical for all features.",
"A fundamental consequence is that sampling the new examples $x_i$ s first and then discretizing has the same distribution as sampling first the bin indices $y_{i}$ s and then un-discretizing."
],
[
"Weights. ",
"We choose to give each example the weight $\\pi _i =\\exp \\left(\\frac{-\\left\\Vert x_i-\\xi \\right\\Vert ^2}{2\\nu ^2}\\right)\\, ,$ where $\\left\\Vert \\cdot \\right\\Vert $ is the Euclidean norm on $\\operatorname{\\mathbb {R}}^d$ and $\\nu > 0$ is a bandwidth parameter.",
"It should be clear that $\\nu $ is a hard parameter to tune: [noitemsep,topsep=0pt] if $\\nu $ is very large, then all the examples receive positive weights: we are trying to build a simple model that captures the complexity of $f$ at a global scale.",
"This cannot work if $f$ is too complicated.",
"if $\\nu $ is too small, then only examples in the immediate neighborhood of $\\xi $ receive positive weights.",
"Given the discretization step, this amounts to choosing $z_i=(1,\\ldots ,1)^\\top $ for all $i$ .",
"Thus the linear model built on top would just be a constant fit, missing all the relevant information.",
"Note that other distances than the Euclidean distance can be used, for instance the cosine distance for text data.",
"The default implementation of LIME uses $\\left\\Vert z_i-t\\right\\Vert $ instead of $\\left\\Vert x_i-\\xi \\right\\Vert $ , with bandwidth set to $0.75d$ .",
"We choose to use the true Euclidean distance between $\\xi $ and the new examples as it can be seen as a smoothed version of the distance to $z_i$ and has the same behavior."
],
[
"Interpretable model. ",
"The final step in TabularLIME is to build a local interpretable model.",
"Given a class of simple, interpretable models $G$ , TabularLIME selects the best of these models by solving $\\operatornamewithlimits{arg\\,min}_{g\\in G} \\biggl \\lbrace L_n(f,g,\\pi _{\\xi }) + \\Omega \\left(g\\right)\\biggr \\rbrace \\, ,$ where $L_n$ is a local loss function evaluated on the new examples $x_1,\\ldots ,x_n$ , and $\\Omega :\\operatorname{\\mathbb {R}}^d\\rightarrow \\operatorname{\\mathbb {R}}$ is a regularizer function.",
"For instance, a natural choice for the local loss function is the weighted squared loss $L_n(f,g,\\pi ) =\\frac{1}{n}\\sum _{i=1}^n \\pi _i \\left(f(x_i)-g(z_i)\\right)^2\\, .$ We saw in Section REF different possibilities for $G$ .",
"In this paper, we will focus exclusively on the linear models, in our opinion the easiest models to interpret.",
"Namely, we set $g(z_i) = \\beta ^\\top z_i + \\beta _0$ , with $\\beta \\in \\operatorname{\\mathbb {R}}^d$ and $\\beta _0\\in \\operatorname{\\mathbb {R}}$ .",
"To get rid of the intercept $\\beta _0$ , we now use the standard approach to introduce a phantom coordinate 0, and $z,\\beta \\in \\operatorname{\\mathbb {R}}^{d+1}$ with $z_0 = 1$ and $\\beta _0 = \\beta _0$ .",
"We also stack the $z_i$ s together to obtain $Z\\in \\lbrace 0,1\\rbrace ^{n\\times (d+1)}$ .",
"The regularization term $\\Omega (g)$ is added to insure further interpretability of the model by reducing the number of non-zero coefficients in the linear model given by TabularLIME.",
"Typically, one uses $L^2$ regularization (ridge regression is the default setting of LIME) or $L^1$ regularization (the Lasso).",
"To simplify the analysis, we will set $\\Omega = 0$ in the following.",
"We believe that many of the results of Section stay true in a regularized setting, especially the switch-off phenomenon that we are going to describe below: coefficients are even more likely to be set to zero when $\\Omega \\ne 0$ .",
"In other words, in our case TabularLIME performs weighted linear regression on the interpretable features $z_i$ s, and outputs a vector $\\widehat{\\beta }\\in \\operatorname{\\mathbb {R}}^{d+1}$ such that $\\widehat{\\beta }\\in \\operatornamewithlimits{arg\\,min}_{\\beta \\in \\operatorname{\\mathbb {R}}^{d+1}} \\left\\lbrace \\frac{1}{n} \\sum _{i=1}^n \\pi _i (y_i - \\beta ^\\top z_i)^2 \\right\\rbrace \\, .$ Note that $\\widehat{\\beta }$ is a random quantity, with randomness coming from the sampling of the new examples $x_1,\\ldots ,x_n$ .",
"It is clear that from a theoretical point of view, a big hurdle for the theoretical analysis is the discretization process (going from the $x_i$ s to the $z_i$ s)."
],
[
"Regression vs. classification. ",
"To conclude, let us note that TabularLIME can be used both for regression and classification.",
"Here we focus on the regression mode: the outputs of the model are real numbers, and not discrete elements.",
"In some sense, this is a more general setting than the classification case, since the classification mode operates as TabularLIME for regression, but with $f$ chosen as the function that gives the likelihood of belonging to a certain class according to the model."
],
[
"Related work",
"Let us mention a few other model-agnostic methods that share some characteristics with LIME.",
"We refer to [3] for a thorough review."
],
[
"Shapley values. ",
"Following [9] the idea is to estimate for each subset of features $S$ the expected prediction difference $\\Delta (S)$ when the value of these features are fixed to those of the example to explain.",
"The contribution of the $j$ th feature is then set to an average of the contribution of $j$ over all possible coalitions (subgroups of features not containing $j$ ).",
"They are used in some recent interpretability work, see [5] for instance.",
"It is extremely costly to compute, and does not provide much information as soon as the number of features is high.",
"Shapley values share with LIME the idea of quantifying how much a feature contributes to the prediction for a given example."
],
[
"Gradient methods. ",
"Also related to LIME, gradient-based methods as in [1] provide local explanations without knowledge of the model.",
"Essentially, these methods compute the partial derivatives of $f$ at a given example.",
"For images, this can yield satisfying plots where, for instance, the contours of the object appear: a saliency map [16].",
"[10], [11] propose to use the “input $\\times $ derivative” product, showing advantages over gradient methods.",
"But in any case, the output of these gradient based methods is not so interpretable since the number of features is so high.",
"LIME gets around this problem by using a local dictionary with much smaller dimensionality than the input space."
],
[
"Theoretical value of the coefficients of the surrogate model",
"We are now ready to state our main result.",
"Let us denote by $\\widehat{\\beta }$ the coefficients of the linear surrogate model obtained by TabularLIME.",
"In a nutshell, when the underlying model $f$ is linear, we can derive the average value $\\beta $ of the $\\widehat{\\beta }$ coefficients.",
"In particular, we will see that the $\\beta _j$ s are proportional to the partial derivatives $\\partial _j f(\\xi )$ .",
"The exact form of the proportionality coefficients is given in the formal statement below, it essentially depends on the scaling parameters $\\tilde{\\mu }=\\frac{\\nu ^2\\mu + \\sigma ^2\\xi }{\\nu ^2+\\sigma ^2}\\in \\operatorname{\\mathbb {R}}^d\\,\\text{and}\\,\\, \\tilde{\\sigma }=\\frac{\\nu ^2\\sigma ^2}{\\nu ^2+\\sigma ^2} >0\\, ,$ and the $q_{j\\pm }$ s, the quantiles left and right of the $\\xi _j$ s. Theorem 3.1 (Coefficients of the surrogate model, theoretical values) Assume that $f$ is of the form $x\\mapsto a^\\top x + b$ , and set $\\beta =\\begin{pmatrix}f(\\tilde{\\mu }) + \\sum _{j=1}^d\\frac{a_j\\theta _j}{1-\\alpha _j} \\\\\\frac{-a_1\\theta _1}{\\alpha _1(1-\\alpha _1)} \\\\\\vdots \\\\\\frac{-a_d\\theta _d}{\\alpha _d(1-\\alpha _d)}\\end{pmatrix}\\in \\operatorname{\\mathbb {R}}^{d+1}\\, ,$ where, for any $1\\le j\\le d$ , we defined $\\alpha _j =\\left[\\frac{1}{2}\\mathrm {erf}\\left(\\frac{x-\\tilde{\\mu }_j}{\\tilde{\\sigma }\\sqrt{2}}\\right)\\right]_{q_{j-}}^{q_{j+}}\\, ,$ and $\\theta _j =\\left[ \\frac{\\tilde{\\sigma }}{\\sqrt{2\\pi }}\\exp \\left(\\frac{-(x-\\tilde{\\mu }_j)^2}{2\\tilde{\\sigma }^2}\\right)\\right]_{q_{j-}}^{q_{j+}}\\, .$ Let $\\eta \\in (0,1)$ .",
"Then, with high probability greater than $1-\\eta $ , it holds that $\\left\\Vert \\widehat{\\beta }-\\beta \\right\\Vert \\lesssim \\max (\\sigma \\left\\Vert \\nabla f\\right\\Vert ,f(\\tilde{\\mu }) + \\tilde{\\sigma }\\left\\Vert \\nabla f\\right\\Vert )\\sqrt{\\frac{\\log 1/\\eta }{n}}\\, .$ A precise statement with the accurate dependencies in the dimension and the constants hidden in the result can be found in the Appendix (Theorem REF ).",
"Before discussing the consequences of Theorem REF in the next section, remark that since $\\xi $ is encoded by $(1,1,\\ldots ,1)^\\top $ , the prediction of the local model at $\\xi $ , $\\hat{f}(\\xi )$ , is just the sum of the $\\widehat{\\beta }_j$ s. According to Theorem REF , $\\hat{f}(\\xi )$ will be close to this value, with high probability.",
"Thus we also have a statement about the error made by the surrogate model in $\\xi $ .",
"Corollary 3.1 (Local error of the surrogate model) Let $\\eta \\in (0,1)$ .",
"Then, under the assumptions of Theorem REF , with probability greater than $1-\\eta $ , it holds that $&\\left|\\hat{f}(\\xi ) - f(\\tilde{\\mu }) +\\sum _{j=1}^d\\frac{a_j\\theta _j}{\\alpha _j}\\right| \\le \\\\&\\phantom{blabla}\\le \\max (\\sigma \\left\\Vert \\nabla f\\right\\Vert ,f(\\tilde{\\mu }) + \\tilde{\\sigma }\\left\\Vert \\nabla f\\right\\Vert )\\sqrt{\\frac{\\log 1/\\eta }{n}}\\, ,$ with hidden constants depending on $d$ and the $\\alpha _j$ s. Obviously the goal of TabularLIME is not to produce a very accurate model, but to provide interpretability.",
"The error of the local model can be seen as a hint about how reliable the interpretation might be.",
"Figure: Example where the true underlying black box model only depends on two features: f(x)=10x 1 -10x 2 f(x)=10x_1-10x_2.",
"For each of the 10 features, we plot the values of the β ^ j \\widehat{\\beta }_js obtained by TabularLIME.The blue line shows the median over all experiments, the red cross the β j \\beta _j theoretical value according to our theorem.",
"The boxplots contain values between first and third quartiles, the whiskers are 1.51.5 times the interquartile ranges, and the black dots mark values outside this range.To produce the figure, we made 20 repetitions of the experiment, with n=10 4 n=10^4 examples and ν=1\\nu =1.We see that TabularLIME finds nonzero coefficients exactly for the first two coordinates, up to noise coming from the sampling.This is the result that one would hope to achieve, and also the result predicted by our theory."
],
[
"Consequences of our main results",
"We now discuss the consequences of Theorem REF and Corollary REF ."
],
[
"Dependency in the partial derivatives. ",
"A first consequence of Theorem REF is that the coefficients of the linear model given by TabularLIME are approximately proportional to the partial derivatives of $f$ at $\\xi $ , with constant depending on our assumptions.",
"An interesting follow-up is that, if $f$ depends only on a few features, then the partial derivatives in the other coordinates are zero, and the coefficients given by TabularLIME for these coordinates will be 0 as well.",
"For instance, if $f(x)=10x_1-10x_2$ as in Figure REF , then $\\beta _1\\simeq 11.4$ , $\\beta _2\\simeq -4.1$ , and $\\beta _j=0$ for all $j\\ge 3$ .",
"In a simple setting, we thus showed that TabularLIME does not produce interpretations with additional erroneous feature dependencies.",
"Indeed, when the number of samples is high, the coordinates which do not influence the prediction will have a coefficient close to the theoretical value 0 in the surrogate linear model.",
"For a bandwidth not too large, this dependency in the partial derivatives seems to hold to some extent for more general functions.",
"See for instance Figure REF , where we demonstrate this phenomenon for a kernel regressor.",
"Figure: Values of the coefficients obtained by TabularLIME on each coordinate in dimension d=13d=13 for a linear model trained on the Boston housing dataset .The β j \\beta _js are concentrated around the red crosses, which denote the β j \\beta _js, the theoretical values predicted by Theorem .To produce the figure, we ran 20 experiments with n=10 3 n=10^3 new samples generated for each run and we set ν=1\\nu =1.Robustness of the explanations.",
"Theorem REF means that, for large $n$ , TabularLIME outputs coefficients that are very close to $\\beta $ with high probability, where $\\beta $ is a vector that can be computed explicitly as per Eq.",
"(REF ).",
"Still without looking too closely at the values of $\\beta $ , this is already interesting and hints that there is some robustness in the interpretations provided by TabularLIME: given enough samples, the explanation will not jump from one feature to the other.",
"This is a desirable property for any interpretable method, since the user does not want explanations to change randomly with different runs of the algorithm.",
"We illustrate this phenomenon in Figure REF .",
"Figure: Values of the coefficients obtained by TabularLIME on each coordinate.",
"We used the same settings as in Figure , but this time we train a kernel ridge regressor on the Boston Housing dataset—a nonlinear function.",
"For the ridge regression, we used the Gaussian kernel with scale parameter set to 5 and default regularization constant (α=1\\alpha =1).We then estimated the partial derivatives of ff at ξ\\xi and reported the corresponding β j \\beta _js in red.For the chosen bandwidth (we took ν=1\\nu =1), the experiments seem to roughly agree with our theory."
],
[
"Influence of the bandwidth. ",
"Unfortunately, Theorem REF does not provide directly a founded way to pick $\\nu $ , which would for instance minimize the variance for a given level of noise.",
"The quest for a founded heuristic is still open.",
"However, we gain some interesting insights on the role of $\\nu $ .",
"Namely, for fixed $\\xi $ , $\\mu $ , and $\\sigma $ , the multiplicative constants $\\theta _j/(\\alpha _j(1-\\alpha _j))$ appearing in Eq.",
"(REF ) depend essentially on $\\nu $ .",
"Without looking too much into these constants, one can already see that they regulate the magnitude of the coefficients of the surrogate model in a non-trivial way.",
"For instance, in the experiment depicted in Figure REF , the partial derivative of $f$ along the two first coordinate has the same magnitude, whereas the interpretable coefficient is much larger for the first coordinate than the second.",
"Thus we believe that the value of the coefficients in the obtained linear model should not be taken too much into account.",
"More disturbing, it is possible to artificially (or by accident) put $\\theta _j$ to zero, therefore forgetting about feature $j$ in the explanation, whereas it could play an important role in the prediction.",
"To see why, we have to return to the definition of the $\\theta _j$ s: since $q_{j-}<q_{j+}$ by construction, to have $\\theta _j=0$ is possible only if $V_{\\text{crit}}=\\sigma ^2 \\frac{2\\xi _j - q_{j-}-q_{j+}}{-2\\mu _j+q_{j-}+q_{j+}} > 0\\, ,$ and $\\nu ^2$ is set to $V_{\\text{crit}}$ .",
"We demonstrate this switching-off phenomenon in Figure REF .",
"An interesting take is that $\\nu $ not only decides at which scale the explanation is made, but also the magnitude of the coefficients in the interpretable model, even for small changes of $\\nu $ .",
"Figure: Values of the coefficients given by LIME.",
"In this experiment, we took exactly the same setting as in Figure , but this time set the bandwidth to ν=0.53\\nu =0.53 instead of 1.In that case, the second feature is switched-off by TabularLIME.Note that it is not the case that ν\\nu is too small and that we are in a degenerated case: TabularLIME still puts a nonzero coefficient on the first coordinate."
],
[
"Error of the surrogate model. ",
"A simple consequence of Corollary REF is that, unless some cancellation happens between in the term $f(\\tilde{\\mu })-\\sum _j \\frac{a_j\\theta _j}{\\alpha _j}$ , the local error of the surrogate model is bounded away from zero.",
"For instance, as soon as $\\tilde{\\mu }\\ne \\mu $ , it is the general situation.",
"Therefore, the surrogate model produced by TabularLIME is not accurate in general.",
"We show some experimental results in Figure REF .",
"Finally, we discuss briefly the limitations of Theorem REF ."
],
[
"Linearity of $f$ . ",
"The linearity of $f$ is a quite restrictive assumption, but we think that it is useful to consider for two reasons.",
"First, the weighted nature of the procedure means that TabularLIME is not considering examples that are too far away from $\\xi $ with respect to the scaling parameter $\\nu $ .",
"Thus it is truly a local assumption on $f$ , that could be replaced by a boundedness assumption on the Hessian of $f$ in the neighborhood of $\\xi $ , at the price of more technicalities and assuming that $\\nu $ is not too large.",
"See, in particular, Lemma REF in the Appendix, after which we discuss an extension of the proof when $f$ is linear with a second degree perturbative term.",
"We show in Figure REF how our theoretical predictions behave for a non-linear function (a kernel ridge regressor).",
"Second, our main concern is to know whether TabularLIME operates correctly in a simple setting, and not to provide bounds for the most general $f$ possible.",
"Indeed, if we can already show imperfect behavior for TabularLIME when $f$ is linear as seen earlier, our guess is that such behavior will only worsen for more complicated $f$ .",
"Figure: Histogram of the errors f ^(ξ)-f(ξ)\\hat{f}(\\xi )-f(\\xi ).The setting is the same as in Figure , but we repeated the experiment 100 times.The red double arrow marks the value given by Corollary around which the local error concentrate.With high probability, the error of the surrogate model is bounded away from 0."
],
[
"Sampling strategy. ",
"In our derivation, we use the theoretical quantiles of the Gaussian distribution along each axis, and not prescribed quantiles.",
"We believe that the proof could eventually be adapted, but that the result would loose in clarity.",
"Indeed, the computations for a truncated Gaussian distribution are far more convoluted than for a Gaussian distribution.",
"For instance, in the proof of Lemma REF in the Appendix, some complicated quantities depending on the prescribed quantiles would appear when computing $\\mathbb {E}\\left[\\pi _i z_{ik}\\right]$ ."
],
[
"Proof of Theorem ",
"In this section, we explain how Theorem REF is obtained.",
"All formal statements and proofs are in the Appendix."
],
[
"Outline. ",
"The main idea underlying the proof is to realize that $\\widehat{\\beta }$ is the solution of a weighted least squares problem.",
"Denote by $\\Pi \\in \\operatorname{\\mathbb {R}}^{n\\times n}$ the diagonal matrix such that $\\Pi _{ii}=\\pi _i$ (the weight matrix), and set $f(x)\\in \\operatorname{\\mathbb {R}}^{d+1}$ the response vector.",
"Then, taking the gradient of Eq.",
"(REF ), one obtains the key equation $(Z^\\top \\Pi Z)\\widehat{\\beta }= Z^\\top \\Pi f(x)\\, .$ Let us define $\\widehat{\\Sigma }=\\frac{1}{n}Z^\\top \\Pi Z$ and $\\widehat{\\Gamma }=\\frac{1}{n}Z^\\top \\Pi f(x)$ , as well as their population counterparts $\\Sigma =\\mathbb {E}[\\widehat{\\Sigma }]$ and $\\Gamma =\\mathbb {E}[\\widehat{\\Gamma }]$ .",
"Intuitively, if we can show that $\\widehat{\\Sigma }$ and $\\widehat{\\Gamma }$ are close to $\\Sigma $ and $\\Gamma $ , assuming that $\\Sigma $ is invertible, then we can show that $\\widehat{\\beta }$ is close to $\\beta =\\Sigma ^{-1}\\Gamma $ .",
"The main difficulties in the proof come from the non-linear nature of the new features $z_i$ , introducing tractable but challenging integrals.",
"Fortunately, the Gaussian sampling of LIME allows us to overcome these challenges (at the price of heavy computations)."
],
[
"Covariance matrix. ",
"The first part of our analysis is thus concerned with the study of the empirical covariance matrix $\\widehat{\\Sigma }$ .",
"Perhaps surprisingly, it is possible to compute the population version of $\\widehat{\\Sigma }$ : $\\Sigma = C_d\\begin{pmatrix}1 & \\alpha _1 & \\cdots & \\alpha _d\\\\\\alpha _1 & \\alpha _1 & & \\alpha _i\\alpha _j \\\\\\vdots & & \\ddots & \\\\\\alpha _d& \\alpha _i\\alpha _j & & \\alpha _d\\end{pmatrix}\\, ,$ where the $\\alpha _j$ s were defined in Section , and $C_d$ is a scaling constant that does not appear in the final result (see Lemma REF ).",
"Since the $\\alpha _j$ s are always distinct from 0 and 1, the special structure of $\\Sigma $ makes it possible to invert it in closed-form.",
"We show in Lemma REF that $C_d^{-1}\\!\\begin{pmatrix}1+\\sum _{j=1}^d\\frac{\\alpha _j}{1-\\alpha _j} & \\frac{-1}{1-\\alpha _1} & \\cdots & \\frac{-1}{1-\\alpha _d} \\\\\\frac{-1}{1-\\alpha _1} & \\frac{1}{\\alpha _1(1-\\alpha _1)} & & 0 \\\\\\vdots & & \\ddots & \\\\\\frac{-1}{1-\\alpha _d} & 0 & & \\frac{1}{\\alpha _d(1-\\alpha _d)}\\end{pmatrix}\\, .$ We then achieve control of $\\left\\Vert \\widehat{\\Sigma }^{-1}- \\Sigma ^{-1}\\right\\Vert _{\\text{op}}$ via standard concentration inequalities, since the new samples are Gaussian and the binary features are bounded (see Proposition REF )."
],
[
"Right-hand side of Eq. (",
"Again, despite the non-linear nature of the new features, it is possible to compute the expected version of $\\widehat{\\Gamma }$ in our setting.",
"In this case, we show in Lemma REF that $\\Gamma = C_d\\begin{pmatrix}f(\\tilde{\\mu }) \\\\\\alpha _1 f(\\tilde{\\mu }) - a_1 \\theta _1 \\\\\\vdots \\\\\\alpha _df(\\tilde{\\mu }) - a_d\\theta _d\\end{pmatrix}\\, ,$ where the $\\theta _j$ s were defined in Section .",
"They play an analogous role to the $\\alpha _j$ s but, as noted before, they are signed quantities.",
"As with the analysis of the covariance matrix, since the weights and the new features are bounded, it is possible to show a concentration result for $\\widehat{\\Gamma }$ (see Lemma REF )."
],
[
"Concluding the proof. ",
"We can now conclude, first upper bounding $\\left\\Vert \\widehat{\\beta }- \\Sigma ^{-1}\\Gamma \\right\\Vert $ by $\\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}}\\left\\Vert \\widehat{\\Gamma }- \\Gamma \\right\\Vert + \\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}}\\left\\Vert \\Gamma \\right\\Vert \\, ,$ and then controlling each of these terms using the previous concentration results.",
"The expression of $\\beta $ is simply obtained by multiplying $\\Sigma ^{-1}$ and $\\Gamma $ ."
],
[
"Conclusion and future directions",
"In this paper we provide the first theoretical analysis of LIME, with some good news (LIME discovers interesting features) and bad news (LIME might forget some important features and the surrogate model is not faithful).",
"All our theoretical results are verified by simulations.",
"For future work, we would like to complement these results in various directions: Our main goal is to extend the current proof to any function by replacing $f$ by its Taylor expansion at $\\xi $ .",
"On a more technical side, we would like to extend our proof to other distance functions (e.g., distances between the $z_i$ s and $\\xi $ , which is the default setting of LIME), to non-isotropic sampling of the $x_i$ s (that is, $\\sigma $ not constant across the dimensions), and to ridge regression."
],
[
"Acknowledgements",
"The authors would like to thank Christophe Biernacki for getting them interested in the topic, as well as Leena Chennuru Vankadara for her careful proofreading.",
"This work has been supported by the German Research Foundation through the Institutional Strategy of the University of Tübingen (DFG, ZUK 63), the Cluster of Excellence “Machine Learning—New Perspectives for Science” (EXC 2064/1 number 390727645), and the BMBF Tuebingen AI Center (FKZ: 01IS18039A).",
"Supplementary material for: Explaining the Explainer: A First Theoretical Analysis of LIME In this supplementary material, we provide the proof of Theorem REF of the main paper.",
"It is a simplified version of Theorem REF .",
"We first recall our setting in Section .",
"Then, following Section of the main paper, we study the covariance matrix in Section , and the right-hand side of the key equation (REF ) in Section .",
"Finally, we state and prove Theorem REF in Section .",
"Some technical results (mainly Gaussian integrals computation) and external concentration results are collected in Section ."
],
[
"Setting",
"Let us recall briefly the main assumptions under which we prove Theorem REF .",
"Recall that they are discussed in details in Section REF of the main paper.",
"H 1 (Linear $f$ ) The black-box model can be written $a^\\top x + b$ , with $a\\in \\operatorname{\\mathbb {R}}^d$ and $b\\in \\operatorname{\\mathbb {R}}$ fixed.",
"H 2 (Gaussian sampling) The random variables $x_1,\\ldots ,x_n$ are i.i.d.",
"$\\operatorname{\\mathcal {N}}\\left(\\mu ,\\sigma ^2\\mathrm {I}_d\\right)$ .",
"Also recall that, for any $1\\le i\\le n$ , we set the weights to $\\pi _i =\\exp \\left(\\frac{-\\left\\Vert x_i-\\xi \\right\\Vert ^2}{2\\nu ^2}\\right)\\, .$ We will need the following scaling constant: $C_d=\\left(\\frac{\\nu ^2}{\\nu ^2+\\sigma ^2}\\right)^{d/2} \\cdot \\exp \\left(\\frac{-\\left\\Vert \\xi -\\mu \\right\\Vert ^2}{2(\\nu ^2+\\sigma ^2)}\\right)\\, ,$ which does not play any role in the final result.",
"One can check that $C_d\\rightarrow 1$ when $\\nu \\gg \\sigma $ , regardless of the dimension.",
"Finally, for any $1\\le j\\le d$ , recall that we defined $\\alpha _j =\\left[\\frac{1}{2}\\mathrm {erf}\\left(\\frac{x-\\tilde{\\mu }_j}{\\tilde{\\sigma }\\sqrt{2}}\\right)\\right]_{q_{j-}}^{q_{j+}}\\, ,$ and $\\theta _j =\\left[ \\frac{\\tilde{\\sigma }}{\\sqrt{2\\pi }}\\exp \\left(\\frac{-(x-\\tilde{\\mu }_j)^2}{2\\tilde{\\sigma }^2}\\right)\\right]_{q_{j-}}^{q_{j+}}\\, ,$ where $q_{j\\pm }$ are the quantile boundaries of $\\xi _j$ .",
"These coefficients are discussed in Section of the main paper.",
"Note that all the expected values are taken with respect to the randomness on the $x_1,\\ldots ,x_n$ ."
],
[
"Covariance matrix",
"In this section, we state and prove the intermediate results used to control the covariance matrix $\\widehat{\\Sigma }$ .",
"The goal of this section is to obtain the control of $\\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}}$ in probability.",
"Intuitively, if this quantity is small enough, then we can inverse Eq.",
"(REF ) and make very precise statements about $\\widehat{\\beta }$ .",
"We first show that it is possible to compute the expected covariance matrix in closed form.",
"Without this result, a concentration result would still hold, but it would be much harder to gain precise insights on the $\\beta _j$ s. Lemma 8.1 (Expected covariance matrix) Under Assumption REF , the expected value of $\\widehat{\\Sigma }$ is given by $\\Sigma =C_d\\begin{pmatrix}1 & \\alpha _1 & \\cdots & \\alpha _d\\\\\\alpha _1 & \\alpha _1 & & \\alpha _i\\alpha _j \\\\\\vdots & & \\ddots & \\\\\\alpha _d& \\alpha _i\\alpha _j & & \\alpha _d\\end{pmatrix}\\, .$ Elementary computations yield $\\widehat{\\Sigma }= \\frac{1}{n}\\begin{pmatrix}\\sum _{i=1}^n \\pi _i & \\sum _{i=1}^n \\pi _i z_{i1} & \\cdots & \\sum _{i=1}^n \\pi _i z_{id} \\\\\\sum _{i=1}^n \\pi _i z_{i1} & \\sum _{i=1}^n \\pi _i z_{i1} & & \\sum _{i=1}^n \\pi _i z_{ik}z_{i\\ell } \\\\\\vdots & & \\ddots & \\\\\\sum _{i=1}^n \\pi _i z_{id} & \\sum _{i=1}^n \\pi _i z_{ik}z_{i\\ell } & & \\sum _{i=1}^n \\pi _i z_{id}\\end{pmatrix}\\, .$ Reading the coefficients of this matrix, we have essentially three computations to complete: $\\mathbb {E}\\left[\\pi _i\\right]$ , $\\mathbb {E}\\left[\\pi _i z_{ik}\\right]$ , and $\\mathbb {E}\\left[\\pi _i z_{ik}z_{i\\ell }\\right]$ ."
],
[
"Computation of $\\mathbb {E}\\left[\\pi _i\\right]$ . ",
"Since the $x_i$ s are Gaussian (Assumption REF ) and using the definition of the weights (Eq.",
"(REF )), we can write $\\mathbb {E}\\left[\\pi _i\\right] = \\int _{\\operatorname{\\mathbb {R}}^d} \\exp \\left(\\frac{-\\left\\Vert x_i-\\xi \\right\\Vert ^2}{2\\nu ^2}\\right)\\exp \\left(\\frac{-\\left\\Vert x_i-\\mu \\right\\Vert ^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x_{i1}\\cdots x_{id}}{(2\\pi \\sigma ^2)^{d/2}}\\, .$ By independence across coordinates, the last display amounts to $\\prod _{j=1}^d\\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi _j)^2}{2\\nu ^2}+ \\frac{-(x-\\mu _j)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\, .$ We then apply Lemma REF to each of the integrals within the product to obtain $\\prod _{j=1}^d\\frac{\\nu }{\\sqrt{\\nu ^2+\\sigma ^2}} \\cdot \\exp \\left(\\frac{-(\\xi _j-\\mu _j)^2}{2(\\nu ^2+\\sigma ^2)}\\right) = \\frac{\\nu ^d}{(\\nu ^2+\\sigma ^2)^{d/ 2}} \\cdot \\exp \\left(\\frac{-\\left\\Vert \\xi -\\mu \\right\\Vert ^2}{2(\\nu ^2+\\sigma ^2)}\\right)\\, .$ We recognize the definition of the scaling constant (Eq.",
"(REF )): we have proved that $\\mathbb {E}\\left[\\pi _i\\right] = C_d$ ."
],
[
"Computation of $\\mathbb {E}\\left[\\pi _i z_{ik}\\right]$ . ",
"Since the $x_i$ s are Gaussian (Assumption REF ) and using the definition of the weights (Eq.",
"(REF )), $\\mathbb {E}\\left[\\pi _i\\right] = \\int _{\\operatorname{\\mathbb {R}}^d} \\exp \\left(\\frac{-\\left\\Vert x_i-\\xi \\right\\Vert ^2}{2\\nu ^2}\\right)\\exp \\left(\\frac{-\\left\\Vert x_i-\\mu \\right\\Vert ^2}{2\\sigma ^2}\\right)\\mathbf {1}_{\\phi (x_i)_k=\\phi (\\xi )_k} \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x_{i1} \\cdots x_{id}}{(2\\pi \\sigma ^2)^{d/2}}\\, .$ By independence across coordinates, the last display amounts to $\\int _{q_{k-}}^{q_{k+}} \\exp \\left(\\frac{-(x-\\xi _k)^2}{2\\nu ^2}+ \\frac{-(x-\\mu _k)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} \\cdot \\prod _{\\begin{array}{c}j=1\\\\j\\ne k\\end{array}}^d\\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi _j)^2}{2\\nu ^2}+ \\frac{-(x-\\mu _j)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\, .$ Using Lemma REF , we obtain $\\frac{\\nu ^d}{(\\nu ^2+\\sigma ^2)^{d/ 2}} \\cdot \\exp \\left(\\frac{-\\left\\Vert \\xi -\\mu \\right\\Vert ^2}{2(\\nu ^2+\\sigma ^2)}\\right) \\cdot \\left[\\frac{1}{2}\\mathrm {erf}\\left(\\frac{\\nu ^2 (x-\\mu _k) + \\sigma ^2 (x-\\xi _k)}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\right)\\right]_{q_{k-}}^{q_{k+}}\\, .$ We recognize the definition of the scaling constant (Eq.",
"(REF )) and of the $\\alpha _k$ coefficient (Eq.",
"(REF )): we have proved that $\\mathbb {E}\\left[\\pi _i z_{ik}\\right] = C_d\\alpha _k$ ."
],
[
"Computation of $\\mathbb {E}\\left[\\pi _i z_{ik}z_{i\\ell }\\right]$ . ",
"Since the $x_i$ s are Gaussian (Assumption REF ) and using the definition of the weights (Eq.",
"(REF )), $\\mathbb {E}\\left[\\pi _iz_{ik}z_{i\\ell }\\right] = \\int _{\\operatorname{\\mathbb {R}}^d} \\exp \\left(\\frac{-\\left\\Vert x_i-\\xi \\right\\Vert ^2}{2\\nu ^2}\\right)\\exp \\left(\\frac{-\\left\\Vert x_i-\\mu \\right\\Vert ^2}{2\\sigma ^2}\\right)\\mathbf {1}_{\\phi (x_i)_k=\\phi (\\xi )_k}\\mathbf {1}_{\\phi (x_i)_\\ell =\\phi (\\xi )_\\ell } \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x_{i1} \\cdots \\mathop {}\\mathopen {}\\mathrm {d}x_{id}}{(2\\pi \\sigma ^2)^{d/2}}\\, .$ By independence across coordinates, the last display amounts to $\\prod _{\\begin{array}{c}j=1\\\\j\\ne k,\\ell \\end{array}}^d\\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi _j)^2}{2\\nu ^2}+ \\frac{-(x-\\mu _j)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} & \\cdot \\int _{q_{k-}}^{q_{k+}} \\exp \\left(\\frac{-(x-\\xi _k)^2}{2\\nu ^2}+ \\frac{-(x-\\mu _k)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} \\\\ &\\cdot \\int _{q_{\\ell -}}^{q_{\\ell +}} \\exp \\left(\\frac{-(x-\\xi _\\ell )^2}{2\\nu ^2}+ \\frac{-(x-\\mu _\\ell )^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\, .$ Using Lemma REF , we obtain $\\frac{\\nu ^d}{(\\nu ^2+\\sigma ^2)^{d/ 2}} \\cdot \\exp \\left(\\frac{-\\left\\Vert \\xi -\\mu \\right\\Vert ^2}{2(\\nu ^2+\\sigma ^2)}\\right) & \\cdot \\left[\\frac{1}{2}\\mathrm {erf}\\left(\\frac{\\nu ^2 (x-\\mu _k) + \\sigma ^2 (x-\\xi _k)}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\right)\\right]_{q_{k-}}^{q_{k+}} \\\\&\\cdot \\left[\\frac{1}{2}\\mathrm {erf}\\left(\\frac{\\nu ^2 (x-\\mu _\\ell ) + \\sigma ^2 (x-\\xi _\\ell )}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\right)\\right]_{q_{\\ell -}}^{q_{\\ell +}}\\, .$ We recognize the definition of the scaling constant (Eq.",
"(REF )) and of the alphas (Eq.",
"(REF )): we have proved that $\\mathbb {E}\\left[\\pi _i z_{ik}z_{i\\ell }\\right] = C_d\\alpha _k\\alpha _{\\ell }$ .",
"As it turns out, we show that it is possible to invert $\\Sigma $ in closed-form, therefore simplifying tremendously our quest for control of $\\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}}$ .",
"Indeed, in most cases, even if concentration could be shown, one would not have a precise idea of the coefficients of $\\Sigma ^{-1}$ .",
"Lemma 8.2 (Inverse of the covariance matrix) If $\\alpha _j\\ne 0,1$ for any $j\\in \\lbrace 1,\\ldots ,d\\rbrace $ , then $\\Sigma $ is invertible, and $\\Sigma ^{-1}= C_d^{-1}\\begin{pmatrix}1+\\sum _{j=1}^d\\frac{\\alpha _j}{1-\\alpha _j} & \\frac{-1}{1-\\alpha _1} & \\cdots & \\frac{-1}{1-\\alpha _d} \\\\\\frac{-1}{1-\\alpha _1} & \\frac{1}{\\alpha _1(1-\\alpha _1)} & & 0 \\\\\\vdots & & \\ddots & \\\\\\frac{-1}{1-\\alpha _d} & 0 & & \\frac{1}{\\alpha _d(1-\\alpha _d)}\\end{pmatrix}\\, .$ Define $\\alpha \\in \\operatorname{\\mathbb {R}}^{d}$ the vector of the $\\alpha _j$ s. Set $A=1$ , $B=\\alpha ^\\top $ , $C=\\alpha $ , and $D =\\begin{pmatrix}\\alpha _1 & & \\alpha _j\\alpha _k \\\\& \\ddots & \\\\\\alpha _j\\alpha _k & & \\alpha _d\\end{pmatrix}\\, .$ Then $\\Sigma $ is a block matrix that can be written $\\Sigma = C_d\\begin{bmatrix} A & B \\\\ C & D\\end{bmatrix}$ .",
"We notice that $D-CA^{-1}B = \\mathrm {Diag}\\left(\\alpha _1(1-\\alpha _1),\\ldots ,\\alpha _d(1-\\alpha _d)\\right)\\, .$ Note that, since $\\mathrm {erf}$ is an increasing function, the $\\alpha _j$ s are always distinct from 0 and 1.",
"Thus $D-CA^{-1}B$ is an invertible matrix, and we can use the block matrix inversion formula to obtain the claimed result.",
"As a direct consequence of the computation of $\\Sigma ^{-1}$ , we can control its largest eigenvalue.",
"Lemma 8.3 (Control of $\\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\text{op}}$ ) We have the following bound on the operator norm of the inverse covariance matrix: $\\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\le \\frac{3dA_{d}}{C_d}\\, ,$ where $A_{d}=\\max _{1\\le j\\le d} \\frac{1}{\\alpha _j(1-\\alpha _j)}$ .",
"We control the operator norm of $\\Sigma ^{-1}$ by its Frobenius norm: Namely, $\\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\text{op}}^2 &\\le \\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\mathrm {F}}^2 \\\\&= C_d^{-2} \\left[\\left(1+\\sum \\frac{\\alpha _j}{1-\\alpha _j}\\right)^2 + \\sum \\frac{1}{(1-\\alpha _j)^2} + \\sum \\frac{1}{\\alpha _j(1-\\alpha _j)}\\right] \\\\\\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\text{op}}^{2} &\\le 6 C_d^{-2} d^{2} \\left(\\max \\frac{1}{\\alpha _j(1-\\alpha _j)}\\right)^{2}\\, ,$ where we used $\\alpha _j\\in (0,1)$ in the last step of the derivation.",
"Remark 8.1 Better bounds can without doubt be obtained.",
"A step in this direction is to notice that $S=C_d\\Sigma ^{-1}$ is an arrowhead matrix [6].",
"Thus the eigenvalues of $S$ are solutions of the secular equation $1+\\sum _{j=1}^d\\frac{\\alpha _j}{1-\\alpha _j}-\\lambda +\\sum _{j=1}^d\\frac{\\alpha _j}{(1-\\alpha _j)(1-\\lambda \\alpha _j (1-\\alpha _j))} = 0\\, .$ Further study of this equation could yield an improved statement for Lemma REF .",
"We now show that the empirical covariance matrix concentrates around $\\Sigma $ .",
"It is interesting to see that the non-linear nature of the new coordinates (the $z_{ij}$ s) calls for complicated computations but allows us to use simple concentration tools since they are, in essence, Bernoulli random variables.",
"Lemma 8.4 (Concentration of the empirical covariance matrix) Let $\\widehat{\\Sigma }$ and $\\Sigma $ be defined as before.",
"Then, for every $t>0$ , $\\operatorname{\\mathbb {P}}\\left(\\left\\Vert \\widehat{\\Sigma }- \\Sigma \\right\\Vert _{\\text{op}} \\ge t\\right) \\le 4d^2 \\exp (-2nt^2)\\, .$ Recall that $\\left\\Vert \\cdot \\right\\Vert _{\\text{op}}\\le \\left\\Vert \\cdot \\right\\Vert _{\\mathrm {F}}$ : it suffices to show the result for the Frobenius norm.",
"Next, we notice that the summands appearing in the entries of $\\widehat{\\Sigma }$ , $X_i^{(1)}=\\pi _i$ , $X_i^{(2,k)}=\\pi _i z_{ik}$ , and $X_i^{(3,k,\\ell )}=\\pi _i z_{ik}z_{i\\ell }$ , are all bounded.",
"Indeed, by the definition of the weights and the definition of the new features, they all take values in $[0,1]$ .",
"Moreover, for given $k,\\ell $ , they are independent random variables.",
"Thus we can apply Hoeffding's inequality (Theorem REF ) to $X_i^{(1)}$ , $X_i^{(2,k)}$ , and $X_i^{(3,k,\\ell )}$ .",
"For any given $t >0$ , we obtain ${\\left\\lbrace \\begin{array}{ll}\\operatorname{\\mathbb {P}}\\left(\\left|\\frac{1}{n}\\sum _{i=1}^n (\\pi _i-\\mathbb {E}\\left[\\pi _i\\right])\\right| \\ge t\\right) \\le 2\\exp (-2nt^2) \\\\\\operatorname{\\mathbb {P}}\\left(\\left|\\frac{1}{n}\\sum _{i=1}^n (\\pi _i z_{ik}-\\mathbb {E}\\left[\\pi _i\\right])\\right| \\ge t\\right) \\le 2\\exp (-2nt^2) \\\\\\operatorname{\\mathbb {P}}\\left(\\left|\\frac{1}{n}\\sum _{i=1}^n (\\pi _i z_{ik}z_{i\\ell }-\\mathbb {E}\\left[\\pi _i\\right])\\right| \\ge t\\right) \\le 2\\exp (-2nt^2)\\end{array}\\right.",
"}$ We conclude by a union bound on the $(d+1)^2\\le 2d^2$ entries of the matrix.",
"As a consequence of the two preceding lemmas, we can control the largest eigenvalue of $\\Sigma ^{-1}$ .",
"Lemma 8.5 (Control of $\\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}}$ ) For every $t\\in \\left(0,\\frac{C_d}{6dA_{d}}\\right]$ , with probability greater than $1-4d^2 \\exp (-2nt^2)$ , $\\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}} \\le \\frac{6dA_{d}}{C_d}\\, .$ Let $t \\in (0,C_d/(6dA_{d})]$ .",
"According to Lemma REF , $\\lambda _{\\max }(\\Sigma ^{-1}) \\le 3dA_{d}/C_d$ .",
"We deduce that $\\lambda _{\\min } (\\Sigma ) \\ge \\frac{C_d}{3dA_{d}} \\, .$ Now let us use Lemma REF with this $t$ : there is an event $\\Omega $ , which has probability greater than $1-4d^2\\exp (-2nt^2)$ , such that $\\left\\Vert \\widehat{\\Sigma }-\\Sigma \\right\\Vert _{\\text{op}} \\le t$ .",
"According to Weyl's inequality [15], on this event, $\\left|\\lambda _{\\min }(\\widehat{\\Sigma }) - \\lambda _{\\min }(\\Sigma )\\right| \\le \\left\\Vert \\widehat{\\Sigma }-\\Sigma \\right\\Vert _{\\text{op}} \\le t\\, .$ In particular, $\\lambda _{\\min }(\\widehat{\\Sigma }) \\ge \\lambda _{\\min }(\\Sigma ) - t \\ge \\frac{C_d}{6dA_{d}}\\, .$ Finally, we deduce that $\\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}} \\le \\frac{6dA_{d}}{C_d}\\, .$ We can now state and prove the main result of this section, controlling the operator norm of $\\widehat{\\Sigma }- \\Sigma $ with high probability.",
"Proposition 8.1 (Control of $\\left\\Vert \\widehat{\\Sigma }^{-1}- \\Sigma ^{-1}\\right\\Vert _{\\text{op}}$ ) For every $t\\in \\left(0,\\frac{3dA_{d}}{C_d}\\right]$ , we have $\\operatorname{\\mathbb {P}}\\left(\\left\\Vert \\widehat{\\Sigma }^{-1}- \\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\ge t\\right) \\le 8d^2 \\exp \\left(\\frac{-C_d^4 nt^2}{162d^4A_{d}^4}\\right)\\, .$ Remark 8.2 Proposition REF is the key tool to invert Eq.",
"(REF ) and gain precise control over $\\widehat{\\beta }$ .",
"In the regime that we consider, the dimension $d$ as well as the number of bins $p$ are fixed, and $d, C_d$ , and $A_{d}$ are essentially numerical constants.",
"We did not optimize these constant with respect to $d$ , since the main message is to consider the behavior for a large number of new examples ($n\\rightarrow +\\infty $ ).",
"We notice that, assuming that $\\widehat{\\Sigma }$ is invertible, $\\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}= \\widehat{\\Sigma }^{-1}(\\Sigma - \\widehat{\\Sigma })\\Sigma ^{-1}$ .",
"Since $\\left\\Vert \\cdot \\right\\Vert _{\\text{op}}$ is sub-multiplicative, we just have to control each term individually.",
"Lemma REF gives us $\\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\le \\frac{3dA_{d}}{C_d}\\, .$ Next, set $t_1=\\frac{C_d^2 t}{18d^2 A_{d}^2}$ .",
"According to Lemma REF , with probability greater than $1-4d^2\\exp (-2nt_1^2)$ , $\\left\\Vert \\widehat{\\Sigma }- \\Sigma \\right\\Vert _{\\text{op}} \\le t_1\\, .$ Finally, set $t_2 =t_1$ .",
"It is easy to check that $t_2 \\le C_d/(6dA_{d})$ .",
"Thus we can use Lemma REF : with probability greater than $1-4d^2\\exp (-2nt_1^2)$ , $\\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}} \\le \\frac{6dA_{d}}{C_d}\\, .$ By the union bound, with probability greater than $1-8d^2 \\exp \\left(\\frac{-C_d^4 nt^2}{162d^4A_{d}^4}\\right)$ , $\\left\\Vert \\widehat{\\Sigma }^{-1}- \\widehat{\\Sigma }\\right\\Vert _{\\text{op}} &\\le \\left\\Vert \\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\cdot \\left\\Vert \\widehat{\\Sigma }- \\Sigma \\right\\Vert _{\\text{op}} \\cdot \\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}} \\\\&\\le \\frac{3dA_{d}}{C_d} \\cdot t_1 \\cdot \\frac{6dA_{d}}{C_d} = t\\, .$"
],
[
"Right-hand side of Eq. (",
"In this section, we state and prove the results in relation to $\\widehat{\\Gamma }$ .",
"We begin with the computation of $\\Gamma $ , the expected value of $\\widehat{\\Gamma }$ .",
"Lemma 9.1 (Computation of $\\Gamma $ ) Under Assumption REF and REF , the expected value of $\\widehat{\\Gamma }$ is given by $\\Gamma = C_d\\begin{pmatrix}f(\\tilde{\\mu }) \\\\\\alpha _1 f(\\tilde{\\mu }) - a_1 \\theta _1 \\\\\\vdots \\\\\\alpha _df(\\tilde{\\mu }) - a_d\\theta _d\\end{pmatrix}\\, ,$ where the $\\theta _j$ s are defined by $\\theta _j =\\left[ \\frac{\\tilde{\\sigma }}{\\sqrt{2\\pi }}\\exp \\left(\\frac{-(x-\\tilde{\\mu }_j)^2}{2\\tilde{\\sigma }^2}\\right)\\right]_{q_{j-}}^{q_{j+}}\\, .$ Given the expression of $\\widehat{\\Gamma }$ , we have essentially two computations to manage: $\\mathbb {E}\\left[\\pi _i f(x_i)\\right]$ and $\\mathbb {E}\\left[\\pi _i z_{ij}f(x_i)\\right]$ ."
],
[
"Computation of $\\mathbb {E}\\left[\\pi _i f(x_i)\\right]$ . ",
"Under Assumption REF , by linearity of the integral, $\\mathbb {E}\\left[\\pi _i f(x_i)\\right] = \\mathbb {E}\\left[\\pi _i (a^\\top +b)\\right] = b\\mathbb {E}\\left[\\pi _i\\right] + \\sum _{j=1}^da_{j} \\mathbb {E}\\left[\\pi _i x_{ij}\\right]\\, .$ Now we have already seen in the proof of Lemma REF that $\\mathbb {E}\\left[\\pi _i\\right] = C_d$ .",
"Thus we can focus on the computation of $\\mathbb {E}\\left[\\pi _i x_{ij}\\right]$ for fixed $i,j$ .",
"Under Assumption REF , we have $\\mathbb {E}\\left[\\pi _i x_{ij}\\right] = \\int _{\\operatorname{\\mathbb {R}}^d} x_j\\cdot \\exp \\left(\\frac{-\\left\\Vert x-\\xi \\right\\Vert ^2}{2\\nu ^2}+\\frac{-\\left\\Vert x-\\mu \\right\\Vert ^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x_1 \\cdots \\mathop {}\\mathopen {}\\mathrm {d}x_d}{(2\\pi \\sigma ^2)^{d/2}}\\, .$ By independence, the last display amounts to $\\int _{-\\infty }^{+\\infty } x\\cdot \\exp \\left(\\frac{-(x-\\xi _j)^2}{2\\nu ^2}+\\frac{-(x-\\mu _j)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} \\cdot \\prod _{k\\ne j} \\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi _k)^2}{2\\nu ^2}+\\frac{-(x-\\mu _k)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\, .$ A straightforward application of Lemmas REF and REF yields $\\mathbb {E}\\left[\\pi _i x_{ij}\\right] = C_d\\cdot \\frac{\\nu ^2 \\mu _j + \\sigma ^2\\xi _j}{\\nu ^2+\\sigma ^2}\\, .$ Back to Eq.",
"(REF ), we have shown that $\\mathbb {E}\\left[\\pi _i f(x_i)\\right] = C_db + \\sum _{j=1}^da_j \\cdot C_d\\frac{\\nu ^2 \\mu _j + \\sigma ^2\\xi _j}{\\nu ^2+\\sigma ^2} = C_df(\\tilde{\\mu })\\, .$"
],
[
"Computation of $\\mathbb {E}\\left[\\pi _i z_{ij}f(x_i\\right])$ . ",
"Under Assumption REF , by linearity of the integral, $\\mathbb {E}\\left[\\pi _i z_{ij}f(x_i)\\right] = b\\mathbb {E}\\left[\\pi _i z_{ij}\\right] + \\sum _{k=1}^da_k \\cdot \\mathbb {E}\\left[\\pi _i z_{ij} x_{ik}\\right]\\, .$ We have already computed $\\mathbb {E}\\left[\\pi _i z_{ij}\\right]$ in the proof of Lemma REF and found that $\\mathbb {E}\\left[\\pi _i z_{ij}\\right] = C_d\\alpha _j\\, .$ Regarding the computation of $\\mathbb {E}\\left[\\pi _i z_{ij} x_{ik}\\right]$ , there are essentially two cases to consider depending whether $k=\\ell $ or not.",
"Let us first consider the case $k=j$ .",
"Then we obtain $\\mathbb {E}\\left[\\pi _i z_{ij} x_{ik}\\right] = \\int _{\\operatorname{\\mathbb {R}}^d} x_{j} \\exp \\left(\\frac{-\\left\\Vert x-\\xi \\right\\Vert ^2}{2\\nu ^2}+\\frac{-\\left\\Vert x-\\mu \\right\\Vert ^2}{2\\sigma ^2}\\right)\\mathbf {1}_{\\phi (x)_j=\\phi (\\xi )_j} \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x_1 \\cdots \\mathop {}\\mathopen {}\\mathrm {d}x_d}{(2\\pi \\sigma ^2)^{d/2}}\\, .$ By independence, the last display amounts to $\\int _{q_{j-}}^{q_{j+}} x \\cdot \\exp \\left(\\frac{-(x-\\xi _j)^2}{2\\nu ^2}+\\frac{-(x-\\mu _j)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} \\cdot \\prod _{k\\ne j} \\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi _k)^2}{2\\nu ^2}+\\frac{-(x-\\mu _k)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\, .$ According to Lemma REF and the definition of $\\alpha _j$ and $\\theta _j$ (Eqs.",
"(REF ) and (REF )), we have $\\mathbb {E}\\left[\\pi _i z_{ij} x_{ij}\\right] = C_d\\frac{\\sigma ^2\\xi _j+\\nu ^2\\mu _j}{\\nu ^2+\\sigma ^2} \\alpha _j - C_d\\theta _j\\, .$ Now if $k\\ne j$ , by independence, $\\mathbb {E}\\left[\\pi _i z_{ij} x_{ik}\\right]$ splits in three parts: $\\mathbb {E}\\left[\\pi _i z_{ij} x_{ik}\\right] &= \\int _{-\\infty }^{+\\infty } x\\cdot \\exp \\left(\\frac{-(x-\\xi _k)^2}{2\\nu ^2}+\\frac{-(x-\\mu _k)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} \\cdot \\int _{q_{j-}}^{q_{j+}} \\exp \\left(\\frac{-(x-\\xi _j)^2}{2\\nu ^2}+\\frac{-(x-\\mu _j)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\cdot \\\\&\\cdot \\prod _{\\ell \\ne j,k} \\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi _k)^2}{2\\nu ^2}+\\frac{-(x-\\mu _k)^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }}\\, .$ Lemma REF and REF yield $\\mathbb {E}\\left[\\pi _i z_{ij} x_{ik}\\right] = C_d\\cdot \\frac{\\sigma ^2\\xi _k+\\nu ^2\\mu _k}{\\nu ^2+\\sigma ^2} \\cdot \\alpha _j\\, .$ In definitive, plugging these results into Eq.",
"(REF ) gives $\\mathbb {E}\\left[\\pi _i z_{ij}f(x_i)\\right] &= C_d\\alpha _j b + a_j\\left(C_d\\frac{\\sigma ^2\\xi _j+\\nu ^2\\mu _j}{\\nu ^2+\\sigma ^2} \\alpha _j - C_d\\theta _j\\right) + \\sum _{k\\ne j} a_k \\cdot C_d\\frac{\\sigma ^2\\xi _k+\\nu ^2\\mu _k}{\\nu ^2+\\sigma ^2} \\alpha _j \\\\&= C_d\\alpha _j f(\\tilde{\\mu }) - C_da_j \\theta _j\\, .$ As a consequence of Lemma REF , we can control $\\left\\Vert \\Gamma \\right\\Vert $ .",
"Lemma 9.2 (Control of $\\left\\Vert \\Gamma \\right\\Vert $ ) Under Assumptions REF and REF , it holds that $\\left\\Vert \\Gamma \\right\\Vert ^2 \\le C_d^2\\left(3df(\\tilde{\\mu })^2 + d\\tilde{\\sigma }^2 \\left\\Vert \\nabla f\\right\\Vert ^2\\right)\\, .$ According to Lemma REF , we have $\\left\\Vert \\Gamma \\right\\Vert ^2 = C_d^2\\left(f(\\tilde{\\mu })^2 + \\sum _{j=1}^{d} (\\alpha _j f(\\tilde{\\mu }) - a_j\\theta _j)^2\\right)\\, .$ Successively using $(x-y)^2\\le 2(x^2+y^2)$ , $\\alpha _j\\in [0,1]$ and $\\theta _j\\in [-\\tilde{\\sigma }/\\sqrt{2\\pi },\\tilde{\\sigma }/\\sqrt{2\\pi }]$ , we write $\\left\\Vert \\Gamma \\right\\Vert ^2 &\\le C_d^2\\left(f(\\tilde{\\mu })^2 + \\sum _{j=1}^{d} 2(\\alpha _j^2 f(\\tilde{\\mu })^2 + a_j^2\\theta _j^2)\\right) \\\\&\\le C_d^2\\left(3df(\\tilde{\\mu })^2 + d\\tilde{\\sigma }^2 \\left\\Vert a\\right\\Vert ^2\\right)\\, ,$ which concludes the proof.",
"Finally, we conclude this section with a concentration result for $\\widehat{\\Gamma }$ .",
"Lemma 9.3 (Concentration of $\\left\\Vert \\widehat{\\Gamma }\\right\\Vert $ ) Under Assumptions REF and REF , for any $t>0$ , we have $\\operatorname{\\mathbb {P}}\\left(\\left\\Vert \\widehat{\\Gamma }- \\Gamma \\right\\Vert > t\\right) \\le 4d\\exp \\left(\\frac{-nt^2}{2\\left\\Vert \\nabla f\\right\\Vert ^2 \\sigma ^2}\\right)\\, .$ Since the $x_i$ are Gaussian with variance $\\sigma ^2$ (Assumption REF ), the random variable $a^\\top x_i+b$ is Gaussian with variance $\\left\\Vert a\\right\\Vert ^2\\sigma ^2$ , and the $X_i^{(1)}=\\pi _i x_{i}$ are sub-Gaussian with parameter $\\left\\Vert a\\right\\Vert ^2\\sigma ^2$ .",
"They are also independent, thus we can apply Theorem REF to the $X_i^{(1)}$ : $\\operatorname{\\mathbb {P}}\\left(\\left|\\frac{1}{n}\\sum _{i=1}^{n} \\pi _if(x_i) - \\mathbb {E}\\left[\\pi _i f(x_i)\\right]\\right| > t\\right) \\le 2\\exp \\left(\\frac{-nt^2}{2\\left\\Vert a\\right\\Vert ^2 \\sigma ^2}\\right)\\, .$ Furthermore, the $z_{ij}$ are $\\lbrace 0,1\\rbrace $ -valued.",
"Thus the random variables $X_i^{(j)}=\\pi _i z_{ij}f(x_i)$ are also sub-Gaussian with parameter $\\left\\Vert a\\right\\Vert ^2\\sigma ^2$ .",
"We use Hoeffding's inequality (Theorem REF ) again, to obtain, for any $j$ , $\\operatorname{\\mathbb {P}}\\left(\\left|\\frac{1}{n}\\sum _{i=1}^{n} \\pi _i z_{ij}f(x_i) - \\mathbb {E}\\left[\\pi _i z_{ij}f(x_i)\\right]\\right| > t\\right) \\le 2\\exp \\left(\\frac{-nt^2}{2\\left\\Vert a\\right\\Vert ^2 \\sigma ^2}\\right)\\, .$ By the union bound, $\\operatorname{\\mathbb {P}}\\left(\\left\\Vert \\widehat{\\Gamma }- \\Gamma \\right\\Vert > t\\right) \\le 2(d+1) \\exp \\left(\\frac{-nt^2}{2\\left\\Vert a\\right\\Vert ^2 \\sigma ^2}\\right)\\, .$ We deduce the result since $d\\ge 1$ ."
],
[
"Proof of the main result",
"In this section, we state and prove our main result, Theorem REF .",
"It is a more precise version than Theorem REF in the main paper.",
"Theorem 10.1 (Concentration of $\\widehat{\\beta }$ ) Let $\\eta \\in (0,1)$ and $\\varepsilon >0$ .",
"Take $n \\ge \\max \\left(\\frac{288\\left\\Vert \\nabla f\\right\\Vert ^2\\sigma ^2d^2A_{d}^2}{\\varepsilon ^2C_d^2}\\log \\frac{12d}{\\eta }, \\frac{18d^2A_{d}^2}{C_d^2}\\log \\frac{24d^2}{\\eta },\\frac{648d^5A_{d}^4(3f(\\tilde{\\mu })^2+\\tilde{\\sigma }^2\\left\\Vert \\nabla f\\right\\Vert ^2)}{C_d^2\\varepsilon ^2}\\log \\frac{24d^2}{\\eta }\\right)\\, .$ Then, under assumptions REF and REF , $\\left\\Vert \\widehat{\\beta }- \\Sigma ^{-1}\\Gamma \\right\\Vert \\le \\varepsilon \\, ,$ with probability greater than $1-\\eta $ .",
"The main idea of the proof is to notice that $\\left\\Vert \\widehat{\\beta }- \\Sigma ^{-1}\\Gamma \\right\\Vert &= \\left\\Vert \\widehat{\\Sigma }^{-1}\\widehat{\\Gamma }- \\Sigma ^{-1}\\Gamma \\right\\Vert \\\\&\\le \\left\\Vert \\widehat{\\Sigma }^{-1}(\\widehat{\\Gamma }- \\Gamma )\\right\\Vert + \\left\\Vert (\\widehat{\\Sigma }^{-1}-\\Sigma ^{-1})\\Gamma \\right\\Vert \\, ,$ and then to control these two terms using the results of Section and ."
],
[
"Control of $\\left\\Vert \\widehat{\\Sigma }^{-1}(\\widehat{\\Gamma }- \\Gamma )\\right\\Vert $ . ",
"We use the upper bound $\\left\\Vert \\widehat{\\Sigma }^{-1}(\\widehat{\\Gamma }- \\Gamma )\\right\\Vert \\le \\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}}\\cdot \\left\\Vert \\widehat{\\Gamma }-\\Gamma \\right\\Vert $ .",
"We then achieve control of the operator norm of the empirical covariance matrix in probability with Lemma REF , and control of the norm of $\\widehat{\\Gamma }-\\Gamma $ in probability with Lemma REF .",
"Set $t_1=\\frac{C_d}{6dA_{d}}\\quad \\text{and}\\quad n_1 =\\frac{18d^2}{C_d^2}\\log \\frac{12d^2}{\\eta }\\, .$ According to Lemma REF , for any $n\\ge n_1$ , there is an event $\\Omega _1^n$ which has probability greater than $1-4d^2\\exp (-2nt_1^2)$ such that $\\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}} \\le \\frac{6dA_{d}}{C_d}$ on this event.",
"It is easy to check that $4d^2\\exp (-2n_1t_1^2) = \\eta / 3$ , thus $\\Omega _1^n$ has probability greater than $1-\\eta / 3$ .",
"Now set $t_2 =\\frac{\\varepsilon C_d}{12dA_{d}} \\quad \\text{and}\\quad n_2 =\\frac{288\\left\\Vert a\\right\\Vert ^2\\sigma ^2d^2A_{d}^2 }{\\varepsilon ^2C_d^2}\\log \\frac{12d}{\\eta }\\, .$ According to Lemma REF , for any $n\\ge n_2$ , there exists an event $\\Omega _2^n$ which has probability greater than $1-4d\\exp \\left(\\frac{-nt_2^2}{2\\left\\Vert a\\right\\Vert ^2\\sigma ^2}\\right)$ such that $\\left\\Vert \\widehat{\\Gamma }-\\Gamma \\right\\Vert \\le t_2$ on that event.",
"One can check that $4d\\exp \\left(\\frac{-n_2t_2^2}{2\\left\\Vert a\\right\\Vert ^2\\sigma ^2}\\right) = \\frac{\\eta }{3}\\, ,$ thus $\\Omega _2^n$ has probability greater than $1-\\eta /3$ .",
"On the event $\\Omega _1^n\\cap \\Omega _2^n$ , we have $\\left\\Vert \\widehat{\\Sigma }^{-1}(\\widehat{\\Gamma }- \\Gamma )\\right\\Vert \\le \\left\\Vert \\widehat{\\Sigma }^{-1}\\right\\Vert _{\\text{op}}\\cdot \\left\\Vert \\widehat{\\Gamma }-\\Gamma \\right\\Vert \\le \\frac{6dA_{d}}{C_d} \\cdot t_2 \\le \\frac{\\varepsilon }{2}\\, ,$ by definition of $t_2$ ."
],
[
"Control of $\\left\\Vert (\\widehat{\\Sigma }^{-1}-\\Sigma ^{-1})\\Gamma \\right\\Vert $ . ",
"We use the upper bound $\\left\\Vert (\\widehat{\\Sigma }^{-1}-\\Sigma ^{-1})\\Gamma \\right\\Vert \\le \\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\cdot \\left\\Vert \\Gamma \\right\\Vert $ .",
"We then achieve control of $\\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}}$ in probability with Proposition REF , whereas we can bound the norm of $\\Gamma $ almost surely with Lemma REF .",
"If $\\left\\Vert \\Gamma \\right\\Vert =0$ , then there is nothing to prove.",
"Otherwise, set $t_3 =\\min \\left(\\frac{\\varepsilon }{2\\left\\Vert \\Gamma \\right\\Vert },\\frac{3dA_{d}}{C_d}\\right),\\,n_3 =\\frac{18d^2A_{d}^2}{C_d^2} \\log \\frac{24d^2}{\\eta },\\, \\quad \\text{and} \\quad n_4 =\\frac{648d^5A_{d}^4(3f(\\tilde{\\mu })^2 + \\tilde{\\sigma }^2\\left\\Vert a\\right\\Vert ^2)}{C_d^2\\varepsilon ^2} \\log \\frac{24d^2}{\\eta }\\, .$ According to Proposition REF , for any $n\\ge \\max (n_3,n_4)$ , there is an event $\\Omega _3^n$ which has probability greater than $1-8d^2\\exp \\left(\\frac{-C_d^3 nt_3^2}{162d^2A_{d}^4}\\right)$ such that $\\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\le t_3$ on this event.",
"With the help of Lemma REF , one can check that $\\max \\left(8d^2\\exp \\left(\\frac{-C_d^3 n_3t_3^2}{162d^2A_{d}^4}\\right),8d^2\\exp \\left(\\frac{-C_d^3 n_4t_3^2}{162d^2A_{d}^4}\\right)\\right) \\le \\frac{\\eta }{3}\\, .$ Therefore, $\\Omega _3^n$ has probability greater than $\\eta / 3$ and, on this event, $\\left\\Vert (\\widehat{\\Sigma }^{-1}-\\Sigma ^{-1})\\Gamma \\right\\Vert \\le \\left\\Vert \\widehat{\\Sigma }^{-1}-\\Sigma ^{-1}\\right\\Vert _{\\text{op}} \\cdot \\left\\Vert \\Gamma \\right\\Vert \\le t_3 \\cdot \\left\\Vert \\Gamma \\right\\Vert \\le \\frac{\\varepsilon }{2}\\, .$"
],
[
"Conclusion. ",
"Set $n \\ge \\max (n_i,i=1\\ldots 4)$ .",
"Define $\\Omega ^n =\\Omega _1^n\\cap \\Omega _2^n\\cap \\Omega _3^n$ , where the $\\Omega _i^n$ are defined as before.",
"According to the previous reasoning, on the event $\\Omega ^n$ , $\\left\\Vert \\widehat{\\beta }- \\Sigma ^{-1}\\Gamma \\right\\Vert &= \\left\\Vert \\widehat{\\Sigma }^{-1}\\widehat{\\Gamma }- \\Sigma ^{-1}\\Gamma \\right\\Vert \\\\&\\le \\left\\Vert \\widehat{\\Sigma }^{-1}(\\widehat{\\Gamma }- \\Gamma )\\right\\Vert + \\left\\Vert (\\widehat{\\Sigma }^{-1}-\\Sigma ^{-1})\\Gamma \\right\\Vert \\\\&\\le \\frac{\\varepsilon }{2}+\\frac{\\varepsilon }{2} = \\varepsilon \\, .$ Moreover, the union bound gives $\\operatorname{\\mathbb {P}}\\left(\\Omega ^n\\right) \\ge 1-\\eta $ .",
"We conclude by noticing that $n_1$ is always smaller than $n_3$ , thus we just have to require $n \\ge \\max (n_2,n_3,n_4)$ , as in the statement of our result."
],
[
"Gaussian integrals",
"In this section, we collect some Gaussian integral computations that are needed in our derivations.",
"We provide succinct proof, since essentially any modern computer algebra system will provide these formulas.",
"Our first result is for zero-th order Gaussian integral.",
"Lemma 11.1 (Gaussian integral, 0-th order) Let $\\xi ,\\mu $ be real numbers, and $\\nu ,\\sigma $ be positive real numbers.",
"Then, it holds that $\\int \\exp \\left(\\frac{-(x-\\xi )^2}{2\\nu ^2}+ \\frac{-(x-\\mu )^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} = \\frac{\\nu }{\\sqrt{\\nu ^2+\\sigma ^2}} \\cdot \\exp \\left(\\frac{-(\\xi -\\mu )^2}{2(\\nu ^2+\\sigma ^2)}\\right) \\cdot \\frac{1}{2}\\mathrm {erf}\\left(\\frac{\\nu ^2 (x-\\mu ) + \\sigma ^2 (x-\\xi )}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\right)\\, .$ In particular, $\\int _{-\\infty }^{+\\infty } \\exp \\left(\\frac{-(x-\\xi )^2}{2\\nu ^2}+ \\frac{-(x-\\mu )^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} = \\frac{\\nu }{\\sqrt{\\nu ^2+\\sigma ^2}} \\cdot \\exp \\left(\\frac{-(\\xi -\\mu )^2}{2(\\nu ^2+\\sigma ^2)}\\right)\\, .$ For any reals $a,b$ , and $c$ , it holds that $\\int \\mathrm {e}^{-ax^2+bx+c} \\mathop {}\\mathopen {}\\mathrm {d}x = \\sqrt{\\frac{\\pi }{a}}\\cdot \\mathrm {e}^{\\frac{b^2}{4a}+c} \\cdot \\frac{1}{2}\\mathrm {erf}\\left(\\frac{2ax-b}{2\\sqrt{a}}\\right)\\, .$ We apply this formula with $a = \\frac{1}{2\\nu ^2}+\\frac{1}{2\\sigma ^2}$ , $b=\\frac{\\xi }{\\nu ^2}+\\frac{\\mu }{\\sigma ^2}$ , and $c=-\\left(\\frac{\\xi ^2}{2\\nu ^2}+\\frac{\\mu ^2}{\\sigma ^2}\\right)$ .",
"We then notice that $b^2/(4a)+c = \\frac{-(\\xi -\\mu )^2}{2(\\nu ^2+\\sigma ^2)}$ and $\\frac{2ax-b}{2\\sqrt{a}} = \\frac{\\nu ^2 (x-\\mu ) + \\sigma ^2 (x-\\xi )}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\, .$ Remark 11.1 We often replace $\\frac{\\nu ^2 (x-\\mu ) + \\sigma ^2 (x-\\xi )}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}$ by the more readable $(x-\\tilde{\\mu })/(\\tilde{\\sigma }\\sqrt{2})$ in the main text of the paper.",
"Since $f$ is assumed to be linear in most of the paper, we need first order computations as well: Lemma 11.2 (Gaussian integral, 1st order) Let $\\xi ,\\mu $ be real numbers, and $\\nu ,\\sigma $ be positive numbers.",
"Then it holds that $\\int & x\\cdot \\exp \\left(\\frac{-(x-\\xi )^2}{2\\nu ^2}+\\frac{-(x-\\mu )^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} = \\frac{\\nu }{\\sqrt{\\nu ^2+\\sigma ^2}} \\cdot \\exp \\left(\\frac{-(\\xi -\\mu )^2}{2(\\nu ^2+\\sigma ^2)}\\right)\\cdot \\\\&\\left[\\frac{\\sigma ^2\\xi + \\nu ^2\\mu }{\\nu ^2+\\sigma ^2} \\cdot \\frac{1}{2}\\mathrm {erf}\\left(\\frac{\\nu ^2(x-\\mu )+\\sigma ^2(x-\\xi )}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\right) - \\frac{\\nu \\sigma }{\\sqrt{2\\pi }\\sqrt{\\nu ^2+\\sigma ^2}}\\cdot \\exp \\left(-\\left(\\frac{\\nu ^2(x-\\mu )+\\sigma ^2(x-\\xi )}{\\nu \\sigma \\sqrt{2(\\nu ^2+\\sigma ^2)}}\\right)^2\\right)\\right]\\, .$ In particular, $\\int _{-\\infty }^{+\\infty } x\\cdot \\exp \\left(\\frac{-(x-\\xi )^2}{2\\nu ^2}+\\frac{-(x-\\mu )^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} = \\frac{\\sigma ^2\\xi + \\nu ^2\\mu }{\\nu ^2+\\sigma ^2} \\cdot \\frac{\\nu }{\\sqrt{\\nu ^2+\\sigma ^2}} \\cdot \\exp \\left(\\frac{-(\\xi -\\mu )^2}{2(\\nu ^2+\\sigma ^2)}\\right)\\, .$ For any $a,b,c$ with $a >0$ , it holds that $\\int x\\cdot \\mathrm {e}^{-ax^2+bx+c} \\mathop {}\\mathopen {}\\mathrm {d}x = \\frac{\\sqrt{\\pi }b}{4a^{3/2}}\\mathrm {e}^{b^2/(4a)+c}\\mathrm {erf}\\left(\\frac{2ax-b}{2\\sqrt{a}}\\right) - \\frac{1}{2a}\\mathrm {e}^{-ax^2+bx+c}\\, .$ Finally we want to mention the following result.",
"Lemma 11.3 (Gaussian integral, 2nd order) Let $\\xi ,\\mu $ be real numbers, and $\\nu ,\\sigma $ be positive real numbers.",
"Then, it holds that $\\int _{-\\infty }^{+\\infty } x^2 \\cdot \\exp \\left(\\frac{-(x-\\xi )^2}{2\\nu ^2} + \\frac{-(x-\\mu )^2}{2\\sigma ^2}\\right) \\frac{\\mathop {}\\mathopen {}\\mathrm {d}x}{\\sigma \\sqrt{2\\pi }} =\\frac{(\\sigma ^2\\xi + \\nu ^2\\mu )^2 + \\nu ^2\\sigma ^2(\\nu ^2+\\sigma ^2)}{(\\nu ^2+\\sigma ^2)^2} \\cdot \\frac{\\nu }{\\sqrt{\\nu ^2+\\sigma ^2}} \\cdot \\exp \\left(\\frac{-(\\xi -\\mu )^2}{2(\\nu ^2+\\sigma ^2)}\\right)\\, .$ Remark 11.2 As a consequence of Lemma REF , it would be possible to further our analysis by adding second degree terms to $f$ .",
"Indeed, quantities depending on $\\left\\Vert x_i-\\xi \\right\\Vert $ , which would have to be computed to extend the proofs of Lemmas REF and REF , can be computed with this lemma.",
"For instance, one can show that $\\mathbb {E}\\left[\\pi _i \\left\\Vert x_i-\\xi \\right\\Vert ^2\\right] = C_d\\cdot \\left[\\frac{\\nu ^4}{(\\nu ^2+\\sigma ^2)^2}\\left\\Vert \\xi -\\mu \\right\\Vert ^2 + \\frac{\\nu ^2\\sigma ^2d}{\\nu ^2+\\sigma ^2}\\right]\\, .$ We use the fact that $\\int x^2 \\cdot \\mathrm {e}^{-ax^2+bx+c} \\mathop {}\\mathopen {}\\mathrm {d}x = \\frac{\\sqrt{\\pi }(2a+b^2)}{8a^{5/2}} \\mathrm {e}^{\\frac{b^2}{4a}+c}\\cdot \\mathrm {erf}\\left(\\frac{2ax-b}{2\\sqrt{a}}\\right) - \\frac{ax+b}{4a^2} \\cdot \\mathrm {e}^{-ax^2+bx+c}\\, .$"
],
[
"Concentration results",
"In this section we collect some concentration results used throughout our proofs.",
"Note that we use rather use the two-sided version of these results.",
"Theorem 11.1 (Hoeffding's inequality) Let $X_1,\\ldots ,X_n$ be independent random variables such that $X_i$ takes its values in $[a_i,b_i]$ almost surely for all $i\\le n$ .",
"Then for every $t>0$ , $\\operatorname{\\mathbb {P}}\\left(\\frac{1}{n}\\sum _{i=1}^n (X_i-\\mathbb {E}\\left[X_i\\right]) \\ge t\\right) \\le \\exp \\left(\\frac{-2t^2n^2}{\\sum _{i=1}^n (b_i-a_i)^2}\\right)\\, .$ This is Theorem 2.8 in [2] in our notation.",
"Theorem 11.2 (Hoeffding's inequality for sub-Gaussian random variables) Let $X_1,\\ldots ,X_n$ be independent random variables such that $X_i$ is sub-Gaussian with parameter $s^2>0$ .",
"Then, for every $t>0$ , $\\operatorname{\\mathbb {P}}\\left(\\frac{1}{n} \\sum _{i=1}^n X_i - \\mathbb {E}\\left[X_i\\right] > t\\right) \\le \\exp \\left(\\frac{-nt^2}{2s^2}\\right)\\, .$ This is Proposition 2.1 in [14]."
]
] | 2001.03447 |
[
[
"Should Artificial Intelligence Governance be Centralised? Design Lessons\n from History"
],
[
"Abstract Can effective international governance for artificial intelligence remain fragmented, or is there a need for a centralised international organisation for AI?",
"We draw on the history of other international regimes to identify advantages and disadvantages in centralising AI governance.",
"Some considerations, such as efficiency and political power, speak in favour of centralisation.",
"Conversely, the risk of creating a slow and brittle institution speaks against it, as does the difficulty in securing participation while creating stringent rules.",
"Other considerations depend on the specific design of a centralised institution.",
"A well-designed body may be able to deter forum shopping and ensure policy coordination.",
"However, forum shopping can be beneficial and a fragmented landscape of institutions can be self-organising.",
"Centralisation entails trade-offs and the details matter.",
"We conclude with two core recommendations.",
"First, the outcome will depend on the exact design of a central institution.",
"A well-designed centralised regime covering a set of coherent issues could be beneficial.",
"But locking-in an inadequate structure may pose a fate worse than fragmentation.",
"Second, for now fragmentation will likely persist.",
"This should be closely monitored to see if it is self-organising or simply inadequate."
],
[
"Introduction",
"In 2018, Canada and France proposed the International Panel on Artificial Intelligence (IPAI).",
"After being rejected at the G7 in 2019, negotiations shifted to the OECD and are presently ongoing.",
"As the field of AI continues to mature and spark public interest and legislative concern [41], the priority of such governance initiatives reflects the growing appreciation that AI has the potential to dramatically change the world for both good and ill [9].",
"Research into AI governance needs to keep pace with policy-making and technological change.",
"Choices made today may have long-lasting impacts on policymakers' ability to address numerous AI policy problems [7].",
"Effective governance can promote safety, accountability, and responsible behaviour in the research, development, and deployment of AI systems.",
"AI governance research to date has predominantly focused at the national and sub-national levels [44], [6], [16].",
"Research into AI global governance remains relatively nascent (though see [5]).",
"[24] [24] have called for specialised, centralised intergovernmental agencies to coordinate policy responses globally, and others have called for a centralised `International Artificial Intelligence Organisation' [14].",
"Others favour more decentralised arrangements based around `Governance Coordinating Committees', global standards, or existing international law instruments [47], [8], [28].",
"No one has taken a step back to inquire: what would the history of multilateralism suggest, given the state and trajectory of AI?",
"Should AI governance be centralised or decentralised?",
"`Centralisation', in this case, refers to the degree to which the coordination, oversight and/or regulation of a set of AI policy issues or technologies are housed under a single (global) institution.",
"This is not a binary choice; it exists across a spectrum.",
"Trade is highly (but not entirely) centralised under the umbrella of the WTO.",
"In contrast, environmental multilateralism is much more decentralised.",
"In this paper, we seek to help the community of researchers, policymakers, and other stakeholders in AI governance understand the advantages and disadvantages of centralisation.",
"This may help set terms and catalyse a much-needed debate to inform governance design decisions.",
"We first outline the international governance challenges of AI, and review early proposed global responses.",
"We then draw on existing literatures on regime fragmentation [3] and `regime complexes' [36] to assess considerations in centralising the international governance of AI.",
"We draw on the history of other international regimesA regime is a set of `implicit or explicit principles, norms, rules and decision-making procedures around which actors' expectations converge in a given area of international relations'[27].",
"to identify considerations that speak in favour or against designing a centralised regime complex for AI.",
"We conclude with two recommendations.",
"First, many trade-offs are contingent on how well-designed a central body would be.",
"An adaptable, powerful institution with a manageable mandate would be beneficial, but a poorly designed body could prove a fate worse than fragmentation.",
"Second, for now there should be structured monitoring of existing efforts to see whether they are they are self-organising or insufficient."
],
[
"The State of AI Governance",
"There is debate as to whether AI is a single policy area or a diverse series of issues.",
"Some claim that AI cannot be cohesively regulated as it is a collection of disparate technologies, with different risk profiles across different applications and industries [45].",
"This is an important but not entirely convincing objection.",
"The technical field has no settled definition for `AI',We define `AI' as any machine system capable of functioning `appropriately and with foresight in its environment' [34]; see too [9].",
"so it should be no surprise that defining a manageable scope for AI governance will be difficult.",
"Yet this challenge is not unique to AI: definitional issues abound in areas such as environment and energy, but have not figured prominently in debates over centralisation.",
"Indeed, energy and environment ministries are common at the domestic level, despite problems in setting the boundaries of natural systems and resources.",
"We contend that there are numerous ways in which a centralised body could be designed for AI governance.",
"For example, a centralised approach could carve out a subset of interlinked AI issues to cover.",
"This could involve focusing on the potentially high-risk applications of AI systems, such as AI-enabled cyberwarfare, lethal autonomous weapons (LAWS), other advanced military applications, or high-level machine intelligence (HLMI).`High-level machine intelligence' has been defined as `unaided machines [that] can accomplish every task better and more cheaply than human workers' [17].",
"Another approach could govern underlying hardware resources (e.g.",
"large-scale compute resources) or software libraries.",
"We are agnostic on the specifics of how centralisation could or should be implemented, and instead focus on the costs and benefits of centralisation in the abstract.",
"The exact advantages and disadvantages of centralisation are likely to vary depending on the institutional design.",
"This is an important area of further study, particularly once more specific proposals are put forward.",
"However, such work must be grounded in a higher-level investigation of trade-offs in centralising AI governance.",
"It is this foundational analysis which we seek to offer.",
"Numerous AI issues could benefit from international cooperation.",
"These include the potentially catastrophic applications mentioned above.",
"It also encompasses more quotidian uses, such as AI-enabled cybercrime; human health applications; safety and regulation of autonomous vehicles and drones; surveillance, privacy and data-use; and labour automation.",
"Multilateral coordination could also use AI to tackle other global problems such as climate change [43], or help meet the Sustainable Development Goals [46].",
"This is an illustrative but not exhaustive list of international AI policy issues.",
"Global regulation across these issues is currently nascent, fragmented, yet evolving.",
"A wide range of UN institutions have begun to undertake some activities on AI [20].",
"The bodies covering AI policy issues range across existing organisations including the International Labour Organisation (ILO), International Telecommunication Union (ITU), and UNESCO.",
"This is complemented by budding regulations and working groups across the International Organisation for Standardisation (ISO), International Maritime Organisation (IMO), International Civil Aviation Organisation (ICAO), and other bodies, as well as treaty amendments, such as the updating of the Vienna Convention on Road Traffic to encompass autonomous vehicles [28], or the ongoing negotiations at the Convention on Certain Conventional Weapons (CCW) on LAWS.",
"The UN System Chief Executives Board (CEB) for Coordination through the High-Level Committee on Programmes has been empowered to draft a system-wide AI capacity building strategy.",
"The High-level Panel on Digital Cooperation has also sought to gather together common principles and ideas for AI relevant areas [19].",
"Whether these initiatives bear fruit, however, remains questionable, as many of the involved international organisations have fragmented membership, were not originally created to address AI issues and lack effective enforcement or compliance mechanisms [32].",
"The trajectory of these initiatives matters.",
"How governance is initially organised can be central to its success.",
"Debates over centralisation and fragmentation are long-lasting and prominent with good reason.",
"How we structure international cooperation can be critical to its success, and most other debates often implicitly hinge on structural debates.",
"Fragmentation and centralisation exist across a spectrum.",
"In a world lacking a global government, some fragmentation will always prevail.",
"But the degree to which it prevails is crucial.",
"We define `fragmentation' as a patchwork of international organisations and institutions which focus on a particular issue area, but differ in scope, membership and often rules [3].",
"We define centralisation as an arrangement in which governance of a particular issue lies under the authority of a single umbrella body.",
"A regime complex is a network of three or more international regimes on a common issue area.",
"These should have overlapping membership and cause potentially problematic interactions [36].",
"These definitions and terms are by nature normatively loaded.",
"For example, some may find `decentralisation' to be a positive framing, while others may see `fragmentation' to possess negative connotations.",
"Recognising this, we seek to use these terms in a primarily analytical manner.",
"We will use findings from each of these theoretical areas to inform our discussion of the history of multilateral fragmentation and its implications for AI governance."
],
[
"Centralisation Criteria: History of Governance Trade-Offs",
"In the following discussion, we explore a series of considerations for AI governance.",
"Political power and efficient participation support centralisation.",
"The breadth vs. depth dilemma, as well as slowness and brittleness support decentralisation.",
"Policy coordination and forum shopping considerations can cut both ways."
],
[
"Political Power",
"Regimes embody power in their authority over rules, norms, and knowledge beyond states' exclusive control.",
"A more centralised regime will see this power concentrated among fewer institutions.",
"A centralised, powerful architecture is likely to be more influential against competing international organisations and with constituent states [36].",
"An absence of centralised authority to manage regime complexes has presented challenges in the past.",
"Across the proliferation of Multilateral Environmental Agreements (MEAs) there is no requirement to cede responsibility to the UN Environmental Programme in the case of overlap or competition.",
"This has led to turf wars, inefficiencies and even contradictory policies [3].",
"One of the most notable examples is that of hydrofluorocarbons (HFCs).",
"HFCs are potent greenhouse gases, and yet their use has been encouraged by the Montreal Protocol since 1987 as a replacement for ozone-depleting substances.",
"This has only recently been resolved via the 2015 Kigali Amendment to the Montreal Protocol, which itself has a prolonged implementation period.",
"Similarly, the internet governance regime complex is diffuse.",
"Multiple venues and norms govern technical standards, cyber crime, human rights, and warfare [35].",
"Although the UN Internet Governance Forum (IGF) discusses several cross-cutting issues, it does not have a mandate to consolidate even principles, let alone negotiate new formal agreements [33].",
"In contrast, other centralised regimes have supported effective management.",
"For example, under the umbrella of the WTO, norms such as the most-favoured-nation principle (equally treating all WTO member states) principle have become the bedrock of international trade.",
"The power and track-record of the WTO is so formidable that it has created a chilling effect: the fear of colliding with WTO norms and rules has led environmental treaties to self-censor and actively avoid discussing or deploying trade-related measures [12].",
"Both the chilling effect and the remarkably powerful application of common trade rules were not a marker of international trade until the establishment of the WTO.",
"The power of these centralised body has stretched beyond influencing states in the domain of trade, to moulding related issues.",
"Political power offers further benefits in governing emerging technologies that are inherently uncertain in both substance and policy impact.",
"Uncertainty in technology and preferences has been associated with some increased centralisation in regimes [25].",
"There may also be benefits to housing a foresight capacity within the regime complex, to allow for accelerated or even proactive efforts [39].",
"Centralised AI governance would enable an empowered organisation to more effectively use foresight analyses to inform policy responses across the regime complex."
],
[
"Supporting Efficiency & Participation",
"Decentralised AI governance may undermine efficiency and inhibit participation.",
"States often create centralised regimes to reduce costs, for instance by eliminating duplicate efforts, yielding economies of scale within secretariats, and simplifying participation [15].",
"Conversely, fragmented regimes may force states to spread resources and funding over many distinct institutions, particularly limiting the ability of less well-resourced states or parties to participate fully [32].",
"Historically, decentralised regimes have presented cost and related participation concerns.",
"Hundreds of related and sometimes overlapping international environmental agreements can create `treaty congestion' [1].",
"This complicates participation and implementation for both developed and developing nations [15].",
"This includes costs associated with travel to different forums, monitoring and reporting for a range of different bodies, and duplication of effort by different secretariats (ibid.).",
"Similar challenges are already being witnessed in AI governance.",
"Simultaneous and globally distributed meetings pose burdensome participation costs for civil society.",
"Fragmented organisations must duplicatively invest in high-demand machine learning subject matter experts to inform their activities.",
"Centralisation would support institutional efficiency and participation."
],
[
"Slowness & Brittleness of Centralised Regimes",
"One potential problem of centralisation lies in the relatively slow process of establishing centralised institutions, which may often be outpaced by the rate of technological change.",
"Another challenge lies in centralised institutions' brittleness after they are established, i.e., their vulnerability to regulatory capture, or failure to react to changes in the problem landscape.",
"Establishing new international institutions is often a slow process.",
"For example, the Kyoto Protocol took three years of negotiations to create and then another eight to enter into force.",
"This becomes even more onerous with higher participation and stakes.",
"Under the GATT, negotiations for a 26% cut in tariffs between 19 countries took 8 months in 1947.",
"The Uruguay round, beginning in 1986, took 91 months to achieve a tariff reduction of 38% between 125 parties [31].",
"International law has been quick to respond to technological changes in some cases, and delayed in others [42].",
"Decentralised efforts may prove quicker to respond to complex, `transversal' issues, if they rely more on informal institutions with a smaller but like-minded membership [32].",
"Centralised AI governance may be particularly vulnerable to sparking lengthy negotiations, because progress on centralised regimes for new technologies tends to be hard if a few states hold clearly unequal stakes in the technology, or if there are significant differences in information and expertise among states or between states and private industry [42].",
"Both these conditions closely match the context of AI technology.",
"Moreover, because AI technology develops rapidly, such slow implementation of rules and principles could lead to certain actors taking advantage by setting de facto arrangements or extant state practice.",
"Even after its creation, a centralised regime can be brittle; the very qualities that provide it with political power may exacerbate the adverse effects of regulatory capture; the features that ensure institutional stability, may also mean that the institution cannot adapt quickly to unanticipated outside stressors outside its established mission.",
"The regime might break before it bends.",
"The first potential risk is regulatory capture.",
"Given the high profile of AI issue areas, political independence is paramount.",
"However, as illustrated by numerous cases, including undue corporate influence in the WHO during the 2009 H1N1 pandemic [11], no institution is fully immune to regime capture, and centralisation may reduce the costs of lobbying, making capture easier by providing a single locus of influence.",
"On the other hand, a regime complex comprising many parallel institutions could find itself vulnerable to capture by powerful actors, who are better positioned than smaller parties to send representatives to every forum.",
"Moreover, centralised regimes entail higher stakes.",
"Many issues are in a single basket and thus failure is more likely to be severe if it does occur.",
"International institutions can be notoriously path-dependent and thus fail to adjust to changing circumstances, as seen with the ILO's considerable difficulties in reforming its participation and rulemaking processes in the 1990s [2].",
"The public failure of a flagship global AI institution or governance effort could have lasting political repercussions.",
"It could strangle subsequent, more well-conceived proposals in the crib, by undermining confidence in multilateral governance generally or capable governance on AI issues specifically.",
"By contrast, for a decentralized regime complex to similarly fail, all of its component institutions would need to simultaneously `break' or fail to innovate at once.We thank Nicolas Moës for this observation.",
"A centralised institution that does not outright collapse, but which remains ineffective, may become a blockade against better efforts.",
"Ultimately, brittleness is not an inherent weakness of centralisation–and indeed depends far more on institutional design details.",
"There may be strategies to `innovation-proof'[29] governance regimes.",
"Periodic renegotiation, modular expansion, `principles based regulation', or sunset clauses can also support ongoing reform [30].",
"Such approaches have often proved successful historically, due partially to decentralisation but, importantly, also to particular designs."
],
[
"The Breadth vs. Depth Dilemma",
"Pursuing centralisation may create an overly high threshold that limits participation.",
"All multilateral agreements face a trade-off between having higher participation (`breadth') or stricter rules and greater ambition of commitments (`depth').",
"The dilemma is particularly evident for centralised institutions that are intended to be powerful and require strong commitments from states.",
"However, the opposite dynamics of sacrificing depth for breadth can also pose risks.",
"The 2015 Paris Agreement on Climate Change was significantly watered down to allow for the legal participation of the US.",
"Anticipated difficulties in ratification through the Senate led to negotiators opting for a `pledge and review' structure with few legal obligations.",
"Thus, the US could join simply through the approval of the executive [23].",
"In this case, inclusion of the US (which at any rate proved temporary) came at the cost of significant cutbacks on the demands which the regime sought to make of all parties.",
"In contrast, decentralisation could allow for major powers to engage in relevant regulatory efforts where they would be deterred from signing up to a more comprehensive package.",
"This has precedence in the history of climate governance.",
"Some claim that the US-led Asia-Pacific Partnership on Clean Development and Climate helped, rather than hindered climate governance, as it bypassed UNFCCC deadlock and secured non-binding commitments from actors not bound by the Kyoto Protocol [49].",
"This matters, as buy-in may prove a thorny issue for AI governance.",
"The actors who lead in AI development include powerful states that are potentially most adverse to global regulation in this area.",
"They have thus far proved recalcitrant in the global governance of security issues such as anti-personnel mines or cyberwarfare.",
"In response, some have already recommended a critical-mass governance approach to the military uses of AI.",
"Rather than seeking a comprehensive agreement, devolving and spinning off certain components into separate treaties (e.g.",
"for LAWS testing standards; liability and responsibility; and limits to operational usage) could instead allow for the powerful to ratify and move forward at least a few of those options [48].",
"The breadth vs. depth dilemma is a trade-off in multilateralism generally.",
"However, it is a particularly pertinent challenge for centralisation.",
"The key benefit of a centralised body would be to be a powerful anchor that ensures policy coordination and coherence, without suffering fragmentation in membership.",
"This dilemma suggests it is unlikely to have both.",
"It will likely need to restrict membership to have teeth, or lose its teeth to have wide participation.",
"A critical mass approach may be able to deliver the best of both worlds.",
"Nonetheless these dilemma poses a difficult knot for centralisation to unravel."
],
[
"Forum Shopping",
"Forum shopping may help or hinder AI governance, depending on the particular circumstances.",
"Fragmentation enables actors to choose where and how to engage.",
"Such `forum shopping' may take one of several forms: moving venues, abandoning one organisation, creating new venues, and working across multiple organisations to sew competition between them [4].",
"Even when there is a natural venue for an issue, actors have reasons to forum-shop.",
"For instance, states may look to maximise their influence, appease domestic pressure [40] and placate constituents by shifting to a toothless forum [18].",
"The ability to successfully forum-shop depends on an actor's power.",
"Most successful examples of forum-shifting have been led by the US [4].",
"Intellectual property rights in trade, for example, was subject to prolonged, contentious forum shopping.",
"Developed states resisted attempts of the UN Conference on Trade and Development (UNCTAD) to address intellectual property rights in trade by trying to push them onto the World Intellectual Property Organization (WIPO) (ibid., 566) and then subsequently to the WTO [18], overruling protests from developing states.",
"Outcomes often reflect power, but weak states and non-state actors can also pursue forum shopping strategies in order to challenge the status-quo [22].",
"Forum shopping may help or hurt governance.",
"This is evident in current efforts to regulate LAWS.",
"While the Group of Governmental Experts has made some progress, on the whole the CCW has taken slow deliberations on LAWS.",
"In response, frustrated activists have threatened to shift to another forum, as happened with the Ottawa Treaty that banned landmines [10].",
"This strategy could catalyse progress, but also brings risks of further forum shopping and weak or unimplemented agreements.",
"Forum shopping may similarly delay, stall, or weaken regulation of time-sensitive AI policy issues, including potential future HLMI development.",
"It is plausible that leading AI firms also have sway when they elect to participate in some venues but not others.",
"The OECD Expert Group on AI included representatives from leading firms, whereas engagement at UN efforts, including the Internet Governance Forum (IGF), do not appear to be similarly prioritised.",
"A decentralised regime will enable forum shopping, though further work is needed to determine whether this will help or hurt governance outcomes on the whole."
],
[
"Policy Coordination",
"There are good reasons to believe that either centralisation or fragmentation could enhance coordination.",
"A centralised regime can enable easier coordination both across and within policy issues, acting as a focal point for states.",
"Others argue that this is not always the case, and that fragmentation can mutually supportive and even more creative institutions.",
"Centralisation reduces the occurrence of conflicting mandates and enables communication.",
"These are the ingredients for policy coherence.",
"As noted previously, the WTO has been remarkably successful in ensuring coherent policy and principles across the realm of trade, and even into other areas such as the environment.",
"However, fragmented regimes can often act as complex adaptive systems.",
"Political requests and communication between secretariats often ensures bottom-up coordination even in the absence of centralisation.",
"Multiple organisations have sought to reduce greenhouse gas emissions within their respective remits, often at the behest of the UNFCCC Conference of Parties.",
"When effective, bottom-up coordination can slowly evolve into centralisation.",
"Indeed, this was the case for the GATT and numerous regional, bilateral and sectoral trade treaties, which all coalesced together into the WTO.",
"While this organic self-organisation has occurred, it has taken decades, with forum shopping and inaction prevailing for many years.",
"Indeed, some have argued that decentralisation does not just deliver `good enough' global governance [38] that reflects a demand for diverse principles in a multipolar world.",
"Instead, they argue `polycentric' governance approaches [37] may be more creative and legitimate than centrally coordinated regimes.",
"Arguments in favour of polycentricity include the notion that it enables governance initiatives to begin having impacts at diverse scales, and that it enables experimentation with diverse policies and approaches, learning from experience and best practices (ibid., 552).",
"Consequently, these scholars assume “that the invisible hand of a market of institutions leads to a better distribution of functions and effects” [50].",
"It is unclear if the different bodies covering AI issues will self-organise or collide.",
"Many of the issues are interdependent and will need to be addressed in tandem.",
"Some particular policy-levers, such as regulating computing power or data, will impact almost all use areas, given that AI progress and use is closely tied to such inputs.",
"Numerous initiatives on AI and robotics are displaying loose coordination [28], but it remains uncertain whether the virtues of a free market of governance will prevail here.",
"Great powers can exercise monopsony-like influence in forum shopping, and the supply of both computing power and machine learning expertise are highly concentrated.",
"In sum, centralisation can reduce competition and enhance coordination, but it may suffocate the creative self-organisation of more fragmented arrangements over time."
],
[
"A Summary of Considerations",
"The multilateral track record and peculiarities of AI yield suggestions and warnings for the future.",
"A centralised regime could lower costs, support participation, and act as a powerful new linchpin within the international system.",
"Yet centralisation presents risks for AI governance.",
"It could simply produce a brittle dinosaur, of symbolic value but with little meaningful impact on underlying political or technological issues.",
"A poorly executed attempt could lock-in a poorly designed centralised body: a fate worse than fragmentation.",
"Accordingly, ongoing efforts at the UN, OECD, and elsewhere could benefit from addressing the considerations presented in this paper, a summary of which is presented in Appendix A."
],
[
"The Limitations of `Centralisation vs. Decentralisation' Debates",
"Structure is not a panacea.",
"Specific provisions such as agendas and decision-making procedures matter greatly, as do the surrounding politics.",
"Underlying political will may be impacted by framing or connecting policy issues [26].",
"The success of a regime is not just a result of fragmentation, but of design details.",
"Moreover, institutions can be dynamic and broaden over time by taking in new members, or deepen in strengthening commitments.",
"Successful multilateral efforts, such as trade and ozone depletion, tend to do both.",
"We are in the early days of global AI governance.",
"Decisions taken early on will constrain and partially determine the future path.",
"This dependency can even take place across regimes.",
"The Kyoto Protocol was largely shaped by the targets and timetables approach of the Montreal Protocol, which in turn drew from the Convention on Long-range Transboundary Air Pollution.",
"This targets and timetables approach continues today in the way that most countries frame their climate pledges to the Paris Agreement.",
"The choices we make on governing short-term AI challenges will likely shape the management of other policy issues in the long term [7].",
"On the other hand, committing to centralisation, even if successful, may amount to solving the wrong problem.",
"The problem may not be structural, but geopolitical.",
"Centralisation could even exacerbate the problem by diluting scarce political attention, incurring heavy transaction costs, and shifting discussions away from bodies which have accumulated experience and practice [21].",
"For example, the Bretton Woods Institutions of the IMF and World Bank, joined later by the WTO, are centralised regimes that engender power.",
"However, those institutions had the express support of the US and may have simply manifested state power in institutional form.",
"Efforts to ban LAWS and create a cyberwarfare convention have been broadly opposed by states with an established technological superiority in these areas [13].",
"A centralised regime may not unpick these power struggles, but just add a layer of complexity.",
"Table: Regime Complex Monitoring Suggestions"
],
[
"Lessons and Conclusions",
"Our framework provides a tool for policy-makers to inform their decisions of whether to join, create, or forgo new institutions that tackle AI policy problems.",
"For instance, the recent choice of whether to support the creation of an independent IPAI involved these considerations.",
"Following the US veto, ongoing negotiations for its replacement at the OECD may similarly benefit from their consideration.",
"For now, it is worth closely monitoring the current landscape of AI governance to see if it exhibits enough policy coordination and political power to effectively deal with mounting AI policy problems.",
"While there are promising initial signs [28] there are also already growing governance failures in LAWS, cyberwarfare, and elsewhere.",
"We outline a suggested monitoring method in Table 1.",
"There are three key areas to monitor: conflict, coordination, and catalyst.",
"First, conflict should measure the extent to which principles, rules, regulations and other outcomes from different bodies in the AI regime complex undermine or contradict each other or are in tension either in their principles or goals.",
"Second, coordination seeks to measure the proactive steps that AI-related regimes take to work with each other.",
"This includes liaison relationships, joint initiatives, as well as the extent to which their rules, outputs and principles tend to reinforce one another.",
"Third, catalyst raises the important question of governance gaps: is the regime complex self-organising to proactively address international AI policy problems?",
"Numerous AI policy problems currently have no clear coverage under international law, including AI-enabled cyber warfare and HLMI.",
"Whether this changes is of vital importance.",
"These areas require investigation through multiple methods.",
"Qualitative surveys of relevant organisations and actors can yield data on expert perceptions of these questions.",
"Surveys can be augmented with quantitative methods, including network analyses of the regime complex relations [36].",
"Natural language processing could be used to examine contradictions and similarities between different regime outputs, e.g., statements, meeting minutes, and more.",
"Monitoring the outcomes of fragmentation can help to determine whether centralisation is needed.",
"One way forward would be to empower the OECD AI Policy Observatory or the UN CEB to regularly review the monitoring outcomes.",
"This could inform a democratic discussion and decision of whether to centralise AI governance further.",
"Our framework and discussion may also be useful for non-state actors.",
"Researchers and leading AI firms can play an important role in sharing technical expertise and informing forecasts of new policy problems on the horizon.",
"The considerations may benefit their decisions of where to engage.",
"Civil society has a key role as participants, watch-dogs, and catalysts.",
"For example, the Campaign to Stop Killer Robots has sought to boost engagement and support for a LAWS ban within the CCW.",
"Given prolonged delays and a pessimistic outlook, some have articulated a strategy of creating an entirely new forum for the ban, inspired by the Ottawa Treaty which outlawed landmines.",
"Our framework can help reveal the potential virtues (allowing for progress while avoiding high-threshold deadlocks) and vices (enabling forum shopping) of such an approach.",
"It could even help inform the structure of a future international institution, such as allowing for a modular, flexible structure with `critical mass' agreements.",
"One cross-cutting consideration is clear: a fractured regime sees higher participation costs that may threaten to exclude many civil society organisations altogether.",
"The international governance of AI is nascent and fragmented.",
"Centralisation under a well-designed, modular, `innovation-proof' framework organisation may be a desirable solution.",
"However, such a move must be approached with caution.",
"How to define its scope and mandate is one problem.",
"Ensuring a politically-acceptable and well-designed body is perhaps a more daunting one.",
"It risks cementing in place a fate worse than fragmentation.",
"Monitoring conflict and coordination in the current AI regime complex, and whether governance gaps are filled, is a prudent way of knowing whether the existing structure can suffice.",
"For now we should closely watch the trajectory of both AI technology and its governance initiatives to determine whether centralisation is worth the risk.",
"The authors would like to express thanks to Seth Baum, Haydn Belfield, Jessica Cussins-Newman, Martina Kunz, Jade Leung, Nicolas Moës, Robert de Neufville, and Nicolas Zahn for valuable comments.",
"Any remaining errors are our own.",
"No conflict of interest is identified."
]
] | 2001.03573 |
[
[
"Are you still with me? Continuous Engagement Assessment from a Robot's\n Point of View"
],
[
"Abstract Continuously measuring the engagement of users with a robot in a Human-Robot Interaction (HRI) setting paves the way towards in-situ reinforcement learning, improve metrics of interaction quality, and can guide interaction design and behaviour optimisation.",
"However, engagement is often considered very multi-faceted and difficult to capture in a workable and generic computational model that can serve as an overall measure of engagement.",
"Building upon the intuitive ways humans successfully can assess situation for a degree of engagement when they see it, we propose a novel regression model (utilising CNN and LSTM networks) enabling robots to compute a single scalar engagement during interactions with humans from standard video streams, obtained from the point of view of an interacting robot.",
"The model is based on a long-term dataset from an autonomous tour guide robot deployed in a public museum, with continuous annotation of a numeric engagement assessment by three independent coders.",
"We show that this model not only can predict engagement very well in our own application domain but show its successful transfer to an entirely different dataset (with different tasks, environment, camera, robot and people).",
"The trained model and the software is available to the HRI community as a tool to measure engagement in a variety of settings."
],
[
"Introduction",
"One of the key challenges for long-term interaction in human-robot interaction (HRI) is to maintain user engagement, and, in particular, to make a robot aware of the level of engagement humans display as part of an interactive act.",
"With engagement being an inherently internal mental state of the human(s) interacting with the robot, robots (and observing humans for that matter) have to resort to the analysis of external cues (vision, speech, audio).",
"In the research program that informed the aims of this paper, we are working to close the loop between the user perception of the robot as well as their engagement with it, and our robot's behavior during real-world interactions, i.e., to improve the robot's planning and action over time using the responses of the interacting humans.",
"The estimation of users' engagement is hence considered an important step in the direction of automatic assessment of the robot's own behaviours in terms of its social and communicative abilities, in order to facilitate in-situ adaptation and learning.",
"In the context of reinforcement learning, a scalar measure of engagement can directly be interpreted as a reinforcement signal that can eventually be used to govern the learning of suitable actions in the robot's operational situation and environment.",
"As a guiding principle (and indeed a working hypothesis), we anticipate that higher and sustained engagement with a robot can be interpreted as a positive reinforcement of the robot's action, allowing it to improve its behavior in the long term.",
"Previous work on robot deployment in museum contexts [7] provide evidence on how user engagement during robot guided tours easily degrades with time when employing an open-loop interactive behavior which does not take into account the engagement state of the other (human) parties.",
"However, we argue that the usefulness of a scalar measure of engagement as presented in the paper stretches far beyond our primary aim to use it to guide learning.",
"Work in many application domains of HRI [23], [3], [4] has focused on a measure of engagement to inform the assessment of the implementation for a specific use-case, or to guide a robot's behavior.",
"However, how engagement is measured and represented varies greatly (see Sec. )",
"and there is yet to be found a generally applicable measure of engagement that readily lends itself to guide the online selection of appropriate behavior, learning, adaptation, and analysis.",
"Based on the observation that engagement as a concept is implicitly often quite intuitive for humans to assess, but inherently difficult to formalize into a simple and universal computational model, we propose to employ a data-driven machine learning approach, to exploit the implicit awareness of humans in assessing an interaction situation.",
"Consequently, instead of aiming to comprehensively model and describe engagement as a multi-factored analysis, we use end-to-end machine learning to directly learn a regression model from video frames onto a scalar in the range of $0\\%$ to $100\\%$ , and use a rich annotated dataset obtained from a long-term deployment of a robot tour guide in a museum to train said model.",
"For a scalar engagement measure to be useful in actual HRI scenarios, we postulate that a few requirements have to be fulfilled.",
"In particular, the proposed solution should demonstrably generalize to new unseen people, environments, and situations; operate from a robot's point of view, forgoing any additional sensors in the environment; employ a sensing modality that is readily available on a variety of robot platforms; have few additional software dependencies to maximize community uptake; and operate with modest computational resources at soft real-time.",
"Consequently, we present our novel engagement model, solely operating on first-person (robot-centric) point of view video of a robot and prove its applicability not only in our own scenario but also on a publicly available dataset (UE-HRI) without any transfer learning or adaptation necessary.",
"We demonstrate that the model can operate at typical video frame rates on average GPU hardware typically found on robots.",
"Hence, the core contributions of this paper can be summarised as the appraisal of a scalar engagement score for the purpose of in-situ learning, adaptation, and behavior generation in HRI; a proposed end-to-end deep learning architecture for the regression of first-person view video stream onto scalar engagement factors in real-time; the comprehensive assessment of the proposed model on our own long-term dataset, and a publicly available HRI dataset proving the generalizing capabilities of the learned model; and the availability of a implementation and trained model to provide the community with an easy to use, out of the box methodology to quantify engagement from first-person view video of an interactive robot."
],
[
"Assessment of Engagement",
"Recognizing the level of engagement of the humans during the interactions is an important capability for social robots.",
"In the first place, we want to recognize the level of engagement as a way to assess the robot behavior.",
"Feeding this information to a learning system we can improve the robot behavior to maximize the level of engagement.",
"In an education scenario, such as a museum, being able to engage the users is a crucial factor.",
"It is known that higher level of engagement generates better learning outcomes [21], while engagement with a robot during a learning activity has also been shown to have a similar effect [13].",
"While there is evidence that the presence of a robot, particularly when novel, is sufficient in itself for higher engagement in educational STEM activities, e.g.",
"[2], the focus in the present work is on engagement between individuals and the robot within a direct (social) interaction, for which there is not a universally agreed definition [12].",
"Within interactions, engagement has been characterized as a process that can be separated in four stages: point of engagement, period of sustained engagement, disengagement, and re-engagement [20].",
"Context has also been identified as being of importance, in terms of the task and environment, as well as the social context [5].",
"For example, [19] proposes a simple model to infer engagement for a robot receptionist based on the person spatial position within some predefined areas around the robot, and [25] studies to what extent is possible to predict the engagement of an entity relying solely on the features of the other parties of the interaction, showing that engagement, and the features needed to detect it, changes with the context of the interaction [26].",
"These examples furthermore suggest that there are multiple, overlapping, and likely interacting timescales involved in the characterization of engagement, from the longer term context to short interaction-orientated behaviours that nevertheless impact social dynamics, and which humans are particularly receptive to [9].",
"In the context of the characterization of engagement above, there are a number of approaches to the automatic assessment of engagement that may be distinguished.",
"On the one hand, there is a focus on individual behavioral cues, which may be integrated to form a characterization of engagement.",
"On the other hand, there is a more holistic perspective of engagement taken, where proxy metrics may be used or direct measures of engagement estimated.",
"Combinations of these perspectives are summarised briefly below.",
"Work on characterizing engagement in both human-human and human-robot interactions has identified human gaze as being of particular significance when determining engagement levels in an interaction, e.g.",
"[22], [16].",
"Gaze thus forms an important behavioral cue when assessing engagement, e.g.",
"[27], [3].",
"For example, Lemaignan et al.",
"[18] do not try to directly define and detect engagement, recognizing that it is a complex and broad concept.",
"Instead, the concept of “with-me-ness” is introduced, which is the extent to which the human is “with” the robot during the interactions, and which is based on the human gaze behavior.",
"Beyond only human gaze behavior, Foster et al.",
"[11], for example, address the task of estimating the engagement state of customers for a robot bartender based on the data from audiovisual sensors.",
"They test different approaches reporting that the rule-based classifier shows competitive performances with the trained ones and could actually be preferred for their stability (and to overcome data-scarcity problems).",
"In addition to these explicitly cue-centred approaches, more recently, attempts have been made to leverage the power of machine learning to discover the important overtly visible features with minimal (or at least sparse) explicit guidance from humans (through cue identification for example).",
"For example, [29] use an active learning approach with Deep RL to automatically (and interactively) learn the engagement level of children interacting with a robot from raw video sequences.",
"The learning is incremental and allows for real-time update of the estimates, so that the results can be adapted to different users or situations.",
"The DQN is initially trained with videos labeled with engagement values.",
"In other work [24] investigate the performance of deep learning models in the task of automated engagement estimation from face images of children with autism using a novel deep learning model, named CultureNet, which efficiently leverages the multi-cultural data when performing the adaptation of the proposed deep architecture to the target culture and child, although this is based on a dataset of static images rather than real-time data.",
"These deep learning methods have the advantage that the constituent features of interest do not have to be explicitly defined a priori by the system designer, rather, only the (hidden) phenomenon needs to be annotated; engagement in this case.",
"Since social engagement within interactions is readily recognized by humans based on visible information (see discussion above), human coding of engagement provides a promising source of ground-truth information.",
"Indeed, in this context, [28] employed human coders to assess the `quality' of observed interactions, demonstrating good agreement between coder on what was a subjective metric.",
"Taken together, the literature indicates that while a precise operational definition of engagement may not be universally agreed, it seems that more holistic perspectives may be more insightful.",
"It is likely that while gaze is an important cue involved in making this assessment, there are other contextual factors that influence the interpretation of engagement.",
"Given that humans are naturally able to accurately assess engagement in interactions, it seems that one promising possibility would be to leverage this to directly inform automated systems."
],
[
"Preliminaries",
"This work is embedded in a research program that seeks to employ online learning and adaptation of an autonomous mobile robot to deliver tours in a museum context.",
"The robotic platform, described below, has been operating autonomously in this environment for an extended period of time, as evidenced by the long term autonomy metrics (Table REF ).",
"The goal is to facilitate the visitor's engagement with the museum's display of art and archaeology.",
"This project provides an opportunity to study methodologies to equip the robot with the ability to interact socially with the visitors.",
"In particular, the research aims to find a good model to allow the robot to do the correct thing at the right moment, in terms of social interaction.",
"The first step in doing so is endowing the robot with a means of assessing its own performance at any given moment to allow adaptation, learning, and to avoid repeating the same errors."
],
[
"Robotic Platform",
"The robot is a Scitos G5 robot manufactured by MetraLabs GmbH.",
"It is equipped with a laser scanner with 270$^{\\circ }$ scan angle on its base and two depth cameras.",
"An Asus xtion depth camera is mounted on a pan-tilt unit above his head and a Realsense D415 is mounted above the touchscreen with an angle of 50$^{\\circ }$ w.r.t.",
"the horizontal plane in order to face the people standing in front of the robot.",
"The interactions with the visitors are mediated through a touch screen, two speakers, a microphones array and a head with two eyes that can move with five degrees of freedom to provide human-like expressions.",
"To ensure safe operations in public environments the robot is equipped with an array of bumpers around the circular base with sensors to detect collisions and two easily reachable emergency buttons that, when activated, cuts the power to the motors.",
"The software framework is based on ROS and uses STRANDS project [14] core modules for topological navigation, people tracking, task scheduling and data collection."
],
[
"Long Term Deployment Analysis",
"The data gathered so far spans the date range between the 24th January 2019 (day on which we started recording data of the robot operations) and the 9th May 2019, with data collection remaining ongoing.",
"The work and data recording exercise has been approved by the University of Lincoln's Ethics Board, under approval ID \"COSREC509\".",
"The ethical approval does not allow the public release of any data that can feature identifiable persons, in particular video data.",
"Table: Long-Term Autonomy metrics: total system lifetime (TSL - how long the system is available for autonomous operation), and autonomy percentage (A% - duration the system was actively performing tasks as a proportion of the time it was allowed to operate autonomously), following .During the current deployment the robot performs mainly two types of interactive task: guided tour and go to exhibit and describe.",
"In the first task the robot guides the users to 5 or 6 exhibits sequentially around the museum, describing what they contain when stopping in front of each.",
"During the second interactive task the robot guides to users to one of the exhibits and, when arrived at the destination, describes the content it is showing."
],
[
"TOGURO Dataset Collection",
"The TOur GUide RObot (TOGURO) dataset was collected from the two cameras mounted on the robot's body and head, each providing a stream of rgb and depth frames.",
"These video streams were collected from the start until the end of each guided tour and go to exhibit and describe task.",
"Considering the large number of videos to be stored each we saved the frame streams as compressed MPEG video files directly from the interaction.",
"Moreover, we store, in an additional file, the ROS timestamp at the time each frame is received by the video recorder node.",
"This allows us to reconstruct afterward the alignment between the different video streams frame by frame.",
"The participants were aware that the robot was recording data during the interactions (by means of visible signs and leaflets), although they were not informed that the purpose of this data was for engagement analysis, thus not biasing their behaviours.",
"In total we collected 703 distinct interactions with a total duration of 40 hours and 17 minutes.",
"As described below (section REF ), only a subset of this total data was coded.",
"Given the unconstrained setting, the interactions varied significantly in duration, with the shortest at 1.2 seconds and the longest at 2 hours 40 minutes.",
"Given that the museum in which the robot is deployed is a public space openly accessible to anyone, the interactions between the robot and the museum's visitors are completely unstructured.",
"People walking in the gallery are allowed to roam around the collection or to interact with the robot.",
"When they choose to do so they do not receive any instruction about how to interact with it explicitly, and are not observed by experimenters when doing so."
],
[
"Dataset Coding",
" In order to address the primary research goal – the assessment of robot-centric group engagement – the dataset was manually coded in order to establish a ground truth.",
"As noted previously, given that there is not a universally accepted operationalized definition of engagement, a human observer response method is employed in the present work, following the prior application of a continuous audience response method [28].",
"Figure: One frame from a video in the TOGURO dataset recorded from the robot's head camera during a guided tour.",
"The red, green and blue plots at the bottom of the frame represent each a distinct annotation sequence.The annotations were performed over only the rgb stream of robot's head camera, and not taking into account all the four video streams available from the collected data.",
"Similarly to [28], the annotators were asked to indicate in real time how engaged people interacting with a robot appeared to be in a video captured by the robot (e.g.",
"Figure REF ).",
"They operated a dial using a game-pad joystick while watching the interaction videos using the NOVA annotation toolhttps://github.com/hcmlab/nova [1].",
"This procedure allowed the generation of per-frame annotations of the provided videos, with very little time spent on software training (around 20 minutes per annotator) and on the annotation process itself (not more than the duration of the videos).",
"The annotators were instructed by providing them with a demonstration, and a set of annotation rules based on a set of typical examplesAvailable at: https://justpaste.it/6p1tb/pdf.",
"Three annotators took part in the coding process: each was familiar with the robot being used and the interaction context.",
"Three subsets of the overall dataset collected were randomly drawn and assigned to the annotators.",
"The subsets were partially overlapping.",
"This was to enable an analysis of inter-rater agreement to assess reliability of the essentially subjective metric, but also to maximize annotation coverage of the dataset.",
"As indicated in Table REF , the total length of the annotated data was over nine hours, with 3 hours 27m of overlap between the annotators (resulting in 5 hours 50m of unique videos annotated).",
"The amount of annotated data is depicted in Table REF .",
"96 unique videos were coded by the three annotators with a total of 146 videos (including repeated annotations) for a total duration of 9 hours and 17 minutes.",
"In total the annotated video set features 227 people (53.74% (122) females and 46.26% (105) males, 60.79% (138) adults and 39.21% (89) minors).",
"The composition of each group of people interacting with the robot is very diverse; on average each videos features $2.41$ people ($min=0, max=9, \\sigma =1.56$ ), $1.32$ females ($min=0, max=6, \\sigma =0.89$ ), $1.14$ males ($min=0, max=5, \\sigma =1.26$ ), $1.5$ adults ($min=0, max=5, \\sigma =0.97$ ) and $0.96$ minors ($min=0, max=6, \\sigma =1.14$ ).",
"Table: Video annotations by annotator (coder): unique indicates length of video coded by a single coderThe annotated engagement rating is a continuous scalar for every frame of video data.",
"As such, Spearman's rank correlation ($\\rho $ ) is employed to assess inter-rater agreement.",
"Table REF shows the correlation values for each pair of annotators.",
"Since every frame is annotated (with a frame-rate of 10 frames-per-second), the continuous values were smoothed over time, using different smoothing constant values, in the range $[0.1s, 40s]$ (Figure REF ).",
"Table REF provides a summary of these, with overall mean agreement rates at selected representative values of the smoothing constant.",
"While there is some variability in the between-coder agreement, mean values of $\\rho $ vary in strength from moderate to strong (0.56 to 0.72).",
"In this regard, there is a trade-off to be made between the smoothing constant size and the apparent agreement between the coders: the larger time window size reduces the real-time relevance of the engagement assessment, even though the agreement over the extended periods of time is greater than in comparatively shorter windows.",
"Overall, these results indicate that the use of the independently coded data can be considered reliable in terms of the highly variable and subjective metric of engagement.",
"Table: Spearman's Correlation ρ\\rho at different smoothing constant values SS.",
"The significance pp-value <0.001< 0.001 and sample size n≥89n \\ge 89 for all coder pairs and smoothing constants.Figure: Spearman correlation averaged over coder pairs and weighted by the overlap rate.",
"Value reported over different smoothing constants SS.",
"Figure: Overview of the proposed model.",
"The input is a video stream of interactions between the robot and humans collected in ww size intervals.",
"The frames x i x_i are passed through the pre-trained CNN (ResNet) producing a per-frame feature vector which is then passed sequentially to the LSTM network.",
"After ww steps the LSTM produces a temporal feature vector which is passed to a FC layer with sigmoid activation to produce an engagement value yy for the temporal window.Given the ground-truth provided by the human-coded engagement levels within interactions with the robot, we propose a deep learning approach for the estimation of human engagement from video sequences.",
"The model is trained end-to-end from the raw images coming from the robot's head camera to predict a high-level engagement score of people interacting with the robot.",
"It should be noted that this model does not model individual humans in the view of the robot but provides an overall holistic engagement score.",
"The network architecture, depicted in Figure REF , is composed of two main modules: a convolutional module which extracts frame-wise image features and a recurrent module that aggregates the frame features over a time to produce a temporal feature vector of the scene.",
"The convolutional module is a ResNetXt-50 Convolutional Neural Network (CNN) [30] pre-trained on the ImageNet dataset [17].",
"We obtain the frame features from the activation of the last fully connected layer of the CNN, with dimension 2048, before the softmax layer.",
"The recurrent module is a single layer Long-Short Term Memory (LSTM) [15] with 2048 units followed by a Fully Connected (FC) layer of size $2048 \\times 1$ .",
"The LSTM takes in input a sequence of $w$ frame features coming from the convolutional module and produces in turn a feature vector that represents the entire frame sequence, to capture temporal behavior of humans within the time window $w$ .",
"The temporal features are passed through the FC layer with a sigmoid activation function at the end to produce values $y^{\\prime } \\in [0,1]$ The recurrent module is trained in our experiments to predict engagement values from the provided annotation values, while the CNN layer is fixed.",
"The proposed framework is implemented in Python using the Keras library [6] and will be freely released as a ready-to-use tool to the HRI community."
],
[
"Experiments",
"We train and test the model presented in Section on our own TOGURO dataset, and assess generalization of this model (without modification) on the public UE-HRI dataset in the following subsections."
],
[
"TOGURO Dataset Processing",
"We used the entire annotated dataset presented in Section REF , composed of 94 videos, for a total duration of 5 hours and 50 minutes of interactions.",
"For each video we randomly choose an annotation, if multiple are available from the different coders (see table REF ), in order to avoid repetitions in the data and biasing the model toward those videos that have been annotated multiple times.",
"Each video is then randomly assigned to either the training, test or validation set with a corresponding probability of 50%, 30% and 20%, respectively, to prevent our model to train and test over data that are closely correlated at the video frames level.",
"Sampling for the dataset split hence operates on full video level, rather than on frame level.",
"Each video $V_k$ is composed of $I_{V_k}$ frames $x_i \\in V_k$ for $i \\in 0, \\dots , I_{V_k}$ and has an associated array of annotations $A_k = [y_0, \\dots , y_{I_{V_k}}]$ , also of dimension $I_{V_k}$ .",
"From all the videos in each set (training/test/validation) we extract all the possible sequences of $w$ consecutive frames $X_i = [x_i, \\dots , x_{i+w-1}]$ to be the input sample for our model.",
"Therefore, each sample $X_i$ has an overlap of $w-1$ frames with the consecutive sample $X_{i+1}$ from the same video.",
"For each sample $X_i$ we assign the ground truth value $y_{i+w-1} \\in A_k$ , in order to relate each sequence of frames with the engagement value set at the end of the sequence.",
"After the pre-processing phase over our dataset we obtain 93271 training samples, 72146 test samples and 44581 validation samples.",
"Each frame is reshaped to $224 \\times 224$ pixel frames, and normalized before being fed to the network."
],
[
"Training and Evaluation",
"For training and evaluation we decided to set the window size $w$ equal to 10 frames in order to have a model that gives evaluations of the engagement in a relative short times (i.e.",
"after 1 second).",
"Even though more temporally extended time windows would provide more coherent ground truth values among the different annotators, as discussed in Section REF , we decide to sacrifice some accuracy in favour of increase realtimeness of our model predictions.",
"During training the weights of the Convolutional module, which is already pre-trained, are kept frozen while the Recurrent module is fully trained from scratch.",
"The model is trained to optimize the Mean Squared Error (MSE) regression loss between the prediction values $y^{\\prime }_i$ and the corresponding ground truth values $y_i$ using the Adagrad optimization algorithm [8] with an initial learning rate $lr = 1e-4$ .",
"At each training epoch we sample uniformly $20\\%$ of the training set samples to be used for training and we collect them in batches of size $bs=16$ .",
"The uniform data sampling of the training data is performed in order to reduce training time and limiting overfitting [10].",
"The model has been trained for a total of 22 epochs using early stopping after no improvement on validation loss."
],
[
"Assessing generalization",
"In order to assess the generalization capabilities of our trained model over different scenarios featuring people interacting with robots, we propose to test the performance of our trained model as a detector of the start and end of interactions over the UE-HRI dataset [4].",
"Similarly to our dataset, it provides video recordings from the robots own cameras allowing for engagement estimation from the robot's point of view.",
"The dataset provides videos of spontaneous interactions between humans and a Pepper (Softbank Robotics) robot alongside annotations of start/end of interactions and various signs of engagement decrease (Sign of Engagement Decrease (SED), Early sign of future engagement BreakDown (EBD), engagement BreakDown (BD) and Temporary Disengagement (TD)).",
"The UE-HRI dataset features 54 interactions with 36 males and 18 females, where 32 are mono-users and 22 are multiparty.",
"For a fair comparison with our proposed method, we evaluate the ability of our model to distinguish between the moments during which an interaction is taking place and those in which there is a breakdown (TD or BD), the interaction is not yet started or it is already ended, in line with the UE-HRI coding scheme.",
"Consequently, we predict engagement values over the RGB image streams from the Pepper robot's front camera.",
"By setting a threshold value $thr$ we convert the predictions $y^{\\prime }$ into a binary classification of $C = \\lbrace \\top , \\bot \\rbrace $ (prediction above or below $thr$ ) which indicates whether there is engagement or not.",
"The categorical predictions are then compared with values from the annotations in the dataset.",
"We consider the ground truth value to be $y_{int}^t = \\top $ if at time $t$ there is a annotation of a Mono or Multi interaction and there are no annotations of BD or TB in the UE-HRI coding.",
"The ground truth value is $y_{int}^t = \\bot $ otherwise."
],
[
"Results",
"With our evaluation we set out to provide evidence that our model is able to predict engagement through regression on our own TOGURO dataset by assessing its accuracy in comparison to the ground-truth annotation, and to assess the generalization ability of the model on newly encountered situations through the analysis of the UE-HRI data.",
"To show the ability of our framework to map short-term human behavioral features from image sequences into engagement scores, we compute the Mean Squared Error (MSE) prediction loss on our test set as $0.126$ (in the context of the $[0,1]$ interval of output expected), also reported in Table REF .",
"Looking back at section , soft real-time operation is seen as a requirement for the applicability of our model.",
"Hence, we measured the duration of a forward pass on our GPU hardware of 10 consecutive frames (1 sample) through the the convolutional module and the recurrent module taking at most $50 ms$ (worst case), allowing real-time estimation of engagement at 20 frames per second.",
"Figure: ROC curve generated using our trained model as a classifier of the interaction sessions for the UE-HRI dataset.Evaluating the power of our approach for binary classification on the UE-HRI as detailed above in section REF , allows us to capture the generalization capabilities.",
"In Figure REF we report the Receiver Operating Characteristic (ROC) curve obtained by varying the threshold with values in the range $thr \\in [0, 1]$ of the binary classification task on the UE-HRI data.",
"The Area Under the Curve ($AUC = 0.88$ in our experiment) reports the probability that our classifier ranks a randomly chosen positive instance $y_{int}^t = \\top $ higher than a randomly chosen negative one $y_{int}^t = \\bot $ , i.e., provides a good assessment of the performance of the model in this completely different dataset.",
"Figure: UE-HRI dataset: two sequences of short timescale sequential frames showing how the temporal diverting of attention is reflected in the model predicting a lower engagement value.",
"Red plot shows the predicted engagement values over the frame sequences, with the prediction y ' y^{\\prime } at the frame shown in picture at time tt being in the center, past predictions on the left and future predictions on the right.Figure: UE-HRI dataset: examples of correct prediction of high engagement (y ' >=0.75y^{\\prime } >= 0.75) in situations difficult to understand using standard face description features.",
"Red plot shows the predicted engagement values over the frame sequences with the prediction y ' y^{\\prime } at the frame shown in picture being in the center, past predictions on the left and future predictions on the right.Figure: UE-HRI dataset: examples of correct low/medium engagement prediction (y ' <=0.6y^{\\prime } <= 0.6) in cases in which the people were not actually engaging with the robot.",
"Red plot shows the predicted engagement values over the frame sequences with the prediction y ' y^{\\prime } at the frame shown in picture being in the center, past predictions on the left and future predictions on the right."
],
[
"Discussion and Conclusion",
"This paper has motivated, developed and validated a novel easy-to-use computational model to assess engagement from a robot's perspective.",
"The results presented in the previous sections lead us to the conclusion that a moderate to strong inter-rater agreement (see table REF ) in measuring engagement on $[0,1]$ interval indicates that human can reasonably and reliably assess the holistic engagement from a robot's point of view solely from video; a two-stage deep-learning architecture as presented in figure REF trained from our TOGURO dataset is a suitable computational regression model to capture the inherent human interpretation of engagement provided by the annotators; and that the trained model is generic enough to be successfully applied in a completely different scenario, here the UE-HRI dataset, showing applicability of the model also in different environments, on a different robot with a different camera, and with different tasks and people.",
"The area under the Receiver-Operator Curve (ROC) of $0.88$ in figure REF evidences that indeed the proposed regression model can serve as a strong discriminator to identify situations of loss of engagement (TD or BD in the UE-HRI coding scheme).",
"Given these encouraging quantitative results, some qualitative assessment of exemplary frames with the corresponding computed engagement score are presented in figures REF , REF and REF .",
"All figures show examples of the UE-HRI dataset, which was been completely absent from the training dataset (Section REF ).",
"Figure REF presents two short sequences (roughly 2 seconds apart between frames), showcasing short-term diversion of attention of subjects resulting in a temporarily lower engagement score, but not leading to a very low engagement.",
"Figure REF exemplifies that our model can cope well with perception challenges which would forgo a correct assessment just using gaze or facial feature analysis.",
"While one could in this context argue that our model has simply learned to detect people, figure REF is providing three examples from different videos of the UE-HRI dataset with people present in the vicinity of the robot, but not engaging with it.",
"The engagement score in these examples are significantly lower across all frames.",
"We hypothesize that the learned model does not solely discriminate only person and/or face presence, but that the temporal aspects of the humans' behavior observable in the video are captured by the LSTM layer in our architecture well enough to successfully deal with these situations.",
"These qualitative reflections are evidently supported by the quantitative analysis on both datasets, providing us with confidence that the trained model is broadly applicable and can serve as a very useful tool to the HRI community with its modest computational requirements and high response speed in assessing videos from a robot's point of view.",
"We thank the annotators, the Lincolnshire County Council and the museum's staff for supporting this research."
]
] | 2001.03515 |
[
[
"Detectability of embedded protoplanets from hydrodynamical simulations"
],
[
"Abstract We predict magnitudes for young planets embedded in transition discs, still affected by extinction due to material in the disc.",
"We focus on Jupiter-size planets at a late stage of their formation, when the planet has carved a deep gap in the gas and dust distributions and the disc starts being transparent to the planet flux in the infrared (IR).",
"Column densities are estimated by means of three-dimensional hydrodynamical models, performed for several planet masses.",
"Expected magnitudes are obtained by using typical extinction properties of the disc material and evolutionary models of giant planets.",
"For the simulated cases located at $5.2$ AU in a disc with local unperturbed surface density of $127$ $\\mathrm{g} \\cdot \\mathrm{cm}^{-2}$, a $1$ $M_J$ planet is highly extincted in J-, H- and K-bands, with predicted absolute magnitudes $\\ge 50$ mag.",
"In L- and M-bands extinction decreases, with planet magnitudes between $25$ and $35$ mag.",
"In the N-band, due to the silicate feature on the dust opacities, the expected magnitude increases to $40$ mag.",
"For a $2$ $M_J$ planet, the magnitudes in J-, H- and K-bands are above $22$ mag, while for L-, M- and N-bands the planet magnitudes are between $15$ and $20$ mag.",
"For the $5$ $M_J$ planet, extinction does not play a role in any IR band, due to its ability to open deep gaps.",
"Contrast curves are derived for the transition discs in CQ Tau, PDS70, HL Tau, TW Hya and HD163296.",
"Planet mass upper-limits are estimated for the known gaps in the last two systems."
],
[
"Introduction",
"For the last two decades a large number of exoplanet detections have remarkably expanded and shaped the prevailing planet formation theories.",
"Most of these discoveries have been accomplished using indirect techniques, although in a few cases detections were achieved using direct imaging , , .",
"Direct observations are only possible for systems close enough to us, with planets far from their host stars .",
"When searching for young planets of only a few $\\mathrm {Myr}$ an additional problem arises: while a young planet might itself be bright enough to be detected, it will still be embedded in the primordial disc, and thus hidden behind large columns of dust and gas.",
"Studying the interaction between the disc material and the young planet during its formation is crucial to understand the ongoing processes of planet formation and its further evolution.",
"After the initial stages of planet formation, the protoplanet is likely accreting material and still surrounded by gas and dust.",
"Several processes –mainly internal and/or external photo-evaporation, accretion onto the central star and planet, magneto-hydrodynamical winds continuously reduce the disc material until its complete dispersal.",
"The typical lifetimes for discs can vary considerably and are uncertain, in general the inner disc (within a fraction of AU from the central star) is expected to disperse within a few $\\mathrm {Myr}$ , even though there is potential evidence for replenishment of the inner disc over longer timescales [7], .",
"The dissipation timescales of the outer disc, which is more relevant for the direct detectability of young protoplanets, is much more uncertain.",
"Initial Atacama Large Millimeter/submillimeter Array (ALMA) surveys suggest that the depletion of the outer disc may also proceed on a similar timescale, following a simple estimate of disc masses based on (sub)-millimetre continuum emission from large dust grains [6].",
"More detailed studies seem to imply that gas and dust depletion may be substantial even at young ages , .",
"As the disc keeps losing material, the extinction is reduced proportionally.",
"If dust extinction has decreased enough, a detection of an embedded planet may be possible with state-of-the-art facilities observing at IR wavelengths, where the planet spectrum peaks.",
"Consequently, detections of protoplanets embedded in discs depend on the properties of the planet, its immediate surroundings, and also on the upper atmospheric layers of the disc.",
"The search for indirect detections in (sub)-millimetre observations has been pursued during the past years , [9], more intensively once ALMA started operating [1].",
"Substructure like cavities, gaps and spirals could be first observed at these wavelengths, suggesting planet-disc interaction as a plausible cause of such features , .",
"The DSHARP large program [5] has confirmed that substructure is ubiquitous in large discs when observed with enough resolution, although these disc features do not necessarily confirm the presence of planets.",
"The first indirect detection of planets at these wavelengths was achieved from detailed analyses of the gas kinematics in the HD 163296 disc , .",
"While these indirect detections with ALMA and other (sub)-millimetre facilities tantalise evidence for young, and in some cases, massive planets in discs, the interpretation is not unique.",
"Direct detection of young planet candidates is required to confirm their presence in discs.",
"Additionally, direct detection from IR and spectroscopy is crucial to further characterise the planet properties (e.g.",
"its atmosphere).",
"Several attempts have been made to detect directly young planet candidates in discs, by means of near and mid-IR high contrast imaging, but the vast majority of these efforts resulted in no detections.",
"searched for planets in HL Tau in $L$ -band, without any point-sources detected but setting upper limits of planet masses at the rings location.",
"Much more stringent upper limits were set for the TW Hya system by with the Keck/NIRC2 instrument, and for the HD 163296 system by .",
"In only a few occasions, point-like sources have been claimed as detections in other protoplanetary discs.",
"and announced direct evidence (as a point-sources) of protoplanets embedded in the HD 169142 and HD 100546 at $L$ - and $M^{\\prime }$ -bands respectively, using the NaCo instrument at the Paranal Observatory of the European Southern Observatory.",
"While apparently convincing, both detections have been disputed and further confirmation is still pending.",
"More recently, an additional candidate, still requiring confirmation, has been detected by in the spiral arm of MWC 758 in $L$ -band.",
"To date, the most convincing direct detection of a young planet in a protoplanetary disc, is the multi-wavelength detection of PDS 70b, confirmed using multiple epoch data ; the system also includes an additional planet, PDS 70c .",
"Hydrodynamical (HD) simulations of planet-disc interactions have greatly improved our theoretical understanding of these systems.",
"and [10] simulated a planet embedded in a disc in 2D, showing the formation of gaps from the planet-disc interaction and deriving crucial properties of protoplanetary discs as mass accretion and viscosity.",
"Many other studies followed, focusing on the characterisation of geometrical properties and the derivation of important disc evolution parameters .",
"The implementation of nested grids alleviated the resolution limitation and computation times, allowing for the first 3D simulations to be performed .",
"The detectability of a protoplanet at different wavelengths can also been inferred from HD-simulations.",
"Indeed, performed mock observations to infer planet signatures that could be detected using ALMA, while focused on the detectability in IR-bands of the circumplanetary disc (CPD) around highly accreting planets.",
"Other indirect observable signatures due to the presence of planets have been also investigated: gap opening effects at various wavelengths , , , , , disc inclination effects , spiral arms in scattered light , , , , planet shadowing , and even the effect of migrating planets , .",
"This study focuses on the early evolutionary stages, when the accretion of disc material onto the planet might still have an incidence on the total planet brightness.",
"We performed 3D HD simulations, using high resolution nested grids in the planet surroundings.",
"Column densities and extinction coefficients at the planet location are derived from the simulations, in order to infer planet magnitudes in $J$ -, $H$ -, $K$ -, $L$ -, $M$ - and $N$ -bands.",
"This model can be used to guide future direct imaging observations of young planets embedded in protoplanetary discs.",
"The work is organised as follows: in Section we describe the simulations set-up and the model.",
"The results of the different simulated systems are presented in Section .",
"The application of the model to known protoplanetary discs is discussed in Section , and the main implications of this work are summarised in Section ."
],
[
"Set-up and Model description",
"The HD simulations were performed with the PLUTO code , , a 3-dimensional grid-code designed for astrophysical fluid dynamics.",
"The details of the simulations set-up, the description of the planet flux model used (REF ) and the derivation of the expected planet magnitudes (REF ) are presented in this section."
],
[
"Simulation Set-up",
"We modelled a gaseous disc in hydrostatic equilibrium around a central star with a protoplanet on a fixed orbit.",
"The disc evolution is determined by the $\\alpha $ -viscosity prescription .",
"The disc is considered to be locally isothermal, for which the Equation-of-State (EoS) is described as: $P = n \\cdot k_{B} \\cdot T = \\frac{\\rho }{m_u \\cdot \\mu } \\cdot k_{B} \\cdot T$ where $P$ is the pressure, $n$ the total particle number density, $k_B$ the Boltzmann constant, $T$ the temperature, $\\rho $ the gas density, $m_u$ the atomic mass unit and $\\mu $ the mean molecular weight.",
"The temperature in the disc varies only radially: $T(R) = T_0 \\bigg ( \\frac{R}{R_0} \\bigg ) ^{q}$ with $R$ being the radius in cylindrical coordinates, $R_0 = 5.2$ AU, $T_0 = 121$ $K$ and $q$ the temperature exponent factor, set to $-1$ .",
"The adopted density distribution of the protoplanetary disc is described as: $\\rho (r, \\theta ) = \\rho _0 {\\bigg ( \\frac{R}{R_0} \\bigg )}^{p} \\exp \\bigg [ \\frac{G M_{\\star }}{c_{\\mathrm {iso}}^{2}} \\cdot \\Big ( \\frac{1}{r} - \\frac{1}{R} \\Big ) \\bigg ]$ where $r$ refers to the radial distance from the centre in spherical coordinates, $\\theta $ the polar angle (thus $R = r \\cdot \\cos (\\theta )$ ), $\\rho _0$ the density at the planet location, $p$ is the density exponent factor, in our case with value $-1.5$ , $G$ is the universal gravitational constant, $M_\\mathrm {\\star }$ the mass of the central star, and $c_{\\mathrm {iso}}$ the isothermal speed of sound.",
"From this equation, the surface density $\\Sigma $ scales radially as: $\\Sigma (R) = \\Sigma _0 \\cdot {\\bigg ( \\frac{R}{R_0} \\bigg )}^{-p^{\\prime }}$ with a power law with index $p^{\\prime }=-0,5$ .",
"The planet is included as a modification in the gravitational potential of the central star in the vicinity of the planet location, which is kept fixed.",
"The gravitational potential $\\phi $ considered in the simulations is: $\\phi = \\phi _{\\star } + \\phi _{\\mathrm {pl}} + \\phi _{\\mathrm {ind}}$ $\\phi _{\\star }$ is the term due to the star, $\\phi _{\\mathrm {pl}}$ the planet potential and $\\phi _{\\mathrm {ind}}$ accounts for the effect of the planet potential onto the central star.",
"In the cells closest to the planet (cells at a distance to the planet lower that the smoothing length $d_{\\mathrm {rsm}}$ ), $\\phi _\\mathrm {pl}$ is introduced with a cubic expansion as in to avoid singularities at the planet location: $\\phi _\\mathrm {pl} (d<d_{\\mathrm {rsm}}) = - \\frac{G M_{\\mathrm {pl}}}{d} \\bigg [ {\\Big ( \\frac{d}{d_{\\mathrm {rsm}}} \\Big )}^{4} -2 {\\Big ( \\frac{d}{d_{\\mathrm {rsm}}} \\Big )}^{3} +2 {\\Big ( \\frac{d}{d_{\\mathrm {rsm}}} \\Big )} \\bigg ]$ with $d$ referring to the distance between cell and planet.",
"We set $d_\\mathrm {rsm}$ to be $0.1$ $R_\\mathrm {Hill}$ in the first 4 simulations.",
"In two additional runs we decreased this value and doubled the grid resolution in order to investigate the behaviour of the density in the cells close to the planet surface.",
"In Table REF all the values of $d_\\mathrm {rsm}$ used in the simulations are shown.",
"The gas rotational speed ($\\Omega $ ) is sub-keplerian; the additional terms arise from the force balance equations in radial and vertical directions .",
"It is described by: $\\Omega (R, z) = \\Omega _K {\\bigg [ (p + q) {\\Big ( \\frac{h}{R} \\Big )}^{2} + (1+q) - \\frac{q R}{\\sqrt{R^2 + z^2}} \\bigg ]}^{1/2}$ with $z$ being the vertical coordinate, $\\Omega _k$ the keplerian orbital speed, and $h$ the vertical scale-height of the disc.",
"The simulated protoplanetary discs had a stellar mass of $1.6$ $M_{\\odot }$ and a surface density at $5.2$ AU of 127 $\\mathrm {g} \\cdot \\mathrm {cm}^{-2}$ , consistent with estimates of the Minimum Mass Solar Nebula and the densest discs in the star forming regions in the Solar neighbourhood.",
"[2], .",
"The model can be adapted to systems with different characteristics by re-scaling our results to different surface densities and other parameters, as shown in section .",
"We modelled three different planetary masses (1, 2 and 5 $M_\\mathrm {J}$ ) embedded in a viscous disc with $\\alpha = 0.003$ .",
"A fourth inviscid run with a 1 $M_\\mathrm {J}$ was performed in order to study the effect of viscosity on planet detectability, since recent studies with non-ideal magneto-hydrodynamical effects have shown that the disc can be laminar at the mid-plane .",
"Besides, two additional runs with doubled grid resolution were done for the 1 and 5 $M_\\mathrm {J}$ cases, in order to improve our understanding of the disc-planet interaction at the upper atmospheric layers of the planet.",
"In total 6 simulations were carried out, summarised in Table REF .",
"During the first 20 orbits the planetary mass was raised until its final value (1, 2 or 5 $M_\\mathrm {J}$ ) following a sinusoidal function, in order to prevent strong disturbances in the disc.",
"The simulations ran for 200 orbits, once a steady-state of the system is reached.",
"Table: Set-up parameters of the simulations and inferred column mass densities σ\\sigma .",
"The disc aspect ratio HH, defined as H=h/RH = h/R was set to 0.050.05 in all the simulations.",
"The columns refer to: the α\\alpha -viscosity parameter for a viscously evolving disc; cell size at planet vicinity, given in Hill radii (R Hill R_\\mathrm {Hill}) and in planet radii (R pl R_\\mathrm {pl}); smoothing and accretion radii; and predicted column mass densities in g· cm -2 \\mathrm {g} \\cdot \\mathrm {cm}^{-2}.",
"σ\\sigma is obtained integrating over every cell above the planet except the cells within the d rsm d_{\\mathrm {rsm}}.",
"The uncertainty is computed from the dispersion of the σ\\sigma value in the last 10 orbits.",
"Planet radii are taken from the evolutionary models of giant planets of .The resolution at the vicinity of the planet is crucial for our study in order to obtain realistic infrared (IR) optical depths due to the disc material close to the planet surface.",
"To fulfil this, we used a 3-level nested grid with a maximum resolution of $0.02$ $R_\\mathrm {Hill}$ .",
"For the 1, 2 and 5 $M_\\mathrm {J}$ planets, and considering planetary radii of $1.74$ , $1.69$ and $1.87$ $R_J$ from the 1 Myr old hot-start models of (details of these models in Section REF ), the maximum resolution corresponds to $7.3$ , $9.5$ and $11.8$ planetary radii respectively.",
"In the 2 additional runs with doubled resolution over the entire grid, the smallest cell-size was set to $0.01$ $R_\\mathrm {Hill}$ .",
"To test the accuracy of our simulations, we inspected the gas streamlines in the vicinity of the planet (Section REF ).",
"The grid is centred at the star location, thus close to the planet the grid appears to be Cartesian.",
"As suggested by , , the grid describes the CPD correctly if the gas streamlines form enclosed circular orbits around the planet location, which is confirmed in our simulations (Figure REF ).",
"Figure: Density map at the mid-plane near a 1 M J M_J planet.",
"The gas streamlines are plotted on top, showing the circular motion of the gas around the planet.",
"A value of 1 in radial code units is equivalent to 5.25.2 AU \\mathrm {AU}.",
"The colour scale shows the logarithm of the density, in g· cm -3 \\mathrm {g} \\cdot \\mathrm {cm}^{-3}.To save computational time, we assumed the disc to be symmetric with respect to the mid-plane.",
"The simulated range for $\\theta $ goes from the mid-plane up to $7^\\circ $ , adequate for a proper representation of the disc dynamics and the column density derivation.",
"At the disc upper layers, density has decreased $\\ge 2$ -3 orders of magnitude, and the contribution to the column densities of these upper layers is negligible.",
"The details for the 3 grid regions are summarised in Table REF .",
"Table: Resolution and extension (as #\\# of cells) for each coordinate (radial distance RR, polar θ\\theta and azimuthal φ\\phi angles) of our 3-levels grid.",
"There is no low-resolution level for the θ\\theta coordinate."
],
[
"Intrinsic, accretion and total planet fluxes",
"The total planet emission is considered as a combination of its intrinsic and accretion flux components.",
"At the wavelengths studied, the intrinsic flux of the planet is expected to dominate, except in those cases with very high accretion rates onto the planet.",
"The intrinsic component of the planet flux is derived from the evolutionary models of .",
"These models provide the absolute magnitudes in $J$ -, $H$ -, $K$ -, $L$ -, $M$ -, $N$ - IR bands for a range of planet masses and as a function of age, up to $100Myr$ .",
"Following their nomenclature we refer to hot-start and cold-start models to the cases of a planet fully formed via disc instability or via core accretion respectively.",
"In a more realistic scenario a planet would be characterised by an intermediate solution.",
"The total accretion luminosity is given by : $L_\\mathrm {acc} \\simeq \\frac{G M_\\mathrm {pl} \\dot{M}_\\mathrm {acc}}{R_\\mathrm {acc}}\\,$ where $M_\\mathrm {pl}$ is the planetary mass, $\\dot{M}_\\mathrm {acc}$ is the mass accretion rate onto the planet and $R_\\mathrm {acc}$ the accretion radius, the distance to the planet at which accretion shocks occur .",
"$R_\\mathrm {acc}$ is typically 2-4 times the planetary radius, we have chosen $R_\\mathrm {acc} \\equiv 4 R_\\mathrm {pl}$ for the estimation of the accretion luminosity.",
"The accretion shocks are not resolved in the simulations, this is out of the scope of this work.",
"Nevertheless, the computation of $\\dot{M}_\\mathrm {acc}$ from the simulations (explained in last paragraph of this section) are independent of $R_\\mathrm {acc}$ .",
"The accretion flux considered in this model accounts only for the accretion shocks' irradiation.",
"Flux irradiated by the CPD is not taken into account.",
"The continuum emission of the accretion shocks is at temperatures ($\\equiv T_{\\mathrm {acc}}$ ) of the order of $\\sim 10^4$ $K$ .",
"To obtain the accretion flux in each band, we approximate the shocks' emission to a black body that emits at a radius $R_{\\mathrm {acc}}$ .",
"The surface area covered by the shocks is a fraction (defined as $b_{\\mathrm {acc}}$ ) of the spherical surface with same radius.",
"In this model, the accretion flux in a given band ($F_\\mathrm {acc}^{\\mathrm {band}}$ ) is computed as a fraction $b_{\\mathrm {acc}}$ of the flux within the same band of a spherical black body ($F_\\mathrm {bb}^{\\mathrm {band}}$ ) with temperature $T_{\\mathrm {acc}}$ and radius $R_{\\mathrm {acc}}$ : $F_\\mathrm {acc}^{\\mathrm {band}} = b_\\mathrm {acc} \\cdot F_\\mathrm {bb}^\\mathrm {band}$ The factor $b_{\\mathrm {acc}}$ can be estimated as the fraction between the total accretion luminosity $L_{\\mathrm {acc}}$ (from Eq.",
"REF ) and the bolometric luminosity of the same black body: $b_\\mathrm {acc} = \\frac{L_\\mathrm {acc}^\\mathrm {tot}}{L_\\mathrm {bb}^\\mathrm {tot}}$ We tested various values of $T_{\\mathrm {acc}}$ , the results shown along this work are obtained using a shock temperature of 20000 $\\mathrm {K}$ .",
"As discussed for the band fluxes results (Section REF ), assuming lower $T_{\\mathrm {acc}}$ does not vary the fluxes results significantly.",
"The gas accretion onto the planet is modelled following the prescription in , also .",
"At each time-step, a fraction $f_\\mathrm {acc} \\cdot \\Delta t \\cdot \\Omega $ of the gas is removed from the cells enclosed by a sphere of radius $r_{\\mathrm {sink}}$ centred at the planet (values for $r_{\\mathrm {sink}}$ in Table REF ).",
"This method mimics the direct accretion onto the planet surface.",
"The values of $r_{\\mathrm {sink}}$ have been chosen to guarantee convergence; as shown in , the estimated accretion rates converge to a stable value if $r_{\\mathrm {sink}} \\lesssim 0.07$ $R_{\\mathrm {Hill}}$ .",
"This is independent of the $f_\\mathrm {acc}$ value, that is set to 1.",
"The mass accreted is removed from the computational domain instead of being added to the planet mass (accreted mass is negligible compared to the total planet mass, about $10^{-5}$ times lower)."
],
[
"Derivation of expected magnitudes",
"The extinction in our model is due to the disc material around the planet.",
"We assume that there are no additional astronomical objects of significant brightness or size between the studied planet and the observer.",
"Extinction in the $V$ -band is derived from the column mass density obtained from the simulations.",
"Using the magnitude-flux conversion formula from , assuming constant gas-to-dust ratio of 100 and a molecular weight of $2.353$ : $A_V [\\mathrm {mag}] = \\frac{N_H [\\mathrm {atoms} \\cdot {\\mathrm {cm^{-2}}}]}{2.2\\cdot 10^{21} [\\mathrm {atoms} \\cdot {\\mathrm {cm^{-2}}}\\cdot {\\mathrm {mag^{-1}}}]}$ The above relation was inferred from observations and it applies to an averaged interstellar medium (ISM) in the Milky Way.",
"While the gas-to-dust ratio used is a good first approximation, variations may be expected, particularly in the vicinity of the planet.",
"Recent surveys in close star-forming regions indicate that this ratio might indeed be different in many discs .",
"From the inferred $A_V$ , the extinction coefficients ($A_{\\mathrm {band}}$ ) and optical depths ($\\tau _{\\mathrm {band}}$ ) are obtained using the diffuse ISM extinction curves of [11] for $J$ -, $H$ -, $K$ -bands, and (which accounts for the silicate feature around $10\\mu m$ ) for $L$ -, $M$ - and $N$ -bands.",
"The expected fluxes are then given by: $F_\\mathrm {expected}^{\\mathrm {band}} = F_{\\mathrm {pl}}^{\\mathrm {band}} \\cdot e^{- \\tau _\\mathrm {band}}$ with $F_{\\mathrm {pl}}^{\\mathrm {band}}$ being the total planet flux in that band.",
"The resulting $F_\\mathrm {expected}^{\\mathrm {band}}$ is the expected band flux that we would observe for a planet embedded in a protoplanetary disc with ongoing accretion onto the planet.",
"In this section we first present the results from the HD simulations (subsection REF ), followed by the results from our model (subsection REF )"
],
[
"Results from the HD simulations",
"Jupiter-size planets embedded in a disc generally carve a gap after several orbits [10].",
"In our simulations, this can be seen in 2D density maps of the disc as a function of time (Figure REF , for the 1 $M_{\\mathrm {J}}$ case).",
"For more massive planets, the gap carving process is faster, as one would expect from planet formation theory.",
"Figure: Evolution in time of the 2D density map at the mid-plane for the viscous disc with an embedded 1 M J M_\\mathrm {J} planet.",
"A value of 1 in radial code units is equivalent to 5.25.2 AU \\mathrm {AU} (Jupiter semi-major axis).",
"Density is represented in logarithmic scale, with values in g· cm -3 \\mathrm {g} \\cdot \\mathrm {cm}^{-3}.",
"From top left to bottom right, each snapshot represents the density ρ\\rho of the disc after 1, 20, 50, 100, 150 and 200 orbits.The gap opened by each planet can be compared to disc models to verify the quality of our simulations.",
"We tested our resulting surface density profiles with an analytical model for gaps in protoplanetary discs, as described in .",
"An algebraic solution of the gap profiles is presented in that work, together with the derivation of a formula for the gap depth.",
"In Figure REF we show the azimuthally averaged surface density radial profiles relative to the unperturbed surface density ($\\Sigma _0$ ) for the 1, 2 and 5 $M_J$ simulated planets in a viscous disc.",
"The solid lines represent the profiles after a steady-state is reached, and the dashed lines denote the surface density after the first 20 orbits.",
"The predicted gap depths from the model are shown in the figure as horizontal dash-dot lines.",
"Figure: Surface density radial profile for 1, 2 and 5 M J M_J planets after the simulations reach a steady-state, shown as solid lines.",
"The dash-dot lines are the respective predicted gap-depths, derived from an analytical model for gaps in protoplanetary discs .",
"The dashed lines represent the surface density after 20 orbits.",
"A value of 1 in radial code units is equivalent to 5.25.2 AU \\mathrm {AU}.Our results for 1 and 2 $M_J$ planets are in very good agreement with the analytical model.",
"The gap for a 5 $M_J$ planet is relatively deeper than the prediction from the model.",
"Nevertheless, the model by fails at reproducing the gap profile produced by planets with very high masses, as discussed in that work.",
"Therefore, we can trust the quality of our simulations from the concordance at low planet masses with analytical models.",
"The column density is obtained by integrating the disc density over the line-of-sight towards the planet.",
"We considered our system to be face-on, since is the most likely geometry for a direct protoplanet detection.",
"The density above the planet for every simulation is shown in Figure REF , together with the unperturbed (initial) density.",
"The highest densities correspond to the 1 $M_\\mathrm {J}$ simulation.",
"For more massive planets, the disc densities are lower, since the planet carves a deeper gap.",
"In the inviscid case, the carved gap can not be refilled with adjacent material, resulting in lower densities compared to the viscously evolving case.",
"Figure: Vertical density above the planet for each simulation.",
"Horizontal axis represents the height from the mid-plane in Jupiter semi-major axis (a J a_J); vertical axis shows density in logarithmic scale, in g· cm -3 \\mathrm {g} \\cdot \\mathrm {cm}^{-3} units.The planet and its atmosphere are not resolved in these simulations, therefore the first cell is assumed to be the planet outer radius.",
"The sharp peak from the vertical density profiles extends over the first 2-3 cells.",
"This is expected to be a combination of several effects, mainly an artefact of the simulations due to the potential smoothing (Eq.",
"REF ), which affects every cell within $d_\\mathrm {rsm}$ .",
"However we cannot exclude the possibility that a fraction of the peak might also be the real density stratification of the material at the layers closest to the planet.",
"Additionally, this over-density is also altered by the accretion radius (see REF ), and limited by the grid resolution.",
"A fully resolved CPD would be necessary to disentangle between the different causes.",
"However, we tested whether the extension of the peak is an artefact due to the potential softening and the mentioned resolution limitations.",
"To this aim we performed 2 additional simulations with doubled resolution over the entire grid.",
"We also decreased both accretion and smoothing radii to $0.03$ -$0.05$ $R_\\mathrm {Hill}$ .",
"The values of the main parameters for the doubled resolution runs are summarised in Table REF .",
"This test was done for the discs with 1 and 5 $M_\\mathrm {J}$ planets.",
"The grid from the last snapshot of the original simulations were readjusted to the new resolution, and we let the system evolve until a new steady-state was reached ($\\le 15$ orbits needed in both cases).",
"The vertical density at the closest cells above the planet for both original and doubled resolution for the 1 $M_\\mathrm {J}$ case are shown in Figure REF .",
"The vertical grid-lines represent $0.01$ $R_\\mathrm {Hill}$ , and the coloured lines illustrate the $d_{\\mathrm {rsm}}$ in both original and doubled resolution runs.",
"The figure shows that the over-density with doubled resolution spans approximately half the original case.",
"This is in accordance with what we expect if the over-density is due to the smoothing within $d_{\\mathrm {rsm}}$ .",
"If we were able to completely remove the potential smoothing we would only see a peak inside the planet radius, i.e.",
"within the first cell.",
"Figure: Vertical density profile close to the 1 M J M_\\mathrm {J} planet, comparing the doubled resolution (blue line) to the original case (red).",
"The vertical grid spacing is 0.010.01 R Hill R_\\mathrm {Hill}, equivalent to the cell-size for the doubled resolution run, and half the cell-size of the original run.",
"The red and blue dashed lines represent the smoothing radii for the original and the new run respectively.From the results of this test, we consider the column mass density $\\sigma $ as the integrated density for all the cells above the smoothing radius.",
"The resulting $\\sigma $ for each of the simulated systems are included in Table REF .",
"The uncertainty considered is the dispersion of the column mass density for the last 10 orbits of each simulation.",
"The predicted magnitudes for 1 and 5 $M_J$ planets are derived using the $\\sigma $ values from the doubled resolution runs, since in these cases the planet-disc interaction is represented more accurately.",
"The values for the original and doubled resolution cases are within their respective uncertainty: for a 1 $M_J$ we obtained $2.8\\pm 0.9$ and $2.7\\pm 1.2$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ , while for the 5 $M_J$ runs the column mass densities were $0.011\\pm 0.007$ and $0.01\\pm 0.01$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ respectively."
],
[
"Mass accretion rates",
"The prescription used to derive the mass accretion rates from the simulations was described in Subsection REF .",
"The final value of $\\dot{M}_\\mathrm {acc}$ for each simulated planet is the optimal value of parameter $c$ when fitting the evolution of $\\dot{M}_\\mathrm {acc}$ in the last 50 orbits by an exponential function defined as $f(x) = a \\cdot e^{-b \\cdot x} + c$ .",
"The resulting accretion rates are of the order of $10^{-8}$ $M_{\\odot } \\cdot \\mathrm {yr}^{-1}$ , summarised in Table REF .",
"The highest $\\dot{M}_\\mathrm {acc}$ is obtained for a system with 2 $M_\\mathrm {J}$ planet.",
"This is somewhat counter-intuitive, as one may expect the least massive planet to have the lowest accretion rate, and the most massive planet to have the highest.",
"There are two effects to consider, the viscosity that refills the gap and the gravitational force of the planet: a more massive planet creates a stronger gravitational potential, a larger CPD and has a larger accretion rate, on the other hand if the disc evolves viscously it can replenish the accreted material with new material, thus keeping a higher $\\dot{M}_\\mathrm {acc}$ value.",
"Beyond a certain planet mass, the gravitational potential clears large regions quicker, which limits the refill of the gap material thus ultimately reducing the $\\dot{M}_\\mathrm {acc}$ .",
"Our results indicate that the mass of the planet for which this occurs is between 2 and 5 $M_\\mathrm {J}$ .",
"The different accretion rates obtained for the 1 $M_\\mathrm {J}$ planet in a viscous and an inviscid disc can be understood by considering that an inviscid disc cannot replenish the gap opened by a planet, and consequently there is less material to feed the CPD.",
"Care should be taken when considering these results because of the limitations posed by isothermal simulations, which do not account for accretion heating in the vicinity of the planet.",
"This results in an overestimation of the accretion rate.",
"These simulations provide an upper limit of the expected accretion rates, and this should be taken into account when the model is applied to real systems."
],
[
"Gas streamlines",
"Gas streamlines provide useful insights on the disc behaviour and the planet-disc interaction.",
"In the close-up top view of the system (Figure REF ), the gas streamlines at the system mid-plane are plotted as vectors.",
"As expected from the accretion models, a CPD is formed and the gas motion is concentric to the planet.",
"The gas streamlines follow closed trajectories in the regions nearest to the planet, which indicates that the grid does describe the accreting planet with a CPD accurately .",
"Accretion onto the planet occurs not only in the orbital plane from the CPD but also vertically .",
"In our simulated systems, the gas is indeed falling onto the planet from its pole or with small inclination angle from the vertical.",
"Figure REF shows that a high fraction of the material is being accreted vertically onto the 1 $M_{J}$ planet.",
"Figure: Edge on view density map with gas streamlines around the planet.",
"An important part of the gas is being accreted vertically.",
"A value of 1 in radial code units is equivalent to 5.25.2 AU \\mathrm {AU}.",
"The scale represents the logarithm of the density, in g· cm -3 \\mathrm {g} \\cdot \\mathrm {cm}^{-3}."
],
[
"Bolometric and band fluxes",
"Accretion and intrinsic bolometric fluxes for each planet considered in this work are shown in Table REF .",
"Mass accretion rates are also included in the table.",
"The difference in $F_\\mathrm {acc}$ between hot and cold-start models arises from the different planet radius $R_{\\mathrm {pl}}$ of each model , since the $R_{\\mathrm {acc}}$ used to compute the accretion flux (Equation REF ) is assumed to be $\\equiv 4 R_{\\mathrm {pl}}$ .",
"The bolometric accretion flux is higher than the planet's intrinsic flux for all cases.",
"Nevertheless, the accretion flux (which peaks at $0.15$ $\\mathrm {\\mu m}$ , with $T_\\mathrm {eff} \\sim 20000$ $\\mathrm {K}$ ) is in most cases lower in the IR bands considered than the intrinsic planet flux, whose spectrum peaks in the IR ($\\sim 1$ -10 $\\mu m$ ).",
"This can be seen in the left panel of Figure REF .",
"The figure shows the accretion and intrinsic fluxes of planets with 1, 2 and 5 $M_\\mathrm {J}$ for hot and cold-start planet models.",
"Intrinsic fluxes are considerably lower for cold-start planets than for the hot models.",
"The accretion contribution can be significant, especially in the cold-start cases in $J$ -, $H$ -, $K$ - and $L$ -bands.",
"Reducing the $T_\\mathrm {acc}$ shifts the accretion maximum to longer wavelengths, but it highly decreases its bolometric value, and consequently the overall picture does not vary significantly.",
"Table: Mass accretion rates and bolometric fluxes for every simulated system.",
"Fluxes are expressed in [W·m -2 \\mathrm {W} \\cdot \\mathrm {m}^{-2}].Figure: Intrinsic (F intr F_{\\mathrm {intr}}) and accretion fluxes (F acc F_{\\mathrm {acc}}) for 1, 2 and 5 M J M_J planets.",
"The results for planets with a hot-start model are shown on the left panel, while cold-start models on the right.",
"The intrinsic fluxes are plotted at the central wavelength of each band.These results indicate that radiation from accretion shocks near the planet is an important factor of the planet flux.",
"Nevertheless, it is worth keeping in mind that these results are for accretion rates inferred from isothermal simulations, which are generally overestimated.",
"For more realistic accretion rates at this stage ($\\sim {10}^{-10}$ $M_{\\odot } \\cdot \\mathrm {yr}^{-1}$ ), intrinsic flux dominates at these wavelengths.",
"When scaling our models to larger distances from the host star, accretion rates are highly reduced due to the scaling for lower disc densities.",
"Consequently intrinsic fluxes dominate in planets further out in the disc for both hot and cold-start scenarios."
],
[
"Extinction coefficients and predicted magnitudes",
"From the column mass densities we derived extinction coefficients for each planet in $J$ - $H$ -, $K$ -, $L$ -, $M$ - and $N$ -bands.",
"The inferred coefficients are included in Table REF , and plotted on the top left panel in Figure REF .",
"Extinction coefficients at wavelengths $\\lesssim 2$ $\\mu m$ are extremely high for 1 and 2 $M_J$ .",
"The effect of extinction decreases for longer wavelengths, but it raises up again at $\\sim 8$ -$12\\mu m$ due to the silicate feature present in the diffuse ISM.",
"For a 5 $M_J$ , extinction coefficients are very low in every IR band.",
"This is due to its ability to open a gap in the disc very effectively, and consequently, disc density around the planet and the inferred column density are very low.",
"Table: Absolute magnitudes for planets at 5.25.2 AU \\mathrm {AU} to the host star, with masses: 1 M J M_\\mathrm {J} –viscous and inviscid scenarios– 2 M J M_\\mathrm {J} and 5 M J M_\\mathrm {J}.",
"Mag pl \\mathrm {Mag}_\\mathrm {pl} is the total magnitude of the planet, including accretion flux; A band A_{\\mathrm {band}} is the extinction coefficient in each band, Mag expected \\mathrm {Mag}_\\mathrm {expected} is the magnitude of the planet considering extinction due to disc material.",
"All values are in mag \\mathrm {mag}.The derived magnitudes for each simulated planet as a function of wavelength are shown on the top right panel in Figure REF .",
"For each planet, the upper and lower curves represent its hot and cold models.",
"The results include extinction from the disc material and radiation from the accretion shocks near the planet.",
"The magnitudes for every IR-band and planet at $5.2$ $\\mathrm {AU}$ are summarised in Table REF .",
"For each simulated planet, the rows in the table show: the absolute magnitude of the planet including the contribution from accretion; the extinction coefficients due to the disc material, and the predicted absolute magnitude including extinction effects.",
"Figure: Top panels: extinction coefficients with uncertainties (left), and predicted magnitudes for the simulated systems (right), both as a function of wavelength.",
"The predicted planet magnitudes are shown as an area delimited by the hot and cold planetary model.",
"Bottom panels: extinction and predicted magnitudes of the 1 M J M_J viscous case using different dust grain models.",
"The results for the various dust models are normalised at A V A_V.",
"For every panel, the vertical dotted lines represent (from left to right) the central wavelength of JJ-, HH-, KK-, LL-, MM-, and NN-bands.The predicted magnitudes for a given planet decrease as a function of wavelength, except at $\\sim 10 \\mu m$ due to the silicate feature in dust grain opacities.",
"The curves are more flattened for more massive planets.",
"This is due to the more efficient depletion of material in the planet vicinity, which yields lower column densities, hence lower extinction.",
"For the most massive planet considered (5 $M_J$ ), the gap clearing is extremely effective and extinction in any IR band is negligible.",
"In the inviscid scenario with a 1 $M_\\mathrm {J}$ , the gap can not be replenished efficiently compared to the viscously evolving case.",
"This results on a gap with lower density and extinction.",
"At $\\lesssim 2$ $\\mu m$ , 1-2 $M_J$ planets are completely obscured by the disc material (with extinction above 30 and 15 $\\mathrm {mag}$ respectively).",
"These results are obtained for an unperturbed surface density of $\\Sigma = 127$ $\\mathrm {g} \\cdot \\mathrm {cm}^{-2}$ at $5.2$ $\\mathrm {AU}$ .",
"We have used the ISM law to estimate the extinction under the assumption that mostly small grains will be present in the disc atmosphere above the planet.",
"The actual value of extinction depends on the assumption on the dust properties.",
"To investigate the implications of our assumption, we evaluated the impact of using different assumptions on the dust composition and size distributions.",
"The results for the 1 $M_J$ viscously evolving case are shown in bottom panels of Figure REF .",
"Two other dust models were investigated: one model of grains with fractional abundances comparable to the expected in protoplanetary discs mid-plane and grain population with number density $n(a)\\propto a^{-3.5}$ (where $a$ is the grain size) between $0.01\\mu m < a < 1 \\mu m$ , and a dust coagulation model for ice-coated silicate-graphite aggregates , , applicable to dust in protoplanetary discs (extinction shown is for grain sizes $a \\sim 1\\mu m$ ).",
"The results of the ice silicate graphite model are in very good agreement with the ISM extinction used (in $M$ -, and $N$ -bands the diffuse ISM extinction becomes larger).",
"The other model provides similar results in the $J$ -, and $H$ -bands, however, for longer wavelengths extinctions are $\\sim 2$ -3 times larger than for the diffuse ISM.",
"Thus, when considering dust with different properties (e.g.",
"composition, size, level of processing), the resulting predicted planet magnitudes might change due to the opacity variations in IR wavelengths.",
"On the other hand, the extinction is obtained assuming a gas-to-dust ratio of 100 along the disc.",
"In the atmospheric layers of the disc above the planet, this ratio might be larger due to dust processing and settling.",
"In this regard, our analysis provides a conservative estimate of extinction, and the presented results can be interpreted as the worst case scenario."
],
[
"Scaling results",
"The simulations performed in this work are locally isothermal, therefore the results can be scaled to account for different disc densities without altering the dynamics of the disc.",
"We re-scaled column mass densities and accretion rates for different planet positions and disc densities.",
"The normalisation of the surface density is readjusted in order to preserve the surface density profile $\\Sigma (R)$ as in Equation REF .",
"In this way we could extend the results of our model to study different systems.",
"From dimensional analysis, the column mass density $\\sigma $ is directly proportional to the surface density, while the accretion rate $\\dot{M}_\\mathrm {acc}$ is proportional to the density and inversely proportional to orbital time.",
"Surface density $\\Sigma $ decreases as $R^{-0.5}$ (Equation REF ), while the density at the planet location $\\rho \\propto R^{-1.5}$ (Equation REF ).",
"These relationships are used to derive the scaling factors for $\\sigma $ and $\\dot{M}_\\mathrm {acc}$ for planets at a distance $\\ne 5.2$ $\\mathrm {AU}$ .",
"In case of re-normalising the surface density (as done for real systems in Section ) the ratio between the unperturbed new and original surface densities at a fiducial distance multiplies the scaling factors of the column mass density and the mass accretion rate.",
"Scaling the distance to the central star would change the temperature at the planet location.",
"While this has no direct impact on the scaling applied to the $\\dot{M}_\\mathrm {acc}$ obtained from the simulations (which has no explicit dependence on temperature), it would also change the disc aspect ratio $ H = h/R$ since discs are generally flared.",
"This would have an indirect effect on $\\dot{M}_\\mathrm {acc}$ .",
"In addition, the disc aspect ratio also determines the gap opening planet mass and therefore the depth of the gap.",
"We neglect these effects and remark that this is a limitation of our approach; the performed scaling provides a valuable understanding of how the planet location influences accretion rates and densities, but it does not capture all possible effects.",
"Figure: Absolute magnitudes at JJ-, HH-, KK-, LL-, MM, and NN-bands for the simulated disc with an embedded planet (1, 2, 5 M J M_J, or a 1 M J M_J planet inviscid case) at different distances to the central star.",
"The results are shown for a hot-start model of the formation scenario.",
"The vertical axis for JJ and HH-bands covers a wider range in order to include all the planets in the same panel.Tables with extinction coefficients and predicted magnitudes of the simulated systems at 10, 20, 50 and $100 \\mathrm {AU}$ for every band are included in the Appendix (Tables REF , REF , REF , REF ).",
"Figure REF shows the expected absolute magnitudes of planets with 1 (for both viscous and inviscid cases), 2 and 5 $M_J$ for different distances to the central star, in $J$ -, $H$ -, $K$ -, $L$ -, $M$ , and $N$ -bands.",
"The coloured area of each planet represents the uncertainty associated to the column density.",
"Extinction decreases for planets further out in the disc due to lower column densities.",
"This behaviour is driven by the surface density profile.",
"Accretion flux is higher at shorter distances but its effect is minor compared to extinction.",
"Further out in the disc, the planet magnitude is dominated by the intrinsic flux since the accretion drops with distance.",
"The dispersion of the results is considerable, it can not be neglected as a source of indeterminacy in our results.",
"Nevertheless, large uncertainties are linked to very high values of the column density, and in such cases extinction is so strong that the planet would be completely hidden.",
"At shorter wavelengths ($J$ - $H$ - and $K$ -bands), the magnitude of a 1 $M_J$ planet is completely dominated by extinction: at the furthermost location considered, 100 $\\mathrm {AU}$ , the extinction is $20.4$ , $13.1$ , and $8.2$ $\\mathrm {mag}$ in each of these bands.",
"For a 2 $M_J$ planet the effect of extinction is lower but still large, e.g.",
"at 100 $\\mathrm {AU}$ the extinction in these bands is $5.9$ , $3.8$ , and $2.4$ $\\mathrm {mag}$ .",
"The magnitude of a 5 $M_J$ planet in the simulated disc is barely affected by extinction in any band (the highest extinction coefficient, $A_J$ , ranges between $0.36$ to $0.08$ between $5.2$ and 100 $\\mathrm {AU}$ ): since the planet is substantially more massive, it has cleared almost all the material at the gap and its vicinity, thus both column density and the extinction coefficients are exceptionally low compared to the other simulated planets.",
"In $L$ -band, the disc material above the least massive planet causes an extinction of 10 $\\mathrm {mag}$ at 20 $\\mathrm {AU}$ , and is reduced to $4.5$ $\\mathrm {mag}$ at 100 $\\mathrm {AU}$ .",
"A 2 $M_J$ planet is less affected by extinction, with $A_L$ of $2.9$ and $1.3$ at those distances.",
"In the $M$ -band, the extinction coefficients of all the planets considered are the lowest out of all bands considered, nevertheless still considerable except for the 5 $M_J$ planet.",
"For instance, a 1 $M_J$ planet at 100 $\\mathrm {AU}$ would be extincted by $3.5$ $\\mathrm {mag}$ .",
"In the $N$ -band, extinction is increased due to the silicate feature in the opacity of ISM dust: the extinction coefficients are almost 2 times larger than the coefficients in the $M$ -band.",
"Our models can as well be used for different values of the stellar mass.",
"Scaling our results for different $M_{\\star }$ is especially useful when applying our detectability model to real systems, as discussed in the next section.",
"The ratio $\\frac{M_{\\star }}{M_\\mathrm {pl}}$ cannot change in order to keep the dynamics of the system valid.",
"The planet mass is re-scaled accordingly to keep this ratio constant.",
"Since the planetary models from only provide data for planets with 1, 2, 5 or 10 $M_{J}$ masses, the planet intrinsic magnitudes were interpolated for the new $M_\\mathrm {pl}$ ."
],
[
"Application to observed systems",
"Our results can be applied to real systems to study the detectability of embedded planets, proceeding as explained in REF .",
"In what follows we present the results of our model for the Class II discs of CQ Tau, PDS 70, HL Tau, TW Hya and HD 163296.",
"For the last three systems, our results are combined with contrast limits from previous IR observations , , .",
"The improvement of this revision is the inclusion of the extinction due to the disc material, and the emission from the shocks due to planet accretion.",
"Additionally, to understand how likely would be to detect the simulated planets directly with ALMA, we estimated the $1 M_J$ planet and CPD fluxes at $890 \\mu m$ wavelength.",
"The expected planet flux is $\\sim 10^{-6}$ mJy, below the ALMA sensitivity limit.",
"For the CPD, simplified to a disc of $1 R_\\mathrm {Hill}$ radius centred at the planet and $0.24$ $R_\\mathrm {Hill}$ high (i.e.",
"the region around the planet with a disc-shaped overdensity), we obtain a dust mass of $M_{\\mathrm {dust}}^{\\mathrm {CPD}} \\approx 0.003 M_{\\oplus }$ , which is comparable to the CPD measurements in PDS 70b .",
"Assuming a constant CPD temperature of 121 $\\mathrm {K}$ , the continuum emission in ALMA Band 7 would be $0.07$ mJy if emission is assumed optically thin, and $0.23$ mJy if optically thick.",
"Thus, the CPD of the simulated disc could be detected by ALMA observations with enough sensitivity."
],
[
"CQ Tau",
"CQ Tau is a young star from the Taurus-Auriga region, spectral type A8 and $M_{\\star } = 1.5$ $M_{\\odot }$ .",
"It has an estimated age of $\\sim 5$ -10 $\\mathrm {Myr}$ [12], and very low disc mass, of the order of $10^{-3}$ -$10^{-4}$ $M_{\\odot }$ .",
"A fiducial surface density of $\\Sigma = 1.6 \\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ at 40 $\\mathrm {AU}$ (a factor $\\times 0.035$ respect the simulated disc) was used for the re-normalisation to our unperturbed profile, derived from ALMA observations .",
"We applied our models to investigate the effects of disc extinction on potential planets embedded in the disc.",
"Due to the re-scaling with the stellar mass, the contrast curves are derived for planets with $0.94$ , $1.88$ and $4.69$ $M_{J}$ .",
"A distance of $163.1$ $\\mathrm {pc}$ was used for the predicted contrast.",
"Figure REF shows the contrast of the planets in the $L$ -band as a function of distance to the central star; the planet contrast is shown relative to the stellar value.",
"The coloured area for each planet represents the associated dispersion.",
"The results in $J$ -, $H$ - and $K$ -bands are included in Appendix .",
"Figure: Application of our model to planets with 0.940.94, 1.881.88 and 4.694.69 M J M_J masses embedded in the CQ Tau disc.",
"The coloured lines represent the contrast in LL-band of each planet as a function of the distance to the central star placed at different distances along the disc.At a fixed distance, the more massive the planet, the less affected by extinction, since the planet is more effective at clearing the gap.",
"Due to the very low surface density inferred from the ALMA observations, extinction in $L$ -band is only relevant for the lightest planets.",
"At 20 $\\mathrm {AU}$ , a $0.94$ $M_J$ planet would have a contrast of $16.16$ $\\mathrm {mag}$ and extinction $A_L = 0.34$ $\\mathrm {mag}$ ($A_V = 5.47$ $\\mathrm {mag}$ ).",
"The contrast of a $1.88$ $M_J$ planet is $14.27$ $\\mathrm {mag}$ with an extinction of only $0.10$ $\\mathrm {mag}$ ($A_V = 1.60$ $\\mathrm {mag}$ ).",
"The most massive planet ($4.69$ $M_J$ ) is barely affected by extinction, its contrast is equivalent to the case of a completely depleted disc, $12.17$ $\\mathrm {mag}$ ."
],
[
"PDS70",
"PDS 70 is a member of the Upper Centaurus-Lupus subgroup , with a central star of $5.4$ $\\mathrm {Myr}$ and mass $0.76$ $M_{\\odot }$ .",
"It is surrounded by a transition disc with estimated total disc mass of $1\\cdot 10^{-3}$ $M_{\\odot }$ .",
"A first companion (PDS 70b) was found combining observations with VLT/SPHERE, VLT/NaCo and Gemini/NICI at various epochs, detected as a point-source in $H$ -, $K$ - and $L$ -bands at a projected averaged separation of $194.7$ $\\mathrm {mas}$ .",
"In $J$ -band, PDS 70b could only be marginally detected when collapsing the $J$ - and $H$ -band channels.",
"Due to the high uncertainties, $J$ -band magnitude was not given.",
"Atmospheric modelling of the planet was used to constrain its properties , with an estimated mass range from 2 to $17 M_J$ .",
"Recent $H_{\\alpha }$ line observations using VLT/MUSE confirmed a $8 \\sigma $ detection from a second companion (PDS 70c) at 240 $\\mathrm {mas}$ .",
"Dust continuum emission (likely from its CPD) has been also observed .",
"This second source is very close to an extended disc feature, consequently its photometry should be done with caution.",
"In , the planetary nature of this companion has been confirmed, and absolute magnitudes in $J$ -, $H$ -, and $K$ -bands could be inferred for two SPHERE epochs.",
"The spectrum in the $J$ -band is very faint, and indistinguishable from the adjacent disc feature, thus the $J$ -band magnitude should be regarded as upper limit.",
"The NaCo $L$ -band map detected emission that is partly covered by the disc, therefore its $L$ -band magnitude should also be taken as an upper limit.",
"Using various atmospheric models, constrained the mass PDS 70c to be between $1.9$ and $4.4$ $M_J$ .",
"Figure: Application of our model to planets with 0.480.48, 0.950.95 and 2.382.38 M J M_J embedded in the PDS 70 disc.",
"The contrast curves shown for HH-, KK- and LL-bands were obtained considering stellar magnitudes of H=8.8H = 8.8 mag \\mathrm {mag}, K=8.5K = 8.5 mag \\mathrm {mag} and L=7.9L = 7.9 mag \\mathrm {mag} , .",
"The two planetary companions , , are shown as black and grey crosses, with the corresponding uncertainties.Our models were re-scaled using a fiducial surface density of $\\Sigma = 12.5$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ at 1 $\\mathrm {AU}$ .",
"This corresponds to a surface density scale factor of $\\times 0.043$ with respect to the simulated disc.",
"From the re-scaling, we obtained contrast curves of planets embedded in the PDS 70 disc with $0.48$ , $0.95$ and $2.38$ $M_J$ (Figure REF ).",
"The results show the effect of a disc with very low surface density: extinction has an incidence in $J$ - and $H$ -bands for $0.95$ and $0.48$ $M_J$ planets located within $\\lesssim 40$ $\\mathrm {AU}$ .",
"In the $L$ -band extinction has only a minor effect on the lightest planet model at distances below 20 $\\mathrm {AU}$ .",
"From the assumed surface density profile, none of the planetary companions would be affected by extinction due to material from the protoplanetary disc in the IR bands.",
"The observed contrast of the primary companion in three bands is considerably higher than the value for the most massive planet of our models, thus setting a mass lower limit of $2.38$ $M_J$ for PDS 70b.",
"The second companion lays on top of the $2.38$ $M_J$ model in $H$ -band, and above it in $K$ -band.",
"The redness of this source can explain the difference in the bands contrast.",
"This reddening might be due to material from its own CPD or from the contiguous disc feature.",
"Our models are in agreement with the previous mass ranges estimated for the two companions; further observations and modelling of the disc and their atmospheres are needed to better constrain their masses.",
"The estimated accretion rates of the companions are of the order of $\\sim 10^{-11}$ $M_{} \\cdot {\\mathrm {yr}}^{-1}$ , thus radiation from accretion shocks near both planets are negligible.",
"From our results, accretion flux would only have an incidence in the modelled IR planet fluxes at distances $\\sim 5$ $\\mathrm {AU}$ , since accretion rates are expected to be higher due to the scaling.",
"This can be appreciated in the contrast curve of the three planet models in $H$ -band: planets' contrasts decrease at these distances.",
"The effect of the accretion shock's radiation becomes negligible at $\\gtrsim 10$ $\\mathrm {AU}$ ."
],
[
"HL Tau",
"HL Tau is one of the most extensively studied protoplanetary discs, with several rings and gaps detected in the dust continuum [1].",
"It is a young stellar object of $\\le 1$ $\\mathrm {Myr}$ at around 140 $\\mathrm {pc}$ to us , with an estimated stellar mass of $\\sim 0.7$ $M_{\\odot }$ , .",
"Observations were carried out using the LBTI L/M IR Camera , , using only one of the two primary mirrors of the LBT telescope.",
"No point-sources were detected.",
"For the normalisation of the surface density, we took the inferred gas surface density from CARMA observations , at a fiducial distance of 40 $\\mathrm {AU}$ , $\\Sigma = 34$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ (a factor $\\times 0.74$ compared to the simulated disc).",
"Figure: Contract curves in LL-band for planets embedded in HL Tau, including the 5σ5\\sigma detection limit of the observation from .",
"The observations were performed using LBTI/LMIRcam.",
"The contrast curves are for planet masses of 0.440.44, 0.880.88 and 2.192.19 M J M_J.",
"The considered apparent magnitude of the central star was L=6.23L = 6.23 mag \\mathrm {mag} .",
"The coloured regions accounts for the uncertainty in the planet contrast.",
"The grey vertical area is delimited by the D5 and D6 rings detected in dust continuum .In Figure REF we show the contrast limit of the LBTI observation in $L$ -band as a function of the angular separation to the central star, together with the derived contrast of the re-scaled models for planets with $0.44$ , $0.88$ and $2.19$ $M_J$ .",
"In Appendix , contrast curves in $J$ -, $H$ - and $K$ -bands are included for completeness.",
"In $L$ -band, a high extinction is predicted for $0.44$ and $0.88$ $M_J$ along the entire disc, especially at distances $\\lesssim 60$ $\\mathrm {AU}$ ; at that distance, $A_L$ values are $4.27$ $\\mathrm {mag}$ ($A_V = 68.29$ $\\mathrm {mag}$ ) and $1.25$ $\\mathrm {mag}$ ($A_V = 19.98$ $\\mathrm {mag}$ ) for these planets respectively.",
"For planets outer in the disc, the extinction contribution is smaller but still significant: $3.02$ $\\mathrm {mag}$ ($A_V = 48.29$ $\\mathrm {mag}$ ) for $0.44$ $M_J$ , and $0.88$ $\\mathrm {mag}$ ($A_V = 14.13$ $\\mathrm {mag}$ ) for $0.88$ $M_J$ at 120 $\\mathrm {AU}$ .",
"These planets are not massive enough to clear the gap efficiently.",
"On the other hand, for the most massive planet ($2.19$ $M_J$ ), extinction is negligible at any distance.",
"Six gaps were observed in the ALMA continuum observation; following the example as in , within the gap delimited by D5 and D6 rings (marked as grey vertical line) the contrast limit of the instrument does not allow us to constrain the mass of the companion that could be responsible for the gap.",
"Nevertheless, from the inferred contrast curves, extinction would have an incidence in a hypothetical point-source detection only for planet masses $\\lesssim 0.88$ $M_{J}$ ."
],
[
"TW Hya",
"Observations using the Keck/NIRC2 vortex coronagraph were performed by searching for point-sources in the TW Hya disc.",
"This system is the closest known protoplanetary disc to us , with a central star of $0.7$ -$0.8$ $M_{\\odot }$ [3], relatively old , and with an estimated total disc mass of $0.05$ $M_{\\odot }$ [8].",
"The surface gas density has been modelled from ALMA line emission observations , .",
"From their unperturbed models, we used a fiducial surface density of $10.9$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ at 40 $\\mathrm {AU}$ for the re-scaling (a surface density factor $\\times 0.24$ of the simulated disc).",
"The instrument allows for IR high-contrast imaging in $L$ -band, using angular differential imaging (ADI) and reference star differential imaging (RDI).",
"In Figure REF , the detection limits for ADI and RDI are shown together with the expected contrast of planets with $0.47$ , $0.94$ and $2.34$ $M_J$ .",
"In this observation, RDI allows for detections of point-sources at distances as low as $\\sim 5$ $\\mathrm {AU}$ .",
"At this distance the accretion flux overcomes the intrinsic flux for a planet of $\\gtrsim 2.34$ $M_J$ , thus the contrast decreases compared to the non-accreting case.",
"The contrast curves of these planets in $J$ -, $H$ - and $K$ -bands are shown in Appendix .",
"Figure: Contract curves in LL-band for planets embedded in TW Hya, including 95%95\\% significance detection limits of Keck/NIR2 observations .",
"The contrast limits are shown for angular differential imaging (ADI) and reference star differential imaging (RDI).",
"The contrast curves shown are for planet masses of 0.470.47, 0.940.94 and 2.342.34 M J M_J.",
"The apparent magnitude of the central star is L=7.01L = 7.01 mag \\mathrm {mag}, taken from the W1 band in the WISE catalogue .",
"The coloured regions for each planet model are delimited by the estimated ages .",
"The grey vertical lines account for the gaps observed in and .Different observations of TW Hya confirmed several gaps in the disc.",
"[4] detected three dark annuli at 24, 41 and 47 $\\mathrm {AU}$ distances to the host star (distances corrected with newest Gaia parallax).",
"An unresolved gap in the inner disc was also seen from 870 $\\mu m $ continuum emission using ALMA.",
"Scattered light using SPHERE detected three gaps in the polarised intensity distribution, at $\\lesssim 7$ , 23, and 88 $\\mathrm {AU}$ .",
"In the figure we show the gaps in the outer disc ($\\sim 23$ , 40, 46 and 87 $\\mathrm {AU}$ ).",
"No point-sources were detected in the Keck/NIRC2 observations.",
"set upper limits for planets located at these gaps ($1.6$ -$2.3$ $M_J$ , $1.1$ -$1.6$ $M_J$ , $1.1$ -$1.5$ $M_J$ , and $1.0$ -$1.2$ $M_J$ from inner to outer distances).",
"Analogously, we can infer upper limits of the planets interpolating our results, since the contrast curves lay between our models.",
"Using the models for 7 $\\mathrm {Myr}$ planets, the upper limits for these gaps would be $2.5$ $M_J$ , $2.1$ $M_J$ , $2.0$ $M_J$ and $1.7$ $M_J$ .",
"Considering an age of 10 $\\mathrm {Myr}$ , the upper limits are marginally higher: $2.8$ $M_J$ , $2.4$ $M_J$ , $2.3$ $M_J$ and $2.1$ $M_J$ .",
"Taking into account the extinction due to the disc above the planet increases the estimated upper limits compared to the previous work, which did not consider this effect.",
"This indicates the importance of extinction when looking for protoplanets with direct imaging methods."
],
[
"HD 163296",
"In , the HD 163296 disc was studied in the $L$ -band using the same instrument (Keck/NIRC2 vortex coronagraph).",
"The scattered polarised emission in the $J$ -band was also studied with the Gemini Planet Imager in , detecting a ring with an offset that can be explained by an inclined flared disc.",
"This system has a central star of $2.3$ $M_{\\odot }$ and estimated age of $\\sim 5$ $\\mathrm {Myr}$ .",
"Observations in the dust continuum using ALMA confirmed the existence of three gaps at distances of $\\sim 50$ , $\\sim 81$ and $\\sim 136$ $\\mathrm {AU}$ (corrected with the new Gaia distance of $101.5$ $\\mathrm {pc}$ ).",
"Kinematical analysis of gas observations suggested the presence of two planets at the second and third gaps .",
"In , HD models showed that a third planet is expected further out.",
"The estimated masses of the three potential planets are 1 $M_J$ (at 83 $\\mathrm {AU}$ ), $1.3$ $M_J$ (at 127 $\\mathrm {AU}$ ) and $\\approx 2$ $M_J$ at ($\\approx 260$ $\\mathrm {AU}$ ).",
"The new DSHARP/ALMA observations confirmed an additional gap at $\\sim 10$ $\\mathrm {AU}$ ; assuming that this gap is caused by a planet, estimated a planet mass between $0.2$ and $1.5$ $M_J$ from 2D HD simulations.",
"Figure: Contract curves in LL-band for planets embedded in HD 163296, including the 5σ5\\sigma detection limits of the observation from .",
"The observations were performed using Keck/NIRC2.",
"The contrast of planets with 1.441.44, 2.882.88 and 7.197.19 M J M_J are shown.",
"The apparent magnitude of the central star is L=3.7L = 3.7 mag \\mathrm {mag}, inferred from the W1 band in the WISE catalogue .",
"The grey vertical lines account for the gaps observed in , .The $L$ -band high-contrast imaging detected a point-like source at a distance of $67.7$ $\\mathrm {AU}$ with $4.7\\sigma $ significance.",
"None of the observations in $L$ - or $J$ -band found any point-sources at the gaps observed in the continuum.",
"Our models allow to set upper limits for planets at the location of the gaps.",
"We used a fiducial surface density of $\\Sigma = 82.8$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ at 40 $\\mathrm {AU}$ , corresponding to a factor $\\times 1.8$ of the simulated disc, to obtain contrast curves for $1.44$ , $2.88$ and $7.19$ $M_J$ ($L$ -band in Figure REF , and $J$ -, $H$ - and $K$ -bands in Figures REF in the Appendix ).",
"The innermost gap in Figure REF is within the masked region in the Keck/NIR2 observations, thus a mass upper limit can not be inferred.",
"At the second gap, the model for our most massive planet lays slightly below the detection limit of the observation.",
"A rough extrapolation would yield an upper-limit of $7.6$ $M_J$ , slightly below the range provided by (8-15 $M_J$ in that work).",
"For the third and fourth gaps, we obtain upper limits of $6.7$ $M_J$ and $5.5$ $M_J$ from interpolating our models.",
"These values are slightly higher than the upper limits inferred in ($4.5$ -$6.5$ $M_J$ , and $2.5$ -4 $M_J$ respectively).",
"Taking into account extinction on the contrast of the planets increase the inferred upper limits of the non-detected planets.",
"In every gap, extinction does have an important effect for planets with masses lower than the inferred upper limits.",
"Compared to the estimates of and from indirect analysis, our inferred upper limits are significantly higher; consequently a direct detection of these companions would only be possible improving the detection limit to much higher contrast.",
"In this work we studied the effect of extinction for direct imaging of young planets embedded in protoplanetary discs.",
"A set of HD simulations were performed to reproduce planet-disc interaction at high resolution for several planet masses.",
"Column densities and extinction coefficients were derived in order to model the planet predicted magnitudes in $J$ -, $H$ -, $K$ -, $L$ -, $M$ -, $N$ -bands.",
"Exploiting properties of locally isothermal discs, we applied the models to planets embedded in CQ Tau, PDS 70 and HL Tau protoplanetary discs, and inferred upper-limits for planets at the gaps observed in TW Hya and HD 163296.",
"The most important results of this work are: For the simulated planets at $5.2$ $\\mathrm {AU}$ , the 5 $M_\\mathrm {J}$ clears its surrounding material very effectively.",
"The resulting column density is extremely low, and, as consequence, extinction is not significant in any band.",
"The 1 and 2 $M_\\mathrm {J}$ planets are completely hidden by the disc at $\\lesssim 2$ $\\mu m$ wavelengths (with respective extinction coefficients of $> 30$ , $> 15$ $\\mathrm {mag}$ ), while at wavelengths between 5 to 8 $\\mu m$ their corresponding coefficients are reduced, below 15 and 4 $\\mathrm {mag}$ .",
"In the $N$ -band, extinction is higher compared to $L$ - and $N$ -bands due to the silicate feature in the assumed ISM dust opacities.",
"Jupiter-like planets embedded in discs with very low unperturbed surface densities (of the order of $\\lesssim 1$ $\\mathrm {g \\cdot \\mathrm {cm}^{-2}}$ ) have very low extinction coefficients in IR at any distance considered.",
"In CQ Tau, only planets with $\\lesssim 2$ $M_J$ are affected by extinction in $J$ - and $H$ -bands at distances $\\lesssim 20$ $\\mathrm {AU}$ .",
"In PDS 70, extinction has an incidence only for the least massive planet model at distances <50 $\\mathrm {AU}$ , more significant at shorter wavelengths.",
"In more dense discs like HL Tau and HD 163296, direct detection of companions is unlikely in $J$ -, $H$ -, $K$ -, and $N$ -bands due to the extinction effects.",
"Only the most massive planet from our models would be detectable, since the extinction is negligible.",
"We inferred upper limits of the gaps in TW Hya and HD 163296, slightly higher than previous work due to the effect of extinction.",
"This points out the importance of extinction from the disc material in high-contrast imaging of protoplanetary discs.",
"Radiation from accretion shocks onto the planet has been considered in our models.",
"It can have an important effect on the total planet emission for accretion rates of the order of $\\sim 10^{-8}$ $M_{\\odot }/{\\mathrm {yr}}$ ; these high rates occur at distances $\\lesssim 10$ $\\mathrm {AU}$ in our models.",
"The scarcity of detections so far might suggest different scenarios: giant planet formation further out in the disc is rare, or perhaps planets formed at these early stages are still not massive enough ($\\lesssim 2$ $M_J$ ) to be detected with current instrumentation."
],
[
"Acknowledgements",
"This work was partly supported by the Italian Ministero dellIstruzione, Università e Ricerca through the grant Progetti Premiali 2012 – iALMA (CUP C52I13000140001), by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) - Ref no.",
"FOR 2634/1 TE 1024/1-1, and by the DFG cluster of excellence Origin and Structure of the Universe (www.universe-cluster.de).",
"This work is part of the research programme VENI with project number 016.Veni.192.233, which is (partly) financed by the Dutch Research Council (NWO).",
"The simulations were partly run on the computing facilities of the Computational Center for Particle and Astrophysics (C2PAP) of the Excellence Cluster Universe.",
"BE acknowledges funding by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2094 – 390783311.",
"GP and BE acknowledge support from the DFG Research Unit \"Transition Disks\" (FOR 2634/1, ER 685/8-1)."
]
] | 2001.03565 |
[
[
"Characteristic Timescales of the Local Moment Dynamics in Hund's-metals"
],
[
"Abstract We study the characteristic time scales of the fluctuating local moments in Hund's metal systems for different degrees of correlation.",
"By analyzing the dynamical spin susceptibility in the real-time domain via the fluctuation-dissipation theorem, we determine the time scales controlling oscillation and damping of on-site fluctuations - a crucial factor for the detection of local moments with different experimental probes.",
"We apply this procedure to realistic many-body calculations of different families of iron-pnictides and chalcogenides, explaining the material-specific trend in the discrepancies reported between experimental and theoretical estimates of the magnetic moments."
],
[
"Computational details",
"Density Functional Theory - For the density functional theory (DFT) calculations we employed the VASP code[1], [2], version 5.3.3.",
"As structural inputs, the experimentally found crystal structures as well as the measured lattice parameters (given in tab:structure) have been used.",
"For all of the atoms in the given structures, we used PBE-GGA functionals.",
"The precise functionals used for each atom are given in tab:potentials.",
"Calculations were performed on a $\\Gamma $ -centered MP k-mesh with $12\\times 12\\times 12$ points and $10\\times 10\\times 12$ points for I4/mmm and P4/nmm structures, respectively; The partial occupancies were calculated using the Blöchl tetrahedron method.",
"The respective cut-off energies were, among other parameters, defined by setting the precision to HIGH, and the DOS was evaluated on 2001 points.",
"Table: Crystal structures for all materials under consideration.",
"For I4/mmm materials, cc is given as the lattice parameter of the tetragonal cell, and zz in relation to this cc.Table: List of PAW PBE functionals used by VASP in the DFT calculations of this study.",
"The functionals are uniquely identified by their creation date and the valence electron configuration given in the functionals by VHRFIN.Wannier Projection - The VASP results were projected onto local orbitals via the wannier90 code[11].",
"At the time of the calculations, wannier90 integration into VAPS was only possible with wannier90 v1.2.",
"Specifically, all of the electronic Bloch functions in the DFT calculation were projected onto the $d$ states of Fe, no bands were marked as excluded in the wannier90.win file.",
"In wannier90, the k-points were identical to those from the respective VASP MP-grids.",
"The electronic bands of predominant Fe character are intertwined with bands of other character, such as the $p$ states of the ligands.",
"The degree of entanglement varied across materials, necessitating different disentanglement window parameters for wannier90, and in the case of BaFe$_2$ As$_2$ and KFe$_2$ As$_2$ also frozen windows.",
"The window positions were tweaked manually with respect to disentanglement convergence as well as agreement between original VASP bands and the bands of the Wannier Hamiltonian, the final values are given in tab:wann90wins.",
"The convergence criterion for the disentanglement as well as the wannierization was a difference in spread between successive iterations lower than $10^{-11}$ .",
"Best results were achieved by enabling guiding centres.",
"The Wannier Hamiltonian served as the single-particle Hamiltonian for the Dynamical Mean Field Theory (DMFT) calculations.",
"Table: Energy windows for disentangling of bands in wannier90.",
"KFe 2 _2As 2 _2 and BaFe 2 _2As 2 _2 additionally required windows defining frozen states.",
"Energies are given relative to the Fermi energy E F _\\text{F}=0 eV.Dynamical Mean Field Theory - To include the effects of strong local interactions on top of the DFT, we performed DMFT simulations of an low-energy model for the entire $3d$ -orbital manifold of Fe.",
"The most general form of an on-site electrostatic repulsion in this manifold reads $H_{\\rm {int}} = \\sum _{{\\bf r} \\sigma \\sigma ^{\\prime }}\\sum _{lmno} \\, U^{\\phantom{\\dagger }}_{lmno} \\, c^\\dagger _{{\\bf r} l \\sigma }\\,c^\\dagger _{{\\bf r} m \\sigma ^{\\prime }}\\, c^{\\phantom{\\dagger }}_{{\\bf r} o \\sigma ^{\\prime }} c^{\\phantom{\\dagger }}_{{\\bf r} n \\sigma },$ where the full-fledged, four-indexed $U-$ tensor describes the projected value of the screened Coulomb interaction on the corresponding orbital configurations.",
"As an ab-initio estimate for the orbital-dependent interaction parameters, we take the results by Miyake et al.",
"[12], where constrained random phase approximation (cRPA) results for the two-orbital interaction matrix $U_{lm}$ and $J_{lm}$ were reported for different compoundsWith the only exception of KFe$_2$ As$_2$ , where we used the same values as for the other family-compound (BaFe$_2$ As$_2$ given in [12]) as they are not available in the literature..",
"Here, the $J_{lm}$ values encode the (orbital-dependent) Hund's coupling, while the $U_{ij}$ diagonal/off-diagonal matrix elements describe the inter-/intra-orbital electrostatic repulsion.",
"The relation, which we exploited to extract the interaction parameters appearing in Eq.",
"REF , is: $\\begin{array}{rcl}U^{\\phantom{\\dagger }}_{ijkl} ={\\left\\lbrace \\begin{array}{ll}U^{\\phantom{\\dagger }}_{ij}, & \\text{if } ijkl=ijij, \\\\J^{\\phantom{\\dagger }}_{ij}, & \\text{if } ijkl=iijj \\text{ and }i\\ne j, \\\\J^{\\phantom{\\dagger }}_{ij}, & \\text{if } ijkl=ijji \\text{ and }i\\ne j, \\\\0, & \\text{otherwise}.\\end{array}\\right.",
"}.\\end{array}$ This leads to the low-energy Hamilonian used for our DMFT calculations $H = \\sum _{{\\bf k} \\sigma lm} \\, H_{lm}^{\\phantom{\\dagger }}({\\bf k}) \\; c^{\\dagger }_{{\\bf k} l \\sigma }\\, c^{\\phantom{\\dagger }}_{{\\bf k} m \\sigma } + H_{\\rm int},$ with $\\begin{array}{rcl}H_{\\rm {int}} &=& \\sum _{{\\bf r} l} \\, U^{\\phantom{\\dagger }}_{ll} \\; n_{{\\bf r} l \\uparrow } \\, n_{{\\bf r} l \\downarrow } +\\sum _{{\\bf r} \\sigma \\sigma ^{\\prime }} \\sum _{l<m} \\, \\left( U^{\\phantom{\\dagger }}_{lm} - J^{\\phantom{\\dagger }}_{lm}\\delta _{\\sigma \\sigma ^{\\prime }} \\right) \\, n_{{\\bf r} l \\sigma } \\; n_{{\\bf r} m \\sigma ^{\\prime }} \\\\&& - \\sum _{{\\bf r}}\\sum _{l \\ne m} \\, J^{\\phantom{\\dagger }}_{lm} \\; c^\\dagger _{{\\bf r} l \\uparrow }\\, c^{\\dagger }_{{\\bf r} l \\downarrow } \\, c^{\\phantom{\\dagger }}_{{\\bf r} m \\uparrow }\\, c^{\\phantom{\\dagger }}_{{\\bf r} m \\downarrow }- \\sum _{{\\bf r}}\\sum _{l \\ne m} \\, J^{\\phantom{\\dagger }}_{lm} \\; c^\\dagger _{{\\bf r} l \\uparrow }\\, c^{\\dagger }_{{\\bf r} m \\downarrow } \\, c^{\\phantom{\\dagger }}_{{\\bf r} m \\uparrow }\\, c^{\\phantom{\\dagger }}_{{\\bf r} l \\downarrow }.\\end{array}$ Physically, this corresponds to an orbital-dependent Kanamori interaction, where one can easily recognize an orbital-dependent pair-hopping (the first term in the second line) and spin-flip contribution (second term).",
"In fact, Eq:HamiltonianintGK can be regarded as an orbital-dependent generalization of the Kanamori interaction, since for the special cases of no-orbital dependence e.g.",
"averaged interaction parameters (where $U^{\\phantom{\\dagger }}_{ll} = U $ , $U^{\\phantom{\\dagger }}_{l\\ne m} = V $ and $J^{\\phantom{\\dagger }}_{l m} = J $ ) we recover the usual expression of the Kanamori Hamiltonian: $\\begin{array}{rcl}H^K_{\\rm {int}} &=& U^{\\phantom{\\dagger }} \\; \\sum _{{\\bf r} l} \\; n_{{\\bf r} l \\uparrow } \\, n_{{\\bf r} l \\downarrow } +\\sum _{{\\bf r} \\sigma \\sigma ^{\\prime }} \\, \\left( V - J\\, \\delta _{\\sigma \\sigma ^{\\prime }} \\right) \\sum _{l<m} \\, n_{{\\bf r} l \\sigma } \\; n_{{\\bf r} m \\sigma ^{\\prime }} \\\\&& - J \\, \\sum _{{\\bf r}}\\sum _{l \\ne m} \\; c^\\dagger _{{\\bf r} l \\uparrow }\\, c^{\\dagger }_{{\\bf r} l \\downarrow } \\, c^{\\phantom{\\dagger }}_{{\\bf r} m \\uparrow }\\, c^{\\phantom{\\dagger }}_{{\\bf r} m \\downarrow }- J \\; \\sum _{{\\bf r}}\\sum _{l \\ne m} \\, c^\\dagger _{{\\bf r} l \\uparrow }\\, c^{\\dagger }_{{\\bf r} m \\downarrow } \\, c^{\\phantom{\\dagger }}_{{\\bf r} m \\uparrow }\\, c^{\\phantom{\\dagger }}_{{\\bf r} l \\downarrow }.\\end{array}$ To illustrate concisely the variation of the screened interaction values in the different materials, the corresponding orbitally average $U$ - and $J$ -values are shown in tab:avgUJV.",
"The DMFT calculations shown in the main text were, however, performed using the orbital-resolved Hamiltonian (REF ).",
"Table: Average effective on-site Coulomb (U) exchange (J) interactions between two electrons on the same iron site in the d-d- model (in eV).Finally, let us mention that in order to check the robustness of our conclusions, we have also performed DMFT calculations using the orbitally-averaged values for the $U$ and $J$ interaction (i.e., corresponding to a “conventional\" Kanamori interaction, not shown), finding only marginal changes to the results shown in Fig.",
"2 and 3 of the main text.",
"Larger quantitative modifications can be found in the results of the most correlated materials, as expected, only if one neglects the spin-flip terms in Eq.",
"(REF ) (e.g., when performing density-density calculations, not shown here[13]).",
"The reason is, that in this approximation one tends to overestimate the high-spin configurations in the strong-coupling regime.",
"The number of electrons in the target ($d$ -) manifold was estimated directly from chemical considerations (constituent electronegativity).",
"Throughout our calculation, we assumed that LaFeAsO, BaFe$_2$ As$_2$ LiFeAs and FeTe have a filling of $\\sum _{l, \\sigma } \\mathinner {\\langle {n^l_\\sigma }\\rangle } = 6.0$ electrons per iron atom.",
"For KFe$_2$ As$_2$ we used, instead, $\\sum _{l, \\sigma } \\mathinner {\\langle {n^l_\\sigma }\\rangle } = 5.5$ per iron atom.",
"To avoid double-counting of the Coulomb interaction between Fe-3d electrons already included in DFT, we used an orbital-dependent double-counting correction of the Fully-Localized-Limit (FLL) type [14] (adopted also for DMFT calculations of elemental Fe[15]).",
"The values used were determined by eqn:DCCorbitaldependent and are shown in tab:DCCd.",
"$\\begin{array}{rcl}\\mu ^{FLL}_{DC}(i) &=& \\mu ^{FLL}_{DC}(i) + \\frac{1}{4} \\left(n^0 - \\frac{1}{2}\\right) \\left( \\sum _{j}(U_{ij} - J_{ij}) \\right),\\end{array}$ In eqn:DCCorbitaldependent $ n^0 = \\frac{1}{2(2l+1)} \\sum _{i,\\sigma } n_{i,\\sigma } $ is the DFT filling and the two-indices U-matrix is related to the four-indices local (screened) Coulomb-tensor by $U_{ij}=U_{ijij}$ and $J_{ij}=U_{ijji}$ (with $i\\ne j$ ).",
"Table: Orbital dependent double counting correction (DCC) in the fully-localized limit.The DMFT simulation was performed with a continuous-time quantum Monte Carlo (QMC) algorithm implemented in the code package w2dynamics[16].",
"All calculations were done at $\\beta =50$ [$\\mathrm {eV}^{-1}$ ] corresponding to approximately $232.1$ K. At this temperature all the materials are experimentally found to be in the paramagnetic phase[17], [18], [19].",
"To achieve convergence in the DMFT cycle with orbital dependent Kanamori interaction for the different materials we first converged the DMFT cycle without the pair-hopping term in Eq.",
"(REF )).",
"This was achieved by performing 30$\\div $ 70 DMFT steps with low statistics (Nmeas=$10^4$ , where Nmeas is then number of QMC measurements, see [16] for details).",
"Up to 100 additional DMFT-steps were performed with higher statistics.",
"For each calculations, the final convergence of the DMFT self-consistency was tested for the one-particle quantities encoded in the self-energy $\\Sigma ^l(\\omega )$ , with respect to the previous five iterations.",
"For the number of steps between measurements (where the minimum value gives a measure of auto-correlation time) we found a value of Ncorr=$1500\\div 2000$ to be sufficient, in line with the estimate of the average “renewal time” of the fermionic trace given in Ref.w2dynamics.",
"On top of the converged one-particle quantities we then calculated the spin-spin susceptibility in imaginary time through a single DMFT step with fixed chemical potential and one-particle properties, using Nmeas$>5\\cdot 10^4$ .",
"As a QMC-sampling algorithm we applied the recently developed States-Sampling [20].",
"The result is show in the upper-right part of Figure 2 of the main text.",
"To obtain the DMFT-bubble susceptibility we used $\\chi ^{\\mathrm {Bubble-DMFT}}(\\tau ) = \\sum _l G^l_{\\mathrm {loc}}(\\tau ) G^l_{\\mathrm {loc}}(\\beta -\\tau )$ , and for the bare-bubble we set the interaction as well as the double counting correction in the DMFT calculation to zero ($U=J=V=\\mu _{DC}=0$ ), corresponding to $G_\\mathrm {loc}=G_0$ .",
"Analytical continuation - Analytical continuation from imaginary time (where the QMC data was obtained) to real frequencies was performed with the Maximum Entropy Method (MaxEnt) with the code package Maxent [21].",
"This way we obtained the imaginary part of the retarded susceptibility $\\chi ^R(t) \\equiv \\frac{\\mathrm {i}}{\\hbar } \\theta (t) \\mathinner {\\langle {[\\hat{S}_z (t),\\hat{S}_z(0)]}\\rangle }$ .",
"The effect of different default-models (Flat, Gaussian, Lorentzian) was tested and found to be small.",
"We chose a broad featureless Lorentzian-default model with a width of $\\gamma _{\\mathrm {Model}}=0.5$ .",
"Model-details are found elsewhere [21].",
"The optimal $\\alpha -$ parameter (weight of the entropy term in MaxEnt) was determined by the maximum of the curvature of $\\chi ^2(\\alpha )$ .",
"(See fig:kink.)",
"This way we could reliably determine the region where neither the data was over-fitted nor the default model was take into account too strongly.",
"One advantage of this method is invariance of the final spectrum under re-scaling the error by a global factor[13].",
"A similar approach was already applied in [22].",
"Figure: Log-log-plot of the quadratic difference between the data and the fit χ 2 \\chi ^2 over the entropy parameter α\\alpha for LaFeAsO.",
"We find overfitting (underfitting) of the data to start at α<10 0 \\alpha <10^0 (α>10 2 \\alpha >10^2).",
"The spectrum corresponding to the maximum of f(α)/χ 2 (α)f(\\alpha )/\\chi ^2(\\alpha ) (at α=2·10 1 \\alpha =2 \\cdot 10^1) could be regarded as a good analytical continuation.One particle time scales- For estimating one particle timescales we assumed that (in the presence of a well defined quasi-particle excitation) the one-particle Green's function $G^i(t)$ for each orbital $i$ decays in the following way: $\\left| G^i(t)\\right|^2 = Z_i^2 \\mathrm {e}^{- \\frac{t \\, 2\\, Z_i \\, \\mathrm {Im} \\Sigma _i(\\omega \\rightarrow 0)}{\\hbar }} \\propto \\mathrm {e}^{- \\frac{t}{t^i_{\\mathrm {1P}}}}$ , with $t^i_{\\mathrm {1P}}\\equiv \\frac{\\hbar }{2 Z_i \\mathrm {Im}\\Sigma _i(\\omega \\rightarrow 0)} $ , where $\\Sigma _i$ is the self-energy of the orbital $i$ .",
"The value of the self-energy at zero frequency as well as the orbital dependent quasi-particle mass re-normalization $Z_i=\\left(1 + \\mathrm {d}/\\mathrm {d} \\omega \\mathrm {Re} \\Sigma _i(\\omega )\\big |_{\\omega \\rightarrow 0}\\right)^{-1}$ was extracted from the DMFT self-energy by linear interpolation of $\\mathrm {Im} \\Sigma (\\mathrm {i}\\omega _n \\rightarrow 0)$ (using the Cauchy-Riemann equations for $Z_i$ ).",
"The one particle time scale $t_{\\mathrm {1P}}$ given in the main text was then estimated as the orbital average of $t^i_{\\mathrm {1P}}$ : $t_{\\mathrm {1P}} = \\frac{1}{5} \\sum _i t^i_{\\mathrm {1P}}$ .",
"Spin-excitation time scales - While the time scales of spin-excitations in iron-based superconductors are determined by an intricate interplay of kinetic energy (hopping) and electron-electron-interaction the main time scales can be effectively described by a much simpler model.",
"The extraction of time scales was done by applying a uniform $\\chi ^2$ -fit to $\\mathrm {Im}\\, \\chi ^R(\\omega )$ with cutoff-values chosen for the grid such that the main-peaks structure is well within the frequency window (1eV).",
"The cutoff excludes high-frequency data, which is usually not as well captured by MaxEnt as the low-frequency data.",
"A variation of the cutoff by 20% leads to a change in the time scales by less than 15%.",
"The fitting function is defined as follows: We consider the absorption spectrum of a damped harmonic oscillator, which can be obtained by the Fourier-transform of the Green's function of the differential equation $\\ddot{\\chi }(t) + 2\\gamma \\dot{\\chi }(t)- \\omega ^2_0 \\chi (t) = -\\delta (t)$ , i.e.",
"$\\chi (\\omega ) = \\frac{1}{ \\omega ^2 - 2 \\mathrm {i} \\gamma \\omega + \\omega ^2_0}$ .",
"We note that the latter has poles only on the lower half-plane, and thus it is a retarded function ($\\chi (t<0)=0$ ).",
"Its imaginary part (up to a proportionality-constant reflecting the material-dependent value of the unscreened local moment) defines our fitting model which, thus, reads $\\mathrm {Im} \\chi ^R(\\omega ) = 2 \\gamma \\omega \\frac{1}{(\\omega ^2 - \\omega ^2_0)^2 + 4 \\omega ^2 \\gamma ^2},$ or correspondingly in real times $ \\chi (t) = \\left\\lbrace \\begin{array}{lll}\\frac{\\mathrm {e}^{-\\gamma t}}{\\sqrt{\\omega _0^2 - \\gamma ^2 }} \\sin (\\sqrt{\\omega _0^2 - \\gamma ^2}t) \\theta (t) & \\phantom{aa}&\\text{if } \\omega _0^2 > \\gamma ^2\\\\\\frac{\\mathrm {e}^{-\\gamma t}}{\\sqrt{\\gamma ^2 - \\omega _0^2 }} \\sinh (\\sqrt{\\gamma ^2 - \\omega _0^2 }t) \\theta (t) & \\phantom{aa}&\\text{if } \\omega _0^2 < \\gamma ^2 {.",
"}\\\\\\end{array}\\right.$ The asymptotic behavior, which determines the main-lifetime is given by $ \\lim \\limits _{t \\rightarrow \\infty }\\chi (t) \\propto \\left\\lbrace \\begin{array}{llll}\\mathrm {e}^{-\\gamma t} &\\equiv \\mathrm {e}^{-t/t^{\\mathrm {under}}_\\gamma } & \\phantom{aa}&\\text{if } \\omega _0^2 > \\gamma ^2\\\\\\mathrm {e}^{-\\left(\\gamma - \\sqrt{\\gamma ^2 - \\omega _0^2 }\\right) t} &\\equiv \\mathrm {e}^{-t/t^{\\mathrm {over}}_{\\gamma }} & \\phantom{aa}&\\text{if } \\omega _0^2 < \\gamma ^2 .\\end{array}\\right.$ The corresponding parameters obtained by fitting the DMFT spectra are summarized in tab:fitparams.",
"Table: Fitting parameters extracted with a harmonic oscillator model (second and third column), effective lifetime χ(t→∞)∝e -t/t γ \\chi (t\\rightarrow \\infty )\\propto \\mathrm {e}^{-t/t_\\gamma } (third column) and effective oscillation frequency t ω ¯ =ℏ ω 0 2 -γ 2 t_{\\bar{\\omega }} = \\frac{\\hbar }{\\sqrt{\\omega _0^2 - \\gamma ^2}}(fourth column)One can also define a harmonic-oscillator anti-commutator through the fluctuation-dissipation theorem as $F(\\omega ) = \\frac{1}{\\pi } \\coth (\\omega \\beta /2) \\mathrm {Im}\\chi ^R(\\omega ) $ .",
"For the latter it is not easy to get an analytical expression for the Fourier-transform ($F^{\\mathrm {harm.",
"\\, osz.",
"}}(t) = \\int _{-\\infty }^{\\infty } \\mathrm {d}\\omega \\, \\mathrm {e}^{- \\mathrm {i} \\omega t} \\frac{1}{\\pi } \\coth (\\omega \\beta /2) $ ).",
"Therefore we preformed the transformation only numerically.",
"To assess the quality of the fit we checked also $\\chi ^{\\mathrm {data}}(t)$ against the analytical expressions given in eqn:Analyticalchit.",
"The results of the transformation of the data as well as the transformation of the fits is shown in fig:fitchecks.",
"For LaFeAsO, BeFe$_2$ As$_2$ , LiFeAs and KFe$_2$ As$_2$ a single peak model was used, while for FeTe the double-peak-structure in the data necessitated a two-peak model.",
"Due to the second peak in FeTe no sign-change is observed in the corresponding $\\chi (t)$ , although the main (first) peak would predict an oscillatory (under-damped) behavior.",
"Figure: Dissipative part of the spin-spin susceptibility obtained by MaxEnt (left figure solid lines).",
"The harmonic-oscillator fits are shown as dashed lines.",
"Center figure: Spin-spin susceptibility in time.",
"Direct transform of MaxEnt data shown as solid lines and the analytic expression for the fitted model as dashed lines.",
"Right figure: Spin-spin anti-commutator correlation function for data (solid lines) and fitted model (dashed lines).A comparison between the right and the center-part of fig:fitchecks shows the same behavior qualitatively, although quantitative differences are observed.",
"One reason for the difference is the additional energy scale (temperature).",
"Comments on previous works - The estimated values of the local fluctuating moment $\\mathinner {\\langle {m^2_{\\text{loc}}}\\rangle }$ in the specific case of KFe$_2$ As$_2$ obtained by our DFT+DMFT study deviates from the results of [23], where $\\mathinner {\\langle {m^2}\\rangle }=0.1 \\pm 0.02 $ [$\\mu _B^2/\\text{Fe}$ ].",
"According to our work KFe$_2$ As$_2$ should not have a significantly different local magnetic moment compared with the other iron-pnictides/-chalcogenides, which is consistent with the more recent theoretical/experimental analysis of [24], [25]."
]
] | 2001.03584 |
[
[
"Optimizing spontaneous parametric down-conversion sources for boson\n sampling"
],
[
"Abstract An important step for photonic quantum technologies is the demonstration of a quantum advantage through boson sampling.",
"In order to prevent classical simulability of boson sampling, the photons need to be almost perfectly identical and almost without losses.",
"These two requirements are connected through spectral filtering, improving one leads to a decrease of the other.",
"A proven method of generating single photons is spontaneous parametric downconversion (SPDC).",
"We show that an optimal trade-off between indistinguishability and losses can always be found for SPDC.",
"We conclude that a 50-photon scattershot boson-sampling experiment using SPDC sources is possible from a computational complexity point of view.",
"To this end, we numerically optimize SPDC sources under the regime of weak pumping and with a single spatial mode."
],
[
"SPDC sources",
"SPDC sources turn a pump photon into two down-converted photons, and hence produce photons in pairs.",
"For Type-II SPDC, the two photons from the pair each emerge in a separate mode.",
"Traditionally these modes are referred to as signal and idler.",
"The SPDC process can be understood by considering energy conservation $\\hbar \\omega _{\\rm p} = \\hbar \\omega _{\\rm s} + \\hbar \\omega _{\\rm i}$ as well as momentum conservation $\\vec{k_{\\rm p}}= \\vec{k_{\\rm s}} + \\vec{k_{\\rm i}}$ , where p, s and i denote the pump, signal and idler photons, respectively.",
"Momentum conservation can be tweaked by quasi phase matching by either periodic or apodized poling.",
"Both energy and momentum conservation only allow certain wavelength combinations and together they specify the spectral-temporal properties of the two-photon state [28].",
"Birefringence results in an asymmetry between the signal and idler photon.",
"This leads to spectral-temporal correlations between the two.",
"Such correlations reduce the spectral purity $P_{\\rm x} = \\rm {Tr}( \\rho _{\\rm x}^2)$ , where $\\rho _{\\rm x}$ is the reduced density matrix of photon x.",
"When no correlations exist, the photon state is factorizable and the photons are spectrally pure [29].",
"A visual representation of the two-photon state is shown in Fig.",
"REF .",
"The spot in the center indicates that the two-photon state with what probability the photons are in this region of the frequency space.",
"This probability is also referred to as the joint spectral intensity (JSI), which is related to the joint spectral amplitude (JSA) by $\\rm {JSI} = |\\rm {JSA}|^2$ .",
"The JSA describes the wavefunction of the photon pair as a function of the wavelength of the photons and follows from energy and momentum conservation.",
"The factorizability of the JSA determines the spectral purity of the source.",
"We now proceed with a mathematical description of the JSA, which follows from energy and momentum conservation.",
"The energy conservation $\\alpha (\\omega _{\\rm s},\\omega _{\\rm i})$ function is a Gaussian pulse with a center wavelength $\\omega _{\\rm p}$ and bandwidth $\\sigma _{\\rm p}$ : $\\alpha (\\omega _{\\rm s},\\omega _{\\rm i}) = \\exp \\biggl (\\frac{-(\\omega _{\\rm s}+\\omega _{\\rm i}-\\omega _{\\rm p})^2}{4\\sigma _{\\rm p}^2}\\biggr ).$ The phase-matching function for a periodically poled crystal is given by: $\\phi (\\omega _{\\rm s},\\omega _{\\rm i}) = {\\rm sinc}\\biggl ( \\frac{k_{\\rm p}-k_{\\rm s}-k_{\\rm i}-\\frac{2\\pi }{\\Lambda }}{2}L\\biggr ),$ with $L$ the length of the nonlinear crystal and $\\Lambda $ the poling period.",
"Another type of quasi phase matching exists, which is Gaussian apodization [21] $\\phi _{\\rm G}(\\omega _{\\rm s},\\omega _{\\rm i}) = \\exp {\\biggl (-\\frac{\\gamma \\Delta k^2 L^2}{4}\\biggr )},$ where $\\gamma \\approx 0.193$ , such that the width of this phase-matching function equals that of Eq.",
"REF .",
"The parameter $\\Delta k$ denotes the phase mismatch and $L$ again the crystal length.",
"The energy conservation function together with the appropriate phase-matching function give the JSA: $f(\\omega _{\\rm s},\\omega _{\\rm i}) = \\alpha (\\omega _{\\rm s},\\omega _{\\rm i}) \\phi (\\omega _{\\rm s},\\omega _{\\rm i}).$ The two-photon state corresponding to this JSA can give rise to distinguishability.",
"This can be mitigated by spectral filtering.",
"The overall two-photon state after filtering can now be written as: $|\\psi \\rangle = \\int \\int d\\omega _{\\rm s} d\\omega _{\\rm i} f(\\omega _{\\rm s},\\omega _{\\rm i}) F_{\\rm s,i}(\\omega _{\\rm s},\\omega _{\\rm i}) |1_{\\rm s}\\rangle |1_{\\rm i}\\rangle ,$ where $F_{\\rm s,i}(\\omega _{\\rm s},\\omega _{\\rm i})$ denotes a possible filter function on the signal and/or idler photon.",
"For simplicity, we ignore the vacuum and multiphoton states.",
"The spectral purity of the photon pair can be found with a Schmidt decomposition of the JSA [30], [31].",
"From this follows a Schmidt number $K$ which determines the spectral purity $P = \\frac{1}{K}.$ Physically, $K$ is the effective number of modes that is required to describe the JSA (e.g., see [32]).",
"When $K=1$ the photon pair is factorizable.",
"In this case, detecting a photon as herald leaves the other photon in a pure state.",
"In Fig.",
"REF this would manifest itself such that the JSA becomes aligned with the axes.",
"In case $K>1$ , detecting one photon leaves the other photon in a mixed state of several modes.",
"Hence, the remaining photon has a lower spectral purity.",
"It is possible to improve the spectral purity by filtering the photons.",
"The effect of filtering can be understood as overlaying the filter function over the JSA.",
"This is shown with the dashed lines in Fig.",
"REF .",
"A well-chosen filter removes the frequency correlations between the photons, but inevitably introduces losses, which in turn are detrimental for boson-sampling experiments."
],
[
"Classical simulation of boson sampling with imperfections",
"The presence of imperfections such as losses [33] and distinguishability [34] of photons reduces the computational complexity of boson sampling.",
"Classical simulation algorithms of boson sampling upper bound the allowed imperfections.",
"These classical simulations approximate the boson sampler outcome with a given error.",
"We now present the model of [17].",
"This model approximates an imperfect $n$ -photon boson sampler where $n-m$ photons are lost, by describing the output as up to $k$ -photon quantum interference ($0\\le k\\le m$ ) and at least $m-k$ classical boson interference.",
"Furthermore, this formalism naturally combines losses and distinguishability into a single simulation strategy, thereby introducing an explicit trade-off between the two.",
"In this model, the error bound $E$ of the classical approximation is given by $E < \\sqrt{ \\frac{\\alpha ^{k+1}}{1-\\alpha }}.$ The parameter $\\alpha $ which we will refer to as the 'source quality' is given by $\\alpha = \\eta x^2,$ with $\\eta =m/n$ denotes the transmission efficiency per photon.",
"Losses in different components are equivalent, so different losses can be combined into a single parameter $\\eta $ [35].",
"The average overlap of the internal part of the wave function between two photons is given by $x=\\langle \\psi _{\\rm i} | \\psi _{\\rm j} \\rangle $ (i$\\ne $ j).",
"Therefore $x^2$ is the visibility of a signal-signal Hong-Ou-Mandel interference dip [36].",
"This indistinguishability equals the spectral purity.",
"This model allows for optimizing the SPDC configuration by optimizing the source quality of Eq.",
"REF , which effectively trades-off the losses and distinguishability.",
"Furthermore, from Eq.",
"REF the maximal number of photons $k$ can be calculated by specifying a desired error bound.",
"In order to find the best SPDC configuration for a selection of crystals, we run an optimization over the SPDC settings to maximize $\\alpha $ while varying the filter bandwidth.",
"Since we consider collinear SPDC, the optimization parameters are the crystal length $L$ and the pump bandwidth $\\sigma _{\\rm p}$ .",
"Note that these parameters determine the shape of the JSA and therefore the separability.",
"The pump center wavelength is set such that group velocity dispersion is matched [37], [38], [39], [26].",
"From our numerical calculations we observe that the optimization problem appears to be convex over the region of the parameter space of interest.",
"We note that the optimization parameters are bounded, e.g., the crystal length cannot be negative.",
"A local optimization routine (L-BFGS-B, Python) was used.",
"Figure: a) The transmission efficiency per photon η\\eta and indistinguishability x 2 x^2 corresponding to the ideal SPDC settings at different filter bandwidths for different crystals (see legend in b).",
"The dashed lines are isolines, indicating how many photons kk can be used for a boson-sampling experiment.",
"The indistinguishability and transmission efficiency together result in the source quality factor α=x 2 η\\alpha = x^2 \\eta .",
"b) The values of α\\alpha and the corresponding number of photons kk (right axis) as function of the filter bandwidth.",
"In the legend R. denotes a rectangular filter, otherwise a Gaussian filter was used.The source quality $\\alpha $ can be calculated from the JSA.",
"The JSA was calculated numerically by discretizing the wavelength range of interest [40].",
"The wavelength range was chosen to include possible side lobes of the sinc phase-matching function.",
"The spectral purity is calculated from the discretized JSA using a singular value decomposition (SVD) [41].",
"The transmission efficiency is calculated by the overlap of the filtered and unfiltered JSA.",
"In other words, only 'intrinsic' losses are considered and experimental limitations such as additional absorption by optical components or absorption losses in the crystal are not taken into account.",
"This is permissible since such experimental losses are constant over the wavelength range.",
"The introduction of wavelength-independent losses does not chance the position of the optimum, as it only reduces the transmission efficiency.",
"Wavelength-dependent losses can be understood as an additional filter.",
"Realistic SPDC settings are guaranteed by constraining the crystal length and pump bandwidth values in the optimizer.",
"The crystal lengths are bounded by what is currently commercially available.",
"The pump bandwidth is bounded to a maximum of roughly $25\\,$ fs ($\\Delta f \\approx 17\\,$ THz) pulses.",
"Such pulses can be realized with commercial Ti:Sapph oscillators.",
"See the supplementary materials for the exact bounds and further details.",
"Furthermore we consider Gaussian-shaped and rectangular-shaped bandpass filters.",
"Rectangular filters are a reasonable approximation of broadband bandpass filters.",
"In the calculations, only the herald photon is filtered.",
"Also filtering the other photon reduces the heralding efficiency.",
"Typically the increase in purity is not worth the additional losses, especially if finite transmission efficiency of filters is included."
],
[
"Results",
"We now proceed by using the metric of [17] to compute the optimal filter bandwidth, pump bandwidth and crystal length for KTP, BBO and KDP sources.",
"The upper bound for the error of the classical approximation (Eq.",
"REF ) is set on the conventional $E=0.1$ .",
"Figure REF a) is a parametric plot of the source quality $\\alpha $ .",
"The transmission efficiency $\\eta $ is shown on the y-axis and signal-signal photon indistinguishability $x^2$ on the x-axis.",
"The ideal boson-sampling experiment is located at the top right.",
"Each point represents an optimal SPDC configuration that maximizes $\\alpha $ for that crystal corresponding to a fixed filter bandwidth.",
"The black dashed isolines indicate the maximum number of photons $k$ one can interfere, i.e., they are solutions of Eq.",
"REF for a fixed $E$ and $\\alpha $ .",
"The weak-filtering regime is in the top left, and the strong-filtering regime is in the bottom right.",
"Figure REF b) represents the source quality $\\alpha $ from Fig.",
"REF a) explicitly as a function of the filter bandwidth.",
"The left y-axis indicates the source quality $\\alpha $ .",
"The right y-axis shows the corresponding maximal number of photons $k$ .",
"Both graphs show that there is a filter bandwidth that maximizes $\\alpha $ .",
"From this maximal $\\alpha _{\\rm {opt}}$ the minimal transmission budget $\\eta _{\\rm TB}$ can be defined $\\eta _{\\rm {TB}} ~\\alpha _{\\rm {opt}} = \\alpha _{\\rm {50}},$ where $\\alpha _{\\rm {50}}$ denotes the required value of $\\alpha $ to perform a 50-photon boson-sampling experiment.",
"The transmission budget defines the minimum required transmission efficiency for all other components together.",
"This includes, for instance, non-unity detector efficiencies.",
"The maximal $\\alpha _{\\rm {opt}}$ for each crystal and the corresponding SPDC settings are shown in table REF .",
"The physical intuition behind the curves in Fig.",
"REF a) is the following.",
"In case of weak to no filtering (top left in Fig.",
"REF a)), the transmission efficiency is the highest and the spectral purity the lowest.",
"In this weak filtering regime the crystal length and pump bandwidth are such that the JSA is as factorizable as it can be without filtering.",
"This can also be seen in Fig REF .",
"Examples of such JSAs can be found in the appendix.",
"If we now increase filtering, we arrive at the regime of moderate filtering, at the center of Fig.",
"REF a).",
"While increasing the filtering, the optimal crystal length increases and the optimal pump bandwidth decreases.",
"This results in a relative increase of the transmission efficiency, since the unfiltered JSA is now smaller and 'fits easier' in the filter bandwidth.",
"The filter also smoothens out the JSA side lobes into a two-dimensional approximate Gaussian.",
"This is the regime with the optimal value for $\\alpha $ .",
"In the case of stronger filtering, the losses start to dominate.",
"The optimal strategy in this regime is to make the JSA as small as possible, such that as much of the photons can get through.",
"By doing so, the 'intrinsic' purity, i.e., before filtering, reduces since this configuration does no longer result in a factorizable state.",
"However, this reduction of purity is compensated by the spectral filter.",
"This is shown in Fig.",
"REF , where the 'intrinsic' purity decreases, but the filtered purity increases.",
"Figure: The spectral purities of a ppKTP source with a sinc phase-matching function.",
"The solid lines describe the spectral purity of the resulting photons before filtering.",
"The dashed lines correspond to the purity after passing through the spectral filter.Furthermore this physical picture also explains the differences between a rectangular and Gaussian filter window.",
"The first difference is that a Gaussian filter allows for higher values of $\\alpha $ and thus for more photons in a boson-sampling experiment.",
"The second difference is the optimal filter bandwidth.",
"Both differences can be explained by noting that a rectangular filter window ideally only filters out the side lobes.",
"As a result it cannot increase the factorability of the 'main' JSA, i.e., the part without the side lobes.",
"The results of the Gaussian apodized source cannot be understood using the aforementioned physical intuition.",
"The filter does not improve the spectral purity since there are no side lobes and the pump bandwidth and crystal length can be chosen such that the JSA is factorizable.",
"The limiting factor here is group-velocity dispersion, which is small around $1582\\,$ nm [20].",
"Table: The values of α opt \\alpha _{\\rm {opt}} and the loss budget for a k=50k=50 photon boson-sampling experiment for different crystals at a center wavelength λ c \\lambda _{\\rm c}.",
"The corresponding SPDC settings (crystal length LL, pump bandwidth σ p \\sigma _{\\rm p} and filter bandwidth σ f \\sigma _{\\rm f}) are also listed.",
"The mentioned bandwidths are FWHM of the fields."
],
[
"Discussion",
"It is well known that the spectral purity of symmetrically group-velocity-matched SPDC sources is invariant to changes of either the crystal length or pump bandwidth, as long as the other one is changed accordingly.",
"However, Fig.",
"REF shows that relation no longer holds when filtering is included.",
"In the regime of strong filtering, $\\alpha $ is dominated by the losses.",
"Therefore, the SPDC configuration which optimizes $\\alpha $ inevitably is the one that minimizes the losses.",
"Hence the spectral purity reduces, but this is compensated by the strong filtering.",
"In an experiment the non-unity transmission efficiency of a filter at the maximum of the transmission window will be an important source of losses.",
"As a consequence, spectral filtering is only useful when the filter's maximum transmission is larger than $\\alpha _{\\rm {f}}/\\alpha _{\\rm {0}}$ , where $\\alpha _{\\rm {f}}$ denotes the filtered $\\alpha $ and $\\alpha _{\\rm {0}}$ the unfiltered case.",
"If the filter's transmission is lower, then the gain in $\\alpha $ is not worth the additional losses.",
"We note that the ideal filter bandwidths of Tab.",
"REF are larger than what is reported in [14].",
"We attribute this difference to two points.",
"Firstly, the model of [14] approximates the sinc phase-matching function as a Gaussian.",
"This eliminates the side lobes and hence reduces the losses.",
"As a consequence, smaller filter bandwidths are optimal.",
"Secondly, the model of [14] focuses on the symmetrized heralding efficiency where both photons are filtered.",
"A final point regarding the spectral filters is that the optimal filter bandwidths for ppKTP sources are rather large ($>100\\,$ nm).",
"Photons with such large bandwidths are typically unpractical for multi-photon experiments since the properties of optical components, e.g., the splitting ratio of a beam splitter, are rarely constant over such a wavelength range.",
"These additional constraints on optical components may result in a better classical simulation of boson sampling.",
"Hence it could increase the required effort to do a boson-sampling experiment."
],
[
"Conclusion",
"In conclusion, we have numerically optimized SPDC sources for scattershot boson sampling.",
"Using the recently found source quality parameter $\\alpha $ [17] we have investigated a number of candidates for building the next generation of SPDC sources.",
"From the results of Tab.",
"REF we conclude that SPDC sources in principle allow the demonstration of a quantum advantage with boson sampling.",
"The most suitable source for boson sampling is an apKTP crystal.",
"Such a source can have a maximal source quality $\\alpha _{\\rm {opt}}=0.99$ and has a corresponding transmission budget of $0.87$ %.",
"This transmission budget is sufficient to incorporate state-of-the-art[42], [43] detector efficiencies and keep a small buffer for additional optical losses.",
"The other, periodically poled, KTP source has an optimal filter bandwidth of more than $100\\,$ nm.",
"The other two sources are asymmetrically group-velocity-matched sources.",
"The KDP source with a maximal source quality of $\\alpha _{\\rm opt} = 0.98$ is a good alternative.",
"The optimal source quality for BBO is found to be comparable with ppKTP and less suited for a boson sampling experiment.",
"The fact that these asymmetrically group-velocity-matched sources perform less than symmetrically matches sources is consistent with previous findings.",
"The limited tolerance for additional losses for the Gaussian apodized KTP source suggests that both waveguide sources and bulk sources without focusing of the pump beam are ideal.",
"Such sources have a single spatial mode and thus do not suffer from an additional reduction of distinguishability which is inevitable with focusing [15].",
"This work can be extended to other SPDC sources such as [44], [45], [46], four-wave mixing sources [47] and to Gaussian boson sampling [48].",
"The latter can be realized by including the distinguishability between the signal and idler photons.",
"The Complex Photonic Systems group acknowledges funding from the Nederlandse Wetenschaps Organsiatie (NWO) via QuantERA QUOMPLEX (no.",
"731473), Veni (Photonic Quantum Simulation) and NWA (No.",
"40017607).",
"The Integrated Quantum Optics group acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No.",
"725366, QuPoPCoRN).",
"*"
],
[
"Optimal SPDC settings",
"The effect of the filter bandwidth on the optimal SPDC configuration (except for apKTP) can be categorized in three different regimes.",
"These regimes are the weak, moderate and strong filtering regime.",
"An example of the JSA of a ppKTP source in all three regimes can be seen in Fig.",
"REF .",
"The corresponding SPDC configuration parameters can be seen in Figure REF .",
"This figure shows that in the weak filtering regime, the bounds on the crystal sizes and pump bandwidth can be reached.",
"Once such a bound is reached, the SPDC configuration loses a parameter to optimize the JSA factorizability with, meaning that the general trend of matching the crystal length and pump bandwidth cannot continue anymore.",
"This limits the purity.",
"In case of ppKTP, the limiting factor is the crystal length, whereas in case of a BBO source the maximum pump bandwidth is the limiting factor.",
"Figure: The optimal pump bandwidth and crystal length as a function of the filter bandwidth.Figure: The real part of the JSA of the optimal ppKTP SPDC configuration in case of weak filtering with (left), optimal filtering (center) and strong filtering (right).",
"The top panels show the JSA before filtering, the bottom panels after filtering."
],
[
"Numerical stability",
"We used a local optimization algorithm to find the optimal SPDC configuration for different filter bandwidths.",
"Each iteration of this algorithm computes the spectral purity and losses by discretizing the (filtered) JSA.",
"Such a numerical approach can fail and/or give wrong results.",
"The algorithm can fail because the problem is not convex or that it finds unphysical results (such as a negative crystal length).",
"The algorithm can give wrong results if the discretization of the JSA is too coarse.",
"By bounding the parameter space we guarantee that the algorithm does not reach unphysical results.",
"Furthermore, we note that optimizing over the whole parameter space, i.e., the filter bandwidths, crystal lengths and pump bandwidths is not a convex problem.",
"This problem is solved by optimizing the crystal and pump properties each time for different filter bandwidths.",
"The discretization of the JSA can cause numerical errors.",
"Increasing the number of grid points, i.e., increasing the resolution, decreases this numerical error.",
"Increasing the resolution results to a convergence of the result.",
"Unfortunately, it is not directly known how our numerical calculation converges to a reliable answer.",
"How to a priori estimate the numerical error for a given discretization is also unclear.",
"In order to show that our calculations have converged, we simply try different discretizations of the JSA.",
"For every discretization, we calculate the corresponding source quality $\\alpha $ and observe how it is varies.",
"Figure REF shows that the numerical error originating from this discretization is small in the limit of more than $2000^2$ (2000 per photon) grid points.",
"This confirms the validity of our calculations.",
"Table REF provides an overview of all relevant parameters for the stability of the simulation.",
"Figure: The convergence of the source quality α\\alpha with the discretization of the frequency space.",
"Each data point is the difference of α\\alpha with the α\\alpha corresponding to 8000 2 8000^2 grid points.",
"All crystals, ppKTP, BBO and KDP, are set in their optimal configuration.Table: The simulation parameters for each crystal.",
"The bounds on the crystal lengths and pump bandwidth are given, just as the range of wavelength over which the JSA is computed.",
"The grid points are the number of steps used to discretize the entire wavelength range"
]
] | 2001.03596 |
[
[
"Numerical relativity simulation of GW150914 in Einstein dilaton\n Gauss-Bonnet gravity"
],
[
"Abstract A present challenge in testing general relativity (GR) with binary black hole gravitational wave detections is the inability to perform model-dependent tests due to the lack of merger waveforms in beyond-GR theories.",
"In this study, we produce the first numerical relativity binary black hole gravitational waveform in Einstein dilaton Gauss-Bonnet (EDGB) gravity, a higher-curvature theory of gravity with motivations in string theory.",
"We evolve a binary black hole system in order-reduced EDGB gravity, with parameters consistent with GW150914.",
"We focus on the merger portion of the waveform, due to the presence of secular growth in the inspiral phase.",
"We compute mismatches with the corresponding general relativity merger waveform, finding that from a post-inspiral-only analysis, we can constrain the EDGB lengthscale to be $\\sqrt{\\alpha_\\mathrm{GB}} \\lesssim 11$ km."
],
[
"Introduction",
"Though Einstein's theory of general relativity (GR) has passed all precision tests to date, at some lengthscale, it must break down and be reconciled with quantum mechanics in a beyond-GR theory of gravity.",
"Binary black hole (BBH) mergers probe the strong-field, non-linear regime of gravity, and thus gravitational wave signals from these systems could contain signatures of a beyond-GR theory.",
"While LIGO presently performs model-independent and parametrized tests of general relativity [1], [2], one important additional avenue of looking for deviations from general relativity is to perform model-dependent tests.",
"Such model-dependent tests require access to numerical waveforms in beyond-GR theories of gravity through merger, the lack of which is currently a severe limitation on constraining beyond-GR physics [3].",
"We produce the first numerical relativity gravitational waveforms in Einstein dilaton Gauss-Bonnet (EDGB) gravity, an effective field theory that modifies the Einstein-Hilbert action of GR through the inclusion of a scalar field coupled to terms quadratic in curvature.",
"These terms are meant to encompass underlying quantum gravity effects, in particular motivated by string theory [4], [5], [6], [7], and the coupling to the scalar field is governed by an EDGB lengthscale parameter $\\sqrt{\\alpha _\\mathrm {GB}}$ .",
"The well-posedness of the initial value problem in full EDGB gravity is unknown [8], [9], [10], [11].",
"We thus work in an order-reduction scheme, in which we perturb the EDGB scalar field and spacetime metric about a GR background.",
"Previously, Witek et al.",
"[12] evolved the leading-order EDGB scalar field on a BBH background, predicting a bound of $\\sqrt{\\alpha _\\mathrm {GB}}\\lesssim 2.7$ km on the EDGB lengthscale, a constraint seven orders of magnitude tighter than observational results from solar-system tests.",
"In this study, we evolve the leading-order EDGB correction to the spacetime metric on a BBH background, thus obtaining the leading-order EDGB modification to the merger gravitational waveform.",
"We compute mismatches between the GR and EDGB-corrected waveforms, aiming to similarly bound the EDGB lengthscale.",
"We focus on an astrophysically-relevant BBH system with spin and mass ratio consistent with GW150914, the loudest LIGO detection to date [13], [14], [15], for which significant model-independent and parametrized tests of GR have been performed [1], [3], [2], [16], [17].",
"This extends our results in [18], where we simulated the same system in dynamical Chern-Simons gravity (dCS), another quadratic beyond-GR theory with motivations in string theory and loop quantum gravity [19], [20], [21], [22]."
],
[
"Setup",
"We set $G = c = 1$ throughout.",
"Quantities are given in terms of units of $M$ , the sum of the Christodolou masses of the background black holes at a given reference time [23].",
"Latin letters in the beginning of the alphabet $\\lbrace a, b, c, d \\ldots \\rbrace $ denote 4-dimensional spacetime indices, and $g_{ab}$ refers to the spacetime metric with covariant derivative $\\nabla _c$ ."
],
[
"Equations of motion",
"The overall form of the EDGB action that we will use in this paper, chosen to be consistent with Witek et al.",
"[12], is $S \\equiv \\int \\frac{m_{\\textrm {\\tiny {pl}}}^2}{2} d^4 x \\sqrt{-g} \\left[R - \\frac{1}{2} (\\partial \\vartheta )^2 + 2 \\alpha _\\mathrm {GB} f(\\vartheta ) \\mathcal {R}_\\mathrm {GB} \\right]\\,,$ where the first term is the Einstein-Hilbert action of GR (where $R$ is the 4-dimensional Ricci scalar), $\\vartheta $ is the EDGB scalar field, and $\\alpha _\\mathrm {GB}$ is the EDGB coupling parameter with dimensions of length squared.",
"We will work with $\\sqrt{\\alpha _\\mathrm {GB}}$ , which has dimensions of length, throughout this paper.",
"Finally, $\\mathcal {R}_\\mathrm {GB}$ is the EDGB scalar, of the form $\\mathcal {R}_\\mathrm {GB} = R^{abcd}R_{abcd} - 4 R^{ab} R_{ab} + R^2\\,.$ It is unknown whether EDGB has a well-posed initial value problem [8], [9], [10], [11].",
"However, as we have done in [24], [25], [26], [18], we perturb the spacetime metric and EDGB scalar field about an arbitrary GR background as $g_{ab} &= g_{ab}^{(0)} + \\sum _{n = 1}^{\\infty } \\varepsilon ^n g_{ab}^{(n)}\\,, \\\\\\vartheta &= \\sum _{n = 0}^\\infty \\varepsilon ^n \\vartheta ^{(n)}\\,,$ where $\\varepsilon $ is an order-counting parameter that counts powers of $\\alpha _\\mathrm {GB}$ , and superscript ${}^{(0)}$ corresponds to the GR solution, which we refer to as the background.",
"At each order, the equations of motion are well-posed.",
"Moreover, the EDGB coupling parameter $\\alpha _\\mathrm {GB}$ scales out at each order, and thus we only need to perform one BBH simulation for each set of GR background parameters.",
"Zeroth order corresponds to pure general relativity.",
"The equation of motion for the zeroth order scalar field, $\\vartheta ^{(0)}$ , corresponds to a scalar field minimally coupled to vacuum GR, and thus $\\vartheta ^{(0)}$ should decay to zero in BH spacetimes by the no-hair theorem.",
"The leading-order EDGB scalar field appears at first-order as $\\vartheta ^{(1)}$ , sourced by the curvature of the GR background, with equation of motion (cf.",
"[12] for a full derivation), $\\square ^{(0)} \\vartheta ^{(1)} &= - M^2 \\mathcal {R}^{(0)}_\\mathrm {GB}\\,, \\\\\\mathcal {R}^{(0)}_\\mathrm {GB} &\\equiv R^{(0)} {}^{abcd}R^{(0)}_{abcd} - 4 R^{(0)} {}^{ab} R^{(0)}_{ab} + R^{(0)} {}^2\\,,$ where the superscript ${}^{(0)}$ refers to quantities computed from the GR background.",
"Here, the leading-order correction to $f(\\vartheta )$ has been set to $\\frac{1}{8}$ , in accordance with [12].",
"Meanwhile, the leading EDGB deformation to the spacetime metric comes in at second order, with the equation of motion (cf.",
"[12]), $G_{ab}^{(0)} [g_{ab}^{(2)}] = - 8 M^2 \\mathcal {G}_{ab}^{(0)}[\\vartheta ^{(1)}] + T_{ab}[\\vartheta ^{(1)}]\\,.$ In the above equations, $T_{ab}[\\vartheta ^{(1)}]$ is the standard Klein-Gordon stress energy tensor associated with $\\vartheta ^{(1)}$ , of the formNote that our definition of $T_{ab}[\\vartheta ^{(1)}]$ in Eq.",
"(REF ) differs from Eq.",
"15 in [12] by a factor of 2, and hence the $T_{ab}[\\vartheta ^{(1)}]$ term in Eq.",
"(REF ) differs by a factor of 2.",
"We have chosen this convention to be in line with the canonical form of the Klein-Gordon stress-energy tensor.",
"$T_{ab}[\\vartheta ^{(1)}] = \\nabla ^{(0)}_a \\vartheta ^{(1)} \\nabla ^{(0)}_b \\vartheta ^{(1)} - \\frac{1}{2} g_{ab}^{(0)} \\nabla ^{(0)}_c \\vartheta ^{(1)} \\nabla ^{(0)}{}^c \\vartheta ^{(1)}\\,,$ and $\\mathcal {G}_{ab}^{(0)}[\\vartheta ^{(1)}] = 2 \\epsilon ^{edfg} g_{c(a}^{(0)} g_{b)d}^{(0)} \\nabla _h^{(0)} \\left[\\frac{1}{8} \\,{}^*\\!R^{(0)}{}^{ch} {}_{fg} \\nabla ^{(0)}_e \\vartheta ^{(1)} \\right]\\,,$ where $\\,{}^*\\!R^{ab} {}_{cd} = \\epsilon ^{abef} R_{efcd}^{(0)}$ and $\\epsilon ^{abcd}$ is the Levi-Citiva pseudo-tensor, with $\\epsilon ^{abcd} = -[abcd]/\\sqrt{-g^{(0)}}$ , where $[abcd]$ is the alternating symbol.",
"Note that we work on a vacuum GR background, and thus terms vanish to give the simplified equations of motion $\\square ^{(0)} \\vartheta ^{(1)} &= - M^2 R^{(0)} {}^{abcd}R^{(0)}_{abcd} \\\\\\nonumber G_{ab}^{(0)} [g_{ab}^{(2)}] &= -2 M^2 \\epsilon ^{edfg} g_{c(a}^{(0)} g_{b)d}^{(0)} \\,{}^*\\!R^{(0)}{}^{ch} {}_{fg}\\nabla _h^{(0)} \\nabla ^{(0)}_e \\vartheta ^{(1)} \\\\& \\quad \\quad \\quad \\quad + T_{ab}[\\vartheta ^{(1)}]\\,.$ To summarize: the order-reduction procedure is illustrated in Fig.",
"1 of [18].",
"We will have a GR binary black hole background.",
"The curvature of this background will then source the leading-order EDGB scalar field (Eq.",
"(REF )).",
"This leading-order scalar field and the GR background will then source the leading-order EDGB correction to the spacetime (Eq.",
"()), which in turn will give us the leading-order EDGB correction to the gravitational waveform."
],
[
"Secular growth during inspiral",
"As we initially noted in [18], the perturbative order-reduction scheme outlined in Sec.",
"REF gives rise to secular growth during the inspiral.",
"In the order-reduction scheme, the rate of inspiral is governed by the GR background.",
"However, in the full, non-linear EDGB theory, we expect the black holes to have a faster rate of inspiral due to energy loss to the scalar field [27].",
"Since we do not backreact on the GR background in the order-reduction scheme, we do not capture this correction to the rate of inspiral, and hence our solution contains secular growth.",
"This is a feature generically found in perturbative treatments [28], including in extreme mass-ratio inspirals [29].",
"When we simulated an inspiraling binary black hole system in order-reduced dCS [18], we indeed observed secular growth during the inspiral.",
"We performed a set of simulations where we ramped on the dCS source terms at various start times during the inspiral, for the same set of background parameters.",
"We found secular growth in the amplitude of the resulting dCS correction to the waveform, with simulations with earlier start times having larger amplitudes.",
"However, this secular growth settled to a quadratic minimum for a start time before the portion of the inspiral-merger present in the LIGO band for a GW150914-like system.",
"Thus, we were able to focus on this portion of the waveform in [18] without having contamination from secular effects.",
"In this study, we apply the same procedure, where we ramp on the EDGB source terms at a variety of start times for the same (long) GR binary black hole background simulation.",
"We search for the start time at which the waveform is no longer contaminated by secular effects, and present the resulting merger waveform.",
"The inspiral in EDGB is more strongly modified from GR, with the modifications to the inspiral occurring at -1 PN order relative to GR due to the presence of dipolar radiation in the scalar field [27].",
"This is 3 PN orders higher than the leading modification in the dCS case, where dipolar radiation is absent during inspiral.",
"Thus we expect the minimum of the secular growth to occur later in the inspiral in EDGB than in dCS for the same physical system."
],
[
"Computational details",
"Eqs.",
"(REF ) and () are precisely the equations that we co-evolve with the GR background.",
"We use the Spectral Einstein Code [30], which uses pseudo-spectral methods and thus guarantees exponential convergence in the fields.",
"All of the technical details are given in [18], [26], [25], [24].",
"The domain decomposition is precisely that of the analogous dCS study [18]."
],
[
"Simulation parameters",
"While there is a distribution of mass and spin parameters consistent with GW150914 [14], [31], we choose to use the parameters of SXS:BBH:0305, as given in the Simulating eXtreme Spacetimes (SXS) catalog [32].",
"This simulation was used in Fig.",
"1 of the GW150914 detection paper [13], as well a host of follow-up studies [33], [34], [35].",
"We additionally used precisely these parameters for our dCS BBH simulation [18].",
"The configuration has initial dimensionless spins $\\chi _A = 0.330 \\hat{z}$ and $\\chi _B = -0.440 \\hat{z}$ , aligned and anti-aligned with the orbital angular momentum.",
"The dominant GR spherical harmonic modes of the gravitational radiation for this system are $(l,m)=(2,\\pm 2)$ .",
"The system has initial masses of $0.5497 \\,M$ and $0.4502 \\,M$ , leading to a mass ratio of $1.221$ .",
"The initial eccentricity is $\\sim 8 \\times 10^{-4}$ .",
"The remnant has final Christodolou mass $0.9525 \\,M$ and dimensionless spin $0.692$ purely in the $\\hat{z}$ direction.",
"The GR background simulation completes 23 orbits before merger."
],
[
"Regime of validity",
"The results that we present for the leading-order EDGB scalar and gravitational waveforms have the EDGB coupling parameter $\\alpha _\\mathrm {GB}$ scaled out.",
"For the perturbative order reduction scheme to be valid, we require that $g_{ab}^{(2)} \\lesssim C g_{ab}^{(0)}$ , for some constant $C < 1$ .",
"This in turn becomes a constraint on $\\alpha _\\mathrm {GB}$ , of the form (cf.",
"[24] for an analogous derivation) $\\frac{\\sqrt{\\alpha _\\mathrm {GB}}}{GM} \\lesssim \\left(C \\frac{\\Vert g_{ab}^{(0)}\\Vert }{\\Vert g_{ab}^{(2)}\\Vert } \\right)^{1/4}\\,.$ We choose $C = 0.1$ , and evaluate Eq.",
"(REF ) on each slice of the numerical relativity simulation.",
"We find the the strongest constraint on the allowed value of $\\sqrt{\\alpha _\\mathrm {GB}}/GM$ comes at merger, when the spacetime is most highly perturbed, with a value of $\\sqrt{\\alpha _\\mathrm {GB}}/GM \\sim 0.17$ for the simulation presented in this paper."
],
[
"EDGB scalar field waveforms",
"In Fig.",
"REF , we show the results for the leading-order EDGB scalar field, $\\vartheta ^{(1)}$ .",
"We decompose the scalar field into spherical harmonics, and find that the dominant modes are $(l, m = l)$ , in accordance with [12], [27].",
"We see the presence of $l = 1$ dipolar radiation in the field during inspiral, in accordance with [12], [27].",
"We see that the monopolar $l = 0$ mode is non-radiative during the inspiral, but that there is a burst of monopolar radiation at merger.",
"This is in agreement with [12], and moreover is similar to the results in dCS [24], where we found that the leading non-radiative mode (the dipole in the dCS case) exhibits a burst of radiation at merger.",
"Figure: Dominant modes of the leading-order EDGB scalar field ϑ (1) \\vartheta ^{(1)}, decomposed into spherical harmonics (l,m)(l, m), as a function of time relative to the peak time of the GR gravitational waveform.",
"The top panel corresponds to the dominant (2,2)(2,2) mode of the GR gravitational radiation for comparison.",
"The bottom three panels correspond to the dominant modes of ϑ (1) \\vartheta ^{(1)}, which are (l,m=l)(l,m=l).",
"We see the presence of l=1l = 1 dipolar radiation during the inspiral.",
"While the l=0l = 0 monopole is non-radiative during the inspiral, we see a burst of monopolar radiation at merger.",
"Compare with Fig.",
"4 of and the dCS case in Fig.",
"1 of .",
"Note that the ϑ (1) \\vartheta ^{(1)} waveforms have the EDGB coupling α GB /GM\\sqrt{\\alpha _\\mathrm {GB}}/GM scaled out, and thus an appropriate value (cf.",
"Sec. )",
"of this coupling parameter must be re-introduced for the results to be physically meaningful."
],
[
"EDGB gravitational waveforms",
"As explained in Sec.",
"REF , because of secular growth during the inspiral, we focus on simulations with EDGB effects ramped on close to merger, in order to mitigate the amount of secular growth from the inspiral (we give more details in Sec.",
"REF ).",
"We thus present these merger waveforms in this section.",
"From the leading-order EDGB metric deformation $ g_{ab}^{(2)}$ , we can compute $\\Psi _4^{(2)}$ , the leading-order modification to the gravitational waveform, given by the Newman-Penrose scalar $\\Psi _4$ .",
"Note that $g_{ab}^{(2)}$ and hence $\\Psi _4^{(2)}$ from the simulation are independent of the EDGB coupling parameter.",
"In order to produce a full, second-order-accurate EDGB gravitational waveform, we must add $\\Psi _4^{(2)}$ to the background GR waveform $\\Psi _4^{(0)}$ as $\\Psi _4 = \\Psi _4^{(0)} + (\\sqrt{\\alpha _\\mathrm {GB}}/GM)^4 \\Psi _4^{(2)} + \\mathcal {O}((\\sqrt{\\alpha _\\mathrm {GB}}/GM)^6)\\,,$ for a given choice for the EDGB coupling parameter $\\sqrt{\\alpha _\\mathrm {GB}}/GM$ .",
"We require that $\\sqrt{\\alpha _\\mathrm {GB}}/GM$ lies within the regime of validity for the perturbative scheme as given in Sec.",
"REF In Fig.",
"REF , we show this total waveform for a variety of values of $\\sqrt{\\alpha _\\mathrm {GB}}/GM$ .",
"We see that the EDGB-corrected waveform has an amplitude shift relative to GR, as well as a phase shift, consistent with the notion that EDGB should have a faster inspiral due to energy loss to the scalar field [27].",
"Figure: EDGB-corrected merger gravitational waveforms, as computed from Eq.",
"(), for a variety of values of the EDGB coupling parameter α GB /GM\\sqrt{\\alpha _\\mathrm {GB}}/GM.",
"The dashed black line, with α GB /GM=0\\sqrt{\\alpha _\\mathrm {GB}}/GM= 0, corresponds to the GR waveform.",
"The value α GB /GM=0.17\\sqrt{\\alpha _\\mathrm {GB}}/GM= 0.17 corresponds to the maximal allowed value in order for the perturbative scheme to be valid (cf.",
"Sec. ).",
"We see that the EDGB-corrected waveform has both an amplitude and phase shift relative to GR."
],
[
"Secular growth",
"As discussed in Sec.",
"REF , the perturbative scheme leads to secular growth in the inspiral waveform.",
"In Fig.",
"REF , we show the leading-order EDGB correction to the gravitational waveform for a variety of simulation lengths (with the same background GR simulation).",
"We ramp on the EDGB source terms at different start times in order to produce different inspiral lengths, as discussed in Sec.",
"REF .",
"We see that the longest simulations have the largest amplitude at merger, consistent with secular growth.",
"In Fig.",
"REF , we take a more quantitative look, plotting the peak amplitude of the waveform as a function of inspiral length.",
"In the dCS case (cf.",
"Fig.",
"7 of [18]), we saw that for the closest start time to merger, the secular growth attained a quadratic minimum.",
"In other words, the merger waveform we presented was not contaminated by secular effects.",
"In Fig.",
"REF , we see a similar quadratic minimum for the EDGB correction to the waveform, although this occurs at a shorter inspiral length (later start time) than in dCS.",
"This higher level of secular growth in EDGB than in dCS is consistent with the theoretical predictions of Sec.",
"REF , as the EDGB inspiral is more heavily modified than in dCS due to the presence of dipolar radiation [27].",
"Figure: Secular growth in leading-order EDGB gravitational waveforms as function of inspiral length of the waveform.",
"Each colored curve corresponds to a simulation with a different start time for the EDGB fields (as discussed in Sec.",
"), with the same GR background simulation for each.",
"We label each curve by the time difference between the peak of the waveform and the start time of ramping on the EDGB field (minus the ramp time).",
"We see that simulations with earlier EDGB start times have higher amplitudes at merger, having had more time to accumulate secular growth.Figure: Peak amplitude of the EDGB correction to the gravitational waveform as a function of inspiral length.",
"We show the length relative to the peak of the waveform (as in Fig. ).",
"The dashed black vertical line corresponds to the length of the EDGB merger simulation we present in this paper.",
"The peak amplitude serves as a measure of the amount of secular growth in the waveform (cf.",
"Fig. ).",
"We see that the secular growth attains a quadratic minimum, and thus for a short enough inspiral length, we can obtain an EDGB gravitational waveform with minimal secular contamination."
],
[
"Constraints on $\\sqrt{\\alpha _\\mathrm {GB}}$ from EdGB merger waveforms",
"As shown in Sec.",
"REF , we have access to the leading-order EDGB merger waveform for a GW150914-like system.",
"What sort of physical constraints on EDGB can we extract from the merger phase?"
],
[
"Merger mismatches",
"The first step that we can take is to perform a merger-only analysis by computing mismatches between the GR waveform and the EDGB waveform using the formulae in Sec. .",
"This involves restricting to a given time (or frequency) range over which to compute the mismatch.",
"When performing tests of general relativity, LIGO performs such merger-only calculations.",
"In [1], the authors performed an inspiral-merger-ringdown consistency test for GW150914 by inferring final mass and spin parameters using GR waveforms from the post-inspiral portion of the waveform only, from the inspiral portion of the waveform only, and comparing the resulting posterior distribution to that from the full waveform analysis.",
"For GW150914, the merger-ringdown region was chosen to be $[132, 1024]$ Hz.",
"In this region, the signal had a signal to noise ratio (SNR) of 16, which is larger than the full-waveform SNR of the other nine BBH detections in GWTC-1 [15].",
"We thus compute mismatches between the GR and EDGB merger waveforms, shown in Fig.",
"REF .",
"We show the mismatch (cf.",
"Sec. )",
"for various values of $\\sqrt{\\alpha _\\mathrm {GB}}/GM$ (cf.",
"Fig.",
"REF ).",
"In particular, for a $1\\%$ mismatch, we find $\\sqrt{\\alpha _\\mathrm {GB}}/GM\\lesssim 0.11$ .",
"For GW150914, we choose $M \\sim 68\\, M_\\odot $ [14], and thus compute $\\sqrt{\\alpha _\\mathrm {GB}}\\lesssim 11$ km.",
"Note that though we shift the waveforms in time and phase to compute a minimum mismatch, we do not vary the GR waveform parameters (mass and spin).",
"Thus our mismatch estimate is optimistic, and performing a full parameter-estimation analysis on our EDGB waveform is the subject of future research.",
"Figure: Mismatch between general relativity GW150914 waveform (cf.",
"Sec. )",
"and the corresponding EDGB-corrected gravitational waveform, as defined in Eq. ().",
"We show the mismatch for our merger waveform as a function of the EDGB coupling parameter, α GB /GM\\sqrt{\\alpha _\\mathrm {GB}}/GM.",
"We show the maximum allowed value of α GB /GM\\sqrt{\\alpha _\\mathrm {GB}}/GM from the regime of validity (cf.",
"Sec. )",
"in dot-dashed gray.",
"The dashed horizontal line corresponds to the LIGO mismatch of 4%4\\% from testing GR with GW150914 .",
"The top vertical axis corresponds to α GB \\sqrt{\\alpha _\\mathrm {GB}} computed from α GB /GM\\sqrt{\\alpha _\\mathrm {GB}}/GM on the bottom axis assuming that M=68M ⊙ M = 68 M_\\odot for GW150914.For heavier BBH systems, such as GW170729 with $M = 84\\,M_\\odot $ , which had 3 cycles in the LIGO band [15], we can in theory use only the merger-ringdown EDGB waveforms from numerical relativity simulations for data analysis, without requiring EDGB inspiral waveforms.",
"Note, however, that with all other parameters held equal, this lead to a lower constraint on $\\sqrt{\\alpha _\\mathrm {GB}}$ from the larger total mass.",
"Moreover, GW170729 has an SNR of $\\sim 10$ , which is less than the merger SNR of 16 for GW150914.",
"Note that the LIGO intermediate mass black hole search [36] which looked for BBHs with $M \\in [120, 800]\\,M_\\odot $ did not detect any signals."
],
[
"Including inspiral",
"How much more could we gain if we additionally included the inspiral phase?",
"Gaining access to the inspiral phase for EDGB waveforms is ongoing work, through either implementing a renormalization scheme to remove secular effects as outlined in [18], or by stitching on post-Newtonian or parametrized post-Einstenian (ppE) EDGB waveforms for the inspiral [27], [37], to obtain a full waveform.",
"In [3], the authors use the ppE formalism to bound $\\sqrt{\\alpha _\\mathrm {GB}}$ with GW150914.",
"Fig.",
"15 of [3] shows the upper bounds on $\\sqrt{\\alpha _\\mathrm {GB}}$ , including values of $\\mathcal {O}(20, 40)$ km, but this is very sensitive to the dimensionless spins of the black holes, which are poorly constrained (cf.",
"[14], [15]).",
"Thus, the authors do not place an upper bound on $\\sqrt{\\alpha _\\mathrm {GB}}$ .",
"In [38], the authors place an upper bound of $\\sqrt{\\alpha _\\mathrm {GB}}\\lesssim 51.5$ km for GW150914 using a ppE analysis, which is higher than our merger-only analysis bound.",
"Including a merger phase to these inspiral-only analyses can thus improve their bounds on $\\sqrt{\\alpha _\\mathrm {GB}}$ ."
],
[
"Comparison to observational and projected constraints",
"Let us now compare the merger-analysis result of $\\sqrt{\\alpha _\\mathrm {GB}}\\lesssim 11$ km with observational and predicted observational constraints in the literature.",
"We summarize these present constraints in Table REF .",
"Most notably, Witek et al.",
"[12] estimate from their scalar field calculations that for a GW151226-like system [39], the constraint would be $\\sqrt{\\alpha _\\mathrm {GB}}\\lesssim 2.7$ km.",
"Note that this signal has $\\sim 15$ cycles in the LIGO band (compared to $\\sim 5$ in the LIGO band for GW150914) [15], and thus the inspiral phase, which is not included in our estimate, plays a greater role for this system.",
"Moreover, this esimate was performed with a mass ratio of $q \\sim 2$ and total mass $\\sim 20 M_\\odot $ , which leads to stronger beyond-GR effects due to the higher curvature of the smaller object.",
"Table: Observed and projected bounds on the EDGB lengthscale from various studies.",
"The first to rows (in bold), correspond to observed bounds, from the Cassini probe constraints on Shapiro time delay and observations of X-ray binaries.",
"Note that all bounds are given in terms of the conventions in our action (cf.",
"Eq.",
"()), chosen to be consistent with ."
],
[
"Conclusion",
"We have produced the first astrophysically-relevant numerical relativity binary black hole gravitational waveform in Einstein dilaton Gauss-Bonnet gravity, a beyond-GR theory of gravity.",
"We have focused on a system with parameters consistent with GW150914, the loudest LIGO detection thus far.",
"This extends our previous work for producing such a waveform for GW150914 in dynamical Chern-Simons gravity [18].",
"In Sec.",
", we laid out our order-reduction scheme, which we use to obtain a well-posed initial value formulation and produce the leading-order EDGB correction to the gravitational waveform.",
"In Sec.",
"REF , we showed the EDGB-corrected waveforms for a system consistent with GW150914 (cf.",
"Sec.",
"REF ).",
"We find that there is secular growth in the inspiral phase (Sec.",
"REF ), and thus present a merger-ringdown waveform that is free of secular growth.",
"We thus focus on a post-inspiral-only analysis, and compute the mismatch between the (background) GR waveform and the EDGB-corrected waveforms, finding a bound on the EDGB coupling parameter of $\\sqrt{\\alpha _\\mathrm {GB}}/GM\\lesssim 11$ km.",
"This is a stronger result than inspiral-only analyses for GW150914, which bound $\\sqrt{\\alpha _\\mathrm {GB}}\\lesssim 51.5$ km.",
"Note that GW150914 has an SNR of 16 in the post-inspiral phase (cf.",
"[1]), which is larger than the total SNR of each other event in GWTC-1 [15].",
"Stitching on a parametrized post-Einstenian EDGB inspiral or removing the inspiral secular growth from our simulations (cf.",
"[18]) to take full advantage of an inspiral-merger-ringdown analysis is the subject of future work.",
"Our ultimate goal is to make these beyond-GR waveforms useful for LIGO and Virgo tests of general relativity [1], [2].",
"We can improve the mismatch analysis by allowing the GR waveform parameters to vary, thus checking for degeneracies in the GR-EDGB parameter space.",
"Moreover, we can perform a more quantitative analysis by injecting our beyond-GR waveforms into LIGO noise and computing posteriors recovered using present LIGO parameter estimation and testing-GR methods [44], [45], [14], [2].",
"Ultimately, we would like to generate enough beyond-GR EDGB waveforms to fill the BBH parameter space.",
"We can then produce a beyond-GR surrogate model [46] and perform model-dependent tests of GR."
],
[
"Acknowledgements",
"We thank Leo Stein, Helvi Witek, and Paolo Pani for useful discussions.",
"The Flatiron Institute is supported by the Simons Foundation.",
"Computations were performed using the Spectral Einstein Code [30].",
"All computations were performed on the Wheeler cluster at Caltech, which is supported by the Sherman Fairchild Foundation and by Caltech."
],
[
"Mismatches",
"Given the GR and EDGB-corrected waveforms (as shown in Fig.",
"REF ), let us consider the mismatch between these waveforms.",
"A more involved calculation would involve computing a mismatch in the presence of gravitational wave detector noise and considering a range of parameters for the GR waveform to test for degeneracies [47].",
"Here, we perform a simpler mismatch calculation between the background GR waveform $\\Psi _4^{(0)}$ and the corresponding EDGB-modified waveform considered in this study (cf.",
"Sec.",
"REF ).",
"Once we have the EDGB correction $\\Psi _4^{(2)}$ from the numerical relativity simulation, we introduce a coupling parameter $\\sqrt{\\alpha _\\mathrm {GB}}/GM$ before adding it to the GR waveform using Eq.",
"(REF ) to obtain $\\Psi _4(\\sqrt{\\alpha _\\mathrm {GB}})$ .",
"We then compute the mismatch as (cf.",
"[48]) $&\\mathrm {Mismatch}(\\sqrt{\\alpha _\\mathrm {GB}}) \\equiv \\\\&\\nonumber \\quad 1 - \\mathrm {Re} \\left(\\frac{\\langle \\Psi _4^{(0)}, \\Psi _4 (\\sqrt{\\alpha _\\mathrm {GB}})\\rangle }{\\sqrt{\\langle \\Psi _4^{(0)}, \\Psi _4^{(0)} \\rangle \\times \\langle \\Psi _4 (\\sqrt{\\alpha _\\mathrm {GB}}) ,\\Psi _4 (\\sqrt{\\alpha _\\mathrm {GB}})\\rangle }} \\right)\\,,$ where we have explicitly shown the dependence on $\\sqrt{\\alpha _\\mathrm {GB}}$ .",
"We define the inner product $\\langle \\,,\\,\\rangle $ between two waveforms as $\\langle \\Psi _4 {}^{[1]}, \\Psi _4 {}^{[2]} \\rangle \\equiv \\int _{t_\\mathrm {start}}^{t_\\mathrm {end}} \\Psi _4 (t) {}^{[2]} \\tilde{\\Psi }_4^* (t) {}^{[1]} dt\\,,$ where ${}^*$ denotes complex conjugation.",
"This is precisely the inner product used in [48].",
"We choose $t_\\mathrm {start}$ to be the section of the waveform where EDGB effects are fully ramped-on, and choose $t_\\mathrm {end}$ to be the end of the numerical waveform.",
"This is equivalent, by Parseval's theorem, to a noise-weighted inner product in the frequency domain with noise power spectral density $S_n(|f|) = 1$ .",
"We shift the waveforms in time and phase when computing this overlap."
]
] | 2001.03571 |
[
[
"An eikonal equation approach to thermodynamics and the gradient flows in\n information geometry"
],
[
"Abstract We can incorporate a \"time\" evolution into thermodynamics as a Hamilton-Jacobi dynamics.",
"A set of the equations of states in thermodynamics is considered as the generalized eikonal equation, which is equivalent to Hamilton-Jacobi equation.",
"We relate the Hamilton-Jacobi dynamics of a simple thermodynamic system to the gradient flows in information geometry."
],
[
"Introduction",
"Since classical thermodynamics studies equilibrium states and transitions between them, it is a common belief that there is no dynamics in thermodynamics, unlike Lagrange or Hamilton dynamics in classical mechanics.",
"Nevertheless there is an analogy between classical mechanics and thermodynamics.",
"For example, Rajeev [1] formulated classical thermodynamics in terms of Hamilton-Jacobi theory, Vaz [2] studied a Lagrangian description of thermodynamics, and Baldiotti et al.",
"[3] showed a Hamiltonian approach to thermodynamics.",
"Similarly to the case of thermodynamics, there is no concept of \"time\" in the information geometry (IG) [5].",
"During our studies [4] on the information geometric structures in generalized thermostatics, we are inspired by a remarkable work [6] by Prof. Pistone.",
"He has derived the Euler Lagrange equations from the Lagrange action integral by computing the \"velocity\" and \"acceleration\" of a one-dimensional statistical model.",
"The quite remarkable points of his results are as follows.",
"i) introducing the concepts of “velocity” as Fisher's score, and “acceleration” as a second order geometry, so that a flat structure in IG is characterized as an orbit along the curve with zero-acceleration.",
"ii) introducing the dynamics of a density $q(t)$ in IG as a gradient flow equation [9], $\\frac{d}{dt} \\ln \\frac{q(t)}{q_2} = -\\nabla \\Big ( q \\rightarrow \\mathrm {D}(q(t) \\Vert q_2) \\Big ) =- \\ln \\frac{q(t)}{q_2} + \\mathrm {D}(q(t) \\Vert q_2),$ where $\\mathrm {D}(q(t) \\Vert q_2)$ is the Kullback-Leibler divergence $\\mathrm {D}(q(t) \\Vert q_2) := {\\mathbb {E}}_{{q(t)}} \\left[ \\ln \\frac{q(t)}{q_2} \\right].$ The solution of Eq.",
"(REF ) was given by $q(t) = \\exp \\Big ( \\mathrm {e}^{-t} \\ln q_0 + (1-\\mathrm {e}^{-t}) \\ln q_2 - \\Psi (t) \\Big ),$ where $\\Psi (t)$ is the normalization function.",
"This solution $q(t)$ evolves smoothly from $ q_0 := q(0)$ to $q_2:= q(\\infty )$ with increasing the parameter $t$ .",
"For more technical details see for example [7], [8], [9].",
"As is well-known, the dynamics of a point particle is described by Newton's mechanics, in which the zero-acceleration of a particle means that there is no external force acting on the particle.",
"Point i) reminds us Einstein's general relativity [10].",
"It states that a particle which is free from an external non-gravitational force always moves along a geodesic in a curved space-time.",
"Indeed, for a curve $\\gamma (t)$ on a smooth manifold $\\mathcal {M}$ with an affine connection $\\nabla $ , the geodesic equation is described by $\\nabla _{\\frac{d \\gamma }{dt}} \\frac{d \\gamma (t)}{d t} = 0,$ which states that there is no tangent component of the acceleration vector of the geodesic curve $\\gamma (t)$ .",
"This consideration leads us to study a relation among analytical mechanics, thermodynamics and IG.",
"On Point ii), we wonder the functional form of the parameter $t$ dependency, i.e., the double exponential $t$ -dependency of $\\exp ( \\exp (-t))$ in (REF ).",
"Since the standard formalism of IG has no concept of time-evolution, a natural question is that where does this double exponential $t$ -dependency come from?",
"In other words, what is the physical meaning of the “time like” parameter $t$ in the gradient-flow equation in IG?",
"This is the main motivation of this contribution.",
"The gradient flows in IG were studied by Fujiwara [11], Fujiwara and Amari [12], and Nakamura [13], in which they found that the gradient systems on manifolds of even dimensions are completely integrable Hamiltonian systems.",
"This relation between the gradient systems and Hamilton systems is somewhat mysterious and incomprehensible.",
"Later, Boumuki and Noda [14] gave a theoretical explanation for this relationship between Hamiltonian flow and gradient flow from the viewpoint of symplectic geometries.",
"In this contribution we consider the differential equations: $\\frac{d \\theta ^i(t)}{dt} &= - \\theta ^i(t),$ which is equivalent to the gradient flow equations $\\frac{d \\eta _i(t)}{dt} &= - g_{ij}(\\mathbf {\\theta }) \\, \\frac{\\partial }{\\partial \\eta _j(t)} \\Psi ^{\\star }(\\mathbf {\\eta }),$ for the potential functions $\\Psi (\\mathbf {\\theta })$ and $\\Psi ^{\\star }(\\mathbf {\\eta })$ in a dually-flat statistical manifold $(\\mathcal {M}, g, \\nabla , \\nabla ^{\\star })$ .",
"We here take a different approach to the gradient flows in IG.",
"Since the relation between the Legendre transformations in a simple thermodynamical system and those in IG are already known (e.g, section 2 in [4]), our discussion is based on this correspondence between thermodynamic macroscopic variables and affine coordinates in IG.",
"Basically in both (equilibrium) thermodynamics and IG, there is no time parameter.",
"Nevertheless we can introduce a \"time\" parameter $\\tau $ In order to avoid possible confusion, we use $\\tau $ as a \"time\" (or mock time) parameter to distinguish it from the parameter $t$ which is used in gradient equations.",
"as the parameter which describes Hamilton-Jacobi (HJ) dynamics.",
"By generalizing the eikonal equation in optics, which is equivalent to the square of HJ equation, we can describe a \"time\" ($\\tau $ ) evolution of thermodynamic systems.",
"The rest of the paper consists as follows.",
"The next section briefly reviews the basic of geometrical optics.",
"The eikonal equation describes a real optical path between two different points in an optical media.",
"The generalized eikonal equation is introduced and the relations with HJ equation are explained.",
"In section 3, a \"time\" parameter $\\tau $ is introduced in the standard settings of equilibrium thermodynamics.",
"Einstein's vielbein formalism (or Cartan's method of moving frame) plays a central role in order to obtain a Riemann metric in an equilibrium thermodynamic system.",
"By studying the gradient flow in the ideal gas model, we will relate the parameter $t$ with the temperature $T$ of the thermodynamic system.",
"As a more realistic gas model, the gas model by van der Waals is also studied.",
"Section 4 discusses two main issues.",
"In the first subsection 4.1 we discuss a simple canonical probability of a thermal system, and show the relation to the gradient flow equation (REF ).",
"The origin of the double exponential $t$ -dependency of the solution (REF ) is explained as the $\\beta $ -dependency of the canonical probability.",
"Relation to Gompertz function is also shown.",
"In the next subsection 4.2, we consider a very simple model described by the specific characteristic function $W$ , and discuss the relation between the gradient flow and Hamilton flow.",
"Final section is devoted to our conclusion.",
"In Appendix A, HJ equation is derived by applying appropriate canonical transformation.",
"Appendix B explains a constant (effective) pressure process in the ideal and van der Waals gas models.",
"We use Einstein summation convention throughout this paper except in Subsection REF ."
],
[
"Geometrical optics and generalized eikonal equation",
"In geometrical optics, the real path of the ray in a media with refractive index $n(\\mathbf {q})$ is characterized by Hamilton's characteristic function, (or point eikonal function), $I(\\mathbf {q}; \\mathbf {q}_0) &= \\int _{s_0}^s n(\\mathbf {q}(s)) \\left|\\frac{d \\mathbf {q}(s)}{d s} \\right|ds=: \\int _{s_0}^s L_{\\rm op} \\left( \\mathbf {q}(s), \\frac{d \\mathbf {q}(s)}{d s} \\right) ds,$ where $s$ is a curve (path) parameter and $L_{\\rm op}$ is the so called optical Lagrangian.",
"The point eikonal function $I(\\mathbf {q}; \\mathbf {q}_0)$ is the length of a real optical path from a point $\\mathbf {q}_0 := \\mathbf {q}(0) = (x_0, y_0, z_0) $ to another point $\\mathbf {q}(s) = (x,y,z)$ .",
"From Fermat's principle (the principle of least time), the real optical path satisfies $\\delta I =0$ .",
"Then the variation $\\delta I$ of $I(\\mathbf {q}; \\mathbf {q}_0)$ with respect to the infinitesimal variations of the two end points $\\mathbf {q}$ and $\\mathbf {q}_0$ is given by $\\delta I = \\left[ \\frac{\\partial L_{\\rm op}}{ \\partial \\left( \\frac{d \\mathbf {q}}{d s} \\right)} \\cdot \\delta \\mathbf {q}\\right]_{s_0}^s= \\Big [ \\mathbf {p} \\cdot \\delta \\mathbf {q} \\Big ]^s_{s_0} = \\mathbf {p}(\\mathbf {q}) \\cdot \\delta \\mathbf {q} - \\mathbf {p}(\\mathbf {q}_0) \\cdot \\delta \\mathbf {q}_0\\; .$ Consequently if we find the point eikonal function $I(\\mathbf {q}; \\mathbf {q}_0)$ , the ray vector $\\mathbf {p}(\\mathbf {q})$ at a position $\\mathbf {q}$ and that $\\mathbf {p}(\\mathbf {q}_0)$ at the other point $\\mathbf {q}_0$ are obtained by $\\mathbf {p}(\\mathbf {q}) = \\frac{\\partial I}{\\partial \\mathbf {q}}, \\quad \\mathbf {p}(\\mathbf {q}_0) = -\\frac{ \\partial I}{\\partial \\mathbf {q}_0},$ respectively.",
"Since $\\vert \\mathbf {p}(\\mathbf {q}) \\vert ^2 = n^2(\\mathbf {q}) $ , the point eikonal function $I(\\mathbf {q}; \\mathbf {q}_0)$ satisfies the eikonal equations $\\left( \\frac{\\partial I}{\\partial x} \\right)^2 + \\left( \\frac{\\partial I}{\\partial y} \\right)^2 + \\left( \\frac{\\partial I}{\\partial z} \\right)^2 &= n^2(\\mathbf {q}),\\\\\\left( \\frac{\\partial I}{\\partial x_0} \\right)^2 + \\left( \\frac{\\partial I}{\\partial y_0} \\right)^2 + \\left( \\frac{\\partial I}{\\partial z_0} \\right)^2 &= n^2(\\mathbf {q}_0).$ In order to determine the real optimal path between $\\mathbf {q}$ and $\\mathbf {q}_0$ the both relations () are needed.",
"Recall that the eikonal equation is obtained from the wave equation in the limit of short wavelength $\\lambda $ , i.e., $\\vert \\nabla n \\vert \\ll k = 2 \\pi / \\lambda $ .",
"Historically Hamilton invented his formalism of classical mechanics from his theory of systems of rays [15].",
"Now let us consider the following generalized eikonal equation $g^{\\mu \\nu }(\\mathbf {q}) \\left( \\frac{\\partial W}{\\partial q^{\\mu }} \\right) \\left( \\frac{\\partial W}{\\partial q^{\\nu }} \\right) = E^2(\\mathbf {q}), \\quad \\mu , \\nu =1,2,\\ldots ,N,$ in a smooth manifold with $N$ -dimension.",
"Here $W=W(\\mathbf {q}, \\mathbf {P})$ is Hamilton's characteristic function, $g^{\\mu \\nu }(\\mathbf {q})$ is the inverse tensor of a metric $g_{\\mu \\nu }(\\mathbf {q})$ of the manifold, and $E(\\mathbf {q})$ takes a real positive value.",
"It is known that $W(\\mathbf {q}, \\mathbf {P})$ is a generating function which relates the original variables $(\\mathbf {q}, \\mathbf {p})$ and new variables $(\\mathbf {Q}, \\mathbf {P})$ by the relations $p_{\\mu } = \\frac{\\partial W}{\\partial q^{\\mu }}, \\quad Q^{\\mu } = \\frac{\\partial W}{\\partial P_{\\mu }},\\quad \\mu =1,2, \\cdots , N.$ Note that if $g^{\\mu \\nu }(\\mathbf {q})=\\delta ^{\\mu \\nu }$ , then the generalized eikonal equation (REF ) reduces to the standard eikonal equation () with $E = n(\\mathbf {q}) = \\vert \\mathbf {p}(\\mathbf {q}) \\vert $ .",
"We see that taking the square root of the generalized eikonal equation (REF ) leads to the Hamilton-Jacobi (HJ) equation [16] $H\\left( \\mathbf {q}, \\frac{\\partial W}{\\partial \\mathbf {q}} \\right) - E(\\mathbf {q}) = 0,$ with the time-independent Hamiltonian $H( \\mathbf {q}, \\mathbf {p}) = \\sqrt{ g^{\\mu \\nu }(\\mathbf {q}) p_{\\mu } p_{\\nu }} \\; .$ It is also known that for any time-independent Hamiltonian, by separation of variables, the corresponding action $S$ is expressed as $S( \\mathbf {q}, \\mathbf {P}, \\tau ) = W( \\mathbf {q}, \\mathbf {P}) - E(\\mathbf {P}) \\, \\tau ,$ where $\\tau $ is a \"time\" parameter, $E(\\mathbf {P})$ is a total energy of the Hamiltonian $H$ as a function of $\\mathbf {P}$ .",
"In addition, the action $S(\\mathbf {q}, \\mathbf {P}, \\tau )$ is a generating function of the (type-2) canonical transformation from the original variables $(\\mathbf {q}, \\mathbf {p})$ to new variables $(\\mathbf {Q}, \\mathbf {P})$ which are conserved, i.e., $\\frac{d Q^{\\mu }}{d\\tau } = 0, \\quad \\frac{d P_{\\mu }}{d\\tau } = 0, \\quad \\mu =1,2, \\cdots , N.$ The transformed new Hamiltonian $K$ is given by $K(\\mathbf {Q}, \\mathbf {P}) = H\\left( \\mathbf {q}, \\frac{\\partial W}{\\partial \\mathbf {q}} \\right) - E = 0,$ which is equivalent to HJ equation (REF ), and the relations (REF ) are satisfied (see ).",
"It is worth noting that $p_{\\mu } \\, \\frac{d q^{\\mu }}{d \\tau } = p_{\\mu }\\, \\frac{\\partial H}{\\partial p_{\\mu }} = \\frac{g^{\\mu \\nu }(\\mathbf {q}) p_{\\mu } p_{\\nu }}{\\sqrt{ g^{\\rho \\lambda }(\\mathbf {q}) p_{\\rho } p_{\\lambda }}} = H,$ where in the first step we used $d q^{\\mu }/ d\\tau = \\partial H / \\partial p_{\\mu }$ .",
"Consequently the corresponding Lagrangian $\\mathcal {L}$ , which is the Legendre dual of the Hamiltonian (REF ), becomes null, $\\mathcal {L} \\left(\\mathbf {q}, \\frac{d \\mathbf {q}}{d \\tau } \\right) := p_{\\mu }\\, \\frac{d q^{\\mu }}{d \\tau } -H( \\mathbf {q}, \\mathbf {p}) = 0.$ This leads to $dS(\\mathbf {q}, \\mathbf {P}, \\tau ) = \\mathcal {L} \\left(\\mathbf {q}, \\frac{d \\mathbf {q}}{d \\tau } \\right) \\, d\\tau = 0,$ and by using Eq.",
"(REF ) we see that $dW(\\mathbf {q}, \\mathbf {P}) = E(\\mathbf {P}) d\\tau .$ In addition, according to Carathéodry [17], we introduce the generating function $G(\\mathbf {q}, \\tau ; \\mathbf {q}_0, \\tau _0) := S(\\mathbf {q}, \\mathbf {P}, \\tau )-S(\\mathbf {q}_0, \\mathbf {P}, \\tau _0),$ which generates the canonical transformation between a set of canonical coordinates $(\\mathbf {q}_0 :=\\mathbf {q}(\\tau _0), \\mathbf {p}_0 :=\\mathbf {p}(\\tau _0))$ and another set of canonical coordinates $(\\mathbf {q}(\\tau ), \\mathbf {p}(\\tau ))$ via the conserved quantities $(\\mathbf {Q}, \\mathbf {P})$ .",
"Note that $S(\\mathbf {q}_0, \\mathbf {P}, \\tau _0)$ is the generating function of the canonical transformation between $(\\mathbf {q}_0, \\mathbf {p}_0)$ and $(\\mathbf {Q}, \\mathbf {P})$ and $-S(\\mathbf {q}, \\mathbf {P}, \\tau )$ is that of the canonical transformation between $(\\mathbf {Q}, \\mathbf {P})$ and $(\\mathbf {q}, \\mathbf {p})$ .",
"Later we shall use this generating function $G$ in order to obtain the \"time\" dependency or dynamics described by the $\\tau $ -parameter."
],
[
"Thermodynamical systems",
"In general, equilibrium thermodynamic systems with $N$ -independent macroscopic variables are completely described by the $N$ -independent equations of states, which can be cast into the following form [2], $e_i{}^{\\mu } ({\\mathbf {q}}) \\, p_{\\mu } = r_i, \\quad i, \\mu =1,2, \\ldots N,$ where each $p_{\\mu }$ is an intensive thermodynamic variable, $q_{\\mu }$ is extensive one, and $r_i$ are independent constants.",
"We assume that the matrix $e_i{}^{\\mu } ({\\mathbf {q}})$ to be invertible so that we have $p_{\\mu } = e_{\\mu }{}^i ({\\mathbf {q}}) \\; r_i,$ where $e_{\\mu }{}^i(\\mathbf {q})$ is the inverse matrix of $e_i{}^{\\mu } ({\\mathbf {q}})$ and the following relations are satisfied.",
"$e_{\\mu }{}^i ({\\mathbf {q}}) \\; e_j{}^{\\mu } ({\\mathbf {q}}) = \\delta ^i_j, \\quad e_{\\nu }{}^i ({\\mathbf {q}}) \\; e_i{}^{\\mu } ({\\mathbf {q}}) = \\delta ^{\\mu }_{\\nu },$ where $\\delta ^i_j$ and $\\delta ^{\\mu }_{\\nu }$ denote Kronecker's delta.",
"Since each $r_i$ is independent, we can assign $\\lbrace r_i \\rbrace $ as the components in an orthogonal system with the invertible constant matrix $\\eta ^{i j}$ , and $ e_i{}^{\\mu } $ can be thought of as a vielbein [19].",
"In other words, the frame consisting of the orthogonal basis $\\lbrace r_i \\rbrace $ is Cartan's moving frame.",
"The vielbein (or N-bein) [19] is used in the field of general relativity [20].",
"In the fields of IG, to the best of our knowledge, the first trial of applying vielbein formalism is Ref.",
"[21].",
"In general the frame of $\\lbrace p_{\\mu } \\rbrace $ on a manifold are non-orthogonal and characterized by a metric tensor $g$ .",
"The vielbein field $e_i{}^{\\mu } ({\\mathbf {q}})$ relates this non-orthogonal frame of $\\lbrace p_{\\mu } \\rbrace $ with the local orthogonal frame of $\\lbrace r_i \\rbrace $ .",
"In order to distinguish the two different frames, Greek and Latin indices are often used, and we follow this convention here.",
"The inner product in the orthogonal frame of $\\lbrace r_i \\rbrace $ with a diagonal metric tensor $\\eta $ and that in the frame of $\\lbrace p_{\\mu } \\rbrace $ with a metric tensor $g$ are related with $g^{\\mu \\nu } \\, p_{\\mu } p_{\\nu } = \\eta ^{ij} \\, r_i r_j.$ Here we regard this relation as the generalized eikonal equation (REF ) with the refractive index $n(\\mathbf {q}) = \\eta ^{ij} r_i r_j$ , which is a constant and consequently this corresponds to a homogeneous isotropic medium.",
"From Eq.",
"(REF ) and Eq.",
"(REF ), it follows the next relation $g^{\\mu \\nu } = \\eta ^{i j} e_i{}^{\\mu } e_j{}^{\\nu },$ where $g^{\\mu \\nu }$ is the inverse matrix of the metric $g_{\\mu \\nu }$ on the $N$ -dimensional smooth manifold $\\mathcal {M}$ , i.e., they satisfy $g^{\\mu \\rho } \\, g_{\\rho \\nu } = \\delta ^{\\mu }_{\\nu }.$ It is worth noting that the matrix elements $e_i{}^{\\mu } $ of a vielbein is determined by $N^2$ components, whereas those $g_{\\mu \\nu }$ of a Riemann metric have $N(N+1)/2$ components.",
"Consequently, the metric $g_{\\mu \\mu }$ is obtained from a given vielbein as $g_{\\mu \\nu } = \\eta _{i j} \\, e_{\\mu }{}^{i} e_{\\nu }{}^{j},$ however, the converse is not possible in general.",
"In this way, by using vielbein field we can regard the equations of states for an equilibrium thermodynamic system as the generalized eikonal equations, which are equivalent to HJ equations.",
"Next, from the Hamiltonian (REF ) and Hamilton's equation of motion, we have [left=]align d qd = Hp = g pE, d pd = -Hq = -12 E gq p p. Substituting Eq.",
"() into the transformed Hamiltonian (REF ), we obtain the relation $\\sqrt{ g_{\\mu \\nu } \\frac{d q^{\\mu }}{d \\tau } \\frac{d q^{\\nu }}{d \\tau } } = 1,$ which implies that $d \\tau ^2 = g_{\\mu \\nu } dq^{\\mu } dq^{\\nu } .$ Consequently, the \"time\" parameter $\\tau $ in this setting gives a natural distance (arc-length) between equilibrium states of the thermodynamic systems on the manifold $\\mathcal {M}$ equipped with the metric $g_{\\mu \\nu }$ .",
"For the sake of simplicity, we only consider thermodynamical gas model with $N=2$ dimensions [2].",
"A generalization for $N>2$ case is straightforward.",
"A thermal equilibrium system characterized by the specific entropy $s$ as a state function of the internal energy $u$ and volume $v$ per a molecule of the gas.",
"We use the so-called entropy representation $s = s(u, v)$ , and the first law of thermodynamics is expressed as $ds = \\frac{1}{T} \\, du + \\frac{P}{T} \\, dv,$ where $T$ is the temperature and $P$ the pressure of the gas, respectively.",
"It is well known that they are related with $\\frac{1}{T} = \\left( \\frac{\\partial s}{\\partial u} \\right)_{v},\\quad \\frac{P}{T} = \\left( \\frac{\\partial s}{\\partial v} \\right)_{u}.$ Mathematically the above physical explanation is equivalent that the Pfaff equation $ds(u,v) -\\frac{1}{T} \\, du - \\frac{P}{T} \\, dv =0,$ is an exact differential, which was originally shown by Carathéodory.",
"Next, introducing the Planck potential $\\Xi $ given by $\\Xi \\left( \\frac{1}{T}, \\frac{P}{T} \\right):= s(u, v) - \\frac{1}{T} \\, u - \\frac{P}{T} \\, v,$ which is the total Legendre transform of the entropy $s(u,v)$ .",
"These thermodynamical potentials and their variables correspond to the potential functions and $\\theta $ - and $\\eta $ -coordinates in IG, respectively as follows: $\\textrm {the \\eta -coordinates: } &\\quad \\eta _1 \\mathrel {\\widehat{=}} u, \\; \\eta _2 \\mathrel {\\widehat{=}} v, \\\\\\textrm {the \\theta -coordinates: }&\\quad \\theta ^1 \\mathrel {\\widehat{=}} -\\frac{1}{T}, \\; \\theta ^2 \\mathrel {\\widehat{=}} -\\frac{P}{T}, \\\\\\textrm {the \\theta -potential: } &\\quad \\Psi (\\mathbf {\\theta }) \\mathrel {\\widehat{=}} \\Xi \\left( \\frac{1}{T}, \\frac{P}{T} \\right), \\\\\\textrm {the \\eta -potential: } &\\quad \\Psi ^{\\star }(\\mathbf {\\eta }) \\mathrel {\\widehat{=}} -s(u, v).$ The two potentials are of course related by the total Legendre transformation $\\Psi (\\theta ^1, \\theta ^2) = \\theta ^{\\mu } \\eta _{\\mu } - \\Psi ^{\\star }(\\eta _1, \\eta _2),$ and $\\eta _{\\mu } = \\frac{\\partial }{\\partial \\theta ^{\\mu }} \\, \\Psi (\\mathbf {\\theta }), \\quad \\theta ^{\\mu } = \\frac{\\partial }{\\partial \\eta _{\\mu }} \\, \\Psi ^{\\star }(\\mathbf {\\eta }), \\quad \\mu =1,2.$ An example of Maxwell's relations in thermodynamics is described by $\\frac{\\partial }{\\partial v} \\, \\left(\\frac{1}{T} \\right)= \\frac{\\partial }{\\partial u} \\, \\left( \\frac{P}{T} \\right),$ which is equivalent to the integrability condition $\\partial \\theta ^1 / \\partial \\eta _2 = \\partial \\theta ^2 / \\partial \\eta _1$ .",
"By using the relation (REF ), this condition leads to the following relation for the $\\eta $ -potensial (REF ), $\\frac{\\partial ^2}{\\partial v \\partial u} \\, \\Big ( -s(u,v) \\Big )= \\frac{\\partial ^2}{\\partial u \\partial v} \\, \\Big ( -s(u,v) \\Big ),$ i.e., the Hessian matrix of the negentropy $-s(u,v)$ must be symmetric.",
"It is known [5] that a metric obtained by the Hessian of a convex potential leads to the dually flat structure in IG.",
"Such a Hessian metric in thermodynamics is proposed by Ruppeiner [18].",
"The Ruppeiner metric is written by $(g^{\\rm R})_{\\mu \\nu } = \\frac{\\partial ^2}{\\partial q^{\\mu } \\partial q^{\\nu }} \\, \\big ( -s \\, \\big ),\\quad \\mu ,\\nu =u, v,$ where $q^u = u$ and $q^v = v$ ."
],
[
"Ideal gas model",
"Ideal gas model is the simplest thermodynamic model of an dilute gas, which consists of a huge number of molecule.",
"It is described by $u = \\frac{f}{2} \\, k_{\\rm B} T, \\quad P \\, v = k_{\\rm B} T,$ where $u$ stands for the specific internal energy, $v$ the specific volume, $ k_{\\rm B}$ Boltzmann constant, $T$ the temperature, $P$ the pressure of the ideal gas.",
"The parameter $f$ stands for the degree of freedom of a molecule of the gas, e.g., for a monatomic molecule $f=3$ .",
"The first equation of (REF ) is the equipartition theorem and the second is the equation of state of the ideal gas model.",
"It is known that the corresponding entropy $s(u,v)$ can be expressed as $s(u, v) = P_u \\ln \\frac{u}{u_0} + P_v \\ln \\frac{v}{v_0}.$ Here we introduced the reference state $(u_0, v_0)$ which satisfy $s(u_0, v_0)=0$ , and $P_u := \\frac{f}{2} \\, k_{\\rm B}, \\quad P_v := k_{\\rm B}.$ Note that $P_u$ and $P_v$ are constants (conserved quantities) and canonical transformed momenta in (REF ).",
"The equations (REF ) can be cast into the form (REF ), i.e., $\\begin{pmatrix}u \\; & 0 \\\\[1ex]0 & v\\end{pmatrix}\\begin{pmatrix}\\frac{1}{T} \\\\[1ex]\\frac{P}{T}\\end{pmatrix}=\\begin{pmatrix}\\frac{f}{2} \\, k_{\\rm B}\\\\[1ex]k_{\\rm B}\\end{pmatrix},$ with $e_i{}^{\\mu } =\\begin{pmatrix}u & 0 \\\\0 & v\\end{pmatrix},\\quad p_{\\mu } =\\begin{pmatrix}\\frac{1}{T} \\\\[1ex]\\frac{P}{T}\\end{pmatrix}, \\quad r_i =\\begin{pmatrix}P_u\\\\[1ex]P_v\\end{pmatrix},$ Let us introduce the diagonal metric tensor $\\eta $ as $\\eta ^{i j} =\\begin{pmatrix}\\frac{1}{\\alpha ^2} & 0 \\\\0 & \\frac{1}{\\beta ^2}\\end{pmatrix}, \\quad \\eta _{i j} =\\begin{pmatrix}\\alpha ^2 & 0 \\\\0 & \\beta ^2\\end{pmatrix},$ where $\\alpha ^2$ and $\\beta ^2$ are scale factors [22].",
"Then, from Eq.",
"(REF ), the corresponding inverse matrix of the metric matrix becomes $g^{\\mu \\nu } = \\begin{pmatrix}\\frac{u^2}{\\alpha ^2} & 0 \\\\[1ex]0 & \\frac{v^2}{\\beta ^2}\\end{pmatrix}.$ Next, from the relations (REF ) and (REF ), we have $H &= \\sqrt{g^{\\mu \\nu } p_{\\mu } p_{\\nu }} = \\sqrt{\\eta ^{i j} e_i{}^{\\mu } p_{\\mu } e_j{}^{\\nu } p_{\\nu }}= \\sqrt{\\eta ^{i j} r_i r_j} = E,$ then we find the explicit expression of constant $E$ as $E(P_u, P_v) = \\sqrt{ \\frac{(P_u)^2}{\\alpha ^2} + \\frac{(P_v)^2}{\\beta ^2} }.$ The corresponding generalized eikonal equation (REF ) becomes $\\frac{u^2}{\\alpha ^2} \\left(\\frac{\\partial W}{\\partial u} \\right)^2 + \\frac{v^2}{\\beta ^2} \\left(\\frac{\\partial W}{\\partial v} \\right)^2 = \\frac{(P_u)^2}{\\alpha ^2} + \\frac{(P_v)^2}{\\beta ^2},$ from which a complete solution $W$ can be obtained as $W(u, v, P_u, P_v) = P_u \\ln u + P_v \\ln v,$ where the constant of integration is set to zero.",
"Consequently the action $S$ of the ideal gas model becomes $S(u,v; P_u, P_v, \\tau ) = P_u \\ln u + P_v \\ln v - E(P_u, P_v) \\tau ,$ and the corresponding generating function (REF ) is $G(u,v, \\tau ; u_0, v_0, \\tau _0) = P_u \\ln \\frac{u}{u_0} + P_v \\ln \\frac{v}{v_0} - E(P_u, P_v) (\\tau -\\tau _0),$ which satisfies the relations $\\frac{\\partial G}{\\partial P_i} = 0, \\quad i=u, v,$ since $\\partial S(\\mathbf {q}, \\mathbf {P}, \\tau ) / \\partial P_i = \\partial S(\\mathbf {q}_0, \\mathbf {P}, \\tau _0) / \\partial P_i = Q^i$ .",
"Using the expression (REF ) of the action $S$ of the ideal gas model, it follows that [left=]align GPu = uu0 - Pu (-0)2 E =0, GPv = vv0 - Pv (- 0) 2 E =0, By choosing $\\alpha ^2=P_u$ and $\\beta ^2=P_v$ , which corresponds to a constant pressure process as explained in , we have $u(\\tau ) &= u_0 \\exp \\left[ \\frac{(\\tau -\\tau _0)}{E} \\right], \\quad v(\\tau ) = v_0 \\exp \\left[ \\frac{(\\tau - \\tau _0)}{E} \\right].$ and $\\frac{d u}{d\\tau } &= \\frac{u}{E}, \\quad \\frac{d v}{d\\tau } = \\frac{v}{E},$ Next let us consider the gradient flow of $u(t)$ and $v(t)$ .",
"From the correspondence relations (), the direct mapping of the gradient flow equations (REF ) in IG to those in the ideal gas model become $\\frac{d u}{dt} &= g^{uu} \\, \\frac{\\partial s(u,v)}{\\partial u}, \\quad \\frac{d v}{dt} = g^{vv} \\, \\frac{\\partial s(u,v)}{\\partial v}.$ Substituting the expression (REF ) and entropy (REF ) into these equations, we find $\\frac{d u}{d t} = \\frac{P_u}{\\alpha ^2} \\, u = u, \\quad \\frac{d v}{d t} = \\frac{P_v}{\\beta ^2} \\, v = v.$ for a constant pressure $P$ process.",
"Comparing the results (REF ) and (REF ) we find $d \\tau = E \\, d t.$ Next, taking the derivative of the specific entropy (REF ), we have $ds(u,v) = \\frac{1}{T} \\, du + \\frac{P}{T} \\, dv,$ which corresponds to the relation $d W(\\mathbf {q}, \\mathbf {P}) = \\frac{\\partial W}{\\partial q^i} \\, dq^i + \\frac{\\partial W}{\\partial P_i} \\, dP_i= p_i dq^i,$ in analytical mechanics, because $p_i = \\partial W / \\partial q^i$ in (REF ) and $dP_i = 0$ in (REF ).",
"The relation (REF ) in analytical mechanics corresponds to $ds(u, v) = E \\, d\\tau ,$ in thermodynamics.",
"From the equations (REF ), we have $\\frac{1}{T} du = - u \\, d \\left(\\frac{1}{T} \\right), \\quad \\frac{P}{T} dv = - v \\, d \\left(\\frac{P}{T} \\right).$ Substituting these relations into Eq.",
"(REF ), it follows that $ds(u,v) = - u \\, d \\left(\\frac{1}{T} \\right) - v \\, d \\left(\\frac{P}{T} \\right).$ For a constant pressure process, this becomes $ds(u,v) = (u + v P) \\, \\frac{dT}{T^2}.$ Since $u = P_u T, v P = P_v T$ and $E = \\sqrt{P_u + P_v}$ , we have $ds(u,v) = (P_u + P_v) \\, \\frac{dT}{T} = E^2 \\, d \\ln T.$ Comparing this relation, (REF ) and (REF ), we finally obtain that $d\\tau = E \\, d\\ln T, \\quad dt = d\\ln T.$ In this way, we find out the relation between the parameter $t$ in the gradient flows in IG and the temperature $T$ in a simple gas model of thermodynamics."
],
[
"Van der Waals gas",
"Since the ideal gas model is simple, we next consider the more realistic gas model by van der Waals.",
"His famous model is characterized by the following equations of states.",
"$u + \\frac{a}{v} = \\frac{f}{2} \\, k_{\\rm B} T, \\quad (P + \\frac{a}{v^2})(v-b) = k_{\\rm B} T.$ The first equation of (REF ) states the equipartition theorem and the second equation states the equation of state by van der Waals [23].",
"The term $a/v^2$ accounts for long-range attractive forces which increase pressure, and the $b$ term accounts for short range repulsive forces which decrease the volume available to molecules.",
"In the limit of $a =b=0$ , the gas model of van der Waals reduces to the ideal gas model.",
"Now, the equations in (REF ) can be cast into the form (REF ) with $(e^{\\rm vw})_i{}^{\\mu } = \\begin{pmatrix}u + \\frac{a}{v} & 0 \\\\[1ex]\\frac{a}{v^2} (v-b) & \\; \\; v-b\\end{pmatrix},\\quad p_{\\mu } =\\begin{pmatrix}\\frac{1}{T} \\\\[1ex]\\frac{P}{T}\\end{pmatrix},$ and the constants $r_i =\\begin{pmatrix}\\frac{f}{2} k_{\\rm B}\\\\[1ex]k_{\\rm B}\\end{pmatrix}=\\begin{pmatrix}P_u\\\\[1ex]P_v\\end{pmatrix},$ where we choose the conserved quantity $P_u = f k_{\\rm B} / 2$ and $P_v = k_{\\rm B}$ .",
"The corresponding entropy $s^{\\rm vw}(u,v)$ can be expressed as $s^{\\rm vw}(u, v) = \\frac{f}{2} k_{\\rm B}\\ln \\left(\\frac{u+\\frac{a}{v}}{u_0 + \\frac{a}{v_0}} \\right)+ k_{\\rm B} \\ln \\left( \\frac{v-b}{v_0-b} \\right),$ where we set $s(u_0, v_0)=0$ .",
"From Eq.",
"(REF ) the corresponding metric becomes $(g^{\\rm vw} )^{\\mu \\nu } = \\begin{pmatrix}\\frac{1}{\\alpha ^2} \\left(u + \\frac{a}{v} \\right)^2 + \\frac{a^2 (v-b)^2}{\\beta ^2 v^4} & \\quad \\frac{a (v-b)^2}{\\beta ^2 v^2} \\\\[1ex]\\frac{a (v-b)^2}{\\beta ^2 v^2} & \\frac{(v-b)^2}{\\beta ^2}\\end{pmatrix}.$ Then the generalized eikonal equation becomes $\\frac{\\left( u + \\frac{a}{v} \\right)^2}{\\alpha ^2} \\left( \\frac{\\partial W}{\\partial u} \\right)^2+ \\frac{(v-b)^2}{\\beta ^2} \\left( \\frac{a}{v^2} \\frac{\\partial W}{\\partial u}+ \\frac{\\partial W}{\\partial v} \\right)^2 = (E^{\\rm vw})^2.$ Recall that the action $S$ is expressed in the form (REF ) and we can choose the scale factors $\\alpha ^2$ and $\\beta ^2$ arbitrarily.",
"By solving generalized eikonal equation (REF ) we can obtain a complete solution $W(u, v, P_u, P_v)$ , and then the action $S$ is calculated as $S(\\mathbf {q}, \\mathbf {P}, \\tau ) = P_u \\ln \\left(u + \\frac{a}{v} \\right) + P_v \\ln (v-b) - E^{\\rm vw}(P_u, P_v) \\; \\tau .$ Then, we obtain the following relations.",
"[left=]align GPu = u+avu0+av0 - Pu (-0)2 Evw =0, GPv = v-bv0-b - Pv (- 0) 2 Evw =0, Hence we obtain the \"time\" $\\tau $ -dependence of $u$ and that of $v$ as [left=]align u() = -av() + (u0 + av0 ) [ Pu (-0)2 Evw ], v() = b + (v0 - b) [ Pv (- 0)2 Evw ], respectively.",
"From these relations we find [left=]align d ud = Pu2 Evw (u + av ) + Pv2 Evw av2 (v-b), d vd = Pv2 Evw (v-b).",
"Next let us consider the gradient flow of $u(t)$ and $v(t)$ .",
"From the correspondence relations (), the direct mapping of the gradient flow equations (REF ) in IG to those in the gas model of van der Waals become [left=]align d udt = (gvw)uu svwu +(gvw)uv svwv, d vdt = (gvw)vu svwu +(gvw)vv svwv.",
"Substituting the metric (REF ) and entropy (REF ) into these equations, we find [left=]align d ud t = Pu2 (u + av ) + Pv2 av2 (v-b), d vd t = Pv2 (v-b).",
"Comparing the results (REF ) and (REF ) we find $d \\tau = E^{\\rm vw} \\, d t.$ It is known that the following canonical transformation [3] from the variables $(\\mathbf {q}, \\mathbf {p})$ to $(\\tilde{\\mathbf {q}}, \\tilde{\\mathbf {p}})$ relates the ideal gas model and the gas model by van der Waals.",
"$q^u = u \\rightarrow \\tilde{q}^u = \\tilde{u} := u + \\frac{a}{v}, \\quad q^v = v \\rightarrow \\tilde{q}^v = \\tilde{v} := v - b, \\\\p^u = \\frac{1}{T} \\rightarrow \\tilde{p}^u = \\frac{1}{\\tilde{T}} := \\frac{1}{T}, \\quad p^v = \\frac{P}{T} \\rightarrow \\tilde{p}^v = \\frac{\\tilde{P}}{\\tilde{T}} := \\frac{P}{T} + \\frac{a}{v^2} \\frac{1}{T}.$ This is so-called Mathieu transformation, which is a subgroup of canonical transformations preserving the differential from $p_i \\, d q^i = \\tilde{p}_i \\, d \\tilde{q}^i.$ Indeed we confirm that $\\tilde{p}_u \\, d \\tilde{q}^u + \\tilde{p}_v \\, d \\tilde{q}^v&= \\frac{1}{T} \\, d\\left( u + \\frac{a}{v} \\right) + \\left(\\frac{P}{T} + \\frac{a}{v^2} \\frac{1}{T} \\right) \\, d(v-b) \\\\&= \\frac{1}{T} \\, du + \\frac{P}{T} \\, dv = p_u \\, d q^u + p_v \\, d q^v.$ By applying the canonical transformation (REF ), the equations (REF ) of the gas model by van der Waals become $\\tilde{u} = \\frac{f}{2} \\, k_{\\rm B} \\tilde{T}, \\quad \\tilde{P} \\, \\tilde{v} = k_{\\rm B} \\tilde{T}.$ Consequently the relation (REF ) is transformed as $(e^{\\rm vw})_i{}^{\\mu } \\rightarrow \\begin{pmatrix}\\tilde{u} & 0 \\\\0 & \\tilde{v}\\end{pmatrix},$ which is equivalent to the first relation in (REF ) of the ideal gas model.",
"$\\frac{d}{d \\tau } \\, \\tilde{u}&= \\frac{P_u}{\\alpha ^2 E^{\\rm vw}} \\, \\tilde{u}, \\quad \\frac{d}{d \\tau } \\, \\tilde{v}= \\frac{P_v}{\\beta ^2 E^{\\rm vw}} \\, \\tilde{v},\\\\\\frac{d}{d \\tau } \\, \\frac{\\tilde{P}}{\\tilde{T}}&= - \\frac{P_v}{\\beta ^2 E^{\\rm vw}} \\left( \\frac{\\tilde{P}}{\\tilde{T}} \\right), \\quad \\frac{d}{d \\tau } \\, \\left( \\frac{1}{\\tilde{T}} \\right)= - \\frac{P_u}{\\alpha ^2 E^{\\rm vw}} \\, \\left( \\frac{1}{\\tilde{T}} \\right).$ As explained in , a constant effective pressure $\\tilde{P} = P + a / v^2$ process is achieved when we choose the scale parameter as $\\alpha ^2 = P_u$ and $\\beta ^2=P_v$ .",
"In this case, the relations (REF ) become $\\frac{d}{d \\tau } \\, \\tilde{u}&= \\frac{\\tilde{u}}{E^{\\rm vw}}, \\quad \\frac{d}{d \\tau } \\, \\tilde{v}= \\frac{\\tilde{v}}{E^{\\rm vw}}, \\\\\\frac{d}{d \\tau } \\, \\frac{\\tilde{P}}{\\tilde{T}}&= - \\frac{1}{E^{\\rm vw}} \\left( \\frac{\\tilde{P}}{\\tilde{T}} \\right), \\quad \\frac{d}{d \\tau } \\, \\left( \\frac{1}{\\tilde{T}} \\right)= - \\frac{1}{E^{\\rm vw}} \\, \\left( \\frac{1}{\\tilde{T}} \\right).$ respectively.",
"Then we see that $ds^{\\rm vw} &= \\frac{1}{\\tilde{T}} \\, d \\tilde{u} + \\frac{\\tilde{P}}{\\tilde{T}} \\, d \\tilde{v}= \\left( \\frac{\\tilde{u}}{\\tilde{T}} + \\frac{\\tilde{v} \\tilde{P}}{\\tilde{T}} \\right) \\frac{d \\tau }{E^{\\rm vw}}= \\left( P_u + P_v \\right) \\frac{d \\tau }{E^{\\rm vw}} = E^{\\rm vw} d \\tau ,$ where in the last step we used $E^{\\rm vw} = \\sqrt{P_u + P_v}$ ."
],
[
"Discussion",
"Having found out the nontrivial relation (REF ) between the time parameter $t$ of the gradient flows in IG and the temperature $T$ of a simple gas model in thermodynamics, let us turn our focus on the gradient flow equation (REF )."
],
[
"On the double exponential $t$ -dependency",
"First we'd like to answer the question concerning the double exponential $t$ -dependency of the solution $q(t)$ in (REF ), i.e., the physical meaning of the parameter $t$ in this solution $q(t)$ from the view point of our results obtained until the previous sections.",
"Let us begin with the well known canonical probability of a thermally equilibrium system, $p_i(\\beta ) = \\frac{1}{Z(\\beta )} \\exp \\left( -\\beta \\mathcal {E}_i \\right), \\quad i=1,2, \\ldots , N,$ where $\\beta := 1/ (k_{\\rm B} T)$ is coldness or the inverse temperature, $Z(\\beta )$ is the partition function, $\\mathcal {E}_i$ is the energy level of $i$ -th state.",
"We assume that each discrete energy level is ordered as $\\mathcal {E}_0 < \\mathcal {E}_1 < \\ldots < \\mathcal {E}_N$ .",
"Consequently in the high temperature limit ($\\beta \\rightarrow 0$ ), every probability $p_i$ becomes equal, i.e., $\\forall i.",
"\\quad p_i (0) = \\frac{1}{N}.$ It is well known that the average energy $U$ is related to the partition function $Z(\\beta )$ as $U := \\sum _{i=1}^N p_i(\\beta ) \\mathcal {E}_i = -\\frac{d}{d \\beta } \\ln Z(\\beta ).$ Taking the logarithm of the both sides of (REF ) we have $\\ln p_i(\\beta ) = -\\beta \\mathcal {E}_i - \\ln Z(\\beta ).$ Taking the expectation of this with respect to $p_i(\\beta )$ leads to $\\sum _{i=1}^n p_i(\\beta ) \\, \\ln p_i(\\beta ) = -\\beta U - \\ln Z(\\beta ).$ Differentiating the both sides of (REF ) with respect to $\\beta $ and using the relation (REF ), if follows that $\\frac{d}{d \\beta } \\ln p_i(\\beta ) = -\\mathcal {E}_i - \\frac{d}{d \\beta } \\ln Z(\\beta )= -\\mathcal {E}_i +U,$ Multiplying the both sides of (REF ) by $\\beta $ , we have $\\beta \\frac{d}{d \\beta } \\ln p_i(\\beta ) =- \\beta \\mathcal {E}_i + \\beta U,$ By subtracting (REF ) from (REF ), the right hand side of (REF ) is rewritten as $\\beta \\frac{d}{d \\beta } \\ln p_i(\\beta ) &= \\ln p_i(\\beta ) -\\sum _{i=1}^N p_i(\\beta ) \\, \\ln p_i(\\beta )$ Since $p_0 := \\lim _{\\beta \\rightarrow 0} p_i(\\beta )$ is constant, we can rewrite this relation as $\\frac{d}{d \\ln \\beta } \\ln \\left(\\frac{p_i(\\beta )}{p_0} \\right) &=\\ln \\left( \\frac{p_i(\\beta )}{p_0} \\right) -\\sum _{i=1}^N p_i(\\beta ) \\, \\ln \\left( \\frac{p_i(\\beta )}{p_0} \\right).$ From the second relation in (REF ) we see that $d \\ln \\beta = -d \\ln \\left( k_{\\rm B} T \\right)= - dt.$ By integrating this relation, we can relate the time parameter $t$ and the coldness $\\beta $ as $\\beta = \\exp ( -t ) + C,$ where $C$ is a constant of integration.",
"Setting $C = 0$ for simplicity, and substituting $\\beta = \\exp (-t)$ into (REF ) we finally obtain that $\\frac{d}{d t} \\ln \\left(\\frac{p_i({\\rm e}^{-t})}{p_0} \\right) &= - \\left\\lbrace \\ln \\left( \\frac{p_i({\\rm e}^{-t})}{p_0} \\right) -\\sum _{i=1}^N p_i({\\rm e}^{-t})) \\, \\ln \\left( \\frac{p_i({\\rm e}^{-t})}{p_0} \\right) \\right\\rbrace .$ This is equivalent to the gradient flow equation in (REF ) when we make the following associations.",
"$\\textrm {evolutional parameter:} \\quad t &\\Leftrightarrow -\\ln \\beta , \\\\\\textrm {density: } q(t) &\\Leftrightarrow \\textrm {probability: } p_i(\\beta ), \\\\q_2: = \\lim _{t \\rightarrow \\infty } q(t) &\\Leftrightarrow p_0 := \\lim _{\\beta \\rightarrow 0} p_i(\\beta ), \\\\\\mathrm {D}(q(t) \\Vert q_2) := {\\mathbb {E}}_{{q(t)}} \\left[ \\ln \\frac{q(t)}{q_2} \\right] &\\Leftrightarrow \\sum _{i=1}^N p_i(\\beta ) \\, \\ln \\left( \\frac{p_i(\\beta )}{p_0} \\right) = \\mathrm {D}(p_i(\\beta ) \\Vert p_0).$ Therefore the double exponential $t$ -dependency $\\exp ( \\exp (-t))$ in the solution $q(t)$ in (REF ) can be explained by these associations.",
"The evolutional parameter $t$ in (REF ) is related to the coldness $\\beta $ evolution through the relation $\\beta = \\exp (-t)$ .",
"Next, let us turn our focus on Gompertz function $f_{\\rm G}(t)$ defined by $f_{\\rm G}(t) := K \\exp \\big [ c \\, \\exp (-t) \\big ],$ where $c$ and $K$ are positive constants.",
"This function $f_{\\rm G}(t)$ is a sigmoid function, and has the double exponential $t$ -dependency.",
"Gompertz [24] studied human mortality for working out a series of tables mortality, and this suggested to him his law of human mortality in which he assumed that the mortality rate decreases exponentially as a person ages.",
"Gompertz function is nowadays used in many areas to model a time evolution where growth is slowest at the start and end of a period.",
"The rule of his model is called Gompertz rule which states that $\\frac{d}{dt} f_{\\rm G}(t) = -f_{\\rm G}(t) \\ln \\frac{f_{\\rm G}(t)}{K},$ The solution of the Gompertz rule is (REF ) if we set $K = \\lim _{t \\rightarrow \\infty } f_{\\rm G}(t)$ and $c = \\ln (f_{\\rm G}(0) / K )$ .",
"Now we observe that $q(t)$ in Eq.",
"(REF ) is proportional to Gompertz function.",
"Indeed, let us define $Q(t)$ by the un-normalized version of $q(t)$ , $Q(t) := q(t) \\, \\mathrm {e}^{\\Psi (t)} = \\exp \\left[ {\\mathrm {e}^{-t} \\ln q_0 + (1-\\mathrm {e}^{-t}) \\ln q_2} \\right]=q_2 \\, \\mathrm {e}^{ \\ln \\left( \\frac{q_0}{q_2} \\right) \\, \\mathrm {e}^{-t}}.$ We see that this function $Q(t)$ is a Gompertz function (REF ) with $K=q_2$ and $c= \\ln (q_0/q_2)$ .",
"Then we have $\\frac{d}{dt} \\ln Q(t) = - \\mathrm {e}^{-t} \\ln \\frac{q_0}{q_2}= - \\ln \\frac{Q(t)}{q_2},$ which is nothing but the Gompertz rule (REF ) $\\frac{d}{dt} Q(t) = - Q(t) \\, \\ln \\frac{Q(t)}{q_2}.$"
],
[
"The gradient flow and Hamilton flow",
"Next we focus our attention on the relation between the gradient flow (REF ) and Hamilton flow.",
"In this subsection, for the sake of clarification, we don't use Einstein's summation convention.",
"Let us consider the simple model described by the following characteristic function $W( \\mathbf {q}, \\mathbf {P})$ , $W( \\mathbf {q}, \\mathbf {P}) = \\sum _{\\mu =1}^m P_{\\mu } \\ln q^{\\mu },$ on a smooth manifold $\\mathcal {M}$ with $m$ -dimension.",
"Here each $P_{\\mu }$ is a constant of motion.",
"The corresponding generalized momentum is given by $p_{\\mu } = \\frac{\\partial W}{\\partial q^{\\mu }} = \\frac{P_{\\mu }}{q^{\\mu }}, \\quad \\mu =1,2, \\ldots , m.$ Consequently each constant of motion $P_{\\mu }$ satisfies that $P_{\\mu } = q^{\\mu } \\, p_{\\mu }, \\quad \\mu =1,2, \\ldots , m.$ The metric tensor is obtained by $g_{\\mu \\nu }(\\mathbf {q}) = - \\frac{ \\partial ^2 W}{\\partial q^{\\mu } \\partial q^{\\nu }} = \\frac{P_{\\mu }}{(q^{\\mu })^2} \\, \\delta ^{\\mu \\nu }, \\quad \\mu , \\nu =1,2, \\ldots , m,$ and the inverse matrix of the metric is $g^{\\mu \\nu }(\\mathbf {q}) = \\frac{(q^{\\mu })^2}{ P_{\\mu }} \\, \\delta ^{\\mu \\nu }, \\quad \\mu , \\nu =1,2, \\ldots , m,$ Then the corresponding Hamiltonian (REF ) becomes $H(\\mathbf {q}, \\mathbf {p}) = \\sqrt{ \\sum _{\\mu , \\nu } g^{\\mu \\nu } (\\mathbf {q}) \\, p_{\\mu } p_{\\nu }}= \\sqrt{ \\sum _{\\mu } \\frac{(q^{\\mu } \\, p_{\\mu } )^2}{P_{\\mu }} },$ and its value $E( \\mathbf {P})$ is constant because this Hamiltonian has no explicit time dependence.",
"Indeed, by using the relations (REF ), it follows that $E( \\mathbf {P}) = \\sqrt{ \\sum _{\\mu } P_{\\mu } }.$ Then Hamilton's equations of motion leads to [left=]align dd q = Hp = (q)2 p P E = q E, dd p = -Hq = -q (p)2 P E = -p E, where we used the relations (REF ) again.",
"Now by using the relation (REF ), the equations of motion (REF ) become [left=]align dd t q = q, dd t p = -p. We remind that the correspondence between IG and analytical mechanics: $\\eta _{\\mu } & \\mathrel {\\widehat{=}} q^{\\mu }, &\\qquad \\theta ^{\\mu } & \\mathrel {\\widehat{=}} -p_{\\mu }, \\\\\\Psi ^{\\star }(\\mathbf {\\eta }) & \\mathrel {\\widehat{=}} -W(\\mathbf {q}, \\mathbf {P}), &\\quad \\theta ^{\\mu } = \\frac{\\partial \\Psi ^{\\star }(\\mathbf {\\eta })}{\\partial \\eta _{\\mu }} & \\mathrel {\\widehat{=}} -p_{\\mu } = -\\frac{\\partial W(\\mathbf {q}, \\mathbf {P})}{\\partial q^{\\mu }}.$ Then we see that the Hamilton flow described by (REF ) is equivalent to the gradient flow in IG described by the gradient equations (REF ).",
"Finally let us comment on the Lemma 3.3 in Ref.",
"[14] by Boumuki and Noda, where they introduced the potential functions for a dually flat space in IG as $\\Psi ^{\\star }(\\mathbf {\\eta }) = - \\sum _i \\ln \\eta _i, \\quad \\eta _i = -1/ \\theta ^i, \\quad i=1,2, \\ldots , m.$ This mathematical model is simple but non-trivial, since its gradient flow equation is reformulated as Hamilton equation.",
"For the details see Ref.",
"[14].",
"Now, from the above correspondence between IG and analytical mechanics, we note that the first relation in (REF ) is the special case in which all $P_{\\mu }$ are set to unity in (REF ), and the second relation comes from the relation (REF ).",
"As a result, their model (REF ) is a special case of the model discussed in this subsection."
],
[
"Conclusions",
"We have studied the gradient flows in IG as the dynamics of a simple thermodynamic system.",
"The equations of states in thermodynamics are regarded as the generalized eikonal equations, and we have incorporated a \"time\" ($\\tau $ ) evolution into thermodynamics as HJ dynamics.",
"Through this \"time\" parameter $\\tau $ in the HJ dynamics of the thermodynamics, we have related the parameter $t$ in the gradient flow equation to the inverse temperature $\\beta $ of the thermodynamic system as shown in (REF ).",
"Based on this fact, we have found the physical origin of the double exponential $t$ -dependency of the solution (REF ) in subsection REF .",
"In this way, the gradient flow in IG is related to the HJ dynamics of the thermodynamical systems.",
"Thermodynamics is powerful and useful in a wide range of scientific fields.",
"Very recently, Ghosh and Bhamidipati [25] studied the thermodynamics of black holes from the contact geometry point of view.",
"They showed that the thermodynamic processes of black holes can be modeled by characteristic curves of a suitable contact Hamilton-flow.",
"We hope our findings help further understandings not only between thermodynamics and IG but also among other different fields."
],
[
"Acknowledgements",
"The authors thank to Prof. G. Pistone for his interesting talk in the SigmaPhi2017 Conference held in Corfu, Greece, 10-14 July 2017, and for organizing a seminar held at Collegio Carlo Alberto, Moncalieri, on 5 September 2017.",
"The first named author (T.W) also thanks to Dr. S. Goto for useful discussion on the early version of this work, and to again Prof. G. Pistone for his kind invitation and discussions in Torino on 23rd January 2020.",
"The first named author (T.W.)",
"is supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number JP17K05341.",
"The third named author (H. M.) is partially supported by the JSPS Grants-in-Aid for Scientific Research (KAKENHI) Grant Number JP15K04842 and JP19K03489."
],
[
"Hamilton-Jacobi equation by canonical transformation",
"Here we briefly review the Carathéodory derivation [17] of HJ equation by canonical transformation.",
"Recall that the non-uniqueness of Lagrangian, i.e., two Lagrangians which differ by a total derivative of some function $f(\\mathbf {q}, t)$ with respect to time, describe the same system.",
"For example consider the following two Lagrangians $L$ and $L^*$ related through $\\int _{t_a}^{t_b} \\left( L(\\mathbf {q}, \\dot{\\mathbf {q}}, t) + \\frac{d f(\\mathbf {q}, t)}{dt} \\right) dt= \\int _{t_a}^{t_b} \\left( L^{\\star }(\\mathbf {Q}, \\dot{\\mathbf {Q}}, t) + \\frac{d g(\\mathbf {Q}, t)}{dt} \\right) dt,$ where $f(\\mathbf {q}, t)$ and $g(\\mathbf {Q}, t)$ are some functions.",
"Both Lagrangians lead to the same Euler-Lagrange equation, and consequently describe the same system.",
"Introducing the function $F(\\mathbf {q}, \\mathbf {Q}, t) := g(\\mathbf {Q}, t) - f(\\mathbf {q}, t)$ , then $L(\\mathbf {q}, \\dot{\\mathbf {q}}, t) = L^{\\star }(\\mathbf {Q}, \\dot{\\mathbf {Q}}, t) + \\frac{d F(\\mathbf {q}, \\mathbf {Q}, t)}{dt}.$ Now, introducing the Hamiltonians $H(\\mathbf {q}, \\mathbf {p}, t)$ and $K(\\mathbf {Q}, \\mathbf {P}, t)$ which are Legendre duals of $ L(\\mathbf {q}, \\dot{\\mathbf {q}}, t)$ and $ L^{\\star }(\\mathbf {Q}, \\dot{\\mathbf {Q}}, t) $ , respectively, i.e., $H(\\mathbf {q}, \\mathbf {p}, t) = p_{\\mu } \\frac{dq^{\\mu }}{dt} - L(\\mathbf {q}, \\dot{\\mathbf {q}}, t), \\quad K(\\mathbf {Q}, \\mathbf {P}, t) = P_{\\mu } \\frac{dQ^{\\mu }}{dt} - L^{\\star }(\\mathbf {Q}, \\dot{\\mathbf {Q}}, t).$ By choosing $F(\\mathbf {q}, \\mathbf {Q}, t) =S(\\mathbf {q}, \\mathbf {P}, t) - P_{\\mu } \\, Q^{\\mu }$ , then (REF ) leads to $p_{\\mu } dq^{\\mu } - H(\\mathbf {q}, \\mathbf {p}, t) dt &=P_{\\mu } dQ^{\\mu } - K(\\mathbf {Q}, \\mathbf {P}, t) dt + d \\big ( S(\\mathbf {q}, \\mathbf {P}, t) - P_{\\mu } Q^{\\mu } \\big ) \\\\&= -Q^{\\mu } dP_{\\mu } - K dt + \\frac{\\partial S}{\\partial q^{\\mu }} dq^{\\mu } + \\frac{\\partial S}{\\partial P_{\\mu }} dP_{\\mu } +\\frac{\\partial S}{\\partial t} dt,$ which describes the canonical transformation from the canonical coordinates $(\\mathbf {q}, \\mathbf {p})$ to $(\\mathbf {Q}, \\mathbf {P})$ .",
"Since both the canonical coordinates are independent we obtain $p_{\\mu } = \\frac{\\partial S}{\\partial q^{\\mu }}, \\quad Q^{\\mu } = \\frac{\\partial S}{\\partial P_{\\mu }}, \\quad K = H + \\frac{\\partial S}{\\partial t},$ and from Hamilton's equation of motion for the transformed Hamiltonian $K(\\mathbf {Q}, \\mathbf {P}, t)$ $\\frac{d}{dt} Q^{\\mu } = \\frac{\\partial K}{\\partial P_{\\mu }}, \\quad \\frac{d}{dt} P_{\\mu } = -\\frac{\\partial K}{\\partial Q^{\\mu }}.$ We see that the transformed canonical coordinates $(\\mathbf {Q}, \\mathbf {P})$ are conserved, i.e., $dQ^{\\mu } / dt = dP_{\\mu } /dt = 0$ when $K=0$ .",
"This condition leads to $K = H( \\mathbf {q}, \\frac{\\partial S}{\\partial \\mathbf {q}}, t) + \\frac{\\partial }{\\partial t} \\, S(\\mathbf {q}, \\mathbf {P}, t) = 0,$ which is HJ equation."
],
[
"Constant pressure process",
"From Hamilton's equation of motion for (REF ) with the metric (REF ) of the ideal gas model, we obtain $\\frac{d}{d \\tau } u&= \\frac{P_u}{\\alpha ^2 E} u, &\\qquad \\qquad \\frac{d}{d \\tau } v&= \\frac{P_v}{\\beta ^2 E} v, \\\\\\frac{d}{d \\tau } \\left( \\frac{P}{T} \\right)&= - \\frac{P_v}{\\beta ^2 E} \\left( \\frac{P}{T} \\right), &\\quad \\frac{d}{d \\tau } \\left( \\frac{1}{T} \\right)&= - \\frac{P_u}{\\alpha ^2 E} \\left( \\frac{1}{T} \\right).$ From the first relation of (), we have $\\frac{d}{d \\tau } P= -T P \\, \\frac{d}{d \\tau } \\left( \\frac{1}{T} \\right) - \\frac{P_v P}{\\beta ^2 E}.$ Substituting the second relation of () into (REF ), we obtain $\\frac{d}{d \\tau } P= \\frac{P}{E}\\left( \\frac{P_u}{\\alpha ^2} - \\frac{P_v}{\\beta ^2} \\right).$ Thus when we choose the scaling factors as $\\alpha ^2 = P_u, \\quad \\beta ^2 = P_v,$ it is a constant pressure ($P$ ) process.",
"Next we consider the gas model by van der Waals.",
"From Hamilton's equation of motion for the Hamiltonian (REF ) with (REF ), we obtain $\\frac{d}{d t} \\left( u + \\frac{a}{v} \\right)&= \\frac{P_u}{\\alpha ^2 E} \\left( u + \\frac{a}{v} \\right), \\\\\\frac{d}{d t} v&= \\frac{P_v}{\\beta ^2 E} (v - b), \\\\\\frac{d}{d t} \\left( \\frac{P}{T} + \\frac{a}{v^2} \\frac{1}{T} \\right)&= - \\frac{P_v}{\\beta ^2 E} \\left( \\frac{P}{T} + \\frac{a}{v^2} \\frac{1}{T} \\right), \\\\\\frac{d}{d t} \\left( \\frac{1}{T} \\right)&= - \\frac{P_u}{\\alpha ^2 E} \\left( \\frac{1}{T} \\right).$ By using these equations, we obtain $\\frac{d}{d \\tau } \\left( P + \\frac{a}{v^2} \\right)&= \\frac{1}{E} \\left( P + \\frac{a}{v^2} \\right)\\left( \\frac{P_u}{\\alpha ^2} - \\frac{P_v}{\\beta ^2} \\right).$ Thus when we choose $\\alpha ^2 = P_u, \\quad \\beta ^2 = P_v,$ it corresponds to a constant $P + a / v^2$ process.",
"In the ideal gas limit, i.e., $a = b = 0$ , it reduces to a constant pressure (isobaric) process."
],
[
"Conflicts of Interest",
"The authors declare no conflict of interest."
]
] | 2001.03437 |
[
[
"Faint and fading tails : the fate of stripped HI gas in Virgo cluster\n galaxies"
],
[
"Abstract Although many galaxies in the Virgo cluster are known to have lost significant amounts of HI gas, only about a dozen features are known where the HI extends significantly outside its parent galaxy.",
"Previous numerical simulations have predicted that HI removed by ram pressure stripping should have column densities far in excess of the sensitivity limits of observational surveys.",
"We construct a simple model to try and quantify how many streams we might expect to detect.",
"This accounts for the expected random orientation of the streams in position and velocity space as well as the expected stream length and mass of stripped HI.",
"Using archival data from the Arecibo Galaxy Environment Survey, we search for any streams which might previously have been missed in earlier analyses.",
"We report the confident detection of ten streams as well as sixteen other less sure detections.",
"We show that these well-match our analytic predictions for which galaxies should be actively losing gas, however the mass of the streams is typically far below the amount of missing HI in their parent galaxies, implying that a phase change and/or dispersal renders the gas undetectable.",
"By estimating the orbital timescales we estimate that dissolution rates of 1-10 Msolar/yr are able to explain both the presence of a few long, massive streams and the greater number of shorter, less massive features."
],
[
"Introduction",
"It is well-established that many late-type galaxies in Virgo are strongly deficient in Hi : that is, they possess less Hi gas than similar field galaxies (e.g.",
"[20], [45], [13]).",
"In some cases this is equivalent to a loss of $>$ 6$\\times $ 10$^{9}$ M$_{\\odot }$ ([49]).",
"It is also well-known that a few galaxies in the cluster are associated with spectacular Hi streams up to 500 kpc in extent ([28]), while others appear to have much shorter features ([7]).",
"There appears to be little or no correlation between which galaxies are deficient and which possess streams.",
"Some of the longest streams are associated with galaxies which are even gas rich, whereas many strongly deficient galaxies apparently lack streams entirely.",
"The dominant mechanism for gas loss in clusters is thought to be ram pressure stripping (e.g.",
"[18], [59], [21], [40], [22], [27], for a detailed review see [3]).",
"This can explain the complete removal of the gas content of a massive galaxy in a few orbits, whilst leaving the stellar component largely unaffected.",
"In contrast, tidal encounters (e.g.",
"[57], [2], [10]) have been shown to more likely result in only small amounts of gas being removed (e.g.",
"[54]).",
"Deficiency alone does not necessarily mean a galaxy is currently losing gas - it may have been stripped in the distant past, and even if close to the cluster centre (where ram pressure is expected to be strongest) in projection, its true 3D distance may be significantly greater.",
"Detecting short streams is hampered by the low resolution of single-dish observations, with limited data available from interferometers with the necessary sensitivity.",
"Physically, once gas is removed from its parent galaxy it may disperse into a larger volume, and/or it might experience a phase change (either by heating or cooling) and so rendered undetectable.",
"Yet collectively, the discrepancy between the numbers of deficient galaxies and those with streams seems too strong to ignore.",
"The Arecibo-based ALFALFA (Arecibo Legacy Fast ALFA survey; [16]) and AGES (Arecibo Galaxy Environment Survey; [1]) projects have both covered parts of the cluster to high sensitivity at 17 kpc resolution.",
"Galaxies of low and high deficiency are found in close proximity to one another, strongly suggesting that at least some galaxies in the surveyed regions should be in the process of actively losing gas ([49], see also phase-space investigations, e.g.",
"[23], [39]).",
"While the earlier VLA Imaging of Virgo in Atomic gas survey (VIVA, [8]) discovered several short streams, both Arecibo surveys have reported few new Hi streams in Virgo - none at all in the case of AGES.",
"In principle, a sufficiently rapid phase change could explain the dearth of streams (e.g [6]).",
"Yet the existence of a few extremely long streams at least shows that this cannot be the complete explanation.",
"If evaporation does account for the lack of most of the expected streams, why are there any long streams at all - especially near the centre of the cluster ?",
"Why is there no correlation between the deficiency of a galaxy and the presence of a stream ?",
"This problem has been remarked upon in [34], [60], and ourselves in [53].",
"[34] describe a particularly interesting stream - it is approximately 100 kpc long, located near the centre of the Virgo cluster, and its length suggests a survival time $\\sim $ 100 Myr.",
"Its parent galaxy is strongly Hi deficient, though the mass of Hi in the plume can only account for about 10% of the missing gas.",
"Both molecular and ionised gas were later detected in the stream, but both are an order of magnitude less massive than the Hi ([58]).",
"In contrast, NGC 4569 is strongly Hi deficient but [4] find that 17-42% of its missing Hi can be explained by a phase change to (detected) H$\\alpha $ .",
"It is therefore unclear if phase changes can explain the lack of Hi streams in the Virgo cluster.",
"In contrast, numerous ram-pressure stripping simulations have shown that stripped Hi can remain detectable at distances $>$ 100 kpc from its parent galaxy (e.g.",
"[40], [56]).",
"Cosmological simulations by [62] found that 30% or more of gas-rich galaxies possess long one-sided gas tails, though this does not account for the phase of the gas.",
"Other simulations have shown that while harassment cannot explain strong deficiencies, it can still produce easily detectable, long, one-sided Hi streams ([10], [54]).",
"[60] proposed that only the warm Hi is stripped by ram pressure, quickly rendering it undetectable due to evaporation and dispersal.",
"But this would still allow a 200 Myr detectability window, and some galaxies are so deficient that it appears that the cold, inner Hi has also been stripped.",
"In short, none of the scenarios proposed are very satisfying solutions to the `missing stream' problem - at least in Virgo.",
"In this paper we attempt to address these issues.",
"Section reviews the known optically dark Hi features in Virgo.",
"In section we attempt to quantify how many streams we expect to detect.",
"In section we use new analysis techniques to re-search AGES data cubes, uncovering a number of streams that were previously missed.",
"We interpret the validity and physical nature of our detections in section .",
"Finally in section we comment on the these results and whether they alleviate the problems discussed above."
],
[
"Known optically dark H",
"In order to quantify the (possible) discreprancy between the actual and expected number of Hi streams in Virgo, we require a catalogue of known features.",
"We compile a catalogue of optically dark Hi features in Virgo with $v_{hel}$ $<$ 3,000 km s$^{-1}$ from a literature search (see also [53] for more details), which is presented in table REF .",
"Table: Properties of known optically dark Hi features in the Virgo cluster.",
"The `name' column gives the parent galaxy (where available).",
"Spatial coordinates are in J2000.",
"All parameters, except coordinates, refer to the optically dark gas and not the parent galaxy (if one is present).",
"We divide the table into three categories, separated by horizontal lines : the uppermost section contains discrete clouds, the middle section short streams, and the lower section long streams.",
"Reference codes are as follows : T12 = ; T13 = ; K09 = ; K10 = ; C07 = ; D07 = ; M07 = ; G89 = ; O05 = ; K08 = , S17 = .",
"The parent galaxy of VIRGOHI21 is believed to be NGC 4254 (VCC 307) while the Koopmann stream is associated with NGC 4534/DDO 137.We have arranged the features in table REF into three categories : isolated clouds; short streams attached to their parent galaxies; and much longer features.",
"While most streams and clouds are typically less than 30 kpc in length, a handful are truly enormous, from 100-500 kpc in extent.",
"Table REF compares how the streams relate to their parent galaxies.",
"The Hi deficiency is quantified using the method of [19], using the parameters of [44] (see their table 2).",
"The intrinsic scatter in the relation is generally taken to be around 0.3.",
"Table: Galaxies with known Hi streams, comparing the parent galaxy Hi deficiency with the tail mass.",
"We use the size and morphology parameters from GOLDMine to compute the expected Hi mass, using the method of .",
"Tail masses are taken from the references in table .",
"The missing mass is computed as the difference between the actual mass in the galaxy (ignoring the tail) and its expected mass.",
"For VCC 1249 see section .",
"VCC 307 (NGC 4254) is the likely parent galaxy of the VIRGOHI21 feature.Of the twelve galaxies in table REF , seven are significantly deficient while the others are non-deficient.",
"The lack of correlation between deficiency and presence of a stream is strengthened given that the vast majority of deficient galaxies in the cluster have no reported streams.",
"Table REF also gives the ratio of the gas detected within a tail compared to the amount of gas lost according to the deficiency parameter (MHi$_{tail}$ /MHi$_{miss}$ column).",
"The majority of tails are simple, linear, one-sided features, which contain much less than the amount of missing gas of their parent galaxies.",
"This suggests that after gas removal a rapid phase change or dispersal of the Hi is necessary and sufficient to explain most of the features.",
"This is a simple, appealing view of the evolution of the streams but there are two serious caveats.",
"Firstly, as discussed in T16, there are a small number of cases where the mass in the stream appears excessively large in relation to the parent galaxy.",
"Secondly, it is unclear whether such a process is compatible with the presence of a few very long, massive streams : are those particular features somehow prevented from dissipating ?",
"We therefore need a way to predict how many galaxies should be actively losing gas and the conditions which can render the gas undetectable."
],
[
"How many streams do we expect to find ?",
"There are three aspects to the problem : whether streams are observationally detectable, how many are currently forming, and the evolution of the streams as they disperse into the ICM.",
"In this section we examine the first two aspects of observational limitations and stream formation rate.",
"We will examine their evolution in section REF ."
],
[
"Observational restrictions",
"Almost all of the shorter streams in table REF are comparable in size to the Arecibo beam, or longer, and thus should be distinguishable from their progenitor galaxies even with the low-resolution Arecibo surveys.",
"Furthermore they are mostly quite massive, $\\sim $ 10$^{8}$ M$_{\\odot }$ .",
"ALFALFA and AGES are both ostensibly far more sensitive than the VIVA survey : AGES reports a 1$\\sigma $ column density sensitivity limit of approximately N$_{\\rm HI}$ = 1.5$\\times $ 10$^{17}$ cm$^{-2}$ (at a line width of a single 10 km s$^{-1}$ channel, [24]); ALFALFA is about 5.0$\\times $ 10$^{17}$ cm$^{-2}$ ([17]); VIVA is almost two orders of magnitude worse, at around 1.0$\\times $ 10$^{19}$ cm$^{-2}$ ([8]).",
"An important caveat is that column density is not necessarily a good sensitivity indicator.",
"What the observations are actually sensitive to is the total mass within the beam.",
"Gas can have an arbitrarily high value of N$_{\\rm HI}$ , but if its mass is too small then the observations may not detect it (e.g.",
"a dense but low mass feature would have a low filling factor within a large telescope beam).",
"Counter-intuitively, a survey with a smaller beam and worse N$_{\\rm HI}$ sensitivity may actually be more suitable for detecting low-mass features, provided their N$_{\\rm HI}$ is sufficient.",
"This `beam dilution' is particularly important for single-dish telescopes, and interferometers are better adapted to detecting structures smaller than the beam scale.",
"With this in mind, we can compute the detectability of a stream if we make the idealised assumption that the stream is a linear, uniform-density cylinder.",
"In this case the signal to noise ratio (S/N) from a given survey will depend on the mass, length, velocity profile, and orientation of the stream with respect to the observer.",
"Orientation has two effects, firstly on the projected length $L_{p}$ : $L_{p} = L\\,sin(i)$ Where $L$ is the intrinsic length of the stream and $i$ is the inclination angle to the line of sight, such that $i$$\\,$ =$\\!$ 0.0$^{\\circ }$ if line of sight is parallel to the longest axis of the stream and 90$^{\\circ }$ if perpendicular.",
"The number of beams the stream spans in projection will be given simply by $L_{p}/B$ where $B$ is the beam size.",
"The stream can never (by definition) be smaller than a point source in any survey - it must always appear to span at least one beam.",
"The form of the stream in observational data will be its true shape convolved with the telescope beam, but a reasonable approximation is given by : $N_{beams} = max\\Big ( 1.0,\\,\\frac{L\\,sin(i)}{B} \\Big )$ Secondly, orientation has a very similar effect on how many velocity channels the stream spans.",
"Recall that the relevant parameter for detectability is the mass contained in each beam in each channel : $M = \\frac{M_{total}}{N_{beams}\\,N_{chans}}$ Where $M_{total}$ is the total mass of the entire stream.",
"Given the standard equation for converting Hi flux to mass : $M_{\\rm {HI}} = 2.36\\times 10^{5}\\,d^{2}\\,F_{\\rm {HI}}$ Where for $M_{\\rm {HI}}$ in solar units, $F_{\\rm {HI}}$ is the total flux in Jy km s$^{-1}$ , which for a single channel is given by $S/N\\,\\sigma _{rms}\\,w$ , where $\\sigma _{rms}$ is the $rms$ noise level of the survey, $w$ is the velocity width of the channel (in km s$^{-1}$ ), and $d$ is the distance in Mpc.",
"We can combine the above equations to calculate the S/N of a stream based on its intrinsic parameters and the survey capabilities : $S/N = \\frac{M_{total}}{N_{beams}\\,N_{chans}\\,2.36\\times 10^{5}\\,d^{2}\\,\\sigma _{rms}\\,w}$ If we disregard the cases where the stream is contained within a single beam or channel, then by equations REF and its velocity counterpart (we assume a velocity gradient of magnitude $V$ along the longest axis of the stream), we can re-write equation REF : $S/N = \\frac{M_{total}}{({L}{B})\\,sin(i)\\,V\\,cos(i)\\,2.36\\times 10^{5}\\,d^{2}\\,\\sigma _{rms}}$ Thus, while the noise level of the survey remains critical, the survey resolution only determines where the S/N is truncated (see below) but has no other influence over the curve.",
"This may seem counter-intuitive : for example, if one smoothes data in velocity, the flux is spread into fewer velocity channels and so the S/N increases.",
"Of course this procedure also improves the $\\sigma _{rms}$ , and in practice the resolution and $\\sigma _{rms}$ are not truly independent : to get the same $\\sigma _{rms}$ when the velocity resolution is higher requires a longer integration time.",
"Using equation REF , we plot how S/N varies with inclination angle for a linear stream of given parameters (figure REF ).",
"At low angles, flux is projected into a short spatial length, giving a high S/N despite a wide spread in velocity.",
"At intermediate angles the flux is spread out both in velocity and space, minimizing the S/N.",
"At high angles, the S/N increases - although it is now highly spatially extended, its projected velocity width is very small.",
"For any given stream, there is a range of inclination angles within which it can be detected.",
"Figure: Expected S/N level of a linear stream of mass M HI M_{HI} = 1.45×\\times 10 8 ^{8}M ⊙ _{\\odot } and velocity width 500 km s -1 ^{-1} as a function of viewing angle, for various lengths, assuming survey capabilities equal to AGES (beam size of 17 kpc, rmsrms of 0.6 mJy, and channel width 10 km s -1 ^{-1}).",
"The black line shows a constant S/N level of 3.0, for reference.",
"The vertical axis has been truncated, with an actual peak S/N >> 100.0.",
"The `cut' at low inclination angles occurs when the stream spans less than 1 beam in projected length - a similar cut occurs at high angles when the stream spans less than 1 velocity channel.Perhaps more usefully, we can also consider the streams as a population.",
"If we assume the streams have a random orientation with respect to the observer, equation REF can be used to find the range of inclination angles at which a stream of any given parameters will be detectable - so giving the detectable fraction.",
"The projected length of a stream should be at least two beams, otherwise they will not be distinguishable from their parent galaxies[43] discuss a search for spectral asymmetries.",
"The effect is generally quite subtle, but may prove fruitful for a future examination of the AGES and/or ALFALFA Virgo data..",
"This means their detectable fraction never reaches 100%.",
"We do not know the properties of the entire population of streams in Virgo, but we do know about those which have been detected in VIVA.",
"Assuming these are representative of the true stream population, then by this method we can determine how many such features should be detectable to the ALFALFA and AGES surveys.",
"Their median mass (see table REF ) is 1.45$\\times $ 10$^{8}$ M$_{\\odot }$ .",
"We estimate their intrinsic lengths and velocity widths from the maximum observed values, 60 kpc and 110 km s$^{-1}$ respectively.",
"The expected detection fraction is rather high, around 70% for both AGES and ALFALFA : projected length is the limiting factor in this regime, not total mass, hence the identical spatial resolution of the surveys gives identical detection fractions.",
"The expected number of detected streams in any given survey depends on 1) the number of Hi-detected galaxies in the survey region (439 for ALFALFA in the VCC region, 105 for AGES); 2) what fraction of those galaxies actually have streams, which we take to be 15% based on VIVA; 3) the geometrical correction for how many streams that exist should also be detectable, i.e.",
"70%.",
"This gives expected stream numbers of 46 for ALFALFA and 11 for AGES.",
"The actual numbers are 5 for ALFALFA and 2 for AGES.",
"Hence the geometric correction is insufficient to explain their low detected numbers.",
"We now consider which galaxies are expected to be currently producing streams in the first place."
],
[
"Stream formation",
"The modelling of [27] provides an analytic model of ram pressure stripping.",
"This considers how much pressure is required to strip the Hi down to its observed radius (P$_{req}$ ) in comparison with an estimate of the local pressure the galaxy is actually experiencing (P$_{loc}$ ), given its position in the cluster.",
"Given the uncertainties, a galaxy may be actively stripping if the ratio P$_{loc}$ /P$_{req}$$\\,$$\\ge $ 0.5.",
"The main advantage to this is that it describes current stripping activity - potentially a much better proxy for the presence of a stream than Hi deficiency.",
"The disadvantage is that the necessary data are only available for a small fraction of the galaxies, so we cannot use it to predict the total number of expected streams in this region.",
"Despite this, the model can make more qualitative predictions.",
"We do not expect every galaxy which is predicted to be an active stripper to have an Hi tail, due to the distance uncertainty (which affects the calculated P$_{req}$ value) and the possible geometrical dilution of the tail described in section REF .",
"If, however, ram pressure stripping is indeed the dominant gas-loss mechanism in the cluster, then we expect tails to be more common among galaxies with higher P$_{loc}$ /P$_{req}$ ratios.",
"We also expect every galaxy (with only rare exceptions due to harassment and ICM density variations) which has an Hi tail to be an active stripper according to the model.",
"We will return to this in section REF ."
],
[
"Data processing",
"We test the models of section using the two AGES Virgo data cubes described in [49] and [50].",
"The much larger ALFALFA data set is not publically available, and the AGES data sets, though smaller, have the advantage of higher sensitivity.",
"Both cubes are available via the AGES website at the following URL : http://www.naic.edu/$\\sim $ ages/.",
"There are two significant improvements to the data processing algorithms developed since the original analysis.",
"The first is an implementation of the spatial bandpass processing algorithm MEDMED, described in [37].",
"AGES is a R.A.-parallel drift scan survey, with the baseline level of the spatial bandpass being nominally estimated as the median level of the entire scan.",
"This is adequate for most scans in which galaxies occupy only a few percent of the bandpass, but where bright, extended sources are present, the baseline average value is over-estimated.",
"This results in `shadows' in the cube in R.A. (see [30]).",
"As in [51], we use a Python-based version of MEDMED that splits the scan into five boxes, measures the median of each, and then uses the median of the medians as the baseline.",
"This almost completely eliminates the shadows which would otherwise hamper the search for extended emission in the affected areas.",
"The second change, also described in [51], is to fit a second-order polynomial to the spectrum along each pixel in the cube.",
"While the $rms$ of each spectrum is not affected, the removal of the baseline variation from pixel to pixel greatly improves the `cleanliness' of the data, making it much easier to search for extended emission and improving the accuracy of flux measurements on extended structures.",
"The combined effect of these cleaning processes is shown in figure REF .",
"Figure: Moment 0 (integrated flux) maps of the AGES VC1 data cube, using the same velocity range (100-3,000 km s -1 ^{-1}) and with the same colour scale in both cases.",
"The upper figure uses the standard data cube which has no additional processing besides hanning smoothing; the lower figure uses the cleaning techniques described in section .",
"The rms of the spatial bandpass in the cleaned image is approximately 33% lower than in the raw image, whereas the mean pixel value is a factor of three lower in the cleaned image."
],
[
"Search technique",
"Another key development in our search for streams is the FITS viewer frelled, described in [52].",
"The main benefit here is the user can interactively create moment maps and contour plots, i.e.",
"to find the most appropriate velocity and spatial range with which to examine each galaxy.",
"We can also examine the data in 3D rather than conventional 2D slices, which can make visual identification of extended features much easier.",
"We found that the best way to detect extended emission was through inspection of renzograms (contour maps in which each velocity channel is represented by a different colour, see [41]) for non-circular features.",
"Integrated flux maps, though of greater sensitivity to diffuse gas, tend to be problematic.",
"The galaxies themselves often have marginally resolved gas discs - while the discs tend to have circular Hi contours in every channel, their centre is not quite at the same pixel position in each channel.",
"Thus, integrating over the whole velocity range produces non-circular features which are not related to genuine extensions - see figure REF .",
"Figure: Different 2D display techniques for the same galaxy, VCC 2070.",
"The background image is the SDSS RGB image from the standard pipeline, the same in all panels.",
"The top panels show moment 0 contours while the lower panels show renzograms.",
"The left side images use the full velocity range of the galaxy while those on the right are restricted to the velocity range we identify as containing the associated stream.",
"The green circle is the size of the Arecibo FWHM, 3.5 ' ^{\\prime }.Viewing renzograms in 3D shows essentially isosurfaces that display constant flux levels as a 3D surface rather than a 2D contour.",
"This gives a powerful advantage in the search for non-circular extensions : since most channels tend to have circular contours, those with non-circular extended features easily stand out (especially if those features are coherent over several channels).",
"Unlike volume renders, isosurfaces have the benefit of displaying the flux at an objective level which does not depend on viewing angle (see [52] for a full discussion), and the 3D display makes it easier to see which channels possess extensions than the case of 2D renzograms (where the superposition of many channels can be confusing).",
"We show an example in figure REF .",
"Figure: Example of using different visualisation techniques to search for extended features.",
"The left panel shows a volume rendering (integrating the flux along the line of sight) of VCC 2070 at an arbitrary angle.",
"The extended gas tail is barely visible.",
"The right panel shows the data cube rendered in exactly the same way, but with a renzogram at 4σ\\sigma overlaid.",
"The extension at the low-velocity tip of the galaxy (right side of the image) is now much more obvious.We limited our search to the velocity range 100 $<$ $v_{hel}$ $<$ 3,000 km s$^{-1}$ , avoiding the Milky Way and high velocity clouds.",
"We constructed renzograms/isosurfaces for every catalogued Hi detection in this region, 106 out of our total of 108 Virgo detections for VC1 and VC2 combined.",
"The two omissions were at such low redshifts that Milky Way contamination would make distinguishing any extended emission extremely difficult.",
"Our procedure was to begin with renzograms at 3$\\sigma $ and then increase the S/N level as appropriate.",
"For the brightest sources, extensions are not visible at 3-5$\\sigma $ simply because the disc emission is very bright, “smearing” the emission into many pixels.",
"Our requirement for a detection was that the non-circular features should be visible at a defined level per volumetric pixel (voxel) across a connected span of at least one beam (in addition to the galaxy's disc) and over at least 3 channels.",
"We catalogued possible streams by the noise level of the connected voxels as either certain ($>$ 6$\\sigma $ per voxel), probable ($>$ 4-6$\\sigma $ per voxel) or possible ($>$ 3-4$\\sigma $ per voxel).",
"We discuss these levels in detail in section REF .",
"Maps of the individual certain or probable detections are shown in figure REF .",
"We do not discuss the less confident detections here but present them in the online appendix.",
"The full catalogue of the stream status of all the galaxies in this region (excluding those with no streams of any kind) is shown in table REF .",
"Figure: Renzograms of the `certain' (VCC 2070, VCC 1555, and VCC 2066 on the top row; EVCC 2234 is in the bottom left panel) and `probable' stream detections.",
"The contour levels are of fixed value with colour indicating the channel.",
"The S/N level of the contour (typically 4 σ\\sigma , equivalent to a column density of N HI N_{HI} = 6×\\times 10 17 ^{17} cm -2 ^{-2}) and velocity range of the renzogram (shown by the colour bars in units of km s -1 ^{-1}) have been manually adjusted in each case to reveal the streams most clearly so do not always show the full velocity range of each galaxy - for exact individual values, consult appendix .",
"The green circles show the Arecibo beam size.Table: Catalogue of all the extended Hi features detected in AGES in the Virgo cluster.",
"All measurements are derived from AGES data except for those of AGESVC2 020 (VCC 2066) - as this is on the southern limit of the data where measurements may not be accurate, we use the values from instead.",
"The sections divide the table according to stream type, where 0-2 denote possible streams (0 means the detection is certain, 1 probable, and 2 possible), and 3 indicates that the galaxy has noisy contours with no preferred direction for the extensions.",
"For completeness we also show category 4, meaning no detected streams, for those objects where the pressure ratio described in section was calculated.We catalogued four streams as certain, six as probable, and sixteen as possible.",
"We also found eight galaxies with Hi contours with no indications of asymmetry but which had a distinctly `noisy' appearance, sometimes even at the 7$\\sigma $ level.",
"None of the remaining 73 galaxies showed any indications of any unusual Hi features, though 20 of these were of rather low S/N ($<$ 10) so extensions would be difficult to detect."
],
[
"Measuring the streams",
"Unlike the galaxies in [51], the Virgo objects are marginally resolved, and thus cannot be measured as point-sources and subtracted to allow objective measurements.",
"This means we are compelled to resort to more subjective procedures.",
"We use FRELLED's capability to define volumes of arbitrary shape and sum the flux within them, manually defining volumes we believe only contain flux from the extended Hi.",
"Thus the estimates of the stream masses in table REF should be treated with caution (the estimated ratios MHi$_{stream}$ /MHi$_{miss}$ are similar to the values for the [7] sample as in table REF , though on the low side).",
"We do not attempt this procedure at all for galaxies with noisy contours.",
"We measure the length of the streams as the distance from the centre of the galaxy to the most distant extension of the stream contours.",
"We do not apply the correction for beam smearing described in [61] as the difference is only a few kpc for our sources, and our errors are dominated by the problems of determining the edge of the parent galaxy's disc.",
"Additionally, the beam size means that we cannot accurately measure the thickness of the streams - the apparent visual difference in thickness of the streams compared to their parent galaxies simply reflects the relative brightness of stream and galaxy, not their dimensions."
],
[
"Comparisons to other data",
"Six galaxies in our sample have interferometric observations.",
"VCC 2070 and VCC 1555 were observed with VIVA, with shorter extensions detected at similar orientations to those detected here (see appendix A).",
"The tail of VCC 2066/2062 has a very similar overall morphology in both AGES and the VLA observations of [9], though the VLA data shows structures within the tail that AGES cannot resolve.",
"The morphology of the gas cloud close to VCC 1249 is very similar in AGES (see T12 figure 21) to the KAT7 observations of [46], hereafter S17 (see their figure A3).",
"In general the AGES data support the existing interferometric observations, in some cases extending the length of the tails significantly.",
"There are two exceptions.",
"One is VCC 1205 (see section REF ), the extension of which is not described by S17.",
"However this is not surprising - their observations are nominally 5 times less sensitive than ours, with a 1 $\\sigma $ column density sensitivity of 8$\\times $ 10$^{17}$ cm$^{-2}$ .",
"This is reduced further at the position of VCC 1205 as that region was only observed with KAT7.",
"Additionally, S17 only detect a small part of the ALFALFA Virgo 7 cloud complex described in [25] (which similarly happens to be at the edge of the field where only KAT7 data was taken), and the authors attribute this to the gas being at low column density and below the sensitivity of the interferometer.",
"More puzzlingly, the feature described in S17 associated with AGESVC1 293 is not visible in the AGES data.",
"Here the KAT7 and WSRT pointings overlap, though the galaxy is near the edge of the survey fields where sensitivity is again somewhat reduced (see S17 figure 3).",
"It has an Hi mass of 2.0$\\times $ 10$^{7}$ M$_{\\odot }$ and a W50 of 87 km s$^{-1}$ .",
"The morphology of the source is irregular, so beam dilution may play some role, but the main feature is comparable in size to the Arecibo beam.",
"If entirely contained within the Arecibo beam, the average column density in each 10 kms AGES velocity channel would be 1.2$\\times $ 10$^{18}$ cm$^{-2}$ , well above the AGES sensitivity limit.",
"Deeper observations are needed to confirm the existence of this source."
],
[
"Estimating the false detection rate",
"Our criteria for “probable” detections being at least 4$\\sigma $ may seem weak, implying that we could expect a high fraction of our results to be spurious.",
"In this section we examine this statistically by three different methods.",
"Throughout, it is crucial to remember that our criteria for identification relies not just on S/N but also on the spatial and velocity extent of the features."
],
[
"The number of similar features in galaxy-free regions of the data",
"If the claimed streams are actually just fluctuations in the noise then they should be present throughout the entirety of the data cube.",
"Although difficult to disentangle from bright, marginally resolved galaxies, in empty regions it is straightforward to find and measure such features using objective, repeatable procedures.",
"We begin by masking the galaxies and the identified streams, which accounts for about 5% of the total volume of the VC1 data cube.",
"For the remaining pixels, we use the stilts package ([48]) to match groups of connected pixels at or above a range of S/N levels.",
"We then quantify the number of groups of pixels based on both the number of connected pixels and the S/N level.",
"Given the detection rate in the galaxy-free regions and the total volume searched for extensions, we can estimate the number of false detections we expect around the galaxies.",
"The results are shown in figure REF .",
"Figure: Expected number of false detections of streams at different S/N levels (x-axis) and number of connected pixels (annotations).",
"This is based on a search of the empty regions of the the VC1 data cube, described in the text.",
"The y-axis value has been scaled according to the volume spanned by the galaxies and their streams, and truncated at 26 (the total number of certain, probable and possible streams).At 4$\\sigma $ per voxel, even groups of only five connected pixels (approximately one beam in one channel) are expected to be so rare that they are unlikely to cause significant contamination.",
"Five pixels is extremely small - ten is more reasonable for our “probable” and “certain” detections, which generally fill at least one beam and are found in three or more channels.",
"With ten or more pixels, streams can be considered reliable even at 3.5$\\sigma $ .",
"Furthermore, the curves plotted in figure REF will significantly overpredict the number of false detections : this figure assumes that a spurious clump found at any location within the searched volume would be mistaken for an extension.",
"In reality, the clump would have to appear in a very specific, much smaller region - if it coincided with a galaxy it would not be visible at all, whereas if it was too far from the galaxy it would be identified as an independent object rather than an extension.",
"We conclude that our “certain” and “probable” identifications suffer a negligible rate of false positives, though doubtless there may be contamination in our “possible” features.",
"It should be remembered that only 36% of the galaxies inspected in the VC1 cube showed any signs of extensions at all - even the most modest potential streams included in our catalogue are notably different from most of the galaxies in the data."
],
[
"Searching for simulated streams of known parameters",
"Using artificial galaxies and streams with real noise, we can also test our subjective search techniques.",
"In this way we can examine (a) whether the noise will allow us to detect features as weak as those we claim to have detected and (b) whether we would visually identify more false positives that the objective procedure suggests.",
"We create an artifical galaxy with parameters (S/N = 55.0 and velocity width of 220 km/s) based on the median values of the galaxies we have identified as having sure or probable streams.",
"For the galaxy we use either a simple point source or a more realistic model based on the radially averaged profile of the real, marginally resolved VCC 975.",
"We create a stream beginning with a grid of 6$\\times $ 2 pixels of uniform S/N level (3, 4 or 5$\\sigma $ ), which we then convolve with a 3.5$^{\\prime }$ Gaussian.",
"This we then add to either the first 1, 2, or 3 consecutive channels in the artifical galaxy data set, our aim being to explore the detectability of the weakest features.",
"Next, we run a Python script that chooses random pixels within the masked VC1 cube and checks for the presence of masked pixels within the appropriate surrounding volume.",
"If any are found, another pixel is chosen and this is repeated until a suitable region is found.",
"We then add the galaxy and stream into this region of pure noise.",
"The data in this region is extracted, and the procedure is repeated 100 times to create galaxies+stream+noise data cubes.",
"The properties of the galaxy and stream do not vary so this procedure tests only the influence of the noise.",
"An example subset of the data is shown in figure REF , plotting renzograms of the point source case, with the stream extending from the centre of the galaxy to the right.",
"Figure: Example subset of artifical galaxies with extensions combined with real noise from the AGES VC1 data cube.",
"In this case the extension has an original S/N level of 3σ\\sigma (boosted slightly by the presence of the galaxy) with the contour at 3.5σ\\sigma .",
"This approximately corresponds to some of our faintest claimed detections.",
"When visible, the extension is usually clear, whereas there are almost no visible false positives even at this low S/N level.We inspect the final data set visually, varying the contour level from 3.0-5.0$\\sigma $ in steps of 0.5.",
"At each level, we record how many galaxies show clear signs of the aritifical stream in their renzograms.",
"We require a detection to span at least one beam, the same number of channels as it was injected in, and be present at the correct location.",
"We also count the number of galaxies with similar features that are not at the correct location, i.e.",
"false positives.",
"Knowing the size and orientation of the stream makes this process very fast, enabling us to explore a large parameter space of detectability (note that the detection criteria deliberately probe very low S/N levels and channel numbers, and do not reproduce the criteria used in the actual search).",
"The results are shown in figure REF .",
"Since the streams are added to a bright galaxy, their final S/N is usually slightly higher than their initial value, hence it is sometimes possible to detect nominally faint streams at surprisingly high S/N values.",
"“Completeness” here is in the usual sense, i.e.",
"the fraction of known streams which were detected.",
"We cannot properly measure reliability here, as this depends on the number of injected streams which is a free parameter in this exercise.",
"Instead the independent parameter is the number of false positives.",
"Figure: Completeness levels (left) and false positive rate as a function of contour level for sources injected into different numbers of channels (1, 2, and 3 from top to bottom).",
"Line colour indicates the original S/N level of the injected source.",
"Solid lines show the case of using a point source for the galaxy while dashed lines use a more realistic galaxy profile.We find that at very low S/N levels, the data is essentially one contiguous set of pixels from which nothing whatsoever can be discerned.",
"At high S/N levels one sees only the bright galaxies.",
"In between these extremes, there does not exist a realm in which one sees significant numbers of features resembling the streams we see around actual galaxies.",
"Such features do exist, but are always rare, and hence unlikely to be found in locations where they would be mistaken for streams.",
"Essentially, the noise can obscure real extensions but it does not create many spurious features."
],
[
"A blind search for simulated streams of unknown parameters",
"An even more realistic approach is to vary the properties of the streams so that we do not know if a stream is even present at all.",
"We modify the injected streams to randomly vary their : (i) direction, so that they are assigned to one of eight different directions (N, NE, E, etc.",
"); (ii) length, ranging from 6 to 8 pixels (we found that below 6 pixels the streams are not recoverable due to the large apparent size of the simulated galaxy); (iii) S/N, ranging from 3.0 to 5.0; (iv) number of channels, from 0 through 5 inclusive so that some streams are not actually injected at all.",
"We also vary the absolute value of the channels used so that the colours of the renzogram provide no information of the presence of a stream.",
"Tests showed that the weakest stream in this parameter space was at the very limit of visual detectability, whereas the strongest is extremely obvious.",
"Here we do not impose the strict requirements on any potential detections as in REF , but simply try to decide if there is anything present which we might classify as a detection.",
"We allow ourselves to vary the contour level but do not demand that any particular level is necessary - we attempt to select on the same basis as we did for the actual search.",
"This is considerably slower than in REF but is far more realistic.",
"As before, we injected 100 potential streams (including those spanning zero channels) into real AGES noise.",
"Due to the random number of channels selected, a total of 76 streams were actually injected.",
"We recovered 56 of these, giving a completeness of 74%.",
"We also recorded 5 false positives.",
"All of these were weak compared to the real sources : the strongest being 4 $\\sigma $ in two channels, which is fainter than any of the real streams indentified as “sure” or “probable”.",
"Thus it seems extremely unlikely that any of our “sure” and “probable” features identified in the real cluster are due to noise : the overwhelming probability is that they are real, physical structures.",
"Again, we emphasise that the reliability level of the “possible” detections is surely lower."
],
[
"Comparison with the stripping predictions",
"As described in section REF , the advantage of the [27] model over Hi deficiency is it directly describes current stripping activity.",
"While it is difficult to predefine an exact value of the ratio P$_{loc}$ /P$_{req}$ to distinguish active from past strippers, the broader predictions of the model are borne out reasonably well.",
"Of the galaxies with streams of all levels of confidence or noisy contours (11 objects), the median P$_{loc}$ /P$_{req}$ ratio is 0.28.",
"Examination of table REF suggests a P$_{loc}$ /P$_{req}$ ratio of 0.2 is a plausible (though rough) value to distinguish active from past strippers.",
"Using this in the model, we can state : of the 10 galaxies with detected tails, 8 should be actively losing gas and 2 should be past strippers; of the 9 galaxies without tails, 6 are predicted to be passive and 3 should be active strippers.",
"This broadly supports the assumption that the bulk of the streams should be produced by ram pressure stripping.",
"While we lack the advantages of resolution from interferometric studies, we can still measure all the same major properties : morphology (at least to say whether a tail is one-sided), length, kinematics, and mass.",
"In our view, all of these tend to support the ram-pressure origin scenario of the tails.",
"They are mostly one-sided, the brightest emission occurs over a relatively short velocity range corresponding to that of the parent galaxy (as in the tails in [7]) and they are of comparable length to the [7] tails, though they are of somewhat lower mass."
],
[
"Alternative explanations",
"Hi asymmetries can be produced by internal and external processes.",
"Tidal structures in Virgo (e.g.",
"[29], [28]) are usually longer and show more complex, haphazard structures and kinematics, which is also seen in numerical simulations (e.g.",
"[54]).",
"Other external processes, such as mergers and accretion, have been invoked in other environments ([33], [36]) but neither of these is likely in the high velocity dispersion of a cluster.",
"Could such asymmetrical features be due to internal processes ?",
"Unfortunately comparable studies of isolated galaxies are very rare (three isolated galaxies were targeted with AGES, none of which show extensions - see [31]).",
"Even then, the authors can seldom (if ever) rule out external processes (e.g.",
"[35]), and sometimes even find that this is the most likely explanation despite the isolation ([38]).",
"There are few catalogues of isolated galaxies with resolved Hi maps, but those which exist broadly suggest that features as long and one-sided as those in the present work are far more likely to result from environmental effects.",
"[33] note that only 7% of isolated galaxies are even `mildly' asymmetric.",
"[11], using unresolved line profiles, find that the rate of asymmetry is very low in isolation (2%) compared to the denser field environment (10-20%).",
"[35] describe a VLA survey of 41 isolated galaxies, of which just two (NGC 895 and IC 5078) have clear optically dark Hi extensions.",
"The extension of NGC 895 is morphologically very different to the tails described here, while the authors propose that mergers are responsbile for the features in IC 5078.",
"[42] detect an extension 12 kpc in length, but even for this relatively short feature the authors believe the cause is an external satellite.",
"In short, while asymmetric extensions are very common in group environments, they are rare in isolation (though there is a lack of of large catalogues to properly quantify this).",
"The only features comparable in both length and morphology to those we have described here are widely attributed to various environmental mechanisms, and we can find no evidence in the literature for internal causes."
],
[
"The evolution of the streams",
"Using the [27] model, and assuming radial infall, we can estimate the point at which the ram pressure becomes sufficient to start removing gas from the galaxy.",
"We assume this occurs when the ratio P$_{loc}$ /P$_{req}$ $\\ge $ 0.5.",
"We can then derive the time of flight from this point to the galaxy's current clustercentric position, allowing us to calculate the dissolution rate.",
"We find rates of typically 1-10 M$_{\\odot }$ yr$^{-1}$ , with the time since stripping began around 200 Myr.",
"Very similar rates are obtained by the objects described in [7].",
"If these are correct, then the detectable lifetime of the streams is highly variable (simply because of their mass), from a few megayears to a few gigayears.",
"Similar survival times have been calculated independently by [25] and [5].",
"The least massive streams are likely in a state of active replenishment, otherwise their lifetime would be so short we would not expect to detect them, whereas the most massive streams can survive long after the stripping event by virtue of their high mass."
],
[
"Relation to the dark clouds",
"Figure REF shows that three of the dark clouds appear to be aligned with two of the streams.",
"In one case (GLADOS 001/AGESVC1 231) the alignment is likely just a projection effect, as the velocity difference between the two is over 700 km s$^{-1}$ .",
"In the other case (VCC 740/AGESVC1 247 and 282) the velocity difference between the stream and the clouds is small, however the stream is rather weak.",
"The conclusion of T16 and [54] that the clouds are unlikely to be the result of gas stripping appears to be sound.",
"The dissolution rates calculated in section REF imply that we are witnessing the clouds in the final few Myr of their existence (given their low masses).",
"This is a similar lifetime calculated as in [54] and [53], and considerably shorter than the $\\sim $ 100 Myr timescales estimated in [55].",
"A caveat is that these rates might strongly vary depending on the density of the objects (i.e.",
"allowing them to self-shield from ionising radiation would reduce evaporation as well as giving them stronger self-gravity), of which we at present do not have direct measurements.",
"Figure: Map of all the stream candidates in the VC1 region.",
"Red arrows show the direction the streams point away from their parent galaxies.",
"Blue squares are clouds with no optical counterparts and magenta squares are galaxies with noisy contours but no clear linear extension."
],
[
"Galaxies with noisy contours",
"Figure REF provides a comparison of galaxies with smooth and irregular contours.",
"It is possible that in the weaker cases some of these irregular features are not real, but some cases, which are seen at 5$\\sigma $ in many pixels and channels, are unambiguous.",
"These galaxies show no obvious coherent distribution within the cluster (figure REF ).",
"One possibility is that they are actively losing gas but with motion mainly along the line of sight.",
"Another is that they are losing gas by different, internal mechanisms, e.g.",
"that the extensions are only due to temporary gas displacement (not complete ejection) via supernovae, for example; many dwarf galaxies show irregular contours ([47]).",
"While mergers are not expected in the cluster environment, IC 5078 in [35] (see their figure 40) shows intriguingly similar contours to these `noisy' galaxies, which the authors attribute to a possible minor merger.",
"Figure: Example renzograms of a galaxy identified to have unusually noisy contours (AGESVC1 225, left) compared with another deemed to have clean contours (AGESVC1 277, right).",
"The S/N level of the contours is 5.0 in both cases.",
"Both are bright sources with peak S/N levels of 113 for AGESVC1 225 and 130 for AGESVC1 277.",
"Note that since both sources are marginally resolved, a slight position shift in the centre of their contours can be seen, especially for VCC 975."
],
[
"Summary and Discussion",
"While approximately 60% of the late-type galaxies in Virgo show significant Hi deficiencies, only 2% have previously documented streams.",
"We might expect to see more galaxies in the process of actively losing gas, but quantification is difficult as many different factors influence the exact number of expected streams : how many galaxies are currently stripping, the detectability of the streams in Hi surveys given the stream geometry, orientation and kinematics, and the gas phase change rate.",
"We found through a geometric model that the orientation of the streams is unlikely to significantly reduce the detected number, assuming that previous detections are representative of the full population.",
"Based on existing detections and the geometrical correction, we estimated that ALFALFA should detect 46 streams and AGES 11.",
"Additionally, the model of [27] predicts that streams should tend to be associated with galaxies of higher P$_{loc}$ /P$_{req}$ (the ratio of predicted local ram pressure and required pressure to explain the observed Hi deficiency), and while not every galaxy with a high pressure ratio should have a tail, most galaxies with tails should have high ratios.",
"We re-examined the two AGES Virgo cubes using upgraded data processing and visualisation techniques.",
"We found a minimum of 10 streams (only 1 of which was previously known) and potentially as many as 26.",
"This included galaxies observed in VIVA for which only much shorter tails were previously seen, suggesting that our higher detection rate is in part due to the greater sensitivity of AGES.",
"We demonstrated using statistical analyses that our 10 most confident detections are too bright, extended and coherent to be a result of noise.",
"The predictions from the [27] model appear to hold true.",
"Furthermore we can make a quantitative prediction for future surveys that streams will be most commonly associated with galaxies with P$_{loc}$ /P$_{req}$ $\\gtrsim $ 0.2.",
"Several factors have contributed to the puzzling lack of previous stream detections, most notably quantitative estimates of : 1) the number of currently active stripping galaxies; 2) the time galaxies have spent thus far in the cluster; 3) the evaporation and dispersal rate of stripped material; and additionally 4) the discovery of previously unknown streams.",
"While a naive comparison of the number of deficient galaxies and those with streams shows a clear mismatch, the more detailed analysis reveals a much better agreement.",
"While a few Virgo cluster streams contain a significant fraction of the missing gas, most contain only a few percent of the missing gas mass of their parent galaxy (in contrast, streams in other environments often contain an appreciable fraction of the Hi mass of the parent, as discussed in [51]).",
"This strongly suggests that in most cases a phase change occurs during the stripping process that renders much of the stream undetectable to Hi surveys.",
"There is supporting evidence for this in other clusters : [14] find that 50-60% of observed late-type galaxies in the Coma cluster have H$\\alpha $ tails.",
"The [27] model allows a prediction of the time each galaxy has been undergoing active stripping.",
"For the galaxies with streams this allows us to estimate the dissolution time independently of the physical processes at work, which are typically 1-10 M$_{\\odot }$ /yr.",
"In combination with the time spent within the cluster thus far, these rapid dissolution rates are consistent with the existence of both a few long, relatively massive streams, and a larger population of shorter, less massive tails.",
"Massive streams can persist for longer due to their greater mass, whereas less massive streams are only detectable due to constant replenishment as material is still being stripped from their parent galaxy.",
"The dissolution rates also suggest that, if the optically dark isolated clouds are of a similar nature to the streams, we must be witnessing them in a very short detection window and they should disappear in the next few Myr.",
"We expect to detect a similar number of streams with WAVES, which has a similar sensitivity and area of coverage to the AGES Virgo fields ([32]).",
"Better statistics will allow further tests of the stream models and place more constraints on the nature of the optically dark clouds, but our information about the features described here is limited by the resolution of Arecibo.",
"Future observations at comparable column density sensitivty and higher resolution may give much more precise information on the nature and formation mechanism of individual streams."
],
[
"Acknowledgments",
"We are grateful to the anonymous referee whose comments improved the manuscript.",
"This work was supported by the Czech Ministry of Education, Youth and Sports from the large lnfrastructures for Research, Experimental Development and Innovations project LM 2015067, the Czech Science Foundation grant CSF 19-18647S, and the institutional project RVO 67985815.",
"This work is based on observations collected at Arecibo Observatory.",
"The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association.",
"The SOFIA Science Center is operated by the Universities Space Research Association under NASA contract NNA17BF53C.",
"This research has made use of the GOLDMine Database.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"This work has made use of the SDSS.",
"Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.",
"The SDSS Web Site is http://www.sdss.org/.",
"The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions.",
"The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.",
"figuresection"
],
[
"Comments on individual streams",
"In this section we describe the individual streams in more detail.",
"We also provide figures of each stream not given in the main text (for those not shown here, see figures REF , REF and REF ).",
"All figures are renzograms unless otherwise stated, with nearby galaxies labelled and the target galaxy unlabelled in the centre of the figure.",
"We also give the contour level shown in the renzogram (for reference, 1$\\sigma $ is equivalent to $N_{HI}$ = 1.5$\\times $ 10$^{17}$ cm$^{-2}$ ) for each galaxy figure and the exact velocity range used for each renzogram."
],
[
"VCC 2070",
"This is our clearest detection of a stream, and perhaps the most surprising.",
"The galaxy is near the edge of cube and about 1.7 Mpc in projection from M87.",
"The edge of the stream (at 4$\\sigma $ ) reaches 4 Arecibo beams from the centre of the galaxy, giving it a maximum extent of about 70 kpc.",
"The stream is visible in at least 12 velocity channels and peaks at around 8$\\sigma $ above the noise.",
"This galaxy is one of only two in the VC1 area observed with VIVA.",
"Although the long stream is not detected, the VIVA moment 0 map clearly shows a ragged edge (with a hint of a short extension) on the same side of the galaxy as the long AGES stream, with the opposite edge being much smoother - see figure REF .",
"This detection clearly demonstrates the capacity of the AGES observations to detect features not visible to VIVA because of its comparatively poor column density sensitivity, bearing in mind the caveats discussed in section REF .",
"The main reason this stream appears to have been missed in our original search is probably beam smearing - as shown in figure REF , the stream appears obvious only if the data is processed in the correct way, with the use of contour maps greatly enhancing the non-circular Hi extension.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,102 - 1,242 km s$^{-1}$ .",
"Figure: The two galaxies detected in the AGES VC1 data cube which were also mapped with the VIVA survey.",
"The RGB colours show the VIVA moment 0 maps while the (logarithmically spaced) contours show the moment 0 maps from AGES."
],
[
"VCC 1555",
"This is our second unambiguous detection of a stream.",
"It appears in figure REF (and in the data cube) almost to resemble a companion galaxy, but we can find no optical counterpart at the position of the extended contours.",
"We searched both the SDSS data (smoothing the FITS files so as to increase sensivity to low surface brightness features) and the deeper NGVS data ([12]).",
"Since this is an exceptionally bright galaxy with a peak S/N of 140, beam smearing is extremely strong and makes it difficult to estimate the length and velocity width of the stream.",
"The furthest point of the extension is about 3.5 Arecibo beams at 4$\\sigma $ , or 60 kpc, from the centre of VCC 1555.",
"It appears to be present in 25 velocity channels, though we stress this is uncertain as it is difficult to rigorously define where the disc ends and the stream begins (in some channels it is unclear if the `stream' is actually entirely detached from the disc).",
"The peak flux of the stream is approximately 14$\\sigma $ .",
"This galaxy was also observed with VIVA, and as with VCC 2070 there are hints of a short extension visible in the VIVA data roughly corresponding to the position of the long AGES stream, though the alignment is not quite as good as the case of VCC 2070 (see figure REF ).",
"This galaxy is shown in figure REF with the contour at 5$\\sigma $ over the velocity range 1948 - 2021 km s$^{-1}$ ."
],
[
"VCC 2062/2066",
"While the stream connecting VCC 2062/2066 is well known and already seen in AGES data ([50]), accurate measurements are difficult since the feature is very close to the southern limit of the data.",
"It appears to be significantly more extended than in VIVA data but not more elongated, as shown in figure REF .",
"While the [27] model predicts the galaxy should be stripping, the origin of the gas in this case is less clear.",
"VCC 2062 is too small, VCC 2066 is a lenticular (which are generally gas-poor in Virgo, see [49] and [50]) - and the bulk of the gas is offset from VCC's stellar component.",
"A detailed discussion on this system is given in [9].",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,088 - 1,291 km s$^{-1}$ .",
"Figure: VCC 2062/2066, the single system in VC2 detected by both AGES and VIVA, shown as in figure .",
"A distortion can be seen in the contours south of declination 10:59:00, where the noise level of the data increases significantly near the edge of the cube."
],
[
"VCC 888",
"This is a borderline sure/possible case as the stream is relatively small and only detected in 3-4 channels, with a peak flux of only 5$\\sigma $ , though the extension itself appears clearly different from the otherwise clean contours.",
"The stream extends to about 2 Arecibo beams from the optical centre of the galaxy, or 45 kpc at the GOLDMine distance of 23 Mpc.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,087 - 1,123 km s$^{-1}$ ."
],
[
"VCC 1699",
"This galaxy is at the very southern edge of the AGES cube so unfortunately we cannot examine the whole galaxy.",
"The extension is short, no more than 2 beams (35 kpc at 17 Mpc distance) but detected at the 7-8$\\sigma $ level.",
"It appears to be somewhat less linear than the other features : while the peak S/N levels are found in the extension leading to the north-west, a possible north-eastern extension is detected at the 6$\\sigma $ level.",
"The north-western extension is detected in at least 8 velocity channels.",
"This galaxy is shown in figure REF with the contour at 5$\\sigma $ over the velocity range 1,556 - 1,735 km s$^{-1}$ ."
],
[
"VCC 393",
"The extension here appears to be quite linear.",
"It is relatively weak with a peak detection of 5$\\sigma $ , found in 4 velocity channels.",
"Its maximum extent is around 2 Arecibo beams or 45 kpc at its assumed 23 Mpc distance.",
"The galaxy appears to be somewhat optically disturbed.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 2,513 - 2,723 km s$^{-1}$ ."
],
[
"VCC 94",
"While the contours of this galaxy are roughly circular for most of its velocity range, over about 10 consecutive velocity channels they are distinctly ellipsoidal, with a coherent north-south alignment.",
"This structure is detected at a peak flux equivalent to 25$\\sigma $ in some channels; only the effects of beam smearing (as the galaxy itself is a bright source) caution us to give it a `sure' rather than `certain' detection rating.",
"At the 4$\\sigma $ level, this source is extended up to 2.5 Arecibo beams from the galaxy's optical centre, equivalent to 80 kpc at its 32 Mpc distance.",
"This galaxy is shown in figure REF with the contour at 6$\\sigma $ over the velocity range 2,493 - 2,587 km s$^{-1}$ ."
],
[
"EVCC 2234 (AGESVC2 025 A and B) and NGC 4746",
"The western extension of EVCC 2234 is not very large, but the non-circular contours persist over 3 or 4 channels and are clearly seen at 5$\\sigma $ .",
"We regard this as a reasonly secure detection This extension was already shown in figure REF but figure REF reveals two other, more tenatative detections in the same system.",
"EVCC 2234 appears to be interacting with the bright spiral NGC 4746, which is just outside the VCC catalogue area but almost certainly a cluster member itself.",
"There is a hint of a possible bridge between the two galaxies, but it is present over only a few channels so we regard this as rather tenative.",
"The south-east extension from NGC 4746 appears as asymmetrical noisy contours over $\\sim $ 20 channels so is more secure.",
"Given the identical line of sight velocities of the two galaxies, they are likely interacting.",
"Both of these galaxies are shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,574 - 1,991 km s$^{-1}$ .",
"Figure: Feature 'B' in the image is a possible bridge between the bright spiral NGC 4746 and the fainter irregular EVCC 234."
],
[
"VCC 1859",
"In contrast to the EVCC 2234 / NGC 4746 pair, the VCC 1859/1868 pair do not appear to be interacting.",
"The velocity difference of the two is 630 km s$^{-1}$ , high but not necessarily forbidding an interaction given the velocity dispersion of the cluster.",
"VCC 1868 shows clean, circular contours, whereas VCC 1859 (a 12$\\sigma $ detection) shows elongated, non-circular contours.",
"The extension is a somewhat tenatative detection given its small size.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,596 - 1,746 km s$^{-1}$ ."
],
[
"VCC 1205",
"This is a borderline noisy/sure case - the galaxy certainly has noisy contours, easily visible at 4 sigma in figure REF , but they are distinctly one-sided.",
"We have therefore assumed in the text that this galaxy is actually another case of a tail, albeit a rather ragged one compared to the others.",
"Contours in the figure are shown over the velocity range 2,184 - 2,475 km s$^{-1}$ .",
"Figure: Renzogram of AGESVC1 212 (VCC 1205), which seems to have more irregular contours on the north-western side compared to the south east.",
"Contours are at the 4σ\\sigma level."
],
[
"VCC 1249",
"This is a unique case of displaced gas, discussed in T12 section 4.7 and figure 21.",
"Briefly, while no gas is detected at the optical position of VCC 1249 itself (table REF gives the upper limit from AGES), an Hi cloud (AGESVC1 281, though it was previously detected in many other observations) is found midway between VCC 1249 and the nearby M49.",
"Tables REF and REF give the Hi mass for the cloud.",
"The expected Hi mass for the galaxy is 4.5$\\times $ 10$^{8}$ M$_{\\odot }$ , so the cloud accounts for only about 10% of the missing gas."
],
[
"VCC 667",
"This is a weak stream only visible at 3$\\sigma $ (figure REF ).",
"Several other galaxies are found in its immediate vicinity : VCC 657 (760 km s$^{-1}$ ), 672 (922 km s$^{-1}$ ), 697 (1,230 km s$^{-1}$ ) and 731 (1,242 km s$^{-1}$ ).",
"The positional alignment of VCC with the stream and its close velocity match to VCC 667 (1,405 km s$^{-1}$ ) suggest a possible tidal interaction .",
"This galaxy is shown in figure REF with the contour at 3$\\sigma $ over the velocity range 1,313 - 1,542 km s$^{-1}$ .",
"Figure: VCC 667 (centre) and immediate environs."
],
[
"AGESVC1 204",
"A flat, edge-on blue galaxy with non-circular contours (visible at 5$\\sigma $ ) that are elongated north-south (figure REF ).",
"Two of the nearby galaxies (VCC 265 and 264) have velocities $>$ 3,000 km s$^{-1}$ and are therefore not physically associated with the galaxy.",
"VCC 222 is at 2,410 km s$^{-1}$ whereas AGESVC1 204 is at 2,210 km s$^{-1}$ , suggesting a possible physical association though the tail from VCC 204 is rather short and one-sided.",
"This galaxy is shown in figure REF with the contour at 3$\\sigma $ over the velocity range 2,121 - 2,299 km s$^{-1}$ .",
"Figure: AGESVC1 204 (centre) and immediate environs."
],
[
"VCC 199",
"VCC 199 is an early-type spiral with a relatively weak Hi detection (figure REF ).",
"At 4$\\sigma $ , its contours tentatively suggest a short extension to the north-east, however, it may be that the galaxy is simply marginally resolved with a slightly asymmetric velocity profile.",
"However the nearby galaxy VCC 222 is at a similar velocity (2,306 km s$^{-1}$ ) to VCC 199 (2,603 km/s) and well-aligned with the long axis of the possible extension.",
"This galaxy is shown in figure REF with the contour at 5$\\sigma $ over the velocity range 2,340 - 2,896 km s$^{-1}$ .",
"Figure: VCC 199."
],
[
"AGESVC1 232",
"This is a small blue dwarf galaxy with a possible north-west extension (figure REF ).",
"This is a very uncertain detection as it is short, weak (4$\\sigma $ ) and only present in a couple of velocity channels.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,196 - 1,295 km s$^{-1}$ .",
"Figure: AGESVC1 232"
],
[
"VCC 1011",
"VCC 1011 is a spiral galaxy with a possible extension aligned with the plane of its disc (figure REF ).",
"The extension is rather pronounced, but only visible at 4$\\sigma $ in a few channels.",
"The nearby galaxy VCC 989 is at 1,846 km s$^{-1}$ so unlikely to be associated with VCC 1011 (at 867 km s$^{-1}$ ).",
"The dwarf galaxy AGESVC1 234 is also nearby and rather closer in velocity at 1,184 km s$^{-1}$ but shows no sign of any extensions itself.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 786 - 968 km s$^{-1}$ .",
"Figure: VCC 1011"
],
[
"VCC 688",
"This spiral galaxy shows a 3-4$\\sigma $ hint of asymmetrically noisy contours on its north-eastern side (figure REF ).",
"It is 20$^{\\prime }$ (100 kpc) due north of AGESVC1 232, which as mentioned also has a hint of an extension and is at a similar velocity, but no other galaxies are visible nearby.",
"This galaxy is shown in figure REF with the contour at 3$\\sigma $ over the velocity range 1,043 - 1,230 km s$^{-1}$ .",
"Figure: VCC 688"
],
[
"VCC 740",
"The extension on this irregular galaxy (figure REF ) is visible at 4$\\sigma $ with hints at 5$\\sigma $ .",
"It appears to be significantly large and asymmetrical in comparison to the galaxy itself but is only visible over a few velocity channels.",
"The bright spiral VCC 713, which has its own hint of an extension, is visible nearby.",
"VCC 713 is at 1,138 km s$^{-1}$ whereas VCC 740 is at 877 km s$^{-1}$ .",
"The stream of VCC 713 runs due south, making it unclear if these two galaxies are interacting.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 797 - 968 km s$^{-1}$ .",
"Figure: VCC 740"
],
[
"VCC 713",
"VCC 713 is a bright, red spiral which is just detected by AGES and strongly Hi deficient.",
"Its stream extends due south (figure REF ), and although it is significantly extended and detected at 5$\\sigma $ , it is only present in a couple of channels.",
"Given the high predicted mass lost from this galaxy and the absence of a massive stream, it might be on its second orbit so that the stream has long since dispersed.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,012 - 1,261 km s$^{-1}$ .",
"Figure: VCC 713"
],
[
"VCC 1725",
"At 4$\\sigma $ and only about 1 beam length across, this stream is a tenatative detection at best.",
"It appears in 3-4 velocity channels, though rather more at 3$\\sigma $ .",
"No other galaxies are visible nearby.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,004 - 1,196 km s$^{-1}$ .",
"Figure: VCC 1725"
],
[
"GLADOS 001",
"The detection of the extension from this dwarf irregular is marginal at best, visible at 4$\\sigma $ in only 2-3 channels (figure REF ).",
"There are no obvious galaxies that could be tidally interacting.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 2,634 - 2,718 km s$^{-1}$ .",
"Figure: GLADOS 001"
],
[
"VCC 514",
"This small, faint spiral has a possible extension to the west, visible at 4$\\sigma $ in 4 channels (figure REF ).",
"No companion galaxies are visible.",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 807 - 903 km s$^{-1}$ .",
"Figure: VCC 514"
],
[
"AGESVC1 235",
"AGESVC1 235 is a small blue dwarf galaxy with a possible roughly north-south extension, visible at 5$\\sigma $ in 3 velocity channels.",
"VCC 393, a disturbed spiral galaxy, is only a few arcminutes directly north but its velocity of 2,618 km s$^{-1}$ is quite different to that of AGESVC1 235 (1,667 km s$^{-1}$ ).",
"AGESVC1 127 is also close on the sky but it a background galaxy with a velocity $>$ 7,000 km s$^{-1}$ .",
"This galaxy is shown in figure REF with the contour at 4$\\sigma $ over the velocity range 1,638 - 1,706 km s$^{-1}$ .",
"Figure: AGESVC1 235"
],
[
"AGESVC2 063",
"While the Hi signal itself is secure, this stream (figure REF ) is only visible at $\\sigma $ and only present over a few channels.",
"If confirmed, this could be evidence of a dwarf galaxy experiencing ram pressure stripping, but we caution that the appearance of a stream could simply be due to noise.",
"This galaxy is shown in figure REF with the contour at 3$\\sigma $ over the velocity range 1,837 - 1,926 km s$^{-1}$ .",
"Figure: Possible stream of the AGESVC2 063 detection.",
"The optical counterpart is an extremely faint dwarf irregular, just visible in the centre of the image."
],
[
"VCC 1972",
"The spiral galaxy VCC 1972 is detected as AGESVC2 022.",
"The Hi appears to be entirely associated with the spiral rather than the nearby elliptical galaxy VCC 1978, which is at 300 km s$^{-1}$ lower velocity.",
"While the main extension(s) appear to point away VCC 1978, a possible weak extension heads in the opposite direction.",
"The nature of the extended emission is somewhat ambiguous as it is unclear whether this represents a stream or noisy contours.",
"This galaxy is shown in figure REF with the contour at 4.5$\\sigma $ over the velocity range 1,205 - 1,572 km s$^{-1}$ .",
"Figure: Possible stream(s) associated with AGESVC2 022.",
"The Hi detection is associated with the spiral galaxy VCC 1972, which is interacting with the giant elliptical galaxy VCC 1978."
]
] | 2001.03385 |
[
[
"Towards Minimal Supervision BERT-based Grammar Error Correction"
],
[
"Abstract Current grammatical error correction (GEC) models typically consider the task as sequence generation, which requires large amounts of annotated data and limit the applications in data-limited settings.",
"We try to incorporate contextual information from pre-trained language model to leverage annotation and benefit multilingual scenarios.",
"Results show strong potential of Bidirectional Encoder Representations from Transformers (BERT) in grammatical error correction task."
],
[
"Introduction",
"The goal of grammatical error correction is to detect and correct all errors in the sentence and return the corrected sentences.",
"Current grammatical error correction approaches require a large amount of training data to achieve reasonable results, which unfortunately cannot be extended to languages with limited data.",
"Recently, unsupervised models pre-trained on large corpora have boosted performance in many natural language processing tasks, which indicates a potential of leveraging such models for GEC in any language.",
"In this work, we try to utilize (multilingual) BERT [3] in order to perform grammatical error correction with minimum supervision."
],
[
"Proposed Method",
"Our approach divides the GEC task into two stages: error identification followed by correction.",
"In the first stage, we try to detect the span in the original text that the edit will apply.",
"We formulate this as a sequence labelling task where tokens are labelled in one of the following labels {remain, substitution, insert, delete}.",
"For the second stage (correction) we employ a pretrained-model like BERT.",
"The labels from the error identification stage guide the masking of the inputs (where we mask any tokens marked with substitution or insert mask tokens for insert labels), and we obtain candidate outputs for every masked token.",
"A shortcoming of our current approach is that it only produces as many corrections as masked input tokens; however, most grammar errors in fact tend to be edits of length 1 or 2, which are captured by our identification labels.",
"The second stage is addressed in future work."
],
[
"Results",
"We report preliminary results on the test part of the English FCE dataset [7].",
"The edits are labelled and scored through ERRANT [1].",
"For our preliminary experiments we focus on a simplified single-edit setting, where we attempt to correct sentences with a single error (assuming oracle error identification annotations).",
"The goal is to assess the capabilities of pre-trained BERT-like models assuming perfect performance for all other components (e.g.",
"error identification).",
"We expand the original dataset to fit our single-error scenario with the following two schemes.",
"(1) each edit: all corrections except one are applied, creating a single-error sentence; (2) last edit: all corrections except for the last one are applied.",
"After the split, we obtain 3,585 and 1,024 corrections respectively.",
"We also employ different strategies for deciding the number of masked tokens: (1) based on span length of the original edit, (2) based on length of the final correction (given from an oracle), and (3) using a single mask and measure whether any token of the correction is predicted.",
"Note that subword predictions like {ad, ##e, ##quate} to adequate are merged in sentence-level, but remains in mask-level evaluation.",
"Our preliminary results under the various settings are outlined in Table REF .",
"We find that different masking strategies have comparable performance, with slightly higher accuracy when using a single mask.",
"Interestingly, BERT seems capable to actually produce corrections with quite high precision of more than 70%.",
"Table: Sentence-level evaluation with different masking strategies for single edit pairs.",
"Subword predictions are merged in sentence generation.Table: A reranking mechanism could lead to better results, as performance@5 is higher than [email protected] also study whether the correct output is among the top-$k$ candidates suggested by BERT.",
"We compare the $F_{0.5}$ metrics based on each mask between the most probable prediction and the top k candidates, where k is set to 5.",
"From the result in Table REF , we observe that an appropriate reranking model could further boost the performance by selecting the appropriate correction."
],
[
"Related and Future work",
"A retrieve-edit model is proposed for text generation [5].",
"However, the edition is one-time and sentences with multiple grammatical errors could further reduce the similarity between the correct form and the oracle sentence.",
"An iterative decoding approach [4] or the neural language model [2] as the scoring function are employed for GEC.",
"To the best of our knowledge, there is no prior work in applying pre-trained contextual model in grammatical error correction.",
"In the future work, we will additionally model error fertility, allowing us to exactly predict the number of necessary [MASK] tokens.",
"Last, we will employ a re-ranking mechanism which scores the candidate outputs from BERT, taking into account larger context and specific grammar properties."
],
[
"Better span detection",
"Although BERT could predict all the missing token in the sentence in a reasonable way, prediction of the correct words could easily fall into redundant editing.",
"Our experiment shows that simply rephrasing the whole sentence using BERT would lead to too diverse an output.",
"Instead, a prior error span detection could be necessary for efficient GEC, and it is part of our future work."
],
[
"Partial masking and fluency measures",
"Multi-masking or masking an informative part in the sentence will lead to loss of original information, and it will allow unwanted freedom in the predictions; see Table REF for examples.",
"Put in plain terms, multi-masking allows BERT to get too creative.",
"Instead, we will investigate partial masking strategies [8] which could alleviate this problem.",
"Fluency is an important measure when employing an iterative approach [6].",
"We plan to explore fluency measures as part of our reranking mechanisms."
]
] | 2001.03521 |
[
[
"A Policy-oriented Agent-based Model of Recruitment into Organized Crime"
],
[
"Abstract Criminal organizations exploit their presence on territories and local communities to recruit new workforce in order to carry out their criminal activities and business.",
"The ability to attract individuals is crucial for maintaining power and control over the territories in which these groups are settled.",
"This study proposes the formalization, development and analysis of an agent-based model (ABM) that simulates a neighborhood of Palermo (Sicily) with the aim to understand the pathways that lead individuals to recruitment into organized crime groups (OCGs).",
"Using empirical data on social, economic and criminal conditions of the area under analysis, we use a multi-layer network approach to simulate this scenario.",
"As the final goal, we test different policies to counter recruitment into OCGs.",
"These scenarios are based on two different dimensions of prevention and intervention: (i) primary and secondary socialization and (ii) law enforcement targeting strategies."
],
[
"Introduction",
"Organized crime (OC hereinafter) is present in a wide number of countries all over the world , , , , .",
"Besides the peculiar differences between different criminal organisations, they all pose social, economic and security challenges to societies, institutions and legal economies.",
"OCGs are capable to attract people and strengthen their presence on a territory.",
"Human workforce is a crucial asset for OCGs: a higher number of affiliates and members is an indicator of the actual resources at their disposal and a measure of the potential strength of the groups itself.",
"Understanding how recruitment works, then, becomes a relevant issue both from the research and the policy standpoints.",
"Scientific inquiry can help in pursuing this goal.",
"However, investigating recruitment dynamics at scale is extremely costly in a real-world setting, involving first and foremost feasibility constraints.",
"Additionally, it also poses ethical questions that are often present when dealing with social experiments.",
"Computer simulations, conversely, can overcome these issues.",
"With this regard, their potential has also started to be investigated in criminology , , , .",
"In light of these considerations, this paper presents the rationale and structure of an agent-based model of recruitment into OC.",
"To best of our knowledge, this is the first simulation model that specifically integrates computational science and criminology to address the problem of recruitment into OC.",
"The ABM will resemble a neighbourhood of Palermo, the main city of Sicily: our artificial society will comprise 10,000 agents with socio-economic, demographic and criminal characteristics derived from empirical data.",
"The society will be modelled through a multiplex network structure.",
"The investigation of recruitment dynamics will be coupled with the final goal of the simulation, which is the testing of potential policies to prevent or reduce recruitment into OCGs, especially for youth.",
"The rest of the paper is organised as follows: the theoretical framework part will briefly cover the main theories that constitute the criminological backbone of the model.",
"The Data section will describe the empirical information used in the ABM and the sources from which data have been extracted.",
"The Main Structure section will thoroughly explain the rationale of the multiplex approach and the two main components of the model.",
"Finally, in the Policy Scenarios section, the two proposed families of counter-policies that respectively deal with socialisation and network disruption will be presented."
],
[
"Theoretical Framework",
"The rationale of our simulated society is rooted in the outcomes of a recent systematic review on the factors leading to recruitment into OC .",
"Furthermore, it also takes into account several theoretical perspectives.",
"Some of these suggest that organised crime is embedded in the social environment and that social relations are crucial for the recruitment into organised crime.",
"With this regard, differential association theory and social learning theory , , posit that crime in its various aspects is learned in a social environment by relating with other criminal agents.",
"Empirical studies have also highlighted that OC is socially and criminally embedded in the surrounding environment , .",
"The position agents occupy within a criminal network determines their possibilities to commit crimes.",
"In this sense, an agent's valuable criminal ties determine his social opportunity structure .",
"Conversely, other theories such as the general theory of crime argue that an individual's low self-control levels determine an inability to compute the negative consequences of one's criminal behaviour, thereby determining persisting patterns of criminality throughout his life.",
"The general theory of crime contends that group crime does not have specific characteristics and that the formation of criminal groups is mostly driven by self-selection processes.",
"The social relations (i.e., social learning and differential association) and self-control (i.e., general theory of crime) perspectives may generate opposing views about the recruitment into organised crime: however, a more comprehensive explanation of criminal activity could be reached via the combination of elements of both frameworks.",
"With this regard, the development of an agent-based model is a convenient way to do so, since its flexibility can allow to integrate both personal and inter-personal components.",
"In light of this, the present model operationalizes criminal involvement both as the result of interaction with others and as emerging from agents’ individual characteristics pushing towards crime.",
"Using empirical data to feed the simulations is fundamental when aiming at setting up a reasonable and grounded model, besides theoretical and formal mechanisms (e.g., the mechanism of crime commission).",
"Furthermore, data shall also be used ex-post to assess whether the model produces reliable and plausible results, especially considering the policy-oriented objectives of the ABM.",
"To develop the model and validate the results, we have retrieved and processed several data from different sources regarding specific demographic, economic, social and criminal aspects.",
"We have chosen the city of Palermo as the specific setting to be resembled by the simulation model, as Palermo is one of the cities with the highest mafia presence in Italy (Table REF )."
],
[
"A Multiplex-Network Approach",
"Simulating the dynamics and processes that lead to the recruitment into OC requires to take into account a wide variety of factors.",
"While certain elements are inherently linked to the individual sphere (e.g.",
"age, gender), others span over the personal characteristics of an agent: making new friends, for instance, is dependent upon the social environment in which an agent is set.",
"Two children at the elementary school are more likely to become friends if they are in the same classroom or if they have the same age, rather than being separate into different classrooms or belong to different years.",
"In real life, every person engages in different types of relations, e.g.",
"as part of a family, in friendships, at work, and - if criminals - in co-offending.",
"A member of an OCG is also part of a wider social environment as embedded in multiple social worlds.",
"The literature proves that relations of different types may drive the involvement and recruitment into organised crime , , .",
"To adequately address the dynamics of individual and social drivers, we opted for an ABM based on a multiplex rationale.",
"A multiplex network includes several networks, each mapping specific social relations.",
"Five relational layers are modelled in the simulations: families, friendship networks, professional and school ties, criminal relations and organised crime groups (Figure REF ).",
"Figure: Graphic Depiction of the Multiplex Network StructureConsidering the need for creating a society that synthetically mirrors the real-world, we have decided to include all the main dimensions through which an individual can realistically act and behave.",
"The structure, topology and characteristics of the networks are empirically grounded using official statistics or replicating mechanics found in existing scientific works.",
"The multiplex network framework also allows for considering individual-level characteristics as agents’ attributes, thus making possible to analyse individual and social factors to simulate realistic recruitment dynamics.",
"Agents in the simulation, regardless of being or not part of an OC groups, can be born, get engaged/married, make children, die, create and break relations, and commit crimes.",
"Specifically focusing on the organised crime dimension, the model considers one single OCG existing in the simulation.",
"Its internal structure, composition (in terms of gender distribution and generation/age distribution) reflects the ones found through analyses of several police investigations on Italian OC groups."
],
[
"Recruitment into Organised Crime",
"For the purpose of this simulation, recruitment occurs when an agent commits a crime with at least another agent who is already a member of the OCG.It has to be noted that the model is initialised with a certain number of agents estimated based on data retrieved from criminal investigation, as showed in Table REF .",
"This option was driven by different considerations.",
"First, it is observable and easily operationalizable.",
"Requiring the commission of a crime with OCG members models in a straightforward way the process of recruitment, avoiding subjective evaluations.",
"Second, it is broadly consistent with the criminal law approaches criminalising organised crime across countries .",
"Two different complementary dimensions contribute to the determination of the recruitment processes in the model, namely the probability of committing a crime (called $C$ ) and the embeddedness into organised crime (called $R$ ) (Figure REF ).",
"Figure: Synthetic Description of the General Structure of the Model"
],
[
"Modelling Crime Commission: the “$C$ ” Function",
"The $C$ function models the probability that agent $i$ will commit a crime at time $t$ .",
"Its structure revolves around two components: the probability of committing a crime given gender and age class and the risk of committing a crime associated with several social and criminal factors.",
"The first component has been derived estimating baseline probabilities of committing a crime in Sicily using official statistics.",
"The probabilities are computed using both the figures on reported crimes and data gathered from victimisation surveys.",
"This complementary computation allows to eliminate the problem of dark figure.",
"The dark figure refers to the difference between the real number of crimes and the actual reported number of crimes .",
"In the model, we have thus reliably estimated the actual real number of crimes occurred in Palermo in the period 2012-2016.",
"Baseline probabilities are reported in Table REF .",
"Table: Gender and age class probabilities ((C|θ(g,a)(C|\\theta (g,a)) and Odds Ratios of committing a crime in PalermoThe second component originates from empirical findings in the existing literature.",
"Specifically, several systematic reviews providing information on the impact of certain factors on the general probability of committing a crime.",
"These sources provide effect sizes (in different forms, e.g.",
"odds ratios) allowing to determine the different probabilities of coming a crime given an agent’s network and individual characteristics.",
"The list of individual factor-based rules that drive the model rules for committing a crime is presented in Table REF .",
"Table: Conclusions"
]
] | 2001.03494 |
[
[
"A simple estimation of the size of the molecules using a pencil lead"
],
[
"Abstract The main aim of this article is that students, at the basic level of education, gain a quantitative understanding of the size of molecules by performing a simple experiment easily designed within the classroom."
],
[
"A simple estimation of the size of the molecules using a pencil lead Ricardo Medel Esquivel [email protected] Isidro Gómez Vargas Instituto Politécnico Nacional, CICATA-Legaria, Ciudad de México, CP 11500, México.",
"J. Alberto Vázquez Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, 62210, México Ricardo García Salcedo Instituto Politécnico Nacional, CICATA-Legaria, Ciudad de México, CP 11500, México.",
"One of the main topics of elementary physics is the idea that every material is composed of \"little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one other\"[1].",
"These particles could be atoms or molecules.",
"Atoms are the smallest part into which any material can be divided.",
"Whereas when several atoms are joined together, molecules are formed.",
"Some interesting experiments to estimate the size of such atoms or molecules have been done that do not involve sophisticated equipment.",
"One of these early experiments was conducted by Lord Rayleigh (1842-1919), which consisted of a small drop of oil spread to form a circular patch on the surface of the water.",
"With a few simple calculations it is possible to determine the size of the oil molecule composition and therefore to provide an estimate of the diameter of the carbon atom[2], [3].",
"The main aim of this article is that students, at the basic level of education, gain a quantitative understanding of the size of molecules by performing a simple experiment easily designed within the classroom.",
"All they need is a pencil lead, millimeter paper and a measuring instrument.",
"Of course, we assume that all molecules are approximately the same size[4].",
"The pencil lead is composed of graphite molecules (the fourth most abundant chemical element in the Universe [5]), which we can imagine as identical spherical particles.",
"These molecules form the macroscopic structure of the pencil, which has the shape of an elongated cylinder and hence its volume is given by: $V_C=\\pi R^2 H,$ where $R$ is the radius of the pencil lead and $H$ its height as we can see in Figure REF .",
"If we draw a line on a sheet of millimeter paper, keeping the pencil straight, part of the pencil material will have moved to the paper matrix, forming a very thin parallelepiped or tiny height box, whose volume is given by: $V_B= 2RLh,$ where $L$ is the length of the line, $2R$ is the width of the line and $h$ is the height of the box.",
"While the volume of graphite spent after drawing the line is: $V_C^{\\prime }=\\pi R^2 (H-H^{\\prime }).$ Under this assumption the material is deposited entirely on the surface of the paper, without loss, then we can match the volume spent on the pencil lead would be equal to the volume deposited on the paper.",
"See Figure REF .",
"Figure: It is assumed that the volume of the graphite cylinder is distributed in nn identical boxes, whose height hh gives us an estimate of the maximum size a molecule could have.",
"In detail, it is considered that hh is not the height of a single molecule but of many.And to make the effect more visible, it is possible to draw $n$ equal lines, of known length.",
"Then, the following equality is satisfied $V_C^{\\prime }=V_B$ , therefore from (REF ) and (REF ): $h=\\pi \\frac{R(H-H^{\\prime })}{2nL}.$ This height $h$ can be considered as an upper bound for the size of the graphite molecules, considered that is not the height of a single molecule but of many.",
"Taking $n$ large enough to significantly reduce the length of the pencil lead is possible to calculate this estimate numerically.",
"According to the manufacturer's specifications the HB pencil lead have $H = 60$ mm and $2R = 0.5$ mm (this is a popular standard measure, however there are other presentations with different length and diameter).",
"Figure: It is shown the \"lead\" (A) before drawing the lines and the \"lead\" (B) after draw some lines in the millimeter paper.",
"The measure of what has been consumed of graphite is that it is used to calculate the volume.We found that for this experiment it is easier to draw short lines, about 10 cm long, in the central columns of the millimeter paper, starting from above.",
"We performed the test with $n=50$ y $L=100$ mm (Fig.",
"2), and obtained that $H^{\\prime }=59.5$ mm, and therefore $h=\\pi \\frac{0.25mm(0.5mm)}{2(50)(100mm)}=0.000039 mm = 3.9 \\times 10^{-8}m.$ This result is reasonable as a higher level for the size of the molecules, the individual size of the so-called graphene Graphene is a single layer of carbon atoms arranged in an hexagonal lattice, with one carbon atom at each vertex.",
"sheets has been systematically measured and varies from 2 to 20 nm ($2\\times 10^{-9}m$ to $2 \\times 10^{-8} m$ ) [7].",
"The order of magnitude of the result does not change significantly when the number of lines drawn or their length is increased.",
"Similar results can be found by performing analog experiments to the one presented here, although those are more sophisticated and require more equipment to carry them out [6]."
]
] | 2001.03492 |
[
[
"Guesswork with Quantum Side Information"
],
[
"Abstract What is the minimum number of guesses needed on average to correctly guess a realization of a random variable?",
"The answer to this question led to the introduction of the notion of a quantity called guesswork by Massey in 1994, which can be viewed as an alternate security criterion to entropy.",
"In this paper, we consider the guesswork in the presence of quantum side information, and show that a general sequential guessing strategy is equivalent to performing a single measurement and choosing a guessing strategy from the outcome.",
"We use this result to deduce entropic one-shot and asymptotic bounds on the guesswork in the presence of quantum side information, and to formulate a semi-definite program (SDP) to calculate the quantity.",
"We evaluate the guesswork for a simple example involving the BB84 states, both numerically and analytically, and prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion."
],
[
"Introduction",
"Information theory, among other things, concerns the security of messages against attacks by malicious agents.",
"Conventionally, it is accepted that that the more unpredictable a message is, and the higher the (Shannon) entropy of the distribution from which it is drawn, the more secure it is to brute force attacks.",
"Therefore, when establishing a secret key or a cipher, the gold standard is to choose a key whose elements are picked uniformly at random from some alphabet.",
"Entropy, however, is not the only such criterion for security.",
"Another relevant quantity, which is also maximized by messages drawn uniformly, is the guesswork.",
"First put forth by Massey [1], the quantity is operationally described by the following guessing game.",
"Consider the problem of guessing a realization of a random variable $X$ , taking values in a finite alphabet $\\mathcal {X}$ , by asking questions of the form “Is $X=x$ ?”.",
"The guesswork $G(X)$ is defined as the minimum value of the average number of questions of this form that needs to be asked until the answer is “yes”.",
"That is, $ G(X) = \\sum _{k=1}^{| \\mathcal {X}|} k \\cdot p_G(k)$ where $p_G(k)$ is the probability of the $k$ th guess being correct.",
"In the real world, questions of this form arise from query access to a resource; for example, if a hacker is attempting to guess a user's password on an online portal, he or she can only ask this kind of question (as opposed to, say, “Is $X \\ge x$ ?”) and is allowed a limited number of guesses before being locked out.",
"Therefore, for someone setting up a password, the number of guesses allowed by the portal provides the operational security criterion against which his or her password must compare.",
"In contrast, the entropy of a distribution is approximately the minimum value of the average number of guesses required to obtain the correct guess when one is allowed to ask questions of the form, “Is $X \\in \\widetilde{\\mathcal {X}}?$ \", where each $\\widetilde{\\mathcal {X}}$ is some subset of the alphabet $\\mathcal {X}$ [2].",
"Qualitatively speaking, entropy can be considered to be the query complexity of a binary search-type algorithm, whereas guesswork corresponds to the query complexity of a linear search-type algorithm [3].",
"Figure REF illustrates this difference.",
"It is well known that binary search has a smaller complexity than linear search, which leads to the simple claim that the entropy of a distribution is always less than the guesswork.",
"Massey [1] proved the following stronger lower bound on the guesswork in terms of the Shannon entropy of the random variable $X$ : $G(X) \\ge \\frac{1}{4} 2^{H(X)} + 1 ,$ provided that $H(X) \\ge 2$ bits.",
"Figure: Consider an unknown value of XX in the set {x 1 ,x 2 ,x 3 ,x 4 ,x 5 }\\lbrace x_1, x_2, x_3, x_4, x_5\\rbrace .",
"The search tree for finding the correct value of XX when asking questions of the form (a) “Is X∈𝒳 ˜?",
"X \\in \\tilde{\\mathcal {X}}?” or of the form (b) “Is X=x?X = x?” The former search tree, similar to that in binary search, has fewer branches and the relevant operational quantity is the entropy of the prior distribution.",
"The specific question we ask in this tree is as follows: “Is X∈𝒳 ˜={x 1 ,x 2 }X \\in \\tilde{\\mathcal {X}} = \\lbrace x_1, x_2\\rbrace ?” The latter tree, similar to that of linear search, characterizes the scenario of guesswork.",
"The tree encapsulates a strategy in which one sequentially guesses x 1 ,x 2 x_1, x_2, and so on, until x 5 x_5.Massey [1] also showed that the optimal algorithm to minimize the guesswork, and even the positive moments of the number of guesses, consists of the intuitive strategy of simply guessing elements in decreasing order of their probability of occurrence.",
"Guesswork can also be considered in the presence of classical side information given in the form of a random variable $Y$ that is correlated with $X$ .",
"In this case, the guesswork is the minimal number of questions of the form “Is $X=x$ ?” that is required on average to obtain the correct answer, given the value of $Y$ .",
"The average is taken over the number of guesses, with each guess number weighted by its probability of being correct.",
"That is, $G(X|Y=y) = \\sum _{k=1}^{| \\mathcal {X}|} k \\cdot p_G(k)$ where in this case, $p_G(k)$ is the probability of the $k$ th guess being correct, given that $Y=y$ .",
"The optimal guessing strategy is simply to guess in decreasing order of conditional probability $p_{X|Y}(x|y)$ .",
"Arikan [4] obtained upper and lower bounds on the guesswork and its positive moments, in this scenario as well as in the case without side information.",
"Further work on guesswork in the classical setting has been done in [5], [6], [7], [8], [9], [10], [11], [12].",
"In this paper, we consider a natural generalization of the above guessing problem to the case in which the classical side information is replaced by quantum side information.",
"This generalization was first considered in [13].",
"In this case, the guesser (say, Bob) holds a quantum system $B$ , instead of a classical random variable (or equivalently, a classical system) $Y$ .",
"Here, the joint state of $X$ and $B$ is given by a classical-quantum (c-q) state, which we denote as $\\rho _{XB}$ (see sec:problemstatement for details).",
"Suppose that Alice possesses the classical system $X$ containing the value of the letter $x$ , which is to be guessed by Bob.",
"We define guesswork in the presence of quantum side information to be the minimum number of guesses needed, on average, for Bob to correctly guess Alice's choice, by performing a general sequential protocol, as follows.",
"Bob acts on his system $B$ with a quantum instrument, yielding a classical outcome $\\hat{x}$ which he guesses, as well as a post-measurement state on system B.",
"If his guess is incorrect, he performs another instrument on his system $B$ (possibly adapted based on his previous guess), and repeats the protocol until he either guesses correctly or runs out of guesses (which might be the case if he is allowed a limited number of guesses $K < |\\mathcal {X}|$ ).",
"While the case of classical side information admits a very simple optimal strategy (which amounts to simply sorting the conditional probabilities $p_{X|Y}(\\cdot |y)$ in non-increasing order and guessing accordingly), the quantum case requires measurement on the quantum system $B$ , which potentially disturbs the state of $B$ , a priori complicating the analysis of the sequence of guesses in the optimal strategy.",
"We show in sec:quantumstrategies, however, that a general sequential strategy is in fact equivalent to performing a single generalized measurement yielding a classical random variable $Y$ of outcomes and then doing the optimal strategy using this $Y$ as the classical side information.",
"The earlier work [13] instead defined guesswork with quantum side information as the latter quantity, i.e., a measured version of the guesswork in the presence of classical side information.",
"We prove here that these definitions are equivalent, and thus there is ultimately no difference between them, but we consider the definition in terms of a sequential protocol to be a more natural one.",
"Moreover, the above-mentioned equivalence is proved via an explicit construction, allowing such a guessing strategy to be implemented sequentially.",
"The single-measurement protocol could in general involve making a measurement with exponentially (in $|\\mathcal {X}|$ ) many outcomes.",
"Hence it may be more efficient to implement it instead as a sequence of (linearly-many) measurements with linearly-many outcomes, as allowed by the above construction.",
"Moreover, we consider a slight generalization of the guesswork in which Bob may make only $K \\le |\\mathcal {X}|$ guesses in total, and in which the “cost” of needing to make $k$ guesses is given by a vector $\\vec{c} = (c_1,c_2, \\cdots , c_K)$ , which could be different from $(1,2, \\cdots , K)$ , the latter of which corresponds to the expected value.",
"These generalizations can be better models for certain situations; in the password-guessing example, e.g., Bob may be locked out after $K$ guesses and hence is limited to a small number of guesses, or perhaps one has to wait after each guess before making another, and the time that one waits increases with the number of incorrect guesses.",
"We show that this generalized situation (including the guesswork as a special case) admits a semi-definite programming (SDP) representation in which the number of variables scales as $|\\mathcal {X}|^K$ , and hence smaller values of $K$ yield smaller problems and better scaling with $|\\mathcal {X}|$ .",
"See sec:SDP for more on the computational aspects of the guesswork.",
"One can consider a related task, in which one wishes to maximize what is known as the “guessing probability” $p_{\\text{guess}}(X|B)$ [14].",
"In this case, the guesser is given only one attempt to guess the value of $X$ (and hence is free to perform any arbitrary measurement on his system $B$ ).",
"The guessing probability is related to the so-called conditional min-entropy $H_\\text{min}(X|B)$ of the c-q state $\\rho _{XB}$ .",
"In some sense, we can consider the guesswork to be an extension of the guessing probability.",
"However, the nature of the optimization being done is different: instead of maximizing the probability of success in one attempt, we minimize the total number of guesses required.",
"Therefore, the operations that a guesser performs to minimize the guesswork may be very different from those needed to maximize the guessing probability.",
"Some of the connections between these two tasks have been investigated in [13].",
"In sec:problemstatement, we formally describe the task of guesswork with quantum side information.",
"In sec:strategies, we define classical and quantum guessing strategies in a unified framework, and thm:ordered-from-adaptive states that three classes of quantum strategies are equivalent.",
"In sec:entropicbounds, we establish one-shot and asymptotic entropic bounds on the guesswork, using analogous bounds developed by Arikan for the case of classical side information [4].",
"In sec:SDP, we revisit the idea that the guesswork may be formulated as a semi-definite optimization problem (SDP) (originally discussed in [13]), and we use such a representation to prove that the guesswork in a concave function (see sec:concavity) and a Lipschitz continuous function (see sec:Lipschitz).",
"We discuss the dual formulation of the SDP in sec:dual, a resulting algorithm to efficiently compute upper bounds in sec:ubalgo, and we present a mixed-integer SDP representation in sec:misdp.",
"sec:example shows two simple examples of the guesswork involving the BB84 states and the Y states, and sec:security provides a robustness result for using guesswork as a security criterion."
],
[
"Statement of the problem",
"Alice chooses a letter $x\\in \\mathcal {X}$ with some probability $p_X(x)$ , where $\\mathcal {X}$ is a finite alphabet.",
"This naturally defines a random variable $X\\sim p_X(x)$ .",
"She then sends a quantum system $B$ to Bob, prepared in the state $\\rho _B^{x}$ , which depends on her choice $x$ .",
"Bob knows the set of states $\\lbrace \\rho _B^x : x \\in \\mathcal {X}\\rbrace $ , and the probability distribution $\\lbrace p_X(x) : x \\in \\mathcal {X}\\rbrace $ , but he does not know which particular state is sent to him by Alice.",
"Bob's task is to guess $x$ correctly with as few guesses as possible.",
"From Bob's perspective, he therefore has access to the $B$ -part of the c-q state $\\rho _{XB} := \\sum _x p_X(x)|x\\rangle \\!\\langle x|_X \\otimes \\rho _B^x.$ In the purely classical case, this task reduces to the following scenario: Alice holds the random variable $X\\sim p_X(x)$ , and Bob holds a correlated random variable $Y$ and knows the joint distribution of $(X,Y)$ .",
"In this case $\\rho _{XB}$ reduces to the state $ \\rho _{XY} = \\sum _x p_X(x)|x\\rangle \\!\\langle x|_X \\otimes \\sum _y p_{Y|X}(y|x) |y\\rangle \\!\\langle y|_Y.$ In this case, if Bob's random variable $Y$ has value $y$ , then an optimal guessing strategy is to sort the conditional distribution $p_{X|Y}(\\cdot |y)$ in non-increasing order so that $p_{X|Y}(x_1 | y) \\ge p_{X|Y}(x_2 | y) \\ge \\cdots \\ge p_{X|Y}(x_{|\\mathcal {X}|} | y)$ and simply guess first $x_1$ , then $x_2$ , etc., until he gets it correct [4].",
"We note that in the absence of side information, Bob simply guesses in non-increasing order of $p_X(\\cdot )$ .",
"In the case in which Bob's system $B$ is quantum, he is allowed to perform any local operations he wishes on $B$ , and then make a first guess $x_1$ .",
"He is told by Alice whether or not his guess is correct; then he can perform local operations on $B$ , and make another guess, and so forth.",
"We are interested in determining the minimal number of guesses needed on average for a given ensemble $\\lbrace p_X(x), \\rho _B^x \\rbrace _{x\\in \\mathcal {X}}$ and the associated optimal strategy.",
"More generally, we allow Bob to make $K$ guesses, with possibly $K < |\\mathcal {X}|$ .",
"Formally, we assume that Bob always makes all $K$ guesses; any guess after the correct guess simply does not factor into the calculation of the minimal number of guesses (see sec:strategies for a more detailed definition of the minimal number of guesses).",
"Thus, Bob makes a sequence of guesses, $g_1,\\cdots ,g_K \\in \\mathcal {X}^K_{\\ne }$ with some probability.",
"We could consider the scenario in which Bob makes a guess $x_1$ , then learns whether or not the guess was correct, and uses that information to make his second guess $x_2$ , and so forth.",
"However, if Bob learns that his $j$ th guess $x_j$ is correct, then it does not matter what he guesses subsequently (it has no bearing on the minimal number of guesses).",
"If the guess is incorrect, then his subsequent guesses do matter, and he should make his next guess accordingly.",
"Hence, in such a protocol, the feedback about whether or not the $j$ th guess is correct does not help, and Bob might as well assume that each guess is incorrect."
],
[
"Guessing strategies",
"When Alice chooses $x^* \\in \\mathcal {X}$ , a guessing strategy for Bob outputs a sequence of guesses $\\vec{g} = (g_1,\\cdots ,g_K) \\in \\mathcal {X}^K$ with some probability $p_{\\vec{G} | X}(\\vec{g} | x^*)$ .",
"Hence, formally, a guessing strategy for $X$ with $K$ guesses is a random variable $\\vec{G}$ on $\\mathcal {X}^K$ that is correlated with $X$ , such that the joint random variable $(X, \\vec{G})$ has marginal $X \\sim p_X$ .",
"Note that the definition of a guessing strategy has no reference to the side information (if any) to which Bob has access; instead, the side information dictates the set of guessing strategies to which Bob has access.",
"This allows various types of side information to be analyzed within a uniform framework; in particular, the set of strategies available when Bob has access to some classical side information $Y$ is described in sec:classicalstrategies, while the case of quantum side information is described in sec:quantumstrategies.",
"We are interested in the minimal number of guesses required to guess $x^*$ correctly with a fixed sequence of guesses $\\vec{g}$ .",
"This is defined as follows: $N(\\vec{g}, x^*) := \\\\{\\left\\lbrace \\begin{array}{ll}\\min \\left\\lbrace j : g_j = x^*\\right\\rbrace & g_j = x^* \\text{ for some }j =1,\\cdots , K\\\\\\infty & \\text{else},\\end{array}\\right.",
"}$ where the outcome $\\infty $ occurs when none of the $K$ guesses are correct.",
"We can view $N$ as a random variable taking values in $\\lbrace 1,2,\\cdots ,K, \\infty \\rbrace $ .",
"Given a guessing strategy $\\vec{G}$ , the quantity of interest is $N(\\vec{G}, X)$ , the corresponding random variable.",
"We define $\\mathcal {S}_K(X) := \\left\\lbrace N(\\vec{G}, X) : X\\vec{G} \\sim p_{X\\vec{G}} \\right\\rbrace $ to be the set of all possible random variables $N$ associated to all guessing strategies $\\vec{G}$ with $K$ guesses.",
"We say two guessing strategies $\\vec{G}$ and $\\vec{G}^{\\prime }$ for $X$ with $K$ guesses are equivalent if $N(\\vec{G}^{\\prime }, X) = N(\\vec{G}, X)$ .",
"Note that if $\\vec{G}$ and $\\vec{G}^{\\prime }$ are two strategies with $K$ guesses for $X$ that differ only in guesses made after guessing the correct answer, then they are equivalent.",
"This formalizes the notion introduced at the end of the previous section: since guesses made after the correct answer do not change the value of $N(\\vec{g}, x^*)$ , feedback of whether or not $g_j = x^*$ can only lead to equivalent strategies."
],
[
"Classical strategies",
"Consider a pair of random variables $(X,Y)$ where $X$ has a finite alphabet $\\mathcal {X}$ and $Y$ has a countable alphabet $\\mathcal {Y}$ .",
"Alice chooses $x^* \\in \\mathcal {X}$ (with probability $p_X(x^*)$ ) and Bob is given $y\\in \\mathcal {Y}$ (with probability $p_{Y|X}(y|x^*)$ ).",
"Bob's task is to guess $x^*$ .",
"Since Bob's sequence of guesses $(g_1,\\cdots ,g_K)$ can only depend on $x^*$ via $y$ , a classical guessing strategy $\\vec{G}$ is any random variable $\\vec{G}$ such that the ordered triple $(X,Y,\\vec{G})$ of random variables forms a Markov chain, which we denote as $X - Y - \\vec{G}$ .",
"Hence, given a joint probability distribution $p_{XY}$ , we define the set of random variables $N$ associated to classical guessing strategies as follows: $\\mathcal {S}_K^\\textnormal {Classical}(p_{XY}) := \\left\\lbrace N(\\vec{G}, X) : X - Y - \\vec{G} \\right\\rbrace \\subseteq \\mathcal {S}_K(X).$"
],
[
"Equivalence of quantum strategies",
"Let us consider three classes of quantum strategies: Measured strategy: Bob performs an arbitrary POVM $\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}$ on the $B$ system.",
"Let $Y$ be the random variable with outcomes in a finite alphabet $\\mathcal {Y}$ corresponding to his measurement outcomes, i.e.",
"$p_{Y|X}(y|x) = \\operatorname{tr}[E_{y} \\rho _B^x], \\quad \\forall x\\in \\mathcal {X},\\, y \\in \\mathcal {Y}.$ Bob then employs a classical guessing strategy on $(X,Y)$ .",
"The set of random variables corresponding to the possible number of guesses under such a strategy is given by $\\mathcal {S}_K^\\textnormal {Measured}(\\rho _{XB}) := \\Big \\lbrace N(\\vec{G}, X) : X - Y - \\vec{G}, \\\\ \\text{ $Y$ satisfies (\\ref {eq:Y-from-X}) for finite alphabet } \\mathcal {Y}\\\\ \\text{\\& POVM } \\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}} \\Big \\rbrace .$ We then observe that $\\mathcal {S}_K^\\textnormal {Measured}(\\rho _{XB}) \\subseteq \\mathcal {S}_K(X).$ Ordered strategy: Bob performs a measurement with outcomes in $\\mathcal {X}^K$ , which are identified with guessing orders; i.e., if the outcome is $(x_1,\\cdots ,x_K) \\in \\mathcal {X}^K$ , Bob first guesses $x_1$ , then $x_2$ , and so forth.",
"In this case, Bob performs a POVM $\\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K}$ and the guessing strategy $\\vec{G}$ is distributed according to $p_{\\vec{G} | X}(\\vec{g} | x) = \\operatorname{tr}[E_{\\vec{g}} \\rho _B^{x}].$ As above, we define $\\mathcal {S}_K^\\textnormal {Ordered}(\\rho _{XB}) := \\Big \\lbrace N(\\vec{G}, X) : (\\vec{G}, X) \\text{ satisfy} \\\\ \\text{(\\ref {eq:G-dist-ordered}) for some POVM } \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K}\\Big \\rbrace \\subseteq \\mathcal {S}_K(X).$ It is evident that $\\mathcal {S}_K^\\textnormal {Ordered}(\\rho _{XB}) \\subseteq \\mathcal {S}_K^\\textnormal {Measured}(\\rho _{XB})$ because any such ordered strategy is a special type of measured strategy (with $Y = \\vec{G}$ ).",
"However, any measured strategy can in fact be simulated by an ordered strategy.",
"Suppose we have a measured strategy with alphabet $\\mathcal {Y}$ , POVM $\\lbrace E_y\\rbrace _{y\\in \\mathcal {Y}}$ , and $\\vec{G}$ satisfying $X-Y-\\vec{G}$ .",
"Then $ p_{\\vec{G}|X}( \\vec{g}|x) &= \\sum _{y \\in \\mathcal {Y}} p_{\\vec{G}|Y}( \\vec{g}|y) p_{Y|X}(y | x) \\\\&= \\sum _{y \\in \\mathcal {Y}} p_{\\vec{G}| Y}(\\vec{g}|y) \\operatorname{tr}[E_{y} \\rho _B^x],$ where we have used the Markov property for the first equality and (REF ) for the second equality.",
"Let $\\tilde{E}_{\\vec{g}} := \\sum _{y\\in \\mathcal {Y}} p_{\\vec{G}|Y}(\\vec{g}|y) E_{y}$ .",
"Note $\\lbrace \\tilde{E}_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K}$ is a POVM: each element is positive semi-definite since $\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}$ is a POVM, and $\\sum _{\\vec{g} \\in \\mathcal {X}^K} E_{\\vec{g}} &=\\sum _{\\vec{g} \\in \\mathcal {X}^K} \\sum _{y\\in \\mathcal {Y}} p_{\\vec{G}|Y}(\\vec{g}|y) E_{y} \\\\&= \\sum _{y\\in \\mathcal {Y}} \\sum _{\\vec{g} \\in \\mathcal {X}^K}p_{\\vec{G}|Y}(\\vec{g}|y)E_{y} = \\sum _{y \\in \\mathcal {Y}} E_y = \\mathbf {1}_B,$ using again that $\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}$ is a POVM.",
"Then substituting the definition of $\\tilde{E}_{\\vec{g}}$ into (REF ) yields $p_{\\vec{G}|X}(\\vec{g}|x) = \\operatorname{tr}[\\tilde{E}_{\\vec{g}} \\rho _B^x]$ and hence (REF ) is satisfied with $E = \\tilde{E}$ .",
"Therefore, $\\mathcal {S}^\\textnormal {Ordered}(\\rho _{XB}) = \\mathcal {S}^\\textnormal {Measured}(\\rho _{XB}).$ Sequential quantum strategy: Suppose that Alice chooses $x$ (which occurs with probability $p_X(x)$ ), and hence Bob has the state $\\rho _B^x$ .",
"To make his first guess, Bob chooses a set of generalized measurement operators $\\lbrace M_x^{(1)}\\rbrace _{x\\in \\mathcal {X}}$ and reports the measurement outcome as his guess.",
"He gets outcome $x_1$ with probability $p_{G_1|X}(x_1|x)= \\operatorname{tr}[ M_{x_1}^{(1)} \\rho _B^x M_{x_1}^{(1)}{}^\\dagger ]$ and his post-measurement state is $\\frac{1}{p_{G_1|X}(x_1|x)} M_{x_1}^{(1)} \\rho _B^x M_{x_1}^{(1)}{}^\\dagger .$ Note: in general, Bob could perform a unitary operation $U_1$ on his state before measuring it.",
"However, this would simply correspond to measuring with $\\lbrace M_{x}^{(1)} U_1\\rbrace _{x \\in \\mathcal {X}}$ instead.",
"Hence, it suffices to simply consider a generalized measurement $\\lbrace M_x^{(1)}\\rbrace _{x\\in \\mathcal {X}}$ .",
"Then, after learning the outcome $x_1$ , Bob chooses a new set of generalized measurement operators $\\lbrace M_x^{(2 | x_1)}\\rbrace _{x\\in \\mathcal {X}}$ .",
"Note that this set of measurement operators can depend on $x_1$ .",
"Without loss of generality, we can keep the same outcome set $\\mathcal {X}$ , since Bob could set, e.g.",
"$M_{x_1}^{(2 | x_1)} = 0$ to avoid guessing the same number twice.",
"Bob measures his state and gets the outcome $x_2$ with probability $p_{G_2|G_1 X}(x_2|x_1, x) \\\\= \\frac{1}{p_{G_1|X}(x_1|x)} \\operatorname{tr}[M_{x_2}^{(2| x_1)} M_{x_1}^{(1)} \\rho _B^x M_{x_1}^{(1)}{}^\\dagger M_{x_2}^{(2| x_1)}{}^\\dagger ].$ Multiplying by $p_{G_1|X}(x_1|x)$ we see the joint distribution is given by $p_{G_1 G_2|X}(x_1, x_2|x) \\\\= \\operatorname{tr}[M_{x_2}^{(2| x_1)} M_{x_1}^{(1)} \\rho _B^x M_{x_1}^{(1)}{}^\\dagger M_{x_2}^{(2| x_1)}{}^\\dagger ].$ To make his $j$ th guess, we allow Bob to choose a new set of generalized measurement operators $\\lbrace M_x^{(j|x_1,\\cdots ,x_{j-1})}\\rbrace _{x\\in \\mathcal {X}}$ , which may depend on the previous $j-1$ outcomes.",
"Repeating the previous logic, in the end we find that $ p_{G_1 G_2 \\cdots G_K |X}(x_1, x_2,\\cdots , x_K|x) = \\\\\\operatorname{tr}[M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})} \\cdots M_{x_2}^{(2| x_1)} M_{x_1}^{(1)} \\rho _B^x \\\\M_{x_1}^{(1)}{}^\\dagger M_{x_2}^{(2| x_1)}{}^\\dagger \\cdots M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})}{}^\\dagger ].$ Under such a strategy, the possible random variables giving the number of guesses is given by $\\mathcal {S}^{\\textnormal {Sequential}}(\\rho _{XB}) := \\Big \\lbrace N(\\vec{G}, X) : \\\\(\\vec{G}, X) \\text{ satisfy (\\ref {eq:general_prob_G}) for some collections of} \\\\\\text{measurement operators } \\lbrace M_{x_j}^{(j| x_1, x_2,\\cdots ,x_{j-1})}\\rbrace _{x_j \\in \\mathcal {X}},\\\\\\, j =1,\\cdots ,K,\\,\\, x_1,x_2,\\cdots , x_K \\in \\mathcal {X}\\Big \\rbrace .$ Theorem 1 Let $\\rho _{XB}$ be a c-q state as defined in (REF ) and $K$ a natural number with $K \\le |\\mathcal {X}|$ .",
"Then $\\mathcal {S}^{\\textnormal {Sequential}}_K(\\rho _{XB}) = \\mathcal {S}^{\\textnormal {Ordered}}_K(\\rho _{XB}) = \\mathcal {S}^{\\textnormal {Measured}}_K(\\rho _{XB}).$ We see that all three sets of random variables of the number of guesses obtained from various classes of strategies all coincide.",
"Hence, we call the single class that of quantum strategies, denoted as $\\mathcal {S}^\\textnormal {Quantum}_K(\\rho _{XB})$ .",
"Proof.",
"The second equality was already stated in (REF ) and proven before that, and so it remains to prove the first equality.",
"Consider a sequential strategy, with the notation of point REF above.",
"Define $ E_{x_1,\\cdots ,x_K} := M_{x_1}^{(1)}{}^\\dagger M_{x_2}^{(2| x_1)}{}^\\dagger \\cdots M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})}{}^\\dagger \\\\M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})} \\cdots M_{x_2}^{(2| x_1)} M_{x_1}^{(1)} .$ We see that $E_{x_1,\\cdots ,x_K} = A^\\dagger A$ for $A=M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})} \\cdots M_{x_2}^{(2| x_1)} M_{x_1}^{(1)}$ , and hence it is positive semi-definite.",
"Moreover, $ \\sum _{x_1,\\cdots ,x_K \\in \\mathcal {X}} E_{x_1,\\cdots ,x_K} = I_{B}$ as can be seen by first summing (REF ) over $x_K$ , using $\\sum _{x_K \\in \\mathcal {X}} M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})}{}^\\dagger M_{x_K}^{(K| x_1, x_2,\\cdots ,x_{K-1})} = I_B$ since $\\lbrace M_{x}^{(K| x_1, x_2,\\cdots ,x_{K-1})}\\rbrace _{x\\in \\mathcal {X}}$ is a set of generalized measurement operators, and then similarly summing over $x_{K-1}$ , $x_{K-2}$ ,..., and finally $x_1$ .",
"Let us write $E_{\\vec{x}}$ where $\\vec{x}=(x_1,\\cdots ,x_K)$ for $E_{x_1,\\cdots , x_K}$ .",
"We have shown that $\\lbrace E_{\\vec{x}} \\rbrace _{\\vec{x} \\in \\mathcal {X}^K}$ is a POVM.",
"Moreover, $p_{G_1 G_2 \\cdots G_K |X}(x_1, x_2 ,\\cdots , x_K|x) = \\operatorname{tr}[E_{x_1,\\cdots ,x_K} \\rho _B^x].$ Hence, Bob's strategy is equivalent to simply performing the single POVM $\\lbrace E_{\\vec{x}} \\rbrace _{\\vec{x} \\in \\mathcal {X}^K}$ once, obtaining an outcome $\\vec{x} = (x_1,\\cdots , x_K)$ , and then making $x_1$ his first guess, $x_2$ his second guess, and so forth.",
"That is, any such strategy can be recast as an ordered strategy.",
"On the other hand, any such ordered strategy can be reformulated as an adaptive strategy, by the following recursive approach.",
"Suppose that we are given $\\left\\lbrace E_{\\vec{y}} \\right\\rbrace _{\\vec{y} \\in \\mathcal {X}^K}$ .",
"For each $x_1 \\in \\mathcal {X}$ , define $M^{(1)}_{x_1} = \\sqrt{\\sum _{x_2,\\cdots ,x_K\\in \\mathcal {X}} E_{x_1,\\cdots ,x_K}}$ where we have chosen the positive semi-definite square root.",
"We have that $\\sum _{x_1\\in \\mathcal {X}} M^{(1)}_{x_1}{}^\\dagger M^{(1)}_{x_1} &= \\sum _{x_1 \\in \\mathcal {X}} (M^{(1)}_{x_1})^2 \\\\&= \\sum _{x_1\\in \\mathcal {X}} \\sum _{x_2,\\cdots ,x_K \\in \\mathcal {X}} E_{x_1,\\cdots ,x_K} = I_B,$ and so $\\left\\lbrace M^{(1)}_{x_1} \\right\\rbrace _{x_1 \\in \\mathcal {X}}$ forms a set of generalized measurement operators with outcomes in $\\mathcal {X}$ .",
"Next, for each $x_1 \\in \\mathcal {X}$ , corresponding to obtaining outcome $x_1$ on the first measurement, we define measurement operators $\\lbrace M^{(2| x_1)}_{x_2}\\rbrace _{x_2 \\in \\mathcal {X}}$ by $M^{(2| x_1)}_{x_2} = \\sqrt{(M^{(1)}_{x_1})^{-1}\\sum _{x_3,\\cdots ,x_K \\in \\mathcal {X}} E_{x_1,\\cdots ,x_K}(M^{(1)}_{x_1})^{-1}}.$ Then $ &\\sum _{x_2\\in \\mathcal {X}} (M^{(2| x_1)}_{x_2})^2 \\\\& \\qquad = (M^{(1)}_{x_1})^{-1}\\sum _{x_2\\in \\mathcal {X}}\\sum _{x_3,\\cdots ,x_K\\in \\mathcal {X}} E_{x_1,\\cdots ,x_K}(M^{(1)}_{x_1})^{-1}\\\\& \\qquad = (M^{(1)}_{x_1})^{-1}(M^{(1)}_{x_1})^2(M^{(1)}_{x_1})^{-1}\\\\& \\qquad = I_B.$ Likewise, we define $M^{(3| x_1, x_2)}_{x_3} = \\Big \\lbrace (M^{(2| x_1)}_{x_2})^{-1}(M^{(1)}_{x_1})^{-1}\\sum _{x_4,\\cdots ,x_K\\in \\mathcal {X}} \\\\ E_{x_1,\\cdots ,x_K}(M^{(1)}_{x_1})^{-1}(M^{(2| x_1)}_{x_2})^{-1}\\Big \\rbrace ^{1/2}.$ Then $& \\sum _{x_3\\in \\mathcal {X}} (M^{(3| x_1, x_2)}_{x_3})^2 \\\\& = \\begin{multlined}[t][5cm] (M^{(2| x_1)}_{x_2})^{-1}(M^{(1)}_{x_1})^{-1}\\left( \\sum _{x_3\\in \\mathcal {X}}\\sum _{x_4,\\cdots ,x_K\\in \\mathcal {X}} E_{x_1,\\cdots ,x_K} \\right) \\\\ (M^{(1)}_{x_1})^{-1}(M^{(2| x_1)}_{x_2})^{-1}\\end{multlined} \\\\& = (M^{(2| x_1)}_{x_2})^{-1}(M^{(2| x_1)}_{x_2})^2 (M^{(2| x_1)}_{x_2})^{-1}\\\\&= I_B.$ Repeating this process, we define $ M^{(\\ell | x_1, x_2, \\cdots , x_{\\ell -1})}_{x_\\ell } \\\\= \\Big \\lbrace (M^{(\\ell -1| x_1, x_2, \\cdots , x_{\\ell -2})}_{x_{\\ell -1}})^{-1}\\cdots (M^{(1)}_{x_1})^{-1}\\\\\\sum _{x_{\\ell +1},\\cdots ,x_K\\in \\mathcal {X}} E_{x_1,\\cdots ,x_K} \\\\(M^{(1)}_{x_1})^{-1}\\cdots M^{(\\ell -1| x_1, x_2, \\cdots , x_{\\ell -2})}_{x_{\\ell -1}})^{-1}\\Big \\rbrace ^{1/2}$ to obtain a generalized measurement operator for step $\\ell $ (to use when having obtained outcomes $x_1,\\cdots , x_{\\ell -1}$ during the previous steps).",
"At the last step, $\\ell = K$ , there is no sum, namely $ M^{(K| x_1, x_2, \\cdots , x_{K-1})}_{x_K} \\\\= \\Big \\lbrace (M^{(K-1| x_1, x_2, \\cdots , x_K-2)}_{x_{K-1}})^{-1}\\cdots (M^{(1)}_{x_1})^{-1}E_{x_1,\\cdots ,x_K} \\\\(M^{(1)}_{x_1})^{-1}\\cdots M^{(K-1| x_1, x_2, \\cdots , x_K-2)}_{x_{K-1}})^{-1}\\Big \\rbrace ^{1/2}.$ Lastly, we check that by design, (REF ) holds.",
"Thus, we can work backwards from that equation and see that our newly created adaptive strategy yields the same outcomes with the same probabilities as the initial ordered strategy."
],
[
"Success metrics",
"Given a random variable $X$ and a maximal number $K$ of allowed guesses, how do we measure the success of a guessing strategy $\\vec{G}$ ?",
"We will focus on expectations of $N(\\vec{G}, X)$ .",
"In particular, we consider the expected number of guesses required to guess correctly: $ \\mathbb {E}[N(\\vec{G}, X)] = {\\left\\lbrace \\begin{array}{ll} \\sum _{k = 1}^K k\\cdot p_{N(\\vec{G}, X)}(k) & \\!\\!\\!\\text{if } p_{N(\\vec{G}, X)}(\\infty ) = 0\\\\\\infty & \\!\\!\\!\\text{if } p_{N(\\vec{G}, X)}(\\infty ) > 0.\\end{array}\\right.",
"}$ Here, $p_{N(\\vec{G}, X)}(k)$ is the probability that, for guessing strategy $\\vec{G}$ , the $k$ th guess is correct.",
"We also consider a general cost vector $\\vec{c}$ $ = \\lbrace c_1, c_2, \\cdots ,c_{| \\mathcal {X}|}\\rbrace $ $\\in (\\mathbb {R}\\cup \\lbrace \\infty \\rbrace )^{|\\mathcal {X}|}$ with $0 \\le c_1 \\le c_2 \\le \\cdots \\le c_{|\\mathcal {X}|}.$ Then we define the modified expectation $E_{\\vec{c}}(N(\\vec{G}, X)) := \\sum _{k = 1}^{|\\mathcal {X}|} c_k\\cdot p_{N(\\vec{G}, X)}(k).$ Imposing a maximal number $K < |\\mathcal {X}|$ of allowed guesses is equivalent to choosing $c_{K+1} = \\cdots = c_{|\\mathcal {X}|} = \\infty $ , using the convention $\\infty \\cdot 0 = 0$ .",
"Accordingly, we implicitly associate $K$ with $\\vec{c}$ in all the following via the rule that $K = |\\mathcal {X}|$ if and only if $c_{|\\mathcal {X}|}<\\infty $ , and otherwise $K=\\min \\lbrace i : c_i = \\infty \\rbrace $ .",
"The case $K=|\\mathcal {X}|$ therefore corresponds to $|\\mathcal {X}|$ guesses being allowed, each with finite cost, and the case $K < |\\mathcal {X}|$ corresponds to a limited number of allowed guesses, with a corresponding infinite cost if the correct answer is not obtained in $K$ guesses.",
"Given a c-q state $\\rho _{XB}$ and a cost vector $\\vec{c}$ as in (REF ), we define the generalized guesswork with quantum side information as $G_{\\vec{c}}(X|B)_\\rho := \\inf _{N \\in \\mathcal {S}_K^\\textnormal {Quantum}(\\rho _{XB})} E_{\\vec{c}}(N).$ Likewise, given a joint distribution $p_{XY}$ , let $G_{\\vec{c}}(X|Y)_p := \\inf _{N \\in \\mathcal {S}_K^\\textnormal {Classical}(p_{XY})} E_{\\vec{c}}(N).$ From the equality $\\mathcal {S}_K^\\textnormal {Quantum}(\\rho _{XB}, K) = \\mathcal {S}_K^\\textnormal {Measured}(\\rho _{XB})$ of thm:ordered-from-adaptive it follows that $ G_{\\vec{c}}(X|B)_\\rho = \\inf _{\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}} G_{\\vec{c}}(X|Y)_p$ where the infimum is over all finite alphabets $\\mathcal {Y}$ and POVMs $\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}$ and $p_{XY}(x, y) = p_X(x) \\operatorname{tr}[E_y \\rho _B^x]$ .",
"In the standard case in which $\\vec{c} = (1,2,\\cdots ,|\\mathcal {X}|)$ , we define the guesswork with quantum side information as $ G(X|B) \\equiv G(X|B)_\\rho := G_{\\vec{c}}(X|B)_\\rho $ and likewise define $G(X|Y)_p = G_{\\vec{c}}(X|Y)_p$ in the case of classical side information $Y$ .",
"Remark 2 In Ref.",
"[13], guesswork with quantum side information was defined by the right-hand side of (REF ) (with $\\vec{c} = (1,2,\\cdots ,|\\mathcal {X}|)$ ).",
"Moreover, Proposition 1 of that work shows that the infimum in (REF ) in that case may be restricted to POVMs whose elements are all rank one."
],
[
"Entropic bounds",
"In this section, we use the results of sec:strategies to obtain one-shot and asymptotic entropic bounds on $G(X|B)$ in terms of measured versions of bounds known in the classical case."
],
[
"One-shot bounds",
"In the case in which $K = |\\mathcal {X}|$ , Arikan [4] showed that $ \\frac{1}{1 + \\ln |\\mathcal {X}|} \\exp (H_{\\frac{1}{2}}^\\uparrow (X|Y)_p) \\le G(X|Y)_p \\le \\exp (H_{\\frac{1}{2}}^\\uparrow (X|Y)_p)$ where $H_{\\alpha }^\\uparrow (X|Y)_p$ for $\\alpha \\in (0,1)\\cup (1,\\infty )$ denotes the following $\\alpha $ -conditional entropy of a joint distribution $p_{XY}$ given by $ H_{\\alpha }^\\uparrow (X|Y) &= \\frac{\\alpha }{1-\\alpha }\\ln \\left( \\sum _{y \\in \\mathcal {Y}} \\left( \\sum _{x\\in \\mathcal {X}} p_{XY}(x,y)^\\alpha \\right)^{1/\\alpha } \\right) \\\\&= \\sup _{q_Y} \\left[- D_\\alpha ( p_{XY} \\Vert \\mathbf {1}_X \\otimes q_Y )\\right]$ where the supremum is over probability distributions $q_Y$ on $\\mathcal {Y}$ , and $D_\\alpha $ is the $\\alpha $ -Rényi relative entropy, $D_\\alpha (p_X \\Vert q_X) = \\frac{1}{\\alpha -1} \\ln \\left( \\sum _x p_X(x)^{\\alpha } q_X(x)^{1-\\alpha } \\right).$ The second equality of (REF ) follows from [15].",
"Arikan's bound (REF ) applies to each $G_{\\vec{c}}(p_{XY},K)$ in (REF ), and hence by minimizing over the POVMs $\\lbrace E_y\\rbrace _{y\\in \\mathcal {Y}}$ , we obtain $ \\frac{1}{1 + \\ln |\\mathcal {X}|} \\exp (H_{\\frac{1}{2}}^{\\uparrow , M}(X|B)_{\\rho }) &\\le G(X|B)_\\rho \\\\&\\le \\exp (H_{\\frac{1}{2}}^{\\uparrow , M}(X|B)_\\rho ),$ where for $\\alpha \\in (0,1)\\cup (1,\\infty )$ , $H_{\\alpha }^{\\uparrow , M}(X|B)_\\rho $ is the $B$ -measured conditional $\\alpha $ -Rényi entropy, defined by $H_{\\alpha }^{\\uparrow , M}(X|B)_\\rho := \\inf _{\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}} H_{\\alpha }^\\uparrow (X|Y)_p,$ where $p_{XY}(x,y) = p_X(x) \\operatorname{tr}[E_y \\rho _B^x]$ is the joint probability distribution obtained by measuring the $B$ part of $\\rho _{XB}$ via $\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}$ .",
"Remark 3 We may expand this quantity as $H_{\\alpha }^{\\uparrow , M}(X|B)_\\rho = \\inf _{\\lbrace E_y\\rbrace _{y \\in \\mathcal {Y}}}\\sup _{q_Y} \\left[ -D_{\\alpha }(p_{XY} \\Vert \\mathbf {1}_X \\otimes q_Y)\\right],$ where $p_{XY}$ is induced by the measurement of $\\rho _{XB}$ .",
"This quantity seems to be different from the conditional entropy induced by the measured Rényi divergence, namely $H_{D_\\alpha ^M}^\\uparrow (X|B)_\\rho := \\sup _{\\sigma _B}-D^M_\\alpha ( \\rho _{XB} \\Vert \\mathbf {1}_X \\otimes \\sigma _B),$ where the supremum is over states on the $B$ system, and for any pair of states $(\\rho , \\sigma )$ , $D^M_\\alpha (\\rho \\Vert \\sigma ) := \\sup _{\\lbrace E_z\\rbrace _{z}} D_\\alpha ( \\lbrace \\operatorname{tr}[E_z \\rho ]\\rbrace _{z} \\Vert \\lbrace \\operatorname{tr}[E_z \\sigma ]\\rbrace _{z} )$ is the measured $\\alpha $ -Rényi divergence.",
"Indeed, the latter quantity may be expanded to obtain $H_{D_\\alpha ^M}^\\uparrow (X|B)_\\rho \\\\= \\sup _{\\sigma _B} \\inf _{\\lbrace E_z\\rbrace _{z}} \\left[ -D_\\alpha ( \\lbrace \\operatorname{tr}[E_z \\rho _{XB}]\\rbrace _{z} \\Vert \\lbrace \\operatorname{tr}[E_z \\mathbf {1}_X \\otimes \\sigma _B]\\rbrace _{z} )\\right].$ From the min-max inequality, and the fact that collective measurements on $XB$ can simulate measurements on $B$ alone, we have $H_{D_\\alpha ^M}^\\uparrow (X|B)_\\rho \\le H_{\\alpha }^{\\uparrow , M}(X|B)_\\rho .$"
],
[
"Asymptotic analysis",
"We can consider the asymptotic setting in which Bob receives a sequence of product states $\\rho _B^{\\vec{x}} := \\rho _B^{x_1} \\otimes \\cdots \\otimes \\rho _B^{x_n}$ , with probability $p_X(x_1)\\cdots p_X(x_n)$ and aims to guess the full sequence $\\vec{x} = (x_1,\\cdots , x_n)$ .",
"In this case, the problem is characterized by the c-q state $\\rho _{XB}^{\\otimes n}$ .",
"The 1-shot bounds (REF ) give us $& -\\frac{1}{n}\\ln \\left(1 + n\\ln (|\\mathcal {X}|)\\right) + \\frac{1}{n}H_{\\frac{1}{2}}^{\\uparrow , M}(X^n | B^n)_{\\rho ^{\\otimes n}} \\\\& \\qquad \\le \\frac{1}{n}\\ln G(X^n|B^n)_{\\rho ^{\\otimes n}} \\\\& \\qquad \\le \\frac{1}{n}H_{\\frac{1}{2}}^{\\uparrow , M}(X^n | B^n)_{\\rho ^{\\otimes n}}$ where $H_{\\frac{1}{2}}^{\\uparrow , M}(X^n | B^n)_{\\rho ^{\\otimes n}}$ can involve collective measurements on the system $B^n$ .",
"Taking $n\\rightarrow \\infty $ , we obtain $ \\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\ln G(X^n|B^n)_{\\rho ^{\\otimes n}} = \\lim _{n\\rightarrow \\infty } \\frac{1}{n} H_{\\frac{1}{2}}^{\\uparrow , M}(X^n | B^n)_{\\rho ^{\\otimes n}},$ assuming that the limit on the right-hand side exists.",
"Note that we can bound $\\frac{1}{n} H_{\\frac{1}{2}}^{\\uparrow , M}(X^n | B^n)_{\\rho ^{\\otimes n}}&\\le \\frac{1}{n} \\inf _{\\lbrace E_{y} \\rbrace _{y\\in \\mathcal {Y}}} H_{\\frac{1}{2}}^\\uparrow (X^n|Y^n)_{p^{\\otimes n}}\\\\&= \\inf _{\\lbrace E_{y} \\rbrace _{y\\in \\mathcal {Y}}} H_{\\frac{1}{2}}^\\uparrow (X|Y)_{p} \\\\&= H_{\\frac{1}{2}}^{\\uparrow , M}(X | B)_{\\rho }$ where the first inequality follows from the fact that product measurements are a special case of collective measurements, and the first equality follows from the additivity of the classical Rényi entropy ([4]), and the third by the definition of $H_{\\frac{1}{2}}^{\\uparrow , M}(X | B)_{\\rho }$ .",
"Moreover, by the data-processing inequality [16], $\\frac{1}{n} H_{\\frac{1}{2}}^{\\uparrow , M}(X^n | B^n)_{\\rho ^{\\otimes n}} \\ge \\frac{1}{n} \\widetilde{H}_{\\frac{1}{2}}^{\\uparrow }(X^n | B^n)_{\\rho ^{\\otimes n}} = \\widetilde{H}_{\\frac{1}{2}}^{\\uparrow }(X | B)_{\\rho },$ where the conditional Rényi entropy $\\widetilde{H}_{\\alpha }^{\\uparrow }(C | D)_{\\sigma }$ of a bipartite state $\\sigma _{CD}$ is defined as $\\widetilde{H}_{\\alpha }^{\\uparrow }(C | D)_{\\sigma }= \\sup _{\\omega _D} \\left[-\\widetilde{D}_{\\alpha } (\\sigma _{CD} \\Vert \\mathbf {1}_C \\otimes \\omega _D)\\right],$ with the optimization with respect to states $\\omega _D$ and the sandwiched Rényi relative entropy defined as [17], [18]: $\\widetilde{D}_{\\alpha }(X \\Vert Y) = \\frac{1}{\\alpha -1}\\ln \\operatorname{tr}[ (Y^{(1-2\\alpha )/\\alpha } X Y^{(1-2\\alpha )/\\alpha })^\\alpha ].$ The equality in (REF ) follows from the additivity of $\\widetilde{H}_{\\frac{1}{2}}^{\\uparrow }$ under tensor products (see, e.g., [19]).",
"Hence, we obtain $ \\widetilde{H}_{\\frac{1}{2}}^{\\uparrow }(X | B)_{\\rho } \\le \\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\ln G(X^n|B^n)_{\\rho ^{\\otimes n}} \\le H_{\\frac{1}{2}}^{\\uparrow , M}(X | B)_{\\rho }.$ In the classical case, (REF ), both the left and right-hand sides reduce to $H_{\\frac{1}{2}}^{\\uparrow }(X | Y)_{p}$ where $p$ is the underlying classical distribution of (REF ).",
"Hence, these bounds recover Proposition 5 of [4]."
],
[
"Semi-definite optimization representations and their consequences",
"The task of calculating $G_{\\vec{c}}(X|B)_\\rho $ as defined in (REF ) can be written as a semi-definite optimization problem, as was found in [13].",
"In this section, we present a different derivation of that fact yielding in (REF ) a representation dual to the one found in [13].",
"In sec:concavity we use this representation to prove that the guesswork $G(X|B)_\\rho $ is a concave function of the c-q state $\\rho _{XB}$ .",
"In sec:Lipschitz we likewise use this representation to obtain a Lipschitz continuity bound on the guesswork.",
"Then in sec:dual we compute the dual SDP, recovering the one obtained in [13].",
"In sec:ubalgo we use this dual representation to develop a simple algorithm to obtain upper bounds on the quantity.",
"Lastly, in sec:misdp, we formulate a mixed-integer SDP representation of the problem, whose number of variables and constraints scales polynomially with all the relevant quantities (at the cost of adding binary constraints).",
"We also provide implementations of these SDP representations [20], using the Julia programming language [21] and the optimization library Convex.jl [22].",
"Consider an ordered strategy $\\vec{G}$ with a set of POVMs $\\lbrace E_{\\vec{g}} \\rbrace _{\\vec{g} \\in \\mathcal {X}^K}$ .",
"Then since $p_{\\vec{G}, X}(\\vec{g}, x) = p_X(x)\\operatorname{tr}[ E_{\\vec{g}} \\rho _B^x]$ , we have $c_k p_{N(\\vec{G}, X)}(k) = c_k \\sum _{x \\in \\mathcal {X}} p_X(x)\\sum _{\\begin{array}{c}\\vec{g} \\in \\mathcal {X}^K\\\\ N(\\vec{g}, x) = k\\end{array}} \\operatorname{tr}[E_{\\vec{g}} \\rho _B^x]$ and hence $E_{\\vec{c}}(N(\\vec{G}, X)) &= \\sum _{k=1}^K c_k \\sum _{x \\in \\mathcal {X}} p_X(x)\\sum _{\\begin{array}{c}\\vec{g} \\in \\mathcal {X}^K\\\\ N(\\vec{g}, x) = k\\end{array}} \\operatorname{tr}[E_{\\vec{g}} \\rho _B^x] \\\\&\\qquad + c_\\infty \\sum _{x \\in \\mathcal {X}} p_X(x)\\sum _{\\begin{array}{c}\\vec{g} \\in \\mathcal {X}^K\\\\ N(\\vec{g}, x) = \\infty \\end{array}} \\operatorname{tr}[E_{\\vec{g}} \\rho _B^x]\\\\&= \\sum _{\\vec{g} \\in \\mathcal {X}^K}\\sum _{x \\in \\mathcal {X}} c_{N(\\vec{g}, x)} p_X(x)\\operatorname{tr}[E_{\\vec{g}} \\rho _B^x]\\\\&= \\sum _{\\vec{g} \\in \\mathcal {X}^K} \\operatorname{tr}[R_{\\vec{g}} E_{\\vec{g}}]$ where we define $R_{\\vec{g}} := \\sum _{x \\in \\mathcal {X}}p_X(x) c_{N(\\vec{g}, x)} \\rho _B^x$ for $\\vec{g}\\in \\mathcal {X}^K$ .",
"Thus, $\\begin{aligned}G_{\\vec{c}}(X|B)_\\rho \\,\\, =\\quad & \\text{minimize} & & \\sum _{\\vec{g}\\in \\mathcal {X}^K}\\operatorname{tr}[R_{\\vec{g}} E_{\\vec{g}}]\\\\& \\text{subject to} & & E_{\\vec{g}} \\ge 0 \\qquad \\forall \\vec{g} \\in \\mathcal {X}^K\\\\& & & \\sum _{\\vec{g} \\in \\mathcal {X}^K} E_{\\vec{g}} = \\mathbf {1}_B.\\end{aligned}$ The expression in (REF ) clarifies that $R_{\\vec{g}}$ has an interpretation as a cost operator corresponding to the guessing outcome $\\vec{g}$ .",
"Since $\\sum _{\\vec{g} \\in \\mathcal {X}^K}\\operatorname{tr}[R_{\\vec{g}} E_{\\vec{g}}]$ is linear in each positive semi-definite (matrix) variable $E_{\\vec{g}}$ , $G_{\\vec{c}}(X|B)_\\rho $ admits an SDP representation, given in (REF ).",
"This program has $|\\mathcal {X}|^K$ variables (each $d_B\\times d_B$ complex positive semi-definite matrices), subject to one constraint.",
"Note, however, since the cost vector $\\vec{c}$ is increasing, any guess $\\vec{h} \\in \\mathcal {X}^K$ with repeated elements is a suboptimal guessing order.",
"That is, if $\\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K}$ is a POVM with $E_{\\vec{h}} \\ne 0$ , and $\\vec{h}^{\\prime } \\in \\mathcal {X}^K$ only differs from $\\vec{h}$ by replacing repeated elements such that $\\vec{h}^{\\prime }$ has no repeated elements, then the POVM defined by $\\tilde{E}_{\\vec{g}} := {\\left\\lbrace \\begin{array}{ll}E_{\\vec{g}} & \\vec{g} \\ne \\vec{h} \\text{ and }\\vec{g} \\ne \\vec{h}^{\\prime } \\\\0 & \\vec{g} = \\vec{h}\\\\E_{\\vec{h}} + E_{\\vec{h}^{\\prime }} & \\vec{g} = \\vec{h}^{\\prime }\\end{array}\\right.",
"}$ has $\\sum _{\\vec{g} \\in \\mathcal {X}^K} \\operatorname{tr}[ R_{\\vec{g}} \\tilde{E}_{\\vec{g}}] \\le \\sum _{\\vec{g} \\in \\mathcal {X}^K} \\operatorname{tr}[ R_{\\vec{g}} E_{\\vec{g}}]$ .",
"Hence, we may restrict to the outcome space $\\mathcal {X}^K_{\\ne } := \\lbrace \\vec{g}\\in \\mathcal {X}^K : g_i \\ne g_j, \\forall i \\ne j \\rbrace \\subseteq \\mathcal {X}^K.$ Note $|\\mathcal {X}^K_{\\ne }| = \\frac{|\\mathcal {X}|!",
"}{(|\\mathcal {X}| - K)!",
"}$ , and in the case in which $K = |\\mathcal {X}|$ , the set $\\mathcal {X}_{K}$ is just the set of permutations of $\\mathcal {X}$ .",
"Hence, (REF ) can be re-written as the following smaller problem (with $\\frac{|\\mathcal {X}|!",
"}{(|\\mathcal {X}| - K)!",
"}$ instead of $|\\mathcal {X}|!$ constraints): $\\begin{aligned}G_{\\vec{c}}(X|B)_\\rho \\,\\, =\\quad & \\text{minimize} & & \\sum _{\\vec{g}\\in \\mathcal {X}^K_{\\ne }}\\operatorname{tr}[R_{\\vec{g}} E_{\\vec{g}}]\\\\& \\text{subject to} & & E_{\\vec{g}} \\ge 0 \\qquad \\forall \\vec{g} \\in \\mathcal {X}^K_{\\ne }\\\\& & & \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} E_{\\vec{g}} = \\mathbf {1}_B.\\end{aligned}$ Note that in the case $c_\\infty = \\infty $ and $K < |\\mathcal {X}|$ , there exists a finite solution if and only if there exists a POVM $\\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}$ such that for all $x\\in \\mathcal {X}$ and $\\vec{g} \\in \\mathcal {X}^K_{\\ne }$ with $x \\notin \\vec{g}$ , we have $\\operatorname{tr}[ E_{\\vec{g}} \\rho _B^x] = 0$ .",
"Whether or not this holds depends on the particular state $\\rho _{XB}$ .",
"However, when $c_\\infty < \\infty $ or $K = |\\mathcal {X}|$ , for any state $\\rho _{XB}$ , the problem (REF ) has a finite solution.",
"Moreover, for any POVM $\\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}$ , the objective $\\sum _{\\vec{g}\\in \\mathcal {X}^K_{\\ne }}\\operatorname{tr}[R_{\\vec{g}} E_{\\vec{g}}]$ is finite.",
"In the following, we restrict to those two cases.",
"Remark 4 This optimization problem has the same form as that of discriminating quantum states in an ensemble, as described in, e.g., [23].",
"Note, however, that (1) the $R_{\\vec{g}}$ are positive semi-definite but not normalized, and (2) the case of having two copies of the unknown state in the guessing framework does not correspond to $R_{\\vec{g}}^{\\otimes 2}$ .",
"Nevertheless, slight modifications to [23] show that a POVM $\\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}$ is optimal for (REF ) if and only if $Y = \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} R_{\\vec{g}} E_{\\vec{g}}$ satisfies $Y \\le R_{\\vec{g}}$ for all $\\vec{g} \\in \\mathcal {X}^K_{\\ne }$ .",
"Remark 5 The set of POVMs is convex and since the objective function is linear, any minimizer for (REF ) may be decomposed into extremal POVMs which are also minimizers.",
"By [24], any extremal POVM on a Hilbert space of size $d_B$ has at most $d_B^2$ non-zero elements.",
"Hence, there exist minimizers of (REF ) with at most $d_B^2$ non-zero elements (even though $|\\mathcal {X}^K_{\\ne }|$ could be far larger than $d_B^2$ ).",
"Let $S \\subseteq \\mathcal {X}^K_{\\ne }$ be a set of $d_B^2$ points such that there exists $\\lbrace \\tilde{E}_{\\vec{g}}\\rbrace _{\\vec{g} \\in S}$ with $\\tilde{E}_{\\vec{g}}\\ge 0$ , $\\sum _{\\vec{g} \\in S} \\tilde{E}_{\\vec{g}} = \\mathbf {1}_B$ , and $G_{\\vec{c}}(X|B)_\\rho = \\sum _{\\vec{g} \\in S} \\operatorname{tr}[\\tilde{E}_{\\vec{g}} R_{\\vec{g}}].$ Then (REF ) holds with $\\mathcal {X}^K_{\\ne }$ replaced by $S$ , namely $\\begin{aligned}G_{\\vec{c}}(X|B)_\\rho \\,\\, =\\quad & \\text{minimize} & & \\sum _{\\vec{g}\\in S}\\operatorname{tr}[R_{\\vec{g}} E_{\\vec{g}}]\\\\& \\text{subject to} & & E_{\\vec{g}} \\ge 0 \\qquad \\forall \\vec{g} \\in S\\\\& & & \\sum _{\\vec{g} \\in S} E_{\\vec{g}} = \\mathbf {1}_B.\\end{aligned}$ Note the “$\\le $ ” direction of the equality (REF ) is trivial, since given a minimizer $\\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in S}$ for (REF ), simply extending it by choosing $E_{\\vec{g}} = 0$ for $\\vec{g} \\notin S$ gives a feasible point for the optimization problem on the right-hand side of (REF ).",
"The “$\\ge $ ” direction follows from the existence of the $\\lbrace \\tilde{E}_{\\vec{g}}\\rbrace _{\\vec{g} \\in S}$ described above.",
"We note here that it is nontrivial to identify the set $S$ .",
"In general, identifying the set $S$ is as difficult as solving the original problem in (REF ).",
"However, applying a heuristic method inspired by choosing a smaller set of constraints can lead to useful upper bounds on the guesswork, which we describe in Section REF ."
],
[
"The dual problem",
"Next, we compute the dual problem to (REF ), in the case $K= |\\mathcal {X}|$ or $c_\\infty < \\infty $ .",
"Consider the Lagrangian $ &\\mathcal {L}((E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, (\\lambda _{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, \\nu ) = \\sum _{\\vec{g}\\in \\mathcal {X}^K_{\\ne }}\\mathinner {\\langle { R_{\\vec{g}}, E_{\\vec{g}}}\\rangle } \\\\& \\qquad - \\sum _{\\vec{g}\\in \\mathcal {X}^K_{\\ne }}\\mathinner {\\langle {\\lambda _{\\vec{g}}, E_{\\vec{g}}}\\rangle } + \\left\\langle {\\nu , \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} E_{\\vec{g}} - \\mathbf {1}_B}\\right\\rangle \\\\&= \\sum _{\\vec{g}\\in \\mathcal {X}^K_{\\ne }}\\mathinner {\\langle {R_{\\vec{g}}-\\lambda _{\\vec{g}} + \\nu , E_{\\vec{g}}}\\rangle } - \\operatorname{tr}[\\nu ]$ where we have introduced the Hilbert–Schmidt product $\\mathinner {\\langle {A,B}\\rangle } = \\operatorname{tr}[A^\\dagger B]$ , and where $\\lambda _{\\vec{g}}\\ge 0$ is the dual variable to the inequality constraint $E_{\\vec{g}}\\ge 0$ , and $\\nu = \\nu ^\\dagger $ is the dual variable to the equality constraint $\\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} E_{\\vec{g}} = \\mathbf {1}_B$ .",
"As shown in, e.g., [25], the primal problem (REF ) can be expressed as $\\min _{(E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}}\\max _{\\lambda _{\\vec{g}} \\ge 0, \\nu }\\mathcal {L}((E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, (\\lambda _{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, \\nu )$ while the dual problem is given by $\\max _{\\lambda _{\\vec{g}} \\ge 0, \\nu }\\min _{(E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}}\\mathcal {L}((E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, (\\lambda _{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, \\nu ).$ If $R_{\\vec{g}} - \\lambda _{\\vec{g}} + \\nu \\ne 0$ for any $\\vec{g}\\in \\mathcal {X}^K_{\\ne }$ , then the inner minimization in (REF ) yields $-\\infty $ .",
"Hence, $\\min _{(E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}}\\mathcal {L}((E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, (\\lambda _{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}, \\nu ) \\\\= {\\left\\lbrace \\begin{array}{ll}- \\infty & R_{\\vec{g}} - \\lambda _{\\vec{g}} + \\nu \\ne 0 \\quad \\exists \\vec{g} \\in \\mathcal {X}^K_{\\ne }\\\\- \\operatorname{tr}[\\nu ] & \\text{else.}\\end{array}\\right.",
"}$ The constraint $\\lambda _{\\vec{g}} \\ge 0$ and $R_{\\vec{g}} - \\lambda _{\\vec{g}} + \\nu = 0$ imply the semi-definite inequality $-\\nu \\le R_{\\vec{g}}$ .",
"Writing $Y = - \\nu $ and maximizing over $\\lambda _{\\vec{g}} \\ge 0$ , (REF ) becomes $ \\begin{aligned}\\text{maximize} \\quad & \\operatorname{tr}[Y]\\\\\\text{subject to} \\quad & Y = Y^\\dagger \\\\& Y \\le R_{\\vec{g}} \\qquad \\forall \\vec{g} \\in \\mathcal {X}^K_{\\ne }\\end{aligned}$ Since (REF ) is strictly feasible (e.g., $E_{\\vec{g}} = \\mathbf {1}_B \\frac{1}{|\\mathcal {X}^K_{\\ne }|}$ is a strictly feasible point) by Slater's condition, strong duality holds.",
"Hence, (REF ) obtains the same optimal value as (REF ).",
"The formulation of the problem as given in (REF ) was previously found in the work [13].",
"In contrast to the primal SDP (REF ), the dual problem has a single variable $Y$ subject to $| \\mathcal {X}^K_{\\ne } | $ constraints."
],
[
"A simple algorithm to compute upper bounds",
"The dual form of the SDP can be used to generate upper bounds on $G_{\\vec{c}}(X|B)_\\rho $ simply by removing constraints.",
"This provides an algorithm to find an upper bound on the objective function: Decide on some number of constraints $\\kappa $ to impose in total.",
"Then, Initialize an empty list $L=\\lbrace \\rbrace $ corresponding to constraints to impose.",
"Set $Y$ to be the identity matrix, as a first guess at the optimal dual variable.",
"If $Y$ satisfies $Y \\le R_{\\vec{g}}$ for all $\\vec{g} \\in \\mathcal {X}^K_{\\ne }$ , then $Y$ is the maximizer of the dual problem (REF ), and the optimization is solved.",
"Otherwise, find $\\vec{g}\\in \\mathcal {X}^K_{\\ne }$ such that $Y \\lnot \\le R_{\\vec{g}}$ , and add $\\vec{g}$ to the list $L$ .",
"Solve the problem $ \\begin{aligned}\\text{maximize} \\quad & \\operatorname{tr}[Y]\\\\\\text{subject to} \\quad & Y = Y^\\dagger \\\\& Y \\le R_{\\vec{g}} \\qquad \\forall \\vec{g} \\in L\\end{aligned}$ and set $Y$ to be its maximizer.",
"Repeat steps 2 and 3 until the list $L$ has length $\\kappa $ .",
"Solve the problem one last time, and return the output.",
"In order to find a constraint that $Y$ violates, a heuristic technique such as simulated annealing can be used.",
"Moreover, in the case that there are too many constraints to fit into memory or check exhaustively, using an iterative technique (such as simulated annealing) is essential.",
"If this algorithm was continued (without imposing a limit on the total number of constraints $\\kappa $ to impose), it would eventually yield the true value $G_{\\vec{c}}(\\rho _{XB},K)$ .",
"When a total number of constraints is limited, it provides an upper bound (since it is a relaxation of (REF )).",
"However, even with a limit $\\kappa $ on the total number of constraints, this algorithm can in theory yield the true value $G_{\\vec{c}}(X|B)_\\rho $ .",
"Note that the dual problem to (REF ) is $ \\begin{aligned}\\text{maximize} \\quad & \\operatorname{tr}[Y]\\\\\\text{subject to} \\quad & Y = Y^\\dagger \\\\& Y \\le R_{\\vec{g}} \\qquad \\forall \\vec{g} \\in S\\end{aligned}$ where $S \\subseteq \\mathcal {X}^K_{\\ne }$ has $|S| = d_B^2$ and is described in the remark above.",
"Hence, if $L$ in (REF ) equals $S$ , then the algorithm finds the true value $G_{\\vec{c}}(X|B)_\\rho $ , not just an upper bound.",
"Thus, $\\kappa = d_B^2$ suffices if the constraints $\\vec{g}$ can be chosen precisely to obtain $L=S$ .",
"In general, finding $S$ is as difficult as solving the original problem.",
"Nonetheless, this motivates why choosing a relatively small value of $\\kappa $ (such as $d_B^2$ ) can still yield good upper bounds."
],
[
"A mixed-integer reformulation",
"The problem of computing $G_{\\vec{c}} (X|B)_{\\rho }$ can be formulated another way as a mixed-integer SDP, i.e., an SDP that has additional integer or binary constraints.",
"Consider a POVM $\\lbrace F_j\\rbrace _{j=1}^M$ with $M$ outcomes.",
"When outcome $j$ is obtained, Bob guesses in some order $\\vec{g}^{(j)} \\in \\mathcal {X}^K_{\\ne }$ .",
"Then consider the problem $ \\begin{aligned}& \\text{minimize} & & \\sum _{x\\in \\mathcal {X}, j = 1,\\cdots , M}p_X(x) c_{N(\\vec{g}^{(j)}, x)} \\operatorname{tr}[ F_j \\rho _B^x]\\\\& \\text{subject to} & & F_j \\ge 0 \\qquad j = 1,\\cdots , M,\\\\& & & \\vec{g}^{(j)} \\in \\mathcal {X}^K_{\\ne }, j = 1,\\cdots , M,\\\\& & & \\sum _{j=1}^M F_j = \\mathbf {1}_B.\\end{aligned}$ We note that in the above, the variables to be optimized over are both the POVM $\\lbrace F_j\\rbrace _{j=1}^M$ and the guessing orders $\\lbrace \\vec{g}^{(j)} \\rbrace _j$ corresponding to each POVM outcome.",
"This optimization is not an SDP, since the dependence on the optimization variables $\\lbrace \\vec{g}^{(j)}\\rbrace _{j=1}^M$ and $\\lbrace F_j\\rbrace $ is not linear, and $\\vec{g}^{(j)} \\in \\mathcal {X}^K_{\\ne }$ is a discrete constraint.",
"Consider, however, the case that $K = |\\mathcal {X}|$ .",
"With this assumption, we will be able to remove the nonlinearity although not the discrete variables.",
"This yields a mixed-integer SDP: an optimization problem such that if all integer constraints were removed, the result would be an SDP.",
"We proceed as follows.",
"Under the condition $K = |\\mathcal {X}|$ , we may restrict to considering guessing orders that are permutations without loss of generality; other guessing orders have repeated guesses, which can only increase the value of the objective function.",
"In this case, the outcome $\\infty $ never occurs, and for each $\\vec{g} \\in S_{|\\mathcal {X}|}$ , the quantity $( c_{N(\\vec{g}, x)} )_{x \\in \\mathcal {X}}$ satisfies $( c_{N(\\vec{g}, x)} )_{x \\in \\mathcal {X}} = \\vec{g} \\, ^{-1}(c),$ where $\\vec{g} ^{-1}$ is the inverse permutation to $\\vec{g}$ , and $c = (c_k)_{k=1}^{K}$ is the cost vector (without $\\infty $ ).",
"Here, $S_n$ is the set of permutations on $\\lbrace 1,\\cdots ,n\\rbrace $ .",
"Let $P^{(j)}$ be an $|\\mathcal {X}|\\times |\\mathcal {X}|$ matrix representation of the permutation $\\vec{g}^{(j)}{}^{-1}$ .",
"Then $(P^{(j)} c)_x = \\sum _{y\\in \\mathcal {X}} P^{(j)}_{xy} c_y = c_{N(\\vec{g}, x)}$ .",
"Hence, the optimization (REF ) can be reformulated as $\\begin{aligned}& \\text{minimize} & & \\sum _{x,y\\in \\mathcal {X}, j = 1,\\cdots , M}p_X(x) P^{(j)}_{xy} c_y \\operatorname{tr}[ F_j \\rho _B^x]\\\\& \\text{subject to} & & F_j \\in \\mathcal {M}_{d_B} \\qquad \\forall \\,j \\in [M],\\\\& & & P^{(j)}_{xy} \\in \\lbrace 0,1\\rbrace , \\quad \\forall \\,j \\in [M], x,y \\in \\mathcal {X}\\\\& & & F_j \\ge 0 \\qquad \\forall \\,j \\in [M],\\\\& & & \\sum _{j=1}^M F_j = \\mathbf {1}_B,\\\\& & & \\sum _{x\\in \\mathcal {X}}P^{(j)}_{xy} = 1, \\quad \\forall \\,j \\in [M], y \\in \\mathcal {X}\\\\& & & \\sum _{y\\in \\mathcal {X}}P^{(j)}_{xy} = 1, \\quad \\forall \\,j \\in [M], x \\in \\mathcal {X}\\\\\\end{aligned}$ Note that all the constraints are semi-definite or linear, except that each element $P^{(j)}_{xy}$ is a binary variable: $P^{(j)}_{xy} \\in \\lbrace 0,1\\rbrace $ , which is a particularly simple type of discrete constraint.",
"The non-linearity in the objective function, however, persists.",
"To remove this, we take advantage of the fact that the $P^{(j)}_{xy}$ are binary.",
"In particular, [26] provide a clever trick to turn objective functions with terms of the form $z x$ where $z$ is a binary variable and $x$ a continuous variable into objective functions of a continuous variable $y$ subject to four affine constraints (in terms of $x$ and $z$ ), as long as $x$ is bounded by known constants.",
"We reproduce this argument in the following.",
"We first write the objective function entirely in terms of scalar quantities: $\\sum _{x,y\\in \\mathcal {X}, j \\in [M]}p_X(x) P^{(j)}_{xy} c_y \\operatorname{tr}[ F_j \\rho _B^x] \\\\= \\sum _{k,\\ell \\in [d_B]}\\sum _{x,y\\in \\mathcal {X}, j \\in [M]}p_X(x)(\\rho _B^x)_{k\\ell } c_y\\, P^{(j)}_{xy} (F_j)_{\\ell k}$ Let $x = (F_j)_{\\ell k} $ and $z = P^{(j)}_{xy} \\in \\lbrace 0,1\\rbrace $ .",
"Then $|x| \\le \\operatorname{tr}[F_j]/2 \\le d_B/2$ .",
"Then $x_L := - d_B/2$ and $x_U := d_B/2$ constitute lower and upper bounds to $x$ , respectively.",
"Hence, the following four inequalities hold trivially: $\\begin{aligned}z (x - x_L) \\ge 0,\\\\(z-1)(x - x_U) \\ge 0,\\\\z(x - x_U) \\le 0,\\\\(z-1)(x - x_L) \\le 0.\\end{aligned}$ Now, let $y = xz$ .",
"Then we have $\\begin{aligned}y - zx_L \\ge 0,\\\\y - z x_U \\ge x - x_U,\\\\y - z x_U \\le 0,\\\\y - zx_L \\le x - x_L.\\end{aligned}$ On the other hand, let us remove the constraint $y= xz$ , and consider $y$ as another variable.",
"Then if $z = 0$ , the first equation of (REF ) implies that $y \\ge 0$ , while the third implies $y\\le 0$ , so $y= 0$ .",
"On the other hand, if $z=1$ , then the second equation of (REF ) implies that $y \\ge x$ while the fourth implies that $y \\le x$ .",
"Hence, either way, $y = xz$ .",
"Thus, (REF ) is equivalent to $y = xz$ .",
"With this transformation, (REF ) can be reformulated as the following.",
"$ \\begin{aligned}& \\text{minimize} & & \\sum _{k,\\ell \\in [d_B]}\\sum _{x,y\\in \\mathcal {X}, j \\in [M]}p_X(x)(\\rho _B^x)_{k\\ell } c_y\\, y_{xy \\ell k j}\\\\& \\text{subject to} & & F_j \\in \\mathcal {M}_{d_B} \\qquad \\forall \\,j \\in [M],\\\\& & & y_{xy \\ell k j} \\in \\mathbb {R}, \\quad \\forall \\,x,y \\in \\mathcal {X}, \\ell ,k \\in [d_B], j \\in [M],\\\\& & & P^{(j)}_{xy} \\in \\lbrace 0,1\\rbrace , \\quad \\forall \\,j \\in [M], x,y \\in \\mathcal {X}\\\\& & & F_j \\ge 0 \\qquad \\forall \\,j \\in [M],\\\\& & & \\sum _{j=1}^M F_j = \\mathbf {1}_B,\\\\& & & \\sum _{x\\in \\mathcal {X}}P^{(j)}_{xy} = 1 \\quad \\forall \\,j \\in [M], y \\in \\mathcal {X}, \\\\& & & \\sum _{y\\in \\mathcal {X}}P^{(j)}_{xy} = 1 \\quad \\forall \\,j \\in [M], x \\in \\mathcal {X}, \\\\& & & y_{xy \\ell k j} + P^{(j)}_{xy} \\frac{d_B}{2} \\ge 0,\\\\& & & y_{xy \\ell k j} - P^{(j)}_{xy} \\frac{d_B}{2} \\ge (F_j)_{\\ell k} - \\frac{d_B}{2},\\\\& & & y_{xy \\ell k j} - P^{(j)}_{xy} \\frac{d_B}{2} \\le 0,\\\\& & & y_{xy \\ell k j} + P^{(j)}_{xy} \\frac{d_B}{2} \\le (F_j)_{\\ell k} + \\frac{d_B}{2}.\\end{aligned}$ where the last four constraints hold for $\\forall \\,x,y \\in \\mathcal {X}, \\ell ,k \\in [d_B], \\text{ and } j \\in [M]$ .",
"This is a mixed-integer SDP, with a number of constraints and variables that is polynomial in $M, d_B, |\\mathcal {X}|$ .",
"Moreover, if $M \\ge d_B^2$ , then as follows from the remark below (REF ), the mixed-integer SDP (REF ) obtains the same optimal value as (REF ), namely $G_{\\vec{c}}(\\rho _{XB}, |\\mathcal {X}|)$ , using that $K = |\\mathcal {X}|$ .",
"Note, however, that mixed-integer SDPs are not in general efficiently solvable; they encompass mixed integer linear programs, which are NP-hard.",
"However, in practice they can sometimes be quickly solved.",
"Since the original SDP formulation (REF ) involves an exponential (in $|\\mathcal {X}|$ ) number of variables (or an exponential number of constraints in its dual formulation (REF )), Eq.",
"(REF ) may provide a more practical approach in some cases because it instead has a polynomial (in $|\\mathcal {X}|$ ) number of variables.",
"Mixed-integer SDPs can be solved in various ways; in the code accompanying this paper [20], the problem (REF ) is solved using the library Pajarito.jl [27], which proceeds by solving an alternating sequence of mixed-integer linear problems and SDPs."
],
[
"The ellipsoid algorithm",
"The ellipsoid algorithm (see, e.g., [28]) provides a theoretical proof that under a strict feasibility assumption, semi-definite programs can be solved in time that scales as a polynomial in: the number of scalar variables and constraints, the logarithm of a 2-norm bound on the feasible points, $\\ln (1/\\varepsilon )$ where $\\varepsilon $ is the solution tolerance, and the maximum bit length of the scalar entries of the objective and constraints (see e.g.",
"[29]).",
"In fact, the ellipsoid algorithm applies quite generally to the optimization of a linear objective function over a convex feasible region (which could be described by a domain and constraint functions).",
"The ellipsoid algorithm only requires a separation oracle for the feasible region, a subroutine which either asserts that a given point lies within the feasible region, or provides a separating hyperplane between the given point and the feasible region.",
"When the separation oracle can be evaluated in polynomial time, the overall ellipsoid algorithm runs in polynomial time as well (see [28]).",
"In the case of a single positive semi-definite constraint, e.g.",
"$Y \\ge 0$ , a simple separation oracle is given by computing the eigendecomposition of $Y$ and checking if all of its eigenvalues are non-negative.",
"If so, it returns that $Y$ is indeed feasible, and otherwise returns the matrix $C := U \\text{diag}(f(\\lambda _1),\\cdots ,f(\\lambda _d)) U^\\dagger $ where $Y = U\\text{diag}(\\lambda _1,\\cdots ,\\lambda _d)U^\\dagger $ is the eigendecomposition of $Y$ , $U$ is unitary, $\\lambda _1,\\cdots ,\\lambda _d$ are the eigenvalues, and $f(x) = 1$ if $x < 0$ and $f(x) = 0$ otherwise.",
"This matrix has the properties that $C\\ge 0$ , $\\Vert C\\Vert _\\infty = 1$ , and $\\operatorname{tr}[C^\\dagger Y] = \\sum _{i=1}^d f(\\lambda _i)\\lambda _i = \\sum _{i: \\lambda _i < 0}\\lambda _i <0$ .",
"In the case of the dual problem (REF ) with $K=|\\mathcal {X}|$ , we have $|\\mathcal {X}|!$ positive semi-definite constraints.",
"Thus, we cannot check all of them together in polynomial time.",
"The feasibility problem $Y\\le R_{\\pi }$ for each $\\pi \\in S_{|\\mathcal {X}|}$ can be written as the following mixed-integer non-linear problem, $ \\begin{aligned}\\eta := \\text{minimize} &\\quad \\mathinner {\\langle {\\psi , \\left(\\sum _{x\\in \\mathcal {X}}(P c)_{x} p_X(x) \\rho _B^x - Y\\right) \\psi }\\rangle }\\\\\\text{subject to} &\\quad \\sum _{i\\in \\mathcal {X}} P_{ij} = \\sum _{j\\in \\mathcal {X}} P_{ij} = 1\\\\&\\quad P_{ij}\\in \\lbrace 0\\rbrace , \\,\\,i,j \\in \\mathcal {X}\\\\&\\quad \\psi \\in \\mathbb {C}^{d_B}\\end{aligned}$ where $\\eta \\ge 0$ if and only if $Y \\le R_{\\pi }$ for all $\\pi \\in S_{|\\mathcal {X}|}$ , using that a matrix $M$ satisfies $M\\ge 0$ if and only if $\\mathinner {\\langle {\\psi , M \\psi }\\rangle }\\ge 0$ for all $\\psi \\in \\mathbb {C}^{d_B}$ .",
"Since the convex hull of the permutation matrices is given by the doubly stochastic matrices, the discrete constraints can be relaxed, yielding the following reformulation $\\begin{aligned}\\eta = \\text{minimize} &\\quad \\mathinner {\\langle {\\psi , \\left(\\sum _{x\\in \\mathcal {X}}(Dc)_{x} p_X(x) \\rho _B^x - Y\\right) \\psi }\\rangle }\\\\\\text{subject to} &\\quad \\sum _{i\\in \\mathcal {X}} D_{ij} = \\sum _{j\\in \\mathcal {X}} D_{ij} = 1\\\\&\\quad D_{ij} \\ge 0, \\,\\,i,j \\in \\mathcal {X}\\\\&\\quad \\psi \\in \\mathbb {C}^{d_B}.\\end{aligned}$ If $\\eta \\ge 0$ , then $Y$ is feasible.",
"Otherwise, the optimal value $D^*$ can be decomposed as a convex combination of permutations, $D^* = \\sum _i \\alpha _i P_i$ , and we must have $\\sum _{x} (P_i c)_x p_X(x) \\rho _B^x \\lnot \\ge Y$ for some $P_i$ , using that the objective is an affine function of $D$ .",
"The problem (REF ) can be solved by global non-linear optimization solvers such as EAGO.jl [30] or SCIP [31], but not in general in polynomial time.",
"At each iteration of the ellipsoid algorithm, one must evaluate the separation oracle for some Hermitian matrix $Y$ .",
"In order to avoid solving (REF ), one may attempt to prove the feasibility or infeasibility of a point $Y$ by other means.",
"For example, one may search over permutations $\\pi $ heuristically, in order to find $R_\\pi $ such that $Y\\lnot \\le R_\\pi $ .",
"If such a permutation can be identified, then $Y$ is not feasible, and the problem (REF ) does not need to be solved.",
"Likewise, if one can show that for some $k \\in (1,\\cdots ,|\\mathcal {X}|)$ , $Y \\le \\sum _{i=1}^k c_{|\\mathcal {X}|-i} p_{x_{i}} \\rho _B^{x_{i}} + c_1 \\sum _{x \\in \\mathcal {X}\\setminus \\lbrace x_{i_1},\\cdots ,x_{i_k}\\rbrace }p_x \\rho _B^x \\\\\\forall (x_1,\\cdots ,x_k)\\in \\mathcal {X}^k_{\\ne }$ then $Y$ must be feasible, and again (REF ) does not need to be solved.",
"The number of comparisons required scales as $|\\mathcal {X}|^k$ ; for small choices of $k$ , this provides an efficient check for feasibility (which may, however, be inconclusive)."
],
[
"Numerical comparisons",
"We compare numerical implementations of several of the above algorithms on a set of 12 test problems.",
"The code for these experiments can be found at [20].",
"Each problem has $p \\equiv u$ , the uniform distribution $u := (1/|\\mathcal {X}|, \\cdots , 1/|\\mathcal {X}|)$ , for simplicity.",
"The states are chosen as Two random qubit density matrices Two random qutrit density matrices Three pure qubits chosen equidistant within one plane of the Bloch sphere (the qubit trine states), i.e.",
"$\\cos \\left(j \\tfrac{2\\pi }{3}\\right) \\mathinner {|{0}\\rangle } + \\sin \\left(j \\tfrac{2\\pi }{3}\\right) \\mathinner {|{1}\\rangle }, \\qquad j = 1,2,3$ Three random qubit density matrices Three random qutrit density matrices The four BB84 states, $\\mathinner {|{0}\\rangle }, \\mathinner {|{1}\\rangle }$ , and $\\mathinner {|{\\pm }\\rangle } = \\frac{1}{\\sqrt{2}}(\\mathinner {|{0}\\rangle } \\pm \\mathinner {|{1}\\rangle })$ as well as the “tensor-2” case of $\\lbrace \\rho \\otimes \\sigma : \\rho , \\sigma \\in S\\rbrace $ for each of the six sets $S$ listed above, corresponding to the guesswork problem with quantum side information associated to $\\rho _{XB}^{\\otimes 2}$ , where $\\rho _{XB}$ is the state associated to the original guesswork problem with quantum side information.",
"The random states were chosen uniformly at random (i.e.",
"according to the Haar measure).",
"The exponentially-large SDP formulation (and its dual), the mixed-integer SDP algorithm, and the active set method were compared, with several choices of parameters and underlying solvers.",
"The mixed-integer SDP formulation was evaluated with $M=d_B$ (yielding an upper bound), $M=d_B^2$ (yielding the optimal value), with the Pajarito mixed-integer SDP solver [27], using Convex.jl (version 0.12.7) [22] to formulate the problem.",
"Pajarito proceeds by solving mixed-integer linear problems (MILP) and SDPs as subproblems, and thus uses both a MILP solver and an SDP solver as subcomponents.",
"Pajarito provides two algorithms: an iterative algorithm, which alternates between solving MILP and SDP subproblems, and solving a single branch-and-cut problem in which SDP subproblems are solved via so-called lazy callbacks to add cuts to the mixed-integer problem.",
"The latter is called “mixed-solver drives” (MSD) in the Pajarito documentation.",
"We tested three configurations of Pajarito (version 0.7.0): (c1) Gurobi (version 9.0.3) as the MILP solver and MOSEK (version 8.1.0.82) as the SDP solver, with Pajarito's MSD algorithm, (c2) Gurobi as the MILP solver and MOSEK as the SDP solver, with Pajarito's iterative algorithm, with a relative optimality gap tolerance of 0, (o) Cbc [32] (version 2.10.3) as the MILP solver, and SCS [33] (version 2.1.1) as the SDP solver, with Pajarito's iterative algorithm.",
"Here, `c' stands for commercial, and `o' for open-source.",
"In the configuration (c1), Gurobi was set to have a relative optimality gap tolerance of $10^{-5}$ and in (c2), a relative optimality gap tolerance of 0.",
"In both configurations, Gurobi was given an absolute linear-constraint-wise feasibility tolerance of $10^{-8}$ , and an integrality tolerance of $10^{-9}$ .",
"These choices of parameters match those made in [27].",
"Cbc was given an integrality tolerance of $10^{-8}$ , and SCS's (normalized) primal, dual residual and relative gap were set to $10^{-6}$ for each problem.",
"The default parameters were used otherwise.",
"Note the MSD option was not used with Cbc, since the solver does not support lazy callbacks.",
"For the (exponentially large) SDP primal and dual formulations, the problems were solved with both MOSEK and SCS, and likewise with the active-set upper bound.",
"The active set method uses simulated annealing to iteratively add violated constraints to the problem to find an upper bound, as described in sec:ubalgo, and uses a maximum-time parameter $t_\\text{max}$ to stop iterating when the estimated time of finding another constraint to add would cause the running time to exceed the maximum-timeThe maximum time can still be exceeded, since at least one iteration must be performed and the estimate can be wrong..",
"This provides a way to compare the improvement (or lack thereof) of running the algorithm for more iterations.",
"The algorithm also terminates when a violated constraint cannot be found after 50 runs of simulated annealing (started each time with different random initial conditions).",
"Here, the problems were solved with three choices of $t_\\text{max}$ , 20 s, 60 s, and 240 s. The exact answer was not known analytically for most of these problems, and so the average relative error was calculated by comparing to the mean of the solutions (excluding the active-set method and the MISDP with $M=d_B$ , which only give an upper bound in general).",
"For the cases of the BB84 states and the Y-states, where the solution is known exactly (see sec:example), the solutions obtained here match the analytic value to a relative tolerance of at least $10^{-7}$ .",
"The problems were run sequentially on a four-core desktop computer (Intel i7-6700K 4.00GHz CPU, with 16 GB of RAM, on Ubuntu-20.04 via Windows Subsystem for Linux, version 2), via the programming language Julia [21] (version 1.5.1), with a 5 minute time limit.",
"The results are summarized in tab:summary, and presented in more detail in tab:firstproblems and tab:secondproblems.",
"One can see that the MISDP problems were harder to solve than the corresponding SDPs for these relatively small problem instances.",
"The MISDPs have the advantage of finding extremal solutions, however, in the case $M=d_B$ , and may scale better to large instances.",
"Additionally, the active-set upper bound performed fairly well, finding feasible points within 20 % of the optimum in all cases, with only $t_\\text{max}=20$ s, and often finding near-optimal solutions.",
"It was also the only method able to scale to the largest instances tested, such as two copies of the BB84 states (which involves 16 quantum states in dimension four, and for which the SDP formulation has 16!",
"variables.).",
"In general, the commercial solvers performed better than the open source solvers, with the notable exception of the active-set upper bound with MOSEK, in which two more problems timed out than with SCS.",
"This could be due to SCS being a first-order solver which can therefore possibly scale to larger problem instances than MOSEK, which is a second-order solver.",
"Table: Comparison of average relative error and average solve time for the 12 problems discussed above.",
"A problem is considered “timed out” if an answer is not obtained in 5 minutes, and “errored out” if the solution was not obtained due to errors (such as running out of RAM).",
"The average relative error, which was rounded to two decimal digits, and the time taken are calculated only over the problems which were solved by the given algorithm and choice of parameters.",
"“MISDP (d B d_B)” refers to the choice M=d B M=d_B, and likewise “MISDP (d B 2 d_B^2)” refers to the choice M=d B 2 M=d_B^2.Table: The individual timings for each algorithm and choice of settings on problems (1)–(3), and the corresponding “tensor-2” problems discussed at ().",
"For each algorithm, the running time of the original problem is given followed by the running time on the “tensor-2” problem, e.g.",
"the SDP formulation with MOSEK on the two random qubits problem was solved in 8.69 seconds, and in 8.84 seconds for the corresponding tensor-2 problem.",
"“timeout” is written whenever the problem was not solved within 5 minutes.",
"For the active set algorithms, the relative error is also given for each problem in parenthesis.",
"Note that the MISDP formulation with M=d B M=d_B is also only known to be an upper bound, but a relative error of less than 10 -5 10^{-5} in each instance, so the relative errors are omitted.",
"Lastly, the relative error is written as ?",
"% in the case that only an upper bound was obtained.Table: The individual timings for each algorithm and choice of settings on problems (4)–(6).",
"See tab:firstproblems for a description of the quantities shown.",
"Here, “error” means the solution was not obtained due to an error (such as running out of memory).Proposition 6 For each cost vector $\\vec{c}$ and $K \\le |\\mathcal {X}|$ , the function $ \\rho _{XB} \\mapsto G_{\\vec{c}}(X|B)_\\rho $ from the set of c-q states of the form (REF ) to $\\mathbb {R}_{\\ge 0}\\cup \\lbrace \\infty \\rbrace $ , is concave.",
"Proof.",
"For $\\vec{g} \\in \\mathcal {X}^K_{\\ne }$ , and $\\rho _{XB}$ a c-q state, the quantity $R_{\\vec{g}}^\\rho := \\sum _{x\\in \\mathcal {X}}p_X(x) c_{N(\\vec{g}, x)} \\rho _B^x$ can be expressed as $R_{\\vec{g}}^\\rho = \\operatorname{tr}_X\\!\\left[\\left(\\sum _{x\\in \\mathcal {X}}c_{N(\\vec{g}, x)} |x\\rangle \\!\\langle x|_X \\otimes I_B \\right)\\rho _{XB}\\right]$ and hence is linear in $\\rho _{XB}$ .",
"Then for each POVM $(E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}$ , $\\rho _{XB} \\mapsto \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} \\operatorname{tr}[ R_{\\vec{g}}^\\rho E_{\\vec{g}}]$ is linear in $\\rho _{XB}$ .",
"The arbitrary infimum of concave functions, and in particular linear functions, is concave, and hence $G_{\\vec{c}}(X|B)_\\rho \\equiv \\min _{(E_{\\vec{g}})_{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}} \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} \\operatorname{tr}[R_{\\vec{g}}^\\rho E_{\\vec{g}}],$ where the minimum is taken over all POVMs on system $B$ with outcomes in $\\mathcal {X}^K_{\\ne }$ , is concave.",
"Remark 7 prop:guessworkconcave carries over to guesswork without side information, $G(X)$ , which simply corresponds to the case that $\\rho _B^x \\equiv \\rho _B$ is independent of $x \\in \\mathcal {X}$ .",
"Since $G(X)$ is manifestly symmetric under permutations of the density $p_X$ , this proves that $G(X)$ is a Schur concave function of the distribution $p_X$ (i.e., decreasing in the majorization pre-order; see, e.g., [34] for an overview of majorization and Schur concave functions).",
"Consequently, the work [35] provides an algorithm to calculate local continuity bounds for $G(X)$ ."
],
[
"Continuity of the guesswork",
"Proposition 8 For each cost vector $\\vec{c}$ and $K \\le |\\mathcal {X}|$ , such that either $c_\\infty < \\infty $ or $K = |\\mathcal {X}|$ , the function $\\rho _{XB} \\mapsto G_{\\vec{c}}(X|B)_\\rho $ from the set of c-q states of the form (REF ) to $\\mathbb {R}_{\\ge 0}$ , is Lipschitz continuous, satisfying the bound $|G_{\\vec{c}}(X|B)_\\rho - G_{\\vec{c}}(X|B)_\\sigma | \\le \\kappa \\Vert \\rho _{XB}-\\sigma _{XB}\\Vert _1.$ for any c-q states $\\rho _{XB}$ and $\\sigma _{XB}$ , where $\\kappa = c_\\infty $ if $K < |\\mathcal {X}|$ , and $\\kappa = c_{|\\mathcal {X}|}$ if $K = |\\mathcal {X}|$ .",
"Proof.",
"Define $f(\\rho _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}) := \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} \\operatorname{tr}[ R_{\\vec{g}}^\\rho E_{\\vec{g}}].$ Then, by linearity (as discussed in the proof of prop:guessworkconcave), $& f(\\rho _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }})-f(\\sigma _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}) \\\\& \\qquad = \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} \\operatorname{tr}[ R_{\\vec{g}}^{\\rho -\\sigma } E_{\\vec{g}}]\\\\& \\qquad = \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} \\operatorname{tr}[ \\operatorname{tr}_X[C^{(\\vec{g})}_{XB} \\Delta _{XB}] E_{\\vec{g}}]$ using (REF ), where $C_{XB}^{(\\vec{g})} := \\sum _{x\\in \\mathcal {X}}c_{N(\\vec{g}, x)} |x\\rangle \\!\\langle x| \\otimes I_B\\ge 0$ and $\\Delta _{XB} := \\rho _{XB}-\\sigma _{XB}$ .",
"Since $C_{XB}^{(\\vec{g})}$ and $\\Delta _{XB}$ commute, using the c-q structure of each, we have $& f(\\rho _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }})-f(\\sigma _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }})\\\\& \\qquad =\\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} \\operatorname{tr}[ C^{(\\vec{g})}_{XB} \\Delta _{XB} (I_X \\otimes E_{\\vec{g}})] \\\\& \\qquad = \\operatorname{tr}\\!\\left[ \\Delta _{XB}\\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}C^{(\\vec{g})}_{XB} (I_X\\otimes E_{\\vec{g}})\\right].$ Set $F_{XB} := \\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}C^{(\\vec{g})}_{XB} (I_X\\otimes E_{\\vec{g}}) =\\!\\!\\sum _{x\\in \\mathcal {X}}\\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} c_{N(\\vec{g}, x)} |x\\rangle \\!\\langle x| \\otimes E_{\\vec{g}}.$ Since $c_{N(\\vec{g}, x)}\\le \\kappa $ for each $x \\in \\mathcal {X}$ and $\\vec{g} \\in \\mathcal {X}^K_{\\ne }$ , we have that $F_{XB} \\le \\kappa \\sum _{x\\in \\mathcal {X}}\\sum _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }} |x\\rangle \\!\\langle x| \\otimes E_{\\vec{g}} $ in semi-definite order.",
"Performing the sums, we have $F_{XB} \\le \\kappa \\, I_X \\otimes I_B$ and hence $\\Vert F_{XB}\\Vert _\\infty \\le \\kappa $ .",
"Thus, $& f(\\rho _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }})-f(\\sigma _{XB}, \\lbrace E_{\\vec{g}}\\rbrace _{\\vec{g} \\in \\mathcal {X}^K_{\\ne }}) \\\\& \\qquad = \\operatorname{tr}\\!\\left[ \\Delta _{XB} F_{XB}\\right]\\\\& \\qquad \\le \\Vert \\Delta _{XB} F_{XB}\\Vert _1\\\\& \\qquad \\le \\Vert \\Delta _{XB}\\Vert _1 \\,\\Vert F_{XB}\\Vert _\\infty \\\\& \\qquad \\le \\kappa \\Vert \\rho _{XB}-\\sigma _{XB}\\Vert _1$ using Hölder's inequality in the second to last inequality.",
"Swapping $\\rho _{XB}$ and $\\sigma _{XB}$ completes the proof."
],
[
"BB84 states as side information",
"As an example, we consider the problem of calculating guesswork when one has four uniformly distributed letters to guess from, each correlated to one of the four BB84 states [36].",
"That is, $\\rho _{XB} = \\frac{1}{4} \\sum _{k = 1}^4 |x_k\\rangle \\!\\langle x_k|_X \\otimes |\\psi _k\\rangle \\!\\langle \\psi _k|_B$ with the four $\\mathinner {|{\\psi _k}\\rangle }$ 's being chosen from $\\left\\lbrace \\mathinner {|{0}\\rangle }, \\mathinner {|{1}\\rangle }, \\mathinner {|{+}\\rangle }, \\mathinner {|{-}\\rangle } \\right\\rbrace $ .",
"This example is firmly in the quantum realm of guesswork, as more information about the side information system $B$ can be obtained via a quantum measurement than a classical one (in the computational basis, that is).",
"We establish an analytic upper bound on the guesswork by considering a particular POVM and associated sequences of guesses.",
"We consider the POVM consisting of two orthogonal projectors $|\\theta \\rangle \\!\\langle \\theta |$ and $|\\theta ^{\\perp }\\rangle \\!\\langle \\theta ^{\\perp }|$ with $\\mathinner {|{\\theta }\\rangle } := \\sin \\theta \\mathinner {|{0}\\rangle } + \\cos \\theta \\mathinner {|{1}\\rangle }$ .",
"If the outcome corresponding to $|\\theta \\rangle \\!\\langle \\theta |$ is obtained, then we guess in the order corresponding to $(1, +, -, 0)$ .",
"Similarly, the guessing order corresponding to the other outcome $\\theta ^{\\perp }$ is $(0, -, +, 1)$ , which is the reverse order.",
"By calculating the objective function of (REF ), we obtain $ G(X|B) &\\le \\begin{multlined}[t][5cm] \\frac{1}{2} \\Big ( 1 \\cdot \\cos ^2 \\theta + 2 \\cdot \\frac{1}{2} \\left(1 + \\sin 2 \\theta \\right) + \\\\ 3 \\cdot \\frac{1}{2} \\left( 1 - \\sin 2 \\theta \\right) + 4 \\cdot \\sin ^2 \\theta \\Big ) \\end{multlined} \\\\&= 1.75 + \\frac{3}{2} \\sin ^2 \\theta - \\frac{1}{4} \\sin 2 \\theta .$ With the aim of minimizing the guesswork, we choose $\\theta = \\frac{1}{2} \\arctan {\\frac{1}{3}}$ , and obtain the right-hand side of (REF ) as $\\frac{1}{4} \\left( 10 - \\sqrt{10} \\right) \\approx 1.709430$ .",
"Moreover, the SDP in (REF ) can be solved numerically to obtain the same value, providing a matching numerical lower bound; see [20] for the code involved, including a high-precision demonstration using the SDP solver SDPA-GMP [37] showing agreement to 200 digits.",
"We also consider a generalization of this example, where the side information states are chosen from the set $\\lbrace \\mathinner {|{0}\\rangle }, \\mathinner {|{1}\\rangle }, \\mathinner {|{\\psi (\\varphi )}\\rangle }, \\mathinner {|{\\psi (-\\varphi )}\\rangle }\\rbrace $ where $\\mathinner {|{\\psi (\\varphi )}\\rangle } = \\cos (\\varphi /2) \\mathinner {|{0}\\rangle } + \\sin (\\varphi /2) \\mathinner {|{1}\\rangle }$ .",
"The BB84 states are a special case of this ensemble with $\\varphi = \\pi /2$ .",
"For each of these ensembles, we compute the guesswork using our SDP formulation in (REF ).",
"The results are shown in Figure REF .",
"Furthermore, we can use this example to delineate the difference, or gap, between guesswork with classical and quantum information.",
"There are two ways of reducing the example of BB84 states to a classical setting: (a) restricting to measurements in the standard basis $\\lbrace \\mathinner {|{0}\\rangle }, \\mathinner {|{1}\\rangle } \\rbrace $ , or (b) replacing the side information states $\\mathinner {|{+}\\rangle }$ and $\\mathinner {|{-}\\rangle }$ with the maximally mixed qubit state $\\pi _2$ .",
"In both of these cases, the side information then takes the form of a random variable $Y$ .",
"The joint probability distribution of variables $XY$ is given as follows: Table: NO_CAPTIONGiven either value of $Y$ , one needs an average of $1.75$ guesses.",
"Hence in this classical analogue of the BB84 states, the guesswork is $1.75$ .",
"This is higher than the lower value ($\\approx 1.709$ ) that can be achieved by quantum measurements."
],
[
"Qubit trine states as side information",
"In this example, we consider the problem of calculating guesswork with three uniformly distributed letters to guess from, each correlated to one of the three qubit trine states [38].",
"That is, $\\rho _{XB} = \\frac{1}{3} \\sum _{k = 1}^3 |x_k\\rangle \\!\\langle x_k|_X \\otimes |\\psi _k\\rangle \\!\\langle \\psi _k|_B$ where $\\mathinner {|{\\psi _k}\\rangle } = \\cos \\left(k \\tfrac{2\\pi }{3}\\right) \\mathinner {|{0}\\rangle } + \\sin \\left(k \\tfrac{2\\pi }{3}\\right) \\mathinner {|{1}\\rangle }$ .",
"As in the previous example of the BB84 states, here too we establish an analytic upper bound on the guesswork by considering a particular POVM.",
"Consider that the measurement is characterized by two orthogonal projectors $|\\theta \\rangle \\!\\langle \\theta |$ and $|\\theta ^{\\perp }\\rangle \\!\\langle \\theta ^{\\perp }|$ with $\\mathinner {|{\\theta }\\rangle } := \\cos \\theta \\mathinner {|{0}\\rangle } + \\sin \\theta \\mathinner {|{1}\\rangle }$ .",
"For the sake of simplicity, we restrict to $\\theta \\in [0, \\pi /2]$ .",
"For the outcome corresponding to $\\theta $ , we guess in the order corresponding to $(\\psi _2, \\psi _3, \\psi _1)$ , and for the outcome corresponding to $\\theta ^\\perp $ , we guess in the order corresponding to $(\\psi _1, \\psi _3, \\psi _2)$ .",
"The objective function in (REF ) leads us to $ \\begin{split}G(X|B) &\\le \\frac{1}{2} \\cdot \\frac{2}{3} \\Big ( 1 \\cdot \\left(\\cos ^2 (\\theta - 4\\pi /3) + \\sin ^2 (\\theta - 2\\pi /3) \\right) \\\\ & \\qquad + 2\\cdot \\left( \\cos ^2 \\theta + \\sin ^2 \\theta \\right) \\\\ & \\qquad + 3 \\cdot \\left( \\cos ^2 (\\theta - 2\\pi /3) + \\sin ^2 (\\theta - 4\\pi /3) \\right) \\Big ) \\\\&= \\frac{4}{3} + \\frac{2}{3} \\left( \\cos ^2 (\\theta - 2\\pi /3) + \\sin ^2 (\\theta - 4\\pi /3) \\right).\\end{split}$ The guesswork $G(X|B)$ is minimized by setting $f^{\\prime }(\\theta ) = 0$ where $f(\\theta ) = \\cos ^2 (\\theta - 2\\pi /3) + \\sin ^2 (\\theta - 4\\pi /3)$ .",
"This leads to $\\theta = \\pi /4$ and $G(X|B) = (2 - 1/\\sqrt{3}) \\approx 1.422649$ .",
"The SDP (REF ) is solved for this example as well, and the numerical result shows agreement with the analytic upper bound up to a relative tolerance of at least $10^{-7}$ .",
"Further, in this example, if we restrict to measuring in the standard basis (mimicking the classical analogue of the side information), then the average number of guesses needed is $1.5$ , in contrast to $\\approx 1.4227$ guesses needed using the optimal quantum measurement."
],
[
"Guesswork as a security criterion: certifying an imperfect key state",
"A primitive in any cryptography scheme is the establishment of a secret key between two communicating parties.",
"Quantum key distribution (QKD) protocols can produce a certifiably secure secret key by using pre-shared entanglement [39].",
"However, if the protocol is not implemented perfectly, as is the case in realistic scenarios, then some information can leak out to an eavesdropper.",
"How secure is the key obtained in this “imperfect” scenario?",
"In other words, if there is a small deviation from the ideal protocol, how does it affect the security of the key?",
"We address this question considering the guesswork as a security criterion.",
"Consider two systems $K$ and $E$ , where $K$ denotes the key system and encodes the secret key, and $E$ is the system held by the eavesdropper.",
"An ideal key state is of the form $\\pi _K \\otimes \\rho _E$ where $\\pi _K$ refers to the maximally mixed state on the key system.",
"This means that the eavesdropper can learn nothing about the key with access to the $E$ system alone.",
"An imperfect key, generally, is the joint state $\\rho _{KE}$ .",
"Consider the promise that the imperfect key state is $\\varepsilon $ -close to an ideal one in normalized trace distance: $ \\frac{1}{2} \\Vert \\rho _{KE} - \\pi _K \\otimes \\rho _E \\Vert _1 \\le \\varepsilon .$ For an ideal key state, the expected guesswork for the eavesdropper is $\\sum _{k} \\frac{1}{|K|} k = \\frac{|K|+1}{2}$ where $|K|$ indicates the cardinality of the alphabet associated to system $K$ .",
"We have the following result: Theorem 9 For an imperfect key state satisfying the promise in (REF ), the following bound on guesswork holds $G(K|E) \\ge \\frac{|K| + 1 }{2} - |K| \\varepsilon .$ Proof.",
"We apply the result of Lemma REF below, which holds for the case of guesswork with classical side information.",
"We know from Theorem REF that a measured strategy for guesswork is equivalent to a quantum strategy.",
"Using that fact, and by combining the promise $\\frac{1}{2} \\Vert \\rho _{KE} - \\pi _K \\otimes \\rho _E \\Vert _1 \\le \\varepsilon $ and the result in Lemma REF , we have (REF ).",
"thm:imperfect-key provides a robustness guarantee that imperfect key states continue to have near-maximal guesswork, if they remain close to an ideal key state in normalized trace distance.",
"Our proof of the lower bound in (REF ), as given above, is a consequence of the following extension of an analogous result pertaining to guesswork, due to Pliam [40].",
"Pliam's inequality states that for any random variable $X$ with probability distribution $p_X$ , $ \\frac{| \\mathcal {X} |+1}{2} - G(X) \\le \\frac{1}{2} | \\mathcal {X} | \\, \\Vert p_X - u_X \\Vert _1,$ where $G(X)$ denotes the guesswork and $u_X$ denotes the uniform distribution.",
"Lemma 10 For random variables $X$ and $Y$ , the following bound holds for the guesswork: $\\frac{\\left|\\mathcal {X}\\right|+1}{2}-G(X|Y)\\le \\frac{\\left|\\mathcal {X}\\right|}{2}\\left\\Vert p_{XY}-u_{X}\\otimes p_{Y}\\right\\Vert _{1}.$ Proof.",
"Consider the case of a joint distribution $p_{XY}$ , with conditional distribution $p_{X|Y}$ and marginal distribution $p_{Y}$ , and suppose that the value of $y$ is fixed.",
"Then we can invoke Pliam's bound (REF ) to find that $\\frac{\\left|\\mathcal {X}\\right|+1}{2}-G(X|Y=y)\\le \\frac{\\left|\\mathcal {X}\\right|}{2}\\left\\Vert p_{X|Y=y}-u_{X}\\right\\Vert _{1},$ where the notation $G(X|Y=y)$ indicates the guesswork (without side information) of a random variable distributed according to $p_{X|Y}(\\cdot |y)$ .",
"Taking the expectation of both sides with respect to the random variable$~Y$ , we find that $&\\frac{\\left|\\mathcal {X}\\right|+1}{2}-\\sum _{y}p_{Y}(y)G(X|Y=y) \\\\& \\qquad \\le \\frac{\\left|\\mathcal {X}\\right|}{2}\\sum _{y}p_{Y}(y)\\left\\Vert p_{X|Y=y}-u_{X}\\right\\Vert _{1}\\\\& \\qquad =\\frac{\\left|\\mathcal {X}\\right|}{2}\\sum _{y}p_{Y}(y)\\sum _{x}\\left|p_{X|Y}(x|y)-u_{X}(x)\\right|\\\\& \\qquad =\\frac{\\left|\\mathcal {X}\\right|}{2}\\sum _{y}\\sum _{x}\\left|p_{X|Y}(x|y)p_{Y}(y)-u_{X}(x)p_{Y}(y)\\right|\\\\&\\qquad =\\frac{\\left|\\mathcal {X}\\right|}{2}\\sum _{y}\\sum _{x}\\left|p_{XY}(x,y)-u_{X}(x)p_{Y}(y)\\right|\\\\& \\qquad =\\frac{\\left|\\mathcal {X}\\right|}{2}\\left\\Vert p_{XY}-u_{X}\\otimes p_{Y}\\right\\Vert _{1}.$ Using the fact that $\\sum _{y}p_{Y}(y)G(X|Y=y)=G(X|Y),$ we can conclude the generalization of (REF ) in the presence of classical side information $\\frac{\\left|\\mathcal {X}\\right|+1}{2}-G(X|Y)\\le \\frac{\\left|\\mathcal {X}\\right|}{2}\\left\\Vert p_{XY}-u_{X}\\otimes p_{Y}\\right\\Vert _{1}.$ This concludes the proof.",
"Remark 11 Note that prop:guessworkLipschitz gives the following continuity bound for the guesswork near $\\pi _K \\otimes \\rho _E$ : $|G(K|E)_{\\rho } - G(K|E)_{\\pi \\otimes \\rho _E}| \\le 2 \\varepsilon |K| ,$ and hence $G(K|E)_{\\rho } \\ge \\frac{|K|+1}{2} - 2 |K| \\varepsilon .$ Thus, the bound in (REF ) is slightly better than what we obtain by employing prop:guessworkLipschitz."
],
[
"Open questions",
"Guesswork presents an operationally-relevant method to quantify uncertainty, and has been relatively unexplored in the presence of quantum side information.",
"We hope our investigation opens the door to further analysis of the guesswork and methods to compute it.",
"In particular, our work leaves open the following questions: Does equality hold in (REF )?",
"If so, the single-letter expression $\\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\ln G(X^n|B^n)_{\\rho ^{\\otimes n}} = \\widetilde{H}_{\\frac{1}{2}}^{\\uparrow }(X | B)_{\\rho }$ holds, matching the classical case [4].",
"Ref.",
"[41] presented variational expressions for the measured Rényi divegerences $D^M_\\alpha $ and showed how those lead to efficient ways to compute the divergences.",
"Are there similar variational formulas for $H^{\\uparrow , M}_\\alpha (X|Y)_\\rho $ ?",
"That could similarly provide an efficient way to compute the quantity."
],
[
"Acknowledgements",
"E.H. would like to thank Harsha Nagarajan for pointing out the transformation in [26].",
"E.H. is supported by the Cantab Capital Institute for the Mathematics of Information (CCIMI).",
"V.K.",
"acknowledges support from the Louisiana State University Economic Development Assistantship.",
"M.M.W.",
"acknowledges support from the US National Science Foundation through grant no.",
"1907615."
]
] | 2001.03598 |
[
[
"Deformable Groupwise Image Registration using Low-Rank and Sparse\n Decomposition"
],
[
"Abstract Low-rank and sparse decompositions and robust PCA (RPCA) are highly successful techniques in image processing and have recently found use in groupwise image registration.",
"In this paper, we investigate the drawbacks of the most common RPCA-dissimi\\-larity metric in image registration and derive an improved version.",
"In particular, this new metric models low-rank requirements through explicit constraints instead of penalties and thus avoids the pitfalls of the established metric.",
"Equipped with total variation regularization, we present a theoretically justified multilevel scheme based on first-order primal-dual optimization to solve the resulting non-parametric registration problem.",
"As confirmed by numerical experiments, our metric especially lends itself to data involving recurring changes in object appearance and potential sparse perturbations.",
"We numerically compare its peformance to a number of related approaches."
],
[
"Groupwise Image Registration",
"The problem of aligning one image with another image of the same object is a well-studied problem in image processing and variational methods have proven successful for the task [23], [32].",
"However, many application scenarios involve data comprised of more than two images, as in the case of image data gathered over time, which necessitates groupwise methods.",
"Naive pairwise techniques, that select one image from the group as a fixed reference and register all other images to the reference have been shown to be inconsistent with respect to registration accuracy (depending on the choice of the reference) and are generally deemed inferior to groupwise methods [22], [18].",
"These allow all images of the group to be deformed simultaneously and therefore operate on an implicit reference.",
"A crucial step in solving any image registration problem is the selection of a suitable dissimilarity metric on pairs or groups of images.",
"In the past, both generalizations of established dissimilarity metrics for the classic two image problem and new concepts have been proposed to measure the distance between a group of $N > 2$ images.",
"Examples for the former case include the variance-measure found in [2], [22] that extends the well-known sum of squared distances, different generalizations of the mutual information from [29], [18] and a multi-image version of the normalized gradient fields-measure in [6].",
"One example of a newly developed metric that is also related to the metric proposed in this work is $D_{\\text{PCA2}}$ from [18].",
"Given $N$ images $T_1, \\ldots , T_N \\in \\mathbb {R}^{m \\times n}$ , this measure operates on the so-called Casorati matrix $M_{T_1, \\ldots , T_N} := [ \\operatorname{vec}(T_1), \\ldots , \\operatorname{vec}(T_N) ] \\in \\mathbb {R}^{m n \\times N},$ where $\\operatorname{vec}(\\cdot )$ denotes a column-major vectorization.",
"In $D_{\\text{PCA2}}$ , one proceeds to penalize a weighted sum of the (nonnegative) eigenvalues $\\lambda _i$ of the correlation matrix $K := \\frac{\\Sigma ^{-1} (M_{T_1, \\ldots , T_N} - \\bar{M})^{\\top } (M_{T_1, \\ldots , T_N} - \\bar{M}) \\Sigma ^{-1}}{N - 1}.$ $\\bar{M}$ is the repeated columnwise mean of $M_{T_1, \\ldots , T_N}$ and $\\Sigma $ is diagonal with diagonal elements given by the standard deviations of the columns of $M_{T_1, \\ldots , T_N}$ .",
"To be exact, the metric is given by $D_{\\text{PCA2}}(T_1, \\ldots , T_N) := \\sum _{i=1}^N i \\lambda _i.$ As the number of nonzero eigenvalues of $K$ is equal to the rank of $M_{T_1, \\ldots , T_N}$ , minimizing $D_{\\text{PCA2}}$ promotes low-rankness of $M_{T_1, \\ldots , T_N}$ and similarity between images is modeled as linear dependency.",
"Note that apart from the $\\Sigma ^{-1}$ -weighting in (REF ), the eigenvalues $\\lambda _i$ correspond to variances along the principal components of $M_{T_1, \\ldots , T_N}$ , which emphasizes the relation to the eponymous PCA.",
"$D_{\\text{PCA2}}$ will serve as a comparison method for our proposed metric in the experiments of section ."
],
[
"Robust PCA",
"As the classic PCA is known for its sensitivity towards sparsely distributed outliers, such methods are prone to fail for datasets involving partially unreliable data or strong changes in image intensity over time.",
"To overcome this issue, different versions of a Robust PCA (RPCA) were proposed in the literature – see [14] for an extensive comparison.",
"The most widely-used RPCA-variant is arguably the Principal Component Pursuit (PCP) from [9], [7].",
"PCP is derived as a convex relaxation of the combinatorial optimization problem $\\min _{L, E \\in \\mathbb {R}^{p \\times q}} \\operatorname{rank}(L) + ||E||_0 \\quad \\mbox{s.t. }",
"M = L + E$ for given data $M \\in \\mathbb {R}^{p \\times q}$ .",
"The term $||E||_0$ denotes the number of non-zero entries of $E$ .",
"Replacing both summands of (REF ) with their convex hulls yields $\\min _{L \\in \\mathbb {R}^{p \\times q}} ||L||_* + ||M - L||_1,$ which is convex in $L$ and thus poses a more tractable optimization problem.",
"$||L||_*$ is the so-called nuclear norm, defined as the sum of all singular values of $L$ (see [12]) and $|| M - L ||_1 = \\sum _{i = 1}^p \\sum _{j = 1}^q | M_{i, j} - L_{i, j} |$ is a $\\ell _1$ -type norm.",
"Especially recall the relationship between the singular values $\\sigma _i$ and the rank of a matrix: $\\operatorname{rank}(A) = \\#\\lbrace \\sigma _i(A) > 0\\rbrace $ (see again [12]).",
"The decomposition of $M$ generated by (REF ) is usually referred to as a low-rank and sparse decomposition, in which $L$ is of low rank and $E = M - L$ is sparse.",
"PCP has previously been used in the context of groupwise image registration by [27], [16], [15], [20].",
"Primarily tackled therein were datasets for which low-dimensional approximations using PCA-based techniques were not applicable due to occlusions, local changes in image intensity (for the case of DCE-MRI data) and irregular pathologies in medical image data.",
"In all these publications, the data matrix $M$ for (REF ) was constructed as a Casorati matrix (REF ).",
"The authors of [15], [20] however only used low-rank and sparse decompositions as preprocessing steps and performed subsequent registrations on the generated low-rank components $L$ with different algorithms.",
"Contrary to that, [27], [16] both used the optimal value of (REF ) as a metric for the similarity of a set of given images ${T_1, \\ldots , T_N}$ .",
"Section of this paper will present a deeper analysis of PCP as a distance measure.",
"We argue that PCP has some inherent drawbacks: Perfect alignments of all $T_i$ often constitute local minimizers of PCP in only very narrow neighborhoods.",
"At the same time, degenerated deformations result in comparatively lower energies.",
"To overcome these issues, we present in this work a modification of PCP that is still convex and therefore easy to optimize."
],
[
"Proposed Approach",
"Precisely, we propose to use the following groupwise dissimilarity measure: $D_{\\delta \\text{-RPCA}}(T_1, \\ldots , T_N) := \\min _{L \\in \\mathbb {R}^{m n \\times N}} || M_{T_1, \\ldots , T_N} - L ||_1 \\quad \\mbox{s.t. }",
"|| L - \\bar{L} ||_* \\le \\nu .$ Here $M_{T_1, \\ldots , T_N}$ is again the Casorati matrix (REF ), $\\bar{L}$ is the repeated columnwise mean of $L$ and $\\nu \\ge 0$ is a suitable threshold for the nuclear norm.",
"The intuition behind (REF ) is to jointly measure the $\\ell _1$ -distance between the input images and their optimal approximations in a low-dimensional linear subspace.",
"Details are given in section .",
"Our main contributions in this work involve the following: A novel technique for low-rank and sparse decompositions that results in a more suitable distance metric for groupwise registration tasks than previous approaches.",
"A less restrictive uniqueness constraint than the one commonly employed in the literature.",
"A multi-level strategy with theoretically justified scaling that solves the registration model in an iterative process and that uses first-order primal-dual optimization techniques to solve the subproblems."
],
[
"Other Related Work",
"Major differences between our approach and related methods for variational groupwise registration are as follows: Besides the fact that the $D_{\\text{PCA2}}$ -measure from [18] is based on the classic PCA (whereas ours is based on RPCA), the authors suggest a parametric deformation model based on B-Splines.",
"Instead, we employ a non-parametric model that is fully deformable and that is explicitly (and flexibly) regularized through a total variation penalty (see section ).",
"Regularization in [18] is handled implicitly through grid point spacing, and the same is true for all of [3], [2], [22], [18], [13], [29], as they use B-Spline deformations in the same manner.",
"Concerning the two PCP-based registration approaches [27], [16], the former is even further restricted to affine deformations, while the latter operates on light-field data, for which a geometric relationship between input images is known a priori and is exploited in the registration process.",
"Another non-parametric approach is presented in [6].",
"While also based on rank minimization, the authors use normalized image gradients as feature vectors and define alignments locally (instead of image intensities as features and global alignments as in this article).",
"A continuation of [6] is found in [5], which generalizes the former approach to different kinds of feature vectors and formulates alignments globally."
],
[
"Outline and Contributions",
"The remainder of this article is organized as follows: In section , we analyze the established PCP-metric and derive our proposed approach as a replacement.",
"In section , the total variation is discussed as a regularizer for our model and a new uniqueness constraint for groupwise image registration algorithms is introduced.",
"In section , an in-depth account of the optimization strategy and its implementation is given, including a multilevel scheme with theoretically justified scaling.",
"In the subsequent sections and , we introduce the benchmark data and present a numerical comparison to related approaches.",
"Section gives concluding remarks."
],
[
"Classical Approach",
"The classical PCP image distance from [27], [16] is given by $D_{\\text{PCP}} (T_1, \\ldots , T_N) := \\min _{L \\in \\mathbb {R}^{m n \\times N}} || L ||_* + \\mu || M - L ||_1$ with $M$ as a Casorati matrixFrom here on, we omit the explicit notation of the dependence of $M$ on $T_1, \\ldots , T_N$ for readability..",
"The parameter $\\mu > 0$ controls the weighting between the requirement on $L$ to be of low rank and the requirement on ${E := M - L}$ to be sparse.",
"Figure: Depicted are five subsequent frames T 1 ,...,T 5 T_1, \\ldots , T_5 from a MRI sequence that serve as input for the experiments in Fig.",
":In order to analyze the behavior of a given dissimilarity measure, its energy is determined while one image is kept fixed and the remaining images are warped uniformly in a prescribed manner.Due to their short temporal offset, the input images T 1 ,...,T 5 T_1, \\ldots , T_5 can be regarded as alignedFigure: Experiments on the distance measures D PCP D_{\\text{PCP}} and D δ-RPCA D_{\\delta \\text{-RPCA}}.The classical D PCP D_{\\text{PCP}} energies (second row) exhibit only very narrow minima at the position u 0 =0u^0 = 0 of perfect alignment.Even worse so, global minimizers for all experiments except the rotation are given by degenerated transformations.The proposed modification D δ-RPCA D_{\\delta \\text{-RPCA}} (third row) resolves these problems: u 0 =0u^0 = 0 constitutes a global minimizer across all experiments.Furthermore, degenerated deformations generally result in high energies and are therefore not favored by this metricIn order to assess the general applicability of (REF ) in the context of non-parametric groupwise registration, we conducted a number of experiments to examine the behavior of $D_{\\text{PCP}}$ under certain predefined deformations of the images $T_1, \\ldots , T_N$ .",
"To this end, we define an image $T \\in \\mathbb {R}^{m \\times n}$ as a function of a deformation $u \\in \\mathbb {R}^{m \\times n \\times 2}$ (given over the same grid) through linear interpolation – see, e.g., [24].",
"Using this convention, the experiments were performed by evaluating $D_{\\text{PCP}}(T_1, T_2(u^{j}), \\ldots , T_N(u^{j}))$ for the four cases of the deformation sequence $(u^{j})_{j=-k, \\ldots , k}$ describing a translation, a rotation, a scaling and a shearing.",
"$T_1$ therefore acted as a fixed reference, while $T_2, \\ldots , T_N$ were warped uniformly by $u^j$ .",
"A schematic depiction of each transformation sequence is given in the first row of Fig.",
"REF .",
"Test data was comprised of the $N = 5$ frames from a cardiac MRI sequence, that are displayed in Fig.",
"REF .",
"As suggested by [7], the weighting parameter for (REF ) was chosen as $\\mu = (m n)^{-1/2}$ .",
"The energy plots of $D_{\\text{PCP}}$ for all four experiments are shown in the second row of Fig.",
"REF .",
"Additionally, their respective decompositions into the two summands $||L^*||_*$ and $\\mu ||M - L^*||_1$ are displayed, where $L^*$ denotes the minimizer of (REF ) over the variable $L$ .",
"The results show that $D_{\\text{PCP}}$ has two major shortcomings that are unfavorable for the purposes of image registration.",
"Firstly, the point $u^0 = 0$ at which the $N$ images are most appropriately aligned only marks a local minimizer of $D_{\\text{PCP}}$ in a very narrow neighborhood of $u^0$ in the translation experiment.",
"In the scaling experiment, $u^0$ even constitutes a global maximizer.",
"Secondly, both the left or the right endpoints of each energy plot, i.e., the most degenerated of all evaluated deformations $u^j$ , represent global minimizers in every experiment except for the rotation.",
"This is especially problematic in case of the translation, since constant translations are also not penalized by any regularizer, that is based on derivatives of deformations.",
"As a result, we consider $D_{\\text{PCP}}$ unsuitable as a distance metric in the general case."
],
[
"Proposed Modified Measure",
"Based on these observations, we propose a new distance metric that modifies $D_{\\text{PCP}}$ in two aspects.",
"As a first step, the $||\\cdot ||_*$ -penalty term in (REF ) is replaced by a hard constraint to the set $\\lbrace || \\cdot ||_* \\le \\nu \\rbrace $ for some suitable threshold $\\nu \\ge 0$ .",
"As a second step, we propose not to constrain the nuclear norm of $L$ itself, but to constrain that of the centered variable $L - \\bar{L}$ instead.",
"To this end, let $\\bar{L} := (\\sum _{i = 1}^{N} \\frac{l_i}{N}) \\cdot \\mathbf {1}_{1 \\times N} \\in \\mathbb {R}^{m n \\times N}$ denote the matrix, in which every column is given by the average of the columns $l_i$ of $L$ .",
"The proposed dissimilarity measure, which we term $\\delta $ -RPCA, is then given by $D_{\\delta \\text{-RPCA}}(T_1, \\ldots , T_N) :=\\min _{L \\in \\mathbb {R}^{m n \\times N}} ||M - L||_1 + \\delta _{\\lbrace ||\\cdot ||_* \\le \\nu \\rbrace }(L - \\bar{L}).$ Following the general convention in convex analysis [30], [31], $\\delta _{S}$ denotes an indicator function for a constraint set $S$ , which is defined as $\\delta _{S}(x) = 0$ for $x \\in S$ and $\\delta _{S}(x) = +\\infty $ otherwise.",
"The first modification is based on the observation that all energy curves of the $||\\cdot ||_*$ -term in Fig.",
"REF exhibit at least a local maximum at $u^0$ and therefore counteract the local minimum of the $||\\cdot ||_1$ -term in the joint $D_{\\text{PCP}}$ -energy.",
"Remodeling the low-rank requirement on $L$ as a hard constraint resolves this issue by removing the nuclear norm from the energy as a summand.",
"The second modification of centering $L$ further acts to model the low-rank requirement appropriately: Consider the nuclear norm of the two matrices $A_1 = a \\cdot (1, 0, \\ldots , 0) \\in \\mathbb {R}^{p \\times q}$ and $A_2 = a \\cdot \\mathbf {1}_{1 \\times q} \\in \\mathbb {R}^{p \\times q}$ for some $a \\in \\mathbb {R}^p \\setminus \\lbrace 0\\rbrace $ .",
"While both matrices are obviously of rank one, a short derivation shows that one has $||A_1||_* = ||a||_2 < \\sqrt{q} ||a||_2 = ||A_2||_*$ for all $q > 1$ .",
"In terms of the registration model, this means that a smaller nuclear norm for the uncentered variable $L$ can be achieved by shifting all deformable images out of the image domain – thereby replacing them with the boundary value of zero – than by aligning them inside the images domain.",
"The continuation of the above example shows that this situation, which is highly undesirable for the purpose of image registration, is reversed when dealing with centered variables.",
"These are given by $A_1 - \\bar{A_1} = a \\cdot (1-q^{-1}, -q^{-1}, \\ldots , -q^{-1})$ and $A_2 - \\bar{A_2} = 0$ respectively and in fact, the equivalent of relation (REF ) now reads $||A_2 - \\bar{A_2}||_* = 0 < \\sqrt{1 - q^{-1}} ||a||_2 = ||A_1 - \\bar{A_1}||_*.$ As a consequence, (REF ) does not favor shifting the deformable images out of the image domain and is therefore more suited for image registration.",
"Crucially, $D_{\\delta \\text{-RPCA}}$ is still convex in the variable $L$ , since $\\lbrace || \\cdot ||_* \\le \\nu \\rbrace $ constitutes the level set of a convex function and is therefore convex [30].",
"Repeating the above experiments for $D_{\\delta \\text{-RPCA}}$ with the choice of $\\nu = 0.9 ||M - \\bar{M}||_*$ for every set of deformed images in $M$ , one obtains the energy plots in the third row of Fig.",
"REF .",
"In contrast to $D_{\\text{PCP}}$ , the modified metric $D_{\\delta \\text{-RPCA}}$ shows global minimizers at the point $u^0 = 0$ across all cases.",
"Additionally, the global maximizer of each curve is found towards its left or right endpoint and consequently results from a degenerated deformation.",
"In conclusion, the two main issues of $D_{\\text{PCP}}$ as a distance function for registration tasks are hence resolved by $D_{\\delta \\text{-RPCA}}$ .",
"Apart from the interpretation of $D_{\\delta \\text{-RPCA}}$ as a modified version of $D_{\\text{PCP}}$ , it can also be interpreted in the sense described as follows.",
"First consider the case $\\nu = 0$ .",
"This implies $L = \\bar{L}$ , which in turn implies constant columns $l_1 = \\ldots = l_N$ of $L$ .",
"Using this, one can solve (REF ) analytically for $L$ by recalling, that $\\ell _1$ -distance minimization problems of the type $\\operatorname{argmin}_{x \\in \\mathbb {R}} \\sum _{i = 1}^K |x - y_i|$ are solved by the median of $(y_1, \\ldots , y_K)$ [4].",
"As a consequence, the constant columns of $L$ in the problem above are given by the pointwise median of $T_1, \\ldots , T_N$ and $D_{\\delta \\text{-RPCA}}$ represents the remaining $\\ell _1$ -distance between the input images and that median.",
"In the case of $\\nu > 0$ , $D_{\\delta \\text{-RPCA}}$ can now more generally be interpreted as the joint $\\ell _1$ -distance between the images $T_1, \\ldots , T_N$ and their individual (optimal) approximations $l_1, \\ldots , l_N$ with deviations from the mean $\\bar{l} := \\sum _{i = 1}^N \\frac{l_i}{N}$ restricted to a low-dimensional linear subspace.",
"Consequently, we deem (REF ) especially suited for image groups with inherent low-dimensional structure such as image sequences with strong or pronounced temporal repetition."
],
[
"Total Variation Regularization",
"Total variation (TV) is a popular choice for regularizing motion fields in applications of both optical flow estimation and image registration due to its distinguishing feature of allowing discontinuities in the solution.",
"TV therefore sets itself apart from other common regularizers such as diffusive, elastic or curvature energies that favor smooth transformations.",
"Exemplary early applications of TV regularization for optical flow estimation can be found in [25], [35] and for image registration in [28], [34].",
"In the context of medical image processing, TV regularization is particularly interesting when modeling non-smooth sliding motions, since it eliminates the necessity to explicitly mask all sliding interfaces beforehand [10].",
"We shortly recapitulate that the total variation for vector fields $\\upsilon \\in L^1(\\Omega , \\mathbb {R}^d)$ over $\\Omega \\subset \\mathbb {R}^d$ can be defined as $\\operatorname{TV}(\\upsilon ) = \\int _{\\Omega } \\mathop {}\\!\\mathrm {d}|D \\upsilon |,$ where $D \\upsilon $ denotes the distributional (measure-valued) derivative of $\\upsilon $ with values in $\\mathbb {R}^{d \\times d}$ .",
"In case of $\\upsilon \\in C^1(\\Omega , \\mathbb {R}^d)$ , (REF ) is equivalent to $\\operatorname{TV}(\\upsilon ) = \\int _{\\Omega } ||\\nabla \\upsilon ||_2 \\mathop {}\\!\\mathrm {d}x.$ For all further details, we refer to [1].",
"In our registration model, we assume rectangular domains $\\Omega \\subset \\mathbb {R}^2$ and employ a standard discretization scheme with cell-centered grids of resolution $m \\times n$ and grid spacings of $(h_1, h_2) \\in \\mathbb {R}_{> 0}^2$ in the two coordinate directions.",
"Optimization is performed over discrete displacement fields $u^k \\in \\mathbb {R}^{m \\times n \\times 2}$ , for which we use finite forward differences and Neumann boundary conditions to discretize (REF ).",
"Following [34], we use the notation $||v||_{2, 1} := \\sum _{i = 1}^{p} || (v_i, v_{i + p}, v_{i + 2 p}, v_{i + 3 p}) ||_2$ for $v \\in \\mathbb {R}^{4 p}$ and obtain a discretization of (REF ) $\\operatorname{TV}^h(u^k) := h_1 h_2 || G \\operatorname{vec}(u^k) ||_{2, 1}.$ Therein, $G \\in \\mathbb {R}^{4 m n \\times 2 m n}$ denotes the finite difference operator with the aforementioned characteristics."
],
[
"Uniqueness Constraint",
"As our model does not make use of an explicit reference image that all other images are aligned to, we need to employ an additional constraint on the displacements $u^1, \\ldots , u^N$ in order to ensure the uniqueness of a solution.",
"This can be seen from the simple example, in which $T_1, \\ldots , T_N$ display uniform objects, e.g., white rectangles, before a black background.",
"Consider the case of a perfect alignment $T_1(u^1) = \\ldots = T_N(u^N)$ of these rectangles inside the image domain $\\Omega $ .",
"If all deformations $u^k$ are simultaneously offset by $t \\in \\mathbb {R}^2$ , such that the new deformations $\\hat{u}^k$ still align $T_1, \\ldots , T_N$ inside the common domainTo be exact, these are defined as $\\hat{u}_{i,j,c}^k := u_{i,j,c}^k + t_c$ for all $i = 1, \\ldots , m$ , $j = 1, \\ldots , n$ , $c = 1, 2$ and $k = 1, \\ldots , N$ ., then $(\\hat{u}^k)_{k=1,\\ldots ,N}$ constitute a solution equal to $(u^k)_{k=1,\\ldots ,N}$ both in terms of $D_{\\delta \\text{-RPCA}}$ and $\\operatorname{TV}^h$ .",
"For $\\operatorname{TV}^h$ , this is explained by (REF ) solely penalizing derivatives of deformation fields which are always invariant to translations.",
"The invariance for $D_{\\delta \\text{-RPCA}}$ is due to the equivalence of an offset by $t$ and a simple reordering of the pixels between $T_k(u^k)$ and $T_k(\\hat{u}^k)$ (due to the zero boundary condition).",
"Clearly, the $\\ell _1$ -term in (REF ) is invariant to any reordering and the same is true for the nuclear norm constraint, since a consistent reordering of all $T_k(u^k)$ results in a row permutation of the Casorati matrix $M = [\\operatorname{vec}(T_1(u^1)) | \\ldots | \\operatorname{vec}(T_N(u^N))]$ .",
"As a short derivation shows, a row permutation does not affect the singular values of a matrix: Let $A \\in \\mathbb {R}^{p \\times q}$ be an arbitrary matrix and let $P \\in \\lbrace 0, 1\\rbrace ^{p \\times p}$ be a permutation.",
"If a singular value decomposition (SVD) of $A$ is given by $A = U \\Sigma V^{\\top } $ , then $P A = (P U) \\Sigma V^{\\top }$ constitutes a valid SVD of $P A$ due to $P U$ still being orthogonal, i.e., $(P U)^{\\top } (P U) = U^{\\top } P^{\\top } P U = U^{\\top } U = I.$ Thus, the singular values on the diagonal of $\\Sigma $ stay unaffected and so does the nuclear norm $||P A||_* = ||A||_*$ .",
"In order to eliminate this remaining degree of freedom from the model, we impose an additional constraint on the deformations $u^1, \\ldots , u^N$ , enforcing the mean (or equivalently the sum) over all deformations and grid points to be zero in each coordinate direction: $\\frac{1}{(N m n)} \\sum _{k = 1}^N \\sum _{i = 1}^m \\sum _{j = 1}^n u_{i, j, c}^k \\overset{!",
"}{=} 0 \\quad \\forall c \\in \\lbrace 1, 2\\rbrace .$ Note that [22], [18], [13], [29] constrain their deformations in a related manner by demanding the mean of all deformations to be zero at every grid point as first introduced by [3].",
"The difference however is, that (REF ) only imposes one constraint per dimension instead of one constraint per grid point and dimension.",
"As a result, (REF ) restricts the space of feasible solutions much less severely while still ensuring uniqueness."
],
[
"Implementation & Optimization",
"In this section, we present an optimization scheme for our groupwise registration model that is strongly related to the work in [16].",
"First we combine all components derived in the previous sections into the complete registration model $\\begin{split}\\min _{\\begin{array}{c}u^1, \\ldots , u^N \\in \\mathbb {R}^{m \\times n \\times 2} \\\\ L \\in \\mathbb {R}^{m n \\times N}\\end{array}} \\ &|| [\\operatorname{vec}(T_1(u^1)), \\ldots , \\operatorname{vec}(T_N(u^N)] - L ||_1 + \\delta _{\\lbrace ||\\cdot ||_* \\le \\nu \\rbrace }( L - \\bar{L} ) \\\\[-4ex]& + \\mu \\sum _{k = 1}^N \\operatorname{TV}^h(u^k) + \\sum _{c = 1}^2 \\delta _{ \\lbrace \\langle \\mathbf {1}, \\cdot \\rangle = 0 \\rbrace } ((u_{\\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\textstyle \\bullet $}}}}\\end{split}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\scriptstyle \\bullet $}}}}$ , , c1, ..., u, , cN)), in which $\\mu > 0$ controls the regularization strength."
],
[
"Linearized Subproblems",
"In order to be able to apply convex optimization methods to (), one needs to deal with the non-linearity of the expressions $T_k(u^k)$ that leads to a non-convexity of the model.",
"As in [28], [35], [27], [16], an iterative linearization of the deformed images is used to overcome this issue.",
"Note that while a one-time linear approximation would also be possible in theory, the strong locality of such an approximation becomes prohibiting when larger deformations are required to align the images.",
"In the following, we assume all variables to be in vector format (including the values of all $T_k$ ) and for brevity's sake omit the explicit notation of reshaping operations like $\\operatorname{vec}(\\cdot )$ .",
"The linearization of $T_k$ can then be expressed as $T_k(u^k) \\approx T_k(\\tilde{u}^k) + \\nabla T_k(\\tilde{u}^k)^{\\top } \\cdot (u^k - \\tilde{u}^k)$ for a suitable point $\\tilde{u}^k$ .",
"This enables one to approximate the first term in () by $\\sum _{k = 1}^N || T_k(\\tilde{u}^k) + \\nabla T_k(\\tilde{u}^k)^{\\top } \\cdot (u^k - \\tilde{u}^k) - l_k ||_1.$ Using vectorized variables further allows one to rewrite the centering of $L$ as a linear operation $K L$ with $K = \\left( I_{N \\times N} - \\frac{\\mathbf {1}_{N \\times N}}{N} \\right) \\otimes I_{m n \\times m n} \\in \\mathbb {R}^{m n N \\times m n N}.$ Since solving () through iterative (re-)linearization amounts to solving a series of subproblems, we propose to treat is as a process, in which the threshold $\\nu $ is successively decreased to the threshold value for which the original problem () is meant to be solved.",
"Assuming a predefined number $n_{iter}$ of linearization steps and denoting the final threshold by $\\nu $ , we therefore employ a series of thresholds $\\nu _1 > \\nu _2 > \\ldots > \\nu _{n_{iter}} = \\nu $ for the iterative solution of the separate subproblems.",
"As a strategy to select these parameters, we propose to choose $\\nu $ relative to the nuclear norm of the centered input images and to progressively decrease $\\nu _t$ to that value by multiplication with a constant factor $\\alpha \\in (0, 1)$ .",
"More specifically, let $M$ denote the Casorati matrix of the input images and let $\\bar{M}$ denote the columnwise repetition of their mean.",
"If one now wants to meet a final threshold of $\\nu = \\beta || M - \\bar{M} ||_*$ for some $\\beta \\in (0, 1)$ and one employs a predefined number of $n_{iter}$ linearization steps, the proposed strategy amounts to choosing $\\nu _t = \\alpha ^t || M - \\bar{M} ||_* \\quad \\mbox{for } t = 1, \\ldots , n_{iter},$ where $\\alpha = \\beta ^{(1 / n_{iter})}$ ."
],
[
"Solving the Convex Subproblem",
"Denoting all entries of $u^k$ corresponding to the $c$ -th coordinate axis by $u^{k, c}$ ($u_{\\mathchoice{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\displaystyle \\bullet $}}}}{}{}{}}{\\mathbin {\\hbox{\\scalebox {.5}{$\\m@th \\textstyle \\bullet $}}}}$ , , ck$ in the non-vectorized notation of (\\ref {eq:full_model})), the $ t$-th subproblem now reads{\\begin{@align}{1}{-1}\\begin{split}\\min _{\\begin{array}{c}u^1, \\ldots , u^N \\\\ L \\end{array}} & \\sum _{k = 1}^N || T_k(\\tilde{u}^k) + \\nabla T_k(\\tilde{u}^k)^{\\top } \\cdot (u^k - \\tilde{u}^k) - l_k ||_1 + \\delta _{\\lbrace ||\\cdot ||_* \\le \\nu _t\\rbrace }( K L )\\\\[-2ex]& + \\mu \\sum _{k = 1}^N h_1 h_2 || G u^k ||_{2, 1} + \\sum _{c = 1}^2 \\delta _{ \\lbrace \\langle \\mathbf {1}, \\cdot \\rangle = 0 \\rbrace } ((u^{1, c}, \\ldots , u^{N, c})).\\end{split}\\end{@align}}We solve these subproblems using the primal-dual optimization algorithm~\\ref {alg:cp} from \\cite {Chambolle2011} that is designed for finding saddle-points of problems of the type\\begin{equation}\\min _{x \\in \\mathbb {R}^p} \\max _{y \\in \\mathbb {R}^q} \\ \\langle A x, y \\rangle + H(x) - F^*(y).\\end{equation}$ H: Rp (R{ } =: R)$, $ F*: Rq R$ are proper, lower-semi\\-continuous, convex functions and $ F*$ denotes the conjugate of another proper, lower-semi\\-continuous, convex function $ F: Rq R$.$ A Rq p$ further denotes a linear operator.As is well-known, (\\ref {eq:saddle_point_problem}) is equivalent to the primal minimization problem\\begin{equation}\\min _{x \\in \\mathbb {R}^p} \\ F(A x) + H(x).\\end{equation}For all details, we refer to \\cite [Chapter 11]{Rockafellar1998}.$ [t] InitializationInitialization WhileIterate over: Alg.",
"Choose $x^0 \\in \\mathbb {R}^{p}$ , $y^0 \\in \\mathbb {R}^{q}$ .",
"Set $\\bar{x}^0 \\leftarrow x^0$ .",
"Choose $\\tau , \\eta > 0$ s.t.",
"$\\tau \\eta || A ||_{\\sigma }^2 < 1$ where $|| A ||_{\\sigma } := \\max \\lbrace || A x ||_2 : x \\in \\mathbb {R}^p, ||x||_2 \\le 1 \\rbrace $ $n \\ge 0$ $&y^{n + 1} \\leftarrow (\\operatorname{id}+ \\eta \\partial F^*)^{-1} (y^n + \\eta A \\bar{x}^n) & \\nonumber \\\\&x^{n + 1} \\leftarrow (\\operatorname{id}+ \\tau \\partial H)^{-1} (x^n - \\tau A^{\\top } y^{n+1}) & \\\\&\\bar{x}^{n+1} \\leftarrow 2 x^{n+1} - x^n & \\nonumber $ Primal-dual Optimization Scheme [8] In our case, we bring () into the form () by assigning the first three terms of () to $F$ and the remaining uniqueness term to $H$ .",
"The primal variables for this problem are given by the union of all variables over which () is minimized, $x^{\\top } = \\left[ (u^1)^{\\top }, \\ldots , (u^N)^{\\top }, (l_1)^{\\top }, \\ldots , (l_N)^{\\top } \\right] \\in \\mathbb {R}^{3 N m n},$ and we define the linear operator $A$ to be $\\mbox{\\LARGE $A := $} \\left[\\begin{array}{c c c c c c}\\nabla T_1(\\tilde{u}^1)^{\\top } & & & -I_{m n} & & \\\\& \\ddots & & & \\ddots & \\\\& & \\nabla T_N(\\tilde{u}^N)^{\\top } & & & -I_{m n} \\\\[.5em]\\end{array}\\multicolumn{3}{c}{ \\mbox{\\LARGE $0$} } & \\multicolumn{3}{c}{ \\mbox{\\LARGE $K$} } \\\\[.5em]\\right.G & & & \\multicolumn{3}{c}{{3}{*}{ \\mbox{\\LARGE $0$} }} \\\\& \\ddots & & & & \\\\& & G & & & \\\\$ ], where $K$ and $G$ are as in (REF ) and (REF ) respectively.",
"This allows for a separable definition of the function $F$ by $F(z) := F_1(z^1) + F_2(z^2) + F_3(z^3)$ and $F_1(z^1) &:= || z^1 + b ||_1, \\\\F_2(z^2) &:= \\delta _{\\lbrace ||\\cdot ||_* \\le \\nu _t\\rbrace }(z^2), \\\\F_3(z^3) &:= \\mu \\sum _{k = 1}^N h_1 h_2 || z^{3, k} ||_{2, 1}, $ where $z^{3, k} := (z^3_{4 (k - 1) m n + 1}, \\ldots , z^3_{4 k m n})^{\\top }$ and where the vector $b = (b_1^{\\top }, \\ldots , b_N^{\\top })^{\\top }$ gathers the constants $b_k := T_k(\\tilde{u}^k) - \\nabla T_k(\\tilde{u}^k)^{\\top } \\tilde{u}^k$ for $k = 1, \\ldots , N$ .",
"As the remaining uniqueness term does not depend on $l_1, \\ldots , l_N$ , we define $H(x) := \\tilde{H}( (u^1, \\ldots , u^N) )$ and $\\tilde{H}( (u^1, \\ldots , u^N) ) := \\sum _{c = 1}^2 \\delta _{ \\lbrace \\langle \\mathbf {1}, \\cdot \\rangle = 0 \\rbrace } ((u^{1, c}, \\ldots , u^{N, c})).$ In order to apply Alg.",
"REF to the problem, the proximal operators $(\\operatorname{id}+ \\eta \\partial F^*)^{-1}$ and $(\\operatorname{id}+ \\tau \\partial H)^{-1}$ are required to compute the updates (REF ).",
"We point out that the separable nature of $F$ implies a decomposability of $(\\operatorname{id}+ \\eta \\partial F^*)^{-1}$ into three terms corresponding to the three summands of $F$ (or equivalently of $F^*$ ).",
"These terms as well as the proximal operator of $H$ are given by standard expressions, for which we refer to [26].",
"We however emphasize the point that the proximal step corresponding to the nuclear norm constraint $(\\operatorname{id}+ \\eta \\partial F_2^*)^{-1}(y) =\\\\U \\operatorname{diag}\\left( \\sigma - \\eta \\nu _t \\Pi _{ \\lbrace ||\\cdot ||_1 \\le 1 \\rbrace } \\left( \\frac{\\sigma }{\\eta \\nu _t} \\right) \\right) V^{\\top }$ requires both an SVD $y = U \\operatorname{diag}(\\sigma ) V^{\\top }$ , $\\sigma \\in \\mathbb {R}^N$ , of the input $y$ (assumed to be $m n \\times N$ -shaped) and a projection $\\Pi _{ \\lbrace ||\\cdot ||_1 \\le 1 \\rbrace }$ onto the $\\ell _1$ -unit ball.",
"While the latter step cannot be solved in a decoupled manner, there exist exact algorithms with time complexity $\\mathcal {O}(N)$ to compute such projections – in our implementation we employ the approach from [11].",
"Finally, we shortly address the problem of determining the spectral norm $||A||_{\\sigma }$ that the primal and dual step sizes $\\tau , \\eta $ for Alg.",
"REF are based on.",
"Since the linear operator $A$ given by (REF ) contains the image gradients $\\nabla T_k(\\tilde{u}^k)$ and is therefore dependent on empirical data, an analytical solution for $||A||_{\\sigma }$ is unattainable.",
"Instead, we apply a simple power iteration scheme to estimate this quantity [12]."
],
[
"Multilevel Scheme and Parameter Scaling",
"[t] InitializationInitialization UpdateUpdate EstimateEstimate SolveSolve ProlongateProlongate ForFor: IfElseIfElseIf: Alg.",
"$\\tilde{x}, \\tilde{y} \\leftarrow 0$ ; $\\nu \\leftarrow 2^{-n_{lev}} || [T_1 | \\ldots | T_N ] - (\\sum _{k = 1}^N \\frac{T_k}{N}) \\cdot \\mathbf {1}_{1 \\times N} ||_*$ Choose $\\alpha , \\mu > 0$ .",
"Outer Iteration: Problem Scaling + Prolongation $j = 1, \\ldots , n_{lev}$ grid widths $h_1, h_2 \\leftarrow 2^{(n_{lev} - j)}$ .",
"threshold scale $\\nu \\leftarrow 2 \\nu $ .",
"Inner Iteration: (Re-)Linearization Process + Solving Convex Subproblems for $(n_{lev} -j)$ -fold downsampled images $T_1, \\ldots , T_N$ $k = 1, \\ldots , n_{iter}^j$ threshold $\\nu \\leftarrow \\alpha \\nu $ .",
"$||A||_2$ $\\rightarrow $ choose $\\tau , \\eta $ s.t.",
"$\\tau \\eta ||A||_{\\sigma }^2 < 1$ .",
"$ \\min _{x} \\max _{y} \\ \\langle A x, y \\rangle + H(x) - F^*(y) $ for $x^*, y^*$ using Alg.",
"REF with starting points $\\tilde{x}, \\tilde{y}$ .",
"starting points $\\tilde{x} \\leftarrow x^*$ , $\\tilde{y} \\leftarrow y^*$ .",
"linearization points $\\tilde{u}^1, \\ldots , \\tilde{u}^N$ from $x^*$ .",
"$j < n_{lev}$ $\\tilde{x}, \\tilde{y}$ as in Fig.",
"REF .",
"Multilevel Scheme As is common in image registration, we couple the techniques discussed in the previous subsections REF and REF with a multilevel scheme.",
"This serves the two purposes of lowering the computational effort of our solution strategy on the one hand and of avoiding local minimizers on the other hand [24].",
"An image pyramid of $n_{lev}$ resolution stages serves as input to our multilevel scheme, where images are downsampled by a factor of 2 in each dimension between consecutive stages (for ease of presentation we assume $2^{(n_{lev} - 1)} \\mid m$ , $2^{(n_{lev} - 1)} \\mid n$ ).",
"The inverse operation, i.e., the prolongation of a variable, is implemented as depicted in Fig.",
"REF .",
"Figure: Prolongation scheme.",
"Variable values for the index (i,j)(i, j) in the low-resolution coordinate system (left) are propagated to the variables indexed by (2i-1,2j-1){(2 i - 1, 2 j - 1)}, (2i-1,2j){(2 i - 1, 2 j)}, (2i,2j-1){(2 i, 2 j - 1)}, (2i,2j){(2 i, 2 j)} in the high-resolution system (right)In order to guarantee a consistent scaling of all parts of the subproblem energy () between the different resolutions, we introduce an additional scaling of the $\\ell _1$ -term from (REF ), i.e., we redefine it as $F_1(z^1) := h_1 h_2 || z^1 + b ||_1.$ Moreover, a consistent scaling of the thresholds $\\nu _t$ is required since the low-rank components change in resolution as well in between stages.",
"To this end, it is useful to derive by what factor the nuclear norm of a matrix $M \\in \\mathbb {R}^{p \\times q}$ scales when it is prolongated to the next higher resolution.",
"Interpreting the columns of $M$ as vectorized images and assuming $p > q$ , we define the prolongation of $M$ as the separate prolongation of these image columns (as in Fig.",
"REF ).",
"The operation can therefore be expressed as $P \\begin{bmatrix}M\\\\M\\\\M\\\\M\\end{bmatrix} \\in \\mathbb {R}^{4 p \\times q},$ where $P \\in \\mathbb {R}^{4 p \\times 4 p}$ is a suitable permutation.",
"As the singular values of a matrix are invariant under row-permutations (see (REF )), one can however restrict the analysis of the singular values for (REF ) to the case $P = I$ .",
"Let an economic SVD of $M$ now be given by $M = U \\Sigma V^{\\top }$ with $U \\in \\mathbb {R}^{p \\times q}$ and $\\Sigma \\in \\mathbb {R}^{p \\times p}$ [12].",
"Then it holds $\\underbrace{\\begin{bmatrix}M\\\\M\\\\M\\\\M\\end{bmatrix}}_{=: \\hat{M}} =\\underbrace{\\begin{bmatrix}U\\\\U\\\\U\\\\U\\end{bmatrix}}_{=: \\hat{U}} \\Sigma V^{\\top }.$ (REF ) however does not constitute a valid (economic) SVD of $\\hat{M}$ , since the columns of $\\hat{U}$ are no longer normalized: One has $(\\hat{U}^{\\top } \\hat{U})_{i, j} = 4$ for $i = j$ and $(\\hat{U}^{\\top } \\hat{U})_{i, j} = 0$ for $i \\ne j$ .",
"In order to regain a valid SVD, a factor of 2 has to be redistributed from $\\hat{U}$ to $\\Sigma $ , i.e., $\\hat{M} = (\\hat{U} / 2) (2 \\Sigma ) V^{\\top }.$ This implies $||\\hat{M}||_* = 2 ||M||_*$ , which in turn implies that the sought factor is given by 2.",
"Also note that this result can easily be generalized to the case of $d$ -dimensional images, where that factor is given $2^{(d / 2)}$ .",
"The overall solution scheme is summarized by Alg.",
"REF , where an image domain of $\\Omega = [0, m] \\times [0, n]$ is assumed for the input images $T_1, \\ldots , T_N \\in \\mathbb {R}^{m \\times n}$ .",
"Further assumed are a predefined number $n_{lev}$ of resolution stages, predefined numbers $n_{iter}^j$ of linearization steps per stage (for $j = 1, \\ldots , n_{lev}$ ), as well as a final relative threshold parameter of $\\beta = \\exp (\\ln (\\alpha ) \\sum _{j = 1}^{n_{lev}} n^j_{iter})$ (see subsection REF )."
],
[
"Synthetic Dataset: Textured Ellipse",
"The purpose of the first synthetic dataset is to illustrate the capacity of our model to correct the motion of objects with recurring changes in texture, exposing the inherent low-dimensional structure of the dataset.",
"The image sequence is comprised of ten frames displaying a textured ellipse moving in a semicircular manner before a black background that further features a fixed white rectangle and a fixed white frame.",
"The texture of the ellipse alternates between vertical stripes for all oddly indexed frames and horizontal stripes for all evenly indexed frames.",
"To quantify the accuracy of registration on this dataset, we equipped each frame with 17 landmarks at the same corresponding (analytically determined) positions.",
"All frames were generated at a resolution of $200 \\times 200$ pixels.",
"As an example, four out of the ten input frames are displayed along with their landmarks in Fig.",
"REF .",
"Figure: Exemplary frames from the textured ellipse-dataset with their respective landmarks.A perfect motion correction is expected to unify the ellipse positions, while keeping the white frames and rectangles stationary"
],
[
"Real-world Dataset I: Cardiac MRI",
"Besides the challenge of motion correction in the presence of recurring changes in object appearance that the first synthetic dataset posed, the first real-world dataset comes with the additional difficulty of irregular disturbances to object appearance.",
"The sequence consists of cardiac MRI data in the so-called two-chamber view, where the left atrium and ventricle are on display.",
"Seven repetitions of the heart cycle with blood flow in and out of the two chambers as well as breathing-induced motions of several structures like the thorax, the diaphragm, and the heart are shown.",
"For this dataset, changes in object appearance relate to different phases of the heart cycle as we selected one frame from each systole, one frame from each diastolic relaxation and one frame from each diastolic filling (making for a total of 21 input frames).",
"Due to the turbulent nature of the blood flow, the visual appearances of these phases are somewhat irregular and pose an interesting test case for the low-rank/sparse decomposition generated by our model.",
"As with the textured ellipse-dataset, we equipped this sequence with 23 handselected landmarks per frame.",
"Each individual image was resolved with $220 \\times 220$ pixels.",
"The respective input frames for the first and last heart cycle are displayed together with their respective landmarks in Fig.",
"REF .",
"Figure: Exemplary frames from the cardiac MRI dataset with their respective landmarks.While clear visual congruences between different images of the same phase exist, they are obscured by the irregularity of the blood flow and pose a particular challenge to distance measures that model similarity as linear dependence"
],
[
"Real-world Dataset II: Challenging Data for Stereo and Optical Flow",
"We also evaluated our model on a variety of test sequences from the “Challenging Data for Stereo and Optical Flow”-dataset (CDSOF) [21].",
"This dataset features eleven sequences captured in real-world traffic situations that are deemed challenging for motion estimation algorithms due to diverse phenomena such as illumination changes from blinking signs, occlusions from snowflakes, and blurs from water spray.",
"We selected subsequences of 10 frames from the datasets entitled “Blinking Arrow”, “Flying Snow” and “Shadow on Truck” as test cases as they all feature different distortions that pose interesting challenges to the robustness of our model.",
"In order to restrict the required computational effort, we downsampled all used frames to a resolution of $271 \\times 328$ pixels.",
"Contrary to the other two datasets, all sequences from [21] come with a predefined reference frame, which is why we drop the uniqueness constraint from section REF for these inputs.",
"Instead, we enforce alignments with the reference through a constraint of the form $\\delta _{\\lbrace 0 \\rbrace }( u^{(ref)} )$ , where $u^{(ref)}$ is the displacement field for the respective reference.",
"The reference images for all selected sequences are shown in Fig.",
"REF ."
],
[
"Results",
"For the former two datasets from section , we compare our registration approach to the following two methods: An approach based on the simple variance dissimilarity measure given by $D_{\\text{VAR}} (T_1, \\ldots , T_N) := \\frac{1}{2} \\sum _{k = 1}^N || T_k - \\bar{T} ||_2^2 \\mbox{ with } \\bar{T} = \\sum _{k = 1}^N \\frac{T_k}{N},$ which has previously been used by [2], [22].",
"We combine (REF ) with the same TV-regularization and the same uniqueness constraint as in our model ().",
"A publicly available implementation of the $D_{\\text{PCA2}}$ -metric from [18] in the elastix software package [19].",
"This method uses a cubic B-spline transformation model and implicit regularization.",
"For each individual landmark, accuracy is measured in terms of mean Euclidean distance to the mean landmark position $\\frac{1}{N} \\sum _{k = 1}^N || y_i^k - \\bar{y}_i ||_2 \\quad \\mbox{with} \\quad \\bar{y}_i := \\sum _{k = 1}^N \\frac{y_i^k}{N}.$ $y_i^k \\in \\mathbb {R}^2$ therein denotes the position of the $i$ -th landmark in the $k$ -th image.",
"For the CDSOF-datasets we compared our approach to the publicly available implementation of the “nonlocal” optical flow estimation method from [33], that was suggested as a referemce by the authors of [21].",
"[33] extends the classical Horn-Schunck model for optical flow estimation [17] by a number of techniques, e.g., an additional nonlocal term derived from median filtering.",
"To give a meaningful comparison, we altered our algorithm to include the median filtering of flow fields in between linearization steps as well.",
"Note that the reference method only operates in a pairwise fashion.",
"All experiments were performed on an Intel Core i7-8700 ($6 \\times $ 3.20 GHz) system with 64 GB of memory, running Matlab R2019a under Ubuntu 18.04 (64-Bit).",
"Our implementation is publicly available at https://github.com/roland1993/d_RPCA.",
"Computation times for the textured ellipse- and cardiac MRI-datasets are given in Tab.",
"REF and for the CDSOF-dataset in Tab.",
"REF .",
"Table: Computation times of competing methods"
],
[
"Textured Ellipse",
"Using the parameters $\\alpha = 0.9$ , $\\mu = 0.2$ as well as $n_{lev} = 3$ with $n_{iter}^1 = 16$ and $n_{iter}^j = 2$ for $j \\ge 2$ in Alg.",
"REF , we achieved the results displayed in fig:ellipseresults,fig:ellipsesv for the textured ellipse-dataset.",
"We shortly note that we observed values of $n_{iter}^j \\ge 2$ for $j \\ge 2$ to be mandatory in order for Alg.",
"REF to generate useful linearization points on higher resolution levels.",
"Furthermore, a small enough $\\alpha $ was crucial in finding low-rank components $L$ that accurately describe the structural changes in texture – higher values of $\\alpha $ on the other hand allowed for unwanted motion artifacts in $L$ .",
"For the variance method based on (REF ), we employed an adapted version of Alg.",
"REF with the regularization strength set to $\\mu = 0.1$ .",
"The number of resolution levels as well as the iterations per level were kept the same as for our method.",
"In the elastix-based implementation of $D_{\\text{PCA2}}$ , we increased the settings recommended by the authors of [18] to three resolution stages (instead of the recommended two), 2.000 iterations per stage (recommended: 1.000) and 25.000 random coordinates per stage (recommended: 2.048) in order to ensure sufficient computational capacity for the method to produce accurate solutions.",
"Fig.",
"REF visualizes both the deformations calculated by our model and the warped images for all four exemplary frames from Fig.",
"REF .",
"Fig.",
"REF further analyzes the generated low-rank components in terms of singular values and (left) singular vectors of $L - \\bar{L}$ , i.e., the matrix whose nuclear norm is constrained by our model ().",
"As the dataset is of synthetic nature and therefore without intensity distortions, the sparse outlier components $E = M - L$ are negligible and are not displayed.",
"A quantitative comparison in terms of landmark accuracy between our approach and the two competing methods is presented in Fig.",
"REF .",
"The comparison shows that our method significantly outperforms the other approaches on this dataset: While it corrects the ellipse positions most accurately out of the three methods, it is also the only one that does not introduce notable motion to the white rectangle and frame (which were already stationary in the input sequence).",
"In terms qualitative results, Fig.",
"REF shows that our method was moreover able to find a near-perfect embedding of the motion-compensated images in a low-dimensional subspace: The negligible magnitudes of the singular values $\\sigma _2, \\ldots , \\sigma _{10}$ of the centered low-rank components $L - \\bar{L}$ indicate that a two-dimensional basis consisting of the mean low-rank component $\\bar{l}$ and the singular vector $s_1$ is largely sufficient to approximate the output images.",
"Figure: Results of proposed approach for the selected frames from Fig.",
".Our model allows to correct the motion of the ellipse through piecewise constant deformations (top row), while automatically detecting and discarding repetitive structural noise (the horizontal and vertical bars) in the registration process.The motion-corrected images (bottom row) exhibit a good visual correspondence between matching landmarks.Quantitative results are given in Fig.",
", in which landmarks are indexed in the same order as in T 1 T_1Figure: Comparison of landmark accuracy for the textured ellipse-dataset as measured by () (lower is better).Landmarks are ordered as in Fig.",
"with landmarks 5-95 - 9 attributed to the moving ellipse and the remaining landmarks positioned around the stationary white rectangle and frame.Note that the latter do not appear in the input curve due to zero error and logarithmic axis scaling.Our method clearly outperforms the two competing approaches as it is able to correct the positions of the “ellipse-landmarks” most accurately without introducing artificial motion to the remaining landmarksFigure: Singular values and vectors of centered low-rank components L-L ¯L - \\bar{L}.The singular value progression (left) over the inner iterations of Alg.",
"(scaling adjusted, see subsection ) shows that the norm ||L-L ¯|| * || L - \\bar{L} ||_* – the quantity constrained by our model – is dominated by the singular value σ 1 \\sigma _1 towards the end of the iteration.This indicates that the columns of LL can be reconstructed from their mean l ¯\\bar{l} (second from left) and the dominating singular vector s 1 s_1 of the centered low-rank components L-L ¯L - \\bar{L} (second from right) with only minor error.Intuitively, s 1 s_1 is added to l ¯\\bar{l} when reconstructing horizontal bars and subtracted from l ¯\\bar{l} when reconstructing vertical bars.In comparison, the second singular vector s 2 s_2 (right) only describes negligible remainders of motion with little influence on L-L ¯L - \\bar{L} (as indicated by the final magnitude of σ 2 \\sigma _2 in the left curve).To summarize, our method was able to automatically detect the low-dimensional texture variations present in the dataset"
],
[
"Cardiac MRI",
"For the cardiac MRI dataset, we only adjusted the regularization strength and threshold-scaling parameters of our method to $\\mu = 0.125$ and $\\alpha = 0.95$ .",
"In the variance registration method, we kept all parameters fixed except for $\\mu = 0.065$ and since the $D_{\\text{PCA2}}$ model does not feature explicit regularization parameters, we left all the above settings unchanged for this method.",
"Results of our approach are presented in fig:cardiacresults,fig:cardiacsv,fig:cardiacsparse.",
"fig:cardiacresults,fig:cardiacsparse show deformations, warped images and sparsity components for all example frames from Fig.",
"REF .",
"In Fig.",
"REF , singular values and singular vectors of $L - \\bar{L}$ are analyzed in a presentation similar to Fig.",
"REF .",
"The quantitative evaluation in terms of landmark accuracy is visualized in Fig.",
"REF .",
"Although results are generally more balanced than was the case for the synthetic data, our approach still outperforms the two competing methods on this real-world dataset in terms of landmark accuracy.",
"We refer to Fig.",
"REF , where our method not only achieves the highest accuracy for the majority of the landmarks, but where it is the only out of the three methods that did not introduce additional motion to any landmarks in the registration process: both competing approaches generate several landmarks with worse accuracy than in the unregistered input sequence.",
"In terms of the low-rank/sparse decomposition, we remark that our model was able to generate meaningful low-rank components alongside the actual motion-compensation: As seen from Fig.",
"REF , the centered low-rank components matrix $L - \\bar{L}$ is dominated by three singular value/vector pairs in which the singular vectors $s_1 - s_3$ exhibit clear visual congruences with the three considered phases of the heart cycle.",
"Congruences thereby consist of highlighted anatomical structures and physiological features such as the mitral valve and the direction of blood flow.",
"Furthermore, granting the method the ability to define sparse outlier components aided the generation of a meaningful low-rank approximation by filtering out highly irregular image feature such as the turbulences of blood flow (see Fig.",
"REF ).",
"Figure: Results of proposed approach on the cardiac MRI-dataset.The difference images per phase between the first and seventh heart cycle before and after registration (left two columns) show a successful motion compensation:Motion in the thorax area as well as the front-facing area of the diaphragm and the heart apex is greatly reduced across all phases.The main intensity differences after registration are located inside the heart itself and are due to irregular turbulences of the blood flow.The two right-hand columns show the actual motion-compensated images including their deformed landmarks – in the upper-left image, an additonal ordering of the landmarks is given that is referred to again in Fig.",
"Figure: Comparison of landmark accuracy for the cardiac MRI-dataset as measured by () (lower is better).Landmarks are ordered as in Fig.",
".While our method still produces the most accurate deformations (performing best for 13 out of 23 landmarks), results are much more balanced than in the textured ellipse-experiment (see Fig. )",
"with D PCA2 D_{\\text{PCA2}} performing remarkably well despite being based on pure PCA.However, one notable drawback which both competing methods exhibit is that landmarks occasionally feature more motion in the registered images than was actually present in the input sequence – this is indicated by crossings of their respective curves with the gray input curve (see for example landmarks 7 and 16 from the thorax area)Figure: Singular values and vectors of the centered low-rank components L-L ¯L - \\bar{L} for the cardiac MRI-dataset.The development of the singular values (left) shows, that the nuclear norm ||L-L ¯|| * ||L - \\bar{L}||_* (which is constrained by our model) is dominated by the three largest singular values σ 1 -σ 3 \\sigma _1 - \\sigma _3.Consequently, the low-rank components of the warped images primarily consist of a linear combination of l ¯\\bar{l} (second from left) and the three corresponding singular vectors s 1 ,s 2 ,s 3 s_1, s_2, s_3 (third, second, first from right).Especially note the visual congruences between these three singular vectors and the characteristics of the three considered heart phases:While s 1 s_1 marks a blood flow into the left ventricle (see the diastolic filling in Fig.",
"), s 2 s_2 highlights the mitral valve, which is clearly visibly closed during the diastolic relaxation phase (see again Fig.",
").Moreover, s 3 s_3 exhibits a high contrast between atrium and ventricel as present during the systole in Fig.",
"Figure: Sparse components e i =T i (u i )-l i e_i = T_i(u^i) - l_i for all frames used in Fig.",
".Nonzero entries are sparsely distributed across all images and primarily serve to correct the irregularities of the blood flow that were not captured by the low-rank components l i l_i, i.e., that were not representable in a low-dimensional linear subspace.Thus, a meaningful low-rank approximation of the motion-corrected images is only enabled by allowing for sparsely distributed outliers"
],
[
"CDSOF",
"A comparison of the motion estimation capabilities of our method with the pairwise operating reference [33] for all sequences from Fig.",
"REF is given in Fig.",
"REF .",
"In our model, we used the parameters $n_{lev} = 4$ , $n_{iter}^1 = 16$ , $ n_{iter}^j$ for $j \\ge 2$ , $\\alpha = 0.91$ across all three examples as well as $\\mu = 0.075$ for the “Blinking Arrow”, $\\mu = 0.045$ for the “Flying Snow” and $\\mu = 0.125$ for the “Shadow on Truck” sequences.",
"In the reference method implementation, we kept all parameters at standard values except for the regularization strengths, which were set to $\\lambda = 10$ for the “Blinking Arrow” and to $\\lambda = 20$ for both the “Flying Snow” and the “Shadow on Truck” sequence.",
"Upon inspection of the results in Fig.",
"REF and Tab.",
"REF , it is clear that our approach was mostly outperformed by the reference method in terms of accuracy and computational efficiency.",
"This is especially true for the sequence entitled “Shadow on Truck”, where our method failed to produce a meaningful motion correction.",
"For the other two sequences, our model was able to generate motion fields that successfully aligned all deformable (non-reference) images in the presence of disturbances such as snow flakes and blinking signs.",
"We however point to the observation, that this alignment was not constructed with respect to the explicitly given reference but to another implicit reference generated by our algorithm.",
"In light of this implicit reference, the explicit one hence appears as an outlier.",
"This phenomenon also explains the large discrepancies observed in Fig.",
"REF between the motion fields generated by our model and those generated by the reference method.",
"We draw two conclusions from the experiments: The proposed method exhibits a notable sensitivity towards the degree to which input data meets the model assumption of decomposability into structural low-rank components and sparse outlier components.",
"If variations in object appearance are too irregular or if distortions are too large in scale (as in the “Shadow on Truck”-sequence), the approach might fail to produce meaningful solutions.",
"Imposing an explicit reference on a groupwise operating method cannot be expected to produce deformations that align all deformable images to that reference.",
"On the contrary, deformable images might rather be aligned to a more suitable implicit reference.",
"We primarily attribute this phenomenon to the small relative weight of one fixed reference when compared to the remaining group of $N - 1$ deformable images.",
"We emphasize that the goal of this experiment is not to compete with a specialized method for a different domain (pairwise registration), but rather to give an indication of its usefulness on challenging non-medical real-world sequences.",
"Figure: Comparison of optical flow estimation of the proposed model with the pairwise operating reference method from .For each sequence in Fig.",
", one representative frame was selected, for which HSV color coded overlays and vector field presentations of the computed displacements are displayed.The “Shadow on Truck”-sequence (right column) represents a clear failure case of our model as it was not able to distinguish the shadow casting from the actual physical motion (which the reference method succeeded in).For the other two sequences, our model was able to generate meaningful motion corrections.We again outline the difficulties involved in these sequences:The selected frame from the “Blinking Arrow”-dataset (left column) features a lit traffic sign that is unlit in the reference frame (see Fig.",
"), while the “Flying Snow”-dataset (center column) features heavy and irregular snowfall (compare the displayed frame to the reference in Fig.",
").The apparent discrepancies between our approach and the method from are explained by the phenomenon discussed in subsection :Rather than aligning all deformable images with the fixed reference image (as the pairwise operating reference method does), our groupwise approach aligns these with another implicit reference.The explicit reference hence appears as an outlier in light of this implicit reference with misaligments being absorbed by the ℓ 1 \\ell _1-term of our distance metric ()"
],
[
"Concluding Remarks",
"In this work, we have investigated a novel dissimilarity metric for groupwise image registration tasks based on low-rank and sparse decompositions.",
"The proposed metric corrects the major drawbacks that the established RPCA-image distance from [27], [16] exhibited in the experiments of Sec. .",
"It is primarily suited for registering image data, that features objects with recurring changes in appearance and that can be represented in a low-dimensional linear subspace with potential sparse outliers.",
"We especially emphasize the advantage in interpretability, when dealing with threshold constraints instead of weighted penalties.",
"We further developed a first-order primal-dual optimization framework for solving non-parametric registration tasks using our metric in conjunction with TV regularization, which can easily be replaced by other regularization techniques suited for the individual application.",
"Experimentally, we were able to show the superiority of our method when compared to two commonly used groupwise registration models.",
"The experiments included both synthetic and real-world image data, that met the assumptions made by our model well.",
"We further investigated the robustness of our model on a number of test sequences from the optical flow community, where we found that albeit it was outperformed by a highly optimized reference method, our model was able to correct motion in presence of distortions like snow flakes obstructing the view and illumination changes from blinking signs.",
"As an interesting phenomenon, providing a reference to a groupwise method did not result in deformable images being aligned to the reference, but rather in the reference being treated as an outlier by our model.",
"It remains to be investigated, to what extent this behavior is a trait of our particular model or of the general groupwise approach.",
"Another avenue for future research is the application of our proposed decomposition model to tasks other than image registration.",
"Acknowledgements The authors thank Allen D. Elster (MRIQuestions.com) for kindly providing the cardiac cine study used in this article.",
"The authors further acknowledge support through DFG grant LE 4064/1-1 “Functional Lifting 2.0: Efficient Convexifications for Imaging and Vision” and NVIDIA Corporation."
]
] | 2001.03509 |
[
[
"Fuel-Efficient Powered Descent Guidance on Large Planetary Bodies via\n Theory of Functional Connections"
],
[
"Abstract In this paper we present a new approach to solve the fuel-efficient powered descent guidance problem on large planetary bodies with no atmosphere (e.g.",
"the Moon or Mars) using the recently developed Theory of Functional Connections.",
"The problem is formulated using the indirect method which casts the optimal guidance problem as a system of nonlinear two-point boundary value problems.",
"Using the Theory of Functional Connections, the problem constraints are analytically embedded into a \"constrained expression,\" which maintains a free-function that is expanded using orthogonal polynomials with unknown coefficients.",
"The constraints are satisfied regardless of the values of the unknown coefficients which convert the two-point boundary value problem into an unconstrained optimization problem.",
"This process casts the solution into the admissible subspace of the problem and therefore simple numerical techniques can be used (i.e.",
"in this paper a nonlinear least-squares method is used).",
"In addition to the derivation of this technique, the method is validated in two scenarios and the results are compared to those obtained by the general purpose optimal control software, GPOPS-II.",
"In general, the proposed technique produces solutions of $\\mathcal{O}(10^{-10})$.",
"Additionally, for the proposed test cases, it is reported that each individual TFC-based inner-loop iteration converges within 6 iterations, each iteration exhibiting a computational time between 72 and 81 milliseconds within the MATLAB legacy implementation.",
"Consequently, the proposed methodology is potentially suitable for on-board generation of optimal trajectories in real-time."
],
[
"Introduction",
"Precision landing on large planetary bodies is a technology of utmost importance for future human and robotic exploration of the solar system.",
"Over the past two decades, landing systems for robotic Mars missions have been developed and successfully deployed robotic assets on the Martian surface (e.g.",
"rovers , landers ).",
"Considering the strong interest in sending humans to Mars within the next few decades, as well as the renewed interest in building infrastructure in the Earth-Moon system for easy access to the Lunar surface, the landing system technology will need to progress to satisfy the demand for more stringent requirements.",
"One of the enabling technology for precision landing is the ability to generate on-board and track in real-time fuel optimal trajectories.",
"Generally, finding optimal guidance solutions requires setting up a cost function (i.e.",
"the objective of the optimization) and the desired control and state constraints.",
"For landing on large planetary bodies, especially for systems that deliver large systems to their surface, it is extremely important to minimize the fuel.",
"State and control constraints usually include the system dynamics (i.e.",
"equations of motion), thrust value limitation as well as state space constraints that need to be enforced to ensure safety (e.g., glide slope constraints to ensure safety of the spacecraft).",
"Usually, two methods are available to solve optimal control problems, direct and indirect methods.",
"Direct methods are based on discretizing the continuous states and controls with the goal of transforming the continuous problem into a Non Linear Programming (NLP) problem , , .",
"The latter can be cast as a finite constrained optimization problem that can be solved via any of the available numerical algorithms that have the potential to find a local minimum (e.g., thrust region method ).",
"Whereas direct methods have been applied to solve a large variety of optimal control problems (e.g.",
", , , ), the general NLP problem is considered to be NP-hard, i.e.",
"non deterministic polynomial time hard.",
"NP-hard problems imply that the required amount of computational time needed to find the optimal solution does not have a predetermined bound (i.e.",
"a bound cannot be apriori determined).",
"NP-hard problems are such that the computational time necessary to converge to the solution is not known.",
"As a consequence, the lack of assured convergence may result in questioning the reliability of the proposed approach.",
"Since for optimal closed loop space guidance, most of the problems require computing numerical solutions on board and in real time, general algorithms that solve NLP problems cannot be reliably implemented.",
"More recently, researchers have been experimenting with transforming optimal control problem from a general non convex formulation into a convex optimization problem , .",
"Here, the goal is to take advantage of the assured convex convergence properties.",
"Indeed, convex optimization problems are shown to be computationally tractable as their related numerical algorithms guarantee convergence to a global optimal solution in a polynomial time.",
"The general convex methodology requires that the optimal guidance problem is formulated as convex optimization whenever appropriate or convexification techniques are applied to transform the problem from a non-convex into a convex one.",
"Such methodologies have been proposed and applied to solve optimal guidance and control via direct method in a large variety of problems including, planetary landing , , entry atmospheric guidance , , rocket ascent guidance , and low thrust .",
"Alternatively, a second approach to solve optimal control and guidance problems has been generally applied to a variety of optimal control problems.",
"Named indirect method, the approach applies optimal control theory (i.e.",
"Pontryagin Minimum Principle) to formally derive the first-order necessary conditions that must be satisfied by the optimal solution (state and control).",
"The problem is cast as a Two Point Boundary Value Problem (TPBVP) that must be solved to determine the time evolution of state and costate from which the control generally depends.",
"For general nonlinear problems, the necessary conditions result in a complicate set of equations and conditions.",
"The resulting TPBVP tends to be highly sensitive to the initial guess on the costates making the problem very hard to solve.",
"Although indirect methods are known to yield more accurate optimal solutions, they are very hard to implement and tend to be less used in practice (with respect to direct methods).",
"Recently, a new approach , called Theory of Functional Connections (TFC)This theory, initially called “Theory of Connections”, has been renamed for two reasons.",
"First, the “Theory of Connections” already identifies a specific theory in differential geometry, and second, what this theory is actually doing is “Functional Interpolation” as it provides all functions satisfying a set of constraints in term of function and any derivative in rectangular domains of $n$ -dimensional space.",
"was developed to derive expressions, called constrained expression, with embedded constraints.",
"This approach has been successfully applied to solve both linear and nonlinear differential equations, and IVP and BVP, at machine error accuracy and in milliseconds.",
"TFC is a general methodology providing functional interpolation expressions with an embedded set of $n$ linear constraints.",
"Such expressions can be expressed in the following general form: $y (t) = g (t) + \\displaystyle \\sum _{k=1}^n \\eta _k \\, p_k (t)$ Here, the $p_k (t)$ are $n$ assigned linearly independent functions and the $g (t)$ is a free function that must be linearly independent from the selected $p_k (t)$ terms.",
"The $\\eta _k$ are coefficient functions that are derived by imposing the set of $n$ constraints.",
"The constraints considered in the TFC are any linear combination of the functions and/or derivatives evaluated at specified values of the variable $t$ .",
"The constraints of the differential equation to be solved enable to directly derive the unknown coefficient functions, $\\eta _k$ .",
"Once the $\\eta _k$ terms are determined, the constraints for the differential equations are satisfied for any possible $g (t)$ .",
"The constrained expressions thus obtained can be used to transform the constrained optimization problem in an unconstrained one.",
"This implies reducing the whole solution search space to just the space of admissible solutions (i.e.",
"those fully complying with all constraints).",
"In the field of optimization, this has been done by expanding the free function by a set of basis functions (e.g., Fourier series or orthogonal polynomials, such as Legendre or Chebyshev polynomials) whose coefficients are found by direct application of a least-squares algorithms.",
"Nonlinear initial or boundary value problems require the implementation of an iterative least-squares approach to converge to the desired solution .",
"In this paper, we propose solving the optimal landing guidance problem via a new method based on TFC that can compute fuel-efficient trajectories both fastly and accurately.",
"After deriving the TPBVP arising from the fuel-efficient powered descent guidance necessary conditions, the TFC is employed to generate the whole set of boundary condition-free equations that can be solved by expanding the solution in Chebyshev polynomials and computing the expansion coefficients using Iterative Least-Squares (ILS) method.",
"The proposed methodology is shown to be fast, accurate and thus potentially suitable for on-board generation of optimal landing trajectories on large planetary bodies.",
"This article is organized in the following manner.",
"First, a summary of the TFC approach to solve a general TPBVP is presented.",
"Next, the equations for the optimal powered descent landing problem are presented and the necessary conditions are derived.",
"Afterward, it is shown how the TFC approach is used to transform the TPBVP and eventually solved via and iterative least-squares approach.",
"Lastly, results for two specific optimal thrust profiles are presented: test #1) min-max profile and test #2) max-min-max profile.",
"These results are compared against those obtained from the general purpose optimal control software GPOPS-II ."
],
[
"The Theory of Functional Connections approach to solve a general TPBVP",
"The uni-variate TFC embeds $n$ linear constraints into the expression, $y(t) = g(t) + \\sum _{k=1}^n \\eta _k \\, p_k(t)$ where $g(t)$ is a free function, $\\eta _k$ are unknown coefficients to be solved, and $p_k(t)$ are user defined linearly independent functions; where $g(t)$ and the set of $p_k(t)$ must be linearly independent.",
"This formulation is easily extended for the use in solving vector equations, by utilizing index notation where $y_i(t) = g_i(t) + \\sum _{k=1}^n \\eta _{k_i} \\, p_k(t) \\quad \\text{for} \\quad i = 1,2,3,$ such that $i$ denotes each component of the vector.",
"Since landing problems are second-order BVPs the following example derivation will follow that form.",
"For this class of problems, we have the conditions $y_i(t_0) = y_{0_i}$ , $\\dot{y}_i(t_0) = \\dot{y}_{0_i}$ , $y_i(t_f) = y_{f_i}$ , and $\\dot{y}_i(t_f) = \\dot{y}_{f_i}$ .",
"For these constraint conditions we can select $p_k (t) = t^{(k-1)}$ (see Ref.",
", , for detailed explanation on this choice) in Eq.",
"(REF ), we obtain, $\\begin{aligned}\\begin{Bmatrix} y_{0_i}-g_{0_i} \\\\ y_{f_i}-g_{f_i} \\\\ \\dot{y}_{0_i}-\\dot{g}_{0_i} \\\\ \\dot{y}_{f_i}-\\dot{g}_{f_i}\\end{Bmatrix} = \\begin{bmatrix} 1 & t_0 & t_0^2 & t_0^3 \\\\ 1 & t_f & t_f^2 & t_f^3 \\\\ 0 & 1 & 2t_0 & 3t_0^2 \\\\ 0 & 1 & 2 t_f & 3t_f^2\\end{bmatrix} \\begin{Bmatrix} \\eta _{1_i} \\\\ \\eta _{2_i} \\\\ \\eta _{3_i} \\\\ \\eta _{4_i}\\end{Bmatrix}\\end{aligned}$ which can be solved for the $\\eta _{k_i}$ terms through matrix inversion which leads to the of the form, $\\begin{aligned}y_i(t) = g_i(t) &+ \\Omega _1(t) \\Big (y_{0_i} - g_{0_i}\\Big ) + \\Omega _2(t) \\Big (y_{f_i} - g_{f_i}\\Big ) + \\\\ &+ \\Omega _3(t) \\Big (\\dot{y}_{0_i} - \\dot{g}_{0_i}\\Big ) + \\Omega _4(t) \\Big (\\dot{y}_{f_i} - \\dot{g}_{f_i}\\Big )\\end{aligned}$ where the $\\Omega _k(t)$ terms are defined as in Tables REF where we define $t_* = t - t_0$ and $\\Delta t = t_f - t_0$ Table: Switching functions for TPBVP constraints for general domain of t∈[t 0 ,t f ]t \\in [t_0, t_f].Eq.",
"(REF ) represents all possible functions satisfying the boundary value constraints.",
"Furthermore, the derivatives follow, ${\\left\\lbrace \\begin{array}{ll}\\dot{y}_i(t) = \\dot{g}_i(t) &+ \\dot{\\Omega }_1(t) \\Big (y_{0_i} - g_{0_i}\\Big ) + \\dot{\\Omega }_2(t) \\Big (y_{f_i} - g_{f_i}\\Big ) + \\\\ &+ \\dot{\\Omega }_3(t) \\Big (\\dot{y}_{0_i} - \\dot{g}_{0_i}\\Big ) + \\dot{\\Omega }_4(t) \\Big (\\dot{y}_{f_i} - \\dot{g}_{f_i}\\Big ) \\\\\\ddot{y}_i(t) = \\ddot{g}_i(t) &+ \\ddot{\\Omega }_1(t) \\Big (y_{0_i} - g_{0_i}\\Big ) + \\ddot{\\Omega }_2(t) \\Big (y_{f_i} - g_{f_i}\\Big ) + \\\\ &+ \\ddot{\\Omega }_3(t) \\Big (\\dot{y}_{0_i} - \\dot{g}_{0_i}\\Big ) + \\ddot{\\Omega }_4(t) \\Big (\\dot{y}_{f_i} - \\dot{g}_{f_i}\\Big )\\end{array}\\right.",
"}$ The defined by Eq.",
"(REF ) and its derivatives can then be applied to a differential equation which, in general, can be expressed as a loss function (i.e.",
"in its implicit form) such that, $F_i\\left(t,y_j,\\dot{y}_j,\\ddot{y}_j\\right) = 0 \\quad \\text{for} \\quad i,j = 1,2,3$ where each component of the differential equation can be a function of all component $j$ .",
"By substituting Eq.",
"(REF ) into Eq.",
"(REF ), the differential equation is transformed to a new differential equation that we define as $\\tilde{F}$ , which is only a function of the independent variable $t$ , the free-function, and its derivatives where, $\\tilde{F}_i\\left(t,g_j,\\dot{g}_j,\\ddot{g}_j\\right) = 0 \\quad \\text{for} \\quad i,j = 1,2,3.$ This differential equation is uniqueThis differential equation, which has no constants of integration, has the boundary conditions embedded through the free function evaluated at the constraints.",
"because it is subject to no constraints and will always satisfy the boundary-values.",
"In order to solve this problem numerically, we expand the free function $g_i(t)$ in terms of some known basis (${\\mathbf {h}} (z)$ ) with unknown coefficients (${\\mathbf {\\xi }}$ ) such that, $g_i (t) = {\\mathbf {\\xi }}_i {h} (z) \\quad \\text{where} \\quad z = z(t)$ where ${\\mathbf {\\xi }}_i$ are $m \\times 1$ vectors of unknown coefficients where $i$ is the dimension and $m$ is the number of basis functions.",
"In general the basis functions are defined on a different domains (Chebyshev and Legendre polynomials are defined on $z\\in [-1,+1]$ , Fourier series is defined on $z\\in [-\\pi ,+\\pi ]$ , etc.)",
"so these functions must be linearly mapped to the independent variable $t$ .",
"This can be done using the equations, $z = z_0 + \\frac{z_f-z_0}{t_f-t_0}(t - t_0) \\quad \\longleftrightarrow \\quad t = t_0 + \\frac{t_f-t_0}{z_f-z_0}(z - z_0),$ where $t_f$ represents the upper integration limit.",
"The subsequent derivatives the the free-function defined in Eq.",
"(REF ) follow, $\\frac{\\; \\text{d}^{n} g_i}{\\; \\text{d}t^{n}} = {\\mathbf {\\xi }}_i {\\; \\text{d}^{n} {\\mathbf {h}}(z)}{\\; \\text{d}z^{n}} \\left(\\frac{\\; \\text{d}z}{\\; \\text{d}t}\\right)^{n},$ where by defining, $c := \\frac{\\; \\text{d}z}{\\; \\text{d}t} = \\frac{z_f - z_0}{t_f - t_0}$ the expression can be simplified to, $\\frac{\\; \\text{d}^{n} g_i}{\\; \\text{d}t^{n}} = c^{n} {\\mathbf {\\xi }}_i {\\; \\text{d}^{n} {\\mathbf {h}}(z)}{\\; \\text{d}z^{n}},$ which defines all mappings of the free-function.",
"Lastly, the domain $t\\in [t_0,t_f]$ must be discretized by $N$ points.",
"In this paper (as in prior papers) we consider the linear basis ${\\mathbf {h}}(z)$ as Chebyshev orthogonal polynomials.",
"The optimal distribution of points is provided by collocation points , , defined as, $z_k = -\\cos \\left(\\frac{k \\pi }{N}\\right) \\quad \\text{for} \\quad k = 1, 2, \\cdots , N,$ As compared to the uniform distribution point, the collocation point distribution allows a much slower increase of the condition number as the number of basis functions, $m$ , increases.",
"By defining the free function in this way and then discretizing the domain of the differential equations, $\\tilde{F}_i$ becomes $\\tilde{F}_i(t, {\\mathbf {\\xi }}_j) = 0 \\qquad \\text{for} \\qquad i,j = 1,2,3,$ which are some set of functions (linear or nonlinear) of the unknown parameters ${\\mathbf {\\xi }}_j$ of which many unconstrained optimization schemes can be applied.",
"For the convenience of the reader, the TFC method is summarized below in Fig.",
"REF Figure: Summary of the relevant steps to use the TFC approach.",
"The major steps of TFC include 1) deriving the constrained expression, 2) defining an appropriate free function, and 3) discretizing the domain."
],
[
"Optimal Powered Descent Pinpoint Landing Problem",
"For the problem of powered descent pinpoint landing guidance on large bodies (e.g.",
"the Moon or Mars) the governing system dynamics during the powered descent phase can be modeled as: $\\dot{{\\mathbf {r}}} &&= \\,{\\mathbf {v}} \\\\\\dot{{\\mathbf {v}}} &&= \\,{\\mathbf {a}}_g + { \\dfrac{{\\mathbf {T}}}{m} } \\\\\\dot{m} &&= \\,-\\alpha \\, T$ where $\\alpha = 1 / v_{ex}$ , $v_{ex}$ is the effective exhaust velocity of the rocket engine that is considered constant , , $T = ||{\\mathbf {T}}||$ , and ${\\mathbf {T}} = T \\, \\hat{{\\mathbf {t}}}$ is the thrust and it is constrained as follow: $\\begin{aligned}0 \\le T_{min} &\\le T \\le T_{max} \\\\|| \\hat{{\\mathbf {t}}} ||&=1\\end{aligned}$ Figure: Coordinate frame definition for optimal powered descent pinpoint landing problem.Additionally, ${\\mathbf {a}}_g$ is the gravity acceleration which is also considered constant.",
"As stated in , this assumption is justified for short flights as it is the case for the powered descent phase of the landing process.",
"A summary of the reference frame for the problem is given in Fig.",
"REF .",
"For this problem, the initial and final position and velocity, and initial mass are given: $\\left\\lbrace \\begin{split} {\\mathbf {r}}(0) = {\\mathbf {r}}_0 \\\\ {\\mathbf {v}}(0) = {\\mathbf {v}}_0\\end{split}\\right., \\qquad \\left\\lbrace \\begin{split} {\\mathbf {r}}(t_f) = {\\mathbf {r}}_f \\\\ {\\mathbf {v}}(t_f) = {\\mathbf {v}}_f\\end{split}\\right., \\qquad \\text{and} \\qquad m(0) = m_0.$ The objective is to minimize the mass of propellant used while satisfying the dynamics constraints of the problem.",
"Therefore the problem can be posed as, $\\underset{t_f,T}{\\text{minimize}} \\quad \\alpha \\int _{0}^{t_f}T \\; \\text{d}\\tau $ $\\begin{aligned}\\text{subject to} \\quad \\dot{{\\mathbf {r}}} = \\,{\\mathbf {v}}, \\quad &\\dot{{\\mathbf {v}}} = \\,{\\mathbf {a}}_g + { \\dfrac{{\\mathbf {T}}}{m} }, \\quad \\dot{m} = \\,-\\alpha \\, T \\\\& 0 \\le T_{min} \\le T \\le T_{max}, \\\\& {\\mathbf {r}}(0) = {\\mathbf {r}}_0, \\quad {\\mathbf {v}}(0) = {\\mathbf {v}}_0, \\quad m(0) = m_0 \\\\& {\\mathbf {r}}(t_f) = {\\mathbf {r}}_f, \\quad {\\mathbf {v}}(t_f) = {\\mathbf {v}}_f\\end{aligned}$"
],
[
"First-order necessary conditions (Pontryagin Minimum Principle)",
"The application of the Pontryagin Minimum Principle (PMP) dictates that the Hamiltonian takes the following form , $H = L + {\\mathbf {\\lambda }}{f} + {\\mathbf {\\mu }}{C}$ which can be expanded to, $H=\\alpha T + {\\mathbf {\\lambda }}_r{v}+ {\\mathbf {\\lambda }}_v( {\\mathbf {a}}_g + \\frac{T}{m} \\hat{{\\mathbf {t}}} - \\lambda _m \\alpha T + \\mu _1 (T-T_{\\max }) + \\mu _2 (T_{\\min }-T)$ where $T-T_{\\max }$ and $T_{\\min }-T \\le 0$ and $\\mu _1, \\mu _2 > 0$ .",
"The control optimality is ensured as follows, $\\frac{\\partial H}{\\partial \\hat{{\\mathbf {t}}}} = 0 \\quad \\rightarrow \\quad \\text{minimum when} \\; {\\mathbf {\\lambda }}_v{{\\mathbf {t}}} = -1 \\quad \\rightarrow \\quad \\hat{{\\mathbf {t}}} = -\\frac{{\\mathbf {\\lambda }}_v}{||{\\mathbf {\\lambda }}_v||}$ Thus Eq.",
"(REF ) can be rewritten as, $H = \\alpha T + {\\mathbf {\\lambda }}_r{v}+ {\\mathbf {\\lambda }}_v{a}_g - \\frac{T}{m} ||{\\mathbf {\\lambda }}_v|| - \\lambda _m \\alpha T + \\mu _1 (T-T_{\\max }) + \\mu _2 (T_{\\min }-T)$ Now, in order to determine optimal thrust magnitude, we impose the following, $\\frac{\\partial H}{\\partial T} = \\alpha - \\frac{1}{m}||{\\mathbf {\\lambda }}_v|| - \\alpha \\lambda _m + \\mu _1 - \\mu _2 = 0$ There are three potential cases: if $\\mu _1 = \\mu _2 = 0 \\quad (T_{\\min } < T < T_{\\max })$ then $\\alpha - \\frac{1}{m}||{\\mathbf {\\lambda }}_v|| - \\alpha \\lambda _m = \\sigma = 0$ if $\\mu _1 = 0$ , $\\mu _2 > 0 \\quad (T=T_{\\min })$ then $\\sigma - \\mu _2 = 0 \\rightarrow \\sigma = \\mu _2 > 0$ if $\\mu _1 > 0$ , $\\mu _2 = 0 \\quad (T = T_{\\max })$ then $\\sigma + \\mu _1 = 0 \\rightarrow \\sigma = -\\mu _1 < 0$ Finally, one can conclude that the thrust magnitude has the following program: $T = {\\left\\lbrace \\begin{array}{ll} = T_{\\max } \\qquad \\text{if} \\qquad \\sigma < 0 \\\\ = T_{\\min } \\qquad \\text{if} \\qquad \\sigma > 0 \\end{array}\\right.",
"}$ It has been demonstrated that the singular case $\\sigma = 0$ corresponds to a constant thrust perpendicular to the gravity vector which is generally not possible for a powered descent problem.",
"Therefore, a singular arc is not part of the sought optimal solution.",
"Furthermore, it has been shown that for the guided powered descent on large planetary bodies, the switching function $\\sigma $ changes signs at most twice .",
"Consequently, the thrust magnitude can switch between min-max twice at the most.",
"That is, in the most general case, the thrust magnitude has a bang-bang profile max-min-max.",
"Hence, we can write the thrust magnitude as function of time with $t_1$ and $t_2$ as parameters, where $t_1$ and $t_2$ are the times where the switches happen; i.e.",
"$T=T(t;t_1,t_2)$ .",
"The additional first-order necessary conditions for the costate equations are written as, $\\dot{{\\mathbf {\\lambda }}}_r &= -\\dfrac{\\partial H }{\\partial {\\mathbf {r}}} &= {\\mathbf {0}} \\\\\\dot{{\\mathbf {\\lambda }}}_v &= -\\dfrac{\\partial H }{\\partial {\\mathbf {v}}} &= -\\dot{{\\mathbf {\\lambda }}}_r \\\\\\dot{\\lambda }_m &= -\\dfrac{\\partial H }{\\partial m} &= -\\frac{T}{m^2}||{\\mathbf {\\lambda }}_v||$ and the transversality condition implies that, $\\lambda _m(t_f) = 0 \\qquad \\text{and} \\qquad H(t_f) = 0.$"
],
[
"Solution via the Theory of Functional Connections",
"With the simplifications introduced in the previous section, in order to find the optimal state and thrust program, the following non-linear TPBVP must be solved: $\\dot{{\\mathbf {r}}} &&= {\\mathbf {v}} \\\\\\dot{{\\mathbf {v}}} &&= {\\mathbf {a}}_g - \\beta (t)\\dfrac{{\\mathbf {\\lambda }_v}}{||{\\mathbf {\\lambda }_v}||}\\\\\\dot{{\\mathbf {\\lambda }}}_r &&= {\\mathbf {0}} \\\\\\dot{{\\mathbf {\\lambda }}}_v &&= -{\\mathbf {\\lambda }_r} \\\\\\dot{\\lambda }_m &&= \\,- \\frac{T (t;t_1,t_2)}{m^2}||{\\mathbf {\\lambda }}_v|| \\\\H(t_f) = 0 &&= \\alpha T(t_f;t_1,t_2) + {\\mathbf {\\lambda }}_v({\\mathbf {a}}_g - \\beta (t_f)\\frac{{\\mathbf {\\lambda }}_v}{||{\\mathbf {\\lambda }}_v||} $ where $\\beta (t) \\triangleq \\frac{T(t;t_1,t_2)}{m(t)}$ subject to ${\\mathbf {r}(0)} = \\,{\\mathbf {r}_0} , {\\mathbf {v}(0)} = \\,{\\mathbf {v}_0}$ and ${\\mathbf {r}(t_f)} = \\,{\\mathbf {r}_f}, {\\mathbf {v}(t_f)} = \\,{\\mathbf {v}_f}$ .",
"It must be noted that $\\lambda _m$ , does not show up in any other equation and therefore Eq.",
"() can be solved independently.",
"Since the the transversality condition gives $\\lambda _m (t_f) = 0$ , Eq.",
"() can be solved by back propagation or by simply using the TFC method.",
"The solution of this equation is provided in the Appendix.",
"Since the solution of this problem exhibits a bang-bang profile for thrust, the original formulation of the TFC method (i.e.",
"that derived in Section ) must be adjusted to accommodate switching behavior in the control.",
"The general theory for this extension to hybrid systems is studied in , and it is applied in the following derivation.",
"Additionally, there are a few equations that are redundant and can be removed completely via the TFC to further simplify the solution of this nonlinear system of equations.",
"First, TFC are analytical expressions meaning that derivative of a for ${\\mathbf {r}}(t)$ is exactly the function ${\\mathbf {v}}(t)$ .",
"Therefore, the differential equation expressed by Eq.",
"(REF ) is unnecessary and can be disregarded.",
"Similarly, the equations for $\\dot{{\\mathbf {\\lambda }}}_r$ and $\\dot{{\\mathbf {\\lambda }}}_v$ can be simplified.",
"First, let us express the vector equations as three scalar equations each where the index $i$ represents the individual components.",
"Using this notation we can expand ${\\mathbf {\\lambda }}_v$ such that, $\\lambda _{v_i} = a_{0_i} + a_{1_i} z = {\\mathbf {h}}_{\\lambda }{\\xi }_{\\lambda _i}, \\quad \\text{for} \\quad i = 1,2,3$ which satisfies Eqs.",
"(-) through $\\dot{\\lambda }_{v_i} &= a_{1_i} \\\\\\dot{\\lambda }_{v_i} = -\\lambda _{r_i} &= -a_{1_i}$ This process reduces the problem to the solution of a single differential equation expressed by Eq.",
"() and an algebraic equation for the Hamiltonian at final time given by Eq.",
"().",
"Rewriting the differential equation in index notation and collecting all terms on one side, a loss function based on the residuals of the differential equation can be defined, ${\\cal L}_i = a_i - a_{g_i} + \\beta (t) \\, \\lambda _{v_i} \\left(\\displaystyle \\sum _{j=1}^3 \\lambda ^2_{v_j}\\right)^{-1/2} \\qquad \\text{for} \\qquad i = 1,2,3$ Now, the only step left is to construct a for the state variables.",
"Importantly, care must be taken in the construction of such .",
"In the above derivation of the thrust structure, we have shown that the thrust switches at most twice leading to a max-min-max profile.",
"Therefore, the function $\\beta (t)$ in Eq.",
"(REF ) jumps twice along the solution trajectory.",
"This switching causes three distinct differential equations that cannot be solved with a single polynomial expansion over the entire domain, as it is done for the energy-optimal guidance .",
"Therefore, a new formulation for the TFC approach has been developed to handle these hybrid systems .",
"This process allows for the continuity between each segment of the domain.",
"Figure: Visual representation of piece-wise approach using the TFC method .",
"In this derivation, the constrained expressions maintain continuity of position and velocity through embedded relative constraints.As shown in Fig.",
"REF , it is apparent that all sub-domains share the same constraint conditions (i.e, the initial and final position and velocity are constrained).",
"Therefore, a single constraint expression can be derived for the case of arbitrary constraints and then incorporated into the sub-domains.",
"The for this specific case was derived in Section and it is captured by Eq.",
"(REF ).",
"Consequently, the position, velocity, and acceleration can be expressed as, $r_i = g_i \\; + & \\!\\!",
"\\Omega _1(r_{0_i} - g_{0_i}) + \\Omega _2 (r_{f_i} - g_{f_i}) + \\nonumber \\\\ ~&+ \\Omega _3 (v_{0_i} - \\dot{g}_{0_i}) + \\Omega _4(v_{f_i} - \\dot{g}_{f_i}) \\\\ v_i = \\dot{g}_i\\; + & \\!\\!",
"\\dot{\\Omega }_1(r_{0_i} - g_{0_i}) + \\dot{\\Omega }_2 (r_{f_i} - g_{f_i}) + \\nonumber \\\\ ~&+ \\dot{\\Omega }_3 (v_{0_i} - \\dot{g}_{0_i}) + \\dot{\\Omega }_4(v_{f_i} -\\dot{g}_{f_i}) \\\\a_i = \\ddot{g}_i \\; + & \\!\\!",
"\\ddot{\\Omega }_1(r_{0_i} - g_{0_i}) + \\ddot{\\Omega }_2 (r_{f_i} - g_{f_i}) + \\nonumber \\\\ ~&+ \\ddot{\\Omega }_3 (v_{0_i} - \\dot{g}_{0_i}) + \\ddot{\\Omega }_4(v_{f_i} -\\dot{g}_{f_i}) $ where the $\\Omega $ parameters are solely a function of the independent variable and act as switching functions to force the expression to always satisfy the specified constraints.",
"These functions and their associated derivatives are summarized in Table REF where $t_*$ and $\\Delta t$ are defined by the specific segment.",
"The detailed by Eq.",
"(REF -) can be used as a template to write the constrained expressions for each segment of the solution trajectory.",
"In order to explicitly identify the segment, the pre-superscript notation will be used.",
"For example, ${(1)}{}{r}_i$ describes the position for the first segment defined on $t\\in [t_0,t_1]$ .",
"For this problem, $s=1$ is defined on $t\\in [t_0,t_1]$ , $s=2$ is defined on $t\\in [t_1,t_2]$ , and $s=3$ is defined on $t\\in [t_2,t_f]$ .",
"Using this formulation, the constrained expressions of position for each segment are, $\\begin{aligned}{(1)}{}{r}_i = {(1)}{}{g}_i &+ {(1)}{}{\\Omega }_1\\left(r_{0_i} - {(1)}{}{g}_{0_i}\\right) + {(1)}{}{\\Omega }_2 \\left(r_{1_i} - {(1)}{}{g}_{f_i}\\right) + \\\\ &+ {(1)}{}{\\Omega }_3 \\left(v_{0_i} - {(1)}{}{\\dot{g}}_{0_i}\\right) + {(1)}{}{\\Omega }_4\\left(v_{1_i} -{(1)}{}{\\dot{g}}_{f_i}\\right) \\\\{(2)}{}{r}_i = {(2)}{}{g}_i &+ {(2)}{}{\\Omega }_1\\left(r_{1_i} - {(2)}{}{g}_{0_i}\\right) + {(2)}{}{\\Omega }_2 \\left(r_{2_i} - {(2)}{}{g}_{f_i}\\right) + \\\\ &+ {(2)}{}{\\Omega }_3 \\left(v_{1_i} - {(2)}{}{\\dot{g}}_{0_i}\\right) + {(2)}{}{\\Omega }_4\\left(v_{2_i} -{(2)}{}{\\dot{g}}_{f_i}\\right) \\\\{(3)}{}{r}_i = {(3)}{}{g}_i &+ {(3)}{}{\\Omega }_1\\left(r_{2_i} - {(3)}{}{g}_{0_i}\\right) + {(3)}{}{\\Omega }_2 \\left(r_{f_i} - {(3)}{}{g}_{f_i}\\right) + \\\\ &+ {(3)}{}{\\Omega }_3 \\left(v_{2_i} - {(3)}{}{\\dot{g}}_{0_i}\\right) + {(3)}{}{\\Omega }_4\\left(v_{f_i} -{(3)}{}{\\dot{g}}_{f_i}\\right)\\end{aligned}$ where the derivative of these functions follow the form of Eq.",
"(REF -).",
"Now, the function of $g_i$ can also be expressed as a linear basis such that, $g_i(t) = {\\mathbf {h}}(z){\\xi }_i \\quad \\text{for} \\quad i = 1,2,3.$ with the subsequent derivatives following according to Eq.",
"(REF ).",
"This allows us to collect the unknown ${\\mathbf {\\xi }}_i$ vectors and write the in the form, $\\begin{aligned}{(1)}{}{r}_i = {(1)}{}{} &\\Big ({\\mathbf {h}} - \\Omega _1{\\mathbf {h}}_0 - \\Omega _2{\\mathbf {h}}_f - \\Omega _3\\dot{{\\mathbf {h}}}_0 - \\Omega _4\\dot{{\\mathbf {h}}}_f \\Big ){1}{{\\mathbf {\\xi }}}_i + \\\\ &+ {(1)}{}{\\Omega }_1 r_{0_i} + {(1)}{}{\\Omega }_2 r_{1_i} + {(1)}{}{\\Omega }_3 v_{0_i} + {(1)}{}{\\Omega }_4 v_{1_i} \\\\{(2)}{}{r}_i = {(2)}{}{} &\\Big ({\\mathbf {h}} - \\Omega _1{\\mathbf {h}}_0 - \\Omega _2{\\mathbf {h}}_f - \\Omega _3\\dot{{\\mathbf {h}}}_0 - \\Omega _4\\dot{{\\mathbf {h}}}_f \\Big ){2}{{\\mathbf {\\xi }}}_i + \\\\ &+ {(2)}{}{\\Omega }_1 r_{1_i} + {(2)}{}{\\Omega }_2 r_{2_i} + {(2)}{}{\\Omega }_3 v_{1_i} + {(2)}{}{\\Omega }_4 v_{2_i} \\\\{(3)}{}{r}_i = {(3)}{}{} &\\Big ({\\mathbf {h}} - \\Omega _1{\\mathbf {h}}_0 - \\Omega _2{\\mathbf {h}}_f - \\Omega _3\\dot{{\\mathbf {h}}}_0 - \\Omega _4\\dot{{\\mathbf {h}}}_f \\Big ){3}{{\\mathbf {\\xi }}}_i + \\\\ &+ {(3)}{}{\\Omega }_1 r_{2_i} + {(3)}{}{\\Omega }_2 r_{f_i} + {(3)}{}{\\Omega }_3 v_{2_i} + {(3)}{}{\\Omega }_4 v_{f_i} \\\\\\end{aligned}$ Along with the linear unknowns in ${\\mathbf {\\xi }}_i$ , the equations share linear unknowns in $r_{1_i},v_{1_i},r_{2_i},v_{2_i}$ which serve as the embedded relative constraints between adjacent segments.",
"With this new formulation, we now have three separate loss functions based on the residual of the differential equation over each segment ($s$ ) which are as follows, ${(s)}{}{{\\cal L}}_i = {(s)}{}{a}_i - a_{g_i} + \\beta (t) \\, \\lambda _{v_i} \\, \\left(\\displaystyle \\sum _{j=1}^3 \\lambda ^2_{v_j}\\right)^{-1/2}.$ Note that although the costate constrained expressions do not need to be split into separate domains, special attention must be paid to discretizing the equations according to the segment time ranges.",
"In order to solve for the unknown ${\\mathbf {\\xi }}_i$ parameters, a nonlinear least-squares technique can be used.",
"The latter requires computing the partials of the loss function to be taken with respect to all of the unknowns.",
"All partial derivatives for each segment and each unknown are provided below: $\\begin{aligned}\\frac{\\partial {(s)}{}{{\\cal L}}_i}{\\partial {(s)}{}{{\\mathbf {\\xi }}_i}} &= {(s)}{}{\\left(\\ddot{{\\mathbf {h}}} - \\ddot{\\Omega }_1{\\mathbf {h}}_0 - \\ddot{\\Omega }_2{\\mathbf {h}}_f - \\ddot{\\Omega }_3\\dot{{\\mathbf {h}}}_0 - \\ddot{\\Omega }_4\\dot{{\\mathbf {h}}}_f \\right) \\\\\\frac{\\partial {(1)}{}{{\\cal L}}_i}{\\partial r_{1_i}} &= {(1)}{}{\\ddot{\\Omega }}_2 \\\\\\frac{\\partial {(1)}{}{{\\cal L}}_i}{\\partial v_{1_i}} &= {(1)}{}{\\ddot{\\Omega }}_4 \\\\ \\frac{\\partial {(2)}{}{{\\cal L}}_i}{\\partial r_{1_i}} &= {(2)}{}{\\ddot{\\Omega }}_1, \\qquad \\frac{\\partial {(2)}{}{{\\cal L}}_i}{\\partial v_{1_i}} = {(2)}{}{\\ddot{\\Omega }}_3 \\\\\\frac{\\partial {(2)}{}{{\\cal L}}_i}{\\partial r_{2_i}} &= {(2)}{}{\\ddot{\\Omega }}_2, \\qquad \\frac{\\partial {(2)}{}{{\\cal L}}_i}{\\partial v_{2_i}} = {(2)}{}{\\ddot{\\Omega }}_4 \\\\\\frac{\\partial {(3)}{}{{\\cal L}}_i}{\\partial r_{2_i}} &= {(3)}{}{\\ddot{\\Omega }}_1 \\\\ \\frac{\\partial {(3)}{}{{\\cal L}}_i}{\\partial v_{2_i}} &= {(3)}{}{\\ddot{\\Omega }}_3}\\end{aligned}For the costate portion, if i = j\\begin{equation*}\\frac{\\partial {\\cal L}_i}{\\partial {\\mathbf {\\xi }_{\\lambda _i}}} = \\beta (t) \\left[ \\left(\\displaystyle \\sum _{j=1}^3 \\lambda ^2_{v_j}\\right)^{-1/2} - \\lambda ^2_{v_i} \\, \\left(\\displaystyle \\sum _{j=1}^3 \\lambda ^2_{v_j}\\right)^{-3/2}\\right] {\\mathbf {h}}_{\\lambda }{equation*}if i \\ne j\\begin{equation*}\\frac{\\partial {\\cal L}_i}{\\partial {\\mathbf {\\xi }_{\\lambda _j}}} = \\beta (t) \\left[ - \\lambda _{v_i} \\, \\lambda _{v_j} \\, \\left(\\displaystyle \\sum _{j=1}^3 \\lambda ^2_{v_j}\\right)^{-3/2}\\right] {\\mathbf {h}}_{\\lambda }{equation*}In addition to the loss functions for the problem dynamics given by Eq.",
"(\\ref {eq:v_dot}), a loss function associated with the transversality conditions for the Hamiltonian needs to be defined,\\begin{equation}\\mathcal {L}_H = \\alpha T_\\text{max} + \\sum _{i=1}^{3}\\lambda _{v_i} a_{g_i} - \\beta (t_f) \\left(\\sum _{i=1}^{3}\\lambda ^2_{v_i}\\right)^{\\frac{1}{2}}\\end{equation}It follows that the only none-zero partial of \\mathcal {L}_H is with respect to {\\mathbf {\\xi }}_\\lambda , that is defined by\\begin{equation}\\frac{\\partial \\mathcal {L}_H}{\\partial {\\mathbf {\\xi }_{\\lambda _i}}} = \\left[a_{g_i} - \\beta (t_f) \\, \\lambda _{v_i} \\, \\left(\\sum _{j=1}^3 \\lambda ^2_{v_j}\\right)^{-1/2}\\right] {\\mathbf {h}}_{\\lambda }{equation}Next, by discretizing the domain over N points, these partials become a vector or matrix where the second dimension is the number of unknowns.",
"All partials can be combined into one augmented matrix and one augment vector for the loss function such that according to,\\begin{equation}\\mathbb {J} = \\begin{bmatrix} {(1)}{}{J}_{{\\mathbf {\\xi }}} & {(1)}{}{J}_{r_1, v_1} & {\\mathbf {0}}_{(3N\\times 3m)} & {\\mathbf {0}}_{(3N\\times 6)} & {\\mathbf {0}}_{(3N\\times 3m)} & {(1)}{}{J}_{{\\mathbf {\\xi }}_\\lambda } \\\\ {\\mathbf {0}}_{(3N\\times 3m} & {(2)}{}{J}_{r_1, v_1} & {(2)}{}{J}_{{\\mathbf {\\xi }}} & {(2)}{}{J}_{r_2, v_2} & {\\mathbf {0}}_{(3N\\times 3m)} & {(2)}{}{J}_{{\\mathbf {\\xi }}_\\lambda } \\\\ {\\mathbf {0}}_{(3N\\times 3m)} & {\\mathbf {0}}_{(3N\\times 6)} & {\\mathbf {0}}_{(3N\\times 3m)} & {(3)}{}{J}_{r_2, v_2} & {(3)}{}{J}_{{\\mathbf {\\xi }}} & {(3)}{}{J}_{{\\mathbf {\\xi }}_\\lambda } \\\\ {\\mathbf {0}}_{(1\\times 3m)} & {\\mathbf {0}}_{(1\\times 6)} & {\\mathbf {0}}_{(1\\times 3m)} & {\\mathbf {0}}_{(1\\times 6)} & {\\mathbf {0}}_{(1\\times 3m)} & J_H \\end{bmatrix}_{(\\lbrace 9N+1\\rbrace \\times \\lbrace 9m + 18\\rbrace )}\\end{equation}with the augmented vector of the loss functions and unknown vector defined as,\\begin{equation*}\\begin{aligned}\\mathbb {L} &= \\begin{Bmatrix} {(1)}{}{{\\cal L}}_1 {(1)}{}{{\\cal L}}_2 {(1)}{}{{\\cal L}}_3 {(2)}{}{{\\cal L}}_1 {(2)}{}{{\\cal L}}_2 {(2)}{}{{\\cal L}}_3 {(3)}{}{{\\cal L}}_1 {(3)}{}{{\\cal L}}_2 {(3)}{}{{\\cal L}}_3 \\mathcal {L}_H\\end{Bmatrix}_{(\\lbrace 9N+1\\rbrace \\times 1)} \\Xi &= \\begin{Bmatrix} {(1)}{}{{\\mathbf {\\xi }}}_1 {(1)}{}{{\\mathbf {\\xi }}}_2 {(1)}{}{{\\mathbf {\\xi }}}_3 {\\mathbf {r}}_1 {\\mathbf {v}}_1 {(2)}{}{{\\mathbf {\\xi }}}_1 {(2)}{}{{\\mathbf {\\xi }}}_2 {(2)}{}{{\\mathbf {\\xi }}}_3 {\\mathbf {r}}_2 {\\mathbf {v}}_2 {(3)}{}{{\\mathbf {\\xi }}}_1 {(3)}{}{{\\mathbf {\\xi }}}_2 {(3)}{}{{\\mathbf {\\xi }}}_3 {\\mathbf {\\xi }}_{\\lambda _1} {\\mathbf {\\xi }}_{\\lambda _2} {\\mathbf {\\xi }}_{\\lambda _3}{Bmatrix}_{(9m + 18)}{aligned}\\end{Bmatrix}The terms of Eq.",
"(\\ref {eq:aug_J}) are defined by the following equations:\\begin{equation*}{(s)}{}{J}_{{\\mathbf {\\xi }}} = \\begin{bmatrix} \\frac{\\partial {(s)}{}{{\\cal L}}_1}{\\partial {(s)}{}{{\\mathbf {\\xi }}_1}} & {\\mathbf {0}} & {\\mathbf {0}} \\\\ {\\mathbf {0}} & \\frac{\\partial {(s)}{}{{\\cal L}}_2}{\\partial {(s)}{}{{\\mathbf {\\xi }}_2}} & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {\\mathbf {0}} & \\frac{\\partial {(s)}{}{{\\cal L}}_3}{\\partial {(s)}{}{{\\mathbf {\\xi }}_3}} \\end{bmatrix}_{(3N\\times 3m)},\\quad {(s)}{}{J}_{{\\mathbf {\\xi }}_\\lambda } = \\begin{bmatrix} J_{{\\mathbf {\\xi }}_{\\lambda _{11}}} & J_{{\\mathbf {\\xi }}_{\\lambda _{12}}} & J_{{\\mathbf {\\xi }}_{\\lambda _{13}}} \\\\ J_{{\\mathbf {\\xi }}_{\\lambda _{21}}} & J_{{\\mathbf {\\xi }}_{\\lambda _{22}}} & J_{{\\mathbf {\\xi }}_{\\lambda _{23}}} \\\\ J_{{\\mathbf {\\xi }}_{\\lambda _{31}}} & J_{{\\mathbf {\\xi }}_{\\lambda _{32}}} & J_{{\\mathbf {\\xi }}_{\\lambda _{33}}} \\end{bmatrix}_{(3N\\times 6)}\\end{equation*}\\end{aligned}\\begin{equation*}\\begin{aligned}{(1)}{}{J}_{r_1,v_1} = \\begin{bmatrix} {(1)}{}{ \\ddot{\\Omega }}_2 & {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_4 & {\\mathbf {0}} & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_2 & {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_4 & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_2 & {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_4 \\end{bmatrix}_{(3N\\times 6)} \\\\ {(2)}{}{J}_{r_1,v_1} = \\begin{bmatrix} {(1)}{}{ \\ddot{\\Omega }}_1 & {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_3 & {\\mathbf {0}} & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_1 & {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_3 & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_1 & {\\mathbf {0}} & {\\mathbf {0}} & {(1)}{}{ \\ddot{\\Omega }}_3 \\end{bmatrix}_{(3N\\times 6)} \\\\{(2)}{}{J}_{r_2,v_2} = \\begin{bmatrix} {(2)}{}{ \\ddot{\\Omega }}_2 & {\\mathbf {0}} & {\\mathbf {0}} & {(2)}{}{ \\ddot{\\Omega }}_4 & {\\mathbf {0}} & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {(2)}{}{ \\ddot{\\Omega }}_2 & {\\mathbf {0}} & {\\mathbf {0}} & {(2)}{}{ \\ddot{\\Omega }}_4 & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {\\mathbf {0}} & {(2)}{}{ \\ddot{\\Omega }}_2 & {\\mathbf {0}} & {\\mathbf {0}} & {(2)}{}{ \\ddot{\\Omega }}_4 \\end{bmatrix}_{(3N\\times 6)} \\\\{(3)}{}{J}_{r_2,v_2} = \\begin{bmatrix} {(3)}{}{ \\ddot{\\Omega }}_1 & {\\mathbf {0}} & {\\mathbf {0}} & {(3)}{}{ \\ddot{\\Omega }}_3 & {\\mathbf {0}} & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {(3)}{}{ \\ddot{\\Omega }}_1 & {\\mathbf {0}} & {\\mathbf {0}} & {(3)}{}{ \\ddot{\\Omega }}_3 & {\\mathbf {0}} \\\\ {\\mathbf {0}} & {\\mathbf {0}} & {(3)}{}{ \\ddot{\\Omega }}_1 & {\\mathbf {0}} & {\\mathbf {0}} & {(3)}{}{ \\ddot{\\Omega }}_3 \\end{bmatrix}_{(3N\\times 6)}\\end{aligned}\\end{equation*}\\begin{equation*}J_H = \\begin{bmatrix} \\dfrac{\\partial \\mathcal {L}_H}{\\partial {\\mathbf {\\xi }_{\\lambda _1}}}, & \\quad \\dfrac{\\partial \\mathcal {L}_H}{\\partial {\\mathbf {\\xi }_{\\lambda _2}}}, & \\quad \\dfrac{\\partial \\mathcal {L}_H}{\\partial {\\mathbf {\\xi }_{\\lambda _3}}}\\end{bmatrix}_{(1\\times 6)}\\end{equation*}\\end{equation*}Finally, using Eq (\\ref {eq:aug_J}) along with the augment loss functions and unknown vector, an iterative least-square approach can be used to update the unknown parameters according to,\\begin{equation*}\\Xi _{k+1}= \\Xi _k - \\; \\text{d}\\Xi _k \\quad \\text{where} \\quad \\; \\text{d}\\Xi _k = \\text{qr}(\\mathbb {J}_k,\\mathbb {L}_k)\\end{equation*}Importantly, an initial estimate of the parameters is needed in order to initialize the iterative least-squares process.",
"Since the problem is a boundary-value problem, a first guess for {(s)}{}{{\\mathbf {\\xi }}}_i, {\\mathbf {r}}_1, {\\mathbf {r}}_2, {\\mathbf {v}}_1, and {\\mathbf {v}}_2 can be determined by simply connecting the initial and final position with a straight line and using this trajectory for a least squares fitting of the constrained expressions\\ describing the {(s)}{}{r}_i terms.",
"Next, since the \\lambda _{v_i} is related to the thrust direction it can be assumed similar to Eq.",
"(51) of Ref.",
"\\cite {upg} such that,\\begin{equation*}{\\mathbf {\\lambda }}_{v_0} = \\frac{{\\mathbf {v}}_0}{||{\\mathbf {v}}_0||}\\end{equation*}However, the initialization of {\\mathbf {\\lambda }}_r = {\\mathbf {0}} is prohibitive in the sense of the TFC method because this involves setting {\\mathbf {\\xi }}_\\lambda coefficients to zeros.",
"Therefore in this paper the coefficients are initialized using the following prescription:\\begin{equation*}{\\mathbf {\\lambda }}_{v_f} = -\\frac{{\\mathbf {r}}_0}{||{\\mathbf {r}}_0||}\\end{equation*}\\end{equation}\\end{equation*}\\section {Summary of Algorithm}Overall, the TFC method was used as a inner-loop function to minimize the residuals of the first-order necessary conditions subject to a prescribed thrust profile T(t;t_1,t_2), i.e.",
"the switching times t_1 and t_2 are assumed to be known by the TFC-based inner-loop routine.",
"Consequently, an outer-loop routine has been developed to optimize the switching times given the L_2-norms of the residual of the first-order conditions, and the Hamiltonian over the first two segments (here, the MATLAB \\cite {MATLAB} {\\tt fsolve} was used) .",
"A flow chart of the relevant inputs and outputs is provided in Fig.",
"\\ref {fig:algorithm}.\\begin{figure}[H]\\centering \\includegraphics [width=]{Figures/algorithm.pdf}\\caption {Summary of the full algorithm used with the TFC approach.",
"}\\end{figure}\\end{equation*}$"
],
[
"Results",
"The proposed method was validated using two specific tests cases based on selected initial conditions defining a powered descent guidance scenario for landing on Mars.",
"Within such framework, in Section REF the algorithm is tested on initial conditions where the optimal trajectory is characterized by a min-max thrust profile.",
"Furthermore, in Section REF the case where the optimal thrust profile is max-min-max is studied.",
"In both cases our results were compared with the GPOPS-II solutions for the boundary conditions.",
"The algorithm was fully implemented in MATLAB R$2019a$ and all test cases were solved on a MacBook Pro (2016) macOS Version 10.15, with a 3.3 GHz Dual-Core Intel® Core™ i7 and with 16 GB of RAM."
],
[
"Constant Test Parameters",
"We consider the trajectory optimization problem for a spacecraft performing powered descent for pinpoint landing on Mars.",
"The gravitational field is assumed constant as generally, the powered descent starts below $1.5$ km.",
"For the numerical test, the lander parameters have been assumed to be similar to the one presented in Ref.",
"and reported in Table REF .",
"Table: Appendix"
]
] | 2001.03572 |
[
[
"Transport-induced suppression of nuclear field fluctuations in\n multi-quantum-dot systems"
],
[
"Abstract Magnetic noise from randomly fluctuating nuclear spin ensembles is the dominating source of decoherence for many multi-quantum-dot multielectron spin qubits.",
"Here we investigate in detail the effect of a DC electric current on the coupled electron-nuclear spin dynamics in double and triple quantum dots tuned to the regime of Pauli spin blockade.",
"We consider both systems with and without significant spin-orbit coupling and find that in all cases the flow of electrons can induce a process of dynamical nuclear spin polarization that effectively suppresses the nuclear polarization gradients over neighboring dots.",
"Since exactly these gradients are the components of the nuclear fields that act harmfully in the qubit subspace, we believe that this presents a straightforward way to extend coherence times in multielectron spin qubits by at least one order of magnitude."
],
[
"Introduction",
"Spin qubits hosted in semiconductor quantum dots form an attractive qubit implementation that promises easily scalable quantum processors [1], [2], [3].",
"One drawback of the originally proposed single-spin single-quantum-dot qubit is that it requires highly localized magnetic fields for qubit control [4], [5].",
"To overcome the practical challenge of creating such fields, qubits can also be encoded in a multielectron spin state hosted in a multi-quantum-dot structure.",
"If one defines a qubit in the unpolarized singlet-triplet subspace of two spins in a double quantum dot, then the field along one axis of the Bloch sphere can be controlled fully electrically, but the second control axis is still set by the magnetic field gradient over the two dots [6], [7].",
"Adding one more spin to the setup, one can create a three-electron double-dot hybrid qubit [8], [9] or a triple-dot exchange-only qubit [10], [11], [12], [13], [14], offering electric control over the full Bloch sphere through exchange interactions [15], [16].",
"An important remaining challenge for many multispin qubit implementations is their rapid decoherence.",
"Its two main sources are (i) hyperfine coupling of the electronic spins to the randomly fluctuating nuclear spin baths in the quantum dots [17], [18], [19], [20] and (ii) charge fluctuations in the environment that interfere with exchange-based qubit control [21], [22].",
"The latter could be mitigated by enhancing device quality or operating the qubit at a (higher-order) sweet spot [23], [24], [25], [26], [27], which leaves the nuclear spin noise as an important intrinsic obstacle for further progress.",
"Several approaches to reducing the harmful effects of nuclear spin fluctuations in exchange-only qubits are being explored: (i) One can host the qubits in quantum dots created in isotopically purified ${}^{28}$ Si, which can be made nearly nuclear-spin-free [28], [29], [30], [31], [32].",
"However, silicon comes with the complication of the extra valley degree of freedom [2], which is hard to control [33], [34], [35] and provides an extra channel for leakage and dephasing [36], [37].",
"(ii) It is possible to encode the qubit in a four-electron singlet-only subspace [38], [39], [40], which makes it intrinsically insensitive to the fluctuating nuclear fields.",
"This, however, presents significant complications for device design and tuning.",
"(iii) One can actively mitigate the nuclear spin noise, e.g., by applying complex spin-echo-like pulse sequences that effectively filter out all peaks from the noise spectrum [41] or with an active feedback cycle that relies on continuous measurement of the magnitude of the nuclear fields [42].",
"In this paper we propose another approach that falls in the last category but is much simpler to implement.",
"A few years ago, experiments on a double quantum dot hosted in an InAs nanowire suggested that when running a DC electric current through the system in the regime of Pauli spin blockade, an interplay between the hyperfine interaction and strong spin-orbit interaction (SOI) in InAs can give rise to a process of dynamical nuclear polarization that effectively quenches the total Zeeman gradient over the two dots [43].",
"Here, we investigate this idea in more detail, and we show how it not only works for double quantum dots with strong SOI, but also in the absence of SOI and—maybe more importantly—can be implemented in a similar way in a linear triple quantum dot, where it results in a suppression of both nuclear field gradients between neighboring dots.",
"For all mechanisms we investigate, we present a simple intuitive picture as well as analytic and numerical results that support this picture and predict a suppression of the fluctuations of the nuclear field gradients of one to two orders of magnitude.",
"Since hyperfine-induced decoherence of both singlet-triplet and exchange-only qubits originates mainly from these gradients, we believe that this current-induced suppression mechanisms yields a straightforward way to significantly extend the coherence time of multielectron qubits.",
"The rest of this paper is separated into two main parts, Secs.",
"and , which discuss the double-dot and triple-dot setup, respectively.",
"Both parts are are organized as follows: In Subsections A we briefly review the definition of the respective qubit and present a description of the system in terms of a simple model Hamiltonian.",
"In Subsections B we then present an intuitive picture of the mechanism behind the suppression of the gradients.",
"Subsections C contain approximate analytic expressions for the current-induced dynamics of the nuclear polarizations, which we corroborate in Subsections D with numerical simulations of the stochastic nuclear spin dynamics.",
"Subsections E contain a short conclusion, and a final general conclusion is presented in Sec. .",
"The singlet-triplet qubit is usually hosted by two electrons residing in a double quantum dot and is defined in two two-particle spin states with total spin projection $S_z = 0$ .",
"Using gate voltages, the double dot is tuned close to the (1,1)–(0,2) charge transition (the gray line in the charge stability diagram shown in Fig.",
"REF a).",
"Here, the low-energy part of the spectrum consists of five states: The large orbital level splitting on the dots (typically $\\sim $ meV) allows us to disregard states involving excited orbital states; the Pauli exclusion principle then dictates that the two electrons in the (0,2) configuration must be in a spin-singlet state, $|{S_{02}}\\rangle $ .",
"In the (1,1) charge configuration all four spin states are accessible; one singlet state $|{S}\\rangle $ , and three triplet states $|{T_\\pm }\\rangle $ and $|{T_0}\\rangle $ .",
"We describe this five-level subspace with a simple model Hamiltonian, $\\hat{H}_0 = \\hat{H}_e + \\hat{H}_t + \\hat{H}_{\\rm Z}.$ Here $\\hat{H}_e = -\\epsilon |{S_{02}}\\rangle \\langle {S_{02}}|,$ describes the relative energy detuning of the (1,1) and (0,2) charge states as a function of the detuning parameter $\\epsilon $ , see Fig.",
"REF .",
"Further, $\\hat{H}_t = t_s \\big [ |{S}\\rangle \\langle {S_{02}}| + |{S_{02}}\\rangle \\langle {S}| \\big ],$ accounts for spin-conserving interdot tunneling, and $\\hat{H}_{\\rm Z} = &g\\mu _{\\rm B}B \\big [ |{T_+}\\rangle \\langle {T_+}|-|{T_-}\\rangle \\langle {T_-}|\\big ],$ describes the Zeeman effect due to a homogeneous magnetic field.",
"A typical spectrum of $\\hat{H}_0$ as a function of $\\epsilon $ is shown in Fig.",
"REF c, where we have set $t_s = 0.6\\, E_{\\rm Z}$ with $E_{\\rm Z} = |g\\mu _{\\rm B}B|$ the Zeeman splitting, and we assumed $g<0$ .",
"Figure: (a) Typical charge stability diagram of a double quantum dot, showing the ground state charge configuration of the system as a function of the local dot potentials V L V_L and V R V_R.",
"(b) Sketch of the double quantum dot.If the system is in a (0,2) charge state, the two electrons must have opposite spin.",
"(c) Energy spectrum along the detuning axis indicated in (a) showing the relevant (1,1) and (0,2) states, where a finite interdot tunnel coupling and Zeeman splitting were included.The blue(red) lines correspond to spin triplet(singlet) states.The qubit is defined in an unpolarized subspace consisting of a triplet, $|{1}\\rangle = |{T_0}\\rangle $ , and the lower of the two singlet branches, $|{0}\\rangle = |{S_2}\\rangle = \\cos \\frac{\\theta }{2} |{S_{02}}\\rangle + \\sin \\frac{\\theta }{2} |{S}\\rangle $ (dashed levels in Fig.",
"REF c) where $\\tan \\theta = 2t_s/\\epsilon $ .",
"From the projected qubit Hamiltonian $\\hat{H}_q = \\frac{\\omega _q}{2} \\hat{\\sigma }_z,$ with $\\omega _q = \\epsilon /2 + \\sqrt{(\\epsilon /2)^2 + t_s^2}$ we see that the qubit has a splitting that is tunable electrically via $V_{L,R}$ , presenting an advantage over the single-spin qubit, which requires magnetic control.",
"In semiconductors with non-zero nuclear spin, such as GaAs and InAs, an important source of decoherence for such a qubit is the hyperfine interaction between the nuclear and electronic spins.",
"The dominating term is the contact interaction, described by $\\hat{H}_\\text{hf}= \\frac{A}{2N}\\sum _{d,k}\\left(2\\hat{S}^z_d\\hat{I}^z_{d,k} + \\hat{S}_d^+\\hat{I}_{d,k}^- + \\hat{S}_d^-\\hat{I}_{d,k}^+\\right),$ where $\\hat{\\bf S}_d$ is the electron spin operator on dot $d$ and $\\hat{\\bf I}_{d,k}$ the nuclear spin operator for nucleus $k$ on dot $d$ .",
"For simplicity we assumed that all nuclei in a dot are coupled equally strongly to the electron spin in that dot and that both dots have the same number of spinful nuclei $N$ , typically $N\\sim 10^5$ –$10^6$ .",
"The coupling constant $A$ is a material parameter and usually of the order $\\sim 100~\\mu $ eV.",
"Due to the small nuclear magnetic moment, the nuclear spin ensemble is in a fully mixed state in equilibrium at typical dilution fridge temperatures, and within a mean-field approximation we can then write $\\hat{H}_\\text{hf,mf}= {\\bf K}_L \\cdot \\hat{\\bf S}_L + {\\bf K}_R \\cdot \\hat{\\bf S}_R,$ where the nuclear fields ${\\bf K}_{L,R}$ are random with an r.m.s.",
"value $\\sim A/\\sqrt{N}$ , typically of the order $\\sim $ mT when translated to an effective magnetic field.",
"Projecting this Hamiltonian to the qubit subspace yields $\\hat{H}_\\text{hf,q}= \\delta K^z \\sin \\frac{\\theta }{2}\\, \\hat{\\sigma }_x,$ where $\\delta K^z = \\frac{1}{2}(K^z_L - K^z_R)$ is a quasistatic random field gradient.",
"For the singlet-triplet qubit this gradient can be used for initialization along the $\\pm \\hat{x}$ -axis of the Bloch sphere [6], but in general its random nature presents a main source of qubit decoherence.",
"Protocols how to control or suppress the gradient $\\delta K^z$ could lead to significant improvement of the qubit coherence time."
],
[
"Transport-induced nuclear spin pumping: Qualitative picture",
"In Ref.",
"[43] it was shown how such a gradient can get suppressed naturally in the presence of strong spin-orbit interaction, when the double dot is embedded in a transport setup.",
"We will first review here the intuitive picture of the underlying mechanism, as outlined in Ref.",
"[43], and then show how it also works in the absence of spin-orbit interaction.",
"In the next sections we will support this with an analytic investigation and numerical simulations of the coupled electron-nuclear spin dynamics.",
"Figure: (a) The double quantum dot is tunnel coupled to source and drain reservoirs, and in the presence of a bias voltage electrons can flow from source to drain.Energy spectrum as a function of Δ\\Delta for the (1,1) spin states: (b) with and (c) without spin-orbit coupling.",
"The thickness of the lines indicates the occupation probability of the eigenstates as given by Eqs. ()–().",
"Preferred electron-nuclear spin flip rates close to Δ=0\\Delta =0 are indicated by the gray arrows.We assume the double dot to be connected in a linear arrangement to source and drain reservoirs, as sketched in Fig.",
"REF a, and to be tuned close to the so-called “triple point” (where three stable charge regions meet) indicated by the red dot in Fig.",
"REF a.",
"Then, a finite bias voltage over source and drain can give rise to a current through the system, via the transport cycle $(0,1)\\rightarrow (1,1)\\rightarrow (0,2)\\rightarrow (0,1)$ .",
"We assume that the system is tuned to the open regime, where the couplings to the reservoirs, characterized by the rates $\\Gamma _{\\rm in,out}$ , are the largest relevant energy scales.",
"This ensures that the tunneling processes $(0,2) \\rightarrow (0,1) \\rightarrow (1,1)$ are effectively instantaneous, and the interesting dynamics happen during the transition $(1,1) \\rightarrow (0,2)$ which involves the same five levels as before, $\\lbrace |{T_{\\pm ,0}}\\rangle , |{S}\\rangle , |{S_{02}}\\rangle \\rbrace $ .",
"In the absence of spin-mixing processes, the only available transport path is $(0,1)\\rightarrow |{S}\\rangle \\rightarrow |{S_{02}}\\rangle \\rightarrow (0,1)$ and population of one of the (1,1) triplet states results in spin blockade of the current.",
"The effect of SOI in this context is twofold: (i) small inhomogeneities in the confining potential can result in different effective $g$ -factors $g_{L,R}$ on the two dots, and (ii) tunneling from one dot to the other can now be accompanied by a spin flip [44].",
"These two effects can be described by the Hamiltonian $\\hat{H}_{so} = {} & {} it^+|{T_-}\\rangle \\langle {S_{02}}| - it^-|{T_+}\\rangle \\langle {S_{02}}|\\nonumber \\\\&+ it_z|{T_0}\\rangle \\langle {S_{02}}| + \\Delta _{so} |{T_0}\\rangle \\langle {S}| + {\\rm H.c.},$ where $t^\\pm =\\frac{1}{\\sqrt{2}}\\left(t_x\\pm it_y\\right)$ , with the real vector ${\\bf t}$ characterizing the spin-orbit induced spin-flip tunnel coupling, and $\\Delta _{so} = \\frac{1}{2}(g_L-g_R)\\mu _{\\rm B}B$ accounting for the difference in $g$ -factors on the dots.",
"The magnitude of the vector ${\\bf t}$ can be estimated as $\\sim (d/l_{so}) t_s$ , where $d$ is the distance between the two dots and $l_{so}$ the spin-orbit length in the direction of the interdot axis.",
"We see that SOI can lift the blockade of the polarized states $|{T_\\pm }\\rangle $ .",
"But if the total Zeeman gradient $\\Delta $ vanishes, $\\Delta = \\Delta _{so} + \\delta K^z = 0$ , the two unpolarized (1,1) states can still be combined into a bright state $|{B}\\rangle = [t_s|{S}\\rangle + it_z |{T_0}\\rangle ] / \\sqrt{t_s^2 + t_z^2 }$ (that is coupled to $|{S_{02}}\\rangle $ with strength $\\sqrt{t_s^2 + t_z^2 }$ ) and a dark state $|{D}\\rangle = [it_z|{S}\\rangle + t_s |{T_0}\\rangle ] / \\sqrt{t_s^2 + t_z^2 }$ (that is not coupled).",
"So in this case there is still one spin-blocked state left, $|{D}\\rangle $ , which, as a consequence, will be populated with high probability, whereas the other three states $|{T_\\pm }\\rangle $ and $|{B}\\rangle $ have vanishing population.",
"Adding a finite Zeeman gradient $\\Delta \\ne 0$ mixes the states $|{T_0}\\rangle $ and $|{S}\\rangle $ , and thus $|{B}\\rangle $ and $|{D}\\rangle $ , lifting the blockade of $|{D}\\rangle $ which results in a more evenly distributed population of the levels.",
"These observations are illustrated in Fig.",
"REF b, where we show the energy spectrum of the four (1,1) states as a function of $\\Delta $ : The thickness of the lines indicates the relative occupation probabilities of the four states when embedded in a transport setup.",
"We have set $t_s = 0.6\\,E_{\\rm Z}$ and ${\\bf t} = \\lbrace 0.4,0.4,0.4 \\rbrace t_s$ , and we assumed the escape rates of every state to be proportional to the modulo square of its total coupling to $|{S_{02}}\\rangle $ given by $\\hat{H}_t + \\hat{H}_{so}$ , which is valid in the limit of large $\\Gamma _{\\rm out}$ .",
"Based on this, we can now develop a qualitative understanding of the resulting coupled electron-nuclear spin dynamics.",
"The hyperfine Hamiltonian (REF ) contains terms $\\hat{S}_d^\\pm \\hat{I}_{d,k}^\\mp $ which can give rise to so-called spin flip-flop processes in which the electron on dot $d$ exchanges one unit of angular momentum with one of the nuclei in the dot, which changes the value of the effective nuclear field $K^z_d$ by a small amount.",
"A non-equilibrium electron spin polarization on the dots can thus be slowly transferred to the nuclear spin ensemble which, in turn, can influence the electron dynamics, potentially yielding an intricate feedback cycle.",
"To see if there is a preferred direction of nuclear spin polarization, we investigate the spin structure of the most strongly occupied electronic state: At $\\Delta = 0$ the state $|{D}\\rangle $ contains equally large components of $|{\\uparrow \\downarrow }\\rangle $ and $|{\\downarrow \\uparrow }\\rangle $ , i.e., $|\\mathinner {\\langle {{D}|{\\uparrow \\downarrow }}\\rangle }|^2 = |\\mathinner {\\langle {{D}|{\\downarrow \\uparrow }}\\rangle }|^2 = \\frac{1}{2}$ , where $|{\\alpha \\beta }\\rangle $ denotes the (1,1) state with a spin-$\\alpha $ electron on the left dot and a spin-$\\beta $ electron on the right dot.",
"Due to these equal weights, all possible hyperfine-induced flip-flop processes are to first approximation equally likely, and the net nuclear spin flip rates on both dots thus vanish.",
"However, when $\\Delta > 0$ the most strongly occupied state acquires a slightly $\\downarrow \\uparrow $ -polarized character (see Fig.",
"REF b) and then the flip-flop processes caused by $\\hat{S}_L^+\\hat{I}_{L,k}^-$ and $\\hat{S}_R^-\\hat{I}_{R,k}^+$ (illustrated by the gray arrows in the figure) are more likely than the opposite ones.",
"This results in a net negative(positive) nuclear spin pumping rate in the left(right) dot, which reduces $\\delta K^z$ and thus $\\Delta $ .",
"Similarly, we see that when $\\Delta < 0$ the small polarization of the most strongly occupied state will drive $\\delta K^z$ and thus $\\Delta $ to larger values.",
"All together, this indeed suggests that the specific manifestation of spin blockade in the presence of strong SOI can result in a self-quenching of the Zeeman gradient over the dots.",
"The experimental results presented in Ref.",
"[43] were consistent with this picture.",
"Let us now turn to the limit of very weak SOI, where we set ${\\bf t}= \\Delta _{so} = 0$ .",
"In that case we see that at $\\Delta = 0$ there are three spin-blocked states, the (1,1) triplet states $|{T_{\\pm ,0}}\\rangle $ .",
"At this special point one thus finds an occupation probability of $\\frac{1}{3}$ for each of the triplet states and zero for the coupled state $|{S}\\rangle $ .",
"But again, due to the symmetric polarization of all four states, there will be no net nuclear spin pumping at this point.",
"Away from the special point $\\Delta = 0$ , the Zeeman gradient mixes the states $|{S}\\rangle $ and $|{T_0}\\rangle $ and both unpolarized eigenstates end up having a finite coupling to $|{S_{02}}\\rangle $ , whereas the polarized triplets remain uncoupled.",
"This results in an occupation probability of approximately $\\frac{1}{2}$ for $|{T_+}\\rangle $ and $|{T_-}\\rangle $ and zero for the two unpolarized states.",
"We first focus on the case $\\Delta > 0$ , where $|{D}\\rangle $ evolves into a state with a slightly stronger $\\downarrow \\uparrow $ -component, whereas $|{B}\\rangle $ acquires a slight $\\uparrow \\downarrow $ -character (see Fig.",
"REF c).",
"Flip-flops from the blocked states can cause transitions to both unpolarized states, but due to its stronger coupling to $|{S_{02}}\\rangle $ transitions to the state $|{B}\\rangle $ at $\\Delta = 0$ are favored.",
"This means that the flip-flop processes caused by $\\hat{S}_L^+\\hat{I}_{L,k}^-$ and $\\hat{S}_R^-\\hat{I}_{R,k}^+$ are most likely, which again result in a pumping of $\\delta K^z$ toward smaller values of $\\Delta $ .",
"At $\\Delta < 0$ a similar reasoning results in positive pumping of $\\delta K^z$ toward higher values of $\\Delta $ .",
"So, we see that also in the case of vanishing SOI a naive qualitative investigation of the spin dynamics predicts a transport-induced self-quenching of the Zeeman gradient.",
"In the next two sections we will present analytic and numerical investigations that support the simple picture presented above."
],
[
"Analytic results",
"We start by deriving evolution equations for the nuclear polarizations in the two dots, similar to those derived in Ref.",
"[43] but now including the effect of the strong couplings $\\Gamma _{\\rm in,out}$ in a more general way and not solely focusing on the case of strong SOI.",
"From the flip-flop rates we thus find, we derive an expression for the fluctuations around the stable point at $\\Delta =0$ using a Fokker-Planck equation to describe the stochastic dynamics of the nuclear fields $K^z_{L,R}$ .",
"We start from a time-evolution equation for the electronic density matrix (we use $\\hbar =1$ ), $\\frac{d\\hat{\\rho }}{dt}= -i\\big [\\hat{H},\\hat{\\rho }\\big ] + \\Gamma \\hat{\\rho },$ where $\\hat{H} = \\hat{H}_0 + \\hat{H}_{so} + \\delta K^z \\big [ |{T_0}\\rangle \\langle {S}| + |{S}\\rangle \\langle {T_0}| \\big ]$ .",
"We neglect all other components of ${\\bf K}_{L,R}$ since they lead to small corrections that are of the order $K/E_{\\rm Z}$ , where $K$ is the typical magnitude of the nuclear fields.",
"The term $\\Gamma \\hat{\\rho }= - \\frac{1}{2}\\Gamma \\lbrace \\hat{P}_{02}, \\hat{\\rho }\\rbrace + \\frac{1}{4} \\Gamma (\\mathbb {1} - \\hat{P}_{02} )\\rho _{02,02}$ describes the transitions $|{S_{02}}\\rangle \\rightarrow (0,1) \\rightarrow (1,1)$ , using the projector onto the (0,2) singlet state $\\hat{P}_{02} = |{S_{02}}\\rangle \\langle {S_{02}}|$ .",
"Assuming that the rate $\\Gamma $ is the largest energy scale in (REF ), we can separate the time scales of the part of $\\hat{\\rho }$ involving $|{S_{02}}\\rangle $ and the part describing the dynamics in the (1,1) subspace.",
"This yields an effective Hamiltonian for that subspace $\\hat{H}^{(1,1)} = \\left( \\begin{array}{cccc}E_{\\rm Z} & 0 & 0 & 0 \\\\0 & E_B & \\Delta & 0 \\\\0 & \\Delta & 0 & 0\\\\0 & 0 & 0 & -E_{\\rm Z}\\end{array} \\right),$ written in the basis $\\lbrace |{T_-}\\rangle , |{B}\\rangle , |{D}\\rangle , |{T_+}\\rangle \\rbrace $ , where we assumed $g<0$ and $B>0$ .",
"The projection onto the (1,1) subspace resulted in exchange terms of the form $(\\hat{H}_{ex})_{ij} = 4\\epsilon T_{ij} /(4\\epsilon ^2+\\Gamma ^2)$ , with $T_{ij} = \\langle {i}|(\\hat{H}_t+\\hat{H}_{so})|{S_{02}}\\rangle \\langle {S_{02}}|(\\hat{H}_t+\\hat{H}_{so})|{j}\\rangle ,$ and thus $E_B=4\\epsilon (t_s^2+t_{z}^2)/(4\\epsilon ^2+\\Gamma ^2)$ .",
"Assuming that $E_{\\rm Z}$ is much larger than all exchange corrections, we neglected the terms coupling $|{T_\\pm }\\rangle $ to $|{B,D}\\rangle $ .",
"The four (1,1) states also acquire a finite life time that can be characterized by the four decay rates $\\Gamma _i = 4\\Gamma T_{ii} /(4\\epsilon ^2+\\Gamma ^2)$ , where we note that $\\Gamma _+ = \\Gamma _- \\equiv \\Gamma _t$ .",
"Using (REF ) and the decay rates $\\Gamma _i$ , we can write a time-evolution equation for $\\hat{\\rho }^{(1,1)}$ similar to (REF ).",
"Solving $d\\hat{\\rho }^{(1,1)}/dt = 0$ we find the equilibrium density matrix, which can be written $\\hat{\\rho }^{(1,1)}_{\\rm eq} = \\sum _i p_i |{i}\\rangle \\langle {i}|$ in the basis $\\lbrace |{T_+}\\rangle ,|{1}\\rangle ,|{2}\\rangle ,|{T_-}\\rangle \\rbrace $ , where $|{1}\\rangle = {} & {} \\cos \\frac{\\theta }{2} |{D}\\rangle + e^{i\\varphi } \\sin \\frac{\\theta }{2} |{B}\\rangle ,\\\\|{2}\\rangle = {} & {} \\cos \\frac{\\theta }{2} |{B}\\rangle - e^{-i\\varphi }\\sin \\frac{\\theta }{2} |{D}\\rangle ,$ in terms of the angles $\\varphi = {\\rm arg}( -i\\Gamma _B\\Delta - 2E_{B} \\Delta )$ and $\\theta = {\\rm arctan}\\big (4|\\Delta |/\\sqrt{\\Gamma _B^2 + 4E_{B}^2}\\big )$ .",
"The occupation probabilities $p_i$ of the four states read $p_\\pm = {} & {} \\frac{4\\Gamma _B \\Delta ^2}{\\Gamma _t E_2^2 + 8 \\Gamma _B\\Delta ^2 },\\\\p_1 = {} & {} \\frac{1}{2} - \\frac{4\\Gamma _B\\Delta ^2 - \\frac{1}{2}\\Gamma _t \\sqrt{(4E_{B}^2+\\Gamma _B^2)E_2^2}}{\\Gamma _t E_2^2 + 8 \\Gamma _B\\Delta ^2 },\\\\p_2 = {} & {} \\frac{1}{2} - \\frac{4\\Gamma _B\\Delta ^2 + \\frac{1}{2}\\Gamma _t \\sqrt{(4E_{B}^2+ \\Gamma _B^2 )E_2^2}}{\\Gamma _t E_2^2 + 8 \\Gamma _B\\Delta ^2 },$ with $E_2 = \\sqrt{4E_B^2 + \\Gamma _B^2 + 16 \\Delta ^2}$ .",
"In contrast to Ref.",
"[43], we included the effect $\\Gamma _\\text{out}$ here, resulting in a different basis of unpolarized states $|{1,2}\\rangle $ .",
"We now add the flip-flop terms in (REF ) in a perturbative way where we use Fermi's golden rule to calculate the rates for the resulting nuclear spin flips.",
"Assuming for simplicity nuclear spin $\\frac{1}{2}$ Using a different nuclear spin, such as $\\frac{3}{2}$ (as it is for both Ga and As), yields unimportant overall numerical prefactors of order 1., we write for the flip rates up and down on dot $d$ $\\gamma ^\\pm _d = \\frac{A^2}{4 N^2} N_d^\\mp \\sum _{i,j} p_i\\frac{\\Gamma _j}{E^2_{\\rm Z}}| \\langle {j}| \\hat{S}^\\mp _d |{i}\\rangle |^2 + \\gamma N_d^\\mp ,$ where $N_d^\\pm $ is the number of nuclei with spin $\\pm \\frac{1}{2}$ on the dot.",
"The factor $\\Gamma _j / E_{\\rm Z}^2$ accounts for the finite life time of the final electronic state $|{j}\\rangle $ , assuming a Lorentzian level broadening in the limit $E_{\\rm Z} \\gg \\Gamma _j$ .",
"We also added a term that describes random nuclear spin flips with a rate $\\gamma $ to account phenomenologically for the slow relaxation of the nuclear spins to their fully-mixed equilibrium state.",
"We can translate these flip rates to evolution equations for the dot polarizations $P_d = (N^+_d - N^-_d) / N$ .",
"For the polarization gradient $P_\\Delta = \\frac{1}{2}\\left(P_L-P_R\\right)$ and the average polarization $P_\\Sigma = \\frac{1}{2}\\left(P_L+P_R\\right)$ we find $\\frac{d P_\\Delta }{dt} \\!",
"= &- \\!\\bigg [ F(\\Delta ) +\\frac{1}{\\tau } \\bigg ]P_\\Delta \\!",
"- \\!",
"\\frac{2F(\\Delta )E_B\\Delta }{E_B^2 + \\frac{1}{4}\\Gamma _B^2 + 4 \\Delta ^2},\\\\\\frac{dP_\\Sigma }{dt} \\!= & - \\!",
"\\bigg [ F(\\Delta ) +\\frac{1}{\\tau } \\bigg ]P_\\Sigma ,$ with $F(\\Delta ) = \\frac{A^2}{4N^2E_{\\rm Z}^2}\\frac{\\Gamma _t^2(4E_B^2 + \\Gamma _B^2 + 16 \\Delta ^2) +4\\Gamma _B^2\\Delta ^2}{\\Gamma _t (4E_B^2 + \\Gamma _B^2 + 16 \\Delta ^2) + 8\\Gamma _B\\Delta ^2},$ and $1/\\tau = 2\\gamma /N$ the phenomenological relaxation rate of the polarizations, usually $\\tau \\sim $ 1–10 s. We note that these equations are non-linear, since $\\Delta = \\Delta _{so} + \\delta K^z = \\Delta _{so} + (A/2)P_\\Delta $ .",
"Figure: Pumping curves for the polarization gradient and average polarization, as given by Eqs.",
"() and ().",
"(a) dP Δ /dtdP_\\Delta /dt as a function of P Δ P_\\Delta .",
"(b) dP Σ /dtdP_\\Sigma /dt as a function of P Σ P_\\Sigma .In both plots we show three curves: without SOI (green), with intermediate SOI (red), and with strong SOI (blue); see the main text for the parameters used.As reference, we also added the result without any spin pumping, i.e., with F(Δ)=0F(\\Delta )=0 (orange dashed line).From Eqs.",
"(REF ) and () we see that both polarizations acquire effectively an enhanced relaxation rate, $\\tau ^{-1} \\rightarrow \\tau ^{-1} + F(\\Delta )$ , which does depend on $P_\\Delta $ but always drives the polarizations toward zero.",
"Furthermore, (REF ) has an extra term that pumps the polarization gradient to the point where the total Zeeman gradient $\\Delta $ is zero.",
"For typical parameters, where $E_B \\sim \\Gamma _B \\ll A$ , this term dominates and the result is a stable polarization close to $\\Delta = 0$ .",
"In the limit of vanishing SOI, we can set $\\Gamma _t \\rightarrow 0$ and then find $F(\\Delta ) = A^2 \\Gamma _B / 8 N^2 E_{\\rm Z}^2$ .",
"These results are illustrated in Fig.",
"REF , where we plot (a) $dP_\\Delta /dt$ as a function of $P_\\Delta $ and (b) $dP_\\Sigma /dt$ as a function of $P_\\Sigma $ for three different strengths of SOI (green, red, and blue lines) as well as without any spin pumping (orange dashed line).",
"We used $A = 250~\\mu $ eV, $E_{\\rm Z} = 5~\\mu $ eV, $N = 4 \\times 10^5$ , and $\\tau = 5$ s. For the curve without SOI (green) we used $E_B = 0.5~\\mu $ eV, $\\Gamma _B = 0.25~\\mu $ eV, and $\\Gamma _t = \\Delta _{so} = 0$ .",
"The other two curves have $\\Gamma _t = 0.01~\\mu $ eV, $\\Delta _{so} = 0.5~\\mu $ eV (red) and $\\Gamma _t = 0.0625~\\mu $ eV, $\\Delta _{so} = 1~\\mu $ eV (blue).",
"In these two cases, we adjusted $\\Gamma _B$ and $E_B$ such that the total coupling $\\sqrt{t_s^2 + |{\\bf t}|^2 }$ remains constant; this amounts to assuming that the SOI “converts” part of the tunnel coupling to a non-spin-conserving coupling but it does not affect the total coupling energy.",
"In the next section, we will show that these analytic results also agree well with numerical simulations of the dynamics of the polarizations, see Fig.",
"REF a.",
"Finally, we investigate the stochastic fluctuations of the polarization gradient around the stable point using a Fokker-Planck equation to describe the (time-dependent) probability distribution function ${\\cal P}(n,t)$ , where the integer $n = N P_\\Delta $ labels the allowed polarization gradients [46], [47].",
"Going to the continuum limit, we can find the equilibrium distribution function to be ${\\cal P}(P_\\Delta ) = \\exp \\left\\lbrace \\int ^{P_\\Delta } dP_\\Delta ^{\\prime }\\, 2N \\frac{\\gamma _\\Delta ^+ - \\gamma _\\Delta ^-}{\\gamma _\\Delta ^+ + \\gamma _\\Delta ^-} \\right\\rbrace ,$ where $\\gamma ^\\pm _\\Delta = \\frac{1}{2}(\\gamma _L^\\pm - \\gamma _R^\\pm )$ in terms of the flip rates as written in (REF ).",
"The slope of the integrand close to the points where $\\gamma _\\Delta ^+ - \\gamma _\\Delta ^-=0$ can thus be used to estimate the equilibrium r.m.s.",
"deviation of $P_\\Delta $ from those stable points.",
"In the absence of pumping, i.e., for $F(\\Delta )\\rightarrow 0$ , we find a peak in the distribution around the point $P_\\Delta = 0$ with a variance $\\sigma _0^2 = 1/2N$ .",
"Including pumping, and assuming that the second term in (REF ) dominates around the stable point, we find a peak in ${\\cal P}(P_\\Delta )$ at $P_\\Delta \\approx -2 \\Delta _{so} / A$ , where $\\sigma ^2 \\approx \\sigma ^2_0\\frac{E_B^2 + \\frac{1}{4}\\Gamma _B^2}{A E_B}\\left(1 + 8 \\frac{E_{\\rm Z}^2N^2}{A^2\\tau }\\frac{\\Gamma _B + 2\\Gamma _t}{\\Gamma _B^2 + 4\\Gamma _t^2}\\right).$ In Fig.",
"REF we show the resulting suppression of the fluctuations $\\sigma ^2/\\sigma _0^2$ as a function of detuning $\\epsilon $ and strength of the SOI, parameterized by $\\eta $ , where we fixed the total tunnel coupling to $t = 7.5~\\mu $ eV and then used $t_x^2 + t_y^2 = t^2 \\sin ^2 \\eta $ and $t_z^2 + t_s^2 = t^2 \\cos ^2 \\eta $ .",
"In this way, $\\eta = 0$ corresponds to having no SOI and $\\eta \\sim \\pi /4$ to strong SOI.",
"We further used $A = 250~\\mu $ eV, $E_{\\rm Z} = 12.5~\\mu $ eV, $\\Gamma = 75~\\mu $ eV, $N = 4 \\times 10^5$ , and $\\tau = 5$ s. For these parameters we observe a significant suppression of the fluctuations in the whole range we plotted.",
"We see that the suppression is most effective for strong SOI (where $\\eta \\rightarrow \\pi /4$ ), but still of the same order of magnitude in the absence of SOI (where $\\eta = 0$ ).",
"Figure: Suppression of the fluctuations of the nuclear field gradient, as given in (), as a function of ϵ\\epsilon and η\\eta , where η\\eta characterizes the strength of the SOI.See the main text for the parameters used and the exact definition of η\\eta ."
],
[
"Numerical simulations",
"We complement our analytic results with a numerical simulation of the electron-nuclear spin dynamics, discretizing time in small steps of $\\Delta t$ .",
"We start with two initial polarizations $P_L(0)$ and $P_R(0)$ on the two dots and then solve for the eigenvalues $\\varepsilon _i$ and eigenmodes $\\hat{\\rho }_i$ of the superoperator $\\Lambda $ that describes the coherent evolution and decay of the density matrix, $\\Lambda \\hat{\\rho }= -i\\big [\\hat{H},\\hat{\\rho }\\big ] - \\frac{1}{2} \\big \\lbrace \\hat{\\Gamma }, \\hat{\\rho }\\big \\rbrace ,$ where $\\hat{\\Gamma }$ is a diagonal matrix containing the decay rates of the five basis states We added an infinitesimal decay rate of $10^{-9}~\\mu $ eV to all (1,1) states to avoid singularities.. Each of the 25 eigenmodes of $\\Lambda $ can then be written as $\\hat{\\rho }_i = |{n}\\rangle \\langle {m}|$ where $|{n}\\rangle $ and $|{m}\\rangle $ are picked from a (new) five-dimensional basis.",
"The corresponding eigenvalue $\\varepsilon _i$ has the form $\\varepsilon _i = -i(E_n - E_m) - \\frac{1}{2}(\\gamma _n + \\gamma _m)$ where $E_{n,m}$ and $\\gamma _{n,m}$ give the effective energies and decay rates of the two states $|{n}\\rangle $ and $|{m}\\rangle $ .",
"From knowing all $\\varepsilon _i$ and $\\hat{\\rho }_i$ we can thus derive the appropriate basis states, their effective energies, and their decay rates.",
"To find the steady-state occupation probabilities for these five states, we evaluate their weight in the (1,1) subspace, $w_n = \\langle {n}| (\\mathbb {1} - \\hat{P}_{02} ) |{n}\\rangle $ , from which the occupation probabilities follow as $p_n = w_n\\gamma _n^{-1} / \\sum _i w_i \\gamma _i^{-1}$ .",
"Now we have all ingredients we need to evaluate the spin flip rates on both dots.",
"We rewrite Eq.",
"(REF ) including the detailed dependence on all energy differences and decay rates, $\\gamma ^\\pm _d = {} & {} \\frac{A^2}{N^2}\\sum _{i,j} \\frac{p_i(\\gamma _i + \\gamma _j)| \\langle {j}| \\hat{S}^\\mp _d |{i}\\rangle |^2}{4(E_i - E_j)^2 + (\\gamma _i + \\gamma _j)^2} N_d^\\mp + \\gamma N_d^\\mp .$ Then we pick random numbers of spin-flip events $k_d^\\pm $ on both dots and in both directions, using a Poisson distribution $(\\gamma _d^\\pm \\Delta t)^{k_d^\\pm } e^{\\gamma _d^\\pm \\Delta t} / (k_d^\\pm )!$ , and we update the polarizations $P_d(\\Delta t) = P_d(0) + (2/N) (k_d^+ - k_d^-)$ .",
"This process can then be repeated as many times as desired to simulate the evolution of $P_{L,R}(t)$ over longer times.",
"We note that we make sure that $\\Delta t$ is small enough so that most of the $k_d^\\pm $ turn out 0 or 1.",
"Figure: Simulation of the polarization gradient P Δ (t)P_\\Delta (t) without SOI (red solid lines), strong SOI (yellow solid lines), and without hyperfine-induced spin pumping (A=0A = 0, green solid lines).The dashed lines show the solution of Eq.",
"() using the same parameters.",
"(a) Short-time evolution.",
"Note that we used different initial conditions for clarity: P Δ (0)=0.0025P_\\Delta (0) = 0.0025 for the red and green lines and P Δ (0)=-0.0025P_\\Delta (0) = -0.0025 for the yellow line; we always set P Σ (0)=0P_\\Sigma (0)=0.",
"(b) Long-time evolution for A=0A=0.",
"The horizontal black lines indicate ±σ 0 \\pm \\sigma _0.",
"(c) Long-time evolution in the presence of spin pumping.",
"The horizontal black lines now show ±σ\\pm \\sigma as found from Eq.",
"() (see inset).See the main text for all other parameters used.We show the results of our simulations as solid lines in Fig.",
"REF , where we plot $P_\\Delta (t)$ for three different cases: (i) strong SOI, where $t_{x,y,z} = 3.12~\\mu $ eV and $t = 5.21~\\mu $ eV (yellow), (ii) no SOI, with $t_{x,y,z} = 0$ and $t = 7.5~\\mu $ eV (red), and (iii) no hyperfine interaction (green).",
"The other parameters used were $A=125~\\mu $ eV, $E_{\\rm Z}=12.5~\\mu $ eV, $\\delta =100~\\mu $ eV, $\\Gamma =75~\\mu $ eV, $N=4\\times 10^5$ , $\\tau =5$ s, and $\\Delta t = 10~\\mu $ s. We used as initial conditions $P_\\Delta (0) = 0.0025$ (red and green), $P_\\Delta (0) = -0.0025$ (yellow), and $P_\\Sigma (0) = 0$ (always).",
"We note that, in order to make comparison more straightforward, we set $\\Delta _{so} = 0$ in all cases, including the case of strong SOI.",
"In Fig.",
"REF (a) we show the first 0.1 s of the evolution.",
"We see that the hyperfine interaction accelerates the dynamics of the polarizations and tends to suppress the gradient to zero.",
"We added dashed lines that show time-dependent solutions of Eq.",
"(REF ), which indeed seems to predict the average dynamics of the polarization gradient to reasonable accuracy.",
"In Figs.",
"REF (b,c) we show longer time traces to illustrate the magnitude of the fluctuations around the stable point $P_\\Delta = 0$ .",
"In Fig.",
"REF (b) the fluctuations are clearly much larger than in REF (c), which is what we expected.",
"The horizontal lines show the magnitude of the fluctuations as predicted by Eq.",
"(REF ): For the parameters used we find $\\sigma _0 = 1.1 \\times 10^{-3}$ (to be compared with the green trace), and $\\sigma = 7.8\\times 10^{-5}$ (red trace) and $\\sigma = 7.5\\times 10^{-5}$ (yellow trace).",
"In both simulations that include spin pumping (red and yellow lines) the average polarization $P_\\Sigma $ tends to drift to negative values, stabilizing at $\\sim -0.02$ .",
"This can be understood in qualitative terms from Fig.",
"REF (b,c): With strong SOI [Fig.",
"REF (b)] the state $|{T_+}\\rangle $ decays more efficiently than $|{T_-}\\rangle $ since it is closer in energy to $|{S_{02}}\\rangle $ and $\\Gamma $ is finite.",
"This makes in general spin flips from $|{D}\\rangle $ slightly more likely to happen to $|{T_+}\\rangle $ , resulting in a net average transfer of negative angular momentum to the nuclear spins.",
"Without SOI [Fig.",
"REF (c)], the bright state $|{B}\\rangle $ is closer in energy to $|{T_-}\\rangle $ than to $|{T_+}\\rangle $ (assuming $\\delta >0$ ), resulting in the flip rate $|{T_-}\\rangle \\rightarrow |{B}\\rangle $ to be larger than $|{T_+}\\rangle \\rightarrow |{B}\\rangle $ .",
"This should indeed also result in a small net negative pumping of the average polarization.",
"These effects are not reflected in Eq.",
"() since in that Section we neglected all energy differences in the (1,1) subspace compared to $E_{\\rm Z}$ , which, in turn, was assumed negligible compared to $\\Gamma $ ."
],
[
"Conclusion",
"We found that embedding a double quantum dot in the spin-blockade regime in a transport setup, the flow of electrons induces dynamic nuclear spin polarization that tends to suppress the polarization gradient over the two dots.",
"This mechanism not only works in the case of strong SOI, but also with weak SOI or in the absence of SOI.",
"We derived simple analytic equations to describe the dynamics of the polarization gradient (which we corroborated with numerical simulations), and we found that, over a large range of parameters, the r.m.s.",
"value of the random polarization gradient can be suppressed by one to two orders of magnitude.",
"This could present a straightforward way to extend the coherence time of double-dot-based spin qubits."
],
[
"The qubit",
"Exchange-only qubits are usually hosted in a linear triple quantum dot, with one electron in each dot.",
"The eight-dimensional (1,1,1) subspace consists of one spin quadruplet $|{Q}\\rangle $ and two doublets $|{D_1}\\rangle $ and $|{D_2}\\rangle $ .",
"An external magnetic field lifts the degeneracy of states with different total $S_z$ , and when the system is then tuned close to the border of the (1,1,1) region, exchange effects due to finite interdot tunneling can lift the remaining degeneracies.",
"The qubit is then commonly defined in the two doublet states with spin projection $S_z = +\\frac{1}{2}$ , and turns out to be fully controllable via electric fields only.",
"Figure: (a) Sketch of the charge stability diagram of a linear triple quantum dot tuned close to the (1,1,1)–(1,0,2)–(2,0,1) triple point.",
"(b) Cartoon of the setup.",
"(c,d) Lowest part of the spectrum along the horizontal and vertical dashed line in (a), respectively.In Fig.",
"REF (a) we sketch the charge stability diagram close to the (1,1,1)–(1,0,2)–(2,0,1) triple point, as a function of the two tuning parameters $\\epsilon =\\frac{1}{2}(V_R-V_L)$ and $\\epsilon _M= V_C-\\frac{1}{2}(V_L+V_R)$ , where $V_{L,C,R}$ denote the gate-induced potentials on the left, central, and right dot, respectively.",
"We include energy offsets such that the triple point is defined to be at $(\\epsilon _M,\\epsilon ) = (0,0)$ .",
"In this regime, the low-energy part of the spectrum consists of 12 states: In addition to the eight (1,1,1) states mentioned above, we also need to include a doublet $|{D_L}\\rangle $ in a (2,0,1) configuration and a doublet $|{D_R}\\rangle $ in a (1,0,2) configuration.",
"We can then write a similar Hamiltonian as before, $\\hat{H}_0 = \\hat{H}_e + \\hat{H}_t + \\hat{H}_{\\rm Z}.$ Now we have $\\hat{H}_e = \\sum _{\\alpha = \\pm } \\Big \\lbrace {} & {} -(\\epsilon _M + \\epsilon ) |{D_L^\\alpha }\\rangle \\langle {D_L^\\alpha }|\\nonumber \\\\ {} & {}\\ -(\\epsilon _M - \\epsilon ) |{D_R^\\alpha }\\rangle \\langle {D_R^\\alpha }| \\Big \\rbrace ,$ where $\\alpha = \\pm $ labels the spin projection $S_z = \\pm \\frac{1}{2}$ of the doublet state.",
"The tunneling Hamiltonian is $\\hat{H}_t = \\frac{t}{2} \\sum _{\\alpha = \\pm } \\alpha \\Big \\lbrace {} & {} \\sqrt{3} |{D_1^\\alpha }\\rangle \\big [ \\langle {D_R^\\alpha }| - \\langle {D_L^\\alpha }| \\big ] \\nonumber \\\\{} & {}+|{D_2^\\alpha }\\rangle \\big [ \\langle {D_R^\\alpha }| + \\langle {D_L^\\alpha }| \\big ] \\Big \\rbrace + \\text{H.c.}$ where we assumed the left and right tunneling couplings equal, for simplicity.",
"The Zeeman term is $\\hat{H}_{\\rm Z} = g\\mu _{\\rm B} B \\hat{S}_z^{\\rm tot},$ in terms of the total spin-$z$ projection operator for the three electrons.",
"In the region marked `RX' in Fig.",
"REF (a) the central electron can become delocalized over the three dots [see Fig.",
"REF (b)], yielding relatively strong exchange effects.",
"To illustrate, we sketch in Fig.",
"REF (c) the spectrum of $\\hat{H}_0$ along the dotted line in (a), where we set $t = 3\\, E_{\\rm Z}$ .",
"The two dashed lines (the lowest doublet states with $S_z^{\\rm tot} = +\\frac{1}{2}$ ) form the qubit subspace, where $|{1}\\rangle =|{D_{2}^+}\\rangle $ and $|{0}\\rangle =|{D_{1}^+}\\rangle $ at $\\epsilon = 0$ .",
"Close to that point, the projected qubit Hamiltonian is $\\hat{H}_q = \\frac{J}{2}\\hat{\\sigma }_z - \\frac{\\sqrt{3} j}{2}\\hat{\\sigma }_x,$ with $J = \\frac{1}{2}(J_L+J_R)$ and $j=\\frac{1}{2}(J_L-J_R)$ , in terms of the exchange energies $J_{L,R}$ associated with virtual tunneling to the left or right dot, respectively.",
"To lowest order in $t$ [valid not too close to the borders of the (1,1,1) region] we have $J_{L,R} = -t^2/(\\epsilon _M\\pm \\epsilon )$ .",
"From this it is clear that the exchange-only qubit allows for electric control of rotations around two different axes of the Bloch sphere, by tuning $J$ and $j$ through $\\epsilon $ and $\\epsilon _M$ , whereas the singlet-triplet qubit offered electric control over only one axis.",
"As in the double-dot system, the main effect of the hyperfine interaction with the nuclear spin bath can be described on a mean-field level using three random effective nuclear fields, $\\hat{H}_\\text{hf,mf}= \\mathbf {K}_L\\cdot \\hat{\\mathbf {S}}_L + \\mathbf {K}_C\\cdot \\hat{\\mathbf {S}}_C + \\mathbf {K}_R\\cdot \\hat{\\mathbf {S}}_R.$ Projected onto the qubit subspace, this yields $\\hat{H}_{\\rm hf,q} = -\\frac{2}{3} \\delta K_M^z\\hat{\\sigma }_z - \\frac{1}{\\sqrt{3}} \\delta K_{LR}^z \\hat{\\sigma }_x,$ where $\\delta K^z_M=-\\frac{1}{2}(\\delta K^z_{LC} - \\delta K^z_{CR})$ and $\\delta K^z_{LR}=\\frac{1}{2}\\left(K_L^z-K_R^z\\right)$ , in terms of the field gradients $\\delta K_{ij}^z = \\frac{1}{2}(K^z_i - K^z_j)$ over neighboring dots.",
"We thus see that, also in this case, the random nuclear fields can be an important source of qubit decoherence.",
"Besides, the quadruplet state $|{Q_{+1/2}}\\rangle $ that cannot be split off by increasing the external field $B$ is coupled to the states $|{0}\\rangle $ and $|{1}\\rangle $ through the same gradients $\\delta K^z_M$ and $\\delta K^z_{LR}$ , which can thus cause leakage out of the qubit subspace.",
"To be able to control or suppress the field gradients could therefore again dramatically increase the qubit quality."
],
[
"Transport-induced nuclear spin pumping: Qualitative picture",
"Inspired by our findings for the double dot, we now investigate possibilities to suppress the nuclear field gradients by running a current through the system while tuning it to some sort of spin-blockade regime.",
"In contrast to the double dot setup, there are several different types of spin blockade in a linear triple dot [49], which differ in the geometry of drains and sources and relative detuning of the three dots.",
"In a simplest setup where source and drain are attached to the outer dots, all regimes of spin blockade effectively behave as a double dot connected to one isolated dot containing one “inert” spin.",
"Transport through such a setup would thus only suppress the field gradient between the two interacting dots.",
"To address both field gradients we use a setup where the source is connected to the central dot and both of the outer dots are connected to a drain, see Fig.",
"REF (a).",
"Applying a source–drain bias voltage in vicinity of the triple point shown in Fig.",
"REF (a) can then give rise to a current through the system via the two transport cycles $(1,1,1)\\rightarrow (2,0,1)/(1,0,2)\\rightarrow (1,0,1)\\rightarrow (1,1,1)$ .",
"Again we will assume that the system is in the open regime where the rates $\\Gamma _\\text{in,out}$ are the largest energy scales, such that the interesting dynamics happen during the $(1,1,1) \\rightarrow (2,0,1)/(1,0,2)$ transitions, which involves the 12 spin states discussed above.",
"For simplicity, we will assume a symmetric situation, where $\\epsilon =0$ and $\\epsilon _M > 0$ [see Fig.",
"REF (d)], $t_l = t_r$ , and $\\Gamma _{{\\rm out},l} = \\Gamma _{{\\rm out},r}$ .",
"Figure: (a) The central dot is connected to a source and the two outer dots are connected to drains; an applied bias voltage then enables electrons to flow through the system to either of the drains.",
"(b,c) Spectrum of the (1,1,1) states in the absence of SOI, as a function of the gradients Δ LR \\Delta _{LR} (b) and Δ M \\Delta _M (c), where the thickness of the lines indicates the occupation probabilities.",
"Preferred spin-flip rates are indicated by gray arrows.In absence of spin-mixing processes, the only (1,1,1) states that couple to $|{D_L}\\rangle $ and $|{D_R}\\rangle $ are the doublets $|{D_{1,2}}\\rangle $ , and the current is spin blocked in either of the four quadruplet states.",
"This blockade may be lifted by SOI, which affects the system in the same way as before: (i) variations in the effective $g$ -factor over the dots yield spin-orbit-induced Zeeman gradients $\\Delta _{so,ij} = \\frac{1}{2}(g_i-g_j)\\mu _{\\rm B}B$ and (ii) tunneling between dots can be accompanied by a spin flip.",
"It is easy to show that, in contrast to the double-dot case, in the presence of SOI there are no dark states, even when all total Zeeman gradients $\\Delta _{ij} = \\Delta _{so,ij}+\\delta K^z_{ij}$ are zero.",
"SOI thus always fully lifts the spin blockade and competes with the flip-flop terms in the hyperfine interaction, thereby reducing the efficiency of spin pumping.",
"We will below only focus on the case without SOI, which is experimentally also most relevant since with strong SOI there is no spin blockade that can be used for initialization or read-out.",
"Let us now develop an intuitive picture of the electron-nuclear spin dynamics in this spin-blockade situation, similar to the discussion in Sec.",
"REF .",
"When the gradients $\\Delta _{LR}$ and $\\Delta _M$ are zero, the electrons are trapped in one of the four quadruplet states with equal probability $\\frac{1}{4}$ .",
"As before, due to the symmetric spin structure of all states at this point there will be no net spin pumping.",
"A non-zero gradient mixes states with the same total $S_z^{\\rm tot}$ , giving all six states with $S_z^{\\rm tot} = \\pm \\frac{1}{2}$ a finite coupling to $|{D^\\pm _{L,R}}\\rangle $ , whereas the two fully polarized quadruplets remain spin blocked, each with occupation probability $\\frac{1}{2}$ .",
"For small gradients, the doublets have a much larger coupling to $|{D^\\pm _{L,R}}\\rangle $ than $|{Q_{\\pm 1/2}}\\rangle $ and spin-flip processes are thus dominated by transitions from $|{Q_{\\pm 3/2}}\\rangle $ to a doublet state.",
"We first show that transitions to $|{D^\\pm _2}\\rangle $ do not contribute strongly to spin pumping.",
"When $\\Delta _{LR}\\ne 0$ , the states $|{D_{2}^\\pm }\\rangle $ develop a dominating $\\uparrow \\downarrow \\uparrow $ - and $\\downarrow \\uparrow \\downarrow $ -character, respectively, see Fig.",
"REF (b).",
"This results in an increased spin-flip rate $\\gamma _C^+$ from transitions $|{Q_{+3/2}}\\rangle \\rightarrow |{D_2^+}\\rangle $ as well as an increased rate $\\gamma _C^-$ from $|{Q_{-3/2}}\\rangle \\rightarrow |{D_2^-}\\rangle $ .",
"One thus does not expect a strong net effect.",
"For $\\Delta _M\\ne 0$ the situation is similar: $|{D_2^+}\\rangle $ ($|{D_2^-}\\rangle $ ) gains a larger weight of $\\uparrow \\uparrow \\downarrow $ and $\\downarrow \\uparrow \\uparrow $ ($\\downarrow \\downarrow \\uparrow $ and $\\uparrow \\downarrow \\downarrow $ ).",
"The spin-flip rates from $|{Q_{+3/2}}\\rangle \\rightarrow |{D_2^+}\\rangle $ and $|{Q_{-3/2}}\\rangle \\rightarrow |{D_2^-}\\rangle $ are thus affected in a symmetric way and there is no net spin pumping.",
"The doublet states $|{D_1^\\pm }\\rangle $ , however, have the largest coupling to the outgoing states $|{D^\\pm _{L,R}}\\rangle $ , and effectively pump the field gradients toward zero.",
"For a positive gradient $\\Delta _{LR}>0$ , the state $|{D_{1}^+}\\rangle $ ($|{D_{1}^-}\\rangle $ ) evolves into a state with slight $\\uparrow \\uparrow \\downarrow $ ($\\uparrow \\downarrow \\downarrow $ )-character, see Fig.",
"REF (b).",
"This increases $\\gamma _R^+$ ($\\gamma _L^-$ ) and thus drives $\\Delta _{LR}$ toward lower values.",
"For a negative gradient $\\Delta _{LR} < 0$ , the situation is exactly opposite, again driving $\\Delta _{LR}$ to zero.",
"A similar argument holds for the other gradient $\\Delta _M$ : When $\\Delta _M>0$ , the state $|{D_1^-}\\rangle $ gets a slight $\\downarrow \\uparrow \\downarrow $ -character and $|{D_1^+}\\rangle $ obtains stronger $\\downarrow \\downarrow \\uparrow $ - and $\\uparrow \\downarrow \\downarrow $ -components, see Fig.",
"REF (c).",
"This increases the rates $\\gamma _L^+$ , $\\gamma _C^-$ , and $\\gamma _R^+$ , thereby effectively reducing $\\Delta _M$ .",
"For $\\Delta _M<0$ the situation is again opposite, yielding a positive pumping of $\\Delta _M$ ."
],
[
"Analytic results",
"We now use the same approach as in Sec.",
"REF to derive time-evolution equations for the three nuclear polarizations, valid for small $P_d$ .",
"The time-evolution equation for the electronic density matrix in the triple dot reads, $\\frac{d\\hat{\\rho }}{dt}= -i\\big [\\hat{H},\\hat{\\rho }\\big ] + \\mathbf {\\Gamma }\\hat{\\rho },$ with $\\hat{H}=\\hat{H}_0+\\hat{H}_{\\rm hf,mf}$ .",
"We describe the transitions $(2,0,1)/(1,0,2)\\rightarrow (1,0,1) \\rightarrow (1,1,1)$ with the term $\\mathbf {\\Gamma }\\hat{\\rho }=-\\frac{1}{2}\\Gamma \\lbrace \\hat{P}_{\\rm dec},\\hat{\\rho }\\rbrace +\\frac{1}{8}\\Gamma (\\mathbb {1} - \\hat{P}_{\\rm dec})\\rho _{\\rm dec}$ , where the operator $P_{\\rm dec}=\\sum _{i = D_{L,R}^\\alpha } |{i}\\rangle \\langle {i}|$ projects to the subspace that is coupled to the drain leads and $\\rho _{\\rm dec}=\\sum _{i = D_{L,R}^\\alpha } \\rho _{i,i}$ .",
"Assuming that $\\Gamma $ is the largest energy scale involved, we again separate time scales and write the effective (1,1,1) Hamiltonian $\\hat{H}^{(1,1,1)} = \\sum _{\\alpha = \\pm } -\\alpha \\frac{3}{2}E_{\\rm Z} |{Q_{\\alpha 3/2}}\\rangle \\langle {Q_{\\alpha 3/2}}| + \\hat{H}^\\alpha _{\\frac{1}{2}},$ using the two $3\\times 3$ blocks $\\hat{H}_\\frac{1}{2}^\\alpha = -\\alpha \\frac{1}{2}E_Z + 3E_D|{D_1^\\alpha }\\rangle \\langle {D_1^\\alpha }| + E_D|{D_2^\\alpha }\\rangle \\langle {D_2^\\alpha }|\\\\+\\alpha \\left(\\begin{array}{ccc}0 &-\\frac{\\sqrt{2}}{3}\\Delta _M &\\sqrt{\\frac{}{}}{2}{3}\\Delta _{LR} \\\\-\\frac{\\sqrt{2}}{3}\\Delta _M &-\\frac{1}{3}\\Delta _M &-\\frac{1}{\\sqrt{3}}\\Delta _{LR} \\\\\\sqrt{\\frac{}{}}{2}{3}\\Delta _{LR} &-\\frac{1}{\\sqrt{3}}\\Delta _{LR} &\\frac{1}{3}\\Delta _M\\end{array}\\right),$ acting on the subspaces $\\lbrace |{Q_{\\alpha 1/2}}\\rangle ,|{D_{1}^\\alpha }\\rangle ,|{D_{2}^\\alpha }\\rangle \\rbrace $ .",
"Here $E_{\\rm Z}$ contains the contribution $\\frac{1}{3}(K^z_L+K^z_C+K_R^z)$ from the average nuclear spin polarization.",
"We assumed $E_{\\rm Z}$ to be large enough that we can neglect the transverse components $K_d^{x,y}$ that couple states with different $S^{\\rm tot}_z$ .",
"The projection to the (1,1,1) subspace introduced the exchange energy $E_D=\\frac{2t^2\\epsilon _M}{4\\epsilon _M^2+\\Gamma ^2},$ and makes the states $|{D_1^\\pm }\\rangle $ and $|{D_2^\\pm }\\rangle $ decay with rates $\\Gamma _1=3\\Gamma _D$ and $\\Gamma _2=\\Gamma _D$ , respectively, where $\\Gamma _D=\\frac{2t^2\\Gamma }{4\\epsilon _M^2+\\Gamma ^2}.$ Assuming that the exchange energy $E_D$ is much larger than the gradients $\\Delta _{LR}$ and $\\Delta _M$ , we diagonalize $\\hat{H}^\\pm _\\frac{1}{2}$ using perturbation theory and thusly find expressions for the eigenstates and their decay rates valid to lowest order in the gradients Here we keep only the effect of $E_D$ on the structure of the basis, i.e., we disregard $\\Gamma _D$ .. For non-zero gradients, the occupation probabilities are approximately $\\frac{1}{2}$ for $|{Q_{\\pm 3/2}}\\rangle $ and zero for the remaining six states.",
"Like for the double dot, we then calculate the hyperfine-induced flip-flop rates perturbatively using Fermi's golden rule (REF ), and translate the resulting flip rates to evolution equations for the average polarization $P_\\Sigma = \\frac{1}{3}\\left(P_L+P_C+P_R\\right)$ , and the two polarization gradients $P_{LR} = \\frac{1}{2}\\left(P_L-P_R\\right)$ and $P_M=P_C-\\frac{1}{2}\\left(P_L+P_R\\right)$ .",
"This gives, to lowest order in the field gradients $\\Delta _{LR}$ and $\\Delta _M$ $\\frac{dP_{LR}}{dt} = {} & {} \\!-\\!\\left[G+\\frac{1}{\\tau }\\right]P_{LR} - \\frac{G}{E_D}\\Delta _{LR},\\\\\\frac{dP_M}{dt} = {} & {}\\!-\\!\\left[\\frac{5}{3}G+\\frac{1}{\\tau }\\right]P_{M}-GP_\\Sigma - \\frac{2G}{3E_D}\\Delta _M,\\\\\\frac{dP_\\Sigma }{dt} = {} & {} \\!-\\!\\left[\\frac{4}{3}G+\\frac{1}{\\tau }\\right]P_\\Sigma - \\frac{2}{9}GP_{M},$ with $G=A^2\\Gamma _D/4N^2E_{\\rm Z}^2$ , where we again assumed equal $N$ on all dots, for simplicity.",
"As in the double dot, all polarization gradients thus acquire an effectively enhanced relaxation rate.",
"We further find that the polarization dynamics of $P_{M}$ and $P_\\Sigma $ are coupled, which is a result of the geometry of the source and drains.",
"However, for typical parameters the last terms in Eqs.",
"(REF ) and () dominate, predicting an efficient suppression of both gradients, similar to the double-dot case.",
"Using these results, we can again investigate the stochastic fluctuations around stable points, using a linear Fokker-Planck equation that describes the time-dependent probability distribution $\\mathcal {P}(n,m,l,t)$ , where $n=\\frac{3}{2}N P_\\Sigma $ , $m=NP_{LR}$ and $l=\\frac{2}{3}NP_{M}$ .",
"In the continuous limit, and to lowest order in the gradients, we find a covariance matrix that reads $\\sigma _{LR}^2 = {} & {} \\frac{1}{2N}\\frac{2E_D}{2E_D+A},\\\\\\sigma _{M}^2 = {} & {} \\frac{3}{2N}\\frac{E_D(81E_D+10A)}{81E_D^2+27AE_D+2A^2},\\\\\\sigma _\\Sigma ^2 = {} & {} \\frac{1}{3N}\\left[1 - \\frac{AE_D}{81E_D^2+27AE_D+2A^2}\\right],\\\\\\sigma _{LR,M}^2 = {} & {} \\frac{2}{N}\\frac{AE_D}{81E_D^2+27AE_D+2A^2}.$ Realistically $A \\gg E_D$ , so the r.m.s.",
"of the fluctuations of the two gradients are suppressed by a factor $\\sim \\sqrt{E_D/A}$ , whereas the fluctuations of $P_\\Sigma $ are barely affected, similar to what we found for the double dot."
],
[
"Numerical simulations",
"Using the same method as in Sec.",
"REF we performed numerical simulations to corroborate our analytic results.",
"In Fig.",
"REF (a,b) we first illustrate the coupled dynamics of $P_{LR}$ and $P_M$ .",
"We set $P_\\Sigma =0$ , $A=125~\\mu $ eV, $E_Z=12.5~\\mu $ eV, $N=4\\times 10^5$ , $\\tau =5$ s, $\\epsilon _M=100~\\mu $ eV, $\\epsilon = 0$ , $\\Gamma =75~\\mu $ eV and $t=7.5~\\mu $ eV, and then we plot in color the rates of change $dP_{LR}/dt$ (a) and $dP_M/dt$ (b) as a function of $P_{LR}$ and $P_M$ as found using Eq.",
"(REF ).",
"In both plots we also included the (same) vector field $(dP_M/dt, dP_{LR}/dt)$ , represented by the black arrows, illustrating how both field gradients are indeed pumped toward zero.",
"The insets show line cuts along the red dotted lines, i.e., they show the rate of change of each polarization gradient as a function of the same gradient, where the other one is set to zero.",
"The dashed orange lines indicate the slope of the pumping curve at the stable point, as predicted by Eqs.",
"(REF )–(), showing indeed good agreement with the numerical results.",
"In Fig.",
"REF (c,d) we then show simulations of the stochastic dynamics of the two polarization gradients, performed in the same way as we did in Sec.",
"REF for the double dot.",
"We started with initial polarizations $P_{LR}(0) = 0.001$ , $P_M(0) = 0.002$ , and $P_\\Sigma = 0$ and performed a simulation with the parameters given above (red lines) and one without spin pumping ($A=0$ , green lines).",
"Panels (i) show the short-time dynamics, where the dashed lines correspond to the result predicted by Eqs.",
"(REF )–(), and panels (ii) and (iii) show the long-time dynamics, where the horizontal solid lines indicate the r.m.s.",
"value of the fluctuations as predicted from Eqs.",
"(REF )–().",
"We see that in all cases our analytic expressions agree reasonably well with the simulated dynamics of the gradients.",
"We further note that, for similar reasons as in the double dot, the average polarization $P_\\Sigma $ drifts toward negative values, stabilizing around $\\sim -0.004$ .",
"Due to the way the dynamics of $P_M$ depend on $P_\\Sigma $ [see Eq.",
"()] one expects that the long-time stable polarization of $P_M$ is not at zero but at a small positive value; a careful look at Fig.",
"REF (d,iii) shows that this is indeed the case in our simulations."
],
[
"Conclusion",
"We found that electron transport through a linear triple quantum dot—with a source connected to the central dot and drains connected to the outer dots—tuned to the regime of Pauli spin blockade can yield a hyperfine-induced feedback cycle that dynamically suppresses the two nuclear polarization gradients in the triple dot.",
"To find the approximate magnitude of the r.m.s.",
"value of the remaining nuclear-field fluctuations, we derived simple perturbative analytical expressions to describe the coupled dynamics of the polarization gradients.",
"This predicts a similar suppression of the fluctuations of the gradients as in the double-dot case, i.e., a suppression of one to two orders of magnitude.",
"We corroborated these analytic results with numerical simulations of the coupled electron-nuclear spin dynamics, finding good agreement between the two."
],
[
"Conclusion",
"In multielectron qubits, such as the double-dot-based two-electron singlet-triplet qubit and triple-dot-based three-electron exchange-only qubits, the main source of decoherence are usually the fluctuating nuclear-spin polarization gradients over neighboring dots.",
"These random gradients couple to the spins of the electrons in the dots and can thereby add to the qubit splitting or couple the two qubit states to each other as well as to other nearby states outside of the computational basis.",
"In this paper, we investigated the effect of running a DC current through such systems on the nuclear polarization gradients, while tuning to a regime of Pauli spin blockade.",
"We found that transport through the dots can give rise to a dynamical feedback cycle between the electronic and nuclear spins that results in an active suppression of the nuclear polarization gradients.",
"We considered a double-dot setup with and without significant spin-orbit interaction as well as a triple-dot setup without spin-orbit interaction.",
"For all cases we derived approximate analytical evolution equations for the nuclear polarization gradients, which all predict the possibility of a significant suppression of the fluctuations of the gradients.",
"We corroborated these results with numerical simulations of the stochastic coupled electron-nuclear spin dynamics which confirmed a reduction in the random fluctuations of the nuclear polarization gradients by one to two orders of magnitude.",
"These suppression mechanisms could thus present a straightforward way to significantly reduce the hyperfine-induced decoherence in multielectron qubits.",
"This work is part of FRIPRO-project 274853, which is funded by the Research Council of Norway (RCN), and was also partly supported by the Centers of Excellence funding scheme of the RCN, project number 262633, QuSpin."
]
] | 2001.03481 |
[
[
"The generation of matter-antimatter asymmetries and hypermagnetic fields\n by the chiral vortical effect of transient fluctuations"
],
[
"Abstract We study the contribution of temperature-dependent chiral vortical effect to the generation and evolution of the hypermagnetic fields and the matter-antimatter asymmetries, in the symmetric phase of the early Universe, in the temperature range $100\\mbox{GeV} \\le T\\le 10\\mbox{TeV}$.",
"Our most important result is that, due to the chiral vortical effect, small overlapping transient fluctuations in the vorticity field in the plasma and temperature of matter degrees of freedom can lead to the generation of strong hypermagnetic fields and matter-antimatter asymmetries, all starting from zero initial values.",
"We show that, either an increase in the amplitudes of the fluctuations of vorticity or temperature, or a decrease in their widths, leads to the production of stronger hypermagnetic fields, and therefore, larger matter-antimatter asymmetries.",
"We have the interesting result that fluctuating vorticity fields are more productive, by many orders of magnitude, as compared to vorticities that are constant in time."
],
[
"Introduction",
"Anomalous transport effects play important roles in particle physics and cosmology, particularly in the early Universe [1].",
"One important effect of this kind is the so-called Chiral Vortical Effect (CVE), which refers to the generation of an electric current parallel to the vorticity field in the chiral plasma [2].",
"This effect was discovered by Vilenkin who showed that a neutrino current density can result from a rotating black hole [2].",
"He obtained the neutrino current density in the direction of the rotation axis as $J(0)=-\\frac{\\Omega }{12}T^{2} -\\frac{\\Omega }{4\\pi ^{2}}\\mu ^{2} -\\frac{\\Omega ^{3}}{48\\pi ^{2}},$ where $\\Omega $ is the angular velocity, $\\mu $ is the chiral chemical potential of the neutrino, and $T$ is its temperature.",
"Thirty years after its discovery, the CVE appeared in the relativistic hydrodynamic equations as an interesting manifestation of anomalies in quantum field theory [3].",
"This effect has attracted much attention and has been investigated extensively in recent years, leading to a deeper understanding of the subject [4], [5], [6], [7], [8], [9], [10].",
"In particular, it has been established that in single species chiral plasma in the broken phase at high temperatures, the CVE shows up in the vector current as $\\vec{J}_{\\mathrm {cv}}=\\frac{1}{4\\pi ^2}(\\mu _{R}^2-\\mu _{L}^2)\\vec{\\Omega }$ , and in the axial current as $\\vec{J}^{5}_{\\mathrm {cv}}=\\left[\\frac{T^2}{6}+\\frac{1}{4\\pi ^2}\\left(\\mu _{R}^2+\\mu _{L}^2\\right)\\right]\\vec{\\Omega }$ , where $\\mu _{R}$ and $\\mu _{L}$ are the right-handed and the left-handed chemical potentials of the species, respectively [2], [3], [4], [5], [6], [7], [8], [9], [10]There are additional contributions of $O(\\mu /T)$ , where $\\mu $ is the chemical potential, which are significant at lower temperatures.",
"See Appendix A for details.. Interestingly, the term proportional to $T^2$ indicates that there can be an axial current, even if $\\mu _{R}=\\mu _{L}=0$ .",
"In this study, we present the correct form of the chiral vortical current in the symmetric phase.",
"Then, we show the prominent effects of the temperature-dependent part of this current in the symmetric phase of the early Universe close to the electroweak phase transition (EWPT).",
"In particular, we show that even very small, but overlapping, transient fluctuations in the vorticity field and temperature of matter degrees of freedom can have important consequences, including the generation of hypermagnetic field in the absence of initial matter asymmetries.",
"The vorticity fluctuations that we consider are about the zero background value, while the temperature fluctuations are about the finite equilibrium temperature of the plasma.",
"The most important role of the CVE in this context is to produce the magnetic fields, either through the chiralities, or through the temperature fluctuation, the latter of which is the main focus of this work.",
"Henceforth, we shall refer to transient fluctuations, which we take to be in the form of short pulses, simply as fluctuations.",
"Another anomalous transport effect is the chiral magnetic effect (CME), which refers to the generation of an electric current parallel to the magnetic field in the imbalanced chiral plasma [11], [12], [13], [14].",
"It is known that, in single species chiral plasma in the broken phase at high temperatures, the CME appears in the vector current as $\\vec{J}_{\\mathrm {cm}}=\\frac{Q}{4\\pi ^2}\\left(\\mu _{R}-\\mu _{L}\\right)\\vec{B}$ , and in the axial current as $\\vec{J}_{\\mathrm {cm}}^5=\\frac{Q}{4\\pi ^2}\\left(\\mu _{R}+\\mu _{L}\\right)\\vec{B}$ , where $Q$ is the electric charge [11], [15], [16], [10] of the species.",
"[1] Later, we will present the correct form of the chiral magnetic current, in the symmetric phase.",
"The chiral magnetic current originating from the electroweak Abelian anomaly, and the chiral vortical current are both non-dissipative currents which can strongly affect the generation and the evolution of the magnetic fields and the matter-antimatter asymmetries in the early Universe [17], [18], [19], [20], [21].",
"Observations clearly show that our Universe is magnetized on all scales [22], [23].",
"Various models have been proposed to explain the origin of these magnetic fields [24], [25], [26], [27], [28], [29], [30], [20], among which the one relying on the electroweak Abelian anomaly has attracted much attention and has been considerably investigated [31], [17], [18], [19].",
"There exists a relationship between the generation and the evolution of the hypermagnetic fields and the fermion number densities in this model, which is due to the chiral coupling of the hypercharge gauge fields to the fermions before the EWPT[27].",
"In case there is a preexisting asymmetry of the right-handed electrons, their number density is almost conserved far from the EWPT, i.e.",
"$T>10\\mbox{TeV}$ , due to their tiny Yukawa coupling.",
"For lower temperatures, this asymmetry can be converted to the hypermagnetic helicity according to the Abelian anomaly equation, $\\partial _{\\mu } j_{{e}_R}^{\\mu }\\sim \\vec{E}_{Y}.\\vec{B}_{Y}$ [27], [32], [31], [17], [18], [19].",
"The anomaly equation shows that, in a reverse process, a strong helical hypermagnetic field can generate the matter-antimatter asymmetries in the Universe, as well [33], [34], [35].",
"Another challenge in particle physics and cosmology is the excess of matter over antimatter, with the measured baryon asymmetry of the Universe being of the order of $\\eta _{\\mathrm {B}}\\sim 10^{-10}$ , , .",
"The three Sakharov conditionsi- baryon number violation, ii- C and CP violation, iii- a departure from thermal equilibrium.",
"should be satisfied in any CPT invariant model used to explain this asymmetry from an initially symmetric Universe [39].",
"In previous studies based on the electroweak Abelian anomalous model, it has been assumed that there is either a significant amount of matter-antimatter asymmetries to produce the hypermagnetic field, or a strong hypermagnetic field to produce the matter-antimatter asymmetries.",
"The most important result of this study is that the matter-antimatter asymmetries and the hypermagnetic field can all be generated simultaneously from zero initial values, by considering the temperature-dependent CVE before the EWPT.",
"To obtain this interesting result, we take into account the effects of the temperature-dependent term of the chiral vortical current on the evolution of the hypermagnetic fields and the matter-antimatter asymmetries, by considering simultaneous small fluctuations, about the background values, in temperature of the right-handed electrons and the vorticity field, close to the EWPT.",
"We also show that fluctuations in the vorticity field are much more productive than vorticity fields that are constant in time.",
"To be more precise, sharp fluctuations yield results comparable to constant vorticities whose amplitudes are many orders of magnitude larger.",
"As mentioned above, an underlying assumption of this work is the presence of fluctuations in the plasma of the early Universe, containing all elementary particles and gauge fields.",
"A general description of this plasma in the context of hydrodynamics, as an effective field theory, has been presented [40], [41].",
"In such a physical system, hydrodynamic variables naturally fluctuate around their statistical averages.",
"Such stochastic and frequent fluctuations are a commonplace in any plasma, including that of the early Universe.",
"These fluctuations, which may stem from a sum of weakly-correlated random local events, can occur for physical observables such as the number density, velocity, and temperature [40], [42], [43], [44].",
"Even though the gauge interactions are strong and their rates are fast in this epoch, the extreme temperatures of the primordial plasma imply that fluctuations could still occur for all species.",
"Moreover, owing to the different Yukawa and gauge couplings, and in particular the chiral nature of the electroweak sector, different chiral fermions could, in principle, experience different fluctuations.",
"All that is required here is that the fluctuations for at least one of the species be non-identical to the rest, at least once.",
"Another justification for our hypothesis is the following.",
"Realistic fluctuations of density, temperature and vorticity are usually local in space and time.",
"Meanwhile, in a multi-component plasma local and independent density fluctuations can occur for different matter components.",
"The concurrent occurrence of density, temperature and vorticity fluctuations leads to the possibility of occurrence of non-identical fluctuations for different species of particles.",
"To be concrete, consider a local density fluctuation leading to a local excess of a particle species, e.g.",
"$e_R$ , in the central region and rarefaction in the surrounding region, followed immediately by a temperature fluctuation which increases the temperature of the central region.",
"The concurrent occurrence of these two fluctuations leads to a temporary local, and to a much lesser degree global average, increase of temperature of that particle species relative to quasi-equilibrium temperature of the plasma.",
"All that is required now is an overlapping vorticity fluctuation to have the necessary conditions for our model.",
"In this work, we make the simplifying assumption, as is usually done, of considering these fluctuations to be global rather than local.",
"It should be noted that the amplitude and duration of these fluctuations are usually small enough such that the equilibrium energy density, pressure and entropy of the system can be defined.",
"Nevertheless, as we shall show, even very small and brief fluctuations can trigger the mechanism that we propose here and affect significantly the dynamical evolution of the system, as is formulated within the equations of anomalous magnetohydrodynamics (AMHD).",
"This paper is organized as follows: In Sec.",
", the anomalous magnetohydrodynamics equations and the evolution equations for the matter-antimatter asymmetries are derived in the expanding Universe.",
"In Sec.",
", the set of coupled differential equations are solved numerically.",
"In Sec.",
", the results are summarized and the conclusion is presented."
],
[
"Anomalous Magnetohydrodynamics Equations",
"In this section, the anomalous magnetohydrodynamics (AMHD) equations are obtained in the Landau-Lifshitz frame in the symmetric phase of the expanding Universe.",
"Taking the CVE and the CME into account, the Maxwell's equations for the hypercharge-neutral plasma in the expanding Universe are given as [3], , , , [20], [47], [49], [48], [46], [45] $\\frac{1}{R}\\vec{\\nabla } .\\vec{E}_{Y}=0,\\qquad \\qquad \\frac{1}{R}\\vec{\\nabla }.\\vec{B}_{Y}=0,$ $\\frac{1}{R}\\vec{\\nabla }\\times \\vec{ E}_{Y}+\\left(\\frac{\\partial \\vec{B}_{Y}}{\\partial t}+2H\\vec{B}_{Y}\\right)=0,$ $\\begin{split}\\frac{1}{R}\\vec{\\nabla }\\times \\vec{B}_{Y}-&\\left(\\frac{\\partial \\vec{E}_{Y}}{\\partial t}+2H\\vec{E}_{Y}\\right)=\\vec{J}\\\\&=\\vec{J}_{\\mathrm {Ohm}}+\\vec{J}_{\\mathrm {cv}}+\\vec{J}_{\\mathrm {cm}},\\end{split}$ $\\vec{J}_{\\mathrm {Ohm}}=\\sigma \\left(\\vec{E}_{Y}+\\vec{v}\\times \\vec{B}_{Y}\\right),$ $\\vec{J}_{\\mathrm {cv}}=c_{\\mathrm {v}}\\vec{\\omega },$ $\\vec{J}_{\\mathrm {cm}}=c_{\\mathrm {B}}\\vec{B}_{Y},$ where $R$ is the scale factor, $H=\\dot{R}/R$ is the Hubble parameter, $\\sigma $ is the electrical hyperconductivity of the plasma, and $\\vec{v}$ and $\\vec{\\omega }=\\frac{1}{R}\\vec{\\nabla }\\times \\vec{v}$ are the bulk velocity and vorticity of the plasma, respectively.",
"Furthermore, the chiral vorticity and helicity coefficients $c_{\\mathrm {v}}$ and $c_{\\mathrm {B}}$ are as follows (see Appendix A)The temperature-independent parts of these coefficients were presented in [20].",
"There are additional terms of $O(\\mu /T)$ which, as we shall show explicitly in Sec.",
", are negligible within the initial conditions and results of our model (see Appendix A)., $\\begin{split}c_{\\mathrm {v}}(t)=&\\sum _{i=1}^{n_{G}}\\Big [\\frac{g^{\\prime }}{48}\\Big (-Y_{R}T_{R_{i}}^{2}+Y_{L}T_{L_{i}}^{2}N_{w}-Y_{d_{R}}T_{d_{R_{i}}}^{2}N_{c}-Y_{u_{R}}T_{u_{R_{i}}}^{2}N_{c}+Y_{Q}T_{Q_{i}}^{2}N_{c}N_{w}\\Big )\\\\&+\\frac{{g^{\\prime }}}{16\\pi ^{2}}\\Big (-Y_{R}\\mu _{R_{i}}^{2}+Y_{L}\\mu _{L_{i}}^{2}N_{w}-Y_{d_{R}}\\mu _{d_{R_{i}}}^{2}N_{c}-Y_{u_{R}}\\mu _{u_{R_{i}}}^{2}N_{c}+Y_{Q}\\mu _{Q_{i}}^{2}N_{c}N_{w}\\Big )\\Big ], \\end{split}$ $\\begin{split}c_{\\mathrm {B}}(t)=&-\\frac{g^{\\prime 2}}{8\\pi ^{2}}\\sum _{i=1}^{n_{G}}\\Big [-\\Big (\\frac{1}{2}\\Big )Y_{R}^{2}\\mu _{R_{i}}-\\Big (\\frac{-1}{2}\\Big )Y_{L}^{2}\\mu _{L_{i}}N_{w}-\\Big (\\frac{1}{2}\\Big )Y_{d_{R}}^{2}\\mu _{d_{R_{i}}}N_{c}-\\Big (\\frac{1}{2}\\Big )Y_{u_{R}}^{2}\\mu _{u_{R_{i}}}N_{c}\\\\&-\\Big (\\frac{-1}{2}\\Big )Y_{Q}^{2}\\mu _{Q_{i}}N_{c}N_{w}\\Big ],\\end{split}$ where $n_{G}$ is the number of generations, and $N_{c}=3$ and $N_{w}=2$ are the ranks of the non-Abelian SU$(3)$ and SU$(2)$ gauge groups, respectively.",
"Moreover, $\\mu _{L_i}$ ($\\mu _{R_i}$ ), $\\mu _{Q_i}$ , and $\\mu _{{u_R}_i}$ ($\\mu _{{d_R}_i}$ ) are the common chemical potentials of left-handed (right-handed) leptons, left-handed quarks with different colors, and up (down) right-handed quarks with different colors, respectively.",
"Furthermore, `i' is the generation index, and the relevant hypercharges are $\\begin{split}&Y_{L}=-1, \\quad Y_{R}=-2,\\\\& Y_{Q}=\\frac{1}{3}, \\quad Y_{u_{R}}=\\frac{4}{3}, \\quad Y_{d_{R}}=-\\frac{2}{3}.\\end{split} $ After substituting the hypercharges in Eqs.",
"(REF ) and (REF ), we obtain $\\begin{split}c_{\\mathrm {v}}(t)=&\\sum _{i=1}^{n_{G}}\\Big [\\frac{g^{\\prime }}{24}\\left(T_{R_{i}}^{2}-T_{L_{i}}^{2}+T_{d_{R_{i}}}^{2}-2T_{u_{R_{i}}}^{2}+T_{Q_{i}}^{2}\\right)+\\frac{{g^{\\prime }}}{8\\pi ^{2}}\\Big (\\mu _{R_{i}}^{2}-\\mu _{L_{i}}^{2}+\\mu _{d_{R_{i}}}^{2}-2\\mu _{u_{R_{i}}}^{2}+\\mu _{Q_{i}}^{2}\\Big )\\Big ],\\end{split} $ $\\begin{split}&c_{\\mathrm {B}}(t)=\\frac{-g^{\\prime 2}}{8\\pi ^{2}} \\sum _{i=1}^{n_{G}}\\left[-2\\mu _{R_{i}}+\\mu _{L_{i}}-\\frac{2}{3}\\mu _{d_{R_{i}}}-\\frac{8}{3}\\mu _{u_{R_{i}}}+\\frac{1}{3}\\mu _{Q_{i}}\\right].\\end{split}$ Let us make the same assumptions as in our previous studies, and simplify $c_{\\mathrm {v}}$ and $c_{\\mathrm {B}}$ accordingly [20], [18].",
"We assume that all quark Yukawa processes are in equilibrium and, because of the flavor mixing in the quark sector, all up or down quarks belonging to different generations with distinct handedness have the same chemical potential [18], [53].",
"For simplicity, we also assume that the Higgs asymmetry is zero and obtain [54], [18] $\\mu _{u_{R}}=\\mu _{d_{R}}=\\mu _{Q}.$ Furthermore, we assume that only the contributions of the baryonic and the first-generation leptonic chemical potentials to $c_{\\mathrm {v}}$ and $c_{\\mathrm {B}}$ are significant.",
"As for the temperature fluctuations, it suffices to consider fluctuations in only one of the matter components, which we take to be $e_{R}$ .",
"Using Eq.",
"(REF ) and the aforementioned assumptions, we simplify Eqs.",
"(REF ) and (REF ) to obtain $c_{\\mathrm {v}}(t)=\\frac{g^{\\prime }}{24}\\left(\\Delta T^{2}\\right)+\\frac{g^{\\prime }}{8\\pi ^{2}}\\left(\\mu _{e_{R}}^{2}-\\mu _{e_{L}}^{2}\\right), $ $c_{\\mathrm {B}}(t)=-\\frac{g^{\\prime 2}}{8\\pi ^{2}}\\left(-2\\mu _{e_{R}}+\\mu _{e_{L}}-\\frac{3}{4}\\mu _{\\mathrm {B}}\\right),$ where $\\mu _{\\mathrm {B}}=12\\mu _{Q}$ , and $\\Delta T^{2}=T_{e_{R}}^{2}- T^2$ is the temperature fluctuation, and $T$ is the equilibrium temperature of the thermal bath, which includes all other components of the plasma.",
"We set ${\\Delta T}^{2}=T^{2} \\beta [x(T)]$ , where $\\beta [x(T)]$ is an arbitrary profile function to be specified later, and $x(T)=t(T)/t_\\mathrm {EW}=\\left(T_\\mathrm {EW}/T\\right)^{2}$ is given by the Friedmann law.",
"In the Landau-Lifshitz frame, the continuity and Navier-Stokes equations are given as follows [47], [49], [48], [20] $\\frac{\\partial \\rho }{\\partial t}+\\frac{1}{R}\\vec{\\nabla }.\\left[\\left(\\rho +p\\right)\\vec{v}\\right]+3H\\left(\\rho +p\\right)=0,$ $\\begin{split}&\\left[\\frac{\\partial }{\\partial t}+\\frac{1}{R}\\left(\\vec{v}.\\vec{\\nabla }\\right)+H\\right]\\vec{v}+\\frac{\\vec{v}}{\\rho +p}\\frac{\\partial p}{\\partial t}=-\\frac{1}{R}\\frac{\\vec{\\nabla } p}{\\rho +p}+\\frac{\\vec{J}\\times \\vec{B}_{Y}}{\\rho +p}+\\frac{\\nu }{{R}^{2}}\\left[\\nabla ^{2}\\vec{v}+\\frac{1}{3}\\vec{\\nabla }\\left(\\vec{\\nabla }.\\vec{v}\\right)\\right],\\end{split}$ where $\\nu $ is the kinematic viscosity, and $\\rho $ and $p$ are the energy density and the pressure of the plasma, respectively.",
"Combining the fluid incompressibility condition in the lab frame, $\\partial _{t}\\rho +3H\\left(\\rho +p\\right)=0$ or equivalently $H\\vec{v}+\\vec{v}\\partial _{t} p/\\left(\\rho +p\\right)=0$ , with Eq.",
"(REF ) leads to the condition $\\vec{\\nabla }.\\vec{v}=0$ [20], [55], [49].",
"In the following, we choose a simple monochromatic Chern-Simons configuration for the hypermagnetic field $\\vec{B}_Y=(1/R)\\vec{\\nabla } \\times \\vec{A}_Y$ , and the velocity field $\\vec{v}=(1/R)\\vec{\\nabla } \\times \\vec{S}$ , [20].",
"To do this, we choose $\\vec{A}_{Y}=\\gamma (t)\\left(\\cos kz , \\sin kz, 0\\right)$ , and $\\vec{S}=r(t)\\left(\\cos kz , \\sin kz, 0\\right)$ , for their corresponding vector potentials [56], [57], [58].",
"Note that we have chosen a fully helical form with the negative helicity for both, the reason for which will be stated later.",
"Let us now obtain the evolution equation for the velocity field.",
"Neglecting the displacement current in the lab frame in Eq.",
"(REF ), the total current becomes $\\vec{J}=(1/R)\\vec{\\nabla }\\times \\vec{B}_{Y}$ , and as a result, $\\vec{J}\\times \\vec{B}_{Y}$ vanishes in Eq.",
"(REF ).",
"Then, using $\\vec{\\nabla } .\\vec{v}=0$ and $H\\vec{v}+\\vec{v}\\partial _{t}p/(\\rho +p)=0$ , as stated earlier, and neglecting the gradient terms in Eq.",
"(REF ), the evolution equation for the velocity field becomes The term $(\\vec{v}.\\vec{\\nabla })\\vec{v}$ is neglected because of being next to leading order.",
"Furthermore, the magnetic pressure $B^{2}/8\\pi $ is negligible compared to the fluid (radiation) pressure $p$ which is homogeneous and isotropic.",
"Indeed, the maximum value of their ratio at the onset of the EWPT is $B^{2}/8\\pi p\\approx 10^{-7} \\ll 1$ .",
"Therefore, to a good approximation, the homogenity and isotropy conditions remain valid and the pressure variations in the fluid $\\vec{\\nabla } p$ can be neglected[59].",
"$\\frac{\\partial \\vec{v}}{\\partial t}=-\\nu {k^{\\prime }}^{2}\\vec{v},$ where the kinematic viscosity $\\nu \\simeq 1/(5\\alpha _{Y}^{2}T)$ , .",
"Neglecting the displacement current in the lab frame and using the aforementioned configurations, the hyperelectric field and the evolution equation for the hypermagnetic field are obtained, as follows: $\\vec{E}_{Y}=-\\frac{k^{\\prime }}{\\sigma }\\vec{B}_{Y}+\\frac{c_{\\mathrm {v}}}{\\sigma }k^{\\prime }\\vec{v}-\\frac{c_{\\mathrm {B}}}{\\sigma }\\vec{B}_{Y},$ $\\begin{split}\\frac{dB_{Y}(t)}{dt}=&\\left[-\\frac{1}{ t}-\\frac{{k^{\\prime }}^{2}}{\\sigma } -\\frac{c_{\\mathrm {B}}k^{\\prime }}{\\sigma } \\right]B_{Y}(t)+\\frac{c_{\\mathrm {v}}}{\\sigma }{k^{\\prime }}^{2}\\langle \\vec{v}(t).\\hat{B}_{Y}(t)\\rangle ,\\end{split}$ where $\\vec{\\omega }=-k^{\\prime }\\vec{v}$ , $\\sigma =100T$ , angle brackets denote spatial average, and $k^{\\prime }=k/R=kT$ .",
"The latter shows the increase of the hypermagnetic length scale, due to the expansion of the Universe.",
"Note that with the choice of vector potentials for $\\vec{v}(t)$ and $\\vec{B}_{Y}(t)$ , the advection term $\\vec{v}\\times \\vec{B}_{Y}$ has been set to zero in the above, and we have the following simplification: $\\langle \\vec{v}(t).\\hat{B}_{Y}(t)\\rangle \\rightarrow v(t)$ .",
"In the next subsection we obtain the evolution equations for the matter-antimatter asymmetries."
],
[
"Evolution equations for the matter-antimatter asymmetries",
"Before the EWPT, the gauge fields of $\\textrm {U}_\\textrm {Y}(1)$ couple to the fermions chirally, in contrast to those of $\\textrm {U}_\\textrm {em}(1)$ in the broken phase, leading to the non-conservation of the matter currents.",
"This shows up in the Abelian anomaly equations , which, for the first-generation leptons, are $\\begin{split}&\\nabla _{\\mu } j_{{e}_R}^{\\mu }=-\\frac{1}{4}(Y_{R}^{2})\\frac{g^{\\prime 2}}{16 \\pi ^2}Y_{\\mu \\nu }\\tilde{Y}^{\\mu \\nu }=\\frac{g^{\\prime 2}}{4\\pi ^{2}}\\vec{E}_{Y}.\\vec{B}_{Y},\\\\&\\nabla _{\\mu } j_{{e}_L}^{\\mu }=\\frac{1}{4}(Y_{L}^{2})\\frac{g^{\\prime 2}}{16\\pi ^2}Y_{\\mu \\nu }\\tilde{Y}^{\\mu \\nu }=-\\frac{g^{\\prime 2}}{16\\pi ^{2}}\\vec{E}_{Y}.\\vec{B}_{Y},\\end{split}$ where $\\nabla _{\\mu }$ is the covariant derivative with respect to Friedmann-Robertson-Walker (FRW) metric $ds^{2}=dt^{2}-R^{2}(t)\\delta _{ij}dx^{i}dx^{j}$ , $t$ is the physical time, and $x^{i}$ s are the comoving coordinates.",
"Integrating the above equations over all space and considering the perturbative chirality flip reactions for the leptons, we obtain (see Refs.",
"[17], [18], [63], [20] and Appendix B for details), $\\begin{split}\\frac{d\\eta _{{e}_{R}}}{dt}=&\\frac{g^{\\prime 2}}{4\\pi ^{2} s}\\langle \\vec{E}_{Y}.\\vec{B}_{Y}\\rangle +\\left(\\frac{\\Gamma _{0}}{t_{EW}}\\right)\\left(\\frac{1-x}{\\sqrt{x}}\\right)\\left(\\eta _{e_{L}}-\\eta _{e_{R}}\\right)\\\\&-\\frac{d}{dt}\\Big [\\frac{g^{\\prime }\\mu _{e_{R}}}{4\\pi ^{2}s}\\langle \\vec{v}.\\vec{B}_{Y}\\rangle \\Big ]-\\frac{d}{dt}\\Big [\\big (\\frac{\\mu _{e_{R}}^{2}}{8\\pi ^{2}s}+\\frac{T^{2}}{24s}\\big )\\langle \\vec{v}.\\vec{\\omega }\\rangle \\Big ]+\\frac{d}{dt}\\Big [\\frac{2\\sigma _{e_{R}}}{g^{\\prime }Y_{R}s}\\langle \\vec{v}.\\vec{E}_{Y}\\rangle \\Big ]\\\\\\frac{d\\eta _{\\nu _{e}^{L}}}{dt}=&\\frac{d\\eta _{{e}_{L}}}{dt}=-\\frac{g^{\\prime 2}}{16\\pi ^{2} s}\\langle \\vec{E}_{Y}.\\vec{B}_{Y}\\rangle +\\left(\\frac{\\Gamma _{0}}{2t_{EW}}\\right)\\left(\\frac{1-x}{\\sqrt{x}}\\right)\\left(\\eta _{e_{R}}-\\eta _{e_{L}}\\right)\\\\&+\\frac{d}{dt}\\Big [\\frac{g^{\\prime }\\mu _{e_{L}}}{8\\pi ^{2}s}\\langle \\vec{v}.\\vec{B}_{Y}\\rangle \\Big ]+\\frac{d}{dt}\\Big [\\big (\\frac{\\mu _{e_{L}}^{2}}{8\\pi ^{2}s}+\\frac{T^{2}}{24s}\\big )\\langle \\vec{v}.\\vec{\\omega }\\rangle \\Big ]+\\frac{d}{dt}\\Big [\\frac{2\\sigma _{e_{L}}}{g^{\\prime }Y_{L}s}\\langle \\vec{v}.\\vec{E}_{Y}\\rangle \\Big ].\\end{split}$ In the equations above, $\\eta _{f}=\\left(n_{f}/s\\right)$ with $f=e_{R},e_{L},\\nu _{e}^{L}$ is the fermion asymmetry and $n_{f}$ is the charge density of the $f$ th species of fermion, $s=2\\pi ^{2}g^{*}T^{3}/45$ is the entropy density and $g^{*}=106.75$ is the effective number of relativistic degrees of freedom, $x=\\left(t/t_\\mathrm {EW}\\right)=\\left(T_\\mathrm {EW}/T\\right)^{2}$ is given by the Friedmann law, $\\Gamma _{0}=121$ , $t_\\mathrm {EW}=\\left(M_{0}/2T_\\mathrm {EW}^{2}\\right)$ , and $M_{0}=\\left(M_\\mathrm {Pl}/1.66\\sqrt{g^{*}}\\right)$ , where $M_\\mathrm {Pl}$ is the Plank mass.",
"Furthermore, the term $\\frac{\\Gamma _{0}}{t_\\mathrm {EW}}\\left(\\frac{1-x}{\\sqrt{x}}\\right)$ appearing in the equations is the chirality flip rate of the right-handed electrons.",
"In a similar manner, the evolution equation for the baryon asymmetry can also be obtained as (see Appendix B) $\\frac{d\\eta _{\\mathrm {B}}}{dt}=\\frac{3g^{\\prime 2}}{8\\pi ^{2} s}\\langle \\vec{E}_{Y}.\\vec{B}_{Y}\\rangle +\\frac{d}{dt}\\left[\\frac{2}{sg^{\\prime }}(\\frac{\\sigma _{d_R}}{Y_{d_R}}+\\frac{\\sigma _{u_R}}{Y_{u_R}}+2\\frac{\\sigma _{Q}}{Y_{Q}})\\langle \\vec{v}.\\vec{E}_{Y}\\rangle \\right].$ Using $\\mu _{f}=(6s/T^2)\\eta _f$ with Eq.",
"(REF ) we obtain $\\begin{split}\\langle \\vec{E}_{Y}.\\vec{B}_{Y}\\rangle =&\\frac{B_{Y}^{2}(t)}{100} \\left[-\\frac{k^{\\prime }}{T}-\\frac{6sg^{\\prime 2}}{4\\pi ^{2}T^3}\\left(\\eta _{e_{R}}-\\frac{\\eta _{e_{L}}}{2}+\\frac{3}{8}\\eta _{\\mathrm {B}}\\right)\\right]\\\\&+\\left[\\frac{g^{\\prime }}{24}\\beta [x(T)]+\\frac{36s^2g^{\\prime }}{8\\pi ^{2}T^6}\\left(\\eta _{e_{R}}^{2}-\\eta _{e_{L}}^{2}\\right)\\right]\\frac{k^{\\prime }T }{100}\\langle \\vec{v}(t).\\vec{B}_{Y}(t)\\rangle .\\end{split}$ With the helical configurations chosen, $\\langle \\vec{v}(t).\\vec{B}_{Y}(t)\\rangle \\rightarrow v(t)B_Y(t)$ .",
"Using $1\\mbox{Gauss}\\simeq 2\\times 10^{-20} \\mbox{GeV}^{2}$ , and setting the kinematic viscosity $\\nu $ to zero for simplicity, we obtain the complete set of evolution equations for the matter-antimatter asymmetries and the amplitudes of the hypermagnetic and velocity fields as $\\begin{split}\\frac{d\\eta _{e_R}}{dx}&=\\frac{1}{\\lambda _{R}(x)}\\Bigg [\\Lambda _{R}(x)+\\left[-C_{1}-C_{2} \\eta _{T}(x)\\right]\\left(\\frac{B_{Y}(x)}{10^{20}G}\\right)^{2}x^{3/2}\\\\&+\\left[C_{3}\\beta (x)+C_{4} \\Delta \\eta ^{2}(x)\\right]v(x)\\left(\\frac{B_{Y}(x)}{10^{20}G}\\right)\\sqrt{x}\\Bigg ]-\\Gamma _{0}\\frac{1-x}{\\sqrt{x}}\\left[\\eta _{e_R}(x)-\\eta _{e_L}(x)\\right],\\end{split}$ $\\begin{split}\\frac{d\\eta _{e_L}}{dx}&=\\frac{1}{\\lambda _{L}(x)}\\Bigg [\\Lambda _{L}(x)-\\frac{1}{4}\\left[-C_{1}-C_{2}\\eta _{T}(x)\\right]\\left(\\frac{B_{Y}(x)}{10^{20}G}\\right)^{2}x^{3/2}\\\\&-\\frac{1}{4}\\left[C_{3}\\beta (x)+C_{4} \\Delta \\eta ^{2}(x)\\right]v(x)\\left(\\frac{B_{Y}(x)}{10^{20}G}\\right)\\sqrt{x}\\Bigg ]+\\Gamma _{0}\\frac{1-x}{2\\sqrt{x}}\\left[\\eta _{e_R}(x)-\\eta _{e_L}(x)\\right],\\end{split}$ $\\begin{split}\\frac{dB_{Y}}{dx}&=\\frac{1}{\\sqrt{x}}\\left[-C_{5} -C_{6}\\eta _{T}(x)\\right]B_{Y}(x)-\\frac{1}{x}B_{Y}(x)+\\left[C_{7}\\beta (x)+C_{8}\\Delta \\eta ^{2}(x)\\right]\\frac{v(x)}{x^{3/2}} ,\\end{split}$ $\\begin{split}\\frac{d\\eta _{\\mathrm {B}}}{dx}&=\\frac{3}{2}\\Bigg [-C_{1}-C_{2}\\eta _{T}(x)\\left(\\frac{B_{Y}(x)}{10^{20}G}\\right)^{2}x^{3/2}+\\left[C_{3}\\beta (x)+C_{4} \\Delta \\eta ^{2}(x)\\right]v(x)\\left(\\frac{B_{Y}(x)}{10^{20}G}\\right)\\sqrt{x}\\Bigg ]\\\\&+\\frac{81x}{100Mg^{\\prime }10^{20} G}\\langle \\vec{E}_{Y}(x).\\partial _ x\\vec{v}(x)\\rangle \\end{split}$ $\\begin{split}&\\Delta \\eta ^{2}(x)= \\eta _{e_R}^{2}(x)-\\eta _{e_L}^{2}(x),\\\\&\\eta _{T}(x)=\\eta _{e_R}(x)-\\frac{\\eta _{e_L}(x)}{2}+\\frac{3}{8}\\eta _{\\mathrm {B}(x)},\\\\&\\lambda _{R}(x)=1-\\frac{6g^{\\prime }Y_{R}}{8\\pi ^{2}}\\frac{\\langle \\vec{v}(x).\\vec{B}_{Y}(x)\\rangle }{10^{20}G}\\frac{x}{5000}-\\frac{36M}{4\\pi ^{2}}\\eta _{e_R}(x)kv^{2}(x),\\\\&\\lambda _{L}(x)=1+\\frac{6g^{\\prime }Y_{L}}{8\\pi ^{2}}\\frac{\\langle \\vec{v}(x).\\vec{B}_{Y}(x)\\rangle }{10^{20}G}\\frac{x}{5000}+\\frac{36M}{4\\pi ^{2}}\\eta _{e_L}(x)kv^{2}(x),\\\\\\Lambda _{R}(x)=&\\frac{6g^{\\prime }Y_{R}}{8\\pi ^{2}}\\frac{x}{5000}\\frac{\\eta _{e_R}(x)}{10^{20}G}\\Big [\\langle \\vec{v}(x).\\partial _{x}\\vec{B}_{Y}(x)\\rangle +\\langle \\frac{\\vec{v}(x).\\vec{B}_{Y}(x)}{x}\\rangle +\\langle \\vec{B}_{Y}(x).\\partial _{x}\\vec{v}(x)\\rangle \\Big ]\\\\&+\\Big [\\frac{36M}{4\\pi ^{2}}\\eta _{e_R}^2(x)+\\frac{1}{12M}\\Big ]k\\vec{v}(x).\\partial _{x}\\vec{v}(x)+\\frac{x}{25Mg^{\\prime }Y_{R}10^{20} G}\\langle \\vec{E}_{Y}(x).\\partial _{x}\\vec{v}(x)\\rangle ,\\\\\\Lambda _{L}(x)=&-\\frac{6g^{\\prime }Y_{L}}{8\\pi ^{2}}\\frac{x}{5000}\\frac{\\eta _{e_L}(x)}{10^{2}G}\\Big [\\langle \\vec{v}(x).\\partial _{x}\\vec{B}_{Y}(x)\\rangle +\\langle \\frac{\\vec{v}(x).\\vec{B}_{Y}(x)}{x}\\rangle +\\langle \\vec{B}_{Y}(x).\\partial _{x}\\vec{v}(x)\\rangle \\Big ]\\\\&-\\Big [\\frac{36M}{4\\pi ^{2}}\\eta _{e_L}^{2}(x)+\\frac{1}{12M}\\Big ]k\\vec{v}(x).\\partial _{x}\\vec{v}(x)+\\frac{x}{25Mg^{\\prime }Y_{L}10^{20} G}\\langle \\vec{E}_{Y}(x).\\partial _{x}\\vec{v}(x)\\rangle ,\\\\&\\end{split}$ where $M=2\\pi ^{2}g^{*}/45$ , and the coefficients $C_{i}, i=1,...,8$ , are $\\begin{split}& C_{1}=0.00096\\left(\\frac{k}{10^{-7}}\\right)\\alpha _{Y},\\\\&C_{2}=865688 \\alpha _{Y}^{2},\\\\&C_{3}=0.71488\\left(\\frac{k}{10^{-7}}\\right)\\alpha _{Y}^{3/2},\\\\&C_{4}= 17152.7\\left(\\frac{k}{10^{-7}}\\right)\\alpha _{Y}^{3/2},\\\\&C_{5}= 0.356\\left(\\frac{k}{10^{-7}}\\right)^{2},\\\\&C_{6}=3.18373 \\times 10^{8} \\alpha _{Y} \\left(\\frac{k}{10^{-7}}\\right), \\\\&C_{7}=262.9\\times 10^{20} \\sqrt{\\alpha _{Y}}\\left(\\frac{k}{10^{-7}}\\right)^{2},\\\\&C_{8}=63\\times 10^{25} \\sqrt{\\alpha _{Y}}\\left(\\frac{k}{10^{-7}}\\right)^{2},\\\\&\\end{split}$ and $\\alpha _{Y}=g^{\\prime 2}/4\\pi \\simeq 0.01$ is the fine-structure constant for the $\\textrm {U}_\\textrm {Y}(1)$ .",
"We now choose the profile of temperature fluctuation $\\beta [x(T)] = \\Delta T^2/T^2$ , as defined in Eq.",
"(REF ) and the paragraph below it, to be a Gaussian function of $x$ : $\\beta (x)=\\frac{\\beta _{0}}{b\\sqrt{2\\pi }}\\exp \\left[-\\frac{(x-x_{0})^{2}}{2b^{2}}\\right],$ where $\\beta _{0}$ is the amplitude multiplying the normalized Gaussian distribution, and $x=\\left(t/t_{EW}\\right)=\\left(T_{EW}/T\\right)^{2}$ , as defined before.",
"The profile of the vorticity fluctuation must have an overlap with that of temperature fluctuation, in order to produce any effect.",
"For simplicity, we choose the two profiles to be identical.",
"That is, $\\omega (x)=k^{\\prime } v(x)=\\frac{k^{\\prime }v_{0}}{b\\sqrt{2\\pi }}\\exp \\left[-\\frac{(x-x_{0})^{2}}{2b^{2}}\\right],$ where $v_{0}$ is the amplitude of the velocity fluctuation.",
"We should mention that the occurrence of any fluctuation in a plasma in a quasi-equilibrium state would normally trigger a restoring response originating from dissipative effects, such as viscous effects.",
"Here, for simplicity we assume that the combined results of the original fluctuations and the ensuing dissipative effects have the profiles given by Eqs.",
"(REF ,REF ).",
"The majority of analysis presented in the following section is for a single pulse for temperature and vorticity, as stated above, which, as we shall show, produce matter-antimatter asymmetries and helical hypermagnetic field.",
"However, at the end of the next section we present the results for two sets of successive pulses, the latter one having the same temperature profile but with negative amplitude, showing that the first pulse is the main determinant of the outcome."
],
[
"Numerical Solution",
"In this section, we obtain the numerical solutions of the evolution equations.",
"As mentioned earlier, we investigate the effects of the temperature fluctuations of right-handed electrons, in the presence of vorticity, on the generation and evolution of the hypermagnetic field and the matter-antimatter asymmetries, in the temperature range $100 \\mbox{ GeV}\\le T \\le 10\\mbox{ TeV}$ .",
"We consider the temperature fluctuations as small Gaussian distributions in $x$ , as shown in Eq.",
"(REF ), that occur close to the EWPT.",
"As for the vorticity field, we consider small fluctuations, whose profiles, as shown in Eq.",
"(REF ), coincide with those of the temperature fluctuations.",
"We also investigate the cases with constant vorticity fields for comparison.",
"As we shall show, the former is much more interesting and will be the focus of our work, since it is not only physically more realistic, but also could yield orders of magnitude larger results for the asymmetries and the hypermagnetic field.",
"In the following, we solve the evolution equations by considering the comoving wave number as $k=10^{-7}$ , and setting the initial values of the hypermagnetic field amplitude and the matter-antimatter asymmetries to zero, i.e.",
"$B_{Y}^{(0)}=0$ , and $\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0$ .",
"For our first case, we solve the coupled differential equations with the initial conditions, $v_0=10^{-5}$ , $b=2\\times 10^{-4}$ , and $x_{0}=45\\times 10^{-5}$ , for various values of the amplitude of temperature fluctuations $\\beta _{0}$ , and present the results in Fig.",
"REF .",
"As can be seen in the figure, the simultaneous occurrence of small vorticity fluctuation and temperature fluctuation for the right handed electrons leads to the generation of strong hypermagnetic fields which then produce the matter-antimatter asymmetries, all starting from zero initial values.",
"It can be seen that by increasing the amplitude of the temperature fluctuation, the maximum and the final values of the hypermagnetic field amplitude, as well as the matter-antimatter asymmetries, increase.",
"We have found that in our model signs of the matter-antimatter asymmetries produced and the helicity of hypermagnetic and vorticity fields, are always opposite.",
"This is a manifestation of the generalized charge conservation as stated in Eq.",
"(REF ) of Appendix B.In fact, the second and fourth terms in Eq.",
"(REF ) are negligible in this study.",
"The Chern-Simons configuration that we have chosen for the hypermagnetic and vorticity fields has negative helicity.",
"Figure REF also shows that the hypermagnetic field amplitude grows to its maximum value of about $10^{21}$ G, then decreases due to the expansion of the Universe.",
"We have also investigated the effects of changing the amplitude of the vorticity fluctuation, and have found similar results.",
"Figure: Time plots of: (a) the right-handed electron asymmetry η e R \\eta _{e_{R}}, (b) the left-handed electron asymmetry η e L \\eta _{e_{L}}, (c) the baryon asymmetry η B \\eta _{B}, and (d) the hypermagnetic field amplitude B Y B_{Y}, for various values of the amplitude of temperature fluctuation of e R e_{R}.",
"The initial conditions are: k=10 -7 k=10^{-7}, B Y (0) =0B_{Y}^{(0)}=0, η e R (0) =η e L (0) =η B (0) =0\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0, v 0 =10 -5 v_0=10^{-5}, b=2×10 -4 b=2\\times 10^{-4}, and x 0 =45×10 -5 x_{0}=45\\times 10^{-5}.",
"The dashed line is for β 0 =3×10 -4 \\beta _{0}=3\\times 10^{-4}, the solid line is for β 0 =5×10 -4 \\beta _{0}=5\\times 10^{-4}, and the dotted line is for β 0 =7×10 -4 \\beta _{0}=7\\times 10^{-4}.For our second case we solve the set of evolution equations with the initial conditions, $v_0=10^{-5}$ , $\\beta _{0}=5\\times 10^{-4}$ , and $x_{0}=45\\times 10^{-5}$ , for various values of the width or duration of both fluctuations $b$ , and show the results in Fig.",
"REF .",
"As can be seen, by decreasing the width of the Gaussian function, the maximum and the final values of the hypermagnetic field amplitude, and the baryon asymmetry increase.We have also used a few other profiles, the results of which we can summarize as follows.",
"For smooth profiles with the same normalization, the results are mainly dependent on the widths and not on the precise functional form of the profiles.",
"However, the results usually increase by an order of magnitude for profiles functions which have discontinuities.",
"Figure: Time plots of: (a) the baryon asymmetry η B \\eta _{B}, and (b) the hypermagnetic field amplitude B Y B_{Y}, for various values of the width of fluctuations.",
"The initial conditions are: k=10 -7 k=10^{-7}, B Y (0) =0B_{Y}^{(0)}=0, η e R (0) =η e L (0) =η B (0) =0\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0, v 0 =10 -5 v_0=10^{-5}, β 0 =5×10 -4 \\beta _{0}=5\\times 10^{-4}, and x 0 =45×10 -5 x_{0}=45\\times 10^{-5}.",
"The dotted line is obtained for b=3×10 -4 b=3\\times 10^{-4}, the solid line for b=2×10 -4 b=2\\times 10^{-4}, the dashed line for b=10 -4 b=10^{-4}.For our third case, we solve the coupled equations with the initial conditions, $v_0=10^{-5}$ , $b=2\\times 10^{-4}$ , and $\\beta _{0}=5\\times 10^{-4}$ , for various values of center time of the fluctuations $x_{0}$ , and present the results in Fig.",
"REF .",
"As can be seen, when the fluctuations occur at an earlier time or higher temperature, the maxima and the final amplitudes of the hypermagnetic fields increase, and as a result, the matter-antimatter asymmetries increase as well.",
"Figure: Time plots of: (a) the baryon asymmetry η B \\eta _{B}, and (b) the hypermagnetic field amplitude B Y B_{Y}, for various values of the time of fluctuations.",
"The initial conditions are: k=10 -7 k=10^{-7}, B Y (0) =0B_{Y}^{(0)}=0, η e R (0) =η e L (0) =η B (0) =0\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0, v 0 =10 -5 v_0=10^{-5}, β 0 =5×10 -4 \\beta _{0}=5\\times 10^{-4}, b=2×10 -4 b=2\\times 10^{-4}.",
"The dotted line is obtained for x 0 =55×10 -5 x_{0}=55\\times 10^{-5}, the solid line for x 0 =45×10 -5 x_{0}=45\\times 10^{-5}, and the dashed line for x 0 =35×10 -5 x_{0}=35\\times 10^{-5}.For our fourth and final case, we solve the set of evolution equations, with the initial conditions $\\beta _{0}=5\\times 10^{-4}$ , $b=2\\times 10^{-4}$ and $x_{0}=45\\times 10^{-5}$ , for two different vorticity configurations.",
"First configuration is a vorticity fluctuation with amplitude $v_0=10^{-5}$ and $x_{0}=45\\times 10^{-5}$ .",
"Second configuration is a constant vorticity with amplitude $v_0=10^{-2}$ .",
"The results are presented in Fig.",
"REF .",
"As can be seen from the figure, the general trends of the evolution curves are similar.",
"The prominent feature of this comparison is the surprising result that a fluctuation with amplitude smaller by three orders of magnitude produces results comparable with the constant vorticity configuration.",
"Figure: Time plots of: (a) the baryon asymmetry η B \\eta _{B}, and (b) the hypermagnetic field amplitude B Y B_{Y}, for two different vorticity configurations.",
"The initial conditions are: k=10 -7 k=10^{-7}, B Y (0) =0B_{Y}^{(0)}=0, η e R (0) =η e L (0) =η B (0) =0\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0, β 0 =5×10 -4 \\beta _{0}=5\\times 10^{-4}, b=2×10 -4 b=2\\times 10^{-4}, and x 0 =45×10 -5 x_{0}=45\\times 10^{-5}.",
"The solid line is for vorticity fluctuation with v 0 =10 -5 v_0=10^{-5}, and the dashed line is for constant vorticity with v 0 =10 -2 v_0=10^{-2}.Now we can address our assertion that $\\mu /T\\ll 1$ within our model, i.e., our initial conditions and results.",
"First, note that upon using the relations $\\mu _{f}=(6s/T^2)\\eta _{f}$ and $s=(2\\pi ^{2}g^{\\star }/45)T^{3}$ , we obtain $(\\mu _{f}/T)=(12\\pi ^{2}g^{\\star }/45)\\eta _{f}=280.95 \\eta _{f}$ .",
"The largest asymmetry in our results is obtained for $\\eta _{B}$ and is shown in the Fig.",
"REF .",
"This figure shows that $(\\eta _{B})_{\\mathrm {max}}\\sim 5\\times 10^{-9}$ , leading to $(\\mu /T)_{\\mathrm {max}}\\sim 10^{-6}$ .",
"The condition $\\mu /T\\ll 1$ has been used, for example, to justify neglecting terms of $O(\\mu /T)$ in Eqs.",
"(REF -REF ) in Appendix A, leading to the expressions for $c_{\\mathrm {v}}$ and $c_{\\mathrm {B}}$ shown in Eqs.",
"(REF ,REF ), respectively.",
"Moreover, as shown in Appendix B, this condition, along with the assumption of non-relativistic velocity of the plasma, imply that the CME and CVE contributions to the temporal components of the four-currents in the AMHD equations are negligible within our model.",
"We finally address the issue of two successive pulses with opposite temperature profiles to see by how much can the second pulse negate the results of the first.",
"For this purpose it suffices to assume that the profile of the vorticities are unchanged.",
"To be specific, we assume $\\beta (x)=\\beta _{+}(x)+\\beta _{-}(x),$ $v(x)=v_{+}(x)+v_{-}(x),$ where $\\beta _{\\pm }(x)=\\frac{\\pm \\beta _{0}}{b\\sqrt{2\\pi }}\\exp \\left[-\\frac{(x-x_{0,\\pm })^{2}}{2b^{2}}\\right],$ and $v_{\\pm }(x)=\\frac{v_{0}}{b\\sqrt{2\\pi }}\\exp \\left[-\\frac{(x-x_{0,\\pm })^{2}}{2b^{2}}\\right].$ We consider three cases in which the time separation of the pulses $\\Delta x_0= x_{0,+}-x_{0,-}$ are $5b$ , $b$ and $0.1b$ , where $b$ denotes the width of the pulses.",
"The results are shown in Fig.",
"REF , where we also show our results for a single pulse for comparison.",
"As can be seen, the final values of the asymmetries generated are reduced, as compared to the single pulse case, by a factor of about 5, 50 and 1000, respectively.",
"The final values of the hypermagnetic field generated are reduced by square root of values stated above.",
"It is interesting to note that even in the case $\\Delta x_0=0.1b$ , the model produced $\\eta _B \\simeq 10^{-13}$ , a value which can be increased easily by increasing $\\beta _0$ , $v_0$ , and/or decreasing $b$ .",
"Figure: Time plots of: (a) the baryon asymmetry η B \\eta _{B}, (b) the hypermagnetic field amplitude B Y B_{Y} for two sets of successive and opposing fluctuations.",
"The initial conditions are: k=10 -7 k=10^{-7}, B Y (0) =0B_{Y}^{(0)}=0, η e R (0) =η e L (0) =η B (0) =0\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0, v 0,+ =v 0,- =10 -5 v_{0,+}=v_{0,-}=10^{-5}, b=2×10 -4 b=2\\times 10^{-4}, β 0,+ =-β 0,- =5×10 -4 \\beta _{0,+}=-\\beta _{0,-}=5\\times 10^{-4}, and x 0,+ =4.5×10 -4 x_{0,+}=4.5\\times 10^{-4}.",
"The large dashed line is for x 0,- =1.45×10 -3 =5b+x 0,+ x_{0,-}=1.45\\times 10^{-3}=5b+x_{0,+}, the medium dashed line is for x 0,- =6.5×10 -4 =b+x 0,+ x_{0,-}=6.5\\times 10^{-4}=b+x_{0,+}, the dotted line is for x 0,- =4.7×10 -4 =0.1b+x 0,+ x_{0,-}=4.7\\times 10^{-4}=0.1b+x_{0,+}, and the solid line is obtained in the absence of the second set of fluctuations."
],
[
"Conclusion",
"In this study, we have investigated the contribution of the temperature-dependent CVE to the generation and evolution of the hypermagnetic fields and the matter-antimatter asymmetries, in the symmetric phase of the early Universe and in the temperature range $100\\mbox{ GeV} \\le T\\le 10\\mbox{ TeV}$ .",
"The CVE has two possible sources in a vortical plasma, one from the chiralities and the other from the temperature of the particles.",
"The former has been investigated in the literature much more than the latter.",
"Here, we have focused on the latter in the form of transient temperature fluctuations, and have shown its important role in the production and evolution of the hypermagnetic fields and the matter-antimatter asymmetries.",
"The transient fluctuations that we have considered are in the form of sharp Gaussian shaped pulses.",
"In particular, we have shown that small simultaneous and transient fluctuations of vorticity about zero background value and temperature of some matter degrees of freedom about the equilibrium temperature of the plasma, close to the EWPT, can generate strong hypermagnetic fields and large matter-antimatter asymmetries, even in the absence of any initial seed for the hypermagnetic field or any initial matter-antimatter asymmetries.",
"Furthermore, we have shown that, an increase in the amplitude of temperature or vorticity fluctuations leads to the production of stronger hypermagnetic fields, and therefore, larger matter-antimatter asymmetries.",
"This outcome has not been observed in any of the previous studies.",
"In some studies which only take the CME into account, either an initially strong hypermagnetic field produces matter-antimatter asymmetries, or initial large matter-antimatter asymmetries strengthen a preexisting seed of hypermagnetic field [17], [18], [19].",
"In some other studies which also include the CVE and assume large initial chiralities in the vortical plasma, a seed of the hypermagnetic field is produced which then grows due to the CME [20].",
"In this work, we have considered a simple monochromatic helical configuration for the vorticity and hypermagnetic fields with a negative helicity, which ensures the production of the desired positive matter-antimatter asymmetries.",
"Furthermore, we have shown that, either an increase in the amplitude of the temperature or vorticity fluctuations, or a decrease in their widths leads to the production of stronger hypermagnetic fields, and therefore, larger matter-antimatter asymmetries.",
"We have also shown that fluctuations in vorticity are several orders of magnitude more productive than constant vorticity.",
"Within our model, the temperature-dependent CVE is the dominant effect as compared to the CME.",
"We have also shown that when there are two sets of successive pulses with opposite temperature profiles, the first pulse is the main determinant of the outcome.",
"A generalization of this work would be a stochastic analysis of fluctuations of both temperature and vorticity."
],
[
"APPENDIX A",
"In this appendix we present the expressions for the chiral vorticity and helicity coefficients, $c_{\\mathrm {v}}$ and $c_{\\mathrm {B}}$ given by Eqs.",
"(REF ), and (REF ).",
"First, we start with the relevant and well known expressions in the broken phase.",
"In relativistic hydrodynamics, the energy and number currents can flow separately in the presence of dissipative processes, therefore the definition of the flow will not be trivial[64].",
"Some common hydrodynamic frames in which the equations of AMHD may be formulated include the Landau-Lifshitz (or energy) frame [65], the Eckart (or conserved charge/particle) frame [66], and the more recently introduced anomalous “no-drag” frame[67].",
"In the latter, as its name suggests, a stationary obstacle experiences no drag, even when the energy and charge currents are present.",
"In the Landau-Lifshitz frame, the energy-momentum tensor $T^{\\mu \\nu }$ and the total electric current $J^{\\mu }$ for a plasma consisting of a single species of massless fermions of both chiralities are given by $T^{\\mu \\nu }=(\\rho +p)u^{\\mu }u^{\\nu }- p g^{\\mu \\nu }+\\frac{1}{4}g^{\\mu \\nu } F^{\\alpha \\beta } F_{\\alpha \\beta }-F^{\\nu \\sigma }{F^{\\mu }}_{\\sigma }+ \\tau ^{\\mu \\nu },$ $J^{\\mu }=\\rho _{\\mathrm {el}} u^{\\mu }+J^{\\mu }_\\mathrm {cm}+J^{\\mu }_\\mathrm {cv}+\\nu ^{\\mu },$ $J^{\\mu }_\\mathrm {cm}=(Q_\\mathrm {R}\\xi _{\\mathrm {B,R}}+Q_\\mathrm {L}\\xi _{\\mathrm {B,L}})B^{\\mu }=c_{\\mathrm {B}}B^{\\mu },$ $J^{\\mu }_\\mathrm {cv}=(Q_\\mathrm {R}\\xi _{\\mathrm {v,R}}+Q_\\mathrm {L}\\xi _{\\mathrm {v,L}})\\omega ^{\\mu }=c_{\\mathrm {v}}\\omega ^{\\mu },$ where $F_{\\alpha \\beta }=\\nabla _{\\alpha }A_{\\beta }-\\nabla _{\\beta }A_{\\alpha }$ is the field strength tensor, $p$ and $\\rho $ are the pressure and the energy density of the plasma, $\\rho _{\\mathrm {el}}$ is the electric charge density, $u^{\\mu }=\\gamma \\left(1,\\vec{v}/R\\right)$ is the four-velocity of the plasma normalized such that $u^{\\mu }u_{\\mu }=1$ , and $\\gamma $ is the Lorentz factor.",
"Note that the self-consistency of our calculation in which the diagonal Einstein tensor obtained from the FRW metric is used implies that not only should the electromagnetic field density be small compared to the energy density of the Universe [68], but also the bulk velocity should obey the condition $\\left|\\vec{v}\\right| \\ll 1$ , or equivalently $\\gamma \\simeq 1$ and $u^{\\mu }\\simeq (1,\\vec{v}/R)$ .",
"In the above equations, $\\nu ^{\\mu }$ and $\\tau ^{\\mu \\nu }$ denote the electric diffusion current and viscous stress tensor, respectively [3], $Q_{R}$ ($Q_{L}$ ) denotes the electric charges of the right-handed (left-handed) fermions, $B^{\\mu }=(\\epsilon ^{\\mu \\nu \\rho \\sigma }/2R^3)u_{\\nu }F_{\\rho \\sigma }$ is the magnetic field four-vector, and $\\omega ^{\\mu }=(\\epsilon ^{\\mu \\nu \\rho \\sigma }/R^3)u_{\\nu }\\nabla _{\\rho }u_{\\sigma }$ is the vorticity four-vector, with the totally anti-symmetric Levi-Civita tensor density specified by $\\epsilon ^{0123}=-\\epsilon _{0123}=1$ .",
"Furthermore, in the Landau-Lifshitz frame, the CME and CVE coefficients for chiral fermions are given as [3], [47], [49], [48], [45], [46] $\\xi _{\\mathrm {B,R}}=\\frac{Q_\\mathrm {R}\\mu _\\mathrm {R}}{4\\pi ^2}\\Big [1-\\frac{1}{2}\\frac{n_\\mathrm {R}\\mu _\\mathrm {R}}{\\rho +p}\\Big ]-\\frac{1}{24}\\frac{n_\\mathrm {R}T^{2}}{\\rho +p},$ $\\xi _\\mathrm {B,L}=-\\frac{Q_\\mathrm {L}\\mu _\\mathrm {L}}{4\\pi ^2}\\Big [1-\\frac{1}{2}\\frac{n_\\mathrm {L}\\mu _\\mathrm {L}}{\\rho +p}\\Big ]+\\frac{1}{24}\\frac{n_\\mathrm {L}T^{2}}{\\rho +p},$ $\\xi _{\\mathrm {v,R}}=\\frac{\\mu _\\mathrm {R}^{2}}{8\\pi ^2}\\Big [1-\\frac{2}{3}\\frac{n_\\mathrm {R}\\mu _\\mathrm {R}}{\\rho +p}\\Big ]+\\frac{1}{24}T^{2}\\Big [1-\\frac{2n_\\mathrm {R}\\mu _\\mathrm {R}}{\\rho +p}\\Big ],$ $\\xi _{\\mathrm {v,L}}=-\\frac{\\mu _\\mathrm {L}^{2}}{8\\pi ^2}\\Big [1-\\frac{2}{3}\\frac{n_\\mathrm {L}\\mu _\\mathrm {L}}{\\rho +p}\\Big ]-\\frac{1}{24}T^{2}\\Big [1-\\frac{2n_\\mathrm {L}\\mu _\\mathrm {L}}{\\rho +p}\\Big ].$ where $T$ is the temperature and $n_\\mathrm {R}$ ($n_\\mathrm {L}$ ) is the right-handed (left-handed) charge density .The charge density $n$ is the difference between the particle and anti-particle charge densities.",
"As we shall show explicitly in Sec.",
", $\\mu _\\mathrm {R,L}/T\\ll 1$ within our model, i.e., our initial conditions and results.",
"Hence, $n_\\mathrm {R,L}\\simeq \\frac{1}{6}\\mu _\\mathrm {R,L}T^{2}$ and $\\rho =3p\\simeq \\frac{\\pi ^{2}}{30}g^{*}T^4$ , and Eqs.",
"(REF -REF ) may be simplified as follows $\\xi _{\\mathrm {B,R}}\\simeq \\frac{Q_\\mathrm {R}\\mu _\\mathrm {R}}{4\\pi ^2},$ $\\xi _\\mathrm {B,L}\\simeq -\\frac{Q_\\mathrm {L}\\mu _\\mathrm {L}}{4\\pi ^2},$ $\\xi _{\\mathrm {v,R}}\\simeq \\frac{\\mu _\\mathrm {R}^{2}}{8\\pi ^2}+\\frac{1}{24}T^{2},$ $\\xi _{\\mathrm {v,L}}\\simeq -\\frac{\\mu _\\mathrm {L}^{2}}{8\\pi ^2}-\\frac{1}{24}T^{2}.$ Moreover, since $\\mu /T\\ll 1$ , we will consider only the Ohmic part of the diffusion current $\\nu ^{\\mu }=\\sigma [E^{\\mu }+T(u^{\\mu }u^{\\nu }-g^{\\mu \\nu })\\nabla _{\\nu }\\big (\\frac{\\mu }{T}\\big )]$ given by the first term, where $\\sigma =\\sigma _{R}+\\sigma _{L}$ is the electrical conductivity, and $E^{\\mu }=F^{\\mu \\nu }u_{\\nu }$ is the electric field four-vector.",
"In order to carry over these results to the symmetric phase of the early Universe, it suffices to replace the electromagnetic field by the hypercharge gauge field and the electric charges of different particle species by their relevant hypercharges.",
"Taking into account all three generations of leptons and quarks, we can easily obtain the chiral vorticity and helicity coefficients $c_{\\mathrm {v}}$ and $c_{\\mathrm {B}}$ , given by Eqs.",
"(REF ), and (REF ), using Eqs.",
"(REF -REF ).",
"The four-vectors $B^{\\mu }$ , $\\omega ^{\\mu }$ , $a^{\\mu }$ , and $E^{\\mu }$ in the limit $v\\ll 1$ are given below, $B^{\\mu }=\\gamma \\left(\\vec{v}.\\vec{B},\\frac{\\vec{B}-\\vec{v}\\times \\vec{E}}{R}\\right)\\simeq \\left(\\vec{v}.",
"\\vec{B},\\frac{\\vec{B}}{R}\\right)$ $\\omega ^{\\mu }=\\gamma \\left(\\vec{v}.\\vec{\\omega },\\frac{\\vec{\\omega }-\\vec{v}\\times \\vec{a}}{R}\\right)\\simeq \\left(\\vec{v}.\\vec{\\omega },\\frac{\\vec{\\omega }}{R}\\right)$ $a^\\mu =\\gamma \\left(\\vec{v}.\\vec{a},\\frac{\\vec{a}+\\vec{v}\\times \\vec{\\omega }}{R}\\right) \\simeq \\left(\\vec{v}.\\vec{a},\\frac{\\vec{a}}{R}\\right)$ $E^{\\mu }=\\gamma \\left(\\vec{v}.\\vec{E},\\frac{\\vec{E}+\\vec{v}\\times \\vec{B}}{R}\\right) \\simeq \\left(\\vec{v}.\\vec{E},\\frac{\\vec{E}+\\vec{v}\\times \\vec{B}}{R}\\right),$ where $a^\\mu =\\Omega ^{\\mu \\nu }u_\\nu $ is acceleration four-vector, $\\Omega _{\\mu \\nu }=\\nabla _{\\mu }u_\\nu -\\nabla _{\\nu }u_\\mu $ is vorticity tensor, $a^{i}=R \\Omega ^{0i}$ is three vector acceleration.",
"We have also used the assumption $\\partial _{t}\\sim \\vec{\\nabla }.\\vec{v}$ in the derivative expansion of the hydrodynamics, so $\\vec{v}\\times \\vec{E}\\simeq v^{2}\\vec{B}$ and we have ignored the terms of $O(v^2)$ ."
],
[
"APPENDIX B",
"In this appendix we start with the basic expression for the anomaly given by Eq.",
"(REF ) and obtain its explicit form which eventually leads to Eqs.",
"(REF ,REF ,REF ).",
"The equations of AMHD consist of energy-momentum conservation, Maxwell's equations and the anomaly relations.",
"These equations may be expressed covariantly as $\\nabla _{\\mu }T^{\\mu \\nu }= 0,$ $\\begin{split}&\\nabla _{\\mu }F^{\\mu \\nu }=J^{\\nu }\\\\&\\nabla _{\\mu }{\\tilde{F}}^{\\mu \\nu }=0\\end{split}$ $\\nabla _{\\mu }j^{\\mu }_\\mathrm {R,L}=C_\\mathrm {R,L} E_{\\mu }B^{\\mu },$ where ${\\tilde{F}}^{\\mu \\nu }=\\frac{1}{2R^3}\\epsilon ^{\\mu \\nu \\rho \\sigma }F_{\\rho \\sigma }$ is the dual field strength tensor, $C_{R,L}$ are the corresponding right- and left- handed anomaly coefficients, and $j^{\\mu }_{R,L}$ are the fermionic currents given as follows $\\begin{split}&j^{\\mu }_\\mathrm {R}=n_\\mathrm {R} u^{\\mu }+\\xi _{\\mathrm {B,R}}B^{\\mu }+\\xi _{\\mathrm {v,R}}\\omega ^{\\mu }+V^{\\mu }_{R},\\\\&j^{\\mu }_\\mathrm {L}=n_\\mathrm {L} u^{\\mu }+\\xi _{\\mathrm {B,L}}B^{\\mu }+\\xi _{\\mathrm {v,L}}\\omega ^{\\mu }+V^{\\mu }_{L},\\end{split}$ where $n_\\mathrm {R,L}$ and $V^{\\mu }_\\mathrm {R,L}$ are the chiral charge density and chiral particle diffusion current, respectively [3], [47], [49], [48].",
"The anomaly equations given above can be written out as $\\partial _{t}j^{0}_\\mathrm {(R,L)}+ \\frac{1}{R}\\vec{\\nabla }.\\vec{j}_{(R,L)}+ 3Hj^{0}_\\mathrm {(R,L)} =C_\\mathrm {(R,L)} E_{\\mu }B^{\\mu },$ where $j^{0}_{(R,L)}=n_\\mathrm {(R,L)}+\\xi _{\\mathrm {B,(R,L)}}\\vec{v}.\\vec{B}+\\xi _{\\mathrm {v,(R,L)}}\\vec{v}.\\vec{\\omega }+V^{0}_{R,L}.$ Upon taking the spatial average of Eq.",
"(REF ), the boundary term vanishes and we obtainNote that $E_{\\mu }B^{\\mu }=(\\vec{v}.\\vec{E})(\\vec{v}.\\vec{B})-(\\vec{E}+\\vec{v}\\times \\vec{B}).",
"(\\vec{B}-\\vec{v}\\times \\vec{E})= -\\vec{E}.\\vec{B}+O(v^2)$ , and we have ignored the terms of $O(v^2)$ .",
"$\\begin{split}\\partial _{t}n_\\mathrm {(R,L)}+ 3H n_\\mathrm {(R,L)} =&-\\Big [\\partial _{t}+3H\\Big ]\\Big [\\xi _\\mathrm {B,(R,L)}\\langle \\vec{v}.\\vec{B}\\rangle +\\xi _\\mathrm {v,(R,L)}\\langle \\vec{v}.\\vec{\\omega }\\rangle +\\frac{\\sigma _\\mathrm {R,L}}{Q_\\mathrm {R,L}}\\langle \\vec{v}.\\vec{E}\\rangle \\Big ] \\\\&-C_{R,L} \\langle \\vec{E}.\\vec{B}\\rangle .\\end{split}$ Using the relation $\\dot{s}/s=-3H$ , the anomaly equation reduces to the simplified form $\\begin{split}\\partial _{t}\\left(\\frac{n_\\mathrm {(R,L)}}{s}\\right)= &-\\partial _{t}\\Big [\\frac{\\xi _\\mathrm {B,(R,L)}}{s}\\langle \\vec{v}.\\vec{B}\\rangle +\\frac{\\xi _\\mathrm {v,(R,L)}}{s}\\langle \\vec{v}.\\vec{\\omega }\\rangle +\\frac{\\sigma _\\mathrm {R,L}}{Q_\\mathrm {R,L}s}\\langle \\vec{v}.\\vec{E}\\rangle \\Big ]-\\frac{C_\\mathrm {R,L}}{s} \\langle \\vec{E}.\\vec{B}\\rangle .\\end{split}$ It is worth mentioning that since $(\\partial _{t}+3H)\\langle \\vec{A}.\\vec{B}\\rangle =-2\\langle \\vec{E}.\\vec{B}\\rangle $ , Eq.",
"(REF ) can be rewitten in the following form $\\begin{split}\\partial _{t}\\Big [\\eta _{R,L}+ \\frac{\\xi _{\\mathrm {B,(R,L)}}}{s}\\langle \\vec{v}.\\vec{B}\\rangle +\\frac{\\xi _{\\mathrm {v,(R,L)}}}{s}\\langle \\vec{v}.\\vec{\\omega }\\rangle +\\frac{\\sigma _{(R,L)}}{Q_{(R,L)}s}\\langle \\vec{v}.\\vec{E}\\rangle -\\frac{C_\\mathrm {R,L}}{2s}\\langle \\vec{A}.\\vec{B}\\rangle \\Big ]=0,\\end{split}$ which clearly reveals the conservation of a generalized charge [7], [47].",
"Using Eqs.",
"(REF , REF -REF ) and $x=(T_\\mathrm {EW}/T)^2=t/t_\\mathrm {EW}$ , we obtain $\\begin{split}\\frac{d\\eta _\\mathrm {R}}{dx}=&\\frac{1}{\\Big [1+\\frac{6Q_\\mathrm {R}}{4\\pi ^{2}}\\frac{\\langle \\vec{v}.\\vec{B}\\rangle }{10^{20}G}\\frac{x}{5000}-\\frac{36M}{4\\pi ^{2}}\\eta _\\mathrm {R}kv^{2}\\Big ]}\\Bigg [-\\frac{6Q_\\mathrm {R}}{4\\pi ^{2}}\\frac{x}{5000}\\frac{\\eta _\\mathrm {R}}{10^{20 }G}\\Big [\\langle \\vec{v}.\\partial _{x}\\vec{B}\\rangle +\\langle \\frac{\\vec{v}.\\vec{B}}{x}\\rangle +\\langle \\vec{B}.\\partial _{x}\\vec{v}\\rangle \\Big ]\\\\&+\\Big [\\frac{36M}{4\\pi ^{2}}\\eta _\\mathrm {R}^{2}+\\frac{1}{12M}\\Big ]k\\vec{v}.\\partial _{x}\\vec{v}-\\frac{x}{50MQ_\\mathrm {R}10^{20} G}\\Big [\\langle \\vec{v}.\\partial _{x}\\vec{E}\\rangle +\\langle \\frac{\\vec{v}.\\vec{E}}{x}\\rangle +\\langle \\vec{E}.\\partial _{x}\\vec{v}\\rangle \\Big ]\\\\&-\\frac{t_\\mathrm {EW}C_\\mathrm {R}}{s}\\langle \\vec{E}.\\vec{B}\\rangle \\Bigg ],\\end{split}$ $\\begin{split}\\frac{d\\eta _\\mathrm {L}}{dx}=&\\frac{1}{\\Big [1-\\frac{6Q_\\mathrm {L}}{4\\pi ^{2}}\\frac{\\langle \\vec{v}.\\vec{B}\\rangle }{10^{20}G}\\frac{x}{5000}+\\frac{36M}{4\\pi ^{2}}\\eta _\\mathrm {L}kv^{2}\\Big ]}\\Bigg [\\frac{6Q_\\mathrm {L}}{4\\pi ^{2}}\\frac{x}{5000}\\frac{\\eta _\\mathrm {L}}{10^{20 }G}\\Big [\\langle \\vec{v}.\\partial _{x}\\vec{B}\\rangle +\\langle \\frac{\\vec{v}.\\vec{B}}{x}\\rangle +\\langle \\vec{B}.\\partial _{x}\\vec{v}\\rangle \\Big ]\\\\&-\\Big [\\frac{36M}{4\\pi ^{2}}\\eta _\\mathrm {L}^{2}+\\frac{1}{12M}\\Big ]k\\vec{v}.\\partial _{x}\\vec{v}-\\frac{x}{50MQ_\\mathrm {L}10^{20} G}\\Big [\\langle \\vec{v}.\\partial _{x}\\vec{E}\\rangle +\\langle \\frac{\\vec{v}.\\vec{E}}{x}\\rangle +\\langle \\vec{E}.\\partial _{x}\\vec{v}\\rangle \\Big ]\\\\&-\\frac{t_\\mathrm {EW}C_\\mathrm {L}}{s}\\langle \\vec{E}.\\vec{B}\\rangle \\Bigg ],\\end{split}$ where $M=2\\pi ^{2}g^{*}/45$ .",
"Here the electric field $\\vec{E}$ is not an independent variable and is determined by Eq.",
"(REF ).",
"From the approximation of a vanishing displacement current, we have $\\partial _{x}\\vec{E}=-\\vec{ E}/x$ .",
"As is well known, in the symmetric phase of the early Universe $T>100$ GeV, and in Sec.",
"we show that the condition $\\mu /T\\le 10^{-6}$ holds within our model.",
"These conditions, in conjunction with the low-velocity limit considered in this work, highly suppress the contributions of the temporal components of the CVE, CME and diffusion currents.",
"To ascertain this claim, we have solved Eqs.",
"(REF )-(REF ) both with and without the additional temporal components and present the differences in Fig.",
"REF .",
"Comparing the scales of this figure and Fig.REF , we observe the following for the differences: $\\Delta \\eta /\\eta \\sim 10^{-4}$ and $\\Delta B_Y/B_Y\\sim 10^{-9}$ .",
"Figure: Time plots of the differences between the results with and without the inclusion of the temporal components of the CVE, CME and diffusion currents given by the last three terms of Eq.",
"(): (a) the right-handed electron asymmetry Δη e R \\Delta \\eta _{e_{R}}, (b) the left-handed electron asymmetry Δη e L \\Delta \\eta _{e_{L}}, (c) the baryon asymmetry Δη B \\Delta \\eta _{B}, and (d) the hypermagnetic field amplitude ΔB Y \\Delta B_{Y}.",
"The initial conditions are: k=10 -7 k=10^{-7}, B Y (0) =0B_{Y}^{(0)}=0, η e R (0) =η e L (0) =η B (0) =0\\eta _{e_R}^{(0)}=\\eta _{e_L}^{(0)}=\\eta _{B}^{(0)}=0, β 0 =5×10 -4 \\beta _{0}=5\\times 10^{-4}, v 0 =10 -5 v_0=10^{-5}, b=2×10 -4 b=2\\times 10^{-4}, and x 0 =45×10 -5 x_{0}=45\\times 10^{-5}. \""
]
] | 2001.03499 |
[
[
"QSOR: Quantum-Safe Onion Routing"
],
[
"Abstract In this work, we propose a study on the use of post-quantum cryptographic primitives for the Tor network in order to make it safe in a quantum world.",
"With this aim, the underlying keying material has first been analysed.",
"We observe that breaking the security of the algorithms/protocols that use long- and medium-term keys (usually RSA keys) have the highest impact in security.",
"Therefore, we investigate the cost of quantum-safe variants.",
"These include key generation, key encapsulation and decapsulation.",
"Six different post-quantum cryptographic algorithms that ensure level 1 NIST security are evaluated.",
"We further target the Tor circuit creation operation and evaluate the overhead of the post-quantum variant.",
"This comparative study is performed through a reference implementation based on SweetOnions that simulates Tor with slight simplifications.",
"We show that a quantum-safe Tor circuit creation is possible and suggest two versions - one that can be used in a purely quantum-safe setting, and one that can be used in a hybrid setting."
],
[
"Introduction",
"Nowadays, information available online is expanding in an unforeseen way, a vast amount of data is uploaded and shared through social media, IoT, etc.",
"[14].",
"However, this data also attracts unwanted attention and might paint a bad image of some stakeholders.",
"Consider the case of Edward Snowden who put the National Security Agency (NSA) in the spotlight by shedding light on how the American population was wiretapped [13].",
"When blowing the whistle on such a large scale one would aim to remain anonymous, as this act can negatively affect the career and freedom of the individual.",
"In oppressive regimes, where the freedom of speech is abused, this is even more serious, as any type of negative speech, whistle-blowing or expressing freedom of information may be recognized as an act of treason resulting in severe punishments.",
"The Onion Router (Tor [8]) aims to obfuscate the anonymity of its users when accessing or communicating over the Internet.",
"In principle, when using Tor, the messages or website connection requests are sent through a network of relays and after multiple 'hops' reach their destination.",
"So, if Alice wants to send Bob a message, but does not want an eavesdropper to know that she initiated the contact, Alice can use Tor.",
"The cryptographic schemes used today and in Tor are based on hard mathematical assumptions e.g., Discrete Logarithm Problem and integer factorization [12].",
"These schemes are assumed to be secure against classical adversaries, as solving them with the currently known algorithms cost exponential time.",
"However, with a quantum computer solving these problems become feasible.",
"The transition from current cryptography to post-quantum cryptography needs to be started as soon as possible as quantum computers pose a threat to current and in particular public-key cryptographic algowithms.",
"It is expected that this will have significant effects on IT infrastructure.",
"This is due to the heavier operations quantum-safe cryptography requires for setting up connections [11].",
"Furthermore, network load is also expected to increase as message sizes are bound to get larger due to the increased encryption sizes.",
"Tor is a volunteer run network all across the globe and both the people running the nodes and the users connecting are going to experience drawbacks.",
"In this work, we investigate the main challenges to build and maintain a quantum-safe Tor network.",
"We first examine the different keying material used in Tor and identify the impact of the compromise of each of them.",
"We observe that the migration towards quantum-safe Tor should start with the update of cryptographic algorithms that involve long-term and medium-term keys such as the identity key.",
"Such a migration naturally results in additional cost in terms of CPU and bandwidth.",
"In order to evaluate the actual overhead resulting from the shift to post-quantum cryptographic algorithms, we have conducted an experimental study while considering six different post-quantum public-key encryption algorithms that are part of NIST's round 2 submissionshttps://csrc.nist.gov/Projects/post-quantum-cryptography/round-2-submissions.",
"The implementation of these algorithms is provided by the Open Quantum Safe library [18].",
"We particularly analyze the cost of key generation, key encapsulation and key decapsulation, and compare them with the currently used algorithms deriving from the RSA cryptosystem or elliptic curve cryptography.",
"We observe that each implementation comes with different advantages and limitations, and that consequently there is no ideal solution that offers optimal CPU and bandwidth overhead.",
"We further focus on a Tor network operation, namely circuit building, and compare the cost of the post-quantum variant operation with the original one.",
"This comparative study is performed through a reference implementation based on SweetOnionshttps://github.com/LeonHeTheFirst/SweetOnions, accessed on 28/11/2019, 11:21am that simulates Tor with slight modifications.",
"Additionally, similar to [11], [9], a hybrid implementation combining the use of post-quantum cryptography with classical schemes is also proposed.",
"Such an implementation protects against potential security flaws of quantum-safe schemes due to their recent publications.",
"We show that while the increase in CPU time is acceptable and similar among different implementations, the bandwidth overhead remains significant and the optimal performance is achieved when Sike [10] is used.",
"The onion routing network, Tor [8], is one of the most popular tools to achieve anonymity for web browsing.",
"When a Tor user accesses a website on the Internet, the traffic encrypted with multiple encryption layers is routed across multiple relays.",
"The use of multiple nodes enroute to the destination helps obfuscate the connection of users and hence achieve anonymity: Each node in the path towards the destination (named a circuit), only has information about the previous node and the next node in the path.",
"Messages are encrypted by the source in a layered fashion whereby each encryption layer is removed by one relay node.",
"Nowadays, Tor counts around 6000 nodeshttps://metrics.torproject.org/networksize.html, accessed on 28/11/2019, 4:33pm and the default number of relay nodes to set up a circuit is three (entry node, middle node, exit node)https://trac.torproject.org/projects/tor/wiki/TorRelayGuide, accessed on 28/11/2019, 4:33pm.",
"Each node has to communicate information called descriptors towards Directory Authorities, who maintain a state of the network.",
"The Directory Authorities vote on the status of the network to obtain a consensus document.",
"The user connects to one of the Directory Authorities, fetches the consensus document and the Tor software will select a path from the available nodes.",
"The overall Tor framework is illustrated in Figure REF .",
"Figure: An overview of Tor containing nine directory authorities (DAs), a bridge authority, the consensus document, Tor nodes, the symmetric keys (sk), and the message (msg).The security and privacy features that Tor ensures rely on the use of essential cryptographic primitives such as encryption, digital signatures and key exchange.",
"Consequently, each Tor node receives and maintains multiple cryptographic keys for different purposes.",
"Table REF provides an overview of the asymmetric keys used in Tor with their lifetime and functionalities.",
"Long-term keys are used at least for one year, medium-term keys are used for three to twelve months, and short-term keys have a lifespan of minutes to a maximum of one day.",
"Table: Function of RSA, Curve25519 and Ed25519 keys in Tor."
],
[
"Post-quantum cryptography",
"The security of the current asymmetric encryption and digital signature standards mostly depend on the hardness of integer factorization (RSA) or discrete logarithm (Diffie-Hellman, Elliptic Curve Discrete Logarithm) [12].",
"As described in [16], such cryptographic schemes can be easily broken in polynomial time when using quantum computers.",
"Hence, researchers are actively developing post-quantum cryptographic solutions to resist quantum attacks [4].",
"In 2016, the National Institute of Standards and Technology (NIST) opened a call for proposals on the topic of quantum-safe cryptographic solutions for new quantum-safe standards [15].",
"The first round contained 69 submissions.",
"On January 30, 2019 the candidates for the second round were announced, consisting of 17 asymmetric key encryption and key-establishment algorithms and 9 digital signature algorithms.",
"The transition to quantum-safe cryptographic schemes is expected to be a lengthy process.",
"The adoption of quantum-safe schemes results in a significant increase in bandwidth and computational cost.",
"Developers adopt the hybrid approach whereby currently used standardized cryptographic schemes are combined with quantum-safe schemes."
],
[
"Related work",
"At the time of writing, there are two papers that consider a quantum-safe Tor network, namely [9] and [11].",
"Both solutions mainly focus on the problem of key exchange.",
"In [9], the proposed solution named HybridOR is a customized key exchange protocol.",
"The solution is reported to be computationally more efficient compared to currently used the ntor protocol.",
"HybridOR is assumed to be secure under the ring-Learning With Error (r-LWE) assumption.",
"In [11], the focus is also on securely establishing the short-term keys in a quantum-safe fashion.",
"The currently used ntor protocol is modified and is called Hybrid.",
"Hybrid uses a combination of long-term keys generated by Diffie-Hellman key exchange, and short-term keys generated by a quantum-safe scheme NTRUEncrypt.",
"We observe that existing solutions focus on the problem of key exchange only.",
"Furthermore, their performance study only focuses on the use of one particular quantum-safe cryptographic scheme.",
"For example, Hybrid is evaluated using the NTRUEncrypt algorithm only.",
"Therefore there seems to be a lack of comparative study among different quantum-safe cryptographic primitives.",
"In a quantum world, the users of Tor need to have the same security and anonymity guarantees as they currently have in a classical setting.",
"A quantum-safe Tor network should provide users security and anonymity against quantum adversaries whilst preserving the security and anonymity claims against classical adversaries.",
"Current attack scenarios on Tor do not target the cryptography used in Tor, but aim to exploit other potential weaknesses.",
"However, powerful enough quantum computers will pose a new threat to Tor as cryptography becomes vulnerable for abuse by quantum adversaries."
],
[
"Challenges",
"With quantum computing emerging, the cryptography of Tor needs to be changed, as the quantum vulnerability of asymmetric key cryptography will open a new attack surface for quantum adversaries.",
"Introducing post-quantum cryptography to Tor must be done in order to keep cryptographic vulnerabilities off the list of attack surfaces.",
"It is pivotal to introduce quantum-safe cryptography to the keys of the nodes.",
"Table REF explains the attacker capabilities in case the RSA, Curve25519 or Ed25519 schemes are broken, thereby compromising the keys of the nodes.",
"Table: Attacker capabilities with compromised asymmetric schemes.Current attacks on Tor do not target the cryptography, but rather focus on vulnerabilities in Tor related software, hidden services, bridge node discovery, disabling the network, and on generic attacks like timing.",
"A common technique of adversaries is to introduce new nodes to the Tor network, but this is a lengthy process due to the policy of the network.",
"New nodes are even more closely monitored than nodes already in the network for malicious patterns and if such is recognized, they are excluded from the network."
],
[
"Cryptographic attacks",
"In this section, we consider, attack scenarios on the keys of the nodes that a quantum adversary possesses.",
"There are four types of keys in Tor (See Table REF ) and all of these can be compromised by an attacker: short-term key.",
"Compromising a short-term key at an entry node would make an adversary capable to follow the entire circuit from sender to recipient.",
"This would lead to deanonymization of the user.",
"After ending a TLS connection, the keys are renewed.",
"Therefore, an attack on a short-term key is performed during the lifetime of its TLS connection.",
"medium-term key.",
"In case an adversary compromises the short-term key and the medium-term key of a node, the attacker can impersonate this node.",
"Since a node can decrypt one layer of symmetric encryption when the messages are passed through it, the previous and next `hop' in the circuit are learned by the adversary.",
"The attack has to be performed before the rotation of the medium-term and short-term keys.",
"long-term key.",
"The long-term key may also be compromised by an adversary.",
"This would enable the adversary to impersonate the node and send forged descriptors to the directory nodes.",
"Furthermore, it allows the adversary to gain indefinite, full access to the node.",
"Moreover, the adversary sees previous and consecutive `hops' in the circuit with the encrypted cells.",
"symmetric key.",
"Symmetric keys are used to encrypt the data sent between nodes.",
"In the current implementation of Tor, AES 128-bit is used for the symmetric encryption [7].",
"The key size of this scheme should be doubled in order to remain safe against quantum adversaries.",
"The AES 256-bit scheme is claimed to achieve 128-bit security against quantum adversaries [4].",
"Compromising the symmetric keys enables an adversary to decrypt layers of encryption and learn the destination of the message; anonymity is at risk.",
"In case the attacker learns nothing but the symmetric keys, the encrypted message must be intercepted before entering the network as the TLS connection will add an extra layer of security.",
"If an adversary does not learn all the symmetric keys, but only a subset, then it cannot fully decrypt the message and thus, the circuit is not fully known, so source and destination remain anonymous.",
"Current attacks on Tor are carried out with colluding adversaries.",
"If adversaries control the entry and exit nodes in the network, they can share information with each other and as a result deanonymize communicating parties.",
"Colluding adversaries at the entry and at the exit node who have the medium-term keys will both know the middle relay in a circuit.",
"Sharing this knowledge enables them to attempt to deanonymize users, as the users using the common middle node have the greatest probability to be communicating with each other.",
"Considering Table REF and the lifetime of the asymmetric keys, it is most urgent to update the long-term keys to a quantum-safe alternative.",
"Long-term keys remain unchanged for a long time-period.",
"Hence, an adversary has more time to compromise long-term keys.",
"The effects of compromising long-term keys are also greater, as an adversary can thereby impersonate a node.",
"The second most urgent, is updating the medium-term keys based on the available time period for compromising is smaller.",
"Finally, the short-term keys must be considered, even though the attacker has limited time to compromise these keys due to the security restrictions of Tor.",
"Furthermore, short-term keys are used with TLS, and there are works on making TLS quantum-safe [5].",
"We do stress that it is crucial to update every asymmetric scheme to a quantum-safe alternative in order to enforce the security and anonymity claims of Tor.",
"Lastly, we note that the symmetric keys must be updated to AES 256 bits to prevent `store now, decrypt later'-attacks and ensure that users of Tor maintain life-long anonymity."
],
[
"Impact of post-quantum cryptography on Tor",
"In this section, we investigate the impact that post-quantum cryptography might have on the Tor network, when following the suggested migration strategy in Section REF .",
"The impact of migrating all asymmetric cryptography to a quantum-safe alternative has an impact on the performance (both computational and network) and reliability of Tor.",
"We focus, in particular, on the key exchanges as updating these has the greatest effect on the overall performance and reliability of Tor."
],
[
"Benchmarking post-quantum cryptography",
"We benchmark the post-quantum cryptography schemes that have been implemented in the Open Quantum Safe library [18].",
"The Open Quantum Safe library contains multiple implementations of post-quantum secure key encapsulation and signature schemes.",
"The schemes reach NIST security levels 1 to 5, however we only tested the schemes that achieve level 1 NIST security.",
"The tested schemes are listed in Table REF ."
],
[
"System setup",
"For the experiment, local and virtual environments are both used.",
"The technical specification of the notebook used for the local experiments is Dell Latitude E7240, with Intel Core i5-4310U CPU @ 2.00 - 2.60GHz processor, 8 GB RAM, Samsung SSD SM841N mSATA 128 GB for storage and Windows 10 Enterprise 64-bit operating system.",
"Furthermore, an Ubuntu 18.04 LTS subsystem was installed.",
"In order to emulate the Tor network, 6 virtual machines were used with Intel Core Processor (Broadwell) @ 2.4 GHz processors, 60 GB storage, a virtual network adapter, and Linux version 4.15.0 operating system."
],
[
"Methods and results",
"We performed measurements and obtained benchmarks for the following properties of the encryption schemes: Public key, private key and ciphertext sizes, CPU cycles for RSA key generation, CPU cycles for quantum-safe key generation, CPU cycles for RSA encryption and decryption, CPU cycles for quantum-safe encapsulation and decapsulation.",
"To get an average result for the CPU cycle measurements, 1000 iterations were run with each test, the number of CPU cycles corresponding to one second is 2399753472 cycles.",
"Table REF contains the public key, private key and ciphertext sizes.",
"Table REF contains the number of CPU cycles needed for encapsulation and decapsulation of a message and CPU cycles required for key generation, Figure REF illustrates this in time (ms) for all the schemes tested.",
"Table: The key sizes for RSA and quantum-safe schemes.Key lengths have an effect on network load as they are sent to the Directory Authorities, and the Directory Authorities distribute them to the clients.",
"Larger ciphertexts have a significant detrimental effect on the stability, reliability and performance of a network as an increase in ciphertext size has a direct consequence on network load.",
"We observe that, both Frodo-640-AES and Frodo-640-SHAKE have a problematic ciphertext size of 9720 bytes.",
"From Table REF , we observe that the lattice-based quantum-safe schemes (Kyber, NewHope, NTRU) require less CPU cycles for generating keys than RSA-1024.",
"Kyber and NewHope drastically outperform the other schemes.",
"We also note that the supersingular isogeny-based quantum-safe scheme Sike, even though slightly less performant than RSA-1024, outperforms RSA-2048.",
"Key generation only affects the nodes as they generate keys based on the generation time defined in Tor.",
"The factor that affects both the nodes and the client is the time/computation needed to encapsulate and decapsulate messages.",
"Benchmark encapsulation and decapsulation measurements are presented in Table REF .",
"Opposed to key generation times, where all lattice-based schemes outperformed RSA-1024, we observe that NTRU requires more CPU cycles for encapsulation.",
"However, decapsulation for lattice-based implementations require less CPU cycles than RSA-1024.",
"Sike requires the most CPU cycles for encapsulations and decapsulations, as it is almost 48 times more computationally heavy than RSA-2048.",
"Table: CPU cycles for encapsulation, decapsulation and key generation averaged over 1000 test runs.Figure: CPU cycles of Table converted to time.Lattice-based schemes (Kyber, NewHope, NTRU) have better performance for CPU cycles than the RSA schemes.",
"This suggests that these are the most fit candidates for replacing classical cryptographic schemes.",
"However, based on ciphertext sizes, Sike is the best fit as the ciphertext size fits within one Tor cell (512 bytes)."
],
[
"Impact",
"A migration of classical cryptography to quantum-safe cryptography can have a big effect on the overall availability, reliability, stability and performance of Tor.",
"An important factor to take into account is network load.",
"An increase in the number of packets needed to transfer the ciphertexts of the schemes has a large effect on the network performance.",
"The factor computation time, on the other end, influences the response time to users as an increase in computation time directly influences response time.",
"The impact of computation time will be felt in the performance of Tor.",
"We note that a trade-off has to be made between network load and computation time, when considering the schemes that we tested."
],
[
"Case study: Circuit creation in Tor",
"To investigate the impact of post-quantum cryptography on Tor, we propose to investigate the performance of one particular protocol, namely circuit creation.",
"Our framework uses the SweetOnions implementationhttps://github.com/LeonHeTheFirst/SweetOnions, accessed on 28/11/2019, 11:21am which is a simplified version of the circuit creation protocol used in Tor.",
"We consider two quantum-safe versions of this protocol: a version in which we only use post-quantum cryptography (QSO), and a hybrid version of the protocol (HSO) in which we combine the currently used cryptography with post-quantum cryptography.",
"The reference implementation (SO) that uses standard cryptographic schemes, namely RSA, is also evaluated.",
"We now provide a detailed description of each protocol."
],
[
"Sweet Onion (SO) protocol",
"[htb]Sweet Onion (SO) 0.6cm 0cm -1.3cm 0cm (m)[matrix of nodes, column sep=0.5cm,row sep=8mm, nodes=draw=none, anchor=base west,text depth=1pt, align=left ] Client $(m)$ Node $(pk_N)$ $K_{AES} \\leftarrow _R \\lbrace 0,1\\rbrace ^{256}$ $c \\leftarrow Enc_{AES}(K_{AES},m)$ $ c^{\\prime } \\leftarrow Enc_{RSA}(pk_N, K_{AES})$ $\\xrightarrow{}$ $K_{AES} \\leftarrow Dec_{RSA}(sk_N, c^{\\prime }) $ $m \\leftarrow Dec_{AES}(K_{AES}, c) $ ; [shorten <=-0.1cm,shorten >=-0.1cm] (m-1-1.south east)–(m-1-1.south west); [shorten <=-0.1cm,shorten >=-0.1cm] (m-1-3.south east)–(m-1-3.south west); In the original SweetOnions protocol, defined in Protocol REF for one layer, the client who aims to send a message $m$ to node $N$ , $N\\in \\mathbf {N}$ , encapsulates the symmetric data encryption key using $N$ 's public RSA key $pk_N$ .",
"To set up a circuit the client has to perform these steps with all the nodes in the circuit.",
"Once the client knows the address of every node, this is done in sequence, each node between the client and destination decrypts one layer of encryption and forwards the message."
],
[
"Quantum-safe Sweet Onion (QSO)",
"[htb]Quantum-safe Sweet Onion (QSO) 0.6cm 0cm -1.3cm 0cm (m)[matrix of nodes, column sep=0.5cm,row sep=8mm, nodes=draw=none, anchor=base west,text depth=1pt, align=left ] Client $(m)$ Node $(pk^{PQ}_N)$ $K_{AES} \\leftarrow _R \\lbrace 0,1\\rbrace ^{256}$ $c \\leftarrow Enc_{AES}(K_{AES},m)$ $c^{\\prime } \\leftarrow Enc_{PQC}(pk^{PQ}_N, K_{AES})$ $\\xrightarrow{}$ $K_{AES} \\leftarrow Dec_{PQC}(sk^{PQ}_N, c^{\\prime }) $ $m \\leftarrow Dec_{AES}(K_{AES}, c) $ ; [shorten <=-0.1cm,shorten >=-0.1cm] (m-1-1.south east)–(m-1-1.south west); [shorten <=-0.1cm,shorten >=-0.1cm] (m-1-3.south east)–(m-1-3.south west); The QSO corresponds to the simple quantum-safe variant of SO: the RSA key encapsulation method (KEM) is exchanged with a post-quantum KEM (PQC), see Protocol REF .",
"The public-private key pair of the node consists of post-quantum keys."
],
[
"Hybrid Sweet Onion (HSO)",
"[htb]Hybrid Sweet Onion (HSO) 0.6cm 0cm -1.3cm 0cm (m)[matrix of nodes, column sep=0.5cm,row sep=8mm, nodes=draw=none, anchor=base west,text depth=1pt, align=left ] Client $(m)$ Node $(pk_N,pk^{PQ}_N)$ $K^1_{AES}, K^2_{AES} \\leftarrow _R \\lbrace 0,1\\rbrace ^{256}$ $K_{AES} = K^1_{AES} \\oplus K^2_{AES}$ $c \\leftarrow Enc_{AES}(K_{AES},m)$ $c^{\\prime } \\leftarrow Enc_{RSA}(pk_N, K^1_{AES})$ $c^{\\prime \\prime } \\leftarrow Enc_{PQC}(pk^{PQ}_N, K^2_{AES})$ $\\xrightarrow{}$ $K^1_{AES} \\leftarrow Dec_{RSA}(sk_N, c^{\\prime }) $ $K^2_{AES} \\leftarrow Dec_{PQC}(sk^{PQ}_N, c^{\\prime \\prime }) $ $K_{AES}=K^1_{AES} \\oplus K^2_{AES} $ $m \\leftarrow Dec_{AES}(K_{AES}, c) $ ; [shorten <=-0.1cm,shorten >=-0.1cm] (m-1-1.south east)–(m-1-1.south west); [shorten <=-0.1cm,shorten >=-0.1cm] (m-1-3.south east)–(m-1-3.south west); In the hybrid SweetOnions (HSO) protocol, the RSA KEM is combined with a post-quantum KEM.",
"Hence, the client randomly generates two symmetric encryption keys.",
"The first key is encapsulated with the standard RSA encryption algorithm and the second key is encapsulated with a post-quantum encryption algorithm.",
"The actual data encryption key is the result of a simple XOR of these two symmetric keys.",
"Hence the receiver should perform two decapsulation operations (one with RSA and one with post-quantum decryption)."
],
[
"Experimental results of quantum-safe circuit builds",
"In this section, we evaluate the performance of each protocol in terms of CPU and bandwidth consumption.",
"The size of one Tor packet is 512 bytes.",
"For the reference SO protocol, the underlying encryption algorithms are RSA-2048 and AES-192.",
"For the two other protocols, the post-quantum cryptographic schemes studied in Section REF are used.",
"Experimental results on CPU consumption and bandwidth overhead are given in Table REF .",
"In particular, we evaluate the cost of wrapping the layers of encryption, decapsulating one layer of encryption, and the overall circuit creation.",
"The table also includes the size of one message and the number of packets needed for this protocol.",
"Table: The CPU cycles needed for building a circuit (averaged over 1000 test runs) and message sizes."
],
[
"Quantum-safe Sweet Onion (QSO) experimental results",
"We observe that QSO based on all lattice-based post-quantum schemes outperforms the original SO.",
"On the other hand, while the integration of Sike increases the overall time significantly, the bandwidth overhead is very close to SO.",
"To summarize, a lattice-based scheme may be considered as a potential cryptographic primitive for circuit building since these schemes require less CPU cycles compared to Sike.",
"Nevertheless, the use of lattice-based schemes significantly increases the number of packets and network load compared to Sike.",
"Therefore, depending on the original communication cost, one can decide whether to choose Sike or a lattice-based PQC."
],
[
"Hybrid Sweet Onion (HSO) experimental results",
"When using the hybrid scheme, we observe that both the computational cost and the bandwidth increases significantly.",
"This is mainly due to the fact that HSO uses one RSA encapsulation and one encapsulation with PQC.",
"Consequently, the cost originating from PQC for lattice-based cryptographic schemes becomes negligible when combined with RSA.",
"Even though CPU consumption remains affordable in the hybrid implementation, the bandwidth overhead is important.",
"The number of packets is at least doubled when switching to the hybrid solutions."
],
[
"Conclusion",
"In this paper, we investigated the main challenges to develop a quantum-safe Tor network and focused on the algorithms that use long-term and medium-term keys.",
"Experimental studies show that among the six post-quantum cryptographic scheme evaluated, there is no single winning solution.",
"Nevertheless, given the current status of the NIST standardisation process, Sike seems the most optimal one when it comes to assessing the communication overhead.",
"As for future work, it may be interesting to test other schemes such as the code-based BIKE.",
"Testing the remaining lattice and isogeny-based schemes is also an interesting future topic as they might have better performance measurements than the ones currently available in the Open Quantum Safe library.",
"As for field experiments, an implementation of Tor called TorLAB [17] is available and simulates Tor on a private network of Raspberry PIs.",
"It would be beneficial to re-create the network and extend the measurements of our research to the network load.",
"This would ensure a more realistic study for the evaluation of expected circuit build times, since in the current setting, network latency is omitted."
]
] | 2001.03418 |
[
[
"Improving Image Autoencoder Embeddings with Perceptual Loss"
],
[
"Abstract Autoencoders are commonly trained using element-wise loss.",
"However, element-wise loss disregards high-level structures in the image which can lead to embeddings that disregard them as well.",
"A recent improvement to autoencoders that helps alleviate this problem is the use of perceptual loss.",
"This work investigates perceptual loss from the perspective of encoder embeddings themselves.",
"Autoencoders are trained to embed images from three different computer vision datasets using perceptual loss based on a pretrained model as well as pixel-wise loss.",
"A host of different predictors are trained to perform object positioning and classification on the datasets given the embedded images as input.",
"The two kinds of losses are evaluated by comparing how the predictors performed with embeddings from the differently trained autoencoders.",
"The results show that, in the image domain, the embeddings generated by autoencoders trained with perceptual loss enable more accurate predictions than those trained with element-wise loss.",
"Furthermore, the results show that, on the task of object positioning of a small-scale feature, perceptual loss can improve the results by a factor 10.",
"The experimental setup is available online: https://github.com/guspih/Perceptual-Autoencoders"
],
[
"Introduction",
"Autoencoders have been in use for decades [1], [2] and are prominently used in machine learning research today [3], [4], [5].",
"Autoencoders have been commonly used for feature learning and dimensionality reduction [6].",
"The reduced dimensions are referred to as the latent space, embedding, or simply as $z$ .",
"However, autoencoders have also been used for a host of other tasks like generative modeling [7], denoising [8], generating sparse representations [9], anomaly detection [10] and more.",
"Traditionally, autoencoders are trained by making the output similar to the input.",
"The difference between output and input is quantified by a loss function, like Mean Squared Error (MSE), applied to the differences between each output unit and their respective target output.",
"This kind of loss, where the goal of each output unit is to reconstruct exactly the corresponding target, is known as element-wise loss.",
"In computer vision, when the targets are pixel values, this is known as pixel-wise loss, a specific form of element-wise loss.",
"A problem related to element-wise loss is that it does not take into account relations between the different elements; it only matters that each output unit is as close as possible to the corresponding target.",
"This problem is visualized in Fig.",
"REF , where the first reconstruction with loss $1.0$ is the correct image shifted horizontally by one pixel, and the second reconstruction with lower loss has only one color given by the mean value of the pixels.",
"While a human would likely say that the first reconstruction is more accurate, pixel-wise loss favors the latter.",
"This is because, for a human, the pattern is likely more important than the values of individual pixels.",
"Pixel-wise loss does, on the other hand, only account for the correctness of individual pixels.",
"Figure: A striped image and two reconstructions with their respective element-wise Mean Squared Error.",
"In the first reconstruction each stripe has been moved one pixel to the side and the other is completely gray.Another problem with element-wise loss is that all elements are weighted equally although some group of elements may be more important, for example when solving computer vision tasks like object detection.",
"This problem is visualized in Fig.",
"REF where an otherwise black and white image has a small gray feature.",
"Despite being perceived as important by humans, element-wise loss gives only a small error for completely omitting the gray feature.",
"This is because element-wise loss considers each element to have the same importance in reconstruction, even though some elements might represent a significant part of the input space.",
"Figure: An image and three reconstructions with their respective element-wise Mean Squared Error.",
"The first is the original image.",
"The second is missing the gray feature.",
"The third has four black pixels removed.",
"In the last the gray feature have been moved one pixel up and one pixel to the right.One method that has been used to alleviate these problems for image reconstruction and generation is perceptual loss.",
"Perceptual loss is calculated using the activations of an independent neural network, called the perceptual loss network.",
"This type of loss was introduced for autoencoders in [11].",
"Despite the success of perceptual loss for autoencoders when it comes to image reconstruction and generation, the method has yet to be tested for its usefulness for maintaining information in embedded data in the encoding-decoding task itself.",
"This work investigates how training autoencoders with perceptual loss affects the usefulness of the embeddings for the tasks of image object positioning and classification.",
"This is done by comparing autoencoders and variational autoencoders (VAE) trained with pixel-wise loss to those trained with perceptual loss."
],
[
"Contribution",
"This work shows that, on three different datasets, embeddings created by autoencoders and VAEs trained with perceptual loss are more useful for solving object positioning and image classification than those created by the same models trained with pixel-wise loss.",
"This work also shows that if an image has a small but important feature, the ability to reconstruct this feature from embeddings can be greatly improved if the encoder has been trained with perceptual loss compared to pixel-wise loss."
],
[
"Related Work",
"The VAE is an autoencoder architecture that has seen much use recently [12].",
"The encoder of a VAE generates a mean and variance of a Gaussian distribution per dimension instead of an embedding.",
"The embedding is then sampled from those distributions.",
"To prevent the model from setting the variance to 0 a regulatory loss based on Kullback-Liebler (KL) divergence [13] is used.",
"This regulatory loss is calculated as the KL divergence between the generated Gaussian distributions and a Gaussian with mean 0 and variance 1.",
"Through this the VAE is incentivized not only to create embeddings that contain information about the data, but that these embeddings closely resemble Gaussians with mean 0 and variance 1.",
"The balance between the reconstruction and KL losses incentivizes the VAE to place similar data close in the latent space which means that if you sample a point in the latent space close to a know point those are likely to be decoded similarly.",
"This sampling quality of the VAE makes it a good generative model in addition to its use as feature learner.",
"The VAE has been combined with another generative model, the GAN [14] to create the VAE-GAN [11].",
"In order to overcome problems with the VAE as generative model the VAE-GAN adds a discriminator to the architecture, which is trained to determine if an image have been generated or comes from the ground truth.",
"The VAE is then given an additional loss for fooling the discriminator and a perceptual loss by comparing the activations of the discriminator when given the ground truth to when it is given the reconstruction.",
"This means that the discriminator network is also used as a perceptual loss network in the VAE-GAN.",
"While the VAE-GAN was the first autoencoder to use perceptual loss, it was not the first use of perceptual loss.",
"Perceptual loss was introduced by the field of explainable AI as a way to visualize the optimal inputs for specific classes or feature detectors in a neural network [15], [16].",
"Soon after, GAN were introduced which used perceptual loss to train a generator network to fool a discriminator network.",
"In order to use perceptual loss without the need for training a discriminator, which can be notoriously difficult, [17] proposed using image classification networks in their place.",
"In that work AlexNet [18] is used as perceptual loss network.",
"While the use of perceptual loss has been primarily to improve image generation it has also been used for image segmentation [19], object detection [20], and super-resolution [21]."
],
[
"Perceptual Loss",
"Perceptual loss is in essence any loss that depends on some activations of a neural network beside the machine learning model that is being trained.",
"This external neural network is referred to as the perceptual loss network.",
"In this work perceptual loss is used to optimize the similarity between the image and its reconstruction as perceived by the perceptual loss network.",
"By comparing feature extractions of the perceptual loss network when it's given the original input compared to the recreation, a measure for the perceived similarity is created.",
"This process is described in detail below.",
"Given an input $X$ of size $n$ an autoencoder can be defined as a function $\\hat{X} = a(X)$ where $\\hat{X}$ is the reconstruction of $X$ .",
"Given a loss function $f$ (like square error or cross entropy) the element-wise loss for $a$ is defined as: $E = \\sum ^n_{k=1} f(X_k, a(X)_k)$ Given a perceptual loss network $y = p(X)$ where $y$ is the relevant features of size $m$ the perceptual loss for $a$ is defined as: $E = \\sum ^m_{k=1} f(p(X)_k, p(a(X))_k)$ Optionally the average can be used instead of the sum.",
"This work, like a previous work [17], uses AlexNet [22] pretrained on ImageNet [23] as perceptual loss network ($p$ ).",
"For this work, feature extraction is done early in the convolutional part of the network since we are interested in retaining positional information which would be lost by passing through too many pooling or fully-connected layers.",
"With that in mind feature extraction of $y$ from AlexNet is done after the second ReLU layer.",
"To normalize the output of the perceptual loss network a sigmoid function was added to the end.",
"The parts of the perceptual loss network that are used are visualized in Fig.",
"REF .",
"Figure: The parts of a pretrained AlexNet that were used for calculating and backpropagating the perceptual loss."
],
[
"Datasets",
"This work makes use of three image datasets each with a task that is either object positioning or classification.",
"These three are a collection of images from the LunarLander-v2 environment of OpenAI Gym [24], STL-10 [25], and SVHN [26]."
],
[
"LunarLander-v2 collection",
"The LunarLander-v2 collection consists of images collected from 1400 rollouts of the LunarLander-v2 environment using a random policy.",
"Each rollout is 150 timesteps long.",
"The images are scaled down to the size of $64\\times 64$ pixels.",
"The first 700 rollouts are unaltered while all images where the lander is outside the screen have been removed from the remaining rollouts.",
"This process removed roughly $10\\%$ of the images in the latter half.",
"The task of the LunarLander-v2 collection is object positioning, specifically to predict the coordinates of the lander in the image."
],
[
"STL-10",
"The STL-10 dataset consists of 100000 unlabeled images, 500 labeled images, and 8000 test images of animals and vehicles.",
"The labeled and test images are divided into 10 classes.",
"The task is to classify the images.",
"Specifically the task is to create a model that uses unsupervised learning on the unlabelled images to complement training on the few labelled samples.",
"The images are of size $96\\times 96$ .",
"NOTE: The original task of STL-10 is to only use the provided training data to train a classifier.",
"However, the AlexNet part of the perceptual loss that this work uses has been pretrained on another dataset.",
"Thus, any results achieved by a network trained with that loss cannot be regarded as actually performing the original task of STL-10."
],
[
"SVHN",
"The SVHN dataset consists of images of house numbers where the individual digits have been cropped out and scaled to $32\\times 32$ pixels.",
"The dataset consists of 73257 training images, 26032 testing images, and 531131 extra images.",
"The task is to train a classifier for the digits.",
"The extra images are intended as additional training data if needed."
],
[
"Autoencoder Architecture",
"The autoencoder architecture used in this paper is the same for all datasets and is based on the architecture in [3] and the full architecture can be seen in Fig.",
"REF .",
"The architecture takes input images of size $3\\times 64\\times 64$ or $3\\times 96\\times 96$ .",
"For the SVHN dataset the input to the architecture is each images duplicated into a 2-by-2 grid of $64\\times 64$ pixels.",
"The stride for all convolutional and deconvolutional layers is 2.",
"When training a standard (non-variational) autoencoder $\\sigma $ and the KLD-loss is set to 0.",
"Figure: The convolutional variational autoencoder used in this work."
],
[
"Testing Procedure",
"For each dataset a number of autoencoders were trained with different numbers of latent dimensions.",
"Since the use of autoencoder embeddings to minimize the input for a task is typically helpful when data or labels are limited this work investigates small sizes of the latent space ($\\sim 100$ ).",
"With the actual sizes ($z$ ) tested being 32, 64, 128, 256, 512.",
"Not all values of $z$ were tested for all datasets, with smaller values used for datasets with lower dimensionality.",
"For each size of the latent space two standard autoencoders and two VAEs were trained.",
"One of each with pixel-wise loss (AE and VAE) and one of each with perceptual loss (P. AE and P. VAE).",
"Then for each trained autoencoder a number of predictors were trained to solve the task of that dataset given the embedding as input.",
"There were two types of predictors; (i) Multilayer Perceptrons (MLP) with varying parameters and (ii) linear regressors.",
"The full system including the predictor is shown in Fig.",
"REF .",
"The encoder, $z$ , and decoder make up the autoencoder which is shown in Fig.",
"REF .",
"The autoencoder is either trained with pixel-wise loss given by MSE between $X$ and $\\hat{X}$ , or perceptual loss given by MSE between $y$ and $\\hat{y}$ .",
"The perceptual loss network is the part of AlexNet that is detailed in Fig.",
"REF .",
"The predictor is either a linear regressor or an MLP as described below.",
"Figure: The system used in this work including both the autoencoding and prediction pathways.The MLPs had 1 or 2 hidden layers with 32, 64, or 128 hidden units each and with ReLU or Sigmoid activation functions.",
"The output layer either lacked activation function or used Softmax.",
"The entire search space of hyperparameters were considered with the restriction that the second hidden layer couldn't be larger than the first.",
"Each dataset is divided into three parts: One for training and validating the autoencoders, one for training and validating the predictors, and one for testing.",
"Table REF shows which parts of each dataset are used for what part of the evaluation.",
"Of the autoencoder and predictor parts 80% is used for training and 20% is used for validation.",
"Table: Which parts of the datasets are used for training the autoencoders and predictors, and which is used for final testing.For each trained autoencoder the MLP and regressor with the lowest validation loss was tested using the test set.",
"For the LunarLander-v2 collection the results are the distance between the predicted position and the actual position averaged over the test set and for the other datasets the results are the accuracy of the predictor on classifying the test set.",
"Additionally the decoders of all autoencoders were retrained with pixel-wise MSE loss to see if the reconstructions using perceptually trained embeddings would be better than with pixel-wise trained embeddings."
],
[
"Results",
"The results are broken into seven tables.",
"Tables REF , REF , and REF show the performance of the MLPs with the lowest validation loss on the test sets.",
"Tables REF , REF , and REF show the performance of the regressors on the test sets.",
"For reference the state-of-the-art accuracy, at the time of writing, on STL-10 and SVHN are $94.4\\%$ [27] and $99.0\\%$ [28] respectively.",
"Table REF shows the relative performance on image reconstruction (as measured with the L1 norm) for the best of each type of autoencoder on each dataset.",
"Actual reconstructed images from the LunarLander-v2 collection are visualized in Fig.",
"REF .",
"The image contains the original image and its reconstructions by a pixel-wise and a perceptually trained autoencoder before and after retraining.",
"For this image the reconstruction by the perceptual autoencoder has higher pixel-wise reconstruction error before as well as after retraining of the decoder.",
"Over all tests, the use of perceptual loss added $12\\pm 3\\%$ to the training time of the autoencoders.",
"Since the perceptual loss is only used during autoencoder training it had no effect on the time for inference or training predictors.",
"Table: Average test distance error in pixels for LunarLander-v2 collection for the MLPs with lowest validation loss.Table: Accuracy on STL-10 test set for various z sizes for the MLPs with lowest validation loss.Table: Accuracy on SVHN test set for various z sizes for the MLPs with lowest validation loss.Table: Average test distance error in pixels for LunarLander-v2 collection for the regressors with lowest validation loss.Table: Accuracy on STL-10 test set for various z sizes for the regressors with lowest validation loss.Table: Accuracy on SVHN test set for various z sizes for the regressors with lowest validation loss.Table: Performance of reconstruction of the best autoencoders after retraining as measured by the performance relative to the autoencoder with the lowest L1-norm error.Figure: A comparison of the reconstructed images from pixel-wise and perceptually trained embeddings."
],
[
"Analysis",
"The most prominent result of the experiments is that for all three datasets and all tested sizes of $z$ the perceptually trained autoencoders performed better than the pixel-wise trained ones.",
"Furthermore, for both the LunarLander-v2 collection and STL-10 the pixel-wise trained autoencoder is outperformed significantly.",
"On the LunarLander-v2 collection the perceptually trained autoencoders have an order of magnitude better performance.",
"While the performance on object positioning and classification is better for perceptually trained autoencoders this is not the case with image reconstruction.",
"On LunarLander-v2 collection the perceptual autoencoder is only slightly better at reconstruction.",
"For the other two datasets the pixel-wise trained autoencoders have a much lower relative reconstruction error.",
"Furthermore in Fig.",
"REF the reconstructed image where the lander is actually visible has higher reconstruction loss.",
"This is an actual demonstration of the problems with pixel-wise reconstruction metrics that were visualized in Fig REF and Fig.",
"REF .",
"This lack of correlation between low reconstruction error and performance on a given task is in line with the findings of [29].",
"The results suggest that perceptual loss gives, for the tasks at hand, better embeddings than pixel-wise loss.",
"Taking it even further however, these results combined with earlier work [11] suggests that pixel-wise reconstruction error is a flawed way of measuring the similarity of two images.",
"However, while the results are better for perceptual loss this comes at the cost of training time.",
"While a 12% increase of training time is not a significant amount, especially since training and inference of downstream tasks is not noticeably affected, this increase depends on the perceptual loss network's size in comparison to the size of the remaining model.",
"If the autoencoder is small or the perceptual loss network significantly large the effect on training time could become significant.",
"Another interesting aspect is the difference in performance when switching from MLPs to linear regression.",
"The error of perceptually trained autoencoders on LunarLander-v2 collection is increased by a factor 5 when switching from MLPs to linear regression.",
"This suggests that while the embeddings of the perceptually trained autoencoders contains much more details as to the location of the lander, this information is not encoded linearly which makes linear regressors unable to extract it properly.",
"This is in contrast to STL-10 on which the performance remains roughly the same for both predictor types, which suggests that all the information needed to make class prediction is encoded linearly.",
"On SVHN performance were similar for all autoencoders with MLP predictiors.",
"However, the performance of pixel-wise trained AE and VAE lose 50 and 40 percentage points respectively when using linear regression.",
"This indicates that the autoencoders manage to encode similarly useful information for solving the task but that the pixel-wise trained embeddings demand a non-linear model to extract that information.",
"All this comes together to suggests that perceptually trained autoencoders either have more useful embeddings or the useful information in the embeddings require less computational resources to make use of.",
"The accessibility of the information is important as one of the primary uses of autoencoders is dimensionality reduction to enable the training of smaller predictors for the task at hand.",
"If the information is computationally heavy to access a significant part of the already small model may be dedicated to unpacking it instead of doing prediction.",
"It is important to note the scope of this work.",
"Only three datasets have been investigated, and for only a single perceptual loss network.",
"The work shows that there is promise in investigating this use of perceptual loss, but further studies are needed.",
"Specifically to see if these results generalize to other datasets and perceptual loss networks."
],
[
"Conclusion",
"Element-wise loss disregards high-level features in images which can lead to embeddings that do not encode the features sufficiently well.",
"This work investigates perceptual loss as an alternative to element-wise loss to improve autoencoder embeddings for downstream prediction tasks.",
"The results show that perceptual loss based on a pretrained model produces better embeddings than pixel-wise loss for the three tasks investigated.",
"This work demonstrates that it is important to research on alternatives to element-wise loss and to directly analyze the learned embeddings.",
"Future investigations of perceptual loss should investigate the importance of which perceptual loss network one chooses, how the features are extracted, and in general apply perceptual loss in other domains than images."
]
] | 2001.03444 |
[
[
"Zooming into chaos for a fast, light and reliable cryptosystem"
],
[
"Abstract In previous work, the $k$-logistic map [Machicao and Bruno, Chaos, vol.",
"27, 053116 (2017)] was introduced as a transformation operating in the $k$ less significant digits of the Logistic map.",
"It exploited the map's pseudo-randomness character that is present in its less significant digits.",
"In this work, we comprehensively analyze the dynamical and ergodic aspects of this transformation, show its applicability to generic chaotic maps or sets, and its potential impact on enabling the creation of a cryptosystem that is fast, light and reliable."
],
[
"Introduction",
"The secrecy in chaos-based cryptosystems relies on mathematical transformations that generate a trajectory whose correlation decays rapidly.",
"The correlation of chaotic trajectories will always decay to zero after a sufficiently long time.",
"This is due to the mixing property that allows nearby points to be quickly mapped anywhere in the transformation domain, and due to the sensibility to the initial condition chaotic transformations have.",
"In fact, the speed of correlation decay and the sensibility to the initial conditions quantified by the Lyapunov exponent are intimately connected [1].",
"A chaotic system with a very large positive Lyapunov exponent is thus desirable for cryptography [2], since it allows for very rapid decay of correlations.",
"Moreover, chaotic signals can be generated by low-powered, small area and simple integrated as well as analog circuits operating in very large frequency bandwidths.",
"Cryptosystems need to perform heavy calculations.",
"For example, chaos-based block ciphers [3], [4], [5] such as those that encrypt images, movies and audio employ a series of complex mathematical transformations over too many bits of information.",
"If one wants a light cryptosystem that can be run in any portable devices or that can be considered even for massive streaming, the use of real numbers with higher precision should be avoided.",
"To improve on the performance of chaos-based cryptosystems, the underlying chaotic transformation has been discretized by considering transformations operating on an integer domain.",
"Discretization can preserve important ergodic properties as the mixing property and the sensibility to the initial conditions, but might also create spurious periodic cycles of low-period [4], [6], which result in correlations weakening the security of cryptosystems that rely on these transformations.",
"Even chaotic transformations (such as the Bernoulli shift map) acting on real numbers with finite resolution might create spurious periodic cycles due to numerical errors.",
"With recent advances, it is relatively easy today to perform numerical computation with arbitrary precision, and thus current efficient cryptosystems can rely on maps with real arithmetics of higher precision.",
"However, any meaningful encoding of the chaotic trajectory that allows decoding, such as those used to create a pseudo-random number (PRN) generator or binary secret keys, would be strongly correlated with the most significant digits of the trajectory.",
"To create a stream cipher based on chaos [7], where a binary information stream is encoded by XOR transformation to a binary secret key created by encoding a chaotic trajectory, Gerard Vidal Cassanya [8] has proposed the use of the less significant digits of a trajectory obtained from a higher-dimensional chaotic system of ODEs to create the binary secret key.",
"The idea of using the less significant digits of chaotic trajectories has appeared before in the work of Ref.",
"[9], however it was in Ref.",
"[8] (and other previous patent applications cited within) that less significant digits were taken by a transformation that this work claims to be optimal to support a fast, light and reliable cryptosystem.",
"Inspired by today huge volume of data that needs to be handled in secrecy, there is a desire to develop not only fast (quick run time) and light (little computational cost) but also reliable (highly entropic, sensitive to the initial conditions, low correlation) cryptosystems.",
"An important aspect of a cryptosystem is its initialization.",
"For example, one might employ a PRN to choose parameters.",
"Secret keys, which can be created from PRNs, are also used to encrypt the information and represent a core operation in any cryptosystem.",
"Any innovative invention that creates reliable PRNs or secret keys with optimized use of computational resources will contribute tremendously to a world that wants to communicate massive amounts of information, but securely.",
"PRNs are not only important for secrecy in communication.",
"It is also fundamental to the functioning of several autonomous machines, toys, and they are essential for several numerical algorithms.",
"This work demonstrates that looking at the less significant digits of chaotic trajectories is indeed a pathway for the creation of fast, light and reliable PRNs.",
"The work of Ref.",
"[10] has analyzed the dynamics and the statistical properties of the deep-zoom transformation to a chaotic trajectory, a transformation that takes up the less significant digits of a real number.",
"This transformation applied to the Logistic map regarded as the $k$ -logistic map [10] was defined by the less significant digits located at $k$ digits to the right of the decimal point.",
"It was shown that a PRN based on the $k$ -logistic map has strong properties regarding statistical randomness tests DIEHARD and NIST, and thus demonstrating from a statistical perspective that the $k$ -logistic map can sustain secure cryptosystems.",
"The $k$ -logistic map takes advantage of not only having trajectory points with arbitrarily large precision, and thus within principle no detectable spurious cycle, but also on hiding the information about the most significant digits, which could reveal information about the algorithm behind the generation of the PRNs.",
"The interest in the present work is to understand how the deep-zoom transformation changes a particular map ergodic properties such as its space partition, density measure, Lyapunov exponent, Topological and Shannon's entropies.",
"Our results, mostly illustrated by how the deep-zoom transformation operates into the Logistic map are valid to generic 1D chaotic maps or a set of numbers generated by any other process.",
"The deep-zoom transformation is a complementary operation to chaos-based cryptosystem: we first quickly generate a chaotic trajectory by a low-dimensional map, and then we use the deep-zoom transformation to quickly and lightly enhance security.",
"This is our strategy for the creation of a fast, light and reliable chaos-based cryptosystem.",
"Our first result is to demonstrate that the $k$ -deep-zoom ($k$ -DZ) transformation to a point $x$ is mathematically equivalent to iterating for $k$ times the decimal shift map (DSM) [11], [12].",
"This map is well known, and it is since decades considered to be a mathematical toy model to demonstrate how a shift into the less significant digits results in strong chaos.",
"Despite its tremendous appeal due to the nice way this map deals with decimal digits, scientists working with encryption based on nonlinear transformations have focused their attention on other more known similar maps, such as the Bernoulli shift map [13] or the Baker's map, instead of the DSM.",
"The main difference is whereas the DSM operates by shifting the decimal numbers, Bernoulli shift and Baker's map shift the binary sequence encoding the real numbers of the trajectory.",
"Then, we demonstrate that by applying the $k$ -DZ transformation only once to generic chaotic trajectories, the mapped trajectories will approach a uniform invariant measure for a sufficiently large but in practice small $k$ , thus requiring much less computational effort to create numbers with uniform statistics, a standard requirement for reliable PRNs.",
"The convergence to the uniform invariant measure also dictates the convergence of the Lyapunov exponent (LE) to the Topological and Shanon's entropy of the mapped trajectories, indicating that the transformed points have achieved the largest sensibility to the initial conditions that is possible.",
"Having a trajectory for which the level of chaos is the same as the level of entropy means that uncertainty about the past and the future is as large as one could wish for the particular chaotic map being considered.",
"Moreover, all these quantities are linearly proportional to $k$ , thus implying that randomness (higher entropy) and the sensibility to the initial conditions (large LE) can be trivially increased by the resolution with which a trajectory is observed, and not by increasing a systems dimension or by considering higher-order iterates of the map onto itself, operations that would require computational resources.",
"Throughout this paper, we will show how this map amazing properties applied to any 1D chaotic systems with finite probability measure allows for a clear path to the creation of fast (quick run time, low number of iterations), light (little computational effort, low-dimension) and reliable (uniform statistics, strongly sensitive to the initial conditions, high entropy) pseudo-random numbers or symbolic secret keys, thus supporting fast, light and reliable chaos-based cryptosystems."
],
[
"The $k$ -deep-zoom (k-DZ) transformation and its equivalence to the Decimal shift map (DSM)",
"Given a 1D map $f(x)$ defined in a domain $[a,b]$ and producing an orbit $\\mathcal {O}(x_0)=\\lbrace x_0, x_1,\\ldots , x_t\\rbrace $ generated by the initial condition $x_0$ , with a given invariant density $\\rho (x)$ and probability measure $\\mu (x)$ , such that for an interval $\\epsilon \\in [a,b]$ we have that $\\mu (\\epsilon )=\\int _{x \\in \\epsilon }\\rho (x)dx$ , the $k$ -DZ transformation $\\phi _k(x)$ was defined in [10] by $\\phi _{k}(x) = x 10^{k} - \\lfloor x 10^k \\rfloor \\,,$ where $\\lfloor $ $\\rfloor $ stands for the floor function.",
"In Ref.",
"[10], and motivated mostly for practical reasons, a parameter $L$ was considered which set the number of less significant digits for the function $\\phi _{k}(x)$ .",
"In here, we drop this definition, and assume that $L \\rightarrow \\infty $ , or is a large number.",
"This map can be analogously described by $\\phi _{k}(x) = 10^k(x, \\mod {10^{-k}}) \\,.\\\\\\phi _{k}(x) = 10^k x, \\mod {1} \\,.",
"$ The DSM map is defined by $D(x) = 10 x, \\mod {1},$ and its $k$ -folded version (the $k$ -th iteration of $D$ ) which we represent by $D^k$ is basically $D^k(x) = 10^k x, \\mod {1},$ which is exactly equal to Eq.",
"().",
"Thus, the $k$ -fold DSM map is mathematically equal to the $k$ -DZ transformation.",
"To illustrate the action of the DZ transformation, given the value $x=0.3923481$ , then $\\phi _{k=2}(x)= 0.23481.$"
],
[
"The $k$ -DZ transformation and others maps in literature",
"The idea of using a cryptosystem based on mod transformations that extract the less significant digits of real numbers generated by chaotic systems was to the best of our knowledge first proposed in Ref.",
"[9].",
"Given a real number $x_n$ generated by a chaotic system (discrete or continuous), this work has proposed to cipher $x$ by $R_n \\equiv Ax_n, \\mbox{\\ mod S},$ with $A$ and $S$ representing arbitrary constants.",
"Equation (REF ) can be seen as a particular case of Eq.",
"(REF ), but not of the Eq.",
"() because the k-DZ transformation introduces the extra parameter $k$ .",
"Our work shows when this parameter can generate suitable PRNs.",
"However, in the work of Ref.",
"[9], only the case for $A=10^7$ and $S=256$ was studied, and without the rigour and deepness presented in the present work to study the ergodic and dynamic manifestations of these transformations.",
"The choice of $S=256$ , which turns the map of Eq.",
"(REF ) not equivalent to the DSM map, was made to organize the number $R_n$ into a two-dimensional gray-scale image for further processing.",
"This choice, however, is not optimal for the security of the encryption, measured in terms of the entropy and sensibility to the initial conditions.",
"The optimal choice, demonstrated further, is obtained for $S=1$ , as in Eqs.",
"() or (REF ).",
"The choice made of $A$ as an arbitrary constant is also per se not always optimal to extract the less significant digits, unless this arbitrary constant is of the form $A=10^k$ , as in Eq.",
"()."
],
[
"The Logistic map",
"The logistic map has been extensively studied over the past years [14].",
"Since it is a well know system and that produces typical chaotic behaviour [15], we focus the application of the $k$ -DZ transformation to trajectory points being iterated by the logistic map, which we refer as the $k$ -logistic map, adopting previously defined terminology.",
"It is described by $f(x_{t+1}) = b x_t(1 - x_t) \\,,$ where $x_t \\in [0,1]$ .",
"In Eq.",
"(REF ), $x_t \\in \\Re $ , and as such each trajectory point is assumed to have infinite precision.",
"However, in practice, $x_t$ has finite precision, but this does not prevent one from solving Eq.",
"(REF ) by numerical means.",
"The shadowing lemma [16] guarantees that numerical solutions of this map are stable even if trajectory points have finite resolution, in the sense that the numerical trajectory will remain close to a true trajectory for a very long time, this time depending on the resolution of the trajectory considered."
],
[
"Phase space, partition, and topological entropy",
"Equation (REF ) is useful because it provides the key to calculate the location of the partition points, where the map becomes discontinuous.",
"The points of discontinuities happen at the boundaries created by the mod function, so at multiples of $10^{-k}$ , more specifically at $m10^{-k}$ , for $m \\in \\mathbb {N}$ and $m \\le 10^k$ .",
"There will be then $10^k$ discontinuous intervals.",
"Figure REF show the original Logistic map with $b$ =4 ($k=0$ , left panel), and the corresponding $k$ -DZ transformation for $k=1$ (second panel to the right), $k=2$ (third panel to the right), and $k=3$ (right-most panel).",
"So, the set containing the points $x^*$ where discontinuities appear can be obtained by solving the following equation $x^*(m)= m 10^{-k}.$ We can define a topological entropy of the $k$ -DZ transformation, which is an upper bound for the Shannon-entropy, by the Boltzmann entropy of gas measuring the entropy of it in terms of the number of observable states.",
"Here, we can define the states as being the fall of a trajectory point into an interval within the partition provided by Eq.",
"(REF ).",
"Regardless of the value of $b$ , and actually regardless of which kind of 1D chaotic map is used, this number is given by the number of partition points of the $k$ -DZ transformation and it is equal to $10^k$ , which is also the number of possible symbolic sequences that the $k$ -logistic map produces.",
"It is given by $H_T = k \\ln {(10)}.$ It is useful to compare this result with the topological entropy of the original Logistic map, defined in terms of the number of subintervals in its generating Markov partition, and equal to 2o, where $o \\in \\mathbb {N}$ is the order of the partition representing the resolution of the subintervals composing the partition (measuring $~2^{-o}$ in length).",
"That results in that $H_T=o\\ln {(2)}$ .",
"Here we see an advantage of the use of the $k$ -logistic map to produce efficient pseudo-random numbers in a light fashion, so without requiring too expensive computational resources.",
"Assuming $o$ and $k$ to be of the same order, the topological entropy of the $k$ -logistic map be $\\ln {(10)}/\\ln {(2)}$ larger than that of the Logistic map.",
"It is also worthwhile to compare the result in Eq.",
"(REF ) with the topological entropy, $H_T^{(S)}$ , obtained by applying Eq.",
"(REF ).",
"Defining $A=10^k$ , we obtain that the topological entropy equals $H_T^{(S)} = k \\ln {(10)} - \\ln {(S)}$ for Eq.",
"(REF ).",
"Thus, the entropy achieved in Eq.",
"(REF ) for the $k$ -DZ transformation in Eq.",
"() can only be achieved by applying Eq.",
"(REF ) to that same chaotic set if $S=1$ ."
],
[
"$k$ -logistic map probability density",
"One of the most important characteristics of a good PRN generator is that successive output values of it, say $u_0,u_1,u_2,\\ldots $ are independent random variables from the uniform distribution over the interval [0, 1].",
"It was shown in [10] that as $k$ increases the probability distribution of the map becomes more and more uniform.",
"This is reproduced in Fig.",
"REF , in terms of the histogram (frequency) analysis.",
"As can be seen in this figure, for $k=0$ (a) this distribution is not uniform as it is to be expected from the Logistic map with $b$ =4, with a high probability of finding points close to 1 and 0.",
"As $k$ grows with $k=1$ , $k=2$ , and $k=3$ , the distribution tends to become increasingly uniform, as can be observed in the Figure REF (b).",
"In (c) we show a magnification of (b) for the region close to zero.",
"Figure: Frequency distribution curves for a) the original logistic map, b) the k-k--logistic map with k=1k=1, k=2k=2 and k=3k=3 and parameter b=4b=4.",
"The horizontal axis shows the x t ∈[0,1]x_t \\in [0, 1] (500 bins) and the vertical axis shows the frequency of the 10 4 10^4 values discarding the first 10 3 10^3 transient values.",
"The curves represent the mean and standard deviation (shaded error bar) for sequences generated over 100 random initial conditions.",
"c) The inset plot depicts a zoom on the windows x∈[0,0.03]x \\in [0, 0.03] for these 4 plots."
],
[
"The natural invariant measure of the $k$ -DZ transformation and its Shannon's entropy",
"To calculate the asymptotic Lyapunov exponent of the $k$ -DZ transformation, which is independent on the choice of the chaotic map, we notice that the $k$ -DZ is piecewise linear, wherein each partition sub-interval the map has a constant derivative function.",
"Arranging the values of $x^*(i)$ in Eq.",
"(REF ) in a ranking of crescent order and indicating it by, i.e., $x^*(m) \\equiv x^*_{i}$ such that $x^*_{i+1}>x^*_{i}$ , each partition subinterval comprises the interval $d_{i} = [x^*_{i}, x^*_{i+1}[,$ for $i \\in \\mathbb {N}$ and $i=[0,1, \\ldots , 10^k-1]$ .",
"The derivative of the piecewise-linear map for each sub-interval $d_i$ can be calculated by $\\omega _i = |d_i|^{-1}=10^{-k},$ since $\\phi _k(d_i) = 1$ , where $| d_i |$ represents the length of the sub-partition $d_i$ .",
"The evolution of an arbitrary initial probability measure to a 1D nonlinear transformation is dictated by the Perron-Frobenious operator.",
"For piecewise linear systems, the Perron-Frobenious operator can be cast in terms of a linear system of equations operating in each subinterval of the map partition.",
"The $k$ -DZ transformation takes as the initial measure generated by the nonlinear Logistic map and then applies $k$ times the $DSM$ .",
"If we assume that the measure in each subinterval of the $k$ -DZ is uniform (which initially will be not) and we represent it by the component $[{\\mathbf {\\mu }}]_i$ of the vector ${\\mathbf {\\mu }}$ ($i=\\lbrace 1, \\ldots ,n\\rbrace $ ) with $n= 10^{k}$ , and we define the density in each interval as given by $\\rho _i = \\frac{\\mu _i}{d_i}$ an equation for the evolution of the non-normalized density at iteration $t$ can be obtained [17].",
"${\\mathcal {Z}}{\\mathbf {\\rho }^{\\prime }}^t = {\\mathbf {\\rho }^{\\prime }}^{t+1},$ where the square matrix ${\\mathcal {Z}}$ with component $[{\\mathcal {Z}}]_{ij}$ is the reciprocal of the absolute value of the slope of the map taking the measure from the interval $j$ to the interval $i$ and can be defined by a matrix with equal rows as ${\\mathcal {Z}} =\\left[ {\\begin{array}{ccccc}\\omega _0^{-1} & \\omega _1^{-1} & \\ldots & \\omega _{n-1}^{-1} & \\omega _n^{-1} \\\\\\omega _0^{-1} & \\omega _1^{-1} & \\ldots & \\omega _{n-1}^{-1} & \\omega _n^{-1} \\\\\\multicolumn{5}{c}{\\dotfill } \\\\\\omega _0^{-1} & \\omega _1^{-1} & \\ldots & \\omega _{n-1}^{-1} & \\omega _n^{-1} \\\\\\end{array} } \\right].$ Equation (REF ), representing how the measure evolves concerning only 10$^{k}$ intervals are valid if the measure and the density is uniform for every sub-partition $d_i$ since it has been derived from the continuous Perron-Frobenious operator integrated over intervals where the measure was assumed to be constant.",
"If the initial measure is not uniform for each subpartition interval, as it is the case since the initial measure was generated by the logistic map, we either should consider the continuous operator (effectively described by an infinite dimension matrix) or alternatively, we can adopt a much simpler strategy.",
"We take Eq.",
"(REF ) and study it in the limit, when $t \\rightarrow \\infty $ .",
"Defining the vector $\\mathbf {d}=\\lbrace d_1,d_2,\\ldots ,d_n\\rbrace $ and the diagonal matrix $D=\\mathbb {I} \\mathbf {d}$ , the element $[]_{ij}$ of the matrix $D{\\mathcal {Z}}D^{-1}$ represents the percentage of the measure in the interval $d_j$ that goes to the interval $d_i$ .",
"The matrix ${\\mathcal {Z}}$ has equal rows because the piecewise equivalent of the $k$ -logistic map takes measure from each interval to all others with the same proportion in each of the intervals $d_j$ .",
"The equilibrium point of Eq.",
"(REF ) is obtained when ${\\mathcal {Z}}{\\mathbf {\\rho }^*} = {\\mathbf {\\rho }^*},$ which means that the time invariant density is a normalized eigenvector of ${\\mathcal {Z}}$ .",
"The matrix ${\\mathcal {Z}}$ is a stochastic matrix, since it is a non-negative matrix and the sum of all elements in a row totals 1.",
"This is easy to see since $\\sum _i \\omega ^{-1}_i = \\sum _i d_i = 1.$ The Perron-Frobenious theorem guarantees that a square stochastic matrix has a unique dominant real unitary eigenvalue, with all other eigenvalues smaller than 1.",
"This means that the density of the $k$ -DZ transformation in the limit of $k \\rightarrow \\infty $ is natural (it is unique), regardless of the initial probability measure that is fed into the $k$ -DZ transformation.",
"The natural density can be recovered by proper normalization dividing ${\\mathbf {\\rho }^*}$ by $\\sum _i [{\\mathbf {\\rho }^*}]_id_i$ so that the physical natural density in each interval is given by $[\\rho ]_i = \\frac{{[\\mathbf {\\rho }^*]_i}}{\\sum _i [{\\mathbf {\\rho }^*}]_id_i}.$ This is to guarantee that the density produces the natural measure by Eq.",
"(REF ).",
"It is also easy to see that the unique unitary eigenvalue has associated to it a uniform eigenvector with all components equal to a constant value $c$ : ${\\mathbf {\\rho }}^* = [c\\, c\\, c\\, \\ldots , c]^T$ , so, the piecewise $k$ -DZ transformation has a uniform density given by $[\\rho ]_i = \\frac{{c}}{\\sum _i [c]d_i}=1.$ This leads us to an invariant natural measure in each interval that equals the Lebesgue measure of the interval, and thus $\\mu _i = d_i=10^{-k}.$ So, for sufficiently large $k$ , it is to be expected that the $k$ -logistic map will have a uniform natural invariant density, although the density of the Logistic map is not uniform for each interval.",
"In practice, this sufficiently large number is around $k$ =4, when this map generates PRNs with all the good statistical characteristics for security [10].",
"Being invariant means that any initial probability measure will eventually evolve to the same invariant measure.",
"Thus, the reliability of the security for the PRNs generated by the $k$ -DZ transformation is substantially more dependable on the properties of the DSM, than on the statistical properties of the chaotic set of points being iterated by the $k$ -DZ transformation, or also on the chaotic map considered to initially generate the chaotic trajectory to be fed into the $k$ -DZ transformation.",
"Since the invariant measure of the $k$ -logistic map is constant (for sufficiently large $k$ ), this means that any encoding supported by the partition defined in Eq.",
"(REF ) will produce equiprobable symbols, this rendering cryptoanalysis based on frequency statistics to be inappropriate.",
"The asymptotic Shannon's entropy of the $k$ -DZ transformation is therefore equal to the Topological entropy: $H_{S} = - \\sum _{i=1}^{n} \\mu _i \\ln {\\mu _i} = - \\sum _{i=1}^{n} d_i \\ln {d_i} = H_T$"
],
[
"The Lyapunov exponent of the $k$ -DZ transformation",
"The Lyapunov exponent (LE) of the $k$ -DZ transformation can always be calculated regardless of the chaotic map being used as the generator of the initial measure.",
"This is so because the map is piecewise linear with constant derivative everywhere (except the partition points).",
"The Lyapunov exponent can be calculated by $\\lambda = \\int \\ln {\\left( { \\frac{d\\phi _{k}(x)}{dx} } \\right)} d \\mu $ where $d\\mu =\\rho (x)\\, dx$ , represents the invariant measure of the $k$ -DZ transformation.",
"The chaotic map has its own domain of validity.",
"This domain must be normalized to fit within the domain of the $k$ -DZ transformation.",
"For the Logistic map, the domain is $[0,1]$ , the same as the domain of the $k$ -DZ transformation.",
"Therefore, its LE is equal to $\\lambda = \\int _0^1 \\ln {\\left( { 10^k } \\right)} dx = k\\ln {(10)} = H_T.$ So, we see that for a sufficiently large $k$ , the $k$ -DZ transformation produces a LE that approaches the topological entropy which is also equal to Shannon's entropy.",
"A light cryptosystem that does not require much computational effort demands the use of transformations that can be as entropic as possible and with the largest as possible sensibility to the initial conditions (which implies in a quick decay of correlation).",
"When compared the LE of the $k$ -DZ transformation in Eq.",
"( (result in Eq.",
"(REF )) with the LE of the transformation Eq.",
"(REF ) (proposed in Ref.",
"[9]), assuming $A=10^k$ , we notice that Eq.",
"(REF ) can be equivalently written as $R_n=S\\left( \\frac{10^k}{S}x_n, \\mbox{\\ mod 1} \\right)\\,,$ which can be rewritten as $R_n=S\\Phi (x_n),$ where $\\Phi (x_n)=\\frac{10^k}{S}x_n, \\mbox{\\ mod 1}$ .",
"Noticing that the LE of the function $\\Phi (x_n)$ is the same as the one obtained if $\\Phi (x_n)$ is multiplied by a constant, then the LE of Eq.",
"(REF ) is equal to $\\lambda ^{(S)}=\\ln {\\left( \\frac{10^k}{S} \\right)}=H_T^{(S)}.$ Thus, the LE of Eq.",
"(REF ) is only equal to the one of Eq.",
"(), if $S=1$ .",
"In the result of Eq.",
"(REF ), we have assumed that the speed of convergence of the probability density measure [18] of Eq.",
"(REF ) is the same as the one of Eq.",
"().",
"This is to be expected, since the second largest eigenvalue of the matrix ${\\mathcal {Z}}$ regulating the evolution of the density measure for Eq.",
"() is the same as the one for this matrix regulating the evolution of the density measure for Eq.",
"(REF ), and both are equal to zero."
],
[
"Enhancement of sensibility to the initial conditions of the $k$ -logistic map",
"The LE of the $k$ -DZ transformation does not depend on the choice of the chaotic map generating the measure.",
"It is nevertheless interesting to understand how much chaos is enhanced by the application of the $k$ -DZ transformation into a chaotic map.",
"Considering this chaotic map to be the logistic map (Eq.",
"(REF )), we then want to understand how much chaos is enhanced if the DZ transformation with $k=1$ is applied not to the trajectory points generated by the logistic map, but to the map itself.",
"So, we calculate the Lyapunov exponent of the map $\\phi _{k}(f(x_t))$ , whose state space ($\\phi _{k}(f(x_t)) \\times x_t$ ) is shown in Fig.",
"REF .",
"Additionally, Fig.",
"REF show a colored version of this previous picture for parameters in region $b \\in [3.6, 4]$ .",
"This map is described by $\\phi _{k}(f(x)) = 10^k(f(x_t), \\mod {10^{-k}})\\,.$ Its LE can be calculated by $\\lambda = \\int \\ln {\\left( { \\frac{d\\phi _{k}(f(x))}{dx}} \\right)} d\\mu $ where $d\\mu =\\rho (x)\\, dx$ now represents the measure of the Logistic map.",
"The first derivative of the map in Eq.",
"(REF ) is $\\frac{d\\phi _{k}(x)}{dx} = 10^k b (1-2x),$ whereas its density for $b=4$ is given by $\\rho (x)=\\frac{\\pi ^{-1}}{[x(1-x)]^{1/2}}$ Placing Eqs.",
"(REF ) and (REF ) in Eq.",
"(REF ) and integrating over the map domain ($x \\in [0,1]$ ), we obtain that $\\lambda = \\int _0^1 \\frac{k \\ln {10} + \\ln {4} +\\ln {|(1-2x)|}}{\\pi \\sqrt{x(1-x)}} dx = k \\ln (10) + \\ln (2),$ since $\\int _0^1 \\frac{\\ln {|(1-2x)|}}{\\pi \\sqrt{x(1-x)}} dx = \\ln {2}.$ So, the first thing to notice is that the LE of the map in Eq.",
"(REF ) is equal to the LE of the 1-logistic map plus the LE of the original logistic map for $b=4$ (which is equal to $\\ln {(2)}$ ).",
"This tells us that when creating a cryptosystem based on a chaotic map, more entropy and sensibility to the initial conditions can be achieved by a simple inspection to the least $k$ significant digits, instead of doing more iterations in the chaotic map generating the initial chaotic sequence.",
"This analysis can be easily extended to the logistic map operating under any parameter $b$ that produces chaotic motion.",
"The Lyapunov exponent of the map in Eq.",
"(REF ) can be calculated using the time approach by $\\lambda (b) = \\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\sum _{t=1}^{T} \\ln {10^k b (1-2x_i)},$ which lead us to $\\lambda (b) = k \\ln {10} + \\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\sum _{t=1}^{T} \\ln {b(1-2x_i)},$ and finally to $\\lambda (b) = k \\ln {10} + \\lambda _0(b),$ where $\\lambda _0(b)$ is just the Lyapunov exponent of the Logistic map for the parameter $b$ .",
"Thus, here it is obvious that the gain for sensibility to the initial conditions is trivially achieved by just choosing a sufficiently large $k$ .",
"Once the output of the $k$ -DZ transformation $\\phi _k(x)$ generates real points in the unit interval, these values can be considered directly as a pseudo-random number that can be re-scaled as required.",
"The security analysis of the so-called $k$ -logistic map PRN was analysed in [10], showing high-quality pseudo-random numbers for $k\\ge 4$ through statistical randomness tests such as DIEHARD [19] and NIST [20].",
"Another strategy to generate PRNs is by means of the symbolic representation of the trajectory of the $k$ -DZ transformation.",
"Thus, a partition that is not the natural partition of the $k$ -DZ transformation needs to be considered.",
"This natural partition is given by ${d}$ whose borders are defined by Eq.",
"(REF ).",
"Then, for a given $k$ , there will be $10^k$ symbols for the natural partition.",
"The point $\\phi _k(x_i) \\in [d_i,d_{i+1}]$ is encoded by the $i$ -th symbol of the alphabet ($i=\\lbrace 0,1,\\ldots ,10^k-1\\rbrace $ ), represented by $s_i$ .",
"A transformed trajectory of length $L$ represented by $\\lbrace \\phi _k(x_1), \\phi _k(x_2), \\ldots , \\phi _k(x_L)\\rbrace $ will have the symbolic representation ${s} = \\lbrace s_1, s_2, \\ldots , s_L\\rbrace $ , where $s_i \\in [0,10^k-1]$ .",
"The vector ${s}$ fully represents the information about the location of the points $x_i$ being mapped (within the resolution of the partition cells), and therefore should be avoided for the creation of the secret key.",
"The partition to create a secret key should have a minimal number of intervals, for example a binary partition where $\\phi _k(x_i) < 0.5$ is encoded by `0' and $\\phi _k(x_i) \\ge 0.5$ is encoded by `1'.",
"In this way, points within $x_i \\in [0,1]$ will be encoded with equal probabilities for `0' and `1'."
],
[
"Conclusions",
"Cryptography relies on the application of several transformations to eliminate all existing correlations between the message and its ciphered version.",
"A preliminary requirement for achieving this relies on the use of highly entropic and non-correlated pseudo-random numbers.",
"The sensitivity to the initial conditions property chaotic systems have is the key to this goal.",
"The interest today is to be able to accomplish such a task for reliable encryption but by relying on transformations that require little computational effort (light) and quick running time (fast).",
"In this work, we characterize the properties of the so-called the $k$ -Deep Zoom ($k$ -DZ) to support reliable cryptosystems that uses pseudo-random numbers or secret keys that were created fast and lightly.",
"Besides the Decimal Shift Map (DSM) is not conceptually equivalent to the k-DZ, we show that the k-DZ is mathematically equivalent to the DSM map iterated k times.",
"More than that, we show that the $k$ -fold DSM can be rewritten into a form completely equivalent to the $k$ -DZ transformation.",
"So, all the good properties of the DSM map such as uniform statistics, high entropy, and sensibility to the initial conditions are inherited by the $k$ -DZ.",
"There is a semantic difference between both maps.",
"Whereas the $k$ -DZ transformation effectively represents an algorithm that simply extracts the less significant digits of a real number, the DSM is a map that transforms a point into another point.",
"This semantic interpretation of the DZ-transformation can be in the future exploited for the creation of dedicated electronic chips operating at the hardware level that only work with less significant digits, thus potentially bringing the encryption process to the physical level.",
"We show that the entropy and the Lyapunov exponent is linearly proportional to $k$ .",
"This means that the trivial and light task of peeking onto the sequence of less significant digits positioned $k$ digits to the decimal floating-point is sufficient to drastically increase the entropy and therefore the uncertainty past and future numbers, at a minimal computational cost.",
"Several of the properties of the $k$ -DZ transformation depend only on the map itself, not on the chaotic system being considered as the generator of the original trajectory being encoded, or any other set of numbers being generated by any other process (e.g.",
"stochastic processes).",
"Thus, one might wonder why to use the $k$ -DZ transformation into a chaotic set of numbers after all?",
"The reason is that chaotic trajectories have several advantages.",
"They are easy to be generated and do not require the use of higher-dimensional systems, in both digital or analog domains, they require less algorithmic complexity, less-power electronics, less CPU dedication and can be generated at impressive large bandwidths.",
"Chaos, however, is deterministic and correlation does decay quickly, but not as quickly as one would wish.",
"The additional application of the $k$ -DZ transformations to chaotic trajectories fast and lightly enhances the already existing wished properties of chaos to cryptography.",
"A transformation that optimizes essential ingredients to a secure cryptosystem, but with minimal computational effort.",
"Our strategy to create pseudo-random numbers or secret keys requires the use of a chaotic system whose simulated trajectory is guaranteed to be chaotic for a long period, and that can be additionally generated using minimal computational resources.",
"For this reason, the Logistic map is a good candidate.",
"The $k$ -DZ transformation is then applied a single time to this stable chaotic trajectory.",
"Our claim is that this strategy quickly generates secure and light PRNs.",
"Another strategy to generate secure PRNs, which might increase the computational cost to some extent, was proposed in Ref.",
"[13], where an approximate true trajectory of the Bernoulli map is calculated directly using real algebraic numbers."
],
[
"Acknowledgments",
"J. M. acknowledges a scholarship from the National Council for Scientific and Technological Development (CNPq grant #155957/2018-0).",
"O. M. B. acknowledges support from CNPq (grant #307897/2018-4) and FAPESP (grant #16/18809-9)."
]
] | 2001.03549 |
[
[
"Portable magnetometry for detection of biomagnetism in ambient\n environments"
],
[
"Abstract We present a method of optical magnetometry with parts-per-billion resolution that is able to detect biomagnetic signals generated from the human brain and heart in Earth's ambient environment.",
"Our magnetically silent sensors measure the total magnetic field by detecting the free-precession frequency of highly spin-polarized alkali metal vapor.",
"A first-order gradiometer is formed from two magnetometers that are separated by a 3 cm baseline.",
"Our gradiometer operates from a laptop consuming 5 W over a USB port, enabled by state-of-the-art micro-fabricated alkali vapor cells, advanced thermal insulation, custom electronics, and laser packages within the sensor head.",
"The gradiometer obtains a sensitivity of 16 fT/cm/Hz$^{1/2}$ outdoors, which we use to detect neuronal electrical currents and magnetic cardiography signals.",
"Recording of neuronal magnetic fields is one of a few available methods for non-invasive functional brain imaging that usually requires extensive magnetic shielding and other infractructure.",
"This work demonstrates the possibility of a dense array of portable biomagnetic sensors that are deployable in a variety of natural environments."
],
[
"Supplementary Information",
"` Figure: Data taken for four different subjects showing auditory evoked fields.`"
]
] | 2001.03534 |
[
[
"NWPU-Crowd: A Large-Scale Benchmark for Crowd Counting and Localization"
],
[
"Abstract In the last decade, crowd counting and localization attract much attention of researchers due to its wide-spread applications, including crowd monitoring, public safety, space design, etc.",
"Many Convolutional Neural Networks (CNN) are designed for tackling this task.",
"However, currently released datasets are so small-scale that they can not meet the needs of the supervised CNN-based algorithms.",
"To remedy this problem, we construct a large-scale congested crowd counting and localization dataset, NWPU-Crowd, consisting of 5,109 images, in a total of 2,133,375 annotated heads with points and boxes.",
"Compared with other real-world datasets, it contains various illumination scenes and has the largest density range (0~20,033).",
"Besides, a benchmark website is developed for impartially evaluating the different methods, which allows researchers to submit the results of the test set.",
"Based on the proposed dataset, we further describe the data characteristics, evaluate the performance of some mainstream state-of-the-art (SOTA) methods, and analyze the new problems that arise on the new data.",
"What's more, the benchmark is deployed at \\url{https://www.crowdbenchmark.com/}, and the dataset/code/models/results are available at \\url{https://gjy3035.github.io/NWPU-Crowd-Sample-Code/}."
],
[
"Introduction",
"Crowd analysis is an essential task in the field of video surveillance.",
"Accurate analysis for crowd motion, human behavior, population density is crucial to public safety, urban space design, etc.",
"Crowd counting and localization are fundamental tasks in the field of crowd analysis, which serve high-level tasks, such as crowd flow estimation [1] and pedestrian tracking [2].",
"Due to the importance of crowd counting, many researchers [3], [4], [5] pay attention to it and achieve quite a few significant improvements in this field.",
"Especially, benefiting from the development of deep learning in computer vision, the counting performance on the datasets [6], [7], [8], [9] is continuously refreshed by Convolutional Neural Networks (CNN)-based methods [10], [11], [12].",
"The CNN-based methods need to learn discriminate features from a multitude of labeled data, so a large-scale dataset can effectively promote the development of visual technologies.",
"It is verified in many existing tasks, such as object detection [13] and semantic segmentation [14].",
"However, the currently released crowd counting datasets are so small-scale that most deep-learning-based methods are prone to overfit the data.",
"According to the statistics, UCF-QNRF [9] is the largest released congested crowd counting dataset.",
"Still, it contains only $1,535$ samples, in a total of $1.25$ million annotated instances, which is still unable to meet the needs of current deep learning methods.",
"Moreover, some works [9], [15] focus on the crowd localization task that produces point-wise predictions for each instance.",
"However, the traditional datasets do not contain box-level labels, which makes it hard to evaluate the localization performance using a uniform metric.",
"Furthermore, there is not an impartial evaluation benchmark, which potentially restricts further development of crowd counting.",
"By the way, some methodshttps://github.com/gjy3035/Awesome-Crowd-Counting/issues/78 may use mistaken labels to evaluate models, which is also not accurate.",
"Reviewing some benchmarks in other fields, CityScapes [16] and Microsoft COCO [13], they allow the researchers to submit their results of the test set and impartially evaluate them, which facilitates the study of methodology.",
"Thus, an equitable evaluation platform is important for the community.",
"Considering the problems mentioned above, in this paper, we construct a large-scale crowd counting and localization dataset, named as NWPU-Crowd, and develop a benchmark website to boost the community of crowd analysis.",
"Compared with the existing congested datasets, the proposed NWPU-Crowd has the following main advantages: 1) This is the largest crowd counting and localization dataset, consisting of $5,109$ images and containing $2,133,375$ annotated instances; 2) It introduces some negative samples like high-density crowd images to assess the robustness of models; 3) In NWPU-Crowd, the number of annotated objects range, $0\\sim 20,033$ .",
"More concrete features are described in Section REF .",
"Table REF illustrates the detailed statistics of ten mainstream real-world datasets and the proposed NWPU-Crowd.",
"Table: Statistics of the ten mainstream crowd counting datasets and NWPU-Crowd.Based on the proposed NWPU-Crowd, several experiments of some classical and state-of-the-art methods are conducted.",
"After further analyzing their results, an interesting phenomenon on the proposed dataset is found: diverse data makes it difficult for counting networks to learn useful and distinguishable features, which does not appear or is ignored in the previous datasets.",
"Specifically, 1) there are many error estimations on negative samples; 2) the data of different scene attributes (density level and luminance) have a significant influence on each other.",
"Therefore, it is a research trend on how to alleviate the above two problems.",
"What's more, for localization task, we design a reasonable metric and provide some simple baseline models.",
"In summary, we believe that the proposed large-scale dataset will promote the application of crowd counting and localization in practice and attract more attention to tackling the aforementioned problems."
],
[
"Related Works",
"The existing crowd counting datasets mainly contain two types: surveillance-scene datasets and general-scene datasets.",
"The former commonly records crowd in particular scenarios, of which the data consistency is obvious.",
"For the latter, the crowd samples are collected from the Internet.",
"Thus, there are more perspective variations, occlusions, and extreme congestion in these datasets.",
"Tabel REF demonstrates a summary of the basic information of the mainstream crowd counting datasets, and in the following parts, their unique characters are briefly introduced."
],
[
"Surveillance-scene Dataset",
"Surveillance view.",
"Surveillance-view datasets aim to collect the crowd images in specific indoor scenes or small-area outdoor locations, such as marketplace, walking street, and station.",
"The number of people usually ranges from 0 to 600.",
"UCSD is a typical dataset for crowd analysis.",
"It contains $2,000$ image sequences, which records a pedestrian walk-way at the University of California at San Diego (UCSD).",
"Mall [17] is captured in a shopping mall with more perspective distortion.",
"However, these two datasets contain only a single scene, lacking data diversity.",
"Thus, Zhang et al.",
"[8] build a multi-scene crowd counting dataset, WorldExpo'10, consisting of 108 surveillance cameras with different locations in Shanghai 2010 WorldExpo, e.g., entrance, ticket office.",
"Considering the poor resolution of traditional surveillance cameras, Zhang et al.",
"[7] construct a high-quality crowd dataset, ShanghaiTech Part B, containing 782 images captured in some famous resorts of Shanghai, China.",
"To remedy the occlusion problem in congested scenes, a multi-view dataset is designed by Zhang and Chan [23].",
"By equipping 5 cameras at different positions for a specific view, the data can be recorded synchronously.",
"For getting rid of the manually labeling process, Wang et al.",
"[20] construct a large-scale synthetic dataset (GCC).",
"By simulating the perspective of a surveillance camera, they capture 400 crowd scenes in a computer game (Grand Theft Auto V, GTA V), a total of $15,212$ images.",
"The main advantage of GCC is that it can provide accurate labels (point and mask) and diverse environments.",
"However, there are many domain shifts/gaps between synthetic and real data, limiting their practical values.",
"Therefore, it is necessary to build a large-scale real-world dataset.",
"Compared with GCC, the advantages of NWPU-Crowd are: more natural person models, crowd scenes and environment (weathers, light, etc.).",
"In addition to the aforementioned datasets, there are also other crowd counting datasets with their specific characteristics.",
"SmartCity [24] focuses on some typical scenes, such as sidewalk and subway.",
"ShanghaiTechRGBD [25] records the RGBD crowd images with a stereo camera for concentrating on pedestrian counts and localization.",
"Fudan-ShanghaiTech [26] and Venice [27] capture the video sequences for temporal crowd counting.",
"Drone view.",
"For some big scenes (such as stadium, plaza) or some large rally events (ceremony, hajj, etc.",
"), the above traditional fixed surveillance camera is not suitable due to its small field of view.",
"To tackle this problem, some other datasets are collected through the Drone or Unmanned Aerial Vehicle (UAV).",
"Benefiting from their higher altitudes, more flexible view and free flight, more large scenes can be recorded compared with the traditional surveillance camera.",
"There are two crowd counting datasets with the drone view, DLR-ACD Dataset [28] and DroneCrowd Dataset [29].",
"The former consists of 33 images with $226,291$ annotated persons, including some mass events: sports, concerts, trade fair, etc.",
"The latter consists of 70 crowd scenes , with a total of $33,600$ drone-view image sequences.",
"Due to the Bird's-Eye View (BEV), the whole body of pedestrians can not be seen except their heads, so the perspective change rarely appears in the above two datasets.",
"Figure: The display of the proposed NWPU-Crowd dataset.",
"Column 1 shows some typical samples with normal lighting.",
"The second and third column demonstrate the crowd scenes under the extreme brightness and low-luminance conditions, respectively.",
"The last column illustrates the negative samples, including some scenes with densely arranged other objects."
],
[
"General-scene Dataset",
"In addition to the above crowd images captured in specific scenes, there are also many general-scene crowd counting datasets, which are collected from the Internet.",
"A remarkable aspect of general-scene is that the crowd density varies significantly, which ranges from 0 to $20,000$ .",
"Besides, diversified scenarios, light and shadow conditions, and uneven crowd distribution in one single image are also distinctive attributes of these datasets.",
"The first general-scene dataset for crowd counting, UCF_CC_50 [19], is presented by Idrees et al.",
"in 2013.",
"It only contains 50 images, which is so small to train a robust deep learning model.",
"Consequently, a larger crowd counting dataset becomes more significant nowadays.",
"Zhang et al.",
"propose ShanghaiTech Part A [7], which is constructed of 482 images crawled from the Internet.",
"Although its average number of labeled heads in each image is smaller than UCV_CC_50, it contains more pictures and larger number of labeled head points.",
"For further research on the extremely congested crowd counting, UCF-QNRF [9] is presented by Idrees et al.",
"It is composed of $1,525$ images with more than $1,251,642$ label points.",
"The average number of pedestrians per image is 815, and the maximum number reaches $12,865$ .",
"Aiming at the small size of crowd images, Crowd Surveillance [18] build a large-scale dataset containing $13,945$ images, which provides regions of interest (ROI) for each image to keep out these blobs that are ambiguous for training or testing.",
"In addition to the above datasets, Sindagi et al.",
"introduce a new dataset for unconstrained crowd counting, JHU-CROWD, including $4,250$ samples.",
"All images are annotated from the image and head level.",
"For the former level, they label the scenario (mall, stadium, etc.)",
"and weather conditions.",
"For the head level, the annotation information includes not only head locations but also occlusion, size, and blur attributes."
],
[
"NWPU-Crowd Dataset",
"This section describes the proposed NWPU-Crowd from four perspectives: data collection/specification, annotation tool, statistical analysis, data split and evaluation protocol."
],
[
"Data Collection and Specification",
"Data Source.",
"Our data are collected from self-shooting and the Internet.",
"For the former, $\\sim \\!2,000$ images and $\\sim \\!200$ video sequences are captured in some populous Chinese cities, including Beijing, Shanghai, Chongqing, Xi'an, and Zhengzhou, containing some typical crowd scenes, such as resort, walking street, campus, mall, plaza, museum, station.",
"However, extremely congested crowd scenes are not the norm in real life, which is hard to capture via self-shooting.",
"Therefore, we also collect $\\sim \\!8,000$ samples from some image search engines (Google, Baidu, Bing, Sougou, etc.)",
"via the typical query keywords related to the crowd.",
"Table REF lists the primary data source websites and the corresponding keywords.",
"The third row in the table records some Chinese websites and keywords.",
"Finally, by the above two methods, $10,257$ raw images are obtained.",
"Table: The query keywords on some typical search engines.Data Deduplication and Cleaning.",
"We employ four individuals to download data from the Internet on non-overlapping websites.",
"Even so, there are still some images that contain the same content.",
"Besides, some congested datasets (UCF_CC_50, Shanghai Tech Part A, and UCF-QNRF), are also crawled from the Internet, e.g., Flickr, Google, etc.",
"For avoiding the problem of data duplication, we perform an effective strategy to measure the similarity between two images, which is inspired by Perceptual Loss [30].",
"Specifically, for each image, the layer-wise VGG-16 [31] features (from conv1 to conv5_3 layer) are extracted.",
"Given two resized samples $i_x$ and $i_y$ with the resolution of $224 \\times 224$ , the similarity is defined as follows: $\\scriptsize \\begin{array}{l}\\begin{aligned}D\\left( {{i_x},{i_y}} \\right) = \\sum \\limits _{j \\in L} {\\frac{1}{{{C_j}{H_j}{W_j}}}} \\left\\Vert {{\\psi _j}({i_x}) - {\\psi _j}({i_y})} \\right\\Vert _2^2,\\end{aligned}\\end{array}$ where $L$ is the set of the last activation layer in five groups of VGG-16 network, namely $L = \\left\\lbrace {{\\psi _j}|j = 1,2,...,5} \\right\\rbrace = \\left\\lbrace {relu1\\_2,relu2\\_2,relu3\\_3,relu4\\_3,relu5\\_3} \\right\\rbrace $ .",
"${\\psi _j}({i_x})$ and ${\\psi _j}({i_y})$ denote layer ${\\psi _j}$ 's outputs (feature maps) for sample $i_x$ and $i_y$ , respectively.",
"${C_j}$ , ${H_j}$ and ${W_j}$ are the size of ${\\psi _j}({i_x})$ at three axes: channel, height and width.",
"If $D\\left( {{i_x},{i_y}} \\right)<5$ , these two samples are considered to have similar contents.",
"As a result, one of the two is removed from the dataset.",
"Then remove excess similar images by computing the distance of the feature between any two samples.",
"Furthermore, some blurred images that are difficult to recognize the head location are also removed.",
"Consequently, we obtain $5,109$ valid images."
],
[
"Data Annotation",
"Annotation tools: For conveniently annotating head points in the crowd images, an online efficient annotation tool is developed based on HTML5 + Javascript + Python.",
"This tool supports two types of label form, namely point and bounding box.",
"During the annotation process, each image is flexibly zoomed in/out to annotate head with different scales, and it is divided into $16 \\times 16$ small blocks at most, which allows annotators to label the head under five scales: $2^i$ (i=0,1,2,3,4) times size of the original image.",
"It effectively prompts annotation speed and quality.",
"The more detailed description is shown in the video demo of our provided supplementary material.",
"Point-wise annotation: The entire annotation process has two stages: labeling and refinement.",
"Firstly, there are 30 annotators involved in the initial labeling process, which costs $2,100$ hours totally to annotate all collected images.",
"After this, 6 individuals are employed to refine the preliminary annotations, which takes 150 hours per refiner.",
"In total, the entire annotation process costs $3,000$ human hours.",
"Box-level annotation and generation: There are three steps to annotate box labels: 1) for each image, manually select $\\sim 10\\%$ typical points to draw their corresponding boxes, which can represent the scale variation in the whole scene; 2) for each point without box label, adopt a linear regression algorithm to obtain its box size based on its 8-nearest box-labeled neighbors; 3) manually refine the prediction box labels.",
"Step 1) and 2) takes $1,000$ human hours in total.",
"Here, the step 2) is described as below: For a head point $P_0$ without box label, its 8-nearest box-labeled neighbors ($P_{1~8}$ ) are utilized to fit a linear regression algorithm [32], in which the vertical axis coordinates are variable, and the box size is the dependent variable.",
"According to the linear function and the vertical axis coordinate of $P_0$ , the box size corresponding to $P_0$ can be obtained.",
"We assume each box has a shape of a square, and the point coordinates are its center, and then the box can be obtained.",
"Obviously, the linear regression is not reliable, so we should manually refine the predicted box labels again in Step 3).",
"Then the linear regression and the manually refine will loop continuously until all boxes seem qualified.",
"In the annotation stage, Step 2) and 3) are repeated four times.",
"Discussion on annotation quality In the field of crowd counting and localization, it is important how to ensure high-quality annotation, especially in some extremely congested scenes.",
"In this work, we attempt to alleviate it from the two aspects: 1) the proposed tools support zooming in or out on an image with 1x 16x online.",
"For congested region, the annotator can easily draw a box on a tiny or occluded object using zooming operation; 2) we conduct two stages of refinement in the point annotation, and repeat four times for linear estimation and refinement to minimize labeling errors in the box annotation."
],
[
"Data Characteristic",
"NWPU-Crowd dataset consists of $5,109$ images, with $2,133,375$ annotated instances.",
"Compared with the existing crowd counting datasets, it is the largest from the perspective of image and instance level.",
"Fig.",
"REF respectively demonstrates four groups of typical samples from Row 1 to 4 in the dataset: normal-light, extreme-light, dark-light, and negative samples.",
"Fig.",
"REF compares the number distribution of different counting range on four datasets: NWPU-Crowd, JHU-CROWD++ [21], [22], UCF-QNRF [9], and ShanghaiTech Part A [7].",
"Except the bin of $(0,100]$ , the number of images on NWPU-Crowd is much larger than that on the other three datasets.",
"Fig.",
"REF shows the distributions of the box area in NWPU-Crowd and JHU-CROWD.",
"From the orange bars, more than $50\\%$ of boxes areas are in the range of $(10^2,10^3]$ pixels.",
"Since the average resolution of NWPU-Crowdis is higher than that of JHU-CROWD, the numbers of large-scale heads are more.",
"The larger scale provides more detailed head-structure information, which will aid the model to achieve better performance.",
"Figure: The distribution of the area (pixels) of head region.",
"In addition to data volume and scale distribution, there are four more advantages in NWPU-Crowd: Negative Samples.",
"NWPU-Crowd introduces 351 negative samples (namely nobody scenes), which are similar to congested crowd scenes in terms of texture features.",
"It effectively improves the generalization of counting models while applied in the real world.",
"These samples contain animal migration, fake crowd scenes (sculpture, Terra-Cotta Warriors, 2-D cartoon figure, etc.",
"), empty hall, and other scenes with densely arranged objects that are not the person.",
"Fair Evaluation.",
"For a fair evaluation, the labels of the test set are not public.",
"Therefore, we develop an online evaluation benchmark website that allows researchers to submit their estimation results of the test set.",
"The benchmark can calculate the error between presented results and ground truth, and list them on a scoreboard.",
"Higher Resolution.",
"The proposed dataset collects high-quality and high-resolution scenes, which is entailed for extremely congested crowd counting.",
"From Table REF , the average resolution of NWPU-Crowd is $2191 \\times 3209$ , which is larger than that of other datasets.",
"Specifically, the maximum image size is $4028 \\times 19044$ .",
"Large Appearance Variation.",
"The number of people ranges from 0 to $20,033$ , which means large appearance variations within the data.",
"Notably, the smallest head occupies only 4 pixels, but the largest head covers $1.2\\!\\times \\!10^7$ pixels.",
"In the whole dataset, the ratio of the area of the largest and smallest head in the same image is $3.8\\!\\times \\!10^5$ .",
"In summary, NWPU-Crowd is one of the largest and most challenging crowd counting/localization datasets at present."
],
[
"Data Split and Evaluation Protocol",
"NWPU-Crowd Dataset is randomly split into three parts, namely training, validation and test sets, which respectively contain $3,109$ , 500 and $1,500$ images.",
"To be specific, each image is randomly assigned to a specific set with the corresponding probability (followed by $0.6$ , $0.1$ and $0.3$ for the three subsets) until the number reaches the upper bound.",
"This strategy ensures that the statistics (such as data distribution, the average value of resolutions/counts) of the subset are almost the same.",
"Counting Metrics Following some previous works, we adopt three metrics to evaluate the counting performance, which are Mean Absolute Error (MAE), Mean Squared Error (MSE), and mean Normalized Absolute Error (NAE).",
"They can be formulated as follows: $\\tiny \\begin{array}{l}\\begin{aligned}MAE = \\frac{1}{N}\\sum \\limits _{i = 1}^N {\\left| {{y_i} - {{\\hat{y}}_i}} \\right|}, MSE = \\sqrt{\\frac{1}{N}\\sum \\limits _{i = 1}^N {{{\\left| {{y_i} - {{\\hat{y}}_i}} \\right|}^2}} }, NAE = \\frac{1}{N}\\sum \\limits _{i = 1}^N {\\frac{\\left| {{y_i} - {{\\hat{y}}_i}} \\right|}{{y_i}}},\\end{aligned}\\end{array}$ where $N$ is the number of images, ${{y_i}}$ is the counting label of people and ${{{\\hat{y}}_i}}$ is the estimated value for the $i$ -th test image.",
"Since NWPU-Crowd contains quite a few negative samples, NAE's calculation does not contain them to avoid zero denominators.",
"In addition to the aforementioned overall evaluation on the test set, we further assess the model from different perspectives: scene level and luminance.",
"The former have five classes according to the number of people: 0, $(0,100]$ , $(100,500]$ , $(500,5000]$ , and more than 5000.",
"The latter have three classes based on luminance value in the YUV color space: $[0,0.25]$ , $(0.25,0.5]$ ,and $(0.5,0.75]$ .",
"The two attribute labels are assigned to each image according to their annotated counting number and image contents.",
"For each class in a specific perspective, MAE, MSE, and NAE are applied to the corresponding samples in the test set.",
"Take the luminance attribute as an example, the average values of MAE, MSE, and NAE at the three categories can reflect counting models' sensitivity to the luminance variation.",
"Similar to the overall metrics, the negative samples are excluded during the calculation of NAE.",
"Localization Metrics For the crowd localization task, we adopt the box-level Precision, Recall and F1-measure to evaluate the localization performance.",
"Given two point sets from prediction results $P_p$ and ground truth $P_g$ , we firstly construct a Bipartite Graph ${G_{p,s}}$ for the two sets.",
"Secondly, we compute the distance matrix of $P_p$ and $P_p$ .",
"If the distance between $p_p \\in P_p$ and $p_g \\in P_g$ is less than the predefined distance threshold $\\sigma $ , we think $p_p$ and $p_g$ are successfully matched.",
"Corresponding to each element of the distance matrix, we obtain a boolean match matrix (True and False denote matched and non-matched).",
"Finally, we can get a Maximum Bipartite Matching for ${G_{p,s}}$ by implementing the Hungarian algorithm Note that Hungarian algorithm's matching result is not unique.",
"However, the number of TP, FP and FN is the same for different matching results.",
"Considering that for saving computation time, we perform Hungarian algorithm on match matrix instead of the distance matrix.",
"the match matrix and count the number of True Positive (TP), False Positive (FP) and False Negative (FN).",
"In our evaluation, for each head with the size of width $w$ and height $h$ , we define two threshold $\\sigma _s=min(w,h)/2$ and $\\sigma _l={\\sqrt{{w^2} + {h^2}}}/2 $ .",
"The former is a stricter criterion than the latter.",
"Similar to the category-wise counting evaluation at the image level, we propose a scale-sensitive evaluation scheme at the box level for the localization task.",
"To be specific, all heads are divided into six categories according to their corresponding box areas: $[10^0,10^1]$ , $(10^1,10^2]$ , $(10^2,10^3]$ , $(10^3,10^4]$ , $(10^4,10^5]$ , and more than $10^5$ .",
"For each category, the Recall is calculated separately.",
"Different from the previous localization metrics [9], [15], the $\\sigma $ in this paper is adaptive, which is defined by the real head area.",
"In addition, the performance on different scale classes are reported, which helps researchers analyze the model more deeply.",
"In summary, our evaluation is more reasonable than the traditional methods.",
"In this section, we train ten mainstream open-sourced methods on the proposed NWPU-Crowd and submit their results on the evaluation benchmark.",
"Besides, the further experimental analysis and visualization results on the validation set are discussed."
],
[
"Mainstream Methods Involved in Evaluation",
"MCNN [7]: Multi-Column Convolutional Neural Network.",
"It is a classical and lightweight counting model, proposed by Zhang et al.",
"in 2016.",
"Different from the original MCNN, the RGB images are fed into the network.",
"SANet [33]: Scale Aggregation Network.",
"SANet is an efficient encoder-decoder network with Instance Normalization for crowd counting, which combines the MSE loss and SSIM loss to output the high-quality density map.",
"PCC Net [34]: Perspective Crowd Counting Network.",
"It is a multi-task network, which tackles the following tasks: density-level classification, head region segmentation, and density map regression.",
"The authors provide two versions, a lightweight from scratch and VGG-16 backbone.",
"Reg+Det Net [35]: a subnet of DecideNet.",
"It consists of two branches: Regression and Detection Network.",
"The former is a light-weight network for density estimation, and the latter focuses on head detection via on Faster R-CNN (ResNet-101) [36].",
"C3F-VGG [37]: A simple baseline based on VGG-16 backbone for crowd counting.",
"C3F-VGG consists of the first 10 layers of VGG-16 [31] as image feature extractor and two convolutional layers with a kernel size of 1 for regressing the density map.",
"CSRNet [10]: Congested Scene Recognition Network.",
"CSRNet is a classical and efficient crowd counter, proposed by Li et al.",
"in 2016.",
"The authors design a Dilatation Module and add it to the top of the VGG-16 backbone.",
"This network significantly improves performance in the field of crowd counting.",
"CANNet [27]: Context-Aware Network.",
"CANNet combines the features of multiple streams using different respective field sizes.",
"It encodes the multi-scale contextual information of the crowd scenes and yields a new record on the mainstream datasets.",
"SCAR [38]: Spatial-/Channel-wise Attention Regression Networks.",
"SCAR utilizes the self-attention module [39] on the spatial and channel axis to encode the large-range contextual information.",
"The well-designed attention models effectively extracts discriminative features and alleviates mistaken estimations.",
"BL [40]: Bayesian Loss for Crowd Count Estimation.",
"Different from the traditional strategy for the generation of ground truth, BL design a loss function to directly using head point supervision.",
"It achieves state-of-the-art performance on the UCF-QNRF dataset.",
"SFCN† [20] Spatial Fully Convolutional Network with ResNet-101 [41].",
"SFCN† is the only crowd counting model that uses ResNet-101 as a backbone, which shows the powerful capacity of density regression on the congested crowd scenes."
],
[
"Implementation Details",
"In the experiments, for PCC Net https://github.com/gjy3035/PCC-Net and BL https://github.com/ZhihengCV/Bayesian-Crowd-Counting, the models are trained using the official codes and the default parameters.",
"For SANet, we implement the $C^{3}$ Framework [37] and follow the corresponding parameters to train them on NWPU-Crowd dataset.",
"For DetNet, we train a head detector using this code https://github.com/ruotianluo/pytorch-faster-rcnn.",
"For other models, namely MCNN, RegNet, CSRNet, C3F-VGG, CANNet, SCAR, and SFCN†, they are reproduced in our counting experiments, which is developed based on $C^{3}$ Framework [37], an open-sourced crowd counting project using PyTorch [42].",
"In the data pre-processing stage, the high-resolution images are resized to the 2048-px scale with the original aspect ratio.",
"The density map is generated by a Gaussian kernel with a fixed size of 15 and the $\\sigma $ of 4.",
"For augmenting the data, during the training process, all images are randomly cropped with the size of $576\\times 768$ , flipped horizontally, transformed to gray-scale images, and gamma corrected with a random value in $[0.4,2]$ .",
"To optimize the above counting networks, Adam algorithm [43] is employed.",
"Other parameters (such as learning rate, batch size) are reported in https://github.com/gjy3035/NWPU-Crowd-Sample-Code."
],
[
"Results Analysis on ",
"Quantitative Results.",
"Here, we list the counting performance and density quality of all participation methods in Table REF .",
"For evaluating the quality of the density map, two popular criteria are adopted, Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity in Image (SSIM) [44].",
"Since BL [40] is supervised by point locations instead of density maps, PSNR and SSIM are not reported.",
"In the calculation of PSNR, the negative samples are excluded to avoid zero denominators.",
"Table: The performance of different models on the val set.From the table, we find SCAR [38] attains the best counting performance, MAE of 81.57 and MSE of 397.92.",
"SFCN† [20] produces the most high-quality density maps, PSNR of 30.591 and SSIM of 0.952.",
"For the three light models (MCNN, SANet, PCC-Net-light), we find that the last achieves the best SSIM (0.937), which even surpasses the SSIMs of some other VGG-based algorithms, such as C3F-VGG, CSRNet, CANNet, and SCAR.",
"Similarly, PCC-Net-VGG is the best SSIM in the VGG-backbone methods.",
"Visualization Results.",
"Fig.",
"REF demonstrates some predicted density maps of the eight methods.",
"The first two columns are negative samples, and others are crowd scenes with different density levels.",
"From the first two columns, almost all models perform poorly for negative samples, especially densely arranged objects.",
"For humans, we can easily recognize that the two samples are mural and stones.",
"But for the counting models, they cannot understand them.",
"For the third column, although the predictions of these methods are good, there are still many mistaken errors in background regions.",
"For the last two images that are extremely congested scenes, the estimation counts are far from the ground truth.",
"SCAR is the most accurate method on the validation set, but it is about $1,900$ and $8,000$ people away from the labels, respectively.",
"For the extreme-luminance scenes (Image 3367, 3250, and 3353), there are quite a few estimation errors in the high-light or dark-light regions.",
"In general, the ability of the current models to cope with the above hard samples needs to be further improved."
],
[
"Leaderboard",
"Table REF reports the results of five methods on the test set.",
"It lists the overall performance (MAE, MSE, and NAE), category-wise MAE on the attribute of scene level and luminance, model size, speed (inference time) and floating-point operations per second (FLOPs) For PCC-Net, we remove the useless layers (classification and segmentation modules) to compute the last three items: model size, speed and FLOPs..",
"Compared with the results of the validation set, we find that the ordering has changed significantly.",
"Although SCAR attains the best results of MAE and MSE on the validation set, the performance on the test set is not good.",
"For the primary key (overall MAE), BL, SFCN† and CANNet occupy the top three on the test set.",
"From the category-wise results of Scene Level, we find that all methods perform poorly in S0 (negative samples), S3 ($(500,5000]$ ) and S4 ($\\ge 5000$ ), which causes that the average value of category-wise MAE is larger than the overall MAE (SFCN†: $712.7$ v.s.",
"$105.7$ ).",
"Besides, this phenomenon shows that negative samples and congested scenes are more challenging than sparse crowd images.",
"Similarly, for the luminance classes, the MAE of $L0$ ($[0,0.25]$ ) is larger than that of $L1$ and $L2$ .",
"In other words, the counters work better under the standard luminance than under the low-luminance scenes.",
"More detailed results are shown in https://www.crowdbenchmark.com/nwpucrowd.html."
],
[
"Performance Impact between Different Scenes",
"From Section REF and REF , we find that two interesting phenomena worth attention: 1) Negative samples are prone to be mistakenly estimated; 2) The data with different scene attributes (namely density level) significantly affect each other.",
"In this section, we conduct two experiments using a simple baseline model, C3F-VGG [37], to explore the above problems.",
"Phenomenon 1.",
"For the firth problem, the main reason is that the negative samples contain densely arranged objects, which is similar to the congested crowd scenes.",
"As we all know, most existing counting models focus on texture information and local patterns for congested regions.",
"To verify our thoughts, we design three groups of experiments to explore which samples affect the performance of negative samples.",
"To be specific, we train three C3F-VGG counters on different combination training data: $S0 + S1$ , $S0 + S2$ , and $S0 + S3 + S4$ (considering that the number of S4 is small, so we integrate S3 and S4).",
"Then the evaluation is performed on the validation set.",
"Finally, the corresponding performance is listed in Table REF .",
"From it, the MAE on Negative Sample ($S0$ ) increases from $18.54$ to $147.53$ as the density of positive samples increases.",
"Table: The MAE of the different training data on the val set.Phenomenon 2.",
"For the second issue, we train the counting models only using the data with a single category, $S1$ , $S2$ , and $S3 + S4$ respectively.",
"Removing the impacts of the negative samples, the model is trained on the data of $S1 + S2 + S3 + S4$ .",
"The concrete performance is illustrated in Table REF .",
"According to the results, training each class individually is far better than training together.",
"To be specific, MAE decreases by 36.6%, 25.7%, 22.2% and 12.7% on the four classes, respectively.",
"The main reason is that NWPU-Crowd contains more diverse crowd scenes than the previous datasets.",
"There are large appearance variations in the dataset, especially the scales of the head.",
"At present, the existing models can not tackle this problem well."
],
[
"The Effectiveness of Negative Samples",
"In Section REF , we mention that the Negative Samples (“NS” for short) can effectively improve the generalization ability of the model.",
"Here, we conduct four groups of comparative experiments using C3F-VGG [37] to verify this opinion.",
"To be specific, there are four types of training data: $S1$ , $S2$ , $S3+S4$ and $S1+...+S4$ .",
"We respectively train the models for them using NS and without NS.",
"In other words, we add $S0$ to the above four types of training data.",
"The concrete results are reported in Table REF .",
"After introducing NS, the category-wise MAEs are significantly reduced.",
"Take the last six rows as the examples, the MAE is respectively decreased by 21.7%, 2.0%, 6.1% and 17.4% on the category-wise evaluation.",
"The main reason is that NS contains diverse background objects with different structured information, which can prompt the counting models to learn more discriminative features than ever before."
],
[
"Impact of Data Volume on Performance",
"Generally speaking, large amounts of diverse training data will prompt the model to learn more robust features, and then perform better in the wild.",
"This is also our original intention to build a large-scale crowd counting and localization dataset.",
"In this section, we explore the impact of different data volumes on counting performance.",
"To be specific, we train ten C3F-VGG models using $10\\%$ , $20\\%$ , $30\\%$ , ..., $100\\%$ training data, respectively.",
"Then evaluate them on the validation set.",
"The performances (MAE and MSE) are demonstrated in Fig.",
"REF.",
"From the figure, with the gradual increase of training data, the errors on the validation set also gradually decrease overall.",
"By comparing the MAEs when using the $10\\%$ and $100\\%$ data, the error is significantly reduced from 158.35 to 105.79 (relative decrease of 33.2%).",
"Therefore, a large-scale dataset is very necessary for the community.",
"Figure: The results under different volumes of the training data on the validation set."
],
[
"Experiments on Localization",
"Considering that the box-level labels are provided, we evaluate four crowd localization methods in this section.",
"What's more, we analyze the quantitative and qualitative results of them."
],
[
"Methods and Implementation Details",
"Faster RCNN [36]: a general object detection framework, based on ResNet-101.",
"It directly detects the head boxes, of which center is the prediction head location.",
"We follow the original training parameters of this code https://github.com/ruotianluo/pytorch-faster-rcnn.",
"In the forward process, the thresholds of confidence and nms are set as $0.8$ and $0.3$ , respectively.",
"TinyFaces [45]: a tiny object detection framework, which focuses on tiny faces detection.",
"We implement the third-party code https://github.com/varunagrawal/tiny-faces-pytorch to train a detector using the default parameters.",
"The thresholds of confidence and nms are set as $0.8$ and $0.3$ , respectively.",
"VGG+GPR [46]: a two-stage method that consists of density map regression and point reconstruction based on Gaussian-kernel priors.",
"C3F-VGG's [37] training is the same as Section and GPR using standard Gaussian kernel with a size of 15.",
"RAZ_Loc [15]: the localization branch of RAZNet, which consists of localization map classification and point post-processing based on finding high-confidence peaks.",
"The training details follows RAZNet, and classification threshold is set as $0.5$ ."
],
[
"Results Analysis on ",
"Table REF lists the localization and counting performance of four methods on the validation set.",
"For each head, its $\\sigma _s$ is less than $\\sigma _l$ , which means that the former is more strict than the latter.",
"Thus, the localization results of $\\sigma _s$ are worse than that of $\\sigma _l$ .",
"From the table, we find that the Precision of Faster RCNN is better than others, but they miss quite a few objects.",
"RAZ_Loc produces the best localization result, but its counting error is far from VGG+GPR.",
"Detection-based methods' counting performance is the poorest in all plans.",
"Table: The performance on the val set.",
"F1-m, Pre, Rec are short for F1-measure, Precision and Recall, respectively.Figure: The three groups of qualitative localization results on the validation set.",
"The Green point is true positive, which is inside the green circle (its center is the groundtruth position and its radius is σ l \\sigma _l); the red points and the corresponding circles are false negative; the magenta points are false positive.",
"(For a better comparison, we transform RGB-color images to gray-scale images.",
")Table: The leaderboard of the localization performance on the NWPU-Crowd test set.",
"In the ranking strategy, the Overall F1-measure is the primary key under σ l \\sigma _l, which is bold font in Row 2 of the table.",
"A0∼A4A0 \\sim A4 respectively indicates six categories according to the different head area ranges: [10 0 ,10 1 ][10^0,10^1], (10 1 ,10 2 ](10^1,10^2], (10 2 ,10 3 ](10^2,10^3], (10 3 ,10 4 ](10^3,10^4], (10 4 ,10 5 ](10^4,10^5], and >10 5 >10^5.",
"More detailed results will be reported in https://www.crowdbenchmark.com/nwpucrowdloc.html, which is under deployment.",
"*: Since the counts is transformed to integer data, the performance is slightly different from Table .To intuitively understand the performance of crowd localization, Fig.",
"REF demonstrates the visualization results of four methods on some typical samples.",
"For the first sample, containing different-scale heads, Faster RCNN almost misses all small objects.",
"The other three methods perform better for this scene than it.",
"For large-scale objects (such as the second sample), Faster RCNN produces the perfect results of 100% precision and 100% recall.",
"TinyFaces is the second, and the other two methods miss quite a few heads to different extents.",
"In congested crowd scenes (e.g., Sample 3), VGG+GPR and RAZ_Loc obtain good results though they produce some false positives in the background regions.",
"Faster RCNN and TinyFaces work not well for this case: the former miss 95.4% heads, and the latter yields many false positives.",
"Besides, we also find TinyFaces produces more false positives than other methods.",
"In summary, there is no method to tackle the crowd localization problem well: 1) the traditional general object detection methods can not detect small-scale objects; 2) TinyFaces outputs quite a few false positives; 3) the existing regression-/classification methods can not handle large-range scale variations and mis-estimations on the background."
],
[
"Leaderboard",
"Table REF reports the results of four methods on the test set.",
"It lists the overall localization performance (F1-measure, Precision, and Recall) under $\\sigma _l$ and counting performance (MAE, MSE, NAE), category-wise Recall on the different head scales (Box Level).",
"Compared with the results of the validation set, we find that the rankings are consistent.",
"For the primary key (overall F1-measure), RAZ_Loc attains the first place.",
"From the category-wise results of Box Level, we find that all methods perform poorly for tiny heads (area range is $[1,10]$ ).",
"The main reason is that the current feature extractor loses spatial information (usually, $8\\!\\times $ or $16\\!\\times $ downsampling are adopted).",
"For extremely large-scale heads (area is more than $10^5$ ), detection-based methods is better than regression-/classification methods.",
"The main reason is the latter's label is so small (VGG+GPR: $15\\!\\times \\!15$ , RAZ_Loc: $3\\!\\times \\!3$ ) that can not cover enough semantic head regions."
],
[
"Conclusion and Outlook",
"In this paper, a large-scale NWPU-Crowd counting dataset is constructed, which has the characteristics of high resolution, negative samples, and large appearance variation.",
"At the same time, we develop an online benchmark website to fairly evaluate the performance of counting models.",
"Based on the proposed dataset, we perform the fourteen typical algorithms and rank them from the perspective of the counting and localization performance, the density map quality, and the time complexity.",
"According to the quantitative and qualitative results, we find some interesting phenomena and some new problems that need to be addressed on the proposed dataset: How to improve the robustness of the models?",
"In the real world, the counters may encounter many unseen data, giving incorrect estimation for background regions.",
"Thus, the performance on negative samples is vital in the counting, which represents the models' robustness.",
"How to remedy the performance impact between different scenes?",
"Due to the large appearance variations, the training with all data results in an obvious performance reduction compared with the individual training for each category.",
"Hence, it is essential to prompt the counting model's capacity for appearance representations.",
"How to reduce the estimation errors in the extremely congested crowd scenes?",
"Because of head occlusions, small objects, and lack of structured information, the existing models can not work well in the high-density regions.",
"How to accurately locate the tiny-size and large-scale heads together?",
"The current detection-/regression-/classification-based methods can not handle the problem of large-range scale variation in real crowd scenes.",
"Perhaps researchers need to design scale-aware models or hybrid methods to locate the head position accurately.",
"In the future, we will continue to focus on handling the above issues and dedicate to improving the performance of crowd counting and localization in the real world."
]
] | 2001.03360 |
[
[
"Wiretap channels with causal and non-causal state information: revisited"
],
[
"Abstract The coding problem for wiretap channels with causal channel state information (CSI) available at the encoder (Alice) and/or the decoder (Bob) is studied.",
"We are concerned here particularily with the problem of achievable secret-message secret-key rate pairs under the semantic security criterion.",
"Our main result extends all the previous results on achievable rates as given by Chia and El Gamal [10], Fujita [11], and Han and Sasaki [23].",
"In order to do this, we first derive a unifying theorem (Theorem 2) with causal CSI at Alice, which follows immediately by leveraging the unifying seminal theorem for wiretap channels with non-causal CSI at Alice as recently established by Bunin et al.",
"[22].",
"The only thing to do here is just to re-interpret the latter non-causal one in a causal manner.",
"A prominent feature of this approach is that we are able to dispense with the block-Markov encoding scheme as used in the previous works.",
"Also, the exact secret-message key capacity region for wiretap channels with non-causal CSI at both Alice and Bob is given."
],
[
"Introduction",
"In this paper we address the coding problem for a wiretap channel (WTC) with causal/non-causal channel state information (CSI) available at the encoder (Alice) and/or the decoder (Bob).",
"The intriguing concept of WTC and secret message (SM) transmission through the WTC originates in Wyner [1] (without CSI) under the weak secrecy criterion.",
"This was then extended to a wider class of WTCs by Csiszár and Körner [2] to provide the more tractable framework.",
"Indeed, these landmark papers have offered the fundamental basis for a diversity of subsequent extensive researches.",
"Early works include Mitrpant, Vinck and Luo [5], Chen and Vinck [6], and Liu and Chen [7] that have studied the capacity-equivocation tradeoff for degraded WTCs with non-causal CSI to establish inner and/or outer bounds on the achievable region.",
"Subsequent developments in this direction with non-causal CSI can be found also in Boche and Schaefer [9], Dai and Luo [18], etc., which are mainly concerned with the problem of SM transmission over the WTC.",
"On the other hand, Khisti, Diggavi and Wornell [11] and Zibaeenejad [29] addressed the problem of secret key (SK) agreement over the WTC with non-causal CSI at Alice (and Bob), and tried to give the exact key-capacity formula.",
"Prabhakaran et al.",
"[17] studied an achievable tradeoff between SM and SK rates over the WTC with non-causal CSI, deriving a benchmark inner bound on the SM-SK capacity region under the weak secrecy criterion.",
"Subsequently, heavily based on the work of Goldfeld et al.",
"[23], Bunin et al.",
"[24], [25] have improved [17] by explicitly leveraging the superposition coding to obtain a unifiying formula (cf.",
"Theorem REF ) for inner bounds on the SM-SK capacity region under the semantic secrecy (SS) criterion for WTCs with non-causal CSI at Alice, from which “all\" the typical previous results can be derived.",
"Thus, [24], [25] are regarded currently as establishing the best known achievable rate pairs with non-causal CSI at Alice.",
"The key idea in [24], [25] (which are substantially due to [23]) is to invoke the likelihood encoder (cf.",
"Song et al.",
"[20]) together with the soft-covering lemma (cf.",
"Cuff [22]) $\\endcsname $This is the notion to denote the achievability part of resolvability [28].",
"on the basis of two layered superposition coding scheme (cf.",
"[17], [23]), which makes it possible to guarantee the semantically secure (SS) information transmission.",
"This is one of the strongest ones among various security criteria.",
"In contrast to extensive studies on WTCs with “non-causal\" CSI mentioned above, there have been less number of literatures on WTCs with “causal\" CSI.",
"To our best knowledge, we can list typically a few causal papers including Chia and El Gamal [12], Fujita [13], and Han and Sasaki [14].",
"They are concerned only with SM rates but not with SK rates.",
"A prominent feature common in these papers is to leverage the block-Markov encoding to invoke the Shannon cipher [3] (Vernam's one-time pad cipher).",
"Although there still remain many open problems, possible extensions/generalizations in this direction do not seem to be very fruitful or may be even formidable.",
"Fortunately, however, to solve these problems we can fully exploit, as they are, all the non-causal techniques/concepts as developed in Bunin et al.",
"[25] to derive the causal version of it.",
"The only thing to do here is simply to restrict the range of auxiliary random variables $(U, V)$ 's intervening in [25] (said to be non-causally achievable) to a subclass of auxiliary random variables $(U, V)$ 's (said to be causally achievable).",
"Then, it suffices to notice only that the encoding scheme given in [25] can be carried out, as it is, in a causal way.",
"This process may be termed “plugging\" of causal WTCs into non-causal WTCs.",
"Thus, it is not necessary to give a separate proof to establish the causal version (Theorem REF ) in this paper.",
"The merits of this approach for proof are to inherit all the advantages in [25] to our causal version.",
"For example, the first one is to inherit the SS property as established in [25]; the second one is to enable us, without any extra arguments, to interpret regions of SM-SK achievable rate pairs in [25] as those valid also in Theorem REF ; the third one is to enable us to dispense with the involved block-Markov encoding scheme (cf.",
"[12], [14]); the fourth one is that all the results as established in [12], [13], [14] follow immediately from Theorem REF ; the fifth one is to be able to derive, in a straightforward manner, a variety of novel causal inner bounds on the SM-SK capacity region; the sixth one is, as a by-product, to enable us to exactly determine the general formula for the SM-SK capacity region for WTCs with non-causal CSI available at both Alice and Bob (Theorem REF ).",
"Furthermore, the arguments that have been used to derive Theorems REF and REF can be further exploited to solve harder problems such as deriving a “tighter\" causal outer bound (Theorem REF ) and finding the causal/non-causal SM-SK capacity region for degraded WTCs (Theorem REF ).",
"The present paper is organized as follows.",
"In Section , we give the problem statement as well as the necessary notions and notation, all of which are borrowed from [25] along with Theorem REF with non-causal CSI at Alice.",
"They are used in the next sections.",
"In particular, in Section , we give Theorem REF to demonstrate the general formula for the exact “non-causal\" SM-SK capacity region when the state information is available at both Alice and Bob.",
"In Section , we give the proof of Theorem REF for WTCs with causal CSI at Alice by using the argument of “plugging,\" which is to put the causal scenario into the non-causal scenario, thereby enabling us to produce a diversity of causal inner bounds in Section .",
"In Section , we develop Theorem REF for each of Case 1) $\\sim $ Case 4) to obtain a new class of inner bounds of SM-SK achievable rate pairs for WTCs with causal CSI at Alice.",
"Here, it is also shown that all the results as established in [12], [13], and [14] can be derived as special cases of Theorem REF .",
"Furthermore, in this section we give Proposition REF for state-reproducing coding schemes (with causal CSI at Alice) to derive an SM-SK outer bound, which is paired with Proposition REF (inner bound).",
"In Section , we establish the exact SM-SK capacity region with causal/non-causal CSI available at both Alice and Bob (Theorem REF for degraded WTCs), which is the first solid result from the viewpoint of “causal\" SM-SK capacity regions.",
"In Section , we conclude the paper with several remarks.",
"Finally, in Appendix , we give an elementary proof of the soft-covering lemma that plays the key role in [25], [23].",
"Also, the proof of Remark REF on typical causal inner bounds is given in Appendix ."
],
[
"Wiretap Channel with Non-Causal CSI",
"In this section, we recapitulate the seminal work for wiretap channels with “non-causal\" channel state information (CSI) available at the encoder (Alice) as in Fig.",
"REF , which was recently established by the group of Bunin, Goldfeld, Permuter, Shamai, Cuff and Piantanida [25].",
"For the reader's convenience, we repeat here their notions and key result as they are.",
"Leveraging them, we derive the “causal\" counterparts in Section .",
"II.",
"A: Problem Statement Let ${\\cal S}, {\\cal X}, {\\cal Y}, {\\cal Z}$ be finite sets and ${\\cal S}^n, {\\cal X}^n, {\\cal Y}^n, {\\cal Z}^n$ be the $n$ times product sets.",
"We let $({\\cal S}, {\\cal X}, {\\cal Y}, {\\cal Z}, W_S,$ $ W_{YZ|SX})$ denote a discrete stationary and memoryless WTC with “non-causal\" stationary memoryless CSI $S$ available at the encoder, where $W_{YZ|SX}: {\\cal S}\\times {\\cal X}\\rightarrow {\\cal P}({\\cal Y}\\times {\\cal Z})$ ${\\cal P}(\\cal D)$ denotes the set of all probability distributions on the set ${\\cal D}$ .",
"Also, we use $p_U$ to denote the probability distribution of a random variable $U$ .",
"Similarly, we use $p_{U|V}$ to denote the conditional probability distribution for $U$ given $V$ .",
"is the transmission probability distribution (under state $S$ ) with input $X$ at Alice, and outputs $Y$ at Bob and $Z$ at Eve, while $W_S$ is the probability distribution of state variable $S$ .",
"A state sequence ${\\bf s}\\in {\\cal S}^n$ is sampled in an i.i.d.",
"manner according to $W_S$ and revealed in a non-causal fashion to Alice.",
"Independently of the observation of ${\\bf s}$ , Alice chooses a message $m$ from the set For integers $r\\le l$ , $[r:l]$ denotes $\\lbrace r, r+1, \\cdots , l-1, l\\rbrace $ .",
"$[1:2^{nR_M}]\\ (R_M\\ge 0)$ and maps the pair $({\\bf s}, m)$ into a channel input sequence ${\\bf x}\\in {\\cal X}^n$ and a key index $k\\in [1:2^{nR_K}]\\ (R_K\\ge 0$ ; the mapping may be stochastic).",
"The sequence ${\\bf x}$ is transmitted over the WTC under state ${\\bf s}$ .",
"The output sequences ${\\bf y}\\in {\\cal Y}^n$ and ${\\bf z}\\in {\\cal Z}^n$ are observed by the legitimate receiver (Bob) and the eavesdropper (Eve), respectively.",
"Based on ${\\bf y}$ , Bob produces the pair $(\\hat{k}, \\hat{m})$ as an estimate of $(k,m)$ .",
"Eve maliciously attempts to decipher the SM-SK rate pair from ${\\bf z}$ as much as possible.",
"The random variables corresponding to ${\\bf s}, {\\bf x}, {\\bf y}, {\\bf z}, m, k$ may be denoted by $S^n, X^n, Y^n, Z^n$ (or also ${\\bf S}, {\\bf X}, {\\bf Y}, {\\bf Z})$ , $M, K$ ; respectively.",
"The following Definitions REF $ \\sim $ REF are borrowed from [25].",
"Figure: WTC with CSI available only at Alice (t=1,2,⋯,nt=1,2,\\cdots , n).Definition 1 (Non-causal code) An $(n, R_M, R_K)$ -code $c_n$ for the WTC with “non-causal\" CSI at Alice and message set ${\\cal M}_n\\stackrel{\\Delta }{=}[1:2^{nR_M}]$ and key set ${\\cal K}_n\\stackrel{\\Delta }{=}[1:2^{nR_K}]$ is a pair of functions $(f_n, \\phi _n)$ such that 1) $f_n: {\\cal M}_n\\times {\\cal S}^n \\rightarrow {\\cal P}( {\\cal X}^n\\times {\\cal K}_n)$ , 2) $\\phi _n: {\\cal Y}^n\\rightarrow {\\cal M}_n\\times {\\cal K}_n$ , where $f_n$ is a stochastic function.",
"The performance of the code $c_n$ is evaluated in terms of its rate pair $(R_M, R_K)$ , the maximum decoding error probability, the key uniformity and independence metric, and SS metric as follows: Definition 2 (Error Probability) The error probability of an $(n, R_M, R_K)$ -code $c_n$ is $e(c_n) \\stackrel{\\Delta }{=} \\max _{m\\in {\\cal M}_n} e_m(c_n),$ where, for every $m\\in {\\cal M}_n$ , $e_m(c_n) \\stackrel{\\Delta }{=} \\Pr \\lbrace (\\hat{M}, \\hat{K}) \\ne (m, K) | M=m\\rbrace $ with the decoder output $(\\hat{M}, \\hat{K})\\stackrel{\\Delta }{=}\\phi _n(Y^n)$ .",
"Definition 3 (Key Uniformity and Independence Metric) The key uniformity and independence (from the message) metric under $(n, R_M, R_K)$ -code $c_n$ is $\\delta (c_n) \\stackrel{\\Delta }{=} \\max _{m\\in {\\cal M}_n}\\delta _m(c_n),$ where, for every $m\\in {\\cal M}_n$ , $\\delta _m(c_n) \\stackrel{\\Delta }{=} ||p^{(c_n)}_{K|M=m} -p_{{\\cal K}_n}^{(U)}||_{{\\sf TV}},$ and $p^{(c_n)}$ denotes the joint probability distribution over the WTC induced by the code $c_n$ ; $p_{{\\cal K}_n}^{(U)}$ is the uniform distribution over ${\\cal K}_n$ , and $||\\cdot ||_{{\\sf TV}}$ denotes the total variation.",
"Definition 4 (Information Leakage and SS-Metric) The information leakage to Eve under $(n, R_M, R_K)$ -code $c_n$ and message distribution $p_M \\in {\\cal P}({\\cal M}_n)$ is $\\ell (p_M, c_n)\\stackrel{\\Delta }{=} I_{p^{(c_n)}}(M, K;{\\bf Z}),$ where $I_{p^{(c_n)}}$ denotes the mutual information with respect to the joint probability $p^{(c_n)}$ .",
"The SS-metric with respect to $c_n$ is $\\ell _{{\\sf Sem}}(c_n) \\stackrel{\\Delta }{=} \\max _{p_M \\in {\\cal P}({\\cal M}_n)}\\ell (p_M, c_n).$ Definition 5 (Achievability) A pair $(R_M, R_K)$ is called an SM-SK achievable rate pair for the WTC with non-causal CSI at Alice, if for every $\\epsilon >0$ and sufficiently large $n$ there exists an $(n, R_M, R_K)$ -code $c_n$ with $\\max [e(c_n), \\delta (c_n), \\ell _{{\\sf Sem}}(c_n)] \\le \\epsilon .$ Definition 6 (Non-causal SM-SK capacity region) Throughout in this paper we use the following notation.",
"The SM-SK capacity region of the WTC with non-causal CSI at Alice, denoted by ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}$ $\\endcsname $E denotes Encoder=E and N of NCSI denotes Non-causal=N.",
", is the set of all SM-SK achievable rate pairs.",
"Furthermore, the supremum of the projection of ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}$ on the $R_M$ -axis, denoted by $C^{{\\scriptsize \\rm M}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}$ , is called the SM capacity, whereas the supremum of the projection of ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}$ on the $R_K$ -axis is called the SK capacity, denoted by $C^{{\\scriptsize \\rm K}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}$ .",
"II.",
"B: Wiretap Channel with Non-causal CSI at Alice We can now describe the unifying key theorem of [25].",
"Let ${\\cal U}, {\\cal V}$ be finite sets and let $U, V$ be random variables taking values in ${\\cal U}, {\\cal V}$ , respectively, where $U, V, S, X$ may be correlated.",
"Define joint probability distributions $p_{YZXSUV}$ on ${\\cal Y}\\times {\\cal Z}\\times {\\cal X}\\times {\\cal S}\\times {\\cal U}\\times {\\cal V}$ (said to be non-causally achievable) so that $UV\\rightarrow SX\\rightarrow YZ$ forms a Markov chain We may use $UV, SX, UV$ instead of $(U,V), (S,X), (U, V)$ , and so on, for notational simplicity.",
"and $p_S=W_S, \\ \\ p_{YZ|SX}=W_{YZ|SX}.$ Notice here that, in view of (REF ), such a distribution $p_{YZXSUV}$ is specified by giving the marginal $p_{SUV}$ (input), so we may use $p_{SUV}$ in short instead of $p_{YZXSUV}$ .",
"Define ${\\cal R}_{\\sf in}(p_{SUV})$ to be the set of all nonnegative rate pairs $(R_M, R_K)$ satisfying the rate constraints: $R_M &\\le & I(UV; Y)-I(UV; S),\\\\R_M+R_K &\\le & I(V;Y|U)-I(V;Z|U) \\nonumber \\\\& &\\qquad - [I(U;S)-I(U;Y)]^+,$ where $[x]^+ = \\max (x, 0)$ and $I(\\cdot ;\\cdot ), I(\\cdot ;\\cdot |\\cdot )$ denotes the (conditional) mutual information.",
"With these definitions, Bunin et al.",
"[25] have established the following non-causal inner bound: Theorem 1 (Non-causal SM-SK inner bound) ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-E}}} \\supset {\\cal R}_{\\sf in}^{\\sf N}\\stackrel{\\Delta }{=}\\bigcup _ {{\\sf N:}p_{SUV}} {\\cal R}_{\\sf in}(p_{SUV}),$ where the union is taken over all “non-causally\" achievable probability distributions $p_{SUV}$ 's.",
"Here, the cardinalities of $U, V$ may be restricted to $|{\\cal U}|\\le (|{\\cal X}|-1) |{\\cal S}|+3$ and $ |{\\cal V}|\\le (|{\\cal X}|-1)^2|{\\cal S}|^2+3(|{\\cal X}|-1)|{\\cal S}|+2$ .",
"Remark 1 In particular, in Section , the inner bound given by Theorem REF is shown to be optimal when the state information is available at both Alice and Bob.",
"Remark 2 It should be emphasized also that the technical crux of the paper [25] (due to [23]) is based on the soft covering lemma $\\endcsname $A “stronger\" version of the soft covering lemma is given in [22], although it is actually not necessary to prove Theorem REF .",
", which is summarized as Lemma 1 ([23]) Let $W:{\\cal U}\\times {\\cal V}\\rightarrow {\\cal S}$ be the memoryless channel induced by joint probability distribution $p_{SUV}$ , and set, with $L_n=2^{nR_1}$ and $N_n=2^{nR_2}$ , $q_S^n({\\bf s}) = \\frac{1}{L_nN_n}\\sum _{i=1}^{L_n}\\sum _{j=1}^{N_n}W({\\bf s}|{\\bf u}_i, {\\bf v}_{ij}).$ Then, for any small $\\varepsilon >0$ and for all sufficiently large $n$ , it holds that ${\\rm E}D(q^n_S||p^n_S) \\le \\varepsilon ,$ provided that rate constraints $R_1> I(U;S), R_1+R_2> I(UV; S)$ are satisfied, where $D(Q||P)$ denotes the Kullback-Leibler divergence between $Q$ and $P$ , and $p^n_S({\\bf s})$ indicates the probability of i.i.d.",
"${\\bf s}=(s_1, s_2, \\cdots , s_n)$ and ${\\rm E}$ denotes the expectation over all random codewords ${\\bf u}_i, {\\bf v}_{ij}$ of Codebook ${\\cal B}_n$ as given later in Section .",
"Although in this paper we do not use explicitly this lemma, in view of its importance, it would be worthy of giving a separate elementary proof, which is stated in Appendix ."
],
[
"Capacity region with non-causal CSI at Alice and Bob",
"In this section, we address the problem of converse part (outer bound) for Theorem REF (inner bound).",
"Specifically, we establish the exact SM-SK capacity region for WTCs with non-causal CSI available at “both\" Alice and Bob as in Fig.",
"REF .",
"To do so, let the corresponding non-causal SM-SK capacity region be denoted by ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ .$\\endcsname $ED denotes Encoder=E and Decoder=D.",
"Moreover, let $\\overline{{\\cal R}}_{\\sf in}(p_{SUV})$ denote the set of all nonnegative rate pairs $(R_M, R_K)$ satisfying the rate constraints: $R_M &\\le & I(UV; Y|S),\\\\R_M+R_K &\\le & I(V;Y|SU)-I(V;Z|SU) \\nonumber \\\\&& \\qquad \\qquad +H(S|ZU),$ where $UV$ may be dependent on $S$ , and $H(\\cdot ), H(\\cdot |\\cdot )$ denote the (conditional) entropy.",
"Then, we have Theorem 2 (Non-causal SM-SK capacity region) ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}} = \\overline{{\\cal R}}_{\\sf in}\\stackrel{\\Delta }{=}\\bigcup _{p_{SUV}} \\overline{{\\cal R}}_{\\sf in}(p_{SUV}),$ where the union is taken over all “non-causally\" achievable probability distributions.",
"Here, the cardinalities of $U, V$ may be restricted to $|{\\cal U}|\\le (|{\\cal X}|-1) |{\\cal S}|+2$ and $ |{\\cal V}|\\le (|{\\cal X}|-1)^2|{\\cal S}|^2+2(|{\\cal X}|-1)|{\\cal S}|+2$ , so that the right-hand side of (REF ) is a compact set.",
"Remark 3 Theorem REF , in particular, means the “optimality\" of the non-causal inner bound (Theorem REF ) given by Bunin et.",
"al [25] when the CSI $S$ is available at both Alice and Bob.",
"Figure: WTC with the same CSI available at Alice and Bob (t=1,2,⋯,nt=1,2,\\cdots , n).Proof of achievability for Theorem REF : The achievabilty immediately follows from Theorem REF with $SV, SY$ instead of $V, Y$ in (REF ) and (), that is, $R_M &\\le & I(USV; SY)-I(USV;S)\\nonumber \\\\& =& I(USV; SY)-H(S)\\nonumber \\\\& =&I(UV;Y|S);\\\\R_M+R_K &\\le & I(SV;SY|U)-I(SV;Z|U)\\nonumber \\\\&& \\qquad \\qquad - [I(U;S)-I(U;SY)]^+\\nonumber \\\\&=& I(SV;SY|U)-I(SV;Z|U)\\nonumber \\\\& =& I(V;Y|SU) - I(V;Z|SU) \\nonumber \\\\& &\\qquad \\qquad +H(S|ZU),$ where we have noticed that $I(U;SY) \\ge I(U;S)$ and hence $[I(U;S)-I(U;SY)]^+ =0$ , and also that $I(USV;S)=H(S).$ Proof of converse for Theorem REF : Suppose that $(R_M, R_K)$ is achievable, and set $\\overline{Y}^n =S^nY^n$ .",
"It suffices here to assume that $M$ is uniformly distributed on ${\\cal M}_n$ .",
"1) We first show (REF ).",
"Observe that $H(M|\\overline{Y}^n) \\le n\\varepsilon _n$ holds by Fano inequality, where $\\varepsilon _n \\rightarrow 0$ as $n$ tends to $\\infty $ .",
"Then, noting that $S^n$ and $M$ are independent, we have ${nR_M}\\nonumber \\\\& = & H(M)\\nonumber \\\\&\\le & H(M)-H(M|\\overline{Y}^n)+n\\varepsilon _n\\nonumber \\\\&=& I(M;\\overline{Y}^n) +n\\varepsilon _n\\nonumber \\\\&=& I(MS^n; \\overline{Y}^n) -I(S^n;\\overline{Y}^n|M)+n\\varepsilon _n\\nonumber \\\\&\\le & I(MS^n; \\overline{Y}^n) -H(S^n|M)+2n\\varepsilon _n\\nonumber \\\\&=& I(MS^n; \\overline{Y}^n) -H(S^n)+2n\\varepsilon _n\\nonumber \\\\&=& \\sum _{t=1}^{n}I(MS^n;\\overline{Y}_t|\\overline{Y}^{t-1}) -\\sum _{t=1}^n H(S_t )+2n\\varepsilon _n\\nonumber \\\\&\\le & \\sum _{t=1}^{n}I(MS^n\\overline{Y}^{t-1};\\overline{Y}_t) -\\sum _{t=1}^n H(S_t )+2n\\varepsilon _n\\nonumber \\\\&\\le & \\sum _{t=1}^{n}I(MS^n\\overline{Y}^{t-1}Z_{t+1}^{n};\\overline{Y}_t) -\\sum _{t=1}^n H(S_t )+2n\\varepsilon _n\\nonumber \\\\&\\le & \\sum _{t=1}^{n}I(MKS^n\\overline{Y}^{t-1}Z_{t+1}^{n};\\overline{Y}_t) -\\sum _{t=1}^n H(S_t )+2n\\varepsilon _n\\nonumber \\\\&=& \\sum _{t=1}^{n}I(U_tS_tV_t;\\overline{Y}_t) -\\sum _{t=1}^n H(S_t )+2n\\varepsilon _n\\\\&=& \\sum _{t=1}^{n}I(U_tS_tV_t;S_tY_t) -\\sum _{t=1}^n H(S_t )+2n\\varepsilon _n,$ where we have set $U_t=\\overline{Y}^{t-1}Z_{t+1}^{n}, \\ V_t =MKS^{t-1}S_{t+1}^{n}.$ Let us now consider the random variable $J$ such that $\\Pr \\lbrace J=t\\rbrace =1/n \\ (t=1,2,\\cdots ,n).$ Then, () is written as $R_M &\\le & I(U_JS_JV_J; S_JY_J|J) - H(S_J|J ) +2\\varepsilon _n\\nonumber \\\\&\\le & I(U_JJS_JV_J; S_JY_J) - H(S_J|J) +2\\varepsilon _n\\nonumber \\\\&= & I(U_JJS_JV_J; S_JY_J) - H(S_J) +2\\varepsilon _n\\nonumber \\\\&= & I(USV; SY) - H(S) +2\\varepsilon _n\\nonumber \\\\&=& I(UV;Y|S) + 2\\varepsilon _n,$ where, noting that $S^n$ is stationary and memoryless and hence $H(S_J|J)=H(S_J)=H(S)$ , we have set $U=U_JJ, \\ V=V_J, \\ S=S_J, \\ Y=Y_J, \\ Z=Z_J.$ Thus, by letting $n\\rightarrow \\infty $ in (REF ), we obtain (REF ).",
"It is obvious here that $UV\\rightarrow XS\\rightarrow YZ$ forms a Markov chain, where we have similarly set $X=X_J$ .",
"2) Next, we show ().",
"First observe that, in view of Definitions REF $\\sim $ REF in Section as well as the uniform continuity of entropy (cf.",
"[26]), we have $|H(K|M=m) - H(U_K)| \\le n\\varepsilon _n \\mbox{ for all } m \\in M_n,\\nonumber $ where $U_K$ denotes the random variable uniformly distributed on ${\\cal K}_n$ .",
"In addition, recall that $M$ is uniformly distributed on ${\\cal M}_n$ , and therefore $nR_M &=& H(M),\\nonumber \\\\nR_K &=& H(U_K) \\le H(K|M=m) +n\\varepsilon _n \\mbox{ for all } m \\in M_n,\\nonumber $ which yields $nR_M = H(M), \\ nR_K \\le H(K|M) +n\\varepsilon _n.",
"\\nonumber $ Since $I(MK;Z^n) \\le n\\varepsilon _n$ by assunption and $H(MK|\\overline{Y}^n) \\le n\\varepsilon _n$ by Fano inequality, we obtain ${n(R_M +R_K)}\\nonumber \\\\&\\le & H(M)+H(K|M) +n\\varepsilon _n\\nonumber \\\\&=& H(MK) +n\\varepsilon _n\\nonumber \\\\& \\le & H(MK) -H(MK|\\overline{Y}^n) +2n\\varepsilon _n\\nonumber \\\\&= & I(MK;\\overline{Y}^n) +2n\\varepsilon _n\\nonumber \\\\&\\le & I(MK;\\overline{Y}^n) -I(MK;Z^n)+3n\\varepsilon _n.$ On the other hand, ${I(MK;\\overline{Y}^n) }\\nonumber \\\\&=& I(MKS^n;\\overline{Y}^n)-I(S^n;\\overline{Y}^n|MK)\\nonumber \\\\&=& I(MKS^n;\\overline{Y}^n)-H(S^n|MK)\\nonumber \\\\&& \\qquad \\qquad +H(S^n|MK\\overline{Y}^n)$ and similarly ${I(MK;Z^n)}\\nonumber \\\\& = & I(MKS^n;Z^n)-H(S^n|MK) \\nonumber \\\\&& \\qquad \\qquad +H(S^n|MKZ^n).$ Thus, inequality (REF ) is continued to ${n(R_M +R_K)}\\nonumber \\\\&\\le & I(MKS^n;\\overline{Y}^n) - I(MKS^n; Z^n)\\nonumber \\\\& &-H(S^n|MKZ^n) +H(S^n|MK\\overline{Y}^n)+3n\\varepsilon _n\\\\&\\le & I(MKS^n;\\overline{Y}^n) - I(MKS^n; Z^n) \\nonumber \\\\&&\\qquad \\qquad \\qquad +H(S^n|MK\\overline{Y}^n) +3n\\varepsilon _n\\\\&\\le & I(MKS^n;\\overline{Y}^n) - I(MKS^n; Z^n) +4n\\varepsilon _n\\\\&=&\\sum _{t=1}^n I(MKS^n; \\overline{Y}_t|\\overline{Y}^{t-1})\\nonumber \\\\& & \\qquad \\qquad -\\sum _{t=1}^n I(MKS^n; Z_t|Z_{t+1}^{n})+4n\\varepsilon _n\\nonumber \\\\&\\stackrel{(c)}{=} & \\sum _{t=1}^n I(MKS^nZ_{t+1}^n;\\overline{Y}_t|\\overline{Y}^{t-1})\\nonumber \\\\& & -\\sum _{t=1}^n I(MKS^n\\overline{Y}^{t-1}; Z_t|Z_{t+1}^{n})+4n\\varepsilon _n\\nonumber \\\\&\\stackrel{(d)}{=} & \\sum _{t=1}^n I(MKS^n;\\overline{Y}_t|\\overline{Y}^{t-1}Z_{t+1}^n)\\nonumber \\\\& & \\qquad \\qquad -\\sum _{t=1}^n I(MKS^n; Z_t|\\overline{Y}^{t-1}Z_{t+1}^{n})+4n\\varepsilon _n\\nonumber \\\\&\\stackrel{(e)}{=} & \\sum _{t=1}^n I(S_tV_t;\\overline{Y}_t|U_t)-\\sum _{t=1}^n I(S_tV_t; Z_t|U_t)+4n\\varepsilon _n,\\nonumber \\\\&& \\\\&=& \\sum _{t=1}^n I(S_tV_t;S_tY_t|U_t)-\\sum _{t=1}^n I(S_tV_t; Z_t|U_t)+4n\\varepsilon _n\\nonumber \\\\&=&\\sum _{t=1}^n I(V_t;Y_t|S_tU_t)-\\sum _{t=1}^n I(V_t; Z_t|S_tU_t)\\nonumber \\\\&& \\qquad \\qquad +\\sum _{t=1}^nH(S_t|Z_tU_t)+4n\\varepsilon _n,$ where $(c)$ and $(d)$ follow from Csiszár identity (cf.",
"[19]); $(e)$ comes from (REF ).",
"Therefore, using (REF ), we have $R_M +R_K \\le I(V;Y|SU)-I(V; Z|SU)+H(S|ZU)+4\\varepsilon _n.$ Thus, letting $n\\rightarrow \\infty $ in (REF ), we conclude (), thereby completing the proof of Theorem REF .",
"An immediate consequence of Theorem REF is the following two corollaries, where we let $C^{{\\scriptsize \\rm M}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ (called the SM capacity) denote the supremum of the projection of ${\\cal C}{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ on the $R_M$ -axis, and $C^{{\\scriptsize \\rm K}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ (called the SK capacity) denote the supremum of the projection of ${\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ on the $R_K$ -axis.",
"Then, we have, with $UV$ and $S$ that may be correlated, Corollary 1 (Non-causal SM capacity) ${C^{{\\scriptsize \\rm M}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}}\\nonumber \\\\&=&\\max _{p_{SUV}}\\min (I(V;Y|SU)-I(V;Z|SU)+H(S|ZU),\\nonumber \\\\& & \\qquad \\qquad \\qquad I(UV;Y|S)).$ Corollary 2 (Non-causal SK capacity) ${C^{{\\scriptsize \\rm K}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}} }\\nonumber \\\\&=&\\max _{p_{SUV}} (I(V;Y|SU)-I(V;Z|SU)+H(S|ZU)).\\nonumber \\\\&&$ Remark 4 The variable $U$ in (REF ) appears to play the role of “time-sharing\" parameter, so one may wonder if this $U$ can be omitted as in Khisti et al.",
"[11] who have, instead of (REF ), given the following formula with the time-sharing parameter $U$ omitted: $C^{\\mbox{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}=\\max _{p_{SV}} \\bigl (I(V;Y|S) - I(V;Z|S)+H(S|Z) \\bigr ).$ It is evident that the achievability in formula (REF ) subsumes that of formula (REF ) in that we can set $U=\\emptyset $ in (REF ) to get (REF ).",
"We notice here also that, as will be seen from the proof of Theorem REF , if the WTC in consideration is a degraded one ($Z$ is a degraded version of $Y$ ), then the right-hand sides of both (REF ) and (REF ) boil down to the right-hand side of () in Corollary REF .",
"Nevertheless, the $U$ cannot be omitted in general, because in maximizing (REF ) the “time-sharing\" parameter $U$ cannot necessarily be selected so as to be independent of the given “state\" $S$ (see “technical flaws\" in the converse proofs of [11] and [10]).",
"We are thus tempted to think about the following conjecture: Conjecture: There exists a WTC with non-causal CSI $S$ at both Alice and Bob such that ${\\max _{p_{SUV}} (I(V;Y|SU)-I(V;Z|SU)+H(S|ZU))}\\nonumber \\\\&>&\\max _{p_{SV}} \\bigl (I(V;Y|S) - I(V;Z|S)+H(S|Z) \\bigr ),$ which then means that formula (REF ) is not tight in general."
],
[
" Wiretap channel with Causal CSI",
"The encoding scheme in [25] used to prove Theorem REF is based on the soft covering lemma as well as the “non-causal\" likelihood encoding [20].",
"Since the re-interpretation of this scheme from the “causal\" viewpoint is the very point to be invoked in this section, we here summarize the (non-causal) encoding scheme given by [25].",
"Codebook ${\\cal B}_n$ : Define the index sets ${\\cal I}_n \\stackrel{\\Delta }{=} [1: 2^{nR_1}]$ and ${\\cal J}_n \\stackrel{\\Delta }{=} [1: 2^{nR_2}]$ .",
"For each $i\\in {\\cal I}_n $ , generate ${\\bf u}_i \\in {\\cal U}^n$ of length $n$ that are i.i.d.",
"according to probability distribution $p_U^n$ for a random variable $U$ denotes the $n$ times product probability distribution of $p_U$ .",
"Similarly for $p_{V|U}^n$ .",
"$p_U^n$ .",
"Next, given $i\\in {\\cal I}_n $ , for each $(j, k, m) \\in {\\cal J}_n\\times {\\cal K}_n \\times {\\cal M}_n$ generate ${\\bf v}_{ijkm} \\in {\\cal V}^n$ that are i.i.d.",
"according to conditional probability distribution $p^n_{V|U}(\\cdot | {\\bf u}_i).$ Likelihood encoder $f_n$ : Given $m\\in {\\cal M}_n$ and ${\\bf s}\\in {\\cal S}^n$ , the encoder “randomly\" chooses $(i, j, k) \\in {\\cal I}_n\\times {\\cal J}_n \\times {\\cal K}_n$ according to the conditional probability ratio “proportional\" to $f_{\\sf LE}(i,j,k|m, {\\bf s}) \\stackrel{\\Delta }{=} p^n_{S|UV}({\\bf s}|{\\bf u}_i, {\\bf v}_{ijkm}),$ where $p_{S|UV}$ is the conditional probability distribution induced from $p_{SUVX}$ .",
"The encoder declares the chosen index $k\\in {\\cal K}_n$ as the key.",
"Given the chosen $({\\bf u}_i, {\\bf v}_{ijkm})$ , the channel input sequence ${\\bf x}\\in {\\cal X}^n$ is generated according to conditional probability distribution $p_{X|SUV}^n(\\cdot |{\\bf s}, {\\bf u}_i, {\\bf v}_{ijkm})$ .",
"Decoder $\\phi _n$ : Upon observing the channel output ${\\bf y}\\in {\\cal Y}^n$ , the decoder searches for a unique $(\\hat{i},\\hat{j},\\hat{k}, \\hat{m})$ $ \\in {\\cal I}_n\\times {\\cal J}_n\\times {\\cal K}_n \\times {\\cal M}_n$ such that $({\\bf u}_{\\hat{i}}, {\\bf v}_{\\hat{i}\\hat{j}\\hat{k}\\hat{m}}, {\\bf y})\\in {\\cal T}_{\\epsilon }^n(p_{UVY}),$ where ${\\cal T}_{\\epsilon }^n(p_{UVY})$ denotes the set of jointly $\\varepsilon $ -typical sequences (cf.",
"[26]).",
"If such a unique quadruple is found, then set $\\phi _n({\\bf y}) = (\\hat{m}, \\hat{k})$ .",
"Otherwise, $\\phi _n({\\bf y}) = (1, 1)$ .",
"Remark 5 Roughly speaking, the likelihood encoder $f_n$ can be regarded as a smoothed version of the joint typicality encoder (cf.",
"Gelfand and Pinsker [21]) that, given ${\\bf s}$ , picks up “at random\" sequences $({\\bf u}_i, {\\bf v}_{ijkm})$ with larger weights on jointly typical (with ${\\bf s})$ sequences and smaller weights on jointly atypical sequences.",
"Theorem REF is of crucial significance in the sense that this provides the “best\" inner bound to subsume, in a unifying way, all the known results in this field for WTCs with “non-causal\" CSI available at Alice.",
"As such, on the other hand, at first glance Theorem REF does not appear to give any insights into WTCs with “causal\" CSI.",
"However, for the region ${\\cal R}_{\\sf in}(p_{SUV})$ with a class of some simple but relevant $UV$ s, it is possible to re-interpret ${\\cal R}_{\\sf in}(p_{SUV})$ as inner bounds for WTCs with “causal\" CSI at Alice.",
"This operation is called plugging, which is developed hereafter.",
"The “causal code\" that we consider in this section is the following, which is the causal counterpart of the non-causal code defined as in Definition REF : Definition 7 (Causal code) An $(n, R_M, R_K)$ -code $c_n$ for the WTC with “causal\" CSI at Alice and message set ${\\cal M}_n$ and key set ${\\cal K}_n$ is a triple of functions $(f_n^{(1)}, f_n^{(2)}, \\phi _n)$ such that 1) $ f_n^{(1)}: {\\cal M}_n\\times {\\cal S}^{t} \\rightarrow {\\cal P}({\\cal X}) \\quad (t=1,2,\\cdots , n)$ ; 2) $f_n^{(2)}: {\\cal M}_n\\times {\\cal S}^n \\rightarrow {\\cal P}({\\cal K}_n)$ , 3) $\\phi _n: {\\cal Y}^n \\rightarrow {\\cal M}_n\\times {\\cal K}_n$ , where $ f_n^{(1)}, f_n^{(2)}$ are stochastic functions.",
"Remark 6 One may wonder if $f_n^{(2)}$ in the above should be $f_n^{(2)}: {\\cal M}_n \\rightarrow {\\cal K}_n$ because we are here considering “causal\" encoders but $f_n^{(2)}$ here looks to require $S^n$ at once before the beginning of encoding at Alice.",
"However, actually, the operation $f_n^{(2)}: {\\cal M}_n\\times {\\cal S}^n \\rightarrow {\\cal P}({\\cal K}_n) $ can be carried out by Alice at the end of the current block (of length $n$ ).",
"This is possible with causal codes.",
"Definition 8 (Causal SM-SK capacity region) The SM-SK capacity region of the WTC with “causal\" CSI at Alice, denoted by ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ , is the set of all causally SM-SK achievable rate pairs with CSI at Alice, and the supremum of the projection of ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ on the $R_M$ -axis, denoted by $C^{{\\scriptsize \\rm M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ , is called the SM capacity, whereas the supremum of the projection of ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ on the $R_K$ -axis is called the SK capacity, denoted by $C^{{\\scriptsize \\rm K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ .",
"Definition 9 (Causal achievability) We now consider the following special class of random variables $UV$ 's such that there exists some $\\tilde{U}\\tilde{V}$ independent of $S$ ($\\tilde{U}$ and $\\tilde{V}$ may be correlated) for which $& Case\\ 1): & V=\\tilde{V},\\ U=\\tilde{U}; \\\\& Case\\ 2): &V= (S, \\tilde{V}),\\ U= \\tilde{U}; \\\\& Case\\ 3): & V=\\tilde{V}, \\ U= (S, \\tilde{U}); \\\\&Case\\ 4): & V= (S, \\tilde{V}),\\ U= (S, \\tilde{U}).",
"$ We say that the probability distribution $p_{YZSXUV}$ (or the corresponding random variable $YZSXUV$ ) is causally achievable if, in addition to (REF ) and the independence of $S$ and $\\tilde{U}\\tilde{V}$ , one of conditions (REF ) $\\sim $ () is satisfied.",
"Figure: Causal SM-SK achievable rate region.With these preparations, we have the following causal version of Theorem REF (cf.",
"Fig.",
"REF ), where ${\\cal R}_{\\sf in}^{{\\sf N}}$ as in Section is replaced here by the causally achievable region ${\\cal R}_{\\sf in}^{\\sf C}$ .",
"Theorem 3 (Causal SM-SK inner bound) ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\supset {\\cal R}_{\\sf in}^{\\sf C}\\stackrel{\\Delta }{=}\\bigcup _{{\\sf C:}p_{SUV}} {\\cal R}_{\\sf in}(p_{SUV}),$ where the union is taken over all “causally\" achievable probability distributions $p_{SUV}$ 's and ${\\cal R}_{\\sf in}(p_{SUV})$ is the same one as in Theorem REF .",
"Proof: In this proof too, under all Definitions REF $\\sim $ REF with Definition REF replaced by Definition REF , we invoke the same Codebook ${\\cal B}_n$ and the likelihood encoder $f_n$ as in Section .",
"The point here is to show that the likelihood encoder $f_n$ can in fact be implemented in a causal way for causally achievable probability distributions $p_{SUV}$ 's.",
"Although it may look to be necessary to give the proofs for each of Case 1) $\\sim $ Case 4), the ways of those proofs are essentially the same, so it suffices, without loss of generality, to show that the likelihood encoder $f_n$ can actually be implemented for Case 2) in a causal way.",
"First, recall that, in Case 2), $p_{S|UV}\\equiv p_{S|US\\tilde{V}}$ is the conditional distribution of $S$ given $UV=US\\tilde{V}$ and hence, irrespective of $u, \\tilde{v}$ , $p_{S|US\\tilde{V}}(s| u, s^{\\prime },\\tilde{v})= \\left\\lbrace \\begin{array}{cl}1 & \\mbox{if} \\ s=s^{\\prime }, \\\\0 & \\mbox{if} \\ s\\ne s^{\\prime }.\\end{array}\\right.$ Then, since $p^n$ is a product probability distribution (i.e., memoryless) of $p$ , setting as ${\\bf v}_{ijkm} = ({\\bf s}_{ijkm}, \\tilde{{\\bf v}}_{ijkm})$ , the conditional probability ratio in (REF ) can be evaluated as follows.",
"${f_{\\sf LE}(i,j,k|m, {\\bf s})}\\nonumber \\\\&=&p^n_{S|UV}({\\bf s}|{\\bf u}_i, {\\bf v}_{ijkm})\\nonumber \\\\&=& p^n_{S|US\\tilde{V}}({\\bf s}|{\\bf u}_i, {\\bf s}_{ijkm}, \\tilde{{\\bf v}}_{ijkm})\\nonumber \\\\&=& \\prod _{t=1}^{n}p_{S|US\\tilde{V}}(s^{(t)}| u_i^{(t)}, s^{(t)}_{ijkm},\\tilde{v}^{(t)}_{ijkm}), $ where we have set ${\\bf s}&=& (s^{(1)}, s^{(2)}, \\cdots , s^{(n)}),\\\\{\\bf u}_i &=& (u_i^{(1)}, u_i^{(2)}, \\cdots , u_i^{(n)}),\\\\{\\bf s}_{ijkm} &=& (s_{ijkm}^{(1)}, s_{ijkm}^{(2)}, \\cdots , s_{ijkm}^{(n)}),\\\\\\tilde{{\\bf v}}_{ijkm} &=& (\\tilde{v}_{ijkm}^{(1)}, \\tilde{v}_{ijkm}^{(2)} \\cdots , \\tilde{v}_{ijkm}^{(n)}).$ Now, in view of (REF ), it turns out that $p_{S|US\\tilde{V}}(s^{(t)}| u_i^{(t)}, s^{(t)}_{ijkm},\\tilde{v}^{(t)}_{ijkm})$ in (REF ) is equal to 1 if $s^{(t)} = s^{(t)}_{ijkm}$ ; otherwise, equal to 0 ($t=1,2,\\cdots , n)$ , so that we have, irrespective of $({\\bf u}, \\tilde{{\\bf v}})$ , $p^n_{S|US\\tilde{V}}({\\bf s}|{\\bf u}, {\\bf s}_{ijkm}, \\tilde{{\\bf v}})= \\left\\lbrace \\begin{array}{cl}1 & \\mbox{if} \\ {\\bf s}_{ijkm}={\\bf s}, \\\\0 & \\mbox{if} \\ {\\bf s}_{ijkm}\\ne {\\bf s}.\\end{array}\\right.$ Therefore, in particular, $p^n_{S|US\\tilde{V}}({\\bf s}|{\\bf u}_{i}, {\\bf s}, \\tilde{{\\bf v}}_{ijkm})=1 \\ \\mbox{for all}\\ (i,j,k)\\in {\\cal I}_n\\times {\\cal J}_n\\times {\\cal K}_n,$ so that, given $(m,{\\bf s})$ , the stochastic (non-causal) likelihood encoder $f_n$ as specified in Section chooses $({\\bf u}_{i}, {\\bf s}, \\tilde{{\\bf v}}_{ijkm})$ uniformly over the set ${\\cal L}(m, {\\bf s})\\stackrel{\\Delta }{=} \\lbrace ({\\bf u}_{i}, {\\bf s}, \\tilde{{\\bf v}}_{ijkm})|(i,j,k)\\in {\\cal I}_n\\times {\\cal J}_n\\times {\\cal K}_n \\rbrace .$ We notice here that, since $U\\tilde{V}$ and $S$ are independent and hence $({\\bf u}_{i}, \\tilde{{\\bf v}}_{ijkm})$ , ${\\bf s}_{ijkm}$ and ${\\bf s}$ are also mutually independent, the set ${\\cal L}(m)\\stackrel{\\Delta }{=} \\lbrace ({\\bf u}_{i}, \\tilde{{\\bf v}}_{ijkm})|(i,j,k)\\in {\\cal I}_n\\times {\\cal J}_n\\times {\\cal K}_n \\rbrace $ can actually be generated in advance of encoding, not depending on $({\\bf s}_{ijkm}, \\, {\\bf s}).$ Up to here, it was assumed that the full state information ${\\bf s}$ is non-causally available at the encoder, so the point here is how this non-causal encoder $f_n$ can be replaced by a causal encoder.",
"This is indeed possible, because ${\\bf s}_{ijkm}= {\\bf s}$ can be written componentwise as $s^{(t)}_{ijkm} =s^{(t)}\\ (t=1,2,\\cdots , n)$ and therefore the encoder can set $s^{(t)}_{ijkm}$ to be $s^{(t)}$ at each time $t$ using the state information $s^{(t)}$ available at time $t$ at the encoder, which clearly can be carried out in the “causal\" way.",
"Moreover, $({\\bf u}_i, \\tilde{{\\bf v}}_{ijkm})$ can also be fed in the causal way (componentwise) according as $(u_i^{(t)},\\tilde{v}_{ijkm}^{(t)})$ ($t=1,2,\\cdots , n)$ , because $({\\bf u}_i, \\tilde{{\\bf v}}_{ijkm})$ was generated in advance of encoding.",
"Thus, given the chosen $({\\bf u}_i, {\\bf s}, \\tilde{{\\bf v}}_{ijkm})$ , the encoder generates the channel input sequence ${\\bf x}=(x^{(1)}, x^{(2)}, \\cdots , x^{(n)}) \\in {\\cal X}^n$ according to the conditional probability: ${p_{X|SUS\\tilde{V}}^n({\\bf x}|{\\bf s}, {\\bf u}_i, {\\bf s}, \\tilde{{\\bf v}}_{ijkm})}\\nonumber \\\\&=&\\prod _{t=1}^{n}p_{X|SUS\\tilde{V}}(x^{(t)}| s^{(t)}, u_i^{(t)}, s^{(t)},\\tilde{v}^{(t)}_{ijkm}),$ which implies that the ${\\bf x}$ can also be generated in the causal way according as $x^{(t)}\\ (t=1,2, \\cdots , n)$ , thereby completing the proof of Theorem REF .",
"So far in this section we have invoked, as a crucial step, the argument of plugging, the logical core of which is schematically summarized as follows: Proposition 1 (Principle of plugging) Consider a channel coding system (memoryless but not necessarily WTCs) with CSI $S$ and auxiliary random variables $U_1, U_2, \\cdots , U_a$ together with rate tuple $(R_1, R_2, \\cdots , R_b)$ to be used for generation of the random code ${{\\cal C}=}\\nonumber \\\\& &\\left\\lbrace ({\\bf u}_{1i_1},{\\bf u}_{2i_2},\\cdots , {\\bf u}_{ai_a})\\right\\rbrace _{i_1\\in [1:2^{nR^{\\prime }_{1}}], i_2\\in [1:2^{nR^{\\prime }_{2}}], \\cdots , i_a\\in [1:2^{nR^{\\prime }_{a}}]}\\nonumber \\\\&&$ where each $R^{\\prime }_k$ ($k=1,2,\\cdots , a$ ) ia a partial sum of $R_1,R_2,\\cdots , R_b$ (for example, $R^{\\prime }_1=R_1+R_3, R^{\\prime }_2=R_2$ , etc.)",
"and each codeword $({\\bf u}_{1 i_1},{\\bf u}_{2i_2},\\cdots , {\\bf u}_{ai_a})$ is generated according to product probability distribution $p^n_{U_1U_2\\cdots U_a}$ (or its marginal conditional distributions).",
"Given message $m$ and state sequence ${\\bf s}$ , the non-causal (likelihood) encoder $f_n$ stochastically picks $f_n$ may also be a joint typicality encoder (cf.",
"Example REF ).",
"an element of ${\\cal C}$ and maps it “componentwise\" to a channel input ${\\bf x}$ according to conditional probability distribution $p^n_{X|SU_1U_2 \\cdots U_a}(\\cdot |{\\bf s},{\\bf u}_{1i_1},{\\bf u}_{2i_2},\\cdots , {\\bf u}_{ai_a})$ .",
"Now suppose that any rate tuple $(R_1, R_2, \\cdots , R_b)$ satisfying the rate constraints $F_1(R_1, R_2,\\cdots , R_b; U_1, U_2,\\cdots , U_a; S) & \\ge & 0,\\\\F_2(R_1, R_2,\\cdots , R_b; U_1, U_2,\\cdots , U_a; S) & \\ge & 0,\\\\\\ \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots & & \\nonumber \\\\F_c(R_1, R_2,\\cdots , R_b; U_1, U_2,\\cdots , U_a; S) & \\ge & 0$ is “non-causally\" SM-SK achievable.",
"Then, any rate tuple $(R_1, R_2, \\cdots , R_b)$ satisfying the rate constraints () $\\sim $ () with $U_1 &=& \\tilde{U}_1 \\mbox{\\ or\\ } (S,\\tilde{U}_1); U_2 = \\tilde{U}_2 \\mbox{\\ or\\ } (S, \\tilde{U}_2);\\nonumber \\\\& &\\qquad \\qquad \\qquad \\cdots ; U_a = \\tilde{U}_a \\mbox{\\ or\\ } (S,\\tilde{U}_a)$ is “causally\" SM-SK achievable, where $\\tilde{U}_1, \\tilde{U}_2, \\cdots , \\tilde{U}_a$ (may be correlated) are independent of $S$ .",
"Example 1 A simple example (with $Z \\equiv \\emptyset $ (constant variable)) is the relation of the Gelfand-Pinsker (non-causal) coding [21] and the Shannon strategy (causal) coding [4].",
"The former gives the formula $C^{\\rm M}_{\\mbox{{\\scriptsize \\rm NCSI-E}}} = \\max _{p_{SU}}(I(U;Y)-I(U;S)),$ while the latter gives the formula $C^{\\rm M}_{\\mbox{{\\scriptsize \\rm CSI-E}}} = \\max _{p_Sp_U}I(U;Y).$ Principle of plugging applied to (REF ) claims that, given independent $S$ and $\\tilde{U}$ , rates $R^{\\prime }=I(\\tilde{U}; Y)-I(\\tilde{U};S)=I(\\tilde{U}; Y)$ and $R^{\\prime \\prime }=I(\\tilde{U}S; Y)-I(\\tilde{U}S; S)$ $=I(\\tilde{U}S; Y)-H(S)$ are “causally\" achievable.",
"It is easy to check that $R^{\\prime } \\ge R^{\\prime \\prime }$ , so in this case $R^{\\prime \\prime }$ is redundant.",
"Thus, the achievablity part of (REF ) is concluded from that of (REF ) without a separate proof."
],
[
"Applications of Theorem ",
"Having established Theorem REF on WTCs with causal CSI at Alice, in this section we develop it for each of Case 1) $\\sim $ Case 4) to demonstrate that, via Theorem REF , we can unifyingly derive the previously known causal “lower\" bounds such as in [12], [13] and [14].",
"In addition, we also demonstrate that a new class of causal “inner\" bounds directly follow from Theorem REF , which could not have been easily obtained without Theorem REF .",
"They are largely classified into Propositions REF and REF .",
"In particular, we emphasize that in this section we are concerned solely with “two-dimensional\" inner/outer bounds of causally achievable rate pairs $(R_M, R_K)$ , which are derived in this paper for the first time.",
"V.A: Causal inner bounds: Let us now scrutinize the claim of Theorem REF .",
"For the convenience of discussion, we record again here the rate constraints (REF ) and () as $R_M &\\le & I(UV; Y)-I(UV; S),\\\\R_M+R_K &\\le & I(V;Y|U)-I(V;Z|U) \\nonumber \\\\& & \\qquad - [I(U;S)-I(U;Y)]^+,$ which is specifically developed according to Cases 1) $\\sim $ 4) as follows.",
"Case 1) : Since $U=\\tilde{U}, V=\\tilde{V}$ and $\\tilde{U}\\tilde{V}$ is independent of $S$ , (REF ) and () reduce to $R_M &\\le & I(\\tilde{U}\\tilde{V}; Y),\\\\R_M+R_K &\\le & I(\\tilde{V};Y|\\tilde{U})-I(\\tilde{V};Z|\\tilde{U}),$ where we have used $I(\\tilde{U}\\tilde{V}; S) =0$ and $[I(\\tilde{U};S)-I(\\tilde{U};Y)]^+=0$ .",
"Clearly, (REF ) is redundant, so only () remains.",
"Hence, removing tilde $\\tilde{}$ to make the notation simpler, we have $R_M+R_K \\le I(V;Y|U)-I(V;Z|U).$ It is not difficult to check that replacing (REF ) by $R_M+R_K \\le I(V;Y)-I(V;Z)$ does not affect the inner region.",
"Thus, ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\supset \\bigcup _{p_{S}p_{V}}\\lbrace \\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:non-p8})}\\rbrace ,$ which implies, in particular, the non-causal SM achievability ($R_K=0$ ) of Dai and Luo [18].",
"Case 2) : Since $U=\\tilde{U}, V=S\\tilde{V}$ and $\\tilde{U}\\tilde{V}$ is independent of $S$ , (REF ) and () are computed as $R_M &\\le &I(\\tilde{U}S\\tilde{V};Y)-I(\\tilde{U}S\\tilde{V};S)\\nonumber \\\\&=& I(\\tilde{U}S\\tilde{V};Y)-H(S);\\\\R_M+R_K &\\le & I(S\\tilde{V};Y|\\tilde{U})-I(S\\tilde{V};Z|\\tilde{U})\\nonumber \\\\& & -[I(\\tilde{U};S)-I(\\tilde{U};Y)]^+\\nonumber \\\\&\\stackrel{(a)}{=}&I(S\\tilde{V};Y|\\tilde{U})-I(S\\tilde{V};Z|\\tilde{U}),$ where $(a)$ follows from $I(\\tilde{U};S)=0.$ Therefore, removing tilde $\\tilde{}$ again to make the notation simpler, we have the rate constraints for Case 2), $R_M &\\le & I(U S V;Y)-H(S);\\\\R_M+R_K &\\le & I(SV;Y|U)-I(SV;Z|U),$ where $UV$ and $S$ are independent.",
"Therefore, any nonnegative rate pair $(R_M, R_K)$ is achievable if rate constraints () and () are satisfied.",
"Thus, we have the following fundamental inner bound: Proposition 2 (Causal SM-SK inner bound: type I) ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\supset \\bigcup _{p_Sp_{UV}}\\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:non-p10}) and\\ } (\\ref {eq:keio-4-1})}\\rbrace .$ An immediate by-product of (REF ) is the following corollary: Corollary 3 (Causal lower bound (1) at Alice) $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}&\\ge & \\max _{p_Sp_{UV}}\\min ( I(SV;Y|U)-I(SV;Z|U), \\nonumber \\\\& & \\qquad \\qquad \\qquad I(U SV;Y)-H(S)),\\\\C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}&\\ge & \\max _{\\stackrel{p_Sp_{UV}:}{I(US V;Y)\\ge H(S)}}(I(SV;Y|U)-I(SV;Z|U)),\\nonumber \\\\& &$ where $UV$ and $S$ are independent.",
"Proof: Setting $R_K=0$ in (REF ) yields (REF ), while setting $R_M=0$ in (REF ) yields ().",
"Let us now consider two special cases of (REF ).",
"A: Let $U=\\emptyset $ (constant variable), then () and () reduce to $R_M &\\le & I(SV;Y)-H(S);\\\\R_M+R_K &\\le & I(SV;Y)-I(SV;Z)$ with independent $V$ and $S$ .",
"Consequently, any nonnegative rate pair $(R_M, R_K)$ is achievable if rate constraints (REF ) and () are satisfied.",
"Thus, we have ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\supset \\bigcup _{p_Sp_{V}}\\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:keio-5}) and\\ } (\\ref {eq:keio-6})\\rbrace }.$ Remark 7 Setting $R_K=0$ in (REF ) yields the SM lower bound: $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}\\ge \\max _{p_Sp_V}\\min ( I(SV;Y)-I(SV;Z), I(SV;Y)-H(S)).$ On the other hand, setting $R_M=0$ in (REF ) yields the SK lower bound: $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}\\ge \\max _{\\stackrel{p_Sp_{V}:}{I(S V;Y)\\ge H(S)}}(I(SV;Y)-I(SV;Z)),$ which was leveraged, without the proof, in Han and Sasaki [14].",
"Next, in order to compare formula (REF ) with the previous result, we develop it in the sequel.",
"First, (REF ) is rewritten as $R_M &\\le & I(SV;Y)-H(S)\\nonumber \\\\&=& I(V;Y) +I(S;Y|V)-H(S)\\nonumber \\\\&\\stackrel{(b)}{=}& I(V;Y) -H(S|VY),$ where $(b)$ follows from the independence of $V$ and $S$ .",
"On the other hand, () is evaluated as follows: ${R_M+R_K}\\nonumber \\\\&\\le &I(SV;Y)-I(SV;Z)\\nonumber \\\\&=& I(V;Y)+I(S;Y|V)-I(S;Z)-I(V;Z|S)\\nonumber \\\\&=& I(V;Y)+H(S|V)-H(S|VY) -H(S)\\nonumber \\\\& & +H(S|Z) -I(V;Z|S)\\nonumber \\\\&=& I(V;Y)-I(V;SZ) +I(V;S)+H(S|V)\\nonumber \\\\& & -H(S|VY)-H(S) +H(S|Z)\\nonumber \\\\&=& I(V;Y)-I(V;SZ)+H(S|Z)-H(S|VY).",
"$ Summarizing, we have, with independent $V$ and $S$ , $R_M &\\le & I(V;Y) -H(S|VY),\\\\R_M+R_K &\\le & I(V;Y)-I(V;SZ)\\nonumber \\\\& & \\qquad +H(S|Z)-H(S|VY).$ Thus, ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\supset \\bigcup _{p_Sp_{V}}\\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:non-p13}) and\\ } (\\ref {eq:keio-7})\\rbrace },$ which is equivalent to (REF ).",
"Now, setting $R_K=0$ in (REF ), it turns out that formula (REF ) is rewritten as $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}& \\ge & \\max _{p_Sp_V}\\min (I(V;Y)-I(V;SZ)\\nonumber \\\\& &\\quad +H(S|Z)-H(S|VY),\\nonumber \\\\& & \\qquad \\qquad I(V;Y) -H(S|VY))$ with independent $V$ and $S$ , which was given as $R_{\\mbox{{\\scriptsize \\rm CSI-1}}}$ by Han and Sasaki [14] (also cf.",
"Fujita [13]).",
"B: Let $V=\\emptyset $ , then () and () reduce to $R_M &\\le & I(US;Y)-H(S),\\\\R_M+R_K &\\le & I(S;Y|U)-I(S;Z|U)$ with independent $U$ and $S$ .",
"It is easy to check that (REF ) and () are rewritten equivalently as $R_M &\\le & I(U;Y)-H(S|UY), \\\\R_M+R_K &\\le & H(S|UZ)-H(S|UY).$ Consequently, any nonnegative pair $(R_M, R_K)$ is achievable if constraints (REF ) and () are satisfied.",
"Thus, ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\supset \\bigcup _{p_Sp_{U}}\\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:keio-10}) and\\ } (\\ref {eq:keio-10-2})\\rbrace }.$ Remark 8 Setting $R_K=0$ in (REF ) yields the lower bound with independent $U$ and $S$ : $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}&\\ge & \\max _{p_Sp_U}\\min (H(S|UZ)-H(S|UY), \\nonumber \\\\& & \\qquad \\qquad \\qquad I(U;Y)-H(S|UY))$ which was given as s $R_{\\mbox{{\\scriptsize \\rm CSI-2}}}$ by Han and Sasaki [14].",
"On the other hand, setting $R_M=0$ in (REF ), we have, for independent $U$ and $S$ , $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}} \\ge \\max _{\\stackrel{p_Sp_U:}{I(U;Y)\\ge H(S|UY)}}(H(S|UZ)-H(S|UY)),$ which is a new type of lower bound.",
"We notice here that either (REF ) or (REF ) does not always outperform the other.",
"Similarly, we can check that either (REF ) or (REF ) does not always outperform the other.",
"The proof of them is given in Appendix .",
"We now have the following two corollaries for WTCs with causal CSI available at “both\" Alice and Bob.",
"Corollary 4 (Causal inner bound (2) at Alice and Bob) Let us consider the WTC with causal CSI at both Alice and Bob, as depicted in Fig.",
"REF .",
"Then, a pair $(R_M, R_K)$ is achievable if the following rate constraints are satisfied: $R_M &\\le & I(V;Y|S);\\\\R_M+R_K &\\le & I(V;Y|S)-I(V;Z|S) \\nonumber \\\\& &\\qquad +H(S|Z),$ where $V$ and $S$ are independent.",
"Thus, ${{\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} \\supset \\bigcup _{p_Sp_{V}}}\\nonumber \\\\& & \\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:keio-12}) and\\ } (\\ref {eq:keio-13-2})}\\rbrace ,\\nonumber \\\\& &$ where ED denotes that the causal CSI $S$ is available at both Alice and Bob.",
"Proof: It is sufficient to replace $Y$ by $SY$ in (REF ) and ().",
".",
"Remark 9 As far as we are concerned with “degraded\" WTCs ($Z$ is a degraded version of $Y$ ), the inclusion $\\supset $ in (REF ) can be replaced by $=$ , so that in this case (REF ) actually gives the causal SM-SK capacity region, as will be explicitly stated later in Theorem REF .",
"Remark 10 Setting $R_M=0$ in (REF ) yields one more new lower bound: $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} \\ge \\max _{p_Sp_V}( I(V;Y|S)-I(V;Z|S) +H(S|Z)).$ where $V$ and $S$ are independent, and $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}}$ denotes the causal SK capacity.",
"On the other hand, setting $R_K =0$ in (REF ) yields the lower bound given by Chia and El Gamal [12]: $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} &\\ge &\\max _{p_Sp_V} \\min ( I(V;Y|S)-I(V;Z|S) \\nonumber \\\\& & \\qquad \\qquad \\quad +H(S|Z),I(V;Y|S)),$ with independent $V$ and $S$ , where $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}}$ denotes the causal SM capacity.",
"Corollary 5 (Causal inner bound (3) at Alice and Bob) Let us consider the WTC with causal CSI at both Alice and Bob, as depicted in Fig.",
"REF .",
"Then, a pair $(R_M, R_K)$ is achievable if the following rate constraints are satisfied: $R_M &\\le & I(U;Y|S)\\\\R_M +R_K &\\le & H(S|UZ),$ where $U$ and $S$ are independent, Thus, ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} \\supset \\bigcup _{p_Sp_{U}}\\lbrace \\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:chia-31}) and\\ (\\ref {eq:chia-41}})\\rbrace .$ Proof: It is sufficient to replace $Y$ by $SY$ in (REF ) and ().",
"Remark 11 Setting $R_K =0$ in (REF ) yields the lower bound given by Chia and El Gamal [12]: $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} \\ge \\max _{p_Sp_U}\\min (H(S|UZ), I(U;Y|S)).$ On the other hand, setting $R_M=0$ in (REF ) yields $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} \\ge H(S|UZ)$ .",
"Also, we can set $U=\\emptyset $ to obtain $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} \\ge \\max _{p_{SX}}H(S|Z),$ which is obviously attained without transmission coding at the encoder, because in this case sharing of common secret key at Alice and Bob is enough without extra transmission of secret message (cf.",
"Ahlswede and Csiszár [16]).",
"Here, in view of (REF ) and [11], it is easy to see that, for reversely degraded ($Y$ is a degraded version of $Z$ ) WTCs, $C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}}=C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}= \\max _{p_{SX}}H(S|Z).$ Remark 12 Comparing (REF ) and (REF ), we see that either one does not necessarily subsume the other, which depends on whether $I(V;Y|S) \\ge I(V;Z|S)$ or not.",
"Specifically, in the case of $I(V;Y|S) \\ge I(V;Z|S)$ coding helps, otherwise coding does not help.",
"Notice that, for example, if $Z$ is a degraded version of $Y$ , then $I(V;Y|S) \\ge I(V;Z|S)$ always holds and so coding helps.",
"Case 3) : Since $U=S\\tilde{U}, V=\\tilde{V}$ and $\\tilde{U}\\tilde{V}$ is independent of $S$ , (REF ) and () are computed as $R_M &\\le & I(\\tilde{U}S\\tilde{V}; Y)-I(\\tilde{U}S\\tilde{V}; S)\\nonumber \\\\&=& I(\\tilde{U}S\\tilde{V}; Y) -H(S);$ $R_M+R_K &\\le & I(\\tilde{V};Y|S \\tilde{U})-I(\\tilde{V};Z|S\\tilde{U})\\nonumber \\\\& & -[I(S\\tilde{U};S) - I(S\\tilde{U};Y)]^+\\nonumber \\\\&=& I(\\tilde{V};Y|S\\tilde{U})-I(\\tilde{V};Z|S\\tilde{U})\\nonumber \\\\& & -[H(S)-I(S\\tilde{U};Y)]^+.",
"$ As a consequence, removing tilde $\\tilde{}$ , we have the rate constraints, with independent $UV$ and $S$ , $R_M &\\le & I(USV;Y) -H(S);\\\\R_M+R_K &\\le & I(V;Y|SU)-I(V;Z|SU)\\nonumber \\\\& & -[H(S)-I(SU;Y)]^+.$ Therefore, any nonnegative rate pair $(R_M, R_K)$ is achievable if rate constraints (REF ) and () are satisfied.",
"Thus, we have the following one more fundamental inner bound (type II), which is paired with Proposition REF (type I): Proposition 3 (Causal SM-SK inner bound: type II) ${{\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}}} \\nonumber \\\\&\\supset & \\bigcup _{p_Sp_{UV}}\\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:le-3}) and\\ } (\\ref {eq:kan-1})}\\rbrace .\\nonumber \\\\& &$ Remark 13 We observe here that (REF ) and () remain invariant under replacement of $Z$ by $SZ$ .",
"This implies that the achievability due to Case 3) is invulnerable to the leakage of state information $S^n$ to Eve, which is in notable contrast with Case 2).",
"An immediate consequence of (REF ) is the following corollary: Corollary 6 (Causal lower bound (4) at Alice) $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}&\\ge &\\max _{p_Sp_{UV}} \\min (I(V;Y|SU)-I(V;Z|SU)\\nonumber \\\\& & -[H(S)-I(SU;Y)]^+, I(USV;Y)-H(S)),\\nonumber \\\\\\\\C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}&\\ge &\\max _{\\stackrel{p_Sp_{UV}:}{I(US V;Y)\\ge H(S)}}(I(V;Y|SU)-I(V;Z|SU) \\nonumber \\\\& &\\qquad \\qquad \\qquad -[H(S)-I(SU;Y)]^+),$ where $UV$ and $S$ are independent.",
"Proof: Setting $R_K=0$ in (REF ) yields (REF ), while setting $R_M=0$ in (REF ) yields ().",
"Remark 14 (Comparison of Case 2) and Case 3)) We first notice that (REF ) is the same as (), and moreover, noting that $H(S)-I(SU;Y)\\nonumber &=& H(S|Y)-I(U;Y|S)\\nonumber \\\\&=& H(S|Y) -I(U;SY)\\nonumber \\\\&=& H(S|Y)-I(U;Y)-I(U;S|Y)\\nonumber \\\\&=& H(S|UY)-I(U;Y)$ and summarizing (REF ), () and (REF ), we have for Case 3).",
"$R_M &\\le & I(USV;Y) -H(S);\\\\R_M+R_K &\\le & I(V;Y|SU)-I(V;Z|SU)\\nonumber \\\\& & -[H(S|UY)-I(U;Y)]^+.$ In order to compare this with that for Case 2), we rewrite () and () as $R_M&\\le & I(USV;Y) -H(S);\\\\R_M+R_K &\\le & I(SV;Y|U)-I(SV;Z|U)\\nonumber \\\\&=& I(S;Y|U)-I(S;Z|U)\\nonumber \\\\& & +I(V;Y|SU)-I(V;Z|SU)\\nonumber \\\\&=& I(V;Y|SU)-I(V;Z|SU) \\nonumber \\\\& & -[H(S|UY)-H(S|UZ)].\\nonumber $ Thus, for Case 2), $R_M &\\le & I(USV;Y) -H(S);\\\\R_M+R_K &\\le & I(V;Y|SU)-I(V;Z|SU)\\nonumber \\\\& & -[H(S|UY)-H(S|UZ)].$ Comparing () and (), we see that the difference consists in that of the terms $[H(S|UY)-I(U;Y)]^+$ and $[H(S|UY)-H(S|UZ)]$ , so either one does not necessarily subsume the other, which depends on the choice of achievable probability distributions $p_{YZSXUV}.$ Remark 15 As such, to get more insight, let us consider the WTC with causal CSI available at both Alice and Eve, as depicted in Fig.",
"REF .",
"Then, since $[H(S|UY)-I(U;Y)]^+\\le H(S|UY)$ and $[H(S|UY)-H(S|UZ)]=H(S|UY)$ , in this case Case 3) outperforms Case 2), where $Z$ was replaced by $SZ$ as the state $S$ is available also at Eve (cf.",
"Remark REF ).",
"This means that Case 3) is preferable to Case 2) when Eve have full access to $S^n$ .",
"On the other hand, consider an opposite case with CSI available at both Alice and Bob as n Fig.",
"REF .",
"Then, since $H(S|UY)=0$ with $SY$ instead of $Y$ and hence $[H(S|UY)-I(U;Y)]^+=0$ and $[H(S|UY)-H(S|UZ)]=-H(S|UZ)$ , we see that, in this case, Case 2) outperforms Case 3).",
"Remark 16 As is seen from the proof of Theorem REF in Bunin et al.",
"[24], [25], in both cases of Case 2) and Case 3) the state information $S^n$ is to be reliably reproduced at Bob, while the crucial difference between Case 2) and Case 3) is that in Case 2) the $S^n$ is used to carry on secure transmission of message and/or key between Alice and Bob, whereas in Case 3) the $S^n$ is not used to convey secure message and/or key but simply to help reliable (secured or unsecured) transmission.",
"On the other hand, in Case 1) the $S^n$ is not to be reproduced at Bob.",
"As was illustrated in Remark REF , favorable choices of these three cases depend on the probabilistic structure of WTCs.",
"Figure: WTC with the same CSI available at Alice and Eve (t=1,2,⋯,nt=1,2,\\cdots , n).Case 4) : Since $U=S\\tilde{U}, V=S\\tilde{V}$ and $\\tilde{U}\\tilde{V}$ is independent of $S$ , (REF ) and () are computed as $R_M &\\le & I(\\tilde{U}S\\tilde{V}; Y)-I(\\tilde{U}S\\tilde{V}; S)\\nonumber \\\\&=& I(\\tilde{U}S\\tilde{V}; Y) -H(S);\\\\R_M+R_K &\\le & I(S\\tilde{V};Y|S \\tilde{U})-I(S\\tilde{V};Z|S\\tilde{U})\\nonumber \\\\& & -[I(S\\tilde{U};S) - I(S\\tilde{U};Y)]^+\\nonumber \\\\&=& I(\\tilde{V};Y|S\\tilde{U})-I(\\tilde{V};Z|S\\tilde{U})\\nonumber \\\\& & -[H(S)-I(S\\tilde{U};Y)]^+, $ which is nothing but (REF ) and (REF ) in Case 3), and therefore Case 4) reduces to Case 3).",
"V.B: Causal outer bound: So far we have discussed a diversity of causal SM-SK inner bounds, but not about outer bounds.",
"This is because, in general, it is much harder with the problem of causal outer bounds, in contrast with non-causal outer bounds.",
"However, we can show an example of causal “tighter\" outer bound, which is a rare case (from the causal viewpoint) and is paired with Proposition REF (achievability part).",
"In passing this section we consider this problem.",
"To do so, we first notice that the coding scheme used to prove Proposition REF required the CSI $S^n$ to be reliably reproduced at Bob, i.e., $H(S^n|Y^n)\\le n\\varepsilon _n$ .",
"This kind of coding scheme is said to be state-reproducing (cf.",
"Han and Sasaki [14]).",
"Then, one may ask what happens if we confine ourselves to within such state-reproducing coding schemes.",
"An answer is: Proposition 4 (Causal/non-causal outer bound) With state-reproducing coding schemes, we have the following outer bound: ${{\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}}\\subset {\\cal C}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}} \\nonumber \\\\&\\subset & \\bigcup _{p_{SUV}}\\lbrace {\\mbox{rate pairs $(R_M, R_K)$ satisfying(\\ref {eq:non-p10}) and\\ } (\\ref {eq:keio-4-1})}\\rbrace \\nonumber \\\\& &$ Notice that the difference between Proposition REF (outer bound) and Proposition REF (inner bound) is that the union in the former is taken over all probability distributions $p_{SUV}$ 's, while in the latter the union is taken over all product probability distributions $p_Sp_{UV}$ 's.",
"Proof: It suffices only to literally parallel the converse part of Theorem REF with $\\overline{Y}^n=S^nY^n$ replaced by $Y^n$ , while using $H(S^n|Y^n) \\le n\\varepsilon _n$ (due to the state-reproducibility) in inequality () of Section , which together with (REF ) (with $Y_t$ instead of $\\overline{Y}_t$ ) brings about the required outer bound.",
"An immediate consequence of (REF ) is the following corollary, which is paired with Corollary REF : Corollary 7 (Causal/non-causal upper bound) With state-reproducing coding schemes, we have the upper bounds: $C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm CSI-E}}} &\\le &C^{{\\scriptsize \\sf M}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}} \\nonumber \\\\&\\le & \\max _{p_{SUV}}\\min ( I(SV;Y|U)-I(SV;Z|U), \\nonumber \\\\& &\\qquad \\qquad \\qquad \\quad I(U SV;Y)-H(S)),\\\\C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}} &\\le &C^{{\\scriptsize \\sf K}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}} \\nonumber \\\\&\\le & \\max _{\\stackrel{p_{USV}:}{I(US V;Y)\\ge H(S)}}(I(SV;Y|U)-I(SV;Z|U)).\\nonumber \\\\& &$"
],
[
"SM-SK Capacity Theorems for Degraded WTCs",
"1) Let us now address the problem of SM-SK capacity regions to provide the exact SM-SK capacity region for degraded WTCs with causal/non-causal CSI available at “both\" Alice and Bob as in Fig.",
"REF .",
"To do so, let the corresponding causal SM-SK capacity region be denoted by ${\\cal C}^{{\\sf d}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}}$ .",
"Similarly, the corresponding non-causal SM-SK capacity region is denoted by ${\\cal C}^{{\\sf d}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ .",
"Moreover, let $\\overline{{\\cal R}}^{{\\sf d}}_{\\sf in}(p_{SX})$ denote the set of all nonnegative rate pairs $(R_M, R_K)$ satisfying the rate constraints: $R_M &\\le & I(X; Y|S),\\\\R_M+R_K &\\le & I(X;Y|S)-I(X;Z|S)+H(S|Z) \\nonumber \\\\& &.$ Then, we have Theorem 4 (Causal/non-causal SM-SK capacity region) Consider a degraded WTC ($Z$ is a degraded version of $Y$ ) with causal/non-causal CSI at Alice and Bob.",
"Then, ${\\cal C}^{{\\sf d}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} &=&{\\cal C}^{{\\sf d}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}} \\nonumber \\\\& =&\\overline{{\\cal R}}^{{\\sf d}}_{\\sf in}\\stackrel{\\Delta }{=}\\bigcup _{p_{SX}} \\overline{{\\cal R}}^{{\\sf d}}_{\\sf in}(p_{SX}),$ where the union is taken over all possible probability distributions $p_{SX}$ 's.",
"Remark 17 Notice, in particular, that Theorem REF means also that the causal and non-causal capacity regions coincide for degraded WTCs.",
"An immediate consequence of Theorem REF is Corollary 8 $C^{{\\sf d,M}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} &=&C^{{\\sf d,M}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}} \\nonumber \\\\&=& \\max _{p_{SX}}\\min (I(X;Y|S)-I(X;Z|S) \\nonumber \\\\& & \\qquad \\qquad +H(S|Z), I(X;Y|S)),\\\\C^{{\\sf d,K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}} &=&C^{{\\sf d,K}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}} \\nonumber \\\\&=&\\max _{p_{SX}}(I(X;Y|S)-I(X;Z|S) \\nonumber \\\\& &\\qquad \\qquad +H(S|Z)),$ where $C^{{\\sf d,M}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}},C^{{\\sf d,M}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ $\\left({\\rm resp}.",
"\\ C^{{\\sf d,K}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}},C^{{\\sf d,K}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}} \\right)$ is the supremum of the projection of ${\\cal C}^{{\\sf d}}_{\\mbox{{\\scriptsize \\rm CSI-ED}}},{\\cal C}^{{\\sf d}}_{\\mbox{{\\scriptsize \\rm NCSI-ED}}}$ on the $R_M$ -axis (resp.",
"$R_K$ -axis).",
"Remark 18 Formula (REF ) has earlier been given by [12] in a quite different manner.",
"Proof of achievability for Theorem REF : Let $(X,S)$ be arbitrarily given, then the functional representation lemma [19] claims that there exist a random variable $V$ and a deterministic function $f: {\\cal V}\\times {\\cal S}\\rightarrow {\\cal X}$ such that $V$ and $S$ are independent and $X=f(V, S)$ .",
"Then, Theorem REF (Case 2) $A$ : with $U=\\emptyset $ ) claims that any rate pair $(R_M, R_K)$ satisfying the rate constraints (REF ) and (), that is, $R_M &\\le & I(V;Y|S);\\\\R_M+R_K &\\le & I(V;Y|S)-I(V;Z|S) \\nonumber \\\\& & \\qquad \\qquad \\qquad +H(S|Z),$ is “causally\" achievable.",
"Then, it suffices to observe that the right-hand sides of (REF ) is rewritten as ${ I(V;Y|S)}\\nonumber \\\\&\\stackrel{(e)}{=}& I(VX;Y|S)\\nonumber \\\\&\\stackrel{(g)}{=}& I(X;Y|S),$ where $(e)$ is because $X$ is a deterministic function of $(V, S)$ ; $(g)$ follows from the Markov chain property $UV\\rightarrow SX \\rightarrow YZ$ .",
"Similarly, ()) can be rewritten as ${ I(V;Y|S)-I(V;Z|S) +H(S|Z)}\\nonumber \\\\&=& I(X;Y|S) -I(X;Z|S)+H(S|Z).$ Proof of converse for Theorem REF : Theorem REF claims that any achievable rate pair $(R_M, R_K)$ must satisfy the rate constraints (REF ) and (), that is, $R_M &\\le & I(UV; Y|S),\\\\R_M+R_K &\\le & I(V;Y|SU)-I(V;Z|SU) \\nonumber \\\\& &\\qquad \\qquad +H(S|ZU)$ with some $UVSXYZ$ .",
"The right-hand sides of (REF ) and () are evaluated as follows: $I(UV;Y|S) &\\le & I(UVX;Y|S)\\nonumber \\\\&=&I(X;Y|S) +I(UV;Y|SX)\\nonumber \\\\&\\stackrel{(v)}{=}&I(X;Y|S),$ where $(v)$ follows from the Markov chain property $UV\\rightarrow SX\\rightarrow Y$ .",
"Hence, $I(UV;Y|S) \\le I(X;Y|S).$ On the other hand, ${I(V;Y|SU) - I(V;Z|SU)}\\nonumber \\\\&=& I(VX;Y|SU)-I(X;Y|SUV)\\nonumber \\\\&& - I(VX;Z|SU)+I(X;Z|SUV)\\nonumber \\\\&=& I(VX;Y|SU)-I(VX;Z|SU) \\nonumber \\\\& & - [I(X;Y|SUV)-I(X;Z|SUV)]\\nonumber \\\\&\\stackrel{(a)}{=}& I(X;Y|SU)-I(X;Z|SU) \\nonumber \\\\& & - [I(X;Y|SUV)-I(X;Z|SUV)]\\nonumber \\\\&\\stackrel{(b)}{\\le } & I(X;Y|SU)-I(X;Z|SU) \\nonumber \\\\&=&I(UX;Y|S)-I(UX;Z|S)\\nonumber \\\\& &-[I(U;Y|S)-I(U;Z|S)]\\nonumber \\\\&\\stackrel{(c)}{\\le } &I(X;Y|S)-I(X;Z|S)\\nonumber \\\\& &-[I(U;Y|S)-I(U;Z|S)],$ where $(a), (c)$ follows from the Markov chain property $UV\\rightarrow SX\\rightarrow YZ$ ; $(b)$ follows from the assumed degradedness.",
"Moreover, since $H(S|ZU) -H(S|Z) = - I(S;U|Z),$ it follows that ${H(S|ZU) -H(S|Z)-[I(U;Y|S)-I(U;Z|S)]}\\nonumber \\\\&=& - I(S;U|Z)-[I(U;Y|S)-I(U;Z|S)]\\nonumber \\\\&=& - I(U;Y|S) - [I(S;U|Z) -I(U;Z|S)]\\nonumber \\\\&=& I(S;U)-I(U;SY) -[ I(S;U)-I(U;Z)]\\nonumber \\\\&=& -I(U;SY)+I(U;Z)\\nonumber \\\\&\\le & -I(U;Y)+I(U;Z)\\nonumber \\\\&\\stackrel{(j)}{=}&- I(U;Y|Z) \\le 0,$ where $(j)$ follows from the assumed degradedness.",
"Therefore, ${H(S|ZU)}\\nonumber \\\\& & -[I(U;Y|S)-I(U;Z|S)]\\nonumber \\\\& &\\le H(S|Z)).$ Thus, by virtue of (REF ) and (REF ), we obtain ${I(V;Y|SU)-I(V;Z|SU)+ H(S|ZU)}\\nonumber \\\\&\\le & I(X;Y|S)-I(X;Z|S) +H(S|Z),$ which together with (REF ) completes the proof of Theorem REF .",
"2) Next let us address the problem of SM-SK capacity region to provide an SM-SK outer bound for degraded WTCs with causal/non-causal CSI available “only\" at Alice as in Fig.REF .",
"To do so, let the corresponding causal SM-SK capacity region be denoted by ${\\cal C}^{{\\sf e}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ .",
"Similarly, the corresponding non-causal SM-SK capacity region is denoted by ${\\cal C}^{{\\sf e}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}$ .",
"Moreover, let $\\overline{{\\cal R}}^{{\\sf e}}_{\\sf out}(p_{SX})$ denote the set of all nonnegative rate pairs $(R_M, R_K)$ satisfying the rate constraints: $R_M &\\le & I(X; Y|S),\\\\R_M+R_K &\\le & I(X;Y|S)-I(X;Z|S) \\nonumber \\\\& & \\qquad +H(S|Z)-H(S|Y).$ Then, we have Theorem 5 (Causal/non-causal SM-SK outer bound) Consider a degraded WTC ($Z$ is a degraded version of $Y$ ) with causal/non-causal CSI at Alice.",
"Then, ${\\cal C}^{{\\sf e}}_{\\mbox{{\\scriptsize \\rm CSI-E}}} & \\subset &{\\cal C}^{{\\sf e}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}} \\nonumber \\\\& \\subset &\\overline{{\\cal R}}^{{\\sf e}}_{\\sf out}\\stackrel{\\Delta }{=}\\bigcup _{p_{SX}} \\overline{{\\cal R}}^{{\\sf e}}_{\\sf out}(p_{SX}),$ where the union is taken over all possible probability distributions $p_{SX}$ 's.",
"Proof: The upper bound (REF ) for WTCs with CSI at both Alice and Bob must hold also for WTCs with CSI only at Alice, yielding (REF ).",
"On the other hand, in order to yield inequality (), it suffices to parallel the converse proof of Theorem REF while keeping in mind $H(S^n|MK\\overline{Y}^n)\\le H(S^n|MKZ^n)$ with $Y^n$ instead of $\\overline{Y}^n=S^nY^n$ (due to the assumed degradedness) in (REF ) and skipping () to () $\\sim $ () (with $Y_t$ instead of $\\overline{Y}_t$ ), which claims that the achievable rate pair $(R_M, R_K)$ needs to satisfy the rate constraints: ${n(R_M+R_K)}\\nonumber \\\\&\\le &\\sum _{t=1}^n I(V_t;Y_t|S_tU_t)-\\sum _{t=1}^nI(V_t;Z_t|S_tU_t) \\nonumber \\\\& & +\\sum _{t=1}^nH(S_t|Z_tU_t)-\\sum _{t=1}^nH(S_t|Y_tU_t)+4n\\varepsilon _n\\nonumber \\\\&=& nI(V;Y|SU)-nI(V;Z|SU)\\nonumber \\\\& & +nH(S|ZU)-nH(S|YU) +4n\\varepsilon .$ Therefore, by dividing by $n$ and letting $n\\rightarrow \\infty $ , we have $R_M+R_K & \\le &I(V;Y|SU)-I(V;Z|SU)\\nonumber \\\\& & +H(S|ZU)-H(S|YU).$ Then, in the same manner as in the converse proof of Theorem REF , we can check that (REF ) yields inequality ().",
"Finally, the following corollary follows from Theorem REF : Corollary 9 (Upper bound on SK rates) For a degraded WTC with causal/non-causal CSI at Alice, we have ${C^{{\\sf e,K}}_{\\mbox{{\\scriptsize \\rm CSI-E}}}\\le C^{{\\sf e,K}}_{\\mbox{{\\scriptsize \\rm NCSI-E}}}}\\nonumber \\\\&\\le & \\max _{p_{SX}}(I(X;Y|S)-I(X;Z|S) +H(S|Z)-H(S|Y)).\\nonumber \\\\& &$"
],
[
"Concluding Remarks",
"So far, we have studied the coding problem for WTCs with causal/non-causal CSI available at Alice and/or Bob under the semantic security criterion, the key part of which was summarized as Theorem REF for WTCs with causal CSI at Alice.",
"As is already clear, all the advantages of Theorem REF are inherited directly from Theorem REF that had been established by Bunin et al.",
"[25] for WTCs with non-causal CSI at Alice, This suggests that it is sometimes useful to deal with the causal problem as a special class of non-causal problems.",
"It is rather surprising to see that all the previous results [12], [13], [14] for WTCs with causal CSI follow immediately from Theorem REF alone.",
"Notice here that the validity of Theorem REF is based heavily on the superiority of the two layered superposition coding scheme (cf.",
"[17], [23]) along with that of soft covering lemma.",
"It is also pleasing to see that Theorem REF , as a by-product of Theorem REF , gives for the first time the exact SM-SK capacity region for WTCs with non-causal CSI at both Alice and Bob.",
"Theorem REF is also regarded as one of the key results from the viewpoint of SK-SM capacity regions for degraded WTCs.",
"Although Theorem REF treats the WTC with causal CSI available only at Alice, it can actually be effective also for investigating general WTCs with three correlated causal CSIs $S_a, S_b, S_e$ (correlated with state $S$ ) available at Alice, Bob and Eve, respectively (cf.",
"Fig.",
"REF ).",
"Figure: WTC with causal CSIs S a ,S b ,S e S_a, S_b, S_e available at Alice, Bob and Eve (t=1,2,⋯,nt=1,2,\\cdots , n).We would like to remind that this seemingly “general” WTCs actually boils down to the so far studied WTC with causal CSI available only at Alice simply by replacing channel $W_{YZ|SX}(y,z|s,x)$ with $W_{YZ|S_aX}(y,z|s_a, x)$ $\\stackrel{\\Delta }{=}\\sum _{s}W_{YZ|SX}(y,z|s,x)p(s|s_a)$ and at the same time by replacing $Y, Z$ with $S_bY, S_eZ$ , respectively, In this connection, the reader may refer, for example, to Khisti, Diggavi and Wornell [11], and Goldfeld, Cuff and Permuter [23]."
],
[
"Proof of Lemma ",
"From the manner of generating the random code, we see that the total joint probability of all $({\\bf u}_i,{\\bf v}_{ij})^{\\prime }$ s is given by $P_{1n}P_{2n}P_{3n},$ where $P_{1n}&=&\\prod _{k=2}^{L_n}\\prod _{\\ell =1}^{N_n} p({\\bf u}_k)p({\\bf v}_{k\\ell } |{\\bf u}_k),\\\\P_{2n}&=&\\prod _{\\ell =2}^{N_n} p({\\bf v}_{1\\ell } |{\\bf u}_1),\\\\P_{3n} &=& p({\\bf u}_1, {\\bf v}_{11}).$ We now directly develop ${\\rm E}D(q^n_S||p^n_S)$ as follows.",
"Here, for simplicity, we set $p({\\bf s})=p^n_S({\\bf s})$ .",
"${{\\rm E}D(q^n_S||p^n_S) }\\nonumber \\\\&=& \\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{L_n}\\sum _{j=1}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{1n}P_{2n}P_{3n}\\nonumber \\\\& &\\cdot \\left(\\frac{1}{L_nN_n}\\sum _{i^{\\prime }=1}^{L_n}\\sum _{j^{\\prime }=1}^{N_n}W({\\bf s}|{\\bf u}_{i^{\\prime }},{\\bf v}_{i^{\\prime }j^{\\prime }})\\right)\\nonumber \\\\& & \\cdot \\log \\left(\\frac{1}{L_nN_np({\\bf s})}\\sum _{k^{\\prime }=1}^{L_n}\\sum _{\\ell ^{\\prime }=1}^{N_n}W({\\bf s}|{\\bf u}_{k^{\\prime }}, {\\bf v}_{k^{\\prime }\\ell ^{\\prime }})\\right)\\nonumber \\\\&\\stackrel{(a)}{=}&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{L_n}\\sum _{j=1}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{1n}P_{21n}P_{3n}\\nonumber \\\\& & \\cdot W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\log \\left(\\frac{1}{L_nN_np({\\bf s})}\\sum _{k^{\\prime }=1}^{L_n}\\sum _{\\ell ^{\\prime }=1}^{N_n}W({\\bf s}|{\\bf u}_{k^{\\prime }}, {\\bf v}_{k^{\\prime }\\ell ^{\\prime }})\\right),\\nonumber \\\\& &$ where $(a)$ follows from the symmetry of codes.",
"We decompose the quantities in (REF ) as $\\sum _{k^{\\prime }=1}^{L_n}\\sum _{\\ell ^{\\prime }=1}^{N_n}W({\\bf s}|{\\bf u}_{k^{\\prime }}, {\\bf v}_{k^{\\prime }\\ell ^{\\prime }})= A_{1n}+A_{2n}+A_{3n},$ where $A_{1n}&=& \\sum _{k^{\\prime }=2}^{L_n}\\sum _{\\ell ^{\\prime }=1}^{N_n}W({\\bf s}|{\\bf u}_{k^{\\prime }}, {\\bf v}_{k^{\\prime }\\ell ^{\\prime }})\\\\A_{2n}&= & \\sum _{\\ell ^{\\prime }=2}^{N_n}W({\\bf s}|{\\bf u}_1, {\\bf v}_{1\\ell ^{\\prime }})\\\\A_{3n}&=&W({\\bf s}|{\\bf u}_1, {\\bf v}_{11}).$ Again, from the manner of generating the random code, we see that $A_{1n}$ and $(A_{2n}, A_{3n})$ are independent, whereas $A_{2n}$ and $A_{3n}$ are conditionally independent given ${\\bf u}_1$ .",
"Thus, ${{\\rm E}D(q^n_S||p^n_S) }\\nonumber \\\\&=&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{L_n}\\sum _{j=1}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{1n}P_{2n}P_{3n} \\nonumber \\\\& & \\cdot W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\log \\left(\\frac{A_{1n}+A_{2n}+A_{3n}}{L_nN_np({\\bf s})}\\right)\\nonumber \\\\&\\stackrel{(b)}{\\le }&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{1}\\sum _{j=1}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{2n}P_{3n}\\nonumber \\\\& & \\cdot W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\log \\left(\\frac{\\sum ^{*} A_{1n}+A_{2n}+A_{3n}}{L_nN_np({\\bf s})}\\right),\\nonumber \\\\& &$ where $(b)$ follows from the concavity of the function $x\\mapsto \\log x$ along with the Jensen's inequality.",
"Here, $\\sum ^{*} A_{1n} &\\stackrel{\\Delta }{=}& \\sum _{i=2}^{L_n}\\sum _{j=1}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{1n}A_{1n}\\nonumber \\\\&=& (L_n-1)N_n p({\\bf s}).$ Hence, ${{\\rm E}D(q^n_S||p^n_S) }\\nonumber \\\\&\\le &\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{1}\\sum _{j=1}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{2n}P_{3n}\\nonumber \\\\& & \\cdot W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\log \\left( 1+\\frac{A_{2n}+A_{3n}}{L_nN_np({\\bf s})}\\right).$ Moreover, ${{\\rm E}D(q^n_S||p^n_S) }\\nonumber \\\\&\\le &\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{1}\\sum _{j=1}^{1}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{3n}\\nonumber \\\\& & \\cdot W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\log \\left( 1+\\frac{\\sum ^*A_{2n}+A_{3n}}{L_nN_np({\\bf s})}\\right),$ where $\\sum ^*A_{2n}&\\stackrel{\\Delta }{=}& \\sum _{i=1}^{1}\\sum _{j=2}^{N_n}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{2n}A_{2n}\\nonumber \\\\&=& (N_n -1)W({\\bf s}|{\\bf u}_1),$ so that, with $0\\le \\rho <1$ , ${{\\rm E}D(q^n_S||p^n_S) }\\nonumber \\\\&\\le &\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{i=1}^{1}\\sum _{j=1}^{1}\\sum _{{\\bf u}_i\\in {\\cal U}^n} \\sum _{{\\bf v}_{ij}\\in {\\cal V}^n}P_{3n}\\nonumber \\\\& & \\cdot W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\nonumber \\\\& & \\cdot \\log \\left( 1+\\frac{W({\\bf s}|{\\bf u}_1)}{L_np({\\bf s})}+\\frac{W({\\bf s}|{\\bf u}_1, {\\bf v}_{11})}{L_nN_np({\\bf s})}\\right)\\nonumber \\\\&=&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}_1\\in {\\cal U}^n} \\sum _{{\\bf v}_{11}\\in {\\cal V}^n}p({\\bf u}_1, {\\bf v}_{11})W({\\bf s}|{\\bf u}_{1},{\\bf v}_{11})\\nonumber \\\\& & \\cdot \\log \\left( 1+\\frac{W({\\bf s}|{\\bf u}_1)}{L_np({\\bf s})}+\\frac{W({\\bf s}|{\\bf u}_1, {\\bf v}_{11})}{L_nN_np({\\bf s})}\\right)\\nonumber \\\\&=&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}_1\\in {\\cal U}^n} \\sum _{{\\bf v}_{11}\\in {\\cal V}^n}p({\\bf s}, {\\bf u}_1, {\\bf v}_{11})\\nonumber \\\\& & \\cdot \\log \\left( 1+\\frac{W({\\bf s}|{\\bf u}_1)}{L_np({\\bf s})}+\\frac{W({\\bf s}|{\\bf u}_1, {\\bf v}_{11})}{L_nN_np({\\bf s})}\\right)\\nonumber \\\\&=&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}_1\\in {\\cal U}^n} \\sum _{{\\bf v}_{11}\\in {\\cal V}^n}\\frac{1}{\\rho }p({\\bf s}, {\\bf u}_1, {\\bf v}_{11})\\nonumber \\\\& & \\cdot \\log \\left( 1+\\frac{W({\\bf s}|{\\bf u}_1)}{L_np({\\bf s})}+\\frac{W({\\bf s}|{\\bf u}_1, {\\bf v}_{11})}{L_nN_np({\\bf s})}\\right)^{\\rho }\\nonumber \\\\&\\stackrel{(c)}{\\le }&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}_1\\in {\\cal U}^n} \\sum _{{\\bf v}_{11}\\in {\\cal V}^n}\\frac{1}{\\rho }p({\\bf s}, {\\bf u}_1, {\\bf v}_{11})\\nonumber \\\\& & \\cdot \\log \\left( 1+\\left(\\frac{W({\\bf s}|{\\bf u}_1)}{L_np({\\bf s})}\\right)^{\\rho }+\\left(\\frac{W({\\bf s}|{\\bf u}_1, {\\bf v}_{11})}{L_nN_np({\\bf s})}\\right)^{\\rho }\\right)\\nonumber \\\\&\\stackrel{(d)}{\\le }&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}_1\\in {\\cal U}^n} \\frac{1}{\\rho }p({\\bf s}, {\\bf u}_1)\\left(\\frac{W({\\bf s}|{\\bf u}_1)}{L_np({\\bf s})}\\right)^{\\rho } \\\\& & +\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}_1\\in {\\cal U}^n} \\sum _{{\\bf v}_{11}\\in {\\cal V}^n}\\frac{1}{\\rho }p({\\bf s}, {\\bf u}_1, {\\bf v}_{11})\\left(\\frac{W({\\bf s}|{\\bf u}_1, {\\bf v}_{11})}{L_nN_np({\\bf s})}\\right)^{\\rho }.\\nonumber \\\\& &$ where $(c)$ follows from $(x+y+z)^{\\rho }\\le x^{\\rho } +y^{\\rho }+z^{\\rho }$ ; $(d)$ follows from $\\log (1+x)\\le x.$ For simplicity, we delete the subscripts $``1, 11\"$ in () and () to obtain $F_{1n} &\\stackrel{\\Delta }{=}&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}\\in {\\cal U}^n} \\frac{1}{\\rho }p({\\bf s}, {\\bf u})\\left(\\frac{W({\\bf s}|{\\bf u})}{L_np({\\bf s})}\\right)^{\\rho }, \\\\F_{2n} &\\stackrel{\\Delta }{=}&\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}\\in {\\cal U}^n} \\sum _{{\\bf v}\\in {\\cal V}^n}\\frac{1}{\\rho }p({\\bf s}, {\\bf u}, {\\bf v})\\left(\\frac{W({\\bf s}|{\\bf u}, {\\bf v})}{L_nN_np({\\bf s})}\\right)^{\\rho }.\\nonumber \\\\& & $ Hereafter, let us show that $F_{1n}\\rightarrow 0$ , $ F_{2n}\\rightarrow 0$ as $n$ tends to $\\infty $ if rate constraints $R_1>I((U;S), R_1+R_2> I(UV;S)$ are satisfied.",
"First, let us show $F_{2n}\\rightarrow 0$ .",
"Since $p({\\bf s}, {\\bf u}, {\\bf v})=p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})$ , $F_{2n}$ can be rewritten as ${F_{2n}}\\nonumber \\\\&=& \\frac{1}{\\rho (L_nN_n)^{\\rho }}\\sum _{{\\bf s}\\in {\\cal S}^n}\\sum _{{\\bf u}\\in {\\cal U}^n} \\sum _{{\\bf v}\\in {\\cal V}^n}p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})^{1+\\rho }p({\\bf s})^{-\\rho }.\\nonumber \\\\& & $ On the other hand, by virtue of Hölder's inequality, ${\\left(\\sum _{({\\bf u}, {\\bf v})\\in {\\cal U}^n\\times {\\cal V}^n}p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})^{1+\\rho }\\right)p({\\bf s})^{-\\rho }} \\nonumber \\\\&=& \\left(\\sum _{({\\bf u}, {\\bf v})\\in {\\cal U}^n\\times {\\cal V}^n}p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})^{1+\\rho }\\right)\\nonumber \\\\& & \\cdot \\left(\\sum _{({\\bf u}, {\\bf v})\\in {\\cal U}^n\\times {\\cal V}^n}p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})\\right)^{-\\rho }\\nonumber \\\\&\\le & \\left(\\sum _{({\\bf u}, {\\bf v})\\in {\\cal U}^n\\times {\\cal V}^n} p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})^{\\frac{1}{1-\\rho }}\\right)^{1-\\rho }$ for $0 < \\rho < 1$ .",
"Therefore, it follows from (REF ) that ${F_{2n}}\\nonumber \\\\&\\le & \\frac{1}{\\rho (L_nN_n)^{\\rho }} \\sum _{{\\bf s}\\in {\\cal S}^n}\\left(\\sum _{({\\bf u}, {\\bf v})\\in {\\cal U}^n\\times {\\cal V}^n}p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})^{\\frac{1}{1-\\rho }}\\right)^{1-\\rho } \\nonumber \\\\&=& \\frac{1}{\\rho }\\exp \\left[-[n\\rho (R_1+R_2)+ E_0(\\rho , p)]\\right],$ where ${E_0(\\rho , p)} \\nonumber \\\\&=& -\\log \\left[\\sum _{{\\bf s}\\in {\\cal S}^n}\\left(\\sum _{({\\bf u}, {\\bf v})\\in {\\cal U}^n\\times {\\cal V}^n}p({\\bf u}, {\\bf v})W({\\bf s}|{\\bf u}, {\\bf v})^{\\frac{1}{1-\\rho }}\\right)^{1-\\rho }\\right].\\nonumber \\\\& &$ Then, by means of Gallager [27], we have $E_0(\\rho , p)|_{\\rho =0}=0$ and $\\left.\\frac{\\partial E_0(\\rho , p)}{\\partial \\rho }\\right|_{\\rho =0}&=&-I(p, W)\\nonumber \\\\&=&-I({\\bf U}{\\bf V}; {\\bf S})\\nonumber \\\\&\\stackrel{(e)}{=}& -nI(UV;S), $ where $(e)$ follows because $({\\bf U}{\\bf V}; {\\bf S})$ is a correlated i.i.d.",
"sequence with generic variable $(UV, S)$ .",
"Thus, for any small constant $\\tau >0$ there exists a $\\rho _0>0$ such that, for all $0<\\rho \\le \\rho _0$ , $E_0(\\rho , p)\\ge -n\\rho (1+\\tau )I(UV;S)$ which is substituted into (REF ) to obtain $F_{2n}&\\le & \\frac{1}{\\rho }\\exp \\left[-n\\rho (R_1+R_2 -(1+\\tau ) I(UV;S))\\right].\\nonumber \\\\& &$ On the other hand, in view of rate constraint $R_1+R_2>I(UV; S)$ , with some $\\delta >0$ we can write $R_1+R_2=I(UV;S)+2\\delta ,$ which leads to ${R_1+R_2 -(1+\\tau )I(UV;S)}\\nonumber \\\\&=& I(UV;S) +2\\delta - I(UV;S) -I(UV;S)\\nonumber \\\\&=& 2\\delta -\\tau I(UV;S).$ We notice here that $\\tau >0$ can be arbitrarily small, so that the last term on the right-hand side of (REF ) can be made larger than $\\delta >0$ .",
"Then, (REF ) yields $F_{2n}\\le \\frac{1}{\\rho }\\exp [-n\\rho \\delta ],$ which implies that with any small $\\varepsilon >0$ it holds that $F_{2n} \\le \\varepsilon $ for all sufficiently large $n$ .",
"Similarily, $F_{1n} \\le \\varepsilon $ with rate constraint $R_1>I(U; S)$ can also be shown.",
"Thus, the proof of Lemma REF has been completed."
],
[
"Proof of Remark ",
"For simplicity, set the right-hand sides of (REF ), REF ), (REF ) and (REF ) as $M_1 &=& \\max _{p_Sp_V}\\min ( I(SV;Y)-I(SV;Z),\\nonumber \\\\& &\\qquad \\qquad \\qquad I(SV;Y)-H(S)),\\\\K_1 &=& \\max _{\\stackrel{p_Sp_{V}:}{I(S V;Y)\\ge H(S)}}(I(SV;Y)-I(SV;Z)),\\\\M_2 &=& \\max _{p_Sp_U}\\min (H(S|UZ)-H(S|UY), \\nonumber \\\\& &\\qquad \\qquad \\qquad I(U;Y)-H(S|UY)), \\\\K_2 &=& \\max _{\\stackrel{p_Sp_U:}{I(U;Y)\\ge H(S|UY)}}(H(S|UZ)-H(S|UY)),\\nonumber \\\\ &&$ which, after some calculation with $Y$ replaced by $SY$ , leads, respectively, to $M_1^{\\prime } &=& \\max _{p_Sp_V}\\min (I(V;Y|S)-I(V;Z|S) +H(S|Z), \\nonumber \\\\& &\\qquad \\qquad \\qquad I(V;Y|S)),\\\\K_1^{\\prime } &=& \\max _{p_Sp_V}(I(V;Y|S)-I(V;Z|S) +H(S|Z)), \\\\M_2^{\\prime } &=& \\max _{p_Sp_U}\\min (H(S|UZ), I(U;Y|S)), \\\\K_2^{\\prime } &=& \\max _{p_Sp_U}H(S|UZ)=\\max _{p_{SX}}H(S|Z).$ Notice here that (REF ) $\\sim $ () give lower bounds for ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-E}}}$ (with CSI $S$ available only at Alice), whereas (REF ) $\\sim $ () give lower bounds for ${\\cal C}_{\\mbox{{\\scriptsize \\rm CSI-ED}}}$ (with CSI $S$ available at both Alice and Bob).",
"1) First, consider the reversely degraded binary WTC as in Fig.REF with $W(y,z|x,s) =W(y,z|x)$ with binary entropy $H(S) =1-h(0.2)<1-h(0.1)$ .",
"It is easy to check that $I(SV;Y)-I(SV;Z)$ in () can be rewritten as ${I(SV;Y)-I(SV;Z)}\\nonumber \\\\&=& (I(V;Y)-I(V;Z)) + H(S|UZ)-H(S|UY).\\nonumber \\\\&&$ Suppose here that $I(V;Y) =0$ , then $I(S V;Y)=I(S;Y|V)=H(S|V)-H(S|VY),$ from which, together with the constraint $I(S V;Y)\\ge H(S)$ in (), it follows that $H(S|V)-H(S|VY)\\ge H(S),$ i.e., $-H(S|VY)\\ge I(S;V)=0$ (owing to the independence of $S$ and $V$ ) and hence $H(S|VY)=0$ .",
"On the other hand, in view of the Markov chain property $SV\\rightarrow Z\\rightarrow Y$ (due to the reverse degradedness) as well as the independence of $S$ and $V$ , it must hold that $H(S|VY) >0$ in that we are here considering the causal WC with CSI $S$ available only at Alice.",
"This is a contradiction.",
"Thus, it should hold that $I(V;Y) >0$ for all $V$ satisfying the constraint $I(S V;Y)\\ge H(S)$ and hence $I(V;Y) \\ge c_0$ for some $c_0>0$ and for all $V$ satisfying the constraint.",
"Furthermore, we can show also that $I(V;Z)-I(V;Y)=I(V;Z|Y) >0$ .",
"To see this, assume $I(V;Z|Y) =0$ to lead to a contradiction.",
"Then, it is easy to check that $I(V;Z|Y) =0$ together with the reverse degradedness implies that $V$ and $ZY$ are independent and hence particularly $I(V;Y)=0$ , which is a contradiction.",
"Thus, $ I(V;Y)-I(V;Z) \\le -d_0$ for some $d_0>0$ and for all $V$ satisfying the constraint, which, together with (), () and (REF ), implies that $K_1<K_2,$ where we have taken account that the constraint in () is tighter than that in ().",
"Figure: Reversely degraded binary WTC with causal CSI SS available only at Alice.2) Now consider the reversely degraded binary WTC as in Fig.REF with $W(y,z|x,s) =W(y,z|x)$ .",
"Then, setting $U=X$ ($\\Pr \\lbrace X=1\\rbrace =\\Pr \\lbrace X=0\\rbrace =1/2$ ) independently of $S$ with binary entropy $H(S)=1-h(0.1)$ in (), we have $M_2^{\\prime } \\ge 1-h(0.1) > M_1^{\\prime },$ where the first inequality in (REF ) follows directly by observing that in this case $H(S|UZ)=I(U;Y|S)=1-h(0.1)$ and the second inequality in (REF ) can be verified as follows.",
"Suppose otherwise, i.e., $1-h(0.1) \\le M_1^{\\prime },$ which then, together with (REF ), means that there exists some $VX$ such that $I(V;Y|S) \\ge 1-h(0.1),$ $I(V;Y|S)-I(V;Z|S) +H(S) \\ge 1-h(0.1).$ On the other hand, (REF ) implies that it must hold that $V=X$ ($\\Pr \\lbrace X=1\\rbrace =\\Pr \\lbrace X=0\\rbrace =1/2$ ) independently of $S$ with $H(S)=1-h(0.1)$ .",
"Thus, in view of (REF ) with this $V=X$ , we must have $I(X;Y|S)-I(X;Z|S) +H(S) \\ge 1-h(0.1),$ which, together with $I(X;Y|S)=1-h(0.1), I(X;Z|S)=1$ , means that $-h(0.1) +H(S)\\ge 1-h(0.1),$ that is, $H(S)=1-h(0.1) \\ge 1,$ which is a contradiction, thus establishing the second inequality in (REF ).",
"Figure: Reversely degraded binary WTC with causal CSI SS available at Alice and Bob.3) We next consider the degraded binary WTC as in Fig.REF with $W(y,z|x,s) =W(y,z|x)$ and $S=\\emptyset $ .",
"Let $\\Pr \\lbrace X=1\\rbrace =\\Pr \\lbrace X=0\\rbrace =1/2$ .",
"Then, since $S=\\emptyset $ , (REF ) and () are evaluated as $M_1^{\\prime } &=& \\max _{p_V}\\min (I(V;Y)-I(V;Z), I(V;Y)),\\nonumber \\\\&=& \\max _{p_V} (I(V;Y) -I(V;Z))\\nonumber \\\\&\\ge & I(X;Y)-I(X;Z) = h(0.1),\\\\M_2^{\\prime } &=&0.$ Hence, $M_1^{\\prime } > M_2^{\\prime }.$ Similarly, again by letting $\\Pr \\lbrace X=1\\rbrace =\\Pr \\lbrace X=0\\rbrace =1/2$ and taking account of $S=\\emptyset $ , we see that () reduces to $K_1^{\\prime } &\\ge & \\max _{p_V}(I(V;Y)-I(V;Z))\\nonumber \\\\&\\ge & I(X;Y)-I(X;Z) \\nonumber \\\\&=& h(0.1) \\nonumber \\\\&>& 0 = K_2^{\\prime },\\nonumber $ that is, $K_1^{\\prime } > K_2^{\\prime }.",
"$ Finally, summarizing up (REF ), (REF ), (REF ) and (REF ), the claim of Remark REF has been proved.",
"Figure: Degraded binary WTC without CSI at Alice and Bob."
],
[
"Acknowledgments",
"The authors are grateful to Hiroyuki Endo for useful discussions.",
"Special thanks go to the reviewers for their stimulating comments which have occasioned to largely improve the quality of the earlier version.",
"This work was funded: by JSPS KAKENHI Grant Number 17H01281, and partly supported by “Research and Development of Quantum Cryptographic Technologies for Satellite Communications (JPJ007462)” of Ministry of Internal Affairs and Communication (MIC), Japan.",
"Te Sun Han (M'80-SM'88-F'90-LF'11) received the B.Eng., M.Eng., and D.Eng.",
"degrees in mathematical engineering from the University of Tokyo, Japan, in 1964, 1966, and 1971, respectively.",
"Since 1993, he has been a Professor with the University of Electro-Communications, where he has been a Professor Emeritus since 2007.",
"He has published papers on information theory, most of which appeared in the IEEE TRANSACTIONS ON INFORMATION THEORY.",
"Also, he has published two books: one of them is Information- Spectrum Methods in Information Theory (Springer Verlag, 2003).",
"Especially, this book was written to try to demonstrate part of the general logics latent in information theory.",
"His research interests include basic problems in Shannon theory, multi-user source/channel coding systems, multiterminal hypothesis testing and parameter estimation under data compression, large-deviation approach to information-theoretic problems, and especially, information spectrum theory.",
"He is a member of the Board of Governors for the IEEE Information Theory Society.",
"He has been an IEICE Fellow and an Honorary Member since 2011.",
"He was a recipient of the 2010 Shannon Award.",
"From 1994 to 1997, he was the Associate Editor for Shannon Theory of the IEEE TRANSACTIONS ON INFORMATION THEORY.",
"Masahide Sasaki received the B.S., M.S., and Ph.D. degrees in physics from Tohoku University, Sendai Japan, in 1986, 1988 and 1992, respectively.",
"During 1992–1996, he worked on the development of semiconductor devices in Nippon-Kokan Company (currently JFE Holdings).",
"In 1996, he joined the Communications Research Laboratory, Ministry of Posts and Telecommunications (since 2004, National Institute of Information and Communications Technology (NICT), Ministry of Internal Affairs and Communications).",
"He has been working on quantum optics, quantum communication and quantum cryptography.",
"He is presently Distinguished Researcher of Advanced ICT Research Institute, and NICT Fellow.",
"Dr. Sasaki is a member of Japanese Society of Physics, and the Institute of Electronics, Information and Communication Engineers of Japan."
]
] | 2001.03482 |
[
[
"Unidirectional tilt of domain walls in equilibrium in biaxial stripes\n with Dzyaloshinskii-Moriya interaction"
],
[
"Abstract The orientation of a chiral magnetic domain wall in a racetrack determines its dynamical properties.",
"In equilibrium, magnetic domain walls are expected to be oriented perpendicular to the stripe axis.",
"We demonstrate the appearance of a unidirectional domain wall tilt in out-of-plane magnetized stripes with biaxial anisotropy and Dzyaloshinskii--Moriya interaction (DMI).",
"The tilt is a result of the interplay between the in-plane easy-axis anisotropy and DMI.",
"We show that the additional anisotropy and DMI prefer different domain wall structure: anisotropy links the magnetization azimuthal angle inside the domain wall with the anisotropy direction in contrast to DMI, which prefers the magnetization perpendicular to the domain wall plane.",
"Their balance with the energy gain due to domain wall extension defines the equilibrium magnetization the domain wall tilting.",
"We demonstrate that the Walker field and the corresponding Walker velocity of the domain wall can be enhanced in the system supporting tilted walls."
],
[
"[rgb]0.00,0.50,0.75Draft by on Unidirectional tilt of domain walls in equilibrium in biaxial stripes with Dzyaloshinskii–Moriya interaction Oleksandr V. Pylypovskyi [email protected] Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine Volodymyr P. Kravchuk [email protected] Institut für Theoretische Festkörperphysik, Karlsruher Institut für Technologie, D-76131 Karlsruhe, Germany Bogolyubov Institute for Theoretical Physics of National Academy of Sciences of Ukraine, 03143 Kyiv, Ukraine Oleksii M. Volkov [email protected] Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany Jürgen Faßbender [email protected] Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany Denis D. Sheka [email protected] Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine Denys Makarov [email protected] Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany The orientation of a chiral magnetic domain wall in a racetrack determines its dynamical properties.",
"In equilibrium, magnetic domain walls are expected to be oriented perpendicular to the stripe axis.",
"We demonstrate the appearance of a unidirectional domain wall tilt in out-of-plane magnetized stripes with biaxial anisotropy and Dzyaloshinskii–Moriya interaction (DMI).",
"The tilt is a result of the interplay between the in-plane easy-axis anisotropy and DMI.",
"We show that the additional anisotropy and DMI prefer different domain wall structure: anisotropy links the magnetization azimuthal angle inside the domain wall with the anisotropy direction in contrast to DMI, which prefers the magnetization perpendicular to the domain wall plane.",
"Their balance with the energy gain due to domain wall extension defines the equilibrium magnetization the domain wall tilting.",
"We demonstrate that the Walker field and the corresponding Walker velocity of the domain wall can be enhanced in the system supporting tilted walls.",
"Spin orbitronics relies on the manipulation of magnetic textures via spin orbit torques and enables new devices ideas for application in magnetic storage and logics[1], [2], [3], [4], [5].",
"The key component of these devices is a stripe with out of plane easy axis of magnetization and featuring the Dzyaloshinskii–Moriya interaction (DMI).",
"Typically, asymmetrically sandwiched ultrathin films of ferromagnets are used, where the DMI originates from the broken symmetry at the film interfaces[6], [7].",
"There are numerous demonstrations of the energy efficient and fast motion of chiral magnetic solitons including skyrmions[8], [9], [10], skyrmion-bubbles[11], [12] and domain walls[13], [14], [15] in stripes.",
"The orientation of the plane of a domain wall with respect to the stripe main axis has major impact on its dynamics including the maximum velocity[16] and Walker limit[17], [18].",
"It is commonly accepted that in equilibrium the plane of a magnetic domain wall is perpendicular to the stripe main axis.",
"This remains true even if the sample possesses DMI.",
"Domain wall can acquire a tilt yet only if exposed to an external magnetic field[19], [20], [21], [22], [23], driven by a current[24], [25], [26], [19], [27], [16], or pinned on edge roughness during current-induced dynamics[28].",
"Being exposed to an in-plane magnetic field, domain wall tilts unidirectionaly with the rotation direction determined by the sign of the DMI.",
"The tilt increases linearly with the field strength and the slope of the resulting dependence was proposed to be used for the determination of the DMI constant[19], [22].",
"Here, we demonstrate that domain walls can acquire a unidirectional tilt even at equilibrium if the out-of-plane magnetized stripe possesses DMI and an additional easy-axis anisotropy in the plane of the stripe.",
"The easy axis direction of the in-plane anisotropy can be given by a crystalline structure of the ferromagnet[29] or induced via exchange bias when the stripe is in proximity to an antiferromagnet[30], [31], [32], [4].",
"In contrast to the shape anisotropy[33], [17] with an easy axis along the stripe, any misalignment between the in-plane anisotropy and the stripe axes breaks the symmetry of the magnetic texture and tilts (i) the magnetization inside the domain wall as well as (ii) the plane of the domain wall.",
"The motion a domain wall in a biaxial magnetic with DMI has strong impact on the Walker field and the domain wall velocity.",
"The obtained results allow to design stripes with stronger Walker field (i.e.",
"extend range of the linear motion of domain walls) and control the domain wall nucleation process in T-junctions by selecting the initial tilt direction[34], [35].",
"Figure: The equilibrium structure and orientation of a domain wall in a biaxial stripe with DMI.",
"(a) Schematics of the out-of-plane magnetized stripe containing a domain wall.",
"The mechanical tilt of the domain wall with respect to the y ^{\\hat{y}} axis is characterized with χ\\chi .",
"Angle ψ\\psi describes the tilt of the magnetization in the wall.",
"(b) Energy profile (Eq.",
"()) for the case of k 2 =0.30k_2 = 0.30, α=45 ∘ \\alpha =45^\\circ , p=+1p=+1 and varying strength of the DMI, d 0 d_0.",
"Arrows show the evolution of the energy minima with the change of DMI strength parameter |d 0 ||d_0|.",
"(c) Domain wall tilt angle (χ\\chi ) and magnetization tilt angle (ψ\\psi ) as a function of the strength of the DMI.",
"Solid line corresponds to the numerically calculated minimum based on Eq.",
"(), symbols represent results of the full-scale micromagnetic simulations (open square marks).",
"The simulation data where magnetostatics was reduced to the effective anisotropy is shown with symbols.",
"(d) A structure of the domain wall for two different DMI parameters, see mark in panel (c).",
"(e) Size of bistability regions d 0 bis d_0^\\text{bis} for different angles α\\alpha of the in-plane easy axis e 2 {e}_2.We consider an infinitely long magnetic thin stripe of thickness $h$ and width $w$ .",
"The total magnetic energy of the stripe is $E = h \\int \\mathrm {d}S \\left[ {E}_x + {E}_\\textsc {a} + {E}_\\textsc {dm} + {E}_\\textsc {z} \\right],$ where the integration is performed over the sample's area in $xy$ plane (${\\hat{x}}$ axis is along the stripe).",
"The first energy term is the exchange energy density ${E}_x = A\\sum _{i=x,y,z} (\\partial _i {m})^2$ with $A$ being the exchange stiffness, ${m} = {M}/M_\\textsc {s}$ being the unit magnetization vector and $M_\\textsc {s}$ being the saturation magnetization.",
"The second term is the anisotropy energy density of a biaxial magnet, ${E}_\\textsc {a} = K_1(1-m_z^2) - K_2 ({m}\\cdot {e}_2)^2$ , with $K_1 > K_2 > 0$ .",
"The easy axis of the in-plane anisotropy ${e}_2$ lies in the stripe's plane at an angle $\\alpha $ to the ${\\hat{x}}$ direction.",
"The third energy term is the energy density of the DMI[36], [37] ${E}_\\textsc {dm} = D \\left[m_z (\\nabla \\cdot {m}) - ({m}\\cdot \\nabla m_z)\\right]$ .",
"The last energy term is the Zeeman energy density ${E}_\\textsc {z} = -M_\\textsc {s}B m_z$ with $B$ being an external magnetic field intensity.",
"We assume, that the magnetostatic interaction can be reduced to a local anisotropy and results in the renormalization of the first anisotropy constant $K_1 = K_0 - 2\\pi M_\\textsc {s}^2$ with $K_0$ being the strength of the magnetic anisotropy with an out-of-plane easy axis.",
"To describe the structure of the domain wall, we apply the following ansatz[19]: $\\cos \\theta = - p \\tanh \\xi , \\phi = \\psi - 90^\\circ , \\xi = \\dfrac{(x-q\\ell ) \\cos \\chi + y \\sin \\chi }{\\Delta \\ell },$ where the magnetization vector is parametrized as ${m} = \\left\\lbrace \\sin \\theta \\cos \\phi , \\sin \\theta \\sin \\phi , \\cos \\theta \\right\\rbrace $ in the local spherical reference frame with $\\theta $ and $\\phi $ being polar and azimuthal angles, respectively, $p = \\pm 1$ is the topological charge of the domain wall (kink or anti-kink), $\\ell = \\sqrt{A/K_1}$ is the magnetic length, $\\Delta $ and $q$ are domain wall width and position of its center, respectively, measured in units of $\\ell $ .",
"The origin is placed in the center of the stripe.",
"The angle $\\psi \\in (-180^\\circ ,180^\\circ ]$ describes the tilt of the magnetization inside the domain wall with respect to ${\\hat{y}}$ axis and the angle $\\chi \\in (-90^\\circ ,90^\\circ )$ characterizes the mechanical tilt of the plane of the domain wall with respect to ${\\hat{y}}$ , see Fig.",
"REF (a).",
"In this notation, $\\psi = 0$ or $180^\\circ $ and $\\psi = 90^\\circ $ or $-90^\\circ $ with $\\chi = 0$ corresponds to Néel and Bloch domain walls, respectively.",
"The plane of the domain wall is perpendicular to the stripe axis when $\\chi = 0$ .",
"The total energy, normalized by $E_0 = 2K_1hw\\ell $ , reads $\\begin{split}\\mathcal {E} = \\dfrac{E}{E_0} & = \\dfrac{1}{\\cos \\chi } \\Bigg \\lbrace \\frac{1}{\\Delta } + \\Delta \\times \\Big [1-k_2 \\sin ^2(\\psi -\\alpha )\\Big ] \\\\& + d_0 \\sin (\\psi -\\chi ) \\Bigg \\rbrace - pbq,\\end{split}$ where $k_2 = K_2/K_1$ is the normalized in-plane anisotropy, $d_0 = \\pi pD/(2\\sqrt{AK_1})$ is the dimensionless parameter characterizing the DMI strength and $b = M_\\textsc {s}B/K_1$ is the normalized magnetic field $B$ along ${\\hat{z}}$ .",
"Note, that the DMI parameter $d_0$ incorporates the topological charge of the domain wall $p$ .",
"The static domain wall configuration ($b = 0$ , and $q = 0$ without loss of generality) is given by the minimum of the energy (REF ) with respect to $\\Delta $ , $\\chi $ and $\\psi $ .",
"The equilibrium domain wall width is $\\Delta _0(\\psi ) = 1/\\sqrt{1 - k_2 \\sin ^2(\\psi -\\alpha )}$ .",
"The relation between values of the angles ${\\chi }$ and ${\\psi }$ in equilibrium reads $2\\sin {\\chi } = d_0 \\Delta _0(\\psi )\\cos {\\psi },$ After the substitution of (REF ) in (REF ), we obtain the expression for the energy as a function of the angle $\\psi $ , characterizing the orientation of the magnetization in the wall: $\\mathcal {E}(\\psi ) = \\sqrt{\\dfrac{4}{\\Delta _0(\\psi )^2}-d_0^2 \\cos ^2\\psi } + d_0 \\sin \\psi .$ Figure: The domain wall structure (tilt of magnetization ψ\\psi and domain wall tilt χ\\chi ) as a function of material parameters.",
"(a) Tilt angles as a function of the strength of the in-plane anisotropy k 2 k_2 with the easy axis direction α=45 ∘ \\alpha = 45^\\circ and different strength of the DMI parameter d 0 d_0.",
"(b) and (c) Domain wall and magnetization tilt angles for different d 0 d_0 and α\\alpha .",
"Values of angles are shown with isolines.",
"The normalized anisotropy coefficient k 2 =0.30k_2 = 0.30, topological charge of the domain wall p=+1p = +1.There are several limiting cases related to the absence of the in-plane anisotropy ($k_2 = 0$ ) or DMI ($d_0 = 0$ ).",
"If $k_2 = 0$ and $d_0 = 0$ , we obtain a classical case when a magnetic stripe with perpendicular anisotropy can support Bloch domain walls ($\\psi = 0,180^\\circ $ as a consequence of minimization of magnetostatic energy), with a plane of the domain wall being perpendicular to the stripe axis ($\\chi = 0$ ).",
"For any finite $k_2$ (still when $d_0 = 0$ ), the magnetization in the domain wall is tilted to its equilibrium value of $\\psi = \\alpha \\pm 90^\\circ $ .",
"This corresponds to the two equivalent minima in the energy (REF ), see red line ($d_0 = 0$ ) in Fig.",
"REF (b).",
"However, the plane of the domain wall remains perpendicular to the stripe axis ($\\chi = 0$ ).",
"This result is expected from the analysis of (REF ), indicating that the mechanical rotation of the plane of the domain wall is possible only if the sample possesses a finite DMI.",
"A nonzero DMI results in the symmetry breaking between the opposite magnetization directions, see blue line ($d_0 = 0.1$ ) in Fig.",
"REF (b).",
"Arrows indicate the shift of the energy minima with the change of the DMI constant.",
"For a sufficiently large DMI, the second energy minimum disappears and only one orientation of the domain wall remains stable, see green line ($d_0 = 0.3$ ) in Fig.",
"REF (b).",
"Fig.",
"REF (c) shows the domain wall structure in a broad range of DMI parameters for $k_2 = 0.30$ and $\\alpha = 45^\\circ $ .",
"Solid lines represent the numerically determined minimum of the energy (REF ) taking into account (REF ).",
"The domain walls acquires the unidirectional tilt for any finite DMI parameter $d_0$ : the energetically preferable state corresponds to $\\chi > 0$ only if $\\alpha > 0$ .",
"The bistability region exists in a vicinity of $d_0 = 0$ .",
"We made two sets of micromagnetic simulationsNumerical analysis is performed using energy minimization in OOMMF[46], [47], [48] for samples of length 1000 nm, width 200 nm and thickness 1 nm with mesh $2\\times 2\\times 1$ nm with the domain wall placed in the center.",
"The material parameters correspond to Co/Pt ultrathin films with the saturation magnetization $M_\\textsc {s} = 1100$ kA/m, exchange stiffness $A = 16$ pJ/m and out-of-plane anisotropy $K_0 = 1.3$ MJ/m$^3$ .",
"The in-plane anisotropy coefficient $K_2 = K_0/8$ is chosen if other is not stated.",
"The domain wall structure is extracted from the inner part of the stripe with $|y| < 40$ nm to avoid boundary effects.",
"All data presented in figures are calculated for the case when the easy axis of the in-plane anisotropy is directed at $\\alpha = 45^\\circ $ , if other is not stated.",
"The difference between simulations and analytical theory for is explained by the influence of non-local magnetostatics contribution in the corresponding simulation series and the difference between ansatz (REF ) and real domain wall structure.",
"The finite stripe length also influences the domain wall structure for very large $\\chi $ if magnetostatics is calculated explicitly (full scale micromagnetic simulations).",
": symbols show the result, where magnetostatics is reduced to a local anisotropy and open squares represent full-scale simulations.",
"The internal domain wall structure, given by the angle $\\psi $ , is governed by the direction of the easy axis of the in-plane anisotropy ${e}_2$ .",
"In an extended film, the domain wall is always oriented perpendicularly to ${e}_2$ and the DMI energy favors $\\psi =\\chi $ or $\\psi = \\chi + 180^\\circ $ (magnetization rotates perpendicularly to the domain wall plane).",
"In a stripe of a finite width, the balance between the domain wall tension energy (proportional to its length and increasing with $\\chi $ ) and the DMI energy results in a certain value of $\\chi $ , which is different from $\\psi $ .",
"Note that the model (REF ) is applicable for relatively narrow stripes, where the domain wall shape can be approximated by a straight line.",
"For wide stripes the curvilinear distortion of the domain wall shape must be taken into account.",
"The domain wall tilt angle $\\chi $ rapidly increases when $d_0$ approaches its critical value.",
"The domain wall structure is shown in Fig.",
"REF (d), where tilt angles $\\chi $ and $\\psi $ as well as the orientation of the easy axis of the in-plane anisotropy are depicted.",
"The size of the bistability region in terms of the DMI parameter $d_0^\\text{bis}$ is shown in Fig.",
"REF (e).",
"Note, that the state $\\alpha = 0$ is degenerated with the domain wall tilt $\\chi \\equiv 0$ .",
"The dependencies of the domain wall ($\\chi $ ) and magnetization ($\\psi $ ) tilt angles on the orientation of the easy axis of the in-plane anisotropy ($\\alpha $ ) is summarized in Fig.",
"REF (b,c).",
"The sign of $\\chi $ is given by the direction of the anisotropy axis $\\alpha $ , while the sign of $\\psi $ is opposite to the sign of $d_0$ .",
"The domain wall tilt angle monotonically increases with the increase of $d_0$ and $k_2$ , while the magnetization tilt angle is mainly determined by the $k_2$ for the case of strong DMI.",
"Figure: Domain wall dynamics.",
"(a) Comparison of the results of micromagnetic simulations (symbols) and collective variables model (lines).",
"Blue and red colors correspond to direction of the easy axis of the in-plane anisotropy α=45 ∘ \\alpha = 45^\\circ and α=-45 ∘ \\alpha = -45^\\circ , respectively.",
"Labels ψ fav \\psi _\\text{fav} and ψ unf \\psi _\\text{unf} indicate parameters, where favorable and unfavorable magnetization tilt exist, gray arrows indicate simulations, represented in panel (e)–(g).",
"(b) and (c) Dimensionless Walker field and velocity (solid blue lines).",
"Red dashed line shows the field and velocity at which the magnetization tilt ψ\\psi changes from unfavorable value to the favorable one (unf to fav).",
"(d) Comparison of the numerical solution of the collective variable model (solid blue line) with asymptotic (dashed black line).",
"(e)–(g) Structure of a moving domain wall for different orientation of the easy axis of the in-plane anisotropy α\\alpha and applied field bb, see also gray arrows in the panel (a).",
"The parameters used for these simulations are k 2 =0.30k_2 = 0.30, d 0 =0.63d_0 = 0.63 , p=+1p = +1, η=0.5\\eta = 0.5.In the following, we address the dynamics of domain walls driven by an external magnetic field applied along ${\\hat{z}}$ .",
"We apply a collective variables approach[39], [40], considering the wall position $q(t)$ , the magnetization tilt $\\psi (t)$ , domain wall tilt $\\chi (t)$ and the domain wall width $\\Delta (t)$ as time-dependent quantities.",
"The solutions of the corresponding Euler–Lagrange–Rayleigh equations are found numerically, see Supplementary Materials for details and compared with micromagnetic simulationsWe consider stripes of length 2000 nm for simulations of domain wall dynamics and domain wall initial position 500 nm far from the stripe end.",
"Only exchange interaction, anisotropies and DMI are taken into account., see Fig.",
"REF (a).",
"The analysis is performed in the fields, which are smaller than the Walker field ($b < b_\\textsc {w}$ )The discussed model is not applicable above Walker fields due to instability of domain wall shape against bending.",
"Micromagnetic simulations show irregular wavy bends of domain boundary during its expansion, see Supplementary Materials..",
"In this case, the tilt angles $\\psi $ and $\\chi $ and the domain wall width quickly relax to their equilibrium values $\\psi _\\infty $ and $\\chi _\\infty $ , respectively (see also Eqns.",
"(S-11)–(S-13) in Supplementary Materials).",
"The domain wall velocity with equilibrium values of its width and angles reads $v = \\dfrac{pb}{2\\eta \\cos \\chi _\\infty \\sqrt{1 - k_2 \\sin (\\psi _\\infty - \\alpha )}},$ where the dimensionless velocity $v$ is measured in units of $2\\gamma \\sqrt{AK_1}/M_\\textsc {s}$ with $\\gamma $ being gyromagnetic ratio and $\\eta $ being Gilbert damping.",
"Note, that the maximum of the Walker field and, hence, the largest velocity is reached at the angle $\\alpha _0 \\approx 60^\\circ $ for the given parameters, which does not coincide with a shape anisotropy along the stripe main axis[33], [17].",
"Asymptotic analysis shows a good coincidence with numerical solution of equations of motion even for large enough fields and material parameters, see Fig.",
"REF (d) and Supplementary Materials for details.",
"Upon the motion of the domain wall, its internal structure changes dependent on the direction of the easy axis of the in-plane anisotropy $\\alpha $ and on the strength of the applied magnetic field $b$ (Fig.",
"REF ).",
"There exists a symmetry break with a favorable magnetization tilt direction (indicated as fav states in Fig.",
"REF ), in a small angular range about the orientation of ${e}_2$ (along or opposite to it).",
"It results in a higher velocity at a given field.",
"The domain wall with unfavorable tilt angle (indicated as unf states in Fig.",
"REF ) will switch the internal magnetization in the wall to the ${e}_2$ direction at the beginning of motion.",
"The red dashed line in Fig.",
"REF (b) indicates the smallest field, which is needed for switching of the magnetization angle in the wall.",
"The positive $b$ results in the appearance of “unf” state for negative $\\alpha $ and vice versa.",
"Sign of $\\chi $ coincides with sign of $b$ for fields, close to $b_\\textsc {w}$ .",
"We note a certain similarity with the texture-induced chirality breaking for moving magnetic domain walls in nanotubes due to non-local magnetostatics[43], [44].",
"To summarize, we investigate the internal domain wall structure and its orientation in an out-of-plane mangetized stripe with biaxial (in-plane and out-of-plane) anisotropy and Dzyaloshinskii-Moriya interaction.",
"The cooperative effect of the DMI and the additional anisotropy with an in-plane easy axis results in a unidirectional tilt of domain wall in equilibrium and in a symmetry break of a domain wall static state with respect to the stripe axis.",
"The domain wall dynamics in an applied out-of-plane magnetic field exhibits slow and fast motion similarly to vortex domain walls in tubes[43], [44].",
"We demonstrate that the Walker field but also the associated Walker velocity strongly depend on the orientation of the easy axis of the in-plane anisotropy.",
"There appears an optimal angle of the orientation of the in-plane easy axis to maximize the Walker field and the Walker velocity.",
"This optimal angle does not coincide with the direction of the easy axis of the shape anisotropy.",
"These results are relevant for the optimization of the domain wall dynamics in data storage and logic devices, relying on spintronic and spin-orbitronic concepts.",
"See Supplementary Material for the details of domain wall dynamics under the action of perpendicular magnetic field.",
"This work was financed in part via the German Research Foundation (DFG) Grants No.",
"MA5144/9-1, No.",
"MA5144/13-1, MA5144/14-1.",
"D.D.S.",
"thanks Helmholtz-Zentrum Dresden-Rossendorf e.V., where part of this work was performed, for their kind hospitality and acknowledges the support from the Alexander von Humboldt Foundation (Research Group Linkage Programme).",
"In part, this work was supported by Taras Shevcheko National University of Kyiv (Project No.",
"19BF052-01).",
"Simulations were performed using a high-performance cluster of the Taras Shevchenko National University of Kyiv[45].",
"in 1,...,8 [pages=,]dw-axis-supp.pdf"
]
] | 2001.03408 |
[
[
"Universal Error Bound for Constrained Quantum Dynamics"
],
[
"Abstract It is well known in quantum mechanics that a large energy gap between a Hilbert subspace of specific interest and the remainder of the spectrum can suppress transitions from the quantum states inside the subspace to those outside due to additional couplings that mix these states, and thus approximately lead to a constrained dynamics within the subspace.",
"While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking.",
"Here we establish an observable-based error bound for such a constrained-dynamics approximation in generic gapped quantum systems.",
"This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former.",
"We demonstrate that either the intercept or the slope in the bound is asymptotically saturable by simple models.",
"We generalize the result to quantum many-body systems with local interactions, for which the coupling strength diverges in the thermodynamic limit while the error is found to grow no faster than a power law $t^{d+1}$ in $d$ dimensions.",
"Our work establishes a universal and rigorous result concerning nonequilibrium quantum dynamics."
],
[
"Universal Error Bound for Constrained Quantum Dynamics Zongping Gong Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Nobuyuki Yoshioka Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Reserach (CPR), Wako-shi, Saitama 351-0198, Japan Naoyuki Shibata Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Ryusuke Hamazaki Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR), RIKEN iTHEMS, Wako, Saitama 351-0198, Japan It is well known in quantum mechanics that a large energy gap between a Hilbert subspace of specific interest and the remainder of the spectrum can suppress transitions from the quantum states inside the subspace to those outside due to additional couplings that mix these states, and thus approximately lead to a constrained dynamics within the subspace.",
"While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking.",
"Here we establish an observable-based error bound for such a constrained-dynamics approximation in generic gapped quantum systems.",
"This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former.",
"We demonstrate that either the intercept or the slope in the bound is asymptotically saturable by simple models.",
"We generalize the result to quantum many-body systems with local interactions, for which the coupling strength diverges in the thermodynamic limit while the error is found to grow no faster than a power law $t^{d+1}$ in $d$ dimensions.",
"Our work establishes a universal and rigorous result concerning nonequilibrium quantum dynamics.",
"Introduction.— Approximations appear ubiquitously in science and their validity should be justified by error estimations [1].",
"In quantum physics, where only a few systems are exactly solvable [2], one of the most widely used approximations is based on the separation of energy scales or the existence of large energy gaps [3].",
"While an entire quantum system can be very complicated, it can dramatically be simplified by keeping only the degrees of freedom with relevant energy scales.",
"Technically, this is achieved by projecting the full Hamiltonian onto a Hilbert subspace.",
"Two prototypical examples are approximating atoms as few-level systems in quantum optics [4] and crystalline materials as few-band systems in condensed matter physics [5].",
"When we add a weak coupling term such as external fields or interactions, it suffices to consider the action restricted in the subspace, provided that the manifold is energetically well-isolated from the remaining.",
"It is well-known from the perturbation theory that the error is of the order of the inverse energy gap, and thus vanishes in the infinite gap limit [6].",
"The suppression of error by energy gap has tacitly been used for approximating not only static quantum states but also quantum dynamics [7].",
"For example, quench dynamics in the Bose- and Fermi-Hubbard models are implemented by ultracold atoms in deep optical lattices such that the projection onto the ground-state band is a good approximation [8], [9], [10], [11].",
"More recently, the peculiarly slow thermalization dynamics observed in a strongly interacting Rydberg-atom chain is actively studied on the basis of the so-called PXP model, where the constrained dynamics is within the Hilbert subspace with adjacent Rydberg excitations forbidden [12], [13].",
"While the approximation of constrained dynamics from large energy gaps has widely been used in the literature, a general and quantitative justification stays lacking.",
"Here, we fill this gap by deriving a universal error bound for constrained dynamics.",
"This bound is simply a linear function of time and depends only on the coupling strength and energy gap.",
"Our main strategy is a general perturbative analysis based on the Schrieffer-Wolff transformation (SWT) [14], which is a unitary transformation that block diagonalizes the perturbed Hamiltonian and has been used to estimate the errors in equilibrium setups [15].",
"We mostly focus on bounded coupling terms, but will also outline the many-body generalization based on locality.",
"In addition to the Lieb-Robinson bound [16], [17], [18], the quantum speed limit [19], [20], [21], [22], the bound on energy absorption for Floquet systems [23], [24] and the bound on chaos [25], our work contributes yet another rigorous and universal bound in nonequilibrium quantum dynamics.",
"Figure: (a) Schematic illustration of the setup.",
"The original Hamiltonian H 0 H_0 has an isolated energy band onto which the projector is PP, whose complement is Q=1-PQ=1-P. A general coupling can be decomposed into V=V P +V Q +V off V=V_P+V_Q+V_{\\rm off}, where V P ≡PVPV_P\\equiv PVP, V Q ≡QVQV_Q\\equiv QVQ and V off ≡PVQ+QVPV_{\\rm off}\\equiv PVQ+QVP.",
"The arrow within PP refers to a constrained dynamics.",
"(b) Typical energy spectrum of H 0 H_0 with an isolated band ℓ 0 \\ell _0.",
"The energy gap is defined as Δ 0 ≡min{Δ u ,Δ d }\\Delta _0\\equiv \\min \\lbrace \\Delta _{\\rm u},\\Delta _{\\rm d}\\rbrace .",
"(c) Error dynamics () in a random model with Δ 0 =10∥V∥\\Delta _0=10\\Vert V\\Vert .",
"Different colors correspond to different realizations.",
"The rectangle and the arrow indicate the initial sudden jump and the subsequent linear growth, respectively.",
"Inset: Zoom in of the initial jump.Setup and numerical trials.— We consider a quantum system with arbitrarily large Hilbert-space dimension described by $H_0$ , which has an isolated energy band $\\ell _0$ separated from the remainder of the spectrum by an energy gap $\\Delta _0$ (see Fig.",
"REF (b)).",
"The projector onto the subspace spanned by all the eigenstates in $\\ell _0$ is denoted as $P$ (see Fig.",
"REF (a)).",
"With an additional perturbation term $V$ added to $H_0$ , the total Hamiltonian becomes $H=H_0+V.$ To quantify the deviation between the constrained dynamics generated by the projected Hamiltonian $H_P\\equiv PHP$ and the actual dynamics starting from a state in $\\ell _0$ , we define the error with respect to an observable $O$ as $\\epsilon (t)\\equiv \\Vert P(e^{iHt}Oe^{-iHt}-e^{iH_Pt}Oe^{-iH_Pt})P\\Vert ,$ where $\\Vert \\cdot \\Vert $ denotes the operator norm, i.e., the largest singular value.",
"Without loss of generality, we assume $O$ to be normalized as $\\Vert O\\Vert =1$ .",
"While only Hermitian observables are directly measurable in experiments, our result applies equally to non-Hermitian operators.",
"The maximal value of $\\epsilon (t)$ over $O$ is actually the superoperator norm $\\Vert {P}(e^{it{\\rm ad}_H}-e^{it{\\rm ad}_{H_P}})\\Vert _{\\infty \\rightarrow \\infty }$ induced by the operator norm [26], where $\\Vert {L}\\Vert _{\\infty \\rightarrow \\infty }\\equiv \\max _{\\Vert O\\Vert =1}\\Vert {L}O\\Vert $ , ${P}O\\equiv POP$ and ${\\rm ad}_AO\\equiv [A,O]$ .",
"It is helpful to first form an intuition on the typical behavior of $\\epsilon (t)$ .",
"We carry out numerical simulations for a randomly constructed system with three bands, each containing four levels and separated from the others by a large gap.",
"The observable is also taken to be random, and $\\ell _0$ is chosen to be the middle band.",
"As shown in Fig.",
"REF (c), while the local fluctuations in $\\epsilon (t)$ differ significantly for different random realizations of the system, there seems to be two universal features.",
"First, $\\epsilon (t)$ initially undergoes a sudden “jump\".",
"Precisely speaking, the “jump\" is a rapid growth within an $\\mathcal {O}(\\Delta _0^{-1})$ time interval (see the inset in Fig.",
"REF (c)).",
"Second, $\\epsilon (t)$ grows linearly despite of irregular fluctuations.",
"It is thus natural to conjecture that $\\epsilon (t)$ is bounded by a linear function of time.",
"Main result and its qualitative explanation.— The above conjecture turns out to be indeed true.",
"In the regime $\\Delta _0\\gg \\Vert V\\Vert $ and for an intermediately long time $t\\ll \\frac{\\Delta _0}{\\Vert V\\Vert ^2}$ (i.e., $\\Vert V\\Vert t$ is considered as order one), we claim the following universal asymptotical bound: $\\epsilon (t)\\lesssim \\frac{4\\Vert V\\Vert }{\\Delta _0}+\\frac{2\\Vert V\\Vert ^2}{\\Delta _0}t,$ where “$\\lesssim $ \" means that there could be a tiny violation up to $\\mathcal {O}(\\frac{\\Vert V\\Vert ^2}{\\Delta _0^2})$ .",
"It is possible to derive a bound valid even when $\\Delta _0$ is comparable with $\\Vert V\\Vert $ , but the form is a bit involved and exactly reproduces Eq.",
"(REF ) in the large-gap regime [27].",
"Two more remarks on the applicability of Eq.",
"(REF ) are in order.",
"First, the energy band can be embedded anywhere in the energy spectrum of $H_0$ .",
"It may consist of the ground states, mid-gap states or even the most excited states.",
"All these situations will later be exemplified.",
"Second, we do not assume any constraint on the width of the energy band.",
"It can be zero (e.g., for ground-state manifolds) or even larger than $\\Delta _0$ .",
"It is rather easy to understand the orders of the intercept and the slope in Eq.",
"(REF ).",
"According to the standard perturbation theory [3], a state in $\\ell _0$ should basically lie in $\\ell $ , the perturbed energy band in $H$ , but also slightly contain some components outside $\\ell $ .",
"These components have $\\mathcal {O}(\\frac{\\Vert V\\Vert }{\\Delta _0})$ weights (amplitudes), and their rapid oscillations owing to the dynamical phases $e^{-it\\mathcal {O}(\\Delta _0)}$ lead to the initial “jump\" of $\\epsilon (t)$ .",
"Also, the effective Hamiltonian in the Green's function (Fourier transform) of the projected unitary $Pe^{-iHt}P$ is known to be $H_{\\rm eff}(\\omega )=H_P+\\Sigma (\\omega )$ , where the self-energy $\\Sigma (\\omega )=PVQ(\\omega -H_Q)^{-1}QVP$ is of the order of $\\frac{\\Vert V\\Vert ^2}{\\Delta _0}$ for $\\omega \\in \\ell _0$ [28], [29], [30].",
"However, it is unclear from the above argument why the factors before $\\frac{\\Vert V\\Vert }{\\Delta _0}$ and $\\frac{\\Vert V\\Vert ^2}{\\Delta _0}$ should be 4 and 2.",
"It is even unclear why these factors can be finite, even though there can be a very large number of levels in or/and outside $\\ell _0$ .",
"“Worst\" models.— Before deriving the main result, let us comment on the tightness of the bound (REF ).",
"We emphasize that the bound is universally valid, so the tightness should be analyzed for the “worst\" models and observables instead of the typical ones like the random model in Fig.",
"REF .",
"It turns out that, separately, both of the constant and the time-linear terms are tight and can asymptotically (in the large gap limit) be saturated in very simple models.",
"Figure: “Worst\" models which realize the saturation of (a) the intercept and (b) the slope in Eq. ().",
"(c) and (d) are the corresponding error dynamics (green solid curves) of (a) and (b) given in Eqs.",
"() and () with Δ 0 =10Ω\\Delta _0=10\\Omega .",
"The black dotted line indicates the intercept and the red dashed lines are the asymptotic bound ().We first demonstrate the saturation of the constant in a two-level atom driven by a classical laser with detuning $\\Delta _0$ and Rabi frequency $\\Omega $ (see Fig.",
"REF (a)).",
"In this case, $H_0=\\frac{\\Delta _0}{2}\\sigma ^z$ and $V=\\frac{\\Omega }{2}\\sigma ^x$ [31] so that $\\Vert V\\Vert =\\frac{\\Omega }{2}$ , where $\\sigma ^x=|e\\rangle \\langle g|+|g\\rangle \\langle e|$ and $\\sigma ^z=|e\\rangle \\langle e|-|g\\rangle \\langle g|$ are the Pauli matrices.",
"Choosing $P=\\frac{1}{2}(1-\\sigma ^z)=|g\\rangle \\langle g|$ to be the ground-state projector and $O=\\sigma ^x$ , we can easily calculate the error to be $\\epsilon (t)=\\frac{\\Delta _0\\Omega }{\\Delta ^2}|1-\\cos (\\Delta t)|,$ where $\\Delta =\\sqrt{\\Delta _0^2+\\Omega ^2}$ .",
"This quantity rapidly reaches its maximum $\\frac{2\\Omega \\Delta _0}{\\Delta ^2}$ at $t=\\frac{\\pi }{\\Delta }$ , which asymptotically saturates $\\frac{4\\Vert V\\Vert }{\\Delta _0}=\\frac{2\\Omega }{\\Delta _0}$ in the large $\\Delta _0$ limit.",
"The error dynamics stays exactly the same for $P=\\frac{1}{2}(1+\\sigma ^z)=|e\\rangle \\langle e|$ .",
"We move on to demonstrate the saturation of the slope.",
"Here we should consider a situation with $\\operatorname{Tr}P>1$ , otherwise $\\epsilon (t)$ never exceeds an $\\mathcal {O}(\\frac{\\Vert V\\Vert }{\\Delta _0})$ constant even in the long-time limit [27].",
"It turns out that a four-level system with $H_0=\\frac{\\Delta _0}{2}(\\sigma ^z_1+\\sigma ^z_2)$ and $V=\\frac{\\Omega }{2}\\sigma ^x_1$ , which describes two identical two-level atoms with only one driven by a classical laser (see Fig.",
"REF (b)), already constitutes such a worst-case scenario.",
"Choosing $P=\\frac{1}{2}(1-\\sigma ^z_1\\sigma ^z_2)$ (with $\\operatorname{Tr}P=2$ ) and $O=\\frac{1}{2}(\\sigma ^x_1\\sigma ^x_2+\\sigma ^y_1\\sigma ^y_2)$ ($\\sigma ^y\\equiv i(|g\\rangle \\langle e|-|e\\rangle \\langle g|)$ ), we have $\\epsilon (t)=\\left|\\left[\\cos \\left(\\frac{\\Delta t}{2}\\right)+i\\frac{\\Delta _0}{\\Delta }\\sin \\left(\\frac{\\Delta t}{2}\\right)\\right]^2-e^{i\\Delta _0 t}\\right|,$ which is well approximated by $|e^{i(\\Delta -\\Delta _0)t}-1|\\simeq \\frac{\\Omega ^2}{2\\Delta _0}t=\\frac{2\\Vert V\\Vert ^2}{\\Delta _0}t$ for $t\\ll \\frac{\\Delta _0}{\\Vert V\\Vert ^2}=\\frac{4\\Delta _0}{\\Omega ^2}$ .",
"We plot Eqs.",
"(REF ) and (REF ) in Figs.",
"REF (c) and (d), respectively, where the asymptotic bound (REF ) is also shown for comparison.",
"Derivation of the main result.— We turn to the derivation of Eq.",
"(REF ).",
"The first step is to perform the SWT [14], [15]: $SHS^\\dag = H_0+V_{\\rm diag}+V^{\\prime },$ where $S=e^T$ is unitary and the anti-Hermitian generator $T$ is determined from ${\\rm ad}_T(H_0+V_{\\rm diag})=-V_{\\rm off}$ , with $V_{\\rm diag}\\equiv PVP+QVQ$ and $V_{\\rm off}\\equiv V-V_{\\rm diag}$ being the block-diagonal and block-off-diagonal components of $V$ , respectively.",
"Since $H_0$ is block diagonalized while $V_{\\rm off}$ is off-block diagonalized, it follows that $T$ can be restricted to be off-block diagonalized to satisfy the following Sylvester equation [32]: $T H_Q-H_PT=-PVQ.$ Provided that the spectra of $H_P$ and $H_Q$ are separated by a gap $\\Delta $ , $\\Vert T\\Vert $ is rigorously upper bounded by [15], [33] $\\Vert T\\Vert \\le \\frac{\\Vert PVQ\\Vert }{\\Delta }\\lesssim \\frac{\\Vert V\\Vert }{\\Delta _0}.$ We recall that “$\\lesssim $ \" in Eq.",
"(REF ) allows a tiny violation with $\\mathcal {O}(\\frac{\\Vert V\\Vert ^2}{\\Delta _0^2})$ , and is validated by $\\Delta \\ge \\Delta _0-2\\Vert V\\Vert $ , a result ensured by Weyl's perturbation theorem [34].",
"Accordingly, the norm of the remaining term $V^{\\prime }=\\sum ^\\infty _{n=1}\\frac{n}{(n+1)!",
"}{\\rm ad}_T^nV_{\\rm off}$ in Eq.",
"(REF ) should asymptotically be bounded by $\\Vert V^{\\prime }\\Vert \\lesssim \\frac{1}{2}\\Vert [T,V_{\\rm off}]\\Vert \\le \\Vert T\\Vert \\Vert V\\Vert \\lesssim \\frac{\\Vert V\\Vert ^2}{\\Delta _0}$ in the large gap regime.",
"Here we have used the fact that $\\Vert T\\Vert $ is small and $\\Vert V_{\\rm off}\\Vert \\le \\Vert V\\Vert $ [27].",
"With the help of the SWT, we can rewrite the error (REF ) as $\\epsilon (t)=\\Vert P[S_{H_1}(t)^\\dag L(t)SOS^\\dag L(t)^\\dag S_{H_1}(t)-O]P\\Vert ,$ where $H_1\\equiv H_0+V_{\\rm diag}$ , $L(t)=e^{-iH_1t}e^{i(H_1+V^{\\prime })t}$ is the Loschmidt-echo operator [35] and $S_{H_1}(t)=e^{-iH_1t}e^Te^{iH_1t}=e^{e^{-iH_1t}Te^{iH_1t}}$ is the SWT in the interacting picture.",
"This rewritten form (REF ) has a crucial property that the generators of $S$ and $S_{H_1}(t)$ both have small norms (at most) of the order of $\\frac{\\Vert V\\Vert }{\\Delta _0}$ , and so is $L(t)=\\overrightarrow{{\\rm T}}e^{i\\int _0^tdt^{\\prime }e^{-iH_1t^{\\prime }}V^{\\prime }e^{iH_1t^{\\prime }}}$ ($\\overrightarrow{{\\rm T}}$ : time ordering) [36] for a time scale of interest (i.e., $\\Vert V\\Vert t$ is of order one).",
"Applying the inequality $\\Vert (\\prod _\\alpha U_\\alpha )O(\\prod _\\alpha U_\\alpha )^\\dag -O\\Vert \\le \\sum _\\alpha \\Vert U_\\alpha OU_\\alpha ^\\dag -O\\Vert $ for unitaries $U_\\alpha $ 's [27] to Eq.",
"(REF ), we obtain $\\begin{split}\\epsilon (t)\\le \\Vert SOS^\\dag -O\\Vert &+\\Vert L(t)OL(t)^\\dag -O\\Vert \\\\&+\\Vert S_{H_1}(t)^\\dag OS_{H_1}(t)-O\\Vert .\\end{split}$ Using the inequality [27] $\\Vert e^{T}Oe^{-T}-O\\Vert \\le \\Vert [T,O]\\Vert \\le 2\\Vert T\\Vert $ for all $T=-T^\\dag $ (and $\\Vert O\\Vert =1$ ), we can bound the first and the third terms in Eq.",
"(REF ) by $2\\Vert T\\Vert $ , and the second term by $\\begin{split}\\Vert L(t)OL(t)^\\dag -O\\Vert &\\le \\int ^t_0dt^{\\prime }\\Vert [e^{-iH_1t^{\\prime }}V^{\\prime }e^{iH_1t^{\\prime }},O]\\Vert \\\\&\\le 2\\Vert V^{\\prime }\\Vert t.\\end{split}$ Substituting these exact bounds in Eqs.",
"(REF ) and (REF ) and those asymptotic ones in Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ), we obtain Eq.",
"(REF ).",
"Now it is clear that the constant in Eq.",
"(REF ) arises from the SWT and the time-evolved SWT, i.e., the first and third terms on the rhs of Eq.",
"(REF ), while the time-linear term arises from the Loschmidt echo, i.e., the second term on the rhs of Eq.",
"(REF ).",
"Generalization to many-body systems.— We next focus on quantum many-body systems defined on a general $d$ -dimensional lattice $\\Lambda $ , where each site is associated with a finite dimensional local Hilbert space.",
"A particularly important case is locally interacting systems, whose many-body Hamiltonians still take the form of Eq.",
"(REF ) while both $H_0$ and $V$ are a sum of Hermitian operators supported on finite regions, whose norms are uniformly bounded.",
"Formally, we can write $V=\\sum _{A\\subseteq \\Lambda }V_A$ with $V_A$ supported on a connected region $A$ and $V_A=0$ if its volume $|A|$ exceeds a threshold.",
"In contrast with the few-body systems, the error bound on the rhs of Eq.",
"(REF ) diverges in the thermodynamic limit.",
"This is because $\\Vert V\\Vert $ grows linearly with respect to the system volume $|\\Lambda |\\sim L^d$ , where $L=l_\\Lambda $ is the diameter of the entire system.",
"However, by further assuming (i) $H_0=\\sum _j H_{0j}$ is commuting and frustration-free in the sense that all the local operators commute, i.e., $[H_{0j},H_{0j^{\\prime }}]=0$ for all $j,j^{\\prime }\\in \\Lambda $ , and all the global ground states minimize local energies everywhere; (ii) $O=O_X$ is a local observable supported on $X$ with $|X|,l_X\\sim \\mathcal {O}(1)$ , we can still derive a meaningful bound: $\\epsilon (t)\\le \\frac{\\Vert V\\Vert _{\\star }}{\\Delta _0}p(t),$ where $\\Vert V\\Vert _{\\star }\\equiv \\max _{j\\in \\Lambda }\\sum _{A\\ni j}\\Vert V_A\\Vert $ is the local interaction strength, which is set to be $\\mathcal {O}(1)$ by rescaling the time scale, and $p(t)$ is a polynomial of $t$ with degree $d+1$ and (at most) order-one coefficients.",
"This result implies that for a prescribed precision $\\epsilon $ , the constrained dynamics is a locally good approximation up to a time scale (at least) of $\\mathcal {O}((\\Delta _0\\epsilon )^{\\frac{1}{d+1}})$ .",
"Let us outline the proof of Eq.",
"(REF ), whose full details are available in Ref. [27].",
"The general idea is to combine the local SWT [15], [37] and the Lieb-Robinson bound [16].",
"By local we mean that the generator $T$ is a sum of local operators.",
"The locality of $T$ and $O_X$ allows us to modify Eq.",
"(REF ) into [27] $\\Vert e^TO_Xe^{-T}-O_X\\Vert \\le 2|X|\\Vert T\\Vert _\\star .$ Similar to Eq.",
"(REF ), $\\Vert T\\Vert _\\star $ can be upper bounded by $\\mathcal {O}(\\frac{\\Vert V\\Vert _\\star }{\\Delta _0})$ , and so can the first term on the rhs of Eq.",
"(REF ).",
"As for the third term, we note that $\\Vert S_{H_1}(t)^\\dag O_XS_{H_1}(t)-O_X\\Vert =\\Vert e^{-T}O_X^{H_1}(t)e^T-O_X^{H_1}(t)\\Vert $ with $O_X^{H_1}(t)\\equiv e^{iH_1t}O_Xe^{-iH_1t}$ being the observable in the Heisenberg picture.",
"While the support of $O_X^{H_1}(t)$ generally covers the entire lattice in a rigorous sense, we can show from the Lieb-Robinson bound [16] that the support volume is effectively of the order of $(l_X+2vt)^d$ [27], where $v$ is the Lieb-Robinson velocity.",
"We emphasize that $v$ is essentially determined by $V$ since $H_0$ is by assumption commuting, i.e., consists of commutative local operators, and thus almost does not contribute to the spreading of operators [38].",
"This fact ensures the finiteness of $v$ even in the infinite-gap limit, where the usual Lieb-Robinson velocity [16] determined from $H$ diverges.",
"Moreover, the locality of $T$ in turn ensures the (quasi-)locality of $V^{\\prime }$ in Eq.",
"(REF ), and we can show that $\\Vert V^{\\prime }\\Vert _\\star $ is no more than $\\mathcal {O}(\\frac{\\Vert V\\Vert _\\star ^2}{\\Delta _0})$ in the large gap regime, just like Eq.",
"(REF ).",
"Following the same argument used for bounding $\\Vert S_{H_1}(t)^\\dag O_XS_{H_1}(t)-O_X\\Vert $ , we can show that the order of the integrand in Eq.",
"(REF ) is no more than $\\Vert V^{\\prime }\\Vert _\\star (l_X+2vt^{\\prime })^d$ , whose integral is a polynomial of $t$ with degree $d+1$ .",
"Combining all the analyses above, we obtain Eq.",
"(REF ) from Eq.",
"(REF ).",
"To illustrate our findings, we consider the error dynamics in the parent Hamiltonian of the PXP model [13].",
"As is graphically illustrated in Fig.",
"REF (a), the Hamiltonian is given as $H_0=\\frac{\\Delta _0}{4}\\sum _{j=1}^{L-1}(\\sigma ^z_j+1)(\\sigma ^z_{j+1}+1),\\;\\;\\;\\;V=\\frac{\\Omega }{2}\\sum _j\\sigma ^x_j,$ where $L$ is the system size.",
"The constrained dynamics concerns $P=\\prod _j\\left[1- \\frac{1}{4}(1+\\sigma _j^z)(1+\\sigma _{j+1}^z)\\right]$ , which is a projector onto the Hilbert subspace with adjacent excitations forbidden.",
"As shown in Fig.",
"REF (b), $\\epsilon (t)$ indeed grows like $t^2$ rather than $t$ after the “sudden\" jump, implying that the power bound $t^{d+1}$ presented in Eq.",
"(REF ) is qualitatively tight.",
"On the other hand, we do not expect quantitative saturation of the many-body bound due to the looseness of the Lieb-Robinson bound [16].",
"We also argue that beyond the Lieb-Robinson time $t^*\\sim L/v$ the error grows linearly.",
"This is because the correlation spreads throughout the system and hence the bound based on the locality argument no longer holds.",
"Figure: (a) An array of two-level atoms with interaction Δ 0 \\Delta _0 between adjacent excited states.Each atom is resonantly driven with an identical Rabi frequency Ω\\Omega .",
"(b) The quadratic growth of ϵ(t)\\epsilon (t) concerning the error dynamics of O=σ j=1 y O=\\sigma ^y_{j=1} in the parent Hamiltonian of the PXP model defined by Eq.",
"() with log 10 Δ 0 =1.0,1.5,2.0,2.5\\log _{10}\\Delta _0=1.0, 1.5, 2.0, 2.5.",
"The best fitting quadratic curves (grey dotted lines) are confirmed to be more accurate than the linear ones, while the growth becomes rather linear after the correlation spreads throughout the system at Ωt∼12\\Omega t\\sim 12.",
"The Rabi frequency is Ω=2\\Omega =2 and the system size is L=12L=12.",
"Inset: Rescaled error Δ 0 ϵ(t)\\Delta _0 \\epsilon (t), whose quasi-independence on Δ 0 \\Delta _0 is consistent with Eq.",
"().Summary and outlook.— In summary, we have established a universal and tight error bound (REF ) for constrained dynamics in generic quantum systems with isolated energy bands.",
"By universal we mean that the bound is generally applicable and only involves a minimal number of parameters (coupling strength and energy gap).",
"By tight we mean that it can partially be saturated in some worst cases.",
"The result has been generalized to many-body systems by means of the Lieb-Robinson bound.",
"It is found that the error of a local observable grows no faster than $t^{d+1}$ , so many-body constrained dynamics can stay locally good approximations.",
"The error bound can readily be generalized to open quantum systems with decoherence-free subspaces [39], [40] subject to coherent perturbations, as has been shown in Ref. [27].",
"The obtained result is actually related to the quantum Zeno effect [41], [42], [43].",
"Our work also raises many open questions such as whether the intercept and the slope in Eq.",
"(REF ) can simultaneously be saturated and whether it is possible to generalize to open quantum many-body systems with an extensive number of dark states [44], [45].",
"Other directions of future studies include the generalizations to higher-order SWTs [15], [27] and long-range interacting systems [46], [47], [48], [49], [50].",
"We acknowledge Takashi Mori for valuable comments.",
"The numerical calculations were carried out with the help of QuTiP [51].",
"Z.G.",
"was supported by MEXT.",
"N.Y. and R.H. were supported by Advanced Leading Graduate Course for Photon Science (ALPS) of Japan Society for the Promotion of Science (JSPS).",
"N.Y. was supported by JSPS KAKENHI Grant-in-Aid for JSPS fellows Grant No.",
"JP17J00743.",
"N.S.",
"acknowledges support of the Materials Education program for the future leaders in Research, Industry, and Technology (MERIT).",
"R.H. was supported by JSPS KAKENHI Grant-in-Aid for JSPS fellows Grant No.",
"JP17J03189)."
]
] | 2001.03419 |
[
[
"Continuous-variable quantum teleportation with vacuum-entangled Rindler\n modes"
],
[
"Abstract We consider a continuous-variable quantum teleportation protocol between a uniformly accelerated sender in the right Rindler wedge, a conformal receiver restricted to the future light cone, and an inertial observer in the Minkowski vacuum.",
"Using a non-perturbative quantum circuit model, the accelerated observer interacts unitarily with the Rindler modes of the field, thereby accessing entanglement of the vacuum as a resource.",
"We find that a Rindler-displaced Minkowski vacuum state prepared and teleported by the accelerated observer appears mixed according to the inertial observer, despite a reduction of the quadrature variances below classical limits.",
"This is a surprising result, since the same state transmitted directly from the accelerated observer appears as a pure coherent state to the inertial observer.",
"The decoherence of the state is caused by an interplay of opposing effects as the acceleration increases: the reduction of vacuum noise in the output state for a stronger entanglement resource, constrained by the amplification of thermal noise due to the presence of Unruh radiation."
],
[
"Introduction",
"Entanglement is a fundamental property of the vacuum state of relativistic quantum field theory.",
"It arises when spacetime is partitioned into distinct regions and appears between the resulting subsystems [1].",
"For example, in flat Minkowski spacetime, the vacuum state can be expressed as an entangled state between the complete sets of modes spanning the left and right Rindler wedges.",
"This gives rise to the Unruh effect, which predicts that a uniformly accelerated detector sees a thermal bath of particles.",
"Since the detector is confined to the right Rindler wedge, the unobserved Rindler modes in the left wedge are traced out, yielding a thermalised vacuum state [2], [3], [4], [5].",
"The extraction of this underlying vacuum entanglement has been well studied in the literature.",
"The seminal work of Reznik et al.",
"[6] established the field of entanglement harvesting, which considers the swapping of vacuum entanglement onto bi-partite quantum systems, such as Unruh-deWitt detectors, through local interactions.",
"Such harvesting protocols have been applied to spacelike [7] and timelike [8], [9] separated detectors, as well as situations involving uniform accelerations [10], black holes [11], [12] and expanding universes [13], [14], [15], [16].",
"A natural question to consider is whether the intrinsic entanglement of spacetime can be utilised as an information-theoretic and physical resource for quantum communication protocols.",
"Previous work by Ralph et al.",
"uses observers coupling to the vacuum-entangled modes of a massless scalar field (by operating detectors whose time-dependent energy scaling imitates a uniformly accelerated observer interacting with the Rindler modes [8], [9], [17]) to implement a quantum key distribution protocol, whilst Reznik has offered proof-of-principle proposals for dense coding and quantum teleportation protocols through vacuum entanglement swapped onto pairs of stationary, spacelike-separated atoms [18].",
"In [19], the authors employ two Unruh-deWitt detectors in relative inertial motion that interact perturbatively with the field to extract entanglement and perform better-than-classical teleportation.",
"The focus of this paper is also vacuum entanglement-enabled quantum teleportation, however here, we adopt a nonperturbative, continuous-variable approach, formulated with observers in relativistic, non-inertial reference frames.",
"Whilst teleportation protocols with accelerated partners have been previously studied [20], [21], [22], [23], they generally feature qubit degrees of freedom (Bell state measurements and two-level detectors) and a pre-existent entanglement resource (such as entangling photon cavities) which is distributed between the observers.",
"The main results from those papers is that the Unruh radiation perceived by the accelerated observer typically leads to the degradation of the fidelity of the protocol.",
"Here, we consider a continuous-variable teleportation protocol in which Rob, who uniformly accelerates in the right Rindler wedge, uses the entanglement of the quantum vacuum to teleport unknown Rindler-displaced Minkowski vacua (displaced thermal states) to a stationary `conformal receiver', Charlie (who resides in the same reference frame as Rob, by virtue of being restricted to the future light-cone of Minkowski spacetime), who then directly transmits the teleported state to Alice, an inertial observer in the Minkowski vacuum.",
"We employ the nonperturbative quantum circuit model developed in [24], [25] to describe the evolution of the initial state between the observers.",
"Our main result indicates that although performance beyond the classical channel limit is possible, the limit of high entanglement does not lead to ideal, continuous-variable teleportation in this scenario.",
"This paper is organised as follows: in Sec.",
"II, we review the continuous-variable teleportation protocol which we will implement in the reference frame of the uniformly accelerated observer.",
"In Sec.",
"III, we discuss the notion of vacuum entanglement between independent regions of spacetime, before reviewing the quantum circuit model and the self-homodyne detection technique for the inertial observer in Sec.",
"IV.",
"In Sec.",
"V, we present analytic results for the purity of output states according to the inertial Minkowski observer.",
"We conclude with some final remarks and opportunities for further research."
],
[
"Continuous-Variable Teleportation Between Inertial Observers",
"Quantum teleportation describes the transfer of an unknown quantum state $\\hat{\\rho }_\\text{in}$ between distant observers using a classical channel and a preexisting entanglement resource [26].",
"Whilst originally formulated using entangled Bell pairs as the resource, a continuous-variable version of the protocol was first introduced by Vaidman in [27] and then by Braunstein and Kimble in [28].",
"We follow the all-optical approach in [29], where the classical channel is enacted via the direct transmission of a classical field wherein the conjugate quadrature operators of the field are largely amplified above the quantum noise limit (QNL).",
"The quantum channel involves the distribution of an entangled bi-partite system between the observers.",
"By performing appropriate local operations on their entangled system, the receiver can retrieve an arbitrarily good version of the initial state $\\hat{\\rho }_\\text{in}$ without any quantum information being directly transmitted.",
"For our purposes, the utility of this approach is that the evolution of field modes can be modeled as unitary operations.",
"Consider the diagram shown in Fig.",
"1.",
"Figure: The all-optical teleportation protocol between inertial observers.",
"Bob prepares an unknown state in the mode a ^ in \\hat{a}_\\text{in} and mixes it at a linear optical amplifier (modeled by a two-mode squeezing unitary) with one half of an EPR-entangled field mode, a ^ i \\hat{a}_i.",
"The amplification of the input state generates an effective classical channel between Rob and Charlie, who combines a ^ ch.",
"\\hat{a}_\\text{ch.}",
"with the other half of the EPR pair at a beam splitter.",
"If Charlie selects η\\eta appropriately, Alice can receive a perfect reconstruction of a ^ in \\hat{a}_\\text{in}, in the limit of high entanglement.A pair of EPR-entangled field modes is distributed between two distant observers, Rob and Charlie.",
"Rob, operating a linear optical amplifier, mixes the input mode $\\hat{a}_\\text{in}$ with the entangled vacuum mode $\\hat{a}_i$ and transmits the output field $\\hat{a}_\\text{ch.",
"}$ to Charlie.",
"For a sufficiently large gain $\\mathcal {G}$ , the conjugate quadrature operators $\\hat{X}_\\text{ch.",
"}^+, \\hat{X}_\\text{ch.",
"}^-$ have uncertainties much larger than the quantum limit of unity.",
"Hence, the quantum noise introduced by a joint measurement of $\\hat{X}_\\text{ch.",
"}^+, \\hat{X}_\\text{ch.",
"}^-$ is negligible compared to the already amplified quadrature amplitudes, allowing for the designation of $\\hat{a}_\\text{ch.",
"}$ as a classical field [29].",
"Charlie receives $\\hat{a}_\\text{ch.",
"}$ and mixes it at a beam splitter with the other half of the entanglement resource, $\\hat{a}_j$ .",
"By attenuating $\\hat{a}_\\text{ch.",
"}$ by $1/\\mathcal {G}$ and then considering the limit of perfect entanglement between the vacuum modes $\\hat{a}_i$ and $\\hat{a}_j$ , Charlie is able to retrieve the input mode $\\hat{a}_\\text{in}$ .",
"To illustrate this mathematically, consider the evolution of the bosonic operator $\\hat{a}_\\text{in}$ through the teleportation protocol, given in the Heisenberg picture by $\\hat{a}_\\text{out} &= \\hat{S}_2^\\dagger (r) \\hat{U}_\\text{BS}^\\dagger (\\eta ) \\hat{a}_\\text{in} \\hat{U}_\\text{BS} (\\eta ) \\hat{S}_2(r)$ where $\\hat{S}_2(r) &= \\exp \\big [\\xi ^\\star \\hat{a}_i\\hat{a}_\\text{in} - \\xi \\hat{a}_\\text{in}^\\dagger \\hat{a}_i^\\dagger \\big ] \\\\\\hat{U}_\\text{BS}(\\eta ) &= \\exp \\big [ i\\theta \\big ( e^{i\\phi } \\hat{a}_\\text{in}^\\dagger \\hat{a}_j + e^{-i\\phi }\\hat{a}_\\text{in} \\hat{a}_j^\\dagger \\big ) \\big ] .$ $\\hat{S}_2(r)$ is a unitary two-mode squeezing operator with amplitude $\\xi = re^{i\\alpha }$ , describing the linear amplification of $\\hat{a}_\\text{in}$ mixed with $\\hat{a}_i$ , whilst $\\hat{U}_\\text{BS}(\\eta )$ represents the beam splitter interaction performed by Charlie.",
"Recalling that in the Heisenberg picture, operators act on the modes in reversed temporal order, we have $\\hat{a}_\\text{out} &= \\hat{S}_2^\\dagger (r) \\big (\\sqrt{\\eta } \\hat{a}_\\text{in} - \\sqrt{1 - \\eta } \\hat{a}_j \\big ) \\hat{S}_2(r) \\\\&= \\sqrt{\\eta } \\hat{S}_2^\\dagger (r) \\hat{a}_\\text{in} \\hat{S}_2(r) - \\sqrt{1 - \\eta } \\hat{a}_j ,$ where we have used the substitution $\\eta = \\cos ^2\\theta $ .",
"Now, $\\hat{S}_2^\\dagger (r) \\hat{a}_\\text{in} \\hat{S}_2(r)$ imitates the linear amplification of the field quadratures of $\\hat{a}_\\text{in}$ mixed with $\\hat{a}_i$ for $r\\gg 1$ , acting as the classical channel between Rob and Charlie, $\\hat{a}_\\text{out} &= \\underbrace{ \\sqrt{\\eta } \\cosh r \\hat{a}_\\text{in} + \\sqrt{\\eta }\\sinh r \\hat{a}_i^\\dagger }_{\\hat{a}_\\text{ch.}}",
"- \\sqrt{1 - \\eta } \\hat{a}_j .$ It can also be regarded as a joint, continuous-variable measurement of the quadrature operators for $\\hat{a}_\\text{in}$ and $\\hat{a}_i$ .",
"Then, by taking the attenuation of Charlie's beam splitter to be $\\eta = (\\cosh r )^{-2}$ , $\\hat{a}_\\text{out}$ reduces to $\\hat{a}_\\text{out} &= \\hat{a}_\\text{in} + \\hat{a}_i^\\dagger \\tanh r - \\hat{a}_j \\sqrt{1 - (\\cosh r)^{-2} }.$ In the limit $r\\gg 1$ , Eq.",
"(REF ) becomes $\\hat{a}_\\text{out} &\\simeq \\hat{a}_\\text{in} + \\hat{a}_i^\\dagger - \\hat{a}_j,$ implying that the output state is polluted by two additional QNL from $\\hat{a}_i, \\hat{a}_j$ .",
"Recall however that $\\hat{a}_i$ and $\\hat{a}_j$ are highly correlated via two-mode squeezing of the vacuum state, related by $\\hat{a}_i &= \\hat{S}_2^\\dagger (r_\\omega ) \\hat{v}_1 \\hat{S}_2(r_\\omega ) = \\hat{v}_1\\cosh r_\\omega + \\hat{v}_2^\\dagger \\sinh r_\\omega \\\\\\hat{a}_j &= \\hat{S}_2^\\dagger (r_\\omega ) \\hat{v}_2\\hat{S}_2(r_\\omega ) = \\hat{v}_2 \\cosh r_\\omega + \\hat{v}_1^\\dagger \\sinh r_\\omega ,$ where $r_\\omega $ is the squeezing amplitude.",
"Expressing $\\hat{a}_i^\\dagger $ and $\\hat{a}_j$ in terms of $\\hat{v}_1,\\hat{v}_2$ yields $\\hat{a}_\\text{out} &= \\hat{a}_\\text{in} + \\left( \\cosh r_\\omega - \\sinh r_\\omega \\right) \\big ( \\hat{v}_1^\\dagger - \\hat{v}_2 \\big ).$ In the limit of perfect entanglement between $\\hat{a}_i$ , $\\hat{a}_j$ ($r_\\omega \\rightarrow \\infty )$ , then $\\hat{a}_\\text{out} &= \\hat{a}_\\text{in} .$ In this limit, the effective channel between the input and output is the identity.",
"This protocol can be classified as quantum teleportation because the only direct link between the input and output is the classical channel through which the highly amplified field propagates [29]."
],
[
"Vacuum Entanglement in (1+1)-Dimensional Quantum Field Theory",
"We now review the entanglement structure of the vacuum state in (1+1)-dimensional quantum field theory.",
"Consider a massless, scalar field $\\hat{\\Phi }$ which can be expanded in a plane wave basis, given by $\\hat{\\Phi } (U,V) &= \\int k̥ \\big ( \\hat{a}_{kl} u_{k}(V) + \\hat{a}_{kr} u_{k}(U) + \\text{h.c} \\big ),$ where h.c denotes the Hermitian conjugate, and $u_{k}(V) (u_{k}(U))$ are left-moving (right-moving) single-frequency mode functions [24] $u_{k} (V) &= \\frac{1}{\\sqrt{4\\pi k}}e^{-ikV} \\\\u_{k} (U) &= \\frac{1}{\\sqrt{4\\pi k }} e^{-ikU},$ where $V = t+z$ and $U= t-z$ are Minkowski light-cone coordinates and $k$ is the frequency of the mode with respect to $t$ .",
"The single-frequency Minkowski operators $\\smash[b]{\\hat{a}_{kl} (\\hat{a}_{kr})}$ and $\\smash[b]{\\hat{a}_{kl}^{\\dagger } (\\hat{a}_{kr}^{\\dagger } )}$ satisfy standard bosonic commutation relations, $\\smash[b]{[\\hat{a}_{ki},a_{k^{\\prime }i^{\\prime }}^{\\dagger }] = \\delta _{i,i^{\\prime }}\\delta (k-k^{\\prime })}$ and $\\smash[b]{[\\hat{a}_{ki},\\hat{a}_{k^{\\prime }i^{\\prime }}] = 0}$ , where $i= l,r$ denotes the directionality.",
"The single-frequency Unruh operators $\\hat{c}_\\omega $ , $\\hat{d}_\\omega $ are related to the Minkowski operator by $\\hat{a}_{kl} &= \\int \\big ( A_{k\\omega } \\hat{c}_{\\omega l} + \\hat{B}_{k\\omega } \\hat{d}_{\\omega l} \\big )\\\\\\hat{a}_{kr} &= \\int \\big ( B_{k\\omega } \\hat{c}_{\\omega r} + \\hat{A}_{k\\omega } \\hat{d}_{\\omega r} \\big ) ,$ where $A_{k\\omega } = B_{k\\omega }^\\star &= \\frac{ i \\sqrt{\\sinh (\\pi \\omega /a)}}{2\\pi \\sqrt{\\omega k }} \\Gamma ( 1- i \\omega /a) \\left( \\frac{k}{a}\\right)^{i\\omega /a}$ are the Bogoliubov transformation coefficients [3].",
"The Unruh operators are a stepping stone between the Rindler and Minkowski operators."
],
[
"Rindler Wedges, Coordinates and Operators",
"Minkowski spacetime can be partitioned into four wedges denoted (R), (L), (F) and (P), representing the right and left Rindler wedges and the future and past light cones respectively (Fig.",
"REF ) [24].",
"Figure: Rob, accelerating uniformly in the right Rindler wedge, interacts with the Rindler wavepacket modes b ^ g IV \\hat{b}_g^\\text{IV} and b ^ g I \\hat{b}_g^\\text{I} in the blue ellipse region.",
"The classical channel is sent to the conformal receiver Charlie, who mixes it with the right-moving Rindler modes b ^ g III \\hat{b}_g^\\text{III} at a beam splitter (graded ellipse), before transmitting the state to Alice, in the Minkowski vacuum.",
"The unitary evolution of the Rindler modes is shown in Fig.",
".coordinates in these wedges are defined as follows, $\\begin{split}\\text{(F)}&\\:\\:\\:{\\left\\lbrace \\begin{array}{ll}t &= a^{-1} e^{a\\eta }\\cosh (a\\zeta ) \\\\z &= a^{-1} e^{a\\eta }\\sinh (a\\zeta )\\end{array}\\right.",
"}\\end{split}\\\\\\begin{split}\\text{(P)}&\\:\\:\\:{\\left\\lbrace \\begin{array}{ll}t &= -a^{-1} e^{a\\bar{\\eta }} \\cosh (a\\bar{\\zeta }) \\\\z &= - a^{-1} e^{a\\bar{\\eta }} \\sinh (a\\bar{\\zeta })\\end{array}\\right.",
"}\\end{split} ,$ where $\\eta $ , $\\bar{\\eta }$ and $\\zeta , \\bar{\\zeta }$ are the conformal time and spatial coordinates of observers restricted to the (F) and (P) light-cones respectively, and $\\begin{split}\\text{(R)}& \\:\\:\\:{\\left\\lbrace \\begin{array}{ll}t &= a^{-1} e^{a\\xi } \\sinh (a\\tau ) \\\\z &= a^{-1} e^{a\\xi } \\cosh (a\\tau )\\end{array}\\right.",
"}\\end{split}\\\\\\begin{split}\\text{(L)}&\\:\\:\\:{\\left\\lbrace \\begin{array}{ll}t &= - a^{-1} e^{a\\bar{\\xi }} \\sinh (a\\bar{\\tau }) \\\\z &= -a^{-1} e^{a\\bar{\\xi }} \\cosh (a\\bar{\\tau })\\end{array}\\right.",
"}\\end{split},$ where $\\tau ,\\bar{\\tau }$ and $\\xi ,\\bar{\\xi }$ are the proper time and the spatial coordinates of a uniformly accelerated observer with proper acceleration $a$ in the (R) and (L) wedge respectively.",
"In Fig.",
"REF , the conformal receiver Charlie is stationary according to an inertial observer but operates a beam splitter with time-dependent reflectivity.",
"Specifically, Olson et al.",
"demonstrated in [8] that an Unruh-deWitt detector whose energy gap is scaled continuously as $1/at$ responds to the Minkowski vacuum identically to one on an accelerated trajectory with fixed energy gap.",
"By using an analogously time-dependent, mode-selective beam splitter, Charlie remains in the causal future of Rob, who is uniformly accelerating, but can still interact with the same modes as him.",
"This proceeds by analytically extending the (left-moving) Rindler mode functions in (R) into (P).",
"To see this, it is useful to introduce the Rindler analogues to the light-cone coordinates, given by $\\begin{split}\\text{(F)} \\:\\:\\nu &= \\eta + \\zeta \\\\\\text{(P)} \\:\\:\\overline{\\nu } &= - \\overline{\\eta } - \\overline{\\zeta } \\\\\\text{(R)} \\:\\: \\chi &= \\tau + \\xi \\\\\\text{(L)} \\:\\:\\overline{\\chi } &= - \\overline{\\tau } - \\overline{\\xi }\\end{split}\\begin{split}\\mu &= \\eta - \\zeta \\\\\\overline{\\mu } &= - \\overline{\\eta } + \\overline{\\zeta } \\\\\\kappa &= \\tau - \\xi \\\\\\overline{\\kappa } &= - \\overline{\\tau } + \\overline{\\xi } .\\end{split}$ In the reference frame of a uniformly accelerated observer, the field operator can be expanded in Rindler modes confined to the (R) and (L) wedges of spacetime $\\hat{\\Phi }(U,V) &= \\sum _\\text{{$\\text{X}=\\text{R,L}$}} \\int ( \\hat{b}_{\\omega l}^\\text{X} g_{\\omega l}^\\text{X} + \\text{h.c} )$ where $\\omega $ is the frequency defined with respect to the proper time co-ordinate in the respective quadrant of Rindler space and $\\text{X}$ denotes the region over which $\\hat{b}_\\omega $ has support.",
"Equivalently, $\\hat{\\Phi }$ can be expanded in the modes restricted to the future (F) and past (P).",
"For example, the left-moving mode functions are $g^\\text{R}_{\\omega l}(\\chi ) &= (4\\pi \\omega )^{-1/2} e^{-i\\omega \\chi } \\\\g_{\\omega l}^\\text{L} (\\bar{\\chi }) &= (4\\pi \\omega )^{-1/2} e^{-i\\omega \\bar{\\chi }}$ with their analogues in the future and past given by, $g_{\\omega l}^\\text{F}(\\nu ) &= (4\\pi \\omega )^{-1/2} e^{-i \\omega \\nu } \\\\g_{\\omega l}^\\text{P}(\\overline{\\nu })&= (4\\pi \\omega )^{-1/2} e^{-i \\omega \\overline{\\nu }}.$ It can be easily shown that $g_{\\omega l}^\\text{R}(V) = g_{\\omega l}^\\text{F}(V) &= (4\\pi \\omega )^{-1/2} (aV)^{-i\\omega /a}\\Theta (V) \\\\g_{\\omega l}^\\text{L}(V) = g_\\omega ^\\text{P,$\\alpha $} (V) &= (4\\pi \\omega )^{-1/2} (-aV)^{i\\omega /a}\\Theta (-V).$ Because $g_{\\omega l}^\\text{R}(\\chi )$ and $g_{\\omega l}^\\text{F}(\\nu )$ are identical as functions of $V$ , as are $g_{\\omega l}^\\text{L}(\\chi )$ and $g_{\\omega l}^\\text{P}(\\nu )$ , the Bogoliubov coefficients relating the Minkowski and Rindler modes in (R) and (L) are duplicated in (F) and (P).",
"Therefore the Rindler mode functions can be extended across the Rindler horizons by a change of coordinates.",
"Using this property, one can define left-moving mode functions which span the entirety of Regions II and IV in Minkowski spacetime [30], [8], with their corresponding bosonic operators given by $ \\hat{b}_\\omega ^\\text{II} &:= \\hat{b}_{\\omega l}^\\text{P} = \\hat{b}_{\\omega l}^\\text{L} ,\\\\ \\hat{b}_\\omega ^\\text{IV} &:= \\hat{b}_{\\omega l}^\\text{R} = \\hat{b}_{\\omega l}^\\text{F} ,$ which have support over the entirety of Regions II and IV (see Fig REF ).",
"For completeness, the following association for the right-moving Rindler operators can also be made, $\\hat{b}_\\omega ^\\text{I} &:= \\hat{b}_{\\omega r}^\\text{L} = \\hat{b}_{\\omega r}^\\text{F} ,\\\\\\hat{b}_\\omega ^\\text{III} &:= \\hat{b}_{\\omega r}^\\text{P} = \\hat{b}_{\\omega r}^\\text{R} .",
"$ Thus, the left-moving sector of the field can be expanded as follows [30] $\\hat{\\Phi } (V) &= \\sum _\\text{$\\chi =$ II,IV} \\int _0^\\infty \\big ( \\hat{b}_{\\omega l}^\\chi g_{\\omega l}^\\chi (V) + \\text{h.c} \\big ) .$ Demonstrating the presence of entanglement between the left-moving modes in II and IV (analogously the right-moving modes in I and III) is identical to the derivation for spacelike entanglement between (R) and (L) with a mere change of labels: that is, (P), (L) $\\Rightarrow $ II and (R), (F) $\\Rightarrow $ IV (for the right-movers, replacing (P), (R) $\\Rightarrow $ I and (L), (F) $\\Rightarrow $ III) [8], [30], [3].",
"This derivation is shown in detail in [8].",
"Eq.",
"(REF )-() imply that the complete sets of right-moving modes in Regions I and III and the left-moving modes in Regions II and IV are entangled, in the same way that spacelike separated Rindler modes in (R) and (L) and timelike separated conformal modes in (F) and (P) are entangled.",
"In the following analysis, we harness this entanglement as the resource for teleportation between the accelerated and stationary observers."
],
[
"Quantum Circuit Model",
"The quantum circuit model is a nonperturbative approach to describing interactions between observers in relativistic, non-inertial reference frames and quantum fields [24], [25].",
"It describes these interactions as the Heisenberg evolution of Unruh operators ($\\hat{c}_{\\omega i},\\hat{d}_{\\omega i}$ ) into Rindler operators $(\\hat{b}_{\\omega i}^\\chi $ ), and back into Unruh operators, which can then be detected as Minkowski modes according to Eq.",
"(REF ).",
"The basis transformation between the Unruh and Rindler operators is defined in [24] and is essentially two-mode squeezing, the `source' of entanglement.",
"The circuit in Fig.",
"REF imposes the all-optical teleportation protocol into the reference frames of accelerated Rob and conformal Charlie.",
"We consider interactions with localised wavepacket modes in the accelerated (and conformal) frames, described by the operator $\\hat{b}_{g}^\\chi &\\equiv \\int _0^\\infty \\: g(\\omega ) \\hat{b}_{\\omega }^\\chi ,$ where $g(\\omega )$ assumes a Gaussian profile.",
"The circuit diagram illustrates transformations of many single-frequency modes and the corresponding interaction in the accelerated frame as acting on a single, localised wavepacket mode, in the continuum limit.",
"Figure: Quantum circuit for teleportation between the accelerated sender and conformal receiver.",
"The time-dependent interaction mixes different single-frequency Rindler modes, and the coloured lines identify Rindler modes confined to different regions of spacetime (these match Fig.",
").",
"The accelerated observer creates the signal using D ^ g (α,β)\\hat{D}_g(\\alpha ,\\beta ), amplifies this through S ^ 2,g (r)\\hat{S}_{2,g}(r), before the conformal receiver attenuates it using a beam splitter, U ^ BS, g\\hat{U}_\\text{BS,$g$}.",
"The classical channel is denoted by the dashed line.For the unitary interaction $\\hat{U}_g$ with an arbitrary Rindler wavepacket $g(\\omega )$ , there exist a complete set of wavepackets $g_{\\perp ,i}(\\omega )$ orthogonal to $g(\\omega )$ which do not interact with $\\hat{U}_g$ .",
"Thus, the single-frequency Rindler operators can be decomposed into an `interacting' and `noninteracting' part [31], $\\hat{b}_\\omega ^\\chi &= g^\\star (\\omega ) \\hat{b}_g^\\chi + \\sum _i g_{\\perp ,i}(\\omega ) \\hat{b}_{g\\perp ,i}^\\chi .$ Acting $\\hat{U}_g$ on a single-frequency Rindler operator yields $\\hat{b}_\\omega ^{\\chi \\prime } = \\hat{U}_g^\\dagger \\hat{b}_\\omega ^\\chi \\hat{U}_g &= \\hat{b}_\\omega ^\\chi + g^\\star (\\omega ) \\big ( \\hat{U}_g^\\dagger \\hat{b}_g^\\chi \\hat{U}_g - \\hat{b}_g^\\chi \\big ) ,$ from Eq.",
"(REF ).",
"Using the relations between the single-frequency Unruh and Rindler operators [24] and Eq.",
"(REF ) and (REF ), we obtain the following expressions for the output left-moving Unruh operators [25] $ \\hat{c}_{\\omega l}^{\\prime }&= \\hat{c}_{\\omega l} + g^\\star (\\omega ) \\cosh r_\\omega \\big ( \\hat{U}_g^\\dagger \\hat{b}_g^\\text{IV} \\hat{U}_g - \\hat{b}_g^\\text{IV} \\big ) \\\\ \\hat{d}_{\\omega l}^{\\prime } &= \\hat{d}_{\\omega l} - g(\\omega ) \\sinh r_\\omega \\big ( \\hat{U}_g^\\dagger \\hat{b}_g^\\text{IV$\\dagger $} \\hat{U}_g - \\hat{b}_g^\\text{IV$\\dagger $} \\big ) .$ where $ r_\\omega = \\text{arctanh}\\: e^{-\\pi \\omega /a}$ is the two-mode squeezing factor.",
"Alice reconstructs the Minkowski modes from $\\hat{c}_{\\omega l}^{\\prime }, \\hat{d}_{\\omega l}^{\\prime }$ .",
"In Fig.",
"REF , $\\hat{D}_g(\\alpha ,\\beta ) = \\hat{D}_g(\\beta ) \\hat{D}_g(\\alpha )$ where $\\hat{D}_g(\\alpha )$ imposes a strong local oscillator to the classical signal $\\beta $ , prepared and teleported by Rob, which is an important feature of the self-homodyne detection model which we employ."
],
[
"Self-Homodyne Detection",
"Since the teleported state is unknown to the inertial observer, we model them as operating an infinite bandwidth detector that integrates over all Minkowski modes, $\\hat{N} &= \\int _{-\\infty }^\\infty \\mathrm {d} k \\:\\hat{a}_k^\\dagger \\hat{a}_k.$ This approximates a physical detector that captures a finite range of frequencies much larger than the spread of frequencies of the signal [32], [25].",
"By displacing the input Rindler mode by a strong local oscillator characterised by the complex amplitude $\\alpha = |\\alpha | e^{i\\phi }$ , the quadrature amplitudes of the input field now exhibit phase-dependent oscillations, $\\hat{X}(\\phi ) &= \\hat{a} e^{-i\\phi } + \\hat{a}^\\dagger e^{i\\phi },$ where $\\phi $ is the phase of the displacement in phase space.",
"In the limit where $|\\alpha |\\gg 1$ , $\\hat{N}(\\phi )$ can be approximated as $\\hat{N}(\\phi ) &\\simeq |\\alpha |^2+ |\\alpha |(\\beta + \\beta ^\\star ).$ The quadrature operator can be written as $\\langle \\hat{X}(\\phi ) \\rangle &\\simeq \\frac{\\langle \\hat{N} ( \\phi ) \\rangle - \\langle \\hat{N}_0\\rangle }{\\sqrt{\\langle \\hat{N}_0 \\rangle }},$ where $\\langle \\hat{N}_0\\rangle $ is the average number of photons when no signal $\\beta $ , is imposed.",
"The variance of the quadrature amplitude is [32], [25] $\\left( \\Delta X(\\phi ) \\right)^2 &= \\frac{\\langle \\hat{N}^2(\\phi ) \\rangle - \\langle \\hat{N} (\\phi ) \\rangle ^2}{\\langle \\hat{N}_0 \\rangle } = \\frac{\\left( \\Delta N(\\phi ) \\right)^2}{\\langle \\hat{N}_0 \\rangle },$ where $\\Delta N ( \\phi )$ is the variance in the photon number according to the inertial detector.",
"Conveniently, the purity of Gaussian states is quantified by Eq.",
"(REF ), the criterion for which is that the product of the $\\phi = 0$ and $\\phi = \\pi /2$ quadrature variances is equal to unity [33]."
],
[
"Analytic Results",
"In the following analysis, we consider the teleportation of the following states from the accelerated frame: a displaced Minkowski vacuum state (a displaced thermal state in the Rindler observer's frame, acting as a classical signal) created by displacing $\\smash{\\hat{b}_{g}^\\text{IV}}$ with $\\smash{\\hat{D}_g(\\beta )}$ (Fig.",
"REF ), and a squeezed thermal state (a quantum signal) generated by single-mode squeezing the input Rindler mode $\\hat{b}_g^\\text{IV}$ , and determine their purity according to Alice.",
"For (a), an ideal teleportation protocol should result in Alice detecting a pure coherent state.",
"We expect this from the result in [25], which found that direct transmission of the Rindler-displaced Minkowski vacuum was detected as a pure coherent state according to an inertial Minkowski observer."
],
[
"Displaced Thermal State",
"From Fig.",
"REF , $\\hat{b}_g^\\text{IV}$ evolves under the action of the unitary $\\smash{\\hat{U}_g = \\hat{U}_\\text{BS,$g$} \\hat{S}_{2,g} \\hat{D}_g(\\alpha ,\\beta )}$ as $\\hat{b}_g^\\text{IV} {}^{\\prime }&= \\hat{b}_g^\\text{IV} + \\hat{b}_g^\\text{III$\\dagger $} \\tanh r - \\hat{b}_g^\\text{I} \\sqrt{1 - (\\cosh r )^{-2}} + \\alpha + \\beta ,$ where $\\alpha $ characterises the strong local oscillator, and $\\beta $ (with $|\\beta |\\ll |\\alpha |$ ) creates the displaced state which we analyse.",
"Using Eq.",
"(REF ) and assuming the limit of strong amplification of the field through the classical channel $(r\\gg 1$ ) the output single-frequency Rindler operator takes the form $ \\hat{b}_\\omega ^\\text{IV} {}^{\\prime } &\\simeq \\hat{b}_\\omega ^\\text{IV} + g^\\star (\\omega ) \\big ( \\hat{b}_g^\\text{III$\\dagger $} - \\hat{b}_g^\\text{I} + \\alpha + \\beta \\big ) .$ Since the left-moving modes originating in the past light cone do not interact with $\\hat{U}_g$ , then $\\hat{b}_g^\\text{II} {}^{\\prime } = \\hat{b}_g^\\text{II}$ .",
"From Eq.",
"(REF ) we find that Charlie, in the conformal reference frame, sees an approximation of the original state, which becomes better as the acceleration increases.",
"This becomes evident after decomposing the Rindler operators into their constituent Unruh operators and assuming $g^\\star (\\omega ) = g(\\omega )$ , such that, $\\hat{b}_\\omega ^\\text{IV} {}^{\\prime } &\\simeq \\hat{b}_{\\omega }^\\text{IV} + g(\\omega ) \\Big [ \\varphi _{cs} (\\hat{d}_{\\omega ^{\\prime } r}^{\\dagger } - \\hat{c}_{\\omega ^{\\prime } r} ) + \\alpha + \\beta \\Big ] ,$ where $\\varphi _{cs} &= \\int ^{\\prime } \\:g(\\omega ^{\\prime }) \\big ( \\cosh r_{\\omega ^{\\prime }} - \\sinh r_{\\omega ^{\\prime }} \\big ) ,$ which vanishes in the limit of infinite acceleration.",
"Eq.",
"(REF ) is formally equivalent to Eq.",
"(REF ) for the static case.",
"In the limit of infinite acceleration, the right-moving Rindler modes become perfectly entangled, reducing the output Rindler operator to an identical reconstruction of the input displaced by the amplitude $g(\\omega )(\\alpha + \\beta )$ .",
"Therefore in the limit of infinite acceleration, our model predicts the ideal teleportation of states sent between the uniformly accelerated sender in the right Rindler wedge and the conformal receiver restricted to the future.",
"We note however, that for large accelerations, the state becomes increasingly thermalised.",
"We now consider the purity of the output state according to an inertial observer with access to the entire Minkowski spacetime, which requires a transformation of the Rindler operators into Minkowski operators.",
"Applying Eq.",
"(REF ) and () to the output Rindler operators yields $ \\hat{c}_{\\omega l}^{\\prime } &= \\hat{c}_{\\omega l} + g(\\omega ) \\cosh r_\\omega \\big ( \\hat{b}_g^\\text{III$\\dagger $} - \\hat{b}_g^\\text{I} + \\alpha + \\beta \\big )\\\\\\hat{d}_{\\omega l}^{\\prime }&= \\hat{d}_{\\omega l} - g(\\omega ) \\sinh r_\\omega \\big ( \\hat{b}_g^\\text{III} - \\hat{b}_g^\\text{I$\\dagger $} + \\alpha ^\\star + \\beta ^\\star \\big ) .$ Eq.",
"(REF ) for the Minkowski photon number operator simplifies to $\\hat{N} &= \\int \\big ( \\hat{c}_{\\omega l}^{\\dagger \\prime } \\hat{c}_{\\omega l}^{\\prime } + \\hat{d}_{\\omega l}^{\\dagger \\prime } \\hat{d}_{\\omega l}^{\\prime } \\big ) ,$ where we have used the identities $\\int k̥ A_{k\\omega }A_{k\\omega ^{\\prime }}^\\star = \\delta ( \\omega - \\omega ^{\\prime })$ and $\\int k̥ \\: A_{k\\omega }A_{k\\omega ^{\\prime }}= 0$ .",
"Similarly, the square of the number operator is $\\hat{N}^2 &= \\int \\int \\big (c_{\\omega l}^{\\dagger \\prime } c_{\\omega l}^{\\prime } + d_{\\omega l}^{\\dagger \\prime } d_{\\omega l}^{\\prime } \\big ) \\big ( c_{\\gamma l}^{\\dagger \\prime } c_{\\gamma l}^{\\prime } + d_{\\gamma l}^{\\dagger \\prime } d_{\\gamma l}^{\\prime }\\big ) .$ By computing the relevant vacuum expectation values of the products of output Unruh operators, the quadrature variance of the output state detected by Alice is found to be (details in Appendix A) $\\left( \\Delta X \\right)^2 &= \\underbrace{ 2 \\mathcal {I}_{cs} \\big ( \\mathcal {I}_c + \\mathcal {I}_s \\big )}_\\text{Thermal noise}\\: + \\underbrace{1\\vphantom{\\big |}}_\\text{QNL}$ where $\\mathcal {I}_c &= \\int \\:g(\\omega )^2 \\cosh ^2r_\\omega \\\\\\mathcal {I}_s &= \\int \\:g(\\omega )^2 \\sinh ^2r_\\omega \\\\\\mathcal {I}_{cs} &= \\int \\: g(\\omega )^2 \\big ( \\cosh r_\\omega - \\sinh r_\\omega \\big )^2 .$ In the limiting case where $g(\\omega )$ is a narrow bandwidth Rindler mode $(\\sigma \\ll \\omega _0)$ , the quadrature variance reduces to $\\left(\\Delta X \\right)^2 &\\simeq \\big ( 1 + \\exp ( - 4r_0 ) \\big ) + 1 .$ For $\\omega _0/a\\gg 1$ , the non-inertial (Charlie and Rob) observers are effectively reduced to a static, inertial frame.",
"As shown in Fig.",
"REF , this occurs in the limits $a\\rightarrow 0$ with fixed $\\omega _0$ (smaller accelerations) or $\\omega _0\\gg 1$ with fixed $a$ .",
"In the latter case, the non-inertial observers interact with sufficiently high-frequency modes such that they are surrounded by vacuum (due to the exponentially decaying tail of the Planck spectrum).",
"The input Rindler modes remain independent, so that two additional QNL are added to the output state, analogous to Eq.",
"(REF ).",
"Figure: (ΔX) 2 (\\Delta X)^2 for the teleported state as detected by Alice in the Minkowski vacuum, as a function of the acceleration.",
"The individual lines represent ω 0 =0.5,1.0,⋯,3.5\\omega _0 = 0.5, 1.0, \\hdots , 3.5 (increasing from blue to yellow).As $\\omega _0/a\\rightarrow 0$ , the right-moving resource modes become perfectly entangled.",
"Curiously, the output state detected by the inertial Minkowski observer is not pure, but carries at best, an extra unit of QNL above the Heisenberg limit of unity.",
"Even for a perfect entanglement resource, the state appears mixed according to the inertial observer, despite a reduction below the classical limit.",
"To understand this behaviour, we notice that Eq.",
"(REF ) contains a thermal noise term in addition to the single QNL that originates from the Rindler mode displaced by Rob and teleported to Charlie.",
"It was found in [25] that the direct transmission of a displaced Rindler mode from the accelerated reference frame was detected as a pure coherent state according to an inertial Minkowski observer.",
"Comparatively, the thermal noise term in Eq.",
"(REF ) approaches one (an additional QNL) rather than vanishing in the limit of infinite acceleration, since the output Rindler mode only reconstructs the input identically at infinite acceleration.",
"Hence, whilst the noise terms from the right-moving vacuum modes become increasingly suppressed with increasing acceleration (due to stronger entanglement between the resource modes $\\hat{b}_g^\\text{I}$ , $\\hat{b}_g^\\text{III}$ ), they are simultaneously amplified due to the increasing thermalisation of the Unruh effect.",
"These competing effects fortuitously balance one another, resulting in (for the ideal case of infinite acceleration) an additional QNL to the output state."
],
[
"Squeezed Thermal State",
"To examine the evolution of nonclassical states through the teleportation protocol, we apply single-mode squeezing to the initial state prepared in the accelerated reference frame.",
"We are also interested in how the teleported state is affected by the so-called decoherence effects found previously in [25], [32].",
"As before, we derive the output single-frequency Rindler operators, given by $\\hat{b}_\\omega ^\\text{IV} {}^{\\prime } &= b_\\omega ^\\text{IV} + g(\\omega ) \\left( b_g^\\text{IV} (\\cosh r_s - 1) \\right.",
"\\nonumber \\\\& \\left.",
"+ b_g^\\text{IV$\\dagger $} \\sinh r_s + b_g^\\text{I$\\dagger $} - b_g^\\text{III} + \\alpha \\right) ,$ where now $\\hat{U}_g = \\hat{U}_\\text{BS} \\hat{S}_{2,g}(r) \\hat{S}_{1,g}(r_s) \\hat{D}_g(\\alpha )$ and $\\hat{S}_{1,g}(r_s)$ squeezes the Rindler vacuum.",
"It is straightforward to show that the output Unruh operators are $\\hat{c}_{\\omega l}^{\\prime } &= \\hat{c}_{\\omega l} + g(\\omega ) \\cosh r_\\omega \\big ( \\hat{b}_g^\\text{IV} (\\cosh r_s - 1) \\nonumber \\\\& + \\hat{b}_g^\\text{IV$\\dagger $} \\sinh r_s + \\hat{b}_g^\\text{III$\\dagger $} - \\hat{b}_g^\\text{I} + \\alpha \\big ) \\\\\\hat{d}_{\\omega l}^{\\prime } &= \\hat{d}_{\\omega l} - g(\\omega ) \\sinh r_\\omega \\big ( \\hat{b}_g^\\text{IV$\\dagger $} (\\cosh r_s - 1) \\nonumber \\\\& + \\hat{b}_g^\\text{IV} \\sinh r_s + b_g^\\text{III} - \\hat{b}_g^\\text{I$\\dagger $} + \\alpha ^\\star \\big ).$ Performing a similar calculation to the previous case, we find that the quadrature variance of the state detected by the inertial Minkowski observer is given by, $\\left( \\Delta X (\\phi ) \\right)^2 &= \\underbrace{2\\mathcal {I}_{cs} \\big ( \\mathcal {I}_c + \\mathcal {I}_s \\big )}_\\text{Thermal noise} + \\underbrace{\\Delta (\\phi ) \\vphantom{\\big |}}_\\text{Decoherence} ,$ where $\\Delta (\\phi ) &= \\cosh 2r_s + 4\\mathcal {I}_c ( \\mathcal {I}_c - 1) ( \\cosh 2r_s - 2 \\cosh r_s + 1 ) \\nonumber \\\\& + 2 \\sinh r_s \\big [( 2 \\mathcal {I}_c - 1)^2 \\cosh r_s - 4\\mathcal {I}_c ( \\mathcal {I}_c - 1) \\big ] \\cos (2\\phi ) .$ The maximum and minimum quadrature variances correspond to the phases $\\phi = 0, \\pi /2$ , $ \\left( \\Delta X (0) \\right)^2 &= 2\\mathcal {I}_{cs} (\\mathcal {I}_c + \\mathcal {I}_s) + \\Delta (0) \\\\ \\left( \\Delta X (\\pi /2) \\right)^2 &= 2\\mathcal {I}_{cs} (\\mathcal {I}_c + \\mathcal {I}_s ) + \\Delta (\\pi /2)$ where $\\Delta (0) &= e^{2r_s} + 4 \\mathcal {I}_c( \\mathcal {I}_c -1) (e^{r_s} - 1)^2 \\\\ \\Delta (\\pi /2) &= e^{-2r_s} + 4 \\mathcal {I}_c (\\mathcal {I}_c -1 ) (e^{-r_s}-1)^2.$ Eq.",
"(REF ) reveals two competing factors affecting the purity of the output state.",
"The first is the thermal noise term found previously for the displaced Rindler vacuum state prepared and teleported by Rob, which is uncorrelated from the noise of the signal itself.",
"The second is the decoherence term $\\Delta (\\phi )$ which also appeared in [25] for a squeezed state sent from the accelerated frame to an inertial observer.",
"There, it was concluded that such terms arise from the transformation of the bipartite Hilbert space of the Rindler and Unruh modes to the single Hilbert space of the Minkowski modes.",
"This leads to a loss of phase information in the Unruh modes when computing the Minkowski particle number and the observed decoherence according to inertial detectors.",
"Figure: Contributions to (ΔX) 2 (\\Delta X)^2 for the squeezed thermal state as detected by the inertial observer in the Minkowski vacuum, with r s =0.5,ω 0 =1r_s= 0.5, \\omega _0 = 1.",
"The decoherence terms dominate for large aa.Fig.",
"REF shows that for small $a$ , the strengthening of entanglement between the Rindler modes reduces the noise on the output state, but as $a$ increases, the Unruh effect becomes significant and the variances grow unbounded.",
"We conclude that the purity of the output state according to Alice, in the Minkowski vacuum, is affected by the thermal noise term, which encodes within it the competing suppression and amplification effects discussed previously, and the squeezing of the input Rindler mode, the source of the decoherence terms first found in [25]."
],
[
"Conclusion",
"The decoherence of quantum states prepared in an accelerated reference frame and detected by inertial observers has been studied previously [25], [32].",
"In [25], the authors argue that any interaction that leads to entanglement between the Unruh modes, as occurs for the squeezed Rindler vacuum state, appears as decoherence according to the measurements of inertial observers, whilst classical signals do not produce such entanglement.",
"Nevertheless, we found that the displaced Rindler vacuum state prepared and teleported by Rob was still polluted by additional thermal noise terms according to the inertial Minkowski observer.",
"We attributed this additional decoherence to the fact that the output state teleported to Charlie only becomes an identical reconstruction of the input at infinite acceleration, which is nullified by the opposing thermalisation effects of Unruh radiation in this limit.",
"As is evident from the simplicity of our calculations, the quantum circuit model presents a powerful tool for analysing quantum information protocols in relativistic, non-inertial settings, possessing some advantages over previous models.",
"A prominent one is that the model is nonperturbative, which allows the evolution of quantum fields to be described by unitary operations.",
"It is also a natural setting to analyse continuous-variable protocols where only discrete-variable versions have been studied, and these can be straightforwardly mapped to quantum optical contexts.",
"An immediate extension to this work would be to implement other entanglement-based protocols using the quantum circuit model, such as continuous-variable dense coding [34], [35], quantum energy teleportation [36], [37], [38], [39], [40] and quantum key distribution [17]."
],
[
"Acknowledgements",
"This research is supported by the Australian Research Council (ARC) under the Centre of Excellence for Quantum Computation and Communication Technology (Grant No.",
"CE170100012)."
],
[
"Rindler-Displaced Minkowski Vacuum State",
"In the main text, we derived expressions for the output Unruh operators, given by $\\hat{c}_{\\omega l}^{\\prime } &= \\hat{c}_{\\omega l} + g (\\omega ) \\cosh r_\\omega \\big [ \\varphi _{cs} \\big ( \\hat{c}_{\\omega ^{\\prime }r}^{\\dagger } - \\hat{d}_{\\omega ^{\\prime }r} \\big ) + \\alpha + \\beta \\big ] \\\\\\hat{d}_{\\omega l}^{\\prime } &= \\hat{d}_{\\omega l}- g(\\omega ) \\sinh r_\\omega \\big [ \\varphi _{cs} \\big ( \\hat{c}_{\\omega ^{\\prime }r} - \\hat{d}_{\\omega ^{\\prime }r}^{\\dagger } \\big ) + \\alpha ^\\star + \\beta ^\\star \\big ].$ We can conveniently express these as $\\hat{c}_{\\omega l}^{\\prime } &= \\alpha g(\\omega ) \\cosh r_\\omega + \\hat{c}_{\\omega l}^{\\prime \\prime } \\\\\\hat{d}_{\\omega l}^{\\prime } &= - \\alpha ^\\star g(\\omega ) \\sinh r_\\omega + \\hat{d}_{\\omega l}^{\\prime \\prime }$ where $c_\\omega ^{\\prime \\prime }$ and $d_\\omega ^{\\prime \\prime }$ are terms not multiplied by $|\\alpha |$ .",
"The Unruh particle numbers can thus be calculated using $c_{\\omega l}^{\\dagger \\prime } c_{\\omega l}^{\\prime } &= |\\alpha |^2 g^2(\\omega ) \\cosh ^2 r_\\omega + |\\alpha |g(\\omega ) \\cosh r_\\omega \\left[ e^{i\\phi } c_{\\omega l}^{\\dagger \\prime \\prime } + e^{-i\\phi } c_{\\omega l}^{\\prime \\prime }\\right] \\\\d_{\\omega l}^{\\dagger \\prime } d_{\\omega l}^{\\prime } &= |\\alpha |^2 g^2(\\omega ) \\sinh ^2 r_\\omega - |\\alpha | g(\\omega ) \\sinh r_\\omega \\left[ e^{-i\\phi }d_{\\omega l}^{\\dagger \\prime \\prime } + e^{i\\phi } d_{\\omega l}^{\\prime \\prime } \\right]$ where we have neglected terms not multiplied by $|\\alpha | \\gg 1$ .",
"The vacuum expectation value of the particle number is thus $\\langle 0 | \\hat{N} | 0 \\rangle &= |\\alpha |^2 \\big ( \\mathcal {I}_c + \\mathcal {I}_s \\big ) .$ The relevant expectation values in $\\langle 0 |\\hat{N}^2|0 \\rangle $ are $\\langle 0| c_{\\omega l}^{\\prime \\prime } c_{\\gamma l}^{\\dagger \\prime \\prime }|0 \\rangle &= \\delta (\\omega - \\gamma ) + g(\\omega ) g(\\gamma ) \\cosh r_{\\omega } \\cosh r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0| c_{\\omega l}^{\\dagger \\prime \\prime } c_{\\gamma l}^{\\prime \\prime } |0 \\rangle &= g(\\omega )g(\\gamma ) \\cosh r_{\\omega } \\cosh r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0|d_{\\omega l}^{\\prime \\prime } d_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= \\delta (\\omega - \\gamma ) + g(\\omega ) g(\\gamma ) \\sinh r_{\\omega }\\sinh r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0| d_{\\omega l}^{\\dagger \\prime \\prime }d_{\\gamma l}^{\\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\sinh r_{\\omega } \\sinh r_{\\gamma }\\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0|c_{\\omega l}^{\\prime \\prime } d_{\\gamma l}^{\\prime \\prime } |0\\rangle &= -g(\\omega ) g(\\gamma ) \\cosh r_{\\omega } \\sinh r_{\\gamma }\\mathcal {I}_{cs}^{\\prime }\\\\\\langle 0|c_{\\omega l}^{\\dagger \\prime \\prime }d_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= -g(\\omega ) g(\\gamma ) \\cosh r_{\\omega }\\sinh r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0| d_{\\omega l}^{\\prime \\prime } c_{\\gamma l}^{\\prime \\prime }|0 \\rangle &= - g(\\omega ) g(\\gamma ) \\sinh r_{\\omega } \\cosh r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0| d_{\\omega l}^{\\dagger \\prime \\prime } c_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= - g(\\omega ) g(\\gamma ) \\sinh r_{\\omega }\\cosh r_{\\gamma } \\mathcal {I}_{cs}^{\\prime }$ where $\\mathcal {I}_{cs}^{\\prime } = \\int ^{\\prime }g^2(\\omega ^{\\prime }) ( \\cosh r_{\\omega ^{\\prime }} - \\sinh r_{\\omega ^{\\prime }})^2$ .",
"The terms in $\\langle 0|\\hat{N}^2|0\\rangle $ are $\\langle 0 | \\hat{c}_{\\omega l}^{\\dagger \\prime } \\hat{c}_{\\omega l}^{\\prime }\\hat{c}_{\\gamma l}^{\\dagger \\prime } \\hat{c}_{\\gamma l}^{\\prime }|0\\rangle &= |\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\cosh r_{\\omega } \\cosh r_{\\gamma } \\left[ \\delta (\\omega - \\gamma ) + 2\\mathcal {I}_{cs}^{\\prime } \\cosh r_{\\omega } \\cosh r_{\\gamma } \\right] \\\\\\langle 0 | \\hat{d}_{\\omega l}^{\\dagger \\prime }\\hat{d}_{\\omega l}^{\\prime }\\hat{d}_{\\gamma l}^{\\dagger \\prime } \\hat{d}_{\\gamma l}^{\\prime }|0\\rangle &= |\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\sinh r_{\\omega } \\sinh r_{\\gamma } \\left[ \\delta (\\omega - \\gamma ) + 2\\mathcal {I}_{cs}^{\\prime } \\sinh r_{\\omega } \\sinh r_{\\gamma } \\right] \\nonumber \\\\\\langle 0 | \\hat{c}_{\\omega l}^{\\dagger \\prime }\\hat{c}_{\\omega l}^{\\prime }\\hat{d}_{\\gamma l}^{\\dagger \\prime } \\hat{d}_{\\gamma l}^{\\prime }|0\\rangle &= 2|\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\cosh ^2 r_{\\omega } \\sinh ^2 r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } \\\\\\langle 0 | \\hat{d}_{\\omega l}^{\\dagger \\prime }\\hat{d}_{\\omega l}^{\\prime }\\hat{c}_{\\gamma l}^{\\dagger \\prime } \\hat{c}_{\\gamma l}^{\\prime }|0\\rangle &= 2|\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\sinh ^2 r_{\\omega } \\cosh ^2 r_{\\gamma } \\mathcal {I}_{cs}^{\\prime } ,$ where we have left out the terms fourth order in $|\\alpha |$ , since they are subtracted away by the $\\langle 0 | \\hat{N} | 0 \\rangle ^2$ term in the variance.",
"Adding these terms together, integrating with respect to $\\omega , \\gamma $ and normalising by the strength of the local oscillator $|\\alpha |^2 ( \\mathcal {I}_c + \\mathcal {I}_s)$ yields Eq.",
"(REF )."
],
[
"Rindler-Squeezed Minkowski Vacuum State",
"Like the displaced thermal state, we derived expressions for the output Unruh operators, given by $\\hat{c}_{\\omega l}^{\\prime } &= \\hat{c}_{\\omega l} + g(\\omega ) \\cosh r_{\\omega } \\Big [ (\\cosh r_s - 1) \\int ^{\\prime }\\:g(\\omega ^{\\prime }) \\big ( \\cosh r_{\\omega ^{\\prime }} \\hat{c}_{\\omega ^{\\prime }l} + \\sinh r_{\\omega ^{\\prime }} \\hat{d}_{\\omega ^{\\prime }l}^{\\dagger } \\big )\\nonumber \\\\& + \\sinh r_s \\int ^{\\prime }g(\\omega ^{\\prime }) \\big ( \\cosh r_{\\omega ^{\\prime }} \\hat{c}_{\\omega ^{\\prime }l}^{\\dagger } + \\sinh r_{\\omega ^{\\prime }} \\hat{d}_{\\omega ^{\\prime }l} \\big ) + \\varphi _{cs}( \\hat{c}_{\\omega ^{\\prime }r}^{\\dagger } - \\hat{d}_{\\omega ^{\\prime }r}\\big ) + \\alpha \\Big ] \\\\\\hat{d}_{\\omega l}^{\\prime }&= \\hat{d}_{\\omega l} - g(\\omega ) \\sinh r_{\\omega } \\Big [ (\\cosh r_s - 1) \\int ^{\\prime }g(\\omega ^{\\prime }) \\big ( \\cosh r_{\\omega ^{\\prime }} \\hat{c}_{\\omega ^{\\prime }l}^{\\dagger } + \\sinh r_{\\omega ^{\\prime }} \\hat{d}_{\\omega ^{\\prime }l} \\big ) \\nonumber \\\\& + \\sinh r_s \\int ^{\\prime }g(\\omega ^{\\prime }) \\big ( \\cosh r_{\\omega ^{\\prime }} \\hat{c}_{\\omega ^{\\prime }l} + \\sinh r_{\\omega ^{\\prime }} \\hat{d}_{\\omega ^{\\prime }l}^{\\dagger } \\big ) + \\varphi _{cs} \\big ( \\hat{c}_{\\omega ^{\\prime }r}- \\hat{d}_{\\omega ^{\\prime }r}^{\\dagger } \\big ) + \\alpha ^\\star \\Big ] .$ $\\langle 0|\\hat{N}|0\\rangle $ is the same as previously derived.",
"As before, the relevant expectation value for the products of Unruh operators are $\\langle 0| c_{\\omega l}^{\\prime \\prime } c_{\\gamma l}^{\\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\cosh r_{\\omega }\\cosh r_{\\gamma } \\psi _{cc} \\\\\\langle 0| c_{\\omega l}^{\\dagger \\prime \\prime } c_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\cosh r_{\\omega }\\cosh r_{\\gamma }\\psi _{cc} \\\\\\langle 0| c_{\\omega l}^{\\prime \\prime } c_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= \\delta (\\omega - \\gamma ) + g(\\omega ) g(\\gamma ) \\cosh r_{\\omega }\\cosh r_{\\gamma } \\bar{\\phi }_{cc} \\\\\\langle 0| c_{\\omega l}^{\\dagger \\prime \\prime } c_{\\gamma l}^{\\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\cosh r_{\\omega }\\cosh r_{\\gamma } \\phi _{cc} \\\\\\langle 0| d_{\\omega l}^{\\prime \\prime } d_{\\gamma l}^{\\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\sinh r_{\\omega } \\sinh r_{\\gamma } \\psi _{dd} \\\\\\langle 0| d_{\\omega l}^{\\dagger \\prime \\prime } d_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\sinh r_{\\omega }\\sinh r_{\\gamma }\\psi _{dd} \\\\\\langle 0| d_{\\omega l}^{\\prime \\prime } d_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= \\delta (\\omega - \\gamma ) + g(\\omega ) g(\\gamma ) \\sinh r_{\\omega }\\sinh r_{\\gamma } \\bar{\\phi }_{dd} \\\\\\langle 0| d_{\\omega l}^{\\dagger \\prime \\prime } d_{\\gamma l}^{\\prime \\prime } |0\\rangle &= g(\\omega ) g(\\gamma ) \\sinh r_{\\omega }\\sinh r_{\\gamma }\\phi _{dd} ,$ where $\\psi _{cc} &= \\sinh r_s \\big [ (\\cosh r_s - 1) ( \\mathcal {I}_c^{\\prime } + \\mathcal {I}_s^{\\prime }) + 1\\big ] \\\\\\phi _{cc} &= (\\cosh r_s - 1)^2 \\mathcal {I}_s^{\\prime } + \\sinh ^2 r_s \\mathcal {I}_c^{\\prime } + \\mathcal {I}_{cs}^{\\prime } \\\\\\bar{\\phi }_{cc} &= 2(\\cosh r_s -1 ) + ( \\cosh r_s - 1)^2 \\mathcal {I}_c^{\\prime } + \\sinh ^2 r_s \\mathcal {I}_s^{\\prime } + \\mathcal {I}_{cs}^{\\prime } \\\\\\psi _{dd} &= \\sinh r_s \\big [ (\\cosh r_s - 1) ( \\mathcal {I}_c^{\\prime } + \\mathcal {I}_s^{\\prime } ) - 1\\big ] \\\\\\phi _{dd} &= (\\cosh r_s -1 )^2 \\mathcal {I}_c^{\\prime } + \\sinh ^2 r_s \\mathcal {I}_s^{\\prime } + \\mathcal {I}_{cs}^{\\prime } \\\\\\bar{\\phi }_{dd} &= -2 (\\cosh r_s - 1) + ( \\cosh r_s - 1)^2 \\mathcal {I}_s^{\\prime } + \\sinh ^2r_s \\mathcal {I}_c^{\\prime } + \\mathcal {I}_{cs}^{\\prime }$ with $\\mathcal {I}_c^{\\prime } = \\int ^{\\prime }g^2(\\omega ^{\\prime })\\cosh ^2r_{\\omega ^{\\prime }}$ and $\\mathcal {I}_s^{\\prime } = \\int ^{\\prime }g^2(\\omega ^{\\prime })\\sinh ^2r_{\\omega ^{\\prime }}$ .",
"For the cross-terms, we have $\\begin{split}\\langle 0| c_{\\omega l}^{\\prime \\prime } d_{\\gamma l}^{\\prime \\prime } |0\\rangle &= - g(\\omega ) g(\\gamma ) \\cosh r_{\\omega } \\sinh r_{\\gamma } \\big [ \\bar{\\phi }_{cc} - (\\cosh r_s - 1) \\big ] \\\\\\langle 0| c_{\\omega l}^{\\dagger \\prime \\prime } d_{\\gamma l}^{\\dagger \\prime \\prime }|0\\rangle &= -g(\\omega ) g (\\gamma ) \\cosh r_{\\omega } \\sinh r_{\\gamma }\\big [ \\bar{\\phi }_{dd} + (\\cosh r_s - 1) \\big ] \\\\\\langle 0| d_{\\omega l}^{\\prime \\prime } c_{\\gamma l}^{\\prime \\prime } |0\\rangle &= -g(\\omega ) g (\\gamma ) \\sinh r_{\\omega } \\cosh r_{\\gamma } \\big [ \\bar{\\phi }_{dd} + (\\cosh r_s - 1) \\big ] \\\\\\langle 0| d_{\\omega l}^{\\dagger \\prime \\prime } c_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= -g (\\omega ) g(\\gamma ) \\sinh r_{\\omega } \\cosh r_{\\gamma }\\big [ \\bar{\\phi }_{cc} - ( \\cosh r_s - 1) \\big ] \\\\\\end{split}\\begin{split}\\langle 0| c_{\\omega l}^{\\prime \\prime } d_{\\gamma l}^{\\dagger \\prime \\prime } |0\\rangle &= -g (\\omega ) g (\\gamma ) \\cosh r_{\\omega } \\sinh r_{\\gamma } \\gamma _{cd} \\\\\\langle 0| c_{\\omega l}^{\\dagger \\prime \\prime } d_{\\gamma l}^{\\prime \\prime } |0\\rangle &= -g(\\omega ) g(\\gamma ) \\cosh r_{\\omega } \\sinh r_{\\gamma }\\gamma _{cd} \\\\\\langle 0| d_{\\omega l}^{\\prime \\prime } c_{\\gamma l}^{\\dagger \\prime \\prime }|0\\rangle &= - g(\\omega ) g(\\gamma ) \\sinh r_{\\omega } \\cosh r_{\\gamma }\\gamma _{cd} \\\\\\langle 0| d_{\\omega l}^{\\dagger \\prime \\prime } c_{\\gamma l}^{\\prime \\prime } |0\\rangle &= - g (\\omega ) g(\\gamma ) \\sinh r_{\\omega } \\cosh r_{\\gamma }\\gamma _{cd}\\end{split}$ where $\\gamma _{cd} = (\\cosh r - 1) \\sinh r ( \\mathcal {I}_c + \\mathcal {I}_s)$ .",
"Thus, the relevant terms in $\\langle 0|\\hat{N}^2|0\\rangle $ are $\\langle 0 | \\hat{c}_{\\omega l}^{\\dagger \\prime }\\hat{c}_{\\omega l}^{\\prime }\\hat{c}_{\\gamma l}^{\\dagger \\prime } \\hat{c}_{\\gamma l}^{\\prime }|0\\rangle &= |\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\cosh r_{\\omega } \\cosh r_{\\gamma } \\big [ \\delta (\\omega - \\gamma ) + 2 \\bar{\\phi }_{cc} \\cosh r_{\\omega } \\cosh r_{\\gamma } + 2 \\cos (2\\phi ) \\cosh r_{\\omega } \\cosh r_{\\gamma } \\psi _{cc} \\big ] \\nonumber \\\\\\langle 0 | \\hat{d}_{\\omega l}^{\\dagger \\prime }\\hat{d}_{\\omega l}^{\\prime }\\hat{d}_{\\gamma l}^{\\dagger \\prime } \\hat{d}_{\\gamma l}^{\\prime }|0\\rangle &= |\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\sinh r_{\\omega } \\sinh r_{\\gamma } \\big [ \\delta (\\omega - \\gamma ) + 2 \\bar{\\phi }_{dd} \\sinh r_{\\omega l} \\sinh r_{\\gamma } + 2 \\cos (2\\phi ) \\sinh r_{\\omega } \\sinh r_{\\gamma } \\psi _{dd} \\big ] \\nonumber \\\\\\langle 0 | \\hat{c}_{\\omega l}^{\\dagger \\prime }\\hat{c}_{\\omega l}^{\\prime }\\hat{d}_{\\gamma l}^{\\dagger \\prime } \\hat{d}_{\\gamma l}^{\\prime }|0\\rangle &= |\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\cosh ^2 r_{\\omega } \\sinh ^2 r_{\\gamma } \\big [ \\bar{\\phi }_{cc} + \\bar{\\phi }_{dd} + 2 \\cos (2\\phi ) \\gamma _{cd} \\big ] \\nonumber \\\\\\langle 0 | \\hat{d}_{\\omega l}^{\\dagger \\prime }\\hat{d}_{\\omega l}^{\\prime }\\hat{c}_{\\gamma l}^{\\dagger \\prime } \\hat{c}_{\\gamma l}^{\\prime }|0\\rangle &= |\\alpha |^2 g^2(\\omega ) g^2(\\gamma ) \\sinh ^2 r_{\\omega } \\cosh ^2 r_{\\gamma } \\big [ \\bar{\\phi }_{cc} + \\bar{\\phi }_{dd} + 2 \\cos (2\\phi ) \\gamma _{cd}\\big ] .$ As before, we add these terms together and integrate over $\\omega , \\gamma $ to obtain the photon number variance.",
"After normalising by the strength of the local oscillator, we obtain Eq.",
"(REF )."
]
] | 2001.03387 |
[
[
"Video Coding for Machines: A Paradigm of Collaborative Compression and\n Intelligent Analytics"
],
[
"Abstract Video coding, which targets to compress and reconstruct the whole frame, and feature compression, which only preserves and transmits the most critical information, stand at two ends of the scale.",
"That is, one is with compactness and efficiency to serve for machine vision, and the other is with full fidelity, bowing to human perception.",
"The recent endeavors in imminent trends of video compression, e.g.",
"deep learning based coding tools and end-to-end image/video coding, and MPEG-7 compact feature descriptor standards, i.e.",
"Compact Descriptors for Visual Search and Compact Descriptors for Video Analysis, promote the sustainable and fast development in their own directions, respectively.",
"In this paper, thanks to booming AI technology, e.g.",
"prediction and generation models, we carry out exploration in the new area, Video Coding for Machines (VCM), arising from the emerging MPEG standardization efforts1.",
"Towards collaborative compression and intelligent analytics, VCM attempts to bridge the gap between feature coding for machine vision and video coding for human vision.",
"Aligning with the rising Analyze then Compress instance Digital Retina, the definition, formulation, and paradigm of VCM are given first.",
"Meanwhile, we systematically review state-of-the-art techniques in video compression and feature compression from the unique perspective of MPEG standardization, which provides the academic and industrial evidence to realize the collaborative compression of video and feature streams in a broad range of AI applications.",
"Finally, we come up with potential VCM solutions, and the preliminary results have demonstrated the performance and efficiency gains.",
"Further direction is discussed as well."
],
[
"Introduction",
"In the big data era, massive videos are fed into machines to realize intelligent analysis in numerous applications of smart cities or Internet of things (IoT).",
"Like the explosion of surveillance systems deployed in urban areas, there arise important concerns on how to efficiently manage massive video data.",
"There is a unique set of challenges (e.g.",
"low latency and high accuracy) regarding efficiently analyzing and searching the target within the millions of objects/events captured everyday.",
"In particular, video compression and transmission constitute the basic infrastructure to support these applications from the perspective of Compress then Analyze.",
"Over the past decades, a series of standards ( e.g.",
"MPEG-4 AVC/H.264 [1] and High Efficiency Video Coding (HEVC) [2]), Audio Video coding Standard (AVS) [3] are built to significantly improve the video coding efficiency, by squeezing out the spatial-temporal pixel-level redundancy of video frames based on the visual signal statistics and the priors of human perception.",
"More recently, deep learning based video coding makes great progress.",
"With the hierarchical model architecture and the large-scale data priors, these methods largely outperform the state-of-the-art codecs by utilizing deep-network aided coding tools.",
"Rather than directly targeting machines, these methods focus on efficiently reconstructing the pixels for human vision, in which the spatial-temporal volume of pixel arrays can be fed into machine learning and pattern recognition algorithms to complete high-level analysis and retrieval tasks.",
"However, when facing big data and video analytics, existing video coding methods (even for the deep learning based) are still questionable, regarding whether such big video data can be efficiently handled by visual signal level compression.",
"Moreover, the full-resolution videos are of low density in practical values.",
"It is prohibitive to compress and store all video data first and then perform analytics over the decompressed video stream.",
"By degrading the quality of compressed videos, it might save more bitrates, but incur the risk of degraded analytics performance due to the poorly extracted features.",
"To facilitate the high-level machine vision tasks in terms of performance and efficiency, lots of research efforts have been dedicated to extracting those pieces of key information, i.e., visual features, from the pixels, which is usually compressed and represented in a very compact form.",
"This poses an alternative strategy Analyze then Compress, which extracts, saves, and transmits compact features to satisfy various intelligent video analytics tasks, by using significantly less data than the compressed video itself.",
"In particular, to meet the demand for large-scale video analysis in smart city applications, the feature stream instead of the video signal stream can be transmitted.",
"In view of the necessity and importance of transmitting feature descriptors, MPEG has finalized the standardization of compact descriptors for visual search (CDVS) (ISO/IEC15938-13) in Sep. 2015 [16] and compact descriptors for video analysis (CDVA) (ISO/IEC15938-15) [17] in July 2019 to enable the interoperability for efficient and effective image/video retrieval and analysis by standardizing the bitstream syntax of compact feature descriptors.",
"In CDVS, hand-crafted local and global descriptors are designed to represent the visual characteristics of images.",
"In CDVA, the deep learning features are adopted to further boost the video analysis performance.",
"Over the course of the standardization process, remarkable improvements are achieved in reducing the size of features while maintaining their discriminative power for machine vision tasks.",
"Such compact features cannot reconstruct the full resolution videos for human observers, thereby incurring two successive stages of analysis and compression for machine and human vision.",
"For either Compress then Analyze or Analyze then Compress, the optimization jobs of video coding and feature coding are separate.",
"Due to the very nature of multi-tasks for machine vision and human vision, the intrusive setup of two separate stages is sub-optimal.",
"It is expected to explore more collaborative operations between video and feature streams, which opens up more space for improving the performance of intelligent analytics, optimizing the video coding efficiency, and thereby reducing the total cost.",
"Good opportunities to bridge the cross-domain research on machine vision and human vision have been there, as deep neural network has demonstrated its excellent capability of multi-task end-to-end optimization as well as abstracting the representations of multiple granularities in a hierarchical architecture.",
"In this paper, we attempt to identify the opportunities and challenges of developing collaborative compression techniques for humans and machines.",
"Through reviewing the advance of two separate tracks of video coding and feature coding, we present the necessity of machine-human collaborative compression, and formulate a new problem of video coding for machines (VCM).",
"Furthermore, to promote the emerging MPEG-VCM standardization efforts and collect for evidences for MPEG Ad hoc Group of VCM, we propose trial exemplar VCM architectures and conduct preliminary experiments, which are also expected to provide insights into bridging the cross-domain research from visual signal processing, computer vision, and machine learning, when AI meets the video big data.",
"The contributions are summarized as follows, We present and formulate a new problem of Video Coding for Machines by identifying three elements of tasks, features, and resources, in which a novel feedback mechanism for collaborative and scalable modes is introduced to improve the coding efficiency and the analytics performance for both human and machine-oriented tasks.",
"We review the state-of-the-art approaches in video compression and feature compression from a unique perspective of standardized technologies, and study the impact of more recent deep image/video prediction and generation models on the potential VCM related techniques.",
"We propose exemplar VCM architectures and provide potential solutions.",
"Preliminary experimental results have shown the advantages of VCM collaborative compression in improving video and feature coding efficiency and performance for human and machine vision.",
"The rest of the article is organized as follows.",
"Section and briefly review previous works on video coding and feature compression, respectively.",
"Section provides the definition, formulation, and paradigm of VCM.",
"Section illustrates the emerging AI technique, which provides useful evidence for VCM, After that, in Section , we provide potential solutions for VCM problem.",
"In Section , the preliminary experimental results are reported.",
"In Section , several issues, and future directions are discussed.",
"In Section , the concluding remarks are provided."
],
[
"Review of Video Compression: From Pixel Feature Perspective",
"Visual information takes up at least 83% of all information [18] that people can feel.",
"It is important for humans to record, store, and view the image/videos efficiently.",
"For past decades, lots of academic and industrial efforts have been devoted to video compression, which is to maximize the compression efficiency from the pixel feature perspective.",
"Below we review the advance of traditional video coding as well as the impact of deep learning based compression on visual data coding for human vision in a general sense."
],
[
"Traditional Hybrid Video Coding",
"Video coding transforms the input video into a compact binary code for more economic and light-weighted storage and transmission, and targets reconstructing videos visually by the decoding process.",
"In 1975, the hybrid spatial-temporal coding architecture [20] is proposed to take the lead and occupy the major proportion during the next few decades.",
"After that, the following video coding standards have evolved through the development of the ITU-T and ISO/IEC standards.",
"The ITU-T produced H.261 [21] and H.263 [22], ISO/IEC produced MPEG-1 [23] and MPEG-4 Visual [24], and the two organizations worked together to produce the H.262/MPEG-2 Video [25], H.264/MPEG-4 Advanced Video Coding (AVC) [26] standards, and H.265/MPEG-H (Part 2) High Efficiency Video Coding (HEVC) [27] standards.",
"The design and development of all these standards follows the block-based video coding approach.",
"The first technical feature is block-wise partition.",
"Each coded picture is partitioned into macroblocks (MB) of luma and chroma samples.",
"MBs will be divided into slices and coded independently.",
"Each slice is further partitioned into coding tree units.",
"After that, the coding unit (CU), prediction unit (PU), and transform unit (TU) are obtained to make the coding, prediction and transform processes more flexible.",
"Based on the block-based design, the intra and inter-predictions are applied based on PU and the corresponding contexts, i.e.",
"neighboring blocks and reference frames in the intra and inter modes, respectively.",
"But these kinds of designed patterns just cover parts of the context information, which limits the modeling capacity in prediction.",
"Moreover, the block-wise prediction, along with transform and quantization, leads to the discontinuity at the block boundaries.",
"With the quantization of the residue or original signal in the transform block, the blockness appears.",
"To suppress the artifacts, the loop filter is applied for smoothing.",
"Another important technical feature is hybrid video coding.",
"Intra and inter-prediction are used to remove temporal and spatial statistical redundancies, respectively.",
"For intra-prediction, HEVC utilizes a line of preceding reconstructed pixels above and on the left side of the current PU as the reference for generating predictions.",
"The number of intra modes is 35, including planar mode, DC mode, and 33 angular modes.",
"It is performed in the transform unit.",
"For inter-prediction, HEVC derives a motion-compensated prediction for a block of image samples.",
"The homogeneous motion inside a block is assumed, and the size of a moving object is usually larger than one block.",
"Reference blocks will be searched from previously coded pictures for inter prediction.",
"For both intra and inter-predictions, the best mode is selected by the Rate-Distortion Optimization (RDO) [28].",
"However, the multi-line prediction scheme and the block-wise reference block might not provide a desirable prediction reference when the structures and motion are complex.",
"Besides, when the RDO boosts the modeling capacity, the overhead of signaled bits and computation occur.",
"The target of optimizing the rate distortion efficiency is to seek for the trade-off in bitrate and signal distortion.",
"It can be solved via Lagrangian optimization techniques.",
"Coder control finally determines a set of coding parameters that affect the encoded bit-streams.",
"With tremendous expert efforts, the coding performance is dramatically improved in the past decade that there is the rule of thumb that, one generation of video coding standards almost surpasses the previous one by up to 50% in coding efficiency."
],
[
"Deep Learning Based Video Coding",
"The deep learning techniques significantly promote the development of video coding.",
"The seminar work [29] in 2015 opened a door to the end-to-end learned video coding.",
"The deep learning based coding tools [30], [31] are developed since 2016.",
"Benefiting from the bonus of big data, powerful architectures, end-to-end optimization, and other advanced techniques, e.g.",
"unsupervised learning, the emerging deep learning excel in learning data-driven priors for effective video coding.",
"First, complex nonlinear mappings can be modeled by the hierarchical structures of neural networks, which improves prediction models to make the reconstructed signal more similar to the original one.",
"Second, deep structures, such as PixelCNN and PixelRNN, are capable to model the pixel probability and provide powerful generation functions to model the visual signal in a more compact form.",
"The deep learning based coding methods do not rely on the partition scheme and support full resolution coding, and thereby removing the blocking artifacts naturally.",
"Since the partition is not required, these modules can access more context in a larger region.",
"As the features are extracted via a hierarchical network and jointly optimized with the reconstruction task, the resulting features tend to comprehensive and powerful for high efficient coding.",
"More efforts are put into increasing the receptive field for better perception of larger regions via recurrent neural network [32], [33] and nonlocal attention [34], [35], leading to improved coding performance.",
"The former infers the latent representations of image/videos progressively.",
"In each iteration, with the previously perceived context, the network removes the unnecessary bits from the latent representations to achieve more compactness.",
"The latter makes efforts to figure out the nonlocal correspondence among endpixels/regions, in order to remove the long-term spatial redundancy.",
"The efforts in improving the performance of neural network-aided coding tools rely on the excellent prediction ability of deep networks.",
"Many works attempt to effectively learn the end-to-end mapping in a series of video coding tasks, e.g., intra-prediction [7], [8], inter-prediction [9], [10], [4], [5], [6], deblocking [11], [12], [13], [14], and fast mode decision [15].",
"With a powerful and unified prediction model, these methods obtain superior R-D performance.",
"For intra-prediction, the diversified modes derived from RDO in the traditional video codec are replaced by a learned general model given a certain kind of context.",
"In [7], based on the multi-line reference context, fully-connected (FC) neural networks are utilized to generate the prediction result.",
"In [36], the strengths of FC and CNN networks are combined.",
"In [8], benefiting from the end-to-end learning and the block-level reference scheme, the proposed predictor employs context information in a large scale to suppress quantization noises.",
"For inter-prediction, deep network-based methods [37], [38] break the limit of block-wise motion compensation and bring in better inter-frame prediction, namely generating better reference frames by using all the reconstructed frames.",
"For loop-filter, the powerful network architectures and the full-resolution input of all reconstructed frames [14], [39] significantly improve the performance of loop filters, say up to 10% BD-rate reduction as reported in many methods.",
"The end-to-end learning based compression leverages deep networks to model the pixel distribution and generate complex signals from a very compact representation.",
"The pioneering work [33] proposes a parametric nonlinear transformation to well Gaussianize data for reducing mutual information between transformed components and show impressive results in image compression.",
"Meanwhile, in [29], a general framework is built upon convolutional and deconvolutional LSTM recurrent networks, trained once, to support variable-rate image compression via reconstructing the image progressively.",
"Later works [32], [40], [41], [42] continue to improve compression efficiency by following the routes.",
"All these methods attempt to reduce the overall R-D cost on a large-scale dataset.",
"Due to the model's flexibility, there are also more practical ways to control the bitrate adaptively, such as applying the attention mask [43] to guide the use of more bits on complex regions.",
"Besides, as the R-D cost is optimized in an end-to-end manner, it is flexible to adapt the rate and distortion to accommodate a variety of end applications, e.g.",
"machine vision tasks.",
"Although video coding performance is improved constantly, some intrinsic problems exist , especially when tremendous volumes of data need to be processed and analyzed .",
"The low-value data volume still constitutes a major part.",
"So these methods of reconstructing whole pictures cannot fulfill the requirement of real-time video content analytics when dealing with large-scale video data.",
"However, the strengths in deep learning based image/video coding, i.e.",
"the excellent prediction and generation capacity of deep models and the flexibility of R-D cost optimization, provide opportunities to develop VCM technology to address these challenges."
],
[
"Review of Feature Compression: From Semantic Feature Perspective",
"The traditional video coding targets high visual fidelity for humans.",
"With the proliferation of applications that capture video for (remote) consumption by a machine, such as connected vehicles, video surveillance systems, and video capture for smart cities, more recent efforts on feature compression target low bitrate intelligent analytics (e.g., image/video recognition, classification, retrieval) for machines, as transmission bandwidth for raw visual features is often at a premium, even for the emerging strategy of Analyze then Compress.",
"However, it is not new to explore feature descriptors for MPEG.",
"Back in 1998, MPEG initiated MPEG-7, formally known as Multimedia Content Description Interface, driven by the needs for tools and systems to index, search, filter, and manage audio-visual content.",
"Towards interoperable interface, MPEG-7 [44] defines the syntax and semantics of various tools for describing color, texture, shape, motion, etc.",
"Such descriptions of streamed or stored media help human or machine users to identify, retrieve, or filter audio-visual information.",
"Early visual descriptors developed in MPEG-7 have limited usage, as those low-level descriptors are sensitive to scale, rotation, lighting, occlusion, noise, etc.",
"More recently, the advance of computer vision and deep learning has significantly pushed forward the standardized visual descriptions in MPEG-7.",
"In particular, CDVS [45] and CDVA [46] have been in the vanguard of the trend of Analyze then Compress by extracting and transmitting compact visual descriptors.",
"Figure: The proposed video coding for machines (VCM) framework by incorporating the feedback mechanism into the collaborative coding of multiple streams including videos, features, and models, targeting the multi-tasks of human perception and machine intelligence."
],
[
"CDVS",
"CDVS can trace back to the requirements for early mobile visual search systems by 2010, such as faster search, higher accuracy, and better user experience.",
"Initial research [47], [48], [49], [50], [51], [52] demonstrated that one could reduce transmission data by at least an order of magnitude by extracting compact visual features on the mobile device and sending descriptors at low bitrates to a remote machine for search.",
"Moreover, a significant reduction in latency could also be achieved when performing all processing on the device itself.",
"Following initial research, an exploratory activity in the MPEG was initiated at the 91st meeting (Kyoto, Jan. 2010).",
"In July 2011, MPEG launched the standardization of CDVS.",
"The CDVS standard (formally known as MPEG-7, Part 13) was published by ISO on Aug. 25, 2015, which specifies a normative bitstream of standardized compact visual descriptors for mobile visual search and augmented reality applications.",
"Over the course of the standardization process, CDVS has made remarkable improvements over a large-scale benchmark in image matching/retrieval performance with very compact feature descriptors (at six predefined descriptor lengths: 512B, 1KB, 2KB, 4KB, 8KB, and 16KB) [16].",
"High performance is achieved while stringent memory and computational complexity requirements are satisfied to make the standard ideally suited for both hardware and software implementations [53].",
"To guarantee the interoperability, CDVS makes a normative feature extraction pipeline including interest point detection, local feature selection, local feature description, local feature descriptor aggregation, local feature descriptor compression, and local feature location compression.",
"Key techniques, e.g.",
"low-degree polynomial detector [54], [55], [45], light-weighted interest point selection [56], [57], scalable compressed fisher vector [58], [59], and location histogram coding [60], [61], have been developed by competitive and collaborative experiments within a rigorous evaluation framework [62].",
"It is worthy to mention that CDVS makes a normative encoder (the feature extraction process is fixed), which is completely distinct from conventional video coding standards in making a normative decoder.",
"The success of CDVS standardization originates from the mature computer vision algorithm ( like SIFT [63]) on the reliable image matching between different views of an object or scene.",
"However, when dealing with more complex analysis tasks in the autonomous drive and video surveillances, the normative encoder process incurs more risks of lower performance than those task-specific fine-tuned features directly derived from an end-to-end learning framework.",
"Fortunately, the collaborative compression of video and feature streams is expected to address this issue, as we may leverage the joint advantages of the normative feature encoding process and the normative video decoding process."
],
[
"CDVA",
"The bold idea of CDVA initiated in the 111th MPEG meeting in Feb. 2015 is to have a normative video feature descriptor based on neural networks for machine vision tasks, targeting an exponential increase in the demand for video analysis in autonomous drive, video surveillance systems, entertainment, and smart cities.",
"The standardization at the encoder requires the deterministic deep network model and parameters.",
"However, due to fast-moving deep learning techniques and their end-to-end optimization nature, there is a lack of a generic deep model sufficing for a broad of video analytics tasks.",
"This is the primary challenge for CDVA.",
"To kick off this standardization, CDVA narrows down to the general task of video matching and retrieval, aims at determining if a pair of videos share the object or scene with similar content, and searching for videos containing similar segments to the one in the query video.",
"Extensive experiments [64] over CDVA benchmark report comparable performances of the deep learning features with different off-the-shelf CNN models ( like VGG16 [65], AlexNet [66], ResNet [67]), which provide useful evidence for normative deep learning features for the task of video matching and retrieval.",
"Due to the hardware friendly merits of uniformly sized small filters (3x3 convolution kernel and 2x2 max pooling), and the competitive performance of combining convolution layers of the small filters by replacing a large filter (5 $\\times $ 5 or 7 $\\times $ 7), VGG16 is adopted by CDVA as the normative backbone network to derive compact deep invariant feature representations.",
"The CDVA standard (formally known as MPEG-7, Part 15) was published by ISO in July 2019.",
"Based on a previously agreed-upon methodology, key technologies are developed by MPEG experts, including Nested Invariance Pooling (NIP) [64] for deep invariance global descriptor, Adaptive Binary Arithmetic Coding (ABAC) based temporal coding of global and local descriptors [46], the integration of hand-crafted and deep learning features [17].",
"The NIP method produces compact global descriptors from a CNN model by progressive pooling operations to improve the translation, scale and rotation invariance over the feature maps of intermediate network layers.",
"The extensive experiments have demonstrated that the NIP (derived from the last pooling layer, i.e., pool5, of VGG16) outperforms the state-of-the-art deep and canonical hand-crafted descriptors with significant gains.",
"In particular, the combination of (deep learning) NIP global descriptors and (hand-crafted) CDVS global descriptors has significantly boosted the performance with a comparable descriptor length.",
"More recently, great efforts are made to extend the CDVA standard for VCM.",
"Smart tiled CDVA [68] targeting video/image analysis of higher resolution input, can be applied to rich tasks including fine-grained feature extraction network, small object detection, segmentation, and other use cases.",
"Smart sensing [69] is an extension of the CDVA standard to directly process raw Bayer pattern data, without the need of a traditional ISP.",
"This enables all Bayer pattern sensors to be natively CDVA compatible.",
"SuperCDVA [70] is an application of the CDVA standard for video understanding by using the temporal information, in which each CDVA vector is embedded as one block in an image, and the multiple blocks are put in a sequential manner to represent the sequence of video frames for classification by a 2-D CNN model.",
"In summary, the new MPEG-7 standard CDVA opens the door to exciting new opportunities in supporting machine-only encoding as well as hybrid (machine and human) encoding formats to unburden the network and storage resources for dramatic benefits in various service models and a broad range of applications with low latency and reduced cost."
],
[
"Video Coding for Machine",
"The emerging requirements of video processing and content analysis in a collaborative manner are expected to support both human vision and machine vision.",
"In practice, two separate streams of compressed video data and compact visual feature descriptors are involved to satisfy the human observers and a variety of low-level and high-level machine vision tasks, respectively.",
"How to further improve the visual signal compression efficiency as well as the visual task performance by leveraging multiple granularities of visual features remains a challenging but promising task, in which the image/video data is considered as a sort of pixel-level feature."
],
[
"Definition and Formulation",
"Traditional video coding targets visual signal fidelity at high bitrates, while feature coding targets high performance of vision tasks at very low bitrates.",
"Like CDVS and CDVA, most of the existing feature coding approaches heavily rely on specific tasks, which significantly save bitrate by ignoring the requirement of full-resolution video reconstruction.",
"By contrast, the video codecs like HEVC focus on the reconstruction of full-resolution pictures from the compressed bit stream.",
"Unfortunately, the pixel-level features from data decompression cannot suffice for large-scale video analysis in a huge or a moderate scale camera network efficiently, due to the bottleneck of computation, communication, and storage [71].",
"We attempt to identify the role of Video Coding for Machines (VCM), as shown in Fig.",
"REF , to bridge the gap between coding semantic features for machine vision tasks and coding pixel features for human vision.",
"The scalability is meant to incrementally improve the performance in a variety of vision tasks as well as the coding efficiency by optimizing bit utility between multiple low-level and high-level feature streams.",
"Moreover, VCM is committed to developing key techniques for economizing the use of a bit budget to collaboratively complete multiple tasks targeting humans and/or machines.",
"VCM aims to jointly maximize the performance of multiple tasks ranging from low-level processing to high-level semantic analysis, but minimize the use of communication and computational resources.",
"VCM relates to three key elements: Tasks.",
"VCM incurs low-level signal fidelity as well as high-level semantics, which may request a complex optimization objective from multiple tasks.",
"It is important to figure out an efficient coding scheme across tasks.",
"Resources.",
"VCM is committed to tackling the practical performance issue, which cannot be well solved by traditional video coding or feature coding approaches solely, subject to the constraints of resources like bandwidth, computation, storage, and energy.",
"Features.",
"VCM is to explore a suite of rate distortion functions and optimization solutions from a unified perspective of leveraging the features at different granularities including the pixels for humans to derive efficient compression functions (e.g., transform and prediction).",
"Here we formulate the VCM problem.",
"The features are denoted by $\\mathbf {F} = \\left\\lbrace F^{0}, F^{1}, ..., F^{z} \\right\\rbrace $ of $(z+1)$ tasks, where $V:=F^{z}$ is the pixel feature, associated with the task of improving visual signal fidelity for human observers.",
"$F^{i}$ denotes more task (non-)specific semantic or syntactic features if $0 \\le i < z$ .",
"A smaller $i$ denotes that the feature is more abstract.",
"The performance of the task $i$ is defined as follows: $q^{i} = \\Phi ^{i}(\\hat{F}^{i}),$ where $\\Phi ^{i}(\\cdot )$ is the quality metric related to task $i$ and $\\hat{F}^{i}$ is the reconstructed feature undergoing the encoding and decoding processes.",
"Note that, certain compressed domain processing or analysis may not require the complete reconstruction process.",
"We use $C(\\cdot | \\theta _c ), D(\\cdot | \\theta _d)$ , and $\\mathcal {G}(\\cdot | \\theta _g)$ to denote the processes of compression, decompression, feature prediction, where $\\theta _c$ , $\\theta _d$ , and $\\theta _g$ are parameters of the corresponding processes.",
"We define $S(\\cdot )$ to measure the resource usage of computing a given feature, namely compressing, transmitting, and storing the feature, as well as further analyzing the feature.",
"By incorporating the elements of tasks, resources, and features, VCM explicitly or non-explicitly models a complex objective function for maximizing the performance and minimizing the resource cost, in which joint optimization applies.",
"Generally, the VCM optimization function is described as follows, $& \\mathop {\\text{argmax} }\\limits _{\\Theta = \\left\\lbrace {{\\theta _c},{\\theta _d},{\\theta _g}} \\right\\rbrace } \\sum \\limits _{0 \\le i \\le z} {{\\omega ^i}{q^i}}, \\nonumber \\\\&\\text{ subject to:} \\sum \\limits _{0 \\le i \\le z} {{\\omega ^i}} = 1, \\\\S\\left( {{R_{{F^0}}}} \\right) + & \\sum \\limits _{i > 0}^{} {\\mathop {\\min }\\limits _{0 \\le j < i} } \\left\\lbrace {S\\left( {{R_{{F_{i \\rightarrow j}}}}} \\right)} \\right\\rbrace + S\\left( {{R_M}} \\right) + S\\left( \\Theta \\right) \\le {S_T}, \\nonumber $ where ${S_T}$ is total resource cost constraint, and ${\\omega ^i}$ balances the importance of different tasks in the optimization.",
"The first two terms in the resource constraint are the resource cost of compressing and transmitting features of all tasks with feature prediction.",
"Note that, ${R_V} = {\\min _j}\\left\\lbrace {S\\left( {{R_{{F_{z \\rightarrow j}}}}} \\right)} \\right\\rbrace $ is the resource cost to encode videos ${F^z}$ with feature prediction.",
"The third term in the resource constraint is the resource overhead from leveraging models and updating models accordingly.",
"The last term $S\\left( \\Theta \\right)$ calculates the resource cost caused by using a more complex model, such as longer training time, response time delay due to feedbacks, larger-scale training dataset.",
"Principle rules apply to the key elements as below, ${R_{{F^0}}} & = C\\left( {{F^0}|{\\theta _c}} \\right), \\\\{R_{{F_{i \\rightarrow j}}}} & = C\\left( {F^i} - {\\rm {{\\cal G}}}\\left( {{F^j},i|{\\theta _g}} \\right)\\right), \\text{ }{\\rm { for }} \\text{ } i \\ne 0, \\\\{\\widehat{F}^0} & = D\\left( {{R_{{F^0}}}|{\\theta _d}} \\right), \\\\{\\widehat{F}^i} & = D\\left( {{R_{{F_{i \\rightarrow j}}}}|{\\theta _d}} \\right) + \\mathcal {{\\cal G}}\\left( {{{\\widehat{F}}^j},i|{\\theta _g}} \\right),\\text{ }{\\rm { for }} \\text{ } i \\ne 0, $ where $\\mathcal {G}(\\cdot , j| \\theta _g)$ projects the input feature to the space of $F^{j}$ .",
"$\\hat{\\cdot }$ denotes that, the feature decoded from the bit stream would be suffering from compression loss.",
"In practice, the VCM problem can be rephrased in Eq.",
"(REF ) by designing more task specific terms $C(\\cdot | \\theta _c )$ , $D(\\cdot | \\theta _d)$ and $\\mathcal {G}(\\cdot | \\theta _g)$ ."
],
[
"A VCM Paradigm: Digital Retina",
"As a VCM instance, digital retina [71], [72], [73] is to solve the problem of real-time surveillance video analysis collaboratively from massive cameras in smart cities.",
"Three streams are established for human vision and machine vision as follows: Video stream: Compressed visual data is transmitted to learn data-driven priors to impact the optimization function in Eq.",
"(REF ).",
"In addition, the fully reconstructed video is optionally for humans as requested on demand.",
"Model stream: To improve the task-specific performance and hybrid video coding efficiency, hand-crafted or learning based models play a significant role.",
"This stream is expected to guide the model updating and the model based prediction subsequently.",
"The model learning works on the task-specific performance metric in Eq.",
"(REF ).",
"Feature stream: This stream consisting of task-specific semantic or syntactic features extracted at the front end devices is transmitted to the server end.",
"As formulated in Eq.",
"(REF ), a joint optimization is expected to reduce the resource cost of video and feature streams.",
"Instead of working on video stream alone, the digital retina may leverage multiple streams of features to reduce the bandwidth cost by transmitting task-specific features, and balance the computation load by moving part of feature computing from the back end to the front end.",
"As an emerging VCM approach, the digital retina is revolutionizing the vision system of smart cities.",
"However, for the current digital retina solution, the optimization in Eq.",
"(REF ) is reduced to minimizing an objective for each single stream separately, without the aid of feature prediction in Eq.",
"(REF ) and (), rather than multiple streams jointly.",
"For example, state-of-the-art video coding standards like HEVC are applied to compress video streams.",
"The compact visual descriptor standards CDVS/CDVA are applied to compress visual features.",
"The collaboration between two different types of streams works in a combination mode.",
"There is a lack of joint optimization across streams in terms of task-specific performance and coding efficiency.",
"Generally speaking, the traditional video coding standardization on HEVC/AVS, together with recent compact feature descriptor standardization on CDVS/CDVA, targets efficient data exchange between human and human, machine and human, and machine and machine.",
"To further boost the functionality of feature streams and open up more room for collaborating between streams, the digital retina introduces a novel model stream, which takes advantage of pixel-level features, semantic features (representation), and even existing models to generate a new model for improving the performance as well as the generality of task-related features.",
"More recent work [72] proposed to reuse multiple models to improve the performance of a target model, and further came up with a collaborative approach [74] to low cost and intelligent visual sensing analysis.",
"However, how to improve the video coding efficiency and/or optimize the performance by collaborating different streams is still an open and challenging issue."
],
[
"Reflection on VCM",
"Prior to injecting model streams, the digital retina [75] has limitations.",
"First, the video stream and feature stream are handled separately, which limits the utilization of more streams.",
"Second, the processes of video coding and feature coding are uni-directional, thereby limiting the optimization performance due to the lack of feedback mechanism, which is crucial from the perspective of a vision system in smart cities.",
"Beyond the basic digital retina solution, VCM is supposed to jointly optimize the compression and utilization of feature, video, and model streams in a scalable way by introducing feedback mechanisms as shown in Fig.",
"REF : Feedback for collaborative mode: The pixel and/or semantic features, equivalently video and feature streams, can be jointly optimized towards higher coding efficiency for humans and/or machines in a collaborative manner.",
"That is, the features can be fed back to each other between streams for improving the task-specific performance for machines, in addition to optimizing the coding efficiency for humans.",
"Advanced learning-based prediction or generation models may be applied to bridge the gap between streams.",
"Feedback for scalable mode: When bit budget cannot suffice for video or feature streams, or namely, the quality of reconstructed feature and video are not desirable, more additional resources are utilized, along with the previously coded streams, to improve the quality of both feature and video streams with a triggered feedback.",
"Therefore, the desired behavior of incrementally improving the performance of those human and machine vision tasks subject to an increasing bit budget can be expected.",
"Figure: The key modules of VCM architecture, as well as the relationship between features and human/machine tasks."
],
[
"New Trends and Technologies",
"The efficient transition between the features of different granularities is essential for VCM.",
"In this section, we review more recent advances in the image/video predictive and generative models."
],
[
"Image Predictive Models",
"Deep convolutional neural networks [66], [67] have been proven to be effective to predict semantic labels of images/videos.",
"This superior capacity has been witnessed in many visual tasks, e.g.",
"image classification [76], [67], object detection [77], [78], semantic segmentation [79], pose estimation [80].",
"The key to the success of these tasks is to extract discriminative features to effectively model critical factors highly related to the final predictions.",
"Deep networks have hierarchical structures to extract features from low-level to high-level progressively.",
"The features at the deep layer are more compact and contain less redundant information, with very sparse activated regions.",
"As such, the image predictive model can capture the compact and critical feature from a single image, which provides an opportunity to develop efficient compression approaches."
],
[
"Video Predictive Models",
"The deep networks designed for video analytics pay additional attention to modeling temporal dynamics and utilizing complex joint spatial and temporal correlations for semantic label prediction of videos.",
"In [81], several approaches extend the temporal connectivity of a CNN to fully make use of local spatio-temporal information for video classification.",
"In [82], a two-stream ConvNet architecture incorporated with spatial appearance and motion information is built for action recognition.",
"Their successive works are based on mixed architectures with both CNN and RNN [83], and 3D convolutional networks [84] for action recognition [85], [86], scene recognition [84], captioning, commenting [87].",
"These models are capable to extract discriminative and compact joint spatial and temporal features, potential to benefit squeezing out the redundancy in videos."
],
[
"Generative Adversarial Networks",
"The advance of generative adversarial networks (GANs) [88] makes a significant impact in machine vision.",
"Recent years have witnessed the prosperity of image generation and its related field [89], [90], [91], [92].",
"In general, most of the existing methods can be categorized into two classes: supervised and unsupervised methods.",
"In supervised methods [93], [94], [95], GANs act as a powerful loss function to capture the visual property that the traditional losses fail to describe.",
"Pix2pix [96] is a milestone work based on conditional GAN to apply the image-to-image translation from the perspective of domain transfer.",
"Later on, more efforts [89], [97] are dedicated to generating high-resolution photo-realistic images with a progressive refinement framework.",
"In unsupervised methods, due to the lack of the paired ground truth, the cycle reconstruction consistency [98], [99], [100] is introduced to model the cross-domain mapping."
],
[
"Guided Image Generation",
"Some works focus on guided image generation, where semantic features, e.g.",
"human pose and semantic map, are taken as the guidance input.",
"The early attempt pays attention to pose-guided image generation and a two-stage network PG$^2$ [101] is built to coarsely generate the output image under the target pose in the first stage, and then refine it in the second stage.",
"In [102], deformable skips are utilized to transform high-level features of each body part to better model shapes and appearances.",
"In [103], the body part segmentation masks are used as guidance for image generation.",
"However, the above-mentioned methods [101], [102], [103] rely on paired data.",
"To address the limitation, in [104], a fully unsupervised GAN is designed, inspired by [98], [105].",
"Furthermore, the works in [106], [107] resort to sampling from the feature space based on the data distribution.",
"These techniques bring in precious opportunities to develop efficient video coding techniques.",
"The semantic feature guidance is much more compact .",
"With the support of the semantic feature, the original video can be well reconstructed economically."
],
[
"Video Prediction and Generation",
"Another branch of generation models are for video prediction and generation.",
"Video prediction aims to produce future frames based on previous frames of a video sequence in a deterministic manner, in which recurrent neural networks are often used to model the temporal dynamics [108], [109], [110].",
"In [108], an LSTM encoder-decoder network is utilized to learn patch-level video representations.",
"In [109], a convolutional LSTM is built to predict video frames.",
"In [110], a multi-layer LSTM is constructed to progressively refine the prediction process.",
"Some methods do not rely on recurrent networks, e.g.",
"3D convolutional neural network [111], [112].",
"Some other methods [113], [114], [115] estimate local and global transforms and then apply these transforms to generate future frames indirectly.",
"Comparatively, video generation methods aim to produce visually authentic video sequences in a probabilistic manner.",
"In the literature, methods based on GAN [116], [117], [118], [119] and Variational AutoEncoder (VAE) [120], [121], [122], [123], [124] are built.",
"The above-mentioned methods just predict a few video frames.",
"Later works [125] target long-term video prediction.",
"In [125], up to 100 future frames of an Atari game are generated.",
"The future frame is generated with the guidance of the encoded features by CNN and LSTM from previous frames.",
"In [126], a new model is proposed to generate real video sequences.",
"The high-level structures are estimated from past frames, which are further propagated to those of the future frame via an LSTM.",
"In [127], an unsupervised method is built to extract a high-level feature which is further used to predict the future high-level feature.",
"After that, the model can predict future frames based on the predicted features and the first frame.",
"Undoubtedly, inter prediction plays a significant role in video coding.",
"The video prediction and generation models are expected to leverage compact features to propagate the context of video for improving coding efficiency.",
"Deep learning brings in the wealth of `deep'.",
"That is, with the hierarchical layer structure, the information is processed and distilled progressively, which benefits both high-level semantic understanding and low-level vision reconstruction.",
"For VCM, an important property is scalability, a capacity of feature prediction ranging from the extreme compactness to the injection of redundant visual data for serving humans and machines, which is closely correlated to the hot topics of deep progressive generation and enhancement.",
"A series of works have been proposed to generate or enhance images progressively.",
"In [144], a cascade of CNN within a Laplacian pyramid framework is built to generate images in a coarse-to-fine fashion.",
"In [145], a similar framework is applied for single-image super-resolution.",
"Later works refining features progressively at the feature-level, like ResNet [67] , or concatenating and fusing features from different levels, like DenseNet [146], lead to better representations of pixels and their contexts for low-level visions.",
"The related beneficial tasks include super-resolution [147], [93], [148], [149], [150], rain removal [151], [152], dehazing [148], inpainting [153], compression artifacts removal [154], and deblurring [155].",
"Zhang et al.",
"[148] combined the structure of ResNet and DenseNet.",
"Dense blocks are used to obtain dense local features.",
"All features in each dense block are connected by skip connections, and then fused in the last layer adaptively in a holistic way.",
"In video coding, there is also a similar tendency to pursue a progressive image and video compression, namely, scalable image/video coding [128], affording to form the bit-stream at any bitrate.",
"The bit-stream usually consists of several code layers (one base layer and several enhancement layers).",
"The base layer is responsible for basic but coarse modeling of image /video.",
"The enhancement layers progressively improve the reconstruction quality with additional bit-streams.",
"A typical example is JPEG2000 [129], where an image pyramid from the wavelet transform is build up for scalable image reconstruction based on compact feature representations.",
"Later on, the extension of the scalable video codec is made in the H.264 standard [130].",
"The base layer bit-stream is formed by compressing the original video frames, and the enhanced layer bit-stream is formed by encoding the residue signal.",
"VCM is expected to incorporate the scalability into the collaborative coding of multiple task-specific data streams for humans and/or machines.",
"It is worthy to note that, the rapid development of DNNs (deep neural networks) is proliferating scalable coding schemes.",
"In [32], an RNN model is utilized to realize variable-bitrate compression.",
"In [131], bidirectional ConvLSTM is adopted to decompose the bit plane by efficiently memorizing long-term dependency.",
"In [132], inspired by the self-attention, a transformer-based decorrelation unit is designed to reduce the feature redundancy at different levels.",
"More recently, several works [133], [134], [135], [137], [138] attempt to jointly compress videos and features in a scalable way, which shows more evidence for the scalability in VCM."
],
[
"Video Coding for ",
"In this section, we present several exemplar VCM solutions: deep intermediate feature compression, predictive coding with collaborative feedback, and enhancing predictive coding with scalable feedback.",
"Based on the fact that the pretrained deep learning networks, e.g.",
"VGG and AlexNet, can support a wide range of tasks, such as classification and object detection, we first investigate the issue of compressing intermediate features (usually before pool5 layer) extracted from off-the-shelf pretrained networks (left part of Fig.",
"REF ), less specific to given tasks, via state-of-the-art video coding techniques.",
"In the next step, we explore the solution that learns to extract key points as a highly discriminative image-level feature (right part of Fig.",
"REF ) to support the motion-related tasks for both machine and human vision with collaborative feedback in a feature assisted predictive way.",
"Finally, we attempt to pursue a preliminary scheme to offer a general feature scalability to derive both pixel and image-level representations and improve the coding performance incrementally .",
"Figure: Two potential VCM solutions: (a) Predictive coding with collaborative feedback, and (b) Enhancing predictive coding with scalable feedback."
],
[
"Deep Intermediate Feature Compression",
"In the VCM, features are the key bridge for both front and back user-ends as well as high and low-level visions.",
"It naturally raises questions about the optimized feature extraction and model updating in the VCM framework.",
"For deep model-based applications, the feature compression is hindered by that, the models are generally tuned for specific tasks, and that, the top-layer features are very task-specific and hard to be generalized.",
"The work in [136] explores a novel problem: the intermediate layer feature compression, reducing the computing burden while being capable to support different kinds of visual analytics applications.",
"In practice, it provides a compromise between the traditional video coding and feature compression and yields a good trade-off among the computational load, communication cost, and the generalization capacity.",
"As illustrated in Fig.",
"REF , VCM attempts to connect the features of different granularities to the human/machine vision tasks from the perspective of a general deep learning framework.",
"The intermediate layer features are compressed and transmitted instead of the original video or top layer features.",
"Compared with the deep layer features, the intermediate features from shallow layers contain more informative cues in a general sense, as the end-to-end learning usually makes the deep layer features more task-specific with a large receptive field.",
"To accommodate a wide range of machine vision tasks, VCM prefers to optimize the compression of intermediate features for general purposes.",
"An interesting view is that, the shallow layers closer to the input image/video, are supposed to be less important than the deep layers, as the semantics play a significant role in data compressing even for humans.",
"As indicated in Fig.",
"REF , VCM prefers to use the deep layers features in improving coding efficiency for humans.",
"Beyond that, a further idea is to propose the problem of feature recomposition.",
"The features for different tasks are with various granularities.",
"It is worthwhile to explore how to evaluate the feature coding performance of all tasks in a unified perspective, and further decide to recompose the feature of different tasks, sharing common features and organizing the related compression in a more or less scalable way."
],
[
"Predictive Coding with Collaborative Feedback",
"Fig.",
"REF (a) shows the overview pipeline of a joint feature and video compression approach [137], [138].",
"At the encoder side, a set of key frames ${{{v}}_k}$ will be first selected and compressed with traditional video codecs to form the bit-stream ${B_I}$ .",
"The coded key frames convey the appearance information and are transmitted to the decoder side to synthesize the reconstructed non-key frames.",
"Then, the network learns to represent $V=\\left\\lbrace v_1, v_2, ..., v_N \\right\\rbrace $ with the learned sparse points $F=\\left\\lbrace f_1, f_2, ..., f_N \\right\\rbrace $ to describe temporal changes and object motion among frames.",
"We employ prediction model $P\\left( \\cdot \\right)$ and generation model $G\\left( \\cdot \\right)$ to implement the feature transition operation $\\mathcal {G}(\\cdot )$ to convert the feature from a redundant form to compact one and vice verse, respectively.",
"More specifically, $P(\\cdot )$ and $G(\\cdot )$ are the processes to extract key points from videos and generate videos based on the learned key points.",
"To extract the key points, we have: $F = P\\left( {V,\\lambda |{\\theta _{gp}}} \\right),$ where ${\\theta _{gp}}$ is a learnable parameter.",
"$F$ is a compact feature, which only requires very fewer bit streams for transmission and storage.",
"$\\lambda $ is a rate control parameter.",
"The compression model ${{{C}}_F}\\left( { \\cdot |{\\theta _{cf}}} \\right)$ compresses $F$ into the feature stream ${B_F}$ : ${B_F}{\\rm { = }}{{{C}}_F}\\left( {F|{\\theta _{cf}}} \\right),$ where ${\\theta _{cf}}$ is a learnable parameter.",
"Then, a motion guided generation network calculates the motion based on these points and then transfers the appearance from the reconstructed key frames to those remaining non-key frames.",
"Specifically, for the $t$ -th frame to be reconstructed, we denote its previous reconstructed key frame, previous reconstructed key points, and the current reconstructed key points by ${{{\\widehat{v}}_{\\psi (t)}}}$ ,${{\\widehat{f}_{\\psi (t)}}}$ and ${{\\widehat{f}_{t}}}$ : where $\\psi (t)$ maps the index of the key frame of the $t$ -th frame.",
"The target frame ${{\\widetilde{v}}_t} \\in \\widetilde{V}$ is synthesized as follows: ${{\\widetilde{v}}_t} = G \\left( {{{\\widehat{v}}_{\\psi (t)}},{{\\widehat{f}}_{\\psi (t)}},{{\\widehat{f}}_t}} | \\theta _{gg} \\right),$ where ${\\theta _{gg}}$ is a learnable parameter.",
"After that, the residual video $R=V-\\widetilde{V}$ can be calculated, where $\\widetilde{V}=\\left\\lbrace \\widetilde{v}_1, \\widetilde{v}_2, ..., \\widetilde{v}_N \\right\\rbrace $ .",
"The sparse point descriptor and residual video will be quantized and compressed to the bit stream ${B_F}$ and ${B_V}$ for transmission.",
"That is ${B_V} = \\left\\lbrace B_I, B_R \\right\\rbrace $ .",
"We can adjust the total bitrate via controlling the bitrates of ${B_F}$ and ${B_V}$ .",
"At the decoder side, the key frames will be first reconstructed from ${{{B}}_{{I}}}$ .",
"The sparse point representations are also decompressed as ${{\\widehat{F}}} = \\left\\lbrace {{{\\widehat{f}}_1},{{\\widehat{f}}_2},...,{{\\widehat{f}}_N}} \\right\\rbrace $ from $B_F$ as follows, $\\widehat{F}{\\rm { = }}{{{D}}_F}\\left( {{B_F}|{\\theta _{df}}} \\right),$ where ${{{D}}_F}\\left( { \\cdot |{\\theta _{df}}} \\right)$ is a feature decompression model, and ${\\theta _{df}}$ is a learnable parameter.",
"The videos are reconstructed via: $\\widehat{V} = \\widetilde{V} +\\widehat{R} $ .",
"Finally, $\\widehat{F}$ along with $\\widehat{V}$ serves machine analysis and human vision, respectively."
],
[
"Enhancing Predictive Coding with Scalable Feedback",
"Fig.",
"REF (b) shows an optimized architecture for VCM, by adding scalable feedback.",
"Similarly, the key points of video frames $F$ are extracted.",
"After that, the redundancy of the video and key frames ${{{\\widehat{v}}_{\\psi (t)}}}$ is removed by applying feature-guided prediction.",
"Then, the residue video $R$ is compressed into the video stream, which is further passed to the decoder.",
"When the feature and video qualities after decompression do not meet the requirements, a scalable feedback is launched and more guidance features are introduced: $\\Delta F & = Y\\left( F, V | \\theta _y \\right), \\\\F^U & = F + \\Delta F,$ where $\\theta _y$ is a learnable parameter.",
"With the key points $F$ in the base layer (the result in the scheme in Fig.",
"REF (a)) and the original video $V$ , we generate the residual feature $\\Delta F$ to effectively utilize additionally allocated bits to encode more abundant information and form the updated feature $F^U$ .",
"Then, $G^U$ is used to refine $V$ by taking the reconstructed key points and video as its input: $\\Delta \\widetilde{V} = G^U\\left( \\widehat{F}, \\Delta \\widehat{F}, \\widehat{V} | \\theta _h \\right),$ where $\\theta _h$ is a learnable parameter.",
"Then, we can infer the incremental residue video: ${R^U} = V - (\\widetilde{V} + \\Delta \\widetilde{V}) - R$ , which is compressed into the bit stream $B_{DV}$ .",
"At the decoder side, ${\\hat{R}^U}$ is decompressed from $B_{DV}$ .",
"Then, the video with a high quality is inferred via: ${\\widehat{V}^U} = \\widetilde{V} + \\Delta \\widetilde{V} + {\\widehat{R}^U} + \\widehat{R}$.",
"This allows to introduce more bits via launching scalable feedback."
],
[
"Preliminary Experimental Results",
"In this section, we provide preliminary experimental results from the perspectives of intermediate deep feature compression and machine-human collaborative compression.",
"Table: Lossy feature compression results for different tasks (Comp.Rate|Fidelity 2 ^2)Table: Lossy feature compression results for different tasks.",
"DD-Channel Concatenation denotes channel concatenation by descending difference.",
"(Comp.Rate|Fidelity 2 ^2)"
],
[
"Compression Results",
"We show the compression performance on the intermediate features for different tasks in Table REF .",
"The compression rate is calculated by the ratio of original intermediate deep features and the compressed bit-streams.",
"As to the fidelity evaluationhttps://github.com/ZoomChen/DeepFeatureCoding/tree/master/Coding_and _Evaluation , the reconstructed features are passed to their birth-layer of the original neural network to infer the network outputs, which will be compared with pristine outputs to evaluate the information loss of the lossy compression methods.",
"More results and details on the evaluation framework can be found in [139].",
"From Table REF, several interesting observations are reached.",
"First, the potential of lossy compression is inspiring.",
"In the extreme case, for example in image retrieval, ResNet conv2 feature achieves at least 1000$\\times $ compression ratio at QP 42, while the lossless methods usually provide 2-5$\\times $ .",
"Second, for each feature type, the fidelity metric decreases with a larger QP value.",
"Third, QP 22 generally provides high fidelity and fair compression ratio.",
"Forth, upper layer features, like conv4 to pool5, tend to be more robust to heavy compression."
],
[
"Channel Packaging",
"Deep features have multiple channels.",
"It needs to arrange these features into single-channel or three-channel maps and then compress them with the existing video codecs.",
"Three modes are studied: channel concatenation, channel concatenation by descending difference, and channel tiling.",
"For channel concatenation, each channel of the feature map corresponds to a frame in the input data of a traditional video encoder.",
"The height and width of the feature map are filled to the height and width that meet the input requirements of the video encoder.",
"The feature map channel order is the original order and remains unchanged.",
"In this mode, inter-coding of HEVC is applied.",
"For channel concatenation by descending difference, to obtain higher compression efficiency, the channel of the feature map is reordered before being fed into a traditional video encoder.",
"The first channel is fixed, and the remaining channels are arranged according to the L2 norm of the different between the current channel to the previous one.",
"For channel tiling, multiple channels are tiled into a two-dimensional map, serving as an input to a video encoder.",
"The result is presented in Table REF .",
"These results are preliminary and more efforts are expected to improve the efficiency of compressing deep feature maps.",
"Let us evaluate the effectiveness of the potential solution: feature assisted predictive coding with the collaborative feedback.",
"We show the results of compression for machine vision, including action recognition, human detection and compression for human vision, video reconstruction."
],
[
"Experimental Settings",
"PKU-MMD dataset [140] is used to generate the testing samples.",
"In total, 3317 clips, each sampling 32 frames, are used for training, and 227 clips, each sampling 32 frames, are used for testing.",
"All frames are cropped and resized to $512\\times 512$ .",
"To verify the coding efficiency, we use HEVC, implemented in FFmpeg version 2.8.15https://www.ffmpeg.org/, as the anchor for comparison by compressing all frames with the HEVC codec in the constant rate factor mode.",
"To evaluate the performance of feature assisted predictive coding, the sparse motion pattern [141] is extracted to serve machine vision.",
"For a given input frame, a U-Net followed by softmax activations is used to extract heatmaps for key point prediction.",
"The covariance matrix is additionally generated to capture the correlations between the key points and its neighbor pixels.",
"For each key point, in total 6 float numbers including two numbers indicating the position and 4 numbers in the covariance matrix are used for description.",
"The selected key frames are compressed and transmitted, along with the sparse motion pattern to generate the full picture for human vision.",
"In the testing process, we select the first frame in each clip as the key frames.",
"At the encoder side, the key frame is coded with the HEVC codec in the constant rate factor mode.",
"Besides the key frame, 20 key points of each frame form the sparse motion representation.",
"Each key point contains 6 float numbers.",
"For the two position numbers, a quantization with the step of 2 is performed for compression.",
"For the other 4 float numbers belonging to the covariance matrix, we calculate the inverse of the matrix in advance, and then quantize the 4 values with a step of 64.",
"Then, the quantized key point values are further losslessly compressed by the Lempel Ziv Markov chain algorithm (LZMA) algorithm [142].",
"Table: SSIM comparison between different methods and corresponding bitrate costs.Figure: Proposed"
],
[
"Action Recognition",
"We first evaluate the efficiency of the learned key points for action recognition.",
"Although each key point is represented with 6 numbers, we only use two quantized position numbers for action recognition.",
"To align with the bitrate cost of the features, the clips are first down-scaled to $256\\times 256$ and then compressed with the constant rate factor 51 with HEVC.",
"All 227 clips are used in the testing.",
"Table REF has shown the action recognition accuracy and corresponding bitrate costs of different kinds of data.",
"Our method can obtain considerable action recognition accuracy with only $5.2$ Kbps bitrate cost, superior to that by HEVC.",
"Figure: Rate distortion curves of HEVC and the proposed method."
],
[
"Human Detection",
"Apart from action recognition, human detection accuracy is also compared.",
"The original skeleton information in the dataset is used to form the ground truth bounding box and a classical detection algorithm YOLO v3 [156] is adopted for comparison.",
"All 227 clips are used in the testing.",
"The testing clips are down-scaled to different scales of $64\\times 64$ , $128\\times 128$ , $256\\times 256$ and $512\\times 512$ and compressed by HEVC with the constant rate factor 51 to form the input of YOLO v3.",
"Fig.",
"REF has shown the Intersection over Union (IoU) of different methods and their corresponding bitrates.",
"Our method can achieve much better detection accuracy with lower bitrate costs.",
"Some subjective results of human detection are shown in Fig.",
"REF , we can see that our method can achieve better detection accuracy with fewer bit costs.",
"Figure: The visual results of the reconstructed videos by HEVC (left panel) and our method (right panel) , respectively.Figure: Proposed"
],
[
"Video Reconstruction",
"The video reconstruction quality of the proposed method is compared with that of HEVC.",
"During the testing phase, we compress the key frames with a constant rate factor 32 to maintain a high appearance quality.",
"The bitrate is calculated by considering the compressed key frames and key points.",
"As for HEVC, we compress all frames with a constant rate factor 44 to achieve an approaching bitrate .",
"Table REF has shown the quantitative reconstruction quality of different methods.",
"SSIM values are adopted for quantitative comparison.",
"It can be observed that, our method can achieve better reconstruction quality than HEVC with lower bitrate cost.",
"The subjective results of different methods are shown in Fig.",
"REF .",
"There are obvious compression artifacts on the reconstruction results of HEVC, which heavily degrade the visual quality.",
"Compared with HEVC, our method can provide far more visually pleasing results.",
"Moreover, we add a rate distortion curve for comparison.",
"HEVC is used as the anchor undergoing four constant rate factors 44, 47, 49 and 51.",
"For our method, the key frames are compressed respectively under constant rate factors 32, 35, 37 and 40.",
"The rate distortion curve is shown in Fig.",
"REF .",
"Our method yields better reconstruction quality at all bitrates."
],
[
"Entropy Bounds for Tasks",
"Machine vision tasks rely on features at different levels of granularity.",
"High-level tasks prefer more discriminative and compact features , while low-level tasks need abundant pixels for fine modeling.",
"As VCM is to explore the collaborative compression and intelligent analytics over multiple tasks, it is valuable to figure out the intrinsic relationship among a variety of tasks in a general sense.",
"There's some preliminary work to reveal the connection between typical vision tasks [143].",
"However, there is no theoretical evidence to measure the information associated with any given task.",
"We may resort to extensive experiments on the optimal compression ratio vs. the desired performance for each specific task.",
"But the empirical study hinders the collaborative optimization across multiple tasks due to the complex objective and the heavy computational cost.",
"Moreover, the proposed VCM solution benefits from the incorporation of collaborative and scalable feedback over tasks.",
"How to mathematically formulate the connection of different tasks helps to pave the path to the completeness in theory on the feedback mechanism in VCM.",
"In particular, the theoretical study on entropy bounds for tasks is important for VCM to improve the performance and efficiency for machine and human vision in a broad range of AI applications."
],
[
"Bio-Inspired Data Acquisition and Coding",
"Recently, inspired by the biological mechanism of human vision, researchers invent the bio-inspired spike camera to continuously accumulate luminance intensity and launch spikes when reaching the dispatch threshold.",
"The spike camera brings about a new capacity of capturing the fast-moving scene in a frame-free manner while reconstructing full texture, which provides new insights into the gap between human vision and machine vision, and new opportunities for addressing the fundamental scientific and technical issues in video coding for machines.",
"Valuable works have been done to investigate the spike firing mechanism [157], spatio-temporal distribution of the spikes [158], and lossy spike coding framework for efficient spike data coding [159].",
"The advance of the spike coding shows other potentials for VCM."
],
[
"Domain Shift in Prediction and Generation",
"By employing the data-driven methods, the VCM makes more compact and informative features.",
"The risk is that those methods might be trapped in the over-fitting due to the domain shift problem.",
"Targeting more reliable VCM, how to improve the domain generalization of the prediction and generation models, and how to realize the domain adaptation (say, via online learning) are important topics."
],
[
"Conclusion",
"As a response to the emerging MPEG standardization efforts VCM, this paper formulates a new problem of video coding for machines, targeting the collaborative optimization of video and feature coding for human and/or machine visions.",
"Potential solutions, preliminary results, and future direction of VCM are presented.",
"The state-of-the-art video coding, feature coding, and general learning approaches from the perspective of predictive and generative models, are reviewed comprehensively as well.",
"As an initial attempt in identifying the roles and principles of VCM, this work is expected to call for more evidence of VCM from both academia and industry, especially when AI meets the big data era."
]
] | 2001.03569 |
[
[
"Gaussian processes with Volterra kernels"
],
[
"Abstract We study Volterra processes $X_t = \\int_0^t K(t,s) dW_s$, where $W$ is a standard Wiener process, and the kernel has the form $K(t,s) = a(s) \\int_s^t b(u) c(u-s) du$.",
"This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index $H>1/2$.",
"We establish smoothness properties of $X$, including continuity and Holder property.",
"It happens that its Holder smoothness is close to well-known Holder smoothness of fBm but is a bit worse.",
"We give a comparison with fBm for any smoothness theorem.",
"Then we investigate the problem of inverse representation of $W$ via $X$ in the case where $c\\in L^1[0,T]$ creates a Sonine pair, i.e.",
"there exists $h\\in L^1[0,T]$ such that $c * h = 1$.",
"It is a natural extension of the respective property of fBm that generates the same filtration with the underlying Wiener process.",
"Since the inverse representation of the Gaussian processes under consideration are based on the properties of Sonine pairs, we provide several examples of Sonine pairs, both well-known and new.",
"Key words: Gaussian process, Volterra process, Sonine pair, continuity, Holder property, inverse representation."
],
[
"Introduction",
"Among various classes of Gaussian processes, consider the class of the processes admitting the integral representation via some Wiener process.",
"Such processes arise in finance, see e.g.",
"[4].",
"They are the natural extension of fractional Brownian motion (fBm) which admits the integral representation via the Wiener process, and the Volterra kernel of its representation consists of power functions.",
"The solution of many problems related to fBm is based on the Hölder properties of its trajectories.",
"Therefore it is interesting to consider the smoothness properties of Gaussian processes admitting the integral representation via some Wiener process, with the representation kernel that generalizes the kernel in the representation of fBm.",
"The next question is what properties should the kernel have in order for the Wiener process and the corresponding Gaussian process to generate the same filtration.",
"It turned out that the functions in the kernel should form, in a specific way, so called Sonine pair, property that the components of the kernel generating fBm have.",
"Thus, the properties of the Gaussian process turned out to be directly related to the analytical properties of the generating kernel.",
"The present work is devoted to the study of these properties.",
"It is organized as follows.",
"Section is devoted to the smoothness properties of the Gaussian processes generated by Volterra kernels.",
"Assumptions which supply the existence and continuity of the Gaussian process are provided.",
"Then the Hölder properties are established.",
"They have certain features.",
"Namely, under reasonable assumptions on the kernel we can establish only Hölder property up to order $1/2$ while fBm with Hurst index $H$ has Hölder property of the trajectories up to order $H$ , and for $H>1/2$ (exactly the case from which we start) fBm has better smoothness properties.",
"In this connection, we establish the conditions of smoothness that is comparable with the one for fBm, but only on any interval separated from zero.",
"Finally, we establish the conditions on the kernel supplying Hölder property at zero.",
"Section describes how the generalized fractional calculus related to a Volterra process with Sonine kernel can be used to invert the corresponding covariance operator.",
"Section contains examples of Sonine pairs, and Section contains all necessary auxiliary results."
],
[
"Gaussian Volterra processes and their smoothness properties",
"Let $(\\Omega , \\mathcal {F}, \\mathbf {F}=\\lbrace \\mathcal {F}_t, t\\ge 0\\rbrace , \\mathbf {P})$ be a stochastic basis with filtration, and let $W=\\lbrace W_t, t\\ge 0\\rbrace $ be a Wiener process adapted to this filtration.",
"Consider a Gaussian process of the form $X_t = \\int _0^t K(t,s) dW_s$ where $K\\in L^2([0,T]^2)$ is a Volterra kernel, i.e.",
"$K(t,s) = 0$ for $s>t$ .",
"Obviously, $X$ is also adapted to the filtration $\\mathbf {F}$ .",
"Recall that a very common example of such process is a fractional Brownian motion (fBm) with Hurst index $H$ , i.e., a Gaussian process $B^H=\\lbrace B^H_t, t\\ge 0\\rbrace $ , admitting a representation $B^H_t=\\int _0^t K(t,s) \\, dW_s,$ with some Wiener process $W$ and Volterra kernel $\\begin{aligned}K(t,s) &= c_Hs^{1/2 - H}\\Big ((t (t-s))^{H-1/2} - (H- 1/2)\\int _s^t u^{H-3/2} (u-s)^{H-1/2}\\, du \\Big )\\operatorname{\\mathbb {1}}_{0<s<t},\\end{aligned}$ where $c_H = \\left(\\frac{2 H \\,\\Gamma (\\frac{3}{2} - H)}{\\Gamma (H+\\frac{1}{2})\\,\\Gamma (2-2H)}\\right)^{1/2}.$ If $H>\\frac{1}{2}$ , then the kernel $K$ from (REF ) can be simplified to $K(t,s) =\\left(H-\\frac{1}{2}\\right)c_Hs^{1/2 - H}\\int _s^t u^{H-1/2} (u-s)^{H-3/2}\\, du.$ Now, motivated by a fractional Brownian motion with $H>1/2$ , we assume that the kernel in the representation (REF ) is given by $K(t,s) = a(s) \\int _s^t b(u) \\, c(u-s) du,$ where $a,b,c:[0,T]\\rightarrow \\mathbb {R}$ are some measurable functions.",
"Since many applications of fBm are based on its smoothness properties, we consider what properties of functions $a,b,c$ provide a certain smoothness of the process $X$ which, in the case under consideration, takes the form $X_t = \\int _0^t \\left(a(s) \\int _s^t b(u) \\, c(u-s) \\, du \\right)dW_s, \\; t\\in [0,T].$ Our first goal is to investigate the assumptions which supply the existence and continuity of process $X$ .",
"Considering $L$ -spaces, we put, as is standard, $1/\\infty =0$ and $1/0=\\infty .$ Assume that (K2) (K1) $a\\in L^{p}[0,T]$ , $b\\in L^{q}[0,T]$ , and $c\\in L^{r}[0,T]$ for $p\\in [2,\\infty ]$ , $q\\in [1,\\infty ]$ , $r\\in [1,\\infty ]$ , such that $1/p + 1/q + 1/r \\le \\frac{3}{2}$ .",
"Then $\\sup _{t\\in [0,T]} \\left\\Vert K(t,\\,\\cdot \\,)\\right\\Vert _{L^2[0,t]}<\\infty ,$ which means that the process $X$ is well defined.",
"If, in addition, $1/p + 1/r < \\frac{3}{2}$ , then the process $X$ has a continuous modification.",
"In the case of fBm with $H>1/2$ we have $a(t) = \\left(H-\\frac{1}{2}\\right) c_H t^{1/2-H}$ ,$b(t) = t^{H-1/2}$ and $c(t) = t^{H-3/2}$ .",
"Therefore, $p$ can be any number such that$\\frac{1}{2}\\mathbin {>}\\frac{1}{p}\\mathbin {>}H{-}\\frac{1}{2}$ , $q$ can be any number from $[1,\\infty ]$ , and $r$ can be any number such that $1 > \\frac{1}{r}> \\frac{3}{2}-H$ .",
"It means that both conditions of Theorem are satisfied if we put $\\frac{1}{p}=H-\\frac{1}{2}+\\frac{\\varepsilon }{3}$ , $\\frac{1}{q}=\\frac{\\varepsilon }{3}$ and $\\frac{1}{r}= \\frac{3}{2}-H+\\frac{\\varepsilon }{3}$ , where $0 \\mathbin {<} \\epsilon \\mathbin {<} \\min \\bigl (3\\bigl (H-\\frac{1}{2}\\bigr ), \\: 3(1-H), \\: \\frac{1}{2}\\bigr )$ .",
"For both statements, without loss of generality, we can assume that $1/q + 1/r \\ge 1$ .",
"Considering statement 2) we can assume that $q<\\infty $ .",
"1) Extend the functions $a$ , $b$ , $c$ onto the entire set $\\mathbb {R}$ assuming $a(s) = b(s) =\\linebreak c(s) = 0$ for all $s\\notin [0,T]$ .",
"Extend the kernel $K(t,s)$ assuming $K(t,s) = 0$ for $s\\notin [0,t]$ .Then we have $K(t,s) = a(s) \\, (b \\operatorname{\\mathbb {1}}_{[0,t]} * \\tilde{c})(s),\\quad \\mbox{for all} \\quad 0\\le t \\le T,\\quad s \\in \\mathbb {R},$ where $\\tilde{c}(v) = c(-v)$ .",
"By Young's convolution inequality (REF ) $\\Vert b \\operatorname{\\mathbb {1}}_{[0,t]} * \\tilde{c}\\Vert _{(1/q + 1/r - 1)^{-1}}\\le \\Vert b \\operatorname{\\mathbb {1}}_{[0,t]}\\Vert _q \\,\\Vert \\tilde{c}\\Vert _r \\le \\Vert b\\Vert _q \\, \\Vert c\\Vert _r.$ (Here we applied inequality $1/q + 1/r \\ge 1$ .)",
"By Hölder inequality (REF ) for non-conjugate exponents $\\Vert K(t,\\,\\cdot \\,)\\Vert _{ (1/p + 1/q + 1/r - 1)^{-1}}&=\\Vert a \\, (b \\operatorname{\\mathbb {1}}_{[0,t]} * \\tilde{c})\\Vert _{ (1/p+1/q+1/r-1)^{-1}} \\nonumber \\\\&\\le \\Vert a\\Vert _p \\,\\Vert b \\operatorname{\\mathbb {1}}_{[0,t]} * \\tilde{c}\\Vert _{(1/q+1/r-1)^{-1}}\\le \\Vert a\\Vert _p \\,\\Vert b\\Vert _q \\,\\Vert c\\Vert _r .$ Hence $K(t,\\,\\cdot \\,) \\in L^{ (1/p + 1/q + 1/r - 1)^{-1}}[0,t]$ .",
"Since $(1/p + 1/q + 1/r - 1)^{-1} > 2$ , we conclude that $K(t,\\,\\cdot \\,) \\in L^2[0,t]$ , and it follows from (REF ) that the norms are uniformly bounded.",
"It completes the proof of the first statement.",
"2) Let $0\\le t_1 < t_2 \\le T$ .",
"It follows from (REF ) that $K(t_2,s) - K(t_1,s) =a(s)\\, (b\\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}) (s),\\qquad s\\in \\mathbb {R}.$ Similarly to (REF ) and (REF ), $\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}\\Vert _{(1/q+1/r-1)^{-1}}\\le \\Vert b \\operatorname{\\mathbb {1}}{(t_1,t_2]}\\Vert _q \\,\\Vert \\tilde{c}\\Vert _r \\le \\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\Vert _q \\, \\Vert c\\Vert _r,$ and $\\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _{(1/p + 1/q + 1/r - 1)^{-1}}=\\Vert a \\, (b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c})\\Vert _{(1/p + 1/q + 1/r - 1)^{-1}} \\\\\\le \\Vert a\\Vert _p \\,\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}\\Vert _{(1/q+1/r-1)^{-1}}\\le \\Vert a\\Vert _p \\,\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\Vert _q \\,\\Vert c\\Vert _r.$ Notice that $2 < (1/p + 1/q + 1/r - 1)^{-1}$ , and the function $K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)$ is zero-valued outside the interval $[0,t_2]$ .",
"Apply the inequality (REF ) between the norms in $L^2[0,t_2]$ and $L^{(1/p + 1/q +1/r - 1)^{-1}}[0,t_2]$ : $\\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _2&\\le \\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _{(1/p + 1/q + 1/r - 1)^{-1}}t_2^{\\frac{3}{2} - 1/p - 1/q - 1/r}\\\\ &\\le \\Vert a\\Vert _p \\,\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\Vert _q \\,\\Vert c\\Vert _rt_2^{\\frac{3}{2} - 1/p - 1/q - 1/r}\\le C \\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\Vert _q,$ with $C = T^{\\frac{3}{2} - 1/p - 1/q - 1/r}\\Vert a\\Vert _p \\, \\Vert c\\Vert _r$ .",
"Hence $\\mathrm {E}\\left[\\, (X_{t_2} - X_{t_1})^2\\,\\right] &=\\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _2^2\\le C^2 \\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\Vert _q^2\\\\ &=C^2 \\left( \\int _{t_1}^{t_2} |b(s)|^q ds\\right)^{\\!2/q}= (F(t_2) - F(t_1))^{2/q},$ where $F(t) = C^q \\int _0^{t} |b(s)|^q\\, ds$ is a nondecreasing function.",
"By Lemma REF , the process $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ has a continuous modification.",
"Now, let us establish the conditions supplying Hölder properties of $X$ .",
"Assume that $a \\in L^p [0,T]$ , $b \\in L^q [0,T]$ , and $c \\in L^r [0,T]$ with $ p\\in [2,\\infty ]$ , $ q\\in (1,\\infty ]$ , $ r\\in [1,\\infty ]$ , so that $1/p + 1/r \\ge \\frac{1}{2}$ and $1/p + 1/q + 1/r < \\frac{3}{2}$ .",
"Then the stochastic process $X$ defined by (REF ) has a modification satisfying Hölder condition up to order $\\frac{3}{2} - 1/p - 1/q - 1/r$ .",
"As it was mentioned in Remark , in the case of fractional Brownian motion, for any small positive $\\varepsilon $ , we have chose $p$ , $q$ and $r$ so that $1\\le 1/p + 1/q+1/r \\le 1+\\varepsilon $ .",
"Therefore in conditions of Lemma we get for fBm Hölder property only up to order $1/2$ while in reality we know Hölder property up to order $H>1/2$ .",
"Extend the functions $a$ , $b$ , $c$ and $K(t,s)$ as it was done in the proof of Theorem .",
"Let $0 \\le t_1 < t_2 \\le T$ .",
"We are going to find an upper bound for $\\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _2$ using a representation (REF ).",
"By Hölder inequality for non-conjugate exponents (REF ), $\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} \\Vert _{ (\\frac{3}{2} - 1/p - 1/r)^{-1}} \\le \\Vert b\\Vert _q \\, \\Vert \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\Vert _{ (\\frac{3}{2} - 1/p - 1/q - 1/r)^{-1}}=\\Vert b\\Vert _q (t_2 - t_1)^{\\frac{3}{2} - 1/p - 1/q - 1/r} .$ Here we use that $1/p + 1/q + 1/r \\le \\frac{3}{2}$ .",
"By Young's convolution inequality (REF ), $\\Vert b \\operatorname{\\mathbb {1}}_{[t_1,t_2]} * \\tilde{c}\\Vert _{ (\\frac{1}{2} - 1/p)^{-1}} &\\le \\Vert b \\operatorname{\\mathbb {1}}_{[t_1,t_2]} \\Vert _{(\\frac{3}{2} - 1/p - 1/r)^{-1}}\\Vert \\tilde{c} \\Vert _r \\\\&\\le \\Vert b\\Vert _q \\Vert c\\Vert _r(t_2 - t_1)^{\\frac{3}{2} - 1/p - 1/q - 1/r} .$ Here $\\tilde{c}(v) = c(-v)$ ; we used inequalities $r\\ge 1$ , $\\frac{1}{2} \\le 1/p + 1/r < \\frac{3}{2}$ so $(\\frac{3}{2} - 1/p - 1/r)^{-1} \\ge 1$ , and $p \\ge 2$ , so $(\\frac{1}{2} - 1/p)^{-1} \\ge 2$ .",
"Again, by Hölder inequality for non-conjugate exponents, $\\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _2&=\\Vert a\\,(b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c})\\Vert _2\\le \\Vert a\\Vert _p\\,\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}\\Vert _{1/(\\frac{1}{2} - 1/p)}\\nonumber \\\\ & \\le \\Vert a\\Vert _p \\, \\Vert b\\Vert _q \\, \\Vert c\\Vert _r \\,(t_2 - t_1)^{\\frac{3}{2} - 1/p - 1/q - 1/r} .$ Hence $\\mathrm {E}\\left[\\, (X_{t_2} - X_{t_1})^2\\,\\right] &=\\Vert K(t_2,\\,\\cdot \\,) - K(t_1,\\,\\cdot \\,)\\Vert _2^2\\\\ &\\le \\Vert a\\Vert ^2_p\\, \\Vert b\\Vert ^2_q\\, \\Vert c\\Vert ^2_r\\,(t_2 - t_1)^{3 - 2(1/p + 1/q + 1/r)}.$ By Corollary REF , the process $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ has a modification that satisfies Hölder condition up to order $\\frac{3}{2} - 1/p - 1/q - 1/r$ .",
"The following statement follows, to some extent, from Lemma .",
"Now we drop the condition $1/p + 1/r \\ge \\frac{1}{2}$ , and simultaneously relax the assertion of the mentioned lemma.",
"Let $a \\in L^{p} [0,T]$ , $b \\in L^{q} [0,T]$ , and $c \\in L^{r} [0,T]$ with $p \\in [2, \\infty ]$ , $q \\in (1, \\infty ]$ , and $r \\in [1, \\infty ]$ , which satisfy the inequality $1/p + 1/q + 1/r < \\frac{3}{2}$ .",
"Then the stochastic process $X$ defined in (REF ) has a modification that satisfies Hölder condition up to order $\\frac{3}{2} - 1/q - \\max (\\frac{1}{2}, \\: 1/p + 1/r)$ .",
"For the fBm with Hurst index $H\\in \\bigl (\\frac{1}{2}, 1\\bigr )$ and functions $a$ , $b$ and $c$ and exponents $p$ , $q$ and $r$ defined in Remark , Theorem provides Hölder condition up to order $\\frac{3}{2} - \\frac{\\epsilon }{3} -\\max \\bigl (\\frac{1}{2}, \\: 1 + \\frac{2\\epsilon }{3}\\bigr ) = \\frac{1}{2} - \\epsilon $ .",
"However, since conditions of Lemma holds true in this case, Lemma gives the same result.",
"Let $r^{\\prime } = \\left(\\max (1/r,\\: \\frac{1}{2} - 1/p)\\right)^{-1}$ .",
"Then $r^{\\prime } \\in [1, + \\infty ]$ , $r^{\\prime } \\le r$ , $c \\in L^{r^{\\prime }}[0,T]$ , $1/p + 1/r^{\\prime } \\ge \\frac{1}{2}$ , $1/p + 1/q + 1/r^{\\prime } < \\frac{3}{2}$ .",
"Applying Lemma to the functions $a$ , $b$ , $c$ and exponents $p$ , $q$ and $r^{\\prime }$ , we obtain that the process $X$ has a modification that satisfies Hölder condition up to order $\\frac{3}{2} - 1/p - 1/q - 1/r^{\\prime } = \\frac{3}{2} - 1/q - \\max (\\frac{1}{2}, \\: 1/p + 1/r)$ .",
"Now, let us formulate stronger conditions on the functions $a$ , $b$ and $c$ , supplying better Hölder properties on any interval, “close” to $[0,T]$ , but not on the whole $[0,T]$ .",
"Let $t_1 \\ge 0$ , $t_2 \\ge 0$ and $t_1 + t_2 < T$ .",
"Let the functions $a$ , $b$ and $c$ and constants $p$ , $p_1$ , $q$ , $q_1$ , $r$ , and $r_1$ satisfy the following assumptions $a&\\in L^p[0,T]\\cap L^{p_1}[t_1,T],\\; where\\;2 \\le p \\le p_1;\\\\b&\\in L^q[0,T]\\cap L^{q_1}[t_1+t_2,T],\\; where\\;1 < q \\le q_1;\\\\c&\\in L^r[0,T]\\cap L^{r_1}[t_2,T],\\; where\\;1 \\le r \\le r_1.$ Also, let $1/p+1/q+1/r \\le \\frac{3}{2}$ , and $1/q_1 + \\max \\left(\\frac{1}{2}, \\: 1/p + 1/r_1, \\: 1/p_1 + 1/r\\right) < \\frac{3}{2}$ .",
"Then the stochastic process $\\lbrace X_t,\\; t\\in [t_1+t_2,\\:T]\\rbrace $ has a modification that satisfies Hölder condition up to order $\\frac{3}{2} - 1/q_1 - \\max \\!\\left(\\frac{1}{2}, \\: 1/p+1/r_1, \\: 1/p_1 + 1/r\\right)$ .",
"Consider the fBm with Hurst index $H \\in \\bigl (\\frac{1}{2}, 1\\bigr )$ on interval $[0, T]$ .",
"Define the functions $a$ , $b$ and $c$ and exponents $p$ , $q$ and $r$ as it is done in Remark .",
"Let $p_1 = q_1 = r_1 = 3/\\epsilon $ , where $\\epsilon $ comes from Remark , and let $t_1 = t_2 = t_0/2$ for some $t_0 \\in (0, T)$ .",
"Then the conditions of Theorem are satisfied, and, according to Theorem the fBm has a modification which satisfies Hölder condition in the interval $[t_0,\\: T]$ up to order $\\frac{3}{2} - \\frac{\\epsilon }{3}- \\max \\bigl (\\frac{1}{2}, \\:\\frac{3}{2} - H + \\frac{2\\epsilon }{3}, \\:H - \\frac{1}{2} + \\frac{2\\epsilon }{3}\\bigr )= H - \\epsilon $ .",
"This is equivalent to the fact that the fBm satisfies Hölder condition in the interval $[t_0, T]$ up to order $H$ .",
"Let us extend the function $a(s)$ , $b(s)$ , $c(s)$ and $K(t,s)$ as it was done in the proof of Theorem .",
"With this extension, (REF ) holds true for all $t\\in [0,T]$ and $s\\in \\mathbb {R}$ .",
"Denote $a_1(s) &= a(s) \\operatorname{\\mathbb {1}}_{[0,t_1)}, & b_1(s) &= b(s) \\operatorname{\\mathbb {1}}_{[t_1+t_2,\\:T]}, \\\\a_2(s) &= a(s) \\operatorname{\\mathbb {1}}_{[t_1,T]}, & c_1(s) &= c(s) \\operatorname{\\mathbb {1}}_{[t_2,\\:T]},\\\\\\tilde{c}(s) &= c(-s), &\\tilde{c}_1(s) &= c_1(-s) = c(-s) \\operatorname{\\mathbb {1}}_{[-T,\\:{-}t_2]}(s) .$ The process $\\lbrace X_t, \\; t\\in [0,T]\\rbrace $ is well-defined according to Theorem .",
"We consider the increments of the process $\\lbrace X_t, \\; t\\in [t_1 + t_2,\\:T]\\rbrace $ .",
"Let $t_3$ and $t_4$ be such that $t_1+t_2 \\le t_3 < t_4 < T$ .",
"Then $K(t_4,s) - K(t_3,s) &= a(s) \\int _{t_3}^{t_4} b(u) c(u-s) \\, du= a(s) \\int _{t_3}^{t_4} b_1(u) c(u-s) \\, du\\\\&\\hspace{176.407pt} \\mbox{for all $s\\in \\mathbb {R}$}; \\\\K(t_4,s) - K(t_3,s)&= a_1(s) \\int _{t_3}^{t_4} b_1(u) c_1(u-s) \\, du\\quad \\mbox{for} \\quad 0\\le s < t_1; \\\\K(t_4,s) - K(t_3,s) &= a_2(s) \\int _{t_3}^{t_4} b_1(u) c(u-s) \\, du\\quad \\mbox{for} \\quad t_1 \\le s \\le T.$ Thus, for all $s\\in \\mathbb {R}$ $K(t_4,s) - K(t_3,s)&= a_1(s) \\int _{t_3}^{t_4} b_1(u) c_1(u-s) \\, du+ a_2(s) \\int _{t_3}^{t_4} b_1(u) c(u-s) \\, du\\\\ &=a_1(s) \\, (b_1 \\operatorname{\\mathbb {1}}_{(t_3,t_4]} * \\tilde{c}_1) (s)+ a_2(s) \\, (b_1 \\operatorname{\\mathbb {1}}_{(t_3,t_4]} * \\tilde{c}) (s).$ Functions $a_1$ , $b_1$ and $c_1$ with exponents $p$ , $q_1$ and $\\left(\\max (1/r_1,\\:\\frac{1}{2} - 1/p)\\right)^{-1}$ satisfy conditions of Lemma .",
"Functions $a_2$ , $b_1$ and $c$ with exponents $p_1$ , $q_1$ and $\\left(\\max (1/r,\\:\\frac{1}{2} - 1/p_1)\\right)^{-1}$ also satisfy conditions of Lemma .",
"By inequality (REF ) in the proof of Lemma , $\\Vert a_1 \\, (b_1 \\operatorname{\\mathbb {1}}_{(t_3,t_4]} * \\tilde{c}_1)\\Vert _2 \\le \\Vert a_1\\Vert _p \\, \\Vert b_1\\Vert _{q_1} \\,\\Vert c_1\\Vert _{1/{\\max (1/r_1,\\:\\frac{1}{2} - 1/p)}} \\,(t_4 - t_3)^{\\lambda _1}, \\\\\\Vert a_2 \\, (b_1 \\operatorname{\\mathbb {1}}_{(t_3,t_4]} * \\tilde{c})\\Vert _2 \\le \\Vert a_2\\Vert _{p_1} \\, \\Vert b_1\\Vert _{q_1} \\,\\Vert c\\Vert _{1/{\\max (1/r,\\:\\frac{1}{2} - 1/p_1)}} \\,(t_4 - t_3)^{\\lambda _2}$ where $\\lambda _1 = \\frac{3}{2} - \\frac{1}{p} - \\frac{1}{q_1} - \\max \\!\\left(\\frac{1}{r_1},\\:\\frac{1}{2} - \\frac{1}{p}\\right)= \\frac{3}{2} - \\frac{1}{q_1} - \\max \\!\\left(\\frac{1}{2}, \\: \\frac{1}{p} + \\frac{1}{r_1}\\right), \\\\\\lambda _2 = \\frac{3}{2} - \\frac{1}{p_1} - \\frac{1}{q_1} - \\max \\!\\left(\\frac{1}{r},\\:\\frac{1}{2} - \\frac{1}{p_1}\\right)= \\frac{3}{2}- \\frac{1}{q_1} - \\max \\!\\left(\\frac{1}{2}, \\: \\frac{1}{p_1} + \\frac{1}{r}\\right).$ Denote $\\lambda = \\min (\\lambda _1, \\lambda _2) =\\frac{3}{2}- \\frac{1}{q_1} - \\max \\!\\left(\\frac{1}{2}, \\: \\frac{1}{p} + \\frac{1}{r_1}, \\: \\frac{1}{p_1} + \\frac{1}{r}\\right).$ Then $\\Vert K(t_4,\\,\\cdot \\,) - K(t_3,\\,\\cdot \\,)\\Vert _2 \\le \\Vert a_1 \\, (b_1 \\operatorname{\\mathbb {1}}_{(t_3,t_4]} * \\tilde{c}_1)\\Vert _2 +\\Vert a_2 \\, (b_1 \\operatorname{\\mathbb {1}}_{(t_3,t_4]} * \\tilde{c})\\Vert _2\\le C \\, (t_4 - t_3)^\\lambda ,$ where $C &=\\Vert a_1\\Vert _p \\, \\Vert b_1\\Vert _{q_1} \\,\\Vert c_1\\Vert _{1/{\\max (r_1,\\:\\frac{1}{2} - 1/p)}} \\,T^{\\lambda _1-\\lambda } \\\\ &\\quad +\\Vert a_2\\Vert _{p_1} \\, \\Vert b_1\\Vert _{q_1} \\,\\Vert c\\Vert _{1/{\\max (r,\\:\\frac{1}{2} - 1/p_1)}} \\,T^{\\lambda _2-\\lambda }.$ Finally, $\\mathrm {E}\\left[\\,\\bigl (X_{t_4}-X_{t_3}\\bigr )^2\\,\\right] &\\le \\int _{t_3}^{t_4} (K(t_4,s) - K(t_3,s))^2 \\, ds\\\\ &=\\Vert K(t_4,\\,\\cdot \\,) - K(t_3,\\,\\cdot \\,)\\Vert _2^2 \\le C^2 (t_4-t_3)^{2\\lambda } .$ By Corollary REF , the stochastic process $\\lbrace X_t,\\; t\\in [t_1+t_2,\\:T]\\rbrace $ has a modification that satisfies Hölder condition up to order $\\lambda $ .",
"The next result, namely, Lemma , generalizes Lemma and Theorem .",
"It allows us to apply the mentioned lemma directly to the power functions $a(s) = s^{-1/p_0}$ and $c(s) = s^{-1/r_0}$ .",
"Let $p_0 \\in (0,{+}\\infty ]$ , $q_0 \\in (1,{+}\\infty ]$ , $r_0 \\in (0,{+}\\infty ]$ with $1/p_0 + 1/q_0 + 1/r_0 < \\frac{3}{2}$ .",
"Also, for any $p\\in (0,p_0)$ let $a \\in L^{\\max (2, p)} [0,T]$ , for any $q\\in [1,q_0)$ let $b \\in L^q [0,T]$ , and for any $r\\in (0,r_0)$ let $c \\in L^{\\max (1, r)} [0,T]$ .",
"Then the stochastic process $X$ defined in (REF ) has a modification that satisfies Hölder condition up to order $\\lambda = \\frac{3}{2} - 1/q_0 - \\max (\\frac{1}{2}, \\: 1/p_0 + 1/r_0)$ .",
"In Remark we applied Lemma and obtained that the fBm with Hurst index $H > \\frac{1}{2}$ has a modification that satisfies Hölder condition up to order $\\frac{1}{2}$ .",
"With Lemma , we can obtain the same result more easily.",
"We just apply Lemma for $p_0 = \\left(H-\\frac{1}{2}\\right)^{\\!-1}\\!$ , $q_0 = \\infty $ and $r_0 = \\left(\\frac{3}{2} - H\\right)^{\\!-1}$ and do not bother with $\\epsilon $ .",
"Notice that $0 < \\lambda \\le 1$ .",
"Denote $A = \\left\\lbrace m \\in \\mathbb {N} \\, : \\,m > \\max \\bigl (3, \\frac{\\lambda q_0}{q_0-1}\\bigr )\\right\\rbrace $ a set of “large enough” positive integers.",
"Let $n \\in A$ .",
"Let $p_n$ , $q_n$ and $r_n$ be such real numbers that $1/p_n = \\min (\\frac{1}{2}, \\: 1/p_0 + \\lambda /n)$ , $1/q_n = 1/q_0 + \\lambda /n$ , and $1/r_n = \\min (1, \\: 1/r_0 + \\lambda /n)$ .",
"Then $p_n \\in \\bigl [\\frac{1}{2}, \\infty \\bigr )$ , $q_n \\in (1,\\infty )$ , $r_n \\in [1,\\infty )$ , and $1/p_n + 1/q_n + 1/r_n < \\frac{3}{2}$ .",
"Apply Lemma for functions $a$ , $b$ , $c$ and exponents $p_n$ , $q_n$ and $r_n$ .",
"By Lemma , the process $X$ has a modification $X^{(n)}$ that satisfies Holder condition up to order $\\frac{3}{2} - 1/q_n - \\max (\\frac{1}{2},\\: 1/p_n + 1/r_n)\\ge (n-3) \\lambda / n$ .",
"For different $n\\in A$ , the processes $X^{(n)}$ coincide almost surely on $[0,T]$ .",
"Let $B$ be a random event which occurs when all these processes coincide: $B = \\lbrace \\forall m\\mathbin {\\in } A \\;\\,\\forall n\\mathbin {\\in } A \\;\\,\\forall t\\mathbin {\\in }[0,T] \\; : \\;X^{(m)}_t = X^{(n)}_t\\rbrace .$ Then $\\mathrm {P}(B) = 1$ , and $\\widetilde{X} = X^{(k)} \\operatorname{\\mathbb {1}}_B$ (where $k = \\min A$ is the least element of the set $A$ ) is a modification of $X$ that satisfies Hölder condition up to order $\\lambda $ .",
"Let $a\\in L^p[0,T]$ , $b\\in L^q[0,T]$ , $c\\in L^r[0,T]$ , where the exponents satisfy relations $ p\\in [2, \\infty ] $ , $ q\\in [1, \\infty ) $ , $ r\\in [1, \\infty ] $ , and $1/p + 1/q + 1/r \\le \\frac{3}{2}$ .",
"Let there exist $\\lambda >0$ and $C\\in \\mathbb {R}$ such that $\\forall t\\in [0,T]\\; : \\;0 \\le \\Vert b\\operatorname{\\mathbb {1}}_{[0,t]}\\!\\Vert _q \\le C t^\\lambda .$ Then the stochastic process $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ has a modification which is continuous on $[0,T]$ and satisfies Hölder condition at point 0 up to order $\\lambda $ .",
"For the fBm with Hurst index $H>\\frac{1}{2}$ , apply Lemma to the functions $a$ , $b$ and $c$ defined in Remark , but for exponents $1/p = H - \\frac{1}{2} + \\frac{\\epsilon }{2}$ , $1/q = \\frac{1}{2} - \\epsilon $ , and $1/r = \\frac{3}{2} - H + \\frac{\\epsilon }{2}$ for some $\\epsilon $ such that $0 < \\epsilon < \\min \\bigl (2(1-H), \\: \\frac{1}{2}, \\: 2\\left(H-\\frac{1}{2}\\right) \\bigr )$ .",
"Verify the conditions of Lemma .",
"We have $H - \\frac{1}{2} < 1/p < \\frac{1}{2}$ , $0 < 1/q < 1$ , $\\frac{3}{2} - H < 1/r < 1$ (whence $a\\in L^p[0,T]$ and $c\\in L^r[0,T]$ ; the relation $b\\in L^q[0,T]$ holds true for all $q\\ge 1$ ) and $1/p + 1/q + 1/r = \\frac{3}{2}$ .",
"Moreover, $\\Vert b \\operatorname{\\mathbb {1}}_{[0,t]}\\Vert _q= C_\\epsilon t^{H-1/2+1/q}$ , where $C_\\epsilon = ((H-\\frac{1}{2})q + 1)^{-1/q}$ .",
"According to Lemma , the fBm satisfies Hölder condition at point 0 up to order $H-\\frac{1}{2}+1/q = H - \\epsilon $ .",
"As this can be proved for any $\\epsilon >0$ small enough, the fBm satisfies Hölder condition at point 0 up to order $H$ .",
"Without loss of generality we can assume that $1/q + 1/r \\ge 1$ .",
"Indeed, under original conditions of the lemma, let $r^{\\prime } = \\min (r,\\: q / (q-1))$ .",
"Then $1 \\le r^{\\prime } \\le r$ , $1/q + 1/r^{\\prime } \\ge 1$ , $1/p + 1/q + 1/r^{\\prime } \\le \\frac{3}{2}$ , and $c\\in L^{r^{\\prime }}[0,T]$ .",
"The inequality $1/p + 1/q + 1/r^{\\prime } \\le \\frac{3}{2}$ can be proved as follows: $\\frac{1}{p} + \\frac{1}{q} + \\frac{1}{r^{\\prime }} = \\frac{1}{p} + \\frac{1}{q} + \\frac{1}{r} \\le \\frac{3}{2} \\quad \\mbox{if} \\quad r \\le \\frac{q}{q-1};\\\\\\frac{1}{p} + \\frac{1}{q} + \\frac{1}{r^{\\prime }} = \\frac{1}{p} + \\frac{1}{q} + \\frac{q-1}{q} = \\frac{1}{p} + 1 \\le \\frac{1}{2} + 1 = \\frac{3}{2}\\quad \\mbox{if} \\quad r \\ge \\frac{q}{q-1}.$ The other relations can be proved easily.",
"Thus, after substitution of $r^{\\prime }$ for $r$ all conditions of Lemma still hold true, as well as $1/q + 1/r \\ge 1$ .",
"Denote $F(t) = \\int _0^t |b(s)|^q \\, dt + t^{\\lambda q}.$ Then $F:[0,T] \\rightarrow [0,+\\infty )$ is a strictly increasing function such that $F(0) = 0, \\qquad \\qquad F(t) \\le C_1 t^{\\lambda q}\\quad \\mbox{if\\quad $0\\le t \\le T$},\\\\\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\!\\Vert _q< (F(t_2) - F(t_1))^{1/q}\\quad \\mbox{if\\quad $0\\le t_1 < t_2 \\le T$}.$ Let $0 \\le t_1 < t_2 \\le T$ .",
"Again, denote $\\tilde{c}(v) = c(-v)$ .",
"Let us construct an upper bound for $\\Vert K_1(t_2,\\,\\cdot \\,) - K_1(t_1,\\,\\cdot \\,)\\Vert _2 =\\Vert a \\, (b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}) \\Vert _2$ , see (REF ).",
"By Young's convolution inequality (REF ), $\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}\\Vert _{ (1/q+1/r-1)^{-1}} \\le \\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]}\\!\\Vert _q \\, \\Vert \\tilde{c}\\Vert _r\\le (F(t_2) - F(t_1))^{1/q} \\, \\Vert c\\Vert _r .$ Here we used that $q\\ge 1$ , $r \\ge 1$ and $1/r+1/q \\ge 1$ .",
"The function $a \\, (b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c})$ is equal to 0 outside the interval $[0, t_2]$ .",
"Noticing that $2 \\le (1/p + 1/q + 1/r - 1)^{-1}$ , using the inequality (REF ) for norms in $L^2[0, t_2]$ and $L^{(1/p + 1/q + 1/r - 1)^{-1}}[0, t_2]$ and Hölder inequality for non-conjugate exponents (REF ), we get $\\Vert a \\, (b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}) \\Vert _2&\\le \\Vert a \\, (b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}) \\Vert _{ (1/p+1/q+1/r-1)^{-1}}\\,t_2^{\\frac{3}{2} - 1/p - 1/q - 1/r}\\\\ &\\le \\Vert a\\Vert _p \\,\\Vert b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}\\Vert _{ (1/q+1/r-1)^{-1}}\\,t_2^{\\frac{3}{2} - 1/p - 1/q - 1/r}\\\\ &\\le \\Vert a\\Vert _p \\,(F(t_2) - F(t_1))^{1/q} \\, \\Vert c\\Vert _r \\,t_2^{\\frac{3}{2} - 1/p - 1/q - 1/r}.$ Hence $\\mathrm {E}\\left[\\,\\bigl (X_{t_2}-X_{t_1}\\bigr )^2\\,\\right] &= \\Vert K(t_2,\\,\\cdot \\,)-K(t_1,\\,\\cdot \\,)\\Vert _2^2= \\Vert a \\, (b \\operatorname{\\mathbb {1}}_{(t_1,t_2]} * \\tilde{c}) \\Vert _2^2 \\\\&\\le \\Vert a\\Vert _p^2 \\,(F(t_2) - F(t_1))^{2/q} \\, \\Vert c\\Vert _r^2 \\,t_2^{3 - 2(1/p + 1/q + 1/r)} .$ Consider stochastic process $Y = \\lbrace Y_s : s\\in [0, F(T)]\\rbrace $ , with $Y_{F(t)} = X_t$ for all $t\\in [0,T]$ .",
"This process $Y$ satisfies inequality $\\mathrm {E}\\left[\\,\\bigl (Y_{s_2}-Y_{s_1}\\bigr )^2\\,\\right]\\le \\Vert a\\Vert _p^2 \\,(s_2 - s_1)^{2/q} \\, \\Vert c\\Vert _r^2T^{3 - 2(1/p + 1/q + 1/r)}\\quad \\mbox{if $0\\le s_1 < s_2 \\le F(T)$}.$ By Corollary REF , the process $Y$ has a modification $\\widetilde{Y}$ that satisfies Hölder condition up to order $1/q$ .",
"Therefore, for any $\\lambda _1 \\in (0, \\lambda )$ $\\exists C_2 \\; \\forall s_1 \\!",
"\\in [0,F(T)] \\;\\,\\forall s_2 \\mathrel {\\in } [0,F(T)] \\; : \\;|\\widetilde{Y}_{s_2} - \\widetilde{Y}_{s_1}| \\le C_2 \\, |s_2 - s_1|^{\\lambda _1 / (\\lambda q)} ,$ where $C_2$ is a random variable; $C_2 < \\infty $ surely.",
"In particular, $\\exists C_2 \\;\\forall s \\mathrel {\\in } [0,F(T)] \\; : \\;|\\widetilde{Y}_{s} - \\widetilde{Y}_0| \\le C_2 \\, s^{\\lambda _1 / (\\lambda q)} .$ The stochastic process $\\widetilde{X} = \\lbrace \\widetilde{X}_t, \\; t \\in [0,T]\\rbrace = \\lbrace \\widetilde{Y}_{F(t)}, \\; t\\in [0,T]\\rbrace $ is a modification of the stochastic process $X$ .",
"It satisfies inequalities $\\exists C_2 \\; \\forall t \\mathrel {\\in } [0,T] \\; : \\;|\\widetilde{X}_t - \\widetilde{X}_0| \\le C_2 \\, F(t)^{\\lambda _1 / (\\lambda q)} ; \\\\\\exists C_3 \\; \\forall t \\mathrel {\\in } [0,T] \\; : \\;|\\widetilde{X}_t - \\widetilde{X}_0| \\le C_3 \\, t^{\\lambda _1} .$ Thus, all the paths of the stochastic process $\\widetilde{X}$ satisfy Hölder condition at point 0 with exponent $\\lambda _1$ .",
"Gaussian Volterra processes with Sonine kernels Fractional Brownian motion and Sonine kernels Consider now a natural question: for which kernels $K$ of the form (REF ) Gaussian process of the form (REF ) with Volterra kernel $K$ generates the same filtration as the Wiener process $W$ .",
"Sufficient condition for this is the representation of the Wiener process $W$ as $W_t = \\int _0^t L(t,s) \\, dX_s$ where $L\\in L^2([0,T]^2)$ is a Volterra kernel, and the integral is well defined, in some sense.",
"As an example, let us consider fractional Brownian motion $B^H, H>1/2 $ admitting a representation (REF ) with Volterra kernel (REF ).",
"For any $0<\\varepsilon <1$ consider the approximation $B^{H,\\varepsilon }_t=d_H\\int _0^t\\left(s^{1/2 - H}\\int _s^t u^{H-1/2} (u-\\varepsilon s)^{H-3/2}\\, du\\right) dW_s, t\\ge 0.$ Unlike the original process, in such approximation we can change the limits of integration and get that $B^{H,\\varepsilon }_t=d_H\\int _0^t \\left(u^{H-1/2}\\int _0^us^{1/2 - H}(u-\\varepsilon s)^{H-3/2}dW_s\\, \\right)du.$ This representation allows to write the equality $\\int _0^tu^{1/2-H}dB^{H}_u=d_H\\int _0^t \\left( \\int _0^us^{1/2 - H}(u-\\varepsilon s)^{H-3/2}dW_s\\, \\right)du,$ and it follows immediately from (REF ) that ${\\begin{array}{c}\\int _0^t(t-u)^{1/2-H}u^{1/2-H}dB^{H,\\varepsilon }_u\\hspace{128.0374pt}\\\\=d_H\\int _0^t (t-u)^{1/2-H}\\left( \\int _0^us^{1/2 - H}(u-\\varepsilon s)^{H-3/2}dW_s\\, \\right)du\\\\= d_H\\int _0^t s^{1/2 - H}\\left( \\int _s^t(t-u)^{1/2-H}(u-\\varepsilon s)^{H-3/2}du\\,\\, \\right)dW_s.\\end{array}}$ Applying Theorem 3.3 from [2], p. 160, we can go to the limit in (REF ) and get that $\\int _0^t(t-u)^{1/2-H}u^{1/2-H}dB^{H}_u=d_H\\int _0^t s^{1/2 - H}\\left( \\int _s^t(t-u)^{1/2-H}(u- s)^{H-3/2}du\\,\\, \\right)dW_s.",
"$ Now the highlight is that the integral $\\int _s^t(t-u)^{1/2-H}(u- s)^{H-3/2}\\,du$ is a constant, namely, $\\int _s^t(t{-}u)^{1/2-H}(u{-} s)^{H-3/2}du=\\int _0^t(t{-}u)^{1/2-H}u^{H-3/2}du=\\mathrm {B}(3/2{-}H, H{-}1/2)$ , where $\\mathrm {B}$ is a beta-function.",
"After we noticed this, then everything is simple: $Y_t:=\\int _0^t(t-u)^{1/2-H}u^{1/2-H}dB^{H}_u=d_H \\mathrm {B}(3/2-H,\\: H-1/2)\\int _0^t s^{1/2 - H} dW_s, $ and finally we get that $W_t=e_H\\int _0^t s^{ H-1/2} dY_s $ with some constant $e_H$ .",
"It means that we have representation (REF ) and, in particular, $W$ and $B^H$ generate the same filtration.",
"Of course, these transformations can be performed much faster, but our goal here was to pay attention on the role of the property of the convolution of two functions to be a constant.",
"This property is a characterization of Sonine kernels.",
"General approach to Volterra processes with Sonine kernels First we give basic information about Sonine kernels, more details can be found in [12].",
"We also consider, in a simplified form, the related generalized fractional calculus introduced in [6].",
"A function $c\\in L^1[0,T]$ is called a Sonine kernel if there exists a function $h\\in L^1[0,T]$ such that $\\int _0^t c(s) h(t-s) \\, ds= 1,\\quad t\\in (0,T].$ Functions $c,h$ are called Sonine pair, or, equivalently, we say that $c$ and $h$ form (or create) a Sonine pair.",
"If $\\hat{c}$ and $\\hat{h}$ denote the Laplace transforms of $c$ and $h$ respectively, then (REF ) is equivalent to $\\hat{c}(\\lambda )\\hat{h}(\\lambda ) = \\lambda ^{-1}$ , $\\lambda >0$ .",
"Since the Laplace transform characterizes a function uniquely, for any $c$ there can be not more than one function $h$ satisfying (REF ).",
"Examples of Sonine pairs are given in Section .",
"Let functions $c$ and $h$ form a Sonine pair.",
"For a function $f\\in L^1[0,T]$ consider the operator $\\mathrm {I}^c_{0+} f (t) =\\int _0^t c(t-s) f(s) ds.$ It is an analogue of forward fractional integration operator.",
"Let us identify an inverse operator.",
"In order to do this, for $g \\in AC[0,T]$ define $\\mathrm {D}^h_{0+} g(t) = \\int _0^t h(t-s)g^{\\prime }(s)ds + h(t)g(0).$ Note that $\\int _0^t \\mathrm {D}^h_{0+} g(u) du = \\int _0^t \\left(\\int _0^u h(u-s)g^{\\prime }(s)ds + h(u)g(0)\\right)\\, du \\\\ = \\int _0^t \\int _0^u h(s)g^{\\prime }(u-s)ds\\, du + g(0)\\int _0^t h(u)du\\\\=\\int _0^t h(s)\\int _s^t g^{\\prime }(u-s)du\\, ds + g(0)\\int _0^t h(u)du \\\\=\\int _0^t h(s)\\big (g(t-s)-g(0)\\big )ds\\, ds + g(0)\\int _0^t h(u)du = \\int _0^t h(s) g(t-s)ds,$ so we can also write $\\mathrm {D}^h_{0+} g(t) = \\frac{d}{dt}\\int _0^t h(s) g(t-s)ds = \\frac{d}{dt}\\int _0^t h(t-s) g(s)ds,$ where the derivative is understood in the weak sense.",
"Similarly, we can define an analogue of backward fractional integral: $\\mathrm {I}^c_{T-} f(s) = \\int _s^T c(t-s) f(t)dt, \\qquad f \\in L^1[0,T]$ and the corresponding differentiation operator $\\mathrm {D}^h_{T-} g(s) = g(T)h(T-s) - \\int _s^T h(t-s)g^{\\prime }(t)dt.$ Let $g\\in AC[0,T]$ .",
"Then $\\displaystyle \\mathrm {I}_{0+}^c\\big ( \\mathrm {D}^h_{0+} g\\big )(t) = g(t)$ and $\\displaystyle \\mathrm {I}_{T-}^c\\big ( \\mathrm {D}^h_{T-} g\\big )(s) = g(s)$ .",
"We have $\\mathrm {I}^c_{0+}\\big ( \\mathrm {D}^h_{0+} g\\big )(t)&=\\int _0^t c(t-s)\\left(\\int _0^s h(s-u) g^{\\prime }(u)du + h(s)g(0)\\right)\\, ds\\\\ &=\\int _0^t \\int _u^t c(t-s)h(s-u)ds\\, g^{\\prime }(u)du + g(0)\\int _{0}^t c(t-s)h(s)ds\\\\ & =\\int _0^t g^{\\prime }(u)du + g(0) = g(t),$ as required.",
"Similarly, $\\mathrm {I}^c_{T-}\\big ( \\mathrm {D}^h_{T-} g\\big )(s) &= \\int _s^T c(t-s)\\left(h(T-t)g(T) - \\int _t^T h(u-t) g^{\\prime }(u)du \\right) ds\\\\&= g(T)\\int _{s}^T c(t-s)h(T-t)dt - \\int _s^T \\int _s^u c(t-s)h(u-t)\\,dt\\, g^{\\prime }(u)\\, du \\\\&= g(T) - \\int _s^T g^{\\prime }(u)du + g(s) = g(s)$ as required.",
"Now consider a Gaussian process $X$ given by the integral transformation of type (REF ) of the form (REF ) satisfying condition REF of Theorem .",
"Define the integral operator $\\mathcal {K} f(t) = \\int _0^t a(s)\\int _s^t b(u)c(u-s) du\\, f(s) ds.$ Note that for $f\\in L^2[0,T]$ , $\\mathcal {K} f(t)\\in AC[0,T]$ .",
"Indeed, by definition, $\\mathcal {K} f (t) = \\int _0^t K(t,s) f(s)ds = \\int _0^t \\int _s^t \\frac{\\partial }{\\partial u}K(u,s) du\\, f(s)ds.$ Since $f$ and $\\frac{\\partial }{\\partial t}K(t,s)$ are square integrable, the product $f \\frac{\\partial }{\\partial u}K$ is integrable on $\\left\\lbrace (s,u): 0\\le s\\le u\\le t \\right\\rbrace $ .",
"Therefore, we can apply Fubini theorem to get $\\mathcal {K} f (t) = \\int _0^t \\int _0^u \\frac{\\partial }{\\partial u}K(u,s) f(s) ds\\, du = \\int _0^t \\alpha (u)du,$ where $\\alpha \\in L^1[0,t]$ for all $t\\in [0,T]$ , so $\\alpha \\in L^1[0,T]$ .",
"Consequently, for $f\\in L^2[0,T]$ we can denote by $\\mathcal {J} f(t) = \\int _0^t \\frac{\\partial }{\\partial t}K(t,s) f(s) ds$ the weak derivative of $\\mathcal {K} f$ .",
"Further, define for a measurable $g\\colon [0,T] \\rightarrow \\mathbb {R}$ such that $\\left\\Vert g\\right\\Vert _{\\mathcal {H}_X}^2 := \\int _0^T \\left(\\int _s^T \\frac{\\partial }{\\partial u} K(u,s) g(u) du\\right)^2 ds<\\infty $ the integral operator $\\mathcal {J}^* g(s) = \\int _s^T \\frac{\\partial }{\\partial u} K(u,s) g(u)du.$ It can be extended to the completion $\\mathcal {H}_X$ of the set of measurable functions with finite norm $\\left\\Vert \\cdot \\right\\Vert _{\\mathcal {H}_X}^2$ so that $\\left\\Vert g\\right\\Vert _{\\mathcal {H}_X}^2 = \\int _0^T \\bigl (\\mathcal {J}^* g (t)\\bigr )^2 dt,\\ g\\in \\mathcal {H}_X.$ The operator $\\mathcal {J}^*$ is related to the adjoint $\\mathcal {K}^*$ of $\\mathcal {K}$ in the following way: for a finite signed measure $\\mu $ on $[0,T]$ , $\\mathcal {K}^* \\mu = \\mathcal {J}^* h\\quad \\mbox{with}\\quad h(t) = \\mu ([t,T]).$ We are going to identify inverse to the operators $\\mathcal {J}$ and $\\mathcal {J}^*$ .",
"Clearly, it is not possible in general, so we will assume that (K2) (S) the function $c$ forms a Sonine pair with some $h\\in L^1[0,T]$ .",
"In this case the operators $\\mathcal {J}$ and $\\mathcal {J}^*$ can be written in terms of “fractional” operators defined above: $\\mathcal {J} f(t) = \\int _0^t \\frac{\\partial }{\\partial t}K(t,s) f(s)ds =\\int _0^t a(s) \\, b(t) \\, c(t-s) f(s) \\, ds= b(t)\\, \\mathrm {I}_{0+}^c (af)(t),$ and $\\mathcal {J}^* g(s) =\\int _s^T \\!",
"a(s) \\, b(t) \\, c(t-s) \\, g(t) \\, dt =a(s)\\,\\mathrm {I}_{T-}^c (bg)(s).$ In order for this operators to be injective, we assume (K2) (K2) the functions $a,b$ are positive a.e.",
"on $[0,T]$ .",
"For $f$ such that $fb^{-1}\\in AC[0,T]$ , define $\\mathcal {L} f(t) = a(t)^{-1}\\mathrm {D}_{0+}^h \\big (f b^{-1}\\big )(t) = a(t)^{-1}\\left(\\int _0^t h(t-s) \\big (fb^{-1}\\big )^{\\prime }(s)ds + h(t) \\big (fb^{-1}\\big )(0)\\right),$ and for $g$ such that $ga^{-1}\\in AC[0,T]$ , define $\\mathcal {L}^* g(s)&=b(s)^{-1}\\mathrm {D}_{T-}^h \\big (g a^{-1}\\big )(s)\\\\ &=b(s)^{-1}\\left(h(T-s) \\big (ga^{-1}\\big )(T) - \\int _s^T h(t-s)\\big (ga^{-1}\\big )^{\\prime }(t)dt\\right).$ Let the assumptions REF , REF and REF hold.",
"Then the operators $\\mathcal {J}$ and $\\mathcal {J}^*$ are injective, and for functions $f,g$ such that $fb^{-1}\\in AC[0,T]$ , $ga^{-1}\\in AC[0,T]$ , $\\mathcal {J} \\mathcal {L} f(t) = f(t), \\qquad \\mathcal {J}^* \\mathcal {L}^* g(s) = g(s).$ Assume that $\\mathcal {J} f = 0$ for some $f\\in L^2[0,T]$ .",
"Then, by REF , $\\mathrm {I}_{0+}^c (bf) = 0$ a.e.",
"on $[0,T]$ .",
"Therefore, for any $t\\in [0,T]$ $0 &= \\int _0^t h(t-s)\\mathrm {I}_{0+}^c (bf)(s)ds = \\int _0^t h(t-s)\\int _0^s c(s-u) b(u)f(u)du \\,ds \\\\&= \\int _0^t \\int _u^t h(t-s)c(s-u)ds\\, b(u)f(u) du = \\int _0^t b(u)f(u) du,$ whence $b f = 0$ a.e.",
"on $[0,T]$ , so, applying to REF once more, $f = 0$ a.e.",
"The injectivity of $\\mathcal {J}^*$ is shown similarly, and the second statement follows from Lemma REF .",
"Now we are in a position to invert the covariance operator $\\mathcal {R} = \\mathcal {K}\\mathcal {K}^*$ of $X$ .",
"We need a further assumption.",
"(K2) (K3) $a^{-1}\\in C^1[0,T]$ , $d:= b^{-1}\\in C^2[0,T]$ and either $d(0) = d^{\\prime }(0) = 0$ or $a^{-2}h\\in C^1[0,T]$ .",
"Let the assumptions REF , REF – REF hold, and $f\\in C^3[0,T]$ with $f(0)= 0$ .",
"Then for $h = \\mathcal {L}^*\\mathcal {L} f^{\\prime }$ , the measure $\\mu ([t,T]) = h(t)$ is such that $\\mathcal {R} \\mu = f$ .",
"Thanks to REF , $f^{\\prime }b^{-1}\\in AC[0,T]$ and $a(t)^{-1}\\mathcal {L} f^{\\prime }(t) = a(t)^{-2}\\left(\\int _0^t h(t-s) \\big (fb^{-1}\\big )^{\\prime }(s)ds + h(t) \\big (f^{\\prime }b^{-1}\\big )(0)\\right).$ Similarly to (REF ), $\\int _0^t h(t-s) \\big (fb^{-1}\\big )^{\\prime }(s)ds$ is absolutely continuous with $\\frac{d}{dt}\\int _0^t h(t-s) \\big (f^{\\prime }b^{-1}\\big )^{\\prime }(s)ds = \\int _0^t h(s) \\big (f^{\\prime }b^{-1}\\big )^{\\prime \\prime }(t-s)ds + h(t) \\big (f^{\\prime }b^{-1}\\big )^{\\prime }(0).$ Then, thanks to REF , both summands in the right-hand side of (REF ) are absolutely continuous with bounded derivatives.",
"So by Proposition REF , $\\mathcal {K}^* \\mu = \\mathcal {J}^* h = \\mathcal {J}^* \\mathcal {L}^* \\mathcal {L} f^{\\prime }= \\mathcal {L} f^{\\prime } .$ and $\\mathcal {J} \\mathcal {K}^* \\mu = \\mathcal {J} \\mathcal {L} f^{\\prime } = f^{\\prime } .$ Therefore, $\\mathcal {R} \\mu \\,(t) =\\mathcal {K} \\mathcal {K}^* \\mu \\, (t) =\\int _0^t \\mathcal {J} \\mathcal {K}^* \\mu \\, (s) \\, ds =\\int _0^t f^{\\prime }(s) \\, ds = f(t)$ as required.",
"Now we recall the definition of integral with respect to the $X$ given by (REF ); for more details see [1].",
"Define $I_X(\\mathbb {1}_{[0,t]}) = \\int _0^T \\mathbb {1}_{[0,t]}(s) \\, dX_s = X_t$ and extend this by linearity to the set $\\mathcal {S}$ of piecewise constant function.",
"Then, for any $g\\in \\mathcal {S}$ , $\\mathrm {E}\\left[\\,I_X(g)^2\\,\\right] = \\left\\Vert g\\right\\Vert ^2_{\\mathcal {H}_X}.$ Therefore, $I_X$ can be extended to isometry between $\\mathcal {H}_X$ and a subspace of $L^2(\\Omega )$ .",
"Moreover, for any $g\\in \\mathcal {H}_X$ , $\\int _0^T \\!",
"g(t) \\, dX_t = \\int _0^T \\!",
"\\mathcal {J}^* g(t) \\, dW_t.$ Let the assumptions $(\\mathrm {S})$ , $(\\mathrm {K}1)-(\\mathrm {K}3)$ be satisfied, and $X$ be given by (REF ).",
"Then $W_t = \\int _0^t k(t,s) \\, dX_s,$ where $k(t,s) = p(t)b(s)^{-1} h(t-s) - b(s)^{-1}\\int _s^t p^{\\prime }(v) h(v-s) \\, dv ,$ and $p = a^{-1}$ .",
"Write $k(t,s) = k_1(t,s) - k_2(t,s)$ , where $ k_1(t,s) = p(t)b(s)^{-1} h(t-s), k_2(t,s) =b(s)^{-1}\\int _s^t p^{\\prime }(v) h(v-s) dv$ , and transform $\\bigl (\\mathcal {J}^* k_1(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) & =\\int _s^T \\frac{\\partial }{\\partial u}K(u,s) p(t)b(u)^{-1} h(t-u)\\mathbb {1}_{[0,t]}(u)du \\\\& = p(t)\\int _s^t a(s)b(u)c(u-s) b(u)^{-1}h(t-u)du\\, \\mathbb {1}_{[0,t]}(s) \\\\& = p(t)\\,a(s)\\int _s^t c(u-s) h(t-u)du\\, \\mathbb {1}_{[0,t]}(s)\\\\& = p(t)\\,a(s)\\mathbb {1}_{[0,t]}(s).$ Similarly, $\\bigl (\\mathcal {J}^* k_2(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) & =\\int _s^t a(s)\\,c(u-s)\\int _u^t p^{\\prime }(v) h(v-u)dv\\, du\\,\\mathbb {1}_{[0,t]}(s)\\\\& = a(s)\\int _s^t p^{\\prime }(v) \\int _s^v c(u-s)h(v-u)du\\, dv \\,\\mathbb {1}_{[0,t]}(s)\\\\& = a(s)\\int _s^t p^{\\prime }(v) \\, dv \\,\\mathbb {1}_{[0,t]}(s) = a(s)\\bigl (p(t) - p(s)\\bigr )\\mathbb {1}_{[0,t]}(s).$ Consequently, $\\bigl (\\mathcal {J}^* k(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) & = p(t)\\,a(s)\\mathbb {1}_{[0,t]}(s) - a(s)\\bigl (p(t) - p(s)\\bigr )\\mathbb {1}_{[0,t]}(s)\\\\& = a(s) \\, p(s)\\mathbb {1}_{[0,t]}(s) = \\mathbb {1}_{[0,t]}(s).$ Therefore, thanks to (REF ), $\\int _0^T k(t,s) \\, dX_s = \\int _0^T \\bigl (\\mathcal {J}^* k(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) \\, dW_s = \\int _0^T \\mathbb {1}_{[0,t]}(s) \\, dW_s = W_t,$ as required.",
"Examples of Sonine kernels Functions $c(s)=s^{-\\alpha }$ and $h(s)=s^{ \\alpha -1}$ with some $\\alpha \\in (0,1/2)$ were considered above in connection with fractional Brownian motion, see subsection REF .",
"For $\\alpha \\in (0,1)$ and $A\\in \\mathbb {R}$ , let $\\gamma =\\Gamma ^{\\prime }(1)$ be Euler-Mascheroni constant, $l = \\gamma -A$ .",
"Then $c(x) = \\frac{1}{\\Gamma (\\alpha )} x^{\\alpha -1}\\left(\\ln \\textstyle {\\frac{1}{x}} + A\\right)$ and $h(x) = \\int _0^\\infty \\frac{x^{t-\\alpha }e^{lt}}{\\Gamma (1-\\alpha + t)}dt$ create a Sonine pair, see [12].",
"This example was proposed by Sonine himself [13]: for $\\nu \\in (0,1)$ , $h(x) = x^{-\\nu /2}J_{-\\nu }(2\\sqrt{x}), \\quad c(x) = x^{(\\nu -1)/2} I_{\\nu -1}(2\\sqrt{x}),$ where $J$ and $I$ are, respectively, Bessel and modified Bessel functions of the first kind, $J_\\nu (y)=\\frac{y^\\nu }{2^\\nu }\\sum _{k=0}^\\infty \\frac{(-1)^ky^{2k}2^{-2k}}{k!\\Gamma (\\nu +k+1)},$ and $I_\\nu (y)=\\frac{y^\\nu }{2^\\nu }\\sum _{k=0}^\\infty \\frac{ y^{2k}2^{-2k}}{k!\\Gamma (\\nu +k+1)}.$ In particular, setting $\\nu = 1/2$ , we get the following Sonine pair: $h(x) = \\frac{\\cos 2\\sqrt{x}}{2\\sqrt{\\pi x}},\\quad c(x) = \\frac{\\cosh 2\\sqrt{x}}{2\\sqrt{\\pi x}}.$ It is interesting that the creation of Sonine pairs allows to get the relations between the special functions (see [9]).",
"Let $c(x)=x^{-1/2}\\cosh (ax^{1/2}),$ and let $h(x)=\\int _0^xs^{\\nu /2}J_\\nu (as^{1/2})\\,(x-s)^\\gamma ds$ be a fractional integral of $s^{\\nu /2}J_\\nu (as^{1/2})$ , where $-1<\\nu <-\\frac{1}{2}$ , $\\gamma +\\nu =-\\frac{3}{2}$ .",
"If we denote $F_y(\\lambda )$ Laplace transform of function $y$ at point $\\lambda $ , then the Laplace transforms of these functions equal $F_c(\\lambda )=(\\pi /\\lambda )^{1/2}\\exp (a^2/4\\lambda ),\\\\\\begin{aligned}F_{h}(\\lambda )&=\\Gamma (\\gamma +1)2^{-\\nu }a^\\nu \\lambda ^{-\\nu -1} \\exp (-a^2/4\\lambda )\\lambda ^{-\\gamma -1} \\\\&=\\Gamma (\\gamma +1)2^{-\\nu }a^\\nu \\lambda ^{-1/2}\\exp (-a^2/4\\lambda ),\\end{aligned}\\\\F_c(\\lambda )F_{h}(\\lambda )=\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi } a^\\nu \\lambda ^{-1},\\qquad \\lambda >0,$ whence their convolution equals $(c\\ast h)_t=\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi }a^\\nu , \\qquad t>0.$ Therefore $c(x)$ and $(\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi }a^\\nu )^{-1}h(x)$ create a Sonine pair.",
"However, comparing with Example with $a=2$ , and taking into account that the pair in Sonine pair is unique, we get that $ 4\\sqrt{\\pi }(\\Gamma (\\gamma +1))^{-1}\\int _0^xs^{\\nu /2}J_\\nu (2s^{1/2})\\,(x-s)^\\gamma ds=\\frac{\\cos 2\\sqrt{x}}{ \\sqrt{ x}}.$ Similarly, let $c(x)=\\int _0^xt^{-1/2}\\cosh (at^{1/2})\\,(x-t)^\\gamma dt,\\ h(x)=x^{\\nu /2}J_\\nu (ax^{1/2})$ with $\\gamma \\in (-1,-\\frac{1}{2})$ , $\\nu \\in (-1,0)$ ,, $\\gamma +\\nu =-\\frac{3}{2}$ .",
"Then $F_{c}(\\lambda )=\\pi ^{1/2}\\Gamma (\\gamma +1)\\lambda ^{-\\gamma -3/2}\\exp (a^2/4\\lambda ),$ and $F_h(\\lambda )=\\frac{a^\\nu }{2^\\nu }\\lambda ^{-\\nu -1}\\exp (-a^2/4\\lambda ),\\;\\text{whence}\\;F_{c}(\\lambda )F_h(\\lambda )=\\pi ^{1/2}\\Gamma (\\gamma +1)\\frac{a^\\nu }{2^\\nu }\\lambda ^{-1}.$ If we put $a=2$ and compare with (REF ), we get the following representation $\\pi ^{-1/2}(\\Gamma (\\gamma +1))^{-1}\\int _0^xt^{-1/2}\\cosh (2t^{1/2})\\,(x-t)^\\gamma dt=x^{(-\\nu -1)/2} I_{-\\nu -1}(2\\sqrt{x}).$ On the way of creation of the new Sonine pairs, a natural idea is to consider $g(s)=e^{\\beta s}s^{ \\alpha -1}$ with $\\beta \\in \\mathbb {R}$ and examine if this function admits a Sonine pair.",
"It happens so that the answer to this question is positive, but far from obvious and not simple.",
"All preliminary results are contained in subsection REF .",
"Let $g(x) = \\frac{\\exp (\\beta x)}{\\Gamma (\\alpha ) x^{1-\\alpha }}, \\quad 0<\\alpha <1, \\quad \\beta <0;\\qquad y(x) = 1.$ Then $h(x) = \\alpha \\beta \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha +1; \\: 2; \\: \\beta x)< 0, \\qquad x\\in [0,T],$ where $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }$ is Kummer hypergeometric function; see Section REF in the Appendix.",
"The conditions of Theorem REF hold true.",
"The equation (REF ) has a unique solution in $L^{1}[0,T]$ (Actually, it has many solutions, but each two solutions are equal almost everywhere.)",
"The solution has a representative that is continuous and attains only positive values on the left-open interval $(0,T]$ , and it is a Sonine pair to $g(s)=e^{\\beta s}s^{ \\alpha -1}$ .",
"Appendix Inequalities for norms of convolutions and products Recall notation $\\Vert f\\Vert _p$ for the norm of function $f\\in L^p(\\mathbb {R})$ , $p \\in [1, \\infty ]$ .",
"The convolution of two measurable functions $f$ and $g$ is defined by integration $(f*g)(t) = \\int _{\\mathbb {R}} f(s) g(t-s) \\, ds .$ Now we state an inequality for the norm of convolution of two functions.",
"If $p\\in [1,\\infty ]$ , $q\\in [1,\\infty ]$ but $1/p+1/q\\ge 1$ , $f\\in L^p(\\mathbb {R})$ , $g\\in L^q(\\mathbb {R})$ , then the convolution $f*g$ is well-defined almost everywhere (that is the integral in (REF ) converges absolutely for almost all $t\\in \\mathbb {R}$ ), $f * g \\in L^{(1/p+1/q-1)^{-1}}(\\mathbb {R})$ , and $\\Vert f*g\\Vert _{(1/p+1/q-1)^{-1}}\\le \\Vert f\\Vert _p \\, \\Vert g\\Vert _q.$ Now we state an inequality for the norm of the product of two functions $(fg)(t) = f(t) g(t)$ .",
"We call it Hölder inequality for non-conjugate exponents.",
"If $p\\in [1,\\infty ]$ , $q\\in [1,\\infty ]$ , $1/p+1/q\\le 1$ , $f\\in L^p(\\mathbb {R})$ , $g\\in L^q(\\mathbb {R})$ , then $fg \\in L^{(1/p+1/q)^{-1}}(\\mathbb {R})$ and $\\Vert fg\\Vert _{(1/p+1/q)^{-1}} \\le \\Vert f\\Vert _p \\, \\Vert g\\Vert _q.$ Now we state an inequality for the norms in $L_p[a,b]$ and $L_q[a,b]$ .",
"If $-\\infty < a < b < \\infty $ , $1 \\le p \\le q \\le \\infty $ , $f \\in L^q(\\mathbb {R})$ and the $f(t)=0$ for all $t\\notin [a,b]$ , then $f\\in L^p(\\mathbb {R})$ and $\\Vert f\\Vert _p \\le (b-a)^{1/p-1/q} \\, \\Vert f\\Vert _q .$ Conditions for inequalities (REF ) and (REF ) are over-restrictive because of restrictive notation $\\Vert f\\Vert _p$ .",
"This notation can be extended to all $p\\in (0,\\infty ]$ and all measurable functions $f$ .",
"Then the conditions for inequalities (REF ) and (REF ) may be relaxed.",
"Inequality (REF ) is proved in [7]; see item (2) in the remarks after this theorem and part (A) of its proof.",
"If $p<\\infty $ and $q<\\infty $ , then inequality (REF ) follows from the conventional Hölder inequality.",
"Otherwise, if $p=\\infty $ or $q=\\infty $ , then inequality (REF ) is trivial.",
"Inequality (REF ) can be rewritten as $\\Vert f \\operatorname{\\mathbb {1}}_{[a,b]}\\!\\Vert _p \\le \\Vert \\!\\operatorname{\\mathbb {1}}_{[a,b]}\\!\\Vert _{(1/p-1/q)^{-1}} \\, \\Vert f\\Vert _q $ , and so follows from (REF ).",
"Continuity of trajectories and Hölder condition Kolmogorov continuity theorem provides sufficiency conditions for a stochastic process to have a continuous modification.",
"The following theorem aggregates Theorems 2, 4 and 5 in [3].",
"[Kolmogorov continuity theorem] Let $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ be a stochastic process.",
"If there exist $K \\ge 0$ , $\\alpha >0$ and $\\beta >0$ such that $\\mathrm {E}\\left[\\,|X_t - X_s|^\\alpha \\,\\right] \\le K\\, |t-s|^{1+\\beta }\\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T,$ then The process $X$ has a continuous modification; Every continuous modification of the process $X$ whose trajectories almost surely satisfies Hölder condition for all exponents $\\gamma \\in (0,\\: \\beta /\\alpha )$ ; There exists a modification of the process $X$ that satisfies Hölder condition for exponent $\\gamma \\in (0, \\: \\beta /\\alpha )$ .",
"This theorem can be applied for Gaussian processes.",
"Let $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ be a centered Gaussian process.",
"If there exist $K\\ge 0$ and $\\delta > 0$ such that $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le K \\, |t-s|^\\delta \\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T,$ then the following holds true: The process $X$ has a modification $\\widetilde{X}$ that has continuous trajectories.",
"For every $\\gamma $ , $0< \\gamma < \\frac{1}{2}\\delta $ , the trajectories of the process $\\widetilde{X}$ satisfy $\\gamma $ -Hölder condition almost surely.",
"The process $X$ has a modification that satisfies Hölder condition for all exponents $\\gamma \\in (0, \\frac{1}{2}\\delta )$ .",
"Since $X_s - X_t$ is a centered Gaussian variable, $\\mathrm {E}\\left[\\,|X_t - X_s|^\\alpha \\,\\right] = \\frac{2^{\\alpha /2}}{\\sqrt{\\pi }}\\Gamma \\!\\left(\\frac{\\alpha +1}{2}\\right)\\left(\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right]\\right)^{\\alpha /2} .$ The first statement of the corollary can be proved by applying Kolmogorov continuity theorem for $\\alpha > 2/\\delta $ and $\\beta = \\frac{1}{2} \\alpha \\delta - 1$ .",
"The second statement of the corollary can be proved by applying Kolmogorov continuity theorem for $\\alpha > \\frac{2}{\\delta - 2\\gamma }$ and $\\beta = \\frac{1}{2} \\alpha \\delta - 1$ .",
"Consider the random event $A &= \\left\\lbrace \\forall \\gamma \\in (0, \\textstyle {\\frac{1}{2}} \\delta ) : \\widetilde{X} \\;\\mbox{satisfies $\\gamma $-Hölder condition} \\right\\rbrace \\\\ &=\\left\\lbrace \\forall n\\in \\mathbb {N} : \\widetilde{X} \\;\\mbox{satisfies $\\frac{1}{2} \\left(1-\\frac{1}{n}\\right) \\delta $-Hölder condition} \\right\\rbrace .$ (The measurability of $A$ follows from the continuity of the process $\\widetilde{X}$ ).",
"By the second statement of Corollary REF $\\mathrm {P}(A) = 1$ .",
"Thus, $\\lbrace \\widetilde{X}_t \\mathbb {1}_{A}, \\;t\\in [0,t]\\rbrace $ is the desired modification which satisfies Hölder condition for all exponents $\\gamma \\in (0, \\frac{1}{2} \\delta )$ .",
"Corollary REF holds true even without assumption that the Gaussian process $X$ is centered.",
"The first statement of Corollary REF can be proved with Xavier Fernique's continuity criterion [5] as well.",
"Let $\\lbrace X_t, \\; t\\in [0, T]\\rbrace $ be a centered Gaussian process.",
"Suppose that there exist $\\delta > 0$ and a nondecreasing continuous function $F : [0, T] \\rightarrow \\mathbb {R}$ such that $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le (F(t) - F(s))^\\delta \\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T.$ Then The process $X$ have a modification $\\widetilde{X}$ that has continuous trajectories.",
"If the function $F$ satisfies Lipschitz condition in an interval $[a,b] \\subset [0,T]$ , then for every $\\gamma $ , $0< \\gamma < \\frac{1}{2}\\delta $ , the process $\\widetilde{X}$ has a modification whose trajectories satisfy $\\gamma $ -Hölder property on the interval $[a, b]$ .",
"Without loss of generality, we can assume that the function $F$ is strictly increasing.",
"Indeed, if the condition (REF ) holds true for $F$ being continuous nondecreasing function $F_1$ , it also holds true for $F=F_2$ with $F_2(t)=F_1(t)+t$ , where $F_2$ is a continuous strictly increasing function.",
"With this additional assumption, the inverse function $F^{-1}$ is one-to-one, strictly increasing continuous function $[F(0), F(T)] \\rightarrow [0, T]$ .",
"Consider a stochastic process $\\lbrace Y_u, \\; u \\in [F(0), F(T)]\\rbrace $ , with $Y_u = Y_{F^{-1}(u)}$ .",
"The stochastic process $Y$ is centered and Gaussian; it satisfies condition $\\mathrm {E}\\left[\\,(Y_v - Y_u)^2\\,\\right]= \\mathrm {E}\\left[\\,(X_{F^{-1}(v)} - X_{F^{-1}(u)})^2\\,\\right]\\le (F(F^{-1}(v)) - F(F^{-1}(u)))^\\delta = (v - u)^\\delta $ for all $F(0) \\le u \\le v \\le F(T)$ .",
"According to Corollary REF , the process $Y$ has a modification $\\widetilde{Y}$ with continuous trajectories.",
"Then $\\widetilde{X}$ with $\\widetilde{X}_t = \\widetilde{Y}_{F(t)}$ is a modification of the process $X$ with continuous trajectories.",
"The second statement of the lemma is a direct consequence of Corollary REF .",
"If the function $F$ satisfies Lipschitz condition with constant $L$ on the interval $[a,b]$ , then $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le L^\\delta (t - s)^\\delta \\quad \\mbox{for all} \\quad a \\le s \\le t \\le b ,$ which is the main condition for Corollary REF .",
"Application of fractional calculus The lower and upper Riemann–Liouville fractional integrals of a function $f\\in L^{1}[a,b]$ are defined as follows: $(I_{a+}^\\alpha f) (x) = \\frac{1}{\\Gamma (\\alpha )}\\int _a^x \\frac{f(t) \\, dt}{(x-t)^{1-\\alpha }}, \\qquad (I_{b-}^\\alpha f) (x) = \\frac{1}{\\Gamma (\\alpha )}\\int _x^b \\frac{f(t) \\, dt}{(t-x)^{1-\\alpha }}.$ The integrals $(I_{a+}^\\alpha f) (x)$ and $(I_{b-}^\\alpha f) (x)$ are well-defined for almost all $x\\in [a,b]$ , and are integrable functions of $x$ , that is $I_{a+}^\\alpha f \\in L^1[a,b]$ and $I_{b-}^\\alpha f \\in L^1[a,b]$ .",
"Thus, $I_{a+}^\\alpha $ and $I_{b-}^\\alpha $ might be considered linear operators $L^1[a,b] \\rightarrow L^1[a,b]$ .",
"A reflection relation for functions $g(x) = f(a+b-x)$ imply the following relation for their fractional integrals: $(I_{b-}^{\\alpha } g)(x) = (I_{a+}^{\\alpha } f) (a+b-x);$ see [11].",
"The integration-by-parts formula is given, e.g., in [11].",
"[integration-by-parts formula] Let $\\alpha > 0$ , $f \\in L^p[a,b]$ , $g \\in L^q[a,b]$ , $p \\in [1, +\\infty ]$ , $q \\in [1, +\\infty ]$ , while $\\frac{1}{p} + \\frac{1}{q} \\le 1 + \\alpha $ and $\\max \\Bigl (1 + \\alpha - \\frac{1}{p} - \\frac{1}{q},\\:\\min \\!\\left(1 - \\frac{1}{p}, \\:1 - \\frac{1}{q}\\right)\\Bigr ) > 0$ .",
"Then $\\int _a^b (I_{a+}^\\alpha f)(t) \\, g(t) \\, dt =\\int _a^b f(t) \\, (I_{b-}^\\alpha g)(t) \\, dt.$ Now we establish conditions for a function to be in the range of the fractional operator $I_{a+}^\\alpha $ , and we provide formulas for the preimage, which is called a fractional derivative.",
"The following statements are the modifications of the Theorem 2.1 and following corollary in [11].",
"The formulas for the fractional derivative are also provided in [8].",
"Let $0 < \\alpha < 1$ .",
"Consider the integral equation $I_{a+}^\\alpha f = g$ with unknown function $f \\in L^1[a,b]$ and known function (i.e., a parameter) $g \\in L^1[a,b]$ .",
"Denote $h(x) = {\\left\\lbrace \\begin{array}{ll} (I_{a+}^{1-\\alpha } g) (x)& \\mbox{if $a < x \\le b$}, \\\\0& \\mbox{if $x=a$}.\\end{array}\\right.",
"}$ If $h \\in \\operatorname{\\!\\textit {AC}}[a,b]$ , then equation (REF ) has a unique (up to equality almost everywhere in $[a,b]$ ) solution $f$ , namely $f(x) = h^{\\prime }(x)$ .",
"Otherwise, if $h \\notin \\operatorname{\\!\\textit {AC}}[a,b]$ , then equation (REF ) has no solutions in $L^1[a,b]$ .",
"If for some $x\\in (a,b]$ the integral $(I_{a+}^{1-\\alpha } g) (x)$ is not well-defined, then equation (REF ) does not have solutions in $L^1[a,b]$ .",
"Let $0 < \\alpha < 1$ .",
"The integral equation (REF ) with unknown function $f \\in L^1[a,b]$ and known function $g \\in \\operatorname{\\!\\textit {AC}}[a,b]$ has a unique solution.",
"The solution is equal to $f(x) &=(I_{a+}^{1-\\alpha }(g^{\\prime }))(x) +\\frac{g(a)}{\\Gamma (1-\\alpha )\\,(x-a)^\\alpha }\\\\ &=\\frac{1}{\\Gamma (1-\\alpha )}\\left( \\int _a^x \\frac{g^{\\prime }(t)\\,dt}{(x-t)^\\alpha }+ \\frac{g(a)}{(x-a)^\\alpha } \\right) .$ Existence of the solution to Volterra integral equation where the integral operator is an operator of convolution with integrable singularity at 0 Consider Volterra integral equation of the first kind $\\int _0^x f(t) \\, g(x-t) \\, dt = y(x), \\qquad x \\in (0,T],$ with $g(x)$ and $y(x)$ known (parameter) functions and $f(x)$ unknown function.",
"Suppose that the function $g(x)$ is integrable in the interval $(0,T]$ but behaves asymptotically as a power function in the neighborhood of 0: $g(x) \\sim \\frac{K}{x^{1-\\alpha }}, \\qquad x \\rightarrow 0,$ where $0 < \\alpha < 1$ .",
"More specifically, assume that $g(x)$ admits a representation $g(x) = \\frac{1}{\\Gamma (\\alpha ) x^{1-\\alpha }}+ (I^{\\alpha }_{0+} h)(x) =\\frac{1}{\\Gamma (\\alpha )}\\left( \\frac{1}{x^{1-\\alpha }} + \\int _0^x \\frac{h(t)\\, dt}{(x-t)^{1-\\alpha }} \\right),$ where $\\Gamma (\\alpha )$ is a gamma function, $I^{\\alpha }_{0+}h$ is a lower Riemann–Liouville fractional integral of $h$ , $(I^{\\alpha }_{0+} h)(x) =\\frac{1}{\\Gamma (\\alpha )}\\int _0^x \\frac{h(t)\\, dt}{(x-t)^{1-\\alpha }},$ and $h(x)$ is a absolutely continuous function.",
"The sufficient conditions for existence and uniqueness of the solution to integral equation claimed in [8] are not satisfied.",
"The kernel of the integration operator in (REF ) is unbounded, and $y(0)$ might be nonzero.",
"But we use Remark 2 in [8].",
"We reduce the Volterra integral equation of the first kind to a Volterra integral equation of the second kind similarly as it is done for regular functions $g(x)$ ; compare with [8] for the case of regular $g(x)$ .",
"For the next theorem we keep in mind that if a function $f$ is a solution to (REF ), then every function that is equal to $f$ almost everywhere on $[0,T]$ is also a solution to (REF ).",
"Let $y,\\,h \\in C^1[0,T]$ and $g$ be defined in (REF ).",
"Then the equation (REF ) has a unique (up to equality almost everywhere) solution $f \\in L^1[0,T]$ .",
"The solution is (more precisely, some of almost-everywhere equal solutions are) continuous in the left-open interval $(0,T]$ .",
"Substitute (REF ) into (REF ): $\\int _0^x f(t)\\left( \\frac{1}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }}+ (I^{\\alpha }_{0+} h)(x-t) \\right) dt= y(x), \\\\(I^{\\alpha }_{0+} f)(x) +\\int _0^x f(t) \\, (I^{\\alpha }_{0+} h)(x-t) \\, dt= y(x) .$ Denote $h_x(t) = h(x-t)$ .",
"According to equation (REF ), the fractional integrals of $h$ and $h_x$ satisfy the relation $(I^{\\alpha }_{0+} h)(x-t)= (I^{\\alpha }_{x-} h_x)(t)$ .",
"Hence, equation (REF ) is equivalent to the following one: $(I^{\\alpha }_{0+} f)(x) +\\int _0^x f(t) \\, (I^{\\alpha }_{x-} h_x)(t) \\, dt= y(x).$ Now apply the integration-by-parts formula.",
"We have $f \\in L^1[0,x]$ , $h_x \\in L^\\infty [0,x]$ , and $1 + 0 < 1 + \\alpha $ .",
"Hence, by Proposition REF , $\\int _0^x f(t) \\, (I^{\\alpha }_{x-} h_x)(t) \\, dt =\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h_x(t) \\, dt.$ It means that equation (REF ) is equivalent to the following ones: $(I^{\\alpha }_{0+} f)(x) +\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h_x(t) \\, dt= y(x), $ and $(I^{\\alpha }_{0+} f)(x) +\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h(x-t) \\, dt= y(x) .$ Denote $F = I^{\\alpha }_{0+} f$ , and obtain a Volterra integral equation of the second kind: $F(x) = y(x) - \\int _0^x F(t) \\, h(x-t) \\, dt.$ Equation (REF ) has a unique solution in $C[0,T]$ , as well as in $L^1[0,T]$ .",
"In other words, (REF ) has a unique integrable solution, and this solution is a continuous function.",
"According to Theorem REF , either unique (up to almost-everywhere equality) function $f$ , or no functions $f$ correspond to the function $F$ .",
"Thus, all integrable solution to integral equation (REF ) are equal almost everywhere.",
"Now we construct a solution to equation (REF ) that is continuous and integrable on $(0,T]$ .",
"Differentiating (REF ), we obtain $F^{\\prime }(x) = y^{\\prime }(x) - F(x)\\, h(0) - \\int _0^x F(t) \\, h^{\\prime }(x-t) \\, dt,$ whence $F \\in C^1[0,T]$ .",
"According to Corollary REF , the integral equation $F = I_{0+}^\\alpha f$ has a unique solution $f\\in L^1[0,T]$ , which is equal to $f(x)= \\frac{1}{\\Gamma (1-\\alpha )}\\left( \\int _0^x \\frac{F^{\\prime }(t)\\,dt}{(x-t)^\\alpha } +\\frac{F(0)}{x^\\alpha } \\right) .$ The constructed function $f(x)$ is continuous and integrable in $(0,T]$ , and $f(x)$ is a solution to (REF ).",
"In Theorem REF the condition $h \\in C^1[0,T]$ can be relaxed and replaced with the condition $h \\in \\operatorname{\\!\\textit {AC}}[0,T]$ .",
"In other words, if the function $h$ is absolutely continuous but is not continuously differentiable, the statement of Theorem REF still holds true.",
"Example: $g(x) = \\exp (\\beta x) x^{\\alpha -1} / \\Gamma (\\alpha )$ and $y(x) = 1$Example: $g(x) = \\exp (\\beta x) x^{\\alpha -1} / \\Gamma (\\alpha )$ and $y(x) = 1$ It is well known that $\\int _0^x \\frac{1}{\\Gamma (1-\\alpha ) t^\\alpha } \\,\\frac{1}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }} \\, dt = 1 .$ In this section, we prove that the equation $\\int _0^x f(t) \\,\\frac{e^{(x-t)\\beta }}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }} \\, dt = 1$ has an integrable solution.",
"According to (REF ), $f(x) = x^{-\\alpha } / \\, \\Gamma (1-\\alpha )$ is a solution to (REF ) if $\\beta = 0$ .",
"Denote $g(x) = \\frac{\\exp (\\beta x)}{\\Gamma (\\alpha ) x^{1-\\alpha }} .$ Demonstrate that $g(x)$ admits a representation (REF ).",
"To construct $h$ , we need Kummer confluent hypergeometric function [14]: $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;z) = \\frac{1}{\\mathrm {B}(a,\\: b-a)}\\int _0^1 e^{zt} t^{a-1} (1-t)^{b-a-1} \\, dt,\\qquad 0<a<b, \\quad z\\in \\mathbb {C} .$ For $a$ and $b$ fixed, $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;\\,\\cdot \\,)$ is an entire function.",
"Its derivative equals $\\frac{\\partial }{\\partial z}\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;z) = \\frac{a}{b}\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a+1;\\:b+1;\\:z) .$ For all $0<a<b$ and $z\\in \\mathbb {R}$ $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a; b; z) > 0, \\qquad \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a; b; 0) = 1.$ Notice that if $0<\\alpha <1$ and $x>0$ , then $\\frac{1}{\\mathrm {B}(\\alpha , \\: 1-\\alpha )}\\int _0^x \\frac{\\exp (zt) \\, dt}{t^{1-\\alpha } (x-t)^\\alpha } =\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; xz) .$ Being considered an equation for unknown $h$ , (REF ) is equivalent to $I_{0+}^\\alpha h = g_0$ , where $g_0(x) = g(x) - \\frac{1}{\\Gamma (\\alpha ) x^{1-\\alpha }}= \\frac{e^{\\beta x} - 1}{\\Gamma (\\alpha ) x^{1-\\alpha }} .$ Then $(I_{0+}^{1-\\alpha } g_0)(x) =\\frac{1}{\\mathrm {B}(\\alpha , \\: 1-\\alpha )}\\int _0^x \\frac{e^{\\beta t} - 1}{t^{1-\\alpha }(1-t)^\\alpha } \\, dt =\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1 .$ Besides, $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1$ is an absolutely continuous function in $x$ , and$\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1;\\beta x) - 1 = 0$ if $x=0$ .",
"According to Theorem REF , the equation $I_{0+}^\\alpha h = g_0$ has the unique solution $h =L^1[0,T]$ , which is equal to $h(x) = \\frac{\\partial (\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1)}{\\partial x} =\\alpha \\beta \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha +1;\\: 2;\\: \\beta x).$ The constructed function $h(x)$ is a solution to (REF ) and is continuously differentiable.",
"In summary, $h \\in C^1[0,T]$ , $y(x) = 1$ , and $y \\in C^1[0,T]$ .",
"According to Theorem REF the integral equation $\\int _0^x f(t) \\, g(x-t) \\, dt = 1, \\qquad x\\in (0,T],$ has a unique solution $f \\in L^1[0,T]$ (up to equality almost everywhere).",
"The solution is continuous in $(0,T]$ .",
"The fact that the functions $g$ and $h$ defined in (REF ) and (REF ), respectively, satisfy (REF ), can be checked directly.",
"For such verification, one can apply Lemma 2.2(i) from [10].",
"Positive solution to the Volterra integral equation Let the conditions of Theorem REF hold true.",
"Additionally, let $y(x)>0, \\quad y^{\\prime }(x)\\ge 0, \\quad h(x) < 0 \\qquad \\mbox{for all $x\\in [0,T]$}.$ Then the continuous solution $f(x)$ to (REF ) attains only positive values in $(0,T]$ .",
"Notice that (REF ) implies $F(0) = y(0) > 0$ .",
"Taking this into account, let's differentiate both sides of (REF ) the other way: $F(x) = y(x) - \\int _0^x F(x-t) \\, h(t) \\, dt,\\nonumber \\\\F^{\\prime }(x) = y^{\\prime }(x) - F(0) \\, h(x)- \\int _0^x F^{\\prime }(x-t) \\, h(t) \\, dt .$ Let us prove that $F^{\\prime }(x)>0$ in $[0,T]$ by contradiction.",
"Assume the contrary, that is $\\exists x \\in [0,1] : F^{\\prime }(x) \\le 0$ .",
"Since the function $F^{\\prime }(x)$ is continuous in $[0,T]$ , the contrary implies the existence of the minimum in $x_0 = \\min \\lbrace x\\in [0,T] : F^{\\prime }(x) \\le 0\\rbrace .$ But for $x=x_0$ the left-hand side in (REF ) is less or equal then zero, while the right-hand side is greater than zero.",
"Thus, (REF ) does not hold true.",
"The proof also works for $x_0 = 0$ .",
"There is a contradiction.",
"Thus, we have proved that $F^{\\prime }(x)>0$ for all $x\\in [0,T]$ .",
"By (REF ), $f(x)>0$ for all $x\\in (0,T]$ .",
"Consider now a natural question: for which kernels $K$ of the form (REF ) Gaussian process of the form (REF ) with Volterra kernel $K$ generates the same filtration as the Wiener process $W$ .",
"Sufficient condition for this is the representation of the Wiener process $W$ as $W_t = \\int _0^t L(t,s) \\, dX_s$ where $L\\in L^2([0,T]^2)$ is a Volterra kernel, and the integral is well defined, in some sense.",
"As an example, let us consider fractional Brownian motion $B^H, H>1/2 $ admitting a representation (REF ) with Volterra kernel (REF ).",
"For any $0<\\varepsilon <1$ consider the approximation $B^{H,\\varepsilon }_t=d_H\\int _0^t\\left(s^{1/2 - H}\\int _s^t u^{H-1/2} (u-\\varepsilon s)^{H-3/2}\\, du\\right) dW_s, t\\ge 0.$ Unlike the original process, in such approximation we can change the limits of integration and get that $B^{H,\\varepsilon }_t=d_H\\int _0^t \\left(u^{H-1/2}\\int _0^us^{1/2 - H}(u-\\varepsilon s)^{H-3/2}dW_s\\, \\right)du.$ This representation allows to write the equality $\\int _0^tu^{1/2-H}dB^{H}_u=d_H\\int _0^t \\left( \\int _0^us^{1/2 - H}(u-\\varepsilon s)^{H-3/2}dW_s\\, \\right)du,$ and it follows immediately from (REF ) that ${\\begin{array}{c}\\int _0^t(t-u)^{1/2-H}u^{1/2-H}dB^{H,\\varepsilon }_u\\hspace{128.0374pt}\\\\=d_H\\int _0^t (t-u)^{1/2-H}\\left( \\int _0^us^{1/2 - H}(u-\\varepsilon s)^{H-3/2}dW_s\\, \\right)du\\\\= d_H\\int _0^t s^{1/2 - H}\\left( \\int _s^t(t-u)^{1/2-H}(u-\\varepsilon s)^{H-3/2}du\\,\\, \\right)dW_s.\\end{array}}$ Applying Theorem 3.3 from [2], p. 160, we can go to the limit in (REF ) and get that $\\int _0^t(t-u)^{1/2-H}u^{1/2-H}dB^{H}_u=d_H\\int _0^t s^{1/2 - H}\\left( \\int _s^t(t-u)^{1/2-H}(u- s)^{H-3/2}du\\,\\, \\right)dW_s.",
"$ Now the highlight is that the integral $\\int _s^t(t-u)^{1/2-H}(u- s)^{H-3/2}\\,du$ is a constant, namely, $\\int _s^t(t{-}u)^{1/2-H}(u{-} s)^{H-3/2}du=\\int _0^t(t{-}u)^{1/2-H}u^{H-3/2}du=\\mathrm {B}(3/2{-}H, H{-}1/2)$ , where $\\mathrm {B}$ is a beta-function.",
"After we noticed this, then everything is simple: $Y_t:=\\int _0^t(t-u)^{1/2-H}u^{1/2-H}dB^{H}_u=d_H \\mathrm {B}(3/2-H,\\: H-1/2)\\int _0^t s^{1/2 - H} dW_s, $ and finally we get that $W_t=e_H\\int _0^t s^{ H-1/2} dY_s $ with some constant $e_H$ .",
"It means that we have representation (REF ) and, in particular, $W$ and $B^H$ generate the same filtration.",
"Of course, these transformations can be performed much faster, but our goal here was to pay attention on the role of the property of the convolution of two functions to be a constant.",
"This property is a characterization of Sonine kernels."
],
[
"General approach to Volterra processes with Sonine kernels",
"First we give basic information about Sonine kernels, more details can be found in [12].",
"We also consider, in a simplified form, the related generalized fractional calculus introduced in [6].",
"A function $c\\in L^1[0,T]$ is called a Sonine kernel if there exists a function $h\\in L^1[0,T]$ such that $\\int _0^t c(s) h(t-s) \\, ds= 1,\\quad t\\in (0,T].$ Functions $c,h$ are called Sonine pair, or, equivalently, we say that $c$ and $h$ form (or create) a Sonine pair.",
"If $\\hat{c}$ and $\\hat{h}$ denote the Laplace transforms of $c$ and $h$ respectively, then (REF ) is equivalent to $\\hat{c}(\\lambda )\\hat{h}(\\lambda ) = \\lambda ^{-1}$ , $\\lambda >0$ .",
"Since the Laplace transform characterizes a function uniquely, for any $c$ there can be not more than one function $h$ satisfying (REF ).",
"Examples of Sonine pairs are given in Section .",
"Let functions $c$ and $h$ form a Sonine pair.",
"For a function $f\\in L^1[0,T]$ consider the operator $\\mathrm {I}^c_{0+} f (t) =\\int _0^t c(t-s) f(s) ds.$ It is an analogue of forward fractional integration operator.",
"Let us identify an inverse operator.",
"In order to do this, for $g \\in AC[0,T]$ define $\\mathrm {D}^h_{0+} g(t) = \\int _0^t h(t-s)g^{\\prime }(s)ds + h(t)g(0).$ Note that $\\int _0^t \\mathrm {D}^h_{0+} g(u) du = \\int _0^t \\left(\\int _0^u h(u-s)g^{\\prime }(s)ds + h(u)g(0)\\right)\\, du \\\\ = \\int _0^t \\int _0^u h(s)g^{\\prime }(u-s)ds\\, du + g(0)\\int _0^t h(u)du\\\\=\\int _0^t h(s)\\int _s^t g^{\\prime }(u-s)du\\, ds + g(0)\\int _0^t h(u)du \\\\=\\int _0^t h(s)\\big (g(t-s)-g(0)\\big )ds\\, ds + g(0)\\int _0^t h(u)du = \\int _0^t h(s) g(t-s)ds,$ so we can also write $\\mathrm {D}^h_{0+} g(t) = \\frac{d}{dt}\\int _0^t h(s) g(t-s)ds = \\frac{d}{dt}\\int _0^t h(t-s) g(s)ds,$ where the derivative is understood in the weak sense.",
"Similarly, we can define an analogue of backward fractional integral: $\\mathrm {I}^c_{T-} f(s) = \\int _s^T c(t-s) f(t)dt, \\qquad f \\in L^1[0,T]$ and the corresponding differentiation operator $\\mathrm {D}^h_{T-} g(s) = g(T)h(T-s) - \\int _s^T h(t-s)g^{\\prime }(t)dt.$ Let $g\\in AC[0,T]$ .",
"Then $\\displaystyle \\mathrm {I}_{0+}^c\\big ( \\mathrm {D}^h_{0+} g\\big )(t) = g(t)$ and $\\displaystyle \\mathrm {I}_{T-}^c\\big ( \\mathrm {D}^h_{T-} g\\big )(s) = g(s)$ .",
"We have $\\mathrm {I}^c_{0+}\\big ( \\mathrm {D}^h_{0+} g\\big )(t)&=\\int _0^t c(t-s)\\left(\\int _0^s h(s-u) g^{\\prime }(u)du + h(s)g(0)\\right)\\, ds\\\\ &=\\int _0^t \\int _u^t c(t-s)h(s-u)ds\\, g^{\\prime }(u)du + g(0)\\int _{0}^t c(t-s)h(s)ds\\\\ & =\\int _0^t g^{\\prime }(u)du + g(0) = g(t),$ as required.",
"Similarly, $\\mathrm {I}^c_{T-}\\big ( \\mathrm {D}^h_{T-} g\\big )(s) &= \\int _s^T c(t-s)\\left(h(T-t)g(T) - \\int _t^T h(u-t) g^{\\prime }(u)du \\right) ds\\\\&= g(T)\\int _{s}^T c(t-s)h(T-t)dt - \\int _s^T \\int _s^u c(t-s)h(u-t)\\,dt\\, g^{\\prime }(u)\\, du \\\\&= g(T) - \\int _s^T g^{\\prime }(u)du + g(s) = g(s)$ as required.",
"Now consider a Gaussian process $X$ given by the integral transformation of type (REF ) of the form (REF ) satisfying condition REF of Theorem .",
"Define the integral operator $\\mathcal {K} f(t) = \\int _0^t a(s)\\int _s^t b(u)c(u-s) du\\, f(s) ds.$ Note that for $f\\in L^2[0,T]$ , $\\mathcal {K} f(t)\\in AC[0,T]$ .",
"Indeed, by definition, $\\mathcal {K} f (t) = \\int _0^t K(t,s) f(s)ds = \\int _0^t \\int _s^t \\frac{\\partial }{\\partial u}K(u,s) du\\, f(s)ds.$ Since $f$ and $\\frac{\\partial }{\\partial t}K(t,s)$ are square integrable, the product $f \\frac{\\partial }{\\partial u}K$ is integrable on $\\left\\lbrace (s,u): 0\\le s\\le u\\le t \\right\\rbrace $ .",
"Therefore, we can apply Fubini theorem to get $\\mathcal {K} f (t) = \\int _0^t \\int _0^u \\frac{\\partial }{\\partial u}K(u,s) f(s) ds\\, du = \\int _0^t \\alpha (u)du,$ where $\\alpha \\in L^1[0,t]$ for all $t\\in [0,T]$ , so $\\alpha \\in L^1[0,T]$ .",
"Consequently, for $f\\in L^2[0,T]$ we can denote by $\\mathcal {J} f(t) = \\int _0^t \\frac{\\partial }{\\partial t}K(t,s) f(s) ds$ the weak derivative of $\\mathcal {K} f$ .",
"Further, define for a measurable $g\\colon [0,T] \\rightarrow \\mathbb {R}$ such that $\\left\\Vert g\\right\\Vert _{\\mathcal {H}_X}^2 := \\int _0^T \\left(\\int _s^T \\frac{\\partial }{\\partial u} K(u,s) g(u) du\\right)^2 ds<\\infty $ the integral operator $\\mathcal {J}^* g(s) = \\int _s^T \\frac{\\partial }{\\partial u} K(u,s) g(u)du.$ It can be extended to the completion $\\mathcal {H}_X$ of the set of measurable functions with finite norm $\\left\\Vert \\cdot \\right\\Vert _{\\mathcal {H}_X}^2$ so that $\\left\\Vert g\\right\\Vert _{\\mathcal {H}_X}^2 = \\int _0^T \\bigl (\\mathcal {J}^* g (t)\\bigr )^2 dt,\\ g\\in \\mathcal {H}_X.$ The operator $\\mathcal {J}^*$ is related to the adjoint $\\mathcal {K}^*$ of $\\mathcal {K}$ in the following way: for a finite signed measure $\\mu $ on $[0,T]$ , $\\mathcal {K}^* \\mu = \\mathcal {J}^* h\\quad \\mbox{with}\\quad h(t) = \\mu ([t,T]).$ We are going to identify inverse to the operators $\\mathcal {J}$ and $\\mathcal {J}^*$ .",
"Clearly, it is not possible in general, so we will assume that (K2) (S) the function $c$ forms a Sonine pair with some $h\\in L^1[0,T]$ .",
"In this case the operators $\\mathcal {J}$ and $\\mathcal {J}^*$ can be written in terms of “fractional” operators defined above: $\\mathcal {J} f(t) = \\int _0^t \\frac{\\partial }{\\partial t}K(t,s) f(s)ds =\\int _0^t a(s) \\, b(t) \\, c(t-s) f(s) \\, ds= b(t)\\, \\mathrm {I}_{0+}^c (af)(t),$ and $\\mathcal {J}^* g(s) =\\int _s^T \\!",
"a(s) \\, b(t) \\, c(t-s) \\, g(t) \\, dt =a(s)\\,\\mathrm {I}_{T-}^c (bg)(s).$ In order for this operators to be injective, we assume (K2) (K2) the functions $a,b$ are positive a.e.",
"on $[0,T]$ .",
"For $f$ such that $fb^{-1}\\in AC[0,T]$ , define $\\mathcal {L} f(t) = a(t)^{-1}\\mathrm {D}_{0+}^h \\big (f b^{-1}\\big )(t) = a(t)^{-1}\\left(\\int _0^t h(t-s) \\big (fb^{-1}\\big )^{\\prime }(s)ds + h(t) \\big (fb^{-1}\\big )(0)\\right),$ and for $g$ such that $ga^{-1}\\in AC[0,T]$ , define $\\mathcal {L}^* g(s)&=b(s)^{-1}\\mathrm {D}_{T-}^h \\big (g a^{-1}\\big )(s)\\\\ &=b(s)^{-1}\\left(h(T-s) \\big (ga^{-1}\\big )(T) - \\int _s^T h(t-s)\\big (ga^{-1}\\big )^{\\prime }(t)dt\\right).$ Let the assumptions REF , REF and REF hold.",
"Then the operators $\\mathcal {J}$ and $\\mathcal {J}^*$ are injective, and for functions $f,g$ such that $fb^{-1}\\in AC[0,T]$ , $ga^{-1}\\in AC[0,T]$ , $\\mathcal {J} \\mathcal {L} f(t) = f(t), \\qquad \\mathcal {J}^* \\mathcal {L}^* g(s) = g(s).$ Assume that $\\mathcal {J} f = 0$ for some $f\\in L^2[0,T]$ .",
"Then, by REF , $\\mathrm {I}_{0+}^c (bf) = 0$ a.e.",
"on $[0,T]$ .",
"Therefore, for any $t\\in [0,T]$ $0 &= \\int _0^t h(t-s)\\mathrm {I}_{0+}^c (bf)(s)ds = \\int _0^t h(t-s)\\int _0^s c(s-u) b(u)f(u)du \\,ds \\\\&= \\int _0^t \\int _u^t h(t-s)c(s-u)ds\\, b(u)f(u) du = \\int _0^t b(u)f(u) du,$ whence $b f = 0$ a.e.",
"on $[0,T]$ , so, applying to REF once more, $f = 0$ a.e.",
"The injectivity of $\\mathcal {J}^*$ is shown similarly, and the second statement follows from Lemma REF .",
"Now we are in a position to invert the covariance operator $\\mathcal {R} = \\mathcal {K}\\mathcal {K}^*$ of $X$ .",
"We need a further assumption.",
"(K2) (K3) $a^{-1}\\in C^1[0,T]$ , $d:= b^{-1}\\in C^2[0,T]$ and either $d(0) = d^{\\prime }(0) = 0$ or $a^{-2}h\\in C^1[0,T]$ .",
"Let the assumptions REF , REF – REF hold, and $f\\in C^3[0,T]$ with $f(0)= 0$ .",
"Then for $h = \\mathcal {L}^*\\mathcal {L} f^{\\prime }$ , the measure $\\mu ([t,T]) = h(t)$ is such that $\\mathcal {R} \\mu = f$ .",
"Thanks to REF , $f^{\\prime }b^{-1}\\in AC[0,T]$ and $a(t)^{-1}\\mathcal {L} f^{\\prime }(t) = a(t)^{-2}\\left(\\int _0^t h(t-s) \\big (fb^{-1}\\big )^{\\prime }(s)ds + h(t) \\big (f^{\\prime }b^{-1}\\big )(0)\\right).$ Similarly to (REF ), $\\int _0^t h(t-s) \\big (fb^{-1}\\big )^{\\prime }(s)ds$ is absolutely continuous with $\\frac{d}{dt}\\int _0^t h(t-s) \\big (f^{\\prime }b^{-1}\\big )^{\\prime }(s)ds = \\int _0^t h(s) \\big (f^{\\prime }b^{-1}\\big )^{\\prime \\prime }(t-s)ds + h(t) \\big (f^{\\prime }b^{-1}\\big )^{\\prime }(0).$ Then, thanks to REF , both summands in the right-hand side of (REF ) are absolutely continuous with bounded derivatives.",
"So by Proposition REF , $\\mathcal {K}^* \\mu = \\mathcal {J}^* h = \\mathcal {J}^* \\mathcal {L}^* \\mathcal {L} f^{\\prime }= \\mathcal {L} f^{\\prime } .$ and $\\mathcal {J} \\mathcal {K}^* \\mu = \\mathcal {J} \\mathcal {L} f^{\\prime } = f^{\\prime } .$ Therefore, $\\mathcal {R} \\mu \\,(t) =\\mathcal {K} \\mathcal {K}^* \\mu \\, (t) =\\int _0^t \\mathcal {J} \\mathcal {K}^* \\mu \\, (s) \\, ds =\\int _0^t f^{\\prime }(s) \\, ds = f(t)$ as required.",
"Now we recall the definition of integral with respect to the $X$ given by (REF ); for more details see [1].",
"Define $I_X(\\mathbb {1}_{[0,t]}) = \\int _0^T \\mathbb {1}_{[0,t]}(s) \\, dX_s = X_t$ and extend this by linearity to the set $\\mathcal {S}$ of piecewise constant function.",
"Then, for any $g\\in \\mathcal {S}$ , $\\mathrm {E}\\left[\\,I_X(g)^2\\,\\right] = \\left\\Vert g\\right\\Vert ^2_{\\mathcal {H}_X}.$ Therefore, $I_X$ can be extended to isometry between $\\mathcal {H}_X$ and a subspace of $L^2(\\Omega )$ .",
"Moreover, for any $g\\in \\mathcal {H}_X$ , $\\int _0^T \\!",
"g(t) \\, dX_t = \\int _0^T \\!",
"\\mathcal {J}^* g(t) \\, dW_t.$ Let the assumptions $(\\mathrm {S})$ , $(\\mathrm {K}1)-(\\mathrm {K}3)$ be satisfied, and $X$ be given by (REF ).",
"Then $W_t = \\int _0^t k(t,s) \\, dX_s,$ where $k(t,s) = p(t)b(s)^{-1} h(t-s) - b(s)^{-1}\\int _s^t p^{\\prime }(v) h(v-s) \\, dv ,$ and $p = a^{-1}$ .",
"Write $k(t,s) = k_1(t,s) - k_2(t,s)$ , where $ k_1(t,s) = p(t)b(s)^{-1} h(t-s), k_2(t,s) =b(s)^{-1}\\int _s^t p^{\\prime }(v) h(v-s) dv$ , and transform $\\bigl (\\mathcal {J}^* k_1(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) & =\\int _s^T \\frac{\\partial }{\\partial u}K(u,s) p(t)b(u)^{-1} h(t-u)\\mathbb {1}_{[0,t]}(u)du \\\\& = p(t)\\int _s^t a(s)b(u)c(u-s) b(u)^{-1}h(t-u)du\\, \\mathbb {1}_{[0,t]}(s) \\\\& = p(t)\\,a(s)\\int _s^t c(u-s) h(t-u)du\\, \\mathbb {1}_{[0,t]}(s)\\\\& = p(t)\\,a(s)\\mathbb {1}_{[0,t]}(s).$ Similarly, $\\bigl (\\mathcal {J}^* k_2(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) & =\\int _s^t a(s)\\,c(u-s)\\int _u^t p^{\\prime }(v) h(v-u)dv\\, du\\,\\mathbb {1}_{[0,t]}(s)\\\\& = a(s)\\int _s^t p^{\\prime }(v) \\int _s^v c(u-s)h(v-u)du\\, dv \\,\\mathbb {1}_{[0,t]}(s)\\\\& = a(s)\\int _s^t p^{\\prime }(v) \\, dv \\,\\mathbb {1}_{[0,t]}(s) = a(s)\\bigl (p(t) - p(s)\\bigr )\\mathbb {1}_{[0,t]}(s).$ Consequently, $\\bigl (\\mathcal {J}^* k(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) & = p(t)\\,a(s)\\mathbb {1}_{[0,t]}(s) - a(s)\\bigl (p(t) - p(s)\\bigr )\\mathbb {1}_{[0,t]}(s)\\\\& = a(s) \\, p(s)\\mathbb {1}_{[0,t]}(s) = \\mathbb {1}_{[0,t]}(s).$ Therefore, thanks to (REF ), $\\int _0^T k(t,s) \\, dX_s = \\int _0^T \\bigl (\\mathcal {J}^* k(t,\\cdot )\\mathbb {1}_{[0,t]}\\bigr )(s) \\, dW_s = \\int _0^T \\mathbb {1}_{[0,t]}(s) \\, dW_s = W_t,$ as required.",
"Examples of Sonine kernels Functions $c(s)=s^{-\\alpha }$ and $h(s)=s^{ \\alpha -1}$ with some $\\alpha \\in (0,1/2)$ were considered above in connection with fractional Brownian motion, see subsection REF .",
"For $\\alpha \\in (0,1)$ and $A\\in \\mathbb {R}$ , let $\\gamma =\\Gamma ^{\\prime }(1)$ be Euler-Mascheroni constant, $l = \\gamma -A$ .",
"Then $c(x) = \\frac{1}{\\Gamma (\\alpha )} x^{\\alpha -1}\\left(\\ln \\textstyle {\\frac{1}{x}} + A\\right)$ and $h(x) = \\int _0^\\infty \\frac{x^{t-\\alpha }e^{lt}}{\\Gamma (1-\\alpha + t)}dt$ create a Sonine pair, see [12].",
"This example was proposed by Sonine himself [13]: for $\\nu \\in (0,1)$ , $h(x) = x^{-\\nu /2}J_{-\\nu }(2\\sqrt{x}), \\quad c(x) = x^{(\\nu -1)/2} I_{\\nu -1}(2\\sqrt{x}),$ where $J$ and $I$ are, respectively, Bessel and modified Bessel functions of the first kind, $J_\\nu (y)=\\frac{y^\\nu }{2^\\nu }\\sum _{k=0}^\\infty \\frac{(-1)^ky^{2k}2^{-2k}}{k!\\Gamma (\\nu +k+1)},$ and $I_\\nu (y)=\\frac{y^\\nu }{2^\\nu }\\sum _{k=0}^\\infty \\frac{ y^{2k}2^{-2k}}{k!\\Gamma (\\nu +k+1)}.$ In particular, setting $\\nu = 1/2$ , we get the following Sonine pair: $h(x) = \\frac{\\cos 2\\sqrt{x}}{2\\sqrt{\\pi x}},\\quad c(x) = \\frac{\\cosh 2\\sqrt{x}}{2\\sqrt{\\pi x}}.$ It is interesting that the creation of Sonine pairs allows to get the relations between the special functions (see [9]).",
"Let $c(x)=x^{-1/2}\\cosh (ax^{1/2}),$ and let $h(x)=\\int _0^xs^{\\nu /2}J_\\nu (as^{1/2})\\,(x-s)^\\gamma ds$ be a fractional integral of $s^{\\nu /2}J_\\nu (as^{1/2})$ , where $-1<\\nu <-\\frac{1}{2}$ , $\\gamma +\\nu =-\\frac{3}{2}$ .",
"If we denote $F_y(\\lambda )$ Laplace transform of function $y$ at point $\\lambda $ , then the Laplace transforms of these functions equal $F_c(\\lambda )=(\\pi /\\lambda )^{1/2}\\exp (a^2/4\\lambda ),\\\\\\begin{aligned}F_{h}(\\lambda )&=\\Gamma (\\gamma +1)2^{-\\nu }a^\\nu \\lambda ^{-\\nu -1} \\exp (-a^2/4\\lambda )\\lambda ^{-\\gamma -1} \\\\&=\\Gamma (\\gamma +1)2^{-\\nu }a^\\nu \\lambda ^{-1/2}\\exp (-a^2/4\\lambda ),\\end{aligned}\\\\F_c(\\lambda )F_{h}(\\lambda )=\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi } a^\\nu \\lambda ^{-1},\\qquad \\lambda >0,$ whence their convolution equals $(c\\ast h)_t=\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi }a^\\nu , \\qquad t>0.$ Therefore $c(x)$ and $(\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi }a^\\nu )^{-1}h(x)$ create a Sonine pair.",
"However, comparing with Example with $a=2$ , and taking into account that the pair in Sonine pair is unique, we get that $ 4\\sqrt{\\pi }(\\Gamma (\\gamma +1))^{-1}\\int _0^xs^{\\nu /2}J_\\nu (2s^{1/2})\\,(x-s)^\\gamma ds=\\frac{\\cos 2\\sqrt{x}}{ \\sqrt{ x}}.$ Similarly, let $c(x)=\\int _0^xt^{-1/2}\\cosh (at^{1/2})\\,(x-t)^\\gamma dt,\\ h(x)=x^{\\nu /2}J_\\nu (ax^{1/2})$ with $\\gamma \\in (-1,-\\frac{1}{2})$ , $\\nu \\in (-1,0)$ ,, $\\gamma +\\nu =-\\frac{3}{2}$ .",
"Then $F_{c}(\\lambda )=\\pi ^{1/2}\\Gamma (\\gamma +1)\\lambda ^{-\\gamma -3/2}\\exp (a^2/4\\lambda ),$ and $F_h(\\lambda )=\\frac{a^\\nu }{2^\\nu }\\lambda ^{-\\nu -1}\\exp (-a^2/4\\lambda ),\\;\\text{whence}\\;F_{c}(\\lambda )F_h(\\lambda )=\\pi ^{1/2}\\Gamma (\\gamma +1)\\frac{a^\\nu }{2^\\nu }\\lambda ^{-1}.$ If we put $a=2$ and compare with (REF ), we get the following representation $\\pi ^{-1/2}(\\Gamma (\\gamma +1))^{-1}\\int _0^xt^{-1/2}\\cosh (2t^{1/2})\\,(x-t)^\\gamma dt=x^{(-\\nu -1)/2} I_{-\\nu -1}(2\\sqrt{x}).$ On the way of creation of the new Sonine pairs, a natural idea is to consider $g(s)=e^{\\beta s}s^{ \\alpha -1}$ with $\\beta \\in \\mathbb {R}$ and examine if this function admits a Sonine pair.",
"It happens so that the answer to this question is positive, but far from obvious and not simple.",
"All preliminary results are contained in subsection REF .",
"Let $g(x) = \\frac{\\exp (\\beta x)}{\\Gamma (\\alpha ) x^{1-\\alpha }}, \\quad 0<\\alpha <1, \\quad \\beta <0;\\qquad y(x) = 1.$ Then $h(x) = \\alpha \\beta \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha +1; \\: 2; \\: \\beta x)< 0, \\qquad x\\in [0,T],$ where $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }$ is Kummer hypergeometric function; see Section REF in the Appendix.",
"The conditions of Theorem REF hold true.",
"The equation (REF ) has a unique solution in $L^{1}[0,T]$ (Actually, it has many solutions, but each two solutions are equal almost everywhere.)",
"The solution has a representative that is continuous and attains only positive values on the left-open interval $(0,T]$ , and it is a Sonine pair to $g(s)=e^{\\beta s}s^{ \\alpha -1}$ .",
"Appendix Inequalities for norms of convolutions and products Recall notation $\\Vert f\\Vert _p$ for the norm of function $f\\in L^p(\\mathbb {R})$ , $p \\in [1, \\infty ]$ .",
"The convolution of two measurable functions $f$ and $g$ is defined by integration $(f*g)(t) = \\int _{\\mathbb {R}} f(s) g(t-s) \\, ds .$ Now we state an inequality for the norm of convolution of two functions.",
"If $p\\in [1,\\infty ]$ , $q\\in [1,\\infty ]$ but $1/p+1/q\\ge 1$ , $f\\in L^p(\\mathbb {R})$ , $g\\in L^q(\\mathbb {R})$ , then the convolution $f*g$ is well-defined almost everywhere (that is the integral in (REF ) converges absolutely for almost all $t\\in \\mathbb {R}$ ), $f * g \\in L^{(1/p+1/q-1)^{-1}}(\\mathbb {R})$ , and $\\Vert f*g\\Vert _{(1/p+1/q-1)^{-1}}\\le \\Vert f\\Vert _p \\, \\Vert g\\Vert _q.$ Now we state an inequality for the norm of the product of two functions $(fg)(t) = f(t) g(t)$ .",
"We call it Hölder inequality for non-conjugate exponents.",
"If $p\\in [1,\\infty ]$ , $q\\in [1,\\infty ]$ , $1/p+1/q\\le 1$ , $f\\in L^p(\\mathbb {R})$ , $g\\in L^q(\\mathbb {R})$ , then $fg \\in L^{(1/p+1/q)^{-1}}(\\mathbb {R})$ and $\\Vert fg\\Vert _{(1/p+1/q)^{-1}} \\le \\Vert f\\Vert _p \\, \\Vert g\\Vert _q.$ Now we state an inequality for the norms in $L_p[a,b]$ and $L_q[a,b]$ .",
"If $-\\infty < a < b < \\infty $ , $1 \\le p \\le q \\le \\infty $ , $f \\in L^q(\\mathbb {R})$ and the $f(t)=0$ for all $t\\notin [a,b]$ , then $f\\in L^p(\\mathbb {R})$ and $\\Vert f\\Vert _p \\le (b-a)^{1/p-1/q} \\, \\Vert f\\Vert _q .$ Conditions for inequalities (REF ) and (REF ) are over-restrictive because of restrictive notation $\\Vert f\\Vert _p$ .",
"This notation can be extended to all $p\\in (0,\\infty ]$ and all measurable functions $f$ .",
"Then the conditions for inequalities (REF ) and (REF ) may be relaxed.",
"Inequality (REF ) is proved in [7]; see item (2) in the remarks after this theorem and part (A) of its proof.",
"If $p<\\infty $ and $q<\\infty $ , then inequality (REF ) follows from the conventional Hölder inequality.",
"Otherwise, if $p=\\infty $ or $q=\\infty $ , then inequality (REF ) is trivial.",
"Inequality (REF ) can be rewritten as $\\Vert f \\operatorname{\\mathbb {1}}_{[a,b]}\\!\\Vert _p \\le \\Vert \\!\\operatorname{\\mathbb {1}}_{[a,b]}\\!\\Vert _{(1/p-1/q)^{-1}} \\, \\Vert f\\Vert _q $ , and so follows from (REF ).",
"Continuity of trajectories and Hölder condition Kolmogorov continuity theorem provides sufficiency conditions for a stochastic process to have a continuous modification.",
"The following theorem aggregates Theorems 2, 4 and 5 in [3].",
"[Kolmogorov continuity theorem] Let $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ be a stochastic process.",
"If there exist $K \\ge 0$ , $\\alpha >0$ and $\\beta >0$ such that $\\mathrm {E}\\left[\\,|X_t - X_s|^\\alpha \\,\\right] \\le K\\, |t-s|^{1+\\beta }\\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T,$ then The process $X$ has a continuous modification; Every continuous modification of the process $X$ whose trajectories almost surely satisfies Hölder condition for all exponents $\\gamma \\in (0,\\: \\beta /\\alpha )$ ; There exists a modification of the process $X$ that satisfies Hölder condition for exponent $\\gamma \\in (0, \\: \\beta /\\alpha )$ .",
"This theorem can be applied for Gaussian processes.",
"Let $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ be a centered Gaussian process.",
"If there exist $K\\ge 0$ and $\\delta > 0$ such that $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le K \\, |t-s|^\\delta \\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T,$ then the following holds true: The process $X$ has a modification $\\widetilde{X}$ that has continuous trajectories.",
"For every $\\gamma $ , $0< \\gamma < \\frac{1}{2}\\delta $ , the trajectories of the process $\\widetilde{X}$ satisfy $\\gamma $ -Hölder condition almost surely.",
"The process $X$ has a modification that satisfies Hölder condition for all exponents $\\gamma \\in (0, \\frac{1}{2}\\delta )$ .",
"Since $X_s - X_t$ is a centered Gaussian variable, $\\mathrm {E}\\left[\\,|X_t - X_s|^\\alpha \\,\\right] = \\frac{2^{\\alpha /2}}{\\sqrt{\\pi }}\\Gamma \\!\\left(\\frac{\\alpha +1}{2}\\right)\\left(\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right]\\right)^{\\alpha /2} .$ The first statement of the corollary can be proved by applying Kolmogorov continuity theorem for $\\alpha > 2/\\delta $ and $\\beta = \\frac{1}{2} \\alpha \\delta - 1$ .",
"The second statement of the corollary can be proved by applying Kolmogorov continuity theorem for $\\alpha > \\frac{2}{\\delta - 2\\gamma }$ and $\\beta = \\frac{1}{2} \\alpha \\delta - 1$ .",
"Consider the random event $A &= \\left\\lbrace \\forall \\gamma \\in (0, \\textstyle {\\frac{1}{2}} \\delta ) : \\widetilde{X} \\;\\mbox{satisfies $\\gamma $-Hölder condition} \\right\\rbrace \\\\ &=\\left\\lbrace \\forall n\\in \\mathbb {N} : \\widetilde{X} \\;\\mbox{satisfies $\\frac{1}{2} \\left(1-\\frac{1}{n}\\right) \\delta $-Hölder condition} \\right\\rbrace .$ (The measurability of $A$ follows from the continuity of the process $\\widetilde{X}$ ).",
"By the second statement of Corollary REF $\\mathrm {P}(A) = 1$ .",
"Thus, $\\lbrace \\widetilde{X}_t \\mathbb {1}_{A}, \\;t\\in [0,t]\\rbrace $ is the desired modification which satisfies Hölder condition for all exponents $\\gamma \\in (0, \\frac{1}{2} \\delta )$ .",
"Corollary REF holds true even without assumption that the Gaussian process $X$ is centered.",
"The first statement of Corollary REF can be proved with Xavier Fernique's continuity criterion [5] as well.",
"Let $\\lbrace X_t, \\; t\\in [0, T]\\rbrace $ be a centered Gaussian process.",
"Suppose that there exist $\\delta > 0$ and a nondecreasing continuous function $F : [0, T] \\rightarrow \\mathbb {R}$ such that $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le (F(t) - F(s))^\\delta \\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T.$ Then The process $X$ have a modification $\\widetilde{X}$ that has continuous trajectories.",
"If the function $F$ satisfies Lipschitz condition in an interval $[a,b] \\subset [0,T]$ , then for every $\\gamma $ , $0< \\gamma < \\frac{1}{2}\\delta $ , the process $\\widetilde{X}$ has a modification whose trajectories satisfy $\\gamma $ -Hölder property on the interval $[a, b]$ .",
"Without loss of generality, we can assume that the function $F$ is strictly increasing.",
"Indeed, if the condition (REF ) holds true for $F$ being continuous nondecreasing function $F_1$ , it also holds true for $F=F_2$ with $F_2(t)=F_1(t)+t$ , where $F_2$ is a continuous strictly increasing function.",
"With this additional assumption, the inverse function $F^{-1}$ is one-to-one, strictly increasing continuous function $[F(0), F(T)] \\rightarrow [0, T]$ .",
"Consider a stochastic process $\\lbrace Y_u, \\; u \\in [F(0), F(T)]\\rbrace $ , with $Y_u = Y_{F^{-1}(u)}$ .",
"The stochastic process $Y$ is centered and Gaussian; it satisfies condition $\\mathrm {E}\\left[\\,(Y_v - Y_u)^2\\,\\right]= \\mathrm {E}\\left[\\,(X_{F^{-1}(v)} - X_{F^{-1}(u)})^2\\,\\right]\\le (F(F^{-1}(v)) - F(F^{-1}(u)))^\\delta = (v - u)^\\delta $ for all $F(0) \\le u \\le v \\le F(T)$ .",
"According to Corollary REF , the process $Y$ has a modification $\\widetilde{Y}$ with continuous trajectories.",
"Then $\\widetilde{X}$ with $\\widetilde{X}_t = \\widetilde{Y}_{F(t)}$ is a modification of the process $X$ with continuous trajectories.",
"The second statement of the lemma is a direct consequence of Corollary REF .",
"If the function $F$ satisfies Lipschitz condition with constant $L$ on the interval $[a,b]$ , then $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le L^\\delta (t - s)^\\delta \\quad \\mbox{for all} \\quad a \\le s \\le t \\le b ,$ which is the main condition for Corollary REF .",
"Application of fractional calculus The lower and upper Riemann–Liouville fractional integrals of a function $f\\in L^{1}[a,b]$ are defined as follows: $(I_{a+}^\\alpha f) (x) = \\frac{1}{\\Gamma (\\alpha )}\\int _a^x \\frac{f(t) \\, dt}{(x-t)^{1-\\alpha }}, \\qquad (I_{b-}^\\alpha f) (x) = \\frac{1}{\\Gamma (\\alpha )}\\int _x^b \\frac{f(t) \\, dt}{(t-x)^{1-\\alpha }}.$ The integrals $(I_{a+}^\\alpha f) (x)$ and $(I_{b-}^\\alpha f) (x)$ are well-defined for almost all $x\\in [a,b]$ , and are integrable functions of $x$ , that is $I_{a+}^\\alpha f \\in L^1[a,b]$ and $I_{b-}^\\alpha f \\in L^1[a,b]$ .",
"Thus, $I_{a+}^\\alpha $ and $I_{b-}^\\alpha $ might be considered linear operators $L^1[a,b] \\rightarrow L^1[a,b]$ .",
"A reflection relation for functions $g(x) = f(a+b-x)$ imply the following relation for their fractional integrals: $(I_{b-}^{\\alpha } g)(x) = (I_{a+}^{\\alpha } f) (a+b-x);$ see [11].",
"The integration-by-parts formula is given, e.g., in [11].",
"[integration-by-parts formula] Let $\\alpha > 0$ , $f \\in L^p[a,b]$ , $g \\in L^q[a,b]$ , $p \\in [1, +\\infty ]$ , $q \\in [1, +\\infty ]$ , while $\\frac{1}{p} + \\frac{1}{q} \\le 1 + \\alpha $ and $\\max \\Bigl (1 + \\alpha - \\frac{1}{p} - \\frac{1}{q},\\:\\min \\!\\left(1 - \\frac{1}{p}, \\:1 - \\frac{1}{q}\\right)\\Bigr ) > 0$ .",
"Then $\\int _a^b (I_{a+}^\\alpha f)(t) \\, g(t) \\, dt =\\int _a^b f(t) \\, (I_{b-}^\\alpha g)(t) \\, dt.$ Now we establish conditions for a function to be in the range of the fractional operator $I_{a+}^\\alpha $ , and we provide formulas for the preimage, which is called a fractional derivative.",
"The following statements are the modifications of the Theorem 2.1 and following corollary in [11].",
"The formulas for the fractional derivative are also provided in [8].",
"Let $0 < \\alpha < 1$ .",
"Consider the integral equation $I_{a+}^\\alpha f = g$ with unknown function $f \\in L^1[a,b]$ and known function (i.e., a parameter) $g \\in L^1[a,b]$ .",
"Denote $h(x) = {\\left\\lbrace \\begin{array}{ll} (I_{a+}^{1-\\alpha } g) (x)& \\mbox{if $a < x \\le b$}, \\\\0& \\mbox{if $x=a$}.\\end{array}\\right.",
"}$ If $h \\in \\operatorname{\\!\\textit {AC}}[a,b]$ , then equation (REF ) has a unique (up to equality almost everywhere in $[a,b]$ ) solution $f$ , namely $f(x) = h^{\\prime }(x)$ .",
"Otherwise, if $h \\notin \\operatorname{\\!\\textit {AC}}[a,b]$ , then equation (REF ) has no solutions in $L^1[a,b]$ .",
"If for some $x\\in (a,b]$ the integral $(I_{a+}^{1-\\alpha } g) (x)$ is not well-defined, then equation (REF ) does not have solutions in $L^1[a,b]$ .",
"Let $0 < \\alpha < 1$ .",
"The integral equation (REF ) with unknown function $f \\in L^1[a,b]$ and known function $g \\in \\operatorname{\\!\\textit {AC}}[a,b]$ has a unique solution.",
"The solution is equal to $f(x) &=(I_{a+}^{1-\\alpha }(g^{\\prime }))(x) +\\frac{g(a)}{\\Gamma (1-\\alpha )\\,(x-a)^\\alpha }\\\\ &=\\frac{1}{\\Gamma (1-\\alpha )}\\left( \\int _a^x \\frac{g^{\\prime }(t)\\,dt}{(x-t)^\\alpha }+ \\frac{g(a)}{(x-a)^\\alpha } \\right) .$ Existence of the solution to Volterra integral equation where the integral operator is an operator of convolution with integrable singularity at 0 Consider Volterra integral equation of the first kind $\\int _0^x f(t) \\, g(x-t) \\, dt = y(x), \\qquad x \\in (0,T],$ with $g(x)$ and $y(x)$ known (parameter) functions and $f(x)$ unknown function.",
"Suppose that the function $g(x)$ is integrable in the interval $(0,T]$ but behaves asymptotically as a power function in the neighborhood of 0: $g(x) \\sim \\frac{K}{x^{1-\\alpha }}, \\qquad x \\rightarrow 0,$ where $0 < \\alpha < 1$ .",
"More specifically, assume that $g(x)$ admits a representation $g(x) = \\frac{1}{\\Gamma (\\alpha ) x^{1-\\alpha }}+ (I^{\\alpha }_{0+} h)(x) =\\frac{1}{\\Gamma (\\alpha )}\\left( \\frac{1}{x^{1-\\alpha }} + \\int _0^x \\frac{h(t)\\, dt}{(x-t)^{1-\\alpha }} \\right),$ where $\\Gamma (\\alpha )$ is a gamma function, $I^{\\alpha }_{0+}h$ is a lower Riemann–Liouville fractional integral of $h$ , $(I^{\\alpha }_{0+} h)(x) =\\frac{1}{\\Gamma (\\alpha )}\\int _0^x \\frac{h(t)\\, dt}{(x-t)^{1-\\alpha }},$ and $h(x)$ is a absolutely continuous function.",
"The sufficient conditions for existence and uniqueness of the solution to integral equation claimed in [8] are not satisfied.",
"The kernel of the integration operator in (REF ) is unbounded, and $y(0)$ might be nonzero.",
"But we use Remark 2 in [8].",
"We reduce the Volterra integral equation of the first kind to a Volterra integral equation of the second kind similarly as it is done for regular functions $g(x)$ ; compare with [8] for the case of regular $g(x)$ .",
"For the next theorem we keep in mind that if a function $f$ is a solution to (REF ), then every function that is equal to $f$ almost everywhere on $[0,T]$ is also a solution to (REF ).",
"Let $y,\\,h \\in C^1[0,T]$ and $g$ be defined in (REF ).",
"Then the equation (REF ) has a unique (up to equality almost everywhere) solution $f \\in L^1[0,T]$ .",
"The solution is (more precisely, some of almost-everywhere equal solutions are) continuous in the left-open interval $(0,T]$ .",
"Substitute (REF ) into (REF ): $\\int _0^x f(t)\\left( \\frac{1}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }}+ (I^{\\alpha }_{0+} h)(x-t) \\right) dt= y(x), \\\\(I^{\\alpha }_{0+} f)(x) +\\int _0^x f(t) \\, (I^{\\alpha }_{0+} h)(x-t) \\, dt= y(x) .$ Denote $h_x(t) = h(x-t)$ .",
"According to equation (REF ), the fractional integrals of $h$ and $h_x$ satisfy the relation $(I^{\\alpha }_{0+} h)(x-t)= (I^{\\alpha }_{x-} h_x)(t)$ .",
"Hence, equation (REF ) is equivalent to the following one: $(I^{\\alpha }_{0+} f)(x) +\\int _0^x f(t) \\, (I^{\\alpha }_{x-} h_x)(t) \\, dt= y(x).$ Now apply the integration-by-parts formula.",
"We have $f \\in L^1[0,x]$ , $h_x \\in L^\\infty [0,x]$ , and $1 + 0 < 1 + \\alpha $ .",
"Hence, by Proposition REF , $\\int _0^x f(t) \\, (I^{\\alpha }_{x-} h_x)(t) \\, dt =\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h_x(t) \\, dt.$ It means that equation (REF ) is equivalent to the following ones: $(I^{\\alpha }_{0+} f)(x) +\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h_x(t) \\, dt= y(x), $ and $(I^{\\alpha }_{0+} f)(x) +\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h(x-t) \\, dt= y(x) .$ Denote $F = I^{\\alpha }_{0+} f$ , and obtain a Volterra integral equation of the second kind: $F(x) = y(x) - \\int _0^x F(t) \\, h(x-t) \\, dt.$ Equation (REF ) has a unique solution in $C[0,T]$ , as well as in $L^1[0,T]$ .",
"In other words, (REF ) has a unique integrable solution, and this solution is a continuous function.",
"According to Theorem REF , either unique (up to almost-everywhere equality) function $f$ , or no functions $f$ correspond to the function $F$ .",
"Thus, all integrable solution to integral equation (REF ) are equal almost everywhere.",
"Now we construct a solution to equation (REF ) that is continuous and integrable on $(0,T]$ .",
"Differentiating (REF ), we obtain $F^{\\prime }(x) = y^{\\prime }(x) - F(x)\\, h(0) - \\int _0^x F(t) \\, h^{\\prime }(x-t) \\, dt,$ whence $F \\in C^1[0,T]$ .",
"According to Corollary REF , the integral equation $F = I_{0+}^\\alpha f$ has a unique solution $f\\in L^1[0,T]$ , which is equal to $f(x)= \\frac{1}{\\Gamma (1-\\alpha )}\\left( \\int _0^x \\frac{F^{\\prime }(t)\\,dt}{(x-t)^\\alpha } +\\frac{F(0)}{x^\\alpha } \\right) .$ The constructed function $f(x)$ is continuous and integrable in $(0,T]$ , and $f(x)$ is a solution to (REF ).",
"In Theorem REF the condition $h \\in C^1[0,T]$ can be relaxed and replaced with the condition $h \\in \\operatorname{\\!\\textit {AC}}[0,T]$ .",
"In other words, if the function $h$ is absolutely continuous but is not continuously differentiable, the statement of Theorem REF still holds true.",
"Example: $g(x) = \\exp (\\beta x) x^{\\alpha -1} / \\Gamma (\\alpha )$ and $y(x) = 1$Example: $g(x) = \\exp (\\beta x) x^{\\alpha -1} / \\Gamma (\\alpha )$ and $y(x) = 1$ It is well known that $\\int _0^x \\frac{1}{\\Gamma (1-\\alpha ) t^\\alpha } \\,\\frac{1}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }} \\, dt = 1 .$ In this section, we prove that the equation $\\int _0^x f(t) \\,\\frac{e^{(x-t)\\beta }}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }} \\, dt = 1$ has an integrable solution.",
"According to (REF ), $f(x) = x^{-\\alpha } / \\, \\Gamma (1-\\alpha )$ is a solution to (REF ) if $\\beta = 0$ .",
"Denote $g(x) = \\frac{\\exp (\\beta x)}{\\Gamma (\\alpha ) x^{1-\\alpha }} .$ Demonstrate that $g(x)$ admits a representation (REF ).",
"To construct $h$ , we need Kummer confluent hypergeometric function [14]: $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;z) = \\frac{1}{\\mathrm {B}(a,\\: b-a)}\\int _0^1 e^{zt} t^{a-1} (1-t)^{b-a-1} \\, dt,\\qquad 0<a<b, \\quad z\\in \\mathbb {C} .$ For $a$ and $b$ fixed, $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;\\,\\cdot \\,)$ is an entire function.",
"Its derivative equals $\\frac{\\partial }{\\partial z}\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;z) = \\frac{a}{b}\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a+1;\\:b+1;\\:z) .$ For all $0<a<b$ and $z\\in \\mathbb {R}$ $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a; b; z) > 0, \\qquad \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a; b; 0) = 1.$ Notice that if $0<\\alpha <1$ and $x>0$ , then $\\frac{1}{\\mathrm {B}(\\alpha , \\: 1-\\alpha )}\\int _0^x \\frac{\\exp (zt) \\, dt}{t^{1-\\alpha } (x-t)^\\alpha } =\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; xz) .$ Being considered an equation for unknown $h$ , (REF ) is equivalent to $I_{0+}^\\alpha h = g_0$ , where $g_0(x) = g(x) - \\frac{1}{\\Gamma (\\alpha ) x^{1-\\alpha }}= \\frac{e^{\\beta x} - 1}{\\Gamma (\\alpha ) x^{1-\\alpha }} .$ Then $(I_{0+}^{1-\\alpha } g_0)(x) =\\frac{1}{\\mathrm {B}(\\alpha , \\: 1-\\alpha )}\\int _0^x \\frac{e^{\\beta t} - 1}{t^{1-\\alpha }(1-t)^\\alpha } \\, dt =\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1 .$ Besides, $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1$ is an absolutely continuous function in $x$ , and$\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1;\\beta x) - 1 = 0$ if $x=0$ .",
"According to Theorem REF , the equation $I_{0+}^\\alpha h = g_0$ has the unique solution $h =L^1[0,T]$ , which is equal to $h(x) = \\frac{\\partial (\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1)}{\\partial x} =\\alpha \\beta \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha +1;\\: 2;\\: \\beta x).$ The constructed function $h(x)$ is a solution to (REF ) and is continuously differentiable.",
"In summary, $h \\in C^1[0,T]$ , $y(x) = 1$ , and $y \\in C^1[0,T]$ .",
"According to Theorem REF the integral equation $\\int _0^x f(t) \\, g(x-t) \\, dt = 1, \\qquad x\\in (0,T],$ has a unique solution $f \\in L^1[0,T]$ (up to equality almost everywhere).",
"The solution is continuous in $(0,T]$ .",
"The fact that the functions $g$ and $h$ defined in (REF ) and (REF ), respectively, satisfy (REF ), can be checked directly.",
"For such verification, one can apply Lemma 2.2(i) from [10].",
"Positive solution to the Volterra integral equation Let the conditions of Theorem REF hold true.",
"Additionally, let $y(x)>0, \\quad y^{\\prime }(x)\\ge 0, \\quad h(x) < 0 \\qquad \\mbox{for all $x\\in [0,T]$}.$ Then the continuous solution $f(x)$ to (REF ) attains only positive values in $(0,T]$ .",
"Notice that (REF ) implies $F(0) = y(0) > 0$ .",
"Taking this into account, let's differentiate both sides of (REF ) the other way: $F(x) = y(x) - \\int _0^x F(x-t) \\, h(t) \\, dt,\\nonumber \\\\F^{\\prime }(x) = y^{\\prime }(x) - F(0) \\, h(x)- \\int _0^x F^{\\prime }(x-t) \\, h(t) \\, dt .$ Let us prove that $F^{\\prime }(x)>0$ in $[0,T]$ by contradiction.",
"Assume the contrary, that is $\\exists x \\in [0,1] : F^{\\prime }(x) \\le 0$ .",
"Since the function $F^{\\prime }(x)$ is continuous in $[0,T]$ , the contrary implies the existence of the minimum in $x_0 = \\min \\lbrace x\\in [0,T] : F^{\\prime }(x) \\le 0\\rbrace .$ But for $x=x_0$ the left-hand side in (REF ) is less or equal then zero, while the right-hand side is greater than zero.",
"Thus, (REF ) does not hold true.",
"The proof also works for $x_0 = 0$ .",
"There is a contradiction.",
"Thus, we have proved that $F^{\\prime }(x)>0$ for all $x\\in [0,T]$ .",
"By (REF ), $f(x)>0$ for all $x\\in (0,T]$ ."
],
[
"Examples of Sonine kernels",
"Functions $c(s)=s^{-\\alpha }$ and $h(s)=s^{ \\alpha -1}$ with some $\\alpha \\in (0,1/2)$ were considered above in connection with fractional Brownian motion, see subsection REF .",
"For $\\alpha \\in (0,1)$ and $A\\in \\mathbb {R}$ , let $\\gamma =\\Gamma ^{\\prime }(1)$ be Euler-Mascheroni constant, $l = \\gamma -A$ .",
"Then $c(x) = \\frac{1}{\\Gamma (\\alpha )} x^{\\alpha -1}\\left(\\ln \\textstyle {\\frac{1}{x}} + A\\right)$ and $h(x) = \\int _0^\\infty \\frac{x^{t-\\alpha }e^{lt}}{\\Gamma (1-\\alpha + t)}dt$ create a Sonine pair, see [12].",
"This example was proposed by Sonine himself [13]: for $\\nu \\in (0,1)$ , $h(x) = x^{-\\nu /2}J_{-\\nu }(2\\sqrt{x}), \\quad c(x) = x^{(\\nu -1)/2} I_{\\nu -1}(2\\sqrt{x}),$ where $J$ and $I$ are, respectively, Bessel and modified Bessel functions of the first kind, $J_\\nu (y)=\\frac{y^\\nu }{2^\\nu }\\sum _{k=0}^\\infty \\frac{(-1)^ky^{2k}2^{-2k}}{k!\\Gamma (\\nu +k+1)},$ and $I_\\nu (y)=\\frac{y^\\nu }{2^\\nu }\\sum _{k=0}^\\infty \\frac{ y^{2k}2^{-2k}}{k!\\Gamma (\\nu +k+1)}.$ In particular, setting $\\nu = 1/2$ , we get the following Sonine pair: $h(x) = \\frac{\\cos 2\\sqrt{x}}{2\\sqrt{\\pi x}},\\quad c(x) = \\frac{\\cosh 2\\sqrt{x}}{2\\sqrt{\\pi x}}.$ It is interesting that the creation of Sonine pairs allows to get the relations between the special functions (see [9]).",
"Let $c(x)=x^{-1/2}\\cosh (ax^{1/2}),$ and let $h(x)=\\int _0^xs^{\\nu /2}J_\\nu (as^{1/2})\\,(x-s)^\\gamma ds$ be a fractional integral of $s^{\\nu /2}J_\\nu (as^{1/2})$ , where $-1<\\nu <-\\frac{1}{2}$ , $\\gamma +\\nu =-\\frac{3}{2}$ .",
"If we denote $F_y(\\lambda )$ Laplace transform of function $y$ at point $\\lambda $ , then the Laplace transforms of these functions equal $F_c(\\lambda )=(\\pi /\\lambda )^{1/2}\\exp (a^2/4\\lambda ),\\\\\\begin{aligned}F_{h}(\\lambda )&=\\Gamma (\\gamma +1)2^{-\\nu }a^\\nu \\lambda ^{-\\nu -1} \\exp (-a^2/4\\lambda )\\lambda ^{-\\gamma -1} \\\\&=\\Gamma (\\gamma +1)2^{-\\nu }a^\\nu \\lambda ^{-1/2}\\exp (-a^2/4\\lambda ),\\end{aligned}\\\\F_c(\\lambda )F_{h}(\\lambda )=\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi } a^\\nu \\lambda ^{-1},\\qquad \\lambda >0,$ whence their convolution equals $(c\\ast h)_t=\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi }a^\\nu , \\qquad t>0.$ Therefore $c(x)$ and $(\\Gamma (\\gamma +1)2^{-\\nu }\\sqrt{\\pi }a^\\nu )^{-1}h(x)$ create a Sonine pair.",
"However, comparing with Example with $a=2$ , and taking into account that the pair in Sonine pair is unique, we get that $ 4\\sqrt{\\pi }(\\Gamma (\\gamma +1))^{-1}\\int _0^xs^{\\nu /2}J_\\nu (2s^{1/2})\\,(x-s)^\\gamma ds=\\frac{\\cos 2\\sqrt{x}}{ \\sqrt{ x}}.$ Similarly, let $c(x)=\\int _0^xt^{-1/2}\\cosh (at^{1/2})\\,(x-t)^\\gamma dt,\\ h(x)=x^{\\nu /2}J_\\nu (ax^{1/2})$ with $\\gamma \\in (-1,-\\frac{1}{2})$ , $\\nu \\in (-1,0)$ ,, $\\gamma +\\nu =-\\frac{3}{2}$ .",
"Then $F_{c}(\\lambda )=\\pi ^{1/2}\\Gamma (\\gamma +1)\\lambda ^{-\\gamma -3/2}\\exp (a^2/4\\lambda ),$ and $F_h(\\lambda )=\\frac{a^\\nu }{2^\\nu }\\lambda ^{-\\nu -1}\\exp (-a^2/4\\lambda ),\\;\\text{whence}\\;F_{c}(\\lambda )F_h(\\lambda )=\\pi ^{1/2}\\Gamma (\\gamma +1)\\frac{a^\\nu }{2^\\nu }\\lambda ^{-1}.$ If we put $a=2$ and compare with (REF ), we get the following representation $\\pi ^{-1/2}(\\Gamma (\\gamma +1))^{-1}\\int _0^xt^{-1/2}\\cosh (2t^{1/2})\\,(x-t)^\\gamma dt=x^{(-\\nu -1)/2} I_{-\\nu -1}(2\\sqrt{x}).$ On the way of creation of the new Sonine pairs, a natural idea is to consider $g(s)=e^{\\beta s}s^{ \\alpha -1}$ with $\\beta \\in \\mathbb {R}$ and examine if this function admits a Sonine pair.",
"It happens so that the answer to this question is positive, but far from obvious and not simple.",
"All preliminary results are contained in subsection REF .",
"Let $g(x) = \\frac{\\exp (\\beta x)}{\\Gamma (\\alpha ) x^{1-\\alpha }}, \\quad 0<\\alpha <1, \\quad \\beta <0;\\qquad y(x) = 1.$ Then $h(x) = \\alpha \\beta \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha +1; \\: 2; \\: \\beta x)< 0, \\qquad x\\in [0,T],$ where $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }$ is Kummer hypergeometric function; see Section REF in the Appendix.",
"The conditions of Theorem REF hold true.",
"The equation (REF ) has a unique solution in $L^{1}[0,T]$ (Actually, it has many solutions, but each two solutions are equal almost everywhere.)",
"The solution has a representative that is continuous and attains only positive values on the left-open interval $(0,T]$ , and it is a Sonine pair to $g(s)=e^{\\beta s}s^{ \\alpha -1}$ ."
],
[
"Inequalities for norms of convolutions and products",
"Recall notation $\\Vert f\\Vert _p$ for the norm of function $f\\in L^p(\\mathbb {R})$ , $p \\in [1, \\infty ]$ .",
"The convolution of two measurable functions $f$ and $g$ is defined by integration $(f*g)(t) = \\int _{\\mathbb {R}} f(s) g(t-s) \\, ds .$ Now we state an inequality for the norm of convolution of two functions.",
"If $p\\in [1,\\infty ]$ , $q\\in [1,\\infty ]$ but $1/p+1/q\\ge 1$ , $f\\in L^p(\\mathbb {R})$ , $g\\in L^q(\\mathbb {R})$ , then the convolution $f*g$ is well-defined almost everywhere (that is the integral in (REF ) converges absolutely for almost all $t\\in \\mathbb {R}$ ), $f * g \\in L^{(1/p+1/q-1)^{-1}}(\\mathbb {R})$ , and $\\Vert f*g\\Vert _{(1/p+1/q-1)^{-1}}\\le \\Vert f\\Vert _p \\, \\Vert g\\Vert _q.$ Now we state an inequality for the norm of the product of two functions $(fg)(t) = f(t) g(t)$ .",
"We call it Hölder inequality for non-conjugate exponents.",
"If $p\\in [1,\\infty ]$ , $q\\in [1,\\infty ]$ , $1/p+1/q\\le 1$ , $f\\in L^p(\\mathbb {R})$ , $g\\in L^q(\\mathbb {R})$ , then $fg \\in L^{(1/p+1/q)^{-1}}(\\mathbb {R})$ and $\\Vert fg\\Vert _{(1/p+1/q)^{-1}} \\le \\Vert f\\Vert _p \\, \\Vert g\\Vert _q.$ Now we state an inequality for the norms in $L_p[a,b]$ and $L_q[a,b]$ .",
"If $-\\infty < a < b < \\infty $ , $1 \\le p \\le q \\le \\infty $ , $f \\in L^q(\\mathbb {R})$ and the $f(t)=0$ for all $t\\notin [a,b]$ , then $f\\in L^p(\\mathbb {R})$ and $\\Vert f\\Vert _p \\le (b-a)^{1/p-1/q} \\, \\Vert f\\Vert _q .$ Conditions for inequalities (REF ) and (REF ) are over-restrictive because of restrictive notation $\\Vert f\\Vert _p$ .",
"This notation can be extended to all $p\\in (0,\\infty ]$ and all measurable functions $f$ .",
"Then the conditions for inequalities (REF ) and (REF ) may be relaxed.",
"Inequality (REF ) is proved in [7]; see item (2) in the remarks after this theorem and part (A) of its proof.",
"If $p<\\infty $ and $q<\\infty $ , then inequality (REF ) follows from the conventional Hölder inequality.",
"Otherwise, if $p=\\infty $ or $q=\\infty $ , then inequality (REF ) is trivial.",
"Inequality (REF ) can be rewritten as $\\Vert f \\operatorname{\\mathbb {1}}_{[a,b]}\\!\\Vert _p \\le \\Vert \\!\\operatorname{\\mathbb {1}}_{[a,b]}\\!\\Vert _{(1/p-1/q)^{-1}} \\, \\Vert f\\Vert _q $ , and so follows from (REF )."
],
[
"Continuity of trajectories and Hölder condition",
"Kolmogorov continuity theorem provides sufficiency conditions for a stochastic process to have a continuous modification.",
"The following theorem aggregates Theorems 2, 4 and 5 in [3].",
"[Kolmogorov continuity theorem] Let $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ be a stochastic process.",
"If there exist $K \\ge 0$ , $\\alpha >0$ and $\\beta >0$ such that $\\mathrm {E}\\left[\\,|X_t - X_s|^\\alpha \\,\\right] \\le K\\, |t-s|^{1+\\beta }\\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T,$ then The process $X$ has a continuous modification; Every continuous modification of the process $X$ whose trajectories almost surely satisfies Hölder condition for all exponents $\\gamma \\in (0,\\: \\beta /\\alpha )$ ; There exists a modification of the process $X$ that satisfies Hölder condition for exponent $\\gamma \\in (0, \\: \\beta /\\alpha )$ .",
"This theorem can be applied for Gaussian processes.",
"Let $\\lbrace X_t,\\; t\\in [0,T]\\rbrace $ be a centered Gaussian process.",
"If there exist $K\\ge 0$ and $\\delta > 0$ such that $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le K \\, |t-s|^\\delta \\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T,$ then the following holds true: The process $X$ has a modification $\\widetilde{X}$ that has continuous trajectories.",
"For every $\\gamma $ , $0< \\gamma < \\frac{1}{2}\\delta $ , the trajectories of the process $\\widetilde{X}$ satisfy $\\gamma $ -Hölder condition almost surely.",
"The process $X$ has a modification that satisfies Hölder condition for all exponents $\\gamma \\in (0, \\frac{1}{2}\\delta )$ .",
"Since $X_s - X_t$ is a centered Gaussian variable, $\\mathrm {E}\\left[\\,|X_t - X_s|^\\alpha \\,\\right] = \\frac{2^{\\alpha /2}}{\\sqrt{\\pi }}\\Gamma \\!\\left(\\frac{\\alpha +1}{2}\\right)\\left(\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right]\\right)^{\\alpha /2} .$ The first statement of the corollary can be proved by applying Kolmogorov continuity theorem for $\\alpha > 2/\\delta $ and $\\beta = \\frac{1}{2} \\alpha \\delta - 1$ .",
"The second statement of the corollary can be proved by applying Kolmogorov continuity theorem for $\\alpha > \\frac{2}{\\delta - 2\\gamma }$ and $\\beta = \\frac{1}{2} \\alpha \\delta - 1$ .",
"Consider the random event $A &= \\left\\lbrace \\forall \\gamma \\in (0, \\textstyle {\\frac{1}{2}} \\delta ) : \\widetilde{X} \\;\\mbox{satisfies $\\gamma $-Hölder condition} \\right\\rbrace \\\\ &=\\left\\lbrace \\forall n\\in \\mathbb {N} : \\widetilde{X} \\;\\mbox{satisfies $\\frac{1}{2} \\left(1-\\frac{1}{n}\\right) \\delta $-Hölder condition} \\right\\rbrace .$ (The measurability of $A$ follows from the continuity of the process $\\widetilde{X}$ ).",
"By the second statement of Corollary REF $\\mathrm {P}(A) = 1$ .",
"Thus, $\\lbrace \\widetilde{X}_t \\mathbb {1}_{A}, \\;t\\in [0,t]\\rbrace $ is the desired modification which satisfies Hölder condition for all exponents $\\gamma \\in (0, \\frac{1}{2} \\delta )$ .",
"Corollary REF holds true even without assumption that the Gaussian process $X$ is centered.",
"The first statement of Corollary REF can be proved with Xavier Fernique's continuity criterion [5] as well.",
"Let $\\lbrace X_t, \\; t\\in [0, T]\\rbrace $ be a centered Gaussian process.",
"Suppose that there exist $\\delta > 0$ and a nondecreasing continuous function $F : [0, T] \\rightarrow \\mathbb {R}$ such that $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le (F(t) - F(s))^\\delta \\quad \\mbox{for all} \\quad 0 \\le s \\le t \\le T.$ Then The process $X$ have a modification $\\widetilde{X}$ that has continuous trajectories.",
"If the function $F$ satisfies Lipschitz condition in an interval $[a,b] \\subset [0,T]$ , then for every $\\gamma $ , $0< \\gamma < \\frac{1}{2}\\delta $ , the process $\\widetilde{X}$ has a modification whose trajectories satisfy $\\gamma $ -Hölder property on the interval $[a, b]$ .",
"Without loss of generality, we can assume that the function $F$ is strictly increasing.",
"Indeed, if the condition (REF ) holds true for $F$ being continuous nondecreasing function $F_1$ , it also holds true for $F=F_2$ with $F_2(t)=F_1(t)+t$ , where $F_2$ is a continuous strictly increasing function.",
"With this additional assumption, the inverse function $F^{-1}$ is one-to-one, strictly increasing continuous function $[F(0), F(T)] \\rightarrow [0, T]$ .",
"Consider a stochastic process $\\lbrace Y_u, \\; u \\in [F(0), F(T)]\\rbrace $ , with $Y_u = Y_{F^{-1}(u)}$ .",
"The stochastic process $Y$ is centered and Gaussian; it satisfies condition $\\mathrm {E}\\left[\\,(Y_v - Y_u)^2\\,\\right]= \\mathrm {E}\\left[\\,(X_{F^{-1}(v)} - X_{F^{-1}(u)})^2\\,\\right]\\le (F(F^{-1}(v)) - F(F^{-1}(u)))^\\delta = (v - u)^\\delta $ for all $F(0) \\le u \\le v \\le F(T)$ .",
"According to Corollary REF , the process $Y$ has a modification $\\widetilde{Y}$ with continuous trajectories.",
"Then $\\widetilde{X}$ with $\\widetilde{X}_t = \\widetilde{Y}_{F(t)}$ is a modification of the process $X$ with continuous trajectories.",
"The second statement of the lemma is a direct consequence of Corollary REF .",
"If the function $F$ satisfies Lipschitz condition with constant $L$ on the interval $[a,b]$ , then $\\mathrm {E}\\left[\\,(X_t - X_s)^2\\,\\right] \\le L^\\delta (t - s)^\\delta \\quad \\mbox{for all} \\quad a \\le s \\le t \\le b ,$ which is the main condition for Corollary REF ."
],
[
"Application of fractional calculus",
"The lower and upper Riemann–Liouville fractional integrals of a function $f\\in L^{1}[a,b]$ are defined as follows: $(I_{a+}^\\alpha f) (x) = \\frac{1}{\\Gamma (\\alpha )}\\int _a^x \\frac{f(t) \\, dt}{(x-t)^{1-\\alpha }}, \\qquad (I_{b-}^\\alpha f) (x) = \\frac{1}{\\Gamma (\\alpha )}\\int _x^b \\frac{f(t) \\, dt}{(t-x)^{1-\\alpha }}.$ The integrals $(I_{a+}^\\alpha f) (x)$ and $(I_{b-}^\\alpha f) (x)$ are well-defined for almost all $x\\in [a,b]$ , and are integrable functions of $x$ , that is $I_{a+}^\\alpha f \\in L^1[a,b]$ and $I_{b-}^\\alpha f \\in L^1[a,b]$ .",
"Thus, $I_{a+}^\\alpha $ and $I_{b-}^\\alpha $ might be considered linear operators $L^1[a,b] \\rightarrow L^1[a,b]$ .",
"A reflection relation for functions $g(x) = f(a+b-x)$ imply the following relation for their fractional integrals: $(I_{b-}^{\\alpha } g)(x) = (I_{a+}^{\\alpha } f) (a+b-x);$ see [11].",
"The integration-by-parts formula is given, e.g., in [11].",
"[integration-by-parts formula] Let $\\alpha > 0$ , $f \\in L^p[a,b]$ , $g \\in L^q[a,b]$ , $p \\in [1, +\\infty ]$ , $q \\in [1, +\\infty ]$ , while $\\frac{1}{p} + \\frac{1}{q} \\le 1 + \\alpha $ and $\\max \\Bigl (1 + \\alpha - \\frac{1}{p} - \\frac{1}{q},\\:\\min \\!\\left(1 - \\frac{1}{p}, \\:1 - \\frac{1}{q}\\right)\\Bigr ) > 0$ .",
"Then $\\int _a^b (I_{a+}^\\alpha f)(t) \\, g(t) \\, dt =\\int _a^b f(t) \\, (I_{b-}^\\alpha g)(t) \\, dt.$ Now we establish conditions for a function to be in the range of the fractional operator $I_{a+}^\\alpha $ , and we provide formulas for the preimage, which is called a fractional derivative.",
"The following statements are the modifications of the Theorem 2.1 and following corollary in [11].",
"The formulas for the fractional derivative are also provided in [8].",
"Let $0 < \\alpha < 1$ .",
"Consider the integral equation $I_{a+}^\\alpha f = g$ with unknown function $f \\in L^1[a,b]$ and known function (i.e., a parameter) $g \\in L^1[a,b]$ .",
"Denote $h(x) = {\\left\\lbrace \\begin{array}{ll} (I_{a+}^{1-\\alpha } g) (x)& \\mbox{if $a < x \\le b$}, \\\\0& \\mbox{if $x=a$}.\\end{array}\\right.",
"}$ If $h \\in \\operatorname{\\!\\textit {AC}}[a,b]$ , then equation (REF ) has a unique (up to equality almost everywhere in $[a,b]$ ) solution $f$ , namely $f(x) = h^{\\prime }(x)$ .",
"Otherwise, if $h \\notin \\operatorname{\\!\\textit {AC}}[a,b]$ , then equation (REF ) has no solutions in $L^1[a,b]$ .",
"If for some $x\\in (a,b]$ the integral $(I_{a+}^{1-\\alpha } g) (x)$ is not well-defined, then equation (REF ) does not have solutions in $L^1[a,b]$ .",
"Let $0 < \\alpha < 1$ .",
"The integral equation (REF ) with unknown function $f \\in L^1[a,b]$ and known function $g \\in \\operatorname{\\!\\textit {AC}}[a,b]$ has a unique solution.",
"The solution is equal to $f(x) &=(I_{a+}^{1-\\alpha }(g^{\\prime }))(x) +\\frac{g(a)}{\\Gamma (1-\\alpha )\\,(x-a)^\\alpha }\\\\ &=\\frac{1}{\\Gamma (1-\\alpha )}\\left( \\int _a^x \\frac{g^{\\prime }(t)\\,dt}{(x-t)^\\alpha }+ \\frac{g(a)}{(x-a)^\\alpha } \\right) .$"
],
[
"Existence of the solution to Volterra integral equation\nwhere the integral operator is an operator\nof convolution with integrable singularity\nat 0",
"Consider Volterra integral equation of the first kind $\\int _0^x f(t) \\, g(x-t) \\, dt = y(x), \\qquad x \\in (0,T],$ with $g(x)$ and $y(x)$ known (parameter) functions and $f(x)$ unknown function.",
"Suppose that the function $g(x)$ is integrable in the interval $(0,T]$ but behaves asymptotically as a power function in the neighborhood of 0: $g(x) \\sim \\frac{K}{x^{1-\\alpha }}, \\qquad x \\rightarrow 0,$ where $0 < \\alpha < 1$ .",
"More specifically, assume that $g(x)$ admits a representation $g(x) = \\frac{1}{\\Gamma (\\alpha ) x^{1-\\alpha }}+ (I^{\\alpha }_{0+} h)(x) =\\frac{1}{\\Gamma (\\alpha )}\\left( \\frac{1}{x^{1-\\alpha }} + \\int _0^x \\frac{h(t)\\, dt}{(x-t)^{1-\\alpha }} \\right),$ where $\\Gamma (\\alpha )$ is a gamma function, $I^{\\alpha }_{0+}h$ is a lower Riemann–Liouville fractional integral of $h$ , $(I^{\\alpha }_{0+} h)(x) =\\frac{1}{\\Gamma (\\alpha )}\\int _0^x \\frac{h(t)\\, dt}{(x-t)^{1-\\alpha }},$ and $h(x)$ is a absolutely continuous function.",
"The sufficient conditions for existence and uniqueness of the solution to integral equation claimed in [8] are not satisfied.",
"The kernel of the integration operator in (REF ) is unbounded, and $y(0)$ might be nonzero.",
"But we use Remark 2 in [8].",
"We reduce the Volterra integral equation of the first kind to a Volterra integral equation of the second kind similarly as it is done for regular functions $g(x)$ ; compare with [8] for the case of regular $g(x)$ .",
"For the next theorem we keep in mind that if a function $f$ is a solution to (REF ), then every function that is equal to $f$ almost everywhere on $[0,T]$ is also a solution to (REF ).",
"Let $y,\\,h \\in C^1[0,T]$ and $g$ be defined in (REF ).",
"Then the equation (REF ) has a unique (up to equality almost everywhere) solution $f \\in L^1[0,T]$ .",
"The solution is (more precisely, some of almost-everywhere equal solutions are) continuous in the left-open interval $(0,T]$ .",
"Substitute (REF ) into (REF ): $\\int _0^x f(t)\\left( \\frac{1}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }}+ (I^{\\alpha }_{0+} h)(x-t) \\right) dt= y(x), \\\\(I^{\\alpha }_{0+} f)(x) +\\int _0^x f(t) \\, (I^{\\alpha }_{0+} h)(x-t) \\, dt= y(x) .$ Denote $h_x(t) = h(x-t)$ .",
"According to equation (REF ), the fractional integrals of $h$ and $h_x$ satisfy the relation $(I^{\\alpha }_{0+} h)(x-t)= (I^{\\alpha }_{x-} h_x)(t)$ .",
"Hence, equation (REF ) is equivalent to the following one: $(I^{\\alpha }_{0+} f)(x) +\\int _0^x f(t) \\, (I^{\\alpha }_{x-} h_x)(t) \\, dt= y(x).$ Now apply the integration-by-parts formula.",
"We have $f \\in L^1[0,x]$ , $h_x \\in L^\\infty [0,x]$ , and $1 + 0 < 1 + \\alpha $ .",
"Hence, by Proposition REF , $\\int _0^x f(t) \\, (I^{\\alpha }_{x-} h_x)(t) \\, dt =\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h_x(t) \\, dt.$ It means that equation (REF ) is equivalent to the following ones: $(I^{\\alpha }_{0+} f)(x) +\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h_x(t) \\, dt= y(x), $ and $(I^{\\alpha }_{0+} f)(x) +\\int _0^x (I^{\\alpha }_{0+} f)(t) \\, h(x-t) \\, dt= y(x) .$ Denote $F = I^{\\alpha }_{0+} f$ , and obtain a Volterra integral equation of the second kind: $F(x) = y(x) - \\int _0^x F(t) \\, h(x-t) \\, dt.$ Equation (REF ) has a unique solution in $C[0,T]$ , as well as in $L^1[0,T]$ .",
"In other words, (REF ) has a unique integrable solution, and this solution is a continuous function.",
"According to Theorem REF , either unique (up to almost-everywhere equality) function $f$ , or no functions $f$ correspond to the function $F$ .",
"Thus, all integrable solution to integral equation (REF ) are equal almost everywhere.",
"Now we construct a solution to equation (REF ) that is continuous and integrable on $(0,T]$ .",
"Differentiating (REF ), we obtain $F^{\\prime }(x) = y^{\\prime }(x) - F(x)\\, h(0) - \\int _0^x F(t) \\, h^{\\prime }(x-t) \\, dt,$ whence $F \\in C^1[0,T]$ .",
"According to Corollary REF , the integral equation $F = I_{0+}^\\alpha f$ has a unique solution $f\\in L^1[0,T]$ , which is equal to $f(x)= \\frac{1}{\\Gamma (1-\\alpha )}\\left( \\int _0^x \\frac{F^{\\prime }(t)\\,dt}{(x-t)^\\alpha } +\\frac{F(0)}{x^\\alpha } \\right) .$ The constructed function $f(x)$ is continuous and integrable in $(0,T]$ , and $f(x)$ is a solution to (REF ).",
"In Theorem REF the condition $h \\in C^1[0,T]$ can be relaxed and replaced with the condition $h \\in \\operatorname{\\!\\textit {AC}}[0,T]$ .",
"In other words, if the function $h$ is absolutely continuous but is not continuously differentiable, the statement of Theorem REF still holds true."
],
[
"Example: $g(x) = \\exp (\\beta x) x^{\\alpha -1} / \\Gamma (\\alpha )$ \nand {{formula:7d38e747-aac0-4b1e-9a5b-db3b315248f0}}",
"Example: $g(x) = \\exp (\\beta x) x^{\\alpha -1} / \\Gamma (\\alpha )$ and $y(x) = 1$ It is well known that $\\int _0^x \\frac{1}{\\Gamma (1-\\alpha ) t^\\alpha } \\,\\frac{1}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }} \\, dt = 1 .$ In this section, we prove that the equation $\\int _0^x f(t) \\,\\frac{e^{(x-t)\\beta }}{\\Gamma (\\alpha ) (x-t)^{1-\\alpha }} \\, dt = 1$ has an integrable solution.",
"According to (REF ), $f(x) = x^{-\\alpha } / \\, \\Gamma (1-\\alpha )$ is a solution to (REF ) if $\\beta = 0$ .",
"Denote $g(x) = \\frac{\\exp (\\beta x)}{\\Gamma (\\alpha ) x^{1-\\alpha }} .$ Demonstrate that $g(x)$ admits a representation (REF ).",
"To construct $h$ , we need Kummer confluent hypergeometric function [14]: $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;z) = \\frac{1}{\\mathrm {B}(a,\\: b-a)}\\int _0^1 e^{zt} t^{a-1} (1-t)^{b-a-1} \\, dt,\\qquad 0<a<b, \\quad z\\in \\mathbb {C} .$ For $a$ and $b$ fixed, $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;\\,\\cdot \\,)$ is an entire function.",
"Its derivative equals $\\frac{\\partial }{\\partial z}\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a;b;z) = \\frac{a}{b}\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a+1;\\:b+1;\\:z) .$ For all $0<a<b$ and $z\\in \\mathbb {R}$ $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a; b; z) > 0, \\qquad \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(a; b; 0) = 1.$ Notice that if $0<\\alpha <1$ and $x>0$ , then $\\frac{1}{\\mathrm {B}(\\alpha , \\: 1-\\alpha )}\\int _0^x \\frac{\\exp (zt) \\, dt}{t^{1-\\alpha } (x-t)^\\alpha } =\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; xz) .$ Being considered an equation for unknown $h$ , (REF ) is equivalent to $I_{0+}^\\alpha h = g_0$ , where $g_0(x) = g(x) - \\frac{1}{\\Gamma (\\alpha ) x^{1-\\alpha }}= \\frac{e^{\\beta x} - 1}{\\Gamma (\\alpha ) x^{1-\\alpha }} .$ Then $(I_{0+}^{1-\\alpha } g_0)(x) =\\frac{1}{\\mathrm {B}(\\alpha , \\: 1-\\alpha )}\\int _0^x \\frac{e^{\\beta t} - 1}{t^{1-\\alpha }(1-t)^\\alpha } \\, dt =\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1 .$ Besides, $\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1$ is an absolutely continuous function in $x$ , and$\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1;\\beta x) - 1 = 0$ if $x=0$ .",
"According to Theorem REF , the equation $I_{0+}^\\alpha h = g_0$ has the unique solution $h =L^1[0,T]$ , which is equal to $h(x) = \\frac{\\partial (\\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha ; 1; \\beta x) - 1)}{\\partial x} =\\alpha \\beta \\operatorname{\\mathop {\\mmlmultiscripts{\\mathop F{1}{\\mmlnone }\\mmlprescripts {1}{\\mmlnone }}}\\limits }(\\alpha +1;\\: 2;\\: \\beta x).$ The constructed function $h(x)$ is a solution to (REF ) and is continuously differentiable.",
"In summary, $h \\in C^1[0,T]$ , $y(x) = 1$ , and $y \\in C^1[0,T]$ .",
"According to Theorem REF the integral equation $\\int _0^x f(t) \\, g(x-t) \\, dt = 1, \\qquad x\\in (0,T],$ has a unique solution $f \\in L^1[0,T]$ (up to equality almost everywhere).",
"The solution is continuous in $(0,T]$ .",
"The fact that the functions $g$ and $h$ defined in (REF ) and (REF ), respectively, satisfy (REF ), can be checked directly.",
"For such verification, one can apply Lemma 2.2(i) from [10]."
],
[
"Positive solution to the Volterra\nintegral equation",
"Let the conditions of Theorem REF hold true.",
"Additionally, let $y(x)>0, \\quad y^{\\prime }(x)\\ge 0, \\quad h(x) < 0 \\qquad \\mbox{for all $x\\in [0,T]$}.$ Then the continuous solution $f(x)$ to (REF ) attains only positive values in $(0,T]$ .",
"Notice that (REF ) implies $F(0) = y(0) > 0$ .",
"Taking this into account, let's differentiate both sides of (REF ) the other way: $F(x) = y(x) - \\int _0^x F(x-t) \\, h(t) \\, dt,\\nonumber \\\\F^{\\prime }(x) = y^{\\prime }(x) - F(0) \\, h(x)- \\int _0^x F^{\\prime }(x-t) \\, h(t) \\, dt .$ Let us prove that $F^{\\prime }(x)>0$ in $[0,T]$ by contradiction.",
"Assume the contrary, that is $\\exists x \\in [0,1] : F^{\\prime }(x) \\le 0$ .",
"Since the function $F^{\\prime }(x)$ is continuous in $[0,T]$ , the contrary implies the existence of the minimum in $x_0 = \\min \\lbrace x\\in [0,T] : F^{\\prime }(x) \\le 0\\rbrace .$ But for $x=x_0$ the left-hand side in (REF ) is less or equal then zero, while the right-hand side is greater than zero.",
"Thus, (REF ) does not hold true.",
"The proof also works for $x_0 = 0$ .",
"There is a contradiction.",
"Thus, we have proved that $F^{\\prime }(x)>0$ for all $x\\in [0,T]$ .",
"By (REF ), $f(x)>0$ for all $x\\in (0,T]$ ."
]
] | 2001.03405 |
[
[
"Canonical maps and integrability in $T\\bar T$ deformed 2d CFTs"
],
[
"Abstract We study $T\\bar T$ deformations of 2d CFTs with periodic boundary conditions.",
"We relate these systems to string models on $\\mathbb{R}\\times {S}^1\\times{\\cal M}$, where $\\cal M$ is the target space of a 2d CFT.",
"The string model in the light cone gauge is identified with the corresponding 2d CFT and in the static gauge it reproduces its $T\\bar T$ deformed system.",
"This relates the deformed system and the initial one by a worldsheet coordinate transformation, which becomes a time dependent canonical map in the Hamiltonian treatment.",
"The deformed Hamiltonian defines the string energy and we express it in terms of the chiral Hamiltonians of the initial 2d CFT.",
"This allows exact quantization of the deformed system, if the spectrum of the initial 2d CFT is known.",
"The generalization to non-conformal 2d field theories is also discussed."
],
[
"Introduction",
"The so-called $T\\bar{T}$ deformation of two-dimensional quantum field theories, which was introduced by Zamolodchikov in 2004 [1], has recently attracted much attention.",
"Being a deformation by an irrelevant operator, one would naively expect that the deformed theory looses any of the nice properties the undeformed theory might have had and that the UV behaviour gets completely out of control.",
"But this is not the case.",
"For instance, in [2] it was shown that if the originial theory is integrable, so is the deformed one.",
"Another remarkable fact is that the spectrum of the deformed theory formulated on a cylinder can be determined exactly from the one of the undeformed theory [1], [2], [3].",
"An interesting observation first made in [3] is the connection between a deformed free boson and string theory.",
"More precisely, it was shown that the classical dynamics of the deformed system is that controlled by the Nambu-Goto action with three-dimensional Minkowski space as target space, after fixing the static gauge.",
"In the same paper this was generalized to several free bosons and also to a single boson with an arbitrary potential.",
"Further generalizations and refinements along these lines (and beyond) were considered in [4], [5] and [6], again at the classical level.",
"The relation between $T\\bar{T}$ deformed CFTs and the quantum string was studied in detail in [7].",
"Here we also consider the connection between deformed field theories and string theory, mainly at the classical level.",
"As a large part of our analysis will be within the Hamiltonian framework, the next section reviews the Hamiltonian treatment of two-dimensional Lagrangian field theories.",
"While the Lagrangian treatment is more familiar and transparent, the Hamiltonian one is more convenient for generalizations.",
"The main examples are non-linear sigma-models with a metric and anti-symmetric tensor background.",
"Classically they are always conformally invariant.",
"Within the context of string theory one needs to impose conditions on the background fields, but this will not play a role in our classical discussion.",
"A simple generalization, which explicitly breaks the conformal symmetry, is adding a potential.",
"In Section 3 we look at the $T\\bar{T}$ deformation of these theories, again in the Hamiltonian framework.",
"A simple formula for the deformed Hamiltonian density for systems with symmetric canonical energy-momentum tensor can be derived.",
"This formula is valid for arbitrary (classical) CFTs which are characterized by two independent components of the energy-momentum tensor whose Poisson brackets generate two copies of the centerless Virasoro algebra.",
"The simplest conformally invariant sigma-model is a free massless scalar field on a cylinder.",
"Its deformation will be reviewed in Section 4, with emphasis on the connection to closed string dynamics in three-dimensional space-time, where one spatial coordinate is compactified on $S^1$ .",
"When the latter is formulated in a diffeomorphism invariant way, the deformed free scalar is obtained by breaking the invariance through fixing the static gauge.",
"This gauge identifies the time and one spatial coordinate of the target space with the worldsheet coordinates.",
"For this reason compactification is necessary.",
"The string energy is, up to an additive constant, equal to the Hamiltonian of the deformed theory.",
"If one chooses light-cone gauge instead, one reaches the undeformed theory.",
"We generalize the light-cone gauge treatment of a closed string dynamics with a compactified spatial coordinate, using as space-time light-cone directions those of the cylinder.",
"This generalization is straightforward.",
"In particular, in this gauge the string energy can be computed explicitly and by using its gauge invariance one obtains an expression for the Hamiltonian – rather than the density – of the deformed theory in terms of the Hamiltonian of the undeformed theory.",
"This result applies, in fact, to more general undeformed theories than just the free massless scalar.",
"The relation between the deformed and the undeformed theory as simply choosing different gauges in the string theory, implies that the undeformed and the deformed theory are related by a (time-dependent) canonical transformation.",
"This will be shown in detail.",
"The worldsheet coordinate transformation between the two gauges depends on the solutions of the equation of motion in the fixed gauge.",
"We use the explicit form of this transformation to obtain the Hamiltonian of the deformed theory without resorting to the gauge invariance of the string energy.",
"In Section 5 we show how the previous discussion extends to general conformally invariant sigma models and to the case when one adds a potential.",
"A remarkable example here is the Liouville model with a negative cosmological constant.",
"We show that the corresponding string model is the SL(2,$\\mathbb {R})$ WZW theory with vanishing stress tensor [8].",
"This string model in the static and light-cone gauges coincides to the $T\\bar{T}$ deformed and the initial Liouville models, respectively.",
"Some of the results reported in this note were obtained but not published about two years ago [9] and they have meanwhile appeared in various papers.",
"We have taken the opportunity of being asked to contribute to this volume to include them, with due reference to the existing literature.",
"Most importantly we point out [7], [10], [11], [12], [13], [14] for extensive discussions of the relation between the $T\\bar{T}$ deformed and the initial 2d field theories in the context of worldsheet gauge transformations."
],
[
"Hamiltonian formulation of 2d field theory",
"We consider two-dimensional classical field theories on a cylinder with circumference $2\\pi $ , described by an action $S[\\phi ]={2\\pi }\\int \\mbox{d}\\tau \\,\\mbox{d}\\sigma \\,\\,{\\cal L}(\\phi , \\dot{\\phi }, \\acute{\\phi })~.$ Here, $\\tau $ and $\\sigma $ are time and space coordinates, respectively, $\\phi :=(\\phi ^1,\\dots , \\phi ^N)$ denotes a set of periodic fields, $\\phi (\\tau ,\\sigma +2\\pi )=\\phi (\\tau ,\\sigma )$ , and we use the notation $\\dot{\\phi }:=\\partial _\\tau \\phi $ , $\\acute{\\phi }:=\\partial _\\sigma \\phi $ .",
"The components of the canonical stress tensor ($a,b\\in \\lbrace \\tau ,\\sigma \\rbrace $ ) $\\begin{aligned}&T^{\\,a}_{\\,~b}={\\partial {\\cal L}\\over \\partial (\\partial _a\\phi ^k)}\\partial _b\\phi ^k-\\delta ^a_{~b}\\,{\\cal L}\\end{aligned}$ satisfy, by Noether's theorem, the local conservation laws $\\partial _a T^{\\,a}_{\\,~b}=0\\,.$ The first order formulation of the same dynamics is obtained from the action $S[\\Pi , \\phi ]=\\int \\mbox{d}\\tau \\int _0^{2\\pi } {2\\pi } \\left[\\Pi _k\\,\\dot{\\phi }^k-{\\cal H}(\\Pi , \\phi ,\\acute{\\phi })\\right]~,$ where $\\Pi _k$ are the periodic canonical momenta, $\\Pi (\\tau ,\\sigma +2\\pi )=\\Pi (\\tau ,\\sigma )$ .",
"We assume that the Lagrangian in (REF ) is non-singular,Singular Lagrangians also lead to the action (REF ) by Hamiltonian reduction, but with a reduced number of target space fields.",
"i.e.",
"the velocities $\\dot{\\phi }^k$ are solvable in terms of the momenta $\\Pi _k$ .",
"The stress tensor components (REF ) are $\\begin{aligned}&T^{\\,\\tau }_{~\\,\\tau }={\\cal H}~, &&T^{\\,\\tau }_{\\,~\\sigma }=\\Pi _{k}\\,\\acute{\\phi }^{\\,k}~,\\\\[1mm]&T^{\\,\\sigma }_{\\,~\\tau } =-{\\partial \\Pi _{k}}\\,{\\partial \\acute{\\phi }^{\\,k}}~, \\qquad &&T^{\\,\\sigma }_{\\,~\\sigma } ={\\cal H}-\\Pi _k\\,{\\partial \\Pi _{k}}-\\acute{\\phi }^{\\,k}\\,{\\partial \\acute{\\phi }^{\\,k}}~,\\end{aligned}$ and the conservation laws (REF ) follow from the Hamilton equations of motion $\\dot{\\phi }^k= {\\partial \\Pi _k}~, \\qquad \\dot{\\Pi }_k= -{\\partial \\phi ^k}+\\partial _\\sigma \\left({\\partial \\acute{\\phi }^{\\,k}}\\right)~.$ Note that the covariant canonical stress tensor $T_{ab}$ in 2d Minkowski space is symmetric ($T_{\\tau \\,\\sigma }=T_{\\sigma \\,\\tau }$ ) when the Hamiltonian density satisfies the condition ${\\partial \\Pi _{k}}\\,{\\partial \\acute{\\phi }^{\\,k}}=\\Pi _k\\,\\acute{\\phi }^{\\,k}\\,.$ Below we assume that (REF ) is fulfilled, without referring to 2d metric structure.While we can always add improvement terms to symmetrize the energy-momentum tensor, here we assume that the canonical one is symmetric.",
"We also assume that the canonical stress tensor (REF ) is traceless, i.e.",
"$\\hat{V}[{\\cal H}]=2\\,{\\cal H}~, \\qquad \\text{where} ~~~\\hat{V}=\\Pi _k\\,{\\partial \\Pi _k}+\\acute{\\phi }^{\\,k}\\,{\\partial \\acute{\\phi }^{\\,k}}~.$ In this case $T^{\\,a}_{\\,~b}=\\left(\\begin{array}{cr}{\\cal H} &{\\cal P}\\\\-{\\cal P} &-{\\cal H}\\end{array}\\right)~,\\qquad \\text{with} \\quad {\\cal P}:=\\Pi _{k}\\,\\acute{\\phi }^{\\,k}~.$ The components $T^{\\,\\tau }_{~\\tau }={\\cal H}$ and $T^{\\,\\tau }_{~\\sigma }={\\cal P}$ are interpreted as the energy and the momentum densities, respectively.",
"They obey the Poisson bracket relations $\\begin{aligned}&\\lbrace {\\cal P}(\\sigma _1), {\\cal P}(\\sigma _2)\\rbrace =\\lbrace {\\cal H}(\\sigma _1), {\\cal H}(\\sigma _2)\\rbrace =2\\pi \\big [{\\cal P}(\\sigma _1)+{\\cal P}(\\sigma _2)\\big ]\\delta ^{\\prime }(\\sigma _2-\\sigma _1), \\\\[1mm]&\\lbrace {\\cal P}(\\sigma _1), {\\cal H}(\\sigma _2)\\rbrace =\\lbrace {\\cal H}(\\sigma _1), {\\cal P}(\\sigma _2)\\rbrace =2\\pi \\big [{\\cal H}(\\sigma _1)+{\\cal H}(\\sigma _2)\\big ]\\delta ^{\\prime }(\\sigma _2-\\sigma _1),\\end{aligned}$ which follow from the canonical Poisson brackets, $\\lbrace \\Pi _k(\\sigma _1), \\phi ^{l}(\\sigma _2)\\rbrace =2\\pi \\,\\delta _{k}^{\\,\\,l}\\,\\delta (\\sigma _1-\\sigma _2)~,$ and the conditions (REF ) and (REF ).",
"The Lie algebra (REF ) is equivalent to $\\begin{aligned}&\\lbrace T(x), T(y)\\rbrace =2\\pi \\big [T(x)+T(y)\\big ]\\delta ^{\\prime }(y-x)\\,,& \\quad &\\lbrace T(x), \\bar{T}(\\bar{x})\\rbrace =0\\,,\\\\[1mm] &\\lbrace \\bar{T}(\\bar{x}),\\bar{T}(\\bar{y})\\rbrace =2\\pi \\big [\\bar{T}(\\bar{x})+\\bar{T}(\\bar{y})\\big ]\\delta ^{\\prime }(\\bar{y}-\\bar{x})\\,,& &\\end{aligned}$ with $T(x)={2}\\big [{\\cal H}(x)+{\\cal P}(x)\\big ]~, \\qquad \\bar{T}(\\bar{x})={2}\\big [{\\cal H}(-\\bar{x})-{\\cal P}(-\\bar{x})]~.$ The conservation laws (REF ) in terms of $T$ and $\\bar{T}$ become $\\partial _{\\bar{x}}T=0~, \\qquad \\partial _{x}\\bar{T}=0~,$ where $x=\\tau +\\sigma $ and $\\bar{x}=\\tau -\\sigma $ are the chiral coordinates, and we arrive at the standard formulation of 2d CFT with zero central charge.",
"In a more general treatment, a 2d CFT on a cylinder is provided by two periodic functions $T(x)$ and $\\bar{T}(\\bar{x})$ , which satisfy the Poisson bracket relations (REF ), without referring to the canonical structure (REF ).",
"Thus, the Hamiltonian density $\\cal H$ that satisfies the conditions (REF ) and (REF ) corresponds to a classical 2d CFT.",
"A standard example is the $\\sigma $ -model $S_{_{G,B}}[\\phi ]={4\\pi }\\int \\mbox{d}\\tau \\,\\mbox{d}\\sigma \\left[\\dot{\\phi }^k\\, G_{kl}(\\phi )\\, \\dot{\\phi }^l-\\acute{\\phi }^{\\,k}\\,G_{kl}(\\phi )\\,\\acute{\\phi }^{\\,l}-2\\dot{\\phi }^k\\, B_{kl}(\\phi )\\,\\acute{\\phi }^{\\,l}\\right]~,$ where $G_{kl}(\\phi )$ is a target space metric tensor and $B_{kl}(\\phi )$ is a 2-form on the target space.",
"This system has stress tensor $\\begin{aligned}&T^{\\,\\tau }_{~\\,\\tau }=-T^{\\,\\sigma }_{~\\,\\sigma }={2}\\left(\\dot{\\phi }^k\\, G_{kl}\\, \\dot{\\phi }^l+ \\acute{\\phi }^{\\,k}\\,G_{kl}\\,\\acute{\\phi }^{\\,l}\\right),\\quad &&T^{\\,\\tau }_{~\\,\\sigma }=-T^{\\,\\sigma }_{~\\,\\tau }=\\dot{\\phi }^{\\,k}\\, G_{kl}\\,\\acute{\\phi }^{\\,l} ~,\\end{aligned}$ and Hamiltonian density ${\\cal H}_{_{G,B}}={2}\\left[\\Pi _k\\, G^{kl}\\, \\Pi _l+\\acute{\\phi }^{\\,k}\\left(G_{kl}-B_{km}\\,G^{mn}\\,B_{nl}\\right)\\acute{\\phi }^{\\,l}\\right]+\\Pi _k\\,G^{kj}\\,B_{jl}\\,\\acute{\\phi }^{\\,l}~,$ which indeed satisfies conditions (REF ) and (REF ).",
"Adding a potential $U(\\phi )$ to a 2d CFT $\\tilde{\\cal H}={\\cal H}+U(\\phi )~,$ leads to a stress tensor with non-zero trace $T^{\\,a}_{\\,~b}=\\left(\\begin{array}{cr}{\\cal H}+U(\\phi ) &{\\cal P}~~~~~~~\\\\-{\\cal P} &-{\\cal H}+U(\\phi )\\end{array}\\right)~.$"
],
[
"$T\\bar{T}$ deformation of 2d Hamiltonian systems",
"The following analysis is usually done in the Lagrangian formulation (cf.",
"e.g.",
"[3], [4], [14]).",
"Here we present a Hamiltonian version of these well-known results.",
"We introduce the $T\\bar{T}$ deformation of the system (REF ) as [1] $S_\\alpha [\\Pi ,\\phi ]=\\int \\mbox{d}\\tau \\int _{0}^{2\\pi }{2\\pi } \\left[\\Pi _k\\,\\dot{\\phi }^k-{\\cal H}_\\alpha (\\Pi , \\phi ,\\acute{\\phi })\\right]~,$ with ${\\cal H}_\\alpha $ defined by the `initial' condition ${\\cal H}_0={\\cal H}$ and the differential equation ${\\partial \\alpha }= 2 \\mbox{det}[T_{(\\alpha )}]\\,.$ Here $T_{(\\alpha )}^{\\,\\,a}\\,_b $ is the canonical stress tensor obtained from (REF ) by the replacement ${\\cal H}\\mapsto {\\cal H}_\\alpha $ .",
"Note that $\\mbox{det}[T^{a}_{~\\,b}]={\\cal P}^2-{\\cal H}^2=-4 T\\bar{T}$ for a 2d CFT.",
"Thus, the first order correction to the Hamiltonian density of a 2d CFT is ${\\cal H}_\\alpha ={\\cal H}-2\\,\\alpha \\, T\\bar{T} +\\cdots ~;$ hence the name $T\\bar{T}$ deformation.",
"However, the higher order terms do not have this structure and are more complicated.",
"From (REF ) and (REF ) follows that ${\\cal H}_\\alpha $ satisfies the equation $2{\\partial \\alpha }={\\cal H}_\\alpha ^2-{\\cal H}_\\alpha \\,\\hat{V}\\left[{\\cal H}_\\alpha \\right]+{\\cal P}\\,{\\partial \\Pi _{\\,k}}\\,{\\partial \\acute{\\phi }^{\\,k}}~,$ and one is looking for solutions which are analytic in $\\alpha $ at $\\alpha =0$ .",
"Using (REF ), one shows by a straightforward but slightly tedious calculation that the variable $Y_\\alpha ={\\partial \\Pi _{\\,k}}\\,{\\partial \\acute{\\phi }^{\\,k}}-{\\cal P}$ satisfies the equation ${\\partial \\alpha }={\\cal H}_\\alpha \\,Y_\\alpha -{2}\\,\\hat{V}\\left({\\cal H}_\\alpha \\,Y_\\alpha \\right)+{2}\\,{\\cal P}\\left({\\partial \\Pi _k}\\,{\\partial \\acute{\\phi }^k}+{\\partial \\acute{\\phi }^{\\,k}}\\,{\\partial \\Pi _{\\,k}}\\right)~.$ From the `initial' condition $Y_{\\alpha =0}=0$ then follows that $Y_\\alpha $ remains zero for all $\\alpha $ .",
"Hence, ${\\cal H}_\\alpha $ satisfies the condition ${\\partial \\Pi _{\\,k}}\\,{\\partial \\acute{\\phi }^{\\,k}}=\\Pi _k\\,\\acute{\\phi }^{\\,k}\\,,$ and (REF ) reduces to $2\\,{\\partial _\\alpha {\\cal H}_\\alpha }={\\cal H}_\\alpha ^2-{\\cal H}_\\alpha \\,\\hat{V}\\left[{\\cal H}_\\alpha \\right]+{\\cal P}^2~.$ This equation can be easily integrated if the stress tensor of the undeformed theory is traceless.",
"Indeed, taking into account (REF ) and $\\hat{V}[{\\cal P}]=2\\cal P$ , one finds that ${\\cal H}_\\alpha $ is expressed in terms of $\\cal H$ and $\\cal P$ only.",
"Dimensional analysis suggests the ansatz ${\\cal H}_\\alpha =F_\\alpha (r\\,{\\cal H}+\\alpha \\,{\\cal P}^2) ~,$ where $r$ is a real number.",
"Inserting it into (REF ) one finds $F^{\\prime }(u)=\\left({r^2+4\\,\\alpha \\, u}\\right)^{-{2}}$ .",
"Integration, requiring the regularity condition at $\\alpha =0$ and that it satisfies (REF ), leads to [14] ${\\cal H}_\\alpha ={\\alpha }\\,\\left(\\sqrt{1+2\\,\\alpha \\, {\\cal H}+\\alpha ^2\\,{\\cal P}^{2}}-1\\right)~.$ The structure of the energy-momentum tensor of the deformed theory is $T_{(\\alpha )\\,b}^{\\,\\,a}=\\left(\\begin{array}{cr}{\\cal H}_\\alpha &{\\cal P}\\\\-{\\cal P} &-{\\cal K}_\\alpha \\end{array}\\right),$ with ${\\cal K}_\\alpha ={1\\over \\alpha }\\left({1-\\alpha ^2\\,{\\cal P}^2\\over \\sqrt{1+2\\,\\alpha \\,{\\cal H}+\\alpha ^2{\\cal P}^2}}-1\\right)={1+\\alpha \\,{\\cal H}_\\alpha }\\,.$ One also verifies $\\mbox{Tr}[T_{(\\alpha )}]=-\\alpha \\,\\mbox{det}[T_{(\\alpha )} ]~$ and, therefore, for a 2d CFT, ${\\cal H}_\\alpha $ satisfies the linear equation $2\\,\\alpha \\,\\partial _\\alpha {\\cal H}_\\alpha +2\\,{\\cal H}_\\alpha -\\hat{V}[{\\cal H}_\\alpha ]=0~.$ The above results, in particular the form of the deformed Hamiltonian density (REF ), were derived for a particular class of conformal field theories, but one wonders how general they are.",
"If we assume that the energy-momentum tensor of the undeformed theory is symmetric, it has only two independent components, $T$ and $\\bar{T}$ .",
"In terms of those ${\\cal H}_\\alpha ={\\alpha }\\left(\\sqrt{1+2\\,\\alpha \\,(T+\\bar{T})+\\alpha ^2(T-\\bar{T})^2}-1\\right)\\,.$ Using the algebra (REF ), which holds for any CFT, one verifies that $\\dot{\\cal H}_\\alpha =\\lbrace H_\\alpha ,{\\cal H}_\\alpha \\rbrace =\\partial _\\sigma (T-\\bar{T}),\\qquad \\hbox{where}\\qquad H_\\alpha =\\int _0^{2\\pi } {2\\pi }\\,{\\cal H}_\\alpha \\,.$ Imposing the $\\tau $ -component of the conservation equation in (REF ) for the deformed theory, this shows that $T^{\\,\\sigma }_{\\,~\\tau }=\\bar{T}-T$ is not deformed.",
"Imposing instead the $\\sigma $ -component and requiring symmetry of $T_{(\\alpha )}$ leads to $T_{(\\alpha )\\,\\sigma }^{\\,\\sigma }={\\cal H}_\\alpha -2\\,{\\partial T}T-2{\\partial \\bar{T}}\\bar{T}~.$ These results are completely general for two-dimensional conformal field theories, in particular the expression (REF ) for the Hamiltonian density.",
"We stress that our discussion so far was classical.",
"In particular, in the quantized theory the algebra (REF ) is modified by a central extension leading to the Virasoro algebra.",
"Even for string theory, when the contribution of the ghosts is included, the above calculation does not go through straightforwardly because of ordering issues in the expression for ${\\cal H}_\\alpha $ .",
"The $T\\bar{T}$ deformation of the model (REF ), with the potential $U(\\phi )$ , can be performed similarly.",
"In this case $\\hat{V}[\\tilde{\\cal H}]=2{\\cal H}$ and $\\tilde{\\cal H}_\\alpha $ becomes a function of ${\\cal H}$ , $\\cal P$ and $U(\\phi )$ only.",
"Repeating the arguments which lead to (REF ), we obtain [4] $\\tilde{\\cal H}_\\alpha ={\\beta }\\,\\left[\\sqrt{1+2\\,\\beta \\,{\\cal H}+\\beta ^2\\,{\\cal P}^{2}}+{2}\\right]-{\\alpha }~,$ with $\\beta =\\alpha \\left(1-{2}\\,U(\\phi )\\right)~.$ The check of (REF ) and (REF ) is again straightforward."
],
[
"Integrability of the deformed 2d massless free field",
"In this section we investigate integrability of the deformed massless free-field model with the undeformed Lagrangian ${\\cal L}={2}\\left(\\dot{\\phi }^2-\\acute{\\phi }^{\\,2}\\right).$ The energy and momentum densities ${\\cal H}={2}\\left(\\Pi ^2+\\acute{\\phi }^{\\,2}\\right), \\qquad {\\cal P}=\\Pi \\,\\acute{\\phi }~,$ lead to the following deformed Hamiltonian density ${\\cal H}_\\alpha ={\\alpha }\\left(\\sqrt{1+\\alpha \\left(\\Pi ^2+\\acute{\\phi }^{\\,2}\\right)+\\alpha ^2 \\Pi ^2\\,\\acute{\\phi }^{\\,2}}-1\\right).$ From the related Lagrangian ${\\cal L}_\\alpha =-{\\alpha }\\left(\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}-1\\right)~,$ one derives a non-linear dynamical equation which is hard to integrate directly.",
"Furthermore the construction of the Hamilton operator by (REF ) seems a highly nontrivial problem due to the non-polynomial dependence of ${\\cal H}_\\alpha $ on the canonical variables.",
"However, the deformed free-field theory is related to a 3d string with one compactified coordinate [3].",
"This enables us to integrate the system both at classical and quantum levels.",
"We first consider the Lagrangian approach to the compactified 3d string dynamics and then turn to its Hamiltonian treatment.",
"For later use we note that $\\Pi $ and $\\dot{\\phi }$ of the deformed theory (REF ) are related by $\\Pi ={\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}}~, \\qquad \\dot{\\phi }=\\Pi \\sqrt{{1+\\alpha \\,\\Pi ^2}}~,$ and the energy and momentum densities in the Lagrangian formulation become ${\\cal H}_\\alpha ={\\alpha }\\left({\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}}-1\\right),\\qquad {\\cal P}={\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}}~.$"
],
[
"Lagrangian approach to a compactified 3d string",
"We start with a review of the connection between the string and the deformed system [3].",
"The Nambu-Goto action for a closed string is $S=-{2\\pi \\alpha }\\int \\mbox{d}\\tau \\int _0^{2\\pi }\\mbox{d}\\sigma \\sqrt{(\\dot{X}\\,\\acute{X})^2-(\\dot{X}\\,\\dot{X})(\\acute{X}\\,\\acute{X})}~.$ $X:=(X^0, X^1, X^2)$ is a vector in 3d Minkowski space and $1/\\alpha $ is proportional to the string tension.",
"We use the notation $(X X)=X^\\mu X^\\nu g_{\\mu \\nu }$ with the target space metric tensor $g_{\\mu \\nu }=\\mbox{diag}(-1,1,1)$ .",
"This theory has two-dimensional diffeomorphism invariance and is classically equivalent to the Polyakov action with a world-sheet metric.",
"To connect the deformed free-field theory to the closed string dynamics, we compactify the coordinate $X^1$ on the unit circle and consider string configurations with winding number one around this circle, i.e.",
"we identify $X^1\\simeq X^1+2\\,\\pi $ .",
"This enables us to parameterize $X^1$ by $\\sigma $ .",
"We then identify $X^0$ with $\\tau $ and parameterize $X^2$ by $\\sqrt{\\alpha }\\,\\phi $ , i.e.",
"we use the static gauge where $X^\\mu =\\left(\\begin{array}{c}\\tau \\\\ \\sigma \\\\ \\sqrt{\\alpha }\\,\\phi \\end{array}\\right)~,\\qquad \\dot{X}^\\mu =\\left(\\begin{array}{c}1\\\\0\\\\ \\sqrt{\\alpha }\\,\\dot{\\phi }\\end{array}\\right)~,\\qquad \\acute{X}^{\\,\\mu }=\\left(\\begin{array}{c}0\\\\ 1 \\\\ \\sqrt{\\alpha }\\,\\acute{\\phi }\\end{array}\\right)~.$ In this gauge the string Lagrangian in (REF ) reduces to the deformed Lagrangian (REF ), up to the additive constant $1/\\alpha $ .",
"The string energy-momentum densities obtained from the Nambu-Goto action (REF ), ${\\cal P}^\\mu ={\\alpha }\\,{\\sqrt{(\\dot{X}\\,\\acute{X})^2-(\\dot{X}\\,\\dot{X})(\\acute{X}\\,\\acute{X})}}~,$ satisfy the (primary) constraints $(\\acute{X}\\,{\\cal P})=0\\,,\\qquad \\alpha ^2\\,(\\acute{X}\\,\\acute{X})+({\\cal P}\\,{\\cal P})=0\\,.$ As in the uncompactified case, the tangent vectors $\\acute{X}$ and $\\dot{X}$ are assumed spacelike and timelike, respectively, and $X^0$ is monotonically increasing in $\\tau ,$ i.e.",
"$(\\acute{X}\\,\\acute{X})>0~, \\qquad (\\dot{X}\\,\\dot{X})<0~, \\qquad \\dot{X}^0>0~.$ The momentum density ${\\cal P}^\\mu $ is then timelike and ${\\cal P}^0$ is positive.",
"In static gauge $\\begin{aligned}&{\\cal P}^0={\\alpha }\\,{\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}}~,\\\\& {\\cal P}^1={\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}}~,\\qquad {\\cal P}^2={\\sqrt{\\alpha }}\\,{\\sqrt{1+\\alpha \\,\\acute{\\phi }^{\\,2}-\\alpha \\,\\dot{\\phi }^2}}~.\\end{aligned}$ Comparing these expressions to (REF )-(REF ), we find ${\\cal P}^0={\\cal H}_\\alpha +{\\alpha }~, \\qquad {\\cal P}^1=-{\\cal P}~,\\qquad {\\cal P}^2={\\sqrt{\\alpha }}\\, \\Pi ~.$ Integrating the densities over $\\sigma $ gives the gauge invariant string energy-momentum.",
"In particular, the string energy reads $E_{\\text{str}}= \\int _0^{2\\pi }{2\\pi }\\,{\\cal P}^0(\\sigma )=H_\\alpha +{\\alpha }~,$ where $H_\\alpha $ is the energy of the deformed system (REF ).",
"Thus, the deformed system (REF ) and the compactified 3d string in the gauge (REF ) are identical dynamical systems.",
"On the other hand, it is well known that the classical string dynamics is integrable in the light-cone gauge.",
"The compactification of the coordinate $X^1$ does not destroy integrability, but rather modifies it, as we show below.",
"The static gauge (REF ) is not a conformal one for which one requires $(\\dot{X}\\,\\acute{X})=0$ and $(\\dot{X}\\,\\dot{X})+(\\acute{X}\\,\\acute{X})=0$ and the equation of motion for $X^\\mu $ becomes the free wave equation.",
"These two constraints have to be imposed on the solutions.",
"We denote the conformal worldsheet coordinates by $(\\tau _c,\\sigma _c)$ , to distinguish them from $(\\tau , \\sigma )$ , and introduce the corresponding chiral coordinates $z=\\tau _c+\\sigma _c$ and $\\bar{z}=\\tau _c-\\sigma _c$ .",
"One then has $\\partial _z\\partial _{\\bar{z}} X^\\mu =0$ , and its solutions $X^\\mu =\\Phi ^\\mu (z)+\\bar{\\Phi }^\\mu (\\bar{z})$ are restricted to satisfy the conformal gauge conditions $(\\Phi ^{\\prime }\\, \\Phi ^{\\prime })=0~, \\qquad (\\bar{\\Phi }^{\\prime }\\,\\bar{\\Phi }^{\\prime })=0~.$ The chiral functions $\\Phi ^{\\prime \\,\\mu }(z)$ and $\\bar{\\Phi }^{\\prime \\,\\mu }(\\bar{z})$ are periodic.",
"Therefore, similarly to the uncompactified case, $\\Phi ^\\mu (z)$ and $\\bar{\\Phi }^\\mu (\\bar{z})$ obey the monodromy conditions $\\Phi ^\\mu (z+2\\pi )=\\Phi ^\\mu (z)+2\\pi \\,\\rho ^\\mu ~, \\qquad \\bar{\\Phi }^\\mu (\\bar{z}+2\\pi )=\\bar{\\Phi }^\\mu (\\bar{z})+2\\pi \\,\\bar{\\rho }^\\mu ~,$ where $\\rho ^\\mu $ and $\\bar{\\rho }^\\mu $ are the zero modes of $\\Phi ^{\\prime \\,\\mu }(z)$ and $\\bar{\\Phi }^{\\prime \\,\\mu }(\\bar{z})$ , respectively.",
"From the periodicity conditions in $\\sigma $ one finds $\\rho ^0=\\bar{\\rho }^{\\,0}~, \\qquad \\rho ^1=\\bar{\\rho }^{\\,1}+L~, \\qquad \\rho ^2=\\bar{\\rho }^{\\,2}~,$ where $L$ is the winding number around the compactified coordinate $X^1$ .",
"For now we analyze the case of general $L$ , though our interest is $L=1$ .",
"To find independent variables on the constraint surface (REF ), we follow the standard scheme and introduce the light-cone coordinates $X^\\pm =X^0\\pm X^1$ .",
"Note that while one usually chooses the space-time light-cone directions along two non-compact coordinates, our definition of $X^\\pm $ involves the compact direction $X^1$ .",
"The remaining freedom of conformal transformations allows us to simplify the chiral components of $X^+$ as in the uncompactified caseThe conditions (REF ) require $\\rho ^+>0$ and $\\bar{\\rho }^+>0$ .",
"We will see in (REF ) that these conditions are indeed fulfilled.",
"$\\Phi ^+(z)=\\rho ^+ z~, \\qquad \\bar{\\Phi }^+=\\bar{\\rho }^{\\,+} \\bar{z}~.$ The constraints (REF ) can then be written as $\\rho ^+ \\,\\Phi ^{\\prime }{\\,^-}(z)={\\alpha }\\, F^{\\prime \\,2}(z)~, \\qquad \\bar{\\rho }^{\\,+}\\,\\bar{\\Phi }^{\\prime }{\\,^-}(\\bar{z})={\\alpha }\\,\\bar{F}^{\\prime \\,2}(\\bar{z})~,$ where $X^2$ is rescaled similarly to (REF ), i.e.",
"$\\Phi ^2(z)=\\sqrt{\\alpha }\\,F(z)$ and $\\bar{\\Phi }^2(\\bar{z})=\\sqrt{\\alpha }\\,\\bar{F}(\\bar{z})$ .",
"As a result, one obtains the following parameterization of the string coordinates $X^\\mu =\\left(\\begin{array}{c} {2}\\left[\\rho ^+z +\\Phi ^-(z)+\\bar{\\rho }^{\\,+}\\bar{z}+\\bar{\\Phi }^{\\,-}(\\bar{z})\\right]\\\\[2mm]{2}\\left[\\rho ^+z - \\Phi ^-(z)+\\bar{\\rho }^{\\,+}\\bar{z}-\\bar{\\Phi }^{\\,-}(\\bar{z}) \\right]\\\\[2mm]\\sqrt{\\alpha }\\left[ F(z)+\\bar{F}(\\bar{z})\\right]\\end{array}\\right).$ The functions $F(z)$ and $\\bar{F}(\\bar{z})$ have the mode expansions $F(z)={2}+{\\sqrt{2}}\\sum _{m\\ne 0}{n}\\,\\text{e}^{-\\mathrm {i}nz}, \\qquad \\bar{F}(\\bar{z})={2}+{\\sqrt{2}}\\sum _{n\\ne 0}{n}\\,\\text{e}^{-\\mathrm {i}n\\bar{z}},$ with $p={\\sqrt{\\alpha }}\\,\\rho ^2$ , and $\\Phi ^-(z)$ and $\\bar{\\Phi }^{\\,-}(\\bar{z})$ are obtained from (REF ) (see Appendix A).",
"In particular, one has $\\rho ^-=\\alpha \\,h̑{\\rho ^+}~, \\qquad \\bar{\\rho }^{\\,-}=\\alpha \\,{\\bar{\\rho }^{\\,+}}~,$ where $h$ and $\\bar{h}$ are the chiral free-field Hamiltonians $h=\\int _0^{2\\pi } {2\\pi }\\, F^{\\prime \\,2}(z)={4}+\\sum _{n >0} |a_n|^2\\,,~~~\\bar{h}=\\int _0^{2\\pi } {2\\pi }\\, \\bar{F}^{\\prime \\,2}(\\bar{z})={4}+\\sum _{n> 0} |\\bar{a}_n|^2\\,.$ Note that we set $\\bar{p}=p$ in (REF ), due to the third relation in (REF ).",
"The other two relations of (REF ), in terms of the light-cone variables, read $\\rho ^+ + \\rho ^- - \\bar{\\rho }^{\\,+}- \\bar{\\rho }^{\\,-}=0~, \\qquad \\rho ^+ - \\rho ^- - \\bar{\\rho }^{\\,+}+ \\bar{\\rho }^{\\,-}=2L~.$ For $L\\ne 0$ this leads to differences for the compactified case as compared to the non-compact one.",
"Indeed, for $L=0$ , the solution of (REF )-(REF ) is $\\rho ^+=\\bar{\\rho }^{\\,+}~,\\qquad \\rho ^-=\\bar{\\rho }^{\\,-}=\\alpha \\,h̑{\\rho ^+}=\\alpha \\,{\\bar{\\rho }^{\\,+}}~, \\qquad h=\\bar{h}~.$ Here, $\\rho ^+$ is a free dynamical variable.",
"The condition $h=\\bar{h}$ becomes, after quantization, the level matching condition in the zero winding sector.",
"When $L\\ne 0$ , we obtain instead the following solution of (REF )-(REF ) $\\begin{aligned}&\\rho ^\\pm ={2}\\left(\\alpha \\,{\\cal E}_{L}\\pm {L}(\\bar{h}-h)\\pm L\\right),\\quad &\\bar{\\rho }^{\\,\\pm }={2}\\left(\\alpha \\,{\\cal E}_{L}\\pm {L}(\\bar{h}-h)\\mp L\\right),\\end{aligned}$ $\\text{with} ~~~~~~~~{\\cal E}_{L}={L\\,\\alpha \\,}\\sqrt{L^4+2L^2\\,\\alpha (h+\\bar{h})+\\alpha ^2(h-\\bar{h})^2}~.$ Here, solving quadratic equations, we choose the positive roots, since they correspond to the physical solutions for which $\\rho ^\\pm >0$ and $\\bar{\\rho }^{\\,\\pm }>0$ .",
"Thus, for $L\\ne 0$ , the string solutions (REF ) are completely parametrized by the chiral free fields $F(z), \\,\\bar{F}(\\bar{z})$ .",
"We now find that the level matching condition is modified to $L(\\rho ^1+\\bar{\\rho }^1)=\\alpha \\,(\\bar{h}-h)\\,.$ According to (REF ), the string energy density in the conformal gauge is given by ${\\alpha }\\,\\partial _{\\tau _c} X^0$ , and from (REF ) we obtain the string energy for winding number $L$ $E_{\\text{str}}^{(L)}={2\\,\\alpha }\\left(\\rho ^+ + \\rho ^- +\\bar{\\rho }^{\\,+}+\\bar{\\rho }^{\\,-}\\right)={\\cal E}_L ~.$ For winding number one, which corresponds to the deformed system, this yields $E_{\\text{str}}={\\alpha }\\sqrt{1+2\\,\\alpha (h+\\bar{h})+\\alpha ^2(h-\\bar{h})^2}~,$ and, due to the gauge invariance of the string energy, we obtain from (REF ) [5] $H_\\alpha ={\\alpha }\\left(\\sqrt{1+2\\,\\alpha (h+\\bar{h})+\\alpha ^2(h-\\bar{h})^2}-1\\right)~.$ This expression for the Hamiltonian should be contrasted with (REF ).",
"There the Hamiltonian density of the deformed theory was expressed in terms of the energy-momentum densities of the undeformed theory while here the relation is between the integrated densities.",
"Furthermore, this expression can be easily quantized as $h$ and $\\bar{h}$ are diagonal in the Fock-space of the undeformed theory.",
"In Section 5.1 we will briefly discuss generalizations to general CFTs.",
"In this case the expression for $H_\\alpha $ is straightforwardly generalized by replacing $(h,\\bar{h})$ by $(L_0,\\bar{L}_0)$ of the undeformed theory.",
"In fact, many of the expressions in the following discussion are generalized if one replaces in the expression in Appendix A the $L_n$ of the free field by the generators of the Virasoro algebra of a general CFT.",
"In Appendix B we derive (REF ) directly (without referring to the gauge invariance), using the map that relates the worldsheet coordinates and the fields in two different gauges.",
"We will now analyze this map in detail.",
"Comparing the string coordinates in the gauges (REF ) and (REF ), we find the map from the coordinates $(z, \\bar{z})$ to $(\\tau , \\sigma )$Recall that $z=\\tau _c+\\sigma _c$ and $\\bar{z}=\\tau _c-\\sigma _c$ .",
"$\\begin{aligned}\\tau = {2}\\left[\\rho ^+z +\\Phi ^-(z)+\\bar{\\rho }^{\\,+}\\bar{z}+\\bar{\\Phi }^{\\,-}(\\bar{z})\\right]~,\\\\[1mm]\\sigma = {2}\\left[\\rho ^+z -\\Phi ^-(z)+\\bar{\\rho }^{\\,+}\\bar{z}-\\bar{\\Phi }^{\\,-}(\\bar{z})\\right]~,\\end{aligned}$ and we also express the solutions of the deformed system by the undeformed one $\\phi (\\tau ,\\sigma )= F(z)+\\bar{F}(\\bar{z})~.$ Differentiating (REF ) in $\\tau , \\,\\sigma $ and using (REF ), we obtain $\\begin{aligned}&\\dot{z}={\\alpha \\left[(\\rho ^+\\,\\bar{F}^{\\prime })^2-(\\bar{\\rho }^{\\,+}\\, F^{\\prime })^2\\right]}~,\\qquad &\\acute{z}={\\alpha \\left[(\\rho ^+\\,\\bar{F}^{\\prime })^2-(\\bar{\\rho }^{\\,+}\\, F^{\\prime })^2\\right]}~,\\\\[1mm]&\\dot{\\bar{z}}=-{\\alpha \\left[(\\rho ^+\\,\\bar{F}^{\\prime })^2-(\\bar{\\rho }^{\\,+}\\, F^{\\prime })^2\\right]}~,&\\acute{\\bar{z}}=-{\\alpha \\left[(\\rho ^+\\,\\bar{F}^{\\prime })^2-(\\bar{\\rho }^{\\,+}\\, F^{\\prime })^2\\right]}~.\\end{aligned}$ A similar differentiation of (REF ), with the help of (REF ), gives $\\dot{\\phi }={\\alpha \\left(\\rho ^+\\,\\bar{F}^{\\prime }+\\bar{\\rho }^+\\,F^{\\prime }\\right)}~, \\qquad \\acute{\\phi }={\\alpha \\left(\\rho ^+\\,\\bar{F}^{\\prime }+\\bar{\\rho }^+\\,F^{\\prime }\\right)}~,$ and they lead to $1+\\alpha \\acute{\\phi }^{\\,2}-\\alpha \\dot{\\phi }^2={\\left(\\rho ^+\\,\\bar{F}^{\\prime }+\\bar{\\rho }^{\\,+}\\,F^{\\prime }\\right)^2}~.$ The left hand side here defines the determinant of the induced worldsheet metric in static gauge and for regular surfaces it has to be positive.",
"Thus, for regular surfaces, the expressions $\\rho ^+\\,\\bar{F}^{\\prime }\\pm \\bar{\\rho }^{\\,+}\\,F^{\\prime }$ have no zeros.",
"Note that these expressions have the same sign for a sufficiently large zero mode $p$ .",
"Assuming this, we get $\\sqrt{1+\\alpha \\acute{\\phi }^{\\,2}-\\alpha \\dot{\\phi }^2}={\\rho ^+\\,\\bar{F}^{\\prime }+\\bar{\\rho }^{\\,+}\\,F^{\\prime }}~.$ From (REF ) then follows $\\Pi ={\\alpha \\left[\\rho ^+\\,\\bar{F}^{\\prime }(\\bar{z})-\\bar{\\rho }^+\\,F^{\\prime }(z)\\right]}~,$ and using (REF ) we obtain ${2}\\left( \\acute{\\phi }+\\Pi \\right)=\\acute{z}\\,F^{\\prime }(z)~,\\qquad {2}\\left( \\acute{\\phi }-\\Pi \\right)=\\acute{\\bar{z}}\\,\\bar{F}^{\\prime }(\\bar{z})~.$ Equation (REF ), for a fixed $\\tau $ , defines $z$ and $\\bar{z}$ as functions of $\\sigma $ .",
"For example, when the non-zero modes of $F^{\\prime }$ and $\\bar{F}^{\\prime }$ are not excited, $z={\\sqrt{1+\\alpha \\,p^2}}+\\sigma ~, \\qquad \\bar{z}={\\sqrt{1+\\alpha \\,p^2}}-\\sigma ~.$ In general, writing these functions as $z=\\zeta (\\sigma )$ , $\\bar{z}=\\bar{\\zeta }(-\\sigma )$ , we find that they are monotonic $\\zeta ^{\\prime }(x)>0$ , $\\bar{\\zeta }^{\\prime }(\\bar{x})>0$ and obey the monodromies $\\zeta (x+2\\pi )=\\zeta (x)+2\\pi ~, \\qquad \\bar{\\zeta }(\\bar{x}+2\\pi )=\\bar{\\zeta }(\\bar{x})+2\\pi ~,$ related to diffeomorphisms of a circle.",
"In the next subsection we show that (REF ) realizes a time dependent canonical map between the two gauges.",
"Concluding this subsection we express the energy-momentum density components in the static gauge (REF ) in terms of the light-cone gauge variables, using (REF ), (REF ) and (REF ).",
"With (REF ), ${\\cal P}^2$ is obtained from (REF ) and $\\nonumber &&{\\cal P}^0={\\alpha ^2\\left[(\\rho ^+\\,\\bar{F}^{\\prime })^2-(\\bar{\\rho }^{\\,+}\\, F^{\\prime })^2\\right]}=\\acute{z} \\left({\\alpha }+{\\rho ^+}\\right)=-\\acute{\\bar{z}} \\left({\\alpha }+{\\bar{\\rho }^{\\,+}}\\right),\\\\[1mm]&&{\\cal P}^1=-{\\alpha ^2\\left[(\\rho ^+\\,\\bar{F}^{\\prime })^2-(\\bar{\\rho }^{\\,+}\\, F^{\\prime })^2\\right]}\\\\ \\nonumber &&\\hspace{142.26378pt}=\\acute{z}\\left({\\alpha }-{\\rho ^+}\\right)-{\\alpha }=-\\acute{\\bar{z}} \\left({\\alpha }-{\\bar{\\rho }^{\\,+}}\\right)+{\\alpha }\\,.$ We will use these relations in the next section to relate the static and light-cone gauges in the Hamiltonian formulation."
],
[
"Hamiltonian approach to the compactified 3d string",
"We now consider the Hamiltonian treatment of the same system.",
"In the first order formulation of 3d string dynamics the action is $S=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[{\\cal P}_\\mu \\,\\dot{X}^\\mu -\\lambda _1\\,{\\cal C}_1 - \\lambda _2\\,{\\cal C}_2\\right]~,$ where $\\lambda _1$ , $\\lambda _2$ are Lagrange multipliers and ${\\cal C}_1$ , ${\\cal C}_2$ are the Virasoro constraints $\\begin{aligned}&{\\cal C}_1=({\\cal P}\\,\\acute{X})~,\\qquad &{\\cal C}_2={2}\\left[\\alpha ^2({\\cal P}\\,{\\cal P})+(\\acute{X}\\, \\acute{X})\\right].\\end{aligned}$ The compact coordinate $X^1$ has the expansion (for $L=1$ ) $X^1=\\sigma +\\sum _{n\\in \\mathbb {Z}}q_n\\,e^{-\\text{i}\\,n\\,\\sigma }~,$ with $q_{-n}=q_n^*$ , while the canonical momenta ${\\cal P}_\\mu $ and the coordinates ($X^0$ , $X^2$ ) remain periodic.",
"They have the standard mode expansion without the $\\sigma $ term in (REF ).",
"It follows from the canonical Poisson brackets on the extended phase space $\\lbrace {\\cal P}_\\mu (\\sigma _1), X^\\nu (\\sigma _2)\\rbrace =2\\pi \\,\\delta _\\mu ^{~\\nu }\\,\\delta (\\sigma _1-\\sigma _2)~,$ that the Poisson brackets of the constraints (REF ) form the algebra (REF ) $\\begin{aligned}&\\lbrace {\\cal C}_1(\\sigma _1), {\\cal C}_1(\\sigma _2)\\rbrace =2\\pi \\big [{\\cal C}_1(\\sigma _1)+{\\cal C}_1(\\sigma _2)\\big ]\\delta ^{\\prime }(\\sigma _1-\\sigma _2)\\,, \\\\[1mm]&\\lbrace {\\cal C}_1(\\sigma _1), {\\cal C}_2(\\sigma )\\rbrace =2\\pi \\big [{\\cal C}_2(\\sigma _1)+{\\cal C}_2(\\sigma _2)\\big ]\\delta ^{\\prime }(\\sigma _1-\\sigma _2)\\,,\\\\[1mm]&\\lbrace {\\cal C}_2(\\sigma _1), {\\cal C}_2(\\sigma _2)\\rbrace =2\\pi \\,\\alpha ^2\\big [{\\cal C}_1(\\sigma _1)+{\\cal C}_1(\\sigma _2)\\big ]\\delta ^{\\prime }(\\sigma _1-\\sigma _2)\\,,\\end{aligned}$ and one has to complete these first class constraints by gauge fixing conditions in order to eliminate non-physical degrees of freedom.",
"This can be done by the Faddeev-Jackiw reduction in static gauge $X^0=\\tau ,\\, X^1=\\sigma $ .",
"For this one computes ${\\cal P}_\\mu \\,\\dot{X}^\\mu $ on the constrained surface ${\\cal C}_1={\\cal C}_2=0$ in this gauge.",
"The action (REF ) then reduces to $S|_{\\text{st.g.",
"}}=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left({\\cal P}_0+{\\cal P}_2\\,\\dot{X}^2 \\right),$ where ${\\cal P}_0$ becomes a function of the reduced canonical variables $({\\cal P}_2 ,{X}^2)$ .",
"Hence, ${\\cal P}^0=-{\\cal P}_0$ plays the role of the Hamiltonian density.",
"In order to relate the reduced Hamiltonian system to the deformed model, we rescale the canonical variables, $\\,{\\cal P}_2={\\sqrt{\\alpha }}~, \\qquad X^2=\\sqrt{\\alpha }\\,\\phi ~,$ and rewrite the constraints (REF ) as ${\\cal C}_1=\\Pi \\,\\phi ^{\\prime }+{\\cal P}_1=0~,\\qquad 2\\,{\\cal C}_2=\\alpha (\\Pi ^2+\\acute{\\phi }^2)+\\alpha ^2{\\cal P}_1^2-\\alpha ^2{\\cal P}_0^2+1=0~.$ These equations define the remaining phase space variables ${\\cal P}_1=-\\Pi \\,\\phi ^{\\prime }~, \\quad {\\cal P}_0=-{\\alpha }\\,\\sqrt{1+\\alpha \\left(\\Pi ^2+\\acute{\\phi }^2\\right)+\\alpha ^2\\left(\\Pi \\,\\acute{\\phi }\\right)^2}~$ and we finally obtain $S|_{\\text{st.g.",
"}}=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[\\Pi \\,\\dot{\\phi }-\\left({\\cal H}_\\alpha +{\\alpha }\\right)\\right]\\,.$ ${\\cal H}_\\alpha $ is the Hamiltonian density of the deformed model (REF ).",
"Thus, the Faddeev-Jackiw reduction of the compactified 3d string in the static gauge leads to the deformed free-field model.",
"We now consider Hamiltonian reduction of (REF ) in light-cone gauge.",
"Introducing the light-cone coordinates $X^\\pm =X^0\\pm X^1~, \\qquad {\\cal P}_\\pm ={2}\\left({\\cal P}_0 \\pm {\\cal P}_1\\right)~,$ the string action (REF ) and the constraints become $S=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[{\\cal P}_+\\,\\dot{X}^++{\\cal P}_{-}\\,\\dot{X}^{-}+{\\cal P}_2\\,\\dot{X}^2-\\lambda _1\\,{\\cal C}_1 - \\lambda _2\\,{\\cal C}_2\\right]~,\\quad \\\\{with}{\\cal C}_1={\\cal P}_+\\,\\acute{X}^+ + {\\cal P}_-\\,\\acute{X}^-+{\\cal P}_2\\,\\acute{X}^2,\\quad {\\cal C}_2={2}\\left[\\alpha ^2{\\cal P}_2^2+\\acute{X}_2^2-4\\,\\alpha ^2\\,{\\cal P}_+{\\cal P}_- -\\acute{X}^+\\,\\acute{X}^-\\right].$ Using the gauge freedom, we can eliminate the non-zero modes of ${\\cal P}_-(\\sigma )$ and $X^+(\\sigma )$ , similarly to the uncompactified case.",
"Taking into account that $X^1$ has winding number one, the light-cone gauge condition reads $X^+(\\sigma )=-2\\,\\alpha \\,{\\cal P}_-\\tau +\\sigma ~, \\qquad \\acute{\\cal P}_-(\\sigma )=0~.$ This provides $\\acute{X}^+(\\sigma )=1$ and ${\\cal P}_-(\\sigma )=p_-$ , where $p_-$ is the zero mode of ${\\cal P}_-(\\sigma )$ .",
"Rescaling then the canonical variables similarly to (REF )Note that the pairs $(\\Pi , \\phi )$ and $(, )$ differ from each other, though they denote the same variables in the initial extended phase space.",
"$\\,{\\cal P}_2={\\sqrt{\\alpha }}~, \\qquad X^2=\\sqrt{\\alpha }\\,~,$ the constraints () can be written as ${\\cal C}_1={\\cal P}_+ + p_-\\,\\acute{X}^-+{\\cal P}=0,\\quad 2\\,{\\cal C}_2=2\\,\\alpha \\,{\\cal H}-4\\,\\alpha ^2\\,p_-\\,{\\cal P}_+ -\\acute{X}^-=0~,$ with ${\\cal P}=\\acute{}~, \\qquad {\\cal H}= 2\\left(^2+\\acute{}^2 \\right)~.$ By (REF ) one finds ${\\cal P}_+$ and $\\acute{X}^-$ in terms of $(,)$ and the zero mode $p_-$ ${\\cal P}_+=-{1-4\\,\\alpha ^2 p_-^2}~, \\qquad \\acute{X}^-={1-4\\,\\alpha ^2 p_-^2}~.$ The zero modes of the constraints (REF ) satisfy $(p_+-p_- )+ P=0~, \\qquad 2\\,\\alpha \\,H+(1-4\\,\\alpha ^2\\,p_-\\,p_+)=0~,$ where $p_+$ is the zero mode of ${\\cal P}_+$ and $P=\\int _0^{2\\pi } {2\\pi }\\, {\\cal P}~, \\qquad H=\\int _0^{2\\pi }{2\\pi }\\,{\\cal H}~.$ The string energy then becomes $E_{\\text{str}}=-(p_+ + p_-)={\\alpha }\\,\\sqrt{1+2\\,\\alpha \\, H+\\alpha ^2 P^2}~.$ Faddeev-Jackiw reduction of the action (REF ) by the constraints ()-(REF ) yields $S|_{\\text{l-c.\\,g.",
"}}=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[(\\sigma )\\,\\dot{}(\\sigma )-2\\,\\alpha \\, p_+(p_-+\\dot{p}_-\\tau )+p_-\\dot{x}^-\\right],$ where $x^-$ is the zero mode of the periodic part of $X^-(\\sigma )$ , and we have used the rescaled variables (REF ).",
"Neglecting the total derivative term ${\\rm {d}{\\tau }}(-2\\,\\alpha \\, p_+ p_- \\tau )$ in (REF ), we obtain $S|_{\\text{l-c.\\,g.",
"}}=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[(\\sigma )\\,\\dot{}(\\sigma )+p_-\\dot{q}^--2\\,\\alpha \\, p_+\\,p_-\\right].$ with $q^-=x^-+2\\,\\alpha \\,p_+ p_-\\tau $ .",
"Using (REF ) and neglecting also the constant term $1/(2 \\alpha )$ , we end up with the action $S|_{\\text{l-c.\\,g.",
"}}=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[\\,\\dot{}+p_-\\dot{q}^- -{\\cal H}\\right]~,$ where ${\\cal H}$ is the free-field Hamiltonian density (REF ) and $p_-$ is obtained from (REF ) $p_{-}={2}\\left(P-{{\\alpha }}\\,\\sqrt{1+2\\,\\alpha \\, H+\\alpha ^2 P^2}\\right)~.$ The situation here is similar to the uncompactified case, where instead of (REF ) one has the level matching condition $P=h-\\bar{h}=0$ .In the Hamiltonian formulation the light-cone gauge is not a complete gauge fixing for the closed string.",
"The constraint corresponding to the remaining gauge freedom is the level matching condition.",
"After complete gauge fixing one arrives at a conformal gauge and the Hamiltonian formulation is then equivalent to the Lagrangian formulation in light cone gauge [17], [18].",
"Further Hamiltonian reduction in both cases is inconvenient.",
"One has to quantize the free-field model together with the particle $(p_-, q^-)$ and impose the condition (REF ) at the quantum level.",
"Note that the right hand side in (REF ) is a well defined operator in the Fock space of the free-field theory.",
"We now discuss the relation between the static and light-cone gauges in the Hamiltonian approach.",
"In general, reduced Hamiltonian systems obtained in two different gauges are related to each other by a canonical transformation generated by the constraints of the initial gauge invariant system.",
"Our aim is to describe the canonical map between the light-cone and the static gauges of the compactified 3d string.",
"First note that the Virasoro constraints (REF ) can be represented in the form ${\\cal C}(\\sigma ):=f_\\mu (\\sigma )\\,f^\\mu (\\sigma )=0~, \\qquad \\bar{\\cal C}(\\sigma ):=\\bar{f}_\\mu (\\sigma )\\,\\bar{f}^\\mu (\\sigma )=0~,$ with $f^\\mu (\\sigma )={2{\\sqrt{\\alpha }}}\\left(\\alpha \\,{\\cal P}^\\mu (\\sigma )+\\acute{X}^\\mu (\\sigma )\\right)\\,, \\quad \\bar{f}^\\mu (\\sigma )={2{\\sqrt{\\alpha }}}\\left(\\alpha {\\cal P}^\\mu (-\\sigma )-\\acute{X}^\\mu (-\\sigma )\\right)\\,.$ From the canonical Poisson brackets (REF ) follows $\\nonumber &&\\lbrace {\\cal C}(\\sigma _1), f^\\mu (\\sigma )\\rbrace =2\\pi \\,\\partial _\\sigma [f^\\mu (\\sigma )\\,\\delta (\\sigma _1-\\sigma )],\\quad \\lbrace \\bar{\\cal C}(\\sigma _1), \\bar{f}^\\mu (\\sigma )\\rbrace =2\\pi \\,\\partial _\\sigma [\\bar{f}^\\mu (\\sigma )\\,\\delta (\\sigma _1-\\sigma )],\\\\[1mm]&&\\lbrace {\\cal C}(\\sigma _1), \\bar{f}^\\mu (\\sigma )\\rbrace =\\lbrace \\bar{\\cal C}(\\sigma _1), f^\\mu (\\sigma )\\rbrace =0.$ The corresponding infinitesimal transformations $f^\\mu (\\sigma )\\mapsto f^\\mu (\\sigma )+\\partial _\\sigma \\left[\\epsilon (\\sigma )\\,f^\\mu (\\sigma )\\right]~, \\qquad \\bar{f}^\\mu (\\sigma )\\mapsto \\bar{f}^\\mu (\\sigma )+\\partial _\\sigma \\left[\\bar{\\epsilon }(\\sigma )\\,\\bar{f}^\\mu (\\sigma )\\right]~,$ lead to the global ones $f^\\mu (\\sigma )\\mapsto \\zeta ^{\\prime }(\\sigma )\\, f^\\mu (\\zeta (\\sigma ))~, \\qquad \\bar{f}^\\mu (\\sigma )\\mapsto \\bar{\\zeta }^{\\prime }(\\sigma )\\, \\bar{f}^\\mu (\\bar{\\zeta }(\\sigma ))~,$ where $\\zeta (\\sigma ),~\\bar{\\zeta }(\\sigma )$ are diffeomorphisms of the unit circle.",
"Note that, in general, the group parameters $\\epsilon (\\sigma ),~\\bar{\\epsilon }(\\sigma )$ could be functions on the phase space, since the transformations are on-shell.",
"The static gauge provides the following parameterization of $f^\\mu $ and $\\bar{f}^\\mu $ $\\begin{aligned}&f^\\mu _{\\text{st}.g.",
"}(\\sigma )={2}\\left(\\begin{array}{c}\\sqrt{\\alpha }\\,{\\cal P}^{0}(\\sigma )\\\\[1mm] \\sqrt{\\alpha }\\,\\,{\\cal P}^1(\\sigma )+{\\sqrt{\\alpha }} \\\\[1mm]\\Pi (\\sigma )+\\phi ^{\\prime }(\\sigma )\\end{array}\\right),\\quad \\bar{f}^\\mu _{\\text{st}.g.",
"}(\\sigma )={2}\\left(\\begin{array}{c}\\sqrt{\\alpha }\\,{\\cal P}^{0}(-\\sigma )\\\\[1mm] \\sqrt{\\alpha }\\,{\\cal P}^1(-\\sigma )-{\\sqrt{\\alpha }} \\\\[1mm]\\Pi (-\\sigma )-\\phi ^{\\prime }(-\\sigma ) \\end{array}\\right),\\end{aligned}$ where ${\\cal P}^{0}$ and ${\\cal P}^1$ are given by (REF ).",
"The light-cone gauge parameterization of $f^\\mu $ and $\\bar{f}^\\mu $ is obtained from (REF )-(REF ) $\\begin{aligned}&f^\\mu _{\\text{l-c.g.",
"}}(\\sigma )={2}\\left(\\begin{array}{c} {\\sqrt{\\alpha }} +{4\\rho ^+}\\\\[1mm] {\\sqrt{\\alpha }} -{4\\rho ^+}\\\\[1mm](\\sigma ) +^{\\prime }(\\sigma )\\end{array}\\right),&\\bar{f}^\\mu _{\\text{l-c}.g.",
"}(\\sigma )={2}\\left(\\begin{array}{c} {\\sqrt{\\alpha }}+{4\\bar{\\rho }^{\\,+}}\\\\[1mm]{\\sqrt{\\alpha }} -{4\\bar{\\rho }^{\\,+}}\\\\[1mm](-\\sigma ) -^{\\prime }(-\\sigma )\\end{array}\\right),\\end{aligned}$ where we have used $2\\rho ^+=1-2\\,\\alpha \\,p_-~, \\qquad 2\\bar{\\rho }^{\\,+}=-1-2\\,\\alpha \\,p_-~.$ Based on (REF ), we introduce the relations $f^\\mu _{\\text{st.g.",
"}}(\\sigma )=\\zeta ^{\\prime }(\\sigma )\\, f^\\mu _{\\text{l-c.g.",
"}}(\\zeta (\\sigma ))~, \\qquad \\bar{f}^\\mu _{\\text{st.g.",
"}}(\\sigma )=\\bar{\\zeta }^{\\prime }(\\sigma )\\, f^\\mu _{\\text{l-c.g.",
"}}(\\bar{\\zeta }(\\sigma ))~,$ which by (REF )-(REF ) are equivalent to $\\begin{aligned}&\\alpha \\left[{\\cal P}^0(\\sigma )+{\\cal P}^1(\\sigma )\\right]+1=2\\rho ^+\\zeta ^{\\prime }(\\sigma )\\,,\\\\[1mm]&\\alpha \\left[{\\cal P}^0(\\sigma )-{\\cal P}^1(\\sigma )\\right]-1={2\\rho ^+}\\,\\zeta ^{\\prime }(\\sigma )[(\\zeta (\\sigma ))+\\acute{}(\\zeta (\\sigma ))]^2\\,,\\\\&\\Pi (\\sigma )+\\acute{\\phi }(\\sigma )=\\zeta ^{\\prime }(s)\\left((\\zeta (\\sigma ))+\\acute{}(\\zeta (\\sigma ))\\right)\\,,\\end{aligned}$ and similarly for the anti-chiral part $\\begin{aligned}&\\alpha \\left[{\\cal P}^0(-\\sigma )+{\\cal P}^1(-\\sigma )\\right]-1=2\\bar{\\rho }^{\\,+}\\bar{\\zeta }^{\\prime }(\\sigma )\\,,\\\\[1mm]&\\alpha \\left[{\\cal P}^0(-\\sigma )-{\\cal P}^1(-\\sigma )\\right]+1={2\\bar{\\rho }^{\\,+}}\\,\\bar{\\zeta }^{\\prime }(\\sigma )[(-\\bar{\\zeta }(\\sigma ))+\\acute{}(-\\bar{\\zeta }(\\sigma ))]^2\\,,\\\\[1mm]&\\Pi (-\\sigma )+\\acute{\\phi }(-\\sigma )=\\bar{\\zeta }^{\\prime }(\\sigma )\\left((-\\bar{\\zeta }(\\sigma ))+\\acute{}(-\\bar{\\zeta }(\\sigma ))\\right)\\,.\\end{aligned}$ The integration in (REF ) over $\\sigma $ provides the relations $\\alpha \\left(E_{\\text{str}}+P^1_{\\text{str}}\\right)+1=2\\rho ^+\\,,\\quad \\alpha \\left(E_{\\text{str}}-P^1_{\\text{str}}\\right)-1={\\rho ^+}\\,,$ which for the string energy leads again to (REF ).",
"The same result is obtained for the antichiral part (REF ).",
"Equations (REF )-(REF ) are equivalent to (REF ) and (REF ) with $z(\\sigma )=\\zeta (\\sigma )$ and $\\bar{z}(\\sigma )=\\bar{\\zeta }(-\\sigma )$ , which indicates that they define a canonical map between the two gauges.",
"The direct computation with the help of (REF )-(REF ) shows that this map preserves the canonical symplectic form $\\begin{aligned}&\\int _{0}^{2\\pi }{2\\pi }\\,\\delta \\Pi (\\sigma )\\wedge \\delta \\phi (\\sigma )=\\int _{0}^{2\\pi }{2\\pi }\\,\\delta (\\sigma )\\wedge \\delta (\\sigma )=\\\\[1mm]&\\int _{0}^{2\\pi }{2\\pi }\\,\\delta F(x)\\wedge \\delta F^{\\prime }(x)+\\int _{0}^{2\\pi }{2\\pi }\\,\\delta \\bar{F}(\\bar{x})\\wedge \\delta \\bar{F}^{\\prime }(\\bar{x})+{2}\\,\\delta p\\wedge \\left[\\delta F(0)+\\delta \\bar{F}(0)\\right]\\,.\\end{aligned}$"
],
[
"Generalization to 2d CFTs and to (non-conformal) models with a potential",
"In this section we first generalize the scheme described in Section 4.2 to other 2d CFTs.",
"Recall that starting from the free field model we had arrived at the $T\\bar{T}$ deformed action.",
"This was identified with the Nambu-Goto action of a 3d string in static gauge.",
"We then wrote the unfixed NG action in Hamiltonian form and fixed the light-cone gauge.",
"Faddeev-Jackiw reduction of the gauge fixed action lead to the original free field Hamiltonian.",
"Guided by this, starting from a 2d CFT with a canonical description, specified by a Hamiltonian density ${\\cal H}(\\Pi ,\\phi ,\\acute{\\phi })$ , we will devise a first order system such that after going to static gauge we recover the deformed Hamiltonian while when working in light-cone gauge we arrive at the undeformed Hamiltonian ${\\cal H}$ .",
"We then apply the same scheme to the model (REF ) with a potential, which explicitly breaks conformal symmetry.",
"Relevant references for this section are [11], [12], [13]."
],
[
"Integrability of the deformed 2d CFTs",
"We introduce a constrained Hamiltonian system with a string type action $S=\\int \\mbox{d}\\tau \\int _0^{2\\pi }{2\\pi }\\left[{\\cal P}_0\\,\\dot{X}^0+{\\cal P}_{1}\\,\\dot{X}^{1}+\\Pi _k\\,\\dot{\\phi }^k-\\lambda _1\\,{\\cal C}_1 - \\lambda _2\\,{\\cal C}_2\\right]~,$ where $\\begin{aligned}&{\\cal C}_1={\\cal P}_0\\,\\acute{X}^0+{\\cal P}_{1}\\,\\acute{X}^{1}+ {\\cal P} ~,\\\\[1mm]&{\\cal C}_2={2}\\left[\\alpha ^2\\left({\\cal P}_{1}^2-{\\cal P}_0^2\\right)+\\left(\\acute{X}_{1}^2-\\acute{X}_0^2\\right)\\right]+\\alpha \\,{\\cal H}(\\Pi ,\\phi ,\\acute{\\phi })~.\\end{aligned}$ ${\\cal H}$ and ${\\cal P}$ are the Hamiltonian and momentum densities of a 2d CFT.",
"We assume that the conditions (REF )-(REF ) are fulfilled.",
"Because of (REF ) the Poisson brackets of the constraints (REF ) satisfy (REF ).",
"The system is reparametrization invariant (with the appropriate transformation properties of $\\lambda _{1,2}$ [16]).",
"This enables us to introduce the static gauge, where again $X^{1}$ is a compact coordinate.",
"Doing this and applying the Faddeev-Jackiw reduction one finds that the action (REF ) reduces to the $T\\bar{T}$ -deformed system (REF ) with the Hamiltonian density ${\\cal H}_\\alpha +{1}/{(2\\alpha )}$ , where ${\\cal H}_\\alpha $ as in eq.",
"(REF ).",
"If we fix instead light-cone gauge (REF ) and use the definitions (REF ), we arrive again at (REF ).",
"The equations (REF )-(REF ) are trivially generalized with the replacements $\\,\\dot{}\\mapsto _k\\,\\dot{}^k~, \\qquad \\ \\,\\acute{}\\mapsto _k\\,\\acute{}^k~,\\qquad 2\\left(^2+\\acute{}^2\\right)\\mapsto {\\cal H}(,,\\acute{})~.$"
],
[
"Generalization to models with a potential",
"We now generalize the above discussion to theories with a conformal symmetry breaking potential $U(\\phi )$ .",
"To this end we introduce a string like dynamical system such that in static gauge it reduces to the deformed theory specified by the Hamiltonian density (REF ).",
"Consider an action of the type (REF ), where the constraint ${\\cal C}_1$ is the same as in (REF ) but with a modified ${\\cal C}_2$ of the form ${\\cal C}_2={2}\\Big [g({\\cal P}_{1}^2-{\\cal P}_0^2)+{g}(\\acute{X}_{1}^2-\\acute{X}_0^2)-2\\,b\\,g({\\cal P}_0 \\acute{X}^{1}+{\\cal P}_{1} \\acute{X}^{0})+2\\,{\\cal H}\\Big ]\\,.$ This has the structure of the Hamiltonian density (REF ) which guarantees that the constraints ${\\cal C}_1$ and ${\\cal C}_2$ satisfy the algebra (REF ).",
"The matrices $G$ and $B$ in the space spanned by ($X^0, X^{1}$ ), are $G^{kl}=g\\left(\\begin{array}{cr} -1&0\\\\ \\phantom{-}0 & 1\\end{array}\\right)~, \\qquad B_{kl}=b\\left(\\begin{array}{cr} \\phantom{-}0&1\\\\ -1& 0\\end{array}\\right).$ As before the system is reparametrization invariant and we can fix either static or light-cone gauge.",
"The Faddeev-Jackiw reduction in the static gauge is again straightforward.",
"If we identify [11], [13] $g=\\beta ~, \\qquad b={2\\,\\beta } ~,\\qquad \\beta =\\alpha \\left(1-{\\alpha \\over 2}U\\right)$ it leads to the Hamiltonian system (REF ) with the deformed Hamiltonian (REF ).",
"We now turn to the reduction in light-cone gauge.",
"The precise form of this gauge choice is less obvious in the non-conformal case and to find it we rewrite the first order system in second order Lagrangian form as a sigma-model with target space coordinates $(X^0,X^1,\\phi ^k)$ : $S=S[\\phi ]+{1\\over 2\\pi }\\int d\\tau d\\sigma \\Big (-{1\\over 2\\beta (\\phi )}\\partial _z X^+\\partial _{\\bar{z}} X^-+{1\\over \\alpha }\\big (\\dot{X}^0\\,\\acute{X}^1-\\acute{X}^0\\dot{X}^1\\big )\\Big )$ where the first term is the 2d CFT action and the last term does not contribute to the equations of motion.",
"For the light-cone fields $X^\\pm $ they are $\\partial _z\\left({1\\over \\beta (\\phi )}\\partial _{\\bar{z}}X^-\\right)=0\\,,\\qquad \\partial _{\\bar{z}}\\left({1\\over \\beta (\\phi )}\\partial _z X^+\\right)=0\\,,$ which can be integrated once ${1\\over \\beta (\\phi )}\\partial _{\\bar{z}}X^-=\\rho ^-(\\bar{z})\\,,\\qquad {1\\over \\beta (\\phi )}\\partial _z X^+=\\rho ^+(z)\\,.$ $\\rho ^+$ and $\\rho ^-$ transform as one-forms under reparametrizations of the circle.",
"Assuming that they have constant sign, which poses a restriction on the potential, one can gauge away the non-constant (oscillator) parts.",
"In light-cone gauge $\\rho ^\\pm $ are (arbitrary) constants.",
"If we insert this into the equation of motion for $\\phi $ , we obtain ${\\delta \\over \\delta \\phi ^k}S[\\phi ] +{1\\over 4}\\alpha ^2\\rho ^+\\!\\rho ^-{\\partial \\over \\partial \\phi ^k} U(\\phi )=0\\,.$ For appropriate choice for $\\rho ^\\pm $ these are the equations of motion of the undeformed theory.",
"In the case of a single scalar field $\\phi $ with a free action and potential $U(\\phi )=2-2\\,e^{2\\,\\phi }$ equation (REF ) becomes the Liouville equation.",
"For the same choice of potential and $\\alpha =1$ , the action (REF ) (before gauge fixing) and ignoring the boundary term is the $SL(2)$ WZW-model [19], [20].",
"Acknowledgements We would like to thank Harald Dorn and Alessandro Sfondrini for useful discussions.",
"G.J.",
"thanks MPI for Gravitational Physics in Potsdam for warm hospitality during his visits in 2018 and 2019 and S.T.",
"is grateful to the Mathematical Institute of TSU for supporting a visit in Tbilisi."
],
[
"Solution for the light-cone chiral fields",
"Due to (REF ), the Fourier mode expansions of $F^{\\prime \\,2}(z)$ and $\\bar{F}^{\\prime \\,2}(\\bar{z})$ $F^{\\prime \\,2}(z)=\\sum _{n\\in \\mathbb {Z}} L_n\\,\\text{e}^{-\\mathrm {i}nz}~, \\qquad \\bar{F}^{\\prime \\,2}(\\bar{z})=\\sum _{n\\in \\mathbb {Z}}\\bar{L}_n\\,\\text{e}^{-\\mathrm {i}n\\bar{z}}~,$ define $L_n$ and $\\bar{L}_n$ as the Virasoro generators in the standard free-field form $L_n={2}\\sum _{n\\in \\mathbb {Z}}a_m\\,a_{n-m}~, \\qquad \\bar{L}_n={2}\\sum _{n\\in \\mathbb {Z}}\\bar{a}_m\\,\\bar{a}_{n-m}~,$ with $a_0=\\bar{a}_0={p}$ .",
"The solution of (REF ) can then be written as $\\Phi ^-(z)=\\rho ^-z+{\\rho ^+}\\sum _{n\\ne 0}{n}\\,\\text{e}^{-\\mathrm {i}nz}~,\\qquad \\bar{\\Phi }^{\\,-}(\\bar{z})=\\bar{\\rho }^{\\,-}z+{\\bar{\\rho }^+}\\sum _{n\\ne 0}{n}\\,\\text{e}^{-\\mathrm {i}n\\bar{z}}~,$ where $\\rho ^-$ and $\\bar{\\rho }^{\\,-}$ are given by (REF ).",
"We neglect the constant zero modes of $\\Phi ^-(z)$ and $\\bar{\\Phi }^{\\,-}(\\bar{z})$ ; they correspond to translations of $X^0$ and $X^2$ ."
],
[
"String energy in the static and light-cone gauges",
"The integration of (REF ) over $\\sigma $ , for a fixed $\\tau $ , yields $\\begin{aligned}{\\alpha }\\int _0^{2\\pi } {2\\pi }\\, {\\sqrt{1+\\alpha \\acute{\\phi }^{\\,2}-\\alpha \\dot{\\phi }^2}}={\\alpha }\\int _0^{2\\pi }{2\\pi } \\left(\\rho ^+ +{\\rho ^+}\\right)={\\alpha }(\\rho ^+ +\\rho ^-)~~~\\\\[1mm]~~~~={\\alpha }\\int _0^{2\\pi }{2\\pi } \\left(\\bar{\\rho }^{\\,+}+{\\bar{\\rho }^{\\,+}}\\right)={\\alpha }(\\bar{\\rho }^{\\,+} +\\bar{\\rho }^{\\,-}).\\end{aligned}$ According to (REF ), the left hand side of this equation is the string energy in the static gauge and the right hand sides correspond to the string energy in the light-cone gauge (REF ).",
"This straightforward calculation confirms the validity of (REF ), without referring to the gauge invariance of the string energy.",
"A similar calculation for the string momentum $P^1$ by (REF ) yields $\\begin{aligned}P^1=\\int _0^{2\\pi } {2\\pi }\\, {\\sqrt{1+\\alpha \\acute{\\phi }^{\\,2}-\\alpha \\dot{\\phi }^2}}={\\alpha }\\int _0^{2\\pi }{2\\pi } \\left(\\rho ^+ -{\\rho ^+}\\right)=\\\\[1mm]{\\alpha }(\\rho ^+ -\\rho ^--1)=\\bar{h} -h.\\end{aligned}$"
]
] | 2001.03563 |
[
[
"Radiation of charge moving through a dielectric spherical target: ray\n optics and aperture methods"
],
[
"Abstract Radiation of charged particles moving in the presence of dielectric targets is of significant interest for various applications in the accelerator and beam physics.",
"The size of these targets is typically much larger than the wavelengths under consideration.",
"This fact gives us an obvious small parameter of the problem and allows developing approximate methods for analysis.",
"We develop two methods, which are called the \"ray optics method\" and the \"aperture method\".",
"In the present paper, we apply these methods to analysis of Cherenkov radiation from a charge moving through a vacuum channel in a solid dielectric sphere.",
"We present the main analytical results and describe the physical effects.",
"In particular, it is shown that the radiation field possesses an expressed maximum at a certain distance from the sphere at the Cherenkov angle.",
"Additionally, we perform simulations in COMSOL Multiphysics and show a good agreement between numerical and analytical results."
],
[
"Introduction",
"Radiation of charged particles moving in the presence of dielectric objects (“targets”) is of vital interest for various applications [1], [2], [3], [4], [5].",
"For example, several experiments have shown that prismatic and conical targets can be prospective for both bunch diagnostics and generation of high-power radiation [2], [3], [4], [5].",
"Further development of these topics requires an accurate calculation of Cherenkov radiation (CR) outside dielectric objects, which is typically impossible to do rigorously due to the complicated geometry of these objects.",
"The only exceptions allowing the construction of rigorous solutions are the simplest geometries like infinite cylinder [6], [7], [8], [9] or sphere [9], [10].",
"However, the obtained formulas (infinite series) can be applied to the field analysis only when the wavelength $\\lambda $ is comparable with the target radius.",
"Moreover, such “practical” modifications of the geometry as a vacuum channel for the charge flight cannot be incorporated in this solution.",
"For the discussed applications, the vacuum channel is needed for the flight of the bunch.",
"Moreover, the target dimensions are typically much larger than the wavelengths of interest.",
"Therefore, both calculations based on the solutions mentioned above and numerical simulations are very complicated.",
"However, the mentioned relation between $\\lambda $ and target size gives us an obvious small parameter allowing the development of approximate (asymptotic) methods for the analysis of radiation.",
"Recently we have offered and successfully verified two such methods called the “ray optics method” [11], [12] and “aperture method” [13], [14], [15], [16], [17], [18], [19].",
"They can be divided into three steps.",
"The first two steps are the same for both methods.",
"First, we solve the specific “etalon” problem, which does not take into account the “external” boundaries of the target.",
"For example, if the charge moves in the vacuum channel inside the target, then in the first step, we consider the problem with the channel in the unbounded medium.",
"In other words, we consider only the boundary nearest to the charge trajectory and solve the problem for the semi-infinite medium.",
"In the second step, we select a part of the external surface of the object which is illuminated by CR and transparent for CR (so that there is no total internal reflection here).",
"This part of the object boundary is called an “aperture” further.",
"Then we use the fact that the object is large in comparison with the wavelengths under consideration.",
"More precisely, we assume that: (i) the size of the aperture $\\Sigma $ is much larger than the wavelength $\\lambda $ ; (ii) the distance from the main part of the aperture to the charge trajectory is also much larger than the wavelength $\\lambda $ .",
"The field obtained in the first step is used as the incident field on the aperture.",
"Due to point (ii), we can neglect the quasi-static (quasi-Coulomb) part of this field and use corresponding asymptotic approximation, which is a quasi-plane wave (more precisely, a cylindrical wave with a small curvature of the wave front).",
"Further, we decompose this wave into a superposition of vertical and horizontal polarizations (concerning the plane of incidence) and calculate the field on the external surface of the aperture using Snell’s law and the Fresnel equations.",
"The third step is different for the two methods.",
"The ray optics method uses the ray optics laws for calculation of the wavefield outside the object [11], [12].",
"However, this technique has essential limitations.",
"First, the so-called “wave parameter” $W \\sim \\lambda L/\\Sigma $ ($L$ is a distance from the aperture to the observation point) should be small $W \\ll 1$ .",
"Note that this means that the distance $L$ can be much larger than $\\lambda $ but cannot be larger than $\\Sigma /\\lambda $ .",
"In particular, we cannot consider the important Fraunhofer area where $W \\gg 1$ .",
"Second, the observation point can not be in the neighborhood of focuses and caustics, where ray optics is not applicable.",
"The aperture method is more general [13], [14], [15], [16], [17], [18], [19] than the ray optics.",
"It is valid for observation points with arbitrary wave parameter $W$ , including the Fraunhofer (far-field) area and neighborhoods of focuses and caustics.",
"In the third step of this technique, we calculate the field outside the target using Stratton-Chu formulas (“aperture integrals”).",
"These formulas allow determining the field in the surrounding space if tangential components of the electric and magnetic fields on the aperture are known.",
"This paper is devoted to the study of CR from a solid dielectric sphere with the radius being much larger than the wavelength and having an axisymmetric vacuum channel where the charge moves.",
"Such a target can be manufactured with high accuracy and can be a prospective candidate for the aforementioned applications.",
"We apply both methods for the spherical target and compare the obtained analytical results with the results of COMSOL Multiphysics simulations."
],
[
"The field on the ball surface",
"Here we consider a dielectric ball with the radius $R_0$ having the cylindrical vacuum channel with the radius $a$ (Fig.",
"REF , left).",
"In accordance with point (ii) in the Introduction, it is assumed that $kR_0 \\gg 1$ , where $k=\\omega /c$ ($\\omega $ is the frequency, $c$ is the speed of light in vacuum).",
"The ball material is characterized by permittivity $\\varepsilon $ , permeability $\\mu $ , and the refractive index $n=\\sqrt{\\varepsilon \\mu }$ (the conductivity is considered to be negligible).",
"The channel axis ($z$ -axis) coincides with the ball diameter.",
"The charge $q$ moves with constant velocity $\\vec{V}=c\\beta {{\\vec{e}}_{z}}$ along the $z$ -axis, and this velocity exceeds the “Cherenkov threshold”, i.e.",
"$\\beta >{1}/{n}$ .",
"Figure: Cross-section of the dielectric ball with vacuum channel (left); the incident and refracted waves (right).",
"Incidence angle θ i {{\\theta }_{i}} and refraction angle θ t {{\\theta }_{t}} are positive for the left ray, and negative for the right ray; the illuminated part of the sphere (aperture) is highlighted by the bold red line.For definiteness, we deal with a point charge having the charge density $\\rho =q\\delta (x)\\delta (y)\\delta (z-Vt)$ where $\\delta (\\xi )$ is the Dirac delta function.",
"However, the results obtained further can be easily generalized for the case of a thin bunch with finite length because we consider Fourier transforms of the field components.",
"Further, we use the spherical ($R, \\theta , \\varphi $ ) and cylindrical ($r, \\varphi , z$ ) coordinate systems.",
"First, we find the “incident” field, i.e., solution of the “etalon” problem (field in the infinite medium with the vacuum channel).",
"For the case under consideration, this field is well known [20].",
"We are interested in the incident field on the ball surface at the point $R_0, \\theta ^{\\prime }, \\varphi ^{\\prime }$ .",
"Considering that $kR_0 \\gg 1$ one can write the Fourier-transform of the magnetic component in the form of the cylindrical wave (we use the Gaussian system of units): $H_{\\varphi ^{\\prime }}^{\\left( i \\right)}\\left(R_0, \\theta ^{\\prime } \\right)\\approx \\frac{q}{c}\\eta \\sqrt{\\frac{s}{2\\pi r^{\\prime }}}\\exp \\left\\lbrace i\\left( sr^{\\prime }+\\frac{\\omega }{V}z^{\\prime }-\\frac{\\pi }{4} \\right) \\right\\rbrace ,$ $\\eta =-\\frac{2i}{\\pi a}{{\\left[ \\kappa \\frac{1-{{n}^{2}}{{\\beta }^{2}}}{\\varepsilon \\left( 1-{{\\beta }^{2}} \\right)}{{I}_{1}}\\left( \\kappa a \\right)H_{0}^{\\left( 1 \\right)}\\left( sa \\right)+s{{I}_{0}}\\left( \\kappa a \\right)H_{1}^{\\left( 1 \\right)}\\left( sa \\right) \\right]}^{-1}},$ where $r^{\\prime }=R_0\\sin \\theta ^{\\prime }$ , $z^{\\prime }=R_0\\cos \\theta ^{\\prime }$ , $s(\\omega )=k{{\\beta }^{-1}}\\sqrt{{{n}^{2}}{{\\beta }^{2}}-1}$ , $\\kappa (\\omega )=k{{\\beta }^{-1}}\\sqrt{1-{{\\beta }^{2}}}$ , ${{I}_{0, 1}}(x)$ are the modified Bessel functions, $H_{0, 1}^{(1)}(x)$ are the Hankel functions.",
"Note that $\\operatorname{Im}s\\left( \\omega \\right)\\ge 0$ if we take into account a small dissipation.",
"If dissipation tends to zero, then this condition results in the rule $\\operatorname{\\textrm {sgn}}\\left( s\\left( \\omega \\right) \\right)=\\operatorname{\\textrm {sgn}}\\left( \\omega \\right)$ (we exclude the exotic case of the so-called “left-handed” medium).",
"The result (REF ) is valid for $\\left| sr^{\\prime } \\right| \\gg 1$ .",
"The electric field ${{\\vec{E}}^{(i)}}$ can be easily found because vectors ${{\\vec{E}}^{(i)}}$ , ${{\\vec{H}}^{(i)}}$ and the wave vector of Cherenkov radiation ${{\\vec{k}}^{(i)}}=s{{\\vec{e}}_{r}}+{{{{\\vec{e}}}_{z}}\\omega }/{V}$ form the right-hand orthogonal triad in this area, thus ${{\\vec{E}}^{(i)}}=-\\sqrt{{\\mu }/{\\varepsilon }}\\left[ {{{{\\vec{k}}}^{(i)}}}/{{{k}^{(i)}}}\\times {{{\\vec{H}}}^{(i)}} \\right]$ .",
"The angle between the wave vector ${{\\vec{k}}^{(i)}}$ and the charge velocity $\\vec{V}$ is ${{\\theta }_{p}}=\\arccos \\left( {1}/{\\left( n\\beta \\right)} \\right)$ .",
"Applying Snell’s law and the Fresnel equations (note that waves have only vertical polarization) one can obtain the following expressions for the field components on the outer surface of the ball: ${{H}_{\\varphi ^{\\prime }}}\\left(R_0, \\theta ^{\\prime } \\right)={{T}_{v}}(\\theta ^{\\prime })H_{\\varphi ^{\\prime }}^{(i)}\\left(R_0, \\theta ^{\\prime } \\right),\\quad {{E}_{\\theta ^{\\prime }}}\\left(R_0, \\theta ^{\\prime } \\right)={{H}_{\\varphi ^{\\prime }}}\\left(R_0, \\theta ^{\\prime } \\right) \\cos {{\\theta }_{t}}(\\theta ^{\\prime }),$ where ${{T}_{v}}(\\theta ^{\\prime })=\\frac{2\\cos {{\\theta }_{i}}(\\theta ^{\\prime })}{\\cos {{\\theta }_{i}}(\\theta ^{\\prime })+\\sqrt{{\\varepsilon }/{\\mu }} \\cos {{\\theta }_{t}}(\\theta ^{\\prime })},$ ${{\\theta }_{t}}(\\theta ^{\\prime })=\\arcsin \\left( n\\sin {{\\theta }_{i}}(\\theta ^{\\prime }) \\right),\\quad {{\\theta }_{i}}(\\theta ^{\\prime })=\\theta ^{\\prime }-{{\\theta }_{p}}.$ Here ${{\\theta }_{i}}(\\theta ^{\\prime })$ is the angle of incidence of the wave on the surface, ${{\\theta }_{t}}(\\theta ^{\\prime })$ is the angle of refraction (see Fig.",
"REF , right, where rays with positive and negative angles ${{\\theta }_{i}}$ and $ \\theta _t $ are shown)."
],
[
"Ray optics method",
"According to the ray optics approach, in order to determine the field at the given observation point $r, \\varphi , z$ , one should first determine the ray which starts at a certain point ${r}^{\\prime }({\\theta }^{\\prime })=R_0\\sin {\\theta }^{\\prime }$ , ${\\varphi }^{\\prime }$ , ${z}^{\\prime }({\\theta }^{\\prime })=R_0\\cos {\\theta }^{\\prime }$ at the aperture and reaches this observation point.",
"Due to the symmetry over $\\varphi $ we obtain $\\varphi ={\\varphi }^{\\prime }$ and $r={r}^{\\prime }(\\theta ^{\\prime })+l\\sin (\\theta ^{\\prime }-{{\\theta }_{t}}), \\quad z={z}^{\\prime }(\\theta ^{\\prime })+l\\cos (\\theta ^{\\prime }-{{\\theta }_{t}}),$ where $l$ is the length of the ray.",
"Therefore for each pair $r, z$ corresponding pairs ${\\theta }^{\\prime }, l$ should be determined from (REF ).",
"This problem can be solved numerically.",
"Figure: The ray picture for ε=2\\varepsilon =2, β=0.8\\beta =0.8.A typical example of the rays structure is shown in Fig.",
"REF .",
"One can see that rays intersect each other (solution of (REF ) is not unique) and form caustics.",
"Moreover, the considerable concentration of the rays occurs near the ray which is not refracted ($\\theta ^{\\prime }={{\\theta }_{p}}$ ) at some distance from the ball.",
"The area of the concentration of the rays corresponds to the area where the field increases.",
"The field along each ray can be written in the following form (for example, we consider Fourier-transform of $\\varphi $ -component of the magnetic field): ${{H}_{\\varphi }}\\left(R, \\theta \\right)={{H}_{\\varphi }}\\left( R_0, {{\\theta }^{\\prime }} \\right)\\sqrt{{D\\left( 0 \\right)}/{D\\left( l \\right)}}\\exp (ikl)$ (the electric field is equal to the magnetic field and orthogonal to it and the ray).",
"Here ${{H}_{\\varphi }}\\left(R_0, \\theta ^{\\prime } \\right)$ is the corresponding Fourier-transform at the point of the ray exit and $D(l)$ is the square of the ray tube cross-section [21].",
"Square root in (REF ) describes the change in the field magnitude due to the divergence (or convergence) of the ray tube.",
"Using (REF ), Cartesian coordinates of the observation point can be obtained as the function of $\\theta ^{\\prime }$ and $\\varphi ^{\\prime }$ : $x({\\theta }^{\\prime }, {\\varphi }^{\\prime })=r(\\theta ^{\\prime }) \\cos {\\varphi }^{\\prime }$ , $y({\\theta }^{\\prime }, {\\varphi }^{\\prime })=r(\\theta ^{\\prime }) \\sin {\\varphi }^{\\prime }$ .",
"$D(l)$ can be calculated as follows [21]: $D\\left( l \\right)=\\frac{1}{\\sqrt{g}}\\left| \\begin{matrix}\\kappa _{x}^{*} & \\kappa _{y}^{*} & \\kappa _{z}^{*} \\\\{\\partial x}/{\\partial \\varphi ^{\\prime }} & {\\partial y}/{\\partial \\varphi ^{\\prime }} & {\\partial z}/{\\partial \\varphi ^{\\prime }} \\\\{\\partial x}/{\\partial \\theta ^{\\prime }} & {\\partial y}/{\\partial \\theta ^{\\prime }} & {\\partial z}/{\\partial \\theta ^{\\prime }} \\\\\\end{matrix} \\right|,$ where $g=R_{0}^{4}\\cdot {{\\sin }^{2}}\\theta ^{\\prime }$ is a determinant of the metric tensor for the sphere, ${{\\vec{\\kappa }}^{*}}$ is a unit vector along refracted ray: $\\kappa _{x}^{*}=\\sin (\\theta ^{\\prime }-{{\\theta }_{t}})\\cos {\\varphi }^{\\prime }$ , $\\kappa _{y}^{*}=\\sin (\\theta ^{\\prime }-{{\\theta }_{t}})\\sin {\\varphi }^{\\prime }$ , $\\kappa _{z}^{*}=\\cos (\\theta ^{\\prime }-{{\\theta }_{t}})$ .",
"After a series of transformations the following expression can be obtained: $D\\left( l \\right)=\\cos {{\\theta }_{t}}-\\frac{l}{R_0}\\left[ \\frac{\\sin \\left( {{\\theta }_{t}}-{{\\theta }_{i}} \\right)}{\\sin {{\\theta }_{i}}\\cos {{\\theta }_{t}}}+\\frac{\\sin \\left( {{\\theta }_{t}}-\\theta ^{\\prime } \\right)\\cos {{\\theta }_{t}}}{\\sin \\theta ^{\\prime }} \\right]+{{\\left( \\frac{l}{R_0} \\right)}^{2}}\\frac{\\sin \\left( {{\\theta }_{t}}-{{\\theta }_{i}} \\right)\\sin \\left( {{\\theta }_{t}}-\\theta ^{\\prime } \\right)}{\\sin {{\\theta }_{i}}\\cos {{\\theta }_{t}}\\sin \\theta ^{\\prime }}.$ Numerical results based on (REF ) will be given in Section ."
],
[
"Aperture method",
"The aperture integrals (Stratton-Chu formulas) for Fourier transform of the electric field can be written in the following general form [15], [16], [17], [18], [19]: $\\begin{aligned}& \\vec{E}\\left( {\\vec{R}} \\right)={{{\\vec{E}}}^{(h)}}\\left( {\\vec{R}} \\right)+{{{\\vec{E}}}^{(e)}}\\left( {\\vec{R}} \\right), \\\\& {{{\\vec{E}}}^{(h)}}\\left( {\\vec{R}} \\right)=\\frac{ik}{4\\pi }\\int \\limits _{\\Sigma }{\\left\\lbrace \\left[ \\vec{n}^{\\prime }\\times \\vec{H}\\left( \\vec{R}^{\\prime } \\right) \\right]G\\left( \\left| \\vec{R}-\\vec{R}^{\\prime } \\right| \\right) \\right.}\\left.",
"+\\frac{1}{{{k}^{2}}}\\left( \\left[ \\vec{n}^{\\prime }\\times \\vec{H}\\left( \\vec{R}^{\\prime } \\right) \\right]\\cdot \\nabla ^{\\prime } \\right)\\nabla ^{\\prime }G\\left( \\left| \\vec{R}-\\vec{R}^{\\prime } \\right| \\right) \\right\\rbrace d\\Sigma ^{\\prime }, \\\\& {{{\\vec{E}}}^{(e)}}\\left( {\\vec{R}} \\right)=\\frac{1}{4\\pi }\\int \\limits _{\\Sigma }{\\left[ \\left[ \\vec{n}^{\\prime }\\times \\vec{E}\\left( \\vec{R}^{\\prime } \\right) \\right]\\times \\nabla ^{\\prime }G\\left( \\left| \\vec{R}-\\vec{R}^{\\prime } \\right| \\right) \\right]d\\Sigma ^{\\prime }, } \\\\\\end{aligned}$ where $\\Sigma $ is the aperture area, $\\vec{E}\\left( \\vec{R}^{\\prime } \\right)$ , $\\vec{H}\\left( \\vec{R}^{\\prime } \\right)$ is the field on the surface of the aperture, the prime sign indicates that operator or coordinate is referred to the surface of an object, $k={\\omega }/{c}$ , $\\vec{n}^{\\prime }$ is the unit external normal to the aperture in the point $\\vec{R}^{\\prime }$ , $G\\left( R \\right)={\\exp \\left( ikR \\right)}/{R}$ is the Green function of Helmholtz equation, and ${\\nabla }^{\\prime }$ is the gradient: $\\nabla ^{\\prime }={{\\vec{e}}_{x}}{\\partial }/{\\partial x^{\\prime }}+{{\\vec{e}}_{y}}{\\partial }/{\\partial y^{\\prime }}+{{\\vec{e}}_{z}}{\\partial }/{\\partial z^{\\prime }}$ .",
"Analogous formulas are known for the magnetic field as well.",
"Note that $\\left| {\\vec{E}} \\right|\\approx \\left| {\\vec{H}} \\right|$ in the region several wavelengths far from the aperture.",
"In the case of the spherical object, it is convenient to write aperture integrals using spherical coordinates $R, \\theta , \\varphi $ .",
"Besides the primary condition $kR_0 \\gg 1$ , we impose for simplicity an additional condition $k(R-R_0) \\gg 1$ , which means that the observation point is located at a distance of no less than several wavelengths from the ball surface.",
"Using the cylindrical symmetry of the problem, we can choose an observation point on the plane $x, z$ ($\\varphi =0$ ).",
"As a result, one can obtain from (REF ) the following expressions for Fourier-transforms of the non-zero electric field components: $\\vec{E}={{\\vec{E}}^{(h)}}+{{\\vec{E}}^{(e)}}={{\\vec{E}}^{(h1)}}+{{\\vec{E}}^{(h2)}}+{{\\vec{E}}^{(e)}},$ $\\left\\lbrace \\begin{aligned}& E_{r}^{(h1)} \\\\& E_{z}^{(h1)} \\\\\\end{aligned} \\right\\rbrace =\\frac{ikR_{0}^{2}}{4\\pi }\\int \\limits _{{{\\Theta }_{1}}}^{{{\\Theta }_{2}}}{d\\theta ^{\\prime }\\int \\limits _{0}^{2\\pi }{d\\varphi ^{\\prime }}}\\left\\lbrace \\begin{aligned}-\\cos &\\theta ^{\\prime }\\cos \\varphi ^{\\prime } \\\\& \\sin \\theta ^{\\prime } \\\\\\end{aligned} \\right\\rbrace \\sin \\theta ^{\\prime }\\frac{\\exp (ik\\tilde{R})}{{\\tilde{R}}}H_{\\varphi ^{\\prime }}\\left(R_0, \\theta ^{\\prime } \\right),$ $\\begin{aligned}& \\left\\lbrace \\begin{aligned}& E_{r}^{(h2)} \\\\& E_{z}^{(h2)} \\\\\\end{aligned} \\right\\rbrace =\\frac{ikR_{0}^{2}R}{4\\pi }\\int \\limits _{{{\\Theta }_{1}}}^{{{\\Theta }_{2}}}{d\\theta ^{\\prime }\\int \\limits _{0}^{2\\pi }{d\\varphi ^{\\prime }}}\\left\\lbrace \\begin{aligned}& R_0\\sin \\theta ^{\\prime }\\cos \\varphi ^{\\prime }-R\\sin \\theta \\\\& R_0\\cos \\theta ^{\\prime }-R\\cos \\theta \\\\\\end{aligned} \\right\\rbrace \\\\& \\times \\sin \\theta ^{\\prime }\\left( \\cos \\theta \\sin \\theta ^{\\prime }-\\sin \\theta \\cos \\theta ^{\\prime }\\cos \\varphi ^{\\prime } \\right)H_{\\varphi ^{\\prime }}\\left(R_0, \\theta ^{\\prime } \\right)\\frac{\\exp (ik\\tilde{R})}{{{{\\tilde{R}}}^{3}}}, \\\\\\end{aligned}$ $\\left\\lbrace \\begin{aligned}& E_{r}^{(e)} \\\\& E_{z}^{(e)} \\\\\\end{aligned} \\right\\rbrace =\\frac{ikR_{0}^{2}}{4\\pi }\\int \\limits _{{{\\Theta }_{1}}}^{{{\\Theta }_{2}}}{d\\theta ^{\\prime }\\int \\limits _{0}^{2\\pi }{d\\varphi ^{\\prime }}\\left\\lbrace \\begin{aligned}& \\left( R_0\\cos \\theta ^{\\prime }-R\\cos \\theta \\right)~\\cos \\varphi ^{\\prime } \\\\& -R_0\\sin \\theta ^{\\prime }+R\\sin \\theta \\cos \\varphi ^{\\prime } \\\\\\end{aligned} \\right\\rbrace }\\sin \\theta ^{\\prime }\\frac{\\exp (ik\\tilde{R})}{{{{\\tilde{R}}}^{2}}}E_{\\theta ^{\\prime }}\\left(R_0, \\theta ^{\\prime } \\right),$ $\\tilde{R}=\\sqrt{R_{0}^{2}+{{R}^{2}}-2RR_0\\left( \\cos \\theta \\cos \\theta ^{\\prime }+\\sin \\theta \\sin \\theta ^{\\prime }\\cos \\varphi ^{\\prime } \\right)}.$ Here $E_{\\theta ^{\\prime }}\\left(R_0, \\theta ^{\\prime } \\right)$ , $H_{\\varphi ^{\\prime }}\\left(R_0, \\theta ^{\\prime } \\right)$ are the tangential components of the transmitted field on the surface of the ball determined by formulas (REF ).",
"The limits of integration in (REF ) – (REF ) are determined by the conditions that the aperture is a part of the object surface illuminated by CR which does not experience total internal reflection, in other words, $\\left| {{\\theta }_{i}} \\right|<{{\\theta }_{*}} \\equiv \\arcsin \\left( {1}/{n} \\right)$ .",
"One can show that ${{\\Theta }_{1}}=\\max \\left\\lbrace {{\\theta }_{p}}-{{\\theta }_{*}}, \\arcsin ({a}/{R_0}) \\right\\rbrace $ , ${{\\Theta }_{2}}= \\min \\left\\lbrace {{\\theta }_{p}}+{{\\theta }_{*}}, 2{{\\theta }_{p}} \\right\\rbrace $ ."
],
[
"Numerical results",
"Figure REF shows the dependencies of the field magnitude on the angle $\\theta $ obtained with the use of the ray optics (green curves) and aperture (blue curves) methods.",
"The results of simulations performed with RF module of COMSOL Multiphysics (red curves) are demonstrated as well.",
"We emphasize that all three methods give similar results even for not very large targets (the sphere radius is approximately ${30}/{(2\\pi )}\\approx 5$ wavelengths in the Figures REF and REF ).",
"As we see, the ray optics results have typical (non-physical) gaps at the boundaries of the illuminated area.",
"Naturally, the agreement between the results is better for the larger radius of the target (see Figures REF and REF , where sphere radius is approximately 50 wavelengths).",
"It is natural as well that the aperture method gives the results which are closer to the COMSOL Multiphysics simulations in comparison with the ray optics.",
"One can conclude that the ray optics method is suitable for estimation of the magnitude of the radiation field from the target with the radius of several wavelengths or larger.",
"However, this technique is not suitable for studying the field behavior.",
"The aperture technique is more general, gives more exact results and allows analyzing the field behaviour.",
"Figure REF shows the dependencies of the field magnitude on the distance $R$ at the angle $\\theta $ which is equal to the CR angle ${{\\theta }_{p}}$ .",
"In this case, the refracted wave propagates normal to the sphere.",
"It can be seen that with an increase in the distance from the sphere, the tendency toward an increase in the field is observed (complicated by the oscillations).",
"The condensation of rays, which was noted within the ray optics examination (see Fig.",
"REF ), causes the observed phenomenon.",
"The field starts to steadily decrease only after certain distance from the sphere surface."
],
[
"Conclusion",
"In this paper, two approaches have been applied for the analysis of radiation from a dielectric ball: the ray optics method and the aperture method.",
"Each of them demonstrated a good coincidence with COMSOL Multiphysics (in the area of their applicability).",
"However, the aperture method gave more exact results for the field structure.",
"Numerous calculations performed for various parameters show that, as a rule, the error of the aperture method in the area of the largest magnitudes of the field is less than $10\\%$ for the objects having the size of the order of 10 wavelengths.",
"For larger objects, the error becomes even smaller.",
"The main physical effects have been described.",
"For example, it has been shown that, in the direction of the Cherenkov angle, the radiation field possesses an expressed maximum."
],
[
"Acknowledgments",
"This research was supported by the Russian Science Foundation, Grant No.",
"18-72-10137.",
"Numerical simulations with COMSOL Multiphysics have been performed in the Computer Center of the Saint Petersburg State University."
]
] | 2001.03512 |
[
[
"A machine learning based plasticity model using proper orthogonal\n decomposition"
],
[
"Abstract Data-driven material models have many advantages over classical numerical approaches, such as the direct utilization of experimental data and the possibility to improve performance of predictions when additional data is available.",
"One approach to develop a data-driven material model is to use machine learning tools.",
"These can be trained offline to fit an observed material behaviour and then be applied in online applications.",
"However, learning and predicting history dependent material models, such as plasticity, is still challenging.",
"In this work, a machine learning based material modelling framework is proposed for both elasticity and plasticity.",
"The machine learning based hyperelasticity model is developed with the Feed forward Neural Network (FNN) directly whereas the machine learning based plasticity model is developed by using of a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN).",
"In order to account for the loading history, the accumulated absolute strain is proposed to be the history variable of the plasticity model.",
"Additionally, the strain-stress sequence data for plasticity is collected from different loading-unloading paths based on the concept of sequence for plasticity.",
"By means of the POD, the multi-dimensional stress sequence is decoupled leading to independent one dimensional coefficient sequences.",
"In this case, the neural network with multiple output is replaced by multiple independent neural networks each possessing a one-dimensional output, which leads to less training time and better training performance.",
"To apply the machine learning based material model in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiation tool AceGen.",
"The effectiveness and generalization of the presented models are investigated by a series of numerical examples using both 2D and 3D finite element analysis."
],
[
"Introduction",
"With the development of data mining technology, machine learning algorithms, high performance computing and robust numerical methods, data-driven computational modelling play an important role in not only accurate but also fast predictions of complex industrial processes.",
"In particular, accurate material models are key parts in structure analysis.",
"In the past years, tremendous effort has been made in developing material models, see e.g.",
"the review of models by [4] for metal forming processes.",
"However, the proposed models show limitations in generalization or accuracy in some cases when the model is applied to engineering problems.",
"As a data-driven approach, the Machine Learning (ML) based material modelling provides an alternative tool to narrow the gap between experimental data and material models.",
"By use of the ML technology, such as artificial neural networks, see e.g.",
"[11], or Gaussian Processes, see e.g.",
"[29], constitutive equations can be approximated by using experimental data without postulation of a specific constitutive model.",
"An advantage of machine learning based material models is that they can iteratively be improved if more experimental data are available, which yields more flexible and sustainable material descriptions.",
"For a review of machine learning in computational mechanics, see [27] and references therein.",
"In order to replace the classical constitutive model in computational mechanics by data-driven modelling, multiple approaches have been proposed in the literature.",
"The model-free data-driven computing paradigm proposed by [16], [15], [5] and [31], conducts the computing directly from experimental material data under the constraints of conservation laws, which bypasses the empirical material modelling step.",
"This approach works without constitutive model and seeks to find the closest possible state from a prespecified material data set.",
"A manifold learning approach is proposed by [12], [13] and [14], where the so-called constitutive manifold is constructed from collected data.",
"A self-consistent clustering approach has been developed to predict the behaviour of heterogeneous materials under inelastic deformation, see [21] and [30].",
"[32] proposed a mapping approach, where one-dimensional data are mapped into three-dimensions for nonlinear elastic material modelling without the construction of an analytic mathematical function for the material equation.",
"Since the performance of the data-driven computing is highly determined by quality and completeness of the available data, data completion and data uncertainties have been investigated, see [2] and [3].",
"In addition to the data-driven approaches mentioned above, the artificial neural network as a machine learning approach has been applied to approximate the constitutive model based on data as well, see [6], [10], and [19].",
"In order to fit a constitutive material equation, the neural network is trained offline using experimental data collected from different loading paths.",
"Afterwards, the network based model is applied online for testing and applications.",
"A nested adaptive neural network has been applied in [7], [6] for modelling the constitutive behaviour of geomaterials.",
"In [10], a feed forward neural network based constitutive model is implemented in finite element analysis to capture the nonlinear material behaviour, where the consistent material tangent matrix is derived and evaluated.",
"Artificial neural networks are also applied as incremental non-linear constitutive models in [19] for finite element applications.",
"Furthermore, this approach has been applied to predict the stress-strain curves and texture evolution of polycrystalline metals by [1].",
"Instead of the offline training, the neural network based constitutive model can be trained online by auto-progressive algorithms as well, see [28] and [7].",
"Lastly, artificial neural networks have been applied to the heterogeneous material modelling, such as [18], [23], [20], [22] and [34].",
"The data-driven model free approach conducts calculations directly from the data, which bypasses the model on one hand but highly relies on the quality and completeness of the data on the other hand.",
"The machine learning approaches mentioned above apply the previous strain and stress as history variables, which introduces extra errors for the elastic stage of inelastic deformation, and thus affects the capabilities to capture the load history in real applications.",
"Additionally, the derivation of the tangent matrix for the neural network based model is complex when changing the network architecture.",
"Thus, there are many issues present in machine learning based material modelling approaches, such as the data collection strategy, the selection of history variables and the applications in finite element analysis.",
"The objective of this work is to develop a machine learning based hyperelastic and plasticity models for finite element applications as well as a corresponding data collection strategy.",
"To simplify the data collection process from experiments, only strain components act as input data and only stress components represent output data.",
"Instead of using the previous strain and stress as history variables in plasticity, in this work, the accumulated absolute total strain is applied as history variable to distinguish different loading paths.",
"This variable can be computed from preexisting input data without additional effort as e.g.",
"different experiments.",
"Due to its history dependence, the training data for plasticity will be sequential data sets obtained under different loading-unloading paths.",
"Since the isotropic plasticity can be formulated in the principle space, the training sequence data is collected only from tension and compression tests, which simplifies the data collection.",
"A novel method called Proper Orthogonal Decomposition Feedforward Neural Network (PODFNN) is proposed in combination with the introduced history variable for predicting the stress sequences in case of plasticity.",
"By means of the Proper Orthogonal Decomposition (POD), the stress sequence is transformed into multiple independent coefficient sequences, where the stress at any time step can be recovered by a linear combination of the coefficients and the basis.",
"The presented approach decomposes the strain-stress relationship into multiple independent neural networks with only one output, which significantly decreases the complexity of the model.",
"In order to apply the machine learning based model in finite element analysis, the tangent matrix has to be computed.",
"It is derived by the symbolic differentiation tool AceGen, see [17].",
"The effectiveness and generalization of the machine learning based plasticity model is validated in 2D and 3D using several applications.",
"This paper is structured as follows: In Section 2, the machine learning based material modelling framework is presented.",
"Then the Feed forward Neural network (FNN) is applied to learn the hyper-elastic material law in Section 3.",
"In Section 4, the data collection strategy for plasticity is proposed.",
"Based on the training data, the machine learning based plasticity model is developed and validated in finite element analysis in Section 5, which is followed by the conclusions in Section 6."
],
[
"Data-driven material modelling framework",
"To develop a data-driven material model by means of machine learning technology, three steps are necessary: data collection, machine learning and validation, see Fig.",
"REF .",
"As a fundamental ingredient for data-driven models, the data, representing the material behaviour, have to be collected firstly.",
"According to the specific problem, the training data can be collected from experiments and simulations.",
"In this work, strain-stress data are employed as the input and output of the data-driven model.",
"For the plasticity model, the strain-stress data will be collected for specific loading paths and stored as sequences.",
"Depending on the problem, the training data usually have to be preprocessed utilizing data scaling, data decomposition and data arrangement.",
"The second step is to fit the constitutive equation related to the data by means of the ML technology.",
"The Artificial Neural Network (ANN) as a machine learning technology will be employed in this work.",
"The hyperparameters of the neural network based model have to be selected according to the data and the accompanying accuracy requirements.",
"Once the model is trained, the describing parameters will be used and stored for the material description of the developed model.",
"The final step is to validate the accurate reproduction of the ML based material model.",
"To do so, the ML based material model is compared with a standard material model within several finite element applications.",
"By deriving the tangent matrix and residual vector, the ML based model can be incorporated into a FEM code.",
"The performance of the developed model will be evaluated by benchmark tests.",
"If the accuracy of the material model can not meet the necessary requirements, the model hyperparameters will be optimized or supplemental data will be collected.",
"Therefore, the machine learning based framework is an open system and the accuracy of the developed model can be improved iteratively during its application.",
"Figure: The data-driven material modelling framework."
],
[
"Machine Learning (ML) based hyperelasticity",
"Before the plasticity model is developed in detail, a ML based hyper-elasticity model is presuited by utilizing feed forward neural networks (FNNs) in this section."
],
[
"Feed forward neural network",
"Feed forward neural network is a fundamental machine learning technology.",
"A deep FNN is composed of several connected layers of artificial neurons and biases, where the data is fed into an input layer and then flows through some hidden layers.",
"The output is finally predicted at an output layer, as shown in Fig.",
"REF .",
"The neurons from different layers are fully connected through the weights $w$ .",
"In the prediction phase, the data flows in one way from the input layer to the output layer.",
"In the training phase, the global error defined by the mean-squared differences between the target value and the FNN output will be back-propagated through the hidden layers.",
"This step is performed in order to update the weights, where the objective is to minimize the global error.",
"Figure: The feed forward neural network.At each neuron, an activation function is attached, see Fig.",
"REF .",
"The output of each neuron is computed by multiplying the outputs from the previous layer with the corresponding weights.",
"For the neuron $j$ in the layer $k$ , the data of the previous layer $k-1$ is summed up and then altered by an activation function.",
"The output of the neuron $j$ in layer $k$ is computed as $o_j^k=f_s\\left( \\sum _{i=1}^{N} w_{ij} o_i^{k-1}+b^{k-1}_i \\right),$ where $N$ is the number of neurons in the previous layer $k-1$ , $w_{ij}$ is the weight connecting neurons $i$ and $j$ , $o_i^{k-1}$ is the output of the neuron $i$ in layer $k-1$ whereas $b_{i}^{k-1}$ is its bias.",
"A common choice for the activation function is the sigmoid function $f_s(x)=\\frac{2}{1+e^{-2x}}-1.$ Figure: The artificial neuron.The specific architecture of the FNN, such as the number of layers and the number of neurons in each layer, has to be determined according to the complexity of the data set."
],
[
"Neural network training",
"In the training phase, the weighs of neural network will be initialized firstly, see [26], which is followed by the weights updating using a training algorithm such that the global error is minimized.",
"The global error, also named as loss function or network performance, is defined according to the difference between the network prediction and the target data as shown in Fig.",
"REF .",
"The mean squared error is used to measure the loss $E(w)=\\frac{1}{N} \\sum _{i=1}^{N} \\left[ o_i(w)-t_i \\right]^2=\\frac{1}{N} \\sum _{i=1}^{N} e_i,$ where $N$ is the number of outputs, $o_i$ is the $i$ -th output, $w$ is the vector that contains the weights of neural network, and $t_i$ is the $i$ -th target value.",
"Training a feed forward neural network is an optimization problem, where the global error is treated as the objective function.",
"To minimize the global error, the Levenberg-Marquardt algorithm is applied to update the weighs, see [9], $w^{n+1} =w^{n}- (J^T J +\\mu I)^{-1} J^T \\cdot e,$ in which $w^{n+1}$ is the weight vector in iteration $n+1$ , $\\mu $ is a parameter to adaptively control the speed of convergence, and $J$ is the Jacobian matrix that contains the derivatives of network errors with respect to the network weights $J_{ij} = \\frac{\\partial e_i}{\\partial w_j^{n}}.$ In the training process, many iterations are required to update the weights until the stopping criteria is fulfilled, where one iteration is also known as one epoch."
],
[
"Data collection for the ML based hyperelasticity",
"To approximate hyperelastic behaviour by the FNN for finite element applications, the first task is to determine the input and output variables for the neural network.",
"Since the loading and the unloading curve coincide for the elastic deformation, as shown in Fig.",
"REF , the relationship between the strain space and stress space can be seen as a one-to-one mapping.",
"Hence, the strain-stress mapping can be approximated by the FNN without considering the loading history.",
"Figure: The loading and unloading curve for hyperelasticity.Instead of using experimental data, the training data are collected here from an analytical model, which allows us to test the performance of the ML based model by comparing its simulation results with that by an analytical model.",
"As an example of hyperelasticity, the non-linear neo-Hookean model is applied as the target model to learn $\\sigma = \\frac{1}{2}\\frac{\\lambda }{J}(J^2 - 1) I + \\frac{\\mu }{J}(b - I),$ where the Cauchy stress $\\sigma $ and the left Cauchy Green tensor $b$ are symmetric tensors.",
"For the 2D problem, the inputs of the model can be chosen as the strain components $(J,b_{11}, b_{22}, b_{12})$ , whereas the outputs are chosen as the stress components $(\\sigma _{11}, \\sigma _{22}, \\sigma _{12})$ .",
"According to the number of input and the output, the architecture of the FNN can be determined as $4-n-3$ for instance, where one hidden layer with $n$ neurons is applied for this hyperelastic law.",
"The input data is generated by taking equally spaced points within the given range of strain space.",
"The stresses as output data can be computed from the neo-Hookean model in equation (REF ) accordingly.",
"Before training the FNN, the generated data is scaled to the range $(-1,1)$ such that training is accelerated.",
"Then the neural network is trained until the stopping criteria is reached.",
"After training, the weights $w$ and bias $b_s$ will be saved as the model parameters.",
"The ML based hyper-elasticity model is thus expressed as $\\sigma ^{NN}&=FNN(b, J, w, b_s),$ where $\\sigma ^{NN}$ is the predicted Cauchy stress by the FNN."
],
[
"The residual and tangent",
"The ML based model can be used in the same way as the classical constitutive model in the finite element analysis.",
"The residual for the static problem is given by $R (u)=f-\\int _\\Omega B^T \\sigma ^{NN} d\\Omega , \\\\f=\\int _\\Omega N \\rho \\hat{b} dv- \\int _{\\partial \\Omega } N \\hat{t} da,$ in which $B$ is the gradient of shape functions $N$ , $\\rho $ is the density, $\\sigma ^{NN}$ is the stress computed from the machine learning based model, $\\hat{b}$ and $\\hat{t}$ are the body force and the surface traction respectively.",
"Due to the non-linearity, the Newton Rapson iterative solution scheme is applied.",
"The tangent matrix is computed by taking the derivative of residual in terms of displacement $K_T = \\frac{\\partial R(u)}{\\partial u}.$ The derivation of the tangent matrix for the neural network based model requires the computation of derivatives by the chain rule, which will be complex if the number of neuron is very large.",
"In this work, the automatic differentiation tool AceGen, see [17], based on the symbolic computing in Mathematica is applied, by which the tangent matrix and residual vector can be derived automatically."
],
[
"Testing the ML based hyperelasticity model in FEM",
"The material parameters for the neo-Hookean model used in the training data collection are set as $E=700N/mm^2,\\nu =0.499$ .",
"An FNN with architecture of 4-10-3 is applied, with 4 neurons in the input layer, 10 neurons in the hidden layer and 3 neurons in the output layer.",
"The Levenberg-Marquardt algorithm [9] is applied as the training optimizer.",
"After 14082 training iterations, the mean squared error decreased to $0.0326$ , which costs training time of $6h40m55s$ .",
"The first example is the uniaxial compression test of a plate in 2D.",
"As shown in Fig.",
"REF , the pressure is imposed on the top surface of the plate, the bottom of the plate is fixed in vertical direction.",
"The distributed load is given as $ q_0=-20MPa$ .",
"Figure: Compression of the plate.The final deformation of the plate computed with the ML based model is compared with the outcome of using the neo-Hooken model in equation (REF ).",
"It can be see from Fig.",
"REF that the displacements in vertical direction are very close.",
"The computation time with analytical hyperelastic model is $8.14s$ , whereas the computation time with the ML based model is $9.75s$ on the same computer.",
"Figure: With ML based modelIn order to further validate the generalization, a second test case, the Cook's membrane problem, is conducted.",
"The tapered beam is clamped at the left end and loaded at the right end by a constant distributed vertical load $q_0=5Mpa$ , as depicted in Fig.",
"REF .",
"The geometric domain of the structure is discretized by 40 quadratic 9-node quadrilateral elements leading to 189 nodes.",
"Figure: Cook's membrane problem.With the same model as trained in the first test, the final deformation of the membrane is computed and compared.",
"As shown in Fig.",
"REF , the vertical displacement in both cases are very close to each other, which highlights the proficient generalization capabilities of the ML based elasticity model.",
"The computation time with analytical hyperelastic model is $15.72s$ , whereas the computation time with the ML based model is $19.88s$ on the same computer.",
"Figure: With ML based modelTo this end, it proves that the FNN works well for approximating the nonlinear elastic behaviour as shown in the above results.",
"Elastic deformation is a history independent process and the stress depends only on current kinematic variables.",
"However, for plastic material behaviour, the response of the deformed material depends not only on the current deformation but also on its loading history.",
"Since the FNN does not have any inherent ability to record loading history, the current approach needs to be improved.",
"Furthermore, the collection process of the training data for plasticity needs to be different from elasticity."
],
[
"Data collection strategy for plasticity",
"The aim of this part is to develop a data-driven material model which can be used to computationally reproduce the plastic material behavior by means of machine learning tools.",
"Collecting data from experiments is a key part for data-driven material model.",
"In experiments, only the total strain and stress data of a specimen can be collected, which means the classical concept of elastic-plastic splitting to total strain can not be applied in the data-driven model.",
"This leads to the questions: how to build the data-driven model using the total strain and stress data available from experiment?",
"and what kind of experiments have to be conducted to collect data?"
],
[
"Concept of sequence for plasticity",
"Since the plastic flow depends not only on the current stress state but also on the loading history, the plastic deformation is a history dependent process.",
"For 1D plastic deformations, the loading and unloading curves do not coincide as shown in Fig.",
"REF , where the strain and stress data are time series of data sets and can be seen as sequences.",
"Figure: The loading and unloading curve for plasticity.Along a 1D loading-unloading path, the strain and stress data sets of one material point have a strict sequential order and can be written as two sets of corresponding sequences $\\lbrace \\varepsilon ^1, \\varepsilon ^2, \\varepsilon ^3, ..., \\varepsilon ^t, ... \\rbrace \\Leftrightarrow \\lbrace \\sigma ^1, \\sigma ^2, \\sigma ^3, ..., \\sigma ^t, ... \\rbrace ,$ where $\\varepsilon ^t$ and $\\sigma ^t$ are the total strain and stress at time $t$ collected from the experiment.",
"Thus, the basic data unit for a plasticity model is not a strain-stress pair but a strain-stress sequence pair.",
"Each strain-stress sequence pair refers to one loading-unloading path and thus sequence data collected from different loading-unloading paths are required to train a data-driven plasticity model.",
"In machine learning, the classical equation of plasticity will be replaced with a ML based plasticity model driven by experimental data as $\\sigma ^t=f^{ML}(\\varepsilon ^t,h^t),$ where $h^t$ is a history variable for distinguishing loading history and $\\varepsilon ^t$ is the total strain.",
"In this data-driven model, both the input and the output are sequence data.",
"To build a ML based plasticity model, the history data as well as the strain-stress data have to be obtained from experiments.",
"The choice of history variable is crucial for a successful prediction of sequences.",
"In the literature, the stress and strain in the last step are applied as the history information together with the current strain in the input, see [10].",
"However, the previous strain and previous stress are not enough to distinguish the loading history in real applications, since any error in the predicted previous stress by the ML tool will introduce an extra error into the system in an accumulate way.",
"In this work, the accumulated absolute strain is applied in the input as history variable.",
"The accumulated absolute strain component $\\varepsilon _{acc,j}^t$ at time step $t$ can be computed for the $j$ -th strain component as $h_j^t:=\\varepsilon _{acc,j}^t={\\left\\lbrace \\begin{array}{ll}\\sum _{i=3}^{t} \\vert \\varepsilon ^{i-1}_j - \\varepsilon ^{i-2}_j \\vert , & \\text{ $t\\geqslant 3$},\\\\0, & \\text{$t=1,2$},\\end{array}\\right.",
"}$ where $\\varepsilon ^{i-1}_j$ and $\\varepsilon ^{i-2}_j$ are total strain components at time step $i-1$ and $i-2$ respectively.",
"This history variable has to be computed for each total strain component.",
"$\\vert \\varepsilon ^{i-1}_j - \\varepsilon ^{i-2}_j \\vert $ is the absolute increment of strain component $j$ from time step $i-2$ to $i-1$ , which is necessary to distinguish the loading history when tension and compression loadings are mixed, such as in loading-unloading path.",
"For the 1D case, depicted in Fig.",
"REF , a monotonic loading (e.g.",
"from $\\sigma ^1$ to $\\sigma ^4$ ) and a mixed loading-unloading (e.g.",
"from $\\sigma ^1$ to $\\sigma ^6$ ) may lead to the same total strain (e.g.",
"$\\varepsilon ^4=\\varepsilon ^6$ ).",
"To distinguish monotonic loading from mixed loading-unloading, the absolute value of strain increment $\\vert \\varepsilon ^{i-1} - \\varepsilon ^{i-2} \\vert $ is applied in equation (REF ), which leads to different accumulated absolute strain for different paths, e.g.",
"$\\varepsilon _{acc}^6&=\\sum _{i=3}^{6} \\vert \\varepsilon ^{i-1} - \\varepsilon ^{i-2} \\vert \\\\&=\\sum _{i=3}^{4} \\vert \\varepsilon ^{i-1} - \\varepsilon ^{i-2} \\vert +\\sum _{i=5}^{6}\\vert \\varepsilon ^{i-1} - \\varepsilon ^{i-2} \\vert \\\\&= \\varepsilon _{acc}^4 +\\sum _{i=5}^{6}\\vert \\varepsilon ^{i-1} - \\varepsilon ^{i-2} \\vert >\\varepsilon _{acc}^4.$ Note that $ \\vert \\varepsilon ^{i-1} - \\varepsilon ^{i-2} \\vert $ is applied instead of $ \\vert \\varepsilon ^{i} - \\varepsilon ^{i-1} \\vert $ in equation (REF ), since $\\sum _{i=3}^{t} \\vert \\varepsilon ^{i} - \\varepsilon ^{i-1} \\vert $ is equal to $\\varepsilon ^{t}$ for monotonic loading, and it is not a history variable but the current input.",
"The advantage of applying the accumulated absolute strain as the history variable is that it can be obtained from the existing experimental data without the effort to collect them additionally."
],
[
"Loading paths to collect data from experiments",
"To collect the strain-stress sequence data from experiments, the loading paths required in experiments have to be investigated.",
"Since isotropic plasticity can be formulated in the principle strain-stress space, the ML based plasticity model can be formulated in the principle space as well, where the input and output of the model are principle stain and principle stress respectively.",
"Therefore only the principle strain and principle stress are required to be collected from the experiments.",
"To collect the principle strain and principle stresses, the multi-axial loading tests can be conducted on special designed specimens, such as the biaxial experiments conducted by [25] and the loading paths suggested by [8].",
"The von Mises yield surface covering different stress states is shown in Fig.",
"REF for the 2D case.",
"In order to learn the plasticity behavior by e.g.",
"artificial neural networks, yielding as well as hardening effects have to be captured implicitly.",
"To fully describe the stress states existed in the deformed structures, the data have to be collected from several tests under different loading-unloading paths.",
"However, only biaxial tests for 2D and triaxial tests for 3D are required to collect data in principle space.",
"Figure: Stress states in 2D.In experiments, the principle strain and stress data can be collected within a homogeneous region at one point within a specimen, which can be descried by a quadrilateral region for 2D case depicted in Fig.",
"REF .",
"The biaxial loading-unloading paths at this region can be characterized by the displacements of the edges connected to point $A$ .",
"Figure: Quadrilateral region within a specimen to collect data with biaxial loading.For each biaxial loading-unloading path, the point $A$ moves from its original position to $A^{\\prime }$ for loading and then goes back to original position for unloading, during which the displacements $(u_x, u_y)$ will increase linearly from zero to a specific value obeying $\\sqrt{u_x^2+u_y^2}=r_i, (i=1,2,3...)$ and then decrease to zero.",
"$max(r_i)$ has to be large enough to capture as much loading range of plastic deformation as possible.",
"As shown in Fig.",
"REF , the loading-unloading paths are just determined by setting a radius $r_i$ and different values of the angles $\\phi $ .",
"Since the unloading can start from different positions, multiple circles with radius $r_i$ have to be applied in data collection.",
"Figure: Loading-unloading paths for data collection in 2D.For the 3D case, data can be collected from a cubic region, as shown in Fig.",
"REF , where the triaxial loading-unloading paths at this point can be characterized by the displacement $(u_x, u_y, u_z)$ at the point $A$ .",
"Figure: Cubic region within a specimen to collect data with triaxial loading.The loading-unloading path for the 3D case can be generated in the spherical coordinate system as shown in Fig.",
"REF , where the displacement of point $A$ obeys $\\sqrt{u_x^2+u_y^2+u_z^2}=r_i, (i=1,2,3,...)$ .",
"The loading path $OP_i$ is distinguished by the angles $\\phi $ and $\\theta $ with radius $r_i$ .",
"By looping the loading path around the sphere, the whole range of stress states can be included.",
"Figure: Loading-unloading paths for data collection in 3D.For example, along the loading path $OP_i$ in Fig.",
"REF , the strain-stress sequence data will be collected firstly as $\\hat{\\varepsilon } = \\begin{bmatrix}\\varepsilon ^1_1 & \\varepsilon ^2_1 & ... & \\varepsilon ^t_1 & ...\\\\\\varepsilon ^1_2 & \\varepsilon ^2_2 & ... & \\varepsilon ^t_2 & ...\\\\\\varepsilon ^1_3 & \\varepsilon ^2_3 & ... & \\varepsilon ^t_3 & ...\\\\\\end{bmatrix}_{3 \\times np}, \\,\\,\\,\\,\\,\\sigma _{opi} = \\begin{bmatrix}\\sigma ^1_1 & \\sigma ^2_1 & ... & \\sigma ^t_1 & ...\\\\\\sigma ^1_2 & \\sigma ^2_2 & ... & \\sigma ^t_2 & ...\\\\\\sigma ^1_3 & \\sigma ^2_3 & ... & \\sigma ^t_3 & ...\\\\\\end{bmatrix}_{3 \\times np},$ where $np$ is the number of data point on the loading path $OP_i$ , $ \\varepsilon ^t_i (i=1,2,3)$ are the principle strains at time $t$ measured in the cubic region within the specimen, $ \\sigma ^t_i (i=1,2,3)$ are the principle stresses at time $t$ in that region, $\\hat{\\varepsilon }$ is the strain sequence and $\\sigma _{opi}$ is the stress sequence of path $OP_i$ .",
"Then the history data, accumulated absolute strain $\\varepsilon _{acc}^t$ , are computed from the strain sequence $\\hat{\\varepsilon }$ using equation (REF ).",
"The final strain sequence data of path $OP_i$ are obtained by combining the total strain sequence and the history variable sequence as $\\varepsilon _{opi}= \\begin{bmatrix}\\varepsilon ^1_1 & \\varepsilon ^2_1 & ... & \\varepsilon ^t_1 & ...\\\\\\varepsilon ^1_2 & \\varepsilon ^2_2 & ... & \\varepsilon ^t_2 & ...\\\\\\varepsilon ^1_3 & \\varepsilon ^2_3 & ... & \\varepsilon ^t_3 & ...\\\\\\varepsilon _{acc,1}^1 & \\varepsilon _{acc,1}^2 & ... & \\varepsilon _{acc,1}^t & ...\\\\\\varepsilon _{acc,2}^1 & \\varepsilon _{acc,2}^2 & ... & \\varepsilon _{acc,2}^t & ...\\\\\\varepsilon _{acc,3}^1 & \\varepsilon _{acc,3}^2 & ... & \\varepsilon _{acc,3}^t & ...\\\\\\end{bmatrix}_{6 \\times np},$ where $\\varepsilon _{acc,1}^t$ , $\\varepsilon _{acc,2}^t$ and $\\varepsilon _{acc,3}^t$ are computed from the strain components $\\varepsilon ^t_1$ , $\\varepsilon ^t_2$ and $\\varepsilon ^t_3$ respectively according to equation (REF ).",
"Finally, the input and output data are obtained by combining the sequences from all of loading paths $OP_i, (i=1,2,...,nl)$ as $I_{\\varepsilon } = \\begin{bmatrix}\\varepsilon _{op1} & \\varepsilon _{op2} & ... & \\varepsilon _{opi} & ...\\end{bmatrix}_{6 \\times M}, \\,\\,\\,\\,\\,O_{\\sigma } = \\begin{bmatrix}\\sigma _{op1} & \\sigma _{op2} & ... & \\sigma _{opi} & ...\\end{bmatrix}_{3 \\times M},$ where $nl$ is the number of loading path and $M=np \\times nl$ ."
],
[
"Machine learning based plasticity",
"After data collection, the feed forward neural network is used to learn the relationship within the data.",
"The neural network will approximate a mapping between inputs and outputs.",
"However, the accuracy of the approximation depends on the complexity of the relationship.",
"As a widely used technique in model order reduction, the Proper Orthogonal Decomposition (POD) provides an approach to decouple the training data, which simplifies the training and increases accuracy."
],
[
"Decouple the stress data by POD",
"The Proper Orthogonal Decomposition (POD) in combination with machine-learning tools, such as Gaussian Processes ([33]) and Long-Short-Term memory network ([24]), as a reduced order model has been used to surrogate model generation of fluid dynamic systems.",
"Here we introduce a novel combination framework, where POD and FNNs are combined for preprocessing and prediction of sequence data.",
"We call this approach Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN).",
"Using POD, the time series vector variables can be represented with a reduced number of modes neglecting higher order modes if the error is acceptable.",
"Thus, by use of the POD, the problem will be decoupled into a combination of several different modes.",
"As time series data, the stress sequence in training data can be rewritten as a snapshot matrix $O_{\\sigma }=\\begin{bmatrix}\\sigma _{op1} & \\sigma _{op2} & ... & \\sigma _{opi} & ...\\end{bmatrix}=\\begin{bmatrix}\\sigma ^1_1 & \\sigma ^2_1 & ... & \\sigma ^k_1 & ...\\\\\\sigma ^1_2 & \\sigma ^2_2 & ... & \\sigma ^k_2 & ...\\\\\\sigma ^1_3 & \\sigma ^2_3 & ... & \\sigma ^k_3 & ...\\end{bmatrix}_{3 \\times M},$ where each column of the matrix is a snapshot and can be written as a vector $o_{\\sigma }^k =\\begin{bmatrix} \\sigma ^k_1 & \\sigma ^k_2 & \\sigma ^k_3 \\end{bmatrix}^T$ .",
"Using POD, any snapshot $o_{\\sigma }^k$ can be represented by a linear combination of the basis $o_{\\sigma }^k = \\bar{\\sigma } + \\sum _{i=1}^{m} \\alpha _i^k \\varphi _i,$ where $\\varphi _i=\\begin{bmatrix} \\phi ^k_1 & \\phi ^k_2 & \\phi ^k_3 \\end{bmatrix}^T$ is the $i$ -th basis vector, $\\alpha ^k_i$ is the coefficient, $m$ is the number of POD mode and $\\bar{\\sigma }=\\begin{bmatrix}\\bar{\\sigma }_1 & \\bar{\\sigma }_2 & \\bar{\\sigma }_3\\end{bmatrix}^T$ is the mean value vector of the snapshot matrix.",
"Since $\\bar{\\sigma }$ , $\\varphi _i$ are constants, and the bases $\\varphi _i$ are independent with each other, the stress sequence can thus be decoupled into independent coefficient sequences.",
"The components of mean value vector $\\bar{\\sigma }$ of the snapshot matrix are computed as $\\bar{\\sigma }_1=\\frac{1}{M}\\sum _{i=1}^{M} \\sigma ^i_1, \\,\\,\\,\\,\\,\\, \\bar{\\sigma }_2=\\frac{1}{M}\\sum _{i=1}^{M} \\sigma ^i_2, \\,\\,\\,\\,\\,\\, \\bar{\\sigma }_3=\\frac{1}{M}\\sum _{i=1}^{M} \\sigma ^i_3.$ To find the basis vectors and its coefficients, the deviation matrix is firstly computed as $O_{\\sigma }^{dev}=\\begin{bmatrix}\\sigma ^1_1-\\bar{\\sigma }_1 & \\sigma ^2_1-\\bar{\\sigma }_1 & ... & \\sigma ^k_1-\\bar{\\sigma }_1 & ...\\\\\\sigma ^1_2-\\bar{\\sigma }_2 & \\sigma ^2_2-\\bar{\\sigma }_2 & ... & \\sigma ^k_2-\\bar{\\sigma }_2 & ...\\\\\\sigma ^1_3-\\bar{\\sigma }_3 & \\sigma ^2_3-\\bar{\\sigma }_3 & ... & \\sigma ^k_3-\\bar{\\sigma }_3 & ...\\end{bmatrix}_{3 \\times M}.$ Then, by applying Singular Value Decomposition (SVD) to the deviation matrix $O_{\\sigma }^{dev}=U S V^T,$ where $U$ and $V$ are the unitary matrices, $S$ is the diagonal matrix with non-negative real numbers on the diagonal, the basis vectors $\\varphi _i$ can be determined from the non-zero columns of matrix $U$ $\\Phi =\\begin{bmatrix}\\varphi _1 & \\varphi _2 & ... &\\varphi _m \\end{bmatrix}=\\begin{bmatrix} u_1 & u_2 & ... & u_m \\end{bmatrix}_{3 \\times m},$ where $u_m$ is the $m$ -th non-zero column of matrix $U$ and $m$ is equal to the rank of $O_{\\sigma }^{dev}$ .",
"The coefficients $\\alpha ^k=\\begin{bmatrix} \\alpha ^k_1 & \\alpha ^k_2& ...& \\alpha ^k_m \\end{bmatrix}^T$ can be obtained by projecting the snapshot $o^k_{\\sigma }$ on the basis matrix $\\Phi $ $\\alpha ^k=\\Phi ^To^k_{\\sigma }.$ Since the stress components are independent, the rank of matrix $O_{\\sigma }^{dev}$ is 3 in this work ($m=3$ ).",
"The stress sequence in equation (REF ) will be represented by 3 independent coefficient sequences $O_{\\sigma }=\\begin{bmatrix}\\sigma ^1_1 & \\sigma ^2_1 & ... & \\sigma ^k_1 & ...\\\\\\sigma ^1_2 & \\sigma ^2_2 & ... & \\sigma ^k_2 & ...\\\\\\sigma ^1_3 & \\sigma ^2_3 & ... & \\sigma ^k_3 & ...\\end{bmatrix}_{3 \\times M}\\frac{POD}{\\rightarrow }\\begin{array}{lcr}\\begin{bmatrix} \\alpha ^1_1 & \\alpha ^2_1 & ... & \\alpha ^k_1& ... \\end{bmatrix}_{1 \\times M}\\\\\\begin{bmatrix} \\alpha ^1_2 & \\alpha ^2_2 & ... & \\alpha ^k_2& ... \\end{bmatrix}_{1 \\times M}\\\\\\begin{bmatrix} \\alpha ^1_3 & \\alpha ^2_3 & ... & \\alpha ^k_3& ... \\end{bmatrix}_{1 \\times M},\\end{array}.$ Therefore, the training data composed by the strain-stress sequences in equation (REF ) for plasticity model is transformed into training data composed by strain-coefficient sequences and can be written as $\\begin{bmatrix}\\varepsilon _{op1} & \\varepsilon _{op2} & ... & \\varepsilon _{opi} & ...\\end{bmatrix}_{6 \\times M} \\Leftrightarrow \\begin{array}{lcr}\\begin{bmatrix} \\alpha ^1_1 & \\alpha ^2_1 & ... & \\alpha ^k_1& ... \\end{bmatrix}_{1 \\times M}\\\\\\begin{bmatrix} \\alpha ^1_2 & \\alpha ^2_2 & ... & \\alpha ^k_2& ... \\end{bmatrix}_{1 \\times M}\\\\\\begin{bmatrix} \\alpha ^1_3 & \\alpha ^2_3 & ... & \\alpha ^k_3& ... \\end{bmatrix}_{1 \\times M},\\end{array}$ where the three coefficient sequences are independent from each other."
],
[
"Prediction of coefficients using FNN",
"Once the training data is prepared, FNNs are applied to learn the mapping between the strain sequence and the coefficient sequences in equation (REF ).",
"Since the coefficients in the POD representation are independent, each coefficient can be predicted by one FNN, as shown in Fig.",
"REF , where the original strain-stress mapping approximated by one complex neural network is decoupled into three independent stain-coefficient mappings approximated by simpler neural networks.",
"Figure: Decoupling the strain-stress mapping (left) into independent strain-coefficient mappings (right) by POD.After training, the coefficient $\\alpha ^t_{NN,i}$ at time $t$ will be predicted by the feed forward neural network $FNN_i$ as $\\alpha ^t_{NN,i} = FNN_i (\\varepsilon ^t,\\varepsilon _{acc}^t, w, b_s), \\,\\,\\, (i=1,2,3),$ where $i$ indicates the number of the coefficient, $\\varepsilon ^t=\\begin{bmatrix} \\varepsilon _1^t &\\varepsilon _2^t & \\varepsilon _3^t\\end{bmatrix}^T$ is the current strain, $\\varepsilon _{acc}^t=\\begin{bmatrix} \\varepsilon _{acc,1}^t&\\varepsilon _{acc,2}^t&\\varepsilon _{acc,3}^t\\end{bmatrix}^T$ is the accumulated absolute strain, $w$ is the weight matrix and $b_s$ is the bias of neural network."
],
[
"PODFNN based plasticity model",
"Once the coefficient $\\alpha ^t_{NN}$ is predicted by FNNs as described in equation (REF ), the principle stress can be recovered from the POD representation as $\\tilde{\\sigma }^t_{PODFNN} = \\bar{\\sigma } + \\sum _{i=1}^{3} \\alpha _{NN,i}^t \\varphi _i.$ Finally, the Cauchy stress is obtained by transforming the principle stress into the general space $\\sigma ^t_{PODFNN} =Q\\tilde{\\sigma }^t_{PODFNN} Q^T.$ The formulation of the ML based plasticity model defines a constitutive function in the following format $\\sigma _{PODFNN}^t=PODFNN (\\varepsilon _t, \\varepsilon _{acc}^t, w, b_s),$ where $\\varepsilon _t$ is the current total strain, $\\varepsilon _{acc}^t$ is the history variable.",
"To apply the ML based plasticity model in the finite element analysis, the residual vector and tangent matrix have to be derived.",
"The automatic symbolic differentiation tool AceGen is applied to derive the tangent matrix again."
],
[
"Testing the ML based plasticity model in FEM",
"In the following, the performance of the developed ML based plasticity model is evaluated using finite element applications."
],
[
"Data collection from analytical model",
"Apart from collecting the training data from experiments, simulation data using the von Mises plasticity model can be collected as well to train the ML tool, in this way the performance of the ML based plasticity model can be verified by comparing to the analytical model.",
"In this work, the strain-stress sequences are collected at a Gauss point of a finite element under specific loading paths described above.",
"The strain is computed as the symmetric part of the displacement gradient for small deformations $\\varepsilon =\\frac{1}{2}(H+H^T),$ where $H$ is the displacement gradient with $H={\\rm Grad}u$ .",
"Using spectral decomposition, the strain can be formulated as $\\varepsilon =Q \\cdot \\Lambda \\cdot Q^T,$ where $\\Lambda $ is the principle strain and $Q$ is the rotation matrix obtained from the eigendirections.",
"The principle strain $\\Lambda $ along different loading paths will serve as input data to train the ML based plasticity model.",
"For small strain plasticity, the additively decomposition of strain into elastic part and plastic part is assumed $\\Lambda =\\Lambda ^e +\\Lambda ^p,$ where $\\Lambda ^e$ and $\\Lambda ^p$ are the elastic strain and plastic strain respectively.",
"The plastically admissible stress is given by $\\Sigma =\\rho \\frac{\\partial \\psi }{\\partial \\Lambda ^e},$ where $\\psi $ is the free energy function.",
"The principle stress $\\Sigma $ will serve as the output data, corresponding to the principle strain as input.",
"By assuming the von Mises yield criteria, the yield function is written as $f=\\sqrt{\\frac{3}{2}} \\Vert \\Sigma ^{dev} \\Vert -\\sigma _y(\\alpha ),$ where $\\Sigma ^{dev}$ is the deviatoric stress with $\\Sigma ^{dev} = \\Sigma - \\frac{1}{3} tr\\Sigma \\cdot 1 $ and $\\alpha $ is the isotropic hardening variable.",
"By use of the associated plastic flow rule, the evolution equations for the principle plastic strain and the hardening variable are formulated as $\\dot{\\Lambda }^p=\\dot{\\gamma }\\frac{\\partial f}{\\partial \\Sigma ^{dev}}, \\,\\,\\,\\,\\, \\dot{\\alpha }=\\dot{\\gamma }\\frac{\\partial f}{\\partial A}$ where $\\gamma $ is the plastic multiplier and $A$ is the thermodynamic force conjugate with $\\alpha $ .",
"The plastic flow has to full fill the Kuhn-Tucker conditions $f \\leqslant 0, \\,\\,\\,\\,\\, \\dot{\\gamma } \\geqslant 0, \\,\\,\\,\\,\\, \\dot{\\gamma } f=0.$"
],
[
"Uniaxial tension and compression",
"To test the performance of the ML based plasticity model, the 1D uniaxial tension and compression test is conducted firstly.",
"The von Mises plasticity with the linear isotropic hardening is applied as the target model.",
"The material parameters of the plasticity model are set as: Young's modulus $E=700N/mm^2$ , yield stress $\\sigma _y=100MPa$ and isotropic hardening parameter $H_{iso}=10$ .",
"To prepare the training data, 11 sets of strain-stress sequence data are collected from the target model with strain increments being increased linearly from $0.02$ to $0.03$ .",
"The stress sequences are then transformed into the coefficient sequence by the POD.",
"The feed forward neural network with the architecture of (2-20-20-1) is applied to predict the coefficient, where the input layer containing 2 neurons is connected with two hidden layers containing 20 neurons each.",
"The output layer contains one neuron.",
"The input of the neural network is the total strain together with the accumulated absolute strain, and the output of the neural network is the coefficient transformed from the stress sequence.",
"The Levenberg-Marquardt algorithm is applied as the optimizer in training.",
"The weights are initialized by the Nguyen-Widrow method.",
"After 4000 epochs, the mean squared error decreased to $0.0393$ which costs the training time of 2m24s.",
"The testing strain sequence is generated by setting the strain increment as 0.15 so that it is different from the training data.",
"The stress computed from the ML based plasticity model is compared with the the stress from the target plasticity model, as shown in Fig.",
"REF .",
"It can be seen that the predicted stress follows the exact solution well, which validates the accuracy of the proposed machine learning approach for plasticity.",
"This test shows that the accumulated absolute total strain as a history variable captures the loading history for the cyclic loading condition.",
"Figure: Uniaxial tension and compression."
],
[
"Applications in 2D finite element analysis",
"To evaluate the ML based plasticity model, benchmark tests in 2D are presented.",
"The von Mises plasticity with an exponential isotropic hardening law $\\sigma _y = y_0+y_0(0.00002+\\gamma )^{0.3}$ is set as the target model.",
"The material parameters are set as: Young's modulus $E=1N/mm^2$ , Poison's ratio $\\nu =0.33$ , and the initial yield stress $y_0=0.05MPa$ .",
"To collect the training data, 122 loading-unloading paths evenly distributed within the circles ($r_1=0.1, r_2=0.075$ ) in Fig.",
"REF are selected to conduct the biaxial tests, where 61 values are assigned to the angle $\\phi $ .",
"Then the collected stress sequence data is transformed into the coefficient sequences using POD.",
"Since there are two coefficients referring to the two principle stress components in the 2D case, two FNNs will be required to predict the coefficients.",
"In this part, the same network architecture (4-20-20-1) is applied for the two FNNs, where the input layer containing 4 neurons is connected with two hidden layers containing 20 neurons each.",
"The output layer contains always one neuron.",
"The total strain together with the accumulated absolute strain are applied as the input of the neural network.",
"The output of the neural network is the coefficient transformed from the stress sequences.",
"The Levenberg-Marquardt algorithm is applied as the optimizer as well.",
"The training progress is terminated when the gradient of error is less than $10^{-7}$ , where the mean squared error is decreased to $7.96 \\times 10^{-9}$ for the first FNN and $6.35 \\times 10^{-9}$ for the second FNN.",
"After the training process, the weighs and biases of the neural network are output as the constant model parameters, by which the Cauchy stress is recovered according to the POD formulation.",
"The tangent matrix and the residual vector are derived using the symbolic differentiation tool AceGen again.",
"Firstly, the 2D ML based plasticity model is tested by the Cook's membrane problem.",
"The beam is clamped at the left end and loaded at the right end by a constant distributed vertical load $q_0=0.03Mpa$ , as depicted in Fig.",
"REF .",
"In the unloading process, the direction of vertical load is changed to be negative.",
"The geometric domain of the structure is discretized by 40 quadratic 9-node quadrilateral elements leading to 189 nodes.",
"Before unloading, the final deformation state of the beam using the proposed ML based plasticity model is compared with the target plasticity model, which is depicted in Fig.",
"REF .",
"It can be observed that the vertical displacement of the structure is almost identical.",
"Figure: With NN modelThe load displacement curve of the upper node $(48, 60)$ at the right end of the cantilever beam is plotted in Fig.",
"REF .",
"The figure shows that the ML based plasticity model captures the loading and unloading behaviour very well.",
"Figure: Load deflection curve of the 2D Cook's membrane.The second example to test the ML based plasticity model is a punch test as shown in Fig.",
"REF , where the vertical displacement boundary condition ($u_0 = 0.07mm$ ) is imposed on the top of the block and the bottom of the block is only fix in the vertical direction.",
"In the unloading process, the direction of vertical displacement boundary is changed to be positive.",
"The block is discretized with 100 quadratic 9-node quadrilateral elements leading to 441 nodes.",
"Figure: 2D punch problem.Before unloading, the final deformation state of the block with ML based plasticity model is compared with that using the target plasticity model, which are depicted in Fig.",
"REF .",
"It can be observed that the horizontal displacement of the structure is very close for the two models.",
"Figure: With NN modelThe load displacement curve of the upper node $(0, 1)$ at the left end of the block is plotted in Fig.",
"REF , where it can be seen that the ML based model follows the plasticity model well both in loading and unloading.",
"Figure: Load deflection curve of the 2D block."
],
[
"Applications in 3D finite element analysis",
"In this section, the ML based plasticity model is extended to 3D applications.",
"To generate the training data, one hexahedron finite element is applied to different loading situations as described in Fig.",
"REF .",
"The von Mises plasticity with an exponential isotropic hardening law $\\sigma _y = y_0+y_0(0.00002+\\gamma )^{0.1}$ is set as the target model.",
"The material parameters are set as: Young's modulus $E=10N/mm^2$ , Poison's ratio $\\nu =0.33$ , and the initial yield stress $y_0=0.3MPa$ .",
"During the training data preparation, strain-stress sequences along 8100 loading paths are generated based on the sphere ($r=0.02$ ) in Fig.",
"REF , where 90 values are assigned to the angles $\\phi $ and $\\theta $ respectively.",
"Since the huge amount of data have to be collected for unloading in 3D, only the loading data is collected here and the unloading is not considered in this part.",
"In the three-dimensional case, three FNNs are required to predict the coefficients, which are corresponding to the principle Cauchy stress components.",
"The same network architecture (6-16-16-1) is employed for all FNNs, where three total strain components together with three accumulated absolute strains are applied as the input of the networks.",
"The output of the neural network is the coefficient.",
"The weights are initialized by the Nguyen-Widrow method.",
"During training, the Levenberg-Marquardt algorithm is applied as the optimizer, where the training progress is terminated when the gradient of global error is less than $10^{-7}$ .",
"The mean squared errors are decreased to $5.70\\times 10^{-8}$ , $1.13 \\times 10^{-9}$ and $6.78 \\times 10^{-10}$ for the first, second and third FNN respectively.",
"The training process costs time of $1h20m21s$ , $1h18m46s$ and $52m47s$ for the first, second and third FNN respectively.",
"To evaluate the performance of the POD representation, the performances of training strain-stress model with one FNN(6-16-16-3) and training three strain-coefficient models with three FNNs(6-16-16-1) are compared.",
"Without POD, only one FNN is required to approximate the mapping, where the output includes 3 stress components.",
"With POD, three independent FNNs will be applied, where the output of FNN includes only one POD coefficient.",
"The training performances within 2000 epochs are shown in Fig.",
"REF .",
"It can be seen that the average of the mean squared errors of the three FNNs is smaller than that without POD.",
"Additionally, training a FNN with architecture of (6-16-16-3) costs computation time of $4h22m43s$ whereas the average training time of the three FNNs (6-16-16-1) is $1h26m27$ .",
"It can be observed that the POD approach leads to less training time and better training performance.",
"Figure: MSE of training FNN(6-16-16-3) and the average MSE of training 3 FNNs(6-16-16-1).The first example to test the 3D machine learning based plasticity model is the necking of a bar as shown in Fig.",
"REF , where the left end of the bar is fixed and the displacement boundary $u_0=0.05mm$ is imposed at the right end along its axial direction.",
"An artificial imperfection is set in the center of the bar to trigger the necking, where the radius at the center is chosen to be $R_c= 0.98R$ .",
"The bar is discretized with 200 quadratic 27-node elements leading to 2193 nodes.",
"Figure: Geometry and boundary conditions of the bar.Fig.",
"REF shows the final deformation of the bar after tension, where only one quarter of the bar is computed due to the symmetry.",
"It can be observed that the amounts of the necking computed by the two models are close to each other.",
"Figure: With NN modelThe load displacement curve of the bar under the uniaxial tension is plotted in Fig.",
"REF , where the neural network based model follows the plasticity model quite well.",
"Figure: Load deflection curve of the cylindrical bar.The second example is the punch test, where the vertical displacement boundary condition $(u_0=0.15mm)$ is imposed on the top of the block and the bottom of the block is only fixed in the vertical direction, as shown in Fig.",
"REF .",
"The block is discretized with 100 quadratic 27-node elements leading to 441 nodes.",
"Figure: 3D punch problem.Fig.",
"REF shows the final deformation of the block after compression.",
"It can be observed that the displacements in the horizontal direction computed by the two models are close to each other.",
"The load displacement curve of the block under compression is plotted in Fig.",
"REF .",
"It can be seen that the neural network based model follows the plasticity model quite well.",
"Figure: With NN modelFigure: Load deflection curve of the 3D punch problem.The last example for the 3D neural network based model is the Cook's membrane problem.",
"The 3D beam is clamped at the left end and loaded at the right end by a constant distributed vertical load $q_0=0.03MPa$ , as depicted in Fig.",
"REF .",
"By use of the quadratic finite element with 27 nodes, the beam is discretized using 1080 elements leading to 10309 nodes.",
"Figure: 3D Cook's membrane problem.Fig.",
"REF shows the final deformation of the Cook's membrane.",
"It can be observed that the displacements in the vertical direction computed by the two models are almost identical.",
"The load displacement curve of the upper node (48,0,60) is plotted in Fig.",
"REF .",
"It can be seen that the machine learning based model follows the plasticity model quite well.",
"Figure: With NN modelFigure: Load deflection curve of the 3D Cook's membrane."
],
[
"Conclusions",
"In this work, a machine learning based material modelling approach for hyper-elasticity and plasticity is proposed.",
"Common tools such as FNNs show proficient performances for capturing the mapping between strain and stress in the case of elasticity.",
"However, history variables are required to distinguish the loading history in case of plasticity.",
"In this context, FNNs show subpar performances.",
"Thus, the accumulated absolute strain is proposed to be the history variable, which captures the loading history well without requirement for additional data.",
"Here we present a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN), which in combination with the introduced history variable is able to overcome this problem.",
"By use of the POD, less training time and better training performance are obtained in the network training.",
"Additionally, it has been shown that the training data collected only from the multi-axial loading tests are enough to capture the von Mises yield surface and the hardening law.",
"The automatic symbolic differentiation tool AceGen provides a very convenient way to derive the tangent matrix for the machine learning based material model.",
"The generalization and accuracy of the presented model as well as the data generation strategy have been verified by finite element applications both in 2D and 3D."
],
[
"Acknowledgements",
"The first author would like to thank the China Scholarship Council (CSC) and the Graduate Academy of Leibniz Universität Hannover for the financial support.",
"The second author acknowledges the financial support from the Deutsche Forschungsgemeinschaft under Germanys Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).",
"The last author acknowledges the support of Deutsche Forschungsgemeinschaft for the project C2 within the collaborative research center/Transregio TR73."
]
] | 2001.03438 |
[
[
"Particle trajectories in Weibel filaments: influence of external field\n obliquity and chaos"
],
[
"Abstract When two collisionless plasma shells collide, they interpenetrate and the overlapping region may turn Weibel unstable for some values of the collision parameters.",
"This instability grows magnetic filaments which, at saturation, have to block the incoming flow if a Weibel shock is to form.",
"In a recent paper [J.",
"Plasma Phys.",
"(2016), vol.",
"82, 905820403], it was found implementing a toy model for the incoming particles trajectories in the filaments, that a strong enough external magnetic field $\\mathbf{B}_0$ can prevent the filaments to block the flow if it is aligned with.",
"Denoting $B_f$ the peak value of the field in the magnetic filaments, all test particles stream through them if $\\alpha=B_0/B_f > 1/2$.",
"Here, this result is extended to the case of an oblique external field $B_0$ making an angle $\\theta$ with the flow.",
"The result, numerically found, is simply $\\alpha > \\kappa(\\theta)/\\cos\\theta$, where $\\kappa(\\theta)$ is of order unity.",
"Noteworthily, test particles exhibit chaotic trajectories."
],
[
"Introduction",
"Collisionless plasmas can sustain shock waves with a front much smaller than the particles mean-free-path [29].",
"These shocks, which are mediated by collective plasma effects rather than binary collisions, have been dubbed “collisionless shocks”.",
"It is well known that the encounter of two collisional fluids generates two counter-propagating shock waves [36].",
"Likewise, the encounter of two collisionless plasmas generate two counter-propagating collisionless shock waves [18], [31], [32], [28].",
"In the collisional case, the shocks are launched when the two fluids make contact.",
"In the collisionless case, the two plasmas start interpenetrating as the long mean-free-path prevent them from “bumping” into each other.",
"As a counter-streaming plasma system, the interpenetrating region quickly turns unstable.",
"The instability grows, saturates, and creates a localized turbulence which stops the incoming flow, initiating the density build-up in the overlapping region [10], [11], [16].",
"Various kind of instabilities do grow in the overlapping region [6].",
"But the fastest growing one takes the lead and eventually defines the ensuing turbulence.",
"When the system is such that the filamentation, or Weibel, instability grows faster, magnetic filaments are generated [24], [34], [23], [20], [21].",
"In a pair plasmaTo our knowledge, there is no systematic study of the hierarchy of unstable modes for two magnetized colliding ion/electron plasma shells.",
"See [23], [35], [30] for works contemplating the un-magnetized case., the conditions required for the Weibel instability to lead the linear phase have been studied in [3] for a flow-aligned field, and in [5] for an oblique field.",
"A mildly relativistic flow is required.",
"Also, accounting for an oblique field supposes the Larmor radius of the particles is large compared to the dimensions of the system.",
"Since the field modifies the hierarchy of unstable modes, it can prompt another mode than Weibel to lead the linear phase.",
"In such cases, studies found so far that a shock still forms [9], [16], [17], mediated by the growth of the non-Weibel leading instability, like two-stream for example.",
"Since the blocking of the flow entering the filaments is key for the shock formation, it is interesting to study under which conditions a test particle will be stopped in these magnetic filaments.",
"Recently, a toy model of the process successfully reproduced the criteria for shock formation in the case of pair plasmas [2].",
"When implemented accounting for a flow-aligned magnetic field [4], the same kind of model could predict how too strong a field can deeply affect the shock formed [9], [7].",
"In view of the many settings involving oblique magnetic fields, we extend here the previous model to the case of an oblique field.",
"Figure: Setup considered.",
"We can consider ϕ=π/2\\varphi =\\pi /2 since the direction of the flow, the field, and the 𝐤\\mathbf {k} of the fastest growing Weibel mode, are coplanar for the fastest growing Weibel modes , .The system considered is sketched on Figure REF .",
"The half space $z\\ge 0$ is filled with the magnetic filaments $\\mathbf {B}_f=B_f\\sin ( kx) ~\\mathbf {e}_y$ .",
"In principle, we should consider any possible orientation for $\\mathbf {B}_0$ , thus having to consider the angles $\\theta $ and $\\varphi $ .",
"Yet, previous works in 3D geometry found that the fastest growing Weibel modes are found with a wave vector coplanar with the field $\\mathbf {B}_0$ and the direction of the flow [1], [25].",
"Here, the initial flow giving rise to Weibel is along $z$ and we set up the axis so that $\\mathbf {k}$ is along $x$ .",
"We can therefore consider $\\varphi =\\pi /2$ so that $\\mathbf {B}_0=B_0(\\sin \\theta ,0,\\cos \\theta )$ .",
"A test particle is injected at $(x_0,0,0)$ with velocity $\\mathbf {v}_0=(0,0,v_0)$ and Lorentz factor $\\gamma _0=(1-v_0^2/c^2)^{-1/2}$ , mimicking a particle of the flow entering the filamented overlapping region.",
"Our goal is to determine under which conditions the particle streams through the magnetic filaments to $z=+\\infty $ , or is trapped inside.",
"As explained in previous articles [2], [4], in the present toy model, this dichotomy comes down to determining whether test particles stream through the filaments, or bounce back to the region $z<0$ .",
"The reason for this is that in a more realistic setting, particles reaching the filaments and turning back will likely be trapped in the turbulent region between the upstream and the downstream.",
"This problem is clearly related to the one of the shock formation, since the density build-up leading to the shock requires the incoming flow to be trapped in the filaments."
],
[
"Equations of motion",
"Since the Lorentz factor $\\gamma _0$ is a constant of the motion, the equation of motion for our test particle reads, $m\\gamma _0 \\ddot{\\mathbf {x}} = q\\frac{\\dot{\\mathbf {x}}}{c}\\times (\\mathbf {B}_f + \\mathbf {B}_0).$ Explaining each component, we find for $z>0$ , $\\ddot{x} &=& \\frac{q B_f}{\\gamma _0 m c} \\left[ \\frac{B_0}{B_f} \\dot{y} \\cos \\theta - \\dot{z} \\sin kx\\right], \\\\\\ddot{y} &=& \\frac{q B_f}{\\gamma _0 m c}\\frac{B_0}{B_f}\\left[\\dot{z} \\sin \\theta - \\dot{x}\\cos \\theta \\right] , \\\\\\ddot{z} &=& \\frac{q B_f}{\\gamma _0 m c} \\left[-\\frac{B_0}{B_f} \\dot{y} \\sin \\theta + \\dot{x} \\sin kx \\right], $ while $B_f=0$ for $z<0$ .",
"We can now define the following dimensionless variables, $\\mathbf {X}=k\\mathbf {x},~~\\alpha = \\frac{B_0}{B_f}, ~~ \\tau = t\\omega _{B_f},~~\\mathrm {with} ~~ \\omega _{B_f} = \\frac{qB_f}{\\gamma _0 mc}.$ With these variables, the system (REF -) reads, $\\ddot{X} =& -\\dot{Z} \\mathcal {H}(Z) \\sin X + & \\alpha \\dot{Y} \\cos \\theta , \\nonumber \\\\\\ddot{Y} =& & \\alpha ( \\dot{Z} \\sin \\theta - \\dot{X} \\cos \\theta ), \\\\\\ddot{Z} =& \\dot{X} \\mathcal {H}(Z) \\sin X - & \\alpha \\dot{Y} \\sin \\theta , \\nonumber $ where $\\mathcal {H}$ is the Heaviside step function $\\mathcal {H}(x)=0$ for $x<0$ and $\\mathcal {H}(x)=1$ for $x \\ge 0$ .",
"The initial conditions are, $\\mathbf {X}(\\tau =0) &=& \\left( X_0,0,0 \\right) ~~ \\mathrm {with} ~~ X_0 \\equiv kx_0, \\nonumber \\\\\\dot{\\mathbf {X}}(\\tau =0) &=& \\left(0,0,\\dot{Z}_0 \\right) ~~ \\mathrm {with} ~~ \\dot{Z}_0 \\equiv \\frac{kv_0}{\\omega _{B_f}}.$"
],
[
"Constants of the motion and chaotic behavior",
"The total field in the region $z>0$ reads $\\mathbf {B}=\\left( B_0 \\sin \\theta , B_f\\sin ( kx), B_0 \\cos \\theta \\right)$ .",
"It can be written as $\\mathbf {B}=\\nabla \\times \\mathbf {A}$ , with the vector potential in the Coulomb gauge, $\\mathbf {A} = \\left(\\begin{array}{c}0 \\\\B_0 x \\cos \\theta \\\\B_0 y \\sin \\theta + B_f\\frac{\\cos (kx)}{k}\\end{array}\\right).$ The canonical momentum then reads [19], $\\mathbf {P} = \\mathbf {p} + \\frac{q}{c}\\mathbf {A} = \\left(\\begin{array}{c}p_x \\\\p_y + \\frac{q}{c} B_0 x \\cos \\theta \\\\p_z + \\frac{q}{c}\\left( B_0 y \\sin \\theta + B_f\\frac{\\cos (kx)}{k} \\right)\\end{array}\\right),$ where $\\mathbf {p}=\\gamma _0 m \\mathbf {v}$ .",
"Since it does not explicitly depend on $z$ , the $z$ component is a constant of the motion.",
"It can equally be obtained time-integrating Eq.",
"().",
"Note that for $\\theta =0$ , the $y$ dependence vanishes so that the $y$ component of the canonical momentum is also a constant of the motion [4].",
"We can derive another constant of the motion from Eq.",
"().",
"Time-integrating it and remembering $\\dot{y}(t=0)=z(t=0)=0$ , we find, $\\dot{y} = \\frac{q B_0}{\\gamma _0 m c}\\left[z \\sin \\theta - (x-x_0)\\cos \\theta \\right].$ Since $x_0$ is obviously a constant, we can express it in terms of the other variables and obtain an invariant, that is $x_0 &=& x - z \\tan \\theta + \\frac{ c}{q B_0 \\cos \\theta }\\gamma _0 m\\dot{y} \\nonumber \\\\&=& x - z \\tan \\theta + \\frac{ c}{q B_0 \\cos \\theta }\\left(P_y -\\frac{q}{c}A_y\\right).$ Finally, the Hamiltonian, $\\mathcal { H} = c \\sqrt{c^2 m^2 + \\left( \\mathbf {P} - \\frac{q}{c}\\mathbf {A} \\right)^2}$ is also a constant of the motion.",
"Replacing $A_y$ in Eq.",
"(REF ) by its expression from (REF ), we therefore have the following constants of the motion, $\\mathcal {H} \\equiv C_1 &=& c \\sqrt{c^2 m^2 + \\left( \\mathbf {P} - \\frac{q}{c}\\mathbf {A} \\right)^2}, \\\\C_2 &=& P_z, \\\\x_0 \\equiv C_3 &=& \\frac{ c}{q B_0 \\cos \\theta }P_y - z \\tan \\theta .$ According to Liouville's theorem on integrable systems, a $n$ -dimensional Hamiltonian system is integrable if it has $n$ constants of motion $C_j(x_i,P_i,t)_{j \\in [1\\ldots n]}$ in involution [26], [22], [14], that is, $\\lbrace C_j,C_k\\rbrace = \\sum _{i=1}^3 \\left(\\frac{\\partial C_j}{\\partial x_i}\\frac{\\partial C_k}{\\partial P_i} - \\frac{\\partial C_k}{\\partial x_i}\\frac{\\partial C_j}{\\partial P_i}\\right) = 0, \\forall (j,k),$ where $\\lbrace f,g\\rbrace $ is the Poisson bracket of $f$ and $g$ .",
"It is easily checked that $C_{2,3}$ being constants of the motion, $\\lbrace \\mathcal { H},C_2\\rbrace =\\lbrace \\mathcal { H},C_3\\rbrace =0$ .",
"However, $\\lbrace C_2,C_3\\rbrace = -\\tan \\theta .$ As a result, the system is integrable only for $\\theta =0$ [4].",
"Otherwise, it is chaotic, as will be checked numerically in the following sections."
],
[
"Reduction of the number of free parameters",
"The free parameters of the system (REF -) with initial conditions (REF ) are $(X_0,\\dot{Z}_0,\\alpha ,\\theta )$ .",
"In order to deal with a more tractable phase space parameter, we now reduce its dimension accounting for the physical context of the problem.",
"Consider the magnetic filaments generated by the growth of the filamentation instability triggered by the counter-streaming of 2 cold (thermal spread $\\Delta v \\ll v$ ) symmetric pair plasmas.",
"Both plasma shells have identical density $n$ in the lab frame, and initial velocities $\\pm v \\mathbf {e}_z$ .",
"We denote $\\beta = v/c$ .",
"The Lorentz factor $\\gamma _0$ previously defined equally reads $\\gamma _0=(1-\\beta ^2)^{-1/2}$ since the test particles entering the magnetic filaments belong to the same plasma shells.",
"The wave vector $\\mathbf {k}$ defining the magnetic filaments is also the wave vector of the fastest growing filamentation modes.",
"We can then set [10], $k = \\frac{\\omega _p}{c\\sqrt{\\gamma _0}},$ where $\\omega _p^2=4\\pi n q^2/m$ , so that, $\\dot{Z}_0 = \\frac{\\beta }{\\sqrt{\\gamma _0}} \\frac{\\omega _p}{\\omega _{B_f} }= \\frac{\\beta }{\\sqrt{\\gamma _0}} \\frac{\\omega _{B_0}}{\\omega _{B_f}} \\frac{\\omega _p}{\\omega _{B_0} }= \\frac{\\beta }{\\sqrt{\\gamma _0}} \\alpha \\frac{\\omega _p}{\\omega _{B_0} },$ where, $\\omega _{B_0}=\\frac{qB_0}{\\gamma _0 mc}.$ The peak field $B_f$ in the filaments can be estimated from the growth rate $\\delta $ of the instability, considering $\\omega _{B_f} \\sim \\delta $ [15].",
"It turns out that over the domain $\\delta \\gg \\omega _{B_0}$ , the growth rate $\\delta $ depends weakly on $\\theta $ and can be well approximated by [33], [1], $\\omega _{B_f} \\sim \\delta = \\omega _p\\sqrt{\\frac{2 \\beta ^2}{\\gamma _0} - \\left( \\frac{\\omega _{B_0}}{\\omega _p}\\right)^2},$ so that, $\\omega _{B_f} = \\frac{\\omega _{B_0}}{\\alpha } = \\omega _p\\sqrt{\\frac{2 \\beta ^2}{\\gamma _0} - \\left( \\frac{\\omega _{B_0}}{\\omega _p}\\right)^2}.$ This expression allows one to express $\\omega _{B_0}/\\omega _p$ as, $\\frac{\\omega _{B_0}}{\\omega _p} = \\sqrt{\\frac{2}{\\gamma _0}} \\frac{\\alpha \\beta }{\\sqrt{1+\\alpha ^2 }},$ so that Eq.",
"(REF ) eventually reads, $\\dot{Z}_0 = \\sqrt{\\frac{1+\\alpha ^2 }{2}} .$ The parameters phase space is thus reduced to 3 dimensions, $(X_0,\\alpha ,\\theta )$ ."
],
[
"Numerical exploration",
"It was previously found that for $\\theta =0$ , all particles stream through the filaments, no matter their initial position and velocity, if $\\alpha > 1/2$ [4].",
"Clearly for $\\theta =\\pi /2$ , no particle can stream to $z=+\\infty $ .",
"As we shall see, the $\\theta $ -dependent threshold value of $\\alpha $ beyond which all particles go to $\\infty $ is simply $\\propto 1/\\cos \\theta $ .",
"The system (REF -) is solved using the Mathematica “NDSolve” function.",
"The equations are invariant under the change $X \\rightarrow X+2 \\pi $ , so that we can restrict the investigation to $X_0 \\in [-\\pi , \\pi ]$ .",
"Unless $\\theta = 0$ , there are no other trivial symmetries.",
"In particular, the transformation $\\theta \\rightarrow -\\theta $ does not leave the system invariant.",
"We shall detail the case $\\theta \\in [0,\\pi /2]$ and only give the results, very similar though not identical, for $\\theta \\in [-\\pi /2,0]$ .",
"Figure: Value of Z(τ max )Z(\\tau _{max}) in terms of X 0 X_0 for τ max =10 3 \\tau _{max}=10^3 and two values of θ\\theta .The numerical exploration is conducted solving the equations and looking for the value of $Z$ at large time $\\tau =\\tau _{max}$ .",
"Figure REF shows the value of $Z(\\tau _{max})$ in terms of $X_0 \\in [0,2\\pi ]$ for the specified values of $(\\alpha ,\\theta )$ and $\\tau _{max}=10^3$ .",
"Similar results have been obtained for larger values of $\\tau $ like $\\tau =10^4$ , or even smaller ones, like $\\tau =500$ .",
"For $\\theta =\\pi /4$ , save a few exceptions for some values of $X_0$ , all particles have bounced back to the $z<0$ region.",
"As explained in [2], [4], this means that in a more realistic setting, they would likely be trapped in the filaments.",
"Figure: Value of Z(τ max )Z(\\tau _{max}) for θ=π/4\\theta =\\pi /4 and increasingly small X 0 X_0 intervals.As evidenced by Figure REF -left, the function $Z(\\tau _{max})$ is smooth for $\\theta =0$ .",
"Yet, for $\\theta =\\pi /4$ (right), the result features regions where $Z(\\tau _{max})$ varies strongly with $X_0$ .",
"In order to identify chaos, Figure REF present a series of successive zooms on Fig.",
"REF -right, where the function $Z(\\tau _{max})$ is plotted over an increasingly small $X_0$ interval inside $X_0 \\in [\\pi /4,\\pi /2]$ .",
"As expected from the analysis conducted in Sec.",
", the system is chaotic.",
"Note that chaotic trajectories in magnetic field lines have already been identified in literature [14], [12], [27], [13].",
"Figure REF has been plotted solving the system for $N+1$ particles shot from $X_0(j) = j 2\\pi /N$ with $j=0\\ldots N$ .",
"We denote $Z_j(\\tau _{max})$ the value of $Z$ reached by the $j^{\\mathrm {th}}$ particle at $\\tau =\\tau _{max}$ .",
"Then we define the following function, $\\phi (\\alpha ,\\theta ) = \\frac{1}{N+1}\\sum _{j=0}^{N} \\mathcal {H}\\left[ -Z_j(\\tau _{max}) \\right],$ where $\\mathcal {H}$ is again the Heaviside function.",
"Figure: Plot on the function φ\\phi defined by Eq.",
"(), in terms of α\\alpha and for θ=0\\theta =0 and π/5\\pi /5.The function $\\phi $ represents therefore the fraction of particles that bounced back against the magnetic filaments.",
"Figure REF -left plots it in terms of $\\alpha $ for $\\theta =0$ .",
"For $\\alpha =0$ , that is $B_0=0$ , about 40% of the particles bounce back, i.e, are trapped in the filaments.",
"As $\\alpha $ is increased, the field $\\mathbf {B}_0$ guides the test particles more and more efficiently until $\\alpha \\sim 0.5$ where all the particles stream through the filaments.",
"In turn, Figure REF -right displays the case $\\theta =\\pi /5$ .",
"Being oblique, the field $\\mathbf {B}_0$ is less efficient to guide the particles through the filaments, and more efficient to trap them inside.",
"As a result, it takes a higher value of $B_0$ , that is, $\\alpha = 1.4$ , to reach $\\phi =0$ .",
"Figure: Top: Numerical computation of α c (θ)\\alpha _c(\\theta ) (blue dots) compared to 1/cosθ1/\\cos \\theta (red line).",
"Bottom: Values of the κ(θ)\\kappa (\\theta ) function entering the expression of α c \\alpha _c in Eq.",
"(), in terms of θ\\theta .Similar numerical calculations have been conducted for various values of $\\theta \\in [0,\\pi /2]$ .",
"We finally define $\\alpha _c(\\theta )$ such as, $\\phi (\\alpha _c)=0.$ $\\alpha _c(\\theta )$ is therefore the threshold value of $\\alpha $ beyond which all particles bounce back against the filamented region, i.e, $\\phi (\\alpha \\ge \\alpha _c)=0$ .",
"It is plotted on Figure REF and can be well approximated by, $\\alpha _c = \\kappa (\\theta ) \\frac{1}{\\cos \\theta },$ where $\\kappa (\\theta )$ is of order unity (see Fig.",
"REF -bottom).",
"Figure: Same as Fig.",
"-right, but for θ=-π/4\\theta =-\\pi /4."
],
[
"Case $\\theta < 0$",
"As already noticed, there is no invariance by the change $\\theta \\rightarrow -\\theta $ .",
"Figure REF plots the counterpart of Fig.",
"REF -right, but for $\\theta =-\\pi /4$ .",
"Though quite similar, the results are not identical.",
"The numerical analysis detailed above for $\\theta \\in [0,\\pi /2]$ has been conducted for $\\theta \\in [-\\pi /2,0]$ .",
"The function $\\alpha _c(\\theta <0)$ again adjusts very well to the right-hand-side of Eq.",
"(REF ), with the values of $\\kappa (\\theta )$ plotted on Fig.",
"REF -bottom."
],
[
"Conclusion",
"A model previously developed to study test particles trapping in magnetic filaments has been extended to the case of an oblique external magnetic field.",
"The result makes perfect physical sense: up to a constant $\\kappa $ of order unity, only the component of the field parallel to the filaments is relevant.",
"The criteria obtained in [4] for a flow-aligned field, namely that particles are trapped if $\\alpha > 1/2$ , now reads, $\\alpha > \\frac{\\kappa (\\theta )}{\\cos \\theta },$ where $\\kappa (\\theta )$ is of order unity.",
"As a consequence, a parallel field affects the shock more than an oblique one.",
"Kinetic effects triggered by a parallel field can significantly modify the shock structure while a perpendicular field will rather “help” the shock formation.",
"This is in agreement with theoretical works which found a parallel field can divide by 2 the density jump expected from the MHD Rankine-Hugoniot conditions [9], [7], while the departure from MHD is less pronounced in the perpendicular case [8]."
],
[
"Acknowledgments",
"A.B.",
"acknowledges support by grants ENE2016-75703-R from the Spanish Ministerio de Educación and SBPLY/17/180501/000264 from the Junta de Comunidades de Castilla-La Mancha.",
"Thanks are due to Ioannis Kourakis and Didier Bénisti for enriching discussions."
]
] | 2001.03473 |
[
[
"Investigating a Deep Learning Method to Analyze Images from Multiple\n Gamma-ray Telescopes"
],
[
"Abstract Imaging atmospheric Cherenkov telescope (IACT) arrays record images from air showers initiated by gamma rays entering the atmosphere, allowing astrophysical sources to be observed at very high energies.",
"To maximize IACT sensitivity, gamma-ray showers must be efficiently distinguished from the dominant background of cosmic-ray showers using images from multiple telescopes.",
"A combination of convolutional neural networks (CNNs) with a recurrent neural network (RNN) has been proposed to perform this task.",
"Using CTLearn, an open source Python package using deep learning to analyze data from IACTs, with simulated data from the upcoming Cherenkov Telescope Array (CTA), we implement a CNN-RNN network and find no evidence that sorting telescope images by total amplitude improves background rejection performance."
],
[
"Motivation",
"Very-high-energy (VHE; from about 20 GeV to 300 TeV) gamma rays provide a critical probe of the Universe's most extreme environments, offering the opportunity to study exotic astrophysics and fundamental physics at high energies and cosmological distances.",
"Gamma rays in this energy range can be indirectly detected on the ground using arrays of imaging atmospheric Cherenkov telescopes (IACTs), which detect the Cherenkov light emitted from air showers produced by VHE gamma rays when they are absorbed by the atmosphere.",
"A wide variety of scientific studies can be performed with VHE gamma rays [1].",
"VHE gamma rays are observed from supernova remnants and pulsar wind nebulae in the Milky Way and supermassive black holes in distant galaxies, providing insight into the nature of these sources, such as how and where in these sources particles are accelerated to relativistic energies.",
"Astrophysicists also search for VHE gamma-ray emission from dark-matter-dominated objects such as dwarf galaxies, looking for gamma rays hypothesized to be produced by dark matter annihilation or decay.",
"In addition, IACTs play a key role in multimessenger astronomy, regularly searching for VHE emission produced by gamma-ray bursts and by the sources of gravitational wave events, and having recently detected TeV gamma-ray emission from a flaring blazar coincident with a highly energetic neutrino detected by the IceCube Neutrino Observatory [2].",
"Figure: Left: An example IACT image from a CTA FlashCam camera simulation, illustrating the hexagonally spaced grid of pixels typical of many IACT cameras.",
"Right: The same image mapped to a square matrix of pixels by rebinning, which preserves the image's overall amplitude.",
"Both images are from .Measurements with IACTs enable these scientific studies by extracting information about VHE particles from the air showers they produce in the atmosphere.",
"In a conventional IACT analysis, images from multiple telescopes are parameterized and stereoscopically combined to extract the spatial, temporal, and calorimetric information of the originating VHE particle."
],
[
"Gamma-ray Image Analysis",
"The sensitivity of IACTs depends strongly on efficiently rejecting the background of much more numerous cosmic-ray showers, which resemble those produced by gamma rays but tend to have a more complex morphology.",
"Using the information contained in the shapes of the shower images is therefore critical to maximizing IACT sensitivity.",
"Supervised learning algorithms, like random forests and boosted decision trees, have been shown to effectively classify IACT events based on event-level parameters constructed using images from multiple telescopes (e.g.",
"[4]).",
"Deep learning techniques, such as convolutional neural networks (CNNs), may be used to improve on these methods because they do not require the images to be parameterized and may therefore access features of these images that would be washed out by the parameterization [5].",
"A deep learning approach that combines CNNs with a recurrent neural network (RNN) has been shown to improve background rejection performance using data from the H.E.S.S.",
"IACT array [6].",
"In previous work, the input images to such a network have been sorted by total amplitude.",
"In this study, we apply a similar model to simulated data from the Cherenkov Telescope Array (CTA) [7], the next-generation observatory for gamma-ray astronomy, to determine the effect of this sorting procedure on classification performance."
],
[
"CTLearn",
"We implement our neural network model using CTLearnhttps://github.com/ctlearn-project/ctlearn [8], an open-source Python package for using deep learning to analyze pixel-wise camera data from arrays of IACTs.",
"CTLearn provides an application-specific framework for configuring and training machine learning models with TensorFlowhttps://www.tensorflow.org and applying the trained models to generate predictions on a test set [9].",
"CTLearn v0.3.0 was used for training the models used in this work.",
"Through the associated DL1-Data-Handler package [10], CTLearn can load and preprocess IACT data from any major current- or next-generation IACT.",
"In particular, because many IACT cameras have pixels arranged in a hexagonal layout, posing a challenge for convolutional neural networks that conventionally require as input a rectangular matrix of input pixels, DL1-Data-Handler provides a number of methods to map hexagonally spaced pixels to a square grid.",
"In this work, the rebinning method was chosen (Fig.",
"REF ), which is one of several mapping methods that provide comparably good performance [3].",
"Figure: Diagram of the CNN-RNN particle classification model implemented in CTLearn, from .",
"The model uses a CNN block (labeled as a deep convolutional network or DCN) to derive a vector representation of each image in an event.",
"The vectors are combined using a Long Short Term Memory network (LSTM), a type of recurrent neural network (RNN)."
],
[
"CNN-RNN Particle Classification Model",
"A challenge of using deep learning methods with IACT data is combining images from multiple telescopes providing different views of an air shower event.",
"Each event triggers multiple telescopes, and the number of triggered telescopes may vary from event to event.",
"One approach to deal with this challenge is to break the problem into two stages.",
"First, each image is processed into a vector representation by a CNN, using the same weight parameters for each image.",
"The vectors are then combined by a recurrent neural network (RNN), a type of neural network that takes as input a sequence of vectors, and, by maintaining an internal state, produces an output vector that depends not only on the most recent input but on all preceding inputs in the sequence.",
"This vector is then fed into a set of densely connected layers that produce the final prediction.",
"Connecting these networks allows a single model trained end-to-end to classify events consisting of images from multiple telescopes.",
"For this work, the built-in CNN-RNN model of CTLearn was used, which implements an architecture similar to the CRNN network presented in [6].",
"More details on the model and the default hyperparameter settings that were used can be found in [9].",
"The RNN in this model is specifically a Long Short-Term Memory (LSTM) network.",
"Recurrent neural networks are capable of processing sequential data in which the ordering of inputs may affect their interpretation.",
"Therefore, having a meaningful ordering of telescope images in a CNN-RNN network may improve performance.",
"In previous work using a CNN-RNN network for classifying Cherenkov air showers as produced by a gamma ray or a cosmic-ray proton, the telescope images were ordered by total image amplitude, or size.",
"As size can be considered to be a proxy for proximity to the shower center, sorting on this parameter may provide an ordering given the absence of temporal information [6].",
"To understand the effect of this ordering on performance, we trained two CNN-RNN networks as described above to classify IACT images as produced by a gamma ray or a cosmic-ray proton, changing only the ordering of the input images.",
"As a control, in one network the images were ordered by telescope ID number, an arbitrary but consistent ordering, while in the other the images were ordered by size.",
"The networks were trained using a sample of 250,000 simulated events from 25 FlashCam telescopes [11], part of a proposed CTA array in Paranal, Chile.",
"Ten percent of the events in the sample were reserved as a validation set, which was not used for training.",
"Figure: Validation accuracy of the CNN-RNN model with images ordered by ID (dark blue) and total brightness (light blue) as a function of number of training steps (batches of 16 events).",
"The models reach respective accuracies of 80.6% and 80.2%.Figure: Validation AUC with images ordered by ID (dark blue) and total brightness (light blue) as a function of number of training steps (batches of 16 events).",
"AUC is the numerically integrated area under the receiver operating characteristic curve, measuring sensitivity and specificity.",
"The models reach respective AUCs of 0.899 and 0.894."
],
[
"Results and Discussion",
"The results of this experiment are shown in Fig.",
"REF and Fig.",
"REF .",
"The validation metrics of the two models were approximately the same, with those of the control model being slightly higher.",
"The control model attained validation accuracy and AUC of 80.6% and 0.899, while the model with images sorted by size reached 80.2% and 0.894.",
"We therefore find no evidence that sorting images by size improves gamma-proton classification performance with a CNN-RNN model.",
"This finding leaves open the possibility that a different ordering of telescope images could result in improved performance.",
"In particular, an ordering which provides sufficient information about the telescopes' position on the ground could help a CNN-RNN to perform stereoscopic reconstruction of Cherenkov air showers.",
"While ordering by size as a proxy for distance to the shower center should provide some relative position information, it is possible this information is too incomplete to be useful to the network.",
"In addition to performing background rejection, deep learning algorithms could be used to determine the arrival direction and energy of the particles initiating Cherenkov air showers [12], tasks for which stereoscopic reconstruction is particularly important.",
"Ensuring that telescope position information is effectively provided to CNN-RNN networks may therefore not only improve their performance on background rejection but also on additional tasks critical for IACT image analysis."
]
] | 2001.03602 |
[
[
"Chance-constrained optimal inflow control in hyperbolic supply systems\n with uncertain demand"
],
[
"Abstract In this paper, we address the task of setting up an optimal production plan taking into account an uncertain demand.",
"The energy system is represented by a system of hyperbolic partial differential equations (PDEs) and the uncertain demand stream is captured by an Ornstein-Uhlenbeck process.",
"We determine the optimal inflow depending on the producer's risk preferences.",
"The resulting output is intended to optimally match the stochastic demand for the given risk criteria.",
"We use uncertainty quantification for an adaptation to different levels of risk aversion.",
"More precisely, we use two types of chance constraints to formulate the requirement of demand satisfaction at a prescribed probability level.",
"In a numerical analysis, we analyze the chance-constrained optimization problem for the Telegrapher's equation and a real-world coupled gas-to-power network."
],
[
"Introduction",
"In recent years, significant attention has been paid to the energy market.",
"On the one hand, this is due to climate protection policies.",
"On the other hand, the decision of the German government on the nuclear phase-out by the end of 2022https://www.bmwi.de/Redaktion/EN/Artikel/Energy/nuclear-energy-nuclear-phase-out.html , last checked: 4th of April, 2019 will cause significant changes in the energy sector.",
"Those changes are to a large extent triggered by the specification to have $65\\%$ of the gross electricity consumption generated out of renewable energy sources by the end of 2030.",
"One major challenge is the handling of uncertainty in the renewable energy production.",
"It heavily depends on the weather conditions and is subject to large fluctuations.",
"Those fluctuations can heavily affect the grid operation or even lead to outages ([5]).",
"Another source of uncertainty in the power grid operation are power demands (see [50]).",
"This makes it difficult to guarantee a sufficient supply thereby influencing reliability and profitability in power system operation (see [19]).",
"It is crucial for all players in the electricity sector to cope with stochastic demand fluctuations.",
"In the context of gas-to-power, those fluctuations might carry over to the gas network being coupled to the electricity grid.",
"The amount of gas converted to power and feeded into the electricity system depends on the power demand and therefore inherits its stochasticity.",
"Furthermore, the gas demand itself is uncertain (see e.g.",
"[27]).",
"There are several ways to formulate optimization problems in the presence of uncertainty.",
"As in [21], we distinguish between robust optimization and chance constrained optimization (CCOPT).",
"A robust formulation is appropriate if the distribution of the uncertain parameter in the system is unknown because it cannot be measured or observed (no historical data), whereas a probabilistic programming (CCOPT) is suitable in the presence of historical observations where a distribution of the unknown quantity can be derived (see [21]).",
"In this paper, we focus on uncertain demands.",
"In this context, it is reasonable to assume access to historical demand data and we therefore focus on CCOPT for the remainder of the present manuscript.",
"Chance constraints have been introduced in 1958 in [7] by Charnes, Cooper and Symonds in the context of production planning, and further developed in [6].",
"A lot of work has been done in CCOPT.",
"It is worthwhile to particular mention the contributions of Prékopa.",
"His monograph on stochastic programming from 1995 [38] nowadays still serves as a standard reference.",
"A general overview on properties, solution methods, and fields of application such as hydro reservoir management ([2], [46]), optimal power flow ([5], [50]), energy management ([45]), and portfolio optimization ([39]) can be found in [19].",
"In this work, we are interested in a constrained optimization problem composed of three constraints: a stochastic differential equation (SDE) that models the uncertain demand, a system of hyperbolic balance laws to describe the energy system (electricity or gas), and a chance constraint ensuring demand satisfaction with a certain probability (risk level).",
"Chance constraints in the context of PDE-constrained optimization are also considered in [34].",
"However, the requirement of a continuously Fréchet-differentiable solution of the PDE rules out most hyperbolic PDEs due to the possibility of discontinuous solutions even for continuously Fréchet-differentiable initial data.",
"In [15], they apply their results on semi-continuity, convexity, and stability in an infinite-dimensional setting to a simple PDE-constrained control problem.",
"Again, the PDE is not of hyperbolic nature.",
"There already exist a few investigations of CCOPT for systems of hyperbolic nature such as gas networks.",
"Accounting for the stochasticity of demand, [25] presents a method to compute the probability of feasible loads in a stationary, passive gas network.",
"Note that the steady state assumption on the network entails constraints formulated in terms of purely algebraic equations instead of full hyperbolic dynamics.",
"In [21], they extend the aforementioned work by also allowing for uncertainty with respect to the roughness coefficient.",
"They analyze the question of the maximal uncertainty allowed such that random loads can be satisfied at a prescribed probability level within a stationary, passive gas network.",
"The feasibility of random loads, again in a stationary gas model, this time with active elements in terms of compressor stations is considered in [26].",
"In [1], they also account for the transient case.",
"However, they do not take into account the full hyperbolic dynamic but the linear wave equation instead.",
"In contrast to the above-mentioned contributions, here, we consider the full isentropic gas dynamics.",
"Note however that the full stochasticity of the demand in the power-to-gas setting presented in Section REF does not enter the nonlinear dynamics of the gas flow directly but is coupled to it via the objective function.",
"Another important difference to the aforementioned contributions is the modelling of the uncertain demand.",
"In many cases, (truncated) Gaussian random vectors are used as proposed in [27] and used in for example [1], [21], [26].",
"Using stochastic processes to model the demand enables to capture time dynamics.",
"Sometimes the demand is modeled by a discrete time stochastic process as e.g.",
"in [2].",
"In [46], they use a continuous time stochastic process additively decoupled in a deterministic trend and a causal process generated by Gaussian innovations.",
"In contrast to that, we consider a dynamic demand model for the random loads via a continuous time stochastic process, the so-called Ornstein-Uhlenbeck process, where the deterministic trend and the stochastic evolution of the process are coupled.",
"This is also a popular choice to model demand in various fields (see e.g.",
"[3] in the context of electricity).",
"In [37], and [48], they use a multivariate Ornstein-Uhlenbeck process from the supply-side point of view.",
"They model uncertain injections into the power network and assess the probability of outages.",
"A desirable feature for both supply and demand modeling is the mean reverting property.",
"The process is always attracted to a certain predefined mean level, which may result from historical supply/demand data.",
"In contrast to those contributions, we use a time-dependent mean level, which enables us to depict seasonal and daily patterns appearing in historical data also in our mean-demand level (no longer assumed to be constant) and set up an optimization framework based thereon.",
"The paper is organized as follows: in Section , we introduce the stochastic optimal control setting.",
"We define the energy system as well as the stochastic demand process, and introduce the considered cost functional.",
"We distinguish between a single chance constraint (SCC) as well as a joint chance constraint (JCC).",
"In Section , we consider a deterministic reformulation of both the cost functional and the SCC, and present a stochastic reformulation of the JCC, which allows to incorporate it in our numerical framework.",
"In Section , we validate the numerical framework for a simple case of a hyperbolic supply system, i.e.",
"the scalar linear advection with source term.",
"It represents a suitable test case setting as we are able to derive the corresponding analytical solution in case of an SCC.",
"We then apply our numerical routine to the Telegrapher's equations, a linear system of hyperbolic balance laws.",
"We conclude with a numerical investigation of a nonlinear system of hyperbolic balance laws in terms of a real-world gas-to-power application."
],
[
"Stochastic optimal control setting",
"In this section, we set up the mathematical framework for the task of finding an optimal injection plan taking into account an uncertain demand for the time period from $t_0=0$ to final time $T$ .",
"This has been done for a linear transport equation in [23].",
"Here, we use the stochastic optimal control framework set up in [23], and extend it to more complex supply dynamics on a network.",
"We model a general energy system by a system of hyperbolic balance laws on a network, and the stochastic demand is described by an Ornstein-Uhlenbeck process (OUP)."
],
[
"Energy system with uncertain demand",
"We consider different types of 2-dimensional energy systems.",
"The network is modeled by a finite, connected, directed graph $\\mathcal {G}=(\\mathcal {V},\\mathcal {E})$ with a non-empty vertex (node) set $\\mathcal {V}$ and a non-empty set of edges $\\mathcal {E}$ .",
"For $v \\in \\mathcal {V}$ , we define the set of all incoming edges by $\\delta ^{-}(v) = \\lbrace e \\in \\mathcal {E}: e=(\\cdot ,v) \\rbrace $ , and the set of all outgoing edges by $\\delta ^{+}(v)=\\lbrace e \\in \\mathcal {E}: e=(v,\\cdot ) \\rbrace $ .",
"In the sequel, we identify an edge $e=(v_{in},v_{out})$ by the interval $[a_e,b_e]$ , where $a_e$ denotes the starting point of the edge, and $b_e$ its end point.",
"We further define the set of inflow vertices by $\\mathcal {V}_{in}=\\lbrace v\\in \\mathcal {V}: \\delta ^{-}(v) = \\emptyset \\rbrace $ , and the set of outflow vertices by $\\mathcal {V}_{out}=\\lbrace v\\in \\mathcal {V}: \\delta ^{+}(v) = \\emptyset \\rbrace $ .",
"As a simplification, here, we restrict our network to $|\\mathcal {V}_{in}|=|\\mathcal {V}_{out}|=1$ .",
"Note however that an extension to several inflow nodes ($|\\mathcal {V}_{in}|>1$ ), and outflow nodes ($|\\mathcal {V}_{out}|>1$ ) is straightforward by considering a vector-valued inflow control, and a multivariate Ornstein-Uhlenbeck process as used in [48].",
"The inflow control $u(t)$ acts on the vertex $v_{in} \\in \\mathcal {V}_{in}$ , and the demand $Y_t$ is realized at the vertex $v_{d} \\in \\mathcal {V}_{out}$ .",
"We require $b^{e}=L$ for all $e \\in \\delta ^{-}(v_{d})$ .",
"Figure: Example of an energy network with uncertain demand at v d v_dThe dynamics of the energy system on one edge $e$ are given by the following hyperbolic balance law with initial condition (IC) and boundary conditions (BCs) $\\partial _t q^{e}+ \\partial _x f^{e}(q^{e}) &= s(q^{e}), \\quad x \\in [a^{e}, b^{e}], \\ t \\in [0,T] \\\\q^{e}(x,0) &= q_0^{e}(x), \\\\\\Gamma _a^{e}(q^{e}(a_e,t)) &= \\gamma _a^{e}(t), \\quad \\Gamma _b^{e}(q^{e}(b_e,t)) = \\gamma _b^{e}(t).",
"$ Thereby, $\\mathbb {R}^{2} \\rightarrow \\mathbb {R}^2$ is a given flux function and $s:\\mathbb {R}^{2} \\rightarrow \\mathbb {R}^{2}$ the source term.",
"$\\rho _0^{e}: [a^{e}, b^{e}]\\rightarrow \\mathbb {R}^{2}$ describes the initial state of the system on edge $e$ .",
"The functions $\\Gamma _{a/b}^{e}:\\mathbb {R}^2\\rightarrow \\mathbb {R}^{m_{l/r}}$ enable to prescribe a certain evaluation of a certain number of components of the density at the boundary.",
"Thereby, $0 \\le m_l,m_r \\le 2$ denote the number of prescribed BCs at the left respectively right boundary.",
"A value of $m_{l/r}=0$ has to be interpreted in a way that no left/right boundary condition is prescribed.",
"For a scalar conservation law, choosing $m_l=1$ , $m_r=0$ , and $\\Gamma _a^{e}= corresponds to prescribing only the flow at the left boundary.$ The boundary conditions (BCs) themselves are given in terms of the functions $\\gamma _a^{e}: [0,T] \\rightarrow \\mathbb {R}^{m_l}$ at the left boundary $a_e$ of edge $e$ , and $\\gamma _b^{e}: [0,T] \\rightarrow \\mathbb {R}^{m_r}$ at the right boundary $b_e$ of edge $e$ .",
"The numbers of BCs $m_{l/r}$ , and the functions $\\gamma _{a/b}^{e}$ need to be chosen carefully such that they are consistent with the characteristics of the conservation law.",
"This will be further specified below for each setting REF -REF .",
"Moreover, for the non-degenerate network case, i.e.",
"$\\exists v \\in \\mathcal {V}: |\\delta ^{-}(v)|+|\\delta ^{+}(v)|\\ge 2$ , suitable coupling conditions $c_v: \\mathbb {R}^{2\\times \\left(|\\delta ^{+}(v)|+|\\delta ^{-}(v)|\\right)} \\rightarrow \\mathbb {R}^{m_c}$ for each vertex need to be imposed.",
"Thereby, $m_c$ denotes the number of coupling conditions.",
"This will be made explicit in Subsection REF .",
"For $e \\in \\delta ^{+}(v_{in})$ , the function $\\gamma _a^{e}(t)$ depends on the inflow control $u(t)$ .",
"To simplify notation, we do not explicitly write down the dependence of the supply on the density at the end points of all ingoing edges and denote the supply at $v_d$ at time $t$ simply by $S_{u(t)}$ meaning $S_{u(t)}=S_u\\left(_{e \\in \\delta ^{-}(v_d)} q^{e}(b_e,t)\\right).$"
],
[
"Discretization scheme",
"For the numerical investigation in Section , the considered hyperbolic energy systems need an appropriate discretization scheme.",
"Motivated by our real-world example, we choose an implicit box scheme (IBOX) [32] for all considered scenarios.",
"For a general system of balance laws (on any edge) $\\partial _t q + \\partial _x f(q) = s(q),$ the considered scheme reads $\\frac{Q^{n+1}_{j-1} + Q^{n+1}_{j}}{2} = \\frac{Q^{n}_{j-1} + Q^{n}_{j}}{2} - \\frac{\\Delta t}{\\Delta x}\\left( f(Q^{n+1}_j)- f(Q^{n+1}_{j-1}) \\right) + \\Delta t \\frac{s(Q^{n+1}_j)+s(Q^{n+1}_{j-1})}{2}.",
"$ Here, $\\Delta t$ and $\\Delta x$ are the temporal and spatial mesh size, respectively, and the numerical approximation is thought in the following sense: $Q_j^n \\approx q(x,t) \\quad \\text{for} \\quad x\\in X_j=\\big [ a+(j-\\tfrac{1}{2})\\Delta x , a+(j+\\tfrac{1}{2})\\Delta x \\big ) \\cap \\big [ a , b \\big ] , \\ t \\in I_i=\\big [ i\\Delta t , (i+1)\\Delta t \\big ).$ To avoid undesired boundary effects, the discretization of the initial condition on bounded domains is done pointwise, i.e., $Q_j^n = q_0(a + j \\Delta x).$ As remarked in [32], for a discretization $x_l<x_{l+1}<\\cdots <x_{r-1}<x_r$ , we obtain $r-l$ equations for $r-l+1$ variables (in the scalar case).",
"This entails the need to prescribe BCs at exactly one boundary specified by the characteristic direction.",
"This also explains the above mentioned assumption of no change in the signature of the characteristic directions on the considered domain.",
"The discrete version of the BCs is given by $\\Gamma _a(Q_0^i) = \\gamma _a(t_i), \\quad \\Gamma _b(Q_{N_{\\Delta x}}^{i}) = \\gamma _b(t_i) \\quad \\forall \\ i \\in \\lbrace 1,\\cdots ,N_{\\Delta t}\\rbrace .",
"$ Note that the implicit box scheme has to obey an inverse CFL condition [32], which is beneficial for problems with large characteristic speeds whereas the solution is merely quasi-stationary.",
"This is usually the case for daily operation tasks in gas networks and therefore motivates the choice within this work.",
"The uncertain demand stream is modeled by an Ornstein-Uhlenbeck process (OUP), which is the unique strong solution of the following stochastic differential equation $dY_t = \\kappa \\left(\\mu (t)-Y_t\\right)dt + \\sigma dW_t,\\ \\quad Y_{t_0}=y_{t_0} $ on the probability space $\\left(\\Omega ,\\mathcal {F},P\\right)$ .",
"$W_t$ is a given one-dimensional Brownian motion on the same probability space, and $y_0$ describes the demand at time $t_0=0$ .",
"The constants $\\sigma >0$ and $\\kappa >0$ describe the speed of mean reversion and the intensity of demand fluctuations.",
"By mean reversion, we refer to the property of the process that it is always attracted by a certain time-dependent level $\\mu (t)$ , called the mean demand level.",
"This is due to the sign of the drift term $\\kappa \\left(\\mu (t)-Y_t\\right)$ , which ensures that being above (below) the mean demand level, the process experiences a reversion back to it.",
"By the OUP (REF ), we are able to capture daily or seasonal patterns in the demand.",
"From a mathematical point of view, this process has some nice analytical properties.",
"For example, we can derive the solution to equation (REF ) explicitly via the Itô formula.",
"It reads as $Y_t &= y_{t_0} e^{-\\kappa (t-t_0)} + \\kappa \\int _{t_0}^t{e^{-\\kappa (t-s)} \\mu (s)ds} + \\sigma \\int _{t_0}^t{e^{-\\kappa (t-s)}dW_s}.",
"$ Moreover, it is possible to derive its distribution explicitly as $Y_t &\\sim N {\\underbrace{\\left(y_{t_0}e^{ - \\kappa (t-t_0)} + \\kappa \\int \\limits _{t_0}^t {e^{ - \\kappa \\left( {t - s} \\right)} \\mu \\left( s \\right)ds}\\right.",
"}_{m_{OUP}(t)} ,\\,\\,\\underbrace{\\left.",
"\\vphantom{y_{t_0}e^{ - \\kappa (t-t_0)} + \\kappa \\int \\limits _{t_0}^t {e^{ - \\kappa \\left( {t - s} \\right)} \\mu \\left( s \\right)ds}}\\frac{\\sigma ^2}{2\\kappa }\\left(1 - e^{ - 2\\kappa \\left( {t - t_0} \\right)}\\right)\\right)}_{v_{OUP}(t)} \\,}.",
"$ For further details on the demand process, we refer the reader to [23]."
],
[
"Chance constraints",
"Requiring demand satisfaction for every realization of the demand process might be too restrictive as, in some cases, it might lead to an infeasible optimization problem ([41]).",
"One possibility to overcome problems of infeasibility and to reduce the average undersupply is to introduce an undersupply penalty term in the cost function.",
"The effect of an undersupply penalty on the optimal output has been analyzed in [24].",
"A comparison of different types of undersupply can be found in [36].",
"Another approach is to guarantee with a certain probability that there is no undersupply within a prescribed time interval $I_{CC} \\subset [t^{*},T]$ , where $t^{*}$ is the first time that a supply is realized at $v_d$ .",
"Mathematically this is formulated in terms of a chance constraint (CC).",
"One possibilty is to require at each point in time that the probability of a demand satisfaction is at least equal to one minus a given risk level $\\theta $ (see (REF )).",
"This results in a so called single chance constraint (SCC).",
"Another possibility is a joint chance constraint (JCC), that is, we require that the probability of a demand satisfaction is at least equal to one minus a given risk level $\\theta $ on a whole interval simultaneously (see ()).",
"$P\\left(Y_t \\le S_{u(t)}\\right) &\\ge 1-\\theta \\ \\forall \\ t \\in I_{CC}, \\\\P\\left(Y_t \\le S_{u(t)}\\ \\forall \\ t \\in I_{CC}\\right) &\\ge 1-\\theta .",
"$"
],
[
"Objective function and stochastic optimal control problem",
"Having formulated all the optimization constraints, we now address the objective function.",
"We aim at minimizing the arising costs.",
"We construct our cost function out of several components.",
"One component consists of deterministic costs $C_{u(t)}$ such as operating costs for a gas compressor ($C1$ ).",
"Another component are tracking type costs that arise from a mismatch between the externally given demand (REF ) and our supply $S_u$ realized at the demand vertex $v_d$ .",
"We measure the tracking type costs in terms of the expected quadratic deviation between the demand and supply ($C2$ ).",
"Particularly accounting for negative mismatches, we introduce undersupply costs as a third cost component ($C3$ ).",
"If a sale of excess supply is possible, the revenue can be included into the cost function as a fourth component ($R$ ).",
"Putting the deterministic costs $C1$ , the tracking costs $C2$ , the undersupply costs $C3$ , and the excess revenue $C4$ together, we obtain a cost function of the following type: $OF(Y_t, t_0,y_{t_0} ,S_{u(t)}) =& \\underbrace{\\vphantom{\\mathbb {E}\\left[\\left(S_{u(t)}-Y_t\\right)^2|Y_{t_0}=y_{t_0}\\right]} w_1\\cdot C_{u(t)}}_{C1} + \\underbrace{w_2\\cdot \\mathbb {E}\\left[\\left(S_{u(t)}-Y_t\\right)^2|Y_{t_0}=y_{t_0}\\right]}_{C2} \\\\&- \\underbrace{\\vphantom{\\mathbb {E}\\left[\\left(S_{u(t)}-Y_t\\right)^2|Y_{t_0}=y_{t_0}\\right]} w_3\\cdot \\mathbb {E}\\left[\\left(S_{u(t)}- Y_t\\right)_{-}|Y_{t_0}=y_{t_0}\\right]}_{C3} - \\underbrace{\\vphantom{\\mathbb {E}\\left[\\left(S_{u(t)}-Y_t\\right)^2|Y_{t_0}=y_{t_0}\\right]} w_4\\cdot \\mathbb {E}\\left[\\left(S_{u(t)}- Y_t\\right)_{+}|Y_{t_0}=y_{t_0}\\right]}_{R} , $ where $\\left(S_{u(t)}- Y_t\\right)_{-} &= \\left\\lbrace \\begin{array}{ll}S_{u(t)}- Y_t & \\text{if} \\ S_{u(t)}<Y_t \\\\0 & \\text{if} \\ S_{u(t)}\\ge Y_t\\end{array}\\right.$ All together, we come up with the stochastic optimal control (SOC) problem $\\min _{u \\in \\mathcal {U}_{ad}} \\int _{t^{\\ast }}^{T} OF(Y_t; t_0;y_{t_0} ;S_{u(t)}) dt \\quad \\text{subject to} \\ (\\ref {eq:IBVP}), (\\ref {eq:OUP}),\\ \\text{and} \\ (\\ref {eq:CC}).",
"$ To determine the optimal control, we need to specify measurability assumptions on the control by defining the space of admissible controls $\\mathcal {U}_{ad}$ .",
"$\\mathcal {U}_{ad} &= \\lbrace u:[t_0,T] \\rightarrow \\mathbb {R}\\ | \\ u \\in L^2\\left([t_0,T]\\right), u(t) \\ge 0, \\ \\text{and} \\ u(t) \\ \\text{is} \\ \\mathcal {F}_{t_0}\\text{--predictable} \\ \\text{for} \\ t \\in [t_0,T]\\rbrace .",
"$ Results for this control method and two other control methods with an objective function of pure tracking type ($w_1=w_3=w_4=0$ , $w_2=1$ ) for the linear advection equation without imposing CCs can be found in [23]."
],
[
"Deterministic reformulation of the stochastic problem",
"Having set up the SOC problem (REF ), the question of how to solve this minimization problem naturally arises.",
"One way is to trace back the SOC problem (REF ) to a deterministic setting, in which we can apply well-known methods from deterministic PDE-constrained optimization such as adjoint calculus.",
"In order to do so, in REF , we analytcially treat both types of CCs presented in Subsection REF .",
"The deterministic expression of the objective function introduced in REF is derived in REF ."
],
[
"Reformulation of chance constraints",
"The reformulation of the CC heavily depends on the type of CC.",
"Whereas the SCC (REF ) can be reformulated via quantiles of a normal distribution, the reformulation of the JCC () is not obvious.",
"Therefore, we need to treat the different types of CCs separately.",
"Single chance constraint For the SCCs (REF ), we use a quantile-based reformulation as mentioned in [2].",
"By using the known distribution (REF ) of the OUP, this results in the deterministic state constraints $S_{u(t)}\\ge m_{OUP}(t) + v_{OUP}(t)\\Phi ^{-1}\\left(1-\\theta \\right) \\ \\forall t \\in I_{CC}, $ where $m_{OUP}(t)=y_{t_0}e^{-\\kappa (t-t_0)} + \\kappa \\int \\limits _{t_0}^t {e^{-\\kappa \\left( {t - s} \\right)} \\mu \\left( s \\right)ds}$ , $v_{OUP}(t)=\\sigma ^2 \\int \\limits _{t_0}^t {e^{ - 2\\kappa \\left( {t - s} \\right)}ds}$ , and $\\Phi $ is the standard normal cumulative distribution function.",
"Joint chance constraint JCCs () are mathematically by far more involved than SCCs (see [19]).",
"No longer considering the constraint pointwise in time, we now have to deal with a joint probability distribution.",
"Unfortunately, we can no longer make use of the deterministic reformulation as state constraint as for (REF ).",
"As the integration of the JCC into the optimization framework is a core issue to tackle the SOC problem (REF ), we need to come up with a different approach.",
"As in [48], we now use a non-deterministic reformulation of the JCC as a first passage time problem.",
"We denote by $\\mathcal {T}_{y_0} &= \\inf _{t\\ge t_0}\\left(t \\ |\\ Y_t > S_{u(t)}\\right)$ for $Y_{t_0}=y_{t_0}<S_{u(t_0)}$ the first passage time of the OUP, and obtain the equivalent formulation of the JCC () as a first passage time problem of the form $P(\\mathcal {T}_{y_0} > t) &\\ge 1-\\theta .",
"$ For further details on first passage times, we refer the reader to [13].",
"We will see that reformulation (REF ) enables to include constraint () into our SOC framework.",
"However, this is not obvious.",
"The drawback is that we have to deal with the distribution of the first passage time of the OUP with time-dependent mean demand level for a time-dependent absorbing boundary.",
"The time-dependency rules out some classical approaches.",
"Furthermore, even for a constant boundary, the task turns out to be by far more complicated than deriving the distribution of the first passage time of a Brownian motion (see [35], [49]).",
"A closed-form solution is only known for a few special cases.",
"One reason making the present case particularly hard is that there is no pure diffusion equation any more that would allow to apply the formula presented in the paper [11].",
"So far, to the best of our knowledge, no closed-form solution of the first passage time density in our case is known.",
"Several semi-analytical approaches exist: In [28], they derive an integral representation of the first-passage time of an inhomogeneous OUP with an arbitrary continuous time-dependent barrier and extend their results to continuous Markov processes.",
"The idea of exploiting transformations among Gauss-Markov processes to relate the problem to the known first passage time density of a Wiener process.",
"However, as stated in [12], the transformation of the OUP to the Brownian motion entails exponentially large times.",
"In an alternative approach presented in [12], they consider this broader class of real continuous Gauss-Markov process with continuous mean and covariance functions.",
"The latter approach is the one that we tailor to our time-dependent OUP.",
"In a first step, we show that the approach in [12] is applicable to our case of the time-dependent OUP, and in a second step, we introduce the method itself.",
"Verification of prerequisites As we want to apply a result on the first passage time density formulated in a general Gauss-Markov setting, we first need to verify that the time-dependent OUP given by equation (REF ) is indeed a Gauss-Markov process.",
"Lemma 3.1 The OUP given by equation (REF ) is a Gauss-Markov process.",
"We first verify that the OUP is a Gauss process, i.e.",
"that for any integer $n\\ge 1$ and times $0\\le t_1<t_2<\\cdots <t_n \\le T$ , the random vector $\\left(Y_{t_1},Y_{t_2},\\cdots ,Y_{t_n}\\right)$ has a joint normal distribution.",
"From (REF ), we know that, for any $n\\ge 1$ , and arbitrary $k \\in \\lbrace 1,\\cdots ,n\\rbrace $ , $Y_{t_k}$ is normally distributed.",
"From [4], we know that, to verify the joint normal distribution of $\\left(Y_{t_1},Y_{t_2},\\cdots ,Y_{t_n}\\right)$ , it is sufficient to show that any linear combination of $Y_{t_1},Y_{t_2},\\cdots ,Y_{t_n}$ is normally distributed.",
"This holds true due to the linearity of the Itô integral.",
"Furthermore, the drift coefficient $b(t,x) = \\kappa \\left(\\mu (t) - x\\right)$ , and the diffusion coefficient $\\sigma (t,x)\\equiv \\sigma $ of (REF ) satisfy the assumptions for existence and uniqueness of a strong solution in [18].",
"Hence, [18] is applicable, which states that the corresponding SDE, in our case (REF ), is a Markov process (see [33]) on the interval $[0,T]$ .",
"We are now in the Gauss-Markov setting of [12].",
"To prepare the numerical computation of the first passage time density, we need to calculate some basic characteristics of our OUP.",
"We recall, that $Y_t$ is normally distributed with mean $m_{OUP}(t) &= y_{t_0}e^{ - \\kappa (t-t_0)} + \\kappa \\int \\limits _{t_0}^t {e^{ - \\kappa \\left( {t - s} \\right)} \\mu \\left( s \\right)ds},$ and variance given by $v_{OUP}(t) &= \\frac{\\sigma ^2}{2\\kappa }\\left(1 - e^{ - 2\\kappa \\left( {t - t_0} \\right)}\\right),$ both being $C^1\\left([0,T]\\right)$ -functions.",
"The probability density function of the OUP starting at time $t_0$ in $y_{t_0}$ coincides with a normal density with mean $m_{OUP}(t)$ , and variance $v_{OUP}(t)$ .",
"We proceed with the covariance function.",
"Note that the covariance is determined by the stochastic integral term $I_t = \\sigma \\int _{t_0}^{t} e^{-\\kappa \\left(t-s\\right)}dW_s$ in (REF ) and the deterministic part can be neglected for its calculation.",
"Moreover, note that $\\mathbb {E}\\left[I_t\\right]=0$ .",
"$\\operatorname{Cov}(Y_s,Y_t) &= \\sigma ^2\\mathbb {E}\\left[\\int _{t_0}^{s} e^{-\\kappa \\left(s-u\\right)}dW_u \\int _{t_0}^{t}e^{-\\kappa \\left(t-v\\right)}dW_v\\right] \\\\&= \\sigma ^2 e^{-\\kappa \\left(s+t\\right)} \\mathbb {E}\\left[\\int _{t_0}^{s} e^{\\kappa u}dW_u \\int _{t_0}^{t} e^{\\kappa v} dW_v\\right] \\\\&= \\sigma ^2 e^{-\\kappa \\left(s+t\\right)} \\mathbb {E}\\left[\\int _{t_0}^{t} e^{\\kappa u} {1}_{[t_0,s]}(u) dW_u \\int _{t_0}^{t} e^{\\kappa v} dW_v\\right] \\\\&= \\sigma ^2 e^{-\\kappa \\left(s+t\\right)} \\mathbb {E}\\left[\\int _{t_0}^{s} e^{2\\kappa u} du \\right] \\\\&= \\sigma ^2 \\frac{e^{-\\kappa \\left(t-s\\right)} - e^{-\\kappa \\left(s + t -2t_0\\right)}}{2\\kappa } $ The covariance function (REF ) can be decomposed in $\\operatorname{Cov}\\left(Y_s,Y_t\\right) &= \\underbrace{ \\vphantom{\\frac{\\sigma ^2}{2\\kappa }} e^{\\kappa s} \\left(1 - e^{-\\kappa \\left(2s - 2t_0\\right)}\\right)}_{h_1(s)} \\cdot \\underbrace{\\frac{\\sigma ^2}{2\\kappa }e^{-\\kappa t}}_{h_2(t)}.$ Also, the functions $h1$ , and $h_2$ are elements of $C^1\\left([0,T]\\right)$ , and their derivatives are $h_1^{\\prime }(t) &= \\kappa e^{\\kappa t} + \\kappa e^{-\\kappa \\left(t-2t_0\\right)}, \\ \\text{and} \\\\h_2^{\\prime }(t) &= -\\frac{1}{2}\\sigma ^2e^{-\\kappa t}.$ Numerical calculation of the first passage time density In [12], it is shown in a first step that the first passage time density for a $C^1$ -barrier satisfies a non-singular Volterra second-kind integral equation.",
"In a second step, this equation is iteratively solved by a repeated Simpson's rule yielding an approximation to the desired first passage time density in discretized form.",
"Below, we state Theorem 3.1. of [12] adapted to our OUP (REF ) and our notation.",
"Theorem 3.2 Let $S(t)$ be a $C^1\\left([0,T]\\right)$ -function.",
"Then, the first passage time density $g\\left(S(t),t|y_{t_0},t_0\\right) = \\frac{\\partial }{\\partial _t} P\\left(\\mathcal {T}_{y_0}<t\\right)$ solves the non-singualr second-kind Volterra integral equation given by $g\\left(S(t),t|y_{t_0},t_0\\right) &= -2\\Psi \\left(S(t),t|y_{t_0},t_0\\right) + 2\\int _{t_0}^{t} g\\left(S(t)s,s|y_{t_0},t_0\\right) \\Psi \\left(S(t),t|S(s),s\\right) ds, \\quad y_{t_0}<S(t_0).",
"$ Thereby, the function $\\Psi $ is defined via $\\Psi \\left(S(t),t|y,s\\right) =& \\left(\\frac{S^{\\prime }(t)-m_{OUP}^{\\prime }(t)}{2} - \\frac{S(t) - m_{OUP}(t)}{2} \\frac{h_1^{\\prime }(t)h_2(s) - h_2^{\\prime }(t)h_1(s)}{h_1(t)h_2(s) - h_2(t)h_1(s)} \\right.\\\\&\\left.- \\frac{y - m_{OUP}(t)}{2} \\frac{h_2^{\\prime }(t)h_1(t) - h_2(t)h_1^{\\prime }(t)}{h_1(t)h_2(s) - h_2(t)h_1(s)}\\right)\\cdot p_{y,s}(S(t),t).$ We adopt the notational short cuts introduced in [12]: $g(t) &:= g\\left(S(t),t|y_{t_0},t_0\\right), \\quad t,t_0 \\in [0,T], t_0<t \\\\\\Psi (t) &:= \\Psi \\left(S(t),t|y_{t_0},t_0\\right), \\quad t,t_0 \\in [0,T], t_0<t \\\\\\Psi \\left(t|s\\right) &:= \\Psi \\left(S(t),t|S(s),s\\right) \\quad t,s \\in [0,T], t_0<s \\le t.$ We introduce a grid $t_0<t_1<...<T_N$ , where $t_k = t_0 + k\\cdot \\Delta t, k \\in \\lbrace 1,\\cdots ,N\\rbrace $ .",
"The iterative procedure based on the repeated Simpson's rule to obtain an approximation $\\tilde{g}(t_k)$ of the first passage time density $g(t_k)$ reads as follows: $\\tilde{g}(t_1) &= -2\\Psi (t_1), \\\\\\tilde{g}(t_k) &= -2\\Psi (t_k) + 2\\Delta t\\sum _{j=1}^{k-1} w_{k,j} \\tilde{g}(t_j)\\Psi (t_k|t_j), \\quad k=2,3,\\cdots ,N $ where the weights are specified by $w_{2n,2j-1} &= \\frac{4}{3}, \\quad j=1,2,\\cdots ,n; n=1,2,\\cdots ,\\frac{N}{2}, \\\\w_{2n,2j} &= \\frac{2}{3}, \\quad j=1,2,\\cdots ,n-1; n=2,3,\\cdots ,\\frac{N}{2}, \\\\w_{2n+1,2j-1} &= \\frac{4}{3}, \\quad j=1,2,\\cdots ,n-1; n=2,3,\\cdots ,\\frac{N}{2}-1, \\\\w_{2n+1,2j} &= \\frac{2}{3}, \\quad j=1,2,\\cdots ,n-2; n=3,4\\cdots ,\\frac{N}{2}-1, \\\\w_{2n+1,2(n-1)} &= \\frac{17}{24}, \\quad n=2,3,\\cdots ,\\frac{N}{2}-1, \\\\w_{2n+1,2n-1} &= w_{2n+1,2n} = \\frac{9}{8}, \\quad n=1,2,\\cdots ,\\frac{N}{2}-1.$ The iterative procedure has been proven to converge in [12].",
"Theorem 3.3 We shall be given the above discretization $t_0<t_1<...<t_N$ , where $t_k = t_0 + k\\cdot \\Delta t, k \\in \\lbrace 1,\\cdots ,N\\rbrace $ , where $\\Delta t$ is the discretization step.",
"The first passage time density obtained by the iterative procedure (REF ) converges to the true first passage time density as the step size tends to zero, i.e.",
"$\\lim \\limits _{\\Delta t \\rightarrow 0} |g(t_k)-\\tilde{g}(t_k)| = 0 \\quad \\text{for all} \\ k \\in \\lbrace 1,\\cdots ,N\\rbrace $ To use the iteratively approximated first passage time density to obtain the risk level corresponding to $S(t)$ , we apply Algorithm REF .",
"Algorithm to calculate the risk level corresponding to $S(t)$ [1] OUP characteristics: mean $m_{OUP}(t)$ , variance $v_{OUP}(t)$ , covariance decomposition in terms of $h_1(t),h_2(t)$ , and its derivatives $h_1^{\\prime }(t),h_2^{\\prime }(t)$ ; boundary $S(t)$ ; discretization step $\\Delta t$ , final active time $T_{CC}$ of CC; risk level $\\theta $ First passage time density $g$ , and cumulative distribution function $G$ Define parameters of OUP.",
"Choose time discretization $\\Delta t$ .",
"Calculate $m_{OUP}(t),h_1(t),h_2(t),S(t),m_{OUP}^{\\prime }(t),h_1^{\\prime }(t),h_2^{\\prime }(t),S^{\\prime }(t)$ for discretized time interval $[0,T_{CC}]$ with step size $\\Delta t$ .",
"Calculate approximation of first passage time density via (REF ).",
"Calculate the corresponding discrete values of the cumulative distribution function $G$ .",
"$G(end)<= \\theta $ JCC fulfilled.",
"JCC violated.",
"$g$ and $G$ .",
"It enables to integrate the JCC () in our optimization procedure to solve the SOC problem (REF ).",
"In our case the optimal inflow control will take the role of the $C^1$ -boundary $S(t)$ .",
"As this control is a result of the optimization procedure, the calculation of the first passage time density needs to be repeated in every optimization iteration.",
"It is therefore worthwhile to mention that the above introduced iterative procedure is well-suited for computational efficiency.",
"This is because the algorithm only requires the characteristics of the OUP in terms of its initial data $(t_0,y_0)$ , its mean $m_{OUP}(t)$ , its variance $v_{OUP}(t)$ , its covariance decomposition in terms of the functions $h_1$ and $h_2$ as well as a prespecified boundary $S(t)$ and a chosen discretization step $\\Delta t$ .",
"No Monte Carlo (MC) methods, and no high-dimension integral computations are involved and no particular software packages are necessary (see [12])."
],
[
"Validation of first passage time simulation",
"For a validation of Algorithm REF , we consider the following test case setting: $t_0=0, \\kappa ={1}{3600}, \\sigma =0.003, \\mu (t) = 0.7 + 0.3 \\cdot \\sin ({1}{7200}\\ \\pi t), y_0=0.8, S(t_i)=m_{OUP}(t_k) + 0.2 + 0.25\\cdot {t_k}{T}$ , where $t_k, k\\in \\lbrace 1,\\cdots ,N_{\\Delta t}\\rbrace $ are the discretization points, and $N_{\\Delta t}+1$ denotes the number of discretization points for the interval $[0,T]$ that corresponds to the chosen time step $\\Delta t$ .",
"We denote by $M$ the number of MC repetitions used.",
"In Table REF , we show the risk of hitting the boundary $S(t)$ based on a MC simulation (risk-MC) and based on the evaluation of the cumulative distribution function $G$ from Algorithm REF (risk-fptd), and calculate the differences (diff).",
"We observe that our algorithm REF gives rather precise results already for large step sizes.",
"This is beneficial when using it within the optimization of large gas networks in Subsection REF .",
"The MC-risk values are obtained using the plain MC method.",
"Note the rare event character of undersupply.",
"This is even more pronounced in real world settings where most likely the chosen risk tolerance is $5\\%$ or lower instead of values around $15\\%$ in our test case.",
"Therefore, a rare event simulation technique as e.g.",
"in [48] in the context of power flow reliability, where the probability of an outage is very small, might lead to more accurate risk estimates even for larger step sizes.",
"However, even with the plain MC method, we observe that the MC-risk and the risk-fptd values approach up to a precision in the range of $10^3$ indicating the correct functioning of the algorithm.",
"Table: Comparison of risk-MC and risk values based on Algorithm"
],
[
"Reformulation of the cost function",
"We benefit from an analytical treatment of the cost function that has been set up in [23] for the tracking type costs $C2$ of (REF ).",
"We use the slightly more general formulation of it allowing for arbitrary $t_0$ instead of only $t_0=0$ : $\\mathbb {E}\\left[\\left(Y_t-S_{u(t)}\\right)^2|Y_{t_0}=y_{t_0}\\right] =& y_{t_0}^2e^{-2\\kappa (t-t_0)} + 2y_0e^{-\\kappa (t-t_0)} \\int _{t_0}^{t}e^{-\\kappa (t-s)}\\kappa \\mu (s) ds + \\left(\\kappa \\int _{t_0}^{t}e^{-\\kappa (t-s)} \\mu (s) ds\\right)^2 \\\\&+ \\frac{\\sigma ^2}{2\\kappa }\\left(1-e^{-2 \\kappa (t-t_0)}\\right) - 2y(t) \\cdot \\left( y_0 e^{-\\kappa (t-t_0)} + \\kappa \\int _{t_0}^{t} e^{-\\kappa (t-s)}\\mu (s) ds \\right) + y(t)^2.",
"$ It remains to extend the approach for the undersupply cost component $C3$ and the excess supply revenue component $R$ .",
"To this end, we make use of the truncated normal distribution.",
"The following definition is taken from [29] and adapted to our notation.",
"Definition 3.1 Let $X$ be a random variable on a probability space $\\left(\\Omega , \\mathcal {A}, P\\right)$ .",
"We say that $X$ follows a doubly truncated normal distribution with lower and upper truncation points $a$ and $b$ respectively ($X \\sim \\mathcal {N}_a^b(\\xi ,\\sigma ^2)$ ) if its probability density function is given by $\\rho _{X,a,b}(x) = \\left(\\frac{1}{\\sigma }\\varphi \\left(\\frac{x-\\xi }{\\sigma }\\right)\\left(\\Phi \\left(\\frac{b-\\xi }{\\sigma }\\right)-\\Phi \\left(\\frac{a-\\xi }{\\sigma }\\right)\\right)^{-1}\\right){1}_{[a,b]}(x),$ where $\\varphi $ is the density, and $\\Phi $ the cumulative distribution function of a standard normally distributed random variable.",
"We denote by $\\xi $ and $\\sigma ^2$ the mean and the variance of the non-truncated normal distribution.",
"We call the distribution singly truncated from above respectively from below if $a$ is replaced by $-\\infty $ respectively $b$ is replaced by $\\infty $ .",
"Proposition 3.4 (cf.",
"[29]) Let $X \\sim \\mathcal {N}_a^b(\\xi ,\\sigma ^2)$ .",
"Then, the expected value of $X$ reads as $\\mathbb {E}\\left[X\\right] = \\xi + \\frac{\\varphi \\left(\\frac{a-\\xi }{\\sigma }\\right) - \\varphi \\left(\\frac{b-\\xi }{\\sigma }\\right)}{\\Phi \\left(\\frac{b-\\xi }{\\sigma }\\right) - \\Phi \\left(\\frac{a-\\xi }{\\sigma }\\right)} \\sigma .",
"$ We use equation (REF ) denoting the expectation of a truncated normally distributed random variable to reformulate the expectations in $C3$ and $R$ in (REF ).",
"For $C3$ , we obtain $\\mathbb {E}\\left[\\left(S_{u(t)}- Y_t\\right)_{-}|Y_{t_0}=y_{t_0}\\right] &= \\int _{S_{u(t)}}^{\\infty } \\left(S_{u(t)}-y\\right)\\rho _{Y_t}(y) dy \\\\&= S_{u(t)}\\cdot \\left(1-\\Phi _{Y_t}(S_{u(t)})\\right) - \\int _{S_{u(t)}}^{\\infty } y\\rho _{Y_t}(y)dy \\\\&= S_{u(t)}\\cdot \\left(1-\\Phi _{Y_t}(S_{u(t)})\\right) - \\int _{\\mathbb {R}} y\\rho _{Y_t,S_{u(t)},\\infty }(y)dy \\cdot P\\left(Y_t > S_{u(t)}\\right) \\\\&= S_{u(t)}\\cdot \\left(1-\\Phi _{Y_t}(S_{u(t)})\\right) - \\left(m_{OUP}(t) + \\frac{\\rho _{Y_t}(S_{u(t)})}{1-\\Phi _{Y_t}(S_{u(t)})}\\sqrt{V_{OUP}(t)}\\right) \\cdot \\left(1-\\Phi _{Y_t}(S_{u(t)})\\right), $ where $\\rho _{Y_t}$ denotes the density, and $\\Phi _{Y_t}$ the cumulative distribution function of $Y_t$ , and $\\rho _{Y_t,S_{u(t)},\\infty }$ denotes the density of the singly truncated random variable from below by $S_{u(t)}$ .",
"The last equation results from formula (REF ) with $a=S_{u(t)}$ , and $b=\\infty $ .",
"In a similar manner, by setting $a=-\\infty $ , and $b=S_{u(t)}$ in (REF ), for the expectation in $R$ , we have $\\mathbb {E}\\left[\\left(S_{u(t)}- Y_t\\right)_{+}|Y_{t_0}=y_{t_0}\\right]&= S_{u(t)}\\Phi _{Y_t}(S_{u(t)}) - \\left(m_{OUP}(t) - \\frac{\\rho _{Y_t}(S_{u(t)})}{\\Phi _{Y_t}(S_{u(t)})}\\sqrt{V_{OUP}(t)}\\right) \\Phi _{Y_t}(S_{u(t)}).",
"$ With equations (REF ), (REF ), and (REF ), we have a completely deterministic reformulation of our cost function (REF ) at hand and denote it by $OF_{\\text{detReform}}(Y_t, t_0,y_{t_0} ,S_{u(t)})$ .",
"This reformulation of the cost function together with the reformulation of the CC as state constraint (REF ) allows us to drop the OUP (REF ) as constraint in the optimization problem (REF ).",
"We are left with a completely deterministic PDE-constrained optimization problem, i.e.",
"(REF ) without the OUP (REF ) as constraint and with (REF ) replaced by $OF_{\\text{detReform}}(Y_t, t_0,y_{t_0} ,S_{u(t)})$ , and (REF ) replaced by (REF ).",
"Hence, we are free to apply a suitable method from deterministic PDE-constrained optimization of our choice to solve the problem.",
"Here, we use a first-discretize-then-optimize approach, and numerically calculate the discrete optimal solution based on discrete adjoints.",
"All together, we come up with the deterministic reformulation of the stochastic optimal control (SOC) problem given by $\\min _{u \\in \\mathcal {U}_{ad}} \\int _{t^{\\ast }}^{T} OF_{\\text{detReform}}(Y_t; t_0;y_{t_0} ;S_{u(t)}) dt \\quad \\text{subject to} \\ (\\ref {eq:IBVP}), (\\ref {eq:stateConstr}),\\ \\text{and/or} \\ (\\ref {eq:reformJCCasFPT}).",
"$ Note that the JCC (REF ) is still a probabilistic constraint, which can, however, be handled in a deterministic way by applying Algorithm REF ."
],
[
"Numerical results",
"We apply deterministic discrete adjoint calculus (see e.g.",
"[10], [20], [31]) to solve the deterministically reformulated SOC problem (REF ).",
"Numerically, this is implemented in ANACONDA, a modularized simulation and optimization environment for hyperbolic balance laws on networks that allows the inclusion of state constraints, which has been developed in [30].",
"Note that the inclusion of state constraint is important for the inclusion of the SCC (REF ).",
"Further note that ANACONDA allows for regularization terms in the objective function to tackle finite horizon effects or oversteering for coarse discretizations.",
"Both effects can be reduced by penalizing control variations, which has been activated within the computations in Sections REF and REF with penalty parameter $10^{-5}$ .",
"To be more precise about the applied schemes, we use the IBOX scheme (REF ) to discretize the energy system (REF ), and the IPOPT solver [47] as well as the DONLP2 method [42], [43] within ANACONDA for the optimization procedure.",
"We start this section with a validation of our numerical routine in Subsection REF for a special case of (REF ), for which we are able to derive an analytical solution of the SOC problem in case of an SCC.",
"In Subsections REF and REF , we apply our numerical routine in case of two different parameter settings.",
"The choices in the energy system (REF ), the uncertain demand (REF ), and the CCs (REF ) stated in Table REF correspond to the settings in Sections REF , and REF .",
"Table: Specification of energy systems"
],
[
"Validation via scalar linear advection with source term",
"For a numerical validation of our optimization routine, we consider the special case of the linear advection on one edge with velocity $\\lambda \\in \\mathbb {R}$ , and nonzero, linear source term $s\\rho $ .",
"As we only consider the dynamics on one edge, we omit the dependence on the edge $e$ .",
"The latter equation results from (REF ) by setting the dimension $n=1$ , the number of prescribed left, and right BCs to $m_l=1$ , and $m_r=0$ , $a^{e}=0$ and by choosing the functional relation of the left BC as $\\Gamma _a= \\operatorname{id}$ , where $\\operatorname{id}$ denotes the identity function, and by setting the flux function to $\\rho ) = \\lambda \\rho $ .",
"It reads as $\\partial _t \\rho + \\lambda \\partial _x \\rho &= s\\rho \\\\\\rho (x,0) &= \\rho _0(x), \\\\\\rho (0,t) &= u(t), \\quad x \\in [0,b], \\ t \\in [0,T].",
"$ This IBVP has the advantage that we are able to derive an analytical solution to the corresponding SOC problem (REF ) with SCC (REF ) and can compare our numerical implementation against it."
],
[
"Analytical solution of the linear advection with source term",
"To derive the analytical solution of the SOC problem, we first need the analytical solution of the IBVP (REF ).",
"Therefore, we divide the time interval $[0,T]$ in two subintervals $I_0 = [0,{x}{\\lambda }]$ , and $I_b = ({x}{\\lambda },T]$ depending on the position $x \\in [0,b]$ .",
"The solution on $I_0$ is determined by the initial condition $\\rho _0(x)$ , and the solution on $I_b$ is determined by the inflow control $u(t)$ acting as a left BC.",
"For the solution of (REF ) on $I_b$ , we can change the role of $x$ , and $t$ as we deal with an empty system at the beginning.",
"Then, we can use the explicit solution of the IVP for the linear advection equation with source term (see [44]).",
"All together, the solution of (REF ) reads as $\\rho (x,t) = \\left\\lbrace \\begin{array}{ll}e^{st} \\rho _0(x-\\lambda t) & t \\in I_0\\\\e^{s{x}{\\lambda }} u(t-{x}{\\lambda }) & t \\in I_b \\end{array}\\right.$ The solution of (REF ) is illustrated graphically in Figure REF based on the choices $\\lambda =4$ , $\\Delta x=0.1$ , $\\Delta t = {\\Delta x}{\\lambda }$ (exact CFL-condition), $T=1$ , $s=-1$ , $\\rho _0(x)=\\sin (x)$ , and $u(t) = \\sin (10\\cdot t)$ .",
"Figure: Regions of influence of IC (I 0 I_0) and inflow control (I b I_b) in solution ()Note that the solution at $t=0$ is the prescribed initial condition.",
"For $x \\in [0,0.4)$ , $t=0.1$ is an element of $I_b$ , i.e.",
"the inflow control determines the solution, and for $x \\in [0.4,1]$ , $t=0.1$ belongs to $I_0$ , which means that, thereon, the solution results from the shifted initial condition.",
"The solution at $t=0.275$ depends only on the inflow control.",
"For the particular case of (REF ), we are able to derive an analytical expression for the optimal control for the particular cost function $OF_{\\text{detReform}}$ with $w_1=w_3=w_4=0$ , and $w_2=1$ .",
"Theorem 4.1 Let the supply dynamics (REF ) be given by (REF ), and the uncertain demand by the OUP (REF ) with existing second moment.",
"Furthermore, set $w_1=w_3=w_4=0$ , and $w_2=1$ in (REF ).",
"Then solving (REF ) with (REF ) being an active SCC (REF ) on the whole interval $[0,T]$ yields the optimal control $u^\\ast (t) = m_{OUP}(t) + v_{OUP}(t)\\Phi ^{-1}\\left(1-\\theta \\right) \\ \\forall t \\in [0,T-{1}{\\lambda }] $ From [23], we know that the optimal supply without imposing a CC in the linear advection setting (REF ) with $s=0$ for CM1 is given by the conditional expectation $S_{u(t)}= \\mathbb {E}[Y_{t+{1}{\\lambda }}|Y_0]$ .",
"As the source term, does not influence the characteristics of the linear advection (see equation (REF )), we again have a constant time delay between inflow and outflow.",
"That enables us to still consider the solution pointwise in time.",
"Due to the analytical solution (REF ), the optimal supply is always reachable as long as $S_{u(t)}\\in [u_{\\text{min}},u_{\\text{max}}]$ , where $u_{\\text{min}}$ , and $u_{\\text{max}}$ are lower and upper bounds on the control ($u_{\\text{min}}=0$ , and $u_{\\text{max}}=+\\infty $ in (REF )).",
"Since we have for the chance constraint level $m_{OUP}(t+{1}{\\lambda }) + v_{OUP}(t+{1}{\\lambda })\\Phi ^{-1}\\left(1-\\theta \\right) \\ge m_{OUP}(t+{1}{\\lambda }) = \\mathbb {E}[Y_{t+{1}{\\lambda }}|Y_0],$ and the objective function is quadratic with vertex in $(t,\\mathbb {E}[Y_{t+{1}{\\lambda }}|Y_0])$ , the optimal control is obtained at the left boundary of the feasible region at each point in time.",
"We now validate our numerical procedure by considering the particular SOC problem given by $\\min _{u(t), t \\in [0,T], u \\in L^2} \\int _{t^{\\ast }}^{T} OF_{\\text{detReform}}(Y_t; t_0;y_{t_0} ;S_{u(t)}) dt \\quad \\text{subject to} \\ (\\ref {eq:SCC}), \\ \\text{and} \\ (\\ref {eq:linAdvWithSource-IBVP}), $ where we set $w_1=w_3=w_4=0$ , and $w_2=1$ in (REF ).",
"We consider the interval $[0,1]$ .",
"For this setting, we derived the analytical optimal control in Theorem REF such that we can compare our numerical solution against the analytical one.",
"We activate the SCC within $I_{CC}=[0.6,1]$ .",
"The transport velocity is $\\lambda =4$ and we consider a rate for the source term of $s=-0.1$ .",
"In the numerical implementation, we have included a nonnegativity constraint $u(t)\\ge 0$ on the inflow control.",
"Note however that this constraint does not affect our test case (see Figure REF ).",
"Therefore, a direct comparison of our numerical solution against the analytical one is possible.",
"In our validation, we used the parameters $y_0=1$ , $\\mu (t)=1 + 2\\sin (8\\pi t)$ , $\\kappa =3$ and $\\sigma =0.1$ for the OUP.",
"In Figure REF , we observe that the numerical optimal control is close to the analytical control (REF ).",
"This finding suggests the correct functioning of our routine.",
"Note that the small spike at the activation time of the SCC in the numerical optimal control is inherited from the jump in the analytical solution.",
"Figure: NO_CAPTION"
],
[
"Linear system of hyperbolic balance laws: Telegrapher's equation ",
"Having seen that our numerical routine provides very good approximations to the analytical solution in the case of the linear advection with source term, in this section, we now consider the Telegrapher's equation on a network, and apply our routine thereon.",
"The corresponding energy system is obtained by specifying the quantities in the IBVP (REF ) according to the values in Table REF according to the parameter setting Tele REF .",
"In the following assertions, we stick to the conventional notation of the Telegrapher's equation and denote $(q^{e}_1,q^{e}_2)^{T} = (U^{e},I^{e})^{T}$ , where $U^{e}$ is the voltage, and $I^{e}$ the current on edge $e$ .",
"We deal with a system of linear hyperbolic balance laws with linear source terms, where we assume the same constant parameters $R,L,C,G$ for resistance, inductance, capacitance, and conductance on each edge.",
"We endow the system with initial conditions (ICs) $U^{e}_0$ , and $I^{e}_0$ .",
"On the bounded domain $X = [a,b]$ , where $a = \\min \\lbrace a^{e}\\ |\\ e \\in \\mathcal {E}\\rbrace $ , and $b = \\max \\lbrace b^{e}\\ |\\ e \\in \\mathcal {E}\\rbrace $ , we prescribe boundary conditions (BCs) $v_{\\text{ext}}(t)$ as an externally given function to prescribe the voltage at the end of the line, and $u(t)$ as the voltage control at the beginning of the line, taking the role of the inflow control into the energy system.",
"$\\partial _t U^{e}+ C^{-1}\\partial _x I^{e}&= -GC^{-1}U^{e}\\\\\\partial _t I^{e}+ L^{-1} \\partial _x U^{e}&= -RL^{-1}I^{e}, \\\\U^{e}(x,0) &= U^{e}_0(x), \\quad I^{e}(x,0) = I^{e}_0(x), \\\\U^{e}(a,t) &= u(t), \\quad U^{e}(b,t) = v_{\\text{ext}}(t), $ Before we present our numerical results for the Telegrapher's equation, we first address the question of well-posedness of the system on one edge, and in a second step the well-posedness of the system in the network case as well as the existence of an optimal control.",
"Note that the system (REF ) is diagonalizable.",
"Hence, we can rewrite it in characteristic variables denoted by $\\xi =\\left(\\xi ^{+},\\xi ^{-}\\right)^T$ (see [22]).",
"In case of lossless transmission ($R=G=0$ ), we trace the system back to a decoupled system of two classical linear advection equations by splitting the dynamics into left- and right-traveling waves with characteristic speeds $\\lambda ^{-}$ and $\\lambda ^{+}$ .",
"In the lossless case for one edge normed to a length of 1, the IBVP (REF ) reads as $\\partial _t \\xi ^{+} + \\lambda ^{+} \\partial _x \\xi ^{+} &= 0 \\\\\\partial _t \\xi ^{-} + \\lambda ^{-} \\partial _x \\xi ^{-} &= 0 \\\\\\xi ^{+}(x,0) = 0, \\quad \\xi ^{-} &= 0 \\\\\\xi ^{+}(0,t) = g^{+}(t), \\quad \\xi ^{-}(1,t) &= g^{-}(t),$ where $\\lambda ^{\\overset{+}{-}} = \\overset{+}{-}\\left(\\sqrt{LC}\\right)^{-1}$ .",
"Note that voltage $U$ , and current $I$ can be expressed in terms of the characteristic variables as $U(x,t) = \\sqrt{\\frac{L}{C}}\\left(\\xi ^{+} - \\xi ^{-}\\right)$ , and $I(x,t) = \\xi ^{+} + \\xi ^{-}$ .",
"As we deal with an empty system at the beginning, we can change the role of $x$ , and $t$ , and obtain for $\\lambda ^{+}>0$ , and $\\lambda ^{-}<0$ ${1}{\\lambda ^{+}} \\partial _t \\xi ^{+} + \\partial _x \\xi ^{+} &= 0 \\\\{1}{\\lambda ^{-}} \\partial _t \\xi ^{-} + \\partial _x \\xi ^{-} &= 0 \\\\\\xi ^{+}(0,t) = g^{+}(\\xi ^{+}(1,t)), \\quad \\xi ^{-}(1,t) &= g^{-}(\\xi ^{-}(0,t)).$ whose solution is $\\xi ^{+}(x,t) &= \\xi ^{+}(0,t-{1}{\\lambda }{+}x) = g^{+}(t-{1}{\\lambda ^{+}}x) \\\\\\xi ^{-}(x,t) &= \\xi ^{-}(1,t-{1}{\\lambda }{-}x) = g^{-}(t-{1}{\\lambda ^{-}}x) \\\\$ By interchanging the role of $x$ and $t$ again (empty system at the beginning), the existence of a unique weak solution of (REF ) with nonzero source term is ensured as a special case of the IVP for the system of hyperbolic balance laws with dissipative source term studied in [8].",
"For further details, we refer the reader to [44].",
"As we consider the dynamics given by the Telgrapher's equation (REF ) on each edge of our network, we need to impose suitable coupling conditions at the inner nodes $v \\in \\mathcal {V}_{int} = \\mathcal {V}\\backslash \\left(\\mathcal {V}_{in} \\cup \\mathcal {V}_{out}\\right)$ .",
"Therefore, we define the set of all edges connected to node $v \\in \\mathcal {V}$ as $\\mathcal {E}^{v}= \\lbrace e^{v}_1,\\cdots ,e^{v}_{k_{\\text{in}}}\\rbrace \\cup \\lbrace e^{v}_{k_{\\text{in}} + 1},\\cdots ,e^{v}_{k_{\\text{in}} + k_{\\text{out}}}\\rbrace .$ We define the coupling function as $c_v: \\mathbb {R}^{2\\times \\left(|\\delta ^{+}(v)|+|\\delta ^{-}(v)|\\right)} \\rightarrow \\mathbb {R}^{m_c}, c\\left(q^{e}, e \\in \\delta ^{-}(v) \\cup \\delta ^{+}(v)\\right) = \\begin{pmatrix}U^{e^{v}_1}(b^{e^{v}_1},t) - U^{e^{v}_{2}}(b^{e^{v}_{2}},t)\\\\\\vdots \\\\U^{e^{v}_1}(b^{e^{v}_1},t) - U^{e^{v}_{k_{\\text{in}}}}(b^{e^{v}_{k_{\\text{in}}}},t)\\\\U^{e^{v}_1}(b^{e^{v}_1},t) - U^{e^{v}_{k_{\\text{in}} + 1}}(a^{e^{v}_{k_{\\text{in}} + 1}},t)\\\\U^{e^{v}_1}(b^{e^{v}_1},t) - U^{e^{v}_{k_{\\text{in}} + k_{\\text{out}}}}(a^{e^{v}_{k_{\\text{in}}} + k_{\\text{out}}},t)\\\\\\sum _{i \\in \\delta ^{-}(v)} I(b^{i},t) - \\sum _{j \\in \\delta ^{+}(v)} I(a^{j},t)\\end{pmatrix}$ for each vertex $v \\in \\mathcal {V}_{int}$ .",
"Thereby, $m_c$ denotes the number of coupling conditions.",
"Now, we simply state the coupling conditions as $c_v = 0$ for all $v \\in \\mathcal {V}_{int}$ .",
"Remark The coupling conditions with respect to $U^{e^{v}}, e^{v}\\in \\mathcal {E}^{v},$ state the equality of the voltages at the corresponding boundaries of all edges with common node $v$ .",
"The coupling condition for $I^{e}$ at node $v$ models the conservation of the flow of current.",
"We are interested in the existence of an optimal inflow control $u(t)$ for the system (REF ).",
"Similar problems have been tackled in [9].",
"There, they derive the well-posedness of a system of nonlinear hyperbolic balance laws on a network with edges represented by the interval $[0,\\infty )$ , and the existence of an optimal control minimizing a given cost functional under certain conditions in the context of gas networks and open canals.",
"As our flow function $f^{e}(\\xi ) = \\Lambda \\xi $ is linear with $\\lambda ^{-} < 0 < \\lambda ^{+}$ , it satisfies the assumption (F) in [9].",
"Moreover, our source term $s(\\xi ) = B$ is a constant function implying its Lipschitz property and its bounded total variation.",
"Hence, condition (G) from [9] is also fulfilled.",
"This gives us the well-posedness of our IBVP (REF ) on a network on the positive half-line (see [9]).",
"We now turn our attention to the full SOC problem (REF ) in consideration.",
"An existence result of an optimal control in a gas network setting in the presence of state constraints has been derived in [14], where the state constraints enter the optimization problem via a barrier method.",
"Note that the SCC (REF ) for our SOC problem can be incorporated as a state constraint.",
"The existence of a minimizer in the presence of a JCC () is more involved as a reformulation as state constraint is no longer possible.",
"Therefore, we focus on the numerical analysis of the SOC problem (REF ) with particular focus on the influence of different types of chance constraints."
],
[
"Telegrapher's equation with single chance constraint",
"We start with a single chance constraint (REF ) in the Telegrapher's setting on the network depicted in Figure REF with parameters specified in Table REF (Tele REF ), and consider the cost function (REF ) with $w_1=w_3=w_4=0$ , and $w_2=1$ .",
"We consider the network topology from Figure REF .",
"The initial conditions are set to $U^{e}(x,0)=1$ on all edges and $I^{e}(x,0) = 1$ for $e_1$ and $e_7$ , $I^{e}(x,0) = 0.5$ for $e_2$ , $e_3$ , $e_5$ and $e_6$ as well as $I^{e}(x,0) = 0$ for $e_4$ .",
"We consider $[a^{e},b^{e}]=[0,1]$ for edges $e \\in \\lbrace e_1,\\dots ,e_6\\rbrace $ and $[a^{e},b^{e}] = [0,2]$ for $e_7$ .",
"We set the model parameters in (REF ) to $R=0.01$ (resistance), $L=0.5$ (inductance), $C=0.125$ (capacitance), and $G=0.01$ (conductance).",
"Our numerical results in Figure REF show that the optimally available current $I$ (purple solid line) at node $v_d$ matches the expected value of the OUP (blue dashed line) till $t = 1.5$ , and follows the course of the optimally available current without SCC (black dotted line).",
"For $t \\in I_{CC}$ , the optimal available current matches the given CC-Level (turquoise solid line).",
"This goes along with our intuitive understanding: the optimal available current matches the $95\\%$ - confidence level.",
"This is because a higher supply results in an even higher tracking-type error in the cost function, and a lower supply violates the SCC at the risk level of $5\\%$ .",
"This numerical finding is in line with the theoretical result in Theorem REF .",
"Figure: NO_CAPTION"
],
[
"Nonlinear system of hyperbolic balance laws: gas-to-power system",
"The energy transition phase comes with a lot of challenges in its realization.",
"Aiming for a high percentage of energy provided from renewable energy sources, one has to somehow cope with the large volatility in the energy generation.",
"One possibility to react to fluctuations to still ensure a stable demand satisfaction are gas turbines.",
"The possibility of gas-to-power, i.e.",
"the withdrawal of gas from the gas system and its transformation to power, seems to be a promising approach.",
"A gas turbine can be booted to full performance within several minutes.",
"Thus, a gas turbine might be appropriate to overcome short-term bottlenecks in energy generation.",
"Therefore, we finally focus on the modeling of gas-to-power in the context of uncertain demands.",
"We particularly pay attention how the withdrawal of gas affects the behavior of the gas network.",
"One major mathematical challenge that the gas network entails is that its governing equations are no longer a linear system of hyperbolic balance laws.",
"The nonlinear gas transport can be described by the so-called isentropic Euler equations.",
"They can be obtained from equation (REF ) by putting in the parameters of GtP REF of Table REF : $\\begin{pmatrix} \\rho ^{e} \\\\ q^{e} \\end{pmatrix}_t &+ \\begin{pmatrix} q^{e} \\\\ p(\\rho ^{e}) + \\frac{(q^{e})^2}{\\rho ^{e}} \\end{pmatrix}_x = \\begin{pmatrix} 0 \\\\ g(\\rho ^{e},q^{e}) \\end{pmatrix} \\\\\\rho ^{e}(x,0) &= \\rho _0^{e}(x), \\quad q^{e}(x,0) = q_0^{e}(x).",
"$ Thereby, $q^{e}_1$ is denoted by $\\rho ^{e}$ and describes the density, $q^{e}_2$ is identified with $q$ and gives the flow, $g$ a given source term chosen as in [17], and $p^{e}$ is the pressure on edge $e$ , which is specified by the pressure law $p^{e}(\\rho ^{e})=d^2\\cdot \\rho ^{\\beta }$ for all edges $e$ (with $\\beta =1$ and $d=340 \\frac{\\text{m}}{\\text{s}}$ in the examples below).",
"Note that the exponent $\\beta =1$ means that we deal with the isothermal Euler equations, a special case of the isentropic Euler equations.",
"However, the analysis is not limited to this choice and results for various pressure laws can for example be found in [16].",
"To ensure the well-posedness of the system, we also need coupling conditions at the nodes.",
"As in [17], we use pressure equality and mass conservation at all nodes at all times (Kirchhoff-type-coupling), i.e.",
"for all $v \\in \\mathcal {V}$ and for all $t \\in [0,T]$ , we have $p_{in}^v(t) &= p_{out}^v(t), \\\\\\sum _{e \\in \\delta ^{-}(v)} q^{e}(t) &= \\sum _{\\tilde{e} \\in \\delta ^{+}(v)} q^{\\tilde{e}}(t).$ To couple the gas network to the power system, we adapt the deterministic coupled gas-to-power-system described in [17], and extend it by an uncertain power demand given by the OUP (REF ).",
"The adapted setting is sketched in Figure REF .",
"Figure: Coupled gas-to-power systemIn our case, the power system is shrinked to only one node $v_d$ , where the aggregated uncertain demand is realized.",
"This is because our focus is on matching this demand $Y_t$ best possible by the power $S_{u(t)}$ provided by the gas-to-power turbine, whereas the distribution within the power system is considered as a separate task.",
"The power demand is attained preferably by the gas-to-power conversion amount $S_{u(t)}$ .",
"The missing power $\\tilde{Y}_t$ needs to be covered by an external power source.",
"The gas consumption to generate the power is described by a quadratic function $\\epsilon (S_{u(t)}) = a_0 + a_1S_{u(t)}+ a_2(S_{u(t)})^2.$ As the amount of gas withdrawn from the network is our control, we have $u(t) = \\epsilon (S_{u(t)}).$ Pressure bounds play an important role in gas networks.",
"In our case, we add a lower pressure bound at node $v_5$ , which is $p^{v_5}(t) \\ge 43 [bar] $ in the first scenario below, appearing as an additional algebraic constraint in the SOC (REF ).",
"There is a pressure drop in the gas network caused by the gas withdrawal for the conversion to power.",
"It can be compensated by a compressor station.",
"The modeling of the compressor station is taken from [17]: the compressor is modeled as edge $e_1$ with time-independent in- and outgoing pressure $p^{v_0}$ and $p^{v_1}$ , and flux values $q^{v_0}$ , and $q^{v_1}$ through the nodes $v_0$ and $v_1$ .",
"We assume that the compressor is run via an external power source only linked to the scenario in Figure REF via the cost component $C1$ in (REF ) accounting for operator costs of the compressor.",
"This entails the flux equality $q^{v_1}=q^{v_0}$ .",
"Note that the operating costs increase if the ratio ${p^{v_1}}{p^{v_0}}$ increases.",
"The compressor can be controlled by the gas network operator.",
"Therefore, a second control variable, i.e.",
"the control of the compressor station $u_{compr}(t)$ , is added to the SOC problem (REF ) in terms of $u_{compr}(t)&= p^{v_1}(t) - p^{v_0}(t).$ It appears in the operating costs $C1^{\\text{compr}}$ in the cost component $C1$ of $OF_{\\text{detReform}}$ .",
"Moreover, costs occur for the gas consumption to satisfy the power demand.",
"These costs complete the deterministic costs $C1$ in (REF ) as $C1 = C1^{\\text{compr}} + 0.0001\\cdot \\int _{t_0}^{T} u(t) dt.$ For our purpose, it is important to note the deterministic nature of the compressor costs $C1^{\\text{compr}}$ and the costs for the gas consumption within the objective function (REF ).",
"Due to the active element in terms of the compressor station, we need to adapt two of the above coupling conditions at $v_1$ and $v_2$ as $p_{out}^{v_1}(t) &= p_{in}^{v_1}(t) + u_{compr}(t), \\ \\text{and} \\\\q_{out}^{v_2}(t) &= q_{in}^{v_2}(t) - u(t).",
"$ Equation (REF ) describes the gas coupling accounting for the possible pressure increase at $v_1$ .",
"The mass conservation at $v_2$ except for the gas withdrawn for the gas-to-power conversion is modeled by equation ().",
"It might well be the case that the power demand cannot completely be served by the gas network due to pressure bounds in combination with the maximal performance of the compressor station, or other physical or economical reasons.",
"In this case, the external power supplier comes into play.",
"Hence, the difference $Y_t-S_{u(t)}$ is covered by external power supply."
],
[
"Gas-to-power system with single chance constraint",
"For the numerical investigation of the impact of an SCC on the optimal amount of gas withdrawn from the network, we consider the parameter setting GtP_s in Table REF .",
"In the cost function (REF ) of the SOC (REF ), we consider a pure tracking type functional by setting $w_1=w_3=w_4=0$ , and $w_2=1$ .",
"Our numerics refer to the setting schematically represented in REF .",
"The gas network is a subgrid of the GasLib-40 network, which approximates a segment of the low-calorific gas network located in the Rhine-Main-Ruhr area in Germany [40].",
"An extension of the gas network by a compressor station in a purely deterministic setting has already been analyzed in [17].",
"Here, we present results including the compressor station in the presence of an uncertain demand stream as well as an SCC.",
"To the best of our knowledge, this has not been considered before.",
"We choose a discretization of $dt=900[\\text{s}]$ and $dx \\approx 1[\\text{km}]$ , slightly adapted to the individual length of the edges.",
"The specifications of the intervals $[a^{e},b^{e}]$ for the edges are due to the network specifications in [40] and can be found in [17] for the subgrid considered here.",
"Figure: NO_CAPTIONFigure: Pressure evolutionIn Figure REF , we see that the optimal amount of gas-to-power conversion (purple solid line) follows well the course of the expected value of the OUP (REF ) (blue dashed line), and from $t=6$ on matches the given SCC level (turquoise solid line) until $T=12$ .",
"This again coincides with the theoretical result for the linear advection with source term in Theorem REF .",
"In Figure REF , the necessity of the compressor station as an active element to keep the lower pressure bound is illustrated.",
"There is a pressure increase at the compressor station with time-shift to ensure the lower pressure bound at $v_5$ , which is set to $p^{v_5}(t) \\equiv 43 [bar]$ here.",
"This behavior has already been observed and investigated in a deterministic setting in [17].",
"As we would expect, this increase is particularly pronounced while the SCC is active.",
"The lower bound on the supply induces the need for a higher power level.",
"This entails a higher gas withdrawal inducing a pressure drop in the gas network.",
"The latter drop needs to be compensated by the compressor station to keep the lower pressure bound at the sink.",
"In a next step, we consider the general objective function (REF ) being able to depict real costs.",
"To do so, we include operating costs of the compressor station (C1), costs for external power supply (C3), as well as profit of selling excess power from the gas conversion (R), and set $w_1=w_3=10^{-4}$ , $w_2=0$ , and $w_4=10^{-6}$ .",
"Since the cost component $C2$ cannot directly be interpreted as real costs, we set $w_2=0$ .",
"To handle the terms in the cost function, we use the deterministic reformulation of the additional cost components from Subsection REF .",
"Furthermore, we impose a JCC on the interval $I_{CC}=[0,4]$ .",
"We consider a discretization of $dt=60[\\text{s}]$ and $dx\\approx 1[\\text{km}]$ , again slightly adapted to the individual length of the edges, applied to the same subgrid of the GasLib-40 network [40] as above.",
"Note that this coarse discretization is possible due to the chosen numerical scheme, the IBOX scheme (REF ).",
"The coarse time discretization is possible due to the properties of the IBOX scheme introduced in Subsection REF .",
"The control grid differs from the one in the simulation.",
"The amount of gas-to-power conversion can be adapted every 15 minutes.",
"We work with a lower pressure bound of $p^{v_5}(t) \\equiv 42.5$ .",
"In Figure REF , we compare the optimal amount of gas withdrawal for the JCC with the one that would be obtained for an SCC at the same risk level of $\\theta =5\\%$ within the same interval $I_{CC}$ .",
"Figure: NO_CAPTIONThe grey scale shows the pointwise quantile levels of the OUP and the blue dashed line indicates the mean of the OUP.",
"Our numerical results reveal that the optimal supply for the JCC (red solid line) evolves in the same structural manner as the SCC (purple dotted line) but at a higher level.",
"This is in line with our intuitive understanding: the JCC acts pathwise and thus represents a more restrictive constraint.",
"Note that a path that once violated the constraint cannot be counted as save path, i.e.",
"a path below the imposed CC, at any later point in time anymore.",
"However, this is possible for an SCC.",
"We conclude our numerical investigation of the impact of chance constraints on the optimal supply with a Monte Carlo (MC) investigation of the obtained optimal supply levels for the JCC.",
"We introduce two time-dependent sets of paths distinguishing those paths that at least once hit the prescribed supply boundary $S(t)$ until time $t$ from those that stay below this level until time $t$ , i.e.",
"$\\Omega _{hit}(t) &= \\lbrace \\omega \\in \\Omega \\ |\\ \\exists s \\in [0,t]: Y_s(\\omega )>S(t)\\rbrace , \\ \\text{and} \\\\\\Omega _{save}(t) &= \\lbrace \\omega \\in \\Omega \\ |\\ \\forall s \\in [0,t]: Y_s(\\omega )\\le S(t)\\rbrace .$ Note that $\\Omega _{hit} \\cup \\Omega _{save} = \\Omega $ .",
"We denote the elements in $\\Omega _{hit}$ hit paths, and the elements in $\\Omega _{save}$ save paths.",
"In Figure REF , we depict the optimal supply level with JCC as the solid blue line.",
"The confidence intervals of the OUP are shown in grey scale.",
"Note that they are calculated pointwise in time corresponding to an SCC.",
"We investigate the running maximum of the save paths (green dotted line) and over all paths (light blue solid line), which we define via the evaluations of the numerical approximation $\\hat{Y}_{j}$ of the OUP obtained for $M=10^3$ MC samples.",
"This leads to $r(j) &= \\max _{ \\lbrace k \\in 1,\\dots ,M \\rbrace } \\hat{Y}_j(\\omega _k), \\quad j \\in \\lbrace 1,\\dots ,N_{\\Delta t}\\rbrace \\quad \\text{and} \\\\r_{save}(j) &= \\max _{\\lbrace k \\in 1,\\dots ,M \\ | \\ \\hat{Y}_j(\\omega _k) \\le S(t_j) \\rbrace } \\hat{Y}_j(\\omega _k), \\quad \\ \\text{for} \\ j \\in \\lbrace 1,\\dots ,N_{\\Delta t}\\rbrace ,$ where $N_{\\Delta t}+1$ is the number of time grid points.",
"Note that the running maximum $r_{save}(j)$ is only well-defined if the index set is nonempty.",
"Furthermore, we depict the instances of the first passage time of each hit paths as black asterisks.",
"We observe that the running maximum $r_{save}(t)$ stays close below the optimal supply $S(t)$ whereas the running maximum $r_{hit}(t)$ evolves slightly above $S(t)$ .",
"This can be interpreted in the following way: we minimize the expected quadratic deviation between our supply and the demand at the market as a major component in our cost function.",
"At the same time, a guarantee that no undersupply occurs with a $95\\%$ probability is given.",
"Hence, it appears logical that we control our energy system in a way being close to this lower probabilistic undersupply bound to avoid costly oversupply occurrences.",
"With respect to the range where we expect the demand to evolve within (confidence intervals in grey scale), our optimal supply with JCC appears by far too large.",
"However, looking at the save distance $d_{save}(j) = S(t_j) - r_{save}(j)$ in Figure REF , we observe primarily values below $0.1$ indicating a very limited buffer explaining the comparably large optimal supply.",
"Figure: MC analysis of optimal supply with JCC"
],
[
"Conclusion",
"In this work, we analyzed the optimal inflow control in hyperbolic energy systems subject to uncertain demand and particularly focused on quantifying the related uncertainty.",
"By imposing chance constraints, we were able to limit the risk of an undersupply to a chosen probability level $\\theta $ .",
"Thereby, we distinguished between SCCs and JCCs.",
"Whereas it turned out that there is a quantile-based reformulation of the SCC enabling us to include the SCC as a state constraint into the optimization, the JCC case was more involved.",
"We came up with a first passage time reformulation of the JCC, which we solved by using an algorithm set up in the general context of Gauss-Markov processes [12].",
"To show that our numerical procedure can be applied to real-world phenomena, we took a real gas network from the GasLib-40 library [40] and calculated the optimal amount of gas withdrawn from the network to cover an uncertain power demand stream described by an OUP in the presence of a JCC.",
"An interesting aspect for further research is the theoretical existence of an optimal control for the above analyzed hyperbolic energy systems in the presence of a JCC."
],
[
"Acknowledgement",
"The authors are grateful for the support of the German Research Foundation (DFG) within the project “Novel models and control for networked problems: from discrete event to continuous dynamics” (GO1920/4-1) and the BMBF within the project “ENets”.",
"Moreover, the author Kerstin Lux would like to thank Hanno Gottschalk for suggesting the consideration of joint chance constraints in this context."
]
] | 2001.03591 |
[
[
"An Ising-Glauber Spin Cluster Model for Temperature Dependent\n Magnetization Noise in SQUIDs"
],
[
"Abstract Clusters of interacting two-level-systems (TLS),likely due to $F^+$ centers at the metal-insulator interface, are shown to self consistently lead to $1/f^{\\alpha }$ magnetization noise in SQUIDs.",
"By introducing a correlation-function calculation method and without any a priori assumptions on the distribution of fluctuation rates, it is shown why the flux noise is only weakly temperature dependent with $\\alpha\\lesssim 1$, while the inductance noise has a huge temperature dependence seen in experiment, even though the mechanism producing both spectra is the same.",
"Though both ferromagnetic- RKKY and short-range-interactions (SRI) lead to strong flux-inductance-noise cross-correlations seen in experiment, the flux noise varies a lot with temperature for SRI.",
"Hence it is unlikely that the TLS's time reversal symmetry is broken by the same mechanism which mediates surface ferromagnetism in nanoparticles and thin films of the same insulator materials."
],
[
"An Ising-Glauber Spin Cluster Model for Temperature Dependent Magnetization Noise in SQUIDs Amrit De Department of Physics and Astronomy, University of California - Riverside,CA 92521 Clusters of interacting two-level-systems (TLS),likely due to $F^+$ centers at the metal-insulator interface, are shown to self consistently lead to $1/f^{\\alpha }$ magnetization noise in SQUIDs.",
"By introducing a correlation-function calculation method and without any a priori assumptions on the distribution of fluctuation rates, it is shown why the flux noise is only weakly temperature dependent with $\\alpha \\lesssim 1$ , while the inductance noise has a huge temperature dependence seen in experiment, even though the mechanism producing both spectra is the same.",
"Though both ferromagnetic- RKKY and short-range-interactions (SRI) lead to strong flux-inductance-noise cross-correlations seen in experiment, the flux noise varies a lot with temperature for SRI.",
"Hence it is unlikely that the TLS's time reversal symmetry is broken by the same mechanism which mediates surface ferromagnetism in nanoparticles and thin films of the same insulator materials.",
"85.25.Dq, 05.40.-a, 75.10.-b, 03.67.Lx Superconducting quantum interference devices (SQUID) are of considerable interest for quantum information as they can replicate natural qubits, such as electron and nuclear spins, using macroscopic devices.",
"However the performance of superconducting qubits is severely impeded by the presence of $1/f$ magnetization noise which limits their quantum coherence.",
"This type of noise was first observed in SQUIDs well over two decades ago[1], [2], however its origins and many of its features remain unexplained.",
"Recent activity in quantum computing has however revived interests to better understand this magnetization noise.",
"[3], [4].",
"Figure: Proposed 1/f1/f noise model consisting of interacting spins that fluctuate within a cluster.",
"The clusters are assumed to form due to random defects at the SQUID's metal-insulator interface and are sufficiently far apart so that only spins within a single cluster interact.",
"Number of spins within a cluster and the lattice constant aa vary.Magnetic noise in SQUIDs has several puzzling features.",
"While the flux noise (the first spectrum) is also weakly dependent on temperature, the choice of the superconducting material and the SQUID's area[1], [5], [6] – the inductance noise (the second spectrum or the noise of the flux noise), surprisingly shows a strong temperature dependence.",
"It decreases with increasing temperature and scales as $1/f^{\\alpha }$[5] where the temperature dependent ($0<\\alpha (T)<1$ )[7].",
"The flux noise is also known to be only weakly dependent on geometry and the noise scales as $l/w$ , in the limit $w/l<<1$ ($l$ is the length and $w$ is the width of the superconducting wire) [8].",
"This along with recent experiments [5] suggests that flux noise arises from unpaired surface spins which reside at the superconductor-insulator interface in thin-film SQUIDs.",
"The estimated areal spin density from the paramagnetic susceptibilities is about $5\\times 10^{17}~m^{-2}$ [5], [9].",
"Experimental evidence also suggests that these surface spins are strongly interacting and that there is a net spin polarization [5] as the $1/f$ inductance noise is highly correlated with the usual $1/f$ flux noise.",
"This cross-correlation is inversely proportional to the temperature and is about the order of unity roughly below 100mK.",
"Since inductance is even under time inversion and flux is odd, their three-point cross-correlation function must vanish unless time reversal symmetry is broken, which indicates the appearance of long range magnetic order.",
"As this further implies that the mechanism producing both the flux- and inductance noise is the same, it is not clear on why only the associated spectrum (inductance noise) should have a large temperature dependence[5].",
"Usually $1/f$ noise is associated with the onset of a spin-glass phase as is believed to account for the many anomalous properties of spin-glasses at low temperatures[10], [11].",
"However recent Monte-Carlo simulations by Chen and Yu [12] have ruled out this out to explain magnetization noise in SQUIDs.",
"They considered an Ising spin glass with random nearest neighbor interactions.",
"Though their model reproduced qualitative features of the inductance noise, it did not show cross-correlations between inductance and flux noise.",
"This is expected as spin glasses preserve time reversal symmetry.",
"The microscopic origin of the magnetization noise is also not known.",
"Phenomenologically, the noise arises from spins bereave like randomly fluctuating two level systems(TLS).",
"Choi et al[13] explained this in terms of metal induced gap states that arise due to the potential disorder at the metal-insulator interface.",
"Another model involved unpaired and non-interacting electrons randomly hopping between traps with different spin orientations and a $1/f$ distribution of trap energies[14].",
"Quite often hopping conductivity models are used for $1/f$ noise in solid state systems[15], [16], [17].",
"Dangling bond states near the Fermi energy in disordered SiO$_{2}$ [18] and fractal spin structures with tunneling interactions[19] have also been suggested for the SQUID noise.",
"Typically, the dc-SQUIDs consitute of an amorphous Al$_{2}$ O$_{3}$ insulating layer deposited on the surface of a metal (commonly Nb[5] or Al[4]).",
"The Al$_{2}$ O$_{3}$ is likely to cluster on the surface before filling in and forming a homogeneous layer due to its higher binding energy which could lead to the Volmer-Weber growth mode.",
"The lattice mismatch between the insulator and the metal could also lead to the formation of clusters.",
"Near the metal surface, the clusters can host a number of point defects in the form of O vacancies that can capture one electron (F$^+$ -center) or two (F-center).",
"In a related development, a few years ago surface ferromagnetism (SFM) was reported in thin-films and nanoparticles of a number of otherwise insulating metallic oxides[20] (including Al$_2$ O$_3$ ) where the materials were not doped with any magnetic impurities.",
"Further recent investigations attribute this room temperature SFM in Al$_2$ O$_3$ nanoparticles[21] to F$^+$ -centers where it was found that amorphous Al$_2$ O$_3$ is more likely to host the number of F$^+$ -centers to cross the magnetic percolation threshold than the crystalline variant.",
"The origin of SFM in these otherwise non-magnetic metal oxides is itself somewhat controversial[22].",
"Some of the suggested mechanisms include exchange coupling from F$^+$ center induced impurity bands[23], F$^+$ center mediated superexchange[24], and spin-triplets at the F-center[25].",
"In addition to this, in the SQUID geometry because of the proximity to the metal, these local magnetic moments can spin polarize the metal's conduction band electrons which can lead to an RKKY type long range interaction mechanism, which was first pointed out by Faoro and Ioffe[26].",
"This can likely lead to competing interaction mechanisms.",
"In this paper, in order to see if the interaction mechanism induces any distinguishing features in the noise spectrum, both ferromagnetic RKKY and ferromagnetic nearest neighbor interaction(NNI) mechanisms are considered for the spin cluster model.",
"While both mechanisms give rise to $1/f$ noise self consistently as shown here, for NNIs the flux noise varies far more with temperature.",
"A new method is introduced in this paper to obtain any arbitrary $n-$ point correlation function.",
"As a result various subsequent spectral functions for the interacting Ising-Glauber spin model can be systematically obtained, both analytically and numerically.",
"The magnetization noise calculations are carried out using this method and the spin cluster model shown in fig.REF .",
"For the sake of self consistency, the usual heuristic assumption on the $1/\\gamma $ distribution of switching rates required for $1/f$ noise[27]) is avoided.",
"Instead it is shown that at low temperatures, the $1/f$ spectrum arises naturally in the model from the spin-spin interactions and a uniform distribution of cluster sizes.",
"All experimentally observed features such as the lack of temperature dependence of the flux noise, the previously unexplained considerable temperature dependence of the inductance noise (or the associated spectrum) and the flux-inductance noise cross-spectrum are shown and explained here.",
"The Model and the Method: A schematic of our model is shown in figureREF where each cluster comprises of an interacting 2D spin-lattice.",
"To model the randomness of the surface defects, randomly varying lattice constants are considered.",
"Individual clusters are assumed to be sufficiently far apart so that there are no interactions across the clusters but there could be an effective mean field.",
"For the infinite range Ising-Glauber model, all $N$ spins within a single cluster interact with every other via an RKKY type mechanism.",
"The overall temporal evolution for $N$ interacting spins is governed by the master equation $\\dot{\\mathbf {W}}(t)=\\mathbf {VW\\mathrm {\\mathit {(t)}}}$ [28], where $\\mathbf {V}$ is a matrix of transition rates (such that the sum of each of its columns is zero) and $\\mathbf {W}$ is the flipping probability matrix for the spins.",
"Each spin's random fluctuation is a temperature (or interaction) driven process governed by Glauber dynamics.",
"This is a Markov process where the new spin distribution depends only on the current spin configuration and that the new and old spin configurations agree everywhere except at a single site.",
"Overall the non-equlibrium spin dynamics for a system of correlated spins can be treated this way[29].",
"The conditional probability for a single spin to flip is determined by the Boltzmann factor and the matrix-elements of $\\mathbf {V}$ are $\\small {{\\mathbf {V}}(\\mathbf {s}\\rightarrow \\mathbf {s}^{\\prime })=\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\gamma e^{-\\beta H(\\mathbf {s^{\\prime }})}}{e^{-\\beta H(\\mathbf {s})}+e^{-\\beta H(\\mathbf {s^{\\prime }})}} &\\mbox{for$\\mathbf {s}\\ne \\mathbf {s}^{\\prime }$ and} \\\\& \\mbox{$\\displaystyle \\sum _i(1-s_is_i^{\\prime })=2$} \\\\-\\displaystyle \\sum _{\\mathbf {s}\\ne \\mathbf {s}^{\\prime }}V(\\mathbf {s}\\rightarrow \\mathbf {s}^{\\prime }) & \\mbox{for $\\mathbf {s}=\\mathbf {s}^{\\prime }$}\\end{array}\\right.}",
"$ Here, $\\mathbf {s}^{\\prime }$ ($\\mathbf {s}$ ) is a vector that denotes the present(earlier) spin configuration and $\\gamma $ is the flipping rate of a spin (all $\\gamma =1$ in this paper).",
"The non negative off-diagonal matrix elements in Eq.REF satisfy the detailed balance condition and the diagonal terms are the just negative sum of the off-diagonal column elements so that the column's zero sum ensures the conservation of probability.",
"To obtain the random temporal dynamics of an individual spin cluster, the $2^N$ dimensional $\\mathbf {W}=\\exp (-{\\bf V}t)$ matrix has to be evaluated.",
"The eigenvalues of $\\mathbf {V}$ are either zero, which corresponds to the equilibrium distribution, or are real and negative, which also eventually tend to the equilibrium distribution as $t\\rightarrow \\infty $ [28].",
"This method is also provides the quasi-Hamiltonian open quantum systems methods[30], [31], [32] with a connection to the underlaying noise microscopics.",
"The general system Hamiltonian is $H(\\mathbf {s})=-\\frac{1}{2}\\sum _{i,j}J_{i,j}s_{i}s_{j}-B\\sum _{i}s_{i}$ where $B$ is the magnetic field(which is set to zero here) and $J_{i,j}$ is the spin-spin interaction between the $i^{th}$ and $j^{th}$ Ising spins.",
"Now, for $N$ interacting spins, the $n^{th}$ order auto- or cross-correlation function can be calculated as follows $\\langle s_{i}(t_{1})s_{j}(t_{2})...s_{\\kappa }(t_{n})\\rangle = \\langle {\\mathbf { f}}|\\sigma _z^{(\\kappa )}\\mathbf {W}(t_n) ... \\sigma _z^{(i)}\\mathbf {W}(t_1)|\\mathbf { i}\\rangle $ where the spin indices $\\lbrace i,j...\\kappa \\rbrace \\in \\lbrace 1,2,...N\\rbrace $ , $|\\mathbf {i}\\rangle $ =$|\\mathbf {f}\\rangle $ are the initial and final state vectors that correspond to the equilibrium distribution (i.e.,$\\mathbf {W}\\left|\\mathbf {i}\\right\\rangle =\\left|\\mathbf {i}\\right\\rangle $ ).",
"It is implied that $\\sigma _z^{(\\kappa )} = \\underset{1}{\\sigma _o}\\otimes \\underset{2}{\\sigma _o}...\\underset{\\kappa -1}{\\sigma _o}\\otimes \\underset{\\kappa }{\\sigma _z}\\otimes ... \\underset{N}{\\sigma _o}$ where $\\sigma _z$ is the z-Pauli matrix and $\\sigma _o$ is the identity.",
"For just two spins, if all $\\gamma _i=1$ (which is subsequently followed for all calculations) the two-point correlation functions are $\\langle {s_i(0)s_j(t)}\\rangle = e^{-2\\Gamma _-|t|} + (2\\delta _{ij}-1)\\exp (4\\beta J)e^{-2\\Gamma _+|t|}$ where $\\Gamma _{\\pm }=[1+\\exp (\\pm 2\\beta J)]^{-1}$ .",
"Whereas if $\\gamma _i$ is retained then $\\displaystyle \\lim _{T\\rightarrow 0}$ $\\langle {s_is_j}\\rangle =\\delta _{ij}e^{-2\\gamma _i|t|}$ is obtained from this model.",
"Within a single cluster, the spins interact via an oscillatory RKKY-like form with a ferromagnetic $J_o$ , $J_{i,j}=J_{o}{[k_FR_{i,j}\\cos (k_FR_{i,j})-\\sin (k_FR_{i,j})]}/{(k_FR_{i,j})^{4}}$ where $R_{i,j}$ is the separation between two spins (on a lattice of lattice constant $a$ ), $k_F$ is a Fermi wavevector type parameter.",
"For the calculations here $J_o=9\\times 10^{10}~Hz/\\hbar $ is taken as a fitting parameter independent of $k_F$ .",
"Figure: (a) Temperature dependent net correlation function for 400 spin clusters, eachwith 6-9 spins with ferromagnetic (+|J o |+|J_o|) RKKY interactions.",
"(b) Corresponding flux noise power-spectrum showing 1/f α 1/f^{\\alpha } noise below ∼0.1\\sim 0.1 Hz and (c) its respective slope (α\\alpha ).",
"Note that α≲1\\alpha \\lesssim 1 below about 0.10.1 Hz (d) Distribution of k F ak_Fa for each cluster, with mean 〈k F a〉\\langle {k_Fa}\\rangle and standard deviation (Σ\\Sigma ) as indicated.Flux Noise: Due to the high estimated areal spin density, the coherent magnetization of the spins strongly flux couples to the SQUID.",
"The fluctuation-dissipation theorem relates the the magnetization noise spectrum to the imaginary part of the susceptibility[33], [34], [6].",
"If all the surface spins couple to the SQUID equally, the flux noise for the $\\ell ^{th}$ spin-cluster is the Fourier transform of the sum of all two-point spin correlation functions $P_{\\phi }^{(\\ell )}(\\omega )=2\\mu _{o}^{2}\\mu _{B}^{2}\\frac{\\rho }{\\pi }\\frac{R}{r}\\displaystyle \\int _{0}^{\\infty }\\displaystyle \\sum _{i,j=1}^{N}\\langle {s_{i}(0)s_{j}(t)}\\rangle e^{\\imath \\omega t}dt~~$ where $R$ is the radius of the loop, $r$ is the radius of the wire, $R/r=10$[6] and $\\rho $ is the surface spin density.",
"All possible combinations of two-point autocorrelation ($i=j$ ) and cross-correlation ($i\\ne j$ ) functions are explicitly calculated within a cluster using Eq.REF .",
"Since the clusters are assumed to be sufficiently far apart and noninteracting, $\\langle {s_{i}(0)s_{j}(t)}\\rangle $ is calculated individually for each cluster.",
"The total flux noise power spectrum for the SQUID is $P_{\\phi }(\\omega )=\\sum _{\\ell }{P_{\\phi }^{(\\ell )}(\\omega )}$ .",
"The temperature dependent net $P_{\\phi }(\\omega )$ , and the respective noise-slopes are shown in Figs.REF (a)-(c), where 400 spin clusters were considered with 6-9 spins each.",
"Each cluster is assigned a random $k_Fa$ and a uniform distribution of $k_Fa$ is considered (see Fig.REF -(a)).",
"Note that from the estimated areal spin density of unpaired surface spins of $\\rho \\sim 5\\times 10^{17}~m^{-2}$ , one can estimate $k_F$ from which we can infer what the average spin separation $\\langle {a}\\rangle $ is.",
"The $1/f^{\\alpha }$ noise spectrum (indicated by the slope $\\sim 1$ ) is shown in Fig.REF (b) and (c), at for the intermediate range of frequencies.",
"At high frequencies, the log noise spectra shows a slope of 2 which corresponds to the Lotrentzian tail of the noise power.",
"While considering various cases, it was found that the upper and lower cutoff frequencies for the $1/f$ type noise depended on the distribution of $k_{F}a$ and the interaction strength (as evident from the temperature dependence).",
"In the absence of interactions the noise spectrum reduces to a simple Lorentzian.",
"And eventually for all temperatures, $\\alpha \\rightarrow 0$ (as determined by the interaction strength) at very low frequencies which corresponds to Gaussian noise.",
"These calculations show that the noise power is only weakly dependent on temperature which is in agreement with experiment[5].",
"Note that in the early experiments of REF.",
"[Wellstood1987], $P_{\\phi }$ was not necessarily independent of temperature under all circumstances.",
"While there was no temperature dependence for the flux noise bellow 1K, there was a strong low temperature dependence for certain parameters/materials – for example, for PbIn/Nb and Pb/Nb [2].",
"Quite strikingly, for the same set of materials (PbIn/Nb for the SQUID's loop/electrode) and depending on the construction, the flux noise can be either completely independent of temperature or inversely proportional to it or even oscillate with temperature.",
"Such conflicting temperature dependencies of $1/f$ noise are known to exist for glassy systems[35], [36].",
"Figure: (a) Sample flux noise power-spectrum P φ P_\\phi and (b) its slope for spin clusters with nearest neighbor ferromagnetic interactions that vary randomly for each cluster.Next consider short range ferromagnetic NNI $\\mathcal {J}$ that randomly vary for each cluster.",
"Now $1/f$ noise (see fig.REF ) can only be obtained in the spin-cluster model if the NNIs have a $1/{\\mathcal {J}}$ type distribution.",
"If the RKKY $J_{i,j}$ is expanded for small $k_F$ , then $J_{i,j}\\propto {1/R_{i,j}}$ – hence a uniform distribution of $R_{i,j}$ results in a $1/J_{i,j}$ distribution of interaction strengths at a certain crossover length $1/k_F$ .",
"Although this ferromagnetic NNI model also self consistently produces $1/f$ noise, the flux noise's variations with temperature are quite large (see Fig.REF ) and is $1/f$ -like only over a short range (compare to fig.REF ).",
"In view of the experiments[5], this likely rules out NNIs being the dominant mechanism even though they are the thought to be the cause of SFM in insulating metal oxide nanoparticles[23], [37], [24].",
"Inductance Noise: In the experiments of Ref.Sendelbach2008, the temperature dependent inductance noise was measured for temperatures bellow $2K$ where the inductance noise($P_L$ ) was mostly dominated by the imaginary part of the susceptibility and varied considerably with temperature.",
"$P_L$ is the associated noise spectrum or the second spectrum – which is a quantitative measure of the spectral wandering of the first spectrum and is interpreted as the noise of the noise [38].",
"The first spectrum (flux noise) is related to the imaginary part of the susceptibility via the fluctuation-dissipation theorem $P(\\omega )\\approx 2{k_{B}T}\\chi ^{\\prime \\prime }/{\\omega }$ .",
"Assuming all spins couple to the SQUID equally [31], the imaginary part of the inductance then relates to the spin susceptibility within a layer of thickness $d=\\rho /\\tilde{n}$ on the surface as $L^{\\prime \\prime }=\\mu _{o}d\\frac{R}{r}\\chi ^{\\prime \\prime }$ , therefore $P_{L}(\\omega ) &=&\\left( \\mu _{o}d\\frac{R}{r}\\right) ^{2}\\int _{0}^{\\infty }\\langle \\chi (0)\\chi (t)\\rangle e^{\\imath \\omega t}dt $ and from the fluctuation dissipation theorem, $\\chi ^{\\prime \\prime }(\\omega )=2\\frac{\\tilde{n}\\mu _{o}\\mu _{B}^{2}\\omega }{k_{B}T}\\displaystyle \\sum _{i,j}\\displaystyle \\int _{0}^{\\infty }\\langle s_{i}(0)s_{j}(t)\\rangle e^{\\imath \\omega t}dt.$ It is argued here that the sum of all two-point correlation functions for the system of interacting spins can always be expressed as $\\sum \\langle {s_{i}(0)s_{j}(t)}\\rangle =\\sum {C_{\\nu }e^{-2\\Gamma _{\\nu }t}}$ , which is shown analytically for two spins (see Eq.REF ) and systematically verified for more numerically.",
"Hence $\\chi ^{\\prime \\prime }(\\omega )=2\\tilde{n}\\mu _{o}\\mu _{B}^{2}\\frac{\\omega }{k_{B}T}\\sum _{\\nu }\\frac{C_{\\nu }\\Gamma _{\\nu }}{\\Gamma _{\\nu }^{2}+\\omega ^{2}}$ and the real part is $\\chi ^{\\prime }(\\omega )=\\frac{2}{\\pi }{\\mathcal {P}}\\displaystyle \\int _{0}^{\\infty }\\frac{\\chi ^{\\prime \\prime }(\\omega ^{\\prime } )}{\\omega ^{\\prime 2}-\\omega ^{2}}d\\omega ^{\\prime }=\\frac{2\\tilde{n}\\mu _{o}\\mu _{B}^{2}}{k_{B}T}\\sum _{\\nu }\\frac{C_{\\nu }\\Gamma _{\\nu }^{2}}{\\Gamma _{\\nu }^{2}+\\omega ^{2}}~$ where, $\\mathcal {P}$ is Cauchy's principal value.",
"Hence from the total susceptibility $\\chi (\\omega )=\\chi ^{\\prime }(\\omega )+i\\chi ^{\\prime \\prime }(\\omega )$ , $\\chi (t)=\\int _{0}^{\\infty }\\chi (\\omega )e^{\\imath \\omega t}d\\omega =\\frac{2\\tilde{n}\\mu _{o}\\mu _{B}^{2}}{k_{B}T}\\sum _{i,j}\\langle s_{i}(0)s_{j}(t)\\rangle .~~~$ The inductance noise can then be explicitly expressed in terms of the spectral density of the dynamical four-point noise correlation functions[27], $P_L(\\omega ) = \\left(2\\rho \\frac{\\mu _o^2\\mu _B^2}{k_BT} \\frac{R}{r}\\right)^2 \\displaystyle \\iint \\limits _{\\omega _{a}}^{~~~~\\omega _{b}}S^{[2]}(\\omega ,\\omega _1,\\omega _2)d\\omega _1 d\\omega _2~~~~~~$ where $S^{[2]}(\\omega ,\\omega _1,\\omega _2)=\\iiint \\limits _0^{~~~~\\infty }\\sum \\langle s_i(t_1)s_j(t_2)s_k(t_3)s_l(t_4)\\rangle $ $e^{\\imath (\\omega _1-\\omega )\\tau ^{\\prime }}e^{\\imath (\\omega _2+\\omega )\\tau ^{\\prime \\prime }}e^{\\imath \\omega \\tau }d\\tau ^{\\prime } d\\tau ^{\\prime \\prime } d\\tau $ here $\\tau ^{\\prime }=t_2-t_1$ , $\\tau ^{\\prime \\prime }=t_4-t_3$ and $\\tau =t_4+t_3-t_2-t_1$ .",
"$\\Delta \\omega =\\omega _{b}-\\omega _{a}$ is the bandwidth within which the second spectrum is observed Figure: (a) Power-spectrum of the inductance noise, P L P_L (associated noise of P φ P_\\phi of fig.)",
"and (b) its respective slope (α\\protect \\alpha ) for the spin cluster model with ferromagnetic (+|J o |)(+|J_o|) RKKY interactions.The integrated α\\left\\langle \\alpha \\right\\rangle between 0.001-0.05 Hz is {1.569,1.415,1.247,1.242} for the respective temperatures in ascending order.The temperature-dependent inductance noise spectrum and its slope is shown in fig.REF , where $\\Delta \\omega $ is set to cover the full spectrum.",
"The noise power spectrum now shows $1/f^{\\alpha }$ behavior at intermediate frequencies where the average integrated $\\alpha $ between $0.001-0.05~Hz$ varies from $\\sim 1.57$ (at $200~mK$ ) to $\\sim 1.24$ (at $500~mK$ ).",
"And for even higher temperatures $\\alpha \\rightarrow 0$ .",
"The $\\alpha =4$ at high frequencies is due to the square of the Lorentzian tail while at the lowest frequencies $\\alpha $ eventually rolls over to zero.",
"This onset of gaussian noise type behavior at low frequencies is again determined by the temperature.",
"Overall the temperature dependent $1/f$ inductance noise behavior for the spin cluster model used here agrees very well with experiment[5], [7].",
"Flux-Inductance-Noise Cross-Correlation: Finally, the SQUID's surface spins spins show a net polarization in the experiments[5] as the $1/f$ inductance noise was found to be highly correlated with the $1/f$ flux noise.",
"The following expression gives the flux- and inductance noise cross power spectrum $P_{L\\phi }(\\omega )=\\frac{1}{k_{B}T}\\left( 2\\rho \\mu _{o}^{2}\\mu _{B}^{2}\\frac{R}{r}\\right)^{\\frac{3}{2}}\\times ~~~~~~~~~~~~~\\\\\\nonumber \\int \\limits _{\\omega _{a}}^{\\omega _{b}}\\iint \\limits _{0}^{~~~~\\infty }\\displaystyle \\sum _{i,j,k}\\langle s_{i}(t_{1})s_{j}(t_{2})s_{k}(t_{3})\\rangle e^{\\imath \\omega _{-}\\tau }e^{\\imath \\omega _{+}\\tau ^{\\prime }}d\\tau d\\tau ^{\\prime }d\\omega ^{\\prime }$ where, $\\tau =t_{2}-t_{1}$ , $\\tau ^{\\prime }=t_{3}-t_{2}$ , $\\omega _{\\pm }=\\omega \\pm \\omega ^{\\prime }$ and $\\omega _{b}-\\omega _{a}$ defines the bandwidth.",
"In the experiments $P_{L\\phi }$ was found to be inversely proportional to temperature and about $\\sim 1$ at temperatures roughly below $100mK$ .",
"Now $P_{L\\phi }$ depends on the sum of all three-point auto- and cross-correlation functions(TPCF).",
"As inductance is even under time inversion and magnetic flux is odd, the flux-inductance-TPCF can only be nonzero if time reversal symmetry is broken – indicating the appearance of long range magnetic order.",
"This indicates that the interactions must be ferromagnetic.",
"To show this, the TPCF (for all possible spin combinations) is calculated by for a single cluster of 10 spins with ferromagnetic RKKY interactions, which gives $\\sum \\langle {s_{i}s_{j}s_{k}}\\rangle _{max}\\sim 1$ at low temperatures and keeps decreasing as the temperature is increased (see Fig.REF ).",
"This is in excellent agreement with experiment and also verifies that the mechanism mechanism produces both the flux noise and inductance noise.",
"Whereas for antiferromagnetic RKKY interactions, $\\sum \\langle {s_{i}s_{j}s_{k}}\\rangle _{max}\\sim 0$ .",
"Figure: Normalized sum of all three-point correlation functions, ∑〈s i (0)s j (t 1 )s k (t 2 )〉/N 3 \\sum \\langle {s_i(0)s_j(t_1)s_k(t_2)}\\rangle /N^3, indicating flux-inductance-noise crosscorrelation for a single cluster of N=10N=10spins with ferromagnetic (+|J o |+|J_o|) RKKY interactions at temperatures of (a) T=200mKT=200~mK (b) T=300mKT=300~mK and (c) T=400mKT=400~mK.Summary: Overall, previously unexplained experimentally observed features of the temperature dependent $1/f^\\alpha $ magnetization noise in SQUIDs is explained by an Ising-Glauber spin-cluster model.",
"The inductance noise is inherently $T^{-2}$ dependent while the flux noise is not.",
"A general method is introduced for obtaining $n-$ point correlation functions and various spectral functions subsequently.",
"Explicit flux-inductance cross-correlation function calculations suggest that ferromagnetic RKKY interactions between $F^+$ centers at the metal-insulator interface are the most likely cause of the observed long range magnetic ordering of the TLSs.",
"I wish to thank Robert Joynt for a number of invaluable discussions and for his comments on this work.",
"I would like to thank Robert McDermott for carefully explaining the experiments.",
"And I would like to thank Leonid Pryadko for his support.",
"This work was done partially under DARPA-QuEst Grant No.",
"MSN118850 and presently with the support of U.S. Army Research Office Grant No.",
"W911NF-11-1-0027 and NSF Grant No.",
"1018935."
]
] | 1403.0124 |
[
[
"Particle methods enable fast and simple approximation of Sobolev\n gradients in image segmentation"
],
[
"Abstract Bio-image analysis is challenging due to inhomogeneous intensity distributions and high levels of noise in the images.",
"Bayesian inference provides a principled way for regularizing the problem using prior knowledge.",
"A fundamental choice is how one measures \"distances\" between shapes in an image.",
"It has been shown that the straightforward geometric L2 distance is degenerate and leads to pathological situations.",
"This is avoided when using Sobolev gradients, rendering the segmentation problem less ill-posed.",
"The high computational cost and implementation overhead of Sobolev gradients, however, have hampered practical applications.",
"We show how particle methods as applied to image segmentation allow for a simple and computationally efficient implementation of Sobolev gradients.",
"We show that the evaluation of Sobolev gradients amounts to particle-particle interactions along the contour in an image.",
"We extend an existing particle-based segmentation algorithm to using Sobolev gradients.",
"Using synthetic and real-world images, we benchmark the results for both 2D and 3D images using piecewise smooth and piecewise constant region models.",
"The present particle approximation of Sobolev gradients is 2.8 to 10 times faster than the previous reference implementation, but retains the known favorable properties of Sobolev gradients.",
"This speedup is achieved by using local particle-particle interactions instead of solving a global Poisson equation at each iteration.",
"The computational time per iteration is higher for Sobolev gradients than for L2 gradients.",
"Since Sobolev gradients precondition the optimization problem, however, a smaller number of overall iterations may be necessary for the algorithm to converge, which can in some cases amortize the higher per-iteration cost."
],
[
"Introduction",
"Computational analysis of microscopy images has become a key step in many biological studies.",
"It enables processing ever-larger sets of images at high throughput, improves reproducibility, and enables image-based modeling and simulation of biological systems [1], [2], [3].",
"Additionally, computational image analysis methods can sometimes detect signals that the human eye cannot see [4].",
"Biological microscopy image data, however, come with their own set of challenges: They are usually diffraction limited and recorded at low signal-to-noise ratios (SNR), in order to minimize photo-toxic effects in the sample.",
"In addition, the trend is to acquire three or even higher-dimensional images using microscopy techniques such as Selective Plane Illumination Microscopy (SPIM) [5].",
"The noise in these images is often not Gaussian.",
"In fluorescence microscopy, only the fluorescently labeled structures are visible in the image, whereas other structures that are present in the sample are not imaged.",
"This limits system observability.",
"For these and other reasons, bio-image analysis tasks tend to be highly ill-posed.",
"The strong ill-posedness of bio-image analysis problems requires additional regularization.",
"This can be done in a biologically meaningful way by including prior knowledge about the imaged system and the imaging process into the analysis algorithms.",
"This prior knowledge constrains the set of possible solutions to biologically and physically feasible ones, hence regularizing the analysis task.",
"Bio-image analysis aims to extract quantitative information from higher-dimensional low-signal images, exploiting strong prior knowledge [2].",
"Bayesian inference provides a principled way for including prior knowledge into image-analysis algorithms.",
"In Bayesian image analysis, one aims to maximize the posterior probability of the analysis result to be correct, given the observed image (see Methods section).",
"Using Bayes' formula, the posterior is expressed as the product of the likelihood of observing the image and the prior probability of the result (see Methods section).",
"In image segmentation, deformable models provide for a straightforward implementation of Bayesian, or “model-based”, methods [6], [7].",
"In this class of methods, the contours of the objects in the image are represented using models from computational geometry, such as splines [8], level sets [9], or triangulated surfaces [10].",
"These models then deform and move over the image so as to maximize the posterior.",
"The evolution is driven by an optimization algorithm.",
"In segmentation of fluorescence microscopy images, the prior knowledge that has been used to improve the quality and robustness of the results includes: (1) the point-spread function of the microscope to describe how the image has been acquired and to yield “deconvolving segmentations” [11], [12], [13], [14], [15]; (2) the statistical distribution of the noise in the image in order to produce optimally denoised segmentations [16], [15]; (3) topological priors about the connectedness of regions in the image [14]; (4) physical priors about the mechanics of the imaged objects [17], [11], [12]; (5) the expected shape of the imaged objects [18], [19]; (6) the expected color, texture, or motion of the imaged objects [20], etc.",
"A fundamental choice in any Bayesian method is how one measures distances between different shapes or segmentations.",
"This is required by the optimization algorithm in order to perturb the deformable model and quantify the magnitude of this perturbation.",
"Defining the gradient of the posterior hence relies on defining an inner product in the set of perturbations of a deformable model.",
"The most common choice is the geometric $L^2$ -type inner product.",
"This has, however, been shown to lead to a pathological metric in which the “distance” between any two curves is zero [21], [22], [23], [24].",
"As a result, optimizing a deformable model via a $L^2$ -type gradient flow is very sensitive to noise and requires length or curvature regularization of the contour [23].",
"Another undesirable feature of $L^2$ -gradient flows is that they do not distinguish between global (rigid-body) motion of the contour and local deformation [24].",
"The pathological nature of the Riemannian metric induced by the $L^2$ inner product on the space of smooth curves is effectively avoided when using Sobolev gradients [25].",
"This considers the Bayesian posterior to be an element of a Sobolev space [26], [27].",
"In a Sobolev space, the definition of neighborhood is naturally adapted to the segmentation problem [26], [27], [24], [23], [28].",
"This leads to results that are more robust against noise, smoother, and allow for more “natural” contour perturbations.",
"It is well known and has previously been demonstrated that Sobolev gradients are useful in a variety of segmentation and tracking algorithms, since they allow one to use model formulations that would be ill-posed or numerically intractable when using $L^2$ -based gradients [29].",
"Here, we show how a recently introduced particle-based deformable model [14] can be extended to Sobolev gradients.",
"The particle-method nature of the algorithm enables a novel, simple and computationally efficient approximation of Sobolev gradients.",
"We exploit the fact that pairwise particle–particle interactions amount to a discrete convolution [30], [31], [32], [33].",
"Since Sobolev gradients can be computed by convolution of the $L^2$ gradient with a decaying kernel function (see Eq.",
"(13) in Ref.",
"[24]), they can be approximated by local interactions between the contour particles within a certain cutoff radius.",
"This effectively avoids solving a global Poisson equation at each iteration, as was previously necessary [28].",
"In the present particle-based algorithm, Sobolev gradients incur virtually no additional implementation overhead, are computationally efficient, and can easily be parallelized.",
"We provide the mathematical formalism for approximating Sobolev gradients using discrete particle methods, and we show that this approximation preserves the known qualitative properties of Sobolev gradient flows as compared to $L^2$ gradient flows.",
"Deformable models consist of a geometry representation and an evolution law [7].",
"The evolution law acts on the degrees of freedom of the geometry representation [6], [7].",
"In Bayesian methods, this is done such as to maximize the posterior or, equivalently (according to a Boltzmann distribution), minimize the energy functional obtained by taking the negative logarithm of the posterior (see Methods section).",
"Deformable models can be continuous or discrete.",
"In either case they can be implicit (also called geometric models, such as level sets [9]) or explicit (also called parametric models, such as splines [8]).",
"The relationship between explicit and implicit models has previously been studied [34].",
"In continuous deformable models, the contour is represented by a mathematical object that is a continuous function of space.",
"The contour is hence not limited to the pixel grid of a digital image, but may also evolve in sub-pixel steps [11], [12], [15].",
"Discrete representations directly store region labels at grid nodes.",
"Grid nodes usually coincide with pixels.",
"This allows straightforward representation of multiple regions and efficient querying of spatial information.",
"Figure: Discrete contour representation using particles.",
"Shown is the pixel grid with two closed foreground regions (dark and light gray) and the open background region (white).",
"Dots represent particles marking discrete contour points.",
"The contour pixels belong to the respective foreground region and contribute to its region statistics.",
"The bold lines illustrate the contour Γ\\Gamma that is represented by the particles.Here, we consider a discrete deformable model where the contour is represented by computational particles placed in the pixels around which the contour passes.",
"This is illustrated in Fig.",
"REF and provides a geometry representation that is somewhere between explicit and implicit [35].",
"Particles migrate to neighboring pixels in order to deform the contour.",
"All foreground regions are defined as closed sets, i.e., the contour pixels belong to the foreground region.",
"There is only one single background region in the entire image, which is an open set.",
"Foreground regions are constrained to be connected sets of pixels, whereas the background region may be disconnected.",
"See Ref.",
"[14] for details.",
"The Region Competition algorithm [14] is a discrete optimization algorithm to drive the evolution of contours represented by discrete particles.",
"The algorithm can advance an arbitrary number of contours simultaneously and provides topological control over the evolving contours.",
"Topological control ensures that contours remain connected (i.e., “intact” according to the region definition used [14]) and provides control over merging and splitting events between different contours.",
"Local energy minimization is done by gradient descent, approximating the gradient as the energy difference between a perturbed and the original set of particles.",
"The result of the algorithm does not depend on the order in which the particles are processed (or, equivalently, the indexing order of the particles), since the particles are ranked in order of decreasing energy difference before the moves are executed.",
"The RC algorithm readily generalizes to 3D images, as the particle contour representation remains unaltered.",
"RC has been demonstrated on 2D and 3D images using a variety of image models, including piecewise constant, piecewise smooth, and deconvolving models [14].",
"The computational performance of RC is competitive with other state-of-the-art discrete methods, such as graph-cuts [36].",
"For details, we refer to the original publication [14].",
"The original work presenting the RC algorithm used a $L^2$ -type discrete gradient approximation, which is known to frequently get trapped in local minima and be sensitive to noise.",
"Here, we show how Sobolev gradients can be approximated in the same algorithm, and we demonstrate the computational efficiency of the resulting implementation.",
"Sobolev active contours are deformable models whose evolution is driven by the Sobolev gradient flow of the underlying Bayesian energy [26], [27], [24], [29], [28].",
"The original application of Sobolev gradients is the numerical solution of nonlinear partial differential equations [25].",
"In image segmentation, Sobolev gradients have successfully been used for continuous deformable models, in particular for level sets methods.",
"There, the Sobolev gradients are either directly computed on the continuous implicit function representation using variational calculus [28], or the implicit representation is intermediately translated to an explicit (linear spline) representation and the Sobolev gradients approximated there [24].",
"Here, we integrate Sobolev gradients into discrete contour models represented by particles.",
"The RC algorithm amounts to a gradient descent, where the gradient is approximated at discrete points.",
"The original work used a $L^2$ -type gradient [14].",
"This implies that one can only consider energies (image models) belonging to the $L^2$ inner-product function space.",
"This inner product has a number of undesirable properties for deformable models [23].",
"For discrete models, the following two of these properties are of special interest: First, the inner product does not contain any regularity terms.",
"There is hence nothing in the metric that would discourage the emergence and evolution of non-smooth contour/time (hyper) surfaces.",
"Hence, in the presence of noise, the contour becomes non-smooth during evolution, which in term reduces the numerical accuracy of the gradient approximation.",
"Curvature regularization via priors is typically required to prevent this.",
"Second, the $L^2$ -type gradient is ignorant with respect to the type of contour motion, such as global translations or local deformations.",
"Intuitively, the contour therefore locally optimizes “on a small scale” and frequently gets trapped in local optima of the Bayesian posterior.",
"A Sobolev function space is equipped with an inner product that contains $L^p$ -terms and derivatives of the function.",
"The metric on that space induced by this inner product hence includes smoothness terms that allow addressing the regularity issues mentioned above.",
"Using such a metric does not affect the global minimum of the function, but it amounts to preconditioning the gradient flow, hence rendering it less ill-posed.",
"While Sobolev gradients are computationally more expensive to compute, they often result in a smaller overall number of iterations required by the segmentation algorithm to converge to a local optimum.",
"Depending on the problem at hand, the cost of Sobolev gradient approximation may hence be amortized.",
"We consider a Sobolev space $W^{1,2}$ , which is a Hilbert space $H^1$, with inner product [24] $\\langle h,k \\rangle _{H^1} := \\bar{h}\\cdot \\bar{k}+\\epsilon \\cdot E^2\\cdot \\langle \\nabla h,\\nabla k\\rangle _{L^2},$ where $h$ and $k$ are elements of the tangent space of the evolving contour $\\Gamma $ .",
"The tangent space is the set of all possible deformations of $\\Gamma $ .",
"The $\\nabla $ -operator in Eq.",
"(REF ) is with respect to the $L^2$ -norm.",
"The scalar $\\epsilon \\in \\mathbb {R}^{+}$ is a hyper parameter for smoothness and $E\\in \\mathbb {R}^{+}$ determines the length scale of the smoothness terms in the inner product [24].",
"The average $\\bar{h}$ of $h$ over $\\Gamma $ is $\\bar{h}=\\frac{1}{|\\Gamma |}\\int _{\\Gamma } h(s)\\, \\mathrm {d}s \\, ,$ where $s$ is the intrinsic position along the contour, in image coordinates (e.g., pixels).",
"The average $\\bar{k}$ is defined similarly.",
"The $L^2$ inner product is $\\langle h,k\\rangle _{L^2}=\\frac{1}{|\\Gamma |}\\int _{\\Gamma } h(s) k(s)\\, \\mathrm {d}s \\, .$ We present a deformable model where the geometry representation uses a discrete particle method and the the evolution law is given by a Sobolev gradient flow.",
"We use the RC algorithm to numerically optimize the resulting system, and we demonstrate and benchmark its behavior on synthetic and real-world images.",
"We adapt the RC algorithm to use an approximate Sobolev gradient flow to maximize the segmentation model posterior.",
"In Bayesian energy minimization, as described in the Methods section, this requires computing the first-order Sobolev gradient $\\nabla _{H^1}\\mathcal {E}$ of the image energy $\\mathcal {E}$ using the metric induced by the inner product in Eq.",
"(REF ).",
"It has been shown [24] that this amounts to a convolution of the $L^2$ -type gradient $\\nabla \\mathcal {E}$ $\\nabla _{H^1} \\mathcal {E}(s) = \\int _{\\Gamma } \\tilde{K}(\\hat{s}-s)\\cdot \\nabla \\mathcal {E}(\\hat{s})\\mathrm {d}\\hat{s} = (\\tilde{K}*\\nabla \\mathcal {E})(s)$ with convolution kernel $\\tilde{K}(r) = \\frac{1}{E}\\left(1 + \\frac{(|r|/E)^2-(|r|/E)+1/6}{2\\epsilon }\\right),\\quad r\\in \\left[-\\frac{E}{2},+\\frac{E}{2}\\right] \\, .$ Figure REF shows $\\tilde{K}$ for different $\\epsilon $ [24].",
"Equation (REF ) enables computing the first-order Sobolev gradient from the $L^2$ gradient.",
"The convolution domain is $\\Gamma $ .",
"All distances $(\\hat{s}-s)$ are hence geodesic distances along the contour.",
"Kernel $\\tilde{K}$ for different $\\epsilon $ .",
"The solid, dotted, and dashed curves show $\\tilde{K}$ for $\\epsilon =1/24$ , $\\epsilon =0.06$ , and $\\epsilon =0.08$ , respectively.",
"The kernel becomes local (decays to zero at $r=\\pm E/2$ ) for $\\epsilon =1/24$ .",
"Figure: Sobolev gradient kernel K ˜\\tilde{K} for different ϵ\\epsilon .",
"The solid, dotted, and dashed curves show K ˜\\tilde{K} for ϵ=1/24\\epsilon =1/24, ϵ=0.06\\epsilon =0.06, and ϵ=0.08\\epsilon =0.08, respectively.",
"The kernel becomes local (decays to zero at r=±E/2r=\\pm E/2) for ϵ=1/24\\epsilon =1/24.The parameter $E$ controls the length scale of the inner product.",
"It is often set equal to the total contour arc-length $|\\Gamma |$ in order to obtain a scale-invariant inner product [27], [24].",
"Gradient information is then shared along the entire contour, which enables global (rigid-body) contour movements, such as translations and rotations.",
"However, this also introduces a global coupling into the computations, requiring the solution of a global Poisson equation [28] or convolution [27], [24] at each iteration of the algorithm, which is computationally expensive.",
"Here, we fix $E$ and thus impose a scale with respect to the image coordinate system.",
"This makes sense in bio-image segmentation, where the length-scale of the objects of interest is often known.",
"$E$ hence becomes a user-defined free parameter of the method.",
"We find $E\\approx 10\\ldots 12$ to work well on the examples presented hereafter.",
"We always use $\\epsilon =\\frac{1}{24}$ in order to get a kernel that decays to zero at $\\pm E/2$ , rendering particle–particle interactions local (see Fig.",
"REF ).",
"Using this kernel, a length scale of $E$ of 10 to 12 hence corresponds to a particle–particle interactions radius of 5 to 6 pixel.",
"As we show below, this leads to a simple and fast approximation of Sobolev gradients, which is accurate enough to retain the favorable properties of Sobolev gradient flows.",
"In a discrete particle representation of the contour, the domain of the convolution in Eq.",
"(REF ) is the set of all particles representing that contour.",
"In RC, the particles store a discrete $L^2$ -type gradient approximation on both sides of the contour $\\Gamma $ , we can thus approximate the Sobolev gradient by discretizing Eq.",
"(REF ) over particles [30].",
"Let $\\mathcal {Q}_p$ be the set of particles that are located in the support of $\\tilde{K}$ and belong to the same contour as particle $p$ , i.e., have region label $l=l_p$ , thus: $\\mathcal {Q}_p =\\lbrace q|\\,d(x_q,x_p)<E/2, \\, l_p = l_q\\rbrace $ .",
"Similarly, let $\\mathcal {Q}^{\\prime }_p$ be the particles within the kernel support that lie on the other side of the contour, i.e., $\\mathcal {Q}^{\\prime }_p =\\lbrace q|\\,d(x_q,x_p)<E/2, \\, l_p = l^{\\prime }_q\\rbrace $ .",
"For each particle $p$ we then compute the energy difference as: $\\Delta \\mathcal {E}_p \\leftarrow \\frac{1}{|\\mathcal {Q}_p|}\\sum _{q\\in \\mathcal {Q}_p} \\tilde{K}(d(x_q,x_p))\\Delta \\mathcal {E}_q - \\frac{1}{|\\mathcal {Q}^{\\prime }_p|}\\sum _{q\\in \\mathcal {Q}^{\\prime }_p} \\tilde{K}(d(x_q,x_p))\\Delta \\mathcal {E}_q \\, .$ This amounts to local pairwise particle–particle interactions, as they are commonplace in particle methods, where a convolution is approximated by a sum over particles [30], [31], [32], [33].",
"We use cell lists [30] with a cell edge-length equal to the interaction cutoff radius of $E/2$ in order to efficiently find the neighbors (interaction partners) of each particle.",
"This reduces the average time complexity of the discrete convolution from $O(N^2)$ to $O(N)$ for a total of $N$ contour particles.",
"If both terms in Eq.",
"(REF ) have the same sign, the approximated Sobolev gradients on both sides of the contour have opposite directions.",
"This happens at an extremum of the energy.",
"Since the discrete contour representation does not allow sub-pixel deformations, the contour then oscillates.",
"This, however, is easily detected [14] and the optimizer switches back to using $L^2$ -type gradients when oscillations occur.",
"According to Eq.",
"(REF ), the distance between two particles on the contour $\\Gamma $ should be the geodesic arc distance $(\\hat{s} - s)$ between the two points where the particles are located.",
"The present discrete representation, however, does not allow computing this quantity.",
"Since we consider a relatively small support of $\\tilde{K}$ ($E/2\\approx 5$ pixel), and objects in biological microscopy images are usually smoother than that, we approximate geodesic distances by Euclidean distances $d(\\cdot ,\\cdot )$ .",
"This approximation breaks down when contours in the image significantly curve on length scales below $E$ .",
"One may then argue, however, that a continuous contour representation is anyway advised [11], [12].",
"We demonstrate the computational efficiency of particle-approximated Sobolev gradients in the RC algorithm [14] by comparing with a previous mesh-based level-set implementation [28].",
"We also illustrate that the present approximation retains the known favorable properties of Sobolev gradient flows by qualitatively comparing with segmentations obtained using approximated $L^2$ gradients for a range of regularization parameters $\\lambda $ (see Methods section for an explanation of the meaning of $\\lambda $ ).",
"All benchmarks are performed using a C++ implementation of RC with $L^2$ and Sobolev gradients, run on a single 2.67 GHz Intel i7 core with 4 GB RAM using the Intel C++ compiler (v. 12.0.2).",
"Figure: Sobolev flows favor smooth contour evolution.",
"We show the region evolution using L 2 L^2 and Sobolev gradient approximations for the piecewise smooth deconvolving energy described in Ref. .",
"(A) Ground-truth data with overlaid Gaussian point-spread function of width σ=15\\sigma =15 pixels.",
"(B) Blurred image convolved with the point-spread function.",
"(C) Convolved image with Poisson noise of PSNR 6.",
"(D) Initialization for both algorithms.",
"(E–H) RC segmentations after 15, 59, 88, and 500 iterations using L 2 L^2-type gradients.",
"(I–L) RC segmentations after 15, 59, 88, and 282 iterations using Sobolev gradients.",
"The Sobolev gradient flow is more regular and converges faster.",
"The same λ=0.04\\lambda =0.04 is used for both flows.",
"The energy landscapes are hence identical, permitting direct comparison between optimization trajectories.The metric induced by the inner product in a Sobolev space includes smoothness terms that favor smooth contour evolution, whereas the $L^2$ -type gradient flow tends to produce non-smooth contours [24], [29].",
"We use an artificial deconvolution problem to illustrate that the present particle approximation retains this property.",
"Figure REF C shows the artificial data, which is generated by blurring the ground-truth scene in Fig.",
"REF A with the point-spread function shown as a bright spot.",
"This yields the noise-free image shown in Fig.",
"REF B.",
"The input image for RC is obtained by adding modulatory Poisson noise (Fig.",
"REF C), modeling a common situation in fluorescence microscopy.",
"From this image, the algorithm should reconstruct the denoised, deblurred, and segmented objects of the ground truth.",
"The RC algorithm is always initialized with 25 bubbles on a Cartesian grid as show in Fig.",
"REF D. Intermediate results during energy minimization are shown in Figs.",
"REF E–G and REF I–K for $L^2$ and Sobolev gradient flows, respectively.",
"During the first 15 iterations, regions evolve almost identically for both gradient types (Figs.",
"REF E and I).",
"After 59 iterations, the Sobolev-gradient approach is closer to the final solution (Figs.",
"REF F and J).",
"After 88 iterations, it starts oscillating and therefore falls back to the $L^2$ -gradient mode (Figs.",
"REF G and K).",
"It converges after 282 iterations (Fig.",
"REF L).",
"The $L^2$ -gradient approach converges after 500 iterations (Fig.",
"REF H).",
"The results for both algorithms are shown in Figs.",
"REF H and REF L. As expected, they are visually indistinguishable, because changing the gradient flow (at constant $\\lambda =0.04$ for both flows) does not change the location of the energy minimum.",
"However, the Sobolev flow uses a different optimization trajectory, which produces intermediate contours that are smoother than those generated by the $L^2$ flow.",
"This leads to faster convergence and lower computational cost, since the the computational cost of the RC algorithm is proportional to the total contour length [14].",
"The average iteration times are 0.74 s with and 0.73 s without Sobolev gradients.",
"Figure: Comparison with the mesh-based implementation of Renka.",
"We compare segmentations obtained using the present particle-based Sobolev gradient approximation with the fully accurate mesh-based method of Renka .",
"We use the complete set of all four example images considered by Renka (A–D).",
"The raw images are shown in the top row, the final segmentation masks in the bottom row.",
"We refer to Ref.",
"for the results obtained with the mesh-based method.",
"In all cases, we initialize the present algorithm using a single rectangular region.",
"Cases (A–C) use a piecewise constant region intensity model, (D) a piecewise smooth one.",
"Curvature regularization is used in all cases with (A) λ=0.01\\lambda = 0.01, (B) λ=0.04\\lambda = 0.04, (C) λ=0.04\\lambda = 0.04, (D) λ=0.05\\lambda = 0.05.Next, we compare the present particle approximation with a fully accurate mesh-based Sobolev solver, both in terms of the solutions found and in terms of computational cost.",
"In Fig.",
"REF we consider the same four test images as were used to benchmark the mesh-based level-set implementation of Renka [28].",
"The results obtained with the present particle method are visually indistinguishable from the results shown in Ref.",
"[28], and even seem to be slightly better for the Airplane image.",
"The present particle approximation does hence not seem to have a detectable adverse effect on the solution quality compared with a fully resolved mesh-based level-set method.",
"The present implementation using particle–particle interactions, however, leads to significant computational savings, as shown in Table REF .",
"Compared with the full-grid method, the present algorithm is between 3.6 and 17 times faster.",
"Compared with the efficient narrow-band level-set implementation, the present particle method is still between 2.8 and 10 times faster.",
"The present method is also easier to implement, as it does not require programming an additional Poisson solver.",
"Table: Runtime comparisons (in seconds per iteration) of the present particle-based implementation as compared with those reported by Renka for a mesh-based level-set implementation of Sobolev gradients .One case where Sobolev gradients are popular is for segmenting very noisy images, where the amount of regularization that would be required in an $L^2$ flow is so large that it would destroy the solution [24], [29].",
"In Fig.",
"REF we show an example of such as case, a microscopy image of fluorescently labeled cells at low signal-to-noise ratio (SNR).",
"The example illustrates that the present implementation retains the known property that Sobolev gradients keep the contour smooth despite the noise, resulting in less noise-sensitive segmentations, as shown in Fig.",
"REF B.",
"The $L^2$ -gradient flow requires strong curve-length penalization in order to sufficiently regularize the contour (here, $\\lambda =3$).",
"However, this fails in the present example, as the regions collapse when the regularization parameter is increased to that level, as shown in Fig.",
"REF A.",
"For the same $\\lambda $ , this is not the case for the Sobolev flow (Fig.",
"REF B).",
"The present example is hence impossible to segment using a piecewise constant image model with $L^2$ -type gradients.",
"Figure: Sobolev gradients allow segmenting very noisy images, as shown here for fluorescently labeled cells using the piecewise constant energy from Ref. .",
"Region fusions are disallowed and the regions are initialized at local intensity maxima after image blurring.",
"(A) Final RC segmentation when using L 2 L^2-type gradient approximations.",
"(B) Final RC segmentation when using Sobolev gradient approximations.",
"In both cases λ=3\\lambda =3.Figure: Sobolev gradient flows are less sensitive to the choice of regularization parameter.",
"Comparison of segmentation results for a noisy part of a Drosophila melanogaster wing disc tissue using the L 2 L^{2}-type gradient approximation (D–F) and the Sobolev gradient approximation (G–I).",
"The RC algorithm is used with a piecewise smooth energy and a Poisson noise model .",
"The initial segmentation is obtained by Otsu thresholding.",
"The final foreground region is the set union of all foreground regions found by RC.",
"(A) Raw image with cell membranes fluorescently labeled.",
"Image courtesy of Prof. Christian Dahmann, Technical University of Dresden.",
"(B) Result using Sobolev gradients until spatial convergence.",
"Oscillations are detected at iteration 23 and the algorithm switches to L 2 L^2 gradients, leading to a less smooth result after the next iteration (C).",
"(D) RC result with L 2 L^{2} gradients and length regularization λ=1\\lambda =1.",
"(E) RC result with L 2 L^{2} gradients and λ=2\\lambda =2.",
"(F) RC result with L 2 L^{2} gradients and λ=3\\lambda =3.",
"(G) RC result with Sobolev gradients and λ=1\\lambda =1.",
"(H) RC result with Sobolev gradients and λ=2\\lambda =2.",
"(I) RC result with Sobolev gradients and λ=3\\lambda =3.Sobolev gradients precondition the optimization problem and lead to less ill-posed inverse problems [24], [29].",
"This renders the result less sensitive to the choice of regularization constant $\\lambda $ , i.e., giving more weight to the image data and less weight to the Bayesian prior (see Methods section).",
"This is desirable since it frees the user of the tedious parameter tuning of finding a “good” $\\lambda $ for a given image.",
"We illustrate that the present fast implementation retains the property of being robust against the choice of $\\lambda $ .",
"We do so by comparing results obtained with the RC algorithm with $L^2$ -type and Sobolev gradient approximations for different values of the regularization constant (higher $\\lambda $ means stronger regularization and more weight to the prior).",
"The results are shown in Fig.",
"REF for fluorescently labeled cell membranes of a small group of four cells in a fly wing disk, acquired at low SNR (raw data courtesy of Prof. Ch.",
"Dahmann, TU Dresden).",
"The fluorescence signal varies inhomogeneously across the tissue (see Fig.",
"REF A), which is why we use a piecewise smooth image model in this case [14].",
"We also assume Poisson noise in the image, since shot noise is the dominant noise source in confocal fluorescence microscopy.",
"Figure REF A shows the raw data.",
"After 23 iterations, the RC algorithm with Sobolev gradient converged to a point where the contour oscillates around a local energy minimum (Fig.",
"REF B).",
"The algorithm hence switches to $L^2$ gradients, leading to a significantly less smooth contour after the subsequent iteration, i.e., after one step of $L^2$ -gradient descent (Fig.",
"REF C).",
"Nevertheless, the result is still smoother than when using $L^2$ gradients from the beginning (Fig.",
"REF D; 39 iterations required to converge), indicating that the Sobolev gradient flow converged to a different local minimum than the $L^2$ flow.",
"When increasing the regularization constant $\\lambda $ , the $L^2$ results become smoother, but image features start to be missed, leading in to the formation of a “hole” in membrane of the second cell from the left (Figs.",
"REF E–F).",
"It is hence very important for $L^2$ flows that the user properly tunes $\\lambda $ by trial and error in order to obtain a topologically correct segmentation.",
"The final segmentations when using Sobolev gradients are less sensitive to changes in $\\lambda $ and yield the correct topology of the segmentation mask in all tested cases (Figs.",
"REF G–I).",
"When using Sobolev gradients, the contour is also less sensitive to noise and good segmentation results are obtained already for smaller regularization constants than when using $L^2$ -type gradients.",
"This is beneficial in practice, as it frees to user of some of the tedious parameter tuning involved in finding a good $\\lambda $ for a given image.",
"The present particle approximation retains this known property of Sobolev flows.",
"The per-iteration CPU time is 0.01 s when using $L^2$ gradients and 0.02 s for Sobolev gradients.",
"Figure: Sobolev gradients relax the tradeoff between solution smoothness and completeness.",
"Comparison of segmentation results for filaments in a cyst in Drosophila melanogaster using the L 2 L^{2}-type gradient approximation (B–C) and the Sobolev gradient approximation (D).",
"The RC algorithm is used with a piecewise smooth energy and a Poisson noise model .",
"The initial segmentation is obtained by Otsu thresholding.",
"(A) Raw data with spectrin fluorescently labeled.",
"Images courtesy of Dr. Guillaume Salbreux, Max Planck Institute for the Physics of Complex Systems, Dresden.",
"(B) RC result with L 2 L^{2} gradients and λ=0.5\\lambda =0.5.",
"(C) RC result with L 2 L^{2} gradients and λ=1.5\\lambda =1.5 (D) RC result with Sobolev gradients and λ=0.5\\lambda =0.5.In any estimation problem, there is a fundamental tradeoff between completeness (fitting the data as completely as possible) and smoothness (generalization; not over-fitting the noise in the data).",
"This is also true in image segmentation.",
"The regularization parameter $\\lambda $ allows the user to tune this tradeoff.",
"It is known that the increased robustness of Sobolev gradients against the choice of $\\lambda $ comes from the fact that the optimization problem is pre-conditioned in a way that relaxes this tradeoff [24], [29].",
"We illustrate next that the present particle approximation shares this property of relaxing the tradeoff between contour smoothness and not missing weak image features (see Fig.",
"REF ).",
"The raw data in Fig.",
"REF A shows a part of a cyst in a fruit fly with the cytoskeletal protein spectrin fluorescently labeled.",
"The RC segmentation results using $L^2$ -type gradients are shown in Figs.",
"REF B–C.",
"For a small $\\lambda $ , the contour is non-smooth, as it delineates noise (Fig.",
"REF B; over-fitting).",
"For even smaller $\\lambda $ the $L^2$ gradient flow does not converge due to insufficient regularization.",
"This is hence the most-connected segmentation one can obtain on this image when using $L^2$ gradients.",
"Increasing the regularization constant $\\lambda $ makes the contour smoother, but causes filaments to break and to go missing, as shown in Fig.",
"REF C. We use a piecewise smooth image model here, which is able to handle intensity variations within a foreground region.",
"The missing and broken filaments in the center of the image in Fig.",
"REF C are hence not a problem of photometry, but are caused by the $L^2$ gradient flow being trapped in a local minimum due to noise.",
"This is the reason for the above-mentioned tradeoff between smoothness and completeness.",
"It is known that this tradeoff is relaxed when using Sobolev gradients, leading to smoother and more connected segmentations already for low regularization constants [24], [29], as confirmed in Fig.",
"REF D. Albeit in the present case the segmentation remains unsatisfactory from a biological point of view, the present fast particle approximation seems to preserve the property of Sobolev flows to relax the tradeoff between smoothness and completeness of a segmentation.",
"For $\\lambda =0.5$ , the per-iteration CPU time was 0.04 s for $L^2$ gradients and 0.08 s for Sobolev gradients.",
"The ratios for other $\\lambda $ 's were similar.",
"Figure: Extension to 3D and segmentation using weaker regularization.",
"Comparison of segmentation results for a 3D confocal microscopy stack of a noisy part of the Drosophila melanogaster wing disc using L 2 L^2-type gradients (B–C) and Sobolev gradients (D).",
"The RC algorithm is used with a piecewise smooth energy and a Poisson noise model .",
"The initial segmentation is obtained by Otsu thresholding.",
"(A) Volume rendering of the 3D raw image with cell membranes fluorescently labeled.",
"Image courtesy of Prof. Christian Dahmann, Technical University of Dresden.",
"(B) RC result with L 2 L^{2} gradients and λ=0.1\\lambda =0.1.",
"(C) RC result with L 2 L^{2} gradients and λ=5\\lambda =5.",
"(D) RC result with Sobolev gradients and λ=0.01\\lambda =0.01.",
"When using L 2 L^2 gradients, the algorithm did not converge for λ=0.01\\lambda =0.01 due to insufficient regularization.We next show that the present particle-based contour representation naturally extends to 3D images and allows segmentation using much less regularization than what would be required by an $L^2$ flow to generate comparable results.",
"This is important in practice, because the regularizer is often chosen ad hoc with no biological meaning.",
"One hence wishes to keep $\\lambda $ as small as possible in order to limit the regularization bias in the result, but still $\\lambda $ has to be chosen large enough so the solution does not over-fit the noise in the image.",
"Figure REF shows a 3D example with a piecewise smooth image model.",
"Using Sobolev gradients yields a smoother and less noisy segmentation with smaller $\\lambda $ , as shown in Fig.",
"REF D. For the same value of $\\lambda =0.01$ , the $L^2$ gradient descent does not converge due to insufficient regularization.",
"The smallest regularization for which the $L^2$ flow produces a stable result is $\\lambda =0.1$ .",
"This result is shown in Fig.",
"REF B.",
"It is completely irregular and dominated by noise.",
"In order to achieve a result that is qualitatively comparable to that in Fig.",
"REF D, the $L^2$ flow requires a 500-fold(!)",
"stronger regularization than the Sobolev flow (Fig.",
"REF C).",
"This massive regularization, however, introduces a visible bias in the final result.",
"Some of the loops and geometric features in the tissue are broken (Fig.",
"REF C).",
"At $\\lambda =0.1$ , RC with Sobolev gradients requires 359 iterations to converge, with $L^2$ gradients 896 iterations.",
"The per-iteration CPU time was 8 s when using $L^2$ gradients and 11 s when using Sobolev gradients.",
"Sobolev gradients hence segment this image about twice faster than $L^2$ gradients.",
"Figure: Sobolev gradients are less prone to converge to noisy local minima, as shown here for shoot apical meristems of Arabidopsis thaliana labeled with a fluorescent protein that localizes to the plasma membrane.",
"(A) Raw image of size 756×622756\\times 622 pixels (source: ).",
"(B) Initial segmentation using Otsu thresholding.",
"(C) Segmentation result using L 2 L^2 gradients.",
"(D) Segmentation result using approximated Sobolev gradients.",
"In both cases λ=3\\lambda =3, hence keeping the energy landscape the same.Another known property of Sobolev gradients is that they lead to final segmentations with less “holes”, as the flow can overcome small local minima that are induced by noise.",
"Figure REF illustrates that the present particle approximation retains this property.",
"The figure shows an image of fluorescently labeled plant tissue at low SNR (data: [37]).",
"Even for the same $\\lambda =3$ , the Sobolev flow leads to contours that are more connected and less broken (Fig.",
"REF D) than those obtained by the $L^2$ flow (Fig.",
"REF C).",
"Since the same $\\lambda $ is used in both cases, this can only be because the Sobolev flow converges to a different, less noisy local minimum.",
"This present implementation hence retains this property.",
"RC with $L^2$ and gradients needed 106 iterations (254 s) to converge, with Sobolev gradients 200 iterations (549 s).",
"This is hence an example of where the Sobolev flow converges slower, but finds a different, better local minimum.",
"Figure: Sobolev gradients require no additional regularization on images with little noise, as shown here for segmentations of proton emission patterns without additional regularization.",
"(A–C) Raw data (images: Josefine Metzkes, Bussmann group, Helmholtz Center Dresden Rossendorf).",
"(D–F) RC segmentations using a piecewise smooth image model and Sobolev gradients with λ=0\\lambda =0.",
"(G–I) Skeletonization of the segmentation results in order to extract filament patterns.Since Sobolev flows require less additional regularization, they are also popular to process images that contain very little or no noise.",
"In those cases, Sobolev flows sometimes allows segmenting the image with no additional regularization at all.",
"We present such an example in Fig.",
"REF .",
"It is not a microscopy example, because microscopy images always contain significant noise.",
"Instead, the image shows proton emission patterns that occur when a high-energy laser beam hits a block of metal (data: Bussmann group, Helmholtz Center Dresden Rossendorf).",
"We initialize the RC algorithm with small bubbles around local intensity maxima.",
"Since intensities are varying within filament patterns, we use a piecewise smooth energy model [14].",
"The images are almost noise-free, and the present fast particle-Sobolev flow allows segmenting them with $\\lambda =0$ , i.e., without any additional regularization.",
"This absence of regularization maximizes the detection ability for dark filaments, a property that is retained by the present particle approximation.",
"However, the final result is sensitive to the initialization, and the topological prior that a region is a connected component is necessary in this case [14].",
"In summary, the examples shown here confirm that the present approximation of Sobolev gradients using local particle interactions qualitatively retains the known qualitative properties of Sobolev flows and does not seem to have a detectable adverse effect on solution quality.",
"However, the local character of the computations and the simple implementation as particle–particle interactions provide savings in both runtime and programmer effort.",
"The use of Sobolev gradients preconditions the energy-minimization problem.",
"This leads to smoother results that are less sensitive to noise and to the value of the regularization constant $\\lambda $ .",
"The Sobolev gradient flow frequently converges to a different local minimum than the $L^2$ -type gradient flow and requires less regularization.",
"A Bayesian model is determined by a likelihood $p(I|\\Gamma ,\\vec{\\theta })$ and a prior $p(\\Gamma )$ , where in image segmentation $I$ is the image data and $\\Gamma $ the segmentation contour.",
"The vector $\\vec{\\theta }$ contains photometric parameters such as region intensities.",
"The posterior probability density function is obtained using Bayes' formula: $p(\\Gamma ,\\vec{\\theta }|I)=\\frac{p(I|\\Gamma ,\\vec{\\theta })\\cdot p(\\Gamma ,\\vec{\\theta })}{p(I)}.$ The likelihood expresses how likely it is to observe the measured image $I$ given a certain segmentation $\\Gamma $ and parameters $\\vec{\\theta }$ .",
"The likelihood therefore formalizes the image-formation model.",
"The image-formation model describes the mapping between a ground-truth object state $(\\Gamma _{\\text{GT}}, \\vec{\\theta }_\\text{GT})$ and an expected image $J$ conditional on that state.",
"For many image-formation models, $\\vec{\\theta }$ is jointly determined by the model and by $I$ .",
"For example, in many popular models the estimated intensity $\\theta _i$ of a region $i$ is equal to the mean intensity of the area enclosed by that region's contour $\\Gamma _i$ in $I$ .",
"We therefore mostly omit $\\vec{\\theta }$ in our notation.",
"Generating $J$ from $\\Gamma $ is called the forward problem.",
"It amounts to simulating the expected (i.e., noise-free) image under a given segmentation.",
"In fluorescence microscopy, the forward model is linear: $J = s_{\\vec{\\theta }}(\\Gamma ) * K$ and amounts to a convolution of the expected intensity distribution $s_{\\vec{\\theta }}(\\Gamma )$ with the point-spread function (impulse-response function) $K$ of the microscope.",
"The function $s_{\\vec{\\theta }}$ is called the image-generating function.",
"It assigns an expected intensity to each pixel according to the image-formation model.",
"Frequently used image-generating functions are the piecewise constant approximation, assigning a constant intensity $\\theta _i$ to all pixels enclosed by contour $\\Gamma _i$ , and the piecewise smooth approximation, assigning shaded intensities within regions.",
"Estimating the region intensities $\\vec{\\theta }$ (and their distribution inside regions) from the image data is called the photometric estimation problem.",
"Finding the contours $\\Gamma _i$ constitutes the geometric estimation problem.",
"Estimating $\\Gamma $ and $\\vec{\\theta }$ from given $I$ , $K$ , and $s_{\\vec{\\theta }}$ is an inverse problem that is addressed using Bayesian inference.",
"Prior terms measure how likely a certain segmentation $\\Gamma $ is, independent of the observed image.",
"The most popular prior term penalizes the contour length $|\\Gamma |$ , favoring short (and hence smooth) region boundaries [38].",
"Other priors may include global shape characteristics and penalize deviations of the segmented shape from a template shape [18], [19].",
"Both the likelihood (i.e., the forward model) and the Bayesian prior can be used to include prior knowledge into the image-analysis algorithm.",
"We hence define prior knowledge more general than just the Bayesian prior.",
"In Bayesian (i.e., model-based) image segmentation, the estimation problem of finding $\\Gamma $ and $\\vec{\\theta }$ given $I$ , $K$ , and $s_{\\vec{\\theta }}$ is formulated as a maximum-a-posteriori (MAP) problem $\\max _{\\Gamma ,\\vec{\\theta }}{ p(\\Gamma , \\vec{\\theta }|I)} \\, .$ The problem is often restated as minimizing the anti-logarithm of the posterior, called the energy function $\\mathcal {E}=-\\log p(\\Gamma ,\\vec{\\theta }| I)$ .",
"The use of a logarithm is motivated by the Boltzmann distribution and changes the product of the likelihood and the prior to a sum of energy terms.",
"The energies corresponding to the likelihood and the prior are called the external and internal energy, respectively.",
"The resulting energy-minimization problem then reads: $\\min _{\\Gamma ,\\vec{\\theta }}\\left[ -\\log p(I|\\Gamma ,\\vec{\\theta }) - \\log p(\\Gamma ,\\vec{\\theta }) \\right] = \\min _{\\Gamma ,\\vec{\\theta }}\\left[ \\mathcal {E}_{\\textrm {external}} + \\lambda \\mathcal {E}_{\\textrm {internal}} \\right].$ The denominator in Bayes' formula (REF ) drops out as a constant shift.",
"The scalar parameter $\\lambda $ is included in order to weight the prior with respect to the likelihood.",
"The larger $\\lambda $ , the more weight is given to the prior and the less weight to the image data.",
"This increases the regularization.",
"Choosing the optimal $\\lambda $ is an open problem.",
"In practice, however, one usually wants to choose $\\lambda $ as small as possible to still get a robust segmentation and give as much weight to the image data $I$ as possible.",
"A plethora of minimization algorithms for the total energy $\\mathcal {E}=\\mathcal {E}_{\\textrm {external}} + \\lambda \\mathcal {E}_{\\textrm {internal}}$ has been presented in the literature.",
"The method of choice depends on the characteristics of $\\mathcal {E}$ .",
"Gradient-based local optimization of $\\mathcal {E}$ is a popular approach for non-convex $\\mathcal {E}$ .",
"In a discrete space, the energy gradient becomes an energy difference $\\Delta \\mathcal {E}$ .",
"In order to evolve the contour along the energy gradient flow, we (only) need to be able to evaluate energy differences, corresponding to posterior ratios $\\frac{p(\\Gamma ^{\\prime }|I)}{p(\\Gamma |I)}=\\exp (-\\Delta \\mathcal {E})$ for the original contour $\\Gamma $ and the perturbed contour $\\Gamma ^{\\prime }$ .",
"The quantity of interest hence is the energy difference $\\Delta \\mathcal {E}$ when deforming $\\Gamma $ to $\\Gamma ^{\\prime }$ .",
"We assume that image noise is realized independently for each pixel.",
"Using Bayes' formula (REF ) we then decompose $\\Delta \\mathcal {E}$ as $\\begin{split}\\Delta \\mathcal {E} = -\\log \\left(\\frac{p(\\Gamma ^{\\prime }|I)}{p(\\Gamma |I)}\\right) = -\\sum _{i=0}^{M-1} &\\left(\\log \\prod _{x\\in \\Omega _i^{\\prime }}{p\\left(I(x)|J^{\\prime }(x)\\right)\\cdot p(\\Gamma ^{\\prime })}-\\log \\prod _{x\\in \\Omega _i}{p\\left(I(x)|J(x)\\right)\\cdot p(\\Gamma )}\\right),\\end{split}$ where the images $J$ and $J^{\\prime }$ are computed using Eq.",
"(REF ).",
"$\\Omega _i$ and $\\Omega ^{\\prime }_i$ are the regions (sets of pixels) enclosed by $\\Gamma _i$ and $\\Gamma _i^{\\prime }$ , respectively.",
"$M$ is the total number of regions.",
"Foreground regions are defined as closed sets, whereas the background region with index 0 is an open set (see Fig.",
"REF ).",
"We have shown how Sobolev gradients can be used to drive the evolution of a discrete particle-based deformable model.",
"The particle nature of the present method allows for a simple and efficient approximation of Sobolev gradients and dispenses with the need to implement additional global solvers.",
"This does not only reduce the runtime of Sobolev codes, but also reduces the programmer burden during software development.",
"Sobolev gradients are known to precondition the optimization problem in Bayesian image segmentation and hence have a number of favorable properties: they are less sensitive to noise, they require less regularization, they favor the evolution of smooth contours, and they distinguish between local contour deformations and global contour motion [24], [29].",
"Sobolev gradients can be approximated from $L^2$ -type gradients by convolution with a local kernel [24].",
"In particle methods, discrete convolution amounts to particle–particle interactions [30], [31], [32], [33], which can efficiently be computed if the kernel is local.",
"Particle methods in image processing are hence naturally suited for more general gradient definitions, such as the Sobolev gradients considered here.",
"The Region Competition (RC) algorithm is a local black-box (i.e., zeroth-order) optimizer for particle-based deformable models [14].",
"Since it only requires point-wise evaluations of energy differences, which are used as discrete gradient approximations, the gradient definition can easily be changed without affecting the rest of the algorithm.",
"In fact, Sobolev gradients can easily be plugged into any discrete optimization algorithm or convexification scheme by simply changing the way the gradient is approximated.",
"This renders them an appealing extension to existing methods.",
"We have benchmarked the RC algorithm with and without Sobolev gradients on a number of artificial and real-world images, showing that the known qualitative properties of Sobolev flows are preserved by the present approximation algorithm.",
"We have also compared the results obtained with the present approximation algorithm with results from the literature that were computed using a fully accurate mesh-based solver.",
"The results obtained with the present particle-based scheme do not visibly differ from those obtained with mesh-based level-sets, but the present particle scheme is between 2.8 and 17 times faster than the level-set solver.",
"Also in the present implementation, Sobolev gradients remain computationally more costly to evaluate than $L^2$ -type gradients.",
"However, they precondition the problem such that less iterations are required for the optimizer to converge to the same solution, or such that better solutions (different local minima, possibly requiring more iterations to be reached) are found.",
"This may amortize the increased cost per iteration.",
"Currently, our implementation is limited by mainly two approximations: First, we have approximated the intrinsic geodesic distance along the contour by the Euclidean distance.",
"This limits the accuracy of the present gradient approximation if the contour is significantly curved on small length scales ($<E$).",
"Future work will consider using the true geodesic distance, exploiting concepts from differential geometry as applied to particle methods on curved Riemannian manifolds [39], [40].",
"Second, we have fixed the length scale $E$ of the Sobolev inner product to a user-defined value.",
"This was required in order to keep particle–particle interactions local and hence computationally efficient.",
"Global all-against-all interactions would incur a nominal computational complexity in $O(N^2)$ for $N$ contour particles, which is not practically feasible.",
"In order to allow global (rigid-body) movements of the contour, it would, however, be desirable to consider all-against-all interactions in future work.",
"The resulting $N$ -body problem could be efficiently (in $O(N)$ ) approximated using fast multipole solvers [41], [42].",
"The present implementation is available from http://mosaic.mpi-cbg.de as open-source as part of the RC filter in the ITK image-processing toolkit [43], implemented in C++.",
"We thank all members of the MOSAIC Group for the many fruitful discussions.",
"Particular thanks go to Dr. Grégory Paul for sharing his expertise in image segmentation and pointing us to the Sobolev gradient literature.",
"We thank Prof. Christian Dahmann (Technical University of Dresden), Dr. Guillaume Salbreux (Max Planck Institute for the Physics of Complex Systems, Dresden), and Dr. Michael Bussmann (Helmholtz Center Dresden Rossendorf) for providing test images.",
"This work was supported by the Swiss SystemsX.ch Initiative under Grant WingX and the German Federal Ministry of Research and Education (BMBF) under funding code 031A099."
]
] | 1403.0240 |
[
[
"Pluricomplex energy classes associated to a positive closed current"
],
[
"Abstract The aim of this paper is to extend the domain of definition of $(dd^c\\centerdot)^q\\wedge T$ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where $T$ is a positive closed current of bidimension $(q,q)$ on an open set $\\Omega$ of $\\Bbb C^n$.",
"We introduce two classes $\\mathcal{F}_{p}^{T}(\\Omega)$ and $\\mathcal{E}_p^T(\\Omega)$ and we show that they belong to the domain of definition of the operator $(dd^c\\centerdot)^q\\wedge T$.",
"We also prove that all functions belong to these classes are $C_T$-quasicontinuous and that the comparison principle is valid in them."
],
[
"Introduction",
"Let $\\Omega $ be a bounded open set of $C^n$ and denote by $PSH(\\Omega )$ the set of psh functions on $\\Omega $ .",
"The definition of the complex Monge-Ampère operator $(dd^c\\centerdot )^n$ on the set of psh functions has been studied by Bedford and Taylor in [1], they showed that this operator is well defined on the set of bounded psh functions and they established the comparaison principle to study the Dirichlet problem on $PSH(\\Omega )\\cap L^{\\infty }(\\Omega )$ .",
"The problem of extending its domain of definition was treated by many other authors, in particular Cegrell has introduced, between 1998 and 2004 (see [2], [3]), a general class $\\mathcal {E}(\\Omega )$ : the class of psh functions which are locally equal to decreasing limits of bounded psh functions vanishing on $\\partial \\Omega $ with bounded Monge-Ampère mass on $\\Omega $ .",
"He showed that the Monge-Ampère operator is well defined on $\\mathcal {E}(\\Omega )$ and this is the largest domain of definition if the operator is required to be continuous under decreasing sequences.",
"The study of this class leads to many results such that the comparaison principle, the solvability of the Dirichlet problem and the convergence in capacity.",
"Throughout this paper, $T$ will be a positive closed current of bidimension $(q,q)$ on $\\Omega $ where $1\\le q\\le n$ .",
"The question is to extend the domain of definition of the operator $(dd^c\\centerdot )^q\\wedge T$ .",
"This problem was studied by Dabbek and Elkhadhra [4] in the case of bounded psh functions.",
"We will extend the domain of definition of this operator to some classes of unbounded psh functions.",
"In this paper we recall the classes $\\mathcal {F}^T(\\Omega )$ and $\\mathcal {E}^T(\\Omega )$ introduced in [7] where the Monge-Ampère operator $(dd^c\\centerdot )^q\\wedge T$ is well defined and we introduce two new classes, the first will be $\\mathcal {F}_p^T(\\Omega ),\\ p\\ge 1$ a subclass of $\\mathcal {F}^T(\\Omega )$ and the second will be $\\mathcal {E}_p^T(\\Omega )$ .",
"In the first part we introduce the class $\\mathcal {E}_p^T(\\Omega )$ and we show that the Monge-Ampère operator $(dd^c\\centerdot )^q\\wedge T$ is well defined on this class then we give some properties of the classes $\\mathcal {E}_p^T(\\Omega )$ and $\\mathcal {F}^T(\\Omega )$ .",
"In the second part we prove that every functions in $\\mathcal {E}_p^T(\\Omega )$ or in $\\mathcal {F}^T(\\Omega )$ are $C_T$ -quasicontinuous; it means that they are continuous outside subsets of small $C_T$ -capacity.",
"The main tool of this result will be an estimate of the growth of $C_T(\\lbrace u<-s\\rbrace )$ .",
"Indeed we prove that $C_T(\\lbrace u<-s\\rbrace )=O\\left(\\frac{1}{s^{p+q}}\\right)\\quad (\\hbox{resp.",
"}C_T(\\lbrace u<-s\\rbrace )=O\\left(\\frac{1}{s^q}\\right))$ for every $u\\in \\mathcal {E}_p^T(\\Omega )$ (resp.",
"$u\\in \\mathcal {F}^T(\\Omega )$ ).",
"Using some analogous Xing's inequalities, we prove in the last part the main result of this paper.",
"Main result (Comparison principle) Let $u\\in \\mathcal {F}^{T}(\\Omega )$ and $v\\in \\mathcal {E}^{T}(\\Omega )$ .",
"Then $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u<v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(dd^cu)^q\\wedge T.$"
],
[
"Preliminary results",
"Let $\\Omega $ be a hyperconvex domain of $C^n$ , that means it is open, bounded, connected and that there exists $h\\in PSH^-(\\Omega )$ such that for all $c<0$ , $\\lbrace z\\in \\Omega ,\\ h(z)<c\\rbrace $ is relatively compact in $\\Omega $ where $PSH^-(\\Omega )$ is the set of negative psh functions.",
"Let us introduce the Cegrell pluricomplex class $\\mathcal {E}_{0}^{T}(\\Omega )$ associated to $T$ , slightly different to a class introduced in [7], as follows: $ \\mathcal {E}_{0}^{T}(\\Omega ):=\\left\\lbrace \\varphi \\in PSH^-(\\Omega )\\cap L^{\\infty }(\\Omega );\\ \\lim _{z\\rightarrow \\partial \\Omega \\cap Supp\\; T}\\varphi (z)=0,\\ \\int _{\\Omega }(dd^c\\varphi )^q\\wedge T<+\\infty \\right\\rbrace .$ Using the same proof as in [7], we can prove easly that this class is a convex cone and that for all $\\psi \\in PSH^-(\\Omega )$ and $\\varphi \\in \\mathcal {E}_{0}^{T}(\\Omega )$ one has $\\max (\\varphi ,\\psi )\\in \\mathcal {E}_{0}^{T}(\\Omega )$ .",
"In this section we introduce new energy classes $\\mathcal {E}_{p}^{T}(\\Omega )$ and $\\mathcal {F}_{p}^{T}(\\Omega )$ , similar to Cegrell's ones and we will show that the Monge-Ampère operator is well defined on them.",
"Definition 1 For every real $p\\ge 1$ we define $\\mathcal {E}_{p}^{T}(\\Omega )$ as the set: $\\mathcal {E}_{p}^{T}(\\Omega ):=\\left\\lbrace \\varphi \\in PSH^-(\\Omega ); \\ \\exists \\ \\mathcal {E}_{0}^{T}(\\Omega )\\ni \\varphi _j\\searrow \\varphi , \\ \\sup _{j\\ge 1}\\int _{\\Omega }(-\\varphi _j)^p(dd^c\\varphi _j)^q\\wedge T<+\\infty \\right\\rbrace .$ When the sequence $(\\varphi _j)_j$ associated to $\\varphi $ can be chosen such that $\\displaystyle \\sup _{j\\ge 1}\\int _{\\Omega }(dd^c\\varphi _j)^q\\wedge T<+\\infty ,$ we say that $\\varphi \\in \\mathcal {F}_{p}^{T}(\\Omega )$ .",
"It's Easy to check that $\\mathcal {E}_{0}^{T}(\\Omega )\\subset \\mathcal {F}_{p}^{T}(\\Omega )\\subset \\mathcal {E}_{p}^{T}(\\Omega )$ and that, using Hölder's Inequality, one has $\\mathcal {F}_{p_1}^{T}(\\Omega )\\subset \\mathcal {F}_{p_2}^{T}(\\Omega )$ for all $p_2\\le p_1$ .",
"We recall the following result which will be useful to prove some properties of our classes.",
"Theorem 1 (See [4]) Suppose that $u,v\\in \\mathcal {E}_{0}^{T}(\\Omega )$ .",
"If $p\\ge 1$ then for every $0\\le s\\le q$ one has $\\begin{array}{l}\\displaystyle \\int _{\\Omega }(-u)^p (dd^c u)^s \\wedge (dd^c v)^{q-s}\\wedge T\\\\\\displaystyle \\le D_{s,p}\\left( \\int _{\\Omega }(-u)^p (dd^c u)^q \\wedge T\\right)^{\\frac{p+s}{p+q}}\\left( \\int _{\\Omega }(-v)^p (dd^c v)^q \\wedge T\\right)^{\\frac{q-s}{p+q}}\\end{array}$ where $D_{s,1}=e^{(j+1)(q-j)}$ and $D_{s,p}=p^{\\frac{(p+s)(q-s)}{p-1}}$ , $p>1$ .",
"We begin by showing that the two introduced classes inherit some properties of the energy class $\\mathcal {E}_{0}^{T}(\\Omega ).$ Theorem 2 The classes $\\mathcal {E}_{p}^{T}(\\Omega )$ and $\\mathcal {F}_{p}^{T}(\\Omega )$ are convex cones.",
"It suffices to prove that $u+v\\in \\mathcal {E}_{p}^{T}(\\Omega )$ for every $u,v\\in \\mathcal {E}_{p}^{T}(\\Omega )$ .",
"Let $(u_j)_j$ and $(v_j)_j$ be two sequences that decrease to $u$ and $v$ respectively as in Definition REF .",
"We want to estimate $\\int _\\Omega (-u_j-v_j)^p (dd^c(u_j+v_j))^q\\wedge T.$ Thanks to Minkowsky Inequality, it is enough to estimate the following terms: $\\int _{\\Omega }(-u_j)^p (dd^c u_j)^s \\wedge (dd^cv_j)^{q-s} \\wedge T$ and $\\int _{\\Omega }(-v_j)^p (dd^c u_j)^s \\wedge (dd^cv_j)^{q-s} \\wedge T$ for all $0<s<q$ .",
"Using Theorem REF , we can estimate last terms by $\\int _{\\Omega }(-u_j)^p (dd^c u_j)^q\\wedge T\\quad \\hbox{and} \\quad \\int _{\\Omega }(-v_j)^p (dd^c v_j)^q\\wedge T.$ As these sequences are uniformly bounded by the definition of $\\mathcal {E}_{p}^{T}(\\Omega )$ , the result follows.",
"Proposition 1 Let $u\\in \\mathcal {E}_{p}^{T}(\\Omega )$ (resp.",
"$\\mathcal {F}_{p}^{T}(\\Omega )$ ) and $v\\in PSH^-(\\Omega )$ .",
"Then the function $w:=\\max (u,v)$ is in $\\mathcal {E}_{p}^{T}(\\Omega )$ (resp.",
"in $\\mathcal {F}_{p}^{T}(\\Omega )$ ).",
"Let $(u_j)_j$ be a sequence that decreases to $u$ as in Definition REF and take $w_j:=\\max (u_j,v)$ .",
"The sequence $(w_j)$ decreases to $w$ .",
"So it's enough to prove that $\\displaystyle \\sup _j\\int _{\\Omega }(-w_j)^p (dd^cw_j)^q\\wedge T<+\\infty .$ Thanks to Theorem REF , one has $\\begin{array}{lcl}\\displaystyle \\int _{\\Omega }(-w_j)^p (dd^c w_j)^q\\wedge T&\\le &\\displaystyle \\int _{\\Omega }(-u_j)^p (dd^c w_j)^q\\wedge T\\\\&\\le &\\displaystyle D_{0,p}\\left( \\int _{\\Omega }(-u_j)^p (dd^c u_j)^q \\wedge T\\right)^{\\frac{p}{p+q}}\\left( \\int _{\\Omega }(-w_j)^p (dd^c w_j)^q \\wedge T\\right)^{\\frac{q}{p+q}}.\\end{array}$ Therefore $\\int _{\\Omega }(-w_j)^p (dd^c w_j)^q\\wedge T\\le D_{0,p}^{\\frac{p+q}{p}}\\int _{\\Omega }(-u_j)^p (dd^c u_j)^q\\wedge T .$ The right-hand side is uniformly bounded because $u\\in \\mathcal {E}_{p}^{T}(\\Omega )$ and the result follows.",
"The most important result of this section is the following theorem which proves that the Monge-Ampère operator $(dd^c\\centerdot )^q\\wedge T$ is well defined on the new classes.",
"Theorem 3 Let $u\\in \\mathcal {E}_p^T(\\Omega )$ and $(u_j)_j$ be a sequence of psh functions that decreases to $u$ as in Definition REF .",
"Then $(dd^cu_j)^q\\wedge T$ converges weakly to a positive measure $\\mu $ and this limit is independent of the choice of the sequence $(u_j)_j$ .",
"We set $(dd^cu)^q\\wedge T:=\\mu $ .",
"Let $0\\le \\chi \\in \\mathcal {D}(\\Omega )$ , $\\delta =\\sup \\lbrace u_1(z);\\ z\\in Supp\\chi \\rbrace $ and $\\varepsilon >0$ .",
"There exists a sequence $(r_j)_j$ such that $0<r_j<r_{j-1}$ and $r_j<dist(\\lbrace u_j<\\frac{\\delta }{2}\\rbrace ,\\Omega ^c).$ Let $u_{r_j}(z):=\\int _{B} u_j(z+r_j\\xi )dV(\\xi )$ where $dV$ is the normalized Lebesgue measure on the unit ball $B$ .",
"Then one has $\\left|\\int _\\Omega \\chi (dd^cu_{r_j})^q \\wedge T-\\chi (dd^cu_j)^q\\wedge T\\right|<\\varepsilon .$ The function $u_{r_j}$ is continuous, psh on $\\lbrace u_j<\\frac{\\delta }{2}\\rbrace $ and $u_j\\le u_{r_j}$ on $\\Omega $ .",
"Let $\\widetilde{u_j}=\\max (u_{r_j}+\\delta ,2u_j)$ .",
"Then the sequence $(\\widetilde{u_j})_j$ decreases to a psh function $\\widetilde{u}$ and $\\widetilde{u_j}\\in \\mathcal {E}_{0}^{T}(\\Omega )$ by Proposition REF .",
"Furthermore, using the same technic of the previous proof, we obtain $\\sup _{j\\ge 1}\\int _{\\Omega }(-\\widetilde{u_j})^p(dd^c\\widetilde{u_j})^q\\wedge T<+\\infty .$ The proof of the theorem will be complete if we show that $\\displaystyle \\lim _{j\\rightarrow +\\infty }\\int _\\Omega \\chi (dd^c\\widetilde{u_j})^q \\wedge T$ exists.",
"Let $h$ be an exhaustion function in $\\mathcal {E}_0^T(\\Omega )$ .",
"Then $\\begin{array}{l}\\displaystyle \\int _\\Omega (-\\widetilde{u})^p(dd^ch)^q\\wedge T=\\displaystyle \\lim _{j\\rightarrow +\\infty }\\int _\\Omega (-\\widetilde{u_j})^p(dd^ch)^q\\wedge T\\\\\\le \\displaystyle D_{0,p}\\sup _{j\\ge 1}\\left( \\int _{\\Omega }(-\\widetilde{u_j})^p (dd^c \\widetilde{u_j})^q \\wedge T\\right)^{\\frac{p}{p+q}}\\left( \\int _{\\Omega }(-h)^p (dd^c h)^q \\wedge T\\right)^{\\frac{q}{p+q}}<+\\infty .\\end{array}$ Thanks to Dabbek-Elkhadhra [4], the sequence of measures $(dd^c\\max (\\widetilde{u_j},-k))^q\\wedge T$ converges weakly for every $k$ .",
"So it is enough to control $\\left|\\int \\chi (dd^cu_{r_j})^q \\wedge T-\\chi (dd^c\\max (\\widetilde{u_j},-k))^q\\wedge T \\right|.$ Since $\\widetilde{u_j}$ is continuous near $Supp\\chi $ then $\\begin{array}{ll}&\\displaystyle \\left|\\int \\chi (dd^cu_j)^q \\wedge T-\\chi (dd^c\\max (\\widetilde{u_j},-k))^q\\wedge T \\right|\\\\=&\\displaystyle \\left|\\int _{\\lbrace \\widetilde{u}\\le -k\\rbrace }\\chi (dd^c\\widetilde{u_j})^q \\wedge T +\\int _{\\lbrace \\widetilde{u}> -k\\rbrace }\\chi (dd^c\\widetilde{u_j})^q \\wedge T\\right.\\\\&\\displaystyle \\left.-\\int _{\\lbrace \\widetilde{u}\\le -k\\rbrace }\\chi (dd^c\\max (\\widetilde{u_j},-k))^q \\wedge T-\\int _{\\lbrace \\widetilde{u}> -k\\rbrace }\\chi (dd^c\\max (\\widetilde{u_j},-k))^q \\wedge T \\right|\\\\\\le &\\displaystyle \\int _{\\lbrace \\widetilde{u}\\le -k\\rbrace }\\chi (dd^c\\widetilde{u_j})^q \\wedge T+\\int _{\\lbrace \\widetilde{u}\\le -k\\rbrace }\\chi (dd^c\\max (\\widetilde{u_j},-k))^q \\wedge T\\\\\\le & \\displaystyle \\frac{\\sup \\chi }{k^p}\\int _{\\lbrace -\\widetilde{u}\\ge k\\rbrace }k^p\\left[(dd^c\\widetilde{u_j})^q \\wedge T+(dd^c\\max (\\widetilde{u_j},-k))^q \\wedge T\\right]\\\\\\le & \\displaystyle \\frac{\\sup \\chi }{k^p} \\int _{\\Omega }(-\\widetilde{u})^p(dd^c\\widetilde{u_j})^q\\wedge T+(-\\max (\\widetilde{u_j},-k))^pdd^c\\max (\\widetilde{u_j},-k))^q\\wedge T\\\\\\le &\\displaystyle C\\frac{\\sup \\chi }{k^p} \\sup _{m\\ge 1}\\int _{\\Omega }(-\\widetilde{u}_m)^p(dd^c\\widetilde{u_m})^q\\wedge T.\\end{array}$ This completes the proof of the theorem.",
"Theorem 4 If $u\\in \\mathcal {E}_1^{T}(\\Omega )$ then $\\int _{\\Omega } u (dd^cu)^q\\wedge T>-\\infty .$ Moreover, if $v_j\\in PSH^-(\\Omega )$ such that $(v_j)_j$ decreases to $u$ then $ \\int _{\\Omega } v_j (dd^cv_j)^q\\wedge T\\hbox{ converges to }\\int _{\\Omega } u (dd^cu)^q\\wedge T .$ Since $u\\in \\mathcal {E}_1^{T}(\\Omega )$ then there exists a sequence $(u_j)_j\\subset \\mathcal {E}_0^{T}$ such that $ \\lim _{j\\rightarrow +\\infty }u_j=u\\ \\ and \\ \\ \\alpha :=\\sup _j \\int -u_j (dd^c u_j)^q\\wedge T<+\\infty .$ Let us prove that $\\lim _{j\\rightarrow +\\infty }\\int _\\Omega u_j(dd^cu_j)^q\\wedge T=\\int _\\Omega u(dd^cu)^q\\wedge T.$ For every $k\\ge j$ and $\\varepsilon >0$ , one has $\\begin{array}{l}\\displaystyle \\int _{\\Omega } -u_j(dd^cu_j)^q\\wedge T\\\\\\le \\displaystyle \\int _{\\Omega } -u_j(dd^cu_k)^q\\wedge T\\\\=\\displaystyle \\int _{\\lbrace u_j\\ge -\\varepsilon \\rbrace } -u_j(dd^cu_k)^q\\wedge T +\\int _{\\lbrace u_j< -\\varepsilon \\rbrace }-u_j(dd^cu_k)^q\\wedge T\\end{array}$ and $\\begin{array}{l}\\displaystyle \\int _{\\lbrace u_j\\ge -\\varepsilon \\rbrace } -u_j(dd^cu_k)^q\\wedge T \\\\=\\displaystyle \\int _{\\lbrace u_j\\ge -\\varepsilon \\rbrace } -\\max (u_j,-\\varepsilon )(dd^cu_k)^q\\wedge T\\\\\\le \\displaystyle \\left(\\int _\\Omega -\\max (u_j,-\\varepsilon )(dd^c\\max (u_j,-\\varepsilon ))^q\\wedge T\\right)^{\\frac{1}{q+1}}\\left(\\int _\\Omega -u_k(dd^cu_k)^q\\wedge T\\right)^{\\frac{q}{q+1}}\\\\\\le \\displaystyle \\left(\\varepsilon \\int _\\Omega (dd^cu_j)^q\\wedge T\\right)^{\\frac{1}{q+1}}\\alpha ^{\\frac{q}{q+1}}\\end{array}$ This goes to 0 when $\\varepsilon \\rightarrow 0$ .",
"By Theorem REF we obtain $\\limsup _{k\\rightarrow +\\infty }\\int _{\\lbrace u_j< -\\varepsilon \\rbrace } -u_j(dd^cu_k)^q\\wedge T\\le \\int _{\\Omega }-u_j (dd^cu)^q\\wedge T .$ Now since $-u_j$ is lower semi-continuous then $\\liminf _{k\\rightarrow +\\infty }\\int _{\\Omega } -u_j(dd^cu_k)^q\\wedge T\\ge \\int _{\\Omega }-u_j (dd^cu)^q\\wedge T .$ Hence for all $j$ , $\\lim _{k\\rightarrow +\\infty }\\int _{\\Omega } u_j(dd^cu_k)^q\\wedge T= \\int _{\\Omega } u_j(dd^cu)^q\\wedge T.$ It follows that $\\begin{array}{l}\\displaystyle \\lim _{j\\rightarrow +\\infty }\\int _{\\Omega } u_j(dd^cu_j)^q\\wedge T\\\\\\ge \\displaystyle \\lim _{j\\rightarrow +\\infty } \\lim _{k\\rightarrow +\\infty }\\int _{\\Omega } u_j(dd^cu_k)^q\\wedge T = \\int _{\\Omega } u(dd^cu)^q\\wedge T\\\\\\ge \\displaystyle \\limsup _{k\\rightarrow +\\infty }\\int _{\\Omega } u(dd^cu_k)^q\\wedge T = \\limsup _{k\\rightarrow +\\infty }\\lim _{j\\rightarrow +\\infty }\\int _{\\Omega } u_j(dd^cu_k)^q\\wedge T\\\\\\ge \\displaystyle \\lim _{j\\rightarrow +\\infty }\\int _{\\Omega } u_j(dd^cu_j)^q\\wedge T.\\end{array}$ Thus $\\lim _{j\\rightarrow +\\infty }\\int _{\\Omega } u_j(dd^cu_j)^q\\wedge T= \\int _{\\Omega } u(dd^cu)^q\\wedge T.$ As $(v_k)_k$ decreases to $u$ then $v_k \\in \\mathcal {E}_1^T(\\Omega )$ .",
"It follows that $\\int _{\\Omega } \\max (u_j,v_k)(dd^c\\max (u_j,v_k))^q\\wedge T\\ge \\int _{\\Omega } u_j (dd^cu_j)^q\\wedge T\\ge -\\alpha .$ Moreover, $(\\max (u_j,v_k))_{j\\in N}\\subset \\mathcal {E}_0^T(\\Omega )$ and decreases to $v_k$ so thanks to Equality (REF ), $\\lim _{j\\rightarrow +\\infty }\\int _{\\Omega } \\max (u_j,v_k)(dd^c\\max (u_j,v_k))^q\\wedge T =\\int _{\\Omega } v_k(dd^cv_k)^q\\wedge T.$ By tending $j\\rightarrow +\\infty $ , Inequality (REF ), Equalities (REF ) and (REF ) give $\\int _{\\Omega } v_k(dd^cv_k)^q\\wedge T\\ge \\int _{\\Omega } u(dd^cu)^q\\wedge T.$ Thus $\\liminf _{k\\rightarrow +\\infty }\\int _{\\Omega } v_k(dd^cv_k)^q\\wedge T\\ge \\int _{\\Omega } u(dd^cu)^q\\wedge T.$ With the same reason, as $(\\max (u_j,v_k))_{k\\in N}$ decreases to $u_j$ then $\\int _{\\Omega } u_j(dd^cu_j)^q\\wedge T\\ge \\limsup _{k\\rightarrow +\\infty }\\int _{\\Omega } v_k(dd^cv_k)^q\\wedge T.$ Hence $\\limsup _{k\\rightarrow +\\infty }\\int _{\\Omega } v_k(dd^cv_k)^q\\wedge T\\le \\int _{\\Omega } u(dd^cu)^q\\wedge T.$ The result follows from Inequalities (REF ) and (REF ).",
"Remark 1 Claim that if $u\\in \\mathcal {E}_1^T(\\Omega )$ and $(u_j)_j$ is a decreasing sequence to $u$ as in Definition REF then $ \\int _{\\Omega } u_j (dd^cu_j)^q\\wedge T\\hbox{ decreases to }\\int _{\\Omega } u (dd^cu)^q\\wedge T .$"
],
[
"Comparaison theorems",
"We recall two classes $\\mathcal {E}^{T}(\\Omega )$ and $\\mathcal {F}^{T}(\\Omega )$ introduced in [7] where authors prove that the Monge-Ampère operator $(dd^c\\centerdot )^q\\wedge T$ is well defined on them.",
"Definition 2 We say that $u\\in \\mathcal {F}^{T}(\\Omega )$ if there exists a sequence $(u_j)_j\\subset \\mathcal {E}_0^{T}(\\Omega )$ which decreases to $u$ such that $\\sup _j\\int _\\Omega (dd^cu_j)^q\\wedge T< +\\infty .$ A function $u$ will belong to $ \\mathcal {E}^{T}(\\Omega )$ if for all $z\\in \\Omega $ there exist a neighborhood $\\omega $ of $z$ and a function $v\\in \\mathcal {F}^{T}(\\Omega )$ such that $u=v$ on $\\omega $ .",
"As a consequence, for every $p\\ge 1$ one has $\\mathcal {F}_p^T(\\Omega )\\subset \\mathcal {F}^T(\\Omega )\\subset \\mathcal {E}^T(\\Omega )$ but we dont know any relationship between $\\mathcal {E}_p^T(\\Omega )$ and $\\mathcal {E}^T(\\Omega )$ .",
"Lemma 1 Let $u,v\\in PSH(\\Omega )\\cap L^{\\infty }(\\Omega )$ and $U$ be an open subset of $\\Omega $ such that $u=v$ near $\\partial U$ .",
"Then $\\int _U(dd^cu)^q\\wedge T=\\int _U(dd^cv)^q\\wedge T$ Let $u_{\\varepsilon }$ and $v_{\\varepsilon }$ be the usual regularization of $u$ and $v$ respectively.",
"Choose $U^{\\prime }\\subset \\subset U$ such that $u=v$ near $\\partial U^{\\prime }$ .",
"If $\\varepsilon >0$ is small enough, one has $u_{\\varepsilon }=v_{\\varepsilon }$ near $\\partial U^{\\prime }$ and if we take $\\chi \\in \\mathcal {D}(U^{\\prime })$ with $\\chi =1$ near $\\lbrace u_{\\varepsilon }\\ne v_{\\varepsilon }\\rbrace $ then $dd^c\\chi =0$ on $\\lbrace u_{\\varepsilon }\\ne v_{\\varepsilon }\\rbrace $ .",
"So $\\begin{array}{lcl}\\displaystyle \\int _{\\Omega }\\chi (dd^cu_{\\varepsilon })^q \\wedge T&=&\\displaystyle \\int _{\\Omega } u_{\\varepsilon } dd^c\\chi \\wedge (dd^cu_{\\varepsilon })^{q-1} \\wedge T\\\\&=&\\displaystyle \\int _{\\Omega } v_{\\varepsilon } dd^c\\chi \\wedge (dd^cu_{\\varepsilon })^{q-1} \\wedge T\\\\&=&\\displaystyle \\int _{\\Omega }\\chi (dd^cv_{\\varepsilon })^q \\wedge T.\\end{array}$ Hence $\\int _{\\Omega }\\chi (dd^cu)^q \\wedge T=\\int _{\\Omega }\\chi (dd^cv)^q \\wedge T.$ The result follows.",
"Corollary 1 Let $u,v\\in \\mathcal {F}^{T}(\\Omega )$ .",
"Assume that there exists an open subset $U$ of $\\Omega $ such that $u=v$ near $\\partial U$ .",
"Then $\\int _U(dd^cu)^q\\wedge T=\\int _U(dd^cv)^q\\wedge T.$ Let $u,v\\in \\mathcal {F}^{T}(\\Omega )$ and $w\\in \\mathcal {E}_0^{T}(\\Omega )$ such that $w(z) \\ne 0$ for all $z$ .",
"Then $u_j:=\\max (u,jw)$ and $v_j=\\max (v,jw)$ belong to $\\mathcal {E}_0^{T}(\\Omega )$ and they are equal on $\\partial U$ .",
"The result follows from the previous lemma.",
"Now we recall a result due to [7] and we give a different proof.",
"Proposition 2 (See [7]) For $u,v\\in \\mathcal {F}^{T}(\\Omega )$ such that $u\\le v$ on $\\Omega $ one has $\\int _{\\Omega }(dd^cv)^q\\wedge T\\le \\int _{\\Omega }(dd^cu)^q\\wedge T.$ Let $(u_j)_j$ and $(v_j)_j$ be the corresponding decreasing sequences to $u$ and $v$ respectively as in Definition REF .",
"Replace $v_j$ by $\\max (u_j,v_j)$ we can assume that $u_j\\le v_j$ for all $j\\in N$ .",
"For $h\\in \\mathcal {E}_0^T(\\Omega )$ and $\\varepsilon >0$ we have $\\begin{array}{lcl}\\displaystyle \\int _{\\Omega }-h(dd^cv_j)^q\\wedge T&\\le &\\displaystyle \\int _{\\Omega }-h(dd^cu_j)^q\\wedge T\\\\&\\le &\\displaystyle \\int _{\\Omega }-h(dd^cu)^q\\wedge T+\\limsup _{j\\rightarrow +\\infty }\\int _{\\lbrace h>-\\varepsilon \\rbrace }-h(dd^cu_j)^q\\wedge T\\\\&\\le &\\displaystyle \\int _{\\Omega }-h(dd^cu)^q\\wedge T+\\varepsilon \\limsup _{j\\rightarrow +\\infty }\\int _{\\Omega }(dd^cu_j)^q\\wedge T.\\end{array}$ By tending $\\varepsilon $ to 0 we obtain $\\int _{\\Omega }-h(dd^cv)^q\\wedge T\\le \\int _{\\Omega }-h(dd^cu)^q\\wedge T$ The result follows by choosing $h$ decreases to $-1$ .",
"Lemma 2 Let $u\\in \\mathcal {F}^{T}(\\Omega )$ then there exists a sequence $ (u_j)_j\\subset \\mathcal {E}_{0}^{T}(\\Omega )\\cap {\\mathcal {C}(\\overline{\\Omega })}$ that decreases to $u$ .",
"We claim that this lemma was cited in [7] with uncompleted proof; in fact authors had used a comparaison theorem, proved by Dabbek-Elkhadhra [4] only for bounded psh functions, in $\\mathcal {F}^T(\\Omega )$ where functions are not in general bounded.",
"We refer to Cegrell [3] for the construction of the sequence $(u_j)_j$ .",
"It remains to show that $\\displaystyle \\int _{\\Omega }(dd^cu_j)^q\\wedge T <\\infty .$ As $u_j\\ge u$ then by Proposition REF one has $\\int _{\\Omega }(dd^cu_j)^q\\wedge T \\le \\int _{\\Omega }(dd^cu)^q\\wedge T<+\\infty .$"
],
[
"$C_T$ -quasicontinuity",
"Now we establish the quasicontinuity of psh functions belong to $\\mathcal {F}^{T}(\\Omega )$ and $\\mathcal {E}_p^{T}(\\Omega )$ .",
"We need to recall some notions given in [4] (see also [9]) about the capacity associated to $T$ which is defined as $C_T(K,\\Omega )=C_T(K)=\\sup \\left\\lbrace \\int _K(dd^cv)^q\\wedge T,\\ v\\in PSH(\\Omega ,[-1,0])\\right\\rbrace .$ for all compact subset $K$ of $\\Omega $ .",
"If $E$ is a subset of $\\Omega $ , we define $C_T(E,\\Omega )=\\sup \\lbrace C_T(K),\\ K\\hbox{ compact subset of }E\\rbrace .$ We refer to [4], [9] for the properties of this capacity.",
"Definition 3 A subset $A$ of $\\Omega $ is said to be $T$ -pluripolar if $C_T(A,\\Omega )=0$ .",
"A psh function $u$ is said to be quasicontinuous with respect to $C_T$ , if for every $\\varepsilon >0$ , there exists an open subset $O_{\\varepsilon }$ such that $C_T(O_{\\varepsilon },\\Omega )<\\varepsilon $ and $u$ is continuous on $\\Omega \\setminus O_{\\varepsilon }$ .",
"Proposition 3 Let $u\\in \\mathcal {F}^T(\\Omega )$ .",
"Then for every $s>0$ one has $s^q C_T(\\lbrace u\\le -s\\rbrace ,\\Omega )\\le \\int _\\Omega (dd^cu)^q\\wedge T.$ In particular, the set $\\lbrace u=-\\infty \\rbrace $ is $T$ -pluripolar.",
"Let $(u_j)_j\\subset \\mathcal {E}_0^T(\\Omega )$ be a decreasing sequence to $u$ on $\\Omega $ as in Definition REF .",
"Take $s>0$ , $v\\in PSH(\\Omega ,[-1,0])$ and $K$ a compact subset in $\\lbrace u_j\\le -s\\rbrace $ .",
"Thanks to the comparaison principle (for bounded psh functions), we have $\\begin{array}{lcl}\\displaystyle \\int _K (dd^cv)^q\\wedge T&\\le &\\displaystyle \\int _{\\lbrace s^{-1}u_j<v\\rbrace }(dd^cv)^q\\wedge T\\le \\frac{1}{s^q}\\int _{\\lbrace s^{-1}u_j<v\\rbrace }(dd^cu_j)^q\\wedge T\\\\&\\le &\\displaystyle \\frac{1}{s^q}\\int _\\Omega (dd^cu_j)^q\\wedge T\\end{array}$ It follows that $C_T(\\lbrace u_j\\le -s\\rbrace ,\\Omega )\\le \\frac{1}{s^q}\\int _\\Omega (dd^cu_j)^q\\wedge T.$ By tending $j$ to infinity, we obtain $C_T(\\lbrace u\\le -s\\rbrace ,\\Omega )\\le \\frac{1}{s^q}\\int _\\Omega (dd^cu)^q\\wedge T.$ Corollary 2 Every $u\\in \\mathcal {F}^{T}(\\Omega )$ is $C_T$ -quasicontinuous.",
"Let $u\\in \\mathcal {F}^{T}(\\Omega )$ and $\\varepsilon >0$ .",
"Denote by $B_u(t):=\\lbrace z\\in \\Omega ;\\ u(z)<t\\rbrace ,\\ t\\le 0$ .",
"By Proposition REF , there is $s_\\varepsilon \\ge 1$ such that $C_T(B_u(-s_\\varepsilon ),\\Omega )<\\frac{\\varepsilon }{2}$ .",
"The function $u_\\varepsilon :=\\max (u,-s_\\varepsilon )$ is bounded on $\\Omega $ so thanks to Dabbek-Elkhadhra [4], there is an open subset $\\mathcal {O}$ in $\\Omega $ such that $C_T(\\mathcal {O},\\Omega )<\\frac{\\varepsilon }{2}$ and $u_\\varepsilon $ is continuous on $\\Omega \\setminus \\mathcal {O}$ .",
"The result follows by taking $\\mathcal {O}_\\varepsilon =\\mathcal {O}\\cup B_u(-s_\\varepsilon )$ .",
"To study the $C_T$ -quasicontinuity on $\\mathcal {E}_p^T(\\Omega )$ , we will proceed as in the previous case.",
"Proposition 4 Let $u\\in \\mathcal {E}_p^T(\\Omega )$ and $(u_j)_j\\subset \\mathcal {E}_0^T(\\Omega )$ decreases to $u$ on $\\Omega $ as in Definition REF .",
"Then for every $s>0$ one has $s^{p+q} C_T(\\lbrace u\\le -2s\\rbrace ,\\Omega )\\le \\sup _{j\\ge 1}\\int _\\Omega (-u_j)^p(dd^cu_j)^q\\wedge T.$ In particular, the set $\\lbrace u=-\\infty \\rbrace $ is $T$ -pluripolar.",
"Let $s>0$ , $v\\in PSH(\\Omega ,[-1,0])$ .",
"Thanks to comparaison principle (for bounded psh functions), we have $\\begin{array}{lcl}\\displaystyle \\int _{\\lbrace u_j\\le -2s\\rbrace } (dd^cv)^q\\wedge T&\\le &\\displaystyle \\int _{\\lbrace u_j<-s+sv\\rbrace }(dd^cv)^q\\wedge T\\le \\frac{1}{s^q}\\int _{\\lbrace s^{-1}u_j<-1+v\\rbrace }(dd^cu_j)^q\\wedge T\\\\&\\le &\\displaystyle \\frac{1}{s^{p+q}}\\int _\\Omega (-u_j)^p(dd^cu_j)^q\\wedge T\\end{array}$ It follows that $C_T(\\lbrace u_j\\le -2s\\rbrace ,\\Omega )\\le \\frac{1}{s^{p+q}}\\sup _{m\\ge 1}\\int _\\Omega (-u_m)^p(dd^cu_m)^q\\wedge T.$ By tending $j$ to infinity, we obtain $C_T(\\lbrace u\\le -2s\\rbrace ,\\Omega )\\le \\frac{1}{s^{p+q}}\\sup _{m\\ge 1}\\int _\\Omega (-u_m)^p(dd^cu_m)^q\\wedge T.$ By the same argument as in corollary REF we can easily deduce the following result: Corollary 3 Every function in $\\mathcal {E}_p^T(\\Omega )$ is $C_T$ -quasicontinuous.",
"Now we need a first version of the comparaison principle where one of the functions will be unbounded.",
"This result was proved in [4] for bounded functions.",
"Theorem 5 Let $u\\in \\mathcal {F}^{T}(\\Omega )$ and $v\\in PSH(\\Omega )\\cap L^\\infty (\\Omega )$ such that $\\liminf _{z\\rightarrow \\partial \\Omega \\cap SuppT} u(z)-v(z)\\ge 0.$ Then $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u<v\\rbrace }(dd^cu)^q\\wedge T.$ Firstly we assume that $u$ and $v$ are continuous on a neighborhood $W$ of $Supp T$ .",
"Without loss of generality we can assume that $u<v$ on $W$ and $u=v$ on $\\partial W$ .",
"Let $v_{\\varepsilon }:=\\max (u,v-\\varepsilon )$ then one has $v_{\\varepsilon }=u$ on $\\partial W$ and $\\int _{\\lbrace u<v\\rbrace }(dd^cv_{\\varepsilon })^q\\wedge T= \\int _{\\lbrace u<v\\rbrace }(dd^cu)^q\\wedge T.$ Since the family of measures $(dd^cv_{\\varepsilon })^q\\wedge T$ converges weakly to $(dd^cu)^q\\wedge T$ as $\\varepsilon \\rightarrow 0$ , then we obtain $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T= \\int _{\\lbrace u<v\\rbrace }(dd^cu)^q\\wedge T.$ Let now treat the general cas.",
"Replace $u$ by $u+\\delta $ if necessary, we can assume that $\\liminf (u-v)\\ge 2\\delta $ ; so there is an open subset $\\mathcal {O}\\subset \\subset \\Omega $ such that $u(z)\\ge v(z)+\\delta $ for all $z\\in \\Omega \\setminus \\mathcal {O}$ .",
"Let $(u_k)_k$ and $(v_j)_j$ be two smooth sequences of psh functions which decrease respectively to $u$ and $v$ on a neighborhood of $\\overline{\\mathcal {O}}$ such that $u_k\\ge v_j$ on $\\partial \\overline{\\mathcal {O}}\\cap Supp T$ for $j\\ge k$ .",
"Using the previous argument we obtain $\\int _{\\lbrace u_k<v_j\\rbrace }(dd^cv_j)^q\\wedge T= \\int _{\\lbrace u_k<v_j\\rbrace }(dd^cu_k)^q\\wedge T.$ For $\\varepsilon > 0$ , there exists an open subset $G$ of $\\Omega $ such that $C_T(G,\\Omega )<\\varepsilon $ and $u,v$ are continuous on $\\Omega \\setminus G$ .",
"We can write $v=\\varphi + \\psi $ where $\\varphi $ is continuous on $\\Omega $ and $\\psi =0$ on $\\Omega \\setminus G.$ Take $U:=\\lbrace u_k<\\varphi \\rbrace $ then $\\int _U(dd^cv)^q\\wedge T\\le \\lim _{j\\rightarrow +\\infty }\\int _U(dd^cv_j)^q\\wedge T.$ Since $U\\cup G=\\lbrace u_k<v\\rbrace \\cup G$ then $\\begin{array}{l}\\displaystyle \\int _{\\lbrace u_k<v\\rbrace }(dd^cv)^q\\wedge T\\\\\\displaystyle \\le \\int _U(dd^cv)^q\\wedge T+\\int _G(dd^cv)^q\\wedge T\\\\\\displaystyle \\le \\lim _{j\\rightarrow +\\infty }\\int _U(dd^cv_j)^q\\wedge T+\\int _G(dd^cv)^q\\wedge T\\\\\\displaystyle \\le \\lim _{j\\rightarrow +\\infty }\\left(\\int _{\\lbrace u_k<v_j\\rbrace }(dd^cv_j)^q\\wedge T+\\int _G(dd^cv_j)^q\\wedge T\\right)+ \\int _G(dd^cv)^q\\wedge T\\\\\\displaystyle \\le \\lim _{j\\rightarrow +\\infty }\\int _{\\lbrace u_k<v_j\\rbrace }(dd^cv_j)^q\\wedge T+2\\varepsilon ||v||_{\\infty }^q\\\\\\displaystyle \\le \\lim _{j\\rightarrow +\\infty }\\int _{\\lbrace u_k<v_j\\rbrace }(dd^cu_k)^q\\wedge T+2\\varepsilon ||v||_{\\infty }^q.\\end{array}$ Now as $\\lbrace u_k<v_j\\rbrace \\downarrow {\\lbrace u_k\\le v\\rbrace }$ , ${\\lbrace u_k<v\\rbrace }\\uparrow {\\lbrace u<v\\rbrace }$ then $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\lim _{k\\rightarrow +\\infty }\\int _{\\lbrace u_k\\le v\\rbrace }(dd^cu_k)^q\\wedge T+2\\varepsilon ||v||_{\\infty }^q.$ The continuity of $u$ and $v$ on $\\Omega \\setminus G$ gives that ${\\lbrace u\\le v\\rbrace }\\setminus G$ is a closed subset of $\\Omega $ .",
"It follows that $\\int _{\\lbrace u\\le v\\rbrace \\setminus G}(dd^cu)^q\\wedge T\\ge \\lim _{k\\rightarrow +\\infty }\\int _{\\lbrace u\\le v\\rbrace \\setminus G}(dd^cu_k)^q\\wedge T.$ Thus $\\begin{array}{lcl}\\displaystyle \\int _{\\lbrace u\\le v\\rbrace }(dd^cu)^q\\wedge T&\\ge &\\displaystyle \\int _{\\lbrace u\\le v\\rbrace \\setminus G}(dd^cu)^q\\wedge T\\\\&\\ge &\\displaystyle \\lim _{k\\rightarrow +\\infty }\\int _{\\lbrace u\\le v\\rbrace \\setminus G}(dd^cu_k)^q\\wedge T\\\\&\\ge &\\displaystyle \\lim _{k\\rightarrow +\\infty }\\left(\\int _{\\lbrace u_k<v\\rbrace }(dd^cu_k)^q\\wedge T-\\int _G(dd^cu_k)^q\\wedge T\\right)\\\\&\\ge &\\displaystyle \\lim _{k\\rightarrow +\\infty }\\int _{\\lbrace u_k<v\\rbrace }(dd^cu_k)^q\\wedge T-\\varepsilon ||v||_{\\infty }^q.\\end{array}$ So $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u\\le v\\rbrace }(dd^cu)^q\\wedge T+3\\varepsilon ||v||_{\\infty }^q.$ By tending $\\varepsilon $ to 0, we obtain $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u\\le v\\rbrace }(dd^cu)^q\\wedge T$ As $\\lbrace u+\\rho <v\\rbrace \\uparrow \\lbrace u<v\\rbrace $ and $\\lbrace u+\\rho \\le v\\rbrace \\uparrow \\lbrace u<v\\rbrace $ when $\\rho \\searrow 0$ then the desired inequality follows by replacing $u$ by $u+\\rho $ .",
"Recall that the Lelong-Demailly number of $T$ with respect to a psh function $\\varphi $ is defined as the limit $\\nu (T,\\varphi ):=\\lim _{t\\rightarrow -\\infty }\\nu (T,\\varphi ,t)$ where $\\nu (T,\\varphi ,t)=\\int _{B_\\varphi (t)}T\\wedge (dd^c\\varphi )^q,\\ t<0\\ .$ The following result was proved in [6] but author has used Stokes formula where a regularity condition on $\\varphi $ is required.",
"Theorem 6 Let $\\varphi \\in \\mathcal {F}^T(\\Omega )$ such that $e^\\varphi $ is continuous on $\\Omega $ .",
"Then for every $s,t>0$ one has $s^q C_T(B_\\varphi (-t-s),\\Omega )\\le \\nu (T,\\varphi ,-t)\\le (s+t)^q C_T(B_\\varphi (-t),\\Omega ).$ In particular, $\\nu (T,\\varphi )=\\int _{\\lbrace \\varphi =-\\infty \\rbrace } T\\wedge (dd^c\\varphi )^q=\\lim _{t\\rightarrow +\\infty } t^q C_T(B_\\varphi (-t),\\Omega ).$ Let $t,s>0$ and $v\\in PSH(\\Omega ,[-1,0])$ .",
"For $\\varepsilon >0$ , we set $v_\\varepsilon =\\max (v,\\frac{\\varphi +t+\\varepsilon }{s})$ .",
"Thanks to Theorem REF we have $\\begin{array}{lcl}\\displaystyle \\int _{B_\\varphi (-t-s-\\varepsilon )}T\\wedge (dd^cv)^q&=& \\displaystyle \\int _{B_\\varphi (-t-s-\\varepsilon )}T\\wedge (dd^cv_\\varepsilon )^q\\\\&\\le &\\displaystyle \\int _{\\lbrace \\varphi <-t+sv-\\varepsilon \\rbrace }T\\wedge (dd^cv_\\varepsilon )^q\\\\&\\le &\\displaystyle \\frac{1}{s^q}\\int _{\\lbrace \\varphi <-t+sv-\\varepsilon \\rbrace }T\\wedge (dd^c\\varphi )^q\\\\&\\le &\\displaystyle \\frac{1}{s^q}\\int _{B_\\varphi (-t)}T\\wedge (dd^c\\varphi )^q.\\end{array}$ By passing to the supremum over all $v\\in PSH(\\Omega ,[-1,0])$ , we obtain the following estimate $s^q C_T(B_\\varphi (-s-t-\\varepsilon ),\\Omega )\\le \\nu (T,\\varphi ,-t).$ By passing to the limit when $\\varepsilon \\rightarrow 0$ , the left inequality in (REF ) is obtained.",
"However, for the right inequality, we remark that the function $\\psi =\\max (\\frac{\\varphi }{s+t},-1)$ is psh and satisfies $-1\\le \\psi \\le 0$ on $\\Omega $ , so by Corollary REF and using the fact that $\\psi >-1$ near $\\partial B_\\varphi (-t)$ we obtain $\\begin{array}{lcl}\\displaystyle \\int _{B_\\varphi (-t)}T\\wedge (dd^c\\varphi )^q&=&\\displaystyle (s+t)^q\\int _{B_\\varphi (-t)}T\\wedge (dd^c\\psi )^q\\\\&\\le &\\displaystyle (s+t)^q C_T(B_\\varphi (-t),\\Omega )\\end{array}$ and the right inequality in (REF ) follows.",
"By the right inequality in (REF ), we have $\\nu (T,\\varphi )=\\lim _{t\\rightarrow +\\infty }\\nu (T,\\varphi ,-t)\\le \\lim _{t\\rightarrow +\\infty } \\frac{(s+t)^q}{t^q}t^q C_T(B_\\varphi (-t),\\Omega )=\\lim _{t\\rightarrow +\\infty } t^q C_T(B_\\varphi (-t),\\Omega ).$ If we take $\\alpha >1$ and $s=\\alpha t$ in the left inequality in (REF ), we obtain $\\begin{array}{lcl}\\nu (T,\\varphi )=\\displaystyle \\lim _{t\\rightarrow +\\infty }\\nu (T,\\varphi ,-t)&\\ge &\\displaystyle \\lim _{t\\rightarrow +\\infty }\\frac{\\alpha ^q}{(1+\\alpha )^q}(1+\\alpha )^qt^qC_T(B_\\varphi (-(1+\\alpha )t),\\Omega )\\\\&=&\\displaystyle \\left(\\frac{\\alpha }{1+\\alpha }\\right)^q \\lim _{t\\rightarrow +\\infty } t^q C_T(B_\\varphi (-t),\\Omega ).\\end{array}$ The result follows by letting $\\alpha \\rightarrow +\\infty $ .",
"Remark 2 Claim that if $\\varphi \\in \\mathcal {F}_p^T(\\Omega )$ where $e^\\varphi $ is continuous on $\\Omega $ , then thanks to Proposition REF and Theorem REF , $\\nu (T,\\varphi )=0.$"
],
[
"Main result",
"The aim of this part is to prove the following main result: Main result (Comparison principle) Let $u\\in \\mathcal {F}^{T}(\\Omega )$ and $v\\in \\mathcal {E}^{T}(\\Omega )$ .",
"Then $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u<v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(dd^cu)^q\\wedge T.$ Before giving the proof, we give some corollaries."
],
[
"Consequences of the main result",
"Corollary 4 Let $u,v\\in \\mathcal {F}_p^{T}(\\Omega )$ such that $e^u$ is continuous on $\\Omega $ .",
"Then $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u<v\\rbrace }(dd^cu)^q\\wedge T.$ Thanks to the comparaison principle, we have $\\int _{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u<v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(dd^cu)^q\\wedge T\\le \\int _{\\lbrace u<v\\rbrace }(dd^cu)^q\\wedge T+\\nu (T,u).$ The result follows by the fact that $\\nu (T,u)=0$ because $u\\in \\mathcal {F}_p^{T}(\\Omega )$ .",
"Corollary 5 Let $u\\in \\mathcal {F}^{T}(\\Omega )$ and $v\\in \\mathcal {F}_p^{T}(\\Omega )$ such that $e^v$ is continuous on $\\Omega $ .",
"We assume that $(dd^cu)^q\\wedge T\\le (dd^cv)^q\\wedge T.$ Then $C_T(\\lbrace u<v\\rbrace ,\\Omega )=0$ .",
"Assume that $C_T(\\lbrace u<v\\rbrace ,\\Omega )>0$ , then there exists $\\psi \\in PSH(\\Omega ,[0,1])$ such that $\\int _{\\lbrace u<v\\rbrace }(dd^c\\psi )^q\\wedge T>0.$ For $\\varepsilon >0$ small enough, one has $v+\\varepsilon \\psi \\in \\mathcal {F}^T(\\Omega )$ so thanks to the comparaison principle, $\\begin{array}{lcl}\\displaystyle \\int _{\\lbrace u<v+\\varepsilon \\psi \\rbrace }(dd^c(v+\\varepsilon \\psi ))^q\\wedge T&\\le &\\displaystyle \\int _{\\lbrace u<v+\\varepsilon \\psi \\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(dd^cu)^q\\wedge T\\\\&\\le &\\displaystyle \\int _{\\lbrace u<v+\\varepsilon \\psi \\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(dd^cv)^q\\wedge T\\\\&\\le &\\displaystyle \\int _{\\lbrace u<v+\\varepsilon \\psi \\rbrace }(dd^cv)^q\\wedge T+\\nu (T,v).\\end{array}$ So: $\\varepsilon ^q\\int _{\\lbrace u<v\\rbrace }(dd^c\\psi )^q\\wedge T+\\int _{\\lbrace u<v+\\varepsilon \\psi \\rbrace }(dd^cv)^q\\wedge T\\le \\int _{\\lbrace u<v+\\varepsilon \\psi \\rbrace }(dd^cv)^q\\wedge T$ which is absurd."
],
[
"Proof of the main result",
"To prove the main result, we shall use a similar Xing's inequalities (see [10], [11] for more details), generalized to $\\mathcal {E}^{T}(\\Omega )$ .",
"We start by recalling the following lemma: Lemma 3 (See [7]) Let $S$ be a positive closed current of bidimension $(1,1)$ on $\\Omega $ and $u,v\\in PSH(\\Omega )\\cap L^{\\infty }(\\Omega )$ .",
"Assume that $u\\le v$ on $\\Omega $ and $\\displaystyle \\lim _{z\\rightarrow \\partial \\Omega }[u(z)-v(z)]=0.$ Then one has $\\int _{\\Omega }(v-u)^k dd^cw\\wedge S\\le k\\int _{\\Omega }(1-w)(v-u)^{k-1}dd^cu\\wedge S$ for all $k\\ge 1$ and $w\\in PSH(\\Omega ,[0,1])$ .",
"Lemma 4 Let $u,v\\in PSH(\\Omega )\\cap L^{\\infty }(\\Omega )$ such that $u\\le v$ on $\\Omega $ and $\\displaystyle \\lim _{z\\rightarrow \\partial \\Omega }[u(z)-v(z)]=0.$ Then one has $\\frac{1}{q!",
"}\\int _{\\Omega }(v-u)^q dd^cw_1\\wedge ...\\wedge dd^cw_q\\wedge T+\\int _{\\Omega }(r-w_1)(dd^cv)^q\\wedge T\\le \\int _{\\Omega }(r-w_1)(dd^cu)^q\\wedge T$ for every $r\\ge 1$ and $w_1,..., w_q\\in PSH(\\Omega ,[0,1])$ .",
"Let $K\\subset \\subset \\Omega $ and assume that $u=v$ on $\\Omega \\setminus K$ .",
"Using Lemma REF we obtain $\\begin{array}{l}\\displaystyle \\int _{\\Omega }(v-u)^q dd^cw_1\\wedge ...\\wedge dd^cw_q\\wedge T\\\\\\displaystyle \\le q\\int _{\\Omega }(v-u)^{q-1}dd^cw_1\\wedge ...\\wedge dd^cw_{q-1}\\wedge dd^cu\\wedge T\\\\\\vdots \\\\\\displaystyle \\le q!\\int _{\\Omega }(v-u)dd^cw_1\\wedge (dd^cu)^{q-1}\\wedge T\\\\\\displaystyle \\le q!\\int _{\\Omega }(w_1-r)dd^c(v-u)\\wedge \\left(\\sum _{i=0}^{q-1}(dd^cu)^i\\wedge (dd^cv)^{q-i-1}\\right)\\wedge T\\\\\\displaystyle = q!\\int _{\\Omega }(r-w_1)dd^c(u-v)\\wedge \\left(\\sum _{i=0}^{q-1}(dd^cu)^i\\wedge (dd^cv)^{q-i-1}\\right)\\wedge T\\\\\\displaystyle = q!\\int _{\\Omega }(r-w_1)((dd^cu)^q-(dd^cv)^q)\\wedge T.\\end{array}$ In the general case, for every $\\varepsilon >0$ we set $v_{\\epsilon }=\\max (u,v-\\varepsilon )$ .",
"Then $v_{\\epsilon }\\nearrow v$ on $\\Omega $ and satisfies $v_{\\epsilon }=u$ on $\\Omega \\setminus K$ for some $K\\subset \\subset \\Omega $ .",
"Hence $\\frac{1}{q!",
"}\\int _{\\Omega }(v_{\\varepsilon }-u)^q dd^cw_1\\wedge ...\\wedge dd^cw_q\\wedge T+\\int _{\\Omega }(r-w_1)(dd^cv_{\\varepsilon })^q\\wedge T\\le \\int _{\\Omega }(r-w_1)(dd^cu)^q\\wedge T$ Since $v_{\\varepsilon }-u\\nearrow v-u$ , the family of measures $(dd^cv_{\\varepsilon })^q\\wedge T$ converges weakly to $(dd^cv)^q\\wedge T$ as $\\varepsilon \\searrow 0$ and the function $r-w_1$ is lower semicontinuous then, by letting $\\varepsilon \\searrow 0$ , we obtain the desired inequality.",
"Proposition 5 Let $r\\ge 1$ and $w\\in PSH(\\Omega ,[0,1])$ .",
"For every $u,v\\in \\mathcal {F}^{T}(\\Omega )$ such that $u\\le v$ on $\\Omega $ one has $\\displaystyle \\frac{1}{q!",
"}\\int _{\\Omega }(v-u)^q (dd^cw)^q\\wedge T+\\int _{\\Omega }(r-w)(dd^cv)^q \\wedge T \\le \\displaystyle \\int _{\\Omega }(r-w)(dd^cu)^q\\wedge T.$ Furthermore, Inequality (REF ) holds for $u,v\\in \\mathcal {E}^{T}(\\Omega )$ such that $u\\le v$ on $\\Omega $ and $u=v$ on $\\Omega \\setminus K$ for some $K\\subset \\subset \\Omega $ .",
"$(a)$ Let $u,v\\in \\mathcal {F}^{T}(\\Omega )$ and $u_m,v_j\\in \\mathcal {E}_{0}^{T}(\\Omega )$ which decrease to $u$ and $v$ respectively as in Definition REF .",
"Replace $v_j$ by $\\max (u_j,v_j)$ we may assume that $u_j\\le v_j$ for $j\\ge 1$ .",
"By lemma REF we have for $m\\ge j\\ge 1$ $\\displaystyle \\frac{1}{q!",
"}\\int _{\\Omega }(v_j-u_m)^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\Omega }(r-w)(dd^cv_j)^q\\wedge T \\displaystyle \\le \\int _{\\Omega }(r-w)(dd^cu_m)^q\\wedge T.$ By approximating $w$ by a sequence of continuous psh functions vanishing on $\\partial \\Omega $ (see [3]) and using Proposition REF , we obtain when $m\\rightarrow +\\infty $ $\\displaystyle \\frac{1}{q!",
"}\\int _{\\Omega }(v_j-u)^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\Omega }(r-w)(dd^cv_j)^q\\wedge T \\le \\displaystyle \\int _{\\Omega }(r-w)(dd^cu)^q\\wedge T.$ Since $ r-w$ is lower semi-continuous then $\\displaystyle \\lim _{j\\rightarrow \\infty }\\int _{\\Omega }(r-w)(dd^cv_j)^q\\wedge T\\ge \\int _{\\Omega }(r-w)(dd^cv)^q\\wedge T.$ Hence by tending $j\\rightarrow +\\infty $ , we obtain the result.",
"$(b)$ Let $G$ and $W$ be open subsets of $\\Omega $ such that $K\\subset \\subset G\\subset \\subset W \\subset \\subset \\Omega $ .",
"There exists $\\widetilde{v}\\in \\mathcal {F}^{T}(\\Omega )$ such that $\\widetilde{v}\\ge v$ on $\\Omega $ and $\\widetilde{v}=v$ on $W$ .",
"Let $\\widetilde{u}$ such that $\\widetilde{u}=u$ on $G$ and $\\widetilde{u}=\\widetilde{v}$ either.",
"Since $u=v=\\widetilde{v}$ on $W\\setminus K$ , we have $\\widetilde{u}\\in PSH^-(\\Omega )$ .",
"It follows that $\\widetilde{u}\\in \\mathcal {F}^{T}(\\Omega )$ , $\\widetilde{u}\\le \\widetilde{v}$ and $\\widetilde{u}=u$ on $W$ .",
"Using $(a)$ we obtain $\\displaystyle \\frac{1}{q!",
"}\\int _{\\Omega }(\\widetilde{v}-\\widetilde{u})^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\Omega }(r-w)(dd^c\\widetilde{v})^q\\wedge T\\displaystyle \\le \\int _{\\Omega }(r-w)(dd^c\\widetilde{u})^q\\wedge T.$ As $\\widetilde{v}=\\widetilde{u}$ on $\\Omega \\setminus G$ then $\\displaystyle \\frac{1}{q!",
"}\\int _{W}(\\widetilde{v}-\\widetilde{u})^q\\wedge (dd^cw)^q \\wedge T+\\int _{W}(r-w)(dd^c\\widetilde{v})^q\\wedge T\\le \\int _{W}(r-w)(dd^c\\widetilde{u})^q\\wedge T.$ Now since $\\widetilde{u}=u$ , $\\widetilde{v}=v$ and $u=v$ on $\\Omega \\setminus K$ we obtain $\\displaystyle \\frac{1}{q!",
"}\\int _{\\Omega }(v-u)^q \\wedge (dd^cw)^q \\wedge T+\\int _{\\Omega }(r-w)(dd^cv)^q\\wedge T \\displaystyle \\le \\int _{\\Omega }(r-w)(dd^cu)^q\\wedge T.$ Remark 3 If we take $w=0$ and $r=1$ in Proposition REF , we obtain another proof of Proposition REF .",
"Theorem 7 Let $u,w_1,..., w_{q-1}\\in \\mathcal {F}^{T}(\\Omega )$ and $v\\in PSH^-(\\Omega )$ .",
"If we set $S=dd^cw_1\\wedge ...\\wedge dd^cw_{q-1}$ then $dd^c\\max (u,v)\\wedge T\\wedge S_{|\\lbrace u>v\\rbrace }=dd^cu\\wedge T\\wedge S_{|\\lbrace u>v\\rbrace }.$ We prove the theorem in two steps, first we assume that $v\\equiv a < 0$ .",
"Thanks to Lemma REF , there exist $u_j, w_{k,j}\\in \\mathcal {E}_{0}^{T}(\\Omega )\\cap \\mathcal {C}(\\overline{\\Omega })$ such that $(u_j)_j$ decreases to $u$ and $(w_{k,j})_j$ decreases to $w_k$ for each $1\\le k\\le q-1$ .",
"Since $\\lbrace u_j>a\\rbrace $ is open, one has $dd^c\\max (u_j,a)\\wedge T\\wedge S^j_{|\\lbrace u_j>a\\rbrace }=dd^cu_j\\wedge T\\wedge S^j_{|\\lbrace u_j>a\\rbrace }$ where $S^j=dd^cw_{1,j}\\wedge ...\\wedge dd^cw_{q-1,j}$ .",
"As $\\lbrace u>a\\rbrace \\subset \\lbrace u_j>a\\rbrace $ we obtain $dd^c\\max (u_j,a)\\wedge T\\wedge S^j_{|\\lbrace u>a\\rbrace }=dd^cu_j\\wedge T\\wedge S^j_{|\\lbrace u>a\\rbrace }$ It follows from [7] that $\\max (u-a,0)dd^c\\max (u_j,a)\\wedge T\\wedge S^j\\underset{j\\rightarrow +\\infty }{\\longrightarrow }\\max (u-a,0)dd^c\\max (u,a)\\wedge T\\wedge S$ $\\max (u-a,0)dd^cu_j\\wedge T\\wedge S^j\\underset{j\\rightarrow +\\infty }{\\longrightarrow }\\max (u-a,0)dd^cu\\wedge T\\wedge S.$ Hence $\\max (u-a,0)[dd^c\\max (u,a)\\wedge T\\wedge S-dd^cu\\wedge T\\wedge S]=0.$ So $dd^c\\max (u,a)\\wedge T\\wedge S=dd^cu\\wedge T\\wedge S\\quad on\\ \\lbrace u>a\\rbrace .$ Now assume that $v\\in PSH^-(\\Omega )$ .",
"Since $\\lbrace u>v\\rbrace =\\cup _{a\\in Q^-}\\lbrace u>a>v\\rbrace $ , it suffices to show that $dd^c\\max (u,v)\\wedge T\\wedge S=dd^cu\\wedge T\\wedge S\\quad on\\ \\lbrace u>a>v\\rbrace $ for all $a\\in Q^-$ .",
"As $\\max (u,v)\\in \\mathcal {F}^{T}(\\Omega )$ then by the first step, we have $dd^c\\max (u,v)\\wedge T\\wedge S_{|\\lbrace \\max (u,v)>a\\rbrace }&=dd^c\\max (\\max (u,v),a)\\wedge T \\wedge S_{|\\lbrace \\max (u,v)>a\\rbrace }\\\\&=dd^c\\max (u,v,a)\\wedge T\\wedge S_{|\\lbrace \\max (u,v)>a\\rbrace }\\\\dd^cu\\wedge T\\wedge S_{|\\lbrace u>a\\rbrace }&=dd^c\\max (u,a)\\wedge T \\wedge S_{|\\lbrace v>a\\rbrace }.$ The fact that $\\max (u,v,a)=\\max (u,a)$ on the open set $\\lbrace a>v\\rbrace $ gives $dd^c\\max (u,v,a)\\wedge T\\wedge S_{|\\lbrace a>v\\rbrace }=dd^c\\max (u,a)\\wedge T\\wedge S_{|\\lbrace a>v\\rbrace }.$ As $\\lbrace u>a>v$ } is contained in $\\lbrace u>a\\rbrace $ , in $\\lbrace \\max (u,v)>v\\rbrace $ and in $\\lbrace a>v\\rbrace $ , then by combining the last equalities we obtain $dd^c\\max (u,v)\\wedge T\\wedge S_{|\\lbrace u>a>v\\rbrace }=dd^c\\max (u,a)\\wedge T\\wedge S_{|\\lbrace u>a>v\\rbrace }.$ We can now prove an inequality analogous to Demailly's one found in [8].",
"Proposition 6 Let $u,v \\in \\mathcal {F}^{T}(\\Omega )$ such that $(dd^cu)^q\\wedge T(\\lbrace u=v=-\\infty \\rbrace )=0$ then $(dd^c\\max (u,v))^q\\wedge T\\ge {\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u\\ge v\\rbrace }(dd^cu)^q\\wedge T+{\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u<v\\rbrace }(dd^cv)^q\\wedge T.$ Let $\\mu $ be a positive measure vanishing on all pluripolar sets of $\\Omega $ and $u,v \\in \\mathcal {E}^{T}(\\Omega )$ such that $(dd^cu)^q\\wedge T\\ge \\mu $ , $(dd^cv)^q\\wedge T\\ge \\mu $ .",
"Then $(dd^cmax(u,v))^q\\wedge T\\ge \\mu $ .",
"a) For each $\\epsilon > 0$ put $A_\\epsilon =\\lbrace u=v-\\epsilon \\rbrace \\setminus \\lbrace u=v=-\\infty \\rbrace $ .",
"Since $A_\\epsilon \\cap A_\\delta = \\emptyset $ for $\\epsilon \\ne \\delta $ then there exists $\\epsilon _j\\searrow 0$ such that $(dd^cu)^q\\wedge T(A_{\\epsilon _j})=0$ for $j\\ge 1$ .",
"On the other hand, since $(dd^cu)^q\\wedge T(\\lbrace u=v=-\\infty \\rbrace )=0$ we have $(dd^cu)^q\\wedge T(\\lbrace u=v-\\epsilon _j\\rbrace )=0$ for $j\\ge 1$ .",
"Using theorem REF it follows that $\\begin{array}{lcl}\\displaystyle (dd^c\\max (u,v-\\epsilon _j))^q\\wedge (dd^cw)^q\\wedge T\\\\\\ge (dd^c\\max (u,v-\\epsilon _j))^q\\wedge T_{|\\lbrace u>v-\\epsilon _j\\rbrace }+(dd^c\\max (u,v-\\epsilon _j))^q\\wedge T_{|\\lbrace u<v-\\epsilon _j\\rbrace }\\\\=(dd^cu)^q\\wedge T_{|\\lbrace u>v-\\epsilon _j\\rbrace }+(dd^cv)^q\\wedge T_{|\\lbrace u<v-\\epsilon _j\\rbrace }\\\\={\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u\\ge v-\\epsilon _j\\rbrace }(dd^cu)^q\\wedge T+{\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u<v-\\epsilon _j\\rbrace }(dd^cv)^q\\wedge T\\\\\\ge {\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u\\ge v\\rbrace }(dd^cu)^q\\wedge T+{\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u<v-\\epsilon _j\\rbrace }(dd^cv)^q\\wedge T.\\end{array}$ Letting $j\\rightarrow +\\infty $ and by Theorem REF , we get $(dd^c\\max (u,v))^q\\wedge T\\ge {\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u\\ge v\\rbrace }(dd^cu)^q\\wedge T+{\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u<v\\rbrace }(dd^cu)^q\\wedge T$ because $\\max (u,v-\\epsilon _j)\\nearrow \\max (u,v)$ and ${\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u<v-\\epsilon _j\\rbrace }\\nearrow {\\mathchoice{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.22214pt}l}{\\rm 1\\hspace{-2.5pt}l}{\\rm 1\\hspace{-2.77771pt}l}}_{\\lbrace u<v\\rbrace }$ as $j\\rightarrow +\\infty $ .",
"b) Argument as a).",
"Proposition 7 Let $u\\in \\mathcal {F}^T(\\Omega ),\\ v\\in \\mathcal {E}^T(\\Omega )$ .",
"Then $\\begin{array}{l}\\displaystyle \\frac{1}{q!",
"}\\int _{\\lbrace u<v\\rbrace }(v-u)^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\lbrace u<v\\rbrace }(r-w)(dd^cv)^q\\wedge T\\\\\\displaystyle \\le \\int _{\\lbrace u<v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(r-w)(dd^cu)^q\\wedge T\\end{array}$ for $w\\in PSH(\\Omega ,[0,1])$ and all $r\\ge 1$ .",
"Let $\\varepsilon >0$ and set $\\widetilde{v}=\\max (u,v-\\varepsilon )$ .",
"By Inequality (REF ) in Proposition REF we have $\\begin{array}{l}\\displaystyle \\frac{1}{q!",
"}\\int _{\\Omega }(\\widetilde{v}-u)^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\Omega }(r-w)(dd^c\\widetilde{v})^q\\wedge T \\le \\displaystyle \\int _{\\Omega }(r-w)(dd^cu)^q\\wedge T.\\end{array}$ Since $\\lbrace u<\\widetilde{v}\\rbrace =\\lbrace u<v-\\varepsilon \\rbrace $ then thanks to Theorem REF , we have $\\begin{array}{l}\\displaystyle \\frac{1}{q!",
"}\\int _{\\lbrace u<v-\\varepsilon \\rbrace }(v-\\varepsilon -u)^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\lbrace u\\le v-\\varepsilon \\rbrace }(r-w) (dd^cv)^q\\wedge T\\\\ \\le \\displaystyle \\int _{\\lbrace u\\le v-\\varepsilon \\rbrace }(r-w)(dd^cu)^q\\wedge T.\\end{array}$ As $\\lbrace u\\le v-\\varepsilon \\rbrace \\subset \\lbrace u<v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace $ so $\\begin{array}{l}\\displaystyle \\frac{1}{q!",
"}\\int _{\\lbrace u<v-\\varepsilon \\rbrace }(v-\\varepsilon -u)^q\\wedge (dd^cw)^q \\wedge T+\\int _{\\lbrace u\\le v-\\varepsilon \\rbrace }(r-w) (dd^cv)^q\\wedge T\\\\ \\le \\displaystyle \\int _{\\lbrace u\\le v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(r-w)(dd^cu)^q\\wedge T.\\end{array}$ Letting $\\varepsilon \\rightarrow 0$ we obtain $\\begin{array}{l}\\displaystyle \\frac{1}{q!",
"}\\int _{\\lbrace u<v\\rbrace }(v-u)^q\\wedge (dd^cw)^q\\wedge T+\\int _{\\lbrace u< v\\rbrace }(r-w) (dd^cv)^q\\wedge T\\\\\\le \\displaystyle \\int _{\\lbrace u< v\\rbrace \\cup \\lbrace u=v=-\\infty \\rbrace }(r-w)(dd^cu)^q\\wedge T.\\end{array}$ To conclude the proof of the main result, it suffices to take $w=0$ and $r=1$ in the previous proposition."
]
] | 1403.0375 |
[
[
"Wave-particle duality from stochastic electrodynamics"
],
[
"Abstract The wave-particle duality is one of the most mysterious phenomena of the quantum theory, in this paper first it's studied the rise of the wave properties of matter from the theory of stochastic electrodynamics (SED), in which de Broglie's idea of a guiding wave that follows the particle and determine its trajectory is taken back.",
"In the frame of the theory (SED) this wave is of electromagnetic character and is a traveling wave that propagates within the zero point field (ZPF) which according the SED is a classic electromagnetic field that permeates all universe.",
"The wave rise from the interaction between the background field and the particle that moves with certain velocity.",
"Then, I expose the results of numerical simulations in which the behavior of the ZPF in a two slit diffraction experiment is calculated and a series of macroscopic experiments that shows behaviors similar to the ones described by the SED.",
"Both, simulations and experiments give us a better physical understanding of the guiding wave and of the wave properties of quantum particles"
],
[
"Introduction",
"The wave particle duality is one of the most important concepts within the quantum mechanics, the fact that particles such as electrons or neutrons exhibit a wave behavior can be seen in experiments such as the double slit experiment.",
"Even that, the quantum theory describes perfectly this phenomena, it doesn't explain the reason why we can study one particle some times as a wave and others as a particle, hence, this duality is not very well understand, and some times misinterpreted In Luis de Broglie's papers where the matter wave mechanics is treated, it is not associated any physical meaning to the matter wave, nether an origin to such behavior [1].",
"In later papers, he searched for a better understanding for the particle associated waves.",
"He did not understand a particle as a wave; rather it is suggested the existence of a \"pilot\" wave that dictates the corpuscle trajectory.",
"However, de Broglie didn't study the behavior of this wave and nether its interaction with the particle As basis for the SED it is assumed the existence of an electromagnetic the zero point field (ZPF) which is a classical background radiation field, i.e.",
"that follows Maxwell's equations, with stochastic amplitude and it is assumed that its stochastic properties are the same for every inertial frame of reference [2].",
"The stochastic electrodynamics (SED) theory suggest an interaction between a moving particle and the ZPF that give rise to an electromagnetic wave that rules the particle's trajectory, giving an explanation to the wave-particle duality as an emergent phenomena It is also assumed an intrinsic particle movement, known as \"Zitterbewegung\" witch is an oscillation for the particles with frequency $\\omega _c$ close to Compton's frequency [3] $\\omega _c = \\frac{m c^2}{\\hbar }$ This oscillation is such that the particle acquire an effective area and acts like it have an \"internal watch\".",
"we notice that this oscillation interacts with the background field vibration's modes of frequency $\\omega _c$"
],
[
"De Broglie's wave",
"Lets take a moving particle with a certain velocity $v$ immerse in the ZPF, as was mentioned before, the particle oscillates with frequency $\\omega _c$ , therefore it radiates, however, for the particle to be in equilibrium, this radiation most be absorbed by the field, i.e.",
"the particle interacts with the field modes of frequency $\\omega _c$ [4] If we consider that the particle moves in the x direction, then, there will be two waves associated to the frequency $\\omega _c$ each in one direction $-x$ and $x$ .",
"By the relativistic Doppler effect, this frequencies will be different.",
"Lets be this frequencies $\\omega _{+}$ and $\\omega _{-}$ , we can write $\\omega _{\\pm } = \\omega _{z} \\pm \\omega _B$ Where $\\omega _{z} = \\gamma \\omega _c$ is associated with the Lorentz's time contraction and $\\omega _B = \\gamma \\beta \\omega _c $ to the Doppler effect.",
"Analogously we can define a wave number $k_{\\pm }$ associated to $\\omega _{\\pm }$ as $k_{\\pm } = \\gamma k_z \\pm k_B$ with $k_B = \\gamma \\beta k_c$ Thus, even if in the electron's frame of reference there is an stationary wave, in the laboratory frame of reference there is a superposition of waves with frequencies $\\omega _{+}$ and $\\omega _{-}$ that travels in opposite directions, lets $\\phi _{-}$ and $\\phi _{+}$ represents this waves, the resultant wave for this superposition will be given by $\\phi (v) = \\phi _{+} + \\phi _{-} = 4 \\cos {(\\omega _z t - k_B x + \\theta _{1}) \\cos (\\omega _B t- \\gamma k_c x + \\theta _{2})}$ Where $\\theta _1$ and $\\theta _2$ are two random independent phases We notice that equation (REF ) represents a traveling wave with space and time modulation.",
"For the spatial case, we notice the traveling wave have wave number $k_B$ , with a modulation for the wave length $\\lambda _B = \\frac{2 \\pi }{k_B} = \\frac{\\lambda _c}{\\gamma \\beta } = \\frac{m \\lambda _c c}{p}$ On the other hand $\\lambda _c = \\frac{2 \\pi c}{\\omega _c}$ Thus, from equations (REF ) and (REF ) we get $\\lambda _B = 2 \\pi \\frac{m c^2}{\\omega _c}\\frac{1}{p}$ Finally, from equation (REF ) it is obtain the following relation $\\lambda _B = \\frac{2 \\pi \\hbar }{p} = \\frac{h}{mv}$ Equation (REF ) is the wave length equation for the matter waves suggested by Luis de Broglie.",
"Thus, it is associated a traveling wave of electromagnetic character to the particle, we also notice that, the previous development does not depend on the particle's charge, which is why, the particle may be neutral.",
"The ZPF polarize the neutral particle, creating an electromagnetic dipole, then, this particle interacts with the field as a particle with charge equal to the dipole charge"
],
[
"Schrödinger's equation",
"From SED and the \"Zitterbewegung\" it is possible to obtain Schrödinger's equation [5], however, most of this treatments requires an ensemble of particles.",
"On the other hand, from the pilot wave from equation (REF ) we can obtain time independent Schrödinger equation considering only one single particle Lets consider a wave equation for $\\phi $ given by $\\nabla ^2 \\phi - \\frac{1}{u^2} \\frac{\\partial ^2 \\phi }{\\partial t^2} =0$ Where $u$ is the wave $\\phi $ phase velocity given by $u = c^2 / v$ with $v$ the particle's velocity from equation (REF ) we get $\\frac{\\partial ^2 \\phi }{\\partial t^2} = -\\omega _{z}^{2} \\phi = - \\frac{\\gamma ^2 m^{2} c^{4}}{\\hbar ^2} \\phi $ Thus, equation (REF ) becomes $\\nabla ^2 \\phi + \\frac{v^2 \\gamma ^2 m^2}{\\hbar ^2} \\phi = \\nabla ^2 \\phi + \\frac{p^2}{\\hbar ^2} \\phi $ rewriting $p^2$ in terms of external potential $V$ , we get $\\nabla ^2 \\phi +\\frac{1}{\\hbar ^2 c^2}[(\\Sigma - V)^2 -m^2 c^4] \\phi $ Where $\\Sigma $ is the particle relativistic energy.",
"we notice that equation (REF ) is Kein-Gordon's equation and in the non relativistic limit, with $E = \\Sigma - m c^2$ we get $\\nabla ^2 \\phi + \\frac{2m}{\\hbar }(E-V)\\phi = 0$ This is time independent Schrödinger's equation for one particle.",
"lets notices that, even if the equation make reference to the particle properties Such as its mass and energy, it was obtain from the equation of the wave that generates upon moving in the background field, showing that, even if they are different entities, there is a relationship between the pilot wave and the particle"
],
[
"Physical meaning",
"The SED bring us a new way to interpret matter waves, as shown before, de Brogle's wave is an electromagnetic wave that follows the moving particle.",
"In this theory, the particle is always consider as a corpuscle, the wave-particle duality is not associated to it, however, its wave properties are given by the traveling wave that \"guides\" the particle within the background field, with this interpretation we obtain a more intuitive idea of the phenomena.",
"Also, interpretations such as saying that \"the particle interfere with it self\" are avoided.",
"Also, we notice that the pilot wave rises from the particle's velocity There are several interference and diffraction experiments with particles [6], in general, this experiments consist in three phases, the fist one is the particle emission, the second one is the so called \"interference screen\" or \"diffraction screen\" and the third one is the detection screen For neutron interference, it is possible to use an arrangement such that neutron emission is one by one, i.e.",
"one neutron is detected before the other is emitted, so that there isn't interference between two different neutrons.",
"In this experiments we can preserve the wave properties of neutrons, and, we can conclude that a single neutron presents by its own, wave properties.",
"Considering the stochastic theory given in this paper, that make sense, because a particle by its self already presents a pilot wave that rules its trajectory, in this sense, the particles don't interfere with their self and, neither they have \"free will\", simply, it's just the electromagnetic wave induced in the background field that presents interference (or is diffracted), this wave will be affected by such interference and, therefore, the particle as well, witch will have a bigger probability of being detected in the spots where the pilot wave's amplitude is bigger The mechanism by which, this wave rules the particle's behavior is not entirely clear, this is because of the difficulty of the mathematical theory, however, we can notice the works from J. Avendaño and L. de la peña [7], [8] in which the double slit experiment is studied and how its spatial conditions interact with the background field are considered.",
"By numerical approximations it is possible to know the effects of the particles moving across the slits over the field, the resultant field is diffracted whit the intensity maximum distributed in a similar way that in a real life double slit experiment maximum, also, if the trajectories of the particles immerse in the ZPF are analyzed, it is notice that particles arrive to isolated spots on the screen (placed several wave lengths apart from the slit) being the regions with higher probability of a particle arriving the ones where the field (being diffracted) amplitudes are higher, this behavior is similar to Young's experiment for photons and for electrons.",
"The results given by this simulations give us a better idea of how the ZPF give its wave properties to the particles and works as a more intuitive model of the wave-particle duality Even that, to date, it is not possible for us to experimentally test the results of the SED given previously in this paper, there are similar macroscopic experiments, some of this experiments are the ones performed by Couder and Ford [9] in which a vertical oscillation is induced to the fluid, over it, little fluid drops bounce and move in directions perpendicular to the fluid vibrations, in this experiments it is notice that when the drops moves with a given velocity, a wave in the fluid surrounding the drop rises.",
"This wave follows the droplet and guide its movement such as is described in the pilot wave theory for quantum particles.",
"Also, by making the double slit experiment using the droplets it was observed a behavior similar to Young's experiment with electrons, each droplet pass trough some of the two slits, but, its associated wave passes trough both of them and it was diffracted.",
"The final position of the droplets was always different and follows a statistic similar to a diffraction experiment.",
"Although this experiments can't be used to verify the pilot wave theory for quantum particles, the behavior of fluid droplets bouncing in a medium with a given oscillation is similar to the particles interacting with the ZPF oscillation modes in witch are immersed"
],
[
"Conclusion",
"Under the frame of SED it was possible to give a more intuitive interpretation to the wave-particle duality, not as phenomena witch associate the particle and wave properties to the same object, instead, as two different and separated things the, wave and the particle related, but with each other.",
"Thus, a particle never loses its corpuscular character and we could talk about a particle's trajectory, however, the particle's behavior within the ZPF is not easy to describe, given that is a stochastic field, is not possible to accurately predict the path that a particle will follow, however, with statistic treatments it can be said which will be the most probable position to find it given by the pilot wave intensity maximum, in that way, we can recover the already familiar results from quantum mechanics, including Schrodinger equation Even that there isn't a way to experimentally prove the pilot wave theory, the numeric simulations, the mentioned experiments in this paper and its similarities with the phenomena foretold by the SED lead us to thing that this description for quantum mechanics may be accurate, at least for the wave-particle duality phenomenon, but it would be necessary to perform more convincing experiments involving quantum particles so that we can certainly assure the existence of a pilot wave as described here."
]
] | 1403.0016 |
[
[
"Casimir Effect of Scalar Massive Field"
],
[
"Abstract The energy momentum tensor is used to introduce the Casimir force of the massive scalar field acting on a nonpenetrating surface.",
"This expression can be used to evaluate the vacuum force by employing the appropriate field operators.",
"To simplify our formalism we also relates the vacuum force expression to the imaginary part of the Green function via the fluctuation dissipation theorem and Kubo formula.",
"This allows one to evaluate the vacuum force without resorting to the process of field quantization.",
"These two approaches are used to calculate the attractive force between two nonpenetrating plates.",
"Special attention is paid to the generalization of the formalism to D + 1 space-time dimensions."
],
[
"Introduction",
"Historically the Casimir effect is known as the small force acting between two parallel uncharged perfect conducting plates.",
"The attractive force per unit area is [1] $ F=\\frac{\\pi ^2 \\hbar c }{240a^4}$ where $\\hbar $ is Plank's constant divided by $2\\pi $ , $c$ is the velocity of light and $a$ is the separation between the two conducting plates.",
"Though the force can be thought of as a limiting case of the Van der Waals interaction when retardation is included [2], it does not depend on the electric charge.",
"This encourages one to look for a new interpretation of the force.",
"It is known that the phenomena stems from the quantum aspect of the electromagnetic field, since the force tends to zero when we try to find the classical limit.",
"The force can, therefore, be considered as the reaction of the vacuum of the quantum field against the presence of the boundary surface or surfaces [3], [4], [5].",
"In other words, the effect is simply the stress on the boundary surfaces due to the confinement of a quantum field in a finite volume of space.",
"The restriction on the modes of the quantum field gives rise to the force acting on the macroscopic bodies [6].",
"Great variety of problems of practical and theoretical interests are intimately connected to various quantum fields having mass or massless, quanta of different spin.",
"The later interpretation of the Casimir force signifies that the effect can not be merely for the electromagnetic field, and all the other quantum fields must display the same effect [7], [8], [9], [10], [11].",
"It is, in fact, the most important characteristic of the Casimir effect in all instances that it depends neither on electric charge nor on any other coupling constants, but it does build on the quantum nature of the fields [12].",
"The effect can be displayed in $D+1$ space-time dimensions for a variety of boundary surfaces and boundary conditions, and manifested on all scales, from the substructure of quarks [13], [14], [15], [16], [17] to the large scale structure of the universe [18], [19], [20], [21], [22], [23], [24], [25], [26].",
"There are different approaches to calculate the Casimir force [3], [5], [6].",
"Perhaps, the most usual one is to calculate the Casimir energy [1].",
"This is simply the difference between the zero point energy in the presence and the absence of the boundaries.",
"The Casimir energy is apparently a function of the position of the boundaries.",
"Once the casimir energy is evaluated, the force is obtained easily by differentiation.",
"This approach can be cast into a more technical framework based upon the use of Green function.",
"The Green function is related to the vacuum expectation value of the time ordered products of the field.",
"It is, therefore, possible to calculate the Casimir energy in terms of the Green function at coincident arguments [6].",
"The equivalence of the sum of the zero point energy of the modes and the vacuum expectation value of the field is straightforward.",
"In an alternative approach one may obtain the variation in the electromagnetic energy when the dielectric function is varied.",
"The vacuum force between two semi-infinite parallel dielectrics can be obtained in this way [27].",
"The wide application of the effect on the one hand and the recent growth on the experimental verification of the phenomena [28], [29], [30], [31] on the other hand, demands some more illuminative and simpler derivation of this effect for different quantum fields.",
"The vacuum force is an indication of the momentum inherent in a quantized field when the field is in its vacuum state.",
"We are thus led to quantize the field for the particular geometry in question, and thereafter, calculate the Casimir force using the energy momentum tensor.",
"This is the most straightforward and clear treatment which provides both the force and the interpretation of the effect.",
"Apart from a few simple configurations [32], considerable difficulties arise in practice when the field is to be quantized.",
"Any simplification in this formalism is therefore of value whenever more complicated geometries are involved.",
"This is achieved by using the fluctuation dissipation theorem and Kubo's formula [33], [34].",
"The aim of the present paper is to develop this approach into the massive scalar field.",
"This may be a step forward to make a simple unified, yet comprehensive treatment of the Casimir force in a wide variety of domains and variously shaped configurations.",
"The paper is organized as follows.",
"In Sec.",
"II we begin with the quantization of the massive scalar field in 1+1 space-time dimensions in the presence of two nonpenetrating plates.",
"The attractive force between the two plates is obtained by the use of the latter field operators.",
"The formalism is developed to 3+1 space-time dimensions in Sec.",
"III.",
"In Sec.",
"IV, we calculate the vacuum force in both 1+1 and 3+1 space-time dimensions by using the fluctuation dissipation theorem and Kubo's formula.",
"The generalization of the two approaches to $D+1$ space-time dimensions are provided in Sec.",
"V. Finally, we summarize and discuss the main results in Sec.",
"VI."
],
[
"One Dimensional Formalism",
"In this section we will make use of the 1+1 space-time dimensions.",
"This is obviously a mathematical idealization that do not exist in the physical world, but it is instructive.",
"The treatment is rather straightforward and, therefore, the basic idea of the formalism can be explored properly."
],
[
"Field Quantization",
"We begin by summarizing the elements of field quantization in an appropriate form which are useful in the following sections.",
"To simplify the calculations we restrict our discussion to neutral spin zero field ${\\hat{\\varphi }} \\left(x, t \\right)$ with mass $m$ .",
"The Klein-Gordon equation governing the field of a massive spin zero particle in one dimension is [35] $ \\left(\\frac{\\partial ^2}{\\partial x^2}-\\frac{1}{c^2}\\frac{\\partial ^2}{\\partial t^2}-\\frac{m^2c^2}{{\\hbar }^2}\\right){\\hat{\\varphi }} \\left(x, t \\right)=0$ The usual technique of Lagrangian mechanics shows that the conjugate momentum of the field ${\\hat{\\varphi }} \\left(x, t\\right)$ is ${\\hat{\\pi }} \\left(x, t \\right)=\\frac{1}{c^2}\\frac{\\partial }{\\partial t}{\\hat{\\varphi }} \\left(x, t\\right)$ The quantization procedure is carried out easily by decomposing the field ${\\hat{\\varphi }} \\left(x, t \\right)$ into positive and negative frequency parts, and expanding each one in terms of an appropriate set of mode functions $u \\left(p,x \\right)$ as ${\\hat{\\varphi }}\\left( x,t \\right) &=&{\\hat{\\varphi }}^{+}\\left(x,t\\right) +{\\hat{\\varphi }}^{-}\\left(x, t\\right)\\nonumber \\\\ &=&\\left(\\frac{\\hbar c^2}{2}\\right)^{1/2}\\int _{-\\infty }^{+\\infty }\\frac{dp}{\\sqrt{\\omega }_p }{\\hat{a}}\\left(p,t\\right)u \\left(p,x \\right)+\\mathrm {H.C.}.$ where ${\\hat{\\varphi }}^{+}\\left(x,t\\right)$ contains all amplitudes which vary as $\\exp (-i {\\omega }_p t)$ and ${\\hat{\\varphi }}^{-}\\left(x,t\\right)$ contains all amplitudes which vary as $\\exp (i {\\omega }_p t)$ and ${\\hat{\\varphi }}^{-}\\left(x,t\\right)=\\left[{\\hat{\\varphi }}^{+}\\left(x,t\\right)\\right]^{\\dagger }$ .",
"In this expression $p$ is the particle momentum and ${\\omega }_p=\\left(p^2 c^2 + m^2 c^4\\right)^{1/2}/\\hbar $ We note that the use of mode function expansion has the effect that the time dependence of annihilation operator ${\\hat{a}}\\left(p,t\\right)$ being characterized by a simple phase factor ${\\hat{a}}\\left(p,t\\right)={\\hat{a}}\\left(p\\right)\\exp (-i {\\omega }_p t)$ we expect that the annihilation and creation operators ${\\hat{a}}\\left(p\\right)$ and ${\\hat{a}}^{\\dagger }\\left(p\\right)$ fulfill the usual algebra of bosonic operators, that is $\\left[{\\hat{a}}\\left(p\\right),{\\hat{a}}^{\\dagger }\\left(p^{\\prime }\\right)\\right]=\\delta (p-p^{\\prime })$ The mode function $u \\left(p,x \\right)$ satisfies the following differential equation $\\left(\\frac{d^2}{dx^2}+\\frac{p^2}{{\\hbar }^2}\\right) u\\left(p,x \\right)=0$ This is obtained by substitution of Eq.",
"(REF ) into Eq.",
"(REF ), and using Eq.",
"(REF ).",
"Regardless of the explicit form of $u \\left(p,x\\right)$ , it is seen from the sturm-Liouville theory that the mode functions obtained from Eq.",
"(REF ) are restricted to the orthogonality and completeness relations if a proper boundary condition is used and the mode functions are appropriately normalized.",
"The particular choice of coefficient in Eq.",
"(REF ) is made for a reason that will become evident shortly.",
"In order to demonstrate the full consistency of the field quantization, one must necessary prove the equal-time canonical commutation relation between the field operator ${\\hat{\\varphi }}\\left( x,t \\right)$ and its complex conjugate ${\\hat{\\pi }} \\left(x, t \\right)$ $ \\left[{\\hat{\\varphi }}\\left( x,t \\right),{\\hat{\\pi }}\\left(x^{\\prime }, t \\right)\\right]=i\\hbar \\delta (x-x^{\\prime })$ This can be verified most easily by combining Eq.",
"(REF ) with Eq.",
"(REF ) to obtain the explicit form of ${\\hat{\\pi }} \\left(x, t\\right)$ , and making use of Eqs. (7).",
"It is seen that the explicit form of $u \\left(p,x \\right)$ is not needed and the orthogonality and completness relations of the mode functions are the only requirements to prove Eq.",
"(REF ).",
"We are now in a stage to apply the presented quantization scheme to geometrical configurations of our interest.",
"Consider two parallel nonpenetrating plates of thickness $d$ located in an empty space.",
"The $x$ axis is chosen to be perpendicular to the interfaces with the origin at a distance $a/2$ from each plate as sketched in Fig.",
"REF .",
"We call the left(right) hand plate as plate 1(2) and refer the different empty regions separated by the plates as region 1, 2 and 3, as shown in Fig.",
"REF .",
"The aim is to obtain the field operator in the regions 2 and 3.",
"It is understood that the complete expression of the field operator in the region 3 is in the form of Eq.",
"(REF ).",
"In this region the appropriate mode functions are the solution of Eq.",
"(REF ) with the boundary condition $u \\left(p,x=d+a/2 \\right)=0$ It is straightforward, without elaborate algebra to show that $u \\left(p,x \\right)=\\frac{1}{\\sqrt{2\\pi \\hbar }}\\left\\lbrace \\exp \\left(i px/\\hbar \\right)-\\exp \\left[-ip(x-a-2d)/\\hbar \\right]\\right\\rbrace $ The mode functions in region 2 are obtained from Eq.",
"(REF ) with the boundary conditions $u \\left(p,x=a/2 \\right)=u\\left(p,x=-a/2 \\right)=0$ .",
"One can easily show that the mode functions are also the eigenfunctions of parity operator, and thus divided into two classes with parity $\\pm $ .",
"Therefore, $u_n^{-} \\left(x \\right)=\\sqrt{\\frac{}{}}{2}{a}{\\sin }\\frac{p_n^{-}}{\\hbar }x,\\hspace{42.67912pt}p_n^{-}=\\frac{2n\\pi \\hbar }{a}$ and $u_n^{+} \\left(x \\right)=\\sqrt{\\frac{}{}}{2}{a}{\\cos }\\frac{p_n^{+}}{\\hbar }x,\\hspace{42.67912pt}p_n^{+}=\\frac{(2n-1)\\pi \\hbar }{a}$ Figure: Spatial configuration of the two plates with the geometricparameters.where the integer $n=1,2,...$ .",
"This is the typical mode functions of a cavity.",
"Since $p=p_n$ takes discrete values, the integration with respect to $p$ in Eq.",
"(REF ) must be replaced with the summation over $p_n$ or, equivalently, $n$ ${\\hat{\\varphi }}\\left(x,t\\right)=\\left(\\frac{\\hbar c^2}{2}\\right)^{1/2}\\sum _{\\Omega , n}\\frac{1}{\\sqrt{{\\omega }_n^{\\Omega } }}{\\hat{a}}_n^{\\Omega }\\left(t\\right)u_n^{\\Omega } \\left(x \\right)+H.C.$ where $\\Omega =(\\pm )$ and ${\\omega }_n^{\\Omega }$ is $\\omega _p$ for $p=p_n^{\\Omega }$ .",
"We note that in this region the Dirac delta function in the right hand side of Eq.",
"(REF ) must be replaced by Kronecker delta functions $\\left[{\\hat{a}}_n^{\\Omega },{\\hat{a}}_{n^{\\prime }}^{{\\Omega }^{\\prime } \\dagger }\\right]={\\delta }_{nn^{\\prime }}{\\delta }_{\\Omega \\Omega ^{\\prime }}.$"
],
[
"Casimir Force",
"The statement of conservation of linear momentum for a classical field shows that the flow per unit area of linear momentum across the surface $S$ into volume $V$ is given by ${\\bf T} \\cdot {\\bf \\tilde{n}}$ , where ${\\bf \\tilde{n}}$ is the unit outward normal vector at the surface $S$ .",
"The three dimensional tensor ${\\bf T}$ is the stress tensor associated with the classical field.",
"It is in fact the space-space components of the four dimensional symmetric mixed canonical energy momentum tensor.",
"The explicit form of ${\\bf T}$ for the massive scalar field is $T_{\\alpha \\beta }=-\\frac{\\partial \\varphi }{\\partial x_{\\alpha }}\\frac{\\partial \\varphi }{\\partial x_{\\beta }}+\\frac{1}{2}\\left[(\\mbox{$\\nabla $}\\varphi )\\cdot (\\mbox{$\\nabla $}\\varphi )-\\frac{1}{c^2}\\left(\\frac{\\partial \\varphi }{\\partial t}\\right)^{2}+\\frac{m^2 c^2}{\\hbar ^2}\\varphi ^2 \\right]\\delta _{\\alpha \\beta }$ where $\\alpha , \\beta =1,2,3$ .",
"To describe the force in the quantum domain, we must replace the classical field by the corresponding field operator.",
"The expectation value of the force operator evaluated for a given state of the field (the vacuum state in our case) will yield the force.",
"Let us now calculate the Casimir force acting on plate 2 in Fig.",
"REF Recalling the symmetry of the present configuration, it is evident that the force per unit area acting on the exterior(interior) interface of plate 2 is given by ${T_{11}}$ (-${T_{11}}$ ).",
"It is seen from Eq.",
"(REF ) that the explicit form of ${T_{11}}$ in 1+1 space-time dimensions is $T_{11}=-\\frac{1}{2}\\left[\\left(\\frac{\\partial \\varphi }{\\partial x}\\right)^2+\\frac{1}{c^2}\\left(\\frac{\\partial \\varphi }{\\partial t}\\right)^{2}-\\frac{m^2 c^2}{\\hbar ^2}\\varphi ^2 \\right]$ To simplify the force expression it is convenient to separate the field operator into positive and negative frequency parts.",
"We, therefore, find that $F=\\mp \\frac{1}{2}\\langle 0|\\left(\\frac{\\partial \\varphi ^+}{\\partial x}\\right)\\left(\\frac{\\partial \\varphi ^-}{\\partial x}\\right)+\\frac{1}{c^2}\\left(\\frac{\\partial \\varphi ^+}{\\partial t}\\right)\\left(\\frac{\\partial \\varphi ^-}{\\partial t}\\right)-\\frac{m^2 c^2}{\\hbar ^2}\\varphi ^+ \\varphi ^-|0\\rangle $ where the upper(lower) sign holds for the force per unit area acting on the exterior(interior) interface of plate 2 and $|0\\rangle $ refers to the vacuum state of the field.",
"Note that the terms in which the annihilation operator acts directly on the vacuum state of the field have been set to zero.",
"The vacuum radiation pressure experienced by the exterior interface of plate 2 as well as its interior interface due to the vacuum field in the regions $x>a/2 +d$ and $-a/2<x<a/2$ respectively, are obtained by the substitution of Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ).",
"These calculations are not provided here for the sake of brevity.",
"The net force per unit area acting on plate 2 can be written down as $F=\\frac{c^2}{2a \\hbar }\\sum _{n=1}^{\\infty }\\frac{{p_n}^2}{{\\omega _n}}-\\frac{c^2}{2 \\pi \\hbar ^2}\\int _{0}^{+\\infty }dp\\frac{p^2}{{\\omega }_p}\\,.$ It is convenient to introduce $n=\\frac{a}{\\pi \\hbar }p$ as the variable of integration and rewrite Eq.",
"(REF ) in the form of $F=\\frac{ \\pi ^2 \\hbar ^2 c}{2a^3}\\left[\\sum _{n=0}^{\\infty }f(n)-\\int _{0}^{+\\infty }dn f(n)\\right]$ where $f(n)=\\frac{n^2}{\\sqrt{\\left(\\frac{n \\pi \\hbar }{a}\\right)^2+m^2 c^2}}$ Note that the term $n=0$ in Eq.",
"(REF ) is zero and thus Eq.",
"(REF ) and Eq.",
"(REF ) are identical.",
"Using the poisson's sum formula [3], [36], the summation of $f(n)$ over $n$ can be treated as the summation of the Fourier cosine transform of $f$ over $n$ , that is $F=\\frac{\\pi ^2 \\hbar ^2 c}{2a^3}\\left[\\sum _{n=-\\infty }^{+\\infty }\\int _{0}^{+\\infty }dx f(x) \\cos (2\\pi nx)-\\int _{0}^{+\\infty }dn f(n)\\right].$ It is seen that in the squared brackets in the right hand side of Eq.",
"(REF ), the first term for $n=0$ and the second term cancel one another and we find that $F=-\\frac{{\\hbar }^2 c}{4 a^3}\\sum _{n=1}^{\\infty }\\frac{\\partial ^2}{\\partial n^2}\\int _{0}^{+\\infty }dx\\frac{ \\cos (2\\pi nx )}{\\sqrt{\\left(\\frac{x \\pi \\hbar }{a}\\right)^2+m^2 c^2}}\\,.$ The integral representation of the modified Bessel function $K_0$ can be used for integration over $x$ and the recurrence relations of modified Bessel functions can be employed for differentiation in Eq.",
"(REF ) [37].",
"We finally obtain the force expression as $F=-\\frac{m^2 c^3}{\\pi \\hbar }\\sum _{n=1}^{\\infty }\\left[K_2 (n \\xi )-\\frac{1}{n \\xi }K_1(n \\xi )\\right].$ Employing the limiting forms of $K_1$ and $K_2$ for small argument, one can easily show that the Casimir force for a neutral massless scalar field in 1+1 dimensions is $F=-\\frac{\\pi \\hbar c}{24 a^2}$ This is half of the similar expression for the electromagnetic field.",
"This is understood in terms of the two states of polarization of the electromagnetic field [3]."
],
[
"Three Dimensional Formalism",
"A great variety of problems of practical and theoretical interest are in three space dimensions.",
"The development of the present formalism to 3+1 space-time dimensions is, therefore, important.",
"The purpose of this section is to set 3+1 dimensional formalism in a context to establish a close contact with the results already obtained in 1+1 dimensions.",
"This allows us to avoid any duplication for similar calculations."
],
[
"Field Quantization",
"The dynamical behaviour of a neutral spin zero field $\\hat{\\varphi }({\\bf r}, t)$ with mass $m$ is obtained from Klein-Gordon equation.",
"$\\left({\\mbox{$\\nabla $}^2}-\\frac{1}{c^2}\\frac{\\partial ^2}{\\partial t^2}-\\frac{m^2c^2}{\\hbar ^2}\\right){\\hat{\\varphi }({\\bf r}, t)}=0$ where ${\\bf r}=(x_1,x_2,x_3)$ .",
"We usually work with a complete continuum of plane wave mode functions in which the ${\\bf p}$ vector are not restricted to the discrete spectrum.",
"As in the 1+1 dimensional formulation it is useful to decompose the field operator into positive and negative frequency parts $\\hat{\\varphi }^{\\pm } ({\\bf r}, t)$ .",
"Expanding each part in terms of an appropriate set of mode functions $u({\\bf p},{\\bf r})$ , we find that ${\\hat{\\varphi }}\\left( {\\bf r},t \\right)=\\left(\\frac{\\hbar c^2}{2}\\right)^{1/2}\\int \\frac{d^3 {\\bf p}}{\\sqrt{\\omega }_p }{\\hat{a}}\\left({\\bf p },t\\right)u \\left({\\bf p},{\\bf r} \\right) +\\mathrm {H.c.}.$ Comparing Eqs.",
"(REF ) and (REF ) with their corresponding 1+1 dimensional forms Eq.",
"(REF ) and (REF ), it is seen that the annihilation operator ${\\hat{a}}\\left({\\bf p },t\\right)$ has the same time dependence as in Eq.",
"(REF ), and possesses the bosonic commutation relation $\\left[{\\hat{a}}\\left({\\bf p}\\right),{\\hat{a}}^{\\dagger }\\left({\\bf p }^{\\prime }\\right)\\right]=\\delta ({\\bf p}-{\\bf p}^{\\prime })$ It can be verified from substitution of Eq.",
"(REF ) into Eq.",
"(REF ) that the mode functions are obtained from the differential equation $\\left({\\mbox{$\\nabla $}^2} +\\frac{p^2}{\\hbar ^2}\\right)u({\\bf p},{\\bf r})=0$ For some purposes which become evident shortly, it is useful to cast the mode functions for free space in the absence of any boundary surface into some what different form $u\\left({\\bf p, {\\bf r}}\\right)=\\frac{1}{2 \\pi \\hbar }\\exp (i{\\bf p}_{\\parallel }\\cdot {\\bf x}_{\\parallel }/\\hbar )u(p_1, x_1)$ where ${\\bf x}_{\\parallel }=(x_2,x_3)$ , ${\\bf p}_{\\parallel }=(p_2,p_3)$ and $u(p_1, x_1)$ is obtained by solving Eq.",
"(REF ) and replacing $x$ and $p$ with $x_1$ and $p_1$ , respectively.",
"We now consider briefly the field quantization in the region 3 of Fig.",
"REF in three space dimensions.",
"We note that the $x_2x_3$ plane is assumed to be parallel to the plane interfaces of the two plates.",
"It is convenient to retain the forms of Eqs.",
"(REF ) and (REF ) for the field operator and the mode functions.",
"This means that the explicit form of $u(p_1, x_1)$ is now given by Eq.",
"(REF ).",
"A more complicated form for the field operator appears in the region 2.",
"In this case we have to work with a complete plane wave mode functions in which the $p_1$ component of the ${\\bf p}$ vectors are restricted to the discrete spectrum, while their ${\\bf p_{\\parallel }}$ are not.",
"This indicates that the proper form for the field operator is ${\\hat{\\varphi }}\\left({\\bf r},t\\right)=\\left(\\frac{\\hbar c^2}{2}\\right)^{1/2}\\int d^2 {\\bf p}_{\\parallel } \\sum _{\\Omega , n}\\frac{1}{\\sqrt{{\\omega }_n^{\\Omega }}}{\\hat{a}}_n^{\\Omega }\\left({\\bf p}_{\\parallel },t\\right)u_n^{\\Omega } \\left({\\bf p}_{\\parallel }, {\\bf r}\\right)+H.c.$ where $u_n^{\\Omega }\\left({{\\bf p}_{\\parallel }, {\\bf r}}\\right)=\\frac{1}{2 \\pi \\hbar }\\exp (i{\\bf p}_{\\parallel }\\cdot {\\bf x}_{\\parallel }/\\hbar )u_n ^{\\Omega }(x_1)$ Note that $u_n ^{\\Omega }(x_1)$ are given by Eqs.",
"(REF ) and (REF ).",
"Recalling the typical algebra of bosonic operators inside the cavity in one dimension, we are led to consider the following algebra in the present case $\\left[{\\hat{a}}_n^{\\Omega }({\\bf p}_{\\parallel }),{\\hat{a}}_{n^{\\prime }}^{{\\Omega }^{\\prime } \\dagger }({\\bf p}_{\\parallel }^{\\prime })\\right]={\\delta }_{nn^{\\prime }}{\\delta }_{\\Omega \\Omega ^{\\prime }}\\delta ({\\bf p}_{\\parallel }-{\\bf p}_{\\parallel }^{\\prime })$ The question such as the orthogonality and completeness relations of the mode functions in each region in three dimensions are easily verified in terms of Eq.",
"(REF ) and the corresponding one dimensional problem."
],
[
"Casimir Force",
"Having established the appropriate forms for the field operator in regions 2 and 3 in 3+1 dimensions, we can now seek to establish the force expression.",
"Decomposing the field into positive and negative frequency parts, it is straightforward to show that the force per unit area acting on the exterior(interior) interface of plate 2 is $F=\\mp \\frac{1}{2}\\langle 0|\\left(\\frac{\\partial \\varphi ^+}{\\partial x_1}\\right)\\left(\\frac{\\partial \\varphi ^-}{\\partial x_1}\\right)-\\left(\\frac{\\partial \\varphi ^+}{\\partial x_{\\parallel }}\\right)\\left(\\frac{\\partial \\varphi ^-}{\\partial x_{\\parallel }}\\right)+\\frac{1}{c^2}\\left(\\frac{\\partial \\varphi ^+}{\\partial t}\\right)\\left(\\frac{\\partial \\varphi ^-}{\\partial t}\\right)-\\frac{m^2 c^2}{\\hbar ^2}\\varphi ^+ \\varphi ^-|0\\rangle $ with the sign convention of Eq.",
"(REF ).",
"We are now in a position to evaluate the force on the exterior and interior interfaces of plate 2 and express the net force as $F=\\frac{1}{(2 \\pi \\hbar )^2} \\int d^2 {\\bf p}_{\\parallel } \\left\\lbrace \\frac{ \\pi ^2 \\hbar ^2 c}{2a^3}\\left[\\sum _{n=0}^{\\infty }f(n, p_{\\parallel })-\\int _{0}^{+\\infty }dn f(n,p_{\\parallel })\\right]\\right\\rbrace $ where $f(n, p_{\\parallel })=\\frac{n^2}{\\sqrt{\\left(\\frac{n\\pi \\hbar }{a}\\right)^2+p_{\\parallel }^2+m^2 c^2}}$ Though the two terms in Eq.",
"(REF ) diverge, it is straightforward with the help of Poisson's sum formula to obtain a finite net force acting on plate 2.",
"Following the steps from Eq.",
"(REF ) to Eq.",
"(REF ), the force can be written as $F=\\frac{1}{(2 \\pi \\hbar )^2} \\int d^2 {\\bf p}_{\\parallel } \\left\\lbrace -\\frac{{\\hbar }^2 c}{4 a^3}\\sum _{n=1}^{\\infty }\\frac{\\partial ^2}{\\partial n^2}\\int _{0}^{+\\infty }dx\\frac{ \\cos (2\\pi nx )}{\\sqrt{\\left(\\frac{x \\pi \\hbar }{a}\\right)^2+p_{\\parallel }^2+m^2 c^2}}\\right\\rbrace $ Employing the integral representation of the modified Bessel function $K_0$ and using the recurrence relations of modified Bessel functions for differentiation in Eq.",
"(REF ) [37], the force can be obtained as $F=-\\frac{m^4 c^5}{2 \\pi ^2 \\hbar ^3}\\sum _{n=1}^{\\infty } \\frac{1}{n\\xi }\\left[K_3 (n \\xi )-\\frac{1}{n \\xi }K_2(n\\xi )\\right]$ The Casimir force for a neutral massless scalar field in 3+1 space-time dimensions is obtained by using the limiting forms of $K_2$ and $K_3$ for small argument $F=-\\frac{\\pi ^2 \\hbar c}{480 a^4}$ As in the one dimensional formalism, Eq.",
"(REF ) is half of the similar expression for massless vector field.",
"We thus introduce a formal approach for the calculation of the Casimir force for massless scalar field.",
"It is, however, useful to cast this formalism into a some what different form."
],
[
"Casimir Effect and Fluctuation Dissipation Theorem",
"It is seen that in this formalism one should necessarily go through the process of the field quantization for the particular geometry in question and then calculate the Casimir force using the stress tensor.",
"However, the field quantization encounters considerable difficulties, except for a few simple geometries.",
"Any simplification in this formulation to avoid the use of explicit form of the field operator is therefore appreciated whenever more complicated geometries are involved.",
"It is convenient to separate the field operator into positive and negative frequency parts in the usual way ${\\hat{\\varphi }}\\left({\\bf r}, t\\right) &=&{\\hat{\\varphi }}^{+}\\left({\\bf r},t\\right)+{\\hat{\\varphi }}^{-}\\left({\\bf r}, t\\right)\\nonumber \\\\ &=&\\frac{1}{\\sqrt{2\\pi }}\\int _{mc^2/\\hbar }^{+\\infty }d\\omega _p { \\hat{\\varphi }} ^{+}\\left( {\\bf r},\\omega _p \\right)\\exp \\left( -i\\omega _p t\\right) +\\mathrm {H.c.},$ It is clear that the positive and negative frequency parts in the integrand involve only the annihilation and creation operators, respectively.",
"Note that the lower limit of integration in Eq.",
"(REF ) is the condition which implies by Eq.",
"(REF ).",
"The quantized field correlation function can be related to the imaginary part of the field Green function in the frequency domain $G({\\bf r}, {\\bf r}^{\\prime }, \\omega _p)$ by using fluctuation dissipation theorem and Kubo's formula [38] $ \\langle 0|{ \\hat{\\varphi }}^{+}\\left({\\bf r,\\omega _p}\\right){ \\hat{\\varphi }}^{-}\\left({\\bf r^{\\prime },\\omega _p^{\\prime }}\\right)|0\\rangle = 4 \\hbar {\\mathrm {Im}}G\\left({\\bf r},{\\bf r}^{\\prime },\\omega _p\\right)\\delta \\left( \\omega _p -\\omega _p^{\\prime }\\right).$ Note that the right hand side of Eq.",
"(REF ) is half of the similar expression for the electromagnetic field.",
"This arises from the two states of polarization of the electromagnetic field.",
"The whole idea in this approach is based on the understanding that we can express Eq.",
"(REF ) or Eq.",
"(REF ) in terms of the field correlation function such that to use Eq.",
"(REF ) for the calculation of the Casimir force.",
"It is useful to treat the problem for both 1+1 and 3+1 space-dimensions separately."
],
[
"Casimir Force in 1+1 dimensions",
"It is seen that in order to calculate the force per unit area acting on plate 2, we need the explicit form of the coordinate space Green function for the case where both the source and observation points are on either side of the plate.",
"One can easily show from Eq.",
"(REF ) that the Fourier-time transformed Green function in 1+1 dimensions is determined by the solution of $\\left(\\frac{d^2}{dx^2}+\\frac{p^2}{{\\hbar }^2}\\right) G\\left(x, x^{\\prime }, \\omega _p\\right)=-\\delta (x-x^{\\prime })$ The boundary conditions on $G\\left(x, x^{\\prime }, \\omega _p\\right)$ are divided into two types.",
"The first are the boundary conditions at $\\pm \\infty $ , which are easily imposed by assuming outgoing travelling waves.",
"The second are those at the plane interfaces of the plates, which are governed by the boundary conditions on the field.",
"The vanishing of the field operator on the plates entails that the Green function should necessarily vanish when the observation point $x$ is on the interface of the plates.",
"The details of these calculation are omitted here for the sake of brevity.",
"The coordinate space Green function is $G(x, {x}^{\\prime }, \\omega _p)=\\frac{i \\hbar }{2p}\\left\\lbrace \\exp [ip|x-x^{\\prime }|/\\hbar ]-\\exp [ip(x+x^{\\prime }-a-2d)/\\hbar ]\\right\\rbrace $ where the two points $x$ and $x^{\\prime }$ are both within region 3.",
"The two terms in Eq.",
"(REF ) are typical of the semi-infinite geometry.",
"The bulk part term with argument $|x-x^{\\prime }|$ is associated with the direct communication between $x$ and $x^{\\prime }$ , while the other term corresponds to the communication between $x$ and $x^{\\prime }$ via reflection in the exterior interface of of plate 2.",
"If $x$ and $x^{\\prime }$ are both within region 3, the appropriate Green function in the coordinate space is $&&G(x, {x}^{\\prime }, \\omega _p)=\\frac{i\\hbar }{2p}\\exp (ip|x-x^{\\prime }|/\\hbar )-\\frac{i\\hbar }{2p} \\frac{\\exp (2ipa/\\hbar )}{1-\\exp (2ipa/\\hbar )}\\big \\lbrace \\exp [-ip(x+x^{\\prime }+a)/\\hbar ]\\nonumber \\\\&&\\hspace{108.405pt}+\\exp [ip(x+x^{\\prime }-a)/\\hbar ]-\\exp [-ip(x-x^{\\prime })/\\hbar ]-\\exp [ip(x-x^{\\prime })/\\hbar ]\\big \\rbrace .$ The structure of this Green function is typical of the cavity systems.",
"The first term, the bulk part, corresponds to the direct communication between $x$ and $x^{\\prime }$ , while the rest is associated with the communication between $x$ and $x^{\\prime }$ via reflections from the cavity walls.",
"Note that in the derivation of Eqs.",
"(REF ) and (REF ), the boundary conditions requires that these two Green function vanish on the exterior and interior interface of plate 2, respectively.",
"Having found the coordinate space Green function on either side of plate 2, the evaluation of the Casimir force acting on plate 2 is straightforward by means of Eqs.",
"(REF ) and (REF ).",
"It seems clear, in view of the typical behaviour of the Green function on the interfaces of plate 2, that the only nonvanishing term of the force expression in 1+1 dimensions is $F=\\mp \\frac{1}{2}\\langle 0|\\frac{\\partial }{\\partial x} \\hat{\\varphi }^+(x,t) \\frac{\\partial }{\\partial x}\\hat{\\varphi }^-(x,t)|0\\rangle $ This means that only the spatial derivatives of the field correlation function is not zero on the interfaces.",
"Substitution of the field operator for 1+1 dimensions from Eq.",
"(REF ) into Eq.",
"(REF ) yields $F=\\mp \\frac{1}{4\\pi }\\int _{mc^2/\\hbar }^{+\\infty }d\\omega _p\\int _{mc^2/\\hbar }^{+\\infty }d\\omega _p^{\\prime }\\left[\\frac{\\partial }{\\partial x}\\frac{\\partial }{\\partial x^{\\prime }}\\langle 0| \\hat{\\varphi }^{+}(x,\\omega _p) \\hat{\\varphi }^{-}(x^{\\prime }, \\omega _p^{\\prime }) |0\\rangle \\right]_{x=x^{\\prime }}\\exp [-i(\\omega _p -\\omega _p^{\\prime })t]$ Employing Eq.",
"(REF ), the force expression can be rewritten as $F=\\mp \\frac{\\hbar }{2\\pi }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p\\left[ \\frac{\\partial }{\\partial x}\\frac{\\partial }{\\partial x^{\\prime }} {\\mathrm {Im}}G\\left(x,x^{\\prime },\\omega _p\\right) \\right]_{x=x^{\\prime }}$ The Casimir force acting on plate 2 is evaluated by taking into account the vacuum radiation pressure on both side of this plate.",
"Using Eq.",
"(REF ), the net force per unit area is $F=\\frac{\\hbar }{2\\pi }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p\\left\\lbrace \\left[ \\frac{\\partial }{\\partial x}\\frac{\\partial }{\\partial x^{\\prime }}{\\mathrm {Im}}G\\left(x, x^{\\prime },\\omega _p\\right) \\right]_{x=x^{\\prime }=a/2}-\\left[ \\frac{\\partial }{\\partial x}\\frac{\\partial }{\\partial x^{\\prime }}{\\mathrm {Im}}G\\left(x, x^{\\prime },\\omega _p\\right) \\right]_{x=x^{\\prime }=a/2+d}\\right\\rbrace $ The coordinate space Green function needed for substitution into the first and second terms of Eq.",
"(REF ) are given by Eqs.",
"(REF ) and (REF ), respectively.",
"The bulk part contributions, which are identical in both Eqs.",
"(REF ) and (REF ), cancel each other in Eq.",
"(REF ).",
"Furthermore, on expanding the prefactor of the square brackets in Eq.",
"(REF ), one can easily show, by an appropriate manipulation of the summation indices, that the four terms of Eqs.",
"(REF ) can be written as summation over $n$ , where in the first term $n$ varies from 0 to $\\infty $ , while in the other three terms it varies from 1 to $\\infty $ .",
"The reflection term in Eq.",
"(REF ) on the exterior interface of plate 2 is identical with the first term in Eq.",
"(REF ) with $n=0$ on the interior interface.",
"These two terms cancel one another as well.",
"The result after some algebra can be rewritten as $F=\\frac{1}{\\pi }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p p\\sum _{n=1}^{\\infty } \\cos (2np a/ \\hbar )$ It is more convenient to choose $p$ as the variable of integration.",
"Therefore $F=\\frac{c}{\\pi \\hbar }\\sum _{n=1}^{\\infty }\\int _{0}^{\\infty }dp \\frac{{p}^2\\cos (2np a/ \\hbar )}{\\sqrt{{p}^2+m^2 c^2}}$ The integration in Eq.",
"(REF ) is all that need be done.",
"To establish contact with the result already obtained, given by Eq.",
"(REF ), it is adequate to introduce $x=ap/ \\pi \\hbar $ as the variable of integration.",
"The treatment is straightforward and the force expression can, therefore, be simplified as Eq.",
"(REF )."
],
[
"Casimir Force in 3+1 dimensions",
"As in 1+1 dimensions, we need the explicit form of the coordinate space Green function for the case where both the source and observation points ${\\bf r}$ and ${\\bf r}^{\\prime }$ are within the gap between the two plates as well as where both ${\\bf r}$ and ${\\bf r}^{\\prime }$ are in region 3.",
"One can easily begin with the usual definition of the Fourier time transformed Green function along with the use of Eq.",
"(REF ) to show that the response function satisfies the following differential equation $\\left({\\mbox{$\\nabla $}^2}+\\frac{p^2}{\\hbar ^2}\\right)G({\\bf r}, {\\bf r}^{\\prime },\\omega _p)=-\\delta ({\\bf r}-{\\bf r}^{\\prime }).$ The symmetry of the present configuration enables us to convert this partial differential equation into an ordinary differential equation in variable $x_1$ .",
"This is obtained by expressing the response function in terms of its Fourier transform as $ G({\\bf r}, {\\bf r}^{\\prime }, \\omega _p)=\\frac{1}{(2\\pi \\hbar )^2} \\int d^2 {\\bf p}_{\\parallel } G({\\bf p}_{\\parallel },x_1, x_1^{\\prime }, \\omega _p) \\exp [i{\\bf p}_{\\parallel }\\cdot ({\\bf x}_{\\parallel }-{\\bf x}_{\\parallel }^{\\prime })/ \\hbar ]$ Substitution of Eq.",
"(REF ) into Eq.",
"(REF ) shows that $G({\\bf p}_{\\parallel }, x_1, x_1^{\\prime }, \\omega _p)$ satisfies $\\left(\\frac{\\partial ^2}{\\partial x_1^2}+\\frac{{p_1}^2}{\\hbar ^2}\\right)G({\\bf p}_{\\parallel }, x_1,x_1^{\\prime }, \\omega _p)=-\\delta (x_1-x_1^{\\prime })$ It is obvious that the boundary conditions on Eq.",
"(REF ) are the same as those applies on Eq.",
"(REF ).",
"It, therefore, follows that the solutions of Eq.",
"(REF ) on the right and left side of plate 2 are in the form of Eqs.",
"(REF ) and (REF ), respectively.",
"The coordinate space Green function corresponding to Eq.",
"(REF ) can be written in the form of $ G({\\bf r}, {\\bf r}^{\\prime }, \\omega _p)={\\cal G}({\\bf r}_{0}^{rel}, \\omega _p)-{\\cal G}({\\bf r}_{1}^{rel}, \\omega _p)$ where the first term corresponds to the bulk part and defined as ${\\cal G}({\\bf r}_{0}^{rel}, \\omega _p)=\\frac{i}{8\\pi ^2 \\hbar }\\int \\frac{d^2 {\\bf p}_{\\parallel }}{p_1}\\exp [ip_1|x_1-x_1^{\\prime }|/ \\hbar ] \\exp [i{\\bf p}_{\\parallel }\\cdot ({\\bf x}_{\\parallel }-{\\bf x}_{\\parallel }^{\\prime })/ \\hbar ]$ where ${\\bf r}_{0}^{rel}=(x_1-x_1^{\\prime }){\\bf \\hat{\\bf x_1}}+{\\bf x}_{\\parallel }^{rel}, \\hspace{42.67912pt} {\\bf x}_{\\parallel }^{rel}={\\bf x}_{\\parallel }-{\\bf x}_{\\parallel }^{\\prime }$ The second term in Eq.",
"(REF ) is associated with the surface term having the following form ${\\cal G}({\\bf r}_{1}^{rel}, \\omega _p)=\\frac{i}{8 \\pi ^2\\hbar }\\int \\frac{d^2 {\\bf p}_{\\parallel }}{p_1}\\exp [ip_1(x_1+x_1^{\\prime }-a-2d)/ \\hbar ]\\exp [i{\\bf p}_{\\parallel }\\cdot ({\\bf x}_{\\parallel }-{\\bf x}_{\\parallel }^{\\prime })/ \\hbar ]$ where ${\\bf r}_{1}^{rel}=(x_1+x_1^{\\prime }-a-2d){\\bf \\hat{x}_1}+{\\bf x}_{\\parallel }^{rel}.$ The integration in Eqs.",
"(REF ) and (REF ) can be done in a straightforward manner to find the coordinate space Green function.",
"The treatment is lengthy and rather boring and conventional.",
"We actually look for a plain, and not elaborate, argument to obtain the coordinate space response function.",
"The bulk part can be thought of as the Green function in the absence of any boundary surfaces.",
"In this sense ${\\cal G}({\\bf r}_{0}^{rel},\\omega _p)$ is the Green function of Helmholtz differential equation, given by Eq.",
"(REF ), in the absence of any boundary surfaces.",
"This is a well known response function which we write it down in the following form for the reason that will be clear shortly $ {\\cal G}({\\bf r}_{i}^{rel}, \\omega _p)=\\frac{1}{4 \\pi {\\bf r}_{i}^{rel}}\\exp (ip{\\bf r}_{i}^{rel}/ \\hbar ), \\hspace{42.67912pt}i=0,1,...$ where ${\\bf r}_{i}^{rel}$ for $i=0$ denotes the position of the observation point relative to the source point.",
"From the similarity of Eq.",
"(REF ) with Eq.",
"(REF ), it is seen that the response function ${\\cal G}({\\bf r}_{1}^{rel}, \\omega _p)$ can also be written in the form of Eq.",
"(REF ) where the position of the observation points relative to the source point is now defined as ${\\bf r}_{1}^{rel}$ , given by Eq.",
"(REF ).",
"The structure of Eq.",
"(REF ) is typical of a semi-infinite free space Green function.",
"The surface part, the second term, that corresponds to the communication between the points ${\\bf r}$ and ${\\bf r}^{\\prime }$ via reflection in the plane interface is associated with the image source.",
"We are now in a position to adopt this mathematical treatment for the calculation of the coordinate space Green function corresponding to Eq.",
"(REF ).",
"In fact the expansion of the prefactor of the square brackets in Eq.",
"(REF ) prepares the ground for the use of latter technique in a straightforward manner.",
"This allows one to cast the response function into the standard form as $ G({\\bf r}, {\\bf r}^{\\prime }, \\omega _p)={\\cal G}({\\bf r}_{0}^{rel}, \\omega _p)-\\sum _{n=1}^{\\infty }\\left[{\\cal G}({\\bf r}_{2}^{rel}, \\omega _p)+{\\cal G}({\\bf r}_{3}^{rel},\\omega _p)-{\\cal G}({\\bf r}_{4}^{rel}, \\omega _p)-{\\cal G}({\\bf r}_{5}^{rel}, \\omega _p)\\right]$ where the position of the observation point relative to the source point ${\\bf r}_{i}^{rel}$ for $i=2,3,4,5$ have the same ${\\bf x}_{\\parallel }^{rel}$ while their ${x}_{i}^{rel}$ coordinates are $&&{x}_{2}^{rel}=(2n-1)a-(x_1+x_1^{\\prime })\\hspace{56.9055pt}{x}_{4}^{rel}=2na-(x_1-x_1^{\\prime })\\nonumber \\\\&&{x}_{3}^{rel}=(2n-1)a+(x_1+x_1^{\\prime })\\hspace{56.9055pt} {x}_{5}^{rel}=2na+(x_1-x_1^{\\prime })$ The structure of Eq.",
"(REF ) is typical of a cavity geometry made up of two perfectly nonpenetrating walls.",
"The first term is the bulk part, while the other terms correspond to the communication between the two points ${\\bf r}$ and ${\\bf r}^{\\prime }$ via a series of infinite reflections in the cavity walls.",
"The coordinate space Green functions Eqs.",
"(REF ) and (REF ) enable us, at least in principle, to calculate the Casimir force acting on plate 2 by the help of fluctuation dissipation theorem.",
"The obvious and fundamental feature of the correlation function, provided by Eq.",
"(REF ), is that only its derivatives with respect to $x_1$ or $x_1^{\\prime }$ may possibly give a nonvanishing value on either side of plate 2.",
"This means that the force expression in 3+1 dimensions can be rewritten in the form of $F=\\mp \\frac{\\hbar }{2\\pi }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p\\left[ \\frac{\\partial }{\\partial x_1}\\frac{\\partial }{\\partial x_1^{\\prime }} {\\mathrm {Im}}G\\left({\\bf r}, {\\bf r}^{\\prime },\\omega _p\\right) \\right]_{{\\bf r}={\\bf r}^{\\prime }}$ on the either side of plate 2.",
"This is obtained by combining Eqs.",
"(REF ), (REF ) and (REF ).",
"In this way, we can calculate the net force per unit area from the analogue of Eq.",
"(REF ) for 3+1 dimensions $F=\\frac{\\hbar }{2\\pi }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p\\left\\lbrace \\left[ \\frac{\\partial }{\\partial x_1}\\frac{\\partial }{\\partial x_1^{\\prime }}{\\mathrm {Im}}G\\left({\\bf r}, {\\bf r }^{\\prime },\\omega _p\\right)\\right]_{\\begin{array}{c}{\\bf r}={\\bf r}^{\\prime }\\\\x_1=x_1^{\\prime }=a/2\\end{array}}-\\left[\\frac{\\partial }{\\partial x_1}\\frac{\\partial }{\\partial x_1^{\\prime }} {\\mathrm {Im}}G\\left({\\bf r}, {\\bf r}^{\\prime },\\omega _p\\right) \\right]_{\\begin{array}{c}{\\bf r}={\\bf r}^{\\prime }\\\\x_1=x_1^{\\prime }=a/2+d \\end{array}}\\right\\rbrace $ It should be obvious that Eq.",
"(REF ) and (REF ) must be substituted into the first and second term of Eq.",
"(REF ), respectively.",
"It is seen that the bulk part contributions cancel each other.",
"Furthermore, the first term in Eq.",
"(REF ) whose relative position vector is ${\\bf r}_{2}^{rel}$ with $n=1$ on the interior interface of plate 2 is identical with the reflection term in Eq.",
"(REF ) on the exterior interface of the plate.",
"These two terms also cancel each other.",
"Changing the index $n$ in the first term as $n\\rightarrow n-1$ , along with taking into account that $\\partial /\\partial x_1=\\partial /\\partial x_1^{\\prime }$ in the second and third terms of Eq.",
"(REF ), while $\\partial /\\partial x_1=-\\partial /\\partial x_1^{\\prime }$ in the forth and fifth terms, it is not difficult after some rearrangement to show that the net force takes the simple form $F=-\\frac{\\hbar }{2\\pi ^2}{\\mathrm {Im}}\\sum _{n=1}^{\\infty }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p\\left[\\frac{(ip/\\hbar )^2}{(2na)}-2\\frac{(ip/\\hbar )}{(2na)^2}+\\frac{2}{(2na)^3}\\right]\\exp [ip(2na)/\\hbar ]$ Taking the imaginary part of the summation along with changing the variable of integration from $\\omega _p$ to $p$ , one can easily show that $F=\\frac{\\hbar c}{32\\pi ^2 a^4 }\\sum _{n=1}^{\\infty }\\left(\\frac{1}{n}\\frac{\\partial ^2}{\\partial n^2}-\\frac{2}{n^2}\\frac{\\partial }{\\partial n}+\\frac{2}{n^3}\\right)\\frac{\\partial }{\\partial n}\\int _{0}^{\\infty }dp \\frac{\\cos (2npa/ \\hbar )}{\\sqrt{p^2+m^2 c^2}}$ The integration in Eq.",
"(REF ) can be done easily by using the integral representation of the modified Bessel function $K_0$ .",
"The result is the same as Eq.",
"(REF )."
],
[
"Casimir force in D+1 dimensions",
"Our consideration so far have applied to 1 and 3 space dimensions.",
"For some purposes it is, however, useful to generalize the preset calculations to the $D+1$ space-time dimensions.",
"To avoid any ambiguity it is easier to develop the abstract concept of $D$ dimensional formalism by comparing it with the 3 dimensional formulation.",
"The boundaries are now nonpenetrating hyperplates of thickness $d$ whose interior and exterior hyperplanes of interfaces are located at $x_1=\\pm a/2$ and $x_1=\\pm (a/2+d)$ , respectively.",
"We label the hyperplates as well as the different regions of this geometry, by analogy with Fig.",
"REF , by 1 and 2 as well as 1, 2 and 3, respectively.",
"It is understood that the field operator in the region 3 is now in the form of ${\\hat{\\varphi }}\\left( {\\bf r},t \\right)=\\left(\\frac{\\hbar c^2}{2}\\right)^{1/2}\\int d^{(D-1)} {\\bf p}_{\\parallel }\\int _{0}^{\\infty } \\frac{d p_1}{\\sqrt{\\omega }_p}{\\hat{a}}\\left({\\bf p }, t\\right)u \\left({\\bf p},{\\bf r} \\right)+\\mathrm {H.c.}$ where ${\\bf r}$ and ${\\bf p}$ are $D$ -vectors in $D$ dimensional coordinate and momentum space.",
"The mode function $u \\left({\\bf p},{\\bf r} \\right)$ is in the form of $u\\left({\\bf p, {\\bf r}}\\right)=(2 \\pi \\hbar )^{(1-D)/2}\\exp (i{\\bf p}_{\\parallel }\\cdot {\\bf x}_{\\parallel }/\\hbar )u(p_1, x_1)$ where ${\\bf x}_{\\parallel }=(x_2, x_3,...x_D)$ , ${\\bf p}_{\\parallel }=(p_2, p_3,...p_D)$ and $u(p_1, x_1)$ is given by Eq.",
"(REF ).",
"Likewise, we can write down the field operator in the region 2 with the help of Eqs.",
"(REF ) and (REF ) along with the obvious changes which are related the dimensional considerations.",
"Therefore ${\\hat{\\varphi }}\\left({\\bf r},t\\right)=\\left(\\frac{\\hbar c^2}{2}\\right)^{1/2}\\int d^{(D-1)}{\\bf p}_{\\parallel } \\sum _{\\Omega , n}\\frac{1}{\\sqrt{{\\omega }_n^{\\Omega } }}{\\hat{a}}_n^{\\Omega }\\left({\\bf p}_{\\parallel }, t\\right)u_n^{\\Omega } \\left({\\bf p}_{\\parallel },{\\bf r} \\right)+H.c.$ where $u_n^{\\Omega } \\left({\\bf p}_{\\parallel }, {\\bf r}\\right)=\\frac{1}{(2 \\pi \\hbar )^{(D-1)/2}}\\exp (i{\\bf p}_{\\parallel }\\cdot {\\bf x}_{\\parallel }/\\hbar )u_n ^{\\Omega }(x_1)$ Note that $u_n ^{\\Omega }(x_1)$ is given by Eqs.",
"(REF ) and (REF ).",
"The field operators show that the calculations are very similar to the 3 dimensional formalism.",
"The net force on the hyperplate 2 can, therefore, be evaluated in the usual manner and expressed as $F=\\frac{1}{(2 \\pi \\hbar )^{(D-1)}} \\int d^{(D-1)} {\\bf p}_{\\parallel } \\left\\lbrace \\frac{ \\pi ^2 \\hbar ^2 c}{2a^3}\\left[\\sum _{n=0}^{\\infty }f(n, p_{\\parallel })-\\int _{0}^{+\\infty }dn f(n,p_{\\parallel })\\right]\\right\\rbrace $ where $f(n)$ is given by Eq.",
"(REF ).",
"Comparing Eq.",
"(REF ) with Eq.",
"(REF ), it is seen that the latter expression is obtained by the well-known replacement $ \\frac{1}{(2 \\pi \\hbar )^{2}}\\int d^2 {\\bf p}_{\\parallel }\\rightarrow \\frac{1}{(2 \\pi \\hbar )^{(D-1)}}\\int d^{(D-1)} {\\bf p}_{\\parallel }$ We can now use the poisson's sum formula along with the integral representation of the modified Bessel function $K_0$ to write the net force in the form of $F=\\frac{1}{(2 \\pi \\hbar )^{(D-1)}} \\int d^{(D-1)} {\\bf p}_{\\parallel } \\left\\lbrace -\\frac{\\hbar c}{4 \\pi a^2}\\sum _{n=1}^{\\infty }\\frac{\\partial ^2}{\\partial n^2}K_0\\left(\\frac{2na}{\\hbar }\\sqrt{p_{\\parallel }^2+m^2c^2}\\right)\\right\\rbrace $ The integration on the hyperplane ${\\bf p}_{\\parallel }$ is all that need be done.",
"Before evaluating the integral, it is instructive to treat the problem by the alternative approach based on the fluctuation dissipation theorem and Kubo's formula.",
"By analogy with the 3 dimensional formalism, it is seen that the symmetry of the present configuration allows us to express the coordinate space Green function in terms of its Fourier transform as $ G({\\bf r}, {\\bf r}^{\\prime }, \\omega _p)=\\frac{1}{(2\\pi \\hbar )^{(D-1)}} \\int d^{(D-1)} {\\bf p}_{\\parallel } G({\\bf p}_{\\parallel }, x_1, x_1^{\\prime }, \\omega _p) \\exp [i{\\bf p}_{\\parallel }\\cdot ({\\bf x}_{\\parallel }-{\\bf x}_{\\parallel }^{\\prime })/ \\hbar ]$ where $G({\\bf p}_{\\parallel }, x_1, x_1^{\\prime }, \\omega _p)$ is given by Eqs.",
"(REF ) and (REF ) for the regions 3 and 2, respectively.",
"Note that this is in agreement with Eq.",
"(REF ).",
"In general, the calculation of the Fourier transform integral over the $D-1$ dimensional hyperplane ${\\bf p}_{\\parallel }$ will be much more difficult than the calculation of the same integral over the plane ${\\bf p}_{\\parallel }$ .",
"The reason for greater difficulty is apparently that, in general, the more number of dimensions involved the more difficulty to perform the integration.",
"Fortunately, it is the imaginary part of the space Green function at ${\\bf r}={\\bf r}^{\\prime }$ , not the explicit form of it, that concern us.",
"Therefore, recalling the discussion above Eq.",
"(REF ), it is seen that regardless of how complicated the space Green functions are in the region 3 and 2, the net force on the hyperplate 2 is obtained by substitution of Eq.",
"(REF ) into Eq.",
"(REF ).",
"We stressed that for the first term of Eq.",
"(REF ), the Fourier transformed Green function is given by Eq.",
"(REF ), while for the second term it is given by Eq.",
"(REF ).",
"By combining Eqs.",
"(REF ) and (REF ) and making use of ${\\bf x}_{\\parallel }={\\bf x}_{\\parallel }^{\\prime }$ on both of the interior and exterior hyperplanes of the hyperplate 2, we find easily that $&&F=\\frac{1}{(2 \\pi \\hbar )^{(D-1)}}\\int d^{(D-1)} {\\bf p}_{\\parallel }\\frac{\\hbar }{2\\pi }\\int _{mc^2/\\hbar }^{\\infty }d\\omega _p \\left\\lbrace \\left[\\frac{\\partial }{\\partial x_1}\\frac{\\partial }{\\partial x_1^{\\prime }} {\\mathrm {Im}}G\\left({\\bf r}, {\\bf r}^{\\prime },\\omega _p\\right) \\right]_{x_1=x_1^{\\prime }=a/2}\\right.\\nonumber \\\\&&\\hspace{216.81pt}\\left.-\\left[\\frac{\\partial }{\\partial x_1}\\frac{\\partial }{\\partial x_1^{\\prime }} {\\mathrm {Im}}G\\left({\\bf r}, {\\bf r}^{\\prime },\\omega _p\\right) \\right]_{x_1=x_1^{\\prime }=a/2+d}\\right\\rbrace .$ Inserting Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ), and taking into account that the bulk part contributions in the first and second terms of Eq.",
"(REF ) cancel each other.",
"Furthermore, on expanding the prefactor of the square brackets in Eq.",
"(REF ) and by comparing with the 1 and 3 dimensional formalism, it is seen that the reflection term in Eq.",
"(REF ) and the first term in Eq.",
"(REF ) with $n=0$ cancel each other as well.",
"The result can, therefore, be written down as $F=\\frac{1}{(2 \\pi \\hbar )^{(D-1)}}\\int d^{(D-1)} {\\bf p}_{\\parallel }\\left(\\frac{c}{ \\pi \\hbar }\\right)\\sum _{n=1}^{\\infty }\\int _{0}^{\\infty }dp_1 \\frac{p_1^2\\cos \\left(\\frac{2np_1 a}{\\hbar }\\right)}{\\sqrt{p_1^2+P_{\\parallel }^2 + m^2 c^2}}$ In writing Eq.",
"(REF ) we apply a suitable change of variable of integration, that is $p_1$ instead of ${\\omega }_p$ .",
"Employing the integral representation of the modified Bessel function $K_0$ , Eq.",
"(REF ) can be written in the form of Eq.",
"(REF ).",
"It is, therefore, seen that the result obtained in this way fully agrees with result that can be obtained using the method of field quantization.",
"The integration in Eq.",
"(REF ) over the hyperplane ${\\bf p}_{\\parallel }$ can be done easily in the usual manner.",
"The method rests on the definition of the solid angle in $D-1$ dimensional space of the hyperplane ${\\bf p}_{\\parallel }$ .",
"Since the integrand possesses spherical symmetry in the hyperplane, it depends only on $p_{\\parallel }$ , the integration over the solid angle in the momentum space ${\\bf p}_{\\parallel }$ is trivial.",
"We find that $F=-\\frac{ \\hbar c}{4 \\pi a^2}\\frac{1}{(2 \\pi \\hbar )^{(D-1)}}\\frac{(\\sqrt{\\pi })^{(D-1)}}{\\Gamma (\\frac{D-1}{2})}\\sum _{n=1}^{\\infty } \\frac{\\partial ^2}{\\partial n^2 }\\int dp_{\\parallel }{p_{\\parallel }}^{(D-2)}K_0\\left(\\frac{2na}{\\hbar }\\sqrt{p_{\\parallel }^2+m^2 c^2}\\right)$ where $\\Gamma $ is the Gamma function.",
"There remains only the integration on $p_{\\parallel }$ .",
"Despite the complicated dependence of the integrand on $p_{\\parallel }$ , the integral is well-known and yields $F=-\\frac{ \\hbar c}{4 \\pi a^2}\\left(\\frac{\\xi }{2a \\sqrt{2 \\pi } }\\right)^{(D-1)}\\sum _{n=1}^{\\infty }\\frac{\\partial ^2}{\\partial n^2 }\\left(\\frac{1}{n\\xi }\\right)^{(D-1)/2}K_{(D-1)/2} (n \\xi )$ Using the recurrence relations of the modified Bessel functions, we can easily show that $F=-2 \\hbar c\\left(\\frac{\\xi }{2a \\sqrt{2 \\pi }}\\right)^{(D+1)}\\sum _{n=1}^{\\infty } \\left(\\frac{1}{n\\xi }\\right)^{(D+1)/2}\\left[(n \\xi )K_{(D+3)/2} (n \\xi )-K_{(D+1)/2}(n \\xi )\\right]$ We can now derive the Casimir force for a neutral massless scalar field.",
"This is obtained by inserting the limiting forms of the modified Bessel functions in Eq.",
"(REF ).",
"The result can be expressed as $F=-\\frac{D \\Gamma [(D+1)/2]\\zeta (D+1) \\hbar c}{(2a\\sqrt{ \\pi })^{(D+1)}}$ where $\\zeta $ is the Zeta function.",
"It is evident that by setting $D=1$ and $D=3$ in Eq.",
"(REF ), one can easily obtain Eqs.",
"(REF ) and (REF ), respectively.",
"The latter equation is a well known result which has been reported in the literature [6]."
],
[
"Conclusion",
"We have presented two approaches to the calculation of the Casimir force for massive scalar field.",
"The more formal and illustrative one relies on the general tool of the field theory, i.e.",
"the explicit form of the quantum field operators, while the other way, perhaps concise and more simple, originates from the fluctuation dissipation theorem and Kubo's formula.",
"Apparently, the latter provides an alternative formulation of the effect that mutually equivalent.",
"To simplify the calculation, we consider the attractive force between two nonpenetrating plates.",
"The scope of the presented paper is extended to the derivation of the formal expression of the force in three different spatial dimensions.",
"We begin with 1+1 space-time dimensions whose force expression is given in Eq.",
"(REF ).",
"The details of the calculations are written in an appropriate form which can easily be extended to 3+1 space-time dimensions whose force expression is provided in Eq.",
"(REF ).",
"It is seen that the latter calculation is the straight forward generalization of the former.",
"This allows one to recast this formalism into $D+1$ space-time dimensions in a similar manner whose force expression is of the form Eq.",
"(REF ).",
"If we restrict our discussion to massless scalar field, we are thus led to cast the main result of the present work into a brief and concise statement that the force between two nonpenetrating plates is always in the form of $C_{D} \\left( \\hbar c/a^{D+1}\\right)$ , where $a$ is the separation between the two plates, $D$ denotes to the spatial dimensions of the problem and the constant factor $C_{D}$ is all that is needed to be obtained by the details of the calculations.",
"This can also be verified by using the dimensional considerations.",
"I would like to thank R. Matloob and H. Safari for very helpful comments and suggestions without which the present work would not have been refined."
]
] | 1403.0501 |
[
[
"Multi-Natural Inflation in Supergravity"
],
[
"Abstract We show that the recently proposed multi-natural inflation can be realized within the framework of 4D ${\\cal N}=1$ supergravity.",
"The inflaton potential mainly consists of two sinusoidal potentials that are comparable in size, but have different periodicity with a possible non-zero relative phase.",
"For a sub-Planckian decay constant, the multi-natural inflation model is reduced to axion hilltop inflation.",
"We show that, taking into account the effect of the relative phase, the spectral index can be increased to give a better fit to the Planck results, with respect to the hilltop quartic inflation.",
"We also consider a possible UV completion based on a string-inspired model.",
"Interestingly, the Hubble parameter during inflation is necessarily smaller than the gravitino mass, avoiding possible moduli destabilization.",
"Reheating processes as well as non-thermal leptogenesis are also discussed."
],
[
"Introduction",
"The recent observations of the cosmic microwave background (CMB) by the Planck satellite [1] showed that $\\Lambda $ CDM cosmology is consistent with the data and fluctuations in the cosmic microwave background (CMB) can be explained by single-field inflation [2], [3], which solves the fine-tuning problems in the early universe.",
"The spectral index $n_s$ and the tensor-to-scalar ratio $r$ are tightly constrained by the Planck data combined with other CMB observations [1]: $n_s &=& 0.9603 \\pm 0.0073, \\\\r &<& 0.11 ~(95\\%~{\\rm CL}).$ The index $n_s$ is determined by the shape of the inflaton potential, whereas the ratio $r$ is done by the energy scale of the potential:After submission of this paper, the BICEP2 collaboration announced the detection of the primordial B-mode polarization, which can be explained by $r = 0.20^{+0.07}_{-0.05}$ [4].",
"See note added at the end of this paper.",
"$H_{\\rm inf} \\simeq 8.5 \\times 10^{13}~{\\rm GeV}\\left( \\frac{r}{0.11} \\right)^{1/2}.$ Here, $H_{\\rm inf}$ is the Hubble scale during inflation.",
"To construct a viable inflation model, the inflaton potential should be under good control so as not to break the slow-roll condition and to suppress the inflation scale compared with the the Planck scale.",
"There have been many attempts to accomplish this.",
"One way is to introduce a certain symmetry which keeps the inflaton potential flat.",
"See Refs.",
"[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] for various chaotic inflation models along this line.",
"In this sense, an axion is a good candidate for the inflaton due to the approximate shift symmetry $\\phi \\rightarrow \\phi + {\\rm const}.",
",$ which controls its potential structure and suppresses the scale of inflation to be consistent with observation.",
"Here, $\\phi $ is an axion.",
"So, it is possible to consider a natural inflation model [20], [21] with an axion potential $V(\\phi )$ given by: $V(\\phi ) = \\Lambda ^4 \\bigg [1 - \\cos \\left(\\frac{\\phi }{f}\\right) \\bigg ] .$ In this model, the shift symmetry is broken non-perturbatively by the dynamical scale $\\Lambda $ much smaller than the Planck scale.",
"However, the decay constant $f$ is required to be larger than the Planck scaleSee [22] for realizing a large decay constant effectively, [23], [24] for other ways to relax the bound on the decay constant, and [25], [27], [26], [13], [14] for other models with axion(s)., $f \\gtrsim 5M_{\\rm Pl}$ [1], for the predicted $n_s$ and $r$ to be consistent with the observed values, where $M_{\\rm Pl} \\simeq 2.4 \\times 10^{18}$ GeV is the reduced Planck mass.Hereafter, we take the Planck unit of $M_{\\rm Pl}=1$ for a simplicity, unless otherwise stated.",
"So, one might worry about the control of the correction $f/M_{\\rm Pl}$ after all.",
"Recently, two of the present authors (MC and FT) proposed an extension of natural inflation, called multi-natural inflation [28], in which the inflaton potential mainly consists of two (or more) sinusoidal functions.",
"Interestingly, multi-natural inflation is versatile enough to realize both large-field and small-field inflation.",
"In the case of large-field inflation with super-Planckian decay constants, the predicted values of the spectral index as well as the tensor-to-scalar ratio can be closer to the center values of the Planck results, with respect to the original natural inflation.",
"In the case of small-field inflation with sub-Planckian decay constants, we arrange those sinusoidal functions so that they conspire to make the inflaton potential sufficiently flat for slow-roll inflation.",
"In a certain limit, this axion hilltop inflation is equivalent to hilltop quartic inflation [3].",
"The hilltop quartic inflation has been studied extensively so far, and it is known that, for the e-folding number $N_e \\simeq 50$ , its predicted spectral index tends to be too low to explain the Planck results [REF ].",
"There are various proposals for resolving this tension in the literature.",
"The purpose of this paper is twofold.",
"First, we will show that the predicted spectral index for the axion hilltop inflation can be increased with respect to the hilltop quartic inflation case by including a relative phase between two sinusoidal functions.",
"This gives a better fit to the Planck data.",
"Second, we consider a UV completion of multi-natural inflation within supergravity(SUGRA)/string theory.",
"This is because a viable inflation model can be easily realized for large decay constants close to the GUT or Planck scale, and because the string theory offers many axions through compactifications [30], [29], [31], some of which could play an important role in inflation.",
"Also, non-perturbative dynamics which explicitly break the axionic shift symmetry can be studied rigorously in a supersymmetric (SUSY) framework.",
"The rest of this paper is organized as follows.",
"In Sec.",
"we build an axion hilltop inflation model with one axion multiplet in the context of SUGRA, taking into account SUSY breaking effects.",
"In Sec., we consider a UV completion of multi-natural inflation in the string-inspired model, which is reduced to the model analyzed in Sec.",
"in the low energy effective theory.",
"The last section is devoted to discussion and conclusions.",
"Let us consider a supergravity realization of multi-natural inflation in which an axion plays the role of the inflaton [28].",
"To this end, we introduce the following Kähler and super-potentials with an axion chiral superfield $\\Phi $ , $K &=& K(\\Phi + \\Phi ^{\\dag }), \\\\W &=& W_0 + Ae^{-a\\Phi } + Be^{-b\\Phi },$ where $a>0$ , $b>0$ and $a \\ne b$ .The case of the irrational ratio of $a/b$ was considered in Ref. [32].",
"The following discussion holds even in this case.",
"In the following we assume $|a^4A| < |b^4B|$ without loss of generality.",
"The scalar potential is given by $V= e^{K}[K^{i\\bar{j}}(D_iW) (\\overline{D_jW})-3|W|^2],$ with $D_i W = (\\partial _i K) W + \\partial _i W$ .",
"For convenience we write the scalar component of $\\Phi $ as $\\Phi = \\sigma + i \\varphi ,$ where $\\sigma $ and $\\varphi $ are the saxion and the axion, respectively.",
"Note that the Kähler potential respects the axionic shift symmetry, $\\varphi \\rightarrow \\varphi + {\\rm const},$ which is explicitly broken by the two exponentials in the superpotential.",
"We assume that the breaking of the shift symmetry is so weak that the axion mass is hierarchically smaller than the saxion mass.",
"As we shall see shortly, this is the case if $|A|, |B| \\ll |W_0| < 1.$ The saxion is then decoupled from the inflaton dynamics.",
"The softly broken shift symmetry is one of the essential ingredients for multi-natural inflation, and such a small breaking naturally arises from non-perturbative effects at low-energy scalesSuch softly broken shift symmetries and the associated light axions may play an important cosmological role in a different context.",
"For instance, the axion with mass 7 keV [33] can explain the recently observed X-ray line at about $3.5$ keV [34], [35].. On the other hand, if the saxion mass were comparable to the axion, one would have to follow the multi-field dynamics during inflation.",
"Although the analysis becomes rather involved, it is possible to realize successful inflation as is the case with racetrack inflation [27]."
],
[
"Saxion stabilization",
"First we study the saxion stabilization in the above setup.",
"For simplicity and concreteness we consider the following Kähler potential to stabilize the saxion: $K = \\frac{f^2}{2}(\\Phi +\\Phi ^\\dag )^2$ with $f \\lesssim 1$ .",
"We will see in the next section that the Kähler potential of this form is indeed obtained in the low energy effective theory of a more realistic string-inspired model.",
"The kinetic term for the saxion and the axion is given by ${\\cal L}_{\\rm kin} \\;=\\; K_{\\Phi {\\bar{\\Phi }}} \\partial \\Phi ^\\dag \\partial \\Phi = f^2(\\partial \\sigma )^2 + f^2(\\partial \\varphi )^2.$ For the moment let us focus on the saxion stabilization by setting $A=B=0$ .",
"Then the saxion potential has a $Z_2$ symmetry, $\\sigma \\rightarrow -\\sigma $ , and therefore the saxion potential has an extremum at the origin $\\sigma =0$ .",
"In fact, the origin can be the potential minimum as shown below.",
"The saxion potential is approximately given by $V &= & e^{2f^2 \\sigma ^2}\\bigg (4 f^2 \\sigma ^2 - 3 \\bigg ) |W_0|^2 + \\Delta V \\\\& \\simeq & 2 f^2 |W_0|^2 \\sigma ^2 + \\cdots ,$ where we have expanded the potential around the origin in the second equality, and we have added a sequestered uplifting potential $\\Delta V$ to cancel the cosmological constantFor instance, one can consider $K= -3\\log \\left[e^{-K(\\Phi + \\Phi ^{\\dag })/3}-\\lbrace XX^{\\dag } - (X X^\\dag )^2/\\Lambda ^2 \\rbrace /3 \\right]$ and $W= W_0 + \\sqrt{3}W_0 X + W(\\Phi )$ to break the SUSY and to obtain a small cosmological constant.",
"In this model the vacuum and the mass are given by $\\langle X \\rangle \\sim \\Lambda ^2 \\ll 1$ and $m_{3/2}/\\Lambda \\gg m_{3/2}$ [36].",
"Then, $\\Delta V = e^{K}|D_X W|^2K^{X \\bar{X}} = 3|W_0|^2 e^{2K(\\Phi + \\Phi ^{\\dag })/3} $ .",
"The SUSY breaking fields can be integrated out during inflation as long as its mass is heavier than or comparable to the gravitino mass.",
"See [37], [38], [39], [40] for related topics., $\\Delta V &=& 3 e^{2K/3} |W_0|^2\\simeq \\bigg (3 + 4f^2 \\sigma ^2 + \\cdots \\bigg )|W_0|^2.$ Thus, the saxion is stabilized at the origin with mass $m_\\sigma \\simeq \\sqrt{2} |W_0|$ .",
"Note that the saxion is stabilized by the SUSY-breaking effect through the equation $\\partial _{\\Phi }K = 0.$ The saxion can be similarly stabilized for a more general Kähler potential; see Refs.",
"[41], [42], [43] for detailed discussions on the saxion stabilization.",
"In general, the saxion mass is considered to be of order the gravitino mass.",
"The axion mass is protected by a shift symmetry.",
"For a sufficiently small breaking of the shift symmetry, therefore, the axion acquires a mass much smaller than the saxion mass, while the saxion stabilization studied above remains almost intact.",
"See Fig.REF for the saxion potential in the presence of small explicit breaking of the shift symmetry, with and without the uplifting potential.",
"We can see that the saxion is stabilized near the origin, when the sequestered up-lifting potential is added.",
"The saxion vacuum is located near the origin as long as the parameters satisfy the relation $\\frac{|A|}{|W_0|f^2} \\sim \\frac{|B|}{|W_0|f^2} \\lesssim 10^{-2},$ i.e., the axion mass is much lighter than that of the saxion by a factor of ten.",
"We shall see that the Hubble parameter during inflation is necessarily smaller than the gravitino mass as long as (REF ) is met.",
"Then the saxion stabilization is hardly affected by inflation, and we can integrate out the saxion during inflation.",
"This makes the inflation dynamics extremely simple: the inflationary epoch is described by single-field inflation driven by the axion.",
"Figure: The saxion potential for A=2.3×10 -12 ,B=A/4,a=2π/10,b=2π/5,f=0.1A=2.3\\times 10^{-12},~B=A/4,~a=2\\pi /10,~b=2\\pi /5,~f=0.1 and W 0 =10 -4 W_0=10^{-4}.We have set ϕ=0\\varphi = 0.",
"The dashed (blue) line shows the saxion potential without the uplifting potential;the saxion is stabilized at σ∼±5\\sigma \\sim \\pm 5, where the vacuum energy 3|W 0 | 2 3|W_0|^2 is added for visualization purpose.The saxion can be stabilized near the origin if the sequestered uplifting potential ΔV\\Delta V is added, as shown bythe solid (red) line."
],
[
"Axion hilltop inflation",
"Let us study the axion potential.",
"Using $U(1)_R$ symmetry and an appropriate shift of $\\varphi $ , we can set $W_0$ and $A$ real and positive, while $B$ is complex in general.",
"To take account of this complex phase, we replace $B$ with $B e^{- i \\theta }$ , where $B$ is a real and positive constant, and $\\theta $ represents the relative phase between the two exponentials.",
"Using $\\left\\langle \\sigma \\right\\rangle \\simeq 0$ , the axion potential can be approximately written as $\\nonumber V_{\\rm axion}(\\phi ) &\\simeq &6 A W_0 \\left[ 1 - \\cos \\left(\\frac{\\phi }{f_1}\\right)\\right] + 6 B W_0 \\left[1- \\cos \\left(\\frac{\\phi }{f_2} + \\theta \\right)\\right] \\\\&& - 2 A B \\left( \\frac{2}{f_1 f_2} -3\\right) \\left[1- \\cos \\bigg [ \\left( \\frac{1}{f_1} -\\frac{1}{f_2} \\right)\\phi -\\theta \\bigg ]\\right] + {\\rm const},$ where $\\phi \\equiv \\sqrt{2} f\\varphi $ is the canonically normalized axion field, and we have defined $f_1 \\equiv \\frac{\\sqrt{2}f}{a},~~~f_2 \\equiv \\frac{\\sqrt{2}f}{b},$ with $f_1 \\ne f_2$ .",
"We have added the vacuum energy from SUSY breaking to obtain the Minkowski spacetime in the true vacuum and the last constant term in Eq.",
"(REF ) depends on $\\theta $ ; it vanishes for $\\theta =0$ .",
"We impose relation (REF ) to realize a hierarchy between the saxion mass and the axion mass.",
"Then, the third term in Eq.",
"(REF ) becomes irrelevantIn the presence of two axion fields, the third term can be responsible for natural inflation with an effective super-Planckian decay constant [22].",
"and the first two terms are equivalent to the inflaton potential for the multi-natural inflation discussed in Ref. [28].",
"A successful multi-natural inflation requires the two sinusoidal functions to have comparable magnitude and periodicity, i.e., $&&A \\sim B \\ll W_0,\\\\&&f_1 \\sim f_2.$ The inflation scale is roughly given by $H_{\\rm inf} \\sim (A W_0)^{1/2} \\ll W_0$ , and so, the saxion mass is generically heavier than the Hubble parameter during inflation, which justifies our assumption.",
"Also, the upper bound on the tensor mode (REF ) places the following condition, $AW_0 \\lesssim 10^{-10},$ for $f_1 \\sim f_2 \\lesssim {\\cal O}(1)$ .",
"The inflaton potential must have a flat plateau in order to realize successful inflation with sub-Planckian decay constants.",
"In the following we show that, in a certain limit, the axion potential is reduced to the hilltop quartic inflation model.",
"For the moment we focus on the first two terms in Eq.",
"(REF ).",
"In the numerical calculations we will include all the terms.",
"Requiring that the first, second and third derivatives of $V_{\\rm axion}$ vanish and the fourth derivative of $V_{\\rm axion}$ is negative at $\\phi = \\phi _{\\rm max}$ , we obtain the following conditions among the parameters: $\\sin \\left(\\frac{\\phi _{\\rm max}}{f_1}\\right) &= \\sin \\left(\\frac{\\phi _{\\rm max}}{f_2} + \\theta \\right) = 0,\\\\-\\cos \\left(\\frac{\\phi _{\\rm max}}{f_1}\\right) &= \\cos \\left(\\frac{\\phi _{\\rm max}}{f_2} + \\theta \\right) = 1,\\\\\\frac{A}{f_1^2} &= \\frac{B}{f_2^2},$ where we have used our assumption $a^4 A < b^4 B$ , i.e., $A/f_1^4 < B/f_2^4$ , to fix the sign of the cosine functions.",
"Note that $A > B$ as well as $f_1 > f_2$ must be satisfied to meet the above conditions.",
"We choose the following solutions without loss of generality, $\\phi _{\\rm max} &=& \\pi f_1,\\\\\\theta &=& - \\pi \\frac{f_1}{f_2}~~~~({\\rm mod}~2\\pi )$ The inflaton potential becomes simple for a particular choice of $f_1 = 2f_2$ (i.e.",
"$A=4B$ ), as the relative phase $\\theta $ vanishes.",
"Expanding the inflaton potential around $\\phi _{\\rm max}$ , we obtain $V_{\\rm axion}(\\hat{\\phi }) \\;\\simeq \\; V_0 - \\lambda \\hat{\\phi }^4 + \\cdots $ with $V_0 &=& {\\cal O}(A W_0),\\\\\\lambda &=& \\frac{W_0}{4} \\left(\\frac{B}{f_2^4} - \\frac{A}{f_1^4} \\right),$ where we have defined $\\hat{\\phi } \\equiv \\phi - \\pi f_1$ , the constant term $V_0$ is fixed so that the potential vanishes at the minimum, and the dots represent the higher order terms.",
"Therefore, for the parameters satisfying (), (REF ), and (), the axion potential is equivalent to the hilltop quartic inflation.",
"In Fig.",
"REF we show the scalar potential for the saxion and the axion and its section along $\\left\\langle \\sigma \\right\\rangle = 0$ .",
"We can see that the saxion is stabilized during inflation and that the axion potential is given by a flat-top potential.",
"Figure: The scalar potential for the saxion and the axion (left) and the axion potential at the section of σ=0\\left\\langle \\sigma \\right\\rangle = 0 (right).In the left panel, we show the logarithm of the scalar potential for the visualization purpose.We use the same model parameters as in Fig.",
".For comparison, the case with B=0B=0 is also shown by the dashed (blue) line in the right panel.For sub-Planckian decay constants, the quartic coupling $\\lambda $ is fixed by the Planck normalization on the curvature perturbation as $\\lambda _{\\rm Planck} \\simeq 6.5 \\times 10^{-14} \\left(\\frac{N_e}{50} \\right)^{-3},$ where $N_e$ is the e-folding number.",
"For instance, the Planck normalization is satisfied if $A W_0& \\simeq & 8.7 \\times 10^{-14}\\, f_1^4 \\left(\\frac{N_e}{50} \\right)^{-3},$ for $f_1 = 2 f_2$ (i.e.",
"$\\theta = 0$ and $A=4B$ ).",
"For this choice of the parameters, the axion mass at the potential minimum is given by $m_{\\phi } &\\simeq & \\frac{2\\sqrt{3 A W_0}}{f_1} \\simeq 2.5 \\times 10^{11}{\\rm \\,GeV} \\left(\\frac{N_e}{50} \\right)^{-\\frac{3}{2}} \\left(\\frac{f_1}{0.1}\\right).$ Assuming the axion coupling with the standard model (SM) gauge bosons ${\\cal L} \\supset c (\\phi /f_1) F_{\\mu \\nu }\\tilde{F}^{\\mu \\nu }$ , the decay rate is given by $\\Gamma (\\phi \\rightarrow A_\\mu A_\\mu ) \\;=\\; N_g \\frac{c^2 m_\\phi ^3 }{4 \\pi f_1^2},$ where $N_g = 8+3+1$ counts the number of SM gauge bosons.",
"The reheating temperature after the inflation is then estimated as $T_R & \\equiv & \\left(\\frac{\\pi ^2 g_*}{90} \\right)^{-\\frac{1}{4}} \\sqrt{\\Gamma } \\simeq 4 \\times 10^{8} \\, c \\left(\\frac{N_e}{50} \\right)^{-\\frac{9}{4}} \\left(\\frac{f_1}{0.1} \\right)^{\\frac{1}{2}} {\\rm GeV},$ where $g_*$ counts the relativistic degrees of freedom in plasma and we have substituted $g_* = 106.75$ in the second equality.",
"So far we have adopted the special case of $f_1 = 2f_2$ .",
"The typical scales of the inflaton mass and the reheating temperature are similar for other choices.",
"Here let us take another case.",
"If the two decay constants are very close to each other, i.e., $(f_1 - f_2)/f_2 \\ll 1$ , we can approximate the inflaton potential by keeping the leading order term in $(f_1-f_2)/f_2 $ : $V_{\\rm axion}(\\phi ) &\\simeq &6AW_0 \\left(\\frac{f_1-f_2}{f_2}\\right) \\left(v_0- 2 \\cos \\frac{\\phi }{f_1} + \\left(\\pi - \\frac{\\phi }{f_1} \\right) \\sin \\frac{\\phi }{f_1} \\right)+\\cdots ,$ where the dots represent higher order terms of ${\\cal O}((f_1-f_2)^2/f_2^2)$ , $v_0 \\approx 4.8206$ , and the potential maximum and minimum are located at $\\phi _{\\rm max}/f_1 = \\pi $ and $\\phi _{\\rm min}/f_1 \\approx -1.3518$ , respectively.",
"This approximation is valid only for $|\\phi /f_1| \\ll \\frac{f_2}{f_1-f_2}$ .",
"The potential height $V_0$ , the quartic coupling $\\lambda $ , and the inflaton mass at the minimum are approximately given by $V_0 &\\approx & 40.9 A W_0 \\left(\\frac{f_1-f_2}{f_2}\\right),\\\\\\lambda &\\approx & \\frac{AW_0}{2 f_1^4 } \\left(\\frac{f_1-f_2}{f_2}\\right), \\\\m_\\phi &\\approx & 5.13 \\frac{\\sqrt{A W_0}}{f_1} \\sqrt{\\frac{f_1-f_2}{f_2}} \\simeq 7.3 \\sqrt{\\lambda } f_1.$ We can see that the inflaton mass is of similar order to Eq.",
"(REF ).",
"As the Planck normalization fixes $\\lambda $ , $AW_0$ scales as $f_2/(f_1-f_2)$ , while the inflaton potential shape itself is not significantly changed even when $f_1 \\approx f_2$ .",
"Indeed, using the Planck-normalized quartic coupling, we can rewrite the inflaton potential as $V_{\\rm axion}(\\phi ) &\\simeq & 12 \\lambda _{\\rm {Planck}} \\,f_1^4\\left(v_0- 2 \\cos \\frac{\\phi }{f_1} + \\left(\\pi - \\frac{\\phi }{f_1} \\right) \\sin \\frac{\\phi }{f_1} \\right),$ in the limit of $f_1 \\approx f_2$ .",
"The inflaton potential is shown in Fig.",
"REF .",
"Note that there are many other potential minima and maxima; the inflation takes place near the hilltop around $\\phi /f_1 \\lesssim \\pi $ .",
"Figure: The scalar potential for the saxion and the axion (left) and the axion potentialat the section of σ=0\\left\\langle \\sigma \\right\\rangle = 0 (right) similarly to Fig..We use A=4.0×10 -11 ,B=(6/7) 2 A,a=π/7,b=π/6,f=0.1,W 0 =10 -4 A=4.0\\times 10^{-11},~ B=(6/7)^2 A,~a=\\pi /7,~b=\\pi /6,~f=0.1,~W_0=10^{-4} and θ=-7π/6\\theta =-7\\pi /6.The case with B=0B=0 is also shown by the dashed (blue) line in the right panel,where the minima are chosen tocoincide for visualization purposes.The spectral index for hilltop quartic inflation is predicted to be $n_s \\simeq 1 - \\frac{3}{N_e} = 0.94 - 0.95 ,$ for $N_e = 50 - 60$ .",
"As is well known, the predicted spectral index tends to be too low to fit the Planck result (REF ).",
"In the context of new inflation in supergravity [44], [45], the resolution of the tension was discussed in detail in the literature, and it is known that the prediction of $n_s$ can be increased to be consistent with the Planck data either by adding a logarithmic correction [46], [47] or a linear term [48], or by considering higher powers of the inflaton coupling [49].",
"As we shall see below, in the axion hilltop inflation, we can easily increase the spectral index by varying the relative phase $\\theta $ around ().",
"Let us study the axion potential by varying the parameters around the solutions (), (REF ), and ().",
"Expanding the potential in terms of ${\\hat{\\phi }} = \\phi - \\pi f_1$ , we obtain $V_{\\rm axion}({\\hat{\\phi }})&=& V_0 + \\frac{6BW_0 \\sin \\Theta }{f_2} \\,{\\hat{\\phi }} - 3 W_0 \\left( \\frac{A}{f_1^2} - \\frac{B}{f_2^2} \\cos \\Theta \\right) {\\hat{\\phi }}^2-\\frac{BW_0 \\sin \\Theta }{f_2^3} \\, {\\hat{\\phi }}^{3}\\nonumber \\\\&& - \\frac{1}{4}W_0 \\left( \\frac{B}{f_2^4} \\cos \\Theta - \\frac{A}{f_1^4} \\right) {\\hat{\\phi }}^4 + \\cdots ,$ where we have defined $\\Theta \\equiv \\theta + \\pi f_1/f_2$ .",
"It is the linear term in $\\hat{\\phi }$ that affects the inflaton dynamics significantly.",
"As pointed out in Ref.",
"[48], if there is a small linear term in the hilltop quartic inflation model, the inflaton field value at the horizon exit of cosmological scales can be closer to the hilltop, making the curvature of the potential smaller and therefore increasing the spectral index.",
"We have numerically solved the inflaton dynamics based on the potential given by Eq.",
"(REF ) to evaluate the predicted values of $n_s$ and $r$ .",
"To be concrete, we have varied the model parameters $B/A$ and $\\theta $ around the solutions (), (REF ), and () with $f_1 = 0.5$ and $f_2 = 0.45$ .",
"The results are shown in Fig.",
"REF .",
"Note that the Planck normalization can be satisfied by varying $W_0$ for fixed $A/W_0$ and $B/W_0$ without affecting the predicted values of $n_s$ and $r$ .",
"From Fig.",
"REF we can see that the spectral index can be increased to fit the 2$\\sigma $ limit of the Planck data shown by the shaded (green) region.",
"We have also confirmed that the spectral index can be similarly increased to give a good fit to the Planck data for different values of $f_1$ and $f_2$ , e.g.",
"$f_1 = 2 f_2$ .",
"In general, for smaller values of the decay constants, the deviation from the solution () must be smaller.",
"On the other hand, as expected, the tensor-to-scalar ratio $r$ is well below the upper bound from the Planck data ($r < 0.11$ ).",
"For a larger value of the decay constant, e.g.",
"$f = 1$ ($f_1 \\approx 2.25$ ) $r$ can be as large as $\\sim 10^{-3}$ in the allowed region of $n_s$ .",
"Fig.",
"REF also shows the behavior of $n_s$ and $r$ as a function of $f_1$ with the same parameters as in Fig.",
"REF .",
"The behavior is similar to a hilltop quartic model as discussed in Ref.",
"[28] for $\\theta = 0$ , but the spectral index can be increased by allowing a non-zero relative phase.",
"Figure: Plots of n s n_s (left) and rr (right) for varying values of BB and θ\\theta for fixed decayconstants f 1 =0.5f_1 = 0.5 and f 2 =0.45f_2 = 0.45, which corresponds to the case of f 1 ≈f 2 f_1 \\approx f_2 studied in the text.The green shaded region corresponds to the 2σ\\sigma allowed regionfor n s n_s from the Planck data.Figure: Plots of n s n_s (left) and rr (right) as a function of f/M p f/M_p.",
"In the left figure, Θ≡θ+πf 1 /f 2 \\Theta \\equiv \\theta + \\pi f_1/f_2.",
"In the right figure, there was no significant difference in the behavior of rr for the two values of Θ\\Theta , hence we chose Θ=-4.1×10 -5 \\Theta = -4.1\\times 10^{-5}.",
"Solid (dotted) lines correspond to N e =60N_e=60 (N e =50N_e=50)."
],
[
"Set-up",
"We now provide a further UV completion of the effective SUGRA model given in the previous section, based on the string-inspired model.",
"Let us consider a model with three Kähler moduli on a Calabi-Yau space with the following Kähler and super-potentials A similar UV completion may be possible in a LARGE volume scenario [50] with string-loop corrections and non-perturbative superpotentials, if a (moderately) big cycle allows a gauge coupling which generates the axion mass through non-perturbative effects.",
"A large mass hierarchy between the saxion and the axion can then be realized: the saxion mass is suppressed by the power of the Calabi-Yau volume while the axion mass is exponentially suppressed by the volume [51].",
"$K &=& -2\\log (t_0^{3/2} - t_1^{3/2}-t_2^{3/2});~~~t_i = (T_i +T_i^{\\dag }) ~~~{\\rm for}~i=0,1,2, \\\\W &=& W_0 - C e^{-\\frac{2\\pi }{N}T_0} - D e^{-\\frac{2\\pi }{M}(T_1+T_2)}+ A e^{-\\frac{2\\pi }{n_1}T_2}+ B e^{-\\frac{2\\pi }{n_2}T_2},$ where $T_i$ are complex Kähler moduli, and $W_0,~A,~B,~C$ and $D$ are determined by the vacuum expectation values (VEVs) of heavy dilaton/complex structure stabilized via three-form flux compactification [52], [29].",
"(See [53] for realization of a small $W_0$ .)",
"The exponential terms in the superpotential are assumed to be generated by gaugino condensations in a pure $SU(N)\\times SU(M)\\times SU(n_1)\\times SU(n_2)$ gauge theory.",
"Those gauge fields are living on the D-branes wrapping on the divisors whose volume is determined by the real part of the moduli, $T_0$ , $T_1+T_2$ , $T_2$ and $T_2$ respectively.",
"In other words, (at least some part of) the gauge coupling of each gauge group is given by the corresponding moduli.",
"We define $T \\equiv T_1 + T_2 $ and $\\Phi \\equiv -T_1 +T_2$ , and express the lowest component of $\\Phi $ as $\\Phi = \\sigma + i \\phi $ for later use.",
"Using the U(1)$_R$ symmetry and an appropriate shift of the imaginary components of the moduli fields, we can take $W_0$ , $A$ , $C$ , and $D$ real and positive without loss of generality.",
"We will include a relative phase in $B$ by replacing it with $B e^{-i \\theta }$ where $B$ is a real and positive constant.",
"We assume that those parameters satisfy $&& A, B, C, D = {\\cal O}(1),~~~W_0 \\ll 1.$ We also assume that $N,~M,~n_1$ and $n_2$ are integers satisfying We shall see that the axion hilltop inflation with $f_1 \\approx f_2$ is realized for this choice of the parameters.",
"The other cases such as $f_1 = 2 f_2$ can also be realized if there is a hierarchy between $A$ and $B$ .",
"See Appendix .",
"$&& n_1 \\sim n_2~~{\\rm and}~~n_1\\ne n_2\\\\&& 2n_1 < M \\lesssim N.$ The mild hierarchy between $(M,N)$ and $(n_1,n_2)$ implies that $T_0$ and $T=T_1+T_2$ are stabilized in a supersymmetric manner by the first two exponentials.",
"On the other hand $\\Phi = -T_1 + T_2$ remains relatively light, and this combination becomes the axion supermultiplet in the previous section.",
"We shall see that, while $\\sigma = {\\rm Re}[\\Phi ]$ can be stabilized by the SUSY breaking effect through the Kähler potential, the axion, $\\phi = {\\rm Im}[\\Phi ]$ , acquires an even lighter mass by the last two exponentials.",
"In order to have successful inflation with sub-Planckian decay constants, the resultant two exponentials expressed in terms of $\\Phi $ must be comparable in size.",
"This is possible for $A \\sim B$ , if $n_1$ is close to $n_2$ within $10\\%$ or so.Precisely speaking, this is the case if $|n_2 - n_1| \\lesssim n_1 n_2/\\pi \\left\\langle T \\right\\rangle $ .",
"If $n_1$ is not close to $n_2$ , some hierarchy between $A$ and $B$ is necessary."
],
[
"Heavy moduli stabilization",
"We first study the stabilization of the heavy moduli, $T_0$ , $T$ and $\\sigma $ .",
"For $A=B=0$ , the model is reduced to the string-theoretic QCD axion model considered in Ref.",
"[42]; there exists a Minkowski vacuum where, while the other moduli are stabilized, ${\\rm Im}[\\Phi ]$ remains (almost) massless and eventually becomes the QCD axion.",
"Because of the assumed mild hierarchy between $(M,N)$ and $(n_1, n_2)$ , our model is similar to this model as long as the heavy moduli stabilization is concerned.",
"The scalar potential of the moduli and the sequestered SUSY-breaking up-lifting potential $V_{\\rm up}$ are given by $V = V_{\\rm moduli} + V_{\\rm up},~~~{\\rm where}~~V_{\\rm up} = \\hat{\\epsilon }\\, e^{2K/3};~~~\\hat{\\epsilon } ={\\cal O}(W_0^2),$ where $V_{\\rm moduli}$ is given by Eq.",
"(REF ) with the above Kähler and super-potentials, and $\\hat{\\epsilon }$ is fixed so that a (nearly) Minkowski vacuum is realized in the low energy.",
"The moduli stabilization is determined by the conditions for extremizing the potential $V$ , $D_{T_0}W \\simeq D_{T}W \\simeq \\partial _{\\Phi }K \\simeq 0$ : $\\frac{2\\pi }{N}T_0 &\\simeq \\frac{2\\pi }{M} T \\simeq \\log \\bigg [\\log (1/W_0)/W_0\\bigg ] \\gg 1,~~{\\rm Re}[\\Phi ]= 0.$ The VEVs of these moduli fields fix the volume of the Calabi-Yau space ${\\cal V}$ as well as the gravitino mass as ${\\cal V} & = & t_0^{3/2} - \\frac{t^{3/2}}{\\sqrt{2}},\\\\m_{3/2} &=& W_0/{\\cal V},$ where $t = T+T^{\\dag }$ .",
"The moduli masses are given by $m_{T_0}& \\simeq m_{T} \\simeq \\log (M_{\\rm Pl}/m_{3/2}) m_{3/2},\\\\m_{\\sigma } &\\simeq \\sqrt{2} m_{3/2},$ where we used a fact that the up-lifting potential after integrating out $T_0$ and $T$ is approximately given by $V_{{\\rm up}L} &\\approx & 3m_{3/2}^2 + f^2 m_{3/2}^2 (\\Phi +\\Phi ^{\\dag })^2 + \\cdots .$ See Appendix for the higher order terms in $V_{{\\rm up}L}$ .",
"Note that the axion $\\phi $ remains massless in this case, which should be contrasted to the original KKLT [37].",
"The SUSY-breaking $F$ -terms of the moduli fields are given by $\\frac{F^{T_{i}}}{T_i+T_i} \\sim \\frac{m_{3/2}}{\\log (M_{\\rm Pl}/m_{3/2})} \\sim m_{\\rm soft} \\lesssim m_{3/2},$ and, therefore, all the SUSY particles generically acquire a soft SUSY breaking slightly lighter than the gravitino mass through the modulus mediation.",
"The anomaly mediation also gives a comparable contribution to the soft mass.",
"Although the mass of the SUSY SM particles are relevant for the observed SM-like Higgs boson mass, they are irrelevant for the inflaton dynamics during inflation.",
"For $A,~B \\ne 0$ , the axion can have a non-zero mass much smaller than the gravitino mass In order to implement the QCD axion, one would need to introduce another moduli field.",
"Alternatively, the QCD axion may originate from an open string mode., while the stabilization of the heavy moduli is not changed drastically.",
"This is because, as long as (REF ) is satisfied, the last two exponential terms in the superpotential (REF ) are much smaller than the others: $ A e^{-\\frac{2\\pi }{n_1}\\langle T_2 \\rangle }\\sim W_0^{\\frac{M}{2n_1}} \\ll W_0 (\\ll 1)$ ."
],
[
"Axion inflation in low energy effective theory",
"In this subsection, we focus on the lightest axion multiplet $\\Phi $ at scales below the heavy moduli masses.",
"We discuss multi-natural inflation within the low energy effective theory of this string-inspired model, using the results obtained in the previous section.",
"After integrating out the heavy moduli $T_0$ and $T $ , we obtain the low energy effective theory for $\\Phi $ , $K_L & \\approx &\\frac{f^2}{2} (\\Phi +\\Phi ^{\\dag })^2+ \\cdots , \\\\W_L & \\approx & W_0 + \\hat{A} e^{-\\frac{\\pi }{n_1}\\Phi }+ \\hat{B} e^{-\\frac{\\pi }{n_2}\\Phi - i \\theta }.$ The higher order terms in $K_L$ are given in Appendix .",
"Here, we have defined $f^2 &\\equiv & \\frac{3}{2 \\sqrt{2} \\sqrt{t} {\\cal V}} \\lesssim 1, ~~~\\hat{A} \\equiv A e^{-\\frac{\\pi }{n_1}\\langle T \\rangle },~~~ \\hat{B} \\equiv B e^{-\\frac{\\pi }{n_2}\\langle T \\rangle }.$ A natural value of $f$ is considered to be of order $0.1$ since it is on the order of the string scale.",
"The above Kähler and super potentials are equivalent to (REF ) and (REF ) studied in the previous section, and successful axion inflation is possible for a certain choice of the parameters.",
"The parameters are related as $&&a = \\frac{\\pi }{n_1},~~~~~~~b = \\frac{\\pi }{n_2}\\\\&&f_1 = n_1 f_a,~~f_2 = n_2 f_a,$ where we have defined $f_a \\equiv \\sqrt{2}f / \\pi $ .",
"Note that the prefactors of the exponentials, ${\\hat{A}}$ and ${\\hat{B}}$ , are comparable to each other, and much smaller than $W_0$ , $\\hat{A} \\sim \\hat{B} \\sim W_0^{M/2n_1} \\ll W_0 (\\ll 1).$ As we have seen before, this hierarchy is one of the essential ingredients for multi-natural inflation.",
"Indeed, the ratio of the axion mass to the saxion mass is much smaller than unity; $\\frac{m_{\\phi }^2}{m_{\\sigma }^2} \\sim \\frac{\\hat{A}W_0}{W_0^2} \\sim W_0^{\\frac{M}{2n_1} -1} \\ll 10^{-2}~~~{\\rm for}~M > 2n_1,$ and therefore the saxion remains stabilized near the origin, $\\left\\langle \\sigma \\right\\rangle \\simeq 0$ .",
"In order to have axion hilltop inflation, the model parameters must satisfy the relations (), (REF ), and () to a high accuracy.",
"In particular, the condition () reads $\\hat{A} \\approx \\bigg (\\frac{n_1}{n_2}\\bigg )^2\\hat{B},$ which implies that the gaugino condensation from $SU(n_1)$ should be comparable to that from $SU(n_2)$ in size.",
"In terms of the gauge couplings at the cut-off scale, this condition can be expressed as $\\frac{g_2^2}{g_1^2} \\sim \\frac{n_1}{n_2}\\bigg [1 + \\frac{n_2 g_2^2}{8\\pi ^2}\\log \\left(\\frac{n_2}{n_1}\\right)\\bigg ],$ where $g_i~(i=1,2)$ is the gauge coupling in $SU(n_i)$ gauge group at the cut-off scale and we have used the fact that $\\hat{A}$ and $\\hat{B}$ are proportional to $n_1$ and $n_2$ , respectively: $W_{SU(n_i)} = n_i \\Lambda _{SU(n_i)}^3 = n_i e^{-8\\pi ^2/n_i g_i^2}$ , where the $\\theta $ -term is omitted.",
"Thus, successful axion hilltop inflation requires a certain relation between the rank of the gauge groups and the value of the gauge couplings of a gauge theory where gaugino condensations form in the low energy.",
"This is equivalent to the relation between the world volume of the relevant D-branes and the number of such branes in string theory.",
"For further discussions on the magnitude of $A$ and $B$ , see Appendix .",
"Lastly let us express the inflaton mass and the Hubble parameter during inflation in terms of the gravitino mass.",
"To this end we consider the case where $n_1$ is not degenerate with $n_2$ .",
"When $n_1$ is close to $n_2$ , the factor proportional to $(n_1-n_2)/n_2$ should be included as discussed in the previous section.",
"The axion mass at the potential minimum, the potential height and the inflation scale are estimated as $m_{\\phi }^2 &\\sim & \\frac{\\hat{A} W_0}{n_1^2 f_a^2}\\sim m_{3/2}^2 \\bigg (\\frac{m_{3/2}}{n_1 f_a}\\bigg )^{2}\\bigg (\\frac{m_{3/2}}{M_{\\rm Pl}}\\bigg )^{\\frac{M}{2n_1}-3}\\bigg [\\frac{1}{\\log (M_{\\rm Pl}/m_{3/2})}\\bigg ]^{\\frac{M}{2n_1}},\\\\V_0 &\\sim & \\hat{A} W_0\\sim m_{3/2}^4 \\bigg (\\frac{m_{3/2}}{M_{\\rm Pl}}\\bigg )^{\\frac{M}{2n_1}-3}\\bigg [\\frac{1}{\\log (M_{\\rm Pl}/m_{3/2})}\\bigg ]^{\\frac{M}{2n_1}}, \\\\H_{\\rm inf} &\\sim & \\frac{\\sqrt{V_0}}{M_{\\rm Pl}} \\sim m_{3/2} \\bigg (\\frac{m_{3/2}}{M_{\\rm Pl}}\\bigg )^{\\frac{M-2n_1}{4n_1}}\\bigg [\\frac{1}{\\log (M_{\\rm Pl}/m_{3/2})}\\bigg ]^{\\frac{M}{4n_1}}$ for $M>2n_1$ .",
"For instance, $m_{\\phi } \\simeq 10^{11}$ GeV is obtained for $m_{3/2} \\simeq 10^{14}$ GeV, $f_a \\simeq 10^{17}$ GeV, $n_1 =6$ and $M= 24$ .",
"The last equation implies that the Hubble parameter during inflation is necessarily smaller than the gravitino mass, $H_{\\rm inf} < m_{3/2},$ which enables us to avoid the moduli destabilization [54].",
"This is because the flatness of the inflaton potential is not due to SUSY, but (mostly) due to both the axionic shift symmetry and the dynamical origin of the potential."
],
[
"Reheating and leptogenesis",
"In order to have successful inflation, the inflaton must transfer its energy to the SM particles.",
"Also, as any pre-existing baryon asymmetry is diluted by the inflationary expansion, the right amount of baryon asymmetry must be created after inflation.",
"Here we study reheating and the baryon number generation through leptogenesis [55].",
"As for reheating, the axion will decay into the SM gauge bosons through its couplings to the SM gauge fields, ${\\cal L}_{\\rm SM} &=& \\frac{1}{16\\pi }\\int d^2 \\theta T_2 {\\cal W}^{\\alpha }_{\\rm SM} {\\cal W}_{\\alpha {\\rm SM}} + {\\rm h.c.}\\supset \\frac{1}{32\\pi }\\int d^2 \\theta \\Phi {\\cal W}^{\\alpha }_{\\rm SM} {\\cal W}_{\\alpha {\\rm SM}}+ {\\rm h.c.} \\\\\\Gamma ^{\\phi }_g &\\equiv &\\Gamma (\\phi \\rightarrow 2A_{\\mu }) \\simeq \\frac{N_g}{32 \\pi t^2}\\frac{{m_\\phi }^3 }{f^2}\\simeq \\frac{N_g g_{\\rm SM}^4 }{4096 \\pi ^5}\\frac{m_{\\phi }^3}{f_a^2},$ where $t \\simeq 16\\pi /g_{\\rm SM}^2$ .",
"We have assumed that the SM is living on the D-brane wrapping on $T_2$ -cycle.",
"Thus, the reheating temperature is given by $T_R^g \\simeq \\left(\\frac{\\pi ^2 g_*(T_R) }{90 }\\right)^{-1/4}\\sqrt{\\Gamma ^{\\phi }_g M_{\\rm Pl}} \\simeq 4\\times 10^{5}~{\\rm GeV}\\bigg (\\frac{m_{\\phi }}{10^{11}{\\rm GeV}}\\bigg )^{3/2} \\bigg (\\frac{f_a}{10^{17}{\\rm GeV}}\\bigg )^{-1},$ if this is the main decay mode.The axion cannot decay into the SM gauginos since their mass is heavier than the axion mass.",
"Even if it is possible, the estimation will not be changed drastically as studied in [56]; the R-parity may have to be violated to avoid the overabundance of dark matter.",
"The moduli-induced baryogenesis [57] may work in this case.",
"Here we have used $N_g=12$ and $g_{\\rm SM}^2/4\\pi =1/25$ and $g_*(T_R) = 106.75$ .",
"Next, let us consider the origin of the baryon asymmetry.",
"Among various baryogenesis scenarios, leptogenesis is a plausible and interesting possibility in the light of the observed neutrino masses and mixings.",
"We focus on non-thermal leptogenesis [58], [59], because it can generate a sufficient amount of baryon asymmetry with a relatively low reheating temperature.",
"The right-handed neutrino ${\\nu }^c$ can be produced by the axion decays via the coupling below $W = C_{{\\nu }^c} e^{-\\frac{2\\pi }{n_1}T_2} \\nu ^c \\nu ^c ,$ in which $C_{\\nu ^c}$ is a constant.",
"Note that the mass of the neutrinos is given by $m_{\\nu ^c} \\simeq 2 C_{{\\nu }^c} M_{\\rm Pl}\\bigg [\\frac{m_{3/2}/M_{\\rm Pl}}{\\log (M_{\\rm Pl}/m_{3/2})}\\bigg ]^{\\frac{M}{2n_1}},$ while $m_{\\phi } \\sim W_0^{(M+2n_1)/4n_1}/f$ ; $m_{{\\nu }^c}/m_{\\phi } \\sim W_0^{\\frac{(M-2n_1)}{4n_1}}f < 1$ .",
"For instance, one obtains $m_{{\\nu }^c} \\simeq 10^{10}$ GeV when taking $W_0 = 10^{-4}$ , $M=24$ , $n_1=6$ and $C_{\\nu ^c}= 30$ .",
"Such a term is generated when the right-handed neutrino is coupled to the gauge field of $SU(n_1)$ : $ \\int d^2 \\theta {\\nu }^c {\\nu }^c [{\\cal W}^{\\alpha }{\\cal W}_{\\alpha }]_{SU(n_1)}$ .",
"(A similar origin is also discussed in the literatures [60], [61], [62].)",
"The decay fraction is given by $\\Gamma _{{\\nu }^c}^{\\phi }\\equiv \\Gamma (\\phi \\rightarrow 2{\\nu }^c) \\simeq \\frac{1}{16\\pi }\\left(\\frac{m_{{\\nu }^c}}{n_1 f_a}\\right)^2 m_{\\phi }.$ The reheating will proceed mainly via the decay into the neutrinos if $\\frac{m_{\\nu ^c}}{m_{\\phi }} \\gtrsim 10^{-2}.$ The reheating temperature is then estimated as $T_R^{\\nu ^c} \\simeq 1 \\times 10^7{\\rm GeV}\\left( \\frac{m_{\\phi }}{10^{11}~{\\rm GeV}}\\right)^{1/2}\\left( \\frac{m_{\\nu ^c}}{10^{10}~{\\rm GeV}}\\right)\\left( \\frac{n_1 f_a}{10^{17}~{\\rm GeV}}\\right)^{-1} .$ Using the reheating temperature $T_R \\simeq \\sqrt{(\\Gamma ^\\phi _{{\\nu }^c} + \\Gamma ^\\phi _g) M_{\\rm Pl}}$ , the net baryon asymmetry is written as $\\frac{n_B}{s} \\simeq \\frac{28}{78}\\cdot \\epsilon \\cdot \\frac{3}{2} \\frac{T_R}{m_{\\phi }}B_{\\nu ^c}^{\\phi },$ with $\\epsilon \\simeq \\frac{3}{16\\pi }\\frac{m_{\\nu _3}m_{\\nu ^c}}{v^2}\\delta _{\\rm eff},$ where $B_{\\nu ^c}^{\\phi }$ is the decay fraction into the right-handed neutrino, $m_{\\nu _3}$ is the heaviest neutrino mass, $v\\simeq 174$ GeV is the Higgs VEV, and $\\delta _{\\rm eff}$ is an effective CP-phase in the neutrino Yukawa couplings.",
"Here we have assumed the SM contribution for the sphaleron process and assumed that the axion decays mainly into lightest right-handed neutrino.",
"Then, we can generate a right amount of the baryon asymmetry, $\\frac{n_B}{s} \\simeq 5.4\\times 10^{-11}\\delta _{\\rm eff}\\left(\\frac{T_R/m_{\\phi }}{10^{-4}}\\right)\\left(\\frac{m_{\\nu _3}}{0.05{\\rm eV}}\\right)\\left( \\frac{m_{\\nu ^c}}{10^{10}~{\\rm GeV}}\\right)\\left( \\frac{B_{\\nu ^c}^{\\phi }}{1}\\right).$ Thus, if the axion decays mainly into the lightest right-handed neutrinos, non-thermal leptogenesis works successfully.",
"In Fig.REF , the plots for the baryon asymmetry are shown with $\\delta _{\\rm eff} = 1$ , using the relation between the mass scales and the gravitino mass of Eq.",
"(REF ) and (REF ).",
"In the successful case, we can obtain $m_{3/2} \\sim 10^{13}$ GeV, $m_{\\nu ^c} \\sim 10^{11}$ GeV and $T_R \\sim 10^{7}$ GeV.",
"Figure: Plots for the baryon asymmetry and the Planck normalization in the (M,m 3/2 )(M,m_{3/2})-plane.The green shaded region shows 0.5×10 -10 ≤n B /s≤1.5×10 -10 0.5 \\times 10^{-10} \\le n_B/s \\le 1.5 \\times 10^{-10}.The blue line shows the Planck normalization corresponding to λ Planck =3.7×10 -14 \\lambda _{\\rm Planck} = 3.7 \\times 10^{-14} for N e ≃60N_e \\simeq 60.In the red shaded region, the saxion vacuum deviates from the origin.We used n 1 =7n_1 = 7, n 2 =6n_2 = 6, f a =2.3×10 17 f_a = 2.3\\times 10^{17} GeV and C ν c =8C_{\\nu ^c} = 8;f 1 =7f a f_1 = 7f_a and f 2 =6f a f_2 =6f_a.Then m φ ≃9.6×10 11 m_{\\phi } \\simeq 9.6 \\times 10^{11} GeV is obtained, using Eq.",
"().We find also m 3/2 ∼10 13 m_{3/2} \\sim 10^{13} GeV, m ν c ∼10 11 m_{\\nu ^c} \\sim 10^{11} GeV and T R ∼10 7 T_R \\sim 10^{7} GeVin the viable region, where we can obtain the correct curvature perturbation and baryon asymmetry.No dark matter candidate has been considered in the setup so far.",
"As the inflation scale is lower than the typical scale of the soft SUSY breaking mass, no SUSY particles are produced during and after reheating.",
"Therefore, the QCD axion or light axion-like particles, light sterile neutrinos, hidden photons, etc., or their combination, are candidates for dark matter.",
"Finally, we give a comment on the SM-like Higgs boson mass.",
"In this model, the soft SUSY-breaking terms will be on the order of the gravitino mass or slightly smaller [38].",
"For $m_{\\rm soft} \\sim m_{3/2}/\\log (M_{\\rm Pl}/m_{3/2}) \\sim 10^{12}$ GeV in the viable region for inflation, the Higgs mass becomes $\\sim 126$ GeV for $\\tan \\beta \\sim 1$ [63].",
"The small value of $\\tan \\beta $ can be realized in the presence of a shift symmetry or an exchange symmetry in the Higgs sector [64], [65], [66].",
"We have studied multi-natural inflation [28] in SUGRA for a UV completion.",
"In this model the inflaton potential mainly consists of two sinusoidal functions that are comparable in size, but have different periodicity.",
"For sub-Planckian values of the decay constants, this model is reduced to the hilltop quartic inflation in a certain limit.",
"It is known however that the predicted spectral index, $n_s \\simeq 0.94$ , for the e-folding number $N_e \\simeq 50$ tends to be too low to explain the Placnk results.",
"We have shown that, allowing a relative phase between the two sinusoidal functions, the spectral index can be increased to give a better fit to the Planck data based on the axion hilltop inflation in SUGRA.",
"We have also considered a further UV completion based on a string-inspired framework, and have shown that the axion hilltop inflation model can be indeed obtained in the low energy limit.",
"The axion hilltop inflation requires a rather flat potential near the (local) potential maximum.",
"For realizing the flat-top potential, there should exist a relation between the ratio of the decay constants and the dynamical scale: $(f_1/f_2)^2 \\approx A/B$ .",
"This in turn implies that in string theory there should be a relation between the world volume of D-branes where the non-perturbative effects occur and the number of such branes.",
"It is also noted that because we have used supergravity, the gravitino mass is related to physical quantities.",
"For successful inflation, the typical scale of the gravitno mass is $m_{3/2} \\sim 10^{13}{\\rm \\,GeV}$ , whereas the soft mass is about one order of magnitude smaller, $m_{\\rm soft} = 10^{12}{\\rm \\,GeV}$ , while the inflaton mass is $m_\\phi \\sim 10^{11}{\\rm \\,GeV}$ .",
"Thus, the SUSY particles are not produced in the Universe after reheating.",
"The dark matter candidates can be considered such as the QCD axion, axion-like particles, or sterile neutrinos, if they exist.",
"In particular, a light dark matter is interesting in light of its longevity.",
"The recently discovered X-ray line at $3.5$ keV [34], [35] may be due to the decay of one of such light dark matterSee the recent works on explaining the $3.5$ keV X-ray line by axions [33] or sterile neutrinos [67]..",
"It may be possible to explain the baryon asymmetry from the inflaton decay into right-handed neutrinos.",
"Note added: After the submission of our paper, the BICEP2 experiment found the primordial B-mode polarization [4], which suggests $r = 0.20^{+0.07}_{-0.05}$ .",
"Although we have focused on the hilltop inflation limit of the multi-natural inflation when we evaluate the spectral index and the tensor-to-scalar ratio, most of the discussion including the realization of the multi-natural inflation in supergravity and string-inspired set-up, the reheating, and leptogenesis is applicable to a more general multi-natural inflation.",
"In particular, for such a large value of $r$ , the inflaton mass will be of order $10^{13}{\\rm \\,GeV}$ , leading to the reheating temperature close to $10^{9}{\\rm \\,GeV}$ (cf.",
"Eq.",
"(REF )).",
"Therefore, thermal leptogenesis will be possible.",
"See also the related papers on the multi-natural inflation [69], [70] that appeared after BICEP2."
],
[
"Acknowledgment",
"This work was supported by Grant-in-Aid for Scientific Research on Innovative Areas (No.24111702, No.",
"21111006, and No.23104008) [FT], Scientific Research (A) (No.",
"22244030 and No.21244033) [FT], and JSPS Grant-in-Aid for Young Scientists (B) (No.",
"24740135 [FT] and No.",
"25800169 [TH]), and Inoue Foundation for Science.",
"This work was also supported by World Premier International Center Initiative (WPI Program), MEXT, Japan [FT]."
],
[
"The tuning for the inflation in the string-inspired model",
"In this section of the Appendix, we will discuss the tuning of $\\hat{B} \\approx \\bigg (\\frac{n_2}{n_1}\\bigg )^2 \\hat{A}$ found in Sec.REF .",
"Here, we define the phase of $B$ : $\\theta \\equiv {\\rm arg}[1/B].$ It should be noted that the phase $\\theta $ will be given by the VEVs of the dilaton and the complex structure moduli stabilized by closed string fluxes.",
"In terms of $A$ and $B$ this relation becomes $B \\approx A \\bigg (\\frac{n_2}{n_1}\\bigg )^2 \\bigg (\\frac{W_0}{\\log [1/W_0]}\\bigg )^{-\\frac{M}{2}\\big (\\frac{1}{n_2}-\\frac{1}{n_1}\\big )}.$ Here we substituted the solutions of moduli VEVs in Eq.",
"(REF ).",
"For instance, one finds $B \\sim 15 A$ for $W_0 =10^{-4},~n_1 =7,~n_2=6$ and $M=22$ .",
"In this case, complex structure moduli can play a role in 1-loop threshold corrections from the heavy modes, which depend on the gauge group.",
"On the other hand, for $W_0 =10^{-4},~n_1 =6,~n_2=3$ and $M=22$ , one finds $B \\sim 3.2\\times 10^8 A$ .",
"In the latter case, the heavy moduli such as the dilaton and complex structure may play an important role in the relevant gauge couplings, e.g., $\\frac{4\\pi }{g_1^2} \\sim T_2,~~~\\frac{4\\pi }{g_2^2} \\sim T_2 - \\Delta f,$ where $\\Delta f$ contains heavy moduli [29] and $ 1- \\Delta f/T_2 \\sim 1/2$ .",
"Then one finds that $ B \\sim e^{\\frac{2\\pi }{3}\\Delta f} \\sim e^{\\frac{2\\pi }{6}T_2} \\gg 1$ [68], using the fact that the size of one gaugino condensation is similar to the other,, $A e^{-2\\pi T_2/6 } \\sim B^{\\prime } e^{-2\\pi (T_2-\\Delta f)/3 }$ , where $B^{\\prime } = B e^{-2\\pi \\Delta f/3 } ={\\cal O}(1)$ .",
"Note that even if $B$ is much larger than unity, the heavy moduli/saxion stabilization does not change as long as $\\bigg (\\frac{m_{\\phi }}{m_{\\sigma }}\\bigg )^2\\simeq \\frac{|\\hat{B}|}{|W_0|f^2} \\lesssim 10^{-2} .$ A relation $M>2n_2$ is important to satisfy the above condition."
],
[
"Higher order terms in Choi-Jeong models",
"We write down higher order terms in $\\Phi $ : $K_L & \\approx &\\frac{f^2}{2} (\\Phi +\\Phi ^{\\dag })^2+ f^2 \\frac{k_4}{4!}",
"(\\Phi +\\Phi ^{\\dag })^4 + f^2 \\frac{k_6}{6!}",
"(\\Phi +\\Phi ^{\\dag })^6+ \\cdots , \\\\W_L & \\approx & W_0 + C e^{-\\frac{\\pi }{n_1}(\\Phi + \\langle T \\rangle )}- D e^{-\\frac{\\pi }{n_2}(\\Phi + \\langle T \\rangle )},\\\\V_{{\\rm up}L} &\\approx & 3m_{3/2}^2 + f^2 m_{3/2}^2 (\\Phi +\\Phi ^{\\dag })^2+f^2 m_{3/2}^2 \\zeta (\\Phi +\\Phi ^{\\dag })^4+ \\cdots .$ Here, we defined $\\nonumber && f^2 \\equiv \\frac{3}{2 \\sqrt{2} \\sqrt{t} {\\cal V}},~~k_4 \\equiv \\frac{9 \\sqrt{2} t^{3/2}+6 {\\cal V}}{8 t^2 {\\cal V}}\\sim f^2 <1 , ~~k_6 \\equiv \\frac{15 \\left(18 t^3+9 \\sqrt{2} t^{3/2} {\\cal V}+14 {\\cal V}^2\\right)}{32 t^4 {\\cal V}^2} \\sim f^4 <1, \\\\&& m_{3/2} = e^{K/2}W = \\frac{W_0}{{\\cal V}},~~~\\zeta = \\frac{ \\left(7 t^{3/2}+\\sqrt{2} {\\cal V}\\right) }{16 \\sqrt{2} t^2 {\\cal V}} \\sim f^2 <1,$ where $t = T+T^{\\dag } $ , ${\\cal V} \\equiv t_0^{3/2} - t^{3/2} /\\sqrt{2}$ and $\\hat{\\epsilon } \\approx 3{W_0}^2/{\\cal V}^{2/3}$ are used, and it is noted that $t$ and ${\\cal V}$ are given by the VEVs.",
"The higher order term of $\\Phi $ in the Kähler potential will become irrelevant for $f \\lesssim 1$ , when the uplifting potential is added."
]
] | 1403.0410 |
[
[
"Photon and photino as Nambu-Goldstone zero modes in an emergent SUSY QED"
],
[
"Abstract We argue that supersymmetry with its well known advantages, such as naturalness, grand unification and dark matter candidate seems to possess one more attractive feature: it may trigger, through its own spontaneous violation in the visible sector, a dynamical generation of gauge fields as massless Nambu-Goldstone modes during which physical Lorentz invariance itself is ultimately preserved.",
"We consider the supersymmetric QED model extended by an arbitrary polynomial potential of massive vector superfield that breaks gauge invariance in the SUSY invariant phase.",
"However, the requirement of vacuum stability in such class of models makes both supersymmetry and Lorentz invariance to become spontaneously broken.",
"As a consequence, massless photino and photon appear as the corresponding Nambu-Goldstone zero modes in an emergent SUSY QED, and also a special gauge invariance is simultaneously generated.",
"Due to this invariance all observable relativistically noninvariant effects appear to be completely cancelled out among themselves and physical Lorentz invariance is recovered.",
"Nevertheless, such theories may have an inevitable observational evidence in terms of the goldstino-photino like state presented in the low-energy particle spectrum.",
"Its study is of a special interest for this class of SUSY models that, apart from some indication of an emergence nature of QED and the Standard Model, may appreciably extend the scope of SUSY breaking physics being actively studied in recent years."
],
[
"Introduction and overview",
"It is long believed that spontaneous Lorentz invariance violation (SLIV) may lead to an emergence of massless Nambu-Goldstone (NG) zero modes [1] which are identified with photons and other gauge fields appearing in the Standard Model.",
"This old idea [2] supported by a close analogy with the dynamical origin of massless particle excitations for spontaneously broken internal symmetries has gained new impetus in recent years.",
"On the other hand, besides its generic implication to a possible origin of physical gauge fields [3], [4], [5], [6], [7] in a conventional quantum field theory (QFT) framework, there are many different contexts in literature where Lorentz violation may stem in itself from string theory [8], quantum gravity [9] or any unspecified dynamics at an ultraviolet scale perhaps related to the Planck scale [10], [11], [12].",
"Though we are mainly related to the spontaneous Lorentz violation in QFT, particularly in QED and Standard Model, we give below some brief comments on other approaches as well to make clearer the aims and results of the present work."
],
[
"Vector NG bosons in gauge theories. Inactive SLIV",
"When speaking about SLIV, one important thing to notice is that, in contrast to the spontaneous violation of internal symmetries, it seems not to necessarily imply a physical breakdown of Lorentz invariance.",
"Rather, when appearing in a gauge theory framework, this may ultimately result in a noncovariant gauge choice in an otherwise gauge invariant and Lorentz invariant theory.",
"In substance, the SLIV ansatz, due to which the vector field develops a vacuum expectation value (VEV) $<A_{\\mu }(x)>\\text{ }=n_{\\mu }M $ (where $n_{\\mu }$ is a properly-oriented unit Lorentz vector, $n^{2}=n_{\\mu }n^{\\mu }=\\pm 1$ , while $M$ is the proposed SLIV scale), may itself be treated as a pure gauge transformation with a gauge function linear in coordinates, $\\omega (x)=$ $n_{\\mu }x^{\\mu }M$ .",
"From this viewpoint gauge invariance in QED leads to the conversion of SLIV into gauge degrees of freedom of the massless photon emerged.",
"A good example for such a kind of SLIV, which we call the \"inactive\" SLIV hereafter, is provided by the nonlinearly realized Lorentz symmetry for underlying vector field $A_{\\mu }(x)$ through the length-fixing constraint $A_{\\mu }A^{\\mu }=n^{2}M^{2}\\text{ .}",
"$ This constraint in the gauge invariant QED framework was first studied by Nambu a long ago [13], and in more detail in recent years [14], [15], [16], [17], [18].",
"The constraint (REF ) is in fact very similar to the constraint appearing in the nonlinear $\\sigma $ -model for pions [19], $\\sigma ^{2}+\\pi ^{2}=f_{\\pi }^{2}$ , where $f_{\\pi }$ is the pion decay constant.",
"Rather than impose by postulate, the constraint (REF ) may be implemented into the standard QED Lagrangian extended by the invariant Lagrange multiplier term $\\mathcal {L}=L_{QED}-\\frac{\\lambda }{2}\\left( A_{\\mu }A^{\\mu }-n^{2}M^{2}\\right) $ provided that initial values for all fields (and their momenta) involved are chosen so as to restrict the phase space to values with a vanishing multiplier function $\\lambda (x)$ , $\\lambda =0$ .",
"Otherwise, as was shown in [20] (see also [17]), it might be problematic to have the ghost-free QED model with a positive HamiltonianNote that this solution with the basic Lagrangian multiplier field $\\lambda (x)$ being vanished can technically be realized by introducing some additional Lagrange multiplier term of the type $\\xi \\lambda ^{2}$ , where $\\xi (x)$ is a new multiplier field.",
"One can now easily confirm that a variation of the modified Lagrangia $\\mathcal {L}+$ $\\xi \\lambda ^{2}$ with respect to the $\\xi $ field leads to the condition $\\lambda =0$ , whereas a variation with respect to the basic multiplier field $\\lambda $ preserves the vector field constraint (REF ).. One way or another, the constraint (REF ) means in essence that the vector field $A_{\\mu }$ develops the VEV (REF ) and Lorentz symmetry $SO(1,3)$ breaks down to $SO(3)$ or $SO(1,2)$ depending on whether the unit vector $n_{\\mu }$ is time-like ($n^{2}>0$ ) or space-like ($n^{2}<0$ ).",
"The point, however, is that, in sharp contrast to the nonlinear $\\sigma $ model for pions, the nonlinear QED theory, due to gauge invariance in the starting Lagrangian $L_{QED}$ , leaves physical Lorentz invariance intact.",
"Indeed, the nonlinear QED contains a plethora of Lorentz and $CPT$ violating couplings when it is expressed in terms of the pure vector NG boson modes ($a_{\\mu }$ ) associated with a physical photon $A_{\\mu }=a_{\\mu }+n_{\\mu }(M^{2}-n^{2}a^{2})^{\\frac{1}{2}}\\text{ , \\ }n_{\\mu }a_{\\mu }=0\\text{ \\ \\ (}a^{2}\\equiv a_{\\mu }a^{\\mu }\\text{) .}",
"$ including that the effective Higgs mode given by the second term in (REF ) is properly expanded in a power series of $a^{2}$ .",
"However, the contributions of all these couplings to physical processes completely cancel out among themselves, as was shown in the tree [13] and one-loop approximations [14].",
"Actually, the nonlinear constraint (REF ) implemented as a supplementary condition can be interpreted in essence as a possible gauge choice for the starting vector field $A_{\\mu }$ .",
"Meanwhile the $S$ -matrix remains unaltered under such a gauge convention unless gauge invariance in the theory turns out to be really broken (see next subsection) rather than merely being restricted by gauge condition (REF ).",
"Later similar result concerning the inactive SLIV in gauge theories was also confirmed for spontaneously broken massive QED [15], non-Abelian theories [16] and tensor field gravity [18].",
"Remarkably enough, the nonlinear QED model (REF ) may be considered in some sense as being originated from a conventional QED Lagrangian extended by the vector field potential energy terms, $\\mathcal {L}^{\\prime }=L_{QED}-\\frac{\\lambda }{4}\\left( A_{\\mu }A^{\\mu }-n^{2}M^{2}\\right) ^{2} $ (where $\\lambda $ is a coupling constant) rather than by the Lagrange multiplier term.",
"This is the simplest example of a theory being sometimes referred to as the “bumblebee” model (see [7] and references therein) where physical Lorentz symmetry could in principle be spontaneously broken due to presence of an active Higgs mode in the model.",
"On the other hand, the Lagrangian (REF ) taken in the limit $\\lambda \\rightarrow \\infty $ can formally be regarded as the nonlinear QED.",
"Actually, both of models are physically equivalent in the infrared energy domain, where the Higgs mode is considered infinitely massive.",
"However, as was argued in [20], a bumblebee-like model appears generally unstable, its Hamiltonian is not bounded from below unless the phase space sector is not limited by the nonlinear vector field constraint $A_{\\mu }A^{\\mu }=n^{2}M^{2}$ (REF ).",
"With this condition imposed, the massive Higgs mode never appears, the Hamiltonian is positive, and the model is physically equivalent to the constraint-based nonlinear QED (REF ) with the inactive SLIV which does not lead to physical Lorentz violationApart from its generic instability, the “bumblebee” model, as we will see it shortly, can not be technically realized in a SUSY context, whereas the nonlinear QED model successfully matches supersymmetry.. To summarize, we have considered above the standard QED with vector field constraint (REF ) being implemented into the Lagrangian through the Lagrange multiplier term (REF ).",
"In crucial contrast to an internal symmetry breaking (say, the breaking of a chiral $SU(2)\\times SU(2)$ symmetry in the nonlinear $\\sigma $ -model for pions) SLIV caused by a similar $\\sigma $ -model type vector field constraint (REF ), does not lead to physical Lorentz violation.",
"Indeed, though SLIV induces the vector Goldstone-like states (REF ), all observable SLIV effects appear to be completely canceled out among themselves due to a generic gauge invariance of QED.",
"We call it the inactive SLIV in the sense that one may have Goldstone-like states in a theory but may have not a nonzero symmetry breaking effect.",
"This is somewhat new and unusual situation that just happens with SLIV in gauge invariant theories (and never in an internal symmetry breaking case).",
"More precisely there are, in essence, two different (though related to each other) aspects regarding the inactive SLIV.",
"The first is a generation of Goldstone modes which inevitably happens once the nonlinear $\\sigma $ -model type constraint (REF ) is put on the vector field.",
"The second is that gauge invariance even being restricted by this constraint (interpreted as a gauge condition) provides a cancellation mechanism for physical Lorentz violation.",
"As a consequence, emergent gauge theories induced by the inactive SLIV mechanism are in fact indistinguishable from conventional gauge theories.",
"Their emergent nature can only be seen when a gauge condition is taken to be the vector field length-fixing constraint (REF ).",
"Any other gauge, e.g.",
"the Coulomb gauge, is not in line with an emergent picture, since it explicitly breaks Lorentz invariance.",
"As to an observational evidence in favor of emergent theories, the only way for SLIV to be activated may appear if gauge invariance in these theories turns out to be broken in an explicit rather than spontaneous way.",
"As a result, the SLIV cancellation mechanism does not work longer and one inevitably comes to physical Lorentz violation."
],
[
"Activating SLIV by gauge symmetry breaking",
"Looking for some appropriate examples of physical Lorentz violation in a QFT framework one necessarily come across a problem of proper suppression of gauge noninvariant high-dimension couplings where such violation can in principle occur.",
"Remarkably enough, for QED type theories with the supplementary vector field constraint (REF ) gauge symmetry breaking naturally appears only for five- and higher-dimensional couplings.",
"Indeed, all dimension-four couplings are generically gauge invariant, if the vector field kinetic term has a standard $F_{\\mu \\nu }F^{\\mu \\nu }$ and, apart from relativistic invariance, the restrictions related to the conservation of parity, charge-conjugation symmetry and fermion number conservation are generally imposed on a theory [21].",
"With these restrictions taken, one can easily confirm that all possible dimension-five couplings are also combined by themselves in some would-be gauge invariant form provided that vector field is constrained by the SLIV condition (REF ).",
"Indeed, for charged matter fermions interacting with vector field such couplings are generally amounted to $L_{\\dim 5}=\\frac{1}{\\mathcal {M}}\\check{D}_{\\mu }^{\\ast }\\overline{\\psi }\\cdot \\check{D}^{\\mu }\\psi +\\frac{G}{\\mathcal {M}}A_{\\mu }A^{\\mu }\\overline{\\psi }\\psi \\text{ , \\ }A_{\\mu }A^{\\mu }=n^{2}M^{2}\\text{ .}",
"$ Such couplings could presumably become significant at an ultraviolet scale $\\mathcal {M}$ probably being close to the Planck scale $M_{P}$ .",
"They, besides covariant derivative terms, also include an independent \"sea-gull\" fermion-vector field term with the coupling constant $G$ being in general of the order 1.",
"The main point regarding the Lagrangian (REF ) is that, while it is gauge invariant in itself, the coupling constant $\\check{e}$ in the covariant derivative $\\check{D}^{\\mu }=\\partial ^{\\mu }+i\\check{e}A^{\\mu }$ differs in general from the coupling $e$ in the covariant derivative $D^{\\mu }=\\partial ^{\\mu }+ieA^{\\mu }$ in the standard Dirac Lagrangian (REF ) $L_{QED}=-\\frac{1}{4}F_{\\mu \\nu }F^{\\mu \\nu }+\\overline{\\psi }(i\\gamma _{\\mu }D^{\\mu }-m)\\psi \\text{ .}",
"$ Therefore, gauge invariance is no longer preserved in the total Lagrangian $L_{QED}+$ $L_{\\dim 5}$ .",
"It is worth noting that, though the high-dimension Lagrangian part $L_{\\dim 5}$ (REF ) usually only gives some small corrections to a conventional QED Lagrangian (REF ), the situation may drastically change when the vector field $A_{\\mu }$ develops a VEV and SLIV occurs.",
"Actually, putting the SLIV parameterization (REF ) into the basic QED Lagrangian (REF ) one comes to the truly emergent model for QED being essentially nonlinear in the vector Goldstone modes $a_{\\mu }$ associated with photons.",
"This model contains, among other terms, the inappropriately large (while false, see below) Lorentz violating fermion bilinear $-eM\\overline{\\psi }(n_{\\mu }\\gamma ^{\\mu })\\psi $ .",
"This term appears when the effective Higgs mode expansion in Goldstone modes $a_{\\mu }$ (as is given in the parametrization (REF )) is applied to the fermion current interaction term $-e\\overline{\\psi }\\gamma _{\\mu }A^{\\mu }\\psi $ in the QED Lagrangian (REF ).",
"However, due to local invariance this bilinear term can be gauged away by making an appropriate redefinition of the fermion field $\\psi \\rightarrow e^{-ie\\omega (x)}\\psi $ with a gauge function $\\omega (x)$ linear in coordinates, $\\omega (x)=$ $(n_{\\mu }x^{\\mu })M$ .",
"Meanwhile, the dimension-five Lagrangian $L_{\\dim 5}$ (REF ) is substantially changed under this redefinition that significantly modifies fermion bilinear terms $L_{\\overline{\\psi }\\psi }=i\\overline{\\psi }\\gamma _{\\mu }\\partial ^{\\mu }\\psi +\\frac{1}{\\mathcal {M}}\\partial _{\\mu }\\overline{\\psi }\\cdot \\partial ^{\\mu }\\psi -i\\Delta e\\frac{M}{\\mathcal {M}}n_{\\mu }\\overline{\\psi }\\overleftrightarrow{\\partial ^{\\mu }}\\psi -m_{f}\\overline{\\psi }\\psi $ where we retained the notation $\\psi $ for the redefined fermion field and denoted, as usually, $\\overline{\\psi }\\overleftrightarrow{\\partial ^{\\mu }}\\psi =\\overline{\\psi }(\\partial ^{\\mu }\\psi )-(\\partial ^{\\mu }\\overline{\\psi })\\psi $ .",
"Note that the extra fermion derivative terms given in (REF ) is produced just due to the gauge invariance breaking that is determined by the electromagnetic charge difference $\\Delta e=\\check{e}-e$ in the total Lagrangian $L_{QED}+$ $L_{\\dim 5}$ .",
"As a result, there appears the entirely new, SLIV inspired, dispersion relation for a charged fermion (taken with 4-momentum $p_{\\mu }$ ) of the type $p_{\\mu }^{2}\\cong [m_{f}+2\\delta (p_{\\mu }n^{\\mu }/n^{2})]^{2},\\text{\\ \\ }m_{f}=\\left( m-G\\frac{M^{2}}{\\mathcal {M}}\\right) -\\delta ^{2}n^{2}\\mathcal {M} $ given to an accuracy of $O(m_{f}^{2}/\\mathcal {M}^{2})$ with a properly modified total fermion mass $m_{f}$ .",
"Here $\\delta $ stands for the small characteristic, positive or negative, parameter $\\delta =(\\Delta e)M/\\mathcal {M}$ of physical Lorentz violation that reflects the joint effect as is given, from the one hand, by the SLIV scale $M$ and, from the other, by the charge difference $\\Delta e$ being a measure of an internal gauge non-invariance.",
"Notably, the space-time in itself still possesses Lorentz invariance, however, fermions with SLIV contributing into their total mass $m_{f}$ (REF ) propagate and interact in it in the Lorentz non-covariant way.",
"At the same time, the photon dispersion relation is still retained in the order $1/\\mathcal {M}$ considered.",
"So, we have shown in the above that SLIV caused by the vector field VEV (REF ), while being superficial in gauge invariant theory, becomes physically significant for some high value of the SLIV scale $M$ being close to the scale $\\mathcal {M}$ , which is proposed to be located near the Planck scale $M_{P}$ .",
"This may happen even at relatively low energies provided the gauge noninvariance caused by high-dimension couplings of matter and vector fields is not vanishingly small.",
"This leads, as was demonstrated in [21], through special dispersion relations appearing for matter charged fermions, to a new class of phenomena which could be of distinctive observational interest in particle physics and astrophysics.",
"They include a significant change in the GZK cutoff for UHE cosmic-ray nucleons, stability of high-energy pions and $W$ bosons, modification of nucleon beta decays, and some others just in the presently accessible energy area in cosmic ray physics.",
"However, though one could speculate about some generically broken or partial gauge symmetry in a QFT framework [21], this seems to be too high price for an actual Lorentz violation which may stem from SLIV.",
"And, what is more, is there really any strong theoretical reason left for Lorentz invariance to be physically broken, if emergent gauge fields are anyway generated through the “safe” inactive SLIV models which recover a conventional Lorentz invariance?"
],
[
"Direct Lorentz noninvariant extensions of SM and gravity",
"Nevertheless, it must not be ruled out that physical Lorentz invariance might be explicitly, rather than spontaneously, broken at high energies.",
"This has attracted considerable attention in recent years as an interesting phenomenological possibility appearing in direct Lorentz noninvariant extensions of SM [10], [11], [12].",
"They are generically regarded as being originated in a more fundamental theory at some large scale probably related to the Planck scale $M_{P}$ .",
"These extensions are in a certain measure motivated [8] by a string theory according to which an explicit (from a QFT point of view) Lorentz violation might be in essence a spontaneous Lorentz violation related to hypothetical tensor-valued fields acquiring non-zero VEVs in some non-perturbative vacuum.",
"These VEVs appear effectively as a set of external background constants so that interactions with these coefficients have preferred spacetime directions in an effective QFT framework.",
"The full SM extension (SME) [11] is then defined as the effective gauge invariant field theory obtained when all such Lorentz violating vector and tensor field backgrounds are contracted term by term with SM (and gravitational) fields.",
"However, without a completely viable string theory, it is not possible to assign definite numerical values to these coefficients.",
"Moreover, not to have disastrous consequences (especially when these coefficients are contracted with non-conserved currents) one also has to additionally propose that observable violating effects in a low-energy theory with a laboratory scale $m$ should be suppressed by some power of the ratio $m/M_{P}$ being depended on dimension of Lorentz breaking couplings.",
"Therefore, one has in this sense a pure phenomenological approach treating the above arbitrary coefficients as quantities to be bounded in experiments as if they would simply appear due to explicit Lorentz violation.",
"Actually, in sharp contrast to the above formulated SLIV in a pure QFT framework, there is nothing in the SME itself that requires that these Lorentz-violation coefficients emerge due to a process of a spontaneous Lorentz violation.",
"Indeed, neither the corresponding massless vector (tensor) NG bosons are required to be generated, nor these bosons have to be associated with photons or any other gauge fields of SM.",
"Apart from Lorentz violation in Standard Model, one can generally think that the vacuum in quantum gravity may also determine a preferred rest frame at the microscopic level.",
"If such a frame exists, it must be very much hidden in low-energy physics since, as was mentioned above, numerous observations severely limit the possibility of Lorentz violating effects for the SM fields [10], [11], [12].",
"However, the constraints on Lorentz violation in the gravitational sector are generally far weaker.",
"This allows to introduce a pure gravitational Lorentz violation having no significant impact on the SM physics.",
"An elegant way being close in spirit to our SLIV model (REF , REF ) seems to appear in the so called Einstein-aether theory [9].",
"This is in essence a general covariant theory in which local Lorentz invariance is broken by some vector “aether” field $u_{\\mu }$ defining the preferred frame.",
"This field is similar to our constrained vector field $A_{\\mu }$ , apart from that this field is taken to be unit $u_{\\mu }u^{\\mu }=1$ .",
"It spontaneously breaks Lorentz symmetry down to a rotation subgroup, just like as our constrained vector field $A_{\\mu }$ does it for a timelike Lorentz violation.",
"So, they both give nonlinear realization of Lorentz symmetry thus leading to its spontaneous violation and induce the corresponding Goldstone-like modes.",
"The crucial difference is that, while modes related to the vector field $A_{\\mu }$ are collected into the physical photon, modes associated with the unit vector field $u_{\\mu }$ (one helicity-0 and two helicity-1 modes) exist by them own appearing in some effective SM and gravitational couplings.",
"Some of them might disappear being absorbed by the corresponding spin-connection fields related to local Lorentz symmetry in the Einstein-aether theory.",
"In any case, while aether field $u_{\\mu }$ can significantly change dispersion relations of fields involved, thus leading to many gravitational and cosmological consequences of preferred frame effects, it certainly can not be a physical gauge field candidate (say, the photon in QED)."
],
[
"Lorentz violation and supersymmetry. The present paper",
"There have been a few active attempts [22], [23] over the last decade to construct Lorentz violating operators for matter and gauge fields in the supersymmetric Standard Model through their interactions with external vector and tensor field backgrounds.",
"These backgrounds, according to the SME approach [11] discussed above, are generated by some Lorentz violating dynamics at an ultraviolet scale of order the Planck scale.",
"As some advantages over the ordinary SME, it was shown that in the supersymmetric Standard Model the lowest possible dimension for such operators is five, just as we had above in the high-dimensional SLIV case (REF ).",
"Therefore, they are suppressed by at least one power of an ultraviolet energy scale, providing a possible explanation for the smallness of Lorentz violation and its stability against radiative corrections.",
"There were classified all possible dimension five and six Lorentz violating operators in the SUSY QED [23], analyzed their properties at the quantum level and described their observational consequences in this theory.",
"These operators, as was confirmed, do not induce destabilizing $D$ -terms, gauge anomaly and the Chern-Simons term for photons.",
"Dimension-five Lorentz violating operators were shown to be constrained by low-energy precision measurements at $10^{-10}-10^{-5}$ level in units of the inverse Planck scale, while the Planck-scale suppressed dimension six operators are allowed by observational data.",
"Also, it has been constructed the supersymmetric extension of the Einstein-aether theory [24] discussed above.",
"It has been found that the dynamics of the super-aether is somewhat richer than of its non-SUSY counterpart.",
"In particular, the model possesses a family of inequivalent vacua exhibiting different symmetry breaking patterns while remaining stable and ghost free.",
"Interestingly enough, as long as the aether VEV preserves spatial supersymmetry (SUSY algebra without boosts), the Lorentz breaking does not propagate into the SM sector at the renormalizable level.",
"The eventual breaking of SUSY, that must be incorporated in any realistic model, is unrelated to the dynamics of the aether.",
"It is assumed to come from a different source characterized by a lower energy scale.",
"However, in spite of its own merits an important final step which would lead to natural accommodation of this super-aether model into the supergravity framework has not yet been done.",
"In contrast, we are strictly focused here on a spontaneous Lorentz violation in an actual gauge QFT framework related to the Standard Model rather than in an effective low energy theory with some hypothetical remnants in terms of external tensor-valued backgrounds originatating somewhere around the Planck scale.",
"In essence, we try to extend emergent gauge theories with SLIV and an associated emergence of gauge bosons as massless vector Nambu-Goldstone modes studied earlier [3], [4], [5], [6], [7] (see also [14], [15], [16], [17], [18]) to their supersymmetric analogs.",
"Generally speaking, it may turn out that SLIV is not the only reason why massless photons could dynamically appear, if spacetime symmetry is further enlarged.",
"In this connection, special interest may be related to supersymmetry, as was recently argued in [25].",
"Actually, the situation is changed remarkably in the SUSY inspired emergent models which, in contrast to non-SUSY theories, could naturally have some clear observational evidence.",
"Indeed, as we discussed above (subsection 1.2), ordinary emergent theories admit some experimental verification only if gauge invariance is properly broken being caused by some high-dimension couplings.",
"Their SUSY counterparts, and primarily emergent SUSY QED, are generically appear with supersymmetry being spontaneously broken in a visible sector to ensure stability of the theory.",
"Therefore, the verification is now related to an inevitable emergence of a goldstino-like photino state in the SUSY particle spectrum at low energies, while physical Lorentz invariant is still left intactOf course, physical Lorentz violation will also appear if one admits some gauge noninvariance in the emergent SUSY theory as well.",
"This may happen, for example, through high-dimension couplings being supersymmetric analogs of the couplings (REF )..",
"In this sense, a generic source for massless photon to appear may be spontaneously broken supersymmetry rather than physically manifested spontaneous Lorentz violation.",
"To see how such a scenario may work, we consider supersymmetric QED model extended by an arbitrary polynomial potential of massive vector superfield that induces the spontaneous SUSY violation in the visible sector.",
"As a consequence, a massless photino emerges as the fermion NG mode in the broken SUSY phase, and a photon as a photino companion to also appear massless in the tree approximation (section 2).",
"However, the requirement of vacuum stability in such class of models makes Lorentz invariance to become spontaneously broken as well.",
"As a consequence, massless photon has now appeared as the vector NG mode, and also a special gauge invariance is simultaneously generated in an emergent SUSY QED.",
"This invariance is only restricted by the supplemented vector field constraint being invariant under supergauge transformations (section 3).",
"Due to this invariance all observable SLIV effects appear to be completely cancelled out among themselves and physical Lorentz invariance is restored.",
"Meanwhile, photino being mixed with another goldstino appearing from a spontaneous SUSY violation in the hidden sector largely turns into the light pseudo-goldstino whose physics seems to be of special observational interest (section 4).",
"And finally, we conclude (section 5)."
],
[
"Extended supersymmetric QED",
"We start by considering a conventional SUSY QED extended by an arbitrary polynomial potential of a general vector superfield $V(x,\\theta ,\\overline{\\theta })$ which in the standard parametrization [26] has a form $V(x,\\theta ,\\overline{\\theta }) &=&C(x)+i\\theta \\chi -i\\overline{\\theta }\\overline{\\chi }+\\frac{i}{2}\\theta \\theta S-\\frac{i}{2}\\overline{\\theta }\\overline{\\theta }S^{\\ast } \\\\&&-\\theta \\sigma ^{\\mu }\\overline{\\theta }A_{\\mu }+i\\theta \\theta \\overline{\\theta }\\overline{\\lambda ^{\\prime }}-i\\overline{\\theta }\\overline{\\theta }\\theta \\lambda ^{\\prime }+\\frac{1}{2}\\theta \\theta \\overline{\\theta }\\overline{\\theta }D^{\\prime }, $ where its vector field component $A_{\\mu }$ is usually associated with a photon.",
"Note that, apart from the conventional photino field $\\lambda $ and the auxiliary $D$ field , the superfield (REF ) contains in general the additional degrees of freedom in terms of the dynamical $C$ and $\\chi $ fields and nondynamical complex scalar field $S$ (we have used the brief notations, $\\lambda ^{\\prime }=\\lambda +\\frac{i}{2}\\sigma ^{\\mu }\\partial _{\\mu }\\overline{\\chi }$ and $D^{\\prime }=D+\\frac{1}{2}\\square C$ with $\\sigma ^{\\mu }=(1,\\overrightarrow{\\sigma })$ and $\\overline{\\sigma }^{\\mu }=(1,-\\overrightarrow{\\sigma })$ ).",
"The corresponding SUSY invariant Lagrangian may be written as $\\mathcal {L}=L_{SQED}+\\sum _{n=1}b_{n}V^{n}|_{D} $ where terms in this sum ($b_{n}$ are some constants) for the vector superfield (REF ) are given through the polynomial $D$ -term $V^{n}|_{D}$ expansion into the component fields .",
"It can readily be checked that the first term in this expansion appears to be the known Fayet-Iliopoulos $D$ -term, while other terms only contain bilinear, trilinear and quadrilinear combination of the superfield components $A_{\\mu }$ , $S$ , $\\lambda $ and $\\chi $ , respectivelyNote that all terms in the sum in (REF ) except Fayet-Iliopoulos $D$ -term explicitly break gauge invariance which is then recovered in the SUSY broken phase (see below).",
"For simplicity, we could restrict ourselves to the third degree superfield polynomial potential in the Lagrangian $\\mathcal {L}$ (REF ) to eventually have a theory with dimesionless coupling constants in interactions of the component fields.",
"However, for completeness sake, we will proceed with a general superfield potential.. Actually, there appear higher-degree terms only for the scalar field component $C(x)$ .",
"Expressing them all in terms of the $C$ field polynomial $P(C)=\\sum _{n=1}\\frac{n}{2}b_{n}C^{n-1}(x) $ and its first three derivatives with respect to the $C$ field $P^{\\prime }\\equiv \\frac{\\partial P}{\\partial C}\\text{ , \\ \\ }P^{\\prime \\prime }\\equiv \\frac{\\partial ^{2}P}{\\partial C^{2}}\\text{ , \\ \\ }P^{\\prime \\prime \\prime }\\equiv \\frac{\\partial ^{3}P}{\\partial C^{3}}\\text{ }$ one has for the whole Lagrangian $\\mathcal {L}$ $\\mathcal {L} &=&-\\text{ }\\frac{1}{4}F^{\\mu \\nu }F_{\\mu \\nu }+i\\lambda \\sigma ^{\\mu }\\partial _{\\mu }\\overline{\\lambda }+\\frac{1}{2}D^{2} \\\\&&+\\text{ }P\\left( D+\\frac{1}{2}\\square C\\right) +P^{\\prime }\\left( \\frac{1}{2}SS^{\\ast }-\\chi \\lambda ^{\\prime }-\\overline{\\chi }\\overline{\\lambda ^{\\prime }}-\\frac{1}{2}A_{\\mu }A^{\\mu }\\right) \\\\&&+\\text{ }\\frac{1}{2}P^{\\prime \\prime }\\left( \\frac{i}{2}\\overline{\\chi }\\overline{\\chi }S-\\frac{i}{2}\\chi \\chi S^{\\ast }-\\chi \\sigma ^{\\mu }\\overline{\\chi }A_{\\mu }\\right) +\\frac{1}{8}P^{\\prime \\prime \\prime }(\\chi \\chi \\overline{\\chi }\\overline{\\chi })\\text{ .}",
"$ where, for more clarity, we still omitted matter superfields in the model reserving them for section 4.",
"One can see that the superfield component fields $C$ and $\\chi $ become dynamical due to the potential terms in (REF ) rather than from the properly constructed supersymmetric field strengths, as appear for the vector field $A_{\\mu }$ and its gaugino companion $\\lambda $ .",
"A very remarkable point is that the vector field $A_{\\mu }$ may only appear with bilinear mass terms in the polynomially extended Lagrangian (REF ).",
"Hence it follows that the “bumblebee” type model mentioned above (REF ) with nontrivial vector field potential containing both a bilinear mass term and a quadrilinear stabilizing term can in no way be realized in a SUSY context.",
"Meanwhile, the nonlinear QED model, as will become clear below, successfully matches supersymmetry.",
"Varying the Lagrangian $\\mathcal {L}$ with respect to the $D$ field we come to $D=-P(C) $ that finally gives the standard potential energy for for the field system considered $U(C)=\\frac{1}{2}P^{2}\\text{ } $ provided that other superfield field components do not develop VEVs.",
"The potential (REF ) may lead to the spontaneous SUSY breaking in the visible sector if the polynomial $P$ (REF ) has no real roots, while its first derivative has, $P\\ne 0\\text{ , \\ }P^{\\prime }=0.\\text{\\ } $ This requires $P(C)$ to be an even degree polynomial with properly chosen coefficients $b_{n}$ in (REF ) that will force its derivative $P^{\\prime }$ to have at least one root, $C=C_{0}$ , in which the potential (REF ) is minimized and supersymmetry is spontaneously broken.",
"As an immediate consequence, one can readily see from the Lagrangian $\\mathcal {L}$ (REF ) that a massless photino $\\lambda $ being Goldstone fermion in the broken SUSY phase make all the other component fields in the superfield $V(x,\\theta ,\\overline{\\theta })$ including the photon to also become massless.",
"However, the question then arises whether this masslessness of photon will be stable against radiative corrections since gauge invariance is explicitly broken in the Lagrangian (REF ).",
"We show below that it could be the case if the vector superfield $V(x,\\theta ,\\overline{\\theta })$ would appear to be properly constrained."
],
[
"Instability of superfield polynomial potential",
"Let us first analyze possible vacuum configurations for the superfield components in the polynomially extended QED case taken above.",
"In general, besides the \"standard\" potential energy expression (REF ) determined solely by the scalar field component $C(x)$ of the vector superfield (REF ), one also has to consider other field component contributions into the potential energy.",
"A possible extension of the potential energy (REF ) seems to appear only due to the pure bosonic field contributions, namely due to couplings of the vector and auxiliary scalar fields, $A_{\\mu }$ and $S$ , in (REF ) $\\mathcal {U}=\\frac{1}{2}P^{2}+\\frac{1}{2}P^{\\prime }(A_{\\mu }A^{\\mu }-SS^{\\ast })\\text{ } $ rather than due to the potential terms containing the superfield fermionic componentsActually, this restriction is not essential for what follows and is taken just for simplicity.",
"Generally, the fermion bilinears involved could also develop VEVs..",
"It can be immediately seen that these new couplings in (REF ) can make the potential unstable since the vector and scalar fields mentioned may in general develop any arbitrary VEVs.",
"This happens, as emphasized above, due the fact that their bilinear term contributions are not properly compensated by appropriate four-linear field terms which are generically absent in a SUSY theory context.",
"For more detail we consider the extremum conditions for the entire potential (REF ) with respect to all fields involved: $C$ , $A_{\\mu }$ and $S$ .",
"They are given by the appropriate first partial derivative equations $\\mathcal {U}_{C}^{\\prime } &=&PP^{\\prime }+\\frac{1}{2}P^{\\prime \\prime }(A_{\\mu }A^{\\mu }-SS^{\\ast })=0,\\text{ } \\\\\\mathcal {U}_{A_{\\mu }}^{\\prime } &=&P^{\\prime }A^{\\mu }=0,\\text{ \\ \\ }\\mathcal {U}_{S}^{\\prime }=-P^{\\prime }S^{\\ast }=0.",
"$ where and hereafter all the VEVs are denoted by the corresponding field symbols (supplied below with the lower index 0).",
"One can see that there can occur a local minimum for the potential (REF ) with the unbroken SUSY solutionHereafter by $P(C_{0})$ and $P^{\\prime }(C_{0})$ are meant the $C$ field polynomial $P$ (REF ) and its functional derivative $P^{\\prime }$ (REF ) taken in the potential extremum point $C_{0}$ .",
"$C=C_{0},\\text{ }P(C_{0})=0\\text{, }P^{\\prime }(C_{0})\\ne 0\\text{ };\\text{ \\ }A_{\\mu 0}=0,\\text{ \\ }S_{0}=0 $ with the vanishing potential energy $\\mathcal {U}_{\\min }^{s}=0 $ provided that the polynomial $P$ (REF ) has some real root $C=C_{0}$ .",
"Otherwise, a local minimum with the broken SUSY solution can occur for some other $C$ field value (though denoted by the same letter $C_{0}$ ) $C=C_{0},\\text{ }P(C_{0})\\ne 0\\text{, }P^{\\prime }(C_{0})=0\\text{ };\\text{ \\ }A_{\\mu 0}\\ne 0,\\text{ \\ }S_{0}\\ne 0,\\text{ }A_{\\mu 0}A_{0}^{\\mu }-S_{0}S_{0}^{\\ast }=0 $ In this case one has the non-zero potential energy $\\mathcal {U}_{\\min }^{as}=\\frac{1}{2}[P(C_{0})]^{2} $ as directly follows from the extremum equations (REF ) and potential energy expression (REF ).",
"However, as shows the standard second partial derivative test, the fact is that the local minima mentioned above are minima with respect to the $C$ field VEV ($C_{0}$ ) only.",
"Actually, for all three fields VEVs included the potential (REF ) has indeed saddle points with \"coordinates\" indicated in (REF ) and (REF ), respectively.",
"For a testing convenience this potential can be rewritten in the form $\\mathcal {U}=\\frac{1}{2}P^{2}+\\frac{1}{2}P^{\\prime }g^{\\Theta \\Theta ^{\\prime }}B_{\\Theta }B_{\\Theta ^{\\prime }}\\text{ , \\ }g^{\\Theta \\Theta ^{\\prime }}=diag\\text{ }(1,-1,-1,-1,-1,-1)\\text{\\ \\ \\ } $ with only two variable fields $C$ and $B_{\\Theta }$ where the new field $B_{\\Theta }$ unifies the $A_{\\mu }$ and $S$ field components, $B_{\\Theta }=(A_{\\mu },S_{a})$ ($\\Theta =\\mu ,a;$ $\\mu =0,1,2,3$ ; $a=1,2$ )Interestingly, the $B_{\\Theta }$ term in the potential (REF ) possesses the accidental $SO(1,5)$ symmetry.",
"This symmetry, though it is not shared by kinetic terms, appears in fact to be stable under radiative corrections since $S$ field is non-dynamical and, therefore, can always be properly arranged..",
"The complex $S$ field is now taken in a real basis, $S_{1}=(S+S^{\\ast })/\\sqrt{2}$ and $S_{2}=(S-S^{\\ast })/i\\sqrt{2}$ , so that the \"vector\" $B_{\\Theta }$ field has one time and five space components.",
"As a result, one finally comes to the following Hessian $7\\times 7$ matrix (being in fact the second-order partial derivatives matrix taken in the extremum point ($C_{0}$ , $A_{\\mu 0}$ , $S_{0}$ ) (REF )) $H(\\mathcal {U}^{s})=\\left[\\begin{array}{cc}[P^{\\prime }(C_{0})]^{2} & 0 \\\\0 & P^{\\prime }(C_{0})g^{\\Theta \\Theta ^{\\prime }}\\end{array}\\right] \\text{, \\ }\\left|H(\\mathcal {U}^{s})\\right|=-\\text{\\ }[P^{\\prime }(C_{0})]^{8}\\text{ .}",
"$ This matrix clearly has the negative determinant $\\left|H(\\mathcal {U}^{s})\\right|$ , as is indicated above, that confirms that the potential definitely has a saddle point for the solution (REF ).",
"This means the VEVs of the $A_{\\mu }$ and $S$ fields can take in fact any arbitrary value making the potential (REF , REF ) to be unbounded from below in the unbroken SUSY case that is certainly inaccessible.",
"One might think that in the broken SUSY case the situation would be better since due to the conditions (REF ) the $B_{\\Theta }$ term completely disappears from the potential $\\mathcal {U}$ (REF , REF ) in the ground state.",
"Unfortunately, the direct second partial derivative test in this case is inconclusive since the determinant of the corresponding Hessian $7\\times 7$ matrix appears to vanish $H(\\mathcal {U}^{as})=\\left[\\begin{array}{cc}P(C_{0})P^{\\prime \\prime }(C_{0}) & P^{\\prime \\prime }(C_{0})g^{\\Theta \\Theta ^{\\prime }}B_{\\Theta ^{\\prime }} \\\\P^{\\prime \\prime }(C_{0})g^{\\Theta \\Theta ^{\\prime }}B_{\\Theta ^{\\prime }} &0\\end{array}\\right] \\text{, \\ \\ }\\left|H(\\mathcal {U}^{as})\\right|=0\\text{ .",
"}$ Nevertheless, since in general the $B_{\\Theta }$ term can take both positive and negative values in small neighborhoods around the vacuum point ($C_{0}$ , $A_{\\mu 0}$ , $S_{0}$ ) where the conditions (REF ) are satisfied, this point is also turned out to be a saddle point.",
"Thus, the potential $\\mathcal {U}$ (REF , REF ) appears generically unstable both in SUSY invariant and SUSY broken phase."
],
[
"Stabilization of vacuum by constraining vector superfield",
"The only possible way to stabilize the ground states (REF ) and (REF ) seems to seek the proper constraints on the superfield component fields ($C$ , $A_{\\mu }$ , $S$ ) themselves rather than on their expectation values.",
"Indeed, if such (potential bounding) constraints are physically realizable, the vacua (REF ) and (REF ) will be automatically stabilized.",
"In a SUSY context a constraint can only be put on the entire vector superfield $V(x,\\theta ,\\overline{\\theta })$ (REF ) rather than individually on its field components.",
"Actually, we can constrain our vector superfield $V(x,\\theta ,\\overline{\\theta })$ by analogy with the constrained vector field in the nonlinear QED model (see (REF )).",
"This will be done again through some invariant Lagrange multiplier coupling simply adding its $D$ term to the above Lagrangian (REF , REF ) $\\mathcal {L}_{tot}=\\mathcal {L}+\\frac{1}{2}{\\large \\Lambda }(V-C_{0})^{2}|_{D}\\text{ ,} $ where ${\\large \\Lambda }(x,\\theta ,\\overline{\\theta })$ is some auxiliary vector superfield, while $C_{0}$ is the constant background value of the $C$ field for which potential $U$ (REF ) vanishes as is required for the supersymmetric minimum or has some nonzero value corresponding to the SUSY breaking minimum (REF ) in the visible sector.",
"We will consider both cases simultaneously using the same notation $C_{0}$ for either of the potential minimizing values of the $C$ field.",
"Note first of all, the Lagrange multiplier term in (REF ) has in fact the simplest possible form that leads to some nontrivial constrained superfield $V(x,\\theta ,\\overline{\\theta })$ .",
"The alternative minimal forms, such as the bilinear form ${\\large \\Lambda }(V-C_{0})$ or trilinear one ${\\large \\Lambda }(V^{2}-C_{0}^{2})$ , appear too restrictive.",
"One can easily confirm that they eliminate most component fields in the superfield $V(x,\\theta ,\\overline{\\theta })$ including the physical photon and photino fields that is definitely inadmissible.",
"As to appropriate non-minimal high linear multiplier forms, they basically lead to the same consequences as follow from the minimal multiplier term taken in the total Lagrangian (REF ).",
"Writing down its invariant $D$ term through the component fields one finds ${\\large \\Lambda }(V-C_{0})^{2}|_{D} &=&{\\large C}_{{\\large \\Lambda }}\\left[\\widetilde{C}D^{\\prime }+\\left( \\frac{1}{2}SS^{\\ast }-\\chi \\lambda ^{\\prime }-\\overline{\\chi }\\overline{\\lambda ^{\\prime }}-\\frac{1}{2}A_{\\mu }A^{\\mu }\\right) \\right] \\\\&&+\\text{ }{\\large \\chi }_{{\\large \\Lambda }}\\left[ 2\\widetilde{C}\\lambda ^{\\prime }+i(\\chi S^{\\ast }+i\\sigma ^{\\mu }\\overline{\\chi }A_{\\mu })\\right] +\\overline{{\\large \\chi }}_{{\\large \\Lambda }}[2\\widetilde{C}\\overline{\\lambda ^{\\prime }}-i(\\overline{\\chi }S-i\\chi \\sigma ^{\\mu }A_{\\mu })]\\\\&&+\\text{ }\\frac{1}{2}{\\large S}_{{\\large \\Lambda }}\\left( \\widetilde{C}S^{\\ast }+\\frac{i}{2}\\overline{\\chi }\\overline{\\chi }\\right) +\\frac{1}{2}{\\large S}_{{\\large \\Lambda }}^{\\ast }\\left( \\widetilde{C}S-\\frac{i}{2}\\chi \\chi \\right) \\\\&&+\\text{ }2{\\large A}_{{\\large \\Lambda }}^{\\mu }(\\widetilde{C}A_{\\mu }-\\chi \\sigma _{\\mu }\\overline{\\chi })+2{\\large \\lambda }_{{\\large \\Lambda }}^{\\prime }(\\widetilde{C}\\chi )+2\\overline{{\\large \\lambda }}_{{\\large \\Lambda }}^{\\prime }(\\widetilde{C}\\overline{\\chi })+\\frac{1}{2}{\\large D}_{{\\large \\Lambda }}^{\\prime }\\widetilde{C}^{2} $ where ${\\large C}_{{\\large \\Lambda }},\\text{ }{\\large \\chi }_{{\\large \\Lambda }},\\text{ }{\\large S}_{{\\large \\Lambda }},\\text{ }{\\large A}_{{\\large \\Lambda }}^{\\mu },\\text{ }{\\large \\lambda }_{{\\large \\Lambda }}^{\\prime }={\\large \\lambda }_{{\\large \\Lambda }}+\\frac{i}{2}\\sigma ^{\\mu }\\partial _{\\mu }\\overline{{\\large \\chi }}_{{\\large \\Lambda }},\\text{ }{\\large D}_{{\\large \\Lambda }}^{\\prime }={\\large D}_{{\\large \\Lambda }}+\\frac{1}{2}\\square {\\large C}_{{\\large \\Lambda }} $ are the component fields of the Lagrange multiplier superfield ${\\large \\Lambda }(x,\\theta ,\\overline{\\theta })$ in the standard parametrization (REF ) and $\\widetilde{C}$ stands for the difference $C(x)-C_{0}$ .",
"Varying the Lagrangian (REF ) with respect to these fields and properly combining their equations of motion $\\frac{\\partial \\mathcal {L}_{tot}}{\\partial \\left( {\\large C}_{{\\large \\Lambda }},{\\large \\chi }_{{\\large \\Lambda }},{\\large S}_{{\\large \\Lambda }},{\\large A}_{{\\large \\Lambda }}^{\\mu },{\\large \\lambda }_{{\\large \\Lambda }},{\\large D}_{{\\large \\Lambda }}\\right) }=0$ we find the constraints which appear to put on the $V$ superfield components $C=C_{0},\\text{ \\ }\\chi =0,\\text{\\ \\ }A_{\\mu }A^{\\mu }=SS^{\\ast }\\text{.",
"}$ Again, as before in non-SUSY case (REF ), we only take a solution with initial values for all fields (and their momenta) chosen so as to restrict the phase space to vanishing values of the multiplier component fields (REF ) that will provide a ghost-free theory with a positive HamiltonianAs in the non-supersymmetric case discussed above (see footnote$^{1}$ ), this solution with all vanishing components of the basic Lagrangian multiplier superfield ${\\large \\Lambda }(x,\\theta ,\\overline{\\theta })$ can be reached by introducing some extra Lagrange multiplier term..",
"Remarkably, the constraints (REF ) does not touch the physical degrees of freedom of the superfield $V(x,\\theta ,\\overline{\\theta })$ related to photon and photino fields.",
"The point is, however, that apart from the constraints (REF ), one has the equations of motion for all fields involved in the basic superfield $V(x,\\theta ,\\overline{\\theta })$ .",
"With vanishing multiplier component fields (REF ), as was proposed above, these equations appear in fact as extra constraints on components of the superfield $V(x,\\theta ,\\overline{\\theta })$ .",
"Indeed, equations of motion for the fields $C$ , $S$ and ${\\large \\chi }$ received by the corresponding variations of the total Lagrangian $\\mathcal {L}$ (REF ) are turned out to be, respectively, $P(C_{0})P^{\\prime }(C_{0})=0,\\text{ \\ }S(x)P^{\\prime }(C_{0})=0\\text{ , \\ }\\lambda (x)P^{\\prime }(C_{0})=0 $ where the basic constraints (REF ) emerging at the potential extremum point $C=C_{0}$ have also been used.",
"One can immediately see now that these equations turn to trivial identities in the broken SUSY case, in which the factor $P^{\\prime }(C_{0})$ in each of them appears to be identically vanished, $P^{\\prime }(C_{0})$ $=0$ (REF ).",
"In the unbroken SUSY case, in which the potential (REF ) vanishes instead, i.e.",
"$P(C_{0})=0$ (REF ), the situation is drastically changed.",
"Indeed, though the first equation in (REF ) still automatically turns into identity at the extremum point $C(x)=C_{0}$ , other two equations require that the auxiliary field $S$ and photino field $\\lambda $ have to be identically vanished as well.",
"This causes in turn that the photon field should also be vanished according to the basic constraints (REF ).",
"Besides, the $D$ field component in the vector superfield is also vanished in the unbroken SUSY case according to the equation (REF ), $D=-P(C_{0})=0 $ .",
"Thus, one is ultimately left with a trivial superfield $V(x,\\theta ,\\overline{\\theta })$ which only contains the constant $C$ field component $C_{0}$ that is unacceptable.",
"So, we have to conclude that the unbroken SUSY fails to provide stability of the potential (REF ) even by constraining the superfield $V(x,\\theta ,\\overline{\\theta })$ .",
"In contrast, in the spontaneously broken SUSY case extra constraints do not appear at all, and one has a physically meaningful theory that we basically consider in what follows.",
"Actually, substituting the constraints (REF ) into the total Lagrangian $\\mathcal {L}_{tot}$ (REF , REF ) we eventually come to the emergent SUSY QED appearing in the broken SUSY phase $\\mathcal {L}_{tot}^{^{\\mathbf {em}}}=-\\text{ }\\frac{1}{4}F^{\\mu \\nu }F_{\\mu \\nu }+i\\lambda \\sigma ^{\\mu }\\partial _{\\mu }\\overline{\\lambda }+\\frac{1}{2}D^{2}+P(C_{0})D\\text{ , \\ }A_{\\mu }A^{\\mu }=SS^{\\ast } $ supplemented by the vector field constraint as its vacuum stability condition.",
"Remarkably, for the constrained vector superfield involved $\\widehat{V}(x,\\theta ,\\overline{\\theta })=C_{0}+\\frac{i}{2}\\theta \\theta S-\\frac{i}{2}\\overline{\\theta }\\overline{\\theta }S^{\\ast }-\\theta \\sigma ^{\\mu }\\overline{\\theta }A_{\\mu }+i\\theta \\theta \\overline{\\theta }\\overline{\\lambda }-i\\overline{\\theta }\\overline{\\theta }\\theta \\lambda +\\frac{1}{2}\\theta \\theta \\overline{\\theta }\\overline{\\theta }D, $ we have the almost standard SUSY QED Lagrangian with the same states - photon, photino and an auxiliary scalar $D$ field - in its gauge supermultiplet, while another auxiliary complex scalar field $S$ gets only involved in the vector field constraint.",
"The linear (Fayet-Iliopoulos) $D$ -term with the effective coupling constant $P(C_{0})$ in (REF ) shows that the supersymmetry in the theory is spontaneously broken due to which the $D$ field acquires the VEV, $D=-P(C_{0})$ .",
"Taking the nondynamical $S$ field in the constraint (REF ) to be some constant background field (for a more formal discussion, see below) we come to the SLIV constraint (REF ) which we discussed above regarding an ordinary non-supersymmetric QED theory (section1).",
"As is seen from this constraint in (REF ), one may only have the time-like SLIV in a SUSY framework but never the space-like one.",
"There also may be a light-like SLIV, if the $S$ field vanishesIndeed, this case, first mentioned in [13], may also mean spontaneous Lorentz violation with a nonzero VEV $<A_{\\mu }>$ $=(\\widetilde{M},0,0,\\widetilde{M})$ and Goldstone modes $A_{1,2}$ and $(A_{0}+A_{3})/2$ $-\\widetilde{M}.$ The \"effective\" Higgs mode $(A_{0}-A_{3})/2$ can be then expressed through Goldstone modes so as the light-like condition $A_{\\mu }^{2}=0$ to be satisfied..",
"So, any possible choice for the $S$ field corresponds to the particular gauge choice for the vector field $A_{\\mu }$ in an otherwise gauge invariant theory."
],
[
"Constrained superfield: a formal view",
"We conclude this section by showing that the extended Lagrangian $\\mathcal {L}_{tot}$ (REF , REF ), underlying the emergent QED model described above, as well as the vacuum stability constraints on the superfield component fields (REF ) appearing due to the Lagrange multiplier term in (REF ) are consistent with supersymmetry.",
"The first part of this assertion is somewhat immediate since the Lagrangian $\\mathcal {L}_{tot}$ , aside from the standard supersymmetric QED part $L_{SQED}$ (REF ), only contains $D$ -terms of various vector superfield products.",
"They are, by definition, invariant under conventional SUSY transformations [26] which for the component fields (REF ) of a general superfield $V(x,\\theta ,\\overline{\\theta })$ (REF ) are written as $\\delta _{\\xi }C &=&i\\xi \\chi -i\\overline{\\xi }\\overline{\\chi }\\text{ , \\ }\\delta _{\\xi }\\chi =\\xi S+\\sigma ^{\\mu }\\overline{\\xi }(\\partial _{\\mu }C+iA_{\\mu })\\text{ , \\ }\\frac{1}{2}\\delta _{\\xi }S=\\overline{\\xi }\\overline{\\lambda }+\\overline{\\sigma }_{\\mu }\\partial ^{\\mu }\\chi \\text{ ,} \\\\\\delta _{\\xi }A_{\\mu } &=&\\xi \\partial _{\\mu }\\chi +\\overline{\\xi }\\partial _{\\mu }\\overline{\\chi }+i\\xi \\sigma _{\\mu }\\overline{\\lambda }-i\\lambda \\sigma _{\\mu }\\overline{\\xi }\\text{ , \\ }\\delta _{\\xi }\\lambda =\\frac{1}{2}\\xi \\sigma ^{\\mu }\\overline{\\sigma }^{\\nu }F_{\\mu \\nu }+\\xi D\\text{ ,}\\\\\\delta _{\\xi }D &=&-\\xi \\sigma ^{\\mu }\\partial _{\\mu }\\overline{\\lambda }+\\overline{\\xi }\\sigma ^{\\mu }\\partial _{\\mu }\\lambda \\text{ .}",
"$ However, there may still be left a question whether supersymmetry remains in force when the constraints (REF ) on the field space are \"switched on\" thus leading to the final Lagrangian $\\mathcal {L}_{tot}^{^{\\mathbf {em}}}$ (REF ) in the broken SUSY phase with both dynamical fields $C$ and $\\chi $ eliminated.",
"This Lagrangian appears similar to the standard supersymmetric QED taken in the Wess-Zumino gauge, except that the supersymmetry is spontaneously broken in our case.",
"In both cases the photon stress tensor $F_{\\mu \\nu }$ , photino $\\lambda $ and nondynamical scalar $D$ field form an irreducible representation of the supersymmetry algebra (the last two line in (REF )).",
"Nevertheless, any reduction of component fields in the vector superfield is not consistent in general with the linear superspace version of supersymmetry transformations, whether it is the Wess-Zumino gauge case or our constrained superfield $\\widehat{V}$ (REF ).",
"Indeed, a general SUSY transformation does not preserve the Wess-Zumino gauge: a vector superfield in this gauge acquires some extra terms when being SUSY transformed.",
"The same occurs with our constrained superfield $\\widehat{V}$ as well.",
"The point, however, is that in both cases a total supergauge transformation $V\\rightarrow V+i(\\Omega -\\Omega ^{\\ast })\\text{ ,} $ where $\\Omega $ is a chiral superfield gauge transformation parameter, can always restore a superfield initial form.",
"Actually, the only difference between these two cases is that whereas the Wess-Zumino supergauge leaves an ordinary gauge freedom untouched, in our case this gauge is unambiguously fixed in terms of the above vector field constraint (REF ).",
"However, this constraint remains under supergauge transformation (REF ) applied to our superfield $\\widehat{V}$ (REF ).",
"Indeed, the essential part of this transformation which directly acts on the constraint (REF ) has the form $\\widehat{V}\\rightarrow \\widehat{V}+i\\theta \\theta F-i\\overline{\\theta }\\overline{\\theta }F^{\\ast }-2\\theta \\sigma ^{\\mu }\\overline{\\theta }\\partial _{\\mu }\\varphi \\text{ .}",
"$ where the real and complex scalar field components, $\\varphi $ and $F$ , in a chiral superfield parameter $\\Omega $ are properly activated.",
"As a result, the vector and scalar fields, $A_{\\mu }$ and $S$ , in the supermultiplet $\\widehat{V}$ (REF ) transform as $A_{\\mu }\\rightarrow A_{\\mu }^{\\prime }=A_{\\mu }-\\partial _{\\mu }(2\\varphi )\\text{ , \\ \\ }S\\rightarrow S^{\\prime }=S+2F\\text{ .}",
"$ It can be immediately seen that our basic Lagrangian $\\mathcal {L}_{tot}^{^{\\mathbf {em}}}$ (REF ) being gauge invariant and containing no the scalar field $S$ is automatically invariant under either of these two transformations individually.",
"In contrast, the supplementary vector field constraint (REF ), though it is also turned out to be invariant under supergauge transformations (REF ), but only if they are made jointly.",
"Indeed, for any choice of the scalar $\\varphi $ in (REF ) there can always be found such a scalar $F$ (and vice versa) that the constraint remains invariant $A_{\\mu }A^{\\mu }=SS^{\\ast }\\rightarrow A_{\\mu }^{\\prime }A^{\\prime \\mu }=S^{\\prime }S^{\\prime \\ast } $ In other words, the vector field constraint is invariant under supergauge transformations (REF ) but not invariant under an ordinary gauge transformation.",
"As a result, in contrast to the Wess-Zumino case, the supergauge fixing in our case will also lead to the ordinary gauge fixing.",
"We will use this supergauge freedom to reduce the $S$ field to some constant background value and find the final equation for the gauge function $\\varphi (x)$ .",
"So, for the parameter field $F$ chosen in such a way to have $S^{\\prime }=S+2F=Me^{i\\alpha (x)}\\text{ }, $ where $M$ is some constant mass parameter (and $\\alpha (x)$ is an arbitrary phase), we come in (REF ) to $(A_{\\mu }-2\\partial _{\\mu }\\varphi )(A^{\\mu }-2\\partial ^{\\mu }\\varphi )=M^{2}\\text{ .}",
"$ that is precisely our old SLIV constraint (REF ) being varied by the gauge transformation (REF ).",
"Recall that this constraint, as was thoroughly discussed in Introduction (subsection 1.1), only fixes gauge (to which such a gauge function $\\varphi (x)$ has to satisfy), rather than physically breaks gauge invariance.",
"Notably, in contrast to the non-SUSY case where this constraint was merely postulated, it now follows from the vacuum stability and supergauge invariance in the emergent SUSY QED.",
"Besides, this constraint, as mentioned above, may only be time-like (and light-like if the mass parameter $M$ is taken to be zero).",
"When such inactive time-like SLIV is properly developed one come to the essentially nonlinear emergent SUSY QED in which the physical photon arises as a three-dimensional Lorentzian NG mode (just as is in non-SUSY case for the time-like SLIV, see subsection 1.1).",
"To finalize, it was shown that the vacuum stability constraints (REF ) on the allowed configurations of the physical fields in a general polynomially extended Lagrangian (REF ) appear entirely consistent with supersymmetry.",
"In the broken SUSY phase one eventually comes to the standard SUSY QED type Lagrangian (REF ) being supplemented by the vector field constraint which is invariant under supergauge transformations.",
"One might think that, unlike the gauge invariant linear (Fayet-Iliopoulos) superfield term, the quadratic and higher order superfield terms in the starting Lagrangian (REF ) would seem to break gauge invariance.",
"However, this fear proves groundless.",
"Actually, as was shown above, this breaking amounts to the gauge fixing determined by the nonlinear vector field constraint (REF ).",
"It is worth noting that this constraint formally follows from the SUSY invariant Lagrange multiplier term in (REF ) for which is required the phase space to be restricted to vanishing values of all the multiplier component fields (REF ).",
"The total vanishing of the multiplier superfield provides the SUSY invariance of such restrictions.",
"Any non-zero multiplier component field left in the Lagrangian would immediately break supersymmetry and, even worse, would eventually lead to ghost modes in the theory and a Hamiltonian unbounded from below."
],
[
"Broken SUSY phase: photino as pseudo-goldstino",
"Let us now turn to matter superfields which have not yet been included in the model.",
"In their presence spontaneous SUSY breaking in the visible sector, which fundamentally underlies our approach, might be phenomenologically ruled out by the well-known supertrace sum rule [26] for actual masses of quarks and leptons and their superpartnersNote that an inclusion of direct soft mass terms for scalar superpartners in the model would mean in general that the visible SUSY sector is explicitly, rather than spontaneosly, broken that could immediately invalidate the whole idea of the massless photons as the zero Lorentzian modes triggered by the spontaneously broken supersymmetry..",
"However, this sum rule is acceptably relaxed when taking into account large radiative corrections to masses of supersymmetric particles that proposedly stems from the hidden sector.",
"This is just what one may expect in conventional supersymmetric theories with the standard two-sector paradigm, according to which SUSY breaking entirely occurs in a hidden sector and then this spontaneous breaking is mediated to the visible sector by some indirect interactions whose nature depends on a particular mediation scenario [26].",
"An emergent QED approach advocated here requires some modification of this idea in such a way that, while a hidden sector is largely responsible for spontaneous SUSY breaking, supersymmetry can also be spontaneously broken in the visible sector that ultimately leads to a double spontaneous SUSY breaking pattern.",
"We may suppose, just for uniformity, only $D$ -term SUSY breaking both in the visible and hidden sectorsIn general, both $D$ - and $F$ -type terms can be simultaneously used in the visible and hidden sectors (usually just $F$ -term SUSY breaking is used in both sectors [26])..",
"Properly, our supersymmetric QED model may be further extended by some extra local $U^{\\prime }(1)$ symmetry which is proposed to be broken at very high energy scale $M^{\\prime }$ (for some appropriate anomaly mediated scenario, see [27] and references therein).",
"It is natural to think that due to the decoupling theorem all effects of the $U^{\\prime }(1)$ are suppressed at energies $E<<M^{\\prime }$ by powers of $1/M^{\\prime }$ and only the $D^{\\prime }$ -term of the corresponding vector superfield $V^{\\prime }(x,\\theta ,\\overline{\\theta })$ remains in essence when going down to low energies.",
"Actually, this term with a proper choice of messenger fields and their couplings naturally provides the $M_{SUSY\\text{ \\ }}$ order contributions to masses of scalar superpartners.",
"As a result, the simplified picture discussed above (in sections 2 and 3) is properly changed: a strictly massless fermion eigenstate, the true goldstino $\\zeta _{g}$ , should now be some mix of the visible sector photino $\\lambda $ and the hidden sector goldstino $\\lambda ^{\\prime }$ $\\zeta _{g}=\\frac{\\left\\langle D\\right\\rangle \\lambda +\\left\\langle D^{\\prime }\\right\\rangle \\lambda ^{\\prime }}{\\sqrt{\\left\\langle D\\right\\rangle ^{2}+\\left\\langle D^{\\prime }\\right\\rangle ^{2}}}\\text{ .}",
"$ where $\\left\\langle D\\right\\rangle $ and $\\left\\langle D^{\\prime }\\right\\rangle $ are the corresponding $D$ -component VEVs in the visible and hidden sectors, respectively.",
"Another orthogonal combination of them may be referred to as the pseudo-goldstino $\\zeta _{pg}$ , $\\zeta _{pg}=\\frac{\\left\\langle D^{\\prime }\\right\\rangle \\lambda -\\left\\langle D\\right\\rangle \\lambda ^{\\prime }}{\\sqrt{\\left\\langle D\\right\\rangle ^{2}+\\left\\langle D^{\\prime }\\right\\rangle ^{2}}}\\text{ .",
"}$ In the supergravity context, the true goldstino $\\zeta _{g}$ is eaten through the super-Higgs mechanism to form the longitudinal component of the gravitino, while the pseudo-goldstino $\\zeta _{pg}$ gets some mass proportional to the gravitino mass from supergravity effects.",
"Due to large soft masses required to be mediated, one may generally expect that SUSY is much stronger broken in the hidden sector than in the visible one, $\\left\\langle D^{\\prime }\\right\\rangle >>$ $\\left\\langle D\\right\\rangle $ , that means in turn the pseudo-goldstino $\\zeta _{pg}$ is largely the photino $\\lambda ,$ $\\zeta _{pg}\\simeq \\lambda \\text{ .",
"}$ These pseudo-goldstonic photinos seem to be of special observational interest in the model that, apart from some indication of the QED emergence nature, may shed light on SUSY breaking physics.",
"The possibility that the supersymmetric Standard Model visible sector might also spontaneously break SUSY thus giving rise to some pseudo-goldstino state was also considered, though in a different context, in [28], [29].",
"Interestingly enough, our polynomially extended SQED Lagrangian (REF ) is not only SUSY invariant but also generically possesses a continuous $R$ -symmetry $U(1)_{R}$ [26].",
"Indeed, vector superfields always have zero $R$ -charge, since they are real.",
"Accordingly, it follows that the physical field components in the constrained vector superfield $\\widehat{V}$ (REF ) transform as $A_{\\mu }\\rightarrow A_{\\mu }\\text{ , \\ }\\lambda \\rightarrow e^{i\\alpha }\\lambda \\text{ , \\ }D\\rightarrow D $ and so have $R$ charges 0, 1 and 0, respectively.",
"Along with that, we assume a suitable $R$ -symmetric matter superfield setup as well making a proper $R$ -charge assignment for basic fermions and scalars (and messenger fields) involved.",
"This will lead to the light pseudo-goldstino in the gauge-mediated scenario.",
"Indeed, if the visible sector possesses an $R$ -symmetry which is preserved in the course of mediation the pseudo-goldstino mass is protected up to the supergravity effects which violate an $R$ -symmetry.",
"As a result, the pseudo-goldstino mass appears proportional to the gravitino mass, and, eventually, the same region of parameter space simultaneously solves both gravitino and pseudo-goldstino overproduction problems in the early universe [29].",
"Apart from cosmological problems, many other sides of new physics related to pseudo-goldstinos appearing through the multiple SUSY breaking were also studied recently (see [28], [29], [30] and references therein).",
"The point, however, is that there have been exclusively used non-vanishing $F$ -terms as the only mechanism of the visible SUSY breaking in models considered.",
"In this connection, our pseudo-goldstonic photinos solely caused by non-vanishing $D$ -terms in the visible SUSY sector may lead to somewhat different observational consequences.",
"One of the most serious differences may be related to the Higgs boson decays when the present SUSY QED is further extended to the supersymmetric Standard Model.",
"For the cosmologically safe masses of pseudo-goldstino and gravitino ($\\lesssim $ $1keV$ , as typically follows from the $R$ -symmetric gauge mediation) these decays are appreciably modified.",
"Actually, the dominant channel becomes the conversion of the Higgs boson (say, the lighter CP-even Higgs boson $h^{0}$ ) into a conjugated pair of corresponding pseudo-sgoldstinos $\\phi _{pg}$ and $\\overline{\\phi }_{pg}$ (being superpartners of pseudo-goldstinos $\\zeta _{pg}$ and $\\overline{\\zeta }_{pg}$ , respectively), $h^{0}\\rightarrow \\phi _{pg}+\\overline{\\phi }_{pg}\\text{ ,}$ once it is kinematically allowed.",
"This means that the Higgs boson will dominantly decay invisibly for $F$ -term SUSY breaking in a visible sector [29].",
"By contrast, for the $D$ -term SUSY breaking case considered here the roles of pseudo-goldstino and pseudo-sgoldstino are just played by photino and photon, respectively, that could make the standard two-photon decay channel of the Higgs boson to be even somewhat enhanced.",
"In the light of recent discovery of the Higgs-like state [31] just through its visible decay modes, the $F$ -term SUSY breaking in the visible sector seems to be disfavored by data, while $D$ -term SUSY breaking is not in trouble with them."
],
[
"Concluding remarks",
"It is well known that spontaneous Lorentz violation in general vector field theories may lead to an appearance of massless Nambu-Goldstone modes which are identified with photons and other gauge fields in the Standard Model.",
"Nonetheless, it may turn out that SLIV is not the only reason for emergent massless photons to appear, if spacetime symmetry is further enlarged.",
"In this connection, a special interest may be related to supersymmetry and its possible theoretical and observational relation to SLIV.",
"To see how such a scenario may work we have considered supersymmetric QED model extended by an arbitrary polynomial potential of a general vector superfield $V(x,\\theta ,\\overline{\\theta })$ whose pure vector field component $A_{\\mu }(x)$ is associated with a photon in the Lorentz invariant phase.",
"Gauge noninvariant couplings other than potential terms are not included into the theory.",
"For the theory in which gauge invariance is not required from the outset this is in fact the simplest generalization of a conventional SUSY QED.",
"This superfield potential (REF ) is turned out to be generically unstable unless SUSY is spontaneously broken.",
"However, it appears not to be enough.",
"To provide an overall stability of the potential one additionally needs the special direct constraint being put on the vector superfield itself that is made by an appropriate SUSY invariant Lagrange multiplier term (REF ).",
"Remarkably enough, when this term is written in field components it leads precisely to the nonlinear $\\sigma $ -model type constraint of type (REF ) which one has had in the non-SUSY case.",
"So, we come again to the picture, which we called the inactive SLIV, with a Goldstone-like photon and special (SLIV restricted) gauge invariance providing the cancellation mechanism for physical Lorentz violation.",
"But now this picture follows from the vacuum stability and supergauge invariance in the extended SUSY QED rather than being postulated as is in the non-SUSY case.",
"This allows to think that a generic trigger for massless photons to dynamically emerge happens to be spontaneously broken supersymmetry rather than physically manifested Lorentz noninvariance.",
"In more exact terms, in the broken SUSY phase one eventually comes to the almost standard SUSY QED Lagrangian (REF ) possessing some special gauge invariance emerged.",
"This invariance is only restricted by the gauge condition put on the vector field, $A_{\\mu }A^{\\mu }=|S|^{2}$ , which appears to be invariant under supergauge transformations.",
"One can use this supergauge freedom to reduce the nondynamical scalar field $S$ to some constant background value so as to eventually come to the nonlinear vector field constraint (REF ).",
"As a result, the inactive time-like SLIV is properly developed, thus leading to essentially nonlinear emergent SUSY QED in which the physical photon arises as a three-dimensional Lorentzian NG mode.",
"So, figuratively speaking, the photon passes through three evolution stages being initially the massive vector field component of a general vector superfield (REF ), then the three-level massless companion of an emergent photino in the broken SUSY stage (REF ) and finally a generically massless state as an emergent Lorentzian mode in the inactive SLIV stage (REF ).",
"As to an observational status of emergent SUSY theories, one can see that, as in an ordinary QED, physical Lorentz invariance is still preserved in the SUSY QED model at the renormalizable level and can only be violated if some extra gauge noninvariant couplings (being supersymmetric analogs of the high-dimension couplings (REF )) are included into the theory.",
"However, one may have some specific observational evidence in favor of the inactive SLIV even in the minimal (gauge invariant) supersymmetric QED and Standard Model.",
"Indeed, since as mentioned above the vacuum stability is only possible in spontaneously broken SUSY case, this evidence is related to an existence of an emergent goldstino-photino type state in the SUSY visible sector.",
"Being mixed with another goldstino appearing from a spontaneous SUSY violation in the hidden sector this state largely turns into the light pseudo-goldstino.",
"Its study seem to be of special observational interest for this class of models that, apart from some indication of an emergence nature of QED and the Standard Model, may appreciably extend the scope of SUSY breaking physics being actively studied in recent years.",
"We may return to this important issue elsewhere."
],
[
"Acknowledgments",
"I thank Colin Froggatt, Alan Kostelecky, Rabi Mohapatra and Holger Nielsen for stimulating discussions and correspondence.",
"Discussions with the participants of the International Workshop ”What Comes Beyond the Standard Models?” (14–21 July 2013, Bled, Slovenia) are also appreciated.",
"This work was supported in part by the Georgian National Science Foundation under grant No.",
"31/89."
]
] | 1403.0436 |
[
[
"Magellan Adaptive Optics first-light observations of the exoplanet\n $\\beta$ Pic b. I. Direct imaging in the far-red optical with MagAO+VisAO and\n in the near-IR with NICI"
],
[
"Abstract We present the first ground-based CCD ($\\lambda < 1\\mu$m) image of an extrasolar planet.",
"Using MagAO's VisAO camera we detected the extrasolar giant planet (EGP) $\\beta$ Pictoris b in $Y$-short ($Y_S$, 0.985 $\\mu$m), at a separation of $0.470 \\pm 0.010''$ and a contrast of $(1.63 \\pm 0.49) \\times 10^{-5}$.",
"This detection has a signal-to-noise ratio of 4.1, with an empirically estimated upper-limit on false alarm probability of 1.0%.",
"We also present new photometry from the NICI instrument on the Gemini-South telescope, in $CH_{4S,1\\%}$ ($1.58$ $\\mu m$), $K_S$ ($2.18\\mu m$), and $K_{cont}$ (2.27 $\\mu m$).",
"A thorough analysis of our photometry combined with previous measurements yields an estimated near-IR spectral type of L$2.5\\pm1.5$, consistent with previous estimates.",
"We estimate log$(L_{bol}/L_{Sun})$ = $-3.86 \\pm 0.04$, which is consistent with prior estimates for $\\beta$ Pic b and with field early-L brown dwarfs.",
"This yields a hot-start mass estimate of $11.9 \\pm 0.7$ $M_{Jup}$ for an age of $21\\pm4$ Myr, with an upper limit below the deuterium burning mass.",
"Our $L_{bol}$ based hot-start estimate for temperature is $T_{eff}=1643\\pm32$ K (not including model dependent uncertainty).",
"Due to the large corresponding model-derived radius of $R=1.43\\pm0.02$ $R_{Jup}$, this $T_{eff}$ is $\\sim$$250$ K cooler than would be expected for a field L2.5 brown dwarf.",
"Other young, low-gravity (large radius), ultracool dwarfs and directly-imaged EGPs also have lower effective temperatures than are implied by their spectral types.",
"However, such objects tend to be anomalously red in the near-IR compared to field brown dwarfs.",
"In contrast, $\\beta$ Pic b has near-IR colors more typical of an early-L dwarf despite its lower inferred temperature."
],
[
"Introduction",
"In contrast to the stellar main sequence, brown dwarfs (BDs) form a true evolutionary sequence.",
"BDs are not massive enough to maintain a constant effective temperature ($T_{eff}$ ) via hydrogen fusion [31].",
"Thus, a BD cools as it ages, radiating away the gravitational potential energy from its formation [29].",
"BDs are classified into spectral types by comparison to anchor objects.",
"Various clues to classification were judiciously chosen such that they should correspond to temperature, at least in a relative sense [69], [25], [24].",
"Temperature is not a readily observable quantity, however it is well established from theory that substellar objects ranging in mass from $\\sim $ 1 to $\\sim $ 75 $M_{Jup}$ will have radius in a narrow range of $0.8-1.1$ $R_{Jup}$ [28], [57].",
"This means that bolometric luminosity, $L_{bol} = 4\\pi \\sigma _B R^2 T_{eff}^4$ , is approximately determined by temperature alone.",
"$L_{bol}$ is observable, so with our theoretical understanding of radius we can infer $T_{eff}$ and find that field brown dwarf spectral types appear to be a well-defined temperature sequence [62], [124], except perhaps for the coolest objects [50].",
"The result is that the SpT of a brown dwarf is a function of both mass and age.",
"The situation is even more challenging for young objects, which have not completed post-formation contraction.",
"The radius of such an object can be significantly larger, depending on how it formed [31], [33], [6], [92], [122].",
"Young objects will also have lower mass than older objects of the same temperature.",
"With lower mass and larger radius these young objects have lower surface gravity (low-g), which changes their spectral morphology [88], [67], but even so their spectra can generally be classified within the ultracool dwarf sequence [44], [2].",
"This population of such low-g BDs is especially interesting because they potentially serve as analogs for young extrasolar giant planets (EGPs).",
"Many of the best studied low-g BDs are companions, such as AB Pic B [35], and 2M0122 B [21].",
"Examples of isolated low-g objects are 2M0355 [54], and PSO318.5 [86].",
"These objects tend to have fainter near-IR absolute magnitudes [54], [82], and have $T_{eff}$ several hundred K cooler than field BDs of the same SpT [21], [86].",
"These low-g BDs also tend to be much redder in near-IR colors, and despite being fainter in the bluer filters and having lower $T_{eff}$ , their bolometric luminosities tend to be consistent with the field for their spectral types [86].",
"The first handful of directly imaged planets show similar properties, highlighting the challenges of studying substellar objects in the new physical regime of low-g. For instance, the EGP HR 8799 b and the planetary mass companion 2M1207 b have L-dwarf like very red near-IR colors, but their luminosities and inferred temperatures (800-1000 K) are more like mid-T dwarfs [34], [95].",
"This has been interpreted as a consequence of the gravity dependence of the L-T transition [98].",
"Thick dust clouds, which cause the redward progression of the L dwarf sequence as temperature drops, persist to even lower temperatures at low-g [116], [9], [93].",
"In this framework extremely red and under-luminous HR 8799 b and 2M1207 b are objects which have yet to make the transition to the cloudless, bluer, T dwarf sequence, hence they are often thought of as extensions of the L dwarf sequence [20], [8], [90].",
"The directly imaged EGP $\\beta \\mbox{ Pic b}$ , in contrast, is much hotter and its near-IR SED is much more typical when compared to field and low-g BDs [78], [107], [17], [45], [19], [47].",
"$\\beta \\mbox{ Pic b}$ is unique among the directly imaged EGPs in that we have a dynamical constraint on its mass from radial velocity (RV).",
"A complete orbit has not yet been observed, so RV monitoring constrains the mass depending on the semi-major axis ($a$ ): for $a < 8, 9, 10, 11, 12$ AU the upper mass limit is $M<10,12,15.5, 20, 25$ $M_{Jup}$ respectively [77].",
"The astrometry currently favors $8 \\lesssim a \\lesssim 9$ AU [36], hence $M\\lesssim 12$ $M_{Jup}$ , though larger values are not ruled out.",
"We can expect to have a good dynamical understanding of $\\beta \\mbox{ Pic b}$ 's mass in the near future.",
"$\\beta \\mbox{ Pic b}$ is also noteworthy in that we have relatively good constraints on the age of its primary star.",
"The age of the $\\beta $ Pictoris moving group, of which $\\beta \\mbox{ Pic A}$ is the eponymous member, has recently been revised upward to $21\\pm 4$ Myr [13] using the lithium depletion boundary technique.",
"Though somewhat larger than the earlier age estimate of $12^{+8}_{-4}$ Myr by [133], these two estimates are consistent at the $1\\sigma $ level.",
"A well determined age and a dynamical mass constraint make $\\beta \\mbox{ Pic b}$ a valuable benchmark for understanding the formation and evolution of both low-mass BDs and giant planets.",
"Here we present the bluest observations of $\\beta \\mbox{ Pic b}$ from the first light of the Magellan Adaptive Optics (MagAO) system, using its visible wavelength imager VisAO.",
"We also present detections with the Gemini Near Infrared Coronagraphic Imager (NICI).",
"In Section we describe MagAO and VisAO, and briefly discuss calibrations of this new high contrast imaging system.",
"In Section we present our observations and data reduction procedures.",
"We analyze the $0.9-2.4$ $\\mu m$ SED of $\\beta \\mbox{ Pic b}$ in Section , showing that this EGP looks like a typical early L dwarf.",
"We explore the ramifications of this for the physical properties ($L_{bol}$ , mass, $T_{eff}$ , and radius) of the planet.",
"Then in Section , we compare these derived properties to field objects, and discuss the relationship of the measured characteristics of EGPs and BDs.",
"Finally, we summarize our conclusions in Section ."
],
[
"The Magellan VisAO Camera ",
"MagAO is a 585-actuator adaptive secondary mirror (ASM) and pyramid wavefront sensor (PWFS) adaptive optics (AO) system, installed at the 6.5 m Magellan Clay Telescope at Las Campanas Observatory (LCO), Chile.",
"The system is a near clone of the Large Binocular Telescope (LBT) AO systems [52], [53].",
"MagAO has two science cameras: the Clio2 $1-5\\mu \\mbox{m}$ camera [58], [115] and the VisAO $0.5-1\\mu \\mbox{m}$ camera.",
"Here we describe characterization and calibration of VisAO relevant to this report.",
"For additional information about MagAO see [38], [91], and [72].",
"For additional information about the on-sky performance of VisAO see [40], [56], and [131].",
"High contrast imaging of $\\beta \\mbox{ Pic b}$ in the thermal-IR with Clio2 is presented in our companion paper [101]."
],
[
"The $Y_S$ Filter",
"The Ys Filter Since this was the first attempt to conduct high-contrast observations with VisAO, we used our longest wavelength bandpass to maximize Strehl ratio (SR) and flux from the thermally self-luminous planet.",
"We refer to this filter as “Y-short”, or $Y_S$ .",
"For a complete characterization of this filter see Appendix .",
"In brief, it is defined by a long-pass dichroic at $950\\mu $ m and the CCD QE cutoff at $\\sim $$1.1\\mu $ m. Including a representative atmosphere in the profile, the central wavelength of $Y_S$ is $0.985\\mu $ m and the width is $0.086\\mu $ m."
],
[
"The VisAO Anti-blooming Occulting Mask",
"The VisAO camera contains a partially transmissive occulting mask, used to prevent saturation of the CCD when observing bright stars.",
"The mask has a radius equivalent to $0.1^{\\prime \\prime }$ , were it in the focal plane, but it is approximately 60mm out of focus in an f/52 beam.",
"This has two consequences of note here.",
"The first is that the mask has an apodized attenuation profile which extends to $\\sim $$0.8^{\\prime \\prime }$ in radius, so at a separation of $\\sim $$0.46^{\\prime \\prime }$ $\\beta \\mbox{ Pic b}$ is under the mask.",
"The second, and more challenging, consequence is that the mask attenuation will depend on wavefront quality.",
"We did not fully characterize the mask using $\\beta $ Pic itself, so we bootstrapped an attenuation profile from other measurements at different levels of wavefront correction.",
"We describe this process in detail in Appendix .",
"We estimate that the mask transmission at the separation of $\\beta \\mbox{ Pic b}$ was $0.60^{+0.05}_{-0.10}$ ."
],
[
"Astrometric Calibration",
"In [40] we describe the calibration of platescale in several filters and North orientation of the VisAO camera.",
"Here we describe our calibrations in $Y_S$ and how we tied VisAO to the Clio2 astrometry.",
"The primary stars used for VisAO calibration were $\\theta ^1$ Ori $B1$ and $B2$ in the Trapezium.",
"These stars were observed repeatedly, and with a separation of $\\sim $$0.94^{\\prime \\prime }$ $B2$ is well within the isoplanatic patch when guiding on $B1$ .",
"[40] recently showed that all four stars in $\\theta ^1$ Ori $B$ are exhibiting orbital motion, so we measured their current astrometry using the wider FOV Clio2 camera.",
"A complete description of these measurements is provided in Paper II.",
"In brief, we used combinations of Trapezium stars other than $B1$ and $B2$ to measure the distortion, platescale, and orientation of Clio2.",
"This was done using the recent astrometry given in [39] which used LBTAO/Pisces referenced to HST astrometry from [110].",
"We then compared measurements of $B1$ and $B2$ with Clio2 to those with VisAO.",
"This was done in Dec. 2012, in the $Y_S$ filter with the occulting mask out of the beam.",
"The extra glass added by the mask substrate changes the focus position slightly, decreasing the platescale.",
"We measured this change in May, 2013, also using $\\theta ^1$ Ori $B1$ and $B2$ .",
"The ratio of platescales with and without the mask is 0.9972 $\\pm $ 0.0003.",
"This was applied to the $Y_S$ -open platescale to determine the $Y_S$ -mask platescale.",
"We also determined the orientation of the CCD with respect to North, which we denote by $NORTH_{VisAO}$ .",
"Our images are derotated counter-clockwise using the equation $DEROT_{VisAO} = ROTOFF + 90^o + NORTH_{VisAO}$ , where $ROTOFF$ is the rotator offset, equal to rotator angle plus parallactic angle.",
"The astrometric calibration of VisAO is summarized in Table REF .",
"Table: VisAO Y S Y_S platescale and rotator calibration.",
"Measurement uncertainty includes both Clio2 and VisAO scatter.",
"Astrometric uncertainty was propagated from the LBTAO/Pisces measurements of .By dithering $\\theta ^1$ Ori $B$ around the detector we diagnosed a slight focal plane tilt, which causes a small change in platescale, $<1$ %, from top to bottom predominantly in the y direction [40].",
"At the $\\sim $$0.47^{\\prime \\prime }$ separation of $\\beta \\mbox{ Pic b}$ this change in platescale amounts to less than 1 mas, so we neglect it."
],
[
"VisAO",
"We observed $\\beta $ Pic with VisAO on the night of 2012 Dec 04 UT in the $Y_S$ filter using the 50/50 beamsplitter, sending half the $\\lambda < 1$ $\\mu \\mbox{m}$ light to the PWFS.",
"The image rotator was fixed to facilitate angular differential imaging [81], [94].",
"Conditions were photometric.",
"$V$ band seeing, as measured by a co-located differential image motion monitor (DIMM), varied from $\\sim 0.45^{\\prime \\prime }$ $\\sim 0.75^{\\prime \\prime }$ during the 4.17 hours of elapsed time included in these observations.",
"Towards the end of the observation we took off-mask calibration data to assess AO performance and calibrate our photometry.",
"We took 1061 0.283 sec frames off the occulting mask, at airmass 1.14.",
"$V$ band seeing as measured by the DIMM was $\\sim 0.65^{\\prime \\prime }$ during these measurements.",
"To avoid saturation this data was taken in our fastest full frame mode ($1024\\times 1024$ pixels at 3.51 frames-per-second) and in the lowest gain setting.",
"Even with these settings, we saturated the peak pixel in roughly a third of the exposures.",
"To compensate we selected frames with peak pixel between 8000 and 9000 ADU, where the detector is linear, such that we are using data between approximately the 75th and 25th percentiles.",
"VisAO and Clio2 are operated simultaneously.",
"The dichroic entrance window of Clio2 transmits light with $\\lambda \\gtrsim 1.05\\mu $ m, while reflecting shorter wavelengths to the PWFS and VisAO camera.",
"When working in the thermal IR with Clio2, we perform small pointing offsets (called “nods”) to facilitate background subtraction.",
"These nods occur at intervals ranging from $\\sim 2$ to $\\sim 5$ minutes depending on wavelength and star brightness.",
"The consequence for VisAO observations is occasionally short (5 to 15 second) periods of unusable data while the AO loop is paused during a nod.",
"Pausing the loop causes wavefront error (WFE) to become much worse.",
"To automatically reject these periods during post-processing we apply a WFE cut using AO telemetry [91].",
"For all of these observations we used 130 nm RMS phase.",
"For the PSF measurement this rejected 84 frames.",
"We then registered and median combined the remaining 491 images, with the result shown in Figure REF .",
"Figure: VisAO PSF measurements using β\\beta Pictoris A, a Y s =3.561Y_s=3.561 mag A6V star.",
"Left: the unsaturated (off the occulting mask) PSF made by shifting and adding 491 0.283 sec exposures.",
"Right: the on-mask PSF, formed by registering and combining 2.15 hours of 2.27 sec individual exposures.",
"Images shown on a log stretch.",
"The observed Y S Y_S-band SR was ∼\\sim 32%32\\%, meaning the actual SR after correcting for the CCD's PRF was ∼\\sim 40%40\\%.",
"The bright signal ∼\\sim 1.3 '' 1.3^{\\prime \\prime } to the right of the star is an in-focus beamsplitter ghost.The unsaturated un-occulted PSF core has a FWHM of 4.73 pixels (37 mas), compared to the 6.5 m diffraction limit at $Y_S$ of 3.87 pixels (31 mas).",
"We have identified two sources of broadening: $\\sim $ 7 mas of residual jitter, most likely due to 60 Hz primary mirror cell fans; and the CCD's charge diffusion pixel response function (PRF).",
"For a near-Nyquist sampled CCD charge diffusion causes a blurring effect, which was well documented for the HST ACS and WFPC cameras.",
"See [73], [4], and the ACS handbookhttp://www.stsci.edu/hst/acs/documents/handbooks/cycle20/c05_imaging7.html.",
"We measured the PRF in the lab, and found that it broadens our PSF by $\\sim $$0.4$ pixels, and lowers the peak such that SR due to PRF alone is 80%.",
"SR measured using WFS telemetry and the unsaturated PSF was $32\\pm 2$ %.",
"Since this includes PRF, we can divide by 0.8 to estimate that the true $Y_S$ SR was $40\\%$ ."
],
[
"High Contrast Observations",
"The observations intended to detect $\\beta \\mbox{ Pic b}$ were conducted in full frame mode (1024x1024 pixels, $8^{\\prime \\prime }\\times 8^{\\prime \\prime }$ ), in the camera's highest sensitivity gain setting, with an individual exposure time of 2.27 sec.",
"The star was behind the occulting mask, so no dithers were conducted.",
"The individual images were bias and dark subtracted, using shutter-closed darks taken at 15 min intervals throughout the observation.",
"We did not flat field.",
"The CCD-47 has very low pixel-to-pixel variation, and with an $8^{\\prime \\prime }\\times 8^{\\prime \\prime }$ FOV we expect only small illumination changes across an image.",
"Clio2 nods were rejected using WFE telemetry as described above.",
"We then examined each image by eye, rejecting those with apparent poor mask alignment or poor AO correction.",
"Finally, to reduce the number of images to process we median coadded the remaining 3399 dark-subtracted exposures until either 30 seconds had elapsed or 0.5 degrees of rotation had occurred.",
"This process resulted in 317 images corresponding to 2.5hrs of open-shutter data, with an elapsed clock-time of 4.17 hrs and 116 degrees of sky rotation from start to finish.",
"Airmass was 1.12 at the beginning, reached 1.08 at transit, and was at a maximum of 1.29 at the end of the observations.",
"The coadded images were first coarsely registered and centered using a beamsplitter ghost.",
"The mask is partially transmissive (ND$\\approx 2.8$ ) in the core.",
"To more finely register the images we located the center of rotational symmetry of the attenuated star using cross correlation, with a tolerance of 0.05 pixels.",
"We median combined the registered images, forming our master PSF.",
"This is shown in Figure REF .",
"We employed a reduction technique based on principal component analysis (PCA), using the Karhunen-Loéve Image Processing (KLIP) algorithm of [121] (see also [3]).",
"We applied KLIP in search regions, as suggested by [121], dividing the image in both radius and azimuth in “optimization regions” using the strategy developed by [76] for the locally optimized combination of images (LOCI) algorithm.",
"In each optimization region we conducted the complete KLIP procedure.",
"Only a subsection of the optimization region, a “subtraction region”, was kept.",
"We used the parameters specifying the regions given by [76], except that instead of having the same azimuthal width, our subtraction regions were 1/3 the width of the optimization regions in azimuth.",
"This provided a noticeable ($\\sim 10\\%$ ) improvement in signal-to-noise ratio (S/N) without significantly increasing the computational burden.",
"The final image is formed by combining the individually reduced subtraction regions.",
"Note that when we choose the number of KL modes, we apply this choice to all subtraction regions.",
"We compare the results of our KLIP reduction to the simultaneously obtained Clio2 $M^{\\prime }$ image, shown in Figure REF and described in Paper II.",
"The Clio2 position is shown in both images as an ellipse corresponding to the $2\\sigma $ uncertainty.",
"Here we use the mean position and error using data taken with Clio2 in four filters on subsequent nights.",
"See Paper II for further details.",
"In Figure REF we zoom in on the VisAO image of $\\beta \\mbox{ Pic b}$ and compare it to the PSF at that position under the occulting mask.",
"This under-the-mask PSF was measured on-sky in closed-loop by scanning a star across the mask.",
"The mask radially elongates a point source, and the image produced by our KLIP analysis matches this expectation.",
"We also show two representative fake planets, injected using the same PSF (the details of this procedure are given below).",
"The signal recovered at the precisely known location of the planet closely matches the expected signal based on our PSF model.",
"Other tests conducted included a basic ADI-only reduction, which yields a lower significance detection (S/N $<$ 3).",
"We also tried many other search region geometries.",
"Though varying reduction parameters modulates various speckles throughout the image and changes algorithm throughput, the signal at the location of $\\beta \\mbox{ Pic b}$ is always detected.",
"We also tested reducing half the data in alternating chunks (to preserve rotation), and even with ADI-only we detected $\\beta \\mbox{ Pic b}$ using only half the data (albeit with much lower S/N).",
"Along with our PSF comparisons, these results give confidence that our detection of $\\beta \\mbox{ Pic b}$ is valid.",
"We quantify the significance of this detection next.",
"Figure: NO_CAPTIONFigure: NO_CAPTIONLeft: The MagAO+VisAO detection of $\\beta \\mbox{ Pic b}$ .",
"Right: The MagAO+Clio2 $M^{\\prime }$ -band detection of $\\beta \\mbox{ Pic b}$ (from Paper II), taken simultaneously with the VisAO image.",
"In each image the red ellipse denotes the $\\pm 2\\sigma $ uncertainty in the Clio2 position.",
"The color scales are relative to the peak of the planet.",
"The inner regions of the images, which are dominated by residuals, have been masked out in post-processing.beta Pic Detections Figure: Comparison between our image of βPicb\\beta \\mbox{ Pic b}, the occulting mask PSF, and two representative simulated planets which were injected into the data using the mask PSF.",
"The mask PSF was measured on-sky by scanning a star across the mask, and the image at the lower left corresponds to the separation of βPicb\\beta \\mbox{ Pic b}.",
"At lower right we zoom in on the VisAO detection of βPicb\\beta \\mbox{ Pic b}.",
"As in Figure the red ellipse corresponds to the 2σ2\\sigma Clio2 position uncertainty.",
"The recovered image of βPicb\\beta \\mbox{ Pic b} matches the PSF.",
"The top two images are of simulated planets, injected with contrast 2×10 -5 2\\times 10^{-5}, and recovered using the same data processing pipeline.",
"These show that our injected planets are correctly modeling the PSF.",
"Each image is plotted on the same spatial scale, and the color table is stretched relative to the peak of the object in each as in Figure ."
],
[
"Detection Significance",
"To assess the statistical significance of the VisAO detection we next calculated a S/N map.",
"For this analysis we used 150 KL modes (we discuss choosing the number of modes in Section REF ).",
"The final image was Gaussian smoothed with a kernel of width 3 pixels.",
"Each pixel was divided by the standard deviation calculated in an annulus 1 pixel in width.",
"Pixels within 1 FWHM radius of the location of the planet were excluded from the standard deviation calculation.",
"The S/N map is shown at left in Figure REF .",
"Within the $2\\sigma $ uncertainty of the Clio2 detection $\\beta \\mbox{ Pic b}$ is detected by VisAO with S/N$ = 4.1$ .",
"Other choices of smoothing kernels, different numbers of modes, and different techniques for calculating the noise can change this value by $\\pm $$\\sim $$25\\%$ (see Figure REF ).",
"Regardless, this is the maximum S/N pixel at or near the separation of $\\beta \\mbox{ Pic b}$ , so these choices have minimal impact on the following analysis.",
"As a starting point we first calculated the histogram of all pixels within the annulus of width $\\pm 2\\sigma $ (Clio2 uncertainty) centered on the location of the planet, excluding the planet location.",
"We restrict ourselves to this annulus to ensure the statistics are representative.",
"The histogram is shown in Figure REF , where we also overplot a Gaussian distribution with $\\sigma =1$ for comparison.",
"There are 2628 pixels included in the histogram, all with S/N$ < 4.1$ .",
"We note that these pixels will tend to be correlated across the PSF width and by the smoothing kernel, so we do not estimate a false alarm probability (FAP) from the histogram of individual pixels.",
"However, this does show that the S/N = 4.1 peak at the location of $\\beta $ Pic b is not expected from the distribution of pixels in the image.",
"In truth, we would have considered any signal close to the location determined by Clio2 as a VisAO detection.",
"We next analyze the 101 unique apertures with a radius of $2\\sigma $ (Clio2 radial uncertainty) which fit in the same annulus, choosing the highest S/N pixel in each.",
"These apertures have a diameter of $\\sim 7$ VisAO pixels, larger than the 4.7 pixel FWHM, so they are uncorrelated with respect to both the PSF and the smoothing kernel.",
"The histogram for these trials is shown at lower right in Figure REF .",
"The probability of having the aperture with the highest S/N pixel occur at the Clio2 position by chance is $1/102 = 1.0\\%$ (adding one aperture at the location of the planet).",
"This then sets an empirical upper limit on FAP.",
"Figure: NO_CAPTIONFigure: NO_CAPTIONLeft: S/N map.",
"The white circle shows the $2\\sigma $ uncertainty in the Clio2 position.",
"We detect $\\beta \\mbox{ Pic b}$ with VisAO at S/N = 4.1 at the location of the Clio2 detection.",
"Right: Pixel S/N histograms.",
"The top panel shows all pixels within the annulus of width $\\pm 2\\sigma $ (Clio2 uncertainty) around the detection.",
"The red curve is the Standard Normal distribution (not fit to the data), which we show for comparison.",
"The arrow shows the detection of $\\beta \\mbox{ Pic b}$ .",
"The bottom panel shows the distribution of the maximum S/N in apertures with radius of the twice the Clio2 position uncertainty ($\\pm 2\\sigma $ ).",
"We estimate a conservative upper limit on false alarm probability of $FAP = 1.0\\%$ .beta Pic b S/N"
],
[
"Photometry with Simulated Planets",
"We calibrated our photometry by injecting simulated planets.",
"We used our on-sky under-the-mask PSF measurement, shown in Figure REF (see also Appendix ) to simulate a planet, which we scaled based on the under-the-mask image of the star in each frame, using a 3 pixel radius photometric aperture.",
"Note that this accounts for Strehl variation and airmass effects.",
"The mask transmission was also applied, taking into account the changing position of $\\beta \\mbox{ Pic A}$ under the mask over the course of the observation due to flexure and re-alignment.",
"This caused the transmission at the location of $\\beta \\mbox{ Pic b}$ to change by roughly $\\pm 3\\%$ .",
"With this procedure we injected planets with contrasts of 1, 2, and $3\\times 10^{-5}$ at 21 locations $13.5^o$ apart in the 270o opposite the location of $\\beta \\mbox{ Pic b}$ , at the same 59 pixel ($\\sim $$0.47^{\\prime \\prime }$ ) separation.",
"This was done prior to registration and then the complete reduction was carried out, meaning an entirely new set of KL modes was calculated in each search region.",
"We conducted aperture photometry on these simulated planets, and on $\\beta \\mbox{ Pic b}$ .",
"Using a reduction with no injected planets but otherwise having the same parameters, we estimated the noise in our photometry by sampling 59 locations spaced by 2 FWHM, at the same separation, avoiding the known location of $\\beta \\mbox{ Pic b}$ .",
"Using a simple grid search strategy, we tested various aperture sizes and Gaussian smoothing kernels.",
"Using the mean photometry of the simulated planets we found the aperture radius and smoothing width which maximized S/N.",
"These values vary depending on the number of modes and contrast.",
"We used the values determined for the $2\\times 10^{-5}$ planets for photometry on the actual $\\beta \\mbox{ Pic b}$ .",
"In Figure REF we show S/N vs. number of KL modes for the mean of the simulated $1\\times 10^{-5}$ and $2\\times 10^{-5}$ planets.",
"In each case, the mean S/N has become nearly constant once 75 modes are included in the reduction.",
"We also show the individual results for each of the 21 injected $2\\times 10^{-5}$ planets, and the result for $\\beta \\mbox{ Pic b}$ .",
"The S/N vs. number of modes curve for $\\beta \\mbox{ Pic b}$ varies significantly up to 200 modes, but we see that this appears rather typical compared to the individual injected planets.",
"Combined with the comparisons shown in Figure REF , it appears that our injected fake planets model the true signal well.",
"In the right-hand panel of Figure REF we show the contrast of $\\beta \\mbox{ Pic b}$ .",
"This was calibrated using the injected planets.",
"We linearly interpolated between the mean values of the injected planet photometry, finding the contrast which corresponds to the photometry of $\\beta \\mbox{ Pic b}$ .",
"The problem we face, illustrated in Figure REF , is that there is no clear choice for number of KL modes.",
"Increasing the number of modes from 75 to 200 does not improve the mean S/N of the injected planets, but it does have a large ($\\sim $$30\\%$ ) impact on the measured contrast of $\\beta \\mbox{ Pic b}$ .",
"Rather than choose a single number of modes, we instead average the contrast measurements from 75 to 200 modes.",
"This gives a contrast of $(1.63 \\pm 0.49)\\times 10^{-5}$ .",
"We have adopted an uncertainty of $\\pm 30\\%$ .",
"Though this is larger than the $\\sim $$25\\%$ expected from the S/N found in Section REF , it accounts for the additional uncertainty in choosing the number of KL modes.",
"We added the uncertainty in mask transmission ($+5\\%,-10\\%$ ) in quadrature, so our total uncertainty in contrast is $+32\\%, -30\\%$ .",
"In magnitudes we have $\\Delta Y_S = 11.97^{+0.34}_{-0.33}$ .",
"Figure: Left: S/N vs. number of KL modes.",
"The thick black curve is the mean of 21 fake planets injected with contrast 2×10 -5 2\\times 10^{-5}.",
"The thick dashed black curve is the same for 1×10 -5 1\\times 10^{-5} fake planets.",
"The red curve is for βPicb\\beta \\mbox{ Pic b}.",
"Also shown are the results for each of the 21 2×10 -5 2\\times 10^{-5} injected planets, showing that the βPicb\\beta \\mbox{ Pic b} result is typical.",
"Right: The resulting contrast measurements.",
"Given that the mean S/N of the fake planets has flattened at 75 modes, we have no clear way to choose how many modes to use.",
"As a result, we average the six contrast measurements from 75 to 200 modes.",
"We adopt an uncertainty of 30%30\\%.",
"Though this is larger than implied by our detection S/N=4.1S/N=4.1, it accounts for the additional uncertainty from choosing reduction parameters.",
"The resulting raw contrast measurement is (1.63±0.49)×10 -5 (1.63 \\pm 0.49)\\times 10^{-5}, indicated by the horizontal lines."
],
[
"VisAO Astrometry",
"We also used the simulated planets to calibrate our astrometric precision, finding that Gaussian centroiding on the injected planets gave unbiased results.",
"We measured the positions of the $1\\times 10^{-5}$ and $2\\times 10^{-5}$ planets by Gaussian centroiding, and used these to estimate the statistical uncertainties.",
"We found $\\sigma _{sep} = 0.82$ pixels and $\\sigma _{PA} = 0.63$ degrees for $\\beta \\mbox{ Pic b}$ .",
"In tests using binary stars under the occulting mask, we found a $\\sim $ 1 pixel scatter in recovered positions (at the separation of $\\beta \\mbox{ Pic b}$ ), which we attribute to uncertainty in centroiding under the mask.",
"Finally, we include the astrometric calibration uncertainty (see Table REF ).",
"We also measured the position of $\\beta \\mbox{ Pic b}$ using Gaussian centroiding.",
"Our astrometry for $\\beta \\mbox{ Pic b}$ is presented in Table REF .",
"Table: VisAO Astrometry of β\\beta Pictoris b"
],
[
"NICI",
"We present observations of $\\beta \\mbox{ Pic b}$ taken during the course of the Gemini Near-Infrared Coronagraphic Imager [37] campaign [85], [130], [12], [102].",
"We observed $\\beta $ Pic on 25 Dec 2010 UT, in $K_S$ , and again on 20 Oct 2011 UT in $CH_{4S,1\\%}$ and $K_{cont}$ .",
"The $K_S$ data was independently analyzed by [14], but with an extrapolated calibration of mask transmission and hence large photometric uncertainties.",
"Here we provide photometry based on a direct measurement of the focal plane mask transmission.",
"We reduced the NICI data using the well tested tools developed for the NICI campaign (see references above), with some differences as noted next.",
"The VisAO reduction techniques described above were developed independently.",
"The images were reduced as described in [129], but with the addition of smart frame selection for PSF building similar to [94] and [76].",
"The standard ADI reduction procedure median combines all the science images in the pupil-aligned orientations to make a source-less PSF and thereby subtract the star from each individual image.",
"In our method, we median-combine only the frames which are similar to the image which is to undergo PSF subtraction.",
"This similarity is measured in the difference of the target image and the candidate reference image using the RMS of pixel values at radii between $0.3^{\\prime \\prime }$ and $0.6^{\\prime \\prime }$ separation.",
"Only the best 20 images are used to make the reference PSF.",
"We differ from [76] and [94] in that we do not reject frames because they have too little rotation relative to the target image.",
"Instead we require that the range of reference frames selected have total rotation $> 2$ times the NICI FWHM.",
"These parameters were optimized by comparing the S/N maps resulting from the reductions, and we found that this algorithm performed better than either basic ADI or LOCI for data taken with NICI.",
"The NICI detections of $\\beta \\mbox{ Pic b}$ are presented in Figure REF .",
"To estimate the S/N of these detections, each pixel was divided by the standard deviation of the pixels in an annulus of width 5 pixels centered on the same separation.",
"No smoothing was applied.",
"A robust standard deviation algorithm was used, so the much higher pixels at the location of the planet were not included in the estimated noise.",
"The peak S/N in $CH_{4S,1\\%}$ was 10.4, in $K_S$ it was 6.8, and in the $K_{cont}$ filter it was 27.6.",
"NICI photometry was calibrated by injecting simulated planets.",
"The simulated planet signals were scaled from the star PSF, and were injected into the data at the same separation as the real planet.",
"The star was nonlinear in the $CH_{4S,1\\%}$ and $K_{cont}$ science exposures, so we instead used short acquisition exposures appropriately scaled.",
"The simulated planets were injected into the data at twenty position angles opposite the planet, one at a time, and a complete reduction was carried out.",
"The stellar PSF halo was occasionally saturated near the radius of the planet in the $K_S$ observations, and such frames were removed from the reduction process.",
"We used aperture photometry with an aperture radius of 3 pixels.",
"ADI self-subtraction was corrected using the difference between the injected and recovered flux of the simulated planets.",
"We estimated the uncertainty in the measured flux as the standard deviation of the recovered fluxes.",
"For $CH_{4S,1\\%}$ and $K_{cont}$ we estimated an additional uncertainty of $8\\%$ due to variability of the star peak.",
"Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONNICI images of $\\beta \\mbox{ Pic b}$ .",
"The color scale is given at right, relative to the peak of the planet in each image.",
"These observations were made using a $0.22^{\\prime \\prime }$ radius coronagraphic focal plane mask.",
"The $0.22^{\\prime \\prime }$ mask opacity falls to zero at $0.4^{\\prime \\prime }$ from the center of the mask, so we have to correct for the mask opacity at the location of $\\beta \\mbox{ Pic b}$ .",
"As described in [128] for the $0.32^{\\prime \\prime }$ mask, we measured the $0.22^{\\prime \\prime }$ mask opacities using a pair of 2MASS stars, both off the mask, and then one under the mask at different separations from the center.",
"The mask opacities in the $K_S$ , $K_{cont}$ and $CH_{4S,1\\%}$ bands were $5.03\\pm 0.03$ , $5.05\\pm 0.03$ and $5.47\\pm 0.09$ mags, respectively.",
"Typically, more than 5 measurements were taken at each position of the target under the mask, so that our opacity measurement uncertainties should be well-estimated [128].",
"The sampling of separations are sparse (in steps of $0.1^{\\prime \\prime }$ ) and any discontinuities could introduce systematics into our estimates, which are based on fitting a smooth curve through the opacities as a function of separation.",
"However, we have no evidence that such discontinuities exist, and the uncertainties are estimated with this assumption.",
"Contrasts were calculated as $-2.5 \\log (\\mbox{star counts}/\\mbox{planet counts}) + \\mbox{mask opacity} $ .",
"The uncertainties described above were added in quadrature.",
"We measured contrasts of $\\Delta CH_{4S,1\\%} = 9.65 \\pm 0.14$ mag, $\\Delta K_S = 8.92 \\pm 0.13$ mag, and $\\Delta K_{cont} = 8.23 \\pm 0.14 $ mag.",
"Astrometry from NICI, as well as an analysis of all available astrometry, is presented in Nielsen et al.",
"(2014, submitted).",
"Figure: The SED of β\\beta Pic A.",
"The synthetic spectrum was normalized to the VV photometry only.",
"We assume that the lack of reddening also holds for the planet.",
"Also note the lack of significant variability implied by these different measurements."
],
[
"The SED of $\\beta $ Pic A",
"The SED of beta Pic A We used archival photometry of $\\beta $ Pictoris A to find its brightness in the filters used here, and also to investigate whether there is any reddening, which could affect our $\\lambda <1\\mu $ m photometry.",
"We used $V,R_C,I_C$ photometry from [43] and [42].",
"Especially useful was photometry in the 13-color system, which spans $0.33\\mu $ m to $1.10\\mu $ m in somewhat narrow passbands, from [99] and [66].",
"In the IR we used Johnson-Glass $J,H$ , and $K$ from [59], and ESO broadband $J,H,K$ and narrowband $H_0$ , $Br_\\gamma $ , $K_0$ , and $CO$ photometry from [126].",
"Finally, following [19] we synthesize an A6V spectrum by averaging the A5V and A7V spectra in the Pickles Spectral Atlas [106].",
"We then normalized this spectrum to Cousins $V$ .",
"We compare this spectrum to the photometry in Figure REF .",
"Based on the good agreement we conclude that reddening is not significant for $\\beta $ Pic A, and assume that this will also be true for the planet (though there could be circumplanetary reddening that this analysis would miss).",
"This exercise also demonstrates that variability of the primary star is not a concern when using it as a photometric reference for AO observations.",
"The central wavelength of the `99' filter of [99] is nearly identical to VisAO $Y_S$ , so we adopt their measurement of $Y_S=3.561\\pm 0.035$ mag for $\\beta \\mbox{ Pic A}$ .",
"Using the interpolated A6V spectrum we find that $CH_{4S,1\\%} - H_{ESO} = 0.024 \\pm 0.05$ , so we have $CH_{4S,1\\%} = 3.526 \\pm 0.05$ mag for $\\beta \\mbox{ Pic A}$ [19].",
"We find $K_{S,NICI} - K_{ESO} = -0.026 \\pm 0.05$ so we use $K_{S,NICI} = 3.468\\pm 0.05$ mag.",
"We find $K_{cont}-K_{ESO}=0.068\\pm 0.05$ mag for an A6V, so for $\\beta \\mbox{ Pic A}$ we have $K_{cont} = 3.563 \\pm 0.05$ mag.",
"We combine these with our contrast measurements, adding uncertainties in quadrature.",
"We present our new photometry of $\\beta \\mbox{ Pic b}$ in Table REF , along with prior measurements in $J$ , $H$ , and $K$ .",
"Measurements at longer wavelengths are considered in Paper II.",
"Table: YJHKYJHK Photometry of β\\beta Pictoris b"
],
[
"Analysis ",
"We now analyze the combined $Y_S,J, H$ , and $K_S$ photometry of $\\beta \\mbox{ Pic b}$ .",
"Our analysis is based on a library of over 500 field BDs, low-g BDs, and low-mass companions, which we describe in Appendix ."
],
[
"Color-Color and Color-Magnitude Comparisons",
"In Figure REF we compare $\\beta \\mbox{ Pic b}$ to field BDs and other low-mass companions in color-color plots.",
"We show $J-K_S$ vs $H-K_S$ colors, where nearly all of the companions have photometry (HR 8799 e is the notable exception in $J$ ).",
"$Y$ band is relatively unexplored territory for low-mass companions, so we present three different permutations of $Y_S$ colors.",
"The field objects are plotted in the NICI system using synthetic photometry as described in Appendix .",
"We found that conversions between photometric systems are typically $<0.1$ mags, especially for spectral type L (see also [123]), so in Figure REF the companion objects are plotted in the $J$ , $H$ , and $K_S$ photometric system in which they were observed.",
"We used available spectra to estimate $Y_S$ photometry for the companions shown.",
"For $\\beta \\mbox{ Pic b}$ we plot our new $K_S$ point from NICI, and the NICI $H$ point from [47].",
"In all cases $\\beta \\mbox{ Pic b}$ has colors consistent with early to mid L-dwarfs.",
"It is also consistent with an early T, a consequence of the blueward progression at the L/T transition.",
"The color degeneracy of the L/T transition is broken by absolute magnitude.",
"In Figure REF we show $Y_S$ and $K_S$ color-magnitude diagrams.",
"These plots firmly place $\\beta \\mbox{ Pic b}$ in the early L-dwarfs.",
"As noted by [19] and [47] it is somewhat redder than the L dwarfs in $K_S$ vs $H-K_S$ .",
"This is consistent with other low surface-gravity L dwarfs such as AB Pic B, 2M0122B, and PSO318.5, though within $1\\sigma $ $\\beta \\mbox{ Pic b}$ is consistent with the field."
],
[
"Spectral Fitting",
"We next fit each of 499 BD spectra collected from various sources to the $Y_S$ through $K_{cont}$ photometry of $\\beta \\mbox{ Pic b}$ (treating each measurement independently).",
"We did this by computing the flux of each spectrum in each of the bandpasses, then finding the single scaling factor which minimized $\\chi ^2$ .",
"Only spectra with a complete $Y$ band measurement were used, which tends to select for high S/N.",
"We adopted an error of 0.05 mag for our synthetic photometry in each bandpass based on the findings of [86].",
"This was added in quadrature to the $\\beta \\mbox{ Pic b}$ measurement errors.",
"The results are presented in Figure REF , where we show $\\chi ^2_{\\nu }$ vs. SpT for each of the field objects, as well as the median in each SpT.",
"The error bars indicate the standard deviation in each SpT.",
"There is a minimum in the early to mid L dwarfs, giving a range of L2 to L5.",
"In Figure REF we show the 5 best fitting spectra, which range from L3 to L5.5.",
"Our new $Y_S$ photometry is well fit by these spectra, and our new $CH_{4S,1\\%}$ and $K_S$ point are consistent with the prior measurements.",
"The $K_{cont}$ point is only consistent with prior measurements at the $2\\sigma $ level, though.",
"A possible culprit is the noted non-linearity of the star, which may have biased the result.",
"However, the same procedure was applied to the $CH_{4S,1\\%}$ measurement from the same night.",
"Based on the many measurements of $\\beta \\mbox{ Pic A}$ we discount the primary as a source of variability at this level.",
"BDs are known to be variable [22] but typically with low amplitude for early L dwarfs [61].",
"We do not have enough evidence to speculate other than to note variability in $\\beta \\mbox{ Pic b}$ as a possible explanation.",
"Though it established that the $Y$ through $K$ photometry of $\\beta \\mbox{ Pic b}$ does not appear to be atypical compared to the field, this exercise ultimately does not yield a well-determined SpT."
],
[
"Near-IR SpT",
"Having established that the photometry of $\\beta \\mbox{ Pic b}$ places it in the early to mid L dwarfs, we next attempt to estimate its SpT.",
"Spectral typing is usually done by comparing the spectral morphology of the object to spectral standards.",
"Since there is not yet a measurement of the $Y$ through $K$ spectrum of $\\beta \\mbox{ Pic b}$ , we instead compare its broadband photometry to field objects.",
"We first plot color vs. SpT for the 499 spectra in our library, shown in Figure REF .",
"For each color we fit a polynomial to the sequence, and then find the root(s) of this polynomial corresponding to the photometry of $\\beta \\mbox{ Pic b}$ .",
"A consequence of the L/T transition is that the BD SpT sequence is dual valued in each permutation of the $YJHK$ colors, hence the double values in most of the panels of Figure REF (and the broad minimum in Figure REF ).",
"The $H-K_S$ color measurements do not intersect the polynomial, though they are consistent with field dwarfs.",
"For these $H-K_S$ measurements we assign the SpT of the turnover in color vs. SpT.",
"Figure: The best five fits from the spectral library, in order of χ ν 2 \\chi ^2_{\\nu } from top to bottom.",
"The best single match is an L4, but all of these have similar values of χ ν 2 ∼1\\chi ^2_{\\nu } \\sim 1.",
"The red circles are our new measurements, and the black open squares are prior measurements.The bluer colors of $\\beta \\mbox{ Pic b}$ (e.g.",
"$Y_S-J$ , $J-H$ ) indicate an SpT of early-L or mid-T.",
"The redder colors — especially $H-K_S$ — favor mid-L. To break the degeneracy from colors we turn to absolute magnitudes.",
"In similar fashion, we fit polynomials to the objects in our library with parallaxes.",
"These fits, shown at right in Figure REF are single valued, and all favor an SpT of early-L.",
"This paints a picture of $\\beta \\mbox{ Pic b}$ as having the luminosity of an early L dwarf but being somewhat redder than typical for the field, much like many well studied low-gravity field BDs and companions.",
"We next compared all of these SpT determinations.",
"It has been shown that low-g BDs do not follow the field BD sequence in near-IR absolute magnitudes [54], [82], so we must be cautious about using the absolute magnitudes of $\\beta \\mbox{ Pic b}$ to directly estimate SpT.",
"However, according to [82] in the SpT range of roughly L0-L3 the absolute J-band magnitudes of low-g objects do match the field.",
"We also note that for spectral types later than L3, the absolute J-band magnitudes tend to be fainter than the field for low-g objects, which does not appear to be true for $\\beta \\mbox{ Pic b}$ .",
"The absolute magnitudes of $\\beta \\mbox{ Pic b}$ all correspond to the early-L dwarf SpTs, and the weighted mean using just the absolute magnitudes is L$2.1\\pm 0.8$ .",
"Turning to the colors next, we reject the T classifications based on the absolute photometry.",
"We form a best estimate of SpT by averaging all of the L values, finding L$3.6\\pm 2.2$ using only the colors — as expected from the range we derived from using Figure REF .",
"Based on the consistency and the findings of [82], we give somewhat more weight to the type estimate derived from absolute magnitudes than the color derived estimate.",
"We also take into account that low-g objects tend to be red compared to the field for their spectral types.",
"Based on all of these considerations we adopt L$2.5\\pm 1.5$ as the near-IR spectral type of $\\beta \\mbox{ Pic b}$ .",
"This is consistent with [107], who found an approximate spectral type of L4, with a range of L0-L7, using $L^{\\prime }$ and narrow-band 4.05 $\\mu $ m color.",
"Our estimate is also consistent with the spectral type of L2$\\gamma \\pm 2$ estimated by [19], and the range of L2-L5 given by [47].",
"Figure: Left: A color by color comparison of βPicb\\beta \\mbox{ Pic b} to the field BDs, with each possible combination of the available photometry.",
"The L/T transition causes the resultant SpT estimate to be dual-valued.",
"As we saw in Figures and the planet is modestly redder than the field in the redder colors, especially H-KH-K.",
"Right: A filter by filter comparison of the absolute magnitude of βPicb\\beta \\mbox{ Pic b} to field BDs.",
"For the early to mid L dwarfs the sequence is single valued in SpT.",
"In terms of its absolute magnitudes βPicb\\beta \\mbox{ Pic b} is quite clearly an early L."
],
[
"Physical Properties",
"Armed with our SpT estimate, we next estimated the bolometric luminosity of $\\beta \\mbox{ Pic b}$ .",
"We first convert to the MKO photometric system, using the relationships we derive in Appendix .",
"We converted each measurement independently, then calculated the uncertainty-weighted mean for each bandpass.",
"The results are presented in Table REF .",
"From there, we determined the BC for each of $J,H,K$ using the polynomials given in [84].",
"We then took the uncertainty-weighted mean value of the bolometric magnitude and its uncertainty, also given in Table REF .",
"The resulting uncertainty in $M_{bol}$ is a relatively small 0.11 mag.",
"There are likely additional systematic errors, for example from using the field relationships on a low-g object.",
"Table: MKO photometry, bolometric corrections, and bolometric magnitudes for βPicb\\beta \\mbox{ Pic b}.Converting to luminosity gives us $\\log (L_{bol}/L_) = -3.86 \\pm 0.04$ dex.",
"This is very close to the value of $-3.87\\pm 0.08$ dex by [19].",
"They used a somewhat different method, determining only the K band based $M_{bol}$ by using the mean BC of two similar field dwarfs.",
"They also reported $-3.83\\pm 0.24$ dex based on model atmosphere fitting, including longer wavelengths.",
"Our value is modestly fainter than the estimate of [47], $-3.80\\pm 0.02$ dex, which was also inferred from model fitting including longer wavelength measurements.",
"Figure: Results of Monte Carlo experiments using our luminosity estimate, log(L bol /L )=-3.86±0.04\\log (L_{bol}/L_{})=-3.86\\pm 0.04, a β\\beta Pic moving group age of 21±421\\pm 4 Myr (solid line) and 12 -4 +8 12^{+8}_{-4} Myr (dashed line), and hot start evolutionary models , .",
"Note the negative skewness in mass.",
"We can place an upper limit at M≈13M\\approx 13 M Jup M_{Jup}.We next calculated estimates of mass, temperature, and radius with Monte Carlo trials based on the hot start evolutionary models [33], [6].",
"We adopt the lithium depletion boundary age of $21\\pm 4$ Myr for the $\\beta $ Pic moving group from [13].",
"We treated our estimate for $\\log (L_{bol})$ and this age estimate as Gaussian distributions, and drew $10^4$ trial values.",
"For each random pair of $\\log (L_{bol})$ and age we calculated the hot start model prediction of mass, temperature, and radius by linear interpolation on the model grid.",
"The resulting distributions are shown in Figure REF , and summarized in Table REF .",
"The mass distribution has mean and standard deviation $11.9 \\pm 0.7$ $M_{Jup}$ , and median 12.0 $M_{Jup}$ .",
"The mass distribution has negative skewness, and there appears to be an upper limit at $M\\approx 13$ $M_{Jup}$ .",
"For temperature we find $T_{eff} = 1643 \\pm 32$ K. For radius we find $R = 1.43 \\pm 0.02$ $R_{Jup}$ with a small positive skewness.",
"These very small uncertainties are the formal errors from our Monte Carlo analysis of the model grid, and do not include any additional terms such as the uncertainties in the models themselves.",
"Repeating the Monte Carlo experiment with the prior age estimate of $12^{+8}_{-4}$ Myr from [133], we found $M=10.5\\pm 1.5$ $M_{Jup}$ , $T_{eff} = 1623\\pm 38$ K, and $R=1.47\\pm 0.04$ $R_{Jup}$ .",
"To account for the asymmetric uncertainty we used the skew-normal distribution for age, choosing its parameters such that the mode was 12 Myr, the left-half-width was 4 Myr, and the right-half-width was 8 Myr.",
"The resulting 12 Myr distributions are also shown in REF and Table REF .",
"There is again an upper limit on mass at roughly 13 $M_{Jup}$ .",
"We surmise that this is due to our luminosity estimate being too faint to allow deuterium burning at these ages according to the models, hence the mass is strongly constrained to be below 13 $M_{Jup}$ regardless of age.",
"Note that we consider additional models, using the complete 1-5$\\mu $ m photometry of $\\beta \\mbox{ Pic b}$ in Paper II.",
"[19] arrive at an estimate $T_{eff} = 1700\\pm 100$ K from fitting PHOENIX based atmosphere models coupled with various cloud models [33], [1], [6] and found $R = 1.4\\pm 0.2$ $R_{Jup}$ .",
"[47] estimated a range of $1575-1650$ K using the various cloud + atmosphere models of [30], [90], and [46], and estimated $R=1.65\\pm 0.06$ $R_{Jup}$ .",
"Both of these efforts assumed the previous age estimate of $12^{+8}_{-4}$ Myr.",
"Table: Estimated Physical Properties of βPicb\\beta \\mbox{ Pic b}."
],
[
"Discussion ",
"It has recently become clear that the SpT to $T_{eff}$ relationship for low-gravity objects is different from the relationship for the field [21], [86].",
"Like other low-g objects, $\\beta \\mbox{ Pic b}$ appears to be cooler than field BDs with the same SpT.",
"The SpT to $T_{eff}$ relationship of [124] gives an estimate of $T_{eff} = 1904$ K for an L2.5 field brown dwarf, which is $\\sim $ 250 K hotter than we calculated assuming hot start evolution for this young object.",
"It was noted by [86] that low-g objects tend to have $L_{bol}$ consistent with the field for their SpT, despite their lower temperatures and red near-IR colors.",
"In Figure REF we show log$(L_{bol}/L_{})$ vs. SpT for field brown dwarfs and low-g brown dwarfs and companions.",
"The field BDs are plotted using BCs from [84], and companions are plotted using published values of $L_{bol}$ (see Table REF for references).",
"Attempts have been made to use spectral indices and morphology to assign SpTs or SpT ranges to 2M1207 b and HR 8799 b [104], [20], [8], [2].",
"As evident in Figures REF and REF , however, these objects do not fit within the field brown dwarf sequence in either colors or absolute magnitudes.",
"Many authors have noted that 2M1207 b and the HR 8799 EGPs appear to be extensions of the L dwarf sequence [95], [20], [8], [90], and in Figure REF we have plotted them with a spectral type of L7.25 which appears to be a likely place to expand the L dwarf sequence to include these objects.",
"Figure: Bolometric luminosity vs. SpT.",
"We expect a temperature sequence to produce a luminosity sequence under the assumption of nearly constant radius.",
"The young low-gravity BDs and EGPs also appear to fall on the same luminosity sequence as the field BDs despite their lower effective temperatures.",
"The HR 8799 planets and 2M1207 b, objects which don't fit in the standard L spectral types, are plotted as L7.25.Interestingly, $\\beta \\mbox{ Pic b}$ 's luminosity is consistent with the field for a spectral type of L2.5.",
"Like the other directly-imaged planets, $\\beta \\mbox{ Pic b}$ is younger than field brown dwarfs of similar luminosity.",
"Its young age corresponds to a lower mass and larger radius, hence it has lower surface gravity.",
"Given its comparable luminosity, these properties imply a lower effective temperature.",
"However, in the case of $\\beta \\mbox{ Pic b}$ , these physical differences do not appear to have a large effect on its overall appearance in near-IR color-color and color-magnitude diagrams.",
"Though it is somewhat red in $H-K$ , $\\beta \\mbox{ Pic b}$ 's colors and luminosity are otherwise consistent with field early-L brown dwarfs, implying that it shares basic atmospheric properties with such objects.",
"We contrast this with other directly-imaged planets: the four HR 8799 planets and 2M1207 b are systematically fainter and redder than the field brown dwarf sequence [34], [95], [96].",
"The difference appears to result from (1) the persistence of clouds at luminosities where field brown dwarfs have transitioned from cloudy to cloud-free [46], [90], [116], and (2) the absence of methane absorption at luminosities where field brown dwarfs are beginning to show strong methane absorption [8], [9], [118].",
"[93] present a phenomenological explanation for why clouds may exist at higher altitudes in low-gravity objects.",
"The methane transition may occur on a different timescale, also as a result of a youth-based mechanism [118].",
"The fact that $\\beta \\mbox{ Pic b}$ has more “normal” near-IR colors separates it from the HR 8799 EGPs, 2M1207 b, and the unusually red low-gravity L dwarfs [54], [21], [86]."
],
[
"Conclusions ",
"We have presented the first high-contrast far-red optical observations of an EGP with MagAO's VisAO CCD camera, detecting $\\beta \\mbox{ Pic b}$ in $Y_S$ at a contrast of $(1.63\\pm 0.49)\\times 10^{-5}$ , at a separation of $0.470\\pm 0.010^{\\prime \\prime }$ .",
"The VisAO detection has S/N = 4.1, and a conservative upper-limit on false alarm probability of 1.0%.",
"We also present observations of $\\beta \\mbox{ Pic b}$ in the near-IR made with the NICI instrument at the Gemini-South Telescope.",
"Combining our VisAO $Y_S$ and NICI $CH_{4S,1\\%}$ , $K_S$ , and $K_{cont}$ photometry with previous measurements in $J,H$ and $K$ , we estimated that $\\beta \\mbox{ Pic b}$ has a spectral type of L$2.5\\pm 1.5$ .",
"In color-color and color-magnitude plots, $\\beta \\mbox{ Pic b}$ fits very well with other early-L dwarfs, perhaps being slightly redder in $H-K$ .",
"We used our spectral type estimate to evaluate the physical properties of $\\beta \\mbox{ Pic b}$ .",
"Using field brown dwarf bolometric corrections, we estimate $\\log (L/L_) = -3.86\\pm 0.04$ dex.",
"This is consistent with previous estimates.",
"Using hot-start evolutionary models at an age of $21\\pm 4$ Myr, our $L_{bol}$ measurement yields a mass estimate of $M=11.9 \\pm 0.7$ $M_{Jup}$ , with an upper limit at $M\\approx 13$ $M_{Jup}$ due to the model treatment of deuterium burning.",
"For temperature we find $T_{eff} = 1643 \\pm 32$ K. For radius we find $R = 1.43 \\pm 0.02$ $R_{Jup}$ .",
"All of these results are consistent with those of prior studies.",
"If we instead used the field BD sequence to estimate temperature, we would find a $T_{eff}$ $\\sim $ 250K hotter than expected from the evolutionary models.",
"The population of low surface gravity ultracool dwarfs and directly-imaged EGPs likewise have low effective temperatures compared to field brown dwarfs of similar spectral type.",
"However, these objects tend to be very red in near-IR colors, and so don't follow the field brown dwarf sequence in color-magnitude diagrams.",
"In contrast to other directly-imaged young EGPs (such as HR 8799 b and 2M 1207 b), $\\beta \\mbox{ Pic b}$ looks much more like a typical early L dwarf in the near-IR, both in terms of its colors and luminosity, despite its inferred low gravity and cooler temperature."
],
[
"Acknowledgements",
"We thank the anonymous referee for many helpful comments and suggestions which greatly improved this manuscript.",
"J.R.M.",
"is grateful for the generous support of the Phoenix ARCS Foundation.",
"J.R.M and K.M.M.",
"were supported under contract with the California Institute of Technology (Caltech) funded by NASA through the Sagan Fellowship Program.",
"L.M.C.",
"'s research was supported by NSF AAG and NASA Origins of Solar Systems grants.",
"A.J.S.",
"was supported by the NASA Origins of Solar Systems Program, grant NNX13AJ17G.",
"The MagAO ASM was developed with support from the NSF MRI program.",
"The MagAO PWFS was developed with help from the NSF TSIP program and the Magellan partners.",
"The Active Optics guider was developed by Carnegie Observatories with custom optics from the MagAO team.",
"The VisAO camera and commissioning were supported by the NSF ATI program.",
"We thank the LCO and Magellan staffs for their outstanding assistance throughout our commissioning runs.",
"We also thank the teams at the Steward Observatory Mirror Lab/CAAO (University of Arizona), Microgate (Italy), and ADS (Italy) for their contributions to the ASM.",
"The NICI campaign was supported in part by NSF grants AST-0713881 and AST-0709484 awarded to M. Liu, NASA Origins grant NNX11 AC31G awarded to M. Liu, and NSF grant AAG-1109114 awarded to L. Close.",
"The Gemini Observatory is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil), and CONICET (Argentina).",
"This work made use of the General Catalogue of Photometric Datahttp://obswww.unige.ch/gcpd/ [97].",
"This research has benefited from the SpeX Prism Spectral Libraries, maintained by Adam Burgasserhttp://pono.ucsd.edu/ adam/browndwarfs/spexprism.",
"We thank Davy Kirkpatrick for providing WISE spectra, Jackie Faherty for providing the 2M0355 spectrum, Travis Barman for the 2M1207b model, and Sasha Hinkley and Laurent Pueyo for the $\\kappa $ And B spectrum.",
"We made use of the Database of Ultracool Parallaxes maintained by Trent Dupuyhttps://www.cfa.harvard.edu/tdupuy/plx/Database_of_Ultracool_Parallaxes.html.",
"This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and NASA's Astrophysical Data System.",
"Synthetic Photometry and Conversions In this appendix we provide details of our synthetic photometry.",
"The primary purpose of this is to verify the methodology used for our analysis, but we also determine transformations between various filter systems used in brown dwarf and exoplanet imaging which may be useful to others.",
"Filters Table: Atmospheres.We obtained a transmission profile and atmospheric transmission profile appropriate for each site.",
"Table REF summarizes the atmosphere assumptions and models used.",
"We converted to photon-weighted “relative spectral response” (RSR) curves, using the following equation [11] $T(\\lambda ) = \\frac{1}{hc}\\lambda T_0(\\lambda ),$ where $T_0$ is the raw energy-weighted profile.",
"We calculate the central wavelength as $\\lambda _0 = \\frac{\\int _0^\\infty \\lambda T(\\lambda )d\\lambda }{ \\int _0^\\infty T(\\lambda )d\\lambda }.$ We calculate the effective width $\\Delta \\lambda $ , such that $F_{\\lambda }(\\lambda _0)\\Delta \\lambda = \\int _0^\\infty F_{\\lambda }(\\lambda )T(\\lambda )d\\lambda .$ To calculate the magnitude of some object with a spectrum given by $F_{\\lambda ,\\: obj}$ we used $m = -2.5\\log \\left[ \\frac{\\int _0^\\infty R(\\lambda )F_{\\lambda ,\\: obj}d\\lambda }{\\int _0^\\infty R(\\lambda )F_{\\lambda ,\\: vega}d\\lambda }\\right]$ using the Vega spectrum of [15], which has an uncertainty of 1.5%.",
"These calculations are summarized in Table REF .",
"We next describe details particular to the different photometric systems.",
"The $Y$ Band: The $Y$ band was first defined in [64].",
"We follow [83] and assume that the UKIDSS $Y$ filter defines the MKO system passband, as the largest number of published observations in this passband are from there (cf.",
"[27]).",
"This is a slightly narrow version of the filter.",
"The UKIDSS $Y$ RSR curve is provided in [63], which is already photo-normalized and includes an atmosphere appropriate for Mauna Kea.",
"We also consider the unfortunately named $Z$ filter used at Subaru/IRCS and Keck/NIRC2, which is actually in the $Y$ window rather than in the traditionally optical $Z/z$ band.",
"To add to the confusion the filter has been labeled with a lower-case $z$ , but the scanned filter curves and Alan Tokunaga's website http://www.ifa.hawaii.edu/~tokunaga/MKO-NIR_filter_set.html indicate that it was meant to be capital $Z$ .",
"In any case, it is a narrow version of the $Y$ passband.",
"Here we follow [83] and refer to it as $z_{1.1}$ to emphasize its location in the $Y$ window, and that it is not related to the optical bandpasses of the same name.",
"We used the same atmospheric assumptions as for the MKO system (see below).",
"Figure: Comparison of the VisAO Y S Y_S bandpass with other YY band filters.",
"Top: raw filter profiles.",
"The YY bandpass is from .",
"The “z 1.1 z_{1.1}” bandpass, used at Subaru/IRCS and Keck/NIRC2 (where it is called either “z” or “Z”).",
"Bottom: the filters after photon-weighting and including models for atmospheric transmission, here we show the UKIDSS “Y” RSR curve from , and the y p1 y_{p1} filter of PAN-STARRS .VisAO $\\mathbf {Y_S}$ : The VisAO Y-short ($Y_S$ ) filter is defined by a Melles-Griot long wavepass filter (LPF-950), which passes $\\lambda \\gtrsim 0.95\\mu \\mbox{m}$ , and the quantum efficiency (QE) limit of our near-IR coated EEV CCD47-20 detector.",
"We convolved the transmission curve with the QE for our EEV CCD47-20 (both provided by the respective manufacturers), and included the effects of 3 Al reflections.",
"We also include the Clio2 entrance window dichroic which reflects visible light to the WFS and VisAO, cutting on at 1.05$\\mu $ m. We next multiply the profile by a model of atmospheric transmission, using the 2.3 mm of precipitable water vapor (PWV), airmass (AM) 1.0 ATRAN model for Cerro Pachon provided by Gemini Observatoryhttp://www.gemini.edu/?q=node/10789 [87].",
"Cerro Pachon, $\\sim 2700\\mbox{m}$ , is slightly higher than the Magellan site at Cerro Manqui, $\\sim 2380\\mbox{m}$ (D. Osip, private communication), so this will slightly underestimate atmospheric absorption.",
"We finally determine the photon-weighted RSR.",
"We refer to this filter as “Y-short”, or $Y_S$ .",
"This is similar to the $y_{p1}$ bandpass of the PAN-STARRS optical survey [125].",
"$Y_S$ is $\\sim 22$ nm redder than $y_{p1}$ , and we prefer to emphasize that we are working in the $Y$ atmospheric window.",
"$Y$ band filter profiles are compared in Figure REF .",
"The $Y_S$ filter is slightly affected by telluric water vapor.",
"Using the ATRAN models we assessed the impact of changes in both AM and PWV.",
"The mean transmission changes by $\\pm 3\\%$ over the ranges $1.0 \\le \\mbox{ AM } \\le 1.5$ and $2.3 \\le \\mbox{ PWV } \\le 10.0$ mm.",
"This change in transmission has little impact on differential photometry so long as PWV does not change significantly between measurements.",
"AM has almost no effect on $\\lambda _0$ but changes in PWV do change it by 2 to 4 nm as expected given the $H_2O$ absorption band at $\\sim 0.94$ $\\mu \\mbox{m}$ .",
"This is relatively small and since we have no contemporaneous PWV measurements for the observations reported here we ignore this effect.",
"The 2MASS System: The 2MASS $J$ , $H$ , and $K_S$ transmission and RSR profiles were collected from the 2MASS websitehttp://www.ipac.caltech.edu/2mass/releases/allsky/doc/sec6_4a.html.",
"The RSR profiles are from [41].",
"They used an atmosphere based on the PLEXUS model for AM 1.0.",
"This model does not use a parameterization corresponding directly to PWV, but according to the website it is equivalent to 5.0 mm of PWV.",
"Table: Synthetic Photometric System CharacteristicsThe MKO System: We used the Mauna Kea filter profiles provided by the IRTF/NSFCam websitehttp://irtfweb.ifa.hawaii.edu/~nsfcam/filters.html for the MKO $J$ , $H$ , $K_S$ , and $K$ passbands which correspond to the 1998 production run of these filters.",
"We again used the ATRAN model atmosphere from Gemini Observatory, now for Mauna Kea with 1.6 mm precipitable water vapor (PWV) at AM 1.0.",
"The NACO System: We obtained transmission profiles for NACO from the instrument websitehttp://www.eso.org/sci/facilities/paranal/instruments/naco/inst/filters.html.",
"We used the “Paranal-like” atmosphere provided by ESOhttp://www.eso.org/sci/facilities/eelt/science/drm/tech_data/data/atm_abs/, which is for AM 1.0 and 2.3 mm of PWV.",
"The NACO filters are close to the 2MASS system, but there are subtle differences, which are somewhat more pronounced once the atmosphere appropriate for each site is included.",
"The NICI System: Profiles for the NICI filters were obtained from the instrument websites for NICIhttp://www.gemini.edu/sciops/instruments/nici/ and NIRIhttp://www.gemini.edu/sciops/instruments/niri/.",
"The Cerro Pachon ATRAN atmosphere was used, with AM 1.0 and 2.3 mm of PWV.",
"The NICI $J$ , $H$ , and $K_S$ bandpasses are intended to be in the MKO system and so should be insensitive to atmospheric conditions, but there are subtle differences between the filter profiles.",
"Photometric Conversions To quantify the differences between these systems and to accurately compare results for objects measured in the different systems, we used the library of brown dwarf spectra we compiled from various sources (described in Section REF ).",
"We calculated the magnitudes in each of the various filters and then fit a 4th or 5th order polynomial to the results.",
"Our notation is $m_1-m_2 = c_0 + c_1 \\mbox{SpT}+ c_2 \\mbox{SpT}^2+ c_3 \\mbox{SpT}^3+ c_4 \\mbox{SpT}^4 + c_5 \\mbox{SpT}^5$ where SpT is the near-IR spectral type given by $\\mbox{SpT} &=& 0 ... 9, \\mbox{ for } \\mbox{M0} ... \\mbox{M9}\\\\\\mbox{SpT} &=& 10 ... 19, \\mbox{ for } \\mbox{L0} ... \\mbox{L9}\\\\\\mbox{SpT} &=& 20 ... 29, \\mbox{ for } \\mbox{T0} ... \\mbox{T9}$ We provide the coefficients determined in this manner for a variety of transformations in Table REF .",
"Figure: Synthetic photometry (red points) in the 2MASS and MKO systems and measurements made in the 2 systems (crosses, from ).",
"We also plot the binned median of the measurements.",
"Our polynomial fit is shown as the solid black line.",
"For comparison we show the fit determined by , who also used synthetic photometry.",
"Our fit to the synthetic photometry appears to be a better match to the actual measurements in J.",
"In the H and K bands both fits appear to be acceptable, with being somewhat better for M and L dwarfs in H.As a check on our calculations consider the conversions from 2MASS to MKO.",
"There are many objects with measurements in both systems, which allows us to directly compare our synthetic photometry to actual measurements.",
"We here use the compilation of [51].",
"This also allows a comparison to the previous work of [123], who employed similar methodology to ours but with fewer objects.",
"The results are shown graphically in Figure REF .",
"In all three bands our synthetic photometry and fit appear to be a good match to the measurements.",
"In $J$ our results appear to be an improvement over [123], and in $H$ and $K$ either fit appears to be reasonable.",
"These results give confidence that our synthetic photometry reproduces the variations in these systems reasonably well.",
"Photometric conversion coefficients.",
"Table: NO_CAPTION Comparison Objects $\\mathbf {Y_S-K_S}$ Spectral Library: We analyzed the $Y_SJHK_S$ photometry of $\\beta $ Pic b by comparison with a library consisting of 499 brown dwarf spectra.",
"Of these: 441 are from the SpeX Prism Spectral libraries maintained by Adam Burgasserhttp://pono.ucsd.edu/~adam/browndwarfs/spexprism/ (from various sources), 23 are WISE brown dwarfs from [70], and 35 are young field BDs from [2].",
"We correlated 115 of these with parallax measurements, either listed in the SpeX Library (from various sources) or from [51].",
"We conducted synthetic photometry on these spectra as described above.",
"For the objects with parallaxes we normalized the spectra to available photometry.",
"In most cases there is a near-IR SpT assigned which we use here.",
"In the few cases where there is no near-IR SpT we use the optical SpT.",
"Figure: Optical and near-IR photometry and models of 2M 1207 b and HR 8799 b.",
"To estimate photometry in the Y atmospheric window, we used the models to calculate colors and then extrapolated or interpolated from the measured photometry.",
"Measured photometry is indicated by filled circles, and our estimated photometry is indicated by asterisks.",
"The 2M 1207 b model is from , and the HR 8799 b model is from .",
"See also Table .We also compiled very late-T and early-Y dwarf photometry from [83] and [79].",
"Here we lack spectra, so instead we use the above transformations from the MKO system determined using our compiled spectra.",
"Y Band Photometry of EGPs: The other young planetary mass companions with $Y$ band photometry are HR 8799 b and 2M 1207 b.",
"[46] detected HR 8799 b at $1.04 \\mu \\mbox{m}$ in the $z_{1.1}$ filter with Subaru/IRCS.",
"[103] also have the ability to work in the $Y$ band with Project 1640, and reported low S/N detections of HR 8799 b and HR8799 c at $1.05 \\mu \\mbox{m}$ .",
"2M 1207 b was observed with the Hubble Space Telescope by [119].",
"To compare our results we must convert these measurements into our $Y_S$ bandpass.",
"The SEDs of these two objects do not correspond to those of field brown dwarfs of comparable temperature, so we turn to published best-fit model spectra: for 2M 1207 b we use the model of [9] and for HR 8799 b we use the best-fit model from [90].",
"We use the models to calculate color between bandpasses.",
"We illustrate this in Figure REF and summarize the results in Table REF .",
"Table: YY-band photometry of 2M 1207 b and HR 8799 bOther Comparison Objects: We also collected a number of low-temperature, low-mass objects from the literature to compare to $\\beta \\mbox{ Pic b}$ , which are listed in Table REF .",
"Most of these are companions.",
"Of particular interest are the low-surface gravity objects.",
"This includes the four HR 8799 planets and the planetary mass BD companion 2M1207b.",
"Two other companions with photometric properties similar to these faint red companions are AB Pic b [35] and 2MASS 0122-2439B [21].",
"Both [19] and [47] noted that the near-IR SED of $\\beta \\mbox{ Pic b}$ resembled that of an early to mid-L dwarf.",
"For comparison then, we use companions such as $\\kappa $ And B [18], [65]) and CD-35 2722 B [128]).",
"We used the Project 1640 spectrum of $\\kappa $ And B from [65] to estimate its Y band photometry.",
"We also highlight several field dwarfs.",
"The L0$\\gamma $ object 2MASS 0608-2753 is believed to be a member of the $\\beta $ Pic moving group [112], giving us a potentially co-eval object to compare with $\\beta \\mbox{ Pic b}$ .",
"We include two field objects which appear to have low surface gravity, dusty photospheres, and are young moving group members: 2MASS 0355+1133 [54], [86] and PSO318.5 [86].",
"PSO318.5 is also a possible member of the $\\beta $ Pic moving group.",
"2MASS 0032-4405, an L0$\\gamma $ , was discussed as a possible proxy for both $\\beta \\mbox{ Pic b}$ and $\\kappa $ And B by [19].",
"Comparison Objects Table: NO_CAPTION VisAO Occulting Mask Transmission We first measured the mask transmission and PSF by scanning an artificial test source across the mask with the instrument off the telescope.",
"Our internal alignment source has a slightly faster f/#, resulting in smaller FWHM (it was originally designed for the LBTAO systems), and delivers a Strehl $>90$ %.",
"The source was scanned along 12 different lines, spaced roughly 30 degrees apart, across the mask in the $Y_S$ filter.",
"We found the center of the mask by determining the x,y position which best symmetrizes all the scans simultaneously.",
"We measured attenuation of the mask using a 3 pixel radius photometric aperture.",
"The results of our laboratory scans are shown in Figure REF .",
"The maximum attenuation is 0.0015, or ND = 2.8.",
"Figure: Transmission of the VisAO occulting mask.",
"The dot-dashed line is the median smoothed profile measured in the laboratory (Strehl >90>90%).",
"The solid black line is a piecewise polynomial fit to the on-sky scans with 300 modes of wavefront correction.",
"The individual points correspond to measurements of β\\beta Pic A and binaries.",
"The red curve is our boot-strapped profile for 200 modes of wavefront correction.",
"The separation of βPicb\\beta \\mbox{ Pic b} is ∼59\\sim 59 pixels ≈0.47 '' \\approx 0.47^{\\prime \\prime }, which gives a mask transmission of 0.60 -0.10 +0.05 0.60^{+0.05}_{-0.10}.During the first-light commissioning run (Comm1, Nov.–Dec.",
"2012), we observed a $0.28^{\\prime \\prime }$ , $\\Delta Y_S \\sim 3.5$ binary both on and off the mask, providing a single transmission measurement.",
"The center of the mask was found using AO-off (seeing-limited) acquisition images, which show a well defined circular shadow.",
"This is also plotted in Figure REF , and is over 200% higher than expected based on the lab scans.",
"The key point is that this measurement was done with a 200 mode reconstructor, with gains applied to only the first 120 modes.",
"We can also use acquisition images of $\\beta $ Pic A, recorded during coronagraph alignment.",
"These sample much closer in than the position of the planet, but provide guidance on the changes in the transmission profile due to wavefront correction.",
"We also include the median transmission of $\\beta $ Pic A centered on the mask, which was measured using the science exposures.",
"During the $\\beta $ Pic observations the same reconstructor was used as for the $0.28^{\\prime \\prime }$ binary, but gains were applied to all 200 modes.",
"We thus expect slightly better correction.",
"During our 2nd commissioning run (Comm2, Mar.–Apr.",
"2013), we scanned a star across the mask with the AO loop closed in the $Y_S$ filter.",
"The caveat to these measurements is that we used a 300 mode reconstructor, significantly improving wavefront correction over the 200 mode matrix in use on Comm1.",
"We took two scans, separated by 90 degrees, and found the best-fit center of symmetry.",
"We corrected for Strehl ratio variations using the un-occulted beamsplitter ghost, measured with the same 3 pixel radius aperture.",
"The result, also shown in Figure REF , is intermediate between the lab and the $0.28^{\\prime \\prime }$ binary.",
"We also observed a $0.64^{\\prime \\prime }$ binary on and off the mask during Comm2, with the 300 mode reconstructor.",
"The mask transmission measured on this binary was higher than for both the lab and on-sky scans.",
"Figure: Ratio of mask transmission at lower wavefront correction quality to the transmission measured on-sky with a 300 modes.",
"The solid line is a piecewise linear function which we use to form a boot-strap estimate of transmission under 200 modes of wavefront correction.",
"This results in the red curve shown in Figure .In Figure REF we show the ratio of coronagraph transmission for $\\beta $ Pic A (200 modes), the $0.28^{\\prime \\prime }$ binary (120 modes), and the $0.64^{\\prime \\prime }$ binary (300 modes) to the transmission profile measured on-sky (300 modes).",
"To form an estimate of the complete transmission profile under 200 modes of wavefront control, we fit this ratio with a piecewise linear function, which we then multiply by the on-sky profile.",
"Our fit is obviously not the only functional form which could be used to describe the transmission change from 300 to 200 modes of correction, but it is the simplest estimate we can make given the data.",
"We note that even significant changes in the ratio close to the center result in relatively small changes in the transmission at 59 pixels, the separation of $\\beta \\mbox{ Pic b}$ , due to the contraints at wider separations.",
"The resultant boot-strapped profile is also shown in REF .",
"Using it, our adopted transmission at the location of $\\beta \\mbox{ Pic b}$ is $0.60^{+0.05}_{-0.10}$ .",
"The final consideration when working under the mask is that it changes the PSF, causing an elongation in the radial direction.",
"We measured this both in the lab and on-sky by fitting an elliptical Gaussian at each point in the scans.",
"The result is shown in Figure REF , along with an on-sky image of a star at roughly the separation of $\\beta \\mbox{ Pic b}$ shown earlier in Figure REF .",
"This change in shape was taken into account when conducting photometry on the planet.",
"Figure: PSF shape measurements.",
"The broad apodized transmission profile results in a PSF which varies with distance from the coronagraph center.",
"We plot the ratio of FWHMs of the elliptical Gaussian which best fits the PSF at each location.",
"On-sky, the ratio does not reach 1.0, as there is usually an elongation in the wind direction.",
"Correction quality also appears to have an effect on shape, as the on-sky measurements have a lower peak FWHM ratio.",
"See also Figure ."
],
[
"Synthetic Photometry and Conversions",
"In this appendix we provide details of our synthetic photometry.",
"The primary purpose of this is to verify the methodology used for our analysis, but we also determine transformations between various filter systems used in brown dwarf and exoplanet imaging which may be useful to others."
],
[
"Filters",
"We obtained a transmission profile and atmospheric transmission profile appropriate for each site.",
"Table REF summarizes the atmosphere assumptions and models used.",
"We converted to photon-weighted “relative spectral response” (RSR) curves, using the following equation [11] $T(\\lambda ) = \\frac{1}{hc}\\lambda T_0(\\lambda ),$ where $T_0$ is the raw energy-weighted profile.",
"We calculate the central wavelength as $\\lambda _0 = \\frac{\\int _0^\\infty \\lambda T(\\lambda )d\\lambda }{ \\int _0^\\infty T(\\lambda )d\\lambda }.$ We calculate the effective width $\\Delta \\lambda $ , such that $F_{\\lambda }(\\lambda _0)\\Delta \\lambda = \\int _0^\\infty F_{\\lambda }(\\lambda )T(\\lambda )d\\lambda .$ To calculate the magnitude of some object with a spectrum given by $F_{\\lambda ,\\: obj}$ we used $m = -2.5\\log \\left[ \\frac{\\int _0^\\infty R(\\lambda )F_{\\lambda ,\\: obj}d\\lambda }{\\int _0^\\infty R(\\lambda )F_{\\lambda ,\\: vega}d\\lambda }\\right]$ using the Vega spectrum of [15], which has an uncertainty of 1.5%.",
"These calculations are summarized in Table REF .",
"We next describe details particular to the different photometric systems.",
"The $Y$ Band: The $Y$ band was first defined in [64].",
"We follow [83] and assume that the UKIDSS $Y$ filter defines the MKO system passband, as the largest number of published observations in this passband are from there (cf.",
"[27]).",
"This is a slightly narrow version of the filter.",
"The UKIDSS $Y$ RSR curve is provided in [63], which is already photo-normalized and includes an atmosphere appropriate for Mauna Kea.",
"We also consider the unfortunately named $Z$ filter used at Subaru/IRCS and Keck/NIRC2, which is actually in the $Y$ window rather than in the traditionally optical $Z/z$ band.",
"To add to the confusion the filter has been labeled with a lower-case $z$ , but the scanned filter curves and Alan Tokunaga's website http://www.ifa.hawaii.edu/~tokunaga/MKO-NIR_filter_set.html indicate that it was meant to be capital $Z$ .",
"In any case, it is a narrow version of the $Y$ passband.",
"Here we follow [83] and refer to it as $z_{1.1}$ to emphasize its location in the $Y$ window, and that it is not related to the optical bandpasses of the same name.",
"We used the same atmospheric assumptions as for the MKO system (see below).",
"Figure: Comparison of the VisAO Y S Y_S bandpass with other YY band filters.",
"Top: raw filter profiles.",
"The YY bandpass is from .",
"The “z 1.1 z_{1.1}” bandpass, used at Subaru/IRCS and Keck/NIRC2 (where it is called either “z” or “Z”).",
"Bottom: the filters after photon-weighting and including models for atmospheric transmission, here we show the UKIDSS “Y” RSR curve from , and the y p1 y_{p1} filter of PAN-STARRS .VisAO $\\mathbf {Y_S}$ : The VisAO Y-short ($Y_S$ ) filter is defined by a Melles-Griot long wavepass filter (LPF-950), which passes $\\lambda \\gtrsim 0.95\\mu \\mbox{m}$ , and the quantum efficiency (QE) limit of our near-IR coated EEV CCD47-20 detector.",
"We convolved the transmission curve with the QE for our EEV CCD47-20 (both provided by the respective manufacturers), and included the effects of 3 Al reflections.",
"We also include the Clio2 entrance window dichroic which reflects visible light to the WFS and VisAO, cutting on at 1.05$\\mu $ m. We next multiply the profile by a model of atmospheric transmission, using the 2.3 mm of precipitable water vapor (PWV), airmass (AM) 1.0 ATRAN model for Cerro Pachon provided by Gemini Observatoryhttp://www.gemini.edu/?q=node/10789 [87].",
"Cerro Pachon, $\\sim 2700\\mbox{m}$ , is slightly higher than the Magellan site at Cerro Manqui, $\\sim 2380\\mbox{m}$ (D. Osip, private communication), so this will slightly underestimate atmospheric absorption.",
"We finally determine the photon-weighted RSR.",
"We refer to this filter as “Y-short”, or $Y_S$ .",
"This is similar to the $y_{p1}$ bandpass of the PAN-STARRS optical survey [125].",
"$Y_S$ is $\\sim 22$ nm redder than $y_{p1}$ , and we prefer to emphasize that we are working in the $Y$ atmospheric window.",
"$Y$ band filter profiles are compared in Figure REF .",
"The $Y_S$ filter is slightly affected by telluric water vapor.",
"Using the ATRAN models we assessed the impact of changes in both AM and PWV.",
"The mean transmission changes by $\\pm 3\\%$ over the ranges $1.0 \\le \\mbox{ AM } \\le 1.5$ and $2.3 \\le \\mbox{ PWV } \\le 10.0$ mm.",
"This change in transmission has little impact on differential photometry so long as PWV does not change significantly between measurements.",
"AM has almost no effect on $\\lambda _0$ but changes in PWV do change it by 2 to 4 nm as expected given the $H_2O$ absorption band at $\\sim 0.94$ $\\mu \\mbox{m}$ .",
"This is relatively small and since we have no contemporaneous PWV measurements for the observations reported here we ignore this effect.",
"The 2MASS System: The 2MASS $J$ , $H$ , and $K_S$ transmission and RSR profiles were collected from the 2MASS websitehttp://www.ipac.caltech.edu/2mass/releases/allsky/doc/sec6_4a.html.",
"The RSR profiles are from [41].",
"They used an atmosphere based on the PLEXUS model for AM 1.0.",
"This model does not use a parameterization corresponding directly to PWV, but according to the website it is equivalent to 5.0 mm of PWV.",
"Table: Synthetic Photometric System CharacteristicsThe MKO System: We used the Mauna Kea filter profiles provided by the IRTF/NSFCam websitehttp://irtfweb.ifa.hawaii.edu/~nsfcam/filters.html for the MKO $J$ , $H$ , $K_S$ , and $K$ passbands which correspond to the 1998 production run of these filters.",
"We again used the ATRAN model atmosphere from Gemini Observatory, now for Mauna Kea with 1.6 mm precipitable water vapor (PWV) at AM 1.0.",
"The NACO System: We obtained transmission profiles for NACO from the instrument websitehttp://www.eso.org/sci/facilities/paranal/instruments/naco/inst/filters.html.",
"We used the “Paranal-like” atmosphere provided by ESOhttp://www.eso.org/sci/facilities/eelt/science/drm/tech_data/data/atm_abs/, which is for AM 1.0 and 2.3 mm of PWV.",
"The NACO filters are close to the 2MASS system, but there are subtle differences, which are somewhat more pronounced once the atmosphere appropriate for each site is included.",
"The NICI System: Profiles for the NICI filters were obtained from the instrument websites for NICIhttp://www.gemini.edu/sciops/instruments/nici/ and NIRIhttp://www.gemini.edu/sciops/instruments/niri/.",
"The Cerro Pachon ATRAN atmosphere was used, with AM 1.0 and 2.3 mm of PWV.",
"The NICI $J$ , $H$ , and $K_S$ bandpasses are intended to be in the MKO system and so should be insensitive to atmospheric conditions, but there are subtle differences between the filter profiles."
],
[
"Photometric Conversions",
"To quantify the differences between these systems and to accurately compare results for objects measured in the different systems, we used the library of brown dwarf spectra we compiled from various sources (described in Section REF ).",
"We calculated the magnitudes in each of the various filters and then fit a 4th or 5th order polynomial to the results.",
"Our notation is $m_1-m_2 = c_0 + c_1 \\mbox{SpT}+ c_2 \\mbox{SpT}^2+ c_3 \\mbox{SpT}^3+ c_4 \\mbox{SpT}^4 + c_5 \\mbox{SpT}^5$ where SpT is the near-IR spectral type given by $\\mbox{SpT} &=& 0 ... 9, \\mbox{ for } \\mbox{M0} ... \\mbox{M9}\\\\\\mbox{SpT} &=& 10 ... 19, \\mbox{ for } \\mbox{L0} ... \\mbox{L9}\\\\\\mbox{SpT} &=& 20 ... 29, \\mbox{ for } \\mbox{T0} ... \\mbox{T9}$ We provide the coefficients determined in this manner for a variety of transformations in Table REF .",
"Figure: Synthetic photometry (red points) in the 2MASS and MKO systems and measurements made in the 2 systems (crosses, from ).",
"We also plot the binned median of the measurements.",
"Our polynomial fit is shown as the solid black line.",
"For comparison we show the fit determined by , who also used synthetic photometry.",
"Our fit to the synthetic photometry appears to be a better match to the actual measurements in J.",
"In the H and K bands both fits appear to be acceptable, with being somewhat better for M and L dwarfs in H.As a check on our calculations consider the conversions from 2MASS to MKO.",
"There are many objects with measurements in both systems, which allows us to directly compare our synthetic photometry to actual measurements.",
"We here use the compilation of [51].",
"This also allows a comparison to the previous work of [123], who employed similar methodology to ours but with fewer objects.",
"The results are shown graphically in Figure REF .",
"In all three bands our synthetic photometry and fit appear to be a good match to the measurements.",
"In $J$ our results appear to be an improvement over [123], and in $H$ and $K$ either fit appears to be reasonable.",
"These results give confidence that our synthetic photometry reproduces the variations in these systems reasonably well.",
"Photometric conversion coefficients.",
"Table: NO_CAPTION"
],
[
"Comparison Objects",
"$\\mathbf {Y_S-K_S}$ Spectral Library: We analyzed the $Y_SJHK_S$ photometry of $\\beta $ Pic b by comparison with a library consisting of 499 brown dwarf spectra.",
"Of these: 441 are from the SpeX Prism Spectral libraries maintained by Adam Burgasserhttp://pono.ucsd.edu/~adam/browndwarfs/spexprism/ (from various sources), 23 are WISE brown dwarfs from [70], and 35 are young field BDs from [2].",
"We correlated 115 of these with parallax measurements, either listed in the SpeX Library (from various sources) or from [51].",
"We conducted synthetic photometry on these spectra as described above.",
"For the objects with parallaxes we normalized the spectra to available photometry.",
"In most cases there is a near-IR SpT assigned which we use here.",
"In the few cases where there is no near-IR SpT we use the optical SpT.",
"Figure: Optical and near-IR photometry and models of 2M 1207 b and HR 8799 b.",
"To estimate photometry in the Y atmospheric window, we used the models to calculate colors and then extrapolated or interpolated from the measured photometry.",
"Measured photometry is indicated by filled circles, and our estimated photometry is indicated by asterisks.",
"The 2M 1207 b model is from , and the HR 8799 b model is from .",
"See also Table .We also compiled very late-T and early-Y dwarf photometry from [83] and [79].",
"Here we lack spectra, so instead we use the above transformations from the MKO system determined using our compiled spectra.",
"Y Band Photometry of EGPs: The other young planetary mass companions with $Y$ band photometry are HR 8799 b and 2M 1207 b.",
"[46] detected HR 8799 b at $1.04 \\mu \\mbox{m}$ in the $z_{1.1}$ filter with Subaru/IRCS.",
"[103] also have the ability to work in the $Y$ band with Project 1640, and reported low S/N detections of HR 8799 b and HR8799 c at $1.05 \\mu \\mbox{m}$ .",
"2M 1207 b was observed with the Hubble Space Telescope by [119].",
"To compare our results we must convert these measurements into our $Y_S$ bandpass.",
"The SEDs of these two objects do not correspond to those of field brown dwarfs of comparable temperature, so we turn to published best-fit model spectra: for 2M 1207 b we use the model of [9] and for HR 8799 b we use the best-fit model from [90].",
"We use the models to calculate color between bandpasses.",
"We illustrate this in Figure REF and summarize the results in Table REF .",
"Table: YY-band photometry of 2M 1207 b and HR 8799 bOther Comparison Objects: We also collected a number of low-temperature, low-mass objects from the literature to compare to $\\beta \\mbox{ Pic b}$ , which are listed in Table REF .",
"Most of these are companions.",
"Of particular interest are the low-surface gravity objects.",
"This includes the four HR 8799 planets and the planetary mass BD companion 2M1207b.",
"Two other companions with photometric properties similar to these faint red companions are AB Pic b [35] and 2MASS 0122-2439B [21].",
"Both [19] and [47] noted that the near-IR SED of $\\beta \\mbox{ Pic b}$ resembled that of an early to mid-L dwarf.",
"For comparison then, we use companions such as $\\kappa $ And B [18], [65]) and CD-35 2722 B [128]).",
"We used the Project 1640 spectrum of $\\kappa $ And B from [65] to estimate its Y band photometry.",
"We also highlight several field dwarfs.",
"The L0$\\gamma $ object 2MASS 0608-2753 is believed to be a member of the $\\beta $ Pic moving group [112], giving us a potentially co-eval object to compare with $\\beta \\mbox{ Pic b}$ .",
"We include two field objects which appear to have low surface gravity, dusty photospheres, and are young moving group members: 2MASS 0355+1133 [54], [86] and PSO318.5 [86].",
"PSO318.5 is also a possible member of the $\\beta $ Pic moving group.",
"2MASS 0032-4405, an L0$\\gamma $ , was discussed as a possible proxy for both $\\beta \\mbox{ Pic b}$ and $\\kappa $ And B by [19].",
"Comparison Objects Table: NO_CAPTION"
],
[
"VisAO Occulting Mask Transmission",
"We first measured the mask transmission and PSF by scanning an artificial test source across the mask with the instrument off the telescope.",
"Our internal alignment source has a slightly faster f/#, resulting in smaller FWHM (it was originally designed for the LBTAO systems), and delivers a Strehl $>90$ %.",
"The source was scanned along 12 different lines, spaced roughly 30 degrees apart, across the mask in the $Y_S$ filter.",
"We found the center of the mask by determining the x,y position which best symmetrizes all the scans simultaneously.",
"We measured attenuation of the mask using a 3 pixel radius photometric aperture.",
"The results of our laboratory scans are shown in Figure REF .",
"The maximum attenuation is 0.0015, or ND = 2.8.",
"Figure: Transmission of the VisAO occulting mask.",
"The dot-dashed line is the median smoothed profile measured in the laboratory (Strehl >90>90%).",
"The solid black line is a piecewise polynomial fit to the on-sky scans with 300 modes of wavefront correction.",
"The individual points correspond to measurements of β\\beta Pic A and binaries.",
"The red curve is our boot-strapped profile for 200 modes of wavefront correction.",
"The separation of βPicb\\beta \\mbox{ Pic b} is ∼59\\sim 59 pixels ≈0.47 '' \\approx 0.47^{\\prime \\prime }, which gives a mask transmission of 0.60 -0.10 +0.05 0.60^{+0.05}_{-0.10}.During the first-light commissioning run (Comm1, Nov.–Dec.",
"2012), we observed a $0.28^{\\prime \\prime }$ , $\\Delta Y_S \\sim 3.5$ binary both on and off the mask, providing a single transmission measurement.",
"The center of the mask was found using AO-off (seeing-limited) acquisition images, which show a well defined circular shadow.",
"This is also plotted in Figure REF , and is over 200% higher than expected based on the lab scans.",
"The key point is that this measurement was done with a 200 mode reconstructor, with gains applied to only the first 120 modes.",
"We can also use acquisition images of $\\beta $ Pic A, recorded during coronagraph alignment.",
"These sample much closer in than the position of the planet, but provide guidance on the changes in the transmission profile due to wavefront correction.",
"We also include the median transmission of $\\beta $ Pic A centered on the mask, which was measured using the science exposures.",
"During the $\\beta $ Pic observations the same reconstructor was used as for the $0.28^{\\prime \\prime }$ binary, but gains were applied to all 200 modes.",
"We thus expect slightly better correction.",
"During our 2nd commissioning run (Comm2, Mar.–Apr.",
"2013), we scanned a star across the mask with the AO loop closed in the $Y_S$ filter.",
"The caveat to these measurements is that we used a 300 mode reconstructor, significantly improving wavefront correction over the 200 mode matrix in use on Comm1.",
"We took two scans, separated by 90 degrees, and found the best-fit center of symmetry.",
"We corrected for Strehl ratio variations using the un-occulted beamsplitter ghost, measured with the same 3 pixel radius aperture.",
"The result, also shown in Figure REF , is intermediate between the lab and the $0.28^{\\prime \\prime }$ binary.",
"We also observed a $0.64^{\\prime \\prime }$ binary on and off the mask during Comm2, with the 300 mode reconstructor.",
"The mask transmission measured on this binary was higher than for both the lab and on-sky scans.",
"Figure: Ratio of mask transmission at lower wavefront correction quality to the transmission measured on-sky with a 300 modes.",
"The solid line is a piecewise linear function which we use to form a boot-strap estimate of transmission under 200 modes of wavefront correction.",
"This results in the red curve shown in Figure .In Figure REF we show the ratio of coronagraph transmission for $\\beta $ Pic A (200 modes), the $0.28^{\\prime \\prime }$ binary (120 modes), and the $0.64^{\\prime \\prime }$ binary (300 modes) to the transmission profile measured on-sky (300 modes).",
"To form an estimate of the complete transmission profile under 200 modes of wavefront control, we fit this ratio with a piecewise linear function, which we then multiply by the on-sky profile.",
"Our fit is obviously not the only functional form which could be used to describe the transmission change from 300 to 200 modes of correction, but it is the simplest estimate we can make given the data.",
"We note that even significant changes in the ratio close to the center result in relatively small changes in the transmission at 59 pixels, the separation of $\\beta \\mbox{ Pic b}$ , due to the contraints at wider separations.",
"The resultant boot-strapped profile is also shown in REF .",
"Using it, our adopted transmission at the location of $\\beta \\mbox{ Pic b}$ is $0.60^{+0.05}_{-0.10}$ .",
"The final consideration when working under the mask is that it changes the PSF, causing an elongation in the radial direction.",
"We measured this both in the lab and on-sky by fitting an elliptical Gaussian at each point in the scans.",
"The result is shown in Figure REF , along with an on-sky image of a star at roughly the separation of $\\beta \\mbox{ Pic b}$ shown earlier in Figure REF .",
"This change in shape was taken into account when conducting photometry on the planet.",
"Figure: PSF shape measurements.",
"The broad apodized transmission profile results in a PSF which varies with distance from the coronagraph center.",
"We plot the ratio of FWHMs of the elliptical Gaussian which best fits the PSF at each location.",
"On-sky, the ratio does not reach 1.0, as there is usually an elongation in the wind direction.",
"Correction quality also appears to have an effect on shape, as the on-sky measurements have a lower peak FWHM ratio.",
"See also Figure ."
]
] | 1403.0560 |
[
[
"Well-posedness and Robust Preconditioners for the Discretized\n Fluid-Structure Interaction Systems"
],
[
"Abstract In this paper we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models.",
"After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness.",
"Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions.",
"The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems.",
"Numerical examples are presented to show the robustness and efficiency of these preconditioners."
],
[
"Introduction",
"Fluid-structure interaction (FSI) is a much studied topic aimed at understanding the interaction between some moving structure and fluid and how their interaction affects the interface between them.",
"FSI has a wide range of applications in many areas including hemodynamics [26], [43], [44], [18] and wind/hydro turbines [8], [32], [7], [6].",
"FSI problems are computationally challenging.",
"The computational domain of FSI consists of fluid and structure subdomains.",
"The position of the interface between fluid domain and structure domain is time dependent.",
"Therefore, the shape of the fluid domain is one of the unknowns, increasing the nonlinearity of the FSI problems.",
"Many numerical approaches have been proposed to tackle the interface problem of FSI.",
"The arbitrary Lagrangian-Eulerian (ALE) method is commonly used.",
"ALE adapts the fluid mesh to match the displacement of structure on interface.",
"Other approaches, such as the fictitious domain method [29], [53] and the immersed boundary method [54], [49], [41], have inconsistent fluid and structure meshes and, therefore, need special treatment at the interface, such as interpolation between different meshes.",
"In this paper, we focus on the ALE method.",
"There is much research focused solving fluid-structure interaction problem numerically using ALE formulation.",
"These studies can be roughly classified into partitioned approaches and monolithic approaches [22].",
"Partitioned approaches employ single-physics solvers to solve the fluid and structure problems separately and then couple them by the interface conditions.",
"Monolithic approaches solve the fluid and structure problems simultaneously.",
"Depending on whether the interface conditions are exactly enforced at every time step, these approaches can also be classified into weakly and strongly coupled algorithms.",
"Weakly coupled partitioned approaches are usually considered unstable due to the added-mass effect [15].",
"A semi-implicit approach proposed in [23] can avoid the added-mass effect for a wide range of applications, but it is subject to pressure boundary conditions.",
"Several types of semi-implicit methods were proposed in [42], [37].",
"Strongly coupled approaches are preferred for their stability.",
"Although it is possible to achieve the strong coupling via partitioned solvers (by fixed-point iteration, for example), they usually introduce prohibitive computational costs due to slow convergence [25].",
"In this paper we consider strongly coupled monolithic approaches and address some solver issues.",
"Monolithic approaches give us larger linear systems, for which efficient solvers are needed.",
"A great deal of work has been carried out to develop monolithic solvers for FSI [27], [47], [14], [5].",
"In [30], a fully-coupled solution strategy is proposed to solve the FSI problem with large structure displacement.",
"The nonlinearity is handled by Newton's method and various approaches to solve the Jacobian system are proposed.",
"Block triangular preconditioners and pressure Schur complement preconditioners are used for the preconditioned Krylov subspace solvers.",
"However, in [27] it is pointed out that block preconditioning for fluid and structure separately cannot resolve the coupling between fields and it is proposed that structure degrees of freedoms on interface be eliminated in order to effectively precondition degrees of freedom at the interface.",
"In [5], [3], [4], a Newton-Krylov-Schwarz method for FSI is developed.",
"Additive Schwarz preconditioners are used for Krylov subspace solvers and two-level methods are also developed.",
"In [1], [2], ILU preconditioners and inexact block-LU preconditioners are proposed to solve FSI problems.",
"In this paper, we reformulate semi-discretized systems of FSI as saddle point problems with fluid velocity, pressure and structure velocity as unknowns.",
"The ALE mapping is decoupled from the solution of the velocity and pressure.",
"Then, we carry out our theoretical analysis and solver design under this framework.",
"With particular choice of norms, we prove that the saddle point problem is well-posed.",
"For the finite element discretization of FSI, we propose two approaches to prove the well-posedness.",
"The first introduces a stabilization term to the fluid equations and the second adopts a norm of the velocity space that depends on the choice of the pressure space.",
"Both of these approaches lead to uniform well-posedness of the finite element discretization of the FSI model under appropriate assumptions.",
"Based on the uniform well-posedness, we propose optimal preconditioners based on the framework in [36], [55] such that the preconditioned linear systems have uniformly bounded condition numbers.",
"Then, we compare the proposed preconditioners with the augmented Lagrangian preconditioners [11], [9], [10], [40].",
"To test the preconditioners, we solve the linear systems coming from the discretization of the Turek and Hron benchmark problems [48].",
"The iteration counts of GMRes with several preconditioners are compared.",
"The rest of this paper is organized as follows.",
"In section 2, we introduce an FSI model and the ALE method.",
"In section 3, we study the proposed time and space discretization and its well-posedness.",
"In section 4, we propose optimal preconditioners for the discretized systems and demonstrate their performance with numerical examples."
],
[
"An FSI model",
"We consider a domain $\\Omega \\subset \\mathbb {R}^N(N=2,3)$ with a fluid occupying the upper half $\\Omega _f$ and a solid occupying the lower half $\\Omega _s$ , as illustrated in Figure REF .",
"Figure: Moving domains of FSILet $\\Gamma :=\\partial \\Omega _f\\cap \\partial \\Omega _s$ be the interface of the fluid domain and the solid domain.",
"On the outer boundary of the solid $\\partial \\Omega _s\\backslash \\Gamma $ , the solid is clamped; namely, the displacement of the solid is zero on $\\partial \\Omega _s\\backslash \\Gamma $ .",
"In this paper, we always assume that both $\\partial \\Omega _s\\backslash \\Gamma $ and $\\partial \\Omega _f\\backslash \\Gamma $ have positive measures.",
"In addition, we assume that the interaction of the fluid and solid only occurs at the interface, and the interface $\\Gamma $ may move over time due to this interaction.",
"We assume that the outer boundary is fixed.",
"In the dynamic setting, we use $\\Omega _f(t)$ and $\\Omega _s(t)$ to denote the domains at time $t\\in [0,T]$ .",
"The domains satisfy $\\bar{\\Omega }=\\bar{\\Omega }_f(t)\\cup \\bar{\\Omega }_s(t)$ and $\\Gamma (t)=\\partial \\Omega _f(t)\\cap \\partial \\Omega _s(t)$ .",
"We denote the reference domains by $\\hat{\\Omega }_f=\\Omega _f(0),\\quad \\hat{\\Omega }_s=\\Omega _s(0)$ and the domains at time $t$ by $\\Omega _f=\\Omega _f(t),\\quad \\Omega _s=\\Omega _s(t).$ The motion in the fluid and structure can be characterized by a flow map $\\mathbf {x}(\\hat{\\mathbf {x}},t)$ ; namely, the position of the particle $\\hat{\\mathbf {x}}$ at time $t$ is $\\mathbf {x}(\\hat{\\mathbf {x}},t)$ .",
"Then, given $t>0$ , $\\mathbf {x}(\\cdot ,t)$ is a diffeomorphism from $\\Omega (0)$ to $\\Omega (t)$ .",
"For $({\\hat{\\mathbf {x}}},t) \\in \\Omega (0)\\times [0,T]$ , we introduce the following variables in Lagrangian coordinates : the displacement $\\hat{\\mathbf {u}}({{\\hat{\\mathbf {x}}}},t)={\\mathbf {x}}({\\hat{\\mathbf {x}}},t)-{\\hat{\\mathbf {x}}}$ , the velocity $\\displaystyle \\hat{\\mathbf {v}}({\\hat{\\mathbf {x}}},t)=\\frac{\\partial \\hat{\\mathbf {x}}}{\\partial t}$ , the deformation tensor $\\displaystyle {F}({\\hat{\\mathbf {x}}},t)=\\frac{\\partial {\\mathbf {x}}}{\\partial {\\hat{\\mathbf {x}}}}({\\hat{\\mathbf {x}}},t)$ , and its determinant $\\displaystyle {J}({\\hat{\\mathbf {x}}},t)=det({F}({\\hat{\\mathbf {x}}},t))$ .",
"Using the relationship $\\mathbf {x}=\\mathbf {x}(\\hat{\\mathbf {x}},t)$ , we also introduce the velocity in Eulerian coordinates: ${\\mathbf {v}}({\\mathbf {x}},t)=\\hat{\\mathbf {v}}({\\hat{\\mathbf {x}}},t)$ .",
"The symmetric part of the gradient is denoted by $\\displaystyle \\epsilon (\\mathbf {v})=\\frac{\\nabla \\mathbf {v}+(\\nabla \\mathbf {v})^T}{2}.",
"$ Let us now introduce a simple FSI model which consists of the incompressible Navier-Stokes equations for the fluid (in Eulerian coordinates) and linear elasticity equations for the structure (in Lagrangian coordinates).",
"For clarity, we start with the momentum equations for fluid and solid both in Eulerian coordinates: $\\rho _f D_t\\mathbf {v}_f-\\nabla \\cdot \\sigma _f=g_f,\\quad \\mbox{in}\\Omega _f,$ and $\\rho _s D_t\\mathbf {v}_s-\\nabla \\cdot \\sigma _s=g_s,\\quad \\mbox{in}\\Omega _s.$ Here $\\sigma _f$ and $\\sigma _s$ are the Cauchy stress tensors for fluid and structure, respectively.",
"Here $D_t\\mathbf {v}_f$ and $D_t\\mathbf {v}_s$ are the material derivatives.",
"On the interface $\\Gamma =\\partial \\Omega _f\\cap \\partial \\Omega _s$ , the interface conditions are given in Eulerian coordinates as $\\mathbf {v}_f=\\mathbf {v}_s\\quad \\mbox{and}\\quad \\sigma _f\\mathbf {n}=\\sigma _s\\mathbf {n}\\quad \\mbox{ on }\\Gamma .$ Note that we neglect some effects such as the surface tension in this model and thus the stress is continuous on interface.",
"While we keep the Eulerian description for the fluid model, we use the Lagrangian description for the structure.",
"Accordingly, we introduce the following Sobolev spaces: $\\mathbb {V}:=\\lbrace (\\mathbf {v}_f,\\hat{\\mathbf {v}}_s)\\in H^1_D(\\Omega _f(t))\\times H^1_D(\\hat{\\Omega }_s) \\text{ such that }\\mathbf {v}_f\\circ \\mathbf {x}_s=\\hat{\\mathbf {v}}_s, \\text{ on }\\hat{\\Gamma }\\rbrace ,$ where $H^1_D(\\Omega _f(t)):=\\lbrace \\mathbf {u}\\in (H^1(\\Omega _f(t)))^N| \\mathbf {u}=0, \\text{ on } \\partial \\Omega \\cap \\partial \\Omega _f\\rbrace ,$ $H^1_D(\\hat{\\Omega }_s):=\\lbrace \\mathbf {u}\\in (H^1(\\hat{\\Omega }_s))^N| \\mathbf {u}=0, \\text{ on } \\partial \\Omega \\cap \\partial \\hat{\\Omega }_s\\rbrace ,$ and $\\mathbb {Q}:=L^2(\\Omega _f(t)).$ $\\mathbb {V}$ is defined for the fluid velocity in Eulerian coordinates and the structure velocity in Lagrangian coordinates.",
"The condition $\\mathbf {v}_f\\circ \\mathbf {x}_s=\\hat{\\mathbf {v}}_s$ is used to enforce continuity of velocity in (REF ).",
"We will discuss the choice of norms for these spaces in the next section.",
"In order to formulate the problem weakly, we use test functions defined on $\\Omega $ , With the test function $\\phi \\in H_0^1(\\Omega )$ , we first write the weak form for the fluid and structure, respectively: $\\int _{\\Omega _f}\\rho _fD_t\\mathbf {v}_f\\phi d\\mathbf {x}+\\int _{\\Omega _f}\\sigma _f:\\epsilon (\\phi )d\\mathbf {x}-\\int _{\\Gamma }\\sigma _f\\mathbf {n}_f\\cdot \\phi d\\mathbf {x}=\\int _{\\Omega _f}g_f\\phi d\\mathbf {x},$ $\\int _{\\Omega _s}\\rho _s D_t\\mathbf {v}_s\\phi d\\mathbf {x}+\\int _{\\Omega _s}\\sigma _s:\\epsilon (\\phi ) d\\mathbf {x}-\\int _{\\Gamma }\\sigma _s\\mathbf {n}_s\\cdot \\phi d\\mathbf {x}=\\int _{\\Omega _s}g_s\\phi d\\mathbf {x}.$ We add these two equations based on interface conditions (REF ): $\\begin{aligned}\\int _{\\Omega _f}\\rho _f D_t\\mathbf {v}_f\\phi d\\mathbf {x}+\\int _{\\Omega _f}\\sigma _f:\\epsilon (\\phi )d\\mathbf {x}+\\int _{\\Omega _s}\\rho _sD_t\\mathbf {v}_s\\phi d\\mathbf {x}&+\\int _{\\Omega _s}\\sigma _s:\\epsilon (\\phi ) d\\mathbf {x}\\\\=&\\int _{\\Omega _f}g_f\\phi d\\mathbf {x}+\\int _{\\Omega _s}g_s\\phi d\\mathbf {x}.\\end{aligned}$ By a change of coordinates $\\mathbf {x}=\\mathbf {x}(\\hat{\\mathbf {x}},t),$ the stress term of structure part can be written in Lagrangian coordinates $\\int _{\\Omega _s}\\sigma _s:\\epsilon (\\phi )d\\mathbf {x}=\\int _{\\hat{\\Omega }_s}\\hat{\\sigma }_s:\\nabla _{\\hat{\\mathbf {x}}}\\hat{\\phi }F^{-1}\\hat{J}d\\hat{\\mathbf {x}}=\\int _{\\hat{\\Omega }_s}(J\\hat{\\sigma }_sF^{-T}):\\nabla _{\\hat{\\mathbf {x}}}\\hat{\\phi }d\\hat{\\mathbf {x}},$ where $\\hat{\\phi }(\\hat{\\mathbf {x}},t)=\\phi (\\mathbf {x}(\\hat{\\mathbf {x}},t),t)$ and $\\hat{\\sigma }_s(\\hat{\\mathbf {x}},t)=\\sigma _s(\\mathbf {x}(\\hat{\\mathbf {x}},t),t)$ .",
"We also change the coordinates for the inertial term and the body force term.",
"Then, we get the following weak form of FSI $\\begin{aligned}\\int _{\\Omega _f}\\rho _f D_t\\mathbf {v}_f\\phi +\\sigma _f:\\epsilon (\\phi ) d\\mathbf {x}+\\int _{\\hat{\\Omega }_s}\\hat{\\rho }_s\\partial _{tt} \\hat{\\mathbf {u}}_s\\hat{\\phi }+&\\mathbf {P}_s:\\nabla \\hat{\\phi }d\\hat{\\mathbf {x}}\\\\&=\\int _{\\Omega _f}g_f\\phi d\\mathbf {x}+\\int _{\\hat{\\Omega }_s}J \\hat{g}_s\\phi d\\hat{\\mathbf {x}},\\end{aligned}$ which holds for any $\\phi \\in \\mathbb {V}$ .",
"Here, the density of the structure $\\hat{\\rho }_s$ is defined as $\\hat{\\rho }_s(\\hat{\\mathbf {x}},t)=J(\\hat{\\mathbf {x}},t) \\rho _s(\\mathbf {x}(\\hat{\\mathbf {x}},t),t)$ and $\\mathbf {P}_s=J\\hat{\\sigma }_sF^{-T}$ is the first Piola-Kirchhoff stress.",
"By the conservation of mass, $\\hat{\\rho }_s$ is independent of $t$ .",
"The variational formulation (REF ) holds for general fluid and structure models described by the Cauchy stresses $\\sigma _f$ and $\\sigma _s$ , respectively.",
"We now make some specific choices for $\\sigma _f$ and $\\sigma _s$ .",
"For the fluid, we use the incompressible Newtonian model, which is given by $\\sigma _f= 2\\mu _f\\epsilon (\\mathbf {v}_f)-p\\mathbf {I}$ and $\\nabla \\cdot \\mathbf {v}_f=0.$ For the structure, we use the linear elasticity model (for small deformations) in Lagrangian coordinates, which corresponds to the following approximation: $\\mathbf {P}_s\\approx \\tilde{\\mathbf {P}}_s:=\\mu _s\\epsilon (\\hat{\\mathbf {u}}_s)+\\lambda _s\\nabla \\cdot \\hat{\\mathbf {u}}_s\\mathbf {I}.$"
],
[
"Initial and boundary conditions",
"We consider the following Dirichlet boundary conditions $\\begin{aligned}\\mathbf {v}_f&=\\mathbf {v}_f^D, &\\text{ on } &\\partial \\Omega _f\\cap \\partial \\Omega ,\\\\\\hat{\\mathbf {u}}_s&=0, &\\text{ on }& \\partial \\Omega _s\\cap \\partial \\Omega ,\\\\\\end{aligned}$ and initial conditions $\\begin{aligned}\\hat{\\mathbf {u}}_s(0)=\\mathbf {u}_{s,0}, \\quad \\partial _t \\hat{\\mathbf {u}}_s(0)&=\\mathbf {u}_{s,1},\\quad \\mathbf {v}_f(0)&=\\mathbf {v}_{f,0}.\\\\\\end{aligned}$ Figure: Computational domains of FSIIn the rest of this paper, we do not rewrite the initial conditions in the weak formulations for brevity.",
"Moreover, we assume $\\mathbf {v}_f^D=0$ .",
"That is, there are only homogeneous Dirichlet boundary conditions for the fluid problem.",
"Together with the continuity equation and interface condition, the weak formulation of FSI is as follows: The weak formulation of FSI: Find $\\mathbf {v}_f$ , $p$ and $\\hat{\\mathbf {u}}_s$ such that for any given $t>0$ , the following equations hold for any $(\\phi ,\\hat{\\phi })\\in \\mathbb {V}$ and $ q\\in \\mathbb {Q}$ $\\left\\lbrace \\begin{aligned}(\\hat{\\rho }_s\\partial _{tt}\\hat{\\mathbf {u}}_s,\\hat{\\phi })_{\\hat{\\Omega }_s}+\\left(\\rho _fD_t\\mathbf {v}_f,\\phi \\right)_{\\Omega _f}+(\\tilde{\\mathbf {P}}_s,\\nabla \\hat{\\phi })_{\\hat{\\Omega }_s}&+(\\sigma _f,\\epsilon (\\phi ))_{\\Omega _f}\\\\&=\\langle J\\hat{g}_s,\\hat{\\phi }\\rangle +\\langle g_f,\\phi \\rangle ,\\\\(\\nabla \\cdot \\mathbf {v}_f,q)_{\\Omega _f}&=0,\\\\\\mathbf {v}_f\\circ \\mathbf {x}_s&=\\partial _t \\hat{\\mathbf {u}}_s,\\quad \\text{on } \\hat{\\Gamma }.\\\\\\end{aligned}\\right.$ Remark The solution $\\mathbf {v}_f$ , $p$ and $\\hat{\\mathbf {u}}_s$ are in some specific function spaces that require sufficient regularity in the time variable.",
"Since the regularity in time variable is not discussed in this paper, we do not introduce these spaces in the weak formulation."
],
[
"Finite element discretization based on the ALE method",
"In this section, we consider both time and space discretizations of Equations (REF ) and discuss the well-posedness.",
"We first discretize the time variable $t$ with uniform time step size $k=\\Delta t$ : $t^n=nk, \\quad n=0, 1, \\ldots ,$ and use the finite difference method to discretize time derivatives.",
"For the space-time formulation of FSI, we refer to [46], [45] and references therein.",
"Since the function spaces usually depend on $t$ , we use the superscript $n$ to indicate that the function space is at time $t^n.$ For example, $\\mathbb {V}^n:=\\lbrace (\\mathbf {v}_f,\\hat{\\mathbf {v}}_s)\\in H^1_D(\\Omega _f(t^n))\\times H^1_D(\\hat{\\Omega }_s) \\text{ such that }\\mathbf {v}_f\\circ \\mathbf {x}_s^n=\\hat{\\mathbf {v}}_s, \\text{ on }\\hat{\\Gamma }\\rbrace .$ We use an ALE approach for the discretization of spatial variable.",
"In this approach, the structure domain is discretized by a fixed mesh on the initial domain $\\hat{\\Omega }_s$ and the fluid domain is discretized by a sequence of moving meshes on the moving domain $\\Omega _f(t)$ ."
],
[
"Time discretization for the structure domain",
"Without loss of generality, we consider for the time discretization of the structure variables the following simple finite difference schemes: $\\begin{aligned}(\\partial _{t} \\hat{\\mathbf {u}}_s)^{n+1}\\approx &(\\partial _{t,h} \\hat{\\mathbf {u}}_s)^{n+1}\\equiv \\frac{\\hat{\\mathbf {u}}_s^{n+1}-\\hat{\\mathbf {u}}_s^{n}}{k}, \\\\(\\partial _{tt}\\hat{\\mathbf {u}}_s)^{n+1}\\approx &(\\partial _{tt,h}\\hat{\\mathbf {u}}_s)^{n+1}\\equiv \\frac{\\hat{\\mathbf {u}}_s^{n+1}-2\\hat{\\mathbf {u}}_s^{n}+\\hat{\\mathbf {u}}_s^{n-1}}{k^2}.\\end{aligned}$ Other popular time discretization schemes such as the Newmark method [38] can also be used."
],
[
"Time discretization for the fluid domain by moving\nmeshes",
"We need to find a mapping to move the fluid mesh such that it matches the structure displacement on $\\hat{\\Gamma }$ and remains non-degenerate in $\\Omega _f$ as time evolves.",
"This mapping is a diffeomorphism in continuous case, and we use piecewise polynomials to approximate it in discrete case.",
"For a triangular mesh, only piecewise linear functions preserve the triangular shape of the elements in the mesh.",
"In the rest of this paper, we assume that the mesh motion is piecewise linear.",
"We denote the image of $\\hat{\\Omega }_f$ under the piecewise linear map $\\mathbf {x}_{h,f}$ by $\\Omega _f^n$ .",
"$\\Omega _f^n$ is discretized by a moving mesh with respect to time, denoted by $T_h(\\Omega _f^n)$ .",
"Note that $\\Omega _f^n$ is a polygonal domain in 2D, and a polyhedral domain in 3D.",
"$\\Omega _f^n$ is a result of numerical discretization, and is, in general, different from the domain shape $\\Omega _f(t^n)$ in the analytic solution of $(\\ref {eq:FSI_weak})$ .",
"The technique we use to determine the mesh motion is the ALE method.",
"First introduced for finite element discretizations of incompressible fluids in [33], [20], the ALE method provides an approach to finding the fluid mesh that can fit the moving domain $\\Omega _f(t)$ .",
"There are two main ingredients in the ALE approach: Defining how the grid is moving with respect to time such that it matches the structure displacement at the fluid-structure interface.",
"Defining how the material derivatives are discretized on the moving grid.",
"Given the structure trajectory $\\mathbf {x}_s^n$ defined on $\\hat{\\Gamma },$ the moving grid can be described by a diffeomorphism ${\\mathcal {A}}^n:\\hat{\\Omega }_f\\mapsto \\Omega _f$ that satisfies $\\left\\lbrace \\begin{aligned}{\\mathcal {A}}^n(\\hat{\\mathbf {x}})&=\\hat{\\mathbf {x}}, &\\text{ on }& \\partial \\hat{\\Omega }_f\\cap \\partial \\hat{\\Omega },\\\\{\\mathcal {A}}^n(\\hat{\\mathbf {x}})&=\\mathbf {x}_s^n(\\hat{\\mathbf {x}},t), &\\text{ on }& \\hat{\\Gamma }.\\\\\\end{aligned}\\right.$ Figure: ALE mappingALE mappings satisfying (REF ) are by no means unique.",
"In the interior of $\\hat{\\Omega }_f$ , the ALE mapping can be “arbitrary\".",
"One popular approach to uniquely determine ${\\mathcal {A}}$ is to solve partial differential equations $\\mathcal {L}{\\mathcal {A}}=0,\\quad \\mbox{in }\\hat{\\Omega }_f.$ A popular choice for the operator $\\mathcal {L}$ is the Laplacian, $\\mathcal {L}=-\\Delta .",
"$ To improve the quality of the fluid mesh with respect to the displacement of the structure near the interface, the following elasticity model is often used [21] $\\mathcal {LA}=-\\mu \\Delta \\mathcal {A}-\\lambda \\nabla (\\nabla \\cdot \\mathcal {A}).$ For more choices of formulating the ALE problem, we refer to [8], [20] and references therein."
],
[
"Discretization of the material derivative",
"With the ALE mapping ${\\mathcal {A}}$ introduced, material derivatives can be written as follows $D_t\\mathbf {v}&=&\\partial _t\\mathbf {v}+(\\mathbf {v}\\cdot \\nabla )\\mathbf {v}\\\\&=&\\partial _t\\mathbf {v}+(\\partial _t{\\mathcal {A}}\\cdot \\nabla )\\mathbf {v}+((\\mathbf {v}-\\partial _t{\\mathcal {A}})\\cdot \\nabla )\\mathbf {v}\\\\&=&\\partial _t\\mathbf {v}({\\mathcal {A}}(\\hat{\\mathbf {x}},t),t))+((\\mathbf {v}-\\partial _t{\\mathcal {A}})\\cdot \\nabla )\\mathbf {v}.$ Using the simple approximation: $\\partial _t\\mathbf {v}({\\mathcal {A}}(\\hat{\\mathbf {x}},t^{n+1}),t^{n+1}))\\approx \\partial ^{\\mathcal {A}}_{t,h}\\mathbf {v}|_{({\\mathcal {A}}(\\hat{\\mathbf {x}},t^{n+1}),t^{n+1})}:=\\frac{v({\\mathcal {A}}(\\hat{\\mathbf {x}}, t^{n+1}), t^{n+1})-v({\\mathcal {A}}(\\hat{\\mathbf {x}}, t^{n}), t^{n})}{k}$ and $(\\partial _t{\\mathcal {A}})(\\hat{\\mathbf {x}},t)\\approx (\\partial _{t,h}{\\mathcal {A}})(\\hat{\\mathbf {x}},t):=\\frac{{\\mathcal {A}}(\\hat{\\mathbf {x}}, t^{n+1})-{\\mathcal {A}}(\\hat{\\mathbf {x}}, t^{n})}{k},$ we obtain an approximation of material derivatives as follows: $(D_t\\mathbf {v})^{n+1}\\approx (D_{t,h}\\mathbf {v})^{n+1}:=\\partial ^{\\mathcal {A}}_{t,h}\\mathbf {v}(\\mathbf {x},t^{n+1})+((\\mathbf {v}-\\partial _{t,h}{\\mathcal {A}})\\cdot \\nabla )\\mathbf {v}(\\mathbf {x},t^{n+1}),$ for $\\mathbf {x}={\\mathcal {A}}(\\hat{\\mathbf {x}},t^{n+1})$ .",
"With the aforementioned discretization of material derivatives, we write the momentum equation of Navier-Stokes equations as $\\rho _f\\partial ^{\\mathcal {A}}_{t,h}\\mathbf {v}_f+\\rho _f((\\mathbf {v}_f-\\partial _{t,h}{\\mathcal {A}})\\cdot \\nabla )\\mathbf {v}_f-\\mu \\nabla \\cdot \\sigma _f=g_f.",
"$ Once the time derivatives are discretized using (REF ) and (REF ), we obtain the fully implicit scheme.",
"Fully implicit (FI) scheme: find $\\mathbf {v}^{n+1}_f\\in \\mathbb {V}_{f}^{n+1}$ , $\\hat{\\mathbf {u}}^{n+1}_s\\in \\hat{\\mathbb {V}}_{s}$ , $p\\in \\mathbb {Q}^{n+1}$ and ${\\mathcal {A}}^{n+1}\\in H^1(\\hat{\\Omega }_f)$ such that for any $(\\phi ,\\hat{\\phi })\\in \\mathbb {V}^{n+1}$ and $q\\in \\mathbb {Q}^{n+1}$ , $\\left\\lbrace \\begin{aligned}( \\hat{\\rho }_s (\\partial _{tt,h}\\hat{\\mathbf {u}}_s)^{n+1},\\hat{\\phi })_{\\hat{\\Omega }_s}+( \\rho _f(D_{t,h}\\mathbf {v}_f)^{n+1},&\\phi )_{\\Omega _f}+(\\sigma _f^{n+1},\\epsilon (\\phi ))_{\\Omega _f}\\\\+(\\tilde{\\mathbf {P}}^{n+1}_s,\\nabla \\hat{\\phi })_{\\hat{\\Omega }_s}&=\\langle J\\hat{g}_s,\\hat{\\phi }\\rangle +\\langle g_f,\\phi \\rangle ,\\\\(\\nabla \\cdot \\mathbf {v}^{n+1}_f,q)_{\\Omega _f}&=0,\\\\\\mathbf {v}_f^{n+1}\\circ {\\mathbf {x}_{s}^{n+1}}&=(\\partial _{t,h}\\hat{\\mathbf {u}}_s)^{n+1},\\quad \\mbox{ on }\\hat{\\Gamma },\\\\\\mathcal {L}{\\mathcal {A}}^{n+1}&=0, ~~~~~~~~\\qquad \\text{ in } \\hat{\\Omega }_f,\\\\{\\mathcal {A}}^{n+1}( \\hat{\\mathbf {x}})&= \\hat{\\mathbf {x}}, ~~~~~~~\\qquad \\text{ on } \\partial \\hat{\\Omega }_f\\cap \\partial \\hat{\\Omega },\\\\{\\mathcal {A}}^{n+1}( \\hat{\\mathbf {x}})&= \\hat{\\mathbf {x}}+\\hat{\\mathbf {u}}_s^{n+1}, ~~~~\\text{ on } \\hat{\\Gamma },\\\\\\end{aligned}\\right.$ The structure displacement $\\hat{\\mathbf {u}}_s^{n+1}$ serves as the boundary condition for the ALE problem.",
"Note that ${\\mathcal {A}}^{n+1}$ has to be a homeomorphism.",
"The fluid stress $\\sigma _f^{n+1}$ is defined by (REF ) in terms of $\\mathbf {v}_f^{n+1}$ and $p^{n+1}$ .",
"The structure stress $\\tilde{\\mathbf {P}}_s^{n+1}$ is defined by (REF ) in terms of $\\hat{\\mathbf {u}}_s^{n+1}$ .",
"In the FI scheme, nonlinearity comes from the convection term and the dependence of the Navier-Stokes (NS) equations on the ALE mapping.",
"To solve (REF ), Newton's method or fixed-point iteration may be used to linearize the problem.",
"Another frequently used linearization of the FI scheme is the following geometry-convective explicit scheme[18], [17], [35] Geometry-convective explicit (GCE) scheme: Find $\\mathbf {v}^{n+1}_f\\in H_D^1(\\Omega _f(t^{n}))$ , $\\hat{\\mathbf {u}}^{n+1}_s\\in H_D^1(\\hat{\\Omega }_s)$ , $p\\in L^2(\\Omega _f(t^{n}))$ and ${\\mathcal {A}}^{n+1}\\in H^1(\\hat{\\Omega }_f)$ such that for any $(\\phi ,\\hat{\\phi })\\in \\mathbb {V}^{n}$ and $q\\in \\mathbb {Q}^{n}$ , $\\footnotesize \\left\\lbrace \\begin{aligned}( \\rho _f(\\partial _{t,h}^{{\\mathcal {A}}}\\mathbf {v}_f)^{n+1},\\phi )_{\\Omega _f}+( \\hat{\\rho }_s & (\\partial _{tt,h}\\hat{\\mathbf {u}}_s)^{n+1},\\hat{\\phi })_{\\hat{\\Omega }_s}+(\\sigma _f^{n+1},\\epsilon (\\phi ))_{\\Omega _f}\\\\+(\\tilde{\\mathbf {P}}_s^{n+1},\\nabla (\\hat{\\phi }))_{\\hat{\\Omega }_s} &=\\langle g_f+((\\mathbf {v}_f^{n}-\\partial _{t,h}{\\mathcal {A}}^{n+1})\\cdot \\nabla ) \\mathbf {v}_f^{n},\\phi \\rangle _{\\Omega _{f}}+\\langle J\\hat{g}_s,\\hat{\\phi }\\rangle _{\\hat{\\Omega }_s}, \\\\(\\nabla \\cdot \\mathbf {v}^{n+1}_f,q)_{\\Omega _f}&=0,\\\\\\mathbf {v}_f^{n+1}\\circ {\\mathbf {x}_{h}^{n}}&=(\\partial _{t,h}\\hat{\\mathbf {u}}_s)^{n+1},\\quad \\qquad \\qquad \\qquad \\mbox{ on }\\hat{\\Gamma },\\\\\\mathcal {L}{\\mathcal {A}}^{n+1}&=0, ~~~~~~~~~~~\\qquad \\qquad \\qquad \\qquad \\text{ in } \\hat{\\Omega }_f,\\\\{\\mathcal {A}}^{n+1}( \\hat{\\mathbf {x}})&= \\hat{\\mathbf {x}}, ~~~~~~~~~~~\\qquad \\qquad \\qquad \\qquad \\text{ on } \\partial \\hat{\\Omega }_f\\cap \\partial \\hat{\\Omega },\\\\{\\mathcal {A}}^{n+1}( \\hat{\\mathbf {x}})&= \\hat{\\mathbf {x}}+\\hat{\\mathbf {u}}_s^{n}(\\hat{\\mathbf {x}})+k\\mathbf {v}_f^{n}\\circ \\mathbf {x}_h^n(\\hat{\\mathbf {x}}), ~~~~~\\text{ on } \\hat{\\Gamma }.\\\\\\end{aligned}\\right.$ The boundary condition for ${\\mathcal {A}}^{n+1}$ is given by $\\hat{\\mathbf {u}}_s^n$ , the structure displacement, and $\\mathbf {v}_f^n$ , the fluid velocity, from the previous time step.",
"Thus, the solution of ${\\mathcal {A}}^{n+1}$ is decoupled from solving momentum and continuity equations.",
"After ${\\mathcal {A}}^{n+1}$ is solved, the mapping from $\\hat{\\Omega }_f$ to $\\Omega _f(t^n)$ is known and $\\partial _{t,h}{\\mathcal {A}}^{n+1}$ can be calculated.",
"In (REF ), the convection term is explicitly calculated using $\\partial _{t,h}{\\mathcal {A}}^{n+1}$ and $\\mathbf {v}_f^{n}$ $(\\mathbf {v}_f^{n+1}-\\partial _t{\\mathcal {A}}^{n+1})\\cdot \\nabla \\mathbf {v}_f^{n+1}\\approx (\\mathbf {v}_f^{n}-\\partial _{t,h}{\\mathcal {A}}^{n+1})\\cdot \\nabla \\mathbf {v}_f^{n}.$ The GCE scheme in the literature has the following linearization of the convection term [18], [17], [35]: $(\\mathbf {v}_f^{n+1}-\\partial _t{\\mathcal {A}}^{n+1})\\cdot \\nabla \\mathbf {v}_f^{n+1}\\approx (\\mathbf {v}_f^{n}-\\partial _{t,h}{\\mathcal {A}}^{n+1})\\cdot \\nabla \\mathbf {v}_f^{n+1}.$ We take (REF ) instead of (REF ) since the former results in symmetric variational problems and facilitates our analysis.",
"However, we also briefly discuss about the unsymmetric cases due to (REF ) in the next section.",
"Since the solution of ${\\mathcal {A}}^{n+1}$ is decoupled from momentum and continuity equations, we do not rewrite the equations about ${\\mathcal {A}}$ in the GCE scheme in the rest of the paper."
],
[
"Change of variables for structure equations",
"Note that the discretized interface condition for the velocity is $\\mathbf {v}_f^n\\circ \\mathbf {x}_{s,h}^n=\\frac{\\hat{\\mathbf {u}}_s^n-\\hat{\\mathbf {u}}_s^{n-1}}{\\Delta t},\\quad \\mbox{ on }\\hat{\\Gamma }.$ The velocities of fluid and structure are assumed to be continuous on the interface $\\hat{\\Gamma }$ .",
"By introducing the structure velocity in the same fashion as in (REF ), $\\hat{\\mathbf {v}}_s^n=\\frac{\\hat{\\mathbf {u}}_s^n-\\hat{\\mathbf {u}}_s^{n-1}}{\\Delta t},$ the interface condition becomes $\\mathbf {v}_f^n\\circ \\mathbf {x}_{s}^n=\\hat{\\mathbf {v}}_s^n,\\quad \\mbox{ on }\\hat{\\Gamma }.$ Therefore, the unknowns $\\mathbf {v}_f$ and $\\hat{\\mathbf {v}}_s$ are continuous on $\\Gamma $ with a change of coordinates for $\\mathbf {v}_f$ and $(\\mathbf {v}_f^n,\\hat{\\mathbf {v}}_s^n)$ belongs to the space $\\mathbb {V}^n$ .",
"Instead of $\\hat{\\mathbf {u}}_s$ , we take $\\hat{\\mathbf {v}}_s$ as one of the unknowns since it facilitates our theoretical analysis in the next section.",
"We change the variables in the GCE scheme and get the modified GCE scheme: Modified GCE scheme: Find $(\\mathbf {v}_f^{n+1},\\hat{\\mathbf {v}}_s^{n+1})\\in \\mathbb {V}^{n}$ and $p\\in \\mathbb {Q}^{n}$ such that $\\forall (\\phi ,\\hat{\\phi })\\in \\mathbb {V}^{n}$ and $\\forall q\\in \\mathbb {Q}^{n}$ , $\\left\\lbrace \\begin{aligned}\\frac{1}{k} ( \\rho _f\\mathbf {v}_f^{n+1},\\phi )_{\\Omega _f}+\\frac{1}{k} ( \\hat{\\rho }_s \\hat{\\mathbf {v}}_s^{n+1},\\hat{\\phi })_{\\hat{\\Omega }_s}+&(\\sigma _f^{n+1},\\epsilon (\\phi ))_{\\Omega _f}\\\\+k(\\tilde{\\mathbf {P}}_s(\\hat{\\mathbf {v}}^{n+1}_s),\\nabla \\hat{\\phi })_{\\hat{\\Omega }_s} &=\\langle \\tilde{g}_f,\\phi \\rangle _{\\Omega _f}+\\langle \\tilde{g}_s,\\hat{\\phi }\\rangle _{\\hat{\\Omega }_s}, \\\\(\\nabla \\cdot \\mathbf {v}^{n+1}_f,q)_{\\Omega _f}&=0,\\\\\\end{aligned}\\right.$ where $ \\tilde{g}_f=g_f+((\\mathbf {v}_f^{n}-\\partial _{t,h}{\\mathcal {A}}^{n+1})\\cdot \\nabla ) \\mathbf {v}_f^{n}+\\rho _f\\mathbf {v}_f^n/k$ $\\tilde{g}_s=J\\hat{g}_s+\\hat{\\rho }_s\\hat{\\mathbf {v}}_s^n/k-\\tilde{\\mathbf {P}}_s(\\hat{\\mathbf {u}}_s^n).$ $\\tilde{\\mathbf {P}}_s(\\hat{\\mathbf {v}}_s^{n+1})$ is in terms of $\\hat{\\mathbf {v}}_s^{n+1}$ instead of $\\hat{\\mathbf {u}}_s^{n+1}$ ; that is, $\\tilde{\\mathbf {P}}_s(\\hat{\\mathbf {v}}_s^{n+1})=\\mu _s\\epsilon (\\hat{\\mathbf {v}}_s^{n+1})+\\lambda _s\\nabla \\cdot \\hat{\\mathbf {v}}_s^{n+1}\\mathbf {I}.$"
],
[
"Space discretization",
"The structure domain $\\hat{\\Omega }_s$ is discretized by a fixed triangulation, denoted by $T_h(\\hat{\\Omega }_s)$ .",
"The corresponding finite element space is defined as: $\\hat{\\mathbb {V}}_{h,s}=\\lbrace \\hat{\\mathbf {u}}\\in H_D^1(\\hat{\\Omega }_s): \\hat{\\mathbf {u}}|_{\\tau }\\in \\mathcal {P}_m,\\forall \\tau \\in T_{h}(\\hat{\\Omega }_s)\\rbrace .$ The fluid domain $\\Omega _f$ is moving over time due to the interaction.",
"At time $t=0$ , we have the initial triangulation $T_h(\\hat{\\Omega }_f)$ on $\\hat{\\Omega }_f$ .",
"In this paper we only consider the case in which $T_h(\\hat{\\Omega }_s)$ and $T_h(\\hat{\\Omega }_f)$ are matching on the interface $\\hat{\\Gamma }$ .",
"For $t>0$ , the fluid domain $\\Omega _f(t)$ evolves due to the motion of interface.",
"Therefore, we discuss the discrete interface motion first.",
"The structure displacement $\\mathbf {u}_s$ provides the motion of the interface.",
"Note that $\\mathbf {u}_s$ is in some finite element space and, therefore, the displacement of the interface $\\Gamma $ is piecewise polynomial.",
"This approximation of interface motion introduces additional error, besides that of approximating velocity in $H^1$ and pressure in $L^2$ with piecewise polynomials.",
"Since only the triangular elements are considered in this paper, we use piecewise linear interface motion, which transforms a triangular element to another triangular element.",
"If higher order elements are used for the structure displacement, like P2, interpolations have to be performed in order to get P1 interface motion.",
"For example, the interface motion of GCE scheme is approximated by $\\mathbf {x}_s^{n+1}(\\hat{\\mathbf {x}})\\approx \\hat{\\mathbf {x}}+\\Pi _h^1(\\hat{\\mathbf {u}}_s^{n}+k\\mathbf {v}_f^{n}\\circ \\mathbf {x}_h^n)(\\hat{\\mathbf {x}}),\\quad \\hat{\\mathbf {x}}\\in \\hat{\\Gamma }.",
"$ Here, $\\Pi _h^1$ is a interpolation operator, the range of which is the space of the continuous and piecewise linear functions."
],
[
"Discrete ALE problem",
"With the discrete boundary motion provided, we solve a discrete version of the ALE equations.",
"We only consider piecewise linear ALE mappings to keep the mesh triangular.",
"Once we obtain the discrete ALE mapping ${\\mathcal {A}}_h$ , the fluid triangulation on the current configuration can be obtained.",
"Denote the set of grid points for the triangulation of $T_h(\\hat{\\Omega }_f) $ by $\\hat{\\cal N}_h=\\lbrace \\hat{\\mathbf {x}}_i; i=1:n_h\\rbrace .$ Then, the set of grid points for the triangulation of $T_h(\\Omega _f^n) $ is given by ${\\cal N}_h^n=\\lbrace x_i^n={\\mathcal {A}}_h(\\hat{\\mathbf {x}}_i, t^n)| i=1:n_h, \\hat{x}_i\\in {\\mathcal {\\hat{N}}}_h\\rbrace .$ Therefore, $T_h(\\Omega _f^n)$ is obtained accordingly.",
"Since the grid points are moved according to ${\\mathcal {A}}_h$ , we know that no interpolation is needed for evaluating the material derivative $D_t\\mathbf {v}$ at grid points.",
"We define the finite element spaces for the fluid velocity and pressure on the triangulation $T_h(\\Omega _f^n)$ : $\\mathbb {V}_{h,f}^{n}=\\lbrace \\mathbf {v}\\in H_D^1(\\Omega _f^n): \\mathbf {v}|_{\\tau }\\in \\mathcal {P}_m,\\forall \\tau \\in T_h(\\Omega _f^n)\\rbrace ,$ and $\\mathbb {Q}_{h}^n=\\lbrace q\\in L^2(\\Omega _f^n): q|_{\\tau }\\in \\mathcal {P}_l,\\forall \\tau \\in T_h(\\Omega _f^n)\\rbrace ,$ where $m$ and $l$ denote the orders of finite elements."
],
[
"Global finite element space",
"We define the finite element approximation of (REF ) as follows: $\\mathbb {V}_h^{n+1}:=\\lbrace (\\mathbf {v}_f,\\hat{\\mathbf {v}}_s): \\mathbf {v}_f\\in \\mathbb {V}_{h,f}^{n+1},~~ \\hat{\\mathbf {v}}_s\\in \\hat{\\mathbb {V}}_{h,s},~~\\mathbf {v}_f\\circ \\mathbf {x}_{h,s}^{n+1}=\\hat{\\mathbf {v}}_s, \\text{ on }\\hat{\\Gamma }\\rbrace .$ Note that the space is for both velocity unknowns and the test functions in the variational problem.",
"Modified GCE finite element scheme: Find $(\\mathbf {v}_f^{n+1},\\hat{\\mathbf {v}}_s^{n+1})\\in \\mathbb {V}_h^{n}$ and $p\\in \\mathbb {Q}_h^{n}$ such that for all $(\\phi ,\\hat{\\phi })\\in \\mathbb {V}_h^{n}$ and $q\\in \\mathbb {Q}_h^{n}$ , $\\left\\lbrace \\begin{aligned}\\frac{1}{k} ( \\rho _f\\mathbf {v}_f^{n+1},\\phi )_{\\Omega _f}+\\frac{1}{k} ( \\hat{\\rho }_s \\hat{\\mathbf {v}}_s^{n+1},\\hat{\\phi })_{\\hat{\\Omega }_s}+&(\\sigma _f^{n+1},\\epsilon (\\phi ))_{\\Omega _f}\\\\+k(\\tilde{\\mathbf {P}}_s(\\hat{\\mathbf {v}}^{n+1}_s),\\nabla \\hat{\\phi })_{\\hat{\\Omega }_s} &=\\langle \\tilde{g}_f,\\phi \\rangle _{\\Omega _f}+\\langle \\tilde{g}_s,\\hat{\\phi }\\rangle _{\\hat{\\Omega }_s}, \\\\(\\nabla \\cdot \\mathbf {v}^{n+1}_f,q)_{\\Omega _f}&=0,\\\\\\end{aligned}\\right.$ Remark GCE can be used not only in weakly coupled explicit algorithms for FSI, but also in fixed-point iteration to achieve strong coupling.",
"Newton's method can also be used to linearize the FI scheme [24], where shape derivatives have to be calculated.",
"We do not consider this type of discretization in this paper.",
"There are many different approaches to enforce interface conditions.",
"Many of them use Lagrange multipliers [18], [19] and this introduces additional degrees of freedom.",
"An approach to avoiding Lagrange multipliers is to consider velocity and displacement in the entire domain [48], [31], [22].",
"The velocity in the structure domain is naturally the time derivative of structure displacement, while the displacement in the fluid domain is the mesh displacement [31].",
"In [42], [1], [2], fluid velocity, pressure, and structure velocity are considered as unknowns.",
"In our approach, we also use this velocity-pressure formulation of FSI to facilitate our analysis.",
"In the next section, we start our theoretical analysis based on the formulation in (REF ) and (REF )."
],
[
"Reformulation as a saddle point problem",
"For brevity, we do not keep the superscript $n$ and we use $\\mathbb {V}_h$ and $\\mathbb {Q}_h$ instead of $\\mathbb {V}_h^n$ and $\\mathbb {Q}_h^n$ .",
"In this section, we focus on the linear systems resulting from (REF ) and formulate them as saddle point problems.",
"For the space $\\mathbb {V}$ , we assume that $\\mathbf {x}_s=\\mathbf {x}_{h,s}$ ; namely, $\\mathbf {x}_s$ in the definition of $\\mathbb {V}$ is assumed to be piecewise linear on the triangulation $T_h(\\hat{\\Omega }_s)$ .",
"As a consequence, $\\mathbb {V}_h$ is a subspace of $\\mathbb {V}$ .",
"Similarly, $\\mathbb {Q}_h\\subset \\mathbb {Q}$ .",
"For $\\mathbf {v}\\in \\mathbb {V}$ , we use $\\mathbf {v}_f$ and $\\hat{\\mathbf {v}}_s$ to denote its fluid and structure components, respectively.",
"This convention applies to other functions in $\\mathbb {V}$ , such as $\\mathbf {u}=(\\mathbf {u}_f,\\hat{\\mathbf {u}}_s)\\in \\mathbb {V}$ and $\\phi =(\\phi _f,\\hat{\\phi }_s)\\in \\mathbb {V}$ .",
"To guarantee the continuity of velocity on interface, we use polynomials of the same order for the fluid velocity and structure velocity.",
"We introduce the following definition of the $H^1$ norm for $\\mathbf {v}=(\\mathbf {v}_f,\\hat{\\mathbf {v}}_s)\\in \\mathbb {V}$ : $\\Vert \\mathbf {v}\\Vert _1^2=\\Vert \\mathbf {v}_f\\Vert _{1,\\Omega _f}^2+\\Vert \\hat{\\mathbf {v}}_s\\Vert _{1,\\hat{\\Omega }_s}^2,$ and define the following bilinear forms for $\\mathbf {v}=(\\mathbf {v}_f,\\hat{\\mathbf {v}}_s)\\in \\mathbb {V}$ , $\\phi =(\\phi _f,\\hat{\\phi }_s)\\in \\mathbb {V}$ and $p\\in \\mathbb {Q}$ $\\begin{aligned}a(\\mathbf {v},\\phi )=&\\frac{1}{k}(\\rho _f\\mathbf {v}_f,\\phi _f)_{\\Omega _f}+\\frac{1}{k}(\\hat{\\rho }_s\\hat{\\mathbf {v}}_s,\\hat{\\phi }_s)_{\\hat{\\Omega }_s}+(\\mu _f\\epsilon (\\mathbf {v}_f),\\epsilon ( \\phi _f))_{\\Omega _f}\\\\&+k(\\mu _s\\epsilon (\\hat{\\mathbf {v}}_s),\\epsilon (\\hat{\\phi }_s))_{\\hat{\\Omega }_s}+k (\\lambda _s\\nabla \\cdot \\hat{\\mathbf {v}}_s,\\nabla \\cdot \\hat{\\phi }_s)_{\\hat{\\Omega }_s}\\end{aligned}$ and $b(\\mathbf {v},p)=(\\nabla \\cdot \\mathbf {v}_f,p)_{\\Omega _f}.$ In this paper, we assume the material parameters to be constant within the fluid domain and the structure domain.",
"With the bilinear forms defined, (REF ) can be reformulated as a saddle point problem: Find $\\mathbf {v}\\in \\mathbb {V}$ and $p\\in \\mathbb {Q}$ such that $\\left\\lbrace \\begin{aligned}&a(\\mathbf {v},\\phi )+b(\\phi ,p)&=&\\langle \\tilde{g},\\phi \\rangle ,&\\forall & \\phi \\in \\mathbb {V},\\\\&b(\\mathbf {v},q) &=&0,&\\forall & q\\in \\mathbb {Q},\\\\\\end{aligned}\\right.$ where $\\langle \\tilde{g},\\phi \\rangle =\\langle \\tilde{g}_f,\\phi _f\\rangle +\\langle \\tilde{g}_s,\\hat{\\phi }_s\\rangle .$ This type of problems has various applications, for example in Stokes equations and constrained optimization, and is well studied [13], [28].",
"In order to study the well-posedness of this problem, we need to carefully define norms for $\\mathbb {V}$ and $\\mathbb {Q}$ as $\\begin{aligned}\\mbox{for all }\\mathbf {v}\\in \\mathbb {V},~~\\Vert \\mathbf {v}\\Vert _V^2&:=a(\\mathbf {v},\\mathbf {v})+r\\Vert \\nabla \\cdot \\mathbf {v}_f\\Vert _{0,\\Omega _f}^2,\\\\\\mbox{for all } q\\in \\mathbb {Q},~~\\Vert q\\Vert ^2_Q&:=r^{-1}\\Vert q\\Vert ^2_0,\\\\\\end{aligned}$ where $r=\\max \\lbrace 1,\\mu _f, \\rho _fk^{-1}, \\hat{\\rho }_sk^{-1}, k\\mu _s,k\\lambda _s\\rbrace .$ It is well known that (REF ) is well-posed if the following conditions can be verified [28] $a(\\cdot ,\\cdot ) \\mbox{ is bounded and coercive in } \\mathbb {Z}:=\\lbrace \\mathbf {v}\\in \\mathbb {V}| \\nabla \\cdot \\mathbf {v}=0\\mbox{ in }\\Omega _f\\rbrace ,$ $\\begin{aligned}b(\\cdot ,\\cdot ) \\mbox{ is bounded }&\\mbox{and satisfies the inf-sup condition } \\\\&\\inf _{p\\in \\mathbb {Q}}\\sup _{\\mathbf {v}\\in \\mathbb {V}}\\frac{b(\\mathbf {v},p)}{\\Vert \\mathbf {v}\\Vert _V\\Vert p\\Vert _Q}\\ge \\beta >0.\\\\\\end{aligned}$ In the rest of the paper, we prove the boundedness and coercivity of $a(\\cdot ,\\cdot )$ and the inf-sup condition of $b(\\cdot ,\\cdot )$ in order to show the well-posedness of saddle point problems, like (REF ).",
"By definition, it is straightforward to prove the conditions on $a(\\cdot ,\\cdot )$ since $a(\\mathbf {v},\\mathbf {v})= \\Vert \\mathbf {v}\\Vert _V^2,\\quad \\forall \\mathbf {v}\\in \\mathbb {Z}.$ The boundedness of $b(\\cdot ,\\cdot )$ follows from the definition: $b(\\mathbf {v},q)\\le \\Vert \\nabla \\cdot \\mathbf {v}\\Vert _{0,\\Omega _f}\\Vert q\\Vert _0\\le r^{1/2}\\Vert \\nabla \\cdot \\mathbf {v}\\Vert _{0,\\Omega _f}r^{-1/2}\\Vert q\\Vert _0 \\le \\Vert \\mathbf {v}\\Vert _V\\Vert q\\Vert _Q.$ Now, we need to prove the inf-sup condition of $b(\\cdot ,\\cdot )$ .",
"First, we have the following lemma.",
"Lemma 1 ([12]) Let $\\partial \\Omega _D\\subset \\partial \\Omega $ satisfy $|\\partial \\Omega _D|>0$ and $|\\partial \\Omega \\setminus \\partial \\Omega _D|>0$ .",
"Then there exists a constant $C$ such that $\\sup _{\\mathbf {v}\\in H_D^1(\\Omega )}\\frac{(\\nabla \\cdot \\mathbf {v},q)}{\\Vert \\mathbf {v}\\Vert _{1,\\Omega }}\\ge C \\Vert q\\Vert _{0,\\Omega },\\quad \\mbox{for all } q\\in L^2(\\Omega ),$ where $H_D^1(\\Omega )=\\lbrace \\mathbf {v}\\in H^1(\\Omega )| \\mathbf {v}(\\mathbf {x})=0,~~\\mbox{ for all }\\mathbf {x}\\in \\partial \\Omega _D\\rbrace .$ The following lemma is the key ingredient in proving the well-posedness of (REF ).",
"In this case, the fluid domain is deformed due to the motion of the structure.",
"In the GCE scheme, $\\mathbf {x}_s$ is treated explicitly and the inf-sup constant depends on $\\mathbf {x}_s$ .",
"Lemma 2 Assume that $\\mathbf {x}_s\\in W^{1,\\infty }(\\hat{\\Omega }_s)\\quad \\mbox{and} \\quad \\inf _{\\hat{\\mathbf {x}}\\in \\hat{\\Omega }_s}\\det (\\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}})) >0.$ Then the following inf-sup condition holds $\\inf _{q\\in \\mathbb {Q}}\\sup _{\\mathbf {v}\\in \\mathbb {V}}\\frac{b(\\mathbf {v},q)}{\\Vert \\mathbf {v}\\Vert _1\\Vert q\\Vert _0}\\gtrsim \\frac{1}{d_0^{N/2+1}d_1},$ where $d_0=\\max \\left\\lbrace \\sup _{\\hat{\\mathbf {x}}\\in \\hat{\\Gamma }}\\Vert \\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}})\\Vert _{2},1\\right\\rbrace ,\\quad d_1=\\max \\left\\lbrace \\sup _{\\hat{\\mathbf {x}}\\in \\hat{\\Gamma }}\\left\\lbrace \\det (\\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}}))^{-1}\\right\\rbrace , 1\\right\\rbrace .$ Note that $N=2,3$ is the dimension of the FSI problem and $\\Vert \\nabla \\mathbf {x}_s\\Vert _2$ is the induced matrix 2-norm.",
"Given $q\\in \\mathbb {Q}=L^2(\\Omega _f)$ , we can find $\\mathbf {v}_f\\in H_D^1(\\Omega _f)=\\lbrace \\mathbf {v}\\in H^1(\\Omega _f) \\mbox{ and } \\mathbf {v}_f|_{\\partial \\Omega _f\\cap \\partial \\Omega }=0\\rbrace $ such that $\\frac{(\\nabla \\cdot \\mathbf {v}_f,q)_{\\Omega _f}}{\\Vert \\mathbf {v}_f\\Vert _{1,\\Omega _f}\\Vert q\\Vert _0}\\gtrsim 1.$ Then, we take $\\hat{\\mathbf {v}}_s\\in \\hat{\\mathbb {V}}_{h,s}$ satisfying $\\mathbf {v}_f\\circ \\mathbf {x}_s=\\hat{\\mathbf {v}}_s$ on $ \\hat{\\Gamma }$ and $\\int _{\\hat{\\Omega }_s}\\nabla \\hat{\\mathbf {v}}_s:\\nabla \\phi =0,\\quad \\mbox{ for all } \\phi \\in H_0^1(\\hat{\\Omega }_s).$ Then, we know that $\\mathbf {v}:=(\\mathbf {v}_f,\\hat{\\mathbf {v}}_s)\\in \\mathbb {V}_h$ and $\\Vert \\hat{\\mathbf {v}}_s\\Vert _{1,\\hat{\\Omega }_s}\\lesssim \\Vert \\hat{\\mathbf {v}}_s\\Vert _{1/2,\\partial \\hat{\\Omega }_s}$ .",
"The structure flow map $\\mathbf {x}_s$ maps from $\\hat{\\Gamma }$ to $\\Gamma $ .",
"By Nanson's formula [8], the following inequality about surface elements $ds$ and $d\\hat{s}$ holds $ds(\\mathbf {x}_s(\\hat{\\mathbf {x}}))\\le \\det (\\nabla \\mathbf {x}_s)\\Vert (\\nabla \\mathbf {x}_s)^{-1}\\Vert _2d\\hat{s}(\\hat{\\mathbf {x}}).$ Given $\\mathbf {x},\\mathbf {y}\\in \\Gamma $ , $|\\mathbf {x}-\\mathbf {y}|$ denotes the distance between $\\mathbf {x}$ and $\\mathbf {y}$ on $\\Gamma $ .",
"It is easy to verify that $|\\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {x}_s(\\hat{\\mathbf {y}})|\\le \\sup _{\\mathbf {z}\\in \\Gamma }\\Vert \\nabla \\mathbf {x}_s(\\mathbf {z})\\Vert _{2}|\\mathbf {x}-\\mathbf {y}|\\le d_0|\\mathbf {x}-\\mathbf {y}|$ and, accordingly, $\\mbox{dist}(\\mathbf {x}_s(\\hat{\\mathbf {x}}),\\Gamma )=\\inf _{\\mathbf {y}\\in \\Gamma }|\\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {y}|=\\inf _{\\hat{\\mathbf {y}}\\in \\hat{\\Gamma }}|\\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {x}_s(\\hat{\\mathbf {y}})|\\le d_0\\inf _{\\hat{\\mathbf {y}}\\in \\hat{\\Gamma }}|\\hat{\\mathbf {x}}-\\hat{\\mathbf {y}}|=d_0\\mbox{dist}(\\hat{\\mathbf {x}},\\hat{\\Gamma }).$ The integral on the interface $\\hat{\\Gamma }$ can be estimated as follows $\\begin{aligned}&|\\mathbf {v}_f\\circ \\mathbf {x}_s|^2_{H_{00}^{1/2}(\\hat{\\Gamma })}\\\\=&\\int _{\\hat{\\Gamma }}\\int _{\\hat{\\Gamma }}\\frac{|\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {y}})|^2}{|\\hat{\\mathbf {x}}-\\hat{\\mathbf {y}}|^N}d\\hat{s}(\\hat{\\mathbf {x}})d\\hat{s}(\\hat{\\mathbf {y}})+\\int _{\\hat{\\Gamma }}\\frac{|\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {x}})|^2}{\\mbox{dist}(\\hat{\\mathbf {x}},\\partial \\hat{\\Gamma })}d\\hat{s}(\\hat{\\mathbf {x}})\\\\=&\\int _{\\hat{\\Gamma }}\\int _{\\hat{\\Gamma }}\\frac{|\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {y}})|^2}{|\\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {x}_s(\\hat{\\mathbf {y}})|^N}\\frac{|\\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {x}_s(\\hat{\\mathbf {y}})|^N}{|\\hat{\\mathbf {x}}-\\hat{\\mathbf {y}}|^N}d\\hat{s}(\\hat{\\mathbf {x}})ds(\\hat{\\mathbf {y}})\\\\&+\\int _{\\hat{\\Gamma }}\\frac{|\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {x}})|^2}{\\mbox{dist}(\\mathbf {x}_s(\\hat{\\mathbf {x}}),\\partial \\Gamma )}\\frac{\\mbox{dist}(\\mathbf {x}_s(\\hat{\\mathbf {x}}),\\partial \\Gamma )}{\\mbox{dist}(\\hat{\\mathbf {x}},\\partial \\hat{\\Gamma })}d\\hat{s}(\\hat{\\mathbf {x}})\\\\\\le &d_0^N\\int _{\\hat{\\Gamma }}\\int _{\\hat{\\Gamma }}\\frac{|\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {y}})|^2}{|\\mathbf {x}_s(\\hat{\\mathbf {x}})-\\mathbf {x}_s(\\hat{\\mathbf {y}})|^N}d\\hat{s}(\\hat{\\mathbf {x}})d\\hat{s}(\\hat{\\mathbf {y}})+d_0\\int _{\\hat{\\Gamma }}\\frac{|\\mathbf {v}_f\\circ \\mathbf {x}_s(\\hat{\\mathbf {x}})|^2}{\\mbox{dist}(\\mathbf {x}_s(\\hat{\\mathbf {x}}),\\partial \\Gamma )}d\\hat{s}(\\hat{\\mathbf {x}})\\\\\\le &d_0^N\\int _{\\Gamma }\\int _{\\Gamma }\\frac{|\\mathbf {v}_f(\\mathbf {x})-\\mathbf {v}_f(\\mathbf {y})|^2}{|\\mathbf {x}-\\mathbf {y}|^N}\\det (\\nabla \\mathbf {x}_s)^{-2}\\Vert \\nabla \\mathbf {x}_s\\Vert _2^{2}ds(\\mathbf {x})ds(\\mathbf {y})\\\\&+d_0\\int _{\\Gamma }\\frac{|\\mathbf {v}_f(\\mathbf {x})|^2}{\\mbox{dist}(\\mathbf {x},\\partial \\Gamma )}\\det (\\nabla \\mathbf {x}_s)^{-1}\\Vert \\nabla \\mathbf {x}_s\\Vert _2ds(\\mathbf {x})\\\\\\le & d_0^{N+2}d_1^2\\int _{\\Gamma }\\int _{\\Gamma }\\frac{|\\mathbf {v}_f(\\mathbf {x})-\\mathbf {v}_f(\\mathbf {y})|^2}{|\\mathbf {x}-\\mathbf {y}|^N}ds(\\mathbf {x})ds(\\mathbf {y})+ d_0^{2}d_1\\int _{\\Gamma }\\frac{|\\mathbf {v}_f(\\mathbf {x})|^2}{\\mbox{dist}(\\mathbf {x},\\partial \\Gamma )}ds(\\mathbf {x})\\\\\\end{aligned}$ and $\\Vert \\mathbf {v}_f\\circ \\mathbf {x}_s\\Vert _{L^2(\\hat{\\Gamma })}^2\\le d_0d_1 \\Vert \\mathbf {v}_f\\Vert ^2_{L^2(\\Gamma )}.$ Therefore, $\\Vert \\mathbf {v}_f\\circ \\mathbf {x}_s\\Vert _{H_{00}^{1/2}(\\hat{\\Gamma })}^2\\le d_0^{N+2}d_1^2 \\Vert \\mathbf {v}_f\\Vert ^2_{H_{00}^{1/2}(\\Gamma )}.$ Based on the intrinsic definition of the semi norm $|\\mathbf {v}_f|^2_{H_{00}^{1/2}(\\Gamma )}= \\int _{\\Gamma }\\int _{\\Gamma }\\frac{|\\mathbf {v}_f(\\mathbf {x})-\\mathbf {v}_f(\\mathbf {y})|^2}{|\\mathbf {x}-\\mathbf {y}|^n}ds(\\mathbf {x})ds(\\mathbf {y})+\\int _{\\Gamma }\\frac{|\\mathbf {v}_f|^2}{\\mbox{dist}(x,\\partial \\Gamma )}ds(x),$ we know that [52] $| \\mathbf {v}_f\\circ \\mathbf {x}_s|_{1/2,\\partial \\hat{\\Omega }_f}\\mathrel {\\hbox{$\\copy \\hspace{0.0pt}$\\sim $$$$}|\\mathbf {v}_f\\circ \\mathbf {x}_s|_{H_{00}^{1/2}(\\hat{\\Gamma })}\\mathrel {\\hbox{$\\copy \\hspace{0.0pt}$\\sim $$$$}|\\hat{\\mathbf {v}}_s|_{1/2,\\partial \\hat{\\Omega }_s}.Then\\Vert \\hat{\\mathbf {v}}_s\\Vert ^2_{1,\\hat{\\Omega }_s}\\lesssim \\Vert \\hat{\\mathbf {v}}_s\\Vert ^2_{1/2,\\partial \\hat{\\Omega }_s}\\lesssim \\Vert \\mathbf {v}_f\\circ \\mathbf {x}_s\\Vert ^2_{H_{00}^{1/2}(\\hat{\\Gamma })}\\lesssim d^{N+2}_0d_1^2\\Vert \\mathbf {v}_f\\Vert ^2_{1/2,\\partial \\Omega _f}\\lesssim d^{N+2}_0d^2_1\\Vert \\mathbf {v}_f\\Vert _{1,\\Omega _f}.",
"}}Therefore, we have$ v21d0N+2d12vf1,f2 $and$ $\\frac{(\\nabla \\cdot \\mathbf {v}_f,q)_{\\Omega _f}}{\\Vert \\mathbf {v}\\Vert _1\\Vert q\\Vert _0}\\gtrsim \\frac{1}{d_0^{N/2+1}d_1}.$ This finishes the proof.",
"With the inf-sup condition of $b(\\cdot ,\\cdot )$ proved, the well-posedness of (REF ) is shown.",
"Theorem 1 Assume that at a given time step $t^n$ , there exist positive constants $C_0$ and $C_1$ such that $\\sup _{\\hat{\\mathbf {x}}\\in \\hat{\\Gamma }}\\Vert \\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}})\\Vert _{2}\\le C_0, \\quad \\sup _{\\hat{\\mathbf {x}}\\in \\hat{\\Gamma }}\\left\\lbrace \\det (\\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}}))^{-1}\\right\\rbrace \\le C_1,$ where the positive constants $C_0$ and $C_1$ are independent of material parameters and time step sizes.",
"Then, under the norms $\\Vert \\cdot \\Vert _V$ and $\\Vert \\cdot \\Vert _Q$ , the variational problem (REF ) is uniformly well-posed with respect to material parameters and time step sizes.",
"We prove this theorem by verifying the Brezzi's conditions (REF ) and (REF ).",
"The boundedness and coercivity of $a(\\cdot ,\\cdot )$ are shown by (REF ) and the boundedness of $b(\\cdot ,\\cdot )$ is shown by (REF ).",
"Therefore, we only need to prove the inf-sup condition of $b(\\cdot ,\\cdot )$ .",
"Due to the choice of the parameter $r$ , the following inequality holds $\\Vert \\mathbf {v}\\Vert _{V}\\lesssim r^{1/2}\\Vert \\mathbf {v}\\Vert _{1,\\Omega },\\quad \\forall \\mathbf {v}\\in \\mathbb {V}.$ Based on Lemma REF , it indicates that $\\inf _{q\\in \\mathbb {Q}}\\sup _{\\mathbf {v}\\in \\mathbb {V}}\\frac{(\\nabla \\cdot \\mathbf {v},q)_{\\Omega _f}}{\\Vert \\mathbf {v}\\Vert _V\\Vert q\\Vert _Q}\\gtrsim \\frac{1}{d_0^{N/2+1}d_1}.$ Since $d_0\\le \\max \\lbrace C_0,1\\rbrace $ , $d_1\\le \\max \\lbrace C_1,1\\rbrace $ and $C_0$ and $C_1$ are independent of material parameters and time step sizes, the inf-sup constant is uniformly bounded below.",
"Therefore, we have shown that (REF ) is uniformly well-posed with respect to material parameters $\\rho _f$ , $\\hat{\\rho }_s$ , $\\mu _f$ , $\\mu _s$ and $\\lambda _s$ and time step size $k$ ."
],
[
"Applications in unsymmetric cases",
"In the GCE scheme we are considering, convection terms are treated explicitly using (REF ).",
"A more stable discretization is to linearize convection terms by Newton's method.",
"This adds unsymmetric terms to the variational problem $c(\\mathbf {u},\\mathbf {v})= \\int _{\\Omega _f}\\rho _f(\\mathbf {w}\\cdot \\nabla )\\mathbf {u}_f\\cdot \\mathbf {v}_f+\\int _{\\Omega _f}\\rho _f(\\mathbf {u}_f\\cdot \\nabla )\\mathbf {z}\\cdot \\mathbf {v}_f,$ where $\\mathbf {w}$ and $\\mathbf {z}$ are functions obtained from previous iteration steps.",
"With the new term $c(\\mathbf {v},\\phi )$ added, the following variational problem is also well-posed under certain assumptions Find $\\mathbf {v}\\in \\mathbb {V}$ and $p\\in \\mathbb {Q}$ such that $\\left\\lbrace \\begin{aligned}&a(\\mathbf {v},\\phi )+c(\\mathbf {v},\\phi )+b(\\phi ,p)&=&\\langle \\tilde{f},\\phi \\rangle ,&\\forall & \\phi \\in \\mathbb {V},\\\\&b(\\mathbf {v},q) &=&0,&\\forall & q\\in \\mathbb {Q}.\\\\\\end{aligned}\\right.$ The well-posedness of (REF ) requires the boundedness and coercivity of $a(\\mathbf {u},\\mathbf {v})+c(\\mathbf {u},\\mathbf {v})$ .",
"First we have $\\begin{aligned}\\int _{\\Omega _f}\\rho _f(\\mathbf {w}\\cdot \\nabla )\\mathbf {u}_f\\cdot \\mathbf {v}_f \\le C \\left(\\frac{k\\rho _f}{\\mu _f}\\right)^{1/2}\\Vert \\mathbf {w}\\Vert _{\\infty }\\Vert \\mathbf {u}\\Vert _{V}\\Vert \\mathbf {v}\\Vert _V\\\\\\end{aligned}$ and $\\begin{aligned}\\int _{\\Omega _f}\\rho _f(\\mathbf {u}_f\\cdot \\nabla )\\mathbf {z}\\cdot \\mathbf {v}_f \\le k \\Vert \\nabla \\mathbf {z}\\Vert _{\\infty }\\Vert \\mathbf {u}\\Vert _V\\Vert \\mathbf {v}\\Vert _V.\\\\\\end{aligned}$ Then $c(\\mathbf {u},\\mathbf {v})\\le \\left( C\\left(k\\rho _f/\\mu _f\\right)^{1/2}\\Vert \\mathbf {w}\\Vert _\\infty + k\\Vert \\nabla \\mathbf {z}\\Vert _\\infty \\right)\\Vert \\mathbf {u}\\Vert _V\\Vert \\mathbf {v}\\Vert _V.$ Assume $k$ is small enough such that $C\\left(k\\rho _f/\\mu _f\\right)^{1/2}\\Vert \\mathbf {w}\\Vert _\\infty + k\\Vert \\nabla \\mathbf {z}\\Vert _\\infty \\le c_0<1,$ where $0<c_0<1$ is a constant.",
"Then we have the boundedness and coercivity of $a(\\mathbf {u},\\mathbf {v})+c(\\mathbf {u},\\mathbf {v})$ $\\begin{aligned}a(\\mathbf {u},\\mathbf {u})+c(\\mathbf {u},\\mathbf {u})\\ge & (1-c_0)\\Vert \\mathbf {u}\\Vert _V^2,\\quad \\forall \\mathbf {u}\\in \\mathbb {V},\\\\a(\\mathbf {u},\\mathbf {v})+c(\\mathbf {u},\\mathbf {v})\\le & (1+c_0)\\Vert \\mathbf {u}\\Vert _V\\Vert \\mathbf {v}\\Vert _V ,\\quad \\forall \\mathbf {u},\\mathbf {v}\\in \\mathbb {V}.\\end{aligned}$ The boundedness and the inf-sup condition of $b(\\cdot ,\\cdot )$ are not affected by $c(\\cdot ,\\cdot )$ .",
"Therefore, the well-posedness of variational problem (REF ) follows based on standard arguments.",
"(See Corollary 4.1 in [28].)",
"We do not show the details here.",
"Although our study can be applied to unsymmetric case, we only deal with the symmetric cases in the rest of this paper.",
"In the next section, we consider the well-posedness of the finite element problem (REF )."
],
[
"Well-posedness of finite element discretization",
"Since we have already assumed $\\mathbb {V}_h\\subset \\mathbb {V}$ and $\\mathbb {Q}_h\\subset \\mathbb {Q}$ , (REF ) can be formulated as follows Find $\\mathbf {v}_h\\in \\mathbb {V}_h$ and $p_h\\in \\mathbb {Q}_h$ such that $\\left\\lbrace \\begin{aligned}&a(\\mathbf {v}_h,\\phi _h)+b(\\phi _h,p_h)&=&\\langle \\tilde{g},\\phi _h\\rangle ,&\\forall & \\phi _h\\in \\mathbb {V}_h,\\\\&b(\\mathbf {v}_h,q_h) &=&0,&\\forall & q_h\\in \\mathbb {Q}_h.\\\\\\end{aligned}\\right.$ The well-posedness of this finite element problem can be proved with some additional assumptions The discrete kernel space is $\\mathbb {Z}_h:=\\lbrace \\mathbf {v}_h=(\\mathbf {v}_{h,f},\\hat{\\mathbf {v}}_{h,s})\\in \\mathbb {V}_h| (\\nabla \\cdot \\mathbf {v}_{h,f},q_h)_{\\Omega _f}=0, ~\\mbox{ for all }q_h\\in \\mathbb {Q}_h\\rbrace .$ As is pointed out in [50], for finite element spaces that do not satisfy $\\mathbb {Z}_h\\subset \\mathbb {Z}$ , the uniform coercivity of $a(\\cdot ,\\cdot )$ in $\\mathbb {Z}_h$ cannot be guaranteed.",
"In fact, if $r(\\nabla \\cdot \\mathbf {v}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}\\le a(\\mathbf {v},\\mathbf {v}), \\quad \\mbox{ for all }\\mathbf {v}\\in \\mathbb {Z}_h$ holds uniformly with respect to $r$ , then it implies that $\\nabla \\cdot \\mathbf {v}_f=0$ in $\\Omega _f$ , i.e.",
"$\\mathbf {v}\\in \\mathbb {Z}$ .",
"However, most commonly used finite element pairs do not satisfy $\\mathbb {Z}_h\\subset \\mathbb {Z}$ .",
"Although there are exceptions like P4-P3 in 2D, the choice is very restricted.",
"We propose two remedies for this issue: the first is to add a stabilization term to $a(\\mathbf {u},\\mathbf {v})$ and the second is to Introduce a new norm for $\\mathbb {V}$ ."
],
[
"Remedy 1: Stabilized formulation for finite elements",
"The first remedy we propose is to add the stabilization term proposed in [50] $\\tilde{a}(\\mathbf {u},\\mathbf {v})=a(\\mathbf {u},\\mathbf {v})+r(\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}.$ Then $\\tilde{a}(\\mathbf {u},\\mathbf {v})$ is uniformly coercive in $\\mathbb {V}_h$ since $\\tilde{a}(\\mathbf {u},\\mathbf {u})\\equiv \\Vert \\mathbf {u}\\Vert _V^2,\\quad \\forall \\mathbf {u}\\in \\mathbb {V}_h.$ The stabilization term $r(\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}$ is one of the key ingredients in our formulation.",
"This term has also been used in [39] to stabilize Stokes equations and the effects of this term on discretization error and preconditioning of the linear system are discussed.",
"Another type of stabilization technique, the orthogonal subgrid scales technique, is applied to FSI in [1], [2] to stabilize the Navier-Stokes equations with equal-order velocity-pressure pairs (like P1-P1).",
"The stabilization parameters of this technique are determined by Fourier analysis in [16].",
"The new FEM problem is as follows: Find $\\mathbf {v}_h\\in \\mathbb {V}_h$ and $p_h\\in \\mathbb {Q}_h$ such that $\\left\\lbrace \\begin{aligned}&\\tilde{a}(\\mathbf {v}_h,\\phi _h)+b(\\phi _h,p_h)&=&\\langle \\tilde{g},\\phi _h\\rangle ,&\\forall & \\phi _h\\in \\mathbb {V}_h,\\\\&b(\\mathbf {v}_h,q_h) &=&0,&\\forall & q_h\\in \\mathbb {Q}_h.\\\\\\end{aligned}\\right.$ For this new formulation, we just need to prove the inf-sup conditions of $b(\\cdot ,\\cdot )$ in order to show that it is well-posed.",
"Similar to Theorem REF , the inf-sup conditions of $b(\\cdot ,\\cdot )$ also depend on $\\mathbf {x}_s$ .",
"Note that $\\mathbf {x}_s$ is the solid trajectory and is calculated based on the solid velocity calculated at previous time steps.",
"Moreover, $\\mathbf {x}_s$ corresponds to mesh motion and thus we assume that $\\mathbf {x}_s$ is piecewise linear on the triangulation.",
"Corollary 1 Assume that $\\mathbf {x}_s$ is continuous and satisfies $\\mathbf {x}_s|_{\\tau }\\in \\mathcal {P}_1, ~~\\forall \\tau \\in T_h(\\hat{\\Omega }_s)~~\\mbox{ and } \\inf _{\\hat{\\mathbf {x}}\\in \\hat{\\Omega }_s}\\det (\\nabla \\mathbf {x}_s) >0,$ and that the finite element pair $(\\mathbb {V}_{h,f},\\mathbb {Q}_h)$ for the fluid variables satisfies that $\\inf _{q\\in \\mathbb {Q}_h}\\sup _{\\mathbf {v}_f\\in \\mathbb {V}_{h,f}}\\frac{(\\nabla \\cdot \\mathbf {v}_f,q)_{\\Omega _f}}{\\Vert \\mathbf {v}_f\\Vert _1\\Vert q\\Vert _0}\\gtrsim 1.$ Then the following inf-sup condition holds $\\inf _{q\\in \\mathbb {Q}_h}\\sup _{\\mathbf {v}\\in \\mathbb {V}_h}\\frac{b( \\mathbf {v},q)}{\\Vert \\mathbf {v}\\Vert _1\\Vert q\\Vert _0}\\gtrsim \\frac{1}{d_0^{N/2+1}d_1}.$ Note that $d_0$ and $d_1$ are defined in (REF ).",
"Based on (REF ), we know that given any $q^h\\in \\mathbb {Q}_h$ , we can find $\\mathbf {v}_f^h\\in \\mathbb {V}_{h,f}$ such that $\\frac{(\\nabla \\cdot \\mathbf {v}^h_f,q^h)_{\\Omega _f}}{\\Vert \\mathbf {v}_f^h\\Vert _1}\\gtrsim \\Vert q^h\\Vert _0.$ We take $\\hat{\\mathbf {v}}_s^h$ such that $\\hat{\\mathbf {v}}_s^h=\\mathbf {v}_f^h\\circ \\mathbf {x}_s^h$ on $\\hat{\\Gamma }$ and $\\int _{\\hat{\\Omega }_s}\\nabla \\hat{\\mathbf {v}}_s^h:\\nabla \\phi _h=0,\\quad \\forall \\phi _h\\in \\mathbb {V}_{h,s}^0,$ where $\\mathbb {V}_{h,s}^0:=\\lbrace \\mathbf {v}\\in \\mathbb {V}_{h,s}| \\mathbf {v}=0, \\mbox{ on }\\partial \\hat{\\Omega }\\rbrace .$ This discrete harmonic extension $\\hat{\\mathbf {v}}_{s}^h$ still satisfies $\\Vert \\hat{\\mathbf {v}}_s^h\\Vert _{1,\\hat{\\Omega }_s}\\lesssim \\Vert \\hat{\\mathbf {v}}_s^h\\Vert _{1/2,\\partial \\hat{\\Omega }_s}$ since $\\hat{\\mathbf {v}}_s^h$ is the projection of the continuous harmonic extension (see (REF )) under the inner product $(\\nabla \\mathbf {u},\\nabla \\mathbf {v}).$ Then, take $\\mathbf {v}^h=(\\mathbf {v}_f^h,\\hat{\\mathbf {v}}_s^h)\\in \\mathbb {V}_h$ .",
"We know that $\\Vert \\mathbf {v}^h\\Vert ^2_1\\lesssim d_0^{N+2}d_1^2\\Vert \\mathbf {v}_f^h\\Vert _{1}^2$ and, therefore, the following inequality holds $\\frac{(\\nabla \\cdot \\mathbf {v}^h,q^h)_{\\Omega _f}}{\\Vert \\mathbf {v}^h\\Vert _1}\\gtrsim \\frac{\\Vert q^h\\Vert _0}{d_0^{N/2+1}d_1} .$ This finishes the proof.",
"With the inf-sup condition of $b(\\cdot ,\\cdot )$ proved, the well-posedness of (REF ) follows.",
"Theorem 2 Assume that the assumptions in Corollary REF hold and that at a given time step $t^n$ , there exist constants $C_0$ and $C_1$ such that $\\sup _{\\hat{\\mathbf {x}}\\in \\hat{\\Gamma }}\\Vert \\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}})\\Vert _{2}\\le C_0, \\quad \\sup _{\\hat{\\mathbf {x}}\\in \\hat{\\Gamma }}\\left\\lbrace \\det (\\nabla \\mathbf {x}_s(\\hat{\\mathbf {x}}))^{-1}\\right\\rbrace \\le C_1.$ Moreover, assume that $C_0$ and $C_1$ are independent of material and discretization parameters.",
"Then, under the norms $\\Vert \\cdot \\Vert _V$ and $\\Vert \\cdot \\Vert _Q$ the stabilized variational problem (REF ) is uniformly well-posed with respect to material and discretization parameters.",
"To prove this theorem we also verify the Brezzi's conditions.",
"The boundedness and coercivity of $\\tilde{a}(\\cdot ,\\cdot )$ is obvious due to (REF ).",
"The boundedness of $b(\\cdot ,\\cdot )$ can be similarly proved by $(\\ref {eq:b_bounded})$ .",
"Corollary REF proves $\\inf _{q\\in \\mathbb {Q}_h}\\sup _{\\mathbf {v}\\in \\mathbb {V}_h}\\frac{b(\\mathbf {v},q)}{\\Vert \\mathbf {v}\\Vert _1\\Vert q\\Vert _0}\\gtrsim \\frac{1}{d_0^{N/2+1}d_1}.$ Since (REF ) still holds for $\\mathbf {v}\\in \\mathbb {V}_h$ , the following inf-sup condition is proved $\\inf _{q\\in \\mathbb {Q}_h}\\sup _{\\mathbf {v}\\in \\mathbb {V}_h}\\frac{b(\\mathbf {v},q)}{\\Vert \\mathbf {v}\\Vert _V\\Vert q\\Vert _Q}\\gtrsim \\frac{1}{d_0^{N/2+1}d_1}.$ Moreover, the inf-sup constant $d_0^{-N/2-1}d_1^{-1}$ is uniformly bounded below due to $d_0\\le \\max \\lbrace C_0,1\\rbrace $ and $d_1\\le \\max \\lbrace C_1,1\\rbrace $ .",
"We have verified all the Brezzi's conditions and all of the inequalities hold uniformly with respect to material parameters $\\rho _f$ , $\\hat{\\rho }_s$ , $\\mu _f$ , $\\mu _s$ and $\\lambda _s$ , time step size $k$ and mesh size.",
"Therefore, (REF ) is uniformly well-posed with respect to material and discretization parameters."
],
[
"Remedy 2: A new norm for $\\mathbb {V}$",
"An equivalent form of the norm $\\Vert \\cdot \\Vert _V$ is $\\mbox{for all }\\mathbf {u}\\in \\mathbb {V},~~\\Vert \\mathbf {u}\\Vert _{V_Q}^2:=a(\\mathbf {u},\\mathbf {u})+r\\Vert \\mathcal {P}_{\\mathbb {Q}}\\nabla \\cdot \\mathbf {u}_f\\Vert _{0,\\Omega _f}^2,$ where $\\mathcal {P}_{\\mathbb {Q}}$ is the $L^2$ projection from $L^2(\\Omega _f)$ to $\\mathbb {Q}$ .",
"This norm was used in [10] to study the well-posedness of linearized Navier-Stokes equations.",
"Note that this norm depends on the choice of space $\\mathbb {Q}$ and we use the subscript $V_Q$ to emphasize that.",
"For $\\mathbb {Q}=L^2(\\Omega _f)$ , we have $\\Vert \\mathbf {u}\\Vert _{V}=\\Vert \\mathbf {u}\\Vert _{V_Q}$ , for all $\\mathbf {u}\\in \\mathbb {V}$ .",
"For finite element pair $(\\mathbb {V}_h,\\mathbb {Q}_h)$ , the norm is $\\forall \\mathbf {u}\\in \\mathbb {V}_h,~~\\Vert \\mathbf {u}\\Vert _{V_{Q}}^2=a(\\mathbf {u},\\mathbf {u})+r\\Vert \\mathcal {P}_{\\mathbb {Q}_h}\\nabla \\cdot \\mathbf {u}_f\\Vert _{0,\\Omega _f}^2.\\\\$ With this new norm, we prove the well-posedness of the original finite element discretization (REF ) without adding the stabilization term $r(\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}$ .",
"Theorem 3 Assume that the assumptions in Theorem REF hold.",
"Then, under the norms $\\Vert \\cdot \\Vert _{V_{Q}}$ and $\\Vert \\cdot \\Vert _Q$ the original variational problem (REF ) is uniformly well-posed with respect to material and discretization parameters.",
"Note that under the new norm $\\Vert \\cdot \\Vert _{V_Q}$ , $a(\\cdot ,\\cdot )$ is uniformly coercive in $\\mathbb {Z}_h$ .",
"In fact, $\\mbox{for all }\\mathbf {u}\\in \\mathbb {Z}_h, ~~ a(\\mathbf {u},\\mathbf {u})=\\Vert \\mathbf {u}\\Vert _{V_Q}^2.$ The boundedness of $a(\\cdot ,\\cdot )$ is obvious.",
"The boundedness of $b(\\cdot ,\\cdot )$ is also easy to show: $b(\\mathbf {v}_f,p)=(\\nabla \\cdot \\mathbf {v}_f,p)_{\\Omega _f}\\le \\Vert p\\Vert _{0,\\Omega _f}\\sup _{q\\in \\mathbb {Q}_h}\\frac{(\\nabla \\cdot \\mathbf {v}_f,q)_{\\Omega _f}}{\\Vert q\\Vert _{0,\\Omega _f}}\\le \\Vert p\\Vert _Q\\Vert \\mathbf {v}\\Vert _{V_{Q_h}}.$ Since $\\Vert \\mathbf {v}\\Vert _{V_Q}\\lesssim r^{1/2}\\Vert \\mathbf {v}\\Vert _{1,\\Omega }$ is still valid, the inf-sup conditions of $b(\\cdot ,\\cdot )$ can be proved by using Corollary REF .",
"This concludes our proof.",
"We have provided two remedies in order to get uniformly well-posed finite element discretizations.",
"In the next section, we introduce how these stable formulations can help us find optimal preconditioners."
],
[
"Solution of linear systems",
"In this section, we consider preconditioners for (REF ).",
"Define $\\mathbb {X}_h=\\mathbb {V}_h\\times \\mathbb {Q}_h$ .",
"The underlying norm is $\\Vert (\\mathbf {v},p)\\Vert ^2_{X}=\\Vert \\mathbf {v}\\Vert _V^2+\\Vert p\\Vert _Q^2,\\quad (\\mathbf {v},p)\\in \\mathbb {X}_h.$ Consider the following saddle point problem: Find $x\\in \\mathbb {X}_h$ , such that $K(x,y)=\\langle \\tilde{g},y\\rangle ,\\quad \\forall y\\in \\mathbb {X}_h,$ where $\\tilde{g}\\in \\mathbb {X}_h^{\\prime }$ .",
"The operator form of (REF ) is ${\\mathcal {K}}_h x=\\tilde{g}.$ Under the assumption that (REF ) is uniformly well-posed, an optimal preconditioner can be found [36], [55], which is the Riesz operator ${\\mathcal {B}}_h:\\mathbb {X}_h^{\\prime }\\mapsto \\mathbb {X}_h$ defined by $({\\mathcal {B}}_h f,y)_X=\\langle f,y\\rangle ,\\quad \\forall y\\in \\mathbb {X}_h, f\\in \\mathbb {X}_h^{\\prime }.$ Thus, ${\\mathcal {B}}_h$ satisfies $\\kappa ({\\mathcal {B}}_h{\\mathcal {K}}_h)\\lesssim 1.$ The uniform boundedness of the condition number $\\kappa ({\\mathcal {B}}_h{\\mathcal {K}}_h)$ results in uniform convergence of Krylov subspace methods, such as MINRES."
],
[
"Two optimal preconditioners for FSI",
"In the previous section, we have introduced two stable finite element formulations, which provide two optimal preconditioners.",
"To facilitate our discussion, we first introduce the block matrices $A_h$ , $D_h$ , $B_h$ , defined by $\\begin{aligned}( A_h\\bar{u}_h,\\bar{v}_h)&=a(\\mathbf {u}_h,\\mathbf {v}_h),\\\\(B_h\\bar{u}_h,\\bar{p}_h)&=b(\\mathbf {u}_h,q_h),\\\\(D_h\\bar{u}_h,\\bar{v}_h)&=(\\nabla \\cdot \\mathbf {u}_{h,f},\\nabla \\cdot \\mathbf {v}_{h,f})_{\\Omega _f},\\\\\\end{aligned}$ for any $\\mathbf {u}_h$ , $\\mathbf {v}_h\\in \\mathbb {V}_h$ and $p_h\\in \\mathbb {Q}_h$ .",
"$\\bar{u}_h, \\bar{v}_h$ and $\\bar{p}_h$ are the corresponding vector representations with given bases for $\\mathbb {V}_h$ and $\\mathbb {Q}_h$ .",
"We also introduce the pressure mass matrix $M_p$ .",
"Now, we introduce two optimal preconditioning strategies (M1) and (M2) based on the uniformly well-posed formulations introduced in the previous section.",
"Note that these two preconditioners are applied to (REF ) and (REF ), respectively.",
"Formulation 1 (M1): With the stabilization term added, (REF ) is uniformly well-posed under the norms $\\Vert \\cdot \\Vert _V$ and $\\Vert \\cdot \\Vert _Q$ .",
"In this case, $K(x,y)=\\tilde{a}(\\mathbf {v},\\phi )+b(\\phi ,p)+b(\\mathbf {v},q),$ where $x=(\\mathbf {v},p)$ and $y=(\\phi ,q)$ .",
"The optimal preconditioner in this case is ${\\mathcal {B}}_h^1=\\left(\\begin{array}{cc}A_h+rD_h &0\\\\0& \\frac{1}{r}M_p\\\\\\end{array}\\right)^{-1}.$ Formulation 2 (M2): With the new norm $\\Vert \\cdot \\Vert _{V_Q}$ introduced, (REF ) is uniformly well-posed under the norms $\\Vert \\cdot \\Vert _{V_Q}$ and $\\Vert \\cdot \\Vert _{Q}.$ In this case, $K(x,y)=a(\\mathbf {v},\\phi )+b(\\phi ,p)+b(\\mathbf {v},q),$ where $x=(\\mathbf {v},p)$ and $y=(\\phi ,q)$ .",
"Given $p_h\\in \\mathbb {Q}_h$ and $\\mathbf {v}_h\\in \\mathbb {V}_h$ satisfying $p_h=\\mathcal {P}_Q(\\nabla \\cdot \\mathbf {v}_h)$ , we know that $M_p\\bar{p}_h=B_h\\bar{v}_h.$ Therefore, $\\Vert p_h\\Vert _{0,\\Omega _f}^2=\\bar{p}_h^TM_p\\bar{p}_h=\\bar{v}_h^T B_h^TM_p^{-1}B_h\\bar{v}_h.$ Then we know that the corresponding optimal preconditioner in this case is ${\\mathcal {B}}_h^2=\\left(\\begin{array}{cc}A_h+rD_h^Q &0\\\\0& \\frac{1}{r}M_p\\\\\\end{array}\\right)^{-1},$ where $D_h^Q:=B_h^TM_p^{-1}B_h$ ."
],
[
"Comparing ${\\mathcal {B}}_h^1$ , {{formula:58612a3e-75ee-4667-949f-5b1c85c6728a}} and the augmented Lagrangian (AL) preconditioner",
"The AL preconditioner was proposed for Oseen problems in [9] and has been extended to the Navier-Stokes equations in [11], [10].",
"The AL preconditioner is designed for saddle point problems of the following form $\\left(\\begin{array}{cc}A & B^T\\\\B&0\\\\\\end{array}\\right)\\left(\\begin{array}{c}u\\\\p\\\\\\end{array}\\right)=\\left(\\begin{array}{c}f\\\\0\\\\\\end{array}\\right).$ The AL preconditioner is applied to the modified saddle point problem $\\left(\\begin{array}{cc}A +\\gamma B^TW^{-1}B& B^T\\\\B&0\\\\\\end{array}\\right)\\left(\\begin{array}{c}u\\\\p\\\\\\end{array}\\right)=\\left(\\begin{array}{c}f\\\\0\\\\\\end{array}\\right),$ and the ideal form of the AL preconditioner is $P_\\gamma =\\left(\\begin{array}{cc}A_\\gamma & B^T\\\\0&\\frac{1}{\\nu +\\gamma }W\\\\\\end{array}\\right)^{-1},$ where $A_\\gamma =A +\\gamma B^TW^{-1}B$ , $\\nu $ is the kinematic viscosity, and the ideal choice of $W$ is the pressure mass matrix $M_p$ .",
"Note that (REF ) and (REF ) have the same solution.",
"Practical choices for the preconditioner $P_{\\gamma }$ are discussed extensively in literature, though we do not discuss this issue here.",
"For the application to the Oseen problem[9], eigenvalue analysis shows that the preconditioned matrix has all the eigenvalues tend to 1 as $\\gamma $ tends to $\\infty $ .",
"In the application to linearized Navier-Stokes problem [10], it is shown that for certain choices of the parameter $\\gamma $ , the convergence rate of AL-preconditioned GMRes is independent of discretization and material parameters.",
"Note that in these applications, convection terms are considered and, therefore, the linear systems are not symmetric.",
"The AL preconditioning technique can also be applied to our FSI problem.",
"By simply adding the term $r(P_Q\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}$ (or $rB^TW^{-1}B$ in matrix form) to the first equation of $(\\ref {eq:saddle_fem})$ , the resultant variational problem Find $\\mathbf {v}_h\\in \\mathbb {V}_h$ and $p_h\\in \\mathbb {Q}_h$ such that $\\left\\lbrace \\begin{aligned}&a(\\mathbf {v}_h,\\phi _h)+r(P_Q\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}+b(\\phi _h,p_h)&=&\\langle \\tilde{g},\\phi _h\\rangle ,&\\forall & \\phi _h\\in \\mathbb {V}_h,\\\\&b(\\mathbf {v}_h,q_h) &=&0,&\\forall & q_h\\in \\mathbb {Q}_h,\\\\\\end{aligned}\\right.$ is also uniformly well-posed under the norms $\\Vert \\cdot \\Vert _{V_Q}$ and $\\Vert \\cdot \\Vert _Q$ since adding this term yields $a(\\mathbf {u},\\mathbf {u})+r(P_Q\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {u}_f)_{\\Omega _f}=\\Vert \\mathbf {u}\\Vert _{V_Q}^2,~~\\forall \\mathbf {u}\\in \\mathbb {V}_h,$ and the boundedness and the inf-sup condition of $b(\\cdot ,\\cdot )$ still hold.",
"Based on this observation, we propose the third optimal preconditioning strategy (M3), which is very similar to the AL preconditioner.",
"Formulation 3 (M3): We take the following bilinear form $K(\\cdot ,\\cdot )$ for the saddle point problem (REF ) $K(x,y)=a(\\mathbf {v},\\phi )+r(P_Q\\nabla \\cdot \\mathbf {v}_f,\\nabla \\cdot \\phi _f)_{\\Omega _f}+b(\\phi ,p)+b(\\mathbf {v},q),$ where $x=(\\mathbf {v},p)$ and $y=(\\phi ,q)$ .",
"The optimal preconditioner in this case is also $\\mathcal {B}_h^2$ .",
"By using $\\mathcal {B}_h^2$ in an upper triangular fashion, it becomes quite similar to the AL preconditioner.",
"Therefore, our analysis can also provide justification for the AL-type preconditioner for FSI in the absence of the convection term.",
"Note that the choice of parameters (in terms of $r$ ) in (REF ) is different from those used in AL precondtioners in the literature.",
"We compare the preconditioning techniques (M1), (M2) and (M3) in the Table REF .",
"All of these three preconditioners are similar to the velocity Schur complement preconditioners.",
"For comparison, we also list a pressure Schur complement (SC) preconditioner in Table REF .",
"Table: Compare M1, M2, M3 and SCNote that in the pressure Schur complement preconditioner (SC), we use the inverse of the diagonal part of $A_h$ to approximate $A_h^{-1}$ .",
"Remark Adding the term $r(\\nabla \\cdot \\mathbf {u}_f,\\nabla \\cdot \\mathbf {v}_f)_{\\Omega _f}$ to the continuous problem (REF ) does not change the solution.",
"But adding it may change the solution of finite element discretized problems; thus, (REF ) and (REF ) may have different solutions, especially when $r$ is large.",
"In comparison, M2 and M3 do not change the solutions of finite element problems.",
"M2 and M3 have very similar forms.",
"They differ in that M2 does not add $rD_h^Q$ to the stiffness matrix.",
"M1, M2 and M3 are all proven to be optimal for FSI based on our analysis.",
"For the practical implementation, the performance of these preconditioners also depends on the efficiency of inverting the diagonal blocks, such as $A_h+rD_h$ and $M_p$ .",
"The mass matrix $M_p$ is easy to invert by iterative methods.",
"The velocity block $A_h$ is symmetric positive definite for the FSI problem; Krylov subspace method preconditioned by multigrid is usually one of the most efficient solvers.",
"However, there are still some difficulties that need special consideration: The different scales of the fluid and structure problems result in large jumps in coefficients.",
"For example, the material parameters $\\mu _s$ and $\\mu _f$ can differ greatly in magnitude.",
"This leads to the following general jump-coefficient problem: $\\mbox{Find } \\mathbf {u}\\in H^1_0(\\Omega ) \\mbox{ such that } ~~a(\\mathbf {u},\\mathbf {v})=\\langle f,\\mathbf {v}\\rangle , ~~\\mbox{ for all } \\mathbf {v}\\in H^1_0(\\Omega ),$ where $a(\\mathbf {u},\\mathbf {v})=(\\alpha (\\mathbf {x})\\epsilon (\\mathbf {u}),\\epsilon (\\mathbf {v}))+(\\beta (\\mathbf {x})\\nabla \\cdot \\mathbf {u},\\nabla \\cdot \\mathbf {v})+(\\gamma (\\mathbf {x})\\mathbf {u},\\mathbf {v})$ .",
"The domain $\\bar{\\Omega }=\\bar{\\Omega }_1\\cup \\bar{\\Omega }_2$ is illustrated in Figure REF .",
"Figure: The domain for the jump-coefficient problemThe coefficients $\\alpha (\\mathbf {x}),\\beta (\\mathbf {x})$ and $\\gamma (\\mathbf {x})$ are piecewise positive constants on $\\Omega _i$ $(i=1,2)$ .",
"The question is how to design solvers that are robust with respect to the jumps of $\\alpha (\\mathbf {x})$ , $\\beta (\\mathbf {x})$ and $\\gamma (\\mathbf {x})$ .",
"There is much research work on solving jump-coefficient problems.",
"We refer to [51] and the references therein for related discussions."
],
[
"Numerical Examples",
"In this section, we present some numerical experiments in order to verify our analysis.",
"Preconditioning techniques M1, M2, M3 and the SC preconditioner are tested.",
"Figure: FSI benchmark problemWe use the data from the FSI benchmark problem in [48].",
"Note that this is a 2D problem.",
"The FSI code is implemented in the framework of FEniCS[34].",
"The computational domain is shown in Figure REF .",
"We have an elastic beam in a channel, where the inflow comes from the left end of the domain.",
"We prescribe zero Dirichlet boundary conditions on the top and bottom of the channel.",
"On the right end we use no-flux boundary condition.",
"We use P2-P0 finite elements for the FSI system.",
"We use three meshes with different sizes.",
"Numbers of degrees of freedom for these meshes are shown in Table REF .",
"Table: DoFs of the meshesThe values of the parameter $r$ in M1, M2 and M3 are the same and are calculated by (REF ).",
"Preconditioned GMRes is used to solve the linear systems.",
"Although M1, M2 and M3 are originally block diagonal preconditioners, we use them in a block upper triangular fashion.",
"Each of the diagonal blocks is solved exactly.",
"The iteration of GMRes stops when the relative residual has magnitude less than $10^{-10}$ .",
"In Table REF , we test the preconditioners for different meshes and time step sizes.",
"In Table REF , we show the test results for different meshes and density ratios.",
"Table: Number of iterations for preconditioned GMRES for different time step sizes (k=0.01,0.001,0.0001k=0.01, 0.001, 0.0001)Table: Number of iterations for preconditioned GMRES for varying density ratiosFrom the data we see that the convergence of preconditioned GMRes for M1, M2 and M3 is almost uniform and quite robust for different mesh sizes, time step sizes, and density ratios.",
"The case with SC shows dependence on mesh sizes and the dependence becomes more significant when the time step size $k$ grows.",
"M1 and M3 in general need significantly fewer number of iterations than M2 and are more stable than M2 for various combinations of material and discretization parameters."
],
[
"Concluding remarks",
"In this paper, we formulate the FSI discretized system as saddle point problems.",
"Under mild assumptions, the uniform well-posedness of the saddle point problems is shown.",
"By adding a stabilization term or adopting a new norm for velocity, the finite element discretization of the FSI problem is also proved to be uniformly well-posed.",
"Two optimal preconditioners are proposed based on the well-posed formulations.",
"Our theoretical framework also provides an alternative justification for the AL-type preconditioners in the absence of the convection term.",
"In the numerical examples, we show the robustness of these preconditioners.",
"We use direct solves for the sub-blocks.",
"In practice, these sub-blocks have to be inverted by iterative methods when their sizes are large.",
"Robust preconditioners for the sub-blocks have to be considered.",
"We appreciate the contributions to the numerical tests from Dr. Xiaozhe Hu, Dr. Pengtao Sun, Feiteng Huang, and Lu Wang and many suggestions from Dr. Shuo Zhang, Dr. Xiaozhe Hu, and Dr. Maximilian Metti, which have greatly improved the presentation of this paper.",
"We also appreciate the helpful suggestions from Professor Alfio Quarteroni and Dr. Simone Depairs during the visit of the second author to EPFL."
]
] | 1403.0046 |
[
[
"On Tamed Euler Approximations of SDEs Driven by L\\'evy Noise with\n Applications to Delay Equations"
],
[
"Abstract We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by L\\'evy noise with super-linearly growing drift coefficients.",
"Strong convergence results are presented for the case of locally Lipschitz coefficients.",
"Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity.",
"Finally, we further extend these techniques to the case of delay equations."
],
[
"Introduction",
"In economics, finance, medical sciences, ecology, engineering, and many other branches of sciences, one often encounters problems which are influenced by event-driven uncertainties.",
"For example, in finance, the unpredictable nature of important events such market crashes, announcements made by central banks, changes in credit ratings, defaults, etc.",
"might have sudden and significant impacts on the stock price movements.",
"Stochastic differential equations (SDEs) with jumps, or more precisely SDEs driven by Lévy noise, have been widely used to model such event-driven phenomena.",
"The interested reader may refer, for example, to [3], [18], [23] and references therein.",
"Many such SDEs do not have explicit solutions and therefore one requires numerical schemes so as to approximate their solutions.",
"Over the past few years, several explicit and implicit schemes of SDEs driven by Lévy noise have been studied and results on their strong and weak convergence were proved.",
"For a comprehensive discussion on these schemes, one could refer to [2], [8], [9], [14], [20], and references therein.",
"It is also known, however, that the computationally efficient explicit Euler schemes of SDEs (even without jumps) may not convergence in strong ($\\mathcal {L}^q$ ) sense when the drift coefficients are allowed to grow super-linearly, see for example [11].",
"The development of tamed Euler schemes was a recent breakthrough in order to address this problem; one may consult [12], [21] as well as [10], [22], [24] and references therein for a thorough investigation of the subject.",
"In this article, we propose explicit tamed Euler schemes to numerically solve SDEs with random coefficients driven by Lévy noise.",
"The taming techniques developed here allow one to approximate these SDEs with drift coefficients that grow super-linearly.",
"By adopting the approach of [21], we prove strong convergence in (uniform) $\\mathcal {L}^q$ sense of these tamed schemes by assuming one-sided local Lipschitz condition on drift and local Lipschitz conditions on both diffusion and jump coefficients.",
"Moreover, our technical calculations are more refined than those of [12], [21] in that we develop new techniques to overcome the challenges arising due to jumps.",
"In addition, explicit formulations of the tamed Euler schemes are presented at the end of Section 3 for the case of SDEs driven by Lévy noise which have non-random coefficients.",
"To the best of the authors' knowledge, the results obtained in this article are the first for the case of super-linear coefficients in this area.",
"Moreover, the techniques developed here allow for further investigation of convergence properties of higher order explicit numerical schemes for SDEs driven by Lévy noise with super-linear coefficients.",
"As an application of our approach which considers random coefficients, we also present in this article uniform $\\mathcal {L}^q$ convergence results of explicit tamed Euler schemes for the case of stochastic delay differential equations (SDDEs) driven by Lévy noise.",
"The link between delay equations and random coefficients utilises ideas from [7].",
"The aforementioned results are derived under the assumptions of one-sided local Lipschitz condition on drift and local Lipschitz conditions on both diffusion and jump coefficients with respect to non-delay variables, whereas these coefficients are only asked to be continuous with respect to arguments corresponding to delay variables.",
"It is worth mentioning here that our approach allows one to use our schemes to approximate SDDEs with jumps when drift coefficients can have super-linear growth in both delay and non-delay arguments.",
"Thus, the proposed tamed Euler schemes provide significant improvements over the existing results available on numerical techniques of SDDEs, for example, [1], [15].",
"It should also be noted that, by adopting the approach of [7], we prove the existence of a unique solution to the SDDEs driven by Lévy noise under more relaxed conditions than those existing in the literature, for example, [13] whereby we ask for the local Lipschitz continuity only with respect to the non-delay variables.",
"Finally, rate of convergence results are obtained (which are in agreement with classical literature) when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity.",
"Similar results are also obtained for delay equations when the following assumptions hold - (a) drift coefficients satisfy one-sided Lipschitz and polynomial Lipschitz conditions in non-delay variables whereas polynomial Lipschitz conditions in delay variables and (b) diffusion and jump coefficients satisfy Lipschitz conditions in non-delay variables whereas polynomial Lipschitz conditions in delay variables.",
"This finding is itself a significant improvement over recent results in the area, see for example [1] and references therein.",
"We conclude this section by introducing some basic notation.",
"For a vector $x\\in \\mathbb {R}^d$ , we write $|x|$ for its Euclidean norm and for a $d\\times m$ matrix $\\sigma $ , we write $|\\sigma |$ for its Hilbert-Schmidt norm and $\\sigma ^*$ for its transpose.",
"Also for $x,y \\in \\mathbb {R}^d$ , $xy$ denotes the inner product of these two vectors.",
"Further, the indicator function of a set $A$ is denoted by $I_A$ , whereas $[x]$ stands for the integer part of a real number $x$ .",
"Let ${P}$ be the predictable sigma-algebra on $\\Omega \\times \\mathbb {R}_+ $ and ${B}(V)$ , the sigma-algebra of Borel sets of a topological space $V$ .",
"Also, let $T>0$ be fixed and $\\mathbb {L}^p$ denote the set of non-negative measurable functions $g$ on $[0,T]$ , such that $\\int _0^T|g_t|^pdt < \\infty $ .",
"Finally, for a random variable $X$ , the notation $X \\in \\mathcal {L}^p$ means $E|X|^p < \\infty $ ."
],
[
"SDE with Random Coefficients Driven by Lévy Noise",
"Let us assume that $(\\Omega , \\lbrace {F}_t\\rbrace _{t \\ge 0}, {F}, P)$ denotes a probability space equipped with a filtration $\\lbrace {F}_t\\rbrace _{t \\ge 0}$ which is assumed to satisfy the usual conditions, i.e.",
"${F}_0$ contains all $P$ -null sets and the filtration is right continuous.",
"Let $w$ be an $\\mathbb {R}^m-$ valued standard Wiener process.",
"Further assume that $(Z, {Z}, \\nu )$ is a $\\sigma -$ finite measure space and $N(dt,dz)$ is a Poisson random measure defined on $(Z, {Z}, \\nu )$ with intensity $\\nu \\lnot \\equiv 0$ (in case $\\nu \\equiv 0$ , one could consult [21]).",
"Also let the compensated poisson random measure be denoted by $\\tilde{N}(dt,dz):=N(dt,dz)-\\nu (dz)dt$ .",
"Let $b_t(x)$ and $\\sigma _t(x)$ be $P \\otimes {B}(\\mathbb {R}^d)$ -measurable functions which respectively take values in $\\mathbb {R}^d$ and $\\mathbb {R}^{d \\times m}$ .",
"Further assume that $\\gamma _t(x,z)$ is $P \\otimes {B}(\\mathbb {R}^d)\\otimes {Z}$ -measurable function which takes values in $\\mathbb {R}^{d}$ .",
"Also assume that $t_0$ and $t_1$ are fixed constants satisfying $0 \\le t_0 <t_1 \\le T$ .",
"We consider the following SDE $ dx_t=b_t(x_t)dt+\\sigma _t(x_t)dw_t+\\int _{Z}\\gamma _t(x_t,z) \\tilde{N}(dt,dz)$ almost surely for any $t \\in [t_0,t_1]$ with initial value $x_{t_0}$ which is an ${F}_{t_0}$ -measurable random variable in $\\mathbb {R}^d$ .",
"Remark 2.1 For notational convenience, we write $x_t$ instead of $x_{t-}$ on the right hand side of the above equation.",
"This does not cause any problem since the compensators of the martingales driving the equation are continuous.",
"This notational convention shall be adopted throughout this article.",
"Remark 2.2 In this article, we use $K>0$ to denote a generic constant which varies at different occurrences.",
"The proof for the following lemma can be found in [16].",
"Lemma 2.1 Let $r \\ge 2$ .",
"There exists a constant $K$ , depending only on $r$ , such that for every real-valued, ${P} \\otimes Z-$ measurable function $g$ satisfying $\\int _0^T\\int _Z |g_t(z)|^2\\nu (dz) dt < \\infty $ almost surely, the following estimate holds, $E\\sup _{0 \\le t \\le T} \\Big |\\int _0^t \\int _Z g_s(z)& \\tilde{N}(ds,dz)\\Big |^r \\le K E\\Big (\\int _0^T \\int _Z|g_t(z)|^2\\nu (dz)dt\\Big )^{r/2} +K E\\int _0^T \\int _Z |g_t(z)|^r \\nu (dz)dt.",
"$ It is known that if $1\\le r \\le 2$ , then the second term in (REF ) can be dropped."
],
[
"Existence and Uniqueness",
"Let $\\mathcal {A}$ denotes the class of non-negative predictable processes $L:=(L_t)_{t \\in [0,T]}$ such that $\\int _0^T L_t dt < \\infty $ for almost every $\\omega \\in \\Omega $ .",
"For the purpose of this section, the set of assumptions are listed below.",
"A- 1 There exists an $\\mathcal {M} \\in \\mathcal {A}$ such that $x b_t(x)+|\\sigma _t(x)|^2 + \\int _Z|\\gamma _t(x, z)|^2 \\nu (dz)& \\le \\mathcal {M}_t(1+|x|^2) $ almost surely for any $t \\in [t_0, t_1]$ and $x \\in \\mathbb {R}^d$ .",
"A- 2 For every $R>0$ , there exists an $\\mathcal {M}(R) \\in \\mathcal {A}$ such that, $(x-\\bar{x}) \\, (b_t(x)-b_t(\\bar{x}))+ |\\sigma _t(x)-\\sigma _t(\\bar{x})|^2 & +\\int _Z |\\gamma _t(x,z)-\\gamma _t(\\bar{x},z)|^2\\nu (dz) \\le \\mathcal {M}_t(R)|x-\\bar{x}|^2 $ almost surely for any $t \\in [t_0, t_1]$ whenever $|x|,|\\bar{x}| \\le R$ .",
"A- 3 For any $t \\in [t_0,t_1]$ and $\\omega \\in \\Omega $ , the function $b_t(x)$ is continuous in $x\\in \\mathbb {R}^d$ .",
"The proof for the following theorem can be found in [6].",
"Theorem 2.1 Let Assumptions A-1 to A-3 be satisfied.",
"Then, there exists a unique solution to SDE (REF )."
],
[
"Moment Bounds",
"We make the following assumptions on the coefficients of SDE (REF ).",
"A- 4 For a fixed $p \\ge 2$ , $E|x_{t_0}|^p < \\infty $ .",
"A- 5 There exist a constant $L>0$ and a non-negative random variable $M$ satisfying $EM^\\frac{p}{2}< \\infty $ such that $x b_t(x) \\vee |\\sigma _t(x)|^2 \\vee \\int _Z|\\gamma _t(x, z)|^2 \\nu (dz)\\le L(M+|x|^2) $ almost surely for any $t \\in [t_0, t_1]$ and $x\\in \\mathbb {R}^d$ .",
"A- 6 There exist a constant $L>0$ and a non-negative random variable $M^{\\prime }$ satisfying $EM^{\\prime } < \\infty $ such that $\\int _Z |\\gamma _t(x,z)|^p \\nu (dz) \\le L(M^{\\prime }+|x|^p)$ almost surely for any $t \\in [t_0,t_1]$ and $x \\in \\mathbb {R}^d$ .",
"The following is probably well-known.",
"However, the proof is provided for the sake of completeness and for the justification of finiteness of the right hand side when applying Gronwall's lemma, something that is missing from the existing literature.",
"Lemma 2.2 Let Assumptions A-2 to A-6 be satisfied.",
"Then there exists a unique solution $(x_t)_{t\\in [t_0,t_1]}$ of SDE (REF ) and the following estimate holds $E\\sup _{t_0 \\le t \\le t_1}|x_t|^p \\le K,$ with $K:=K(t_0, t_1, L, p, E|x_{t_0}|^p, EM^\\frac{p}{2}, EM^{\\prime })$ .",
"The existence and uniqueness of solution to SDE (REF ) follows immediately from Theorem REF by noting that due to Assumption A-5, Assumption A-1 is satisfied.",
"Let us first define the stopping time $\\pi _R:= \\inf \\lbrace t \\ge t_0: |x_t| >R\\rbrace \\wedge t_1$ , and notice that $|x_{t-}| \\le R$ for $ t_0 \\le t \\le \\pi _R$ .",
"By Itô's formula, $ |x_t|^p &= |x_{t_0}|^p+ p \\int _{t_0}^{t} |x_s|^{p-2} x_s b_s( x_s) ds + p\\int _{t_0}^{t} |x_s|^{p-2} x_s \\sigma _s( x_s) dw_s \\\\& + \\frac{p(p-2)}{2} \\int _{t_0}^{t} |x_s|^{p-4}|\\sigma _s^{*}(x_s) x_s|^2ds +\\frac{p}{2}\\int _{t_0}^{t} |x_s|^{p-2}|\\sigma _s(x_s)|^2 ds \\\\&+ p\\int _{t_0}^{t} \\int _{Z} |x_s|^{p-2} x_{s} \\gamma _s( x_{s},z) \\tilde{N}(ds,dz) \\\\+\\int _{t_0}^{t} &\\int _{Z}\\lbrace |x_{s}+\\gamma _s( x_{s},z)|^p-|x_{s}|^p-p|x_{s}|^{p-2} x_{s}\\gamma _s( x_{s},z) \\rbrace N(ds,dz)$ almost surely for any $t \\in [t_0,t_1]$ .",
"By virtue of Assumption A-5 and Young's inequality, one can estimate the second, fourth and fifth terms of equation (REF ) by $ K M^\\frac{p}{2}+K \\int _{t_0}^{t} |x_s|^{p} ds.$ Further, since the map $y \\rightarrow |y|^p$ is of class $C^2$ , by the formula for the remainder, for any $y_1, y_2 \\in \\mathbb {R}^d$ , one gets $|y_1+y_2|^p-|y_1|^p-p|y_1|^{p-2}y_1y_2& \\le K\\int _0^1| y_1+q y_2|^{p-2}|y_2|^2 dq \\\\& \\le K(|y_1|^{p-2}|y_2|^2+|y_2|^p).",
"$ Hence the last term of (REF ) can be estimated by $ K\\int _{t_0}^t \\int _Z \\lbrace |x_{s}|^{p-2}|\\gamma _s( x_{s},z)|^2+|\\gamma _s( x_{s},z)|^p \\rbrace \\ N(ds,dz).$ One substitutes the estimates from (REF ) and (REF ) in equation (REF ) which by taking suprema over $[t_0,u \\wedge \\pi _R]$ for $u \\in [t_0, t_1]$ and expectations gives $ & E\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }|x_t|^p \\le E|x_{t_0}|^p+ K EM^\\frac{p}{2}+ K E\\int _{t_0}^{u \\wedge \\pi _R}|x_s|^p ds \\\\&+ pE\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }\\Big |\\int _{t_0}^{t} |x_s|^{p-2} x_s \\sigma _s( x_s) dw_s\\Big | \\\\& + pE\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }\\Big |\\int _{t_0}^{t} \\int _{Z} |x_s|^{p-2} x_{s} \\gamma _s(x_{s},z) \\tilde{N}(ds,dz)\\Big | \\\\&+ KE\\int _{t_0}^{u \\wedge \\pi _R} \\int _Z \\lbrace |x_{s}|^{p-2} |\\gamma _s( x_{s},z)|^2+|\\gamma _s( x_{s},z)|^p \\rbrace \\ N(ds,dz) \\\\&=:C_1+C_2+C_3+C_4+C_5.$ Here $C_1:=E|x_{t_0}|^p+K EM^\\frac{p}{2}$ .",
"By the Burkholder-Davis-Gundy inequality, $C_3$ can be estimated as $C_3 & = pE\\sup _{t_0 \\le t < u \\wedge \\pi _R }\\Big |\\int _{t_0}^{t} |x_{s-}|^{p-2} x_{s-} \\sigma _s( x_{s-}) dw_s\\Big |\\\\& \\le KE\\sup _{t_0 \\le t \\le u\\wedge \\pi _R}|x_{t-}|^{p-1}\\left(\\int _{t_0}^{u \\wedge \\pi _R} |\\sigma _s(x_{s-})|^2ds\\right)^{1/2}$ which on the application of Young's inequality gives $C_3 \\le \\frac{1}{4} E\\sup _{t_0 \\le t \\le u\\wedge \\pi _R}|x_{t-}|^p+K E\\left(\\int _{t_0}^{u \\wedge \\pi _R} |\\sigma _s(x_s)|^2ds\\right)^{p/2}$ and then due to Hölder's inequality and Assumption A-5, one has $ C_3 \\le \\frac{1}{4} E\\sup _{t_0 \\le t \\le u\\wedge \\pi _R}|x_{t-}|^p+KEM^\\frac{p}{2}+K E\\int _{t_0}^{u \\wedge \\pi _R} |x_r|^pds< \\infty .$ To estimate $C_4$ , one uses Lemma REF to write $C_4 &:=pE\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }\\Big |\\int _{t_0}^{t} \\int _{Z} |x_{s-}|^{p-2} x_{s-} \\gamma _s(x_{s-},z) \\tilde{N}(ds,dz)\\Big |\\\\& \\le K E\\Big (\\int _{t_0}^{u \\wedge \\pi _R} \\int _{Z} |x_{s-}|^{2p-2} |\\gamma _s(x_{s-},z)|^2 \\nu (dz) ds \\Big )^\\frac{1}{2}\\\\& \\le K E\\sup _{t_0 \\le t \\le u\\wedge \\pi _R}|x_{t-}|^{p-1}\\Big (\\int _{t_0}^{u \\wedge \\pi _R} \\int _{Z}|\\gamma _s(x_{s},z)|^2 \\nu (dz) ds \\Big )^\\frac{1}{2}$ which due to Young's inequality, Assumption A-5 and Hölder's inequality implies $ C_4 \\le \\frac{1}{4} E\\sup _{t_0 \\le t \\le u\\wedge \\pi _R}|x_{t-}|^p+KEM^\\frac{p}{2}+K E\\int _{t_0}^{u \\wedge \\pi _R} |x_r|^pds< \\infty .$ For $C_5$ , by Assumptions A-5, A-6 and Young's inequality, $ C_5&:= KE\\int _{t_0}^{u \\wedge \\pi _R} \\int _Z \\left(|x_{s}|^{p-2} |\\gamma _s( x_s,z)|^2+|\\gamma _s( x_s,z)|^p \\right)\\ \\nu (dz)ds \\\\& \\le KE\\int _{t_0}^{u \\wedge \\pi _R} \\left\\lbrace |x_{s}|^{p-2}(M+|x_{s}|^2)+M^{\\prime }+|x_s|^p \\right\\rbrace ds \\\\& \\le K EM^\\frac{p}{2}+K EM^{\\prime } +EK\\int _{t_0}^{u \\wedge \\pi _R} |x_r|^pds < \\infty .$ By substituting the estimates from (REF )-(REF ) in (REF ), one has $ E\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }|x_t|^p \\le K+\\frac{1}{2} E\\sup _{t_0 \\le t \\le u\\wedge \\pi _R}|x_{t-}|^p+K E\\int _{t_0}^{u \\wedge \\pi _R} |x_r|^pds < \\infty $ for any $u \\in [t_0, t_1]$ .",
"In particular we obtain $E\\sup _{t_0 \\le t \\le t_1 \\wedge \\pi _R }|x_t|^p< \\infty .$ Since it holds that $E\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }|x_{t-}|^p \\le E\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }|x_t|^p,$ by rearrenging in (REF ), we obtain $ \\nonumber E\\sup _{t_0 \\le t \\le u \\wedge \\pi _R }|x_t|^p &\\le K+K E\\int _{t_0}^{u \\wedge \\pi _R} |x_r|^pds \\\\& \\le K +E\\int _{t_0}^u \\sup _{t_0 \\le t \\le s \\wedge \\pi _R }|x_t|^pds < \\infty .$ From here we can finish the proof by Gronwall's and Fatou's lemmas."
],
[
"Tamed Euler Scheme",
"For every $n \\in \\mathbb {N}$ , let $b_t^n(x)$ and $\\sigma _t^n(x)$ are $P \\otimes {B}(\\mathbb {R}^d)$ -measurable functions which respectively take values in $\\mathbb {R}^d$ and $\\mathbb {R}^{d \\times m}$ .",
"Also, for every $n \\in \\mathbb {N}$ , let $\\gamma _t^n(x,z)$ be $P \\otimes {B}(\\mathbb {R}^d)\\otimes {Z}$ -measurable function which takes values in $\\mathbb {R}^{d}$ .",
"For every $n \\in \\mathbb {N}$ , we consider a scheme of SDE (REF ) as defined below, $ dx_t^n=b_t^n(x^n_{\\kappa (n,t)})dt + \\sigma _t^n(x^n_{\\kappa (n,t)})dw_t+\\int _{Z}\\gamma _t^n(x^n_{\\kappa (n,t)},z)\\tilde{N}(dt, dz), \\,$ almost surely for any $t \\in [t_0, t_1]$ where the initial value $x_{t_0}^n$ is an ${F}_{t_0}$ -measurable random variable which takes values in $\\mathbb {R}^d$ and function $\\kappa $ is defined by $ \\kappa (n, t):=\\frac{[n(t-t_0)]}{n}+t_0$ for any $t \\in [t_0, t_1]$ ."
],
[
"Moment Bounds",
"We make the following assumptions on the coefficients of the scheme (REF ).",
"B- 1 We have $\\sup _{n \\in \\mathbb {N}}E|x^n_{t_0}|^p < \\infty $ .",
"B- 2 There exist a constant $L>0$ and a sequence $(M_n)_{n \\in \\mathbb {N}}$ of non-negative random variables satisfying $\\sup _{n \\in \\mathbb {N}} EM_n^\\frac{p}{2} < \\infty $ such that $xb_t^n(x) \\vee |\\sigma _t^n(x)|^2 \\vee \\int _Z|\\gamma _t^n(x, z)|^2 \\nu (dz)\\le L(M_n+|x|^2) $ almost surely for any $t \\in [t_0, t_1]$ , $n \\in \\mathbb {N}$ and $x\\in \\mathbb {R}^d$ .",
"B- 3 There exist a constant $L>0$ and a sequence $(M_n^{\\prime })_{n \\in \\mathbb {N}}$ of non-negative random variables satisfying $\\sup _{n \\in \\mathbb {N}} EM_n^{\\prime } < \\infty $ such that $\\int _Z |\\gamma _t^n(x,z)|^p \\nu (dz) \\le L(M_n^{\\prime }+|x|^p)$ almost surely for any $t \\in [t_0,t_1]$ , $n \\in \\mathbb {N}$ and $x \\in \\mathbb {R}^d$ .",
"Below is our taming assumption on drift coefficient of scheme (REF ) following the approach of [21].",
"B- 4 For any $t \\in [t_0, t_1]$ and $x \\in \\mathbb {R}^d$ , $|b_t^n(x)| \\le n^\\theta $ almost surely with $\\theta \\in (0,\\frac{1}{2}]$ for every $n \\in \\mathbb {N}$ .",
"Remark 3.1 Note that due to assumption B-4, for each $n \\ge 1$ , the norm of $b^n$ is a bounded function of $t$ and $x$ which along with B-1 and B-2 guarantee the existence of a unique solution to (REF ).",
"Moreover, they also guarantee that for each $n\\ge 1$ , $E\\sup _{0\\le t\\le T}|x^n_t|^p < \\infty .$ Clearly, one cannot claim at this point that this bound is independent of $n$ .",
"Nevertheless, as a result of this observation, one needs not to apply stopping time arguments, similar to the one used in the proof of Lemma REF , in the proofs of Lemma REF and Lemma REF mentioned below.",
"Lemma 3.1 Let Assumptions B-2 to B-4 hold.",
"Then $\\int _{t_0}^{u}E|x^n_t-x^n_{\\kappa (n,t)}|^p dt & \\le K n^{-1} + K n^{-1} \\int _{t_0}^{u} E|x^n_{\\kappa (n,t)}|^p dt$ for any $u \\in [t_0, t_1]$ with $K:=K\\big (t_0,t_1,L,p,\\sup _{n \\in \\mathbb {N}}EM_n^\\frac{p}{2}, \\sup _{n \\in \\mathbb {N}}EM_n^{\\prime }\\big )$ which does not depend on $n$ .",
"From the definition of scheme (REF ), one writes, $E&|x^n_t-x^n_{\\kappa (n,t)}|^p \\le K E \\Big |\\int _{\\kappa (n,t)}^{t} b_s^n(x^n_{\\kappa (n,s)}) ds \\Big |^p+ K E \\Big |\\int _{\\kappa (n,t)}^{t} \\sigma ^{n}_s(x^n_{\\kappa (n,s)}) dw_s \\Big |^p\\\\& \\qquad \\qquad + KE\\Big |\\int _{\\kappa (n,t)}^{t} \\int _{Z} \\gamma _s^n(x^n_{\\kappa (n,s)},z) \\tilde{N}(ds,dz)\\Big |^p.$ which on the application of Hölder's inequality and an elementary stochastic inequalities gives $E|x^n_{t}-x^n_{\\kappa (n,t)}|^p & \\le Kn^{-(p-1)} E \\int _{\\kappa (n,t)}^{t} |b_s^n(x^n_{\\kappa (n,s)})|^p ds + K E \\Big (\\int _{\\kappa (n,t)}^{t} |\\sigma ^{n}_s(x^n_{\\kappa (n,s)}) |^2 ds\\Big )^\\frac{p}{2}\\\\+ K E\\Big (\\int _{\\kappa (n,t)}^{t} & \\int _{Z} | \\gamma _s^n(x^n_{\\kappa (n,s)},z)|^2 \\nu (dz) ds\\Big )^\\frac{p}{2} + K E\\int _{\\kappa (n,t)}^{t} \\int _{Z} | \\gamma _s^n(x^n_{\\kappa (n,s)},z)|^p \\nu (dz) ds.$ On using Assumptions B-2, B-3 and B-4, one obtains, $E|x^n_{t}-x^n_{\\kappa (n,t)}|^p & \\le K\\Big (n^{-p(1-\\theta )} + n^{-\\frac{p}{2}}E (M_n+|x^n_{\\kappa (n,t)}|^2)^\\frac{p}{2} + n^{-1}E (M_n^{\\prime }+|x^n_{\\kappa (n,t)}|^p)\\Big )$ which completes the proof by noticing that $\\theta \\in (0,\\frac{1}{2}]$ and $p\\ge 2$ .",
"Lemma 3.2 Let Assumptions B-1 to B-4 be satisfied.",
"Then, $\\sup _{n \\in \\mathbb {N}} E\\sup _{t_0 \\le t \\le t_1}|x_t^n|^p \\le K,$ with $K:=K\\big (t_0,t_1,L,p, \\sup _{n\\in \\mathbb {N}} E|x_{t_0}^n|^p, \\sup _{n\\in \\mathbb {N}}EM_n^\\frac{p}{2},\\sup _{n\\in \\mathbb {N}}EM_n^{\\prime } \\big )$ which is independent of $n$ .",
"By the application of Itô formula, one gets $|x_t^n|^p &= |x_{t_0}^n|^p+ p \\int _{t_0}^{t} |x_s^n|^{p-2} x_s^n b_s^n( x_{\\kappa (n,s)}^n) ds + p\\int _{t_0}^{t} |x_s^n|^{p-2} x_s^n \\sigma _s^n(x_{\\kappa (n,s)}^n) dw_s \\\\& + \\frac{p(p-2)}{2} \\int _{t_0}^{t} |x_s^n|^{p-4}|\\sigma _s^{n*}(x_{\\kappa (n,s)}^n) x_s^n|^2ds +\\frac{p}{2}\\int _{t_0}^{t} |x_s^n|^{p-2}|\\sigma _s^n(x_{\\kappa (n,s)}^n)|^2 ds \\\\&+ p\\int _{t_0}^{t} \\int _{Z} |x_s^n|^{p-2} x_{s}^n \\gamma _s^n( x_{\\kappa (n,s)}^n,z) \\tilde{N}(ds,dz) \\\\+\\int _{t_0}^{t} &\\int _{Z}\\lbrace |x_{s}^n+\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^p-|x_{s}^n|^p-p|x_{s}^n|^{p-2} x_{s}^n\\gamma _s^n(x_{\\kappa (n,s)}^n,z) \\rbrace N(ds,dz) $ almost surely for any $t \\in [t_0,t_1]$ .",
"In order to estimate second term of (REF ), one writes $x_s^n b_s^n( x_{\\kappa (n,s)}^n) =(x_s^n-x_{\\kappa (n,s)}^n) b_s^n( x_{\\kappa (n,s)}^n) + x_{\\kappa (n,s)}^n b_s^n( x_{\\kappa (n,s)}^n)$ which due to Assumption B-2 and equation (REF ) gives $x_s^n b_s^n( x_{\\kappa (n,s)}^n) \\le & |b_s^n( x_{\\kappa (n,s)}^n)|\\Big \\lbrace \\Big |\\int ^{s}_{\\kappa (n,s)}b_r^n(x^n_{\\kappa (n,r)})dr \\Big |+ \\Big |\\int ^{s}_{\\kappa (n,s)}\\sigma _r^n(x^n_{\\kappa (n,r)})dw_r\\Big |\\\\& +\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big |\\Big \\rbrace + K(M_n+|x_{\\kappa (n,s)}^n|^2)$ and then Assumption B-4 implies, $x_s^n b_s^n( x_{\\kappa (n,s)}^n) \\le & n^{2\\theta -1}+ n^\\theta \\Big |\\int ^{s}_{\\kappa (n,s)}\\sigma _r^n(x^n_{\\kappa (n,r)})dw_r\\Big | \\\\&+n^\\theta \\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | + K(M_n+|x_{\\kappa (n,s)}^n|^2)$ almost surely for any $s \\in [t_0,t_1]$ .",
"By using the fact that $\\theta \\in (0,1/2]$ implies $2\\theta -1 \\le 0$ , one obtains $|x_s^n|^{p-2} & x_s^n b_s^n( x_{\\kappa (n,s)}^n) \\le |x_s^n|^{p-2} + n^\\theta |x_s^n|^{p-2} \\Big |\\int ^{s}_{\\kappa (n,s)}\\sigma _r^n(x^n_{\\kappa (n,r)})dw_r\\Big |\\\\& \\,\\,\\, +n^\\theta |x_s^n|^{p-2}\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | + K |x_s^n|^{p-2}(M_n+|x_{\\kappa (n,s)}^n|^2)$ which on using Young's inequality along with the inequality $|x_s^n|^{p-2} \\le 2^{p-3} |x_s^n-x_{\\kappa (n,s)}^n|^{p-2}+2^{p-3}|x_{\\kappa (n,s)}^n|^{p-2}$ gives $|x_s^n|^{p-2} x_s^n b_s^n( x_{\\kappa (n,s)}^n) & \\le 1+ K |x_s^n|^p + K n^{\\theta \\frac{p}{2}} \\Big |\\int ^{s}_{\\kappa (n,s)}\\sigma _r^n(x^n_{\\kappa (n,r)})dw_r\\Big |^\\frac{p}{2} \\\\& + K n^{\\theta } |x_{\\kappa (n,s)}^n|^{p-2}\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | \\\\& + K n^{\\theta } |x_s^n-x_{\\kappa (n,s)}^n|^{p-2}\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | \\\\& + K (M_n^{\\frac{p}{2}}+|x_{\\kappa (n,s)}^n|^p) $ almost surely for any $s \\in [t_0,t_1]$ .",
"Therefore, by substituting estimates from equations (REF ) and (REF ) in equation (REF ), one obtains for $u \\in [t_0,t_1]$ , $E \\sup _{t_0 \\le t \\le u} &|x_t^n|^p \\le E|x_{t_0}^n|^p+ K + K E \\int _{t_0}^{u} |x_s^n|^p ds \\\\& + K n^{\\theta \\frac{p}{2}} E \\int _{t_0}^{u} \\Big |\\int ^{s}_{\\kappa (n,s)}\\sigma _r^n(x^n_{\\kappa (n,r)})dw_r\\Big |^\\frac{p}{2} ds \\\\& +K n^{\\theta } E\\int _{t_0}^{u} \\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}|x_{\\kappa (n,s)}^n|^{p-2} \\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | ds \\\\&+ K n^{\\theta } E \\int _{t_0}^{u} |x_s^n-x_{\\kappa (n,s)}^n|^{p-2}\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | ds \\\\&+KE\\int _{t_0}^{u} (M_n^{\\frac{p}{2}}+|x_{\\kappa (n,s)}^n|^p) ds+ pE \\sup _{t_0 \\le t \\le u} \\Big |\\int _{t_0}^{t} |x_s^n|^{p-2} x_s^n \\sigma _s^n(x_{\\kappa (n,s)}^n) dw_s \\Big | \\\\& + K E \\int _{t_0}^{u} |x_s^n|^{p-2}|\\sigma _s^{n}(x_{\\kappa (n,s)}^n)|^2ds \\\\& + pE \\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t} \\int _{Z} |x_s^n|^{p-2} x_{s}^n \\gamma _s^n( x_{\\kappa (n,s)}^n,z) \\tilde{N}(ds,dz) \\Big | \\\\& +E \\int _{t_0}^{u} \\int _{Z}\\lbrace |x_{s}^n|^{p-2}|\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^2+ |\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^p \\rbrace N(ds,dz) \\\\& \\quad =: E_1+E_2+E_3+E_4+E_5+E_6+E_7+E_8+E_9+E_{10}.",
"$ Here $E_1:=E|x_{t_0}^n|^p+K$ .",
"One estimates $E_2$ by $ E_2 := K E \\int _{t_0}^{u} |x_s^n|^p ds \\le K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|x_r^n|^p ds.$ In order to estimate $E_3$ , one applies an elementary stochastic inequality to obtain $E_3 & := K n^{\\theta \\frac{p}{2}} E \\int _{t_0}^{u} \\Big |\\int ^{s}_{\\kappa (n,s)}\\sigma _r^n(x^n_{\\kappa (n,r)})dw_r\\Big |^\\frac{p}{2} ds \\\\& \\le K n^{\\theta \\frac{p}{2}} \\int _{t_0}^{u} E \\Big (\\int ^{s}_{\\kappa (n,s)}|\\sigma _r^n(x^n_{\\kappa (n,r)})|^2dr\\Big )^\\frac{p}{4} ds $ and then on the application of Assumption B-2, one obtains $E_3 \\le K n^{\\frac{p}{2}(\\theta -\\frac{1}{2})} \\int _{t_0}^{u} E (M_n+|x^n_{\\kappa (n,s)}|^2)^\\frac{p}{4} ds$ which by noticing that $\\frac{p}{2}(\\theta -\\frac{1}{2}) \\in (-\\frac{p}{4},0]$ implies $ E_3 \\le K\\int _{t_0}^{u} E \\lbrace 1+(M_n+|x^n_{\\kappa (n,s)}|^2)^\\frac{p}{2}\\rbrace ds \\le K +\\int _{t_0}^{u} E \\sup _{t_0 \\le r \\le s} |x^n_{r}|^p ds.$ By Lemma REF , one estimates $E_4$ as $E_4 & :=K n^{\\theta } E\\int _{t_0}^{u} \\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}|x_{\\kappa (n,s)}^n|^{p-2} \\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | ds \\\\& \\le K n^{\\theta } \\int _{t_0}^{u} E\\Big (\\int ^{s}_{\\kappa (n,s)}\\int _{Z}|x_{\\kappa (n,s)}^n|^{2p-4} |\\gamma _r^n(x^n_{\\kappa (n,r)},z)|^2 \\nu (dz) dr \\Big )^\\frac{1}{2} ds $ which due to Assumption B-2 gives $E_4 & \\le KE\\sup _{t_0 \\le s \\le u}|x_s^n|^{p-2} n^{\\theta -\\frac{1}{2}} \\int _{t_0}^{u} (M_n+ |x^n_{\\kappa (n,s)}|^2)^\\frac{1}{2} ds $ and then on using Young's inequality and Hölder's inequality, one obtains $E_4 \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|x_s^n|^ p+ K n^{\\frac{p}{2}(\\theta -\\frac{1}{2})} E \\int _{t_0}^{u} (M_n+ |x^n_{\\kappa (n,s)}|^2)^\\frac{p}{4} ds.",
"$ By noticing that $\\theta \\in (0,\\frac{1}{2}]$ , one has $ E_4 &\\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|x_s^n|^ p+ K E \\int _{t_0}^{u} \\lbrace 1+(M_n+ |x^n_{\\kappa (n,s)}|^2)^\\frac{p}{2} \\rbrace ds.",
"\\\\& \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|x_s^n|^ p+ K +K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|x^n_r|^p ds.$ Further, to estimate $E_5$ , one uses Young's inequality and Hölder's inequality to write $& E_5:=K n^{\\theta } E \\int _{t_0}^{u} |x_s^n-x_{\\kappa (n,s)}^n|^{p-2}\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big | ds \\\\&\\le K n^{\\theta } \\int _{t_0}^{u}E|x_s^n-x_{\\kappa (n,s)}^n|^p ds + K n^{\\theta }\\int _{t_0}^{u}E\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big |^\\frac{p}{2} ds \\\\&\\le K n^{\\theta } \\int _{t_0}^{u}E|x_s^n-x_{\\kappa (n,s)}^n|^p ds +1+K n^{2\\theta }\\int _{t_0}^{u}E\\Big |\\int ^{s}_{\\kappa (n,s)}\\int _{Z}\\gamma _r^n(x^n_{\\kappa (n,r)},z)\\tilde{N}(dr, dz)\\Big |^p ds.",
"$ which on the application of Lemma REF and Lemma REF implies, $E_5 &\\le 1+K n^{\\theta -1} + K n^{\\theta -1} \\int _{t_0}^{u} E|x^n_{\\kappa (n,s)}|^p ds \\\\& + K n^{2\\theta }\\int _{t_0}^{u}E\\Big (\\int ^{s}_{\\kappa (n,s)}\\int _{Z}|\\gamma _r^n(x^n_{\\kappa (n,r)},z)|^2\\nu (dz) dr\\Big )^\\frac{p}{2} ds\\\\& + K n^{2\\theta }\\int _{t_0}^{u}E\\int ^{s}_{\\kappa (n,s)}\\int _{Z}|\\gamma _r^n(x^n_{\\kappa (n,r)},z)|^p\\nu (dz) dr ds.$ By using Assumptions B-2 and B-3, one obtains $E_5 &\\le 1+ K n^{\\theta -1} + K n^{\\theta -1} \\int _{t_0}^{u} E|x^n_{\\kappa (n,s)}|^p ds + K n^{2\\theta -\\frac{p}{2}}\\int _{t_0}^{u}E\\big (M_n+|x^n_{\\kappa (n,s)}|^2\\big )^\\frac{p}{2} ds\\\\& \\qquad + K n^{2\\theta -1}\\int _{t_0}^{u}E(M_n^{\\prime }+|x^n_{\\kappa (n,s)}|^p) ds.$ Notice that $2\\theta -1 \\in (-1,0]$ and $p\\ge 2$ .",
"Hence one has $ E_5 &\\le K+ KE M_n^\\frac{p}{2} +EM_n^{\\prime } + K \\int _{t_0}^{u} E|x^n_{\\kappa (n,s)}|^p ds \\le K + K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|x^n_r|^p ds.$ It is easy to observe that $E_6$ can be estimated by $ E_6:=KE\\int _{t_0}^{u} (M_n^{\\frac{p}{2}}+|x_{\\kappa (n,s)}^n|^p) ds \\le K + K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|x^n_r|^p ds.$ By using Burkholder-Davis-Gundy inequality and Assumption B-2, one obtains the following estimates of $E_7$ , $E_7&:=pE \\sup _{t_0 \\le t \\le u} \\Big |\\int _{t_0}^{t} |x_s^n|^{p-2} x_s^n \\sigma _s^n(x_{\\kappa (n,s)}^n) dw_s \\Big |\\\\& \\le K E \\Big (\\int _{t_0}^{u} |x_s^n|^{2p-2} | \\sigma _s^n(x_{\\kappa (n,s)}^n)|^2 ds \\Big )^\\frac{1}{2} \\le K E \\Big (\\int _{t_0}^{u} |x_s^n|^{2p-2} (M_n+|x_{\\kappa (n,s)}^n|^2) ds \\Big )^\\frac{1}{2} \\\\& \\le K E \\sup _{t_0 \\le s \\le u}|x_s^n|^{p-1} \\Big (\\int _{t_0}^{u} (M_n+|x_{\\kappa (n,s)}^n|^2) ds \\Big )^\\frac{1}{2}$ which due to Young's inequality and Hölder's inequality gives $ E_7 \\le \\frac{1}{8} E \\sup _{t_0 \\le s \\le u}|x_s^n|^p+K+K E \\int _{t_0}^{u} E \\sup _{t_0 \\le r \\le s}|x_r^n|^p ds.$ Similarly, by using Assumption B-2 and Young's inequality, $E_8$ can be estimated by $ E_8 & := K E \\int _{t_0}^{u} |x_s^n|^{p-2}|\\sigma _s^{n}(x_{\\kappa (n,s)}^n)|^2ds \\le K+K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|x_r^n|^p ds.$ Further one uses Lemma REF and Assumption B-2 to estimate $E_9$ by, $E_9 &:= pE \\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t} \\int _{Z} |x_s^n|^{p-2} x_{s}^n \\gamma _s^n( x_{\\kappa (n,s)}^n,z) \\tilde{N}(ds,dz) \\Big | \\\\& \\le K E \\Big (\\int _{t_0}^{u} \\int _{Z} |x_s^n|^{2p-2} |\\gamma _s^n( x_{\\kappa (n,s)}^n,z)|^2 \\nu (dz) ds \\Big )^\\frac{1}{2}\\\\&\\le K E \\Big (\\int _{t_0}^{u} |x_s^n|^{2p-2} (M_n+| x_{\\kappa (n,s)}^n|^2) ds \\Big )^\\frac{1}{2} $ which due to Young's inequality and Hölder's inequality gives $ E_9 & \\le \\frac{1}{8} E \\sup _{t_0 \\le s \\le u}|x_s^n|^{p} + K + K E \\int _{t_0}^{u} E \\sup _{t_0 \\le r \\le s}|x_r^n|^{p} ds.$ Finally, due to Assumptions B-2 and B-3, $E_{10}$ can be estimated as follow, $E_{10}&:=E \\int _{t_0}^{u} \\int _{Z}\\lbrace |x_{s}^n|^{p-2}|\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^2+ |\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^p \\rbrace N(ds,dz)\\\\& =E \\int _{t_0}^{u} \\int _{Z}\\lbrace |x_{s}^n|^{p-2}|\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^2+ |\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^p \\rbrace \\nu (dz) ds\\\\& =E \\int _{t_0}^{u} |x_{s}^n|^{p-2}(M_n+|x_{\\kappa (n,s)}^n|^2) ds+ E \\int _{t_0}^{u} (M_n^{\\prime }+|x_{\\kappa (n,s)}^n|^p) ds$ and then Young's inequality implies $ E_{10} \\le K+ \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|x_{r}^n|^{p} ds.$ By substituting estimates from (REF ) - (REF ) in equation (REF ), one obtains $E \\sup _{t_0 \\le t \\le u}|x_t^n|^{p} \\le \\frac{1}{2} E \\sup _{t_0 \\le t \\le u}|x_t^n|^{p} +K + K E \\int _{t_0}^{u} E \\sup _{t_0 \\le r \\le s}|x_r^n|^{p} ds.$ The application of Gronwall's Lemma completes the proof.",
"Remark 3.2 Due to Assumptions B-2 and B-3, there exist a constant $L>0$ and a sequence $(M_n^{\\prime })_{n \\in \\mathbb {N}}$ of non-negative random variables satisfying $\\sup _{n \\in \\mathbb {N}} EM_n^{\\prime } < \\infty $ such that $\\int _Z |\\gamma _t^n(x,z)|^r \\nu (dz) \\le L(M_n^{\\prime }+|x|^r)$ almost surely for any $2 \\le r \\le p$ , $t \\in [t_0,t_1]$ , $n \\in \\mathbb {N}$ and $x \\in \\mathbb {R}^d$ .",
"Lemma 3.3 Let Assumptions B-1 to B-4 be satisfied.",
"Then $\\sup _{t_0\\le t\\le t_1}E|x^n_t-x^n_{\\kappa (n,t)}|^r & \\le K n^{-1}$ for any $2 \\le r \\le p$ with $K:=K\\big (t_0,t_1,L,p, \\sup _{n\\in \\mathbb {N}} E|x_{t_0}^n|^p, \\sup _{n\\in \\mathbb {N}}EM_n^\\frac{p}{2},\\sup _{n\\in \\mathbb {N}}EM_n^{\\prime } \\big )$ which does not depend on $n$ .",
"The lemma follows immediately from Lemma REF and Lemma REF ."
],
[
"Convergence in $\\mathcal {L}^q$",
"For every $R>0 $ , we consider $F_{t_0}$ -measurable random variables $C_R$ which satisfy, $\\lim _{R \\rightarrow \\infty }P (C_R > f(R)) =0, $ for a non-decreasing function $f: \\mathbb {R_+} \\rightarrow \\mathbb {R_+}$ .",
"This notation for the family of random variables with the above property will be used throughout this article.",
"A- 7 For every $R>0$ and $t \\in [t_0,t_1]$ , $(x-\\bar{x})(b_t(x)-b_t(\\bar{x})) \\vee |\\sigma _t(x)-\\sigma _t(\\bar{x})|^2 \\vee \\int _{Z} |\\gamma _t(x,z)-\\gamma _t(\\bar{x},z)|^2 \\nu (dz)\\le C_R |x-\\bar{x}|^2 $ almost surely whenever $|x|, |\\bar{x}| \\le R$ .",
"A- 8 For every $R>0$ and $t \\in [t_0,t_1]$ , $\\sup _{|x| \\le R}| b_t(x)| \\le {C}_R,$ almost surely.",
"B- 5 For every $R>0$ and $B(R):= \\lbrace \\omega \\in \\Omega : C_R \\le f(R)\\rbrace $ , $\\lim _{n \\rightarrow \\infty }E\\int _{t_0}^{t_1}I_{B(R)}\\sup _{|x| \\le R}\\lbrace |b^n_t(x)-b_t(x)|^2+|\\sigma ^n_t(x)-\\sigma _t(x)|^2\\rbrace dt &=0\\\\\\lim _{n \\rightarrow \\infty }E \\int _{t_0}^{t_1}I_{B(R)}\\sup _{|x| \\le R}\\int _{Z}|\\gamma ^n_t(x,z)-\\gamma _t(x,z)|^2\\nu (dz) dt &=0.$ B- 6 For every $n \\in \\mathbb {N}$ , the initial values of SDE (REF ) and scheme (REF ) satisfy $|x_{t_0}-x_{t_0}^n| \\stackrel{P}{\\rightarrow } 0$ as $n \\rightarrow \\infty $ .",
"We introduce families of stopping times that shall be used frequently in this report.",
"For every $R>0$ and $n \\in \\mathbb {N}$ , let $ \\pi _R:=\\inf \\lbrace t \\ge t_0: |x_t| \\ge R\\rbrace , \\,\\, \\pi _{nR}:=\\inf \\lbrace t \\ge t_0:|x_t^n| \\ge R\\rbrace , \\,\\,\\tau _{nR}:=\\pi _R \\wedge \\pi _{nR}$ almost surely.",
"Theorem 3.1 Let Assumptions A-3 to A-8 be satisfied.",
"Also assume that B-1 to B-6 hold.",
"Then, $\\lim _{n \\rightarrow \\infty } E \\sup _{t_0 \\le t \\le t_1}|x_t-x_t^n|^q =0 $ for all $q < p$ .",
"Let $e^n_t:=x_t-x^n_t$ and define $\\bar{b}^n_t:= b_t(x_t)-b^n_t(x^n_{\\kappa (n,t)}) , \\bar{\\sigma }^n_t:=\\sigma _t(x_t)-\\sigma _t^n(x^n_{\\kappa (n,t)}), \\bar{\\gamma }^n_{t}(z):=\\gamma _t(x_{t},z)-\\gamma _t^n(x^n_{\\kappa (n,t)},z) $ almost surely for any $t \\in [t_0,t_1]$ .",
"In this simplified notation, $e_t^n$ can be written as $ e^n_t = e^n_{t_0} + \\int _{t_0}^t \\bar{b}_s^n ds+\\int _{t_0}^t \\bar{\\sigma }_s^n dw_s +\\int _{t_0}^t \\int _Z \\bar{\\gamma }_s^n(z) \\tilde{N}(ds, dz)$ almost surely for any $t \\in [t_0, t_1]$ .",
"Further, by using the stopping times defined in equation (REF ) and random variables defined in (REF ), let us partition the sample space $\\Omega $ into two parts $\\Omega _1$ and $\\Omega _2$ where $\\Omega _1& =\\lbrace \\omega \\in \\Omega :\\pi _R \\le t_1 \\, \\text{or} \\, \\pi _{nR} \\le t_1 \\, \\text{or} \\, C_R >f(R)\\ \\rbrace \\\\& =\\lbrace \\omega \\in \\Omega : \\pi _R \\le t_1\\rbrace \\cup \\lbrace \\omega \\in \\Omega : \\pi _{nR} \\le t_1\\rbrace \\cup \\lbrace \\omega \\in \\Omega : C_R >f(R)\\rbrace \\\\\\Omega _2& =\\Omega \\backslash \\Omega _1=\\lbrace \\omega \\in \\Omega : \\pi _R > t_1\\rbrace \\cap \\lbrace \\omega \\in \\Omega : \\pi _{nR} > t_1\\rbrace \\cap B(R)$ where $ B(R):= \\lbrace \\omega \\in \\Omega : C_R \\le f(R)\\rbrace $ as defined in Assumption B-5.",
"Also note that $I_\\Omega =I_{\\Omega _1 \\cup \\Omega _2} \\le I_{\\Omega _1}+I_{\\Omega _2}$ .",
"By using this fact, for any $q <p$ , one could write the following, $ E\\sup _{t_0 \\le t \\le t_1}|e_t^n|^q = E\\sup _{t_0 \\le t \\le t_1}|e_t^n|^q I_{\\Omega _1}+ E\\sup _{t_0 \\le t \\le t_1}|e_t^n|^q I_{\\Omega _2} =: D_1+D_2.$ By the application of Hölder's inequality, Lemma REF and Lemma REF one could write, $ D_1& :=E\\sup _{t_0 \\le t \\le t_1}|e_t^n|^q I_{\\Omega _1} \\le \\Big (E\\sup _{t_0 \\le t \\le t_1}|e_t^n|^{q\\frac{p}{q}}\\Big )^\\frac{q}{p} \\Big (EI_{\\Omega _1}\\Big )^\\frac{p-q}{p} \\\\& \\le K \\Big (\\frac{E|x_{\\pi _R}|^p}{R^p} + \\frac{E|x^n_{\\pi _{nR}}|^p}{R^p} + P( \\lbrace \\omega \\in \\Omega : C_R >f(R)\\rbrace ) \\Big )^\\frac{p-q}{p} \\\\& \\le K \\Big (\\frac{1}{R^p} + P( \\lbrace \\omega \\in \\Omega : C_R >f(R)\\rbrace ) \\Big )^\\frac{p-q}{p}$ where the constant $K>0$ does not depend on $n$ .",
"Having obtained estimates for $D_1$ , we now proceed to obtain the estimates for $D_2$ .",
"For this, we recall equation (REF ) and use Itô formula to obtain the following, $ |e_t^n|^2 & = |e_{t_0}^n|^2+2 \\int _{t_0}^{t} e^n_s \\bar{b}_s^n ds + 2\\int _{t_0}^{t} e^n_s \\bar{\\sigma }^{n}_s dw_s +\\int _{t_0}^{t}|\\bar{\\sigma }^n_s|^2ds \\\\&+ 2\\int _{t_0}^{t} \\int _{Z} e^n_{s} \\bar{\\gamma }_s^n(z) \\tilde{N}(ds,dz) +\\int _{t_0}^{t} \\int _{Z}|\\bar{\\gamma }_s^n(z)|^2 N(ds,dz)$ almost surely for any $t \\in [t_0,t_1]$ .",
"Also, to estimate the second term of (REF ), one uses the following splitting, $ e^n_s \\bar{b}^n_s=(x_s-x_{\\kappa (n,s)}^n) & (b_s(x_s)-b_s(x_{\\kappa (n,s)}^n))+ (x_s-x_{\\kappa (n,s)}^n) (b_s(x_{\\kappa (n,s)}^n)-b_s^n(x_{\\kappa (n,s)}^n)) \\\\&+ (x_{\\kappa (n,s)}^n-x_s^n)(b_s(x_s)-b_s(x_{\\kappa (n,s)}^n))\\\\& +(x_{\\kappa (n,s)}^n-x_s^n)(b_s(x_{\\kappa (n,s)}^n)-b_s^n(x_{\\kappa (n,s)}^n))$ almost surely for any $s \\in [t_0,t_1]$ .",
"Notice that $D_2$ is non-zero only on $\\Omega _2$ , thus one can henceforth restrict all the calculations in the estimation of $D_2$ on the interval $[t_0, t_1 \\wedge \\tau _{nR})$ which also means that $|x_t| \\vee |x_t^n| < R$ for any $t \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"As a consequence, on the application of Assumption A-7 and Cauchy-Schwarz inequality, one obtains $e^n_s \\bar{b}^n_s & \\le C_R |x_s-x_{\\kappa (n,s)}^n|^2 + |x_s-x_{\\kappa (n,s)}^n| |b_s(x_{\\kappa (n,s)}^n)-b_s^n(x_{\\kappa (n,s)}^n)|\\\\&+ |x_{\\kappa (n,s)}^n-x_s^n||b_s(x_s)-b_s(x_{\\kappa (n,s)}^n)|+|x_{\\kappa (n,s)}^n-x_s^n||b_s(x_{\\kappa (n,s)}^n)-b_s^n(x_{\\kappa (n,s)}^n)|$ almost surely for any $s \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"By using Assumption A-8, this can further be estimated as $ e^n_s \\bar{b}^n_s & \\le (2C_R+1) |x_s-x_s^n|^2+(2C_R+\\frac{3}{2})|x_s^n-x_{\\kappa (n,s)}^n|^2 + |b_s(x_{\\kappa (n,s)}^n)-b_s^n(x_{\\kappa (n,s)}^n)|^2 \\\\& \\qquad \\qquad + 2C_R|x_s^n-x_{\\kappa (n,s)}^n|$ almost surely for any $s \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"Now, by using the definition of $\\Omega _2$ and of $\\tau _{nR}$ in equation (REF ), one has $ D_2& :=E\\sup _{t_0 \\le t \\le t_1}|e_t^n|^q I_{\\Omega _2} \\le E\\sup _{t_0 \\le t \\le t_1}|e_{t \\wedge \\tau _{nR}}^n|^q I_{B(R)}.$ Thus using the estimate obtained in (REF ), one obtains $E \\sup _{t_0 \\le t \\le u} & |e_{t \\wedge \\tau _{nR}}^n|^2 I_{B(R)} \\le E|e^n_{t_0}|^2+ E(2C_R+1)\\int _{t_0}^{u \\wedge \\tau _{nR}}|e^n_s|^2 I_{B(R)} ds \\\\& + E(2C_R+\\frac{3}{2})\\int _{t_0}^{u \\wedge \\tau _{nR}}|x^n_s-x^n_{\\kappa (n,s)}|^2I_{B(R)}ds \\\\& + 2E C_R \\int _{t_0}^{u \\wedge \\tau _{nR}} |x_s^n-x_{\\kappa (n,s)}^n|I_{B(R)}ds \\\\&+E\\int _{t_0}^{u \\wedge \\tau _{nR}}|b_s(x^n_{\\kappa (n,s)})-b^n_s(x^n_{\\kappa (n,s)})|^2 I_{B(R)} ds \\\\& + 2E\\sup _{t_0 \\le t \\le u} \\Big |\\int _{t_0}^{t \\wedge \\tau _{nR}} I_{B(R)} e^n_s \\bar{\\sigma }^{n}_s dw_s \\Big | +E\\int _{t_0}^{u \\wedge \\tau _{nR}}|\\bar{\\sigma }^n_s|^2I_{B(R)}ds \\\\& + 2E\\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t \\wedge \\tau _{nR}} \\int _{Z} I_{B(R)} e^n_{s} \\bar{\\gamma }_s^n(z) \\tilde{N}(ds,dz)\\Big | \\\\& +E \\sup _{t_0 \\le t \\le u}\\int _{t_0}^{t \\wedge \\tau _{nR}} \\int _{Z} I_{B(R)}|\\bar{\\gamma }_s^n(z)|^2 N(ds,dz) \\\\&=:F_1+F_2+F_3+F_4+F_5+F_6+F_7+F_8+F_9 $ for every $R>0$ and $u \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"Here $F_1:=E|e^n_{t_0}|^2$ .",
"$F_2$ is estimated easily by $ F_2 & :=E(2C_R+1)\\int _{t_0}^{u \\wedge \\tau _{nR}}|e^n_s|^2I_{B(R)}ds \\\\&\\le (2f(R)+1)\\int _{t_0}^{u}E\\sup _{t_0 \\le r \\le s}|e_{r \\wedge \\tau _{nR}}^n|^2 I_{B(R)}ds$ for every $R>0$ and $u \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"Further, $ F_3 & :=E(2C_R+\\frac{3}{2})\\int _{t_0}^{u \\wedge \\tau _{nR}}|x^n_s-x^n_{\\kappa (n,s)}|^2I_{B(R)}ds \\\\& \\le (f(R)+1)K\\sup _{t_0 \\le t \\le t_1} E|x^n_t-x^n_{\\kappa (n,t)}|^2$ and similarly, term $F_4$ can be estimated by $ F_4&:=2E C_R \\int _{t_0}^{u \\wedge \\tau _{nR}} |x_s^n-x_{\\kappa (n,s)}^n|I_{B(R)} ds \\le f(R) K \\sup _{t_0 \\le t \\le t_1} E|x_t^n-x_{\\kappa (n,t)}^n|$ for every $R>0$ .",
"Again, term $F_5$ has following estimate, $ F_5& := E\\int _{t_0}^{u \\wedge \\tau _{nR}}|b_s(x^n_{\\kappa (n,s)})-b^n_s(x^n_{\\kappa (n,s)})|^2 I_{B(R)} ds \\\\&\\le E\\int _{t_0}^{t_1}I_{\\lbrace t_0 \\le s <\\tau _{nR}\\rbrace } I_{B(R)}|b_s(x^n_{\\kappa (n,s)})-b^n_s(x^n_{\\kappa (n,s)})|^2 ds.$ To estimate the term $F_6$ , one uses Burkholder-Davis-Gundy inequality to write $F_6 & :=2E\\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t \\wedge \\tau _{nR}} I_{B(R)} e^n_s \\bar{\\sigma }^{n}_s dw_s \\Big | \\le K E\\Big (\\int _{t_0}^{u \\wedge \\tau _{nR}} I_{B(R)}|e^n_s|^2 |\\bar{\\sigma }^{n}_s|^2 ds \\Big )^\\frac{1}{2}\\\\& \\le K E\\sup _{t_0 \\le s \\le u}|e^n_{s\\wedge \\tau _{nR}}|I_{B(R)} \\Big (\\int _{t_0}^{u \\wedge \\tau _{nR} } I_{B(R)} |\\bar{\\sigma }^{n}_s|^2 ds \\Big )^\\frac{1}{2}$ which on the application of Young's inequality gives $ F_6+F_7 \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|e^n_{s\\wedge \\tau _{nR}}|^2 I_{B(R)} + K E\\int _{t_0}^{u \\wedge \\tau _{nR}} I_{B(R)} |\\bar{\\sigma }^{n}_s|^2 ds$ for any $R>0$ and $u \\in [t_0, t_1]$ where constant $K>0$ does not depend on $R$ and $n$ .",
"In order to estimate the second term of the above inequality, one uses the following splitting of $\\bar{\\sigma }_s^n$ , $ \\bar{\\sigma }^n_s & = (\\sigma _s(x_s)-\\sigma _s(x_s^n))+(\\sigma _s(x_s^n)-\\sigma _s(x^n_{\\kappa (n,s)}))+(\\sigma _s(x^n_{\\kappa (n,s)})-\\sigma _s^n(x^n_{\\kappa (n,s)}))$ almost surely for any $s \\in [t_0, t_2]$ .",
"As before, one again notices that $|x_s|\\le R$ and $|x_s^n|\\le R$ whenever $s \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"Thus on the application of Assumption A-7, one obtains $|\\bar{\\sigma }^n_s|^2 \\le 3C_R|e_s^n|^2+3C_R|x_s^n-x^n_{\\kappa (n,s)}|^2+3|\\sigma _s(x^n_{\\kappa (n,s)})-\\sigma _s^n(x^n_{\\kappa (n,s)})|^2$ almost surely $s \\in [t_0, t_1 \\wedge \\tau _{nR})$ .",
"Hence substituting this estimate in inequality (REF ) gives $ F_6+F_7 & \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|e^n_{s\\wedge \\tau _{nR}}|^2 I_{B(R)} + K f(R) \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|e^n_{r\\wedge \\tau _{nR}}|^2 I_{B(R)} ds \\\\&+ K f(R) \\sup _{t_0 \\le s \\le t_1} E |x_s^n-x^n_{\\kappa (n,s)}|^2 \\\\& + K E\\int _{t_0}^{t_1} I_{\\lbrace t_0 \\le s < \\tau _{nR}\\rbrace } I_{B(R)} |\\sigma _s(x^n_{\\kappa (n,s)})-\\sigma _s^n(x^n_{\\kappa (n,s)})|^2 ds$ for any $u \\in [t_0, t_1]$ .",
"Further, one proceeds as above in the similar way to the derivation of (REF ) and uses Lemma REF to obtain $ F_8+F_9 \\le & \\frac{1}{8} E\\sup _{t_0 \\le s \\le u} |e^n_{s \\wedge \\tau _{nR}}|^2 I_{B(R)} \\\\& + K E\\int _{t_0}^{u \\wedge \\tau _{nR} }\\int _{Z} I_{B(R)} |\\bar{\\gamma }_s^n(z)|^2 \\nu (dz) ds$ for any $u \\in [t_0, t_1]$ .",
"In order to estimate the second term of the above inequality, one uses the following splitting, $ \\hspace{-2.84526pt}\\bar{\\gamma }^n_s(z) = & (\\gamma (x_s,z)-\\gamma _s(x_s^n,z))+(\\gamma _s(x_s^n,z)-\\gamma _s(x^n_{\\kappa (n,s)},z) \\\\&+(\\gamma _s(x^n_{\\kappa (n,s)},z)-\\gamma _s^n(x^n_{\\kappa (n,s)},z))$ almost surely for any $s \\in [t_0,t_1]$ .",
"Thus, by using the Assumption A-7, one has $ F_8+F_9 \\le & \\frac{1}{8} E\\sup _{t_0 \\le s \\le u} |e^n_{s \\wedge \\tau _{nR}}|^2 I_{B(R)} + K f(R) E \\int _{t_0}^{u } E\\sup _{t_0 \\le r \\le s} |e^n_{r \\wedge \\tau _{nR}}|^2 I_{B(R)} ds \\\\& + K f(R) \\sup _{t_0\\le s \\le t_1}E|x_s^n-x^n_{\\kappa (n,s)}|^2 \\\\& + K E \\int _{t_0}^{t_1 } \\int _Z I_{\\lbrace t_0 \\le s < \\tau _{nR}\\rbrace } I_{B(R)} |\\gamma _s(x^n_{\\kappa (n,s)},z)-\\gamma _s^n(x^n_{\\kappa (n,s)},z)|^2 \\nu (dz) ds$ for any $u \\in [t_0, t_1]$ .",
"On combining estimates obtained in (REF ), (REF ), (REF ), (REF ) and (REF ) in (REF ) and then applying Gronwall's inequality, one obtains $E &\\sup _{t_0 \\le t \\le t_1}|e_{t \\wedge \\tau _{nR}}^n|^2 I_{B(R)} \\le \\exp (Kf(R)) \\Big \\lbrace E|e_{t_0}^n|^2 + Kf(R)\\sup _{t_0\\le s \\le t_1}E|x^n_{s}-x^n_{\\kappa (n,s)}|^2 \\\\& \\qquad \\qquad + Kf(R)\\big (\\sup _{t_0\\le s \\le t_1}E|x^n_{s}-x^n_{\\kappa (n,s)}|^2\\big )^\\frac{1}{2} \\\\&+K E\\int _{t_0}^{t_1}I_{\\lbrace t_0 \\le s < \\tau _{nR}\\rbrace }I_{B(R)}|b_s(x^n_{\\kappa (n,s)})-b^n_s(x^n_{\\kappa (n,s)})|^2 ds \\\\&+ K E\\int _{t_0}^{t_1}I_{\\lbrace t_0 \\le s < \\tau _{nR}\\rbrace }I_{B(R)}|\\sigma _s(x^n_{\\kappa (n,s)})-\\sigma ^n_s(x^n_{\\kappa (n,s)})|^2 ds \\\\& + K E\\int _{t_0}^{t_1} \\int _Z I_{\\lbrace t_0 \\le s < \\tau _{nR}\\rbrace }I_{B(R)}|\\gamma _s(x^n_{\\kappa (n,s)}, z)-\\gamma ^n_s(x^n_{\\kappa (n,s)},z)|^2 \\nu (dz)ds \\Big \\rbrace .$ Hence, by the application of Lemma REF , Assumptions B-5 and B-6, one obtains $E \\sup _{t_0 \\le t \\le t_1}|e_{t \\wedge \\tau _{nR}}^n|^2 I_{B(R)} \\rightarrow 0 \\mbox{\\,\\,as\\,\\,} n \\rightarrow \\infty $ for every $R>0$ .",
"Consequently $\\sup _{t_0 \\le t \\le t_1}|e_{t \\wedge \\tau _{nR}}^n|I_{B(R)} \\rightarrow 0$ in probability, as $n\\rightarrow \\infty $ .",
"By Lemma REF and Lemma REF , we have that the sequence of random variables, $(\\sup _{t_0 \\le t \\le t_1}|e_{t \\wedge \\tau _{nR}}^n|^q I_{B(R)} )_{n \\in \\mathbb {N}}$ is uniformly integrable for any $q<p$ .",
"Hence, for each $R >0 $ we have $E \\sup _{t_0 \\le t \\le t_1}|e_{t \\wedge \\tau _{nR}}^n|^q I_{B(R)} \\rightarrow 0, \\text{as $n\\rightarrow \\infty $}$ which implies from inequality (REF ) that $D_2 \\rightarrow 0$ as $n\\rightarrow \\infty $ for every $R>0$ .",
"Also by choosing sufficiently large $R>0$ in inequality (REF ) along with equation (REF ), one obtains $D_1 \\rightarrow 0$ .",
"This complete the proof."
],
[
"Rate of Convergence",
"In order to obtain rate of convergence of the scheme (REF ), one replaces Assumption A-7 by the following assumptions.",
"A- 9 There exist constants $C>0$ , $q \\ge 2$ and $\\chi >0$ such that $(x-\\bar{x})( b_t(x)-b_t(\\bar{x})) \\vee |\\sigma _t(x)-\\sigma _t(\\bar{x})|^2 &\\vee \\int _{Z} |\\gamma _t(x,z)-\\gamma _t(\\bar{x},z)|^2 \\nu (dz) \\le C |x-\\bar{x}|^2 \\\\\\int _{Z} |\\gamma _t(x,z)-\\gamma _t(\\bar{x},z)|^q \\nu (dz) & \\le C |x-\\bar{x}|^q \\\\|b_t(x)-b_t(\\bar{x})|^2 & \\le C(1+|x|^\\chi +|\\bar{x}|^\\chi )|x-\\bar{x}|^2 $ almost surely for any $t \\in [t_0,t_1]$ , $x,\\bar{x} \\in \\mathbb {R}^d$ and a $\\delta \\in (0,1)$ such that $\\max \\big \\lbrace (\\chi +2)q,\\frac{q \\chi }{2}\\frac{q+\\delta }{\\delta }\\big \\rbrace \\le p$ .",
"Remark 3.3 Due to (REF ) and Assumption A-8, one immediately obtains $|b_t(x)|^2 \\le K(1+|x|^{\\chi +2})$ almost surely for any $t \\in [t_0, t_1]$ and $x \\in \\mathbb {R}^d$ .",
"Furthermore, one replaces Assumption B-5 by the following assumption.",
"B- 7 There exists a constant $C>0$ such that $E\\int _{t_0}^{t_1}\\lbrace |b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q +|\\sigma ^n_t(x_{\\kappa (n,t)}^n)-\\sigma _t(x_{\\kappa (n,t)}^n)|^q\\rbrace dt &\\le C n^{-\\frac{q}{q+\\delta }} \\\\E \\int _{t_0}^{t_1}\\Big (\\int _{Z}|\\gamma ^n_t(x_{\\kappa (n,t)}^n,z)-\\gamma _t(x_{\\kappa (n,t)}^n,z)|^{\\zeta }\\nu (dz)\\Big )^{\\frac{q}{\\zeta }} dt &\\le C n^{-\\frac{q}{q+\\delta }} $ for $\\zeta =2,q$ .",
"Finally, Assumption B-6 is replaced by the following assumption.",
"B- 8 There exists a constant $C>0$ such that $E|x_{t_0}-x_{t_0}^n|^q \\le C n^{-\\frac{q}{q+\\delta }}.",
"$ Theorem 3.2 Let Assumptions A-3 to A-6, A-8 and A-9 be satisfied.",
"Also suppose that Assumptions B-1 to B-4, B-7 and B-8 hold.",
"Then $E\\sup _{t_0 \\le t \\le t_1}|x_{t}-x_{t}^n|^q \\le K n^{-\\frac{q}{q+\\delta }} $ where constant $K>0$ does not depend on $n$ .",
"First of all, let us recall the notations used in the proof of Theorem REF .",
"By the application of Itô formula, one obtains $|e_t^n|^q &= |e_{t_0}^n|^q+ q \\int _{t_0}^{t} |e_s^n|^{q-2} e_s^n \\bar{b}_s^n ds + q\\int _{t_0}^{t} |e_s^n|^{q-2} e_s^n \\bar{\\sigma }_s^n dw_s \\\\& + \\frac{q(q-2)}{2} \\int _{t_0}^{t} |e_s^n|^{q-4}|\\bar{\\sigma }_s^{n*} e_s^n|^2ds +\\frac{q}{2}\\int _{t_0}^{t} |e_s^n|^{q-2}|\\bar{\\sigma }_s^n|^2 ds \\\\&+ q\\int _{t_0}^{t} \\int _{Z} |e_s^n|^{q-2} e_s^n \\bar{\\gamma }_s^n(z) \\tilde{N}(ds,dz) \\\\+\\int _{t_0}^{t} &\\int _{Z}\\lbrace |e_s^n+\\bar{\\gamma }_s^n(z)|^q-|e_s^n|^q-q|e_s^n|^{q-2} e_s^n \\bar{\\gamma }_s^n(z) \\rbrace N(ds,dz) $ almost surely for any $t \\in [t_0,t_1]$ .",
"In Theorem REF , the splitting given in (REF ) is used to prove the $\\mathcal {L}^q$ convergence of the scheme (REF ).",
"In order to obtain a rate of convergence of scheme (REF ), one uses the following splitting, $ e^n_s \\bar{b}^n_s =& (x_s-x_s^n)(b_s(x_s)-b_s(x^n_s)) + (x_s-x^n_s)(b_s(x^n_s)-b_s(x^n_{\\kappa (n,s)})) \\\\& +(x_s-x^n_s)(b_s(x^n_{\\kappa (n,s)})-b_s^n(x^n_{\\kappa (n,s)}))$ which on the application of Assumption A-9, Cauchy-Schwarz inequality and Young's inequality gives $ |e^n_s|^{q-2}e^n_s \\bar{b}^n_s & \\le K|e^n_s|^q + K|b_s(x^n_s)-b_s(x^n_{\\kappa (n,s)})|^q + K|b_s(x^n_{\\kappa (n,s)})-b_s^n(x^n_{\\kappa (n,s)})|^q$ almost surely for any $s \\in [t_0,t_1]$ .",
"Therefore by taking suprema over $[t_0, u]$ for any $u \\in [t_0,t_1]$ and expectations, one has $ E\\sup _{t_0 \\le t \\le u}|e_t^n|^q \\le & E|e_{t_0}^n|^q+ K E\\int _{t_0}^{u} |e_s^n|^q ds + K E\\int _{t_0}^{u} |b_s(x^n_s)-b_s(x^n_{\\kappa (n,s)})|^q ds \\\\& \\hspace{-56.9055pt} + K E\\int _{t_0}^{u} |b_s(x^n_{\\kappa (n,s)})-b_s^n(x^n_{\\kappa (n,s)})|^q ds + qE\\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t} |e_s^n|^{q-2} e_s^n \\bar{\\sigma }_s^n dw_s\\Big | \\\\& + \\frac{q(q-2)}{2} E\\int _{t_0}^{u} |e_s^n|^{q-4}|\\bar{\\sigma }_s^{n*} e_s^n|^2ds +\\frac{q}{2}E\\int _{t_0}^{u} |e_s^n|^{q-2}|\\bar{\\sigma }_s^n|^2 ds \\\\&+ qE\\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t} \\int _{Z} |e_s^n|^{q-2} e_s^n \\bar{\\gamma }_s^n(z) \\tilde{N}(ds,dz)\\Big | \\\\&+E\\sup _{t_0 \\le t \\le u}\\int _{t_0}^{t} \\int _{Z}\\lbrace |e_s^n|^{q-2}|\\bar{\\gamma }_s^n(z)|^2+|\\bar{\\gamma }_s^n(z)|^q \\rbrace N(ds,dz) \\\\=& G_1+G_2+G_3+G_4+G_5+G_6+G_7+G_8+G_9$ for any $u \\in [t_0, t_1]$ .",
"Here $G_1:=E|e_{t_0}^n|^q$ and $G_2$ can be estimated by $ G_2:= K E\\int _{t_0}^{u} |e_s^n|^q ds \\le K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|e_r^n|^q ds$ for any $u \\in [t_0, t_1]$ .",
"By the application of Assumption A-9, Hölder's inequality and Lemma REF , $G_3$ can be estimated by $ G_3 :=K E\\int _{t_0}^{u} &|b_s(x^n_s)-b_s(x^n_{\\kappa (n,s)})|^q ds \\\\&\\hspace{-56.9055pt}\\le K \\int _{t_0}^{u} \\left( 1+E|x^n_s|^{\\chi \\frac{q}{2}\\frac{q+\\delta }{\\delta }}+E|x^n_{\\kappa (n,s)}|^{\\chi \\frac{q}{2}\\frac{q+\\delta }{\\delta }}\\right)^\\frac{\\delta }{q+\\delta } \\left( E|x^n_s-x^n_{\\kappa (n,s)}|^{q+\\delta }\\right)^\\frac{q}{q+\\delta } ds \\\\& \\le K \\int _{t_0}^{t_1} \\left( E|x^n_s-x^n_{\\kappa (n,s)}|^{q+\\delta }\\right)^\\frac{q}{q+\\delta } ds.$ Further, $G_4$ can be estimated by $ G_4&:=K E\\int _{t_0}^{u} |b_s(x^n_{\\kappa (n,s)})-b_s^n(x^n_{\\kappa (n,s)})|^q ds \\\\& \\le K E\\int _{t_0}^{t_1} |b_s(x^n_{\\kappa (n,s)})-b_s^n(x^n_{\\kappa (n,s)})|^q ds.$ By the application of Burkholder-Davis-Gundy inequality, one obtains $G_5 & := qE\\sup _{t_0 \\le t \\le u}\\Big |\\int _{t_0}^{t} |e_s^n|^{q-2} e_s^n \\bar{\\sigma }_s^n dw_s\\Big | \\le K E\\Big (\\int _{t_0}^{u} |e_s^n|^{2q-2}|\\bar{\\sigma }_s^n|^2 ds\\Big )^\\frac{1}{2}\\\\& \\le K E\\sup _{t_0 \\le s \\le u}|e_s^n|^{q-1}\\Big (\\int _{t_0}^{u} |\\bar{\\sigma }_s^n|^2 ds\\Big )^\\frac{1}{2}$ which due to Young's inequality and Hölder's inequality gives $ G_5 \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|e_s^n|^{q} + K E\\int _{t_0}^{u} |\\bar{\\sigma }_s^n|^q ds$ for any $u \\in [t_0, t_1]$ .",
"Further, due to Cauchy-Schwarz inequality and Young's inequality, $G_6$ and $G_7$ can be estimated together by $ G_6+G_7&:=\\frac{q(q-2)}{2} E\\int _{t_0}^{u} |e_s^n|^{q-4}|\\bar{\\sigma }_s^{n*} e_s^n|^2ds +\\frac{q}{2}E\\int _{t_0}^{u} |e_s^n|^{q-2}|\\bar{\\sigma }_s^n|^2 ds \\\\& \\le K E\\int _{t_0}^{u} |e_s^n|^{q-2}|\\bar{\\sigma }_s^n|^2 ds \\le K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|e_r^n|^{q} ds+ K E\\int _{t_0}^{u}|\\bar{\\sigma }_s^n|^q ds$ for any $u \\in [t_0, t_1]$ .",
"On combining the estimated from (REF ) and (REF ), one has $ G_5+G_6+G_7 \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|e_s^n|^{q} + K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|e_r^n|^{q} ds+ K E\\int _{t_0}^{u}|\\bar{\\sigma }_s^n|^q ds$ for any $u \\in [t_0, t_1]$ .",
"Now, one uses the splitting of $\\bar{\\sigma }_s^n$ given in (REF ) along with Assumption A-9 to write $ G_5+G_6+G_7 & \\le \\frac{1}{8} E\\sup _{t_0 \\le s \\le u}|e_s^n|^{q} + K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|e_r^n|^{q} ds+ K \\int _{t_0}^{t_1}E|x_s^n-x_{\\kappa (n,s)}^n|^q ds \\\\&+ K E\\int _{t_0}^{t_1}|\\sigma _s(x_{\\kappa (n,s)}^n)-\\sigma _s^n(x_{\\kappa (n,s)}^n)|^q ds$ for any $u \\in [t_0, t_1]$ .",
"Further, for estimating $G_8$ , one uses the splitting of $\\bar{\\gamma }_s^n(z)$ given in (REF ) to write $G_8 & \\le E \\sup _{t_0 \\le t \\le u} \\Big | \\int _{t_0}^{t} \\int _{Z} |e_s^n|^{q-2} e_s^n \\lbrace \\gamma _s(x_s,z)-\\gamma _s(x_s^n,z)\\rbrace \\tilde{N}(ds,dz) \\Big | \\\\& +E \\sup _{t_0 \\le t \\le u} \\Big | \\int _{t_0}^{t} \\int _{Z} |e_s^n|^{q-2} e_s^n \\lbrace \\gamma _s(x_s^n,z)-\\gamma _s(x_{\\kappa (n,s)}^n,z) \\rbrace \\tilde{N}(ds,dz) \\Big | \\\\& + E \\sup _{t_0 \\le t \\le u} \\Big | \\int _{t_0}^{t} \\int _{Z} |e_s^n|^{q-2} e_s^n \\lbrace \\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) \\rbrace \\tilde{N}(ds,dz) \\Big | $ which due to Lemma REF gives $G_8 & \\le E \\Big ( \\int _{t_0}^{u} \\int _{Z} |e_s^n|^{2q-2} |\\gamma _s(x_s,z)-\\gamma _s(x_s^n,z)|^2 \\nu (dz)ds \\Big )^\\frac{1}{2} \\\\& +E \\Big ( \\int _{t_0}^{u} \\int _{Z} |e_s^n|^{2q-2} |\\gamma _s(x_s^n,z)-\\gamma _s(x_{\\kappa (n,s)}^n,z) |^2 \\nu (dz)ds \\Big )^\\frac{1}{2} \\\\& + E \\Big ( \\int _{t_0}^{u} \\int _{Z} |e_s^n|^{2q-2} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^2 \\nu (dz)ds \\Big )^\\frac{1}{2} $ for any $u \\in [t_0,t_1]$ .",
"Then on the application of Young's inequality and Hölder's inequality, one obtains, $G_8 & \\le \\frac{1}{8} E \\sup _{t_0 \\le s \\le u} |e_s^n|^q + E \\int _{t_0}^{u} \\Big (\\int _{Z} |\\gamma _s(x_s,z)-\\gamma _s(x_s^n,z)|^2 \\nu (dz)\\Big )^\\frac{q}{2} ds \\\\& + E \\int _{t_0}^{u} \\Big (\\int _{Z} |\\gamma _s(x_s^n,z)-\\gamma _s(x_{\\kappa (n,s)}^n,z) |^2 \\nu (dz) \\Big )^\\frac{q}{2} ds\\\\& + E\\int _{t_0}^{u}\\Big ( \\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^2 \\nu (dz) \\Big )^\\frac{q}{2}ds.",
"$ Thus by using Assumption A-9, one has $ G_8 & \\le \\frac{1}{8} E \\sup _{t_0 \\le s \\le u} |e_s^n|^q + \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s}|e_s^n|^q ds + \\int _{t_0}^{t_1} E|x_s^n-x_{\\kappa (n,s)}^n|^q ds\\\\& + E\\int _{t_0}^{t_1}\\Big ( \\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z)|^2 \\nu (dz) \\Big )^\\frac{q}{2}ds$ for any $u \\in [t_0, t_1]$ .",
"Finally, one could write $G_9$ as $ G_9 & := E\\sup _{t_0 \\le t \\le u}\\int _{t_0}^{t} \\int _{Z}\\lbrace |e_s^n|^{q-2}|\\bar{\\gamma }_s^n(z)|^2+|\\bar{\\gamma }_s^n(z)|^q \\rbrace N(ds,dz) \\\\&= E\\int _{t_0}^{u} \\int _{Z} |e_s^n|^{q-2}|\\bar{\\gamma }_s^n(z)|^2 \\nu (dz) ds +E\\int _{t_0}^{u} \\int _{Z} |\\bar{\\gamma }_s^n(z)|^q \\nu (dz) ds =:H_1+H_2$ for any $u \\in [t_0,t_1]$ .",
"In order to estimate the first term $H_1$ on the right hand side of the inequality (REF ) along with Assumption A-9, one recalls the splitting of $\\gamma _s^n(z)$ given in (REF ) to get the following estimate, $H_1 \\le & K E\\int _{t_0}^{u} |e_s^n|^{q} ds + K E\\int _{t_0}^{u} |e_s^n|^{q-2}|x_s^n-x_{\\kappa (n,s)}^n|^2 ds \\\\&+ E\\int _{t_0}^{u} \\int _{Z} |e_s^n|^{q-2}|\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^2 \\nu (dz)ds $ for any $u \\in [t_0, t_1]$ .",
"By the application of Young's inequality, one obtains $ H_1 & \\le K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s} |e_r^n|^{q} ds + K\\int _{t_0}^{t_1} E|x_s^n-x_{\\kappa (n,s)}^n|^q ds \\\\&+ K E\\int _{t_0}^{t_1} \\Big (\\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^2 \\nu (dz)\\Big )^\\frac{q}{2} ds$ for any $u \\in [t_0, t_1]$ .",
"For the second term $H_2$ on the right hand side of the inequality (REF ) along with Assumption A-9, one again uses the splitting of $\\bar{\\gamma }_s^n(z)$ given in equation (REF ) to get the following estimate, $ H_2 & \\le K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s} |e_r^n|^{q} ds + K\\int _{t_0}^{t_1} E|x_s^n-x_{\\kappa (n,s)}^n|^q ds \\\\&+ K E\\int _{t_0}^{t_1} \\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^q \\nu (dz) ds$ for any $u \\in [t_0, t_1]$ .",
"Hence on combining the estimates obtained in (REF ) and (REF ) in (REF ), one obtains $ G_9 & \\le K \\int _{t_0}^{u} E\\sup _{t_0 \\le r \\le s} |e_r^n|^{q} ds + K\\int _{t_0}^{t_1} E|x_s^n-x_{\\kappa (n,s)}^n|^q ds \\\\& + K E\\int _{t_0}^{t_1} \\Big (\\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^2 \\nu (dz)\\Big )^\\frac{q}{2} ds \\\\&+ K E\\int _{t_0}^{t_1} \\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^q \\nu (dz) ds$ for any $u \\in [t_0, t_1]$ .",
"Thus one can substitute estimates from (REF ), (REF ), (REF ), (REF ), (REF ) and (REF ) in (REF ) and then apply Gronwall's inequality to obtain $E\\sup _{t_0 \\le t \\le t_1}&|e_t^n|^q \\le E|e_{t_0}^n|^q + K \\int _{t_0}^{t_1} \\left( E|x^n_s-x^n_{\\kappa (n,s)}|^{q+\\delta }\\right)^\\frac{q}{q+\\delta }ds\\\\& + K\\int _{t_0}^{t_1} E|x_s^n-x_{\\kappa (n,s)}^n|^q ds + K E\\int _{t_0}^{t_1} |b_s(x^n_{\\kappa (n,s)})-b_s^n(x^n_{\\kappa (n,s)})|^q ds \\\\& + K E\\int _{t_0}^{t_1} |\\sigma _s(x^n_{\\kappa (n,s)})-\\sigma _s^n(x^n_{\\kappa (n,s)})|^q ds\\\\& + K E\\int _{t_0}^{t_1} \\Big (\\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^2 \\nu (dz)\\Big )^\\frac{q}{2} ds \\\\&+ K E\\int _{t_0}^{t_1} \\int _{Z} |\\gamma _s(x_{\\kappa (n,s)}^n,z)-\\gamma _s^n(x_{\\kappa (n,s)}^n,z) |^q \\nu (dz) ds.$ By the application of Assumptions B-7, B-8 and Lemma REF , one obtains, $E\\sup _{t_0 \\le t \\le t_1}&|e_t^n|^q \\le Kn^{-\\frac{q}{q+\\delta }}$ which completes the proof."
],
[
"A Simple Example",
"We now introduce a tamed Euler scheme of SDEs driven by Lévy noise which have coefficients that are not random.",
"For this purpose, we only highlight the modifications needed in the settings of our previous discussion.",
"In SDE (REF ), $b_t(x)$ and $\\sigma _t(x)$ are ${B}([0,T]) \\otimes {B}(\\mathbb {R}^d)$ -measurable functions with values in $\\mathbb {R}^d$ and $\\mathbb {R}^{d\\times m}$ respectively.",
"Also $\\gamma _t(x,z)$ is a ${B}([0,T]) \\otimes {B}(\\mathbb {R}^d)\\otimes {Z}$ -measurable function with values in $\\mathbb {R}^d$ .",
"Moreover, one modifies Assumptions A-5 and A-6 by assigning $M=M^{\\prime }=1$ .",
"Further, for every $n \\in \\mathbb {N}$ , the scheme (REF ) is given by defining $ b^n_t(x)=\\frac{b_t(x)}{1+n^{-\\theta }|b_t(x)|}, \\sigma _t^n(x)=\\sigma _t(x) \\mbox{ and } \\gamma _t^n(x,z)=\\gamma _t(x,z)$ with $\\theta \\in (0,\\frac{1}{2}]$ for any $t \\in [t_0,t_1]$ , $x \\in \\mathbb {R}^d$ and $z \\in Z$ .",
"Then, it is easy to observe that Assumptions B-2 to B-4 hold since $M_n=M_n^{\\prime }=1$ and $\\theta \\in (0,\\frac{1}{2}]$ .",
"Hence Lemmas [REF , REF , REF , REF ] follow immediately.",
"Finally, ${F}_{t_0}$ measurable random variable $C_R$ in Assumptions A-7 and A-8 is a constant for every $R$ .",
"In this new settings, one obtains the following corollaries for SDE (REF ) and scheme (REF ) with coefficients given by (REF ).",
"Corollary 3.1 Let Assumptions A-3 to A-8 be satisfied by the coefficients of SDE given immediately above.",
"Also assume that B-1 and B-6 hold.",
"Then, the numerical scheme (REF ) with coefficients given by (REF ) converges to the solution of SDE (REF ) in $\\mathcal {L}^q$ sense i.e.",
"$\\lim _{n \\rightarrow \\infty } E \\sup _{t_0 \\le t \\le t_1}|x_t-x_t^n|^q =0 $ for all $q < p$ .",
"Assumption A-7 and A-8 are satisfied on taking $f(R)=C_R$ in equation (REF ).",
"For Assumption B-5, one observes due to (REF ) and Assumption A-8, $E\\int _{t_0}^{t_1} I_{B(R)} \\sup _{|x| \\le R}|b_t^n(x)-b_t(x)|^2dt& \\le n^{-2\\theta } E\\int _{t_0}^{t_1} I_{B(R)} \\sup _{|x| \\le R}|b_t(x)|^4dt \\le K f(R)^4 n^{-2\\theta } \\rightarrow 0 $ as $ n \\rightarrow \\infty $ for every $R$ .",
"Also for diffusion and jump coefficients, Assumption B-5 holds trivially.",
"Thus, Theorem REF completes the proof.",
"For rate of convergence of scheme (REF ), one takes $\\theta =\\frac{1}{2}$ in equation (REF ).",
"Corollary 3.2 Let Assumptions A-3 to A-6, A-8 and A-9 be satisfied by the coefficients of SDE given immediately above.",
"Also suppose that Assumptions B-1 and B-8 hold.",
"Then, the numerical scheme (REF ) with coefficients given by (REF ) achieves the classical rate (of Euler scheme) in $\\mathcal {L}^q$ sense i.e.",
"$E\\sup _{t_0 \\le t \\le t_1}|x_{t}-x_{t}^n|^q \\le K n^{-\\frac{q}{q+\\delta }}$ where constant $K>0$ does not depend on $n$ .",
"By using equation (REF ) and Remark REF , one obtains $E\\int _{t_0}^{t_1} |b_t^n(x^n_{\\kappa (n,t)})-b_t(x^n_{\\kappa (n,t)})|^q dt & \\le n^{-2\\theta } E\\int _{t_0}^{t_1} |b_t(x^n_{\\kappa (n,t)})|^{2q} dt \\le K n^{-1} (1+E\\sup _{t_0 \\le t \\le t_1}|x^n_t|^{q(\\chi +2)})$ since $\\theta =\\frac{1}{2}$ .",
"Hence Assumption B-7 for drift coefficients follows due to Lemma REF .",
"For diffusion and jump coefficients, Assumption B-7 holds trivially.",
"The proof is completed by Theorem REF ."
],
[
"Application to Delay Equations",
"Let us assume that $\\beta _t(y_1, \\ldots , y_k, x)$ and $\\alpha _t(y_1, \\ldots , y_k, x)$ are $B([0,T]) \\otimes B(\\mathbb {R}^{d\\times k}) \\otimes B(\\mathbb {R}^d)$ -measurable functions and take values in $\\mathbb {R}^d$ and $\\mathbb {R}^{d \\times m}$ respectively.",
"Also let $\\lambda _t( y_1, \\ldots , y_k, x, z)$ be $B([0,T]) \\otimes B(\\mathbb {R}^{d\\times k}) \\otimes B(\\mathbb {R}^d) \\otimes {Z}$ -measurable function and takes values in $\\mathbb {R}^{d}$ .",
"For fixed $H>0$ , we consider a $d$ -dimensional stochastic delay differential equations (SDDEs) on $(\\Omega , \\lbrace F_t\\rbrace _{t \\ge 0}, F, P)$ defined by, $ dx_t &= \\beta _t(y_t, x_t)dt +\\alpha _t(y_t, x_t)dw_t + \\int _{Z} \\lambda _t(y_{t},x_{t},z)\\tilde{N}(dt,dz), \\,\\,\\, t \\in [0,T], \\\\x_t& = \\xi _t, \\,\\,\\, t \\in [-H, 0 ],$ where $\\xi :[-H,0]\\times \\Omega \\rightarrow \\mathbb {R}^d$ and $y_t:=(x_{\\delta _1(t)}, \\ldots , x_{\\delta _k(t)})$ .",
"The delay parameters $\\delta _1(t), \\ldots , \\delta _k(t)$ are increasing functions of $t$ and satisfy $-H \\le \\delta _j(t) \\le [t/h]h$ for some $h>0$ and $j=1,\\ldots , k$ .",
"Remark 4.1 In the following, we assume, without loss of generality, that $T$ is a multiple of $h$ .",
"If not, then SDDE (REF ) can be defined for $T^{\\prime } > T$ so that $T^{\\prime }=N^{\\prime } h$ , where $N^{\\prime }$ is a positive integer.",
"The results proved in this article are then recovered for the original SDDE (REF ) by choosing parameters as $\\beta I_{t\\le T}$ , $\\alpha I_{t \\le T}$ and $\\lambda I_{t \\le T}$ .",
"Remark 4.2 We remark that two popular cases of delay viz.",
"$\\delta _i(t)=t-h$ and $\\delta _i(t)=[t/h]h$ can be addressed by our findings which have been widely used in literature, for example, [4], [5], [17], [19] and references therein."
],
[
"Existence and Uniqueness",
"To prove the existence and uniqueness of the solution of SDDE (REF ), we make the following assumptions.",
"C- 1 For every $R>0$ , there exists an $M(R) \\in \\mathbb {L}^1$ such that $x \\beta _t(y,x) + |\\alpha _t(y,x)|^2 +\\int _Z |\\lambda _t(y,x,z)|^2 \\nu (dz) \\le M_t(R)(1+|x|^2) $ for any $t \\in [0,T]$ whenever $|y| \\le R$ and $x \\in \\mathbb {R}^d$ .",
"C- 2 For every $R>0$ , there exists an $M(R) \\in \\mathbb {L}^1$ such that $(x-\\bar{x}) (\\beta _t(y, x)-\\beta _t(y, \\bar{x})) &+ |\\alpha _t(y,x)-\\alpha _t(y,\\bar{x})|^2 + \\int _Z |\\lambda _t(y, x,z)-\\lambda _t(y, \\bar{x},z)|^2 \\nu (dz) \\le M_t(R)|x-\\bar{x}|^2 $ for any $t \\in [0,T]$ whenever $|x|,|\\bar{x}|, |y| \\le R$ .",
"C- 3 The function $\\beta _t(y,x)$ is continuous in $x$ for any $t$ and $y$ .",
"Theorem 4.1 Let Assumptions C-1 to C-3 be satisfied.",
"Then there exists a unique solution to SDDE (REF ).",
"We adopt the approach of [7] and consider SDDE (REF ) as a special case of SDE (REF ) by assigning the following values to the coefficients, $b_t(x)=\\beta _t( y_t, x), \\sigma _t(x)=\\alpha _t(y_t, x), \\gamma _t(x,z)=\\lambda _t(y_t, x, z) $ almost surely for any $t \\in [0,T]$ .",
"Then the proof is a straightforward generalization of Theorem 2.1 of [7] and follows due to Theorem REF ."
],
[
"Tamed Euler Scheme",
"For every $n \\in \\mathbb {N}$ , define the following tamed Euler scheme $dx_t^n &= \\beta _t^n(y_t^n, x_{\\kappa (n,t)}^n) dt + \\alpha _t(y_t^n, x_{\\kappa (n,t)}^n) dw_t +\\int _{Z} \\lambda _t( y_t^n, x_{\\kappa (n,t)}^n, z)\\tilde{N}(dt,dz), \\,\\,\\,t \\in [0, T], \\\\x_t^n &= \\xi _t, \\,\\,\\, t \\in [-H, 0], $ where $y_t^n:=(x_{\\delta _1(t)}^n, \\ldots , x_{\\delta _k(t)}^n)$ and $\\kappa $ is defined by (REF ) with $t_0=0$ .",
"Furthermore, for every $n\\in \\mathbb {N}$ , the drift coefficient is given by $\\beta _t^n(y, x):=\\frac{\\beta _t(y, x)}{1+n^{-\\theta }|\\beta _t(y, x)|} $ which satisfies $|\\beta _t^n(y, x)| \\le \\min (n^\\theta , |\\beta _t(y, x)|) $ for any $t \\in [0,T]$ , $x \\in \\mathbb {R}^d$ and $y \\in \\mathbb {R}^{d \\times k}$ .",
"C- 4 For a fixed $p \\ge 2$ , $E\\sup _{-H \\le t \\le 0}|\\xi _t|^p< \\infty $ .",
"C- 5 There exist constants $G >0$ and $\\chi \\ge 2$ such that $x \\beta _t(y, x)\\vee |\\alpha _t(y, x)|^2 \\vee \\int _{Z} |\\lambda _t(y,x,z)|^2\\nu (dz)\\le G(1+|y|^\\chi +|x|^2) $ for any $t \\in [0,T]$ , $x \\in \\mathbb {R}^d$ and $y \\in \\mathbb {R}^{d \\times k}$ .",
"C- 6 There exist constants $G >0$ and $\\chi \\ge 2$ such that $&\\int _{Z} |\\lambda _t(y,x,z)|^p\\nu (dz) \\le G(1+|y|^{\\chi \\frac{p}{2}}+|x|^p) $ for any $t \\in [0,T]$ , $x \\in \\mathbb {R}^d$ and $y \\in \\mathbb {R}^{d \\times k}$ .",
"C- 7 For every $R>0$ , there exists a constant $K_R >0 $ such that $(x-\\bar{x})(\\beta _t(y, x)-\\beta _t(y, \\bar{x}) ) \\vee |\\alpha _t(y, x)&-\\alpha _t(y, \\bar{x})|^2 \\vee \\int _{Z} |\\lambda _t(y,x,z)-\\lambda _t(y,\\bar{x},z)|^2 \\nu (dz)\\le K_R |x-\\bar{x}|^2 $ for any $t \\in [0,T]$ whenever $|x|, |y|, |\\bar{x}| < R$ .",
"C- 8 For every $R>0$ , there exists a constant $K_R>0$ such that $\\sup _{|x| \\le R} \\sup _{|y|\\le R} |\\beta _t(y, x)|^2 \\le K_R $ for any $t \\in [0, T]$ .",
"C- 9 For every $R>0$ and $t \\in [0,T]$ , $\\sup _{|x| \\le R}\\Big \\lbrace |\\beta _t(y, x)-&\\beta _t(y^{\\prime }, x)|^2+|\\alpha _t(y, x)-\\alpha _t(y^{\\prime }, x)|^2 + \\int _{Z} |\\lambda _t(y, x, z)-\\lambda _t(y^{\\prime }, x, z)|^2 \\nu (dz)\\Big \\rbrace \\rightarrow 0 $ when $y^{\\prime } \\rightarrow y$ .",
"Let us also define, $p_i=\\Big (\\frac{2}{\\chi }\\Big )^i p $ for $i=1, \\ldots , N^{\\prime }$ , where $\\chi $ and $p$ satisfy $p/2 \\ge (\\chi /2)^{N^{\\prime }}$ .",
"Also $p^*=\\min _{i}p_i= \\Big (\\frac{2}{\\chi }\\Big )^{N^{\\prime }} p. $ The following corollary is a consequence of Theorem REF .",
"Corollary 4.1 Let Assumptions C-3 to C-9 hold, then $\\lim _{n \\rightarrow \\infty }E \\sup _{0 \\le t \\le T}|x_t-x_t^n|^q =0$ for any $q <p^*$ .",
"First as before, one observes that SDDE (REF ) can be regarded as a special case of SDE (REF ) with coefficients given by (REF ).",
"Moreover, tamed Euler scheme (REF ) is a special of (REF ) with coefficients given by $ b^n_t(x)=\\frac{\\beta _t(y^n_t, x)}{1+n^{-\\theta }|\\beta _t(y^n_t, x)|}, \\sigma _t^n(x)=\\alpha _t(y^n_t, x), \\gamma _t^n(x,z)=\\lambda _t(y_t^n, x,z)$ almost surely for any $t \\in [0, T]$ and $x\\in \\mathbb {R}^d$ .",
"We shall use inductive arguments to show $\\lim _{n \\rightarrow \\infty }E\\sup _{(i-1)h \\le t \\le i h} |x_t-x_t^n|^q =0 $ for any $q < p_{i}$ and for every $i \\in \\lbrace 1, \\ldots , N^{\\prime }\\rbrace $ .",
"Case: $\\mathbf {t \\in [0, h]}$ .",
"For $t \\in [0, h]$ , one could consider SDDE (REF ) and their tamed Euler scheme (REF ) as SDE (REF ) and scheme (REF ) respectively with $t_0=0$ , $t_1=h$ , $x_0=x^n_0=\\xi _0$ and with coefficients given in (REF ) and (REF ).",
"Further, one observes that Assumptions A-3 to A-8 and B-1 to B-6 hold due to Assumptions C-3 to C-9.",
"In particular, Assumption A-3 holds due to Assumption C-3 while Assumptions A-4 and B-1 due to Assumption C-4.",
"Further Assumptions A-5, A-6, B-2 and B-3 hold due to Assumptions C-5 and C-6 with $L=G$ , $M=M_n=1+\\Psi ^{\\chi } \\in \\mathcal {L}^{\\frac{p_1}{2}}$ and $M^{\\prime }=M_n^{\\prime }=1+\\Psi ^{\\chi \\frac{p_1}{2}} \\in \\mathcal {L}^1$ , where $\\Psi :=\\sup _{t \\in [0, h]}|(\\xi _{\\delta _1(t)}, \\ldots , \\xi _{\\delta _k(t)})| \\in \\mathcal {L}^p$ .",
"Also Assumption A-7 holds due to Assumption C-7 with $C_R:=K_R I_{\\Omega _R} + \\sum _{j=R}^{\\infty } K_{j+1} I_{\\Omega _{j+1} \\backslash {\\Omega }_{j}}$ where $\\Omega _j:=\\lbrace \\omega \\in \\Omega :\\Psi \\le j \\rbrace $ .",
"Further one takes $f(R):=K_R$ and then $P(C_R >f(R)) \\le P(\\Psi >R) \\le \\frac{E\\Psi }{R} \\rightarrow 0 $ as $R \\rightarrow \\infty $ .",
"This also implies that Assumption A-8 holds due to Assumption C-8.",
"To verify Assumption B-5, one observes that $b_t^n(x)=\\frac{\\beta _t(\\xi _{\\delta _1(t)}, \\ldots , \\xi _{\\delta _k(t)}, x)}{1+n^{-\\theta }|\\beta _t(\\xi _{\\delta _1(t)}, \\ldots , \\xi _{\\delta _k(t)}, x)|} \\rightarrow \\beta _t(\\xi _{\\delta _1(t)}, \\ldots , \\xi _{\\delta _k(t)}, x)=b_t(x) $ as $n \\rightarrow \\infty $ and sequence $\\Big \\lbrace I_{B(R)}\\sup _{|x| \\le R}|b_t^n(x)-b_t(x)|^2\\Big \\rbrace _{\\lbrace n \\in \\mathbb {N}\\rbrace } $ is uniformly integrable which implies $\\lim _{n \\rightarrow \\infty } E\\int _{t_0}^{t_1}I_{B(R)}\\sup _{|x| \\le R}|b_t^n(x)-b_t(x)|^2 dt =0 $ and similarly for diffusion and jump coefficients.",
"Finally Assumption B-6 holds trivially.",
"Therefore equation (REF ) holds due to Theorem REF , Lemma REF and Lemma REF when $i=1$ .",
"We note that the convergence here is achieved for all $q < p_1$ and as we proceed to the next interval $[h, 2h]$ , the convergence is achieved in the lower space i.e.",
"$q < p_2$ due to Assumptions C-5 and C-6.",
"Therefore for the inductive arguments, we assume that the convergence in the interval $[(r-1)h, rh]$ is achieved for all $q < p_r$ i.e.",
"we assume that Theorem REF , Lemma REF and Lemma REF hold for any $q < p_r$ when $i=r$ .",
"Case: $\\mathbf {t \\in [r h, (r+1)h]}$ .",
"When $t \\in [rh, (r+1)h]$ , then SDDE (REF ) and scheme (REF ) become SDE (REF ) and scheme (REF ) respectively with $t_0=rh$ , $t_1=(r+1)h$ , $x_{t_0}=x_{rh}$ , $x^n_{t_0}=x^n_{rh}$ and coefficients given by (REF ) and (REF ).",
"Verify A-3.",
"Assumption A-3 holds due to Assumption C-3 trivially.",
"Verify A-4 and B-1.",
"Assumptions A-4 and B-1 hold due to Lemma REF , Lemma REF and inductive assumptions.",
"Verify A-5, A-6, B-2 and B-3.",
"Assumptions A-5 and B-2 hold due to Assumption C-5 with $M:=1+\\sup _{rh \\le t \\le (r+1)h}|y_t|^\\chi $ and $M_n:=1+\\sup _{rh \\le t \\le (r+1)h}|y_t^n|^\\chi $ which are bounded in $\\mathcal {L}^\\frac{p_{r+1}}{2}$ due to Lemma REF , Lemma REF and inductive assumptions.",
"Furthermore Assumptions A-6 and B-3 hold with $M^{\\prime }:=1+\\sup _{rh \\le t \\le (r+1)h}|y_t|^{\\chi \\frac{p_{r+1}}{2}}$ and $M_n^{\\prime }:=1+\\sup _{rh \\le t \\le (r+1)h}|y_t^n|^{\\chi \\frac{p_{r+1}}{2}}$ which are bounded in $\\mathcal {L}^1$ due to Lemma REF , Lemma REF and inductive assumptions.",
"Verify A-7.",
"For every $R>0$ , $|x|, |\\bar{x}| \\le R$ and $t \\in [rh, (r+1)h]$ , Assumption A-7 holds due to Assumption C-7 with $\\mathcal {F}_{rh}$ -measurable random variable $C_R$ given by $C_R:=K_R I_{\\Omega _R} + \\sum _{j=R}^{\\infty } K_{j+1} I_{\\Omega _{j+1} \\backslash {\\Omega }_{j}} $ where $\\Omega _j:=\\lbrace \\omega \\in \\Omega :\\sup _{t \\in [r h, (r+1) h]}|y_t| \\le j \\rbrace $ .",
"Further one takes $f(R):=K_R$ and then $P(C_R >f(R)) \\le P\\big (\\sup _{r h \\le t <(r+1)h}|y_t|>R\\big ) \\rightarrow 0 \\, \\mbox{as} \\, R \\rightarrow \\infty .$ Verify A-8.",
"For every $R>0$ and any $t \\in [rh, (r+1)h]$ , we take $C_R$ as defined in (REF ), $f(R)=K_R$ .",
"Then one uses (REF ) to establish A-8.",
"Verify B-5.",
"The inductive assumption implies $|y_t^n-y_t| \\rightarrow 0$ in probability and thus due to Assumption C-9, $\\sup _{|x|\\le R}|\\beta _t^n(y_t^n,x)-\\beta _t(y_t,x)| \\rightarrow 0$ in probability.",
"Furthermore the sequence $I_{B(R)}\\big \\lbrace \\sup _{|x|\\le R}|\\beta _t^n(y_t^n,x)-\\beta _t(y_t,x)|^2\\big \\rbrace _{\\lbrace n \\in \\mathbb {N}\\rbrace }$ is uniformly integrable due to Assumption C-8 and inductive assumption, which implies $\\lim _{n \\rightarrow \\infty }E \\int _{rh}^{(r+1)h} \\sup _{|x| \\le R} |b_t^n(x)-b_t(x)|^2 =0.",
"$ For diffusion coefficient, due to the inductive assumption $\\Big \\lbrace \\sup _{rh \\le t \\le (r+1)h}|y_t^n-y_t|^\\chi \\Big \\rbrace _{\\lbrace n \\in \\mathbb {N}\\rbrace } \\, \\mbox{and hence} \\, \\, \\Big \\lbrace \\sup _{rh \\le t \\le (r+1)h}|y_t^n|^\\chi \\Big \\rbrace _{\\lbrace n \\in \\mathbb {N}\\rbrace }$ are uniformly integrable which on using Assumptions C-5 to C-6 imply $\\Big \\lbrace \\sup _{|x| \\le R}|\\alpha _t(y_t^n,x)-\\alpha _t(y_t,x)|^2\\Big \\rbrace _{\\lbrace n \\in \\mathbb {N}\\rbrace }$ is uniformly integrable.",
"Moreover due to Assumption C-9, $\\sup _{|x| \\le R}|\\alpha _t(y_t^n,x)-\\alpha _t(y_t,x)|^2 \\rightarrow 0$ in probability as $n \\rightarrow \\infty $ and therefore Assumption B-5 holds for the diffusion coefficients.",
"One adopts similar arguments for jump coefficients.",
"Verify B-6.",
"This follows due to the inductive assumptions.",
"This completes the proof.",
"We now proceed to obtain the rate of convergence of the scheme (REF ).",
"For this purpose, we replace Assumptions C-7 and C-9 by the following assumptions.",
"C- 10 There exist constants $C>0$ , $q \\ge 2$ and $\\chi >0$ such that, $(x-\\bar{x})(\\beta _t(y, x)-\\beta _t( y, \\bar{x})) \\vee |\\alpha _t( y, x)-\\alpha _t( y, \\bar{x})|^2 \\vee & \\\\\\int _{Z} |\\lambda _t( y, x, z)-\\lambda _t( y, \\bar{x}, z)|^2 \\nu (dz) &\\le C|x-\\bar{x}|^2 \\\\\\int _{Z} |\\lambda _t( y, x, z)-\\lambda _t( y, \\bar{x}, z)|^q \\nu (dz)& \\le C|x-\\bar{x}|^q \\\\|\\beta _t( y, x)-\\beta _t( y, \\bar{x}) |^2 &\\le C( 1+|x|^{\\chi }+|\\bar{x}|^{\\chi }) |x-\\bar{x}|^2 $ for any $t \\in [0, T]$ , $ x , \\bar{x} \\in \\mathbb {R}^d$ , $y \\in \\mathbb {R}^{d \\times k}$ and a $\\delta \\in (0,1)$ such that $\\max \\lbrace (\\chi +2)q, \\frac{q \\chi }{2}\\frac{q+\\delta }{\\delta }\\rbrace \\le p^*$ .",
"C- 11 Assume that $|\\beta _t( y, x)-\\beta _t( \\bar{y}, x) |^2 & \\vee |\\alpha _t( y, x)-\\alpha _t( \\bar{y}, x) |^2 \\vee \\Big (\\int _Z |\\lambda _t( y, x)-\\lambda _t( \\bar{y}, x) |^\\zeta \\nu (dz)\\Big )^\\frac{q}{\\zeta } \\le C(1+|y|^{\\chi }+|\\bar{y}|^{\\chi })| y-\\bar{y}|^2 $ where $\\zeta =2,q$ , for any $t \\in [0,T]$ , $ x \\in \\mathbb {R}^d$ and $y , \\bar{y} \\in \\mathbb {R}^{d \\times k}$ .",
"Remark 4.3 Due to Assumptions C-8, C-10 and C-11, there exists a constant $C>0$ such that $|\\beta _t(y,x)|^2 \\le C(1+|y|^{\\chi +2}+|x|^{\\chi +2}) $ for any $t \\in [0,T]$ , $x \\in \\mathbb {R}^d$ and $y \\in \\mathbb {R}^{d \\times k}$ .",
"In the following corollary, we obtain a convergence rate for the tamed Euler scheme (REF ) which is equal to the classical convergence rate of Euler scheme.",
"For this purpose, one can take $\\theta =\\frac{1}{2}$ .",
"Corollary 4.2 Let Assumptions C-3 to C-6, C-8, C-10 and C-11 be satisfied.",
"Then $ E \\sup _{0 \\le t \\le T}|x_t-x_t^n|^q \\le K n^{-\\frac{q}{q+N^{\\prime }\\delta }}$ for any $q <p^*$ where constant $K>0$ does not depend on $n$ .",
"The corollary can be proved by adopting similar arguments as used in the proof of Corollary REF .",
"For this purpose, one can use Theorem REF inductively to show that for every $i=1,\\ldots ,N^{\\prime }$ , $E \\sup _{(i-1)h \\le t \\le ih}|x_t-x_t^n|^q \\le K n^{-\\frac{q}{q+i\\delta }}$ for any $q <p_i$ where constant $K>0$ does not depend on $n$ .",
"Now, notice that Assumptions A-3 through A-6, A-8 and B-1 through B-3 have already been verified in the proof of Corollary REF .",
"Hence, one only needs to verify Assumptions A-9, B-7 and B-8.",
"Case $\\mathbf {t \\in [0,h]}.$ As before, one considers SDDE (REF ) as a special case of SDE (REF ) with $t_0=0$ , $t_1=h$ , $x_{t_0}=\\xi _0$ and coefficients given by equation (REF ).",
"Also, scheme (REF ) can be considered as a special case of scheme (REF ) with $t_0=0$ , $t_1=h$ , $x_{t_0}=\\xi _0$ and coefficients given by (REF ).",
"Verify A-9.",
"Assumption A-9 follows from Assumption C-10 trivially.",
"Verify B-7.",
"Notice that $y_t=y_t^n=:\\Phi _t$ for $t \\in [0,h]$ which implies $E\\int _{0}^{h}|b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q dt \\le n^{-q \\theta } E\\int _{0}^{h}\\big |\\beta _t(\\Phi _t, x_{\\kappa (n,t)}^n)\\big |^{2q} dt$ which on using Remark REF , Assumption C-4 and Lemma REF gives $E\\int _{0}^{h}|b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q dt & \\le n^{-q \\theta } K \\big (1+ E\\Psi ^{(\\chi +2) q}+ E\\sup _{0 \\le t \\le h}| x_{\\kappa (n,t)}^n|^{(\\chi +2) q}\\big ) \\le K n^{-\\frac{q}{2}}$ for any $q<p_1$ because $\\theta =\\frac{1}{2}$ .",
"Verify B-8.",
"This holds trivially.",
"Thus, by Theorem REF , one obtains that equation (REF ) holds for $i=1$ .",
"For inductive arguments, one assumes that equation (REF ) holds for $i=r$ and then verifies it for $i=1+r$ .",
"Case $\\mathbf {t \\in [rh,(r+1)h]}.$ Again, consider SDDE (REF ) as a special case of SDE (REF ) with $t_0=rh$ , $t_1=(r+1)h$ , $x_{t_0}=x_{rh}$ and coefficients given by equation (REF ).",
"Similarly, consider scheme (REF ) as a special case of scheme (REF ) with $t_0=rh$ , $t_1=(r+1)h$ , $x_{t_0}=x_{rh}$ and coefficients given by (REF ).",
"Verify A-9.",
"Assumption A-9 follows from Assumption C-10 trivially.",
"Verify B-7.",
"One observes that $& \\qquad \\qquad E\\int _{0}^{h}|b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q dt\\\\&\\le K E\\int _{0}^{h}\\big |\\frac{\\beta _t(y_t^n, x_{\\kappa (n,t)}^n)}{1+n^{-\\theta }|\\beta _t(y_t^n, x_{\\kappa (n,t)}^n)|}-\\beta _t(y_t^n, x_{\\kappa (n,t)}^n)\\big |^q dt\\\\& + K E\\int _{0}^{h} \\big |\\beta _t(y_t^n, x_{\\kappa (n,t)}^n)-\\beta _t(y_t, x_{\\kappa (n,t)}^n)\\big |^q dt\\\\& \\le K n^{-q\\theta }E\\int _{0}^{h}|\\beta _t(y_t^n,x_{\\kappa (n,t)}^n)|^{2q} dt + K E\\int _{0}^{h} \\big |\\beta _t(y_t^n, x_{\\kappa (n,t)}^n)-\\beta _t(y_t,x_{\\kappa (n,t)}^n)\\big |^q dt$ which on the application of Remark REF and Assumption C-11 gives $E\\int _{0}^{h}&|b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q dt \\le K n^{-q\\theta } E\\int _{0}^{h} (1+|y_t^n|^{(\\chi +2)q}+|x_{\\kappa (n,t)}^n|^{(\\chi +2)q}) dt\\\\&+ K E\\int _{0}^{h} (1+|y_t|^\\frac{q\\chi }{2}+|y_t^n|^\\frac{q\\chi }{2}) |y_t-y_t^n|^q dt$ and then on the application of Hölder's inequality, Lemma REF and Lemma REF along with inductive assumptions gives $E\\int _{0}^{h}|b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q dt \\le K n^{-q\\theta } + K E\\int _{0}^{h} (E|y_t-y_t^n|^{q+\\delta })^\\frac{q}{q+\\delta }.$ Finally on using the inductive assumption and $\\theta =\\frac{1}{2}$ , one obtains $E\\int _{0}^{h}|b^n_t(x_{\\kappa (n,t)}^n)-b_t(x_{\\kappa (n,t)}^n)|^q dt \\le K n^{-\\frac{q}{2}} + K n^{-\\frac{q}{q+(r+1)\\delta }}$ and hence (REF ) holds for $i=r+1$ .",
"Verify B-8.",
"This holds due to inductive assumptions.",
"Thus, by Theorem REF , one obtains that equation (REF ) holds for $i=r+1$ .",
"This completes the proof."
],
[
"Numerical Illustrations",
"We demonstrate our results numerically with the help of following examples.",
"Example 1.",
"Consider the following SDE, $ dx_t =& -x_t^5 dt + x_t dw_t + \\int _{\\mathbb {R}} x_t z \\tilde{N}(dt, dz)$ for any $t \\in [0, 1]$ with initial value $x_0 =1$ .",
"The jump size follows standard normal distribution and jump intensity is 3.",
"The tamed Euler scheme with step-size $2^{-21}$ is taken as true solution.",
"Table REF and Figure REF are based on 1000 simulations.",
"Table: SDE: Errors in the tamed Euler scheme.Example 2.",
"Consider the following SDDE, $ dx_t = (x_t-x_t^3+ y_t^2)dt+(x_t+y_t^3)dw_t+ \\int _{\\mathbb {R}} (x_t+y_t) z \\tilde{N}(dt,dz)$ where $y_t=x_{t-1}$ for $t\\in [0,2]$ with initial data $\\xi _t=t+1$ for $t \\in [-1,0]$ .",
"Figure: Tamed Euler Schemes of SDE () and SDDE ()The jump size follows standard Normal distribution and jump intensity is 3.",
"The tamed scheme with step size $2^{-23}$ is taken as the true solution.",
"Figure REF is based on 300 sample paths."
]
] | 1403.0498 |
[
[
"Green's function formalism for a condensed Bose gas consistent with\n infrared-divergent longitudinal susceptibility and\n Nepomnyashchii-Nepomnyashchii identity"
],
[
"Abstract We present a Green's function formalism for an interacting Bose-Einstein condensate (BEC) satisfying the two required conditions: (i) the infrared-divergent longitudinal susceptibility with respect to the BEC order parameter, and (ii) the Nepomnyashchii-Nepomnyashchii identity stating the vanishing off-diagonal self-energy in the low-energy and low-momentum limit.",
"These conditions cannot be described by the ordinary mean-field Bogoliubov theory, the many-body $T$-matrix theory, as well as the random-phase approximation with the vertex correction.",
"In this paper, we show that these required conditions can be satisfied, when we divide many-body corrections into singular and non-singular parts, and separately treat them as different self-energy corrections.",
"The resulting Green's function may be viewed as an extension of the Popov's hydrodynamic theory to the region at finite temperatures.",
"Our results would be useful in constructing a consistent theory of BECs satisfying various required conditions, beyond the mean-field level."
],
[
"Introduction",
"The Bogoliubov-type mean-field theory [1] has successfully clarified various superfluid phenomena of ultracold Bose gases.",
"However, the theory of Bose–Einstein condensates (BECs) still has room for improvement.",
"The Hartree-Fock-Bogoliubov (HFB) approximation gives a finite energy gap [2], and the HFB-Popov (Shohno) theory [3], [4], [5], [2], [6], [7] unphysically concludes the first-order phase transition [6], [8] (whereas a real Bose gas is expected to exhibit the second-order phase transition).",
"These mean-field type BEC theories also assume a static self-energy $\\Sigma $ , where its off-diagonal self-energy part $\\Sigma _{12}$ characterized by two outgoing particle lines is specific to the BEC phase.",
"This static result contradicts with the exact identity proved by Nepomnyashchii and Nepomnyashchii [9], [10], stating the vanishing off-diagonal self-energy $\\Sigma _{12}$ in the low-energy and low-momentum limit.",
"When we try to go beyond the mean-field approximation to include many-body correlations, we suffer from the infrared divergence associated with fluctuations of BECs.",
"The infrared singularity directly appears in some quantities such as the correlation functions of the phase and amplitude fluctuations of a BEC order parameter [11], [16], [12], [17], [18], [13], [14], [15], [19], [20] (that are also referred to as the transverse and longitudinal response functions in the literature, respectively).",
"On the other hand, in some cases such as the density-density correlation function [21], the infrared divergence does not appear in the final result.",
"The singularity only appears on the way of calculation.",
"Indeed, the density response function satisfies the compressibility sum-rule [21].",
"One thus needs to carefully treat the infrared divergence, depending on what we are considering.",
"For curing the infrared divergences in BEC theories, a number of ideas have been proposed.",
"Instead of bosonic fields, hydrodynamic variables (such as density and phase) have been adopted to describe BECs in the low momentum region at $T=0$ [22], [23], [5].",
"This so-called Popov's hydrodynamic approach correctly describes the long-range correlations.",
"A renormalization group technique has also been applied to the BEC phase at $T=0$ [13], [25], [24], [15].",
"To obtain correct infrared behaviors in this approach, the Ward–Takahashi identities associated with the gauge symmetry play an important role.",
"The infrared divergences are also removed by an artificial field that breaks $U(1)$ gauge symmetry [26].",
"In this approach, one takes the limit of vanishing symmetry breaking terms after calculating physical quantities, and long-range correlations are corrected by incorporating the Popov's hydrodynamic theory.",
"In this paper, we construct a Green's function formalism that can correctly describe the low-energy singularity of the longitudinal response function $\\chi _\\parallel ({\\bf p},\\omega )$ .",
"This function is a typical quantity exhibiting the infrared divergence below the BEC phase transition temperature $T_{\\rm c}$ .",
"This infrared behavior is strongly related to the so-called Nepomnyashchii–Nepomnyashchii (NN) identity [9], [10].",
"Indeed, it has been shown that $\\chi _\\parallel (0,0)$ is proportional to $\\Sigma _{12}^{-1}(0,0)$ [16].",
"The longitudinal response function is also a useful quantity in constructing a consistent theory of BECs.",
"The Bogoliubov mean-field theory incorrectly gives a finite value of $\\chi _\\parallel (0,0)$ [16], [12].",
"This approximation only includes fluctuations around the mean-field order parameter to the second-order, where longitudinal (amplitude) and transverse (phase) fluctuations of the BEC order parameter are decoupled from each other in the Bogoliubov Hamiltonian.",
"Anharmonic effects beyond such a Gaussian approximation have been pointed out to be important for the NN identity [12].",
"However, the so-called many-body $T$ -matrix theory, which involves interaction effects beyond the mean-field level, still cannot reproduce the infrared singularity of the longitudinal response function, nor the NN identity [6], [27].",
"We note that the infrared divergence of the longitudinal susceptibility associated with an order parameter is a general phenomenon in a system with spontaneously broken continuous symmetry.",
"This divergence was originally discussed in a Heisenberg ferromagnet [28], and was extended to a general system described by a multi-component ordering field [11].",
"Although the longitudinal susceptibility has not been observed in an ultracold Bose gas, the singularity of the longitudinal dynamical susceptibility is observable in Bose–Einstein condensation of magnons in a quantum Heisenberg antiferromagnet via neutron scattering [29].",
"Here, we explain our strategy in this paper.",
"Effects of a particle-particle interaction can be conveniently included in the single-particle thermal Green's function $G$ through the Dyson equation $G({\\bf p},i\\omega _n) = {1 \\over G^{-1}_0({\\bf p},i\\omega _n)-\\Sigma ({\\bf p},i\\omega _n)}.$ Here, $G_0 $ is the Green's function for a free Bose gas, $\\Sigma $ is the irreducible self-energy, and $\\omega _n$ is the boson Matsubara frequency.",
"Equation (REF ) is the most conventional expression in considering an interacting Bose gas.",
"This equation is actually a $(2\\times 2)$ -matrix in the BEC phase.",
"One may also include many-body corrections into $G$ through the reducible self-energy $\\Sigma ^{\\prime }$ by using the expression $G({\\bf p},i\\omega _n)=G_0({\\bf p},i\\omega _n) + G_0({\\bf p},i\\omega _n) \\Sigma ^{\\prime } ({\\bf p},i\\omega _n)G_0({\\bf p},i\\omega _n).$ The reducible self-energy $\\Sigma ^{\\prime }$ is related to the irreducible self-energy $\\Sigma $ through $\\Sigma ^{\\prime } ({\\bf p},i\\omega _n)={1 \\over 1-\\Sigma ({\\bf p},i\\omega _n)G_0({\\bf p},i\\omega _n)} \\Sigma ({\\bf p},i\\omega _n).$ Equations (REF ) and (REF ) are equivalent to each other.",
"We may also employ the hybrid version of Eqs.",
"(REF ) and (REF ), given by $G({\\bf p},i\\omega _n)={\\tilde{G}}({\\bf p},i\\omega _n) + {\\tilde{G}}({\\bf p},i\\omega _n){\\tilde{\\Sigma }}({\\bf p},i\\omega _n){\\tilde{G}}({\\bf p},i\\omega _n),$ where we divide the self-energy $\\Sigma $ into two parts, i.e., $\\Sigma =\\Sigma _{\\rm a} +\\Sigma _{\\rm b}$ .",
"In Eq.",
"(REF ), $\\Sigma _{\\rm a}$ is treated as the self-energy correction to $G_{0}$ , which provides ${\\tilde{G}}^{-1}=G_0^{-1}-\\Sigma _{\\rm a}$ .",
"On the other hand, $\\tilde{\\Sigma }$ is given by Eq.",
"(REF ), where $\\Sigma $ and $G_0$ are replaced by $\\Sigma _{\\rm b}$ and ${\\tilde{G}}$ , respectively.",
"If the sum of $\\Sigma _{\\rm a}$ and $\\Sigma _{\\rm b}$ provides the exact irreducible self-energy possessing all orders of interactions, Eq.",
"(REF ) is a rewriting of Eq.",
"(REF ).",
"In most cases, we need an approximate treatment of many-body effects.",
"In an extreme case, one may introduce different approximations between ${\\tilde{G}}$ in the first term of (REF ) and those in the second term, giving the form $G({\\bf p},i\\omega _n)={\\tilde{G}}({\\bf p},i\\omega _n) + \\tilde{G}_{\\rm L}({\\bf p},i\\omega _n){\\tilde{\\Sigma }}({\\bf p},i\\omega _n) \\tilde{G}_{\\rm R}({\\bf p},i\\omega _n).$ Equation (REF ) is more flexible than Eq.",
"(REF ) in the sense that one may employ different approximations in the first and the second terms.",
"This flexibility is particularly useful in constructing the BEC theory that satisfies various required conditions, such as the infrared divergence of the longitudinal response function, the NN identity, the Hugenholtz-Pines relation [30] as well as the second-order phase transition.",
"In this paper, we employ the hybrid expression in Eq.",
"(REF ).",
"We treat the first term in Eq.",
"(REF ) within the many-body $T$ -matrix approximation (MBTA), as well as the random phase approximation (RPA) with the vertex correction.",
"In a previous paper [27], we examined a weakly interacting Bose gas within the framework of the ordinary Green's function formalism in Eq.",
"(REF ).",
"Evaluating the self-energy within the MBTA as well as the RPA with the vertex correction, we found that these many-body theories describe the enhancement of $T_{\\rm c}$ as predicted by various methods [31], whereas they do not meet the NN identity.",
"We overcome this problem in this paper, by determining the second term in Eq.",
"(REF ) so as to cure the broken NN identity.",
"The resulting Green's function is found to also reproduce the infrared divergence of the longitudinal response function, as well as the Hugenholtz-Pines relation.",
"We determine ${\\tilde{\\Sigma }}$ in Eq.",
"(REF ) on the basis of the hydrodynamic theory developed by Popov [22], [23], [5].",
"Indeed, our approach based on Eq.",
"(REF ) is strongly related to the Popov's hydrodynamic theory.",
"The Green's function in Eq.",
"(REF ) has formally the same structure as that given in the Popov's hydrodynamic theory.",
"In this hydrodynamic theory, a factor corresponding ${\\tilde{\\Sigma }}$ in Eq.",
"(REF ) exhibits infrared divergence that originates from phase fluctuations of the BEC order parameter.",
"Using this, Popov obtained the vanishing off-diagonal self-energy in the low-energy and low-momentum limit (NN identity).",
"This result provides a crucial key in determining ${\\tilde{\\Sigma }}$ .",
"Section presents our Green's function formalism.",
"In Sec.",
", we examine low-energy properties of the longitudinal response function in our formalism.",
"We explicitly show that our formalism satisfies the NN identity.",
"We also discuss how our approach is related to the Popov's hydrodynamic theory.",
"In Sec.",
", we examine the condensate fraction as a function of the temperature, to see how the present theory affects the previous results based on the ordinary Green's function formalism in Eq.",
"(REF ).",
"Throughout this paper, we set $\\hbar = k_{\\rm B} = 1$ , and the system volume $V$ is taken to be unity."
],
[
"Framework",
"We consider a three-dimensional Bose gas with an atomic mass $m$ .",
"The Hamiltonian is given by $H=\\sum _{\\bf p} (\\varepsilon _{\\bf p} - \\mu ) a_{\\bf p}^{\\dag } a_{\\bf p} + \\frac{U}{2} \\sum _{{\\bf p},{\\bf p}^{\\prime },{\\bf q}} a_{{\\bf p} + {\\bf q}}^{\\dag } a_{{\\bf p}^{\\prime }-{\\bf q}}^{\\dag } a_{{\\bf p}^{\\prime }} a_{{\\bf p}},$ where $a_{\\bf p}$ is the annihilator of a Bose atom with the kinetic energy $\\varepsilon _{\\bf p}-\\mu = {\\bf p}^{2}/(2m)-\\mu $ , measured from the chemical potential $\\mu $ .",
"We consider a weak repulsive interaction $U (> 0)$ , which is related to the $s$ -wave scattering length $a$ as $\\frac{4 \\pi a}{m} = \\frac{U}{1+ \\displaystyle {U} \\sum _{{\\bf p}}^{p_{\\rm c}} \\displaystyle { \\frac{1}{2 \\varepsilon _{\\bf p}} } },$ where $p_{\\rm c}$ is a cutoff momentum.",
"The BEC phase is conveniently characterized by the BEC order parameter $\\langle a_{{\\bf p}=0}\\rangle $ .",
"It is related to the condensate fraction $n_0$ through $\\langle a_{{\\bf p}=0}\\rangle =\\sqrt{n_0}$ [1].",
"In this paper, we take $\\langle a_{{\\bf p}=0}\\rangle $ as a real number, without loss of generality.",
"We consider the $(2\\times 2)$ -matrix single-particle thermal Green's function having the form in Eq.",
"(REF ).",
"We divide a self-energy $\\Sigma ({\\bf p},i\\omega _n)$ into the sum of the singular part $\\Sigma ^{\\rm IR}({\\bf p},i\\omega _n)$ (which exhibits infrared divergence) and the regular part $\\Sigma ^{\\rm R}({\\bf p},i\\omega _n)$ (which remains finite even in the low-energy and low-momentum limit).",
"For ${\\tilde{G}}$ and ${\\tilde{\\Sigma }}$ in Eq.",
"(REF ), we take ${\\tilde{G}}({\\bf p},i\\omega _n)= {1 \\over i\\omega _n\\sigma _3- \\varepsilon _{\\bf p}+\\mu -{\\Sigma }^{\\rm R}({\\bf p},i\\omega _n)},$ ${\\tilde{\\Sigma }}({\\bf p},i\\omega _n) = \\Sigma ^{\\rm IR}({\\bf p},i\\omega _n).$ Here, $\\sigma _i$ ($i=1,2,3$ ) are Pauli matrices.",
"We determine the chemical potential $\\mu $ , so as to satisfy the Hugenholtz-Pines relation $\\mu = \\Sigma ^{\\rm R}_{11}(0,0) - \\Sigma ^{\\rm R}_{12}(0,0)$ [30].",
"To explain how to divide the self-energy into two parts in the the MBTA and the RPA, we conveniently introduce the $(4\\times 4)$ -matrix generalized correlation function [27] $\\Pi (p) = -T\\sum _q g(p+q) \\otimes g(-q),$ where $\\otimes $ is the Kronecker product.",
"Here, we have used the simplified notation $p=({\\bf p},i\\omega _n)$ .",
"The correlation function $\\Pi (p)$ is diagrammatically given in Fig.",
"REF (a).",
"In Eq.",
"(REF ), $g_{} (p)$ is the $(2\\times 2)$ -matrix single-particle Green's function in the HFB–Popov approximation, given by $g(p) = \\frac{ 1 }{i \\omega _{n} \\sigma _{3} - \\xi _{\\bf p} - U n_{0} \\sigma _{1}},$ where $\\xi _{\\bf p} = \\varepsilon _{\\bf p} + U n_{0}$ .",
"Using the symmetry properties $g_{22}(p) = g_{11} (-p)$ and $g_{12}(p) = g_{12}(-p)$ , we reduce Eq.",
"(REF ) to $\\Pi (p) = &\\begin{pmatrix}\\Pi _{11} (p) & \\Pi _{12} (p) & \\Pi _{12} (p) & \\Pi _{14} (p) \\\\\\Pi _{12} (p) & \\Pi _{22} (p) & \\Pi _{14} (p) & \\Pi _{12}^{*} (p) \\\\\\Pi _{12} (p) & \\Pi _{14} (p) & \\Pi _{22} (p) & \\Pi _{12}^{*} (p) \\\\\\Pi _{14} (p) & \\Pi _{12}^{*} (p) & \\Pi _{12}^{*} (p)& \\Pi _{11}^{*} (p)\\end{pmatrix}.$ The detailed expressions of $\\Pi _{ij}$ are summarized in Appendix REF .",
"Figure: (a) Generalized polarization function Π\\Pi .",
"(b) Single particle Green's function g(p)g(p) used in (a).We divide (REF ) into the sum $\\Pi (p)=\\Pi ^{\\rm IR}(p)+\\Pi ^{\\rm R}(p)$ of the singular part $\\Pi ^{\\rm IR}(p)$ (which exhibits infrared divergence) and the regular part $\\Pi ^{\\rm R}(p)$ (which remains finite in the low-energy and low-momentum limit).",
"In this case, the singular part $\\Pi ^{\\rm IR}(p)$ can be written so as to be proportional to $\\Pi _{14}(p)$ , giving the form ${ \\Pi }^{\\rm IR} (p) = \\Pi _{14} (p) \\hat{C},$ with $\\hat{C} =\\begin{pmatrix}1 & - 1 & -1 & 1 \\\\- 1 & 1 & 1 & - 1 \\\\- 1 & 1 & 1 & - 1 \\\\1 & - 1 & -1 & 1\\end{pmatrix}.$ The infrared singularity of (analytic continued) $\\Pi _{14}$ is given by [21], [16], [18], [19], [20], [17], [29], [32] $\\Pi _{14}({\\bf p},i\\omega _n\\rightarrow \\omega +i\\delta ) \\propto \\left\\lbrace \\begin{array}{ll}\\ln ( c_{0}^{2} {\\bf p}^{2} - \\omega ^{2}) & (T = 0)\\\\1/|{\\bf p}| & (T \\ne 0) .\\end{array}\\right.$ (For the derivation of (REF ), see Appendix REF .)",
"In Eq.",
"(REF ), $c_{0} = \\sqrt{n_{0} U/m}$ is the Bogoliubov sound speed, and $\\delta $ is an infinitesimally small positive number.",
"The singular part $\\Pi ^{\\rm IR}$ only appears in the BEC phase below $T_{\\rm c}$ .",
"Indeed, the singular part $\\Pi _{14}$ is constructed from the off-diagonal Green's functions $g_{12}$ and $g_{21}$ .",
"The regular part $\\Pi ^{\\rm R}$ is free from the infrared divergence.",
"In fact, the singular part $\\Pi _{14}$ is completely eliminated from $\\Pi ^{\\rm R}$ .",
"Using $\\Pi ^{\\rm IR}(p)$ , we construct $\\tilde{\\Sigma }$ in Eq.",
"(REF ).",
"We consider the single bubble diagram in Fig.",
"REF , which provides ${ \\Sigma }^{\\rm IR} (p) = & - \\frac{1}{2} { G}_{1/2} U \\langle f_{0} |{ \\Pi }^{\\rm IR} (p) | f_{0} \\rangle U { G}_{1/2}^{\\dag },$ where $|f_0\\rangle = (0, 1,1, 0)^{\\rm T}$ .",
"In Eq.",
"(REF ), $G_{1/2} = \\sqrt{-n_{0}} (1,1)^{\\rm T}$ and $G_{1/2}^{\\dag } = \\sqrt{-n_{0}} (1,1)$ are the condensate Green's functions.",
"Figure: Self-energy Σ IR \\Sigma ^{\\rm IR} used for Σ ˜\\tilde{\\Sigma }.",
"We take the single-bubble structure for Σ IR \\Sigma ^{\\rm IR}.",
"The wavy line describes the repulsive interaction UU.",
"The dashed arrow describes the condensate Green's functions G 1/2 G_{1/2} and G 1/2 † G_{1/2}^{\\dag }.We calculate the regular part $\\Sigma ^{\\rm R}(p)$ in Eq.",
"(REF ) so as to be free from the infrared divergence.",
"In the MBTA, summing up the diagrams in Fig.",
"REF , we obtain $\\Sigma ^{\\rm R}_{11}(p) = & 2 n_{0} \\Gamma ^{\\rm R}_{11}(p) - 2 T \\sum _{q} \\Gamma _{11} (q) g_{11} (-p+q),\\\\\\Sigma ^{\\rm R}_{12}(p) = & n_{0} \\Gamma ^{\\rm R}_{11}(0),$ where $\\Gamma ^{\\rm R}_{ij} (p)$ is the $(4 \\times 4)$ -matrix four-point vertex, given by ${ \\Gamma }^{\\rm R} (p) = & \\frac{U}{1-U{ \\Pi }^{\\rm R}(p)}.$ The four-point vertex $\\Gamma $ in the second term of Eq.",
"(REF ) involves the singular part $\\Pi ^{\\rm IR}$ , giving the form ${ \\Gamma }(p) = & \\frac{U}{1-U[\\Pi ^{\\rm R}(p)+\\Pi ^{\\rm IR}(p)]}.$ The regular part $\\Sigma ^{\\rm R}$ does not exhibit infrared divergence.",
"The first terms in Eqs.",
"(REF ) and () involve $\\Pi ^{\\rm R}$ , which are free from the infrared divergence.",
"The second term in Eq.",
"(REF ) does not exhibit the infrared divergence of $\\Pi ^{\\rm IR}$ , after carrying out the summation with respect to the internal momentum $q=({\\bf q},\\omega _m)$ .",
"Figure: Self-energy Σ R \\Sigma ^{\\rm R} in the many-body TT-matrix approximation.",
"(a) Diagonal component Σ 11 R \\Sigma _{11}^{\\rm R}.",
"(b) Off-diagonal component Σ 12 R \\Sigma _{12}^{\\rm R}.",
"The dashed arrows describe n 0 \\sqrt{n_0}.",
"(c) Bethe-Salpeter equation of the four-point vertex function Γ R \\Gamma ^{\\rm R}.The regular part of the RPA self-energy $\\Sigma ^{\\rm R}$ is also obtained in the same manner.",
"Summing up the diagrams in Fig.",
"REF , one has $\\Sigma ^{\\rm R}_{11}(p) =& (n_{0}+n^{\\prime }) U^{\\rm R}_{\\rm eff}(0) + n_{0} U^{\\rm R}_{\\rm eff}(p) - T \\sum \\limits _{q} U_{\\rm eff} (q) g_{11} (p-q),\\\\\\Sigma ^{\\rm R}_{12}(p) = & n_{0} U^{\\rm R}_{\\rm eff}(p),$ where $n^{\\prime }=-T \\sum _{p} g_{11}(p) e^{i \\omega _{n} \\delta }$ is the non-condensate density, and $U^{\\rm R}_{\\rm eff} (p) = & \\frac{U}{1 - U \\chi ^{\\rm R}(p) }.$ Here, the correlation function $\\chi ^{\\rm R}$ is given by [27], $\\chi ^{\\rm R} (p) = & \\frac{1}{2} \\langle f_0 | [ \\Pi ^{\\rm R}(p) + \\Pi ^{\\rm R}(p) \\Gamma ^{\\rm R}(p)\\Pi ^{\\rm R}(p) ] |f_0 \\rangle .$ In Eq.",
"(REF ), $U_{\\rm eff}$ is also given by Eq.",
"(REF ), where both $\\Pi ^{\\rm R}$ and $\\Gamma ^{\\rm R}$ in Eq.",
"(REF ) are replaced by $\\Pi =\\Pi ^{\\rm R}+\\Pi ^{\\rm IR}$ and $\\Gamma $ in Eq.",
"(REF ), respectively.",
"As in the MBTA, the infrared singularity in $U_{\\rm eff}$ does not remain in the final result $\\Sigma _{11}^{\\rm R}$ after taking the $q$ -summation.",
"Figure: Self-energy Σ R \\Sigma ^{\\rm R} in the random phase approximation with the vertex correction.",
"(a) Diagonal component Σ 11 R \\Sigma _{11}^{\\rm R}.",
"(b) Off-diagonal component Σ 12 R \\Sigma _{12}^{\\rm R}.",
"(c) Effective interaction U eff R (p)U_{\\rm eff}^{\\rm R} (p), which involves the density fluctuation effects.",
"(d) The correlation function χ R \\chi ^{\\rm R}.The factors $\\tilde{G}_{\\rm L}(p)$ and $\\tilde{G}_{\\rm R}(p)$ in Eq.",
"(REF ) are also evaluated in a diagrammatic manner.",
"In this paper, we consider the following two cases as typical examples.",
"Recalling the expression in Eq.",
"(REF ), one may take, as the first example, $\\tilde{G}^{(1)}_{\\rm L,\\rm R}(p)= {\\tilde{G}}(p).$ As the second example, we may employ the simple version that involves the self-energy in the Hartree-Fock (HF) approximation, given by $\\tilde{G}^{(2)}_{\\rm L,\\rm R}(p)= {1 \\over i\\omega _n\\sigma _3- \\varepsilon _{\\bf p} + \\mu -\\Sigma _{\\rm HF}(p)},$ where $\\Sigma _{\\rm HF}(p)=2U (n_0+n^{\\prime })$ is the HF self-energy.",
"These two examples lead to the same infrared singularity in the second term of Eq.",
"(REF ), giving the form $\\tilde{G}_{\\rm L}^{(1)}(p){\\tilde{\\Sigma }}(p) \\tilde{G}^{(1)}_{\\rm R}(p)\\Bigr |_{p\\rightarrow 0} = \\frac{n_{0} U^{2} \\Pi _{14} (p)}{2 [\\Sigma ^{\\rm R}_{12}(0)]^{2}}\\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} ,$ $\\tilde{G}_{\\rm L}^{(2)}(p){\\tilde{\\Sigma }}(p) \\tilde{G}^{(2)}_{\\rm R}(p)\\Bigr |_{p\\rightarrow 0} = {2n_{0}U^{2} \\Pi _{14} (p) \\over [\\mu - \\Sigma _{\\rm HF}(0)]^{2}}\\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix}.$ Both Eqs.",
"(REF ) and (REF ) diverge reflecting the infrared singularity of the correlation function $\\Pi _{\\rm 14}(p\\rightarrow 0)$ in Eq.",
"(REF ).",
"Indeed, we have $\\Sigma _{12}^{\\rm R} (0) \\ne 0$ , as well as $\\mu -\\Sigma ^{\\rm HF} \\ne 0$ below $T_{\\rm c}$ (at least in the MBTA and the RPA with the vertex correction we are using).",
"As will be shown in the next section, this infrared divergence is a crucial key to reproduce the infrared divergence of the longitudinal response function $\\chi _\\parallel (p)$ , as well as the NN identity.",
"The present approach separately evaluates each term in Eq.",
"(REF ) in a diagrammatic manner, so that one needs to be careful about double counting of many-body corrections.",
"In this regard, we emphasize that this problem is safely avoided in our formalism, because $\\Sigma _{\\rm a}$ and $\\tilde{\\Sigma }$ involve qualitatively different diagrams with respect to the infrared behavior.",
"We briefly note that, although the second term in Eq.",
"(REF ) exhibits the infrared divergence in our approach, it is expected that this contribution is still weaker than the first term in Eq.",
"(REF ).",
"Indeed, for a small ${\\bf p}$ , the first term in Eq.",
"(REF ) exhibits $\\tilde{G} (0, {\\bf p}) \\propto {\\bf p}^{-2}$ .",
"This divergence is stronger than that of the second term in Eq.",
"(REF ), (which is proportional to $\\Pi _{14}$ in Eq.",
"(REF ))."
],
[
"Longitudinal susceptibility",
"The longitudinal response function $\\chi _{\\parallel }$ and the transverse response function $\\chi _{\\perp }$ are given by [12], [17] $\\chi _{\\nu } (p) = & \\int _{0}^{1/T} d\\tau e^{i \\omega _{n} \\tau } \\langle T_{\\tau } a_{\\nu \\bf p} (\\tau ) a_{\\nu -\\bf p} (0) \\rangle ,$ where $T_{\\tau }$ denotes a $\\tau $ -ordering operation, and $\\nu \\equiv (\\parallel , \\perp )$ .",
"In Eq.",
"(REF ), $a_{\\parallel {\\bf p}}$ and $a_{\\perp {\\bf p}}$ are longitudinal and transverse operators, respectively.",
"When the BEC order parameter is taken to be real, they are respectively given by $a_{\\parallel {\\bf p}} = \\frac{1}{2 } ( a_{\\bf p} +a_{- {\\bf p}}^{\\dag } ) ,\\quad a_{\\perp {\\bf p}} = \\frac{1}{2 i } (a_{\\bf p} - a_{- {\\bf p}}^{\\dag } ).$ Equation (REF ) can be also written as $\\chi _{\\parallel } (p) = - \\frac{1}{4} \\langle + | G (p) | + \\rangle ,\\quad \\chi _{\\perp } (p) = - \\frac{1}{4 } \\langle - | G (p) | - \\rangle ,$ where $| \\pm \\rangle \\equiv (1, \\pm 1)^{\\rm T}$ .",
"We here summarize exact properties of these static susceptibilities obtained from the exact Green's function with the self-energy that satisfies the NN identity [10], as well as the Hugenholtz-Pines relation [30].",
"The transverse susceptibility exhibits the infrared divergence as $\\chi _{\\perp } (0, {\\bf p}) \\simeq \\frac{ n_{0} m }{n |{\\bf p}|^{2} }.$ This indicates the instability of this state against an infinitesimal perturbation in the transverse direction (phase fluctuations) of the BEC order parameter.",
"The static longitudinal susceptibility in the low-momentum region is dominated by the off-diagonal self-energy [16], [32], [19], given by $\\chi _{\\parallel } (0, {\\bf p}) \\simeq \\frac{1}{4 \\Sigma _{12} (0, {\\bf p}) }.$ Because of the NN identity $\\Sigma _{12} (0) = 0$ , Eq.",
"(REF ) diverges when ${\\bf p}=0$ .",
"This infrared divergence is, however, weaker than that of the transverse susceptibility, i.e., [33] $\\chi _{\\perp } (0, {\\bf p}) \\gg \\chi _{\\parallel } (0, {\\bf p}).$ The infrared divergence of the longitudinal susceptibility can be correctly described by our approach (Fig.",
"REF ).",
"This result is quite different from the cases of the HFB–Popov approximation, the MBTA, as well as the RPA with the vertex corrections, based on the standard formalism in Eq.",
"(REF ) [27], [34].",
"Figure: (Color online) Static longitudinal susceptibility χ ∥ (iω n =0,𝐩)\\chi _{\\parallel } (i\\omega _{n} = 0, {\\bf p}).",
"(a) T=0.1T c 0 T = 0.1T_{\\rm c}^{0}.",
"(b) T=0.5T c 0 T = 0.5T_{\\rm c}^{0}.",
"Here, T c 0 T_{\\rm c}^{0} is the critical temperature of an ideal Bose gas.",
"We consider two approximations (the many-body TT-matrix approximation (MBTA) as well as the random-phase approximation (RPA) with the vertex correction).",
"For each approximation, we apply two different formalisms.",
"One is our formalism in Eq.",
"() with G ˜ L,R =G ˜ L,R (1) =G ˜\\tilde{G}_{\\rm L,R} = \\tilde{G}_{\\rm L,R}^{(1)}= \\tilde{G}, given in Eq. ().",
"The other is the Green's function obtained from the standard formalism in Eq. ().",
"In self-energies in the standard formalism in Eq.",
"(), we replace Γ 11 R \\Gamma _{11}^{\\rm R} with Γ 11 \\Gamma _{11} in the MBTA in Eqs.",
"() and (), and also replace U eff R U_{\\rm eff}^{\\rm R} with U eff U_{\\rm eff} in the RPA in Eqs.",
"() and ().",
"We set an 1/3 =10 -2 an^{1/3} = 10^{-2} and p c =5p 0 p_{\\rm c} = 5 p_{0}, where p 0 =2mT c 0 p_{0} = \\sqrt{2m T_{\\rm c}^{0}}.In the present case, the longitudinal susceptibility in the limit $p\\rightarrow 0$ behaves as $\\chi _{\\parallel }^{(1)} (p) \\simeq - \\frac{n_{0} U^{2} \\Pi _{14} (p)}{2 [\\Sigma ^{\\rm R}_{12}(0)]^{2}},$ $\\chi _{\\parallel }^{(2)} (p) \\simeq - {2n_{0}U^{2} \\Pi _{14} (p) \\over [\\mu - \\Sigma _{\\rm HF}(0)]^{2}}.$ Here, $\\chi _{\\parallel }^{(1)}$ and $\\chi _{\\parallel }^{(2)}$ are the longitudinal susceptibility in the cases of Eqs.",
"(REF ) and (REF ), respectively.",
"The second term in Eq.",
"(REF ) provides the infrared divergence of $\\chi _{\\parallel }$ thanks to the singularity of $\\Pi _{14}(p)$ .",
"(See also Eqs.",
"(REF ) and (REF ).)",
"This result is consistent with the previous work dealing with the infrared divergence of the longitudinal susceptibility [11], [16], [12], [17], [18], [13], [14], [15], [19].",
"The Bogoliubov approximation fails to reproduce this infrared-divergent longitudinal susceptibility.",
"In this approximation, one obtains [16], [12] $\\chi _{\\perp } (0, {\\bf p}) \\simeq \\frac{m}{{\\bf p}^{2}}, \\quad \\chi _{\\parallel } (0, {\\bf p}) \\simeq \\frac{1}{4 m c_{0}^{2}}.$ The origin of the finite longitudinal susceptibility is considered to be decoupling of transverse and longitudinal fluctuations in the mean-field theory [12].",
"Indeed, the Hamiltonian in the Bogoliubov theory has the form [12] $H_{2} = & \\sum \\limits _{{\\bf p} \\ne {\\bf 0}} \\biggl [ F_{0} a_{\\perp {\\bf p}} a_{\\perp - {\\bf p}} + F_{2} a_{\\parallel {\\bf p}} a_{\\parallel - {\\bf p}} + \\frac{1}{2} F_{1} \\biggr ],$ where $F_{j} = \\varepsilon _{\\bf p} + j U n_{0}$ .",
"Anharmonic effects of fluctuations were pointed out to be important to obtain the infrared-divergent longitudinal susceptibility [12]."
],
[
"Nepomnyashchii–Nepomnyashchii identity",
"Our Green's function also satisfies the NN identity [9], [10], as well as the Hugenholtz-Pines relation [30].",
"These two required conditions are conveniently summarized as ${ \\Sigma } (0) = \\mu .$ Given the hybrid version of the Green's function in Eq.",
"(REF ) in the standard expression in Eq.",
"(REF ), one finds that the self-energy in Eq.",
"(REF ) has the form ${ \\Sigma } (p) = & G_{0}^{-1} (p) - \\frac{1}{1 + \\tilde{G}^{-1} (p) \\tilde{G}_{\\rm L}(p) \\tilde{\\Sigma }(p) \\tilde{G}_{\\rm R} (p)} \\tilde{G}^{-1} (p) .$ In Eq.",
"(REF ), the factor $\\tilde{G}_{\\rm L} \\tilde{\\Sigma } \\tilde{G}_{\\rm R}$ involves the infrared divergence (as seen in Eqs.",
"(REF ) and (REF )), whereas $\\tilde{G}^{-1}(p\\rightarrow 0)$ safely converges in the MBTA as well as the RPA with the vertex correction.",
"The second term thus vanishes when $p\\rightarrow 0$ , leading to Eq.",
"(REF ) through the relation ${ \\Sigma } (0) = { G}_{0}^{-1} (0)=\\mu $ .",
"The present approach in Eq.",
"(REF ) may be viewed as an extension of the Popov's hydrodynamic approach at $T=0$ [22], [23], [5] to the finite temperature region.",
"Far below $T_{\\rm c}$ in the weak coupling regime where the non-condensate density is negligible, one may retain the regular self-energy $\\Sigma ^{\\rm R}$ to the lowest order, giving the form $\\Sigma _{11}^{\\rm R} = 2 U n_{0}, \\quad \\Sigma _{12}^{\\rm R} = U n_{0}.$ When we apply Eq.",
"(REF ) to $\\tilde{G}_{\\rm L,R}$ in Eq.",
"(REF ), one has $\\tilde{G}_{\\rm L,R} (p) = g (p).$ For $\\tilde{\\Sigma }$ , we use Eq.",
"(REF ).",
"In addition, we assume the hydrodynamic regime $|{\\bf p}| \\ll \\sqrt{2} m c_{0}$ (where $c_{0}$ is the Bogoliubov sound speed).",
"Then, the Green's function (REF ) is reduced to $G (p) = - \\frac{ m c_{0}^{2}}{\\omega _{n}^{2} + c_{0}^{2} {\\bf p}^{2}} \\begin{pmatrix} 1 & -1 \\\\ -1 & 1 \\end{pmatrix} + \\frac{\\Pi _{14} (p)}{2 n_{0}} \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} .$ Equation (REF ) equals the Green's function in the Popov's hydrodynamic theory (which is explained in Appendix ).",
"We note that the phase (transverse) fluctuation affects the amplitude (longitudinal) fluctuation, and leads to the infrared divergence of the longitudinal susceptibility [20].",
"Indeed, according to the Popov's hydrodynamic theory, the second term in Eq.",
"(REF ) providing the infrared divergence of $\\chi _{\\parallel }$ originates from the convolution of the phase-phase correlation.",
"(See also Appendix .)",
"We also note that Eq.",
"(REF ) has the same structure as the exact Green's function in the low-energy and low-momentum limit obtained by Nepomnyashchii and Nepomnyashchii [10], $G (p) = & \\frac{n_{0} m c^{2}}{n} \\frac{ 1 }{ \\omega _{}^{2} - c^{2} |{\\bf p}|^{2} } \\begin{pmatrix} 1 & -1 \\\\ -1 & 1 \\end{pmatrix} - \\frac{1}{4 \\Sigma _{12} (p) } \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix},$ where $c$ is the macroscopic sound velocity determined from the compressibility.",
"The single bubble diagram giving Eq.",
"(REF ) is the primitive many-body correction to the self-energy $\\tilde{\\Sigma }$ to reproduce the NN identity.",
"Any other $p$ -dependent second-order corrections do not contribute to $\\tilde{\\Sigma }$ .",
"To explicitly see this, we conveniently write $\\tilde{\\Sigma }$ in the form $\\tilde{\\Sigma }(p) = & \\Sigma ^{\\rm IR} (p) + \\delta \\Sigma _{}^{\\rm IR} (p),$ where $\\Sigma ^{\\rm IR} (p)$ is given in Eq.",
"(REF ), and $\\delta \\Sigma ^{\\rm IR} (p)$ is diagrammatically described as Fig.",
"REF , which gives $\\delta \\Sigma ^{\\rm IR} (p) = & - {\\mathcal {G}}_{1/2}^{\\dag } \\hat{T} U { \\Pi }^{\\rm IR} (p) U \\hat{T} {\\mathcal {G}}_{1/2}\\nonumber \\\\ &- {\\mathcal {G}}_{1/2}^{\\dag } U { \\Pi }^{\\rm IR} (p) U \\hat{T} {\\mathcal {G}}_{1/2}\\nonumber \\\\ &- G_{1/2} U \\langle f_{0} | { \\Pi }^{\\rm IR} (p) U \\hat{T} {\\mathcal {G}}_{1/2}\\nonumber \\\\ &- {\\mathcal {G}}_{1/2}^{\\dag } \\hat{T} U { \\Pi }^{\\rm IR} (p) | f_{0} \\rangle U G_{1/2}^{\\dag }.$ Here, we have introduced matrix condensate Green's functions ${\\mathcal {G}}_{1/2} = \\sqrt{-n_{0}} \\hat{\\eta }_{g}$ and ${\\mathcal {G}}_{1/2}^{\\dag } = \\sqrt{-n_{0}} \\hat{\\eta }_{g}^{\\dag }$ .",
"The matrices $\\hat{T}$ and $\\hat{\\eta }_{g}^{}$ are given by, respectively, $\\hat{T}= \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix},\\qquad \\hat{\\eta }_{g}^{} = \\begin{pmatrix} 1 & 0 \\\\ 1 & 0 \\\\ 0 & 1 \\\\ 0 & 1 \\end{pmatrix} .$ Using $\\hat{C} \\hat{T} \\hat{\\eta }_{g} = \\hat{C} \\hat{\\eta }_{g} = 0$ , we find that $\\delta { \\Sigma }^{\\rm IR} (p) = 0$ , as expected.",
"Figure: Self-energy δΣ IR \\delta \\Sigma ^{\\rm IR}.We point out that the NN identity still obtains even if one includes vertex corrections to the single bubble contribution in Eq.",
"(REF ).",
"Indeed, considering the self-energy corrections given in Fig.",
"REF , we have $\\tilde{\\Sigma }(p) = & - T \\sum \\limits _{q} P^{\\dag } (q;p) K_{0}^{\\rm IR} (q;p) U \\frac{1}{\\sqrt{-1}} \\hat{T} {\\mathcal {G}}_{1/2}\\\\&- \\frac{1}{2 } T \\sum \\limits _{q} P^{\\dag } (q;p) K_{0}^{\\rm IR} (q;p) |f_0 \\rangle U \\frac{1}{\\sqrt{-1}} G_{1/2}^{\\dag },\\nonumber $ where $P^{\\dag } (q;p)$ is a $(2\\times 4)$ -matrix three-point vertex, and ${ K}_{0}^{\\rm IR} (q;p) \\equiv K^{0}_{1212} (q;p) \\hat{C}$ with $K^{0}_{1212} (q;p) \\equiv g_{12}(p+q) g_{12} (-q)$ .",
"Although $K_{0}^{\\rm IR}$ provides the singular part $\\Pi ^{\\rm IR}$ in Eq.",
"(REF ), the first term in Eq.",
"(REF ) actually does not exhibit the infrared divergence, because $\\hat{C} \\hat{T} \\hat{\\eta }_{g} = 0$ .",
"Figure: Self-energy Σ ˜\\tilde{\\Sigma } with the vertex correction P † P^{\\dag }.To examine the infrared behavior of the second term in (REF ), it is convenient to use the Ward-Takahashi identity with respect to the three-point vertex $P^{\\dag }$ and the off-diagonal self-energy $\\Sigma _{12} (0)$ in the limit $p\\rightarrow 0$ [10], giving the form $P^{\\dag } (0;0) = & 2 \\frac{\\Sigma _{12}(0)}{\\sqrt{n_{0}}} \\eta ^{\\dag } + n_{0}^{3/2} \\frac{\\partial }{\\partial n_{0}} \\left( \\frac{\\Sigma _{12}(0)}{n_{0}} \\right) \\eta _{\\rm a}^{\\dag } ,$ where $\\eta ^{\\dag } = \\begin{pmatrix} 1 & 1 & 1 & 0 \\\\ 0 & 1 & 1 & 1 \\end{pmatrix} ,\\quad \\eta _{\\rm a}^{\\dag } = \\begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\end{pmatrix} .$ (For the derivation, see Appendix REF .)",
"The contribution $P^{\\dag } (0;0)$ can be clearly extracted from Eq.",
"(REF ), if we apply the second mean value theorem for integrals [35].",
"Using this theorem, the second term in (REF ) can be divided into two terms: the term involving the infrared divergence and the term which is finite in the limit $p\\rightarrow 0$ .",
"The former involves $P^{\\dag } (0 ;0) \\Pi _{14}^{\\Lambda } \\hat{C}$ , where $\\Pi _{14}^{\\Lambda } = - T \\sum _{{\\bf q}=0}^{\\Lambda } K_{1212}^{0}(i \\omega _{n} = 0,{\\bf q};0)$ , and $\\Lambda $ is a cutoff determined from the mean value theorem.",
"Then, using the relation $\\eta ^{\\dag } \\hat{C} | f_{0} \\rangle = 2 | + \\rangle $ , one finds $\\tilde{ \\Sigma } (0) = & 2 U \\Pi _{14}^{\\Lambda } \\Sigma _{12} (0) \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} + \\delta \\Sigma ,$ where $\\delta \\Sigma _{} $ is the non-singular part of $\\tilde{\\Sigma }$ .",
"When we evaluate the self-energy $\\Sigma = G_{0}^{-1} - G^{-1}$ using Eq.",
"(REF ) together with $\\Pi _{14}^{\\Lambda }(0) = \\infty $ , we reach the expected result $\\Sigma (0) = \\mu $ .",
"Nepomnyashchii and Nepomnyashchii derived the NN identity in a similar manner [10].",
"The NN identity is ascribed to the infrared divergence in the bubble structure self-energy with the vertex correction.",
"Diagrams providing required infrared behaviors of $\\chi _{\\parallel }(0)$ as well as $\\Sigma _{12}(0)$ are common between our formalism and the exact results studied by Nepomnyashchii and Nepomnyashchii [10].",
"The original derivation of the NN identity as well as the relation to the phase fluctuation are summarized in Appendices REF and REF , respectively."
],
[
"Condensate fraction",
"Figure REF shows the condensate fraction $n_0$ in the BEC phase, calculated from the equation, $n_0=n-n^{\\prime }=n+T \\sum _{p} G_{11} (p) e^{i\\omega _{n} \\delta },$ where the second term $n^{\\prime }$ is the non-condensate density.",
"Many-body corrections to $T_{\\rm c}$ are dominated by the first term in Eq.",
"(REF ), when $T_{\\rm c}$ is determined from the theory above $T_{\\rm c}$ .",
"Indeed, both $\\delta \\tilde{G}^{(1)} \\equiv \\tilde{G}_{\\rm L}^{(1)} \\tilde{\\Sigma }\\tilde{G}_{\\rm R}^{(1)}$ and $\\delta \\tilde{G}^{(2)} \\equiv \\tilde{G}_{\\rm L}^{(2)} \\tilde{\\Sigma }\\tilde{G}_{\\rm R}^{(2)}$ are absent in the normal state above $T_{\\rm c}$ .",
"As a result, the enhancement of $T_{\\rm c}$ in each case of the MBTA and the RPA with the vertex correction is the same as our previous results based on the standard Green's function formalism in Eq.",
"(REF ) [27].",
"Given the shift of $T_{\\rm c}$ as $\\frac{ T_{\\rm c}-T_{\\rm c}^{0} }{ T_{\\rm c}^{0} } = c_{1} an^{1/3},$ one finds $c_{1} \\simeq 3.9$ in the MBTA and $c_{1} \\simeq 1.1$ in the RPA with the vertex correction [27] (where $T_{\\rm c}^{0}$ is the phase transition temperature in an ideal Bose gas).",
"The RPA result is close to the Monte-Carlo result $c_1\\simeq 1.3$ [36], [38], [37], whereas the MBTA overestimates the coefficient $c_{1}$ .",
"When $T_{\\rm c}$ is evaluated by the theory below $T_{\\rm c}$ , $\\delta \\tilde{G}^{(1)}$ and $\\delta \\tilde{G}^{(2)}$ give different results.",
"In the case of $\\delta \\tilde{G}^{(2)}$ , the contribution of $\\delta G_{}^{(2)}$ smoothly vanishes in the limit $n_{0} \\rightarrow 0$ , so that the value of $T_{\\rm c}$ coincides with that evaluated from the region above $T_{\\rm c}$ .",
"The order of the phase transition is also the same as the result based on the standard Green's function formalism in Eq.",
"(REF ) [27].",
"When the self-energy in the first term of Eq.",
"(REF ) is treated within the RPA with the vertex correction, the weak first-order phase transition is obtained.",
"In the case of the MBTA, one obtains the second-order phase transition.",
"(See the inset of Fig.",
"REF .)",
"In the case of $\\delta \\tilde{G}^{(1)}$ , on the other hand, $T_{\\rm c}$ determined by the temperature in the limit $n_0\\rightarrow +0$ does not coincide with $T_{\\rm c}$ determined by the theory above $T_{\\rm c}$ .",
"In addition, in both the cases of the MBTA and the RPA with the vertex correction, the condensate fraction $n_0$ exhibits a remarkable reentrant behavior near $T_{\\rm c}$ .",
"(See the inset in Fig.",
"REF .)",
"These are ascribed to large contribution of Eq.",
"(REF ).",
"Given the regular part of the off-diagonal self-energy $\\Sigma _{12}^{\\rm R} (0)=n_0 V_{\\rm eff},$ (where $V_{\\rm eff} = \\Gamma _{11}^{\\rm R}(0)$ in the MBTA, and $V_{\\rm eff} = U_{\\rm eff}^{\\rm R}(0)$ in the RPA), we find that in the limit $p\\rightarrow 0$ , Eq.",
"(REF ) is reduced to $\\tilde{G}_{\\rm L}^{(1)} (p) \\tilde{\\Sigma }(p) \\tilde{G}_{\\rm R}^{(1)} (p) \\simeq \\frac{ U^{2} \\Pi _{14} (p)}{2 n_{0} V_{\\rm eff}^{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} .$ Equation (REF ) becomes very large, when $n_{0}\\rightarrow +0$ [39].",
"Thus, the temperature has to decrease near $T_{\\rm c}$ , so as to satisfy the number equation $n = n_{0} + n^{\\prime }$ .",
"This leads to the reentrant behavior of $n_0$ seen in the inset of Fig.",
"REF .",
"In addition, while Eq.",
"(REF ) is very large in the limit $n_{0} \\rightarrow + 0$ , it is absent in the normal state above $T_{\\rm c}$ because $\\Pi _{14} = 0$ .",
"One thus obtains the discrepancy of the critical temperature between formalisms above and below $T_{\\rm c}$ .",
"The reentrant behavior of $n_0$ is more remarkable in the MBTA than in the RPA.",
"In the former MBTA, the effective interaction $V_{\\rm eff}$ vanishes at $T_{\\rm c}$ [6], [27], because $\\Gamma _{11} (p) = \\Gamma _{11}^{\\rm R} (p) = \\frac{U}{1 - U \\Pi _{11} (p)} \\quad (T = T_{\\rm c}),$ where $\\Pi _{11} (0) = \\infty $ .",
"On the other hand, in the latter RPA with the vertex correction [27], one finds $V_{\\rm eff} = U/2$ at $T_{\\rm c}$ , because $U_{\\rm eff} (p) = U_{\\rm eff}^{\\rm R} (p) = \\frac{U [ 1- U \\Pi _{22} (p) ]}{1 - 2 U \\Pi _{22} (p)} \\quad (T = T_{\\rm c}),$ and $\\Pi _{22} (0) = \\infty $ .",
"As a result, Eq.",
"(REF ) is larger in the MBTA than in the RPA with the vertex correction.",
"Although the present approach can correctly describe the infrared behavior of the longitudinal response function, as well as the NN identity, the above results indicate that it still has room for improvement in the fluctuation region near $T_{\\rm c}$ .",
"In this region, strong fluctuations dominate over the phase transition behavior [26], [38], [40], [41], [42], [43]."
],
[
"Summary",
"We have presented a Green's function formalism, which can correctly describe two required conditions for any consistent theory of Bose–Einstein condensates (BECs): (i) the infrared divergence of the longitudinal susceptibility in the low-energy and low-momentum limit, as well as (ii) the Nepomnyashchii–Nepomnyashchii (NN) identity, which states the vanishing off-diagonal self-energy in the same limit.",
"These conditions cannot be satisfied in the Bogoliubov mean-field theory, the many-body $T$ -matrix theory (MBTA), as well as the random-phase approximation (RPA) with the vertex correction.",
"Our key idea is to divide the irreducible self-energy contribution into the singular and non-singular parts with respect to the infrared divergence.",
"These self-energies are separately included in the Green's function so as to satisfy various conditions that are required for any consistent theory of BECs.",
"In this paper, we treated the non-singular self-energy as the ordinary self-energy correction in the Green's function.",
"On the other hand, we dealt with the singular self-energy to the first-order.",
"The resulting Green's function consists of two terms, which is similar to the Green's function in the Popov's hydrodynamic theory [22], [23], [5].",
"The singular component mentioned above enables us to correctly describe the infrared divergence of the longitudinal susceptibility, the Hugenholtz-Pines relation, as well as the NN identity.",
"In addition, we showed that the non-singular part of the self-energy provides the enhancement of the BEC phase transition temperature $T_{\\rm c}$ (which has been predicted by various methods).",
"The value of the enhancement depends on to what extent we take into account many-body corrections in the non-singular self-energy.",
"The present approach can describe various required conditions that are not satisfied in the previous theories, such the Bogoliubov-type mean-field theory, the MBTA, as well as the RPA with the vertex correction.",
"On the other hand, it still has room for improvement in considering the region near $T_{\\rm c}$ .",
"When the non-singular self-energy is treated within the MBTA, the expected second-order phase transition may be obtained, whereas the enhancement of $T_{\\rm c}$ is overestimated compared with the Monte-Carlo simulation result.",
"When the non-singular part is calculated within the RPA with the vertex correction, the enhancement of $T_{\\rm c}$ is close to the Monte-Carlo result compared with the MBTA, whereas it incorrectly gives the first-order phase transition.",
"The further improvement of the present approach to overcome this problem is a remaining issue.",
"We thank M. Ueda for valuable discussions and comments.",
"We also thank A. Leggett and Y. Takada for discussions.",
"S.W.",
"was supported by JSPS KAKENHI Grant Number (249416).",
"Y.O.",
"was supported by Grant-in-Aid for Scientific research from MEXT in Japan (25400418, 25105511, 23500056)."
],
[
"list of $\\Pi $",
"The polarization functions used in this paper are summarized as follows: $\\Pi _{11} (p) = &- \\sum \\limits _{\\bf q}\\frac{1}{2} \\left[ (E_{{\\bf p}+{\\bf q}} - E_{\\bf q}) \\left( 1 - \\frac{\\xi _{{\\bf p}+{\\bf q}}\\xi _{\\bf q}}{E_{{\\bf p}+{\\bf q}}E_{\\bf q}} \\right) + i \\omega _{n} \\left( \\frac{\\xi _{{\\bf p}+{\\bf q}}}{E_{{\\bf p}+{\\bf q}}} - \\frac{\\xi _{\\bf q}}{E_{\\bf q}} \\right) \\right]\\frac{n_{{\\bf p}+{\\bf q}} - n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} - E_{\\bf q})^{2}}\\nonumber \\\\& \\quad - \\sum \\limits _{\\bf q}\\frac{1}{2} \\left[ (E_{{\\bf p}+{\\bf q}} + E_{\\bf q}) \\left( 1 + \\frac{\\xi _{{\\bf p}+{\\bf q}}\\xi _{\\bf q}}{E_{{\\bf p}+{\\bf q}}E_{\\bf q}} \\right)+ i \\omega _{n} \\left( \\frac{\\xi _{{\\bf p}+{\\bf q}}}{E_{{\\bf p}+{\\bf q}}} + \\frac{\\xi _{\\bf q}}{E_{\\bf q}} \\right) \\right]\\frac{1 + n_{{\\bf p}+{\\bf q}} + n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} + E_{\\bf q})^{2}} ,\\\\\\Pi _{12} (p) = &- \\sum \\limits _{\\bf q}\\frac{1}{2} \\Delta \\left[ \\frac{\\xi _{{\\bf p}+{\\bf q}}}{E_{{\\bf p}+{\\bf q}}E_{\\bf q}} (E_{{\\bf p}+{\\bf q}} - E_{\\bf q}) + \\frac{ i \\omega _{n} }{E_{\\bf q}} \\right]\\frac{n_{{\\bf p}+{\\bf q}} - n_{\\bf q}}{\\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} - E_{\\bf q})^{2}}\\nonumber \\\\& \\quad + \\sum \\limits _{\\bf q}\\frac{1}{2} \\Delta \\left[ \\frac{\\xi _{{\\bf p}+{\\bf q}}}{E_{{\\bf p}+{\\bf q}}E_{\\bf p}} (E_{{\\bf p}+{\\bf q}} + E_{\\bf q}) + \\frac{ i \\omega _{n} }{E_{\\bf q}} \\right]\\frac{1 + n_{{\\bf p}+{\\bf q}} + n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} + E_{\\bf q})^{2}} ,\\\\\\Pi _{14} (p) = &\\sum \\limits _{\\bf q}\\frac{1}{2} \\frac{\\Delta ^{2}}{E_{{\\bf p}+{\\bf q}}E_{\\bf q}}\\left[(E_{{\\bf p}+{\\bf q}} - E_{\\bf q})\\frac{n_{{\\bf p}+{\\bf q}} - n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} - E_{\\bf q})^{2}}- (E_{{\\bf p}+{\\bf q}} + E_{\\bf q})\\frac{1 + n_{{\\bf p}+{\\bf q}} + n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} + E_{\\bf q})^{2}}\\right] ,\\\\\\Pi _{22} (p) = &\\sum \\limits _{\\bf q}\\frac{1}{2} \\left[ (E_{{\\bf p}+{\\bf q}} - E_{\\bf q}) \\left( 1 + \\frac{\\xi _{{\\bf p}+{\\bf q}}\\xi _{\\bf q}}{E_{{\\bf p}+{\\bf q}}E_{\\bf q}} \\right)+ i \\omega _{n} \\left( \\frac{\\xi _{{\\bf p}+{\\bf q}}}{E_{{\\bf p}+{\\bf q}}} + \\frac{\\xi _{\\bf q}}{E_{\\bf q}} \\right) \\right]\\frac{n_{{\\bf p}+{\\bf q}} - n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} - E_{\\bf q})^{2}}\\nonumber \\\\& \\quad +\\sum \\limits _{\\bf q}\\frac{1}{2} \\left[ (E_{{\\bf p}+{\\bf q}} + E_{\\bf q}) \\left( 1 - \\frac{\\xi _{{\\bf p}+{\\bf q}}\\xi _{\\bf q}}{E_{{\\bf p}+{\\bf q}}E_{\\bf q}} \\right)+ i \\omega _{n} \\left( \\frac{\\xi _{{\\bf p}+{\\bf q}}}{E_{{\\bf p}+{\\bf q}}} - \\frac{\\xi _{\\bf q}}{E_{\\bf q}} \\right) \\right]\\frac{1 + n_{{\\bf p}+{\\bf q}} + n_{\\bf q}}{ \\omega _{n}^{2} + (E_{{\\bf p}+{\\bf q}} + E_{\\bf q})^{2}} ,$ where $\\xi _{\\bf p} \\equiv \\varepsilon _{\\bf p} + \\Delta $ , $\\Delta \\equiv U n_{0}$ , $E_{\\bf p} \\equiv \\sqrt{\\varepsilon _{\\bf p} (\\varepsilon _{\\bf p} + 2 \\Delta )}$ , and $n_{\\bf p}$ is the Bose distribution function $n_{\\bf p} \\equiv 1/ ( e^{\\beta E_{\\bf p}} - 1)$ with $\\beta = 1/T$ ."
],
[
"Infrared behaviors of $\\Pi _{14}$",
"We discuss the infrared properties of the polarization function $\\Pi _{14}$ for the system dimensionality $d = 3$ .",
"We are going to derive the relation $\\Pi _{14} (p) \\propto \\left\\lbrace \\begin{array}{ll}\\ln (c_{0}^{2} |{\\bf p}|^{2} - \\omega ^{2} ) & \\qquad (T = 0)\\\\ 1/ |{\\bf p}| & \\qquad (T \\ne 0) .\\end{array}\\right.$ For simplicity, we use the dimensionless quantities.",
"We scale the energy by the critical temperature of an ideal Bose gas $T_{\\rm c}^{0}$ .",
"In the dimensionless formula, we use $\\tilde{E}_{\\bf p} = E_{\\bf p}/T_{\\rm c}^{0}$ , $\\tilde{\\Delta }= \\Delta /T_{\\rm c}^{0}$ , $\\tilde{\\Pi }_{14} (q) = \\Pi _{14} (q) T_{\\rm c}^{0}$ , and $\\tilde{\\varepsilon }_{\\bf p} = \\varepsilon _{\\bf p}/T_{\\rm c}^{0} = \\tilde{p}^{2}$ .",
"We wrote the modulus of the momentum in the dimensionless form as $\\tilde{p} = |\\tilde{\\bf p}|$ .",
"In the following, we omit the tilde for simplicity.",
"After lengthy calculation, we reduce the polarization function $\\Pi _{14}$ as $\\Pi _{14} (p) = - A_3 2 \\pi \\int _{0}^{{p}_{\\rm c}} d q q^2 \\frac{ \\Delta ^2}{2 E_{\\bf q} } \\Xi _{14} g_{\\bf q} ,$ where $g_{\\bf q} \\equiv \\coth \\left( \\beta E_{\\bf q} / 2 \\right)$ .",
"Here, $\\Xi _{14}$ is given by $\\Xi _{14} = {\\rm Re}\\left[ \\frac{1}{2 p q R } \\ln { \\frac{ ( P_{+} + \\Delta - R ) ( P_{-} + \\Delta + R) }{ ( P_{-} + \\Delta - R ) ( P_{+} + \\Delta + R ) } } \\right] ,$ where $P_{\\pm } = ( p \\pm q)^2$ , $R = \\sqrt{A^2 + \\Delta ^2}$ and $A = i \\omega _n - E_{\\bf q}$ .",
"The coefficient $A_{3}$ is given by $A_{3} = 1/[\\pi ^{3/2} \\zeta (3/2)]$ , and $\\zeta $ is the Riemann zeta function.",
"For the small $q$ and $i\\omega _{n}$ , we have $A \\simeq i \\omega _{n} - c_{0} q$ and $R \\simeq \\Delta + c_{0}^{-2} (i \\omega _{n} - c_{0} q)^{2}$ .",
"In this case, the main contribution of $\\Xi _{14}$ reads as $\\Xi _{14} \\simeq & \\frac{1}{c_{0}^{2} p q} {\\rm Re} \\left[ \\ln \\left( \\frac{A_{+} q + B}{A_{-} q + B} \\right) \\right],$ where $A_{\\pm } = 2 \\left( \\pm p + i \\omega _{n} / c_{0} \\right)$ , and $B = p^{2} - \\left( i \\omega _{n} / c_{0} \\right)^{2}$ .",
"At $T = 0$ , we replace $i \\omega _{n}$ with $\\omega $ , and take $g_{\\bf q} = 1$ .",
"We have $\\Pi _{14} \\simeq & - A_{3} \\frac{\\pi c_{0}}{4 p} {\\rm Re} [ F( p_{\\rm c}) - F (0) ] ,$ where $F (q) = & q \\ln \\left( \\frac{B + A_{+} q}{B + A_{-} q} \\right) + \\sum \\limits _{j = \\pm } \\frac{j B}{A_{j}} \\ln (B + A_{j} q) .$ The main contribution originates from $F (0)$ , and we end with $\\Pi _{14} \\simeq & A_{3} \\frac{\\pi c_{0}}{4} \\ln \\left( p^{2} - \\frac{ \\omega ^{2}}{c_{0}^{2}} \\right).$ This leads to (REF ) for $T=0$ .",
"At $T \\ne 0$ , we take $\\omega _{n} = 0$ and $g_{\\bf q} = 2 T / c_{0} q$ .",
"In this case, we have $\\Pi _{14} \\simeq & - A_{3} \\frac{\\pi T }{2 p} {\\rm Re}[ F ( p_{\\rm c}) ] ,$ where $F ( q ) = & - {\\rm Li}_{2} \\left( 4 q / A_{-} \\right) + {\\rm Li}_{2} \\left( 4 q / A_{+} \\right) .$ Here, ${\\rm Li}_{n}(z)$ is the polylogarithm (Jonquière's function).",
"We also used $F (0) = 0$ .",
"For large $ p_{\\rm c}$ , ${\\rm Re} [ F ( p_{\\rm c}) ] \\simeq \\pi ^{2} / 2 - p / p_{\\rm c}$ holds.",
"We thus end with $\\Pi _{14} \\simeq & - A_{3} \\frac{\\pi ^{3} T}{4} \\frac{1}{ p}.$ This leads to (REF ) for $T\\ne 0$ ."
],
[
"Popov's hydrodynamic theory",
"We derive the single-particle Green's function in the Popov's hydrodynamic theory.",
"We suppose that the system is (a) in the weak coupling regime, (b) at $T \\simeq 0$ , as well as (c) in the hydrodynamic regime $|{\\bf p}| \\ll \\sqrt{2} m c$ .",
"In the Popov's hydrodynamic theory, the bosonic field operator $\\Psi (x)$ is written in the hydrodynamic variables, i.e., $\\Psi (x) = \\sqrt{n_{0} + \\pi (x)} e^{i \\varphi (x)}$ with $x = ({\\bf r}, \\tau )$ .",
"Here, $\\pi (x)$ and $\\varphi (x)$ are density and phase fluctuation operators.",
"Green's functions in the hydrodynamic picture and the standard picture are related each other [22], [23], [5], [19], [32], giving the form $G (p) = & - \\frac{1}{4 n_{0}} G_{\\pi \\pi } (p) \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} + i G_{\\pi \\varphi } (p) \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix}\\nonumber \\\\& - n_{0} G_{\\varphi \\varphi } (p) \\begin{pmatrix} 1 & -1 \\\\ -1 & 1 \\end{pmatrix} + \\delta G (p) \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix},$ where $\\delta G (p) = - \\frac{n_{0}}{2} T \\sum \\limits _{q} G_{\\varphi \\varphi } (p+q) G_{\\varphi \\varphi } (q) .$ Here, $G_{AB}$ with $A, B = \\pi , \\varphi $ is the non-perturbed correlation function for the hydrodynamic variables, given by [22], [23], [5], [19], [32] $\\begin{pmatrix} G_{\\pi \\pi } (p) & G_{\\pi \\varphi } (p) \\\\ G_{\\varphi \\pi } (p) & G_{\\varphi \\varphi } (p) \\end{pmatrix}=\\frac{1}{\\omega _{n}^{2} + c_{0}^{2} {\\bf p}^{2}}\\begin{pmatrix} n_{0}{\\bf p}^{2}/m & - \\omega _{n} \\\\ \\omega _{n} & m c_{0}^{2} / n_{0} \\end{pmatrix}.$ In (REF ), we used the conditions (a) and (b), which leads an approximate equality between the mean density and the condensate density, i.e., $n \\simeq n_{0}$ [19].",
"To obtain (REF ), the bosonic field operator $\\Psi (x)$ is expanded by the fluctuation operators $\\pi (x)$ and $\\varphi (x)$ .",
"Fluctuations are considered up to the second-order.",
"According to (REF ), the phase fluctuation is stronger than the density fluctuation.",
"Thus, the phase fluctuation effect alone is taken as the second-order fluctuation.",
"In the hydrodynamic regime $|{\\bf p}| \\ll \\sqrt{2} m c_{0} $ , by using (REF ), we end with $G (p) = - \\frac{ m c_{0}^{2}}{\\omega _{n}^{2} + c_{0}^{2} {\\bf p}^{2}} \\begin{pmatrix} 1 & -1 \\\\ -1 & 1 \\end{pmatrix} + \\delta G (p) \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix}.$ The first term in (REF ) is equivalent to the first term in (REF ).",
"The second term in (REF ) becomes also equal to the second term in (REF ) for small $p$ .",
"Indeed, in the hydrodynamic regime, a relation $g_{12} (p) \\simeq n_{0} G_{\\varphi \\varphi } (p)$ holds.",
"As a result, $\\delta G (p)$ in (REF ) is reduced into $\\Pi _{14} (p)/(2n_{0})$ , which reproduces the second term in (REF ).",
"To summarize, in the hydrodynamic regime, the Green's function in our approach (REF ) reproduces the Green's function obtained in the Popov's hydrodynamic approach.",
"In particular, the term (REF ) including the convolution of the phase-phase correlation is reproduced from the second term in (REF ), where $\\tilde{\\Sigma }$ involves the single bubble self-energy (REF ).",
"One of the highlights in the Popov's hydrodynamic approach is to meet the NN identity [23].",
"We substitute the Green's function (REF ) into the Dyson–Beliaev equation.",
"Inversely solving this Dyson–Beliaev equation $\\Sigma (p) = G_{0}^{-1} (p) - G^{-1} (p)$ , we obtain $\\Sigma (0) = {\\bf \\mu }$ .",
"This result meets the Hugenholtz-Pines relation [30] as well as the NN identity [9], [10].",
"This equality $\\Sigma (0) = {\\bf \\mu }$ originates from the fact that the term $\\delta G$ has the infrared divergence.",
"Our Green's function approach employs the same procedure to obtain (REF )."
],
[
"Derivation of Eq. (",
"To derive the equality (REF ), it is convenient to refer to an exact many-line vertex $M ( r_{\\rm out}, r_{\\rm in}, r_{U})$ , given by Nepomnyashchii and Nepomnyashchii [10].",
"Here, $r_{\\rm in}$ and $r_{\\rm out}$ are numbers of incoming and outgoing external particle lines, respectively.",
"$r_{U}$ is the number of an external potential line $U$ .",
"In this vertex $M$ , momentum and frequency are taken to be zeros with respect to the external particle line and the external interaction potential line.",
"The exact many-line vertex, which is constructed from diagrams irreducible in the particle lines, reads as [10] $M ( r_{\\rm out}, r_{\\rm in}, r_{U} )= & n_{0}^{(r_{\\rm out} - r_{\\rm in}) /2 } \\left( - \\frac{\\partial }{\\partial \\mu } \\right)_{n_{0}}^{r_{U}} \\left( \\frac{\\partial }{\\partial n_{0}} \\right)_{\\mu }^{r_{\\rm out}}n_{0}^{r_{\\rm in}} \\left( \\frac{\\partial }{\\partial n_{0}} \\right)_{\\mu }^{r_{\\rm in}} E^{\\prime } (T, \\mu , n_{0}),$ where $E^{\\prime }$ is the thermodynamic potential given by $E^{\\prime } = - T \\ln {\\rm Tr}[\\exp ({-\\beta H^{\\prime }})]$ .",
"For the Hamiltonian $H^{\\prime }$ , we subtract the contribution $- \\mu n_{0}$ from the original Hamiltonian (REF ), where the Bogoliubov prescription is applied.",
"Indeed, we are considering diagrams irreducible in the particle lines.",
"An operator $\\partial / \\partial \\mu $ creates a vertex point connecting to an external potential line $U$ .",
"An operator $ \\sqrt{n_{0}} \\partial / \\partial n_{0}$ creates a vertex point connecting to an external particle line by eliminating one condensate line.",
"Using (REF ), we obtain matrix forms of the two-point vertex (the self-energy) $\\Sigma $ and the three-point vertex $P^{\\dag }$ with respect to the external particle line, which are respectively given by [10] $\\Sigma (0) = \\frac{\\partial E^{\\prime }}{\\partial n_{0}} \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} + n_{0} \\frac{\\partial ^{2} E^{\\prime }}{\\partial n_{0}^{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} ,$ $P^{\\dag } (0;0) = 2 \\sqrt{n_{0}} \\frac{\\partial ^{2} E^{\\prime }}{\\partial n_{0}^{2}} \\eta ^{\\dag } + n_{0}^{3/2} \\frac{\\partial ^{3} E^{\\prime }}{\\partial n_{0}^{3}} \\eta _{\\rm a}^{\\dag } .$ We thus obtain the relation (REF ).",
"We note that the Hugenholtz-Pines relation $\\mu = \\Sigma _{11} (0) - \\Sigma _{12} (0)$ is also obtained from (REF ), when we apply $\\mu = \\partial E^{\\prime } / \\partial n_{0}$ [10]."
],
[
"Original derivation of Nepomnyashchii–Nepomnyashchii identity",
"Nepomnyashchii and Nepomnyashchii considered the full self-energy contribution by using vertex functions [10] (diagrammatically described in Fig.",
"REF ), giving the form $\\Sigma (p) = & \\Sigma ^{(0)} (p) + \\Sigma ^{(1)} (p) + \\Sigma ^{(2)} (p) + \\Sigma ^{(3)} (p),$ where $\\Sigma ^{(0)} (p) = & - U \\frac{1}{2} G_{1/2}^{\\dag } G_{1/2} - U \\frac{\\partial E^{\\prime }}{\\partial \\mu } = U n,\\\\\\Sigma ^{(1)} (p) = & - G_{1/2} U G_{1/2}^{\\dag } - T \\sum \\limits _{q} U \\gamma (q;p) G (q) ,\\\\\\Sigma ^{(2)} (p) = & - T \\sum \\limits _{q} P^{\\dag } (q;p) K_{} (q;p) U \\frac{ 1 }{ \\sqrt{-1} } \\hat{T} {\\mathcal {G}}_{1/2} ,\\\\\\Sigma ^{(3)} (p) = & - \\frac{T}{2} \\sum \\limits _{q} P^{\\dag } (q;p) K_{} (q;p) | f_{0}\\rangle U \\frac{1}{\\sqrt{-1}} G_{1/2}^{\\dag }.$ Here, $K_{} (q;p)$ is a bare part of the $(4\\times 4)$ -matrix two-particle Green's function, given by $K (p+q) = G (p+q) \\otimes G(-q).$ The Green's function $G_{}$ is the full Green's function, where all the diagrammatic contributions are included to the self-energy.",
"In the small-$p$ regime, the leading term of $G_{}$ is reduced to [21], [10] $G (p)= & - \\frac{n_{0} m c^{2}}{n} \\frac{1}{\\omega _{n}^{2} + c^{2} |{\\bf p}|^{2}} \\begin{pmatrix} 1 & -1 \\\\ - 1 & 1 \\end{pmatrix}.$ In (), $\\gamma (p;q)$ is the $(2\\times 2)$ -matrix three point vertex that has an external potential line and two external particle lines.",
"In (REF ), $\\partial E^{\\prime } / \\partial \\mu $ corresponds to the one point vertex connecting to an external potential line.",
"We used the fact that this vertex is equivalent to the non-condensate density, i.e., $n^{\\prime } = - \\partial E^{\\prime } / \\partial \\mu $ .",
"The contributions (REF ) and () converge.",
"On the other hand, for () and (), the bare part of the two-particle Green's function $K$ provides the infrared divergences.",
"Since the infrared divergences are strongly related to each other, these are simply extracted by using $K^{\\rm IR} (q;p) = G_{12} (p+q) G_{12} (q) \\hat{C}$ , where we used the symmetry relation $G_{12} (p) = G_{12} (-p)$ .",
"Indeed, according to (REF ), we have the infrared-divergent relation $\\lim _{p\\rightarrow 0} G_{11,22} (p) = - \\lim _{p\\rightarrow 0} G_{12,21} (p)$ [21].",
"The infrared divergences in () are exactly canceled out, because we have a relation $\\hat{C} \\hat{T} \\hat{\\eta }_{\\rm g} = 0$ , as discussed in the case of the first term of (REF ).",
"On the other hand, for (), the infrared divergence remains as discussed in the second term in (REF ).",
"Using the identity (REF ) as well as the second mean value theorem for integrals [35], we can reduce the self-energy $\\Sigma $ to $\\Sigma (0) = & 2 U \\Pi _{\\rm IR}^{\\Lambda } \\Sigma _{12} (0) \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} + \\delta \\Sigma $ Here, $\\Pi _{\\rm IR}^{\\Lambda } \\equiv - T \\sum _{{\\bf q}=0}^{\\Lambda } [ G_{12} (i \\omega _{n} = 0, {\\bf q}) ]^{2}$ provides the infrared divergence.",
"The $(2\\times 2)$ -matrix $\\delta \\Sigma $ is a remaining converging part of $\\Sigma $ .",
"Solving (REF ) for $\\Sigma _{12} (0)$ , we end with $\\Sigma _{12} (0) = 0$ , where we used $\\Pi _{\\rm IR}^{\\Lambda } = \\infty $ .",
"This is the original derivation of the NN identity.",
"To summarize, the NN identity was originally derived from (a) the infrared divergence of the Green's function due to the spontaneously broken continuous symmetry, as well as (b) the Ward-Takahashi identity with respect to two and three-point vertices.",
"The diagrammatic contribution essential to the NN identity is the bubble structure diagram with the vertex correction (), which is the same diagram as the second term of (REF ) in our formalism."
],
[
"Gauge invariance",
"To derive the NN identity, Nepomnyashchii and Nepomnyashchii used the fact that the Green's function exhibits the infrared divergence, which originates from the phase fluctuation thanks to the spontaneously broken continuous gauge symmetry.",
"If we apply the idea that physical quantities are gauge invariant, we may more simply understand the NN identity as well as the finiteness of the chemical potential $\\mu $ , the diagonal self-energy $\\Sigma _{11}$ , and the macroscopic sound speed $c$ .",
"We apply the exact many-line vertex (REF ) at $p=0$ .",
"Note that if the system is not at the critical temperature, the exact many-line vertex $M$ does not diverge.",
"Indeed, it is given by thermodynamic derivative of the condensate density and the chemical potential.",
"This vertex $M$ is thus finite not at $T_{\\rm c}$ .",
"We also note that the thermodynamic potential $E^{\\prime }$ is gauge invariant.",
"In the representation (REF ), we assumed that the BEC order parameter is a real number.",
"We now consider that the BEC order parameter is a complex number, given by $\\sqrt{n_{0}} e^{i \\varphi _{0}}$ .",
"In this case, the exact many-line vertex (REF ) has the factor given by $\\exp {[ i \\varphi _{0} (r_{\\rm out} - r_{\\rm in}) ]}$ .",
"The off-diagonal self-energy $\\Sigma _{12}$ at $p=0$ is now given by $\\Sigma _{12} (0) = e^{2 i \\varphi _{0} }n_{0} \\frac{\\partial ^{2} E^{\\prime }}{\\partial n_{0}^{2}}.$ The phase $\\varphi _{0}$ is chosen to be arbitrary in the ordered phase with the spontaneously broken continuous symmetry.",
"We thus take the average with respect to $\\varphi _{0}$ over the range $\\varphi _{0} \\in [0, 2 \\pi )$ .",
"The averaged off-diagonal self-energy $\\langle \\Sigma _{12} (0) \\rangle $ is now given by $\\langle \\Sigma _{12} (0) \\rangle = 0$ , which leads to the NN identity.",
"On the other hand, the diagonal self-energy is given by $\\Sigma _{11} (0) = & \\frac{\\partial E^{\\prime }}{\\partial n_{0}} + n_{0} \\frac{\\partial ^{2} E^{\\prime }}{\\partial n_{0}^{2}}.$ The chemical potential is given by $\\mu = \\partial E^{\\prime } / \\partial n_{0}$ [10].",
"The density-density correlation function $\\chi _{}$ , which has two external potential lines, reads as $\\chi _{} (0) = & \\frac{\\partial ^{2} E^{\\prime }}{\\partial \\mu ^{2}} = - \\frac{\\partial n^{\\prime }}{\\partial \\mu }.$ As shown in Ref.",
"[10], we have $\\chi (0) = - n /(mc^{2})$ .",
"These quantities do not involve a factor given by $\\exp {(i \\varphi _{0})}$ , and their values are unchanged even if we take the average with respect to the phase $\\varphi _{0}$ .",
"To summarize, averaged quantities of gauge invariant operators are not affected by uncertainty of the phase $\\varphi _{0}$ .",
"The gauge invariance protects their finiteness against the phase fluctuations.",
"On the other hand, the gauge-dependent quantities are affected by the phase fluctuations, and their values vanish at $p=0$ .",
"It may lead to the NN identity $\\Sigma _{12} (0) = 0$ [9], [10]."
]
] | 1403.0005 |
[
[
"On the Influence of Spatial Dispersion on the Performance of\n Graphene-Based Plasmonic Devices"
],
[
"Abstract We investigate the effect of spatial dispersion phenomenon on the performance of graphene-based plasmonic devices at THz.",
"For this purpose, two different components, namely a phase shifter and a low-pass filter, are taken from the literature, implemented in different graphene-based host waveguides, and analyzed as a function of the surrounding media.",
"In the analysis, graphene conductivity is modeled first using the Kubo formalism and then employing a full-$k_\\rho$ model which accurately takes into account spatial dispersion.",
"Our study demonstrates that spatial dispersion up-shifts the frequency response of the devices, limits their maximum tunable range, and degrades their frequency response.",
"Importantly, the influence of this phenomenon significantly increases with higher permittivity values of the surrounding media, which is related to the large impact of spatial dispersion in very slow waves.",
"These results confirm the necessity of accurately assessing non-local effects in the development of practical plasmonic THz devices."
],
[
"Graphene has recently attracted large attention as a potential platform for surface plasmon polaritons (SPPs) at terahertz (THz) frequencies [1].",
"Graphene electrical properties, which include high carrier mobility and saturation speed, electrostatic and magnetostatic reconfiguration, and relatively low losses among many others [2], have triggered the theoretical development of tunable plasmonic devices such as waveguides [3], leaky-wave antennas [4], [5], reflectarrays [6], switches [7], phase shifters [8], or filters [9].",
"Importantly, the propagation of SPPs along graphene has just been experimentally confirmed [10] and some initial devices, including THz modulators [11] and Faraday rotators [12], have been demonstrated.",
"Consequently, it is expected that novel plasmonic devices with unprecedent performance in the THz band will be developed in the coming years.",
"Figure: Schematic diagram of two graphene-based waveguides.",
"Graphene conductivity may be tuned at each waveguide section by applying a DC bias to the poliysilicon gating pads.",
"(a) Monolayer implementation.",
"(b) PPW implementation.In the theoretical development of non-magnetic plasmonic devices, graphene has usually been modeled as an infinitesimally thin layer characterized by a scalar conductivity obtained through the Kubo formalism [13].",
"This formulation provides a large scale model of graphene, valid from DC to optical frequencies, but neglects the possible presence of spatial dispersion (non-local) effects, i.e.",
"the dependence of graphene conductivity with the propagating wavenumber.",
"Several authors have recently investigated how this phenomenon modifies the propagation characteristics of SPPs [13], [14], [15], [16], [17], concluding that spatial dispersion becomes significant in case of very large propagating wavenumbers.",
"In this context, this letter studies the influence of spatial dispersion effects on the performance of graphene-based plasmonic devices at THz.",
"For this purpose, two examples of such devices, namely a phase shifter [8] and a low-pass filter [9], have been chosen from the literature, implemented in graphene-based single layer and parallel-plate (PPW) host waveguides [14], [17], and analyzed as a function of the surrounding media.",
"In the analysis, graphene is modeled first using the Kubo formalism [13] and then employing an accurate conductivity model which takes into account spatial dispersion in the THz band [16].",
"Our study demonstrates that (a) spatial dispersion modifies the behavior of plasmonic devices by up-shifting their operation frequency, degrading their frequency response, and limiting their maximum tunable range; and (b) the influence of spatial dispersion increases with higher permittivities of the surrounding media.",
"These results clearly confirm the necessity of accurately assessing spatial dispersion phenomenon in the development of plasmonic THz devices."
],
[
"This section briefly describes the behavior of two well-known 2D graphene-based waveguides [13], [8] (see Fig.",
"REF ) which are employed below to design the plasmonic devices under investigation.",
"The lateral dimensions of the waveguides are assumed to be much larger than the guided wavelength.",
"The first configuration (see Fig.",
"REF ) consists of a single graphene layer transferred onto a dielectric, and it supports a unique TM SPP [13] at the frequency band of interest.",
"The second structure (see Fig.",
"REF ) is composed of a graphene-based parallel-plate waveguide with a separation distance $d$ between the graphene sheets much smaller than their width.",
"Even though this waveguide may support two modes [3], [14], we focus here on the quasi-TEM mode due to its extreme confinement to the graphene layers.",
"In both waveguides, gating pads are located close to the graphene layers to dynamically control the guiding properties of each waveguide section by exploiting graphene's field effect [2], [8].",
"In the analysis of the host waveguides, graphene is modeled as an infinitesimally thin layer characterized by a scalar conductivity which depends on frequency, graphene's electron relaxation time $\\tau $ and chemical potential $\\mu _c$ , and temperature $T$ .",
"Neglecting the influence of spatial dispersion, graphene's conductivity is obtained using the well-known Kubo formalism [13].",
"Then, the propagation constant $k_{\\rho _L}$ and characteristic impedance $Z_{C_L}$ of the propagating modes can be easily obtained using standard techniques [13], [8].",
"When spatial dispersion effects are taken into account, graphene is characterized using the full-$k_\\rho $ relaxation-time-approximation (RTA) conductivity model introduced in [16], which provides accurate representation for any plasmon wavenumber and is valid in the whole THz band.",
"Importantly, we have numerically verified that the Bhatnagar-Gross-Krook conductivity model also proposed in [16], which enforces charge conservation and is expected to be more accurate, leads to extremely similar results [17] while adding substantial mathematical complexity.",
"The propagation constant $k_{\\rho _{NL}}$ and characteristic impedance $Z_{C_{NL}}$ of the host waveguides can then be obtained in closed-form following the approach developed in [17].",
"For the sake of illustration, Figs.",
"REF -REF show the normalized phase constant of the considered spatially-dispersive graphene waveguides embedded in silicon (Si) ($\\varepsilon _{r} = 11.9$ ) as a function of frequency and chemical potential.",
"Figs.",
"REF -REF show the relative error in these computations when neglecting spatial dispersion.",
"As can be observed, the Kubo conductivity model systematically overestimates the propagation constants of the waves, especially when graphene chemical potential is low.",
"In addition, the non-linear distribution of this error as a function of $\\mu _c$ will inevitably lead to unexpected behavior of graphene-based plasmonic devices.",
"Very large wavenumbers which were possible to obtain in the local case using low chemical potentials, can no longer be obtained due to spatial dispersion.",
"This can be understood in terms of the finite quantum capacitance of graphene [16], [9], which impairs large values of the wavevector.",
"This limitation implies that plasmonic devices that rely on abrupt variations of graphene conductivity may become more difficult to realize in practice than initially expected.",
"Figure: Normalized phase constant of (a) TM mode in a monolayer graphene waveguide (see Fig. )",
"and (b) quasi-TEM mode in a graphene-based PPW (see Fig.",
").",
"(c)-(d) show the relative error 100k ρ L -k ρ NL k ρ NL 100\\left|\\frac{k_{\\rho _L}-k_{\\rho _{NL}}}{k_{\\rho _{NL}}}\\right| in the calculation of panels (a)-(b) when using a local conductivity model.",
"Parameters are d=100d = 100 nm, τ=1\\tau = 1 ps, ε r1 =ε r2 =11.9\\varepsilon _{r1} = \\varepsilon _{r2} = 11.9 and T=300T=300 K."
],
[
"This section studies the influence of spatial dispersion on the performance of graphene-based plasmonic phase shifters [8] and low-pass filters [9].",
"Results have been obtained using a transmission line model combined with a transfer-matrix approach [8], [9], considering graphene with $\\tau =1.0$ ps at T=300 K. Without loss of generality, we assume $\\varepsilon _{r1} = \\varepsilon _{r2} = \\varepsilon _r$ throughout this section.",
"In the local case, results have been validated using data from full-wave simulations (not shown here for the sake of compactness).",
"However, note that current commercial full-wave software packages are not able to simulate spatially-dispersive graphene structures."
],
[
"Let us consider a digital load-line graphene-based phase shifter where $N$ gated sections of the waveguide allow for $2^N$ relative phase shift states [8].",
"Specifically, we implement a 3-bit phase shifter with shifts of $45^\\circ $ , $90^\\circ $ and $180^\\circ $ at the operation frequency $f=2$ THz.",
"The characteristics of each waveguide section, in terms of chemical potential and pad length, are obtained following the approach described in [8], and are shown in Table REF considering a local graphene conductivity model.",
"This design assumes that the host waveguides are embedded in Si ($\\varepsilon _r=11.9$ ), which is a realistic situation.",
"Note that in [8] spatial dispersion was safely neglected because the structures were standing in free-space.",
"Figs.",
"REF -REF illustrate the performance of this phase shifter in the single layer implementation (Fig.",
"REF ).",
"Ignoring spatial dispersion leads to large errors in the relative phase shifts between the ports, although no significant error occurs in the transmitted and reflected power.",
"Fig REF illustrates the error in the phase shift at the operation frequency for various phase shifters designed with different surrounding permittivity.",
"As can be observed, the influence of neglecting spatial dispersion significantly increases for higher permittivities values.",
"Figs.",
"REF -REF show the same information for the PPW implementation (Fig.",
"REF ).",
"A larger error in the phase shift is observed in this case, which is due to the higher confinement of the quasi-TEM mode.",
"Importantly, note that these errors can be easily compensated by simply using the correct spatially dispersive conductivity in the design procedure."
],
[
"Let us consider graphene-based low-pass filters in the THz, as proposed in [9].",
"Specifically, we implement a 7th degree filter with cutoff frequency $f_c=3$ THz.",
"For the sake of comparison, we design this filter in free-space ($\\varepsilon _r=1$ ) and embedded in Si ($\\varepsilon _r=11.9$ ).",
"The design parameters of the Si-embedded filter, computed following [9], are shown in Table REF .",
"Figs.",
"REF -REF show the frequency response corresponding to the single graphene sheet filters, standing in free-space and embedded in Si, respectively.",
"As expected, for $\\varepsilon _r = 1$ spatial dispersion proves to be irrelevant.",
"On the other hand, for $\\varepsilon _r = 11.9$ the filter response severely deteriorates, up-shifting its cutoff frequency and unevenly increasing the reflection Table: Parameters of the 7th degree filter (ε r =11.9\\varepsilon _r = 11.9).Figure: Influence of spatial dispersion in the response of 7th degree graphene-based low-pass filters.",
"Scattering parameters of the single sheet implementation in (a) free-space and (b) embedded in Si (ε r1 =ε r2 =11.9\\varepsilon _{r1} = \\varepsilon _{r2} = 11.9, see Table ).",
"(c) Error in the cutoff frequency and maximum in-band reflection due to spatial dispersion as a function of surrounding permittivity.",
"(d)-(f) show the same data for the PPW implementation (see Table ).",
"Parameters are d=100d = 100 nm, τ=1\\tau = 1 ps and T=300T=300 K (solid line - results neglectingspatial dispersion effects, dashed line - results including spatial dispersion effects).throughout the passband.",
"Note that the presence of spatial dispersion prevents the total compensation of this latter effect, due to the higher non-linear nature of the mode's propagation constant and characteristics impedance.",
"Fig REF illustrates the error in the filter's cutoff frequency and the maximum return loss in the filter's passband for various low-pass filters designed using dielectrics with increasing permittivity values.",
"Similarly to the case of the phase-shifters, the influence of spatial dispersion increases when the permittivity of the surrounding medium increases.",
"Figs.",
"REF -REF show the same study for the PPW implementation.",
"As expected, a larger shift in the cutoff frequency is observed in this case, but interestingly, the maximum error of the return loss within the passband is lower.",
"Contrary to the single sheet implementation, we have verified that a uniform level of in-band return loss can be achieved using graphene-based PPW, because the characteristic impedance of each spatially-dispersive transmission line section remains more linear with frequency.",
"This indicates that the use of graphene PPW structures could be advantageous over the use of single sheet structures, when low return losses are essential in lowpass filter applications."
],
[
"We have studied the influence of non-local effects in the response and performance of plasmonic graphene THz devices.",
"Following previous works, we have focused on graphene-based phase shifters and low pass filters, necessary elements for THz communication and sensing systems.",
"Due to the extremely slow waves supported by graphene-based waveguides in the presence of high permittivity media, spatial dispersion becomes a significant mechanism of propagation that modifies the expected behavior of these devices by up-shifting their operation frequency, limiting their tunable range, and degrading their frequency response.",
"Consequently, spatial dispersion must be accurately taken into account in the development of graphene-based plasmonic THz devices."
],
[
"This work was supported by the EU FP7 Marie-Curie IEF grant Marconi, with ref.",
"300966, Spanish Ministry of Education under grant TEC2010-21520-C04-04, and European Feder Fundings.",
"References F. H. L. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: A plaftform for strong light-matter interactions,” Nano Letters, vol.",
"11, pp.",
"3370–3377, 2011.",
"K. Geim and K. S. Novoselov, “The rise of graphene,” Nature materials, vol.",
"6, pp.",
"183–91, 2007.",
"J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano, vol.",
"6, pp.",
"431–440, 2012.",
"J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Optic Express, vol.",
"21, pp.",
"24 856–24 872, 2013.",
"M. Esquius-Morote, J. S. Gómez-Díaz, and J. Perruisseau-Carrier, “Sinusoidally-modulated graphene leaky-wave antenna for electronic beam-scanning at thz,” IEEE Transactions on Terahertz Science and Technology, vol.",
"4, pp.",
"116–122, January 2014.",
"E. Carrasco and J. Perruisseau-Carrier, “Reflectarray antenna at terahertz using graphene,” IEEE Antennas and Wireless Propagation Letters, vol.",
"12, pp.",
"253–256, 2013.",
"J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Optics Express, vol.",
"21, no.",
"13, pp.",
"15 490–15 504, 2013.",
"P. Y. Chen, C. Argyropoulos, and A. Alù, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE Transactions on Antennas and Propagation, vol.",
"61, pp.",
"1528 – 1537, 2013.",
"D. Correas-Serrano, J. S. Gómez-Díaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene based plasmonic tunable low pass filters in the thz band,” Preprint arXiv:1304.6320v1 [cond-mat.mes-hall], 2013.",
"J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza-Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature, vol.",
"487, pp.",
"77–81, 2012.",
"B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nature Communications, vol.",
"3, p. 780, 2012.",
"D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically-biased graphene at microwave frequencies,” Appl.",
"Phys.",
"Lett., vol.",
"102, p. 191901, 2013.",
"G. W. Hanson, “Dyadic green's functions for an anisotropic non-local model of biased graphene,” IEEE Transactions on Antennas and Propagation, vol.",
"56, no.",
"3, pp.",
"747–757, March 2008.",
"G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene nanowaveguides,” IEEE Transactions on Electromagnetic Compatibility, vol.",
"11, pp.",
"1489–1492, 2013.",
"J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Effect of spatial dispersion on surfaces waves propagating along graphene sheets,” IEEE Transactions on Antennas and Propagation, vol.",
"61, pp.",
"3589–3596, 2013.",
"G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Physical Review B, vol.",
"87, p. 115429, 2013.",
"D. Correas-Serrano, J. S. Gómez-Díaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially dispersive graphene single and parallel plate waveguides: analysis and circuit model,” IEEE Transactions on Microwave Theory and Techniques, vol.",
"61, no.",
"12, pp.",
"4333–4344, December 2013."
]
] | 1403.0130 |
[
[
"An alternative derivation of a new Lanczos-type algorithm for systems of\n linear equations"
],
[
"Abstract Various recurrence relations between formal orthogonal polynomials can be used to derive Lanczos-type algorithms.",
"In this paper, we consider recurrence relation $A_{12}$ for the choice $U_i(x)=P_i(x)$, where $U_i$ is an auxiliary family of polynomials of exact degree $i$.",
"It leads to a Lanczos-type algorithm that shows superior stability when compared to existing Lanczos-type algorithms.",
"The new algorithm is derived and described.",
"It is then computationally compared to the most robust algorithms of this type, namely $A_{12}$, $A_5/B_{10}$ and $A_8/B_{10}$, on the same test problems.",
"Numerical results are included."
],
[
"Introduction",
"The Lanczos algorithm, [1], [2], has been designed to find the eigenvalues of a matrix.",
"However, it has found application in the area of Systems of Linear Equations (SLE's) where it is now well established.",
"It is an iterative process which, in exact arithmetic, finds the exact solution in at most $n$ number of steps [3], where $n$ is the dimension of the problem.",
"Several Lanczos-type algorithms have been designed and among them, the famous conjugate gradient algorithm of Hestenes and Stiefel [4], when the matrix is Hermitian and the bi-conjugate gradient algorithm of Fletcher [5], in the general case.",
"In the last few decades, Lanczos-type algorithms have evolved and different variants have been derived, which can be found in [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].",
"Lanczos-type algorithms are commonly derived using Formal Orthogonal Polynomials (FOP's), [8].",
"The connection between the Lanczos algorithm, [3] and orthogonal polynomials, [27] has been studied extensively in [6], [8], [11], [12], [13], [28], [29], [30], [31]."
],
[
"Notation",
"The notation introduced by Baheux, in [6], [32], for recurrence relations with three terms is adopted here.",
"It puts recurrence relations involving FOP's $P_k(x)$ (the polynomials of degree at most $k$ with regard to the linear functional $c$ ) and/or FOP's $P^{(1)}_k(x)$ (the monic polynomials of degree at most $k$ with regard to linear functional $c^{(1)}$ , [30]) into two groups: $A_i$ and $B_j$ .",
"Although relations $A_i$ , when they exist, rarely lead to Lanczos-type algorithms on their own (the exceptions being $A_4$ , [6], [32], and $A_{12}$ , [23]), relations $B_j$ never lead to such algorithms for obvious reasons.",
"It is the combination of recurrence relations $A_i$ and $B_j$ , denoted as $A_i/B_j$ , when both exist, that lead to Lanczos-type algorithms.",
"In the following we will refer to algorithms by the relation(s) that lead to them.",
"Hence, we will have, potentially, algorithms $A_i$ and algorithms $A_i/B_j$ , for some $i=1,2,\\dots $ and some $j=1,2,\\dots $ .",
"The paper is organized as follows.",
"In the next section, the background to the Lanczos process is presented.",
"Section 3 is on FOP's.",
"Section 4 is on algorithm $A_{12}$ , [23] and the estimation of the coefficients of the recurrence relations $A_{12}$ used to derive it.",
"Section 5 is the estimation of the coefficients of recurrence relation $A_{12}$ , [23], used to derive the new algorithm of the same name $i.e$ $A_{12}(new)$ .",
"Section 6 describes the test problems and reports the numerical results.",
"Section 7 is the conclusion and further work."
],
[
"The Lanczos Process",
"Consider the system of linear equations, $\\textit {A}\\textbf {x}=\\textbf {b},$ where $\\textit {A}$ is $n\\times n$ real matrix, $\\textbf {b}$ and $\\textbf {x}$ are vectors in $\\textit {R}^{n}.$ Choose $\\textbf {x}_{0}$ and $\\textbf {y}$ , two arbitrary vectors in $\\textit {R}^{n}$ , such that $\\textbf {y}\\ne 0$ .",
"Then, Lanczos process [3] consists in generating a sequence of vectors $\\textbf {x}_{k}\\in \\textit {R}^n$ , such that $\\textbf {x}_{k}-\\textbf {x}_{0}\\in \\textit {F}_{k}(\\textit {A}, \\textbf {r}_{0})=\\texttt {span}(\\textbf {r}_{0}, \\textit {A}\\textbf {r}_{0},\\ldots ,\\textit {A}^{k-1}\\textbf {r}_{0}),$ and $\\textbf {r}_{k}=\\textbf {b}-\\textit {A}\\textbf {x}_{k}\\bot \\textit {G}_{k}(\\textit {A}^{T}, \\textbf {y})=\\texttt {span}(\\textbf {y}, \\textit {A}^{T}\\textbf {y},\\ldots ,\\textit {(}{A}^{T})^{k-1}\\textbf {y}),$ where $\\textit {A}^{T}$ is the transpose of matrix $\\textit {A}$ .",
"Equation $(2)$ implies, $\\textbf {x}_{k}-\\textbf {x}_{0}=-\\beta _{1}\\textbf {r}_{0}-\\beta _{2}\\textit {A}\\textbf {r}_{0}-\\dots -\\beta _{k}\\textit {A}^{k-1}\\textbf {r}_{0}.$ Multiplying both sides by $\\textit {A}$ and adding and subtracting $\\textbf {b}$ on the left hand side of (4) gives $\\textbf {r}_{k}=\\textbf {r}_{0}+\\beta _{1}\\textit {A}\\textbf {r}_{0}+\\beta _{2}\\textit {A}^{2}\\textbf {r}_{0}+\\dots +\\beta _{k}\\textit {A}^{k}\\textbf {r}_{0}.$ If we set $P_{k}(x)=1+\\beta _{1}x+\\dots +\\beta _{k}x^{k},$ then we can write from $(5)$ $\\textbf {r}_{k}=P_{k}(\\textit {A})\\textbf {r}_{0}.$ From $(3)$ , since $(\\it {A^T})^{i}\\bf {y}$ and $\\bf {r}_k$ are each in orthogonal subspaces, we can write, $((\\it {A^T})^{i}\\bf {y},\\bf {r}_{k}) =(\\bf {y}, \\it {A}^i\\bf {r}_{k})=(\\bf {y}, \\it {A}^i\\it {P}_{k}(\\it {A})\\bf {r}_{0})=0$ , $\\mbox{ for } i=0,\\dots ,k-1.$ Thus, the coefficients $\\beta _{1}$ ,...,$\\beta _{k}$ form a solution of system of linear equations, $\\beta _{1}(\\textbf {y}, \\textit {A}^{i+1}\\textbf {r}_{0})+\\dots +\\beta _{k}(\\textbf {y}, \\textit {A}^{i+k}\\textbf {r}_{0})=-(\\textbf {y}, \\textit {A}^{i}\\textbf {r}_{0}), \\mbox{ for } i=0,\\dots ,k-1.$ If the determinant of the above system is not zero then its solution exists and allows to obtain $\\textbf {x}_{k}$ and $\\textbf {r}_{k}$ .",
"Obviously, in practice, solving the above system directly for increasing values of $k$ is not feasible; $k$ is the order of the iterate in the solution process.",
"We shall see now how to solve this system for increasing values of $k$ recursively, that is, if polynomials $P_{k}$ can be computed recursively.",
"Such computation is feasible as the polynomials $P_{k}$ form a family of FOP's and will now be explained.",
"In exact arithmetic, $k$ should not exceed $n$ , where $n$ is the dimension of the problem."
],
[
"Formal Orthogonal Polynomials",
"Let $c$ be a linear functional on the space of complex polynomials defined by $c(x^i)=c_{i}$ for $i=0,1,\\dots $ where $c_{i}=({(\\textit {A}^T)}^{i}\\textbf {y},\\textbf {r}_{k})=(\\textbf {y}, \\textit {A}^i\\textbf {r}_{k})$ for $i=0,1,\\dots $ Again, because of (3) above, an orthogonality condition can be written as, $c(x^iP_{k})=0 \\mbox{ for } i=0,\\dots ,k-1.$ This condition shows that $P_{k}$ is the polynomial of degree at most $k$ which is a FOP with respect to the functional $c$ , [28].",
"Given the expression of $P_k(x)$ above, $P_{k}(0)=1, \\forall k$ is a normalization condition for these polynomials; $P_{k}$ exists and is unique if the following Hankel determinant $\\textit {H}^{(1)}_{k}=\\left|\\begin{array}{cccc}c_{1} & c_{2} & \\cdots & c_{k}\\\\c_{2} & c_{3} & \\cdots & c_{k+1}\\\\\\vdots & \\vdots & & \\vdots \\\\c_{k} & c_{k+1} & \\cdots & c_{2k-1}\\\\\\end{array}\\right|$ is not zero, in which case we can write $P_{k}(x)$ as $P_{k}(x)=\\frac{\\left|\\begin{array}{cccc}1 & x & \\cdots & x^k\\\\c_{0} & c_{1} & \\cdots & c_{k}\\\\\\vdots & \\vdots & & \\vdots \\\\c_{k-1} & c_{k} & \\cdots & c_{2k-1}\\\\\\end{array}\\right|}{\\left|\\begin{array}{cccc}c_{1} & \\cdots & c_{k}\\\\\\vdots & & \\vdots \\\\c_{k} & \\cdots & c_{2k-1}\\\\\\end{array}\\right|},$ where the denominator of this polynomial is $\\textit {H}^{(1)}_{k}$ , the determinant of the system (7).",
"We assume that $\\forall $ $k$ , $\\textit {H}^{(1)}_{k}\\ne 0$ and therefore all the polynomials $P_{k}$ exist for all $k$ .",
"If for some $k$ , $\\textit {H}^{(1)}_{k}=0$ , then $P_{k}$ does not exist and breakdown occurs in the algorithm, [8], [11], [12], [13], [29].",
"A Lanczos-type method consists in computing $P_{k}$ recursively, then $\\textbf {r}_{k}$ and finally $\\textbf {x}_{k}$ such that $\\textbf {r}_{k}=\\textbf {b}-\\textit {A}\\textbf {x}_{k}$ , without inverting $A$ .",
"This gives the solution of the system $(1)$ in at most $n$ steps, where $n$ is the dimension of the SLE.",
"For more details, see [8], [10].",
"FOPs can be put together into recurrence relations.",
"Such relations give rise to various procedures for the recursive computation of $P_{k}$ and hence we get different Lanczos-type algorithms for computing $\\textbf {r}_{k}$ and, therefore, $\\textbf {x}_{k}$ .",
"These algorithms have been studied in [6], [8], [10], [11], [12], [13], [29], [32].",
"They differ by the recurrence relationships used to express the polynomials $P_{k}, k=2,3,...$ ."
],
[
"Recurrence relation $A_{12} based algorithm$",
"Algorithms $A_5/B_{10}$ , $A_8/B_{10}$ and $A_{12}$ are the most robust algorithms as found in [6], [32], [23], [25], [26], on the same problems considered here.",
"We, therefore compare our results with these algorithms.",
"Since the algorithm we introduce here is also based on the recurrence relation $A_{12}$ [23], [24], according to the notation of [6], it is really a modification of algorithm $A_{12}$ that can be found in [23].",
"Indeed, $A_{12}$ is derived using the auxiliary polynomial $U_i(x)=x^i$ , of exact degree $i$ , while here we derive a new algorithm $A_{12}$ but for $U_i(x)=P_i(x)$ .",
"For completeness, we recall algorithm $A_{12}$ here but leave out its derivation which can be found by the interested reader in [23]."
],
[
"Algorithm $A_{12}$",
"Algorithm $A_{12}$ [23] can be described as follows.",
"[H] : Lanczos-type algorithm $A_{12}$ Choose $\\textbf {x}_{0}$ and $\\textbf {y}$ such that $\\textbf {y}\\ne 0$ , Set $r_{0}=b-Ax_{0}$ , $y_{0}=y$ , $p=Ar_{0}$ , $p_{1}=Ap$ , $c_{0}=(y, r_{0})$ , $c_{1}=(y, p)$ , $c_{2}=(y, p_{1})$ , $c_{3}=(y, Ap_{1})$ , $\\delta =c_{1}c_{3}-c_{2}^2$ , $\\alpha =\\frac{c_{0}c_{3}-c_{1}c_{2}}{\\delta }$ , $\\beta =\\frac{c_{0}c_{2}-c_{1}^2}{\\delta }$ , $r_{1}=r_{0}-\\frac{c_{0}}{c_{1}}p$ , $x_{1}=x_{0}+\\frac{c_{0}}{c_{1}}r_{0}$ , $r_{2}=r_{0}-\\alpha p+\\beta p_{1}$ , $x_{2}=x_{0}+\\alpha r_{0}-\\beta p$ , $y_{1}=A^{T}y_{0}$ , $y_{2}=A^{T}y_{1}$ , $y_{3}=A^{T}y_{2}$ .",
"k = 3, 4,..., $y_{k+1}=A^{T}y_{k}$ , $q_{1}=Ar_{k-1}$ , $q_{2}=Aq_{1}$ , $q_{3}=Ar_{k-2}$ , $a_{11}=(y_{k-2}, r_{k-2})$ , $a_{13}=(y_{k-3}, r_{k-3})$ , $a_{21}=(y_{k-1}, r_{k-2})$ , $a_{22}=a_{11}$ ,$a_{23}=(y_{k-2}, r_{k-3})$ , $a_{31}=(y_{k}, r_{k-2})$ , $a_{32}=a_{21}$ , $a_{33}=(y_{k-1}, r_{k-3})$ , $s=(y_{k+1}, r_{k-2})$ , $t=(y_{k}, r_{k-3})$ , $F_{k}=-\\frac{a_{11}}{a_{13}}$ ,$b_{1}=-a_{21}-a_{23}F_{k}$ , $b_{2}=-a_{31}-a_{33}F_{k}$ , $b_{3}=-s-tF_{k}$ ,$\\Delta _{k}=a_{11}(a_{22}a_{33}-a_{32}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})$ , $B_{k}=\\frac{b_{1}(a_{22}a_{33}-a_{32}a_{23})+a_{13}(b_{2}a_{32}-b_{3}a_{22})}{\\Delta _{k}}$ , $G_{k}=\\frac{b_{1}-a_{11}B_{k}}{a_{13}}$ , $C_{k}=\\frac{b_{2}-a_{21}B_{k}-a_{23}G_{k}}{a_{22}}$ , $A_{k}=\\frac{1}{C_{k}+G_{k}}$ , $r_{k}=A_{k}\\lbrace q_{2}+B_{k}q_{1}+C_{k}r_{k-2}+F_{k}q_{3}+G_{k}r_{k-3}\\rbrace $ , $x_{k}=A_{k}\\lbrace C_{k}x_{k-2}+G_{k}x_{k-3}-(q_{1}+B_{k}r_{k-2}+F_{k}r_{k-3})\\rbrace $ ; If $||r_{k}|| \\le \\epsilon $ , then $x=x_{k}$ , Stop."
],
[
"The new algorithm $A_{12}$ and its derivation",
"As said above, in [23], relation $A_{12}$ is derived using the auxiliary polynomial $U_i(x)=x^i$ , of exact degree $i$ .",
"Here, we discuss the same relation, but for $U_i(x)=P_i(x)$ .",
"Coefficients are estimated for the new case.",
"The Lanczos-type algorithm based on $A_{12}$ for the new choice of $U_i$ , is called $A_{12}(new)$ .",
"This new algorithm is described below.",
"Before deriving and discussing it, we recall the definition of an orthogonal polynomials sequence, [23].",
"Definition 1.",
"A sequence ${P_m}$ is called an orthogonal polynomial sequence, [33] with respect to the linear functional $c$ if, for all nonnegative integers $n$ and $m$ , (i) $P_m$ is a polynomial of degree $m$ , (ii) $c(x^nP_m) = 0$ , for $m \\ne n$ , (iii) $c(x^mP_m)$ $\\ne 0.$"
],
[
"Relation $A_{12}$ for the choice {{formula:e23395be-f66c-4504-b019-f44b2fbe6459}} .",
"Consider the following recurrence relationship, [23], $P_k(x)=A_k\\lbrace (x^2+B_kx+C_k)P_{k-2}(x)+(D_kx^3+E_kx^2+F_kx+G_k)P_{k-3}(x)\\rbrace \\\\$ where $P_k$ , $P_{k-2}$ , and $P_{k-3}$ are polynomials of degree $k$ , $k-2$ , and $k-3$ , respectively.",
"$A_k$ , $B_k$ , $C_k$ , $D_k$ , $E_k$ , $F_k$ and $G_k$ are the coefficients to be determined using the normality and the orthogonality conditions given in Section 3.",
"Let, again, $c$ be a linear functional defined by $c(x^i)=c_i$ .",
"The orthogonality condition gives $c(U_iP_k)=0, i=0,1\\cdots , k-1.$ For $x=0$ , and applying the normality condition, (REF ) becomes $1=A_k\\lbrace C_k+G_k\\rbrace .$ Now multiply (REF ) by $U_i$ .",
"Applying `c' on both sides and using the orthogonality condition, we get $c(x^2U_iP_{k-2})+B_kc(xU_iP_{k-2})+C_kc(U_iP_{k-2})+D_kc(x^3U_iP_{k-3})\\nonumber \\\\+E_kc(x^2U_iP_{k-3})+F_kc(xU_iP_{k-3})+G_kc(U_iP_{k-3})=0.$ The orthogonality condition holds for $i=0,1,2,\\cdots ,k-7.$ For $i=k-6$ , equation $(\\ref {5})$ gives $D_kc(x^3U_{k-6}P_{k-3})=0$ , which implies that $D_k=0$ , since $c(x^3U_{k-6}P_{k-3}) \\ne 0$ .",
"For $i=k-5$ , $(\\ref {5})$ becomes $E_kc(x^2U_{k-5}P_{k-3})=0$ .",
"Since $c(x^2 U_{k-5}P_{k-3})\\ne 0$ , $E_k=0$ .",
"For $i=k-4$ , we get $c(x^2U_{k-4}P_{k-2})+F_kc(xU_{k-4}P_{k-3})=0$ , which gives $F_k=-\\frac{c(x^2U_{k-4}P_{k-2})}{c(xU_{k-4}P_{k-3})}.$ For $i=k-3$ , $i=k-2$ and $i=k-1$ equation $(\\ref {5})$ can be respectively written as, $B_kc(xU_{k-3}P_{k-2})+G_kc(U_{k-3}P_{k-3})=-c(x^2U_{k-3}P_{k-2})-F_kc(xU_{k-3}P_{k-3}),$ $B_kc(xU_{k-2}P_{k-2})+C_kc(U_{k-2}P_{k-2})+G_kc(U_{k-2}P_{k-3})=\\nonumber \\\\-c(x^2U_{k-2}P_{k-2})-F_kc(xU_{k-2}P_{k-3}),$ $B_kc(xU_{k-1}P_{k-2})+C_kc(U_{k-1}P_{k-2})+G_kc(U_{k-1}P_{k-3})=\\nonumber \\\\-c(x^2U_{k-1}P_{k-2})-F_kc(xU_{k-1}P_{k-3}).$ Now for simplicity let us denote the right sides of equations $(\\ref {11})$ , $(\\ref {12})$ and $(\\ref {13})$ by $b_1$ ,$b_2$ and $b_3$ respectively then we get the following system of equations, $B_kc(xU_{k-3}P_{k-2})+G_kc(U_{k-3}P_{k-3})=b_1,$ $B_kc(xU_{k-2}P_{k-2})+C_kc(U_{k-2}P_{k-2})+G_kc(U_{k-2}P_{k-3})=b_2,$ $B_kc(xU_{k-1}P_{k-2})+C_kc(U_{k-1}P_{k-2})+G_kc(U_{k-1}P_{k-3})=b_3.$ If $\\Delta _k$ denotes the determinant of the coefficient matrix of the above system of equations then, $\\Delta _k=c(xU_{k-3}P_{k-2})\\lbrace c(U_{k-2}P_{k-2})c(U_{k-1}P_{k-3})-c(U_{k-2}P_{k-3})c(U_{k-1}P_{k-2})\\rbrace +\\nonumber \\\\c(U_{k-3}P_{k-3})\\lbrace c(xU_{k-2}P_{k-2})c(U_{k-1}P_{k-2})-c(U_{k-2}P_{k-2})c(xU_{k-1}P_{k-2})\\rbrace .$ If $\\Delta _k\\ne 0$ then, $B_k=\\frac{b_1}{\\Delta _k}\\lbrace c(U_{k-2}P_{k-2})c(U_{k-1}P_{k-3})-c(U_{k-2}P_{k-3})c(U_{k-1}P_{k-2})\\rbrace \\\\\\\\ +\\frac{c(U_{k-3}P_{k-3})\\lbrace b_2c(U_{k-1}P_{k-2})-b_3c(U_{k-2}P_{k-2})\\rbrace }{\\Delta _k},$ $G_k=\\frac{b_1-c(xU_{k-3}P_{k-2})B_k}{c(U_{k-3}P_{k-3})},$ $C_k=\\frac{b_2-c(xU_{k-2}P_{k-2})B_k-c(U_{k-2}P_{k-3})G_k}{c(U_{k-2}P_{k-2})}.$ With the above new estimated coefficients, the expression of polynomials $P_k(x)$ can be written as, $P_k(x)=A_k\\lbrace (x^2+B_kx+C_k)P_{k-2}(x)+(F_kx+G_k)P_{k-3}(x)\\rbrace .$ Now, for $U_i(x)=P_k(x)$ , and from equation (REF ), $F_k$ becomes $F_k=-\\frac{c(x^2P_{k-4}P_{k-2})}{c(xP_{k-4}P_{k-3})}.$ Similarly, from equation (REF ) $\\Delta _k$ becomes, $\\Delta _k=c(xP_{k-3}P_{k-2})\\lbrace c(P_{k-2}^2)c(P_{k-1}P_{k-3})-c(P_{k-2}P_{k-3})c(P_{k-1}P_{k-2})\\rbrace +\\\\c(P_{k-3}^2)\\lbrace c(xP_{k-2}^2)c(P_{k-1}P_{k-2})-c(P_{k-2}^2)c(xP_{k-1}P_{k-2})\\rbrace .$ Using (definition 1), [23], $\\Delta _k$ simplifies to $\\Delta _k=-c(P_{k-3}^2)c(P_{k-2}^2)c(xP_{k-1}P_{k-2}).$ Using again definition 1 and $U_i(x)=P_k(x)$ , the rest of the coefficients can be determined as follows.",
"Let $b_1=-c(x^2P_{k-3}P_{k-2})-F_kc(xP_{k-3}^2),$ $b_2=-c(x^2P_{k-2}^2)-F_kc(xP_{k-2}P_{k-3}),$ $b_3=-c(x^2P_{k-1}P_{k-2})-F_kc(xP_{k-1}P_{k-3}),$ then $B_k = \\frac{1}{\\Delta _k}\\lbrace b_1\\lbrace c(P_{k-2}P_{k-2})c(P_{k-1}P_{k-3})-c(P_{k-2}P_{k-3})c(P_{k-1}P_{k-2})\\rbrace \\\\+c(P_{k-3}P_{k-3})\\lbrace b_2c(P_{k-1}P_{k-2})-b_3c(P_{k-2}P_{k-2})\\rbrace \\rbrace ,$ or, $B_k = -\\frac{b_3c(P_{k-3}^2)c(P_{k-2}^2)}{\\Delta _k} = -\\frac{b_3}{c(xP_{k-1}P_{k-2})},$ $G_k = \\frac{b_1-c(xP_{k-3}P_{k-2})B_k}{c(P_{k-3}^2)},$ $C_k=\\frac{b_2-c(xP_{k-2}P_{k-2})B_k-c(P_{k-2}P_{k-3})G_k}{c(P_{k-2}P_{k-2})}= \\frac{b_2-c(xP_{k-2}^2)B_k}{c(P_{k-2}^2)},$ and $A_k=\\frac{1}{C_k+G_k}.$ As in [23], we can write, $\\textbf {r}_k=A_k\\lbrace \\textit {A}^2\\textbf {r}_{k-2}+B_k \\textit {A}\\textbf {r}_{k-2}+C_k \\textbf {r}_{k-2}+F_k \\textit {A}\\textbf {r}_{k-3}+G_k \\textbf {r}_{k-3}\\rbrace ,$ $\\textbf {x}_k=A_k\\lbrace C_k\\textbf {x}_{k-2}+G_k \\textbf {x}_{k-3}-(\\textit {A}\\textbf {r}_{k-2}+B_k \\textbf {r}_{k-2}+F_k\\textbf {r}_{k-3})\\rbrace .$ As we know from [6], [32], ${\\left\\lbrace \\begin{array}{ll} setting U_k(x)=P_k(x) and \\textbf {z}_k=P_k(A^T)\\textbf {y}, we get \\\\c(U_kP_k)=(y,U_k(A)P_(A)\\textbf {r}_0)=(U_k(A^T)\\textbf {y},P_k(A)\\textbf {r}_0)=(\\textbf {z}_k,\\textbf {r}_k).\\end{array}\\right.",
"}$ So, from relation (REF ), after replacing $\\textbf {x}$ by $A^T$ , multiplying by $\\textbf {y}$ on both sides and using (REF ) we can write, $ \\textbf {z}_k=A_k\\lbrace (\\textit {A}^T)^2\\textbf {z}_{k-2}+B_k \\textit {A}^T\\textbf {z}_{k-2}+C_k \\textbf {z}_{k-2}+F_k \\textit {A}^T\\textbf {z}_{k-3}+G_k \\textbf {z}_{k-3}\\rbrace .$ Similarly using (REF ) all coefficients become, $F_k=-\\frac{c(x^2P_{k-4}P_{k-2})}{c(xP_{k-4}P_{k-3})}$ =$-\\frac{(\\textit {A}^T \\textbf {z}_{k-2},\\textit {A}\\textbf {r}_{k-4})}{(\\textbf {z}_{k-3},\\textit {A}\\textbf {r}_{k-4})}$ , $\\Delta _k=-c(P_{k-3}^2) c(P_{k-2}^2)c(xP_{k-1}P_{k-2})$ =$-(\\textbf {z}_{k-3},\\textbf {r}_{k-3})(\\textbf {z}_{k-2},\\textbf {r}_{k-2})(\\textbf {z}_{k-1},\\textit {A}\\textbf {r}_{k-2})$ .",
"$b_1=-(A^T \\textbf {z}_{k-3},A\\textbf {r}_{k-2})- F_k(\\textbf {z}_{k-3},A \\textbf {r}_{k-3}),$ $b_2=-(A^T \\textbf {z}_{k-2},A\\textbf {r}_{k-2})- F_k(\\textbf {z}_{k-2},A \\textbf {r}_{k-3}),$ $b_3=-(A^T\\textbf {z}_{k-1},A\\textbf {r}_{k-2})- F_k(\\textbf {z}_{k-1},A \\textbf {r}_{k-3}),$ $B_k$ = $\\frac{b_3}{c(xP_{k-1}P_{k-2})}$ = $\\frac{b_3}{(\\textbf {z}_{k-1},A\\textbf {r}_{k-2})}$ , $G_k$ =$\\frac{b_1-c(xP_{k-3}P_{k-2})B_k}{c(P_{k-3}^2)}$ =$\\frac{b_1-(\\textbf {z}_{k-3},A\\textbf {r}_{k-2})B_k}{(\\textbf {z}_{k-3},\\textbf {r}_{k-3})}$ , $C_k$ =$\\frac{b_2-c(xP_{k-2}^2)B_k}{c(P_{k-2}^2)}$ =$\\frac{b_2-(\\textbf {z}_{k-2},A\\textbf {r}_{k-2})B_k}{(\\textbf {z}_{k-2},\\textbf {r}_{k-2})}$ , $A_k$ = $\\frac{1}{C_k+G_k}$ .",
"All previous formulae are valid for $k\\ge 4$ .",
"So we need $\\textbf {r}_1$ , $\\textbf {r}_2$ , $\\textbf {r}_3$ and $\\textbf {z}_1$ , $\\textbf {z}_2$ , $\\textbf {z}_3$ to calculate $\\textbf {r}_k$ and $\\textbf {z}_k$ recursively.",
"$\\textbf {r}_1$ , $\\textbf {r}_2$ and $\\textbf {z}_1$ , $\\textbf {z}_2$ are found differently in [23], while $\\textbf {r}_3$ and $\\textbf {z}_3$ can be determined in a similar way giving, $\\textbf {r}_3=\\textbf {r}_0-\\frac{\\acute{\\alpha }}{\\Delta } \\textbf {p}+\\frac{\\acute{\\beta }}{\\Delta }\\textbf {p}_1-\\frac{\\acute{\\gamma }}{\\Delta } \\textbf {p}_2$ , $\\textbf {z}_3=\\textbf {z}_0-\\frac{\\acute{\\alpha }}{\\Delta } \\textbf {y}_1+\\frac{\\acute{\\beta }}{\\Delta }\\textbf {y}_2-\\frac{\\acute{\\gamma }}{\\Delta } \\textbf {y}_3$ .",
"Using $\\textbf {r}_k=\\textbf {b}-\\emph {A}\\textbf {x}_k$ , we get from $\\textbf {r}_3$ , $\\textbf {x}_3=\\textbf {x}_0+\\frac{\\acute{\\alpha }}{\\Delta } \\textbf {r}_0-\\frac{\\acute{\\beta }}{\\Delta }\\textbf {p}+\\frac{\\acute{\\gamma }}{\\Delta } \\textbf {p}_1$ , where $\\Delta =c_1(c_3c_5-c_4^{2})-c_2(c_2c_5-c_3c_4)+c_3(c_2c_4-c_3^{2})$ , $\\acute{\\alpha }=c_0(c_3c_5-c_4^{2})-c_2(c_1c_5-c_2c_4)+c_3(c_1c_4-c_3c_2)$ , $\\acute{\\beta }=c_0(c_2c_5-c_4c_3)-c_1(c_1c_5-c_2c_4)+c_3(c_1c_3-c_2^{2})$ , $\\acute{\\gamma }=c_0(c_2c_4-c_3^{2})-c_1(c_1c_4-c_2c_3)+c_2(c_1c_3-c_2^{2})$ .",
"Note that parameters p,p$_1$ ,p$_2$ ,y$_1$ ,y$_2$ ,y$_3$, $\\Delta , \\alpha , \\beta ,$ and $\\gamma $ are temporary and defined in the algorithm below."
],
[
"Algorithm $A_{12}(new)$",
"We can now describe the new variant of algorithm $A_{12}(new)$ as follows.",
"[H] : Lanczos-type Algorithm $A_{12}(new)$ .",
"Choose $\\textbf {x}_0$ and $\\textbf {y}$ such that $\\textbf {y}\\ne 0$ .",
"Set $\\textbf {r}_0=\\textbf {b}-A\\textbf {x}_0$ , $\\textbf {z}_0=\\textbf {y}$ , $\\textbf {p}=A \\textbf {r}_0$ , $\\textbf {p}_1=A \\textbf {p}$ , $\\textbf {p}_2=A \\textbf {p}_1$ , $\\textbf {p}_3=A\\textbf {p}_2$ , $\\textbf {p}_4=A \\textbf {p}_3$ , $c_0=(\\textbf {y},\\textbf {r}_0)$ , $c_1=(\\textbf {y},\\textbf {p})$ , $c_2=(\\textbf {y},\\textbf {p}_1)$ , $c_3=(\\textbf {y},\\textbf {p}_2)$ , $ c_4=(\\textbf {y},\\textbf {p}_3)$ , $c_5=(\\textbf {y},\\textbf {p}_4)$ , $\\delta =c_1c_3-c_2^2$ , $\\alpha =\\frac{c_0c_3-c_1c_2}{\\delta }$ , $\\beta =\\frac{c_0c_2-c_1^2}{\\delta }$ , $\\textbf {r}_1=\\textbf {r}_0-(\\frac{c_0}{c_1})\\textbf {p}$ , $\\textbf {x}_1=\\textbf {x}_0+(\\frac{c_0}{c_1})\\textbf {r}_0$ , $\\textbf {r}_2=\\textbf {r}_0-\\alpha \\textbf {p}+\\beta \\textbf {p}_1$ , $\\textbf {x}_2=\\textbf {x}_0+\\alpha \\textbf {r}_0-\\beta \\textbf {p}$ , $\\textbf {y}_1=A^T\\textbf {y}$ , $\\textbf {y}_2=A^T \\textbf {y}_1$ , $\\textbf {y}_3=A^T \\textbf {y}_2$ , $\\textbf {z}_1=\\textbf {z}_0-(\\frac{c_0}{c_1})\\textbf {y}_1$ , $\\textbf {z}_2=\\textbf {z}_0-\\alpha \\textbf {y}_1+\\beta \\textbf {y}_2$ , $\\Delta =c_1(c_3c_5-c_4^{2})-c_2(c_2c_5-c_3c_4)+c_3(c_2c_4-c_3^{2})$ , $\\acute{\\alpha }=c_0(c_3c_5-c_4^{2})-c_2(c_1c_5-c_2c_4)+c_3(c_1c_4-c_3c_2)$ , $\\acute{\\beta }=c_0(c_2c_5-c_4c_3)-c_1(c_1c_5-c_2c_4)+c_3(c_1c_3-c_2^{2})$ , $\\acute{\\gamma }=c_0(c_2c_4-c_3^{2})-c_1(c_1c_4-c_2c_3)+c_2(c_1c_3-c_2^{2})$ , $\\textbf {r}_3=\\textbf {r}_0-\\frac{\\acute{\\alpha }}{\\Delta } \\textbf {p}+\\frac{\\acute{\\beta }}{\\Delta }\\textbf {p}_1-\\frac{\\acute{\\gamma }}{\\Delta } \\textbf {p}_2$ , $\\textbf {z}_3=\\textbf {z}_0-\\frac{\\acute{\\alpha }}{\\Delta } \\textbf {y}_1+\\frac{\\acute{\\beta }}{\\Delta }\\textbf {y}_2-\\frac{\\acute{\\gamma }}{\\Delta } \\textbf {y}_3$ , $\\textbf {x}_3=\\textbf {x}_0+\\frac{\\acute{\\alpha }}{\\Delta } \\textbf {r}_0-\\frac{\\acute{\\beta }}{\\Delta }\\textbf {p}+\\frac{\\acute{\\gamma }}{\\Delta } \\textbf {p}_1$ .",
"k = 4,5..., $ q_1=A \\textbf {r}_{k-2}$ , $q_2=A \\textbf {q}_1$ , $q_3=A \\textbf {r}_{k-3}$ , $ \\textbf {s}_1=A^T \\textbf {z}_{k-2}$ , $\\textbf {s}_2=A^T \\textbf {s}_1$ , $\\textbf {s}_3=A^T \\textbf {z}_{k-3}$ , $\\Delta _k=-(\\textbf {z}_{k-3},\\textbf {r}_{k-3})(\\textbf {z}_{k-2},\\textbf {r}_{k-2})(\\textbf {z}_{k-1},A\\textbf {r}_{k-2})$ , $F_k=-\\frac{(A^T \\textbf {z}_{k-2},A\\textbf {r}_{k-4})}{(\\textbf {z}_{k-3},A\\textbf {r}_{k-4})}$ , $b_{1}=-(A^T \\textbf {z}_{k-3},A\\textbf {r}_{k-2})- F_k(\\textbf {z}_{k-3},A \\textbf {r}_{k-3})$ , $b_2=-(A^T \\textbf {z}_{k-2},A\\textbf {r}_{k-2})- F_k(\\textbf {z}_{k-2},A \\textbf {r}_{k-3}),$ $b_3=-(A^T \\textbf {z}_{k-1},A\\textbf {r}_{k-2})- F_k(\\textbf {z}_{k-1},A \\textbf {r}_{k-3})$ , $B_{k}=\\frac{b_3}{(\\textbf {z}_{k-1},A\\textbf {r}_{k-2})}$ , $G_k=\\frac{b_1-(\\textbf {z}_{k-3},A\\textbf {r}_{k-2})B_k}{(\\textbf {z}_{k-3},\\textbf {r}_{k-3})}$ , $C_k=\\frac{b_2-(\\textbf {z}_{k-2},A\\textbf {r}_{k-2})B_k}{(\\textbf {z}_{k-2},\\textbf {r}_{k-2})}$ , $A_k=\\frac{1}{C_k+G_k}$ , $\\textbf {r}_k=A_k\\lbrace \\textbf {q}_2+B_k \\textbf {q}_1+C_k \\textbf {r}_{k-2}+F_k \\textbf {q}_{3}+G_k \\textbf {r}_{k-3}\\rbrace $ , $x_k=A_k\\lbrace C_k\\textbf {x}_{k-2}+G_k \\textbf {x}_{k-3}-(\\textbf {q}_1+B_k \\textbf {r}_{k-2}+F_k\\textbf {r}_{k-3})\\rbrace $ , $\\textbf {z}_k=A_k\\lbrace \\textbf {s}_2+B_k \\textbf {s}_1+C_k \\textbf {z}+F_k \\textbf {s}_{3}+G_k \\textbf {z}_{k-3}\\rbrace $ .",
"If $||r_{k}|| \\le \\epsilon $ , then $x=x_{k}$ , Stop."
],
[
"Numerical Tests",
"$A_{12}(new)$ has been tested against $A_{12}$ , $A_5/B_{10}$ and $A_8/B_{10}$ , the best Lanczos-type algorithms according to [6], [23], [32].",
"The test problems arise in the 5-point discretisation of the operator $\\frac{-\\partial ^{2}}{\\partial x^2}-\\frac{\\partial ^{2}}{\\partial y^2}+\\gamma \\frac{\\partial }{\\partial x}$ on a rectangular region [32].",
"Comparative results on instances of the following problem ranging from dimension 10 to 100 for parameter $\\delta $ taking values $0.0$ and for the tolerance $eps=1.0e-05$ , are recorded in Table 1.",
"$A=\\left(\\begin{array}{ccccccc}B & -I & \\cdots & \\cdots & 0\\\\-I & B & -I & & \\vdots \\\\\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\\vdots & & -I & B & -I\\\\0 & \\cdots & \\cdots & -I & B\\\\\\end{array}\\right),$ with $B=\\left(\\begin{array}{ccccccc}4 & \\alpha & \\cdots & \\cdots & 0\\\\\\beta & 4 & \\alpha & & \\vdots \\\\\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\\vdots & & \\beta & 4 & \\alpha \\\\0 & \\cdots & & \\beta & 4\\\\\\end{array}\\right),$ and $\\alpha =-1+\\delta $ , $\\beta =-1-\\delta $ .",
"The right hand side $\\textbf {b}$ is taken to be $\\textbf {b}=\\textit {A}\\textbf {X}$ , where $\\textbf {X}=(1, 1,\\dots , 1)^{T}$ , is the solution of the system.",
"The dimension of $\\textit {B}$ is 10.",
"Table: Experimental results for problems when δ=0\\delta =0Table 1 records the computational results obtained with algorithms $A_{12}$ (new), $A_{12}$ , $A_5/B_{10}$ and $A_8/B_{10}$ .",
"Clearly, $A_{12}$ (new) is an improvement on $A_{12}$ on both robustness/stability and efficiency accounts.",
"Compared to the well established $A_5/B_{10}$ and $A_8/B_{10}$ , it is definitely more robust/stable; indeed, all problems have been solved to the required accuracy by $A_{12}$ (new), and the other two algorithms failed to do so in one case as evidenced by the \"NaN\" outputs which point to breakdown or lack of robustness and stability, on the problem of dimension n=90.",
"On efficiency, however, as expected, algorithms $A_5/B_{10}$ and $A_8/B_{10}$ are faster since they rely on recurrence relations involving lower order FOP's requiring few coefficients to estimate; unlike $A_{12}$ and $A_{12}$ (new)."
],
[
"Conclusion ",
"In this paper we have shown that, if the recurrence relation $A_{12}$ [23], is determined for the choice of $U_i(x)=P_i(x)$ , other than $x^i$ which is discussed in [23], then a more robust algorithm $A_{12}(new)$ can be derived.",
"The numerical performance of this algorithm compares well to that of three existing Lanczos-type algorithms, which were found to be the best among a number of such algorithms, [6], [23], [32], on the same set of problems as considered here.",
"Another achievement of $A_{12}(new)$ is that it can solve the above problem when its dimension is up to 500, while the rest of algorithms give results for problems with dimensions less or equal to 100."
]
] | 1403.0326 |
[
[
"A tree-valued Markov processes associated with an admissible family of\n branching mechanisms"
],
[
"Abstract By studying an admissible family of branching mechanisms introduced in Li (2014), we obtain a pruning procedure on L\\'evy trees.",
"Then we could construct a decreasing L\\'evy-CRT-valued process $\\{{\\mathcal T}_t\\}$ by pruning L\\'evy trees and an analogous process $\\{{\\mathcal T}^*_t\\}$ by pruning a critical L\\'evy tree conditioned to be infinite.",
"Under a regular condition on the admissible family of branching mechanisms, we show that the law of $\\{{\\mathcal T}_t\\}$ at the ascension time can be represented by $\\{{\\mathcal T}^*_t\\}$.",
"The results generalize those studied in Abraham and Delmas (2012)."
],
[
"Introduction",
"By pruning Lévy trees, Abraham and Delmas in [1] constructed and studied decreasing continuum-tree-valued Markov processes which correspond to a family of branching mechanisms obtained by shifting a given branching mechanism.",
"More precisely, let $\\psi $ be a branching mechanism defined by $\\psi (\\lambda )=b\\lambda +c\\lambda ^2+\\int _{(0,\\infty )}\\left(\\mathop {\\mathrm {e}^{ -\\lambda z}}-1+\\lambda z\\right)m(dz),\\quad \\lambda \\ge 0,$ where $b\\in {\\mathbb {R}}$ , $c\\ge 0$ and $m$ is a $\\sigma $ -finite measure on $(0,+\\infty )$ such that $\\int _0^{\\infty }(z\\wedge z^2)m(dz)<+\\infty $ .",
"Define $\\psi ^{\\theta }(\\lambda )=\\psi (\\theta +\\lambda )-\\psi (\\theta )$ .",
"Denote by $\\Theta ^{\\psi }$ the set of $\\theta $ such that $\\int _1^{\\infty }ze^{-\\theta z}m(dz)<\\infty $ .",
"The family of branching mechanisms $\\lbrace \\psi ^{\\theta }: \\theta \\in \\Theta ^{\\psi }\\rbrace $ was considered in [1] .",
"Li in [20] introduced the admissible family of branching mechanisms.",
"Roughly, the model is described as follows: Given a time interval ${\\mathfrak {T}}\\subset {\\mathbb {R}}$ , let $(\\theta ,\\lambda )\\mapsto \\zeta _{\\theta }(\\lambda )$ be a continuous function on ${\\mathfrak {T}}\\times [0,\\infty )$ with the representation $\\zeta _{\\theta }(\\lambda )=\\beta _{\\theta }(\\lambda )+\\int _{(0,\\infty )}(1-e^{-z\\lambda })n_{\\theta }(dz),\\quad \\theta \\in {\\mathfrak {T}},\\,\\lambda \\ge 0,$ where $\\beta _{\\theta }\\ge 0$ and $(1\\wedge z)n_{\\theta }(dz)$ is a finite kernel from ${\\mathfrak {T}}$ to $(0,\\infty )$ .",
"Then $\\lbrace \\psi _{\\theta }: \\theta \\in {\\mathfrak {T}}\\rbrace $ is called an admissible family if $\\psi _{q}(\\lambda )=\\psi _t({\\lambda })+\\int _{t}^{q}\\zeta _{\\theta }(\\lambda )d\\theta ,\\quad q\\ge t\\in {\\mathfrak {T}},\\, \\lambda \\ge 0.$ Then it is easy to see that $\\lbrace \\psi ^{\\theta }: \\theta \\in \\Theta ^{\\psi }\\rbrace $ considered in [1] is an admissible family with $\\zeta _{\\theta }(\\lambda )=2c\\lambda +\\int _{(0,\\infty )}(1-e^{-z\\lambda } )e^{-z\\theta }zm(dz).$ By using the techniques of stochastic equations and measure-valued processes, Li [20] studied a class of increasing path-valued Markov processes which correspond to the admissible family.",
"Those path-valued processes could be regarded as counterparts of the tree-valued processes constructed in [1] (However, to the best of our knowledge, no link is actually pointed out between tree-valued processes and path-valued branching processes).",
"It is natural to ask whether there exists a continuum-tree-valued process corresponding to a given admissible family by pruning Lévy trees.",
"The second motivation of our work is the study of so-called ascension process and its representation.",
"The pioneer works were given by Aldous and Pitman in [9] where, according to the marks on edges, they constructed a tree-valued Markov process $\\lbrace {\\cal G}(u)\\rbrace $ by pruning Galton-Watson trees and an analogous process $\\lbrace {\\cal G}^*(u)\\rbrace $ by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite.",
"It was shown in [9] that the process $\\lbrace {\\cal G}(u)\\rbrace $ run until its ascension time has a representation in terms of $\\lbrace {\\cal G}^*(u)\\rbrace $ in the special case of Poisson offspring distributions.",
"By using the pruning procedure defined in [6] and exploration processes introduced in [17], Abraham and Delmas in [1] extended the above results to Lévy trees, where a decreasing Lévy-tree-valued process $\\lbrace {\\theta }: \\theta \\in \\Theta ^{\\psi }\\rbrace $ was constructed such that ${\\theta }$ is a $\\psi ^{\\theta }$ -Lévy tree.",
"They also showed that $\\lbrace {\\theta }\\rbrace $ run until its ascension time can be represented in terms of another tree-valued process obtained by applying the same pruning procedure to a Lévy tree conditioned on non-extinction.",
"Similar results can be found in [2] for Galton-Watson trees where the trees are pruned according to the marks on nodes.",
"Similar cases for sub-trees of Lévy trees were also studied in [3].",
"Motivated by the above mentioned works, in this paper, we shall show that given an admissible family of branching mechanisms $\\lbrace \\psi _t: t\\in {\\mathfrak {T}}\\rbrace $ , one can define a pruning procedure on Lévy trees which will lead to a deceasing tree-valued Markov process $\\lbrace t: t\\in {\\mathfrak {T}}\\rbrace $ such that $t$ is a $\\psi _t$ -Lévy tree; see Theorem in Section .",
"Furthermore, under a regular condition on the admissible family, we also prove that the law of the tree-valued process at its ascension time can be represented in terms of another tree-valued process obtained by pruning a critical Lévy tree conditioned to be infinite; see Theorem and Corollary in this paper.",
"Since $\\lbrace \\psi ^{\\theta }: \\theta \\in \\Theta ^{\\psi }\\rbrace $ considered in [1] is a special case of admissible family, our results generalize those in [1].",
"However, we use the framework of real trees but not exploration processes.",
"Let us mention that the study of theory of continuum random trees was initiated by Aldous in [7] and [8].",
"Lévy trees, also known as Lévy continuum random trees or Lévy CRTs, were first studied by Le Gall and Le Jan in [17] and [18], where it was shown that Lévy trees code the genealogy of continuous state branching processes (CSBPs).",
"Later, in [10], it was shown that Galton-Watson trees which code the genealogy of Galton-Watson processes, suitably rescaled, converge to Lévy trees, as rescaled Galton-Watson processes converge to CSBPs.",
"Then based on [20] and the present work, one may expect to introduce the notation “admissible family” to study the Galton-Watson processes and Galton-Watson trees.",
"And a general pruning procedure on Galton-Watson trees may be developed.",
"Possibly such a pruning procedure is a combination of Aldous and Pitman's pruning procedure in [9] and Abraham et.al.",
"'s pruning procedure in [2].",
"This gives the third motivation of the present work.",
"We will explore those questions in future.",
"The remaining of this paper is organized as follows.",
"In Section , we introduce and study the admissible family of branching mechanisms.",
"In Section , we recall some notation and collect some known results on real trees and Lévy trees.",
"In Section , based on the study of admissible family, the pruning procedure will be given and the marginal distributions of the pruning process are studied.",
"The evolution of the tree-valued process will be explored in Section .",
"In the last section, Section , we construct a tree-valued process by pruning a critical Lévy tree conditioned to be infinite and get the representation of the tree at the ascension time."
],
[
" Admissible family of branching mechanisms",
"Throughout the paper, for $-\\infty \\le a\\le b\\le +\\infty $ , we make the convention $\\int _a^b=\\int _{(a,b)}.$ The admissible family of branching mechanisms was first introduced by Li in [20].",
"Suppose that $\\mathfrak {T}\\subset {\\mathbb {R}}$ is an interval and $\\Psi =\\lbrace \\psi _q: q\\in {\\mathfrak {T}}\\rbrace $ is a family of branching mechanisms, where $\\psi _q$ is given by $\\psi _q(\\lambda )=b_q\\lambda +c\\lambda ^2+\\int _0^{\\infty }(e^{-\\lambda z}-1+\\lambda z)m_q(dz),\\quad \\lambda \\ge 0,$ with the parameters $(b, m)=(b_q, m_q)$ depending on $q\\in {\\mathfrak {T}}$ such that $b_q\\in {\\mathbb {R}}$ and $\\int z\\wedge z^2m_q(dz)<\\infty $ .",
"Definition 2.1 [Li(2014)] We call $\\lbrace \\psi _q:q\\in {\\mathfrak {T}}\\rbrace $ an admissible family if for each $\\lambda >0$ the function $q\\mapsto \\psi _q(\\lambda )$ is increasing and continuously differentiable with $\\zeta _q(\\lambda ):=\\frac{\\partial }{\\partial q}\\psi _q(\\lambda )=\\beta _q\\lambda +\\int _0^{\\infty }(1-e^{-z\\lambda })n_q(dz), \\quad q\\in {\\mathfrak {T}},\\quad \\lambda >0,$ where $\\beta _q\\ge 0$ and $(1\\wedge z)n_q(dz)$ is a finite kernel from ${\\mathfrak {T}}$ to $(0, \\infty )$ satisfying $\\int _t^q \\beta _{\\theta }d\\theta +\\int _t^qd\\theta \\int _0^{\\infty }zn_{\\theta }(dz)<\\infty ,\\quad q\\ge t\\in {\\mathfrak {T}}.$ Remark 2.2 In fact, in Li [20], it is assumed that $q\\mapsto \\psi _q(\\lambda )$ is decreasing and $\\zeta _q(\\lambda )=-\\frac{\\partial }{\\partial q}\\psi _q(\\lambda ).$ In that case, we will get an increasing tree-valued process.",
"Remark 2.3 For the purpose in this work, we also weaken the assumptions on $\\beta _{q}$ and $n_q(dz)$ .",
"In [20], it is assumed that $\\sup _{t\\le \\theta \\le q}\\left(\\beta _{\\theta }+\\int _0^{\\infty }zn_{\\theta }(dz)\\right)<\\infty ,\\quad q\\ge t\\in {\\mathfrak {T}},$ which is essential there.",
"If we assume (REF ), some interesting cases of pruning Lévy trees may be excluded.",
"See Example REF below for some cases.",
"Remark 2.4 It is also possible to assume that $\\psi _q(\\lambda )=b_q\\lambda +c\\lambda ^2+\\int _0^{\\infty }(e^{-\\lambda z}-1+\\lambda z{\\bf 1}_{\\lbrace z\\le 1\\rbrace })m_q(dz),$ with the parameters $(b, m)=(b_q, m_q)$ depending on $q\\in {\\mathfrak {T}}$ such that $b_q\\in {\\mathbb {R}}$ and $\\int 1\\wedge z^2m_q(dz)<\\infty $ .",
"Then (REF ) would be replaced by $\\int _t^q \\beta _{\\theta }d\\theta +\\int _t^qd\\theta \\int _{(0,1]}zn_{\\theta }(dz)<\\infty ,\\quad q\\ge t\\in {\\mathfrak {T}}.$ We assume further that $\\psi _q$ is conservative; i.e.",
"; $\\int _{(0,\\epsilon ]}\\frac{d\\lambda }{|\\psi _q(\\lambda )|}=+\\infty $ for all $\\epsilon >0$ .",
"We conjecture that all results in this work could be deduced in this framework.",
"In the following we shall give some examples of admissible family of branching mechanisms.",
"Example 2.5 Let $\\psi $ be defined in (REF ).",
"Abraham and Delmas in [1] considered the case of $\\psi _{q}(\\lambda )=\\psi (q+\\lambda )-\\psi (q), q\\in \\Theta ^{\\psi }$ , where $\\Theta ^{\\psi }$ is set of $\\theta \\in {\\mathbb {R}}$ such that $\\int _1^{\\infty }ze^{-\\theta z}m(dz)<\\infty $ .",
"Then $\\lbrace \\psi _{q}: q\\in \\Theta ^{\\psi }\\rbrace $ is an admissible family with $b_q=b+2cq+\\int _0^{\\infty }(1-e^{-zq})zm(dz),\\quad m_q(dz)=e^{-zq}m(dz).$ and $\\beta _q=2c,\\quad n_{q}(dz)=e^{-zq}zm(dz),\\quad m_z(t, dq)=z{\\bf 1}_{\\lbrace q\\ge t\\rbrace }dq.$ Note that $\\Theta ^{\\psi }=[\\theta _{\\infty }, +\\infty )$ or $(\\theta _{\\infty }, +\\infty )$ for some $\\theta _{\\infty }\\in [-\\infty ,0].$ However, in the case of $\\Theta ^{\\psi }=[\\theta _{\\infty }, +\\infty ) $ , $n_{\\theta _{\\infty }}(dz)$ may fail to satisfy (REF ).",
"A sufficient condition for that (REF ) holds is $\\int _1^{\\infty }z^2e^{-\\theta _{\\infty } z}m(dz)<\\infty .$ We also remark here that for the study of the ascension process, we always exclude the case of $\\Theta ^{\\psi }=[\\theta _{\\infty }, +\\infty )$ ; see Remark in Section below.",
"Example 2.6 Let $\\psi $ be defined in (REF ).",
"Let $f\\ge 0$ be a bounded decreasing function on ${\\mathbb {R}}$ with bounded derivative and $\\sup _{x\\ge 0}|xf^{\\prime }(x)|<+\\infty $ .",
"Let $g$ be a differentiable increasing function on ${\\mathbb {R}}$ .",
"For $q\\in {\\mathbb {R}}$ , let $\\psi _q$ be a branching mechanism with parameters $(b_q, m_q)$ defined by $b_q=b+g(q)+\\int _0^{\\infty }(f(0)-f(zq))zm(dz),\\quad m_q(dz)=f(qz)m(dz).$ Then one can check that $\\lbrace \\psi _q: q\\in {\\mathbb {R}}\\rbrace $ is an admissible family of branching mechanisms with $\\frac{\\partial }{\\partial q}\\psi _q(\\lambda )=g^{\\prime }(q)\\lambda -\\int _0^{\\infty }(1-e^{-z\\lambda })zf^{\\prime }(qz)m(dz),\\quad q\\in {\\mathbb {R}},\\; \\lambda \\ge 0,$ and $\\beta _q=g^{\\prime }(q),\\quad n_q(dz)=-zf^{\\prime }(qz)m(dz).$ Typically, if $m=0$ , then $\\psi _q(\\lambda )=(b+g(q))\\lambda +c\\lambda ^2.$ If $f\\equiv 1$ , then $\\psi _q(\\lambda )=\\psi (\\lambda )+g(q)\\lambda .$ Example 2.7 Let $\\psi $ be defined in (REF ) with $m(dz)=f(z)dz$ for some nonnegative measurable function $f$ on ${\\mathbb {R}}$ , where $dz$ is the Lebesgue measure.",
"Let $g$ be a differentiable increasing function on ${\\mathbb {R}}$ .",
"Let $h$ be a differentiable decreasing function on ${\\mathbb {R}}$ such that $h(z)>0$ for all $z\\in {\\mathbb {R}}$ .",
"For $q\\in {\\mathbb {R}}$ , define $b_q=g(q)+\\int _{h(q)}^{\\infty }zm(dz),\\quad m_q={\\bf 1}_{\\lbrace z\\le h(q)\\rbrace }m(dz).$ Then $\\lbrace \\psi _q: q\\in {\\mathbb {R}}\\rbrace $ is an admissible family of branching mechanisms with parameters $(b_q, m_q)$ such that $\\frac{\\partial }{\\partial q}\\psi _q(\\lambda )=g^{\\prime }(q)\\lambda +(e^{-\\lambda h(q)}-1)h^{\\prime }(q)f(h(q)),\\quad q\\in {\\mathbb {R}},\\; \\lambda \\ge 0,$ and $\\beta _q=g^{\\prime }(q),\\quad n_q(dz)=-h^{\\prime }(q)f(h(q))\\delta _{h(q)}(dz).$ Example 2.8 In the above example, if $h(\\cdot )=-(\\cdot \\wedge a)$ for some constant $a\\le 0$ , then $\\lbrace \\psi _q: q\\in (-\\infty , a)\\rbrace $ is an admissible family of branching mechanisms.",
"Example 2.9 Let ${\\mathfrak {T}}_-\\subset (-\\infty , 0]$ be an interval and let $\\lbrace \\psi _q: q\\in {\\mathfrak {T}}_-\\rbrace $ be an admissible family of branching mechanisms with the parameters $(b_q, m_q)$ .",
"Assume that $0\\in {\\mathfrak {T}}_-$ and $\\psi _0$ is critical.",
"Let $\\eta _q$ denote the largest root of $\\psi _q(s)=0$ .",
"For $q\\in -{\\mathfrak {T}}_-:=\\lbrace -t: t\\in {\\mathfrak {T}}_-\\rbrace $ , define $\\psi _q(\\cdot )=\\psi _{-q}(\\eta _{-q}+\\cdot )$ .",
"Then we have $\\lbrace \\psi _q: q\\in {\\mathfrak {T}}_-\\cup (-{\\mathfrak {T}}_-)\\rbrace $ is an admissible family of branching mechanisms such that for $q\\in -{\\mathfrak {T}}_-$ $b_q=b_{-q}+2c\\eta _{-q}+\\int _0^{\\infty }(1-e^{-z\\eta _{-q}})zm_{-q}(dz),\\quad m_q=e^{-z\\eta _{-q}}m_{-q}(dz).$ Next, we shall show how to get pruning parameters from a given admissible family of branching mechanisms.",
"Without loss of generality, we always assume that $\\psi _t\\ne \\psi _q$ for $t\\ne q\\in {\\mathfrak {T}}.$ It follows from the Definition REF that for $q\\ge t\\in {\\mathfrak {T}},$ $b_q=b_t+\\int _t^q \\beta _{\\theta }d\\theta +\\int _t^qd\\theta \\int _0^{\\infty }zn_{\\theta }(dz)$ and $m_t(dz)=m_q(dz)+\\int _{\\lbrace t\\le \\theta < q\\rbrace }n_{\\theta }(dz)d\\theta .$ Remark 2.10 By (REF ) one can see $q\\mapsto b_q$ is a continuous decreasing function on ${\\mathfrak {T}}$ .",
"Typically, $b_t=b_q$ implies $\\psi _t=\\psi _q$ and vice versa.",
"For $t\\in {\\mathfrak {T}}$ , define ${\\mathfrak {T}}_t={\\mathfrak {T}}\\cap [t,+\\infty )$ .",
"Then we shall see from (REF ) that for any $q\\in {\\mathfrak {T}}_t$ , $m_q(dz)\\ll m_t(dz),\\quad \\text{on }(0,\\infty ).$ Denote by $m_z(t, q)$ the corresponding Radon-Nikodym derivative; i.e.",
"; $m_q(dz)=m_z({t,q})m_t(dz).$ Then we see $\\int _{t\\le \\theta <q}n_{\\theta }(dz)d\\theta =\\left(1-m_z({t,q})\\right)m_t(dz),\\quad q\\in {\\mathfrak {T}}_t,$ which implies $m_{t}(dz)$ -a.e.",
"$m_z({t,q})\\le 1 \\text{ and } q\\mapsto m_z({t,q})(z) \\text{ is decreasing.",
"}$ Furthermore, we have for $ t\\le \\theta \\le q$ , $m_{t}(dz)$ -a.e.",
"$m_z(t, q)=m_z(t, \\theta )m_z(\\theta ,q).$ Then we make the following assumptions: (H1) For every $z\\in (0,\\infty )$ and $t\\in {\\mathfrak {T}}$ , $m_z({t,q})\\le 1 \\text{ and } q\\mapsto m_z({t,q})(z) \\text{ is decreasing.",
"}$ (H2) For every $z\\in (0,\\infty )$ and $ t\\le \\theta \\le q\\in {\\mathfrak {T}}$ , $m_z(t, q)=m_z(t, \\theta )m_z(\\theta ,q).$ (H3) For every $q\\in {\\mathfrak {T}}$ , $\\int ^{\\infty }\\frac{d\\lambda }{\\psi _q(\\lambda )}<+\\infty .$ By (H1), we could define a measure $m_z(t, dq)$ on ${\\mathfrak {T}}_t$ by $m_z(t, [t,q])=-\\ln (m_z(t, q)),$ where induces the pruning measure on branching nodes of infinite degree of a $\\psi _t$ -Lévy tree.",
"By (H2) we could have a tree-valued Markov processes on ${\\mathfrak {T}}$ .",
"(H3) is used to ensure that all trees are locally compact.",
"From now on, we assume that (H1-3) are in force."
],
[
"Notations",
"Let $(E,d)$ be a metric Polish space.",
"We denote by $M_{f}(E)$ (resp.",
"$M_f^{\\text{loc}}(E)$ ) the space of all finite (resp.",
"locally finite) Borel measures on $E$ .",
"For $x\\in E$ , let $\\delta _x$ denote the Dirac measure at point $x$ .",
"For $\\mu \\in M_f^{\\text{loc}}(E)$ and $f$ a non-negative measurable function, we set $\\langle \\mu ,f \\rangle =\\int f(x) \\, \\mu (dx)= \\mu (f)$ ."
],
[
"Real trees",
"We refer to [13] or [16] for a general presentation of random real trees.",
"A metric space $(d)$ is a real tree if the following properties are satisfied: For every $s,t\\in , there is a unique isometric map $ fs,t$from $ [0,d(s,t)]$ to $ such that $f_{s,t}(0)=s$ and $f_{s,t}(d(s,t))=t$ .",
"For every $s,t\\in , if $ q$ is a continuous injective map from$ [0,1]$ to $ such that $q(0)=s$ and $q(1)=t$ , then $q([0,1])=f_{s,t}([0,d(s,t)])$ .",
"If $s,t\\in , we will note $ s,t $the range of theisometric map $ fs,t$ described above and $ s,t $ for $ s,t {t}$.$ We say that $(d,\\emptyset )$ is a rooted real tree with root $\\emptyset $ if $(d)$ is a real tree and $\\emptyset \\in is adistinguished vertex.",
"For every $ x, $\\llbracket \\emptyset ,x\\rrbracket $ is interpreted as the ancestral line of vertex $x$ in the tree.",
"Let $(d,\\emptyset )$ be a rooted real tree.",
"The degree $n(x)$ of $x\\in is the number of connected components of $ {x}$ and thenumber of children of $ x$ is $ x=n(x)-1$ and of theroot is $ =n()$.",
"We shall consider the set ofleaves $ Lf(={xT, x=0}$, the set of branching points$ Br(={x x2}$ and the set of infinite branchingpoints $ Br( = { x x = } $.The skeleton of $ is the set of points in the tree that aren't leaves: ${\\rm Sk}(={\\rm Lf}($ .",
"The trace of the Borel $\\sigma $ -field of $ restricted to $ Sk($ is generated by thesets $ s,s' $; $ s,s' Sk($.",
"Onedefines uniquely a $$-finite Borel measure $$ on $ , called the length measure of $, such that$$\\ell ^{({\\rm Lf}() = 0\\quad \\text{and}\\quad \\ell ^{(\\llbracket s,s^{\\prime }\\rrbracket )=d(s,s^{\\prime }).",
"}}\\subsection {Measured rooted real trees}According to \\cite {ADH0b}, one can define aGromov-Hausdorff-Prohorov metric on the space of rooted measured metricspace as follows; see also \\cite {DW07} and \\cite {GPW08} for some related works.$ Let $(X,d)$ be a Polish metric space.",
"For $A,B\\in {\\mathcal {B}}(X)$ , set $d_\\text{H}(A,B)= \\inf \\lbrace \\varepsilon >0,\\ A\\subset B^\\varepsilon \\ \\mathrm {and}\\ B\\subset A^\\varepsilon \\rbrace ,$ the Hausdorff distance between $A$ and $B$ , where $A^\\varepsilon = \\lbrace x\\in X,\\inf _{y\\in A} d(x,y) < \\varepsilon \\rbrace $ .",
"If $\\mu ,\\nu \\in M_f(X)$ , we define $d_\\text{P}(\\mu ,\\nu ) = \\inf \\lbrace \\varepsilon >0,\\ \\mu (A)\\le \\nu (A^\\varepsilon ) +\\varepsilon \\text{ and }\\nu (A)\\le \\mu (A^\\varepsilon )+\\varepsilon \\ \\text{ for all closed set } A \\rbrace ,$ the Prohorov distance between $\\mu $ and $\\nu $ .",
"A rooted measured metric space ${\\mathcal {X}}= (X, d, \\emptyset ,\\mu )$ is a metric space $(X, d)$ with a distinguished element $\\emptyset \\in X$ and a locally finite Borel measure $\\mu \\in M_f^{\\text{loc}}(X)$ .",
"Let ${\\mathcal {X}}=(X,d,\\emptyset ,\\mu )$ and ${\\mathcal {X}}^{\\prime }=(X^{\\prime },d^{\\prime },\\emptyset ^{\\prime },\\mu ^{\\prime })$ be two compact rooted measured metric spaces, and define: $d_{\\text{GHP}}^c({\\mathcal {X}},{\\mathcal {X}}^{\\prime }) = \\inf _{\\Phi ,\\Phi ^{\\prime },Z} \\left(d_\\text{H}^Z(\\Phi (X),\\Phi ^{\\prime }(X^{\\prime })) +d^Z(\\Phi (\\emptyset ),\\Phi ^{\\prime }(\\emptyset ^{\\prime })) + d_\\text{P}^Z(\\Phi _* \\mu ,\\Phi _*^{\\prime }\\mu ^{\\prime }) \\right),$ where the infimum is taken over all isometric embeddings $\\Phi :X\\hookrightarrow Z$ and $\\Phi ^{\\prime }:X^{\\prime }\\hookrightarrow Z$ into some common Polish metric space $(Z,d^Z)$ and $\\Phi _* \\mu $ is the measure $\\mu $ transported by $\\Phi $ .",
"If ${\\mathcal {X}}=(X,d,\\emptyset ,\\mu )$ is a rooted measured metric space, then for $r\\ge 0$ we will consider its restriction to the ball of radius $r$ centered at $\\emptyset $ , ${\\mathcal {X}}^{(r)}=(X^{(r)}, d^{(r)}, \\emptyset ,\\mu ^{(r)})$ , where $X^{(r)}=\\lbrace x\\in X; d(\\emptyset ,x)\\le r\\rbrace ,$ with $d^{(r)}$ and $\\mu ^{(r)}$ defined in an obvious way.",
"By a measured rooted real tree $( d, \\emptyset , \\mathbf {)}$ , we mean $(d, \\emptyset )$ is a locally compact rooted real tree and $ \\mathbf {\\in }{\\mathcal {M}}_f^\\text{loc}($ is a locally finite measure on $.",
"When there is no confusion, we will simplywrite $ for $( d, \\emptyset , \\mathbf {)}$ .",
"We define for two measured rooted real trees $1,2$ : $d_{\\text{GHP}}(1,2) = \\int _0^\\infty \\mathop {\\mathrm {e}^{ -r}} \\left(1 \\wedge d^c_{\\text{GHP}}\\left(1^{(r)},2^{(r)}\\right)\\right) \\ dr.$ 1 and 2 are said GHP-isometric if $ d_{\\text{GHP}}(1,2)=0.$ Denote by ${\\mathbb {T}}$ the set of (GHP-isometry classes of) measured rooted real trees $( d, \\emptyset , \\mathbf {)}$ .",
"According to Corollary 2.8 in [5], $({\\mathbb {T}}, d_{\\text{GHP}})$ is a Polish metric space."
],
[
"Grafting procedure",
"Let $ be a measured rooted real tree and let$ ((i,xi),iI)$ be a finite or countable family of elements of$ T. We define the real tree obtained by grafting the trees $i$ on $ at point $ xi$.",
"We set $ = ( iI i{i} ) $ wherethe symbol $$ means that we choose for the sets $ (i)iI$representatives of GHP-isometry classes in $ T$ which are disjointsubsets of some common set and that we perform the disjoint union of allthese sets.",
"We set $ =$.",
"The set $$ isendowed with the following metric $ d$: if $ s,t$,\\begin{equation*}d^{\\hat{} (s,t) ={\\left\\lbrace \\begin{array}{ll}d^{(s,t)\\ & \\text{if}\\ s,t\\in \\\\d^{(s,x_i)+d^{{_i}(\\emptyset ^{{_i},t)\\ & \\text{if}\\ s\\in {,\\ t\\in {_i\\backslash \\lbrace \\emptyset ^{{_i}\\rbrace , \\\\d^{{_i}(s,t)\\ & \\text{if}\\ s,t\\in T_i\\backslash \\lbrace \\emptyset ^{{_i}\\rbrace ,\\\\d^{(x_i,x_j)+d^{{_j}(\\emptyset ^{{_j},s)+d^{{_i}(\\emptyset ^{{_i},t)\\ & \\text{if}\\ i\\ne j \\ \\text{and}\\ s\\in {_j\\backslash \\lbrace \\emptyset ^{{_j}\\rbrace ,\\ t\\in {_i\\backslash \\lbrace \\emptyset ^{{_i}\\rbrace .",
"}}}We define the mass measure on \\hat{ by:\\mathbf {m}^{\\hat{}=\\mathbf {m}^{+\\sum _{i\\in I}\\left({\\bf 1}_{{_i\\backslash \\lbrace \\emptyset ^{{_i}\\rbrace } \\mathbf {m}^{{_i}+\\mathbf {m}^{{_i}(\\lbrace \\emptyset ^{{_i}\\rbrace ) \\delta _{x_i}.Then (\\hat{{},d^{\\hat{},\\emptyset ^{\\hat{}) is still a rooted completereal tree.",
"(Notice that it is not always true that \\hat{T} remains locallycompact or that\\mathbf {m}^{\\hat{} is alocally finite measure on \\hat{).We will use the following notation for the grafted tree:\\begin{equation}_{i\\in I}(i,x_i) = (\\hat{,d^{\\hat{},\\emptyset ^{\\hat{},\\mathbf {m}^{\\hat{} ) ,}with convention that {\\otimes _{i\\in I}({_i,x_i)=for I=\\emptyset .",
"If \\varphi is an isometry from { onto {^{\\prime }, then{ \\otimes _{i\\in I}({_i,x_i) and {^{\\prime } \\otimes _{i\\in I}({_i,\\varphi (x_i)) are also isometric.",
"Therefore, the graftingprocedure is well defined on {\\mathbb {T}}.",
"}}\\subsection {Sub-trees above a given level}For {\\mathbb {T}}, define H_{\\text{max}}(=\\sup _{x\\in d^\\emptyset ^x) the height of and fora\\ge 0:{(a)}=\\lbrace x\\in \\, d(\\emptyset ,x)\\le a\\rbrace \\quad \\mbox{and}\\quad a)=\\lbrace x\\in \\, d(\\emptyset ,x)=a\\rbrace the restriction of the tree under level a and the set ofvertices of at level a respectively.",
"We denote by ({i,\\circ },i\\in I) the connected components of{(a)}.",
"Let \\emptyset _i be the most recent common ancestor of all thevertices of {i,\\circ }.",
"We consider the real treei={i,\\circ }\\cup \\lbrace \\emptyset _i\\rbrace rooted at point \\emptyset _i withmass measure \\mathbf {^}{i} defined as the restriction of \\mathbf {^} to{i,\\circ } and \\mathbf {^}{i}(\\emptyset _i)=0.",
"Notice that {(a)} \\circledast _{i\\in I}(i,\\emptyset _i) .We will consider the point measure on {\\mathbb {T}}:\\begin{equation}{\\mathcal {N}}_a^\\sum _{i\\in I}\\delta _{(\\emptyset _i,i)}.\\end{equation}}}}}\\subsection {Excursion measure of a Lévy tree}Recall (\\ref {eq:psi}).",
"We say \\psi is subcritical, critical or super-critical if b>0, b=0 or b<0, respectively.",
"Typically, we say \\psi is (sub)critical, if b\\ge 0.",
"We assume the Grey condition holds:\\begin{equation}\\int ^{+\\infty }\\frac{d\\lambda }{\\psi (\\lambda )}<+\\infty .\\end{equation}}}\\begin{remark}The Grey condition is used toensure that the corresponding Lévy tree is locally compact and also implies c>0 or \\int _{(0,1)} \\ell m(d\\ell )=+\\infty which is equivalent to the fact that the Lévy process with index\\psi is of infinite variation.\\end{remark}}Let v^{\\psi } bethe unique non-negative solution of the equation:\\begin{eqnarray}\\int _{v^{\\psi }(a)}^{+\\infty }\\frac{d\\lambda }{\\psi (\\lambda )}=a.\\end{eqnarray}}Results from \\cite {DL05} in the (sub)critical cases, where height functions are introduced tocode the compact real trees, can be extended to thesuper-critical cases; see \\cite {ADH0a}.",
"We state those results in thefollowing form.",
"There exists a \\sigma -finite measure {\\mathbb {N}}^\\psi [d{\\mathcal {T}}]on {\\mathbb {T}}, or excursion measure of a Lévy tree.",
"A \\psi -Lévy tree is a ``random^{\\prime \\prime } tree with law {\\mathbb {N}}^{\\psi } and the followingproperties.\\begin{enumerate}\\item [(i)] \\textbf {Height.}",
"For all a>0,{\\mathbb {N}}^\\psi [H_{\\text{max}}({\\mathcal {T}})>a]=v^{\\psi }(a).\\end{enumerate}\\item [(ii)] \\textbf {Mass measure.}",
"The mass measure \\mathbf {^}{\\mathcal {T}} is supported by{\\rm Lf}({\\mathcal {T}}), {\\mathbb {N}}^\\psi [d{\\mathcal {T}}]-a.e.",
"}\\item [(iii)] \\textbf {Local time.",
"}There exists a -measure-valued process (\\ell ^a, a\\ge 0)which is càdlàg for the weak topology on the set of finite measures on such that{\\mathbb {N}}^\\psi [d{\\mathcal {T}}]-a.e.",
":\\begin{equation}\\mathbf {m}^{{\\mathcal {T}}}(dx) = \\int _0^\\infty \\ell ^a(dx) \\, da,\\end{equation}\\ell ^0=0, \\inf \\lbrace a > 0 ; \\ell ^a = 0\\rbrace =\\sup \\lbrace a \\ge 0 ; \\ell ^a\\ne 0\\rbrace =H_{\\text{max}}({\\mathcal {T}}) and for every fixed a\\ge 0,{\\mathbb {N}}^\\psi [d{\\mathcal {T}}]-a.e.",
":\\begin{itemize}\\item The measure \\ell ^a is supported on{\\mathcal {T}}(a).\\item We have for every boundedcontinuous function \\phi on {\\mathcal {T}}:{\\begin{@align*}{1}{-1}\\langle \\ell ^a,\\phi \\rangle & = \\lim _{\\varepsilon \\downarrow 0}\\frac{1}{v^{\\psi }(\\varepsilon )} \\int \\phi (x) {\\bf 1}_{\\lbrace H_{\\text{max}}({\\mathcal {T}}^{\\prime })\\ge \\varepsilon \\rbrace } {\\mathcal {N}}_a^{{\\mathcal {T}}}(dx, d{\\mathcal {T}}^{\\prime }) \\\\& = \\lim _{\\varepsilon \\downarrow 0} \\frac{1}{v^{\\psi }(\\varepsilon )} \\int \\phi (x){\\bf 1}_{\\lbrace H_{\\text{max}}({\\mathcal {T}}^{\\prime })\\ge \\varepsilon \\rbrace }{\\mathcal {N}}_{a-\\varepsilon }^{{\\mathcal {T}}}(dx, d{\\mathcal {T}}^{\\prime }),\\ \\text{if}\\ a>0.\\end{@align*}}\\end{itemize}Under {\\mathbb {N}}^\\psi , the real valued process (\\langle \\ell ^a,1 \\rangle ,a\\ge 0) is distributed as a CSBP with branching mechanism \\psi under its canonical measure.\\item [(iv)] \\textbf {Branching property.",
"}For every a>0, the conditionaldistribution of the point measure {\\mathcal {N}}_a^{{\\mathcal {T}}}(dx,d{\\mathcal {T}}^{\\prime }) under{\\mathbb {N}}^\\psi [d{\\mathcal {T}}|H_{\\text{max}}({\\mathcal {T}})>a], given {\\mathcal {T}}^{(a)}, is that of a Poissonpoint measure on {\\mathcal {T}}(a)\\times {\\mathbb {T}} with intensity\\ell ^a(dx){\\mathbb {N}}^\\psi [d{\\mathcal {T}}^{\\prime }].\\item [(v)] \\textbf {Branching points.",
"}\\begin{itemize}\\item {\\mathbb {N}}^\\psi [d{\\mathcal {T}}]-a.e., the branching points of {\\mathcal {T}} have 2children or aninfinity number of children.\\item The set of binary branching points (i.e.",
"with 2 children) is empty{\\mathbb {N}}^\\psi -a.e if c=0 and is a countable dense subset of {\\mathcal {T}}if c>0.\\item The set {\\rm Br}_\\infty ({\\mathcal {T}}) of infinite branching points isnonempty with {\\mathbb {N}}^\\psi -positive measure if and only if m\\ne 0.",
"If\\langle m,1\\rangle =+\\infty , the set {\\rm Br}_\\infty ({\\mathcal {T}}) is{\\mathbb {N}}^\\psi -a.e.",
"a countable dense subset of {\\mathcal {T}}.\\end{itemize}\\item [(vi)] \\textbf {Mass of the nodes.",
"}The set \\lbrace d(\\emptyset ,x),\\ x\\in {\\rm Br}_\\infty ({\\mathcal {T}}) \\rbrace coincides{\\mathbb {N}}^\\psi -a.e.",
"with the set of discontinuity times of the mapping a\\mapsto \\ell ^a.",
"Moreover, {\\mathbb {N}}^\\psi -a.e., for every such discontinuity time b, thereis a unique x_b\\in {\\rm Br}_\\infty ({\\mathcal {T}}) with d(\\emptyset , x_b)=b and\\Delta _b>0, such that:\\ell ^b = \\ell ^{b-} + \\Delta _b \\delta _{x_b},where \\Delta _b is called the mass of the node x_b.",
"Furthermore\\Delta _b can be obtainedby the approximation:\\begin{equation}\\Delta _b = \\lim _{\\varepsilon \\rightarrow 0}\\frac{1}{v^{\\psi }(\\varepsilon )}n(x_b,\\varepsilon ),\\end{equation}where n(x_b,\\varepsilon )=\\int {\\bf 1}_{\\lbrace x_b\\rbrace }(x){\\bf 1}_{\\lbrace H_{\\text{max}}({\\mathcal {T}}^{\\prime }) >\\varepsilon \\rbrace } {\\mathcal {N}}_b^{\\mathcal {T}}(dx,d{\\mathcal {T}}^{\\prime }) is the number of sub-trees with MRCAx_b and height larger than \\varepsilon .",
"}\\end{equation}In order to stress the dependence in {\\mathcal {T}}, we may write \\ell ^{a, {\\mathcal {T}}}for \\ell ^a.We set \\sigma ^{\\mathcal {T}} or simply \\sigma when there is no confusion, forthetotal mass of the mass measure on {\\mathcal {T}}:\\begin{equation}\\sigma =\\mathbf {^}{{\\mathcal {T}}}({\\mathcal {T}}).\\end{equation}Notice that (\\ref {eq:int-la}) readily implies that \\mathbf {^}{\\mathcal {T}}(\\lbrace x\\rbrace )=0 forall x\\in {\\mathcal {T}}.",
"}\\subsection {Related measures on Lévy trees}}We define a probability measure on {\\mathbb {T}} as follow.",
"Let r>0 and\\sum _{k\\in {\\mathcal {K}}}\\delta _{{\\mathcal {T}}^k} be a Poisson random measure on {\\mathbb {T}} withintensity r{\\mathbb {N}}^\\psi .",
"Consider \\emptyset as the trivial measuredrooted real tree reduced to the root with null mass measure.",
"Define{\\mathcal {T}}=\\emptyset \\circledast _{k\\in {\\mathcal {K}}}( {\\mathcal {T}}^k, \\emptyset ).",
"UsingProperty (i) as well as (\\ref {sigma}) below, one easily get that {\\mathcal {T}}is a measured locally compact rooted real tree, and thus belongs to{\\mathbb {T}}.",
"We denote by {\\mathbb {P}}^\\psi _r its distribution.",
"Its corresponding localtime and mass measure are respectively defined by \\ell ^a=\\sum _{k\\in {\\mathcal {K}}} \\ell ^{a, {\\mathcal {T}}^k} for a\\ge 0, and \\mathbf {^}{\\mathcal {T}}=\\sum _{k\\in {\\mathcal {K}}}\\mathbf {^}{{\\mathcal {T}}^k}.",
"Furthermore, its total mass is defined by\\sigma =\\sum _{k\\in {\\mathcal {K}}} \\sigma ^{{\\mathcal {T}}^k}.",
"By construction, we have{\\mathbb {P}}^\\psi _r(d{\\mathcal {T}})-a.s. \\emptyset \\in {\\rm Br}_\\infty ({\\mathcal {T}}),\\Delta _\\emptyset =r (see definition (\\ref {DefMas}) with b=0) and\\ell ^0=r\\delta _\\emptyset .",
"Under {\\mathbb {P}}^\\psi _r or under {\\mathbb {N}}^\\psi , wedefine the process {\\mathcal {Z}}=({\\mathcal {Z}}_a,a\\ge 0) by:\\begin{eqnarray}{\\mathcal {Z}}_a=\\langle \\ell ^a,1 \\rangle .\\end{eqnarray}Noticethat (under {\\mathbb {N}} or {\\mathbb {P}}_r^\\psi ):\\begin{equation}\\sigma =\\int _0^{+\\infty } {\\mathcal {Z}}_a \\; da =\\mathbf {^}{\\mathcal {T}}({\\mathcal {T}}).\\end{equation}In particular, as \\sigma is distributed as the total mass of a CSBPunder its canonical measure, we have thatfor \\lambda \\ge 0\\begin{equation}{\\mathbb {N}}^\\psi \\left[1-\\mathop {\\mathrm {e}^{ -\\lambda \\sigma }} \\right]=\\psi ^{-1}(\\lambda ),\\quad {\\mathbb {N}}^{\\psi }[1-e^{-\\lambda {\\mathcal {Z}}_a}]=u^{\\psi }(a,\\lambda ),\\end{equation}where (u^{\\psi }(a, \\lambda ), a\\ge 0, \\lambda >0) is the unique non-negative solution to\\begin{eqnarray}\\int _{u^{\\psi }(a,\\lambda )}^{\\lambda }\\frac{dr}{\\psi (r)}=a;\\quad u^{\\psi }(0,\\lambda )=\\lambda .\\end{eqnarray}By results in Chapter 3 in \\cite {Li12}, we further have \\begin{eqnarray}u^{\\psi }(a, u^{\\psi }(a^{\\prime },\\lambda ))=u^{\\psi }(a+a^{\\prime },\\lambda ), \\quad \\lim _{\\lambda \\rightarrow 0}u^{\\psi }(a,\\lambda )=v^{\\psi }(a).\\end{eqnarray}Finally, we recall the Girsanov transformation from \\cite {AD12}.",
"Let \\theta \\in \\Theta ^\\psi anda>0.",
"We set:M_a^{\\psi , \\theta }=\\exp \\left\\lbrace \\theta {\\mathcal {Z}}_0-\\theta {\\mathcal {Z}}_a-\\psi (\\theta )\\int _0^a{\\mathcal {Z}}_sds\\right\\rbrace .Recall that {\\mathcal {Z}}_0=\\langle \\ell _0,0\\rangle =0 under {\\mathbb {N}}^\\psi .For any non-negative measurable functional F defined on {\\mathbb {T}}, we havefor \\theta \\in \\Theta ^\\psi and a\\ge 0:\\begin{equation}{\\mathbb {E}}_r^{\\psi ^\\theta }[F({\\mathcal {T}}^{(a)})]= {\\mathbb {E}}_r^{\\psi } \\left[F({\\mathcal {T}}^{(a)})M_a^{\\psi ,\\theta } \\right]\\quad \\text{and}\\quad {\\mathbb {N}}^{\\psi ^\\theta }[F({\\mathcal {T}}^{(a)})]= {\\mathbb {N}}^{\\psi } \\left[F({\\mathcal {T}}^{(a)})M_a^{\\psi ,\\theta } \\right].\\end{equation}Typically,\\begin{eqnarray}{\\mathbb {N}}^{\\psi ^\\theta }[F({\\mathcal {T}})]&= {\\mathbb {N}}^{\\psi } \\left[F({\\mathcal {T}})\\mathop {\\mathrm {e}^{ - \\psi (\\theta ) \\sigma }}{\\bf 1}_{\\lbrace \\sigma <+\\infty \\rbrace }\\right].\\end{eqnarray}}We have that under {\\mathbb {P}}_r^{\\psi }(d{\\mathcal {T}}), the random measure{{\\mathcal {N}}}_0^{{\\mathcal {T}}}(dx, d{\\mathcal {T}}^{\\prime }), defined by (\\ref {eq:def-cna}) with a=0, isa Poisson point measure on \\lbrace \\emptyset \\rbrace \\times {\\mathbb {T}} with intensityr\\delta _{\\emptyset }(dx){\\mathbb {N}}^{\\psi }[d{\\mathcal {T}}^{\\prime }].Then, using the firstequality in (\\ref {Gir}) with F=1, we get that for\\theta >0 such that \\psi (\\theta )\\ge 0,\\begin{equation}{\\mathbb {N}}^{\\psi ^{\\theta }}\\left[1-\\exp \\left\\lbrace \\theta {\\mathcal {Z}}_{a}+\\psi (\\theta )\\int _0^a{\\mathcal {Z}}_sds\\right\\rbrace \\right]=-\\theta .\\end{equation}}}}}}}}\\section { A general pruning procedure}In this section we define a pruning procedure on a Lévy tree according to the admissible family of branching mechanisms in Definition \\ref {defbranching}.\\right.Recall (\\ref {zeta}) and (\\ref {prunmeasure}).",
"Also recall that {\\mathfrak {T}}_t={\\mathfrak {T}}\\cap [t,\\infty ) for t\\in {\\mathfrak {T}}.",
"For {\\mathbb {T}}, weconsider two Poisson random measures M_t^{ske}(d\\theta ,dy) and M_t^{nod}(d\\theta , dy) on {\\mathfrak {T}}_t\\times whose intensity is\\beta _{\\theta }d\\theta \\ell ^dy)\\quad \\text{ and }\\quad \\sum _{x\\in \\text{Br}_\\infty (\\setminus \\lbrace \\emptyset \\rbrace }m_{\\Delta _x} (t, d\\theta )\\delta _x(dy),respectively.",
"Then M_t^{ske}(d\\theta ,dy) gives the marks on the skeleton and M_t^{nod}(d\\theta , dy) gives the marks on the nodes of infinite degree.",
"}We define a new Poisson random measure on {\\mathfrak {T}}_t\\times byM_t^{(d\\theta , dy)=M_t^{ske}(d\\theta ,dy)+M^{nod}_t(d\\theta , dy).",
"}}}Using this measure of marks, we definethe pruned tree at time q as:\\begin{eqnarray}q^t:=\\lbrace x\\in {\\mathcal {T}},\\ M_t^[t, q]\\times \\llbracket \\emptyset ,x\\llbracket )=0\\rbrace , \\quad q\\in {\\mathfrak {T}}_t,\\end{eqnarray}with the induced metric, root \\emptyset and mass measure which is therestriction of the mass measure \\mathbf {^}{.",
"}\\begin{remark}Note that {\\mathbb {N}}^{\\psi _t}-a.e n(\\emptyset )=1 and {\\mathbb {P}}_r^{\\psi _t}-a.s. n(\\emptyset )=\\infty with \\Delta _{\\emptyset }=r.",
"The above definition of q^t means that we do not put marks on the root even if \\emptyset is a node of infinite degree with mass r. Or {\\mathbb {P}}_r^{\\psi _t}(q^t=\\emptyset )>0.",
"However, {\\mathbb {P}}_r^{\\psi _q}(\\emptyset )=0.\\end{remark}}For fixed q\\in {\\mathfrak {T}}_t, M_t^{([t, q], dy)=M_t^{ske}([t,q],dy)+M_t^{nod}([t, q], dy) is also a point measure on tree :\\begin{itemize}\\item [(i)] M_t^{ske}([t,q],dy) is a Poisson point measure on the skeleton of with intensity \\int _t^q\\beta _{\\theta }d\\theta \\ell ^{(dy);}\\item [(ii)] The atoms of M_t^{nod}([t, q], dy) give the marked nodes: each node of infinite degree is marked (or pruned) independently from the others with probability \\begin{eqnarray}{\\mathbb {P}}\\left(M_{t}^{nod}([t, q], \\lbrace y\\rbrace )>0\\right)=1-\\exp \\lbrace -m_{\\Delta _y}(t, [t, q])\\rbrace =1-m_{\\Delta _y}(t,q),\\end{eqnarray}where \\Delta _y is the mass associated with the node.\\end{itemize}Thus for fixed q\\in {\\mathfrak {T}}_t, there exists a measurable functional {\\mathcal {M}}_{\\alpha _{t,q}, p_{t,q}} on {\\mathbb {T}} such that:\\begin{eqnarray}q^t={\\mathcal {M}}_{\\alpha _{t,q}, p_{t,q}}(,\\end{eqnarray}where \\alpha _{t,q}=\\int _t^q\\beta _{\\theta }d\\theta , p_{t,q}=1-m_z(t,q).Our main result in this section is the following theorem.",
"}\\begin{theo}Assume that \\lbrace \\psi _t:t\\in {\\mathfrak {T}}\\rbrace is an admissible family satisfying (H1-3).",
"Then we have\\begin{itemize}\\item [(a)]The tree-valued process (q^t,\\; q\\in {\\mathfrak {T}}_t) is aMarkov process under {\\mathbb {N}}^{\\psi _t};\\end{itemize}\\item [(b)]For fixed q\\in {\\mathfrak {T}}_t, the distribution of q^t under {\\mathbb {N}}^{\\psi _t}is {\\mathbb {N}}^{\\psi _q};\\end{theo}\\item [(c)] Given ({\\cal T}, {\\bf m}^{{\\cal T}})\\in {\\mathbb {T}}, let \\cal M(dx, d{\\cal T})=\\sum _{i\\in I}\\delta _{(x_i, i)} be a Poisson random measure on {\\cal T}\\times {\\mathbb {T}} with intensity\\begin{eqnarray}{\\bf m}^{{\\cal T}}(dx)\\left(\\int _t^q\\beta _{\\theta }d\\theta {\\mathbb {N}}^{\\psi _{t}}[d{\\cal T}]+\\int _t^qd\\theta \\int _0^{\\infty }n_{\\theta }(dz){\\mathbb {P}}_z^{\\psi _{t}}(d{\\cal T})\\right).\\end{eqnarray}Then for q\\in {\\mathfrak {T}}_t,( {\\mathcal {M}}_{\\alpha _{t,q}, p_{t,q}}() under {\\mathbb {N}}^{\\psi _t} has the same distribution as (\\tilde{, {) under {\\mathbb {N}}^{\\psi _q},where \\begin{eqnarray}\\tilde{=_{i\\in I}(i, x_i).",
"}\\end{eqnarray}}\\begin{remark}(c) in Theorem \\ref {MainA} is the so-called special Markov property.",
"One may follow the proof in Appendix A in \\cite {H12} to extend (c) to have pruning times in (sub)critical cases and then follow the arguments in {\\it Step 4} below to extend the result to super-critical cases.\\end{remark}}\\begin{xmlelement*}{proof}The proof will be divided into five steps: \\\\{\\it Step 1:} We shall prove (a).",
"We first study the behavior of M_t^{nod}(d\\theta , dy).",
"Given a branching node y\\in Br_{\\infty }(, for t\\le \\theta \\le q\\in {\\mathfrak {T}}, we have\\begin{eqnarray}\\!\\!\\!&\\!\\!\\!&{\\mathbb {P}}\\left(M_t^{nod}([\\theta , q], \\lbrace y\\rbrace )>0\\big {|}M_t^{nod}([t, \\theta ], \\lbrace y\\rbrace )=0\\right)\\cr \\!\\!\\!&\\!\\!\\!&\\quad =\\frac{{\\mathbb {P}}\\left(M_t^{nod}([\\theta , q], \\lbrace y\\rbrace )>0,\\, M_t^{nod}([t, \\theta ], \\lbrace y\\rbrace )=0\\right)}{{\\mathbb {P}}\\left(M_t^{nod}([t, \\theta ], \\lbrace y\\rbrace )=0\\right)}\\cr \\!\\!\\!&\\!\\!\\!&\\quad =\\frac{{\\mathbb {P}}\\left(M_t^{nod}([t, q], \\lbrace y\\rbrace )>0)-{\\mathbb {P}}( M_t^{nod}([t, \\theta ], \\lbrace y\\rbrace )>0\\right)}{{\\mathbb {P}}\\left(M_t^{nod}([t, \\theta ], \\lbrace y\\rbrace )=0\\right)}\\cr \\!\\!\\!&\\!\\!\\!&\\quad =\\frac{m_{\\Delta _y}(t, \\theta )-m_{\\Delta _y}(t,q)}{m_{\\Delta _y}(t, \\theta )}\\cr \\!\\!\\!&\\!\\!\\!&\\quad =1-m_{\\Delta _y}(\\theta , q)\\cr \\!\\!\\!&\\!\\!\\!&\\quad ={\\mathbb {P}}\\left(M_{\\theta }^{nod}([\\theta , q], \\lbrace y\\rbrace )>0\\right),\\end{eqnarray}where we used (\\ref {Pmark}) for the third equality and assumption (H2) for the fourth equality.",
"Similarly, one can prove that for x\\in and t\\le \\theta \\le q\\in {\\mathfrak {T}}.",
"{\\mathbb {P}}\\left(M_t^{ske}([\\theta , q], \\llbracket \\emptyset , x\\llbracket )>0\\big {|}M_t^{ske}([t, \\theta ], \\llbracket \\emptyset , x\\llbracket )=0\\right)={\\mathbb {P}}\\left(M_{\\theta }^{ske}([\\theta , q], \\llbracket \\emptyset , x\\llbracket )>0\\right).Then (a) follows readily.\\end{xmlelement*}\\medskip }{\\it Step 2:}For the second and the third assertions, if \\psi _t is (sub)critical, then an application of Theorem 1.1 in \\cite {ADV10} gives the desired results.",
"So we only need to study the super-critical case.",
"Without loss of generality, we may assume t=0.",
"From now on we shall assume that \\psi _0 is super-critical.",
"For this proof only, we set |x|=d^{(\\emptyset , x),\\quad x\\in We also write q for q^0.",
"}In this step, we prove the desired results for a special sub-critical case.",
"Let \\eta _0 be the maximum root of \\psi _0(s)=0.",
"Define\\begin{eqnarray}\\psi _q^{\\eta _0}(\\lambda )=\\psi _q(\\lambda +\\eta _0)-\\psi _q(\\eta _0),\\quad \\lambda \\ge 0,\\quad q\\in {\\mathfrak {T}}_0.\\end{eqnarray}One can check that if \\lbrace \\psi _q: q\\in {\\mathfrak {T}}_0\\rbrace is an admissible family satisfying (H1-3), then \\lbrace \\psi _q^{\\eta _0}: q\\in {\\mathfrak {T}}_t\\rbrace is also an admissible family with parameter (b_q^{\\eta _0}, m_q^{\\eta _0}, q\\in {\\mathfrak {T}}_0) satisfying (H1-3) such that\\begin{eqnarray}b_q^{\\eta _0}=b_q+2c\\eta _0+\\int _0^{\\infty }(1-e^{-\\eta _0z})zm_q(dz),\\quad m_q^{\\eta _0}(dz)=e^{-\\eta _0 z}m_q(dz).\\end{eqnarray}Typically, by (\\ref {b_q}) and (\\ref {m_q}),\\begin{eqnarray*}b_q^{\\eta _0}\\!\\!\\!&=\\!\\!\\!&b_0^{\\eta _0}+\\int _0^q\\beta _{\\theta }d\\theta +\\int _0^qd{\\theta }\\int _0^{\\infty }zn_{\\theta }(dz)-\\int _0^{q}d\\theta \\int _0^{\\infty }(1-e^{-\\eta _0z})zn_{\\theta }(dz)\\cr \\!\\!\\!&=\\!\\!\\!&b_0^{\\eta _0}+\\int _0^q\\beta _{\\theta }d\\theta +\\int _0^{q}d\\theta \\int _0^{\\infty }e^{-\\eta _0z}zn_{\\theta }(dz)\\cr \\!\\!\\!&\\ge \\!\\!\\!&b_0^{\\eta _0}.\\end{eqnarray*}Since b_0^{\\eta _0}=\\psi ^{\\prime }_0(\\eta _0)>0, \\psi _q^{\\eta _0} is subcritical for all q\\in {\\mathfrak {T}}_0.",
"Moreover, by (\\ref {psit_0}) and (\\ref {bmt_0}), we have\\begin{eqnarray}\\frac{\\partial }{\\partial q}\\psi _q^{\\eta _0}(\\lambda )\\!\\!\\!&=\\!\\!\\!&\\zeta _q(\\lambda +\\eta _0)-\\zeta _q(\\eta _0)\\cr \\!\\!\\!&=\\!\\!\\!&\\beta _q\\lambda +\\int _0^{\\infty }(1-e^{-\\lambda z})e^{-\\eta _0z}n_q(dz)\\end{eqnarray}and\\frac{m_q^{\\eta _0}(dz)}{m_0^{\\eta _0}(dz)}=\\frac{m_q(dz)}{m_0(dz)}=m_z(0,q),\\quad q\\in {\\mathfrak {T}}_0.Thus \\lbrace \\psi _q: q\\in {\\mathfrak {T}}_0\\rbrace and \\lbrace \\psi _q^{\\eta _0}: q\\in {\\mathfrak {T}}_0\\rbrace induce the same pruning parameters \\beta _q and m_z(0,q).",
"Typically according to assertions (b) and (c) for (sub)critical case, we have\\begin{itemize}\\item [(b^{\\prime })]q={\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}( is a \\psi _q^{\\eta _0}-Lévy tree under {\\mathbb {N}}^{\\psi _0^{\\eta _0}};\\end{itemize}\\item [(c^{\\prime })] Given ({\\cal T}, {\\bf m}^{{\\cal T}})\\in {\\mathbb {T}}, let \\cal M^{\\eta _0}(dx,d{\\cal T})=\\sum _{i\\in I_{\\eta _0}}\\delta _{(x_i, i)} be a Poisson point measure on {\\cal T}\\times {\\mathbb {T}} with intensity\\begin{eqnarray}{\\bf m}^{{\\cal T}}(dx)\\left(\\int _0^q\\beta _{\\theta }d\\theta {\\mathbb {N}}^{\\psi _{0}^{\\eta _0}}[d{\\cal T}]+\\int _0^qd\\theta \\int _0^{\\infty }e^{-\\eta _0z}n_{\\theta }(dz){\\mathbb {P}}_z^{\\psi _{0}^{\\eta _0}}(d{\\cal T})\\right).\\end{eqnarray}Then for q\\in {\\mathfrak {T}}_0,( {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}() under {\\mathbb {N}}^{\\psi _0^{\\eta _0}} has the same distribution as (\\hat{_{\\eta _0}, {) under {\\mathbb {N}}^{\\psi _q^{\\eta _0}},where \\begin{eqnarray}\\hat{=_{i\\in I_{\\eta _0}}(i, x_i).",
"}\\end{eqnarray}}{\\it Step 3:} We shall prove (b) for \\psi _0 is super-critical.Recall {(a)}=\\lbrace x\\in d^{(\\emptyset , x)\\le a\\rbrace .By Girsanov transformation (\\ref {Gir}), for any nonnegative function F on {\\mathbb {T}}, we have\\begin{eqnarray}{\\mathbb {N}}^{\\psi _0}[F(q^{(a)})]&=&{\\mathbb {N}}^{\\psi _0}[F({\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)}))]\\cr &=&{\\mathbb {N}}^{\\psi _0^{\\eta _0}}[e^{\\eta _0{\\mathcal {Z}}_a}F({\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)}))]\\cr &=&{\\mathbb {N}}^{\\psi _q^{\\eta _0}}[e^{\\eta _0\\hat{{\\mathcal {Z}}}_a}F({(a)})],\\end{eqnarray}where the last equality follows from Special Markov property (c^{\\prime }) and \\hat{{\\mathcal {Z}}}_a=\\langle \\ell ^{a,\\, \\hat{},1\\rangle ={\\mathcal {Z}}_a+\\sum _{i\\in I_{\\eta _0}}{\\bf 1}_{\\lbrace |x_i|\\le a\\rbrace }{\\mathcal {Z}}_{a-x_i}^{i}, with {\\mathcal {Z}}_a^{i}=\\langle \\ell ^{a, i}, 1\\rangle .",
"Thus\\begin{eqnarray}{\\mathbb {N}}^{\\psi _q^{\\eta _0}}[e^{\\eta _0 \\hat{{\\mathcal {Z}}}_a}F({(a)})]={\\mathbb {N}}^{\\psi _q^{\\eta _0}}[e^{\\eta _0 {\\mathcal {Z}}_a}F({(a)})H(a,\\eta _0)],\\end{eqnarray}where, by property of Poisson random measure,\\begin{eqnarray*}H(a, \\eta _0)\\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0\\sum _{i\\in I_{\\eta _0}}{\\bf 1}_{\\lbrace |x_i|\\le a\\rbrace }{\\mathcal {Z}}_a^{i}}\\bigg {|}]\\\\\\!\\!\\!&=\\!\\!\\!&\\exp \\bigg {\\lbrace }-\\int _{{(a)}}{\\bf m}^{(dx)\\bigg {(}\\int _0^q\\beta _{\\theta }d\\theta {\\mathbb {N}}^{\\psi _0^{\\eta _0}}\\left[1-e^{\\eta _0{\\mathcal {Z}}_{a-|x|}}\\right]\\cr \\!\\!\\!&\\!\\!\\!&\\qquad \\qquad +\\int _0^{q}d{\\theta }\\int _0^{\\infty }e^{-\\eta _0z}n_{\\theta }(dz)\\left(1-e^{-z{\\mathbb {N}}^{\\psi _0^{\\eta _0}}[1-e^{\\eta _0{\\mathcal {Z}}_{a-|x|}}]}\\right)\\bigg {)}\\bigg {\\rbrace }.",
"}Noting that \\right.\\psi _0(\\eta _0)=0, together with (\\ref {Gir2}), yields\\begin{eqnarray}{\\mathbb {N}}^{\\psi _0^{\\eta _0}}\\left[1-e^{\\eta _0{\\mathcal {Z}}_{a-|x|}}\\right]=-\\eta _0.\\end{eqnarray}Then (\\ref {m_q}) and (\\ref {m_q1}) imply\\begin{eqnarray*}H(a, \\eta _0)=\\exp \\bigg {\\lbrace }-\\int _{{(a)}}{\\bf m}^{(dx)\\bigg {(}-\\int _0^q\\beta _{\\theta }d\\theta \\eta _0+\\int _0^{q}d{\\theta }\\int _0^{\\infty }e^{-\\eta _0z}n_{\\theta }(dz)(1-e^{z\\eta _0})\\bigg {)}\\bigg {\\rbrace }.",
"}Since \\mathbf {^}{{(a)}}({(a)})=\\int _0^{a}{\\mathcal {Z}}_sds and\\begin{eqnarray}\\psi _q(\\eta _0)\\!\\!\\!&=\\!\\!\\!&\\psi _q(\\eta _0)-\\psi _0(\\eta _0)\\cr \\!\\!\\!&=\\!\\!\\!&\\int _0^q\\beta _{\\theta }d\\theta \\eta _0-\\int _0^{q}d{\\theta }\\int _0^{\\infty }e^{-\\eta _0z}n_{\\theta }(dz)(1-e^{z\\eta _0}),\\end{eqnarray}we haveH(a, \\eta _0)=\\exp \\left\\lbrace \\psi _q(\\eta _0){\\bf m}^{({(a)})=\\exp \\left\\lbrace \\psi _q(\\eta _0)\\int _0^a{\\mathcal {Z}}_sds\\right\\rbrace .By (\\ref {MainA01}), (\\ref {MainA02}) and Girsanov transformation (\\ref {Gir}), one can see that\\begin{eqnarray*}{\\mathbb {N}}^{\\psi _0}[F(q^{(a)})]\\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _q^{\\eta _0}}[e^{\\eta _0{\\mathcal {Z}}_a+\\psi _q(\\eta _0)\\int _0^a{\\mathcal {Z}}_sds}F({(a)})]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _q}[F({(a)})],\\end{eqnarray*}which implies that under {\\mathbb {N}}^{\\psi _0}, q is a \\psi _q-Lévy tree.",
"We have completed the proof of assertion (b).",
"}\\medskip \\right.",
"{\\it Step 4:} We shall prove (c) for \\psi _0 is super-critical.",
"Recall (\\ref {Ttilde}) and (\\ref {Ttilde0}).Note that\\tilde{^{(a)}={(a)}\\otimes _{i\\in I, |x_i|\\le a}(i^{(a-|x_i|)}, x_i).We only need to show that for all a\\ge 0, ({(a)}, {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)})) under {\\mathbb {N}}^{\\psi _0} has the same distribution as (\\tilde{^{(a)}, {(a)}) under {\\mathbb {N}}^{\\psi _q}.",
"By (\\ref {Gir}), we have for any nonnegative functional F on {\\mathbb {T}}^2:\\begin{eqnarray}{\\mathbb {N}}^{\\psi _0}\\left[F({(a)}, {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)}))\\right]={\\mathbb {N}}^{\\psi _0^{\\eta _0}}\\left[e^{\\eta _0{\\mathcal {Z}}_a}F({(a)}, {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)}))\\right]\\end{eqnarray}By (c^{\\prime }) in {\\it Step 2}, we have ({(a)}, {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)})) under {\\mathbb {N}}^{\\psi _0^{\\eta _0}} has the same distribution as (\\hat{^{(a)}, {(a)}) under {\\mathbb {N}}^{\\psi _q^{\\eta _0}}, where\\hat{^{(a)}={(a)}\\otimes _{i\\in I_{\\eta _0}, |x_i|\\le a}(i^{(a-|x_i|)}, x_i).Thus (\\ref {MainA03}) implies\\begin{eqnarray}{\\mathbb {N}}^{\\psi _0}\\left[F({(a)}, {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)}))\\right]={\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0\\hat{{\\mathcal {Z}}}_a}F(\\hat{^{(a)}, {(a)}).",
"}We CLAIM that for all \\right.a>0 and any nonnegative measurable functional \\Phi on {(a)}\\times {\\mathbb {T}},\\begin{eqnarray}{\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0\\hat{{\\mathcal {Z}}}_a}F({(a)})\\exp \\lbrace -\\langle {\\mathcal {M}}_a^{\\eta _0},\\Phi \\rangle \\rbrace \\right]={\\mathbb {N}}^{\\psi _q}\\left[F({(a)})\\exp \\lbrace -\\langle {\\mathcal {M}}_a,\\Phi \\rangle \\rbrace \\right],\\end{eqnarray}where{\\mathcal {M}}_a^{\\eta _0}(dx, d=\\sum _{i\\in I_{\\eta _0}} {\\bf 1}_{|x_i|\\le a}\\delta _{(x_i, i^{(a-|x_i|)})}(dx, dand{\\mathcal {M}}_a(dx, d=\\sum _{i\\in I}{\\bf 1}_{|x_i|\\le a}\\delta _{(x_i, i^{(a-|x_i|)})}(dx, d.Then with (\\ref {MainA04}) in hand, we have\\begin{eqnarray*}{\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0\\hat{{\\mathcal {Z}}}_a}F(\\hat{^{(a)}, {(a)})\\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0\\hat{{\\mathcal {Z}}}_a}F({(a)}\\otimes _{i\\in I_{\\eta _0}, |x_i|\\le a}(i^{(a-|x_i|)}, x_i), {(a)})\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _q}\\left[F({(a)}\\otimes _{i\\in I, |x_i|\\le a}(i^{(a-|x_i|)}, x_i), {(a)})\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _q}\\left[F(\\tilde{^{(a)}, {(a)}),}which, together with (\\ref {MainA05}), gives that\\right.",
"{\\mathbb {N}}^{\\psi _0}\\left[F({(a)}, {\\mathcal {M}}_{\\alpha _{0,q}, p_{0,q}}({(a)}))\\right]={\\mathbb {N}}^{\\psi _q}\\left[F(\\tilde{^{(a)}, {(a)}).Since a is arbitrary, assertion (c) follows readily.",
"}\\bigskip \\right.",
"{\\it Step 5:} The remainder of this proof is devoted to (\\ref {MainA04}).Defineg(a, x)={\\mathbb {N}}^{\\psi _0}\\left[1-e^{-\\Phi (x, {a-|x|})}\\right].Then we have{\\mathbb {P}}_z^{\\psi _0}\\left(1-e^{-\\Phi (x, {(a-|x|)})}\\right)=1-e^{-zg(a, x)}.First, by property of Poisson random measure,\\begin{eqnarray}\\!\\!\\!&\\!\\!\\!&{\\mathbb {N}}^{\\psi _q}\\left[F({(a)})\\exp \\lbrace -\\langle {\\mathcal {M}}_a, \\Phi \\rangle \\rbrace \\right]\\cr \\!\\!\\!&\\!\\!\\!&\\qquad = {\\mathbb {N}}^{\\psi _q}\\left[F({(a)})\\exp \\left\\lbrace -\\int _{{(a)}}\\mathbf {^}{{(a)}}dxG(a,x)\\right\\rbrace \\right],\\end{eqnarray}where\\begin{eqnarray}G(a,x)=\\left[\\int _0^q\\beta _{\\theta }d\\theta g(a, x)+\\int _0^qd\\theta \\int _0^{\\infty }n_{\\theta }(dz)\\left(1-e^{-zg(a, x)}\\right)\\right].\\end{eqnarray}By (\\ref {Gir}),\\begin{eqnarray}g(a,\\theta , x)\\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _0}\\left[1-e^{-\\Phi (x, {a-|x|})}\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _0^{\\eta _0}}\\left[e^{\\eta _0{\\mathcal {Z}}_{a-|x|}}\\left(1-e^{-\\Phi (x, {a-|x|})}\\right)\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _0^{\\eta _0}}\\left[e^{\\eta _0{\\mathcal {Z}}_{a-|x|}}-1+1-e^{-\\Phi (x, {a-|x|})+\\eta _0{\\mathcal {Z}}_{a-|x|}}\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\eta _0+{\\mathbb {N}}^{\\psi _0^{\\eta _0}}\\left[1-e^{-\\Phi (x, {a-|x|})+\\eta _0{\\mathcal {Z}}_{a-|x|}}\\right]=:\\eta _0+g_{\\eta _0}(a,x),\\end{eqnarray}where the last equality follows from (\\ref {Nt0}).",
"With (\\ref {MainApsiq}) in hand, substituting (\\ref {MainA04d}) into (\\ref {MainA04c}) yields\\begin{eqnarray}G(a,x)\\!\\!\\!&=\\!\\!\\!&\\int _0^q\\beta _{\\theta }d\\theta \\eta _0-\\int _0^qd\\theta \\int _0^{\\infty }e^{-z\\eta _0}n_{\\theta }(dz)\\left(1-e^{z\\eta _0}\\right)\\cr \\!\\!\\!&\\!\\!\\!&\\quad +\\int _0^q\\beta _{\\theta }d\\theta g_{\\eta _0}(a,x)+\\int _0^qd\\theta \\int _0^{\\infty }e^{-z\\eta _0}n_{\\theta }(dz)\\left(1-e^{-zg_{\\eta _0}(a,x)}\\right)\\cr \\!\\!\\!&=\\!\\!\\!&\\psi _q(\\eta _0)+\\int _0^q\\beta _{\\theta }d\\theta g_{\\eta _0}(a,x)+\\int _0^qd\\theta \\int _0^{\\infty }e^{-z\\eta _0}n_{\\theta }(dz)\\left(1-e^{-zg_{\\eta _0}(a,x)}\\right)\\cr \\!\\!\\!&=:\\!\\!\\!&\\psi _q(\\eta _0)+G(\\eta _0,a,x).\\end{eqnarray}Applying (\\ref {Gir}) to (\\ref {MainA04a}) gives\\begin{eqnarray}\\!\\!\\!&\\!\\!\\!&{\\mathbb {N}}^{\\psi _q}\\left[F({(a)})\\exp \\lbrace -\\langle {\\mathcal {M}}_a, \\Phi \\rangle \\rbrace \\right]\\cr \\!\\!\\!&\\!\\!\\!&\\qquad = {\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[\\exp \\left\\lbrace {\\eta _0{\\mathcal {Z}}_{a}}+\\psi _q(\\eta _0)\\int _0^{a}{\\mathcal {Z}}_sds\\right\\rbrace F({(a)})\\exp \\left\\lbrace -\\int _{{(a)}}\\mathbf {^}{{(a)}}(dx)G(a,x)\\right\\rbrace \\right]\\cr \\!\\!\\!&\\!\\!\\!&\\qquad ={\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0{\\mathcal {Z}}_{a}}F({(a)})\\exp \\left\\lbrace -\\int _{{(a)}}\\mathbf {^}{{(a)}}(dx)G(\\eta _0,a,x)\\right\\rbrace \\right],\\end{eqnarray}where the last equality follows from (\\ref {MainA04b}) and the fact \\mathbf {^}{{(a)}}({(a)})=\\int _0^{a}{\\mathcal {Z}}_sds.On the other hand, by similar reasonings, we also have\\begin{eqnarray*}\\!\\!\\!&\\!\\!\\!&{\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0\\hat{{\\mathcal {Z}}}_a}F({(a)})\\exp \\lbrace -\\langle \\Phi , {\\mathcal {M}}_a^{\\eta _0}\\rbrace \\right]\\cr \\!\\!\\!&\\!\\!\\!&\\quad ={\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0{Z}_a}F({(a)})\\exp \\left\\lbrace -\\langle \\Phi , {\\mathcal {M}}_a^{\\eta _0}\\rangle +\\eta _0\\sum _{i\\in I_{\\eta _0}}{\\bf 1}_{|x_i|\\le a}{\\mathcal {Z}}_{a-|x_i|}^{i}\\right\\rbrace \\right]\\cr \\!\\!\\!&\\!\\!\\!&\\quad ={\\mathbb {N}}^{\\psi _q^{\\eta _0}}\\left[e^{\\eta _0{\\mathcal {Z}}_{a}}F({(a)})\\exp \\left\\lbrace -\\int _{{(a)}}\\mathbf {^}{{(a)}}(dx)G(\\eta _0,a,x)\\right\\rbrace \\right],\\end{eqnarray*}which, together with (\\ref {MainA04e}), implies (\\ref {MainA04}).",
"We have completed the proof.",
"\\hfill \\Box \\medskip }A direct consequence of Theorem \\ref {MainA} is that\\right.\\begin{corollary}Assume that \\lbrace \\psi _t:t\\in {\\mathfrak {T}}\\rbrace is an admissible family satisfying (H1-3).",
"Then for r>0 we havethe tree-valued process (q^t,\\; q\\in {\\mathfrak {T}}_t) is aMarkov process under {\\mathbb {P}}^{\\psi _t}_r and for fixed q\\in {\\mathfrak {T}}_t, the distribution of q^t under {\\mathbb {P}}_r^{\\psi _t}is {\\mathbb {P}}_r^{\\psi _q}.\\end{corollary}\\end{eqnarray*}\\end{eqnarray}}}}}\\end{eqnarray*}\\end{eqnarray*}}\\section {a tree-valued process}Because of the pruning procedure, we have t_{q}\\subset t_{p}for p\\le q\\in {\\mathfrak {T}}_t.",
"The process(q^t, q\\in {\\mathfrak {T}}_t) is a non-increasing process (for the inclusion oftrees) and is càdlàg.",
"Theorem \\ref {MainA} implies that for t_1\\le t_2\\in {\\mathfrak {T}}, \\lbrace q^{t_2}: q\\in {\\mathfrak {T}}_{t_2}\\rbrace under {\\mathbb {N}}^{\\psi _{t_2}} has the same distribution as \\lbrace q^{t_1}: q\\in {\\mathfrak {T}}_{t_2}\\rbrace under {\\mathbb {N}}^{\\psi _{t_1}}.",
"By Kolmogorov^{\\prime }s theorem, there exists a tree-valued Markov process \\lbrace t: t\\in {\\mathfrak {T}}\\rbrace such that \\lbrace q: q\\in {\\mathfrak {T}}_t\\rbrace has the same finite dimensional distribution as \\lbrace q^t: q\\in {\\mathfrak {T}}_t\\rbrace under {\\mathbb {N}}^{\\psi _t}.",
"Denote by {\\mathbf {N}}^{\\Psi } the law of (t: t\\in {\\mathfrak {T}}).",
"We have for any nonnegative measurable functional F,{\\mathbf {N}}^{\\Psi }[F(q)]={\\mathbb {N}}^{\\psi _q}[F(].Set\\sigma _{t}=\\mathbf {^}{{\\mathcal {T}}_t}({\\mathcal {T}}_t),\\quad t\\in {\\mathfrak {T}}.Then one can check that \\lbrace \\sigma _t: t\\in {\\mathfrak {T}}\\rbrace is a non-increasing [0,\\infty ]-Markov process.",
"In the sequel, we will use the version of \\lbrace t: t\\in {\\mathfrak {T}}\\rbrace such that \\lbrace \\sigma _t: t\\in {\\mathfrak {T}}\\rbrace is càdlàg.",
"For t\\in {\\mathfrak {T}}, set \\Psi _t=\\lbrace \\psi _q: q\\in {\\mathfrak {T}}_t\\rbrace and \\Psi _t^{\\eta _t}=\\lbrace \\psi _q^{\\eta _t}: q\\in {\\mathfrak {T}}_t\\rbrace .\\begin{proposition}For t\\in {\\mathfrak {T}} and any non-negative measurable functional F,{\\mathbf {N}}^{\\Psi _t}[F({q}: q\\in {\\mathfrak {T}}_t){\\bf 1}_{\\lbrace \\sigma _t<\\infty \\rbrace }]={\\mathbf {N}}^{\\Psi _t^{\\eta _t}}[F(q: q\\in {\\mathfrak {T}}_t)].\\end{proposition}\\begin{xmlelement*}{proof}According to arguments in {\\it Step 2} in the proof of Theorem \\ref {MainA}, \\lbrace \\psi _q: q\\in {\\mathfrak {T}}_t\\rbrace and \\lbrace \\psi _q^{\\eta _t}: q\\in {\\mathfrak {T}}_t\\rbrace induce the same pruning parameters.",
"Then the desired result is a direct consequence of the fact {\\mathbb {N}}^{\\psi _t}[F({\\bf 1}_{\\lbrace \\sigma <\\infty \\rbrace }]={\\mathbb {N}}^{\\psi _t^{\\eta _t}}[F(]; see (\\ref {Gir3}).",
"\\hfill \\Box \\medskip \\end{xmlelement*}We then study the behavior of \\lbrace \\sigma _t: t\\in {\\mathfrak {T}}\\rbrace .",
"}\\begin{lemma} For t\\le q\\in {\\mathfrak {T}} and \\lambda \\ge 0,{\\mathbf {N}}^{\\Psi }[e^{-\\lambda \\sigma _t}|{q}]=\\exp \\lbrace {-\\psi _{q}(\\psi _t^{-1}(\\lambda ))\\sigma _{q}}\\rbrace and {\\mathbf {N}}^{\\Psi }[\\sigma _t<+\\infty |{q}]=\\exp \\lbrace {-\\psi _{q}(\\psi _t^{-1}(0))\\sigma _{q}}\\rbrace .",
"Moreover, if \\psi _t is subcritical, then\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }[\\sigma _t|q]=\\psi ^{\\prime }_q(0)\\sigma _q/\\psi ^{\\prime }_t(0).\\end{eqnarray}\\end{lemma}\\begin{xmlelement*}{proof}Recall (\\ref {zeta}) and (\\ref {sigma}).",
"By (c) in Theorem \\ref {MainA},\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }\\left[e^{-\\lambda \\sigma _t}|{q}\\right]\\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[e^{-\\lambda \\sigma _{q}-\\lambda \\sum _{i\\in I}\\sigma _i}|{q}\\right]\\cr \\!\\!\\!&=\\!\\!\\!&e^{-\\lambda \\sigma _{q}}e^{-\\lambda \\int _{{q}}\\mathbf {^}{{q}}(dx)G(\\lambda )},\\end{eqnarray*}where \\sigma _i=\\mathbf {^}{i}(i) and\\begin{eqnarray*}G(\\lambda )\\!\\!\\!&=\\!\\!\\!&\\int _t^{q}\\beta _{\\theta }d{\\theta }{\\mathbb {N}}^{\\psi _t}\\left[1-e^{-\\lambda \\sigma }\\right]+\\int _t^{q}n_{\\theta }(dz){\\mathbb {P}}^{\\psi _t}_z(1-e^{-\\lambda \\sigma })\\cr \\!\\!\\!&=\\!\\!\\!&\\int _t^{q}\\beta _{\\theta }d{\\theta } \\psi _t^{-1}(\\lambda )+\\int _t^{q}n_{\\theta }(dz)\\left(1-e^{-z\\psi _t^{-1}(\\lambda )}\\right)\\cr \\!\\!\\!&=\\!\\!\\!&\\int _t^q\\zeta _{\\theta }(\\psi _t^{-1}(\\lambda ))\\cr \\!\\!\\!&=\\!\\!\\!&\\psi _{q}(\\psi _t^{-1}(\\lambda ))-\\psi _{t}(\\psi _t^{-1}(\\lambda )).\\end{eqnarray*}Thus\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }[e^{-\\lambda \\sigma _t}|{q}]=\\exp \\lbrace {-\\psi _{q}(\\psi _t^{-1}(\\lambda ))\\sigma _{q}}\\rbrace .\\end{eqnarray}Then,{\\mathbf {N}}^{\\Psi }[\\sigma _t<+\\infty |{q}]=\\lim _{\\lambda \\rightarrow 0}{\\mathbf {N}}^{\\Psi }[e^{-\\lambda \\sigma _t}|{q}]=\\exp \\lbrace {-\\psi _{q}(\\psi _t^{-1}(0))\\sigma _{q}}\\rbrace .If \\psi _t is subcritical, then {\\mathbf {N}}^{\\Psi }-a.e.",
"\\sigma _t<\\infty .",
"We obtain {\\mathbf {N}}^{\\Psi }[\\sigma _t|q]=\\frac{d}{d\\lambda }{\\mathbf {N}}^{\\Psi }[e^{-\\lambda \\sigma _t}|{q}]\\big {|}_{\\lambda =0}=\\psi ^{\\prime }_q(0)\\sigma _q/\\psi _t^{\\prime }(0).\\hfill \\Box \\medskip \\end{xmlelement*}}}}Recall that \\eta _q the largest root of \\psi _{q}(\\lambda )=0.",
"Thus \\eta _q=\\lim _{\\lambda \\rightarrow 0+}\\psi _q^{-1}(\\lambda )=\\psi _q^{-1}(0).",
"Define the {\\it ascension time}A=\\inf \\lbrace t\\in {\\mathfrak {T}}: \\sigma _{t}=+\\infty \\rbrace with the convention that \\inf \\lbrace \\emptyset \\rbrace =\\inf {\\mathfrak {T}}=:t_{\\infty }\\in [-\\infty ,0].",
"In the sequel of this paper, we always assume that0\\in {\\mathfrak {T}},\\quad t_{\\infty }<0.Recall (\\ref {zeta}).",
"Let us consider the following condition:}\\begin{eqnarray}\\lim _{t\\rightarrow t_{\\infty }+}\\int ^0_{t}\\zeta _{\\theta }(\\lambda )d\\theta =\\psi _0(\\lambda )-\\lim _{t\\rightarrow t_{\\infty }+}\\psi _t(\\lambda )<+\\infty ,\\quad \\text{ for some }\\lambda >0.\\end{eqnarray}\\begin{proposition} \\lim _{q\\rightarrow t_{\\infty }+}\\psi _q^{-1}(0)<\\infty if and only if (\\ref {bmcon1}) holds.\\end{proposition}\\begin{xmlelement*}{proof}{\\it ``if^{\\prime \\prime } part:} (\\ref {bmcon1}) implies\\begin{eqnarray}\\lim _{t\\rightarrow t_{\\infty }+}\\int ^0_{t}\\beta _{\\theta }d\\theta +\\lim _{t\\rightarrow t_{\\infty }+}\\int ^0_{t}\\int _0^{\\infty }(1\\wedge z)n_{\\theta }(dz)d\\theta <+\\infty .\\end{eqnarray}Since 1\\wedge z^2\\le 1\\wedge z, by (\\ref {m_q}) and (\\ref {bmcon2}), we have \\sup _{q\\in {\\mathfrak {T}}}\\int _0^{\\infty }(1\\wedge z^2)m_q(dz)<\\infty .Then monotonicity of q\\mapsto (1\\wedge z^2)m_q(dz) yields that there exists a \\sigma -finite measure m_{t_{\\infty }}(dz) on (0, \\infty ) such that\\int (1\\wedge z^2)m_{t_{\\infty }}(dz)<+\\infty and as {q\\rightarrow t_{\\infty }},(1\\wedge z^2)m_q(dz){\\rightarrow } (1\\wedge z^2)m_{t_{\\infty }}(dz)\\quad \\text{in } M_f((0,\\infty )).Then we can define\\psi _{t_{\\infty }}(\\lambda )=b_{t_{\\infty }}\\lambda +c\\lambda ^2+\\int _0^{\\infty }(e^{-\\lambda z}-1+\\lambda z{\\bf 1}_{\\lbrace z\\le 1\\rbrace })m_{t_{\\infty }}(dz),\\quad \\lambda \\ge 0,where b_{t_{\\infty }}=b_0+\\int _1^{\\infty }zm_0(dz)-\\lim _{q\\rightarrow t_{\\infty }+}\\left(\\int ^0_{q}\\beta _{\\theta }d\\theta +\\int ^0_q\\int _0^1z n_{\\theta }(dz)d\\theta \\right).\\psi _{t_{\\infty }}(\\lambda ) is a convex function.",
"Since \\psi _q satisfies (H3) which implies c>0 or \\int _0^1zm_q(dz)=\\infty , we have \\lim _{\\lambda \\rightarrow \\infty }\\psi _{t_{\\infty }}(\\lambda )=\\infty and \\psi ^{-1}_{t_{\\infty }}(0)<\\infty which is the largest root of \\psi _{t_{\\infty }}(\\lambda )=0.Meanwhile, note that e^{-\\lambda z}-1+\\lambda z{\\bf 1}_{\\lbrace z\\le 1\\rbrace }\\le 1\\wedge z^2.",
"By (\\ref {b_q}) and (\\ref {m_q}),\\begin{eqnarray*}\\psi _q(\\lambda )\\!\\!\\!&=\\!\\!\\!&\\left(b_0+\\int _1^{\\infty }zm_0(dz)-\\int ^0_q\\beta _{\\theta }d\\theta -\\int ^0_q\\int _0^{1}zn_{\\theta }(dz)d\\theta \\right)\\lambda \\cr \\!\\!\\!&\\!\\!\\!&\\quad +c\\lambda ^2+\\int _0^{\\infty }(e^{-\\lambda z}-1+\\lambda z{\\bf 1}_{\\lbrace z\\le 1\\rbrace })m_q(dz)\\cr \\!\\!\\!&\\rightarrow \\!\\!\\!&\\psi _{t_{\\infty }}(\\lambda ),\\quad \\text{as }q\\rightarrow t_{\\infty }+.\\end{eqnarray*}\\end{xmlelement*}Then the fact \\lim _{q\\rightarrow t_{\\infty }+}\\psi _q(\\lambda )\\rightarrow \\psi _{t_{\\infty }}(\\lambda ) yields \\psi _{t_{\\infty }}^{-1}(0)=\\lim _{q\\rightarrow t_{\\infty }+}\\psi _q^{-1}(0)<\\infty .",
"}\\smallskip }{\\it ``only if^{\\prime \\prime } part:} If \\int _{t_{\\infty }-}\\zeta _{\\theta }(\\lambda )d\\theta =+\\infty for some \\lambda >0 ( hence for all \\lambda >0 ), by (\\ref {b_q}) and (\\ref {m_q}),\\begin{eqnarray*}\\psi _q(\\lambda )=\\psi _0(\\lambda )-\\int ^0_q\\zeta _{\\theta }(\\lambda )d\\theta \\rightarrow -\\infty ,\\quad \\text{as }q\\rightarrow t_{\\infty }.\\end{eqnarray*}Then we have\\lim _{q\\rightarrow t_{\\infty }}\\psi _q^{-1}(0)=+\\infty .",
"\\hfill \\Box \\medskip }Define \\psi _{t_{\\infty }}^{-1}(0)=\\lim _{q\\rightarrow t_{\\infty }}\\psi _q^{-1}(0) and\\begin{eqnarray*}{\\mathfrak {T}}_{\\infty }={\\left\\lbrace \\begin{array}{ll}{\\mathfrak {T}}\\cup \\lbrace t_{\\infty }\\rbrace ,&\\psi _{t_{\\infty }}^{-1}(0)<+\\infty \\\\{\\mathfrak {T}},& \\psi _{t_{\\infty }}^{-1}(0)=+\\infty .\\end{array}\\right.",
"}\\end{eqnarray*}\\begin{remark}From the proof of Proposition \\ref {lemtinf} we see that it is possible to extend the definition of a given admissible family to {\\mathfrak {T}}_{\\infty }.",
"For some results in the sequel of this paper, we need to avoid this case by assuming t_{\\infty }\\notin {\\mathfrak {T}}_{\\infty } (hence t_{\\infty }\\notin {\\mathfrak {T}}).\\end{remark}Next, we study the distribution of A and A.\\begin{lemma} For q\\in {\\mathfrak {T}}\\cup \\lbrace t_{\\infty }\\rbrace ,{\\mathbf {N}}^{\\Psi }[A>q]=\\psi _{q}^{-1}(0).and\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }[A=t_{\\infty }]={\\left\\lbrace \\begin{array}{ll} 0,& t_{\\infty }\\notin {\\mathfrak {T}}_{\\infty }\\\\\\infty ,& t_{\\infty }\\in {\\mathfrak {T}}_{\\infty }.\\end{array}\\right.",
"}\\end{eqnarray}\\end{lemma}\\begin{xmlelement*}{proof}Recall (\\ref {sigma}).",
"By Lemma \\ref {lemsigma}, for q>t_{\\infty }\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }[A>q]\\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[\\sigma _{q}=+\\infty \\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _{q}}\\left[\\sigma =+\\infty \\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\lim _{\\lambda \\rightarrow 0}{\\mathbb {N}}^{\\psi _{q}}\\left[1-e^{-\\lambda \\sigma }\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\lim _{\\lambda \\rightarrow 0}\\psi _{q}^{-1}(\\lambda )\\cr \\!\\!\\!&=\\!\\!\\!&\\psi _{q}^{-1}(0).\\end{eqnarray*}And letting q\\rightarrow t_{\\infty } gives the case of q=t_{\\infty }.",
"By applying Lemma \\ref {lemsigma} again, we obtain\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }\\left[A=t_{\\infty }|0\\right]\\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[\\forall q>t_{\\infty }, \\sigma _q<+\\infty |0\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\lim _{q\\rightarrow t_{\\infty }}{\\mathbf {N}}^{\\Psi }\\left[ \\sigma _q<+\\infty |0\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\lim _{q\\rightarrow t_{\\infty }}e^{-\\psi _0(\\psi _q^{-1}(0))\\sigma _0}\\cr \\!\\!\\!&=\\!\\!\\!&{\\left\\lbrace \\begin{array}{ll}0,& \\text{ if }t_{\\infty }\\notin {\\mathfrak {T}}_{\\infty }\\\\e^{-\\sigma _0\\psi _0(\\psi _{t_{\\infty }}^{-1}(0))},& \\text{ if }t_{\\infty }\\in {\\mathfrak {T}}_{\\infty }.\\end{array}\\right.",
"}\\end{eqnarray*}Then noting that \\forall \\lambda >0, {\\mathbb {N}}^{\\psi _0}[e^{-\\lambda \\sigma }]=+\\infty gives the desired result.",
"\\hfill \\Box \\medskip \\end{xmlelement*}\\begin{remark}(\\ref {lemA1}) implies that if t_{\\infty }\\in {\\mathfrak {T}}_{\\infty }, then {\\mathbf {N}}^{\\Psi }[t \\text{ is compact for all }t>t_{\\infty }]=+\\infty .",
"If t_{\\infty }\\notin {\\mathfrak {T}}_{\\infty }, then {\\mathbf {N}}^{\\Psi }-a.e.",
"there exists t\\in {\\mathfrak {T}} such that q is not compact (\\sigma _q=\\infty ) for t>q\\in {\\mathfrak {T}}.\\end{remark}}}\\begin{theo} Assume that \\psi _0 is critical.",
"For q\\in (t_{\\infty }, 0) and any nonnegative measurable functional F on {\\mathbb {T}},\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }[F(A)|A=q]=\\psi ^{\\prime }_{q}(\\eta _{q}){\\mathbb {N}}^{\\psi _{q}}[F(\\sigma {\\bf 1}_{\\lbrace \\sigma <\\infty \\rbrace }]\\end{eqnarray}and for \\lambda \\ge 0\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }[e^{-\\lambda \\sigma _A}|A=q]=\\frac{\\psi ^{\\prime }_{q}(\\eta _{q})}{\\psi ^{\\prime }_{q}(\\psi _{q}^{-1}(\\lambda ))}.\\end{eqnarray}In particular, we have\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }[\\sigma _A<\\infty |A=q]=1.\\end{eqnarray}\\end{theo}\\begin{xmlelement*}{proof}By Lemma \\ref {lemsigma}, we have for every t_{\\infty }<t\\le q<0,\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }\\left[F({q}){\\bf 1}_{\\lbrace A>t\\rbrace }\\right]\\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[F({q}){\\bf 1}_{\\lbrace \\sigma _t=+\\infty \\rbrace }\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[F({q}){\\mathbf {N}}^{\\Psi }[\\sigma _t=+\\infty |{q}]\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[F({q})\\left(1-e^{-\\sigma _{q}\\psi _{q}(\\psi _t^{-1}(0))}\\right)\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[F({q})\\left(1-e^{-\\sigma _{q}\\psi _{q}(\\eta _{t})}\\right)\\right].\\end{eqnarray*}Since \\eta _t is the largest root of \\psi _t(s)=0, we have the mapping t\\mapsto \\eta _{t} is differentiable with\\begin{eqnarray}\\frac{d\\eta _{t}}{dt}=-\\frac{\\zeta _{t}(\\eta _{t})}{\\psi ^{\\prime }_{t}(\\eta _{t})}.\\end{eqnarray}Then we get\\begin{eqnarray*}\\!\\!\\!&\\!\\!\\!&\\frac{d}{dt}{\\mathbf {N}}^{\\Psi }\\left[F({q}){\\bf 1}_{\\lbrace A>t\\rbrace }\\right]\\cr \\!\\!\\!&\\!\\!\\!&\\qquad ={\\mathbf {N}}^{\\Psi }\\left[F({q})\\sigma _{q}e^{-\\sigma _{q}\\psi _{q}(\\eta _{t})} \\right]\\frac{d\\psi _{q}(\\eta _{t})}{dt}\\cr \\!\\!\\!&\\!\\!\\!&\\qquad = -{\\mathbf {N}}^{\\Psi }\\left[F({q})\\sigma _{q}e^{-\\sigma _{q}\\psi _{q}(\\eta _{t})} \\right]\\frac{\\psi ^{\\prime }_{q}(\\eta _{t})\\zeta _{t}(\\eta _{t})}{\\psi ^{\\prime }_{t}(\\eta _{t})}.\\end{eqnarray*}The right-continuity of \\sigma gives\\begin{eqnarray*}\\frac{{\\mathbf {N}}^{\\Psi }\\left[F(A), A\\in dq\\right]}{dq}\\!\\!\\!&=\\!\\!\\!&-\\frac{d}{dt}\\left({\\mathbf {N}}^{\\Psi }\\left[F({q}){\\bf 1}_{\\lbrace A>t\\rbrace }\\right]\\right)\\bigg {|}_{t=q}\\cr \\!\\!\\!&=\\!\\!\\!&\\zeta _{t}(\\eta _{t}){\\mathbf {N}}^{\\Psi }\\left[F({q})\\sigma _{q}{\\bf 1}_{\\lbrace \\sigma _{q}<+\\infty \\rbrace }\\right].\\end{eqnarray*}Thus\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }\\left[F(A)| A=q\\right]=\\frac{ {\\mathbf {N}}^{\\Psi }\\left[F({q})\\sigma _{q}{\\bf 1}_{\\lbrace \\sigma _{q}<+\\infty \\rbrace }\\right]}{{\\mathbf {N}}^{\\Psi }\\left[\\sigma _{q}{\\bf 1}_{\\lbrace \\sigma _{q}<+\\infty \\rbrace }\\right]}=\\frac{ {\\mathbb {N}}^{\\psi _{q}}\\left[F(\\sigma {\\bf 1}_{\\lbrace \\sigma <+\\infty \\rbrace }\\right]}{{\\mathbb {N}}^{\\psi _{q}}\\left[\\sigma {\\bf 1}_{\\lbrace \\sigma <+\\infty \\rbrace }\\right]}.\\end{eqnarray}Meanwhile, note that{\\mathbb {N}}^{\\psi _{q}}\\left[\\sigma e^{-r\\sigma }\\right]=\\frac{d}{dr}{\\mathbb {N}}^{\\psi _{q}}\\left[1-e^{-r\\sigma }\\right]=\\frac{d}{dr}\\psi _{q}^{-1}(r)=\\frac{1}{\\psi ^{\\prime }_{q}(\\psi _{q}^{-1}(r))}.Then we have{\\mathbb {N}}^{\\psi _{q}}\\left[\\sigma {\\bf 1}_{\\lbrace \\sigma <+\\infty \\rbrace }\\right]=\\lim _{r\\rightarrow 0}{\\mathbb {N}}^{\\psi _{q}}\\left[\\sigma e^{-r\\sigma }\\right]=\\frac{1}{\\psi ^{\\prime }_{q}(\\eta _{q})},which, together with (\\ref {TA1}), implies (\\ref {TA}).",
"Using (\\ref {TA1}) again, we also get\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }\\left[e^{-\\lambda \\sigma _A}|A=q\\right]=\\frac{ {\\mathbb {N}}^{\\psi _{q}}\\left[e^{-\\lambda \\sigma }\\sigma \\right]}{{\\mathbb {N}}^{\\psi _{q}}\\left[\\sigma {\\bf 1}_{\\lbrace \\sigma <+\\infty \\rbrace }\\right]}=\\frac{\\psi ^{\\prime }_{q}(\\eta _{q})}{\\psi ^{\\prime }_{q}(\\psi _{q}^{-1}(\\lambda ))},\\end{eqnarray}which is just (\\ref {sigmaA}).",
"Then (\\ref {sigmaA1}) follows readily by letting \\lambda in (\\ref {sigmaA2}) go to 0.",
"\\hfill \\Box \\medskip \\end{xmlelement*}}}\\bigskip \\begin{proposition}Assume t_{\\infty }\\in {\\mathfrak {T}} and \\psi _0 is critical.",
"Then for any nonnegative measurable functional F on {\\mathbb {T}},{\\mathbf {N}}^{\\Psi }\\left[F(A){\\bf 1}_{\\lbrace A=t_{\\infty }\\rbrace }\\right]={\\mathbb {N}}^{\\psi _{t_{\\infty }}^{\\eta _{t_{\\infty }}}}\\left[F(\\right],where \\psi _{t_{\\infty }}^{\\eta _{t_{\\infty }}}(\\cdot )= \\psi _{t_{\\infty }}(\\eta _{t_{\\infty }}+\\cdot ).Typically, for \\lambda \\ge 0{\\mathbf {N}}^{\\Psi }\\left[(1-e^{-\\lambda \\sigma _A}){\\bf 1}_{\\lbrace A=t_{\\infty }\\rbrace }\\right] =\\psi _{t_{\\infty }}^{-1}(\\lambda )-\\eta _{t_{\\infty }}.\\end{proposition}\\begin{xmlelement*}{proof}\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }\\left[F(A){\\bf 1}_{\\lbrace A=t_{\\infty }\\rbrace }\\right]\\!\\!\\!&=\\!\\!\\!&{\\mathbf {N}}^{\\Psi }\\left[F({t_{\\infty }}){\\bf 1}_{\\lbrace \\sigma _{t_{\\infty }}<+\\infty \\rbrace }\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _{t_{\\infty }}}\\left[F({\\bf 1}_{\\lbrace \\sigma <+\\infty \\rbrace }\\right]\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _{t_{\\infty }}^{\\eta _{t_{\\infty }}}}\\left[F(\\right],\\end{eqnarray}where the last equality follows from (\\ref {Gir3}).Thus\\begin{eqnarray}{\\mathbf {N}}^{\\Psi }\\left[(1-e^{-\\lambda \\sigma _A}){\\bf 1}_{\\lbrace A=t_{\\infty }\\rbrace }\\right]\\!\\!\\!&=\\!\\!\\!&{\\mathbb {N}}^{\\psi _{t_{\\infty }}^{\\eta _{t_{\\infty }}}}\\left[1-e^{-\\lambda \\sigma }\\right]\\cr \\!\\!\\!&=\\!\\!\\!&(\\psi _{t_{\\infty }}^{\\eta _{t_{\\infty }}})^{-1}(\\lambda )\\cr \\!\\!\\!&=\\!\\!\\!&\\psi _{t_{\\infty }}^{-1}(\\lambda )-\\eta _{t_{\\infty }}.\\end{eqnarray}\\hfill \\Box \\medskip \\end{xmlelement*}Recall that {\\mathfrak {T}}_t={\\mathfrak {T}}\\cap [t,\\infty ) for t\\in {\\mathfrak {T}}.",
"According to the proof in {\\it Step 2} in Theorem \\ref {MainA}, for q\\in {\\mathfrak {T}}, \\Psi _q^{\\eta _q}=\\lbrace \\psi _t^{\\eta _q}: t\\in {\\mathfrak {T}}_q\\rbrace is also an admissible family satisfying (H1-3), where \\psi _t^{\\eta _q}(\\cdot )=\\psi _t(\\eta _q+\\cdot )-\\psi _t(\\eta _q).",
"Set {\\mathfrak {T}}^q_0=\\lbrace \\theta \\ge 0: \\theta +q\\in {\\mathfrak {T}}_q\\rbrace and \\Psi ^q=\\lbrace \\psi _{\\theta +q}^{\\eta _q}: \\theta \\in {\\mathfrak {T}}_0^q\\rbrace .",
"Then we have the following corollary.\\begin{corollary}Assume that \\psi _0 is critical.",
"For q\\in (t_{\\infty }, 0), for any nonnegative measurable functional F{\\mathbf {N}}^{\\Psi }[F({A+t}: t\\in {\\mathfrak {T}}^q_0)|A=q]=\\psi ^{\\prime }_{q}(\\eta _{q}){\\mathbf {N}}^{\\Psi ^q}[F(t: t\\in {\\mathfrak {T}}^q_0)\\sigma _0].\\end{corollary}\\begin{xmlelement*}{proof}Applying (\\ref {Gir3}) to (\\ref {TA}), we have for any nonnegative measurable functional F on {\\mathbb {T}},{\\mathbf {N}}^{\\Psi }[F(A)|A=q]=\\psi ^{\\prime }_{q}(\\eta _{q}){\\mathbb {N}}^{\\psi _{q}^{\\eta _q}}[F(\\sigma ].Note that {\\mathfrak {T}}_0^q={\\mathfrak {T}}_q-q.",
"Then the desired result follows from the fact that \\lbrace \\psi _t: t\\in {\\mathfrak {T}}_q\\rbrace and \\lbrace \\psi _t^{\\eta _q}: t\\in {\\mathfrak {T}}_q\\rbrace induce the same pruning parameters.",
"\\hfill \\Box \\medskip \\end{xmlelement*}An application of Corollary \\ref {coroTA+} is to study the distribution of exit times.",
"DefineA_h=\\sup \\lbrace t\\in {\\mathfrak {T}}: H_{max}(t)>h\\rbrace ,\\quad h>0,with convention \\sup \\emptyset =t_{\\infty }.",
"Then A_h is the first time that the height of the trees is larger than h.\\begin{proposition}Assume that \\psi _0 is critical.",
"For t_{\\infty }<q<q_0<0, we have\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }[A_h>q_0|A=q]\\!\\!\\!&=\\!\\!\\!&1-\\psi ^{\\prime }_{q_0}(\\eta _q)\\psi _q^{\\eta _q}(v^{\\psi _q^{\\eta _q}}(h))\\int _{v^{\\psi _q^{\\eta _q}}(h)}^{\\infty }\\frac{dr}{\\psi _q^{\\eta _q}(r)^2};\\cr {\\mathbf {N}}^{\\Psi }[A_h=A|A=q]\\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_{q}(\\eta _q)\\psi _q^{\\eta _q}(v^{\\psi _q^{\\eta _q}}(h))\\int _{v^{\\psi _q^{\\eta _q}}(h)}^{\\infty }\\frac{dr}{\\psi _q^{\\eta _q}(r)^2}.\\end{eqnarray*}\\end{proposition}\\begin{xmlelement*}{proof}The second equality follows from the fact {\\mathbf {N}}^{\\Psi }[A_h\\ge q|A=q]=1 and the first equality as q_0\\rightarrow q.We only need to prove the first one.",
"Recall (\\ref {Za}).",
"We write {\\mathcal {Z}}_a( to emphasize that {\\mathcal {Z}}_a corresponds to the tree .",
"Note that\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }[A_h>q_0|A=q]={\\mathbf {N}}^{\\Psi }[{\\mathcal {Z}}_h({q_0})>0|A=q]={\\mathbf {N}}^{\\Psi }[{\\mathcal {Z}}_h({{A+q_0-q}})>0|A=q]\\end{eqnarray*}which, by Corollary \\ref {coroTA+}, is equal to\\psi _q^{\\prime }(\\eta _q){\\mathbf {N}}^{\\Psi ^q}[{\\bf 1}_{\\lbrace {\\mathcal {Z}}_h({q_0-q})>0\\rbrace }\\sigma _0].Since for every t\\in {\\mathfrak {T}}_q, \\psi _t^{\\eta _q} is subcritical, by (\\ref {lemsigma01}), we have\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }[A_h>q_0|A=q]\\!\\!\\!&=\\!\\!\\!&\\psi _q^{\\prime }(\\eta _q){\\mathbf {N}}^{\\Psi ^q}\\left[{\\bf 1}_{\\lbrace {\\mathcal {Z}}_h({q_0-q})>0\\rbrace }{\\mathbf {N}}^{\\Psi ^q}[\\sigma _0|{q_0-q}]\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_{q_0}(\\eta _q){\\mathbf {N}}^{\\Psi ^q}\\left[{\\bf 1}_{\\lbrace {\\mathcal {Z}}_h({q_0-q})>0\\rbrace }\\sigma _{q_0-q}\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_{q_0}(\\eta _q){\\mathbb {N}}^{\\psi _q^{\\eta _q}}\\left[{\\bf 1}_{\\lbrace {\\mathcal {Z}}_h>0\\rbrace }\\sigma \\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_{q_0}(\\eta _q){\\mathbb {N}}^{\\psi _q^{\\eta _q}}[\\sigma ]-\\psi ^{\\prime }_{q_0}(\\eta _q){\\mathbb {N}}^{\\psi _q^{\\eta _q}}\\left[{\\bf 1}_{\\lbrace {\\mathcal {Z}}_h=0\\rbrace }\\int _0^h{\\mathcal {Z}}_ada\\right]\\cr \\!\\!\\!&=\\!\\!\\!&1-\\psi ^{\\prime }_{q_0}(\\eta _q)\\int _0^hda\\lim _{\\lambda \\rightarrow 0}{\\mathbb {N}}^{\\psi _q^{\\eta _q}}\\left[{\\mathcal {Z}}_ae^{-\\lambda {\\mathcal {Z}}_h}\\right].\\end{eqnarray*}Recall (\\ref {va}), (\\ref {ua}) and (\\ref {uva}).",
"Thenby (\\ref {sigma}) and branching property (iv), conditioning on {\\mathcal {Z}}_a, we see\\lim _{\\lambda \\rightarrow 0}{\\mathbb {N}}^{\\psi _q^{\\eta _q}}\\left[{\\mathcal {Z}}_ae^{-\\lambda {\\mathcal {Z}}_h}\\right]=\\lim _{\\lambda \\rightarrow 0}{\\mathbb {N}}^{\\psi _q^{\\eta _q}}\\left[{\\mathcal {Z}}_ae^{-{\\mathcal {Z}}_au^{\\psi _q^{\\eta _q}}(h-a,\\lambda )}\\right]=\\frac{\\partial }{\\partial \\lambda }u^{\\psi _q^{\\eta _q}}(a, v^{\\psi _q^{\\eta _q}}(h-a)).Typically,\\frac{\\partial }{\\partial \\lambda }u^{\\psi _q^{\\eta _q}}(a, v^{\\psi _q^{\\eta _q}}(h-a))=\\frac{\\psi _q^{\\eta _q}(u^{\\psi _q^{\\eta _q}}(a, v^{\\psi _q^{\\eta _q}}(h-a)))}{\\psi _q^{\\eta _q}(v^{\\psi _q^{\\eta _q}}(h-a))}=\\frac{\\psi _q^{\\eta _q}(v^{\\psi _q^{\\eta _q}}(h))}{\\psi _q^{\\eta _q}(v^{\\psi _q^{\\eta _q}}(h-a))^2}\\frac{\\partial }{\\partial a}v^{\\psi _q^{\\eta _q}}(h-a).Thus\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }[A_h>q_0|A=q]\\!\\!\\!&=\\!\\!\\!&1-\\psi ^{\\prime }_{q_0}(\\eta _q)\\int _0^h\\frac{\\partial }{\\partial \\lambda }u^{\\psi _q^{\\eta _q}}(a, v^{\\psi _q^{\\eta _q}}(h-a))da\\cr \\!\\!\\!&=\\!\\!\\!&1-\\psi ^{\\prime }_{q_0}(\\eta _q)\\psi _q^{\\eta _q}(v^{\\psi _q^{\\eta _q}}(h))\\int _{v^{\\psi _q^{\\eta _q}}(h)}^{\\infty }\\frac{dr}{\\psi _q^{\\eta _q}(r)^2}.\\end{eqnarray*}We have completed the proof.",
"\\hfill \\Box \\medskip \\begin{remark}It is easy to see that {\\mathbf {N}}^{\\Psi }[A_h\\ge q]=v^{\\psi _q}(h).\\end{remark}\\begin{remark}With Theorem \\ref {MainA} and Remark \\ref {remincreasing} in hand, similarly to the work in \\cite {ADH0a}, one may give an explicit construction of an increasing tree-valued process which has the same distribution as \\lbrace q: q\\in {\\mathfrak {T}}_t\\rbrace under {\\mathbf {N}}^{\\Psi }.",
"Then by this construction and similar arguments as in \\cite {ADH0a} ( Theorem 4.6 there), one could know the joint distribution of ({A_h-}, {A_h}) (and hence ({A-}, {A})).",
"We left these to interested readers.\\end{remark}\\end{xmlelement*}\\section {Tree at the ascension time}In this section, we study the representation of the tree at the ascension time.",
"We shall always assume that0\\in {\\mathfrak {T}},\\quad t_{\\infty }<0,\\quad \\sup {\\mathfrak {T}}>0.We first consider an infinite CRT and its pruning.",
"An infinite CRT wasconstructed in \\cite {AD12} which is the Lévy CRT conditioned to have infinite height.",
"Before recallingits construction, we stress that under {\\mathbb {P}}_r^\\psi , the root \\emptyset belongs to {\\rm Br}_\\infty and has mass \\Delta _\\emptyset =r.",
"Weidentify the half real line [0,+\\infty ) with a real tree denoted by\\llbracket 0,\\infty \\llbracket with the null mass measure.",
"We denoteby dx the length measure on \\llbracket 0,\\infty \\llbracket .Let\\sum _{i\\in I_1^*} \\delta _{(x^{*}_i, {*,i})} and \\sum _{i\\in I_2^*} \\delta _{(x^{*}_i, {*,i})} be two independent Poisson random measures on\\llbracket 0,\\infty \\llbracket \\times {{\\mathbb {T}}} with intensitiesdx \\, 2c{\\mathbb {N}}^{\\psi _0}[d \\quad \\text{and}\\quad dx\\int _0^{\\infty }lm_0(dl){\\mathbb {P}}_l^{\\psi _0}(d,respectively.",
"The infinite CRT from \\cite {AD12} is defined as:\\begin{equation}{\\mathcal {T}}^*=\\llbracket \\emptyset ,\\infty \\llbracket \\circledast _{i\\in I_1^*\\cup I_2^*}(x^{*,i}, T^{*,i}).\\end{equation}We denote by {\\mathbb {P}}^{*,\\psi _0}(d{\\mathcal {T}}^*) the distribution of {\\mathcal {T}}^*.Similarly to the setting in Section\\ref {sec:prune}, we consider on {\\mathcal {T}}^* the mark processesM^{{\\mathcal {T}}^*}_{ske}(dq, dy) and M^{{\\mathcal {T}}^*}_{node}(dq, dy) which are Poisson random measures on {\\mathfrak {T}}_0\\times * with intensities:{\\bf 1}_{\\lbrace q\\in {\\mathfrak {T}}_0\\rbrace }\\beta _q dq \\ell ^{{\\mathcal {T}}^*}(dy)\\quad \\text{and}\\quad {\\bf 1}_{\\lbrace q\\in {\\mathfrak {T}}_0\\rbrace }\\sum _{i\\in I_1^*\\cup I_2^*}\\sum _{x\\in {\\rm Br}_{\\infty }(T^{*,i })}m_{\\Delta _x}(0, dq)\\delta _x(dy),respectively, with the identification of x^{*, i} as the root of {\\mathcal {T}}^{*,i}.",
"Inparticular nodes in \\llbracket \\emptyset ,\\infty \\llbracket with infinite degree willbe charged by M^{{\\mathcal {T}}^*}_{node}.",
"Then setM^{{\\mathcal {T}}^*}(dq, dy)=M^{{\\mathcal {T}}^*}_{ske}(dq, dy)+M^{{\\mathcal {T}}^*}_{node}(dq, dy).For every q\\in {\\mathfrak {T}}_0, the pruned tree at time q is defined as:{\\mathcal {T}}_q^*=\\lbrace x\\in *,\\ M^{*}([0,q]\\times \\llbracket \\emptyset ,x\\llbracket )=0\\rbrace ,with the induced metric, root \\emptyset and massmeasure which is the restriction to {\\mathcal {T}}_q^* of the mass measure \\mathbf {^}{{\\mathcal {T}}^*}.",
"Our main result in this section is the following theorem whose proof will be given at the end of this section.\\begin{theo}Suppose \\psi _0 is critical.",
"Given q\\in (t_{\\infty }, 0), if there exists \\bar{q}\\in {\\mathfrak {T}}_0 such that \\psi _{\\bar{q}}(\\cdot )=\\psi _q(\\eta _q+\\cdot ), then conditioned on \\lbrace A=q\\rbrace , A is distributed as {\\bar{q}}^*.\\end{theo}\\begin{xmlelement*}{proof}Delayed.",
"\\hfill \\Box \\medskip \\end{xmlelement*}\\medskip }Before proving Theorem \\ref {therep}, we give some applications of it.",
"Recall {\\mathfrak {T}}^q_0=\\lbrace \\theta \\ge 0: \\theta +q\\in {\\mathfrak {T}}_q\\rbrace .",
"By a similar reasoning as Corollary \\ref {coroTA+} we have the following corollary.\\begin{corollary}Suppose \\psi _0 is critical.",
"Given q\\in (t_{\\infty }, 0), if there exists \\bar{q}\\in {\\mathfrak {T}}_0 such that \\psi _{\\bar{q}+t}(\\cdot )=\\psi _{q+t}^{\\eta _q}(\\cdot ) for all t\\in {\\mathfrak {T}}^{\\bar{q}}_0, then conditioned on \\lbrace A=q\\rbrace , \\lbrace {A+t}: t\\in {\\mathfrak {T}}^{\\bar{q}}_0\\rbrace is distributed as \\lbrace {\\bar{q}+t}^*:t\\in {\\mathfrak {T}}^{\\bar{q}}_0\\rbrace .\\end{corollary}\\begin{lemma}Assume that \\psi _0 is critical and for every t\\in (t_{\\infty }, 0), there exists \\bar{t}\\in {\\mathfrak {T}} such that \\psi _{\\bar{t}}(\\cdot )=\\psi _{t}(\\eta _t+\\cdot ).",
"We further suppose t_{\\infty }\\notin {\\mathfrak {T}}_{\\infty }.Then t\\rightarrow \\bar{t} is differentiable and\\frac{d\\bar{t}}{dt}=\\frac{\\zeta ^{\\prime }_t(\\eta _t)\\psi ^{\\prime }_t(\\eta _t)-\\psi ^{\\prime \\prime }_t(\\eta _t)\\zeta _t(\\eta _t)}{\\zeta ^{\\prime }_{\\bar{t}}(0)\\psi ^{\\prime }_t(\\eta _t)}=:\\frac{-\\gamma _t}{\\zeta ^{\\prime }_{\\bar{t}}(0)},\\qquad t\\in (t_{\\infty }, 0).\\end{lemma}\\begin{xmlelement*}{proof}It is obvious that t\\rightarrow \\bar{t} is differentiable.",
"By \\psi _{\\bar{t}}(\\cdot )=\\psi _t(\\eta _t+\\cdot ) and (\\ref {detat}), we have for all \\lambda >0,\\begin{eqnarray*}\\frac{d\\bar{t}}{dt}=\\frac{\\zeta _t(\\eta _t+\\lambda )\\psi ^{\\prime }_t(\\eta _t)-\\psi ^{\\prime }_t(\\eta _t+\\lambda )\\zeta _t(\\eta _t)}{\\zeta _{\\bar{t}}(\\lambda )\\psi ^{\\prime }_t(\\eta _t)}.\\end{eqnarray*}Letting \\lambda \\rightarrow 0 in above equality gives the desired result.\\hfill \\Box \\medskip \\end{xmlelement*}Define \\bar{t}_{\\infty }=\\sup \\lbrace \\bar{t}: t\\in {\\mathfrak {T}}, b_t<0\\rbrace .",
"For t\\in (0, \\bar{t}_{\\infty }), let \\hat{t} be the unique negative number such that\\bar{\\hat{t}}=t.Let U be a positive ``random^{\\prime \\prime } variable with nonnegative ``density^{\\prime \\prime } with respect to the Lebesgue measure given by{\\bf 1}_{\\lbrace t\\in (0, \\bar{t}_{\\infty })\\rbrace }\\frac{\\zeta _{\\hat{t}}(\\eta _{\\hat{t}})\\zeta ^{\\prime }_t(0)}{\\psi ^{\\prime }_{{\\hat{t}}}(\\eta _{\\hat{t}})\\gamma _{\\hat{t}}}.Assume that U is independent of *.\\begin{corollary}Suppose that all assumptions in Lemma \\ref {lemrep} hold.",
"Then A is distributed under {\\mathbf {N}}^{\\Psi } as U^*.\\end{corollary}\\begin{remark}If U has the same distribution as A, then we have A is distributed as {\\bar{U}}^*.\\end{remark}\\begin{xmlelement*}{proof}Recall (\\ref {detat}).",
"By Lemma \\ref {lemA}, if we have the law of A under {\\mathbf {N}}^{\\Psi } has a density with respect to the Lebesgue measure on \\mathbb {R} give by{\\bf 1}_{\\lbrace t\\in (t_{\\infty }, 0)\\rbrace }\\frac{\\zeta _{t}(\\eta _{t})}{\\psi ^{\\prime }_{t}(\\eta _{t})}.Thus for any nonnegative measurable function F on {\\mathbb {T}}, by Theorem \\ref {therep},\\begin{eqnarray*}{\\mathbf {N}}^{\\Psi }[F(A)]\\!\\!\\!&=\\!\\!\\!&\\int _{(t_{\\infty }, 0)}{\\mathbb {E}}^{*, \\psi _0}[F(*_{\\bar{t}})]\\frac{\\zeta _{t}(\\eta _{t})}{\\psi ^{\\prime }_{t}(\\eta _{t})}dt\\cr \\!\\!\\!&=\\!\\!\\!&\\int _{(t_{\\infty }, 0)}{\\mathbb {E}}^{*, \\psi _0}[F(*_{t})]\\frac{\\zeta _{\\hat{t}}(\\eta _{\\hat{t}})}{\\psi ^{\\prime }_{\\hat{t}}(\\eta _{\\hat{t}})}d\\hat{t}\\cr \\!\\!\\!&=\\!\\!\\!&\\int _{(t_{\\infty }, 0)}{\\mathbb {E}}^{*, \\psi _0}[F(*_{t})]\\frac{\\zeta _{\\hat{t}}(\\eta _{\\hat{t}})}{\\psi ^{\\prime }_{\\hat{t}}(\\eta _{\\hat{t}})}\\frac{d\\hat{t}}{d\\bar{\\hat{t}}}dt\\cr \\!\\!\\!&=\\!\\!\\!&\\int _{(0, \\bar{t}_{\\infty })}{\\mathbb {E}}^{*, \\psi _0}[F(*_{t})]\\frac{\\zeta _{\\hat{t}}(\\eta _{\\hat{t}})}{\\psi ^{\\prime }_{\\hat{t}}(\\eta _{\\hat{t}})}\\frac{\\zeta ^{\\prime }_t(0)}{\\gamma _{\\hat{t}}}dt\\cr \\!\\!\\!&=\\!\\!\\!&{\\mathbb {E}}^{*, \\psi _0}[F(U^*)],\\end{eqnarray*}where the fourth equality follows from Lemma \\ref {lemrep}.",
"We have completed the proof.\\hfill \\Box \\medskip \\end{xmlelement*}By Corollary \\ref {correpb} and Corollary \\ref {correp}, we have the following result which is a generalization of Corollary 8.2 in \\cite {AD12}.\\begin{corollary}Suppose that \\psi _0 is critical and [0,\\infty )\\subset {\\mathfrak {T}}.",
"If for every q\\in (t_{\\infty }, 0), there exists \\bar{q}\\in {\\mathfrak {T}}_0 such that\\psi _{\\bar{q}+t}(\\cdot )=\\psi _{q+t}^{\\eta _q}(\\cdot ) for all t\\in {\\mathfrak {T}}_0.",
"We further assume that t_{\\infty }\\notin {\\mathfrak {T}}_{\\infty }.",
"Then\\lbrace {A+t}: t\\ge 0\\rbrace is distributed under {\\mathbf {N}}^{\\Psi } as \\lbrace {U+t}^*:t\\ge 0\\rbrace .\\end{corollary}}}In the following we give some examples.\\begin{example}We first consider the case studied by Abraham and Delmas in \\cite {AD12}.Let \\psi be defined in (\\ref {eq:psi}).Recall \\psi ^{q}(\\lambda )=\\psi (q+\\lambda )-\\psi (q), q\\in \\Theta ^{\\psi }, where \\Theta ^{\\psi } is set of q\\in {\\mathbb {R}} such that \\int _{(1,\\infty )}re^{-q r}m(dr)<\\infty .",
"In \\cite {AD12}, it was assumed that 0>\\theta _{\\infty }:=\\inf \\Theta ^{\\psi }\\notin \\Theta ^{\\psi } and \\psi is critical.",
"\\lbrace \\psi ^q: q\\in \\Theta ^{\\psi }\\rbrace satisfies all assumptions in Corollary \\ref {correpa}.",
"Then for t\\in (\\theta _{\\infty }, 0),\\eta _t=\\bar{t}-tand\\zeta _t(\\lambda )=2c\\lambda +\\int _0^{\\infty }(1- e^{-z\\lambda })e^{-zt}zm(dz)=\\psi ^{\\prime }(t+\\lambda )-\\psi ^{\\prime }(t).Using \\eta _{\\hat{t}}=t-\\hat{t}, we shall see{\\bf 1}_{\\lbrace t\\in (0, \\bar{\\theta }_{\\infty })\\rbrace }\\frac{\\zeta _{\\hat{t}}(\\eta _{\\hat{t}})\\zeta ^{\\prime }_t(0)}{\\psi ^{\\prime }_{{\\hat{t}}}(\\eta _{\\hat{t}})\\gamma _{\\hat{t}}}={\\bf 1}_{\\lbrace t\\in (0, \\bar{\\theta }_{\\infty })\\rbrace }\\left(1-\\frac{\\psi ^{\\prime }(t)}{\\psi ^{\\prime }(\\hat{t})}\\right).Then we go back to Corollary 8.2 in \\cite {AD12}.\\end{example}\\begin{example}Let b>0 and c>0 be two constants.",
"Define \\psi _q({\\lambda })=qb\\lambda +c\\lambda ^2, q\\in {\\mathbb {R}}, \\lambda \\ge 0.",
"Then \\lbrace \\psi _q: q\\in {\\mathbb {R}}\\rbrace satisfies all assumptions in Corollary \\ref {correpa}.",
"Typically, if b=2c, we have \\psi _q(\\cdot )=\\psi _0(q+\\cdot )-\\psi _0(q).\\end{example}\\begin{example}Let {\\mathfrak {T}}^-\\subset (-\\infty , 0] be an interval and let \\lbrace \\psi _q: q\\in {\\mathfrak {T}}^-\\rbrace be an admissible family of branching mechanisms with the parameters (b_q, m_q).",
"Assume that 0\\in {\\mathfrak {T}}^-, \\psi _0 is critical and \\inf {\\mathfrak {T}}^-\\notin {\\mathfrak {T}}^-_{\\infty }.",
"Let \\eta _q denote the largest root of \\psi _q(s)=0.For q\\in -{\\mathfrak {T}}^-:=\\lbrace -t: t\\in {\\mathfrak {T}}^-\\rbrace , define \\psi _q(\\cdot )=\\psi _{-q}(\\eta _{-q}+\\cdot ).",
"Then we have \\lbrace \\psi _q: q\\in {\\mathfrak {T}}^-\\cup (-{\\mathfrak {T}}^-)\\rbrace is an admissible family of branching mechanisms such that for q\\in -{\\mathfrak {T}}^-b_q=b_{-q}+2c\\eta _{-q}+\\int _0^{\\infty }(1-e^{-z\\eta _{-q}})zm_{-q}(dz),\\quad m_q=e^{-z\\eta _{-q}}m_{-q}(dz).Typically, \\lbrace \\psi _q: q\\in {\\mathfrak {T}}^-\\cup (-{\\mathfrak {T}}^-)\\rbrace satisfies all assumptions in Corollary \\ref {correp}.\\end{example}\\end{array}\\medskip \\right.The end of this section is devoted to the proof of Theorem \\ref {therep}.",
"}}\\end{equation*}\\begin{proposition} Suppose \\psi _0 is critical.",
"Then for any nonnegative measurable functional F on {\\mathbb {T}} and for every q\\in {\\mathfrak {T}}_0,\\begin{eqnarray}\\psi ^{\\prime }_{q}(0){\\mathbf {N}}^{\\Psi }\\left[\\sigma _{q}F({q})\\right]={{\\mathbb {E}}}^{*,\\psi _0}\\left[F({q}^*)\\right].\\end{eqnarray}\\end{proposition}\\begin{xmlelement*}{proof}First, by Bismut decomposition (Theorem 4.5 in \\cite {DL05} or Theorem 2.17 in \\cite {ADH12b}), we have that there exists some measurable functional \\bar{F} on [0,\\infty )\\times {\\mathbb {T}} such that\\begin{eqnarray}\\psi ^{\\prime }_{q}(0){\\mathbf {N}}^{\\Psi }\\left[\\sigma _{q}F({q})\\right]\\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_q(0){\\mathbb {N}}^{\\psi _q}\\left[\\sigma F(\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_q(0)\\int _0^{\\infty }dae^{-\\psi ^{\\prime }_q(0)a}{\\mathbb {E}}\\left[\\bar{F}\\left(a, \\sum _{i\\in \\tilde{I}}{\\bf 1}_{\\lbrace z_i\\le a\\rbrace }\\delta _{(z_i, \\tilde{T}_i)}\\right)\\right],\\end{eqnarray}where under {\\mathbb {E}}, \\sum _{i\\in \\tilde{I}}\\delta _{(z_i, \\tilde{T}_i)}(dz, d is a Poisson random measure on [0, \\infty )\\times {\\mathbb {T}} with intensitydz\\left(2c{\\mathbb {N}}^{\\psi _q}[d+\\int _0^{\\infty }lm_q(dr){\\mathbb {P}}_l^{\\psi _q}(d\\right).For i\\in I_1^*\\cup I_2^*, defineq^{*,i}=\\lbrace x\\in {*,i}: M^{*}([0,q]\\times \\llbracket \\emptyset , x\\llbracket )=0\\rbrace ,\\quad q\\in {\\mathfrak {T}}_0.With abuse of notation, we have \\begin{eqnarray}q^*=\\llbracket \\emptyset ,\\xi \\rrbracket \\circledast _{i\\in I_1^*\\cup I_2^*,\\; x^{*,i}< \\xi }(x^{*,i}, q^{*,i}),\\end{eqnarray}where\\begin{eqnarray*}\\xi \\!\\!\\!&:=\\!\\!\\!&\\sup \\lbrace x\\in \\llbracket \\emptyset , +\\infty \\llbracket : M^{*}([0,q]\\times \\llbracket \\emptyset , x\\llbracket )=0\\rbrace \\cr \\!\\!\\!&=\\!\\!\\!&\\sup \\lbrace x\\in \\llbracket \\emptyset , +\\infty \\llbracket : M^{*}_{ske}([0,q]\\times \\llbracket \\emptyset , x\\llbracket )=0\\rbrace \\wedge \\inf \\lbrace x^{*,i}: M^{*}_{node}([0,q]\\times \\lbrace x^{*,i}\\rbrace )>0\\rbrace \\cr \\!\\!\\!&=:\\!\\!\\!&\\xi _1\\wedge \\xi _2.\\end{eqnarray*}\\end{xmlelement*}By (\\ref {proprep1}) and (\\ref {Tqstar}), to prove (\\ref {therep2}), we only need to show that $$ is exponentially distributed with parameter $ 'q(0)$.Obviously, $ 1$ is exponentially distributed with parameter $ 0qd$.According to Corollary \\ref {coromain1} and property of Poisson random measure, we have$$\\sum _{i\\in I_2^*}{\\bf 1}_{\\lbrace M^{*}_{node}([0,q]\\times \\lbrace x^{*,i}\\rbrace )>0\\rbrace }\\delta _{(x^{*,i}, q^{*, i})}(dx, d$$is a Poisson random measure with intensity $ dx 0qd0zn(dz)Pzq(d.$ Then one can deduce that$ 2$ is exponentially distributed with parameter $ 0qd0zn(dz)$.",
"Hence $$ isexponentially distributed with parameter$$\\int _0^q\\beta _{\\theta }d\\theta + \\int _{0}^qd\\theta \\int _0^{\\infty }zn_{\\theta }(dz),$$which, by (\\ref {b_q}), is just $ bq='q(0).$ We have completed the proof.\\hfill $$\\medskip $ Now we are in position to prove Theorem .",
"Proof of Theorem : For any nonnegative measurable function $F$ on ${\\mathbb {T}}$ , by (), we have for $q<0$ , ${\\mathbf {N}}^{\\Psi }[F(A)|A=q]\\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_{q}(\\eta _{q}){\\mathbb {N}}^{\\psi _{q}}\\left[F(\\sigma {\\bf 1}_{\\lbrace \\sigma <\\infty \\rbrace }\\right]\\cr \\!\\!\\!&=\\!\\!\\!&\\psi ^{\\prime }_{q}(\\eta _{q}){\\mathbb {N}}^{\\psi _{q}^{\\eta _q}}[F(\\sigma ],$ where the last equality follows from ().",
"Since $\\psi _{\\bar{q}}(\\cdot )=\\psi _{q}(\\eta _{q}+\\cdot )=\\psi _q^{\\eta _q}(\\cdot )$ and $\\psi ^{\\prime }_q(\\eta _q)=\\psi ^{\\prime }_{\\bar{q}}(0)$ , Proposition yields ${\\mathbf {N}}^{\\Psi }[F(A)|A=q]=\\psi ^{\\prime }_{\\bar{q}}(0){\\mathbf {N}}^{\\Psi }\\left[\\sigma _{\\bar{q}}F({\\bar{q}})\\right]={\\mathbb {E}}^{*,\\psi _0}[F({\\bar{q}}^*)].$ We have completed the proof.",
"$\\Box $"
]
] | 1403.0397 |
[
[
"Efficient Representation for Online Suffix Tree Construction"
],
[
"Abstract Suffix tree construction algorithms based on suffix links are popular because they are simple to implement, can operate online in linear time, and because the suffix links are often convenient for pattern matching.",
"We present an approach using edge-oriented suffix links, which reduces the number of branch lookup operations (known to be a bottleneck in construction time) with some additional techniques to reduce construction cost.",
"We discuss various effects of our approach and compare it to previous techniques.",
"An experimental evaluation shows that we are able to reduce construction time to around half that of the original algorithm, and about two thirds that of previously known branch-reduced construction."
],
[
"Introduction",
"The suffix tree is arguably the most important data structure in string processing, with a wide variety of applications [1], [10], [14], and with a number of available construction algorithms [23], [17], [22], [5], [9], [3], each with its benefits.",
"Improvements in its efficiency of construction and representation continues to be a lively area of research, despite the fact that from a classical asymptotic time complexity perspective, optimal solutions have been known for decades.",
"Pushing the edge of efficiency is critical for indexing large inputs, and make large amounts of experiments feasible, e.g., in genetics, where lengths of available genomes increase.",
"Much work has been dedicated to reducing the memory footprint with representations that are compact [13] or compressed (see Cánovas and Navarro [3] for a practical view, with references to theoretical work), and to alternatives requiring less space, such as suffix arrays [15].",
"Other work adresses the growing performance-gap between cache and main memory, frequently using algorithms originally designed for secondary storage [4], [6], [21], [20].",
"While memory-reduction is important, it typically requires elaborate operations to access individual fields, with time overhead that can be deterring for some applications.",
"Furthermore, compaction by a reduced number of pointers per node is ineffective in applications that use those pointers for pattern matching.",
"Our work ties in with the more direct approach to improving performance of the conventional primary storage suffix tree representation, taken by Senft and Dvořák [19].",
"Classical representations required in Ukkonen's algorithm [22] and the closely related predecessor of McCreight [17] remain important in application areas such as genetics, data compression and data mining, since they allow online construction as well as provide suffix links, a feature useful not only in construction, but also for string matching tasks [10], [12].",
"In these algorithms, a critically time-consuming operation is branch: identifying the correct outgoing edge of a given node for a given character [19].",
"This work introduces and evaluates several representation techniques to help reduce both the number of branch operations and the cost of each such operation, focusing on running time, and taking an overall liberal view on space usage.",
"Our experimental evaluation of runtime, memory locality, and the counts for critical operations, shows that a well chosen combination of our presented techniques consistently produce a significant advantage over the original Ukkonen scheme as well as the branch-reduction technique of Senft and Dvořák."
],
[
"Suffix Trees and Ukkonen's Algorithm",
"We denote the suffix tree (illustrated in fig.",
"REF ) over a string $T=t_0\\cdots t_{N-1}$ of length $|T|=N$ by ${\\mathcal {ST}}$ .",
"Each edge in ${\\mathcal {ST}}$ , directed downwards from the root, is labeled with a substring of $T$ , represented in constant space by reference to position and length in $T$ .",
"We define a point on an ${\\mathcal {ST}}$ edge as the position between two characters of its label, or – when the point coincides with a node – after the whole edge label.",
"Each point in the tree corresponds to precisely one nonempty substring $t_i\\cdots t_j$ , $0\\le i\\le j<N$ , obtained by reading edge labels on the path from the root to that point.",
"A consequence is that the first character of an edge label uniquely identifies it among the outgoing edges of a node.",
"The point corresponding to an arbitrary pattern can be located (or found non-existent) by scanning characters left to right, matching edge labels from the root down.",
"For convenience, we add an auxiliary node $\\topnode $ above the root (following Ukkonen), with a single edge to the root.",
"We denote this edge $\\topedge $ and label it with the empty string, which is denoted by ${\\epsilon }$ .",
"(Although $\\topnode $ is the topmost node of the augmented tree, we consistently refer to the root of the unaugmented tree as the root of ${\\mathcal {ST}}$ .)",
"Each leaf corresponds to some suffix $t_i\\cdots t_{N-1}$ , $0\\le i<N$ .",
"Hence, the label endpoint of a leaf edge can be defined implicitly, rather than updated during construction.",
"Note, however, that any suffix that is not a unique substring of $T$ corresponds to a point higher up in the tree.",
"(We do not, as is otherwise common, require that $t_{N-1}$ is a unique character, since this clashes with online construction.)",
"Except for $\\topedge $ , all edges are labeled with nonempty strings, and the tree represents exactly the substrings of $T$ in the minimum number of nodes.",
"This implies that each node is either $\\topnode $ , the root, a leaf, or a non-root node with at least two outgoing edges.",
"Since the number of leaves is at most $N$ (one for each suffix), the total number of nodes cannot exceed $2N+1$ (with equality for $N=1$ ).",
"We generalize the definition to ${\\mathcal {ST}_{\\hspace{-2.22pt}i}}$ over string $T_i=t_0 \\cdots t_{i-1}$ , where ${\\mathcal {ST}_{\\hspace{-2.22pt}N}}={\\mathcal {ST}}$ .",
"An online construction algorithm constructs ${\\mathcal {ST}}$ in $N$ updates, where update $i$ reshapes ${\\mathcal {ST}_{\\hspace{-2.22pt}i-1}}$ into ${\\mathcal {ST}_{\\hspace{-2.22pt}i}}$ , without looking ahead any further than $t_{i-1}$ .",
"We describe suffix tree construction based on Ukkonen's algorithm [22].",
"Please refer to Ukkonen's original, closer to an actual implementation, for details such as correctness arguments.",
"Define the active point before update $i>1$ as the point corresponding to the longest suffix of $T_{i-1}$ that is not a unique substring of $T_{i-1}$ .",
"Thanks to the implicit label endpoint of leaf edges, this is the point of the longest string where update $i$ might alter the tree.",
"The active point is moved once or more in update $i$ , to reach the corresponding start position for update $i+1$ .",
"(This diverges slightly from Ukkonen's use, where the active point is only defined as the start point of the update.)",
"Since any leaf corresponds to a suffix, the label end position of any point coinciding with a leaf in ${\\mathcal {ST}_{\\hspace{-2.22pt}i}}$ is $i-1$ .",
"The tree is augmented with suffix links, pointing upwards in the tree: Let $v$ be a non-leaf node that coincides with the string $aA$ for some character $a$ and string $A$ .",
"Then the suffix link of $v$ points to the node coinciding with the point of $A$ .",
"The suffix link of the root leads to $\\topnode $ , which has no suffix link.",
"Before the first update, the tree is initialized to ${\\mathcal {ST}_{\\hspace{-2.22pt}0}}$ consisting only of $\\topnode $ and the root, joined by $\\topedge $ , and the active point is set to the endpoint of $\\topedge $ (i.e.",
"the root).",
"Update $i$ then procedes as follows: If the active point coincides with $\\topnode $ , move it down one step to the root, and finish the update.",
"Otherwise, attempt to move the active point one step down, by scanning over character $t_i$ .",
"If the active point is at the end of an edge, this requires a branch operation, where we choose among the outgoing edges of the node.",
"Otherwise, simply try matching the character following the point with $t_i$ .",
"If the move down succeeds, i.e., $t_i$ is present just below the active point, the update is finished.",
"Otherwise, keep the current active point for now, and continue with the next step.",
"Unless the active point is at the end of an edge, split the edge at the active point and introduce a new node.",
"If there is a saved node $v_p$ (from step REF ), let $v_p$ 's suffix link point to the new node.",
"The active point now coincides with a node, which we denote $v$ .",
"Create a new leaf $w$ and make it a child of $v$ .",
"Set the start pointer of the label on the edge from $v$ to $w$ to $i$ (the end pointer of leaf labels being implicit).",
"If the active point corresponds to the root, move it to $\\topnode $ .",
"Otherwise, we should move the active point to correspond to the string $A$ , where $aA$ is the string corresponding to $v$ for some character $a$ .",
"There are two cases: Unless $v$ was just created, it has a suffix link, which we can simply follow to directly arrive at a node that coincides with the point we seek.",
"Otherwise, i.e.",
"if $v$ 's suffix link is not yet set, let $u$ be the parent of $v$ , and follow the suffix link of $u$ to $u^{\\prime }$ .",
"Then locate the edge below $u^{\\prime }$ containing the point that corresponds to $A$ .",
"Set this as the active point.",
"Moving down from $u^{\\prime }$ requires one or more branch operations, a process referred to as rescanning (see fig.",
"REF ).",
"If the active point now coincides with a node $v^{\\prime }$ , set the suffix link of $v$ to point to $v^{\\prime }$ .",
"Otherwise, save $v$ as $v_p$ to have its suffix link set to the node created next.",
"Continue from step 1."
],
[
"Reduced Branching Schemes",
"Senft and Dvořák [19] observe that the branch operation, searching for the right outgoing edge of a node, typically dominates execution time in Ukkonen's algorithm.",
"Reducing the cost of branch can significantly improve construction efficiency.",
"Two paths are possible: attacking the cost of the branch operation itself, through the data structures that support it, which we consider in section , and reducing the number of branch operations in step REF of the update algorithm.",
"We refer to Ukkonen's original method of maintaining and using suffix links as node-oriented top-down (notd).",
"Section REF discusses the bottom-up approach (nobu) of Senft and Dvořák, and sections REF –REF present our novel approach of edge-oriented suffix links, in two variants top-down (eotd) and variable (eov)."
],
[
"Node-Oriented Bottom-Up",
"A branch operation comprises the rather expensive task of locating, given a node $v$ and character $c$ , $v$ 's outgoing edge whose edge label begins with $c$ , if one exists.",
"By contrast, following an edge in the opposite direction can be made much cheaper, through a parent pointer.",
"Senft and Dvořák [19] suggests the following simple modification to suffix tree representation and construction: Maintain parents of nodes, and suffix links for leaves as well as non-leaves.",
"In step REF of update, follow the suffix link of $v$ to $v^{\\prime }$ rather than that of its parent $u$ to $u^{\\prime }$ , and locate the point corresponding to $A$ moving up, climbing from $v^{\\prime }$ rather than rescanning from $u^{\\prime }$ (see fig.",
"REF ).",
"Senft and Dvořák experimentally demonstrate a runtime improvement across a range of typical inputs.",
"A drawback is that worst case time complexity is not linear: a class of inputs with time complexity $\\Omega (N^{1.5})$ is easily constructed, and it is unknown whether actual worst case complexity is even higher.",
"To circumvent degenerate cases, Senft and Dvořák suggest a hybrid scheme where climbing stops after $c$ steps, for constant $c$ , falling back to rescan.",
"(As an alternative, we suggest bounding the number of edges to climb to by using rescan iff the remaining edge label length below the active point exceeds constant $c^{\\prime }$ .)",
"Some of the space overhead can be avoided in a representation using clever leaf numbering."
],
[
"Edge-Oriented Top-Down",
"We consider an alternative branch-saving strategy, slightly modifying suffix links.",
"For each split edge, the notd update algorithm follows a suffix link from $u$ to $u^{\\prime }$ , and immediately obtains the outgoing edge $e^{\\prime }$ of $u^{\\prime }$ whose edge label starts with the same character as the edge just visited.",
"We can avoid this first branch operation in rescan (which constitutes a large part of rescan work) , by having $e^{\\prime }$ available from $e$ directly, without taking the detour via $u$ and $u^{\\prime }$ .",
"Define the string that marks an edge as the shortest string represented by the edge (corresponding to the point after one character in its label).",
"For edge $e$ , let $aA$ , for character $a$ and string $A$ , be the shortest string represented $e$ such that $A$ marks some other edge $e^{\\prime }$ .",
"(The same as saying that $aA$ marks $e$ , except when $e$ is an outgoing edge of the root and $|A|=1$ , in which case $a$ marks $e$ .)",
"Let the edge oriented suffix link of $e$ point to $e^{\\prime }$ (illustrated i fig.",
"REF ).",
"Figure: Suffix tree over the string abcabdaabcabda, with dotted linesshowing node-oriented suffix links for internal nodes only, as in Ukkonen'soriginal scheme (left), andedge-oriented suffix links (right).Modifying the update algorithm for this variant of suffix links, we obtain an edge-oriented top-down (eotd) variant.",
"The update algorithm is analogous to the original, except that edge suffix links are set and followed rather than node suffix links, and the first branch operation of each rescan avoided as a result.",
"The following points deserve special attention: When an edge is split, the top part should remain the destination of incoming suffix links, i.e., the new edge is the bottom part.",
"After splitting one or more edges in an update, finding the correct destination for the suffix link of the last new edge (the bottom part of the last edge split) requires a sibling lookup branch operation, not necessary in notd.",
"Following a suffix link from the endpoint of an edge occasionally requires one or more extra rescan operation, in relation to following the node-oriented suffix link of the endpoint.",
"The first point raises some implementation issues.",
"Efficient representations (see e.g.",
"Kurtz's [13]) do not implement nodes and edges as separate records in memory.",
"Instead, they use a single record for a node and its incoming edge.",
"Not only does this reduce the memory overhead, it cuts down the number of pointers followed on traversing a path roughly by half.",
"The effect of our splitting rule is that while the top part of the split edge should retain incoming suffix links, the new record, tied to the bottom part should inherit the children.",
"We solve this by adding a level of indirection, allowing all children to be moved in a single assignment.",
"In some settings (e.g., if parent pointers are needed), this induces a one pointer per node overhead, but it also has two potential efficiency benefits.",
"First, new node/edge pairs become siblings, which makes for a natural memory-locality among siblings (cf.",
"child inlining in section ).",
"Second, the original bottom node stays where it was in the branching data structure, saving one replace operation.",
"These properties are important for the efficiency of the eotd representation.",
"The latter two points go against the reduction of branch operations that motivated edge-oriented suffix links, but does not cancel it out.",
"(Cf.",
"table REF .)",
"These assertions are supported by experimental data in section .",
"Furthermore, eotd retains the $O(N)$ total construction time of notd.",
"To see this, note first that the modification to edge-oriented suffix links clearly adds at most constant-time operation to each operation, except possibly with regards to the extra rescan operations after following a suffix link from the endpoint of an edge.",
"But Ukkonen's proof of total $O(N)$ rescan time still applies: Consider the string $t_j\\cdots t_i$ , whose end corresponds to the active point, and whose beginning is the beginning of the currently scanned edge.",
"Each downward move in rescanning deletes a nonempty string from the left of this string, and characters are only added to the right as $i$ is incremented, once for each online suffix tree update.",
"Hence the number of downward moves are bounded by $N$ , the total number of characters added.",
"Figure: Examples of moving the active point across a suffix link infour schemes, where in each case xx is a node thatcoincides with the active point after the move.",
"For eov, we see two cases,before (i) and after (ii) the destination of the suffixlink is moved"
],
[
"Edge-Oriented Variable",
"Let $e$ be an edge from node $u$ to $v$ , and let $v^{\\prime }$ and $u^{\\prime }$ be nodes such that node suffix links would point from $u$ to $u^{\\prime }$ and from $v$ to $v^{\\prime }$ .",
"If the path from $u^{\\prime }$ to $v^{\\prime }$ is longer than one edge, the eotd suffix link from $e$ would point to the first one, an outgoing edge of $u^{\\prime }$ .",
"Another edge-oriented approach, more closely resembling nobu, would be to let $e$ 's suffix link to point to the last edge on the path, the incoming edge of $v^{\\prime }$ , and use climb rather than rescan for locating the right edge after following a suffix link.",
"But this approach does not promise any performance gain over nobu.",
"An approach worth investigating, however, is to allow some freedom in where to on the path between $u^{\\prime }$ and $v^{\\prime }$ to point $e$ 's suffix link.",
"We refer to the path from $u^{\\prime }$ to $v^{\\prime }$ as the destination path of $e$ 's suffix link.",
"Given that an edge maintains the length of the longest string it represents (which is a normal edge representation in any case), we can use climb or rescan as required.",
"We suggest the following edge-oriented variable (eov) scheme: When an edge is split, let the bottom part remain the destination of incoming suffix links, i.e., let the top part be the new edge.",
"(The opposite of the eotd splitting rule.)",
"This sets a suffix link to the last edge on its destination path, possibly requiring climb operations after the link is followed.",
"When a suffix link is followed and $c$ edges on its destination path climbed, if $c>k$ for a constant $k$ , move the suffix link $c-k$ edges up.",
"Intuitively, this approach looks promising, in that it avoids spending time on adjusting suffix links that are never used, while eliminating the $\\Omega (N^{1.5})$ degeneration case demonstrated for nobu [19].",
"Any node further than $k$ edges away from the top of the destination path is followed only once per suffix link, and hence the same destination path can only be climbed multiple times when multiple suffix links point to the same path, and each corresponds to a separate occurrence of the string corresponding to the climbed edge labels.",
"We conjecture that the amortized number of climbs per edge is thus $O(1)$ .",
"However, our experimental evaluation indicates that the typical savings gained by the eov approach are relatively small, and are surpassed by careful application of eotd."
],
[
"Branching Data Structure",
"Branching efficiency depends on the input alphabet size.",
"Ukkonen proves $O(N)$ time complexity only under the assumption that characters are drawn from an alphabet ${\\mathrm {\\Sigma }}$ where $|{\\mathrm {\\Sigma }}|$ is $O(1)$ .",
"If $|{\\mathrm {\\Sigma }}|$ is not constant, expected linear time can be achieved by hashing, as suggested in McCreight's seminal work [17], and more recent dictionary data structures [2], [11] can be applied for bounds very close to deterministic linear time.",
"Recursive suffix tree construction, originally presented by Farach [5] achieves the same asymptotic time bound as character sorting, but does not support online construction.",
"We limit our treatment to simple schemes based on linked lists or hashing since, to our knowledge, asymptotically stronger results have not been shown to yield a practical improvement.",
"Kurtz [13] observed in 1999 that linked lists appear faster for practical inputs when $|{\\mathrm {\\Sigma }}|\\le 100$ and $N\\le 150\\,000$ .",
"For a lower bound estimate of the alphabet size breaking point, we tested suffix tree construction on random strings of different size alphabets.",
"We used a hash table of size $3N$ with linear probing for collision resolution, which resulted in an average of less than two hash table probes per insert or lookup across all files.",
"The results, shown in table REF , indicate that hashing can outperform linked lists for alphabet sizes at least as low as 16, and our experiments did indeed show hashing to be advantageous for the protein file, with this size of alphabet.",
"However, for many practical input types that produce a much lower average suffix tree node out-degree, the breaking point would be at a much larger $|{\\mathrm {\\Sigma }}|$ .",
"Figure: Comparing linked list (ll) to hash tableimplementations (ht) for random filesdifferent size alphabets.",
"Each file is 50 million characters long, andthe vertical axis shows runtime in seconds."
],
[
"Child Inlining",
"An internal node has, by definition, at least two children.",
"Naturally occurring data is typically repetitive, causing some children to be accessed more frequently than others.",
"(This is the basis of the ppm compression method, which has a direct connection to the suffix tree [14].)",
"By a simple probabilistic argument, children with a high traversal probability also have a high probability of being encountered first.",
"Hence, we obtain a possible efficiency gain by storing the first two children of each node, those that cause the node to be created, as inline fields of the node record together with their first character, instead of including them in the overall child retrieval data structure.",
"The effect should be particularly strong for eotd, which, as noted in section REF , eliminates the replace child operation that otherwise occurs when an edge is split, and the record of the original child hence remains a child forever.",
"Furthermore, if nodes are laid out in memory in creation order, eotd's consecutive creation of the first two children can produce an additional caching advantage.",
"Note that inline space use is compensated by space savings in the non-inlined child-retrieval data structure.",
"When linked lists are used for branching, we can achieve an effect similar to inlining by always inserting new children at the back of the list.",
"This change has no significant cost, since an addition is made only after an unsuccessful list scan."
],
[
"Performance Evaluation on Input Corpora",
"Our target is to keep the number of branch operations low, and their cost low through lookup data structures with low overhead and good cache utilization.",
"The overall goal is reducing construction time.",
"Hence, we evaluate these factors."
],
[
"Models, Measures, and Data",
"Practical runtime measurement is, on the one hand, clearly directly relevant for evaluating algorithm behavior.",
"On the other hand, there is a risk of exaggerated effects dependent on specific hardware characteristics, resulting in limited relevance for future hardware development.",
"Hence, we are reluctant to use execution time as the sole performance measure.",
"Another important measure, less dependent on conditions at the time of measuring, is memory probing during execution.",
"Given the central role of main memory access and caching in modern architectures, we expect this to be directly relevant to the runtime, and include several measures to capture it in our evaluation.",
"We measure level 3 cache misses using the Perf performance counter subsystem in Linux [18], which reports hardware events using the performance monitoring unit of the cpu.",
"Clearly, with this hardware measure, we are again at the mercy of hardware characteristics, not necessarily relevant on a universal scale.",
"Measuring cache misses in a theoretically justified model such as the ideal-cache model [8] would be attractive, but such a model does not easily lend itself to experiments.",
"Attempts of measuring emulated cache performance using a virtual machine (Valgrind) produced spurious results, and the overhead made large-scale experiments infeasible.",
"Instead, we concocted two simple cache models to evaluate the locality of memory access: one minimal cache of only ten 64 byte cache lines with a least recently used replacement scheme (intended as a baseline for the level one cache of any reasonable cpu), and one with a larger amount of cache lines with a simplistic direct mapping without usage estimation (providing a baseline expected to be at least matched by any practical hardware).",
"We measure runtimes of Java implementations kept as similar as possible in regards to other aspects than the techniques tested, with the 1.6.0_27 Open jdk runtime, a common contemporary software environment.",
"With current hotspot code generation, we achieve performance on par with compiled languages by freeing critical code sections of language constructs that allocate objects (which would trigger garbage collection) or produce unnecessary pointer dereference.",
"We repeat critical sections ten times per test run, to even out fluctuation in time and caching.",
"Experiment hardware was a Xeon E3-1230 v2 3.3GHz quadcore with 32 kB per core for each of data and instructions level 1 cache, 256 kB level 2 cache per core, 8 MB shared level 3 cache, and 16 GB 1600 MHz ddr3 memory.",
"Note that this configuration influences only runtime and physical cache (table REF and the first two bars in each group of fig.",
"REF ); other measures are system independent.",
"We evaluate over a variety of data in common use for testing string processing performance, from the Pizza & Chili [7] and lightweight [16] corpora.",
"In order to evaluate a degenerate case for nobu, we also include an adversary input constructed for the pattern $T=ab^{m^2}abab^2ab^3\\cdots ab^ma$ (with $m=4082$ for a 25 million character file), which has $\\Omega (N^{1.5})$ performance in this scheme [19]."
],
[
"Results",
"Fig.",
"REF shows performance across seven implementations and five performance measures (explained in section REF ), which we deem to be relevant for comparison.",
"It summarizes the runtimes (also in table REF ) and memory access measures by taking averages across all files except adversary, with equal weight per file.",
"The bars are scaled to show percentages of the measures for the basic notd implementation, which is used as the benchmark.",
"The order of the implementations when ranked by performance is fairly consistent across the different measures, with some deviation in particular for the hardware cache measure and smaller-cache models.",
"The hardware cache measurement comes out as a relatively poor predictor of performance; by the numbers reported by Perf, the hardware cache even appears to be outperformed by our simplistic theoretical cache model.",
"We detect only a minor improvement of eotd ll implementations in relation to nobu ll, while inline eotd ht provides a more significant improvement.",
"Note, however that for nobu, the ht implementation is much worse than the ll implementation, while the reverse is true for eotd.",
"This can be attributed to the different hash table use and the particular significance of inlining, noted in section .",
"The fact that eotd ht without inlining (not in the diagram) is not clearly better than nobu ht stands to confirm this.",
"Although table REF shows that eotd ll beats its ht counterpart for files producing a low average out-degree in ${\\mathcal {ST}}$ (because of a small alphabet and/or high repetitiveness), the robustness of hashing (cf.",
"fig REF ) has the greater impact on average.",
"We have included results to show the impact of the add to back heuristic in eotd ll, which also produced a slight improvement for nobu (not shown in diagram), as expected.",
"The operation counts shown in table REF generally confirm our expectations.",
"(Branch counts include moves down from $\\topnode $ to the root, in order to match Senft and Dvořák's corresponding counts [19].)",
"eov yields a large rescan reduction, even for the adversary file, which makes it an attractive alternative to nobu when branching is very expensive.",
"We found the exact choice of the $k$ parameter of eov not to be overly delicate.",
"All values shown were obtained with $k=5$ ."
],
[
"Conclusion",
"It is possible to significantly improve online suffix tree construction time through modifications that target reducing branch operations and cache utilization, while maintaining linear worst-case time complexity.",
"In many applications, our representation variants should be directly applicable for runtime reduction.",
"Interesting topics remaining to explore are how our techniques for, e.g., suffix link orientation, fit into the compromise game of time versus space in succinct representations such as compressed suffix trees, and comparison to off-line construction.",
"Table: Running times in seconds for the same files astable"
]
] | 1403.0457 |
[
[
"Towards holographic duals for anomalous supercurrents"
],
[
"Abstract In this paper, based on the usual techniques of Gauge/gravity duality, we derive the Ginzburg-Landau current for $ s $- wave superconductors in the presence of higher derivative corrections to the abelian gauge sector in the bulk.",
"It has been observed that at sufficiently low temperatures the $U(1)$ current thus computed at the boundary varies inversely with the sixth power of the temperature ($ T $) which therefore gives rise to the phenomenon like anomalous superconductivity for the boundary theory.",
"Interestingly we note that this anomalous effect is associated with the leading order higher derivative corrections to the $ U(1) $ sector in the bulk."
],
[
"Overview and Motivation",
"For the past several years the AdS/CFT correspondence [1]-[3] has been found to play a significant role in order to understand several crucial properties for type II superconductors those are believed to be strongly coupled.",
"The dual gravitational description for such superconductors/superfluids essentially consists of an abelian Higgs model coupled to gravity in an asymptotically anti de-Sitter (AdS) space time [4]-[16].",
"Till date several crucial properties of these holographic superconductors have been investigated under different circumstances for example, the effect of external magnetic field on holographic superconductors have been investigated extensively in [17]-[33].",
"To be more precise, the holographic computation of the London equation as well as the magnetic penetration depth was first carried out in [17] where the authors had shown that the holographic superconductors are basically of type II in nature.",
"Based on the AdS/CFT framework, the vortex lattice structure for $ s $ - wave superconductors was first investigated by Maeda et al in [18] where the authors had shown that for $ T<T_c $ the triangular vortex configuration is the thermodynamically most favorable one.",
"On top of it, in their analysis [18] the authors had also shown that using the AdS/CFT prescription one can in fact arrive at some local expression for the super-current associated with type II vortices that has the remarkable structural similarity to that with the standard Ginzburg-Landau (GL) expression for the supercurrent in ordinary type II superconductors [34]-[35].",
"This observation is quite important in the sense that it establishes the precise connection between two apparently different looking pieces, namely the superconductivity and the AdS/CFT correspondence.",
"Keeping the spirit of these earlier analysis, and based on the $AdS_4/CFT_3$ framework, the purpose of the present article is to go beyond the usual framework of the GL theory and construct the local version of the $ U(1) $ current in the presence of various higher derivative corrections to the dual (gravitational) description in the bulk.",
"In other words, in the present work we explore the effect of adding higher derivative corrections to the gauge sector of the abelian Higgs model [5] and compute the $ U(1) $ current for the boundary theory.",
"From our computations we note that, at sufficiently low temperatures the $ U(1) $ current thus computed at the boundary of the $ AdS_4 $ varies inversely with the sixth power of the temperature ($ T $ ) and thereby diverges as $ T\\rightarrow 0 $ .",
"This observation at the first place gives some intuitive theoretical understanding of the low temperature anomalous behaviour in superconducting materials based on the $AdS_4/CFT_3$ duality.",
"The crucial fact that emerges from our analysis is that the higher derivative corrections on the $ U(1) $ sector of the bulk theory eventually acts as a source for the low temperature anomalous effects at the boundary.",
"$\\bullet $ Note Added: The existence of the low temperature anomalous superconductivity across the junction of certain anisotropic superconducting materials [36]-[39] has been known for a long time.",
"The crucial fact about our analysis is that in our present calculation we do not consider any anisotropic model for superconductors.",
"Rather we claim that our model plays a significant role in the holographic understanding of such low temperature anomalous behaviour in supercurrents flowing across the junction of two superconducting materials.",
"The reason behind this claim rests on the fact that the superconducting gap placed in between the junction of two superconducting materials essentially behaves like a weak superconductor [40].",
"This therefore suggests the fact that the order parameter does not actually vanish inside the gap and therefore the Josephson current is fundamentally no different from the usual GL current in superconductors [41].",
"With this construction in hand, one can in fact go a step further to build a holographic model for anomalous supercurrents across the junction.",
"We leave this project for future investigations.",
"The organization of the paper is the following: In Section 2 we provide all the essential details of the dynamics of the scalar as well as the gauge fields considering them as a probe on the background $ AdS_{4} $ space time.",
"Based on the $AdS_4/CFT_3$ prescription, the computation of the boundary current up to leading order in the BI coupling ($ b $ ) has been performed in Section 3.",
"In Section 4, for the sake of completeness of our analysis we compute the free energy of the vortex configuration.",
"Finally, we conclude in Section 5."
],
[
"The set up",
"We start our analysis by considering the abelian Higgs model coupled to gravity in $(3+1)$ dimensions in the presence of the negative cosmological constant, $S= \\frac{1}{16\\pi G_{4}}\\int d^{4}x\\sqrt{-g}\\left[R-2\\Lambda +\\frac{1}{b}\\left(1-\\sqrt{1+\\frac{bF^{2}}{2}}\\right)-|\\nabla _{\\mu }\\Psi -iA_{\\mu }\\Psi |^{2}-m^{2}|\\Psi |^{2} \\right].$ The above action is the non linear generalization of the abelian Higgs model proposed originally in [5], where one replaces the usual Maxwell action by the Born Infeld (BI) term Here $ b $ is the BI coupling parameter and $ F^{2}=F^{\\mu \\nu }F_{\\mu \\nu } $ ..",
"It is in fact quite easy to check that in the limit $ b\\rightarrow 0 $ one recovers the usual Maxwell actionThe quantity $ \\Lambda (=-3/l^{2}) $ is the cosmological constant.",
"In the present analysis we set $ l=1 $ ..",
"In our analysis we consider the effect of the BI term (which is nothing but the higher derivative corrections to the usual Maxwell action) perturbatively in the BI parameter ($ b $ ) and all our expressions are valid upto leading order in $ b $ .",
"The background over which the analysis is performed is an asymptotically $ AdS_4 $ black brane solution, $ds^{2}=-f(u)dt^{2}+\\frac{r_+^{2}}{u^{4}}f^{-1}(u)du^{2}+\\frac{r_+^{2}}{u^{2}}d\\textbf {x}^{2}$ where, $f(u)=\\frac{r_+^{2}}{u^{2}}(1-u^{3}).$ Note that in these coordinates the horizon is located at $ u=1 $ whereas on the other hand, the boundary of the $AdS_4$ is located at $ u=0 $ .",
"The temperature of the black brane is given by, $T=\\frac{3r_{+}}{4\\pi }$ which is considered to be fixed for the present analysis.",
"Therefore the boundary field theory could also be considered to be at the same temperature as that of the black brane.",
"In order to proceed further we first define a parameter $ \\varepsilon (=\\frac{H_{c2}-H}{H_{c2}}) $ such that $ \\varepsilon $ is positive definite and $ |\\varepsilon |\\ll 1 $ .",
"Here $ H_{c2} $ is the (upper) critical magnetic field strength above which the charge condensate ($ \\Psi $ ) vanishes.",
"We expand both the gauge field as well as the scalar field in this parameter as, $A_{\\mu }&=& A_{\\mu }^{(0)}+\\varepsilon A_{\\mu }^{(1)}(u,\\textbf {x})+\\mathcal {O}(\\varepsilon ^{2})\\\\\\Psi &=& \\varepsilon ^{1/2}\\psi _{1}(u,\\textbf {x})+\\mathcal {O}(\\varepsilon ^{3/2}).$ Note that here $ A^{(0)}_{\\mu } $ is the solution of the Maxwell's equation when the charge condensate ($ \\Psi $ ) is zero.",
"In the present analysis we choose the following ansatz for $ A^{(0)}_{\\mu } $ namely, $A_{\\mu }^{(0)}=(A_{t}^{(0)}(u),0,0,A_{y}^{(0)}(x))$ where the spatial component of the gauge field acts as a source for some non zero magnetic field ($ H_{c2} $ ) at the boundary.",
"Furthermore here $ A^{(i)}_{\\mu }(i=1,2,..) $ s are the fluctuations of the $ U(1) $ gauge field in the presence of the non zero charge condensate.",
"The quantity $ \\psi _1 $ denotes the first non trivial fluctuation in the charge condensate.",
"Our next goal would be to provide a detail of the dynamics of both the scalar field as well as the gauge field considering only the leading order terms in $ \\varepsilon $ ."
],
[
"Dynamics of scalar field: vortex structure",
"In this part of our analysis, considering the probe limit we would like to explore the effect of BI correction on the dynamics of the scalar field.",
"We start our analysis considering the following ansatz for the scalar field namely, $\\Psi = \\Psi (u,x,y).$ Using the above ansatz (REF ) and considering the leading order fluctuations () the equation for the scalar field turns out to be, $\\partial _{u}^{2}\\psi _{1}+\\frac{f^{^{\\prime }}(u)}{f(u)}\\partial _{u}\\psi _{1}+\\frac{r_{+}^{2}A^{(0)2}_{t}}{u^{4}f^{2}(u)}\\psi _{1}-\\frac{m^{2}r_{+}^{2}}{u^{4}f(u)}\\psi _{1}+ \\frac{1}{u^{2}f(u)}(\\Delta -2iH_{c2}x\\partial _{y}-H_{c2}^{2}x^{2})\\psi _{1}=0.$ Our next goal would be to solve (REF ) using the separation of variable.",
"In order to do so we choose the following ansatz [18], $\\psi _{1}(u,\\textbf {x})=\\rho _{1}(u)e^{ik_{y}y}X(x)=\\rho _{1}(u)\\mathcal {V}(x,y).$ Substituting (REF ) into (REF ) we are essentially left with the following two sets of equations namely, $\\partial _{u}^{2}\\rho _{1}+\\frac{f^{^{\\prime }}(u)}{f(u)}\\partial _{u}\\rho _{1}+\\frac{r_{+}^{2}A^{(0)2}_{t}}{u^{4}f^{2}(u)}\\rho _{1}(u)-\\frac{m^{2}r_{+}^{2}}{u^{4}f(u)}\\rho _{1}(u)=\\frac{\\rho _{1}(u)}{\\xi ^{2}u^{2}f(u)}$ and, $-X^{^{\\prime \\prime }}(x)+H_{c2}^{2}\\left( x-\\frac{k_{y}}{H_{c2}}\\right)^{2}X(x)=\\frac{X(x)}{\\xi ^{2}}$ where $ \\xi $ is a constant that appears during the separation of variables which eventually plays the role of the correlation length [34]-[35] and which is also related to the strength of the external magnetic field ($ H \\sim \\frac{1}{\\xi ^{2}} $ )[17].",
"Eq.",
"(REF ) has in fact a remarkable structural similarity to that with the vortex structure appearing in type II superconductors in a traditional GL theory See Appendix for details..",
"Using elliptic theta function, the above solution (REF ) could be expressed in its standard form as [18], $\\psi _{1}(u,\\textbf {x})=\\rho _{1}(u)e^{-\\frac{x^{2}}{2\\xi ^{2}}}\\vartheta _{3}(v,\\tau )$ where the elliptic theta function could be formally expressed as, $\\vartheta _{3}(v,\\tau )=\\sum _{l=-\\infty }^{l=\\infty }q^{l^{2}}z^{2l}$ with, $q&=&e^{i\\pi \\tau }= e^{i\\pi \\left( \\xi ^{2}\\frac{2\\pi i - a_x}{a_y^{2}}\\right)},~~~\\tau = \\xi ^{2}\\frac{2\\pi i - a_x}{a_y^{2}} \\nonumber \\\\z&=&e^{i\\pi v}=e^{i\\pi \\left( \\frac{y-ix}{a_y}\\right)},~~~v= \\frac{y-ix}{a_y}$ where we have introduced two arbitrary parameters namely $ a_x $ and $ a_y $ where $ a_y $ is in particular associated with the periodicity along the $ y $ direction.",
"Note that the solution (REF ) has a Gaussian fall off along the $ x $ direction which corresponds to the fact that the vortex structure eventually dies out for $ |x|\\gg \\xi $ .",
"Thus the correlation length ($ \\xi $ ) acts as a natural length scale in order to determine the size of a single vortex lattice."
],
[
"Dynamics of gauge fields",
"In this section we explore the dynamics of the abelian gauge field in the presence of zeroth as well as the first order fluctuations in the charge condensation.",
"Our aim would be to solve the equations perturbatively both for the scalar field fluctuations ($ \\varepsilon $ ) as well as for the BI coupling ($ b $ ).",
"To start with we note that the Maxwell's equation turns out to be, $\\nabla _{\\mu }\\left(\\frac{F^{\\nu \\mu }}{\\sqrt{1+\\frac{bF^{2}}{2}}} \\right)=j^{\\nu }$ where, $j^{\\nu }=i(\\Psi (D^{\\nu }\\Psi )^{\\dagger }-\\Psi ^{\\dagger }D^{\\nu }\\Psi )$ with $ D_{\\mu } = \\partial _{\\mu }-iA_{\\mu }$ as the gauge covariant derivative.",
"Since we are interested in solving these equations only up to leading order in the BI coupling ($ b $ ), therefore we expand the l.h.s.",
"of the above equation (REF ) up to leading order in $ b $ which finally yields, $\\nabla _{\\mu }F^{\\nu \\mu }-\\frac{b}{4}F^{\\nu \\mu }\\partial _{\\mu }F^{2}=j^{\\nu }\\left( 1+ \\frac{bF^{2}}{4}\\right).$ Considering the perturbative expansions (REF ) and (), we expand both the l.h.s.",
"as well as the r.h.s.",
"of (REF ) perturbatively in the parameter $ \\varepsilon $ which finally yields, $\\nabla _{\\mu }F^{\\nu \\mu (0)}-\\frac{b}{4}F^{\\nu \\mu (0)}\\partial _{\\mu }(F^{\\lambda \\sigma (0)}F^{(0)}_{\\lambda \\sigma })&=&0\\\\\\nabla _{\\mu }F^{\\nu \\mu (1)}-\\frac{b}{4}[2F^{\\nu \\mu (0)}\\partial _{\\mu }(F^{\\lambda \\sigma (0)}F^{(1)}_{\\lambda \\sigma })+F^{\\nu \\mu (1)}\\partial _{\\mu }(F^{\\lambda \\sigma (0)}F^{(0)}_{\\lambda \\sigma })]&=&\\left(1+\\frac{b}{4}F^{\\lambda \\sigma (0)}F^{(0)}_{\\lambda \\sigma } \\right)j^{\\nu (1)}.\\nonumber \\\\$ Since we want to solve the above set of equations (REF ) and () upto leading order in the BI coupling ($ b $ ) therefore we expand the $ U(1) $ gauge field pertubatively in the BI parameter as, $A^{(m)}_{\\mu }=\\mathcal {A}^{(m)(b^{(0)})}_{\\mu }+b\\mathcal {A}^{(m)(b^{(1)})}_{\\mu }+\\mathcal {O}(b^{2}).$ Note that in the above expansion (REF ) we have used two different indices in order to incorporate the effect of fluctuations of two different kinds.",
"The index ($ m $ ) corresponds to the fluctuations in the order parameter ($ \\Psi $ ).",
"In other words, various terms corresponding to different values of ($ m $ ) stand for different terms in the perturbative $ \\varepsilon $ expansion of (REF ).",
"On the other hand, the indices $ b^{n} (n=0,1,2,..)$ stand for the perturbations at different levels in the BI coupling ($ b $ ).",
"With the above machinery in hand, we are now in a position to solve the equations (REF ) and () perturbatively in the BI parameter ($ b $ ).",
"Let us first consider (REF ).",
"Using (REF ) one can in fact show that it leads to the following set of equations namely, $\\nabla _{\\mu }\\mathcal {F}^{\\nu \\mu (0)(b^{(0)})}&=&0\\\\\\nabla _{\\mu }\\mathcal {F}^{\\nu \\mu (0)(b^{(1)})}-\\frac{1}{4}\\mathcal {F}^{\\nu \\mu (0)(b^{(0)})}\\partial _{\\mu }(\\mathcal {F}^{\\lambda \\sigma (0)(b^{(0)})}\\mathcal {F}^{(0)(b^{(0)})}_{\\lambda \\sigma })&=&0$ whose solutions could be expressed as, $A^{(0)}_{t}(u)&=&\\mu (1-u)\\left[ 1-\\frac{b}{10 r_{+}^{4}}(\\mu ^{2}r_+^{2}-H_{c2}^{2})\\zeta (u)\\right]+\\mathcal {O}(b^{2})\\\\A^{(0)}_{y}(x)&=&H_{c2}x$ where $ \\zeta (u)=u(1+u+u^{2}+u^{3}) $ .",
"In order to solve () we first split Eq.",
"() into different components which could be enumerated as followsHere we choose a particular gauge $ A_u =0 $ and also exploit the residual gauge symmetry $ A_i^{(1)}\\rightarrow A_i^{(1)} -\\partial _i\\varpi (\\textbf {x}) $ to fix the gauge $ \\partial _x A_x^{(1)} +\\partial _y A_y^{(1)} =0 $ ., $L_t A^{(1)}_{t}-\\frac{b u^{2}f(u)}{4}\\left[ 2\\partial _{u}A^{(0)}_{t}\\partial _{u}(F^{\\lambda \\sigma (0)}F^{(1)}_{\\lambda \\sigma })+ \\partial _u A^{(1)}_{t}\\partial _{u}F^{2(0)}\\right]=\\frac{2r_+^{2}A^{(0)}_{t}|\\psi _{1}|^{2}}{u^{2}}\\left(1+\\frac{b}{4}F^{2(0)}\\right)$ $L_s A^{(1)}_{x}+\\frac{b}{4}\\left( 2H_{c2}\\partial _y (F^{\\lambda \\sigma (0)}F^{(1)}_{\\lambda \\sigma })+u^{2}f(u)F_{xu}^{(1)}\\partial _{u}F^{2(0)}\\right)=-\\frac{r_+^{2}}{u^{2}}\\left(1+\\frac{b}{4}F^{2(0)}\\right)j^{(1)}_{x}$ $L_s A^{(1)}_{y}-\\frac{b}{4}\\left( 2H_{c2}\\partial _x (F^{\\lambda \\sigma (0)}F^{(1)}_{\\lambda \\sigma })-u^{2}f(u)F_{yu}^{(1)}\\partial _{u}F^{2(0)}\\right)=-\\frac{r_+^{2}}{u^{2}}\\left(1+\\frac{b}{4}F^{2(0)}\\right)j^{(1)}_{y}$ where, $ L_t = u^{2}f(u)\\partial _u^{2}+\\Delta $ and $ L_s = \\partial _u (u^{2}f(u)\\partial _u)+\\Delta $ are the differential operators and $ \\Delta = \\partial _{x}^{2}+\\partial _{y}^{2}$ is the usual Laplacian.",
"At this stage one might note that the solution corresponding to the radial equation (REF ) could be further expressed as a perturbation in the BI coupling ($ b $ ) as, $\\rho _{1}(u)=\\rho _{1}^{(b^{(0)})}+b\\rho _{1}^{(b^{(1)})}+\\mathcal {O}(b^{2}).$ With the above prescriptions (REF ) and (REF ) in hand we are now in a position to solve the above set of equations (REF )-(REF ) order by order as a perturbation in the BI coupling ($ b $ ).",
"Let us first note down the equations at the at the zeroth order level which turns out to be, $L_t \\mathcal {A}^{(1)(b^{(0)})}_{t}&=&\\frac{2r_+^{2}\\rho _{1}^{2(b^{(0)})}}{u^{2}}\\mathcal {A}^{(0)(b^{(0)})}_{t}\\sigma (\\textbf {x})\\\\L_s \\mathcal {A}^{(1)(b^{(0)})}_{x}&=&\\frac{r_+^{2}\\rho _{1}^{2(b^{(0)})}}{u^{2}}\\epsilon _{x}\\ ^{y}\\partial _{y}\\sigma (\\textbf {x})\\\\L_s \\mathcal {A}^{(1)(b^{(0)})}_{y}&=&\\frac{r_+^{2}\\rho _{1}^{2(b^{(0)})}}{u^{2}}\\epsilon _{y}\\ ^{x}\\partial _{x}\\sigma (\\textbf {x})$ where, $ \\sigma (\\textbf {x})=(=|e^{-\\frac{x^{2}}{2\\xi ^{2}}}\\vartheta _{3}(v,\\tau )|^{2})$ corresponds to the triangular vortex solution in the ($ x,y $ ) plane andHere $ i(=x,y) $ denotes the spatial coordinates.",
"$ j^{(1)}_{i}=\\rho _{1}^{2}\\epsilon _{i}\\ ^{j}\\partial _j\\sigma (\\textbf {x}) $ .",
"Note that here $ \\epsilon _{ij} $ is an anti symmetric tensor with the property $ \\epsilon _{xy}=- \\epsilon _{yx}=1$ .",
"Note that the above set of equations (REF )-() essentially corresponds to a set of inhomogeneous differential equations with a source term on the r.h.s.",
"of it.",
"Therefore in general the solutions to these equations could be expressed in terms of Green's functions that satisfy certain boundary conditions near the boundary of the $ AdS_4 $ .",
"These solutions could be formally expressed as, $A^{(1)(b^{(0)})}_{t}&=&-2r_+^{2}\\int _{0}^{1}du^{^{\\prime }}\\frac{\\rho _{1}^{2(b^{(0)})}(u^{\\prime })}{u^{\\prime 2}}\\mathcal {A}^{(0)(b^{(0)})}_{t}(u^{\\prime })\\int d\\textbf {x}^{\\prime }\\mathcal {G}_{t}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })\\sigma (\\textbf {x}^{\\prime })\\nonumber \\\\A^{(1)(b^{(0)})}_{i}&=&a_{i}(\\textbf {x})-r_+^{2}\\epsilon _{i}^{j}\\int _{0}^{1}du^{^{\\prime }}\\frac{\\rho _{1}^{2(b^{(0)})}(u^{\\prime })}{u^{\\prime 2}}\\int d\\textbf {x}^{\\prime }\\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })\\partial _{j}\\sigma (\\textbf {x}^{\\prime }).$ Here $ a_{i}(\\textbf {x})$ is the homogeneous part of the solution of ()-() which is the only term that contributes to a uniform magnetic field ($ H_{c2}=\\epsilon _{ij}\\partial _ia_j $ ) at the boundary of the $ AdS_4 $ .",
"On the other hand, $ \\mathcal {G}_{t}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime }) $ and $ \\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime }) $ are the Green's functions corresponding to the above set of equations (REF )-() that obey the following differential equations namely, $L_t \\mathcal {G}_t(u,u^{^{\\prime }};\\textbf {x},\\textbf {x}^{\\prime })&=&-\\delta (u-u^{\\prime })\\delta (\\textbf {x}-\\textbf {x}^{\\prime })\\nonumber \\\\L_s \\mathcal {G}_s(u,u^{^{\\prime }};\\textbf {x},\\textbf {x}^{\\prime })&=&-\\delta (u-u^{\\prime })\\delta (\\textbf {x}-\\textbf {x}^{\\prime })$ along with the following (Dirichlet) boundary conditionsThe above boundary conditions (REF ) eventually clarify the following two things : Firstly, we are working with a fixed chemical potential ($ \\mu $ ) at the boundary which is reflected in the fact that any non trivial fluctuation of $ A_t $ eventually vanishes near the boundary of the $AdS_4$ .",
"Secondly, we have a uniform magnetic field ($ H_{c2} $ ) at the boundary since any corrections appearing to the spatial components of the gauge field eventually dye out at the boundary.",
"near the boundary of the $ AdS_4 $ [18], $\\mathcal {G}_t(u,u^{^{\\prime }};\\textbf {x},\\textbf {x}^{\\prime })|_{u=0}=\\mathcal {G}_t(u,u^{^{\\prime }};\\textbf {x},\\textbf {x}^{^{\\prime }})|_{u=1}=0\\nonumber \\\\\\mathcal {G}_{s}(u,u^{^{\\prime }};\\textbf {x},\\textbf {x}^{\\prime })|_{u=0}=u^{2}f(u)\\partial _u\\mathcal {G}_{s}(u,u^{^{\\prime }};\\textbf {x},\\textbf {x}^{^{\\prime }})|_{u=1}=0.$ Next, we almost follow the same procedure in order to solve the equations (REF )-(REF ) for the leading order in the BI coupling ($ b $ ).",
"Let us first note down the equations at leading order in $ b $ namely, $L_t \\mathcal {A}^{(1)(b^{(1)})}_{t}-\\frac{u^{2}f(u)}{4}\\left[ 2\\partial _{u}\\mathcal {A}^{(0)(b^{(0)})}_{t}\\partial _{u}(\\mathcal {F}^{\\lambda \\sigma (0)(b^{(0)})}\\mathcal {F}^{(1)(b^{(0)})}_{\\lambda \\sigma })+ \\partial _u \\mathcal {A}^{(1)(b^{(0)})}_{t}\\partial _{u}\\mathcal {F}^{2(0)(b^{(0)})}\\right]&=&\\frac{2r_+^{2}}{u^{2}}\\mathcal {J}(u)\\sigma (\\textbf {x})\\nonumber \\\\L_s \\mathcal {A}^{(1)(b^{(1)})}_{x}+\\frac{1}{4}\\left[ 2H_{c2}\\partial _y (\\mathcal {F}^{\\lambda \\sigma (0)(b^{0})}\\mathcal {F}^{(1)(b^{(0)})}_{\\lambda \\sigma })+u^{2}f(u)\\mathcal {F}_{xu}^{(1)(b^{(0)})}\\partial _{u}\\mathcal {F}^{2(0)(b^{(0)})}\\right]&=&-\\frac{r_+^{2}}{u^{2}}\\mathcal {I}(u) \\epsilon _{x}\\ ^{y}\\partial _{y}\\sigma \\nonumber \\\\L_s \\mathcal {A}^{(1)(b^{(1)})}_{y}-\\frac{1}{4}\\left[ 2H_{c2}\\partial _x (\\mathcal {F}^{\\lambda \\sigma (0)(b^{(0)})}\\mathcal {F}^{(1)(b^{(0)})}_{\\lambda \\sigma })-u^{2}f(u)\\mathcal {F}_{yu}^{(1)(b^{(0)})}\\partial _{u}\\mathcal {F}^{2(0)(b^{(0)})}\\right]&=&-\\frac{r_+^{2}}{u^{2}}\\mathcal {I}(u) \\epsilon _{y}\\ ^{x}\\partial _{x}\\sigma \\nonumber \\\\.$ where $ \\mathcal {J}(u)$ and $ \\mathcal {I}(u)$ are some radial functions which could be expressed as, $\\mathcal {J}(u)&=&\\frac{1}{4}\\mathcal {A}^{(0)(b^{(0)})}_{t}\\mathcal {F}^{2(0)(b^{(0)})}\\rho _{1}^{2(b^{(0)})}+\\mathcal {A}^{(0)(b^{(1)})}_{t}\\rho _{1}^{2(b^{(0)})}+2\\rho _{1}^{(b^{(1)})}\\rho _{1}^{(b^{(0)})}\\mathcal {A}^{(0)(b^{(0)})}_{t}\\nonumber \\\\\\mathcal {I}(u) &=& \\frac{\\rho _{1}^{2(b^{(0)})}}{4}\\mathcal {F}^{2(0)(b^{(0)})}+2\\rho _{1}^{(b^{(0)})}\\rho _{1}^{(b^{(1)})}.$ These are again set of inhomogeneous differential equations whose solutions could be expressed in terms of Green's functions as, $\\mathcal {A}^{(1)(b^{(1)})}_{t}=-\\int _{0}^{1}du^{\\prime }\\int d\\textbf {x}^{\\prime } \\mathcal {P}_{i}(u^{\\prime },\\textbf {x}^{\\prime })\\mathcal {G}_{t}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })\\nonumber \\\\\\mathcal {A}^{(1)(b^{(1)})}_{i}=-\\int _{0}^{1}du^{\\prime }\\int d\\textbf {x}^{\\prime } \\mathcal {Q}_{i}(u^{\\prime },\\textbf {x}^{\\prime })\\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })$ where $ \\mathcal {P}_{i}(u,\\textbf {x}) $ and $ \\mathcal {Q}_{i}(u,\\textbf {x}) $ are some nontrivial functions that could be expressed as, $\\mathcal {P}_{i}&=&\\frac{2r_+^{2}}{u^{2}}\\mathcal {J}(u)\\sigma (\\textbf {x})+\\frac{u^{2}f(u)}{4}\\left[ 2\\partial _{u}\\mathcal {A}^{(0)(b^{(0)})}_{t}\\partial _{u}(\\mathcal {F}^{\\lambda \\sigma (0)(b^{(0)})}\\mathcal {F}^{(1)(b^{(0)})}_{\\lambda \\sigma })+ \\partial _u \\mathcal {A}^{(1)(b^{(0)})}_{t}\\partial _{u}\\mathcal {F}^{2(0)(b^{(0)})}\\right]\\nonumber \\\\\\mathcal {Q}_{i}&=&-\\epsilon _{i}\\ ^{j}\\partial _{j}\\Re (u,\\textbf {x})$ with, $\\Re (u,\\textbf {x})= \\frac{r_+^{2}}{u^{2}}\\mathcal {I}(u)\\sigma (\\textbf {x})+\\frac{H_{c2}}{2}\\mathcal {F}^{\\lambda \\sigma (0)(b^{(0)})}\\mathcal {F}^{(1)(b^{(0)})}_{\\lambda \\sigma }\\nonumber \\\\+\\frac{r_+^{2}u^{2}f(u)}{4}\\partial _u \\mathcal {F}^{2(0)(b^{(0)})}\\int _{0}^{1}du^{\\prime } \\frac{\\rho _{1}^{2(b^{(0)})}(u^{\\prime })}{u^{\\prime 2}}\\partial _u \\int d\\textbf {x}^{\\prime }\\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })\\sigma (\\textbf {x}^{\\prime }).\\nonumber \\\\$"
],
[
"The $ U(1) $ current",
"From the $ AdS_{4}/CFT_{3} $ duality, it is indeed quite evident that the $ U(1) $ gauge field in the bulk acts as the source for some global $ U(1) $ operator at the boundary.",
"Following the holographic prescription [6] the $ U(1) $ current for our present case turns out to beWe have set $ 16\\pi G_{4}=1 $ .",
", $\\langle J^{\\mu }\\rangle =\\lim _{u\\rightarrow 0}\\frac{\\delta S^{(os)}}{\\delta A_{\\mu }}=\\lim _{u\\rightarrow 0} \\frac{\\sqrt{-g}F^{\\mu u}}{\\sqrt{1+\\frac{b F^{2}}{2}}}.$ The goal of the present paper is to compute the above current (REF ) for the leading order in the gauge fluctuations.",
"Keeping terms up to leading order in $ b $ and considering only the spatial components of the current we finally have, $\\langle J_{i}\\rangle = \\left[ \\mathcal {F}_{iu}^{(1)(b^{(0)})}+b\\mathcal {F}_{iu}^{(1)(b^{(1)})} \\right]_{u=0}+ \\mathcal {O}(b^{2}).$ where we have re-scaled the current by a factor of $ \\varepsilon r_{+} $ .",
"Substituting the above set of solutions (REF ) and (REF ) into (REF ) we finally obtain, $\\langle J_{i}\\rangle = \\epsilon _{i}^{j}\\partial _{j}\\Theta (\\textbf {x})$ where, $\\Theta (\\textbf {x}) =r_+^{2} \\int _{0}^{1}du^{^{\\prime }}\\frac{\\rho _{1}^{2(b^{(0)})}(u^{\\prime })}{u^{\\prime 2}}\\partial _u \\int d\\textbf {x}^{\\prime }\\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })\\sigma (\\textbf {x}^{\\prime })|_{u=0}\\nonumber \\\\-b\\int _{0}^{1}du^{\\prime }\\partial _u \\int d\\textbf {x}^{\\prime } \\Re (u^{\\prime },\\textbf {x}^{\\prime })\\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })|_{u=0}+\\mathcal {O}(b^{2}).$ Eq.",
"(REF ) is the exact expression for the $ U(1) $ current at the leading order in the BI coupling ($ b $ ).",
"Note that the boundary current ($ J_i $ ) is a non local function of the vortex solution $ \\sigma (\\textbf {x}) $ in the sense that in order to evaluate the above current one needs to integrate the function ($ \\sigma (\\textbf {x}) $ ) over a region around the point ($ \\textbf {x} $ ) where vortex is localized.",
"Our next step would be to remove the above non localityThe Ginzburg Landau expression for the super-current in type II superconductors are usually expressed as a local function of the vortex solution [34]-[35].",
"and express the current as a local function of the vortex solution $ \\sigma (\\textbf {x}) $ .",
"In order to remove the above non locality we take the following steps.",
"As a first step, following the prescription of [18], we explicitly decompose the full Green's functions into following two pieces namely, $\\mathcal {G}_{t}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })&=&\\sum _{\\alpha }\\vartheta _{\\alpha }(u)\\vartheta _{\\alpha }^{\\dagger }(u^{\\prime })\\tilde{\\mathcal {G}_{t}}(\\textbf {x}-\\textbf {x}^{\\prime },\\alpha )\\nonumber \\\\\\mathcal {G}_{s}(u,u^{\\prime };\\textbf {x},\\textbf {x}^{\\prime })&=&\\sum _{\\lambda }\\zeta _{\\lambda }(u)\\zeta _{\\lambda }^{\\dagger }(u^{\\prime })\\tilde{\\mathcal {G}_{s}}(\\textbf {x}-\\textbf {x}^{\\prime },\\lambda )$ where, $ \\vartheta _{\\alpha }(u) $ and $ \\zeta _{\\lambda }(u) $ are the radial functions that satisfy the following eigen value equations namelyFollowing the boundary conditions (REF ), the value of these radial functions near the boundary of the $ AdS_{4} $ could be set as, $ \\vartheta _{\\alpha }(0)=0 $ and $ \\zeta _{\\lambda }(0)=0 $ ., $\\mathcal {L}_{t}\\vartheta _{\\alpha }(u)=\\alpha \\vartheta _{\\alpha }(u);~~~\\sum \\limits _{\\alpha }\\vartheta _{\\alpha }(u)\\vartheta _{\\alpha }^{\\dagger }(u^{\\prime })=\\delta (u-u^{\\prime });~~~\\langle \\vartheta _{\\alpha }|\\vartheta _{\\alpha ^{\\prime }}\\rangle = \\delta _{\\alpha \\alpha ^{\\prime }}\\nonumber \\\\\\mathcal {L}_{s}\\zeta _{\\lambda }(u)=\\lambda \\zeta _{\\lambda }(u);~~~\\sum \\limits _{\\lambda }\\zeta _{\\lambda }(u)\\zeta _{\\lambda }^{\\dagger }(u^{\\prime })=\\delta (u-u^{\\prime });~~~\\langle \\zeta _{\\lambda }|\\zeta _{\\lambda ^{\\prime }}\\rangle = \\delta _{\\lambda \\lambda ^{\\prime }}.$ where, $\\mathcal {L}_{t}= -u^{2}f(u)\\partial _u^{2} $ and $ \\mathcal {L}_{s}=-\\partial _u (u^{2}f(u)\\partial _u) $ are the two differential operators that solely depend on the radial coordinate ($ u $ ).",
"Note that here $ \\tilde{\\mathcal {G}_{t}}(\\textbf {x}-\\textbf {x}^{\\prime },\\alpha ) $ and $ \\tilde{\\mathcal {G}_{s}}(\\textbf {x}-\\textbf {x}^{\\prime },\\lambda ) $ are the Green's functions defined on the two dimensional ($ x,y $ ) plane over which the condensate forms.",
"Following the definitions (REF ) and (REF ), it is indeed quite trivial to show that these two dimensional Green's functions satisfy the differential equation of the following form, $(\\Delta - \\mathbb {k}^{2})\\tilde{\\mathcal {G}}(\\textbf {x},\\mathbb {k}^{2}) &=& -\\delta (\\textbf {x})$ for any real positive value of $ \\mathbb {k}^{2} $ .",
"The solution of (REF ) could be expressed in terms of the modified Bessel function namely, $\\tilde{\\mathcal {G}}(\\textbf {x},\\mathbb {k}^{2})=\\frac{1}{2\\pi }K_0 (\\mathbb {k}|x|)$ which satisfies the boundary condition $ lim_{|\\textbf {x}|\\rightarrow \\infty }|\\tilde{\\mathcal {G}}(\\textbf {x})|<\\infty $ .",
"Note that we have two length scales in our theory.",
"One is the length scale $\\frac{1}{\\sqrt{\\lambda }} $ (or $ \\frac{1}{\\sqrt{\\alpha }} $ ) over which the Green's function $ \\mathcal {G}_{s}(\\textbf {x}-\\textbf {x}^{\\prime },\\lambda ) $ (or $ \\mathcal {G}_{t}(\\textbf {x}-\\textbf {x}^{\\prime },\\alpha ) $ ) variesOne can check that as we move away from the origin, the special Bessel function of the kind $ K_0 (\\mathbb {k}|x|) $ has a sharp fall off that is determined by the factor $ \\frac{1}{\\mathbb {k}} $ which essentially measures the width of the curve about the origin.",
"Therefore as the value of $ \\mathbb {k}$ increases the width decreases which results in a faster fall off of the function.",
"and the other one is the correlation length ($ \\xi $ ) which eventually determines the size of the vortex and thereby defines a scale over which the vortex could exist.",
"In order to remove the non localities associated with (REF ) we assume that the scale over which the Green's function fluctuates is quite small compared to that of the correlation length ($ \\xi $ ) over which the vortex could fluctuate i.e; $\\frac{1}{\\sqrt{\\lambda }} (or \\frac{1}{\\sqrt{\\alpha }}) \\ll \\xi $ .",
"In other words, with the above assumption in mind we can take the condensate to be almost uniform over the scale $\\frac{1}{\\sqrt{\\lambda }} $ (or $ \\frac{1}{\\sqrt{\\alpha }} $ ) which eventually results in the following mathematical identities namely, $\\int d\\textbf {x}^{\\prime }\\tilde{\\mathcal {G}_{t}}(\\textbf {x}-\\textbf {x}^{\\prime },\\alpha )\\sigma (\\textbf {x}^{\\prime })&=&\\frac{\\sigma (\\textbf {x})}{\\alpha }+\\mathcal {O}(\\frac{1}{\\alpha ^{3/2}})\\nonumber \\\\\\int d\\textbf {x}^{\\prime }\\tilde{\\mathcal {G}_{s}}(\\textbf {x}-\\textbf {x}^{\\prime },\\lambda )\\sigma (\\textbf {x}^{\\prime })&=&\\frac{\\sigma (\\textbf {x})}{\\lambda }+\\mathcal {O}(\\frac{1}{\\lambda ^{3/2}}).$ where we have used (REF ) in order to arrive at the above identityIn order to arrive at the above identity (REF ) one needs to consider the integral version of (REF ).. Also we have ignored all the sub leading terms in the Taylor expansion of $ \\sigma (\\textbf {x}^{\\prime }) $ about the point $ \\textbf {x}^{\\prime }=\\textbf {x} $ since they are highly suppressed compared with the leading term in the large $ \\lambda $ (or $ \\alpha $ ) limit.",
"Finally, using (REF ) at sufficiently low temperatures ($ T\\rightarrow T_{min} $ ) the most dominant part of the super-current turns out to beHere we have subtracted out the effects of regular terms since they are insignificant as $ T\\rightarrow 0 $ .",
"More over in order to arrive at (REF ) we have replaced $\\lambda $ (or $ \\alpha $ ) in the series (REF ) by $ \\lambda _{min} $ (or $ \\alpha _{min} $ ) which could be termed as the large $ \\lambda $ (or $ \\alpha $ ) approximation [18]., $\\langle \\Delta J_i\\rangle &=& \\mathcal {D}\\ \\epsilon _i\\ ^{j}\\partial _j \\Delta \\sigma (\\textbf {x})\\nonumber \\\\&\\approx & \\frac{9bH_{c2}^{2} \\mathcal {Z}}{16 \\pi ^{2} T^{2}\\lambda _{min}^{2}}\\epsilon _i\\ ^{j}\\partial _j \\Delta \\sigma (\\textbf {x})$ where the coefficients $ \\mathcal {D} $ and $ \\mathcal {Z} $ are respectively given byNote that our final expression (REF ) essentially corresponds to the most dominant term in the series $ \\mathcal {D} $ (REF ).",
"This is the due to the fact that at low temperatures we replace $ T\\rightarrow T_{min} $ and as a result $ \\lambda \\rightarrow \\lambda _{min} $ since $ \\lambda \\sim T^{2} $ .",
"Therefore in the above framework the low temperature limit essentially corresponds to the fact that we are basically considering the most dominant term in the series $ \\mathcal {D} $ (REF )., $\\mathcal {D}&=&- \\frac{bH_{c2}^{2}}{r_+^{2}}\\sum _{\\lambda ,\\lambda ^{\\prime }}\\frac{\\zeta ^{\\prime }_{\\lambda }(0)}{\\lambda \\lambda ^{\\prime }}\\int _{0}^{1}u^{\\prime 4}\\zeta _{\\lambda ^{\\prime }}(u^{\\prime })\\zeta ^{\\dagger }_{\\lambda }(u^{\\prime })du^{\\prime }\\int _{0}^{1}du^{\\prime \\prime } \\frac{\\rho _{1}^{2(b^{(0)})}(u^{\\prime \\prime })}{u^{\\prime \\prime 2}}\\zeta ^{\\dagger }_{\\lambda ^{\\prime }}(u^{\\prime \\prime })\\nonumber \\\\\\mathcal {Z}&=&- \\zeta ^{\\prime }_{\\lambda _{min}}(0)\\int _{0}^{1}u^{\\prime 4}\\zeta _{\\lambda _{min}^{\\prime }}(u^{\\prime })\\zeta ^{\\dagger }_{\\lambda _{min}}(u^{\\prime })du^{\\prime }\\int _{0}^{1}du^{\\prime \\prime } \\frac{\\rho _{1}^{2(b^{(0)})}(u^{\\prime \\prime })}{u^{\\prime \\prime 2}}\\zeta ^{\\dagger }_{\\lambda _{min}^{\\prime }}(u^{\\prime \\prime }).$ Considering (REF ), (REF ) and (REF ) and noting the fact that $ u(=r_+/r) $ is a dimensionless parameter, one can see that $ \\lambda \\sim T^{2} $ where $ T $ is the temperature of the system.",
"Therefore the leading term appearing in (REF ) effectively goes as ($ \\sim \\frac{1}{T^{6}} $ ) and thereby exhibits anomalous behaviour as $ T\\rightarrow 0 $ ."
],
[
"Free energy",
"We would like to conclude our analysis of the present paper by computing the free energy for the dual theory living in the boundary of the $ AdS_{4} $ .",
"In this section, considering the large $ \\lambda $ approximation, our goal is to study the effect of BI corrections on the free energy of the system in the presence of (triangular) vortex lattice.",
"In order to carry out our analysis we consider that the scalar condensation is confined with in a compact region of volume $ V $ whose size is much bigger than the size of a single unit cell of the triangular lattice.",
"The free energy of the system could be computed from the knowledge of the onshell action evaluated at the boundary namely, $F=-S_{(os)}.$ The full onshell action consists of two parts.",
"Let us first consider the the onshell action corresponding to the scalar field ($ \\Psi $ ).",
"Using the equation of motion of the scalar field, one can in fact show that the onshell action for the scalar field turns out to be, $S_{\\psi }|_{(os)}=-\\frac{1}{2}\\int _{\\partial M}d\\Sigma _{\\mu }\\sqrt{-g}(\\nabla ^{\\mu }-iA^{\\mu })|\\Psi |^{2}$ where $ d\\Sigma _{\\mu } $ is the volume measured at the boundary of the $ AdS_{4} $ .",
"In order to evaluate the above integral one needs to take into account different choices for the hyper surface ($ \\partial M $ ) corresponding to different values of $ \\mu $ which are the following: For $ \\mu = t $ , the above integral vanishes if we take our field configuration to be stationary at the past and future space like surfaces.",
"Considering the radial behavior ($ \\rho _{1}\\sim u^{2} $ ), the above integral (REF ) also vanishes for $ \\mu =u $ .",
"Finally for $ \\mu =i $ , at large values of the spatial coordinates, the above integral will vanish due to the Gaussian behavior of the vortex solution (See Eq (REF )).",
"Therefore from the above discussion we conclude that the scalar field does not directly contribute to the free energy of the system.",
"Thus we are only left with the onshell action corresponding to the $U(1)$ gauge sector.",
"Since we are interested to calculate the free energy in the presence of the vortex solution, therefore our next aim would be to compute the onshell action corresponding to the fluctuations in the $ U(1) $ gauge field.",
"This is how the scalar condensation could indirectly affect the free energy of the system through its interaction with the gauge fields.",
"In order to proceed further we first consider the following perturbative expansion for the onshell action in the parameter $ \\varepsilon $ namely, $S_{(os)}= S_{(os)}^{(0)}+\\varepsilon S_{(os)}^{(1)}+\\varepsilon ^{2}S_{(os)}^{(2)}+\\mathcal {O}(\\varepsilon ^{3}).$ Let us now consider the action corresponding to the first order in the gauge fluctuations which turns out to be, $S_{(os)}^{(1)}&=& -\\frac{1}{2}\\int d^{4}x\\sqrt{-g}F^{\\nu \\mu (0)}F_{\\nu \\mu }^{(1)}\\left(1-\\frac{b}{4}F^{2(0)} \\right)\\nonumber \\\\&=& -\\int _{\\partial M}d\\Sigma _{u}\\sqrt{-g}F^{u\\mu (0)}\\left(1-\\frac{b}{4}F^{2(0)} \\right)A^{(1)}_{\\mu }|_{u=0}.$ Since our boundary theory has been kept at a fixed chemical potential ($ \\mu $ ), therefore the above integral yields a vanishing contribution to the free energy of the system.",
"At this stage of our analysis it is now quite reasonable to expect that the first non trivial contribution to the free energy of the system might arise at the quadratic level in the gauge fluctuations.",
"The onshell action at the quadratic level in the gauge fluctuations turns out to be, $S_{(os)}^{(2)}= -\\frac{1}{2}\\int d^{4}x \\sqrt{-g}F^{\\nu \\mu (0)}F_{\\nu \\mu }^{(2)}\\left(1-\\frac{b}{4}F^{2(0)} \\right)\\nonumber \\\\-\\frac{1}{4}\\int d^{4}x\\sqrt{-g}\\left[ F^{\\nu \\mu (1)}F_{\\nu \\mu }^{(1)}-\\frac{b}{4}F^{\\nu \\mu (1)}F_{\\nu \\mu }^{(1)}F^{2(0)}-\\frac{b}{2}F^{\\nu \\mu (0)}F_{\\nu \\mu }^{(1)}F^{\\lambda \\sigma (0)}F_{\\lambda \\sigma }^{(1)}\\right].$ After using the equation of motion () it takes the following form, $S_{(os)}^{(2)}=-\\int _{\\partial M}d\\Sigma _{u}\\sqrt{-g}F^{u\\mu (0)}\\left(1-\\frac{b}{4}F^{2(0)} \\right)A^{(2)}_{\\mu }|_{u=0}+\\frac{r_+}{2}\\int _{\\partial M} d\\Sigma _{u}\\langle J^{i}\\rangle a_i(\\textbf {x})$ where in order to arrive at the above relation we have used the following orthogonality condition namely, $\\int _{M}d^{4}x \\sqrt{-g}A_{\\mu }^{(1)}j^{\\mu (1)}=0.$ Following our previous arguments we note that the first term on the r.h.s.",
"of (REF ) vanishes identically.",
"Finally using (REF ), the expression for the free energy turns out to be, $F=-\\frac{\\varepsilon ^{2}r_+H_{c2}}{2}\\int _{\\Re ^{2}}d\\textbf {x}\\Theta (\\textbf {x}).$ Note that this is again a non local expression in the vortex function $ \\sigma (\\textbf {x}) $ .",
"Following our previous arguments namely the large $ \\lambda $ approximation, we may convert the above equation (REF ) into a local function of the vortex solution $ \\sigma (\\textbf {x}) $ averaged over the finite volume $ V $ on which the condensate forms.",
"This eventually leads us to the following local expression for the free energy, $F/V=-\\frac{\\varepsilon ^{2}H_{c2}}{2}\\left( \\mathcal {Z}_{1}\\overline{\\sigma (\\textbf {x})}+\\frac{9bH_{c2}^{2} \\mathcal {Z}_{2}}{16 \\pi ^{2} T^{2}\\lambda _{min}^{2}}\\overline{\\Delta \\sigma (\\textbf {x})} \\right) +\\mathcal {O}(b^{2})$ where the bar indicates the average value of the condensate over the region under consideration.",
"From the above expression (REF ), we note that the free energy for the vortex configuration is negative.",
"This suggests that the (triangular) lattice configuration is more stable than the normal phase with the vanishing order parameter ($ \\Psi =0 $ ).",
"Furthermore we note that due to the presence of the BI term in the dual gravitational description, the corresponding free energy of the boundary theory also receives a nontrivial finite temperature correction that we have already found during the computation of the supercurrent in the previous section."
],
[
"Summary and final remarks",
"In the present article based on the $ AdS_{4}/CFT_3 $ duality we discuss the holographic framework for the low temperature anomalous superconductivity.",
"From our analysis we note that the higher derivative corrections to the $ U(1) $ sector of the abelian Higgs model acts as a source for the low temperature anomaly in the boundary current.",
"At sufficiently low temperatures the most dominant part of the $ U(1) $ current (thus computed at the boundary of the $ AdS_4 $ ) is found to vary inversely with the sixth power of the temperature.",
"Such low temperature anomalies in the usual superconducting materials indicate the presence of zero energy delta peaks in the quasi particle spectrum.",
"As far as the $ AdS_4/CFT_3 $ framework is concerned the origin of such low temperature peaks is not very much transparent.",
"This is something which is certainly beyond the scope of the present article and therefore we leave it as a part of the future investigations.",
"Acknowledgements : Author would like to acknowledge the financial support from CHEP, Indian Institute of Science, Bangalore.",
"Appendix"
],
[
"Ginzburg-Landau theory for type II vortices",
"Here we would like to briefly outline the structure vortices in type II superconductors in a typical Ginzburg-Landau (GL) theory.",
"In GL theory one generally starts with a non vanishing order parameter that depends on the spatial coordinates namely $ \\psi = \\psi (\\textbf {x}) $ .",
"In the presence of a constant magnetic field $\\textbf {B}$ along $\\hat{z}$ direction the linearised Ginzburg-Landau equation turns out to be, $\\frac{1}{2m^*}\\left( {1\\over i} \\nabla + {2 e H}\\hat{y} x \\right) ^2\\psi +\\alpha \\psi =0 \\nonumber \\\\\\left[ -\\nabla ^2 -\\frac{4\\pi i}{\\Phi _0}H x {\\partial \\over \\partial y} +\\left( \\frac{2\\pi H}{\\Phi _0} \\right) ^2 x^2 \\right] \\psi = \\frac{1}{\\xi ^2} \\psi $ where the coherence length could be formally expressed as, $\\xi ^2(T) = - \\frac{1}{2 m^* \\alpha (T)}.$ Eq.",
"(REF ) is a kind of reminiscent of the so called Schrödinger equation in the presence of a nontrivial potential along $x$ direction.",
"From the equation itself it is in fact quite evident that the particle does not experience any force along $y$ and $z$ directions and therefore behaves like a free particle.",
"Based on these observations we consider the following ansatz for the order parameter $\\psi $ namely, $\\psi = e^{i k_y y} e^{i k_z z} g(x).$ Substituting (REF ) in to (REF ) we find, $-g^{\\prime \\prime }(x) + \\left( \\frac{2\\pi H}{\\Phi _0}\\right) ^2 (x-x_0)^2 g(x) =\\left( {1\\over \\xi ^2} - k_z^2 \\right) g(x)$ where, $x_0 = \\frac{k_y \\Phi _0}{2 \\pi H},~~\\Phi _{0}=\\frac{\\pi }{e} .$ Eq.",
"(REF ) has the remarkable structural similarity to that with Eq.",
"(REF ) and it is nothing but the equation of an one dimensional quantum harmonic oscillator with frequency $\\omega _c = 2H e/m^* c$ .",
"The solution to the above eigen value equation (REF ) exists if, $H = \\frac{\\Phi _0}{2\\pi (2n+1)}\\left( {\\frac{1}{\\xi ^2}-k_{z}^{2}}\\right).$ According to (REF ) there exists an upper limit for the external magnetic field corresponding to $n=0$ and $k_z=0$ which gives the upper critical magnetic field $H_{c2}$ namely, $H_{c2} = \\frac{\\Phi _0}{2\\pi \\xi ^2(T)}.$"
]
] | 1403.0085 |
[
[
"Parametrizing the moduli space of curves and applications to smooth\n plane quartics over finite fields"
],
[
"Abstract We study new families of curves that are suitable for efficiently parametrizing their moduli spaces.",
"We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields.",
"In this way, we can visualize the distributions of their traces of Frobenius.",
"This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak."
],
[
"Introduction",
"One of the central notions in arithmetic geometry is the (coarse) moduli space of curves of a given genus $g$ , denoted $\\mathsf {M}_g$ .",
"These are algebraic varieties whose geometric points classify these curves up to isomorphism.",
"The main difficulty when dealing with moduli spaces – without extra structure – is the non-existence of universal families, whose construction would allow one to explicitly write down the curve corresponding to a point of this space.",
"Over finite fields, the existence of a universal family would lead to optimal algorithms to write down isomorphism classes of curves.",
"Having these classes at one's disposal is useful in many applications.",
"For instance, it serves for constructing curves with many points using class field theory [31] or for enlarging the set of curves useful for pairing-based cryptography as illustrated in genus 2 by [9], [14], [32].",
"More theoretically, it was used in [5] to compute the cohomology of moduli spaces.",
"We were ourselves drawn to this subject by the study of Serre's obstruction for smooth plane quartics (see Section REF ).",
"The purpose of this paper is to introduce three substitutes for the notion of a universal family.",
"The best replacement for a universal family seems to be that of a representative family, which we define in Section .",
"This is a family of curves $\\mathcal {C}\\rightarrow \\mathcal {S}$ whose points are in natural bijection with those of a given subvariety $S$ of the moduli space.",
"Often the scheme $\\mathcal {S}$ turns out to be isomorphic to $S$ , but the notion is flexible enough to still give worthwhile results when this is not the case.",
"Another interesting feature of these families is that they can be made explicit in many cases when $S$ is a stratum of curves with a given automorphism group.",
"We focus here on the case of non-hyperelliptic genus 3 curves, canonically realized as smooth plane quartics.",
"The overview of this paper is as follows.",
"In Section we introduce and study three new notions of families of curves.",
"We indicate the connections with known constructions from the literature.",
"In Proposition REF and Proposition REF , we also uncover a link between the existence of a representative family and the question of whether the field of moduli of a curve is a field of definition.",
"In Section we restrict our considerations to the moduli space of smooth plane quartics.",
"After a review of the stratification of this moduli space by automorphism groups, our main result in this section is Theorem REF .",
"There we construct representative families for all but the two largest of these strata by applying the technique of Galois descent.",
"For the remaining strata we improve on the results in the literature by constructing families with fewer parameters, but here much room for improvement remains.",
"In particular, it would be nice to see an explicit representative (and in this case universal) family over the stratum of smooth plane quartics with trivial automorphism group.",
"Parametrizing by using our families, we get one representative curve per $\\bar{k}$ -isomorphism class.",
"Section refines these into $k$ -isomorphism classes by constructing the twists of the corresponding curves over finite fields $k$ .",
"Finally, Section concludes the paper by describing the implementation of our enumeration of smooth plane quartics over finite fields, along with the experimental results obtained on distributions of traces of Frobenius for these curves over $\\mathbb {F}_p$ with $11 \\le p \\le 53$ .",
"In order to obtain exactly one representative for every isomorphism class of curves, we use the previous results combined with an iterative strategy that constructs a complete database of such representatives by ascending up the automorphism strataDatabases and statistics summarizing our results can be found at http://perso.univ-rennes1.fr/christophe.ritzenthaler/programme/qdbstats-v3_0.tgz..",
"Notations.",
"Throughout, we denote by $k$ an arbitrary field of characteristic $p \\ge 0$ , with algebraic closure $\\overline{k}$ .",
"We use $K$ to denote a general algebraically closed field.",
"By $\\zeta _n$ , we denote a fixed choice of $n$ -th root of unity in $\\overline{k}$ or $K$ ; these roots are chosen in such a way to respect the standard compatibility conditions when raising to powers.",
"Given $k$ , a curve over $k$ will be a smooth and proper absolutely irreducible variety of dimension 1 and genus $g>1$ over $k$ .",
"In agreement with [23], we keep the notation $\\mathbf {C}_n$ (resp.",
"$\\mathbf {D}_{2n}$ , resp.",
"$\\mathbf {A}_n$ , resp.",
"$\\mathbf {S}_n$ ) for the cyclic group of order $n$ (resp.",
"the dihedral group of order $2n$ , resp.",
"the alternating group of order $n!/2$ , resp.",
"the symmetric group of order $n!$ ).",
"We will also encounter $\\mathbf {G}_{16}$ , a group of 16 elements that is a direct product $\\mathbf {C}_4 \\times \\mathbf {D}_4$ , $\\mathbf {G}_{48}$ , a group of 48 elements that is a central extension of $\\mathbf {A}_4$ by $\\mathbf {C}_4$ , $\\mathbf {G}_{96}$ , a group of 96 elements that is a semidirect product $(\\mathbf {C}_4 \\times \\mathbf {C}_4) \\rtimes \\mathbf {S}_3$ and $\\mathbf {G}_{168}$ , which is a group of 168 elements isomorphic to $\\operatorname{PSL}_2 (\\mathbb {F}_7)$ .",
"Acknowledgments.",
"We would like to thank Jonas Bergström, Bas Edixhoven, Everett Howe, Frans Oort and Matthieu Romagny for their generous help during the writing of this paper.",
"Also, we warmly thank the anonymous referees for carefully reading this work and for suggestions."
],
[
"Families of curves",
"Let $g>1$ be an integer, and let $k$ be a field of characteristic $p = 0$ or $p>2g+1$ .",
"For $\\mathcal {S}$ a scheme over $k$ , we define a curve of genus $g$ over $\\mathcal {S}$ to be a morphism of schemes $\\mathcal {C}\\rightarrow \\mathcal {S}$ that is proper and smooth with geometrically irreducible fibers of dimension 1 and genus $g$ .",
"Let $\\mathsf {M}_g$ be the coarse moduli space of curves of genus $g$ whose geometric points over algebraically closed extensions $K$ of $k$ correspond with the $K$ -isomorphism classes of curves $C$ over $K$ .",
"We are interested in studying the subvarieties of $\\mathsf {M}_g$ where the corresponding curves have an automorphism group isomorphic with a given group.",
"The subtlety then arises that these subvarieties are not necessarily irreducible.",
"This problem was also mentioned and studied in [25], and resolved by using Hurwitz schemes; but in this section we prefer another way around the problem, due to Lønsted in [24].",
"In [24] the moduli space $\\mathsf {M}_g$ is stratified in a finer way, namely by using `rigidified actions' of automorphism groups.",
"Given an automorphism group $G$ , Lønsted defines subschemes of $\\mathsf {M}_g$ that we shall call strata.",
"Let $\\ell $ be a prime different from $p$ , and let $\\Gamma _{\\ell } = \\operatorname{Sp}_{2g} (\\mathbb {F}_{\\ell })$ .",
"Then the points of a given stratum $S$ correspond to those curves $C$ for which the induced embedding of $G$ into the group ($\\cong \\Gamma _{\\ell }$ ) of polarized automorphisms of $\\operatorname{Jac}(C)[\\ell ]$ is $\\Gamma _{\\ell }$ -conjugate to a given group.",
"Combining [17] with [24] now shows that under our hypotheses on $p$ , such a stratum is a locally closed, connected and smooth subscheme of $\\mathsf {M}_g$ .",
"If $k$ is perfect, such a connected stratum is therefore defined over $k$ if only one rigidification is possible for a given abstract automorphism group.",
"As was also observed in [25], this is not always the case; and as we will see in Remark REF , in the case of plane quartics these subtleties are only narrowly avoided.",
"We return to the general theory.",
"Over the strata $S$ of $\\mathsf {M}_g$ with non-trivial automorphism group, the usual notion of a universal family (as in [27]) is of little use.",
"Indeed, no universal family can exist on the non-trivial strata; by [1], $S$ is a fine moduli space (and hence admits a universal family) if and only if the automorphism group is trivial.",
"In the definition that follows, we weaken this notion to that of a representative family.",
"While such families coincide with the usual universal family on the trivial stratum, it will turn out (see Theorem REF ) that they can also be constructed for the strata with non-trivial automorphism group.",
"Moreover, they still have sufficiently strong properties to enable us to effectively parametrize the moduli space.",
"Definition 2.1 Let $S \\subset \\mathsf {M}_g$ be a subvariety of $\\mathsf {M}_g$ that is defined over $k$ .",
"Let $\\mathcal {C}\\rightarrow \\mathcal {S}$ be a family of curves whose geometric fibers correspond to points of the subvariety $S$ , and let $f_{\\mathcal {C}}: \\mathcal {S}\\rightarrow S$ be the associated morphism.",
"The family $\\mathcal {C}\\rightarrow \\mathcal {S}$ is geometrically surjective (for $S$ ) if the map $f_{\\mathcal {C}}$ is surjective on $K$ -points for every algebraically closed extension $K$ of $k$ .",
"The family $\\mathcal {C}\\rightarrow \\mathcal {S}$ is arithmetically surjective (for $S$ ) if the map $f_{\\mathcal {C}}$ is surjective on $k^{\\prime }$ -points for every finite extension $k^{\\prime }$ of $k$ .",
"The family $\\mathcal {C}\\rightarrow \\mathcal {S}$ is quasifinite (for $S$ ) if it is geometrically surjective and $f_{\\mathcal {C}}$ is quasifinite.",
"The family $\\mathcal {C}\\rightarrow \\mathcal {S}$ is representative (for $S$ ) if $f_{\\mathcal {C}}$ is bijective on $K$ -points for every algebraically closed extension $K$ of $k$ .",
"Remark 2.2 A family $\\mathcal {C}\\rightarrow \\mathcal {S}$ is geometrically surjective if and only if the corresponding morphism of schemes $\\mathcal {S}\\rightarrow S$ is surjective.",
"Due to inseparability issues, the morphism $f_{\\mathcal {C}}$ associated to a representative family need not induce bijections on points over arbitrary extensions of $k$ .",
"Note that if a representative family $\\mathcal {S}$ is absolutely irreducible, then since $S$ is normal, we actually get that $f_{\\mathcal {C}}$ is an isomorphism by Zariski's Main Theorem.",
"However, there are cases where we were unable to find such an $\\mathcal {S}$ given a stratum $S$ (see Remark REF ).",
"The notions of being geometrically surjective, quasifinite and representative are stable under extension of the base field $k$ .",
"On the other hand, being arithmetically surjective can strongly depend on the base field, as for example in Proposition REF .",
"To prove that quasifinite families exist, one typically considers the universal family over $\\mathsf {M}_g^{(\\ell )}$ (the moduli space of curves of genus $g$ with full level-$\\ell $ structure, for a prime $\\ell >2$ different from $p$ , see [1]).",
"This gives a quasifinite family over $\\mathsf {M}_g$ by the forgetful (and in fact quotient) map $\\mathsf {M}_g^{(\\ell )} \\rightarrow \\mathsf {M}_g$ that we will denote $\\pi _{\\ell }$ when using it in our constructions below.",
"Let $K$ be an algebraically closed extension of $k$ .",
"Given a curve $C$ over $K$ , recall that an intermediate field $k \\subset L \\subset K$ is a field of definition of $C$ if there exists a curve $C_0/L$ such that $C_0$ is $K$ -isomorphic to $C$ .",
"The concept of representative families is related with the question of whether the field of moduli $\\mathbf {M}_C$ of the curve $C$ , which is by definition the intersection of the fields of definition of $C$ , is itself a field of definition.",
"Since we assumed that $p>2g+1$ or $p = 0$ , the field $\\mathbf {M}_C$ then can be recovered more classically as the residue field of the moduli space $\\mathsf {M}_g$ at the point $[C]$ corresponding to $C$ by [33].",
"This allows us to prove the following.",
"Proposition 2.3 Let $S$ be a subvariety of $\\mathsf {M}_g$ defined over $k$ that admits a representative family $\\mathcal {C}\\rightarrow \\mathcal {S}$ .",
"Let $C$ be a curve over an algebraically closed extension $K$ of $k$ such that the point $[C]$ of $\\mathsf {M}_g (K)$ belongs to $S$ .",
"Then $C$ descends to its field of moduli $\\mathbf {M}_C$ .",
"In case $k$ is perfect and $K = \\overline{k}$ , then $C$ even corresponds to an element of $\\mathcal {S}(\\mathbf {M}_C)$ .",
"First we consider the case where $k = \\mathbf {M}_C$ and $K$ is a Galois extension of $k$ .",
"Let $x \\in \\mathcal {S}(K)$ be the preimage of $[C]$ under $f_{\\mathcal {C}}$ .",
"For every $\\sigma \\in \\operatorname{Gal}(K / k)$ it makes sense to consider $x^{\\sigma } \\in \\mathcal {S}(K)$ , since the family $\\mathcal {C}$ is defined over $k$ .",
"Now since $f_{\\mathcal {C}}$ is defined over $k$ , we get $f_{\\mathcal {C}}(x)=f_{\\mathcal {C}}(x^{\\sigma })=s$ .",
"By uniqueness of the representative in the family, we get $x=x^{\\sigma }$ .",
"Since $\\sigma $ was arbitrary and $K / k$ is Galois, we therefore have $x \\in \\mathcal {S}(k)$ , which gives a model for $C$ over $k$ by taking the corresponding fiber for the family $\\mathcal {C}\\rightarrow \\mathcal {S}$ .",
"This already proves the final statement of the proposition.",
"Since the notion of being representative is stable under changing the base field $k$ , the argument in the Galois case gives us enough leverage to treat the general case (where $K / k$ is possibly transcendental or inseparable) by appealing to [19].",
"Conversely, we have the following result.",
"A construction similar to it will be used in the proof of Theorem REF .",
"Proposition 2.4 Let $S$ be a stratum defined over a field $k$ .",
"Suppose that for every finite Galois extension $F \\supset E$ of field extensions of $k$ , the field of moduli of the curve corresponding to a point in $S(E)$ equals $E$ .",
"Then there exists a representative family $\\mathcal {C}_U \\rightarrow U$ over a dense open subset of $S$ .",
"If $k$ is perfect, this family extends to a possibly disconnected representative family $\\mathcal {C}\\rightarrow \\mathcal {S}$ for the stratum $S$ .",
"Let $\\eta $ be the generic point of $S$ and again let $\\pi _{\\ell } :\\mathsf {M}_g^{(\\ell )} \\rightarrow \\mathsf {M}_g$ be the forgetful map obtained by adding level structure at a prime $\\ell >2$ different from $p$ .",
"Note that as a quotient by a finite group, $\\pi _{\\ell }$ is a finite Galois cover.",
"Let $\\nu $ be a generic point in the preimage of $\\eta $ by $\\pi _{\\ell }$ and $\\mathcal {C}\\rightarrow \\nu $ be the universal family defined over $k(\\nu )$ .",
"By definition the field of moduli $\\mathbf {M}_{\\mathcal {C}}$ is equal to $k(\\nu )$ and as $k(\\nu )$ is a field of definition there exists a family $\\mathcal {C}_0 \\rightarrow k(\\nu )$ geometrically isomorphic to $\\mathcal {C}$ .",
"Since $k(\\nu ) \\supset k(\\eta )$ is a Galois extension, we can argue as in the proof of Proposition REF to descend to $k(\\eta )$ , and hence by a spreading-out argument we can conclude that $\\mathcal {C}_0$ is a representative family on a dense open subset $U$ of $S$ .",
"Proceeding by induction over the (finite) union of the Galois conjugates of the finitely many irreducible components of the complement of $U$ , which is again defined over $k$ , one obtains the second part of the proposition.",
"Whereas the universal family $\\mathcal {C}\\rightarrow \\mathsf {M}_g^{(\\ell )}$ is sometimes easy to construct, it seems hard to work out $\\mathcal {C}_0$ directly by explicit Galois descent; the Galois group of the covering $\\mathsf {M}_g^{(\\ell )} \\rightarrow \\mathsf {M}_g$ is $\\operatorname{Sp}_{2g}(\\mathbb {F}_{\\ell })$ , which is a group of large cardinality $\\ell ^{g^2}\\prod _{i=1}^g (\\ell ^{2i}-1)$ whose quotient by its center is simple.",
"Moreover, for enumeration purposes, it is necessary for the scheme $\\mathcal {S}$ to be as simple as possible.",
"Typically one would wish for it to be rational, as fortunately turns out always to be the case for plane quartics.",
"On the other hand, for moduli spaces of general type that admit no rational curves, such as $\\mathsf {M}_g$ with $g> 23$ , there does not even exist a rational family of curves with a single parameter [15]."
],
[
"Review : automorphism groups",
"Let $C$ be a smooth plane quartic over an algebraically closed field $K$ of characteristic $p \\ge 0$ .",
"Then since $C$ coincides up to a choice of basis with its canonical embedding, the automorphism $\\operatorname{Aut}(C)$ can be considered as a conjugacy class of subgroups $\\operatorname{PGL}_3 (K)$ (and in fact of $\\operatorname{GL}_3 (K)$ ) by using the action on its non-zero differentials.",
"The classification of the possible automorphism groups of $C$ as subgroup of $\\operatorname{PGL}_3(K)$ , as well as the construction of some geometrically complete families, can be found in several articles, such as [16], [38], [25], [3] and [8] (in chronological order), in which it is often assumed that $p = 0$ .",
"We have verified these results independently, essentially by checking which finite subgroups of $\\operatorname{PGL}_3 (K)$ (as classified in [19]) can occur for plane quartics.",
"It turns out that the classification in characteristic 0 extends to algebraically closed fields $K$ of prime characteristic $p \\ge 5$ .",
"In the following theorem, we do not indicate the open non-degeneracy conditions on the affine parameters, since we shall not have need of them.",
"Theorem 3.1 Let $K$ be an algebraically closed field whose characteristic $p$ satisfies $p= 0$ or $p \\ge 5$ .",
"Let $C$ be a genus 3 non-hyperelliptic curve over $K$ .",
"The following are the possible automorphism groups of $C$ , along with geometrically surjective families for the corresponding strata: $ \\lbrace 1\\rbrace $ , with family $q_4(x,y,z)=0,$ where $q_4$ is a homogeneous polynomial of degree 4; $ \\mathbf {C}_2$ , with family $x^4 + x^2 q_2(y,z) + q_4(y,z) = 0$ , where $q_2$ and $q_4$ are homogeneous polynomials in $y$ and $z$ of degree 2 and 4; $ \\mathbf {D}_4$ , with family $x^4 + y^4 + z^4 + r x^2 y^2 + s y^2 z^2 + t z^2x^2 = 0$ ; $\\mathbf {C}_3$ , with family $x^3 z + y (y - z) (y - r z) (y - s z)=0$ ; $ \\mathbf {D}_8$ , with family $x^4 + y^4 + z^4 + r x^2 y z + s y^2 z^2 = 0$ ; $\\mathbf {S}_3$ , with family $x (y^3 + z^3) + y^2 z^2 + r x^2 y z + s x^4 =0$ ; $ \\mathbf {C}_6$ , with family $x^3 z + y^4 + r y^2 z^2 + z^4 = 0$ ; $ \\mathbf {G}_{16}$ , with family $x^4 + y^4 + z^4 + r y^2 z^2 = 0$ ; $ \\mathbf {S}_4$ , with family $x^4 + y^4 + z^4 + r (x^2 y^2 + y^2 z^2 + z^2 x^2)= 0$ ; $ \\mathbf {C}_9$ , represented by the quartic $x^3 y + y^3 z + z^4 = 0$ ; $ \\mathbf {G}_{48}$ , represented by the quartic $x^4 + (y^3 - z^3) z = 0$ ; $ \\mathbf {G}_{96}$ , represented by the Fermat quartic $x^4 + y^4 + z^4 = 0$ ; (if $p \\ne 7$ ) $ \\mathbf {G}_{168}$ , represented by the Klein quartic $x^3 y +y^3 z + z^3 x = 0$ .",
"The families in Theorem REF are geometrically surjective.",
"Moreover, they are irreducible and quasifinite (as we will see in the proof of Theorem REF ) for all groups except the trivial group and $\\mathbf {C}_2$ .",
"The embeddings of the automorphism group of these curves into $\\operatorname{PGL}_3 (K)$ can be found in Theorem REF in Appendix .",
"Because of the irreducibility properties mentioned in the previous paragraph, each of the corresponding subvarieties serendipitously describes an actual stratum in the moduli space $\\mathsf {M}_3^{\\textrm {nh}} \\subset \\mathsf {M}_3$ of genus 3 non-hyperelliptic curves as defined in Section (see Remark REF below).",
"From the descriptions in REF , one derives the inclusions between the strata indicated in Figure REF , as also obtained in [38].",
"Figure: Automorphism groupsRemark 3.2 As promised at the beginning of Section , we now indicate two different possible rigidifications of an action of a finite group on plane quartics.",
"Consider the group $\\mathbf {C}_3$ .",
"Up to conjugation, this group can be embedded into $\\operatorname{PGL}_3 (K)$ in exactly two ways; as a diagonal matrix with entries proportional to $(\\zeta _3 , 1 , 1)$ or $(\\zeta _3^2 , \\zeta _3 , 1)$ .",
"This gives rise to two rigidifications in the sense of Lønsted.",
"While for plane curves of sufficiently high degree, this indeed leads to two families with generic automorphism group $\\mathbf {C}_3$ , the plane quartics admitting the latter rigidification always admit an extra involution, so that the full automorphism group contains $\\mathbf {S}_3$ .",
"It is this fortunate phenomenon that still makes a naive stratification by automorphism groups possible for plane quartics.",
"For the same reason, the stratum for the group $\\mathbf {S}_3$ is not included in that for $\\mathbf {C}_3$ , as is claimed incorrectly in [3]."
],
[
"Construction of representative families",
"We now describe how to apply Galois descent to extensions of function fields to determine representative families for the strata in Theorem REF with $|G| > 2$ .",
"By Proposition REF , this shows that the descent obstruction always vanishes for these strata.",
"Our constructions lead to families that parametrize the strata much more efficiently; for the case $\\mathbf {D}_4$ , the family in Theorem REF contains as much as 24 distinct fibers isomorphic with a given curve.",
"Moreover, by Proposition REF , in order to write down a complete list of the $\\overline{k}$ -isomorphism classes of smooth plane quartics defined over a perfect field $k$ we need only consider the $k$ -rational fibers of the new families.",
"As in Theorem REF , we do not specify the condition on the parameters that avoid degenerations (i.e.",
"singular curves or a larger automorphism group), but such degenerations will be taken into account in our enumeration strategy in Section .",
"Theorem 3.3 Let $k$ be a field whose characteristic $p$ satisfies $p = 0$ or $p \\ge 7$ .",
"The following are representative families for the strata of smooth plane quartics with $| G | > 2$ .",
"$G \\simeq \\mathbf {D}_4$ : [compact,style=,spread=-3pt] (a + 3) x4 + (4 a2 - 8 b + 4 a) x3 y + (12 c + 4 b) x3 z + (6 a3 - 18 a b + 18 c + 2 a2) x2 y2 + (12 a c + 4 a b) x2 y z + (6 b c + 2 b2) x2 z2 + (4 a4 - 16 a2 b + 8 b2 + 16 a c + 2 a b - 6 c) x y3 + (12 a2 c - 24 b c + 2 a2 b - 4 b2 + 6 a c) x y2 z + (36 c2 + 2 a b2 - 4 a2 c + 6 b c) x y z2 + (4 b2 c - 8 a c2 + 2 a b c - 6 c2) x z3 + (a5 - 5 a3 b + 5 a b2 + 5 a2 c - 5 b c + b2 - 2 a c) y4 + (4 a3 c - 12 a b c + 12 c2 + 4 a2 c - 8 b c) y3 z + (6 a c2 + a2 b2 - 2 b3 - 2 a3 c + 4 a b c + 9 c2) y2 z2 + (4 b c2 + 4 b2 c - 8 a c2) y z3 + (b3 c - 3 a b c2 + 3 c3 + a2 c2 - 2 b c2) z4 = 0 along with [compact,style=,spread=-3pt] x4 + 2 x2 y2 + 2 a x2 y z + (a2 - 2 b) x2 z2 + a y4 + 4 (a2 - 2 b) y3 z + 6 (a3 - 3 a b) y2 z2 + 4 (a4 - 4 a2 b + 2 b2) y z3 + (a5 - 5 a3 b + 5 a b2) z4 = 0 .",
"$G \\simeq \\mathbf {C}_3$ : $x^3 z + y^4 + a y^2 z^2 + a y z^3 + b z^4 = 0$ along with $x^3 z + y^4 + a y z^3 + a z^4 = 0$ ; $G \\simeq \\mathbf {D}_8$ : $x^4 + x^2 y z + y^4 + a y^2 z^2 + b z^4 = 0$ ; $G \\simeq \\mathbf {S}_3$ : $x^3 z + y^3 z + x^2 y^2 + a x y z^2 + b z^4 = 0$ ; $G \\simeq \\mathbf {C}_6$ : $x^3 z + a y^4 + a y^2 z^2 + z^4 = 0$ ; $G \\simeq \\mathbf {G}_{16}$ : $x^4 + (y^3 + a y z^2 + a z^3) z = 0$ ; $G \\simeq \\mathbf {S}_4$ : $x^4 + y^4 + z^4 + a (x^2 y^2 + y^2 z^2 + z^2 x^2) =0$ ; $G \\simeq \\mathbf {C}_9$ : $x^3 y + y^3 z + z^4 = 0$ ; $G \\simeq \\mathbf {G}_{48}$ : $x^4 + (y^3 - z^3) z = 0$ ; $G \\simeq \\mathbf {G}_{96}$ : $x^4 + y^4 + z^4 = 0$ ; (if $p \\ne 7$ ) $G \\simeq \\mathbf {G}_{168}$ : $x^3 y + y^3 z + z^3 x = 0$ .",
"We do not give the full proof of this theorem, but content ourselves with some families that illustrate the most important ideas therein.",
"Let $K$ be an algebraically closed extension of $k$ .",
"The key fact that we use, which can be observed from the description in Theorem REF , is that the fibers of the families in Theorem REF all have the same automorphism group $G$ as a subgroup of $\\operatorname{PGL}_3 (K)$ .",
"Except for the zero-dimensional cases, which are a one-off verification, one then proceeds as follows.",
"The key fact above implies that any isomorphism between two curves in the family is necessarily induced by an element of the normalizer $N$ of $G$ in $\\operatorname{PGL}_3 (K)$ .",
"So one considers the action of this group on the family given in Theorem REF .",
"One determines the subgroup $N^{\\prime }$ of $N$ that sends the family to itself again.",
"The action of $N^{\\prime }$ factors through a faithful action of $Q = N^{\\prime } / G$ .",
"By explicit calculation, it turns out that $Q$ is finite for the families in Theorem REF with $|G| > 2$ .",
"This shows in particular that these families are already quasifinite on these strata.",
"One then takes the quotient by the finite action of $Q$ , which is done one the level of function fields over $K$ by using Galois descent.",
"By construction, the resulting family will be representative.",
"For the general theory of Galois descent, we refer to [40] and [37].",
"We now treat some representative cases to illustrate this procedure.",
"In what follows, we use the notation from Theorem REF to denote elements and subgroups of the normalizers involved.",
"The case $G \\simeq \\mathbf {S}_3.$ Here $N = T(K) \\widetilde{\\mathbf {S}}_3$ contains the group of diagonal matrices $T(K)$ .",
"Transforming, one verifies that the subgroup $N^{\\prime } \\subsetneq N$ equals $\\widetilde{\\mathbf {S}_3}$ ; indeed, since $\\widetilde{\\mathbf {S}}_3$ fixes the family pointwise, we can restrict to the elements $T(K)$ .",
"But then preserving the trivial proportionality of the coefficients in front of $x^3 z$ , $y^3 z$ , and $x^2 y^2$ forces such a diagonal matrix to be scalar.",
"This implies the result; the group $Q$ is trivial in $\\operatorname{PGL}_3 (K)$ , so we need not adapt our old family since it is geometrically surjective and contains no geometrically isomorphic fibers.",
"A similar argument works for the case $G \\cong \\mathbf {S}_4$ .",
"The case $G \\simeq \\mathbf {C}_6$ .",
"This time we have to consider the action of the group $D (K)$ on the family $x^3 z + y^4 + r y^2 z^2 + z^4 = 0$ from Theorem REF .",
"After the action of a diagonal matrix with entries $\\lambda , \\mu , 1$ , one obtains the curve $\\lambda ^3 x^3 z + \\mu ^4 y^4 + \\mu ^2 r y^2 z^2 + z^4 = 0$ .",
"We see that we get a new curve in the family if $\\lambda ^3 = 1$ and $\\mu ^4 = 1$ , in which case the new value for $r$ equals $\\mu r$ .",
"But this equals $\\pm r$ since $(\\mu ^2)^2 = 1$ .",
"The degree of the morphism to $\\mathsf {M}_3$ induced by this family therefore equals 2.",
"This also follows from the fact that the subgroup $N^{\\prime }$ that we just described contains $G$ as a subgroup of index 2, so that $Q \\cong \\mathbf {C}_2$ .",
"We have a family over $L = K (r)$ whose fibers over $r$ and $-r$ are isomorphic, and we want to descend this family to $K(a)$ , where $a = r^2$ generates the invariant subfield under the automorphism $r \\rightarrow -r$ .",
"This is a problem of Galois descent for the group $Q \\cong \\mathbf {C}_2$ and the field extension $M \\supset L$ , with $M = K (r)$ and $L = K(a)$ .",
"The curve $C$ over $M$ that we wish to descend to $L$ is given by $x^3 z + y^4 + r y^2 z^2 +z^4 = 0$ .",
"Consider the conjugate curve $C^{\\sigma } : x^3 z + y^4 - r y^2 z^2 +z^4 = 0$ and the isomorphism $\\varphi : C \\rightarrow C^{\\sigma }$ given by $(x,y,z) \\rightarrow (x,i y,z)$ .",
"Then we do not have $\\varphi ^{\\sigma } \\varphi = \\textrm {id}$ .",
"To trivialize the cocycle, we need a larger extension of our function field $L$ .",
"Take $M^{\\prime } \\supset M$ to be $M^{\\prime } = M(\\rho )$ , with $\\rho ^2 = r$ .",
"Let $\\tau $ be a generator of the cyclic Galois group of order 4 of the extension $M^{\\prime } \\supset L$ .",
"Then $\\tau $ restricts to $\\sigma $ in the extension $M \\supset L$ , and for $M^{\\prime } \\supset L$ one now indeed obtains a Weil cocycle determined by the isomorphism $C \\mapsto C^{\\tau } = C^{\\sigma }$ sending $(x,y,z)$ to $(x,i y,z)$ .",
"The corresponding coboundary is given by $(x,y,z) \\mapsto ( x , \\rho y , z)$ .",
"Transforming, we end up with $x^3 z + (\\rho y)^4 + r (\\rho y)^2 z^2 + z^4 = x^3 z + a y^4 + a y^2 z^2 + z^4= 0,$ which is what we wanted to show.",
"The case $G \\cong \\mathbf {D}_8$ can be dealt with in a similar way.",
"The case $G \\simeq \\mathbf {D}_4$ .",
"We start with the usual Ciani family from Theorem REF , given by $x^4 + y^4 + z^4 + r x^2 y^2 + s y^2 z^2 + t z^2 x^2 = 0 .$ Using the $\\widetilde{\\mathbf {S}}_3$ -elements from the normalizer $N = D(K)\\widetilde{\\mathbf {S}}_3$ induces the corresponding permutation group on $(r,s,t)$ .",
"The diagonal matrices in $D(K)$ then remain, and they give rise to the transformations $(r,s,t) \\mapsto (\\pm r , \\pm s , \\pm t)$ with an even number of minus signs.",
"This is slightly awkward, so we try to eliminate the latter transformations.",
"This can be accomplished by moving the parameters in front of the factors $x^4$ , $y^4$ , $z^4$ .",
"So we instead split up $\\mathcal {S}$ into a disjoint union of two irreducible subvarieties by considering the family $r x^4 + s y^4 + t z^4 + x^2 y^2 + y^2 z^2 + z^2 x^2 = 0 ,$ and its lower-dimensional complement $r x^4 + s y^4 + z^4 + x^2 y^2 + y^2 z^2 = 0 .$ Here the trivial coefficient in front of $z^4$ is obtained by scaling $x,y,z$ by an appropriate factor in the family $r x^4 + s y^4 + t z^4 + x^2 y^2 + y^2 z^2 =0$ .",
"Note that because of our description of the normalizer, the number of non-zero coefficients in front of the terms with quadratic factors depends only on the isomorphism class of the curve, and not on the given equation for it in the geometrically surjective Ciani family.",
"This implies that the two families above do not have isomorphic fibers.",
"Moreover, the a priori remaining family $rx^4 + y^4 + z^4 + y^2 z^2 = 0$ has larger automorphism group, so we can discard it.",
"We only consider the first family, which is the most difficult case.",
"As in the previous example, after our modification the elements of $N^{\\prime } \\cap D(K)$ are in fact already in $G$ .",
"Therefore $Q = N^{\\prime } / G \\subset D(K)\\widetilde{\\mathbf {S}}_3$ is a quotient of the remaining factor $\\widetilde{\\mathbf {S}}_3$ , which clearly acts freely and is therefore isomorphic with $Q$ .",
"We obtain the invariant subfield $L = K(a,b,c)$ of $M = K (r,s,t)$ , with $a = r + s +t$ , $b = r s + s t + t r$ and $c = r s t$ the usual elementary symmetric functions.",
"The cocycle for this extension is given by sending a permutation of $(r,s,t)$ to its associated permutation matrix on $(x,y,z)$ .",
"A coboundary is given by the isomorphism $(x,y,z)\\mapsto (x+y+z, r\\,x+s\\,y+t\\,z, st\\,x+tr\\,y+rs\\,z)\\,.$ Note that this isomorphism is invertible as long as $r,s,t$ are distinct, which we may assume since otherwise the automorphism group of the curve would be larger.",
"Transforming by this coboundary, we get our result.",
"The case $G \\simeq \\mathbf {C}_3$ .",
"This case needs a slightly different argument.",
"Consider the eigenspace decomposition of the space of quartic monomials in $x,y,z$ under the action of the diagonal generator $(\\zeta _3,1,1)$ of $\\mathbf {C}_3$ .",
"The curves with this automorphism correspond to those quartic forms that are eigenforms for this automorphism, which is the case if and only if it is contained in one of the aforementioned eigenspaces.",
"We only need consider the eigenspace spanned by the monomials $x^3 y$ , $x^3 z$ , $y^4$ , $y^3 z$ , $y^2 z^2$ , $y z^3$ , $z^4$ ; indeed, the quartic forms in the other eigenspaces are all multiples of $x$ and hence give rise to reducible curves.",
"Using a linear transformation, one eliminates the term with $x^3 y$ , and a non-singularity argument shows that we can scale to the case $x^3 z + y^4 + r y^3 z + s y^2 z^2 + t y z^3 + u z^4 = 0 .$ We can set $r = 0$ by another linear transformation, which then reduces $N^{\\prime }$ to $D (K)$ .",
"Depending on whether $s = 0$ or not, one can then scale by these scalar matrices to an equation as in the theorem, which one verifies to be unique by using the same methods as above.",
"The case $G \\simeq \\mathbf {G}_{16}$ can be proved in a completely similar way.",
"Remark 3.4 As mentioned in Remark REF , these constructions give rise to isomorphisms $\\mathcal {S}\\rightarrow S$ in all cases except $\\mathbf {D}_4$ , and $\\mathbf {C}_3$ .",
"In these remaining cases, we have constructed a morphism $\\mathcal {S}\\rightarrow S$ that is bijective on points but not an isomorphism.",
"It is possible that no family $\\mathcal {C}\\rightarrow S$ inducing such an isomorphism exists; see [12] for results in this direction for hyperelliptic curves."
],
[
"Remaining cases",
"We have seen in Proposition REF that if there exist a representative family over $k$ over a given stratum, then the field of moduli needs to be a field of definition for all the curves in this stratum.",
"In [2], it is shown that there exist $\\mathbb {R}$ -points in the stratum $\\mathbf {C}_2$ for which the corresponding curve cannot be defined over $\\mathbb {R}$ .",
"In fact we suspect that this argument can be adapted to show that representative families for this stratum fail to exist even if $k$ is a finite field.",
"However, we can still find arithmetically surjective families over finite fields.",
"Proposition 3.5 Let $C$ be a smooth plane quartic with automorphism group $\\mathbf {C}_2$ over a finite field $k$ of characteristic different from 2.",
"Let $\\alpha $ be a non-square element in $k$ .",
"Then $C$ is $k$ -isomorphic to a curve of one of the following forms: $& x^4+\\epsilon x^2 y^2 + a y^4+\\mu y^3 z+b y^2 z^2+c y z^3+d z^4 = 0 &\\textrm {with} \\; \\epsilon =1 \\; \\textrm {or} \\; \\alpha \\; \\textrm {and} \\;\\mu =0 \\; \\textrm {or} \\; 1, \\\\& x^4+x^2 y z + a y^4+\\epsilon y^3 z+b y^2 z^2+c y z^3+d z^4 = 0 &\\textrm {with} \\; \\epsilon =0,1 \\; \\textrm {or} \\; \\alpha , \\\\& x^4+x^2 (y^2-\\alpha z^2) + a y^4+b y^3 z+c y^2 z^2+d y z^3+e z^4 = 0 \\,.&$ The involution on the quartic, being unique, is defined over $k$ .",
"Hence by choosing a basis in which this involution is a diagonal matrix, we can assume that it is given by $(x,y,z) \\mapsto (-x,y,z)$ .",
"This shows that the family $x^4+ x^2 q_2(y,z) + q_4(y,z) = 0$ of Theorem REF is arithmetically surjective.",
"We have $q_2(y,z) \\ne 0$ since otherwise more automorphisms would exist over $K$ .",
"We now distinguish cases depending on the factorization of $q_2$ over $k$ .",
"If $q_2$ has a multiple root, then we may assume that $q_2(y,z)=r y^2$ where $r$ equals 1 or $\\alpha $ .",
"Then either the coefficient $b$ of $y^3 z$ in $q_4$ is 0, in which case we are done, or we can normalize it to 1 using the change of variable $z \\mapsto z/b $ .",
"If $q_2$ splits over $k$ , then we may assume that $q_2(y,z)=yz$ .",
"Then either the coefficient $b$ of $y^3 z$ in $q_4$ is 0, in which case we are done, or we attempt to normalize it by a change of variables $y \\mapsto \\lambda y$ and $z \\mapsto z/\\lambda $ .",
"This transforms $b y^3 z$ into $b\\lambda ^2 y^3 z$ .",
"Hence we can assume $b$ equals 1 or $\\alpha $ .",
"If $q_2$ is irreducible over $k$ , then we can normalize $q_2(y,z)$ as $y^2- \\alpha z^2$ where $\\alpha $ is a non-square in $k$ .",
"This gives us the final family with 5 coefficients.",
"Remark 3.6 The same proof shows the existence of a quasifinite family for the stratum in Proposition REF , since over algebraically fields we can always reduce to the first or second case.",
"We have seen in Section that a universal family exists for the stratum with trivial automorphism group.",
"Moreover, as $\\mathsf {M}_3$ is rational [20], this family depends on 6 rational parameters.",
"However, no representative (hence in this case universal) family seems to have been written down so far.",
"Classically, when the characteristic $p$ is different from 2 or 3, there are at least two ways to construct quasifinite families for the generic stratum.",
"The first method fixes bitangents of the quartic and leads to the so-called Riemann model; see [13], [29], [39] for relations between this construction, the moduli of 7 points in the projective plane and the moduli space $\\mathsf {M}_3^{(2)}$ .",
"The other method uses flex points, as in [35].",
"In neither case can we get such models over the base field $k$ , since for a general quartic, neither its bitangents nor its flex points are defined over $k$ .",
"We therefore content ourselves with the following result which was kindly provided to us by J. Bergström.",
"Proposition 3.7 ((Bergström)) Let $C$ be a smooth plane quartic over a field $k$ admitting a rational point over a field of characteristic $\\ne 2$ .",
"Then $C$ is isomorphic to a curve of one of the following forms: $m_1 x^4 + m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + m_7 x y^3 + x y^2 z +m_{11} y^4 + m_{12} y^3 z + y^2 z^2 + y z^3 = 0, \\\\m_1 x^4 + m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + x y^3 + m_{11} y^4 +m_{12} y^3 z + y^2 z^2 + y z^3 = 0, \\\\m_1 x^4 + m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + m_{11} y^4 + m_{12} y^3 z+ y^2 z^2 + y z^3 = 0, \\\\m_1 x^4 + m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + x y^3 + x y^2 z + m_{11}y^4 + m_{12} y^3 z + y z^3 = 0, \\\\m_1 x^4 + m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + x y^2 z + m_{11} y^4 +m_{12} y^3 z + y z^3 = 0, \\\\x^4 + m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + m_7 x y^3 + m_{11} y^4 +m_{12} y^3 z + y z^3 = 0, \\\\m_2 x^3 y + m_4 x^2 y^2 + m_6 x^2 z^2 + m_7 x y^3 + m_{11} y^4 + m_{12} y^3z + y z^3 = 0, \\\\x^3 z + m_4 x^2 y^2 + m_7 x y^3 + m_8 x y^2 z + x y z^2 + m_{11} y^4 +m_{12} y^3 z + m_{13} y^2 z^2 + y z^3 = 0, \\\\x^3 z + m_4 x^2 y^2 + m_7 x y^3 + m_8 x y^2 z + m_{11} y^4 + m_{12} y^3 z +m_{13} y^2 z^2 + y z^3 = 0, \\\\x^4 + m_4 x^2 y^2 + m_5 x^2 y z + m_7 x y^3 + m_8 x y^2 z + m_{11} y^4 +m_{12} y^3 z + y z^3 = 0.$ We denote by $m_1,\\ldots ,m_{15}$ the coefficients of the quartic $C$ , with its monomials ordered as $x^4,x^3 y,x^3 z,x^2 y^2,x^2 y z,x^2 z^2,x y^3,x y^2 z, x y z^2,x z^3,y^4,y^3z,y^2 z^2,y z^3,z^4.$ As there is a rational point on the curve, we can transform this point to be $(0:0:1)$ with tangent equal to $y=0$ .",
"We then have $m_{15}=m_{10}=0$ , and we can scale to ensure that $m_{14}=1$ .",
"The proof now divides into cases.",
"Case 1: $m_6 \\ne 0$ .",
"Consider the terms $m_6 x^2 (z^2 + m_3/m_6 x z)$ .",
"Then by a further change of variables $z \\rightarrow z+m_3\\, x/(2 m_6) $ we can assume $m_3=0$ without perturbing the previous conditions.",
"Starting with this new equation, we can now cancel $m_5$ in the same way, and finally $m_9$ (note that the order in which we cancel the coefficients $m_3,m_5,m_9$ is important, so as to avoid re-introducing non-zero coefficients).",
"If $m_8$ and $m_{13}$ are non-zero, then we can ensure that $m_{8}=m_{13}=1$ by changing variables $(x:y:z) \\rightarrow (r x:s y:t z)$ such that $m_8 r s^2 t= \\alpha , \\quad m_{13} s^2 t^2=\\alpha , \\quad s t^3=\\alpha $ for a given $\\alpha \\ne 0$ and then divide the whole equation by $\\alpha $ .",
"One calculates that it is indeed possible to find a solution $(r,s,t,\\alpha )$ to these equations in $k^4$ .",
"If $m_8=0 , m_{13} \\ne 0 , m_7 \\ne 0$ , then we can transform to $m_{13}=m_7=1$ as above; If $m_8=0,m_{13} \\ne 0 , m_7=0$ , then we can transform to $m_{13}=1$ ; If $m_8 \\ne 0, m_{13} = 0, m_7 \\ne 0$ , then we can transform to $m_8=m_7=1$ ; If $m_8 \\ne 0 , m_7 = m_{13} = 0$ , then we can transform to $m_8=1$ ; If $m_{13} = m_8 = 0, m_1 \\ne 0$ , then we can transform to $m_1=1$ ; If $m_{13} = m_8 = m_1 = 0$ , then we need not do anything.",
"enumerate Case 2: $m_6 = 0 , m_3 \\ne 0$ .",
"As before, working in the correct order we can ensure that $m_1=m_2=m_5=0$ by using the non-zero coefficient $m_3$ .",
"enumerate If $m_9 \\ne 0$ , we can transform to $m_3=m_9=1$ ; If $m_9=0$ , we can transform to $m_3=1$ .",
"enumerate Case 3: $m_6 = m_3 = 0$ .",
"enumerate If $m_1 \\ne 0$ , then put $m_1=1$ .",
"Using $m_{14}$ , we can transform to $m_9=m_{13}=0$ and using $m_1$ , we can transform to $m_2=0$ .",
"The proof is now concluded by noting that if $m_1=m_3=m_6=m_{10}=m_{15}=0$ , then the quartic is reducible.",
"Bergström has also found models when rational points are not available, but these depend on as many as 9 coefficients.",
"Using the Hasse-Weil-Serre bound, one shows that when $k$ is a finite field with $\\# k>29$ , the models in Proposition REF constitute an arithmetically surjective family of dimension 7, one more than the dimension of the moduli space.",
"Over finite fields $k$ of characteristic $>7$ and with $\\# k \\le 29$ there are always pointless curves [18].",
"Our experiments showed that except for one single example, these curves all have non-trivial automorphism group.",
"As such, they already appear in the non-generic family.",
"The exceptional pointless curve, defined over $\\mathbb {F}_{11}$ , is $7 x^4 + 3 x^3 y + 10 x^3 z + 10 x^2 y^2 + 10 x^2 y z + 6 x^2 z^2 +7 x y^2 z\\\\+ x y z^2 + 4 x z^3 + 9 y^4 + 5 y^3 z + 8 y^2 z^2 + 9 y z^3 + 9 z^4 = 0 \\,.$"
],
[
"Computation of twists",
"Let $C$ be a smooth plane quartic defined over a finite field $k=\\mathbb {F}_q$ of characteristic $p$ .",
"In this section we will explain how to compute the twists of $C$ , i.e.",
"the $k$ -isomorphism classes of the curves isomorphic with $C$ over $\\overline{k}$ .",
"Let $\\operatorname{Twist}(C)$ be the set of twists of $C$ .",
"This set is in bijection with the cohomology set $H^1(\\operatorname{Gal}( \\overline{k}/k ),\\operatorname{Aut}( C ))$ , (see [36]).",
"More precisely, if $\\beta : C^{\\prime } \\rightarrow C$ is any $\\overline{k}$ -isomorphism, the corresponding element in $H^1(\\operatorname{Gal}(\\overline{k}/k ),\\operatorname{Aut}( C ))$ is given by $\\sigma \\mapsto \\beta ^{\\sigma }\\beta ^{-1}$ .",
"Using the fact that $\\operatorname{Gal}(\\overline{k}/k)$ is pro-cyclic generated by the Frobenius morphism $\\varphi : x \\mapsto x^q$ , computing $H^1(\\operatorname{Gal}(\\overline{k}/k ),\\operatorname{Aut}( C ))$ boils down to computing the equivalence classes of $\\operatorname{Aut}( C )$ for the relation $g \\sim h \\iff \\exists \\alpha \\in \\operatorname{Aut}(C), \\; g\\alpha =\\alpha ^{\\varphi }h ,$ as in [26].",
"For a representative $\\alpha $ of such a Frobenius conjugacy class, there will then exist a curve $C_{\\alpha }$ and an isomorphism $\\beta : C_{\\alpha } \\rightarrow C$ such that $\\beta ^{\\varphi } \\beta ^{-1}= \\alpha $ .",
"As isomorphisms between smooth plane quartics are linear [8], $\\beta $ lifts to an automorphism of $\\mathbb {P}^2$ , represented by an element $B$ of $\\operatorname{GL}_3(\\overline{k})$ , and we will then have that $C_{\\alpha } = B^{-1} (C)$ as subvarieties of $\\mathbb {P}^2$ .",
"This is the curve defined by the equation obtained by substituting $B (x,y,z)^t$ for the transposed vector $(x,y,z)^t$ in the quartic relations defining $C$ ."
],
[
"Algorithm to compute the twists of a smooth plane quartic",
"We first introduce a probabilistic algorithm to calculate the twists of $C$ .",
"It is based on the explicit form of Hilbert 90 (see [34] and [11]).",
"Let $\\alpha \\in \\operatorname{Aut}(C)$ defined over a minimal extension $\\mathbb {F}_{q^n}$ of $k = \\mathbb {F}_q$ for some $n \\ge 1$ , and let $C_{\\alpha }$ be the twist of $C$ corresponding to $\\alpha $ .",
"We construct the transformation $B$ from the previous section by solving the equation $B^{\\varphi } = A B$ for a suitable matrix representation $A$ of $\\alpha $ .",
"Since the curve is canonically embedded in $\\mathbb {P}^2$ , the representation of the action of $\\operatorname{Aut}(C)$ on the regular differentials gives a natural embedding of $\\operatorname{Aut}(C)$ in $\\operatorname{GL}_3(\\mathbb {F}_{q^n})$ .",
"We let $A$ be the corresponding lift of $\\alpha $ in this representation.",
"As $\\operatorname{Gal}(\\overline{\\mathbb {F}}_q / \\mathbb {F}_q)$ is topologically generated by $\\varphi $ and $\\alpha $ is defined over a finite extension of $\\mathbb {F}_q$ , there exists an integer $m$ such that the cocycle relation $\\alpha _{\\sigma \\tau }= \\alpha _{\\tau }^{\\sigma } \\alpha _{\\sigma }$ reduces to the equality $A^{\\varphi ^{m-1}} \\cdots A^{\\varphi } A =\\mathrm {Id}$ .",
"Using the multiplicative form of Hilbert's Theorem 90, we let $B = P + \\sum _{i=1}^{m-1} P^{\\varphi ^i} A^{\\varphi ^{i-1}} \\cdots A^{\\varphi } A$ with $P$ a random matrix $3\\times 3$ with coefficients in $\\mathbb {F}_{q^m}$ chosen in such a way that at the end $B$ is invertible.",
"We will then have $B^{\\varphi } = B A^{-1}$ , the inverse of the relation above, so that we can apply $B$ directly to the defining equation of the quartic.",
"Note that the probability of success of the algorithm is bigger than ${1}/{4}$ (see [11]).",
"To estimate the complexity, we need to show that $m$ is not too large compared with $n$ .",
"We have the following estimate.",
"Lemma 4.1 Let $e$ be the exponent of $\\operatorname{Aut}( C )$ .",
"Then $m\\le ne$ .",
"By definition of $n$ we have $\\alpha ^{\\varphi ^n} = \\alpha $ .",
"Let $\\gamma =\\alpha ^{\\varphi ^{n-1}} \\cdots \\alpha ^{\\varphi } \\alpha $ , and let $N$ be the order of $\\gamma $ in $\\operatorname{Aut}_{\\mathbb {F}_{q^n}}( C )$ .",
"Since $\\gamma ^{\\varphi ^n} = \\gamma $ and $\\mathrm {Id} = \\gamma ^N =\\alpha ^{\\varphi ^{N n-1}} \\cdots \\alpha ^{\\varphi } \\alpha $ , we can take $m \\le n N \\le ne$ .",
"In practice we compute $m$ as the smallest integer such that $\\alpha ^{\\varphi ^{m-1}} \\cdots \\alpha ^{\\varphi } \\alpha $ is the identity."
],
[
"How to compute the twists by hand when $\\#\\operatorname{Aut}( C )$ is small",
"When the automorphism group is not too complicated, it is often possible to obtain representatives of the classes in $H^1(\\operatorname{Gal}( \\overline{\\mathbb {F}}_q/\\mathbb {F}_q ) ,\\operatorname{Aut}( C ))$ and then to compute the twists by hand, a method used in genus 2 in [6].",
"We did this for $\\operatorname{Aut}(C)=\\mathbf {C}_2,\\mathbf {D}_4,\\mathbf {C}_3,\\mathbf {D}_8,\\mathbf {S}_3$ .",
"Let us illustrate this in the case of $\\mathbf {D}_8$ .",
"As we have seen in Theorem REF , any curve $C/\\mathbb {F}_q$ with $\\operatorname{Aut}(C) \\simeq \\mathbf {D}_8$ is $\\overline{\\mathbb {F}}_q$ -isomorphic with some curve $x^4 + x^2 y z + y^4 + a y^2 z^2 +b z^4$ with $a,b\\in \\mathbb {F}_q$ .",
"The problem splits up into several cases according to congruences of $q-1 \\pmod {4}$ and the class of $b \\in \\mathbb {F}_q^*/(\\mathbb {F}_q^*)^4$ .",
"We will assume that $4\\,|\\,(q-1)$ and $b$ is a fourth power, say $b=r^4$ in $\\mathbb {F}_q$ .",
"The 8 automorphisms are then defined over $\\mathbb {F}_q$ : if $i$ is a square root of $-1$ , the automorphism group is generated by $S= \\left[{\\begin{matrix}1 & 0 & 0 \\\\0 & i & 0 \\\\0 & 0 & -i\\end{matrix}}\\right]\\:\\textrm {and}\\:\\:\\:T = \\left[{\\begin{matrix}1 & 0 & 0 \\\\0 & 0 & r \\\\0 & r^{-1} & 0\\end{matrix}}\\right]\\,.$ Representatives of the Frobenius conjugacy classes (which in this case reduce to the usual conjugacy classes) are then $\\mathrm {Id}$ , $S$ , $S^2$ , $T$ and $ST$ .",
"So there are 5 twists.",
"Let us give details for the computation of the twist corresponding to the class of $T$ .",
"We are looking for a matrix $B$ such that $T B=B^{\\varphi }$ up to scalars.",
"We choose $B$ such that $B (x,y,z)^t = (x, \\alpha y + \\beta z , \\gamma y + \\delta z )^t$ .",
"Then we need to solve the following system: $\\alpha ^{\\varphi } = r \\gamma \\,,\\beta ^{\\varphi } = r \\delta \\,,\\gamma ^{\\varphi } = r^{-1} \\alpha \\,,\\delta ^{\\varphi } = r^{-1} \\beta .$ The first equation already determines $\\gamma $ in terms of $\\alpha $ .",
"So we need only satisfy the compatibility condition given by the second equation.",
"Applying $\\varphi $ , we get $\\alpha ^{\\varphi ^2}= (r \\gamma )^{\\varphi } = r \\gamma ^{\\varphi }= r (\\alpha / r) = \\alpha $ .",
"Reasoning similarly for $\\beta $ and $\\delta $ , we see that it suffices to find $\\alpha $ and $\\beta $ in $\\mathbb {F}_{q^2}$ such that $\\det \\left({\\begin{matrix}\\alpha & \\beta \\\\\\alpha ^{\\varphi } / r & \\beta ^{\\varphi } / r\\end{matrix}}\\right)\\ne 0.$ We can take $\\alpha = \\sqrt{\\tau }$ and $\\beta =1$ , with $\\tau $ a primitive element of $\\mathbb {F}_q^*$ .",
"Transforming, we get the twist $x^4 + r x^2 y^2 - r \\tau x^2 z^2 + (a r^2 + 2 r^4) y^4 + (-2 a r^2 \\tau + 12r^4 \\tau ) y^2 z^2 + (a r^2 \\tau ^2 + 2 r^4 \\tau ^2) z^4 = 0 .$"
],
[
"Implementation and experiments",
"We combine the results obtained in Sections and to compute a database of representatives of $k$ -isomorphism classes of genus 3 non-hyperelliptic curves when $k = \\mathbb {F}_p$ is a prime field of small characteristic $p > 7$ ."
],
[
"The general strategy",
"We proceed in two steps.",
"The hardest one is to compute one representative defined over $k$ for each $\\bar{k}$ -isomorphism class, keeping track of its automorphism group.",
"Once this is done, one can apply the techniques of Section to get one representative for each isomorphism class.",
"In order to work out the computation of representatives for the $\\bar{k}$ -isomorphism classes, the naive approach would start by enumerating all plane quartics over $k$ by using the 15 monomial coefficients $m_1$ , ..., $m_{15}$ ordered as in Equation (REF ) and for each new curve to check whether it is smooth and not $\\bar{k}$ -isomorphic to the curves we already kept as representatives.",
"This would have to be done for up to $p^{15}$ curves.",
"For $p>29$ , a better option is to use Proposition REF to reduce to a family with 7 parameters.",
"In both cases, checking for $\\bar{k}$ -isomorphism is relatively fast as we make use of the so-called 13 Dixmier-Ohno invariants.",
"These are generators for the algebra of invariants of ternary quartics forms under the action of $\\operatorname{SL}_3(\\mathbb {C})$ .",
"Among them 7 are denoted $I_3$ , $I_6$ , $I_9$ , $I_{12}$ , $I_{15}$ , $I_{18}$ and $I_{27}$ (of respective degree 3, 6, ..., 27 in the $m_i$ 's) and are due to Dixmier [7]; one also needs 6 additional invariants that are denoted $J_{9}$ , $J_{12}$ , $J_{15}$ , $J_{18}$ , $I_{21}$ and $J_{21}$ (of respective degree 9, 12, ..., 21 in the $m_i$ 's) and that are due to Ohno [28], [10].",
"These invariants behave well after reduction to $\\mathbb {F}_p$ for $p>7$ and the discriminant $I_{27}$ is 0 if and only if the quartic is singular.",
"Moreover, if two quartics have different Dixmier-Ohno invariants (seen as points in the corresponding weighted projective space, see for instance [22]) then they are not $\\bar{k}$ -isomorphic.",
"We suspect that the converse is also true (as it is over $\\mathbb {C}$ ).",
"This is at least confirmed for our values of $p$ since at the end we obtain $p^6+1$ $\\overline{\\mathbb {F}}_p$ -isomorphism classes, as predicted by [4].",
"The real drawback of this approach is that we cannot keep track of the automorphism groups of the curves, which we need in order to compute the twists.",
"Unlike the hyperelliptic curves of genus 3 [22], for which one can read off the automorphism group from the invariants of the curve, we lack such a dictionary for the larger strata of plane smooth quartics.",
"We therefore proceed by ascending up the strata, as summarized in Algorithm REF .",
"In light of Proposition REF , we first determine the $\\bar{k}$ -isomorphism classes for quartics in the small strata by using the representative families of Theorem REF .",
"In this case, the parametrizing is done in an optimal way and the automorphism group is explicitly known.",
"Once a stratum is enumerated, we consider a higher one and keep a curve in this new stratum if and only if its Dixmier-Ohno invariants have not already appeared.",
"As mentioned at the end of Section , this approach still finds all pointless curves (except one for $\\mathbb {F}_{11}$ ) for $p \\le 29$ .",
"We can then use the generic families in Proposition REF and Proposition REF .",
"Figure: Database of representatives for𝔽 p {\\mathbb {F}}_p-isomorphism classes of smooth plane quartics"
],
[
"Implementation details",
"We split our implementation of Algorithm REF into two parts.",
"The first one, developed with the Magma computer algebra software, handles quartics in the strata of dimension 0, 1, 2 and 3.",
"These strata have many fewer points than the ones with geometric automorphism group $\\mathbf {C}_2$ and $\\lbrace 1\\rbrace $ but need linear algebra routines to compute twists.",
"The second part has been developed in the C-language for two reasons: to efficiently compute the Dixmier-Ohno invariants in the corresponding strata and to decrease the memory needed.",
"We now discuss these two issues."
],
[
"Data structures.",
"We decided to encode elements of $\\mathbb {F}_p$ in bytes.",
"This limits us to $p < 256$ , but this is not a real constraint since larger $p$ seem as yet infeasible (even considering the storage issue).",
"As most of the time is spent computing Dixmier-Ohno invariants, we group the multiplications and additions that occur in these calculations as much as possible in 64-bit microprocessor words before reducing modulo $p$ .",
"This decreases the number of divisions as much as possible.",
"To deal with storage issues in Step 6 of Algorithm REF , only the 13 Dixmier-Ohno invariants of the quartics are made fully accessible in memory; we store the full entries in a compressed file.",
"These entries are sorted by these invariants and additionally list the automorphism group, the number of twists, and for each twist, the coefficients of a representative quartic, its automorphism group and its number of points."
],
[
"Size of the hash table.",
"We make use of an open addressing hash table to store the list ${\\mathcal {L}}_p$ from Algorithm REF .",
"This hash table indexes $p^5$ buckets, all of equal size $(1+\\varepsilon ) \\times p $ for some overhead $\\varepsilon $ .",
"Given a Dixmier-Ohno 13-tuple of invariants, its first five elements (eventually modified by a bijective linear combination of the others to get a more uniform distribution) give us the address of one bucket of the table of invariants.",
"We then store the last eight elements of the Dixmier-Ohno 13-tuple at the first free slot in this bucket.",
"The total size of the table is thus $8\\,(1+\\varepsilon ) \\times p^6$ bytes.",
"All the buckets do not contain the same number of invariants at the end of the enumeration, and we need to fix $\\varepsilon $ such that it is very unlikely that one bucket in the hash table goes over its allocated room.",
"To this end, we assume that Dixmier-Ohno invariants behave like random 13-tuples, i.e.",
"each of them has probability $1/p^5$ to address a bucket.",
"Experimentally, this assumption seems to be true.",
"Therefore the probability that one bucket $\\mathcal {B}$ contains $n$ invariants after $k$ trials follows a binomial distribution, $\\operatorname{P}({\\mathcal {B}} = n) = \\binom{n}{k}\\times \\frac{(p^5-1)^{k-n}}{(p^5)^k} =\\binom{n}{k} \\times \\left(\\frac{1}{p^5}\\right)^n\\times \\left(1-\\frac{1}{p^5}\\right)^{k-n} \\,.$ Figure: Overhead ε\\varepsilon Now let $k \\approx p^6$ .",
"Then $k\\times (1/p^5) \\approx p$ , which is a fixed small parameter.",
"In this setting, Poisson approximation yields $\\operatorname{P}({\\mathcal {B}} = n) \\simeq p^n\\, e^{-p} / n\\,!$ , so the average number of buckets that contain $n$ entries at the end is about $p^5\\,\\operatorname{P}({\\mathcal {B}} = n) \\simeq p^{5+n}\\,e^{-p} / n\\,!$ and it remains to choose $n = (1+\\varepsilon )\\,p$ , and thus $\\varepsilon $ , such that this probability is negligible.",
"We draw $\\varepsilon $ as a function of $p$ when this probability is smaller than $10^{-3}$ in Figure REF .",
"For $p=53$ , this yields a hash table of 340 gigabytes."
],
[
"Results and first observations",
"We have used our implementation of Algorithm REF to compute the qylist ${\\mathcal {L}}_p$ for primes $p$ between 11 and 53.",
"Table REF gives the corresponding timings and database sizes (once stored in a compressed file).",
"Because of their size, only the databases $\\mathcal {L}_p$ for $p = 11$ or $p =13$ , and a program to use them, are available onlinehttp://perso.univ-rennes1.fr/christophe.ritzenthaler/programme/qdbstats-v3_0.tgz.. Table: Calculation of ℒ p {\\mathcal {L}}_p on a 32 AMD-Opteron 6272 based serverAs a first use of our database, and sanity check, we can try to interpolate formulas for the number of $\\mathbb {F}_p$ - or $\\overline{\\mathbb {F}}_p$ -isomorphism classes of genus 3 plane quartics over $\\mathbb {F}_p$ with given automorphism group.",
"The resulting polynomials in $p$ are given in Table REF .",
"The `$+[a]_{\\ \\mathrm {condition}}$ ' notation means that $a$ should be added if the `condition' holds.",
"Table: Number of isomorphism classes of plane quarticswith given automorphism groupMost of these formulas can actually be proved (we emphasize the ones we are able to prove in Table REF ).",
"In particular, it is possible to derive the number of most of the #$\\overline{\\mathbb {F}}_p$ -isomorphic classes from the representative families given in Theorem REF ; one merely needs to consider the degeneration conditions between the strata.",
"For example, for the strata of dimension 1, the singularities at the boundaries of the strata of dimension 1 corresponding to strata with larger automorphism group are given by $\\mathbb {F}_p$ -points, except for the stratum $\\mathbf {S}_4$ .",
"The latter stratum corresponds to singular curves for $a \\in \\left\\lbrace -2,-1,2 \\right\\rbrace $ , and the Klein quartic corresponds to $a = 0$ .",
"But the Fermat quartic corresponds to both roots of the equation $a^2 + 3 a + 18$ (note that the family for the stratum $\\mathbf {S}_4$ is no longer representative at that boundary point).",
"The number of roots of this equation in $\\mathbb {F}_p$ depends on the congruence class of $p$ modulo 7.",
"One proceeds similarly for the other strata of small dimension; the above degeneration turns out to be the only one that gives a dependence on $p$ .",
"To our knowledge, the point counts for the strata $\\mathbf {C}_2$ and $\\lbrace 1\\rbrace $ are still unproved.",
"Note that the total number of $\\overline{\\mathbb {F}}_p$ -isomorphism classes is known to be $p^6+1$ by [4], so the number of points on one determines the one on the other.",
"Determining the number of twists is a much more cumbersome task, but can still be done by hand by making explicit the cohomology classes of Section .",
"For the automorphism groups $\\mathbf {\\mathbf {G}_{168}}$ , $\\mathbf {\\mathbf {G}_{96}}$ , $\\mathbf {G}_{48}$ and $\\mathbf {\\mathbf {S}_4}$ , we have recovered the results published by Meagher and Top in [26] (a small subset of the curves defined over $\\mathbb {F}_p$ with automorphism group $\\mathbf {G}_{16}$ was studied there as well)."
],
[
"Distribution according to the number of points",
"Once the lists $\\mathcal {L}_p$ are determined, the most obvious invariant function on this set of isomorphism classes is the number of rational points of a representative of the class.",
"To observe the distributions of these classes according to their number of points was the main motivation of our extensive computation.",
"In Appendix , we give some graphical interpretations of the results for prime field $\\mathbb {F}_p$ with $11 \\le p \\le 53$The numerical values we exploited can be found at http://perso.univ-rennes1.fr/christophe.ritzenthaler/programme/qdbstats-v3_0.tgz..",
"Although we are still at an early stage of exploiting the data, we can make the following remarks: Among the curves whose number of points is maximal or minimal, there are only curves with non-trivial automorphism group, except for a pointless curve over $\\mathbb {F}_{11}$ mentioned at the end of Section REF .",
"While this phenomenon is not true in general (see for instance [30] using the form $43,\\# 1$ over $\\mathbb {F}_{167}$ ), it shows that the usual recipe to construct maximal curves, namely by looking in families with large non-trivial automorphism groups, makes sense over small finite fields.",
"It also shows that to observe the behavior of our distribution at the borders of the Hasse-Weil interval, we have to deal with curves with many automorphisms, which justifies the exhaustive search we made.",
"Defining the trace $t$ of a curve $C/\\mathbb {F}_q$ by the usual formula $t=q+1-\\#C(\\mathbb {F}_q)$ , one sees in Fig.",
"REF that the “normalized trace” $\\tau =t/ \\sqrt{q} $ accurately follows the asymptotic distribution predicted by the general theory of Katz-Sarnak [21].",
"For instance, the theory predicts that the mean normalized trace should converge to zero when $q$ tends to infinity.",
"We found the following estimates for $q=11,17,23,29,37,53$ : $4 \\cdot 10^{-3},\\quad 1 \\cdot 10^{-3}, \\quad 4 \\cdot 10^{-4}, \\quad 2 \\cdot 10^{-4}, \\quad 6\\cdot 10^{-5}, \\quad 3 \\cdot 10^{-5}.$ Our extensive computations enable us to spot possible fluctuations with respect to the symmetry of the limit distribution of the trace, a phenomenon that to our knowledge has not been encountered before (see Fig.",
"REF ).",
"These fluctuations are related to the Serre's obstruction for genus 3 [30] and do not appear for genus $\\le 2$ curves.",
"Indeed, for these curves (and more generally for hyperelliptic curves of any genus), the existence of a quadratic twist makes the distribution completely symmetric.",
"The fluctuations also cannot be predicted by the general theory of Katz and Sarnak, since this theory depends only on the monodromy group, which is the same for curves, hyperelliptic curves or abelian varieties of a given genus or dimension.",
"Trying to understand this new phenomenon is a challenging task and indeed the initial purpose of constructing our database."
],
[
"Generators and normalizers",
"As mentioned in Remark REF , the automorphism groups in Theorem REF have the property that their isomorphism class determines their conjugacy class in $\\operatorname{PGL}_3 (K)$ .",
"Accordingly, the families of curves in Theorem REF have been chosen in such a way that they allow a common automorphism group as subgroup of $\\operatorname{PGL}_3 (K)$ .",
"We proceed to describe the generators and normalizers of these subgroups, that can be computed directly or by using [19].",
"In what follows, we consider $\\operatorname{GL}_2 (K)$ as a subgroup of $\\operatorname{PGL}_3 (K)$ via the map $A \\mapsto \\left[ {\\begin{matrix} 1 & 0 \\\\ 0 & A \\end{matrix}}\\right]$ .",
"The group $D (K)$ is the group of diagonal matrices in $\\operatorname{PGL}_3 (K)$ , and $T (K)$ is its subgroup consisting of those matrices in $D (K)$ that are non-trivial only in the upper left corner.",
"We consider $\\mathbf {S}_3$ as a subgroup $\\widetilde{\\mathbf {S}}_3$ of $\\operatorname{GL}_3 (K)$ by the permutation action that it induces on the coordinate functions, and we denote by $\\widetilde{\\mathbf {S}}_4$ the degree 2 lift of $\\mathbf {S}_4$ to $\\operatorname{GL}_3 (K)$ generated by the matrices $\\begin{bmatrix}1 & 0 & 0 \\\\0 & \\zeta _8 & 0 \\\\0 & 0 & \\zeta _8^{-1}\\end{bmatrix}\\,,\\ \\ \\ \\frac{-1}{i + 1}\\begin{bmatrix}1 & 0 & 0 \\\\0 & i & -i \\\\0 & 1 & 1\\end{bmatrix}\\,.$ Theorem 1.1 The following are generators for the automorphism groups $G$ in Theorem REF , along with the isomorphism classes and generators of their normalizers $N$ in $\\operatorname{PGL}_3 (K)$ .",
"$\\lbrace 1 \\rbrace $ is generated by the unit element.",
"$N = \\operatorname{PGL}_3 (K)$ .",
"$\\mathbf {C}_2 = \\langle \\alpha \\rangle $ , where $\\alpha (x,y,z) = (-x,y,z)$ .",
"$N = \\operatorname{GL}_2 (K)$ .",
"$\\mathbf {D}_4 = \\langle \\alpha , \\beta \\rangle $ , where $\\alpha (x,y,z) =(-x,y,z)$ and $\\beta (x,y,z) = (x,-y,z)$ .",
"$N = D (K) \\widetilde{\\mathbf {S}}_3$ .",
"$\\mathbf {C}_3 = \\langle \\alpha \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _3x,y,z)$ .",
"$N = \\operatorname{GL}_2 (K)$ .",
"$\\mathbf {D}_8 = \\langle \\alpha , \\beta \\rangle $ , where $\\alpha (x,y,z) =(x,\\zeta _4 y,\\zeta _4^{-1} z)$ and $\\beta (x,y,z) = (x,z,y)$ .",
"$N = T(K)\\widetilde{\\mathbf {S}}_4$ .",
"$\\mathbf {S}_3 = \\langle \\alpha , \\beta \\rangle $ , where $\\alpha (x,y,z) =(x,\\zeta _3 y,\\zeta _3^{-1} z)$ and $\\beta (x,y,z) = (x,z,y)$ .",
"$N = T(K)\\widetilde{\\mathbf {S}_3}$ .",
"$\\mathbf {C}_6 = \\langle \\alpha \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _3x,-y,z)$ .",
"$N = D (K)$ .",
"$\\mathbf {G}_{16} = \\langle \\alpha , \\beta , \\gamma \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _4 x,y,z)$ , $\\beta (x,y,z) = (x,-y,z)$ , and $\\gamma (x,y,z) = (x,z,y)$ .",
"$N = T(K) \\widetilde{\\mathbf {S}}_4$ .",
"$\\mathbf {S}_4 = \\langle \\alpha , \\beta , \\gamma \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _4 x,y,z)$ , $\\beta (x,y,z) = (x,\\zeta _3 y ,z)$ , and $\\gamma (x,y,z) = (x,y + 2 z , y - z)$ .",
"$N$ is $\\operatorname{PGL}_3 (K)$ -conjugate to $N = T(K) \\widetilde{\\mathbf {S}}_4$ .",
"$\\mathbf {C}_9 = \\langle \\alpha \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _9x,\\zeta _9^3 y,\\zeta _9^{-3} z)$ .",
"$N = D(K)$ .",
"$\\mathbf {G}_{48} = \\langle \\alpha , \\beta , \\gamma , \\delta \\rangle $ , where $\\alpha (x,y,z) = (-x,y,z)$ , $\\beta (x,y,z) = (x,-y,z)$ , $\\gamma (x,y,z) =(y,z,x)$ , and $\\delta (x,y,z) = (y,x,z)$ .",
"$N = G$ .",
"$\\mathbf {G}_{96} = \\langle \\alpha , \\beta , \\gamma , \\delta \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _4 x,y,z)$ , $\\beta (x,y,z) = (x, \\zeta _4 y,z)$ , $\\gamma (x,y,z) = (y,z,x)$ , and $\\delta (x,y,z) = (y,x,z)$ .",
"$N = G$ .",
"$\\mathbf {G}_{168} = \\langle \\alpha , \\beta , \\gamma \\rangle $ , where $\\alpha (x,y,z) = (\\zeta _7 x , \\zeta _7^2 y , \\zeta _7^4 z)$ , $\\beta (x,y,z) =(y,z,x)$ , and $\\gamma (x,y,z) = ( &(\\zeta _7^4 - \\zeta _7^3) x + (\\zeta _7^2 - \\zeta _7^5) y+ (\\zeta _7 - \\zeta _7^6) z , \\\\& (\\zeta _7^2 - \\zeta _7^5) x + (\\zeta _7 - \\zeta _7^6) y+ (\\zeta _7^4 - \\zeta _7^3) z , \\\\& (\\zeta _7 - \\zeta _7^6) x + (\\zeta _7^4 - \\zeta _7^3) y+ (\\zeta _7^2 - \\zeta _7^5) z) .$ $N = G$ .",
"For lack of space, we do not give the mutual automorphism inclusions or the degenerations between the strata.",
"Most of these can be found in [25]."
],
[
"Numerical results",
"Given a prime number $p$ , we let $N_{p,3}(t)$ denote the number of $\\mathbb {F}_p$ -isomorphism classes of non-hyperelliptic curves of genus 3 over $\\mathbb {F}_p$ whose trace equals $t$ .",
"Define $N^{\\textrm {KS}}_{p,3}(\\tau ) = \\frac{\\sqrt{p}}{\\# \\mathsf {M}_3(\\mathbb {F}_p)} \\cdot N_{p,3}(t), \\quad t=\\lfloor \\sqrt{p} \\cdot \\tau \\rfloor , \\quad \\tau \\in [-6,6]$ which is the normalization of the distribution of the trace as in [21].",
"Our numerical results are summarized on Fig.",
"REF .",
"Figure: Trace distribution"
]
] | 1403.0562 |
[
[
"A Family of Quasisymmetry Models"
],
[
"Abstract We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue.",
"Algebraically, this corresponds to deformations of toric ideals associated with graphs.",
"Our discussion of the statistical issues centers around maximum likelihood estimation."
],
[
"Introduction",
"Consider a square contingency table with commensurable row and column classification variables $X$ and $Y$ .",
"Such tables can arise from cross-classifying repeated measurements of a categorical response variable.",
"They are common in panel and social mobility studies.",
"One of the most cited examples, taken from [12], is shown in Table REF .",
"It cross-classifies 7477 female subjects according to the distance vision levels of their right and left eyes.",
"Table: Cross classification of 7477 women by unaided distancevision of right and left eyes.The most parsimonious model for such tables is the symmetry (S) model, due to [3].",
"While the S model is easy to interpret, it is too restrictive and rarely fits well.",
"An important model that is often of adequate fit is the quasi-symmetry (QS) model of [4].",
"[7] studied the QS model from the information-theoretic point of view and generalized it to a family of models based on the $\\phi $ -divergence [10].",
"In their framework, classical QS is closest to the S model under the Kullback-Leibler divergence.",
"However, by changing the divergence used to measure proximity of distributions, alternative QS models are found.",
"For instance, the Pearsonian divergence yields the Pearsonian QS model.",
"For the data in Table REF , [2] applied the QS model, while [7] applied the Pearsonian QS model, and here these two lead to estimates of similar fit.",
"However, there are other data sets where only one of them performs well.",
"Our goal is to link these two models.",
"We shall construct a one-parameter family of QS models that connects these two.",
"In this way, more options for data analysis are available.",
"In case of a single square contingency table, the optimal choice of this model parameter would be of interest.",
"However, the more interesting practical application lies in analyzing and comparing independent square tables of the same set-up, when they cannot be modeled adequately all by the same (classical or Pearsonian) QS model.",
"For example, consider the same panel study carried out at two independent centers, with one of them being modeled only by the classical QS and the other only by the Pearsonian QS.",
"In this scenario, the two fitted models are not as comparable as we would like.",
"Our approach furnishes in-between compromise models.",
"Our family exhibits interesting properties when viewed from the perspective of algebraic statistics [5].",
"It interpolates between two fundamental classes of discrete variable models, namely, toric models and linear models [9].",
"Indeed, the QS model is toric, and its Markov basis is well-known, by work of [11] and Latunszynski-Trenado [5].",
"The Pearsonian QS model reduces to a linear model, specified by the second factors in (REF ).",
"Its ML degree is the number of bounded regions in the arrangement of hyperplanes $\\lbrace a_i - a_j = 1 \\rbrace $ , by Varchenko's formula [9].",
"This paper is organized as follows.",
"Our parametric family of QS models is introduced in Section 2.",
"In Section 3 we derive the implicit representation of our model by polynomial equations in the cell entries.",
"That section is written in the algebraic language of ideals and varieties.",
"It will be of independent interest to scholars in combinatorial commutative algebra [8], [13].",
"Maximum likelihood estimation (MLE) and the fit of the model are discussed in Section 4.",
"Section 5 examines a natural submodel given by independence constraints.",
"Section 6 discusses statistical applications and presents computations with concrete data sets.",
"Section 7 offers an information-theoretic characterization in terms of $\\phi $ -divergence, following [7] and [10]."
],
[
"Quasisymmetry Models",
"We consider models for square contingency tables of format $I \\times I$ .",
"Probability tables ${\\bf p} = (p_{ij})$ are points in the simplex $\\Delta _{I^2-1}$ .",
"Here $p_{ij }$ is the probability that an observation falls in the $(i, j)$ cell.",
"We write ${\\bf n} = (n_{ij})$ for the table of observed frequencies.",
"The model of symmetry (S) is $\\qquad p_{ij}=s_{ij} \\ \\ \\ \\ \\text{with parameters} \\ \\ s_{ij}=s_{ji} \\ \\ \\ \\ {\\rm for} \\,\\,1 \\le i \\le j \\le I .$ Here, and in what follows, the table $(s_{ij})$ is non-negative and its entries sum to 1.",
"Geometrically, the S model is a simplex of dimension $\\binom{I+1}{2}-1$ inside the ambient probability simplex $\\Delta _{I^2-1}$ .",
"The classical QS model can be defined, as a model of divergence from S, by $p_{ij}=s_{ij}\\frac{2c_i}{c_i+c_j} \\ , \\ \\ \\ \\ i,j=1, \\ldots , I .$ The Pearsonian QS model is defined by the parametrization $p_{ij}=s_{ij}(1+a_i-a_j) \\ , \\ \\ \\ \\ i,j=1, \\ldots , I .$ Both models are semialgebraic subsets of dimension $\\binom{I+1}{2}+I-2$ in the simplex $\\Delta _{I^2-1}$ .",
"The S model is the subset obtained respectively for $c_1 = \\cdots = c_I$ in (REF ) or $a_1 = \\cdots = a_I$ in (REF ).",
"We here study the following quasisymmetry model (${\\rm QS}_t$ ), where $t\\in [0, 1]$ is a parameter: $p_{ij}=s_{ij}\\left(1+\\frac{(1+t)(a_i-a_j)}{2+(1-t)(a_i+a_j)}\\right) \\ , \\ \\ \\ i\\ne j, \\ \\ \\ i,j=1, \\ldots , I .$ In all three models, the matrix entries on the diagonal are set to $p_{ii} = s_{ii}$ for $i=1,\\ldots ,I$ .",
"For $t = 1$ , the model (REF ) specializes to the Pearsonian QS model (REF ).",
"For $t=0$ , it specializes to the QS model (REF ), if we set $a_i = c_i - 1$ .",
"The parameters $a_i$ will be assumed to satisfy the restriction $t \\cdot {\\rm max}_i a_i-{\\rm min}_i a_i \\, \\le \\,1.$ Since we had assumed $0 \\le s_{ij} \\le 1/2$ , the constraint (REF ) on the $a_i$ ensures that the $p_{ij}$ are probabilities (i.e.",
"lie in the interval $[0,1]$ ).",
"Furthermore, if we change the parameters via $s_{ii} = x_{ii}\\, \\,\\,\\,\\hbox{for}\\, \\,i=j, \\quad \\hbox{and} \\quad s_{ij}=x_{ij}\\left(1+(1-t)\\frac{a_i+a_j}{2}\\right) \\ \\ \\, \\hbox{for} \\,\\, i \\ne j, $ then the model (${\\rm QS}_t$ ), defined in (REF ), is rewritten in the simpler form $p_{ij}=x_{ij}(1+a_i-t a_j) \\ , \\ \\ \\ i\\ne j, \\ \\ \\ i,j=1, \\ldots , I .$ Note that $x_{i+} = \\sum _{j=1}^I x_{ij} = \\sum _{j=1}^I x_{ji} = x_{+i}$ , since the table $(x_{ij})$ is also symmetric.",
"For $t=1$ , the probabilities defined by (REF ) satisfy $\\sum _{i=1}^I p_{ij}=1$ for all $j$ .",
"In order to ensure that $\\sum _{i=1}^I p_{ij}=1$ for $t\\ne 1$ as well, we use the `weighted sum to zero' constraint $\\sum _{i=1}^I (x_{i+}-x_{ii})a_i = 0 .$ The expressions (REF ) and (REF ) are equivalent.",
"Whether one or the other is preferred is a matter of convenience.",
"Maximum likelihood estimation is easier with (REF ), since the MLEs of the $s_{ij}$ are rational functions of the observed frequencies $n_{ij}$ .",
"The estimates of the $a_i$ depend algebraically on ${\\bf n}$ , and they generally have to be computed by an iterative method.",
"In the formulation (REF ), none of the parameters have estimates that are rational in ${\\bf n}$ .",
"We shall see this in Section 4.",
"On the other hand, for our algebraic analysis of the ${\\rm QS}_t$ model, it is more convenient to use (REF ).",
"Example 2.1 Fix $I= 3$ .",
"For any fixed $t$ , the model (REF ) is a hypersurface in the simplex $\\Delta _8$ of all $3 \\times 3$ probability tables.",
"This hypersurface is the zero set of the cubic polynomial $\\begin{matrix} (1+t+t^2)(p_{12}p_{23}p_{31}-p_{21}p_{32}p_{13}) \\,+\\, \\\\t(p_{12}p_{23}p_{13}+p_{12}p_{32}p_{31}+p_{21}p_{23}p_{31}-p_{12}p_{32}p_{13}-p_{21}p_{23}p_{13}-p_{21}p_{32}p_{31}).\\end{matrix}$ For $t = 0$ , we recover the familiar binomial relation that encodes the cycle of length three [5].",
"Thus, our family of ${\\rm QS}_t$ models represents a deformation of that Markov basis: $ p_{12}p_{23}p_{31}-p_{21}p_{32}p_{13} + O(t) .$ The generalization of the relation (REF ) to higher values of $I$ will be presented in Section 3.",
"$\\diamondsuit $ Another characteristic model for square tables with commensurable classification variables is the model of marginal homogeneity (MH).",
"This is specified by the equations $p_{i+}\\,=\\,p_{+i} \\, \\quad \\hbox{for} \\,\\,\\,\\, i=1, \\ldots , I .$ The model of symmetry S implies MH and QS, i.e.",
"(REF ) with $c_1 = \\cdots = c_I$ .",
"By [2], if the models MH and QS hold simultaneously, then S is implied.",
"In symbols, $\\text{S}=\\text{MH} \\cap \\text{QS}$ .",
"This identity is important in that it underlines the role of the parameters $c_i$ in the QS model.",
"These express the contribution of the classification category $i$ to marginal inhomogeneity.",
"We shall prove next that the same identity holds for our generalized ${\\rm QS}_t$ model.",
"Proposition 2.2 For any $t \\in [0, 1]$ , we have $\\text{S}=\\text{MH} \\cap QS_t$ .",
"It is straightforward to verify that S implies MH and ${\\rm QS}_t$ with $a_i=0$ , for all $i$ , which leads to $p_{ij}=x_{ij}=s_{ij}$ , for all $i,j$ .",
"On the other hand, under ${\\rm QS}_t$ as defined by (REF ), we have $p_{i+}-p_{+i}\\,\\,=\\,\\,(1+t)\\biggl (a_i(x_{i+}-x_{ii})-\\sum _{j\\ne i} a_j x_{ij}\\biggr ) \\, \\qquad \\hbox{for} \\,\\,\\,\\, i=1, \\ldots , I .$ Combining this with MH as in (REF ), and setting $y_i := x_{ii}-x_{i+}$ , the equation (REF ) implies $\\sum _{j\\ne i} a_j x_{ij} + a_i y_i \\,\\,= \\,\\,0 \\, \\qquad \\hbox{for} \\,\\,\\,\\, i=1, \\ldots , I .$ This can be written in the matrix form ${\\bf B}{\\bf a}={\\bf 0}$ , where ${\\bf a}=(a_1, \\ldots , a_I)^T$ , ${\\bf x}=(x_{ij})$ , and $ \\qquad \\qquad {\\bf B} \\,= {\\bf x}-\\text{diag}({\\bf x}{\\bf 1}) \\, =\\begin{bmatrix}& & & x_{1I} \\\\& \\tilde{\\bf B} & & \\vdots \\\\& & & x_{I-1,I} \\\\x_{I1} & x_{I2} & \\ldots & y_I \\end{bmatrix} .$ The matrix $\\tilde{\\bf B}$ is strictly diagonally dominant, provided $|y_i|=x_{i+}-x_{ii}> \\sum _{j\\ne i}^{I-1} x_{ij}$ .",
"This is ensured if all $x_{il}$ are positive, as in Remark REF ; otherwise a separate argument is needed.",
"By the Levy-Desplanques Theorem, the matrix $\\tilde{\\bf B}$ is invertible and $\\operatorname{rank}(\\tilde{\\bf B})=I-1$ .",
"Hence $\\operatorname{rank}({\\bf B})=I-1$ , since ${\\bf B}{\\bf 1}={\\bf 0}$ .",
"Therefore, all solutions of ${\\bf B}{\\bf a}={\\bf 0}$ have the form ${\\bf a}=a{\\bf 1}$ for some $a \\in \\mathbb {R}$ .",
"For $t=1$ , equation (REF ) now implies $p_{ij}=x_{ij}=s_{ij}$ , for all $i, j$ .",
"For $t\\ne 1$ , combining (REF ) with the positivity of $x_{i+}-x_{ii}$ , we get $a=0$ .",
"Hence symmetry S holds and the proof is complete.",
"Remark 2.3 Contingency tables with structural zeros, i.e., cells of zero probability, are rare.",
"If they exist, they usually have a specific pattern (zero diagonal, triangular table).",
"In our set-up it is realistic to assume that there exists an index $j$ such that $p_{ij}>0$ for all $i=1, \\ldots , I$ .",
"Thus, without loss of generality, we can assume that $p_{iI}>0$ and therefore $x_{iI}>0$ for all $i=1, \\ldots , I$ .",
"Example 2.4 ($I=3$ ) Marginal homogeneity defines a linear space of codimension 2, via $p_{11}+p_{12}+p_{13} &=& p_{11}+p_{21}+p_{31},\\\\p_{21}+p_{22}+p_{23} &=& p_{12}+p_{22}+p_{32},\\\\p_{31}+p_{32}+p_{33} &=& p_{13}+p_{23}+p_{33}.$ Inside that linear subspace, the cubic (REF ) factors into a hyperplane, which is the S model $\\lbrace p_{12} = p_{21}, \\ p_{13} = p_{31}, \\ p_{23} = p_{32}\\rbrace $ , and a quadric, which has no points with positive coordinates.",
"$\\diamondsuit $ In the light of Proposition REF , the parameter $a_i$ of the ${\\rm QS}_t$ model can be interpreted as the contribution of each category $i$ to the marginal inhomogeneity.",
"By this we mean the difference of $a_i$ minus the weighted average of all $a_i$ 's.",
"This is the parenthesized expression in the identity $p_{i+}-p_{+i}\\,\\,=\\,\\,(1+t)x_{i+}\\left(a_i-\\sum _{j} \\frac{x_{ij}}{x_{i+}} a_j \\right) \\ , \\qquad i,j=1, \\ldots , I .$"
],
[
"Implicit Equations",
"We now examine the quasisymmetry models ${\\rm QS}_t$ through the lens of algebraic statistics [5], [9], [11].",
"To achieve more generality and flexibility, we fix an undirected simple graph $G$ with vertex set $\\lbrace 1,2,\\ldots ,I\\rbrace $ .",
"Let $\\mathcal {I}_{G}$ denote the prime ideal of algebraic relations among the quantities $p_{ij} = x_{ij} (1+a_i-ta_j)$ in (REF ), where $\\lbrace i,j\\rbrace $ runs over the edge set $E(G)$ of the graph $G$ .",
"The ideal $\\mathcal {I}_G$ lives in the polynomial ring $\\mathbb {K}[\\,p_{ij},p_{ji}: \\lbrace i,j\\rbrace \\in E(G) \\,]$ .",
"Here we take $\\mathbb {K} = \\mathbb {Q}[[t]]$ to be the local ring of formal Laurent series in one unknown $t$ .",
"Our main result in this section is the derivation of a generating set for the ideal $\\mathcal {I}_G$ .",
"One motivation for studying this ideal is the constrained formulation of the MLE problem in Section 4.",
"The model in Section 2 corresponds to the complete graph on $I$ nodes, denoted $G = K_I$ .",
"In particular, for $I=3$ , the ideal $\\mathcal {I}_{K_3}$ is the principal ideal generated by the cubic in (REF ).",
"Here we work with arbitrary graphs $G$ , not just $K_I$ , so as to allow for sparseness in the models.",
"We disregard the `weighted sum to 0' constraint (REF ), as this does not affect the homogeneous relations in $\\mathcal {I}_{G}$ .",
"Let ${\\mathbb {E}}(G)$ denote the set of oriented edges of $G$ .",
"For each edge $\\lbrace i,j\\rbrace $ in $E(G)$ there are two edges $ij$ and $ji$ in ${\\mathbb {E}}(G)$ .",
"So we have $|{\\mathbb {E}}(G)|=2|E(G)|$ .",
"An orientation of $G$ is the choice of a subset ${\\mathcal {O}}\\subset {\\mathbb {E}}(G)$ such that, for each edge $\\lbrace i,j\\rbrace $ in $E(G)$ , either $ij$ or $ji$ belongs to ${\\mathcal {O}}$ .",
"An orientation of $G$ is called acyclic if it contains no directed cycle.",
"Let $C$ denote the undirected $n$ -cycle, with $E(C) = \\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 2,3\\rbrace , \\ldots ,\\lbrace n,1\\rbrace \\rbrace $ .",
"Then $C$ has $2^n$ orientations, shown in Figure REF for $n=3$ .",
"Precisely two of these orientations are cyclic.",
"These two directed cycles are denoted by $o_C$ and $\\bar{o}_C$ .",
"Their edge sets are ${\\mathbb {E}}(o_C)=\\lbrace 12,23,\\ldots ,n1\\rbrace $ and ${\\mathbb {E}}(\\bar{o}_C)=\\lbrace 21,32,\\ldots ,1n\\rbrace $ .",
"Any orientation $\\delta _C$ of $C$ defines a monomial of degree $n$ via $ p^{\\delta _C} \\,\\,=\\prod _{ij\\in {\\mathbb {E}}(\\delta _C)}p_{ij}.",
"$ We also define the integer ${\\rm c}(\\delta _C)=2|{\\mathbb {E}}(o_C)\\cap {\\mathbb {E}}(\\delta _C)|-n$ .",
"Note that ${\\rm c}(o_C)=n$ and ${\\rm c}(\\bar{o}_C)=-n$ .",
"We associate with the $n$ -cycle $C$ the following polynomial of degree $n$ with $2^n$ terms: $P^{C}\\,\\,=\\,\\,\\sum _{\\delta _C} {\\rm coeff}(\\delta _C) \\cdot p^{\\delta _C}.$ The sum is over all orientations $\\delta _C$ of $C$ , and the coefficients are the scalars in $\\mathbb {K}$ defined by ${\\rm coeff}(\\delta _{C})\\,= \\,{\\left\\lbrace \\begin{array}{ll}\\frac{{\\rm c}(\\delta _C)}{|{\\rm c}(\\delta _C)|} \\cdot \\big (t^{r-\\frac{{|{\\rm c}(\\delta _C)|}}{2}}+t^{r+2-\\frac{{|{\\rm c}(\\delta _C)|}}{2}}+\\cdots +t^{r+\\frac{{|{\\rm c}(\\delta _C)|}}{2}-2} \\big )\\quad \\quad \\quad \\quad {\\rm if}\\quad n=2r,\\smallskip \\\\\\frac{{\\rm c}(\\delta _C)}{|{\\rm c}(\\delta _C)|} \\cdot \\big ( t^{r-\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}}+t^{r+1-\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}}+\\cdots +t^{r+\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}-1}\\big ) \\quad \\ {\\rm if} \\quad n=2r-1.\\end{array}\\right.",
"}$ Figure: The eight orientations δ 1 ,δ 2 ,...,δ 8 \\delta _1,\\delta _2,\\ldots ,\\delta _8 of C=K 3 C=K_3.Example 3.1 We consider the cycle $C = K_3$ of length $n=3$ .",
"It has eight orientations, depicted in Figure REF .",
"The corresponding monomials and their coefficients are as follows: Table: NO_CAPTIONThus, the polynomial $P^{C}$ defined in (REF ) is the cubic (REF ) seen in Example REF .",
"$\\diamondsuit $ We define the classical QS model on the graph $G$ by the parametrization (REF ) where $\\lbrace i,j\\rbrace $ runs over the set $E(G)$ of edges of $G$ .",
"We write $\\mathcal {T}_G$ for the ideal of this model.",
"This is a toric ideal whose Markov basis is obtained from the cycle polynomials $P^C$ by setting $t = 0$ : Lemma 3.2 The ideal $\\,\\mathcal {T}_G$ has a universal Gröbner basis consisting of the binomials $\\qquad \\qquad P^C|_{t=0} \\,\\, = \\,\\, p^{o(C)}-p^{\\bar{o}(C)} \\qquad \\hbox{for all cycles $C$ in $G$.",
"}$ The identity in (REF ) is straightforward from the definition of $c(\\delta _C)$ and ${\\rm coeff}(\\delta _C)$ .",
"It was shown in [5] that the binomials $p^{o(C)}-p^{\\bar{o}(C)}$ form a Markov basis for $QS$ .",
"Since the underlying model matrix is totally unimodular, the Markov basis is also a Graver basis, and hence it is a universal Gröbner basis, by [13].",
"Example 3.3 For $I=4$ , the model ${\\rm QS}_t$ corresponds to the complete graph $K_4$ .",
"This graph has seven undirected cycles $C$ , four of length 3 and three of length 4.",
"Its defining prime ideal $\\mathcal {I}_{K_4}$ is generated by four cubics and three quartics, all of the form $P^C$ .",
"For $t = 0$ , we recover the binomials corresponding to the seven moves that are listed in [11].",
"$\\diamondsuit $ This example is explained by the following theorem, which is our main result in Section 3.",
"Theorem 3.4 The prime ideal $\\,\\mathcal {I}_G$ of the quasisymmetry model associated with an undirected graph $G$ is generated by the cycle polynomials $P^C$ where $C$ runs over all cycles in $G$ .",
"We begin by proving that $P^C$ lies in $\\mathcal {I}_G$ .",
"The image of $P^C$ under the substitution $p_{ij} \\mapsto x_{ij} (1+a_i-ta_j)$ can be written as $\\,Q^C\\times \\prod _{\\lbrace i,j\\rbrace \\in E(C)} x_{ij} $ , where $Q^C$ is a polynomial in $\\mathbb {K}[a_1,\\ldots ,a_n]$ .",
"Since each term $p^{\\delta _C}$ of $P^C$ is divisible by either $p_{1n}$ or $p_{n1}$ , we can write $ Q^C\\,\\,=\\,\\,(1+a_1-ta_n)T_{1n}+(1+a_n-ta_1)T_{n1}.$ We need to show that $Q^C$ is zero.",
"To do this, we shall establish the following identities: $ \\begin{matrix}& T_{1n} & = & (-1)^{[\\frac{n-1}{2}]+1}(t + 1)^{2r-2}(1+a_n-ta_1)\\prod _{i=2}^{n-1}(1+a_i-t a_i) \\\\\\hbox{and} \\quad & T_{n1} & = & (-1)^{[\\frac{n-1}{2}]}(t +1)^{2r-2}(1+a_1-ta_n)\\prod _{i=2}^{n-1}(1+a_i-t a_i).\\end{matrix} $ To prove these, we shall use the decompositions $ \\begin{matrix}& T_{1n} & = &(1+a_1-ta_2)T_{1n,12} + (1+a_2-ta_1)T_{1n,21} \\\\\\hbox{and} \\quad &T_{n1} & = &(1+a_1-ta_2)T_{n1,12} + (1+a_2-ta_1)T_{n1,21}.",
"\\\\\\end{matrix} $ With this notation, we claim that the following holds for a suitable integer $r$ : (i) $T_{1n,12}\\,=\\,(-1)^{[\\frac{n-2}{2}]}t(t +1)^{2r-3}(a_2-a_n)\\prod _{i=3}^{n-1}(1+a_i-t a_i)$ , (ii) $T_{1n,21} \\,=\\,(-1)^{[\\frac{n-2}{2}]}(t +1)^{2r-3}(t^2a_2-t-a_n-1)\\prod _{i=3}^{n-1}(1+a_i-t a_i)$ .",
"Let $C^{\\prime }$ be the cycle $2-3-\\cdots -n-2$ .",
"In analogy to (REF ), we write $Q^{C^{\\prime }}=(1+a_2-ta_n)S_{2n}+(1+a_n-ta_2)S_{n2}.$ Note that for any orientation $\\delta _C$ of $C$ in which $1n$ and 12 belong to ${\\mathbb {E}}(\\delta _C)$ , we have $c(\\delta _C)\\,\\, =\\,\\,{\\left\\lbrace \\begin{array}{ll}c(\\delta _{C^{\\prime }})-1 \\quad \\quad \\quad \\ {\\rm if} \\quad n2\\in {\\mathbb {E}}(\\delta _{C^{\\prime }}),\\smallskip \\\\c(\\delta _{C^{\\prime }})+1 \\quad \\quad \\quad \\ {\\rm if}\\quad 2n\\in {\\mathbb {E}}(\\delta _{C^{\\prime }}).&\\\\\\end{array}\\right.",
"}$ Also note that $\\frac{{\\rm c}(\\delta _C)}{|{\\rm c}(\\delta _C)|}=\\frac{{\\rm c}(\\delta _{C^{\\prime }})}{|{\\rm c}(\\delta _{C^{\\prime }})|}$ .",
"In order to prove (i) we consider the following two cases: Case 1.",
"$n = 2r-1$ is an odd number: We claim that $T_{1n,12}=t(S_{n2}+S_{2n})$ .",
"Note that $C^{\\prime }$ is an even cycle with $n-1=2(r-1)$ .",
"The coefficient for $\\delta _C$ can be written as $t\\times \\frac{{\\rm c}(\\delta _C)}{|{\\rm c}(\\delta _C)|}\\big ((t^{r-1-\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}}+t^{r+1-\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}}+\\cdots +t^{r+\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}-2})&+&\\\\ (t^{r-\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}}+t^{r+2-\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}}+\\cdots +t^{r+\\frac{{|{\\rm c}(\\delta _C)|-1}}{2}-3})\\big ) .$ The first summand corresponds to the orientation $\\delta _{C^{\\prime }}$ with $n2\\in {\\mathbb {E}}(\\delta _{C^{\\prime }})$ .",
"The second summand corresponds to the orientation $\\delta _{C^{\\prime }}$ with $2n\\in {\\mathbb {E}}(\\delta _{C^{\\prime }})$ .",
"By induction on $n$ , we have $\\begin{matrix}& S_{2n} & = & (-1)^{[\\frac{n-2}{2}]+1}(t + 1)^{2r-4}(1+a_n-ta_2)\\prod _{i=3}^{n-1}(1+a_i-t a_i) ,\\\\\\hbox{and} \\quad &S_{n2} & = & (-1)^{[\\frac{n-2}{2}]}(t +1)^{2r-4}(1+a_2-ta_n)\\prod _{i=3}^{n-1}(1+a_i-t a_i) .\\end{matrix}$ Since $-(1+a_n-ta_2)+(1+a_2-ta_n)=(1+t)(a_2-a_n)$ , the claim (i) holds for $n$ odd.",
"Case 2.",
"$n = 2r$ is an even number: We will first show that $T_{1n,12}=t(S_{n2}+S_{2n})/(1+t)^2$ .",
"Here $C^{\\prime }$ is an odd cycle on $n-1=2r-1$ vertices.",
"The coefficient for $\\delta _C$ equals $\\frac{t}{(1+t)^2}\\times \\frac{{\\rm c}(\\delta _C)}{|{\\rm c}(\\delta _C)|}\\big (t^{r-\\frac{|{\\rm c}(\\delta _C)|}{2}-1}+2t^{r-\\frac{|{\\rm c}(\\delta _C)|}{2}}+\\cdots +2t^{r+\\frac{|{\\rm c}(\\delta _C)|}{2}-2}+t^{r+\\frac{|{\\rm c}(\\delta _C)|}{2}-1}\\big ).$ This sum can be decomposed as $(t^{r-\\frac{|{\\rm c}(\\delta _C)|}{2}-1}+t^{r-\\frac{|{\\rm c}(\\delta _C)|}{2}}+\\cdots +t^{r+\\frac{|{\\rm c}(\\delta _C)|}{2}-1})\\\\+(t^{r-\\frac{|{\\rm c}(\\delta _C)|}{2}}+t^{r-\\frac{|{\\rm c}(\\delta _C)|}{2}+1}+\\cdots +t^{r+\\frac{|{\\rm c}(\\delta _C)|}{2}-2}),$ where the first summand corresponds to the orientation $\\delta _{C^{\\prime }}$ with $n2\\in {\\mathbb {E}}(\\delta _{C^{\\prime }})$ , and the second summand corresponds to the orientation $\\delta _{C^{\\prime }}$ with $2n\\in {\\mathbb {E}}(\\delta _{C^{\\prime }})$ .",
"Therefore $T_{1n,12}=\\frac{t(S_{n2}+S_{2n})}{(1+t)^2}$ .",
"By induction on $n$ , we have $\\begin{matrix}& S_{2n} & = & (-1)^{[\\frac{n-2}{2}]+1}(t + 1)^{2r-2}(1+a_n-ta_2)\\prod _{i=3}^{n-1}(1+a_i-t a_i) \\\\\\hbox{and} \\quad &S_{n2} & = & (-1)^{[\\frac{n-2}{2}]}(t +1)^{2r-2}(1+a_2-ta_n)\\prod _{i=3}^{n-1}(1+a_i-t a_i)\\end{matrix}$ Since $-(1+a_n-ta_2)+(1+a_2-ta_n)=(1+t)(a_2-a_n)$ , the result holds for even $n$ as well.",
"By a similar argument one can prove (ii).",
"Now applying (i) and (ii) and the equality $-(1+a_2-ta_2)(1+a_n-ta_1)(1+t)=(1+a_1-ta_2)(a_2-a_n)t+(1+a_2-ta_1)(t^2a_2-t-a_n-1),$ we obtain $T_{1n}\\,\\,= \\,\\,(-1)^{[\\frac{n-2}{2}]+1}(t +1)^{2r-2}(1+a_n-ta_1)\\prod _{i=2}^{n-1}(1+a_i-t a_i)\\ .$ The identity for $T_{n1}$ is analogous.",
"It follows that $P^C \\in \\mathcal {I}_G$ for all cycles of $G$ .",
"It remains to be shown that the $P^C$ generate the homogeneous ideal $\\mathcal {I}_G$ .",
"Recall that, by Lemma REF , the images of the $P^C$ generate this ideal after we tensor, over the local ring $\\mathbb {K}$ , with the residue field $\\mathbb {Q} = \\mathbb {K}/\\langle t \\rangle $ .",
"Hence, by Nakayama's Lemma, the $P^C$ generate $\\mathcal {I}_G$ .",
"Remark 3.5 In Theorem REF we can replace the local ring $\\mathbb {K}= {\\mathbb {Q}}[[t]]$ with the polynomial ring ${\\mathbb {Q}}[t]$ because no $t$ appears in the leading forms $(P^C)|_{t=0}$ .",
"This ensures that ${\\mathbb {Q}}[t][p_{ij}]$ modulo the ideal $\\langle P^C: C\\, \\hbox{cycle in}\\, G \\rangle $ is torsion-free, hence free, and therefore flat over ${\\mathbb {Q}}[t]$ .",
"In statistical applications, the quantity $t$ will always take on a particular real value.",
"In the remainder of this paper, we assume $t \\in {\\mathbb {R}}$ , and we identify $\\mathcal {I}_G$ with its image in ${\\mathbb {R}}[p_{ij}]$ .",
"Corollary 3.6 For any $t \\in {\\mathbb {R}}$ , the cycle polynomials $P^C$ generate the ideal $\\,\\mathcal {I}_G$ in ${\\mathbb {R}}[p_{ij}]$ .",
"Theorem REF furnishes a (flat) degeneration from $\\mathcal {I}_G$ to the toric ideal $\\mathcal {T}_G$ .",
"Geometrically, we view this as a degeneration of varieties (or semialgebraic sets) from $t > 0$ to $t = 0$ .",
"Lemma REF concerns further degenerations from the toric ideal $\\mathcal {T}_G$ to its initial monomial ideals $\\mathcal {M}_G$ .",
"Any such $\\mathcal {M}_G$ is squarefree and serves as a combinatorial model for both $\\mathcal {T}_G$ and $\\mathcal {I}_G$ .",
"We describe one particular choice and draw some combinatorial conclusions.",
"Fix a term order on ${\\mathbb {R}}[p_{ij}]$ with the property that $p_{ij} \\succ p_{k\\ell }$ whenever $i < k$ , or $i=k$ and $j<\\ell $ .",
"For any cycle $C$ , we label the two directed orientations $o_C$ and $\\bar{o}_C$ so that $p^{o(C)} \\succ p^{\\bar{o}(C)} $ .",
"Fix a spanning tree $T$ of $G$ .",
"Let $\\mathfrak {P}_T$ denote the monomial prime ideal generated by all unknowns $p_{ij}$ where $\\lbrace i,j\\rbrace \\in E(G)\\backslash E(T)$ and $p_{ij}$ divides $p^{o_C}$ , where $C$ is the unique cycle in $E(T) \\cup \\lbrace \\lbrace i,j\\rbrace \\rbrace $ .",
"The squarefree monomial ideal $\\mathcal {M}_G \\,\\, =\\,\\,{\\rm in}_\\succ (\\mathcal {T}_G) \\,\\, = \\,\\,\\bigl \\langle \\,p^{o_C} \\,: \\,C \\,\\,\\hbox{cycle in} \\,\\, G \\,\\bigr \\rangle \\,\\, = \\,\\,\\bigcap _T \\mathfrak {P}_T\\ ,$ is obtained by taking the intersection over all spanning trees $T$ of $G$ .",
"The simplicial complex with Stanley-Reisner ideal $\\mathcal {M}_G$ is a regular triangulation of the Lawrence polytope of the graph $G$ .",
"This triangulation is shellable and hence our ideals are Cohen-Macaulay.",
"We record the following fact.",
"Proposition 3.7 The ideals $\\,\\mathcal {M}_G, \\,\\mathcal {T}_G\\,$ and $\\,\\mathcal {I}_G\\,$ define varieties of dimension $|E(G)|+I-1$ in affine space, and their common degree is the number of spanning trees of the graph $G$ .",
"Each of the components $\\mathfrak {P}_T$ in (REF ) has codimension $|E(G)\\backslash E(T)| = |E(G)| {-} I {+} 1$ .",
"Figure: A graph GG on I=4I=4 nodes and its eight spanning trees TTExample 3.8 Consider the graph $G$ depicted in Figure REF .",
"The associated toric ideal equals $\\mathcal {T}_G \\,\\,\\,= \\,\\,\\, \\langle \\,\\underline{p_{12}p_{23}p_{31}}-p_{21}p_{32}p_{13} \\,,\\ \\underline{p_{12}p_{24}p_{41}}-p_{21}p_{42}p_{14} \\,,\\ \\underline{p_{13}p_{32} p_{24}p_{41}}-p_{31}p_{23} p_{42}p_{14}\\,\\rangle .$ This has codimension 2 and degree 8.",
"Its (underlined) initial monomial ideal $\\mathcal {M}_G$ equals $\\langle p_{12},p_{13} \\rangle \\,\\cap \\,\\langle p_{12},p_{32} \\rangle \\,\\cap \\,\\langle p_{12},p_{24} \\rangle \\,\\cap \\,\\langle p_{12},p_{41} \\rangle \\,\\cap \\,\\langle p_{23},p_{41} \\rangle \\,\\cap \\,\\langle p_{23},p_{24} \\rangle \\,\\cap \\,\\langle p_{24},p_{31} \\rangle \\,\\cap \\,\\langle p_{31},p_{41} \\rangle .$ These eight monomial prime ideals correspond to the eight spanning trees in Figure REF .",
"The ideal $\\mathcal {I}_G$ has three generators, two cubics with 8 terms and one quartic with 16 terms, as in (REF ).",
"These are obtained from the Markov basis of $\\mathcal {T}_G$ by adding additional terms that are divisible by $t$ .",
"$\\diamondsuit $"
],
[
"Maximum Likelihood Estimation",
"A data table ${\\bf n} = (n_{ij})$ of format $I \\times I$ can arise either by multinomial sampling or by sampling from $I^2$ independent Poisson distributions, one for each of its cells.",
"In both cases, the log-likelihood function, up to an additive constant, is equal to $\\ell _{\\bf n}({\\bf p}) \\quad = \\quad \\sum _{i=1}^I \\sum _{j=1}^I n_{ij} \\cdot {\\rm log}(p_{ij}).$ Maximum likelihood estimation (MLE) is the problem of maximizing $\\ell _{\\bf n}$ over all probability tables ${\\bf p} = (p_{ij})$ in the model of interest.",
"For us, that model is the quasisymmetry model $(QS_t)$ , where $t$ is a fixed constant in the interval $[0,1]$ .",
"This optimization problem can be expressed in either constrained form or in unconstrained form.",
"The constrained MLE problem is written as $\\hbox{Maximize} \\,\\, \\,\\ell _{\\bf n}({\\bf p}) \\quad \\hbox{subject to} \\quad {\\bf p} \\,\\in \\, V(\\mathcal {I}_{G} ) \\cap \\Delta _{I^2-1},$ where $G = K_I$ is the complete graph on $I$ nodes, and $V(\\mathcal {I}_G)$ is the zero set of the cycle polynomials $P^C$ constructed in Section 3.",
"The unconstrained MLE problem is written as $\\hbox{Maximize} \\,\\, \\,\\ell _{\\bf n}({\\bf a} , {\\bf s}).$ The decision variables in (REF ) are the vector ${\\bf a} = (a_1,\\ldots ,a_I)$ and the symmetric probability matrix ${\\bf s} = (s_{ij})$ .",
"The objective function in (REF ) is obtained by substituting (REF ) into (REF ).",
"We shall discuss both formulations, starting with a simple numerical example for the formulation (REF ).",
"Example 4.1 Let $I = 3, \\,t = 2/3$ and consider the data table $ \\qquad \\qquad {\\bf n} \\,= \\begin{bmatrix}2 & 3 & 5 \\\\11 & 13 & 17 \\\\19 & 23 & 29 \\end{bmatrix} \\qquad \\hbox{ with sample size $\\,n_{++} = 122$}.$ Our aim is to maximize $\\ell _{\\bf n}({\\bf p})$ subject to the cubic equation (REF ) and $p_{11} + p_{12} + \\cdots + p_{33} = 1$ .",
"Using Lagrange multipliers for these two constraints, we derive the likelihood equations by way of [5].",
"These polynomial equations in the nine unknowns $p_{ij}$ have 15 complex solutions.",
"Two of the complex solutions are non-real.",
"Of the 13 real solutions, 12 have at least one negative coordinate.",
"Only one solution lies in the probability simplex $\\Delta _8$ : $\\begin{matrix}\\hat{p}_{11} &=& 1/61, & \\qquad \\hat{p}_{12} &=& 0.0286294, & \\qquad \\hat{p}_{13} &=& 0.0376289, \\\\\\hat{p}_{21} &=& 0.0861247, & \\qquad \\hat{p}_{22} &=& 13/122, & \\qquad \\hat{p}_{23} &=& 0.1446119, \\\\\\hat{p}_{31} &=& 0.1590924, & \\qquad \\hat{p}_{32} &=& 0.1832569, & \\qquad \\hat{p}_{33} &=& 29/122.\\end{matrix}$ This is the global maximum of the constrained MLE problem for this instance.",
"$\\diamondsuit $ The benefit of the constrained formulation is that we can take advantage of the combinatorial results in Section 3, and we do not have to deal with issues of identifiability and singularities arising from the map (REF ).",
"On the other hand, most statisticians would prefer the unconstrained formulation because this corresponds more directly to the fitting of model parameters to data.",
"To solve the unconstrained MLE problem (REF ), we take the partial derivations of the objective function $\\ell _n({\\bf a},{\\bf s})$ with respect to all model parameters $a_i$ and $s_{ij}$ .",
"The resulting system of equations decouples into a system for ${\\bf a}$ and a system for ${\\bf s}$ .",
"The latter is trivial to solve.",
"Using the requirement that the entries of ${\\bf s}$ sum to 1, it has the closed form solution $\\hat{s}_{ij} \\,\\,= \\,\\,\\frac{n_{ij}+n_{ji}}{2n_{++}} , \\quad i,j=1, \\ldots , I .$ After dividing by $1+t$ , the partial derivatives of $\\ell _{\\bf n}({\\bf a},{\\bf s})$ with respect to $a_1,a_2,\\ldots ,a_I$ are $\\sum _{j =1 \\atop j \\ne i }^I \\frac{(1+a_j-ta_j)[n_{ij}(1+a_j-ta_i)-n_{ji}(1+a_i-ta_j)]}{(1+a_i-t a_j)(1+a_j-ta_i)[2+(1-t)(a_i+a_j)]}\\qquad \\hbox{for} \\,\\, i = 1,2,\\ldots ,I.$ This system of equations has infinitely many solutions, because the model ${\\rm QS}_t$ is not identifiable.",
"The general fiber of the map (REF ) is a line in ${\\bf a}$ -space.",
"Hence only $I-1$ of the $I$ parameters $a_i$ can be estimated.",
"One way to fix this is to simply add the constraint $\\hat{a}_I = 0$ .",
"Example 4.2 Let us return to the numerical instance in Example REF .",
"Here we have $\\hat{s}_{11} = 1/61, \\,\\,\\hat{s}_{12} = 7/122, \\,\\,\\hat{s}_{13} = 6/61, \\,\\,\\hat{s}_{22} = 13/122, \\,\\,\\hat{s}_{23} = 10/61, \\,\\,\\hat{s}_{33} = 29/122.$ The equations (REF ) can be solved in a computer algebra system by clearing denominators and then saturating the ideal of numerators with respect to those denominators.",
"As before, there are precisely 15 complex solutions, of which 13 are real.",
"The MLE is given by $\\hat{a}_1 = -0.65948848999731861332,\\,\\,\\hat{a}_2 = -0.13818331109451658084,\\,\\,\\hat{a}_3 = 0.$ These are floating point approximations to algebraic numbers of degree 15 over $\\mathbb {Q}$ .",
"An exact representation is given by their minimal polynomials.",
"For the first coordinate, this is $ \\begin{small}\\begin{matrix}62031304 a_1^{15}+2201861910 a_1^{14}+30829909776 a_1^{13}+206135547000 a_1^{12}+528436383696 a_1^{11} \\\\-1126661553720 a_1^{10}-9740892273264 a_1^9-4305524252579 a_1^8+26533957305582 a_1^7 \\\\ +88281552626154 a_1^6+44254830057030 a_1^5-76332701171853 a_1^4 -83490498412056 a_1^3 \\\\ +1857597611688 a_1^2+29825005557312 a_1+9354112703280 \\qquad = \\,\\, 0.\\end{matrix}\\end{small}$ With this, the second coordinate $\\hat{a}_2$ is a certain rational expression in $\\mathbb {Q}(\\hat{a}_1)$ .",
"By plugging (REF ) and (REF ) into (REF ) with $t = 2/3$ , we recover the estimated probability table in (REF ).",
"$\\diamondsuit $ For larger cases, solutions to the likelihood equations (REF ) are computed by iterative numerical methods, such as the unidimensional Newton's method.",
"The updating equations at the $q$ -th step of this iterative method are $a_i^{(q)} \\,=\\,a_i^{(q-1)} -\\frac{\\partial \\ell _{\\bf n}({\\bf a})/\\partial a_i}{\\partial ^2 \\ell _{\\bf n}({\\bf a})/\\partial a_i^2}\\bigl | {_{{\\bf a}={\\bf a}^{(q-1)}}}\\quad \\hbox{for} \\quad i=1, \\ldots , I-1 , \\ q=1,2,\\ldots \\ .$ We find it convenient to rewrite the first derivatives (REF ) as $\\frac{\\partial \\ell _{\\bf n}({\\bf a})}{\\partial a_i} = (1+t)\\sum _{j=1}^I { \\frac{s_{ij}}{2+(1-t)(a_i+a_j)}\\left(1-\\frac{1-t}{1+t}c_{ij}\\right)\\left(\\frac{n_{ij}}{p_{ij}}-\\frac{n_{ji}}{p_{ji}}\\right) } .$ The second derivative equals $\\frac{\\partial ^2 \\ell _{\\bf n}({\\bf a})}{\\partial a_i^2} &=& -(1+t)\\sum _{j=1}^I { \\frac{2(1-t)s_{ij}}{[2+(1-t)(a_i+a_j)]^2}\\left(1-\\frac{1-t}{1+t}c_{ij}\\right)\\left(\\frac{n_{ij}}{p_{ij}}-\\frac{n_{ji}}{p_{ji}}\\right) }\\\\&& -(1+t)\\sum _{j\\ne i} { \\frac{(1+t)s_{ij}^2}{[2+(1-t)(a_i+a_j)]^2}\\left(1-\\frac{1-t}{1+t}c_{ij}\\right)^2\\left(\\frac{n_{ij}}{p_{ij}^2}+\\frac{n_{ji}}{p_{ji}^2}\\right) } .\\nonumber $ Here $i=1, \\ldots , I-1$ , the $p_{ij}$ are the expressions in (REF ), and $c_{ij}=\\frac{(1+t)(a_i-a_j)}{2+(1-t)(a_i+a_j)} \\ .$ We believe that the numerical solution found by this iteration is always the global maximum in (REF ).",
"This would be implied by the following conjecture, which holds for $t=0$ and $t=1$ .",
"Conjecture 4.3 The Hessian $\\,{\\bf H}({\\bf a}) = \\left(\\frac{\\partial ^2 \\ell _{\\bf n}({\\bf a})}{\\partial a_i \\partial a_j}\\right)$ is negative definite for all ${\\bf a} \\in \\mathbb {R}^I$ with (REF ).",
"We verified this conjecture for many examples with $t \\in (0,1)$ .",
"In each case, we also ran our iterative algorithm for many starting values, and it always converged to the same solution.",
"The diagonal entries of the Hessian matrix are given in (REF ), while the non-diagonal are $\\frac{\\partial ^2 \\ell _{\\bf n}({\\bf a})}{\\partial a_i \\partial a_j} &=& \\frac{2(1-t)^2 s_{ij} c_{ij}}{[2+(1-t)(a_i+a_j)]^2}\\left(\\frac{n_{ij}}{p_{ij}}-\\frac{n_{ji}}{p_{ji}}\\right) \\\\&& + \\ \\frac{(1+t)^2 s_{ij}^2}{[2+(1-t)(a_i+a_j)]^2} \\left[1-\\left(\\frac{1-t}{1+t} c_{ij}\\right)^2\\right]\\left(\\frac{n_{ij}}{p_{ij}^2}+\\frac{n_{ji}}{p_{ji}^2}\\right) .",
"\\nonumber $ In the iterative algorithm described above, we had fixed the last parameter $a_I$ at zero.",
"This ensures identifiability, and it is done for simplicity.",
"The constraint $a_I = 0$ defines a reference point for the other parameters $a_1,\\ldots ,a_{I-1}$ .",
"Under this constraint, (REF ) leads to $a_i = \\frac{1}{1+t}\\left(\\frac{p_{i+}-p_{+i}}{x_{i+}} - \\frac{p_{I+}-p_{+I}}{x_{I+}}\\right) \\quad \\hbox{for} \\quad i=1, \\ldots , I-1 .$ This means that the contribution of category $i$ to marginal inhomogeneity is compared to the last category's contribution.",
"Hence, in view of (REF ), a reasonable alternative constraint could be $\\sum _{j=1}^I \\frac{x_{ij}}{x_{i+}}a_j = 0$ .",
"This constraint calibrates each category's contribution to marginal inhomogeneity relative to the weighted average of all $I$ categories.",
"Remark 4.4 The iterative procedure described above for fitting the ${\\rm QS_t}$ models was implemented by us in R. The algorithm works regardless of whether we impose the restriction $a_I=0$ or not.",
"We noticed that when imposing this constraint, the algorithm requires more iterations to converge.",
"The convergence is also affected by the initial values ${\\bf a}^{(0)}$ we used.",
"A classical choice would be $a_i=0$ for all $i$ , as this corresponds to complete symmetry.",
"However, we observed that for ${\\bf a}^{(0)}$ with coordinates $\\frac{n_{i+}-n_{+i}}{n_{i+}+n_{+i}}$ , $i=1,\\ldots , I$ , the convergence is faster.",
"Remark 4.5 Here we consider the model parameter $t$ as fixed.",
"Alternatively, it could be estimated from the data, as for the power-divergence logistic regression model in [6]."
],
[
"Quasisymmetric Independence",
"A natural submodel of (REF ) is the symmetric independence model (SI), which is given by $p_{ij}=s_i s_j \\ , \\quad i,j=1, \\ldots , I .$ The $I$ parameters $s_i$ are non-negative and sum to 1.",
"The corresponding probability tables ${\\bf p} = (p_{ij})$ are symmetric and have rank 1.",
"The models of quasisymmetric independence (${\\rm QSI}_t$ ) can be defined analogously to the ${\\rm QS}_t$ models, by measuring departure from (REF ).",
"Namely, replacing the symmetric probabilities $s_{ij}$ in (REF ) by the factored form in (REF ), we get $p_{ij}=s_i s_j\\left(1+\\frac{(1+t)(a_i-a_j)}{2+(1-t)(a_i+a_j)}\\right) \\ , \\ \\ \\ i\\ne j, \\ \\ \\ i,j=1, \\ldots , I .$ The MLEs of the parameters of the SI model in (REF ) are $\\hat{s}_i =\\frac{n_{i+}+n_{+i}}{2n} \\quad \\hbox{for} \\quad i=1, \\ldots , I.$ These are also the MLEs of the $s_i$ parameters in the ${\\rm QSI}_t$ model.",
"The likelihood equations for ${\\bf a}$ are as before, but with $p_{ij}$ 's in (REF ) as defined in (REF ) and (REF ).",
"Their numerical solution can be computed with the iterative procedure described in Section 4, adjusted accordingly.",
"Remark 5.1 In Proposition REF , if we replace the models S and ${\\rm QS}_t$ by SI and ${\\rm QSI}_t$ , then an analogous statement holds.",
"Thus, we have $\\text{SI}=\\text{MH} \\cap {\\rm QSI}_t$ for each $t\\in [0, 1]$ .",
"Following the discussion in Section 3, it would be interesting to derive the implicit equations for the model ${\\rm QSI}_t$ .",
"At present, we have a complete solution only for the special case $t=1$ .",
"The quasisymmetric independence model ${\\rm QSI}_1$ is defined by the parametrization $p_{ij} \\, = \\, s_i s_j \\cdot (1 + a_i - a_j) , \\qquad 1 \\le i,j \\le I.$ Alternatively, $\\lbrace i,j\\rbrace $ could range over the edges of a graph $G$ , as in Section 3.",
"In the following result, whose proof we omit, we restrict ourselves to the case of the complete graph $K_I$ .",
"Proposition 5.2 The prime ideal of the ${\\rm QSI}_1$ model in (REF ) is generated by the following homogeneous quadratic polynomials (for any choices of indices $i,j,k,\\ell $ among $1,\\ldots ,I$ ): $(p_{ij}+p_{ji})^2-4p_{ii}p_{jj}$ , $p_{kk}(p_{ij}-p_{ji})+p_{ki}p_{jk}-p_{ik}p_{kj}$ , $(p_{ij}-p_{ji})(p_{jk}-p_{kj})+4(p_{jj}p_{ki}-p_{ji}p_{kj})$ , $p_{\\ell i}(p_{jk}-p_{kj})+p_{\\ell j}(p_{ki}-p_{ik})+ p_{\\ell k}(p_{ij}-p_{ji})$ , $p_{i\\ell }(p_{jk}-p_{kj})+p_{j\\ell }(p_{ki}-p_{ik})+ p_{k\\ell }(p_{ij}-p_{ji})$ .",
"The general case where $t < 1$ differs from the $t=1$ case in that the prime ideal of $ {\\rm QSI}_1$ is no longer generated by quadrics.",
"Even for $I=3$ , a minimal generator of degree 3 is needed: Example 5.3 Fix $I= 3$ .",
"For general $t \\in \\mathbb {R}$ , we consider the model (REF ) with $p_{ii} = s_i s_i$ for $i=1,2,3$ .",
"Its ideal is minimally generated by 7 polynomials: 6 quadrics and one cubic.",
"$\\diamondsuit $"
],
[
"Fitting the Models to Data",
"We next illustrate the new models and their features on some characteristic data sets.",
"The goodness-of-fit of a model is tested asymptotically by the likelihood ratio statistic.",
"The associated degrees of freedom for ${\\rm QS}_t$ and ${\\rm QSI}_t$ are $\\,df({\\rm QS}_t)=(I-1)(I-2)/2\\,$ and $\\,df({\\rm QSI}_t)=(I-1)^2$ , respectively.",
"As we shell see, the models in each family can perform either quite similar or differ significantly, depending on the specific data under consideration.",
"A case of similar behavior is the classical vision example of Table REF .",
"The model of QS ($t=0$ ) has been applied on this data often in the literature, while [7] applied Pearsonian QS.",
"Both models provide a quite similar fit, namely ($G^2=7.27076$ , $p$ -value $=0.06375$ ) for ${\\rm QS}_0$ and ($G^2=7.26199$ , $p$ -value $=0.06400$ ) for ${\\rm QS}_1$ .",
"Here, $df=3$ .",
"The behavior of the ${\\rm QS}_t$ models for $t \\in (0, 1)$ is similar.",
"The log-likelihood values vary from $-16388.11444$ ($t=0$ ) to $-16388.11006$ ($t=1$ ) while the saturated log-likelihood is $-16384.47906$ (see Figure REF , left).",
"Table REF gives the MLEs of the expected cell frequencies under the models $QS_0$ , $QS_1$ and $QS_{2/3}$ .",
"For $t=2/3$ we get $G^2=7.26234$ , with $p$ -value $=0.06399$ .",
"Figure: Log-likelihood values of QS t {\\rm QS}_t for tt in [0,1][0,1] for data in Tables (left) and (c) (right).Table: Unaided distancevision of right and left eyes for 7477 women.Parenthesized values are ML estimates of the expected frequencies under models (aa) QS 0 QS_0, (b) QS 2/3 QS_{2/3}, and (cc) QS 1 QS_1.Examples for which the members of the ${\\rm QS}_t$ family are not of similar performance are the two $3\\times 3$ tables of [7], displayed in Table REF (a) and (b).",
"Table: Simulated 3×33\\times 3 examples of , generated by the models (a) QS 0 {\\rm QS}_0 and (b) QS 1 {\\rm QS}_1 (their Tables 3 and 4, respectively).A toy example in (c).Here, the models ${\\rm QS}_0$ and ${\\rm QS}_1$ differ considerably in their fit.",
"In particular, the data in Table REF (a) are modeled well by ${\\rm QS}_0$ but not by ${\\rm QS}_1$ ($G^2_0=0.18572$ and $G^2_1=5.29006$ ), while the opposite holds for Table REF (b), since $G^2_0=6.29035$ and $G^2_1=0.29215$ .",
"In such situations, the question arises whether some $t$ is appropriate for both data sets.",
"Finding $t$ such that ${\\rm QS}_t$ works for two or more $I\\times I$ tables of the same set-up is of special interest in the study of stratified tables.",
"Using the same model on all strata makes parameter estimates among models comparable.",
"This is a major advantage of the proposed family.",
"Models that lie `in-between' the two extreme cases ($t=0$ and $t=1$ ) may lead to a consensus.",
"Even if that consensus model does not perform as well as ${\\rm QS}_0$ and ${\\rm QS}_1$ on each table separately, it can provide a reasonable fit for both tables.",
"To visualize this, Figure REF (left) shows the $p$ -values of the fit of the ${\\rm QS}_t$ models with $t\\in [0, 1]$ , for Tables REF (a) and (b), by solid and dashed curves, respectively, along with the significance level of $\\alpha =0.05$ .",
"The consensus model ${\\rm QS}_t$ would have $t\\in (0.061, 0.302)$ .",
"Among these models, we propose ${\\rm QS}_{0.14}$ , since the intersection of the two curves happens around $t=0.137$ .",
"The fit of this model for Table REF (a) is $G^2=2.27614$ ($p$ -value=0.1314) while for (b) it is $G^2=2.16744$ ($p$ -value=0.1409).",
"The vector of MLEs for parameters $a_i$ is $(-0.5458, 1.8555, 0)$ and $(2.1247, -0.5406, 0)$ , respectively.",
"We note that, in deriving the consensus model, the $G^2$ values could have been used as an alternative to the $p$ -values in Figure REF .",
"Figure: pp-values for the G 2 G^2 goodness-of-fit test of QS t {\\rm QS}_t (left) and QSI t {\\rm QSI}_t (right) for t∈[0,1]t \\in [0,1], along with the significance level α=0.05\\alpha =0.05.Data are from Table : (a) solid and (b) dashed.Figure: Log-likelihood values of QS t {\\rm QS}_t (upper) and QSI t {\\rm QSI}_t (lower) with t∈[0,1]t \\in [0,1] forthe data in Table (a, left) and (b, right).",
"The straight line marks the saturated log-likelihood value.In all examples treated so far, the log-likelihood under ${\\rm QS}_t$ was monotone in $t$ (see Figure REF , left, and Figure REF , upper), suggesting that the `best' model will be achieved at either $t=0$ or $t=1$ .",
"This is not always the case.",
"For example, for the data in Table REF (c), the best fit occurs for $t=0.036$ (see also Figure REF , right), giving $G^2=1.742943\\cdot 10^{-6}$ ($p$ -value=0.9989) while for $t=0$ and $t=1$ , it is $G^2=0.0610$ ($p$ -value= 0.8049) and $G^2=1.1131$ ($p$ -value=0.2914), respectively.",
"Furthermore, even when the best model is for $t=0$ or $t=1$ , we may still want to use some $t\\in (0, 1)$ , e.g.",
"for stratified tables with different optimal model at each level of the stratifying variable, as explained above.",
"Applying the quasisymmetric independence models to Tables REF (a) and (b), we observe that $QSI_0$ fits well on Table REF (a) but not on (b), while model $QS_1$ is of acceptable fit for both data sets.",
"Indeed, we have $G_a^2(QSI_0)=1.3600$ ($p$ -value=0.8511), $G_b^2(QSI_0)=11.8622$ ($p$ -value=0.0184), $G_a^2(QSI_1)=6.4643$ ($p$ -value=0.1671) and $G_b^2(QSI_1)=5.8640$ ($p$ -value=0.2095).",
"For the performance of the ${\\rm QSI}_t$ model for $t \\in [0, 1]$ , see Figure REF (right) and Figure REF (lower).",
"For $t=0.532$ , the $p$ -value of the fit of the model is equal to 0.1983 for both data sets.",
"All the examples of this section were worked out with R functions we developed for fitting the ${\\rm QS}_t$ and ${\\rm QSI}_t$ models via the unidimensional Newton's method.",
"The adopted inferential approach is asymptotic.",
"In cases of small sample size, exact inference can be carried out via algebraic computations along the lines described in Section 3, and demonstrated in Examples REF and REF ."
],
[
"Divergence Measures",
"The one-parameter family of QS models we proposed, ${\\rm QS}_t$ , $t \\in [0, 1]$ , connects the classical QS model ($t=0$ ) and the Pearsonian QS model ($t=1$ ).",
"These two belong both to a broader class of generalized QS models that are derived using the concept of $\\phi $ –divergence [7], [10].",
"Measures of divergence quantify the distance between two probability distributions and play an important role in information theory and statistical inference.",
"A well known divergence measure is the Kullback-Leibler (KL) divergence.",
"However there exist broader classes of divergences.",
"Such a class, including the KL as a special case, is the $\\phi $ -divergence.",
"In the framework of two-dimensional contingency tables, this class is defined as follows.",
"Let ${\\mathbf {p}}=(p_{ij})$ and ${\\mathbf {q}}=(q_{ij})$ be two discrete bivariate probability distributions.",
"The $\\phi $ –divergence between ${\\mathbf {p}}$ and ${\\mathbf {q}}$ (or Csiszar's measure of information in ${\\mathbf {q}}$ about ${\\mathbf {p}}$ ) is defined by $D_{\\phi }({\\mathbf {p}}, {\\mathbf {q}})\\,\\,=\\,\\,\\sum _{i,j}{q_{ij}\\phi (p_{ij}/q_{ij})} .$ Here $\\phi : [0, \\infty ) \\rightarrow \\mathbb {R}^+$ is a convex function such that $\\,\\phi (1)=\\phi ^{\\prime }(1)=0$ , $\\,0 \\cdot \\phi (0/0)=0$ , and $\\,0 \\cdot \\phi (x/0)= x \\cdot \\lim _{u\\rightarrow \\infty } \\phi (u)/u$ .",
"For $\\phi (u)=u\\log (u)-u+1$ and $\\phi (u)=(u-1)^2/2$ , the divergence (REF ) becomes the KL and the Pearson's divergence, respectively.",
"We adopt the notation in [10].",
"For properties of $\\phi $ -divergence, as well as a list of well-known divergences belonging to this family, we refer to [10].",
"The differential geometric structure of the Riemannian metric induced by such a divergence function is studied by [1].",
"The generalized QS models introduced by [7] are based on the $\\phi $ -divergence and are characterized by the fact that each model in this class is the closest model to symmetry S, when the distance is measured by the corresponding divergence measure.",
"The classical QS model corresponds to the KL divergence, while the Pearsonian QS corresponds to Pearson's distance.",
"We shall prove in Theorem REF that the other members of the ${\\rm QS}_t$ family, i.e.",
"for $t \\in (0,1)$ , are $\\phi $ -divergence QS models as well, and we identify the corresponding $\\phi $ function.",
"Theorem 7.1 Fix $t \\in (0, 1)$ and consider the class of models that preserve the given row (or column) marginals $p_{i+}$ (or $p_{+i}$ ) for $i=1,\\ldots , I$ , and also preserve the given sums $p_{ij}+p_{ji}=2s_{ij}$ for $i, j=1,\\ldots , I$ .",
"In this class, the ${\\rm QS}_t$ model (REF ) is the closest model to the complete symmetry model S in (REF ), where `closest' refers the $\\phi $ -divergence defined by $\\begin{matrix}& \\phi (u) & = & f_t(u)-f_t(1)-f_t^{\\prime }(1)(u-1) ,\\smallskip \\\\\\hbox{where} & f_t(u)& =&(u+\\frac{2t}{1-t}) \\log (u+\\frac{2t}{1-t}).\\quad \\end{matrix}$ We set $F_t(u)=\\phi ^{\\prime }(u)=\\log (u+\\frac{2t}{1-t})-\\ell _t$ , where $\\ell _t=\\log (1+\\frac{2t}{1-t})$ is just a constant for given $t$ .",
"This choice of constant ensures $\\phi ^{\\prime }(1)=0$ .",
"Then the inverse function to $F_t$ is $F_t^{-1}(x)=(\\frac{-2t}{1-t})+e^{x+\\ell _t} .",
"\\vspace{-11.38092pt}$ With this, we can write $p_{ij}&=&s_{ij}F_t^{-1}(\\alpha _i+\\gamma _{ij})\\,=\\,s_{ij}(\\frac{-2t}{1-t}+e^{\\alpha _i+\\gamma _{ij}+\\ell _t})\\,=\\,s_{ij}\\big (\\frac{-2t}{1-t}+ \\frac{\\beta _i(\\frac{2(1+t)}{1-t})}{\\beta _i+\\beta _j}\\big ) \\ ,$ where $\\beta _i\\,=\\,e^{\\alpha _i+\\ell _t}\\quad {\\rm and }\\quad e^{\\gamma _{ij}}\\,=\\, \\frac{\\frac{2(1+t)}{1-t}}{e^{\\alpha _i+\\ell _t}+e^{\\alpha _j+\\ell _t}}\\ .$ We next rewrite $p_{ij}$ as $p_{ij}\\,\\,=\\,\\,s_{ij}\\big (1+\\frac{-(1+t)}{1-t}+ \\frac{\\beta _i(\\frac{2(1+t)}{1-t})}{\\beta _i+\\beta _j}\\big )=s_{ij}\\big (1+ \\frac{\\frac{(1+t)}{1-t}(\\beta _i-\\beta _j)}{\\beta _i+\\beta _j}\\big )\\ .$ Setting $\\beta _i=1+(1-t)a_i$ and $\\beta _j=1+(1-t)a_j$ , this translates into our parametrization (REF ).",
"Now the result follows from [7].",
"For a probability table ${\\bf s}$ with symmetry S, the quantity $D_{\\phi }({\\bf p},{\\bf s})$ is minimized when ${\\bf p}$ is the probability table satisfying ${\\rm QS}_t$ .",
"The fact that the ${\\rm QS}_t$ models are $\\phi $ -divergence QS models implies that they share all the desirable properties of the $\\phi $ -divergence QS models [7].",
"This includes the properties that highlight the physical interpretation issues of these models.",
"As far as we know, the $\\phi $ -divergence for the parametric $\\phi _t$ function (REF ) has not been considered so far.",
"Its study can be the subject of further research.",
"Such a future project has the potential to build a bridge between information geometry [1] and algebraic statistics [5].",
"Acknowledgements.",
"Fatemeh Mohammadi was supported by the Alexander von Humboldt Foundation.",
"Bernd Sturmfels was supported by the NSF (DMS-0968882) and DARPA (HR0011-12-1-0011).",
"Authors' addresses: Maria Kateri, Institute of Statistics, RWTH Aachen University, 52056 Aachen, Germany, [email protected] Fatemeh Mohammadi, Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany, [email protected] Bernd Sturmfels, University of California, Berkeley, CA 94720, USA, [email protected]"
]
] | 1403.0547 |
[
[
"Unification of bosonic and fermionic theories of spin liquids on the\n kagome lattice"
],
[
"Abstract Recent numerical studies have provided strong evidence for a gapped $Z_2$ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model.",
"A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases.",
"The symmetry fractionalization pattern for the kagome spin liquid remains to be determined.",
"A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either bose (Schwinger bosons) or fermi (Abrikosov fermions) statstics, which are then treated within the mean-field theory.",
"A longstanding question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways.",
"Here we show that all 8 $Z_2$ spin liquid phases in Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct.",
"The key idea is that for $Z_2$ spin liquid states that admit a SBMF description on kagome lattice, the symmetry fractionalization of visions is uniquely fixed.",
"Two promising candidate states for kagome Heisenberg model, Sachdev's $Q_{1}=Q_{2}$ SBMF state and Lu-Ran-Lee's $Z_2[0,\\pi]\\beta$ Abrikosov fermion state, are found to describe the same symmetric spin liquid phase.",
"We expect these results to aid in a complete specification of the numerically observed spin liquid phase.",
"We also discuss a set of $Z_2$ spin liquid phases in fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states, that do not admit a SBMF description."
],
[
"Introduction",
"$Z_2$ spin liquids (SLs) are a class of disordered many-spin states which have a finite energy gap for all bulk excitations.",
"They differ fundamentally from symmetry breaking groundstates such as magnetically ordered phases and valence bond solids, since, in their simplest form, they preserve all the symmetries including spin rotation, time reversal and crystal symmetries.",
"More importantly they possess bulk quasiparticles carrying fractional statistics[1].",
"For example in the most common $Z_2$ SL , there are three distinct types of fractionalized bulk excitations[2], [3], [4], [5], [6], [7]: bosonic spinon $b$ with half-integer spin, fermionic spinon $f$ with half-integer spin and bosonic vison $v$ (a vortex excitation of $Z_2$ gauge theory) with integer spin.",
"They all obey mutual semion statistics[7]: i.e.",
"a bosonic spinon acquires a $-1$ Berry phase when it adiabatically encircles a fermionic spinon or a vison.",
"These statistical properties are identical to those of excitations in $Z_2$ gauge theory[8], hence the name “$Z_2$ spin liquid”.",
"Recently, interest in $Z_2$ SLs has been recharged by numerical studies on the spin-$1/2$ Heisenberg model on kagome[9], [10], [11], [12] lattice, where this state is strongly indicated.",
"In particular a topological entanglement entropy[13], [14] of $\\gamma =\\log 2$ is observed in the ground state.",
"Just like local order parameters are used to describe symmetry breaking phases[15], here fractional statistics and topological entanglement entropy serve as fingerprints of topological order[16] in $Z_2$ spin liquids.",
"Analogous results have been reported for the frustrated square lattice, although the correlation lengths in that case are not as small as in the kagome lattice[17], [18].",
"Intriguingly, the experimentally studied spin-$1/2$ kagome materials - such as herbertsmithite[19] - also remain quantum disordered down to the lowest temperature scales studied, well below the exchange energy scales.",
"However, in contrast to the numerical studies, it does not appear to be gapped.",
"It is currently unclear if the gaplessness is an intrinsic feature or a consequence of impurities that are known to be present in these materials.",
"Furthermore the magnetic Hamiltonian of the material may depart from the pure Heisenberg limit.",
"Relating the numerical results to experiments remains an important open question.",
"Since it preserves all symmetries of the system, is a $Z_2$ SL fully characterized by its topological order?",
"The answer is no.",
"In fact, the interplay of symmetry and topological order leads to a very rich structure.",
"There are many different $Z_2$ spin liquids with the same $Z_2$ topological order and the same symmetry group, but they cannot be continuously connected to each other without breaking the symmetry: they are dubbed “symmetry enriched topological (SET)” phases[20], [21], [22], [23], [24], [25], [26].",
"In a SET phase the quasiparticles not only have fractional statistics, but can also carry fractional symmetry quantum numbers: different SET phases are characterized by different patterns of symmetry fractionalization[23].",
"In the literature $Z_2$ SLs have been constructed in various slave-particle frameworks: the most predominant two approaches fractionalize physical spins into bosonic spinons[27], [2], [28], [29] and fermionic spinons[30], [31], [32], [33], [3], [5], [20].",
"Both approaches yield variational wavefunctions with good energetics[34], [35], [36] for the kagome lattice model.",
"It was proposed that symmetric $Z_2$ SLs are classified by the projective symmetry groups[20] (PSGs) of bosonic/fermionic spinons.",
"However it has been a long-time puzzle to understand the relation between different PSGs in bosonic-spinon representation (bSR) and fermionic-spinon representations (fSR)[37].",
"To be specific in the kagome lattice Heisenberg model, in bSR (Schwinger-boson approach) there are 8 different $Z_2$ SLs[29] among which the so-called $Q_1=Q_2$ state[28] is considered a promising candidate according to variational calculations[34].",
"Meanwhile there are 20 distinct $Z_2$ SLs[38] in fSR (Abrikosov-fermion approach), including the so-called $Z_2[0,\\pi ]\\beta $ state[38] which is in the neighborhood of energetically favorable $U(1)$ Dirac SL[35].",
"Are these two candidate states actually two different descriptions of the same gapped phase?",
"If not, what are their counterparts in the other representation?",
"In this paper we establish the general connection between different $Z_2$ SLs in bSR and fSR.",
"We show that $Z_2$ SLs constructed by projecting bilinear mean-field ansatz in bSR (Schwinger-boson representation) never have symmetry-protected gapless edge states.",
"This important observation allows us to determine how visons transform under symmetry in Schwinger-boson $Z_2$ SLs, and to further relate a Schwinger-boson state to one in fSR.",
"Since a bosonic spinon and vison fuse to a fermionic spinon, the corresponding PSGs, roughly speaking, follow a product rule.",
"However, crucially, in some cases such as the PSG relating to inversion symmetry [23], extra phase factors enter, modifying the naive fusion rule.",
"Here we identify two additional instances where such nontrivial PSG fusion rules occur, as explained in Section REF .",
"(In forthcoming work, related results are established using different techniques [77], [78]).",
"This allows us to show that indeed the bosonic $Q_1=Q_2$ state[28] is equivalent to the fermionic $Z_2[0,\\pi ]\\beta $ state[38].",
"Indeed all eight SBMF states have Abrikosov fermion counterparts (see Table REF ).",
"Part of these correspondences can also be inferred from Ref.",
"Yang2012 using wavefunction methods.",
"Next, we demonstrate that knowledge of just the bosonic (or just the fermionic) spinon PSG, with no further information such as the existence of a SBMF ansatz, is not enough to fully characterize a $Z_2$ SL.",
"For example, two distinct $Z_2$ SLs in fSR can have the same PSG for fermionic spinons while only one of them has symmetry-protected gapless edge modes.",
"However, they differ in the transformation properties of visons under symmetry (vison PSG), which provides an interesting link between symmetry implementation and topological edge states.",
"Applying these general principles to $Z_2$ SLs on kagome lattice, we show that all 8 different Schwinger-boson (bSR) states have their partners in the Abrikosov-fermion (fSR) representation.",
"In particular $Q_1=Q_2$ state[28] in Schwinger-boson representation belongs to the same phase as $Z_2[0,\\pi ]\\beta $ state[38] in Abrikosov-fermion representation.",
"This correspondence allows us to identify the possible symmetry-breaking phases in proximity to $Z_2$ SLs on kagome lattice.",
"Moreover these results serve as a useful guide in future studies of $Z_2$ SLs.",
"Spinons and visons in a $Z_2$ SL satisfy the following Abelian fusion rules[7]: $&b\\times f=v,~~~b\\times v=f,~~~f\\times v=b,\\\\& b\\times b=f\\times f=v\\times v=1.$ Here 1 stands for local excitations carrying integer spins, and the fractionalized excitations will not pick up any nontrivial Berry phase when they encircle a local excitation.",
"In other words the bound state of a bosonic spinon and a fermionic spinon is a vison, and the bound state of two bosonic (fermionic) spinons or two visons is a local excitation.",
"The fusion rules (REF ) have important implications on the symmetry properties of spinons and visons.",
"More precisely, the symmetry quantum numbers of spinons and visons must be compatible with the fusion rules[23].",
"Take $SU(2)$ spin rotational symmetryIn this work we will focus on a spin-$1/2$ system, though generalizations to higher spins are straightforward.",
"for a simple example.",
"Both bosonic ($b$ ) and fermionic ($f$ ) spinons carry spin-$1/2$ each, and vison $v$ is a spin-singlet excitation.",
"According to rules of angular momentum addition, one can immediately see fusion rules (REF ) are consistent with spin quantum numbers.",
"Figure: Crystal symmetries of kagome lattice with 4 generators {T 1,2 ,R π/3 ,R y }\\lbrace T_{1,2},R_{\\pi /3},R_y\\rbrace .",
"Translations (T 1 T_1, T 2 T_2) along the direction 1 and 2 are drawn as the directed arrow.",
"R π/3 R_{\\pi /3} stands for 60 degree rotation about a hexagon center.",
"The mirror reflection 1 is denoted by R y R_y, while reflection 2 corresponds to R π/3 R y R_{\\pi /3}R_y.Now let us apply this principle to other symmetries, i.e.",
"time reversal ${T}$ and crystal symmetries of kagome lattice (see FIG.",
"REF ).",
"Note that in a topologically ordered phase, the fractional excitations (anyons) always couple to emergent gauge fields[6], [39].",
"A direct consequence is that symmetry transformations on these anyons are not well-defined, in the sense that they are not invariant under gauge transformations.",
"The gauge-invariant “good” symmetry quantum numbers are the phase factors that anyons pick up through a series of symmetry operations which add up to identity operation.",
"They correspond to different algebraic identities built up by the generators of symmetry group, as summarized in TABLE REF for kagome lattice.",
"In a gapped $Z_2$ SL phase, these gauge-invariant phase factors are quantized to be $\\pm 1$ .",
"This is simply because any local excitation, as a bound state of an even number of spinons/visons, can only pick up a trivial (+1) phase factor after these symmetry operations.",
"These symmetry quantum numbers are characterized by their “projective symmetry group” (PSG)[20], and two symmetric $Z_2$ SLs with different PSGs cannot be smoothly connected without breaking symmetry.",
"TABLE REF shows the spinon and vison PSGs for $Z_2$ SLs on kagome lattice constructed in Schwinger-boson (bSR) and Abrikosov-fermion (fSR) approach, which will be discussed in detail later in section .",
"The vison PSGs are fully determined by PSGs of bosonic/fermionic spinons due to fusion rule (REF ).",
"To be specific, in many cases the phase factor $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}$ picked up by a bosonic spinon equals the product of phase factor $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}$ acquired by a vison and $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ by a fermionic spinon in the same process.",
"These cases will be referred to as the trivial fusion rule of PSGs.",
"For example in the case of time reversal symmetry ${T}$ , it is easy to see that ${T}^2=-1$ for bosonic/fermionic spinons and ${T}^2=+1$ for visons, in accordance with their spin quantum numbers.",
"However in a few cases $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}$ can differ from $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ by an extra minus sign, which originates from the mutual semion statistics between a spinon and a vison.",
"We refer to those cases as nontrivial fusion rules of PSGs.",
"These cases will be discussed in detailed below."
],
[
"Examples",
"In the following we discuss three nontrivial fusion rules of PSGs, which generally applies to any symmetric $Z_2$ spin liquid on a two-dimensional (2d) lattice.",
"We'll first present a general physical picture for these three cases based on toric code model[7] of $Z_2$ topological order, and then demonstrate them in a simple projective wavefucntion.",
"We notice that aside from nontrivial fusion rule associated with $I^2=(R_{\\pi /3})^6$ , the other two nontrivial fusion rules related to reflection symmetry and time reversal symmetry are missed in Ref.",
"Essin2013."
],
[
"Square of inversion operation: $(R_{\\pi /3})^6=I^2$",
"In the case of kagome lattice here, (hexagon-centered) inversion symmetry operation $I=(R_{\\pi /3})^3$ is the triple action of $\\pi /3$ rotation $R_{\\pi /3}$ (see 8th row in TABLE REF ).",
"Clearly when inversion acts twice, all particles rotate counterclockwise around a hexagon center by a full circle.",
"Being a bound state of a bosonic and fermionic spinon, a bosonic spinon would collect an extra $-1$ phase factor[23] because the fermionic spinon encircles the vison once in this process.",
"To be more precise, let's introduce toric code model[7] as a concrete demonstration of $Z_2$ topological order (or $Z_2$ spin liquids).",
"In the toric code model, ends of various open strings represent fractionalized excitations such as spinons and visons.",
"There are three different types of strings, corresponding to three different anyons (bosonic spinon $b$ , fermionic spinon $f$ and vison $v$ ) in a $Z_2$ spin liquid.",
"In the figures we use solid line to represent fermionic spinon (solid red circle) string, and dashed line for vison (blue cross) string.",
"Anyons of different types obey mutual semion statistics, which means each crossing of two different types of strings will yield a $-1$ phase factor.",
"As illustrated in FIG.",
"REF , we consider a fermionic spinon $f$ on the end of a (black) solid string, and another vison $v$ on the end of a (black) dashed string.",
"The bound state of these two object is a bosonic spinon $b=f\\times v$ .",
"When the black string operators act on the ground state (vacuum), such an excited state is created.",
"As $\\pi /3$ rotation acts for six times or equivalently inversion symmetry operation acts twice ($(R_{\\pi /3})^6=I^2$ ), the phase factor acquired in this process is given by vacuum expectation value of the $I^2$ symmetry operator, which is the solid (dashed) closed hexagon string operator[23] with red (blue) color for fermionic spinon $f$ (vison $v$ ).",
"As the dashed blue string crosses with the solid black string, an extra $-1$ sign will appear as we commute the $I^2$ symmetry operator for vison and the string operator for fermionic spinon.",
"As a result the PSGs associated with inversion square operation follows a nontrivial fusion rule.",
"This conclusion remains true no matter the inversion symmetry is plaquette-centered or site-centered.",
"Now let's simply demonstrate this conclusion using a projective wavefunction in the Abrikosov-fermion representation.",
"Consider an excited state with a pair of fermionic spinons $f_{1,2}$ related by inversion symmetry $\\hat{I}$ $|f_{1,2}\\rangle \\equiv f_1 f_2|G\\rangle ,~~~f_2=\\hat{I} f_1 \\hat{I}^{-1}.$ where $|G\\rangle $ represents the (mean-field) ground state.",
"By definition of symmetry fractionalization and PSGs we know that $\\hat{I}^2 f_{1,2}\\hat{I}^{-2}= e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}f_{1,2},~~~e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}=\\pm 1.$ It's straightforward to check that $\\frac{\\langle f_{1,2}|\\hat{I}|f_{1,2}\\rangle }{\\langle G|\\hat{I}|G\\rangle }=-e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ where the $-1$ sign shows up because two fermionic spinon operators $f_1$ and $f_2$ are exchanged under inversion operation $\\hat{I}$ .",
"Similarly for excited state $|v_{1,2}\\rangle \\equiv v_1v_2|G\\rangle ,~~~\\hat{I}v_1\\hat{I}^{-1}=v_2.$ with a pair of visons $v_{1,2}$ on top of mean-field ground state, we have $\\frac{\\langle v_{1,2}|\\hat{I}|v_{1,2}\\rangle }{\\langle G|\\hat{I}|G\\rangle }=e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v},~~~\\hat{I}^2v_{1,2}\\hat{I}^{-2}=e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}v_{1,2}$ And the excited state with a pair of bosonic spinons $b_i=f_i\\times v_i$ is created by $|b_{1,2}\\rangle =f_1f_2v_1v_2|G\\rangle .$ Clearly we have $\\frac{\\langle b_{1,2}|\\hat{I}|b_{1,2}\\rangle }{\\langle G|\\hat{I}|G\\rangle }=e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}=-e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ And this proves the nontrivial fusion rule of $I^2=(R_{\\pi /3})^6$ PSGs."
],
[
"Square of mirror reflection operation: $(R_y)^2$ and {{formula:a8cffaaf-0656-470d-a800-7b94c3722240}}",
"In this case let's consider a pair of fermionic spinons $f_{1,2}$ connected by a solid black string, and another pair of visons $v_{1,2}$ connected by a dashed black string as depicted in FIG.",
"REF .",
"We assume all these anyons lie on the reflection axis so that they are symmetric under reflection operation $R$ .",
"We'll reveal the nontrivial fusion rule of PSGs associated with reflection square operation $R^2$ , by studying the reflection quantum number of these anyons.",
"A key ingredient our discussion is that we also require the pair of fermionic spinons $f_{1,2}$ (and visons $v_{1,2}$ ) to be related by translation $T_a$ , as shown in FIG.",
"REF .",
"This guarantees two anyons (of the same species) in a pair share the same symmetry quantum numbers.",
"Figure: (Color online) Nontrivial fusion rule of PSGs associated with reflection square operation R 2 R^2, as discussed in section .",
"Reflection axis of RR is denoted by the dotted green line.",
"The symmetry operator associated with reflection RR is illustrated by the solid (dashed) closed hexagon string with red (blue) color for fermionic spinons f 1,2 f_{1,2} (vison v 1,2 v_{1,2}), in the bottom of the figure.",
"The phase factor acquired by each bosonic spinon b i =f i ×v i ,i=1,2b_i=f_i\\times v_i,~i=1,2 in the process of R 2 R^2 is e iφ b =-e iφ f ·e iφ v e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}=-e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}, where the extra -1-1 sign comes from the crossing of fermionic spinon string (black solid line) and vison string (blue dashed line).",
"Note that the pair of fermionic spinons f 1,2 f_{1,2} (and visons v 1,2 v_{1,2}) are related by translation T a T_a (parallel to reflection axis), to guarantee that they share the same symmetry quantum number.As illustrated in FIG.",
"REF , the total reflection ($R$ ) quantum number for the pair of fermionic spinons $f_{1,2}$ (visons $v_{1,2}$ ) is the vacuum expectation value of solid (dashed) closed string operator with red (blue) color.",
"Without loss of generality, we can assume this excited state is created by first applying the solid black string on the ground state (vacuum) and then the dashed black string.",
"Since the dashed blue string anti-commute with black solid string, the reflection quantum number of bosonic spinon pair in FIG.",
"REF equals the reflection quantum number of fermionic spinon pair and that of vison pair, with an extra $-1$ sign.",
"Since the pair of anyons are related by translation symmetry, this $-1$ sign should be evenly split into two halves: i.e.",
"each bosonic spinon acquires an extra $\\pm \\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}$ phase (in addition to ) upon reflection operation $R$ .",
"As a result when reflection $R$ acts twice, the phase factor acquired by each bosonic spinon $b_i=f_i\\times v_i,~~i=1,2$ in this process is given by $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}=(\\pm \\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt})^2\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}=-e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}$ Therefore the PSGs associated with reflection square $R^2$ obey the nontrivial fusion rule.",
"Demonstration of this conclusion for reflection square $R^2$ in the Abrikosov-fermion formalism follows the previous section for inversion square $I^2$ , simply by replacing inversion $I$ with reflection $R$ ."
],
[
"Commutation relation between time reversal and mirror reflection operation: $R_y^{-1}T^{-1}R_yT$ and {{formula:ba548ce5-cb51-41b6-8d34-22d76f56cd38}}",
"Unlike other global (on-site) symmetries, time reversal is an anti-unitary symmetry operation involving a complex conjugation.",
"Below we show a nontrivial fusion rule for PSGs associated with commutation relation between time reversal $T$ and mirror reflection $R$ .",
"Let's first consider the combination of time reversal $T$ and mirror reflection $R$ , which is an anti-unitary spatial symmetry.",
"When this symmetry acts twice, does the phase factor $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}$ acquired by a bosonic spinon differ from $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ by an extra $-1$ sign or not?",
"The answer is no.",
"In the same setup as in FIG.",
"REF , each bosonic spinon gets an extra $\\pm \\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}$ phase (in addition to the phase factors acquired by fermionic spinon and vison) under reflection $R$ .",
"However when time reversal acts, it takes complex conjugation and hence we have $(\\pm \\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt})^\\ast (\\pm \\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt})=1$ .",
"This extra phase cancels out upon symmetry operation $(TR)^2$ .",
"Therefore PSGs associated with $(TR)^2$ obey the trivial fusion rule.",
"As an algebraic identity we have $(TR)^2=(R^{-1}T^{-1}RT)\\cdot (T^2)\\cdot (R^2).$ We know that PSGs of $T^2$ and $(TR)^2$ obey trivial fusion rule, while PSGs of $R^2$ obey nontrivial fusion rule (previous section).",
"Therefore the PSGs associated with $R^{-1}T^{-1}RT$ must also obey nontrivial fusion rule.",
"To demonstrate it in the Abrikosov-fermion formulation, let's consider a superconducting ground state of Abrokosov fermions on a simplest YC4 cylinder (in notation of Ref. Yan2011).",
"As shown on FIG.",
"REF , the two sites (on left and right hand sides) labeled by $f_2$ are the same one.",
"Again we have a pair of visons $v_{1,2}$ and a pair of fermionic spinons $f_{1,2}$ , each related by translation $T_1$ .",
"For simplicity but without loss of generality, we consider a mean-field ansatz with on-site chemical potential terms and various-range real pairings of Abrikosov fermions.",
"The visons are created by the following action on the mean-field ansatz: all pairing amplitudes (pointing upwards) that cross the dashed red line acquire a $\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}$ sign while every pairing amplitude (pointing upwards) that crosses the dashed blue line acquires a $-\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}$ sign.",
"Therefore in the presence of this vison pair, the reflection symmetry operation on Abrikosov fermions is $&\\hat{R}^\\prime =e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\frac{\\pi }{2}\\sum _{a=1,2}(-1)^{a}n_a}\\hat{R},\\\\&\\hat{R}^\\prime f_a (\\hat{R}^\\prime )^{-1}=(-1)^a\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\hat{R} f_a\\hat{R}^{-1},~~~a=1,2.$ where $\\hat{R}$ is the original reflection symmetry operation on Abrikosov fermions, in the absence of visons.",
"We used $n_a$ to denote the fermion number on site $a=1,2$ .",
"From the above relation, it's straightforward to check the nontrivial fusion rule of PSGs for $R^2$ and $R^{-1}T^{-1}RT$ .",
"Notice that inversion symmetry $I$ is a combination of two reflection symmetry i.e.",
"$I=R_xR_y$ .",
"Therefore the PSGs associated with $I^{-1}T^{-1}IT$ simply follow the trivial fusion rule.",
"Before closing of this section, we want to mention that all arguments used here can be made more rigorous by considering a thin cylinder geometry which relates the PSGs to one-dimensional invariants of symmetry protected topological (SPT) phases[40].",
"These results will be published elsewhere[78].",
"Table: The algebraic identities and correspondence between bosonic spinon, fermionic spinon and vison PSGs on kagome lattice.",
"Mirror reflection R x R_x is defined as R x ≡(R π/3 ) 2 R y (R π/3 ) -1 R_x\\equiv (R_{\\pi /3})^2R_y(R_{\\pi /3})^{-1}.",
"Here bosonic spinon (b σ b_\\sigma ) PSGs are labeled by three integers p i =0,1(i=1,2,3)p_i=0,1~(i=1,2,3), while fermionic spinon (f σ f_\\sigma ) PSGs are labeled by seven integers (η 12 ,η σ ,η σC 6 ,η C 6 ,η σT ,η C 6 T )(\\eta _{12},\\eta _{\\sigma },\\eta _{{\\sigma }{C_6}},\\eta _{{C_6}},\\eta _{{\\sigma }{T}},\\eta _{{C_6}{T}}) where η=±1\\eta =\\pm 1.Choosing a proper gauge we can always fix T 1 -1 R π/3 -1 T 2 R π/3 =T 1 -1 T 2 R π/3 -1 T 1 R π/3 =1T^{-1}_{1}R^{-1}_{\\pi /3}T_{2}R_{\\pi /3}=T^{-1}_{1}T_{2}R^{-1}_{\\pi /3}T_{1}R_{\\pi /3}=1 for both spinons and visons.",
"If a Z 2 Z_2 SL in bSR and one in fSR correspond to the same state, their PSGs must satisfy conditions ().",
"Meanwhile the vison PSG for any Z 2 Z_2 SL in bSR is completely fixed as shown above.Previously we discussed why the “good” symmetry quantum numbers for spinons/visons are their PSGs, and how the vison PSGs can be determined once we know PSGs of both bosonic and fermionic spinons.",
"But one important question still remains to be answered: what are the physical manifestations of the vison PSG?",
"An important measurable property of topological phases are their edge states.",
"Although $Z_2$ SLs in the absence of symmetries are expected to have gapped edges, the edge may be gapless in the presence of symmetries[42], [43], [25], or, in the case of discrete symmetries, spontaneously break symmetry at the edge.",
"In particular, the criterion for nontrivial edges of $Z_2$ SLs with $SU(2)$ spin rotation symmetry will be established below and shown to be particularly simple.",
"The edge modes of a $Z_2$ SL can always be fermionized[44] with the same number of right ($\\psi _{R,a}$ ) and left ($\\psi _{L,a}$ ) movers (velocity is set to unity): $&\\mathcal {L}_0=\\sum _{a}\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\psi ^\\dagger _{R,a}(\\partial _t-\\partial _x)\\psi _{R,a}-\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\psi ^\\dagger _{L,a}(\\partial _t+\\partial _x)\\psi _{L,a}$ where $a$ denotes different branches of left/right movers.",
"One can always add backscattering terms to gap out the edge modes $\\lbrace \\psi _{R/L,a}\\rbrace $ $\\mathcal {L}_1=\\sum _{a,b}\\psi ^\\dagger _{R,a}{\\mathbf {M}}_{a,b}\\psi _{L,b}+\\psi _{R,a}{\\mathbf {\\Delta }}_{a,b}\\psi _{L,b}+~h.c.$ if they are not forbidden by symmetry.",
"In a different language, the above “mass” terms correspond to condensing either bosonic spinons $b$ or visons $v$ on the edge[45] of a $Z_2$ SL.",
"Since the spinons carry spin-$1/2$ each, condensing them will necessarily break spin rotational symmetry on the edge.",
"Therefore the only way to obtain a gapped edge without breaking the symmetry is to condense visons, unless their crystal symmetry quantum numbers (PSGs) will not allow it.",
"The relevant symmetries here are the ones that leave the physical edge unchanged, e.g.",
"at least one translation symmetry among $T_{1,2}$ will be broken by the edge.",
"Therefore the existence or absence of symmetry protected edge states is a probe of the vison PSGs.",
"Take kagome lattice for instance, on a cylinder with open boundaries parallel to $T_1$ direction (X-edge in FIG.",
"REF ), the remaining symmetries are translation $T_1$ , time reversal $T$ and mirror reflection $R_x\\equiv (R_{\\pi /3})^2R_y(R_{\\pi /3})^{-1}$ .",
"If there are no symmetry protected edge states, then the remaining symmetries must act trivially on visons: $&T_1^{-1}T^{-1}T_1T=R_x^{-1}T^{-1}R_xT=1,\\\\&R_x^2=T_1R_x^{-1}T_1R_x=1.$ so no backscattering term is forbidden by symmetry.",
"Another inequivalent edge is perpendicular to $T_1$ direction (Y-edge in FIG.",
"REF ), which preserves translation $T_1^{-1}T_2^2$ , time reversal $T$ and mirror reflection $R_y$ .",
"Similarly, absence of protected edge modes necessarily implies $&T_2^{-2}T_1T^{-1}T_1^{-1}T_2^2T=R_y^{-1}T^{-1}R_yT=1,\\\\&R_y^2=T_1^{-1}T_2^2R_y^{-1}T_1^{-1}T_2^2R_y=1.$ i.e.",
"these symmetries also have trivial actions on visons.",
"Table: Correspondence between Schwinger-boson Z 2 Z_2 SLs and Abrikosov-fermion states on kagome lattice.",
"All 8 unequal Schwinger-boson (bSR) states have their counterparts among 20 different Abrikosov-fermion (fSR) states.",
"Q 1 =Q 2 Q_1=Q_2 state in Schwinger-boson representation is equivalent to Z 2 [0,π]βZ_2[0,\\pi ]\\beta state, the only gapped Z 2 Z_2 SL near energetically favorable U(1)U(1) Dirac state.",
"On the other hand Q 1 =-Q 2 Q_1=-Q_2 state or (0,0,1)(0,0,1) state in Schwinger-boson representation corresponds to Z 2 [0,π]αZ_2[0,\\pi ]\\alpha state in Abrikosov-fermion representation.In addition to edge states, symmetry-protected in-gap bound state in crystal defects[46], [47] (such as dislocations and disclinations) is another signature to probe crystal symmetry quantum numbers.",
"Just like gapless edge modes, the existence of defect bound states must require non-trivial vison PSGs of the associated crystal symmetry.",
"In our case of kagome lattice, the crystal defect associated with $R_{\\pi /3}$ rotation is a disclination, centered on the hexagon with Frank angle $\\Omega =n\\pi /3,~n\\in {Z}$ .",
"Now let us imagine a vison circles around an elementary disclination with $\\Omega =\\pi /3$ for six times counterclockwise, and the phase factor it picks up in this process equals to the vison PSG $(R_{\\pi /3})^6$ .",
"Note that a vison only acquires a trivial (+1) phase factor when it encircles any number of spinons six times.",
"Therefore if $(R_{\\pi /3})^6=-1$ for visons, there must be a nontrivial in-gap bound state localized at the $\\Omega =\\pi /3$ disclination.",
"And the absence of in-gap bound state inside the disclination dictates $(R_{\\pi /3})^6=R_{\\pi /3}^{-1}T^{-1}R_{\\pi /3}T=1.$ Notice that this argument only applies to a plaquette-centered rotation (such as $R_{\\pi /3}$ here), because visons live on plaquettes.",
"For a site-centered crystal rotation, similar conclusions are not valid anymore.",
"To summarize, if a $Z_2$ SL does not host any gapless edge states or in-gap defect bound state protected by symmetry, vison PSGs must satisfy conditions (REF )-(REF ).",
"Furthermore, for a Mott insulator (with an odd number of spin 1/2 moments per unit cell) like the kagome S=1/2 model, a confined phase with a spin gap must double the unit cell.",
"Therefore the visions PSG under translations must satisfy: $T_1T_2=-T_2T_1$ This may be thought of as visons acquiring an extra phase factor on going around a unit cell with an odd number of spinons.",
"All together this leads to the vison PSGs shown in TABLE REF , assuming the absence of gapless edge states or in-gap disclination bound states which are protected by symmetries.",
"The detailed derivations are summarized in Appendix ."
],
[
"Unification of slave-particle mean-field theories on the kagome lattice",
"In bSR or Schwinger-boson approach[48], a spin-$1/2$ on lattice site ${\\mathbf {r}}$ is decomposed into two species of bosonic spinons $\\lbrace b_{{\\mathbf {r}},\\alpha }|\\alpha =\\uparrow /\\downarrow \\rbrace $ : $\\vec{S}_{\\mathbf {r}}=\\frac{1}{2}\\sum _{\\alpha ,\\beta =\\uparrow /\\downarrow }b^\\dagger _{{\\mathbf {r}},\\alpha }\\vec{\\sigma }_{\\alpha ,\\beta } b_{{\\mathbf {r}},\\beta }$ where $\\vec{\\sigma }$ are Pauli matrices.",
"Meanwhile in fSR or Abrikosov-fermion approach[30] spin-$1/2$ is represented by two flavors of fermionic spinons[33] $\\lbrace f_{{\\mathbf {r}},\\alpha }|\\alpha =\\uparrow /\\downarrow \\rbrace $ $\\vec{S}_{\\mathbf {r}}=\\frac{1}{2}\\sum _{\\alpha ,\\beta =\\uparrow /\\downarrow }f^\\dagger _{{\\mathbf {r}},\\alpha }\\vec{\\sigma }_{\\alpha ,\\beta } f_{{\\mathbf {r}},\\beta }$ To faithfully reproduce the 2-dimensional Hilbert space for spin-$1/2$ , there is a single-occupancy constraint: $\\sum _\\alpha b^\\dagger _{{\\mathbf {r}},\\alpha }b_{{\\mathbf {r}},\\alpha }=\\sum _\\alpha f^\\dagger _{{\\mathbf {r}},\\alpha }f_{{\\mathbf {r}},\\alpha }=1$ on every lattice site $\\forall ~{\\mathbf {r}}$ .",
"The variational wavefunctions are obtained by implementing Gutzwiller projections[49] on spinon mean-field ground state $|MF\\rangle $ , in order to enforce the single-occupancy constraint.",
"Here $|MF\\rangle $ is the ground state of (quadratic) mean-field ansatz for bosonic spinons[28], [29]: $\\hat{H}_{MF}^b=\\sum _{{\\mathbf {x}},{\\mathbf {y}}}\\sum _{\\alpha ,\\beta }A_{{\\mathbf {x},y}}b^\\dagger _{{\\mathbf {x}},\\alpha }b_{{\\mathbf {y}},\\alpha }+B_{{\\mathbf {x},y}}b_{{\\mathbf {x}},\\alpha }\\epsilon ^{\\alpha \\beta }b_{{\\mathbf {y}},\\beta }+~h.c.$ and similarly $\\hat{H}_{MF}^f=\\sum _{{\\mathbf {x}},{\\mathbf {y}}}\\begin{pmatrix}f^\\dagger _{{\\mathbf {x}},\\uparrow }\\\\f_{{\\mathbf {x}},\\downarrow }\\end{pmatrix}^T\\begin{pmatrix}t_{{\\mathbf {x},y}}&\\Delta _{\\mathbf {x},y}\\\\ \\Delta ^\\ast _{\\mathbf {x},y}&-t^\\ast _{\\mathbf {x},y}\\end{pmatrix}\\begin{pmatrix}f_{{\\mathbf {y}},\\uparrow }\\\\f^\\dagger _{{\\mathbf {y}},\\downarrow }\\end{pmatrix}+~h.c.$ for fermionic spinons[3], [20].",
"Proper on-site chemical potentials guarantee single-occupancy in $|MF\\rangle $ on average.",
"The physical properties of a gapped $Z_2$ SL described by a projected wavefunction can be understood in terms of its mean-field ansatz.",
"Specifically in bSR and fSR of $Z_2$ SLs, different PSGs for bosonic and fermionic spinons lead to distinct hopping/pairing patterns in mean-field ansatz $H^b_{MF}$ anf $H^f_{Mf}$ .",
"As pointed out in Ref.",
"Wang2006 there are 8 different Schwinger-boson (bSR) mean-field ansatz of $Z_2$ SLs on kagome lattice, while 20 distinct mean-field ansatz exists in fSR as shown in Ref. Lu2011.",
"A natural question is: what is the relation between the $Z_2$ SLs in bSR and those in fSR?",
"Can they describe the same $Z_2$ SL phase or not?",
"To answer this question, we use their vison symmetry quantum numbers (PSGs) to determine the (in)equivalence of the two representations.",
"To be precise, a Schwinger-boson ansatz corresponds to the same phase as an Abrikosov-fermion ansatz if and only if they share the same vison PSG.",
"As mentioned earlier, vison PSGs of a $Z_2$ SL can be probed by checking whether symmetry-protected edge states or in-gap defect (e.g.",
"disclination) bound states exist or not.",
"One important observation is that none of $Z_2$ SLs constructed in Schwinger-boson approach supports any gapless edge state or in-gap defect bound state.",
"This can be verified by computing the edge spectrum or defect spectrum in a Schwinger-boson mean-field ansatz.",
"Any gapped Schwinger-boson $Z_2$ SL ansatz can be tuned continuously to a limit that on-site chemical potential dominates over pairing/hopping terms, where it is clear no in-gap modes exist in edge/defect spectra.",
"Therefore the vison PSGs in any Schwinger-boson ansatz is fully fixed as in TABLE REF .",
"Amazingly this result (last column in TABLE REF ) agrees with vison PSGs derived microscopically from Schwinger-boson ansatz[41].",
"As discussed previously fermionic/bosnic spinon and vison PSGs are restricted by fusion rules (REF ).",
"More concretely vison PSGs always equal the product of bosonic and fermionic spinon PSGs i.e.",
"$e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}=e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ , except for the three situations discussed in section REF where $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _v}=-e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _b}\\cdot e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _f}$ .",
"These nontrivial fusion rules of PSGs take place in row 6-8 and 11-12 in TABLE REF , for algebraic relations $I^2$ , $R_{x,y}^2$ and $R_{x,y}^{-1}T^{-1}R_{x,y}T$ .",
"Therefore from TABLE REF , we can decide the relation between a Schwinger-boson ansatz and an Abrikosov-fermion ansatz, if they describes the same gapped $Z_2$ SL.",
"This leads to the following conditions: $&\\eta _{12}=(-1)^{p_1+1},~\\eta _{\\sigma }=\\eta _{{\\sigma }{T}}=(-1)^{p_2+p_3+1},\\\\&\\eta _{{\\sigma }{C_6}}=\\eta _{{C_6}{T}}=(-1)^{p_3},~\\eta _{C_6}=(-1)^{p_1+p_3+1}.$ It turns out all 8 distinct gaped $Z_2$ SLs described by bSR(Schwinger-boson approach) can also represented by fSR (Abrikosov-fermion approach) as summarized in TABLE REF .",
"Moreover, the most promising Schwinger-boson state for kagome Heisenberg model, $Q_1=Q_2$ state[28], describes the same spin liquid phase as the most promising $Z_2[0,\\pi ]\\beta $ state[38] in Abrikosov-fermion approach.",
"We notice that 4 of these 8 correspondences (the top 4 rows in TABLE REF ) have been obtained in Ref.",
"Yang2012."
],
[
"Candidate SETs for kagome Heisenberg antiferromagnet",
"We have noted that there are 20 different Abrikosov-fermion (fSR) mean-field states and 8 different Schwinger-boson (bSR) mean-field states of $Z_2$ spin liquids on the kagome lattice.",
"Here we will argue that one state is the most promising candidate from the following two criteria (i) energetics, as elaborated below and (ii) the requirement that the phase to be connected to a $q=0$ magnetically ordered state via a continuous transition.",
"This candidate is $Q_1=Q_2$ Schwinger-boson state, or equivalently $Z_2[0,\\pi ]\\beta $ Abrikosov-fermion state.",
"Numerical studies on the kagome lattice Heisenberg antiferromagnet supplemented by a 2nd neighbor antiferromagnetic coupling ($J_2$ ) reveal that[51] on increasing $J_2$ , the quantum spin liquid phase is initially further stabilized, before undergoing a transition[51] into a $q=0$ magnetic order around $J_2\\simeq 0.2 J_1$ .",
"Since the correlation length increases as the transition is approached, it is likely to be second order.",
"The $q=0$ magnetic order is a coplanar magnetic ordered state in which the three sublattices have spins aligned along three directions at 120 degrees to one another.",
"The Schwinger-boson state that naturally accounts for this is the $Q_1=Q_2$ state in Ref.",
"Sachdev1992, where under condensation of bosonic spinons the $q=0$ magnetic order develops via a continuous transition in the O(4) universality class[52].",
"Furthermore, variational permanent wavefunctions[34] based on the $Q_1=Q_2$ state were shown to give very competitive energies in particular for small and positive $J_2$ , establishing it as a contending state.",
"The $Z_ 2[0,\\pi ]\\beta $ mean-field state[38] of Abrikosov fermions (fSR), also satisfies the desirable properties above.",
"It can be thought as the $s$ -wave paired superconductor of fermionic spinons near the energetically favorable $U(1)$ Dirac SL[35].",
"The $s$ -wave pairing opens up a gap at the Dirac point of the underlying $U(1)$ Dirac SL and this implies that the $Z_ 2[0,\\pi ]\\beta $ state could be energetically competing with the $U(1)$ Dirac SL[38].",
"Although variational wave functions that include pairing are often found to have higher energy[36] than the underlying $U(1)$ Dirac SL, we note that this is a restricted class of states accessible via the parton construction, and a more complete search may land in the superconducting phase.",
"For our purposes we will be content that it is proximate to the energetically favorable $U(1)$ spin liquid state.",
"One can also describe a continuous transition from this Z$_2$ SL to the coplanar $q=0$ magnetically ordered state, although the argument here is more involved than in the case of the Schwinger-boson representation.",
"We make this argument[37] in two parts - first by ignoring the effects of the gauge field and recalling[53] a seemingly unrelated transition between an s-wave superconductor and a quantum spin Hall phase of the fermionic partons.",
"The latter spontaneously breaks the $SU(2)$ spin rotation symmetry down to $U(1)$ that defines the direction of the conserved spin component.",
"On including gauge fluctuations we will argue that the quantum spin Hall phase is to be identified with the $q=0$ magnetic order.",
"Starting with the $U(1)$ Dirac spin liquid, the s-wave superconductor representing the Z$_2$ SL is obtained by including a superconducting `mass' term that gaps the Dirac dispersion.",
"Similarly, the quantum spin Hall phase is obtained by introducing a distinct `mass' term that also gaps out the Dirac nodes.",
"There are three such mass terms indexed by the direction of the conserved spin in the quantum spin Hall state.",
"All of these anti-commute with the superconducting mass term, which implies that a continuous transition is possible between these phases[53].",
"On integrating out the fermions, the coefficients of the mass terms form an $O(5)$ vector of order parameters (real and imaginary parts of the pairing and the three quantum spin Hall mass terms), described by $O(5)$ non-linear sigma model with a Wess-Zumino-Witten (WZW) term[54], [55], [56], [53], [37].",
"Then the presence of the WZW term implies a continuous phase transition from the superconductor to the quantum spin hall state with a spontaneously chosen orientation.",
"This is most readily seen by noting that skyrmions of the quantum spin Hall director carry charge 2, which when condensed lead to a superconductor with spin rotation symmetry [53].",
"Now, on including gauge couplings the superconductor is converted into the $Z_ 2[0,\\pi ]\\beta $ SL state.",
"The quantum spin Hall state is a gapped insulator coupled to a compact U(1) gauge field, which is expected to confine and lead to a conventional ordered state.",
"This is seen to be the non-collinear magnetically ordered phase with the vector chirality at $q=0$ (for details see Appendix ).",
"The photon is identified as the additional Goldstone mode that appears since the $q=0$ state completely breaks spin rotation symmetry.",
"This further confirms the identification between $Q_1=Q_2$ Schwinger-boson state and $Z_2[0,\\pi ]\\beta $ Abrikosov-fermion state, since they are in the neighborhood of the same magnetic order.",
"Table: 20 different Abrikosov-fermion Z 2 Z_2SLs on a kagome lattice in the notation of Ref. Lu2011.",
"Among them state #2\\#2 or Z 2 [0,π]βZ_2[0,\\pi ]\\beta state corresponds to the same phase as Q 1 =Q 2 Q_1=Q_2 state in Schwinger-boson representation with (p 1 ,p 2 ,p 3 )=(0,1,0)(p_1,p_2,p_3)=(0,1,0).",
"Meanwhile #6\\#6 or Z 2 [0,π]αZ_2[0,\\pi ]\\alpha state belongs to the same phase as the so-called Q 1 =-Q 2 Q_1=-Q_2 state in Schwinger-boson representation with (p 1 ,p 2 ,p 3 )=(0,0,1)(p_1,p_2,p_3)=(0,0,1).",
"“Perturbatively gapped” means that fermion spinons can reach a fully-gapped superconducting ground state by perturbing the nearest neighbor (NN) hopping ansatz.",
"“0” is a trivial topological index, indicating the absence of symmetry protected gapless modes on the edge.",
"Meanwhile ℤ{\\mathbb {Z}} is a nontrivial integer topological index for possible symmetry protected gapless edge states.",
"Among the 20 Abrikosov-fermion states, 6 can host protected edge states on X-edge, while 6 can support protected edge modes on Y-edge.",
"None of the 8 Abrikosov-fermion states that have counterparts in Schwinger-boson representation support gapless edge states."
],
[
"Fermionic $Z_2$ spin liquids with symmetry protected edge states",
"In the Schwinger-boson representation, due to the absence of protected gapless edge states (or in-gap disclination bound states), the vison PSG is completely fixed.",
"Therefore the PSG of bosonic spinons ($b$ ) fully determines a symmetric $Z_2$ SL phase in Schwinger-boson approach.",
"In other words, two Schwinger-boson mean-field ansatz correspond to the same $Z_2$ SL phase if and only if they share the same bosonic-spinon PSG.",
"However this is not true in the Abrikosov-fermion representation: i.e.",
"two distinct $Z_2$ SL phases can share the same fermionic-spinon PSG in their Abrikosov-fermion mean-field ansatz.",
"This is essentially because the band topology[57], [58], [59] in an Abrikosov-fermion mean-field ansatz, which determines the vison PSG of a $Z_2$ SL, is not captured by the fermionic-spinon PSG.",
"In particular, certain fermionic-spinon PSGs allow for a nontrivial band topology: this is manifested by the symmetry protected edge states in a reflection-symmetric topological superconductor of Abrikosov-fermions."
],
[
"Reflection protected X-edge states",
"There are two types of open edges in a cylinder geometry, i.e.",
"X-edge and Y-edge in FIG.",
"REF .",
"Other translational-symmetric open edges can be obtained by multiples of $R_{\\pi /3}$ rotation acting on these two prototype edges.",
"As mentioned earlier, a cylinder with open X-edge preserves a symmetry group generated by $\\lbrace T,T_1,R_x\\equiv (R_{\\pi /3})^2R_y(R_{\\pi /3})^{-1}\\rbrace $ and $SU(2)$ spin rotations.",
"It's straightforward to see that $T_1^{-1}T^{-1}T_1T=T_1R_x^{-1}T_1R_x=1.$ for all Abrikosov-fermion states.",
"Since translational symmetry $T_1$ also commutes with spin rotations, it can be disentangled from other symmetries.",
"As discussed in Appendix , one can futher show that translational symmetry $T_1$ won't give rise to any nontrivial topological index.",
"Focusing on time reversal $T$ , mirror reflection $R_x$ (and $SU(2)$ spin rotations which commute with both $T$ and $R_x$ ), we have $(R_x)^2=\\eta _{\\sigma },~~~R_x^{-1}T^{-1}R_xT=\\eta _{{\\sigma }{T}}.$ for Abrikosov fermions $f$ .",
"As proven in Appendix , only when $\\eta _{{\\sigma }}=-1,~~\\eta _{{\\sigma }{T}}=+1.$ will there be a nontrivial integer index ${\\mathbb {Z}}$ for topological superconductors.",
"As summarized in TABLE REF , there are 6 fermionic-spinon PSGs ($\\#7\\sim \\#12$ ) that can support such a topological superconductor."
],
[
"Reflection protected Y-edge states",
"On a cylinder with open Y-edge the symmetry group is generated by $\\lbrace T,T_y\\equiv T_1^{-1}T_2^2,R_y\\rbrace $ and $SU(2)$ spin rotations.",
"Again one can easily show that $T_y^{-1}T^{-1}T_yT=T_yR_y^{-1}T_yR_y=1.$ for all Abrikosov-fermion states and we can disentangle translation $T_y$ from other symmetries.",
"With reflection $R_y$ and time reversal $T$ on Y-edge we have $R_y^2=\\eta _{\\sigma }\\eta _{{\\sigma }{C_6}},~~~R_y^{-1}T^{-1}R_yT=\\eta _{{\\sigma }{T}}\\eta _{{C_6}{T}}.$ Similarly a nontrivial integer index ${\\mathbb {Z}}$ can only happen when $\\eta _{\\sigma }\\eta _{{\\sigma }{C_6}}=-1,~~~\\eta _{{\\sigma }{T}}\\eta _{{C_6}{T}}=+1.$ It turns out that only 6 fermionic-spinon PSGs ($\\#3,\\#4,\\#9,\\#10,\\#19,\\#20$ ) in TABLE REF can support topological superconductors with protected gapless modes on Y-edge."
],
[
"An example and effective field theory",
"As one example, in the root state ($\\nu =1$ ) which generates the integer ($\\nu \\in {\\mathbb {Z}}$ ) topological index, the low-energy edge excitations are described by 2 pairs of counter-propagating fermion modes (see Appendix for derivations) $\\mathcal {L}^0_{edge}=\\sum _{a=\\uparrow ,\\downarrow }\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\psi ^\\dagger _{R,a}(\\partial _t-\\partial _x)\\psi _{R,a}-\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\psi ^\\dagger _{L,a}(\\partial _t+\\partial _x)\\psi _{L,a}$ where the velocity is normalized as unity.",
"The fermion modes transform under symmetries (time reversal $T$ , reflection $R$ and spin rotations) in the following way: $&\\psi _{\\alpha ,a}\\overset{T}{\\longrightarrow }\\sum _{\\beta ,b}[\\tau _x]_{\\alpha ,\\beta }[\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\sigma _y]_{a,b}\\psi _{\\beta ,b}\\\\&\\psi _{\\alpha ,a}\\overset{R}{\\longrightarrow }\\sum _{\\beta ,b}[\\tau _x]_{\\alpha ,\\beta }[\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\sigma _y]_{a,b}\\psi ^\\dagger _{\\beta ,b}\\\\&\\psi _{\\alpha ,a}\\overset{\\exp ({\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\theta \\hat{n}\\cdot \\vec{S}})}{\\longrightarrow }\\sum _b\\big [e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\frac{\\theta }{2}\\hat{n}\\cdot \\vec{\\sigma }}\\big ]_{a,b}\\psi _{\\alpha ,b}$ where we use index $\\alpha =R/L$ and $\\vec{\\tau }$ matrices for chirality (right/left movers), index $a=\\uparrow /\\downarrow $ and $\\vec{\\sigma }$ matrices for spin.",
"It's straightforward to see that $TR=RT$ and $R^2=-1$ .",
"There are two kinds of backscattering terms between right and left movers, which preserve $SU(2)$ spin rotational symmetry: $&\\mathcal {H}_{hop}=\\sum _a\\Big (t~\\psi ^\\dagger _{R,a}\\psi _{L,a}+t^\\ast ~\\psi ^\\dagger _{L,a}\\psi _{R,a}\\Big ),\\\\&\\mathcal {H}_{pair}=\\sum _{a,b}\\Big (\\Delta \\psi _{R,a}[\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\sigma _y]_{a,b}\\psi _{L,b}+\\Delta ^\\ast \\psi ^\\dagger _{L,b}[\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\sigma _y]_{a,b}\\psi ^\\dagger _{R,a}\\Big ).$ Among them, imaginary pairing is forbidden by time reversal $T$ , while hopping and real pairing are both forbidden by mirror reflection $R$ .",
"Such a gapped $Z_2$ SL with symmetry protected edge states is described by a multi-component Chern-Simons theory $\\mathcal {L}_{bulk}=\\frac{\\epsilon _{\\mu \\nu \\rho }}{4\\pi }\\sum _{I,J=1}^4a^I_\\mu {\\mathbf {K}}_{I,J}\\partial _\\nu a^J_\\rho -\\sum _{I=1}^4a_\\mu ^Ij^\\mu _I$ with a $4\\times 4$ ${\\mathbf {K}}$ matrix[25] ${\\mathbf {K}}=\\begin{pmatrix}0&0&1&1\\\\0&0&1&-1\\\\1&1&0&0\\\\1&-1&0&0\\end{pmatrix}=2{\\mathbf {K}}^{-1}$ The associated edge excitations are encoded by chiral boson fields $\\lbrace \\phi _I|1\\le I\\le 4\\rbrace $ with effective theory: $\\mathcal {L}_{edge}=\\frac{1}{4\\pi }\\sum _{I,J=1}^4\\Big (\\partial _t\\phi _I{\\mathbf {K}}_{I,J}\\partial _x\\phi _J-\\partial _x\\phi _I{\\mathbf {V}}_{I,J}\\partial _x\\phi _J\\Big )$ where ${\\mathbf {V}}$ is a positive-definite real symmetric matrix depending on details of the open edge.",
"In particular there are two species of visons ($v$ ) created by vertex operators $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _1}$ and $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _2}$ separately, while bosonic spinons ($b$ ) are created by $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _3}$ and $e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\phi _4}$ operators.",
"Under a symmetry operation $g$ the chiral bosons transform as[60], [61], [25] $\\phi _I\\overset{g}{\\longrightarrow }\\tilde{\\phi }_I=\\sum _Js_g\\big [{\\mathbf {W}}^g]_{I,J}\\phi _J+\\delta \\phi ^g_I.$ where the sign $s_g=+1$ for a unitary symmetry (such as mirror reflection $R$ and spin rotations) and $s_g=-1$ for an anti-unitary symmetry (such as time reversal $T$ ).",
"In our case of a $Z_2$ SL with symmetry protected edge states, for spin rotation $S^z_\\theta \\equiv \\prod _i\\exp (\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\frac{\\theta }{2}\\sigma _i^z)$ by angle $\\theta $ along $\\hat{z}$ -axis ${\\mathbf {W}}^{S^z_\\theta }=1_{4\\times 4},~~~\\delta \\phi ^{S^z_\\theta }=\\frac{\\theta }{2}\\begin{pmatrix}0\\\\0\\\\0\\\\1\\end{pmatrix}.$ For Ising symmetry $S^x\\equiv \\prod _{i}\\sigma ^x_i$ (spin rotation by angle $\\pi $ along $\\hat{x}$ -axis) we have ${\\mathbf {W}}^{S^x}=\\begin{pmatrix}0&1&0&0\\\\1&0&0&0\\\\0&0&1&0\\\\0&0&0&-1\\end{pmatrix},~~~\\delta \\phi ^{S^x}=\\frac{\\pi }{2}\\begin{pmatrix}0\\\\0\\\\1\\\\0\\end{pmatrix}.$ For anti-unitary time reversal symmetry $T$ ${\\mathbf {W}}^{T}=\\begin{pmatrix}0&-1&0&0\\\\-1&0&0&0\\\\0&0&1&0\\\\0&0&0&-1\\end{pmatrix},~~~\\delta \\phi ^{T}=\\frac{\\pi }{2}\\begin{pmatrix}0\\\\0\\\\0\\\\1\\end{pmatrix}.$ For mirror reflection symmetry $R$ we have ${\\mathbf {W}}^{R}=\\begin{pmatrix}0&1&0&0\\\\1&0&0&0\\\\0&0&-1&0\\\\0&0&0&1\\end{pmatrix},~~~\\delta \\phi ^{R}=\\begin{pmatrix}0\\\\ \\pi \\\\0\\\\0\\end{pmatrix}.$ Notice that under either time reversal $T$ or mirror reflection $R$ , the ${\\mathbf {K}}$ matrix changes sign i.e.",
"$\\Big ({\\mathbf {W}}^{g}\\Big )^T{\\mathbf {K}}{\\mathbf {W}}^g=-{\\mathbf {K}},~~~g=T,R$ The two pairs of fermionic modes are given in terms of chiral bosons as $&\\psi _{R,\\uparrow }\\sim e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}(\\phi _1+\\phi _4)},~~~\\psi _{R,\\downarrow }\\sim e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}(\\phi _2-\\phi _4)},\\\\&\\psi _{L,\\uparrow }\\sim e^{\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}(\\phi _4-\\phi _1)},~~~\\psi _{L,\\downarrow }\\sim e^{-\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}(\\phi _2+\\phi _4)}.$ up to Klein factors.",
"The spin rotational symmetric backscattering terms between right and left movers are given by $&\\mathcal {H}_{hop}\\sim |t|\\cdot \\Big [\\cos (2\\phi _1+C_1)+\\cos (2\\phi _2+C_2)\\Big ],\\\\&\\mathcal {H}_{pair}\\sim \\text{Re}\\Delta \\cdot \\cos (\\phi _1-\\phi _2)+\\text{Im}\\Delta \\cdot \\sin (\\phi _1-\\phi _2).$ where $C_1,C_2$ are constants.",
"As discussed earlier they are forbidden by time reversal and mirror reflection symmetries."
],
[
"Conclusion and outlook",
"In this work we systematically establish a connection between two different representations of $Z_2$ SLs, i.e.",
"Schwinger-boson representation (bSR) and Abrikosov-fermion representation (fSR).",
"In the presence of a symmetry group, projective symmetry groups (PSGs) are symmetry quantum numbers of the quasiparticle excitations in a $Z_2$ SL.",
"We show that the vison symmetry quantum numbers can be detected by existence/absence of gapless edge modes and in-gap bound states localized at crystal defect in $Z_2$ SLs.",
"Observing that there are no symmetry-protected edge modes or defect bound states in any Schwinger-boson state, and utilizing the relation between bosonic/fermionic spinon PSGs and vison PSGs, a correspondence between Schwinger-boson states and Abrikosov-fermion states is fully achieved.",
"Applying this general framework to kagome lattice $Z_2$ SLs, we showed that all 8 distinct Schwinger-boson (bSR) states have their counterparts in Abrikosov-fermion (fSR) representation.",
"In particular we found that two energetically favorable states, $Q_1=Q_2$ Schwinger-boson state[28] and $Z_2[0,\\pi ]\\beta $ state[38], in fact belong to the same spin liquid phase in proximity to $q=0$ non-collinear magnetic order.",
"We argue that this phase is the most promising candidate for the observed $Z_2$ SL ground state in kagome Heisenberg model.",
"Note, we have implicitly assumed that the topological order of $Z_2$ SLs is the same as that of the deconfined $Z_2$ gauge theory, based on the measured topological entanglement entropy of $\\gamma =\\log 2$ .",
"However, a distinct topological order, the double semion theory[62], [63], which is a twisted version of the $Z_2$ gauge theory, also yields the same $\\gamma $ .",
"While there is no direct evidence to support such a twisted $Z_2$ theory (in contrast to the energetic arguments for $Z_2$ spin liquids), it would be interesting in future to investigate candidate states.",
"With this connection in hand, we can potentially have a full understanding of the possible proximate phases and quantum phase transitions out of a $Z_2$ SL.",
"It is well-known that the Schwinger-boson approach allows one to identify quantum phase transitions (QPT) into neighboring magnetic-ordered phases from a $Z_2$ SL, though condensation of bosonic spinons[28].",
"Meanwhile, knowing the vison PSGs one can study possible QPTs between paramagnetic valence-bond-solid (VBS) phases and $Z_2$ SLs.",
"On the other hand, in fSR it's straightforward to track down gapless phases connected to a $Z_2$ SL through a phase transition, as well as proximate superconducting ground states upon doping a quantum SL[64].",
"Therefore the identification between fSR and bSR can lead to a full phase diagram around a gapped $Z_2$ SL.",
"The correspondence obtained here also serves as important guidance towards a complete specification of the $Z_2$ spin liquid on the kagome lattice.",
"If one of the two promising states we identified is indeed the ground state, then that provides a clear target for future studies to look for “smoking gun” signatures of these two states.",
"Finally, we point out that similar studies can be applied to $Z_2$ SLs on the square lattice [17], [18], which we leave for future work.",
"We thank M. Hermele, Y.",
"Ran, C. Xu, Y.B Kim, T. Grover, M. Lawler, P. Hosur, F. Wang, S. White, T. Senthil, P.A.",
"Lee, D.N Sheng and S. Sachdev for helpful discussions.",
"We thank Mike Zaletel for penetrating comments and for collaborations on a related paper[78].",
"GYC thanks Joel E. Moore for support and encouragement during this work.",
"YML is indebted to Shenghan Jiang for pointing out a typo in TABLE REF and for bringing Ref.",
"Yang2012 to our attention.",
"The authors acknowledge support from Office of BES, Materials Sciences Division of the U.S. DOE under contract No.",
"DE-AC02-05CH11231 (YML,AV), NSF DMR-1206515, DMR-1064319 and ICMT postdoctoral fellowship at UIUC (GYC) and in part from the National Science Foundation under Grant No.",
"PHYS-1066293(YML).",
"YML thanks Aspen Center for Physics for hospitality where part of the work is finished.",
"GYC is especially thankful to M. Punk, Y. Huh and S. Sachdev for bringing Ref.",
"Huh2011 to our attention and for the helpful discussions and suggestions."
],
[
"Deriving vison PSGs in TABLE ",
"In this section we explicitly show how to determine the vison PSGs in the last column of TABLE REF .",
"First of all we can always choose a proper gauge by multiplying a proper $\\pm 1$ sign to symmetry actions $T_{1,2}$ , so that $T_2R_{\\pi /3}=R_{\\pi /3}T_1,~~~T_1R_{\\pi /3}=R_{\\pi /3}T_2^{-1}T_1.$ In other words both the 2nd and 3rd rows of TABLE REF are $+1$ .",
"Meanwhile as discussed in the end of section , we have $T_1T_2=-T_2T_1.$ for visons in a spin-$1/2$ $Z_2$ SL on kagome lattice.",
"The absence of symmetry protected edge states along X-edge leads to conditions (REF ).",
"In particular we have $&T_1^{-1}T^{-1}T_1T=1,\\\\&R_x^2=(R_{\\pi /3}R_y)^2=1.$ and $R_x^{-1}T^{-1}R_xT=(R_{\\pi /3}^{-1}T^{-1}R_{\\pi /3}T)\\cdot (R_y^{-1}T^{-1}R_yT)=1.$ and $&T_1R_x^{-1}T_1R_x=(T_1^{-1}T_2R_y^{-1}T_2R_y)\\cdot (T_2^{-1}T_1^{-1}T_2T_1)\\\\&=-T_1^{-1}T_2R_y^{-1}T_2R_y=1.$ At the same time, conditions (REF ) come from the absence of protected edge states along Y-edge.",
"Therefore we have $R_y^2=R_y^{-1}T^{-1}R_yT=1.$ and $&T_1^{-1}T_2^2R_y^{-1}T_1^{-1}T_2^2R_y=\\\\&(T_1^{-1}R_y^{-1}T_1R_y)\\cdot (T_2^{-1}T_1^{-1}T_2T_1)=-T_1^{-1}R_y^{-1}T_1R_y=1.$ These conditions fix all the vison PSGs except for $(R_{\\pi /3})^6$ and $T_2^{-1}T^{-1}T_2T$ .",
"The latter one is easily determined as $T_2^{-1}T^{-1}T_2T=1.$ by the absence of protected edge states in a cylinder whose edges are parallel to the direction of translation $T_2$ .",
"As discussed in section , $(R_{\\pi /3})^6=1$ is determined by the absence of protected mid-gap states in a disclination."
],
[
"Vison PSGs obtained by explicit calculations{{cite:73993af6335d835d477cb27ebee18c3db76e67e8}}",
"In this section we deduce the vison PSGs from the dual frustrated Ising model obtained in Ref.",
"Huh2011, which describes vison fluctuations of Schwinger-boson $Z_2$ SLs on kagome lattice.",
"The 4-component vison modes $\\lbrace v_n|1\\le n\\le 4\\rbrace $ in section III A of Ref.",
"Huh2011 transform under symmetry $g$ as $v_m\\overset{g}{\\longrightarrow }\\sum _{n=1}^4~v_n\\cdot \\big [O_\\phi (g)\\big ]_{n,m}$ where the matrices $\\lbrace O_\\phi (g)\\rbrace $ are given by $&O_{\\phi }(T_{1}) =-\\left[\\begin{array}{cccc}0& 0&0 &1 \\\\0 & 0&-1 &0\\\\0&1&0&0 \\\\-1&0&0&0\\end{array}\\right],\\\\&O_{\\phi }(T_{2}) =\\left[\\begin{array}{cccc}0& 0&-1&0 \\\\0& 0&0 &-1\\\\1&0&0&0 \\\\0&1&0&0\\end{array}\\right],\\\\&O_{\\phi }(R_y) =\\left[\\begin{array}{cccc}1& 0&0 &0 \\\\0& 0&1 &0\\\\0&1&0&0 \\\\0&0&0&-1\\end{array}\\right],\\\\&O_{\\phi }(R_{\\pi /3}) =\\left[\\begin{array}{cccc}0& 0&1 &0 \\\\1& 0&0 &0\\\\0& 1&0&0 \\\\0&0&0&1\\end{array}\\right].$ Note that Ref.",
"Huh2011 considers mirror reflection $I_x=R_{\\pi /3}R_y$ with $O_{\\phi }(I_x) =O_{\\phi }(R_{\\pi /3})O_{\\phi }(R_y)=\\left[\\begin{array}{cccc}0& 1&0 &0 \\\\1& 0&0 &0\\\\0&0&1&0 \\\\0&0&0&-1\\end{array}\\right]$ It's straightforward to check the vison PSGs $&O_{\\phi }(T_{1})O_{\\phi }(T_{2})O_{\\phi }(T_{1})^{-1}O_{\\phi }(T_{2})^{-1} =-1; \\nonumber \\\\&O_{\\phi }(R_{\\pi /3})^{-1}O_{\\phi }(T_{1})O_{\\phi }(R_{\\pi /3})O_{\\phi }(T_{2}) =1; \\nonumber \\\\&O_{\\phi }(R_{\\pi /3})^{-1}O_{\\phi }(T_{2})O_{\\phi }(R_{\\pi /3})O_{\\phi }(T_{2})^{-1}O_{\\phi }(T_{1})=1; \\nonumber \\\\&O_{\\phi }(T_{1})O_{\\phi }(T_{2})^{-1}O_{\\phi }(R_{y})O_{\\phi }(T_{2})^{-1}O_{\\phi }(R_{y})^{-1}=-1; \\nonumber \\\\&O_{\\phi }(T_{1})O_{\\phi }(R_{y})O_{\\phi }(T_{1})^{-1}O_{\\phi }(R_{y})^{-1}=-1; \\nonumber \\\\&O_{\\phi }(R_{\\pi /3})^{6} = 1\\nonumber \\\\&O_{\\phi }(R_{y})^{2} = 1\\nonumber \\\\&O_{\\phi }(R_{\\pi /3})O_{\\phi }(R_{y})O_{\\phi }(R_{\\pi /3})O_{\\phi }^{-1}(R_{y}) =1$ which agree with the last column of TABLE REF ."
],
[
"Two-dimensional TRI singlet superconductors (Class CI) with mirror reflection symmetry",
"In this section we discuss possible symmetry-protected edge states of a time-reversal-invariant (TRI) singlet superconductor with mirror reflection symmetry in two dimensions.",
"In a cylinder geometry, the symmetry group is generated by translation $T_e$ along the open edge (or cylinder circumference), mirror reflection $R$ , time reversal $T$ and $SU(2)$ spin rotations.",
"Without loss of generality, let's assume that $SU(2)$ spin rotations commute with all other symmetries $\\lbrace T,T_e,R\\rbrace $ .",
"We further assume that translation $T_e$ acts as $T_e^{-1}T^{-1}T_eT=T_eR^{-1}T_3R=+1.$ for fermions."
],
[
"Classification",
"Since translation $T_e$ has trivial commutation relation with other symmetry operations, it can be disentangled from the full symmetry group.",
"Are there any gapless edge states protected by translation symmetry?",
"This corresponds to “weak index”[65] of 2d topological superconductors in class CI (with time reversal and $SU(2)$ spin rotations), which is nothing but 1d topological index of the same symmetry class.",
"Class CI has trivial classification (0) in 1d, therefore we don't have translation-protected edge states.",
"Due to absence of 2d topological index in class CI, any protected edge states must come from mirror reflection symmetry $R$ .",
"The classification of mirror reflection protected topological insulators/superconductors is resolved in Ref.",
"Chiu2013,Morimoto2013 in the framework of K-theory[68].",
"The classification of non-interacting topological phases of fermions in class CI with mirror reflection $R$ depends on the commutation relation $R^2=s_1,~~~R^{-1}T^{-1}RT=s_2;~~~s_i=\\pm 1.$ with time reversal $T$ .",
"When $s_1=s_2=+1$ , the K-theory classification is given by $\\pi _0(R_6)=0$ , i.e.",
"no topological superconductors with protected edge states.",
"When $s_1=s_2=-1$ , the classification is $\\pi _0(C_5)=0$ i.e.",
"no topological superconductors.",
"When $s_1=+1,s_2=-1$ the classification is $\\big [\\pi _0(R_5)\\big ]^2=0^2=0$ i.e.",
"no topological superconductors.",
"Only when $s_1=-1,~~~s_2=+1.$ the classification is given by $\\pi _0(R_4)={\\mathbb {Z}}$ and there are topological superconductors with an integer index (${\\mathbb {Z}}$ )."
],
[
"Minimal Dirac model and protected edge states",
"Here we construct a Dirac model for the root state ($\\nu =1$ ) of topological superconductors with integer index $\\nu \\in {\\mathbb {Z}}$ and mirror reflection $R$ satisfying $R^2=-1,~~~RT=TR.$ Writing spin-$1/2$ electrons in the Nambu basis $\\psi _k\\equiv (c_{k,\\uparrow },c^\\dagger _{-k,\\downarrow })^T$ , we use Pauli matrices $\\vec{\\tau }$ for Nambu index and $\\vec{\\mu },\\vec{\\rho }$ for orbital index.",
"The 8-band massless Dirac Hamiltonian is given by $H_{Dirac}=\\sum _k(k_x\\mu _x+k_y\\mu _z)\\rho _y\\tau _x$ The Dirac fermion transforms as $&\\psi _k\\overset{T}{\\longrightarrow }\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\tau _y\\psi _{k}^\\ast ,\\\\&\\psi _{(k_x,k_y)}\\overset{R}{\\longrightarrow }\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\mu _x\\rho _y\\psi _{(k_x,-k_y)}$ The only symmetry-allowed mass term is $M=\\tau _z$ Now let's create a mass domain wall across an open edge at $x=0$ along $y$ -axis.",
"The gapless edge states is captured by zero-energy solution of differential equation $-\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\partial _x\\mu _x\\rho _y\\tau _x+m(x)\\tau _z=0,~~~m(x)=|m(x)|\\cdot \\text{Sgn}(x)$ which is $|edge\\rangle \\sim e^{-\\int _0^xm(\\lambda )\\text{d}\\lambda }|\\mu _x\\rho _y\\tau _y=+1\\rangle $ .",
"Therefore $\\rho _y=\\mu _x\\tau _y$ for the protected gapless edge modes, localized on the edge between the root topological superconductor and the vacuum.",
"The Hamiltonian for the protected edge states is given by $H_k=k_y\\mu _z\\rho _y\\tau _x\\rightarrow -k_y\\mu _y\\tau _z$ Apparently there are 4 gapless modes: 2 right movers and 2 left movers.",
"It's straightforward to simplify such edge states to the form in section REF ."
],
[
"$Z_ 2[0,\\pi ]\\beta $ mean-field state and its proximate phases",
"We begin by revisiting $U(1)$ Dirac spin liquid [69], [35], [70] and $Z_{2}[0,\\pi ]\\beta $ state[38] and on a kagome lattice.",
"The spin liquid states are proposed to be the ground state of nearest-neighbor spin-$1/2$ Heisenberg model on kagome lattice: $H = J \\sum _{<ij>}{\\hat{S}}_{i}\\cdot {\\hat{S}}_{j}$ The low-energy theory of the $U(1)$ Dirac spin liquid [70] is described by a 8-component spinor $\\psi $ of fermionic spinons in Dirac spectrum and a strongly fluctuating $U(1)$ gauge field $a_{\\mu }$ : $H =\\sum _{\\mathbf {k}} \\psi ^{\\dagger }_{{\\mathbf {k}}}\\sigma ^{0}\\mu ^{0}(\\tau _{x}k_{x} + \\tau _{y}k_{y})\\psi _{{\\mathbf {k}}}.$ where $\\tau ^\\nu $ , $\\sigma ^{\\nu }$ and $\\mu ^{\\nu }$ are the Pauli matrices acting on Dirac, spin and valley indices of $\\psi _{{\\mathbf {k}}}$ (there are two nodes or two “valleys\" : hence $\\psi _{{\\mathbf {k}}}$ is a 8-component spinor).",
"Here we temporarily ignore the compact $U(1)$ gauge theory for clarity of discussion.",
"To obtain a $Z_{2}$ spin liquid from this $U(1)$ Dirac spin liquid, a BCS-type pairing term is introduced for the fermion $\\psi $ .",
"We require the pairing term to be invariant under spin rotational symmetry, lattice symmetry operations (see Fig.REF for the symmetries of kagome lattice) and time-reversal symmetry, because the spin liquid found in DMRG study does not break any of the symmetries.",
"Furthermore, the pairing should gap out the Dirac spectrum as the $Z_{2}$ spin liquid found numerically is fully gapped.",
"$\\delta H = \\Delta \\psi ^{\\dagger } (\\sigma ^{y}\\mu ^{y}\\tau ^{y})\\psi ^{*} + h.c.,$ which is a singlet of spin, valley and Dirac spinor indices.",
"Being a singlet under spin and valley indices gaurantees that the pairing is invariant under spin rotational symmetry and lattice symmetry operations.",
"The low-energy physics of $Z_{2}[0,\\pi ]\\beta $ state is described by $H + \\delta H$ in Eq.",
"(REF ) and Eq.",
"(REF ), i.e.",
"a gapped singlet superconductor of fermionic spinons coupled to a dynamical $Z_2$ gauge field.",
"It is known that Dirac fermions are unstable (with sufficiently large interactions) to open up gap in various channels.",
"Each channel is called a “mass” and is represented by a constant matrix in the Dirac spinor representation.",
"For example in Dirac Hamiltonian (REF ), $\\tau ^z\\mu ^\\alpha \\sigma ^\\beta $ are all mass terms, with $\\alpha ,\\beta =0,x,y,z$ .",
"For a 2+1-D Dirac fermion, when we can find five such mass matrices anti-commuting with each other[54], [56], [71], we obtain a non-linear sigma model supplemented with a topological WZW term[72], [73] after integrating out massive fermions.",
"This non-linear sigma model describes fluctuating order parameters of the Dirac fermions.",
"Most importantly, the theory can describe a Landau-forbidden second-order transition between the two phases[74], where the transition is driven by condensing the topological defects[55], [56], [53], [37].",
"The mass terms associated to the $Z_2$ spin liquid are real and imaginary parings in (REF ).",
"Because we seek for the nearby phases of the $Z_{2}$ spin liquid, the relevant mass terms should anti-commute with the pairing term (REF ) and the kinetic term (REF ).",
"We immediately find two $O(3)$ vector mass terms among 26 mass terms of the $U(1)$ Dirac spin liquid[70], anti-commuting with the pairing term (REF ).",
"Among the two $O(3)$ vector mass terms, we consider only the $O(3)$ vector chirality operator[70] ${\\hat{V}} \\sim <\\psi ^{\\dagger } \\tau ^{z}{\\vec{\\sigma }}\\psi >$ to examine the magnetically ordered proximate phase of the $Z_{2}[0,\\pi ]\\beta $ state.",
"The operator ${\\hat{V}}$ is spin-triplet and time-reversal symmetric.",
"This order parameter represents the sum of the vector chirality around honeycomb plaquette $H$ on Kagome lattice.",
"${\\hat{V}}^{a} \\sim \\sum _{<ij> \\in H} ({\\vec{S}_{i}} \\times {\\vec{S}_{j}})^{a},$ Because the vector chirality ${\\hat{V}}$ is spin-triplet, we expect the non-linear sigma model for the unit $O(5)$ vector $= (\\Delta _{x},\\Delta _{y}, {\\hat{V}})$ to describe the transition between the spin liquid and a magnetically ordered phase with the non-zero $\\langle {\\hat{V}} \\rangle $ .",
"To see this, we approach the critical point between the spin liquid and the magnetically ordered phase from the ordered phase.",
"The low-energy effective theory for the symmetry-broken phase, including the compact $U(1)$ gauge field $a_{\\mu }$ , is $\\mathcal {L} = \\psi ^{\\dagger }&\\sigma ^{0}\\mu ^{0}\\tau ^{\\mu }\\cdot (\\hspace{1.0pt}\\mathrm {i}\\hspace{1.0pt}\\partial _{\\mu }+a_{\\mu })\\psi + m{\\hat{V}} \\cdot \\psi ^{\\dagger } \\tau ^{z}{\\vec{\\sigma }}\\psi \\nonumber \\\\& + \\frac{1}{g^{2}} (\\partial _{\\mu } {\\vec{V}})^{2} + \\frac{1}{2{\\tilde{e}}^{2}}(\\varepsilon ^{\\mu \\nu \\lambda }\\partial _{\\nu }a_{\\lambda })^{2} +\\cdots $ In the symmetry-broken phase, the $O(3)$ vector order parameter ${\\hat{V}}$ develops a finite expectation value, and we assume $\\langle {\\hat{V}}\\rangle = (0,0, 1)$ without losing generality (other direction of $\\langle {\\hat{V}}\\rangle $ can be generated by the spin rotations).",
"With the expectation value ${\\hat{V}}$ , it is not difficult to see that the spin-up fermions and the spin-down fermions have an energy gap $|m|$ at the Dirac points with the opposite sign, and the mass gap consequently generates the “spin Hall effect” for the fermions.",
"The quantum spin Hall effect has an important implication[75] on the fate of the compact gauge field $a_{\\mu }$ : it ties the gauge fluctuation to the spin fluctuation, and thus the Goldstone mode ($\\sim $ spin fluctuation) of the spin ordered phase becomes a photon ($\\sim $ gauge fluctuation) of $a_{\\mu }$ .",
"This implies that the gauge field $a_{\\mu }$ is in the Coulomb phase and the photon of $a_{\\mu }$ is free to propagate.",
"Hence there are three Goldstone modes in the magnetically ordered phase, one photon mode from the non-compact $U(1)$ guage field $a_{\\mu }$ and two Goldstone modes from the ordering of the $O(3)$ vector ${\\hat{V}}$ .",
"Meanwhile accompanying the proliferation of $a_\\mu $ photons, fermionic spinons will be confined [76] due to instanton effect of 2+1-D $U(1)$ gauge theory.",
"Therefore indeed it is a non-collinear magnetic ordered phase with three Goldstone modes, which does not support fractionalized excitations.",
"Upon integrating out the massive Dirac fermion, we obtain the effective theory[54], [37] for the fluctuating ${\\hat{V}}$ in the presence of the gauge field $a_{\\mu }$ $\\mathcal {L}= \\frac{1}{g^{2}} (\\partial _{\\mu } {\\hat{V}})^{2} + 2a_{\\mu } J^{\\mu }_{skyr} + \\frac{1}{2{\\tilde{e}}^{2}}(\\varepsilon ^{\\mu \\nu \\lambda }\\partial _{\\nu }a_{\\lambda })^{2} +\\cdots $ where $J^{\\mu }_{skyr}$ is the skyrmion current of ${\\hat{V}}$ , e.g.",
"$J^{0}_{skyr} \\propto {\\hat{V}}\\cdot (\\partial _{x} {\\hat{V}}\\times \\partial _{y} {\\hat{V}})$ is the skyrmion density of ${\\hat{V}}$ .",
"From the coupling between $J^{\\mu }_{skyr}$ and $a_{\\mu }$ , it is clear that the skyrmion carries the charge-2 of the gauge field $a_{\\mu }$ .",
"Hence, condensing the skyrmion of ${\\hat{V}}$ breaks $U(1)$ gauge group down to $Z_{2}$ and the skyrmion can be thought as the pairing $\\sim \\langle \\psi ^{\\dagger }\\psi ^{\\dagger }\\rangle $ of the fermionic spinons $\\psi $ in (REF ).",
"As the condensation of the skyrmion would destroy the ordering in ${\\hat{V}}$ and induce the pairing between the fermionic spinons, we will enter the $Z_{2}$ spin liquid phase next to the symmetry-broken phase, i.e.",
"$Z_{2}[0,\\pi ]\\beta $ state.",
"Thus we have established that the magnetically ordered phase next to the $Z_{2}[0,\\pi ]\\beta $ state is a non-collinear magnetically ordered phase with the non-zero vector chirality at $q=0$ .",
"Given that the $q=0$ magnetically ordered state is also a non-collinear magnetically ordered phase with the non-zero vector chirality at $q=0$ , the $Z_{2}[0,\\pi ]\\beta $ state is a natural candidate for the $Z_2$ SL proximate to the $q=0$ magnetically ordered state."
]
] | 1403.0575 |
[
[
"Sudden reversal in the pressure dependence of Tc in the iron-based\n superconductor CsFe2As2: A possible link between inelastic scattering and\n pairing symmetry"
],
[
"Abstract We report a sudden reversal in the pressure dependence of Tc in the iron-based superconductor CsFe2As2, similar to that discovered recently in KFe2As2 [Tafti et al., Nat.",
"Phys.",
"9, 349 (2013)].",
"As in KFe2As2, we observe no change in the Hall coefficient at the zero temperature limit, again ruling out a Lifshitz transition across the critical pressure Pc.",
"We interpret the Tc reversal in the two materials as a phase transition from one pairing state to another, tuned by pressure, and investigate what parameters control this transition.",
"Comparing samples of different residual resistivity, we find that a 6-fold increase in impurity scattering does not shift Pc.",
"From a study of X-ray diffraction on KFe2As2 under pressure, we report the pressure dependence of lattice constants and As-Fe-As bond angle.",
"The pressure dependence of these lattice parameters suggests that Pc should be significantly higher in CsFe2As2 than in KFe2As2, but we find on the contrary that Pc is lower in CsFe2As2.",
"Resistivity measurements under pressure reveal a change of regime across Pc, suggesting a possible link between inelastic scattering and pairing symmetry."
],
[
"Introduction",
"To understand what controls $T_{\\rm c}$ in high temperature superconductors remains a major challenge.",
"Several studies suggest that in contrast to cuprates where chemical substitution controls electron concentration, the dominant effect of chemical substitution in iron-based superconductors is to tune the structural parameters – such as the As-Fe-As bond angle – which in turn control $T_{\\rm c}$ .",
"[1], [2] This idea is supported by the parallel tuning of $T_c$ and the structural parameters of the 122 parent compounds BaFe$_2$ As$_2$ and SrFe$_2$ As$_2$ .",
"[3], [4] In the case of Ba$_{1-x}$ K$_{x}$ Fe$_2$ As$_2$ , at optimal doping ($x=0.4$ , $T_{\\rm c}$ = 38 K) the As-Fe-As bond angle is $\\alpha =109.5\\,^{\\circ }$ , the ideal angle of a non-distorted FeAs$_4$ tetrahedral coordination.",
"Underdoping, overdoping, or pressure would tune the bond angle away from this ideal value and reduce $T_{\\rm c}$ by changing the electronic bandwidth and the nesting conditions.",
"[3] CsFe$_2$ As$_2$ is an iron-based superconductor with $T_{\\rm c}$ = 1.8 K and $H_{\\rm c2}$ = 1.4 T. [5], [6], [7] Based on the available X-ray data, [5] the As-Fe-As bond angle in CsFe$_2$ As$_2$ is $109.58\\,^{\\circ }$ , close to the ideal bond angle that yields $T_{\\rm c}$ = 38 K in optimally-doped Ba$_{0.6}$ K$_{0.4}$ Fe$_2$ As$_2$ .",
"If the bond angle were the key tuning factor for $T_{\\rm c}$ , CsFe$_2$ As$_2$ should have a much higher transition temperature than 1.8 K. In this article, we show evidence that $T_{\\rm c}$ in (K,Cs)Fe$_2$ As$_2$ may be controlled by details of the inelastic scattering processes that are not directly related to structural parameters, but are encoded in the electrical resistivity $\\rho (T)$ .",
"The importance of inter- and intra-band inelastic scattering processes in determining $T_{\\rm c}$ and the pairing symmetry of iron pnictides has been emphasized in several theoretical works.",
"[8], [9], [10] Recently, it was shown that a change of pairing symmetry can be induced by tuning the relative strength of different competing inelastic scattering processes, i.e.",
"different magnetic fluctuation wavevectors.",
"[11] In a previous paper, we reported the discovery of a sharp reversal in the pressure dependence of $T_{\\rm c}$ in KFe$_2$ As$_2$ , the fully hole-doped member of the Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ series.",
"[12] No sudden change was observed in the Hall coefficient or resistivity across the critical pressure $P_{\\rm c}$ = 17.5 kbar, indicating that the transition is not triggered by a change in the Fermi surface.",
"Recent dHvA experiments under pressure confirm that the Fermi surface is the same on both sides of $P_{\\rm c}$ , ruling out a Lifshitz transition and strengthening the case for a change of pairing state.",
"[13] We interpret the sharp $T_{\\rm c}$ reversal as a phase transition from $d$ -wave to $s$ -wave symmetry.",
"Bulk measurements such as thermal conductivity[14], [15] and penetration depth[16] favor $d$ -wave symmetry at zero pressure.",
"Because the high-pressure phase is very sensitive to disorder, a likely $s$ -wave state is one that changes sign around the Fermi surface, as in the $s_{\\pm }$ state that changes sign between the $\\Gamma $ -centered hole pockets, as proposed by Maiti et al.",
"[10] It appears that in KFe$_2$ As$_2$ $s$ -wave and $d$ -wave states are nearly degenerate, and a small pressure is enough the push the system from one state to the other.",
"In this article, we report the discovery of a similar $T_{\\rm c}$ reversal in CsFe$_2$ As$_2$ .",
"The two systems have the same tetragonal structure, but their lattice parameters are notably different.",
"[5] Our high-pressure X-ray data reveal that at least 30 kbar of pressure is required for the lattice parameters of CsFe$_2$ As$_2$ to match those of KFe$_2$ As$_2$ .",
"Yet, surprisingly, we find that $P_{\\rm c}$ is smaller in CsFe$_2$ As$_2$ than in KFe$_2$ As$_2$ .",
"This observation clearly shows that structural parameters alone are not the controlling factors for $P_{\\rm c}$ in (K,Cs)Fe$_2$ As$_2$ .",
"Instead, we propose that competing inelastic scattering processes are responsible for tipping the balance between pairing symmetries.",
"Figure: a) Pressure dependence of T c T_{\\rm c} in CsFe 2 _2As 2 _2.The blue and red circles represent data from samples 1 and 2, respectively.T c T_{\\rm c} is defined as the temperature where the zero-field resistivity ρ(T)\\rho (T) goes to zero.The critical pressure P c P_{\\rm c} marks a change of behaviour from decreasing to increasing T c T_{\\rm c}.Dotted red lines are linear fits to the data from sample 2 in the range P c P_{\\rm c} -10- 10 kbar and P c P_{\\rm c} +5+ 5 kbar.The critical pressure P c P_{\\rm c} =14±1= 14 \\pm 1 kbar is defined as the intersection of the two linear fits.b) Low-temperature ρ(T)\\rho (T) data, from sample 2, normalized to unity at T=2.5T=2.5 K.Three isobars are shown at P<P< P c P_{\\rm c}, with pressure values as indicated.The arrow shows that T c T_{\\rm c} decreases with increasing pressure.c) Same as in b), but for P>P> P c P_{\\rm c}, with ρ\\rho normalized to unity at T=1.5T=1.5 K.The arrow shows that T c T_{\\rm c} now increases with increasing pressure."
],
[
"Experiments",
"Single crystals of CsFe$_2$ As$_2$ were grown using a self-flux method.",
"[7] Resistivity and Hall measurements were performed in in an adiabatic demagnetization refrigerator, on samples placed inside a clamp cell, using a six-contact configuration.",
"Hall voltage is measured at plus and minus 10 T from $T=20$ to 0.2 K and antisymmetrized to calculate the Hall coefficient $R_{\\rm H}$ .",
"Pressures up to 20 kbar were applied and measured with a precision of $\\pm ~0.1$ kbar by monitoring the superconducting transition temperature of a lead gauge placed besides the samples inside the clamp cell.",
"A pentane mixture was used as the pressure medium.",
"Two samples of CsFe$_2$ As$_2$ , labelled “sample 1\" and “sample 2\", were measured and excellent reproducibility was observed.",
"High pressure X-ray experiments were performed on polycrystalline powder specimens of KFe$_2$ As$_2$ up to 60 kbar with the HXMA beam line at the Canadian Light Source, using a diamond anvil cell with silicon oil as the pressure medium.",
"Pressure was tuned blue with a precision of $\\pm ~2$ kbar using the R$_1$ fluorescent line of a ruby chip placed inside the sample space.",
"XRD data were collected using angle-dispersive techniques, employing high energy X-rays ($E_i = 24.35$ keV) and a Mar345 image plate detector.",
"Structural parameters were extracted from full profile Rietveld refinement using the GSAS software.",
"[17] Representative refinements of the X-ray data are presented in appendix ."
],
[
"Results",
"Fig.",
"REF a shows our discovery of a sudden reversal in the pressure dependence of $T_{\\rm c}$ in CsFe$_2$ As$_2$ at a critical pressure $P_{\\rm c}$ = $14\\pm 1$ kbar.",
"The shift of $T_{\\rm c}$ as a function of pressure clearly changes direction from decreasing (Fig.",
"REF b) to increasing (Fig.",
"REF c) across the critical pressure $P_{\\rm c}$ .",
"$T_{\\rm c}$ varies linearly near $P_{\\rm c}$ , resulting in a $V$ -shaped phase diagram similar to that of KFe$_2$ As$_2$ .",
"[12] Figure: Temperature dependence of the Hall coefficient R H R_{\\rm H}(T)(T) in CsFe 2 _2As 2 _2 (sample 2), at five selected pressures, as indicated.The low-temperature data converge to the same value for all pressures, whether below or above P c P_{\\rm c}.Inset:The value of R H R_{\\rm H} extrapolated to T=0T=0 is plotted at different pressures.Horizontal and vertical error bars are smaller than symbol dimensions.R H R_{\\rm H}(T=0)(T=0) is seen to remain unchanged across P c P_{\\rm c}.Measurements of the Hall coefficient $R_{\\rm H}$ allow us to rule out the possibility of a Lifshitz transition, i.e.",
"a sudden change in the Fermi surface topology.",
"Fig.",
"REF shows the temperature dependence of $R_{\\rm H}$ at five different pressures.",
"In the zero-temperature limit, $R_{\\rm H}(T\\rightarrow 0)$ is seen to remain unchanged across $P_{\\rm c}$ (Fig.",
"REF , inset).",
"If the Fermi surface underwent a change, such as the disappearance of one sheet, this would affect $R_{\\rm H}(T\\rightarrow 0)$ , which is a weighted average of the Hall response of the various sheets.",
"Similar Hall measurements were also used to rule out a Lifshitz transition in KFe$_2$ As$_2$ ,[12] in agreement with the lack of any change in dHvA frequencies.",
"[13] Several studies on the Ba$_{1-x}$ K$_{x}$ Fe$_2$ As$_2$ series suggest that lattice parameters, in particular the As-Fe-As bond angle, control $T_{\\rm c}$ .",
"[2], [3], [4], [18] To explore this hypothesis, we measured the lattice parameters of KFe$_2$ As$_2$ as a function of pressure, up to 60 kbar, in order to find out how much pressure is required to tune the lattice parameters of CsFe$_2$ As$_2$ so they match those of KFe$_2$ As$_2$ .",
"Cs has a larger atomic size than K, hence one can view CsFe$_2$ As$_2$ as a negative-pressure version of KFe$_2$ As$_2$ .",
"The four panels of Fig.",
"REF show the pressure variation of the lattice constants $a$ and $c$ , the unit cell volume ($V=a^2c$ ), and the intra-planar As-Fe-As bond angle ($\\alpha $ ) in KFe$_2$ As$_2$ .",
"The red horizontal line in each panel marks the value of the corresponding lattice parameter in CsFe$_2$ As$_2$ .",
"[5] In order to tune $a$ , $c$ , $V$ , and $\\alpha $ in KFe$_2$ As$_2$ to match the corresponding values in CsFe$_2$ As$_2$ , a negative pressure of approximately $-10$ , $-75$ , $-30$ , and $-30$ kbar is required, respectively.",
"Adding these numbers to the critical pressure for KFe$_2$ As$_2$ ($P_{\\rm c}$ = 17.5 kbar), we would naively estimate that the critical pressure in CsFe$_2$ As$_2$ should be $P_{\\rm c}$ $\\simeq 30$ kbar or higher.",
"We find instead that $P_{\\rm c}$ = 14 kbar, showing that other factors are involved in controlling $P_{\\rm c}$ .",
"Figure: Structural parameters of KFe 2 _2As 2 _2 as a function of pressure, up to 60 kbar:a) lattice constant aa;b) lattice constant cc;c) unit cell volume V=a 2 cV=a^2c;d) the intra-planar As-Fe-As bond angle α\\alpha as defined in the inset (See appendix for the inter-planar bond angle).Experimental errors on lattice parameters are smaller than symbol dimensions.The black dotted line in panel a, b, and c is a fit to the standard Murnaghan equation of state extended smoothly to negative pressures.",
"From the fits, we extract the moduli of elasticity and report them in appendix .The black dotted line in panel d is a third order power law fit.In each panel, the horizontal red line marks the lattice parameter of CsFe 2 _2As 2 _2,and the vertical red line gives the negative pressure required for the lattice parameter of KFe 2 _2As 2 _2 to reach the value in CsFe 2 _2As 2 _2.Figure: Pressure dependence of T c T_{\\rm c} in three samples:pure KFe 2 _2As 2 _2 (black circles),less pure KFe 2 _2As 2 _2 (grey circles),and CsFe 2 _2As 2 _2 (sample 2, red circles).Even though the T c T_{\\rm c} values for the two KFe 2 _2As 2 _2 samples are different due to different disorder levels,measured by their different residual resistivity ρ 0 \\rho _0,the critical pressure is the same (P c P_{\\rm c} = 17.5 kbar).This shows that the effect of disorder on P c P_{\\rm c} in KFe 2 _2As 2 _2 is negligible.For comparable ρ 0 \\rho _0, the critical pressure in CsFe 2 _2As 2 _2, P c P_{\\rm c} = 14 kbar, is clearly smaller than in KFe 2 _2As 2 _2.Figure: a) Resistivity data for the KFe 2 _2As 2 _2 sample with ρ 0 =1.3\\rho _0=1.3 μΩ cm \\mathrm {\\mu \\Omega \\, cm} at five selected pressures.The black vertical arrow shows a cut through each curve at T=20T=20 K and the dashed line is a power law fit to the curve at P=23.8P=23.8 kbar from 5 to 15 K that is used to extract the residual resistivity ρ 0 \\rho _0.Inelastic resistivity, defined as ρ(T=20K)-ρ 0 \\rho (T = 20~\\rm {K}) - \\rho _0 is plotted vs P/P c P / P_{\\rm c} inb) the less pure KFe 2 _2As 2 _2 sample,c) the purer KFe 2 _2As 2 _2 sample, andd) CsFe 2 _2As 2 _2 (sample 2)where P c P_{\\rm c} = 17.5 kbar for KFe 2 _2As 2 _2 and P c P_{\\rm c} = 14 kbar for CsFe 2 _2As 2 _2.In panel (b), (c), and (d) the dashed black line is a linear fit to the data above P/P c =1P / P_{\\rm c} = 1.It is possible that the lower $P_{\\rm c}$ in CsFe$_2$ As$_2$ could be due to the fact that $T_{\\rm c}$ itself is lower than in KFe$_2$ As$_2$ at zero pressure, i.e.",
"that the low-pressure phase is weaker in CsFe$_2$ As$_2$ .",
"One hypothesis for the lower $T_{\\rm c}$ in CsFe$_2$ As$_2$ is a higher level of disorder.",
"To test this idea, we studied the pressure dependence of $T_{\\rm c}$ in a less pure KFe$_2$ As$_2$ sample.",
"Fig.",
"REF compares the $T$ -$P$ phase diagram in three samples: 1) a high-purity KFe$_2$ As$_2$ sample, with $\\rho _0=0.2~\\mu \\Omega $ cm (from ref.",
"taftisudden2013); 2) a less pure KFe$_2$ As$_2$ sample, with $\\rho _0=1.3~\\mu \\Omega $ cm, measured here; 3) a CsFe$_2$ As$_2$ sample (sample 2), with $\\rho _0=1.5~\\mu \\Omega $ cm.",
"Different disorder levels in our samples are due to growth conditions, not to deliberate chemical substitution or impurity inclusions.",
"First, we observe that a 6-fold increase of $\\rho _0$ has negligible impact on $P_{\\rm c}$ in KFe$_2$ As$_2$ .",
"Secondly, we observe that $P_{\\rm c}$ is 4 kbar smaller in CsFe$_2$ As$_2$ than in KFe$_2$ As$_2$ , for samples of comparable $\\rho _0$ .",
"These observations rule out the idea that disorder could be responsible for the lower value of $P_{\\rm c}$ in CsFe$_2$ As$_2$ compared to KFe$_2$ As$_2$ ."
],
[
"Discussion",
"We have established a common trait in CsFe$_2$ As$_2$ and KFe$_2$ As$_2$ : both systems have a sudden reversal in the pressure dependence of $T_{\\rm c}$ , with no change in the underlying Fermi surface.",
"The question is: what controls that transition?",
"Why does the low-pressure superconducting state become unstable against the high-pressure state?",
"In a recent theoretical work by Fernandes and Millis, it is demonstrated that different pairing interactions in 122 systems can favour different pairing symmetries.",
"[11] In their model, SDW-type magnetic fluctuations, with wavevector $(\\pi ,0)$ , favour $s_{\\pm }$ pairing, whereas Néel-type fluctuations, with wavevector $(\\pi ,\\pi )$ , strongly suppress the $s_{\\pm }$ state and favour $d$ -wave pairing.",
"A gradual increase in the $(\\pi ,\\pi )$ fluctuations eventually causes a phase transition from an $s_{\\pm }$ superconducting state to a $d$ -wave state, producing a V-shaped $T_{\\rm c}$ vs $P$ curve.",
"[11] In KFe$_2$ As$_2$ and CsFe$_2$ As$_2$ , it is conceivable that two such competing interactions are at play, with pressure tilting the balance in favor of one versus the other.",
"We explore such a scenario by looking at how the inelastic scattering evolves with pressure, measured via the inelastic resistivity, defined as $\\rho (T) - \\rho _0$ , where $\\rho _0$ is the residual resistivity.",
"Fig.",
"REF (a) shows raw resistivity data from the KFe$_2$ As$_2$ sample with $\\rho _0=1.3$ $\\mathrm {\\mu \\Omega \\, cm}$ below 30 K. To extract $\\rho (T) - \\rho _0$ at each pressure, we make a cut through each curve at $T=20$ K and subtract from it the residual resistivity $\\rho _0$ that comes from a power-law fit $\\rho =\\rho _0+AT^n$ to each curve.",
"$\\rho _0$ is determined by disorder level and does not change as a function of pressure.",
"The resulting $\\rho (T=20~\\rm {K})-\\rho _0$ values for this sample are then plotted as a function of normalized pressure $P / $$P_{\\rm c}$ in Fig.",
"REF (b).",
"Through a similar process we extract the pressure dependence of $\\rho (20~\\rm {K})-\\rho _0$ in CsFe$_2$ As$_2$ and the purer KFe$_2$ As$_2$ sample with $\\rho _0 = 0.2$ $\\mathrm {\\mu \\Omega \\, cm}$ in Fig.",
"REF (c) and (d).",
"In all three samples, at $P / $$P_{\\rm c}$ $>1$ , the inelastic resistivity varies linearly with pressure.",
"As $P$ drops below $P_{\\rm c}$ , the inelastic resistivity in (K,Cs)Fe$_2$ As$_2$ shows a clear rise below their respective $P_{\\rm c}$ , over and above the linear regime.",
"Fig.",
"REF therefore suggests a connection between the transition in the pressure dependence of $T_{\\rm c}$ and the appearance of an additional inelastic scattering process.",
"Note that our choice of $T=20$ K for the inelastic resistivity is arbitrary.",
"Resistivity cuts at any finite temperature above $T_c$ give qualitatively similar results.",
"The Fermi surface of KFe$_2$ As$_2$ includes three $\\Gamma $ -centered hole-like cylinders.",
"A possible pairing state is an $s_{\\pm }$ state where the change of sign occurs between the inner cylinder and the middle cylinder, favored by a small-$Q$ interaction.",
"[10] By contrast, the intraband inelastic scattering wavevectors that favour $d$ -wave pairing are large-$Q$ processes.",
"[20] Therefore, one scenario in which to understand the evolution in the inelastic resistivity with pressure (Fig.",
"5), and its link to the $T_{\\rm c}$ reversal, is the following.",
"At low pressure, the large-$Q$ scattering processes that favor $d$ -wave pairing make a substantial contribution to the resistivity, as they produce a large change in momentum.",
"These weaken with pressure, causing a decrease in both $T_{\\rm c}$ and the resistivity.",
"This decrease persists until the low-$Q$ processes that favor $s_{\\pm }$ pairing, less visible in the resistivity, come to dominate, above $P_{\\rm c}$ .",
"In summary, we discovered a pressure-induced reversal in the dependence of the transition temperature $T_{\\rm c}$ on pressure in the iron-based superconductor CsFe$_2$ As$_2$ , similar to a our previous finding in KFe$_2$ As$_2$ .",
"We interpret the $T_{\\rm c}$ reversal at the critical pressure $P_{\\rm c}$ as a transition from one pairing state to another.",
"The fact that $P_{\\rm c}$ in CsFe$_2$ As$_2$ is smaller than in KFe$_2$ As$_2$ , even though all lattice parameters would suggest otherwise, shows that structural parameters alone do not control $P_{\\rm c}$ .",
"We also demonstrate that disorder has negligible effect on $P_{\\rm c}$ .",
"Our study of the pressure dependence of resistivity in CsFe$_2$ As$_2$ and KFe$_2$ As$_2$ reveals a possible link between $T_{\\rm c}$ and inelastic scattering.",
"Our proposal is that the high-pressure phase in both materials is an $s_{\\pm }$ state that changes sign between $\\Gamma $ -centered pockets.",
"As the pressure is lowered, the large-$Q$ inelastic scattering processes that favor $d$ -wave pairing in pure KFe$_2$ As$_2$ and CsFe$_2$ As$_2$ grow until at a critical pressure $P_{\\rm c}$ they cause a transition from one superconducting state to another, with a change of pairing symmetry from $s$ -wave to $d$ -wave.",
"The experimental evidence for this is the fact that below $P_{\\rm c}$ the inelastic resistivity, measured as the difference $\\rho (20~\\mathrm {K})-\\rho _0$ , deviates upwards from its linear pressure dependence at high pressure."
],
[
"ACKNOWLEDGMENTS",
"We thank A. V. Chubukov, R. M. Fernandes and A. J. Millis for helpful discussions, and S. Fortier for his assistance with the experiments.",
"The work at Sherbrooke was supported by the Canadian Institute for Advanced Research and a Canada Research Chair and it was funded by NSERC, FRQNT and CFI.",
"Work done in China was supported by the National Natural Science Foundation of China (Grant No.",
"11190021), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences, and the National Basic Research Program of China.",
"Research at the University of Toronto was supported by the NSERC, CFI, Onatrio Ministry of Research and Innovation, and Canada Research Chair program.",
"The Canadian Light Source is funded by CFI, NSERC, the National Research Council Canada, the Canadian Institutes of Health Research, the Government of Saskatchewan, Western Economic Diversification Canada, and the University of Saskatchewan."
],
[
"X-ray data",
"All our X-ray measurements are performed at room temperature using angle-dispersive technique with the HXMA beam line at CLS.",
"Figure REF includes two representative structural refinements of the X-ray diffraction data at $P=1.6$ kbar and $P=51$ kbar.",
"2D diffraction data from the image plate detector were reduced to 1D using the FIT2D program [21] and plotted as intensity vs $2\\Theta $ .",
"The structural refinements were performed using the GSAS software package.",
"[17] The experimental data points are illustrated by red crosses, the best fit to the diffraction pattern is illustrated by the solid black line, and the difference between the two curves is denoted by the solid blue line.",
"The Bragg reflections corresponding to the tetragonal $I4/mmm$ structure of KFe$_2$ As$_2$ are indicated by the black tick marks below the data."
],
[
"Bond angles",
"Within the tetragonal structure of KFe$_2$ As$_2$ , there are two bond angles in each FeAs$_4$ tetrahedron [22] as indicated in the inset of Fig.",
"REF : The intra-planar bond angle ($\\alpha $ ) that spans over the bond from one As plane to an Fe atom and back to an As atom in the original plane and the inter-planar bond angle ($\\beta $ ) that spans over the bond from one As plane through an Fe atom to the next As plane.",
"In the case of an ideal undistorted tetrahedron $\\alpha = \\beta = 109.47^{\\circ }$ .",
"In Fig.",
"REF (d) we present only the intra-planar bond angle $\\alpha $ to show that about $-30$ kbar is required to tune $\\alpha $ from its value in KFe$_2$ As$_2$ to CsFe$_2$ As$_2$ .",
"For completeness, here we plot the pressure evolution of both bond angles in Fig.",
"REF .",
"$\\alpha $ decreases as a function of pressure while $\\beta $ increases, hence, the size of the tetragonal distortion in KFe$_2$ As$_2$ grows progressively larger as the pressure increases.",
"Interestingly, the form of this tetragonal distortion is opposite to that observed in Ca$_{0.67}$ Sr$_{0.33}$ Fe$_2$ As$_2$ where applied pressure causes intra-layer bond angles to increase and inter-layer bond angles to decrease.",
"[22]"
],
[
"Anisotropic compressibility in KFe$_2$ As{{formula:9abaf381-8d92-4209-8d92-db7a3a9affb8}}",
"In Fig.",
"REF , we fit our data to the Murnaghan equation of state: [19] $P(V) = \\frac{K}{K^{\\prime }}\\left[ \\left( \\frac{V}{V_0} \\right)^{-K^{\\prime }}-1 \\right]$ and extend it smoothly to negative pressures to find how much pressure is required to tune the lattice parameters of KFe$_2$ As$_2$ to those of CsFe$_2$ As$_2$ .",
"Note that the compressibility of KFe$_2$ As$_2$ appears to be anisotropic.",
"The fits also allow us to extract the bulk modulus $K$ and its pressure derivative $K^{\\prime }=\\partial K/ \\partial P$ in KFe$_2$ As$_2$ .",
"Table REF summarizes the values of the bulk modulus $K$ as well as the moduli of elasticity along the $a$ - and $c$ -axes.",
"The modulus of elasticity appears to be almost identical along the $a$ - and the $c$ -axes, but the first derivative of the modulus is over an order of magnitude larger along the $a$ -axis.",
"This accounts for the roughly 40% smaller compression observed for the in-plane lattice constant."
]
] | 1403.0110 |
[
[
"Polyhedra, Complexes, Nets and Symmetry"
],
[
"Abstract Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties.",
"They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig-zag, or helical.",
"A polyhedron or complex is \"regular\" if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples).",
"There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra.",
"Their edge graphs are nets well-known to crystallographers, and we identify them explicitly.",
"There also are 6 infinite families of \"chiral\" apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits."
],
[
"Introduction",
"Polyhedra and polyhedra-like structures in ordinary euclidean 3-space $\\mathbb {E}^3$ have been studied since the early days of geometry (Coxeter, 1973).",
"However, with the passage of time, the underlying mathematical concepts and treatments have undergone fundamental changes.",
"Over the past 100 years we can observe a shift from the classical approach viewing a polyhedron as a solid in space, to topological approaches focussing on the underlying maps on surfaces (Coxeter & Moser, 1980), to combinatorial approaches highlighting the basic incidence structure but deliberately suppressing the membranes that customarily span the faces to give a surface.",
"These topological and combinatorial perspectives are already appearing in the well-known Kepler-Poinsot polyhedra and Petrie-Coxeter polyhedra (Coxeter, 1937, 1973).",
"They underlie the skeletal approach to polyhedra proposed in (Grünbaum, 1977a, 1977b) that has inspired a rich new theory of geometric polyhedra and symmetry (Dress, 1981, 1985; McMullen & Schulte, 1997, 2002; McMullen, in prep.).",
"The polygonal complexes described in this paper form an even broader class of discrete skeletal space structures than polyhedra.",
"Like polyhedra they are comprised of vertices, joined by edges, assembled in careful fashion into polygons, the faces, allowed to be skew or infinite (Pellicer & Schulte, 2010, 2013).",
"However, unlike in polyhedra, more than two faces are permitted to meet at an edge.",
"The regular polygonal complexes in $\\mathbb {E}^3$ are finite structures or infinite 3-periodic structures with crystallographic symmetry groups exhibiting interesting geometric, combinatorial and algebraic properties.",
"This class includes the regular polyhedra but also many unfamiliar figures, once the planarity or finiteness of the polygonal faces is abandoned.",
"Because of their skeletal structure, polygonal complexes are of natural interest to crystallographers.",
"Regular polyhedra traditionally play a prominent role in crystal chemistry (Wells, 1977; O'Keeffe & Hyde, 1996; Delgado-Friedrichs et al.",
"2005; O'Keeffe et al, 2008; O'Keeffe, 2008).",
"There is considerable interest in the study of 3-periodic nets and their relationships to polyhedra.",
"Nets are 3-periodic geometric graphs in space that represent crystal structures, in the simplest form with vertices corresponding to atoms and edges to bonds.",
"The edge graphs of almost all regular polygonal complexes in $\\mathbb {E}^3$ are highly-symmetric nets, with the only exceptions arising from the polyhedra which are not 3-periodic.",
"We explicitly identify the nets by building on the work of (O'Keeffe, 2008) and exploiting the methods developed in (Delgado-Friedrichs et al., 2003).",
"Symmetry of discrete geometric structures is a frequently recurring theme in science.",
"Polyhedral structures occur in nature and art in many contexts that a priori have little apparent relation to regularity (Fejes Tóth, 1964; Senechal, 2013).",
"Their occurrence in crystallography as crystal nets is a prominent example.",
"See also (Wachman et al, 1974) for an interesting architecture inspired study of polyhedral structures that features a wealth of beautiful illustrations of figures reminiscent of skeletal polyhedral structures.",
"The present paper is organized as follows.",
"In Section we investigate basic properties of polygonal complexes, polyhedra, and nets, in particular focussing on structures with high symmetry.",
"Section is devoted to the study of the symmetry groups of regular polygonal complexes as well as chiral polyhedra.",
"In Sections and we review the complete classification of the regular and chiral polyhedra following (McMullen & Schulte, 2002, Ch.",
"7E) and (Schulte, 2004, 2005), respectively.",
"The final Section describes the classification of the regular polygonal complexes (Pellicer & Schulte, 2010, 2013).",
"Along the way we determine the nets of all 3-periodic regular polygonal complexes."
],
[
"Polyhedra, Complexes and Nets",
"Polygonal complexes are geometric realizations in $\\mathbb {E}^3$ of abstract incidence complexes of rank 3 with polygon faces, that is, of abstract polygonal complexes (Danzer & Schulte, 1982).",
"We elaborate on this aspect in Section REF .",
"Here we will not require familiarity with incidence complexes.",
"However, occasionally it is useful to bear this perspective in mind.",
"Polyhedra are the polygonal complexes with just two faces meeting at an edge."
],
[
"Polygonal complexes",
"Following (Grünbaum, 1977a), a finite polygon, or an $n$ -gon (with $n\\ge 3$ ), consists of a sequence $(v_1, v_2, \\dots , v_n)$ of $n$ distinct points in $\\mathbb {E}^3$ , as well as of the line segments $(v_1, v_2), (v_2,v_3), \\ldots , (v_{n-1},v_n)$ and $(v_n, v_1)$ .",
"Note that we are not making a topological disc spanned into the polygon part of the definition of a polygon.",
"In particular, unless stated otherwise, the term “convex polygon\" refers only to the boundary edge path of what is usually called a convex polygon; that is, we ignore the interior.",
"A (discrete) infinite polygon, or apeirogon, similarly consists of an infinite sequence of distinct points $(\\dots , v_{-2},v_{-1}, v_0, v_1, v_2, \\dots )$ in $\\mathbb {E}^3$ , as well as of the line segments $(v_i, v_{i+1})$ for each $i$ , such that each compact subset of $\\mathbb {E}^3$ meets only finitely many line segments.",
"In either case the points are the vertices and the line segments the edges of the polygon.",
"Following (Pellicer & Schulte, 2010), a polygonal complex, or simply a complex, $\\mathcal {K}$ in $\\mathbb {E}^{3}$ is a triple $(\\mathcal {V},\\mathcal {E},\\mathcal {F}$ ) consisting of a set $\\mathcal {V}$ of points, called vertices, a set $\\mathcal {E}$ of line segments, called edges, and a set $\\mathcal {F}$ of polygons, called faces, satisfying the following properties.",
"(a) The graph $(\\mathcal {V},\\mathcal {E})$ , the edge graph of $\\mathcal {K}$ , is connected.",
"(b) The vertex-figure of $\\mathcal {K}$ at each vertex of $\\mathcal {K}$ is connected.",
"By the vertex-figure of $\\mathcal {K}$ at a vertex $v$ we mean the graph, possibly with multiple edges, whose vertices are the vertices of $\\mathcal {K}$ adjacent to $v$ and whose edges are the line segments $(u,w)$ , where $(u, v)$ and $(v, w)$ are edges of a common face of $\\mathcal {K}$ .",
"(There may be more than one such face in $\\mathcal {K}$ , in which case the corresponding edge $(u,w)$ of the vertex-figure at $v$ has multiplicity given by the number of such faces.)",
"(c) Each edge of $\\mathcal {K}$ is contained in exactly $r$ faces of $\\mathcal {K}$ , for a fixed number $r \\ge 2$ .",
"(d) $\\mathcal {K}$ is discrete, in the sense that each compact subset of $\\mathbb {E}^{3}$ meets only finitely many faces of $\\mathcal {K}$ .",
"A (geometric) polyhedron in $\\mathbb {E}^{3}$ is a polygonal complex with $r=2$ .",
"Thus, a polyhedron is a complex in which each edge lies in exactly two faces.",
"The vertex-figures of a polyhedron are finite (simple) polygonal cycles.",
"An infinite polyhedron in $\\mathbb {E}^3$ is also called an apeirohedron.",
"A simple example of a polyhedron is shown in Figure REF .",
"This is the “Petrie-dual\" of the cube obtained from the ordinary cube by replacing the square faces with the Petrie polygons while retaining the vertices and edges.",
"Recall here that a Petrie polygon, or 1-zigzag, of a polyhedron is a path along the edges such that any two, but no three, consecutive edges belong to a common face.",
"The Petrie polygons of the cube are skew hexagons, and there are four of them.",
"Hence the polyhedron in Figure REF has 8 vertices, 12 edges, and 4 skew hexagonal faces, with 3 faces coming together at each vertex.",
"Figure: The Petrie dual of the cube.",
"Shown are its four faces (in red, blue, green and black).",
"The faces are the Petrie polygons of the cube.Polyhedra are the best studied class of polygonal complexes and include the traditional convex polyhedra and star-polyhedra in $\\mathbb {E}^{3}$ (Coxeter, 1973; Grünbaum, 1977a; McMullen & Schulte, 2002; McMullen, in prep.).",
"When viewed purely combinatorially, the set of vertices, edges, and faces of a geometric polyhedron, ordered by inclusion, is an abstract polyhedron, or an abstract polytope of rank 3.",
"More generally, the underlying combinatorial “complex\" determined by the vertices, edges, and faces of any polygonal complex $\\mathcal {K}$ (given by the triple $(\\mathcal {V},\\mathcal {E},\\mathcal {F}$ )), ordered by inclusion, is an incidence complex of rank 3 in the sense of (Danzer & Schulte, 1982).",
"Here the term “rank\" refers to the “combinatorial dimension\" of the object; thus, rank 3 complexes are incidence structures made up of objects called vertices (of rank 0), edges (of rank 1), and faces (of rank 2), in a purely combinatorial sense.",
"An easy example of an infinite polygonal complex in $\\mathbb {E}^{3}$ which is not a polyhedron is given by the vertices, edges, and square faces of the standard cubical tessellation $\\mathcal {C}$ (see Figure REF ).",
"This complex $\\mathcal {K}$ is called the 2-skeleton of $\\mathcal {C}$ ; each edge lies in four square faces so $r=4$ .",
"The tiles (cubes) of $\\mathcal {C}$ are irrelevant in this context.",
"A finite complex with $r=3$ can similarly be derived from the 2-skeleton of the 4-cube projected (as a Schlegel diagram) into $\\mathbb {E}^{3}$ .",
"Figure: The 2-skeleton of the cubical tessellation.",
"Each edge lies in four square faces.The cubical tessellation $\\mathcal {C}$ in $\\mathbb {E}^3$ gives rise to several other interesting complexes.",
"For example, the family of all Petrie polygons of all cubes in $\\mathcal {C}$ gives the (hexagonal) faces of a polygonal complex in which every edge belongs to exactly eight faces; the vertices and edges are just those of $\\mathcal {C}$ .",
"This is the complex $\\mathcal {K}_{6}(1,2)$ appearing in Table REF below.",
"Note here that every edge of a cube in $\\mathcal {C}$ belongs to precisely two Petrie polygons of this cube; since every edge of $\\mathcal {C}$ belongs to four cubes in $\\mathcal {C}$ , every edge of $\\mathcal {C}$ must belong to eight Petrie polygons of cubes so $r=8$ .",
"If only the Petrie polygons of alternate cubes in $\\mathcal {C}$ are taken as faces, we obtain a “subcomplex\" in which every edge belongs to exactly four hexagonal faces so then $r=4$ .",
"The set of vertices, edges, and triangular faces of a cuboctahedron is not a polygonal complex; each edge lies in only one face, and the vertex-figures are not connected.",
"Although our terminology could be adapted to cover polygonal structures in which some edges lie only in one face (as in this last example), we will explicitly exclude them here.",
"Moreover, as we are mainly interested in highly symmetric structures, our definition includes the homogeneity condition (c).",
"This condition is automatically satisfied for any polygonal structures with sufficiently high symmetry (for example, as given by the edge-transitivity of the symmetry group), provided at least two faces meet at an edge.",
"However, for an investigation of polygonal structures regardless of symmetry it is useful to replace part (c) in the definition of a polygonal complex by the following weaker requirement: (c') Each edge of $\\mathcal {K}$ is contained in at least two faces of $\\mathcal {K}$ .",
"In this paper we will not require any of these modifications."
],
[
"Highly symmetric complexes",
"There are several distinguished classes of highly symmetric polygonal complexes, each characterized by a distinguished transitivity property of the symmetry group.",
"Some of these classes have analogues in the traditional theory of polyhedra but others feature characteristics that do not occur in the classical theory.",
"We let $G(\\mathcal {K})$ denote the symmetry group of a polygonal complex $\\mathcal {K}$ , that is, the group of all Euclidean isometries of the affine hull of $\\mathcal {K}$ that map $\\mathcal {K}$ to itself.",
"(Except when $\\mathcal {K}$ is planar, this affine hull is $\\mathbb {E}^{3}$ itself.)",
"The most highly symmetric polygonal complexes $\\mathcal {K}$ are those that we call regular, meaning that the symmetry group $G(\\mathcal {K})$ is transitive on the flags of $\\mathcal {K}$ .",
"A flag of $\\mathcal {K}$ is an incident triple consisting of a vertex, an edge and a face of $\\mathcal {K}$ .",
"Two flags of $\\mathcal {K}$ are called $j$ -adjacent if they differ precisely in their elements of rank $j$ , that is, their vertices, edges, or faces if $j=0$ , 1 or 2, respectively.",
"Flags are $j$ -adjacent to only one flag if $j=0$ or 1, or precisely $r-1$ flags if $j=2$ .",
"For example, in the 2-skeleton of the cubical tessellation (with $r=4$ ) shown in Figure REF every flag has exactly three 2-adjacent flags.",
"For polyhedra, every flag has one $j$ -adjacent flag for every rank $j$ .",
"The faces of a regular polygonal complex are (finite or infinite) congruent regular polygons in $\\mathbb {E}^3$ , with “regular\" meaning that their geometric symmetry group is transitive on the flags of the polygon.",
"(A flag of a polygon is an incident vertex-edge pair.)",
"Note that regular polygons in $\\mathbb {E}^{3}$ are necessarily of one of the following kinds: finite, planar (convex or star-) polygons or non-planar (skew) polygons; (infinite) apeirogons, either planar zigzags or helical polygons; or linear polygons, either a line segment or a linear apeirogon with equal-sized edges (Grünbaum, 1977a; Coxeter, 1991).",
"We can show that linear regular polygons do not occur as faces of polygonal complexes.",
"We call a polygonal complex $\\mathcal {K}$ semiregular (or uniform) if the faces of $\\mathcal {K}$ are regular polygons (allowed to be non-planar or infinite) and $G(\\mathcal {K})$ is transitive on the vertices.",
"A polygonal complex $\\mathcal {K}$ is said to be vertex-transitive, edge-transitive, or face-transitive if $G(\\mathcal {K})$ is transitive on the vertices, edges, or faces, respectively.",
"A complex which is vertex-transitive, edge-transitive, and face-transitive is called totally transitive.",
"Every regular complex is totally transitive, but not vice versa.",
"We call $\\mathcal {K}$ a 2-orbit polygonal complex if $\\mathcal {K}$ has precisely two flag orbits under the symmetry group (Hubard, 2010; Cutler & Schulte, 2011)."
],
[
"Chiral polyhedra",
"Chiral polyhedra are arguably the most important class of 2-orbit complexes.",
"A (geometric) polyhedron $\\mathcal {K}$ is chiral if $G(\\mathcal {K})$ has exactly two orbits on the flags such that any two adjacent flags are in distinct orbits (Schulte, 2004, 2005).",
"This notion of chirality for polyhedra is different from the standard notion of chirality used in crystallography, but is inspired by it.",
"The proper setting is that of a “chiral realization\" in $\\mathbb {E}^{3}$ of an abstractly chiral or regular abstract polyhedron, where here abstract chirality or regularity are defined as above, but now in terms of the combinatorial automorphism group of the abstract polyhedron, not the geometric symmetry group.",
"In a sense that can be made precise, abstract chiral polyhedra occur in a “left-handed\" and a “right-handed\" version (Schulte & Weiss, 1991, 1994), although this handedness is combinatorial and not geometric.",
"Thus a chiral geometric polyhedron has maximum symmetry by “combinatorial rotation\" (but not by “combinatorial reflection\"), and has all its “rotational\" combinatorial symmetries realized by euclidean isometries (but not in general by euclidean rotations).",
"A regular geometric polyhedron has maximum symmetry by “combinatorial reflection\", and has all its combinatorial symmetries realized by euclidean isometries."
],
[
"Nets",
"Nets are important tools used in the modeling of 3-periodic structures in crystal chemistry and materials science.",
"A net in $\\mathbb {E}^3$ is a 3-periodic connected (simple) graph with straight edges (Delgado-Friedrichs & O'Keeffe, 2005; Wells, 1977).",
"Recall here that a figure in $\\mathbb {E}^3$ is said to be 3-periodic if the translation subgroup of its symmetry group is generated by translations in 3 independent directions.",
"Highly symmetric nets often arise as the edge graphs (the graphs formed by the vertices and edges) of 3-periodic higher rank structures in $\\mathbb {E}^3$ such as 3-dimensional tilings, apeirohedra, and infinite polygonal complexes.",
"Note that the combinatorial automorphism group of a net $\\mathcal {N}$ can be larger than its geometric symmetry group $G(\\mathcal {N})$ .",
"The underlying abstract infinite graph of a net can often be realized in several different ways as a net in $\\mathbb {E}^3$ , and there is a natural interest in finding the maximum symmetry realization of this graph as a net.",
"The Reticular Chemistry Structure Resource (RCSR) database located at the website http://rcsr.anu.edu.au contains a rich collection of crystal nets including in particular the most symmetric examples (O'Keeffe et al., 2008).",
"It has become common practice to denote a net by a bold-faced 3-letter symbol such as abc.",
"Examples are described below.",
"The symbol is often a short-hand for a “famous\" compound represented by the net, or for the finer geometry of the net.",
"A net can represent many compounds.",
"Newly discovered nets tend to be named after the person(s) who discovered the net (using initials etc.).",
"Many nets have “alternative symbols\" but we will not require them here.",
"While the RCSR database contains information about 2000 named nets, TOPOS is a more recent research tool for the geometrical and topological analysis of crystal structures with a database of more than 70,000 nets (Blatov, 2012); it is currently under further development by Vladislav Blatov and Davide Proserpio.",
"The nets $\\mathcal {N}$ occurring in this paper are uninodal, meaning that $G(\\mathcal {N})$ is transitive on the vertices (nodes) of $\\mathcal {N}$ .",
"For a vertex $v$ of a net $\\mathcal {N}$ , the convex hull of the neighbors of $v$ in $\\mathcal {N}$ is called the coordination figure of $\\mathcal {N}$ at $v$ .",
"The edge graph of each regular polygonal complex $\\mathcal {K}$ is a net referred to as the net of the complex.",
"The identification of the nets arising as edge graphs of regular polygonal complexes is greatly aided by the fact that there already exists a classification of the nets in $\\mathbb {E}^3$ that are called regular or quasiregular in the chemistry literature (Delgado-Friedrichs et al., 2003, 2005).",
"Although this terminology for nets is not consistent with our terminology for polyhedral complexes, we will maintain it for the convenience of the reader.",
"Note that a net of a regular complex may have symmetries which are not symmetries of the complex.",
"A net $\\mathcal {N}$ in $\\mathbb {E}^3$ is called regular if $\\mathcal {N}$ is uninodal and if, for each vertex $v$ of $\\mathcal {N}$ , the coordination figure of $\\mathcal {N}$ at $v$ is a regular convex polygon (in the ordinary sense) or a Platonic solid whose own rotation symmetry group is a subgroup of the stabilizer of $v$ in $G(\\mathcal {N})$ (the site symmetry group of $v$ in $\\mathcal {N}$ ).",
"Here the rotation symmetry group of a regular convex polygon is taken relative to $\\mathbb {E}^3$ and is generated by two half-turns in $\\mathbb {E}^3$ .",
"As pentagonal symmetry is impossible, the coordination figures of a regular net must necessarily be triangles, squares, tetrahedra, octahedra or cubes.",
"A net $\\mathcal {N}$ is called quasiregular if $\\mathcal {N}$ is uninodal and the coordination figure of $\\mathcal {N}$ at every vertex is a quasiregular convex polyhedron.",
"Recall that a quasiregular convex polyhedron in $\\mathbb {E}^3$ is a semiregular convex polyhedron (with regular faces and a vertex-transitive symmetry group) with exactly two kinds of faces alternating around each vertex.",
"There are only two quasiregular convex polyhedra in $\\mathbb {E}^3$ (Coxeter, 1973), namely the well-known cuboctahedron $3.4.3.4$ and icosidodecahedron $3.5.3.5$ , of which the latter cannot occur because of its icosahedral symmetry.",
"Thus the coordination figures of a quasiregular net are cuboctahedra.",
"Conversely, a uninodal net with cuboctahedra as coordination figures is necessarily quasiregular.",
"There are exactly five regular nets in $\\mathbb {E}^3$ , one per possible coordination figure.",
"Following (O'Keeffe et al., 2008; Delgado-Friedrichs et al., 2003) these nets are denoted by srs, nbo, dia, pcu and bcu; their coordination figures are triangles, squares, tetrahedra, octahedra or cubes, respectively.",
"It is also known that there is just one quasiregular net in $\\mathbb {E}^3$ , denoted fcu, with coordination figure a cuboctahedron.",
"These nets have appeared in many publications, often under different names (O'Keeffe & Hyde, 1996; Delgado Friedrichs et al, 2003).",
"The net srs is the net of Strontium Silicide $\\rm {SrSi}_2$ , hence the notation.",
"It coincides with the “net (10,3)-a\" of (Wells, 1977), the “Laves net\" of (Pearce, 1978), the “Y$^*$ lattice complex\" of (Koch & Fischer, 1983), the net “3/10/c1\" of (Koch & Fischer, 1995), as well as the “labyrinth graph of the gyroid surface\" (Hyde & Ramsden, 2000).",
"For an account on the history of the srs net and its gyroid surface see also (Hyde et al., 2008).",
"The notation nbo signifies the net of Niobium Monoxide NbO and coincides with the lattice complex J$^*$ of (Koch & Fischer, 1983).",
"The net dia is the famous diamond net, or “lattice complex D\" (Koch & Fischer, 1983), which is the net of the diamond form of carbon.",
"The names pcu, fcu, and bcu stand for the “primitive cubic lattice\" (the standard cubic lattice), the “face-centered cubic lattice\", and the “body-centered\" cubic lattice in $\\mathbb {E}^3$ , respectively; these are also known as the lattice complexes $c$ P, $cI$ and $c$ F, respectively.",
"In addition we will also meet the nets denoted acs, sod, crs and hxg (O'Keeffe et al., 2008; O'Keeffe, 2008).",
"The net acs is named after Andrea C. Sudik (Sudik et al., 2005) and is observed in at least 177 compounds (according to the TOPOS database).",
"The net sod represents the sodalite structure and can be viewed as the edge-graph of the familiar tiling of $\\mathbb {E}^3$ by truncated octahedra also known as the Kelvin-structure (Delgado-Friedrichs et al., 2005).",
"The symbol crs labels the net of the oxygens coordination in idealized beta-cristobalite, but also represents at least 10 other compounds (according to the TOPOS database); the net is also known as dia-e and 3d-kagomè, and appears as the 6-coordinated sphere packing net corresponding to the cubic invariant lattice complex T (Koch & Fischer, 1983).",
"Finally, hxg has a regular hexagon as its coordination figure; its “augmented\" net (obtained by replacing the original vertices by the edge graph of the coordination figures) gives a structure representing polybenzene."
],
[
"Symmetry Groups of Complexes and Polyhedra",
"The symmetry groups $G:=G(\\mathcal {K})$ hold the key to the structure of regular polygonal complexes $\\mathcal {K}$ .",
"They have a distinguished generating set of subgroups $G_0,G_1,G_2$ obtained as follows.",
"Via a variant of Wythoff's construction (Coxeter, 1973), these subgroups enable us to recover a regular complex from its group."
],
[
"Distinguished generators",
"Choose a fixed, or base, flag $\\Phi := \\lbrace F_0, F_1, F_2\\rbrace $ of $\\mathcal {K}$ , where $F_{0},F_{1},F_{2}$ , respectively, denote the vertex, edge, or face of the flag.",
"For $i=0,1,2$ let $G_i$ denote the stabilizer of $\\Phi \\setminus \\lbrace F_i\\rbrace $ in $G$ ; this is the subgroup of $G$ stabilizing every element of $\\Phi $ except $F_i$ .",
"For example, $G_2$ consists of all symmetries of $\\mathcal {K}$ fixing $F_0$ and $F_1$ and thus fixing the entire line through $F_1$ pointwise.",
"Also, for $\\Psi \\subseteq \\Phi $ define $G_{\\Psi }$ to be the stabilizer of $\\Psi $ in $G$ , that is, the subgroup of $G$ stabilizing every element of $\\Psi $ .",
"Then $G_i$ is just $G_{\\lbrace F_j, F_k\\rbrace }$ for each $i$ , where here $\\lbrace i, j, k\\rbrace =\\lbrace 0,1,2\\rbrace $ .",
"Moreover, $G_\\Phi $ is the stabilizer of the base flag $\\Phi $ itself.",
"We also write $G_{F_i}:=G_{\\lbrace F_i\\rbrace }$ for $i=0,1,2$ ; this is the stabilizer of $F_i$ in $G$ .",
"The subgroups $G_0,G_1,G_2$ have remarkable properties.",
"In particular, $ G = \\langle G_0,G_1,G_2 \\rangle $ and $ G_0 \\cap G_1 = G_0 \\cap G_2 = G_1 \\cap G_2 = G_0 \\cap G_1 \\cap G_2 = G_\\Phi .$ While these properties already hold at the abstract level of incidence complexes (Schulte, 1983), the euclidean geometry of 3-space comes into play when we investigate the possible size of the flag stabilizer $G_\\Phi $ .",
"It turns out that there are two possible scenarios.",
"In fact, either $G_{\\Phi }$ is trivial, and then the (full) symmetry group $G$ is simply flag-transitive; or $G_{\\Phi }$ has order 2, the complex $\\mathcal {K}$ has planar faces, and $G_{\\Phi }$ is generated by the reflection in the plane of $\\mathbb {E}^{3}$ containing the base face $F_2$ of $\\mathcal {K}$ .",
"In the former case we call $\\mathcal {K}$ a simply flag-transitive complex.",
"In the latter case we say that $\\mathcal {K}$ is non-simply flag-transitive, or that $\\mathcal {K}$ has face mirrors since then the planes in $\\mathbb {E}^3$ through faces of $\\mathcal {K}$ are mirrors (fixed point sets) of reflection symmetries.",
"If $\\mathcal {K}$ is a simply flag-transitive complex, then $G_0 = \\langle R_0 \\rangle $ and $G_1 = \\langle R_1 \\rangle $ , for some point, line or plane reflection $R_0$ and some line or plane reflection $R_1$ ; and $G_2$ is a cyclic or dihedral group of order $r$ .",
"(A reflection in a line is a half-turn about the line.)",
"Moreover, $ G_{F_0}=\\langle R_1, G_2 \\rangle ,\\; G_{F_1}=\\langle R_0, G_2 \\rangle , \\;G_{F_2}=\\langle R_0, R_1 \\rangle \\cong D_p ,$ where $p$ is the number of vertices in a face of $\\mathcal {K}$ and $D_p$ denotes the dihedral group of order $2p$ (allowing $p=\\infty $ ).",
"Since $\\mathcal {K}$ is discrete, the stabilizer $G_{F_0}$ of $F_0$ in $G$ is necessarily a finite group called the vertex-figure group of $\\mathcal {K}$ at $F_0$ .",
"In particular, the vertex-figure group $G_{F_0}$ acts simply flag-transitively on the finite graph forming the vertex-figure of $\\mathcal {K}$ at $F_0$ .",
"(A flag of a graph is just an incident vertex-edge pair.)",
"If $\\mathcal {K}$ is a non-simply flag-transitive complex, then $ G_0 \\cong C_2 \\times C_2 \\cong G_1, \\; G_2\\cong D_r $ and the vertex-figure group $G_{F_0}=\\langle G_1,G_2 \\rangle $ is again finite as $\\mathcal {K}$ is discrete."
],
[
"The case of polyhedra",
"The theory is particularly elegant in the case when $\\mathcal {K}$ is a regular polyhedron.",
"Then $\\mathcal {K}$ is necessarily simply flag-transitive and $G_2$ is also generated by a reflection $R_2$ in a point, line or plane.",
"Thus $ G=\\langle R_0,R_1,R_2\\rangle $ and $G$ is a discrete (generalized) reflection group in $\\mathbb {E}^{3}$ , where here the term “reflection group\" refers to a group generated by reflections in points, lines or planes.",
"A regular polyhedron has a (basic) Schläfli type $\\lbrace p,q\\rbrace $ , where as above $p$ is the number of vertices in a given face and $q$ denotes the number of faces containing a given vertex; here $p$ , but not $q$ , may be $\\infty $ .",
"The distinguished involutory generators $R_0,R_1,R_2$ of $G$ satisfy the standard Coxeter-type relations $ R_{0}^{2} = R_{1}^{2} = R_{2}^{2} = (R_{0}R_{1})^{p} = (R_{1}R_{2})^{q} = (R_{0}R_{2})^{2} = 1, $ but in general there are other independent relations too; these additional relations are determined by the cycle structure of the edge graph, or net, of $\\mathcal {K}$ (McMullen & Schulte, 2002, Ch. 7E).",
"For a chiral polyhedron $\\mathcal {K}$ , the symmetry group $G$ has two non-involutory distinguished generators $S_1,S_2$ obtained as follows.",
"Let again $\\Phi := \\lbrace F_{0},F_{1},F_{2}\\rbrace $ be a base flag, let $F_{0}^{\\prime }$ be the vertex of $F_1$ distinct from $F_0$ , let $F_{1}^{\\prime }$ be the edge of $F_2$ with vertex $F_0$ distinct from $F_1$ , and let $F_{2}^{\\prime }$ be the face containing $F_1$ distinct from $F_2$ .",
"Then the generator $S_{1}$ stabilizes the base face $F_2$ and cyclically permutes the vertices of $F_2$ in such a manner that $F_{1}S_{1} = F^{\\prime }_{1}$ (and $F^{\\prime }_{0}S_{1} = F_{0}$ ), while $S_{2}$ fixes the base vertex $F_0$ and cyclically permutes the vertices in the vertex-figure at $F_0$ in such a way that $F_{2}S_{2} = F^{\\prime }_{2}$ (and $F^{\\prime }_{1}S_{2} = F_{1}$ ).",
"These generators $S_1,S_2$ satisfy (among others) the relations $S_{1}^p = S_{2}^q = (S_{1}S_{2})^{2} = 1,$ where again $\\lbrace p,q\\rbrace $ is the Schläfli type of $\\mathcal {K}$ .",
"Two alternative sets of generators for $G$ are given by $\\lbrace S_1,T\\rbrace $ and $\\lbrace T,S_2\\rbrace $ , where $T:= S_{1}S_{2}$ is the involutory symmetry of $\\mathcal {K}$ that interchanges simultaneously the two vertices of the base edge $F_1$ and the two faces meeting at $F_1$ .",
"Combinatorially speaking, $T$ acts like a half-turn about the midpoint of the base edge (but geometrically, $T$ may not be a half-turn about a line)."
],
[
"Wythoff's construction",
"Regular polygonal complexes and chiral polyhedra can be recovered from their symmetry groups $G$ by variants of the classical Wythoff construction (Coxeter, 1973).",
"Two variants are needed, one essentially based on the generating subgroups $G_0,G_1,G_2$ of $G$ and applying only to polygonal complexes which are regular (McMullen & Schulte, 2002, Ch.",
"5; Pellicer & Schulte, 2010), and the other based on the generators $S_1,S_2$ and applying to both regular and chiral polyhedra.",
"(To subsume regular polyhedra under the latter case it is convenient to set $S_{1}:=R_{0}R_{1}$ , $S_{2}:=R_{1}R_{2}$ and $T:=S_{1}S_{2}=R_{0}R_{2}$ .)",
"For regular polygonal complexes (including polyhedra), Wythoff's construction proceeds from the base vertex $v:=F_0$ , the initial vertex, and builds the complex (or polyhedron) $\\mathcal {K}$ as an orbit structure, beginning with the construction of the base flag.",
"Relative to the set of generating subgroups, the essential property of $v$ is that it is invariant under all generating subgroups but the first; that is, $v$ is invariant under $G_{1}$ and $G_{2}$ but not $G_0$ .",
"The base vertex, $v$ , is already given.",
"The vertex sets of the base edge and base face of $\\mathcal {K}$ are the orbits of $v$ under the subgroups $G_0$ and $\\langle G_{0},G_{1}\\rangle $ , respectively.",
"This determines the base edge as the line segment joining its vertices, and the base face as an edge path joining its vertices in succession.",
"Once the base flag has been constructed, we simply obtain the vertices, edges and faces of $\\mathcal {K}$ as the images under $G$ of the base vertex, base edge or base face, respectively.",
"For chiral polyhedra $\\mathcal {K}$ we can similarly proceed from the alternative generators $T,S_2$ of $G$ , again choosing $v:=F_0$ as the initial (or base) vertex.",
"Now the base edge and base face of $\\mathcal {K}$ are given by the orbits of $v$ under $\\langle T\\rangle $ and $\\langle S_1\\rangle $ , respectively; and as before, the vertices, edges and faces of $\\mathcal {K}$ are just the images under $G$ of the base vertex, base edge or base face, respectively.",
"In practice, Wythoff's construction is often applied to groups that “look like\" symmetry groups of regular complexes or chiral polyhedra.",
"In fact, this approach then often enables us to establish the existence of such structures.",
"A necessary condition in this case is the existence of a common fixed point of $G_1$ and $G_2$ , which then becomes the initial vertex."
],
[
"Regular Polyhedra",
"Loosely speaking there are 48 regular polyhedra in $\\mathbb {E}^3$ , up to similarity (that is, congruence and scaling).",
"They comprise 18 finite polyhedra and 30 (infinite) apeirohedra.",
"We follow the classification scheme described in (McMullen & Schulte, 2002, Ch.",
"7E) and designate these polyhedra by generalized Schläfli symbols that usually are obtained by padding the basic symbol $\\lbrace p,q\\rbrace $ with additional symbols signifying specific information (such as extra defining relations for the symmetry group in terms of the distinguished generators)."
],
[
"Finite polyhedra",
"The finite regular polyhedra are all derived from the five Platonic solids: the tetrahedron $\\lbrace 3,3\\rbrace $ , the octahedron $\\lbrace 3,4\\rbrace $ , the cube $\\lbrace 4,3\\rbrace $ , the icosahedron $\\lbrace 3,5\\rbrace $ , and the dodecahedron $\\lbrace 5,3\\rbrace $ .",
"In addition to the Platonic solids, there are the four regular star-polyhedra, also known as Kepler-Poinsot polyhedra: the great icosahedron $\\lbrace 3,\\frac{5}{2}\\rbrace $ , the great stellated dodecahedron $\\lbrace \\frac{5}{2},3\\rbrace $ , the great dodecahedron $\\lbrace 5,\\frac{5}{2}\\rbrace $ , and the small stellated dodecahedron$\\lbrace \\frac{5}{2},5\\rbrace $ .",
"(The fractional entries $\\frac{5}{2}$ indicate that the corresponding faces or vertex-figures are star-pentagons.)",
"These nine examples are the classical regular polyhedra.",
"The remaining nine finite regular polyhedra are the Petrie duals of the nine classical regular polyhedra.",
"The Petrie dual of a regular polyhedron is (usually) a new regular polyhedron with the same vertices and edges, obtained by replacing the faces by the Petrie polygons.",
"The Petrie dual of the Petrie dual of a regular polyhedron is the original polyhedron.",
"For example, the four (skew hexagonal) Petrie polygons of the cube form the faces of the Petrie dual of the cube, which is usually denoted $\\lbrace 6,3\\rbrace _4$ and is shown in Figure REF .",
"(The suffix indicates the length of the Petrie polygon, 4 in this case.)",
"The underlying abstract polyhedron corresponds to a map with four hexagonal faces on the torus."
],
[
"Apeirohedra",
"The 30 apeirohedra fall into three families comprised of the 6 planar, the 12 “reducible\", and the 12 “irreducible\" examples.",
"Their symmetry groups are crystallographic groups.",
"The use of the terms “reducible\" and “irreducible\" for apeirohedra is consistent with what we observe at the group level: the symmetry group is affinely reducible or affinely irreducible, respectively.",
"In saying that a group of isometries of $\\mathbb {E}^3$ is affinely reducible, we mean that there is a line $l$ in $\\mathbb {E}^3$ such that the group permutes the lines parallel $l$ ; the group then also permutes the planes perpendicular to $l$ .",
"The six planar apeirohedra can be disposed off quickly: they are just the regular tessellations $ \\lbrace 3,6\\rbrace ,\\; \\lbrace 6,3\\rbrace ,\\; \\lbrace 4,4\\rbrace $ in the plane $\\mathbb {E}^2$ by triangles, hexagons, and squares, respectively, as well as their Petrie duals, $ \\lbrace \\infty ,6\\rbrace _{3},\\; \\lbrace \\infty ,3\\rbrace _{6},\\; \\lbrace \\infty ,4\\rbrace _{4},$ which have zig-zag faces."
],
[
"Blended apeirohedra",
"The “reducible\" apeirohedra are blends, in the sense that they are obtained by “blending\" a plane apeirohedron with a linear polygon (that is, a line segment $\\lbrace \\;\\rbrace $ or an apeirogon $\\lbrace \\infty \\rbrace $ ) contained in a line perpendicular to the plane.",
"Thus there are $6\\times 2 =12$ such blends.",
"The two projections of a blended apeirohedron onto its component subspaces recover the two original components, that is, the original plane apeirohedron as well as the line segment or apeirogon.",
"For example, the blend of the square tessellation in $\\mathbb {E}^2$ with a line segment $[-1,1]$ positioned along the $z$ -axis in $\\mathbb {E}^3$ has its vertices in the planes $z=-1$ and $z=1$ parallel to $\\mathbb {E}^2$ , and is obtained from the square tessellation $\\lbrace 4,4\\rbrace $ in $\\mathbb {E}^2$ by alternately raising or lowering (alternate) vertices (see Figure REF ).",
"Its faces are tetragons (skew squares), with vertices alternating between the two planes.",
"Its designation is $\\lbrace 4,4\\rbrace \\#\\lbrace \\;\\rbrace $ , where $\\#$ indicates the blending operation.",
"Figure: The blend of the square tessellation with the line segment.",
"The vertices lie in two parallel planes, and over each square of the original square tessellation lies one skew square (tetragon) of the blend.The blend of the square tessellation in $\\mathbb {E}^2$ with a linear apeirogon positioned along the $z$ -axis is more complicated.",
"Its faces are helical polygons rising in two-sided infinite vertical towers above the squares of the tessellation in such a way that the helical polygons over adjacent squares have opposite orientations (left-handed or right-handed) and meet along every fourth edge as they spiral around the towers.",
"The designation in this case is $\\lbrace 4,4\\rbrace \\#\\lbrace \\infty \\rbrace $ .",
"Strictly speaking, each blended apeirohedron belongs to an infinite family of apeirohedra obtained by (relative) rescaling of the two components of the blend; that is, each blended regular apeirohedron really represents a one-parameter family (of mutually non-similar) regular apeirohedra with the same combinatorial characteristics.",
"Figure: The four helical facets of the blended apeirohedron {4,4}#{∞}\\lbrace 4,4\\rbrace \\#\\lbrace \\infty \\rbrace that share a vertex.",
"Each vertical column over a square of the square tessellation is occupied by exactly one facet of {4,4}#{∞}\\lbrace 4,4\\rbrace \\#\\lbrace \\infty \\rbrace , spiraling around the column.The six blends of a planar apeirohedron with a line segment $\\lbrace \\;\\rbrace $ are listed in Table REF .",
"They have their vertices in two parallel planes and hence are not 3-periodic.",
"Thus their edge graph cannot be a net.",
"The edge graphs of the apeirohedra in the first two rows of Table REF are planar graphs, but they do not lie in a plane.",
"In fact, the graphs are isomorphic to the edge graphs of the regular square tessellation $\\lbrace 4,4\\rbrace $ or triangle tessellation $\\lbrace 3,6\\rbrace $ , respectively; this can be seen be projecting them onto the plane of the planar component of the blend.",
"It follows that, under certain conditions, these apeirohedra have an edge graph that occurs as the contact graph of a sphere packing arrangement with spheres centered at the vertices.",
"In fact, in order to have a faithful representation of the edge graph as a contact graph, the edges must be short enough to avoid forbidden contacts or overlaps between two spheres centered at vertices which are not joined by an edge.",
"This condition depends on the relative scaling of the components of the blend but is satisfied if the line segment in the blend is short compared with the edge length of the planar apeirohedron in the blend (that is, if the two parallel planes containing the vertices of the blend are close to each other).",
"The resulting sphere packings then are the sphere packings $4^{4}\\rm {IV}$ and $3^{6}\\rm {III}$ of (Koch & Fischer, 1978, pp.",
"131-133).",
"On the other hand, the edge graphs of the apeirohedra in the third row of Table REF cannot be contact graphs of sphere packing arrangements (for any relative scaling of the components of the blend), as there always are forbidden overlaps.",
"Table: The edge graphs of the blends of a planar regular apeirohedron with a line segment.",
"Petrie dual apeirohedra have the same edge graph and are listed in the same row.",
"Where applicable, the third column lists the corresponding sphere packing of (Koch & Fischer, 1978).On the other hand, the blends with the linear apeirogon $\\lbrace \\infty \\rbrace $ have 3-periodic edge graphs and hence yield highly symmetric nets.",
"These nets were already identified in (O'Keeffe, 2008) and are listed in Table REF using the notation for nets described earlier.",
"Table: The nets of the blends of a planar regular apeirohedron with a linear apeirogon.",
"Petrie dual apeirohedra have the same net and are listed in the same row."
],
[
"Pure apeirohedra",
"The twelve irreducible, or pure, regular apeirohedra fall into a single family, derived from the cubical tessellation in $\\mathbb {E}^3$ and illustrated in the diagram of Figure REF taken from (McMullen & Schulte, 2002, Ch.",
"7E).",
"There are a number of relationships between these apeirohedra indicated on the diagram such as duality $\\delta $ , Petrie duality $\\pi $ , and facetting $\\varphi _2$ , as well as the operations $\\eta $ , $\\sigma $ and $\\delta \\sigma $ which are not further discussed here.",
"The facetting operation $\\varphi _2$ applied to a regular polyhedron is reminiscent of the Petrie duality operation, in that the vertices and edges of the polyhedron are retained and the faces are replaced by certain edge-paths, in this case the 2-holes; here, a 2-hole, or simply hole, of a polyhedron is an edge path which leaves a vertex by the second edge from which it entered, always in the same sense (on the left, say, in some local orientation).",
"Figure: Relationships among the twelve pure regular apeirohedra.The most prominent apeirohedra of Figure REF are the three Petrie-Coxeter polyhedra $\\lbrace 4,6\\,|\\,4\\rbrace $ , $\\lbrace 6,4\\,|\\,4\\rbrace $ and $\\lbrace 6,6\\,|\\,3\\rbrace $ , occurring in the top and bottom row; the last entry in the symbols records the length of the holes, 4 or 3 in this case, while the first two entries give the standard Schläfli symbol.",
"These well-known apeirohedra are the only regular polyhedra in $\\mathbb {E}^3$ with convex faces and skew vertex-figures.",
"The Petrie duals of the Petrie-Coxeter polyhedra (related to the Petrie-Coxeter polyhedra under $\\pi $ ) have helical faces given by the Petrie polygons of the original polyhedron.",
"The first subscript in their designation gives the length of their own Petrie polygons, and the second subscript that of their 2-zigzags (edge paths leaving a vertex by the second edge, alternately on the right or left).",
"Table REF is a breakdown of the pure regular apeirohedra by mirror vectors, which also helps understanding why there are exactly 12 examples (McMullen-Schulte, 2002, Ch.",
"7E).",
"If $\\mathcal {K}$ is a regular polyhedron and $R_0,R_1,R_2$ denote the distinguished involutory generators for its symmetry group $G$ , the mirror vector $(d_0,d_1,d_2)$ of $\\mathcal {K}$ records the dimensions $d_0$ , $d_1$ and $d_2$ of the mirrors (fixed point sets) of $R_0$ , $R_1$ and $R_2$ , respectively.",
"It turns out that, mostly due to the irreducibility, only four mirror vectors can occur, namely $(2,1,2)$ , $(1,1,2)$ , $(1,2,1)$ and $(1,1,1)$ .",
"For example, the three apeirohedra with mirror vector $(1,1,1)$ in the last row have a symmetry group generated by three half-turns (reflections in lines) and therefore have only proper isometries as symmetries; these helix-faced apeirohedra occur geometrically in two enantiomorphic forms, yet they are geometrically regular, not chiral!",
"Table: Breakdown of the pure regular apeirohedra by mirror vector.While the rows in Table REF represent a breakdown of the pure apeirohedra by mirror vector, the first three columns can similarly be seen as grouping the apeirohedra by the crystallographic Platonic solid (the tetrahedron, octahedron or cube) with which each is associated in a manner described below; or, equivalently, as grouping by the corresponding (Platonic) symmetry group.",
"To explain this, suppose $\\mathcal {K}$ is a pure regular apeirohedron and $G= \\langle R_0,R_1,R_2\\rangle $ , where again $R_0,R_1,R_2$ are the distinguished generators.",
"Then $R_1$ and $R_2$ , but not $R_0$ , fix the base vertex, the origin $o$ (say), of $\\mathcal {K}$ .",
"Now consider the translate of the mirror of $R_0$ that passes through $o$ , and let $R_0^{\\prime }$ denote the reflection in this translate.",
"If $T$ denotes the translation subgroup of $G$ , then $G^{\\prime }:=\\langle R_0^{\\prime },R_1,R_2\\rangle $ is a finite irreducible group of isometries isomorphic to the special group $G\\slash T$ of $G$ , the quotient of $G$ by its translation subgroup.",
"Now alter (if needed) the generators $R_0^{\\prime },R_1,R_2$ as follows.",
"If a generator is a half-turn (with 1-dimensional mirror), replace it by the reflection in the plane through $o$ perpendicular to its rotation axis; otherwise leave the generator unchanged.",
"Let $\\widehat{R}_{0},\\widehat{R}_{1},\\widehat{R}_{2}$ , respectively, denote the plane reflections derived in this manner from $R_0^{\\prime },R_1,R_2$ , and let $\\widehat{G}:=\\langle \\widehat{R}_{0},\\widehat{R}_{1},\\widehat{R}_{2}\\rangle $ denote the finite irreducible reflection group in $\\mathbb {E}^3$ generated by them.",
"Now since $G$ has to be discrete, $\\widehat{G}$ cannot contain 5-fold rotations.",
"Hence there are only three possibilities for $\\widehat{G}$ and its generators, namely $\\widehat{G}$ is the symmetry group of the tetrahedron $\\lbrace 3,3\\rbrace $ , octahedron $\\lbrace 3,4\\rbrace $ or cube $\\lbrace 4,3\\rbrace $ .",
"Bearing in mind that there are just four possible mirror vectors, this then establishes that there are only $12=4\\times 3$ pure regular apeirohedra.",
"Note that $G^{\\prime }$ is either again one of these finite reflection groups, or the rotation subgroup of one of these groups (the latter happens only when the mirror vector is $(1,1,1)$ ).",
"Table REF also gives details about the geometry of the faces and vertex-figures.",
"It is quite remarkable that in a pure regular apeirohedron with finite faces, the faces and vertex-figures cannot both be planar or both be skew.",
"As we will see, this is very different for chiral polyhedra.",
"The nets of the pure regular apeirohedra were identified in (O'Keeffe, 2008) and are listed in Table REF .",
"Table: The nets of the pure regular apeirohedra.",
"Pairs of Petrie duals share the same net and are listed in the same row.",
"The last two apeirohedra are self-Petrie."
],
[
"Chiral Polyhedra",
"The classification of chiral polyhedra in ordinary space $\\mathbb {E}^3$ is rather involved.",
"It begins with the observation that chirality, as defined here, does not occur in the classical theory, so in particular there are no convex polyhedra that are chiral.",
"The chiral polyhedra in $\\mathbb {E}^3$ fall into six infinite families (Schulte, 2004, 2005), each with two or one free parameters depending on whether the classification is up to congruence or similarity, respectively.",
"Each chiral polyhedron is a non-planar “irreducible\" apeirohedron (with an affinely irrreducible symmetry group), so in particular there are no finite, planar, or “reducible\" examples.",
"The six families comprise three families of apeirohedra with finite skew faces and three families of apeirohedra with infinite helical faces.",
"It is convenient to slightly enlarge each family by allowing the parameters to take certain exceptional values which would make the respective polyhedron regular and, in some cases, finite.",
"The resulting larger families will then contain exactly two regular polyhedra, while all other polyhedra are chiral apeirohedra."
],
[
"Finite-faced polyhedra",
"The three families with finite faces only contain apeirohedra and are summarized in Table REF .",
"These apeirohedra are parametrized by two integers which are relatively prime (and not both zero).",
"In fact, the corresponding apeirohedra exist also when the parameters are real, but they are discrete only when the parameters are integers.",
"Membership in these families is determined by the basic Schläfli symbol, namely $\\lbrace 6,6\\rbrace $ , $\\lbrace 4,6\\rbrace $ or $\\lbrace 6,4\\rbrace $ .",
"The corresponding apeirohedra are denoted $P(a,b)$ , $Q(c,d)$ and $Q(c,d)^*$ , respectively, where the star indicates that the apeirohedon $Q(c,d)^*$ in the third family is the dual of the apeirohedron $Q(c,d)$ in the second family.",
"The duality of the apeirohedra $Q(c,d)$ and $Q(c,d)^*$ is geometric, in that the face centers of one are the vertices of the other.",
"The apeirohedra $P(a,b)$ and $P(b,a)$ similarly are geometric duals of each other (again with the roles of vertices and face centers interchanged), and $P(b,a)$ is congruent to $P(a,b)$ .",
"Thus $P(a,b)$ is geometrically self-dual, in the sense that its dual $P(a,b)^{*}=P(b,a)$ is congruent to $P(a,b)$ .",
"Table: The three families of finite-faced chiral apeirohedra and their related regular apeirohedra.The chiral apeirohedra in each family have skew faces and skew vertex-figures.",
"However, the two regular apeirohedra in each family have either planar faces or planar vertex-figures.",
"In either case, the distinguished generators $S_1,S_2$ of the symmetry groups $G$ of an apeirohedron in the family are rotatory reflections defined in terms of the parameters $a,b$ or $c,d$ , and the symmetry $T:=S_{1}S_{2}$ is a half-turn.",
"For example, for the apeirohedron $P(a,b)$ of type $\\lbrace 6,6\\rbrace $ the symmetries $S_1$ , $S_2$ and $T$ are given by $ \\begin{array}{rccl}S_{1}\\colon & (x_{1},x_{2},x_{3}) & \\mapsto & (-x_{2},x_{3},x_{1}) + (0,-b,-a),\\\\S_{2}\\colon & (x_{1},x_{2},x_{3}) & \\mapsto & -(x_{3},x_{1},x_{2}), \\\\T\\colon & (x_{1},x_{2},x_{3}) & \\mapsto & (-x_{1},x_{2},-x_{3}) + (a,0,b).\\end{array} $ Then the apeirohedron $P(a,b)$ itself is obtained from the group $G$ generated by $S_1,S_2$ by means of Wythoff's construction as explained above.",
"The base vertex $F_0$ in this case is the origin $o$ (fixed under $S_2$ ), and the base edge $F_1$ is given by $\\lbrace o,u\\rbrace $ with $u:= T(o) = (a,0,b).$ The base edge $F_1$ lies in the $x_{1}x_{3}$ -plane and is perpendicular to the rotation axes of $T$ , which in turn is parallel to the $x_2$ -axis and passes through $\\frac{1}{2}u$ .",
"The base face $F_2$ of $P$ is determined by the orbit of $o$ under $\\langle S_1\\rangle $ and is given by the generally skew hexagon with vertex-set $ \\begin{array}{l}\\lbrace (0,0,0),(0,-b,-a),(b,-a-b,-a),\\\\(a+b,-a-b,-a+b),(a+b,-a,b),(a,0,b)\\rbrace ,\\end{array}$ where the vertices are listed in cyclic order.",
"The vertices of $P(a,b)$ adjacent to $o$ are given by the orbit of $u$ under $\\langle S_2\\rangle $ , namely $\\begin{array}{l}\\lbrace (a,0,b),(-b,-a,0),(0,b,a),\\\\(-a,0,-b),(b,a,0),(0,-b,-a)\\rbrace ;\\end{array}$ these are the vertices of the generally skew hexagonal vertex-figure of $P(a,b)$ at $o$ , listed in the order in which they occur in the vertex-figure.",
"The faces of $P(a,b)$ containing the vertex $o$ are the images of $F_2$ under the elements of $\\langle S_2\\rangle $ .",
"Each face is a generally skew hexagon with vertices given by one half of the vertices of a hexagonal prism.",
"As mentioned earlier, both the faces and vertex-figures are skew if $P(a,b)$ is chiral.",
"The apeirohedra $P(a,b)$ are chiral except when $b=\\pm a$ .",
"If $b=a$ we arrive at the Petrie-Coxeter polyhedron $\\lbrace 6,6\\,|\\,3\\rbrace $ , which has planar (convex) faces but skew vertex-figures.",
"If $b=-a$ we obtain the regular polyhedron $\\lbrace 6,6\\rbrace _4$ , which has skew faces but planar (convex) vertex-figures.",
"The vertices of $\\lbrace 6,6\\rbrace _4$ comprise the vertices in one set of alternate vertices of the Petrie-Coxeter polyhedron $\\lbrace 4,6\\,|\\,4\\rbrace $ , while the faces of $\\lbrace 6,6\\rbrace _4$ are the vertex-figures at the vertices in the other set of alternate vertices of $\\lbrace 4,6\\,|\\,4\\rbrace $ .",
"Table REF also lists the structure of the special groups.",
"Here $[p,q]$ denotes the full symmetry group of a Platonic solid $\\lbrace p,q\\rbrace $ , and $[p,q]^+$ its rotation subgroup; also $-I$ stands for the point reflection in $o$ ."
],
[
"Helix-faced polyhedra",
"The three families of helix-faced chiral apeirohedra and their related regular polyhedra are summarized in Table REF .",
"The corresponding apeirohedra or polyhedra are denoted by $P_1(a,b)$ , $P_2(c,d)$ and $P_3(c,d)$ .",
"Each family has two real-valued parameters that cannot both be zero.",
"Now the discreteness assumption does not impose any further restrictions on the parameters.",
"Membership in these families is determined by the basic Schläfli symbol as well as the basic geometry of the helical faces.",
"There are two families of type $\\lbrace \\infty ,3\\rbrace $ and one family of type $\\lbrace \\infty ,4\\rbrace $ .",
"In the first family of type $\\lbrace \\infty ,3\\rbrace $ the apeirohedra have helical faces over triangles, and in the second family they have helical faces over squares.",
"Each family contains two regular polyhedra, namely one pure regular apeirohedron as well as one (finite crystallographic) Platonic solid, that is, the tetrahedron, cube or octahedron, respectively, Table: The three families of helix-faced chiral apeirohedra and their related regular polyhedra.The symmetry groups $G$ of the polyhedra $P_1(a,b)$ , $P_2(c,d)$ and $P_3(c,d)$ is generated by a screw motion $S_1$ (a rotation followed by a translation along the rotation axis) and an ordinary rotation $S_2$ in an axis through the base vertex $F_{0}:=o$ .",
"The screw motion $S_1$ moves the vertices of the helical base face $F_2$ (in an apeirohedron) one step along the face, and $S_2$ applied to the vertex $u$ of the base edge $F_1$ distinct from $o$ produces the planar vertex-figure at $o$ .",
"The symmetry $T=S_{1}S_{2}$ is again a half-turn with a rotation axis passing through $\\frac{1}{2}u$ and perpendicular to $F_1$ .",
"For example, for the polyhedron $P_2(c,d)$ the symmetries $S_1$ , $S_2$ and $T$ are given by $ \\begin{array}{rccl}S_{1}\\colon & (x_{1},x_{2},x_{3}) & \\mapsto & (-x_{3},x_{2},x_{1}) + (d,c,-c),\\\\S_{2}\\colon & (x_{1},x_{2},x_{3}) & \\mapsto & (x_{2},x_{3},x_{1}), \\\\T\\colon & (x_{1},x_{2},x_{3}) & \\mapsto & (x_{2},x_{1},-x_{3}) + (c,-c,d).\\end{array} $ The polyhedron $P_{2}(c,d)$ itself is again obtained from $G=\\langle S_1,S_2\\rangle $ by Wythoff's construction with base vertex $F_{0}=o$ .",
"The base edge $F_1$ is given by $\\lbrace 0,u\\rbrace $ with $ u:=T(o) = (c,-c,d),$ and lies in the plane $x_{2}=-x_{1}$ .",
"The half-turn $T$ interchanges the vertices $o$ and $u$ of $F_1$ , and its rotation axis is parallel (in $\\mathbb {E}^3$ ) to the line $x_{2}=x_{1}$ in the $x_{1}x_{2}$ -plane and perpendicular to the plane $x_{2}=-x_{1}$ .",
"The screw motion $S_1$ shifts the base face $F_2$ one step along itself, and since $S_{1}^4$ is the translation along the $x_2$ -axis by $(0,4c,0)$ , we have helical faces over squares (when $c\\ne 0$ ) spiraling around an axis parallel to the $x_2$ -axis.",
"In particular, the vertex-set of $F_2$ is the orbit of $o$ under $\\langle S_1\\rangle $ and is given by $ \\lbrace (c,-c,d),(0,0,0),(d,c,-c),(c+d,2c,-c+d)\\rbrace + \\mathbb {Z}\\!\\cdot \\!",
"t$ with $t:=(0,4c,0)$ .",
"Here the notation means that the four vectors listed on the left side are successive vertices of $F_2$ whose translates by integral multiples of $t$ comprise all the vertices of $F_2$ .",
"When $c=0$ we obtain a finite polyhedron $P_{2}(0,d)$ , a cube $\\lbrace 4,3\\rbrace $ with a finite group $G$ , namely the rotation subgroup of the symmetry group of this cube.",
"In fact, in this case $S_{1}^4$ is the identity mapping and the base face $F_2$ itself is a square (not a helical polygon over a square).",
"In the general case, as usual, all other vertices, edges and faces of the apeirohedron (or polyhedron) are the images of $F_{0}$ , $F_{1}$ and $F_{2}$ under the group $G$ .",
"Moreover, the vertices adjacent to $o$ form the triangular vertex-figure at $o$ and are given by the three points $ u = (c,-c,d),\\, S_{2}(u)=(-c,d,c),\\, S_{2}^2(u)=(d,c,-c)$ in the plane $x_{1}+x_{2}+x_{3}=d$ ."
],
[
"General properties",
"The six families of chiral apeirohedra and related regular polyhedra have stunning geometric and combinatorial properties and exhibit some rather unexpected phenomena.",
"For all three families of apeirohedra with finite faces it is almost true that different parameter values give combinatorially non-isomorphic apeirohedra (that is, the underlying abstract apeirohedra are non-isomorphic).",
"More explicitly, $P(a,b)$ and $P(a^{\\prime },b^{\\prime })$ are combinatorially isomorphic if and only if $ (a^{\\prime },b^{\\prime })=\\pm (a,b),\\pm (b,a); $ and similarly, $Q(c,d)$ and $Q(c^{\\prime },d^{\\prime })$ , and hence $Q(c,d)^*$ and $Q(c^{\\prime },d^{\\prime })^*$ , are combinatorially isomorphic if and only if $ (c^{\\prime },d^{\\prime })=\\pm (c,d),\\pm (-c,d).$ This phenomenon is perhaps even more surprising when expressed in terms of the similarity classes of the apeirohedra within each family, which are parametrized by a single rational parameter, namely $a/b$ or $c/d$ (when $b,d\\ne 0$ ).",
"These similarity classes exhibit a very strong discontinuity, in that any small positive change in the rational parameter produces a new similarity class in which the apeirohedra are not combinatorially isomorphic to those in the original similarity class.",
"By contrast, in the three families of helix-faced apeirohedra and related polyhedra, each chiral apeirohedron is combinatorially isomorphic to the regular apeirohedron in its family, so in particular it is combinatorially regular, but not geometrically regular (Pellicer & Weiss, 2010).",
"In fact, the chiral apeirohedra in each family can be viewed as “chiral deformations\" of the regular apeirohedron in this family.",
"At the other extreme they also allow a “deformation\" to the finite regular polyhedron (Platonic solid) in the family.",
"On the other hand, the finite-faced chiral apeirohedra are also combinatorially chiral (Pellicer & Weiss, 2010).",
"In other words, these apeirohedra are intrinsically chiral and thus not combinatorially isomorphic to a regular apeirohedron in their family.",
"In some sense, each chiral helix-faced apeirohedron can be thought of as unraveling the Platonic solid in its family.",
"In fact, for all parameter values $a,b$ and $c,d$ we have the following coverings of polyhedra: $ P_{1}(a,b)\\!",
"\\mapsto \\!",
"\\lbrace 3,3\\rbrace ,\\; P_{2}(c,d)\\!",
"\\mapsto \\!",
"\\lbrace 4,3\\rbrace ,\\; P_{3}(c,d)\\!",
"\\mapsto \\!",
"\\lbrace 3,4\\rbrace .",
"$ Informally speaking, under these coverings each helical face is “compressed\" (like a spring) to become a polygon (triangle or square) over which it has been rising.",
"The nets arising as edge graphs of chiral polyhedra in $\\mathbb {E}^3$ will be analyzed in a forthcoming paper."
],
[
"Regular Polygonal Complexes",
"We now turn to polygonal complexes with possibly more than two faces meeting at an edge.",
"All regular polygonal complexes that are not polyhedra turn out to be infinite and have an affinely irreducible symmetry group.",
"Unlike a regular polyhedron, a regular polygonal complex $\\mathcal {K}$ can have a symmetry group that is transitive, but not simply transitive, on the flags.",
"As we mentioned earlier, $\\mathcal {K}$ then has face mirrors, meaning that $\\mathcal {K}$ has planar faces and that each flag stabilizer is generated by the reflection in the plane through the (planar) face in the flag.",
"The classification of regular polygonal complexes naturally breaks down into two cases, namely the enumeration of the simply flag-transitive complexes and that of the non-simply flag-transitive complexes (Pellicer & Schulte, 2010, 2013).",
"All regular polyhedra, finite or infinite, are simply flag-transitive polygonal complexes."
],
[
"Non-simply flag-transitive complexes as 2-skeletons",
"For a regular polygonal complex $\\mathcal {K}$ with a non-simply flag transitive symmetry group $G$ , the existence of face mirrors allows us to recognize $\\mathcal {K}$ as the 2-skeleton of a certain type of incidence structure of rank 4 in $\\mathbb {E}^3$ , called a 4-apeirotope (McMullen & Schulte, 2002, Ch.",
"7F).",
"Thus $\\mathcal {K}$ consists of the vertices, edges, and faces (of rank 2) of this 4-apeirotope.",
"The 2-skeleton of the cubical tessellation shown in Figure REF is an example of a regular polygonal complex of this kind, and the underlying cubical tessellation is the corresponding 4-apeirotope.",
"The 4-apeirotopes involved are themselves regular, in the sense that they have a flag-transitive symmetry group on their own (coinciding with $G$ ); in fact, the generating reflection of the base flag stabilizer for $\\mathcal {K}$ is the fourth involutory generator needed to suitably generate the symmetry group of this rank 4 structure.",
"There are eight regular 4-apeirotopes in $\\mathbb {E}^3$ , occurring in four pairs of “Petrie-duals\" (McMullen & Schulte, 2002, Ch.",
"7F).",
"The apeirotopes in each pair have the same 2-skeleton and produce the same polygonal complex.",
"Thus up to similarity there are four non-simply flag-transitive complexes in $\\mathbb {E}^3$ , each with the same symmetry group as its two respective apeirotopes.",
"The 2-skeleton of the cubical tessellation has square faces and is the only example of a non-simply flag-transitive regular polygonal complex with finite faces.",
"The three other complexes all have (planar) zigzag faces, with either 3 or 4 faces meeting at each edge.",
"Table: The four nets arising as edge graphs of regular 4-apeirotopes in 𝔼 3 \\mathbb {E}^3.",
"Pairs of Petrie-dual apeirotopes are listed in the same row.In Table REF we list the four pairs of Petrie dual regular 4-apeirotopes in $\\mathbb {E}^3$ , along with information about the regular polygonal complexes $\\mathcal {K}$ arising as their 2-skeletons.",
"In particular we give the number $r$ of faces at an edge, as well as the structure of the vertex-figure of $\\mathcal {K}$ ; here an entry in the vertex-figure column of Table REF listing a Platonic solid is meant to represent the (geometric) edge-graph of this solid.",
"Note that in each case the vertex-figure of the polygonal complex $\\mathcal {K}$ is simply the edge graph of the Platonic solid which occurs as the vertex-figure of the regular 4-apeirotope listed in the first column.",
"Thus the vertex-figure column gives the structure of the vertex-figures of both the regular polygonal complex and the corresponding regular 4-apeirotope in the first column.",
"Similarly, the number $r$ for $\\mathcal {K}$ also coincides with the number of faces at an edge of the corresponding 4-apeirotope.",
"Table REF also identifies the underlying net of each complex, which here coincides with the edge graphs of the two related regular 4-apeirotopes."
],
[
"Simply flag-transitive complexes",
"In addition to the 48 regular polyhedra described in Section there are 21 simply flag-transitive regular polygonal complexes in $\\mathbb {E}^3$ which are not polyhedra, up to similarity.",
"This gives a total of 69 simply flag-transitive regular complexes, up to similarity and relative scaling of components for blended polyhedra.",
"Let $\\mathcal {K}$ be a simply flag-transitive complex which is not a polyhedron, that is, $\\mathcal {K}$ has $r\\ge 3$ faces at each edge.",
"Then the generating subgroups $G_0,G_1,G_2$ of its symmetry group $G$ described earlier take a very specific form: $G_0$ is generated by a point, line, or plane reflection $R_0$ ; the subgroup $G_1$ is generated by a line or plane reflection $R_1$ ; and $G_2$ is a cyclic or dihedral group of order $r$ .",
"The mirror vector $(d_0,d_1)$ of $\\mathcal {K}$ records the dimensions $d_0$ and $d_1$ of the mirrors of $R_0$ and $R_1$ , respectively.",
"(Note that, for a polyhedron, $G_2$ would also be generated by a reflection and the complete mirror vector introduced earlier would list the dimensions of all three mirrors.)",
"Table REF summarizes the 21 simply flag-transitive complexes and some of their properties.",
"In writing $\\mathcal {K}_i(j,k)$ for a complex, $(j,k)$ indicates its mirror vector and $i$ is its label (serial number) in the list of regular complexes with the same mirror vector $(j,k)$ .",
"There are columns for the pointwise edge stabilizer $G_2$ , the number $r$ of faces at each edge, the types of faces and vertex-figures, the vertex-set, the special group $G^*$ , and the corresponding net.",
"In the face column we use symbols like $p_c$ , $p_s$ , $\\infty _2$ , or $\\infty _k$ with $k=3$ or 4, respectively, to indicate that the faces are convex $p$ -gons, skew $p$ -gons, planar zigzags, or helical polygons over $k$ -gons.",
"(A planar zigzag is viewed as a helix over a “2-gon\", where here a 2-gon is a line segment traversed in both directions.",
"Hence the use of $\\infty _2$ .)",
"An entry in the vertex-figure column describing a solid figure in $\\mathbb {E}^3$ is meant to represent the geometric edge-graph of this figure, with “double\" indicating the double edge-graph (the edges have multiplicity 2).",
"The entry “ns-cuboctahedron\" in the vertex-figure column stands for the edge graph of a “non-standard cuboctahedron\", a certain realization with skew square faces of the ordinary cuboctahedron.",
"Table: The 21 simply flag-transitive regular complexes in 𝔼 3 \\mathbb {E}^3 which are not polyhedra, and their nets.For all but three complexes, the special group $G^*$ is the full octahedral group $[3,4]$ .",
"In the three exceptional cases $G^*$ is the octahedral rotation group $[3,4]^+$ ; note that in these cases we must have a mirror vector $(1,1)$ and a cyclic group $G_{2}$ .",
"For all but five complexes of Table REF the vertex-set is a lattice, namely one of the following, up to scaling: the standard (“primitive\") cubic lattice $\\Lambda _{1}:=\\mathbb {Z}^{3}$ ; the face-centered cubic lattice $\\Lambda _{2}$ , with basis $(1,1,0)$ , $(-1,1,0)$ , $(0,-1,1)$ , consisting of all integral vectors with even coordinate sum; or the body-centered cubic lattice $\\Lambda _3$ , with basis $(2,0,0)$ , $(0,2,0)$ , $(1,1,1)$ .",
"In the five exceptional cases the vertex-set is either $ V:=\\Lambda _{1}\\!\\setminus \\!",
"((0,0,1)\\!+\\!\\Lambda _{3})$ or $ W:= 2\\Lambda _{2} \\cup ((1,-1,1)\\!+\\!2\\Lambda _{2}), $ again up to scaling.",
"The last column of Table REF lists the nets of the simply flag-transitive regular complexes which are not polyhedra.",
"The coordination figures of the nets are just the convex hulls of the vertex-figures of the corresponding regular complex, ignoring multiplicity of the edges in the vertex-figure if it occurs.",
"For example, the vertex-figure of the complex $\\mathcal {K}_{1}(1,2)$ is the graph of an ordinary cuboctahedron and so the coordination figure of the net for $\\mathcal {K}_{1}(1,2)$ is a cuboctahedron; hence the net is quasiregular and must coincide with the face-centered cubic lattice $\\bf fcu$ , which we denoted by $\\Lambda _2$ .",
"The nets ${\\bf pcu}$ and ${\\bf bcu}$ similarly correspond to the lattices $\\Lambda _1$ and $\\Lambda _3$ , respectively.",
"Figure REF illustrates a local picture of $\\mathcal {K}_{1}(1,2)$ around an edge, showing the four skew square faces meeting at the edge.",
"The entire complex can be thought of as being built from an infinite family of Petrie duals of regular tetrahedra inscribed in cubes, one per cube of the cubical tessellation, such that the Petrie duals in adjacent cubes share a common edge and have opposite orientation.",
"Thus the complex has infinitely many “small\" finite subcomplexes each a regular polyhedron in itself.",
"Figure: The four skew square faces of 𝒦 1 (1,2)\\mathcal {K}_{1}(1,2) sharing an edge.",
"Each face is a Petrie polygon of a regular tetrahedron inscribed in a cube of the cubical tessellation.",
"The tetrahedra in adjacent cubes have different orientations.",
"The net is fcu.Figures REF and REF depict the local structure around a vertex for two further simply flag-transitive regular complexes with mirror vector $(1,2)$ , namely $\\mathcal {K}_{4}(1,2)$ and $\\mathcal {K}_{5}(1,2)$ .",
"They also are related to the cubical tessellation and their nets are pcu and nbo, respectively.",
"It turns out that the edge graph of each simply-flag-transitive regular complex $\\mathcal {K}$ is a regular or quasiregular net in the sense described earlier.",
"This can be seen as follows.",
"It is clear that the net is uninodal, since the symmetry group of $\\mathcal {K}$ acts transitively on the vertices of $\\mathcal {K}$ and is a subgroup of the symmetry group of the net.",
"If the vertex-figure of $\\mathcal {K}$ is a cuboctahedron or non-standard (ns) cuboctahedron, then the coordination figures of the net are convex cuboctahedra.",
"Hence the net is quasiregular and must coincide with fcu.",
"Note here that the argument also applies if the vertex-figures of $\\mathcal {K}$ are ns-cuboctahedra, that is, when $\\mathcal {K}$ is one of the four complexes $\\mathcal {K}_5(1,1)$ , $\\mathcal {K}_8(1,1)$ , $\\mathcal {K}(0,1)$ and $\\mathcal {K}(2,1)$ .",
"In these four cases, the edges in the vertex-figure connect pairs of vertices of the standard cuboctahedron which are midpoints of edges two steps apart on a Petrie polygon of the cube used to construct the cuboctahedron as the convex hull of the midpoints of its edges.",
"However, an edge in the vertex-figure of a polygonal complex at a given vertex represents a face of the complex that contains this vertex, in that it joins the two vertices of the face that are adjacent to the given vertex in the complex.",
"Thus the edges of the vertex-figure capture the faces of the complex, not its underlying net.",
"The local structure of the net is only determined by the vertices of the vertex-figure, not its edges.",
"For all other regular complexes $\\mathcal {K}$ , except $\\mathcal {K}_2(1,2)$ and $\\mathcal {K}_{4}(1,2)$ , we can appeal to the classification of regular nets.",
"In fact, the coordination figures of the net of $\\mathcal {K}$ are easily seen to be squares, tetrahedra, octahedra or cubes, and the vertex-figure subgroup $G_{F_0}=\\langle R_{1},G_2\\rangle $ of the symmetry group $G$ at the base vertex of $\\mathcal {K}$ contains (at least) the rotation symmetry group of the coordination figure at this vertex of the net.",
"In the case of the square, the rotational symmetries are taken relative to $\\mathbb {E}^3$ .",
"Thus the complex $\\mathcal {K}$ has enough symmetries to guarantee that its edge graph is a regular net.",
"The situation changes for the complexes $\\mathcal {K}_2(1,2)$ and $\\mathcal {K}_{4}(1,2)$ with cubes and octahedra as coordination figures, respectively.",
"Now the vertex-figure subgroups are too small to imply regularity of the net based on symmetries of $\\mathcal {K}$ ; these subgroups are given by $[3,3]^{+}\\times \\langle -I\\rangle $ and $[3,3]$ , respectively.",
"However, $\\mathcal {K}_{2}(1,2)$ is a subcomplex of $\\mathcal {K}_{3}(1,2)$ with the same edge graph, and the latter is the regular net bcu by our previous analysis.",
"Thus the net of $\\mathcal {K}_2(1,2)$ also is bcu.",
"Similarly, $\\mathcal {K}_{4}(1,2)$ is a subcomplex of $\\mathcal {K}_{6}(1,2)$ with the same edge graph, namely the edge graph of the cubical tessellation of $\\mathbb {E}^3$ (see Figure REF ), which is the regular net pcu.",
"Thus the edge graphs of all regular polygonal complexes which are not polyhedra, simply flag-transitive or not, are regular or quasiregular nets.",
"The regular net srs does not occur in this but all others do; however, srs is the net of two pure regular polyhedra with helical faces (see Table REF ).",
"Figure: The faces of 𝒦 4 (1,2)\\mathcal {K}_{4}(1,2) are the Petrie polygons of alternate cubes in the cubical tessellation of 𝔼 3 \\mathbb {E}^3.",
"For every cube occupied, all its Petrie polygons occur as faces of 𝒦 4 (1,2)\\mathcal {K}_{4}(1,2).",
"Shown are the twelve faces of 𝒦 4 (1,2)\\mathcal {K}_{4}(1,2) that have a vertex in common, here located at the center.",
"Each edge containing this vertex lies in four faces.",
"The vertex-figure of 𝒦 4 (1,2)\\mathcal {K}_{4}(1,2) at the central vertex is the octahedron spanned by the six outer black nodes.",
"The net is pcu.Figure: The faces of 𝒦 5 (1,2)\\mathcal {K}_{5}(1,2) are Petrie polygons of cubes in the cubical tessellation of 𝔼 3 \\mathbb {E}^3.",
"For every cube of the tessellation, only one of its Petrie polygons is a face of 𝒦 5 (1,2)\\mathcal {K}_{5}(1,2).",
"Shown are the eight faces of 𝒦 5 (1,2)\\mathcal {K}_{5}(1,2) that have a vertex in common, here located at the center.",
"Each edge containing this vertex lies in four faces.",
"The vertex-figure of 𝒦 5 (1,2)\\mathcal {K}_{5}(1,2) at the central vertex is the double square spanned by the four outer black nodes in the horizontal plane through the center.",
"The net is nbo.Acknowledgements.",
"I am grateful to Davide Proserpio for making me aware of the extensive work on crystal nets that has appeared in the chemistry literature, as well as for valuable comments about the history of nets research and the notation for nets used by crystallographers.",
"I am also indebted to the anonymous referees for a number of helpful suggestions that have improved the article."
]
] | 1403.0045 |
[
[
"Finite-Difference Time-Domain simulation of spacetime cloak"
],
[
"Abstract In this work, we present a numerical method that remedies the instabilities of the conventional FDTD approach for solving Maxwell's equations in a space-time dependent magneto-electric medium with direct application to the simulation of the recently proposed spacetime cloak.",
"We utilize a dual grid FDTD method overlapped to the time domain to provide a stable approach for the simulation of magneto-electric medium with time and space varying permittivity, permeability and coupling coefficient.",
"The developed method can be applied to explore other new physical possibilities offered by spacetime cloaking, metamaterials, and transformation optics."
],
[
"Introduction",
"In recent years, Transformation Optics (TO) has become one of more interesting topics in science.",
"Utilizing transformation optics and metamaterials, spatial invisibility cloaks [1], [2] have been developed.",
"A spatial cloak functions by manipulating the spatial path of electromagnetic waves within the cloaked region in such a way that waves are bent around the object being cloaked.",
"Transformation optics allows for a description of the medium (ie: material parameters) needed for the construction of such devices.",
"For this particular type of cloak, the implementation requires the use of inhomogeneous media where the inhomogeneity is due to the use of spatially dependent anisotropic permeability and permittivity.",
"Within the past three years, a new type of cloak has been developed [3], [4], the spacetime cloak.",
"Contrary to the spatial cloak, a spacetime cloak functions by altering the speed of the electromagnetic wave as it propagates through the cloaked region.",
"Initially, the speed of the front of the wave is increased while the speed of the back of the wave is decreased.",
"This allows for a gap to occur in which an event can be carried out undetected.",
"To close the gap, the back of the wave is sped up and the front is slowed down allowing the wave to continue on at its original speed.",
"The design of such a cloak can be achieved by transformation optics, and it results in a magneto-electric medium with time and space varying permittivity, permeability, and coupling coefficient.",
"Due to the interesting nature of these devices, simulation is desirable.",
"To this end, we turn to the Finite-Difference Time-Domain (FDTD) method.",
"The FDTD method [5], [6], [7] is a very successful method for simulating electromagnetic wave propagation.",
"It has been applied to the study of various types of materials [7], [8], including dielectrics, linear dispersive materials, nonlinear Raman and Kerr materials, nonlinear dispersive materials [9], bi-isotropic media [10], [11], etc.",
"In this paper, we are interested in space and time dependent magneto-electric materials and their direct application to the spacetime cloak.",
"In [4], the FDTD method has been applied to simulate the spacetime cloak but the authors point out that there are some difficulties near the closing process of the spacetime cloak.",
"Our numerical simulations demonstrate that there are instabilities for conventional FDTD simulation due to the temporal extrapolation of the time dependent magneto-electric constitutive equations.",
"To stably solve Maxwell's equations in time dependent magneto-electric medium, we propose a modified FDTD method based on the use of time overlapped grids.",
"Our method uses two Yee grids that are offset in time by a half time step so that collocated fields are provided in the time dependent magneto-electric constitutive equations.",
"The resulting numerical update equations do not require time extrapolation, therefore the instability due to time extrapolation is avoided.",
"Our method is applied to simulate the spacetime cloak."
],
[
"Transformation Optics and spacetime cloak",
"In this section, we briefly review the derivation of the spacetime cloak using transformation optics.",
"Consider the following covariant form of the Maxwell's equations (Ampère's Law and the Gauss' Law of electric fields) $\\tilde{\\nabla }\\tilde{M}=\\tilde{\\nabla }\\left( \\begin{array}{cccc} 0&D_x&D_y&D_z \\\\ -D_x&0&H_z&-H_y \\\\ -D_y&-H_z&0&H_x \\\\ -D_z&H_y&-H_x&0 \\end{array} \\right)=0,$ where $\\tilde{\\nabla }= (\\frac{\\partial }{\\partial t},\\frac{\\partial }{\\partial x},\\frac{\\partial }{\\partial y},\\frac{\\partial }{\\partial z})$ .",
"Applying a spacetime transformation from $(t,x)$ to $(\\tau ,\\xi )$ and keeping $y$ and $z$ coordinates unchanged, we have the following covariant form of the Maxwell's equation in the new coordinates $(\\tau ,\\xi ,\\eta ,\\zeta )$ : $\\tilde{\\nabla }^{\\prime } \\tilde{M}^{\\prime } = \\tilde{\\nabla }^{\\prime } \\left( \\begin{array}{cccc} 0&D_\\xi &D_\\eta &D_\\zeta \\\\ -D_\\xi &0&H_\\zeta &-H_\\eta \\\\ -D_\\eta &-H_\\zeta &0&H_\\xi \\\\ -D_\\zeta &H_\\eta &-H_\\xi &0 \\end{array} \\right) = 0,$ where $\\tilde{\\nabla }^{\\prime }= (\\frac{\\partial }{\\partial \\tau },\\frac{\\partial }{\\partial \\xi },\\frac{\\partial }{\\partial \\eta },\\frac{\\partial }{\\partial \\zeta })$ , $\\tilde{M} =|\\tilde{\\Lambda }|\\tilde{\\Lambda }^{-1} \\tilde{M}^{\\prime } \\tilde{\\Lambda }^{-T}$ , and $\\tilde{\\Lambda }$ is the Jacobian matrix ($I_2$ represents the $2\\times 2$ identity matrix): $\\tilde{\\Lambda }= \\frac{\\partial (\\tau ,\\xi ,\\eta ,\\zeta )}{\\partial (t,x,y,z)} =\\left( \\begin{array}{cccc} \\tau _t&\\tau _x&0&0 \\\\ \\xi _t&\\xi _x&0&0 \\\\ 0&0&1&0 \\\\ 0&0&0&1 \\end{array} \\right) = \\left( \\begin{array}{cc} \\Lambda &0 \\\\ 0&I_2 \\end{array} \\right).$ Similar matrix equations are obtained for Faraday's Law together with the Gauss' Law of magnetic fields.",
"Figure: (a) A spacetime cloak in the (t,xt,x) domain.",
"(b) Free space after the transformation to the (τ,ξ\\tau ,\\xi ) domain.We consider a spacetime cloak design similar to the spacetime cloaks proposed in [3], [4].",
"Figure REF shows a diamond shaped spacetime cloak in the ($t, x$ ) domain and the transformed ($\\tau , \\xi $ ) free space.",
"The original and the transformed regions are the same parallelograms centered at $(t_0, x_0)$ so we have $\\xi _0 = x_0$ , $\\tau _0=t_0$ , $\\tau _p=t_p$ , and $\\tau _q=t_q$ .",
"In the parallelogram we apply the following coordinate transformation to create such spacetime cloak: $\\tau &=& t, \\\\\\xi &=& \\frac{\\sigma (x-x_S)}{\\sigma - \\delta (t-t_S)/(t_0-t_S)} + x_S,$ where the values of $x_S$ and $t_S$ depend on the regions where $(t,x)$ lies in: $t_S =\\left\\lbrace \\begin{array}{ll}t_p ,& \\mbox{if $(t,x)$ lies in region I or II},\\\\t_q, & \\mbox{if $(t, x)$ lies in region III or IV},\\end{array}\\right.$ and $x_S =\\left\\lbrace \\begin{array}{ll}x_0+(t - t_0)c+\\sigma ,& \\mbox{if $(t, x)$ lies in region I or IV},\\\\x_0+(t - t_0)c-\\sigma , & \\mbox{if $(t, x)$ lies in region II or III}.\\end{array}\\right.$ Since the transformed ($\\tau ,\\xi $ ) domain is free space, we have the constitutive equations $D=\\epsilon E$ and $B = \\mu H$ .",
"The corresponding constitutive equations in ($t,x$ ) domain are magneto-electric: $E_z &=& \\alpha D_z + \\beta B_y, \\\\H_y &=& \\beta D_z + \\gamma B_y, $ where $\\alpha $ , $\\beta $ , and $\\gamma $ are space and time dependent where $\\left\\lbrace \\begin{array}{rcl}\\alpha &=& 1/a_{12}, \\\\\\beta &=& -a_{22}/a_{12} , \\\\\\gamma &=& (a_{12}a_{21}-a_{11}a_{22})/a_{12}, \\end{array}\\right.$ and $\\left( \\begin{array}{cc} a_{11}&a_{12} \\\\ a_{21}&a_{22} \\end{array} \\right)= \\Lambda ^{-1}\\left( \\begin{array}{cc} 0&\\epsilon \\\\ 1/\\mu &0 \\end{array} \\right) \\Lambda .$"
],
[
"The conventional FDTD method for time dependent magneto-electric medium",
"Consider the following 1+1 D Maxwell's equations $\\frac{\\partial D_z}{\\partial t} &=&{\\frac{\\partial H_y}{\\partial x}}, \\\\\\frac{\\partial B_y}{\\partial t} &=&{\\frac{\\partial E_z}{\\partial x}},$ where the constitutive equations are given in Eqs.",
"(REF ) and ().",
"For simplicity, we let $E = E_z$ , $D = D_z$ , $H = H_y$ , and $B = B_y$ .",
"We note that the primary issue with simulating magneto-electric media arises from the more complicated constitutive relations given in Eqs.",
"(REF ) and ().",
"Using the conventional FDTD method, the discretized Maxwell's equations (the $D$ first scheme) are $D_{i}^{n+\\frac{1}{2} } = D_{i}^{n-\\frac{1}{2} } + \\frac{\\Delta t}{\\Delta x} \\left(H_{i+\\frac{1}{2}}^{n} - H_{i-\\frac{1}{2}}^{n} \\right),$ $B^{n+1}_{i+\\frac{1}{2}} = B^{n}_{i+\\frac{1}{2}} + \\frac{\\Delta t}{\\Delta x} \\left(E^{n+\\frac{1}{2}}_{i+1} - E^{n+\\frac{1}{2}}_{i} \\right),$ and the constitutive relations (assuming that the coupled constitutive relations respond locally in space and time) become $ E^{n+\\frac{1}{2}}_{i} = \\alpha ^{n+\\frac{1}{2}}_{i} D^{n+\\frac{1}{2}}_{i} + \\beta ^{n+\\frac{1}{2}}_{i} B^{n+\\frac{1}{2}}_{i},$ $ H^{n+1}_{i+\\frac{1}{2}} = \\beta ^{n+1}_{i+\\frac{1}{2}} D^{n+1}_{i+\\frac{1}{2}} + \\gamma ^{n+1}_{i+\\frac{1}{2}} B^{n+1}_{i+\\frac{1}{2}}.$ Parameters $\\alpha $ , $\\beta $ , and $\\gamma $ are computed analytically from Eq.",
"(REF ) and are thus available at any point $(t,x)$ in the computational domain.",
"However, the quantities $B^{n+\\frac{1}{2}}_{i}$ in Eq.",
"(REF ) and $D^{n+1}_{i+\\frac{1}{2}}$ in Eq.",
"(REF ) are not computed in the discretized Maxwell's Eqs.",
"(REF ) and (REF ).",
"As a result, the conventional Yee approach cannot be directly used in the simulation of magneto-electric media.",
"One method for resolving these issues is to use time extrapolation and space interpolation.",
"To compute $B^{n+\\frac{1}{2}}_{i}$ in Eq.",
"(REF ), we have, $B^{n+\\frac{1}{2}}_{i} &\\approx & \\frac{3B^{n}_{i}-B^{n-1}_{i} }{2}\\\\&\\approx & \\frac{\\frac{3}{2}(B^n_{i+\\frac{1}{2}}+B^n_{i-\\frac{1}{2}}) - \\frac{1}{2}(B^{n-1}_{i+\\frac{1}{2}}+B^{n-1}_{i-\\frac{1}{2}})}{2} \\\\&\\approx & \\frac{ 3B^{n}_{i+\\frac{1}{2} }+ 3B^{n}_{i - \\frac{1}{2}} - B^{n-1}_{i+\\frac{1}{2}}-B^{n-1}_{i-\\frac{1}{2}} }{4},$ which is valid after the first timestep.",
"For the initial timestep, we use $B^{n}_{i}$ as an approximation for $B^{n+\\frac{1}{2}}_{i}$ .",
"Similarly, we compute $D^{n+1}_{i+ \\frac{i}{2}}$ in Eq.",
"(REF ) as follows $ D^{n+1}_{i+\\frac{1}{2}} \\approx \\frac{3D^{n+\\frac{1}{2}}_{i+1}+3D^{n+\\frac{1}{2}}_{i} - D^{n-\\frac{1}{2}}_{i+1}-D^{n-\\frac{1}{2}}_{i}}{4}.$ Numerical simulation on spacetime cloak shows that the conventional FDTD method presented in this section is unstable.",
"This is primarily due to the use of time extrapolation.",
"To avoid such extrapolation, we propose an overlapping Yee algorithm for time dependent magneto-electric media as described in the next section."
],
[
"The overlapping Yee Algorithm for time dependent magneto-electric media",
"Rather than a single $D$ first scheme or $B$ first scheme, we use a combination of both.",
"While computationally more expensive, this produces $D$ and $B$ values at every half time step and spatial step within our computational domain.",
"As a result, we are able to perform the required magneto-electric Maxwell constitutive relation updates without extrapolation.",
"Figure: Computation of E z E_z using (a) the time extrapolation under a conventional FDTD method and(b) the overlapping Yee algorithm without extrapolation.Figure REF illustrates the fundamental differences between utilizing a conventional FDTD approach and the overlapping grid approach.",
"Figure REF depicts that, due to the coupled constitutive relations, at any given step, $E_z$ has a dependence on the quantity $\\overline{B_y}$ .",
"The dotted arrows leading to this value indicate that $\\overline{B_y}$ must be extrapolated in time and interpolated in space from values known on the conventional FDTD grid.",
"On the other hand, Fig.",
"REF illustrates that by utilizing a dual overlapping grid approach, the dependence on time extrapolation and spatial interpolation can be eliminated.",
"A new superscript notation is used to indicate the usage of two overlapping grids and to specify which grid each value of $B_y$ and $E_z$ comes from.",
"All values needed to compute $E_z$ at a particular instance in time exists on one of the two grids.",
"In presenting an algorithm for the adjusted scheme we utilize the convention that $n$ represents the time coordinate of the grid point and $i$ represents the spatial coordinate.",
"Indices of $n\\pm \\frac{1}{2}$ and $i \\pm \\frac{1}{2}$ are use to represent locations that are a half time step or spatial step from the gridpoint.",
"With this established, we present an algorithm for an overlapping Yee FDTD method.",
"The algorithm in 1D case can be summarized by the following steps: Update $D^{n+\\frac{1}{2}}_{i}$ on the first Yee grid and $B^{n+\\frac{1}{2}}_{i}$ on the second Yee grid: $D^{n+\\frac{1}{2}}_{i} = D^{n-\\frac{1}{2}}_{i} + \\frac{\\Delta t}{\\Delta x} \\left[ H^{n}_{i+\\frac{1}{2}} - H^{n}_{i-\\frac{1}{2}} \\right],$ $B^{n+\\frac{1}{2}}_{i} = B^{n-\\frac{1}{2}}_{i} + \\frac{\\Delta t}{\\Delta x} \\left[ E^{n}_{i+\\frac{1}{2}} - E^{n}_{i-\\frac{1}{2}} \\right].$ Update $E^{n+\\frac{1}{2}}_{i}$ and $H^{n+\\frac{1}{2}}_{i}$ : $E^{n+\\frac{1}{2}}_{i} = \\alpha ^{n+\\frac{1}{2}}_{i} D^{n+\\frac{1}{2}}_{i} + \\beta ^{n+\\frac{1}{2}}_{i} B^{n+\\frac{1}{2}}_{i},$ $H^{n+\\frac{1}{2}}_{i} = \\beta ^{n+\\frac{1}{2}}_{i} D^{n+\\frac{1}{2}}_{i} + \\gamma ^{n+\\frac{1}{2}}_{i} B^{n+\\frac{1}{2}}_{i}.$ Update $B^{n+1}_{i+\\frac{1}{2}}$ on the first Yee grid and $D^{n+1}_{i + \\frac{1}{2}}$ on the second Yee grid: $B^{n+1}_{i+\\frac{1}{2}} = B^{n}_{i+\\frac{1}{2} } + \\frac{\\Delta t}{\\Delta x} \\left[ E^{n+\\frac{1}{2}}_{i+1} - E^{n+\\frac{1}{2}}_{i} \\right],$ $D^{n+1}_{i + \\frac{1}{2}} = D^{n}_{i+\\frac{1}{2} } + \\frac{\\Delta t}{\\Delta x} \\left[ H^{n+\\frac{1}{2}}_{i+1} - H^{n+\\frac{1}{2}}_i \\right].$ Update $E^{n+1}_{i+\\frac{1}{2}}$ and $H^{n+1}_{i+\\frac{1}{2}}$ : $E^{n+1}_{i+\\frac{1}{2}} = \\alpha ^{n+1}_{i+\\frac{1}{2}} D^{n+1}_{i+\\frac{1}{2}} +\\beta ^{n+1}_{i+\\frac{1}{2}} B^{n+1}_{i+\\frac{1}{2}},$ $H^{n+1}_{i+\\frac{1}{2}} = \\beta ^{n+1}_{i+\\frac{1}{2}} D^{n+1}_{i+\\frac{1}{2}} +\\gamma ^{n+1}_{i+\\frac{1}{2}} B^{n+1}_{i+\\frac{1}{2}}.$ We note that the extension to 2-D and 3-D should be feasible by utilizing a time collocated grid (where $B$ and $E$ are computed at the same instance in time, but not necessarily in space).",
"Doing so eliminates time extrapolation in 2-D and 3-D but still uses interpolation in space."
],
[
"Spacetime cloak simulation",
"The simulation of magneto-electric media has direct applications to the spacetime cloak presented in [3].",
"We create a similar diamond shaped spacetime cloak using the transformation depicted in Fig.",
"REF .",
"In simulating the above cloak, we utilize a 10 $\\mu m$ by 15 $fs$ long computational domain with a plane wave (with $\\lambda = 600$ $nm$ ) traveling in the positive $x$ direction.",
"Our spacetime cloak is 1200 $nm$ wide and lasts 6 $fs$ .",
"With regards to Fig.",
"REF we have $x_0 = 1.5$ $\\mu m$ , $t_0 = 9$ $fs$ , $t_q = 6$ $fs$ , $t_p=12$ $fs$ , $\\sigma = 600$ $nm$ and $\\delta = 180$ $nm$ .",
"The computational domain is terminated by perfectly matched layer (PML) absorbing boundary conditions [7].",
"Due to the fast and slow phase velocities in the spacetime cloak [3], $\\Delta t$ is chosen by $\\Delta t \\le \\Delta x / v_{max}$ where $v_{max}$ is the largest wave speed computed in the cloaking region.",
"We perform the simulation 3 times with the number of spatial cells, $N$ , equal to 1600, 2000, and 2400 and the same number of time steps Figure: Contour plots of the electric field intensity for spacetime cloak simulations.Left column: the overlapping Yee FDTD results with grid sizes (a) N=1600N = 1600, (c) N=2000N = 2000, and (e) N=2400N = 2400, respectively.Right column: the corresponding results using a conventional time extrapolation based FDTD method.The numerical simulation results are shown in Fig.",
"REF .",
"The figures shown are contour plots of the intensity of the electric field generated by the interaction between the incident source and the spacetime cloak.",
"We compare the conventional FDTD method with time extrapolation and the proposed overlapping Yee FDTD method for various spatial resolutions.",
"The result shows the bending of the electric field (and consequently the electromagnetic wave) in space and time around the cloaked event.",
"The shown simulation results verify that an extrapolation approach is unstable (near the closing process of the cloaked event) and the proposed overlapping FDTD approach provides stable results.",
"Figure: Evolution of the Electric field as the wave propagates through the spacetime cloak.We now examine the behavior of the electric field as the wave propagates through the diamond shaped space-time region under the same simulation parameters.",
"As the wave enters the cloak it begins to split as the front portion of the wave speeds up and the back portion of the wave slows down.",
"This effect is made possible due to the space-time dependence of the media within the cloaking region.",
"As this separation occurs, it is mirrored in the electric field producing a gap in the electric field.",
"The early development of this gap is shown in Fig.",
"REF near $x = 1000~nm$ .",
"As the wave progresses through the media, the cloak widens in the $x$ direction allowing for the size of this gap to increase as time progresses.",
"The gap in the electric field, as shown in Fig.",
"REF , is at a maximum at the center of the cloaking region.",
"As the wave propagates past this point, the cloak (shrink in the $x$ direction) begins to close.",
"Due to the closing of the cloak, the front of the wave begins to slow down while the back speeds up.",
"This action allows for the wave to remerge just as it exits the cloak.",
"The field behavior moments before the wave exits the cloak is shown in Fig.",
"REF .",
"At this point a small gap near $x = 2200~nm$ still exists.",
"Once the wave has propagated through the cloak region, it is fully rejoined and the propagation behavior and resulting field behavior returns to that of propagation in an isotropic media, as shown in Fig.",
"REF ."
],
[
"Stability Analysis",
"To conduct a stability analysis, we compare the distribution of eigenvalues of the conventional FDTD method to that of our overlapping grid approach with respect to the unit circle.",
"For both tests we utilize similar simulation parameters as in the previous section, but with a smaller mesh size of $N = 200$ .",
"We compute the eigenvalues of the numerical update equations at a time instant when $t \\in (t_q, t_p)$ and plot them in the complex plane.",
"The results when $t = 8.34~fs$ are shown in Fig.",
"REF .",
"Figure: Distribution of the eigenvalues in the complex plane for (a) the overlapping Yee FDTD method and (b) the conventional FDTD method.As indicated in the Fig.",
"REF all the eigenvalues of the overlapping approach lie within or on the unit circle with the ones lying on unit circle being simple eigenvalues.",
"On the other hand, Fig.",
"REF shows that some of eigenvalues (near the upper and lower right corners) of the conventional FDTD approach lie outside the unit circle.",
"In particular the modulus of the largest eigenvalue is 1 for the overlapping method and 1.046 for the conventional FDTD method.",
"This indicates that the overlapping Yee FDTD method is stable while the conventional FDTD approach (with extrapolation) would experience instability."
],
[
"Conclusion",
"In conclusion, we have proposed a stable FDTD method for simulation of space and time dependent magneto-electric medium based on the use of two sets of overlapping Yee grids that are offset in time by a half time step.",
"Additionally, we have shown the direct application of this method to the simulation of the spacetime cloak.",
"The proposed method is useful in exploration of other new physical possibilities offered by metamaterials and transformation optics."
],
[
"Acknowledgments",
"We thank Dr. Yong Zeng for invaluable discussions.",
"This work was supported in part by the US Air Force Office of Scientific Research Grant FA9550-10-1-0127, US ARO Grant W911NF-11-2-0046, and NSF Grant HRD-1242067.",
"M. Brio was also supported by US AFOSR MURI Grant FA9550-10-1-0561."
]
] | 1403.0137 |
[
[
"Asymmetric exclusion processes on a closed network with bottlenecks"
],
[
"Abstract We study the generic non-equilibrium steady states in asymmetric exclusion processes on a closed network with bottlenecks.",
"To this end we proposes and study closed simple networks with multiply-connected non-identical junctions.",
"Depending upon the parameters that define the network junctions and the particle number density, the models display phase transitions with both static and moving density inhomogeneities.",
"The currents in the models can be tuned by the junction parameters.",
"Our models highlight how extended and point defects may affect the density profiles in a closed directed network.",
"Phenomenological implications of our results are discussed."
],
[
"Introduction",
"Simplest physical modeling of directed or active classical transports in one dimensions (1D) (e.g., along narrow channels) are often provided in terms of asymmetric simple exclusion processes.",
"A well-known example of directed transport in 1D is the system of unidirectionally moving vehicular traffic along roads [1], [2].",
"Narrow roads with excluded volume interactions between vehicles having no possibility of overtaking, are well-described by the totally asymmetric simple exclusion process (TASEP) [3].",
"Important phenomenological questions about traffic networks include how defects (both extended and point) along a road may control the density profiles and currents in the steady states.",
"We concern ourselves in exploring these issues in terms of a simple model in this paper.",
"In order to focus on the essential physics of the system, we consider a minimal closed directed network (hereafter Model I) consisting of just three segments, with each executing TASEP, two of them (marked $T_A$ and $T_B$ in Fig.",
"REF ) being parallelly and the third one ($T_C$ ) anti-parallelly attached to the multiply connected left ($LJ$ ) and right ($RJ$ ) junctions.",
"Junctions $LJ$ and $RJ$ are non-identical, since the system is directed.",
"Notice that $T_C$ , the antiparallel TASEP channel may be viewed as an extended defect or an extended bottleneck in the system, since its maximum current carrying capacity is less than the total maximum current carrying capacities of the two parallel channels.",
"Thus, $T_C$ restricts the maximum permissible system current.",
"Therefore, $T_C$ acts like an (extended) bottleneck in the system.",
"While in general the length of an extended defect is not necessarily same as the lengths of either of the parallel channels, but for simplicity we take all of them to have the same length with equal number of lattice sites.",
"Although the branching of particles at $LJ$ is controlled by a parameter $\\theta _L$ , at $RJ$ there is no analogous control parameter.",
"There have been a number of applications of TASEP and TASEP-like models in studies on directed vehicular traffic along roads, studying various physical aspects of traffic jam described as TASEP or TASEP-like systems.",
"For instance, the Nagel-Schreckenberg model, closely related to TASEP, has been proposed as a theoretical description for freeway traffic.",
"The model displays traffic jam when the vehicle density is high [4].",
"Shock propagation in traffic systems has also been studied using TASEP [5], [6].",
"In a recent work, Ref.",
"[7] has studied interactions between vehicles and pedestrians in terms of a TASEP with a bottleneck at a boundary caused by interactions.",
"Furthermore, Ref.",
"[8] has considered an “optimal velocity model\" with two kinds of vehicles (fast and slow).",
"With specific lane change rules and under periodic boundary conditions, the traffic states change with increasing densities.",
"Further, Ref.",
"[8] finds a new phenomenon of ratio inversion.",
"In a 1D traffic model, Ref.",
"[9] studies the impact of disruptions on road networks, and the recovery process after the disruption is removed from the system.",
"Extensive reviews on applications of TASEP-like models in the studies of vehicular traffic and related areas are available in Refs.",
"[1], [2].",
"Our studies on Model I are complementary to these model studies.",
"We have strict particle number conservation and an extended defect.",
"Thus Model I should be useful for studying traffic problems where the vehicle current is controlled by an extended defect for a fixed number of vehicles.",
"In the of this article we discuss an extension of Model I, by including additional junction parameters $\\theta _{RA}$ and $\\theta _{RB}$ that control the relative flow of particles from $T_A$ and $T_B$ to $T_C$ at the junction $RJ$ , which appear as point defects in the model, can be considered as traffic signals at that junction.",
"We call this Model II; see Fig.",
"REF .",
"This further allows us to find how the steady state densities are affected by the point defects at $RJ$ .",
"Our results reveal interesting interplay between the extended and point defects, which ultimately controls the steady state densities.",
"We now compare Model I with a recently proposed model by us [10], where two asymmetric exclusion processes in the form of two TASEPs are coupled with $1D$ diffusive motion or SEP.",
"The principal physical difference between them is that, the diffusive channel in Ref.",
"[10] does not control the maximum current in the model, allowing each of the TASEP channels there to reach their individual maximum currents.",
"In contrast, the maximum current in the present models is limited by $T_C,$ the extended bottleneck, which is absent in Ref. [10].",
"As we shall see below, the extended bottleneck has major consequences on the phases and density profiles in Model I; some phases individually accessible to an open TASEP or accessed by the model of Ref.",
"[10] are ruled out here.",
"Here is a summary of our specific results.",
"Both Model I and Model II display generic nonequilibrium phase transitions associated with a variety of density profiles ranging from uniform (flat) profiles to localised (LDW) or static and delocalised (DDW) or moving domain walls, controlled by the particle number and the junction parameters.",
"Nonetheless, there are significant differences between the density profiles of Model I and Model II.",
"Due to the absence of any point defect at $RJ$ , the DDWs in $T_A$ and $T_B$ in Model I are always overlapping, a consequence of a special symmetry at the delocalisation point; in contrast, the DDWs do not overlap in Model II.",
"Furthermore, the presence of the point defect at $RJ$ in Model II allows for a class of phases, which are not permitted in Model I.",
"For example, in Model II, the extended bottleneck $T_C$ can be in its LD-HD coexistence or LDW phase, whereas such a possibility is ruled out in Model I on the ground of current conservation and symmetry.",
"On the whole, our models here serve as good candidates to study the interplay between extended and point defects, and number conservation in closed, simple, directed networks.",
"Our calculational framework may be systematically extended to a network with a larger number of segments/joints.",
"Despite the simplicity and the minimalist nature of our models, the above results should be potentially relevant in the contexts of defect-controlled vehicular traffic in closed network of roads.",
"The rest of the paper is organised as follows.",
"In Sec.",
"we construct Model I.",
"In Sec.",
"we obtain the steady state density profiles and the phase diagram of Model I.",
"In Sec.",
"we conclude.",
"In we introduce Model II.",
"The corresponding phase diagram and the density profiles are analysed, respectively, in and ."
],
[
"Construction of Model I",
"In Model I, each channel consists of equal number of sites designated by $i=1,2,...N.$ As shown in Fig.",
"REF , the left (right) most site of $T_A$ is labeled as $T_A(1)$ ($T_A(N)$ ).",
"Similar labeling follow for $T_B$ and $T_C$ .",
"Particles from $T_C$ enter at the left end of $T_A$ ($T_B$ ) if empty with probability $\\theta _L$ ($1-\\theta _L$ ) and exit to the right end of $T_C$ if vacant from the right end of $T_A$ and $T_B$ with rate unity.",
"In each of $T_A$ , $T_B$ ($T_C$ ) particles can only hop with rate unity to the right (left) neighbor if it is empty.",
"The global particle density is $n_p = N_p/3N$ , with $N_p$ being the total particle number in the system.",
"In this model, the phases are parametrized and tunable by $n_p$ and $\\theta _L.$ The presence of $LJ$ and $RJ$ breaks the translational invariance, and hence, non-trivial steady states are expected [11]."
],
[
"Steady state density profiles",
"We use mean-field theory (MFT) together with extensive Monte-Carlo simulation (MCS) by using random sequential updates to obtain the steady state density profiles in our model.",
"In the MFT, the system is considered as a collection of three separate TASEP channels with effective entry and exit rates [11], [12] ($\\alpha _m$ and $\\beta _m$ , respectively for channel $T_m,\\,m=A,B,C$ ), to be determined by applying the condition of constancy of particle currents at the junctions $LJ$ and $RJ$ and in the bulk.",
"These immediately allow us to obtain the phases of the individual channels and hence of the whole model in terms of the known results for TASEP with open boundaries.",
"In the discrete lattice description of our model, we denote the density at a particular site $i$ in channel $T_m$ by $\\rho _m^i=\\langle n_m^i \\rangle $ , $m=A,B,C$ , $n_m^i = 0$ or 1 and $\\langle ...\\rangle $ denotes time and configuration averages, whereas in MFT considering continuum limit the density is defined as $\\rho _m(x)$ , where $x=i/N,$ and in the thermodynamic limit (TL) $N\\gg 1,$ $x$ in the range $0\\le x\\le 1.$ Given the symmetry between $T_A$ and $T_B$ , $\\rho _A(x)$ and $\\rho _B(x)$ interchange when $\\theta _L$ interchanges with ($1-\\theta _L$ ).",
"This symmetry is identical to the one in the model of Ref. [10].",
"Recall that an isolated TASEP can be in four different phases in its steady states: Low density (LD), high density (HD), maximal current (MC) and coexistence or domain wall (DW) phases [13], [14]; clearly there are almost 64 possibilities for the overall density profiles of our model.",
"For the ease of notation and compactness, we denote a phase by (X-Y-Z) where X/Y/Z refers to the phase of $T_A/T_B/T_C$ for a given choice of $n_p$ and $\\theta _L.$ Obviously, when $n_p$ is very low, all of $T_A,T_B,T_C$ will be in their LD phases for any values of $\\theta _L$ and thus the system adopts the phase LD-LD-LD.",
"Similarly, for a very high $n_p$ all the channels will be in their HD phases and the system is in its HD-HD-HD phase.",
"We now discuss the admissibility of the intermediate phases as $n_p$ varies.",
"Bulk current conservation in the steady states, for any value of $\\theta _L$ , yields (in the MFT), $\\rho _A(x)[1-\\rho _A(x)] + \\rho _B(x) [1-\\rho _B(x)] = \\rho _C(x)[1-\\rho _C(x)].",
"$ The maximum of the right hand side of Eq.",
"(REF ) is $1/4,$ corresponding to the MC phase in $T_C.$ This immediately rules out MC phases in $T_A$ or $T_B$ for $\\theta _L\\ne 0,1$ .",
"Thus, phases ($X,MC,Z$ ) or ($MC,Y,Z$ ) are not allowed.",
"In addition, phases LD-LD-HD and HD-HD-LD are prohibited for any $\\theta _L \\ne 0$ or $1.$ If $T_C$ is in its MC phase for a given $(n_p,\\theta _L)$ value, increasing $n_p$ will lead to addition of more particles with no change in $\\rho _C(x)=1/2$ in the bulk.",
"Hence, the extra particles should accumulate in either $T_A$ or $T_B$ or both, without any change in the total current given by Eq.",
"(REF ).",
"Hence, any addition of particles is then expected to manifest in the form of LDWs in $T_A$ or $T_B$ or both, which leave the currents unchanged.",
"Our detailed steady state density profiles confirm this physically intuitive expectation.",
"The results from Model I are summarized in the phase diagram Fig.",
"REF where different phases are marked in the $(n_p,\\theta _L)$ -plane.",
"Figure: (Color online) Phase diagram in (n p ,θ L )(n_p,\\theta _L) -plane forModel I, phase boundaries are represented by black continuous lines,the line-phases are denoted by green dashed-dotted lines and DDWappears on red dotted line as obtained from MFT, whereascorresponding data points are from MCS with N=500.N = 500.The phase diagram is quite complex in having a large number of phases, as expected.",
"Notice that it is symmetric about the line $\\theta _L=1/2$ , a consequence of the interchange between $T_A$ and $T_B$ with an interchange between $\\theta _L$ and $(1-\\theta _L)$ .",
"Notice that in (REF ) some of the phases are represented by finite areas, whereas others appear just as lines.",
"We discuss the physical principles for the calculation of the density profiles below with some illustrative examples.",
"Full calculations of the steady state densities $\\rho _A,\\,\\rho _B,\\,\\rho _C$ for Model II are given in Appendix as reference.",
"Consider the phase HD-DW-HD, with an LDW in $T_B$ at $x_B^w$ as shown in Fig.",
"REF .",
"Since $T_A$ and $T_C$ are in their HD phases, their densities are $\\rho _A=1-\\beta _{_A}$ and $\\rho _C=1-\\beta _{_C}$ , neglecting the boundary layers (BL).",
"At $RJ$ , both $T_A$ and $T_B$ have no BL; current conservation at $RJ$ yields $\\beta _{_A}= \\beta _{_B}.$ Noting that there is an LDW in $T_B$ , current conservation at $LJ$ gives, $\\beta _{_B}(1-\\beta _{_B}) = (1-\\beta _{_C})(1-\\theta _L)\\beta _{_B}.$ Overall current conservation in the bulk gives, $\\beta _{_A}(1-\\beta _{_A}) + \\beta _{_B}(1-\\beta _{_B}) = \\beta _{_C}(1-\\beta _{_C}).$ Using particle number conservation and neglecting BLs in TL we have, $3n_p = 1-\\beta _{_A}+ \\beta _{_B}+ (1-x_B^w)(1-2\\beta _{_B}) + 1-\\beta _{_C}.$ The boundary lines between HD-LD-HD and HD-DW-HD phases and HD-DW-HD and HD-HD-HD phases may be obtained respectively by setting $x_B^w=1$ and $x_B^w=0;$ $3n_p &=& 3-\\beta _{_A}- \\beta _{_B}- \\beta _{_C}, \\nonumber \\\\3n_p &=& 2 - \\beta _{_A}+ \\beta _{_B}- \\beta _{_C}.$ However, these boundary lines do not span over the entire range $0\\le \\theta _L\\le 1$ , but get cut off at $\\theta _L<1$ at which another phase HD-DW-MC appears; see Fig.",
"REF for the corresponding density profiles.",
"The phase boundary between HD-DW-HD and HD-DW-MC is obtained by setting $\\beta _{_C}=1/2.$ Now putting that in Eqs.",
"(REF ) and (REF ) we have the equation of the horizontal boundary line as $\\theta _L=1/\\sqrt{2}.$ Figure: (Color online) (left) Density profiles for T A ,T B ,T C T_A,T_B,T_Cobtained with n p =0.70n_p = 0.70, θ L =0.90\\theta _L = 0.90 and N=500,N = 500,displaying (HD-DW-HD) phase.",
"LDW is found at x B w =0.661x_B^w=0.661 from MFT which matches with the numerical value 0.672.0.672.",
"(right) Density profiles for (HD-DW-MC) phaseobtained with n p =0.70n_p = 0.70, θ L =0.55\\theta _L = 0.55 and N=500.N = 500.LDW is found at x B w =0.143x_B^w=0.143 from MFT which matches with the numerical value0.149.0.149.",
"Numerical data displayed by points whereas solid lines denote MFT results.The other boundaries of HD-DW-MC phase may be obtained as follows.",
"Current conservation at the $RJ$ gives $\\beta _{_A}=\\beta _{_B}.$ Again from overall current in this phase yields $\\beta _{_A}(1-\\beta _{_A}) + \\beta _{_B}(1-\\beta _{_B}) = 1/4.$ Considering $T_A$ in its HD phase, we get $\\beta _{_A}=0.146=\\beta _{_B}$ from Eq.",
"(REF ).",
"Following the calculation logic outside above, the boundary lines between HD-LD-MC and HD-DW-MC phases and HD-DW-MC and HD-HD-MC phases obtain respectively as $3n_p &=& 3/2, \\nonumber \\\\3n_p &=& 5/2-2\\beta _{_A}.",
"$ The phase HD-DW-MC Again, HD-DW-MC does not span over the whole range of $\\theta _L$ but gets cut by another phase DW-HD-MC as shown in the phase diagram.",
"In the special case with $\\theta _L=1/2$ , $\\rho _A$ and $\\rho _B$ should be statistically symmetric.",
"Therefore, if one of them satisfies the condition for an LDW, the other also must satisfy the same.",
"Thus, we should find a pair of domain walls one each in $T_A$ and $T_B$ , at locations $x_w^A$ and $x_w^B$ , respectively.",
"If there are indeed two LDWs in $T_A$ and $T_B$ , then $\\alpha _{_A}=\\beta _{_A}=\\alpha _{_B}=\\beta _{_B}.$ Furthermore, in this case $T_C$ must be in its MC phase, since once the LDWs are formed in $T_A/T_B$ , any addition of particles in the system will lead to shifting of the LDW positions keeping the currents same.",
"Using current conservation in the bulk, we have, $\\alpha _{_A}(1-\\alpha _{_A}) + \\alpha _{_B}(1-\\alpha _{_B}) = 1/4,$ solving which we have $\\alpha _{_A}=0.146=\\alpha _{_B}.$ Now from particle number conservation we get, $3n_p=\\alpha _{_A}+ (1-x_A^w)(1-2\\alpha _{_A}) + \\alpha _{_B}+ (1-x_B^w)(1-2\\alpha _{_B}) + 1/2.$ Thus, the LDW positions $x_A^w$ and $x_B^w$ can no longer be determined uniquely.",
"Since, all (pairwise) values of $x_A^w$ and $x_B^w$ that satisfy Eq.",
"(REF ), LDWs at each of such pairs of solutions for $x_A^w$ and $x_B^w$ are physically valid solutions for the density profiles in $T_A$ and $T_B$ .",
"In course of time the system should display all these solutions, over long time averages $\\rho _A$ and $\\rho _B$ will essentially appear as two inclined lines, which are the envelopes of the allowed LDW solutions.",
"In other words, we will observe two DDWs.",
"Our MFT analysis and physical arguments for DDWs are verified by our MCS studies.",
"Figure REF shows the DDW-DDW-MC line (red dotted line) as a borderline between the DW-LD-MC and LD-DW-MC phases.",
"In addition, representative DDW profiles for $\\rho _A$ and $\\rho _B$ are shown in Fig.",
"REF .",
"Figure: (Color online) Density profiles for T A ,T B ,T C T_A,T_B,T_Cobtained numerically with θ L =0.50,\\theta _L = 0.50, n p =0.40n_p=0.40 and N=500,N = 500,displaying DDW's in T A T_A and T B T_B with MC phase in T C .T_C.Two DDWs instead of LDWs for $\\theta _L=1/2$ , can be argued from particle number conservation.",
"Notice that for an LDW, its position is uniquely determined by the particle number conservation; see, e.g., Eq.",
"(REF ) that gives $x_B^w$ , the position of the LDW in $T_B$ .",
"However, if there are two LDWs in the system, it is clear that an arbitrary shift in the position of one of the LDWs, together with a compensating reverse shift of the position of the second LDW keeps the total particle number conserved.",
"Thus, the LDW positions are not uniquely determined.",
"Hence the model displays DDWs.",
"This also explains why for a single LDW, its position in the steady state is a quantity that remains stable by the dynamics of the system, where as for DDWs, it is the sum of their positions that remain stable.",
"Understandably, for a very low (high) $\\theta _L$ , $T_A$ ($T_B$ ) is in its LD phase and $\\rho _B >(<)\\rho _A$ in general.",
"For a fixed $\\theta _L.$ when $n_p$ is very low, all of $T_A,T_B,T_C$ are in the LD phases, regardless of the value of $\\theta _L$ .",
"As $n_p$ increases, $T_B$ and $T_C$ move to their DW/HD phase and $T_C$ to MC phases, while $T_A$ remains in its LD phase for small $\\theta _L$ .",
"This is due to the fact that for a small $\\theta _L,$ very few particles enter $T_A$ in comparison with $T_B,$ regardless of $n_p.$ Eventually, as $n_p$ approaches unity (the system is nearly filled), all of $T_A,T_B,T_C$ should be in their HD phases, for any $\\theta _L$ .",
"In general, the densities for $T_A,T_B,T_C$ always change continuously across the phase boundaries.",
"Hence, with channel densities as the order parameter, the transitions are second order in nature.",
"The phases LD-LD-MC and HD-HD-MC are just lines as represented by green dashed-dotted lines in Fig.",
"REF .",
"The fact that they do not cover any area in the $(n_p,\\theta _L)$ -plane can be understood in simple physical terms.",
"Since for these phases, $T_C$ is in MC phase, $\\rho _C=1/2$ and the current through it is $1/4.$ Thus any putative change in $n_p$ is to be reflected by appropriate changes in $\\rho _A$ and $\\rho _B$ .",
"Since $T_A$ and $T_B$ are assumed to have uniform densities (LD or HD phases), any change in $n_p$ automatically leads to changes in $\\rho _A$ and $\\rho _B$ with associated changes in their currents as well.",
"This in turn spoils the bulk current conservation, as the sum of their currents must be $1/4$ (= current in $T_C$ ).",
"Thus, these particular phases can be realized only for one value of $n_p$ for a given $\\theta _L$ , which explains why they appear as lines.",
"For instance, when both $T_A$ and $T_B$ are in their LD phases with densities $\\alpha _{_A}$ and $\\alpha _{_B}$ , respectively, with $\\alpha _{_A}/(1-\\alpha _{_A})=\\theta _L/(1-\\theta _L)$ , the bulk currents here satisfy the same Eq.",
"REF .",
"Now, if $n_p$ is changed, say increased, then both $\\alpha _{_A}$ and $\\alpha _{_B}$ rise keeping their ratio unchanged.",
"This, however, will spoil Eq.",
"REF .",
"Hence, these phases can survive only for one particular value of $n_p$ for a given $\\theta _L$ , which is consistent with their appearance as a line in phase diagram Fig.",
"REF ."
],
[
"Summary and outlook",
"To summarise, Model I reveals interesting interplay between multiple links connecting non-identical junctions and the junctions themselves, that determines the resulting macroscopic steady states density profiles of the overall closed system.",
"Since $T_C$ is effectively an extended bottleneck in the system, our results in fact show the role of extended bottleneck in controlling the phases.",
"Our scheme of MFT for obtaining the steady state density profiles by using current and total particle number conservations are generic and may be extended in straightforward ways to more complex closed systems having larger number of junctions and branches.",
"As discussed in Sec.",
"below, an additional point defect in Model II brings in new macroscopic steady state behaviour, including the possibility of an LDW in $T_C$ and non-overlapping DDWs in $T_A$ and $T_B$ .",
"The steady state bulk current in each of the segments may be easily determined from the knowledge of the corresponding densities; thus we show how the steady state currents may be controlled by the extended and point bottlenecks.",
"Useful modifications in the context of vehicular transport includes unequal lengths of the channels, allowing unequal channel lengths and different velocities for different particles and/or along different channels, impurities on the tracks and possibilities of change in speeds (acceleration and braking) [15].",
"Other possible relevant extensions of theoretical interests include (i) one or some of the branches allow bidirectional motion [10], [16],(ii) when there are local particle non conserving processes, e.g., random attachment and detachment of particles [17], and (iii) the presence of active and inactive agents (particles) [18].",
"We hope our work will motivate further works along these lines."
],
[
"Model II",
"We now introduce Model II, a generalization of Model I, that now includes a point defect at the $RJ$ , defined by two new parameters $\\theta _{RA}$ and $\\theta _{RB}$ ; see Fig.",
"REF .",
"Figure: (Color online) Schematic diagram of Model II with twoadditional parameters θ RA \\theta _{RA} and θ RB \\theta _{RB} at the RJ.RJ.Here, $\\theta _{RA}$ and $\\theta _{RB}$ control the effective particle flow into the the two channels $T_A$ and $T_B$ as point defects at the junction position of regular TASEP channels.",
"Obviously, if we set $\\theta _{RA}= 1 = \\theta _{RB},$ Model II should reduce structurally to Model I.",
"Analysis of the phases in Model II and their differences with their counterparts in Model I allows us to draw conclusions about the effects of the point defects at $RJ.$ As a model for vehicular traffic in a network of roads, these may potentially model dynamic obstacles (e.g., a traffic signal or sudden pedestrian crossings) at $RJ$ in a simple way.",
"We make a detailed comparison between Model I and Model II in Appendix .",
"To reduce the number of tunable parameters, we set $\\theta _{RA}=1-\\theta _{RB}= \\theta _R$ (say), and thereafter all results for Model II are obtained keeping this relation."
],
[
"Phase diagram for Model II",
"The phase diagram for Model II is shown for $\\theta _R=0.20.$ This is quite different from the phase diagram of Model I in Fig.",
"REF regarding the structure and the total number of phases, as new phases emerge in comparison with Model I."
],
[
"$T_A$ in LD-HD coexisting phase, {{formula:d2742bc9-56c7-47e7-b556-1cd1f922bb26}} and {{formula:6b0be851-cb15-42cb-985b-92c6beb2b95c}} both in LD phase",
"Here, $\\rho _B=\\alpha _B,\\,\\rho _C=\\alpha _C$ in the bulk, $\\rho _A=\\alpha _{_A}$ near $LJ$ and $\\rho _A=1-\\beta _{_A}$ near $RJ$ with $\\alpha _{_A}=\\beta _{_A}$ , which meet in the bulk to form an LDW; see Fig.",
"REF .",
"At $LJ$ and $RJ$ the current conservation yields, $\\frac{\\alpha _A}{\\alpha _B}=\\frac{\\theta _L}{1-\\theta _L},\\nonumber \\\\\\beta _{_A}(1-\\beta _{_A})=(1-\\beta _{_A})(1-\\alpha _{_C})\\theta _R.$ Figure: (Color online) Density profile for the phase DW-LD-LD with n p =0.30,θ L =0.70,θ R =0.20.n_p=0.30,\\theta _L=0.70,\\theta _R=0.20.Now particle number conservation yields (neglecting BLs), $3n_p=\\alpha _{_A}+(1-x_A^w)(1-\\alpha _{_A}-\\beta _{_A})+\\alpha _{_B}+\\alpha _{_C},$ where $x_A^w$ is the position of the LDW ($0 < x_A^w < 1$ ) in $T_A$ .",
"From Eqs.",
"(REF -REF $\\alpha _A=\\beta _A,\\alpha _B,\\alpha _B$ may be solved.",
"The boundaries between the LD, LD-HD phases and LD-HD, HD phases of $T_A$ (with $T_B$ and $T_C$ in their LD phases) are obtained by setting $x_A^w=0$ and $x_A^w=1,$ respectively in Eq.",
"(REF ), as shown in Fig.",
"REF for $\\theta _R=0.20$ ."
],
[
"$T_A$ in LD-HD coexisting phase, {{formula:90d58c35-b91c-4526-b490-4e7d8320cd16}} in LD and {{formula:351744fd-f581-40c4-88e6-41f803c8e26b}} in MC phase",
"In this case, in the bulk $\\rho _C=1/2$ , $\\rho _B=\\alpha _{_B}$ ; $T_A$ continues to have an LDW at $x_A^w$ as before.",
"Use now current conservation to write, $\\frac{\\alpha _A}{1-\\alpha _A}=\\frac{\\theta _L}{1-\\theta _L},\\nonumber \\\\\\alpha _{_A}(1-\\alpha _{_A})+\\alpha _{_B}(1-\\alpha _{_B})=1/4.$ Figure: (Color online) Density profile for the phase DW-LD-MC with n p =0.30,θ L =0.25.n_p=0.30,\\theta _L=0.25.Now from particle number conservation we have, $3n_p=\\alpha _{_A}+(1-x_A^w)(1-\\alpha _{_A}-\\beta _{_A})+\\alpha _{_B}+1/2.$ Equations (REF ,REF ) yield $\\alpha _A,\\alpha _B$ and $x_A^w$ .",
"The phase boundaries between LD, LD-HD and LD-HD, HD phases of $T_A$ are obtained by setting $x_A^w=0$ and $x_A^w=1$ respectively with $T_B$ in LD and $T_C$ in MC phase, and are shown in the phase diagram Fig.",
"REF ."
],
[
"$T_A$ in LD-HD coexisting phase, {{formula:c0d6677e-0659-4803-a2e4-79802cccd65a}} and {{formula:ddd30109-0ee6-4f1a-8ac9-dd19c4e8d8ab}} both in HD phase",
"In this case $\\rho _B=1-\\beta _{_B}$ and $\\rho _C=1-\\beta _{_C}$ in the bulk, and $T_A$ has an LDW.",
"Thus $\\alpha _{_A}=\\beta _{_A}$ .",
"From current conservation $\\frac{\\beta _A}{\\beta _B}=\\frac{\\theta _R}{1-\\theta _R},\\nonumber \\\\\\alpha _{_A}(1-\\alpha _{_A})=(1-\\alpha _{_A})\\theta _L(1-\\beta _{_C}).$ Figure: (Color online) Density profile for the phase DW-HD-HD with n p =0.70,θ L =0.10,θ R =0.20.n_p=0.70,\\theta _L=0.10,\\theta _R=0.20.Particle number conservation yields $3n_p=\\alpha _{_A}+(1-x_A^w)(1-\\alpha _{_A}-\\beta _{_A})+1-\\beta _{_B}+1-\\beta _{_C}.$ where $0<x_A^w<1$ is the position of the LDW in $T_A$ .",
"Eqs.",
"(REF -REF ) yield $\\beta _C, \\alpha _{_A}$ and $x_A^w$ .",
"The phase boundaries between LD,LD-HD and LD-HD,HD phases of $T_A$ are, as usual, obtained by setting $x_A^w=0$ and $x_A^w=1$ respectively with both $T_B$ and $T_C$ are in their HD phases.",
"For $\\theta _R=0.20$ the boundaries are shown in Fig.",
"REF ."
],
[
"$T_A$ in LD-HD coexisting phase, {{formula:3ec04121-f623-4ebb-a46a-0855cd88e58d}} in HD and {{formula:8609f05c-470c-4cbd-b6df-1d73a4b3978b}} in MC phase",
"Here $\\rho _C=1/2$ in the bulk.",
"Current conservation at $RJ$ gives, $\\frac{\\beta _A}{\\beta _B}=\\frac{\\theta _R}{1-\\theta _R}.$ Figure: (Color online) Density profile for the phase DW-HD-MC with n p =0.60,θ L =0.15,θ R =0.20.n_p=0.60,\\theta _L=0.15,\\theta _R=0.20.From particle number conservation we have, $3n_p=\\alpha _{_A}+(1-x_A^w)(1-\\alpha _{_A}-\\beta _{_A})+1-\\beta _{_B}+1/2.$ From Eq.",
"REF and REF we $\\alpha _{_A},\\beta _{_B}$ and $x_A^w$ .",
"The phase boundaries between LD,LD-HD and LD-HD,HD phases are obtained by setting $x_A^w=0$ and $x_A^w=1$ respectively with $T_B$ in HD and $T_C$ in MC phase, and for $\\theta _R=0.20$ the boundaries are shown in Fig.",
"REF ."
],
[
"$T_A$ in HD, {{formula:bd610dd2-0275-4b56-9784-4f6da5e1d447}} in LD-HD coexisting phase and {{formula:7e0bfa4a-de4c-4436-bbbd-baaac807dffc}} in HD phase",
"In this case $\\rho _A=1-\\beta _{_A}$ and $\\rho _C=1-\\beta _{_C}$ , respectively.",
"Since $T_B$ is assumed to be in its LD-HD coexistence phase thus $\\alpha _{_B}=\\beta _{_B}$ .",
"From the current conservation we have, $\\alpha _{_B}=(1-\\beta _{_C})(1-\\theta _L), \\nonumber \\\\\\frac{\\beta _A}{\\beta _B}=\\frac{\\theta _R}{1-\\theta _R}.$ Figure: (Color online) Density profile for the phase HD-DW-HD with n p =0.70,θ L =0.80,θ R =0.20.n_p=0.70,\\theta _L=0.80,\\theta _R=0.20.Again from particle number conservation, $3n_p=1-\\beta _{_A}+\\alpha _{_B}+(1-x_B^w)(1-\\alpha _{_B}-\\beta _{_B})+1-\\beta _{_C}.$ Here $0<x_B^w<1$ is the position of the localised DW in $T_B$ , while $T_A$ and $T_C$ both are in HD phase.",
"Eq.",
"(REF -REF ) yield $\\alpha _{_B}, \\beta _{_C}$ and $x_B^w$ .",
"The phase boundaries are obtained by setting $x_B^w=0$ and $x_B^w=1,$ and for $\\theta _R=0.20$ the boundaries are shown in Fig.",
"REF ."
],
[
"$T_A$ in HD, {{formula:a82dddcd-9835-4f99-946f-ea8f445cbfaf}} in LD and {{formula:40347653-2c1d-43fe-9066-eac78748049f}} in LD-HD coexisting phase",
"In this case $\\rho _A=1-\\beta _{_A}$ and $\\rho _B=\\alpha _{_B}$ in the bulk, and $T_C$ is in DW phase.",
"Thus $\\alpha _{_C}=\\beta _{_C}$ , and $1-\\beta _{_C}$ give the bulk densities on the entry ($RJ$ ) and exit ($LJ$ ) sides of $T_C$ .",
"Now the current matching conditions at $LJ$ and $RJ$ give, $\\alpha _B=(1-\\beta _C)(1-\\theta _L),\\nonumber \\\\\\beta _A=\\theta _R(1-\\alpha _C).$ Figure: (Color online) Density profile for the phase HD-LD-DW with n p =0.50,θ L =0.90,θ R =0.20.n_p=0.50,\\theta _L=0.90,\\theta _R=0.20.Now from particle number conservation we have, $3n_p=1-\\beta _{_A}+\\alpha _{_B}+\\alpha _{_C}+(1-x_C^w)(1-\\alpha _{_C}-\\beta _{_C}).$ where $0<x_C^w<1$ is the position of the DW in $T_C$ .",
"From Eq.",
"REF and REF we get $\\alpha _{_C}$ , $\\alpha _{_B}$ and $x_C^w$ .",
"The phase boundaries between LD,LD-HD and LD-HD,HD phases of $T_C$ are obtained by setting $x_C^w=0$ and $x_C^w=1$ respectively, and for $\\theta _R=0.20$ the boundaries are shown in Fig.",
"REF ."
],
[
"Delocalised domain walls",
"Now, assume both $T_A$ and $T_B$ have domain walls.",
"Then,we must have $\\alpha _{_A}=\\beta _{_A}$ and $\\alpha _{_B}=\\beta _{_B}.$ Again from current matching conditions at $LJ$ and $RJ$ we have, $\\alpha _{_A}/\\alpha _{_B}=\\theta _L/(1-\\theta _L)$ and $\\beta _{_A}/\\beta _{_B}=\\theta _R/(1-\\theta _R)$ respectively.",
"These gives the condition for having domain walls in both channels as, $\\theta _L=\\theta _R.",
"$ By the similar argument as given for Model I we can show that, both the domain walls are delocalised with the sum of their positions remain stable and in that case $T_C$ remains in the MC phase.",
"The density profiles for the channels are shown in Fig.",
"REF , while the DDW boundary line is shown by red dotted line in phase diagram Fig.",
"REF .",
"Note that the DDWs in $T_A$ and $T_B$ are non-overlapping.",
"Figure: (Color online) Density profiles when both T A T_A and T B T_B has DDWs with n p =0.50,θ L =0.20=θ R n_p=0.50,\\theta _L=0.20 = \\theta _R and N=500.N=500."
],
[
"Boundary between DW-LD-LD and DW-LD-MC phases",
"Here $\\alpha _{_C}=1/2$ at this phase boundary.",
"Now from Eq.",
"(REF ) we have $\\alpha _{_A}=\\theta _R/2$ and using the overall current conservation the solutions for $\\theta _L$ are given by, $\\theta _L=\\frac{\\theta _R^2}{\\theta _R^2+\\theta _R\\pm \\sqrt{\\theta _R^3(2-\\theta _R)}}.$ Clearly, $\\theta _L$ is independent of $n_p$ ; for $\\theta _R=0.20$ the boundary line is shown in the phase diagram Fig.",
"REF ."
],
[
"Boundary between DW-HD-MC and DW-HD-HD phases",
"At the boundary $1-\\beta _{_C}=1/2.$ Again from Eq.",
"(REF ) we have $\\alpha _{_A}=\\theta _L/2$ and $\\beta _{_B}= q\\alpha _{_A}$ , where $q=(1-\\theta _R)/\\theta _R.$ Now putting these values in overall current conservation equation we have a quadratic equation in $\\theta _L$ with the solutions given as, $\\theta _L=\\frac{1+q\\pm \\sqrt{2q}}{1+q^2}.$ Now enforcing that $T_B$ is in its HD phase, Eq.",
"(REF ) yields the phase boundary as a horizontal line in ($n_p,\\theta _L$ )-plane.",
"For $\\theta _R=0.20$ it is shown in Fig.",
"REF ."
],
[
"Boundary between HD-DW-MC and HD-DW-HD phases",
"At the boundary $1-\\beta _{_C}=1/2,$ and from Eq.",
"(REF ) $\\alpha _{_B}=(1-\\theta _L)/2$ and $\\beta _{_A}= q^{\\prime }\\alpha _{_B}$ , where $q^{\\prime } =\\theta _R/(1-\\theta _R)$ .",
"Now putting those values in the current conservation equation we have, $\\theta _L =\\frac{1+q^{\\prime }\\pm \\sqrt{2q^{\\prime }}}{1+{q^{\\prime }}^2}.$ Now enforcing HD phase for $T_A$ , we have the phase boundary (a horizontal line) in the ($n_p,\\theta _L$ )-plane, as shown for $\\theta _R=0.20$ in Fig.",
"REF ."
],
[
"Boundary between HD-LD-MC and HD-LD-HD phases",
"At the transition $\\beta _{_C}=1/2,$ and at $LJ$ we have $\\alpha _{_B}=(1-\\theta _L)/2.$ Now from overall current conservation, $4\\beta _{_A}(1-\\beta _{_A}) + (1-\\theta _L)(1+\\theta _L)=1.$ From Eq.",
"(REF ) and particle number conservation $6n_p=3-\\theta _L+\\sqrt{1-\\theta _L^2}.$ Equation (REF ) gives the corresponding phase boundary in the ($n_p,\\theta _L$ )-plane as shown Fig.",
"REF ."
],
[
"Boundary between LD-HD-MC and LD-HD-HD phases",
"Set $\\beta _{_C}=1/2,$ for this transition.",
"Then at the $LJ$ $\\alpha _{_A}=\\theta _L/2.$ From overall current conservation we write, $4\\beta _{_B}(1-\\beta _{_B}) + \\theta _L(2-\\theta _L)=1.$ Particle number conservation together with Eq.",
"(REF ) yield $6n_p=2+\\theta _L+\\sqrt{\\theta _L^2-2\\theta _L}.$ Equation (REF ) gives the corresponding phase boundary in ($n_p,\\theta _L$ )-plane as shown in Fig.",
"REF ."
],
[
"Comparison between Model I and Model II",
"We now compare and contrast the results from Model I and Model II.",
"This will allow us to understand the extent to which the point defects at $RJ$ can affect the phases.",
"First of all, the phase diagram Fig.",
"REF for Model I is symmetric about the line $\\theta _L=1/2,$ whereas there is no such symmetry in the phase diagram Fig.",
"REF for Model II in general, due to the point defects at $RJ.$ In Model II there are some phases which has no analogue in Model I, for example DW-LD-LD phase.",
"Let us consider this phase where the BLs are at the two opposite junctions, at $RJ$ for $T_B$ and at $LJ$ for $T_C.$ At $RJ$ the current matching conditions between $T_A$ and $T_C$ gives, $\\alpha _{_A}(1-\\alpha _{_A})=(1-\\alpha _{_A})(1-\\alpha _{_C})\\theta _{RA}.$ Now use the condition for retrieving Model I from Model II given by $\\theta _{RA}=\\theta _{RB}=1,$ then from Eq.",
"(REF ) $\\alpha _{_A}=(1-\\alpha _{_C}).$ Put this in the overall current conservation equation, $\\alpha _{_A}(1-\\alpha _{_A}) + \\alpha _{_B}(1-\\alpha _{_B}) = \\alpha _{_C}(1-\\alpha _{_C})$ to get, $\\alpha _{_B}(1-\\alpha _{_B}) = 0$ , implying $\\alpha _{_B}=0$ or 1, which is unfeasible.",
"Thus, the assumption of the existence of the phase DW-LD-LD in Model I is not correct; therefore, we conclude that DW-LD-LD phase does not appear for Model I.",
"Similar arguments show that HD-LD-LD and HD-LD-DW phases are also absent for Model I.",
"These observations are clearly validated by our MCS studies on Model I, as displayed in Fig.",
"(REF ).",
"Evidently, the phases which appear only in Model II are solely due to the point defect at $RJ.$ Furthermore, the DDWs in Model I should be fully overlapping (under long time averages; not shown in Fig REF ), a feature consistent with the symmetry of Model I about $\\theta _L=1/2.$ In contrast, the DDWs in Model II are generally non-overlapping (even under long time averages), due to the lack of any special symmetry in Model II as seen in Fig.",
"REF .",
"These differences are connected to the fact that in Model I by construction, DDWs in $T_A$ and $T_B$ correspond to $\\alpha _{_A}=\\beta _{_A}=\\beta _{_B}=\\alpha _{_B}.$ In Model II however there are no such equalities, due to the presence of the point defect at $RJ$ .",
"A related consequence is that Model I displays DDWs only for $\\theta _L=1/2,$ where as in Model II, it is possible to have DDWs for arbitrary $\\theta _L$ so long as the general conditions for DDWs are met.",
"Overall, thus, the effect of introducing point defects at $RJ$ is not only to change the locations of the phases in the $(n_p,\\theta _L)$ phase diagram qualitatively, but also to introduce new phases which were absent in Model I.",
"Thus, the topology the phase diagram gets affected."
],
[
"Limiting cases of Model II",
"The limiting cases of Model II reveal interesting features.",
"For instance, when $\\theta _L$ and $\\theta _R$ are either 0 or 1 simultaneously then by construction either $T_A$ or $T_B$ is fully blocked.",
"The remaining system then has equal hopping rate at every site.",
"Thus, the average density at every site is just $n_p$ .",
"Furthermore, if say $\\theta _L$ is 1 or 0 with $\\theta _R$ having a value between zero and unity, then the junction $RJ$ effectively serves as a point defect in an otherwise homogeneous ring executing TASEP.",
"For instance, consider $\\theta _L\\rightarrow 1,\\,0<\\theta _R<1,$ thus eliminating $T_B$ and allowing for a point defect at $RJ$ , given by a reduced hopping rate $\\theta _R\\prime <1.$ In this limit our model is identical to the model in Ref.",
"[19] and our results in this limit $\\theta _L\\rightarrow 1$ are in agreement with that.",
"Acknowledgement:- AB wishes to thank the Max-Planck-Gesellschaft (Germany) and Department of Science and Technology/Indo-German Science and Technology Centre (India) for partial financial support through the Partner Group programme (2009).",
"A.K.C.",
"acknowledges the financial support from DST (India) under the SERC Fast Track Scheme for Young Scientists [Sanction no.",
"SR/FTP/PS-090/2010(G)]."
]
] | 1403.0206 |
[
[
"Charge Gaps at Fractional Fillings in Boson Hubbard Ladders"
],
[
"Abstract The Bose-Hubbard Hamiltonian describes the competition between superfluidity and Mott insulating behavior at zero temperature and commensurate filling as the strength of the on-site repulsion is varied.",
"Gapped insulating phases also occur at non-integer densities as a consequence of longer ranged repulsive interactions.",
"In this paper we explore the formation of gapped phases in coupled chains due instead to anisotropies $t_x \\neq t_y$ in the bosonic hopping, extending the work of Crepin {\\it et al.}",
"[Phys.",
"Rev.",
"B 84, 054517 (2011)] on two coupled chains, where a gap was shown to occur at half filling for arbitrarily small interchain hopping $t_y$.",
"Our main result is that, unlike the two-leg chains, for three- and four-leg chains, a charge gap requires a finite nonzero critical $t_y$ to open.",
"However, these finite values are surprisingly small, well below the analogous values required for a fermionic band gap to open."
],
[
"INTRODUCTION",
"Experiments on ultracold atomic gases have opened new possibilities in the exploration of strongly correlated quantum physics[1].",
"Most significantly, the ratio of interaction to kinetic energy can be readily tuned, something which is possible only with substantial effort in condensed matter systems, e.g.",
"through the application of high pressure in a diamond anvil cell.",
"The flexible nature of the optical lattices that can be generated using interfering lasers has also allowed the study of different dimensionality and dimensional crossover, as well as the systematic interpolation between different geometries in a given dimension, e.g.",
"triangular to kagome[2] or square to triangular[3].",
"In the case of bosonic systems, a key focus has been on the superfluid (SF) to Mott insulator (MI) quantum phase transition, which occurs at zero temperature and commensurate filling as the ratio of kinetic energy to on-site repulsion $U$ is tuned [4].",
"The MI phase is characterized by a nonzero charge gap $\\Delta _c$ : A plot of density, $\\rho $ , versus chemical potential, $\\mu $ , exhibits plateaux when $\\rho $ takes on integer values.",
"The critical interaction strength is now known to very high precision, e.g.",
"taking the value $(t/U)_c = 0.05974(3)$ for a square lattice [5].",
"With longer range interactions, these plateaux can develop at non-integer fillings as well.",
"For example, a sufficiently large near-neighbor repulsion $V_1$ drives a checkerboard solid order at $\\rho =\\frac{1}{2}$ on a square lattice[6].",
"Likewise, a next-near-neighbor repulsion $V_2$ can cause stripe ordering at the same filling.",
"Both patterns of charge ordering are accompanied by a non-zero gap.",
"Long range (e.g.",
"dipolar) interactions give rise to a complex \"devil's staircase\" structure in $\\rho (\\mu )$ [7], [8].",
"The number of plateaux which develop increases with system size, indicating the presence of frustration [8].",
"Another fascinating, and experimentally realizable, mechanism which produces MI phases at fractional fillings is the presence of a superlattice superimposed on the optical lattice.",
"These superlattice fractional filling MI phases are expected in one[9], [10], [11] and two[12], [13] dimensions and in one-dimensional chains of tilted double well potentials[14], [15].",
"In this paper we study the appearance of fractional filling MI plateaux in the Bose-Hubbard model on coupled chains which originate instead due only to anisotropic hopping, $&&{\\hat{\\mathcal {}H}=- \\mu \\sum _{i} n_{i}+ \\frac{1}{2} U \\sum _{i} n_{i} (n_{i} -1 )\\\\&&- t_x \\sum _{ i }(a_{i}^\\dagger a_{i+\\hat{x}}^{\\phantom{\\dagger }} +a_{i+\\hat{x}}^\\dagger a_{i}^{\\phantom{\\dagger }} )- t_y \\sum _{ i }(a_{i}^\\dagger a_{i+\\hat{y}}^{\\phantom{\\dagger }} +a_{i+\\hat{y}}^\\dagger a_{i}^{\\phantom{\\dagger }} )\\nonumber }Here a_{i}^\\dagger , a_{i}^{\\phantom{\\dagger }}, andn_{i}^{\\phantom{\\dagger }} are boson creation, destruction, andnumber operators on sites i of an L_x\\times L_y rectangularlattice.",
"t_{x} and t_{y} are hopping parameters between sites{i} and neighoring sites in the \\hat{x} and \\hat{y} directionsrespectively.",
"We will focus on \"coupled chain\" geometries in whichL_x \\gg L_y.",
"U is an on-site repulsion, We shall consider hereonly the hard-core limit U=\\infty .",
"We note the absence of longerrange interactions and superlattice potentials, so that the onlypossible mechanism for the fractional filling MI is the new one(anisotropic hopping) considered here.$ As noted above, the presence of density plateaux in the ground state of the Hamiltonian, Eq.",
"(1), is expected at commensurate filling for sufficiently large $U$ , but otherwise a compressible SF phase is the most natural low temperature phase.",
"In contrast, for fermions, additional band insulator plateaux can arise from the structure of the dispersion relation $\\epsilon ({\\bf k})$ .",
"For example, for a two-chain ($L_y=2$ ) system $\\epsilon (k_x,k_y) = -2t_x \\,{\\rm cos}k_x -t_y \\,{\\rm cos}k_y$ where $k_y$ takes on only the two values $k_y=0,\\pi $ .",
"This dispersion relation can equivalently be viewed as that of a single one-dimensional (1D) chain with two bands, $\\epsilon (k_x) = -2t_x \\,{\\rm cos}k_x \\pm t_y$ .",
"If $t_y > 2 t_x$ the two bands are separated by a gap, and a plateau in $\\rho (\\mu )$ will occur at $\\rho =\\frac{1}{2}$ .",
"Such gaps arising from the structure of $\\epsilon ({\\bf k})$ are typically to be expected only of fermionic systems, since they occur as a consequence of the Pauli principle and the complete filling of a lower energy band.",
"In one dimension, however, hard-core ($U=\\infty $ ) bosons (the \"Tonks-Girardeau gas\")[16] share many similarities with fermions, via a Jordan-Wigner transformation[17].",
"Hence such band insulating behavior might be expected for hard-core bosons as well.",
"Indeed, such 1D \"bosonic band insulators\" have been observed when a superlattice potential $\\Delta \\sum _i (-1)^i n_i$ is added to the 1D hard core boson Hubbard Hamiltonian [11].",
"As in the two-chain case, a superlattice potential opens a gap in the dispersion relation which can be measured by the plateau in $\\rho (\\mu )$ and which is accompanied by a vanishing of the SF density $\\rho _s$ .",
"Interestingly, these insulators are found to extend deep into the soft-core limit, where the Jordan-Wigner mapping to fermions is no longer valid.",
"Crepin et al.$\\,$[18] have recently studied another example of the persistence of the 1D Jordan-Wigner analogy between hard-core bosons and fermions by examining two-leg ladders.",
"Here the possibility of particle exchange destroys the formal mapping and one might expect features of the fermionic case not to have bosonic analogs.",
"Nevertheless, they found that in the large $t_y$ limit the bosonic system exhibits a charge gap $\\Delta = 2t_y - 4 t_x +2t_x^2/t_y$ , where the first two terms are precisely the fermionic result.",
"The last term represents an increase in the gap for hard-core bosons relative to fermions.",
"In addition, for weakly coupled chains where $t_y \\ll t_x$ , the fermionic case would have overlapping bands and no charge gap.",
"Yet for bosons $\\Delta $ remains non-zero, although exponentially small, with $\\Delta \\propto e^{-\\alpha t_x/t_y}$ .",
"It is natural to consider the extension of the question of insulating behavior in bosonic systems to cases where there are more than two chains, and even to the two-dimensional limit.",
"In this paper we present results for the charge gap and SF response of the Hamiltonian, Eq.",
"(1), on three- and four-leg ladders.",
"Our primary methodology is quantum Monte Carlo (QMC) simulations using the ALPS[19] stochastic series expansion (SSE) code.",
"We supplement this with density matrix renormalization group (DMRG) calculations also using the ALPS library.",
"Our major result is that incommensurate gaps persist beyond the two leg case, although a finite, but surprisingly small, $t_y$ seems to be required.",
"For the three- and four-chain systems we find that even for $t_y/t_x = {\\cal O}(1) $ , gaps form at fractional fillings.",
"As the number of chains increases, the finite critical value increases.",
"Figure: (Color online) (a) ρ(μ)\\rho (\\mu ) and (b) SF density for a 64×264\\times 2 ladder of hard-core bosons with t x =t y =1t_x=t_y=1 (black) andt x =1t_x=1, t y =2t_y=2 (blue).",
"For t y =2t_y=2, a gap is visible atρ=1 2\\rho =\\frac{1}{2}, and the corresponding SF density vanishes,signaling the insulating state.",
"At t y =1t_y=1 the gap is not visiblein the ρ(μ)\\rho (\\mu ) plot but the dip in ρ s (μ=0)\\rho _s(\\mu =0) hints atits presence.",
"Careful finite size scaling demonstrates this clearly.",
"Here and in subsequent figureserror bars are smaller than the symbol size and hence are not shown."
],
[
"Methodology",
"Our simulations are performed using the SSE algorithm[20], [21], [22], a powerful and elegant QMC method to study quantum spin or bosonic lattice models.",
"SSE is a generalization of Handscomb's algorithm [23] for the Heisenberg model.",
"It starts from a Taylor expansion of the partition function in orders of inverse temperature $\\beta $ , and corresponds to a perturbation expansion in all terms of the Hamiltonian.",
"There are no “Trotter errors\" associated with discretization of imaginary time.",
"We use a grand-canonical formulation of the code, but also restrict some measurements to just one particle number sector in order to generate results in the canonical ensemble.",
"To characterize the phases of the Hamiltonian, Eq.",
"(1), we will examine the energy, $E=\\langle H \\rangle $ , the variation of density as a function of chemical potential and the SF density $\\rho _s$ .",
"The SSE code computes the SF density via the relation[24] $\\rho _s=\\frac{\\langle W^2\\rangle }{2td\\beta L^{d-2}},$ where $W$ is the winding number of the particle world lines, $d$ is the dimensionality and $\\beta $ is the inverse temperature.",
"In the coupled-chain problem we address here, we take $L_y\\ll L_x$ , $t_x\\ne t_y$ , and the boundary conditions are open (periodic) in the $\\hat{y}$ ($\\hat{x}$ ) direction.",
"Note, however, that taking periodic boundary conditions in the $\\hat{y}$ direction does not change the physics qualitatively; the fractional filling gaps are still present but the values of the critical $t_y$ change slightly.",
"We focus, therefore, on the SF density in the $\\hat{x}$ direction so that the hopping parameter in Eq.",
"(REF ) is $t_x$ and the length $L$ is $L_x$ .",
"In what follows, we typically take $\\beta =2L_x$ to access ground state properties.",
"However, for small $L_x$ where $2L_x \\le 128$ , we set $\\beta =128$ .",
"In addition, for some cases of $2L_x> 128$ , we verified that putting $\\beta =3L_x$ yields the same results as $\\beta =2L_x$ .",
"We also take $t_x=1$ to set the energy scale."
],
[
"Results",
"We begin by studying the two-chain case considered by Crepin et al.$\\,$[18] in order to confirm our methodology.",
"Figure REF demonstrates that our SSE results are fully consistent with this earlier work.",
"A plateau in the density at half filling is clearly visible for $t_y=2$ when $-0.5\\lesssim \\mu /t \\lesssim 0.5$ , signaling the incompressible insulating state where the SF density vanishes.",
"For $t_y=t_x=1$ , the plateau at half filling is not easily visible, but a sharp minimum appears in the SF density hinting at its presence.",
"A careful finite size scaling study [18] shows its presence clearly.",
"We now explore whether these plateaux persist in geometries having more than two chains, and if so, at what fillings.",
"Figure REF (a) shows the density as a function of chemical potential for a three-leg ladder of 64 sites in the $\\hat{x}$ direction.",
"Plateaux are seen to develop at $\\rho =\\frac{1}{3}, \\frac{2}{3}$ for sufficiently large $t_y$ .",
"Figure REF (b) gives the SF density, and confirms that it vanishes in the gapped phase.",
"At first glance, such incommensurate insulators might seem surprising for a model with only on-site repulsion, since empty sites exist to which the bosons can hop without large energy cost.",
"As discussed already in the two-chain case, these insulators are not a surprise if the particles are fermionic: For the noninteracting system, $U=0$ , the Hamiltonian can be diagonalized by going to momentum space giving rise to the dispersion relation given in the Introduction.",
"In the three leg case (with open boundary conditions in the $\\hat{y}$ direction), there are three bands, $\\epsilon (k_x,k_y) = -2t_x\\,{\\rm cos}k_x + \\lbrace -\\sqrt{2}t_y, 0, \\sqrt{2}t_y$ }.",
"For sufficiently large $t_y$ , a gap $\\Delta _f(\\rho =\\frac{1}{3})=0-2t_x-(-\\sqrt{2}t_y+2t_x)=\\sqrt{2}t_y-4$ is present between the top of the lowest band and the bottom of the middle band.",
"In a fermionic model at $T=0$ a band insulator would be present at $\\rho =\\frac{1}{3}$ .",
"An identical gap would appear at $\\rho =\\frac{2}{3}$ as a consequence of particle-hole symmetry.",
"Thus noninteracting fermions hopping on this three chain geometry would be band insulators at $\\rho =\\frac{1}{3}, \\frac{2}{3}$ for $t_y \\ge 4/\\sqrt{2}\\sim 2.829$ (see Fig.",
"REF ).",
"The gaps at large values of $t_y/t_x$ are easy to understand in the hard core limit.",
"When $t_x\\ll t_y$ , the particles delocalize much more efficiently along the $\\hat{y}$ axis, and hence spread out in that direction.",
"Thus, when $\\rho =1/3$ , the particles are in an effective one-dimensional system with a density $\\rho _{\\rm eff}= 1$ .",
"This system, due to the hard core nature of the particles, becomes an incompressible insulator.",
"This intuitive picture applies equally to the four-chain system.",
"We examine below how small $t_y$ should be for such behavior to change.",
"Figure: (Color online) (a) The gap Δ\\Delta at ρ=1 3\\rho = \\frac{1}{3} in athree-leg ladder as a function of t y t_y.",
"The values of Δ\\Delta areextrapolated from lattice sizes up to L x =250L_x =250.",
"For large t y t_y,Δ\\Delta approaches the value appropriate for a collection ofnoninteracting fermions (dashed red line).",
"(b) Semilogarithmic plot of thegaps in (a) as a function of 1/(t y -0.94)1/(t_y-0.94).The dependence of the bosonic gap on $t_y$ at $\\rho =\\frac{1}{3}$ in a three-leg system is shown in Fig.",
"REF (a).",
"The gap $\\Delta (N)$ is computed from the jump in the chemical potential $\\mu (N)=E(N)-E(N-1)$ at the critical density, i.e.",
"$\\Delta (N)=\\mu (N+1)-\\mu (N)=E(N+1)+E(N-1)-2E(N)$ .",
"We use both the ALPS SSE and DMRG codes.",
"The results are extrapolated using lattice sizes up to $L_x = 250$ for the smaller values of the gap.",
"In the DMRG calculations we keep 800 states, which we have verified to be converged.",
"The gap for a noninteracting three-leg fermion chain is also presented.",
"The convergence of the bosonic results to the fermionic ones at large $t_y$ suggests a fermion mapping is appropriate even in the absence of a formal Jordan-Wigner transformation.",
"The boson $\\Delta $ is larger than its fermionic counterpart and is unambiguously non-zero below the critical $t_y$ where fermions would become metallic.",
"A key issue is whether finite $\\Delta $ persists all the way down to $t_y=0$ as occurs for the two-chain case [18].",
"To try to answer this question, Fig.",
"REF (b) shows a semilogarithmic plot of the gap as a function of $1/(t_y-t_y^c)$ where $t_y^c$ is the putative critical value at which the gap vanishes.",
"A fit of the form $\\Delta =a\\, {\\rm exp}(-b/(t_y-t_y^c))$ yields $t_y^c \\approx 0.94$ and this is shown as the dashed (blue) line in the lower panel.",
"In the two-chain case, Crepin et al.",
"have found [18] an exponential behavior all the way to very small (vanishing) $t_y$ (large $1/t_y$ ).",
"As a consequence, they argued that an exponentially small gap persists all the way to $t_y=0$ .",
"In the present three-leg case, the evidence for a finite value for $t_y^c$ is clear.",
"The value is considerably less than the fermionic band structure value $4/\\sqrt{2}$ and even suggests that when the system is isotropic, $t_x=t_y$ , the three-leg system is gapped at $\\rho =\\frac{1}{3}$ (and also $\\rho =\\frac{2}{3}$ ).",
"Figure: (Color online) SF density at ρ=1 3\\rho =\\frac{1}{3} as a function oft y t_y.",
"The SF density remains nonzero for small t y t_y, even in thelargest lattice size we study here.Figure: (Color online) Semilogarithmic plot of the SF density at ρ=1 3\\rho =\\frac{1}{3} for different t y t_y, as a function of L x L_x.",
"In a non-SFphase, we expect ρ s ∼ exp (-L x /ξ)\\rho _s \\sim {\\rm exp}(\\,-L_x/\\xi \\,) and hencelinear behavior here.Figure: (Color online) Collapse of the data inFig.",
", by rescaling the xx axis,from L x →L x /ξL_x \\rightarrow L_x/\\xi , with the correlation length ξ\\xi obtainedfrom Fig.",
".Figure: (Color online) Correlation length ξ\\xi of the three-legladder versus the y ^\\hat{y} direction hopping t y t_y.",
"The points obtainedfrom the data collapse (black circles) are well fitted by theexponential form (red curve).To confirm the above result for $t_y^c$ , we turn to an examination of the SF density at $\\rho =\\frac{1}{3}$ to see if it becomes nonzero at finite, small $t_y$ .",
"Figure REF shows $\\rho _s$ at $\\rho =\\frac{1}{3}$ for lattice sizes $L_x = 16$ to $L_x =200$ .",
"The SF density decreases with increasing $t_y$ , and for $t_y\\gtrsim 4/\\sqrt{2}$ , the fermionic critical value for the three-band case previously discussed, the behavior seems unambiguously insulating ($\\rho _s=0$ ), consistent with the analysis of the charge gap.",
"Indeed $\\rho _s$ vanishes even for somewhat smaller $t_y$ , in agreement with the observation that the bosonic charge gap lies above the fermion one.",
"However, for smaller $t_y$ , the SF density is nonzero even for the largest lattice size we consider here.",
"The persistence of nonzero $\\rho _s$ as the lattice size increases in Fig.",
"REF can be taken as an indication of a nonzero $t_y^c$ .",
"To verify this possibility quantitatively, we make a semilogarithmic plot of the SF density versus $L_x$ for different $t_y$ (Fig.",
"REF ).",
"For large $t_y$ , the SF density decays exponentially with $L_x$ and we can compute correlation length $\\xi (t_y)$ from $\\rho _s \\simeq {\\rm exp}(\\,-L_x/\\xi \\,)$ .",
"This analysis parallels that presented in [crepin11].",
"The values of $\\xi $ are presented in the legend of Fig.",
"REF .",
"They are fairly well determined down to $t_y \\sim 1.7$ where $\\xi $ is still on the order of the size of our simulation box.",
"For smaller $t_y$ the correlation length becomes much larger than $L_x$ , as indicated by the nearly horizontal traces in Fig.",
"REF .",
"A consistency check on our extraction of $\\xi $ is provided by collapsing the data for $\\rho _s$ , that is, via plotting results for different $t_y$ as a function of a rescaled horizontal axis $L_x/\\xi $ as in Fig.",
"REF .",
"Figure: (Color online) Kinetic, potential, and total energies of thethree-leg ladder as functions of the filling ρ\\rho for (a) t y =1t_y=1and (b) t y =6t_y=6.",
"The kinetic energy is maximized (in absolutevalue) at half filling.",
"While all the energy curves are smooth fort y =1t_y=1, for t y =6t_y=6 the curves of potential energy and totalenergy have vertical jumps at ρ=1 3\\rho =\\frac{1}{3} and 2 3\\frac{2}{3}, alsodisplayed by the kinetic energy although barely visibleon the scale of the figure.Figure: (Color online) (a) ρ(μ)\\rho (\\mu ) and (b) SF density for a fourleg ladder of hard-core bosons, similar to those inFig. .",
"As the transversehopping t y t_y grows, gaps develop at ρ=1 4\\rho =\\frac{1}{4}, 1 2\\frac{1}{2} and3 4\\frac{3}{4}, and SF density vanishes there.Figure REF provides a fit of these values of $\\xi $ to the functional form $\\xi = \\xi _0 \\, {\\rm exp}[\\,a/(t_y-t_y^c)\\,]$ .",
"The best fit is obtained for $t_y^c\\approx 1.07$ .",
"This value is in good agreement with the value, $t_y^c\\approx 0.94$ , obtained above from a study of the charge gap and confirms that a finite, but not very large, value of the transverse hopping, $t_y$ , is necessary to establish a gapped insulating phase at $\\rho =\\frac{1}{3}$ .",
"In Fig.",
"REF , we investigate the individual components of the energy.",
"The kinetic energy is maximized (in absolute value) at half filling, because we are studying the case with no intersite interactions.",
"For $t_y=1$ , all the energy curves are smooth; for $t_y=6$ , the curves of potential energy and total energy have vertical jumps at $\\rho =\\frac{1}{3}$ and $\\frac{2}{3}$ , which correspond to the insulating gap in Fig.",
"REF .",
"The curve of the kinetic energy also has kinks at those densities, although these are barely visible at the scale of the figure.",
"Figure: (Color online) (a) The extrapolated gaps at ρ=1 4\\rho = \\frac{1}{4}in the four-leg ladder as a function of t y t_y from QMC and DMRG.",
"(b)Semilogarithmic plot of the gap versus (t y -1.69) -1 (t_y-1.69)^{-1}.",
"Exponentialdecay over three decades is seen giving the critical valuet y c ≈1.69t_y^c\\approx 1.69.Figure: (Color online) (a) The extrapolated gaps at ρ=1 2\\rho = \\frac{1}{2}in the four-leg ladder as a function of t y t_y from QMC and DMRG.",
"(b)Semilogarithmic plot of the gap versus (t y -1.70) -1 (t_y-1.70)^{-1}.",
"Exponentialdecay over three decades is seen giving the critical valuet y c ≈1.70t_y^c\\approx 1.70.Figure REF is the same as Fig.",
"REF but for the four-leg ladder and shows (a) the density as a function of chemical potential for a system of $64\\times 4$ sites.",
"Plateaux are seen to develop at $\\rho =\\frac{1}{4}, \\frac{1}{2}, \\frac{3}{4}$ for sufficiently large $t_y$ .",
"Figure REF (b) shows the SF density, and confirms that it vanishes in the gapped phase.",
"In this case the fermionic band structure consists of four bands (two of which are degenerate) with $\\epsilon ({\\bf k}) = -2 t_x\\,{\\rm cos}k_x + 2 t_y \\lbrace -1, 0, 0, +1 \\rbrace $ , and the fermionic gaps $\\Delta _f(\\rho =\\frac{1}{4})=\\Delta _f(\\rho =\\frac{3}{4})=t_y-4$ , $\\Delta _f(\\rho =\\frac{1}{2})=\\sqrt{2(3-\\sqrt{5})}t_y-4$ .",
"As opposed to Fig.",
"REF which showed two gaps between the three bands, we have here three gaps at $\\rho =\\frac{1}{4},\\frac{1}{2}, \\frac{3}{4}$ between the four bands.",
"Figures REF and REF show in the (a) panels gaps at $\\rho =\\frac{1}{4}$ and $\\rho =\\frac{1}{2}$ respectively as functions of $t_y$ for the four-leg lattice, similar to Fig.",
"REF .",
"As for the three-leg ladder, these gaps are obtained with DMRG and QMC using the definition $\\Delta (N)=E(N+1)+E(N-1)-2E(N)$ and then extrapolated from lattice sites $L_x = 16$ to 128 (with DMRG $L_x$ is taken up to 250).",
"The fermionic gaps are also included in these figures.",
"Compared with the three-leg ladder, the gaps in the four-leg system need somewhat larger $t_y$ to form and, as expected, the gaps at $\\rho = \\frac{1}{2}$ are wider than at $\\rho = \\frac{1}{4}$ .",
"The fermionic band gaps provide a reasonable estimate of those seen in the bosonic case.",
"Figure: (Color online) The SF density, ρ s \\rho _s, versus thescaled system size, L/ξL/\\xi .",
"ξ\\xi is the correlation length at thevalues of t y t_y shown in the legend.",
"These values of ξ\\xi are usedin Fig.",
"to obtain the critical value of thehopping, t y c t_y^c.Figure: (Color online) Correlation length at ρ=1 4\\rho =\\frac{1}{4} for thefour-leg ladder versus the transverse hopping t y t_y.",
"The pointsobtained from the data collapse (black circles) are fitted well bythe exponential form (red curve) indicating a finite critical valuet y c ≈1.83t_y^c\\approx 1.83.Figure: (Color online) As in Fig.",
"butfor ρ=1 2\\rho =\\frac{1}{2}.",
"The values of ξ\\xi are used inFig.",
"to obtain the critical value of thehopping, t y c t_y^c.Figure: (Color online) Correlation length at ρ=1 2\\rho =\\frac{1}{2} of thefour-leg ladder versus the transverse hopping t y t_y.",
"The pointsobtained from the data collapse (black circles) are fitted well bythe exponential form (red curve).",
"This gives the finite criticalvalue for the hopping, t y c ≈1.93t_y^c\\approx 1.93.We have also examined the exponential behavior of the gap at $\\rho =\\frac{1}{4}$ and $\\rho =\\frac{1}{2}$ in the four-leg ladder as described for the three-leg system.",
"The (b) panels of Figs.",
"REF and REF show semilogarithmic plots of the gaps as functions of $(t_y-t_y^c)^{-1}$ with $t_y^c=1.69$ for $\\rho =\\frac{1}{4}$ and $t_y^c=1.70$ for $\\rho =\\frac{1}{2}$ .",
"Exponential decay is seen over three decades yielding finite values for the critical hoppings.",
"As in the three-leg case the critical values of $t_y$ at which gaps appear are reduced from their fermion counterparts.",
"As for the three-leg ladder, we confirm the above results for $t_y^c$ by examining the SF density at $\\rho =\\frac{1}{4}$ and $\\rho =\\frac{3}{4}$ to see if it becomes nonzero at finite, small $t_y$ .",
"The lattice size is up to $L_x=200$ , as in the three-leg case (Fig.",
"REF ).",
"We follow the same analysis which led to Figs.",
"REF and REF but only show figures for the scaled SF density.",
"Figure REF gives, for $\\rho =\\frac{1}{4}$ , $\\rho _s$ versus the scaled size of the system, $L_x/\\xi $ , where $\\xi $ is the correlation length at the various values of $t_y$ shown in the legend of the figure.",
"The data collapse is excellent and the values of $\\xi $ thus obtained are shown versus $t_y$ in Fig.",
"REF .",
"The exponential fit, also shown in the figure, yields the critical value $t_y^c=1.83$ , which is in reasonable agreement with the value obtained from the gap, $t_y^c=1.69$ .",
"Figures REF and REF show the corresponding figures at $\\rho =\\frac{1}{2}$ and yield $t_y^c=1.93$ in good agreement with the value obtained directly from the gaps, $t_y^c = 1.70$ ."
],
[
"Conclusions",
"The most simple physical picture of the origin of Mott insulating phases focuses on the (large) repulsive interaction energy, and the filling at which the addition of quantum particles first causes that energy to appear.",
"In a model with only a strong on-site repulsion, particles can be added without any potential energy cost up until a commensurate filling¡ª¡ªone particle per site.",
"At that point, double occupancy becomes unavoidable, and the cost to increase the density jumps.",
"The precise form of the kinetic energy, e.g., whether it is isotropic in different spatial directions, or connects only near-neighbor sites, is irrelevant to this simplistic argument: As long as any empty sites remain in the lattice there is no charge gap and, at low temperatures, a collection of bosonic particles will form a SF phase.",
"However, recent work by Crepin has demonstrated that in two-leg ladders insulating phases can arise at half-filling, underlining the need to refine this picture.",
"In this paper we studied hard-core bosons in three- and four-leg ladders with anisotropic hopping parameters in the $\\hat{x}$ and $\\hat{y}$ directions, $t_x$ and $t_y$ .",
"To this end, we used both QMC and DMRG from the ALPS library[19].",
"Our focus was the possibility of the appearance of gaps at fractional fillings as happens for the two-leg ladder[18], a possibility which seems surprising in light of the argument above.",
"At the other extreme, when the number of coupled chains is large and the system approaches the limit of a two-dimensional system, one expects gaps to appear at fractional fillings but at very large values of $t_y/t_x$[25].",
"We have shown above that for both the three-leg and the four-leg systems gaps appear at values of $t_y/t_x$ , which are smaller than those resulting from an argument based on fermionic bands.",
"While in the case of the two-leg system, the gap at $\\rho =\\frac{1}{2}$ persists[18] all the way down to $t_y=0$ , we found that, in the three-leg case, a gap appears at $\\rho =\\frac{1}{3}$ starting at the isotropic value, $t_y/t_x\\approx 1$ .",
"For the four-leg system, a gap at $\\rho =\\frac{1}{4}$ appears starting at $t_y/t_x \\approx 1.8$ and for $\\rho =\\frac{1}{2}$ at $t_y/t_x\\approx 1.9$ ."
],
[
"ACKNOWLEDGMENTS",
"This work was supported by: the National Key Basic Research Program of China (Grant No.",
"2013CB328702), the National Natural Science Foundation of China (Grant No.",
"11374074), a CNRS-UC Davis EPOCAL LIA joint research grant, and by the University of California Office of the President.",
"We thank J. McCrea for useful input."
]
] | 1403.0249 |
[
[
"A conservation formulation and a numerical algorithm for the double-gyre\n nonlinear shallow-water model"
],
[
"Abstract We present a conservation formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean basin.",
"The novelty of the paper is that we reformulate the governing equations into a nonlinear hyperbolic conservation law plus source terms.",
"A second-order fractional-step algorithm is used to solve the reformulated equations.",
"In the first step of the fractional-step algorithm, we solve the homogeneous hyperbolic shallow-water equations by the wave-propagation finite volume method.",
"The resulting intermediate solution is then used as the initial condition for the initial-boundary value problem in the second step.",
"As a result, the proposed method is not sensitive to the choice of viscosity and gives high-resolution results for coarse grids, as long as the Rossby deformation radius is resolved.",
"We discuss the boundary conditions in each step, when no-slip boundary conditions are imposed to the problem.",
"We validate the algorithm by a periodic flow on an f-plane with exact solutions.",
"The order-of-accuracy for the proposed algorithm is tested numerically.",
"We illustrate a quasi-steady-state solution of the double-gyre model via the height anomaly and the contour of stream function for the formation of double-gyre circulation in a closed basin.",
"Our calculations are highly consistent with the results reported in the literature.",
"Finally, we present an application, in which the double-gyre model is coupled with the advection equation for modeling transport of a pollutant in a closed ocean basin."
],
[
"Introduction",
"The two-dimensional shallow-water equations govern the fluid motion in a thin layer.",
"They can be used as a rational approximation to the three-dimensional Euler equations, with the assumption of hydrostaticity and shallow water depth (compared with the horizontal length scale).",
"When wind forcing and latitude-dependent Coriolis forces are included, these equations represent a simple model for describing the depth-average dynamics of the oceans.",
"Furthermore, if we include a Laplacian diffusion in the equations and impose Dirichlet boundary conditions on the velocity field, in particular the no-slip conditions, the equations are often used to simulate a mid-latitude closed ocean basin.",
"In this paper we focus on a reduced-gravity shallow-water model formulated for studying the behavior of western boundary currents (WBCs) in mid latitudes [3].",
"In this ocean model water is assumed to consist of two layers of fluid, a single active layer of fluid of constant density $\\rho $ and variable thickness $h(x, y, t)$ , overlying a deep and motionless layer of density $\\rho +\\Delta \\rho $ .",
"Consequently, the motion of the upper layer represents the gravest baroclinic mode [3].",
"The model equations in non-conservation form are $\\begin{split}&\\frac{\\partial h}{\\partial t}+\\frac{\\partial (uh)}{\\partial x} +\\frac{\\partial (vh)}{\\partial y}=0,\\\\&\\frac{\\partial u}{\\partial t} +u\\frac{\\partial u}{\\partial x} + v\\frac{\\partial u}{\\partial y}=-g_r\\frac{\\partial h}{\\partial x} + (f_0+\\beta y)v + \\nu \\nabla ^2 u +F^{u},\\\\&\\frac{\\partial v}{\\partial t} +u\\frac{\\partial v}{\\partial x} + v\\frac{\\partial v}{\\partial y}=-g_r\\frac{\\partial h}{\\partial y} - (f_0+\\beta y)u + \\nu \\nabla ^2 v +F^{v},\\\\\\end{split}$ where $(u, v)$ is the velocity filed, $h$ is the height field, $g_r=(\\Delta \\rho / \\rho )g$ is the reduced gravity, and $g$ is the acceleration of gravity.",
"$(F^{u}, F^{v})$ is the external forcing term, such as the wind forcing [3], [4], [9], [10].",
"With the imposition of no-slip boundary conditions on the velocity field (the height field is allowed to assume any value on the boundaries), equations (REF ) describe a wind-driven, closed basin on a $\\beta $ plane.",
"The equations are normally referred to as the double-gyre, wind-driven shallow-water model.",
"This model is a convenient test bed for studying mid-latitude ocean dynamics [3], [4], [10].",
"The numerical algorithm MPDATA (Multidimensional Positive Definite Advection Transport Algorithm) has long been used to solve geophysical flows, such as flow governed by Eq.",
"(REF ).",
"MPDATA is a two-pass scheme that preserves positive definite scalar transport functions with small oscillations [14], [15], [16].",
"Technically, the method belongs to the same class of non-oscillatory Lax-Wendroff algorithms such as FCT [20], TVD[17], and ENO [2].",
"Nevertheless, MPDATA was primarily developed for meteorological applications.",
"The method focuses on reducing the implicit viscosity of the donor cell scheme, while retaining the virtues of positivity, low phase error, and simplicity of upstream differencing.",
"However, the disadvantage of MPDATA is that the basic MPDATA is too diffusive, and enhanced MPDATA is too expensive [16].",
"We compare a basic MPDATA implementation described in [11] with the proposed algorithm for the double-gyre model in Section .",
"For a thorough review of MPDATA, we refer the readers to [16].",
"Aimed at improving the resolution and accuracy, a type of multi-scale finite difference method was developed in [4], [10] for solving equations (REF ).",
"The multi-scale method, or enslaved finite-difference method makes use of properties of the governing equations in the absence of time derivatives to reduce the overall truncation errors without changing the order of spatial discretization, nor the time step restriction of the time integrator.",
"This means that the enslaved scheme effectively increases the spatial resolution of the given algorithm without changing its temporal stability or memory requirements.",
"However, the enslaved scheme could be sensitive to the viscosity values used in the calculation of solving the shallow-water double-gyre model for some time integrators.",
"Especially for numerical approximations with resolution near the Rossby deformation radius.",
"For example, it is reported in [4] that for Rossby deformation radius $\\approx 52-75$ km, the implementation of an enslaved scheme using the leapfrog time integrator could be numerically unstable for explicit viscosity values less than $\\nu =1000$ $\\text{m}^2\\,\\text{s}^{-1}$ for the resolution $\\Delta x =40$ km, and $\\nu =750$ $\\text{m}^2\\,\\text{s}^{-1}$ for the resolution $\\Delta x =20$ km.",
"For solving the double-gyre model it is common for this class of schemes that to maintain numerical stability, the viscosity needs to be increased as the grid resolution is decreased.",
"[4].",
"In this paper, we propose a stable method for solving the double-gyre model.",
"We rewrite the governing equations into a conservation form with source terms.",
"A fractional-step algorithm is used to solves the new formulation.",
"In the first step, the hyperbolic equations are solved by the high-resolution wave-propagation method developed by LeVeque [7].",
"Then the resulting intermediate values are used as the initial conditions for the initial-boundary problem.",
"The fractional-step strategy has proven to be efficient and stable for solving the Navier-Stokes equations and other fluid models [5], [6].",
"We organize the rest of the paper as follows.",
"In Section , we present the conservation form of the double-gyre shallow-water model.",
"Then we introduce a fractional-step method to solve the equations and discuss the boundary conditions in each step.",
"In Section , we verify the algorithm by an exact solution of a period flow on an f-plane.",
"We show that numerically the method is second-order accurate.",
"Then we use the algorithm to study an upper-ocean double-gyre circulation in a closed ocean basin.",
"We compute the height anomaly for the formation of double-gyre circulation, and compare the results with the literature values computed by the enslaved finite-difference schemes [10] and the traditional methods of backward Euler and centered finite-difference [3].",
"The results are highly consistent.",
"Finally, we present an example, in which the double-gyre model is coupled with the advection equation for modeling transport of a pollutant in a closed ocean basin.",
"This example demonstrates the flexibility of the proposed method to couple with other equations that require high-resolution results for the monitored quantity, such as a passive tracer in fluid."
],
[
"The fractional-step algorithm",
"The model equations (REF ) can be written in their conservation form $\\begin{split}&\\frac{\\partial h}{\\partial t}+\\frac{\\partial (uh)}{\\partial x} +\\frac{\\partial (vh)}{\\partial y}=0,\\\\&\\frac{\\partial (hu)}{\\partial t} +\\frac{\\partial }{\\partial x}\\left(hu^2+\\frac{1}{2}g_rh^2\\right)+\\frac{\\partial }{\\partial y}\\left(huv\\right) = (f_0+\\beta y) hv + h (\\nu \\nabla ^2 u) +h F^{u},\\\\&\\frac{\\partial (hv)}{\\partial t} +\\frac{\\partial }{\\partial x}\\left(huv\\right)+\\frac{\\partial }{\\partial y}\\left(hv^2+\\frac{1}{2}g_rh^2\\right) =- (f_0+\\beta y) hu + h (\\nu \\nabla ^2 v) +h F^{v},\\end{split}$ where $hu$ and $hv$ are the momenta in $x$ and $y$ directions, and $P(h)= \\frac{1}{2}g_rh^2$ is the hydrostatic equation of state with a reduced gravity.",
"Equations (REF ) represent a system of two-dimensional hyperbolic conservation law with a source term, $q_t+f(q)_x+g(q)_y=\\psi (q, \\tilde{q}),$ where $\\begin{split}&q=\\begin{bmatrix}h \\\\ hu \\\\ hv\\end{bmatrix}, \\qquad f(q)=\\begin{bmatrix}hu \\\\ hu^2+\\frac{1}{2}g_rh^2 \\\\ huv\\end{bmatrix}, \\qquad g(q)=\\begin{bmatrix}hv \\\\ huv \\\\ hv^2+\\frac{1}{2}g_rh^2\\end{bmatrix},\\\\&\\tilde{q}=\\begin{bmatrix}h\\\\u\\\\v\\end{bmatrix}, \\qquad \\psi (q, \\tilde{q})=\\begin{bmatrix}0\\\\ (f_0+\\beta y)hv + h(\\nu \\nabla ^2 u) +h F^{u}\\\\- (f_0+\\beta y)hu + h(\\nu \\nabla ^2 v) +h F^{v}\\end{bmatrix}.\\end{split}$ At first glance, equations (REF ) seem to be inconsistent in the treatment of the stress tensor parametrization.",
"The natural variable for the momentum equations is $q$ , so in principle the assumed eddy viscosity parametrization should also be expressed in terms of $q$ , instead of $u$ and $v$ .",
"However, scaling the non-conservation “advective” form of the shallow-water equations (REF ) leads to the geostrophic balance between the horizontal velocity and the horizontal pressure gradient, i.e.",
"the gradient of height field.",
"In other words, the principal geostrophic balance is between the Coriolis force and the height (pressure) gradient, not the dissipative term[9].",
"The conservation formulation (REF ) preserves the principal geostrophic balance, and the balance is now in the form of the momentum variable $q$ .",
"We propose a fractional-step method, also known as operator splitting, for Eq.",
"(REF ) that simply alternates solving the following two problems: $\\begin{split}&\\text{Problem A:}\\quad q_t + f(q)_x + g(q)_y = 0;\\\\&\\text{Problem B:}\\quad q_t = \\psi (q, \\tilde{q}).\\\\\\end{split}$ Problem A is a homogeneous conservation law that can be solved by the high-resolution finite volume method developed in [7].",
"After spatial discretization, Problem B is treated as a simple system of ordinary differential equations (ODEs) that can be solved by a standard time integrator.",
"Since $h_t = 0$ in Problem B, we can further simplify Problem B by letting $q_1=\\begin{bmatrix}hu \\\\ hv\\end{bmatrix},\\quad \\tilde{q}_1=\\begin{bmatrix}u \\\\ v\\end{bmatrix},$ and Problem B becomes $\\frac{\\partial q_1}{\\partial t}= R q_1 + S (\\tilde{q}_1, h),$ where $R$ is a $2\\times 2$ constant matrix and $S$ is a vector function of $\\tilde{q}_1$ and $h$ .",
"The forms of $R$ and $S$ are explicitly written in Eq.",
"(REF ).",
"If both Problem A and B are solved over one time step $\\Delta t$ , this is the so-called Godunov splitting for a fractional-step method.",
"The splitting error of the Godunov splitting is $O(\\Delta t)$ in theory.",
"In practice, however, the error is smaller than $O(\\Delta t)$ [1].",
"The Strang splitting is a slight modification of the Godunov splitting and yields second-order accuracy generally [7].",
"The difference between the Godunov splitting and the Strang splitting is that the Strang splitting starts and ends with a half time step $\\Delta t /2$ on Problem A.",
"In between the first and the last time steps, the Strang splitting is the same as the Godunov splitting.",
"That is, Problem B and A are solved alternately over one time step $\\Delta t$ .",
"The splitting error of Strang splitting is $O(\\Delta t^2)$ .",
"To be more specific, basically for the Godunov splitting we solve the two sub-problems sequentially, like (A) $\\longrightarrow $ (B), by using the time increments $\\lbrace \\Delta t ,\\,\\Delta t\\rbrace $ in each time step, respectively, and for the Strang splitting in each time step we solve the two sub-problems in a sequence of (A) $\\longrightarrow $ (B) $\\longrightarrow $ (A), by using the time increments $\\lbrace \\frac{\\Delta t}{2},\\,\\Delta t,\\,\\frac{\\Delta t}{2}\\rbrace $ .",
"After combining the the cycles, $\\lbrace \\frac{\\Delta t}{2},\\,\\Delta t,\\,\\frac{\\Delta t}{2}\\rbrace $ , $\\lbrace \\frac{\\Delta t}{2},\\,\\Delta t,\\,\\frac{\\Delta t}{2}\\rbrace $ ,..., $\\lbrace \\frac{\\Delta t}{2},\\,\\Delta t,\\,\\frac{\\Delta t}{2}\\rbrace $ , the Strang splitting is the same as the Godunov splitting, except the Strang splitting uses $\\frac{\\Delta t}{2}$ for solving Problem A in the very beginning, as well as the very end.",
"Moreover, Yoshida [19] introduced a systematic method to construct arbitrary even-order time accurate splitting schemes.",
"The Strang splitting is a modification of the fist member of the Yoshida's method.",
"Let the computational domain be $\\Omega $ and the boundary of the domain be $\\partial \\Omega $ .",
"Let $C$ be a two-dimensional grid cell $\\Delta x \\times \\Delta y$ and $q$ be the solution the partial differential equations.",
"Let $Q_{i,j}^{n}$ be an approximation to the cell average of $q$ over the cell $C_{i,j}$ at time $t=t^{n}$ , i.e.",
"$Q_{i,j}^{n}=\\frac{1}{\\Delta x\\Delta y}\\int _{C_{i,j}} q(x, y, t^{n})dx dy.$ The cell averaged value is placed at the cell center.",
"Suppose that the boundary conditions for the velocity field $u$ and $v$ are prescribed, the fractional-step method is described as follows: Step 1: Given $Q^{n}$ , the semi-discrete system of equations arising from Problem A has the form $\\begin{split}&\\frac{Q^{m}-Q^{n}}{\\Delta t} + F(Q^{m}, Q^{n}) + G(Q^{m}, Q^{n}) = 0,\\\\&Q^{m}\\,\\,\\text{on}\\,\\,{\\partial \\Omega },\\,\\,\\text{the boundary conditions are given}.\\end{split}$ Solve the above system by the wave-prorogation finite volume method to obtain $Q^{m}$ .",
"We briefly describe the multidimensional wave-prorogation finite volume method as follows.",
"Problem A, the hyperbolic shallow-water equations, can be written as a quasi-linear equations $q_t +f^{\\prime }(q)q_x +g^{\\prime }(q)q_y=0,$ where $f^{\\prime }(q) = A(h,u,v) =\\left( \\begin{array}{ccc}0 & 1 & 0 \\\\-u^2+g_rh& 2u & 0 \\\\-uv& v & u\\end{array} \\right),$ and $g^{\\prime }(q) = B(h,u,v) =\\left( \\begin{array}{ccc}0 & 0 & 1 \\\\-uv & v & u \\\\-v^2 + g_rh& 0 & 2v\\end{array} \\right).$ Let $c=\\sqrt{g_rh}$ be the speed of gravity wave.",
"The matrix $A$ has eigenvalues and eigenvectors $\\begin{split}& \\lambda ^{x_1} =u-c,\\quad \\lambda ^{x_2} =u,\\quad \\lambda ^{x_3} =u+c \\\\& r^{x_1}=\\left[ \\begin{array}{c}1\\\\u-c\\\\v\\end{array} \\right],\\quad r^{x_2}=\\left[ \\begin{array}{c}0\\\\0\\\\1\\end{array} \\right],\\quad r^{x_3}=\\left[ \\begin{array}{c}1\\\\u+c\\\\v\\end{array} \\right],\\end{split}$ while the matrix $B$ has eigenvalues and eigenvectors $\\begin{split}& \\lambda ^{x_1} =v-c,\\quad \\lambda ^{x_2} =v,\\quad \\lambda ^{x_3} =v+c \\\\& r^{x_1}=\\left[ \\begin{array}{c}1\\\\u\\\\v-c\\end{array} \\right],\\quad r^{x_2}=\\left[ \\begin{array}{c}0\\\\-1\\\\0\\end{array} \\right],\\quad r^{x_3}=\\left[ \\begin{array}{c}1\\\\u\\\\v+c\\end{array} \\right].\\end{split}$ For the wave-propagation algorithm, the updating formula over a time step $\\Delta t$ is $\\begin{split}Q^{m}_{i, j} = Q^{n}_{i, j} &- \\frac{\\Delta t}{\\Delta x}\\left(\\mathcal {A}^{+}\\Delta Q^{n}_{i-1/2,j}+ \\mathcal {A}^{-}\\Delta Q^{n}_{i+1/2,j} \\right)\\\\& - \\frac{\\Delta t}{\\Delta y}\\left(\\mathcal {B}^{+}\\Delta Q^{n}_{i,j-1/2}+ \\mathcal {B}^{-}\\Delta Q^{n}_{i,j+1/2} \\right)\\\\& - \\frac{\\Delta t}{\\Delta x}\\left(\\tilde{F}_{i+1/2, j} - \\tilde{F}_{i-1/2, j} \\right) - \\frac{\\Delta t}{\\Delta y}\\left(\\tilde{G}_{i, j+1/2} - \\tilde{G}_{i, j-1/2} \\right).\\end{split}$ The second and the third terms on the right-hand-side of Eq.",
"(REF ) are the fluctuations, while the fourth and the fifth terms are the correction fluxes.",
"Both fluctuations and correction fluxes are computed by using the approximate Riemann solver (or the Roe solver) that averags the waves and speeds (corresponding to the eigenvectors and eigenvalues in Eqs.",
"(REF ) & (REF )) by the Roe average.",
"Detailed information about the actual representations of the fluctuations and correction fluxes can be found in [7], pp 471–474.",
"Step 2: From Step 1, we obtain $Q^{m}=\\begin{bmatrix}H^{m} \\\\ (HU)^{m} \\\\ (HV)^{m},\\end{bmatrix}, \\qquad H^{n+1}=H^{m}.$ The semi-discretized equations for (REF ), arising by using the centered-difference scheme for the spatial derivatives, has the form $\\begin{split}\\frac{\\partial (HU)_{i, j}}{\\partial t}&=(f_0+\\beta y)(HV)_{i,j} + \\\\ &\\nu H_{i,j}\\left(\\frac{U_{i-1, j} - 2 U_{i, j} + U_{i+1, j} }{(\\Delta x)^2} + \\frac{U_{i, j-1} - 2 U_{i, j} + U_{i, j+1} }{(\\Delta y)^2} \\right)+ H_{i,j} F^{u}_{i.j},\\\\\\frac{\\partial (HV)_{i, j}}{\\partial t} &= (f_0+\\beta y)(HU)_{i,j} +\\\\ & \\nu H_{i,j}\\left(\\frac{V_{i-1, j} - 2 V_{i, j} + V_{i+1, j} }{(\\Delta x)^2} + \\frac{V_{i, j-1} - 2 V_{i, j} + V_{i, j+1} }{(\\Delta y)^2} \\right)+ H_{i,j} F^{v}_{i.j},\\end{split}$ where $U_{i,j}=\\frac{(HU)_{i,j}}{H_{i,j}},\\quad V_{i,j}=\\frac{(HV)_{i,j}}{H_{i,j}},\\quad \\text{for}\\,\\, H_{i,j}\\ne 0,\\,\\,i,\\,j=1\\cdots N.$ If $H_{i,j}=0$ , it means that the water depth is zero, which is not physically meaningful.",
"Equation (REF ) is a system of ODEs with $2N$ dimensions, where $N$ is the number of cells used in (REF ).",
"The initial conditions are $(HU)_{i.j}=(HU)^m_{i,j}$ and $(HV)_{i.j}=(HV)^m_{i,j}$ and the final time is $t^{n+1}=t^{n}+\\Delta t$ .",
"The prescribed boundary conditions for the velocity field are employed in this step.",
"Note that with a sufficiently small time step, an explicit $p$ -stage, $p^{th}$ -order Runge-Kutta method, $p > 1$ , is a $A$ -stable method for solving equation (REF ), for which $U_{i,j}=U^{m}_{i,j}$ , $V_{i, j}=V^{m}_{i,j}$ , and $H_{i, j}=H^{m}_{i,j}$ [8].",
"It is worth noting that in order to simulate a closed ocean basin, no-slip boundary conditions ($u=0$ and $v=0$ ) are usually prescribed for the non-conservation model equation (REF ).",
"For the fractional-step algorithm, however, we require two sets of boundary conditions: one for $Q^{m}$ in the first step and one for $U$ , $V$ in the second step.",
"Naturally, the no-slip boundary conditions are employed in the second step (REF ), while the choice of the boundary conditions for $Q^{m}$ in the first step must reflect the physical interpretation of no-slip boundary conditions.",
"We choose solid-wall boundary conditions for $Q^{m}$ .",
"The key observation of a solid wall is that at the boundary $x=a$ , $u(a, y, t)= 0,\\quad hu(a,y,t)=0.$ Similarly, a solid wall at the boundary $y=b$ is $v(x, b, t)= 0,\\quad hv(x,b,t)=0.$ To achieve the solid-wall conditions (REF ) and (REF ), in each time step the ghost-cell values in the second-order finite volume wave-propagation algorithm are set to be $\\begin{split}\\text{For}\\,\\, Q^{m}_0&:\\quad H^{m}_0 =H^{m}_{1},\\quad (HU)^{m}_0=-(HU)^{m}_1, \\quad (HV)^{m}_0=-(HV)^{m}_1\\\\\\text{For}\\,\\, Q^{m}_{-1}&:\\quad H^{m}_{-1} =H^{m}_{2},\\quad (HU)^{m}_{-1}=-(HU)^{m}_{2}, \\quad (HV)^{m}_{-1}=-(HV)^{m}_{2}.\\end{split}$ Formula (REF ) imposes a necessary symmetry for achieving the solid-wall conditions (REF ) and (REF ) [7].",
"We remark that because we discretize the Laplacian by the five-point centered-difference scheme, and both $U$ and $V$ are computed at cell centers, we are not using the no-slip conditions $V=0$ at the vertical walls and $U=0$ at the horizontal walls.",
"Instead, we use the ghost cell values of $U$ and $V$ , which are computed based on Eq.",
"(REF ) and Eq (REF ).",
"Our choice of the ghost cell values enforces the boundary condition prescribed for the non-conservation equations.",
"We also remark that the homogeneous hyperbolic shallow-water equations (REF ) is solved by the high-resolution wave propagation algorithms developed by LeVeque [7] in this study.",
"The algorithms can easily be replaced by other efficient anti-diffusion shock-capturing schemes, such as the algorithms developed in [12], [13] and many others, for which we do not attempt to provide a detailed list.",
"We hope to emphasize that this study focuses on introducing a new formulation for the double-gyre shallow-water model and a numerical implementation for solving the formulation.",
"It is not our intention to develop a new efficient anti-diffusion shock-capturing scheme, neither to develop a new algorithm for solving the hyperbolic conservation laws with source terms.",
"We demonstrate that as a result of combining the new formulation and the fractional-step algorithm, we obtain a stable method that is not sensitive to the kinematic viscosity and the grid refinement for the double-gyre shallow-water model."
],
[
"Periodic flows on an f-plane",
"We validate the proposed algorithm by examining a periodic flow on a constant f-plane (i.e., the Coriolis force does not depend on latitude and thus $\\beta =0$ ).",
"We introduce the following dimensionless variables: $u^{*}=\\frac{u}{U},\\,\\,v^{*}=\\frac{v}{U},\\,\\,h^{*}=\\frac{h}{H_0},\\,\\,x^{*}=\\frac{x}{L},\\,\\,y^{*}=\\frac{y}{L},\\,\\,t^{*}=\\frac{t}{L/U},$ where $U$ is the scale of velocity, $L$ is the typical length scale, and $H_0$ is the scale of water height.",
"Substituting the above dimensionless variables into equations (REF ) results in the following scaled system of equations for the double-gyre shallow-water model (we drop `*' herein and after) : $\\begin{split}&\\frac{\\partial h}{\\partial t}+\\frac{\\partial (uh)}{\\partial x} +\\frac{\\partial (vh)}{\\partial y}=0,\\\\&\\frac{\\partial (hu)}{\\partial t} +\\frac{\\partial }{\\partial x}\\left(hu^2+\\frac{1}{2}F_{r}^{-2} h^2\\right)+\\frac{\\partial }{\\partial y}\\left(huv\\right) = \\frac{1}{R_{0}} hv + \\frac{1}{Re}h \\nabla ^2 u +h F^{u},\\\\&\\frac{\\partial (hv)}{\\partial t} +\\frac{\\partial }{\\partial x}\\left(huv\\right)+\\frac{\\partial }{\\partial y}\\left(hv^2+\\frac{1}{2}F_{r}^{-2} h^2\\right) =- \\frac{1}{R_{0}} hu + \\frac{1}{Re}h \\nabla ^2 v +h F^{v},\\end{split}$ where $F_{r}=U / \\sqrt{g_rH_0}$ is the Froude number, $Re=LU / \\nu $ is the Reynolds number, and $R_0=U / Lf$ is the Rossby number.",
"Consider the solution ansatz $\\begin{split}u(x,y,t)&=(\\eta +\\epsilon \\sin (\\omega t))\\cos (2\\pi x)\\sin (2\\pi y),\\\\v(x,y,t)&=-(\\eta +\\epsilon \\sin (\\omega t))\\sin (2 \\pi x)\\cos (2\\pi y),\\\\h(x,y)&=\\exp (\\cos (2 \\pi x)\\cos (2\\pi y)),\\end{split}$ where the parameters $\\eta $ , $\\epsilon $ , and $\\omega $ control the contribution of spatial and temporal derivatives in the solution.",
"Substituting the ansatz into equation (REF ), we obtain the forcing terms $F^u$ , $\\begin{split}F^{u}= & \\epsilon \\omega \\cos (\\omega t)\\cos (2\\pi x)\\sin (2\\pi y)-2\\pi (\\eta +\\epsilon \\sin (\\omega t))^2\\sin (2\\pi x)\\cos (2\\pi x)\\\\ & + \\frac{8\\pi ^2}{Re}(\\eta +\\epsilon \\sin (\\omega t))\\cos (2\\pi x)\\sin (2\\pi y) + \\frac{1}{R_0}(\\eta +\\epsilon \\sin (\\omega t))\\sin (2\\pi x)\\cos (2\\pi y)\\\\ & -\\frac{2\\pi }{Fr^2}\\sin (2\\pi x)\\cos (2\\pi y)\\exp (\\cos (2\\pi x)\\cos (2\\pi y)),\\end{split}$ and $F^v$ , $\\begin{split}F^{v}= & - \\epsilon \\omega \\cos (\\omega t)\\sin (2\\pi x)\\cos (2\\pi y)-2\\pi (\\eta +\\epsilon \\sin (\\omega t))^2\\sin (2\\pi y)\\cos (2\\pi y)\\\\ & - \\frac{8\\pi ^2}{Re}(\\eta +\\epsilon \\sin (\\omega t))\\sin (2\\pi x)\\cos (2\\pi y) + \\frac{1}{R_0}(\\eta +\\epsilon \\sin (\\omega t))\\cos (2\\pi x)\\sin (2\\pi y)\\\\ & -\\frac{2\\pi }{Fr^2}\\cos (2\\pi x)\\sin (2\\pi y)\\exp (\\cos (2\\pi x)\\cos (2\\pi y)).\\end{split}$ Note that the solution ansatz is independent of the dimensionless parameters $F_r$ , $R_0$ , and $Re$ .",
"In principle, for this test problem, the solution behavior of the proposed fractional-step method should be insensitive to the choice of these parameters, if the following conditions are satisfied: 1) the CFL condition in the first step of solving the hyperbolic equation and 2) the stability restriction of the 2-stage, second-order Runge-Kutta method used to solve equation (REF ).",
"We choose $Re=100$ , $R_0=0.1$ , and $Fr=2$ for our simulations.",
"For the required boundary conditions in (REF ) and (REF ), we impose periodic boundary conditions in both steps for periodic flow.",
"For computational domain $[ 0,1]\\times [0,1]$ , Table REF shows the grid refinement study for the fractional-step method.",
"We compute the error of the height field between the exact solution and the numerical solution at the final time $t=1$ , with the (finite) $l_2$ -norm $||\\epsilon ||= \\sqrt{\\frac{1}{N^2}\\sum _{i=1}^{N }\\sum _{j=1}^{N}\\epsilon _{i,j}^2},$ where $N$ is the number of grid cells in one direction.",
"Since the error goes down by four (on average) when we refine the grid, it provides evidence that the proposed method is second-order accurate.",
"We use the parameters $\\eta =0.1$ , $\\epsilon =0.9$ , and $\\omega =\\pi / 20$ .",
"Note that the time step $\\Delta t$ in this calculation is chosen so that when we refine the grid in both $x$ and $y$ directions, the time step used for the fine gird is $1/4$ of that used for the coarse grid.",
"We start with $\\Delta t =0.025$ for $N=10$ .",
"The Strang-splitting method is used for the calculation.",
"No limiters are used in the first step.",
"Figure REF (a) is the exact solution of the height field.",
"30 contour lines are used for values between 0.36794 and 2.7179.",
"Figure REF (b) is the numerical solution at $t=1$ with $250\\times 250$ cells.",
"30 contour lines are used for values between 0.36796 and 2.7183.",
"Table: Convergence rate for the fractional-step algorithm.Figure: The height field of a periodic flow on an f plane.",
"(a) Exact solution of the height field.",
"30 contour lines are used for values between 0.36794 and 2.7179.",
"(b) Numerical solution at t=1t=1 with 250×250250\\times 250 cells.",
"30 contour lines are used for values between 0.36796 and 2.7183.Unlike the enslaved schemes developed in [10], [4], the proposed numerical method is independent of the magnitude of the time dependent contribution to the solution.",
"That is, the accuracy of numerical solution and the efficiency of the algorithm are independent of the choice of $\\eta $ and $\\epsilon $ .",
"Table REF shows errors of the computed solutions in the $l_2$ -norm for the horizontal velocity $u$ , and the elapsed CPU times for various choices of $\\eta $ and $\\epsilon $ .",
"As expected, the numerical experiments show that the solution behavior of the proposed algorithm is insensitive to the choice of $\\eta $ and $\\epsilon $ .",
"Note that the absolute error increases as $\\eta $ increases, due to the fact that the magnitude of $u$ increases as $\\eta $ increases.",
"The numerical experiments use a $50\\times 50$ grid, while the parameter $\\omega = \\pi /10$ and the final run time is $t=5$ .",
"Table: Errors of computed solutions for uu and the elapsed CPU times for various choices of η\\eta and ϵ\\epsilon ."
],
[
"Upper-ocean double-gyre model",
"To demonstrate the strength of the method that combines the new formulation and the fractional-step algorithm, we examine the geophysical flow that describes a closed basin flow on a $\\beta $ -plane, subjected to zonal winds.",
"With reduced gravity, the model resembles a two-layer ocean basin whose upper layer is driven by a zonal wind stress [3], e.g.",
"the external forcing term in equation (REF ) is the imposed wind forcing given by the curl of the wind stress, $\\begin{split}F^{u}& = -\\frac{\\tau _0}{\\rho H_0}\\cos \\left(\\frac{2\\pi y}{L}\\right),\\\\F^{v}& = 0,\\end{split}$ where $\\tau _0$ is the wind stress, $\\rho $ is the water density, $L$ is the domain length in the North-South direction, and $H_0$ is the initial upper-layer depth.",
"The parameter values used in the simulations are listed in Table REF .",
"These values are chosen to closely match of those in [3], [10] for comparison.",
"Table: Model parameters.Figure: Grid refinement study for the proposed formulation and the fractional-step algorithm.",
"The height anomaly of double-gyre model at t=16t=16 years.",
"The grid resolutions, from (a) to (c), are Δx=40\\Delta x=40 km, 20 km, and 10 km, respectivelyFigure REF shows the height anomaly of the double-gyre model at $t=16$ years calculated by using the new formulation and the fractional-step algorithm.",
"The grid resolutions, from (a) to (c), are $\\Delta x=40$ km, 20 km, and 10 km, respectively.",
"The dynamics of Figure REF (a) looks different from that of (b) or (c).",
"This is because dynamically the important length scales are only marginally resolved for (a).",
"The length scales are dominated by the first Rossby deformation radius.",
"The Rossby deformation radius for this choice of parameter varies between $45 \\sim 80$ , from the definition $L_{D} = \\frac{1}{f}\\left(g_rH_{0}\\right)^{1/2}\\approx 45 \\sim 80 \\,\\,\\text{km}.$ The dynamics of the model are dominated by Rossby waves with wave number defined by $\\kappa _{R}=1/L_{D}$ .",
"If we require that the smallest waves are resolved by the grid spacing $\\Delta x$ , we must have $\\kappa _{R} = \\frac{1}{2\\Delta x}$ [10].",
"Hence in this case, the resolution of the grid size must satisfy $\\Delta x \\le 20$ km in order to resolve the Rossby waves.",
"The dynamics of height anomaly, $h-H_0$ , of the double-gyre model quickly settles into a quasi-steady-state solution and exhibits strong western boundary currents, as shown in Figure REF .",
"Figure REF closely match with Figure 5(a) with $\\Delta x=17$ km reported in [10] and Figure 7 reported in [3].",
"We note that the grid resolution is set to be $\\Delta y=\\Delta x$ for all our simulations.",
"In addition to the height anomaly, we also monitor the velocity field.",
"From left to right, Figure REF shows the contour plots of stream function at year 1, 5, 10, and 20, respectively, for the double-gyre model.",
"The wind forcing is described by (REF ).",
"The computational domain is again $[0, 1000]\\times [0, 2000]$ km$^2$ .",
"The grid resolution is $\\Delta x=10$ km, and the time step is $\\Delta t=6$ minutes.",
"For each simulation figure, 20 contour lines are plotted.",
"We note that the stream-line structures show little difference after year 5 (including year 5).",
"The stream-line structures for year 10 and 20 are almost identical, which provides evidence that the velocity field has reached a quasi-steady-state solution.",
"Figure: From left to right, simulation figures show the contour plots of stream function at year 1, 5, 10, and 20, respectively for a double-gyre model that is under a constant wind forcing described by ().",
"The computational domain is [0,1000]×[0,2000][0, 1000]\\times [0, 2000] km 2 ^2.",
"The grid resolution is Δx=Δy=10\\Delta x=\\Delta y=10 km, and the time step is Δt=6\\Delta t=6 minutes.",
"For each figure, equally spacing 20 contour lines between [-30270,15944][ -30270, 15944] are plotted.Finally, we implement a basic MPDATA algorithm described in [11] for the double-gyre model.",
"Figure REF is the comparison of the height anomaly of the double-gyre model after 365 days between the proposed algorithm and the MPDATA implementation.",
"While the structures of the two contour plots are similar, we see that the result from the MPDATA algorithm is more diffusive, even with a mesh that is four-times finer than that for the proposed algorithm.",
"Figure: The height anomaly of the double-gyre model after 365 days.",
"The enclosed basin with no-slip boundary conditions all around is [0,1000]×[0,2000][0, 1000]\\times [0, 2000] km 2 \\text{km}^2.",
"(a) The proposed conservation formulation and the fractional-step algorithm.",
"The grid resolution is Δx=Δy=10\\Delta x= \\Delta y =10 km.",
"Δt=6\\Delta t= 6 minutes.",
"(b) Basic MPDATA implementation for the double-gyre model described in .",
"The grid resolution is Δx=Δy=2.5\\Delta x= \\Delta y =2.5 km.",
"Δt=0.375\\Delta t= 0.375 minutes."
],
[
"Double-gyre model with transport of a pollutant",
"The double-gyre shallow-water model has been used as an underlying ocean model for data assimilation [11].",
"In this section, we use this model to study the circulation of a substance that initially is randomly distributed in certain areas of a closed ocean basin.",
"This problem is related to transport of pollutant in the ocean, and was previously studied by Xu and Shu [18], using only the hyperbolic shallow-water equations.",
"To study this problem, we couple the double-gyre shallow-water equations (REF ) with a two-dimensional scalar advection (transport) equation $C_t+u\\cdot \\nabla C = 0,$ where $C$ is a substance concentration and $u=[u, v]^{T}$ is the velocity field of the double-gyre shallow-water equations.",
"The concentration of the substance is advected by the velocity field of the double-grye shallow-water equations, acting like a scalar tracer.",
"The diffusivity for the scalar tracer is assumed to be very small, so that the diffusion effect of the concentration is negligible.",
"The transport equation is solved by the high-resolution wave-propagation algorithm developed in [7].",
"Because the governing equations are solved in two steps, other than augmenting a conservation equation in the hyperbolic shallow-water equations in Problem A, as suggested in [7], we solve the color equation( REF ) in its non-conservation form.",
"The cell-centered value of $C$ is advected by the edge value of the velocity calculated by averaging the adjacent cell-centered velocities.",
"We consider a closed ocean basin with dimensions $[0, 1000]\\times [0, 2000]$ km$^2$ .",
"The basin has been under a constant wind forcing ( REF ) for 20 years before the substance is present, and is under the same wind forcing after the substance is present.",
"That is, the quasi-steady-state velocity field, shown in Figure REF , is used as the initial velocity field, and the height field, shown Figure REF (b), is used as the initial height field for the double-gyre shallow-water equations.",
"The same parameter values in Table REF are used to evolve the double-gyre shallow-water equations.",
"Suppose that the initial values of the concentration are Gaussian random numbers $0 < C(x,y) \\le 1$ .",
"We distribute the initial concentration in the following way: Consider two circles with the same radius, $r=150$ km.",
"The centers of the circles are at (500 km, 500 km) and (500 km, 1500 km), respectively.",
"We divide the whole domain into $100\\times 200$ grid cells, and assign a random number between 0 and 1 to the center of each grid cell inside the two circles.",
"Figure REF shows the transport of a substance in the basin under the quasi-steady-state velocity field.",
"In the top row, from left to right, the simulation figures show the distribution of concentration at day 0, 50, and 100.",
"In the bottom row, from left to right, the simulation figures show the distribution of concentration at day 150, 240, and 360.",
"Taken as a whole, Figure REF shows that the strong western boundary current drives most of the substance to an area near the western bank.",
"The grid resolution for the simulation is $\\Delta x=\\Delta y = 10$ km, and the time step is 12 minutes.",
"Figure: In the top row, from left to right, the simulation figures show the distribution of a concentration at day 0, 50, and 100.",
"In the bottom row, from left to right, the simulation figures show the distribution of the concentration at day 150, 240, and 360.",
"The concentration is driven by the velocity field of the shallow-water equations.",
"At day 0, the initial concentration is distributed inside the two circles.",
"The concentration values are between zero and one, and are randomly assigned to the center of grid cells inside the circles.",
"Figure show that the strong western boundary current drives most of the substance to an area near the western bank.",
"The grid resolution for the simulation is Δx=Δy=10\\Delta x=\\Delta y = 10 km, and the time step is 12 minutes.",
"The domain of the basin is [0,1000]×[0,2000][0, 1000]\\times [0, 2000] km 2 ^2."
],
[
"Conclusion ",
"We present a new formulation for the double-gyre shallow-water model.",
"A fractional-step method is provided to solve the new formulation.",
"The combination of the formulation and the numerical algorithm is proved to be stable and not sensitive to the kinematic viscosity and grid refinement.",
"For traditional methods, stability of the finite difference scheme often depends on the magnitude of kinematic viscosity.",
"In practice, it is not unusual that to maintain stability, the viscosity needs to be increased as the grid resolution is decreased for those methods [4].",
"The enslaved finite-difference methods that improves the accuracy for MPDATA could also be sensitive to the viscosity value for certain time integrators when refining meshes.",
"The proposed formulation and the fractional-step method remains stable at a fixed viscosity throughout the gird refinement study.",
"The proposed method is second-order accurate.",
"In the constant wind-forcing example, we demonstrate that the numerical solution converges rather quickly to a quasi-steady-state solution, as long as the Rossby deformation radius is resolved.",
"Since the high-resolution wave-propagation method that solves the hyperbolic shallow-water equations introduces little numerical dissipation, the proposed fractional-step method is suitable for applications that require small artifical diffusion.",
"Finally, in the last example, we illustrate the flexibility of the proposed method to incorporate other equations for application, such as the transport equation.",
"Especially, when high-resolution is preferable for the monitored quantity in the transport equation."
],
[
"Acknowledgments",
"The authors thank Zhi (George) Lin for pointing out an error in our earlier numerical implementation."
]
] | 1403.0140 |
[
[
"On the class of weak almost limited operators"
],
[
"Abstract We introduce and study the class of weak almost limited operators.",
"We establish a characterization of pairs of Banach lattices $E$, $F$ for which every positive weak almost limited operator $T:E\\rightarrow F$ is almost limited (resp.",
"almost Dunford-Pettis).",
"As consequences, we will give some interesting results."
],
[
"Introduction",
"Throughout this paper $X,$ $Y$ will denote real Banach spaces, and $E,\\,F$ will denote real Banach lattices.",
"$B_{X}$ is the closed unit ball of $X$ and $\\mathrm {sol}\\left( A\\right) $ denotes the solid hull of a subset $A$ of a Banach lattice.",
"We will use the term operator $T:X\\rightarrow Y$ between two Banach spaces to mean a bounded linear mapping.",
"Let us recall that a norm bounded subset $A$ of $X$ is called a Dunford-Pettis set (resp.a limited set) if each weakly null sequence in $X^{\\ast }$ (resp.",
"weak* null sequence in $X^{\\ast }$ ) converges uniformly to zero on $A$ .",
"An operator $T:X\\rightarrow Y$ is called Dunford-Pettis if $x_{n}\\overset{w}{\\rightarrow }0$ in $X$ implies $\\left\\Vert Tx_{n}\\right\\Vert \\rightarrow 0$ , equivalently, if $T$ carries relatively weakly compact subsets of $X$ onto relatively compact subsets of $Y$ .",
"An operator $T:X\\rightarrow Y$ is said to be limited whenever $T\\left( B_{X}\\right) $ is a limited set in $Y$ , equivalently, whenever $\\left\\Vert T^{\\ast }\\left( f_{n}\\right) \\right\\Vert \\rightarrow 0$ for every weak* null sequence $\\left( f_{n}\\right) \\subset Y^{\\ast }$ .",
"Aliprantis and Burkinshaw [1] introduced the class of weak Dunford–Pettis operators.",
"An operator $T:X\\rightarrow Y$ is said to be weak Dunford–Pettis whenever $x_{n}\\overset{w}{\\rightarrow }0$ in $X$ and $f_{n}\\overset{w}{\\rightarrow }0$ in $Y^{\\ast }$ imply $f_{n}\\left(Tx_{n}\\right) \\rightarrow 0$ , equivalently, whenever $T$ carries weakly compact subsets of $X$ to Dunford–Pettis subsets of $Y$ [2].",
"Next H'michane et al.",
"[8] introduced the class of weak* Dunford-Pettis operators, and characterized this class of operators and studied some of its properties in [9].",
"An operator $T:X\\rightarrow Y$ is called weak* Dunford-Pettis whenever $x_{n}\\overset{w}{\\rightarrow }0$ in $X$ and $f_{n}\\overset{w^{\\ast }}{\\rightarrow }0$ in $Y^{\\ast }$ imply $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ , equivalently, whenever $T$ carries relatively weakly compact subsets of $X$ onto limited subsets of $Y$ [9].",
"Recently, two classes of norm bounded sets are considered in the theory of Banach lattices.",
"From [3] (resp.",
"[5]), a norm bounded subset $A$ of a Banach lattice $E$ is said to be an almost Dunford-Pettis set (resp.",
"an almost limited set), if every disjoint weak null (resp.",
"weak* null) sequence $(f_{n})$ in $E^{\\ast }$ converges uniformly to zero on $A$ .",
"Clearly, all Dunford-Pettis sets (resp.",
"limited sets) in a Banach lattice are almost Dunford-Pettis (resp.",
"almost limited).",
"Also, every almost limited set is almost Dunford-Pettis.",
"But the converse does not hold in general.",
"Let us recall that an operator $T:E\\rightarrow X$ is said to be almost Dunford-Pettis, if $T$ carries every disjoint weakly null sequence to a norm null sequence, or equivalently, if $T$ carries every disjoint weakly null sequence consisting of positive terms to a norm null sequence [12].",
"From [10], an operator $T:X\\rightarrow E$ is called almost limited whenever $T\\left( B_{E}\\right) $ is an almost limited set in $E$ , equivalently, whenever $\\left\\Vert T^{\\ast }\\left( f_{n}\\right)\\right\\Vert \\rightarrow 0$ for every disjoint weak* null sequence $\\left(f_{n}\\right) \\subset E^{\\ast }$ .",
"Using the almost Dunford-Pettis sets, Bouras and Moussa [4] introduced the class of weak almost Dunford-Pettis operators.",
"An operator $T:X\\rightarrow E$ is called weak almost Dunford-Pettis operator whenever $T$ carries relatively weakly compact subsets of $X$ to almost Dunford–Pettis subsets of $E$ , equivalently, whenever $f_{n}(T(x_{n}))\\rightarrow 0$ for all weakly null sequences $(x_{n})$ in $X$ and for all weakly null sequences $(f_{n})$ in $E^{\\ast }$ consisting of pairwise disjoint terms [4].",
"In this paper, using the almost limited sets, we introduce the class of weak almost limited operators $T:X\\rightarrow E$ , which carries relatively weakly compact subsets of $X$ to almost limited subsets of $E$ (Definition REF ).",
"It is a class which contains that of weak* Dunford-Pettis (resp.",
"almost limited).",
"We establish some characterizations of weak almost limited operators.",
"After that, we derive the domination property of this class of operators (Corollary REF ).",
"Next, we characterize pairs of Banach lattices $E$ , $F$ for which every positive weak almost limited operator $T:E\\rightarrow F$ is almost limited (resp.",
"almost Dunford-Pettis).",
"As consequences, we will give some interesting results.",
"To show our results we need to recall some definitions that will be used in this paper.",
"A Banach lattice $E$ has $\\emph {-}$ the Dunford-Pettis property, if $x_{n}\\overset{w}{\\rightarrow }0$ in $E$ and $f_{n}\\overset{w}{\\rightarrow }0$ in $E^{\\ast }$ imply $f_{n}\\left( x_{n}\\right) \\rightarrow 0$ as $n\\rightarrow \\infty $ , equivalently, each relatively weakly compact subset of $E$ is Dunford–Pettis.",
"$\\emph {-}$ the Dunford-Pettis* property (DP* property for short), if $x_{n}\\overset{w}{\\rightarrow }0$ in $E$ and $f_{n}\\overset{w^{\\ast }}{\\rightarrow }0$ in $E^{\\ast }$ imply $f_{n}\\left( x_{n}\\right) \\rightarrow 0$ , equivalently, each relatively weakly compact subset of $E$ is limited.",
"$\\emph {-}$ the weak Dunford-Pettis* property (wDP* property), if $f_{n}\\left( x_{n}\\right) \\rightarrow 0$ for every weakly null sequence $\\left( x_{n}\\right) $ in $E$ and for every disjoint weak* null sequence $\\left( f_{n}\\right) $ in $E^{\\ast }$ , equivalently, each relatively weakly compact subset of $E$ is almost limited [5].",
"$\\emph {-}$ the Schur (resp.",
"positive Schur) property, if $\\left\\Vert x_{n}\\right\\Vert \\rightarrow 0$ for every weak null sequence $\\left(x_{n}\\right) \\subset E$ (resp.",
"$\\left( x_{n}\\right) \\subset E^{+}$ ).",
"$\\emph {-}$ the positive dual Schur property, if $\\left\\Vert f_{n}\\right\\Vert \\rightarrow 0$ for every weak* null sequence $\\left(f_{n}\\right) \\subset \\left( E^{\\ast }\\right) ^{+}$ , equivalently, $\\left\\Vert f_{n}\\right\\Vert \\rightarrow 0$ for every weak* null sequence $\\left( f_{n}\\right) \\subset \\left( E^{\\ast }\\right) ^{+}$ consisting of pairwise disjoint terms [13].",
"$\\emph {-}$ the property (d) whenever $\\left|f_{n}\\right|\\wedge \\left|f_{m}\\right|=0$ and $f_{n}\\overset{w^{\\ast }}{\\rightarrow }0$ in $E^{\\ast }$ imply $\\left|f_{n}\\right|\\overset{w^{\\ast }}{\\rightarrow }0$ .",
"It should be noted, by Proposition 1.4 of [13], that every $\\sigma $ -Dedekind complete Banach lattice has the property $\\mathrm {(d)}$ but the converse is not true in general.",
"In fact, the Banach lattice $\\ell ^{\\infty }/c_{0}$ has the property $\\mathrm {(d)}$ but it is not $\\sigma $ -Dedekind complete [13].",
"Our notions are standard.",
"For the theory of Banach lattices and operators, we refer the reader to the monographs [2], [11]."
],
[
"Main results",
"We start this section by the following definition.",
"Definition 2.1 An operator $T:X\\rightarrow E$ from a Banach space $X$ into a Banach lattice $E$ is called weak almost limited if $T$ carries each relatively weakly compact set in $X$ to an almost limited set in $E$ .",
"Clearly, a Banach lattice $E$ has the DP* property (resp.",
"wDP* property) if and only if the identity operator $I:E\\rightarrow E$ is weak* Dunford-Pettis (resp.",
"weak almost limited).",
"Also, every weak* Dunford-Pettis (resp.",
"almost limited) operator $T:X\\rightarrow E$ is weak almost limited, but the converse is not true in general.",
"In fact, the identity operator $I:L^{1}\\left[ 0,1\\right] \\rightarrow L^{1}\\left[ 0,1\\right] $ (resp.",
"$I:\\ell ^{1}\\rightarrow \\ell ^{1}$ ) is weak almost limited but it fail to be weak* Dunford-Pettis (resp.",
"almost limited).",
"In terms of weakly compact and almost limited operators the weak almost limited operators are characterized as follows.",
"Theorem 2.2 For an operator $T:X\\rightarrow E$ , the following assertions are equivalents: $T$ is weak almost limited.",
"If $S:Z\\rightarrow X$ is a weakly compact operator, where $Z$ is an arbitrary Banach space, then the operator $T\\circ S$ is almost limited.",
"If $S:\\ell ^{1}\\rightarrow X$ is a weakly compact operator then the operator $T\\circ S$ is almost limited.",
"For every weakly null sequence $\\left( x_{n}\\right) \\subset X$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset E^{\\ast }$ we have $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ .",
"$\\left( 1\\right) \\Rightarrow \\left( 2\\right) $ Let $\\left( f_{n}\\right)\\subset E^{\\ast }$ be a disjoint weak* null sequence.",
"We shall proof that $\\left\\Vert \\left( T\\circ S\\right) ^{\\ast }\\left( f_{n}\\right) \\right\\Vert \\rightarrow 0$ .",
"Otherwise, by choosing a subsequence we may suppose that there is $\\varepsilon $ with $\\left\\Vert \\left( T\\circ S\\right) ^{\\ast }\\left( f_{n}\\right) \\right\\Vert >\\varepsilon >0$ for all $n\\in \\mathbb {N}$ .",
"So for every $n$ there exists some $x_{n}\\in B_{Z}$ with $\\left( T\\circ S\\right) ^{\\ast }\\left( f_{n}\\right) \\left( x_{n}\\right) =f_{n}\\left(T\\left( S\\left( x_{n}\\right) \\right) \\right) \\ge \\varepsilon $ .",
"On the other hand, as $S$ is weakly compact, $\\left\\lbrace S\\left( x_{k}\\right):k\\in \\mathbb {N}\\right\\rbrace $ is a relatively weakly compact subset of $X$ .",
"Then $\\left\\lbrace T\\left( S\\left( x_{k}\\right) \\right) :k\\in \\mathbb {N}\\right\\rbrace $ is an almost limited set in $E$ (as $T$ is weak almost limited).",
"So $\\sup \\left\\lbrace \\left|f_{n}\\left( T\\left( S\\left( x_{k}\\right) \\right) \\right)\\right|:k\\in \\mathbb {N}\\right\\rbrace \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"But this implies that $f_{n}\\left( T\\left( S\\left( x_{n}\\right) \\right)\\right) \\rightarrow 0$ , which is impossible.",
"Thus, $\\left\\Vert \\left( T\\circ S\\right) ^{\\ast }\\left( f_{n}\\right) \\right\\Vert \\rightarrow 0$ , and hence $T\\circ S$ is almost limited.",
"$\\left( 2\\right) \\Rightarrow \\left( 3\\right) $ Obvious.",
"$\\left( 3\\right) \\Rightarrow \\left( 4\\right) $ Let $\\left( x_{n}\\right)\\subset X$ be a weakly null sequence and let $\\left( f_{n}\\right) \\subset E^{\\ast }$ be a disjoint weak* null sequence.",
"By Theorem 5.26 [2], the operator $S:\\ell ^{1}\\rightarrow X$ defined by $S\\left( \\left( \\lambda _{i}\\right) \\right) =\\sum _{i=1}^{\\infty }\\lambda _{i}x_{i}$ , is weakly compact.",
"Thus, by our hypothesis $T\\circ S$ is almost limited and hence $\\left\\Vert \\left( T\\circ S\\right) ^{\\ast }\\left( f_{n}\\right) \\right\\Vert \\rightarrow 0$ .",
"But $\\left\\Vert \\left( T\\circ S\\right) ^{\\ast }\\left( f_{n}\\right) \\right\\Vert &=&\\sup \\left\\lbrace \\left|f_{n}\\left( T\\left( S\\left( \\left( \\lambda _{i}\\right) \\right) \\right) \\right) \\right|:\\left( \\lambda _{i}\\right)\\in B_{\\ell ^{1}}\\right\\rbrace \\\\&\\ge &\\left|f_{n}\\left( T\\left( S\\left( e_{n}\\right) \\right) \\right)\\right|=\\left|f_{n}\\left( T\\left( x_{n}\\right) \\right) \\right|$ for every $n$ , where $\\left( e_{n}\\right) $ is the canonical basis of $\\ell ^{1}$ .",
"Then $f_{n}\\left( T\\left( x_{n}\\right) \\right) \\rightarrow 0$ , as desired.",
"$\\left( 4\\right) \\Rightarrow \\left( 1\\right) $ Let $W$ be a relatively weakly compact subset of $X$ .",
"If $T\\left( W\\right) $ is not almost limited set in $E$ then there exists a disjoint weak* null sequence $\\left(f_{n}\\right) \\subset E^{\\ast }$ such that $\\sup \\left\\lbrace \\left|f_{n}\\left( T\\left( x\\right) \\right) \\right|:x\\in W\\right\\rbrace \\nrightarrow 0$ .",
"By choosing a subsequence we may suppose that there is $\\varepsilon $ with $\\sup \\left\\lbrace \\left|f_{n}\\left( T\\left( x\\right)\\right) \\right|:x\\in W\\right\\rbrace >\\varepsilon >0$ for all $n\\in \\mathbb {N}$ .",
"So for every $n$ there exists some $x_{n}\\in W$ with $\\left|f_{n}\\left( T\\left( x_{n}\\right) \\right) \\right|\\ge \\varepsilon $ .",
"On the other hand, since $W$ is a relatively weakly compact subset of $X$ , there exists a subsequence $\\left( x_{n_{k}}\\right) $ of $\\left(x_{n}\\right) $ such that $x_{n_{k}}\\overset{w}{\\rightarrow }x$ holds in $X$ .",
"By hypothesis, $f_{n_{k}}\\left( T\\left( x_{n_{k}}-x\\right) \\right)\\rightarrow 0$ and clearly $f_{n_{k}}\\left( T\\left( x\\right) \\right)\\rightarrow 0$ .",
"Now from $f_{n_{k}}\\left( T\\left( x_{n_{k}}\\right) \\right)=f_{n_{k}}\\left( T\\left( x_{n_{k}}-x\\right) \\right) +f_{n_{k}}\\left( T\\left(x\\right) \\right) $ we see that $f_{n_{k}}\\left( T\\left( x_{n_{k}}\\right)\\right) \\rightarrow 0$ , which is impossible.",
"Thus, $T\\left( W\\right) $ is an almost limited set in $E$ , and so $T$ is weak almost limited.",
"Remark 2.3 Every operator $T:X\\rightarrow E$ that admits a factorization through the Banach lattice $\\ell ^{\\infty }$ , is weak almost limited.",
"In fact, let $R:X\\rightarrow \\ell ^{\\infty }$ and $S:\\ell ^{\\infty }\\rightarrow E$ be two operators such that $T=S\\circ R$ .",
"Let $\\left(x_{n}\\right) \\subset X$ be a weakly null sequence and let $\\left(f_{n}\\right) \\subset E^{\\ast }$ be a disjoint weak* null sequence.",
"Clearly $R\\left( x_{n}\\right) \\overset{w}{\\rightarrow }0$ holds in $\\ell ^{\\infty }$ and $S^{\\ast }f_{n}\\overset{w^{\\ast }}{\\rightarrow }0$ holds in $\\left( \\ell ^{\\infty }\\right) ^{\\ast }$ .",
"Since $\\ell ^{\\infty }$ has the Dunford-Pettis* property then $f_{n}\\left( Tx_{n}\\right) =\\left( S^{\\ast }f_{n}\\right)\\left( R\\left( x_{n}\\right) \\right) \\rightarrow 0$ .",
"Thus $T$ is weak almost limited.",
"The next result characterizes, under some conditions, the order bounded weak almost limited operators between two Banach lattices.",
"Theorem 2.4 Let $E$ and $F$ be two Banach lattices such that the lattice operations of $E^{\\ast }$ are sequentially weak* continuous or $F$ satisfy the property $\\mathrm {(d)}$ .",
"Then for an order bounded operator $T:E\\rightarrow F$ , the following assertions are equivalents: $T$ is weak almost limited.",
"For every weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset F^{\\ast }$ we have $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ .",
"For every disjoint weakly null sequence $\\left( x_{n}\\right) \\subset E$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset F^{\\ast } $ we have $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ .",
"For every disjoint weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset F^{\\ast }$ we have $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ .",
"$T$ carries the solid hull of each relatively weakly compact subset of $E$ to an almost limited subset of $F$ .",
"If $F$ has the property $(\\text{d})$ , we may add: $f_{n}\\left( T\\left( x_{n}\\right) \\right)\\rightarrow 0$ for every weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak$^{\\ast }$ null sequence $\\left( f_{n}\\right)\\subset \\left( F^{\\ast }\\right) ^{+}$ .",
"$f_{n}\\left( T\\left( x_{n}\\right) \\right)\\rightarrow 0$ for every disjoint weakly null sequence $\\left( x_{n}\\right)\\subset E^{+}$ and every disjoint weak$^{\\ast }$ null sequence $\\left(f_{n}\\right) \\subset \\left( F^{\\ast }\\right) ^{+}$ .",
"$\\left( 1\\right) \\Rightarrow \\left( 2\\right) $ and $\\left( 1\\right)\\Rightarrow \\left( 3\\right) $ Follows from Theorem REF .",
"$\\left( 2\\right) \\Rightarrow \\left( 4\\right) $ and $\\left( 3\\right)\\Rightarrow \\left( 4\\right) $ are obvious.",
"$\\left( 4\\right) \\Rightarrow \\left( 5\\right) $ Let $W$ be a relatively weakly compact subset of $E$ and let $\\left( f_{n}\\right) \\subset F^{\\ast }$ be a disjoint weak* null sequence.",
"Put $A=\\mathrm {sol}\\left( W\\right) $ and note that if $\\left( z_{n}\\right) \\subset A^{+}:=A\\cap E^{+}$ is a disjoint sequence then by Theorem 4.34 of [2] $z_{n}\\overset{w}{\\rightarrow }0$ .",
"Thus, by our hypothesis $f_{n}\\left( Tz_{n}\\right) \\rightarrow 0$ for every disjoint sequence $\\left( z_{n}\\right) \\subset A^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset F^{\\ast }$ .",
"Now, by Theorem 2.7 of [10] $T\\left( A\\right) $ is almost limited.",
"$\\left( 5\\right) \\Rightarrow \\left( 1\\right) $ Obvious.",
"$\\left( 2\\right) \\Rightarrow \\left( 6\\right) $ and $\\left( 4\\right)\\Rightarrow \\left( 7\\right) $ are obvious.",
"$\\left( 6\\right) \\Rightarrow \\left( 2\\right) $ and $\\left( 7\\right)\\Rightarrow \\left( 4\\right) $ Let $\\left( x_{n}\\right) \\subset E^{+}$ be a weakly null (resp.",
"disjoint weakly null) sequence and let $\\left(f_{n}\\right) \\subset F^{\\ast }$ be a disjoint weak* null sequence.",
"If $F$ has the property $(\\text{d})$ then $\\left|f_{n}\\right|\\overset{w^{\\ast }}{\\rightarrow }0$ .",
"So from the inequalities $f_{n}^{\\,+}\\le \\left|f_{n}\\right|$ and $f_{n}^{\\,-}\\le \\left|f_{n}\\right|$ , the sequences $(f_{n}^{\\,+})$ , $(f_{n}^{\\,-})$ are weak* null.",
"Finally, by $\\left( 6\\right) $ (resp.",
"$\\left( 7\\right) $ ), $\\lim f_{n}\\left( T\\left( x_{n}\\right) \\right) =\\lim \\left[ f_{n}^{\\,+}\\left(T\\left( x_{n}\\right) \\right) -f_{n}^{\\,-}\\left( T\\left( x_{n}\\right) \\right) \\right] =0$ .",
"As consequence of Theorem REF we obtain the following characterization of the wDP* property which is a generalization of Theorem 3.2 of [5].",
"Corollary 2.5 Let $E$ be a Banach lattice with the property $\\mathrm {(d)}$ .",
"Then the following assertions are equivalents: $E$ has the wDP* property.",
"For every weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset E^{\\ast }$ we have $f_{n}\\left( x_{n}\\right) \\rightarrow 0$ .",
"For every disjoint weakly null sequence $\\left( x_{n}\\right) \\subset E$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset E^{\\ast } $ we have $f_{n}\\left( x_{n}\\right) \\rightarrow 0$ .",
"For every disjoint weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset E^{\\ast }$ we have $f_{n}\\left( x_{n}\\right) \\rightarrow 0$ .",
"The solid hull of every relatively weakly compact set in $E$ is almost limited.",
"$f_{n}\\left( x_{n}\\right) \\rightarrow 0$ for every weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset \\left( E^{\\ast }\\right) ^{+}$ .",
"$f_{n}\\left( x_{n}\\right) \\rightarrow 0$ for every disjoint weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset \\left( E^{\\ast }\\right) ^{+}$ .",
"Recently, the authors in [6] demonstrated that if a positive weak* Dunford-Pettis operator $T:E\\rightarrow F$ has its range in $\\sigma $ -Dedekind complete Banach lattice, then every positive operator $S:E\\rightarrow F$ that it dominates (i.e., $0\\le S\\le T$ ) is also weak* Dunford-Pettis [6].",
"For the positive weak almost limited operators, the situation still hold when $F$ satisfy the property (d).",
"Corollary 2.6 Let $E$ and $F$ be two Banach lattices such that $F$ satisfy the property $\\mathrm {(d)}$ .",
"Let $S,\\ T:E\\rightarrow F$ be two positive operators such that $0\\le S\\le T$ .",
"Then $S$ is a weak almost limited operator whenever $T$ is one.",
"Follows immediately from Theorem REF by noting that $0\\le f_{n}\\left( Sx_{n}\\right) \\le f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ for every disjoint weakly null sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset \\left(E^{\\ast }\\right) ^{+}$ .",
"Note that, clearly, every almost limited operator $T:X\\rightarrow E$ , from a Banach space into a Banach lattice, is weak almost limited.",
"But the converse is not true in general.",
"Indeed, the identity operator $I:\\ell ^{1}\\rightarrow \\ell ^{1}$ is Dunford-Pettis (and hence weak almost limited) but it is not almost limited.",
"The next result characterizes pairs of Banach lattices $E$ , $F$ for which every positive weak almost limited operator $T:E\\rightarrow F$ is almost limited.",
"Theorem 2.7 Let $E$ and $F$ be two Banach lattices such that $F$ has the property $\\mathrm {(d)}$ .",
"Then, the following statements are equivalents: Every order bounded weak almost limited operator $T:E\\rightarrow F$ is almost limited.",
"Every positive weak almost limited operator $T:E\\rightarrow F$ is almost limited.",
"One of the following assertions is valid: $F$ has the positive dual Schur property.",
"The norm of $E^{\\ast }$ is order continuous.",
"$\\left( 1\\right) \\Rightarrow \\left( 2\\right) $ Obvious.",
"$\\left( 2\\right) \\Rightarrow \\left( 3\\right) $ Assume by way of contradiction that $F$ does not have the positive dual Schur property and the norm of $E^{\\ast }$ is not order continuous.",
"We have to construct a positive weak almost limited operator $T:E\\rightarrow F$ which is not almost limited.",
"To this end, since the norm of $E^{\\ast }$ is not order continuous, there exists a disjoint sequence $(f_{n})\\subset \\left( E^{\\ast }\\right)^{+} $ satisfying $\\left\\Vert f_{n}\\right\\Vert =1$ and $0\\le f_{n}\\le f$ for all $n$ and for some $f\\in \\left( E^{\\ast }\\right) ^{+}$ (see Theorem 4.14 of [2]).",
"On the other hand, since $F$ does not have the positive dual Schur property, then there is a disjoint weak* null sequence $(g_{n})\\subset (F^{\\ast })^{+}$ such that $(g_{n})$ is not norm null.",
"By choosing a subsequence we may suppose that there is $\\varepsilon $ with $\\left\\Vert g_{n}\\right\\Vert >\\varepsilon >0$ for all $n$ .",
"From the equality $\\left\\Vert g_{n}\\right\\Vert =\\sup \\lbrace g_{n}(y):y\\in B_{F}^{+}\\rbrace $ , there exists a sequence $\\left(y_{n}\\right) \\subset B_{F}^{+}$ such that $g_{n}(y_{n})\\ge \\varepsilon $ holds for all $n$ .",
"Now, consider the operators $P:E\\rightarrow \\ell ^{1}$ and $S:\\ell ^{1}\\rightarrow F$ defined by $P(x)=\\left( f_{n}(x)\\right) _{n}$ and $S\\left( (\\lambda _{n})_{n}\\right) =\\sum \\limits _{n=1}^{\\infty }\\lambda _{n}y_{n}$ for each $x\\in E$ and each $(\\lambda _{n})_{n}\\in \\ell ^{1}$ .",
"Since $\\sum \\limits _{n=1}^{N}\\left|f_{n}\\left( x\\right) \\right|\\le \\sum \\limits _{n=1}^{N}f_{n}\\left( \\left|x\\right|\\right) =\\left(\\vee _{n=1}^{N}f_{n}\\right) \\left( \\left|x\\right|\\right) \\le f\\left( \\left|x\\right|\\right)$ for each $x\\in E$ and each $N\\in \\mathbb {N}$ , the operator $P$ is well defined, and clearly $P$ and $S$ are positives.",
"Now, consider the positive operator $T=S\\circ P:E\\rightarrow \\ell ^{1}\\rightarrow F$ , and note that $T(x)=\\sum \\limits _{n=1}^{\\infty }f_{n}(x)y_{n}$ for each $x\\in E$ .",
"Clealry, as $\\ell ^{1}$ has the Schur property, then $T$ is Dunford-Pettis and hence $T$ is weak almost limited.",
"However, for the disjoint weak* null sequence $(g_{n})\\subset (F^{\\ast })^{+}$ , we have for every $n$ , $T^{\\ast }\\left( g_{n}\\right) =\\sum \\limits _{k=1}^{\\infty }g_{n}\\left(y_{k}\\right) f_{k}\\ge g_{n}\\left( y_{n}\\right) f_{n}\\ge 0\\text{.",
"}$ Thus $\\left\\Vert T^{\\ast }\\left( g_{n}\\right) \\right\\Vert \\ge \\left\\Vert g_{n}\\left( y_{n}\\right) f_{n}\\right\\Vert =g_{n}\\left( y_{n}\\right) \\ge \\varepsilon $ for every $n$ .",
"This show that $T$ is not almost limited, and we are done.",
"$(i)\\Rightarrow \\left( 1\\right) $ In this case, every operator $T:E\\rightarrow F$ is almost limited.",
"In fact, let $\\left( f_{n}\\right)\\subset F^{\\ast }$ be a disjoint weak* null sequence.",
"Since $F$ has the property $(\\text{d})$ then the positive disjoint sequence $\\left( \\left|f_{n}\\right|\\right) \\subset F^{\\ast }$ is weak* null.",
"So by the positive dual Schur property of $F$ , $\\left\\Vert f_{n}\\right\\Vert \\rightarrow 0$ , and hence $\\left\\Vert T^{\\ast }\\left( f_{n}\\right)\\right\\Vert \\rightarrow 0$ , as desired.",
"$(ii)\\Rightarrow \\left( 1\\right) $ According to Proposition 4.4 of [10] it is sufficient to show $f_{n}\\left( T\\left( x_{n}\\right) \\right)\\rightarrow 0$ for every norm bounded disjoint sequence $\\left( x_{n}\\right)\\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right)\\subset \\left( F^{\\ast }\\right) ^{+}$ .",
"As the norm of $E^{\\ast }$ is order continuous then every norm bounded disjoint sequence $\\left( x_{n}\\right)\\subset E^{+}$ is weakly null [11].",
"Now, since $T$ is an order bounded weak almost limited operator then by Theorem REF $f_{n}\\left( T\\left( x_{n}\\right) \\right) \\rightarrow 0$ for every norm bounded disjoint sequence $\\left( x_{n}\\right) \\subset E^{+}$ and every disjoint weak* null sequence $\\left( f_{n}\\right) \\subset \\left(F^{\\ast }\\right) ^{+}$ .",
"This complete the proof.",
"As consequence of Theorem REF , we obtain the following corollary.",
"Corollary 2.8 For a Banach lattices $E$ with the property $\\mathrm {(d)}$ , $E^{\\ast }$ has an order continuous norm if and only if every positive weak almost limited operator $T:E\\rightarrow E$ is almost limited.",
"Follows from Theorem REF by noting that if $E$ has the positive dual Schur property then the norm of $E^{\\ast }$ is order continuous.",
"As the Banach space $\\ell ^{1}$ has the Schur property then every operator $T:\\ell ^{1}\\rightarrow E$ is weak almost limited.",
"Another consequence of Theorem REF is the following characterization of the positive dual Schur property.",
"Corollary 2.9 A Banach lattices $E$ with the property $\\mathrm {(d)}$ , has the positive dual Schur property if and only if every positive operator $T:\\ell ^{1}\\rightarrow E$ is almost limited.",
"Recall that a reflexive Banach space with the Dunford-Pettis property is finite dimensional [2].",
"We can prove a similar result for Banach lattices as follows.",
"Proposition 2.10 Let $E$ be a Banach lattice with the wDP* property.",
"If the norms of $E$ and $E^{\\ast }$ are order continuous then $E$ is finite dimensional.",
"In particular, a reflexive Banach lattice with the wDP* property is finite dimensional.",
"Assume that the norms of $E$ and $E^{\\ast }$ are order continuous.",
"As $E$ has the wDP* property the identity operator $I:E\\rightarrow E$ is weak almost limited.",
"Since the norm of $E^{\\ast }$ is order continuous then, by Theorem REF , $I$ is almost limited.",
"So $E$ has the positive dual Schur property.",
"Now, as the norm of $E$ is order continuous then $E$ is finite dimensional [13].",
"For the second part, it is enough to note that if $E$ is a reflexive Banach lattice then the norms of $E$ and $E^{\\ast }$ are order continuous [2].",
"Note that from Theorem REF , it is easy to see that if $F$ is a Banach lattice with property $\\mathrm {(d)}$ then every order bounded almost Dunford-Pettis operator $T:E\\rightarrow F$ is weak almost limited.",
"But the convers is false in general.",
"In fact, the identity operator $I:\\ell ^{\\infty }\\rightarrow \\ell ^{\\infty }$ is weak almost limited operator but it fail to be almost Dunford-Pettis.",
"The following result characterizes pairs of Banach lattices $E$ , $F$ for which every positive weak almost limited operator operator $T:E\\rightarrow F$ is almost Dunford-Pettis.",
"Theorem 2.11 Let $E$ and $F$ be two Banach lattices such that $F$ is $\\sigma $ -Dedekind complete.",
"Then, the following statements are equivalents: Every order bounded weak almost limited operator $T:E\\rightarrow F$ is almost Dunford-Pettis.",
"Every positive weak almost limited operator $T:E\\rightarrow F$ is almost Dunford-Pettis.",
"One of the following assertions is valid: $E$ has the positive Schur property.",
"The norm of $F$ is order continuous.",
"$\\left( 1\\right) \\Rightarrow \\left( 2\\right) $ Obvious.",
"$\\left( 2\\right) \\Rightarrow \\left( 3\\right) $ Assume by way of contradiction that $E$ does not have the positive Schur property and the norm of $F$ is not order continuous.",
"We have to construct a positive weak almost limited operator $T:E\\rightarrow F$ which is not almost Dunford-Pettis.",
"As $E$ does not have the positive Schur property, then there exists a disjoint weakly null sequence $(x_{n})$ in $E^{+}$ which is not norm null.",
"By choosing a subsequence we may suppose that there is $\\varepsilon $ with $\\left\\Vert x_{n}\\right\\Vert >\\varepsilon >0$ for all $n$ .",
"From the equality $\\left\\Vert x_{n}\\right\\Vert =\\sup \\lbrace f(x_{n}):f\\in (E^{\\ast })^{+},\\quad \\left\\Vert f\\right\\Vert =1\\rbrace $ , there exists a sequence $\\left( f_{n}\\right) \\subset (E^{\\ast })^{+}$ such that $\\left\\Vert f_{n}\\right\\Vert =1$ and $f_{n}(x_{n})\\ge \\varepsilon $ holds for all $n$ .",
"Now, consider the operator $R:E\\rightarrow \\ell ^{\\infty }$ defined by $R(x)=(f_{n}(x))_{n}$ On the other hand, since the norm of $F$ is not order continuous, it follows from Theorem 4.51 of [2] that $\\ell ^{\\infty }$ is lattice embeddable in $F$ , i.e., there exists a lattice homomorphism $S:\\ell ^{\\infty }\\rightarrow F$ and there exist tow positive constants $M$ and $m$ satisfying $m\\left\\Vert \\left( \\lambda _{k}\\right) _{k}\\right\\Vert _{\\infty }\\le S\\left( \\left( \\lambda _{k}\\right) _{k}\\right) \\le M\\left\\Vert \\left(\\lambda _{k}\\right) _{k}\\right\\Vert _{\\infty }$ for all $\\left( \\lambda _{k}\\right) _{k}\\in \\ell ^{\\infty }$ .",
"Put $T=S\\circ R $ , and note that $T$ is a positive weak almost limited operator (see Remark REF ).",
"However, for the disjoint weakly null sequence $(x_{n})\\subset E^{+}$ , we have $\\left\\Vert T\\left( x_{n}\\right) \\right\\Vert =\\left\\Vert S\\left( \\left(f_{k}\\left( x_{n}\\right) \\right) _{k}\\right) \\right\\Vert \\ge m\\left\\Vert \\left( f_{k}\\left( x_{n}\\right) \\right) _{k}\\right\\Vert _{\\infty }\\ge mf_{n}\\left( x_{n}\\right) \\ge m\\varepsilon $ for every $n$ .",
"This show that $T$ is not almost Dunford-Pettis, and we are done.",
"$\\left( i\\right) \\Rightarrow \\left( 1\\right) $ In this case, every operator $T:E\\rightarrow F$ is almost Dunford-Pettis.",
"$\\left( ii\\right) \\Rightarrow \\left( 1\\right) $ Let $T:E\\rightarrow F$ be an order bounded weak almost limited operator and let $\\left( x_{n}\\right)\\subset E$ be a positive disjoint weakly null sequence.",
"We shall show that $\\left\\Vert Tx_{n}\\right\\Vert \\rightarrow 0$ .",
"By corollary 2.6 of [7], it suffices to proof that $\\left|Tx_{n}\\right|\\overset{w}{\\rightarrow }0$ and $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ for every disjoint and norm bounded sequence $\\left( f_{n}\\right) \\subset \\left(F^{\\ast }\\right) ^{+}$ .",
"Indeed - Let $f\\in \\left( F^{\\ast }\\right) ^{+}$ .",
"By Theorem 1.23 of [2], for each $n$ there exists some $g_{n}\\in \\left[ -f,f\\right] $ with $f\\left|Tx_{n}\\right|=g_{n}\\left( Tx_{n}\\right) $ .",
"Note that the adjoint operator $T^{\\ast }:F^{\\ast }\\rightarrow E^{\\ast }$ is order bounded [2], and pick some $h\\in \\left( E^{\\ast }\\right) ^{+}$ with $T^{\\ast }\\left[ -f,f\\right] \\subseteq \\left[ -h,h\\right] $ .",
"So $0\\le f\\left|Tx_{n}\\right|=\\left( T^{\\ast }g_{n}\\right)\\left( x_{n}\\right) \\le h\\left( x_{n}\\right) $ for all $n$ .",
"Since $x_{n}\\overset{w}{\\rightarrow }0$ , then $h\\left( x_{n}\\right) \\rightarrow 0$ and hence $f\\left|Tx_{n}\\right|\\rightarrow 0$ .",
"Thus $\\left|Tx_{n}\\right|\\overset{w}{\\rightarrow }0$ .",
"- Let $\\left( f_{n}\\right) \\subset \\left( F^{\\ast }\\right) ^{+}$ be a disjoint and norm bounded sequence.",
"As the norm of $F$ is order continuous, then by corollary 2.4.3 of [11] $f_{n}\\overset{w^{\\ast }}{\\rightarrow }0 $ .",
"Now, since $T$ is an order bounded weak almost limited then $f_{n}\\left( Tx_{n}\\right) \\rightarrow 0$ (Theorem REF ).",
"This complete the proof.",
"As consequence of Theorem REF , we obtain the following corollary.",
"Corollary 2.12 A $\\sigma $ -Dedekind complete Banach lattices $E$ has an order continuous norm if and only if every order bounded weak almost limited operator $T:E\\rightarrow E$ is almost Dunford-Pettis.",
"By Remark REF every operator $T:E\\rightarrow \\ell ^{\\infty }$ is weak almost limited.",
"As another consequence of Theorem REF , we obtain the following characterization of the positive Schur property.",
"Corollary 2.13 A Banach lattices $E$ has the positive Schur property if and only if every positive operator $T:E\\rightarrow \\ell ^{\\infty }$ is almost Dunford-Pettis."
]
] | 1403.0136 |
[
[
"Pressure-induced ferromagnetism with strong Ising-type anisotropy in\n YbCu$_2$Si$_2$"
],
[
"Abstract We report dc magnetic measurements on YbCu$_2$Si$_2$ at pressures above 10 GPa using a miniature ceramic anvil cell.",
"YbCu$_2$Si$_2$ shows a pressure-induced transition from the non-magnetic to a magnetic phase at 8 GPa.",
"We find a spontaneous dc magnetization in the pressure-induced phase above 9.4 GPa.",
"The pressure dependence of the ferromagnetic transition temperature T_C and the spontaneous magnetic moment m_0 at 2.0 K have been determined.",
"The value of m_0 in the present macroscopic measurement is less than half of that determined via Mossbauer experiment.",
"The difference may be attributed to spatial phase separation between the ferromagnetic and paramagnetic phases.",
"This separation suggests that the pressure-induced phase boundary between the paramagnetic and ferromagnetic states is of first order.",
"Further, we have studied the magnetic anisotropy in the pressure-induced ferromagnetic state.",
"The effect of pressure on the magnetization with magnetic field along the magnetic easy $c$-axis is much larger than for field along the hard $a$-axis in the tetragonal structure.",
"The pressure-induced phase has strong Ising-type uniaxial anisotropy, consistent with the two crystal electric field (CEF) models proposed for YbCu$_2$Si$_2$."
],
[
"Introduction",
"In recent years there has been growing interest in strongly correlated electron systems of rare earth and actinide compounds located at or close to a magnetic quantum critical point (QCP)[1].",
"The electronic state of such systems can often be tuned with pressure or magnetic field.",
"Unconventional superconductivity and non-Fermi liquid behavior have been observed near pressure-induced magnetic to non-magnetic phase boundaries in many cerium compounds such as CeIn$_3$[2].",
"The novel physical phenomena have been studied from the view point of the quantum criticality.",
"Such phenomena might be expected in ytterbium compounds since the Yb is considered to be a “hole\" equivalent of Ce.",
"Indeed, anomalous physical properties have been reported and extensively studied in YbRh$_2$ Si$_2$ and $\\beta $ -YbAlB$_4$[3], [4].",
"Application of pressure tends to drive the Yb ion from nonmagnetic Yb$^{2+}$ ($4f^{14}$ ) to magnetic Yb$^{3+}$ ($4f^{13}$ ) states.",
"A magnetic ordered state is stabilized at higher pressures.",
"A pressure-induced magnetic phase has been reported in a number of Yb compounds.",
"In most cases the pressure-induced magnetic phase has been detected via ac magnetic susceptibility measurements.",
"There have been few studies of detailed magnetic properties of a pressure-induced phase using dc magnetization measurements.",
"This is due to the common experimental constraint that the maximum pressure is only 1.5 GPa for the most commonly used piston cylinder type cell in a commercial superconducting quantum interference device (SQUID)[8].",
"Recently, we have developed a miniature ceramic anvil cell (mCAC) for magnetic measurements at pressures above 10 GPa with use of the SQUID magnetometer[9], [10], [11].",
"Thanks to the simplicity of cell structure, the mCAC can detect the ferromagnetic ordered state whose spontaneous magnetic moment is significantly less than 1.0 ${\\mu }_{\\rm B}$ per a magnetic ion.",
"The cell enables us to make a quantitative study of the pressure-induced phases in Yb compounds.",
"We report here a study of the anisotropic magnetic properties of the pressure-induced phase in YbCu$_2$ Si$_2$ .",
"YbCu$_2$ Si$_2$ crystallizes in the tetragonal ThCr$_2$ Si$_2$ - structure.",
"This is a paramagnetic compound with a moderately high value of the linear specific heat coefficient ${\\gamma }{\\,}{\\simeq }$ 135 mJK$^{-2}$ mol${^{-1}}$[12], [13].",
"Previous high pressure studies suggested a pressure-induced, possibly ferromagnetic ordered state above 8 GPa from ac magnetic susceptibility measurements and Mössbauer experiment[14], [15], [16], [17].",
"It is therefore important to detect the ferromagnetic component from dc magnetic measurement at high pressure.",
"In this study we have measured the magnetization of YbCu$_2$ Si$_2$ with our mCAC."
],
[
"Experimental",
"Single crystals of YbCu$_2$ Si$_2$ were grown from Sn flux[12], [13].",
"We have used our miniature ceramic-anvil high-pressure cell mCAC with the 0.6 mm culet anvils[9], [10], [11].",
"The Cu-Be gasket was preindented to 0.08 mm from the initial thickness of 0.30 mm.",
"The diameter of the sample space in the gasket was 0.20 mm.",
"To study the anisotropy of the magnetic properties in YbCu$_2$ Si$_2$ , two single crystals were measured with magnetic field applied parallel to the magnetic easy $c$ -axis (the [001] direction) and the hard $a$ -axis ([100] direction) in the tetragonal crystal structure.",
"The sizes of the single crystal samples were 0.11 $\\times $ 0.09 $\\times $ 0.03 mm$^3$ and 0.10 $\\times $ 0.09 $\\times $ 0.02 mm$^3$ for magnetic measurements with magnetic field along the $c$ -axis and the $a$ -axis, respectively.",
"The sample and a Pb pressure sensor were placed in the sample space filled with glycerin as pressure-transmitting medium[18].",
"The pressure values at low temperatures were determined by the pressure dependence of the superconducting transition temperature of Pb[19], [20], [21].",
"The pressure medium glycerin solidifies at 5 GPa at room temperature.",
"The pressure inhomogeneity was estimated as ${\\Delta }P$ $\\sim $ 1 GPa above 10 GPa.",
"The demagnetization effect needs to be taken into account in the pressure-induced ferromagnetic state.",
"The internal field values $H_{int}$ were determined by subtracting the demagnetizing field given by $H_{int}$ = $H_{appl}$ - $DM$ .",
"Here, $H_{appl}$ is the external magnetic field and $D$ is the demagnetizing factor.",
"Error bars in Figure 1 (b) indicate possible errors in the estimation of the magnetization.",
"Figure: (Color online)(a) Temperature dependence of the magnetic susceptibility χ{\\chi } under magnetic field of 1 kOe applied along the magnetic easy cc-axis (HH |||| [001]).",
"(b) magnetic field dependence of the magnetization and (c) Arrott-plots of the magnetization measured at 2.0 K and at 1 bar, 2.5, 3.1, 5.2, 7.2, 9.4, 10.5, and 11.5 GPa."
],
[
"Results and Discussions",
"Figure 1 shows (a) temperature dependence of the magnetic susceptibility ${\\chi }$ in a magnetic field of 1 kOe applied along the magnetic easy $c$ -axis ($H$ $||$ [001]) and (b) magnetic field dependence of the magnetization measured at 2.0 K and at 1 bar, 2.5, 3.1, 5.2, 7.2, 9.4, 10.5, and 11.5 GPa.",
"At 1 bar, ${\\chi }$ shows an almost temperature independent value of ${\\chi }$ = 0.03 emu/mole below 10 K and the magnetization increases linearly with increasing magnetic field, consistent with the previous study[12].",
"Application of pressure above 5 GPa induces a low temperature upturn in ${\\chi }$ and a non-linear increase of the magnetization in low fields.",
"At 9.4, 10.5, and 11.5 GPa, the magnetization shows typical ferromagnetic behavior with the magnetic susceptibility, diverging at low temperatures.",
"These results indicate that the pressure-induced magnetic transition in YbCu$_2$ Si$_2$ is ferromagnetic.",
"Figure: (Color online)(a)Temperature-pressure phase diagram of YbCu 2 _2Si 2 _2.",
"Circles indicate the ferromagnetic transition temperature T C T_{\\rm C}.",
"Dotted and dashed-dotted lines indicate the pressure dependences of T C T_{\\rm C} for the sample with RRR = 200 in the previous study.",
"The former and the latter lines were determined by the ac magnetic susceptibility measurement and the ac calorimetry, respectively.",
"(b) Pressure dependence of the spontaneous magnetic moment μ 0 {\\mu }_0 determined at 2.0 K.The spontaneous magnetic moment ${\\mu }_0$ is determined above 9.4 GPa from the Arrott-plot shown in Fig.1 (c).",
"The values of ${\\mu }_0$ at 2.0 K are estimated as 0.16 $\\pm $ 0.08, 0.30 $\\pm $ 0.08, and 0.42 $\\pm $ 0.05 ${\\mu }_{\\rm B}$ /Yb at 9.4, 10.2, and 11.5 GPa, respectively.",
"The ferromagnetic transition temperatures $T_{\\rm C}$ at 9.4, 10.5, and 11.5 GPa are estimated as 3.5 $\\pm $ 0.5, 4.3 $\\pm $ 0.5, and 4.7 $\\pm $ 0.5 K, respectively, from the peak position in the temperature derivative of the magnetic susceptibility ${\\partial {\\chi }}/{\\partial T}$ .",
"Figure 2 shows the pressure dependences of (a) the ferromagnetic transition temperature $T_{\\rm C}$ and (b) the spontaneous magnetic moment ${\\mu }_0$ at 2.0 K in YbCu$_2$ Si$_2$ .",
"A ferromagnetic transition was not observed down to 2.0 K at 8.8 GPa (data not shown).",
"The transition may occur below 2.0 K. The critical pressure $P_c$ for the ferromagnetic state may be located between 8.0 and 8.5 GPa.",
"Reference 17 reported that the pressure effect on $T_{\\rm C}$ depends largely on the sample quality[17].",
"The present pressure dependence of $T_{\\rm C}$ is consistent with those for samples with the similar quality (RRR = 200) in the previous study, shown as dotted and dashed-dotted lines in Fig.",
"2 (a).",
"The former and the latter lines were determined by the ac magnetic susceptibility measurement and the ac calorimetry, respectively.",
"It has been established that, above critical pressure $P_c$ , the transition to the ferromagnetic phase in YbCu$_2$ Si$_2$ is of first order[16], [17].",
"Indeed, the ac magnetic susceptibility measurement showed a sudden appearance of the ferromagnetic transition above 1 K[17].",
"However, no sharp anomaly at $T_{\\rm C}$ is observed in the temperature dependence of the magnetization at any pressure above 9.4 GPa, indicating a second order phase transition.",
"We suggest that the ferromagnetic transition changes from the first to the second order phase transition at a somewhat higher pressure than $P_c$ in YbCu$_2$ Si$_2$ .",
"$P_c$ may be a weakly first-order critical point.",
"This may be a reason for absence of the Non-Fermi liquid behavior in the resistivity $\\rho $ .",
"It shows a typical Fermi liquid behavior ${\\rho }={{\\rho }_0}+AT^2$ down to 30 mK around $P_c$ , where ${\\rho }_0$ is the residual resistivity[14].",
"The value of $A$ increases continuously as a function of the pressure but it does not show a divergent behavior around $P_c$ .",
"Several ferromagnets such as ZrZn$_2$[22], Co(S$_{1-x}$ Se$_x$ )$_2$[23], MnSi[24] and UGe$_2$[25] have a tricritical point where the paramagnet to ferromagnet transition changes from a second-order to a first order phase transition when driven toward the QCP by applying external pressure or chemical pressure.",
"This seems to be a general property of ferromagnets as has been theoretically discussed[26].",
"We discuss the pressure-induced ferromagnetism in YbCu$_2$ Si$_2$ from two points of views.",
"There are two crystal electric field (CEF) models (I and I') proposed for YbCu$_2$ Si$_2$ in previous studies[12].",
"The values of the magnetic moment expected from the doublet ground state are 2.70 and 2.29 ${\\mu }_{\\rm B}$ /Yb for the CEF models I and I', respectively.",
"We compare the values in the CEF models with that determined in the Mössbauer experiment (i) and that determined in the present macroscopic measurement (ii).",
"(i) The values of the magnetic moment in the CEF models are more than two times larger than that (1.25 ${\\mu }_{\\rm B}$ /Yb) determined with the Mössbauer experiment at 8.9 GPa at 1.8 K[15].",
"The reduced magnetic moment in the Mössbauer experiment may be due to the Kondo effect.",
"Resonant inelastic x-ray scattering measurement showed that the value of the Yb valence is 2.88 at 7 K near $P_c$[27].",
"Thus, ferromagnetism appears in the mixed valence state in YbCu$_2$ Si$_2$ .",
"Contrary to cerium compounds, a magnetic ordering can appear in the intermediate valence state (${n_{4f}}$ $\\ll $ 1) of the Yb systems, where ${n_{4f}}$ is the occupation number of the $4f$ level[5], [28], [29].",
"Differences in the magnetic properties between the Ce and Yb systems arise from differing hierarchies of the energy scales of the Kondo temperature $T_{\\rm K}$ , the $4f$ -band width ${\\Delta }_{4f}$ and the splitting energy between ground and first excited states in the CEF levels ${\\Delta }_{CEF}$[5], [28], [29].",
"$T_{\\rm K}$ of Yb systems could be smaller or comparable to ${\\Delta }_{CEF}$ because of the smaller ${\\Delta }_{4f}$ in Yb systems than that in cerium systems.",
"In YbCu$_2$ Si$_2$ , the electrical resistivity under high pressure suggests that $T_{\\rm K}$ is less than 50 K at around $P_c$[14].",
"The value of $T_{\\rm K}$ is lower than that of ${\\Delta }_{CEF}$ in the model I and I'[12].",
"The linear specific heat coefficient $\\gamma $ is estimated as $\\gamma $ $\\sim $ 1 J/mol$\\cdot $ K$^2$ at $P_c$ from the coefficient $A$ of the $T^2$ -term in the resistivity with the Kadowaki-Woods relation[14], [30].",
"The pressure-induced phase in YbCu$_2$ Si$_2$ is a ferromagnetic heavy fermion system with the intermediate valence of the Yb ion.",
"This is opposed to Ce-systems where the magnetic ordering or heavy fermium states are usually restricted to the trivalent configuration (${n_{4f}}$ $\\sim $ 1.0)[5], [28], [29].",
"Figure: (Color online)(a)Temperature dependence of the magnetic susceptibility χ{\\chi } under magnetic field of 1 kOe applied along the magnetic hard aa-axis (HH |||| [100]) and (b) magnetic field dependence of the magnetization measured at 2.0 K and at 1 bar, 5.0, 9.0, 11.0, and 12.2 GPa.",
"(ii) In the present macroscopic magnetic measurement, the spontaneous magnetic moment ${\\mu }_0$ at 2.0 K and 9.4 GPa is less than half of that with the Mössbauer experiment[15].",
"This difference may be due to a spatial phase separation between the paramagnetic and ferromagnetic states suggested in the Mössbauer spectrum.",
"The value of ${\\mu }_0$ increases with increasing pressure.",
"The continuous change in ${\\mu }_0$ around the critical pressure $P_c$ is difficult to understand because the pressure induced change is of first order[16], [17].",
"We point out two possibilities.",
"One is that the volume fraction of the ferromagnetic phase increases as a function of pressure and the other is that the pressure change of ${\\mu }_0$ reflects the increase of the Yb valence above $P_c$ seen in the resonant inelastic x-ray scattering measurement[27].",
"The phase separation suggests a first order phase boundary between the paramagnetic and the ferromagnetic phases.",
"Figure 3 shows (a) temperature dependence of the magnetic susceptibility ${\\chi }$ in magnetic field of 1 kOe applied along the magnetic hard $a$ -axis ($H$ $||$ [100]) and (b) magnetic field dependence of the magnetization measured at 2.0 K and at 1 bar, 5.0, 9.0, 11.0, and 12.2 GPa.",
"Compared with the magnetization data for $H$ $||$ $c$ , the pressure effect on the magnetization for $H$ $||$ $a$ is significantly smaller.",
"The value of ${\\chi }$ at 2.0 K is increased from 0.01 emu/mole at 1 bar to 0.03 emu/mole at 12.2 GPa.",
"The magnetization curve does not show a ferromagnetic behavior at higher pressures.",
"The magnetic field induced moment at 10 kOe is 0.038 ${\\mu }_{\\rm B}$ /Yb at 12.2 GPa, one order of magnitude smaller than that (0.57 ${\\mu }_{\\rm B}$ /Yb) with magnetic field applied along the easy $c$ -axis at 11.5 GPa.",
"The pressure-induced ferromagnetic phase has strong uniaxial anisotropy.",
"The Ising character of the magnetic property is suggested from the two CEF models proposed for YbCu$_2$ Si$_2$[12].",
"The Ising-type magnetic fluctuation can induce the spin-triplet $p$ -wave superconductivity around the ferromagnetic QCP[1].",
"A motivation for the previous high pressure studies on YbCu$_2$ Si$_2$ was to search for the superconductivity around $P_c$ .",
"However, the superconductivity has not been found in resistivity measurements down to 30 mK[14].",
"Theoretically, the superconducting transition temperature for spin triplet $p$ -wave pairing around the ferromagnetic QCP is largely lower than that of the spin singlet $d$ -wave superconductivity around the antiferromagnetic QCP[31].",
"In Ce systems, CeIn$_3$ and CeRhIn$_5$ exhibit superconductivity under high pressure where $T_{sc}$ attains maximum values of 0.2 K and 2.2 K, respectively[2], [32].",
"In uranium systems, the superconductivity appears in the ferromagnetic state of UGe$_2$[33].",
"The value of $T_{sc}$ is 0.8 K at 1.2 GPa.",
"URhGe and UCoGe show the superconducting transition at $T_{sc}$ = 0.2 and 0.7 K, respectively, at ambient pressure[34], [35].",
"The characteristic temperature of the electronic state in Yb systems is lower than those in Ce and U systems due to the smaller band width of $4f$ -band as mentioned before.",
"If the superconductivity existed in YbCu$_2$ Si$_2$ , the transition temperature would be very low.",
"This may be a reason why the heavy fermion superconductivity of the 4$f$ electrons is elusive in Yb systems.",
"Also, spatial phase separation in YbCu$_2$ Si$_2$ may be harmful for the appearance of the superconductivity.",
"The present study shows convincing evidence of ferromagnetism in the pressure-induced phase of YbCu$_2$ Si$_2$ from dc magnetization measurements.",
"Ferromagnetism has been found in a number of Yb compounds such as YbRhSb, YbInNi$_4$ , and YbNiSn at ambient pressure[36], [37], [38], YbInCu$_4$ and YbIr$_2$ Si$_2$ at high pressure[39], [40].",
"On the other hand, there are only a few cerium based compounds such as CeRh$_3$ B$_2$ and CeAg which show a ferromagnetic ground state[41], [42].",
"The origin of this difference is an interesting question.",
"The hierarchies of the energy scales of $T_{\\rm K}$ , ${\\Delta }_{4f}$ and ${\\Delta }_{CEF}$ in the Ce and Yb system are different as mentioned before[5], [28], [29].",
"This leads to the lager change of valence of Yb-ions from the non-magnetic 2+ to magnetic 3+ in real lattices, compared with that of the Ce ions.",
"The resonant inelastic x-ray scattering experiment shows a wider valence change in YbCu$_2$ Si$_2$ , compared with that in its Ce-couterpart CeCu$_2$ Si$_2$[27].",
"The valence transition or instability of the Yb ion has been detected by the X-ray absorption or emission spectroscopy in YbAgCu$_4$[43], YbInCu$_4$[44], and YbCu$_{5-x}$ Al$_x$[45].",
"Recently, new aspects in strongly correlated electron system originating from valence fluctuation of the rare earth ion have been theoretically discussed[6], [7].",
"Anomalous physical properties in $\\beta $ -YbAlB$_4$ and YbRh$_2$ Si$_2$ have been re-considered from this point of view[7].",
"The theoretical study also shows a simultaneous divergence of the valence susceptibility and the uniform spin susceptibility at the quantum critical point of the valence transition under a magnetic field.",
"This strengthens a ferromagnetic tendency in Yb systems under finite magnetic field.",
"Careful future theoretical study is necessary for a realization of the ferromagnetism under zero magnetic field[46].",
"For the experimental point of view, comprehensive studies on the Yb systems should be done to reveal the valence state of the Yb ions in the wide temperature, magnetic field and pressure regions."
],
[
"Conclusion",
"In conclusion, dc magnetic measurements have been done to study the magnetic property of the pressure-induced phase in YbCu$_2$ Si$_2$ with a miniature ceramic anvil high pressure cell.",
"The ferromagnetic ordered state is confirmed from the observation of the dc spontaneous magnetization.",
"The pressure dependences of the ferromagnetic transition temperature $T_{\\rm C}$ and the spontaneous magnetic moment ${\\mu }_0$ at 2.0 K have been determined.",
"The value of ${\\mu }_0$ in the present macroscopic measurement is less than half of that determined via Mössbauer experiment, which may be attributed to spatial phase separation between the ferromagnetic and paramagnetic phases.",
"Peculiar features in the pressure-induced ferromagnetic state are discussed in comparison with cerium compounds.",
"The effect of pressure on the magnetization with magnetic field along the magnetic easy $c$ -axis is much larger than for field along the hard $a$ -axis in the tetragonal structure.",
"The pressure-induced phase in YbCu$_2$ Si$_2$ has strong Ising-type uniaxial anisotropy, consistent with the two crystal electric field (CEF) models proposed for YbCu$_2$ Si$_2$ ."
],
[
"Acknowledgments",
"We would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.",
"We also acknowledge Prof. S. Watanabe for critical reading of this paper and giving us useful comments.",
"This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Heavy Electrons (Nos.",
"20102002 and 23102726), Scientific Research S (No.",
"20224015), A(No.",
"23246174), C (Nos.",
"21540373, 22540378 and 25400386), and Young Scientists (B) (No.",
"22740241) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society of the Promotion of Science (JSPS).",
"J. Flouquet, Progress in Low Temperature Physics 15, p. 139 (Elsevier, Amsterdam, 2005).",
"N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer and G. G. Lonzarich: Nature 394, 39 (1998).",
"P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel, F. Steglich, E. Abrahams, and Q. Si, Science 315, 969 (2007).",
"Y. Matsumoto, S. Nakatsuji, K. Kuga, Y. Karaki, N. Horie, Y. Shimura, T. Sakakibara, A. H. Nevidomsky, and P. Coleman, Science 331, 316 (2011).",
"J. Flouquet and H. Harima: arXiv: 0910.3110; Kotai Butsuri (Solid State Physics) 47, 47 (2012)[in Japanese].",
"S. Watanabe and K. Miyake, Phys.",
"Rev.",
"Lett.",
"105, 186403 (2010).",
"S. Watanabe and K. Miyake, J.",
"Phys.",
": Condens.",
"Matter 24, 294208 (2012).",
"J. Kamarád, Z. Machátová, Z. Arnold, Rev.",
"Sci.",
"Instrum.",
"75, 5022 (2004).",
"N. Tateiwa, Y. Haga, Z. Fisk, and Y. Ōnuki, Rev.",
"Sci.",
"Instrum.",
"82, 053906 (2011).",
"N. Tateiwa, Y. Haga, T. D. Matsuda and Z. Fisk, Rev.",
"Sci.",
"Instrum.",
"83, 053906 (2012).",
"N. Tateiwa, Y. Haga, T. D. Matsuda, Z. Fisk, S. Ikeda, and H. Kobayashi, Rev.",
"Sci.",
"Instrum.",
"84, 046105 (2013).",
"N. D. Dung, T. D. Matsuda, Y. Haga, S. Ikeda, E. Yamamoto, T. Ishikura, T. Endo, S. Tatsuoka, Y. Aoki, H. Sato, T. Takeuchi, R. Settai, H. Harima, and Y. Ōnuki, J. Phys.",
"Soc.",
"Jpn.",
"78, 084711 (2009).",
"T. D. Matsuda, N. D. Dung, Y. Haga, S. Ikeda, E. Yamamoto, T. Ishikura, T. Endo, T. Takeuchi, R. Settai, and Y. Ōnuki, Physica status solidi.",
"B-A basic solid state physics 247, 757 (2010).",
"K. Alami-Yadri and D. Jaccard, Eur.",
"Phys.",
"J.",
"B 6, 5 (1998).",
"H. Winkelmann, M. M. Abd-Elmeguid, H. Micklitz, J. P. Sanchez, P. Vulliet, K. Alami-Yadri, and D. Jaccard, Phys.",
"Rev.",
"B 60, 3324 (1999).",
"E. Colombier, D. Braithwaite, G. Lapertot, B. Salce, and G. Knebel, Phys.",
"Rev.",
"B 79, 245113 (2009).",
"A. Fernandez-Pañella, D. Braithwaite, B. Salce, G. Lapertot, and J. Flouquet, Phys.",
"Rev.",
"B 84, 134416 (2011).",
"N. Tateiwa and Y. Haga, Rev.",
"Sci.",
"Instrum.",
"80, 123901 (2009).",
"T. F. Smith, C. W. Chu, and M. B.",
"Maple, Cryogenics 9, 53 (1969).",
"A. Eiling and J. S. Schilling, J. Phys.",
"F : Metal Phys., 11, 623 (1981).",
"B. Bireckoven and J. Wittig, J. Phys.",
"E: Sci.",
"Instrum.",
"21, 841 (1988).",
"M. Uhlarz, C. Pfliderer, and J. Flouquet, Phys.",
"Rev.",
"Lett.",
"93, 256404 (2004).",
"T. Goto, Y. Shindo, and H. Takahashi, Phys.",
"Rev.",
"B 56, 14019 (1997).",
"C. Thessieu, C. Pfleiderer, and J. Flouquet, Physica B 237-238, 467 (1997).",
"C. Pfleiderer and A. D. Huxley, Phys.",
"Rev.",
"Lett.",
"89,147005 (2002).",
"D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev.",
"Mod.",
"Phys.",
"77, 579 (2005).",
"A. Fernandez-Pañella, V. Balédent, D. Braithwaite, L. Paolasini, R. Verbeni, G. Lapertot, and J.",
"-P. Rueff, Phys.",
"Rev.",
"B 86, 125104 (2012).",
"J. Flouquet, A. Barla, R. Boursier, J. Derr, and G. Knebel, J. Phys.",
"Soc.",
"Jpn.",
"74, 178 (2005).",
"A. Miyake, K. Kasano, T. Kagayama, K. Shimizu, R. Takahashi, Y. Wakabayashi, T. Kimura, and T. Ebihara, J. Phys.",
"Soc.",
"Jpn.",
"82, 084706 (2013).",
"K. Kadowaki and S. B.",
"Woods, Solid State Commun.",
"58, 507 (1986).",
"Z. Wang, W. Mao, and K. Bedell, Phys.",
"Rev.",
"Lett.",
"87, 257001 (2001).",
"H. Hegger, C. Petrovic, E. G. Moshopoulou, M. F. Hundley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, Phys.",
"Rev.",
"Lett.",
"84, 4986 (2000).",
"S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite, and J. Flouquet, Nature 406, 587 (2000).",
"D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J.-P. Brison, E. Lhotel, and C. Paulsen, Nature 413, 613 (2001).",
"N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Klaasse, T. Gortenmulder, A. de Visser, A. Hamann, T. Görlach, and H. v. Löhneysen, Phys.",
"Rev.",
"Lett.",
"99, 067006 (2007).",
"Y. Muro, Y. Haizaki, M. S. Kim, K. Umeo, H. Tou, M. Sera, and T. Takabatake, Phys.",
"Rev.",
"B 69, 020401(R) (2004).",
"J. L. Sarrao, R. Modler, R. Movshovich, A. H. Lacerda, D. Hristova, A. L. Cornelius, M. F. Hundley, J. D. Thompson, C. L. Benton, C. D. Immer, M. E. Torelli, G. B. Martins, and Z. Fisk, and S. B. Oseroff, Phys.",
"Rev.",
"B 57, 7785 (1998).",
"M. Kasaya, T. Tani, K. Kawate, T. Mizushima, Y. Isikawa, and K. Kiyoo, J. Phys.",
"Soc.",
"Jpn.",
"60, 3145 (1991).",
"T. Mito, T. Koyama, M. Shimoide, S. Wada, T. Muramatsu, T. C. Kobayashi, and J. L. Sarrao, Phys.",
"Rev.",
"B 67, 224409 (2003).",
"H. Q. Yuan, M. Nicklas, Z. Hossain, C. Geibel, and F. Steglich, Phys.",
"Rev.",
"B 74, 212403 (2006).",
"S. K. Dhar, S. K. Malik, and R. Vijayaraghavan, J. Phys.",
"C: Solid State Phys.",
"14, L321 (1981).",
"A. L. Cornelius, A. K. Gangopadhyay, J. S. Schilling, and W. Assmus, Phys.",
"Rev.",
"B 55, 14109 (1997).",
"T. Nakamura, Y. H. Matsuda, J-L.",
"Her, K. Kindo, S. Michimura, T. Inami, M. Mizumaki, N. Kawamura, M. Suzuki, B. Chen, H. Ohta, K. Yoshimura, and A. Kotani, J. Phys.",
"Soc.",
"Jpn.",
"81, 114702 (2012).",
"Y. H. Matsuda, T. Inami, L. Ohwada, Y. Murata, H. Nojiri, Y. Murakami, H. Ohta, W. Zhang, and Y. Yoshimura, J. Phys.",
"Soc.",
"Jpn.",
"76, 034702 (2007).",
"H. Yamaoka, I. Jarrige, N. Tsujii, N. Hiraoka, H. Ishii, and K-D. Tsuei, Phys.",
"Rev.",
"B 80, 035120 (2009).",
"S. Watanabe, privation communication."
]
] | 1403.0290 |
[
[
"Detiding DART buoy data for real-time extraction of source coefficients\n for operational tsunami forecasting"
],
[
"Abstract U.S. Tsunami Warning Centers use real-time bottom pressure (BP) data transmitted from a network of buoys deployed in the Pacific and Atlantic Oceans to tune source coefficients of tsunami forecast models.",
"For accurate coefficients and therefore forecasts, tides at the buoys must be accounted for.",
"In this study, five methods for coefficient estimation are compared, each of which accounts for tides differently.",
"The first three subtract off a tidal prediction based on (1) a localized harmonic analysis involving 29 days of data immediately preceding the tsunami event, (2) 68 pre-existing harmonic constituents specific to each buoy, and (3) an empirical orthogonal function fit to the previous 25 hrs of data.",
"Method (4) is a Kalman smoother that uses method (1) as its input.",
"These four methods estimate source coefficients after detiding.",
"Method (5) estimates the coefficients simultaneously with a two-component harmonic model that accounts for the tides.",
"The five methods are evaluated using archived data from eleven DART buoys, to which selected artificial tsunami signals are superimposed.",
"These buoys represent a full range of observed tidal conditions and background BP noise in the Pacific and Atlantic, and the artificial signals have a variety of patterns and induce varying signal-to-noise ratios.",
"The root-mean-square errors (RMSEs) of least squares estimates of sources coefficients using varying amounts of data are used to compare the five detiding methods.",
"The RMSE varies over two orders of magnitude between detiding methods, generally decreasing in the order listed, with method (5) yielding the most accurate estimate of source coefficient.",
"The RMSE is substantially reduced by waiting for the first full wave of the tsunami signal to arrive.",
"As a case study, the five method are compared using data recorded from the devastating 2011 Japan tsunami."
],
[
"Introduction",
"To collect data needed to provide coastal communities with timely tsunami warnings, the National Oceanic and Atmospheric Administration (NOAA) has deployed an array of Deep-ocean Assessment and Reporting of Tsunamis (DART®) buoys at strategic locations in the Pacific and Atlantic Oceans (González et al., 2005; Titov et al., 2005; Spillane et al., 2008; Mofjeld, 2009).",
"When a tsunami event occurs, data from these buoys are analyzed at U.S. Tsunami Warning Centers (TWCs) using the Short-term Inundation Forecast for Tsunamis (SIFT) application (Gica et al., 2008; Titov, 2009).",
"The SIFT application was developed by the NOAA Center for Tsunami Research to rapidly and efficiently forecast tsunami heights at specific coastal communities.",
"SIFT compares DART® buoy data with precomputed models as one step in creating the forecast.",
"Matching precomputed models with data requires that any tidal components in the data be either removed or compensated for in some manner, an operation that we refer to as detiding.",
"To facilitate the operational needs of SIFT, detiding of data from DART® buoys must be done as soon as possible after the data become available.",
"In the course of developing the SIFT application, we have entertained multiple methods for detiding DART® buoy bottom pressure (BP) data nearly in real time.",
"In this paper we compare five such methods using archived data collected by eleven DART® buoys.",
"These buoys are deployed in both the Pacific and Atlantic Oceans in places with different tidal regimes.",
"Besides the dominant tides, the archived data contain other BP fluctuations of the kind that would be present during an actual tsunami event.",
"We take data from periods when no known significant tsunamis occurred and introduce an artificial tsunami signal.",
"Each signal is patterned after a precomputed computer model for an actual tsunami event.",
"The magnitude of the artificial event is controlled by a source coefficient $\\alpha $ .",
"We consider five different methods for extracting the artificial tsunami signal, each of which handles the tidal component in a different manner.",
"We assess how well each of the five detiding methods allows us to extract the known $\\alpha $ .",
"By repeating this scheme for many different combinations of DART® data and artificial tsunami signals, we can evaluate how well the five detiding methods work under idealized conditions (thus, while we make use of observed tidally-dominated data, we do not take into consideration confounding factors, an important one being a mismatch between the actual tsunami event and our model for it).",
"The remainder of the article is organized as follows.",
"In Sect.",
"we review the format of BP measurements from DART® buoys as received by U.S. TWCs.",
"We next describe construction of simulated tsunami events, which are formed by adding archived data from eleven representative buoys to associated models for tsunami signals (Sect. ).",
"We give details about the five methods for extracting tsunami signals in Sect. .",
"The first two are well-known methods based on harmonic models.",
"The next two are non-standard linear filters that utilize either empirical orthogonal functions or Kalman smoothing in conjunction with a local-level state space model.",
"The final method, which is based on a regression model that includes terms for both the tsunami signal and tidal components, proves to be the method of choice in study described in Sect. .",
"We compare the five methods using DART® buoy data collected during the devastating 2011 Japan tsunami in Sect. .",
"We state our conclusions and discuss our results in Sect.",
"."
],
[
"Bottom pressure measurements from DART",
"A DART® buoy actually consists of two units: a surface buoy and a unit located at the bottom of the ocean with a pressure recorder (González et al., 2005; NOAA Data Management Committee, 2008; Mofjeld, 2009).",
"The bottom unit stores measurements of water pressure integrated over non-overlapping 15-sec time windows, for a total of $60 \\times 4 = 240$ measurements every hour.",
"These internally recorded measurements only become fully available when the bottom unit returns to the surface and is recovered for servicing (time between servicings can be as long as two or three years).",
"We refer to the internally recorded data as the 15-sec stream.",
"Normally the buoy operates in standard reporting mode in which the bottom unit packages together one measurement every 15 minutes (a 60 fold reduction in data) over a 6-h block for transmission via acoustic telemetry up to the surface buoy once every 6 h. The surface buoy then relays the data up to the Iridium Satellite System for dissemination to the outside world.",
"We refer to data collected in standard reporting mode as the 15-min stream.",
"When a typical tsunami event occurs, the bottom unit detects a seismic event, which causes the DART® buoy to go into event reporting mode.",
"The transmission of the 15-min stream is suspended while the DART® buoy is in event reporting mode.",
"As part of this mode, the bottom unit averages together four consecutive 15-sec measurements and transmits these averages up to the surface buoy.",
"We refer to these data as the 1-min stream.",
"An additional 2 hours of 1-min data are transmitted on the hour during the event reporting mode.",
"When the outside world first gets access to the 1-min stream, there can be a gap between it and the most recent value of the 15-min stream ranging up to almost 6 h. After the on-hour data transmission an additional 1 to 2 hours of 1-min data before the event will be available.",
"If we assume that tsunami events commence at random within the 6-h reporting cycle for the 15-min stream, then the average size of this gap will be 3 h. The data that are thus typically available during a tsunami event are a portion of the 15-min stream prior to the event and a 1-min stream that becomes available piece-by-piece in real time as the event evolves and that contains at least 1 h of data occurring before the event."
],
[
"Scenarios and artificial tsunami signals",
"The goal of this paper is to objectively compare different detiding methods for extracting tsunami signals in near real-time.",
"A model for what is recorded by a DART® buoy during a tsunami event has three components: a tsunami signal, tidal fluctuations and background noise.",
"The latter is due to seismic, meteorological, measurement and other nontidal effects (Cartwright et al., 1987; Niiler et al., 1993; Webb, 1998; Cummins et al., 2001; Mofjeld et al., 2001; Zhao and Alford, 2009).",
"The purpose of detiding is to compensate for tidal fluctuations in the recorded data (Consoli et al., 2014).",
"Although DART® buoys have recorded a number of tsunami events, use of these data to evaluate different detiding methods is problematic: we ideally need to know the true tidal fluctuations, and these are not known to sufficient accuracy for actual data due to the presence of background noise.",
"One solution is to simulate each of the three components.",
"In combining these components to simulate tsunami events, we would then know the true tidal fluctuations and thus be in a position to quantify how well different detiding methods did; however, simulating tidal fluctuations and background noise in a manner that does not give an unfair advantage to certain detiding methods is tricky.",
"We can bypass simulation of both tidal fluctuations and background noise by making use of actual BP measurements from DART® buoys under the realistic assumption that significant tsunami signals are rare events.",
"In this paper we use archived 15-sec streams retrieved from eleven representative DART® buoys.",
"The buoys are in the Atlantic and Pacific Oceans and are listed in Table REF .",
"Their locations are displayed in Fig.",
"REF .",
"These particular buoys were chosen because the data they collect represent four well-known types of tides, hence offering the study of detiding over a broad range of ocean tidal conditions.",
"All data were obtained from NOAA's National Geophysical Data Center (NGDC; the reader is referred to Mungov, et al., 2012, for details about data collection and processing).",
"The units for the data archived by NGDC are in pounds per square inch absolute (psia), which we convert to water depth in meters by multiplication by $0.67$ m/psia (this conversion factor is based on a standard ocean, but its value does not impact any of the results we present).",
"As documented in Table REF , we have from 321 to 998 days of 15-sec streams for each of the eleven buoys.",
"We use these streams to construct `scenarios' mimicking the tidal fluctuations and background noise that might have been present in 15-min and 1-min streams available during an actual tsunami event.",
"To do so for a particular buoy, we start by selecting a random starting time $t_0$ .",
"The first part of the scenario associated with $t_0$ consists of a 29-day segment of a 15-min stream extracted by subsampling from the 15-sec stream immediately prior to $t_0$ .",
"To mimic typical operational conditions, we create a 3-h gap prior to $t_0$ by eliminating part of the constructed 15-sec stream.",
"Using data from the 15-sec stream occurring immediately after $t_0$ , we form a 1-day segment of a 1-min stream by averaging four adjacent values.",
"The constructed 15-min and 1-min streams constitute one scenario.",
"For each buoy we form a total of 1000 scenarios with starting times $t_0$ chosen at random without replacement from the set of all possible starting times (for example, as indicated in Table REF for buoy 21416, this set ranges from 29 days after 07/25/2007 up to 1 day prior to 07/01/2009).",
"Figure REF shows an example of one scenario.",
"(There are occasional missing values in the 15-sec streams archived by NGDC.",
"These can lead to gaps in a constructed 15-min stream in additional to the usual 3-h gap prior to $t_0$ .",
"All of the detiding methods in our study can handle these gaps.",
"If, however, the missing values lead to gaps in the constructed 1-min stream, we have elected to disregard the chosen $t_0$ and randomly pick a new one merely to simplify evaluation of the performance of the various detiding methods.",
"Thus, a given scenario can have gaps at arbitrary times in its 15-min stream, but none in its 1-min stream.)",
"With tidal fluctuations and background noise being handled using actual data from DART® buoys, we now turn our attention to simulating tsunami signals and combining these with the other two components to simulate the 1-min streams observed during a tsunami event.",
"Let $\\bar{\\bf y}$ be a vector containing a simulated 1-min stream observed during a tsunami event starting at time $t_0$ and lasting for one day (the bar over ${\\bf y}$ is a reminder that the 1-min stream is produced from four-point averages of the 15-s stream).",
"We model this vector as $\\bar{\\bf y}={\\bf x} + \\alpha _1 {\\bf g}_1 + \\cdots + \\alpha _K {\\bf g}_K+ {\\bf e},$ where ${\\bf x}$ , ${\\bf g}_1$ , ..., ${\\bf g}_K$ and ${\\bf e}$ are vectors representing, respectively, tidal fluctuations, components derived from $K\\ge 1$ unit sources (these collectively serve to model the tsunami signal) and background noise, while $\\alpha _1$ , ..., $\\alpha _K$ are $K$ nonnegative scalars known as source coefficients (Percival et al., 2011).",
"Forming the ${\\bf g}_k$ 's is discussed in detail in Titov et al.",
"(1999) and Gica et al.",
"(2008), and the following overview is taken from Percival et al. (2011).",
"Tsunami source regions are defined along subduction zones and other portions of oceans from which earthquake-generated tsunamis are likely to occur (see the solid curves on Fig.",
"REF ).",
"Each source region is divided up into a number of unit sources, each of which has a fault area of $100\\times 50$ $\\hbox{km}^2$ .",
"For each pairing of a given DART® buoy with a particular unit source, a model ${\\bf g}_k$ has been calculated predicting what would be observed at the buoy from time $t_0$ and onwards based upon the assumption that the tsunami event was caused by a reverse-thrust earthquake with standardized moment magnitude $M_W=7.5$ starting at $t_0$ and located within the unit source.",
"The source coefficient $\\alpha _k$ is used to adjust the standardized magnitude in the model to reflect the magnitude in an actual tsunami event.",
"The sum of the $\\alpha _k$ 's is a measure of the overall strength of the tsunami event and provides initial conditions for models that predict coastal inundation.",
"As discussed in Percival et al.",
"(2011), given $K$ candidate unit sources and based on varying amounts of data, a least squares procedure can be used to obtain statistically tractable estimates of the $\\alpha _k$ 's under the assumption that the tidal fluctuations ${\\bf x}$ are known to reasonable accuracy.",
"To focus on detiding, we assume a simplified version of Equation REF , namely, $\\bar{\\bf y}={\\bf x} + \\alpha {\\bf g} + {\\bf e};$ i.e., the simulated 1-min stream involves just a single unit source and a single source coefficient.",
"The scenarios for a given buoy are stand-ins for ${\\bf x} + {\\bf e}$ .",
"We take the unit source-based ${\\bf g}$ and multiply it by $\\alpha $ to form an artificial tsunami signal $\\alpha {\\bf g}$ .",
"From an examination of actual tsunami events, we set $\\alpha = 6$ as a representative source coefficient.",
"Figure REF shows an example of constructing a simulated tsunami event based upon the 1-min stream shown in Fig.",
"REF and a unit source chosen for DART® buoy 52402.",
"For each of the eleven buoys in our study, we picked from three to seven unit sources with different orientations with respect to the buoy – these choices are listed in Table REF and depicted as solid circles in Fig.",
"REF .",
"For DART® buoys near a subduction zone, we selected the closest unit source so that the buoy would be in the main beam of the radiated tsunami energy.",
"We then added two additional unit sources, bearing 30 to 45 degrees to either side of the first, to represent waves traveling obliquely toward the buoy.",
"In the case of DART® buoy 21416, which is representative of the northwest Pacific and proximate to both the Aleutian and Kamchatka source regions, we selected unit sources from both regions.",
"Buoy 32411 (located close to both Central and South America) has a similar geometry, so we chose three unit sources from Central America and three from South America.",
"For mid-ocean buoys (51406, 51407 and 44401), we selected unit sources from several subduction zones to which the buoy might respond.",
"There are 42 unit sources in all, 5 of which are used by two buoys, for a total of 47 pairings of buoys and unit sources.",
"Each pairing leads to a different artificial tsunami signal $\\alpha \\bf g$ .",
"Twelve representative signals are shown in Fig.",
"REF (here the coefficient $\\alpha $ is adjusted separately for each signal merely for plotting purposes – the actual range for all 47 $\\bf g$ 's is listed in Table REF and varies from 0.2 to 13.1 cm).",
"Each plot shows a 120-min segment of a given artificial tsunami signal $\\alpha \\bf g$ , one point for each minute, but a different segment for each signal.",
"For use later on, five hand-picked points are colored red.",
"From left to right, the first four of these points mark the approximate occurrence of a quarter, a half, three-quarters and all of the first full wave comprising the signal (Table REF lists the actual times associated with these points).",
"The final (right-most) point marks one hour past the end of the first full wave.",
"Each of the eleven buoys is represented once in Fig.",
"REF , except for buoy 32411, which has two signals associated with it that are visually quite different (left-hand and middle plots on second row).",
"One of these two depicted signals (cs027b, left-hand plot, second row) is generated from the same $100\\times 50$ $\\hbox{km}^2$ region as the signal depicted for 51406 (left-hand plot, bottom row).",
"Again these two signals are visually quite different.",
"These duplicate-buoy/duplicate-unit-source pairings illustrate the fact that each signal depends upon both the location of the buoy and the location of the unit source.",
"Figure REF shows 120-min segments of twelve simulated tsunami events, each of which make use of the artificial tsunami signals shown in corresponding plots of Figure REF .",
"These events were formed by adding $\\alpha \\bf g$ (with $\\alpha $ now set to 6) to the 1-min stream from a randomly chosen scenario for each buoy (the scenario chosen for buoy 52402 is the same one used in Fig.",
"REF – thus the left-most plot in the third row of Fig.",
"REF is a zoomed-in version of bottom plot of Fig.",
"REF ).",
"The vertical axis for each plot in Fig.",
"REF spans 1.2m.",
"It is easy to visually pick out the tsunami signal in some plots, but harder in others, which illustrates the fact that, even though $\\alpha =6$ in all cases, the signal-to-noise ratio varies substantially."
],
[
"Methods for handling tides in bottom pressure measurements",
"Here we describe five methods that take BP measurements $\\bar{\\bf y}$ and use them to estimate the source coefficient $\\alpha $ in Equation REF .",
"Each method deals with the tidal fluctuations $\\bf x$ in a different manner.",
"The first four methods do so by predicting or estimating the fluctuations using, say, $\\hat{\\bf x}$ .",
"This prediction is then subtracted from $\\bar{\\bf y}$ to yield detided BP measurements: ${\\bf d} = \\bar{\\bf y} - \\hat{\\bf x} = \\alpha {\\bf g} + {\\epsilon },$ where ${\\epsilon } = {\\bf e} + {\\bf x} - \\hat{\\bf x}$ is an error term encompassing both background noise and inaccuracies in predicting the tidal fluctuations.",
"Given the detided measurements, we then use the ordinary least squares (OLS) method to estimate $\\alpha $ , yielding the estimator $\\hat{\\alpha }= \\frac{{\\bf g}^T{\\bf d}}{{\\bf g}^T{\\bf g}},$ where ${\\bf g}^T$ denotes the transpose of the vector ${\\bf g}$ .",
"The fifth method is different from the other four in that it uses OLS to create $\\bf d$ and to estimate $\\alpha $ jointly.",
"The inversion algorithms for estimating source coefficients from multiple buoys currently in SIFT (Percival, et al., 2011) and under development (Percival, et al., 2014) are based on least squares methods, but go beyond OLS by including nonnegativity constraints and penalties to induce automatic unit source selection.",
"These algorithms assume the model of Equation REF for the multiple buoys, but proceed under the assumption that the BP measurements have been adjusted so that tidal fluctuations have been removed as much as possible.",
"For studying how best to deal with tides, it suffices to use a single buoy and the simplified model of Equation REF , and there is no real gain in using anything other than the OLS estimator of Equation REF .",
"The first two of the five methods are based on harmonic modeling, which we describe in Sect.",
"REF .",
"The third method is based on empirical orthogonal functions (Sect.",
"REF ), while the fourth employs Kalman smoothing, but makes use of harmonic modeling for initialization purposes (Sect.",
"REF ).",
"Sect.",
"REF describes the fifth method, which in part involves a simplified version of harmonic modeling."
],
[
"Harmonic modeling methods",
"The harmonic prediction method is the standard one used by NOAA to predict the tides at coastal stations.",
"For detiding DART® data, it can be thought of as the classic method of tidal prediction.",
"It assumes that the tides are sums of sinusoidal constituents, each with its own frequency.",
"To make tidal predictions at a DART® station, the amplitude and phase lag of each constituent are determined via a tidal analysis of observed BP data.",
"The first detiding method is a harmonic prediction method that carries out the tidal analysis in the following manner.",
"Consider one of the scenarios described in Section .",
"Each scenario consists of a one-day segment of a 1-min stream starting at time $t_0$ , which is preceded by 29 days (less 3 hours) of a 15-min stream.",
"Without loss of generality, set $t_0=0$ to simplify the discussion, and let $y_n$ denote an observation from the 15-min stream, but with $n$ indexing the underlying 15-sec stream from which the 15-min stream is subsampled; i.e., the actual time associated with $y_n$ is $n\\,\\Delta $ , where $\\Delta =15$ s. Assuming $y_n$ to be tidally dominated, we entertain a harmonic model of the form $y_n &=& \\mu + \\sum _{m=1}^M A_m \\cos (\\omega _m n\\,\\Delta - \\phi _m) + e_n \\\\&=& \\mu + \\sum _{m=1}^M \\left[ B_m \\cos (\\omega _m n\\,\\Delta ) + C_m \\sin (\\omega _m n\\,\\Delta )\\right] + e_n $ for $n = -780, -840, \\ldots , -166920, -166980$ (note that index $n = -780$ corresponds to the last value in the observed 15-min stream, which occurs prior to the 3-h gap, while $n = -166980$ indexes the first value, which occurs 29 days prior to $t_0=0$ ).",
"In the above $\\mu $ is an unknown overall mean level; $\\omega _m$ is one of $M$ known tidal frequencies; $B_m$ and $C_m$ are unknown coefficients that can be used to deduce the amplitudes $A_m$ and phase lags $\\phi _m$ ; and $e_n$ is a residual term (hopefully small).",
"We use an OLS fitting procedure to estimate $\\mu $ , $B_m$ and $C_m$ via, say, $\\hat{\\mu }$ , $\\hat{B}_m$ and $\\hat{C}_m$ .",
"If we replace $\\mu $ , $B_m$ and $C_m$ by their estimates, we can then use the right-hand side of Equation with the residual term set to zero to predict what the tidal fluctuation should be at any desired time index $n$ .",
"After an artificial tsunami signal is added to the 1-min stream of the scenario to form a tsunami event, we detide this stream by forming $d_n =\\bar{y}_n - \\hat{\\mu }-\\frac{1}{4} \\sum _{k=0}^3 \\sum _{m=1}^M\\left[ \\hat{B}_m \\cos (\\omega _m [n+k]\\,\\Delta ) + C_m \\sin (\\omega _m [n+k]\\,\\Delta ) \\right]$ for $n=0, 4, \\ldots , 5756, 5760$ , where $\\bar{y}_n$ is an element from the 1-min stream $\\bar{\\bf y}$ .",
"For 29 days of prior data, we make use of $M=6$ tidal frequencies $\\omega _1$ , ..., $\\omega _6$ commonly referred to as N2, M2, S2, Q1, O1 and K1 (see, e.g., Table 1 in Ray and Luthcke, 2006).",
"The $d_n$ 's given above form the elements of the vector $\\bf d$ used to form the estimator $\\hat{\\alpha }$ of Equation REF .",
"The first row of Fig.",
"REF shows detided BP measurements $d_n$ corresponding to the simulated tsunami event shown in the bottom plot of Fig.",
"REF .",
"Visually there is evidence that this detiding is not entirely satisfactory: there is a systematic drift downwards over the first hour that arguably is due to tidal fluctuations and thus should not be present in $d_n$ .",
"As described in the caption to the figure, the five black circles mark various time points associated with the artificial tsunami signal.",
"If we place the data from $t_0$ up to one of these five time points into the vector $\\bf d$ and create the corresponding vector $\\bf g$ , we can obtain an estimate of the source coefficient $\\alpha $ using Equation REF .",
"The resulting estimates $\\hat{\\alpha }$ are listed in the first row of Table REF .",
"Recalling that the true value of $\\alpha $ is 6, we see that, not unexpectedly, the better estimates are associated with larger amounts of data.",
"Looking at estimates based on varying amounts of data is of considerable operational interest.",
"As more BP measurements become available as a tsunami event evolves, we can expect in general to get better estimates of $\\alpha $ , but at the expense of a delay in issuing timely warnings.",
"Determining how much an estimate of $\\alpha $ is likely to improve by waiting for more data is vital for managing the trade-off between accuracy and timeliness.",
"For this example, there is improvement in waiting until the first full wave occurs, but none in waiting an hour past that time.",
"The second detiding method is based on a harmonic analysis that, for a given buoy, is based on all the 15-s data listed for it in Table REF .",
"This `blanket' harmonic model has the same form as Equation REF , but now the time index $n$ increases in steps of one rather than 60, and we use $M=68$ sinusoidal constituents.",
"For optimal accuracy of this type of an analysis, the measurements should span at least one year, which Table REF indicates is true for all buoys with the exception of 41420 (this buoy has 321 days of data, slightly less than a year).",
"The tidal predictions are made by adjusting the lunar harmonic constants for perigean (8.85-year) and nodal (18.6-year) variations in the moon's orbit, computing the height associated with each constituent at the times of interest, and then summing these heights to yield the prediction.",
"Detiding of the 1-min series in a scenario is accomplished using an equation similar to Equation REF , the only difference being that the overall mean level $\\mu $ is estimated using the scenario's 15-min stream rather than being pre-specified.",
"A particular strength of the second detiding method is that it requires minimal use of the 15-min stream (it is only used to estimate $\\mu $ ).",
"In one extreme case, this stream was entirely missing for buoy 21416 over a six-week period prior to 1/15/2009, when a Kuril Islands event triggered reporting of the 1-min stream.",
"If the 15-min stream is not available, it is possible to estimate $\\mu $ using just the 1-min stream, but care would be needed to ensure that the tsunami signal does not unduly distort the estimate.",
"A number of software packages are available to do tidal analyses of observations and to make the tidal predictions, a standard one being the Foreman FORTRAN 77 package (Foreman, 1977, revised 2004).",
"For our study, we used tidal predictions generated by NGDC (see Mungov et al., 2012, for details).",
"The second row of Fig.",
"REF shows detided BP measurements $d_n$ produced by this second harmonic method.",
"In contrast to the first method, we no longer see a systematic drift downwards over the first hour; however, the $d_n$ 's during that hour are systematically elevated above zero, which is questionable.",
"The five estimates $\\hat{\\alpha }$ corresponding to different amounts of data are listed in the second row of Table REF .",
"These estimates are worse than the ones we obtained from the first harmonic method except when using the largest amount of data; however, we shouldn't rely on this single example to draw conclusions about the relative merits of the two harmonic methods – see the discussion in Section .",
"When tidal predictions are subtracted from BP measurements, fluctuations always remain in the tidal bands.",
"They are due to non-stationary fluctuations, non-linear tides, and tidal constituents not accounted for in the tidal analysis.",
"Of these, internal tides are certainly significant.",
"They are generated around the ocean margins and shallow ridges and then propagate elsewhere in the ocean (e.g., Cummins et al., 2001; Zhou and Alford, 2009).",
"The residual tides limit the degree to which the total tide can be removed from BP data through simple subtraction of a predicted tide, even when the tidal analysis is performed on the same time series (as is the case here)."
],
[
"Empirical orthogonal function method",
"The two best known approaches for detiding data are to subtract off a prediction from a harmonic model (as described in the previous section) and to subject the data to a linear time-invariant (LTI) high-pass filter.",
"To isolate a tsunami signal without distortion, the high-pass filter should retain components with periods as long as 2 hrs.",
"The so-called `edge effects' of such a filter distort at least 1-hr sections at the beginning and at the end of the tsunami signal.",
"Thus an LTI filter cannot reliably isolate a tsunami signal immediately after it is registered by a DART® buoy, whereas it is desirable to make use of this data as soon as possible following the start of a tsunami event.",
"Moreover, most digital filters are designed to work with regularly sampled data and are not easy to adapt if there are gaps in the data.",
"While the BP unit in a DART® buoy internally records a measurement once every 15 s, only gappy segments of these data (or 1-min averages thereof) are typically available externally following an earthquake event.",
"In this section and the next, we explore two approaches to detiding involving linear (but not time-invariant) filters.",
"These approaches are tolerant of data gaps and are less prone to edge effects, thus overcoming the two problems we noted about standard LTI high-pass filters.",
"The first approach is based on extracting the tidal component in a segment of data by back and forth projection onto a specific sub-space in an $N$ -dimensional space of vectors.",
"The sub-space is spanned by the empirical orthogonal functions (EOFs) of segments of length $N$ of archived 15-min streams from DART® buoys (Tolkova, 2009; Tolkova, 2010).",
"The following description of this approach is based on Tolkova (2010), to which we refer the reader for more details.",
"The EOF method relies on the premise that, due to the structure of tides in the deep ocean, the sub-space spanned by the leading EOFs of tidally dominated data segments of up to 3-days in length is fairly universal across various DART® buoys.",
"Oceanic tidal energy is concentrated in the long-period, diurnal, and semidiurnal frequency bands centered around 0, 1, and 2 cycles per day (cpd).",
"The effective diurnal band is from 0.8 to 1.1 cpd, and the semidiurnal, from 1.75 to 2.05 cpd, so the bandwidth for both bands is 0.3 cpd (Munk and Cartwright, 1966).",
"Tidal motion thus has two inherent time scales: one day (the apparent tidal quasi-period) and 3.3 days (the shortest tidal segment from which we can in theory resolve individual constituents within either of the two major bands).",
"Since tides at different locations differ only in the fine structure of the tidal bands, the premise behind the EOF method says that the sub-space spanned by the leading EOFs of tidal segments is essentially the same at all DART® locations as long as the segment length is so short as to not allow resolution of the fine structure in the tidal bands.",
"Tolkova (2010) computed the basis of the EOF sub-space of tidally dominated data encompassing one lunar-day (24 h 50 min) using an ensemble of 250 segments taken from DART® buoy 46412 in 2007.",
"Each segment consists of 99 readings sampled at 15 min intervals.",
"The EOFs associated with the seven largest eigenvalues, complemented with a constant vector with elements all equal to $1/\\surd 99$ , provide the basis for a sub-space spanned by $M=8$ vectors of dimension 99.",
"This sub-space is sufficient to capture the bulk of tidal variations in segments of length 99 from a 15-min stream recorded by any DART® buoy.",
"To process 1-min streams within the SIFT system, the $M$ vectors were re-sampled to a 1-min interval, yielding vectors of length $N=15\\times 98+1=1471$ .",
"Re-orthogonalization and re-normalization produces vectors ${\\bf f}_m$ , $m=0,1, \\ldots , M-1$ , satisfying ${\\bf f}^T_m{\\bf f}_n=0$ for $m\\ne n$ and $=1$ for $m=n$ .",
"Figure REF shows the resulting eight vectors.",
"Three steps are needed to accomplish detiding using the ${\\bf f}_m$ vectors.",
"For the sake of argument, suppose the vector $\\bar{\\bf y}$ contains a segment from a 1-min stream of length $N=1471$ (i.e., approximately one lunar day).",
"First, we project this segment onto the vectors ${\\bf f}_m$ to obtain $M$ coefficients $c_m = {\\bf f}_m^T \\bar{\\bf y}$ .",
"Second, we estimate the tidal component by taking the $M$ vectors, multiplying them by their corresponding coefficients and then adding together the resulting scaled vectors.",
"Finally, this estimate of the tidal component is subtracted from $\\bar{\\bf y}$ , yielding the detided data $\\bf d$ .",
"Mathematically, we can write ${\\bf d}=\\bar{\\bf y} - F{\\bf c}\\hbox{with}{\\bf c} = F^T \\bar{\\bf y},$ where $F$ is a $N \\times M$ matrix whose columns are ${\\bf f}_0, \\dots , {\\bf f}_{M-1}$ , and ${\\bf c}$ is a vector containing the $M$ coefficients.",
"Letting $\\bar{y}_n$ and $F_{m,n}$ denote, respectively, the $n$ th element of $\\bar{\\bf y}$ and the $(m,n)$ th element of $F$ and momentarily regarding $\\sum _{n=0}^{N-1} \\left( \\bar{y}_n - \\sum _{m=0}^{M-1} c_m f_{m,n} \\right)^2$ as a function of $c_0, \\ldots c_{M-1}$ , we note that setting $c_m$ equal to ${\\bf f}_m^T \\bar{\\bf y}$ results in minimizing the above sum of squares.",
"We can generalize this technique to handle the case of irregular sampling by the following simple procedure, which ignores the distinction between the 1-min and 15-min streams.",
"For a span of $N=1471$ minutes of interest, construct a vector $\\tilde{\\bf y}$ of length $N$ that contains all available values from either one of the streams.",
"Let $w_n=1$ if the $n$ th element $\\tilde{y}_n$ of $\\tilde{\\bf y}$ is actually available from one of the streams, and let $w_n=0$ if it is not available.",
"We set $\\bf c$ such that $\\sum _{n=0}^{N-1} w_n \\left( \\tilde{y}_n - \\sum _{m=0}^{M-1} c_m f_{m,n} \\right)^2$ is minimized with respect to $c_0, \\ldots , c_{M-1}$ .",
"Minimization of (REF ) is a least squares problem whose associated normal equations are $H^T H {\\bf c}=H^T \\tilde{\\bf y},$ where $H$ is an $N \\times M$ matrix whose $(n,m)$ th element is $w_n f_{m,n}$ .",
"The system (REF ) has a numerically viable solution as long as the symmetric $M \\times M$ matrix $H^TH$ is not poorly conditioned.",
"In this case the detided data are contained in ${\\bf d}= \\tilde{\\bf y} - H{\\bf c}\\hbox{with}{\\bf c}= (H^TH)^{-1}H^T\\tilde{\\bf y}.$ The third row of Fig.",
"REF shows detided BP measurements $d_n$ produced by the EOF method.",
"In contrast to the two methods based on harmonic analysis, the values that are subtracted from the 1-min stream to accomplish EOF detiding depend upon the stream itself and hence can change as different amounts of this stream are utilized, as this example illustrates.",
"In particular we see a systematic upward drift when utilizing data less than or equal to the first full wave, but this drift disappears when we use data one hour past the end of the first full wave.",
"The five estimates $\\hat{\\alpha }$ corresponding to different amounts of data are listed in the third row of Table REF .",
"For the three largest amounts of data, these estimates are an improvement in this example over the ones we obtained from the two harmonic methods."
],
[
"Kalman smoothing method",
"Kalman smoothing (KS) has been used in numerous applications as a method for optimally smoothing time series as new values of the series become available over time.",
"The optimality of this procedure is contingent upon our ability to adequately describe the underlying dynamics of the time series in terms of a so-called state space model.",
"KS-based detiding has been advocated before (see Consoli et al., 2014, and references therein), but the approach we describe here for detiding data from DART® buoys differs from previous approaches in important aspects (for detailed expositions on KS, see Brockwell and Davis, 2002; Durbin and Koopman, 2012; Shumway and Stoffer, 2011).",
"Our KS approach is a two-stage procedure.",
"As before, let $\\bar{y}_n$ represent the 1-min stream from a given scenario to which we have added an artificial tsunami signal.",
"The first stage is to detide $\\bar{y}_n$ using the first harmonic modeling method, yielding the detided series $d_n$ , $n= 0, 4, 8, \\ldots $ , via Equation REF .",
"Merely to simplify equations that follow, we reindex this first-stage detided series by defining $\\tilde{d}_n = d_{4n}$ , $n=0,1,2,\\ldots $ .",
"The second stage applies KS to $\\tilde{d}_n$ , for which we assume a local level model (also called a random walk plus noise model; see Brockwell and Davis, 2002; Durbin and Koopman, 2012).",
"This model consists of two equations, the first of which is known as the state equation, and the second, the observation equation.",
"The state equation takes the form $\\mu _{n+1} = \\mu _n + \\zeta _n,\\quad n=0,1,2,\\ldots ,$ where $\\mu _0$ is the initial state variable, and $\\zeta _n$ is a white noise sequence with mean zero and variance $\\sigma ^2_\\zeta $ .",
"The observation equation takes the form $\\tilde{d}_n = \\mu _n + \\delta _n,\\quad n=0,1,2,\\ldots ,$ where $\\delta _n$ is another white noise sequence with mean zero, but now with time-varying variance $\\sigma ^2_{\\delta ,n}$ (the sequences $\\delta _n$ and $\\zeta _n$ are uncorrelated).",
"The intent with this model is to use $\\mu _n$ to track any tidal component left over in the first-stage detiding $\\tilde{d}_n$ and to compensate for the presence of the tsunami signal by adjusting $\\sigma ^2_{\\delta ,n}$ appropriately.",
"Based upon the initial state variable $\\mu _0$ , the first-stage detided data $\\tilde{d}_0, \\ldots , \\tilde{d}_n$ and the parameters $\\sigma ^2_\\zeta $ and $\\sigma ^2_{\\delta ,n}$ , there is an elegant set of equations known as the Kalman recursions that give the best (in the sense of minimum mean square error) linear estimates of the unknown state variables $\\mu _1, \\ldots , \\mu _n$ .",
"We denote these estimates as $\\hat{\\mu }_{1\\vert n}, \\ldots , \\hat{\\mu }_{n\\vert n}$ .",
"For a fixed index $m\\le n$ , the estimate $\\hat{\\mu }_{m\\vert n}$ of $\\mu _m$ changes as $n$ increases, i.e., as more and more of the first-stage detided series becomes available.",
"Given $\\tilde{d}_0, \\ldots , \\tilde{d}_n$ , we define the KS-based detided series to be $\\hat{\\delta }_m = \\tilde{d}_m - \\hat{\\mu }_{m\\vert n},\\quad m = 0, 1, \\ldots , n.$ The corresponding estimate of the source coefficient $\\alpha $ is given by Equation REF , where now the vector $\\bf d$ contains the $\\hat{\\delta }_m$ variables.",
"The unknown parameters that we must set to implement KS-based detiding are $\\mu _0$ , $\\sigma ^2_{\\delta ,n}$ and $\\sigma ^2_\\zeta $ .",
"For purposes of this paper, we just use $\\tilde{d}_0$ to estimate $\\mu _0$ ; however, to offer some protection against rogue values, the operational version of SIFT has an option for using the median of $\\tilde{d}_0, \\ldots , \\tilde{d}_4$ .",
"Our estimate $\\hat{\\sigma }^2_{\\delta ,n}$ of $\\sigma ^2_{\\delta ,n}$ is the sample variance of the seven variables $\\tilde{d}_{n-3}, \\ldots , \\tilde{d}_{n+3}$ , with estimates for the first three variances $\\sigma ^2_{\\delta ,0}$ , $\\sigma ^2_{\\delta ,1}$ and $\\sigma ^2_{\\delta ,2}$ being set to $\\hat{\\sigma }^2_{\\delta ,3}$ (estimates for the last three variances are handled in an analogous manner).",
"We determined a setting for the final parameter $\\sigma ^2_\\zeta $ in the following manner.",
"For all 1000 scenarios for a given pairing of a buoy and an artificial tsunami signal with the source coefficient $\\alpha =6$ , we used the KS detiding method to compute source coefficient estimates $\\hat{\\alpha }_i$ , $i=1,\\ldots , 1000$ , over a grid of values for $\\sigma ^2_\\zeta $ and for various amounts of data.",
"We then determined which value minimized $\\sum _i ( \\hat{\\alpha }_i - 6)^2$ .",
"Different pairings of buoys and signals and different data lengths led to different minimizing values.",
"We set $\\sigma ^2_\\zeta =6.25\\times 10^{-13}$ after considering a large collection of representative pairings and data lengths – while this value was not optimal for all such pairings and lengths, it performed well overall.",
"The fourth row of Fig.",
"REF shows detided BP measurements $\\hat{\\delta }_n$ produced by the KS method.",
"The starting point for this method is the detided series shown in the first row, which was produced by the first harmonic modeling method.",
"A comparison of these two detided series shows that the KS method has eliminated the downward trend that is evident in the first seventy minutes of the detrended series produced by harmonic modeling.",
"In contrast to the first three methods, there is no evidence in the KS detided series of a drift or offset that might be attributable to lingering tidal fluctuations.",
"Similar to EOF detiding, KS detided values can change as different amounts of the 1-min stream are utilized, but the changes in the KS method are smaller than those for the EOF method in this example.",
"The five KS-based estimates of $\\hat{\\alpha }$ corresponding to different amounts of data are listed in the fourth row of Table REF .",
"For this example, these estimates are closer to the true value $\\alpha =6$ than the ones corresponding to the first three methods."
],
[
"Harmonic modeling method with joint source coefficient estimation",
"The four detiding methods we have considered so far are similar in that they all produce a detided series $\\bf d$ .",
"We then use $\\bf d$ to produce an estimate $\\hat{\\alpha }$ of the source coefficient via the OLS estimator of Equation REF .",
"The fifth and final method estimates the tidal component jointly with the source coefficient based on just the 1-min stream (merely to simplify the description below, we assume this stream to be gap-free, but it is easy to reformulate this method to handle gaps).",
"This joint estimation method is based on the model $\\bar{\\bf y} = {\\mu }{\\bf 1}+ \\sum _{m=1}^2 \\left(B_m {\\bf {c}}_m + C_m {\\bf {s}}_m\\right)+ \\alpha {\\bf g} + {\\bf e},$ where $\\bar{\\bf y}$ is an $N$ dimensional vector containing a portion of the 1-min stream from a scenario to which the artificial tsunami signal $6{\\bf g}$ has been added; ${\\bf 1}$ is a vector of ones; ${\\bf {c}}_m$ is a vector whose elements are $\\cos \\,(\\omega _m n\\,\\Delta )$ , $n=0,\\ldots ,N-1$ , with $\\omega _2$ and $\\omega _1$ being, respectively, the tidal frequency M2 and half that frequency and with $\\Delta = 1$ min; ${\\bf {s}}_m$ is analogous to ${\\bf {c}}_m$ , but with sines replacing cosines; $\\bf e$ is a vector of errors (presumed to have mean zero and a common variance); and $\\mu $ , $B_1$ , $C_1$ , $B_2$ , $C_2$ and $\\alpha $ are unknown parameters.",
"In essence, this method estimates the tidal component via a two-constituent harmonic model (as alternatives to this model, we also considered harmonic models with other than two constituents and polynomial models of various orders, but the two-constituent harmonic model worked best overall for joint estimation of $\\alpha $ ).",
"Equation REF can be rewritten as $\\bar{\\bf y} = X{\\beta } + {\\bf e},$ where $X$ is an $N\\times 6$ design matrix whose columns are ${\\bf 1}$ , ${\\bf {c}}_1$ , ${\\bf {s}}_1$ , ${\\bf {c}}_2$ , ${\\bf {s}}_2$ and ${\\bf g}$ , while ${\\beta } = (\\mu , B_1, C_1, B_1, C_1, \\alpha )^T$ is a vector of coefficients.",
"The OLS estimate $\\hat{\\beta } = (\\hat{\\mu }, \\hat{B}_1, \\hat{C}_1, \\hat{B}_1, \\hat{C}_1, \\hat{\\alpha })^T$ of ${\\beta }$ satisfies the normal equations $X^T X \\hat{\\beta } = X^T \\bar{\\bf y}\\hbox{and hence}\\hat{\\beta } = (X^T X)^{-1} X^T \\bar{\\bf y},$ subject to the invertibility of $X^T X$ (in the study discussed in Sect.",
", no instances of non-invertibility were encountered).",
"We can take the detided series for this method to be ${\\bf d} = \\bar{\\bf y} - \\hat{\\mu }{\\bf 1}- \\sum _{m=1}^2 \\left(\\hat{B}_m {\\bf {c}}_m + \\hat{C}_m {\\bf {s}}_m\\right).$ A strength of this detiding method is that, in contrast to the other four methods, it does not make any use of the 15-min stream, which, as we've noted before, was entirely missing in one actual tsunami event.",
"The fifth row of Fig.",
"REF shows detided BP measurements ${\\bf d}$ produced by the joint estimation method, with the five estimates of $\\hat{\\alpha }$ corresponding to different amounts of data being listed in the fifth row of Table REF .",
"Although in this example the detided series for this and the KS method are visually similar to each other, the joint estimation-based $\\hat{\\alpha }$ estimates are always closer to the true value of $\\alpha =6$ than the KS-based estimates."
],
[
"Comparison of five detiding methods",
"Here we compare the five detiding methods described in the previous section by considering how well each method estimates the source coefficient $\\alpha $ from simulated tsunami events constructed as per Equation REF (an example of one such event is shown in Fig.",
"REF ).",
"For each of the 47 buoy/unit source pairings listed in Table REF , we constructed 1000 simulated events based upon the 1000 scenarios created for each buoy.",
"For each such event and for five different amounts of data from the 1-min stream ranging from a quarter of the first full wave (1/4 FFW) of the tsunami signal up to an hour past the end of the FFW, we estimated $\\alpha $ using the five detiding methods, thus yielding $47\\times 1000\\times 5=235,000$ estimated coefficients for each method.",
"The task at hand is to summarize how well each method did.",
"We start by considering results for the pairing of buoy 52402 with unit source ki060b (this same combination is used in all or part of Figs.",
"REF to REF and in Table REF ).",
"Figure REF has five rows, one for each of the five methods.",
"The dots in a given row show the 1000 estimates $\\hat{\\alpha }$ derived from data just up to 3/4 FFW.",
"Ideally we would like to see estimates that cluster tightly around the true value $\\alpha = 6$ (indicated by a blue dashed line).",
"The scatter in the estimates for the two methods based on harmonic analysis (top two rows) is much larger than that for the three remaining methods.",
"With only a few exceptions, this pattern persists for all 47 buoy/unit source pairings and for all five amounts of data and tells us that these two methods are not competitive with the other methods.",
"The distribution of the estimates in each row of Fig.",
"REF is summarized on the right-hand side by a boxplot (Chambers et al., 1983).",
"The central box in each boxplot has three horizontal lines, which indicate, from bottom to top, the lower quartile, the median and the upper quartile of the data.",
"The short horizontal line below the central box is the lower hinge, which indicates the estimate $\\hat{\\alpha }$ that is closest to – but not less than – the value of the lower quartile minus 1.5 times the interquartile distance (the upper quartile minus the lower quartile).",
"The upper hinge has a similar interpretation, with `not less than' replaced by `not greater than' and with `lower quartile minus' replaced by `upper quartile plus'.",
"Any estimates $\\hat{\\alpha }$ that happen to be either smaller than the lower hinge or greater than the upper hinge are indicated by circles.",
"Because the variability in the estimates for the first two methods is so much greater than those for the three remaining methods, only the boxplots for the former are clearly visible in Fig.",
"REF .",
"Figure REF shows boxplots for the just latter three methods on a common scale, but now for all five amounts of data rather than just data up to 3/4 FFW.",
"These boxplots show that the EOF estimates tend to be biased low when using data less than or equal to 3/4 FFW, whereas the KS estimates are biased high, but to a lesser degree (the boxplots show medians, but there would be no noticeable differences had we shown sample means rather medians in Fig.",
"REF ).",
"By contrast there is little evidence of bias in the joint-estimation estimates.",
"For estimates based on data up to the FFW or smaller amounts, the spreads of the distributions are generally greatest for the EOF estimates and smallest for the joint-estimation estimates.",
"Figure REF summarizes the spreads in the distributions involved in Figs.",
"REF and REF via root-mean-square errors (RMSEs), i.e., $\\surd (\\sum _i(\\hat{\\alpha }_i-6)^2/1000$ ).",
"By this measure the two methods based on harmonic analysis are about an order of magnitude worse than the best method (joint estimation).",
"With almost no exceptions, this poor RMSE performance persists through all 47 buoy/unit source pairings and all five amounts of data under study.",
"The joint estimation method outperforms the EOF and KS methods, but the former becomes competitive when the largest amount of data is used, and the latter, for the smallest and two largest amounts.",
"Increasing the amount of data from 1/4 FFW to FFW results in approximately half an order of magnitude drop in RMSE for all five methods (Table REF indicates that the time it takes to collect the extra data from 1/4 FFW to FFW is 14 min for this particular buoy/unit source pairing).",
"Figure REF is similar to Fig.",
"REF , but now shows RMSE plots for the twelve representative buoy/unit source pairings shown in Figs.",
"REF and REF .",
"To simplify this figure, we do not show results for the two non-competitive methods based upon harmonic analysis.",
"The joint estimation method tends to outperform the EOF and KS methods when more than 1/4 FFW of data is involved, but not uniformly so (the pairing of buoy 51407 with unit source cs100b in the middle plot on the bottom row offers a counterexample).",
"The KS method generally outperforms the EOF method.",
"There are a number of instances in which RMSE increases for the EOF and KS methods when the amount of data increases from 1/4 FFW to 1/2 FFW.",
"This pattern is counterintuitive since more data should imply a more stable estimate of $\\alpha $ ; however, this behavior is confirmed by an analytic theory in which the tsunami's partial waves are regarded as a filter on the tides.",
"Figure REF has five plots, one for each amount of data under study, with points indicating ratios of RMSEs involving all 47 twelve buoy/unit source pairings.",
"The ratios are formed by taking the RMSE for either the EOF, KS or joint estimation method and dividing it by the best RMSE amongst all five detiding methods.",
"A ratio of one for a particular method indicates that it is the best method.",
"The top plot is for 1/4 FFW and shows the KS and joint estimation methods about evenly divided for best-method honors.",
"The buoys are ordered such that those that are separated most in distance from their unit sources (51406, 51407 and 44401) are on the right-hand side of the plots.",
"The KS method generally outperforms the joint estimation method for these buoys, which tend to have lower signal-to-noise ratios than the buoy/unit source pairings on the left-hand side of the plot.",
"As the amount of data increases beyond 1/4 FFW (four bottom plots), there are increasingly fewer pairings where the joint estimation method fails to be the method of choice.",
"We also note that the disparity amongst the three methods tends to decrease as the amount of data increases.",
"Finally, we note that, as alternatives to RMSE as a summary measure, we also considered the mean absolute error $\\sum _i|\\hat{\\alpha }_i-6|/1000$ , the maximum absolute error $\\max \\lbrace |\\hat{\\alpha }_i-6|\\rbrace $ and the maximum absolute negative error, i.e., $\\max \\lbrace |\\hat{\\alpha }_i-6|\\rbrace $ with $i$ ranging over values such that $\\hat{\\alpha }_i < 6$ .",
"These last two measures are based on worst-case scenarios, which are of considerable operational interest; in particular, the last measure focuses on worst-case underestimation of $\\alpha $ , which can lead to forecasting coastal inundations that are too small.",
"All three additional measures lead to the same conclusions we drew using the RMSE: the joint estimation method is generally to be preferred over the EOF and KS methods, and the two methods involving only harmonic analyses are not competitive."
],
[
"Example: March 2011 Japan tsunami",
"Here we apply the five detiding methods to data collected during the devastating 2011 Japan tsunami, which was generated by the great Mw 9.0 earthquake that occurred on March 3rd at 05:46:23 UT ($t_0$ ) (Tang et al., 2012; Wei et al., 2012; Wei et al., 2014).",
"Several DART® buoys recorded this event, including buoy 52402.",
"Figure REF shows the data from 52402 we make use of here.",
"The data consist of a 15-min stream starting 29 days prior to $t_0$ and ending 2.0 hrs before $t_0$ , and a 1-min stream starting 1.8 hrs prior to $t_0$ and ending a day after $t_0$ (there is a small gap in the 15-min stream around 14 days prior to $t_0$ ).",
"The first method uses a 6 constituent harmonic analysis based on the 29 days of data prior to the event time $t_0$ (these are mostly from the 15-min stream, but there are 1.8 hours from the 1-min stream).",
"The top plot in Fig.",
"REF shows the corresponding detided series, which is the difference between data from the 1-min stream recorded after $t_0$ and predicted values based on the fitted harmonic model.",
"This series shows rapid fluctuations starting about 10 min after $t_0$ and dissipating after about 90 min.",
"These are evidently due to seismic noise from the earthquake.",
"Ignoring this noise, the detided series starts at a positive intercept and then rises almost linearly until about 210 minutes, at which point the tsunami signal becomes evident.",
"The five black circles are subjectively chosen markers of a quarter of the first full wave (FFW), a half, three quarters, the end of the FFW and one hour past the end of the FFW (these occur at, respectively, 225, 232, 236, 245 and 305 min after $t_0$ ).",
"The true tidal fluctuations are of course unknown, but a reasonable conjecture is that the linear increase in evidence here is actually a tidal component that the first method failed to properly extract.",
"The second method uses a 68 constituent harmonic analysis based on 465 days of data collected by buoy 52402 between 12/13/2006 and 3/21/2008, i.e., well before the 2011 Japan tsunami.",
"The detided series, shown below the top plot in Fig.",
"REF , is again the difference between data from the 1-min stream and predicted values based on the fitted harmonic model, but with the mean level of the predictions adjusted using the 29 days of data prior to $t_0$ .",
"In comparison to the first method, there is now only a slight linear increase in the first three hours, but the positive intercept is larger.",
"The third method is EOF-based and yields detided series that depend on the amount of the 1-min stream after $t_0$ we wish to detide.",
"The middle plot in Fig.",
"REF shows five detided series color coded to indicate the amount of data used (red, green, cyan, magenta and black for data ending at, respectively from left to right, the locations of the five solid circles, and starting 1470 minutes earlier).",
"Differences between the displayed detided series at the same location in time are typically small, but do get as large as 5.7 cm.",
"Ignoring seismic noise, all five detided series are relatively flat out to 150 min, after which they exhibit a noticeable dip prior to the arrival of the tsunami signal.",
"While we might be tempted to regard this dip as a remanent tidal component, caution is in order since true tidal fluctuations are unknown.",
"The fourth method – Kalman smoothing – is similar to the EOF method in that the detided series depends on the amount of the 1-min stream to be detided, but is dissimilar in that detiding involves data prior to $t_0$ only indirectly because the starting point for the KS method is the detided series provided by the first method.",
"The next-to-bottom plot in Fig.",
"REF shows five detided series color coded in the same manner as before.",
"The differences between the detided series at the same point in time are again typically small, with the largest difference now being 0.7 cm.",
"There is only a slight hint here of the dip readily apparent in the EOF detided series.",
"Similar to the EOF and KS methods, the final method (a 2 constituent local harmonic analysis with joint source coefficient estimation) yields detided series that depend upon the amount of the 1-min stream to be detided, but, in contrast, it makes no direct or indirect use of any data prior to $t_0$ .",
"This method, however, does depend upon a suitable model for the tsunami signal.",
"Percival et al.",
"(2014) discuss selection of unit sources for the Japan tsunami using an objective automatic method.",
"The selection is based on data from three DART® buoys (21401, 21413 and 21418) located much closer to the epicenter of the earthquake than buoy 52402 is.",
"These buoys registered the tsunami signal within 5 min after $t_0$ , well before it arrived at buoy 52402 more than 3 h later.",
"The automatic method selected seven unit sources ${\\bf g}_1$ , ..., ${\\bf g}_7$ to model the tsunami signal.",
"Accordingly, we need to adjust the joint estimation method to make use of seven unit sources rather than just one.",
"We do so by suitably redefining the design matrix $X$ and the vector of coefficients $\\alpha $ in Equation REF .",
"Thus $X$ is now of dimension $N\\times 12$ , with its first five columns being as before and with its next seven columns now being ${\\bf g}_1$ , ..., ${\\bf g}_7$ ; correspondingly, $\\beta $ is augmented to dimension 12, with its last seven elements being the source coefficients for the unit sources.",
"After these adjustments to $X$ and $\\beta $ , we produce the detided series using the same equations as before (REF and REF ).",
"The bottom plot in Fig.",
"REF shows detided series for the joint estimation method (five in all corresponding to different amounts of data).",
"These series are visually quite similar to the KS detided series, but the largest difference between the joint detided series is larger (3.6 cm) than for the KS series (0.7 cm).",
"It is important to note that, in contrast to the other four methods, the joint estimation method is dependent upon a suitable model for the tsunami signal.",
"To demonstrate this fact, the two plots in Fig.",
"REF show detided series using the joint estimation method, but based upon different models.",
"The top plot is the same as the bottom plot in Fig.",
"REF , for which the detided series utilize a model involving seven unit sources.",
"In the bottom plot the detided series use a model based on only one of these seven sources (ki026b).",
"The magnitude of the earthquake that generated the 2011 Japan tsunami was so large that it is physically unrealistic for the signal to be well-modeled by a single unit source (Papazachos et al., 2004).",
"The detided series from this presumably inadequate model have low-frequency fluctuations in the first three hours that are not evident in the top plot (or in the KS-based detided series, which are shown in the next-to-bottom plot of Fig.",
"REF ).",
"These fluctuations are best attributed to a failure on the part of the joint estimation method due to an inappropriate model."
],
[
"Conclusions and discussion",
"We have undertaken a comprehensive comparison of five methods for estimating the source coefficient $\\alpha $ based upon DART® buoy bottom pressure (BP) data collected during a tsunami event (this coefficient reflects the strength of the event and is used to provide initial conditions for predicting coastal inundation).",
"Any method for estimating $\\alpha $ must deal with the fact that the variability in BP data is typically dominated by tidal fluctuations, and hence all viable methods must detide the data in some explicit or implicit manner.",
"The five methods under study have been entertained as part of the on-going development of the SIFT application, a tool developed at NOAA for use by U.S. Tsunami Warning Centers for real-time assessment of tsunami events.",
"The clear method of choice is a scheme by which $\\alpha $ is estimated jointly in a regression model that accounts for the tidal components using sinusoidal constituents involving the tidal frequency M2 and half that frequency.",
"This method is particularly convenient from an operational point of view in that it does not make direct use of data occurring prior to a tsunami event, as is true in varying degrees for the four other methods under study.",
"Amongst the four remaining methods, the Kalman smoothing (KS) method performed best overall in our study.",
"In some cases, the EOF method is competitive with the joint estimation and KS methods, but the two methods based on only harmonic analyses proved to be markedly inferior to the other three methods.",
"Note that we evaluated the performance of the EOF method using a specific set of eight basis vectors adapted for use within the SIFT system.",
"We limited this set to eight vectors due to the fact that only this number of vectors defines a location-independent tidal sub-space, should the set be derived using data from a single buoy, as was the case here (Tolkova, 2010).",
"Expanding the set of vectors by deriving them from multiple buoys might allow for more accurate detiding.",
"Despite the fact that our study points to the joint estimation method as the method of choice, some caution is in order.",
"A presumption behind our study is that the model for the tsunami signal is known perfectly, which is obviously unrealistic in practice.",
"The issue of model mismatch must therefore temper our conclusions, a point that is reinforced by the discussion surrounding Fig.",
"REF .",
"For graphical presentation of detided series within SIFT, the fact that the KS method does not depend on an assumed model for the tsunami signal – but often compares favorably with the joint estimation method – suggests its use.",
"As currently implemented, the KS method uses the output from the 29-day harmonic analysis method as its starting point.",
"In cases where the 15-min stream is not available over much of the preceding 29 days, the blanket harmonic method could provide a model-free detiding for display purposes, as could the EOF method if the 15-min stream were available going back about a lunar day from the start of the tsunami signal.",
"There is thus potential use within SIFT for all five methods we have studied.",
"As of this writing, the current version of SIFT supports the 29-day harmonic analysis, EOF and KS methods, and there are plans to implement the joint estimation method within an overall scheme for automatically selecting unit sources to serve as models for the tsunami signal.",
"This work was funded by the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) under NOAA Cooperative Agreement No.",
"NA17RJ1232 and is JISAO Contribution No. 2185.",
"This work is also Contribution No.",
"4089 from NOAA/Pacific Marine Environmental Laboratory.",
"The authors thank George Mungov of NOAA's National Geophysical Data Center for supplying DART® buoy data and predictions based upon harmonic analyses with 68 sinusoidal constituents."
]
] | 1403.0528 |
[
[
"Modeling X-ray Emission Around Galaxies"
],
[
"Abstract Extended X-ray emission can be studied either spatially (through its surface brightness profile) or spectrally (by analyzing the spectrum at various locations in the field).",
"Both techniques have advantages and disadvantages, and when the emission becomes particularly faint and/or extended, the two methods can disagree.",
"We argue that an ideal approach would be to model the events file directly, and therefore to use both the spectral and spatial information which are simultaneously available for each event.",
"In this work we propose a first step in this direction, introducing a method for spatial analysis which can be extended to leverage spectral information simultaneously.",
"We construct a model for the entire X-ray image in a given energy band, and generate a likelihood function to compare the model to the data.",
"A critical goal of this modeling is disentangling vignetted and unvignetted backgrounds through their different spatial distributions.",
"Employing either maximum likelihood or Markov Chain Monte Carlo, we can derive probability distribution functions for the source and background parameters together, or we can fit and subtract the background, leaving the description of the source non-parametric.",
"We calibrate and demonstrate this method against a variety of simulated images, and then apply it to Chandra observations of the hot gaseous halo around the elliptical galaxy NGC 720.",
"We are able to follow the X-ray emission below a tenth of the background, and to infer a hot gas mass within 35 kpc of 4-5x10^9 Msun, with some indication that the profile continues to at least 50 kpc and that it steepens as the radius increases.",
"We derive much stronger constraints on the surface brightness profile than previous studies, which employed the spectral method, and we show that the density profiles inferred from these studies are in conflict with the observed surface brightness profile.",
"(abridged)"
],
[
"Introduction",
"The study of very extended emission comprises a large portion of the work of extragalactic X-ray astronomy.",
"All galaxy clusters ([32], [25], [61]) and most galaxy groups ([52], [16], [58], [51]) are suffused with a hot ($kT > 10^6$ K), X-ray emitting gaseous medium.",
"In all clusters and many groups, this medium contains the majority of the baryons in the system ([19], [31], [30], [4], [14], [63]), and extends to hundreds of kpc.",
"In recent years, X-ray observations have even been able to push outwards to the virial radius of some nearby clusters (e.g.",
"[29], [7], [60], [36], [40], [65], [1], [9], [50], [64], [76], [75], [39], [77], [74]).",
"Individual galaxies are also surrounded by extended X-ray emitting halos.",
"Around elliptical galaxies, these hot gaseous halos have been studied for decades (e.g.",
"[27], [26], [22], [49], [55], [53]).",
"Starbursting spirals have extended coronae above and below the disk extending to a few tens of kpc ([70], [44], [72], [56], [79], [43]).",
"We also recently reported the detection of hot gaseous halos around more quiescent massive spiral galaxies, extending out to $\\sim 50$ kpc ([2], [13], [3]); [8] have confirmed one of these detections and have discovered another hot halo as well.",
"Very extended hot gas is also detected around merging galaxies such as NGC 6240 [54].",
"As both of these fields continue to detect emission at larger radii and lower X-ray surface brightness, it is becoming increasingly important to have effective observational techniques for studying faint, extended X-ray emission.",
"At present, there are two major approaches to this analysis: spectral fitting and spatial binning.",
"Spatial binning is conceptually simple: one measures the X-ray radial surface brightness profile in a given band, and infers a gas density profile from the surface brightness profile.",
"The major uncertainties with this method are flat-fielding the image and estimating the background.",
"For bright sources, blank-sky backgrounds are sufficient, but for faint emission the background should be estimated in-field, which requires accurate flat-fielding.",
"A secondary concern is separating various components of emission, if they exist; examples of this separation are illustrated in [2], [13], and [8].",
"For the spectral method, one measures the X-ray spectrum in various regions, instead of the surface brightness profile.",
"Analyzing a spectrum requires more photons than measuring a broad-band surface brightness, in part because most realistic spectra have many more free parameters than a surface brightness profile.",
"Thus, the image is usually broken into large radial annuli, which sacrifices some location information.",
"On the other hand, the various instrumental and background components are included in the spectral model, so in theory this method does not require separate flat-fielding or background subtraction.",
"Also additional source components can be included, obviating concerns about confusion between hot gas emission and other X-ray sources such as X-ray binaries or background point sources.",
"The primary downsides to this method are model specification and the need for more photons.",
"The aforementioned issues with these methods can be quite important, and can lead to conflicting results.",
"For the spatial method, one major failure mode in flat-fielding is incorrect estimation of the “vignetting” - the decrease in sensitivity of the detectors as off-axis angle increases.",
"The vignetting profile varies as a function of time (typically getting worse as the telescope degrades), and must be computed separately for each observation.",
"It also varies as a function of energy, so a given exposure map (which contains the information about the vignetting) should only be computed for and applied to a narrow energy band over which the vignetting effects do not vary significantly.",
"An example of this difficulty occurs in [57], where the authors found evidence for a hot halo around the giant spiral NGC 5746 at $4\\sigma $ significance.",
"Later, after applying an updated calibration file which accounted more correctly for the time-dependent degradation of the instruments, the vignetting profile changed and the signal disappeared [59].",
"For the spectral method, model specification is particularly important.",
"In the observations of interest today, the signal from hot gas is lower than the background, so the model for the background components in the spectrum can significantly influence the inferred properties of the signal.",
"It is not trivial to construct a model for the X-ray background, and most studies use slightly different prescriptions.",
"The components generally include the extragalactic AGN background at hard energies, and Solar wind X-rays (time variable), emission from the Local Hot Bubble, and the Galactic hot halo at softer energies.",
"These components are all variable (spatially and sometimes temporally) and so their normalizations are not typically known a priori.",
"Moreover, the emission itself can be quite spectrally complex - especially for the Local Bubble and Solar wind X-rays, where charge exchange can affect the signal and the gas is not necessarily in collisional equilibrium.",
"Without sufficient photons to fit all of these spectral components, this method is very susceptible to systematic errors, either from degeneracies between free model parameters, or as a result of fitting the data with an inappropriate model and/or overfitting the data.",
"An example of this issue can be seen with the isolated elliptical galaxy NGC 720.",
"This galaxy has a massive hot halo that has been observed once with Suzaku, twice with XMM-Newton, and several times with Chandra.",
"One of the Chandra observations was studied by [38], who used the spectral method in eight annuli to measure the hot gas density profile out to 90 kpc; extrapolating their density profile to 300 kpc yields a hot halo mass of $1\\times 10^{11} M_{\\odot }$ , which implies that the galaxy is missing about 1/2 of its expected cosmological allotment of baryons.",
"The rest of the data is analyzed in [37], with slight changes in their model, yielding a hot halo mass of $3\\times 10^{11} M_{\\odot }$ within $R_{200}$ , which implies that the galaxy is baryon complete.",
"Based on the statistical errors quoted in these papers, this is about a $3\\sigma $ discrepancy in the mass.",
"About half the discrepancy is caused by the addition of the Suzaku data, and half by the changes in their modeling; however, at a smaller radius like 100 kpc, the discrepancy is still nearly $3\\sigma $ , and this difference is caused almost entirely by their modeling.",
"We will discuss this galaxy in much more detail in section 5.",
"Finally, there are also situations where the spectral method and the spatial method yield different conclusions.",
"A notable example is in the estimation of galaxy cluster density profiles near the virial radius - where the cluster emission is much fainter and the systematic uncertainties in these methods become more important.",
"There has been some debate over the putative detection of a flattening in the radial decrease of the hot gas density profile, and this debate seems largely to fall along the lines between these two methods.",
"The flattening was first observed with Suzaku, using spectral methods ([29], [65]); spatial methods, using ROSAT, have often not confirmed this result ([20], [17], [18]).",
"Thus, while seems to be general consensus about a number of other features in the gas properties near the virial radius (such as decreasing temperature and flattening entropy), more work needs to be done to understand the behavior of the gas surface brightness and density (and therefore the total gas mass and baryon fraction of the cluster).",
"Relatedly, discrepancies at the 5%-15% level for derived parameters such as luminosity, temperature, and pressure have also been noted by [62] in samples of clusters analyzed with different techniques (including combinations of spectral and spatial).",
"In this paper we explore a potential improvement to the spatial method, taking into account both vignetted and unvignetted backgrounds based entirely on in-field data.",
"This approach is similar to the use of the off field-of-view (OFOV) events for XMM-Newton, but can also be used for Chandra and Suzaku imaging, for which these events are not generally available.",
"Adding an unvignetted component to the background model allows the entire image to be flat-fielded simultaneously, which is important for precise measurements of faint signals.",
"The rest of the paper is devoted to taking the first steps towards full image modeling for X-ray astronomy.",
"X-ray observations possess an unusual advantage over other wavelengths in that one records time, position, and energy measurements for each event.",
"Most of this information is discarded when producing an image or a spectrum from an events file, but we argue that computational power has evolved to the point where it is no longer necessary to discard this information.",
"We think the ultimate goal, which is outside the scope of this paper, is to study the X-ray events file directly instead of binning it spatially or spectrally to produce an image or a spectrum.",
"With a good model for the events file that includes both energy and position (and potentially time), one could combine the best features of both spatial and spectral analysis, while minimizing the systematic errors associated with each.",
"Here we implement a much more limited form of image modeling, using only spatial information in one energy band (typically 0.5-2.0 keV).",
"We construct a simple model with both vignetted and unvignetted backgrounds.",
"We then discuss the likelihood function for the image, and compare several different forms for the likelihood function before settling on one function.",
"We show that this likelihood function is able to recover the input parameters in a variety of simulated images.",
"Finally, we apply the method to the case of NGC 720 and show that even this limited form of image modeling offers significant advantages over traditional spectral fitting.",
"Errors are quoted at $1\\sigma $ unless otherwise noted."
],
[
"The Unvignetted Background",
"In an infinitely long X-ray observation, most of the hard cosmic X-ray background could be resolved into individual AGNs, with a $\\sim 10\\%$ contribution from the intracluster medium ([12], [73], [6], [34], [33]).",
"Below about 2 keV, star-forming galaxies begin to contribute background X-ray point sources as well, also at about the 10%-20% level.",
"There is also a diffuse X-ray component to the soft emission, due to the local hot ISM (i.e.",
"the Local Hot Bubble) and the Galactic hot halo ([66], [67]).",
"These various components are all spatially variable across the sky, but in practice this is not as much of a concern as one might expect.",
"The faint point sources, which are not resolved in typical X-ray observations, are so numerous as to wash out most statistical fluctuations in a typical field of view.",
"The bright point sources are treated separately in our model (section 3.3).",
"And the diffuse soft emission typically shows features on large (i.e.",
"degree) scales, which are not very important in a single field of view, although it is still unclear to what extent the diffuse emission also contains substructure on smaller scales ([69], [28]).",
"All of these backgrounds are focused by the telescope's mirrors, which introduces a vignetting effect across the image: emission that falls at larger radii from the aimpoint is slightly attenuated compared to emission right on the aimpoint.",
"The magnitude of the attenuation is energy dependent; in the soft band the attenuation reduces the signal by roughly a quarter to a third at large off-axis angles.",
"There is another type of background which is not focused by the telescope optics.",
"The particle background – mostly soft protons from the Solar Wind – impinges directly on the X-ray detectors, and also triggers X-ray fluorescence from the instruments themselves.",
"This background also registers as events on the detectors.",
"Many of the cosmic rays are automatically flagged by the standard data reduction scripts, and during periods of especially high particle flux the entire event file is generally excluded as well.",
"However, some particles cannot be automatically distinguished from X-rays and remain in the image as false positives; instrumental X-rays are also not automatically removed and will appear in the events file.",
"Since this background is not focused by the optics, it does not follow the same vignetting profile as the cosmic and Galactic X-ray backgrounds.",
"The unfocused background is not appreciably vignetted at all.",
"In Figure 1, we present an image of the stowed background in the 0.5-2.0 keV band for Chandra ACIS, taken from the most recent observations of the stowed backgroundEvents file produced by M. Markevitch, available at http://cxc.harvard.edu/contrib/maxim/acisbg.",
"The background is very nearly uniform across all the frontside-illuminated ACIS-I chips.",
"On the ACIS-S chips there is a top-to-bottom gradient which may be due to charge transfer inefficiency in the readout from these chips and seems to have a fairly persistent spatial distribution.",
"For XMM-Newton EPIC images, the instrumental background is distributed significantly differently for each instrumental emission line, and standard background images have been created which model the overall unfocused background ([45], [41], [68]).",
"The broad-band unfocused background is largely uniform, however.",
"Figure: 0.5-2.0 keV image of the most recent (period E) study of the Chandra stowed background, which should be similar to the overall unfocused background.",
"The image has been smoothed with a Gaussian kernel of radius 3 pixels.",
"The background is very nearly uniform across all the ACIS-I chips.",
"There is a gradient visible across the ACIS-S chips, of order 20-30%, whose shape does not seem to vary significantly with time.",
"In our analysis, when examining the ACIS-S chips we construct a model for the unvignetted Chandra background based on a smoothed version of this image.The spectral shape of the unfocused background is quite complex, and somewhat variable.",
"In the soft bands the flux is much higher at $E $ <$$ 0.5$ keV, and otherwise is dominated by prominent emission lines from Aluminum, Silicon, and Gold.",
"In the hard bands the spectrum is fairly flat from 3-5 keV, and then turns upwards and begins to show a number of emission lines from various elements in the detectors.",
"An excellent comparison of the instrumental spectra of Chandra, XMM-Newton, and Suzaku can be seen in Figure 8.2 of \\cite {Arnaud2012}.$ With XMM-Newton, the two backgrounds can be disentangled through use of the unexposed off-field of view (OFOV) events, which are by definition entirely composed of unfocused background.",
"This can be used to set the normalization for this component, and then one can apply standard maps of the unfocused background to an observation in order to account for this component.",
"For Chandra and Suzaku, there is currently no straightforward way to separate the two backgrounds.",
"Regardless of instrument, the general practice in reducing X-ray data is to remove as much of this background as possibleAs an aside, we note that the presence of OFOV data is extremely useful in modeling the background, and it might be worthwhile to include these regions as a design consideration in future X-ray missions.. Filtering periods of the observation with higher count rates reduces the particle background significantly, as does strict event grade filtering such as VFAINT mode on Chandra.",
"For the instrumental background, one typically excludes events with energies near prominent instrumental lines, or at low energies ($E $ <$$ 0.5$ keV) where the instrumental background turns upward.",
"These can be effective at removing a good fraction of the unfocused background, but at the cost of discarding a considerable fraction (tens of percent) of the total events.",
"Moreover, in observations of the faint emission around bright sources, VFAINT mode filtering could conceivably introduce a bias, since it can exclude real events in regions where the count rate is high.$ Even with the standard reduction techniques, in most observations some unfocused background inevitably remains, and failure to explicitly account for this can lead to incorrect results.",
"As an illustration we examine a recent 20 ks XMM-Newton EPIC observation of NGC 1961 (obs id 0723180101), a giant spiral galaxy surrounded by a hot gaseous halo ([2]; [8]).",
"The galaxy is placed at the aimpoint of the telescope for this observation.",
"We reduce the data according to the procedure of Snowden and Kuntz (2013) to produce a PN image in the 0.4-1.25 keV energy band (a band which avoids the instrumental line at 1.5 keV and the uptick in the instrumental background below 0.4 keV).",
"The X-ray emission from the galaxy and its corona is visible to $\\sim 3$ arcminutes, at which point it becomes indistinguishable from the background.",
"The background is not flat however; it increases roughly from about 5 arcminutes out to the edge of the image.",
"This is caused by the unfocused background.",
"Dividing the counts per pixel by the exposure map (units of cm$^2$ s count photon$^{-1}$ ) implicitly assumes all the counts are distributed according to the exposure map, i.e.",
"vignetted.",
"Applying this vignetting correction to events associated with the unfocused background results in an overcorrection, causing the apparent increase in the background at large off-axis angle.",
"Figure: XMM-Newton PN 0.4-1.25 keV radial surface brightness profile for a 20 ks observation of the giant spiral galaxy NGC 1961.",
"Black points are the vignetting-corrected observed data; note the overcorrection at r>5r>5'.",
"The green line is the estimated unfocused background (particle background + soft protons) estimated from the OFOV data + standard blank field templates.",
"Subtracting the green line from the data before applying the vignetting correction yields a properly flat background (green points).",
"The red points are the result if we assume the unfocused background is completely uniform, and they are not significantly different from the green points, suggesting this assumption is fairly good for this observation.If one naively divides by the exposure map without accounting for the unvignetted background, one would overestimate the background by as much as 50% at 3 arcminutes.",
"More likely, one could estimate the background based on, say, the 5'-7' annulus where the signal is fairly flat.",
"This should theoretically produce the same background-subtracted profile as one obtains with the correction for the unvignetted background, at the cost of a less precise estimate of the level of the background due to the use of fewer photons.",
"In this image, our band contains 538 photons in the 5'-7' annulus, so statistical uncertainties limit the precision of the estimation of the background to 4.3%, even before considering any systematics.",
"Since this is an XMM-Newton observation, we can use the pn_back and proton utilities to estimate the unfocused background in-field.",
"We illustrate this component with the green curve in Figure 2.",
"We can subtract it from the total emission (black) before the vignetting correction, and the resulting surface brightness profile (green points) no longer shows the uptick at large radii.",
"Using the full 5'-14' region, after subtracting the unfocused background, the statistical uncertainty on the background is reduced to 1.6%.",
"In most situations, statistical uncertainty of 4% for the background is perfectly adequate, but there are important exceptions.",
"For example, in measurements of particularly faint emission (i.e.",
"at $\\;<\\over {\\scriptstyle \\sim }\\;$ 10% of the background), uncertainties in the background can be significant.",
"Also, if the emission is particularly diffuse (extending more than 4', for example), there may be nowhere in the image where a flat in-field background can be measured.",
"Finally, the particle and instrumental background are variable, and observations with higher unvignetted backgrounds will have correspondingly larger deviations from flatness when divided by the exposure map.",
"Note that this unfocused background is nearly uniform.",
"If we perform the same analysis with the unfocused background modeled instead as a uniform distribution with the same average value, we obtain the red points in Figure 2, which are equal to the green points within the statistical errors.",
"This simplifying assumption therefore seems fairly reasonable for this observation.",
"As Figure 1 shows, the unfocused background is essentially uniform on the four Chandra ACIS-I chips, although it exhibits some structure on the ACIS-S chips.",
"The shape is very different than the vignetting profile, however, so in the rest of this paper we will refer to the unfocused background as the “unvignetted” background.",
"Since the unvignetted background is in general not known a priori, one must estimate it in-field.",
"In the absence of OFOV data, this can be done by fitting to the observed surface brightness profile.",
"One can then subtract it out, or include it in a model of the image.",
"We discuss the latter option more in the next section."
],
[
"Image Modeling",
"Instead of fitting and subtracting the unvignetted background, the more sophisticated approach towards accounting for the vignetted and unvignetted backgrounds is image modeling.",
"In this study we construct a simple generic model for an X-ray image with an extended source.",
"The model is constructed in the 0.5-2.0 keV energy band (which comprises the 'soft' and 'medium' bands as defined by the Chandra Source Catalog; [21]), and uses only spatial information (no spectral or temporal information, except for excluding events outside the energy band and excluding flaring events).",
"Our model consists of three major components: the background, the source emission, and X-ray point sources.",
"In this paper we use our model to study data taken with the Chandra ACIS instrument, so the emphasis will be on this instrument.",
"It is fairly straightforward to implement this general model for XMM-Newton or for Suzaku, which may be done in future work.",
"Below, we discuss each component of our model in turn, including details of applying each component to Chandra ACIS."
],
[
"Background",
"The model background has two components: the vignetted background and the unvignetted background.",
"Each of these components is itself a mixture of a number of spectral components (as discussed in Section 2), but here we focus on their spatial distribution.",
"The vignetted background (Cosmic + Galactic + Solar wind charge exchange X-rays) are focused by the telescope's optics and are therefore vignetted at large off-axis angles.",
"The magnitude of this vignetting is energy-dependent, and we assume the vignetting profile is proportional to the exposure map in our energy band.",
"The shape of the exposure map has a weak dependence on the assumed spectrum of the incident photons; we assume a power-law distribution but we tried other simple models and found no significant differencesThe values in the effective exposure map can vary significantly for different assumed spectra (by tens of percent), but we found that the shape (i.e.",
"the vignetting profile) hardly changes at all..",
"In general this is important to check for each energy band, since a difference in the vignetting profile for the background and the source photons could significantly bias the results of a spatial analysis.",
"In practice, as long as the energy band is narrow enough that the instrument response is fairly uniform within the band, this should not be a major issue.",
"Since the long-term goal is to extend this method to leverage spectral information as well, the expectation is that exposure maps will be constructed in many different energy bands across the energy range of interest, and it will be important to ensure that each energy band is sufficiently narrow.",
"The unvignetted background was discussed in Section 2 above.",
"It consists of cosmic rays, soft protons, and the instrumental background.",
"In detail these backgrounds all have some spatial structure, but they are nearly flat when averaged over the 0.5-2.0 keV band (Figures 1 and 2).",
"In general we will treat this component as a uniform background, except when dealing with the ACIS-S chips where we use a heavily smoothed version of the image in Figure 1 as a template for the unvignetted background on these chips."
],
[
"Source and Point Spread Function",
"It is not necessarily required to parametrize the source emission, depending on the research objective.",
"We develop two implementations of our image model: the parametric model, where we model the entire image and include a parameterization of the extended source emission, and the nonparametric model, where we model the background in the image but exclude the region around the source.",
"The latter method therefore does not include a component for the source or the point spread function.",
"It also has fewer data available for constructing the model, since the source region is masked, but once the model is constructed it can be subtracted from the full image, and the difference between the model and the data is the source emission and can be studied independently of any particular parameterization.",
"For the parametric method, we opt to describe the source emission with a $\\beta $ -model [11], which describes an isothermal, azimuthally symmetric gas distribution.",
"In this model, the gas has an assumed density distribution $\\rho (r) \\equiv \\rho _0 \\left[1 + \\left(\\frac{r}{r_0}\\right)^2\\right]^{-\\frac{3}{2} \\beta }$ and the projected surface brightness profile of the X-ray emission as a function of projected radius $r$ is $S_x(r) \\equiv S_0 \\left[1 + \\left(\\frac{r}{r_0}\\right)^2\\right]^{\\frac{1}{2} - 3 \\beta }$ This profile is standard for describing the observed X-ray emission from gas in galaxy clusters and in hot halos around individual galaxies.",
"It generally gives a good fit to the data and can be extended if necessary to account for radial temperature gradients, although for hot halos these gradients tend not to be very large.",
"Another possible class of density profiles are the adiabatic profiles ([46], [23]), which assume the gas is adiabatic instead of isothermal.",
"This tends to produce a profile with a much flatter slope, which yields much lower surface brightness and is therefore difficult to constrain observationally.",
"These profiles are employed less frequently than beta models, but for galactic hot halos they are still consistent with observations, and may be considered in future work.",
"The source emission is convolved with the instrumental point spread function (psf), and for extended emission when we are trying to measure a surface brightness profile this can be an important effect.",
"The Chandra psf is much smaller than the XMM-Newton psf or the Suzaku psf, subtending less than an arcsecond at the aimpoint, but it grows to several times this size at larger off-axis angles.",
"In order to convolve our source emission model with the psf, we therefore need a functional form for the psf which is defined at every location on the detector.",
"For this purpose, we specify a two-dimensional, azimuthally symmetric Gaussian at each point, so that the psf can be parametrized as a function of its standard deviation $\\sigma $ at each point.",
"We estimate $\\sigma $ using the mkpsfmap function, which estimates the radius corresponding to a given ecf (encircled counts fraction) at every location across an imageThe function mkpsfmap does neglect aspect variations over the image, which can also broaden the effective psf, but these variations are uniform, random, and on the order of arcseconds, so they are not a major source of concern for this sort of analysis.. We set the ecf to 0.393 so that the resulting radius is equal to $\\sigma $ .",
"As with the exposure map, the psf depends on the spectral shape of the emission.",
"We are generally interested in extended sources which are composed of hot gas and/or compact objects, but as long as the energy band is narrow the effect of the assumed spectral shape is minorIn fact, for our 0.5-2.0 keV energy band, and for the particular extended sources we study in this paper, we find that our results are actually the same within $1\\sigma $ whether or not we convolve the image model with the psf at all..",
"In Appendix 1, we examine the nongaussianity of the psf, and we find that the Chandra psf in our waveband is very nearly Gaussian except at the aimpoint, where its small size makes the deviations from Gaussianity unimportant.",
"We convolve our source emission model with the psf model to generate the total image modelIn detail, we take the fast Fourier transform of the image and of the model for the spatially varying psf, multiply the two together, and then take the inverse Fourier transform.",
"This is mathematically very similar to a convolution (identical in the limit of infinite spatial resolution), and is much quicker to compute.. We also multiply the image model by the vignetting profile obtained from the exposure map.",
"We do not convolve the psf model with the background, since the background is assumed to be uniform, and we do not convolve the psf model with the point sources described in the next subsection, since the point sources are computed with a more exact method, and so the approximation of Gaussianity is not needed.",
"Therefore, the non-parametric method makes no use of the psf at all."
],
[
"Point sources",
"In addition to the diffuse backgrounds, X-ray images are also contaminated by resolved point sources, and any image model should account for these sources as well.",
"We take a two-pronged approach towards point sources in our image model.",
"We attempt to mask the bright point sources where possible, and we study the completeness of our masking algorithm so that we can include a component in our image model to account for undetected point sources."
],
[
"Masking Sources",
"To detect bright point sources for masking, we run wavdetect on the simulated images.",
"We set the wavelet radii to 1, 2, 4, 8, and 16 pixels (with the image binning at 2, so one pixel is 0.984 arcseconds on a side), and the significance threshold to $10^{-6}$ , corresponding to approximately one false positive per chip.",
"The one non-standard setting we use with wavdetect is to increase the size of the ellipse to mask (ellsigma) from $3\\sigma $ to $8\\sigma $ (the exact number here is not important, as long as it is large).",
"This excludes nearly all of the emission from the point source, as well as a larger region around the point source than is typical.",
"Our goal is to measure the background precisely, not to capture all of the background photons, so it is an acceptable tradeoff to lose some background photons in order to exclude as many of the photons from the bright point sources as possible.",
"During this process, we noticed that wavdetect was failing to detect some point sources at large off-axis angles (especially on the S2 chip).",
"In order to study this effect further, we also tried an alternative method of detecting point sources.",
"For every photon on the detector, we computed the 90% encircled counts radius at the location of that event (using mkpsfmap and assuming the same $\\Gamma = 2$ powerlaw).",
"We compared the number of counts within this radius to the estimated background (estimated either assuming a 100% unvignetted fraction or an 100% vignetted fraction; both methods give very similar results).",
"If any circle has an excess of photons significant at $10^{-6}$ or stronger, we masked that circle.",
"This method is much more computationally intensive than wavdetect, since it has to perform aperture photometry tens of thousands of times (once per photon).",
"It also has no ability to find the centroid of a point source; instead it just masks out any photons that could conceivably be associated with the source.",
"While these constraints are not ideal for a robust multi-purpose point source detection algorithm, for our purposes we found this method to be complementary to wavdetect.",
"In Appendix 2, we show a side-by-side comparison of point sources detected with the two methods in a single 100 ks observation of the Chandra Deep Field-South (CDF-S).",
"In Appendix 3, we consider the effect of the choice of point source detection algorithm on our ability to recover input parameters.",
"We find that both methods yield fairly similar results, but that the best results are obtained by using both methods on the image and masking sources with detections by either method."
],
[
"Accounting for Undetected Sources",
"We account for undetected sources by simulating point sources in our image with a realistic N($>S$ ) distribution.",
"We used the point source number counts function from the CDF-S 4 megasecond field [42] for this distribution, which agrees well with previous estimates but has better sensitivity to sources with fluxes below $10^{-17}$ erg s$^{-1}$ cm$^{-2}$ .",
"Integrating their $dN/dS$ and plugging in the length of a given Chandra observation, we estimate the expected number of point sources in the image as a function of counts (assuming a $\\Gamma = 2$ powerlaw spectrum to convert from flux to count rate).",
"We generate a psf map using mkpsfmap (using the 90% ecf as the characteristic psf radius).",
"For every integer number of counts in a given point source, we use the psf map to determine where in the image a point source with that number of counts could be detected above a given probability threshold (which we set to the value used by wavdetect - $1\\times 10^{-6}$ ).",
"We distribute the expected counts from undetected point sources over the area in which the point source would not be detected, and add together all the point sources expected in the image to get a map of the expected number of counts from undetected point sources at any given location in the field.",
"An example of such a map is shown in Figure 3.",
"Figure: Example map of the average number of counts per pixel corresponding to undetected point sources.",
"This map is included as one component of our full image model, and is almost entirely determined by the length of the observation, the ACIS chip configuration, and the level of the background.",
"The number of counts in this image corresponding to undetected point sources is 1136, and in any particular pixel the undetected point sources are a small correction (the total number of counts in the image is 52434).",
"When binning the outermost pixels together, however, this component can be important.Note that this map is almost entirely determined by the length of the observation, the ACIS chip configuration, and the level of the background.",
"In our experiments, we found that the dependence on the shape of the background is extremely weak: the mean difference in the undetected flux per pixel between images with $f_{\\text{unvig}} = 1$ and $f_{\\text{unvig}} = 0$ is 0.3%.",
"This difference is so small because the two backgrounds only differ significantly at the largest off-axis angles, and at these angles most point sources are undetected everywhere so their flux is diluted across the entire image.",
"Thus, we neglect the effect of the shape of the background on the distribution of undetected sources, so that this component is entirely determined by the length of the observation and the ACIS chip configuration, and therefore contains no free parameters.",
"There are not enough point sources in a real observation to distribute their counts smoothly across every pixel, so we also explore the effect of binning the counts from regions with significant undetected point sources together (Appendix 2).",
"This offers a modest improvement if performed correctly."
],
[
"Simulated Chandra Images",
"Here we generate simulated Chandra ACIS images from our model, and study how well the model is able to recover the input parameters."
],
[
"Generating Simulated Images",
"Our fiducial model has five free parameters - two normalizations for the vignetted and unvignetted backgrounds, and the three parameters that define our source emission ($\\beta $ , $r_0$ , and $S_0$ ).",
"We also need to specify shapes for the vignetted and unvignetted backgrounds, although these are completely determined by the details of the observation and contain no free parameters.",
"In practice, we also fix the average flux per pixel to the measured value in a given observation, which removes one of the free parameters from the normalizations of the two backgrounds (since their sum is fixed).",
"We therefore instead parameterize the backgrounds in terms of $f_{\\text{unvig}}$ , the fraction of the diffuse background counts that we attribute to the unvignetted component.",
"In addition to these model components, we also add random point sources to simulated images as discussed below.",
"The vignetted background follows the shape of the exposure map generated with mkexpmap (either in units of cm$^2$ count photon$^{-1}$ or cm$^2$ s count photon$^{-1}$ , although we use the latter).",
"For a given observation, mkexpmap computes the effective exposure time and/or area across the image, so we can use its output as a vignetting profile for the observation.",
"The inputs are an aspect histogram and an instrument map file.",
"The aspect histogram encodes the movement of the telescope during the observation, and the instrument map encodes the quantum efficiency and detector response as a function of energy for the specified source spectrum (see the CXC helpfile or [15] for more details).",
"We used the fluximage script to create an exposure map, with the non-standard setting of expmapthresh = 20 We also add point sources to the image in order to evaluate any systematic effects they may introduce to the analysis.",
"While we rely on mkpsfmap for modeling the psf, this is necessarily an approximation, so when constructing simulated images we attempt to simulate point sources as accurately as possible.",
"The shape of the Chandra psf varies as a function of location on the detector and of energy, so it is not trivial to generate a simulated point source in detail, but this process has been automated with the Chandra Ray Tracer (ChaRT; [10]) and the Model of AXAF Response to X-rays (MARX; http://space.mit.edu/cxc/marx/) software.",
"We used ChaRT to construct a library of psf shapes across the ACIS field of view.",
"We generated simulated psfs in polar coordinates, recording a psf at every arcminute of off-axis angle (also sampling every half arcminute near the aimpoint) and every $30^{\\circ }$ of position angle.",
"We used a $\\Gamma = 2$ powerlaw for the spectral shape of the point sources over the 0.5-2.0 keV band.",
"We then passed the simulated psfs to MARX, and traced $10^4$ photons through the telescope optics in order to construct an events file for a point source at each location.",
"We repeated this process for both the ACIS-I and ACIS-S configuration, as described in the MARX help files, in order to be able to simulate point sources on the ACIS-S chips in an ACIS-I observation, or vice versa.",
"Armed with this library of events files, we can simulate a point source at any location in an observation.",
"We find the nearest point source events file from our library, center it on the location we want to simulate a point source, and draw events from the events file at random in order to build up the desired count rate.",
"We use the point source number counts function estimated from [42] to estimate the number and count rate of point sources to generate for a given image.",
"We can now simulate an X-ray image.",
"The background is generated by multiplying the vignetted background flux by the vignetting profile specified by the exposure map, multiplying the unvignetted flux by the area of the image specified by the exposure map, and combining the two.",
"For simulated images which include the ACIS-S chips, we use a smoothed version of the image in Figure 1 as a template for the shape of the unvignetted background across the ACIS-S chips.",
"We then add an extended source component to the image, convolved with the spatially varying Gaussian Chandra psf.",
"Finally, we add point sources randomly to the image with fluxes and shapes determined as described above.",
"We can derive a simple prediction for $f_{\\text{unvig}}$ from estimates of the cosmic X-ray background and the average instrumental background.",
"From [47], the average instrumental flux rate in the 1.0-2.0 keV band is about $1.1\\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ arcmin$^{-2}$ (there is additional flux below 1 keV, but it is highly variable).",
"Comparing this to the average cosmic X-ray background of $2.1\\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ arcmin$^{-2}$ [6], the first-order estimate for the unvignetted fraction in a random field is at least 0.34, plus an additional component based on the $<1.0$ keV instrumental flux at that time.",
"However, the unvignetted background and the SWCX component of the vignetted background are both variable, so significant variations in $f_{\\text{uvig}}$ are expected.",
"The sensitivity of the flaring correction can also affect $f_{\\text{uvig}}$ , since a higher fraction of the events during flares are from the non-X-ray background and therefore unvignetted.",
"Additionally, the backside-illuminated chips are more sensitive to low-energy instrumental photons, so observations in the ACIS-S configuration might be expected to have a higher value of $f_{\\text{unvig}}$ .",
"Finally, the use of VFAINT mode filtering will generally reduce $f_{\\text{unvig}}$ .",
"In Figure 4 we illustrate three example simulated images.",
"Two images have been produced with the aimpoint on the ACIS-I3 chip, and one with the aimpoint on the ACIS-S3 chip.",
"We also add an extended source to each image.",
"For one of the simulated images in the ACIS-I configuration, we place the extended source far off-axis, on the I0 chip.",
"For the other two simulated images, the extended source is near the aimpoint.",
"Each image has the same parameters: $f_{\\text{uvig}} = 0.4$ , $\\beta = 0.5$ , $r_0 = 10$ arcsec, and $N = 2000$$N$ is defined as the number of counts in the extended source, integrated out to a radius of 400 pixels.. We apply the same average background flux to each image of $0.035$ ct pix$^{-1}$ , corresponding to the typical 0.5-2.0 keV background in an observation of $\\sim 100$ ks.",
"Figure: Simulated images with mock point sources added at random, and an extended source with parameters as described in section 4.1.",
"Images (a) and (b) have configurations with the aimpoint on ACIS-I3, image (c) uses a configuration with the aimpoint on ACIS-S3.",
"The extended source is nearly on-axis for images (a) and (c), and the source is far off-axis in image (b).",
"All three images have been smoothed with a 3-pixel Gaussian kernel."
],
[
"Likelihood Function",
"We need a likelihood function to relate this (simulated) data to image models.",
"In Appendix 3, we test a variety of likelihood functions and settle on a likelihood function which evaluates the likelihood at every pixel, with some binning at large off-axis angles.",
"The full likelihood function is: $L \\equiv \\prod _{\\text{pixels}} p(c_{\\text{pix}} | m_{\\text{pix}}) + p(c_{\\text{bin}} | m_{\\text{bin}}) $ where every pixel is evaluated separately, except for the 10% of pixels for which the psf is largest.",
"Those 10% of pixels are combined into one bin.",
"Then $p(c|m)$ is the Poisson probability of obtaining the observed number of counts $c$ in a pixel/bin, given a model prediction $m$ for that pixel/bin."
],
[
"Recovering Input Parameters - Nonparametric Method",
"We first examine the data nonparametrically.",
"We mask out point sources using a combination of wavdetect and our manual method, and we exclude everything out to a radius of 5' around the extended source in our image.",
"The goal is to use the remaining data to estimate the vignetted and unvignetted backgrounds, which can then be added to the estimated undetected point source component to produce a full model background image.",
"We can then subtract this background model from the real image (with point sources again masked, but the extended source unmasked) in order to obtain an estimate of the background-subtracted extended emission.",
"We use the likelihood function from section 4.2 to determine the background model.",
"Using maximum likelihood, we find the best-fit value of $f_{\\text{unvig}}$ for each image, and then generate a model background for the full image, consisting of vignetted and unvignetted backgrounds, and the parameter-free component accounting for undetected point sources at large off-axis angles.",
"In Figure 5 we show the results of this analysis.",
"We measure the surface brightness profiles around each of our three simulated images, divide by the exposure map, and then subtract the estimated composite background.",
"We have fit an error-weighted smoothing spline to the profile data in order to minimize the effects of the binning and to show the magnitude of the negative points at large radius.",
"For comparison, the true source emission is shown by the green line, and clearly the non-parametric method can recover the emission very well.",
"Figure: Background-subtracted surface brightness profiles of the extended sources in Figure 4.",
"The background was estimated non-parametrically (section 4.3), and includes vignetted and unvignetted components; it is indicated with the dotted black line.",
"The red curve shows the true surface brightness profile, i.e.",
"the model profile used to generate the simulated image.",
"Accounting for the shape of the background is necessary to match the model once the flux falls significantly below the background.",
"The error bars account (in quadrature) for both uncertainties in the number of source counts (Poisson errors) and in the level of the background (based on the range of acceptable values for f unvig f_{\\text{unvig}}), and are displayed at 1σ\\sigma .",
"A smoothing spline (black line) has been fit to the recovered profile data in order to minimize the effects of the binning and to show the magnitude of the negative points at large radius.The simulated images in the ACIS-I configuration return values of $0.38_{-0.02}^{+0.01}$ and $0.40 \\pm 0.02$ for $f_{\\text{unvig}}$ in the off-axis and on-axis sources respectively.",
"For the simulated image in the ACIS-S configuration, however, the non-parametric method returned an incorrect value of $0.17\\pm 0.02$ for $f_{\\text{unvig}}$ (recall the true value in these simulated images is 0.40).",
"This could be due to the bright regions of the backside-illuminated (BI) S1 chip being excluded due to point sources, or to missed point sources on the frontside-illuminated (FI) chips, either of which would bias $f_{\\text{unvig}}$ downwards; however, when we perform a parametric analysis of this simulated image in the next section, the recovered $f_{\\text{unvig}}$ is much closer to the true value.",
"This suggests that the issue is related to the additional exclusion region imposed by the non-parametric analysis.",
"Almost the entire BI S3 chip is excluded by our 5' source exclusion region, and most of the BI S1 chip is excluded due to point source emission, so the dynamic range over which to discriminate the vignetted and unvignetted background is more limited, and subtler biases related to point sources can play a larger role here.",
"Fortunately, since the difference between the unvignetted and vignetted backgrounds is smaller in the field of view for this configuration, the value of $f_{\\text{unvig}}$ is less important.",
"On the other hand, the uncertainty in the background subtraction is poorer, as can be seen by the higher average value for the noise at large radii for the ACIS-S configuration in Figure 5.",
"We compare the results above to a standard analysis which does not account for the shape of the background.",
"We divide the images (Figure 4) by the exposure map, yielding curves like those in Figure 2 which turn upwards at large radii due to the unvignetted background.",
"We select an annulus where the background is at a minimum and appears closest to flat, and estimate the background in-field within this annulus.",
"The surface brightness of this annulus is assumed to be the surface brightness of the background across the image.",
"We then subtract this value from the image, yielding the profiles in Figure 6.",
"Figure: Background-subtracted surface brightness profiles of the extended sources in Figure 4.",
"Here the background was estimated using a standard in-field analysis which does not account for the unvignetted background; it is indicated with the dotted black line.",
"The red curve shows the true surface brightness profile, i.e.",
"the model profile used to generate the simulated image.",
"This causes systematic failures in matching the true surface brightness profile once the signal falls roughly below the background.",
"The error bars account (in quadrature) for both uncertainties in the number of source counts (Poisson errors) and in the level of the background (Poisson errors from the number of counts within the in-field background annulus), and are displayed at 1σ\\sigma .",
"A smoothing spline (black line) has been fit to the recovered profile data in order to minimize the effects of the binning and to show the magnitude of the negative points at large radius.",
"Comparing to Figure 5, the standard in-field analysis performs noticeably worse if one tries to follow the signal below the background.The differences between the source and the recovered profile are negligible within an arcminute, where the source emission is well above the background.",
"Once the source becomes comparable to, or below, the background, these differences are more important.",
"For the ACIS-I on-axis source, the source emission is lost at 3', while there is still evidence of signal at 5' when we model the background.",
"The off-axis source yields a false upturn in the source emission at 4-5', which is not present when we model the background.",
"In the ACIS-S configuration, the background is over-estimated, so the recovered signal is systematically below the true signal at radii beyond 1'.",
"We can quantify these differences by calculating the $\\chi ^2$ goodness of fit parameter for each simulation.",
"We calculate $\\chi ^2$ out to the radius at which the source is entirely lost to the background for both the modeled and the flat background analyses.",
"This is 6' for the ACIS-I sources and 4.5' for the ACIS-S source.",
"For the ACIS-I on-axis source, we find $\\chi ^2 / $ d.o.f.",
"= 26.4/24 and 40.7/24 for the modeled and flat backgrounds, respectively.",
"For the ACIS-I off-axis source, $\\chi ^2 / $ d.o.f.",
"= 18.4/24 and 35.7/24 for the modeled and flat backgrounds, respectively.",
"For the ACIS-S source, $\\chi ^2 / $ d.o.f.",
"= 16.9/17 and 21.6/17 for the modeled and flat backgrounds, respectively.",
"In each case, we find an acceptable match between the data and the source model if we model the background, and a much less acceptable match if the background is allowed to be flat (although the difference between the two is much smaller for the simulation in the ACIS-S configuration).",
"The major drawback to the non-parametric method is that it requires the source emission to be fairly localized within the image so that it can be masked out in order to fit the background.",
"This is not always possible, and motivates the use of a parametrized model for the source in such cases, so that the background can be fit from the full image without having to mask the region around the source."
],
[
"Recovering Input Parameters - Parametric Method",
"Here we perform a similar analysis, but without masking the region around the extended source.",
"Now the model has four free parameters instead of just one, so running the maximum likelihood analysis becomes much more cumbersome.",
"We found that a fixed grid search was too inefficient for exploring this space, so we opted for Markov Chain Monte Carlo (MCMC) instead.",
"An MCMC search is a Bayesian technique designed to efficiently sample likelihoods in multidimensional space.",
"For long enough chains, the distribution of samples in multidimensional space is supposed to approximate the posterior probability distribution for the fit parameters.",
"Obviously, since it is a Bayesian technique, MCMC requires specification of priors for each fit parameter as well.",
"In this analysis, we use the open-source Python package emcee [24], which implements an affine-invariant MCMC method (Goodman and Weare 2010).",
"This implementation of MCMC uses a user-specified number of independent “walkers” which separately explore the parameter space.",
"The prior sets a uniform region, from within which each walker is randomly assigned an initial position.",
"But as long as the chain is run for a sufficiently long time, and the likelihood function is well-behaved, the walkers can explore the entire space and the result is nearly completely insensitive to the choice of prior.",
"We used uniform broad priors on $\\beta $ , $r_0$ , $N$ , and $f_{\\text{unvig}}$ , and used the same priors for each simulation.",
"We also ran MCMC chains with different priors, including narrower priors around incorrect values, and recovered the same posterior probability distributions.",
"This allows us to verify that our choice of priors has no effect on the posterior probability distribution.",
"We use 50 walkers and compute a chain of 1100 elements for each walker, discarding the first 1000 and keeping the final 100 for analysis.",
"In Figures 7-9 we show the posterior probability distributions for these four parameters, for each simulation.",
"The full image modeling routine is successfully able to recover the input parameters, even for the case with the source on the ACIS-S chips.",
"Interestingly, the full image model performs worse when the source is located on the aimpoint for an ACIS-I configuration (Figure 7) as compared to placing the source off-axis (Figure 8).",
"This is probably due to the degeneracy between the vignetting profile and the surface brightness profile of the source, which makes it difficult to disentangle the shape of the source surface brightness profile.",
"For comparison, we also generate image models with $f_{\\text{unvig}} \\equiv 0$ , which simulates the effect of ignoring the unvignetted background and simply dividing the cleaned image by the exposure map.",
"For the ACIS-I configurations, models with $f_{\\text{unvig}} = 0$ perform significantly worse than models where $f_{\\text{unvig}}$ is fit along with the source parameters.",
"The difference is much smaller for the ACIS-S configuration (Figure 9), because our source is largely confined to the BI chip where the difference between the vignetted and unvignetted backgrounds is small.",
"A larger source would probably present more difficulty for a model with $f_{\\text{unvig}} \\equiv 0$ .",
"There is a discrepancy between the recovered value of $N$ and the input value for the simulated image with the ACIS-S configuration, significant at about the $1\\sigma $ level.",
"Since we present three simulated images, a $1\\sigma $ discrepancy in one image is not very statistically meaningful, and in our experience we have noticed no systematic discrepancy in the recovered values of $N$ with the ACIS-S configuration.",
"Figure: Posterior probability distribution functions (pdfs) for the image parameters corresponding to the ACIS-I on-axis simulated image in Figure 4a.",
"The dashed lines are the true values of the parameter.",
"The blue lines are the pdfs if we include f unvig f_{\\text{unvig}} in our fit; the red lines illustrate the effect of neglecting f unvig f_{\\text{unvig}} and setting it to zero, as in a standard spatial analysis.",
"Our method, which accounts for the unvignetted background, is strongly favored.Figure: Posterior probability distribution functions (pdfs) for the image parameters corresponding to the ACIS-I off-axis simulated image in Figure 4b.",
"The dashed lines are the true values of the parameter.",
"The blue lines are the pdfs if we include f unvig f_{\\text{unvig}} in our fit; the red lines illustrate the effect of neglecting f unvig f_{\\text{unvig}} and setting it to zero, as in a standard spatial analysis.",
"Our method, which accounts for the unvignetted background, is strongly favored.Figure: Posterior probability distribution functions (pdfs) for the image parameters corresponding to the ACIS-S simulated image in Figure 4c.",
"The dashed lines are the true values of the parameter.",
"The blue lines are the pdfs if we include f unvig f_{\\text{unvig}} in our fit; the red lines illustrate the effect of neglecting f unvig f_{\\text{unvig}} and setting it to zero, as in a standard spatial analysis.",
"Our method, which accounts for the unvignetted background, is slightly favored, but the difference is not very significant.",
"For a source of larger radial extent, we expect the difference would be larger."
],
[
"Summary of Simulations",
"In this section we examined the behavior of both parametric and non-parametric image analysis methods, for simulated images of faint extended emission in three different Chandra ACIS configurations (two on ACIS-I, one on ACIS-S).",
"The non-parametric method successfully recovered $f_{\\text{unvig}}$ for both simulated images with the ACIS-I configuration, but it underestimated $f_{\\text{unvig}}$ significantly for the simulated image with the ACIS-S configuration.",
"The effect of $f_{\\text{unvig}}$ does not seem very important for the simulated image in the ACIS-S configuration.",
"However, in all three images, the non-parametric method provides a statistically acceptable match to the simulated source out to the maximum radius at which the source is detectable.",
"This radius is smaller for the ACIS-S simulated image (4.5', compared to 6' for the ACIS-I simulated image).",
"For the parametric method, we specified a likelihood function for comparing an image model to data, and we showed that this likelihood function allows us to recover the parameters used to generate the simulated image.",
"Parameters are recovered within the $1\\sigma $ uncertainties in most cases, and these $1\\sigma $ uncertainties are typically 10-20% of the size of the parameter.",
"The parametric method is better able to recover $f_{\\text{unvig}}$ for the simulated image in the ACIS-S configuration, although this parameter has little effect on the other parameters for this configuration.",
"For the ACIS-I configurations, on the other hand, mis-specifying $f_{\\text{unvig}}$ causes very significant biases in the recovered values of the other image parameters.",
"Fortunately, $f_{\\text{unvig}}$ is recovered accurately in these images as well.",
"We have seen that this method can recover an extended surface brightness profile robustly in realistic simulations, for both ACIS-S and ACIS-I configurations.",
"In this section we apply our analysis to real data.",
"We examine the slightly larger than $L*$ elliptical galaxy NGC 720 (d = 25 Mpc, $M_K = -24.6$ ) .",
"This galaxy was discussed briefly in Section 1 as an example of an object for which spectral fitting has produced inconclusive results.",
"[38] (hereafter H06) inferred a hot halo mass of $\\sim 1\\times 10^{11} M_{\\odot }$ around this galaxy, while [37] (hereafter H11) inferred a hot halo mass about three times larger.",
"It is notable that such different results were obtained for the same galaxy.",
"The earlier paper is based on Chandra obs id 492, a 40-ks ACIS-S observation of the galaxy, of which only 17 ks were used due to flaring.",
"The later paper uses different observations: four other Chandra pointings with ACIS-S (obs id 7062, 7372, 8448, 8449) adding to 100.5 ks, and a 177 ks Suzaku pointing (obs id 80009010).",
"According to H11, if they exclude the Suzaku data, the inferred hot halo mass decreases by $50\\%$ , which suggests that much of the discrepancy is driven by the Suzaku data (they emphasize XIS1 in particular).",
"However, at smaller radii the discrepancy between Chandra and Suzaku essentially disappears, while the discrepancy between the two papers remains.",
"We investigate this discrepancy in more detail by examining the radial density profiles for the hot gas, as derived in both H06 and H11.",
"In Figure 3 of H06, they present their inferred deprojected gas density profile, extending out to 90 kpc.",
"We read in this profile from their figures, including their $1\\sigma $ uncertainties, and used emcee to fit $\\beta $ models to the data.",
"The best-fit profile had $\\beta =0.45$ , $n_0 = 0.04$ cm$^{-3}$ , $r_0 = 1.1$ kpc, yielding a hot gas mass within 300 kpc of $1.5\\times 10^{11} M_{\\odot }$ ; this figure accords with their enclosed gas mass as displayed in their Figure 6.",
"We retained the full range of fits from the MCMC chain in order to estimate the uncertainties around the best-fit profile as well.",
"H11 provides a projected density profile, although their definition of “projected density” is fairly difficult to deproject and analyze.",
"We instead are able to obtain a density profile from their Figure 6, which shows the cumulative enclosed gas mass as a function of radius, extending from about 30 kpc to 400 kpc.",
"The cumulative enclosed mass profile in this range is very well fit with a power-law with slope $1.76\\pm 0.02$ , corresponding to a $\\beta $ -model with $\\beta = 0.41\\pm 0.01$ .",
"The core is not resolved, so we leave this as a free parameter.",
"The normalization can be obtained from the enclosed mass in H11 Figure 6, for any given value of $r_0$ .",
"Unfortunately, the profile in their Figure 6 is just their best-fit profile, and the uncertainties around it need to be estimated as well.",
"We estimate the uncertainties in this profile by scaling the uncertainties from H06 by the size of the error bars on the points in the projected density profile in H11.",
"As H11 point out, the errors are much smaller for the density profile in their newer analysis, partially because they do not deproject the profile, but they are also subject to considerably larger systematics.",
"Next, we analyze this galaxy with our methods.",
"NGC 720 has six different Chandra observations - five in the ACIS-S configuration (obs ids 492, 7062, 7372, 8448, and 8449) with the galaxy on the S3 chip like our simulated images, and one (obs id 11868) in the ACIS-I configuration with the galaxy at a similar off-axis position to our ACIS-I off-axis simulated image.",
"We reduced these six observations in the standard way, using chandra_repro to reprocess, and then fluximage to generate the images and exposure maps.",
"As above, we ran fluximage with the parameter expmapthresh set to 20% and we generated images in the 0.5-2.0 keV band.",
"We run both wavdetect and our manual point source detection algorithms on each image, and mask out point sources detected with either method."
],
[
"Analysis with the non-parametric method",
"NGC 720 is somewhat larger and brighter than our simulated images, and subtends nearly an entire chip.",
"This makes it more difficult to perform the non-parametric analysis when NGC 720 is placed on the S3 chip.",
"We therefore only examine observation 11868, where the galaxy is placed off-axis on the I1 chip.",
"We mask out a circle of radius 7' around the galaxy.",
"This is a larger radius than in our simulations but a larger radius is necessary because of NGC 720's brightness and extent.",
"We find $f_{\\text{unvig}} = 0.05_{-0.03}^{+0.04}$ , so this observation has nearly no unvignetted background.",
"This observation has no significant flaring, and ACIS-I is less sensitive to the softer energies where the instrumental background is higher, which may explain the low value of $f_{\\text{unvig}}$ in this observation.",
"The background-subtracted surface brightness profile, and the best-fit level of the background, are shown in Figure 10a.",
"For comparison we also show a surface brightness profile estimated using a uniform vignetted background estimated in-field.",
"Due to the size of the excluded region, there is no annulus available in-field around the galaxy which can be readily used for the background, so we instead select an in-field background from a roughly conjugate region on the I2 chip, in a similar method to [2].",
"The surface brightness profile measured in this way is shown in Figure 10b.",
"The two profiles are nearly identical, which is not surprising given the low value of $f_{\\text{unvig}}$ in this observation (note that if $f_{\\text{unvig}}$ = 0, the non-parametric method reduces to a standard in-field background subtraction).",
"Emission is detected securely out to at least 4 arcminutes (30 kpc), and possibly up to 6 arcminutes (45 kpc), depending on the size of the spatial bins.",
"Note that these radii are much smaller than the outermost annulus at 11' in H06, and are much closer to the 6' outermost annulus in H11.",
"Figure: Background-subtracted surface brightness profiles of the emission around NGC 720, based on the off-axis observation in the ACIS-I configuration (obs id 11868) In (a), the background was estimated using the non-parametric method in section 4.3, and has f unvig =0.05 -0.03 +0.04 f_{\\text{unvig}} = 0.05_{-0.03}^{+0.04}.",
"In (b), the background was assumed to be entirely vignetted and was determined using an in-field annulus.",
"For both plots, the error bars account (in quadrature) for both uncertainties in the number of source counts (Poisson errors) and in the background, and are displayed at 1σ\\sigma .",
"A smoothing spline has been fit to the profile data in order to minimize the effects of the binning and to show the magnitude of the negative points at large radius.",
"The two plots are essentially equivalent, since f unvig f_{\\text{unvig}} is so close to zero that it has little effect on this observation."
],
[
"Analysis with the parametric method",
"With the parametric method, we can now make use of each observation, not just the observation in the ACIS-I configuration.",
"However, there are a few additional complications with modeling these data, as compared to the simulations.",
"First, the source emission includes an additional contribution from unresolved X-ray binaries (XRBs).",
"X-ray binaries have a different, harder, spectrum than hot gas, and in theory the two could be distinguished even in a spatial analysis by their hardness ratios.",
"For this analysis, since we only examine the 0.5-2.0 keV energy band, we treat the XRBs spatially.",
"We assume the XRBs are distributed like the K-band light is distributed, and we resample the K-band image of the galaxy from 2MASS into the same pixel coordinates as each Chandra image.",
"Using two different methods, H11 estimated the unresolved XRB luminosity to be $2.95\\times 10^{40}$ erg s$^{-1}$ and $2.8\\times 10^{40}$ (significant figures quoted as listed in H11) in the 0.5-7.0 keV band.",
"We use a value of $2.9\\times 10^{40}$ erg $s^{-1}$ (converted into the 0.5-2.0 keV band, assuming an absorbed $\\Gamma =1.6$ powerlaw) for the total luminosity of the XRB component.",
"Second, we found that within the core of the galaxy, even after accounting for XRBs the profile does not quite match the flat profile predicted by the $\\beta $ -model.",
"Since the goal of this method is to study the faint diffuse emission at large radii, for which the behavior of the core is unimportant, we mask out the central 0.7 kpc (12 pixels) of the galaxy and fix the core radius $r_0$ to this value in the fit.",
"Finally, we detect and mask point sources separately for each observation.",
"The deeper observations therefore typically have more point sources resolved.",
"The radii of the masked regions also vary between observations, since these radii depend on the size of the psf, which varies depending on the location of the aimpoint.",
"In the future it should be possible to combine the observations into a single analysis, but that requires additional simulations and testing which may be explored in future work.",
"Because of the different point source masks in each observation, the luminosity of unresolved XRBs will vary between observations as well, so we include a fourth parameter $f_{\\text{XRB}}$ which defines the fraction of XRB emission that is not masked out (the total $L_{\\text{XRB}}$ is assumed to be $2.9\\times 10^{40}$ erg $s^{-1}$ ).",
"We use emcee again to do the MCMC computations.",
"We use 50 walkers, and for each walker discard the first 1000 elements of the chain before collecting the next 100 for analysis.",
"The pdfs for the four fit parameters are displayed below (Figure 11), for all six observations.",
"We also include the pdfs for $\\beta $ and $N$ , as estimated from H11 and H06.",
"Since each observation has a different duration and configuration, we convert $N$ for each observation into the “effective” $N$ that would be measured if the source were positioned as in observation 11868 (the ACIS-I off-axis observation) To do this, we generate exposure maps for thermal $0.5$ keV plasma emission in the 0.5-2.0 keV energy band for each observation, and use the ratios of the exposure maps for the conversion.",
"Figure: Posterior probability distribution functions (pdfs) for the model parameters in our parametric fits to six observations of NGC 720.",
"The four model parameters are f unvig f_{\\text{unvig}} (the fraction of the background counts attributed to the unvignetted background), N (the number of source counts in the model, within a radius of 400 pixels), β\\beta (the slope of the surface brightness profile), and f XRB f_{\\text{XRB}} (the fraction of the total XRB emission which is not masked out by the point source detection algorithms).",
"In each plot, the black curve represents obs id 11868, the observation taken with the ACIS-I configuration, and the red curves represent observations taken in the ACIS-S configuration (solid = 7372, dotted = 8448, dot-dashed = 8449, dashed = 7062, thin dashed = 492).",
"For N and β\\beta , we also present the pdfs for these parameters as estimated from our fits to the profiles of H06 (blue) and H11 (green), discussed in the text.",
"There is strong convergence between all of our model fits, with the exception of obs id 492 which is very significantly affected by flaring.",
"Our results are also consistent with the results of H06 but with much smaller uncertainties.Five of the six observations give essentially equivalent results for the source parameters $N$ and $\\beta $ , which is a strong sign of success for our method.",
"The sixth observation, with obs id 492, is significantly affected by flaring and the overall count rate is significantly elevated compared to the other observations.",
"The effect of this high background is evident from the best-fit value of $f_{\\text{unvig}} = 0.99 \\pm 0.01$ .",
"If other observations were not available and we needed to extract a more reliable measurement from this observation, we could filter the lightcurve more stringently, as did H06, in order to remove more of the contamination from flaring at the cost of reducing the amount of useable time.",
"For this study, we just present the results as-is, and note that the flaring makes this observation unreliable.",
"As we will show, this observation is an obvious outlier compared to the other five observations.",
"The other observations cover a wide range of $f_{\\text{unvig}}$ , showing the need to fit for this parameter in each individual observation Observation 8448 is much shorter than the other observations, and so it has the least certain determination of $f_{\\text{unvig}}$ .",
"We find higher values of $f_{\\text{unvig}}$ for the observations in the ACIS-S configuration, which is likely related to the better sensitivity of the ACIS-S BI chips to low-energy instrumental X-rays.",
"None of the observations yield very tight constraints on $f_{\\text{XRB}}$ , suggesting that broad constraints on the unresolved XRBs are sufficient for galaxies with hot gas as luminous as NGC 720.",
"In Figure 12 we compare the results from the parametric and non-parametric methods for this galaxy.",
"We take the pdfs from Figure 11 and generate surface brightness profiles for the source emission (gas + XRBs) from the model parameters, and we compare these profiles to the surface brightness profile as inferred from the non-parametric method for obs id 11868.",
"Emission from the central 0.7 kpc was excluded from the parametric fit, although the non-parametric data includes this emission, so the leftmost nonparametric data point falls above the parametric fits.",
"Beyond $\\sim 3$ ' (23 kpc) the profile seems to steepen, suggesting that a second, steeper, component might improve the fit in the outer regions where the signal is below $\\sim 1/5$ of the background.",
"At about 35 kpc the surface brightness profile becomes consistent with zero, although this radius is somewhat dependent on the choice of binning.",
"Fitting a smoothing spline to the profile shows the average behavior at larger radii, and the spline remains above zero until about 50 kpc, where systematic uncertainties (probably a slight underestimate of $f_{\\text{unvig}}$ ) take over the profile.",
"For comparison, the outermost bin in H11 is 40-60 kpc, in which the claimed signal is at least $3\\sigma $ .",
"This could be due to the deeper effective integration time in H11, since they analyze six observations (observation 492 is excluded) simultaneously, while the spline in Figure 12 is just fit to the results from a single observation.",
"On the other hand, H06 has a $2.5\\sigma $ detection of emission in their 62-90 kpc bin, which is difficult to reconcile with our results.",
"One potential issue is that H06 relied solely on observation 492, and the background in this observation is particularly poorly-behaved due to all the flaring events throughout the integration.",
"H11 (in their section 2.1.3) has some discussion of the effect of the deprojection procedure on the outermost annulus in H06 as well.",
"Figure: Comparison of background-subtracted surface brightness profiles for NGC 720 from the non-parametric method (data points) and parametric method (shaded regions).",
"The red shaded regions correspond to the 68% confidence regions for each of the six observations of NGC 720 (the outlier is observation 492 which is significantly affected by flaring).",
"The normalizations have been rescaled into the units of observation 11868 as discussed in the text.",
"Blue and green shaded regions correspond to the 68% confidence regions inferred from H06 and H11 respectively; H06 provides a reasonable fit to the data but H11 significantly overestimates the surface brightness profile.",
"Since they employed the spectral method, both H06 and H11 have larger uncertainties than would be warranted by a fit to the surface brightness profile.",
"The dotted black line shows the level of the background in observation 11868, as determined with the non-parametric method.",
"The solid black line is a smoothing spline fit to the black data points; note that the profile appears to steepen faster than allowed by a β\\beta -model at rr>25kpc.",
"kpc.",
"In this figure we also present the surface brightness profiles inferred from the density profiles of H06 and H11 (see section 5).",
"In order to convert the density profiles into 0.5-2.0 keV X-ray surface brightnesses, we assume values of $Z = 0.6 Z_{\\odot }$ and $kT = 0.5$ keV for the hot gas.",
"Our choice of the $\\beta $ -model already assumes the temperature and abundance profiles are spatially uniform as well; this assumption is not quite true in detail, as H06 and H11 show, but their measured deviations from uniformity do not dramatically affect the shape of the surface brightness profile.",
"We add an X-ray binary component to the surface brightness profiles as well, with the same range of luminosities as that of our fit to observation 11868, although the uncertainties in the gas density profile from H06 and H11 dominate over the XRB uncertainties.",
"The H11 profiles are noticeably discrepant with the data.",
"They are much closer to the anomalous fits to obs id 492, although H11 discarded this observation from their sample due to its flaring contamination.",
"H06, which is actually fit to the data from observation 492 (after stringent flaring correction), provides a much better prediction for the surface brightness profile.",
"The uncertainties in the surface brightness profiles from H06 and H11 are both significantly larger than our uncertainty, and seem larger than are warranted by the data.",
"This can be explained by the small number of annuli used to derive the density profile, since H06 and H11 derive the density in each annulus from the X-ray spectrum.",
"With only a few radii at which the density is measured, the range of profiles which can fit the data is much larger than the range of profiles which are consistent with the (much better spatially resolved) surface brightness data.",
"We also compute the mass enclosed within each profile as a function of radius (Figure 13).",
"In order to do this, we again assume values of $Z = 0.6 Z_{\\odot }$ and $kT = 0.5$ keV for the hot gas, as well as assuming that these parameters are spatially uniform.",
"We indicate with solid colors the region within which each model claims to measure the mass profile, and then we extrapolate the mass model out to 300 kpc and indicate the $1\\sigma $ uncertainties around the extrapolated model.",
"We can see that, as H11 claimed, their mass model does indeed predict the hot halo contains most, if not all, of the missing baryons from the galaxy within 300 kpc (which is close to $R_{200}$ for this galaxy).",
"This is also about a factor of two greater than the prediction of H06, and the two fits are inconsistent by a little more than $1\\sigma $ at radii $$ >$$ 80$ kpc.",
"Our model for the hot gas is consistent with the low end of the range inferred by H06, although our uncertainties are about 50\\% smaller.",
"It is surprising that H11 is inconsistent with both H06 and our model, at all radii.",
"There is some convergence at smaller radii where the background is less important, which is what we would expect if the issue in H11 were confusion between the spectrally similar background and source emission in H11, but there is still a discrepancy at all radii.$ Figure: 1σ1\\sigma regions corresponding to the estimated mass in the hot halo around NGC 720, from three different analyses of the galaxy.",
"Blue corresponds to H06, green to H11, and red to our analysis.",
"The regions are denoted with solid colors out to the radius of the outermost spectral bin (for green and blue), or at the outermost radius where the surface brightness is 1σ1\\sigma above the background (red); at larger radii the profiles are extrapolated outwards and the 1σ1\\sigma region is outlined.",
"Discrepancies between H11 and the other profiles persist at all radii, even for the same assumed temperatures and metallicities for the hot gas.",
"Observation 492, which was significantly affected by flaring, has been excluded from this plot.We note that the uncertainties here reflect only the uncertainties in the determination of the surface brightness profile.",
"There are many additional uncertainties which bear on the determination of the mass of the hot halo as well, such as possible temperature and/or abundance gradients, gas clumping, multiphase effects, etc.",
"The measured temperature and abundance gradients are fairly small, so the gas masses inferred within the dark shaded regions (where the gas is securely detected and measured) are fairly trustworthy.",
"If the profile changes at larger radii (where most of the mass is inferred to lie), the true hot gas content at larger radii could be significantly different from the extrapolated value, so these values are much more uncertain and are illustrated solely to show the effects of the method of determining the profile on the inferred gas mass.",
"The true mass in hot gas is unlikely to be higher than our extrapolated values, though, since most plausible systematic errors (i.e.",
"clumping or incomplete filling factor at large radii) reduce the total gas mass.",
"Moreover, we note that the profile appears to steepen at $r $ >$$$ 3 arcmin in Figure 12, which suggests the density profile declines faster at large radii than our single-component $$-model predicts.$ The results in Figure 13 have significant implications for the baryon budget of this galaxy.",
"The mass of NGC 720 within $R_{200}$H11 uses an alternative definition of the virial radius which puts $R_{vir}$ at about 400 kpc, but here we use the more common choice of $R_{200}$ , which is about 300 kpc for this galaxy.",
"The inferred mass budget looks the same for either choice.",
"is about $2.9\\pm 0.3\\times 10^{12} M_{\\odot }$ [37] , so multiplying by the Cosmic baryon fraction of 0.17 [35] yields an expected total baryonic mass of $5\\times 10^{11} M_{\\odot }$ .",
"The stars comprise $1\\times 10^{11} M_{\\odot }$ [37], leaving $4\\times 10^{11} M_{\\odot }$ of baryons missing from the budget.",
"The extrapolated profile of H11 would therefore place most of these missing baryons in the hot gas, but we have argued that H11 overestimated the mass in hot gas, and the true mass is unlikely to be higher than about $1.5\\times 10^{11} M_{\\odot }$ .",
"The “missing” baryons could lie in a cooler phase, such as the $$ <$$ 105$ K gas probed by \\cite {Thom2012}, or outside the virial radius.",
"We also cannot rule out the possibility of a second component in the million-degree gas, such as a low-metallicity, nearly uniform density diffuse medium, though we see no evidence for it in the surface brightness profile, and in fact the profile seems to steepen at $ r$\\;>\\over {\\scriptstyle \\sim }\\;$ 25$ kpc instead of flattening as one would expect from such a model.$"
],
[
"Conclusion",
"In this paper, we have accomplished several goals.",
"We argued that neither spectral fitting nor spatial fitting, at their current levels of sophistication, are adequate for the analysis of extended faint X-ray emission below about 1/5 of the background.",
"We have also argued that an ideal approach would study X-ray observations at the level of the events file, and therefore make use of the spectral and spatial information simultaneously.",
"In order to take the first steps towards such an approach, we presented an improvement to spatial fitting, wherein we model the entire image within a single energy band (0.5-2.0 keV).",
"This discards more detailed energy information but can be extended in this direction in future work.",
"We showed that a typical X-ray background can be decomposed into vignetted and unvignetted components.",
"These components have different spectral shapes as well as different spatial distributions; in a given energy band, the different spatial distributions of these backgrounds can be used to constrain their relative contributions to an image.",
"We introduced two methods of performing this decomposition.",
"The nonparametric method excises a region around the source and fits the rest of the image in order to estimate the ratio of vignetted and unvignetted backgrounds, with no assumptions about the spatial distribution of the source emission other than that it must be localized to one region in the image which can be excluded.",
"The nonparametric method will therefore not work for observations of diffuse emission which fills all or most of the field.",
"The parametric method introduces a parametric model for the extended source (we examine $\\beta $ -models in this paper), and therefore is not limited to sources of small angular extent.",
"We explored a number of possible likelihood functions for comparing the models to data.",
"We decided upon a hybrid pixel-by-pixel likelihood function, with pixels binned where the psf is largest (and the correspond point source detectability the poorest).",
"We showed that this likelihood function can reliably recover the shape of the background across the full spectrum from 100% unvignetted to 100% vignetted.",
"This method works with either wavdetect or with our manual point source detection algorithm, although we find the best results by masking point sources if either method detects them.",
"We tested both methods for simulated extended sources observed with Chandra, in both the ACIS-I and ACIS-S configurations.",
"We showed that we can recover the source emission well, recovering $\\beta $ (a measure of the slope of the surface brightness profile) to 10% accuracy and the total number of source counts to 25% accuracy, for a source about half as bright as NGC 720.",
"The method works better if the source is placed somewhat off-axis, so that its surface brightness profile can more easily be distinguished from the instrumental vignetting profile.",
"Finally we applied our method to the isolated elliptical galaxy NGC 720.",
"Out of two previous studies of the density profile of its hot halo, one study (Humphrey et al.",
"2006) seems to predict an X-ray surface brightness profile that matches reasonably well with the observed data, while the other study (Humphrey et al.",
"2011) systematically over-predicts the X-ray surface brightness.",
"Both studies only measure density or surface brightness in a handful of annuli ($8-11$ ), however, so the inferred surface brightness profiles have large uncertainties - much larger than are warranted by examination of the surface brightness profiles directly.",
"We argue this is an inherent feature of the spectral method, and is one reason why combined spatial and spectral analysis is preferable.",
"The surface brightness profile predicted by our models offers a much better fit to the data.",
"With this method, we are also able to trace the source emission to well below a tenth of the background - a significant improvement over previous methods.",
"Both our parametric and non-parametric methods largely agree with each other, although our parametric model is insufficiently detailed to capture deviations from the single-component $\\beta $ -model which we observe at radii $$ >$$ 3$^{\\prime }.$ The implied mass of the hot halo extrapolated to 300 kpc is $$ <$$ 1.51011 M$.",
"This is insufficient by about a factor of three to bring the galaxy to baryonic closure, in contrast to the conclusions of \\cite {Humphrey2011}.",
"We instead estimate that, after accounting for the stars and the hot gas, NGC 720 is missing over half of its baryons.",
"This brings the galaxy into closer agreement with our estimates in \\cite {Anderson2013} of the amount of hot gas around typical L* galaxies based on stacked images from the ROSAT All-Sky Survey.$"
],
[
"Acknowledgements",
"The authors would like to thank the anonymous referee for a very helpful report that substantially improved the clarity and generality of the manuscript.",
"The authors would also like to thank E. Bell, E. Churazov, A. Evrard, O. Gnedin, C. Miller, J. Miller, and M. Ruszkowski for helpful suggestions and comments on draft versions of the manuscript, as well as M. Miller and C. Slater for useful conversations relating to this work.",
"J. Davis also kindly explained some of the features in MARX, and the documentation for all of the Chandra analysis software was uniformly excellent and informative.",
"The authors would also like to thank T. Gaetz and P. Plucinsky for help interpreting the stowed Chandra background.",
"Since the original data from H06 and H11 were unavailable, the free software WebPlotDigitizer (http://arohatgi.info/WebPlotDigitizer/), which is available under a GNU General Public License, was used to read the density profiles from these papers.",
"This research has been funded by NASA ADAP grant NNX11AJ55G.",
"This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory.",
"This research has made use of NASA's Astrophysics Data System.",
"figuresection"
],
[
"Appendix 1: Nongaussianity of the Chandra psf",
"The true Chandra psf is not exactly Gaussian.",
"It is close to Gaussian in the core, but has wings which are more extended than a Gaussian.",
"We can check how important these wings are by running mkpsfmap with larger ecfs, using the 0.5-2.0 keV energy band.",
"We compare the radii corresponding to higher ecfs to the expected radii if the emission were Gaussian with $\\sigma $ equal to the radius enclosing 39.3% of the emission.",
"The results are below, in Figure A1, for ecfs of 75%, 90%, and 95%.",
"Figure: Ratios of the radii corresponding to the (a) 75% ecf, (b) 90% ecf, and (c) 95% ecf, each divided by the radius corresponding to the 39.3% ecf.",
"This entire ratio is then divided by the expected value of the ratio if the psf were perfectly Gaussian; thus the plots illustrate the amount of nongaussianity in the psf at the given ecf.",
"There is little nongaussianity in the psf until radii corresponding to the 95% ecf, suggesting that a Gaussian is a good approximation to the psf to about the 5% level.",
"There is a region in the center of the ACIS-I array where the psf is more strongly non-Gaussian, but the size of the psf here is small enough that deviations do not effect the image substantially.Overall the psf does seem very close to Gaussian, with deviations becoming more visible at higher ecfs as the extended wings start to dominate.",
"The deviations are not very significant across most of the detector even at an ecf of 95%, suggesting that the deviations from Gaussianity are only at about the 5% level.",
"The psf is more non-Gaussian at the center of the detector, and a roughly 1-arcminute region near the very center is more extended than Gaussian even in the 75% ecf comparison.",
"Fortunately, the size of the psf in this region is still less than an arcsecond, so the effect of these nongaussianities is negligible, and we neglect them in this work."
],
[
"Appendix 2: Point Source Detection Algorithm Comparison",
"Here we examine a 110 ks observation (obs id 8595) of the CDF-S. We reduce the image as described in section 3.",
"We then run wavdetect on the image, masking out the $8\\sigma $ ellipses around each point source.",
"In Figure B1, we show the results from wavdetect with our parameters, and compare them to the CDF-S 4 megasecond catalog of point sources [78].",
"We do not recover all these point sources in the 110 ks image from observation 8595, which is unsurprising, but we do recover the bright point sources.",
"There is one false positive in the image as well.",
"Note that false positives are not a significant problem for us: they cost us a few background photons that are unnecessarily masked out, and probably bias downwards our estimate of the background flux rate slightly, but this effect is negligible for the low number of false positives generated by our significance threshhold.",
"Figure: Comparison of point sources in observation 8595 detected with wavdetect (green; note our unusually large ellipses) to all the known point sources in the field (red; for these sources the size of the circles is arbitrary) from the 4 megasecond analysis (Xue et al.",
"2011).",
"We detect most of the bright point sources, and have one false positive (slightly to the left of the center of the ACIS-I array).As a comparison, in Figure B2 we also illustrate the point sources detected in this image using our manual point source detection algorithm (section 3.3.1).",
"This method measures the likelihood of obtaining the observed number of counts within a radius of the 90% enclosed counts fraction (ecf) of the psf, at the location of every photon on the image.",
"It is therefore more computationally intensive than wavdetect.",
"It also detects slightly fewer point sources overall, but at large off-axis angles it appears to outperform wavdetect; this is notable in Figure B2 on the left side of the S2 chip where wavdetect misses a fairly bright and extended source.",
"None of the sources detected using the manual method are false positives.",
"We compare the effects of the two different methods on recovering the correct unvignetted fraction of the background in Appendix 3.",
"Figure: Comparison of point sources detected with wavdetect (green; note our unusually large ellipses) and our manual point source detection algorithm (red).",
"The two methods generally agree with one another, although each catches a few point sources missed by the other, and the manual method performs better on the ACIS-S chip very far off-axis."
],
[
"Appendix 3: The Likelihood Function",
"We explored a number of different likelihood functions for comparing observed and simulated images.",
"A natural choice is the pixel-by-pixel function, defined as $L \\equiv \\prod _{\\text{all pixels}} p(c_{\\text{pix}} | m_{\\text{pix}}) $ where $p(c|m)$ is the Poisson probability of obtaining the observed number of counts $c$ in a pixel, given a model prediction $m$ for that pixel.",
"We can also bin the data in various ways $L \\equiv \\prod _{\\text{all bins}} p(c_{\\text{bin}} | m_{\\text{bin}}) $ where now we are adding pixels together within a bin before evaluating the likelihood.",
"We explored binning by off-axis angle and by the size of the psf, in various combinations.",
"We also included an additional component in the model to account for undetected point sources, which are more likely to be undetected at large radii.",
"We discussed how to generate this component above, in section 3.3.2.",
"To examine which of these likelihood functions gives the most reliable estimator, we simulated 11 images in each of the ACIS-I and ACIS-S configurations, based on the exposure maps for the CDF-S (obs id 8595, also examined in Appendix 2) and NGC 720 (obs id 7372) respectively.",
"We used the 0.5-2.0 keV band and gave each image the same average flux per pixel as observation 8595 (i.e.",
"0.035 counts pix$^{-1}$ in the diffuse background).",
"The 11 images used different values of $f_{\\text{unvig}}$ ranging from 0% to 100%, in steps of 10%.",
"We also added simulated point sources to each image as described in section 3.3.2, and then applied our wavdetect + manual point source masking algorithms as described in section 3.3.1.",
"For each image, we computed the likelihood as a function of $f_{\\text{unvig}}$ , using each of our different likelihood functions.",
"We estimate uncertainties in the recovered value of the unvignetted fraction using the procedure discussed in [48].",
"As long as the likelihood is well-behaved near the maximum, then the k$\\sigma $ confidence interval around the maximum is “bounded by the values that that correspond to the function $2ln(L)$ dropping by $k^2$ .” So the $1-\\sigma $ region around the best-fit value of the unvignetted background fraction is just the region where the log-likelihood is within $0.61$ of the maximum value.",
"We evaluate the effectiveness of recovering $f_{\\text{unvig}}$ for each likelihood function by calculating the $\\chi ^2$ , i.e.",
"(recovered $f_{\\text{unvig}}$ - true $f_{\\text{unvig}}$ )$^2$ / ($\\sigma f_{\\text{unvig}}$ (recovered)).",
"This is not technically correct at the extreme values, since $f_{\\text{unvig}}$ cannot go below 0 or above 1 and the errors are therefore not Gaussian at the edges.",
"However, we checked the subset of simulations with the extreme points excluded (values of $f_{\\text{unvig}}$ between 0.1 and 0.9) and obtained essentially the same results, so this effect does not appear to bias our results in any meaningful way.",
"We consider the following likelihood functions: - pixel-by-pixel, described in equation 1.",
"- uniform radial bins, where we divide the image into 20 annuli of the same radius, centered around the aimpoint.",
"We bin together all the counts within each annulus before evaluating the likelihood.",
"- $r^{-1/2}$ radial bins, where we divide the image into 20 annuli with radii decreasing as $r^{-1/2}$ (yielding approximately constant area per annulus), centered around the aimpoint.",
"We bin together all the counts within each annulus before evaluating the likelihood.",
"- psf size bins, where we construct 20 bins based on the size of the at each pixel.",
"Pixels with similar values for the radius of the 90% ecf are binned together before evaluating the likelihood.",
"- hybrid, which is a pixel-by-pixel likelihood function for the $N$ % of pixels where the psf is smallest, and we bin the remaining (100-$N$ )% pixels into $M$ bins determined by the size of the psf within that pixel.",
"We explore three versions of hybrid function, with (N,M) = (10%, 1 bin), (20%, 2 bins), and (30%, 3 bins).",
"We study each of these likelihood functions both with (+ps) and without (-ps) the correction for undetected point sources (section 3.3.2) included in the model.",
"For reference, with 10 d.o.f., the fits from a likelihood function can be excluded at 95% confidence if $\\chi ^2 > 18.31$ and at 99% confidence if $\\chi ^2 > 23.21$ .",
"In Table 1 we list the values of $\\chi ^2$ for each of the various likelihood functions we examined.",
"To make the table easier to read, we have averaged the $\\chi ^2$ from ACIS-S and ACIS-I for each likelihood function.",
"There were no large or systematic differences between the results in the two configurations, so no important information is lost by averaging the two values together.",
"lcccc 5 Likelihood Function Comparison Likelihood function $\\chi ^2 $ $\\chi ^2$ $\\chi ^2$ $\\chi ^2$ (no masking) (manual method) (wavdetect) (manual & wavdetect) pixel-by-pixel -ps $>99$ 30.6 31.4 25.6 pixel-by-pixel +ps $>99$ 19.6 14.3 16.7 uniform radial bins -ps $>99$ 24.0 26.5 16.9 uniform radial bins +ps $>99$ 35.1 20.0 21.2 $r^{-1/2}$ radial bins -ps $>99$ 24.6 50.6 18.2 $r^{-1/2}$ radial bins +ps $>99$ 61.2 22.0 22.0 psf size bins -ps $>99$ 22.1 35.3 23.4 psf size bins +ps $>99$ 35.6 20.0 26.1 hybrid, 1 bin -ps $>99$ $>99$ $>99$ $>99$ hybrid, 1 bin +ps $>99$ 17.7 14.5 14.1 hybrid, 2 bins -ps $>99$ $>99$ $>99$ $>99$ hybrid, 2 bins +ps $>99$ 15.2 16.2 16.9 hybrid, 3 bins -ps $>99$ $>99$ $>99$ $>99$ hybrid, 3 bins +ps $>99$ 22.8 19.4 24.1 List of $\\chi ^2$ goodness of fit statistics for recovered values of $f_{\\text{unvig}}$ using different likelihood functions and point source detection algorithms.",
"Formally, the fits from a likelihood function can be excluded at 95% confidence if $\\chi ^2 > 18.3$ and at 99% confidence if $\\chi ^2 > 23.2$ .",
"The wide variation in $\\chi ^2$ shows that the choice of likelihood function is important.",
"The fit with the best average $\\chi ^2$ is the hybrid pixel-by-pixel method, with both wavedtect and our manual point source masking method, and a component included in the model to account for undetected point sources.",
"This method also gives reasonably-sized uncertainties in $f_{\\text{unvig}}$ of a few percent for an observation of $\\sim 100$ ks.",
"We therefore adopt this method for most of the analysis in this paper."
]
] | 1403.0584 |
[
[
"A Six-Point Ceva-Menelaus Theorem"
],
[
"Abstract We provide a companion to the recent Benyi-Curgus generalization of the well-known theorems of Ceva and Menelaus, so as to characterize both the collinearity of points and the concurrence of lines determined by six points on the edges of a triangle.",
"A companion for the generalized area formula of Routh appears, as well."
],
[
"letterpaper justification=centering [subfigure]labelformat=parens, labelsep=space, labelfont=normalfont We provide a companion to the recent Bényi-Ćurgus generalization of the well-known theorems of Ceva and Menelaus, so as to characterize both the collinearity of points and the concurrence of lines determined by six points on the edges of a triangle.",
"A companion for the generalized area formula of Routh appears, as well.",
"The venerable theorems of (Giovanni) Ceva and Menelaus (of Alexandria) concern points on the edge-lines of a triangle.",
"Each point defines —and is defined by— the ratio of lengthsThroughout, we consider segment lengths to be signed, with each of $\\overrightarrow{AB}$ , $\\overrightarrow{BC}$ , $\\overrightarrow{CA}$ —for distinct $A$ , $B$ , $C$ — indicating the direction of a positively-signed segment on the corresponding (extended) side of the triangle.",
"Moreover, we adopt these conventions regarding ratios of these lengths: $\\frac{|PP|}{|PQ|} = 0 \\qquad \\qquad \\frac{|PQ|}{|QQ|} = \\infty \\qquad \\qquad \\frac{|PX|}{|XQ|} = -1, \\;\\; \\text{for $X$ the point at infinity on $\\overleftrightarrow{PQ}$}$ of collinear segments joining it to two of the triangle's vertices, and the theorems use a trio of such ratios to neatly characterize the special configurations in Figure 1.",
"Throughout, we consider segment lengths to be signed, with each of $\\overrightarrow{AB}$ , $\\overrightarrow{BC}$ , $\\overrightarrow{CA}$ —for distinct $A$ , $B$ , $C$ — indicating the direction of a positively-signed segment on the corresponding (extended) side of the triangle.",
"Moreover, we adopt these conventions regarding ratios of these lengths: $\\frac{|PP|}{|PQ|} = 0 \\qquad \\qquad \\frac{|PQ|}{|QQ|} = \\infty \\qquad \\qquad \\frac{|PX|}{|XQ|} = -1, \\;\\; \\text{for $X$ the point at infinity on $\\overleftrightarrow{PQ}$}$ Figure: Points on a triangle's edges,lying on a common line.",
"(Menelaus) Specifically, with $D$ , $E$ , $F$ on edge-lines opposite respective vertices $A$ , $B$ , $C$ , we write $d := \\frac{|BD|}{|DC|} \\qquad e := \\frac{|CE|}{|EA|} \\qquad f := \\frac{|AF|}{|FB|}$ and express the theorems as follows: Theorem 1 (Ceva) Lines $\\overleftrightarrow{AD}$ , $\\overleftrightarrow{BE}$ , $\\overleftrightarrow{CF}$ pass through a common point if and only if $d e f = \\phantom{-}1$ Theorem 2 (Menelaus) Points $D$ , $E$ , $F$ lie on a common line if and only if $d e f = -1$ Bényi and Ćurgus [1], and this author, independently (and nearly-simultaneously) considered separate aspects of the same approach to generalizing the above —namely, doubling the number of points on the triangle's edges— arriving at equations whose terms, in the grand Ceva-Menelaus tradition, differ only in sign.",
"Interestingly, the Bényi-Ćurgus result concerns Ceva-like elements (lines through vertices) and a Menelaus-like phenomenon (collinearity of points).",
"This author's contribution, on the other hand, concerns Menelaus-like elements (points on edges) and a Ceva-like phenomenon (concurrence of lines).",
"Figure: Points on lines through vertices,lying on a common line.",
"(Bényi-Ćurgus)Place points $A^{+}$ and $A^{-}$ on the edge-line opposite vertex $A$ ; likewise, $B^{+}$ and $B^{-}$ opposite $B$ , and $C^{+}$ and $C^{-}$ opposite $C$ .",
"Define these ratios:Observe that the superscripts emphasize an opposing directionality in the definitions of the ratios.",
"For instance, the points in ratio $a^{+}$ trace the path $B$ -$A^{+}$ -$C$ , with endpoints oriented in the positive direction; in $a^{-}$ , the path $C$ -$A^{-}$ -$B$ has endpoints oriented in the negative direction.",
"Were we to define all six ratios in “matching” orientations —as was done in [1]— the resulting formulas would lose some clarity and symmetry.",
"$\\begin{array}{c}\\displaystyle a^{+} := \\frac{|BA^{+}|}{|A^{+}C|} \\qquad b^{+} := \\frac{|CB^{+}|}{|B^{+}A|} \\qquad c^{+} := \\frac{|AC^{+}|}{|C^{+}B|} \\\\[10pt]\\displaystyle a^{-} := \\frac{|CA^{-}|}{|A^{-}B|} \\qquad b^{-} := \\frac{|AB^{-}|}{|B^{-}C|} \\qquad c^{-} := \\frac{|BC^{-}|}{|C^{-}A|}\\end{array}$ Theorem 3 (Six-Point Ceva-Menelaus Theorem) $\\phantom{xyzzy}$ Lines $\\overleftrightarrow{B^{+}C^{-}}$ , $\\overleftrightarrow{C^{+}A^{-}}$ , $\\overleftrightarrow{A^{+}B^{-}}$ pass through a common point if and only if $a^{+} b^{+} c^{+} \\;+\\; a^{-} b^{-} c^{-} \\;=\\; \\phantom{-}1 \\;-\\; a^{+} a^{-} \\;-\\; b^{+} b^{-} \\;-\\; c^{+} c^{-}$ (Bényi-Ćurgus) PointsTo amplify the duality with (a), we write “$\\widehat{P^{-}Q^{+}}$ ” for the point of intersection of lines $\\overleftrightarrow{PP^{-}}$ and $\\overleftrightarrow{QQ^{+}}$ .",
"$\\widehat{B^{-}C^{+}}$ , $\\widehat{C^{-}A^{+}}$ , $\\widehat{A^{-}B^{+}}$ lie on a common line if and only if $a^{+} b^{+} c^{+} \\;+\\; a^{-}b^{-}c^{-} \\;=\\; -1 \\;+\\; a^{+} a^{-} \\;+\\; b^{+} b^{-} \\;+\\; c^{+} c^{-}$ Lines $\\overleftrightarrow{B^{+}C^{-}}$ , $\\overleftrightarrow{C^{+}A^{-}}$ , $\\overleftrightarrow{A^{+}B^{-}}$ pass through a common point if and only if $a^{+} b^{+} c^{+} \\;+\\; a^{-} b^{-} c^{-} \\;=\\; \\phantom{-}1 \\;-\\; a^{+} a^{-} \\;-\\; b^{+} b^{-} \\;-\\; c^{+} c^{-}$ (Bényi-Ćurgus) PointsTo amplify the duality with (a), we write “$\\widehat{P^{-}Q^{+}}$ ” for the point of intersection of lines $\\overleftrightarrow{PP^{-}}$ and $\\overleftrightarrow{QQ^{+}}$ .",
"$\\widehat{B^{-}C^{+}}$ , $\\widehat{C^{-}A^{+}}$ , $\\widehat{A^{-}B^{+}}$ lie on a common line if and only if $a^{+} b^{+} c^{+} \\;+\\; a^{-}b^{-}c^{-} \\;=\\; -1 \\;+\\; a^{+} a^{-} \\;+\\; b^{+} b^{-} \\;+\\; c^{+} c^{-}$ Note: Identifying $A^{-}$ , $B^{-}$ , $C^{-}$ with $C$ , $B$ , $A$ yields $a^{-} = b^{-} = c^{-} = 0$ , so that (REF ) and (REF ) reduce to (REF ) and (REF ).",
"The Six-Point Theorem generalizes the traditional results.",
"For proof, one can invoke vector techniques, as indicated with Theorem REF below.",
"When Ceva's lines fail to concur, and when Menelaus' points fail to “colline”, they determine triangles.",
"One might well ask how the areaAs with length, we consider triangle area to be signed.",
"Areas $|\\triangle ABC|$ and $|\\triangle DEF|$ agree in sign when vertex paths $A$ -$B$ -$C$ -$A$ and $D$ -$E$ -$F$ -$D$ trace their respective figures in the same direction; similarly for triangles defined by their edge-lines.",
"of each resulting triangle compares to that of the original figure.",
"(Edward John) Routh provided answers.",
"(See [1].)",
"Figure: Triangle from points on edges.Theorem 4 (Routh's Formulas) (“Routh's Theorem”).",
"The triangle with (non-parallel) edge-lines $\\overleftrightarrow{AD}$ , $\\overleftrightarrow{BE}$ , $\\overleftrightarrow{CF}$ has area $|\\triangle ABC| \\cdot \\frac{\\left(\\; d e f - 1 \\;\\right)^2}{\\left(\\; 1 + d + d e \\;\\right)\\left(\\; 1 + e + e f \\;\\right) \\left(\\; 1 + f + f d \\;\\right)}$ The triangle with (finite) vertices $D$ , $E$ , $F$ has area $|\\triangle ABC| \\cdot \\frac{\\; d e f + 1 \\;}{\\left(\\; 1 + d \\;\\right) \\left(\\; 1 + e \\;\\right) \\left(\\; 1 + f \\;\\right)}$ (“Routh's Theorem”).",
"The triangle with (non-parallel) edge-lines $\\overleftrightarrow{AD}$ , $\\overleftrightarrow{BE}$ , $\\overleftrightarrow{CF}$ has area $|\\triangle ABC| \\cdot \\frac{\\left(\\; d e f - 1 \\;\\right)^2}{\\left(\\; 1 + d + d e \\;\\right)\\left(\\; 1 + e + e f \\;\\right) \\left(\\; 1 + f + f d \\;\\right)}$ The triangle with (finite) vertices $D$ , $E$ , $F$ has area $|\\triangle ABC| \\cdot \\frac{\\; d e f + 1 \\;}{\\left(\\; 1 + d \\;\\right) \\left(\\; 1 + e \\;\\right) \\left(\\; 1 + f \\;\\right)}$ Observe that the numerator in each of these formulas —and, thus, the area of the triangle in question— vanishes, as it should, when (and only when) the conditions for Ceva's or Menelaus' theorems indicate that the triangle degenerates into a point or a line.",
"The reader may verify that a denominator vanishes when (and only when) the triangle becomes unbounded, having non-finite vertices and parallel edges.",
"Bényi and Ćurgus [1] specifically address the six-point generalization of Theorem 4b.",
"At the suggestion of Mr. Ćurgus, this author derived the counterpart generalization of 4a.",
"Figure: Triangle from points on lines through vertices.",
"(Bényi-Ćurgus)Theorem 5 (Six-Point Routh Formulas) $\\phantom{xyzzy}$ The triangle with (non-parallel) edge-lines $\\overleftrightarrow{B^{+}C^{-}}$ , $\\overleftrightarrow{C^{+}A^{-}}$ , $\\overleftrightarrow{A^{+}B^{-}}$ has area $|\\triangle ABC| \\cdot \\frac{\\left(\\; a^{+} b^{+} c^{+} + a^{-} b^{-} c^{-} + a^{+} a^{-} + b^{+} b^{-} +c^{+} c^{-} - 1 \\;\\right)^2}{\\begin{array}{c}\\phantom{\\cdot }\\left(\\; 1 - a^{+} a^{-} + b^{-} ( 1 + a^{-} ) + c^{+} ( 1 + a^{+} ) \\;\\right) \\\\[3pt]\\cdot \\left(\\; 1 - b^{+} b^{-} + c^{-} ( 1 + b^{-} ) + a^{+} ( 1 + b^{+} ) \\;\\right) \\\\[3pt]\\cdot \\left(\\; 1 - c^{+} c^{-} + a^{-} ( 1 + c^{-} ) + b^{+} ( 1 + c^{+} ) \\;\\right)\\end{array}}$ (Bényi-Ćurgus).",
"The triangle with (finite) vertices $\\widehat{B^{-}C^{+}}$ , $\\widehat{C^{-}A^{+}}$ , $\\widehat{A^{-}B^{+}}$ has area $|\\triangle ABC| \\cdot \\frac{\\; a^{+} b^{+} c^{+} + a^{-} b^{-} c^{-} - a^{+} a^{-} - b^{+} b^{-} -c^{+} c^{-} + 1 \\;}{\\left(\\; 1 + b^{-} + c^{+} \\;\\right)\\left(\\; 1 + c^{-} + a^{+} \\;\\right) \\left(\\; 1 + a^{-} + b^{+} \\;\\right)}$ The triangle with (non-parallel) edge-lines $\\overleftrightarrow{B^{+}C^{-}}$ , $\\overleftrightarrow{C^{+}A^{-}}$ , $\\overleftrightarrow{A^{+}B^{-}}$ has area $|\\triangle ABC| \\cdot \\frac{\\left(\\; a^{+} b^{+} c^{+} + a^{-} b^{-} c^{-} + a^{+} a^{-} + b^{+} b^{-} +c^{+} c^{-} - 1 \\;\\right)^2}{\\begin{array}{c}\\phantom{\\cdot }\\left(\\; 1 - a^{+} a^{-} + b^{-} ( 1 + a^{-} ) + c^{+} ( 1 + a^{+} ) \\;\\right) \\\\[3pt]\\cdot \\left(\\; 1 - b^{+} b^{-} + c^{-} ( 1 + b^{-} ) + a^{+} ( 1 + b^{+} ) \\;\\right) \\\\[3pt]\\cdot \\left(\\; 1 - c^{+} c^{-} + a^{-} ( 1 + c^{-} ) + b^{+} ( 1 + c^{+} ) \\;\\right)\\end{array}}$ (Bényi-Ćurgus).",
"The triangle with (finite) vertices $\\widehat{B^{-}C^{+}}$ , $\\widehat{C^{-}A^{+}}$ , $\\widehat{A^{-}B^{+}}$ has area $|\\triangle ABC| \\cdot \\frac{\\; a^{+} b^{+} c^{+} + a^{-} b^{-} c^{-} - a^{+} a^{-} - b^{+} b^{-} -c^{+} c^{-} + 1 \\;}{\\left(\\; 1 + b^{-} + c^{+} \\;\\right)\\left(\\; 1 + c^{-} + a^{+} \\;\\right) \\left(\\; 1 + a^{-} + b^{+} \\;\\right)}$ Treating points as vectors, we can write $A^{+} = \\frac{B (1+a^{+}) + Ca^{+}}{1+a^{+}} \\qquad A^{-} = \\frac{Ba^{-} + C(1+a^{-})}{1+a^{-}} \\qquad \\text{, etc.",
"}$ to find, after a bit of tedious algebra, that the vertices of the triangles in parts (a) and (b) of the theorem have the respective forms $\\frac{A ( 1 - a^{-} a^{+}) + B ( c^{+} + a^{-} b^{-} ) + C ( b^{-} + a^{+} c^{+} )}{1 - a^{-} a^{+} + b^{-} ( 1 + a^{-} ) + c^{+} ( 1 + a^{+} )}\\qquad \\text{and}\\qquad \\frac{A + B c^{+} + Cb^{-}}{1 + c^{+} + b^{-}}$ The area formulas follow from a bit more —and more-tedious— algebra.",
"Of course, since ratios of lengths of collinear segments, and of areas of coplanar triangles, are preserved under affine transformation, one could simplify this analysis somewhat by assuming, say, $A = (0,0)$ , $B=(1,0)$ , $C=(0,1)$ ; even in generality, however, verification of these formulas amounts to just a few seconds' effort from a computer algebra system.",
"Treating points as vectors, we can write $A^{+} = \\frac{B (1+a^{+}) + Ca^{+}}{1+a^{+}} \\qquad A^{-} = \\frac{Ba^{-} + C(1+a^{-})}{1+a^{-}} \\qquad \\text{, etc.",
"}$ to find, after a bit of tedious algebra, that the vertices of the triangles in parts (a) and (b) of the theorem have the respective forms $\\frac{A ( 1 - a^{-} a^{+}) + B ( c^{+} + a^{-} b^{-} ) + C ( b^{-} + a^{+} c^{+} )}{1 - a^{-} a^{+} + b^{-} ( 1 + a^{-} ) + c^{+} ( 1 + a^{+} )}\\qquad \\text{and}\\qquad \\frac{A + B c^{+} + Cb^{-}}{1 + c^{+} + b^{-}}$ The area formulas follow from a bit more —and more-tedious— algebra.",
"Of course, since ratios of lengths of collinear segments, and of areas of coplanar triangles, are preserved under affine transformation, one could simplify this analysis somewhat by assuming, say, $A = (0,0)$ , $B=(1,0)$ , $C=(0,1)$ ; even in generality, however, verification of these formulas amounts to just a few seconds' effort from a computer algebra system.",
"We can accentuate the duality of the traditional results of Ceva and Menelaus by reciting them thusly: “Points determined by pairs of lines through the (vertex-)points of a triangle coincide if and only if ...” versus “Lines determined by pairs of points on the (edge-)lines of a triangle coincide if and only if ...”.",
"Taking a deep breath, we can do likewise for the parts of the Six-Point Theorem: “Points determined by pairs of lines determined by pairs of points on the (edge-)lines of a triangle coincide if and only if ...” versus “Lines determined by pairs of points determined by pairs of lines through the (vertex-)points of a triangle coincide if and only if ...”.",
"What formulas characterize the coincidence of lines and/or points at the next order of complexity?",
"For that matter, what strategy for pairing lines and/or points best constitutes the next order of complexity?"
]
] | 1403.0478 |
[
[
"Extracting baryon-antibaryon strong interaction potentials from\n p$\\bar{\\Lambda}$ femtoscopic correlation function"
],
[
"Abstract The STAR experiment has measured $p\\Lambda$, $\\bar{p}\\bar{\\Lambda}$, $\\bar{p}\\Lambda$, and $p\\bar{\\Lambda}$ femtoscopic correlation functions in central Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.",
"The system size extracted for $p\\Lambda$ and $\\bar{p}\\bar{\\Lambda}$ is consistent with model expectations and results for other pair types, while for $p\\bar{\\Lambda}$ and $\\bar{p}\\Lambda$ it is not consistent with the other two and significantly lower.",
"In addition an attempt was made to extract the unknown parameters of the strong interaction potential for this baryon-antibaryon ($B\\bar{B}$) pair.",
"In this work we reanalyze the STAR data, taking into account residual femtoscopic correlations from heavier $B\\bar{B}$ pairs.",
"We obtain new estimates for the system size, consistent with the results for $p\\Lambda$ and $\\bar{p}\\bar{\\Lambda}$ pairs and with model expectations.",
"We give new estimates for the strong interaction potential parameters for $p\\bar{\\Lambda}$ and show that similar constraints can be given for parameters for other, heavier $B\\bar{B}$ pairs."
],
[
"Introduction",
"Strong interaction in a two-baryon system is one of the fundamental problems in QCD [1], [2].",
"Such processes are measured in dedicated experiments [3], [4], [5] and significant body of data exists for baryon-baryon (BB) interactions [6].",
"Baryon-antibaryon ($B\\bar{B}$ ) interaction includes a contribution from matter-antimatter annihilation.",
"This process for $p$$\\bar{p}$ was studied in great detail theoretically [7], [8], [9], [10] and is measured with good precision [6].",
"However no measurement exist for any $B\\bar{B}$ system other than $p$$\\bar{p}$ , $p$$\\bar{n}$ and $\\bar{p}$$d$ .",
"There is also little theoretical guidance on what to expect for $B\\bar{B}$ interaction for other baryon types.",
"The standard hadronic rescattering code used in heavy-ion collision modeling, UrQMD [11], assumes that any $B\\bar{B}$ interaction has the same parameters as the $p$$\\bar{p}$ , expressed either as a function of relative momentum or $\\sqrt{s}$ of the pair.",
"The STAR experiment has measured $p$$\\bar{\\Lambda }$ femtoscopic correlation [12] in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.",
"In that work a novel method was proposed to determine the parameters of the strong interaction potential for $B\\bar{B}$ pairs, using such correlations [13].",
"An estimate for the real and imaginary part of the scattering length $f_{0}$ was given, showing significant imaginary component, reflecting $B\\bar{B}$ annihilation in this channel.",
"At the same time femtoscopic system size (radius) was extracted.",
"Surprisingly it was 50% lower then the one for regular $BB$ pairs at similar pair transverse mass $m_{T}$ .",
"It was also inconsistent with hydrodynamic model predictions, which give approximate scaling of the radii with $1/\\sqrt{m_{T}}$ .",
"This scaling is in agreement with all other femtoscopic measurements performed at RHIC, for meson and baryon pairs.",
"Seen in this light, the validity of the $p$$\\bar{\\Lambda }$ analysis should be reconsidered if any significant new effects contributing to such functions are identified.",
"The issue of the residual correlations (RC) in femtoscopic correlations of $BB$ pairs is mentioned in [12], but the work explicitly states that it is not addressed and acknowledges this fact as a weak aspect of the analysis method.",
"In this work we show that proper treatment of RC is of central importance for any $BB$ measurement, but in particular in the $B\\bar{B}$ analysis.",
"On the example of the STAR data we show how the extracted radius and scattering length change when RC are properly taken into account.",
"We reanalyze the STAR data with the formalism which includes the RC contribution.",
"We test, whether the extracted radius is then compatible with other measurements and model expectations.",
"In the process we make assumptions on the strong interaction parameters for several $B\\bar{B}$ pairs, and show if the extracted values are sensitive to those assumptions.",
"As a results we put constraints on the $B\\bar{B}$ strong interaction parameters, particularly on the imaginary part of the scattering length, which parametrizes the $B\\bar{B}$ annihilation process at low relative momentum.",
"The paper is organized as follows.",
"In Sec.",
"we describe the femtoscopic formalism, including the RC treatment.",
"In Sec.",
"we discuss various theoretical assumptions needed for the reanalysis of the data, and define four reasonable parameter sets for the theoretical description of the $B\\bar{B}$ interaction.",
"In Sec.",
"we examine the STAR data from [12] and show how they should be reanalyzed in the frame of the formalism taking into account the RC.",
"In Sec.",
"we apply the formalism to the STAR data and discuss the results.",
"In Sec.",
"we provide the conclusions and give recommendations for future measurements."
],
[
"Femtoscopic formalism",
"The femtoscopic correlation function is defined as a ratio of the conditional probability to observe two particles together, divided by the product of probabilities to observe each of them separately.",
"Experimentally it is measured by dividing the distribution of relative momentum of pairs of particles detected in the same collision (event) by an equivalent distribution for pairs where each particle is taken from a different collision.",
"This is the procedure used by STAR, details are given in [12].",
"The femtoscopy technique focuses on the mutual two-particle interaction.",
"It can come from wave-function (anti-)symmetrization for pairs of identical particles, the measurement in this case is sometimes referred to as “HBT correlations”.",
"Another source is the Final State Interaction (FSI), that is Coulomb or strong.",
"In this work the Coulomb FSI is only present for $p$$\\bar{p}$ pairs, all others are correlated due to the strong FSI only.",
"The interaction for $p$$\\bar{p}$ system is measured in detail and well described theoretically, we will use existing calculations for this system and will not vary any of its parameters in the fits.",
"For details please see [14].",
"For all other pairs the strong FSI is the only source of femtoscopic correlation.",
"Below we will describe the formalism for the strong interaction only, as it is the focus of this work.",
"In femtoscopy an assumption is made that the FSI of the pairs of particles is independent from their production.",
"The two-particle correlation can then be written as [13]: $C(\\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}}) = {{\\int S(\\bf {r^{*}}, \\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}})|\\Psi ^{S(+)}_{-k^{*}}(\\bf {r^{*}}, \\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}})|^{2}} \\over {\\int S(\\bf {r^{*}}, \\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}})}}$ where $\\bf {r^{*}}={\\bf x}_{1}-{\\bf x}_{2}$ is a relative space-time separation of the two particles at the moment of their creation.",
"$\\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}}$ is the momentum of the first particle in the Pair Rest Frame (PRF), so it is half of the pair relative momentum in this frame.",
"$S$ is the source emission function and can be interpreted as a probability to emit a given particle pair from a given set of emission points with given momenta.",
"The source of the correlation is the Bethe-Salpeter amplitude $\\Psi ^{S(+)}_{-k^{*}}$ , which in this case corresponds to the solution of the quantum scattering problem taken with the inverse time direction.",
"When particles interact with the strong FSI only it can be written as: $\\Psi ^{S(+)}_{-k^{*}}({\\bf r^{*}}, \\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}}) = e^{i\\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}}\\mathchoice{\\mbox{$\\displaystyle r^{*}$}}{\\mbox{$\\textstyle r^{*}$}}{\\mbox{$\\scriptstyle r^{*}$}}{\\mbox{$\\scriptscriptstyle r^{*}$}}} + f^{S}(k^{*}){{e^{i{k^{*}}{r^{*}}}} \\over {r^{*}}}$ where $f^{S}$ is the S-wave strong interaction amplitude.",
"In the effective range approximation it can be expressed as: $f^{S}(k^{*}) = \\left({{1} \\over {f^{S}_{0}}} + {{1} \\over {2}} {d_{0}^{S}{k^{*}}^{2}} - ik^{*}\\right)^{-1}$ where $f_{0}^{S}$ is the scattering length and $d_{0}^{S}$ is the effective radius of the strong interaction.",
"These are the essential parameters of the strong interaction, which can be extracted from the fit to the experimental correlation function.",
"Both are complex numbers; the imaginary part of $f_{0}$ is especially interesting as it corresponds to the annihilation process.",
"In the relative momentum range where the effective range approximation is valid they are also directly related to the interaction cross-section: $\\sigma =4\\pi |f^{S}|^{2}$ .",
"Therefore their knowledge is of fundamental importance.",
"For one-dimensional correlation function the source function $S$ has one parameter.",
"Usually a spherically symmetric source in PRF with size $r_{0}$ is taken: $S(\\mathchoice{\\mbox{$\\displaystyle r^{*}$}}{\\mbox{$\\textstyle r^{*}$}}{\\mbox{$\\scriptstyle r^{*}$}}{\\mbox{$\\scriptscriptstyle r^{*}$}}) \\approx \\exp \\left( -{{{r^{*}}^{2}} \\over {4 r_{0}^{2}}} \\right)$ which gives the final form of the analytical correlation function depending on the strong FSI only [13], [12]: $C(\\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}}) &=& 1+\\sum _{S} \\rho _{S} \\left[ {{1} \\over {2}} \\left|{{f^{S}(k^{*})} \\over {r_{0}}} \\right|^{2} \\left( 1-{{d_{0}^{S}} \\over {2\\sqrt{\\pi }r_{0}}} \\right) + \\right.\\nonumber \\\\& & \\left.",
"{{2\\Re f^{S}(\\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}})} \\over {\\sqrt{\\pi }r_{0}}}F_{1}(Qr_{0}) -{{\\Im f^{S}(\\mathchoice{\\mbox{$\\displaystyle k^{*}$}}{\\mbox{$\\textstyle k^{*}$}}{\\mbox{$\\scriptstyle k^{*}$}}{\\mbox{$\\scriptscriptstyle k^{*}$}})} \\over {r_{0}}}F_{2}(Qr_{0})\\right] ,$ where $Q=2k^{*}$ , $F_{1}(z) = \\int _{0}^{z} dx e^{x^{2} - z^{2}}/z$ and $F_{2} =(1-e^{-z^{2}})/z$ .",
"Summation is done over possible pair spin orientations, with $\\rho _{S}$ the corresponding pair spin fractions.",
"Since the data considered in this work is always for unpolarized pairs, the spin dependence of the correlation will be neglected.",
"In this formula the dependence of the correlation function on the real and imaginary part of the scattering length $f_{0}$ is expressed directly.",
"For pairs where only the strong FSI contributes to the correlation, such as $p\\Lambda $ and $p\\bar{\\Lambda }$ , this formula can be fitted directly to extract the source size $r_{0}$ as well as the scattering length and effective radius.",
"In realistic scenarios it is rarely possible to independently determine all parameters.",
"In particular in case of the STAR data discussed here the $d_{0}$ was fixed at zero and only the remaining three were fitted."
],
[
"Residual correlations",
"In experiments conducted at colliders such as STAR experiment at RHIC, all particles propagate to the detector radially from the interaction point located in the center of the detector.",
"A baryon coming from a weak decay often travels in a direction very similar to the parent baryon.",
"The particle's trajectory does not point precisely to the interaction point, but this difference (called the Distance of the Closest Approach or DCA) is often comparable to the spatial resolution of the experiment.",
"As a result significant number of particles identified as protons in STAR are not primary and come from the decay of heavier baryons.",
"The same mechanism applies to $\\Lambda $ baryons.",
"In particular protons can come from a decay of $\\Lambda $ and $\\Sigma ^{+}$ baryons, while $\\Lambda $ baryons can come from decays of $\\Sigma ^{0}$ or $\\Xi ^{0}$ .",
"Table: List of possible parent pairs for the pΛp\\Lambda (andpΛ ¯p\\bar{\\Lambda }) system, with their relative contribution to theSTAR sample and the decay momenta values.STAR experiment has applied the DCA cut to reduce the number of such secondaries and has estimated its effectiveness based on the Monte-Carlo simulation of the detector response.",
"The fraction of true primary pairs, as well as a fraction of all other parent particle pair combinations is taken from [12] and given in Tab.",
"REF .",
"In addition to the effect mentioned above it is also possible that a primary proton is randomly associated to a pion and reconstructed as a fake $\\Lambda $ baryon.",
"For that reason a pair of two protons also appears in Tab.",
"REF .",
"The strong FSI affects the behavior of the two particles in the pair just after their production, on a time scale of fm/$c$ .",
"For particles coming from a weak decay, which occurs on timescales of $10^{-10}$ s, the FSI applies to the parent pair, not the daughter.",
"However it is the daughters that are measured in the detector.",
"For such a scenario Eq.",
"(REF ) cannot be used directly.",
"In this case $\\Psi $ must be taken for the parent pair and calculated for $k^{*}$ and $r^{*}$ between the parent particles.",
"Then one or both of the parent particles must decay and a new $k^{*}$ must be calculated for the daughter pair.",
"This one is measured in the detector, the correlation is measured as a function of this relative momentum.",
"Such scenario is called “residual correlations” (RC) [15], [16].",
"Obviously the random nature of the weak decay will dilute the original correlation.",
"However if the decay momentum is comparable to the width of the correlation effect in relative momentum, some correlation might be preserved for the daughter particles.",
"The RC are important if three conditions are met simultaneously: a) the original correlation for parent particles is large, b) the fraction of daughter pairs coming from a particular parent pair is significant and c) the decay momentum (or momenta) are comparable to the expected correlation width in $k^{*}$ .",
"For $p\\bar{\\Lambda }$ pairs all three conditions are met.",
"The strong FSI for baryon-antibaryon pairs is dominated by annihilation, which appears in Eq.",
"(REF ) as $\\Im f^{S}$ .",
"It causes a negative correlation (anticorrelation), which is wide in $k^{*}$ , even on the order of 300 MeV/$c$ .",
"Decay momenta for all residual pairs listed in Tab.",
"REF are of that order or smaller.",
"Comparing contributions to the sample from all pairs one can see that all listed are of the same order as primary $p\\bar{\\Lambda }$ pairs which constitute only 15% of the sample.",
"As for the strength of the correlation, it is in principle unknown for all pairs, except $p\\bar{p}$ .",
"Estimating its strength is one of the goals of this work.",
"However it is often assumed that at least the annihilation cross-section for $B\\bar{B}$ pairs is very similar for all pairs, comparable to $p$$\\bar{p}$ [11].",
"In that case it is certainly strong enough to induce RC and contributions from all pairs listed in Tab.",
"REF must be considered in the analysis of the $p$$\\bar{\\Lambda }$ correlations.",
"The RC can be calculated for any combination of parent and daughter pairs.",
"The correlation is expressed as a function of the relative momentum of the daughter pair, in our case it is $k^{*}_{p\\Lambda }$ .",
"However Eq.",
"(REF ) is then used for the parent pair (let's call it $XY$ ), and gives the correlation as a function of $k^{*}_{XY}$ .",
"Baryon $X$ is a proton or decays into a proton and baryon $Y$ is a $\\Lambda $ or decays into a $\\Lambda $ .",
"The daughter momenta will differ from the parents' by the decay momentum, listed in Tab.",
"REF .",
"The direction of the decay momentum is random in the parents' rest frame, and it is independent from the direction of $k^{*}$ of the pair.",
"Therefore $k^{*}_{p\\Lambda }$ will differ, in a random way for each pair, from $k^{*}_{XY}$ .",
"The difference is limited by the value of the decay momentum and is a non-trivial consequence of the decay kinematics.",
"Figure: The unnormalized transformation matrix W for ΛΛ\\Lambda \\Lambda (left) and Σ + Σ 0 \\Sigma ^{+}\\Sigma ^{0} (right) pairs decaying intopΛp\\Lambda pairs, as a function of relative momentum of both pair types.One can determine what is the probability that a parent particle pair with a given $k^{*}_{XY}$ will decay into a daughter pair with a given $k^{*}_{p\\Lambda }$ .",
"Let's call such probability distribution $W(k^{*}_{XY}, k^{*}_{p\\Lambda })$ .",
"In this work we have calculated it for all pairs listed in Tab.",
"REF .",
"We have used the Therminator model [17], [18], with parameters describing central Au+Au collisions at $\\sqrt{s_{NN}} =200$ GeV.",
"All the pairs of type $XY$ in a given event were found and their relative momentum $k^{*}_{XY}$ was calculated.",
"Than both baryons $X$ and $Y$ were allowed to decay and $k^{*}_{p\\Lambda }$ was calculated for the daughters.",
"The pair was then inserted in a two-dimensional histogram.",
"As a result an unnormalized probability distribution $W$ was obtained for each pair type.",
"Fig.",
"REF shows two examples of this function, one for a pair where only one particle decays, the other for a pair where both particles decay.",
"In the first case the function has a characteristic rectangular shape at low relative momentum [15], [16].",
"It touches both axes at the value roughly equal to half of the decay momentum.",
"The vertical width of the function is roughly equal to the decay momentum, as discussed above.",
"In the second case the shape at low momentum is not as sharp, and the width is equal to the sum of decay momenta.",
"$W$ depends only on decay kinematics, so it is the same for $BB$ and the corresponding $B\\bar{B}$ pair.",
"Figure: Theoretical correlation function for a given source size forpΛ ¯p\\bar{\\Lambda } and two examples of residual correlation functions forΛΛ ¯\\Lambda \\bar{\\Lambda } and Σ + Σ 0 ¯\\Sigma ^{+}\\bar{\\Sigma ^{0}} pairs.Having defined $W$ one can write the formula for the RC for any type of the parent pair $X\\bar{Y}$ , contributing to the $p$$\\bar{\\Lambda }$ correlation function: $C^{X\\bar{Y} \\rightarrow p\\bar{\\Lambda }}(k^{*}_{p\\bar{\\Lambda }}) = {{\\int C^{X\\bar{Y}}(k^{*}_{X\\bar{Y}})W(k^{*}_{X\\bar{Y}},k^{*}_{p\\bar{\\Lambda }}) d k^{*}_{X\\bar{Y}} } \\over {\\int W(k^{*}_{X\\bar{Y}},k^{*}_{p\\bar{\\Lambda }})d k^{*}_{X\\bar{Y}} }} .$ Examples of correlation functions transformed in this way are shown in Fig.",
"REF .",
"The $p$$\\bar{\\Lambda }$ function for a given source size, calculated according to Eq.",
"(REF ) is given for comparison.",
"It has positive correlation at very low $k^{*}$ coming from the positive real part of the scattering length $f_{0}$ and a wide anticorrelation coming from the positive imaginary part of $f_{0}$ .",
"This anticorrelation is wide, extending beyond $0.4$ GeV/$c$ , so its width is larger than any combination of decay momenta given in Tab.",
"REF .",
"The residual correlations are calculated for the same source size and radius parameters, so in their respective $k^{*}$ variables they look identical to $C^{p\\bar{\\Lambda }}(k^{*}_{p\\bar{\\Lambda }})$ .",
"After the transformation given by Eq.",
"(REF ) the correlation is diluted at low $k^{*}$ .",
"However at higher values the shape of the function changes very little and is almost the same for the parent and residual correlation.",
"The spike at $k^{*}=0$ is transformed with the matrix in Fig.",
"REF to a slight bump in $k^{*}_{p\\bar{\\Lambda }}$ where $W$ touches the $x$ axis, that is around 50 MeV/$c$ , half of the decay momentum of $\\Lambda $ into proton.",
"The same function is diluted twice as strong at low $k^{*}$ for the pair where both particles decay ($\\Sigma ^{+}\\bar{\\Sigma ^{0}}$ ).",
"This difference persists up to around 100 MeV/$c$ , above this value both functions are similar to each other and to the original correlation.",
"In terms of the physics picture the contribution of the $p$$\\bar{p}$ correlation to the $p$$\\bar{\\Lambda }$ one is not RC, instead it comes from fake association of primary proton to a $\\Lambda $ particle.",
"However the formalism to deal with such situation is exactly the same as in the case of RC and Eq.",
"(REF ) can be used.",
"The difference is that the $p$$\\bar{p}$ correlation function has a Coulomb FSI component in addition to the strong FSI, which must be taken into account when calculating $C^{p\\bar{p}}$ .",
"The $W$ matrix for $p\\Lambda $ to $pp$ pair transformation can be used.",
"Once each of the RC components is determined, the complete correlation function for the $p$$\\bar{\\Lambda }$ system can be written: $C(k^{*}_{p\\bar{\\Lambda }}) &=& 1 +\\lambda _{p\\Lambda }\\left(C^{p\\bar{\\Lambda }}(k^{*}_{p\\bar{\\Lambda }})-1\\right) \\nonumber \\\\& &+ \\sum _{X\\bar{Y}} \\lambda _{XY}\\left(C^{X\\bar{Y}}(k^{*}_{p\\bar{\\Lambda }})-1\\right)$ where the $\\lambda $ values are equivalent to the pair fractions given in Tab.",
"REF .",
"It is an additional factor that decreases the correlation for the RC, however for some pairs it is almost as large as $\\lambda $ for true $p$$\\bar{\\Lambda }$ pairs.",
"Eq.",
"(REF ) is the final formula that can be fitted directly to experimental data.",
"In principle each $C^{X\\bar{Y}}$ depends on four independent parameters: the source size $r_{0}$ , real and imaginary part of $f_{0}$ and the value of $d_{0}$ , giving 37 independent parameters ($f_{0}$ and $d_{0}$ for the $p$$\\bar{p}$ pair is known).",
"Some assumptions are obviously needed to reduce this number, we will propose several options in Sec.",
"."
],
[
"Theoretical scenarios",
"Following the procedure employed by STAR in [12] we put the effective range $d_{0} = 0$ fm for all calculations.",
"Radius for the various systems in central Au+Au collisions at RHIC energies is expected to follow hydrodynamic predictions, which give $r_{0} \\sim 1/\\sqrt{\\left<m_{T}\\right>}$ , where $m_{T}$ is the transverse mass of the pair.",
"For baryons $m_{T}$ is large, and the decrease is not expected to be steep (see Fig.",
"5 in [12] for illustration).",
"$\\left< m_{T} \\right>$ for a given pair depends on the momentum spectra of particles taken for this analysis, which is not specified in [12].",
"We expect that $\\left<m_{T} \\right>$ for the pairs considered here will be within 20% of each other, giving little variation of the scaling factor.",
"Therefore we make a simplifying assumption that system size $r_{0}$ for each pair is the same.",
"With these assumptions 18 components of $f_{0}$ remain for the nine pairs.",
"Little theoretical guidance is given for those values.",
"An approach adopted in [11] equates all annihilation cross-sections for the $B\\bar{B}$ pairs and assumes they are equal to the one for $p\\bar{p}$ .",
"In [7] the value of $\\Im {f_{0}} = 0.88 \\pm 0.09$ fm is given.",
"This value is used to calculate the $p$$\\bar{p}$ correlation functions.",
"No such assumption is made for $\\Re {f_{0}}$ , which can vary significantly between various $B\\bar{B}$ pairs.",
"Therefore, we make two assumptions.",
"$\\Im {f_0}$ is assumed to be the same for all $B\\bar{B}$ pairs, but it is not fixed to the $p$$\\bar{p}$ value - it is treated as free in the fit.",
"Similarly $\\Re {f_0}$ is also assumed to be the same for all pairs and is free in the fit.",
"In [11] an alternative scenario for annihilation cross-sections is given.",
"Namely that they are the same as in $p\\bar{p}$ , but at the same $\\sqrt{s}$ of the pair, not relative momentum.",
"In UrQMD these assumptions differ little, the majority of hadronic rescatterings happen at large relative momentum, where the difference between cross-sections scaled with $k^{*}$ and $\\sqrt{s}$ is small.",
"However in the case of femtoscopic correlations, which by definition are concentrated at low relative momentum, the two scenarios differ strongly.",
"For example a $\\Sigma ^{+}\\Xi ^{0}$ pair at $k^{*}=10$ MeV/$c$ taken at the same $\\sqrt{s}$ corresponds to a $pp$ pair at $k^{*}=831.6$ MeV/$c$ .",
"This assumption would then significantly decrease the correlation for higher-mass pairs, including $p\\bar{\\Lambda }$ .",
"We test whether it is consistent with the data.",
"The next scenario comes from the fact, that we have just shown that both $p$$\\bar{p}$ and $\\Lambda \\bar{\\Lambda }$ RC will contribute to the $p$$\\bar{\\Lambda }$ correlation function.",
"One can then ask if it is possible that the observed correlation is explained by annihilation of particle-antiparticle pairs only, not all $B\\bar{B}$ pairs.",
"In such scenario the imaginary part of the scattering length should be put to zero for all pairs except the ones in which the two particles have exactly opposite quark content.",
"The last scenario is the repetition of the STAR procedure, where no RC is included and correlation is present for $p$$\\bar{\\Lambda }$ pairs only.",
"In each of these four scenarios, all $C^{X\\bar{Y}}$ functions can be calculated from Eq.",
"(REF ).",
"The last function remaining to be calculated is then $C^{p\\bar{p}}$ .",
"A dedicated procedure is used.",
"First the relative momenta distributions are taken from Therminator, from collisions simulated with parameters corresponding to central Au+Au collision at $\\sqrt{s_{NN}}=200$ GeV.",
"Then a source size is assumed, equal to the one used for all other pairs.",
"This allows the generation of $r^{*}$ for each pair, according to the probability distribution from Eq.",
"(REF ).",
"This gives pairs, each with its $k^{*}$ and $r^{*}$ , which enables the calculation of $\\Psi $ .",
"It is performed with a dedicated code from Lednicky [14], where the known $p$$\\bar{p}$ interaction parameters are used.",
"The resulting correlation function is then calculated according to Eq.",
"(REF ), corresponding to the value of the source size $r_{0}$ .",
"As mentioned earlier, the $W$ functions are also calculated from Therminator, for all parent pair types."
],
[
"Analysis of STAR data",
"The STAR data on $p$$\\bar{\\Lambda }$ correlation function [12] has been corrected for several effects, most of them experimental in nature.",
"Two of those corrections must be reexamined for this analysis.",
"The correlation was normalized “above 0.35 GeV/$c$ ” [12].",
"As can be see in Fig.",
"REF the femtoscopic correlation is small but non-negligible in this region.",
"However the upper range for the normalization is not given.",
"The number of pairs increases with $k^{*}$ , so if the upper normalization range is large, pairs with negligible correlation will dominate the normalization factor.",
"We will assume this is the case, which means that the experimental correlation is properly normalized.",
"The data was also corrected for “purity”, that is the fraction of true $p$$\\bar{\\Lambda }$ pairs, given in Tab.",
"REF .",
"The procedure used by STAR is correct only if all other pairs are not correlated.",
"In Eq.",
"(REF ) it would correspond to the scenario where all $C^{X\\bar{Y}}$ are at 1.0 in the full $k^{*}$ range.",
"We have just shown that this assumption is explicitly violated by the RC effect, which is expected to be significant for $p$$\\bar{\\Lambda }$ correlations measured by STAR.",
"Therefore the experimental correlation function analyzed later will be “uncorrected” for purity, with the purity factor equal to 0.15, taken from Tab.",
"REF , so that a fit according to Eq.",
"(REF ) can be properly applied.",
"The fitting range was set to $0.45$ GeV/$c$ , the maximum range for which experimental data is available."
],
[
"Fitting the experimental correlation function",
"Formula (REF ) is fitted to the STAR experimental data, with the theoretical assumptions mentioned above.",
"Standard $\\chi ^2$ minimization procedure is used.",
"The result of the fit is shown in Fig.",
"REF .",
"It gives the value of the source size $r_{0} = 2.83 \\pm 0.12$ fm, and the scattering length $f_{0} = 0.49\\pm 0.21 + i (1.00 \\pm 0.21)$ fm.",
"The value of $r_{0}$ is significantly larger than given in [12], indicating that the RC play a critical role in the extraction of physical quantities.",
"The value extracted here is in good agreement with the values obtained for the $p$$\\Lambda $ system.",
"This consistency is naturally expected in practically all realistic models of heavy-ion collisions, while the previous STAR result was violating this consistency without providing any viable explanation.",
"It is also consistent with expectation from hydrodynamical models, which are in good agreement with all other femtoscopic measurements at RHIC.",
"Taking all those arguments into account we claim that the result presented in this work is the correct one, and that the result for $p$$\\bar{\\Lambda }$ from [12] should be considered obsolete.",
"The extracted imaginary part of the scattering length is significant and in agreement with the value given for the $p$$\\bar{p}$ system.",
"This means that the assumption that the annihilation process for any $B\\bar{B}$ system is similar to that process for $p$$\\bar{p}$ , taken at the same relative momentum is consistent with data.",
"Figure: Comparison of all residual correlation components for thepΛ ¯p\\bar{\\Lambda } correlation function (thin black line).",
"For betterillustration the inverse of the correlation effect is plotted.In Fig.",
"REF all the residual correlation components of the fit are shown.",
"The absolute value of the correlation effect $1-C$ is plotted, the logarithmic scale is needed to distinguish the small contributions.",
"No single component is dominating the function, all 10 components are needed to describe the correlation.",
"The largest ones are, as expected, the ones which have large pair fractions and small decay momenta, that is $p\\bar{\\Lambda }$ , $p\\bar{\\Sigma ^{0}}$ , $\\Lambda \\bar{\\Lambda }$ and $p\\bar{\\Xi ^{0}}$ .",
"The systems where both particles decay and the systems where the fraction is small contribute less.",
"All the RC contributions are relevant through the whole $k^{*}$ range.",
"Figure: Fit to the STAR pΛ ¯p\\bar{\\Lambda } correlation function withEq.",
"(), no residual correlation components included.In order to validate the procedure and the new important result, several scenarios, described in Sec.",
", have been tested.",
"In Fig.",
"REF the fit was performed, where no residual correlations were included.",
"This is equivalent to the STAR procedure.",
"The result from [12] is reproduced, the resulting radius is small.",
"$\\Re {f_0}$ changes sign with respect to the default case, but interestingly $\\Im {f_0}$ is consistent with the full RC fit.",
"The next scenario assumes annihilation for particle-antiparticle pairs only.",
"By testing it we check if the annihilation is really necessary for all $B\\bar{B}$ pairs, or is enough if it happens only with baryons having exactly the opposite quark content.",
"A fit is performed, where only $p$$\\bar{p}$ and $\\Lambda \\bar{\\Lambda }$ RC is included, while for all other $B\\bar{B}$ pairs (including $p\\bar{\\Lambda }$ ) there is no correlation.",
"Result similar to the previous test is obtained - the radius is $1.5 \\pm 0.1$ fm.",
"Both scenarios are therefore unlikely.",
"In other words the analysis shows that the annihilation happens between all $B\\bar{B}$ pairs, not just the ones with exactly opposite quark content and that this effect must be taken into account, via the RC formalism in any analysis of $B\\bar{B}$ femtoscopic correlations.",
"Figure: Calculation of correlation function, with f S f^{S} taken thesame as the f S f^{S} for ppΛ ¯\\bar{\\Lambda } pair at the corresponding s\\sqrt{s}.In the last scenario, following the idea from [11] it was proposed that the annihilation cross-section for $B\\bar{B}$ pairs is the same for all pairs, but taken at the same $\\sqrt{s}$ instead of the relative momentum.",
"In femtoscopy such scaling would be reflected in Eq.",
"(REF ) by taking $f^{S}$ at a different $k^{*}$ .",
"In this work we treat the imaginary and real parts of $f_{0}$ for the $p$$\\bar{\\Lambda }$ system as fit parameters and scale the $f^{S}$ for all other pairs, by taking the same $f_{0}$ parameters, but calculating $f^{S}$ at: $k^{*} = \\left( {{s^{2} +m_{p}^{4} + m_{\\Lambda }^{4} - 2 s m_{p}^{2} -2 s m_{\\Lambda }^{2} - 2m_{p}^{2}m_{\\Lambda }^{2}} \\over {4 s}}\\right)^{1/2}$ according to Eq.",
"(REF ).",
"$s$ is the square of the total energy in PRF for the pair $X\\bar{Y}$ .",
"$f^{S}$ is a function rapidly decreasing with $k^{*}$ .",
"By taking $s$ for the baryon pair, where one or both baryons have a mass higher than the proton or the $\\Lambda $ , one gets from Eq.",
"(REF ) $k^{*}$ higher than for the original pair, so $f^{S}$ will be smaller.",
"In Fig.",
"REF the result of such calculation is shown for a pair with smallest and largest mass difference to the $p$$\\bar{\\Lambda }$ pair.",
"The strength of the correlation is visibly decreased.",
"However the shape is only slightly affected.",
"In fact the functions can be described by Eq.",
"(REF ), with altered values of $f_{0}$ .",
"The $\\Re {f_{0}}$ is scaled to approximately 20% of the original value, while $\\Im {f_{0}}$ is scaled to 60% (32%) of the original value for the pair with smallest (largest) mass difference, that is $p\\Sigma ^{0}$ ($\\Sigma ^{+}\\Xi ^{0}$ ).",
"These scaling factors provide the needed constraints on the fit parameters, and the fit can be performed as in the previous cases.",
"Figure: Fit the the STAR pΛ ¯p\\bar{\\Lambda } correlation function withEq.",
"(), all residual correlations included, strengthof the interaction scaled according to the s\\sqrt{s} of the pair(see text for details).Fig.",
"REF shows the result of the fit, with the scaling of $f^{S}$ with $\\sqrt{s}$ of the pair.",
"The resulting source size is comparable to the default case.",
"$\\Im {f_{0}}$ is significantly larger than for the default fit and larger than the measured $p$$\\bar{p}$ value.",
"While this scenario is not ruled out by the data, it is internally inconsistent.",
"It would mean that moving from $p$$\\bar{p}$ to heavier pairs, the cross-section first increases sharply and then decreases for heavier pairs.",
"If one takes the $p$$\\bar{p}$ $f_{0}$ as the starting point, instead of $p$$\\bar{\\Lambda }$ (which would be a more literate implementation of the scenario proposed in [11]), then $f_{0}$ cannot be a free parameter.",
"A fit gives $r_{0}=2.23 \\pm 0.09$ fm which is lower than the expected value.",
"Such scenario cannot be ruled out, but is less likely, due to the disagreement of this value with $r_{0}$ for $p$$\\Lambda $ pairs."
],
[
"Systematic uncertainty discussion",
"All the values given above were obtained with certain assumptions, spelled above, both related to the STAR data treatment as well as the methodology itself and the unknown strong interaction parameters.",
"By varying those assumptions in a reasonable range one can estimate the systematic uncertainty on the extracted parameters coming from the application of the RC method and the assumptions made.",
"Restricting the fitting range to $0.35$ GeV/$c$ (beginning of the normalization range) gives 5% variation in radius, while $\\Im {f_{0}}$ decreases to $0.6 \\pm 0.2$ fm and $\\Re {f_{0}}$ is positive but consistent with zero.",
"Performing the fit separately for $p$$\\bar{\\Lambda }$ and $\\bar{p}$$\\Lambda $ pairs gives $r_{0}$ statistically consistent with the default fit.",
"$\\Im {f_{0}}$ varies by up to 20%, and $\\Re {f_{0}}$ by up to 50%.",
"That is expected - $\\Re {f_{0}}$ affects the function mostly at low $k^{*}$ , where data is less precise, while $\\Im {f_{0}}$ produces the wide anticorrelation which is better constrained by the data.",
"With the statistical power of the STAR data we were unable to test the influence of the $d_{0}$ parameter variation, or independent variation of $f_{0}$ parameters for heavier $B\\bar{B}$ pair types.",
"In conclusion the source size $r_{0}$ is well constrained and comparable to $r_{0}$ measured by STAR [12] for $p$$\\Lambda $ and $\\bar{p}$$\\bar{\\Lambda }$ within the statistical and systematic uncertainty of this work.",
"$\\Im {f_{0}}$ is determined to be finite and positive, consistent with the hypothesis that its value for all $B\\bar{B}$ pairs considered is similar to the value for $p\\bar{p}$ .",
"The systematic uncertainty of the method is at least 20%.",
"$\\Re {f_{0}}$ is consistent with being finite and positive, although the systematic uncertainty of the method is at least 50%.",
"There is also no theoretical expectation that $\\Re {f_{0}}$ is similar for different $B\\bar{B}$ pairs, so this measurement can be interpreted as “average effective” $\\Re {f_{0}}$ for the considered $B\\bar{B}$ pairs.",
"Certain other systematic uncertainties depend on the detail of the experimental treatment.",
"These include, among others, the variation of the normalization range, variation of the pair fractions and the variation of the DCA cuts.",
"Their estimation is beyond the scope of this work, as it requires direct access to experimental raw data and procedures."
],
[
"Summary",
"We have presented the theoretical formalism for dealing with residual correlations in baryon-antibaryon femtoscopic correlations.",
"We have shown that for realistic scenario of heavy-ion collision at $\\sqrt{s_{NN}}=200$ GeV such correlations are critical for the correct interpretation of data.",
"The formalism has been applied to $p$$\\bar{\\Lambda }$ and $\\bar{p}$$\\Lambda $ femtoscopic correlations measured by STAR [12].",
"New estimates for system size $r_{0}$ as well as real and imaginary parts of the scattering length $f_{0}$ have been obtained.",
"New system size is consistent with results for $p$$\\Lambda $ and $\\bar{p}$$\\bar{\\Lambda }$ pairs and model expectations.",
"Therefore the puzzle of unexpectedly small $p$$\\bar{\\Lambda }$ system size reported by STAR in [12] is solved.",
"In addition new, more robust estimates for $f_{0}$ parameter is obtained, not only for the $p$$\\bar{\\Lambda }$ system, but also for a number of heavier $B\\bar{B}$ pairs.",
"A scenario where all $B\\bar{B}$ pairs have similar annihilation cross-section (expressed as a function of pair relative momentum) is judged to be most likely, as it gives the expected source size and is internally consistent.",
"Other scenarios have been explored, but were judged to be less likely.",
"With the new methodology it is possible to measure strong interaction potential for a number of $B\\bar{B}$ pair types, including $\\Lambda $ and $\\Xi $ baryons.",
"More precise data, differential in centrality and pair momentum and obtained for other pair types (e.g.",
"$p\\Xi ^{0}$ , $\\Lambda \\Lambda $ , $\\Lambda \\Xi ^{0}$ ) would help constrain this interesting, unknown quantities.",
"In particular high statistics runs of Au+Au collisions at RHIC, as well as Pb–Pb collisions at the LHC promise better quality data and give hope for more precise measurement in the near future."
]
] | 1403.0433 |
[
[
"A space-time FEM for PDEs on evolving surfaces"
],
[
"Abstract The paper studies a finite element method for computing transport and diffusion along evolving surfaces.",
"The method does not require a parametrization of a surface or an extension of a PDE from a surface into a bulk outer domain.",
"The surface and its evolution may be given implicitly, e.g., as the solution of a level set equation.",
"This approach naturally allows a surface to undergo topological changes and experience local geometric singularities.",
"The numerical method uses space-time finite elements and is provably second order accurate.",
"The paper reviews the method, error estimates and shows results for computing the diffusion of a surfactant on surfaces of two colliding droplets."
],
[
"INTRODUCTION",
"Partial differential equations posed on evolving surfaces appear in a number of applications.",
"Recently, several numerical approaches for handling such type of problems have been introduced, cf. [1].",
"In [2], [3] Dziuk and Elliott developed and analyzed a finite element method for computing transport and diffusion on a surface which is based on a Lagrangian tracking of the surface evolution.",
"Methods using an Eulerian approach were developed in [4], [5], [6], based on an extension of the surface PDE into a bulk domain that contains the surface.",
"Recently, in [7], [8], [9] another Eulerian method, which does not use an extension of the PDE into the bulk domain, has been introduced and analyzed.",
"The key idea of this method is to use restrictions of (usual) space-time volumetric finite element functions to the space-time manifold.",
"This trace finite element technique has been studied for stationary surfaces in [10], [11], [12].",
"In this paper we summarize the key ideas of this space-time trace-FEM and some main results of the error analysis, in particular a result on second order accuracy of the method in space and time.",
"For details we refer to [7], [9].",
"In the numerical experiments in [7], [8], [9] only relatively simple model problems with smoothly evolving surfaces are considered.",
"As a new contribution in this paper we present results of a numerical experiment for a surfactant transport equation on an evolving manifold with a topological singularity, which resembles a droplet collision.",
"The method that we study uses volumetric finite element spaces which are continuous piecewise linear in space and discontinuous piecewise linear in time.",
"This allows a natural time-marching procedure, in which the numerical approximation is computed on one time slab after another.",
"Spatial triangulations may vary per time slab.",
"The results of the numerical experiment show that the method is extremely robust and that even for the case with a topological singularity (droplet collision) accurate results can be obtained on a fixed Eulerian (space-time) grid with a large time step.",
"As a model problem we use the following one.",
"Consider a surface $\\Gamma (t)$ passively advected by a given smooth velocity field $\\mathbf {w}=\\mathbf {w}(x,t)$ , i.e.",
"the normal velocity of $\\Gamma (t)$ is given by $\\mathbf {w}\\cdot \\mathbf {n}$ , with $\\mathbf {n}$ the unit normal on $\\Gamma (t)$ .",
"We assume that for all $t \\in [0,T] $ , $\\Gamma (t)$ is a hypersurface that is closed ($\\partial \\Gamma =\\emptyset $ ), connected, oriented, and contained in a fixed domain $\\Omega \\subset {R}^d$ , $d=2,3$ .",
"In the remainder we consider $d=3$ , but all results have analogs for the case $d=2$ .",
"The convection-diffusion equation on the surface that we consider is given by: $\\dot{u} + ({\\mathop {\\rm div}}_\\Gamma \\mathbf {w})u -{ \\nu _d}\\Delta _{\\Gamma } u= f\\qquad \\text{on}~~\\Gamma (t), ~~t\\in (0,T],$ with a prescribed source term $f= f(x,t)$ and homogeneous initial condition $u(x,0)=u_0(x)=0$ for $x \\in \\Gamma _0:=\\Gamma (0)$ .",
"Here $\\dot{u}= \\frac{\\partial u}{\\partial t} + \\mathbf {w}\\cdot \\nabla u$ denotes the advective material derivative, ${\\mathop {\\rm div}}_\\Gamma :=\\operatorname{tr}\\left( (I-\\mathbf {n}\\mathbf {n}^T)\\nabla \\right)$ is the surface divergence and $\\Delta _\\Gamma $ is the Laplace-Beltrami operator, $\\nu _d>0$ is the constant diffusion coefficient.",
"If we take $f=0$ and an initial condition $u_0 \\ne 0$ , this surface PDE is obtained from mass conservation of the scalar quantity $u$ with a diffusive flux on $\\Gamma (t)$ (cf.",
"[13], [14]).",
"A standard transformation to a homogeneous initial condition, which is convenient for a theoretical analysis, leads to (REF )."
],
[
"WELL-POSED SPACE-TIME WEAK FORMULATION",
"Several weak formulations of (REF ) are known in the literature, see [2], [14].",
"The most appropriate for our purposes is a integral space-time formulation proposed in [7].",
"In this section we outline this formulation.",
"Consider the space-time manifold $\\mathcal {S}= \\bigcup \\limits _{t \\in (0,T)} \\Gamma (t) \\times \\lbrace t\\rbrace ,\\quad \\mathcal {S}\\subset {R}^{4}.$ On $L^2(\\mathcal {S})$ we use the scalar product $(v,w)_0=\\int _0^T \\int _{\\Gamma (t)} v w \\, ds \\, dt$ .",
"Let $\\nabla _\\Gamma $ denote the tangential gradient for $\\Gamma (t)$ and introduce the space $H=\\lbrace \\, v \\in L^2(\\mathcal {S})~|~ \\Vert \\nabla _\\Gamma v\\Vert _{L^2(\\mathcal {S})} <\\infty \\, \\rbrace $ endowed with the scalar product $(u,v)_H=(u,v)_0+ (\\nabla _\\Gamma u, \\nabla _\\Gamma v)_0.",
"$ We consider the material derivative $\\dot{u}$ of $u \\in H$ as a distribution on $\\mathcal {S}$ : $\\left\\langle \\dot{u},\\phi \\right\\rangle = - \\int _0^T \\int _{\\Gamma (t)} u \\dot{\\phi }+ u \\phi {\\,\\operatorname{div_\\Gamma }}\\mathbf {w}\\, ds \\, dt \\quad \\text{for all}~~ \\phi \\in C_0^1(\\mathcal {S}).$ In [7] it is shown that $C_0^1(\\mathcal {S})$ is dense in $H$ .",
"If $\\dot{u}$ can be extended to a bounded linear functional on $H$ , we write $\\dot{u} \\in H^{\\prime }$ .",
"Define the space $W= \\lbrace \\, u\\in H~|~\\dot{u} \\in H^{\\prime } \\,\\rbrace , \\quad \\text{with}~~\\Vert u\\Vert _W^2 := \\Vert u\\Vert _H^2 +\\Vert \\dot{u}\\Vert _{H^{\\prime }}^2.$ In [7] properties of $H$ and $W$ are derived.",
"Both spaces are Hilbert spaces and smooth functions are dense in $H$ and $W$ .",
"Define $\\overset{\\circ }{W}:=\\lbrace \\, v \\in W~|~v(\\cdot , 0)=0 \\quad \\text{on}~\\Gamma _0\\,\\rbrace .$ The space $\\overset{\\circ }{W}$ is well-defined, since functions from $W$ have well-defined traces in $L^2(\\Gamma (t))$ for any $t\\in [0,T]$ .",
"We introduce the symmetric bilinear form $a(u,v)= \\nu _d (\\nabla _\\Gamma u, \\nabla _\\Gamma v)_0 + ({\\,\\operatorname{div_\\Gamma }}\\mathbf {w}\\, u,v)_0, \\quad u, v \\in H,$ which is continuous on $H\\times H$ : $a(u,v)\\le (\\nu _d+\\alpha _{\\infty }) \\Vert u\\Vert _H\\Vert v\\Vert _H,\\quad \\text{with}~\\alpha _{\\infty }:=\\Vert {\\,\\operatorname{div_\\Gamma }}\\mathbf {w}\\Vert _{L^\\infty (\\mathcal {S})}.$ The weak space-time formulation of (REF ) reads: For given $f \\in L^2(\\mathcal {S})$ find $u \\in \\overset{\\circ }{W}$ such that $ \\left\\langle \\dot{u},v\\right\\rangle +a (u,v) = (f,v)_0 \\quad \\text{for all}~~v \\in H.$ In [7] the inf-sup property $ \\inf _{0\\ne u \\in \\overset{\\circ }{W}}~\\sup _{ 0\\ne v \\in \\overset{\\phantom{.",
"}}{H}} \\frac{\\left\\langle \\dot{u},v\\right\\rangle + a(u,v)}{\\Vert u\\Vert _W\\Vert v\\Vert _H} \\ge c_s>0$ is proved.",
"Using this in combination with the continuity result one can show that the weak formulation (REF ) is well-posed.",
"We introduce a similar “time-discontinuous” weak formulation that is better suited for the finite element method that we consider.",
"We take a partitioning of the time interval: $0=t_0 <t_1 < \\ldots < t_N=T$ , with a uniform time step $\\Delta t = T/N$ .",
"The assumption of a uniform time step is made to simplify the presentation, but is not essential.",
"A time interval is denoted by $I_n:=(t_{n-1},t_n]$ .",
"The symbol $\\mathcal {S}^n$ denotes the space-time interface corresponding to $I_n$ , i.e., $\\mathcal {S}^n:=\\cup _{t \\in I_n}\\Gamma (t)\\times \\lbrace t\\rbrace $ , and $\\mathcal {S}:= \\cup _{1 \\le n \\le N} \\mathcal {S}^n $ .",
"We introduce the following subspaces of $H$ : $H_n:=\\lbrace \\, v \\in H~|~v=0 \\quad \\text{on}~~\\mathcal {S}\\setminus \\mathcal {S}^n\\, \\rbrace ,$ and define the spaces $W_n & = \\lbrace \\, v\\in H_n~|~\\dot{v} \\in H_n^{\\prime } \\,\\rbrace , \\quad \\Vert v\\Vert _{W_n}^2 = \\Vert v\\Vert _{H}^2 +\\Vert \\dot{v}\\Vert _{H_n^{\\prime }}^2, \\\\W^b & := \\oplus _{n=1}^N W_n,~~\\text{with norm}~~ \\Vert v\\Vert _{W^b}^2= \\sum _{n=1}^N \\Vert v\\Vert _{W_n}^2.",
"$ For $u \\in W_n$ , the one-sided limits $u_+^{n}=u_+(\\cdot ,t_{n})$ (i.e., $t \\downarrow t_n$ ) and ${u}_{-}^n=u_{-}(\\cdot ,t_n)$ (i.e., $t \\uparrow t_n$ ) are well-defined in $L^2(\\Gamma (t_n))$ .",
"At $t_0$ and $t_N$ only $u_+^{0}$ and $u_{-}^{N}$ are defined.",
"For $v \\in W^b$ , a jump operator is defined by $[v]^n= v_+^n-v_{-}^n \\in L^2(\\Gamma (t_n))$ , $n=1,\\dots ,N-1$ .",
"For $n=0$ , we define $[v]^0=v_+^0$ .",
"On the cross sections $\\Gamma (t_n)$ , $0 \\le n \\le N$ , of $\\mathcal {S}$ the $L^2$ scalar product is denoted by $(\\psi ,\\phi )_{t_n}:= \\int _{\\Gamma (t_n)} \\psi \\phi \\, ds .$ In addition to $a(\\cdot ,\\cdot )$ , we define on the broken space $W^b$ the following bilinear forms: $d(u,v) = \\sum _{n=1}^N d^n(u,v), \\quad d^n(u,v)=([u]^{n-1},v_+^{n-1})_{t_{n-1}},\\quad \\left\\langle \\dot{u} ,v\\right\\rangle _b =\\sum _{n=1}^N \\left\\langle \\dot{u}_n, v_n\\right\\rangle .$ One can show that the unique solution to (REF ) is also the unique solution of the following variational problem in the broken space: Find $u \\in W^b$ such that $ \\left\\langle \\dot{u} ,v\\right\\rangle _b +a(u,v)+d(u,v) =( f,v )_0 \\quad \\text{for all}~~v \\in W^b.$ For this time discontinuous weak formulation an inf-sup stability result (that is weaker than the one in (REF )) can be derived.",
"The variational formulation uses $W^b$ , instead of $H$ , as test space, since the term $d(u,v)$ is not well-defined for an arbitrary $v\\in H$ .",
"Also note that the initial condition $u(\\cdot ,0)=0$ is not an essential condition in the space $W^b$ but is treated in a weak sense (as is standard in DG methods for time dependent problems).",
"From an algorithmic point of view the formulation (REF ) has the advantage that due to the use of the broken space $W^b= \\oplus _{n=1}^N W_n$ it can be solved in a time stepping manner."
],
[
"SPACE-TIME FINITE ELEMENT METHOD ",
"We introduce a finite element method which is a Galerkin method with $W_h \\subset W^b$ applied to the variational formulation (REF ).",
"To define this $W_{h}$ , consider the partitioning of the space-time volume domain $Q= \\Omega \\times (0,T] \\subset {R}^{3+1}$ into time slabs $Q_n:= \\Omega \\times I_n$ .",
"Corresponding to each time interval $I_n:=(t_{n-1},t_n]$ we assume a given shape regular tetrahedral triangulation $n$ of the spatial domain $\\Omega $ .",
"The corresponding spatial mesh size parameter is denoted by $h$ .",
"Then $\\mathcal {Q}_h=\\bigcup \\limits _{n=1,\\dots ,N}n\\times I_n$ is a subdivision of $Q$ into space-time prismatic nonintersecting elements.",
"We shall call $\\mathcal {Q}_h$ a space-time triangulation of $Q$ .",
"Note that this triangulation is not necessarily fitted to the surface $\\mathcal {S}$ .",
"We allow $n$ to vary with $n$ (in practice, during time integration one may wish to adapt the space triangulation depending on the changing local geometric properties of the surface) and so the elements of $\\mathcal {Q}_h$ may not match at $t=t_n$ .",
"For any $n\\in \\lbrace 1,\\dots ,N\\rbrace $ , let $V_n$ be the finite element space of continuous piecewise linear functions on $n$ .",
"We define the volume space-time finite element space: $ V_{h}:= \\lbrace \\, v: Q \\rightarrow {R} ~|~ v(x,t)= \\phi _0(x) + t \\phi _1(x)~\\text{on every}~Q_n,~\\text{with}~\\phi _0,\\, \\phi _1 \\in V_n\\,\\rbrace .$ Thus, $V_{h}$ is a space of piecewise P1 functions with respect to $\\mathcal {Q}_h$ , continuous in space and discontinuous in time.",
"Now we define our surface finite element space as the space of traces of functions from $V_{h}$ on $\\mathcal {S}$ : $ W_{h} := \\lbrace \\, w:\\mathcal {S}\\rightarrow {R}~|~ w=v_{|\\mathcal {S}}, ~~v \\in V_{h} \\, \\rbrace .$ The finite element method reads: Find $u_h \\in W_{h}$ such that $ \\left\\langle \\dot{u}_h ,v_h\\right\\rangle _b +a(u_h,v_h)+d(u_h,v_h) = (f,v_h)_0 \\quad \\text{for all}~~v_h \\in W_{h}.$ As usual in time-DG methods, the initial condition for $u_h(\\cdot ,0)$ is treated in a weak sense.",
"Due to $u_h\\in H^1(Q_n)$ for all $n=1,\\dots ,N$ , the first term in (REF ) can be written as $\\left\\langle \\dot{u}_h ,v_h\\right\\rangle _b=\\sum _{n=1}^N\\int _{t_{n-1}}^{t_n}\\int _{\\Gamma (t)} (\\frac{\\partial u_h}{\\partial t} +\\mathbf {w}\\cdot \\nabla u_h)v_h ds\\,dt.$ The method can be implemented with a time marching strategy.",
"Of course, for the implementation of the method one needs a quadrature rule to approximate the integrals over $\\mathcal {S}^n$ .",
"This issue is briefly addressed in Section ."
],
[
"DISCRETIZATION ERROR ANALYSIS",
"In this section we briefly address the discretization error analysis of the method (REF ), which is presented in [9].",
"We first explain a discrete mass conservation property of the scheme (REF ).",
"We consider the case that (REF ) is derived from mass conservation of a scalar quantity with a diffusive flux on $\\Gamma (t)$ .",
"The original problem then has a nonzero initial condition $u_0$ and a source term $f \\equiv 0$ .",
"The solution $u$ of the original problem has the mass conservation property $\\bar{u}(t):=\\int _{\\Gamma (t)} u \\, ds =\\int _{\\Gamma (0)} u_0 \\, ds $ for all $t \\in [0,T]$ .",
"After a suitable transformation one obtains the equation (REF ) with a zero initial condition $u_0$ and a right hand-side $f$ which satisfies $\\int _{\\Gamma (t)} f \\, ds =0 $ for all $t \\in [0,T]$ .",
"The solution $u$ of (REF ) then has the “shifted” mass conservation property $\\bar{u}(t)=0$ for all $t \\in [0,T]$ .",
"Tak ing suitable test functions in the discrete problem (REF ) we obtain that the discrete solution $u_h$ has the following weaker mass conservation property, with $\\bar{u}_h(t):=\\int _{\\Gamma (t)} u_h \\, ds$ : $\\bar{u}_{h,-}(t_{n})=0\\quad \\text{and}\\quad \\int _{t_{n-1}}^{t_n} \\bar{u}_h(t)\\, dt=0,\\quad n=1,2,\\dots N.$ For a stationary surface, $\\bar{u}_h(t)$ is a piecewise affine function and thus (REF ) implies $\\bar{u}_h(t)\\equiv 0$ , i.e,.",
"we have exact mass conservation on the discrete level.",
"If the surface evolves, the finite element method is not necessarily mass conserving: (REF ) holds, but $\\bar{u}_h(t) \\ne 0$ may occur for $ t_{n-1} \\le t < t_n$ .",
"In the discretization error analysis we use a consistent stabilizing term involving the quantity $\\bar{u}_h (t)$ .",
"More precisely, define $ a_\\sigma (u,v):= a(u,v)+\\sigma \\int _0^T \\bar{u}(t) \\bar{v}(t) \\, dt, \\quad \\sigma \\ge 0.$ Instead of (REF ) we consider the stabilized version: Find $u_h \\in W_{h}$ such that $ \\left\\langle \\dot{u}_h ,v_h\\right\\rangle _b +a_\\sigma (u_h,v_h)+d(u_h,v_h) =(f,v_h)_0 \\quad \\text{for all}~~v_h \\in W_{h}.$ Taking $\\sigma >0$ we expect both a stabilizing effect and an improved discrete mass conservation property.",
"Ellipticity of finite element method bilinear form and error bounds are derived in the mesh-dependent norm: $|\\!|\\!| u |\\!|\\!|_h:=\\left(\\Vert u_{-}^N\\Vert _T^2 + \\sum _{n=1}^N \\Vert [u]^{n-1}\\Vert _{t_{n-1}}^2+\\Vert u\\Vert _H^2\\right)^\\frac{1}{2}.$ In the error analysis we need a condition which plays a similar role as the condition “$ c - \\frac{1}{2} \\mathop {\\rm div}b >0$ ” used in standard analyses of variational formulations of the convection-diffusion equation $- \\Delta u + b\\cdot \\nabla u + c u=f$ in an Euclidean domain $\\Omega \\subset {R}^n$ , cf.",
"[15].",
"This condition is as follows: there exists a $c_0 >0$ such that $ {\\,\\operatorname{div_\\Gamma }}\\mathbf {w}(x,t) +\\nu _d c_F(t) \\ge c_0 \\quad \\text{for all}~~x \\in \\Gamma (t),~t \\in [0,T].$ Here $ c_F(t) >0$ results from the Poincare inequality $\\int _{\\Gamma (t)} |\\nabla _\\Gamma u|^2 \\, ds \\ge c_F(t) \\int _{\\Gamma (t)} ( u- \\frac{1}{|\\Gamma (t)|} \\bar{u})^2 \\, ds\\quad \\forall ~t\\in [0,T],~~\\forall ~u \\in H.$ A main result derived in [9] is given in the following theorem.",
"We assume that the time step $\\Delta t$ and the spatial mesh size parameter $h$ have comparable size: $\\Delta t \\sim h$ .",
"Theorem 1.",
"Assume (REF ) and take $ \\sigma \\ge \\frac{\\nu _d}{2}\\max \\limits _{t\\in [0,T]}\\frac{c_F(t)}{|\\Gamma (t)|}$ , where $c_F(t)$ is defined in (REF ).",
"Then the ellipticity estimate $ \\left\\langle \\dot{u},u\\right\\rangle _b +a_\\sigma (u,u)+d(u,u) \\ge c_s |\\!|\\!| u |\\!|\\!|_h^2 \\quad \\text{for all}~~u \\in W^b$ holds, with $c_s=\\frac{1}{2}\\min \\lbrace 1,\\nu _d,c_0\\rbrace $ and $c_0$ from (REF ).",
"Let $u \\in \\overset{\\circ }{W}$ be the solution of (REF ) and assume $u \\in H^2(\\mathcal {S})$ .",
"For the solution $u_h \\in W_h$ of the discrete problem (REF ) the following error bound holds: $|\\!|\\!| u-u_h |\\!|\\!|_h \\le c h \\Vert u\\Vert _{H^2(\\mathcal {S})}.$ A further main result derived in [9] is related to second order convergence.",
"Denote by $\\Vert \\cdot \\Vert _{-1}$ the norm dual to the $H^1_0(\\mathcal {S})$ norm with respect to the $L^2$ -duality.",
"Under the conditions given in Theorem 1 and some further mild assumptions the error bound $\\Vert u-u_h\\Vert _{-1} \\le c h^2 \\Vert u\\Vert _{H^2(\\mathcal {S})}$ holds.",
"This second order convergence is derived in a norm weaker than the commonly considered $L^2(\\mathcal {S})$ norm.",
"The reason is that our arguments use isotropic polynomial interpolation error bounds on 4D space-time elements.",
"Naturally, such bounds call for isotropic space-time $H^2$ -regularity bounds for the solution.",
"For our problem class such regularity is more restrictive than in an elliptic case, since the solution is generally less regular in time than in space.",
"We can overcome this by measuring the error in the weaker $\\Vert \\cdot \\Vert _{-1}$ -norm."
],
[
"NUMERICAL EXPERIMENT",
"In [7], [8] results of numerical experiments are presented.",
"The examples considered there have smoothly evolving surfaces (e.g.",
"a shrinking sphere) and the results show a convergence of order 1 in an $L^2(H^1)$ -norm (i.e.",
"$L^2$ w.r.t time and $H^1$ w.r.t space) and of order 2 in an $L^\\infty (L^2)$ norm.",
"This convergence behavior occurs already on relatively coarse meshes and there is no (CFL-type) condition on $\\Delta t$ .",
"In the example in this paper we consider an evolving surface $\\Gamma (t)$ which undergoes a change of topology and experiences a local singularity.",
"The computational domain is $x \\in \\Omega =(-3,3)\\times (-2,2)^2$ , $t \\in [0,1]$ .",
"For representation of the evolving surface we use a level set function $\\phi $ defined as: $\\phi (x,t) = 1 - \\frac{1}{\\Vert x -c_+(t)\\Vert ^3} - \\frac{1}{\\Vert x -c_-(t)\\Vert ^3},$ with $c_\\pm (t)= \\pm \\frac{3}{2}(t - 1, 0 , 0)^T$ .",
"The surface $\\Gamma (t)$ is defined as the zero level of $\\phi (x,t)$ , $t \\in [0,1]$ .",
"Take $t=0$ .",
"Then for $x \\in B(c_+(0);1)$ we have $\\Vert x -c_+(0)\\Vert ^{-3} =1$ and $\\Vert x -c_-(0)\\Vert ^{-3} \\ll 1$ .",
"For $ x \\in B(c_-(0);1)$ we have $\\Vert x -c_+(0)\\Vert ^{-3} \\ll 1$ and $\\Vert x -c_-(0)\\Vert ^{-3} =1$ .",
"Hence, the initial configuration $\\Gamma (0)$ is (very) close to two balls of radius 1, centered at $\\pm (1.5, 0, 0)^T$ .",
"For $t=1$ the surface $\\Gamma (1)$ is the ball around 0 with radius $2^{1/3}$ .",
"For $t >0 $ the two spheres approach each other until time $\\tilde{t}= 1-\\tfrac{2}{3} 2^{1/3}\\approx 0.160$ , when they touch at the origin.",
"For $t \\in (\\tilde{t},1]$ the surface $\\Gamma (t)$ is simply connected and smoothly deforms into the sphere $\\Gamma (1)$ .",
"In the vicinity of $\\Gamma (t)$ , the gradient $\\nabla \\phi $ and the time derivative $\\partial _t\\phi $ are well-defined and given by simple algebraic expressions.",
"We construct the normal wind field, which transports $\\Gamma (t)$ , by inserting the ansatz $\\mathbf {w}(x,t)=\\alpha (x,t)\\nabla \\phi (x,t)$ into the level set equation $\\partial _t\\phi + \\mathbf {w}\\cdot \\nabla \\phi =0$ .",
"This yields $\\mathbf {w}= -\\frac{\\partial _t\\phi }{\\vert \\nabla \\phi \\vert ^2}\\nabla \\phi .$ We consider the surfactant advection-diffusion equation $ {\\left\\lbrace \\begin{array}{ll}\\dot{u} + \\operatorname{div}_\\Gamma \\mathbf {w}\\, u - \\Delta _\\Gamma u= 0 & \\text{on }\\Gamma (t),~t\\in (0, 1],\\\\u(\\cdot , 0)= u_0 & \\text{on }\\Gamma (0).\\end{array}\\right.",
"}$ The initial surfactant distribution is given by $u_0(x)= {\\left\\lbrace \\begin{array}{ll} 3 - x_1 & \\text{for }x_1 \\ge 0,\\\\0 & \\text{else}.\\end{array}\\right.",
"}$ The initial configuration is illustrated in Figure REF .",
"Figure: Initial condition as color on the initial zero level Γ(0)\\Gamma (0).For the construction of a volume space-time finite element space we proceed as follows.",
"On $\\Omega $ we start with a level $l=0$ Kuhn-triangulation with mesh width $h_0=2$ .",
"We use regular refinement in the vicinity of the interface $\\Gamma (t)$ to ensure that the interface is embedded in tetrahedra with refinement level $l \\ge 1$ .",
"These tetrahedra have the mesh width $h_l=2^{1-l}$ .",
"On each time slab a level $l$ triangulation is used to define the volume space-time finite element space as in (REF ).",
"For simplicity we use the same value for $l$ on all time slabs.",
"The outer space induces a surface finite element space $W_h$ as in (REF ).",
"This space is used for a Galerkin discretization of (REF ), as given in (REF ) (note that in this experiment we take $f=0$ and a nonhomogeneous initial condition $u_0$ ).",
"We outline the quadrature method for the approximation of integrals over $\\mathcal {S}^n$ .",
"More details are given in [8].",
"Consider a single space-time prism $T \\times I_n$ that is intersected by the space time manifold $\\mathcal {S}^n$ .",
"Here $T$ is a level $l$ tetrahedron from the spatial triangulation.",
"First $T$ is regularly refined into 8 tetrahedra $T_j$ , $j=1,\\ldots ,8$ .",
"Each of the resulting space-time prisms $T_j \\times I_n$ is partitioned into 4 pentatopes by inserting adequate diagonals.",
"On each of these pentatopes the linear interpolant (in ${R}^4$ ) of the level set function $\\phi $ is computed.",
"The zero level of this interpolant is (if not degenerated) a 3-dimensional convex polytope, which can be partitioned into tetrahedra.",
"On these tetrahedra standard quadrature rules can be used.",
"Note that in this approximation procedure we have a geometric error due to the approximation of the zero level of $\\phi $ (which is the surface) by the zero level of its linear interpolant.",
"This assembling procedure is completely local and can be done prism per prism.",
"Figure: Snapshots of discrete solution, l=5l=5, Δt=2 -7 \\Delta t=2^{-7}.We present some results of numerical experiments.",
"In Figure REF we show a few snapshots of the surface and the computed surfactant distribution on a relatively fine space-time mesh, namely level $l=5$ and $\\Delta t =2^{-7}$ .",
"As a measure of accuracy we computed the discrete mass on the space-time manifold: $I_{l,dt}(t_n)= \\int _{\\Gamma _h(t_n)} u\\,d\\sigma ,\\quad n=0,1,\\ldots , N.$ where $\\Gamma _h(t_n)$ is the approximation of $\\Gamma (t_n)$ obtained as zero level of the piecewise linear interpolant of $\\phi $ , cf.",
"explanation above.",
"For $l=5$ , $\\Delta t=1/128$ the result is shown in Figure REF .",
"We interpolated the values $I_{l,dt}(t_n)$ , $n=0, \\ldots , N$ , resulting in the discrete mass quantity as a function of $t \\in [0,1]$ .",
"There is a mass loss of about $0.018$ , which corresponds to a relative error of $\\sim ~ 9\\cdot 10^{-4}$ .",
"Figure: The total amount of surfactant I 5,1/128 I_{5,1/128} over time.In Figure REF we show the result for $l=4$ , $\\Delta t=1/64$ .",
"The mass loss is $\\sim ~0.065$ , which is about a factor $3.6$ more than for the case $l=5$ , $\\Delta t=1/128$ .",
"Figure: The total amount of surfactant I 4,1/64 I_{4,1/64} over time.Finally, we show result for level $l=4$ , but with a large time step size $\\Delta t=1/4$ , i.e.",
"we use use only four time steps to approximate the solution at $t=1$ .",
"The four discrete solutions at $t=0.25,0.5,0.75,1.0$ are shown in Figure REF .",
"The total amount of surfactant is shown in Figure REF .",
"Figure: Snapshots of discrete solution, l=4l=4, Δt=1 4\\Delta t=\\frac{1}{4}, at t=0.25,0.5,0.75,1t=0.25,0.5,0.75,1.Figure: The total amount of surfactant I 4,1/4 I_{4,1/4} over time."
],
[
"DISCUSSION",
"We presented a space-time finite element method for solving PDEs on evolving surfaces.",
"The method is based on traces of outer finite element spaces, is Eulerian in the sense that $\\Gamma (t)$ is not tracked by a mesh, and can easily be combined with both space and time adaptivity.",
"No extension of the equation away from the surface is needed and thus the number of d.o.f.",
"involved in computations is optimal and comparable to methods in which $\\Gamma $ is meshed directly.",
"The computations are done in a time-marching manner as common for parabolic equations.",
"The method has second order convergence in space and time and conserves the mass in a weak sense, cf.",
"(REF ).",
"In practice, an artificial mass flux can be experienced due to geometric errors resulting from the approximation of $\\Gamma (t)$ .",
"In experiments, the loss of mass was found to be small and quickly vanishing if the mesh is refined.",
"The implicit definition of the surface evolution with the help of a level set function is well suited for numerical treatment of surfaces which undergo topological changes and experience singularities.",
"This report shows that the present space-time surface finite element method perfectly complements this property and provides a robust technique for computing diffusion and transport along colliding surfaces."
]
] | 1403.0277 |
[
[
"Enumerating Copies in the First Gribov Region on the Lattice in up to\n four Dimensions"
],
[
"Abstract The covariant gauges are known to suffer from the Gribov problem: even after fixing a gauge non-perturbatively, there may still exist residual copies which are physically equivalent to each other, called Gribov copies.",
"While the influence of Gribov copies in the relevant quantities such as gluon propagators has been heavily debated in recent studies, the significance of the role they play in the Faddeev--Popov procedure is hardly doubted.",
"We concentrate on Gribov copies in the first Gribov region, i.e., the space of Gribov copies at which the Faddeev--Popov operator is strictly positive (semi)definite.",
"We investigate compact U($1$) as the prototypical model of the more complicated standard model group SU($N_{c}$).",
"With our Graphical Processing Unit (GPU) implementation of the relaxation method we collect up to a few million Gribov copies per orbit.",
"We show that the numbers of Gribov copies even in the first Gribov region increase exponentially in two, three and four dimensions.",
"Furthermore, we provide strong indication that the number of Gribov copies is gauge orbit dependent."
],
[
"Introduction",
"The most successful way of studying gauge field theories non-perturbatively is to put them on a finite space-time lattice, this approach is commonly referred to as lattice field theory [1].",
"In the continuum, promising non-perturbative approaches are functional methods, in particular Dyson-Schwinger equations (DSEs) [2] and functional renormalization group equations (FRGs).",
"The DSE approach, for example, can be useful in the low momentum region of quantum chromodynamics (QCD), whereas using lattice QCD one can perform first principles calculations of non-perturbative quantities in QCD.",
"The approximations involved in lattice QCD can be systematically removedIn practice, due to limited computer power, an extrapolation to the infinite volume and continuum limits has to be performed of which the analytical form is unknown., whereas a systematic removal of the truncations in DSEs is much more involved.",
"In other words, lattice simulations can provide an independent check on the results obtained in the DSE approach.",
"There is however a subtle difference in the two approaches.",
"A lattice field theory is manifestly gauge invariant, hence one does not need to fix a gauge on the lattice to calculate gauge invariant observables.",
"In the continuum approaches, each gauge configuration comes with infinitely many equivalent physical copies, the set of which is called a gauge-orbit.",
"Hence, to remove the redundant degrees of freedom, one requires gauge fixing.",
"Thus to compare with DSE results, a corresponding gauge fixing is necessary on the lattice.",
"In the continuum, the standard way to fix a gauge in the perturbative limit is the so-called Faddeev–Popov (FP) procedure [3] which amounts to formulate a gauge fixing device which is called the gauge fixing partition function, $Z_{GF}$ .",
"In the perturbative limit, it can be shown that for an ideal gauge fixing condition, $Z_{GF} = 1$ .",
"Then, this unity is inserted in the measure of the generating functional so that the redundant degrees of freedom are removed after appropriate integration.",
"Becchi, Rouet, Stora and Tyutin (BRST) generalised the FP procedure [4].",
"Gribov found that in non-Abelian gauge theories a generalised Landau gauge fixing condition, if treated non-perturbatively, has multiple solutions, called Gribov or Gribov–Singer copies [5], [6], [2].",
"Thus, the above assumption of the ideal gauge fixing condition became a subtle point in generalizing the FP procedure for non-perturbative field theories.",
"Furthermore, Neuberger showed that on the lattice, the corresponding $Z_{GF}=0$ [7], [8], i.e., the expectation value of a gauge fixed observable awkwardly turns out to be $0/0$ , known as the Neuberger $0/0$ problem.",
"It yields that BRST formulations can not be constructed on the lattice, a situation which may severely hamper any comparison of gauge-dependent quantities on the lattice with those in the continuum.",
"It is argued that Gribov copies may influence the infrared behaviour of the gauge dependent propagators of gauge theories both on the lattice [9], [10], [11], and in the continuum [12], [13].",
"In [14] SU(2) Yang–Mills theory has been investigated in the strong coupling limit on the lattice: it has been shown that the Gribov ambiguity is rather strong in that case and especially affects the ghost propagator.",
"There have also been efforts to count Gribov copies in the continuum in Refs.",
"[15], [16] where the counting was restricted for the static spherically symmetric configurations only, for the SU(2) case.",
"Interestingly, recently, a deep relation between lattice gauge fixing and lattice supersymmetry has been proposed [17], [18]: the partition functions of a class of supersymmetric Yang-Mills theories can be viewed as a gauge fixing partition function a la Faddeev–Popov and the `Gribov copies' are then nothing but the classical configurations of the theory.",
"On the lattice, gauge fixing is reformulated as an optimization problem.",
"With the gauge fields defined through link variables $U_{i,\\mu }\\in G$ where the discrete variable $i$ denotes the lattice-site index, $\\mu =1,\\hdots ,d$ is a directional index and $G$ is the corresponding group of the theory.",
"The standard choice of the lattice Landau gauge (LLG) fixing functional to be optimized with respect to the corresponding gauge transformations $g_{i}\\in G$ , is $F_{U}(g)=\\sum _{i,\\mu }(1-\\frac{1}{N_{c}}\\operatorname{Re}\\operatorname{Tr}g_{i}^{\\dagger }U_{i,\\mu }g_{i+\\hat{\\mu }}),$ for SU($N_{c}$ ) gauge groups.",
"Choosing $f_{i}(g):=\\frac{\\partial F_{U}(g)}{\\partial g_{i}}=0$ for each lattice site $i$ gives the lattice divergence of the lattice gauge fields and in the naive continuum limit recovers the continuum Landau gauge condition, i.e., $\\partial _{\\mu }A_{\\mu }=0$ , where $A_{\\mu }$ is the gauge potential.",
"The corresponding FP operator $M_{FP}$ is then the Hessian matrix of $F_{U}(g)$ with respect to the gauge transformations.",
"The stationary points of $F_{U}[g]$ are the Gribov copies.",
"Neuberger showed [7], [8] that when all the stationary points of $F_{U}[g]$ are taken into account, the gauge fixing partition function $Z_{GF}$ à la FP procedure turns out to be zero and the expectation value of a gauge fixed variable is then $0/0$ .",
"The Morse theory interpretation of this problem was given by Schaden [19] who showed that $Z_{GF}$ calculates the Euler character $\\chi $ of the group manifold $G$ at each site of the lattice.",
"In particular, for a lattice with $N$ lattice sites, $Z_{GF}=\\sum _{i}\\mbox{sign}(\\det \\, M_{FP}(g))=(\\chi (G))^{N},$ where the sum runs over all the Gribov copies.",
"Since $\\chi $ of the group manifold for compact U(1), $S^1$ , and of the group manifold of SU($N_c$ ), $S^3\\times S^5 \\times \\dots \\times S^{2 N_c-1}$ , is zero, the corresponding $Z_{GF} = 0$ .",
"To evade this problem, for an SU(2) gauge theory, Schaden proposed to construct a BRST formulation only for the coset space SU(2)/U(1) for which $\\chi \\ne 0$ .",
"The procedure can be generalised to fix the gauge of an SU($N_{c}$ ) lattice gauge theory to the maximal Abelian subgroup $($ U(1)$)^{N_{c}-1}$ [20], [21].",
"Thus, the Neuberger $0/0$ problem for an SU($N_{c}$ ) lattice gauge theory actually lies in (U(1)$)^{N_{c}-1}$ .",
"For this reason, we concentrate on the compact U(1) case in the rest of the paper.",
"There are other ways proposed to avoid the Neuberger $0/0$ problem by modifying the gauge fixing condition while taking into account that the corresponding $Z_{GF}$ should be orbit-independent, and, for technical convenience, it should be possible to efficiently implement the corresponding gauge fixing numerically.",
"Renormalization, in contrast, is not required: unitary gauge in gauge-Higgs models, for example, is even perturbatively non-renormalizable but still yields the correct physics.",
"In minimal lattice Landau gauge, one focuses on the first Gribov region [22], i.e., the space of minima, in which there is no cancelation among the signs of $M_{FP}$ .",
"Hence, $Z_{GF}$ just counts the number of minima of $F_{U}[G]$ , and the Neuberger $0/0$ is avoided.",
"It is yet to be shown if the corresponding $Z_{GF}$ is orbit-independent in general.",
"However, in the one-dimensional [23], [24] and two-dimensional [25] compact U(1) cases, it was already shown that $Z_{GF}$ is in fact an orbit-dependent quantity.",
"In the present paper, one of our goals is to verify this in higher dimensional cases.",
"In absolute lattice Landau gauge, one focuses on the space of global minima, called the fundamental modular region (FMR).",
"The Neuberger $0/0$ problem is again avoided here.",
"It is anticipated that there are no Gribov copies inside the FMR [26], [27], which was verified to be true in one- and two-dimensional compact U(1) cases [23], [24], [25].",
"Other approaches to evade the Neuberger $0/0$ problem were recently put forward in [28], [29], [30], [31], [32], [33], [34], [35], [36], [55] and reviewed in [37]."
],
[
"Lattice Landau gauge for compact U(1):",
"Following the notations of Ref.",
"[25], for compact U(1) the gauge fields and gauge transformations are $U_{i,\\mu }=e^{i\\phi _{i,\\mu }}$ and $g_{i}=e^{i\\theta _{i}}$ , respectively, where the angles $\\theta _i$ and $\\phi _{i,\\mu }$ take values from $(-\\pi ,\\pi ]$ .",
"Hence, Eq.",
"(REF ) becomes $F_{\\phi }(\\theta )=\\sum _{i,\\mu }\\big (1-\\cos (\\phi _{i,\\mu }+\\theta _{i+\\hat{\\mu }}-\\theta _{i})\\big )\\equiv \\sum _{i,\\mu }(1-\\cos \\phi _{i,\\mu }^{\\theta }),$ where $\\phi _{i,\\mu }^{\\theta }:=\\phi _{i,\\mu }+\\theta _{i+\\hat{\\mu }}-\\theta _{i}$ .",
"When $\\phi _{i,\\mu }$ is picked randomly, it is refereed to as a random or hot orbit and when all the $\\phi $ -angles are zero, it is called the trivial or cold orbit.",
"We concentrate on periodic boundary conditions (PBC), i.e., $\\theta _{i+N\\hat{\\mu }}=\\theta _{i}$ and $\\phi _{i+N\\hat{\\mu },\\mu }=\\phi _{i,\\mu }$ , which is the most natural choice in lattice gauge theories.",
"We remove the global gauge degree of freedom by fixing the angle $\\theta _{(N,...,N)}$ to zero.",
"Furthermore, we let $\\lbrace \\phi _{i,\\mu }\\rbrace $ take random values independent of the action, corresponding to the strong coupling limit $\\beta =0$ , which is sufficient to answer the basic questions of counting Gribov copies and their orbit-dependence as every gauge orbit has a non-vanishing weight for any finite $\\beta $ .",
"The global minimum of $F_{\\phi }(\\theta )$ is usually thought to be unique modulo possible accidental degeneracies which are expected to form a set of measure zero (and non-accidental degeneracies on the boundary of the FMR).",
"Therefore, we focus on the minimal lattice Landau gauge in this work.",
"All the Gribov copies for the one-dimensional LLG for compact U(1) have been found analytically for periodic [23], [24] and antiperiodic [23], [28], [29] boundary conditions.",
"However, solving the stationary equations in more than one dimension turns out to be a difficult task and has not been done so far.",
"The main difficulty here is that the stationary equations are highly nonlinear in higher dimensions.",
"In Ref.",
"[23] it was shown how these equations could be viewed as a system of polynomial equations, and then the numerical polynomial homotopy continuation method (NPHC) was used to find all the stationary points for small lattices in two dimensions.",
"The method was used extensively afterwards to study similar problems of finding stationary points or minima of a multivariate function arising in statistical mechanics and particle physics [38], [25], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50].",
"Interestingly, in Ref.",
"[42], two types of singular solutions were observed for the trivial orbit case: isolated singular solutions at which the Hessian matrix is singular (these solutions are in fact multiple solutions) and a continuous family of singular solutions.",
"It was shown that one can construct one-, two-, etc.",
"parameter solutions, even after fixing the global $O(2)$ freedom.",
"The authors of [51] studied the continuum limit of lattice U(1) theory in two dimensions and found that in that limit, the absolute and local minima become more and more degenerate.",
"In Ref.",
"[25], for the two-dimensional case, among other results using the Conjugate Gradient method it was conjectured that the number of Gribov copies in the first Gribov region increases exponentially.",
"In Ref.",
"[52], [53], the problem of finding minima of the compact U(1) LLG in two dimensions was studied for the trivial orbit case, which is nothing but the two-dimensional XY model without disorder.",
"There, many minima were found using potential energy landscape methods [56], [57], and it was shown that the number of minima increased exponentially in this case.",
"Moreover, using disconnectivity diagrams, it was shown how the minima were connected to each other via the saddles of index 1 (called transitions states in theoretical chemistry).",
"In the current paper, we want to verify this conjectured exponential increase in three and four dimensions.",
"As a byproduct, we also improve on the previous results for two dimensions.",
"In a separate work, we develop a novel and efficient method to find many Gribov copies, if not all, starting from a maximum of the lattice Landau gauge fixing functional and moving towards lower index saddles [54].",
"Figure: The number of distinct minima versus the number of samples (initial guesses) for different orbits for d=2d=2 and N=6,8,10N=6, 8, 10.Figure: The number of distinct minima found versus the numberof samples (random starts) for the three different orbits for d=3d=3 and N=4,6N=4, 6.Figure: The number of distinct minima found versus the numberof samples (random starts) for a single orbit under investigation for d=4d=4 and N=4N=4"
],
[
"A GPU Implementation of the Relaxation Method",
"In this section we describe our numerical approach.",
"We adopt the relaxation algorithm and execute it on graphics processing units (GPUs) which offer a high level of parallelism and thus enable us to gather a large number of samples within a practical amount of computer time.",
"The idea of the relaxation algorithm is to sweep over the lattice while optimizing the gauge functional (REF ) locally on each lattice site.",
"Thus, on each site $i$ the maximum of $\\frac{1}{2}\\mathrm {Re}\\left[ g_i K_i \\right]$ is to be found.",
"Here we introduced $K_i \\equiv \\sum _\\mu \\left[ U_{i,\\mu }\\;g_{i+\\hat{\\mu }}^\\dagger + U_{i-\\hat{\\mu },\\mu }^\\dagger \\; g_{i-\\hat{\\mu }}^\\dagger \\right].$ It is easy to see that Eq.",
"(REF ) becomes maximized if we simply set $\\theta _i$ equal to the phase of $K_i^*$ .",
"Note that the local optimization depends for each lattice site on the nearest neighbors only, hence we can perform a checker board decomposition of the lattice and the local optimization of all lattice sites of one of the two sublattices will be independent of all other lattice sites of the same sublattice.",
"Our implementation will benefit therefrom by optimizing all lattice sites of a given sublattice concurrently instead of performing serial loops over the members of each sublattice.",
"We do not perform overrelaxation since we have found that, while overrelaxation decreases the convergence time, it also decreases the chance of converging to larger minima and thus introduces a bias.",
"NVIDIA offers with CUDA (Compute Unified Device Architecture) a parallel programming model that enables the programmer to run so-called kernels on the GPU.",
"These kernels are specialised functions that perform a sequence of tasks in a highly parallel fashion.",
"The user defines a grid of thread blocks and a number of threads per thread block in such a way that the kernel call replaces serial loops over memory addresses by concurrent calculations on all corresponding addresses.",
"Our implementation is based on the cuLGThttp://www.cuLGT.com code for lattice gauge fixing on GPUs [58].",
"Here we assign one thread to one lattice site and a whole lattice will be encapsulated in one thread block.",
"The GPU can handle several thread blocks per multiprocessor concurrently and we launch a grid of thread blocks where each thread block contains a lattice initialised with random numbers which we generate with the Philox random number generator [59].",
"In this way we minimize in parallel as many samples as we launch thread blocks.",
"Moreover, we add another layer of parallelism by adopting multiple GPUs: therefore we loop over the kernel calls in the main function of the execution code while switching between the multi-GPUs.",
"In practice we adopt four cards of the NVIDIA Tesla C2070 and we launch 1024 thread blocks (i.e.",
"random start samples) per GPU.",
"Hence we run 4096 samples at once.",
"While CUDA allows for a much larger number of thread blocks to be launched per kernel call this can be counterproductive since the runtime depends on the slowest converging sample among all running samples.",
"A smaller number than 1024 thread blocks per GPU, on the other hand, would not fully occupy the GPU and thus result again in a performance loss.",
"Therefore, we keep the grid size as 1024 blocks per GPU fixed.",
"For each sample we store the value of the minimum to which the relaxation algorithm has converged and subsequently sort these values via bitonic sort, again accelerated by the GPU.",
"The execution time of the code depends on the lattice size and the number of iterations until convergence which varies from sample to sample.",
"In practice, the time to minimize one mebisample ($1024^2$ samples) adopting four NVIDIA Tesla C2070 GPUs varies from a few seconds (e.g.",
"$d=2$ up to $N=6$ lattices) over several minutes (e.g.",
"$10^2$ and $4^3$ ) up to a few hours ($8^3$ and $4^4$ ).",
"As a stopping criterion we require the largest gradient over all lattice sites and all concurrently running samples to be smaller than $10^{-12}$ .",
"We have found that this criterion is sufficient to ensure that the values of the minima to which the relaxation algorithm converges to, reach plateaus to a precision of at least $10^{-10}$ .",
"The whole simulation is performed in double precision and we store the values of the minima in double precision.In [58] it was shown that the accumulation of numerical errors of typical observables in double precision in lattice gauge fixing is smaller then $10^{-12}$ even for extensively long simulation runs.",
"Subsequently we transform each minimum $x\\in [0,1.0]$ to an integer $X\\in [0,10^8]$ .",
"These integers can then be unambiguously compared using the bitonic sort algorithm.",
"The inverse of the upper bound of the integer interval defines the resolution with which we choose to distinguish minima.",
"Hence we consider values of minima as the same when they agree within eight decimal places.",
"It is likely that with our resolution of $10^{-8}$ we count some minima as the same which are distinct at finer resolution, i.e.",
"finer resolution may eventually allow higher distinction of otherwise non-distinguishable minima.",
"Adopting this rather conservative resolution we assure that we obtain real lowest bounds of the number of Gribov copies per orbit which has highest priority for our study.",
"More details on numerical distinction of Gribov copies, including a discussion of renormalization effects, can be found in [60].",
"In the present work, we sample orbits randomly, i.e., we consider the theory at the strong coupling limit where the inverse coupling $\\beta =0$ .",
"This choice of $\\beta $ is sufficient to make general conclusions for the number of Gribov copies for the purpose of this work."
],
[
"Number of Gribov copies",
"In Tab.",
"REF we list for each lattice size the number of orbits and the number of samples per orbit for which we have minimized the gauge functional Eq.",
"(REF ).",
"Note that the number of samples is given in units of mebisample ($1024^2$ samples).",
"In Figs.",
"REF –REF we plot the number of distinct minima that we found as a function of the number of random initial guesses.",
"Due to the nature of the bitonic sorting algorithm, we measure the number of minima only at stages of powers of two in the number of samples.",
"In the figures, the resulting points are connected by straight lines to guide the eye.",
"In two dimensions (Fig.",
"REF ), we find for $N=6$ that all orbits have converged to plateaus, indicating that we are very close to having found all minima in that case.",
"Similarly, the plot for $N=8$ (same figure) reveals that still a relatively large fraction of the orbits have converged.",
"The curves in the plot for $N=10$ , in contrast, have not converged and consequently we are further away from having collected all minima here.",
"Analogously, Fig.",
"REF shows the data for $d=3$ where we reach our limit for $N=6$ : one gibisample ($1024^3$ samples) per orbit is not sufficient to get close to finding all minima.",
"In four dimensions, we sample one hundred $2^4$ orbits and a single orbit of $N=4$ with a gibisample initial guesses for which we obtain only a relatively weak lower bound on the number of distinct minima.",
"We conclude that even though we have not found every single minimum for each orbit, Figs.",
"REF –REF provide clear evidence that the number of Gribov copies is orbit-dependent.",
"Figure: Lower bounds of the number of minima as a function of NN for d=2,3,4d=2,3,4 averaged over the different random orbits.The curves correspond to best fits to the function Eq.",
"() and the corresponding fit parameters aresummarized in Tab.",
".Table: The fit parameters of the curves Eq.",
"() shown in Fig.",
".The averages and standard deviations for the lower bounds of the number of minima per orbit and lattice size are summarised in Tab.",
"REF .",
"The lower bounds on the number of minima as a function of $N$ for all dimensions is plotted in Fig.",
"REF .",
"Additionally, best fits to a function $h(x)=a\\exp \\left(bx^c\\right)$ are shown and the corresponding fit parameters are listed in Tab.",
"REF .",
"The data for $d=2$ indicate that the number of distinct gauge functional minima depends exponentially on $N^2$ .",
"The data for $d=3$ do not confirm an exponent $N^d$ , but this is probably because our lowest bound for $6^3$ severely underestimates the true number of copies.The $N=6$ curves in Fig.",
"REF are still rising by more then 6% when increasing the number of samples from 512 to 1024 mebisamples.",
"Moreover, the three curves are rather close to each other compared to, e.g, the curves for $N=4$ .",
"Table: A complementary set of gauge functional minima: more orbits per lattice size with four mebisamples (4,194,304 samples) per orbit.",
"The averages of the values of the gauge functional at the global minimum and the corresponding standard deviations are listed (cp.",
"Fig.",
").Figure: In the upper part of the plot we show the ratio of each of the bins (resolution 10 -4 10^{-4}) of the distribution of the histograms of the minima of ten orbits of a 10 2 10^2 lattice, taking 256 vs. four mebisamples into account.",
"The lower part of the plot shows the bins from 256 mebisamples when the corresponding bin of four mebisamples was empty (i.e.",
"the ratio in the upper plot was not defined).",
"Points on top of the bin bars are plotted for better visibility."
],
[
"The values of the gauge functional at the minima and their distribution",
"In order to investigate how many mebisamples we need to sufficiently sample nearly all minima with reasonable statistics, we compare the set of minima we obtained from ten orbits of a $10^2$ lattice from four mebisamples per orbit to the set of minima when we apply 256 mebisample per orbit to the same ten orbits.",
"We assign each minimum to a bin of resolution $10^{-4}$ and plot the ratios of the entries of the bins from 256 vs. four mebisamples in Fig.",
"REF .",
"The plot reveals that the ratios of the low-lying and midrange minima is very close to the expected factor $256/4$ whereas the ratios fluctuate much stronger for high-lying minima.",
"Moreover, only very high-lying bins of the four mebisample run are empty while the corresponding bins of the 256 mebisample run have entries, as presented in the lower plot of the figure.",
"This indicates that four mebisamples are sufficient to obtain reasonable statistics and running more samples will improve mainly in the range most distant to the global minimum which appears to be less attractive for the relaxation algorithm.",
"Figure: The gauge functional evaluated at the global minimum as a function of NN for d=2,3,4d=2, 3, 4.With the aim of studying the dependence of the value of the gauge functional at the global minimum on $N$ and $d$ , it is desirable to investigate more orbits to increase the statistics.",
"Motivated by the conclusion of the previous paragraph, we limit the number of samples per orbit to four mebisamples which renders increasing the number of orbits affordable.",
"Nevertheless we are confident that the smallest minimum we find on each orbit is at least numerically very close to the global minimum, if not equal to it.",
"Hence we study the global minima of the orbits listed in Tab.",
"REF which additionally include $8^3$ lattices.",
"In Fig.",
"REF , the gauge functional Eq.",
"(REF ) evaluated at the global minimum as a function of $N$ for $d=2, 3, 4$ is shown.",
"For constant $N$ , the value of the global minimum is higher for higher dimension $d$ and for fixed $d$ the global minima seem to converge to plateaus for $N\\gtrsim 6$ .",
"Figure: The normalised distribution of the minima of all orbits and samples of d=2,3,4d=2, 3, 4 (see Tab.",
").Figure: The averages and standard deviations of the relative deviation of the local minima F i F_i from the global minimum F g F_g on the corresponding orbit: (F i -F g )/F g (F_i-F_g)/F_g.In Fig.",
"REF histograms for the distribution of the functional values for all lattice sizes, each superimposing data from all orbits of Tab.",
"REF , are shown.",
"It is evident that the distribution becomes narrower with increasing lattice size $N$ .",
"It is important to stress, however, that the wide spread for lower $N$ is mainly due to the large variance of the value of the global minimum, compare with the data in the table.",
"Fig.",
"REF , in contrast, shows the distribution of the minima $F_i$ relative to the global minimum $F_g$ on that orbit: $(F_i-F_g)/F_g$ .",
"Subsequently, the data has been averaged over all orbits.",
"As a consequence of this strategy, the aforementioned effect of the variance of the global minima is factored out.",
"The deviation (Fig.",
"REF ) appears to increase with $N$ , not decrease as it should if global and local minima became equivalent.",
"In summary, our data does not indicate that global and local minima become equivalent for large $N$ (in the strong coupling limit)."
],
[
"Conclusions",
"On the lattice, gauge fixing is formulated as a minimization problem.",
"The stationary points of the gauge fixing functional are the Gribov copies which are physical replications of a gauge configuration and exist even after fixing the gauge non-perturbatively.",
"In this paper, we aimed at enumerating the number of Gribov copies in the first Gribov region on the lattice in order to address different modifications of the gauge fixing procedure which can be affected by the potential orbit-dependence of the number of Gribov copies.",
"We studied compact U(1) gauge theory.",
"The latter not only can serve as laboratory for testing our computational efforts, but is a very important model in its own right: it has been shown that the origin of the Neuberger $0/0$ problem lies in compact U(1), and the problem is evaded for any SU($N$ ) when it is evaded for compact U(1).",
"This holds even though Gribov copies in the compact U(1) case are just lattice artifacts.",
"We performed a brute force analysis of the first Gribov region for the compact U(1) case in $d=2, 3, 4$ dimensions.",
"We started the relaxation algorithm from up to more than a billion random points on the gauge orbits and collected up to millions of distinct gauge functional minima per orbit.",
"Even though our GPU implementation has proven to be a powerful tool for counting Gribov copies, we observed that the problem of counting Gribov copies becomes increasingly difficult with increasing lattice sizes.",
"In particular, for the biggest volumes of our runs ($10^2$ , $6^3$ and $4^4$ ), the convergence to the full number of distinct minima with an increasing number of minimization attempts (“samples”) could not be achieved.",
"In $d=4$ we reached our limits with a single orbit of the modest lattice size $4^4$ .",
"We were able to show that the number of Gribov copies in the first Gribov region increases exponentially in two, three and four dimensions.",
"More specifically, we found that the number of distinct minima per orbit increases at least with $\\exp \\left(\\sim N^2\\right)$ , an $\\exp \\left(\\sim N^d\\right)$ dependence is likely, though could not definitely be shown with the currently available data.",
"Moreover, we have found strong indication that the number of minima is orbit dependent, i.e., strong indication that the gauge fixing partition function for the minimal Landau gauge on the lattice is orbit dependent.",
"Finally, in the continuum it was conjectured that the local minima of the corresponding gauge fixing functional tend to be degenerate with the global minimum [22].",
"A direct comparison with this conjecture can not be done using our results on the lattice with $\\beta =0$ .",
"However, while our data exhibits narrowing of the distribution of the values of the gauge fixing functional at the minima when increasing the lattice size (taking data from several orbits per lattice size into account); we cannot observe that local minima tend to get closer to the global minimum of the corresponding orbit.",
"The authors are very grateful to Reinhard Alkofer, Maartin Golterman, Axel Maas, Jonivar Skullerud, Martin Schaden and Yigal Shamir for helpful discussions and comments on the manuscript.",
"DM was supported by an ERC and a DARPA Young Faculty Award.",
"The calculations have been performed on the “mephisto” cluster at the University of Graz."
]
] | 1403.0555 |
[
[
"LHC Constraints on the Lee-Wick Higgs Sector"
],
[
"Abstract We determine constraints on the Lee-Wick Higgs sector obtained from the full LHC Higgs boson data set.",
"We determine the current lower bound on the heavy neutral Lee-Wick scalar, as well as projected bounds at a 14 TeV LHC with 300 and 3000 inverse femtobarns of integrated luminosity.",
"We point out that the first sign of new physics in this model may be the observation of a deviation from standard model expectations of the lighter neutral Higgs signal strengths corresponding to production via gluon-gluon fusion and decay to either tau or $Z$ pairs.",
"The signal strength of the latter is greater than the standard model expectation, unlike most extensions of the standard model."
],
[
"Introduction",
"Over the past three decades, the most popular approach to addressing the hierarchy problem of the standard model has been to introduce additional particles whose virtual effects lead to a cancellation of quadratic divergences.",
"Supersymmetry has been the most studied scenario of this type; only a few years ago, there was much anticipation that colored superparticles would be revealed early in the first run of the Large Hadron Collider (LHC).",
"Unfortunately, this expectation has not been realized.",
"Since theories with partner particles have a decoupling limit, it is possible that the colored partners, which the LHC is most capable of detecting, may lie just beyond the reach of the initial $\\sim $ 8 TeV run.",
"It also follows that alternatives to supersymmetry, with their own distinct set of partner particles, remain in play as possible solutions to the hierarchy problem.",
"Here we determine how effectively current LHC data on the Higgs boson can constrain one such possibility, and explore the reach attainable in the future.",
"We assume the framework of the Lee-Wick Standard Model (LWSM) [1].",
"In the LWSM, a higher-derivative term quadratic in the fields is introduced for each standard model particle.",
"An additional pole in each propagator corresponds to a new physical state, the Lee-Wick partner.",
"Quadratic divergences in the theory are eliminated due to the faster fall-off of the momentum-space propagators in the higher-derivative formulation of the theory.",
"The presence of twice as many time derivatives in the theory implies that twice as much initial-value data is needed to specify solutions to the classical equations of motion.",
"Hence, one anticipates that the theory can be reformulated in terms of an equivalent one with twice as many fields, but kinetic terms with only two derivatives.",
"This is precisely what happens in the auxiliary-field formulation of the LWSM [1], as we will illustrate in the next section.",
"The additional field corresponds to the Lee-Wick partner particle, and the elimination of quadratic divergences emerges via cancellations between diagrams involving ordinary and Lee-Wick particles, respectively [1].",
"The LWSM is unusual in that the Lee-Wick partner fields have wrong-sign quadratic terms; this implies that the Lee-Wick states have negative norm.",
"In the original papers of Lee and Wick [2], as well as Cutkosky et al.",
"[3], it was argued that the unitarity of such a theory could be maintained provided that the Lee-Wick partners are unstable (i.e., are excluded from the set of possible asymptotic scattering states) and that a specific pole prescription is used in evaluating loop diagrams.",
"This approach has proven effective at the level it has been checked (one loop) and it is generally taken as a working assumption that some viable prescription exists at higher order.",
"While Lee-Wick theories violate causality at a microscopic level, it has been argued that this may not lead to logical paradoxes [4].",
"In the context of scattering experiments, this has been supported by a study of the large-$N$ limit of the Lee-Wick O($N$ ) model, where the unitarity and Lorentz-invariance of the S-matrix could be explicitly confirmed [5].",
"While the phenomenological implications of microscopic acausality are of substantial interest [6], they will not be the subject of this paper.",
"Other phenomenological studies of Lee-Wick theories can be found in Ref. [7].",
"We focus instead on how the most current LHC data constrains the possibility of Lee-Wick partners.",
"Specifically, we focus on a Lee-Wick extension of the Higgs sector, an effective theory in which the Lee-Wick partner to the Higgs doublet is retained, while all the other Lee-Wick partners are assumed to be heavy and decoupled [8], [9].",
"This approximation is justified for the following reason: the Lee-Wick partners to the Higgs field, the electroweak gauge bosons and the top quark are the most important in the cancellation of quadratic divergences; these would be expected to be the lightest to minimize fine tuning.",
"Of this set, however, all but the partner to the Higgs doublet are forced up to multi-TeV energy scales by existing electroweak constraints [11].",
"As we will show in the next section, the Lee-Wick Higgs sector presents itself as an unusual, constrained two-Higgs doublet model, one that is specified by a single free parameter once the lightest scalar mass eigenvalue is fixed.",
"Current data on the 125 GeV Higgs boson at the LHC can then be used to determine bounds on the masses of the other neutral and charged scalar mass eigenstates in the theory.",
"We note that past studies of the Lee-Wick Higgs sector [8], [9], [10] were undertaken before LHC Higgs boson data was available; in this letter we take into account all such data available to date and determine projected bounds based on current assessments of the integrated luminosities that may be realistically obtained.",
"Our letter is organized as follows: In Section , we define our effective theory.",
"In Section , we determine bounds on the heavier neutral scalar by fitting the model's predictions for the 125 GeV mass eigenstate, using the full data set currently available from the LHC.",
"In the second part of this section, we determine projected bounds based on the assumption of 300 to 3000 fb$^{-1}$ of integrated luminosity at a 14 TeV LHC.",
"In Section , we summarize our results and compare them to other existing bounds on the model."
],
[
"The Lee-Wick Higgs Sector",
"In the Higgs sector of our model, a higher-derivative kinetic term is included in the Higgs field Lagrangian ${\\cal L} = (D_\\mu \\hat{H})^\\dagger (D^\\mu \\hat{H})- \\frac{1}{m_{\\tilde{h}}^2} (D_\\mu D^\\mu \\hat{H})^\\dagger (D_\\nu D^\\nu \\hat{H}) - V(\\hat{H}) \\,\\, .$ Here $D_\\mu =\\partial _\\mu - i g W^a_\\mu T^a - i g^{\\prime } B_\\mu Y$ is the usual covariant derivative for the standard model gauge group and a hat denotes a field in the higher-derivative formulation of the theory.",
"The Higgs potential is given by $V(\\hat{H}) = \\frac{\\lambda }{4} \\left( \\hat{H}^\\dagger \\hat{H} - \\frac{v^2}{2} \\right)^2 \\,\\, .$ Eq.",
"(REF ) is reproduced from the following Lagrangian, ${\\cal L}= (D_\\mu \\hat{H})^\\dagger (D^\\mu \\hat{H}) + [(D_\\mu \\hat{H})^\\dagger (D^\\mu \\tilde{H}) + h.c.]+ m_{\\tilde{h}}^2 \\tilde{H}^\\dagger \\tilde{H} - V(\\hat{H}),$ if one eliminates the auxiliary field $\\tilde{H}$ using its equation of motion.",
"If instead, one uses the field redefinition $\\hat{H}=H-\\tilde{H}$ , Eq.",
"(REF ) takes the standard Lee-Wick form ${\\cal L}_{LW}=(D_\\mu H)^\\dagger (D^\\mu H) - (D_\\mu \\tilde{H})^\\dagger (D^\\mu \\tilde{H}) + m_{\\tilde{h}}^2 \\tilde{H}^\\dagger \\tilde{H}-V(H-\\tilde{H}) \\, .$ In unitary gauge, the Higgs doublet can be decomposed $H = \\left(\\begin{array}{c} 0 \\\\ \\frac{v+h}{\\sqrt{2}} \\end{array}\\right) \\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tilde{H} = \\left(\\begin{array}{c} \\tilde{h}^+ \\\\ \\frac{\\tilde{h} + i \\tilde{P}}{\\sqrt{2}} \\end{array}\\right) \\,,$ where $v\\approx 246$ GeV is the electroweak scale.",
"Expanding the potential in terms of its quadratic, cubic and quartic parts, we find: $V^{(2)} = \\frac{\\lambda \\, v^2}{4} (h-\\tilde{h})^2 - \\frac{m_{\\tilde{h}}^2}{2} (\\tilde{h}^2 + \\tilde{P}^2 + 2 \\tilde{h}^+ \\tilde{h}^-) \\, ,$ $V^{(3)} = \\frac{\\lambda \\, v}{4} (h-\\tilde{h}) \\left[ (h-\\tilde{h})^2 + \\tilde{P}^2 + 2 \\tilde{h}^- \\tilde{h}^+\\right] \\, ,$ $V^{(4)} = \\frac{\\lambda }{16} \\left[ (h-\\tilde{h})^2 + \\tilde{P}^2 + 2 \\tilde{h}^- \\tilde{h}^+\\right]^2 \\,.$ Note that the Lee-Wick charged scalar and pseudoscalar Higgs fields have mass $m_{\\tilde{h}}$ , while there is mixing between the neutral scalar states $h$ and $\\tilde{h}$ .",
"Indicating the neutral mass eigenstates with the subscript 0, we define the mixing angle $\\left(\\begin{array}{c} h \\\\ \\tilde{h} \\end{array} \\right) = \\left(\\begin{array}{cc} \\cosh \\alpha & \\sinh \\alpha \\\\ \\sinh \\alpha & \\cosh \\alpha \\end{array} \\right) \\left(\\begin{array}{c} h_0 \\\\ \\tilde{h}_0 \\end{array} \\right) \\, .$ The symplectic rotation is necessary to preserve the relative sign between the ordinary and Lee-Wick kinetic terms.",
"It follows from Eq.",
"(REF ) that $\\tanh 2 \\alpha = -\\frac{2 m_h^2/m_{\\tilde{h}}^2}{1-2 m_h^2/m_{\\tilde{h}}^2} \\,\\,\\,\\,\\, \\mbox{ or } \\,\\,\\,\\,\\, \\tanh \\alpha = - m_{h_0}^2 / m_{\\tilde{h}_0}^2 \\,\\, ,$ where $m_h^2 \\equiv \\lambda \\, v^2 /2$ is the mass of the lighter Higgs scalar in the absence of mixing.",
"The mass squared eigenvalues are defined by $m_{h_0}^2$ and $-m_{\\tilde{h}_0}^2$ , so that the squared mass parameters appearing in Eq.",
"(REF ) are all positive.",
"Note that $\\alpha $ is always negative.",
"The same steps that led to Eq.",
"(REF ) determine the form of the Yukawa couplings ${\\cal L}&=&\\frac{\\sqrt{2}}{v}\\, \\overline{u}_R \\, m_u^{diag} (H-\\tilde{H}) i \\sigma ^2 Q_L-\\frac{\\sqrt{2}}{v} \\, \\overline{d}_R \\, m_d^{diag} (H-\\tilde{H})^\\dagger V^\\dagger _{\\rm CKM} Q_L \\nonumber \\\\&-& \\frac{\\sqrt{2}}{v} \\, \\overline{e}_R \\, m_e^{diag} (H-\\tilde{H})^\\dagger \\ell _L + \\mbox{ h.c.},$ where we have suppressed generation indices.",
"Here $Q_L \\equiv ( u_L \\, , \\,V_{\\rm CKM} d_L)$ , $\\ell _L \\equiv (\\nu _L \\, , \\, e_L)$ , and all the fermion fields shown are in the mass eigenstate basis.",
"The couplings of the neutral scalar mass eigenstates to fermions can now easily be extracted using Eqs.",
"(REF ) and (REF ).",
"We define the quantity $g_{XY}$ to be the ratio of a neutral scalar coupling in the Lee-Wick theory that we have defined to the same coupling of the Higgs boson in the standard model.",
"Here $X$ designates the scalar state (either $h_0$ or $\\tilde{h}_0$ ) and $Y$ specifies the coupling of interest (for example, $t\\overline{t}$ , $b\\overline{b}$ , $\\tau ^+\\tau ^-$ , $W^+W^-$ or $ZZ$ ).",
"The neutral Higgs couplings to gauge boson pairs can be extracted from Eq.",
"(REF ) and the couplings to fermions from Eq.",
"(REF ).",
"For example, we find $g_{h_0 t \\overline{t}} = g_{h_0 b \\overline{b}}=g_{h_0 \\tau \\tau } = e^{-\\alpha } \\,\\,\\, ,$ $g_{h_0 WW} = g_{h_0 ZZ} = \\cosh \\alpha \\,\\,\\,\\, ,$ $g_{\\tilde{h}_0 t \\overline{t}} = g_{\\tilde{h}_0 b \\overline{b}}=g_{\\tilde{h}_0 \\tau \\tau } = - e^{-\\alpha } \\,\\,\\, ,$ $g_{\\tilde{h}_0 WW} = g_{\\tilde{h}_0 ZZ} = \\sinh \\alpha \\,\\,\\,\\, .$ Note that the couplings $g_{h_0 WW}$ and $g_{h_0 ZZ}$ are bigger than one, unlike most extensions of the standard model.",
"These results provide most of what we need to modify known theoretical results for Higgs boson properties in the standard model to obtain those appropriate to the scalar states in the present theory.",
"The one coupling that is more complicated to modify is the effective Higgs coupling to two photons; the relevant one-loop amplitude depends on a sum of terms that are modified by different $\\alpha $ -dependent factors.",
"To proceed, we write the relevant Lee-Wick Lagrangian terms as ${\\cal L} &=& -\\frac{g \\, m_f}{2 m_W} e^{-\\alpha } (h_0 - \\tilde{h}_0) \\overline{f} f+ (\\cosh \\alpha \\, h_0 + \\sinh \\alpha \\, \\tilde{h}_0) \\, g \\, m_W W^+ W^- \\nonumber \\\\&& - \\left(\\frac{1}{2} \\frac{m_h^2}{m_{\\tilde{h}}^2} e^{-\\alpha } \\right) \\frac{g \\, m_{\\tilde{h}}^2}{m_W} (h_0 - \\tilde{h}_0) \\,\\tilde{h}^- \\tilde{h}^+ \\,\\,\\, .$ Presented in this form, coefficients can be easily matched to those of the effective Lagrangian assumed in Ref.",
"[13] to compute contributions to $h_0 \\rightarrow \\gamma \\gamma $ from intermediate loop particles of various spins.",
"After identifying the appropriate coupling factors, the only other modification that needs to be made to these generic formulae is that an additional minus sign must be included in the amplitude term corresponding to the charged Higgs loop; this takes into account the overall sign difference between ordinary and Lee-Wick propagators.",
"Table: Measured Higgs Signal Strengths"
],
[
"Bounds",
" The quantities that we compute for purpose of comparison to the experimental data are the signal strengths $R^{\\rm LW}_i$ , each a specified Higgs boson production cross section times branching fraction normalized to the standard model expectation for the same quantity.",
"We consider production via gluon-gluon fusion (ggF), vector-boson fusion (VBF), associated production with a W or Z boson (Vh) and production via the top quark coupling (tth), as well as combinations of these possibilities.",
"In most cases, the ratio of Lee-Wick to standard model Higgs production cross sections reduces to a simple factor (for example, $e^{-2\\alpha }$ for ggF).",
"In the case of inclusive production at the LHC, we find that the ratio is well approximated by $\\frac{\\sigma ^{\\rm LW}}{\\sigma ^{\\rm SM} }= 0.88 \\, e^{-2 \\alpha } + 0.12 \\, \\cosh ^2\\alpha \\, ,$ for a center-of-mass energy of either 8 or 14 TeV.",
"The coefficients in this expression were determined using numerical predictions for the different contributions to the standard model Higgs production cross section, given in Ref. [12].",
"A total of 33 signal strengths measured at ATLAS, CMS and the Tevatron were collected for analysis; they correspond to different channels of Higgs production and decay, and include the final states $\\gamma \\gamma $ , $ZZ$ , $WW$ , $bb$ and $\\tau \\tau $ (Tables REF and REF ).",
"The analysis performed here is analogous to others found in the literature [14], [15], [16].",
"These references considered conventional two-Higgs doublet models, with results plotted as a function of $\\alpha $ and $\\tan \\beta $ .",
"We have seen, however, that the Lee-Wick Higgs sector is determined by a single parameter $\\alpha $ ; as indicated by Eq.",
"(REF ), this mixing angle is in one-to-one correspondence with the value of the heavy scalar mass $m_{\\tilde{h}_0}$ after one fixes $m_{h_0}$ at its experimental value.",
"Hence, we will present our results as 95% C.L.",
"lower bounds on the heavy Lee-Wick scalar mass.",
"This analysis is presented in two parts: We first determine bounds using the most recent data for the Higgs boson signal strengths shown in Tables REF and REF .",
"We then determine projected bounds at a 14 TeV LHC by assuming that the experimental data will converge on standard model central values and that the errors will scale in a simple way with the integrated luminosity.",
"Table: Measured Higgs Signal StrengthsTo find a lower bound on $m_{\\tilde{h}_0}$ from the current signal strengths, we construct the $\\chi ^2$ function $\\chi ^2=\\sum _{i=1}^{33}\\left(\\frac{R^{\\text{LW}}_i-R_i^{\\text{meas}}}{\\sigma ^{\\text{meas}}_i}\\right)^2,$ where $i$ runs over the 33 channels in Tables REF and REF .",
"$R^{\\text{LW}}_i$ stands for the predicted strength in the model presented here, $R^{\\text{meas}}_i$ is the measured strength and $\\sigma ^{\\text{meas}}_i$ is the corresponding error.",
"Asymmetric errors were averaged in quadrature, $\\sigma =\\sqrt{(\\sigma _+^2+\\sigma _-^2)/2}$ .",
"Note that only experimental errors were taken into account; in most cases, theoretical errors cancel in the ratio of a given observable with its standard model expectation.",
"In the cases where the cancellation is not exact, the theoretical uncertainty remains small.",
"For example, an ${\\cal O}(10\\%)$ theoretical uncertainty in the ggF production cross section does not entirely scale out the ratio of inclusive production cross sections; however, this translates into an ${\\cal O}(1\\%)$ theoretical uncertainty in the ratio, which is much smaller than the current experimental error bars.",
"We determine the 95% C.L.",
"lower bound on the heavy neutral Lee-Wick scalar mass using Eq.",
"(REF ) and the $\\chi ^2$ probability distribution corresponding to 32 degrees of freedom.",
"For the data in Tables REF and REF , which includes $\\sim 25$ fb$^{-1}$ of LHC data at $\\sim 8$ TeV, we find $m_{\\tilde{h}_0}>255\\mbox{ GeV} \\,\\,\\,\\,\\, \\mbox{ 95\\% C.L.",
"}$ This corresponds to a mixing parameter $\\alpha \\approx -0.25$ .",
"Figure: Projected lower bound on the heavy Lee-Wick scalar mass m h ˜ 0 m_{\\tilde{h}_0} as a functionof the LHC integrated luminosity.To estimate the future reach of the LHC, we follow the same procedure as in Ref. [15].",
"We assume that the experimental signal strengths will converge to their standard model values, namely $R_i^{\\rm meas} =1$ , and that the experimental errors bars will shrink relative to their current values by a factor of $1/\\sqrt{N}$ where $N=\\frac{\\sigma _{\\text{14}}}{\\sigma _{\\text{8}}}\\frac{L_{14}}{L_8}.$ Here, $\\sigma _{X}$ is the total Higgs production cross section at center-of-mass energy $X$ , and $L_X$ is the corresponding integrated luminosity.",
"This scaling of errors as the inverse square root of the number of events was also done in Ref.",
"[15] , and corresponds to “scheme 2” of the CMS [17] high luminosity projectionsThe assumption that the uncertainty scales as one over the square root of the number of events is true for the statistical error.",
"Here we assume that a comparable reduction in the systematic errors is possible with increasing luminosity.",
"This assumption may be optimistic, but is the one used by CMS for the European Strategy Report [17]..",
"Figure: Model predictions for the signal strengths R i LW R_i^{\\rm LW} as a function of m h ˜ 0 m_{\\tilde{h}_0}.The results of our projection are shown in Fig.",
"REF .",
"As one might expect, the lower bound on $m_{\\tilde{h}_0}$ increases monotonically with integrated luminosity; the left-most point on the curve corresponds to the current bound in Eq.",
"(REF ), while the rest follow from our procedure for determining projected bounds at a 14 TeV LHC.",
"For two benchmark points, we find $m_{\\tilde{h}_0}&>420\\ \\text{GeV} \\mbox{ 95\\% C.L.}",
"\\,\\,\\,\\,\\, (L_{14} = 300\\ \\text{fb}^{-1}) \\\\m_{\\tilde{h}_0}&>720\\ \\text{GeV} \\mbox{ 95\\% C.L.}",
"\\,\\,\\,\\,\\, (L_{14} = 3000\\ \\text{fb}^{-1})$ corresponding to the mixing angles $\\alpha \\approx -0.09$ and $-0.03$ , respectively.",
"We discuss the implications of these bounds in the final section.",
"As the experimental uncertainties on the Higgs boson signal strengths become smaller, new physics in this model should become manifest by an emerging pattern of deviations from the standard model expectations.",
"To illustrate this, we show in Fig.",
"REF some of the signal strengths expected in the Lee-Wick theory as a function of the heavy Lee-Wick scalar mass.",
"The $\\tau \\tau $ mode via ggF shows the greatest deviation from the standard model since both the production and decay width are each modified by the factor $\\exp (-2 \\alpha )$ which is larger than one.",
"The signal strength for $H \\rightarrow VV$ decays is also enhanced.",
"Although the deviation is not as great as the $\\tau \\tau $ channel shown, there are very few extensions of the standard model that would lead to such an enhancement.",
"Hence, this effect is a distinctive feature of the model that might be identified if the underlying physics is realized in nature."
],
[
"Conclusions",
" Now that the LHC has discovered a light, standard model-like Higgs boson and begun a study of its properties, one can examine the current and future constraints that can be placed on standard model extensions.",
"In this letter, we have considered such constraints on a Lee-Wick extension of the Higgs sector.",
"Although most of the partners in the LWSM must be heavy, due to various low-energy constraints, the partners of the Higgs boson need not be.",
"The resulting effective theory is a constrained two-Higgs doublet model, one in which some propagators and vertices have unusual signs.",
"In addition, the mixing between the light Higgs and the heavy neutral scalar is described by a symplectic rotation, leading to hyperbolic functions of a mixing angle at the vertices.",
"The mixing angle itself is related to the two neutral Higgs masses, and thus the heavy neutral scalar mass can be taken as the only free parameter.",
"The charged and pseudoscalar Higgs masses are degenerate at tree level and are also determined once the heavy scalar mass has been specified.",
"We first considered the bounds from current LHC data, looking at 33 different signals, and found a $95\\%$ confidence level lower bound of 255 GeV on the heavy scalar mass.",
"Extrapolating to the next runs at the LHC (at 14 TeV), we found that the bound will increase to 420 GeV (720 GeV) for an integrated luminosity of 300 (3000) inverse femtobarns.",
"The first signature of a deviation will come from light Higgs boson decays to either tau or massive gauge boson pairs.",
"Unlike most extensions of the standard model, both of these signal strengths are greater than in the standard model.",
"In Ref.",
"[8], it was shown that flavor constraints on the Lee-Wick charged Higgs provide a lower bound on the heavy neutral scalar mass.",
"The 95% C.L.",
"bounds on the charged Higgs from $B_d-\\bar{B_d}$ , $B_s-\\bar{B_s}$ mixing and $b\\rightarrow X_s\\gamma $ were found to be 303 GeV, 354 GeV and 463 GeV, respectively [8].",
"The most stringent of these bounds translates into a lower bound on the heavy neutral scalar of 445 GeV.",
"We thus see that the current bound from $b\\rightarrow X_s\\gamma $ is more stringent than those from current Higgs data, and that it will require approximately 400 femtobarns at a 14 TeV LHC in order to supersede this bound.",
"The work of CC and RR was supported by the NSF under Grant PHY-1068008.",
"The opinions and conclusions expressed herein are those of the authors, and do not represent the National Science Foundation."
]
] | 1403.0011 |
[
[
"Source-specific routing"
],
[
"Abstract Source-specific routing (not to be confused with source routing) is a routing technique where routing decisions depend on both the source and the destination address of a packet.",
"Source-specific routing solves some difficult problems related to multihoming, notably in edge networks, and is therefore a useful addition to the multihoming toolbox.",
"In this paper, we describe the semantics of source-specific packet forwarding, and describe the design and implementation of a source-specific extension to the Babel routing protocol as well as its implementation - to our knowledge, the first complete implementation of a source-specific dynamic routing protocol, including a disambiguation algorithm that makes our implementation work over widely available networking APIs.",
"We further discuss interoperability between ordinary next-hop and source-specific dynamic routing protocols.",
"Our implementation has seen a moderate amount of deployment, notably as a testbed for the IETF Homenet working group."
],
[
"Introduction",
"The routing paradigm deployed on the Internet is next-hop routing.",
"In next-hop routing, per-packet forwarding decisions are performed by examining a packet's destination address only, and mapping it to a next-hop router.",
"Next-hop routing is a simple, well understood paradigm that works satisfactorily in a large number of cases.",
"The use of next-hop routing restricts the flexibility of the routing system in two ways.",
"First, since a router only controls the next hop, a route $A\\cdot B\\cdot C\\cdots Z$ can only be selected by the router $A$ if its suffix $B\\cdot C\\cdots Z$ has already been selected by a neighbouring router $B$ , which makes some forms of optimisation difficult or impossible.",
"Other routing paradigms, such as circuit switching, label switching and source routing, do not have this limitation.",
"(Source routing, in particular, has been proposed multiple times as a suitable routing paradigm for the Internet [11], but has been discouraged due to claimed security reasons [1]).",
"Second, the only decision criterion used by a router is the destination address: two packets with the same destination are always routed in the same manner.",
"Yet, there are other data in the IP header that can reasonably be used for making a routing decision — the TOS octet, the IPv6 flow-id, and, of course, the source address.",
"We call source-specific routing the modest extension of classical next-hop routing where the forwarding decision is allowed to take into account the source of a packet in addition to its destination.",
"Source-specific routing gives a modest amount of control over routing to the sending host, which can choose among different routes by picking different source addresses.",
"The higher layers (transport or application) are therefore able to choose a route using standard networking APIs (collecting the host's local addresses and binding a socket to a specific address).",
"Unlike source routing, however, source-specific routing remains a hop-by-hop mechanism, and therefore leaves local forwarding decisions firmly in the control of the routers.",
"Two things are needed in order to make source-specific routing practical: a forwarding mechanism that can discriminate on both source and destination addresses, and a dynamic routing protocol that is able to distribute source-specific routes.",
"In this paper, we describe our experiences with the design and implementation of a source-specific extension to the Babel routing protocol [6], including a disambiguation algorithm that allows implementing source-specific routing over existing forwarding mechanisms."
],
[
"Applications",
"The main application of source-specific routing is the implementation of multihoming."
],
[
"Classical multihoming",
"A multihomed network is one that is connected to the Internet through two or more physical links.",
"This is usually done in order to improve a network's fault tolerance, but can also be done in order to improve throughput or reduce cost.",
"Classically, multihoming is performed by assigning Provider-Independent addresses to the multihomed network and announcing them globally (in the Default-Free Zone (DFZ)) over the routing protocol.",
"The dynamic nature of the routing protocol automatically provides for fault-tolerance; improvements in throughput and reductions in cost can be achieved by careful engineering of the routing protocol.",
"While classical multihoming works reasonably well in the network core, it does not apply to the edge.",
"In order to perform classical multihoming, a network needs to be allocated a “Provider-Independent” prefix that is reannounced by some or all of a network's peers.",
"This setup is usually impossible to achieve for home and small business networks.",
"Note that it is not in general possible to implement classical multihoming using a single “Provider-Dependent” prefix.",
"If a network is connected to two providers $A$ and $B$ , a packet with a source address in an address range allocated to $A$ will usually not be accepted by $B$ , which will treat it as a packet with a spoofed source address and discard it [8].",
"What is more, $A$ 's prefix will not be reannounced by $B$ , and hence destinations in $A$ 's prefix will not be reachable over the link to $B$ .",
"There is some concern that classical multihoming, even when restricted to the large networks of the core, is causing uncontrolled growth of the “default-free routing table”.",
"Since we have only experimented with source-specific routing in edge networks, we hold no opinion on the usefulness of our techniques in the network core, and in particular on the desirability of adding it to the BGP external routing protocol."
],
[
"Multihoming with multiple source addresses",
"Since announcing the same Provider-Dependent (PD) prefix to multiple ISPs is not always possible, it is a natural proposition to announce multiple PD prefixes, one per provider.",
"In this approach, every host is assigned multiple addresses, one per provider, and extra mechanisms are needed (i) to choose a suitable source and destination address for each packet, and (ii) to properly route each outgoing packet according to both its source and its destination.",
"In a sense, using multiple addresses splits the difficult problem of multihoming into two simpler problems that are handled at different layers of the network stack."
],
[
"Choosing addresses",
"The choice of source and destination addresses is typically left to the application layer.",
"All destination addresses are stored within the DNS (or explicitly carried by the application protocol), and the sending host tries them all, either in turn [7] or in parallel [12]; similarly, all possible source addresses are tried in turn.",
"Once a flow is established, it is no longer possible to change the source and destination addresses — from the user's point of view, all TCP connections are broken whenever a link outage forces a change of address.",
"Address selection can be implemented in the operating system's kernel and libraries, or by the application itself, which is notably done by most modern web browsers.",
"A different approach is to use a transport layer that has built-in support for multiple addresses and for dynamically renegotiating the set of source and destination addresses.",
"One such transport layer is MPTCP [10]; we describe our experience with MPTCP in Section REF ."
],
[
"Source-specific routing",
"As mentioned above, a provider will discard packets with a source address that is in a different provider's prefix.",
"In a network that is connected to multiple providers, each outgoing packet must therefore be routed through the link corresponding to its source address.",
"When the outgoing links are all connected to a single router, it is feasible to set up traffic engineering rules to ensure that this happens.",
"There can be good reasons, however, why it is desirable to connect each provider to a different router (Figure REF ): avoiding a single point of failure, load balancing, or simply that the various links use different link technologies that are not available in a single piece of hardware.",
"In a home networking environment, the edge routers might be provided by the different service providers, with no possibility to consolidate their functionality in a single device.",
"Figure: A network connected to two providersWith multiple edge routers, it is necessary that the routing protocol itself be able to route according to source addresses.",
"We say that a routing protocol performs source-specific routing when it is able to take both source and destination addresses into account in its routing decisions."
],
[
"Other applications",
"In addition to multihoming with multiple addresses, we are aware of two problematic networking problems that source-specific routing solves cleanly and elegantly."
],
[
"Overlay networks",
"Tunnels and VPNs are commonly used to establish a network-layer topology that is different from the physical topology, notably for security reasons.",
"In many tunnel or VPN deployments, the end network uses its native default route, and only routes some set of prefixes through the tunnel or VPN.",
"In some deployments, however, the default route points at the tunnel.",
"If this is done naively, the network stack attempts to route the encapsulated packets through the tunnel itself, which causes the tunnel to break.",
"Many workarounds are possible, the simplest being to point a host route towards the tunnel endpoint through the native interface.",
"Source-specific routing provides a clean solution to that problem.",
"The native default route is kept unchanged, while a source-specific default route is installed through the tunnel.",
"The source-specific route being more specific than the native default route, packets from the user network are routed through the tunnel, while the encapsulated packets sourced at the edge router follow the native, non-specific route."
],
[
"Controlled anycast",
"Anycast is a technique by which a single destination address is used to represent multiple network endpoints.",
"A packet destined to an anycast address is routed to whichever endpoint is nearest to the source according to the routing protocol's metric.",
"Anycast is useful for load balancing — for example, the DNS root servers are each multiple physical servers, represented by a single anycast address.",
"For most applications of anycast, all of the endpoints are equivalent and it does not matter which endpoint is accessed by a given client.",
"Some applications, however, require that a given user population access a well-defined endpoint — for example, in a Content Distribution Network (CDN), a provider might not want to serve nodes that are not its customers.",
"Ensuring that this is the case by tweaking the routing protocol's metric (or “prepending” in BGP parlance) is fragile and error-prone.",
"Source-specific routing provides an elegant solution to this problem.",
"With source-specific routing, each instance of the distributed server is announced using a source-specific route, and will therefore only receive packets from a given network prefix."
],
[
"Related work",
"Multihoming is a difficult problem, and, unsurprisingly, there are many techniques available to implement it, none of which are fully general.",
"In addition to classical network-layer multihoming, already mentioned above, there are a number of lower-layer techniques, the use of which is usually completely transparent to the network layer; we are aware of Multi-Link PPP, of Ethernet link aggregation (port trunking), of the use of MPLS to provide multiple paths across a rich link layer, as well as of proprietary techniques used by vendors of cable modems.",
"Since these techniques work at the link layer, they are usually restricted to multihoming with a single provider.",
"All of these techniques are compatible, in the sense that they can be used at the same time.",
"We imagine a home network where source-specific routing is used to access two providers, each of which is classically multihomed, over links that consist of multiple physical links combined at the link layer.",
"Source-specific packet forwarding itself is not a new idea [3], and implementing it manually on a single router using traffic engineering interfaces is a well-documented technique [9].",
"Implementing source-specific routing within the routing protocol has been proposed by Bagnulo et al.",
"[2], but the techniques used differ significantly from ours.",
"First, the authors only deal with the non-overlapping case — where the different possible sources are disjoint —, which avoids the need for the disambiguation algorithm which is one of our main concerns.",
"Second, they use a more general facility of an existing routing protocol (BGP Communities) rather than explicitly implementing source-specific routing.",
"We find our more direct approach to be more intuitive, and expect it to be more reliable, since it doesn't require out-of-band agreement on the meaning of the labels carried by the routing protocol.",
"More generally, there are other applications of routing based on more information from the packet header than just the destination address.",
"The traffic-engineering community has been experimenting with routing based on the TOS octet of the IPv4 header for many years, and the ability to do that is part of the OSPFv2 protocol.",
"TOS-based routing is somewhat analoguous to source-specific routing, and many of the issues raised are similar; both can be seen as particular cases of “multi-dimensional routing”.",
"Equal Cost Multipath (ECMP) is somewhat different.",
"A router performing ECMP has multiple routes to the same destination, and chooses among them according to the value of a hash of the packet header.",
"While ECMP does route on multiple header fields, the choice of fields used to choose a route in ECMP is a purely local matter, and does not need to be carried by the routing protocol."
],
[
"Next-hop routing tables",
"Ordinary next-hop routing consists in mapping a destination address to a next-hop.",
"Obviously, it is not practical to maintain a mapping for each possible destination address, so the mapping table must be compressed in some manner.",
"The standard compressed data structure is the routing table (or Forwarding Information Base, FIB), which ranges over prefixes, ranges of addresses the size of which is a power of two.",
"The routing table can be constructed manually, but is usually populated by a routing protocol.",
"Since prefixes can overlap, the routing table is an ambiguous data structure: a packet's destination address can match multiple routing entries.",
"This ambiguity is resolved by the so-called longest-prefix rule: when multiple routing table entries match a given destination address, the most specific matching entry is the one that is used.",
"More precisely, a prefix is a pair $P=p/\\mathit {plen}$ , where $p$ is the first address in the prefix and $\\mathit {plen}$ is the prefix length.",
"An address $a$ is in $P$ when the first $\\mathit {plen}$ bits of $a$ match the first $\\mathit {plen}$ bits of $p$ .",
"We say that a prefix $P=p/\\mathit {plen}$ is more specific than a prefix $P^{\\prime }=p^{\\prime }/\\mathit {plen}^{\\prime }$ , written $P\\le P^{\\prime }$ , when the set of addresses in $P$ is included in the set of addresses in $P^{\\prime }$ .",
"Clearly, $P\\le P^{\\prime }$ if and only if $\\mathit {plen}\\ge \\mathit {plen}^{\\prime }$ , and the first $\\mathit {plen^{\\prime }}$ bits of $p$ and $p^{\\prime }$ match.",
"The specificity ordering defined above has an important property: given two prefixes $P$ and $P^{\\prime }$ , they are either disjoint ($P \\cap P^{\\prime } = \\emptyset $ ), or one is more specific than the other ($P\\le P^{\\prime }$ or $P^{\\prime }\\le P$ ).",
"A routing table is a set of pairs $(P, \\mathit {nh})$ , where $P$ is a prefix and $\\mathit {nh}$ , the next hop, is a pair of an interface and a (link-local) address; we further require that all the prefixes in a routing table be distinct.",
"Because of the particular structure of prefixes, given an address $a$ , either the set of prefixes in the routing table containing $a$ is empty, or it is a chain (a totally ordered set); hence, there exists a most specific prefix $P$ in the routing table containing $a$ .",
"The longest-prefix rule specifies that the next hop chosen for routing a packet with destination $a$ is the one corresponding to this most specific prefix, if any."
],
[
"Source-specific routing tables",
"Source-specific routing is an extension to next-hop routing where both the destination and the source of a packet can be used to perform a routing decision.",
"Source-specific routers use a source-specific routing table, which is a set of triples $(d, s, \\mathit {nh})$ , where $d$ is a destination prefix, $s$ a source prefix, and $\\mathit {nh}$ is a next hop (note the ordering — destination comes first).",
"Such an entry matches a packet with destination address $a_d$ and source address $a_s$ if $a_d$ is in $d$ and $a_s$ is in $s$ .",
"The specificity ordering generalises easily to pairs: a pair of prefixes $(d,s)$ is more specific than a pair $(d^{\\prime },s^{\\prime })$ when all pairs of addresses $(a_d,a_s)$ which are in $(d,s)$ are also in $(d^{\\prime },s^{\\prime })$ ; clearly, $(d,s)\\le (d^{\\prime },s^{\\prime })$ when $d\\le d^{\\prime }$ and $s\\le s^{\\prime }$ .",
"Unfortunately, the set of destination-source pairs of prefixes equipped with the specificity ordering does not have the same structure as the set of single prefixes: given a pair of addresses $(a_d,a_s)$ , the set of pairs of prefixes containing $(a_d,a_s)$ might not be a chain.",
"Consider the pairs $(2001{:}\\mathrm {db8}{:}1{::}/48,{::}/0)$ and $({::}/0,2001{:}\\mathrm {db8}{:}2{::}/48)$ .",
"Clearly, these two pairs are not disjoint (the pair of addresses $(2001{:}\\mathrm {db8}{:}1{::}1,2001{:}\\mathrm {db8}{:}2{::}1)$ is matched by both), but neither is one more specific than the other — the pair $(2001{:}\\mathrm {db8}{:}1{::}1,2001{:}\\mathrm {db8}{:}3{::}1)$ is matched by the first but not the second, and, symmetrically, the pair $(2001{:}\\mathrm {db8}{:}4{::}1,2001{:}\\mathrm {db8}{:}2{::}1)$ is matched by just the second.",
"From a practical point of view, this means that a source-specific routing table can contain multiple most-specific entries, and thus fail to unambiguously specify a forwarding behaviour.",
"We say that a source-specific routing table is ambiguous when it contains multiple non-disjoint most-specific entries.",
"Two entries $r_1$ and $r_2$ that are neither disjoint nor ordered are said to be conflicting, written $r_1\\mathop {\\#}r_2$ .",
"If $r_1=(d_1,s_1)$ and $r_2=(d_2,s_2)$ , then this is equivalent to saying that either $d_1<d_2$ and $s_1>s_2$ or $d_1>d_2$ and $s_1<s_2$ .",
"We call the conflict zone of $r_1$ and $r_2$ the set of $(a_d,a_s)$ that are matched by both $r_1$ and $r_2$ ."
],
[
"Forwarding behaviour",
"In the presence of an ambiguous routing table, there exist packets that are matched by distinct most-specific entries.",
"An arbitrary choice must be made in order to decide how to route such a packet.",
"Let us first remark that all routers in a single routing domain must make a consistent choice — having different routers follow different policies within conflict zones may lead to persistent routing loops.",
"Consider the topology in Figure REF , with two source-specific routes indexed by the pairs $(d_1, s_1)$ and $(d_2, s_2)$ respectively, where packets matching $(d_1,s_1)$ are sent towards the left of the diagram, and packets matching $(d_2,s_2)$ are sent towards the right.",
"If the two pairs are in conflict, and router $A$ chooses $(d_2,s_2)$ while $B$ chooses $(d_1,s_1)$ , then a packet matching both pairs will loop between $A$ and $B$ indefinitely.",
"Figure: A routing loop due to incoherent orderingsIt is therefore necessary to choose a disambiguation rule that is uniform across the routing domain.",
"There are two natural choices: discriminating on the destination first, and comparing sources if destinations are equal, or discriminating on source first.",
"More precisely, the destination-first ordering is defined by: $ (d,s)\\preceq (d^{\\prime },s^{\\prime })\\mbox{ if }d < d^{\\prime }\\mbox{ or }d=d^{\\prime }\\mbox{ and }s\\le s^{\\prime },$ while the source-first ordering is defined by $ (d,s)\\preceq _s (d^{\\prime },s^{\\prime })\\mbox{ if }s<s^{\\prime }\\mbox{ or }s=s^{\\prime }\\mbox{ and }d\\le d^{\\prime }.$ These orderings are isomorphic — hence, there is no theoretical argument that allows us to choose between them.",
"An engineering choice must be made, based on usefulness alone.",
"The current consensus, both within the IETF Homenet group and outside it, appears to be that the destination-first ordering is the more useful of the two.",
"Consider the (fairly realistic) topology in Figure REF , where an edge router $A$ announces a source-specific route towards the Internet, and a stub network $N$ announces a (non-specific) route to itself.",
"A packet matching both routes must follow the route towards $N$ , since it is obviously the only route that can reach the destination, which implies that $A$ must use the destination-first ordering.",
"On the other hand, we know of no such compelling examples of the usefulness of the source-first ordering.",
"Figure: A stub network behind a source-specific routerIn the following sections, we describe our experience with source-specific routing using the destination-first ordering.",
"However, nothing in this article depends on the particular ordering being used, and our techniques would apply just as well to any structure that is a refinement of the specificity ordering and that is totally ordered on route entries containing a given address."
],
[
"Implementing source-specific routing",
"In the previous sections, we have described source-specific routing and shown how all routers in a routing domain must make the same choices with respect to ambiguous routing tables, and have argued in favour of the destination-first semantics.",
"Whichever particular choice is made by an implementation of a routing protocol, however, must be implementable in terms of the primitives made available by the lower layers (the operating system kernel and the hardware).",
"In this section, we describe the two techniques that we have used to implement a source-specific extension to the Babel routing protocol [4].",
"We first describe the technique that we use when running over a lower layer that natively implements destination-first source-specific routing (Section REF ).",
"We then describe our so-called “disambiguation” algorithm (Section REF ) which we use to implement destination-first source-specific routing over any source-specific facility provided by the lower layers, as long as it is compatible with the specificity ordering — a very mild hypothesis that is satisfied by a number of widely available implementations."
],
[
"Native source-specific FIB",
"Ideally, we would like the lower layers of the system (the OS kernel, the line cards, etc.)",
"to implement destination-first source-specific routing tables out of the box.",
"Such native support for source-specific routing is preferable to the algorithm described below, since no additional routes will need to be installed.",
"In practice, while many systems have a facility for source-specific traffic engineering, this lower-layer support often has a behaviour different from the one that we require.",
"The Linux kernel, when compiled with the relevant options (“ipv6-subtrees”), supports source-specific FIBs natively, albeit for IPv6 only.",
"Unfortunately, this support is only functional since Linux 3.11 (source-specific routes were treated as unreachable in earlier versions), and only for IPv6 (for IPv4, the “source” datum is silently ignored).",
"We know of no other TCP/IP stacks with native support for destination-first source-specific routing — other techniques must be used on most systems."
],
[
"Disambiguation of a routing table",
"All versions of Linux, some versions of FreeBSD, and a number of other networking stacks implement a facility to manipulate multiple routing tables and to select a particular one depending on the source address of a packet.",
"Since the table is selected before the destination address is examined, these API implement the source-first behaviour, which is not what we aim to implement.",
"In this section, we describe a disambiguation algorithm that can be used to maintain a routing table that is free of ambiguities, and will therefore yield the same behaviour as long as the underlying forwarding mechanism implements a behaviour that is compatible with the specificity ordering (Section REF ).",
"All the forwarding mechanisms known to us satisfy this very mild hypothesis.",
"Recall that a routing table is ambiguous if there exists a packet that is matched by at least one entry in the table and such that there is no most-specific entry among the matching entries.",
"A necessary and sufficient property for a routing table to be non-ambiguous is that every conflict zone is equal to the union of more specific route entries.",
"The algorithm that we propose maintains, for each conflict, exactly one route entry that covers exactly the conflict zone.",
"While a more parsimonious solution would be possible in some cases, it would greatly complicate the algorithm."
],
[
"Weak completeness",
"We say that a routing table is weakly complete if each conflict zone is covered by more specific entries.",
"More formally, $T$ is weakly complete if $\\forall r_1, r_2 \\in T, r_1 \\cap r_2 = \\bigcup \\lbrace r \\in T \\mid r \\le r_1 \\cap r_2\\rbrace $ .",
"Theorem 1 A routing table is non-ambiguous if and only if it is weakly complete.",
"Let $U_x^y = \\bigcup \\lbrace r \\in T \\,|\\,r \\le x \\cap y\\rbrace $ .",
"We need to show that $T$ is non-ambiguous iff $\\forall r_1, r_2 \\in T, r_1 \\cap r_2 = U_{r_1}^{r_2}$ .",
"($\\Leftarrow $ ) Suppose $T$ is weakly complete, and consider two route entries $x, y \\in T$ in conflict.",
"By weak completeness, $U_x^y = x \\cap y$ , so for all addresses $a \\in x \\cap y$ , there exists a route $r \\in U_x^y$ such that $a \\in U_x^y$ .",
"Since $r \\in x \\cap y$ , we have $r < x$ and $r < y$ , and $r$ is more specific than $x \\cap y$ .",
"Since this is true for all conflicts, the table is not ambiguous.",
"($\\Rightarrow $ ) Suppose $T$ is non-ambiguous and not weakly complete.",
"Then there exist two entries $x, y \\in T$ in conflict such that $x \\cap y \\ne U_x^y$ .",
"Consider an address $a \\in x \\cap y \\setminus U_x^y$ , and an entry $r \\in T$ matching $a$ .",
"Clearly, $r \\supsetneq x \\cap y$ , and so either $r \\mathop {\\#}x$ or $r \\mathop {\\#}y$ , or $r > x$ and $r > y$ .",
"In all cases, $r$ is not more specific than both $x$ and $y$ , so there is no minimum for the set of entries matching $a$ .",
"This contradicts the hypothesis, so if $T$ is not ambiguous, it is weakly complete.",
"Disambiguation with weak completeness is not convenient, since it may require adding multiple route entries to solve a single conflict, and the disambiguation routes added may generate additional conflicts.",
"Suppose for example that the FIB first contains two entries $r_1 > r_2$ , and we add $r_3 > r_2$ which conflicts with $r_1$ (see figure below).",
"Since $r_2 < r_3$ , there is no conflict within $r_2$ , but we need disambiguation routes $d_1$ and $d_2$ .",
"The FIB is now weakly complete.",
"Suppose now that we add $r_4 < r_3$ in conflict both with $r_1$ and the disambiguation route $d_2$ .",
"We install a new disambiguation entry $d_3$ .",
"Note also that since $r_4 < r_3$ , we need to use the next-hop of $r_4$ for the former region covered by $d_1$ : we need to change the currently installed disambiguation route entry.",
"[scale=0.65] (0,0) – (0,4) – (2,4) – (2,0) – cycle; (0, 0) node[above right] $r_1$ ; (0,2) – (0,3) – (1,3) – (1,2) – cycle; (1, 2) node[above left] $r_2$ ; (2.5, 2) node $\\rightarrow $ ; [xshift=3cm] (0,0) – (0,4) – (2,4) – (2,0) – cycle; (0, 0) node[above right] $r_1$ ; (0,2) – (0,3) – (1,3) – (1,2) – cycle; (1, 2) node[above left] $r_2$ ; [very thick] (0,2) – (0,4) – (4,4) – (4,2) – cycle; (4,2) node[above left] $r_3$ ; [dashed] (0 +0.1, 3 +0.1) – (0 +0.1, 4 -0.1) – (1 -0.1, 4 -0.1) – (1 -0.1, 3 +0.1) – cycle; (0.5, 3.5) node $d_1$ ; [dashed] (1 +0.1, 2 +0.1) – (1 +0.1, 4 -0.1) – (2 -0.1, 4 -0.1) – (2 -0.1, 2 +0.1) – cycle; (1.5, 3) node $d_2$ ; (7.5, 2) node $\\rightarrow $ ; [xshift=8cm] (0,0) – (0,4) – (2,4) – (2,0) – cycle; (0, 0) node[above right] $r_1$ ; (0,2) – (0,3) – (1,3) – (1,2) – cycle; (1, 2) node[above left] $r_2$ ; (0,2) – (0,4) – (4,4) – (4,2) – cycle; (4,2) node[above left] $r_3$ ; (0 +0.01, 3 +0.01) – (0 +0.01, 4 -0.01) – (1 -0.01, 4 -0.01) – (1 -0.01, 3 +0.01) – cycle; (1 +0.01, 2 +0.01) – (1 +0.01, 4 -0.01) – (2 -0.01, 4 -0.01) – (2 -0.01, 2 +0.01) – cycle; [very thick] (0,3) – (0,4) – (3,4) – (3,3) – cycle; (3,3) node[above left] $r_4$ ; [dashed] (0 +0.1, 3 +0.1) – (0 +0.1, 4 -0.1) – (1 -0.1, 4 -0.1) – (1 -0.1, 3 +0.1) – cycle; (0.5, 3.5) node $d_1$ ; [dashed] (1 +0.1, 3 +0.1) – (1 +0.1, 4 -0.1) – (2 -0.1, 4 -0.1) – (2 -0.1, 3 +0.1) – cycle; (1.5, 3.5) node $d_3$ ; Some of this complexity can be avoided by requiring a stronger notion of completeness."
],
[
"Completeness",
"A routing table is (strongly) complete if each conflict zone is covered by one route entry.",
"More formally, $T$ is complete if $\\forall r_1, r_2 \\in T, r_1 \\cap r_2 \\in T$ .",
"This obviously implies weak-completeness, and therefore a complete routing table is not ambiguous.",
"Our algorithm maintains the completeness of the routing table.",
"Theorem 2 Adding routes to achieve completeness does not lead to another conflict.",
"Suppose that $r_1 = (d_1, s_1)$ and $r_2 = (d_2, s_2)$ are two route entries in conflict, where $d_1 < d_2$ and $s_1 > s_2$ .",
"Consider the disambiguation entry $r_{sol} = (d_1, s_2)$ which disambiguates this conflict.",
"Suppose now that $r_{sol}$ is in conflict with another route entry $r_3 = (d_3, s_3)$ .",
"We have either $d_1 < d_3$ and $s_1 > s_2 > s_3$ , in which case $r_3 \\mathop {\\#}r_1$ ; or $d_2 > d_1 > d_3$ and $s_2 < s_3$ , in which case $r_3 \\mathop {\\#}r_2$ .",
"In either case, the conflict existed beforehand, and must therefore already have been resolved.",
"Take the previous example again.",
"When adding $r_3$ , we add one route entry to cover the area $d_1$ ($r_1 \\cap r_3$ ).",
"Since $r_2$ is more specific, the new route entry does not affect the routing decision for addresses in $r_2$ .",
"When adding $r_4$ , it is in conflict with both $r_1$ and the disambiguation route $d_1$ , but for the same conflict zone $r_4 \\cap r_1$ .",
"The disambiguation route inserted is thus not an additional conflict.",
"[scale=0.65] (0,0) – (0,4) – (2,4) – (2,0) – cycle; (0, 0) node[above right] $r_1$ ; (0,2) – (0,3) – (1,3) – (1,2) – cycle; (1, 2) node[above left] $r_2$ ; (2.5, 2) node $\\rightarrow $ ; [xshift=3cm] (0,0) – (0,4) – (2,4) – (2,0) – cycle; (0, 0) node[above right] $r_1$ ; (0,2) – (0,3) – (1,3) – (1,2) – cycle; (1, 2) node[above left] $r_2$ ; [very thick] (0,2) – (0,4) – (4,4) – (4,2) – cycle; (4,2) node[above left] $r_3$ ; [dashed] (0 +0.1, 2 +0.1) – (0 +0.1, 4 -0.1) – (2 -0.1, 4 -0.1) – (2 -0.1, 2 +0.1) – cycle; (1.5, 3.5) node $d_1$ ; (7.5, 2) node $\\rightarrow $ ; [xshift=8cm] (0,0) – (0,4) – (2,4) – (2,0) – cycle; (0, 0) node[above right] $r_1$ ; (0,2) – (0,3) – (1,3) – (1,2) – cycle; (1, 2) node[above left] $r_2$ ; (0,2) – (0,4) – (4,4) – (4,2) – cycle; (4,2) node[above left] $r_3$ ; (0 +0.01, 2 +0.01) – (0 +0.01, 4 -0.01) – (2 -0.01, 4 -0.01) – (2 -0.01, 2 +0.01) – cycle; (1.5, 2.5) node $d_1$ ; [very thick] (0,3) – (0,4) – (3,4) – (3,3) – cycle; (3,3) node[above left] $r_4$ ; [dashed] (0 +0.1, 3 +0.1) – (0 +0.1, 4 -0.1) – (2 -0.1, 4 -0.1) – (2 -0.1, 3 +0.1) – cycle; (1, 3.5) node $d_2$ ;"
],
[
"Preliminaries",
"We write $\\min (r_1, r_2)$ for the minimum according to $\\preceq $ .",
"We define two auxiliary functions.",
"The function $\\mathrm {min\\_conflict}(\\mathit {zone}, r)$ (Algorithm REF ) returns, if it exists, the minimum route entry in conflict with $r$ for the conflict zone $\\mathit {zone}$ .",
"The function $\\mathrm {conflict\\_solution}(\\mathit {zone})$ (Algorithm REF ) returns, if it exists, the minimum route entry participating in a conflict for the zone $\\mathit {zone}$ .",
"search for mininum conflicting route (min_conflict($\\mathit {zone}, r$ )) $\\mathit {min} \\leftarrow \\bot $ $r_1 \\in T$ s.t.",
"$r \\mathop {\\#}r_1$ and $r \\cap r_1 = \\mathit {zone}$ $\\mathit {min} \\leftarrow \\min (r_1, \\mathit {min})$ return $\\mathit {min}$ Search for conflict solution (conflict_solution($\\mathit {zone}$ )) $\\mathit {min} \\leftarrow \\bot $ $r_1, r_2 \\in T$ s.t.",
"$r_1 \\mathop {\\#}r_2$ and $r_1 \\cap r_2 = \\mathit {zone}$ and $r_1 \\prec r_2$ $\\mathit {min} \\leftarrow \\min (r_1, \\mathit {min})$ return $\\mathit {min}$ We write $\\mathrm {nh}(r)$ for the next hop of a route $r$ .",
"We use three primitives for manipulating the routing table.",
"Let $r=(d, s, \\mathit {nh})$ be a route entry, and $\\mathit {nh}^{\\prime }$ a nexthop.",
"Then $\\mathrm {install}(r, \\mathit {nh}^{\\prime })$ adds the route entry $(d, s, \\mathit {nh}^{\\prime })$ , $\\mathrm {uninstall}(r, \\mathit {nh}^{\\prime })$ removes the route entry $(d, s, \\mathit {nh}^{\\prime })$ , and $\\mathrm {switch}(r, \\mathit {nh}^{\\prime }, \\mathit {nh}^{\\prime \\prime })$ changes the FIB's route entry $(d, s, \\mathit {nh}^{\\prime })$ to $(d, s, \\mathit {nh}^{\\prime \\prime })$ .",
"Calling $\\mathrm {switch}(r,\\mathit {nh}^{\\prime }, \\mathit {nh}^{\\prime \\prime })$ is equivalent to calling $\\mathrm {uninstall}(r, \\mathit {nh}^{\\prime })$ followed by $\\mathrm {install}(r, \\mathit {nh}^{\\prime \\prime })$ ."
],
[
"Relevant conflicts",
"Consider a route entry $r$ , and a set $E$ of routing entries in conflict with $r$ for the same conflict zone; all of these conflicts will have the same resolution.",
"Moreover, if the resolution was caused by a route in $E$ , then that was necessarily the more specific of the entries in $E$ .",
"Note that the minimum exists because elements of $E$ have either the same destination, or the same source, and match at least one address in $r$ .",
"Given a route entry $r$ , we define the equivalence $\\mathcal {\\sim }_r$ by $r_1 \\mathcal {\\sim }_r r_2\\Leftrightarrow r_1 \\cap r = r_2 \\cap r$ , i.e.",
"two route entries are equivalent for $\\mathcal {\\sim }_r$ if they have the same intersection with $r$ .",
"If two equivalent route entries are in conflict with $r$ , this means that they have the same conflict zone.",
"Quotienting a set of routing entries in conflict with $r$ by this equivalence, and taking the minimum of each of the class of equivalence gives us exactly the routes that we care about."
],
[
"Adding a route entry (Algorithm ",
"Installing a new route entry in the FIB may make it ambiguous.",
"For this reason, we must install the most specific routing entries first.",
"In particular, we must install disambiguation entries (lines 2 to 9) before the route itself (lines 10 to 14).",
"Let $r$ be the route to install, and $C$ the set of route entries in conflict with $r$ , for which there is no natural solution, i.e.",
"$C = \\lbrace r^{\\prime }\\in T \\mid r^{\\prime } \\mathop {\\#}r\\ \\mathrm {and}\\ r^{\\prime } \\cap r \\notin T\\rbrace $ (line 3).",
"We only consider the relevant conflicts upon this set (line 4): $C^{\\prime } =\\lbrace \\min (E) \\mid E \\in C/\\!\\raisebox {-.65ex}{\\mathcal {\\sim }_{r}}\\rbrace $ .",
"For each route entry $r_1 \\in C^{\\prime }$ (considering the most specific first), we first search (line 5), if it exists, the minimum route entry $r_2$ such that $r_2 \\mathop {\\#}r_1$ and $r_2 \\cap r_1 = r \\cap r_1$ .",
"If $r_2$ does not exist, then there was no conflict for this zone before, and we must add $((r_1 \\cap r), \\mathit {nh})$ to the FIB (line 7).",
"Otherwise, a routing entry has been installed for this conflict, and we must decide if the new route entry $r$ is or not the new candidate, which is true if it is more desirable $(\\preceq )$ than both $r_2$ and $r_1$ (line 8).",
"If it is the case, then the previous next-hop installed was the one of $r_2$ : we replace $((r_1 \\cap r), \\mathit {nh}_2)$ by $((r_1 \\cap r), \\mathit {nh})$ (line 9).",
"Finally, we must search if there exists two route entries in conflict for the zone of $r$ (line 10).",
"In that case, a disambiguation route entry has been installed, so $r$ must replace it (line 12).",
"Otherwise, $r$ can be added normally (line 14).",
"We end the procedure by adding $r$ to our local RIB (line 15).",
"Route addition (add_route($r$ )) $r_1 \\in T$ s.t.",
"$r \\mathop {\\#}r_1$ and $r \\cap r_1 \\notin T$ and $r_1 = \\mathrm {min\\_conflict}(r \\cap r_1, r)$ $r_2 \\leftarrow \\mathrm {min\\_conflict}(r \\cap r_1, r_1)$ $r_2 = \\bot $ $\\mathrm {install}(r \\cap r_1, \\mathrm {nh}(\\min (r, r_1)))$ $r \\prec r_2$ and $r \\prec r_1$ $\\mathrm {switch}(r \\cap r_1, \\mathrm {nh}(r_2), \\mathrm {nh}(r))$ $r_1 \\leftarrow \\mathrm {conflict\\_solution}(r)$ $r_1 = \\bot $ $\\mathrm {install}(r, \\mathrm {nh}(r))$ $\\mathrm {switch}(r, \\mathrm {nh}(r_1), \\mathrm {nh}(r))$ $T \\leftarrow T \\cup \\lbrace r\\rbrace $"
],
[
"Removing a route entry (Algorithm ",
"This time, we must first remove the less specific route first to keep the routing table unambiguous.",
"Again, we write $r$ for the route to be removed.",
"First, remove $r$ from the RIB (line 2).",
"As for the addition, $r$ may be solving a conflict, in which case we cannot just remove it, but must first search for the entry covering that conflict (line 3), and if it exists replace $r$ 's next-hop (line 7).",
"Otherwise, we just remove $r$ from the FIB (line 5).",
"We consider $C^{\\prime }$ as previously defined (lines 9 and 10).",
"For each route entry $r_1 \\in C^{\\prime }$ (considering the less specific first), we first search, as we did for the adding process, for the minimum route entry $r_2$ such that $r_2 \\mathop {\\#}r_1$ and $r_2 \\cap r_1 = r \\cap r_1$ (line 11).",
"If $r_2$ does not exist, we remove $((r_1 \\cap r), \\mathit {nh})$ from the FIB (line 13).",
"Otherwise, for the same reasons above, if $r$ is more desirable than both $r_1$ and $r_2$ , then we replace in the FIB the next-hop of $r$ assigned for $r \\cap r_1$ by the one of $r_2$ (line 15).",
"Route deletion (delete_route($r$ )) $T \\leftarrow T \\setminus \\lbrace r\\rbrace $ $r_1 \\leftarrow \\mathrm {conflict\\_solution}(r)$ $r_1 = \\bot $ $\\mathrm {uninstall}(r, \\mathrm {nh}(r))$ $\\mathrm {switch}(r, \\mathrm {nh}(r), \\mathrm {nh}(r_1))$ $r_1 \\in T$ s.t.",
"$r \\mathop {\\#}r_1$ and $r \\cap r_1 \\notin T$ and $r_1 = \\mathrm {min\\_conflict}(r \\cap r_1, r)$ $r_2 \\leftarrow \\mathrm {min\\_conflict}(r \\cap r_1, r_1)$ $r_2 = \\bot $ $\\mathrm {uninstall}(r \\cap r_1, \\mathrm {nh}(\\min (r, r_1)))$ $r \\prec r_2$ and $r \\prec r_1$ $\\mathrm {switch}(r \\cap r_1, \\mathrm {nh}(r), \\mathrm {nh}(r_2))$"
],
[
"Changing a route entry (Algorithm ",
"This is the simplest case, since disambiguation routes must be maintained, and changed only if the route that we want to change has been selected for disambiguation.",
"The order in which we change the route entries does not matter.",
"Let $r$ the route entry to change by $r_{\\mathit {new}}$ .",
"Here, we choose to first replace $r$ by $r_{\\mathit {new}}$ (line 2).",
"We consider $C^{\\prime }$ as previously defined (lines 3 and 4).",
"For each route entry $r_1 \\in C^{\\prime }$ , we search for the minimum route entry $r_2$ such that $r_2 \\mathop {\\#}r_1$ and $r_2 \\cap r_1 = r \\cap r_1$ .",
"If both $r \\prec r_1$ and $r_2$ is $r$ (line 6), then we replace the next-hop $\\mathit {nh}$ of the corresponding disambiguation route entry by the new one $\\mathit {nh}_{\\mathit {new}}$ (line 7).",
"Route modification (change_route($r, r_{\\mathit {new}}$ )) $\\mathrm {switch}(r, \\mathrm {nh}(r), \\mathrm {nh}(r_{\\mathit {new}}))$ $r_1 \\in T$ s.t.",
"$r \\mathop {\\#}r_1$ and $r \\cap r_1 \\notin T$ and $r_1 = \\mathrm {min\\_conflict}(r \\cap r_1, r)$ and $r \\prec r_1$ and $r = \\mathrm {min\\_conflict}(r \\cap r_1, r_1)$ $\\mathrm {switch}(r \\cap r_1, \\mathrm {nh}(r), \\mathrm {nh}(r_{\\mathit {new}}))$"
],
[
"External changes to the routing table",
"In the description above, we have asssumed that only our algorithm ever needs to manipulate the routing table.",
"In practice, however, the routing table is also manipulated by other agents — other routing protocols or human operators.",
"In principle, the same algorithm should be applied to externally changed routes; however, this is not implemented yet."
],
[
"Source-specific Bellman-Ford",
"The distributed Bellman-Ford algorithm is the foundation of a number of more or less widely deployed routing protocols, such as the venerable RIP, EIGRP, Babel and, arguably, BGP.",
"In order to experiment with source-specific routing, we have implemented a source-specific variant of the Babel routing protocol [6]; the exact details of the packet format of our extension are described in [4].",
"Our implementation has seen a moderate amount of deployment, most notably as a testbed for the IETF Homenet working group [5].",
"Ordinary (next-hop) distributed Bellman-Ford maintains a routing table which associates, to each known destination prefix, a next-hop router and a metric; each prefix and metric pair is advertised to neighbours in periodic update messages.",
"In source-specific Bellman-Ford, the routing table is indexed by pairs of a destination prefix and a source prefix, and (source-specific) updates advertise a triple of a destination prefix, a source prefix, and a metric.",
"The source-specific extension to Babel adds a new kind of source-specific update message in addition to the original, non-specific update.",
"Since Babel's loop-avoidance mechanism relies on two kinds of request messages, it also adds two new kinds of source-specific requests.",
"All of these are encoded as new kinds of messages rather than extensions to existing messages, which causes them to be silently ignored by unextended Babel routers, and ensures that our extension interoperates with the original Babel protocol."
],
[
"Bootstrapping",
"In distributed Bellman-Ford, a prefix is reannounced after it has been learnt from a neighbour.",
"This process is bootstrapped by announcing prefixes learned from a different source (typically a different routing protocol or a static route); in Babel, this is known as redistribution.",
"Just like ordinary routes, source-specific routes are originated by performing redistribution.",
"In case a source-specific route is already present, our implementation is able to redistribute it; more generally, the filtering language allows attaching a source prefix to a non-specific route at redistribution time.",
"While careless use of this facility may cause persistent routing loops to occur, this is expected with careless redistribution."
],
[
"Interoperability",
"The Babel protocol has seen a moderate amount of deployment in production networks, and is usually deployed within cheap routers that can be difficult to update with a source-specific version of the protocol.",
"We have therefore paid particular attention to the issue of interoperability between routers running the source-specific and unextended protocols.",
"The extended version of the protocol uses both non-specific and specific update messages.",
"In principle, a non-specific route could be announced in two manners: by using a non-specific update carrying the destination prefix $\\mathit {d}$ , or by using a source-specific update carrying the pair $(\\mathit {d}, {::}/0)$ .",
"As we want non-specific routes to be propagated between source-specific and non-specific routers, source-specific routers interpret a non-specific update as a source-specific update with a source prefix of ${::}/0$ , and, conversely, source-specific routers never send source-specific updates of the form $(\\mathit {d}, {::}/0)$ , preferring the non-specific form instead.",
"A more difficult issue is how a non-specific router should interpret a source-specific update.",
"There are two possibilities: the source can be discarded and the update treated as non-specific, or the entire update can be discarded.",
"The first of these possibilities can cause persistent routing loops.",
"Consider two nodes A and B, with A source-specific announcing a route to $(\\mathit {d}, \\mathit {s})$ (Figure REF ).",
"Suppose that B ignores the source information when it receives the update, and reannounces it as $\\mathit {d}$ .",
"This is reannounced to A, which treats it as $(\\mathit {d}, {::}/0)$ .",
"Packets destined to $\\mathit {d}$ but not sourced in $\\mathit {s}$ will be forwarded by A to B, and by B to A, causing a persistent routing loop.",
"Figure: Non-specific routers cannot accept specific routesOn the other hand, if non-source-specific nodes reject source-specific updates, but source-specific nodes accept non-specific updates, then source-specific nodes can communicate entries of the form $(\\mathit {d}, {::}/0)$ and are completely compatible with non-source-specific nodes.",
"In this case, Bellman-Ford will eventually converge to a loop-free configuration.",
"In general, discarding of source-specific routes by non-specific routers will cause routing blackholes.",
"Intuitively, unless there are enough non-specific routes in the network, non-specific routers will suffer starvation, and discard packets for destinations that are only announced by source-specific routers.",
"A simple yet sufficient condition for avoiding blackholes is to build a connected source-specific backbone that includes all of the edge routers, and announce a (non-specific) default route towards the backbone."
],
[
"Experimental results",
"We have implemented both schemes described in Sections REF and REF within babeld, a Linux implementation of the Babel routing protocol.",
"This has allowed us to perform a number of experiments which we describe in this section.",
"Our experimental network consists of a mesh network consisting of a dozen OpenWRT routers and a single server running Debian Linux.",
"Two of the mesh routers have a wired connection to the Internet, and are connected to the server through VPNs (over IPv4).",
"All of the routers run our modified version of the Babel protocol.",
"IPv4 connectivity for the mesh is provided by the Debian server, which acts as a NAT box.",
"The IPv6 connectivity is more interesting: there are two IPv6 prefixes, one of which is a native prefix provided by our employer's network, the other one being routed through the VPN.",
"The network therefore has two source-specific default IPv6 routes."
],
[
"Routing table for VPN connectivity",
"Figure REF shows an excerpt of the routing tables of one of the two wired routers.",
"The modified babeld daemon has allocated a non-default routing table, table 11, and inserted routes (marked as proto 42) into both the default main table and table 11.",
"The former contains non-specific routes: the default route and the /20 subnet announced by our local DHCP server, and host routes to individual mesh nodes.",
"The encapsulated VPN packets are routed through the default route.",
"Table 11 contains routes for locally originated packets, sourced in 192.168.4.0/24.",
"The only “real” route in this table is the default route, which prevents the VPN from attempting to “enter itself”.",
"The other routes are disambiguation routes, automatically generated by the algorithm described in Section REF .",
"These entries are copies of those present in the main routing table, and prevent locally generated packets destined to local subnets from leaving through the native default route.",
"Figure: IPv4 routing table on a router using a VPN"
],
[
"Multipath TCP",
"Multipath TCP [10] is an extension to TCP which multiplexes a single application-layer flow over multiple network layer sub-flows, and attempts to use as many distinct routes as possible, and to either carry traffic over the most efficient one or to perform load balancing.",
"An obvious application is a mobile node (a telephone) with permanent connectivity to a cellular network and intermittent WiFi connectivity: MPTCP is able to use the cellular link when WiFi is not available, and switch to WiFi when available without dropping already established connections.",
"Multipath TCP and source-specific routing turn out to be a surprisingly good match.",
"MPTCP is able to use all of the addresses of the local host, and to dynamically probe the reliability and performance of packets sourced from each.",
"We have performed two tests that both consist in downloading a 110 MB file over MPTCP from the MPTCP website.",
"In the first test (Figure REF ), a desktop computer is directly connected to the source-specifically routed wired network, and is configured with two IPv4 addresses.",
"The Linux tc subsystem is used to limit each of the addresses to 100 kB/s traffic; MPTCP is able to reliably download at 200 kB/s.",
"Figure: Download using MPTCP and traffic controlIn the second test (Figure REF ), a laptop's WiFi interface is configured with three addresses (one IPv4 and two IPv6).",
"MPTCP multiplexes the traffic across the three routes, and balances their throughput dynamically.",
"Figure: Download using MPTCP"
],
[
"Conclusion and further work",
"Source-specific routing is a modest extension to next-hop routing that keeps the forwarding decisions firmly within control of the routers while allowing end hosts a moderate and clearly defined amount of control over the choice of routes.",
"Since source-specific routing can cause ambiguous routing tables, we have defined the behaviour that we believe source-specific routers should have, and shown how combining different behaviours in the same network can cause persistent routing loops.",
"Similar care must be taken when combining non-specific with source-specific routers in the same network.",
"We have proposed two ways to implement source-specific routing, and obtained experimental results that show that source-specific routing can be usefully exploited by the transport layer protocol MPTCP.",
"Our implementation is of production quality, and has seen a modest amount of deployment, notably as a testbed for the ideas of the IETF Homenet working group.",
"While we enjoy working with distance-vector protocols, much of the networking community appears to have converged on using the OSPF protocol for internal routing.",
"OSPF is a rich and complex protocol, and while many of our techniques should apply without difficulty to it, actually implementing a full source-specific variant of OSPF without sacrificing any of its flexibility remains a challenging endeavour.",
"It was a pleasant surprise to discover that unmodified MPTCP can use source-specific routes without any manual configuration.",
"However, we claim that source-specific routing can also be exploited at the application layer, and we are currently working on an extension to the Mosh [13] UDP-based remote shell that is able to dynamically balance over multiple source-specific routes.",
"Finally, we have only considered the applicability of source-specific routing to edge networks, which tend to carry only a moderate number of distinct routes.",
"However, there is nothing in principle that would prevent source-specific routing from being applicable to BGP and to core networks, where it could perhaps be used for some forms of multihoming and traffic engineering without the routing table growth due to classical multihoming.",
"Extending our results to core networks, with their large routing tables, will require careful analysis of the complexity of our techniques, and a carefully optimised implementation."
],
[
"Code availability",
"The source-specific version of Babel is available from https://github.com/jech/babeld.",
"We are grateful to Benoît Valiron for his help with the presentation of the disambiguation algorithm."
]
] | 1403.0445 |
[
[
"Exact Mapping from Singular Value Spectrum of Fractal Images to\n Entanglement Spectrum of One-Dimensional Quantum Systems"
],
[
"Abstract We examine the snapshot entropy of general fractal images defined by their singular values.",
"Remarkably, the singular values for a large class of fractals are in exact correspondence with the entanglement spectrum of free fermions in one dimension.",
"These fermions allow for a holographic interpretation of the logarithmic scaling of the snapshot entropy, which is in agreement with the Calabrese-Cardy formula.",
"However, the coarse-grained entropy exhibits a linear scaling due to the degeneracy of the spectrum, in contrast with the logarithmic scaling behavior in one-dimensional quantum near-critical systems."
],
[
"Exact Mapping from Singular Value Spectrum of Fractal Images to Entanglement Spectrum of One-Dimensional Quantum Systems Ching Hua Lee${}^{a}$ Yuki Yamada${}^{b}$ Tatsuya Kumamoto${}^{b}$ Hiroaki Matsueda${}^{b}$ ${}^{a}$ Department of Physics, Stanford University, CA 94305, USA ${}^{b}$ Sendai National College of Technology, Sendai 989-3128, Japan We examine the snapshot entropy of general fractal images defined by their singular values.",
"Remarkably, the singular values for a large class of fractals are in exact correspondence with the entanglement spectrum of free fermions in one dimension.",
"These fermions allow for a holographic interpretation of the logarithmic scaling of the snapshot entropy, which is in agreement with the Calabrese-Cardy formula.",
"However, the coarse-grained entropy exhibits a linear scaling due to the degeneracy of the spectrum, in contrast with the logarithmic scaling behavior in one-dimensional quantum near-critical systems.",
"05.10.Cc, 07.05.Pj, 11.25.Hf, 11.25.Tq, 89.70.Cf The study of quantum entanglement has continuously attracted enormous attention.",
"The most important aspect concerns the scaling relation of the entanglement entropy in various quantum systems.",
"Well-known scaling relations are the area law, the Calabrese-Cardy formula [1], [2], and the finite-entanglement scaling (we call this as finite-$\\chi $ scaling for simplicity) [3], [4].",
"Since the latter two formulae enable us to estimate the central charge, the entropy is a poweful tool for the study of critical phenomena.",
"On the other hand, quantum entanglement is also deeply intertwined with the theme of holography.",
"Quantum entanglement holds the key to surprising holographic correspondence between completely different physical systems.",
"Two important manifestations are the anti-de Sitter space / conformal field theory correspondence in string theory and the multiscale entanglement renormalization ansatz in statistical physics.",
"In order to compare between different systems, a crucial factor is the amount of information behind these systems, not their detailed physical properties.",
"Thus, the entropy plays a central role in the comparison.",
"Since any classical system does not have entanglement, we must reconsider the meaning of entanglement entropy in the classical side, if there exists possible classical representation of entanglement.",
"In the conformal field theory language, the entanglement entropy is logarithm of two point correlation function of scaling operators, and this indicates the entropy contains the information from the physics at different length scales.",
"Then, it should be possible to encode these degrees of freedom in an emergent space with an additional dimension.",
"One of the solutions for the encoding is the so-called Ryu-Takayanagi formula [5].",
"Furthermore, the Suzuki-Trotter decomposition is also a well-known quantum-classical correspondence.",
"A typical example is transformation of the transverse-field Ising chain into the anisotropic two-dimensional (2D) classical spin model.",
"One of the authors has found that the entropy of the spin snapshot in the classical system corresponds to the holographic entanglemenet entropy of the original quantum 1D system [6].",
"Remarkably, the snapshot entropy contains an equivalent amount of information as the Calabrese-Cardy and finite-$\\chi $ scaling formulae combined.",
"There, the singular value decomposition (SVD) of the snapshot data is a quite essential procedure for defining the snapshot entropy.",
"A discretized holographic space emerges from this decomposition in the sense that the decomposed data have actually their own length scales.",
"The holographic entropy scaling looks at such sequence of different length-scale information.",
"The purpose of this letter is to derive the exact mapping of the singular values of fractal images to the entanglement spectrum in 1D quantum systems.",
"This is because the snapshot entropy, in spite of its potential applicability, is not still a well-defined quantity in some sense, and thus it's quite necessary to connect it to more physical systems.",
"We find that the mapping produces free fermionic and more exotic particles, and then the discrete scaling symmetry is mapped onto the degeneracy of their entanglement spectra.",
"In this paper, we focus on the mapping onto free fermions.",
"We would like to also discuss about the finite-$\\chi $ scaling.",
"We will prove that the finite-$\\chi $ scaling should be linear in the fractal cases in contrast to $\\ln \\chi $ in quantum near-critical 1D systems.",
"Compared to the Calabrese-Cardy formula, the finite-$\\chi $ scaling depends more delicately on the entanglement spectrum, and is thus a good measure to detect the difference between simple scale and full conformal symmetries.",
"We start from matrix data of a $L\\times L$ image $M(x,y)$ in which each element takes an integer value ranging from 0 to 1.",
"The value 0 denotes a white pixel, and 1 black.",
"We apply SVD to $M(x,y)$ as $M(x,y)&=&\\sum _{l=1}^{L}M^{(l)}(x,y), \\\\M^{(l)}(x,y)&=&U_{l}(x)\\sqrt{\\Lambda _{l}}V_{l}(y),$ where $\\Lambda _{l}$ denote the singular values and $U_{l}(x)$ and $V_{l}(y)$ are column unitary matrices.",
"We normalize the singular values as $\\lambda _{l}=\\Lambda _{l}/\\sum _{l}\\Lambda _{l}$ , and we arrange the order of $\\lambda _{l}$ so that $\\lambda _{1}\\ge \\lambda _{2}\\ge \\cdots $ .",
"Since all of the singular values are non-negative ones, this normalization leads to a probability distribution.",
"We also define the coarse-grained snapshot and entropy with $\\chi $ states kept as $M_{\\chi }&=&\\sum _{l=1}^{\\chi }M^{(l)}(x,y) , \\\\S_{\\chi }&=&-\\sum _{l=1}^{\\chi }\\lambda _{l}\\ln \\lambda _{l}.",
"$ Here we abbreviate the full entropy $S_{L}$ as $S$ .",
"The entropy measures the entanglement (take care about this terminology, this is not quantum entanglement) between the vertical and horizontal components $U_{l}(x)$ and $V_{l}(y)$ .",
"The singular values are the eigenvalues of the density matrices defined by $\\rho _{X}(x,x^{\\prime })&=&\\sum _{y}M(x,y)M(x^{\\prime },y)=\\sum _{l}U_{l}(x)\\Lambda _{l}U_{l}(x^{\\prime }), \\nonumber \\\\&& \\\\\\rho _{Y}(y,y^{\\prime })&=&\\sum _{x}M(x,y)M(x,y^{\\prime })=\\sum _{l}V_{l}(y)\\Lambda _{l}V_{l}(y^{\\prime }).",
"\\nonumber \\\\$ Their eigenvalues are the same.",
"This property is similar to quantum entanglement between two subsystems.",
"Figure: (left upper panel) Sierpinski carpet, (left lower panel) Sierpinski triangle, (right panel) full snapshot entropy as a function of the fractal level NN.",
"The open squares (filled triangles) represent SS for the carpet (triangle).As typical examples, we show a Sierpinski carpet and a Sierpinski triangle in Fig.",
"REF .",
"In the carpet case, we first devide a whole area into nine blocks, and occupy the center block by white color.",
"In the next step, the eight black blocks are respectively devided into nine blocks, and each center of them is occupied by white color.",
"We repeat this process until the minimal length scale comes to our finite system size $L$ .",
"We call this fractal as white-centered one.",
"The fractal dimension $D$ of this image is calculated as $D=\\ln 8 / \\ln 3=1.893$ .",
"A method of general mapping is quite simple, but yet very powerful.",
"Let us start with the $h\\times h$ unit cell of a given fractal.",
"In the Sierpinski carpet and triangle, the unit cells are respectively given by the following matrices $H=\\left(\\begin{array}{ccc}1&1&1\\\\ 1&0&1\\\\ 1&1&1\\end{array}\\right)\\; , \\; H=\\left(\\begin{array}{cc}1&0\\\\0.9&1.5\\end{array}\\right).$ The entries of the fractals can be continuously tuned to obtain the desired luminance, like in the case of the Sierpinski triangle shown.",
"Now, we construct a $h^{N}\\times h^{N}$ fractal matrix $M$ ($L=h^{N}$ ).",
"This can be easily done by taking the tensor product of $N$ copies $M=H\\otimes H\\otimes \\cdots \\otimes H$ .",
"Usually if this factorization occurs in a quantum state, such a state is called pure state.",
"On the other hand, due to its multiple scales of self-similarity, a fractal image cannot be decomposed into the direct product form of the vertical and horizontal components by SVD, and in that sense the fractal is entangled.",
"It is thus interesting that the product form represents such entangled structure.",
"If the eigenvalues of $H$ are $\\gamma _{1},\\gamma _{2},\\cdots ,\\gamma _{h}$ , the non-zero eigenvalues of $M$ are given by $\\Gamma _{\\mbox{$a$}}=\\prod _{j=1}^{N}\\gamma _{a_{j}},$ where each $a_{j}$ takes values from 1 to $r={\\rm rank}H$ ($r\\le h$ ).",
"In the next paragraph, we show that the system is transformed into free fermions when $r=2$ as in the cases of Sierpinski carpet and triangle.",
"This fact does not depend on $h$ .",
"The condition, $r=2$ , is true for a large class of fractals.",
"We normalize $\\Gamma _{\\mbox{$a$}}$ as $\\lambda _{\\mbox{$a$}}=\\Gamma _{\\mbox{$a$}}^{2}/\\sum _{\\mbox{$a$}}\\Gamma _{\\mbox{$a$}}^{2}$ , and we write $c_{k}=\\gamma _{k}^{2}/\\sum _{k=1}^{r}\\gamma _{k}^{2}$ .",
"Then, the entropy is given by $S=-\\sum _{\\mbox{$a$}}\\lambda _{\\mbox{$a$}}\\ln \\lambda _{\\mbox{$a$}}=-N\\sum _{k=1}^{r}c_{k}\\ln c_{k}\\propto \\ln L. $ This result guarantees the logarithmic entropy formula for any $r$ and $h$ values.",
"Here, it should be noted that there are fractals that cannot be written in the tensor-product form, even though they still look self-similar.",
"One explicit counterexample is the black-centered Sierpinski carpet.",
"The unit cell $H^{\\prime }$ is given by $H^{\\prime }=\\left(\\begin{array}{ccc}0&0&0\\\\ 0&1&0\\\\ 0&0&0\\end{array}\\right)=B-H, \\;\\; B=\\left(\\begin{array}{ccc}1&1&1\\\\ 1&1&1\\\\ 1&1&1\\end{array}\\right),$ where $H$ is the unit cell of the white-centered carpet.",
"The matrix $B$ disturbs the factorized form.",
"In general, the entropy will not scale logarithmically when $B$ and $H$ do not commute, as we shall discuss at length in future works.",
"Our entropy is asymmetric with respect to the exchange of white and black pixels.",
"Let us take a more precise look at the white-centered Sierpinski carpet.",
"In this case, $H$ has two non-zero eigenvalues $\\gamma _{\\pm }=1\\pm \\sqrt{3}$ ($r=2$ ).",
"The total number of the non-zero eigenvalues of $M$ is then $2^{N}$ .",
"The independent eigenvalues are given by $\\Gamma _{j}=\\gamma _{+}^{j}\\gamma _{-}^{N-j},$ with $j$ running from 0 to $N$ , and the degeneracy is represented by the binomial coefficient $\\alpha _{j}=N!/j!",
"(N-j)!$ with $\\sum _{j=0}^{N}\\alpha _{j}=2^{N}$ .",
"The characteristic polynomial of $M$ can be represented by $p(z)=\\left|zI-M\\right|=z^{3^{N}-2^{N}}\\prod _{j=0}^{N}\\left(z-\\Gamma _{j}\\right)^{\\alpha _{j}},$ where $I$ is the unit matrix of $L\\times L$ .",
"Then, the normalized singular values of $M$ are given by $\\bar{\\lambda }_{j}=\\frac{\\Gamma _{j}^{2}}{\\sum _{j=0}^{N}\\alpha _{j}\\Gamma _{j}^{2}}=\\left(\\frac{1}{2}+\\frac{\\sqrt{3}}{4}\\right)^{j}\\left(\\frac{1}{2}-\\frac{\\sqrt{3}}{4}\\right)^{N-j}.",
"$ where we have done re-labeling of the index so that $\\bar{\\lambda }_{N}=\\lambda _{1}$ , $\\bar{\\lambda }_{N-1}=\\lbrace \\lambda _{2},...,\\lambda _{N+1}\\rbrace $ , etc.",
"On the other hand, let us consider $N$ sites of an infinite free-fermion chain.",
"According to Refs.",
"[7], [8], [9], the $2^{N}$ eigenvalues of the reduced density matrix are given by the products $\\lambda =\\prod _{k\\in A}\\nu _{k}\\prod _{k\\in B}\\left(1-\\nu _{k}\\right).",
"$ where we consider that $A$ is a subset of $\\lbrace 1,2,...,N\\rbrace $ and $B$ is th rest of the subset.",
"Here $\\nu _{k}$ correspond to the eigenvalues of a $N\\times N$ single-particle correlator matrix $C_{ij}={\\rm Tr}(\\rho c_{j}^{\\dagger }c_{j})$ , when the reduced density matrix is given by $\\rho \\propto \\exp \\left(-\\sum _{1\\le i,j\\le l}h_{ij}c_{i}^{\\dagger }c_{j}\\right)$ with a $N\\times N$ matrix $h=\\ln [(I-C)C^{-1}]$ .",
"The inverse transformation actually gives $\\nu _{k}=(e^{\\epsilon _{k}}+1)^{-1}$ with the entanglement energy $\\epsilon _{k}$ .",
"Comparing Eq.",
"(REF ) with Eq.",
"(REF ), we clearly observe one-to-one correspondence between them $\\nu _{k}=\\frac{1}{2}+\\frac{\\sqrt{3}}{4}.$ It is straightforward to write down the entropy formula from Eq.",
"(REF ) as $S &=& -\\sum _{k=0}^{N}\\left\\lbrace \\nu _{k}\\ln \\nu _{k}+(1-\\nu _{k})\\ln (1-\\nu _{k})\\right\\rbrace \\nonumber \\\\&=& 0.245775\\ln L. $ Therefore, the snapshot entropy should obey this logarithmic scaling, and the scaling agrees with the Calabrese-Cardy formula $S=\\frac{c}{3}\\ln L,$ where $c$ is the central charge.",
"In Fig.",
"REF , we present numerical data of the full snapshot entropy $S$ as a function of the fractal level $N$ .",
"For the Sierpinski carpet, we actually find that the entropy perfectly matches with Eq.",
"(REF ) ($S=0.245775\\ln L=0.270011N$ ).",
"Figure: Induced patterns: (left) δ=-4\\delta =-4, (right) δ=-3\\delta =-\\sqrt{3}.The exact result provides us with several important pieces of information.",
"Firstly, the fractal level $N$ is the degeneracy of the electron's spectrum.",
"In other words, the scaling symmetry is manifested in the degeneracy of the fermion system.",
"Secondly, the quality of the fractal image is measured by how close the value $\\gamma _{-}^{2}/\\gamma _{+}^{2}$ is zero.",
"Here, $\\gamma _{-}^{2}/\\gamma _{+}^{2}=(2-\\sqrt{3})/(2+\\sqrt{3})\\simeq 0.07$ .",
"As shown in Eq.",
"(REF ), the snapshot entropy increases as $\\gamma _{-}^{2}/\\gamma _{+}^{2}$ increases from zero, and reaches a maximum when $\\gamma _{+}^{2}=\\gamma _{-}^{2}$ .",
"The significance of $\\gamma _{\\pm }$ will be further discussed in the paragraph of the finite-$\\chi $ scaling.",
"The third thing is about creation of new fractal images from this algorithm.",
"It is possible to take a reverse process of the present approach to make various beautiful fractals.",
"If $\\gamma _{\\pm }=1\\pm \\delta $ in general, the eigenvalues of the single-particle correlator matrix are given by $\\nu _{\\pm }=\\frac{1}{2}\\pm \\frac{\\delta }{1+\\delta ^{2}}.$ Then, changing the $\\delta $ value creates a family of various fractal images.",
"We show two examples in Fig.",
"REF .",
"Our result suggests that scale-invariance itself is sufficient for the entropy to exhibit a Calabrese-Cardy-type logarithmic scaling.",
"This is reasonable when we remember that the entanglement entropy is logarithm of two point correlation function.",
"The two-point function only with scale invariance is represented as $\\left<O_{1}(x_{1})O_{2}(x_{2})\\right>=c_{12}/\\left(x_{1}-x_{2}\\right)^{\\Delta _{1}+\\Delta _{2}}$ with scaling demensions $\\Delta _{1}$ and $\\Delta _{2}$ , but that of full conformal symmetry has the form $\\left<O_{1}(x_{1})O_{2}(x_{2})\\right>=c_{12}\\delta _{\\Delta _{1}\\Delta _{2}}/\\left(x_{1}-x_{2}\\right)^{2\\Delta _{1}}$ .",
"Although the constraint coming from only scale invariance is somehow weaker than that from the conformal invariance, the form of two-point function does not change so much.",
"We gain a more intuitive understanding from a holographic perspective reminiscent of the Ryu-Takayanagi formula.",
"The entropy is proportional to the fractal level $N$ , as we have already seen in Eq.",
"(REF ).",
"This supports the fact that we have fractal data spanning $N$ different length scales, which can be hierarchically organized into an emergent space with an extra dimension associated with the scale.",
"One paticular finding from this analysis is that there is no constant correction in Eq.",
"(REF ).",
"Going back to the previous work on the snapshot of the Ising spin model, we have observed the negative constant contribution to the entropy $S\\simeq \\ln L -2$ at $T_{c}$ , and the origin of this term was unresolved [6].",
"When we take $L\\rightarrow 0$ , we naively think that there is no information.",
"In the fractal case, this seems to be correct.",
"The Ising spin result reminds us the presence of the negative entropy contribution coming from possible topological effects in the quantum side.",
"Since we are now looking at the holography side, the negative term would come from symmetry breaking, not symmetry protection.",
"Thus, we think that the negative term in the Ising case comes from breaking of $Z_{2}$ zymmetry due to the presence of the ferromagnetic large back ground characterized by $M^{(1)}(x,y)$ .",
"Next, we discuss about finite-$\\chi $ scaling.",
"As we have already proven analytically, the singular value spectrum is degenerate with the degeneracy of the $j$ -th independent singular value $\\bar{\\lambda }_{j}$ being the binomial coefficient $\\alpha _{j}$ .",
"If we focus on the first $(N+1)$ -singular values, the coarse-grained snapshot entropy for $1\\le \\chi \\le N+1$ can be represented by $S_{\\chi }=-\\lambda _{1}\\ln \\lambda _{1}-\\left(\\chi -1\\right)\\lambda _{2}\\ln \\lambda _{2},$ since $\\lambda _{2}=\\lambda _{3}=\\cdots =\\lambda _{N+1}$ .",
"Since the entropy becomes zero for the extrapolation $\\chi \\rightarrow 0$ , we obtain $-\\lambda _{1}\\ln \\lambda _{1}=-\\lambda _{2}\\ln \\lambda _{2}.$ This leads to the following linear-scaling formula $S_{\\chi }=S_{1}\\chi .$ Therefore, this is quite in contrast to the standard finite-$\\chi $ scaling in the quantum 1D near-critical system $S_{\\chi }=\\frac{c\\kappa }{6}\\ln \\chi =\\frac{1}{\\sqrt{12/c}+1}\\ln \\chi ,$ with the finite-$\\chi $ scaling exponent $\\kappa $ .",
"Figure: (upper panels) M χ M_{\\chi } for χ=1,2,...,8\\chi =1,2,...,8, (lower left) Coarse-grained snapshot entropy as a function of χ\\chi , (lower right) Singular value spectrum.",
"We find that λ 1 =0.6155\\lambda _{1}=0.6155, λ 2 =⋯=λ8=0.0442\\lambda _{2}=\\cdots =\\lambda {8}=0.0442, and λ 9 =⋯=λ 29 =0.0032\\lambda _{9}=\\cdots =\\lambda _{29}=0.0032.Figure REF shows numerical results for $M_{\\chi }(x,y)$ , $S_{\\chi }$ , and $\\lambda _{l}$ of the white-centered Sierpinski carpet with $L=3^{7}$ .",
"It is noted that the hierarchy of the fractal process terminates within a finite level.",
"As shown in Fig, REF , the entropy is really a linear function with $\\chi $ , and has kink structure at $\\chi =8$ .",
"By compaing $M_{\\chi }$ with $S_{\\chi }$ , it is clear that the spatial resolution increases with $\\chi $ , and at the kink of the entropy we reach at the minimal length scale.",
"Increasing the $\\chi $ value thus corresponds to the repetition of the scale transformation.",
"Above the kink, we also observe the linear scaling, and this is also due to the next plateau in the degenerate singular value spectrum.",
"When we observe $M_{\\chi }$ , the insides of the white regions have somehow bright fine structures.",
"Deeper layers of SVD with large indices, that is information above the kink, remove these extra structures.",
"We also show the singular value spectrum in Fig.",
"REF .",
"We find that the spectrum is completely degenerate in the linear-$\\chi $ scaling region.",
"The number of the degenerate singular values is exactly $\\alpha _{N-1}(=7)$ for the first plateau, and $\\alpha _{N-2}(=21)$ for the second plateau.",
"The degeneracy is a direct evidence of scale invariance.",
"Therefore, the scaling formula is not logarithmic, but linear due to the scale invariance.",
"As we have suggested many times, the fractal images lack the full conformal symmetry.",
"We have scale invariance, but do not have Poincare and special conformal symmetries.",
"The conformal transformation does not change the local angle, but horizontal and vertical lines in the image are not kept.",
"This is bad for SVD, since the degeneracy of the spectrum changes completely.",
"In the spin model case at $T_{c}$ , however, the global rotation does not change the overall spin structure in the continuous limit.",
"It is curious whether it is possible to physically rotate the spin snapshot of the Ising model and whether we can actually confirm the invariance of the logarithmic scaling after the rotation.",
"Unfortunately, this is technically hard.",
"This is because the physical rotation changes the square-lattice structure of matrix data, and then re-mapping of the rotated data onto a new matrix induces extra entropy that modifies the essential information.",
"This will be a subject of future work.",
"Summarizing, we have examined the snapshot entropy and spectrum of fractal images.",
"We have mathematically proved and numerically confirmed that the logarithmic scaling of the snapshot entropy agrees with the Calabrese-Cardy formula in 1D free fermions.",
"However, the finite-$\\chi $ scaling for the fractal images is different from that in quantum near-critical 1D systems, and this comes from difference between scale and full conformal symmetries.",
"Our conclusion is that the snapshot is a very powerful tool for intuitive understanding of holography, and more detailed examinations will give us meaningful information.",
"CH is supported by the Agency of Science, Technology and Research of Singapore.",
"HM wrote this paper during his stay in Berkeley and Stanford.",
"HM is greatful for hospitality and comments from Joel Moore and Xiaoliang Qi.",
"HM acknowledges financial support from institute of national colleges of technology, Japan."
]
] | 1403.0163 |
[
[
"Partial differential equations from integrable vertex models"
],
[
"Abstract In this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations.",
"This mechanism is illustrated for the $U_q[\\widehat{\\mathfrak{sl}}(2)]$ invariant six-vertex model and the resulting partial differential equation is studied for particular values of the lattice length."
],
[
"Introduction",
"Vertex models of Statistical Mechanics are prominent examples where the computation of the model partition function can be described as an eigenvalue problem.",
"This possibility dates back to Kramers and Wannier transfer matrix technique [1], [2] originally devised for the Ising model.",
"Within that approach the partition function of the model is expressed in terms of the eigenvalues of a matrix usually refereed to as transfer matrix.",
"This technique has been successfully applied to a large variety of models although one has no guarantee a priori that the diagonalisation of the transfer matrix can be achieved.",
"An important class of non-trivial models whose transfer matrix has been exactly diagonalized is formed by those possessing a parameter $\\lambda $ such that its transfer matrix $T(\\lambda )$ satisfies the commutativity condition $[T(\\lambda _1), T(\\lambda _2)]=0$ for general values of $\\lambda _1$ and $\\lambda _2$ [3].",
"This latter property paves the way for a variety of non-perturbative methods such as the coordinate Bethe ansatz [4], the algebraic Bethe ansatz [5], $T\\;$ -$\\; Q$ relations [6], analytical Bethe ansatz [7] and separation of variables [8], [9] among others.",
"Within the approach of the coordinate Bethe ansatz for instance, the transfer matrices diagonalisation process involves the explicit computation of the action of the transfer matrix on a finite dimensional vector in terms of its components.",
"However, suppose one would like to study the spectral problem for an operator constituted by generators of the $\\mathfrak {sl}(2)$ algebra.",
"In that case we could also consider a differential representation of the $\\mathfrak {sl}(2)$ algebra [10] and study the same eigenvalue problem through the corresponding differential equation.",
"It is worth remarking that we use this same methodology in Quantum Mechanics when we convert the spectral problem for a Hamiltonian in the Heisenberg formulation into the solution of a stationary Schrödinger equation.",
"In this work we devise an analogous approach for the transfer matrix of the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ six-vertex model.",
"More precisely, we obtain a partial differential equation describing the spectral problem of the aforementioned operator.",
"The main ingredient of our derivation is the Yang-Baxter algebra employed within the lines of the algebraic-functional approach introduced in [11] and subsequently refined in the series of papers [12], [13], [14], [15].",
"It is also worth remarking that a connection between the spectral problem of transfer matrices and differential equations had appeared previously in the literature under the name ODE/IM correspondence.",
"See for instance the review [16] and references therein.",
"However, it is not clear if there is any relation between the ODE/IM correspondence and the approach considered here.",
"The main reason for that lies in the fact that the ODE/IM correspondence describes a relation between ordinary differential equations and integrable models while here we shall obtain partial differential equations.",
"Moreover, our results are valid for finite lattices while the ODE/IM correspondence emerges in the continuum limit.",
"This paper is organized as follows.",
"In Section we describe the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ transfer matrix and its spectral problem.",
"Section is devoted to the analysis of the aforementioned spectral problem in terms of a functional equation derived as a direct consequence of the Yang-Baxter algebra.",
"This functional equation is converted into a partial differential equation in Section and its analysis is performed for particular values of the lattice length.",
"Concluding remarks are presented in Section and extra results for the case of domain wall boundaries are given in Appendix ."
],
[
"The transfer matrix spectral problem",
"In this section we shall briefly recall some standard definitions and introduce a convenient notation to describe the eigenvalue problem for the transfer matrix associated with the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ solution of the Yang-Baxter equation.",
"Although this case corresponds to the well known trigonometric six-vertex model, here we shall consider it from the perspective described in [15]."
],
[
"Monodromy and transfer matrices.",
"Let $\\mathcal {T} \\in \\mbox{End}(\\mathbb {V}_{\\mathcal {A}} \\otimes \\mathbb {V}_{\\mathcal {Q}})$ be an operator which we shall refer to as monodromy matrix.",
"Here we shall consider the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ vertex model and in that case we have $\\mathbb {V}_{\\mathcal {A}} \\cong \\mathbb {C}^2$ and $\\mathbb {V}_{\\mathcal {Q}} \\cong (\\mathbb {C}^2)^{\\otimes L}$ for $L \\in \\mathbb {N}$ .",
"Hence the monodromy matrix $\\mathcal {T}$ can be recasted as $\\mathcal {T} = \\left( \\begin{matrix}A & B \\\\C & D \\end{matrix} \\right) \\;$ with entries $A, B, C, D \\in \\mathbb {V}_{\\mathcal {Q}}$ .",
"We then define the transfer matrix $T$ as the following operator, $T = A + D \\; .$"
],
[
"Yang-Baxter algebra.",
"Integrable vertex models in the sense of Baxter are characterized by a monodromy matrix $\\mathcal {T}$ satisfying the following algebraic relation, $\\mathcal {R}(x-y) \\left[ \\mathcal {T}(x) \\otimes \\mathcal {T}(y) \\right] = \\left[ \\mathcal {T}(y) \\otimes \\mathcal {T}(x) \\right] \\mathcal {R}(x-y) \\; .$ In (REF ) we have spectral parameters $x, y \\in \\mathbb {C}$ and $\\mathcal {R} \\in \\mbox{End}( \\mathbb {C}^2 \\otimes \\mathbb {C}^2 )$ in the case of the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ vertex model.",
"The algebraic relation (REF ) is associative for $\\mathcal {R}$ -matrices satisfying the Yang-Baxter equation, namely $\\left[ \\mathcal {R}(x) \\otimes \\mathbb {1} \\right] \\left[ \\mathbb {1} \\otimes \\mathcal {R}(x+y) \\right] \\left[ \\mathcal {R}(y) \\otimes \\mathbb {1} \\right] = \\left[ \\mathbb {1} \\otimes \\mathcal {R}(y) \\right] \\left[ \\mathcal {R}(x+y) \\otimes \\mathbb {1} \\right] \\left[ \\mathbb {1} \\otimes \\mathcal {R}(x) \\right] \\; ,$ with symbol $\\mathbb {1}$ denoting the $2 \\times 2$ identity matrix.",
"The matrix $\\mathcal {R}$ plays the role of structure constant for the Yang-Baxter algebra (REF ) and the solution of (REF ) invariant under the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ algebra explicitly reads $\\mathcal {R}(x) = \\left( \\begin{matrix}a(x) & 0 & 0 & 0 \\\\0 & c(x) & b(x) & 0 \\\\0 & b(x) & c(x) & 0 \\\\0 & 0 & 0 & a(x) \\end{matrix} \\right) \\; .$ The non-null entries of (REF ) are given by functions $a(x) = \\sinh {(x + \\gamma )}$ , $b(x) = \\sinh {(x)}$ and $c(x) = \\sinh {(\\gamma )}$ .",
"Remark 1 Since the $\\mathcal {R}$ -matrix (REF ) is invertible, the relation (REF ) implies that the transfer matrix (REF ) forms a commutative family, i.e.",
"$[T(x) , T(y)] = 0$ .",
"Now let $\\mathcal {M}$ be the set $\\mathcal {M}(x) = \\lbrace A, B , C, D\\rbrace (x)$ parameterized by a continuous complex variable $x$ .",
"The elements of $\\mathcal {M}$ obey the Yang-Baxter algebra and among the commutation rules encoded in (REF ) we shall make use of the following ones: $A(x_1) B(x_2) &=& \\frac{a(x_2 - x_1)}{b(x_2 - x_1)} B(x_2) A(x_1) - \\frac{c(x_2 - x_1)}{b(x_2 - x_1)} B(x_1) A(x_2) \\nonumber \\\\D(x_1) B(x_2) &=& \\frac{a(x_1 - x_2)}{b(x_1 - x_2)} B(x_2) D(x_1) - \\frac{c(x_1 - x_2)}{b(x_1 - x_2)} B(x_1) D(x_2) \\nonumber \\\\B(x_1) B(x_2) &=& B(x_2) B(x_1) \\; .$ Remark 2 The elements of $\\mathcal {M}$ are subjected to the Yang-Baxter algebra (REF ) which is in general non-abelian.",
"In this way the 2-tuple $(\\xi _1 , \\xi _2) : \\; \\xi _{i} \\in \\mathcal {M}(\\lambda _i)$ originated from the Cartesian product $\\mathcal {M}(\\lambda _1) \\times \\mathcal {M}(\\lambda _2)$ will be simply understood as the non-commutative product $\\xi _1 \\xi _2$ .",
"This convention is naturally extended for the $n$ -tuples $(\\xi _1 , \\dots , \\xi _n)$ generated by the products $\\mathcal {M}(\\lambda _1) \\times \\dots \\times \\mathcal {M}(\\lambda _n)$ ."
],
[
"Monodromy matrix representation.",
"The ordered product $\\mathcal {T}(\\lambda ) = \\mathop {\\overrightarrow{\\prod }}\\limits _{1 \\le j \\le L} \\mathrm {P}_{\\mathcal {A} j} \\mathcal {R}_{\\mathcal {A} j}(\\lambda - \\mu _j)$ with $\\mathcal {R}$ -matrix given by (REF ) is a representation of (REF ) due to the Yang-Baxter relation (REF ).",
"In (REF ) $\\mathrm {P}$ denotes the standard permutation matrix $\\mathrm {P}_{l m} : \\mathbb {V}_l \\otimes \\mathbb {V}_m \\mapsto \\mathbb {V}_m \\otimes \\mathbb {V}_l$ for $\\mathbb {V}_{l,m} \\cong \\mathbb {C}^2$ , while $\\lambda , \\mu _j \\in \\mathbb {C}$ are respectively the spectral and inhomogeneity parameters.",
"In its turn the subscripts in $\\mathcal {R}_{\\mathcal {A} j}$ indicate that we have a $\\mathcal {R}$ -matrix acting on the tensor product space $\\mathbb {V}_{\\mathcal {A}} \\otimes \\mathbb {V}_{j}$ .",
"More precisely, we have $\\mathcal {R}_{\\mathcal {A} j} \\in \\mbox{End}(\\mathbb {V}_{\\mathcal {A}} \\otimes \\mathbb {V}_{j})$ ."
],
[
"Highest weight vectors.",
"The vector $\\left|0\\right\\rangle = \\left( \\begin{matrix} 1 \\\\ 0 \\end{matrix} \\right)^{\\otimes L}$ is a $\\mathfrak {sl}(2)$ highest weight vector and the action of $\\mathcal {M}$ built from the representation (REF ) can be straightforwardly computed due to the structure of (REF ).",
"They are given by the following expressions: $A(\\lambda ) \\left|0\\right\\rangle &= \\prod _{j=1}^L a(\\lambda - \\mu _j) \\left|0\\right\\rangle & B(\\lambda ) \\left|0\\right\\rangle &\\ne 0 \\nonumber \\\\C(\\lambda ) \\left|0\\right\\rangle &= 0 & D(\\lambda ) \\left|0\\right\\rangle &= \\prod _{j=1}^L b(\\lambda - \\mu _j) \\left|0\\right\\rangle \\;\\;\\; .$"
],
[
"Yang-Baxter relations of higher degrees.",
"The spectrum of the transfer matrix $T$ can be encoded in a set of functional relations along the lines described in [11], [15], [17].",
"Those functional relations are derived as a direct consequence of the Yang-Baxter algebra.",
"In order to simplify our notation we introduce the symbol $[ \\lambda _1 , \\dots , \\lambda _n ]$ defined as the following product of operators, $[ \\lambda _1 , \\dots , \\lambda _n ] = \\mathop {\\overrightarrow{\\prod }}\\limits _{1 \\le j \\le n} B(\\lambda _j) \\; .$ Remark 3 Due to the last relation of (REF ) we have that $[\\lambda _1 , \\dots , \\lambda _n]$ is symmetric under the permutation of variables, i.e.",
"$[\\dots , \\lambda _i , \\dots , \\lambda _j, \\dots ] = [\\dots , \\lambda _j , \\dots , \\lambda _i, \\dots ]$ .",
"This property motivates the set theoretic notation $X^{a,b} = \\lbrace \\lambda _j \\; : \\; a \\le j \\le b \\rbrace $ and we can write $[ \\lambda _1 , \\dots , \\lambda _n ] = [ X^{1,n} ]$ .",
"We shall also employ the notation $X^{a,b}_{\\lambda } = X^{a,b} \\backslash \\lbrace \\lambda \\rbrace $ .",
"Now the products $A(\\lambda _0) [ X^{1,n} ]$ and $D(\\lambda _0) [ X^{1,n} ]$ can be investigated under the light of (REF ) taking into account the previous definitions.",
"By doing so we are left with the following Yang-Baxter relations of degree $n+1$ , $A(\\lambda _0) [ X^{1,n} ] &=& \\prod _{\\lambda \\in X^{1,n}} \\frac{a(\\lambda - \\lambda _0)}{b(\\lambda - \\lambda _0)} [X^{1,n}] A(\\lambda _0) \\nonumber \\\\&& - \\; \\sum _{\\lambda \\in X^{1,n}} \\frac{c(\\lambda - \\lambda _0)}{b(\\lambda - \\lambda _0)} \\prod _{\\tilde{\\lambda } \\in X^{1,n}_{\\lambda }} \\frac{a(\\tilde{\\lambda } - \\lambda )}{b(\\tilde{\\lambda } - \\lambda )} [X^{0,n}_{\\lambda }] A(\\lambda ) \\nonumber \\\\D(\\lambda _0) [ X^{1,n} ] &=& \\prod _{\\lambda \\in X^{1,n}} \\frac{a(\\lambda _0 - \\lambda )}{b(\\lambda _0 - \\lambda )} [X^{1,n}] D(\\lambda _0) \\nonumber \\\\&& - \\; \\sum _{\\lambda \\in X^{1,n}} \\frac{c(\\lambda _0 - \\lambda )}{b(\\lambda _0 - \\lambda )} \\prod _{\\tilde{\\lambda } \\in X^{1,n}_{\\lambda }} \\frac{a(\\lambda - \\tilde{\\lambda })}{b(\\lambda - \\tilde{\\lambda })} [X^{0,n}_{\\lambda }] D(\\lambda ) \\; .$ Here we intend to explore the relations (REF ) in order to describe the spectrum of the transfer matrix (REF ).",
"With that goal in mind we add up both expressions in (REF ) to obtain the identity $T(\\lambda _0) [ X^{1,n} ] &=& [X^{1,n}] ( M^{A}_0 A(\\lambda _0) + M^{D}_0 D(\\lambda _0) ) \\nonumber \\\\&& - \\; \\sum _{\\lambda \\in X^{1,n}} [X^{0,n}_{\\lambda }] ( M^{A}_{\\lambda } A(\\lambda ) + M^{D}_{\\lambda } D(\\lambda ) ) \\; .$ The coefficients in (REF ) explicitly read $M^{A}_0 &= \\prod _{\\lambda \\in X^{1,n}} \\frac{a(\\lambda - \\lambda _0)}{b(\\lambda - \\lambda _0)} & M^{D}_0 &= \\prod _{\\lambda \\in X^{1,n}} \\frac{a(\\lambda _0 - \\lambda )}{b(\\lambda _0 - \\lambda )} \\nonumber \\\\M^{A}_{\\lambda } &= \\frac{c(\\lambda - \\lambda _0)}{b(\\lambda - \\lambda _0)} \\prod _{\\tilde{\\lambda } \\in X^{1,n}_{\\lambda }} \\frac{a(\\tilde{\\lambda } - \\lambda )}{b(\\tilde{\\lambda } - \\lambda )} & M^{D}_{\\lambda } &= \\frac{c(\\lambda _0 - \\lambda )}{b(\\lambda _0 - \\lambda )} \\prod _{\\tilde{\\lambda } \\in X^{1,n}_{\\lambda }} \\frac{a(\\lambda - \\tilde{\\lambda })}{b(\\lambda - \\tilde{\\lambda })} \\; .$ Within the framework of the algebraic Bethe ansatz [5] one would then consider the action of (REF ) on the highest weight vector $\\left|0\\right\\rangle $ and try to fix the set of parameters $X^{1,n}$ in such a way that the transfer matrix eigenvalues can be directly read from the resulting expression.",
"Here we follow a different strategy and we shall demonstrate how the Yang-Baxter relation (REF ) can be converted into a functional equation."
],
[
"Algebraic-functional approach",
"In this section we aim to show that the eigenvalue problem for the transfer matrix (REF ) can be described in terms of certain functional equations originated from the Yang-Baxter algebra.",
"This statement is made precise in Theorem REF and its proof will require the following definitions.",
"Definition 1 (Algebras into functions) Considering the mechanism described in [15] we define the following continuous and additive map $\\pi $ , $\\pi _{n+1} \\; : \\quad \\mathcal {M}(\\lambda _0) \\times \\mathcal {M}(\\lambda _1) \\times \\dots \\times \\mathcal {M}(\\lambda _n) \\mapsto \\mathbb {C} [\\lambda _0^{\\pm 1} , \\lambda _1^{\\pm 1} , \\dots , \\lambda _n^{\\pm 1} ] \\; .$ The map $\\pi _{n+1}$ essentially associates a complex function to the elements of $\\mathcal {M}(\\lambda _0) \\times \\mathcal {M}(\\lambda _1) \\times \\dots \\times \\mathcal {M}(\\lambda _n)$ .",
"The proof of the announced Theorem REF will require the application of the map $\\pi _{n+1}$ over the higher order Yang-Baxter relation (REF ).",
"We shall also need to build a suitable realization of (REF ) and here we consider the recipe given in [17].",
"Lemma 1 Let $\\left| \\mathrm {\\Lambda } \\right\\rangle \\in \\mathrm {span}(\\mathbb {V}_{\\mathcal {Q}})$ be an eigenvector of the transfer matrix (REF ) and let $\\left\\langle \\mathrm {\\Lambda } \\right| $ denote its dual.",
"Also consider the $\\mathfrak {sl}(2)$ highest weight vector $\\left|0\\right\\rangle $ as previously defined and $\\mathcal {W}_{n+1} = \\mathcal {M}(\\lambda _0) \\times \\mathcal {M}(\\lambda _1) \\times \\dots \\times \\mathcal {M}(\\lambda _n)$ .",
"Hence we have that $\\pi _{n+1} (\\mathcal {A}) = \\left\\langle \\mathrm {\\Lambda } \\right| \\mathcal {A} \\left|0\\right\\rangle \\qquad \\qquad \\forall \\; \\mathcal {A} \\in \\mathcal {W}_{n+1}$ is a realization of (REF ).",
"The proof is straightforward and follows from the fact that both $\\left|0\\right\\rangle $ and $\\left| \\mathrm {\\Lambda } \\right\\rangle $ do not depend on the variables $\\lambda _j$ .",
"The independence of $\\left|0\\right\\rangle $ with $\\lambda _j$ is clear from its definition while this same property for $\\left| \\mathrm {\\Lambda } \\right\\rangle $ is due to the fact that the transfer matrix (REF ) forms a commutative family.",
"Theorem 1 (Functional equation) Let $\\Lambda $ be an eigenvalue of the transfer matrix $T$ associated with the eigenvector $\\left| \\mathrm {\\Lambda } \\right\\rangle \\in \\mathrm {span}(\\mathbb {V}_{\\mathcal {Q}})$ , i.e.",
"$T(\\lambda ) \\left|\\mathrm {\\Lambda }\\right\\rangle = \\Lambda (\\lambda ) \\left|\\mathrm {\\Lambda }\\right\\rangle $ .",
"Then $\\exists \\; \\mathcal {F}_n \\; : \\; \\mathbb {C}^n \\mapsto \\mathbb {C}$ characterizing the eigenvalue $\\Lambda $ through the equation $J_0 \\mathcal {F}_n (X^{1,n}) - \\sum _{\\lambda \\in X^{1,n}} K_{\\lambda } \\mathcal {F}_n (X^{0,n}_{\\lambda }) = \\Lambda (\\lambda _0) \\mathcal {F}_n (X^{1,n})$ with coefficients $J_0 &=& \\prod _{j=1}^L a(\\lambda _0 - \\mu _j) \\; M^{A}_0 + \\prod _{j=1}^L b(\\lambda _0 - \\mu _j) \\; M^{D}_0 \\nonumber \\\\K_{\\lambda } &=& \\prod _{j=1}^L a(\\lambda - \\mu _j) \\; M^{A}_{\\lambda } + \\prod _{j=1}^L b(\\lambda - \\mu _j) \\; M^{D}_{\\lambda } \\; .$ The realization (REF ) exhibits useful properties which will aid us in extracting information about the transfer matrix spectrum from the higher degree relation (REF ).",
"For instance, the application of $\\pi _{n+1}$ to the LHS of (REF ) will produce the term $\\pi _{n+1} (T(\\lambda _0) [ X^{1,n} ])$ which simplifies to $\\pi _{n+1} (T(\\lambda _0) [ X^{1,n} ]) = \\Lambda (\\lambda _0) \\pi _{n} ([ X^{1,n} ]) \\; .$ Similarly, we find that the application of the map $\\pi _{n+1}$ over the RHS of (REF ) only yields terms of the form $\\pi _{n+1} ([ Z^{1,n} ] A(x) )$ and $\\pi _{n+1} ([ Z^{1,n} ] D(x) )$ for generic variables $x$ and $Z^{1,n} = \\lbrace z_j \\in \\mathbb {C} \\; : \\; 1 \\le j \\le n \\rbrace $ .",
"Due to (REF ) and (REF ) those terms exhibit the following reduction properties, $\\pi _{n+1} ([ Z^{1,n} ] A(x) ) &=& \\prod _{j=1}^{L} a(x-\\mu _j) \\pi _{n} ([ Z^{1,n} ] ) \\nonumber \\\\\\pi _{n+1} ([ Z^{1,n} ] D(x) ) &=& \\prod _{j=1}^{L} b(x-\\mu _j) \\pi _{n} ([ Z^{1,n} ] ) \\; .$ In their turn the relations (REF ) and (REF ) tell us that the map $\\pi $ given by (REF ) obeys recurrence relations of type $\\pi _{n+1} \\mapsto \\pi _n$ over the elements of (REF ).",
"Next we introduce the notation $\\mathcal {F}_n( X^{1,n} ) = \\pi _n ( [X^{1,n}] )$ in such a way that the application of (REF ) on (REF ), taking into account the properties (REF ) and (REF ), yields the functional relation $J_0 \\mathcal {F}_n (X^{1,n}) - \\sum _{\\lambda \\in X^{1,n}} K_{\\lambda } \\mathcal {F}_n (X^{0,n}_{\\lambda }) = \\Lambda (\\lambda _0) \\mathcal {F}_n (X^{1,n}) \\; ,$ with coefficients $J_0 &=& \\prod _{j=1}^L a(\\lambda _0 - \\mu _j) \\; M^{A}_0 + \\prod _{j=1}^L b(\\lambda _0 - \\mu _j) \\; M^{D}_0 \\nonumber \\\\K_{\\lambda } &=& \\prod _{j=1}^L a(\\lambda - \\mu _j) \\; M^{A}_{\\lambda } + \\prod _{j=1}^L b(\\lambda - \\mu _j) \\; M^{D}_{\\lambda } \\; .$ This completes the proof of Theorem REF ."
],
[
"Operatorial description",
"The transfer matrix eigenvalue problem has been described in Theorem REF as the solution of a functional equation.",
"In this section we intend to show that the obtained functional equation, namely (REF ), can be recasted in an operatorial form which allows us to identify the action of the transfer matrix (REF ) on a particular function space.",
"For that we introduce the operator $D_{z_i}^{z_{\\alpha }}$ whose properties are described as follows.",
"Definition 2 Let $n \\in \\mathbb {N}$ and $\\alpha \\notin \\lbrace k \\in \\mathbb {N} \\; : \\; 1 \\le k \\le n \\rbrace $ .",
"Also, let $f$ be a complex function $f(z) \\in \\mathbb {C}[z]$ where $z=(z_1 , \\dots , z_n) \\in \\mathbb {C}^n$ .",
"Then we define the action of the operator $D_{z_i}^{z_{\\alpha }}$ on $\\mathbb {C}[z]$ as, $D_{z_i}^{z_{\\alpha }} \\; : \\qquad \\quad f(z_1, \\dots , z_i , \\dots , z_n) \\;\\; \\mapsto \\;\\; f(z_1, \\dots , z_{\\alpha } , \\dots , z_n) \\; .$ The operator $D_{z_i}^{z_{\\alpha }}$ has been previously introduced in [18] and it basically replaces a given variable $z_i$ by a variable $z_{\\alpha }$ .",
"In terms of the operator $D_{z_i}^{z_{\\alpha }}$ we can rewrite Eq.",
"(REF ) as $\\mathfrak {L}(\\lambda _0) \\mathcal {F}_n (X^{1,n}) = \\Lambda (\\lambda _0) \\mathcal {F}_n (X^{1,n})$ with operator $\\mathfrak {L}$ reading $\\mathfrak {L}(\\lambda _0) = J_0 - \\sum _{\\lambda \\in X^{1,n}} K_{\\lambda } D^{\\lambda _0}_{\\lambda } \\; .$ Now we can immediately recognize Eq.",
"(REF ) as an eigenvalue equation and some comments are in order at this stage.",
"For instance, the introduction of the operator $D_{z_i}^{z_{\\alpha }}$ is able to localize the whole dependence of the LHS of (REF ) with the spectral parameter $\\lambda _0$ in the operator $\\mathfrak {L}$ .",
"In fact, we can identify $\\mathfrak {L}$ with the transfer matrix (REF ) acting on a particular function space spanned by functions $\\mathcal {F}_n$ .",
"This function space will be described in the next section."
],
[
"The function space $\\mathrm {\\Xi }( \\mathbb {C}^n )$",
"The functions $\\mathcal {F}_n$ solving Eq.",
"(REF ) consist of the projection of the dual transfer matrix eigenvector onto a particular set of vectors usually refereed to as Bethe vectors.",
"Although we are mainly interested in the eigenvalues $\\Lambda $ we still need to restrict our solutions $\\mathcal {F}_n$ to a class of functions preserving certain representation theoretic properties exhibited by the elements involved in the derivation of (REF ).",
"This fact motivates the definition of the function space $\\mathrm {\\Xi }( \\mathbb {C}^n )$ whose properties will be discussed in what follows.",
"Definition 3 Let the functions $\\mathcal {F}_n \\; : \\;\\; \\mathbb {C}^n \\mapsto \\mathbb {C}$ be of the form $\\mathcal {F}_n (X^{1,n}) = \\left\\langle \\mathrm {\\Lambda }\\right| [\\lambda _1 , \\dots , \\lambda _n] \\left|0\\right\\rangle \\; ,$ where each operator $B$ in the product $[\\lambda _1 , \\dots , \\lambda _n]$ is built according to (REF ), (REF ) and (REF ).",
"In its turn $\\left\\langle \\mathrm {\\Lambda }\\right|$ is a dual eigenvector of the transfer matrix (REF ) whilst $\\left|0\\right\\rangle $ is the $\\mathfrak {sl}(2)$ highest weight vector as previously defined.",
"We can see from Lemma REF that the whole dependence of $\\mathcal {F}_n$ with a given variable $\\lambda _j$ comes from the operator $B(\\lambda _j)$ .",
"Thus the characterization of $\\mathrm {\\Xi }( \\mathbb {C}^n )$ can be performed with the help of the following Proposition.",
"Proposition 1 (Polynomial structure) The operator $B(\\lambda _i)$ is of the form $B(\\lambda _i) = x_i^{\\frac{1-L}{2}} P_B (x_i)$ where $x_i = e^{2 \\lambda _i}$ and $P_B$ is a polynomial of degree $L-1$ .",
"The proof follows from induction and it can be found with details in [11].",
"Definition 4 Let $\\mathbb {K} [x_1 , x_2 , \\dots , x_n]$ be the polynomial ring in $n$ variables $x_1, \\dots , x_n$ which we shall simply denote as $\\mathbb {K} [x]$ .",
"Then we define $\\mathbb {K}^m [x] \\subset \\mathbb {K} [x]$ as the subset of $\\mathbb {K} [x]$ formed by polynomials of degree $m$ in each variable $x_i$ .",
"Due to the Proposition REF we can conclude that $\\mathcal {F}_n$ are of the form $\\mathcal {F}_n (X^{1,n}) = \\prod _{i=1}^{n} x_i^{\\frac{1-L}{2}} \\; \\bar{\\mathcal {F}}_n (x_1, x_2 , \\dots , x_n) \\; ,$ where $\\bar{\\mathcal {F}}_n$ is a multivariate polynomial of degree $L-1$ in each one of its variables.",
"Taking into account the Definition REF we can write $x=(x_1, \\dots , x_n) \\in \\mathbb {C}^n$ and conclude that $\\bar{\\mathcal {F}}_n = \\bar{\\mathcal {F}}_n (x) \\in \\mathbb {K}^{L-1} [x]$ .",
"The function space $\\mathrm {\\Xi }$ is then defined as follows.",
"Definition 5 (Space $\\mathrm {\\Xi }$ ) The function space $\\mathrm {\\Xi }(\\mathbb {C}^n)$ consists of the following set of functions, $\\mathrm {\\Xi }(\\mathbb {C}^n) = \\left\\lbrace \\mathcal {F}_n \\; : \\; \\mathfrak {L}(\\lambda ) \\mathcal {F}_n = \\Lambda (\\lambda ) \\mathcal {F}_n , \\; \\mathcal {F}_n = \\mathbf {x}^{\\frac{1-L}{2}} \\; \\bar{\\mathcal {F}}_n (x), \\; \\bar{\\mathcal {F}}_n \\in \\mathbb {K}^{L-1} [x] \\right\\rbrace \\; ,$ where $\\mathbf {x}^{\\frac{1-L}{2}} = \\prod _{i=1}^n x_i^{\\frac{1-L}{2}}$ .",
"Remark 4 We can readily see from Definition 4 that $\\mathrm {\\Xi }(\\mathbb {C}^n) \\subset \\mathbf {x}^{\\frac{1-L}{2}} \\mathbb {K}^{L-1} [x]$ ."
],
[
"Partial differential equations",
"The operator $\\mathfrak {L}$ defined in (REF ) corresponds to the transfer matrix in the function space $\\mathrm {\\Xi }(\\mathbb {C}^n)$ .",
"In its turn $\\mathfrak {L}$ is given in terms of operators $D_{z_i}^{z_{\\alpha }}$ and here we intend to demonstrate that those operators admit a differential realization when their action is restricted to the set $\\mathbb {K}^m [x]$ .",
"This realization could not be immediately employed for (REF ) as we are interested in solutions $\\mathcal {F}_n \\in \\mathrm {\\Xi }(\\mathbb {C}^n)$ .",
"Nevertheless, in what follows we shall see how this differential structure can still be incorporated into Eq.",
"(REF ).",
"This approach has been previously employed in [15], where we have derived a set of partial differential equations (PDEs) satisfied by the partition function of the six-vertex model with domain wall boundaries.",
"For completeness' sake, in the Appendix we also discuss one of the PDEs explicitly obtained in [15].",
"Lemma 2 (Differential realization) Let $\\mathbb {K}^m [z] \\subset \\mathbb {K} [z]$ with $z=(z_1, \\dots , z_n) \\in \\mathbb {C}^n$ be a subset of the polynomial ring according to the Definition REF .",
"The operator $D_{z_i}^{z_{\\alpha }}$ in $\\mathbb {K}^m [z]$ is then given by $D_{z_i}^{z_{\\alpha }} = \\sum _{k=0}^{m} \\frac{(z_{\\alpha } - z_i)^k}{k!}",
"\\frac{\\partial ^k}{\\partial z_i^k} \\; .$ A detailed proof is given in [15].",
"As we have previously remarked we can not immediately substitute the realization (REF ) into (REF ) as the functions $\\mathcal {F}_n$ belong to the function space $\\mathrm {\\Xi }(\\mathbb {C}^n)$ .",
"However, as the non-polynomial part of $\\mathrm {\\Xi }(\\mathbb {C}^n)$ consists of an overall multiplicative factor, we can still rewrite (REF ) in terms of functions $\\bar{\\mathcal {F}}_n \\in \\mathbb {K}^{L-1}[x]$ defined through (REF ).",
"For that we introduce the variable $z=e^{2 \\lambda }$ and define the functions $\\bar{J}_0 = J_0 x_0^{\\frac{L}{2}} \\; , \\qquad \\bar{K}_{z} = K_{\\lambda } x_0^{\\frac{1}{2}} z^{\\frac{L-1}{2}} \\quad \\mbox{and} \\quad \\bar{\\Lambda }(x_0) = \\Lambda (\\lambda _0) x_0^{\\frac{L}{2}} \\; .$ By doing so we are left with the equation $\\bar{\\mathfrak {L}}(x_0) \\bar{\\mathcal {F}}_n (\\bar{X}^{1,n}) = \\bar{\\Lambda }(x_0) \\bar{\\mathcal {F}}_n (\\bar{X}^{1,n})$ where $\\bar{X}^{a,b} = \\lbrace x_k \\; : \\; a \\le k \\le b \\rbrace $ .",
"In its turn the operator $\\bar{\\mathfrak {L}}$ reads $\\bar{\\mathfrak {L}}(x_0) = \\bar{J}_0 - \\sum _{x \\in \\bar{X}^{1,n}} \\bar{K}_{x} D^{x_0}_{x}$ and now it acts on functions $\\bar{\\mathcal {F}}_n \\in \\mathbb {K}^{L-1}[x]$ .",
"Hence we can employ the realization (REF ) and this procedure reveals that $\\bar{\\mathfrak {L}}$ is of the form $\\bar{\\mathfrak {L}}(x_0) = \\sum _{k=0}^{L} x_0^k \\; \\mathrm {\\Omega }_k \\; .$ Here $\\lbrace \\mathrm {\\Omega }_k \\rbrace $ is a set of differential operators and the expression (REF ) implies that the LHS of (REF ) is a polynomial of degree $L$ in the variable $x_0$ .",
"Remark 5 The operator $\\bar{\\mathfrak {L}}$ corresponds to the transfer matrix $T$ in the function space $\\mathbb {K}^{L-1}[x]$ .",
"Thus, since $[T(\\lambda _1) , T(\\lambda _2)]=0$ as matricial operators, we can conclude that $[ \\bar{\\mathfrak {L}}(x_1) , \\bar{\\mathfrak {L}}(x_2) ] = 0$ which implies the condition $[ \\mathrm {\\Omega }_i , \\mathrm {\\Omega }_j ] = 0$ .",
"The RHS of (REF ) is also a polynomial of degree $L$ in the variable $x_0$ and this feature prevents that any operator $\\mathrm {\\Omega }_k$ vanishes identically by construction.",
"This property can be demonstrated with the help of Proposition REF .",
"Proposition 2 The operators $A(\\lambda )$ and $D(\\lambda )$ are of the form $A(\\lambda ) = z^{-\\frac{L}{2}} P_A (z) \\qquad \\mbox{and} \\qquad D(\\lambda ) = z^{-\\frac{L}{2}} P_D (z) \\; ,$ where $P_A$ and $P_D$ are polynomials of degree $L$ .",
"The proof follows from induction and the details can be found in [11].",
"Now from equations (REF ) and (REF ) we can conclude that $T(\\lambda )= z^{-\\frac{L}{2}} P_T (z)$ where $P_T$ is a polynomial of degree $L$ .",
"Moreover, since our transfer matrix forms a commutative family we can conclude that its eigenvalues will be of the form $\\Lambda (\\lambda )= z^{-\\frac{L}{2}} P_{\\Lambda } (z)$ where $P_{\\Lambda }$ is also a polynomial of degree $L$ .",
"Hence, the RHS of (REF ) consists of a polynomial of degree $L$ in the variable $x_0$ and we can write $\\bar{\\Lambda }(x_0) = \\sum _{k=0}^L x_0^k \\; \\Delta _k \\; .$ In this way Eq.",
"(REF ) must be satisfied independently by each power in $x_0$ and we are left with the following system of differential equations, $\\mathrm {\\Omega }_k \\bar{\\mathcal {F}}_n (x_1, \\dots , x_n) = \\Delta _k \\bar{\\mathcal {F}}_n (x_1, \\dots , x_n) \\qquad \\quad 0 \\le k \\le L \\; .$ The system of Eqs.",
"(REF ) comprises a total of $L+1$ eigenvalue problems, i.e.",
"an eigenvalue equation for each operator $\\mathrm {\\Omega }_k$ , being solved by the same eigenfunction $\\bar{\\mathcal {F}}_n$ .",
"Moreover, the direct inspection of (REF ) for small values of $n$ and $L$ shows that each equation is solely able to determine the eigenfunctions $\\bar{\\mathcal {F}}_n$ in addition to its eigenvalue $\\Delta _k$ .",
"Thus the system of differential equations (REF ) can be simultaneously integrated.",
"The explicit form of the operators $\\mathrm {\\Omega }_k$ can be straightforwardly obtained from (REF ) and (REF ).",
"Although their form for general values of $n$ and $L$ can be rather cumbersome we still find compact expressions for some of them.",
"For instance, the operator $\\mathrm {\\Omega }_{L}$ is trivial in consonance with the fact that the leading term coefficient of the transfer matrix corresponds to the Cartan element of the $U_q[\\widehat{\\mathfrak {sl}}(2)]$ algebra [7].",
"Fortunately, the situation is more interesting for the operator $\\mathrm {\\Omega }_{L-1}$ and we find a compact structure containing only derivatives $\\frac{\\partial ^{L-1}}{\\partial x_i^{L-1}}$ .",
"In what follows we present the explicit differential equation associated with the operator $\\mathrm {\\Omega }_{L-1}$ spectral problem."
],
[
"The operator $\\mathrm {\\Omega }_{L-1}$ .",
"Equation (REF ) for $k=L-1$ and arbitrary values of $n$ and $L$ corresponds to the following partial differential equation, $\\left[ \\mathcal {V}^{(n)} + \\sum _{i=1}^n \\mathcal {Q}^{(n)}_i \\frac{\\partial ^{L-1}}{\\partial x_i^{L-1}} \\right] \\bar{\\mathcal {F}}_n (\\bar{X}^{1,n} ) = \\Delta _{L-1} \\bar{\\mathcal {F}}_n ( \\bar{X}^{1,n} ) \\; ,$ with functions $\\mathcal {V}^{(n)} = \\mathcal {V}^{(n)}( \\bar{X}^{1,n} )$ and $\\mathcal {Q}^{(n)}_i = \\mathcal {Q}^{(n)}(x_i ; \\bar{X}_i^{1,n})$ defined over the sets $\\bar{X}^{a,b}$ and $\\bar{X}_i^{a,b} = \\bar{X}^{a,b} \\backslash \\lbrace x_i \\rbrace $ .",
"By writing $\\mathcal {V}^{(n)} = - 2^{-L} \\prod _{k=1}^L y_k^{-\\frac{1}{2}} [\\mathcal {V}_1^{(n)} + \\mathcal {V}_2^{(n)}]$ we then have $\\mathcal {V}_1^{(n)} = (q^n + q^{L-n-2}) \\sum _{k=1}^{L} y_k \\; ,$ and $\\mathcal {V}_2^{(n)} = {\\left\\lbrace \\begin{array}{ll}q^{n-2} (q-1)^2 (q+1) \\sum _{k=0}^{L+1-2n} q^k \\sum _{i=1}^{n} x_i \\qquad \\quad \\quad \\; L \\ge 2(n-1) \\nonumber \\\\-q^{L-n} (q-1)^2 (q+1) \\sum _{k=0}^{2n - 3 - L} q^k \\sum _{i=1}^{n} x_i \\qquad \\quad \\; L < 2(n-1)\\end{array}\\right.}",
"\\; .",
"\\nonumber \\\\$ In their turn the functions $\\mathcal {Q}^{(n)}_i$ are given by $\\mathcal {Q}^{(n)}_i = \\frac{(q-1)^2 (q+1)}{2^L q^{L+n} (L-1)!}",
"\\frac{\\prod _{k=1}^{L} y_k^{-\\frac{1}{2}}}{\\prod _{\\stackrel{j=1}{j \\ne i}}^{n}(x_j - x_i)} \\left[ \\sum _{m=0}^{L} \\mathcal {G}^{(n)}_m (x_i ; \\bar{X}_i^{1,n}) \\sum _{1 \\le j_1 < \\dots < j_m \\le L} \\prod _{\\alpha =1}^{m} y_{j_{\\alpha }} \\right] \\; , \\nonumber \\\\$ where $\\mathcal {G}^{(n)}_{L-d} (x_i ; \\bar{X}_i^{1,n}) = x_i^d \\sum _{l=0}^{n-1} x_i^l \\; \\psi _{l,d} \\sum _{\\stackrel{1 \\le j_1 < \\dots < j_{n-1-l} \\le n}{j_{\\alpha } \\ne i}} \\prod _{\\alpha =1}^{n-1-l} x_{j_{\\alpha }}$ and $&& \\psi _{l,d} = \\nonumber \\\\&& {\\left\\lbrace \\begin{array}{ll}(-1)^{L+d+l} q^{L+2l} \\sum _{k=0}^{2d + 2n - 3 - L - 4l} q^k \\qquad \\qquad \\qquad \\; \\; d > L - (n+1) + 2l \\nonumber \\\\(-1)^{3l - n -1} q^{L+2l} \\sum _{k=0}^{L-5} q^k \\qquad \\qquad \\qquad \\qquad \\qquad d = L - (n+1) + 2l , \\; L \\ge 5 \\nonumber \\\\(-1)^{3l - n} q^{2L + 2 l -4} \\sum _{k=0}^{3-L} q^k \\qquad \\qquad \\qquad \\qquad \\quad \\; \\; d = L - (n+1) + 2l , \\; L < 5 \\nonumber \\\\(-1)^{L+d+l+1} q^{2d+2n-2-2l} \\sum _{k=0}^{L-2d-2n+1+4l} q^k \\qquad \\quad d < L - (n+1) + 2l\\end{array}\\right.}",
"\\; .",
"\\nonumber \\\\$ Clearly the terms $\\sum _{k=0}^{l} q^k$ in (REF ) and (REF ) can be simplified with the help of the geometric sum formula $\\sum _{k=0}^{l} q^k = \\frac{1-q^{l+1}}{1-q}$ .",
"However, we prefer to keep the summation symbol in order to make more explicit that the summation vanishes for $l<0$ .",
"Eq.",
"(REF ) is a partial differential equation of order $L-1$ and in what follows we shall demonstrate how it can be translated into a system of first order equations.",
"We shall also discuss its solutions and properties for particular values of $n$ and $L$ ."
],
[
"Reduction of order",
"Linear differential equations of higher order can be conveniently written as a system of first order equations.",
"In our case we have (REF ), which is a linear partial differential equation of order $L-1$ , and here we intend to embed that equation into a system of first order equations.",
"The resulting system of partial differential equations is explicitly given in Lemma REF .",
"Lemma 3 Let $\\partial _i \\equiv \\frac{\\partial }{\\partial x_i}$ and let $\\vec{\\psi }$ be a $(L-2)n+1$ dimensional vector denoted as $\\vec{\\psi } = \\left( \\begin{matrix}\\psi ^{(0)} \\\\ \\psi ^{(1)} \\\\ \\vdots \\\\ \\psi ^{(L-2)}\\end{matrix} \\right) \\; .$ Also define $\\psi ^{(0)} = \\psi _0 = \\bar{\\mathcal {F}}_n (\\bar{X}^{1,n} )$ while the remaining entries $\\psi ^{(k)}$ are $n$ -dimensional column vectors with components $\\psi _i^{(1)} = \\partial _i \\psi _0$ and $\\psi _i^{(k)} = \\partial _i \\psi _i^{(k-1)}$ for $k > 1$ .",
"The equation (REF ) then reads $(\\mathcal {Q}^{(n)} - \\Delta _{L-1} ) \\psi _0 + \\sum _{i=1}^n \\mathcal {V}^{(n)}_i \\partial _i \\psi _i^{(L-2)} = 0 \\; .$ Straightforward substitution of (REF ) into (REF )."
],
[
"Matricial form.",
"The system of equations described in Lemma REF can be conveniently written as a matrix equation.",
"More precisely, the aforementioned system consists of the following equations $\\partial _i \\psi _0 - \\psi _i^{(1)} &=& 0 \\nonumber \\\\\\partial _i \\psi _i^{(k-1)} - \\psi _i^{(k)} &=& 0 \\qquad 1 < k \\le L-2 \\; ,$ in addition to (REF ).",
"Here we intend to rewrite (REF ) and (REF ) as $\\mathrm {\\Upsilon } \\vec{\\psi } = 0$ for a given matrix $\\mathrm {\\Upsilon }$ .",
"For that we introduce the $n$ -dimensional vectors $\\vec{\\omega }_0 = \\left( \\mathcal {Q}^{(n)}_1 \\partial _1 , \\dots , \\mathcal {Q}^{(n)}_n \\partial _n \\right)\\quad \\mbox{and} \\quad \\vec{\\nabla }_0 = \\left( \\begin{matrix} \\partial _1 \\\\ \\vdots \\\\ \\partial _n \\end{matrix} \\right) \\; .$ Also let $\\mathbf {0}_{r \\times s}$ denote the null matrix with dimensions $r \\times s$ and define vectors $\\vec{\\omega } = \\left( \\mathbf {0}_{1 \\times n(L-3)} , \\vec{\\omega }_0 \\right)\\quad \\mbox{and} \\quad \\vec{\\nabla } = \\left( \\begin{matrix} \\vec{\\nabla }_0 \\\\ \\mathbf {0}_{n(L-3) \\times 1} \\end{matrix} \\right) \\; .$ Next we define $\\hat{\\mathfrak {D}}$ as a matrix of dimensions $(L-2)\\times (L-2)$ with entries $\\hat{\\mathfrak {D}}_{ij} = {\\left\\lbrace \\begin{array}{ll}- \\mathbb {1}_{n \\times n} \\qquad \\quad i=j \\;\\;\\; \\quad ; \\; 1 \\le j \\le L-2 \\cr \\mathcal {D} \\qquad \\qquad \\quad i=j+1 \\; ; \\; 1 \\le j < L-2 \\cr \\mathbf {0}_{n \\times n} \\qquad \\qquad \\mbox{otherwise}\\end{array}\\right.}",
"\\; ,$ where $\\mathbb {1}_{n \\times n}$ denotes the $n \\times n$ identify matrix and $\\mathcal {D}$ is also a $n \\times n$ diagonal matrix given by $\\mathcal {D}= \\mbox{diag}(\\partial _1, \\partial _2 , \\dots , \\partial _n)$ .",
"In this way the system of Eqs.",
"formed by (REF ) and (REF ) can be written as $\\mathrm {\\Upsilon } \\vec{\\psi } = 0$ with matrix $\\mathrm {\\Upsilon }$ given by $\\mathrm {\\Upsilon } = \\left( \\begin{matrix}\\mathcal {Q}^{(n)} - \\Delta _{L-1} & \\vec{\\omega } \\\\\\vec{\\nabla } & \\hat{\\mathfrak {D}} \\end{matrix} \\right) \\; .$ As previously remarked in Section , the above discussed relation between functional equations of type (REF ) and partial differential equations (PDEs) was firstly proposed in [19] for the partition function of the six-vertex model with domain wall boundaries.",
"This relation was subsequently made precise in [15].",
"Despite their similarity, Eq.",
"(REF ) describes an eigenvalue problem while the equation derived in [15] can not be regarded in that way.",
"Nevertheless, in both cases the associated PDEs exhibit a similar structure, and one can wonder if the equation derived in [15] can also be recasted as a system of first order PDEs.",
"In Appendix we address this question and show that this is indeed the case for the PDE describing the partition function of the six-vertex model with domain wall boundaries."
],
[
"Some particular solutions",
"In this section we study the solutions of Eq.",
"(REF ) for the cases $n=0,1,2$ and particular values of the lattice length $L$ .",
"Interestingly, for the case $L=2$ and $n=2$ some geometric features of our equation emerge through the method of characteristics [20]."
],
[
"Case $n=0$",
"Although the case $n=0$ is trivial, we present it here for completeness reasons.",
"By definition we have that $\\bar{\\mathcal {F}}_0$ is a constant and we can conclude that $\\Delta _{L-1} = \\mathcal {V}^{(0)}$ .",
"Thus from (REF ) and (REF ) we find $\\Delta _{L-1} = - \\frac{(1+q^{L-2}) \\sum _{k=1}^{L} y_k}{2^L \\prod _{k=1}^{L} y_k^{\\frac{1}{2}}} \\; .$"
],
[
"Case $n=1$",
"Equation (REF ) for $n=1$ is actually an ordinary differential equation reading $\\frac{d^{L-1} \\bar{\\mathcal {F}}_1}{d x_1^{L-1}} = \\left( \\frac{\\Delta _{L-1} - \\mathcal {V}^{(1)}}{\\mathcal {Q}^{(1)}_1} \\right) \\bar{\\mathcal {F}}_1 \\; .$ Moreover, when $L=2$ we can see that (REF ) is a first order equation and the general solution can be obtained by direct integration.",
"In that case we obtain $\\bar{\\mathcal {F}}_1 (x_1) &=& \\mathcal {C}_1 (q^2 x_1^2 - y_1 y_2)^{\\frac{1}{2}} \\exp { \\left\\lbrace - \\frac{q}{(q^2 -1)^2} \\left[ (1+q^2)\\frac{(y_1 + y_2)}{(y_1 y_2)^{\\frac{1}{2}}} + 4 q \\Delta _{L-1} \\right] \\xi (x_1)\\right\\rbrace } \\; , \\nonumber \\\\$ where $\\mathcal {C}_1$ is an integration constant and $\\xi (x_1) = \\operatorname{arctanh}{( q x_1 (y_1 y_2)^{-\\frac{1}{2}})}$ .",
"We are interested in solutions $\\bar{\\mathcal {F}}_1 \\in \\mathbb {K}^1 [x_1]$ while (REF ) consists of a square root multiplied by an exponential.",
"At first glance this structure does not resemble the desired class of solutions but we then notice that $\\exp {[ \\xi (x_1) ]} = - \\mathrm {i}\\frac{[q x_1 + (y_1 y_2)^{\\frac{1}{2}}]}{(q^2 x_1^2 - y_1 y_2)^{\\frac{1}{2}}} \\; .$ In this way we find that the condition $- \\frac{q}{(q^2 -1)^2} \\left[ (1+q^2)\\frac{(y_1 + y_2)}{(y_1 y_2)^{\\frac{1}{2}}} + 4 q \\Delta _{L-1} \\right] = \\pm 1$ leave us with the desired type of solution.",
"Hence, the requirement $\\bar{\\mathcal {F}}_1 \\in \\mathbb {K}^1 [x_1]$ yields a constraint for the eigenvalues $\\Delta _{L-1}$ which is solved by $\\Delta _{L-1} = - \\frac{(1+q^2)}{4 q} \\frac{(y_1 + y_2)}{(y_1 y_2)^{\\frac{1}{2}}} \\mp \\frac{(q^2 - 1)^2}{4 q^2} \\; .$"
],
[
"Case $n=2$",
"Here we shall address the case $n=2$ and $L=2$ where (REF ) reads $\\mathcal {Q}^{(2)}_1 \\frac{\\partial \\bar{\\mathcal {F}}_2 }{\\partial x_1} + \\mathcal {Q}^{(2)}_2 \\frac{\\partial \\bar{\\mathcal {F}}_2 }{\\partial x_2} = (\\Delta _{L-1} - \\mathcal {V}^{(2)}) \\bar{\\mathcal {F}}_2 \\; .$ For this particular case it is worth remarking that $\\mathcal {V}^{(2)}$ does not depend on the variables $x_1$ and $x_2$ .",
"Now let $\\mathcal {S} = \\lbrace (x_1 , x_2 , \\bar{\\mathcal {F}}_2 ) \\rbrace $ be the surface generated by the solution of (REF ) and let $\\mathcal {C}$ be a curve lying on $\\mathcal {S}$ .",
"Also, let $s$ be a variable parameterizing the curve $\\mathcal {C}$ such that the vector $\\left( \\mathcal {Q}^{(2)}_1 (x_1(s) , x_2(s)) , \\mathcal {Q}^{(2)}_2 (x_1(s) , x_2(s)) , (\\Delta _{L-1} - \\mathcal {V}^{(2)}) \\bar{\\mathcal {F}}_2 (x_1(s) , x_2(s)) \\right)$ is tangent to $\\mathcal {C}$ at each point of the curve.",
"Then the curve $\\mathcal {C} = \\lbrace ( x_1 (s), x_2 (s) , \\bar{\\mathcal {F}}_2 (x_1(s) , x_2(s)) ) \\rbrace $ satisfy the following system of ordinary differential equations, $\\frac{d x_1}{ds} &=& \\mathcal {Q}^{(2)}_1 (x_1(s) , x_2(s)) \\nonumber \\\\\\frac{d x_2}{ds} &=& \\mathcal {Q}^{(2)}_2 (x_1(s) , x_2(s)) \\nonumber \\\\\\frac{d \\bar{\\mathcal {F}}_2}{ds} &=& (\\Delta _{L-1} - \\mathcal {V}^{(2)}) \\bar{\\mathcal {F}}_2 (x_1(s) , x_2(s)) \\; .$ The curve $\\mathcal {C}$ is called characteristic curve for the vector field (REF ) and it is determined by the solution of the system (REF ).",
"The characteristic equations (REF ) can also be written without fixing a particular parameterization variable as $\\frac{d x_1}{\\mathcal {Q}^{(2)}_1} = \\frac{d x_2}{\\mathcal {Q}^{(2)}_2} = \\frac{d \\bar{\\mathcal {F}}_2}{ (\\Delta _{L-1} - \\mathcal {V}^{(2)}) \\bar{\\mathcal {F}}_2} \\; .$ Now we can form any two equations by combining the terms of (REF ) and for convenience we choose $\\frac{d x_1}{d x_2} &=& \\frac{\\mathcal {Q}^{(2)}_1}{\\mathcal {Q}^{(2)}_2} = - \\frac{x_1}{x_2} \\nonumber \\\\\\frac{d \\bar{\\mathcal {F}}_2}{d x_2} &=& (\\Delta _{L-1} - \\mathcal {V}^{(2)}) \\frac{\\bar{\\mathcal {F}}_2}{\\mathcal {V}^{(2)}_2} \\; .$ The integration of (REF ) yields the solution $\\bar{\\mathcal {F}}_2 (x_1 , x_2) = \\kappa (x_1 x_2) \\exp { \\left\\lbrace \\frac{[(1+q^4)(y_1 + y_2) + 4q^2 \\sqrt{y_1 y_2} \\Delta _{L-1}]}{(q^2 -1)^2 (y_1 + y_2)} \\log {\\zeta (x_1 , x_2)} \\right\\rbrace } \\; ,$ where $\\kappa (x_1 x_2)$ is an arbitrary function of the product $x_1 x_2$ and $\\zeta (x_1 , x_2) = q^2 (y_1 + y_2) (x_1 + x_2) - (1+q^2)( y_1 y_2 + q^2 x_1 x_2 ) \\; .$ Hence, in order to having $\\bar{\\mathcal {F}}_2 \\in \\mathbb {K}^1 [x_1 , x_2]$ we choose $\\kappa $ as a constant function and impose the condition $\\frac{[(1+q^4)(y_1 + y_2) + 4q^2 \\sqrt{y_1 y_2} \\Delta _{L-1}]}{(q^2 -1)^2 (y_1 + y_2)} = 1 \\; .$ The resolution of (REF ) for $\\Delta _{L-1}$ yields the eigenvalue $\\Delta _{L-1} = - \\frac{(y_1 + y_2)}{2 \\sqrt{y_1 y_2}} \\; .$"
],
[
"Concluding remarks",
"In this work we have presented a mechanism allowing to associate a linear partial differential equation with the eigenvalue problem of the six-vertex model transfer matrix.",
"This mechanism has its roots in the algebraic-functional approach described in details in [15], and a crucial step in this program is the identification of the function space $\\mathrm {\\Xi } (\\mathbb {C}^n)$ defined in Section REF .",
"In Section REF we identify the action of the transfer matrix on the function space $\\mathrm {\\Xi } (\\mathbb {C}^n)$ and we find that it is given in terms of certain operators $D_{z_i}^{z_{\\alpha }}$ exhibiting a simple action even on larger spaces such as $\\mathbb {C}[z]$ with $z = (z_1 , \\dots , z_n) \\in \\mathbb {C}^n$ .",
"Also, the operators $D_{z_i}^{z_{\\alpha }}$ play a fundamental role in establishing the aforementioned partial differential equations as they possess a differential realization in the desired function space.",
"The six-vertex model transfer matrix admits the series expansion $T(\\lambda ) = T(0) (\\mathbb {1} + \\mathcal {H} \\lambda + \\dots )$ where $T(0)$ is the discrete translation operator, i.e.",
"momentum operator exponentiated, and $\\mathcal {H}$ is the hamiltonian of the $XXZ$ spin chain [21].",
"Although $\\mathcal {H}$ contains only next-neighbors interactions, the higher order terms contain highly non-local terms.",
"On the other hand, our transfer matrix in the appropriate variable consists essentially of a polynomial whose degree scales linearly with the lattice length.",
"Thus the number of independent commuting quantities having $T(\\lambda )$ as its former are finite and, in particular, we find the operator $\\mathrm {\\Omega }_{L-1}$ which exhibits a local structure in terms of differentials.",
"The partial differential equation corresponding to the operator $\\mathrm {\\Omega }_{L-1}$ spectral problem is explicitly given in Section and it consists of a linear equation of order $L-1$ .",
"Although we have an equation whose order scales with the lattice length $L$ , in Section REF we also show how this equation can be translated into a system of first order partial differential equations by standard methods.",
"In Section REF we analyze the solutions of our equation for particular values of the lattice length.",
"Interestingly, we find that the spectrum of eigenvalues is fixed by the condition that the eigenfunctions belong to $\\mathrm {\\Xi } (\\mathbb {C}^n)$ ."
],
[
"Acknowledgements",
"This work is supported by the Netherlands Organization for Scientific Research (NWO) under the VICI grant 680-47-602 and by the ERC Advanced grant research programme No.",
"246974, “Supersymmetry: a window to non-perturbative physics\".",
"The work of W.G.",
"is also supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the Early Universe."
],
[
"Domain wall boundaries",
"The possibility of extracting partial differential equations from the algebraic-functional method described in Section has been first put forward in [19] and subsequently refined in [15].",
"Those works have considered the six-vertex model with domain wall boundary conditions and a compact partial differential equation describing the model partition function has been derived in [15].",
"This equation reads $\\left[ \\sum _{i=1}^{L} \\bar{a}(x_i , y_i) - \\frac{1}{(L-1)!}",
"\\sum _{i=1}^{L} \\prod _{j=1}^{L} \\bar{a}(x_i , y_j) \\prod _{\\stackrel{j=1}{j \\ne i}}^{L} \\frac{\\bar{a}(x_j , x_i)}{\\bar{b}(x_j , x_i)} \\frac{\\partial ^{L-1}}{\\partial x_i^{L-1}} \\right] \\bar{Z} (\\bar{X}^{1,L}) = 0 \\; ,$ where $\\bar{Z}$ is essentially the partition function of the model.",
"In (REF ) we are considering the conventions $\\bar{a}(x,y) = x q - y q^{-1}$ and $\\bar{b}(x,y) = x - y$ which is slightly different from the ones used in [15].",
"Also, the structure of (REF ) is closely related to the structure of (REF ) and to make this feature more apparent we rewrite (REF ) as $\\left[ \\mathcal {V}^{DW} + \\sum _{i=1}^L \\mathcal {Q}^{DW}_i \\frac{\\partial ^{L-1}}{\\partial x_i^{L-1}} \\right] \\bar{Z} (\\bar{X}^{1,L} ) = 0 \\; ,$ where $\\mathcal {V}^{DW} &=& \\sum _{i=1}^{L} \\bar{a}(x_i , y_i) \\nonumber \\\\\\mathcal {Q}^{DW}_i &=& - \\frac{1}{(L-1)!}",
"\\prod _{j=1}^{L} \\bar{a}(x_i , y_j) \\prod _{\\stackrel{j=1}{j \\ne i}}^{L} \\frac{\\bar{a}(x_j , x_i)}{\\bar{b}(x_j , x_i)} \\; .$ Here we intend to translate (REF ) into a system of first order partial differential equations in the same lines of Section REF .",
"Since there are no significant modifications compared to the reduction of order used for Eq.",
"(REF ), we restrict ourselves to presenting only the final result.",
"Lemma 4 Let $\\vec{\\phi }$ be the following $L(L-2)+1$ dimensional vector $\\vec{\\phi } = \\left( \\begin{matrix}\\phi ^{(0)} \\\\ \\phi ^{(1)} \\\\ \\vdots \\\\ \\phi ^{(L-2)}\\end{matrix} \\right) \\; ,$ where $\\phi ^{(0)} = \\phi _0 = \\bar{Z} (\\bar{X}^{1,L} )$ and the remaining entries are $L$ -dimensional column vectors with components $\\phi _i^{(1)} = \\partial _i \\phi _0$ and $\\phi _i^{(k)} = \\partial _i \\phi _i^{(k-1)}$ for $k > 1$ .",
"Then Eq.",
"(REF ) is equivalent to $\\mathrm {\\Upsilon }_{DW} \\vec{\\phi } = 0$ with $\\mathrm {\\Upsilon }_{DW}$ being obtained from $\\mathrm {\\Upsilon }$ given in (REF ) under the mappings $\\Delta _{L-1} \\mapsto 0$ , $\\mathcal {V}^{(n)} \\mapsto \\mathcal {V}^{DW}$ , $\\mathcal {Q}_i^{(n)} \\mapsto \\mathcal {Q}_i^{DW}$ and $n \\mapsto L$ .",
"Direct comparison of (REF ), (REF ) and (REF )."
]
] | 1403.0425 |
[
[
"Imaging the environment of a z = 6.3 submillimeter galaxy with SCUBA-2"
],
[
"Abstract We describe a search for submillimeter emission in the vicinity of one of the most distant, luminous galaxies known, HerMES FLS3 at z=6.34, exploiting it as a signpost to a potentially biased region of the early Universe, as might be expected in hierarchical structure formation models.",
"Imaging to the confusion limit with the innovative, wide-field submillimeter bolometer camera, SCUBA-2, we are sensitive to colder and/or less luminous galaxies in the surroundings of HFLS3.",
"We use the Millennium Simulation to illustrate that HFLS3 may be expected to have companions if it is as massive as claimed, but find no significant evidence from the surface density of SCUBA-2 galaxies in its vicinity, or their colors, that HFLS3 marks an over-density of dusty, star-forming galaxies.",
"We cannot rule out the presence of dusty neighbours with confidence, but deeper 450-um imaging has the potential to more tightly constrain the redshifts of nearby galaxies, at least one of which likely lies at z>~5.",
"If associations with HFLS3 can be ruled out, this could be taken as evidence that HFLS3 is less biased than a simple extrapolation of the Millennium Simulation may imply.",
"This could suggest either that it represents a rare short-lived, but highly luminous, phase in the evolution of an otherwise typical galaxy, or that this system has suffered amplification due to a foreground gravitational lens and so is not as intrinsically luminous as claimed."
],
[
"Introduction",
"Dust extinction and a profusion of less luminous foreground galaxies makes it difficult to select high-redshift ultraluminous star-forming galaxies ($L_{\\rm IR}\\ge 10^{12}$ L$_\\odot $ ) at rest-frame ultraviolet/optical wavelengths.",
"Although extinction is not an issue at radio wavelengths, an unfavourable $K$ -correction works against detecting the highest redshift examples, $z\\gg 3$ .",
"Since the advent of large-format submillimeter (submm) cameras such as the Submillimeter Common-User Bolometer Array [19], however, it has been possible to exploit the negative $K$ -correction in the submm waveband to select dusty, star-forming galaxies (submm-selected galaxies, or SMGs) almost independently of their redshift [16], [2].",
"The scope of this field has been substantially expanded by Herschel [28] which has surveyed approximately a hundred square degrees of extragalactic sky to the confusion limit at 500 $\\mu $ m [26], with simultaneous imaging at 250 and 350 $\\mu $ m, using the SPIRE instrument [18].",
"A SPIRE image of the Spitzer First Look Survey (FLS) field, obtained as part of the Herschel Multi-Tiered Extragalactic Survey [27], led to the discovery of 1HERMES S350 J170647.8+584623 [32], [15] as an unusually red SPIRE source with $S_{250}<S_{350}<S_{500}$ , i.e.",
"with its thermal dust peak within or beyond the 500-$\\mu $ m band [12], [9], [31].",
"Some of these “500-$\\mu $ m risers” are in fact due to synchrotron emission from bright, flat-spectrum radio quasars [24], but HFLS3 does not exhibit such powerful AGN-driven radio emission.",
"Panchromatic spectral-line observations place HFLS3 at $z=6.34$ via the detection of H$_2$ O, CO, OH, OH$^+$ , NH$_3$ , [C i] and [C ii] emission and absorption lines.",
"Its continuum spectral energy distribution (SED) is consistent with a characteristic dust temperature, $T_{\\rm d}$ = 56 , and a dust mass of $1.1\\times 10^9$ M$_\\odot $ .",
"Its infrared luminosity, $L_{\\rm IR}$ = $2.9\\times 10^{13}$ L$_\\odot $ , suggests a star-formation rate (SFR) of $2900\\,\\mu _{\\rm L}^{-1}$ M$_\\odot $ yr$^{-1}$ for a [6] initial mass function, where the lensing magnification suffered by HFLS3 due to a foreground galaxy less than 2$^{\\prime \\prime }$ away has been estimated to be in the range $\\mu _{\\rm L}=1.2$ –1.5 [32].",
"It is expected that the most massive galaxies found at very high redshifts grew in (and thus signpost) the densest peaks in the early Universe, making them useful tracers of distant proto-clusters.",
"Above $z \\sim 6$ , such sources may also contribute to the rapid evolution of the neutral fraction of the Universe, during the so-called `era of reionisation', and to the earliest phase of enrichment of the interstellar medium in galaxies, less than 1 Gyr after the Big Bang.",
"They may also host the highest redshift quasars.",
"In the submm regime, to explore distant galaxies and their environments we have observed radio galaxies and quasars, typically detecting factor $\\sim 2$ –4$\\times $ over-densities of submm companions around these signposts [22], [37], [35], [33], [30].",
"Here, we continue this tradition, targeting the most distant known submm galaxy, HFLS3 at $z= 6.34$ , with the 10,000-pixel SCUBA-2 bolometer camera [20], which is more sensitive than Herschel to cold dust in high-redshift galaxies.",
"In § we describe our SCUBA-2 observations of the field surrounding HFLS3, after its discovery with SPIRE aboard Herschel, and our reduction of those data.",
"In § we analyze the surface density of SCUBA-2 galaxies in the field, and their color, and discuss whether there is any evidence that HFLS3 inhabits an over-dense region of the Universe, as might be expected in hierarchical structure-formation models [25], [34].",
"We finish with our conclusions in §.",
"Throughout, we adopt a cosmology with $H_0 =71$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _{\\rm m}=0.27$ and $\\Omega _\\Lambda = 0.73$ , so 1$^{\\prime \\prime }$ equates to 5.7 kpc at $z=6.34$ .",
"Data were obtained simultaneously at 450 and 850 $\\mu $ m in 2011 September 23-24 and 2012 March 14 (project M11BGT01) using SCUBA-2 on the 15-m James Clerk Maxwell Telescope (JCMT).",
"The observations were taken with the constant speed daisy pattern, which provides uniform exposure-time coverage in the central 3$^\\prime $ -diameter region of a field, and useful coverage over 12$^\\prime $ .",
"A total of 6.9 hr was spent integrating on HFLS3.",
"Observing conditions were good or excellent, with precipitable-water-vapor levels typically 1 mm or less, corresponding to a 225-GHz optical depth of 0.05.",
"The data were calibrated in flux density against the primary calibrators Uranus and Mars, and also secondary sources CRL 618 and CRL 2688 from the JCMT calibrator list [14], with estimated calibration uncertainties amounting to 5% at 850 $\\mu $ m and 10% at 450 $\\mu $ m. The data were reduced using the Dynamic Iterative Map-Maker within the starlink smurf package [7] called from the orac-dr automated pipeline [5].",
"The chosen recipe accounted for attenuation of the signal as a result of time-series filtering and removed residual low-frequency noise from the map using a jack-knife method.",
"The maps were made using inverse-variance weighting, with 1$^{\\prime \\prime }$ pixels at both wavelengths, before application of a matched filter [8].",
"The map-maker used a `blank field' configuration file, optimized for faint, unresolved or compact sources.",
"This applies a high-pass filter with a spatial cutoff of 200$^{\\prime \\prime }$ , corresponding to about 0.8 Hz for a typical scanning speed of 155$^{\\prime \\prime }$ s$^{-1}$ .",
"This removes the majority of low-frequency (large spatial scale) noise, while the remainder is removed using a Fourier-space whitening filter.",
"This is derived from the power spectrum of the central 9$^\\prime $ region of a jack-knife map, produced from two independent halves of the total dataset.",
"This filtering attenuates the peak signal of sources in the map.",
"To estimate the magnitude of this effect, the pipeline re-makes each map with a fake 10-Jy Gaussian added to the raw data, offset from the nominal map centre by 30$^{\\prime \\prime }$ to avoid contamination by any target at the map centre.",
"The amplitude of the fake Gaussian in the output map is measured to determine a correction factor.",
"The standard flux conversion factor (FCF), as determined from observations of primary and secondary calibrators, is then multiplied by this factor (1.17 and 1.15 at 850 and 450 $\\mu $ m, respectively) and applied to the final image to give a map calibrated in Jy beam$^{-1}$ .",
"The maps with the fake Gaussian are also used to form the point spread function (PSF) for the matched filter since they reflect the effective point-source transfer function of the map-maker.",
"The SCUBA-2 850- and 450-$\\mu $ m images shown in Fig.",
"REF reach noise levels of 0.9 and 5.0 mJy beam$^{-1}$ over the central 3$^\\prime $ -diameter regions, yielding detections of HFLS3 at approximately the 41- and 7-$\\sigma $ levels, respectively.",
"At 850 $\\mu $ m, the central 67.2 arcmin$^2$ of the map has a noise level of 1.5 mJy beam$^{-1}$ or better.",
"The astrometry of the SCUBA-2 images was found to be accurate to better than 1$^{\\prime \\prime }$ by stacking at the positions of 3.6-$\\mu $ m and 1.4-GHz sources in the field.",
"Following [17], we create a catalogue of sources from the 450- and 850-$\\mu $ m images by searching for peaks in the beam-convolved signal-to-noise ratio maps, recording their coordinates, flux densities and local noise levels.",
"We then mask a region $1.5\\times $ the beam size and then repeat the search.",
"Above a signal-to-noise level of 3.75 the contamination rate due to false detections is below 5%.",
"We adopt this as our detection threshold, listing the 26 sources with 850-$\\mu $ m flux density uncertainties below 1.5 mJy in Table REF , alongside 450-$\\mu $ m sources selected from the same area at the same significance threshold.",
"We calculate our completeness levels and flux boosting following [17], who followed [39], injecting $10^5$ artificial point sources into a map with the same noise properties as the real image.",
"We correct for false positives using the jack-knife map.",
"lcc Sources detected at 850 and 450 $\\mu $ m near HFLS3.",
"0pt IAU name $S$ /mJya S/N 850 $\\mu $ m: S2FLS850 J170647.67+584623.0b $35.4\\pm 0.9$ 40.9 S2FLS850 J170631.07+584812.9 $11.4\\pm 1.3$ 8.6 S2FLS850 J170621.93+584826.8 $9.3\\pm 1.4$ 6.5 S2FLS850 J170646.64+584816.0 $5.6\\pm 1.0$ 5.5 S2FLS850 J170647.80+584735.0 $4.9\\pm 0.9$ 5.3 S2FLS850 J170701.41+584318.0 $7.5\\pm 1.4$ 5.3 S2FLS850 J170625.54+584725.9 $6.3\\pm 1.3$ 4.8 S2FLS850 J170642.92+584456.0 $4.7\\pm 1.0$ 4.6 S2FLS850 J170717.24+584535.8 $5.8\\pm 1.3$ 4.6 S2FLS850 J170659.77+584859.0 $5.6\\pm 1.2$ 4.6 S2FLS850 J170723.05+584719.7 $6.1\\pm 1.4$ 4.5 S2FLS850 J170642.00+585004.0 $5.4\\pm 1.2$ 4.5 S2FLS850 J170630.54+585015.9 $6.2\\pm 1.4$ 4.4 S2FLS850 J170710.97+584811.9 $5.4\\pm 1.3$ 4.2 S2FLS850 J170653.32+584246.0 $5.5\\pm 1.3$ 4.2 S2FLS850 J170713.78+584618.8 $4.9\\pm 1.2$ 4.2 S2FLS850 J170638.27+585009.0 $5.3\\pm 1.3$ 4.2 S2FLS850 J170710.54+584403.9 $6.1\\pm 1.5$ 4.1 S2FLS850 J170724.30+584514.7 $5.9\\pm 1.5$ 4.0 S2FLS850 J170700.38+584225.0 $5.9\\pm 1.5$ 4.0 S2FLS850 J170715.48+584859.8 $5.4\\pm 1.3$ 4.0 S2FLS850 J170614.39+584458.7 $6.0\\pm 1.5$ 4.0 S2FLS850 J170641.49+585039.0 $5.3\\pm 1.3$ 4.0 S2FLS850 J170637.26+584616.0 $3.9\\pm 1.0$ 3.9 S2FLS850 J170633.28+584430.0 $5.2\\pm 1.3$ 3.8 S2FLS850 J170654.87+584614.0 $3.3\\pm 0.9$ 3.8 450 $\\mu $ m: S2FLS450 J170647.80+584620.0b $39.8\\pm 5.5$ 7.3 S2FLS450 J170701.05+584715.0 $29.4\\pm 6.9$ 4.3 S2FLS450 J170631.33+584813.9 $36.3\\pm 8.8$ 4.1 S2FLS450 J170636.12+584224.0 $32.5\\pm 8.3$ 3.9 S2FLS450 J170658.59+584419.0 $30.2\\pm 7.8$ 3.9 S2FLS450 J170711.99+584734.9 $34.5\\pm 8.9$ 3.9 aDeboosted flux densities; errors exclude 5 and 10% calibration uncertainties at 850 and 450 $\\mu $ m, respectively.",
"bHFLS3."
],
[
"The acquisition and reduction of 16.8 hrs of Herschel SPIRE and (shallow) PACS data for the FLS field (OD159, 164) as part of HerMES is described in detail by [27].",
"The SPIRE data, which are confusion limited, are shown as a three-color image in Fig.",
"REF .",
"We have obtained much deeper data from PACS [29] via a 3.9-hr integration as part of programme ot2_driecher_3 (OD1329) [32].",
"Observations were carried out on 2013 January 01 in mini-scan mapping mode ($4\\times 15$ repeats), using the 70- plus 160-$\\mu $ m parallel mode and the 110- plus 160-$\\mu $ m parallel mode for one orthogonal cross-scan pair each.",
"In the 70-, 110-, and 160-$\\mu $ m bands, the r.m.s.",
"sensitivities at the position of HFLS3 are 0.67, 0.73, and 1.35 mJy beam$^{-1}$ , respectively.",
"Data reduction and mosaicing were carried out using standard procedures.",
"The absolute flux density scale is accurate to 5%.",
"The 160-$\\mu $ m PACS image, the only one potentially useful in the context of faint, distant galaxies, is shown in Fig.",
"REF .",
"The 250-, 350- and 500-$\\mu $ m flux densities, $S_{250}$ , $S_{350}$ and $S_{500}$ , at the positionsPositions are known to $\\sigma _{\\rm pos}\\approx 2.2^{\\prime \\prime }$ even for the least significant SMGs, a small fraction of the beam-convolved SPIRE point spread function.",
"of the 19 SMGs discussed in §REF were determined using beam-convolved SPIRE maps.",
"None of our SMGs lie near bright SPIRE sources so we expect the uncertainties associated with these flux densities should be close to the typical confusion levels, $\\approx 6$ mJy [26]."
],
[
"Results, analysis and discussion",
"HFLS3 dominates the submm sky in the 67.2-arcmin$^2$ (8 Mpc$^2$ ) region we have mapped at 850 $\\mu $ m with SCUBA-2, being three times brighter than the next-brightest submm emitter (Fig.",
"REF ; Table REF ).",
"At 450 $\\mu $ m, HFLS3 is the brightest SMG in the region, despite the peak of its SED having moved beyond that filter; it is one of two sources detected formally at both 450 and 850 $\\mu $ m. Perhaps unsurprisingly, there are no sources in common between the SCUBA-2 and PACS images.",
"We see no evidence for an over-density of SMGs on $<$ 1.5-Mpc scales around the position of HFLS3 in either our 450 or 850-$\\mu $ m maps (Fig.",
"REF )."
],
[
"Number counts relative to blank fields",
"Although no obvious cluster of submm emitters is visible near HFLS3 in Fig.",
"REF , we must ask whether the entire 8-Mpc$^2$ field might be over-dense in SMGs?",
"Fig.",
"REF shows the density of sources brighter than $S_\\nu $ at 450 and 850 $\\mu $ m – extracted at the 3.75-$\\sigma $ level and corrected for incompleteness using the analysis discussed in §REF , excluding HFLS3 itself – relative to the source density seen in typical blank-field surveys, where the same techniques have been used to construct catalogues and correct for incompleteness [11], [17].",
"The only hint of an over-density comes in the 850-$\\mu $ m bin at 5 mJy, but a $\\approx 2$ -$\\sigma $ deviation is not unusual when plotting seven points.",
"Simplifying matters, then, by taking only one large bin above our detection threshold at 850 $\\mu $ m, we find 26 sources (25 if we ignore HFLS3) against an expectation of 20 from [11] – on the face of it, a $\\approx $ 1-$\\sigma $ over-density.",
"When we take into account the errors associated with the [11] counts, we expect to see 26 (25) sources in a random field of the same size 13% (19%) of the time, so the result obtained intuitively from studying Fig.",
"REF is confirmed – any excess is statistically unconvincing.",
"No significant over-density of SMGs is apparent in the vicinity of HFLS3, at least not on Mpc scales at flux densities above the JCMT confusion limit."
],
[
"Redshift constraints for sources in the field",
"The $K$ -correction in the submm waveband means our SCUBA-2 maps are sensitive to SMGs across a very wide redshift range, reducing the contrast of any potential structure around HFLS3.",
"However, with SCUBA-2, SPIRE and PACS photometry in hand, we are able to crudely constrain the likely redshifts of the galaxies detected in the field surrounding HFLS3, using their far-infrared/submm colors.",
"Fig.",
"REF shows color–color plots for HFLS3 and its neighbouring SMGs, designed to exploit information from SCUBA-2 at 850 $\\mu $ m to constrain the redshifts of galaxies at $z>2$ [23], probing their colors across the rest-frame $\\approx $ 100-$\\mu $ m bump seen in the SEDs of all dusty, star-forming galaxies.",
"The colored backgrounds in the upper and lower panels of Fig.",
"REF indicate the typical redshift of the subset of $10^7$ model SEDs that fall in each pixel, where we have adopted a flat redshift distribution ($z=0$ –7), a flat distribution for the spectral dependence of the dust emissivity ($\\beta =1.8$ –2.0, centered on the $\\beta $ measured for HFLS3) and 10% flux density uncertainties.",
"For the upper panel of Fig.",
"REF we adopt the dust temperature of HFLS3 ($T_{\\rm d}$ = 56 K).",
"We concentrate only on those galaxies detected by SCUBA-2, since SPIRE-detected galaxy with a typical SED in the vicinity of HFLS3 could not have evaded detection at 850 $\\mu $ m. Despite its relatively high $T_{\\rm d}$ , HFLS3 is the reddest source detected in the three bands used to make Fig.",
"REF .",
"For SMGs with SPIRE flux densities below $2\\sigma _{\\rm conf}$ , we plot limits based on the measured flux density (zero, if negative) plus $\\sigma _{\\rm conf}$ .",
"We have placed those sources without detections at 350 and 500 $\\mu $ m at $S_{350}/S_{500} = 2$ , arbitrarily; some could be considerably redder than this in $S_{350}/S_{500}$ , but we cannot constrain this color reliably with the relatively shallow Herschel data at our disposal.",
"Several SMGs may also be as red as HFLS3 in $S_{850}/S_{500}$ , perhaps redder.",
"One particularly interesting example is S2FLS850 J170647.80+584735.0, a 5.3-$\\sigma $ SCUBA-2 source with no significant SPIRE emission.",
"With $S_{850}/S_{500}>0.9$ , this SMG likely lies at $z>5$ , with a lower $T_{\\rm d}$ and luminosity than HFLS3.",
"The lower panel of Fig.",
"REF shows the effect of lowering $T_{\\rm d}$ , illustrating an issue long-known to hamper studies of this kind: far-infrared/submm colors are sensitive only to $(1+z)/T_{\\rm d}$ [1], i.e.",
"redshift and $T_{\\rm d}$ are degenerate.",
"As a result, our current data does not allow us to conclude with certainty that the environment surrounding HFLS3 contains other luminous, dusty starbursts; neither can we rule it out.",
"Single-dish imaging of this field at 450, 850, 1100 and 2000 $\\mu $ m is possible from the ground, reaching $\\sigma _{450}\\sim 2.5$ mJy and $\\sigma _{2000}\\sim 0.1$ mJy over tens of arcmin$^2$ with existing facilities in a few tens of hours.",
"Would these data be capable of further constraining the redshifts of the SMGs discovered here?",
"Fig.",
"REF shows $S_{850}/S_{450}$ versus $S_{2000}/S_{850}$ and we see that the latter color offers little insight.",
"For $S_{850}/S_{450}\\mathrel {\\scriptstyle <}\\hspace{-6.00006pt}{\\scriptstyle \\sim }$ 0.3$ we can rule out $ z$\\scriptstyle >$$\\scriptstyle \\sim $ 2$ for all but thewarmest dust; $ S850/S450$\\scriptstyle >$$\\scriptstyle \\sim $ 1$ suggests $ z$\\scriptstyle >$$\\scriptstyle \\sim $ 5$, even for $ Td$\\ = 35\\,K, with significantly cooler dust unlikely in this redshiftregime.",
"Deeper 450-$$m imaging would therefore be useful.$ Figure: S 850 /S 450 S_{850}/S_{450} versus S 2000 /S 850 S_{2000}/S_{850}, ratios for whichdeep, relatively unconfused and unbiased data can be obtained fromthe ground, covering tens of square arcminutes.",
"The observed colorsof HFLS3 are shown.",
"The colored background indicates the typicalredshift of the subset of 10 7 10^7 model SEDs that fall in each pixel,where we have adopted the same dust parameters asFig. .",
"Inset: Same plot, having changed only T d T_{\\rm d},to 35 K, illustrating the degeneracy between redshift and T d T_{\\rm d}.S 850 /S 450 <∼S_{850}/S_{450}\\mathrel {\\scriptstyle <}\\hspace{-6.00006pt}{\\scriptstyle \\sim }0.3suggests suggests z<\\scriptstyle <∼\\scriptstyle \\sim 2forallbutthewarmestdust; for all but the warmestdust; S850/S450>\\scriptstyle >∼\\scriptstyle \\sim 1suggests suggests z>\\scriptstyle >∼\\scriptstyle \\sim 5,withdustmuchcoolerthan35Kunlikelyattheseredshifts;, with dust muchcooler than 35\\,K unlikely at these redshifts; S2000/S850offerslessinsight.offers less insight."
],
[
"Predictions from the Millennium Simulation",
"Is the field surrounding HFLS3 less over-dense in submm sources than expected for such a massive galaxy living in a biased environment at high redshift, similar to those found around high-redshift radio galaxies and radio-loud quasars at $z = 2$ –4 [37], [36]?",
"The answer to this question may have implications for the potential gravitational amplification suffered by HFLS3 (see §1) or for investigating the potential presence of a buried AGN and its role in supporting its high IR luminosity.",
"By necessity this comparison will be crude.",
"We therefore selected the implementation of the [3] galaxy-formation recipe in the Millennium Simulation [34] and searched the $z=6.2$ output for galaxies with a total baryonic mass in excess of $1.3\\times 10^{11}$ M$_{\\odot }$ , consistent with the combined mass of gas and stars estimated for HFLS3 [32].",
"By adopting a total baryonic mass cut we are less sensitive to details of the early star-formation histories of galaxies in the model.",
"We find just one galaxy in the $3.2\\times 10^8$ Mpc$^3$ volume at $z=6.2$ with a total baryonic mass above $1.3\\times 10^{11}$ M$_{\\odot }$ .",
"All its baryonic mass is in stars; it hosts a $2\\times 10^8$ -M$_{\\odot }$ black hole and is the central galaxy of a $6\\times 10^{12}$ -M$_{\\odot }$ halo, $4\\times $ more massive than the next most massive galaxy's halo within the volume, and the optimal environment to find merging galaxies according to the simulations of [21].",
"Another 16 galaxies are spread across a $\\sim $ 0.7-comoving-Mpc-diameter region around the most massive galaxy, but most of these are dwarf galaxies with baryonic masses, $\\mathrel {\\scriptstyle <}\\hspace{-6.00006pt}{\\scriptstyle \\sim }$ 109$\\,M$$.",
"Inside a sphere with an angular size of 9$ '$diameter, centered on the $ 1.31011$-M$$ galaxy, onlytwo galaxies have total baryonic masses of $$\\scriptstyle >$$\\scriptstyle \\sim $$\\,15\\% of the mass ofHFLS3 (we choose this limit as the faintest submm emitters in thisfield have 850-$$m flux densities of around 15\\% that of HFLS3);the next most massive galaxy has half this mass.",
"The total masses ofthese two companion galaxies in stars and gas are approximately 3 and$ 4 1010$\\,M$$ and their predicted $ KVega$magnitudes are 23.5 and 25.0.",
"The HFLS3 clone is predicted to have$ KVega=22.5$ for a distance modulus of 49.0, about a magnitudefainter than observed.$ From this simple theoretical comparison, we thus expect $2\\pm 2$ detectable galaxies in the vicinity of HFLS3 if, as expected, their starburst lifetimes are a significant fraction of the time available at this early epoch.",
"This is consistent with the fact that some high-redshift SMGs do have submm-bright companions [13] while others have Lyman-break galaxies nearby but no submm-bright companions [4].",
"Having found no clear evidence for or against the level of over-density expected in simulations, we can draw no strong conclusions regarding the likely gravitational amplification suffered by HFLS3, or for the likely fraction of its luminosity provided by a buried AGN."
],
[
"Conclusions",
"We have detected the most distant, dusty starburst galaxy, HFLS3, at high significance with SCUBA-2.",
"We detect another 29 dusty galaxies within an area of 67.2 arcmin$^2$ surrounding HFLS3, most of them likely at lower redshift.",
"We find no compelling evidence, from surface density or color, for an over-density of SMGs around HFLS3, although applying similar selection criteria to theoretical models suggests that a modest excess could be expected, as is found for some other high-redshift SMGs [13].",
"We can therefore draw no strong conclusions regarding the likely gravitational amplification suffered by HFLS3, or for the likely fraction of its luminosity provided by a buried AGN."
],
[
"Acknowledgements",
"We thank John Helly for help with the Millennium Simulation.",
"RJI and IRS acknowledge support from the European Research Council (ERC) in the form of Advanced Investigator programs, cosmicism and dustygal, respectively.",
"IRS also acknowledges support from the UK's Science and Technology Facilities Council (STFC, ST/I001573/1), a Leverhulme Fellowship and a Royal Society/Wolfson Merit Award.",
"JEG acknowledges the Royal Society for support.",
"The Dark Cosmology Centre is funded by the Danish National Research Foundation.",
"The JCMT is operated by the Joint Astronomy Centre on behalf of STFC, the National Research Council of Canada and (until 31 March 2013) the Netherlands Organisation for Scientific Research.",
"Additional funds for the construction of SCUBA-2 were provided by the Canada Foundation for Innovation.",
"Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.",
"SPIRE was developed by a consortium of institutes led by Cardiff Univ.",
"(UK) and including: Univ.",
"Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ.",
"Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ.",
"Sussex (UK); and Caltech, JPL, NHSC, Univ.",
"Colorado (USA).",
"This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).",
"The JCMT and Herschel data used in this paper can be obtained from the JCMT archive (www.jach.hawaii.edu/JCMT/archive) and the Herschel Database in Marseille, HeDaM, (hedam.oamp.fr/HerMES), respectively.",
"Facilities: JCMT, Herschel"
]
] | 1403.0247 |
[
[
"Solving fuzzy two-point boundary value problem using fuzzy Laplace\n transform"
],
[
"Abstract A natural way to model dynamic systems under uncertainty is to use fuzzy boundary value problems (FBVPs) and related uncertain systems.",
"In this paper we use fuzzy Laplace transform to find the solution of two-point boundary value under generalized Hukuhara differentiability.",
"We illustrate the method for the solution of the well known two-point boundary value problem Schrodinger equation, and homogeneous boundary value problem.",
"Consequently, we investigate the solutions of FBVPs under as a new application of fuzzy Laplace transform."
],
[
"Introduction",
"“The theory of fuzzy differential equations (FDEs) has attracted much attention in recent years because this theory represents a natural way to model dynamical systems under uncertainty”, Jamshidi and Avazpour [1].The concept of fuzzy set was introduce by Zadeh in 1965 [2].",
"The derivative of fuzzy-valued function was introduced by Chang and Zadeh in 1972 [3].",
"The integration of fuzzy valued function is presented in [9].",
"Kaleva and Seikala presented fuzzy differential equations (FDEs) in [4], [5].",
"Many authors discussed the applications of FDEs in [6], [7], [8].",
"Two-point boundary value problem is investigated in [10].",
"In case of Hukuhara derivative the funding Green's function helps to find the solution of boundary value problem of first order linear fuzzy differential equations with impulses [11].",
"Wintner-type and superlinear-type results for fuzzy initial value problems (FIVPs) and fuzzy boundary value problems (FBVPs) are presented in [12].",
"The solution of FBVPs must be a fuzzy-valued function under the Hukuhara derivative [13], [14], [15], [16], [17].",
"Also two-point boundary value problem (BVP) is equivalent to fuzzy integral equation [18].",
"Recently in [19], [20], [21] the fuzzy Laplace transform is applied to find the analytical solution of FIVPs.",
"According to [22] the fuzzy solution is different from the crisp solution as presented in [13], [14], [15], [16], [23], [24].",
"In [22] they solved the $Schr\\ddot{o}dinger$ equation with fuzzy boundary conditions.",
"Further in [19] it was discussed that under what conditions the fuzzy Laplace transform (FLT) can be applied to FIVPs.",
"For two-point BVP some of the analytical methods are illustrated in [22], [25], [26] while some of the numerical methods are presented in [1], [27].",
"But every method has its own advantages and disadvantages for the solution of such types of fuzzy differential equation (FDE).",
"In this paper we are going to apply the FLT on two-point BVP [22].",
"Moreover we investigate the solution of second order $Schr\\ddot{o}dinger$ equation and other homogeneous boundary value problems [22].",
"After applying the FLT to BVP we replace one or more missing terms by any constant and then apply the boundary conditions which eliminates the constants.",
"The crisp solution of fuzzy boundary value problem (FBVP) always lies between the upper and lower solutions.",
"If the lower solution is not monotonically increasing and the upper solution is not monotonically decreasing then the solution of the FDE is not a valid level set.",
"This paper is organized as follows: In section 2, we recall some basics definitions and theorems.",
"FLT is defined in section 3 and in this section the FBVP is briefly reviewed.",
"In section 4, constructing solution of FBVP by FLT is explained.",
"To illustrate the method, several examples are given in section 5.",
"Conclusion is given in section 6."
],
[
"Basic concepts",
"In this section we will recall some basics definitions and theorems needed throughout the paper such as fuzzy number, fuzzy-valued function and the derivative of the fuzzy-valued functions.",
"Definition 2.1 A fuzzy number is defined in [2] as the mapping such that $u:R\\rightarrow [0,1]$ , which satisfies the following four properties $u$ is upper semi-continuous.",
"$u$ is fuzzy convex that is $u(\\lambda x+(1-\\lambda )y) \\ge \\min {\\lbrace u(x), u(y)\\rbrace }, x, y\\in R$ and $\\lambda \\in [0,1]$ .",
"$u$ is normal that is $\\exists $ $x_0\\in R$ , where $u(x_0)=1$ .",
"$A=\\lbrace \\overline{x \\in \\mathbb {R}: u(x)>0}\\rbrace $ is compact, where $\\overline{A}$ is closure of $A$ .",
"Definition 2.2 A fuzzy number in parametric form given in [3], [4], [5] is an order pair of the form $u=(\\underline{u}(r), \\overline{u}(r))$ , where $0\\le r\\le 1$ satisfying the following conditions.",
"$\\underline{u}(r)$ is a bounded left continuous increasing function in the interval $[0,1]$ .",
"$\\overline{u}(r)$ is a bounded left continuous decreasing function in the interval $[0,1]$ .",
"$\\underline{u}{(r)\\le \\overline{u}(r)}$ .",
"If $\\underline{u}(r)=\\overline{u}(r)=r$ , then $r$ is called crisp number.",
"Now we recall a triangular fuzzy from [2], [19], [20] number which must be in the form of $u=(l, c, r),$ where $l,c,r\\in R$ and $l\\le c\\le r$ , then $\\underline{u}(\\alpha )=l+(c-r)\\alpha $ and $\\overline{u}(\\alpha )=r-(r-c)\\alpha $ are the end points of the $\\alpha $ level set.",
"Since each $y\\in R$ can be regarded as a fuzzy number if $\\widetilde{y}(t)={\\left\\lbrace \\begin{array}{ll}1, \\;\\;\\; if \\;\\; y=t,\\\\ 0, \\;\\;\\; if \\;\\; 1\\ne t.\\end{array}\\right.",
"}$ For arbitrary fuzzy numbers $u=(\\underline{u}(\\alpha ), \\overline{u}(\\alpha ))$ and $v=(\\underline{v}(\\alpha ), \\overline{v}(\\alpha ))$ and an arbitrary crisp number $j$ , we define addition and scalar multiplication as: $(\\underline{u+v})(\\alpha )=(\\underline{u}(\\alpha )+\\underline{v}(\\alpha ))$ .",
"$(\\overline{u+v})(\\alpha )=(\\overline{u}(\\alpha )+\\overline{v}(\\alpha ))$ .",
"$(j\\underline{u})(\\alpha )=j\\underline{u}(\\alpha )$ , $(j\\overline{u})(\\alpha )=j\\overline{u}(\\alpha )$ $j\\ge 0$ .",
"$(j\\underline{u})(\\alpha )=j\\overline{u}(\\alpha )\\alpha , (j\\overline{u})(\\alpha )=j\\underline{u}(\\alpha )\\alpha $ , $j<0$ .",
"Definition 2.3 (See Salahshour & Allahviranloo, and Allahviranloo & Barkhordari [19], [20]) Let us suppose that x, y $\\in E$ , if $\\exists $ $z\\in E$ such that $x=y+z$ .",
"Then, $z$ is called the H-difference of $x$ and $y$ and is given by $x\\ominus y$ .",
"Remark 2.4 (see Salahshour & Allahviranloo [19]).",
"Let $X$ be a cartesian product of the universes, $X_1$ , $X_1, \\cdots , X_n$ , that is $X=X_1 \\times X_2 \\times \\cdots \\times X_n$ and $A_{1},\\cdots ,A_{n}$ be $n$ fuzzy numbers in $X_1, \\cdots , X_n$ respectively.",
"Then, $f$ is a mapping from $X$ to a universe $Y$ , and $y=f(x_{1},x_{2},\\cdots ,x_{n})$ , then the Zadeh extension principle allows us to define a fuzzy set $B$ in $Y$ as; $B=\\lbrace (y, u_B(y))|y=f(x_1,\\cdots ,x_{n}),(x_{1},\\cdots ,x_{n})\\in X\\rbrace ,$ where $u_B(y)={\\left\\lbrace \\begin{array}{ll}\\sup _{{(x_1,\\cdots ,x_n)} \\in f^{-1}(y)} \\min \\lbrace u_{A_1}(x_1),\\cdots u_{A_n}(x_n)\\rbrace , \\;\\;\\; if \\;\\;\\; f^{-1}(y)\\ne 0,\\\\ 0, \\;\\;\\;\\; otherwise,\\end{array}\\right.",
"}$ where $f^{-1}$ is the inverse of $f$ .",
"The extension principle reduces in the case if $n=1$ and is given as follows: $B=\\lbrace (y, u_B(y)|y=f(x), \\mbox{ } x \\in X)\\rbrace ,$ where $u_B(y)={\\left\\lbrace \\begin{array}{ll}\\sup _{x\\in f^{-1}(y)} \\lbrace u_A(x)\\rbrace , \\mbox{ if } f^{-1}(y)\\ne 0,\\\\0, \\;\\;\\;\\; otherwise.",
"\\end{array}\\right.}",
"$ By Zadeh extension principle the approximation of addition of $E$ is defined by $(u\\oplus v)(x)=\\sup _{y\\in R} \\min (u(y), v(x-y))$ , $x \\in R$ and scalar multiplication of a fuzzy number is defined by $(k\\odot u)(x)={\\left\\lbrace \\begin{array}{ll}u(\\frac{x}{k}), \\;\\;\\; k > 0,\\\\ 0 \\;\\;\\; \\mbox{ otherwise }, \\end{array}\\right.}",
"$ where $\\widetilde{0}\\in E$ .",
"The Housdorff distance between the fuzzy numbers [7], [13], [19], [20] defined by $d:E\\times E\\longrightarrow R^{+}\\cup \\lbrace {0}\\rbrace ,$ $d(u,v)=\\sup _{r\\in [0,1]}\\max \\lbrace |\\underline{u}(r)-\\underline{v}(r)|, |\\overline{u}(r)-\\overline{v}(r)|\\rbrace ,$ where $u=(\\underline{u}(r), \\overline{u}(r))$ and $v=(\\underline{v}(r), \\overline{v}(r))\\subset R$ .",
"We know that if $d$ is a metric in $E$ , then it will satisfies the following properties, introduced by Puri and Ralescu [28]: $d(u+w,v+w)=d(u,v)$ , $\\forall $ u, v, w $\\in $ E. $(k \\odot u, k \\odot v)=|k|d(u, v)$ , $\\forall $ k $\\in $ R, and u, v $\\in $ E. $d(u \\oplus v, w \\oplus e)\\le d(u,w)+d(v,e)$ , $\\forall $ u, v, w, e $\\in $ E. Definition 2.5 (see Song and Wu [29]).",
"If $f:R\\times E \\longrightarrow E$ , then $f$ is continuous at point $(t_0,x_0) \\in R \\times E$ provided that for any fixed number $r \\in [0,1]$ and any $\\epsilon > 0$ , $\\exists $ $\\delta (\\epsilon ,r)$ such that $d([f(t,x)]^{r}, [f(t_{0},x_{0})]^{r}) < \\epsilon $ whenever $|t-t_{0}|<\\delta (\\epsilon , r)$ and $d([x]^{r}, [x_{0}]^{r})<\\delta (\\epsilon ,r)$ $\\forall $ t $\\in $ R, x $\\in E$ .",
"Theorem 2.6 (see Wu [30]).",
"Let $f$ be a fuzzy-valued function on $[a,\\infty )$ given in the parametric form as $(\\underline{f}(x,r), \\overline{f}(x,r))$ for any constant number $r\\in [0,1]$ .",
"Here we assume that $\\underline{f}(x,r)$ and $\\overline{f}(x,r)$ are Riemann-Integral on $[a,b]$ for every $b\\ge a$ .",
"Also we assume that $\\underline{M}(r)$ and $\\overline{M}(r)$ are two positive functions, such that $\\int _a^b|\\underline{f}(x,r)| dx \\le \\underline{M}(r)$ and $\\int _a^b |\\overline{f}(x,r)| dx \\le \\overline{M}(r)$ for every $b\\ge a$ , then $f(x)$ is improper integral on $[{a}, \\infty )$ .",
"Thus an improper integral will always be a fuzzy number.",
"In short $ \\int _a^r f(x) dx = ( \\int _a^b|\\underline{f}(x,r)| dx, \\int _a^b |\\overline{f}(x,r)| dx).$ It is will known that Hukuhare differentiability for fuzzy function was introduced by Puri & Ralescu in 1983.",
"Definition 2.7 (see Chalco-Cano and Román-Flores [31]).",
"Let $f:(a,b)\\rightarrow E$ where $x_{0}\\in (a,b)$ .",
"Then, we say that $f$ is strongly generalized differentiable at $x_0$ (Beds and Gal differentiability).",
"If $\\exists $ an element $f^{\\prime }(x_0)\\in E$ such that $\\forall h>0$ sufficiently small $\\exists $ $f(x_0+h)\\ominus f(x_0)$ , $f(x_0)\\ominus f(x_0-h)$ , then the following limits hold (in the metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f(x_0+h)\\ominus f(x_0)}{h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0)\\ominus f(x_0-h)}{h}=f^{\\prime }(x_0)$ , Or $\\forall h>0$ sufficiently small, $\\exists $ $f(x_0)\\ominus f(x_0+h)$ , $f(x_0-h)\\ominus f(x_0)$ , then the following limits hold (in the metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f(x_0)\\ominus f(x_0+h)}{-h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0-h)\\ominus f(x_0)}{-h}=f^{\\prime }(x_0)$ , Or $\\forall h>0$ sufficiently small $\\exists $ $f(x_0+h)\\ominus f(x_0)$ , $f(x_0-h)\\ominus f(x_0)$ and the following limits hold (in metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{(x_0+h)\\ominus f(x_0)}{h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0-h)\\ominus f(x_0)}{-h}=f^{\\prime }(x_0)$ , Or $\\forall h>0$ sufficiently small $\\exists $ $f(x_0)\\ominus f(x_0+h)$ , $f(x_0)\\ominus f(x_0-h)$ , then the following limits holds(in metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f(x_0)\\ominus f(x_0+h)}{-h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0-h)\\ominus f(x_0)}{h}=f^{\\prime }(x_0)$ .",
"The denominators $h$ and $-h$ denote multiplication by $\\frac{1}{h}$ and $\\frac{-1}{h}$ respectively.",
"Theorem 2.8 (Ses Chalco-Cano and Román-Flores [31]).",
"Let $f:R\\rightarrow E$ be a function denoted by $f(t)=(\\underline{f}(t,r),\\overline{f}(t,r))$ for each $r\\in [0,1]$ .",
"Then If $f$ is $(i)$ -differentiable, then $\\underline{f}(t,r)$ and $\\overline{f}(t,r)$ are differentiable functions and $f^{\\prime }(t)=(\\underline{f}^{\\prime }(t,r), \\overline{f}^{\\prime }(t,r))$ , If $f$ is $(ii)$ -differentiable, then $\\underline{f}(t,r)$ and $\\overline{f}(t,r)$ are differentiable functions and $f^{\\prime }(t)=(\\overline{f}^{\\prime }(t,r), \\underline{f}^{\\prime }(t,r))$ .",
"Lemma 2.9 (see Bede and Gal [32], [33]).",
"Let $x_0\\in R$ .",
"Then, the FDE $y^{\\prime }=f(x,y)$ , $y(x_0)=y_0\\in R$ and $f:R\\times E\\rightarrow E$ is supposed to be a continuous and equivalent to one of the following integral equations.",
"$y(x)=y_0+\\int _{x_0}^x f(t, y(t))dt \\;\\;\\; \\forall \\;\\;\\; x\\in [x_0, x_1],$ or $y(0)=y^1(x)+(-1)\\odot \\int _{x_0}^x f(t,y(t))dt\\;\\;\\; \\forall \\;\\;\\; x\\in [x_0, x_1],$ on some interval $(x_0, x_1)\\subset R$ depending on the strongly generalized differentiability.",
"Integral equivalency shows that if one solution satisfies the given equation, then the other will also satisfy.",
"Remark 2.10 (see Bede and Gal [32], [33]).",
"In the case of strongly generalized differentiability to the FDE's $y^{\\prime }=f(x,y)$ we use two different integral equations.",
"But in the case of differentiability as the definition of H-derivative, we use only one integral.",
"The second integral equation as in Lemma $2.10$ will be in the form of $y^{1}(t)=y^{1}_0\\ominus (-1)\\int _{x_0}^x f(t,y(t))dt$ .",
"The following theorem related to the existence of solution of FIVP under the generalized differentiability.",
"Theorem 2.11 Let us suppose that the following conditions are satisfied.",
"Let $R_0=[x_0, x_0+s]\\times B(y_0, q), s,q>0, y\\in E$ , where $B(y_0,q)=\\lbrace y\\in E: B(y,y_0)\\le q\\rbrace $ which denotes a closed ball in $E$ and let $f:R_0\\rightarrow E$ be continuous functions such that $D(0, f(x,y))\\le M$ , $\\forall (x,y) \\in R_0$ and $0\\in E$ .",
"Let $g:[x_0, x_0+s]\\times [0,q]\\rightarrow R$ such that $g(x, 0)\\equiv 0$ and $0\\le g(x,u)\\le M$ , $\\forall x \\in [x_0, x_0+s], 0\\le u\\le q$ , such that $g(x,u)$ is increasing in u, and g is such that the FIVP $u^{\\prime }(x)=g(x, u(x)), u(x)\\equiv 0$ on $[x_0, x_0+s].$ We have $D[f(x,y),f(x,z)\\le g(x, D(y,z))]$ ,$\\forall $ (x,y), (x, z)$\\in R_0$ and $D(y,z)\\le q.$ $\\exists d>0$ such that for $x\\in [x_0, x_0+d]$ , the sequence $y^1_n:[x_0, x_0+d]\\rightarrow E$ given by $y^1_0(x)=y_0$ , $y^1_{n+1}(x)=y_0\\ominus (-1)\\int _{x_0}^{x} f(t, y^{1}_n)dt$ defined for any $n\\in N$ .",
"Then the FIVP $y^{\\prime }=f(x,y)$ , $y(x_0)=y_0$ has two solutions that is (1)-differentiable and the other one is (2)-differentiable for $y$ .",
"$y^{1}=[x_0, x_0+r]\\rightarrow B(y_0, q)$ , where $r=\\min \\lbrace s,\\frac{q}{M},\\frac{q}{M_1},d\\rbrace $ and the successive iterations $y_0(x)=y_0$ , $y_{n+1}(x)=y_{0}+\\int _{x_0}^{x}f(t,y_{n}(t))dt$ and $y^{1}_{n+1}=y_0$ , $y^{1}_{n+1}(x)=y_0\\ominus (-1)\\int _{x_0}^{x}f(t, y^{1}_{n}(t))dt$ converge to these two solutions respectively.",
"Now according to theorem (2.11), we restrict our attention to function which are (1) or (2)-differentiable on their domain except on a finite number of points as discussed in [33]."
],
[
"Fuzzy Laplace Transform",
"Suppose that $f$ is a fuzzy-valued function and $p$ is a real parameter, then according to [19], [20] FLT of the function $f$ is defined as follows: Definition 3.1 The FLT of fuzzy-valued function is [19], [20] $\\widehat{F}(p)=L[f(t)]=\\int _{0}^{\\infty }e^{-pt}f(t)dt,$ $\\widehat{F}(p)=L[f(t)]=\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}f(t)dt,$ $\\widehat{F}(p)=[\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\\underline{f}(t)dt,\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\\overline{f}(t)dt],$ whenever the limits exist.",
"Definition 3.2 Classical Fuzzy Laplace Transform: Now consider the fuzzy-valued function in which the lower and upper FLT of the function are represented by $\\widehat{F}(p;r)=L[f(t;r)]=[l(\\underline{f}(t;r)),l(\\overline{f}(t;r))],$ where $l[\\underline{f}(t;r)]=\\int _{0}^{\\infty }e^{-pt}\\underline{f}(t;r)dt=\\lim _{\\tau \\rightarrow \\infty } \\int _{0}^{\\tau }e^{-pt}\\underline{f}(t;r)dt,$ $l[\\overline{f}(t;r)]=\\int _{0}^{\\infty }e^{-pt}\\overline{f}(t;r)dt=\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\\overline{f}(t;r)dt.",
"$"
],
[
"Fuzzy Boundary Value problem",
"The concept of fuzzy numbers and fuzzy set was first introduced by Zadeh [2].",
"Detail information of fuzzy numbers and fuzzy arithmetic can be found in [13], [14], [15].",
"In this section we review the fuzzy boundary valued problem (FBVP) with crisp linear differential equation but having fuzzy boundary values.",
"For example we consider the second order fuzzy boundary problem as [10], [11], [22], [1].",
"$\\begin{split}\\psi ^{^{\\prime \\prime }}(t)+c_1(t)\\psi ^{^{\\prime }}(t)+c_2(t)\\psi (t)=f(t),\\\\\\psi (0)=\\tilde{A},\\\\\\psi (l)=\\tilde{B}.\\end{split}$"
],
[
"Constructing Solutions Via FBVP",
"In this section we consider the following second order FBVP in general form under generalized H-differentiability proposed in [22].",
"We define $y^{\\prime \\prime }(t)=f(t, y(t),y^{\\prime }(t)), $ subject to two-point boundary conditions $y(0)=(\\underline{y}(0;r), \\overline{y}(0;r)),$ $y(l)=(\\underline{{y}}(l;r), \\overline{y}(l;r)).$ Taking FLT of (REF ) $L[y^{\\prime \\prime }(t)]=L[f(t, y(t),y^{\\prime }(t))], $ which can be written as $p^2L[y(t)]\\ominus py(0)\\ominus y^{\\prime }(0)=L[f(t, y(t),y^{\\prime }(t))].$ The classical form of FLT is given below: $\\begin{split}p^{2}l[\\underline{y}(t;r)]-p\\underline{y}(0;r)-\\underline{y}^{\\prime }(0;r)=l[\\underline{f}(t, y(0;r),y^{\\prime }(0;r))],\\end{split}$ $\\begin{split}p^{2}l[\\overline{y}(t;r)]-p\\overline{y}(0;r)-\\overline{y}^{\\prime }(0;r)=l[\\overline{f}(t, y(0;r),y^{\\prime }(0;r))].\\end{split}$ Here we have to replace the unknown value $y^{\\prime }(0,r)$ by constant $F_1$ in lower case and by $F_2$ in upper case.",
"Then we can find these values by applying the given boundary conditions.",
"In order to solve equations (REF ) and (REF ) we assume that $A(p;r)$ and $B(p;r)$ are the solutions of (REF ) and (REF ) respectively.",
"Then the above system becomes $l[\\underline{y}(t;r)]=A(p;r),$ $l[\\overline{y}(t;r)]=B(p;r).$ Using inverse Laplace transform, we get the upper and lower solution for given problem as: $[\\underline{y}(t;r)]=l^{-1}[A(p;r)],$ $[\\overline{y}(t;r)]=l^{-1}[B(p;r)].$"
],
[
"Examples",
"In this section first we consider the $Schr\\ddot{o}dinger$ equation [22] with fuzzy boundary conditions under Hukuhara differentiability.",
"Example 5.1 The $Schr\\ddot{o}dinger$ FBVP [22] is as follows: $(\\frac{h^2}{2m})u^{^{\\prime \\prime }}(x)+V(x)u(x)=Eu(x),$ where $V(x)$ is potential and is defined as $V(x)={\\left\\lbrace \\begin{array}{ll}0, \\;\\;\\; if \\;\\; x < 0,\\\\ l, \\;\\;\\; if \\;\\; x>0,\\end{array}\\right.",
"}$ subject to the following boundary conditions $u(0)=(1+r, 3-r),$ $u(l)=(4+r, 6-r).$ Now let $a=\\frac{h^2}{2m}$ , $b=E$ .",
"Then, (REF ) becomes $au^{^{\\prime \\prime }}(x)+V(x)u(x)=bu(x).$ In (REF ) for $x<0$ , we discuss (1,1) and (2,2)-differentiability while in the case $x>0$ we will discuss (1,2) and (2,1)-differentiability."
],
[
"Case-I: (1,1) and (2,2)-differentiability",
"For $x<0$ , (REF ) becomes $au^{^{\\prime \\prime }}=bu,$ $au^{^{\\prime \\prime }}-bu=0.$ Now applying FLT on both sides of equation (REF ), we get $aL[u^{^{\\prime \\prime }}(x)]-bL[u(x)]=0,$ where $L[u^{\\prime \\prime }(x)]=p^{2}L[u(x)]\\ominus pu(0)\\ominus u^{\\prime }(0).$ The classical FLT form of the above equation is $l[\\underline{u}^{\\prime \\prime }(x,r)]=p^{2}l[\\underline{u}(x,r)]-p\\underline{u}(0,r)-\\underline{u}^{\\prime }(0,r),$ $l[\\overline{u}^{\\prime \\prime }(x,r)]=p^{2}l[\\overline{u}(x,r)]-p\\overline{u}(0,r)-\\overline{u}^{\\prime }(0,r).$ Solving the above classical equations for lower and upper solutions, we have $\\begin{split}a\\lbrace p^{2}l[\\underline{u}(x,r)]-p\\underline{u}(0,r)-\\underline{u}^{\\prime }(0,r)\\rbrace -bl[\\underline{u}(x,r)]=0,\\end{split}$ or $\\begin{split}(ap^{2}-b)l[\\underline{u}(x,r)]=a\\lbrace p\\underline{u}(0,r)+\\underline{u}^{\\prime }(0,r)\\rbrace .\\end{split}$ Applying the boundary conditions, we have $\\begin{split}(ap^{2}-b)l[\\underline{u}(x,r)]=a\\lbrace p(1+r)+F_1\\rbrace ,\\end{split}$ where $F_1=\\underline{u}^{\\prime }(0,r).$ Simplifying and applying inverse Laplace we get $\\begin{split}\\underline{u}(x,r)=(\\frac{1+r}{2})l^{-1}\\lbrace \\frac{p}{p^2-\\frac{b}{a}}\\rbrace +F_1l^{-1}\\lbrace \\frac{1}{p^2-\\frac{b}{a}}\\rbrace .\\end{split}$ Using partial fraction $\\begin{split}\\underline{u}(x,r)=(\\frac{1+r}{2})\\lbrace e^{\\sqrt{\\frac{b}{a}}x}+e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace +\\frac{F_1}{2\\sqrt{\\frac{b}{a}}}\\lbrace e^{\\sqrt{\\frac{b}{a}}x}-e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace .\\end{split}$ Now applying boundary conditions on (REF ) we get $\\begin{split}F_1=\\frac{4+r-\\frac{1+r}{2}\\lbrace e^{\\sqrt{\\frac{b}{a}}l}+e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }{\\frac{1}{2}\\sqrt{\\frac{a}{b}}\\lbrace e^{\\sqrt{\\frac{b}{a}}l}-e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }.\\end{split}$ Putting value of $F_1$ in (REF ) we get $\\begin{split}\\underline{u}(x,r)=(\\frac{1+r}{2})\\lbrace e^{\\sqrt{\\frac{b}{a}}x}+e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace +\\frac{4+r-\\frac{1+r}{2}\\lbrace e^{\\sqrt{\\frac{b}{a}}l}+e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }{\\lbrace e^{\\sqrt{\\frac{b}{a}}l}-e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }\\lbrace e^{\\sqrt{\\frac{b}{a}}x}-e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace .\\end{split}$ Now solving the classical FLT form for $\\overline{u}(x,r)$ , we have $\\begin{split}a\\lbrace p^{2}l[\\overline{u}(x,r)]-p\\overline{u}(0,r)-\\overline{u}^{\\prime }(0,r)\\rbrace -bl[\\overline{u}(x,r)]=0,\\end{split}$ $\\begin{split}(ap^{2}-b)l[\\overline{u}(x,r)]=a\\lbrace p\\overline{u}(0,r)+\\overline{u}^{\\prime }(0,r)\\rbrace .\\end{split}$ Using the boundary conditions, we have $\\begin{split}(ap^{2}-b)l[\\overline{u}(x,r)]=a\\lbrace p(3-r)+F_2\\rbrace ,\\end{split}$ where $F_2=\\overline{u}^{\\prime }(0,r).$ Simplifying and applying inverse laplace we get $\\begin{split}\\overline{u}(x,r)=(\\frac{3-r}{2})l^{-1}\\lbrace \\frac{p}{p^2-\\frac{b}{a}}\\rbrace +F_2l^{-1}\\lbrace \\frac{1}{ap^2-\\frac{b}{a}}\\rbrace .\\end{split}$ Using partial fraction $\\begin{split}\\overline{u}(x,r)=(\\frac{3-r}{2})\\lbrace e^{\\sqrt{\\frac{b}{a}}x}+e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace +\\frac{F_2}{2\\sqrt{\\frac{b}{a}}}\\lbrace e^{\\sqrt{\\frac{b}{a}}x}-e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace .\\end{split}$ Now applying boundary conditions on (REF ) we have $\\begin{split}F_2=\\frac{6-r-\\frac{3-r}{2}\\lbrace e^{\\sqrt{\\frac{b}{a}}l}+e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }{\\frac{1}{2}\\sqrt{\\frac{a}{b}}\\lbrace e^{\\sqrt{\\frac{b}{a}}l}-e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }.\\end{split}$ Putting value of $F_2$ in (REF ) we get $\\begin{split}\\overline{u}(x,r)=(\\frac{3-r}{2})\\lbrace e^{\\sqrt{\\frac{b}{a}}x}+e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace +\\frac{6-r-\\frac{3-r}{2}\\lbrace e^{\\sqrt{\\frac{b}{a}}l}+e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }{\\lbrace e^{\\sqrt{\\frac{b}{a}}l}-e^{-\\sqrt{\\frac{b}{a}}l}\\rbrace }\\lbrace e^{\\sqrt{\\frac{b}{a}}x}-e^{-\\sqrt{\\frac{b}{a}}x}\\rbrace .\\end{split}$"
],
[
"Case-II: (1) and (2)-differentiability, (2) and (1)-differentiability",
"For $x>0$ , (REF ) becomes $au^{^{\\prime \\prime }}+(l-b)u=0.$ Applying FLT and inverse Laplace transform and then simplifying we get the following lower solution.",
"$\\begin{split}\\underline{u}(x,r)=(\\frac{1+r}{2})\\bigg [\\cos \\frac{x\\sqrt{b-l}}{\\sqrt{a}}+\\cosh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]+\\frac{H_1\\sqrt{a}}{2\\sqrt{b-l}}\\bigg [\\sin \\frac{x\\sqrt{b-l}}{\\sqrt{a}}+\\sinh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]\\\\-\\frac{(3-r)}{2}\\bigg [\\cos \\frac{x\\sqrt{b-l}}{\\sqrt{a}}-\\cosh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]-\\frac{H_2\\sqrt{a}}{2\\sqrt{b-l}}\\bigg [\\sin \\frac{x\\sqrt{b-l}}{\\sqrt{a}}-\\sinh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ],\\end{split}$ or $\\begin{split}\\underline{u}(x,r)=\\frac{1+r}{2}(c_1)+\\frac{H_1\\sqrt{a}}{2\\sqrt{b-l}}(c_2)-\\frac{3-r}{2}(c_3)-\\frac{H_2\\sqrt{a}}{2\\sqrt{b-l}}(c_4).\\end{split}$ The upper solution will be as follows: $\\begin{split}\\overline{u}(x,r)=(\\frac{3-r}{2})\\bigg [\\cos \\frac{x\\sqrt{b-l}}{\\sqrt{a}}+\\cosh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]+\\frac{H_2\\sqrt{a}}{2\\sqrt{b-l}}\\bigg [\\sin \\frac{x\\sqrt{b-l}}{\\sqrt{a}}+\\sinh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]\\\\-\\frac{(1+r)}{2}\\bigg [\\cos \\frac{x\\sqrt{b-l}}{\\sqrt{a}}-\\cosh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]-\\frac{H_1\\sqrt{a}}{2\\sqrt{b-l}}\\bigg [\\sin \\frac{x\\sqrt{b-l}}{\\sqrt{a}}-\\sinh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}\\bigg ]\\end{split}$ $\\begin{split}\\overline{u}(x,r)=\\frac{3-r}{2}(c_1)+\\frac{H_2\\sqrt{a}}{2\\sqrt{b-l}}(c_2)-\\frac{1+r}{2}(c_3)-\\frac{H_1\\sqrt{a}}{2\\sqrt{b-l}}(c_4)\\end{split}$ where $c_1=\\cos \\frac{x\\sqrt{b-l}}{\\sqrt{a}}+\\cosh \\frac{x\\sqrt{b-l}}{\\sqrt{a}},$ $c_2=\\sin \\frac{x\\sqrt{b-l}}{\\sqrt{a}}+\\sinh \\frac{x\\sqrt{b-l}}{\\sqrt{a}},$ $c_3=\\cos \\frac{x\\sqrt{b-l}}{\\sqrt{a}}-\\cosh \\frac{x\\sqrt{b-l}}{\\sqrt{a}},$ $c_4=\\sin \\frac{x\\sqrt{b-l}}{\\sqrt{a}}-\\sinh \\frac{x\\sqrt{b-l}}{\\sqrt{a}}.$ $H_1=\\frac{2c_2}{c^2_2-c^2_4}\\bigg [4+r-\\frac{r+1}{2}c_1+\\frac{3-r}{2}c_3\\bigg ]+\\frac{2c_4}{c^2_2-c^2_4}\\bigg [6-r-\\frac{3-r}{2}c_1+\\frac{1+r}{2}c_3\\bigg ],$ and $H_2=\\frac{2c_4}{c^2_2-c^2_4}\\bigg [4+r-\\frac{r+1}{2}c_1+\\frac{3-r}{2}c_3\\bigg ]+\\frac{2c_2}{c^2_2-c^2_4}\\bigg [6-r-\\frac{3-r}{2}c_1+\\frac{1+r}{2}c_3\\bigg ].$ Example 5.2 Consider the following fuzzy homogenous boundary value problem $x^{^{\\prime \\prime }}(t)-3x^{^{\\prime }}(t)+2x(t)=0,$ subject to the following boundary conditions $x(0)=(0.5r-0.5, 1-r),$ $x(1)=(r-1, 1-r).$ Now applying fuzzy Laplace transform on both sides of equation (REF ), we get $L[x^{^{\\prime \\prime }}(t)]=3L[x^{\\prime }(t)]-2L[x(t)].$ We know that $L[x^{\\prime \\prime }(t)]=p^{2}L[x(t)]\\ominus px(0)\\ominus x^{\\prime }(0).$ The classical FLT form of the above equation is $l[\\underline{x}^{\\prime \\prime }(t,r)]=p^{2}l[\\underline{x}(t,r)]-p\\underline{x}(0,r)-\\underline{x}^{\\prime }(0,r),$ $l[\\overline{x}^{\\prime \\prime }(t,r)]=p^{2}l[\\overline{x}(t,r)]-p\\overline{x}(0,r)-\\overline{x}^{\\prime }(0,r).$ Now on putting in (REF ), we have $p^{2}l[\\underline{x}(t,r)]-p\\underline{x}(0,r)-\\underline{x}^{\\prime }(0,r)\\rbrace -3pl[\\underline{x}(t,r)]+3\\underline{x}(0,r)+2l[\\underline{x}(t,r)]=0,$ $p^{2}l[\\overline{x}(t,r)]-p\\overline{x}(0,r)-\\overline{x}^{\\prime }(0,r)\\rbrace -3pl[\\overline{x}(t,r)]+3\\overline{x}(0,r)+2l[\\overline{x}(t,r)]=0.$ Solving (REF ) for $l[\\underline{x}(t,r)]$ , we get $\\begin{split}(p^{2}-3p+2)l[\\underline{x}(t,r)]=p\\underline{x}(0,r)+\\underline{x}^{\\prime }(0,r)+3[\\underline{x}(0,r)].\\end{split}$ Applying boundary conditions, we have $l[\\underline{x}(t,r)]=\\frac{(0.5r-0.5)p}{p^2-3p+2}-\\frac{3(0.5r-0.5)}{p^2-3p+2}+\\frac{A}{p^2-3p+2}.$ Using partial fraction and then applying inverse Laplace, we get $\\underline{x}(t,r)=(0.5r-0.5)[-e^t+2e^{2t}]-3(0.5r-0.5)[-e^t+2e^{2t}]+A[-e^t+e^{2t}].$ Using boundary values, we get $\\underline{x}(1,r)=r-1=(0.5r-0.5)[-e+2e^2]-3(0.5r-0.5)[-e+2e^2]+A[-e+e^2],$ $A=\\frac{r-1+(0.5r-0.5)[-2e+e^2]}{e^2-e}.$ Finally on putting value of A we have $\\underline{x}(t,r)=(0.5r-0.5)(-e^t+2e^{2t})-3(0.5r-0.5)(-e^t+2e^{2t})+\\frac{r-1+(0.5r-0.5)(-2e+e^2)}{e^2-e}(-e+e^2) $ Now solving (REF ) for $l[\\overline{x}(t,r)]$ , we have $(p^{2}-3p+2)l[\\overline{x}(t,r)]=p\\overline{x}(0,r)+\\overline{x}^{\\prime }(0,r)\\rbrace +3[\\overline{x}(0,r)].$ Applying boundary condition we get $l[\\overline{x}(t,r)]=\\frac{(1-r)p}{p^2-3p+2}-\\frac{3(1-r)}{p^2-3p+2}+\\frac{A}{p^2-3p+2}.$ Using partial fraction and then applying inverse Laplace $\\overline{x}(t,r)=(1-r)[-e^t+2e^{2t}]-3(1-r)[-e^t+2e^{2t}]+A[-e^t+e^{2t}].$ Using boundary values $\\overline{x}(1,r)=1-r=(1-r)[-e+2e^2]-3(1-r)[-e+2e^2]+A[-e+e^2],$ $A=\\frac{1-r+(1-r)[-2e+e^2]}{e^2-e}.$ Putting value of A in (REF ) we get $\\overline{x}(t,r)=(1-r)[-e^t+2e^{2t}]-3(1-r)[-e^t+2e^{2t}]+\\frac{1-r+(1-r)[-2e+e^2]}{e^2-e}[-e+e^{2t}].$"
],
[
"Conclusion",
"In this paper, we applied the fuzzy Laplace transform to solve FBVPs under generalized H-differentiability, in particular, solving $Schr\\ddot{o}dinger$ FBVP.",
"We also used FLT to solve homogenous FBVP.",
"This is another application of FLT.",
"Thus FLT can also be used to solve FBVPs analytically.",
"The method can be extended for an $nth$ order FBVP.",
"This work is in progress.",
"In this paper, we applied the fuzzy Laplace transform to solve FBVPs under generalized H-differentiability, in particular, solving $Schr\\ddot{o}dinger$ FBVP.",
"We also used FLT to solve homogenous FBVP.",
"This is another application of FLT.",
"Thus FLT can also be used to solve FBVPs analytically.",
"The method can be extended for an $nth$ order FBVP.",
"This work is in progress."
]
] | 1403.0571 |
[
[
"Discussion of \"Estimating the Distribution of Dietary Consumption\n Patterns\""
],
[
"Abstract Discussion of \"Estimating the Distribution of Dietary Consumption Patterns\" by Raymond J. Carroll [arXiv:1405.4667]."
],
[
"Model Validity",
"The model of zhang2011 is highly complex—how, without something like sensitivity analysis, are we to know that it is valid?",
"As for inference, the original authors rely on the well-known (Bernstein–von Mises) asymptotic convergence of Bayesian posterior means and maximum likelihood estimates to develop standard errors using balanced repeated replication (BRR).",
"We agree that their sample size is large for many purposes, however, when the inverse Fisher information is large, convergence can be slow.",
"Moreover, this standard convergence result is known to slow down as the number of parameters grows, failing completely for nonparametric models.",
"Can we rely on Bernstein–von Mises, at these sample sizes, for this very complex (and only semi-parametric) model?",
"This is not clear to us."
],
[
"Survey Weights",
"In Section 3.3, carroll2013 notes that the use of survey weights in Bayesian analyses is controversial, and then he proceeds to use them as reported in by the National Center of Health Statistics (NCHS) nonetheless to do a weighted analysis.",
"fienberg2009 reminds us that in the NCHS survey context, weights are not just used to adjust for unequal selection probabilities, but are the product of at least three factors: $w_k &=&\\frac{1}{\\pi _k}\\times \\mbox{(nonresponse adjustment)}\\\\&&{} \\times \\mbox{(post-stratification adjustment)}.$ The first factor is the inverse of the probability of selection, for example, taking into account stratification and clustering.",
"The second factor inflates the sample results to adjust for nonresponse, typically by invoking the assumption that the missing data are missing at random, at least within chosen strata or post-strata.",
"The third factor re-weights the population totals to add up to control totals coming from another source such as a census.",
"gelman2007 rightly states: “Survey weighting is a mess,” and this is especially so from a Bayesian perspective.",
"What weights if any should be used in a Bayesian analysis?",
"In a simple stratification setting, and where we are estimating a mean or a total, weighting using $1/\\pi _k$ has a Bayesian justification.",
"For more complex situations, such as the one Carroll describes, the role of the survey weights is unclear.",
"Bayesian benchmarking is a way to deal with the third component in the weight formula above, but ghosh2013 point out the tricky nature of the choice of both loss function and benchmarking weights for small area estimation of complex surveys.",
"In essence, Carroll and his collaborators appear to be creating a pseudo-likelihood that adjusts individual contributions by the weights and then they use a survey-weighted MCMC calculation with uncertainty estimation coming from balanced repeated replication.",
"This seems unusually strange to us, and decidedly non-Bayesian in character.",
"Even if this pseudo-likelihood structure is correct, to be fully Bayesian, the weight $w_k$ associated with the $k$ th child should be a random variable.",
"The weights should then have a prior distribution and a likelihood, and be estimated together with the other unknown parameters.",
"At the very least, ignoring the variability in the weights will cause the estimate of population distributions to seem unduly precise.",
"It may be that a proper Bayesian weighting justification of carroll2013 exists, but simply hoping that the frequentist approach to survey weighting carries over to the Bayesian setting without change seems problematic."
],
[
"Acknowledgment",
"Supported in part by NSF Grant SES-1130706 to Carnegie Mellon University."
]
] | 1403.0566 |
[
[
"Multichannel Interference In High-Order Harmonic Generation From Ne(+)\n Driven By Ultrashort Intense Laser Pulse"
],
[
"Abstract We apply the time-dependent R-matrix method to investigate harmonic generation from Ne(+) at a wavelength of 390 nm and intensities up to 10(15) Wcm(-2).",
"The 1s(2)2s(2)2p(4) (3)P(e), (1)D(e), and (1)S(e) states of Ne(2+) are included as residual-ion states to assess the influence of interference between photoionization channels associated with these thresholds.",
"The harmonic spectrum is well approximated by calculations in which only the (3)P(e) and (1)D(e) thresholds are taken into account, but no satisfactory spectrum is obtained when a single threshold is taken into account.",
"Within the harmonic plateau, extending to about 100 eV, individual harmonics can be suppressed at particular intensities when all Ne(2+) thresholds are taken into account.",
"The suppression is not observed when only a single threshold is accounted for.",
"Since the suppression is dependent on intensity, it may be difficult to observe experimentally."
],
[
"Introduction",
"High-order harmonic generation (HG) has been investigated intensively since its observation more than three decades ago [1].",
"It has remained a process of great interest due to its potential as a source of XUV radiation [2].",
"Since the light generated by HG is coherent, HG is the key for the production of ultrashort pulses of light attosecond duration [3], [4].",
"These ultrashort pulses have been applied to a variety of time-resolved studies, enabling researchers to explore ultrafast electron dynamics occurring on the sub-femtosecond time scale [5], [6].",
"Examples of these explorations include the investigation of dynamics in inner shell vacancies for Kr atoms [7], the time-resolved investigation of laser induced tunnel ionization [8] and the study of electron dynamics during photoemission from a tungsten surface [9].",
"Harmonic generation is also of interest as a measurement tool in the study of molecular dynamics and structure.",
"Different harmonics are generated at slightly different times, and this characteristic has been used to demonstrate differences in the period of molecular vibrations [10].",
"More recently, a harmonic spectroscopy technique allows photochemical reactions to be measured in real time [11]; this method has the ability to monitor the dynamic electron density as the electrons are transferred between atoms in a molecule during chemical reactions.",
"Furthermore, HG has been applied to investigate how Br atoms move apart in the dissociation of Br$_{2}$ molecules [12].",
"HG is generally understood in terms of the quasi-classical, three-step model [13].",
"Three different stages of electron dynamics need to be included: first, the ejection of an electron from the atom, second, the motion of this ejected electron in the laser field and, third, its photo-recombination with the parent ion.",
"The three-step model has proven to be very useful for understanding many of the features of HG.",
"It provides intuitive understanding about the highest harmonic that can be observed and the timing delays between different harmonics.",
"One of the key findings in recent experiments is that HG in molecules is significantly affected by interferences between ionization channels associated with different ionization thresholds [14].",
"Although the three step model is suitable for noble-gas targets which have an isolated lowest ionization threshold, it may be less suitable for systems with several, closely spaced low-lying ionization thresholds.",
"For these systems, the electronic dynamics may become quite complicated, leading to amplitude and phase differences between different ionization pathways, phase differences between electron trajectories in the laser field associated with different ionization thresholds, and amplitude differences in the recombination step.",
"Hence, although the underlying mechanism of HG does not change for systems with multiple low-lying ionization thresholds, additional physics merits consideration.",
"Recent theoretical studies on HG in atoms have demonstrated that interference effects between states associated with different ionization thresholds can play a notable role in atomic systems as well [15], [16], [17], [18], [19].",
"Atomic systems present significant advantages in developing understanding of HG in systems with multiple ionization thresholds.",
"Many atomic systems have multiple low-lying ionization thresholds.",
"Atomic systems can be described with very good accuracy, enabling a detailed investigation of the effect electronic interactions have on the competition between different pathways.",
"Accurate theoretical methods are available: for example, time-dependent R-matrix (TDRM) theory was developed at Queen's University Belfast to investigate the influence of electron interactions on atomic dynamics in intense fields [20], [21].",
"Pabst and Santra [15], [16] have developed the TD-CIS approach, whereas Ngoko Djiokap and Starace [19] have investigated harmonic generation in the two-electron He atom by solving the full-dimensional Schrödinger equation.",
"The accuracy of the TDRM approach to HG processes was verified by the comparison of HG spectra for He with those obtained by the HELIUM code [22], [18].",
"It has since been applied to investigate the multielectron response for several systems: the effects of resonances on HG in Ar [17], and the effect of multiple ionization thresholds on HG in Ar$^{+}$ [23], [24].",
"In these Ar$^{+}$ studies, only a few harmonics appeared for photon energies exceeding the Ar$^+$ binding energy, and it was therefore difficult to identify clear interference effects in the plateau region.",
"The Ar$^{+}$ studies were carried out at $4\\times 10^{14}$ Wcm$^{-2}$ .",
"At higher intensities, high ionization probabilities made the accurate determination of the hamonic spectrum impossible.",
"The study of HG from ions is not only of theoretical interest.",
"It has been suggested that the very highest harmonics seen in experiment are generated by ionised atoms rather than neutral atoms [25], [26], [27].",
"In the present study, we have chosen to investigate Ne$^{+}$ .",
"Due to its higher ionization potential, the intensity can be increased significantly before ionization leads to a loss of accuracy in the harmonic spectrum.",
"Ne$^{+}$ is therefore a more suitable ion for investigating interference effects between pathways associated with different threshold.",
"The energy gap between the Ne$^{+}$ ionization thresholds is about twice as large as the energy difference between Ar$^{+}$ thresholds.",
"However, the energy difference remains comparable to the photon energy, so the interaction between channels associated with different thresholds should still be strong.",
"The organization of the paper is as follows.",
"In the following section, we present briefly the theoretical background, giving an overview of TDRM theory, and details on the computations.",
"In Sec III we present harmonic spectra including the three low-lying ionization thresholds, the $2s{^2}2p{^4}$ $^{3}P^{e}$ , $^{1}D^{e}$ , $^{1}S^{e}$ states of Ne$^{2+}$ , in the calculations for several intensities.",
"We then look in detail at the role of the different ionization thresholds on HG by presenting harmonic spectra for calculations in which subsets of the ionization thresholds have been included.",
"Since the total magnetic quantum number significantly affects harmonic generation [24], results for $M=0$ are presented in section REF and those for $M=1$ in section REF .",
"Finally, we will summarize our results and conclusions.",
"In this report, we employ TDRM theory to study HG in Ne$^{+}$ .",
"TDRM theory is a fully non-perturbative ab initio theory, which has been developed to describe the interaction of an intense ultrashort light pulse with general multielectron atoms and atomic ions.",
"In the theory, it is assumed that the light field can be treated classically in the dipole approximation, that it is linearly polarized along the $\\hat{z}$ -axis and that it is spatially homogeneous.",
"At present, relativistic effects are not taken into account.",
"TDRM theory is based upon R-matrix theory, in which space is divided into two distinct regions: an inner region and an outer region.",
"In the inner region, all electrons are contained within a distance $a_{r}$ of the nucleus.",
"In the outer region, one electron has separated from the residual ion, and is at a distance greater than $a_{r}$ from the nucleus, while the other electrons remain confined within a distance $a_{r}$ of the nucleus.",
"In the inner region, electron exchange and correlation effects between all pairs of electrons are described in full.",
"However, in the outer region, exchange interactions between the ejected electron and the electrons remaining near the residual ion can be neglected.",
"Hence, in this region only the laser field and the long-range (multipole) potential of the residual ion are included for the motion of the outer electron.",
"To obtain the wavefunction for the initial state of Ne$^+$ , which is fully contained within the inner R-matrix region at time $t=0$ , we solve the time-independent field-free Schrödinger equation.",
"The inner-region wavefunction is expanded in terms of an R-matrix basis $\\psi _{k}(X_{N+1})$ , given by [28] $\\nonumber \\psi _k(\\mathbf {X}_{N+1}) &=& \\mathcal {A} \\sum _{pj}{\\phi _{p}(\\mathbf {X}_{N};\\hat{r}_{N+1})r^{-1}_{N+1} c_{pjk} u_{j}(r_{N+1})}\\\\&&+\\sum _j \\chi _j(\\mathbf {X}_{N+1})d_{jk}.$ $\\mathcal {A}$ is the antisymmetrization operator, $\\phi _p$ are channel functions in which residual Ne$^{2+}$ ion states are coupled with the spin and angular parameters of the outer electron.",
"$\\mathbf {X}_{N+1}$ = $\\mathbf {x}_{1}, \\mathbf {x}_2, \\ldots , \\mathbf {x}_{N+1}$ , where $\\mathbf {x}_{i} = \\mathbf {r}_{i}\\sigma _{i}$ are the space and spin coordinates of the $ith$ electron.",
"The functions $u_j(r_{N+1})$ form a continuum basis set for the radial wavefunction of the outer electron inside the inner region.",
"Correlation functions $\\chi _j$ are $N+1$ -electron basis functions which vanish at the boundary.",
"The residual-ion states $\\phi _p$ and correlation functions $\\chi _j$ are constructed from Hartree Fock, Ne$^{2+}$ orbitals [30], and the functions $u_j$ are orthogonalised with respect to these input orbitals.",
"The coefficients $c_{pjk}$ and $d_{jk}$ are obtained through diagonalization of the field-free Hamiltonian.",
"Once we have obtained the initial state, we solve the time-dependent Schrödinger equation for an atom in a light field on a discrete time grid of step size $\\Delta {t}$ using the Crank-Nicolson technique described by Lysaght et al [20].",
"We obtain the wavefunction at time $t=t_{m+1}$ from the solution at $t=t_m$ , by rewriting the Schrödinger equation using the unitary Cayley form of the time evolution operator exp(-$itH(t)$ ).",
"This gives, correct to ${O(\\Delta {t}}^3)$ , $[H(t_{m+1/2} )-E]\\Psi (\\mathbf {X}_{N+1},t_{m+1} )=\\Theta (\\mathbf {X}_{N+1},t_{m}),$ where $\\Theta (\\mathbf {X}_{N+1},t_{m})=-[H(t_{m+1/2} )+E]\\Psi (\\mathbf {X}_{N+1},t_{m}).$ In Eqs.",
"(2) and (3), $E =2i/\\Delta {t}$ , with $\\Delta {t}=t_{m+1}- t_{m}$ .",
"$H(t_{m+1/2})$ is the time-dependent Hamiltonian at the midpoint time of $t_{m}$ and $t_{m+1}$ .",
"In this time-dependent Hamiltonian, the laser field is described in the dipole-length gauge.",
"In order to solve Eq.",
"(REF ), we apply different approaches to the inner region and the outer region [20].",
"Within the inner region, we expand the time-dependent wavefunction in terms of the field-free R-matrix basis: $\\Psi (\\mathbf {X}_{N+1},t_{m+1}) = \\sum _{k}{\\psi _{k}(\\mathbf {X}_{N+1}) A_{k}(t_{m})},$ so that the time-dependence is contained entirely within the coefficients $A_k$ .",
"However, the continuum functions $u_j$ in the R-matrix basis expansion (REF ) are non-vanishing at the boundary.",
"Hence the Hamiltonian $H(t_{m+1/2})$ is not Hermitian in the inner region due to surface terms arising from the kinetic energy operator, $-\\frac{1}{2}{\\nabla _{i}}^{2}$ .",
"We introduce the Bloch operator, $L$ , to cancel these terms, such that $H{(t_{m+1/2})}+L $ is Hermitian in the internal region.",
"$L = \\frac{1}{2}\\delta (r-a)\\frac{d}{dr},$ Using this result we can rewrite Eq.",
"(REF ) in the internal region as: $\\Psi = {(H + L - E)}^{-1} L\\Psi + {(H + L - E)}^{-1}\\Theta .$ Similar to standard R-matrix theory [28], we now need outer-region information to set the boundary conditions for $\\Psi $ to solve Eq.",
"(REF ) in the inner region [20].",
"Hence, the inner and outer regions are linked to each other through the so-called R-matrix: $\\mathbf {R}_{pp^{\\prime }}(E)=\\frac{1}{(2a_r)}\\sum _{kk^{\\prime }}{\\omega _{pk}\\left(\\frac{1}{(H+L)_{(kk^{\\prime })}-E}\\right)\\omega _{p^{\\prime }k^{\\prime }}}.$ Here, $p$ and $p^{\\prime }$ indicate channel functions $\\phi _{p}$ , whereas $k$ and $k^{\\prime }$ indicate field-free eigenfunctions $\\psi _k$ .",
"$\\omega _{pk}$ indicates the surface amplitude of the field-free eigenfunctions at $a_r$ with respect to each channel function $\\phi _p$ .",
"In the present computational scheme, the R-matrix is obtained through solving a system of linear equations rather than a diagonalization of $H+ L$ .",
"The wavefunctions in the inner and the outer regions are then connected to each other at the boundary according to [28], [20] by $\\mathbf {F}(a_{r} )=\\mathbf {R}a_{r}\\mathbf {\\bar{F}}(a_{r} )+ \\mathbf {T}(a_{r}),$ where the vector $\\mathbf {F}$ is the reduced radial wavefunction of the scattered electron, and $\\mathbf {\\bar{F}}$ its first derivative.",
"Compared to standard R-matrix theory, an additional inhomogenous term appears on the right hand side, which arises from the $\\Theta $ -term in Eq.",
"(REF ).",
"This so-called T-vector is given by: $\\mathbf {T}_{p}(a_{r})=\\sum _{kk^{\\prime }}{\\omega _{pk}\\left({\\frac{1}{(H+L)_{(kk^{\\prime })}-E}}\\right) \\langle \\psi _{k}(\\mathbf {X}_{N+1})\\vert \\Theta \\rangle }.$ The T-vector is determined together with the R-matrix in the linear solver step.",
"With this equation, we can determine the full wavefunction in the inner region once we know the vector $F(a_r)$ .",
"In standard R-matrix theory, this is achieved through setting boundary conditions at infinity.",
"Within time-dependent R-matrix theory, on the other hand, it takes time for the wavefunction to evolve, and consequently, the boundary condition on the F-vector is that at a sufficiently large distance the wavefunction, $F$ , equals zero.",
"In order to obtain the time-dependent wavefunction, we need to consider the outer-region wavefunction at a sufficiently large distance.",
"Although Eq.",
"(REF ) is given at the inner-region boundary, it is a general equation that holds throughout the outer region.",
"The equation also hold at a large distance where it can be assumed that the wavefunction, $F$ , has vanished.",
"We thus need to obtain the R-matrix and T-vector at this large distance.",
"This is achieved by dividing the outer region into subsectors, ranging from the inner-region boundary $a_{r}$ out to this large distance $a_{p}$ .",
"The Hamiltonian in each subsector is calculated in a similar way as in the internal region by including Bloch operators $L_L$ and $L_R$ for the left-hand and right-hand boundaries.",
"We can then obtain the time-dependent Green's function for each subsector, and use these Green's functions to propagate the R-matrix and T-vector from the inner-region boundary $a_r$ to the outer boundary $a_p$ [28], [20].",
"Subsequently, we can use the R-matrix and T-vector to propagate the F-vector inward from $a_p$ to $a_r$ .",
"Once we have obtained $F$ across all subsector boundaries, we can determine the wavefunction in the inner region and within all outer region subsectors, and initiate the computation for the next time step.",
"Repeating this procedure at each time step, we can follow the behaviour of the wavefunction across the full range of times.",
"For more details on the propagation method, see [20].",
"Calculation of the harmonic spectrum through the TDRM approach follows from determination of dipole moment of the wavefunction at each time step.",
"The harmonic radiation emitted from the atoms and ions in an intense laser field can be expressed in terms of the Fourier transform of the time-dependent expectation value of either the dipole moment, the dipole acceleration or the dipole velocity.",
"The relative merits of each of these operators is still an active subject of discussion [29].",
"In the TDRM approach, we have a choice of using the length form, $\\mathbf {d}(t)=\\langle \\Psi (t)\\vert -e\\mathbf {z} \\vert \\Psi (t)\\rangle ,$ or the velocity form, $\\mathbf {\\dot{d}}(t)=\\frac{d}{dt}\\langle \\Psi (t)\\vert -e\\mathbf {z} \\vert \\Psi (t)\\rangle .$ The acceleration form is less appropriate for the TDRM approach, as restrictions on the basis set mean that inner-shell electrons, such as the 1s electrons, are normally kept frozen.",
"As a consequence, the calculations include the action of the 1s electrons on valence electrons, but the back-action on the 1s electrons is not taken into account.",
"This limitation prevents the use of the dipole acceleration in the determination of the harmonic spectrum.",
"In the present calculations, the harmonic spectrum is calculated using both the dipole length and dipole velocity form, and we check for consistency between both spectra."
],
[
"Calculation Parameters",
"As described above, basis functions for the description of Ne$^{+}$ states are expressed as Ne$^{2+}$ residual-ion states plus an additional electron.",
"We describe Ne$^{2+}$ using Hartree-Fock orbitals for $1s, 2s$ , and $2p$ of the Ne$^{2+}$ ground state, as given by Clementi and Roetti [30].",
"The R-matrix inner region has a radius of 15 a.u.",
"The continuum functions are described using a set of 60 B-splines of order $k=17$ , for each available angular momentum $\\ell $ of the outgoing electron.",
"We include all three $1s^{2}2s^{2}2p^{4}$ states, $ ^{3}P^{e}$ , $^{1}D^{e}$ and $^{1}$ Se, as residual-ion states.",
"The description of Ne$^{+}$ includes all $1s^{2}2s^{2}2p^{4} n/\\epsilon \\ell $ Ne$^{+}$ channels up to a maximum total angular momentum $L_{\\rm max}$ =23.",
"In order to test the spectra for convergence some calculations were also carried out for a angular momentum $L_{\\rm max}$ =27.",
"The Ne$^+$ ground-state energy has not been shifted to its experimental value.",
"In the TDRM calculations, the time step in the wavefunction propagation for this calculation is normally set to 0.05 a.u.",
"Additional calculations were carried out at time steps of 0.04 a.u., and 0.06 a.u.",
"with no significant change in the overall spectrum.",
"In the outer region we set the outer boundary to 1000 a.u.",
"to prevent any unphysical reflections of the wavefunction.",
"The outer region is divided into subsectors of width 2 a.u.",
"Here, the radial wave function for each channel is described using a set of 35 B-splines of order 11.",
"The laser pulse wavelength is chosen to be 390 nm.",
"The pulse profile is given by a three-cycle sin$^{2}$ turn-on followed by four cycles at peak intensity and a three-cycle sin$^{2}$ turn-off (3-4-3)."
],
[
"Results",
"In this report, we investigate HG from Ne$^+$ ions irradiated by laser light with a wavelength of 390 nm.",
"Ne$^{+}$ has been chosen for this current study due to its higher ionization potential, 41 eV, compared to 27 eV for Ar$^{+}$ .",
"Due to the higher ionization potential, the same level of ionization requires higher intensities, and it is thus possible to investigate HG at higher intensities for Ne$^+$ compared to Ar$^+$ .",
"The higher intensity leads to an extended plateau region for HG, and Ne$^+$ should therefore show in more detail how interference due to channels associated with different ionization thresholds affects the harmonic spectra.",
"Table: Energies of the three ionization thresholds of Ne 2+ ^{2+} with respect to the Ne 2+ ^{2+}ground state, and compared to literature values .The energies for the lowest three ionization thresholds of Ne$^{+}$ , corresponding to the three different $2s^{2}2p^{4}$ Ne$^{2+}$ states, as calculated in the present study, are listed in Table REF , and compared to literature values.",
"The three ionization thresholds are separated from each other by just over 3 eV.",
"This energy difference is comparable to the photon energy and, hence, the interplay between channels associated with different thresholds cannot be neglected.",
"The energy spacings in the present study differ from the literature values, with the largest difference seen for the $^{1}D^e$ - $^{1}S^e$ gap which is 3.71 eV experimentally compared to 3.13 eV in the present study.",
"The most important energy gap is the $^{3}P^e$ - $^{1}D^e$ gap with a gap difference of 0.22 eV.",
"These differences are sufficiently small for identification of the most important effects of the interplay between channels.",
"Figure: The harmonic spectrum obtained by 390 nm laser pulse with a total durationof 10 cycles generatedfrom Ne + ^+ calculated from the dipole length, at different laser peak intensities:(a) 7×10 14 7\\times 10^{14}Wcm -2 ^{-2} (b) 8×10 14 8\\times 10^{14} Wcm -2 ^{-2} (c) 9×10 14 9\\times 10^{14} Wcm -2 ^{-2} and(d) 10 15 10^{15} Wcm -2 ^{-2}."
],
[
"High Harmonic generation from Ne$^{+}$ aligned with {{formula:04023532-7585-4b35-be3d-0e6bbbfc67f6}}",
"The harmonic response of Ne$^{+}$ , as calculated from the expectation value of the dipole length operator, is shown in Fig.",
"REF , for peak laser intensities between $7\\times 10^{14}$ Wcm$^{-2}$ and $10^{15}$ Wcm$^{-2}$ , and a 3-4-3 pulse profile.",
"The spectra have the expected structure with a plateau containing eight odd harmonics up to a cut-off energy around 90 eV.",
"Beyond this cut-off energy, an exponential decay in the harmonic yield is seen.",
"The plateau is more extensive than observed for Ar$^{+}$ at $4\\times 10^{14}$ Wcm$^{-2}$ , for which the plateau contained 3 harmonic peaks [18], [23].",
"Although the figure shows only the harmonic spectrum obtained through the dipole operator, the harmonic spectrum has also been obtained using the expectation value of the dipole velocity.",
"The spectra calculated through the dipole and the dipole velocity show very good agreement.",
"The convergence is typically within 20$\\%$ in the overall harmonics up to 120 eV where the spectra associated from both dipoles start to diverge clearly.",
"The main reason for the overall difference is the limited basis expansion used for the description of the Ne$^{2+}$ states in the present calculations.",
"Table REF gives indicative intensities of the harmonic peaks, normalized to the harmonic spectrum obtained at $10^{15}$ Wcm$^{-2}$ , and the cutoff energy of the plateau as determined from the spectra.",
"The straight lines in Fig.",
"REF demonstrate the origin of these values.",
"Table REF also gives the population in the outer region.",
"In the three step model ionization is the first step of HG.",
"The table shows that the increase in the harmonic yield approximately follows the increase in population in the outer region, with a factor 13 increase in the harmonic yield going from 7 $\\times $ 10$^{14}$ Wcm$^{-2}$ to 10$^{15}$ Wcm$^{-2}$ matched by an increase of a factor 15 in the outer-region population.",
"Table: Typical harmonic yield for Ne + ^+ irradiated by laser light with a wavelength of 390 nm,normalized to the typical harmonic yield at a peak intensity of 10 15 10^{15} Wcm -2 ^{-2}, as a functionof peak intensity.",
"The harmonic yield is calculated through the dipole operator, and the pulseprofile is a 3-4-3 pulse.",
"The cut-off energy of the harmonic plateau is given as well and compared with theprediction of the cut-off formula, 1.25I p +3.17U p 1.25I_p + 3.17 U_p.",
"The final population in the outer region isgiven as well.Table REF shows the variation of the cut-off energy of the plateau with peak intensity.",
"The determination of the cut-off energy is shown in the graphs.",
"These cut-off values are expected to have an energy uncertainty $\\le $ 1.5 eV.",
"The standard cut-off formula for the energy of the cut-off is given by $\\alpha I_{p} +U_{p}$ [32], where $U_{p}$ is the ponderomotive potential of a free electron in a laser field, $I_{p}$ is the ionization potential and $\\alpha $ is a parameter, which depends on the ratio between $I_p$ and $U_p$ .",
"For the present range of intensities, the parameter ranges between 1.235 and 1.25.",
"For simplicity, we adopt $\\alpha =1.25$ .",
"Values obtained from this formula are also given in the table.",
"It can be seen that for all intensities, the observed cut-off energy values differ by about 2 eV from the predictions of the formula, although at an intensity of $0.8\\times 10^{15}$ Wcm$^{-2}$ the cut-off formula underestimates the observed cut-off energy by 5.1 eV.",
"Figure REF shows great variation of peak intensity within the plateau region.",
"In Fig.",
"REF (b), harmonic 19 at 60 eV is a factor 40 more intense than harmonic 21.",
"In Fig.",
"REF (d), relatively little harmonic response is observed for harmonics 23 and 27, at 72 eV and 86 eV, respectively.",
"The basis set in the present calculations is chosen specifically to exclude resonances above the $2s^{2}2p^{4}$ $^{1}S^{e}$ ionization threshold, so that this reduction in magnitude cannot ascribed to atomic structure at these harmonic energies.",
"Harmonics up to a photon energy of around 50- 55 eV, on the other hand, can be strongly affected by resonances due to the Rydberg series leading up to the $2s^{2}2p^{4}$ thresholds.",
"The HG spectra can be affected by interferences arising from several sources.",
"Below the cut-off energy, harmonics can be created by electrons returning on so-called short and long trajectories [33].",
"For multi-threshold systems, interferences between channels associated with different thresholds can arise as well.",
"Within the TDRM approach, it is difficult to unambiguously separate short and long trajectories.",
"Therefore, in order to understand whether the interplay between channels associated with different thresholds could be responsible for the interference structure, as well as the discrepancy in the cut-off energy, we have carried out additional calculations in which only subsets of the $2s^{2}2p^{4}$ thresholds are taken into account.",
"Figure: Harmonic spectra for Ne + ^+ irradiated by 390 nm laser pulse as calculated throughthe dipole operator, at peak intensity of 10 15 10^{15} W cm -2 ^{-2}.",
"The Ne 2+ ^{2+} 2s 2 ^22p 4 ^4residual-ion states retained are: (a) 3 P e ^{3}P^e,(b) 1 D e ^{1}D^e, (c) 1 S e ^{1}S^e, and(d) all three states.First, we consider the harmonic spectra when only a single residual ion state is included in the calculation.",
"Since harmonics 23 and 27 show a significant reduction in magnitude for an intensity of $10^{15}$ W cm$^{-2}$ , we have chosen this intensity for the comparison.",
"Figure REF shows the harmonic spectra when only an individual $2s^{2}2p^{4}$ threshold of Ne$^{2+}$ , ($^{3}P^e, ^{1}D^e$ or $^{1}S^e$ ) is included.",
"For comparison, the figure also includes the full spectrum.",
"Figure REF shows significant variation in the harmonic efficiency between the different individual threshold calculations.",
"In the calculation in which only the $^{1}D^{e}$ threshold is retained, the observed harmonic intensities are about an order of magnitude greater than those observed when only the $^{1}S^e$ threshold is retained.",
"The harmonic intensities when only the $^{3}P^{e}$ threshold, the Ne$^{2+}$ ground state, is retained are about one order of magnitude smaller than those when only the $^{1}D^{e}$ threshold is retained at 45 eV, and about two orders of magnitude smaller at 90 eV.",
"The spectrum obtained when all thresholds are included shows harmonic intensities one order of magnitude smaller than the spectrum obtained when only the $^{1}D^{e}$ threshold is retained.",
"This is a significant variation in harmonic yields depending on the symmetry of the ionization threshold.",
"Since the smallest harmonic yields are retained when only the Ne$^{2+}$ ground state is obtained, this variation cannot be explained solely by the variation in binding energy of the different thresholds.",
"Table: Typical harmonic yields for Ne + ^+ irradiated by 390 nm laser light at an intensity of 10 15 ^{15}Wcm 2 ^2 as a function of thresholds retained, normalized to the spectrum obtained when all three thresholdsare included.",
"The cut-off energy of the plateau region and the final outer region population are also shownfor each subset of Ne 2+ ^{2+} thresholds retained.",
"No typical harmonic yield can be given when onlythe 3 P e ^3P^e threshold is retained (see text).As demonstrated earlier, the typical harmonic yield behaves similarly to the population in the outer region.",
"We focus our attention therefore first on these populations, shown in Table REF .",
"Note that the table does not provide a typical harmonic yield when only the $^{3}P^e$ threshold is retained.",
"Figure REF (a) shows a decrease of about an order of magnitude across the plateau region, and it is therefore impossible to identify a typical value that applies to the entire plateau.",
"Table REF shows that the variation in the typical harmonic yields is reflected to some extent in the outer-region populations.",
"The outer region population when only the $^1D^e$ threshold is retained is about a factor 3.7-5.5 larger than when either the $^{3}P^e$ or $^1S^e$ threshold is retained.",
"It is, however, also a factor 4 larger than when all three thresholds are retained.",
"The increase in the population in the outer region may therefore be the fundamental reason for the higher harmonic yields when only the $^{1}D^e$ threshold is retained.",
"However, the population in the outer region does not explain the difference between the harmonic yields obtained when only the $^{1}S^e$ or the $^{3}P^e$ threshold is retained.",
"Table REF also shows how the cut-off energies for the plateau region depend on the thresholds retained in the calculation.",
"The cut-off energies when only the $^{1}D^e$ or $^{1}S^e$ threshold is retained lie within 1 eV of the cut-off energy observed when all three thresholds are retained.",
"However, the cut-off energy obtained when only the $^{3}P^e$ threshold is retained is nearly 6 eV smaller than the cut-off energies obtained in the other calculations.",
"This is clear demonstration that there are fundamental differences between the process of HG in Ne$^+$ with $M=0$ when all three thresholds are accounted for and when only the Ne$^{2+}$ ground state is accounted for.",
"Figure REF (d) shows the full harmonic spectrum, in which harmonics 23 and 27 are reduced in intensity by about 1.5 order of magnitude.",
"However, Fig.",
"REF (a), (b) and (c), show no significant reduction in intensity at either of these harmonics.",
"Instead, the spectra show smooth variations in the intensities of the different harmonic peaks.",
"These single-threshold spectra are still affected by interferences between the short and long trajectories.",
"No sign of these interferences is seen in the individual spectra, although there is some variation in the magnitude of individual harmonic peaks.",
"Hence, the reduction in the harmonic intensity seen in Fig.",
"REF (d) cannot be assigned to dynamics associated with a single ionization threshold.",
"It is therefore necessary to consider harmonic spectra obtained when multiple ionization thresholds are retained.",
"Figure: The harmonic spectra obtained for Ne + ^+ irradiated by 390 nm laser pulse,as obtained through the dipole length operator, at peak intensity of 10 15 10^{15} W cm -2 ^{-2}.The spectra are obtained when the following 2s 2 ^22p 4 ^4 thresholds are retained:(a) 3 P e ^{3}P^e and 1 S e ^{1}S^e, (b) 3 P e ^{3}P^e and 1 D e ^{1}D^e, (c) 1 S e ^{1} S^e and 1 D e ^{1} D^e thresholds and (d) all three thresholds.The next stage in the analysis is to investigate harmonic spectra when pairs of Ne$^{2+}$ residual-ion states, ($^{1}D^e$ , $^{3}P^e$ ), ($^{1}S^e$ , $^{3}P^e$ ) and ($^{1}D^e$ , $^{1}S^e$ ), are retained in the calculations.",
"These harmonic spectra are shown in Fig.",
"REF and compared to the spectrum obtained in the full calculation.",
"The harmonic spectrum obtained when both the $^{1}D^e$ and $^{3}P^e$ thresholds are retained shows great similarity to the harmonic spectrum obtained when all three target states are retained.",
"The other spectra show noticeable differences with the full spectrum.",
"The spectrum obtained when both the $^{3}P^e$ and $^{1}S^e$ thresholds are retained shows harmonic intensities, which are about a factor of 4 smaller than those of the full spectrum, especially at higher energies.",
"The ($^{1}D^e$ , $^{1}S^e$ ) spectrum has harmonic intensities which are slightly larger than the full spectrum in the plateau region.",
"However, the yield for harmonics below the ionization threshold is larger than obtained in the full calculation and these peaks are more pronounced compared to the full spectrum.",
"No sign of interference is seen for harmonics 23 and 27.",
"Once again, Table REF lists the population in the outer region when each pair of Ne$^{2+}$ residual-ion states ($^{1}D^e$ , $^{3}P^e$ ), ($^{1}S^e$ , $^{3}P^e$ ) and ($^{1}D^e$ , $^{1}S^e$ ) is retained in the calculation.",
"When the $^{3}P^e$ threshold is included in the calculation, the total population in the outer region is within 10% of the outer-region population when all three thresholds are included.",
"However, when only the $^{1}D^e$ and $^{1}S^e$ thresholds are accounted for, the population in the outer region is larger than the population in the full calculation by more than a factor 3.",
"Hence the inclusion of the $^{3}P^e$ threshold is essential to obtain an accurate population in the outer region.",
"The cut-off energies, given in Table REF , show agreement with the full spectrum within 1 eV.",
"The interaction between channels associated with the $^3P^e$ threshold and channels associated with either the $^{1}D^e$ or $^{1}S^e$ threshold leads to the cut-off energy of the harmonic spectrum being shifted upward by about 6 eV.",
"Signs of destructive interference can be observed in the spectra obtained in both the ($^{3}P^e$ , $^{1}S^e$ ) and ($^{3}P^e$ , $^{1}D^e$ ) calculation.",
"In the former case, harmonics 25 and 27 are suppressed, whereas harmonics 23 and 27 are suppressed in the latter case.",
"For the ($^{3}P^e$ , $^{1}D^e$ ) case, the decrease in magnitude corresponds well to the decrease observed in the full calculation.",
"Overall, these harmonic spectra demonstrate that HG from ground-state Ne$^+$ with $M=0$ at a wavelength of 390 nm requires the inclusion of at least the 2s$^2$ 2p$^4$ $^{3}P^e$ and $^{1}D^e$ thresholds of Ne$^{2+}$ .",
"The harmonic spectrum is dominated by the harmonic response of the excited $^{1}D^e$ threshold, as demonstrated, for example, by the cut-off energy in the full calculation.",
"However, if just the $^{1}D^e$ threshold is accounted for, the ionization rate is too high, and the spectra do not show the right level of variation in harmonic intensities.",
"Inclusion of the (lower-lying) $^{3}P^e$ threshold reduces the ionization rate, and the correct magnitude of the harmonic yield is obtained.",
"Interference between pathways associated with these ionization thresholds then leads to the destructive interference for certain harmonics.",
"It is counterintuitive that inclusion of a lower-lying ionization threshold reduces the ionization rate.",
"The lower ionization rate associated with the $^{3}P^e$ threshold can be explained through the allowed $m$ -values of the ejected electron [23].",
"For $M=0$ , ejection of an electron with $m=0$ is only allowed for the $^{1}D^e$ threshold.",
"The $^{3}P^e$ threshold can only be reached through the emission of $m=1$ electrons.",
"The Rydberg series converging to the $^{1}D^e$ and the $^{3}P^e$ thresholds overlap.",
"When an $m=0$ electron is excited towards the $^{1}D^e$ threshold, electron-electron interactions between the two channels can `re-route' an electron from the $^{1}D^e$ path to the $^{3}P^e$ path changing the $m$ -value of the electron.",
"Since below the $^3P^e$ state the density of states in the Rydberg series converging to the $^{3}P^e$ threshold is higher than that for the series converging to the $^{1}D^e$ threshold, it may be difficult for the electron to return from the $^3P^e$ Rydberg series to the $^{1}D^e$ Rydberg series.",
"This process can slow down ionization and, through reduction of the $^{1}D^e$ ionization pathway, reduce the harmonic yield."
],
[
"High Harmonic generation from Ne$^{+}$ aligned with {{formula:ce136a42-8a78-4187-ab9b-0bf9fca69b94}}",
"In the previous study of HG in Ar$^+$ , a difference of about a factor of 4 was observed between the harmonic yields for $M=0$ and $M=1$ .",
"It is therefore valuable to investigate whether a similar difference is observed for Ne$^+$ as well, and to see whether the magnetic quantum number leads to changes in the interference pattern.",
"The total magnetic quantum number $M$ has a significant effect on the calculations: it affects the allowed radiative transitions.",
"For systems with $M=0$ , the selection rules state that only radiative transitions with $\\Delta L =\\pm 1$ are allowed, but, for $M=1$ , $\\Delta {L} = 0, \\pm 1$ radiative transitions are allowed.",
"Hence, transitions between states with different parity but the same angular momentum ($>0$ ) are now allowed.",
"This doubles the number of total symmetries that need to be retained in the calculation, with a corresponding increase in the size of the calculations.",
"We have therefore performed only a limited number of calculations for $M=1$ , focussing primarily on the interplay between the $^3P^e$ and $^1D^e$ thresholds.",
"Figure REF (d) shows the harmonic spectrum for Ne$^+$ , with $M=1$ initially, obtained when all three residual Ne$^{2+}$ states are retained in the calculation.",
"The $M=0$ spectrum is included for comparison.",
"The figure shows an increase for $M = 1$ of about a factor 26 compared to $M = 0$ .",
"This reflects, in part, an increase in the population in the outer region by a factor of 12.5 for $M = 1$ compared to $M = 0$ .",
"This change in the harmonic yield is a factor 6 larger than the relative change seen for Ar$^+$ , demonstrating that the initial alignment is a more critical factor for Ne$^+$ than for Ar$^+$ .",
"Figure REF (d) also shows that harmonics 17 - 29 have a very similar magnitude, apart from harmonic 25, which has been reduced by over one order of magnitude, compared to the other harmonics in the plateau region.",
"It is noteworthy that, for $M=0$ , both neighbouring harmonics, harmonics 23 and 27- were the ones that decreased noticeably in magnitude.",
"To verify that the reason for this decrease is the interplay between channels associated with the $^3P^e$ and $^1D^e$ thresholds, as for $M=0$ , we have carried out additional calculations in which combinations of these states were included as residual-ion states of Ne$^{2+}$ .",
"Table: Typical harmonic yields for Ne + ^+ with M=1M=1 irradiated by 390 nm laser light at an intensity of 10 15 ^{15}Wcm -2 ^{-2} as a function of thresholds retained, normalized to the corresponding spectrum obtained with M=0M=0.",
"The cut-off energy of the plateau region and the final outer region population are also shownfor each subset of Ne 2+ ^{2+} thresholds retained.Figure REF (a), REF (b) and REF (c) show the harmonic spectra obtained when just the $^1D^e$ threshold is retained, just the $^3P^e$ threshold is retained and when both the $^3P^e$ and $^1D^e$ thresholds are retained, respectively.",
"For Figs.",
"REF (a), REF (b) and REF (c), the corresponding spectrum for $M=0$ is also presented.",
"Figure REF (c) and REF (d) demonstrate that the total harmonic spectrum obtained for the ($^3P^e$ , $^1D^e$ ) case is very similar to the full spectrum, as observed for $M=0$ .",
"The assumption that the $^1S^e$ threshold is less important is therefore justified.",
"The comparison of the harmonic spectra for $M=1$ and $M=0$ is dramatically different for the calculation including the $^3P^e$ threshold only and for the calculation including the $^1D^e$ threshold only.",
"When only the $^1D^e$ threshold is included in the calculations, the harmonic yields for $M=1$ and $M=0$ are very similar.",
"The population in the outer region is also similar, differing by a factor 1.3.",
"On the other hand, when only the $^3P^e$ threshold is retained, the harmonic intensities increase by over 2 orders of magnitude.",
"The intensities of the harmonics across the plateau region show no obvious decrease with photon energy for $M=1$ , whereas they did for $M=0$ .",
"This marked increase in harmonic yield is matched by an increase of a factor 12 in the population in the outer region.",
"This behaviour is in line with the behaviour seen for Ar$^+$ [24].",
"The spectrum obtained when both the $^3P^e$ and $^1D^e$ thresholds are retained in the calculation shows harmonic intensities which have decreased from the typical intensity shown in the $^3P^e$ spectrum by approximately a factor of 3.",
"However, apart from harmonic 25, harmonics 17 - 29 appear with very similar intensity in the ($^3P^e$ , $^1D^e$ ) spectrum, whereas the $^3P^e$ spectrum shows more substantial variation across the harmonics.",
"Harmonic 25 shows a significant reduction of about an order of magnitude compared to harmonics 17 - 29.",
"In the individual $^3P^e$ and $^1D^e$ spectra, harmonic 25 is similar in appearance as harmonic 23.",
"Destructive interference between the two pathways to HG leads to the significant reduction of the yield of harmonic 25 in the combined spectrum.",
"The significant change seen in harmonic yield associated with the $^3P^e$ threshold has the same origin as explained for Ar$^+$ [24].",
"For $M=0$ the initial $2p^5$ configuration only contains a single electron with $m=0$ .",
"Its emission leaves $2p^4$ in a singlet state, so the $^3P^e$ state can not be a state of the residual ion following ejection of an $m=0$ electron.",
"For $M=1$ , two electrons have $m=0$ .",
"Emission of one of these electrons leaves a $2p$ shell with holes at $m=-1$ and $m=0$ , which can combine to form a triplet state.",
"Thus, the emission of an $m=0$ electron can leave the Ne$^{2+}$ residual ion in the $^{3}P^e$ ground state for $M=1$ .",
"The ionization step in the recollision model is dominated by a single $m=0$ electron escaping towards an excited threshold for $M=0$ , whereas it is dominated by one out $m=0$ electron (out of two available) escaping towards the lowest threshold for $M=1$ .",
"Ionization should therefore be significantly stronger for $M=1$ , and the harmonic yield should be higher.",
"The $^1D^e$ threshold can be reached through emission of an $m=0$ electron for both $M=0$ and $M=1$ .",
"In this case, the harmonic yields and populations in the outer region are of similar magnitude.",
"In addition, Table REF also shows that the cut-off energy, when only $^{3}P^e$ threshold is included in the $M=1$ calculation, is about 96 eV.",
"On the other hand, it is about 99 eV when the $^1D^e$ threshold taken into account as well.",
"The additional thresholds therefore appear to increase the cutoff energy for the harmonic plateau.",
"For both $M=0$ and $M=1$ , the inclusion of the $^1D^e$ threshold raises the cut-off energy.",
"The role of the $^1D^e$ threshold differs in these cases: for $M=0$ , it is the primary threshold for HG, but for $M=1$ it is the secondary threshold.",
"Overall, the HG spectra for both $M=0$ and $M=1$ demonstrate the necessity to include at least the lowest two thresholds for the reliable determination of the HG spectrum, for the isolated atom.",
"Interplay between channels associated with these thresholds affects the spectra greatly: the overall yield is reduced by a factor 4 from the most efficient channel.",
"The cut-off energy is increased beyond the cut-off associated with the lowest threshold.",
"Interference between pathways can cause specific harmonics to be significantly reduced at specific intensities."
],
[
"Conclusions",
"We have applied time-dependent R-matrix theory to investigate HG in Ne$^{+}$ at a laser wavelength of 390 nm.",
"Due to the large binding energy of Ne$^{+}$ , harmonic spectra could be obtained for laser intensities up to $10^{15}$ Wcm$^{-2}$ , enabling the determination of the role of different ionization thresholds in HG within the plateau region.",
"To assess the influence of channels associated with individual thresholds and of the interactions between channels associated with different ionization thresholds, calculations were performed using all possible combinations of the three $1s^{2}2s^{2}2p^{4}$ $^3P^{e}$ , $^1D^{e}$ and $^{1}S^{e}$ thresholds of Ne$^{2+}$ .",
"A good approximation to the full spectrum is obtained when both the $^{1}D^e$ and $^{3}P^e$ threshold are included, indicating that inclusion of the $^{1}S^{e}$ threshold is less critical.",
"For $M=0$ , we find that inclusion of just the $^{3}P^e$ threshold on its own underestimates the harmonic yield by up to an order of magnitude, and gives too small a cut-off energy.",
"On the other hand, inclusion of the $^{1}D^e$ threshold on its own overestimates the harmonic yield by an order of magnitude due to an overestimation of the Ne$^+$ ionization rate.",
"Hence both thresholds are essential for a correct description of HG for Ne$^{+}$ with $M=0$ .",
"For $M=1$ , we find that inclusion of just the $^3P^e$ threshold on its own slightly overestimates the harmonic yield by about a factor of 2, whereas, the cut-off energy value underestimates by 3 eV the value when all the three thresholds are retained.",
"On the other hand, inclusion of only the $^{1}D^e$ threshold now underestimates the harmonic yield by an order of magnitude.",
"Hence both thresholds are essential for a correct description of HG for Ne$^{+}$ with $M=1$ .",
"Interactions between the pathways leading up to the $^3P^e$ and $^1D^e$ thresholds affect more than just the overall magnitude of the harmonics.",
"At an intensity of 10$^{15}$ Wcm$^{-2}$ , we observe noticeable decreases in the harmonic yield for specific harmonics.",
"For $M=0$ harmonics 23 and 27 are affected in particular, whereas for $M=1$ , harmonic 25 is affected the most.",
"These decreases in the harmonic yield are only present when both the $^3P^e$ and the $^1D^e$ thresholds are included in the calculations, and are not observed when only a single threshold is included.",
"We thus ascribe this decrease to interference between different ionization channels.",
"It should be noted however that this particular interference is only observed at 10$^{15}$ Wcm$^{-2}$ , and is not observed at 9$\\times $ 10$^{14}$ Wcm$^{-2}$ .",
"Therefore this interference may be strongly intensity dependent.",
"In experiment, where different atoms (or ions) will experience different peak intensities, the effects of this interference may therefore not be apparent.",
"Although the present work demonstrates that the TDRM approach can describe HG in ions successfully, including an extensive plateau up to photon energies $\\sim $ 100 eV, using intensities up to $10^{15}$ Wcm$^{-2}$ at a wavelength of 390 nm, it may be difficult to extend the present approach to longer wavelengths due to an increase in the number of angular momenta that need to be retained in the calculations.",
"The recently developed R-matrix incorporating time-dependence (RMT) codes [34] should be more amenable to the inclusion of many angular momenta.",
"It will therefore be interesting to explore HG using the RMT approach."
],
[
"ACKNOWLEDGMENTS",
"O. Hassouneh acknowledges the University of Jordan for financial support.",
"H. W. H acknowledges financial support from UK EPSRC under Grant No.",
"G/055416/1, and A. C. B support from DEL under the programme for government."
]
] | 1403.0447 |
[
[
"Nuclear Spin Diffusion Mediated by Heavy Hole Hyperfine Non-Collinear\n Interactions"
],
[
"Abstract We show that the hyperfine mediated dynamics of heavy hole states confined in neutral self- assembled quantum dots leads to a nuclear spin diffusion mechanism.",
"It is found that the oftentimes neglected effective heavy hole hyperfine non-collinear interaction is responsible for the low degree of nuclear spin polarization in neutral quantum dots.",
"Moreover, our results demonstrate that after pumping the nuclear spin state is left in a complex mixed state, from which it is not straightforward to deduce the sign of the Ising-like interactions."
],
[
"Nuclear Spin Diffusion Mediated by Heavy Hole Hyperfine Non-Collinear Interactions Hugo Ribeiro Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Franziska Maier Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland We show that the hyperfine mediated dynamics of heavy hole states confined in neutral self-assembled quantum dots leads to a nuclear spin diffusion mechanism.",
"It is found that the oftentimes neglected effective heavy hole hyperfine non-collinear interaction is responsible for the low degree of nuclear spin polarization in neutral quantum dots.",
"Moreover, our results demonstrate that after pumping the nuclear spin state is left in a complex mixed state, from which it is not straightforward to deduce the sign of the Ising-like interactions.",
"We are currently in the midst of an effort to develop reliable nanostructures that can be used to host qubits.",
"Among the possible architectures [1], [2], [3], [4], the progress made with spin-based qubits confined in semiconductor structures [5] has been the most impressive [6].",
"In only a decade, it became possible to efficiently initialize [7], manipulate coherently [8], [9], [10], [11], [12], [13], [14], and measure the state of a single spin confined in both electrically defined and self-assembled quantum dots.",
"All of these remarkable achievements are, however, mitigated by poor coherence times on the order of tens of nanoseconds [8], [9], [10], [11].",
"In quantum dots made out of III-V materials, the fluctuations of the nuclear spin felt by the electronic spin through the hyperfine interaction are the main source of decoherence [8], [9], [15], [16], [17], [18], [19], [20].",
"Nevertheless, dynamical decoupling schemes have improved the situation and revealed longer dephasing times [21], [22], [23], [24].",
"From another perspective, nuclear spins are a helpful resource for quantum computing.",
"In gate defined dots, coherent manipulation of electron spin states via the hyperfine interaction has been demonstrated [13], [25], [26].",
"In self-assembled dots, direct control of nuclear spins has been realized via nuclear magnetic resonance (NMR) [27], [28], which allows to control the direction of the Overhauser field and consequently can be used to control an electron (hole) spin-based qubit.",
"In spite of the efforts made to harness nuclear spins, the role of heavy holes in the dynamics is not yet fully understood.",
"The first theories suggested an Ising-like type of interaction with a strength on the order of $10\\%$ of the one of the electron and with opposite sign [30], [31], which was experimentally verified [32], [33].",
"However, subsequent experiments seem to contradict these early results.",
"It has recently been claimed that the sign of the coupling strength is opposite for cations and anions [29].",
"Some other recent experiments [20], [34] report results which indicate a feedback mechanism between heavy holes and nuclear spins.",
"Theories based on $p$ -symmetric Bloch functions for hole states predict that flip-flop terms similar to those of the electronic hyperfine Hamiltonian are very weak [30], [31].",
"Consequently, it was proposed that non-collinear hyperfine interactions could account for the joint heavy hole nuclear spin dynamics [35], [36].",
"However, non-collinear interactions were predicted to only affect the dynamics if the laser frequency is not on resonance with the electronic transition which is being driven.",
"An alternative explanation would be that hole states have to be described by both $p$ - and $d$ -type Bloch functions [29] leading to a stronger flip-flop exchange mechanism.",
"Figure: (Color online).",
"Level scheme of the excitonic states in a neutral quantum dotshowing optically driven transitions with Rabi frequency Ω\\Omega under the absorption ofσ + \\sigma _+ polarized light (magenta).",
"The nuclear states are described with the totalangular momentum jj and magnetization mm.",
"Hyperfine mediated transitions via theelectron are shown in orange and in purple for the hole spin.",
"The excited states relaxvia spontaneous emission with rates Γ sp ↓⇑ ≈Γ sp ↑⇓ ≫Γ sp ↑⇑ ≈Γ sp ↓⇓ \\Gamma _{\\mathrm {sp}}^{\\downarrow \\Uparrow } \\approx \\Gamma _{\\mathrm {sp}}^{\\uparrow \\Downarrow } \\gg \\Gamma _{\\mathrm {sp}}^{\\uparrow \\Uparrow } \\approx \\Gamma _{\\mathrm {sp}}^{\\downarrow \\Downarrow }.In this letter, by focusing on optical pumping of nuclear spins in neutral quantum dots, we show that the effective hyperfine interaction for heavy hole states, described with $p$ -symmetric Bloch functions, via the non-collinear term leads to an effective nuclear spin diffusion mechanism.",
"Opposite to earlier theories, we find that non-collinear interactions influence the nuclear spin dynamics even when the laser frequency is on resonance with the optically allowed electronic transition.",
"Ironically, nuclear spin diffusion mediated by heavy holes is allowed due to the electron hyperfine interaction which drives the system to a quasi optical dark state.",
"The longer the system stays in the dark state the more efficient diffusion becomes.",
"Our results not only provide an explanation for the experimentally observed low degrees of nuclear spin polarization, but they also offer an alternative explanation to the results found in Ref.",
"[29] since the orientation of nuclear spins cannot be assumed to be solely defined by the pumping scheme.",
"Finally, we simultaneously propose a simple experiment aiming at detecting and cancelling the effective heavy hole non-collinear interaction.",
"The effective Hamiltonian [37], $H = H^{\\prime }_0 + H^{\\prime }_{\\mathrm {L}} + H_{\\mathrm {Z}}^{\\mathrm {nuc}} + H_{\\mathrm {HF},z}^{\\mathrm {e}} +H_{\\mathrm {HF},z}^{\\mathrm {h}} + H_{\\mathrm {HF},\\mathrm {nc}}^{\\mathrm {h}},$ describes the coherent dynamics of the system in the presence of an external magnetic field oriented along the growth axis of the quantum dot (Faraday geometry) and when the laser frequency is close to resonance with the transition $|0\\rangle \\leftrightarrow |\\!\\!\\downarrow \\Uparrow \\rangle $ [c.f.",
"Fig.",
"REF ].",
"The Hamiltonian $H^{\\prime }_0$ describes the evolution of the exciton states, $\\begin{aligned}H^{\\prime }_0 &= \\frac{\\hbar \\Delta }{2}\\left(-|0\\rangle \\langle 0| + |\\!\\!\\downarrow \\Uparrow \\rangle \\langle \\downarrow \\Uparrow \\!\\!|\\right)+ \\left(\\frac{\\hbar \\Delta }{2} + E^{\\uparrow \\Downarrow }_{\\downarrow \\Uparrow }\\right)|\\!\\!\\uparrow \\Downarrow \\rangle \\langle \\uparrow \\Downarrow \\!\\!|\\\\&\\phantom{=}+ \\left(\\frac{\\hbar \\Delta }{2} + E^{\\uparrow \\Uparrow }_{\\downarrow \\Uparrow }\\right) |\\!\\!\\uparrow \\Uparrow \\rangle \\langle \\uparrow \\Uparrow \\!\\!|+ \\left(\\frac{\\hbar \\Delta }{2} + E^{\\downarrow \\Downarrow }_{\\downarrow \\Uparrow }\\right)|\\!\\!\\downarrow \\Downarrow \\rangle \\langle \\downarrow \\Downarrow \\!\\!|,\\end{aligned}$ where $\\Delta $ is the laser detuning and we have $\\begin{aligned}E^{\\uparrow \\Downarrow }_{\\downarrow \\Uparrow } &= -\\sqrt{\\delta _1^2 + g_-^2\\mu _{\\mathrm {B}}^2 B^2},\\\\E^{\\uparrow \\Uparrow }_{\\downarrow \\Uparrow } &= -\\delta _0 + \\sqrt{\\delta _2^2 +g_+^2\\mu _{\\mathrm {B}}^2 B^2} - \\sqrt{\\delta _1^2 + g_-^2\\mu _{\\mathrm {B}}^2 B^2},\\\\E^{\\downarrow \\Downarrow }_{\\downarrow \\Uparrow } &= -\\delta _0 -\\sqrt{\\delta _2^2 + g_+^2\\mu _{\\mathrm {B}}^2 B^2} - \\sqrt{\\delta _1^2 + g_-^2\\mu _{\\mathrm {B}}^2 B^2}.\\end{aligned}$ Here, we have defined $g_+ = g_{\\mathrm {e}} + 3 g_{\\mathrm {h}}$ and $g_- = g_{\\mathrm {e}} - 3 g_{\\mathrm {h}}$ with $g_{\\mathrm {e}}$ ($g_{\\mathrm {h}}$ ) the electron (heavy hole) Landé $g$ -factor, and $\\mu _{\\mathrm {B}}$ is the Bohr magneton.",
"The coefficients $\\delta _0$ , $\\delta _1$ , and $\\delta _2$ describe respectively the fine structure splitting between bright and dark excitons, among bright, and among dark excitons [38].",
"Since we are considering $\\sigma _+$ circularly polarized light and working in a Faraday geometry, the evolution of $|\\!\\!\\uparrow \\Downarrow \\rangle $ is trivial.",
"We can therefore reduce the complexity of the problem by omitting this state.",
"The laser Hamiltonian reads $H^{\\prime }_{\\mathrm {L}} = \\hbar \\Omega \\left(|0\\rangle \\langle \\downarrow \\Uparrow \\!\\!| +|\\!\\!\\downarrow \\Uparrow \\rangle \\langle 0|\\right).$ where $\\Omega $ is the Rabi frequency.",
"The nuclear Zeeman Hamiltonian is given by $H_{\\mathrm {n}}^{\\mathrm {Z}} = g_{\\mathrm {n}} \\mu _{\\mathrm {n}} B I_z,$ with $g_{\\mathrm {n}}$ the nuclear Landé $g$ -factor and $\\mu _{\\mathrm {n}}$ the nuclear Bohr magneton.",
"The electronic hyperfine Hamiltonian within the homogeneous coupling approximation is given by, $H_{\\mathrm {HF}}^{\\mathrm {e}} = H_{\\mathrm {HF},z}^{\\mathrm {e}} + H_{\\mathrm {HF},\\perp }^{\\mathrm {e}} = A^{\\mathrm {e}}\\left(S_z I_z + \\frac{1}{2}\\left(S_+ I_- + S_- I_+\\right)\\right).$ Here, $S_z$ ($I_z$ ) is the electron (nuclear) spin operator in $z$ direction and we have introduced the ladder operators, $S_\\pm = S_x \\pm \\mathrm {i}S_y$ and $I_\\pm = I_x \\pm \\mathrm {i}I_y$ .",
"We denote the average hyperfine coupling constant as $A^{\\mathrm {e}}$ , the longitudinal part of Eq.",
"(REF ) by $H_{\\mathrm {HF},z}^{\\mathrm {e}}$ , and the transverse one by $H_{\\mathrm {HF},\\perp }^{\\mathrm {e}}$ .",
"The effective hyperfine Hamiltonian for heavy holes can be written as [37], $\\begin{aligned}H_{\\mathrm {HF}}^{\\mathrm {h}} &= H_{\\mathrm {HF},z}^{\\mathrm {h}} + H_{\\mathrm {HF},\\perp 1}^{\\mathrm {h}} +H_{\\mathrm {HF},\\perp 2}^{\\mathrm {h}} + H_{\\mathrm {HF},\\mathrm {nc}}^{\\mathrm {h}} \\\\& = A^{\\mathrm {h}}_z S^{\\mathrm {h}}_z I_z + A_{\\perp 1}^{\\mathrm {h}} S^{\\mathrm {h}}_+ I_- + A_{\\perp 1}^{\\mathrm {h}\\ast } S^{\\mathrm {h}}_- I_+ \\\\&\\phantom{=} + A_{\\perp 2}^{\\mathrm {h}} S^{\\mathrm {h}}_+ I_+ + A_{\\perp 2}^{\\mathrm {h}\\ast } S^{\\mathrm {h}}_-I_- + A^{\\mathrm {h}}_{\\mathrm {nc}} S^{\\mathrm {h}}_z I_+ + A^{\\mathrm {h}\\ast }_{\\mathrm {nc}} S^{\\mathrm {h}}_z I_-,\\end{aligned}$ where $S^{\\mathrm {h}}_i$ ($i=z,\\pm $ ) are pseudospin operators for the effective heavy hole states [37].",
"We use a similar notation to the one introduced in Eq.",
"(REF ) for longitudinal and transverse interactions.",
"We denote the non-collinear term by $H_{\\mathrm {Hf},\\mathrm {nc}}^{\\mathrm {h}}$ .",
"Here, we follow the procedure of Refs.",
"[39], [40] and describe the contribution of the transverse (flip-flop) terms of Eqs.",
"(REF ) and (REF ) on the dynamics as a dissipative process.",
"The evolution of the system is then described by the Lindblad master equation [41], $\\dot{\\rho } =-\\frac{\\mathrm {i}}{\\hbar } [H,\\rho ] + \\sum _{j=1}^{d^2 -1} ( [L_j \\rho , L_j^{\\dag }] +[L_j, \\rho L_j^{\\dag }])/2$ , with $d$ the dimension of the Hilbert space.",
"Since we only consider dissipative processes within the electronic subspace, we get by with less Lindblad operators.",
"We take into account spontaneous emission from the bright exciton $|\\!\\!\\downarrow \\Uparrow \\rangle $ and from both (quasi) dark excitons to the ground state [29].",
"These are respectively described by $L_1 = \\sqrt{\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Uparrow }}|0\\rangle \\langle \\downarrow \\Uparrow \\!\\!|$ , $L_2 =\\sqrt{\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Uparrow }} |0\\rangle \\langle \\uparrow \\Uparrow \\!\\!|$ , and $L_3= \\sqrt{\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Downarrow }} |0\\rangle \\langle \\downarrow \\Downarrow \\!\\!|$ , where $\\Gamma _{\\mathrm {sp}}^j$ , $j=\\downarrow \\Uparrow , \\uparrow \\Uparrow ,\\downarrow \\Downarrow $ , is the spontaneous decay rate.",
"We describe the nuclear spin state with its total angular momentum $j$ and its projection along the magnetic field given by $m$ [c.f.",
"Fig.",
"REF ].",
"The Lindblad operators $L_4 =\\sqrt{\\Gamma _{\\mathrm {e}}^{\\downarrow \\Uparrow }} |0,j,m-1\\rangle \\langle \\downarrow \\Uparrow ,j,m|$ and $L_5 = \\sqrt{\\Gamma _{\\mathrm {\\mathrm {h}}}^{\\downarrow \\Uparrow }} |0,j,m+1\\rangle \\langle \\downarrow \\Uparrow ,j,m|$ respectively describe electron and hole flip-flop processes.",
"The rates $\\Gamma _{\\mathrm {e}}^{\\downarrow \\Uparrow }$ and $\\Gamma _{\\mathrm {h}}^{\\downarrow \\Uparrow }$ are calculated with the same method as in Ref.",
"[40], we find $\\Gamma _{\\mathrm {e}}^{\\downarrow \\Uparrow } \\simeq \\frac{\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Uparrow }}{4}\\left|\\frac{A^{\\mathrm {e}}\\sqrt{j(j+1)-m(m-1)}}{E^{\\downarrow \\Uparrow }_{\\uparrow \\Uparrow }-A^{\\mathrm {e}}(m-\\frac{1}{2}) + \\frac{3}{2}A_z^{\\mathrm {h}} +g_{\\mathrm {n}}\\mu _{\\mathrm {n}}B}\\right|^2,$ where we have neglected the contribution coming from $H_{\\mathrm {HF},\\perp 2}^{\\mathrm {h}}$ since $\\left|A_{\\perp ,2}^{\\mathrm {h}}\\right|/A^{\\mathrm {e}}\\ll 1$ and $\\Gamma _{\\mathrm {h}}^{\\downarrow \\Uparrow } = \\frac{\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Downarrow }}{4}\\left|\\frac{A^{\\mathrm {h}}_{\\perp ,1}\\sqrt{j(j+1)-m(m+1)}}{E^{\\downarrow \\Uparrow }_{\\downarrow \\Downarrow } + \\frac{1}{2}A^{\\mathrm {e}} + 3A_z^{\\mathrm {h}}(m+\\frac{1}{2}) - g_{\\mathrm {n}}\\mu _{\\mathrm {n}}B}\\right|^2.$ Figure: (Color online).",
"(a) Nuclear spin polarization PP as a function of pumping timet p t_{\\mathrm {p}} and spontaneous decay rate Γ sp ↑⇑ ≈Γ sp ↓⇓ =Γ sp d \\Gamma _{\\mathrm {sp}}^{\\uparrow \\Uparrow }\\approx \\Gamma _{\\mathrm {sp}}^{\\downarrow \\Downarrow } = \\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}.",
"Values of other parametersare given in the main text.",
"(b) Traces taken along Γ sp d =2·10 8 ,2·10 7 , and 2·10 6 Hz \\Gamma _{\\mathrm {sp}}^{\\mathrm {d}} = 2\\cdot 10^8,\\,2\\cdot 10^7,\\,\\mathrm {and}\\,2\\cdot 10^{6}\\,\\mathrm {Hz}.",
"PP saturates atlower values for smaller Γ sp d \\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}'s.",
"(c) Trace taken along t p =30st_{\\mathrm {p}}= 30\\,\\mathrm {s} showing the dependence on Γ sp d \\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}.",
"(d) Comparison ofPP as a function of t p t_{\\mathrm {p}} between H HF h =H HF ,z h H_{\\mathrm {HF}}^{\\mathrm {h}}=H_{\\mathrm {HF},z}^{\\mathrm {h}}(gray), for which saturation corresponds to formation of a nuclear spin dark state, andthe full effective Hamiltonian Eq.",
"() (red) forΓ sp d =2·10 7 Hz \\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}=2\\cdot 10^7\\,\\mathrm {Hz}.",
"(e) Same as (d) but compared withH HF h =H HF ,z h +H HF ,⊥1 h H_{\\mathrm {HF}}^{\\mathrm {h}}=H_{\\mathrm {HF},z}^{\\mathrm {h}} + H_{\\mathrm {HF},\\perp 1}^{\\mathrm {h}}.",
"The result showsthat heavy hole mediated flip-flop processes are negligible.",
"(f) Same as (d) but comparedwith H HF h =H HF ,z h +H HF , nc h H_{\\mathrm {HF}}^{\\mathrm {h}}=H_{\\mathrm {HF},z}^{\\mathrm {h}} + H_{\\mathrm {HF},\\mathrm {nc}}^{\\mathrm {h}}.",
"The heavyhole hyperfine non-collinear interaction is the origin of the lower values of PP sinceit leads to an effective nuclear spin diffusion mechanism.We assume nuclear spins to be initially in a thermal state.",
"This is a reasonable assumptions even for experiments performed at low temperatures, where the thermal energy is larger than the nuclear Zeeman, $k_{\\mathrm {B}} T \\gg E_{\\mathrm {Z}}^{\\mathrm {nuc}}$ , with $k_{\\mathrm {B}}$ the Boltzmann's constant.",
"Thus, at $t=0$ , the nuclear spins are assumed to be in a fully mixed state.",
"Further assuming spin-1/2 for the nuclei, we have [40] $\\rho _{\\mathrm {nuc}} = \\sum _{j,m}\\frac{(2j+1)N!\\left[\\Theta (j+m)-\\Theta (j-m)\\right]}{\\left(\\frac{N}{2} + j + 1\\right)!\\left(\\frac{N}{2} -j\\right)!",
"2^N}|j,m\\rangle \\langle j,m|,$ with $N$ the number of nuclear spins and $\\Theta (x)$ is the Heaviside theta function.",
"The initial electronic state is given by the quantum dot vacuum, i.e.",
"$\\rho _{\\mathrm {e}} =|0\\rangle \\langle 0|$ .",
"Thus, the density matrix describing the whole system at $t=0$ is written as $\\rho = \\rho _{\\mathrm {e}} \\otimes \\rho _{\\mathrm {nuc}}$ .",
"In Fig.",
"REF (a), we present the degree of nuclear spin polarization $P=\\mathrm {Tr}\\left[\\rho _{\\mathrm {nuc}}(t_{\\mathrm {p}}) I_z\\right]/P_{\\mathrm {max}}$ as a function of pumping time $t_{\\mathrm {p}}$ and spontaneous decay rate of the dark states $\\Gamma ^{\\mathrm {d}}_{\\mathrm {sp}}$ , where $P_{\\mathrm {max}}=N/2$ .",
"Since the ratio of the dark states energy is nearly one, $E_{\\uparrow \\Uparrow }/E_{\\downarrow \\Downarrow }\\approx 1$ [37], we have $\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Uparrow } \\simeq \\Gamma _{\\mathrm {sp}}^{\\downarrow \\Downarrow } = \\Gamma ^{\\mathrm {d}}_{\\mathrm {sp}}$ .",
"The calculations were performed with $\\delta _0=5.6213\\cdot 10^{11}\\,\\mathrm {Hz}$ , $\\delta _1 = \\delta _2 =5.3174\\cdot 10^{10}\\,\\mathrm {Hz}$ , $B=8\\,\\mathrm {T}$ , $g_{\\mathrm {e}}=-0.35$ , $g_{\\mathrm {h}}=0.63$ , $g_{\\mathrm {n}}\\mu _{\\mathrm {n}} = 3.3\\cdot 10^{-8}\\,\\mathrm {eV/T}$ , $\\Delta =0\\,\\mathrm {Hz}$ , $\\Omega =2.03\\cdot 10^{10}\\,\\mathrm {Hz}$ , $A^{\\mathrm {e}}=10^8\\,\\mathrm {Hz}$ , $A_z^{\\mathrm {h}}=-10^7\\,\\mathrm {Hz}$ , $\\left|A_{\\perp 1}^{\\mathrm {h}}\\right|= 3\\cdot 10^5\\,\\mathrm {Hz}$ , $A_{\\mathrm {nc}}^{\\mathrm {h}}= 3\\cdot 10^5\\,\\mathrm {Hz}$ , $\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Uparrow } = 2\\cdot 10^9\\,\\mathrm {Hz}$ , and $N=30$ .",
"We have cancelled out the imaginary part of $H_{\\mathrm {HF},\\mathrm {nc}}^{\\mathrm {h}}$ by performing a suitable rotation of angle $\\theta $ , $U=\\exp [\\mathrm {i}\\theta I_z]$ .",
"We notice that the saturation of the nuclear polarization depends strongly on the lifetime of the dark states.",
"To demonstrate clearly this behavior, we present traces taken respectively for different values of $\\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}$ [Fig.",
"REF (b)] and for $t_{\\mathrm {p}}=30\\,\\mathrm {s}$ [Fig.",
"REF (c)], for which $P$ has reached saturation.",
"Both of these traces show that $P$ saturates at smaller values for slower $\\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}$ .",
"To come to a clear mechanism that explains this result, we compare $P$ as function of $t_{\\mathrm {p}}$ for $\\Gamma _{\\mathrm {d}}= 10^{7}\\,\\mathrm {s^{-1}}$ between an Ising-like ($H_{\\mathrm {HF},z}^\\mathrm {h}$ ) and other forms of the effective heavy hole hyperfine Hamiltonian.",
"In Fig.",
"REF (d), we compare $P$ obtained with $H_{\\mathrm {HF},z}^{\\mathrm {h}}$ (gray) to the effective Hamiltonian given by Eq.",
"(REF ) (red).",
"The different values of $P$ at saturation indicate that the relatively small corrections to the Ising-like hyperfine Hamiltonian influence the dynamics.",
"For $H_{\\mathrm {HF},z}^{\\mathrm {h}}$ , $P$ saturates due to formation of a nuclear spin dark state [40].",
"To tell apart the contribution of $H_{\\mathrm {HF},\\perp 1}^{\\mathrm {h}}$ and $H_{\\mathrm {HF},\\mathrm {nc}}^{\\mathrm {h}}$ , we plot $P$ obtained with $H_{\\mathrm {HF},z}^{\\mathrm {h}}$ (gray) and $H_{\\mathrm {HF},z}^{\\mathrm {h}} + H_{\\mathrm {HF},\\perp 1}^{\\mathrm {h}}$ (orange) in Fig.",
"REF (e), which show that heavy hole flip-flop processes are irrelevant for the nuclear spin dynamics.",
"In the presence of a large magnetic field and due to the smallness of $\\left|A_{\\perp ,1}^{\\mathrm {h}}\\right|$ , the heavy hole forbidden relaxation rate $\\Gamma _{\\mathrm {h}}^{\\downarrow \\Uparrow }$ is too slow compared to electron forbidden relaxation rate $\\Gamma _{\\mathrm {e}}^{\\downarrow \\Uparrow }$ and to spontaneous emission $\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Uparrow }$ to have an impact on the dynamics.",
"Finally, we verify that the non-collinear interaction is responsible for saturations lower than the nuclear dark state limit.",
"In Fig.",
"REF (f), we show $P$ calculated with $H_{\\mathrm {HF},z}^{\\mathrm {h}}$ (gray) and $H_{\\mathrm {HF},z}^{\\mathrm {h}} + H_{\\mathrm {HF},\\mathrm {nc}}^{\\mathrm {h}}$ (blue).",
"Our results demonstrate that the heavy hole non-collinear hyperfine interaction leads to an effective nuclear spin diffusion mechanism that hinders $P$ .",
"As it can be observed from Figs.",
"REF (a), (b), and (c), the diffusion becomes more prominent when the system is hold for a relatively long time in one of the optical dark states.",
"It has been reported that the oscillator strength for optical dark states is a hundred to a thousand times smaller than the oscillator strength of bright states [29], which implies $\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Uparrow }/\\Gamma _{\\mathrm {sp}}^{\\mathrm {d}} \\approx 100-1000$ .",
"Finally, our results indicate that upon reaching saturation most of the nuclear spin states are still populated and the system is left in a mixed state.",
"Thus, our findings suggest that there could be an alternative interpretation of recent experimental results about the sign of the Ising-like interaction [29].",
"The unexpected shift of the Overhauser field could simply originate from nuclear spin diffusion, which lowers $P$ , when measuring the spectral position of the optical dark states.",
"In the following, we propose a simple experiment to detect and simultaneously cancel the presence of non-collinear interactions.",
"The idea is to change the orientation of the external magnetic field to transform the nuclear Zeeman Hamiltonian into, $H_{\\mathrm {n}}^Z = g_{\\mathrm {n}} \\mu _{\\mathrm {n}} B \\cos (\\varphi ) I_z + g_{\\mathrm {n}} \\mu _{\\mathrm {n}} B\\sin (\\varphi )(I_+ + I_-)/2$ , with $\\varphi $ the rotation angle.",
"In our coordinate system the magnetic field has to be rotated around the $y$ -axis, i.e.",
"$\\varphi $ is the angle between the $z$ -axis and $\\mbox{$B$}$ .",
"We solve again a Lindblad master equation, but with a Hamiltonian that takes into account that $\\mbox{$B$}$ is not necessarily aligned with the growth axis of the quantum dot.",
"In addition to the trivial change $B \\rightarrow B\\cos (\\varphi ) \\equiv B_z$ in Eqs.",
"(REF ), (REF ), and (REF ) as well as the discussed modification of the nuclear Zeeman Hamiltonian, we also need to take into account that misalignment of $\\mbox{$B$}$ leads to mixing of bright and dark excitons via $H_{\\mathrm {bd}} = g_{\\mathrm {e}} \\mu _{\\mathrm {B}} B \\sin (\\varphi ) (S_++ S_-)/4 + g_{\\mathrm {h}}^{xx} \\mu _{\\mathrm {B}} B \\sin (\\varphi )(S_+^{\\mathrm {h}} + S_-^{\\mathrm {h}})/4$ , with $g_{\\mathrm {h}}^{xx} \\simeq g_{\\mathrm {h}}/10$ [42], [43] the heavy hole Landé $g$ -factor along the $x$ -axis.",
"We also add to the dissipative part of the Lindblad equation spontaneous relaxation from $|\\!\\!\\uparrow \\Downarrow \\rangle $ to the ground state with rate $\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Downarrow }$ and two non-conserving nuclear spin relaxation mechanisms.",
"These are described by $L_6 = \\sqrt{\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Downarrow }} |0\\rangle \\langle \\uparrow \\Downarrow \\!\\!|$ , $L_7 = \\sqrt{\\Gamma _{\\mathrm {e}}^{\\uparrow \\Downarrow }}|0, j, m+1\\rangle \\langle \\uparrow \\Downarrow , j, m|$ , and $L_8 =\\sqrt{\\Gamma _{\\mathrm {h}}^{\\uparrow \\Downarrow }}|0, j, m-1\\rangle \\langle \\uparrow \\Downarrow , j, m|$ with $\\Gamma _{\\mathrm {e}}^{\\uparrow \\Downarrow } \\simeq \\frac{\\Gamma _{\\mathrm {sp}}^{\\downarrow \\Downarrow }}{4}\\left|\\frac{A^{\\mathrm {e}}\\sqrt{j(j+1)-m(m+1)}}{E^{\\uparrow \\Downarrow }_{\\downarrow \\Downarrow }+A^{\\mathrm {e}}(m+\\frac{1}{2}) +\\frac{3}{2}A_z^{\\mathrm {h}} - g_{\\mathrm {n}}\\mu _{\\mathrm {n}}B_z}\\right|^2,$ and $\\Gamma _{\\mathrm {h}}^{\\uparrow \\Downarrow } = \\frac{\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Uparrow }}{4}\\left|\\frac{A^{\\mathrm {h}}_{\\perp 1}\\sqrt{j(j+1)-m(m-1)}}{E^{\\uparrow \\Downarrow }_{\\uparrow \\Uparrow } + \\frac{1}{2}A^{\\mathrm {e}} +\\frac{3}{2}A_z^{\\mathrm {h}}(m-\\frac{1}{2}) + g_{\\mathrm {n}}\\mu _{\\mathrm {n}}B_z}\\right|^2.$ Figure: (Color online).",
"(a) Nuclear spin polarization PP as a function of t p t_{\\mathrm {p}}and ϕ\\varphi (angle between the external magnetic field B\\mbox{$B$} and the zz-axis) forΓ sp d =2·10 7 s -1 \\Gamma _{\\mathrm {sp}}^{\\mathrm {d}}= 2\\cdot 10^7\\,\\mathrm {s^{-1}}.",
"(b) Trace taken alongt p =30st_{\\mathrm {p}}=30\\,\\mathrm {s}.",
"The effect of the non-collinear interaction can be cancelled byorienting the magnetic field opposite to the effective Overhauser field defined by thehyperfine non-collinear Hamiltonian.In Fig.",
"REF (a), we plot the nuclear spin polarization $P$ as a function of $t_{\\mathrm {p}}$ and $\\varphi $ .",
"We use the same set of parameters as before and $\\Gamma _{\\mathrm {sp}}^{\\mathrm {d}} = 10^7\\,\\mathrm {s^{-1}}$ .",
"As for the optical dark state, we have $E_{\\downarrow \\Uparrow }/E_{\\uparrow \\Downarrow } \\approx 1$ which allows us to write $\\Gamma _{\\mathrm {sp}}^{\\uparrow \\Downarrow } \\simeq \\Gamma _{\\mathrm {sp}}^{\\downarrow \\Uparrow } =\\Gamma _{\\mathrm {sp}}^{\\mathrm {b}}$ .",
"The results show that the non-collinear heavy hole hyperfine interaction is fully cancelled at $\\varphi \\simeq -0.014\\,\\mathrm {rad}$ , for which we retrieve the saturation limit set by the nuclear spin dark state.",
"In Fig.",
"REF (b), we show a trace taken for $t_{\\mathrm {p}} = 30\\,\\mathrm {s}$ .",
"In conclusion, we have shown that the effective heavy hole hyperfine interaction via non-collinear terms influences nuclear spin dynamics.",
"In particular, we have shown how to experimentally detect and cancel the effects of such interaction.",
"We expect the described effects to be stronger when considering an inhomogeneous hyperfine Hamiltonian since the statistical weight of the states contributing the most to $P$ are not suppressed [40].",
"Moreover, when trying to cancel the heavy hole non-collinear interaction, a series of maximums should be observed as a function of the rotation angles.",
"Each maximum corresponds to a different nuclear species.",
"This also implies that none of the maximums correspond to the limit set by the formation of a nuclear dark state.",
"We are thankful to J. Hildmann and R. J. Warburton for fruitful discussions.",
"We acknowledge funding from the Swiss NF, NCCR QSIT, and $\\mathrm {S}^3$ NANO."
]
] | 1403.0490 |
[
[
"An overview of neV probes of PeV scale physics --- and of what's in\n between"
],
[
"Abstract Low-energy experiments which would identify departures from the Standard Model (SM) rely either on the unexpected observation of symmetry breaking, such as of CP or B, or on an observed significant deviation from a precise SM prediction.",
"We discuss examples of each search strategy, and show that low-energy experiments can open windows on physics far beyond accessible collider energies.",
"We consider how the use of a frequentist analysis framework can redress the impact of theoretical uncertainties in such searches --- and how lattice QCD can help control them."
],
[
"Context",
"Direct searches for new physics at the LHC has yielded the discovery of the Higgs boson [1], [2], but no unanticipated, new particles — as yet.",
"On the other hand, observational cosmology, analyzed in the framework of general relativity, tells us that only 4% of the energy density of the universe is in the matter we know [3], so that the SM of particle physics, successful though it is, is probably incomplete.",
"The lack of evidence thus far for new physics and interactions through collider studies at the highest energies motivates broader thinking in the search for new physics.",
"For example, the missing matter could be weakly coupled, making it more challenging if not impossible to identify in a collider environment.",
"Low-energy, precision searches for new physics can also probe this alternative possibility and thus play a key role in the search for new physics.",
"In this contribution we offer a terse overview of the diverse program these experiments comprise.",
"Generally, there are two distinct search strategies.",
"That is, one can either make null tests of the breaking of SM symmetries, or refine the measurement of quantities which can be computed, or assessed, precisely with the SM.",
"In the former case, one can test, e.g., B-L invariance by searching for $n{\\bar{n}}$ oscillations or neutrinoless double-$\\beta $ decay.",
"Although CP is not a symmetry of the SM, there are nevertheless observables for which the SM prediction is so small that searches at current levels of sensitivity also constitute null tests.",
"Searches for permanent electric dipole moments (EDMs) of the neutron or electron, e.g., or for CP violation in the charm sector, be it through $D{\\bar{D}}$ mixing or decay rate asymmetries, are examples of such tests.",
"There are also a variety of nonzero observables whose value can be tested precisely within the SM.",
"Example of this latter class include (i) parity-violating electron scattering from electrons, protons, or light nuclei, in varying kinematics, (ii) the anomalous magnetic moment of the muon (or electron), and (iii) final-state angular correlations in neutron and nuclear $\\beta $ decay.",
"All these studies probe the possibility of new degrees of freedom, including those which couple so weakly to known matter that they are effectively “hidden”."
],
[
"Motivation",
"The SM leaves many questions unanswered; e.g., it cannot explain the nature of dark matter or dark energy, nor can it explain the magnitude, or even existence, of the cosmic baryon asymmetry (BAU).",
"The BAU itself can be determined by confronting the observed $^2$ H abundance with big-bang nucleosynthesis, yielding $\\eta _{\\rm B}\\equiv n_{\\rm baryon}/n_{\\rm photon}=(5.96\\pm 0.28)\\times 10^{-10}$ [4].",
"As demonstrated long ago by Sakharov, the particle physics of the early universe can explain this asymmetry if B, C, and CP violation exist in a non-equilibrium environment [5].",
"Nominally the SM would seem to possess all the conditions required to generate the BAU.",
"However, with the discovery of a Higgs boson of 125 GeV in mass, the phase transition associated with electroweak symmetry breaking is no longer of first order [6], and the SM cannot explain a nonzero BAU.",
"Thus our existence is itself evidence of physics BSM!",
"The mechanism of CP violation in the SM has also been faulted, because an estimate of the BAU (now moot) with a sufficiently light Higgs mass yields a BAU orders of magnitude too small, namely, $\\eta _{\\rm B} < 10^{-26}$ [7].",
"In the SM nonzero CP-violating effects require the participation of three generations of quarks of differing mass [8].",
"Consequently, the small value of the computed BAU follows, in part, from the smallness of SU(3)$_f$ breaking compared to the electroweak scale.",
"This special way in which CP violation appears in the SM makes it seem that new sources of CP violation are needed to explain the BAU; however, searches for such effects at the $B$ factories and through improved EDM limits have thus far failed to discover them.",
"A BAU could potentially be generated in very different ways, and low-energy experiments can help select the underlying mechanism.",
"For example, the discovery of a nonzero EDM at current levels of sensitivity would speak to new CP phases and the possibility of electroweak baryogenesis.",
"The discovery of neutrinoless $\\beta \\beta $ decay would tell us that neutrinos are Majorana particles [9], and would make various models of leptogenesis possible [10].",
"The discovery of $n{\\bar{n}}$ oscillations would reveal that neutrons are also Majorana particles and would support alternate models for baryogenesis [11].",
"Finally, the discovery of a dark-matter asymmetry [12] would tell us that DM carries “baryon” number, suggesting that the key to the nature of dark matter and the origin of the BAU could be tied [13], [14].",
"But only EDMs searches are directly connected to the possibility of new physics at the weak scale."
],
[
"Analysis Framework",
"It is natural to think of the SM as the low-energy limit of a more fundamental theory, and to use an effective theory framework to analyze its possible extensions.",
"To illustrate, suppose new physics enters at an energy scale $E> \\Lambda _{\\rm BSM}$ .",
"Then for energies below the new-physics scale $\\Lambda _{\\rm BSM}$ we can extend the SM through the appearance of effective operators of mass dimension $D>4$ ; specifically, ${\\cal L} = {\\cal L}_{\\rm SM}+ \\sum _i \\frac{c_i}{\\Lambda _{i}^{D-4}}{\\cal O}_i^D\\,.$ Noting the severe empirical constraints on new physics from flavor-changing processes [15], [16], [17], it is efficient to impose SU(2)$_L\\times $ U(1) gauge invariance on the operator basis.",
"If we assume that an experimental bound is saturated by a single term and that the associated $c_i$ is of ${\\cal O}(1)$ , we can estimate the scale $\\Lambda _i$ ; this indicates the rough energy reach of the experiment.",
"For example, a neutrino mass of $0.1\\,{\\rm eV}$ , the expected minimum mass accessible to near future neutrinoless $\\beta \\beta $ experiments [18], if generated via the seesaw mechanism implies $\\Lambda _{\\rm BSM} \\sim 10^{14-15}\\,{\\rm GeV}$ [19].",
"Such estimates should be used with care."
],
[
"The QCD Challenge",
"Estimates of the energy reach of a particular experimental measurement can require non-perturbative QCD input in the form of a hadronic matrix element.",
"In this lattice QCD can play a crucial role.",
"There are examples, however, where lattice QCD calculations are not yet good enough to meet experimental needs.",
"A prominent example of this is the determination of the axial coupling constant of the nucleon $g_A$ .",
"In this specific case, $g_A$ can be determined directly from experiment, specifically from the measured angular-correlation coefficients in neutron $\\beta $ decay.",
"The existing lattice-QCD calculations do not agree well with each other.",
"Moreover, the lattice results typically lie some 5-15% below the values from $\\beta $ decay, albeit with much larger errors [20]."
],
[
"Examples",
"We now turn to specific examples of low-energy experimental probes of new physics."
],
[
"Heavy-atom EDMs",
"Currently, the most stringent experimental EDM limit comes from the study of the diamagnetic atom $^{199}$ Hg, for which $|d| < 3.1\\times 10^{-29}\\,{\\rm e-cm}$ at 95% C.L.",
"[21], a result roughly a thousand times more sensitive than the current experimental limit on the neutron EDM [22].",
"However, the atom's electrons shield any nonzero EDM which the nucleus may possess and weaken the constraint thereby placed on the existence of new sources of CP violation.",
"It has become possible to study the EDMs of very heavy atoms, such as $^{225}$ Ra [23] or $^{221/223}$ Rn [24], that mitigate the cancelling effect of electron shielding through their large $Z$ , finite nuclear size, and octupole deformation [25].",
"The evasion of electron shielding in $^{225}$ Ra is estimated to be some seven hundred times bigger than that in $^{199}$ Hg [26], making these systems excellent candidates for the discovery of a nonzero EDM.",
"Recently the permanent octupole deformation of $^{224}$ Ra has been established through Coulomb excitation studies at REX-ISOLDE (CERN) [27]; this makes the nucleus more “rigid” and the computation of the associated Schiff moment more robust [28].",
"With improved isotope yields, as possible, e.g., through direct production at a proton linac, one expects greatly improved sensitivity to EDMs [24]."
],
[
"Resolving the limits of the $V-A$ law in {{formula:5a851f33-7b3d-4428-8bde-f31a967eb9c5}} decay",
"The possibility of non-$(V-A)$ interactions in $\\beta $ decay can be probed through the angular correlations of the final-state particles.",
"Notably the differential decay rate $d^3\\Gamma /dE_e d\\Omega _{e\\nu }$ can contain a Fierz interference term $b$ ; this quantity vanishes at tree-level in the SM but is nonzero if scalar or tensor interactions are present.",
"Adopting an effective operator analysis of $\\beta $ -decay, working in a SU(2)$_L\\times $ U(1)-invariant basis in dimension six [29], [30], we have, at the quark level, at low energies [31], [32], [33], ${\\cal L}_{\\rm CC} &=&- \\frac{G_F^{(0)} V_{ud}}{\\sqrt{2}} \\ \\Big [\\ \\Big ( 1 + \\delta _{\\beta } \\Big ) \\ \\bar{e} \\gamma _\\mu (1 - \\gamma _5) \\nu _{e} \\cdot \\bar{u} \\gamma ^\\mu (1 - \\gamma _5) d\\\\&+& \\epsilon _S \\ \\bar{e} (1 - \\gamma _5) \\nu _{\\ell } \\cdot \\bar{u} d+\\epsilon _T \\ \\bar{e} \\sigma _{\\mu \\nu } (1 - \\gamma _5) \\nu _{\\ell } \\cdot \\bar{u} \\sigma ^{\\mu \\nu } (1 - \\gamma _5) d + \\ \\dots \\ + {\\rm h.c.}~.\\Big ]\\nonumber $ The first term represents the famous $V-A$ law of the SM, and the others, including the scalar and tensor terms controlled by $\\epsilon _S$ and $\\epsilon _T$ , respectively, reflect the appearance of non-SM physics.",
"The tree-level coupling $G_F^{(0)}$ is fixed through the measurement of muon decay and an analysis of its electroweak radiative corrections, and $\\delta _\\beta $ reflect those to semi-leptonic transitions.",
"The matching to an effective theory in nucleon degrees of freedom requires the computation of hadronic matrix elements; the result maps to the familiar ${\\cal H}_{\\rm eff}$ of Lee and Yang [34] employed in Ref. [35].",
"We refer to Ref.",
"[36] for a detailed review.",
"In neutron $\\beta $ decay, we have $&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\langle p(p^{\\prime }) | \\bar{u} \\gamma ^\\mu d | n (p) \\rangle \\equiv \\overline{u}_p (p^{\\prime })\\left[ f_1(q^2)\\gamma ^\\mu - i \\frac{f_2(q^2)}{M}\\sigma ^{\\mu \\nu }q_\\nu +\\frac{f_3(q^2)}{M}q^\\mu \\right]u_n(p) \\,,\\nonumber \\\\&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\langle p(p^{\\prime }) | \\bar{u} \\gamma ^\\mu \\gamma _5 d | n (p) \\rangle \\equiv \\overline{u}_p (p^{\\prime }) \\left[g_1(q^2)\\gamma ^\\mu \\gamma _5 - i\\frac{g_2(q^2)}{M}\\sigma ^{\\mu \\nu }\\gamma _5 q_\\nu + \\dots \\right]u_n(p) \\,, \\\\&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\langle p(p^{\\prime }) | \\bar{u} d | n (p) \\rangle \\equiv \\overline{u}_p (p^{\\prime }) g_S(q^2) u_n(p) \\,, \\quad \\langle p(p^{\\prime }) | \\bar{u} \\sigma _{\\mu \\nu } d | n (p) \\rangle \\equiv \\overline{u}_p (p^{\\prime })\\left[ g_T(q^2)\\sigma ^{\\mu \\nu } + \\dots \\right]u_n(p) \\,,\\nonumber $ where $q\\equiv p^{\\prime }-p$ denotes the momentum transfer and $M$ is the neutron mass.",
"Working at leading order (LO) in the recoil expansion (i.e., neglecting terms of ${\\cal O}(\\varepsilon /M)$ , where $|\\varepsilon |\\ll M$ ) in new physics and at NLO in the SM terms, all $q^2$ dependence is negligible — and other negligible terms appear as “$\\dots $ ” in Eq.",
"(REF ).",
"Thus we have $f_1(0) \\equiv g_V$ , $g_1(0) \\equiv g_A$ with $g_V = 1$ and $f_2(0)=(\\kappa _p - \\kappa _n)/2$ , noting $\\kappa _{p(n)}$ is the anomalous magnetic moment of the proton (neutron), in the SM, up to ${\\cal O}(\\varepsilon /M)$ corrections.",
"The quantities $f_3(0) \\equiv f_3$ and $g_2(0) \\equiv g_2$ are second-class-current contributions, in that they vanish in the SM in the isospin-symmetric limit.",
"Bhattacharya et al.",
"have computed $g_S(0)=g_S$ and $g_T(0)=g_T$ in lattice QCD and have shown that their results sharpen the limits on $\\epsilon _{S,T}$ considerably [32].",
"Although all the mentioned matrix elements could be computed in lattice QCD, not all of the precise matrix elements needed have been — and we have already noted the problem with $g_A$ .",
"Consequently, to resolve the limits of the $V-A$ law in $\\beta $ decay we must fit for SM physics, specifically for $\\lambda \\equiv g_A/g_V$ , and BSM physics simultaneously [37].",
"There are poorly known recoil-order matrix elements, notably $g_2$ and $f_3$ ; they enter in recoil order and can mimic the appearance of scalar and tensor effects.",
"Let us consider the prospects for finding BSM physics through $b$ ; we can access this quantity either through a measurement of the electron energy spectrum or through its impact on the asymmetry measurements which determine the correlation coefficients $a$ and $A$ .",
"Many systematic errors cancel using this latter approach, and we will use it here.",
"We address the analysis problem we have posed in the frequentist $R$ fit (maximum likelihood) framework adopted by CKMFitter for the analysis of flavor-changing processes for the parameters of the CKM matrix [38], [15].",
"Most importantly this method provides a means of removing the impact of (SM) theoretical errors on the allowed new-physics phase space.",
"We have used Monte Carlo pseudodata of neutron decay observables to illustrate our implementation of this method [37].",
"For concreteness we recap our methodology.",
"We employ a pseudodata set of measurements of $a$ and $A$ as a function of the electron energy $E_e$ , along with values of the neutron lifetime.",
"These results, collectively $\\lbrace x_{\\rm exp}\\rbrace $ , are to be compared with the theoretical computations of the same quantities, collectively $\\lbrace x_{\\rm theo}(y_{\\rm mod})\\rbrace $ , determined by the parameters $\\lbrace y_{\\rm mod}\\rbrace $ .",
"A fraction of the set $\\lbrace y_{\\rm mod}\\rbrace $ can only be determined from theory; this subset is labelled $\\lbrace y_{\\rm calc}\\rbrace $ .",
"The underlying distribution of the $\\lbrace y_{\\rm calc}\\rbrace $ parameters is ill-known; the test statistic $\\chi ^2$ is thus modified so that the theoretical likelihood does not contribute to the $\\chi ^2$ .",
"With this we fit a “New Physics” data set for $\\lambda $ and $b_{\\rm BSM}$ in which $\\lambda =1.2701$ and $b_{\\rm BSM}=-0.00522$ for a value of $g_T \\epsilon _T=1.0\\times 10^{-3}$ just below experimental bounds.",
"The results as a function of the theoretical values of $f_3$ and $g_2$ are shown in Fig.",
"REF .",
"We see that the best-fit ellipses soften in the presence of the second-class current terms.",
"The method also allows us to construct a test statistic for the validity of the SM; the essential role the neutron lifetime plays in realizing it is shown in Fig.",
"REF .",
"For reference, we note that the lattice-QCD result is determined by an extrapolation from the form factors computed in a $|\\Delta S|=1$ transition [39], yielding a result at odds with a QCD sum rule calculation [40].",
"For “Lattice” we use $f_3 \\in (-0.002,0.016)$ and $g_2 \\in (0.020,0.066)$ , replacing $g_2$ with $g_2 \\in (-0.033,0.066)$ for the union of both.",
"We advocate for a lattice QCD calculation of $g_2$ and $f_3$ in neutron decay.",
"Figure: (Left) Illustration of the impact of theoretical certainties(second-class currents) on the search fornon-(V-A)(V-A) currents in neutron β\\beta -decay.",
"We show the results ofthe two-parameter (λ,b BSM )(\\lambda ,b_{\\rm BSM}) simultaneous fit to the{a,A}\\lbrace a,A \\rbrace New Physics data set.",
"The bands indicate the68.3% CL allowed regions.",
"(Right) An illustration of the essential role the neutron lifetimewould play in falsifyingthe SM.",
"We refer to Ref.",
"for all details."
],
[
"Spin-independent CP violation in radiative $\\beta $ decay",
"In radiative $\\beta $ decay one can form a T-odd correlation from momenta alone.",
"This is a pseudo-T-odd observable, so that it can be mimicked by final-state interactions (FSI) in the SM.",
"The energy release associated with neutron and nuclear $\\beta $ decay is sufficiently small that only electromagnetic FSI can possibly generate a mimicking effect.",
"These have been computed up to recoil order terms [41], so that we can determine the SM background rather well.",
"The interaction which generates the primary effect comes from the gauging of the Wess-Zumino-Witten term under SM electroweak gauge invariance[42], [43], [44].",
"A direct measurement of this correlation constrains the phase of this interaction from physics BSM, possibly from “strong” hidden sector interactions [45]."
],
[
"Summary",
"We believe the analysis framework we have espoused in $\\beta $ decay should benefit the analysis of other low-energy experiments.",
"It should be possible to discover physics BSM through the low energy, precision measurements — the game is afoot!",
"S.G. would like to thank the organizers for the invitation to speak; she acknowledges partial support from the U.S. Department of Energy Office of Nuclear Physics under grant number DE-FG02-96ER40989.",
"B.P.",
"acknowledges partial support from the U.S. Department of Energy Office of Nuclear Physics under grant number DE-FG02-08ER41557."
]
] | 1403.0521 |
[
[
"Cross-Scale Cost Aggregation for Stereo Matching"
],
[
"Abstract Human beings process stereoscopic correspondence across multiple scales.",
"However, this bio-inspiration is ignored by state-of-the-art cost aggregation methods for dense stereo correspondence.",
"In this paper, a generic cross-scale cost aggregation framework is proposed to allow multi-scale interaction in cost aggregation.",
"We firstly reformulate cost aggregation from a unified optimization perspective and show that different cost aggregation methods essentially differ in the choices of similarity kernels.",
"Then, an inter-scale regularizer is introduced into optimization and solving this new optimization problem leads to the proposed framework.",
"Since the regularization term is independent of the similarity kernel, various cost aggregation methods can be integrated into the proposed general framework.",
"We show that the cross-scale framework is important as it effectively and efficiently expands state-of-the-art cost aggregation methods and leads to significant improvements, when evaluated on Middlebury, KITTI and New Tsukuba datasets."
],
[
"Introduction",
" Figure: Cross-Scale Cost Aggregation.",
"Top-Left: enlarged-view of a scan-line subsegment from Middlebury Teddy stereo pair; Top-Right: cost volumes ({𝐂 s } s=0 S \\lbrace \\mathbf {C}^s\\rbrace _{s=0}^{S}) after cost computation at different scales, where the intensity + gradient cost function is adopted as in , , .",
"Horizontal axis xx indicates different pixels along the subsegment, and vertical axis LL represents different disparity labels.",
"Red dot indicates disparity generated by current cost volume while green dot is the ground truth; Bottom-Right: cost volumes after applying different cost aggregation methods at the finest scale (from top to bottom: NL , ST , BF and GF ); Bottom-Left: cost volumes after integrating different methods into our cross-scale cost aggregation framework, where cost volumes at different scales are adopted for aggregation.",
"(Best viewed in color.",
")Dense correspondence between two images is a key problem in computer vision [12].",
"Adding a constraint that the two images are a stereo pair of the same scene, the dense correspondence problem degenerates into the stereo matching problem [23].",
"A stereo matching algorithm generally takes four steps: cost computation, cost (support) aggregation, disparity computation and disparity refinement [23].",
"In cost computation, a 3D cost volume (also known as disparity space image [23]) is generated by computing matching costs for each pixel at all possible disparity levels.",
"In cost aggregation, the costs are aggregated, enforcing piecewise constancy of disparity, over the support region of each pixel.",
"Then, disparity for each pixel is computed with local or global optimization methods and refined by various post-processing methods in the last two steps respectively.",
"Among these steps, the quality of cost aggregation has a significant impact on the success of stereo algorithms.",
"It is a key ingredient for state-of-the-art local algorithms [36], [21], [33], [16] and a primary building block for some top-performing global algorithms [34], [31].",
"Therefore, in this paper, we mainly concentrate on cost aggregation.",
"Most cost aggregation methods can be viewed as joint filtering over the cost volume [21].",
"Actually, even simple linear image filters such as box or Gaussian filter can be used for cost aggregation, but as isotropic diffusion filters, they tend to blur the depth boundaries [23].",
"Thus, a number of edge-preserving filters such as bilateral filter [28] and guided image filter [7] were introduced for cost aggregation.",
"Yoon and Kweon [36] adopted the bilateral filter into cost aggregation, which generated appealing disparity maps on the Middlebury dataset [23].",
"However, their method is computationally expensive, since a large kernel size ($35 \\times 35$ ) is typically used for the sake of high disparity accuracy.",
"To address the computational limitation of the bilateral filter, Rhemann [21] introduced the guided image filter into cost aggregation, whose computational complexity is independent of the kernel size.",
"Recently, Yang [33] proposed a non-local cost aggregation method, which extends the kernel size to the entire image.",
"By computing a minimum spanning tree (MST) over the image graph, the non-local cost aggregation can be performed extremely fast.",
"Mei [16] followed the non-local cost aggregation idea and showed that by enforcing the disparity consistency using segment tree instead of MST, better disparity maps can be achieved than [33].",
"All these state-of-the-art cost aggregation methods have made great contributions to stereo vision.",
"A common property of these methods is that costs are aggregated at the finest scale of the input stereo images.",
"However, human beings generally process stereoscopic correspondence across multiple scales [17], [15], [14].",
"According to [14], information at coarse and fine scales is processed interactively in the correspondence search of the human stereo vision system.",
"Thus, from this bio-inspiration, it is reasonable that costs should be aggregated across multiple scales rather than the finest scale as done in conventional methods (Figure REF ).",
"In this paper, a general cross-scale cost aggregation framework is proposed.",
"Firstly, inspired by the formulation of image filters in [18], we show that various cost aggregation methods can be formulated uniformly as weighted least square (WLS) optimization problems.",
"Then, from this unified optimization perspective, by adding a Generalized Tikhonov regularizer into the WLS optimization objective, we enforce the consistency of the cost volume among the neighboring scales, i.e.",
"inter-scale consistency.",
"The new optimization objective with inter-scale regularization is convex and can be easily and analytically solved.",
"As the intra-scale consistency of the cost volume is still maintained by conventional cost aggregation methods, many of them can be integrated into our framework to generate more robust cost volume and better disparity map.",
"Figure REF shows the effect of the proposed framework.",
"Slices of the cost volumes of four representative cost aggregation methods, including the non-local method [33] (NL), the segment tree method [16] (ST), the bilateral filter method [36] (BF) and the guided filter method [21] (GF), are visualized.",
"We use red dots to denote disparities generated by local winner-take-all (WTA) optimization in each cost volume and green dots to denote ground truth disparities.",
"It can be found that more robust cost volumes and more accurate disparities are produced by adopting cross-scale cost aggregation.",
"Extensive experiments on Middlebury [23], KITTI [4] and New Tsukuba [20] datasets also reveal that better disparity maps can be obtained using cross-scale cost aggregation.",
"In summary, the contributions of this paper are three folds: A unified WLS formulation of various cost aggregation methods from an optimization perspective.",
"A novel and effective cross-scale cost aggregation framework.",
"Quantitative evaluation of representative cost aggregation methods on three datasets.",
"The remainder of this paper is organized as follows.",
"In Section , we summarize the related work.",
"The WLS formulation for cost aggregation is given in Section .",
"Our inter-scale regularization is described in Section .",
"Then we detail the implementation of our framework in Section .",
"Finally experimental results and analyses are presented in Section and the conclusive remarks are made in Section ."
],
[
"Related Work",
"Recent surveys [9], [29] give sufficient comparison and analysis for various cost aggregation methods.",
"We refer readers to these surveys to get an overview of different cost aggregation methods and we will focus on stereo matching methods involving multi-scale information, which are very relevant to our idea but have substantial differences.",
"Early researchers of stereo vision adopted the coarse-to-fine (CTF) strategy for stereo matching [15].",
"Disparity of a coarse resolution was assigned firstly, and coarser disparity was used to reduce the search space for calculating finer disparity.",
"This CTF (hierarchical) strategy has been widely used in global stereo methods such as dynamic programming [30], semi-global matching [25], and belief propagation [3], [34] for the purpose of accelerating convergence and avoiding unexpected local minima.",
"Not only global methods but also local methods adopt the CTF strategy.",
"Unlike global stereo methods, the main purpose of adopting the CTF strategy in local stereo methods is to reduce the search space [35], [11], [10] or take the advantage of multi-scale related image representations [26], [27].",
"While, there is one exception in local CTF approaches.",
"Min and Sohn [19] modeled the cost aggregation by anisotropic diffusion and solved the proposed variational model efficiently by the multi-scale approach.",
"The motivation of their model is to denoise the cost volume which is very similar with us, but our method enforces the inter-scale consistency of cost volumes by regularization.",
"Overall, most CTF approaches share a similar property.",
"They explicitly or implicitly model the disparity evolution process in the scale space [27], i.e.",
"disparity consistency across multiple scales.",
"Different from previous CTF methods, our method models the evolution of the cost volume in the scale space, i.e.",
"cost volume consistency across multiple scales.",
"From optimization perspective, CTF approaches narrow down the solution space, while our method does not alter the solution space but adds inter-scale regularization into the optimization objective.",
"Thus, incorporating multi-scale prior by regularization is the originality of our approach.",
"Another point worth mentioning is that local CTF approaches perform no better than state-of-the-art cost aggregation methods [10], [11], while our method can significantly improve those cost aggregation methods [21], [33], [16]."
],
[
"Cost Aggregation as Optimization",
" Figure: The flowchart of cross-scale cost aggregation: {𝐂 ^ s } s=0 S \\lbrace \\mathbf {\\hat{C}}^s\\rbrace _{s=0}^{S}is obtained by utilizing a set of input cost volumes, {𝐂 s } s=0 S \\lbrace \\mathbf {C}^s\\rbrace _{s=0}^{S}, together.Corresponding variables {i s } s=0 S ,{j s } s=0 S \\lbrace i^s\\rbrace _{s=0}^{S}, \\lbrace j^s\\rbrace _{s=0}^{S} and {l s } s=0 S \\lbrace l^s\\rbrace _{s=0}^{S} arevisualized.",
"The blue arrow represents an intra-scale consistency (commonly used in the conventional cost aggregation approaches),while the green dash arrow denotes an inter-scaleconsistency.",
"(Best viewed in color.",
")In this section, we show that the cost aggregation can be formulated as a weighted least square optimization problem.",
"Under this formulation, different choices of similarity kernels [18] in the optimization objective lead to different cost aggregation methods.",
"Firstly, the cost computation step is formulated as a function $f \\colon \\mathbb {R}^{W \\times H \\times 3} \\times \\mathbb {R}^{ W \\times H \\times 3 } \\mapsto \\mathbb {R}^{W \\times H\\times L}$ , where $W$ , $H$ are the width and height of input images, 3 represents color channels and $L$ denotes the number of disparity levels.",
"Thus, for a stereo color pair: $\\mathbf {I}, \\mathbf {I^{\\prime }} \\in \\mathbb {R}^{ W \\times H \\times 3 }$ , by applying cost computation: $\\mathbf {C} = f( \\mathbf {I}, \\mathbf {I^{\\prime }} ),\\vspace{-5.69054pt}$ we can get the cost volume $\\mathbf {C} \\in \\mathbb {R}^{W \\times H\\times L}$ , which represents matching costs for each pixel at all possible disparity levels.",
"For a single pixel $i = (x_i, y_i)$ , where $x_i,y_i$ are pixel locations, its cost at disparity level $l$ can be denoted as a scalar, $\\mathbf {C}(i,l)$ .",
"Various methods can be used to compute the cost volume.",
"For example, the intensity + gradient cost function [21], [33], [16] can be formulated as: $\\mathbf {C}(i,l) & = & (1-\\alpha ) \\cdot \\min ( \\Vert \\mathbf {I}(i) - \\mathbf {I^{\\prime }}(i_l) \\Vert , \\tau _1 ) \\nonumber \\\\& & + \\alpha \\cdot \\min ( \\Vert \\nabla _x \\mathbf {I}(i) - \\nabla _x \\mathbf {I^{\\prime }}(i_l) \\Vert , \\tau _2).\\vspace{-5.69054pt}$ Here $\\mathbf {I}(i)$ denotes the color vector of pixel $i$ .",
"$\\nabla _x$ is the grayscale gradient in $x$ direction.",
"$i_l$ is the corresponding pixel of $i$ with a disparity $l$ , i.e.",
"$i_l = (x_i-l,y_i)$ .",
"$\\alpha $ balances the color and gradient terms and $\\tau _1,\\tau _2$ are truncation values.",
"The cost volume $\\mathbf {C}$ is typically very noisy (Figure REF ).",
"Inspired by the WLS formulation of the denoising problem [18], the cost aggregation can be formulated with the noisy input $\\mathbf {C}$ as: $\\mathbf {\\tilde{C}}(i,l)= \\mathop {\\arg \\min }_{z}{ \\frac{1}{Z_i}\\sum _{j \\in N_i}{K(i,j) \\Vert z - \\mathbf {C}(j,l) \\Vert ^2} },\\vspace{-5.69054pt}$ where $N_i$ defines a neighboring system of $i$ .",
"$K(i,j)$ is the similarity kernel [18], which measures the similarity between pixels $i$ and $j$ , and $\\mathbf {\\tilde{C}}$ is the (denoised) cost volume.",
"$Z_i = \\sum _{j \\in N_i}{K(i,j)}$ is a normalization constant.",
"The solution of this WLS problem is: $\\mathbf {\\tilde{C}}(i,l) = \\frac{1}{Z_i} \\sum _{ j \\in N_i}{K(i,j)\\mathbf {C}(j,l)}.\\vspace{-5.69054pt}$ Thus, like image filters [18], a cost aggregation method corresponds to a particular instance of the similarity kernel.",
"For example, the BF method [36] adopted the spatial and photometric distances between two pixels to measure the similarity, which is the same as the kernel function used in the bilateral filter [28].",
"Rhemann [21] (GF) adopted the kernel defined in the guided filter [7], whose computational complexity is independent of the kernel size.",
"The NL method [33] defines a kernel based on a geodesic distance between two pixels in a tree structure.",
"This approach was further enhanced by making use of color segments, called a segment-tree (ST) approach [16].",
"A major difference between filter-based [36], [21] and tree-based [33], [16] aggregation approaches is the action scope of the similarity kernel, i.e.",
"$N_i$ in Equation (REF ).",
"In filter-based methods, $N_i$ is a local window centered at $i$ , but in tree-based methods, $N_i$ is a whole image.",
"Figure REF visualizes the effect of different action scope.",
"The filter-based methods hold some local similarity after the cost aggregation, while tree-based methods tend to produce hard edges between different regions in the cost volume.",
"Having shown that representative cost aggregation methods can be formulated within a unified framework, let us recheck the cost volume slices in Figure REF .",
"The slice, coming from Teddy stereo pair in the Middlebury dataset [24], consists of three typical scenarios: low-texture, high-texture and near textureless regions (from left to right).",
"The four state-of-the-art cost aggregation methods all perform very well in the high-texture area, but most of them fail in either low-texture or near textureless region.",
"For yielding highly accurate correspondence in those low-texture and near textureless regions, the correspondence search should be performed at the coarse scale [17].",
"However, under the formulation of Equation (REF ), costs are always aggregated at the finest scale, making it impossible to adaptively utilize information from multiple scales.",
"Hence, we need to reformulate the WLS optimization objective from the scale space perspective."
],
[
"Cross-Scale Cost Aggregation Framework",
"It is straightforward to show that directly using Equation (REF ) to tackle multi-scale cost volumes is equivalent to performing cost aggregation at each scale separately.",
"Firstly, we add a superscript $s$ to $\\mathbf {C}$ , denoting cost volumes at different scales of a stereo pair, as $\\mathbf {C}^s$ , where $s \\in \\lbrace 0, 1, \\ldots , S\\rbrace $ is the scale parameter.",
"$\\mathbf {C}^0$ represents the cost volume at the finest scale.",
"The multi-scale cost volume $\\mathbf {C}^s$ is computed using the downsampled images with a factor of $\\eta ^s$ .",
"Note that this approach also reduces the search range of the disparity.",
"The multi-scale version of Equation (REF ) can be easily expressed as: $\\mathbf {\\tilde{v}} = \\mathop {\\arg \\min }_{\\lbrace z^s\\rbrace _{s=0}^{S}}{\\sum _{s=0}^{S}{\\frac{1}{Z^s_{i^s}}\\!\\sum _{j^s \\in N_{i^s}}{K(i^s,j^s) \\Vert z^s - \\mathbf {C}^s(j^s,l^s) \\Vert ^2} }}.\\vspace{-5.69054pt}$ Here, $Z^s_{i^s} = \\sum _{j^s \\in N_{i^s}}{K(i^s,j^s)}$ is a normalization constant.",
"$\\lbrace i^s\\rbrace _{s=0}^{S}$ and $\\lbrace l^s\\rbrace _{s=0}^{S}$ denote a sequence of corresponding variables at each scale (Figure REF ), i.e.",
"$i^{s+1}=i^s/\\eta $ and $l^{s+1}=l^s/\\eta $ .",
"$N_{i^s}$ is a set of neighboring pixels on the $s^{th}$ scale.",
"In our work, the size of $N_{i^s}$ remains the same for all scales, meaning that more amount of smoothing is enforced on the coarser scale.",
"We use the vector $\\mathbf {\\tilde{v}}\\!=\\!",
"[\\mathbf {\\tilde{C}}^0(i^0,l^0), \\mathbf {\\tilde{C}}^1(i^1,l^1), \\cdots , \\mathbf {\\tilde{C}}^S(i^S,l^S)]^T$ with $S+1$ components to denote the aggregated cost at each scale.",
"The solution of Equation (REF ) is obtained by performing cost aggregation at each scale independently as follows: $\\forall s, \\mathbf {\\tilde{C}}^s(i^s,l^s) = \\frac{1}{Z^s_{i^s}} \\sum _{ j^s \\in N_{i^s}}{K(i^s,j^s)\\mathbf {C}^s(j^s,l^s)}.\\vspace{-5.69054pt}$ Previous CTF approaches typically reduce the disparity search space at the current scale by using a disparity map estimated from the cost volume at the coarser scale, often provoking the loss of small disparity details.",
"Alternatively, we directly enforce the inter-scale consistency on the cost volume by adding a Generalized Tikhonov regularizer into Equation (REF ), leading to the following optimization objective: $\\mathbf {\\hat{v}}\\!",
"& = & \\!\\mathop {\\arg \\min }_{\\lbrace z^s\\rbrace _{s=0}^{S}}{(\\sum _{s\\!=\\!0}^{S}{\\!\\frac{1}{Z^s_{i^s}}\\!\\sum _{j^s \\in N_{i^s}}{\\!K(i^s,j^s)\\!\\Vert \\!z^s\\!-\\!\\mathbf {C}^s(j^s,l^s)\\!\\Vert ^2}}}\\nonumber \\\\& & +\\lambda \\sum _{s=1}^{S}{\\Vert z^s-z^{s-1}\\Vert ^2}),\\vspace{-5.69054pt}$ where $\\lambda $ is a constant parameter to control the strength of regularization.",
"Besides, similar with $\\mathbf {\\tilde{v}}$ , the vector $\\mathbf {\\hat{v}}\\!=\\!",
"[\\mathbf {\\hat{C}}^0(i^0,l^0), \\mathbf {\\hat{C}}^1(i^1,l^1), \\cdots , \\mathbf {\\hat{C}}^S(i^S,l^S)]^T$ also has $S+1$ components to denote the cost at each scale.",
"The above optimization problem is convex.",
"Hence, we can get the solution by finding the stationary point of the optimization objective.",
"Let $F(\\lbrace z^s\\rbrace _{s=0}^{S})$ represent the optimization objective in Equation (REF ).",
"For $s \\in \\lbrace 1, 2, \\ldots ,S-1\\rbrace $ , the partial derivative of $F$ with respect to $z^s$ is: $\\frac{\\partial F}{\\partial z^s}\\!&\\!=\\!&\\!\\frac{2}{Z^s_{i^s}}\\!\\sum _{j^s \\in N_{i^s}}{\\!K(i^s,j^s)(z^s-\\mathbf {C}^s(j^s,l^s))}\\nonumber \\\\& & + 2\\lambda (z^s-z^{s-1}) - 2\\lambda (z^{s+1}-z^s)\\nonumber \\\\\\!&\\!=\\!&\\!2(\\!-\\!\\lambda \\!z^{s-1}\\!+\\!",
"( 1\\!+\\!2\\lambda ) z^s\\!-\\!\\lambda \\!z^{s+1}\\!-\\!\\mathbf {\\tilde{C}}^s(i^s,l^s)).\\vspace{-5.69054pt}$ Setting $\\frac{\\partial F}{\\partial z^s} = 0$ , we get: $-\\lambda z^{s-1} + ( 1 + 2\\lambda ) z^s - \\lambda z^{s+1} = \\mathbf {\\tilde{C}}^s(i^s,l^s).\\vspace{-5.69054pt}$ It is easy to get similar equations for $s=0$ and $s=S$ .",
"Thus, we have $S+1$ linear equations in total, which can be expressed concisely as: $A \\mathbf {\\hat{v}} = \\mathbf {\\tilde{v}}.$ The matrix $A$ is an $(S+1) \\times (S+1)$ tridiagonal constant matrix, which can be easily derived from Equation (REF ).",
"Since $A$ is tridiagonal, its inverse always exists.",
"Thus, $\\mathbf {\\hat{v}} = A^{-1}\\mathbf {\\tilde{v}}.\\vspace{-5.69054pt}$ The final cost volume is obtained through the adaptive combination of the results of cost aggregation performed at different scales.",
"Such adaptive combination enables the multi-scale interaction of the cost aggregation in the context of optimization.",
"Finally, we use an example to show the effect of inter-scale regularization in Figure REF .",
"In this example, without cross-scale cost aggregation, there are similar local minima in the cost vector, yielding erroneous disparity.",
"Information from the finest scale is not enough but when inter-scale regularization is adopted, useful information from coarse scales reshapes the cost vector, generating disparity closer to the ground truth.",
"Figure: The effect of inter-scale regularization.",
"On the right side, we visualize two cost vectors of a single pixel (pixel location (295,49)(295,49)) of Teddy stereo pair.",
"The blue line denotes the cost vector computed by NL method.",
"The green line is the cost vector after applying cross-scale cost aggregation (S+NL).",
"The red cross represents the minimal cost location for each cost vector and the vertical dash line denotes the ground truth disparity.",
"On the left side, image and disparity patches centering on this pixel are shown.",
"(Best viewed in color.)"
],
[
"Implementation and Complexity",
" To build cost volumes for different scales (Figure REF ), we need to extract stereo image pairs at different scales.",
"In our implementation, we choose the Gaussian Pyramid [2], which is a classical representation in the scale space theory.",
"The Gaussian Pyramid is obtained by successive smoothing and subsampling ($\\eta =2$ ).",
"One advantage of this representation is that the image size decreases exponentially as the scale level increases, which reduces the computational cost of cost aggregation on the coarser scale exponentially.",
"[htb] [INOUT]InputOutput [1]Input: 1 [1]Output: 1 Build Gaussian Pyramid $\\mathbf {I}^s$ , $\\mathbf {I^{\\prime }}^s$ , $s \\in \\lbrace 0, 1, \\ldots , S\\rbrace $ .",
"Generate initial cost volume $\\mathbf {C}^s$ for each scale by cost computation according to Equation (REF ).",
"Aggregate costs at each scale separately according to Equation (REF ) to get cost volume $\\mathbf {\\tilde{C}}^s$ .",
"Aggregate costs across multiple scales according to Equation (REF ) to get final cost volume $\\mathbf {\\hat{C}}^s$ .",
"Robust cost volume: $\\mathbf {\\hat{C}}^0$ .",
"Cross-Scale Cost Aggregation The basic workflow of the cross-scale cost aggregation is shown in Algorithm , where we can utilize any existing cost aggregation method in Step 3.",
"The computational complexity of our algorithm just increases by a small constant factor, compared to conventional cost aggregation methods.",
"Specifically, let us denote the computational complexity for conventional cost aggregation methods as $O(mWHL)$ , where $m$ differs with different cost aggregation methods.",
"The number of pixels and disparities at scale $s$ are $\\left\\lfloor \\frac{WH}{4^{s}} \\right\\rfloor $ and $\\left\\lfloor \\frac{L}{2^{s}}\\right\\rfloor $ respectively.",
"Thus the computational complexity of Step 3 increases at most by $\\frac{1}{7}$ , compared to conventional cost aggregation methods, as explained below: $\\sum _{s=0}^{S}{\\!\\left(\\!m\\left\\lfloor \\!\\frac{WHL}{8^{s}}\\!\\right\\rfloor \\!\\right)}\\!\\le \\!\\lim _{S \\rightarrow \\infty }{\\left(\\sum _{s=0}^{S}{\\frac{mWHL}{8^{s}}}\\right)} = \\frac{8}{7} mWHL.$ Step 4 involves the inversion of the matrix $A$ with a size of $(S+1)\\times (S+1)$ , but $A$ is a spatially invariant matrix, with each row consisting of at most three nonzero elements, and thus its inverse can be pre-computed.",
"Also, in Equation (REF ), the cost volume on the finest scale, $\\mathbf {\\hat{C}}^0(i^0,l^0)$ , is used to yield a final disparity map, and thus we need to compute only $\\mathbf {\\hat{C}}^0(i^0,l^0)=\\sum \\limits _{s = 0}^S{A^{-1}(0,s)\\mathbf {\\tilde{C}}^s(i^s,l^s)},$ not $\\mathbf {\\hat{v}} = A^{-1}\\mathbf {\\tilde{v}}$ .",
"This cost aggregation across multiple scales requires only a small amount of extra computational load.",
"In the following section, we will analyze the runtime efficiency of our method in more details."
],
[
"Experimental Result and Analysis",
" Figure: S+GF (6.99%6.99\\%)In this section, we use Middlebury [23], KITTI [4] and New Tsukuba [20] datasets to validate that when integrating state-of-the-art cost aggregation methods, such as BF [36], GF [21], NL [33] and ST [16], into our framework, there will be significant performance improvements.",
"Furthermore, we also implement the simple box filter aggregation method (named as BOX, window size is $7 \\times 7$ ) to serve as a baseline, which also becomes very powerful when integrated into our framework.",
"For NL and ST, we directly use the C++ codes provided by the authorshttp://www.cs.cityu.edu.hk/~qiyang/publications/cvpr-12/code/,http://xing-mei.net/resource/page/segment-tree.html, and thus all the parameter settings are identical to those used in their implementations.",
"For GF, we implemented our own C++ code by referring to the author-provided software (implemented in MATLABhttps://www.ims.tuwien.ac.at/publications/tuw-202088) in order to process high-resolution images from KITTI and New Tsukuba datasets efficiently.",
"For BF, we implemeted the asymmetric version as suggested by [9].",
"The local WTA strategy is adopted to generate a disparity map.",
"In order to compare different cost aggregation methods fairly, no disparity refinement technique is employed, unless we explicitly declare.",
"$S$ is set to 4, i.e.",
"totally five scales are used in our framework.",
"For the regularization parameter $\\lambda $ , we set it to $0.3$ for the Middlebury dataset, while setting it to $1.0$ on the KITTI and New Tsukuba datasets for more regularization, considering these two datasets contain a large portion of textureless regions."
],
[
"Middlebury Dataset",
"The Middlebury benchmark [24] is a de facto standard for comparing existing stereo matching algorithms.",
"In the benchmark [24], four stereo pairs (Tsukuba, Venus, Teddy, Cones) are used to rank more than 100 stereo matching algorithms.",
"In our experiment, we adopt these four stereo pairs.",
"In addition, we use `Middlebury 2005' [22] (6 stereo pairs) and `Middlebury 2006' [8] (21 stereo pairs) datasets, which involve more complex scenes.",
"Thus, we have 31 stereo pairs in total, denoted as M31.",
"It is worth mentioning that during our experiments, all local cost aggregation methods perform rather bad (error rate of non-occlusion (non-occ) area is more than $20\\%$ ) in 4 stereo pairs from Middlebury 2006 dataset, i.e.",
"Midd1, Midd2, Monopoly and Plastic.",
"A common property of these 4 stereo pairs is that they all contain large textureless regions, making local stereo methods fragile.",
"In order to alleviate bias towards these four stereo pairs, we exclude them from M31 to generate another collection of stereo pairs, which we call M27.",
"We make statistics on both M31 and M27 (Table REF ).",
"We adopt the intensity + gradient cost function in Equation (REF ), which is widely used in state-of-the-art cost aggregation methods [21], [16], [33].",
"In Table REF , we show the average error rates of non-occ region for different cost aggregation methods on both M31 and M27 datasets.",
"We use the prefix `S+' to denote the integration of existing cost aggregation methods into cross-scale cost aggregation framework.",
"Avg Non-occ is an average percentage of bad matching pixels in non-occ regions, where the absolute disparity error is larger than 1.",
"The results are encouraging: all cost aggregation methods see an improvement when using cross-scale cost aggregation, and even the simple BOX method becomes very powerful (comparable to state-of-the-art on M27) when using cross-scale cost aggregation.",
"Disparity maps of Teddy stereo pair for all these methods are shown in Figure REF , while others are shown in the supplementary material due to space limit.",
"Furthermore, to follow the standard evaluation metric of the Middlebury benchmark [24], we show each cost aggregation method's rank on the website (as of October 2013) in Table REF .",
"Avg Rank and Avg Err indicate the average rank and error rate measured using Tsukuba, Venus, Teddy and Cones images [24].",
"Here each method is combined with the state-of-the-art disparity refinement technique from [33] (For ST [16], we list its original rank reported in the Middlebury benchmark [24], since the same results was not reproduced using the author's C++ code).",
"The rank also validates the effectiveness of our framework.",
"We also reported the running time for Tsukuba stereo pair on a PC with a 2.83 GHz CPU and 8 GB of memory.",
"As mentioned before, the computational overhead is relatively small.",
"To be specific, it consists of the cost aggregation of $\\mathbf {\\tilde{C}}^s$ ($s\\in \\lbrace 0, 1, \\cdots ,S\\rbrace $ ) and the computation of Equation (REF ).",
"Table: Quantitative evaluation of cost aggregation methods on theMiddlebury dataset.",
"The prefix `S+' denotes our cross-scalecost aggregation framework.",
"For the rank part (column 4 and 5), the disparity results were refined with the same disparity refinement technique ."
],
[
"KITTI Dataset",
"The KITTI dataset [4] contains 194 training image pairs and 195 test image pairs for evaluating stereo matching algorithms.",
"For the KITTI dataset, image pairs are captured under real-world illumination condition and almost all image pairs have a large portion of textureless regions, walls and roads [4].",
"During our experiment, we use the whole 194 training image pairs with ground truth disparity maps available.",
"The evaluation metric is the same as the KITTI benchmark [5] with an error threshold 3.",
"Besides, since BF is too slow for high resolution images (requiring more than one hour to process one stereo pair), we omit BF from evaluation.",
"Considering the illumination variation on the KITTI dataset, we adopt Census Transform [37], which is proved to be powerful for robust optical flow computation [6].",
"We show the performance of different methods when integrated into cross-scale cost aggregation in Table REF .",
"Some interesting points are worth noting.",
"Firstly, for BOX and GF, there are significant improvements when using cross-scale cost aggregation.",
"Again, like the Middlebury dataset, the simple BOX method becomes very powerful by using cross-scale cost aggregation.",
"However, for S+NL and S+ST, their performances are almost the same as those without cross-scale cost aggregation, which are even worse than that of S+BOX.",
"This may be due to the non-local property of tree-based cost aggregation methods.",
"For textureless slant planes, roads, tree-based methods tend to overuse the piecewise constancy assumption and may generate erroneous fronto-parallel planes.",
"Thus, even though the cross-scale cost aggregation is adopted, errors in textureless slant planes are not fully addressed.",
"Disparity maps for all methods are presented in the supplementary material, which also validate our analysis."
],
[
"New Tsukuba Dataset",
"The New Tsukuba Dataset [20] contains 1800 stereo pairs with ground truth disparity maps.",
"These pairs consist of a one minute photorealistic stereo video, generated by moving a stereo camera in a computer generated 3D scene.",
"Besides, there are 4 different illumination conditions: Daylight, Fluorescent, Lamps and Flashlight.",
"In our experiments, we use the Daylight scene, which has a challenging real world illumination condition [20].",
"Since neighboring frames usually share similar scenes, we sample the 1800 frames every second to get a subset of 60 stereo pairs, which saves the evaluation time.",
"We test both intensity + gradient and Census Transform cost functions, and intensity + gradient cost function gives better results in this dataset.",
"Disparity level of this dataset is the same as the KITTI dataset, i.e.",
"256 disparity levels, making BF [36] too slow, so we omit BF from evaluation.",
"Table REF shows evaluation results for different cost aggregation methods on New Tsukuba dataset.",
"We use the same evaluation metric as the KITTI benchmark [5] (error threshold is 3).",
"Again, all cost aggregation methods see an improvement when using cross-scale cost aggregation."
],
[
"Regularization Parameter Study",
"The key parameter in Equation (REF ) is the regularization parameter $\\lambda $ .",
"By adjusting this parameter, we can control the strength of inter-scale regularization as shown in Figure REF .",
"The error rate is evaluated on M31.",
"When $\\lambda $ is set to 0, inter-scale regularization is prohibited, which is equivalent to performing cost aggregation at the finest scale.",
"When regularization is introduced, there are improvements for all methods.",
"As $\\lambda $ becomes large, the regularization term dominates the optimization, causing the cost volume of each scale to be purely identical.",
"As a result, fine details of disparity maps are missing and error rate increases.",
"One may note that it will generate better results by choosing different $\\lambda $ for different cost aggregation methods, though we use consistent $\\lambda $ for all methods."
],
[
"Conclusions and Future Work",
"In this paper, we have proposed a cross-scale cost aggregation framework for stereo matching.",
"This paper is not intended to present a completely new cost aggregation method that yields a highly accurate disparity map.",
"Rather, we investigate the scale space behavior of various cost aggregation methods.",
"Extensive experiments on three datasets validated the effect of cross-scale cost aggregation.",
"Almost all methods saw improvements and even the simple box filtering method combined with our framework achieved very good performance.",
"Recently, a new trend in stereo vision is to solve the correspondence problem in continuous plane parameter space rather than in discrete disparity label space [1], [13], [32].",
"These methods can handle slant planes very well and one probable future direction is to investigate the scale space behavior of these methods."
],
[
"Acknowledgement",
"This work was supported by XXXXXXXXXXXX."
]
] | 1403.0316 |
[
[
"Hardy Type Inequalities for $\\Delta_\\lambda$-Laplacians"
],
[
"Abstract We derive Hardy type inequalities for a large class of sub-elliptic operators that belong to the class of $\\Delta_\\lambda$-Laplacians and find explicit values for the constants involved.",
"Our results generalize previous inequalities obtained for Grushin type operators $$ \\Delta_{x}+ |x|^{2\\alpha}\\Delta_{y},\\qquad\\ (x,y)\\in\\mathbb{R}^{N_1}\\times\\mathbb{R}^{N_2},\\ \\alpha\\geq 0, $$ which were proved to be sharp."
],
[
"Introduction",
"Let $\\Omega \\subset \\mathbb {R}^N$ be a domain, where $N\\ge 3.$ The $N$ -dimensional version of the classical Hardy inequality states that there exists a constant $c>0$ such that $c\\int _\\Omega \\frac{|u(x)|^2}{|x|^2}dx\\le \\int _\\Omega |\\triangledown u(x)|^2dx,$ for all $u\\in H_0^1(\\Omega ).$ If the origin $\\lbrace 0\\rbrace $ belongs to the set $\\Omega ,$ the optimal constant is $c=\\left(\\frac{N-2}{2}\\right)^2,$ but not attained in $H_0^1(\\Omega ).$ Hardy originally proved this inequality in 1920 for the one-dimensional case.",
"Hardy inequalities are an important tool in the analysis of linear and non-linear PDEs (see, e.g., [6],[4],[21]), and over the years the classical Hardy inequality has been improved and extended in many directions.",
"Our aim is to derive Hardy type inequalities for a class of degenerate elliptic operators extending previous results by D'Ambrosio in [3].",
"He obtained a family of Hardy type inequalities for the Grushin type operator $\\Delta _{x}+ |x|^{2\\alpha }\\Delta _{y},\\qquad \\ (x,y)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2},$ where $\\alpha $ is a real positive constant.",
"The class of operators we consider contains Grushin type operators and, e.g., operators of the form $\\Delta _{x}+ |x|^{2\\alpha }\\Delta _{y} + |x|^{2\\beta }|y|^{2\\gamma } \\Delta _{z},\\qquad (x,y,z)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}\\times \\mathbb {R}^{N_3},$ where $\\alpha ,\\beta $ and $\\gamma $ are real positive constants.",
"Recently, for Grushin type operators improved Hardy inequalities were obtained in [20], [23], Hardy inequalities involving the control distance in [22] and Hardy inequalities in half spaces with the degeneracy at the boundary in [18].",
"After the seminal paper [10] by Garofalo and Lanconelli, where the Hardy inequality for the Kohn Laplacian on the Heisenberg group was proved, a large amount of work has been devoted to Hardy type inequalities in sub-elliptic settings.",
"For a wide bibliography regarding this topics we directly refer to the paper [4] by D'Ambrosio.",
"The proof of our inequalities is based on an approach introduced by Mitidieri in [19] for the classical Laplacian.",
"Our results coincide for the particular case of Grushin type operators with the inequalities D'Ambrosio obtained in [3], where he proved that the inequalities are sharp.",
"We derive explicit values for the constants in the inequalities, but are currently not able to show its optimality in the general case.",
"The outline of our paper is as follows: We first introduce the class of operators we consider and formulate several examples.",
"In Section we explain our approach to derive Hardy type inequalities and give a motivation for the weights appearing in the inequalities.",
"The main results are stated and proved in Section .",
"In the appendix we illustrate the relation between the fundamental solution and Hardy inequalities and comment on the difficulties we encounter proving the optimality of the constant in our inequalities."
],
[
" $\\Delta _\\lambda $ -Laplacians",
"Here and in the sequel, we use the following notations.",
"We split $\\mathbb {R}^N$ into $\\mathbb {R}^N=\\mathbb {R}^{N_1}\\times \\cdots \\times \\mathbb {R}^{N_k}, \\qquad $ and write $x=(x^{(1)},\\dots ,x^{(k)})\\in \\mathbb {R}^N,\\qquad x^{(i)}= (x^{(i)}_1,\\dots ,x^{(i)}_{N_i}),\\qquad i=1,\\dots ,k.$ The degenerate elliptic operators we consider are of the form $\\Delta _\\lambda =\\lambda _1^2\\Delta _{x^{(1)}}+\\cdots +\\lambda _k^2\\Delta _{x^{(k)}},$ where the functions $\\lambda _i:\\mathbb {R}^{N}\\rightarrow \\mathbb {R}$ are pairwise different and $\\Delta _{x^{(i)}}$ denotes the classical Laplacian in $\\mathbb {R}^{N_i}.$ We denote by $|x|$ the euclidean norm of $x\\in \\mathbb {R}^m,\\ m\\in \\mathbb {N},$ and assume the functions $\\lambda _i$ are of the form $\\lambda _1(x)&= 1,\\\\\\lambda _2(x)&=|x^{(1)}|^{\\alpha _{21}},\\\\\\lambda _3(x)&=|x^{(1)}|^{\\alpha _{31}}|x^{(2)}|^{\\alpha _{32}},\\\\&\\ \\, \\vdots \\\\\\lambda _k(x)&= |x^{(1)}|^{\\alpha _{k1}}|x^{(2)}|^{\\alpha _{k2}}\\cdots |x^{(k-1)}|^{\\alpha _{k k-1}},\\qquad x\\in \\mathbb {R}^N,$ where $\\alpha _{ij}\\ge 0$ for $i=2,\\dots ,k, j=1,\\dots ,i-1.$ Setting $\\alpha _{ij}=0$ for $j\\ge i$ we can write $\\lambda _i(x)=\\prod _{j=1}^{k}|x^{(j)}|^{\\alpha _{ij}}, \\qquad i=1,\\dots ,k.$ This implies that there exists a group of dilations $(\\delta _r)_{r>0},$ $\\delta _r:\\mathbb {R}^N\\rightarrow \\mathbb {R}^N,\\quad \\delta _r(x)=\\delta _r(x^{(1)},\\dots ,x^{(k)})=(r^{\\sigma _1}x^{(1)},\\dots ,r^{\\sigma _k}x^{(k)}),$ where $1=\\sigma _1\\le \\sigma _i$ such that $\\lambda _i$ is $\\delta _r$ -homogeneous of degree $\\sigma _i-1$ , i.e., $\\lambda _i(\\delta _r(x))=r^{\\sigma _i-1}\\lambda _i(x),\\qquad \\forall x\\in \\mathbb {R}^N,\\ r>0,\\ i=1,\\dots ,k,$ and the operator $\\Delta _\\lambda $ is $\\delta _r$ -homogeneous of degree two, i.e., $\\Delta _\\lambda (u(\\delta _r(x)))=r^2 (\\Delta _\\lambda u) (\\delta _r(x))\\qquad \\forall u\\in C^\\infty (\\mathbb {R}^N).$ We denote by $Q$ the homogeneous dimension of $\\mathbb {R}^N$ with respect to the group of dilations $(\\delta _r)_{r>0}$ , i.e., $Q:=\\sigma _1N_1+\\dots +\\sigma _k N_k.$ $Q$ will play the same role as the dimension $N$ for the classical Laplacian in our Hardy type inequalities.",
"For functions $\\lambda _i$ of the form (REF ) we find $\\sigma _1&= 1,\\\\\\sigma _2&=1+\\sigma _1\\alpha _{21},\\\\\\sigma _3&=1+\\sigma _1\\alpha _{31}+\\sigma _2\\alpha _{32}, \\\\&\\ \\vdots \\\\\\sigma _k&= 1+\\sigma _1\\alpha _{k1}+ \\sigma _2\\alpha _{k2} +\\cdots +\\sigma _{k-1} \\alpha _{k k-1}.$ If the functions $\\lambda _i$ are smooth, i.e., if the exponents $\\alpha _{ji}$ are integers, the operator $\\Delta _\\lambda $ belongs to the general class of operators studied by Hörmander in [13] and it is hypoelliptic (see Remark 1.3, [14]).",
"The simplest example is the operator $\\partial _{x_1}^2+ |x_1|^{2\\alpha }\\partial _{x_2}^2,\\qquad x=(x_1,x_2)\\in \\mathbb {R}^2,\\ \\alpha \\in \\mathbb {N},$ where $\\partial _{x_i}=\\frac{\\partial }{\\partial _{x_i}}, \\ i=1,2,$ that Grushin studied in [12].",
"He provided a complete characterization of the hypoellipticity for such operators when lower terms with complex coefficients are added.",
"For real $\\alpha > 0$ the operator is commonly called of Grushin-type.",
"Operators $\\Delta _\\lambda $ with functions $\\lambda _i$ of the form (REF ) belong to the class of $\\Delta _\\lambda $-Laplacians.",
"Franchi and Lanconelli introduced operators of $\\Delta _\\lambda $ -Laplacian type in 1982 and studied their properties in a series of papers.",
"In [7] they defined a metric associated to these operators that plays the same role as the euclidian metric for the standard Laplacian.",
"Using this metric in [8] and [9] they extended the classical De Giorgi theorem and obtained Sobolev type embedding theorems for such operators.",
"Recently, adding the assumption that the operators are homogeneous of degree two, they were named $\\Delta _\\lambda $ -Laplacians by Kogoj and Lanconelli in [14], where existence, non-existence and regularity results for solutions of the semilinear $\\Delta _\\lambda $ -Laplace equation were analyzed.",
"The global well-posedness and longtime behavior of solutions of semilinear degenerate parabolic equations involving $\\Delta _\\lambda $ -Laplacians were studied in [15], and this result was extended in [16], where also hyperbolic problems were considered.",
"We finally remark that the $\\Delta _\\lambda $ -Laplacians belong to the more general class of $X$-elliptic operators introduced in [17].",
"For these operators Hardy inequalities of other kind with weights determined by the control distance were proved by Grillo in [11].",
"To conclude this section we recall some of the examples in our previous paper [15].",
"Example 1 Let $\\alpha $ be a real positive constant and $k=2$ .",
"We consider the Grushin-type operator $\\Delta _\\lambda =\\Delta _{x^{(1)}}+ |x^{(1)}|^{2\\alpha } \\Delta _{x^{(2)}},$ where $\\lambda =(\\lambda _1,\\lambda _2)$ , with $\\lambda _1(x)=1$ and $\\lambda _2(x) = |x^{(1)}|^{\\alpha },$ $x\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}$ .",
"Our group of dilations is $\\delta _r\\left(x^{(1)},x^{(2)}\\right)=\\left(r x^{(1)}, r^{\\alpha +1} x^{(2)}\\right),$ and the homogenous dimension with respect to $(\\delta _r)_{r>0}$ is $Q=N_1 +N_2(\\alpha +1)$ .",
"More generally, for a given multi-index $\\alpha =(\\alpha _1,\\ldots ,\\alpha _{k-1})$ with real constants $\\alpha _i> 0$ , $i=1,\\ldots ,k-1,$ we consider $\\Delta _\\lambda = \\Delta _{x^{(1)}} + |x^{(1)}|^{2\\alpha _1} \\Delta _{x^{(2)}} +\\ldots + |x^{(1)}|^{2\\alpha _{k-1}} \\Delta _{x^{(k)}}.$ The group of dilations is given by $\\delta _r\\left(x^{(1)}, \\ldots ,x^{(k)}\\right)=\\left(r x^{(1)},r^{1+\\alpha _1}x^{(2)},\\ldots , r^{1+\\alpha _{k-1}} x^{(k)}\\right),$ and the homogeneous dimension is $Q=N+\\alpha _1N_2+\\alpha _2N_3+\\cdots +\\alpha _{k-1}N_k.$ Example 2 For a given multi-index $\\alpha =(\\alpha _1,\\ldots ,\\alpha _{k-1})$ with real constants $\\alpha _i> 0$ , $i=1,\\ldots ,k-1,$ we define $\\Delta _\\lambda = \\Delta _{x^{(1)}} + |x^{(1)}|^{2\\alpha _1} \\Delta _{x^{(2)}} + |x^{(2)}|^{2\\alpha _2} \\Delta _{x^{(3)}} +\\ldots + |x^{(k-1)}|^{2\\alpha _{k-1}} \\Delta _{x^{(k)}}.$ Then, in our notation $\\lambda =\\left(\\lambda _1,\\ldots ,\\lambda _k\\right)$ with $\\lambda _1 (x)&= 1,\\quad \\lambda _i(x)= |x^{(i-1)}|^{\\alpha _{i-1}}, \\ i=2,\\ldots , k,\\quad x\\in \\mathbb {R}^{N_1}\\times \\cdots \\times \\mathbb {R}^{N_k},$ and the group of dilations is given by $\\delta _r\\left(x^{(1)}, \\ldots ,x^{(k)}\\right)=\\left(r^{\\sigma _1} x^{(1)},\\ldots , r^{\\sigma _k} x^{(k)}\\right)$ with $\\sigma _1 =1$ and $\\sigma _i =\\alpha _{i-1} \\sigma _{i-1} +1$ for $i=2,\\ldots ,k$ .",
"In particular, if $\\alpha _1=\\ldots =\\alpha _{k-1} =\\alpha $ , the dilations become $\\delta _r \\left(x^{(1)}, \\ldots , x^{(k)}\\right) = \\left( r x^{(1)}, r^{\\alpha +1} x^{(2)},\\ldots , r^{\\alpha ^{k-1}+\\ldots +\\alpha +1} x^{(k)}\\right).$ Example 3 Let $\\alpha , \\beta $ and $\\gamma $ be positive real constants.",
"For the operator $\\Delta _\\lambda =\\Delta _{x^{(1)}} + |x^{(1)}|^{2\\alpha } \\Delta _{x^{(2)}} + |x^{(1)}|^{2\\beta } |x^{(2)}|^{2\\gamma } \\Delta _{x^{(3)}},$ where $\\lambda = (\\lambda _1,\\lambda _2,\\lambda _3)$ with $\\lambda _1 (x)= 1,\\quad \\lambda _2 (x)= |x^{(1)}|^{\\alpha },\\quad \\lambda _3(x) = |x^{(1)}|^{\\beta }|x^{(2)}|^{\\gamma },\\quad x\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}\\times \\mathbb {R}^{N_3},$ we find the group of dilations $\\delta _r\\left(x^{(1)},x^{(2)},x^{(3)}\\right)=\\left( r x^{(1)}, r^{\\alpha +1} x^{(2)}, r^{\\beta + (\\alpha +1)\\gamma +1} x^{(3)}\\right).$"
],
[
"How we approach Hardy-type inequalities",
"Our Hardy type inequalities are based on the following approach indicated by Mitidieri in [19].",
"Let $\\Omega \\subset \\mathbb {R}^N,$ $N\\ge 3,$ be an open subset and $p>1.$ We assume $u\\in C_0^1(\\Omega ),$ and the vector field $h\\in C^1(\\Omega ;\\mathbb {R}^N)$ satisfies $\\textnormal {div} h>0.$ The divergence theorem implies $\\int _{\\Omega }|u(x)|^p \\textnormal {div} h(x)\\,dx = -p\\int _{\\Omega } |u(x)|^{p-2}u(x)\\triangledown u(x)\\cdot h(x)\\,dx,$ where $\\cdot $ denotes the inner product in $\\mathbb {R}^N.$ Taking the absolute value and using Hölder's inequality we obtain $\\int _{\\Omega }|u(x)|^p \\textnormal {div} h(x)dx &=-p\\int _{\\Omega } |u(x)|^{p-2}u(x)\\triangledown u(x)\\cdot h(x)dx\\\\&\\le p\\left( \\int _{\\Omega } |u(x)|^{p} \\textnormal {div} h(x)dx\\right)^{\\frac{p-1}{p}}\\left( \\int _{\\Omega } \\frac{|h(x)|^p}{ (\\textnormal {div} h(x))^{p-1}}|\\triangledown u(x)|^{p}dx\\right)^{\\frac{1}{p}},$ and it follows that $\\int _{\\Omega }|u(x)|^p \\textnormal {div} h(x)dx\\le p^p \\int _{\\Omega } \\frac{|h(x)|^p}{ (\\textnormal {div} h(x))^{p-1}}|\\triangledown u(x)|^{p}dx.$ If we choose the vector field $h_\\varepsilon (x):=\\frac{x}{(|x|^2+\\varepsilon )^{\\frac{p}{2}}},$ where $\\varepsilon >0,$ then $\\textnormal {div} h_\\varepsilon (x) = \\frac{N-p\\frac{|x|^2}{|x|^2+\\varepsilon }}{(|x|^2+\\varepsilon )^{\\frac{p}{2}}},\\qquad |h_\\varepsilon (x)|=\\frac{|x|}{(|x|^{2}+\\varepsilon )^{\\frac{p}{2}}}.$ Assuming that $N>p$ we have $\\textnormal {div} h_\\varepsilon >0,$ and from inequality (REF ) we obtain $\\frac{1}{p^p}\\int _\\Omega \\left(N- p\\frac{|x|^2}{|x|^2+\\varepsilon }\\right)\\frac{|u(x)|^p}{(|x|^2+\\varepsilon )^{\\frac{p}{2}}}\\, dx\\le \\int _{\\Omega }\\left(N- p\\frac{|x|^2}{|x|^2+\\varepsilon }\\right)^{-(p-1)}\\frac{|x|^p}{(|x|^2+\\varepsilon )^{\\frac{p}{2}}} |\\triangledown u(x)|^{p}\\, dx.$ Taking the limit $\\varepsilon $ tends to zero, the classical Hardy inequality follows from the dominated convergence theorem, $\\left( \\frac{N-p}{p}\\right)^p\\int _{\\Omega }\\frac{|u(x)|^p}{|x|^p} \\, dx\\le \\int _{\\Omega } |\\triangledown u(x)|^{p}\\, dx,$ and by a density argument it is satisfied for all functions $u\\in H_0^1(\\Omega ).$ If the origin $\\lbrace 0\\rbrace $ belongs to the domain $\\Omega ,$ the constant $\\frac{N-p}{p}$ is optimal, but not attained in $H_0^1(\\Omega ).$ This approach can be generalized to deduce Hardy type inequalities for degenerate elliptic operators.",
"For the operators $\\Delta _\\lambda $ with functions $\\lambda _i$ of the form (REF ) and a function $u$ of class $C^1(\\Omega )$ we define $\\triangledown _\\lambda u:=(\\lambda _1\\triangledown _{x^{(1)}}u,\\dots ,\\lambda _k\\triangledown _{x^{(k)}}u),\\qquad \\lambda _i\\triangledown _{x^{(i)}}:=(\\lambda _i\\partial _{x_1^{(i)}},\\dots ,\\lambda _i\\partial _{x_{N_i}^{(i)}}),\\ i=1,\\dots ,k.$ We will obtain a wide family of Hardy type inequalities, that include as particular cases inequalities of the form $\\left(\\frac{Q-p}{p}\\right)^p\\int _\\Omega \\frac{|u(x)|^{p}}{[[x]]_\\lambda ^p}dx&\\le \\int _\\Omega \\psi (x)|\\triangledown _\\lambda u(x)|^pdx,\\\\\\left(\\frac{Q-p}{p}\\right)^p\\int _\\Omega \\varphi (x)\\frac{|u(x)|^{p}}{[[x]]_\\lambda ^p}dx&\\le \\int _\\Omega |\\triangledown _\\lambda u(x)|^p dx,$ where $Q$ is the homogeneous dimension, and $\\varphi $ and $\\psi $ are suitable weight functions.",
"Moreover, $[[\\cdot ]]_\\lambda $ is a homogeneous norm that replaces the euclidean norm in the classical Hardy inequality.",
"We introduce the following notation.",
"For a vector field $h$ of class $C^1(\\Omega ;\\mathbb {R}^N)$ we define $\\textnormal {div}_\\lambda h :=\\sum _{i=1}^k\\lambda _i\\textnormal {div}_{x^{(i)}} h,\\qquad \\textnormal {div}_{x^{(i)}} h :=\\sum _{j=1}^{N_i}\\partial _{x_j^{(i)}} h.$ The subsequent lemma follows from the divergence theorem and can be shown similarly as inequality (REF ).",
"See also Theorem 3.5 in [3] for the particular case of Grushin-type operators.",
"Lemma 1 Let $h\\in C^1(\\Omega ;\\mathbb {R}^N)$ be such that $\\textnormal {div}_{\\lambda } h\\ge 0.$ Then, for every $p>1$ and $u\\in C_0^1(\\Omega )$ such that $\\frac{|h|}{(\\textnormal {div}_{\\lambda } h)^{\\frac{p}{p-1}}}|\\triangledown _{\\lambda }u|\\in L^p(\\Omega )$ we have $\\int _\\Omega |u(x)|^p \\textnormal {div}_{\\lambda } h(x)\\, dx \\le p^p \\int _\\Omega \\frac{|h(x)|^p}{(\\textnormal {div}_{\\lambda } h(x))^{p-1}} |\\triangledown _{\\lambda } u(x)|^p\\, dx.$ We define $ \\sigma :=\\left(\\begin{matrix}I_{1} & 0 & \\cdots &0 \\\\0 & \\lambda _2 I_{2} & & \\vdots \\\\\\vdots &&\\ddots &0\\\\[1.2ex]0 & \\cdots &0& \\lambda _k I_{k}\\end{matrix}\\right),$ where $I_{i}$ denotes the identity matrix in $\\mathbb {R}^{N_i},$ $i=1,\\dots ,k.$ The divergence theorem implies $0&=\\int _{\\partial \\Omega }|u|^p h\\cdot \\sigma \\nu \\, d\\zeta =\\int _\\Omega \\textnormal {div}_{\\lambda }(|u|^p h)\\,dx=\\int _\\Omega p|u|^{p-2}u\\triangledown _{\\lambda } u \\cdot h\\, dx+\\int _\\Omega |u|^p \\textnormal {div}_{\\lambda }h\\, dx,$ where $\\nu $ denotes the outward unit normal at $\\zeta \\in \\partial \\Omega .$ Applying Hölder's inequality we obtain $\\int _\\Omega |u|^p \\textnormal {div}_{\\lambda }h\\, dx&=-\\int _\\Omega p|u|^{p-2}u\\triangledown _{\\lambda } u \\cdot h\\, dx\\le \\int _\\Omega p|u|^{p-1}|\\triangledown _{\\lambda } u| |h| \\,dx\\\\&\\le p\\left( \\int _\\Omega |u|^{p} (\\textnormal {div}_{\\lambda }h+\\epsilon ) \\, dx\\right)^{\\frac{p-1}{p}} \\left(\\int _\\Omega \\frac{|h|^p}{ (\\textnormal {div}_{\\lambda }h +\\epsilon )^{p-1}} |\\triangledown _{\\lambda } u|^p \\,dx\\right)^{\\frac{1}{p}},$ and consequently, $\\frac{\\int _\\Omega |u|^p \\textnormal {div}_{\\lambda }h\\, dx}{\\left( \\int _\\Omega |u|^{p} (\\textnormal {div}_{\\lambda }h+\\epsilon ) \\, dx\\right)^{\\frac{p-1}{p}}}&\\le p\\left(\\int _\\Omega \\frac{|h|^p}{ (\\textnormal {div}_{\\lambda }h +\\epsilon )^{p-1}} |\\triangledown _{\\lambda } u|^p \\,dx\\right)^{\\frac{1}{p}}.$ The statement of the lemma now follows from the dominated convergence theorem.",
"To illustrate our approach we first consider Hardy type inequalities of the form (REF ), i.e., $\\left(\\frac{Q-p}{p}\\right)^p\\int _\\Omega \\frac{|u(x)|^{p}}{[[x]]_\\lambda ^p}\\,dx&\\le \\int _\\Omega \\psi (x)|\\triangledown _\\lambda u(x)|^p\\,dx,$ with a certain weight function $\\psi $ and homogeneous norm $[[ \\cdot ]]_\\lambda .$ Motivated by Lemma REF we look for a function $h$ satisfying $\\textnormal {div}_\\lambda h(x)=\\frac{Q-p}{[[x]]_\\lambda ^p}.$ If we choose $h(x)=\\frac{1}{[[x]]_\\lambda ^p}\\left(\\frac{\\sigma _1 x^{(1)}}{\\lambda _1(x)},\\dots ,\\frac{\\sigma _k x^{(k)}}{\\lambda _k(x)} \\right),$ and since $\\lambda _i$ does not depend on $x^{(i)}$ we obtain $\\textnormal {div}_\\lambda h(x)=\\frac{Q}{[[x]]_\\lambda ^p}-p\\frac{1}{[[x]]_\\lambda ^{p+1}}\\sum _{i=1}^k\\sigma _ix^{(i)}\\cdot \\triangledown _{x^{(i)}}(\\Vert x\\Vert _\\lambda ).$ Consequently, the homogeneous norm $[[\\cdot ]]_\\lambda $ should fulfill the relation $\\sum _{i=1}^k\\sigma _ix^{(i)}\\cdot \\triangledown _{x^{(i)}}([[x]]_\\lambda )=[[x]]_\\lambda .$ On the other hand, computing the norm of $h$ we obtain $|h(x)|^2=\\frac{1}{[[x]]_\\lambda ^{2p}}\\frac{1}{\\prod _{i=1}^k\\lambda _i(x)^2}\\left(\\prod _{j\\ne 1}\\lambda _j(x)^2 \\sigma _1^2 |x^{(1)}|^2 + \\dots + \\prod _{j\\ne k}\\lambda _j(x)^2 \\sigma _k^2 |x^{(k)}|^2\\right),$ which motivates to consider the homogeneous norm $[[x]]_\\lambda =\\left(\\prod _{j\\ne 1}\\lambda _j(x)^2 \\sigma _1^2 |x^{(1)}|^2 + \\dots + \\prod _{j\\ne k}\\lambda _j(x)^2 \\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}.$ The exponent is determined by requiring $[[\\cdot ]]_\\lambda $ to be $\\delta _r$ -homogeneous of degree one.",
"Since the functions $\\lambda _i$ are of the form (REF ), the relation (REF ) is satisfied."
],
[
"Our homogeneous norms",
"We recall that $\\Delta _\\lambda =\\lambda _1^2\\Delta _{x^{(1)}}+\\cdots +\\lambda _k^2\\Delta _{x^{(k)}}$ with functions $\\lambda _i$ of the form $\\lambda _i(x)=\\prod _{j=1}^{k}|x^{(j)}|^{\\alpha _{ij}}, \\qquad i=1,\\dots ,k,$ which are $\\delta _r$ -homogeneous of degree $\\sigma _i-1$ with respect to a group of dilations $\\delta _r(x)=(r^{\\sigma _1}x^{(1)},\\dots , r^{\\sigma _k}x^{(k)}),\\qquad x\\in \\mathbb {R}^N,\\ r>0.$ Using our previous notations follow the relations $\\sum _{j=1}^k\\alpha _{ij}\\sigma _j&=\\sigma _i-1,&\\prod _{i=1}^k\\lambda _i(x) &=\\prod _{j=1}^k|x^{(j)}|^{\\sum _{i=1}^k\\alpha _{ij}}.$ Definition 1 We define the homogenous norm $[[\\cdot ]]_\\lambda $ associated to the $\\Delta _\\lambda $ -Laplacian by relation (REF ), $[[x]]_\\lambda :=\\left(\\prod _{i \\ne 1}\\lambda _i(x)^2\\sigma _1^2|x^{(1)}|^2+\\cdots + \\prod _{i\\ne k}\\lambda _i(x)^2\\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}},\\qquad x\\in \\mathbb {R}^N.$ Under our hypotheses $[[\\cdot ]]_\\lambda $ can be written as $[[x]]_\\lambda &=\\left(\\prod _{j= 1}^k |x^{(j)}|^{\\sum _{i\\ne 1}2\\alpha _{ij}} \\sigma _1^2|x^{(1)}|^2+\\cdots +\\prod _{j= 1}^k |x^{(j)}|^{\\sum _{i\\ne k}2\\alpha _{ij}} \\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}.$ We compute the homogeneous norm $[[\\cdot ]]_\\lambda $ for some of the operators in our previous examples.",
"For Grushin-type operators $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y, \\qquad (x,y)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2},$ where the constant $\\alpha $ is non-negative, the definition leads to the same distance from the origin that D'Ambrosio considered in [3], $[[(x,y)]]_\\lambda =\\left(|x|^{2(1+\\alpha )}+(1+\\alpha )^2 |y|^2\\right)^{\\frac{1}{2(1+\\alpha )}}.$ For operators of the form $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y+|x|^{2\\beta }\\Delta _z,\\qquad (x,y,z)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}\\times \\mathbb {R}^{N_3},$ with non-negative constants $\\alpha $ and $\\beta ,$ we obtain $[[(x,y,z)]]_\\lambda =\\left(|x|^{2(1+\\alpha +\\beta )}+(1+\\alpha )^2|x|^{2\\beta } |y|^2+(1+\\beta )^2|x|^{2\\alpha } |z|^2\\right)^{\\frac{1}{2(1+\\alpha +\\beta )}}.$ For $\\Delta _\\lambda $ -Laplacians of the form $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y+|x|^{2\\beta }|y|^{2\\gamma }\\Delta _z,\\qquad (x,y,z)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}\\times \\mathbb {R}^{N_3},$ where the constants $\\alpha , \\beta $ and $\\gamma $ are non-negative, we get $[[(x,y,z)]]_\\lambda = \\left(|y|^{2\\gamma }|x|^{2(1+\\alpha +\\beta )}+(1+\\alpha )^2|x|^{2\\beta }|y|^{2(1+\\gamma )}+(1+\\mu )^2|x|^{2\\alpha } |z|^2\\right)^{\\frac{1}{2(1+\\alpha +\\mu )}},$ where $\\mu =\\beta +(1+\\alpha )\\gamma .$ Proposition 1 Our homogeneous norm $[[\\cdot ]]_\\lambda $ satisfies the following properties: (1) It is $\\delta _r$ -homogeneous of degree one, i.e., $[[\\delta _r(x)]]_\\lambda =r[[x]]_\\lambda .$ (2) It fulfills the relation $\\sum _{i=1}^k\\sigma _i\\left(x^{(i)}\\cdot \\triangledown _{x^{(i)}}\\right)[[x]]_\\lambda =[[x]]_\\lambda .$ $(1)$ Let $x\\in \\mathbb {R}^N.$ The homogeneity of the functions $\\lambda _i$ implies that $&\\ [[\\delta _r(x)]]_\\lambda \\\\=&\\ \\left(\\prod _{i \\ne 1}(\\lambda _i(\\delta _r(x)))^2\\sigma _1^2|r^{\\sigma _1}x^{(1)}|^2+\\cdots +\\prod _{i\\ne k}(\\lambda _i(\\delta _r(x)))^2\\sigma _k^2|r^{\\sigma _k}x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}\\\\=&\\ \\left(\\prod _{i \\ne 1}r^{2\\sigma _1}r^{2(\\sigma _i-1)}(\\lambda _i(x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots +\\prod _{i\\ne k}r^{2\\sigma _k}r^{2(\\sigma _i-1)}(\\lambda _i(x))^2\\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}\\\\=&\\ \\left(r^{2+\\sum _{i=1}^k2(\\sigma _i-1)}\\prod _{i \\ne 1}(\\lambda _i(x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + \\prod _{i\\ne k}(\\lambda _i(x))^2\\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}= r[[x]]_\\lambda .$ $(2)$ We observe $&\\ x^{(l)}\\cdot \\triangledown _{x^{(l)}}[[x]]_\\lambda \\\\=&\\ \\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}\\left(\\prod _{i \\ne 1}(\\lambda _i(x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + \\prod _{i\\ne k}(\\lambda _i(x))^2\\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}-1}\\\\&\\ \\left( (2\\sum _{j\\ne 1}\\alpha _{jl})\\prod _{i \\ne 1}(\\lambda _i(x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + (2\\sum _{j\\ne k}\\alpha _{jl}) \\prod _{i\\ne k}(\\lambda _i(x))^2\\sigma _k^2|x^{(k)}|^2+2\\prod _{i \\ne l}(\\lambda _i(x))^2\\sigma _l^2|x^{(l)}|^2 \\right),$ and using the relation $\\sum _{l=1}^k\\sigma _l\\alpha _{jl}=\\sigma _j-1$ it follows that $&\\ \\sum _{l=1}^k\\sigma _l\\left(x^{(l)}\\cdot \\triangledown _{x^{(l)}}\\right)[[x]]_\\lambda \\\\=&\\ \\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}\\left(\\prod _{i \\ne 1}(\\lambda _i(x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + \\prod _{i\\ne k}(\\lambda _i(x))^2\\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}-1}\\\\&\\ 2\\left( ((\\sum _{j\\ne 1}\\sum _{l=1}^k\\sigma _l\\alpha _{jl})+\\sigma _1)\\prod _{i \\ne 1}(\\lambda _i(x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + ((\\sum _{j\\ne k}\\sum _{l=1}^k\\sigma _l\\alpha _{jl})+\\sigma _k) \\prod _{i\\ne k}(\\lambda _i(x))^2\\sigma _k^2|x^{(k)}|^2 \\right) \\\\=&\\ [[x]]_\\lambda .$"
],
[
"Main results",
"We denote by $\\mathring{W}_\\lambda ^{1,p}(\\Omega )$ the closure of $C_0^1(\\Omega )$ with respect to the norm $\\Vert u\\Vert _{\\mathring{W}_\\lambda ^{1,p}(\\Omega )}:=\\left(\\int _\\Omega |\\triangledown _\\lambda u(x)|^p dx\\right)^{\\frac{1}{p}},$ and for $\\psi \\in L^1_{loc}(\\Omega )$ such that $\\psi >0$ a.e.",
"in $\\Omega $ we define the space $\\mathring{W}_\\lambda ^{1,p}(\\Omega ,\\psi )$ as the closure of $C_0^1(\\Omega )$ with respect to the norm $\\Vert u\\Vert _{\\mathring{W}_\\lambda ^{1,p}(\\Omega ;\\psi )}:=\\left(\\int _\\Omega |\\triangledown _\\lambda u(x)|^p \\psi (x) dx\\right)^{\\frac{1}{p}}.$ Theorem 1 Let $p>1$ and $\\mu _1,\\dots ,\\mu _k,\\ s\\in \\mathbb {R}$ be such that $s<N_1+\\mu _1$ and $-p\\min \\lbrace \\alpha _{1i},\\dots \\alpha _{ki},1 \\rbrace +s<N_i+\\mu _i \\qquad i=1,\\dots ,k.$ Then, for every $u\\in \\mathring{W}_\\lambda ^{1,p}(\\Omega , \\psi ),$ we have $\\left(\\frac{Q-s+\\sum _{i=1}^k\\sigma _i\\mu _i}{p}\\right)^p\\int _\\Omega \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_\\lambda ^s}|u(x)|^p\\ dx\\le \\int _\\Omega \\psi (x)\\left|\\triangledown _\\lambda u(x)\\right|^p\\ dx,$ where $ \\psi (x)=\\frac{[[x]]_\\lambda ^{p(1+\\sum _{i=1}^k(\\sigma _i-1))-s}}{\\prod _{i=1}^k |x^{(i)}|^{p(\\sum _{j=1}^k\\alpha _{ji})- \\mu _i} }.$ In particular, for $s=p$ and $\\mu _1=\\cdots =\\mu _k=0$ we get $\\left(\\frac{Q-p}{p}\\right)^p\\int _\\Omega \\frac{|u(x)|^p}{[[x]]_\\lambda ^p}\\ dx\\le \\int _\\Omega \\frac{[[x]]_\\lambda ^{\\sum _{i=1}^k(\\sigma _i-1)}}{\\prod _{i=1}^k\\lambda _i(x)^p}\\left|\\triangledown _\\lambda u(x)\\right|^p\\ dx,$ and choosing $s=p(1+\\sum _{i=1}^k(\\sigma _i-1))$ and $\\mu _i=p\\sum _{j=1}^k\\alpha _{ji}$ we obtain $\\left(\\frac{Q-p}{p}\\right)^p\\int _\\Omega \\frac{\\prod _{i=1}^k\\lambda _i(x)^p}{[[x]]_\\lambda ^{p(1+\\sum _{i=1}^k(\\sigma _i-1))}}|u(x)|^p\\ dx\\le \\int _\\Omega \\left|\\triangledown _\\lambda u(x)\\right|^p\\ dx.$ We deduce the inequalities from Lemma REF .",
"To this end for $\\varepsilon >0$ we define $\\lambda ^\\varepsilon &:=(\\lambda _1^\\varepsilon ,\\dots ,\\lambda _k^\\varepsilon ),\\qquad \\lambda _i^\\varepsilon (x):=\\prod _{j=1}^k\\left(|x^{(j)}|^2+\\varepsilon \\right)^{\\frac{\\alpha _{ij}}{2}}, \\quad i=1,\\dots , k,\\\\[[x]]_{\\varepsilon ,\\lambda }&:=\\left(\\sum _{j=1}^k\\Big (\\prod _{i\\ne j}\\lambda _i^\\varepsilon (x)^2\\sigma _j^2|x^{(j)}|^2\\Big )\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}$ and consider the function $h_\\varepsilon (x):=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s}\\left(\\frac{\\sigma _1x^{(1)}}{\\lambda ^\\varepsilon _1(x)},\\dots , \\frac{\\sigma _k x^{(k)}}{\\lambda ^\\varepsilon _k(x)}\\right).$ We obtain $\\textnormal {div}_{\\lambda } h_\\varepsilon (x)=&\\ \\sum _{i=1}^k\\frac{\\lambda _i(x)}{\\lambda _i^\\varepsilon (x)}\\triangledown _{x^{(i)}}\\cdot \\left(\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s} \\sigma _i x^{(i)} \\right)\\\\=&\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s}\\left( \\sum _{i=1}^k\\frac{\\lambda _i(x)}{\\lambda _i^\\varepsilon (x)}\\left(N_i\\sigma _i+\\sigma _i\\mu _i-s \\frac{1}{[[x]]_{\\varepsilon ,\\lambda }}\\sigma _i x^{(i)} \\cdot \\triangledown _{x^{(i)}}([[x]]_{\\varepsilon ,\\lambda })\\right)\\right)\\\\=&\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s} c_\\varepsilon (x),$ where $c_\\varepsilon (x):=\\sum _{i=1}^k\\frac{\\lambda _i(x)}{\\lambda _i^\\varepsilon (x)}\\left(N_i\\sigma _i+\\sigma _i\\mu _i-s \\frac{1}{[[x]]_{\\varepsilon ,\\lambda }}\\sigma _i x^{(i)} \\cdot \\triangledown _{x^{(i)}}([[x]]_{\\varepsilon ,\\lambda })\\right).$ Using Proposition REF we observe that $\\lim _{\\varepsilon \\rightarrow 0} c_\\varepsilon (x)=\\sum _{i=1}^k\\left(N_i\\sigma _i+\\sigma _i\\mu _i-s \\right)= Q-s+\\sum _{i=1}^k\\sigma _i\\mu _i,$ which is positive by our hypothesis.",
"Moreover, there exist positive constants $\\alpha _1$ and $\\alpha _2$ such that $0<\\alpha _1\\le c_\\varepsilon (x)\\le \\alpha _2<\\infty \\qquad \\forall x\\in \\Omega .$ Indeed, we compute $&\\ x^{(l)}\\cdot \\triangledown _{x^{(l)}}[[x]]_{\\varepsilon ,\\lambda }\\\\=&\\ \\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}\\left(\\prod _{i \\ne 1}(\\lambda _i^\\varepsilon (x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + \\prod _{i\\ne k}(\\lambda _i^\\varepsilon (x))^2\\sigma _k^2|x^{(k)}|^2\\right)^{\\frac{1}{2(1+\\sum _{i=1}^k(\\sigma _i-1))}-1}\\\\=&\\ \\Big \\lbrace \\frac{|x^{(l)}|^2}{|x^{(l)}|^2+\\varepsilon }\\Big [(2\\sum _{j\\ne 1}\\alpha _{jl})\\prod _{i \\ne 1}(\\lambda _i^\\varepsilon (x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + (2\\sum _{j\\ne k}\\alpha _{jl}) \\prod _{i\\ne k}(\\lambda _i^\\varepsilon (x))^2\\sigma _k^2|x^{(k)}|^2\\Big ]\\\\& +2\\prod _{i \\ne l}(\\lambda _i^\\varepsilon (x))^2\\sigma _l^2|x^{(l)}|^2 \\Big \\rbrace ,$ and consequently, using the relation $\\sum _{l=1}^k\\sigma _l\\alpha _{jl}=\\sigma _j-1$ it follows that $c_\\varepsilon (x)=&\\sum _{l=1}^k\\frac{\\lambda _l(x)}{\\lambda _l^\\varepsilon (x)}\\left(N_l\\sigma _l+\\sigma _l\\mu _l\\right)-s\\frac{1}{[[x]]_{\\varepsilon ,\\lambda }}\\sum _{l=1}^k\\frac{\\lambda _l(x)}{\\lambda _l^\\varepsilon (x)} \\sigma _lx^{(l)} \\cdot \\triangledown _{x^{(l)}}([[x]]_{\\varepsilon ,\\lambda })\\\\&\\sum _{l=1}^k\\frac{\\lambda _l(x)}{\\lambda _l^\\varepsilon (x)}\\left(N_l\\sigma _l+\\sigma _l\\mu _l\\right)-s\\frac{1}{[[x]]_{\\varepsilon ,\\lambda }}[[x]]_{\\varepsilon ,\\lambda }^{1-2(1+\\sum _{i=1}^k(\\sigma _i-1))}\\sum _{l=1}^k \\sigma _lx^{(l)} \\cdot \\triangledown _{x^{(l)}}([[x]]_{\\varepsilon ,\\lambda })\\\\\\ge &\\sum _{l=1}^k\\frac{\\lambda _l(x)}{\\lambda _l^\\varepsilon (x)}\\left(N_l\\sigma _l+\\sigma _l\\mu _l\\right)-s\\frac{[[x]]_{\\varepsilon ,\\lambda }^{-2(1+\\sum _{i=1}^k(\\sigma _i-1))}}{(1+\\sum _{i=1}^k(\\sigma _i-1))}\\cdot \\\\&\\ \\Big \\lbrace ((\\sum _{j\\ne 1}\\sum _{l=1}^k\\sigma _l\\alpha _{jl})+\\sigma _1)\\prod _{i \\ne 1}(\\lambda _i^\\varepsilon (x))^2\\sigma _1^2|x^{(1)}|^2+\\cdots + ((\\sum _{j\\ne k}\\sum _{l=1}^k\\sigma _l\\alpha _{jl}) +\\sigma _k)\\prod _{i\\ne k}(\\lambda _i^\\varepsilon (x))^2\\sigma _k^2|x^{(k)}|^2 \\Big \\rbrace \\\\=&\\sum _{l=1}^k\\frac{\\lambda _l(x)}{\\lambda _l^\\varepsilon (x)}\\left(N_l\\sigma _l+\\sigma _l\\mu _l\\right)-s\\ge N_1+\\mu _1-s>0.$ On the other hand, $c_\\varepsilon (x)=&\\sum _{l=1}^k\\frac{\\lambda _l(x)}{\\lambda _l^\\varepsilon (x)}\\left(N_l\\sigma _l+\\sigma _l\\mu _l-s \\frac{1}{[[x]]_{\\varepsilon ,\\lambda }}\\sigma _lx^{(l)} \\cdot \\triangledown _{x^{(l)}}([[x]]_{\\varepsilon ,\\lambda })\\right)\\\\\\le & \\sum _{l=1}^k\\left(N_l\\sigma _l+\\sigma _l\\mu _l\\right)<\\infty ,$ which concludes the proof of property (REF ).",
"Moreover, we compute $| h_\\varepsilon (x) | & =\\ \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s}\\left(\\sum _{i=1}^k\\frac{\\sigma _i^2|x^{(i)}|^2}{\\lambda ^\\varepsilon _i(x)^2}\\right)^{\\frac{1}{2}}\\\\&=\\ \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s}\\frac{\\left( \\sum _{i=1}^k \\prod _{j\\ne i} \\lambda ^\\varepsilon _j(x)^2 \\sigma _i^2|x^{(i)}|^2 \\right)^{\\frac{1}{2}}}{\\prod _{i=1}^k \\lambda ^\\varepsilon _i(x)}\\\\&=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}[[x]]_{\\varepsilon ,\\lambda }^{(1+\\sum _{i=1}^k(\\sigma _i-1))-s}}{\\prod _{i=1}^k \\lambda ^\\varepsilon _i(x)},$ and Lemma REF applied to $h_\\varepsilon $ yields $&\\frac{1}{p^p}\\int _\\Omega c_\\varepsilon (x) \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^s} |u(x)|^p dx\\\\\\le &\\ \\int _\\Omega \\frac{1}{c_\\varepsilon (x)^{(p-1)}}\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\varepsilon ,\\lambda }^{s}}\\left(\\sum _{i=1}^k\\frac{\\sigma _i^2|x^{(i)}|^2}{\\lambda ^\\varepsilon _i(x)^2}\\right)^{\\frac{p}{2}}|\\triangledown _{\\lambda } u(x)|^p\\ dx,\\\\\\le &\\ \\frac{1}{\\alpha _1^{(p-1)}}\\int _\\Omega \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_{\\lambda }^{s}}\\left(\\sum _{i=1}^k\\frac{\\sigma _i^2|x^{(i)}|^2}{\\lambda _i(x)^2}\\right)^{\\frac{p}{2}}|\\triangledown _{\\lambda } u(x)|^p\\ dx\\\\=&\\ \\frac{1}{\\alpha _1^{(p-1)}}\\int _\\Omega \\psi (x)|\\triangledown _{\\lambda } u(x)|^p\\ dx.$ Since $\\lim _{\\varepsilon \\rightarrow 0}c_{\\varepsilon }(x)=Q+\\sum _{i=1}^k\\sigma _i\\mu _i-s,$ the theorem now follows from the dominated convergence theorem by taking the limit $\\varepsilon $ tends to zero.",
"Remark 1 The first condition on the exponents in Theorem REF allows to derive the uniform estimates for $c_\\epsilon (x)$ in the proof, while the condition (REF ) ensures that $\\psi $ belongs to $L^1_{loc}(\\Omega ).$ We formulated a very general family of Hardy-type inequalities, the parameters allow to adjust the weights and to move them from one side of the inequality to the other.",
"Particular choices lead to inequalities of the form (REF ) or ().",
"Remark 2 For Grushin-type operators $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y,$ $\\alpha \\ge 0, (x,y)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2},$ we recover the Hardy inequalities of Theorem 3.1 in [3], where it was proved that the constants are optimal.",
"For the convenience of the reader we first formulated Hardy type inequalities for the particular case of our homogeneous norms $[[\\cdot ]]_\\lambda .$ We now generalize Theorem REF and consider homogeneous distances from the origin $\\Vert \\cdot \\Vert _\\lambda $ that satisfy the relation $\\sum _{j=1}^k\\sigma _j\\left( x^{(j)}\\cdot \\triangledown _{x^{(j)}} \\right) \\Vert x\\Vert _{\\lambda } = \\Vert x\\Vert _{\\lambda },\\qquad x\\in \\mathbb {R}^N.$ For instance, we could choose $&&\\Vert x\\Vert _{\\lambda }&:=\\left(\\sum _{j=1}^k |x^{(j)}|^{2\\prod _{i\\ne j}\\sigma _i}\\right)^{\\frac{1}{2\\prod _{i=1}^k\\sigma _i}},&& x\\in \\mathbb {R}^N,\\\\\\textnormal {or}&&\\Vert x\\Vert _{\\lambda }&:=\\left(\\sum _{j=1}^k(\\sigma _j |x^{(j)}|)^{2\\prod _{i\\ne j}\\sigma _i}\\right)^{\\frac{1}{2\\prod _{i=1}^k\\sigma _i}},&& x\\in \\mathbb {R}^N.$ Remark 3 For Grushin-type operators the second distance $\\Vert \\cdot \\Vert _{\\lambda }$ coincides with our homogeneous norm $[[\\cdot ]]_\\lambda $ and with the distance considered by D'Ambrosio in [3].",
"We compute the first of the homogeneous distances for our previous examples.",
"For operators of the form $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y+|x|^{2\\beta }\\Delta _z,\\qquad (x,y,z)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}\\times \\mathbb {R}^{N_3},$ with non-negative constants $\\alpha $ and $\\beta ,$ we obtain $\\Vert (x,y,z)\\Vert _\\lambda =\\left(|x|^{2(1+\\alpha )(1+\\beta )}+ |y|^{2(1+\\beta )}+|z|^{2(1+\\alpha )}\\right)^{\\frac{1}{2(1+\\alpha )(1+\\beta )}}.$ For $\\Delta _\\lambda $ -Laplacians of the form $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y+|x|^{2\\beta }|y|^{2\\gamma }\\Delta _z,\\qquad (x,y,z)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}\\times \\mathbb {R}^{N_3},$ where the constants $\\alpha , \\beta $ and $\\gamma $ are non-negative, we get $\\Vert (x,y,z)\\Vert _\\lambda = \\left(|x|^{2(1+\\alpha )(1+\\mu )}+|y|^{2(1+\\mu )}+|z|^{2(1+\\alpha )}\\right)^{\\frac{1}{2(1+\\alpha )(1+\\mu )}},$ where $\\mu =\\beta +(1+\\alpha )\\gamma .$ Theorem 2 Let $p>1$ and $\\mu _1,\\dots ,\\mu _k, s,t\\in \\mathbb {R}$ be such that $s+t<N_1+\\mu _1$ and $-p\\min \\lbrace \\alpha _{1i},\\dots \\alpha _{ki},1 \\rbrace +s+\\frac{t}{\\sigma _i}<N_i+\\mu _i \\qquad i=1,\\dots ,k.$ Then, for every $u\\in \\mathring{W}_\\lambda ^{1,p}(\\Omega ,\\psi )$ we have $\\left(\\frac{Q-s-t+\\sum _{i=1}^k\\sigma _i\\mu _i}{p}\\right)^p\\int _\\Omega \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\lambda }^t[[x]]_\\lambda ^s}|u(x)|^p\\ dx& \\le \\ \\int _\\Omega \\psi (x)\\left|\\triangledown _\\lambda u(x)\\right|^p\\ dx,$ where $\\psi (x)=\\frac{\\prod _{i=1}^k|x^{(i)}|^{ \\mu _i-p\\sum _{j=1}^k\\alpha _{ji}}}{\\Vert x\\Vert _{\\lambda }^t [[x]]_\\lambda ^{s-p(1+\\sum _{i=1}^k(\\sigma _i-1))}},$ and $\\Vert \\cdot \\Vert _{\\lambda }$ denotes the homogeneous norm (REF ) or ().",
"In particular, for $s=0,$ $\\mu _i=0$ and $t=p$ we obtain $\\left(\\frac{Q-p}{p}\\right)^p\\int _\\Omega \\frac{|u(x)|^p}{\\Vert x\\Vert _{\\lambda }^p}\\ dx&\\le \\int _\\Omega \\frac{[[x]]_\\lambda ^{p(1+\\sum _{i=1}^k(\\sigma _i-1))}}{\\Vert x\\Vert _{\\lambda }^p\\prod _{j=1}^k\\lambda _j(x)^p}\\left|\\triangledown _\\lambda u(x)\\right|^p\\ dx.$ For $t=0$ we recover the Hardy inequalities in Theorem REF with our homogeneous norms $[[\\cdot ]]_\\lambda .$ We prove the statement for the homogeneous norm (REF ).",
"The result for the distance () follows analogously.",
"We deduce the inequalities from Lemma REF .",
"To this end we define the function $h_\\varepsilon (x):=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }^t [[x]]_{\\varepsilon ,\\lambda }^s}\\left(\\frac{\\sigma _1x^{(1)}}{\\lambda _1^\\varepsilon (x)},\\dots , \\frac{\\sigma _kx^{(k)}}{\\lambda _k^\\varepsilon (x)}\\right),$ where $||\\cdot ||_{\\varepsilon ,\\lambda }$ is a smooth approximation of $||\\cdot ||_\\lambda ,$ $||x||_{\\varepsilon ,\\lambda }&=\\left(\\sum _{j=1}^k (|x^{(j)}|^2+\\varepsilon )^{\\prod _{i\\ne j}\\sigma _i}\\right)^{\\frac{1}{2\\prod _{i=1}^k\\sigma _i}}.$ We obtain $\\left| h_\\varepsilon (x)\\right|&=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}\\ [[x]]_{\\varepsilon ,\\lambda }^{(1+\\sum _{i=1}^k(\\sigma _i-1))-s}}{\\prod _{i=1}^k \\lambda ^\\varepsilon _i(x)\\ \\Vert x\\Vert _{\\varepsilon ,\\lambda }^t},\\\\\\textnormal {div}_{\\lambda } h_\\varepsilon (x)&=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }^t [[x]]_{\\varepsilon ,\\lambda }^s}\\bigg (c_\\varepsilon (x)-t \\frac{1}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }} \\sum _{i=1}^k\\frac{\\lambda _i(x)}{\\lambda _i^\\varepsilon (x)}\\sigma _i x^{(i)} \\cdot \\triangledown _{x^{(i)}}(\\Vert x\\Vert _{\\varepsilon ,\\lambda } )\\bigg )\\\\&=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }^t [[x]]_{\\varepsilon ,\\lambda }^s}\\bigg (c_\\varepsilon (x)-\\eta _\\varepsilon (x)\\bigg ),$ where $c_\\varepsilon $ was defined in the proof of Theorem REF and $\\eta _\\varepsilon (x):=t \\frac{1}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }} \\sum _{i=1}^k\\frac{\\lambda _i(x)}{\\lambda _i^\\varepsilon (x)}\\sigma _i x^{(i)} \\cdot \\triangledown _{x^{(i)}}(\\Vert x\\Vert _{\\varepsilon ,\\lambda } ).$ We observe that $0\\le \\eta _\\varepsilon (x)&=t \\frac{1}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }} \\bigg (\\sum _{i=1}^k\\frac{\\lambda _i(x)}{\\lambda _i^\\varepsilon (x)}\\frac{|x^{(i)}|^2}{|x^{(i)}|^2+\\varepsilon }(|x^{(i)}|^2+\\varepsilon )^{\\prod _{j\\ne i}\\sigma _j }\\bigg )\\Vert x\\Vert _{\\varepsilon ,\\lambda }^{1-2\\prod _{j=1}^k\\sigma _k}\\\\&\\le t\\frac{1}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }}\\bigg ( \\sum _{i=1}^k(|x^{(i)}|^2+\\varepsilon )^{\\prod _{j\\ne i}\\sigma _j }\\bigg )\\Vert x\\Vert _{\\varepsilon ,\\lambda }^{1-2\\prod _{j=1}^k\\sigma _k}=t$ and consequently, it follows from the proof of Theorem REF that $c_\\varepsilon (x)-\\eta _\\varepsilon (x) \\ge N_1+\\mu _1-s -t>0.$ Moreover, we have $\\lim _{\\varepsilon \\rightarrow 0}(c_\\varepsilon (x)-\\eta _\\varepsilon (x))=Q+ \\sum _{i=1}^k\\sigma _i\\mu _i-s-t.$ By our assumptions $Q>s+t-\\sum _{i=1}^k\\sigma _i\\mu _i,$ which implies that $\\textnormal {div}_{\\lambda } h_\\varepsilon >0$ for all sufficiently small $\\varepsilon >0.$ Lemma REF applied to the function $h_\\varepsilon $ leads to the inequality $&\\frac{1}{p^p}\\int _\\Omega (c_\\varepsilon (x)-\\eta _\\varepsilon (x))\\frac{\\prod _{i=1}^k(|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }^t[[x]]_{\\varepsilon ,\\lambda }^s}|u(x)|^p\\ dx\\\\\\le &\\ \\int _\\Omega \\frac{1}{(c_\\varepsilon (x)-\\eta _\\varepsilon (x))^{(p-1)}}\\psi _\\varepsilon (x)\\left|\\triangledown _{\\lambda } u(x)\\right|^p\\ dx,$ where $\\psi _\\varepsilon (x)&=\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\varepsilon ,\\lambda }^t [[x]]_{\\varepsilon ,\\lambda }^{s}}\\bigg (\\sum _{i=1}^k\\frac{\\sigma _i^2|x^{(i)}|^2}{\\lambda _i^\\varepsilon (x)^2}\\bigg )^{\\frac{p}{2}}\\\\&\\le \\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{\\Vert x\\Vert _{\\lambda }^t [[x]]_{\\lambda }^{s}}\\bigg (\\sum _{i=1}^k\\frac{\\sigma _i^2|x^{(i)}|^2}{\\lambda _i(x)^2}\\bigg )^{\\frac{p}{2}}=\\psi (x).$ By taking the limit $\\varepsilon $ tends to zero the statement of the theorem follows from the dominated convergence theorem.",
"Remark 4 The first condition on the exponents in Theorem REF allows to derive the uniform estimates for $\\eta _\\epsilon (x)$ in the proof, while the condition (REF ) ensures that $\\psi $ belongs to $L^1_{loc}(\\Omega ).$ Finally, we formulate Hardy type inequalities without weights.",
"Theorem 3 Let $N_1>p>1.$ Then, for every $u\\in \\mathring{W}_\\lambda ^{1,p}(\\Omega )$ we have $\\left(\\frac{N_1-p}{p}\\right)^p\\int _\\Omega \\frac{|u(x)|^p }{|x^{(1)}|^{p}}\\ dx&\\le \\int _\\Omega |\\triangledown _{\\lambda } u(x)|^p\\ dx,\\\\\\left(\\frac{N_1-p}{p}\\right)^p\\int _\\Omega \\frac{|u(x)|^p }{\\Vert x\\Vert _{\\lambda }^{p}}\\ dx&\\le \\int _\\Omega |\\triangledown _{\\lambda } u(x)|^p\\ dx.$ It suffices to prove the first inequality.",
"The second inequality is an immediate consequence of the first, since the norms satisfy $\\Vert x\\Vert _{\\lambda }\\ge |x^{(1)}|,$ $x\\in \\mathbb {R}^N.$ We define the function $h_\\varepsilon (x):=\\frac{1}{(|x^{(1)}|^2+\\varepsilon )^{\\frac{p}{2}}}\\left(x^{(1)},0,\\dots ,0\\right)$ and compute $\\textnormal {div}_{\\lambda } h_\\varepsilon (x)&=\\frac{N_1-p\\frac{|x^{(1)}|^2}{|x^{(1)}|^2+\\varepsilon }}{(|x^{(1)}|^2+\\varepsilon )^{\\frac{p}{2}}}>0,\\\\|h_\\varepsilon (x)|&=\\frac{|x^{(1)}|}{(|x^{(1)}|^2+\\varepsilon )^{\\frac{p}{2}}}.$ Since $N_1>p$ we have $\\textnormal {div}_{\\lambda } h_\\varepsilon >0,$ and Lemma REF applied to $h_\\varepsilon $ yields the inequality $&\\frac{1}{p^p}\\int _\\Omega \\left(N_1-p \\frac{|x^{(1)}|^2}{|x^{(1)}|^2+\\varepsilon }\\right)\\frac{|u(x)|^p }{(|x^{(1)}|^2+\\varepsilon )^{\\frac{p}{2}}}\\ dx \\\\\\le & \\int _\\Omega \\left(N_1-p \\frac{|x^{(1)}|^2}{|x^{(1)}|^2+\\varepsilon }\\right)^{-(p-1)} \\frac{|x^{(1)}|^p}{(|x^{(1)}|^2+\\varepsilon )^{\\frac{p}{2}}}|\\triangledown _{\\lambda } u(x)|^p\\ dx.$ The first inequality of the theorem now follows from the dominated convergence theorem by taking the limit $\\varepsilon $ tends to zero."
],
[
" Some Remarks on the Optimality of the Constant",
"For the particular case of Grushin type operators D'Ambrosio proved in [3] that the constants in the inequalities in Theorem REF are optimal.",
"The optimality was shown similarly to the classical case using the explicit form of the function for which the Hardy inequality becomes an equality.",
"This function does not belong to the Sobolev space $H_0^1(\\Omega ),$ but an approximating sequence in $H_0^1(\\Omega )$ is used in the proof.",
"Moreover, the function is strongly related to the fundamental solution at the origin.",
"For more general $\\Delta _\\lambda $ -Laplacians this function as well as the fundamental solution are unknown, and at present we are not able to prove that our Hardy type inequalities are sharp.",
"Using the fundamental solution at the origin the following observations yield a simple proof for Hardy inequalities.",
"We will only consider the case $p=2$ here.",
"Let $\\lambda $ be of the form (REF ), $\\Omega \\subset \\mathbb {R}^N$ be a domain, $N\\ge 3,$ and $\\Phi $ be the fundamental solution at the origin of $-\\Delta _\\lambda $ on $\\Omega ,$ i.e., $-\\Delta _\\lambda \\Phi &=c\\delta _0,\\\\\\Phi &>0,$ for some constant $c>0,$ where $\\delta _0$ denotes the Dirac delta function.",
"Moreover, let $u\\in C^1_0(\\Omega )$ and $v:=u\\Phi ^{-\\frac{1}{2}}.$ Then, the following identities follow from integration by parts and the properties of the fundamental solution (see [2] for the case of the classical Laplacian), $\\begin{split}\\int _\\Omega |\\triangledown _\\lambda u|^2dx&=\\frac{1}{4} \\int _\\Omega \\frac{|\\triangledown _\\lambda \\Phi |^2}{|\\Phi |^2}u^2dx+\\frac{1}{2} \\int _\\Omega \\triangledown _\\lambda \\Phi \\triangledown _\\lambda (v^2)dx+\\int _\\Omega |\\triangledown _\\lambda v|^2\\Phi dx \\\\&=\\frac{1}{4} \\int _\\Omega \\frac{|\\triangledown _\\lambda \\Phi |^2}{|\\Phi |^2}u^2dx+\\frac{1}{2} cv^2(0)+\\int _\\Omega |\\triangledown _\\lambda v|^2\\Phi dx \\\\&=\\frac{1}{4} \\int _\\Omega \\frac{|\\triangledown _\\lambda \\Phi |^2}{|\\Phi |^2}u^2dx+\\int _\\Omega |\\triangledown _\\lambda v|^2\\Phi dx\\ge \\frac{1}{4} \\int _\\Omega \\frac{|\\triangledown _\\lambda \\Phi |^2}{|\\Phi |^2}u^2dx,\\end{split}$ where we used that $v(0)=u(0)\\Phi (0)^{-\\frac{1}{2}}=0.$ The fundamental solution at the origin for the Grushin-type operator $\\Delta _\\lambda =\\Delta _x+|x|^{2\\alpha }\\Delta _y,\\qquad \\alpha \\ge 0,\\ z= (x,y)\\in \\mathbb {R}^{N_1}\\times \\mathbb {R}^{N_2}$ is of the form $\\Phi (x,y)=\\frac{c}{[[(x,y)]]_\\lambda ^{Q-2}},$ for some constant $c\\ge 0$ (see [5]).",
"The estimate (REF ) implies the weighted Hardy type inequality $\\int _\\Omega |\\triangledown _\\lambda u(z)|^2dz&\\ge \\frac{1}{4} \\int _\\Omega \\frac{|\\triangledown _\\lambda \\Phi (z)|^2}{|\\Phi (z)|^2}u(z)^2\\, dz=\\frac{(Q-2)^2}{4} \\int _\\Omega \\frac{|x|^{2\\alpha }}{[[(x,y)]]_\\lambda ^{2(1+\\alpha )}}u(z)^2dz,$ which is a particular case of the inequalities in Theorem REF .",
"To show the optimality of the constant we consider the identity $\\int _\\Omega \\left| \\triangledown _\\lambda u(z) -\\varphi (z)u(z)\\right|^2 dz&=\\int _\\Omega \\left| \\triangledown _\\lambda u(z)\\right|^2 +|u(z)|^2 \\left( |\\varphi (z)|^2 + \\textnormal {div}_\\lambda \\varphi (z) \\right) dz$ and observe that the function $\\varphi (x,y)=-\\frac{Q-2}{2}\\frac{|x|^{2\\alpha }}{[[(x,y)]]_\\lambda ^{2(1+\\alpha )}}\\left(x,\\frac{(1+\\alpha )y}{|x|^\\alpha }\\right),$ which we applied in the proof of Theorem REF , satisfies $|\\varphi (x,y)|^2 + \\textnormal {div}_\\lambda \\varphi (x,y)=-\\left(\\frac{Q-2}{2}\\right)^2\\frac{|x|^{2\\alpha }}{[[(x,y)]]_\\lambda ^{2(1+\\alpha )}}.$ A solution of the equation $\\triangledown _\\lambda u(x,y) =-\\frac{Q-2}{2}\\frac{|x|^{2\\alpha }}{[[(x,y)]]_\\lambda ^{2(1+\\alpha )}}\\left(x,\\frac{(1+\\alpha )y}{|x|^\\alpha }\\right)u(x,y)$ is the function $u(x,y)=\\frac{1}{[[(x,y)]]_\\lambda ^{\\frac{Q-2}{2}}},$ which was used in [3] to prove the optimality of the constant.",
"It transforms the Hardy inequality into an equality, but does not belong to the class $\\mathring{W}_\\lambda ^{1,2}(\\Omega )$ if the domain $\\Omega $ contains the origin (see [3], p.728).",
"The fundamental solution for general $\\Delta _\\lambda $ -Laplacians is unknown.",
"Assuming that there exists a homogeneous distance from the origin $d_\\lambda $ such that the fundamental solution is given by $\\Phi =d_\\lambda ^{2-Q}$ we obtain $\\frac{|\\triangledown _\\lambda \\Phi (x)|^2}{|\\Phi (x)|^2}=(Q-2)^2\\frac{|\\triangledown _\\lambda d_\\lambda (x)|^2}{|d_\\lambda (x)|^{2}},$ and (REF ) implies the Hardy type inequality $\\int _\\Omega |\\triangledown _\\lambda u(x)|^2dx&\\ge \\frac{(Q-2)^2}{4} \\int _\\Omega \\frac{|\\triangledown _\\lambda d_\\lambda (x)|^2}{|d_\\lambda (x)|^2}|u(x)|^2\\, dx.$ Consequently, if the fundamental solution was known we could define the distance $d_\\lambda :=\\Phi ^{\\frac{1}{2-Q}}$ and compute explicit, weighted Hardy inequalities.",
"On the other hand, suitable to analyze the optimality of the constants in our family of Hardy type inequalities is the relation $\\int _\\Omega \\left|\\frac{\\varphi (x)}{\\psi (x)}u(x)-\\psi (x) \\triangledown _\\lambda u(x)\\right|^2 dx =\\int _\\Omega \\psi (x)^2 \\left| \\triangledown _\\lambda u(x)\\right|^2 +u(x)^2\\left( \\frac{|\\varphi (x)|^2}{\\psi (x)^2} + \\textnormal {div}_\\lambda \\varphi (x)\\right)dx,$ which follows from integration by parts, where $\\varphi :\\mathbb {R}\\rightarrow \\mathbb {R}^N$ is a vector field and $\\psi :\\mathbb {R}\\rightarrow \\mathbb {R}$ a scalar function.",
"Comparing with the first inequality in Theorem REF we choose $\\psi (x)^2 =\\frac{[[x]]_\\lambda ^{2(1+\\sum _{i=1}^k(\\sigma _i-1))-s}}{\\prod _{i=1}^k|x^{(i)}|^{2\\sum _{j=1}^k\\alpha _{ji}- \\mu _i}},$ and observe that the function $\\varphi (x)=-\\frac{Q-s+\\sum _{i=1}^k\\sigma _i\\mu _i}{2}\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_\\lambda ^s}\\left(\\frac{\\sigma _1x^{(1)}}{\\lambda _1(x)},\\dots , \\frac{\\sigma _kx^{(k)}}{\\lambda _k(x)}\\right),$ which we used to prove the theorem, satisfies $\\left( \\frac{|\\varphi (x)|^2}{\\psi (x)^2} + \\textnormal {div}_\\lambda \\varphi (x)\\right)=-\\left(\\frac{Q-s+\\sum _{i=1}^k\\sigma _i\\mu _i}{2}\\right)^2\\frac{\\prod _{i=1}^k|x^{(i)}|^{\\mu _i}}{[[x]]_\\lambda ^s}.$ Consequently, the Hardy type inequality in Theorem REF is an equality if $u$ is a solution of the equation $\\triangledown _\\lambda u(x)=\\frac{\\varphi (x)}{\\psi (x)^2}u(x),$ i.e., $\\triangledown _{x^{(i)}}u(x)=-\\frac{Q-s+\\sum _{i=1}^k\\sigma _i\\mu _i}{2} \\frac{\\prod _{j\\ne i}\\lambda _j(x)^2}{\\Vert x\\Vert _\\lambda ^{2(1+\\sum _{i=1}^k(\\sigma _i-1))}}\\sigma _i x^{(i)}u(x),\\qquad i=1,\\dots ,k.$ Except for Grushin type operators we are unable to solve this equation, not even for the particular $\\Delta _\\lambda $ -Laplacians in Examples REF to REF ."
],
[
"Acknowledgements",
"We would like to thank Prof. Enrique Zuazua for valuable discussions and remarks and Prof. Ermanno Lanconelli and Prof. Enzo Mitidieri for some helpful comments."
]
] | 1403.0215 |
[
[
"Elasticity of Twist-Bend Nematic Phases"
],
[
"Abstract The ground state of twist-bend nematic liquid crystals is a heliconical molecular arrangement in which the nematic director precesses uniformly about an axis, making a fixed angle with it.",
"Both precession senses are allowed in the ground state of these phases.",
"When one of the two \\emph{helicities} is prescribed, a single helical nematic phase emerges.",
"A quadratic elastic theory is proposed here for each of these phases which features the same elastic constants as the classical theory of the nematic phase, requiring all of them to be positive.",
"To describe the helix axis, it introduces an extra director field which becomes redundant for ordinary nematics.",
"Putting together helical nematics with opposite helicities, we reconstruct a twist-bend nematic, for which the quadratic elastic energies of the two helical variants are combined in a non-convex energy."
],
[
"Introduction",
"A new liquid crystal equilibrium phase was recently discovered [6], [9], [5], [8].",
"Any such discovery is per se a rare event, but this was even more striking as in some specific materials an achiral phase which had long been known was shown to conceal a chiral mutant, attainable upon cooling through a weakly first-order transition.",
"This is known as the twist-bend nematic ($\\mathrm {N_{tb}}$ ) phase.The name twist-bend was introduced by Dozov [10] together with splay-bend to indicate the two alternative nematic distortions which, unlike pure bend and pure splay, can fill the three-dimensional space, as previously observed by Meyer [24].",
"Molecular bend seems to be necessary for such a phase to be displayed, but it is not sufficient, as most bent-core molecules fail to exhibit it [17].",
"In the molecular architecture capable of inducing the $\\mathrm {N_{tb}}$ phase, dimers with rigid cores are connected by sufficiently flexible linkers.Very recently, experimental evidence has also been provided for $\\mathrm {N_{tb}}$ phases arising from rigid bent-core molecules [8].",
"The molecular effective curvature, while inducing no microscopic twist, allegedly favors a chiral collective arrangement in which bow-shaped molecules uniformly precess along an ideal cylindrical helix.",
"Figure REF sketches the picture envisaged here.",
"Figure: (a) Molecular achiral model with two symmetry axes, one polar, 𝐩\\mathbf {p}, and one apolar, 𝐦\\mathbf {m}.",
"(b) One variant of the helical nematic phase with helix axis 𝐭\\mathbf {t}.",
"In the other variant, not shown here, the helix winds downwards, in the direction opposite to 𝐭\\mathbf {t}.A single bow-shaped molecule exhibits two symmetry axes, represented by the unit vectors $\\mathbf {p}$ and $\\mathbf {m}$ , polar the former and apolar the latter.Despite a visual illusion caused by the curved arc, in Fig.",
"REF (a) the lengths of $\\mathbf {p}$ and $\\mathbf {m}$ are just the same.",
"The local symmetry point-group is $\\mathsf {C}_{2v}$ , but, as explained by Lorman and Mettout [18], [19], by combining the symmetries broken in the helical arrangement in Fig.",
"REF (b), namely, the continuous translations along the helix axis and the continuous rotations around that axis, a symmetry is recovered which involves any given translation along the helix axis, provided it is accompanied by an appropriately tuned rotation.",
"This forbids any smectic modulation in the mass density, rendering the helical phase purely nematic.",
"While the nematic director $\\mathbf {n}$ is defined as the ensemble average $\\mathbf {n}:=\\langle \\mathbf {m}\\rangle $ , no polar order survives in a helical phase, as $\\langle \\mathbf {p}\\rangle =\\mathbf {0}$ .For this reason, no macroscopic analogue of $\\mathbf {p}$ will be introduced in the theory, and the phase will be treated as macroscopically apolar.",
"Fig.",
"REF (b) shows only one of the chiral variants that symmetry allows in a $\\mathrm {N_{tb}}$ phase; the other winds in the direction opposite to $\\mathbf {t}$ .",
"In both cases, $\\mathbf {n}$ makes a fixed cone angle $\\vartheta $ with $\\mathbf {t}$ .",
"For definiteness, we shall call each chiral variant a helical nematic phase.",
"In a different language, each helical nematic phase is precisely the $C$ -phase predicted by Lorman and Mettout [18], [19], which breaks spontaneously the molecular chiral symmetry, producing two equivalent macroscopic variants with opposite chiralities (see Fig.",
"REF ).",
"This paper is intended to study separately each helical nematic variant hosted in a $\\mathrm {N_{tb}}$ phase.",
"How opposite variants may be brought in contact within a purely elastic theory will be the subject of a forthcoming paper [36].",
"Though, in retrospect, a number of experimental studies had already anticipated $\\mathrm {N_{tb}}$ phases (see, for example, [31], [16], [30], to cite just a few), a clear identification of the new phase was achieved in [6] by a combination of methods (see also [15], [7], [28], [29], [4]), and an impressive direct evidence for it has been provided in [9], [5], [8] (see also [34]), with accurate measurements of both the helix pitch ($\\approx 10\\,\\mathrm {nm}$ ) and cone angle ($\\approx 20^\\circ $ ).",
"Theoretically, $\\mathrm {N_{tb}}$ phases had already been predicted by Meyer [24] and Dozov [10], from two different perspectives, the former inspired by the symmetry of polar electrostatic interactions (a line of thought recently further pursued in [32]) and the latter starting from purely elastic (and steric) considerations.",
"A helical molecular arrangement was also seen in the molecular simulations of Memmer [20], who considered bent-core Gay-Berne molecules with no polar electrostatic interactions, though featuring an effective shape polarity.",
"Dozov's $\\mathrm {N_{tb}}$ theory requires a negative bend elastic constant, which is compatible with the boundedness (from below) of the total energy only if appropriate quartic corrections are introduced in the energy density.",
"However, a large number of these terms are allowed by symmetry [10], and the theory may realistically be applied only by choosing just a few of them and neglecting all the others [21].",
"Moreover, recent experiments [1] have reported an increase in the (positive) bend constant measured in the nematic phase near the transition to the $\\mathrm {N_{tb}}$ phase.",
"In an attempt to justify the negative elastic bend constant required by Dozov's theory, a recent study has replaced it with an effective bend constant resulting from the coupling with the polarization characteristic of flexo-elasticity [32].",
"As a consequence, however, the twist-bend modulated phase envisaged in [32] is locally ferroelectric, whereas, as explained by the symmetry argument of Lorman and Mettout [19], [18], each helical nematic phase is expected to be apolar.",
"The theories in [10] and [32] are not in contradiction with one another, the only difference being that the latter justifies a negative bend elastic constant as commanded by the very bend-polarization coupling that gives rise to a modulated polar phase.",
"There is still a conceptual difference between these theories: the former is purely elastic but quartic, whereas the latter is quadratic but flexo-electric.",
"Being highly localized, the ferroelectricity envisaged in [32] does not produce any average macroscopic polarization, and so it would be compatible with the current experimental observations which have not found so far any trace of macroscopic ferroelectricity.",
"This, however, leaves the question unanswered as to whether an intrinsically quadratic, purely elastic theory could also explain $\\mathrm {N_{tb}}$ phases.",
"In this work, we propose such a theory; it will feature the same elastic constants as Frank's classical theory [13], with a positive bend constant and an extra director field.",
"This theory reduces to Frank's for ordinary nematics, for which the extra director becomes redundant.",
"This paper is organized as follows.",
"In Section , we shall formally introduce a helical nematic phase, defined as each $\\mathrm {N_{tb}}$ variant with a prescribed helicity in its ground state.See (REF ) for a precise definition.",
"A quadratic elastic free energy will be considered for each helical nematic variant, under the (temporary) assumption that they can be thought of as separate manifestations of one and the same $\\mathrm {N_{tb}}$ phase.",
"For a given sign of the helicity, negative for definiteness, we shall apply the proposed elastic theory to two classical instabilities, namely, the helix unwinding first encountered in chiral nematics (in Section ), and the Freedericks transition that has long made it possible to measure the elastic constants of ordinary nematics (in Section ).",
"Both these applications will acquire some new nuances within the present theory.",
"In Section , we first derive the quadratic elastic energy density for a helical nematic phase with positive helicity.",
"The helical nematic variants with both helicities are then combined together in a $\\mathrm {N_{tb}}$ phase; the corresponding elastic energy density need to attain one and the same minimum in two separate wells.",
"There are several ways to construct such an energy, which by necessity will not be convex; we shall build upon the quadratic elastic energy for a single helical phase arrived at in Section .",
"In the two ways that we consider in detail, the elastic energy density for a $\\mathrm {N_{tb}}$ phase features only four elastic constants.",
"Finally, in Section , we collect the conclusions reached in this work and comment on some possible avenues for future research."
],
[
"Helical nematic phases",
"A $\\mathrm {N_{tb}}$ differs from classical nematics in its ground state, the state relative to which the elastic cost of all distortions is to be accounted for.",
"The ground state of a classical nematic is the uniform alignment (in an arbitrary direction) of the nematic director $\\mathbf {n}$ .",
"The $\\mathrm {N_{tb}}$ ground state is a heliconical twist, in which $\\mathbf {n}$ performs a uniform precession, making everywhere the angle $\\vartheta $ with a helix axis, $\\mathbf {t}$ , arbitrarily oriented in space.",
"Such a ground state should reflect the intrinsically less distorted molecular arrangement that results from minimizing the interaction energy of the achiral, bent molecules that comprise the medium.",
"By symmetry, there are indeed two such states, distinguished by the sense of precession (either clockwise or anticlockwise around $\\mathbf {t}$ ).",
"In general, borrowing a definition from Fluid Dynamics (see, for example, [25]), we call the pseudoscalar $\\eta :=\\mathbf {n}\\cdot \\operatorname{curl}\\mathbf {n},$ the helicity of the director field $\\mathbf {n}$ .",
"We shall see now that it is precisely the sign of $\\eta $ that differentiates the two variants of the ground state of a $\\mathrm {N_{tb}}$ .",
"Letting $\\mathbf {t}$ coincide with the unit vector $\\mathbf {e}_z$ of a Cartesian frame $(x,y,z)$ , the fields $\\mathbf {n}_0^\\pm $ representing the ground states can be written in the form $\\begin{split}\\mathbf {n}_0^\\pm =&\\sin \\vartheta \\cos (\\pm qz+\\varphi _0)\\,\\mathbf {e}_x\\\\+&\\sin \\vartheta \\sin (\\pm qz+\\varphi _0)\\,\\mathbf {e}_y+\\cos \\vartheta \\,\\mathbf {e}_z,\\end{split}$ where $\\varphi _0$ is an arbitrary phase angle, $q$ is a prescribed wave parameter, taken to be non-negative, as characteristic of the condensed phase as the cone angle $\\vartheta $ (see Fig.",
"REF ).",
"Figure: The angles ϑ\\vartheta and ϕ=±qz+ϕ 0 \\varphi =\\pm qz+\\varphi _0 that in () represent the fields 𝐧 0 ± \\mathbf {n}_0^\\pm are illustrated in a Cartesian frame (x,y,z)(x,y,z).",
"Both ϑ\\vartheta and qq are fixed parameters characteristic of the phase.The pitch $p$ corresponding to $q$ is given byNo confusion should arise here between the pitch $p$ of the $\\mathrm {N_{tb}}$ phase and the polar vector $\\mathbf {p}$ mentioned in the Introduction.",
"The former is macroscopic in nature, whereas the latter is microscopic.",
"$p:=\\frac{2\\pi }{q}.$ On every two planes orthogonal to $\\mathbf {e}_z$ and $p$ apart, each field $\\mathbf {n}_0^\\pm $ in (REF ) delivers one and the same nematic director.",
"A simple computation shows that $\\eta ^\\pm :=\\mathbf {n}_0^\\pm \\cdot \\operatorname{curl}\\mathbf {n}_0^\\pm =\\mp q\\sin ^2\\vartheta .$ We shall call helical nematic each of the two phases for which either $\\mathbf {n}_0^+$ or $\\mathbf {n}_0^-$ is the nematic field representing the ground state.",
"Here, for definiteness, we shall develop our theory as if only $\\mathbf {n}_0^+$ represented the ground state.",
"By (REF ), such a state has negative helicity.",
"The corresponding case of positive helicity will be studied in Section REF .",
"Until then we shall drop the superscript $^+$ from $\\mathbf {n}_0^+$ to avoid clatter."
],
[
"Negative helicity",
"It readily follows from (REF ) thatHere and below $\\times $ and $\\otimes $ denote the vector and tensor products of vectors.",
"In particular, for any two vectors, $\\mathbf {a}$ and $\\mathbf {b}$ , $\\mathbf {a}\\otimes \\mathbf {b}$ is the second-rank tensor that acts as follows on a generic vector $\\mathbf {v}$ , $(\\mathbf {a}\\otimes \\mathbf {b})\\mathbf {v}:=(\\mathbf {b}\\cdot \\mathbf {v})\\mathbf {a}$ , where $\\cdot $ denotes the inner product of vectors.",
"An alternative way of denoting the dyadic product $\\mathbf {a}\\otimes \\mathbf {b}$ would be simply $\\mathbf {a}\\mathbf {b}$ .",
"In Cartesian components, they are both represented as $a_ib_j$ .",
"$\\nabla \\mathbf {n}_0=q\\left(\\mathbf {e}_z\\times \\mathbf {n}_0\\right)\\otimes \\mathbf {e}_z.$ More generally, for $\\mathbf {n}$ prescribed at a point in space, the tensor $\\mathbf {T}:=q(\\mathbf {t}\\times \\mathbf {n})\\otimes \\mathbf {t}$ expresses the natural distortionA natural distortion is a distortion present in the ground state, when the latter fails to be the uniform nematic field $\\mathbf {n}$ for which $\\nabla \\mathbf {n}\\equiv \\mathbf {0}$ .",
"that would be associated at that point with the preferred helical configuration agreeing with the prescribed nematic director $\\mathbf {n}$ and having $\\mathbf {t}$ as helix axis.",
"We imagine that in the absence of any frustrating cause, given $\\mathbf {n}$ at a point, the director field would attain in its vicinity a spatial arrangement such that $\\nabla \\mathbf {n}=\\mathbf {T},$ where $\\mathbf {T}$ is as in (REF ) and $\\mathbf {t}$ is any unit vector such that $\\mathbf {n}\\cdot \\mathbf {t}=\\cos \\vartheta .$ This would make (REF ) locally satisfied, even though $\\mathbf {n}$ does not coincide with $\\mathbf {n}_0$ in the large.",
"Correspondingly, the elastic energy that would locally measure the distortional cost should be measured relative to the whole class of natural distortions, vanishing whenever any of the latter is attained.",
"With this in mind, we write the elastic energy density $f^-_e$ in the formThe superscript $^-$ will remind us that the ground state of the helical nematic phase we are considering here has a negative helicity $\\eta $ , $f^-_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})=\\frac{1}{2}(\\nabla \\mathbf {n}-\\mathbf {T})\\cdot \\mathbb {K}(\\mathbf {n})[\\nabla \\mathbf {n}-\\mathbf {T}],$ where $\\mathbb {K}(\\mathbf {n})$ is the most general positive-definite, symmetric fourth-order tensor invariant under rotations about $\\mathbf {n}$ .",
"Clearly, if for given $\\mathbf {n}$ and $\\nabla \\mathbf {n}$ , $\\mathbf {t}$ can be chosen in (REF ) so that (REF ) is satisfied, $f^-_e$ vanishes, attaining its absolute minimum.",
"On the contrary, if there is no such $\\mathbf {t}$ , then minimizing $f^-_e$ in $\\mathbf {t}$ would identify the natural, undistorted state closest to the nematic distortion represented by $\\nabla \\mathbf {n}$ in the metric induced by $\\mathbb {K}(\\mathbf {n})$ .",
"For this reason, here both $\\mathbf {n}$ and $\\mathbf {t}$ are to be considered as unknown fields linked by (REF ): at equilibrium, the free-energy functional that we shall construct is to be minimized in both these fields.",
"Combining the general representation formula for $\\mathbb {K}(\\mathbf {n})$ and the identitiesThe superscript $ means tensor transposition.$ $(\\nabla \\mathbf {n})=\\mathbf {0},\\qquad \\mathbf {T}=\\mathbf {0},$ we can reduce $\\mathbb {K}$ in (REF ) to the following equivalent form $\\mathbb {K}_{ijhk}=k_1\\delta _{ih}\\delta _{jk}+k_2\\delta _{ij}\\delta _{hk}+k_3\\delta _{ih}n_jn_k+k_4\\delta _{ik}\\delta _{jh},$ where $k_1$ , $k_2$ , $k_3$ , and $k_4$ are elastic moduli (as in [35]).",
"By use of (REF ), of (REF ), and $\\operatorname{tr}\\mathbf {T}=0,\\qquad \\nabla \\mathbf {n}\\cdot \\mathbf {T}=q(\\mathbf {t}\\times \\mathbf {n})\\cdot (\\nabla \\mathbf {n})\\mathbf {t},$ we transform (REF ) into $\\begin{split}f^-_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})&=\\frac{1}{2}\\Bigl \\lbrace K_{11}(\\operatorname{div}\\mathbf {n})^2+K_{22}(\\mathbf {n}\\cdot \\operatorname{curl}\\mathbf {n}+q|\\mathbf {t}\\times \\mathbf {n}|^2)^2\\\\&+K_{33}|\\mathbf {n}\\times \\operatorname{curl}\\mathbf {n}+q(\\mathbf {n}\\cdot \\mathbf {t})\\,\\mathbf {t}\\times \\mathbf {n}|^2\\\\&+K_{24}[\\operatorname{tr}(\\nabla \\mathbf {n})^2-(\\operatorname{div}\\mathbf {n})^2]\\Bigr \\rbrace -K_{24}q\\,\\mathbf {t}\\times \\mathbf {n}\\cdot (\\nabla \\mathbf {n}),\\end{split}$ where $K_{11}$ , $K_{22}$ , $K_{33}$ , and $K_{24}$ , which are analogous to the classical Frank's constants [13], are related to the moduli $k_1$ , $k_2$ , $k_3$ , and $k_4$ through the equations $K_{11}=k_1+k_2+k_4,\\quad K_{22}=k_1,\\quad K_{33}=k_1+k_3,\\quad K_{24}=k_1+k_4.$ For $f^-_e$ in (REF ) to be positive definite, the elastic constants $K_{11}$ , $K_{22}$ , $K_{33}$ , and $K_{24}$ must obey the inequalities, $2K_{11}\\geqq K_{24},\\quad 2K_{22}\\geqq K_{24},\\quad K_{33}\\geqq 0,\\quad K_{24}\\geqq 0,$ which coincide with the classical Ericksen's inequalities for ordinary nematics [12].",
"The total elastic energy ${F}^-_e$ is represented by the functional ${F}^-_e[\\mathbf {t},\\mathbf {n}]:=\\int _{B}f^-_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})dV,$ where both $\\mathbf {t}$ and $\\mathbf {n}$ are subject to the pointwise constraint (REF ) and either of them is prescribed on the boundary ${\\partial {B}}$ of the region in space occupied by the medium.",
"It is worth recalling that both $\\mathbf {t}$ and $\\mathbf {n}$ are fields that need to be determined so as to obey (REF ) and to minimize ${F}^-_e$ .",
"In this theory, the helix axis of the preferred conical state and the nematic director of the actual molecular organization equally participate in the energy minimization with the objective of reducing the discrepancy between natural and actual nematic distortions.",
"Physically, $\\mathbf {t}$ represents the optic axis of the medium, likely to be the only optic observable when the pitch $p$ ranges in the the nanometric domain.",
"Such an abundance of state variables is a direct consequence of the degeneracy in the ground state admitted for helical nematics.",
"The vacuum manifold, as is often called the set of distortions that minimize $f^-_e$ , is indeed a three-dimensional orbit of congruent cones, identifiable with ${\\mathbb {S}^2}\\times {\\mathbb {S}^1}$ , where ${\\mathbb {S}^2}$ is the unit sphere and ${\\mathbb {S}^1}$ the unit circle in three space dimensions.",
"By (REF ), all natural distortions $\\mathbf {T}$ are represented by $\\mathbf {t}\\in {\\mathbb {S}^2}$ and any $\\mathbf {n}$ in the cone of semi-amplitude $\\vartheta $ around $\\mathbf {t}$ .",
"This shows, yet in another language, how rich in states is the single well where $f^-_e$ vanishes.",
"A number of remarks are suggested by formula (REF ).",
"First, it reduces to the elastic free-energy density of ordinary nematics, which features only $\\mathbf {n}$ , when either the wave parameter $q$ or the cone angle $\\vartheta $ vanish, thus indicating two possible mechanisms to induce helicity in an ordinary nematic.",
"Second, for $\\vartheta =\\frac{\\pi }{2}$ , it delivers an alternative energy density for chiral nematics, which is positive-definite for all $K_{24}\\geqq 0$ (whereas, to ensure energy positive-definiteness, the classical theory requires that $K_{24}=0$ , see [35]).",
"Finally, for arbitrary $q>0$ and $0<\\vartheta <\\frac{\\pi }{2}$ , $f^-_e$ in (REF ) is invariant under the reversal of either $\\mathbf {t}$ or $\\mathbf {n}$ , showing that both fields enjoy the nematic symmetry."
],
[
"Helix unwinding",
"In the presence of an external field, say an electric field $\\mathbf {E}$ , the free-energy density acquires an extra term, which we write as $f_E(\\mathbf {n})=-\\frac{1}{2}\\varepsilon _0\\varepsilon _\\mathrm {a}E^2(\\mathbf {n}\\cdot \\mathbf {e})^2,$ where $\\varepsilon _0$ is the vacuum permittivity, $\\varepsilon _\\mathrm {a}$ is the (relative) dielectric anisotropy, and we have set $\\mathbf {E}=E\\mathbf {e}$ , with $E>0$ and $\\mathbf {e}$ a unit vector.",
"Accordingly, the total free-energy functional ${F}^-$ is defined as ${F}^-[\\mathbf {t},\\mathbf {n}]:=\\int _{B}\\lbrace f^-_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})+f_E(\\mathbf {n})\\rbrace dV.$ To minimize ${F}^-$ when ${B}$ is the whole space, we shall assume that $\\mathbf {t}$ is uniform and $\\mathbf {n}$ is spatially periodic with period $L$ and we shall compute the average $F^-$ of ${F}^-$ over an infinite slab of thickness $L$ orthogonal to $\\mathbf {t}$ .",
"Letting $\\mathbf {t}$ coincide with the unit vector $\\mathbf {e}_z$ of a given Cartesian frame $(x,y,z)$ , we represent $\\mathbf {n}$ in precisely the same form adopted in (REF ) for $\\mathbf {n}_0$ , but with $qz+\\varphi _0$ replaced by a function $\\varphi =\\varphi (z)$ such that $\\varphi (0)=0\\quad \\text{and}\\quad \\varphi (L)=2\\pi .$ With no loss of generality, we may choose $\\mathbf {e}$ in the $(y,z)$ plane and represent it as $\\mathbf {e}=\\cos \\psi \\,\\mathbf {e}_z+\\sin \\psi \\,\\mathbf {e}_y.$ Computing $F^-$ on the ground state $\\varphi =2\\pi z/p$ (where $L=p$ ), one easily sees that the average energy is smaller for $\\psi =\\frac{\\pi }{2}$ than for $\\psi =0$ , whenever either $\\varepsilon _\\mathrm {a}<0$ and $\\vartheta <\\vartheta _\\mathrm {c}:=\\arctan \\sqrt{2}\\doteq 54.7^\\circ $ , or $\\varepsilon _\\mathrm {a}>0$ and $\\vartheta >\\vartheta _\\mathrm {c}$ , which are the only cases we shall consider here.",
"In case (REF ) (and for $\\psi =\\frac{\\pi }{2}$ ), $F^-$ reduces to $\\begin{split}F^-[L,\\varphi ]=K\\sin ^2\\vartheta \\left\\lbrace \\frac{1}{2L}\\int _0^L\\left[\\varphi ^{\\prime 2}+\\frac{1}{\\xi _E^2}\\sin ^2\\varphi \\right]dz -\\frac{2\\pi q}{L} \\right\\rbrace ,\\end{split}$ where $K:=K_{22}\\sin ^2\\vartheta +K_{33}\\cos ^2\\vartheta $ is an effective twist-bend elastic constant and $\\xi _E:=\\frac{1}{E}\\sqrt{\\frac{K}{\\varepsilon _0|\\varepsilon _\\mathrm {a}|}}$ is a field coherence length.",
"Minimizing $F^-$ in both $L$ and $\\varphi $ is a problem formally akin to the classical problem of unwinding the cholesteric helix [14], [22], [23].",
"The minimizing $\\varphi $ is determined implicitly by $\\frac{z}{L}=\\frac{1}{4}\\left(\\frac{\\operatorname{F}(\\varphi +\\frac{\\pi }{2},k)}{\\operatorname{K}(k)}-1\\right),$ where $\\operatorname{F}$ and $\\operatorname{K}$ are the elliptic and complete elliptic integrals of the first kind, respectively, and $k$ is the root in the interval $[0,1]$ of the equation $\\frac{\\operatorname{E}(k)}{k}=\\pi ^2\\frac{\\xi _E}{p},$ where $\\operatorname{E}$ is the complete elliptic integral of the second kind [27].",
"Equation (REF ) has a (unique) root only for $\\xi _E\\geqq \\xi _E^\\mathrm {(c)}:=\\frac{p}{\\pi ^2},$ for which $L$ is correspondingly delivered by $L=4\\xi _Ek\\operatorname{K}(k).$ Figure: The graph of the spatial period LL of the distorted helix under field (scaled to the undistorted pitch pp) against the field coherence length ξ E \\xi _E in ()(equally scaled to pp).",
"The complete unwinding takes place for ξ E /p=1/π 2 ≐0.101\\xi _E/p=1/\\pi ^2\\doteq 0.101, where the graph diverges.Figure REF shows the graph of $L$ against $\\xi _E$ , which diverges as $\\xi _E$ approaches $\\xi _E^\\mathrm {(c)}$ .",
"As for ordinary chiral nematics, a field with coherence length $\\xi _E^\\mathrm {(c)}$ unwinds completely the helix of a $\\mathrm {N_{tb}}$ , but its actual strength now depends on the elastic constants $K_{22}$ and $K_{33}$ , and the cone angle $\\vartheta $ .",
"The measured values of $p$ range in the domain of tens of nanometers.",
"Thus, the steepness of the graph in Fig.",
"REF along with (REF ) suggest that in actual terms the field should be very strong for any unwinding to be noticed.A simple estimate based on (REF ) would show that a field strength larger than $100\\,\\mathrm {V}/\\mu \\mathrm {m}$ is needed to unwind the $\\mathrm {N_{tb}}$ helix even if we assume $\\varepsilon _\\mathrm {a}$ as large as 10 and $K$ as small as $1\\,\\mathrm {pN}$ ."
],
[
"Freedericks transition",
"In ordinary nematics, the Freedericks transition is an instability that enables one to measure the classical Frank's elastic constants.",
"For a $\\mathrm {N_{tb}}$ , the setting is complicated by the role played by the additional field $\\mathbf {t}$ .",
"We consider a $\\mathrm {N_{tb}}$ liquid crystal confined between two parallel plates, placed in a Cartesian frame $(x,y,z)$ at $z=0$ and $z=d$ , respectively.",
"On both plates, we subject $\\mathbf {n}$ to a conical anchoring with respect to the plates' normal $\\mathbf {e}_z$ at precisely the cone angle $\\vartheta $ , so that, by the constraint (REF ), $\\mathbf {t}$ is there prescribed to coincide with $\\mathbf {e}_z$ .",
"Within the infinite cell bounded by these plates we allow $\\mathbf {t}$ to vary in the $(y,z)$ plane, so that $\\mathbf {t}=\\cos \\psi \\,\\mathbf {e}_z+\\sin \\psi \\,\\mathbf {e}_y,$ where $\\psi =\\psi (z)$ .",
"Letting $\\mathbf {t}_\\perp :=\\sin \\psi \\,\\mathbf {e}_z-\\cos \\psi \\,\\mathbf {e}_y,$ we represent the nematic director as $\\mathbf {n}=\\cos \\vartheta \\,\\mathbf {t}+\\sin \\vartheta \\cos \\varphi \\,\\mathbf {t}_\\perp +\\sin \\vartheta \\sin \\varphi \\,\\mathbf {e}_x,$ where $\\varphi =\\varphi (z)$ is the precession angle.",
"The function $\\psi $ is subject to the conditions $\\psi (0)=\\psi (d)=0,$ while $\\varphi $ is free in the whole of $[0,d]$ .",
"The system is further subjected to an external field, $\\mathbf {E}=E\\mathbf {e}_z$ .",
"In the following analysis, we shall assume that either of the aforementioned cases (REF ) or (REF ) occurs here.",
"Taking $p/d\\ll 1$ and treating $\\psi $ as a perturbation, of which we only retain quadratic terms in the energy, when the precession angle represents the undistorted helix, $\\varphi =2\\pi z/p$ , we obtain the average $F^-$ of the free-energy functional ${F}^-$ in (REF ) (per unit area of the plates) as $F^-[\\psi ]=\\frac{1}{2} K_{33}\\int _0^d\\left\\lbrace A\\psi ^{\\prime 2}+Bq^2\\psi ^2\\right\\rbrace dz,$ where $A:=\\frac{1}{2}\\Big [\\sin ^2\\vartheta (k_{11}+k_{22}\\cos ^2\\vartheta )+\\cos ^2\\vartheta (1+\\cos ^2\\vartheta )\\Big ],\\\\B:=\\frac{1}{2}\\Big [\\sin ^2\\vartheta (k_{11}+k_{22}\\cos ^2\\vartheta +\\sin ^2\\vartheta )-\\frac{1}{(\\xi _Eq)^2}|2\\cos ^2\\vartheta -\\sin ^2\\vartheta |\\Big ],\\\\k_{11}:=\\frac{K_{11}}{K_{33}},\\quad k_{22}:=\\frac{K_{22}}{K_{33}},\\quad \\xi _E:=\\frac{1}{E}\\sqrt{\\frac{K_{33}}{\\varepsilon _0|\\varepsilon _\\mathrm {a}|}}.$ It is now easily seen that $\\psi \\equiv 0$ is a locally stable extremum of the functional $F^-$ in (REF ) subject to (REF ) for $\\xi _E>\\xi _E^\\mathrm {(c)}$ , whereas it is locally unstable for $\\xi _E<\\xi _E^\\mathrm {(c)}$ , where $\\frac{\\xi _E^\\mathrm {(c)}}{p}:=\\frac{\\sqrt{|2-t^2|(1+t^2)}}{\\pi \\sqrt{4[(1+k_{11})t^2+k_{11}+k_{22}]t^2+ \\frac{p^2}{d^2}[k_{11}t^4+(1+k_{11}+k_{22})t^2+2]}}$ and $t:=\\tan \\vartheta $ .",
"Both this and the helix unwinding treated in Section are continuous transitions.",
"Since in this theory $p/d$ is small, $\\xi _E^\\mathrm {(c)}$ is only weakly dependent on the cell thickness $d$ .",
"It should be noted however that for $\\vartheta =0$ , (REF ) reduces to $\\xi _E^\\mathrm {(c)}=d/\\pi $ , which reproduces the classical Freedericks threshold [35].",
"In the special, hypothetical case of equal elastic constants (for which $k_{11}=k_{22}=1$ ), $\\xi _E^\\mathrm {(c)}$ acquires a simpler form, which retains the qualitative features of (REF ): $\\frac{\\xi _E^\\mathrm {(c)}}{p}=\\frac{\\sqrt{|2-t^2|}}{\\pi \\sqrt{8t^2+\\frac{p^2}{d^2}\\frac{t^4+3t^2+2}{1+t^2}}}.$ Figure: The graphs of ξ E ( c)\\xi _E^\\mathrm {(c)} (scaled to the pitch pp of the undistorted helix) against the cone angle ϑ\\vartheta (expressed in degrees), as delivered by () under the assumption of equal elastic constants, for p/d=0p/d=0 (solid line) and p/d=1p/d=1 (dotted line).",
"For ϑ=0\\vartheta =0, the former graph diverges, while the latter reaches the value ξ E ( c)/p=1/π≐0.318\\xi _E^\\mathrm {(c)}/p=1/\\pi \\doteq 0.318.",
"For 0<p/d<10<p/d<1, the graphs of ξ E ( c)\\xi _E^\\mathrm {(c)}, none of which diverges, fill the region bounded by the graphs shown here.The graph of $\\xi _E^\\mathrm {(c)}$ as delivered by (REF ) is plotted in Fig.",
"REF for $p/d=0$ and $p/d=1$ , the graphs for all intermediate values of $p/d$ being sandwiched between them.",
"These graphs show that for $\\vartheta $ sufficiently large, in particular for $\\vartheta >\\vartheta _\\mathrm {c}$ (and $\\varepsilon _\\mathrm {a}>0$ ), $\\xi _E^\\mathrm {(c)}$ is virtually independent of $d$ , whereas it is not so for $\\vartheta $ small.",
"Moreover, for moderate values of $\\vartheta $ and $d$ much larger than $p$ , $\\xi _E^\\mathrm {(c)}$ can easily be made equal to several times the pitch of the undistorted helix.",
"If, as contemplated by (REF ), the actual field required to ignite the Freedericks transition is weakly dependent on the cell thickness $d$ , the corresponding critical potential $U_\\mathrm {c}$ would scale almost linearly with $d$ .In the experiments performed in [5], it was found that $U_\\mathrm {c}\\sqrt{d}$ , but the boundary conditions imposed there seem to differ from the conical boundary conditions considered here.",
"Moreover, in [5] the transition nucleated locally from the inside of the cell instead of happening uniformly, as presumed here.",
"This might suggest that in the real experiment both helical variants present in a $\\mathrm {N_{tb}}$ are participating in the transition.",
"It would then be advisable taking with a grain of salt any direct comparison of our theory for helical nematics with experiments available for the whole $\\mathrm {N_{tb}}$ phase."
],
[
"Double-well energy",
"A $\\mathrm {N_{tb}}$ phase can be regarded as a mixture of two helical nematic phases with opposite helicities.",
"The ground states of these phases corresponding to all admissible natural distortions are the members of two symmteric energy wells.",
"In a way, this is reminiscent of the mixture of martensite twins in some solid crystals, which are equi-energetic variants with symmetrically sheared lattices (see, for example, [26]).",
"A thorough mathematical theory of these solid phases is based on a non-convex energy functional in the elastic deformation, featuring a multiplicity of energy wells [2], [3].",
"Below, adapting these ideas to the new context envisaged here, in which the energy depends on the local distortion of the molecular arrangement, and no deformation from a reference configuration is involved, we show how to construct a double-well elastic energy density $f_e$ for a $\\mathrm {N_{tb}}$ phase starting from the energy densities for helical nematic phases of opposite helicities.",
"To this end we need first supplement $f_e^-$ in (REF ) with the appropriate energy density $f_e^+$ for a helical nematic phase of positive helicity."
],
[
"Positive helicity",
"Prescribing the helicity of the ground state of a helical nematic phase to be positive, instead of negative as above, would amount to replacing (REF ) by $f^+_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})=\\frac{1}{2}(\\nabla \\mathbf {n}+\\mathbf {T})\\cdot \\mathbb {K}(\\mathbf {n})[\\nabla \\mathbf {n}+\\mathbf {T}],$ still with $\\mathbf {T}$ as in (REF ) and $q>0$ .",
"Our development following (REF ) could be repeated verbatim here and it would lead us to the same conclusion obtained by subjecting (REF ) to the formal change of $q$ into $-q$ : $\\begin{split}f^+_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})&=\\frac{1}{2}\\Bigl \\lbrace K_{11}(\\operatorname{div}\\mathbf {n})^2+K_{22}(\\mathbf {n}\\cdot \\operatorname{curl}\\mathbf {n}-q|\\mathbf {t}\\times \\mathbf {n}|^2)^2\\\\&+K_{33}|\\mathbf {n}\\times \\operatorname{curl}\\mathbf {n}-q(\\mathbf {n}\\cdot \\mathbf {t})\\,\\mathbf {t}\\times \\mathbf {n}|^2\\\\&+K_{24}[\\operatorname{tr}(\\nabla \\mathbf {n})^2-(\\operatorname{div}\\mathbf {n})^2]\\Bigr \\rbrace +K_{24}q\\,\\mathbf {t}\\times \\mathbf {n}\\cdot (\\nabla \\mathbf {n}),\\end{split}$ where $q$ remains a positive parameter.",
"Clearly, the energy well of (REF ) and (REF ) is formally obtained from the corresponding well of (REF ) and (REF ) by a sign inversion."
],
[
"$\\mathrm {N_{tb}}$ free energy density",
"Following [33], which studied systematically how to extend the non-convex energy first proposed by Ericksen [11] for a one-dimensional elastic bar, we consider two possible choices for the elastic energy density $f_e$ of a $\\mathrm {N_{tb}}$ phase: $f_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})=\\min \\lbrace f_e^+(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n}),f_e^-(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})\\rbrace ,$ $f_e(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})=\\frac{1}{f_0}f_e^+(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n})f_e^-(\\mathbf {t},\\mathbf {n},\\nabla \\mathbf {n}),$ where $f_0:=\\frac{1}{2}\\sin ^2\\vartheta (K_{22}\\sin ^2\\vartheta + K_{33}\\cos ^2\\vartheta )$ is a normalization constant chosen so as to ensure that $f_e(\\mathbf {t},\\mathbf {n},\\mathbf {0})=f_e^\\pm (\\mathbf {t},\\mathbf {n},\\mathbf {0})$ .",
"While $f_e$ in (REF ) is quadratic around each well, it fails to be smooth for $\\nabla \\mathbf {n}=\\mathbf {0}$ .",
"On the other hand, $f_e$ in (REF ) is everywhere smooth, but it is quartic.",
"The differences between these energy densities are illustrated pictorially in Fig.",
"REF .",
"Figure: One-dimensional pictures for f e f_e in () (solid line) and () (dashed line).",
"Here ϕ=ϕ(z)\\varphi =\\varphi (z) would represent the precession angle in a molecular arrangement such as that described by () and ϕ ' \\varphi ^{\\prime } is its spatial derivative.",
"Each minimum is representative for a three-dimensional well described by () and its mirror image (with qq replaced by -q-q).We shall not explore other possible forms $f_e$ .",
"We only heed that both (REF ) and (REF ) inherit the simple structure of the elastic energy density of a helical nematic phase, which features only four positive elastic constants, as in the classical Frank's theory of ordinary nematics.",
"General considerations on how to match ground states extracted from two different wells of $f_e$ (at zero energy cost) are independent of the peculiar form assumed for this function, as they are only consequences of the structure of each well.",
"A study of the geometric compatibility conditions that arise in this case will be presented elsewhere [36]."
],
[
"Conclusion",
"The elastic energy density proposed in (REF ) and (REF ) to describe the equilibrium distortions of each helical variant of a $\\mathrm {N_{tb}}$ phase featured just the classical four elastic constants and introduced the helix axis $\\mathbf {t}$ in addition to the nematic director $\\mathbf {n}$ .",
"The instabilities studied above illustrated two second-order transitions that differ also qualitatively from their classical analogues.",
"Only experiments may decide at this stage whether a quadratic elastic theory based on either (REF ) or (REF ) aboveEven if globally quartic, the energy density in (REF ) is quadratic about each energy well.",
"Moreover, it features only the four elastic constants required by the quadratic energy density of a helical phase with prescribed helicity.",
"is better suited to describe the novel $\\mathrm {N_{tb}}$ phases than the quartic theory proposed in [10].",
"The instabilities described in this paper for each $\\mathrm {N_{tb}}$ variant just provide a theoretical means to set the quadratic theory to the test.",
"Two director fields, namely, $\\mathbf {n}$ and $\\mathbf {t}$ , were deemed necessary here to describe the local distortion of a $\\mathrm {N_{tb}}$ phase.",
"This poses the question as to which defects both fields may exhibit and how they are interwoven, in view of the constraint (REF ).",
"An extra field also requires extra boundary conditions.",
"The question is how to set general boundary conditions for both fields to grant existence to the energy minimizers.",
"Finally, no hydrodynamic considerations have entered our study, but the question should be asked as to whether the relaxation in time of $\\mathbf {t}$ represents a further source of dissipation."
],
[
"Acknowledgements",
"I wish to thank Oleg D. Lavrentovich for his encouragement to pursue this study and for his kindness in providing me with some of his results prior to their publication.",
"I am also greatful to Mikhail A. Osipov for having suggested studying the work of Lorman and Mettout [19] on the symmetry of helical nematics.",
"I am finally indebted to the kindness of two anonymous Reviewers whose critical remarks and constructive suggestions improved considerably this manuscript."
]
] | 1403.0119 |
[
[
"Calculation of photoelectron spectra within the time-dependent\n configuration interaction singles scheme"
],
[
"Abstract We present the extension of the time-dependent configuration interaction singles (TDCIS) method to the computation of the electron kinetic-energy spectrum in photoionization processes.",
"Especially for strong and long ionizing light pulses the detection of the photoelectron poses a computational challenge because propagating the outgoing photoelectron wavepacket requires large grid sizes.",
"Two different methods which allow for the extraction of the asymptotic photoelectron momentum are compared regarding their methodological and computational performance.",
"The first method follows the scheme of Tong et al.",
"\\cite{tong} where the photoelectron wavefunction is absorbed by a real splitting function.",
"The second method after Tao and Scrinzi \\cite{scrinzi} measures the flux of the electron wavepacket through a surface at a fixed radius.",
"With both methods the full angle- and energy-resolved photoelectron spectrum is obtained.",
"Combined with the TDCIS scheme it is possible to analyze the dynamics of the outgoing electron in a channel-resolved way and, additionally, to study the dynamics of the bound electrons in the parent ion.",
"As an application, one-photon and above-threshold ionization (ATI) of argon following strong XUV irradiation are studied via energy- and angle-resolved photoelectron spectra."
],
[
"Introduction",
"With the development of new light sources such as free-electron lasers (FELs) and attosecond laser sources the interest in strong-field physics and multiphoton processes has grown, because they provide the experimental means to control and image atomic and molecular systems and to test theoretical predictions of nonlinear processes [3], [4], [5], [6], [7].",
"The photon energies of FELs extend from the UV to the X-ray range, and the intensities are such that they permit the investigation and control of inner-shell processes, Auger decay or above-threshold ionization (ATI) [8], [9].",
"Above-threshold ionization, first observed in the 1970s [10], is a highly nonlinear phenomenon in which more photons are absorbed than are needed for ionization.",
"The pulse durations of FELs can be as short as a few femtoseconds [11].",
"With these pulse properties typical atomic timescales which extend from a few attoseconds to tens of femtoseconds can be accessed in order to study electronic dynamics in atoms, molecules and clusters [12], [13], [14].",
"In the strong-field regime, multiphoton processes play a significant role, especially if the photon energies lie in the UV to X-ray range [15], [16], [17].",
"In general, in this frequency range a diversity of processes must be faced.",
"The removal of a deep inner-shell electron is followed by various processes depending on the atomic states and the photon energy [18].",
"If the laser pulse is strong enough, also multiphoton inner-shell ionization [19] as well as ATI processes can occur.",
"Experimentally, photoelectron spectroscopy is a powerful tool to analyze and quantify the processes that happen due to the irradiation of complex systems [20], [21].",
"For instance, in early experiments with intense light sources in the 1980s the angular distribution in ATI of xenon was measured [22] in order to understand the ATI phenomenon.",
"Synchrotron radiation was used to obtain high-quality angular distributions of electrons in photoionization of atoms [23], [24].",
"Also in recent experiments photoelectron spectroscopy has been used to reveal decay mechanisms and multiphoton excitations in deep shells of atoms [25] and to understand the origin of the low-energy structure in strong-field ionization [26], [27].",
"The process of photoionization has been studied extensively [28], [29], [30], [31], [32], e.g., in argon or xenon [33], [34].",
"In the weak-field limit, where the light-matter interaction can be treated perturbatively, the photoelectron spectrum has been calculated with methods that also include correlation effects.",
"Most prominent examples are post-Hartree-Fock methods that use reference states, e.g.",
"correlation methods like the configuration interaction [35], [29], the coupled clusters method [36] and the random-phase approximation [37], [38], [39].",
"Furthermore, approaches constructing continuum wavefunctions, like R-matrix theory [40], have been applied to calculate photoionization cross sections [41] and photoelectron angular distributions [42]; taking into account the interaction of the liberated electron with other atomic orbitals has led to the explanation of the giant dipole resonance of the $4d$ subshell in xenon [29].",
"In the strong-field regime the description of the ionized wavepacket is challenging due to the nonperturbative interaction between the electrons and the light pulse.",
"Therefore, the calculation of photoelectron spectra is numerically more demanding than in the weak-field limit.",
"Furthermore, many-body processes are often neglected in the strong-field regime and single-active electron (SAE) approaches have become a standard tool [43], [1], [44], [45] where correlation effects are omitted.",
"Nevertheless, recently, extensions to many-body dynamics have been presented, e.g., R-matrix theory [46], [47], [48], [49], [50], two-active electron [51], [52], [53] and time-dependent restricted-active-space configuration interaction theory [54], [55].",
"Generally, the calculation of the photoelectron spectrum can be done after the pulse is over by projecting the photoelectron wavepacket onto the eigenstates of the field-free continuum.",
"However, this approach requires large numerical grids and its application is very limited even in the SAE cases.",
"For this reason, new methods were developed to calculate the spectrum using wavepacket information in a fixed spatial volume much smaller than the volume that would be needed to fully encapsulate the wavepacket at the end of the strong-field pulse.",
"There exist several approaches to overcome the obstacle of large grids, e.g., by measuring the electronic flux through a sphere at a fixed radius [56] or splitting the wavefunction into an internal and an asymptotic part [57], [58] where the latter is then analyzed to yield the spectrum.",
"The first implementation of the flux method in the strong-field case is the time-dependent surface flux (“tsurff”) method introduced by Tao and Scrinzi [2].",
"It has recently been extended to the description of dissociation in molecules [59].",
"Tong et al.",
"[1] applied the splitting approach to strong-field scenarios.",
"With both methods double-differential photoelectron spectra can be calculated.",
"Our method for treating the electron dynamics within atoms is based on the time-dependent configuration interaction singles (TDCIS) scheme [60].",
"The Schrödinger equation is solved exactly by wavepacket propagation in the configuration interaction singles (CIS) basis.",
"The TDCIS approach [61] includes interchannel coupling and allows investigating the wavepacket dynamics and, in particular, the impact of correlation effects between the photoelectron wavepacket and the remaining ion as discussed e.g.",
"in Refs.",
"[62], [63], [64].",
"It is versatile with respect to the electric field properties (also multiple pulses can be chosen) and it has proven especially successful for strong-field studies [63], [64], [65], [66].",
"In Sec.",
", we present the theoretical details of how the photoelectron spectrum is obtained within the TDCIS scheme: The wavefunction splitting method [1] is described in Sec.",
"REF and the time-dependent surface flux method [2] in Sec.",
"REF .",
"In Sec.",
"the two methods are analyzed with respect to their efficiency within TDCIS and are compared briefly.",
"As an application we calculate and study the angle- and energy-resolved photoelectron spectrum of argon irradiated by strong XUV radiation.",
"A summary and short outlook in Sec.",
"conclude the article.",
"Atomic units are used throughout except otherwise indicated.",
"The time dependent Schrödinger equation of an $N$ -electron system is given by $i \\frac{\\partial }{\\partial t} |\\Psi ^N(t)\\rangle = \\hat{H} (t) |\\Psi ^N(t)\\rangle .",
"$ Considering linearly polarized light, the Hamiltonian takes the form $\\hat{H}(t)= \\hat{H}_0 +\\hat{H}_1+\\hat{\\vec{p}} \\cdot \\vec{A}(t), $ where $\\vec{A}(t)$ is the vector potentialUnlike in previous work on the TDCIS method [67], [60], we use the velocity form at this point.",
"Furthermore, the charge of the electron is negative, $q_e=-1$ , so that $|q_e|=1$ ..",
"Here, $\\hat{H}(t)$ is the full $N$ -electron Hamiltonian, $\\hat{H}_0=\\hat{T}+\\hat{V}_{\\rm nuc}+\\hat{V}_{\\rm MF}-E_{\\rm HF}$ contains the kinetic energy $\\hat{T}$ , the nuclear potential $\\hat{V}_{\\rm nuc}$ , the potential at the mean-field level $\\hat{V}_{\\rm MF}$ and the Hartree-Fock energy $E_{\\rm HF}$ , $\\hat{H}_1=\\frac{1}{|r_{12}|}-\\hat{V}_{\\rm MF}$ describes the Coulomb interactions beyond the mean-field level, and $\\hat{\\vec{p}}\\cdot \\vec{A}(t)$ is the light-matter interaction within the velocity form in the dipole approximation.",
"Within the CIS approach only one-particle–one-hole excitations $|\\Phi _i^a\\rangle $ with respect to the Hartree-Fock ground state $|\\Phi _0\\rangle $ are considered.",
"Therefore, the wavefunction (now omitting the superscript $N$ ) is expanded in the CIS basis as $|\\Psi (t)\\rangle &=\\alpha _0(t)|\\Phi _0\\rangle +\\sum _{i,a}\\alpha _i^a(t)|\\Phi _i^a\\rangle ,$ where the index $i$ symbolizes an initially occupied orbital and $a$ denotes an unoccupied (virtual) orbital to which the particle can be excited: $|\\Phi _i^a\\rangle =\\frac{1}{\\sqrt{2}}\\left( \\hat{c}_{a+}^\\dagger \\hat{c}_{i+}+ \\hat{c}_{a-}^\\dagger \\hat{c}_{i-} \\right) |\\Phi _0\\rangle $ .",
"The operators $\\hat{c}_{p\\sigma }^\\dagger $ and $\\hat{c}_{p\\sigma }$ create and annihilate electrons, respectively, in the spin orbitals $|\\varphi _{p\\sigma }\\rangle $ .",
"The total spin is not altered in the considered processes ($S=0$ ), so that only spin singlets occur.",
"Therefore, and for the sake of readability, we drop the spin index and treat the spatial part of the orbitals $|\\varphi _p\\rangle $ .",
"Inserting the wavefunction expansion (REF ) into the Schrödinger equation (REF ) and projecting onto the states $|\\Phi _0\\rangle $ and $|\\Phi _i^a\\rangle $ yields the following equations of motion for the expansion coefficients $\\alpha _i^a(t)$ : $i\\dot{\\alpha }_0(t) &= \\vec{A}(t)\\cdot \\sum _{i,a}\\langle \\Phi _0 | \\, \\hat{\\vec{p}}\\, | \\Phi _i^a\\rangle \\alpha _i^a(t),\\\\i\\dot{\\alpha }_i^a(t) &= (\\varepsilon _a-\\varepsilon _i) \\alpha _i^a(t) + \\sum _{j,b} \\langle \\Phi _i^a |\\hat{H}_1|\\Phi _j^b \\rangle \\alpha _j^b(t) \\nonumber \\\\& +\\vec{A}(t)\\cdot \\left( \\langle \\Phi _i^a | \\, \\hat{\\vec{p}}\\, |\\Phi _0 \\rangle \\alpha _0(t) + \\sum _{j,b}\\langle \\Phi _i^a |\\, \\hat{\\vec{p}}\\,|\\Phi _j^b \\rangle \\alpha _j^b(t) \\right),$ where $\\varepsilon _p$ denotes the energy of the orbital $|\\varphi _p\\rangle $ ($\\hat{H}_0|\\varphi _p\\rangle = \\varepsilon _p |\\varphi _p\\rangle $ ).",
"As introduced in Ref.",
"[67], for each ionization channel all single excitations from the occupied orbital $|\\varphi _i\\rangle $ may be collected in one “channel wavefunction”: $|\\chi _i(t)\\rangle = \\sum _a \\alpha _i^a(t)|\\varphi _a \\rangle .",
"$ These channel wavefunctions may now be used to calculate all quantities in a channel-resolved manner.",
"In this way, effectively one-particle wavefunctions are obtained, which will be used in the following to derive the formulae for the photoelectron spectra.",
"A detailed description of the TDCIS method can be found in Refs.",
"[60], [67].",
"The TDCIS method provides the coefficients of the wavefunction in the CIS basis, which are propagated in time.",
"During the propagation, quantities that are needed for the calculation of the photoelectron spectrum are prepared using the channel wavefunction coefficients.",
"After the propagation, these quantities are then used in the subsequent analysis step to determine the spectral components of the channel wavefunctions.",
"At the end, an incoherent summation over all ionization channels is performed to obtain the photoelectron spectrum.",
"The two analysis methods are described in the following."
],
[
"Wavefunction splitting method",
"We describe the concrete implementation of the splitting method introduced by Tong et al.",
"in Ref.",
"[1] within our time-dependent propagation scheme.",
"A real radial splitting function of the form $\\hat{S}= \\left[1+e^{-(\\hat{r}-r_c)/\\Delta }\\right]^{-1}$ is used to smoothly split the channel wavefunction (REF ).",
"The parameter $r_c$ denotes the radius where the splitting function is centered, and $\\Delta $ is a “smoothing” parameter controlling the slope of the function.",
"At the first splitting time step $t_0$ the channel wavefunction is split into two parts (for each channel $i$ ): $|\\chi _i(t_0)\\rangle = (1-\\hat{S})|\\chi _i(t_0)\\rangle + \\hat{S} |\\chi _i(t_0)\\rangle \\equiv |\\chi _{i, \\rm in}(t_0)\\rangle + |\\chi _{i, \\rm out}(t_0)\\rangle .$ $|\\chi _{i,\\rm in}(t)\\rangle $ is the wavefunction in the inner region $0<r \\lesssim r_c$ and $|\\chi _{i,\\rm out}(t)\\rangle $ is the wavefunction in the outer region $r_c\\lesssim r\\le r_{\\rm max} $ .",
"Then, the following procedure is performed at $t_0$ : The outer part of the wavefunction $|\\chi _{i,\\rm out}(t_0)\\rangle $ is analytically propagated to a long time $T$ after the laser pulse is over using the Volkov Hamiltonian $\\hat{H}_V(\\tau )$ with the time propagator $\\hat{ U}_V(t_2,t_1)=\\exp \\left(-i \\int _{t_1}^{t_2} \\hat{H}_V(\\tau ) d\\tau \\right),\\ \\ \\ \\ \\hat{H}_V(\\tau )=\\frac{1}{2} \\left[ \\hat{\\vec{p}}+\\vec{A}(\\tau ) \\right]^2,$ under the assumption that far from the atom the electron experiences only the laser field and not the Coulomb field of the parent ion.",
"It is also assumed that, at the splitting radius, the electron is sufficiently far away to not return to the ion.",
"The inner part of the wavefunction $|\\chi _{i, \\rm in}(t_0)\\rangle $ is propagated on a numerical grid using the full CIS Hamiltonian [see Eqs.",
"(REF ) and ()].",
"For the splitting function the ratio $r_c/\\Delta \\gg 1$ must be chosen such that the ground state $|\\Phi _0\\rangle $ is not affected by the splitting: $\\hat{S}|\\Phi _0\\rangle =0$ .",
"At the next splitting time $t_1$ the inner part of the wavefunction which was propagated from $t_0$ to $t_1$ is split again.",
"Thus, the following prescription is obtained: $|\\chi _{i, \\rm in}(t_j)\\rangle \\rightarrow |\\tilde{\\chi }_{i}(t_{j+1})\\rangle = |\\chi _{i, \\rm in}(t_{j+1})\\rangle +|\\chi _{i, \\rm out}(t_{j+1})\\rangle .$ This is now repeated for every splitting time $t_j$ , until all parts of the electron wavepacket that are of interest have reached the outer region.",
"Each $|\\chi _{i, \\rm out}(t_{j+1})\\rangle $ is again propagated analytically to $t=T$ .",
"Computationally, $|\\chi _{i,\\rm out}(t_j)\\rangle $ is initially expressed in the CIS basis.",
"For this purpose, we define new expansion coefficients for the outer wavefunction $\\beta _i^a(t_j)=\\langle \\varphi _a|\\hat{S}|\\tilde{\\chi }_i(t_j)\\rangle ,$ and express the wavefunction in the outer region as $|\\chi _{i,\\rm out}(t_j)\\rangle =\\sum _a\\beta _i^a(t_j)|\\varphi _a\\rangle $ .",
"During the propagation, at every splitting time step $t_j$ , which can be —and for computational efficiency should be— a multiple of the actual propagation time step, the splitting function $\\hat{S}$ is applied and the expansion coefficients (REF ) are calculated and stored.",
"Since the outer wavefunction is split from the inner part and treated analytically, the grid size needed for the description of the wavefunction is automatically reduced.",
"Later, when the spectrum is calculated, the coefficients $\\beta _i^a$ are inserted and used for the analysis.",
"The Volkov states $| \\Psi _{\\vec{p}}^{V}\\rangle \\equiv | \\vec{p}\\,^V\\rangle $ are eigenstates of the Volkov Hamiltonian and form a basis set in which the channel wavepacket at time $T$ can be expanded: $|\\chi _{i,\\rm out}(T)\\rangle = \\int \\!",
"d^3p \\sum _{t_j}C_{i}(\\vec{p},t_j) \\,| \\vec{p}\\,^V\\rangle \\equiv \\int \\!d^3p\\ \\tilde{C}_i(\\vec{p}\\,)\\,| \\vec{p}\\,^V\\rangle .",
"$ In the velocity form the Volkov states are nothing but plane waves $\\Psi _{\\vec{p}}^{V}(\\vec{r})= (2\\pi )^{-3/2}e^{i\\vec{p}\\cdot \\vec{r}}$ .",
"The photoelectron spectrum is obtained by calculating the spectral components of the outer wavefunction.",
"For this purpose, the following coefficients are evaluated: $C_i(\\vec{p},t_j)=\\int d^3p^{\\prime }\\langle \\vec{p}\\,^V|\\hat{ U}_V(T,t_j)| \\vec{p}\\,^{\\prime }\\,\\!^V\\rangle \\underbrace{\\langle \\vec{p}\\,^{\\prime }\\,\\!^V | \\chi _{i,\\rm out}(t_j) \\rangle }_{c_i(\\vec{p}\\,^{\\prime },t_j)}.$ First, we calculate the $c_i(\\vec{p},t_j)$ for each splitting time $t_j$ $c_i(\\vec{p},t_j)=(2\\pi )^{-3/2}\\sum _a \\beta _i^a(t_j)\\!\\int d^3 r e^{-i\\vec{p}\\cdot \\vec{r}} \\varphi _a(\\vec{r}),$ where the orbital is now explicitly given in the spatial representation by $\\langle \\vec{r}|\\hat{c}_a^\\dagger |0 \\rangle =\\langle \\vec{r}|\\varphi _a\\rangle = \\varphi _a(\\vec{r})=\\frac{u_{n_a,l_a}(r)}{r}Y_{l_a,m_a}(\\Omega _{\\vec{r}})$ and, thus, possesses a radial and an angular part.",
"We use the multipole expansion for the exponential function $e^{i\\vec{p}\\cdot \\vec{r}}=4\\pi \\sum _{l=0}^\\infty i^l j_l(pr)\\sum _{m=-l}^l Y^*_{lm}(\\Omega _{\\vec{p}})Y_{lm}(\\Omega _{\\vec{r}}),$ where $j_l(pr)$ denotes the spherical Bessel function of order $l$ .",
"The orthonormality relations of the spherical harmonics reduce the three-dimensional integrals in Eq.",
"(REF ) to one-dimensional radial integrals.",
"Finally, propagating to a long time $T$ after the pulse, we obtain the coefficients $C_i(\\vec{p},t_j)&=\\langle \\vec{p}\\,^V|\\hat{ U}_V(T,t_j)| \\chi _{i,\\rm out}(t_j) \\rangle \\\\&= \\sqrt{\\frac{2}{\\pi }}\\exp \\left( -\\frac{i}{2} \\int _{t_j}^{T}\\!\\!d\\tau \\left[\\vec{p}\\!+\\!\\vec{A}(\\tau ) \\right]^2 \\right) \\sum _a (-i)^{l_a}\\beta _i^a(t_j)Y_{l_a,m_a}\\!",
"(\\Omega _{\\vec{p}})\\int \\!",
"dr\\,r\\,u_{n_a,l_a}\\!(r)j_{l_a}\\!",
"(pr) .",
"\\nonumber $ These coefficients can be used to calculate the angle and energy distribution of the ejected electron because at time $T$ the canonical momentum equals the kinetic momentum.",
"One can choose now a homogeneous momentum grid and calculate these coefficients for each splitting time step.",
"In order to obtain the full electron wavepacket at time $T$ all contributions from splitting times $t_j$ must be summed up coherently to obtain the coefficients $\\tilde{C}_i(\\vec{p}\\,)$ in Eq.",
"(REF ) for each ionization channel $i$ .",
"Then, incoherent summation over all possible ionization channels yields the photoelectron spectrum: $\\frac{d^2P(\\vec{p})}{dE d\\Omega }=p \\sum _i \\big | \\tilde{ C}_i(\\vec{p}\\, ) \\big |^2.$ The extra factor of $p$ results from the conversion from the momentum to the energy differential.",
"As long as the time $T$ is chosen to be after the pulse the result is $T$ independent.",
"Of course, one needs to choose a sufficiently large $T$ such that the parts of the electron wavefunction that one wants to record have entered the outer region and can be analyzed."
],
[
"Time-dependent surface flux method (“tsurff”)",
"The second method for the calculation of photoelectron spectra is based on the approach presented by Tao and Scrinzi in Ref.",
"[2] where it was used to calculate strong-field infrared photoionization spectra in combination with infinite-range exterior complex scaling [68].",
"In this approach the electron wavefunction is analyzed during its evolution when crossing the surface of a sphere of a given radius $r_c$ .",
"Again, it is assumed that the wavefunction can be split into two parts: One part is bound to the atom and is a solution to the full Hamiltonian, the other part can be viewed as free from the parent ion and is a solution to the Volkov Hamiltonian.",
"Therefore, the method also relies conceptually on a splitting procedure.",
"Nevertheless, and in contrast to the splitting method, the wavefunction is not altered in this process.",
"As above, the key idea is to obtain the spectral components of the wavefunction by projecting onto plane waves.",
"The surface radius $r_c$ is chosen such that the electron can be considered to be free, and a sufficiently large time $T$ after the pulse is over is picked by which the electron with the kinetic energy of interest has passed this surface.",
"(For very low-energy electrons a correspondingly larger time has to be chosen.)",
"At this time, the channel wavefunction $|\\chi _i\\rangle $ for each ionization channel $i$ can be split into a bound part (corresponding to the inner wavefunction in the splitting method) and an asymptotic part, which describes the ionized contribution: $|\\chi _i(T)\\rangle =|\\chi _{i, \\rm in}(T)\\rangle +|\\chi _{i ,\\rm out}(T)\\rangle $ .",
"As in Sec.",
"REF , the system Hamiltonian for distances larger than $r_c$ is approximated by the Volkov Hamiltonian.",
"Using the Volkov states of Sec.",
"REF and $|\\Psi _{\\vec{p}}^{V}(T)\\rangle =\\hat{U}_V(T,-\\infty )|\\Psi _{\\vec{p}}^{V}\\rangle $ , the outer wavefunction is represented as follows: $|\\chi _{i,\\rm out}(T)\\rangle =\\int \\!",
"d^3 p\\ b_i(\\vec{p})\\ |\\Psi _{\\vec{p}}^{V}(T)\\rangle ,$ which vanishes for $r\\le r_c$ .",
"Thus, the photoelectron spectrum is the sum, over all channels, of the $|b_i(\\vec{p})|^2$ , where $|b_i(\\vec{p})|^2 =\\bigg | \\int _{r>r_c}\\!\\!\\!\\!\\!\\!\\!",
"d^3r\\ \\Psi _{\\vec{p}}^{V*}(\\vec{r},T)\\ \\chi _{i,\\rm out}(\\vec{r},T) \\bigg |^2=: \\big |\\langle \\Psi _{\\vec{p}}^{V}(T)|\\theta (\\hat{r}-r_c)|\\chi _{i,\\rm out}(T)\\rangle \\big |^2.$ Here, the Heaviside step function $\\theta $ enters (we adopted the notation by Tao and Scrinzi [2]).",
"In order to avoid the need for a representation of $\\chi _{i,\\rm out}(\\vec{r},T)$ at large $r$ (because $T$ is large, a fast electron moves far out during this time), this $3D$ -integral is converted into a time integral involving the wavefunction only at $r=r_c$ .",
"For that, we must know the time evolution of the asymptotic part of the wavefunction after it has passed the surface.",
"Inserting the Schrödinger equation where necessary and using the Volkov solutions in the velocity form outside the sphere with radius $r_c$ we obtain [2] $\\langle \\Psi _{\\vec{p}}^{V}(T)|\\theta (\\hat{r}-r_c) |\\chi _{i,\\rm out}(T)\\rangle = i\\int _{-\\infty }^T\\!\\!\\!",
"dt\\,\\langle \\Psi _{\\vec{p}}^{V}(t)|\\left[ -\\frac{1}{2}\\Delta -i \\vec{A} (t)\\cdot \\vec{\\nabla }, \\theta (\\hat{r}-r_c) \\right]|\\chi _{i,\\rm out }(t)\\rangle .$ The commutator, which vanishes everywhere except at $r=r_c$ , is easily evaluated in polar coordinates (assuming linear polarization) and we obtain $\\left[ -\\frac{1}{2}\\Delta -i \\vec{A} (t)\\cdot \\vec{\\nabla }, \\theta (\\hat{r}-r_c) \\right] = -\\frac{1}{2r^2} \\partial _r r^2 \\delta (r-r_c) -\\frac{1}{2}\\delta (r-r_c) \\partial _r+iA(t)\\cos (\\theta ) \\delta (r-r_c).$ More details can be found in Ref.",
"[2] as well as in Ref. [69].",
"We shuffle the derivative in the first operator term to the left, via integration by parts, and obtain the operator $-\\frac{1}{r} \\delta (r-r_c) +\\overleftarrow{\\partial _r} \\frac{1}{2}\\delta (r-r_c) -\\frac{1}{2}\\delta (r-r_c) \\vec{\\partial }_r+iA(t)\\cos (\\theta ) \\delta (r-r_c),$ where “$\\overleftarrow{\\partial } $ ” means that the derivative acts to the left on the Volkov state and “$ \\vec{\\partial } $ ” means that the derivative acts to the right on the channel wavefunction.",
"In order to implement this operator acting on the channel wavefunctions we have to calculate also the first derivative of the wavefunctions with respect to $r$ at the radius $r_c$ .",
"After the propagation, during which the coefficients of the channel wavefunctions $\\chi _i(r_c,t)$ as well as of their first derivatives $[\\partial _r\\chi _i(r,t)|_{r=r_c}]$ have been calculated, the expression (REF ) can be computed.",
"Since we introduce the multipole expansion [see Eq.",
"(REF )] for the Volkov states we have to calculate also derivatives of the spherical Bessel functions at the radius $r_c$ .",
"This calculation is performed during the analysis step for each angular momentum $l$ .",
"In the last term we express the cosine as a spherical harmonic and use the identity for the integral over three spherical harmonics $\\int d\\Omega \\ Y_{l_3,m_3}^*\\!",
"(\\Omega )Y_{l_2,m_2}\\!",
"(\\Omega ) Y_{l_1, m_1}\\!",
"(\\Omega )\\!=\\!",
"\\frac{\\sqrt{(2l_1\\!+\\!1)(2l_2\\!+\\!1)}}{4\\pi (2l_3\\!+\\!1)}C^{l_3m_3}_{l_1m_1,l_2m_2}C^{l_30}_{l_10,l_20}, $ where the Clebsch-Gordan coefficients are given by $C^{l_3m_3}_{l_1m_1,l_2m_2}=\\langle l_1m_1,l_2m_2| l_3m_3 \\rangle $ .",
"Thus, we obtain the spectral components in their final form: $\\langle \\Psi _{\\vec{p}}^{V}(T)&|\\theta (\\hat{r}-r_c)|\\chi _{i,\\rm out}(T)\\rangle = i \\sqrt{\\frac{2}{\\pi }} \\int _{-\\infty }^T\\!\\!\\!",
"dt\\ \\!\\exp \\left( -\\frac{i}{2} \\int _{-\\infty }^{t}\\!\\!\\!d\\tau \\left[\\vec{p}+\\vec{A}(\\tau ) \\right]^2 \\right)\\sum _{a}\\\\&\\left\\lbrace (-i)^{l_a}\\left[-j_{l_a}\\!",
"(pr_c)+ \\frac{pr_c}{2}\\,j^{\\prime }_{l_a}(pr_c) -\\frac{1}{2} j_{l_a}\\!",
"(pr_c) \\right] Y_{l_a,m_a}\\!",
"(\\Omega _p)\\,u_{n_a,l_a}\\!",
"(r_c)\\,\\alpha _i^a(t)\\right.\\nonumber \\\\&\\left.-\\frac{(-i)^{l_a}}{2} j_{l_a}\\!(pr_c)\\,Y_{l_a,m_a}\\!",
"(\\Omega _p)\\, u^{\\prime }_{n_a,l_a}\\!",
"(r_c)\\,\\alpha _i^a(t)\\right.\\nonumber \\\\&\\left.+\\frac{i}{2\\sqrt{\\pi }}\\,r_c\\, u_{n_a,l_a}\\!",
"(r_c)\\,A(t)\\,\\alpha _i^a(t) \\sum _{l=0}^\\infty (-i)^lj_l(pr_c)\\frac{\\sqrt{2l_a\\!+\\!1}}{2l+1} C^{l,m_a}_{l_a,m_a;1,0} C^{l,0}_{l_a,0;1,0} Y_{l,m_a}\\!",
"(\\Omega _p)\\right\\rbrace , \\nonumber $ where $j^{\\prime }_{l_a}(pr_c)=\\partial _z j_{l_a}\\!",
"(z)\\big |_{z=pr_c}$ and $u^{\\prime }_{n_a,l_a}\\!",
"(r_c)=\\partial _ru_{n_a,l_a}\\!",
"(r)\\big |_{r=r_c}$ .",
"Although this expression may seem fairly complicated, it involves only quantities evaluated at one single radius $r=r_c$ .",
"The photoelectron spectrum is then obtained as $\\frac{d^2P(\\vec{p})}{dE d\\Omega }= p\\sum _i \\big | \\langle \\Psi _{\\vec{p}}^{V}(T)|\\theta (\\hat{r}-r_c)|\\chi _{i,\\rm out}(T)\\rangle \\big | ^2,$ where, as in the splitting method, the incoherent sum over all ionization channels $i$ is performed.",
"In the present implementation, a complex absorbing potential (CAP) absorbs the wavefunction near the end of the numerical grid [70], [71], [72].",
"We use a CAP of the form $W(r)=\\theta (r-r_{\\rm CAP})(r-r_{\\rm CAP})^2$ , where $\\theta $ is again the Heaviside step function and $r_{\\rm CAP}$ is the radius where the CAP starts absorbing.",
"It is added to the Hamiltonian in Eq.",
"(REF ) in the form $-i\\,\\eta \\, \\hat{W}$ , where $\\eta $ is the CAP strength.",
"As will be discussed in Sec.",
"REF the absorption via a CAP has to be optimized carefully, because reflections from the end of the numerical grid as well as from the CAP itself have to be minimized in order to obtain an accurate photoelectron spectrum.",
"With two methods for the calculation of photoelectron spectra implemented in TDCIS, we investigate one-photon and above-threshold ionization processes of argon in the XUV regime.",
"Motivated by a recent experiment carried out at the free-electron laser facility FLASH in Hamburg [73] we assume a photon energy of 105 eV, which is far above the threshold for the ionization out of the 3p and 3s subshells.",
"In the following we examine the functionality of the splitting and the surface flux methods by means of the specific example of ionization of argon in the XUV."
],
[
"Wavefunction splitting method",
"In the splitting method, three parameters have to be adjusted: the splitting radius, the smoothness of the splitting function and the rate at which the absorption is applied.",
"A first criterion for verifying that the absorption through the masking function is performed correctly is the comparison of the total ground state population obtained via splitting with the population obtained with the CAP.",
"We use a Gaussian pulse with $9\\times 10^{13}$ Wcm$^{-2}$ peak intensity and $1.2$ fs duration (full width at half maximum, FWHM) at a photon energy of 105 eV.",
"For this pulse a converged result for the CAP strength $\\eta =1\\times 10^{-3}$ , $r_{\\rm max}=150$ a.u., and $r_{\\rm max}\\!-\\!r_{\\rm CAP}=30$ a.u.",
"gives an ionization probability of $7.197\\times 10^{-3}$ after the pulse.",
"In the studied parameter cases the agreement between the splitting results and that CAP result is better than $3\\times 10^{-3}$ relative difference (choosing, e.g., $r_{\\rm max}=150$ a.u., $r_c=80$ a.u., $\\Delta =10$ a.u., and varying the splitting time step between $0.2$ a.u.",
"and 10 a.u.).",
"With more frequent absorption the agreement gets slightly better.",
"Analyzing the splitting method, we find that the splitting radius $r_c$ and the smoothing parameter $\\Delta $ can be varied rather freely without changing the (physical) spectrum.",
"Although the total radial grid size can be chosen as small as 100 a.u.",
"(cf.",
"Fig.",
"REF ) we choose also a larger radial grid extension with $r_{\\rm max}~=250$ a.u.",
"and vary the splitting radius in the wide range from 80 a.u.",
"to 230 a.u.",
"Exemplarily, results in the direction $\\theta =0$ (along the XUV polarization axis) when the radial grid size and splitting radius are varied are shown in Fig.",
"REF (a).",
"The spectrum shows the one-photon absorption peaks at the energy corresponding to the difference between photon energy and binding energy of the corresponding orbital ($3s$ and $3p$ , respectively).",
"The second part of the spectrum, in Fig.",
"REF (b), is separated from the first part by the photon energy and is, therefore, attributed to above-threshold ionization.",
"The width of the peaks corresponds to the Fourier-limited energy width according to $\\tau \\Delta \\omega = 2.765$ (all quantities in atomic units), where $\\tau $ is the duration of the pulse intensity envelope (FWHM) and $\\Delta \\omega $ is the bandwidth of the power spectrum (FWHM).",
"The figure shows that the spectrum is independent of the splitting radius as long as around 30 a.u.",
"are left to the end of the numerical grid for absorption.",
"Reducing the difference $r_{\\rm max}-r_c$ to 20 a.u.",
"produces artificial peaks near the physical peaks.",
"To estimate how large the absorption range must be let us consider an electron with 200 eV kinetic energy.",
"It covers a distance of roughly 4 a.u.",
"per atomic unit of time.",
"The numerical results show that the range over which the wavefunction is absorbed by the splitting function must be much larger than this distance (almost 10 times larger) in order to avoid reflections.",
"This can be understood if one considers that the slope of the splitting function at a smoothing parameter of $\\Delta =10$ a.u.",
"extends over a range of around 30 a.u.",
"beyond the splitting radius to reach $95\\%$ absorption of the wavefunction.",
"Figure: The photoelectron spectrum of argon for a pulse with 105 eV photon energy, 9×10 13 9\\times 10^{13} Wcm -2 ^{-2} intensity and 1.21.2 fs duration is shown for different radial grid sizes r max r_{\\rm max} and splitting radii r c r_c.",
"The smoothing parameter is Δ=10\\Delta =10 a.u.",
"and the splitting time step is dt spl =0.2{\\rm d }t_{\\rm spl}=0.2 a.u.",
"Panel a) shows the one-photon absorption lines, panel b) shows the energetically lowest ATI lines for different splitting radii.",
"All radii are given in atomic units.",
"The spectrum does not change under variation of the splitting radius as long as around 30 a.u.",
"units are left for absorption.Since the splitting radius is not very crucial for the spectrum we proceed to the variation of the other parameters.",
"Spectra for various smoothing parameters $\\Delta $ are shown in Fig.",
"REF .",
"Here, the radial grid size is kept fixed at $r_{\\rm max}=150$ a.u., the splitting radius is 80 a.u., and absorption is performed every 10 a.u.",
"of time.",
"The physical peaks are reproduced correctly for all $\\Delta $ , the noise amplitude, however, is changing.",
"A value of $\\Delta =10$ seems to be the optimum, for $\\Delta =15$ the amplitude of unphysical peaks is higher, while the steeper slope corresponding to $\\Delta =5$ a.u.",
"produces higher oscillations near the physical peaks, which should be avoided.",
"Figure: The argon photoelectron spectrum is shown for different smoothing parameters Δ\\Delta .",
"The pulse parameters are the same as for Fig. .",
"The radial grid size is r max =150r_{\\rm max}=150 a.u., the splitting radius is 80 a.u., and the splitting is applied every 10 a.u.",
"of time.",
"The 3p and 3s peaks are not affected by the change of the slope of the splitting function, although the numerical noise resulting from reflections from the splitting function changes.The method is particularly sensitive to the splitting rate, i.e., how often the splitting is applied.",
"The more frequently the splitting function is applied the less noise is obtained.",
"This is shown in Fig.",
"REF , where only the splitting time step is varied, while the radial grid size is kept constant at 150 a.u., the splitting radius is set to 80 a.u.",
"and the smoothing parameter is 10 a.u.",
"We find that the unphysical peaks or artifacts do not contribute to the physical observables because they are orders of magnitude smaller.",
"The noisy oscillations result from numerical issues, e.g., the higher the frequency of splitting the more reflections are accumulated from the slope of the splitting function.",
"For this reason, the choice of the slope of the splitting function is coupled to the frequency of splitting.",
"For more frequent absorption of the wavefunction the steepness should be reduced.",
"Since for every splitting time step the new coefficients $\\beta _i^a(t_j)$ have to be calculated and stored during the propagation and the quantities (REF ) have to be evaluated during the analysis step, it is not convenient to perform the splitting at every propagation time step as mentioned in Sec.",
"REF .",
"In the calculations shown the propagation time step is $0.05$ a.u.",
"Figure: The argon photoelectron spectrum in the polarization direction is shown.",
"The variation of the splitting time step results in significant changes in the (numerical) oscillations.",
"The pulse parameters are the same as for Figs.",
"and .",
"At a fixed smoothing parameter Δ=10\\Delta =10 a.u., r max =150r_{\\rm max}=150 a.u., r c =80r_c=80 a.u., the noise is suppressed by several orders of magnitude for more frequent splitting.",
"The one-photon peaks and ATI peaks do not change significantly.From the derivation of the splitting method in subsection REF it can be seen that the electron spectrum is normalized to the total ionization probability (because only normalized wavefunctions are used).",
"Therefore, the integrated spectrum represents a good measure of the quality of the spectrum; the fully integrated spectrum must agree with the total ionization probability.",
"This can be verified for different parameter specifications.",
"The relative difference to the CAP result is found to be smaller than $2\\%$ in all studied parameter cases.",
"In the following we apply a strong XUV pulse centered at 105 eV with $0.7$ eV bandwidth (FWHM), which corresponds to a Fourier-transform limited pulse with 108 a.u.",
"($2.6$ fs) duration.",
"The peak intensity of the pulse is $1.0\\times 10^{15}$ Wcm$^{-2}$ .",
"In the upper left panel of Fig.",
"REF the full angle- and energy-resolved photoelectron spectrum of argon after one-photon absorption is shown.",
"The angle denotes the direction with respect to the polarization axis.",
"The peaks arise from ionization out of the 3s and 3p shells, respectively.",
"The lower left panel shows the corresponding ATI spectrum.",
"As expected, the angular distributions feature the corresponding contributions from the different channels, which can be seen on the right in the four cuts along the fixed peak energies: The one-photon peak from the 3s shows a p-wave character, the 3p peak has both an s- and a d-wave contribution.",
"Analogously, the two-photon peak of 3s exhibits an s- and d-wave character and the 3p peak a p- and f-wave character.",
"Figure: The energy- and angle-resolved argon photoelectron spectrum produced with the splitting method is shown for an XUV pulse at 105 eV photon energy, 1.0×10 15 1.0\\times 10^{15} Wcm -2 ^{-2} intensity and 2.62.6 fs pulse duration.",
"The grid size is r max =100r_{\\rm max}=100 a.u., r c =20r_c=20 a.u., Δ=5\\Delta =5 a.u., and dt spl =10t_{\\rm spl}=10 a.u.",
"The angle denotes the direction with respect to the polarization axis of the pulse.",
"The angular distribution reflects the change in angular momentum by multiphoton absorption.In a nutshell, the splitting method is a well-working tool for the calculation of photoelectron spectra although the requirement to optimize three parameters ($r_c,$ $\\Delta $ , d$t_{\\rm spl}$ ) can render calculations time-consuming."
],
[
"Time-dependent surface flux method (tsurff)",
"We turn now to the tsurff method.",
"The method depends on the radius $r_c$ where the surface measuring the flux is placed and on the parameters of the absorption method.",
"As already mentioned in Sec.",
"REF , in the present work the absorption is performed with a CAP, which depends on two parameters: the CAP strength $\\eta $ and the radius $r_{\\rm CAP}$ where the CAP starts absorbing.",
"For tsurff also the total propagation time plays an important role.",
"While the splitting method is not affected by a variation of the propagation time (as long as it is longer than the pulse and long enough for the electronic wavepacket of interest to enter the absorption region), tsurff requires a long propagation.",
"This is shown in Fig.",
"REF .",
"The noise level decreases dramatically with longer time propagation.",
"Figure: The argon photoelectron spectrum calculated via tsurff is shown for different propagation times (in a.u.).",
"The pulse parameters are the same as in Figs.",
"to .",
"The computational parameter specifications are: r max =250r_{\\rm max}=250 a.u., r CAP =230r_{\\rm CAP}=230 a.u., η=1×10 -3 \\eta =1\\times 10^{-3}, and r c =180r_c=180 a.u.",
"The oscillations decrease by orders of magnitude for longer propagation.On the other hand, the calculation of the spectrum itself can be performed faster than with the splitting method, because no radial integrals are involved.",
"Instead, all quantities are evaluated at the radius $r=r_c$ .",
"The method relies on an optimized CAP for the energy range of interest.",
"However, the CAP cannot guarantee a perfect absorption.",
"Since the optimized CAP strength is energy dependent [70] the tsurff spectrum can be optimized only for a limited energy range.",
"In Fig.",
"REF the energy spectrum for $\\theta =0$ is shown for different CAP strengths $\\eta $ .",
"It is clear that reflections from the CAP as well as from the end of the radial grid leave a trace in the spectrum.",
"A weak CAP cannot fully absorb a fast electron before the end of the radial grid.",
"On the other hand, a strong CAP will reflect the electron.",
"For the kinetic energies of the electrons considered here the optimized CAP parameter lies at a value of about $10^{-3}$ .",
"Of course, the other parameter that must be optimized is the CAP radius $r_{\\rm CAP}$ .",
"We find that the optimum is an absorption range of $r_{\\rm max}\\!-\\!r_{\\rm CAP}=30$ a.u.",
"For tsurff also the distance of $r_c$ to $r_{\\rm CAP}$ plays a role.",
"In Fig.",
"REF the spectrum is shown for different $r_{\\rm CAP}-r_c$ values.",
"For a distance of $r_{\\rm CAP}-r_c=20$ a.u.",
"the spectrum becomes less oscillatory and the noise level decreases significantly in comparison to shorter ranges $r_{\\rm CAP}-r_c$ .",
"Figure: The argon photoelectron spectrum along the polarization axis of the field, calculated with tsurff, is shown for a pulse with 105 eV photon energy, 9×10 13 9\\times 10^{13} Wcm -2 ^{-2} intensity and 1.21.2 fs duration for different CAP strengths.",
"The radial grid size is r max =150r_{\\rm max}=150 a.u., the CAP radius is r CAP =120r_{\\rm CAP}=120 a.u., and r c =100r_c=100 a.u.",
"The oscillations are due to reflections from the end of the radial grid and/or the CAP.Figure: The argon photoelectron spectrum for the same pulse as in Figs.",
"to and Figs.",
"to is shown along the polarization direction.",
"The numerical parameters are r max =250r_{\\rm max}=250 a.u., η=1×10 -3 \\eta =1\\times 10^{-3}, r CAP =220r_{\\rm CAP}=220 a.u., and the propagation time is 1000 a.u.",
"The distance r max -r CAP r_{\\rm max}\\!-\\!r_{\\rm CAP} is varied in the range from 0 to 20 a.u.A direct comparison of the spectrum in the direction $\\theta =0$ obtained by splitting and tsurff, respectively, is shown in Fig.",
"REF .",
"The pulse characteristics are the same as for the Figs.",
"REF to REF and REF to REF .",
"The radial grid size is $r_{\\rm max}=250$ a.u.",
"The splitting parameters are $r_c=200$ a.u., $\\Delta =10$ a.u., ${\\rm d} t_{spl}=0.2$ a.u., and the propagation time is 400 a.u.",
"For the surface flux method a CAP strength of $10^{-3}$ , a CAP radius of $r_{\\rm CAP}=220$ a.u., a sphere radius of $r_c=200$ a.u.",
"(according to the optimum found for $r_{\\rm CAP}-r_c=20$ a.u.)",
"and a propagation time of 1000 a.u.",
"are chosen.",
"The spectra agree quite nicely.",
"The one-photon peaks exhibit a nearly perfect agreement.",
"The slight deviation in the two-photon spectrum calculated with tsurff indicates that the CAP could be reoptimized for this energy range.",
"However, for both methods the spectrum has a very low noise level, up to ten orders of magnitude smaller than the physical signal.",
"Figure: The argon photoelectron spectra obtained with the splitting and the tsurff methods for a pulse with 105 eV photon energy, 9×10 13 9\\times 10^{13} Wcm -2 ^{-2} intensity and 1.21.2 fs duration along the polarization direction are compared.",
"The radial grid size is r max =250r_{\\rm max}=250 a.u.",
"for both methods.Summarizing, the tsurff-method is in principle applicable with a CAP, although it requires a good quality absorption over a broad energy range.",
"Qualitatively, the tsurff method reproduces exactly the same results as obtained with the splitting method."
],
[
"Conclusion",
"We have implemented two computational methods for the calculation of photoelectron spectra within the TDCIS scheme.",
"Both methods can produce reasonable and quantitative energy- and angle-resolved spectra within our model.",
"Subshell ionization can be quantified.",
"We have applied and compared these methods for the high-intensity XUV regime.",
"Advantages of the splitting method are the good absorption characteristics through the splitting function and the short propagation time that is needed.",
"A disadvantage is the long evaluation time of the radial integrals.",
"The tsurff method needs a longer propagation time.",
"However, the calculation of the photoelectron spectrum in the analysis step is much faster than with the splitting method due to the evaluation at one point.",
"The comparison of the two methods shows that, in principle, the same spectra can be obtained after the appropriate optimization of the computational parameters.",
"Although our application in the present work focuses on the XUV range, it is of course possible to study also processes in the strong-field regime in the infrared range by analyzing the photoelectron spectrum.",
"Interesting applications arise from the fact that information about the coherence and the entanglement of the ionic state and the photoelectron can be extracted from the properties of the outgoing wavepacket."
],
[
"Acknowledgments",
"This work has been supported by the Deutsche Forschungsgemeinschaft under Grant No.",
"SFB 925/A5."
]
] | 1403.0352 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.