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+{"text":"\\section{Introduction}\nGeometrically branched molecules (small-molecule dendrimers\n\\refto{discovery} \n and \ndendritic polymers \\refto {dendrimer_polymer_review}) \nare seeing progressively sophisticated \napplications  \\refto{dendrimer_review_2,dendrimer_review_1},\nparticularly as biomedical and drug-delivery agents\n\\refto{dendritic_polymer_medicine}.\nThe proliferation of free ends (possibly functionalized) and\na rich design space in the interior of these single molecules\nmakes them ideal as engineering platforms.\n\nThis work is particularly focused on determining the collective \nstructure of large collections of dendritic \nor hyper-branched \\refto{hyperbranched}, \npolymers end-grafted on a surface\n\\refto{brush}.\nEven ordinary, linear polymers gain interesting properties when they \nare crowded together onto a surface by one end.\nIn making an analogy between the Edwards single-chain free energy and\nan electrostatic system, Semenov \\refto{semenov}, and subsequently\nMilner, Witten, and Cates \\refto{mwc} (in making an analogy to the classical\nmechanics of a particle falling in an external potential), determined that\nthe monomer chemical potential took on a universal profile - the\n``parabolic'' potential (and density profile, when the polymer brush was\nsolvated).\nMonodispersity of the chains is a very powerful constraint producing this\nbehavior, as is the unique  self-consistent potential thus determined.\nWhen the chains are grafted on a convex surface \\refto{marko}\nthe parabolic potential profile is no longer the self-consistent solution\nas such an {\\it ansatz} would require monomers to overfill space near the\ngrafting surface (so that the self-consistently determined distribution\nof free ends would be {\\it negative} in some regions).\nA considerably more complex analysis is required to determine the \nself-consistent potential in the presence of regions with zero end-density \n(so-called ``dead'' or ``exclusion'' zones).\n\nHowever, when the polymers in the brush\n are branched (star-like \\refto{katya} \nor dendritic \\refto{galen_forrest}-\\refto{galen_zook})\nthe architectural increase \nof free ends counteracts the ``overfilling'' effect, and it can \noccur that non-zero end-densities are achieved even for \nthe extreme case of a single dendritic polymer in a good solvent.\nHere \\refto{galen_zook}, the absence of dead-zones and the consequent \nparabolic potential and therefore monomer density causes \na single dendritic polymer to have a dense core of monomers that \ndecreases monotonically to its edge in good solvent.\nThis is in contrast to the original prediction \\refto{degennes_dendrimer}\nthat the tips of the dendritic polymer would all extend to the\nsame spherical surface producing a large, characteristic ``hollow core''\nwhich had eluded detection in early simulations \\refto{filled_core}.\nThe monodispersity of the hyper-branched polymer is again the culprit, as\na massive free-energy degeneracy is a consequence of the parabolic\npotential  (and is the cause of the isochronous behavior of the harmonic\noscillator \\refto{goldstein}).\n\nIn this work, we generalize a previous theory \n\\refto{galen_forrest,dendrimer_copoly} as in Ref.~\\refto{galen_zook} in a\n``continuous branching'' model for the dendrimer brush.  \nVastly different architectures (regularly branched, randomly branched,\nand various power-law schemes of branching) can all be handled in the\nsame theory with few assumptions and fitting parameters.\nThe parabolic shape of the insertion potential survives the\ncontinuous branching treatment, and is the exact analytic solution for\nan unreasonably large class of polymer brush architectures.\n\nThe paper is organized as follows.  First, we introduce the strong \nstretching self-consistent theory for continuously branched brushes, \nthen we analyze the end-distributions for various architectures.  Finally we\ndiscuss the implications for the near-universal stabilization of the\nparabolic profile by branching, and make our conclusions.\n\n\n\\omit{\nDendritic molecules have received much attention since their \ninvention \\refto{discovery},\npartly because of their intriguing architecture, but more importantly\nfor their promise in producing materials of decidedly interesting\nproperties \\refto{dendrimer_review_2,dendrimer_review_1}.\nSingle dendrimers can be synthesized with very extremely regular\narchitecture, and are capable of forming thermally tunable complexes\nas drug-delivery agents \\refto{thermal_tune}.\nTheir well-characterized size and relative stiffness makes them suited\nto forming complex two-dimensionally packed arrays that can then be\ndecorated or etched to produce nanoscopically patterned surfaces \n\\refto{patterns}.\nAnd, the geometric proliferation of chain tips, each capable of being\ntagged with a bio-specific functionality, makes them an ideal material\nfor engineering smart surfaces \\refto{darrel_tips}.\nIt should be pointed out that most naturally occurring biologically\nrelevant molecules are either lightly branched (the three-armed fatty-acids)\nor linear (phospholipids, DNA, and proteins), and it is therefore\nentirely possible that hyperbranching in and of itself can grant\nunusual single-molecules properties with no natural analog.\n\nRelated to these dendrimers are dendritic polymers, where the \nbranching points are connected by flexible spacer polymers of a controllable\nmolecular weight and composition.\nIn some respects, these dendritic polymers resemble \npolymer stars \\refto{daoud}, with\na general splaying of their many free arms away from a dense core.\nIndeed, miktoarm stars with an unequal number of $A$ and $B$ arms have been\npredicted to drastically reshape the $AB$ diblock copolymer \nphase diagram \\refto{milner_stars},\nand hyperbranched dendritic polymers are predicted to have the same\neffect \\refto{fredrickson,dendrimer_copoly}.\nThe microphases are consistently skewed toward having the component\nwith the most branches on the exterior of a curved surface,\nwith the most pronounced effects occurring for stars with, for example, \na single $A$ type arm and many $B$ arms.\nSimilarly, when the $A$ block is a linear flexible homopolymer, and the\n$B$ block is a $G$-generation dendritic polymer, the phase diagram\nis skewed considerably toward keeping the branched block on the\nexterior of the cylindrical and spherical domains.\nFor example, it was found that for compositionally symmetric ``tadpole''\ncopolymers \\refto{tadpole}, the lamellar \nphase is stable when the branched block \nis $G1-5$, while the cylindrical phase obtains for $G6-8$, and the spherical\nphase occurs for $G9$ and more branched \\refto{dendrimer_copoly}.\n\nThe purpose of the present work is to extend earlier treatments of this system\nwhich had been limited to this most dramatic case where\nthe $A$ and $B$ blocks of the copolymer are both dendrimers of independent\ngeneration.\nThus, we investigate the strong-segregation limit for $GA-GB$ block \ncopolymers, and determine the phase boundaries between the ``classical''\nblock copolymer phases.\nIn the lamellar phase (L), each of the branched blocks stretches\naway from the flat $AB$ interface, producing a layered material as in \nFigure~1.\nThere are two cylindrical phases, one (CA) with the $A$ material forming\na cylindrical core with the $B$ phase forming a continuous matrix in \nwhich the $A$ cores are hexagonally packed.\nThe second cylindrical phase (CB) has the $B$ material confined to the cores,\nwhile the $A$ material forms the continuous matrix.\nLikewise there are two spherical phases, (SA) and (SB) in which one material \nis confined to spherical domains packed on a bcc lattice, while the\nother material forms a continuous matrix.\nDenoting $\\phi_A$ and $\\phi_B$ the overall compositions of the single\ndendriblock copolymers, the phase boundaries will be controlled by the\noverall composition and the generations of the $A$ and $B$\nblocks, $\\phi_B(G_A,G_B)$.\nThe transitions between these phases will always follow the sequence\n(SA) - (CA) - (L) - (CB) - (SB) as $\\phi_B$ is increased.\nThe calculations we make are in the spherical\/cylindrical unit cell \napproximation, and rule out  a priori exotic bicontinuous\nphases.\n\nThe first set of calculations we employ involve the Alexander-de Gennes\napproximation \\refto{fredrickson,alexander,degennes}.\nWe assume that all of the $A$ tips and the $B$ tips reside \non the same surfaces.  \nIn the (L) phase, this results in the $A$ tips being segregated to a single\nsurface extending away from the $AB$ interface, and similarly for the $B$\ntips.\nIn the (C) phases, the outer block tips all reside on a cylindrical surface\nenclosing the core cylinder, while the inner tips\nare brought toward the center axis of the core,\nand similarly for the (S) phases.\nIt should be noted that the (L) phase thus resembles back-to-back brushes\nof dendritic molecules, while the (C) phase resembles the \nconformation of single dendrimer-comb copolymers, where dendrimers are\ngrafted to a single flexible backbone chain at regular lengths \n\\refto{dendrimer_review_2}.\nThe (S) phase resembles the single-molecule conformation of an individual\ndendrimer molecule.\nThus, the Alexander approximation can be seen as being similar to\nthe de Gennes and Hervet ansatz for a ``hollow-core'' \ndendrimer \\refto{degennes_dendrimer}.\nThe ``filled-core'' picture of Muthukumar and \nLescansec \\refto{filled_core}, however, seems\nto be theoretically and experimentally a more sound description.\nIndeed, allowing the tips of the internal and external dendrimer to occupy\nthe entire lamellar\/cylindrical\/spherical domain turns out to be a more\nrealistic assumption.\nIn the strong-segregation limit, relaxing the confinement of the\nchain free ends is achieved in the so-called ``classical limit'' \n\\refto{semenov,mwc} of\nthe Edwards self-consistent field.\n\n\n\nThe paper is organized as follows.  We initially describe the\nAlexander-de Gennes calculation, and then the classical path approximation.\nThen, the phase diagrams for the dendrimer-dendrimer \ncopolymer will be developed.\nThe results herein rest on the stong-segregation limit, the robustness\nof which can be determined in a simple Random Phase Approximation\n(RPA) calculation.\nFinally, our conclusions will be offered.\n}\n\n\n\\section{Model}\nWe use $n$ as a ``chemical index'' marking the fewest monomers required from\na given monomer to arrive at a non-grafted free end.\nThus, each free tip has a chemical index of $0$, and the unique grafted monomer\nhas a chemical index of $N$ (analogous to the overall\nmolecular weight for a linear polymer).\nWe denote the branching profile, $f(n)$ to stand for the number of statistically\nequivalent monomers with the chemical index $n$.\nFor an unbranched polymer, $f(n)=1$, while for a regular \n(double tip-splitting) dendritic polymer \nof generation $G$, \n$f(n)$ is an exponentially decreasoing step function with $f(0) = 2^G$ and discontinuities at \n$n=1\/G, 2\/G ... (G-1)\/G$.  This branching profile has been used to\nanalyze brushes of dendritic polymers \\refto{galen_forrest} and \ncopolymers of dendritic polymers \\refto{dendrimer_copoly} and brushes\nof star polymers \\refto{katya}.\nWe can model polymers, however, with continuous branching profiles, such as\n\\begin{equation}\n\\label{geometric}\nf(n) = 2^{-nG+G} \\equiv e^{-b n}\n\\end{equation}\nas in \\refto{galen_zook} where the conformation of a single dendrimer\nwas considered.  \nThe ``branching index'', $b$ can be chosen so that we model polymers with\n$G$ generations and a junction functionality of 3 modeling tip-splitting\ndendrimers ... but any positive value of $b$ makes physical sense.\nThis ``continuously branched'' polymer model has significant analytic advantages\nand loses only some of the (admittedly interesting) self-similar details of\nthe exact free-end density.\nWe will consider another class of branching profiles:\n\\begin{equation}\n\\label{powerlaw}\nf(n) = \\left[\\frac{1 - \\alpha n\/N}{1-\\alpha}\\right]^b\n\\end{equation}\nmodeling polymers which are branched so that the average number of \nstatistically equivalent monomers at any chemical index \ndecreases as a power law from the free tips.\nHere, $\\alpha$ is a measure of how strongly branched the polymers are, with the degree of branching diverging at $\\alpha \\rightarrow 1$.\nThese two scenarios are depicted in Figure~\\ref{figure1} (top panel). \n\nAt any rate, we consider a set of polymers characterized by a maximal \nchemical index $N$, of monomers of volume $a^3$ with the $n=N$ monomer \nirreversibly grafted to a flat surface with a grafting density of $\\sigma$\nchains per unit area.\nWe consider a melt brush thus formed so that the overall height of the \ngrafted layer is consistent with the incompressibility of the monomers:\n\\begin{equation}\n\\frac{h}{\\sigma} = a^3 \\int_0^N dn f(n).\n\\end{equation}\nIf $\\sigma$ is large enough, then the chains in the brush will be considerably\nstretched from the grafting surface. \nAs in Figure~\\ref{figure1}, the $n=N$ monomer is located at $z=0$, and \nthe $f(0)$ equivalent chain tips are located at $z=z_o < h$.\nThe single-chain Edwards free energy of this chain is\n\\begin{equation}\nF_{single} = \\int_0^N dn f(n) \\left[ \\frac{1}{2 a^2} \\left|\\frac{dz}{dn}\\right|^2 +\n  a^3 P(z(n)) \\right]\n\\label{single}\n\\end{equation}\nUnder conditions in which $F_{single} >> 1$ (in natural energy units of\n$k_B T$), fluctuations of the chain conformations around the\nconfiguration which minimizes Eq.~\\ref{single} are negligible, and all\nstatistical averages can be calculated from the configurations minimizing \nthe single chain free energy itself.\nHere, the first term counts the Gaussian elastic energy of $f(n)$ equivalent\nchains each consisting of $dn$ monomers and stretched a distance $dz$.\nThe second term counts the energy required to evacuate a volume $a^3$ for\neach monomer in these test chains, at a cost in free energy of $P(z)a^3$.\nThus, $P(z)$ is the monomer pressure in the layer, created by the crowding and \nthe conformations of all of the other chains.\n\nThe minimization of Eq.~\\ref{single} is effected through the usual calculus\nof variations, easily recognizable if the kernel of the integral is\ninterpreted as the Lagrangian of a particle with a time-dependent mass in \na gravitational field $-P(z)$:\n\\begin{equation}\nL[z;\\frac{dz}{dn}] = f(n) \\left[ \\frac{1}{2 a^2} |\\frac{dz}{dn}|^2 +\n  a^3 P(z(n)) \\right],\n\\end{equation}\nwith the momentum conjugate to $z(n)$ given by\n\\begin{equation}\np = \\frac{\\partial}{\\partial dz\/dn} L = \\frac{f(n)}{a^2} \\frac{dz}{dn},\n\\end{equation}\nand the generalized force given by \n\\begin{equation}\nf = \\frac{\\partial L}{\\partial z} = f(n) a^2 \\frac{d}{dz} P(z(n)).\n\\end{equation}\nThus, the Euler-Lagrange equation governing the minimzation of $F_{single}$ is\n\\begin{equation}\n\\frac{d}{dn} p = f \n\\end{equation}\nor\n\\begin{equation}\n\\frac{d}{dn} f(n) \\frac{dz}{dn} = f(n) a^5 \\frac{dP}{dz}.\n\\label{eqmot}\n\\end{equation}\nThe physical initial condition on this equation of motion is that \n\\begin{equation}\n\\left. \\frac{dz}{dn}\\right|_{n=0} =0,\n\\end{equation} \nthus requiring that there be no tension on the free chain end.\n\nThe pressure field, $P$, can be calculated self-consistently from the \nsolutions to Eq.~\\ref{eqmot}.\nFirst, we need that $P$ satisfies an isochronous property.\nFollowing the time-dependent mass analogy, a particle of mass $f(t)$ released at rest \nfrom a position $z_o$ must hit the grafting surface $z=0$ when $n=N$.\nThis condition must obtain from every position with a non-zero density of \nfree ends in the layer:\n\\begin{equation}\n\\label{sc1}\nN = \\int_o^N dn = \\int_0^{z_o} dz \\frac{1}{dz\/dn}.\n\\end{equation}\nAlso, given such a $P$ it must be possible to create a layer completely\nfilled with monomers at every $z$ without overfilling space for the melt brush we consider here.  The\ncommon formulation of this condition is that the height density of free\ntips $d \\sigma \/ dz$ be postive:\n\\begin{equation}\n\\label{sc2}\n\\Phi(z) \\equiv 1 = \\int_z^h dz_o \\frac{d \\sigma (z_o)}{dz_o} \\phi(z;z_o)\n\\end{equation}\nwith the volume fraction of monomers in the vicinity of $z$ for a chain with its \n$f(0)$ ends located at $z_o$:\n\\begin{equation}\n\\phi(z;z_o) = \\sigma a^3 f(n) \\frac{dn}{dz}.\n\\end{equation}\nWhen $f(n)\\equiv1$, (an ordinary polymer brush) self-consistency and the isochronous condition are satisfied \nwhen \\refto{mwc}\n\\begin{equation}\nP(z) = P_o (1-z^2\/h^2)\n\\end{equation}\nwith $P_o = \\pi^2\/8  a \\sigma^2$ and\n\\begin{equation}\n\\frac{d \\sigma}{dz} = \\sigma \\frac{z\/h}{\\sqrt{h^2 - z^2}},\n\\end{equation}\nremarkably compact and elegant expressions.\n\nOne appraoch to solving the self-consistency equations \nEqs~\\ref{sc1}-\\ref{sc2} when $f(n) \\ne 1$ is\nto solve this set of integral equations for the unknown fields $P(z)$ and\n$d \\sigma \/dz$ simultaneously and numerically.  \nHowever, knowing that an arbitrary $f(n)$ gives the same physics as a time\ndependent mass in a gravitational field, we can be confident that the \n{\\em parabolic} pressure profile is a fruitful {\\it ansatz}.\nA time-dependent mass in a harmonic gravitational potential will have the\nsame falling time no matter where it is released from rest.\nThe equation of motion under these conditions becomes:\n\\begin{equation}\n\\label{parabolic_eq}\n\\frac{df}{dn} \\frac{dz}{dn} + f(t) \\frac{d^2 z}{dn^2} = -2 f(n) a^5 P_o z\n\\end{equation}\nwith $dz\/dn|_0=0$ where the pressure scale $P_o$ still has to be \ndetermined self-consistently by enforcing $z(N)=0$ numerically.\n\nTo make further progress, we need to specify $f(t)$, but there  are two\ngeneral results that can be gleaned at once.\nFirst, if the distribution of ends is non-zero for all $0<z<h$, the equation\nof motion is linear in $z(n)$.\nThus, the general solution with $z(0) = z_o$ and $z'(0)=0$ satisfies:\n\\begin{equation}\nz(n) = z_o  \\xi(n),\n\\end{equation}\nwhere $\\xi(n)$ is a solution to Eq~\\ref{parabolic_eq} with\n\\begin{equation}\n\\xi(0)=1 \\mbox{   and   } \\xi'(0) =0.\n\\end{equation}\nThe final condition on $\\xi(n)$ is satisfied by choosing $P_o$ such that\n\\begin{equation}\n\\xi(N) = 0.\n\\label{po_eq}\n\\end{equation}\nThus, the existence and uniqueness of $\\xi$ guarantees that the equal-time \nand minimum free energy conditions are satisfied.\nThe non-negativity of the end-distribution must be checked separately.\n\nThe second is a general degeneracy theorem.\nThe trajectory given by the solution to Eq~\\ref{parabolic_eq} minimizes the\noverall single-chain free energy, Eq~\\ref{single}.  \nWe can very quickly show that {\\em all} chain trajectories, regardless of \nthe location of their free ends, $z_o$, have the {\\em same single chain free\nenergy}.\nThis massive degeneracy is the source of the ``filled-core'' dendrimer \n(as all sub-branch free energies with the subbranches starting at any\nposition from the center out to the edge of the dendrimer have the same\nenergy, and therefore add equally to the statistical averages determining the\nconcentration profile).\nThe minimum single-chain free energy is\n\\begin{equation}\nF_{min}[z(n)] = \\int_0^N dn f(n) \\left[ \\frac{1}{2 a^2} \n\\left|\\frac{dz}{dn}\\right|^2 +\n  a^3 P_o(1- \\frac{z^2}{h^2}) \\right].\n\\end{equation}\nRewriting the first term,\n\\begin{equation}\n\\int_0^N dn f(n) \\frac{1}{2 a^2} \n\\left|\\frac{dz}{dn}\\right|^2 = \n\\int_0^N dn \\left[f(n) \\frac{1}{2 a^2}\\frac{dz}{dn}\\right]\\frac{dz}{dn},\n\\end{equation}\nand integrating by parts:\n\\begin{equation}\n=\\left. z(n) f(n) \\frac{1}{2 a^2}\\frac{dz}{dn}  \\right|_0^N -\n\\int_0^N dn \\frac{d}{dn} \\left[ f(n) \\frac{1}{2 a^2}\\frac{dz}{dn}\\right] z =\n- \\int_0^N a^3 P_o \\frac{z}{h^2} z,\n\\end{equation} using the initial condition, the isochronous condition, and\nthe equation of motion.\nThus, the first term exactly cancels the $z^2$ term in the minimum free \nenergy yielding:\n\\begin{equation}\nF_{min} = P_o a^3 \\int_0^N f(n),\n\\end{equation}\nso that an energy $P_o a^3$ must be paid to insert all of the $\\int dn f(n)$\nmonomers on the branched chain.\nThe free energy of a chain with its free ends located at $z_o$ is thus independent\nof $z_o$, so all of the chains contribute equally to the total layer energy.\nThus, the parameter $P_o$ is the overall monomer pressure to insert \nany monomer of volume $a^3$ into the brush.\n\nIt remains to calculate $P_o$ given $f(n)$ and to verify that the end-density \nis non-zero to verify the parabolic {\\it ansatz} is a solution to the problem.\nWhen the brush is composed of geometrically branched polymers, \nand Eq~\\ref{geometric}\nprevails, and the equation of motion becomes ($f(t) = e^{-b t}$):\n\\begin{equation}\n-b \\frac{dz}{dt} + \\frac{d^2 z}{dt^2} = -2 a^5 P_o z\n\\end{equation}\na ``time-reversed'' linear drag harmonic oscillator.\nWhen Eq~\\ref{powerlaw} obtains (with the substitution $u = (-1+ \\alpha n\/N)$)\nthe equation of motion is\n\\begin{equation}\nu z''(u) +  z'(u) + \\frac{2 P_o a^5 N^2}{\\alpha^2 h^2} u z =0,\n\\end{equation}\na Bessel equation.\n\n\\section{Results}\nWe have determined both $P_o$ and the end-density profiles by numerically\nsolving Eq.~\\ref{po_eq} and by numerically inverting the integral equation\nEq.~\\ref{sc2}.\nFigure~\\ref{figure2} and Figure~\\ref{figure3} shows how the insertion \npotential per monomer depends on the relative degree of branching of the\npolymers in the brush.\nHere we refer to the ``continuous branching'' model of Ref.~\\refto{galen_zook}\nas a continuous version of a tip-splitting dendritic polymer and the\n``power-law'' brush corresponds to choice of $f(n)$ corresponding to \nEq.~\\ref{powerlaw}, with the the power-law parameter $b=1,2,3$ \nin these calculations.  When \n$\\alpha \\rightarrow 1$ the polymer has more and more equivalent ends.\n\nWe have scaled $P_o$ to its value for a linear brush in the classical limit,\n$ \\sigma^2 a \\pi^2 \/8$.\nIn all of the cases we consider, it is apparent that the cost per monomer \nto establish the branched grafted layer decreases, and approaches zero \nin the limit of very high stretching.\nThis is not to say that the overall free energy per chain decreases when\nthe branching increases, as the energy per chain is porportional to the\nnumber of monomers.  For chains of equivalent overall total chemical index\n$N$ the cost at each $n$ is $f(n) P_o$, and thus is a rapidly increasing\nfunction with branching.\n\nThis does imply, however, that the stretching for most of the monomers\nwith low $n$ (and hence high $f(n)$) is very low.  \nFor hyper branched chains, this implies that the vicinity of the free\nends acts essentially as densely cross-linked gel ``buoy'' tethered to \nthe grafting surface by a set of very strongly stretched, but low overall\nmolecular weight, tethers.\nThe symptom of this in the free-end distributions would be a dramatic \nflattening and an enhancement of the end-density near the grafting\nsurface, as in the lower panel of the schematic, Fig.~\\ref{figure1}.\nThis is exactly what we note in Fig.~\\ref{figure4} and Fig.~\\ref{figure5}.\nThe $G=4$ continuous branched layer and the $\\alpha= 3\/4$ linear power\nlaw brush have essentially uniform end-density across the layer.\nWe note that the requirement for self-consistency is that\n$\\frac{d \\sigma}{dn} > 0$ throughout the brush, so that the parabolic\n{\\it ansatz} is indeed the correct self-consistent inter-monomer interaction\npotential.\n\n\n\\section{Discussion}\nGiven how successful the parabolic {\\it ansatz} is, it is tempting to\njump to the erroneous conclusion that the parabolic potential is the\nself-consistent potential for {\\em any} choice of $f(n)$.\nThe two choices we have made result in nicely analytic minimum-action\ntrajectories, $z(n;z_o)$, but the critical feature required is that\n$f(n)$ be a decreasing function of chemical index.\nThat is, the chains are unbranched at the grafting surface, and become more\nand more branched as they continue away from the grafting surface.\nIf the chains started highly branched at the grafting surface, with\nsubchains combining into loops as one left the grafting surface so that the\nchain became less and less branched toward the free ends \nthen the brush will resemble a layer or Alexander-deGennes \n\\refto{alexander} \\refto{degennes} brushes \nof loops, with the free ends evacuated from a large region of the \nbrush \\refto{fredrickson}.  The self-consistent potential is then known\nto be radically different from the parabolic potential here, although the\nreal potential still satisfies an isochronous, or monodisperse, condition.\n\nIndeed, if we start with an unbranched set of chains, $f(n)=1$,\nand alter the branching a bit by\n\\begin{equation}\nf(n) = 1 + \\epsilon(n)\n\\end{equation}\nwith $\\epsilon(n)$ a small function that is very strongly peaked at $n=0$, \nit is easy to show that the surface end-density must be negative, and\nspace is overfilled if the parabolic potential is followed.\nEssentially, we need to increase the number of chain free ends located with\n$z_o$ near the full brush height, $h$, above the unbranched limit. This causes \nthe number of monomers entrained into the layer to be slightly larger than they would \nhave been had $\\epsilon(n)=0$, with the result that building the brush from the\nouter edge to the surface maintaining $\\phi(z)=1$ will overfill space with \nmonomers at the grafting surface.\n\nSo, if $f(n)$ is a {\\em rapidly} decreasing function of $n$, each chain ``dumps''\nalmost all of its monomers right in the vicinity of the free end location,\n$z_o$.\nIn this case, the molten layer resembles a set of large, dense microgels \ntethered by much lower molecular weight net of branched chains.  \nThe minimum-action chain trajectories become (essentially) particles at rest\nfor most of $n<N$, which then use their last few remaining monomers to fully\nstretch back to the grafting surface.  \nThus, the layer loses its character as an anisotropic stress environment, and\nbecomes essentially a molten simple fluid of these free ``heads'' constrained to\nexist between $0<z<h$, with a uniform density of ``heads''.\nThe rest of the brush is merely there to maintain the connectivity of the\nchains.\nThe flattening of the trajectories, and concomitant flattening of the \nchain end density distributions is the result.\n\nThe enrichment of the end-density at $z=0$ has one more important consequence.\nIf the grafting surface is convexly curved, the tendency of the branching to enhance the surface end-density and the curvature to decrease it will cancel in favor of non negative grafting surface density ... the critical ``parabolic'' curvature will of course depend upon the properties of the grafted and branched chains, but there is the possibility that copolymer phases of branched polymers (cylindrical, and spherical in particular) will be devoid completely of free-end ``dead-zones,'' making the estimate of the free energies of the phases elementary.\nIndeed, dead-zones of this type occur for energetic reasons as well, as in brushes of gradient copolymers \\refto{galen_gradient}.  It is an open question as to whether branching of the gradient copolymers will close that dead-zone.\n\n\\section{Conclusion}\nIn the self-consistent classical path analysis with a continuous\nbranching profile, we have shown that the\n parabolic potential profile for hyper branched and statistically hyper\nbranched polymers is the self-consistent chain interaction potential.\nThis parabolic brush is remarkably robust to changing the chain topology\nand the melt-brush thus formed is (in the limit of strong branching) a \nsimple liquid of ``heads'' tethered weakly to the surface by the last\ntether.  Thus, branching dramatically simplifies the structure and analysis\nof these molten layers.\n\n\n\\pagebreak\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe region of the solar system \nbeyond the planets is populated by a vast \nswarm of objects in what is called the Kuiper belt. The objects\nin this swarm -- Kuiper belt objects (KBOs) -- formed on\nthe outer edge of the planetary system where nebular densities were\ntoo low and accretional timescale too high to allow objects to\naccumulate into a single dominant planet, as happened throughout\nthe planetary system. Instead, in the region beyond the planets, \naccretion proceeded up to objects at least as large as the 2700 km diameter\nof Triton but then stalled as dynamical excitation of the region\ncaused collision velocities to be too high to allow continued accretion. \nToday, encounters between objects in the outer solar system lead\nto erosion to smaller diameters.\n\nMuch of we have learned from the Kuiper belt has come from \nstudy of the dynamics of these bodies \\citep[i.e.][]{2008ssbn.book..275M}. Understanding\nthe dynamical state of a KBO is straightforward: after an\nobject is discovered, a few observations spaced over several years\nare usually sufficient to define a precise orbit for the object\n\\citep{2000AJ....120.3323B}. Since their formation\nKBOs have been moved around by the gravitational tugs and perturbations\nof the giant planets, and their current locations preserve a record\nof these billions of years of interactions. Our current understanding\nof the rearrangement and migration of the giant planets is based \nlargely on this record. \n\nIn principle, the objects in the Kuiper belt also contain a wealth of\ninformation on the chemical conditions in the earliest solar system.\nThe materials in these objects have, for example, avoided much of\nthe high temperature and pressure processing to which materials in\nthe inner solar system have been subjected. \nIn practice, however, understanding the detailed composition\nof KBOs in their native environment has proved difficult.\nMost objects are small, far away, and thus difficult to study\neven with the largest telescopes available. For the small number\nof objects large enough for detailed compositional study to\nbe feasible, only the surface chemistry, which is clearly\nnot representative of the entire body, is accessible to observation.\nFor detailed understanding of ices and minerals incorporated into\nbodies in the early outer solar system, short-period comets, \nwhich are generally derived from the Kuiper belt, are much better\ntargets for study. \n\nFor the first decade of the study of compositions of KBOs, most effort\nwas put into simple cataloging and classifying the surfaces, with\nlittle insight into underlying causes. The past decade, however, has\nseen a maturation of both the data sets available and the \nphysical and chemical insights into their meaning. \nThe chemical composition of KBOs serves a tracer of the\nphysical and chemical \nprocesses to which these bodies were subjected  at formation\nand in the time period since. Using studies of surface composition\nwe have gained insight into atmospheric processes,\ndifferentiation and cryovolcanism, radiation processing, the\neffects of giant impacts, and the early\ndynamical excitation of the Kuiper belt. \n\nIn this review we attempt to formulate a coherent picture\nexplaining the compositions of Kuiper belt objects. \nWe show that KBO surface composition divides most strongly\nby size, with large, medium, and small objects having\ndistinct surface types, and we propose physically and\nchemically plausible hypotheses for all of these surfaces.\nWe attempt to formulate similar hypotheses for\nthe bulk compositions of KBOs, but we find no satisfactory\nhypothesis that fits the current data. We end with \nsome thoughts on important open questions that would best\nallow progress to be made in these areas.\n\n\\section{The largest Kuiper belt objects}\n\\subsection{Volatiles}\nSpectroscopic study of some of the earliest discovered KBOs \nrevealed surfaces that were either spectrally featureless or\ncontained only small amounts of absorption due to water ice. \nThese early spectra of relatively small objects were in marked\ncontrast to the reflectance spectrum of the much larger Pluto, \nwhich is rich in absorption\nfeatures due to CH$_4$, N$_2$, and CO \\citep[see][for reviews]{1997plch.book..221C,2002AREPS..30..307B}, all\nof which are volatile at the 30-50 K surface temperatures of Pluto. \nTriton, a body even larger than Pluto and\nthought to be a captured KBO, has a similar compliment of \nvolatile chemicals on its surface \\citep{brown_triton}.\nThe advent of wide field surveys of the outer solar system \n\\citep{2003EM&P...92...99T, 2008ssbn.book..335B, \n2010ApJ...720.1691S}, however, led to the discovery of objects approaching and \neven comparable in size to Pluto, many of which are also covered\nin some of these same volatile chemicals. \n\nDefinitive detections of CH$_4$ (a molecule with strong absorption\nbands in the near-infrared which make the species easy to detect) have\nnow been made on 6 of the 8 largest known KBOs: Triton, Eris, Pluto,\nMakemake, Sedna, and Quaoar \n\\citep{brown_triton, 1997plch.book..221C, 2005ApJ...635L..97B,2006A&A...445L..35L, 2005A&A...439L...1B,2007ApJ...670L..49S}. A 7th -- 2007 OR10 -- \nis suspected from indirect evidence to also contain\nCH$_4$, but no confirming spectrum has yet been obtained \\citep{2011ApJ...738L..26B}.\nN$_2$ and CO have weaker absorption bands and are therefore more\ndifficult to detect, so no absorptions due to these species have\nbeen detected on any objects other than Pluto or Triton. \nFigure 1 \nshows a composite of spectra of some of the volatile-containing\nKBOs.\n\\begin{figure}\n\\includegraphics*{fig1.eps}\n\\caption{\nSpectra of four of the largest and most volatile rich KBOs.\nThe prominent CH$_4$ absorption lines can be seen\non all objects. The weaker CO and N$_2$ lines can only be detected\non Pluto.\n}\n\\end{figure}\n\nThe specific volatile species detected on these largest KBOs are not \nsurprising. Of the molecules know to be abundant in comets,\nCH$_4$, N$_2$, and CO are the only ones that have moderate \nvapor pressures at the temperatures in the Kuiper belt. Most other\nmajor molecules remain in the frozen state on the surface, while some\nsuper-volatiles, like noble gasses, have such high vapor pressures that, if\nthey were present, they would be exclusively in the gas phase. \nCH$_4$, N$_2$, and CO, in contrast, can reach an equilibrium between\nfrost on the surface and a molecular atmosphere. This surface-atmosphere\nexchange ensures that, when volatile abundances are sufficiently high,\nthe volatile frosts will coat the surface, masking the composition\nof whatever lies beneath.\n\nThe relative abundances on the volatile containing KBOs are not identical.\nOn Makemake, the brightest and most easily studied of the newly\ndiscovered large objects,  \n the CH$_4$ absorption features appear significantly\ndifferent from those on the other volatile-rich KBOs. The absorption\nlines are broad and saturated, as occurs when the optical path lengths\nthrough the solid CH$_4$ are long. The surface producing this\nspectrum has been modeled as a solid slab of CH$_4$ with voids \napproximately $\\sim$1 cm apart \\citep{2007AJ....133..284B,2007JGRE..11206003E}. On Pluto and Triton, in contrast, \nsmall wavelength\nshifts in the position of the \nnear-infrared absorption features of the CH$_4$ show that the molecule\nprimarily occurs as a minor constituent diluted in an N$_2$\nmatrix \\citep{1993Sci...261..742C, 1993Sci...261..745O}\nthough regions of pure CH$_4$ frost are also seen to exist \\citep{1999Icar..142..421D}.\nCH$_4$ is at least an\norder of magnitude more abundant compared to N$_2$ than it is\non Pluto \\citep{2007AJ....133..284B}.\nThe presence of N$_2$ has\nnonetheless \nbeen indirectly inferred\nby noting small shifts in the wavelengths of some of the far red\nCH$_4$ absorption features \\citep{2008Icar..195..844T}. \n\nQuaoar, even more dramatically, has such a low surface abundance of CH$_4$\nthat only the strongest absorption features are\ndetectable, superimposed on a background of water ice absorption\n\\citep{2007ApJ...670L..49S}. \nThe CH$_4$ seems to be a patchy covering on an involatile water\nice substrate. The more recently discovered 2007 OR10 is thought to \nhave a similar surface composition to Quaoar, based on its size and\nits optical colors (see below). \n\nThe presence of volatiles on these large objects and even the differences\nin relative abundances between the volatiles can be explained by\na simple model of volatile loss and retention in the Kuiper belt\ndeveloped by \\citet{2007ApJ...659L..61S}. In this model all KBOs are assumed to\nstart with typical cometary abundances but then volatile escape to space\noccurs at a rate determined by the \nsurface temperature and gravity. Volatile loss\ncan occur through a variety of mechanisms on a small body with \na vapor-pressure controlled atmosphere, but the slowest mechanism\npossible is simple Jeans escape from an atmosphere in hydrostatic\nequilibrium at its surface equilibrium temperature. An updated\nversion of the model results \nof \\citet{2007ApJ...659L..61S} (from \\citet{2011ApJ...738L..26B}) is shown in Figure 2. In short, volatiles\n\\begin{figure}\n\\includegraphics*{fig2.eps}\n\\caption{Volatile retention and loss in the Kuiper belt.\nObjects to the left of the CH$_4$, CO, and\nN$_2$ lines are too small and too hot to retain any of those\nsurface volatiles\nover the age of the solar system, while objects to the right can retain those\nvolatiles. All objects shown in purple have had CH$_4$ measured on their\nsurfaces. Some have additionally had N$_2$ or CO detected. No objects to \nthe left of the lines have had any of these volatiles detected. Methane\nis suspected, but not confirmed, on 2007 OR10.\n}\n\\end{figure}\ncan be retained on objects which are either massive enough to prevent\nsignificant Jeans escape or cold enough to prevent significant vapor\npressure of the frosts. The three main volatiles behave slightly\ndifferently because of differing vapor pressures and molecular masses. \nN$_2$ or CO have identical molecular masses, so escape identically\nfor identical vapor pressures, but N$_2$ is more volatile, so N$_2$ always\nhas a higher vapor pressure and escapes more quickly. CH$_4$ has the \nlowest vapor pressure of the three, so generally escapes more slowly,\nbut for high enough temperatures the vapor pressure is sufficiently high\nthat the escape is controlled more by molecular weight, and the lighter\nCH$_4$ escapes the quickest.\n\nIn the model, four important regimes are seen, corresponding nearly \nprecisely to the observations. The largest and coldest objects --\nTriton, Eris, Pluto, and Sedna -- have the ability to maintain\ntheir full complement of volatiles. All three volatiles have\nindeed been detected on Triton and Pluto. On Eris and Sedna\ndirect detection of N$_2$ is difficult owing to the faintness of\nthe objects, the weakness of the single quadrapole N$_2$ absorption\nline, and the fact that at the low temperatures of the very distant\nEris and Sedna, N$_2$ should be in its $\\alpha$ state leading to\nan extremely narrow absorption feature \\citep{1995Icar..116..409T}. \nOn Eris, however,\nsubtle shifts in the far red portion of the spectrum again show\nevidence that some CH$_4$ is diluted within a N$_2$ matrix\n\\citep{2010ApJ...725.1296T}. \nFor Sedna, early observations suggested the \npresence of an N$_2$ absorption feature, but this feature appeared\nsimilar to that on Triton, a broad feature of $\\beta$-nitrogen, which\nwould be difficult to explain on the frigid Sedna \\citep{2005A&A...439L...1B}. \nOther observations\nhave failed to confirm this feature \\citep{2010AJ....140.2095B}. It\nappears likely that N$_2$, even if abundant on Sedna, would be \nnearly impossible to detect spectroscopically. CO would be extremely\ndifficult to detect on Eris or Sedna, and, to date, no meaningful upper\nlimits have been placed.\n\nThe next major regime in the model is occupied only by Makemake. Makemake\nhas just the right size and temperature to be able to retain its CH$_4$,\nbut to be on the edge of not being able to retain N$_2$ and CO. The\nunusual surface of Makemake appears well explained by this model. \nWith a low abundance of N$_2$ on the surface, CH$_4$ becomes the major\nconstituent and anneals into large slabs, giving rise to the long optical\npath lengths which saturate the infrared spectrum. \n\nThe third regime in the model is the transition where CH$_4$ can only\nbarely be retained and other volatiles are nonexistent. \nQuaoar and potentially\n2007 OR10 are both in \nthis regime. These objects are both sufficiently depleted even in methane\nthat their water ice substrate is visible in regions not covered by \nmethane. \n\nThe object 2004 VN112, though likely small, has a perihelion of 47 AU and thus\nstays cold enough to potentially retain at least CH$_4$. \nThis object is too faint for infrared \nspectroscopy, so we have no direct\nindication of its surface composition, and its albedo, and thus, size\nis unknown. Its optical colors are relatively\nneutral, however, suggesting that it is not dominated by CH$_4$ irradiation\nproducts (see below). Further physical study of this object is clearly warranted.\n\nThe final regime within this model is the region in which most of the\nKuiper belt resides, where temperatures are too high and \nmasses are too low, so that even with slow Jeans escape, the three main\nvolatiles must be depleted over solar system time scales.\n\nTo date, this model has been flawless for determining which\nobjects do and do not retain volatiles in the Kuiper belt (with \nthe important exception of Haumea, which is discussed \nbelow). It is interesting -- and even unexpected -- that \nsuch a simple model would work so well. The well-studied \natmospheres of\nPluto and Triton, for example, do not conform to the simple assumptions\nof surface temperature hydrostatic equilibrium and Jeans escape. Perhaps,\nhowever, the power of the model is that as volatiles begin to become\ndepleted through faster mechanisms such as hydrodynamic escape, the atmosphere\neventually becomes tenuous enough that Jeans escape is the only remaining\nmechanism\nfunctioning. Jeans escape is then so slow compared to other processes\nthat it eventually\ndominates the total amount of time that it takes for volatile loss.\n\nWhile Kuiper belt volatiles have mostly been studied from infrared\nspectroscopy, stellar occultations provide another alternative for\nstudy of volatile compositions. \nSuch occultations should be able to\nprovide insights into N$_2$ abundances on Eris, Makemake and Sedna\nand occultations should provide the best means of looking for\nvolatiles around small distant objects such as 2004 VN112.\n\n\\subsection{Radiation processing}\n\nAll bare surfaces in the solar system are subject to irradiation by\nsolar wind, UV photons, and cosmic rays, all of which are \ncapable of inducing chemical changes in the surfaces. The surface\ncolors of most KBOs are likely set by this irradiation (see below), but,\nfor many years, the only radiation products thought to be seen\nin the outer solar system were\ntholins --  the reddish involatile \nresidues of long term irradiation \nof hydrocarbon and nitrogen containing compounds.\n Many other\nsimpler molecular\nradiation products of N$_2$, CO, and CH$_4$ were synthesized\nin laboratories and predicted to be present on Pluto or Triton\n\\citep{2008ssbn.book..507H}, but the low abundances, difficult-to-observe spectral\nfeatures, and lack of adequate data prevented any positive detections.\n\nThe discovery of the large slabs of CH$_4$ on Makemake, however, \nalso led to the first direct detection of simple\nradiation products. The saturation of the strong\nCH$_4$ absorption \nline around 2.3$\\mu$m allows deviations from the CH$_4$ spectrum\nto be easily identified. The strongest deviations are clearly due to the\npresence of C$_2$H$_6$ (Fig 3). C$_2$H$_6$ forms\n\\begin{figure}\n\\plotone{fig3.eps}\n\\caption{\nDetection of pure ethane on Makemake. The data show the difference between\nthe data and a CH$_4$ model of the surface. The deviations from the CH$_4$\nmodel show the clear signature of pure ethane ice, an expected \nirradiation product of CH$_4$. Evidence for additional absorption\nis present, but no positive identification of the species has yet\noccurred.}\n\\end{figure}\nfrom the combination of two CH$_4$ molecules which have each had a \nhydrogen atom removed by irradiation, and it is the first stable molecule\nto form in the presence of CH$_4$ irradiation \\citep{2006ApJ...653..792B}. \n\nMakemake, which is dominated by CH$_4$ rather than N$_2$, as on Pluto,\nis an ideal laboratory for understanding this radiation processing.\nOn Pluto, creation of C$_2$H$_6$ is stymied by the dilution of the\nCH$_4$ in a N$_2$ matrix. Radiation can remove a hydrogen atom, \nbut another CH$_3$ radical will rarely be present to form C$_2$H$_6$.\nNonetheless, recent observations have suggested a small amount\nof C$_2$H$_6$ on Pluto, as well \\citep{2010Icar..208..412D,2010Icar..210..930M}, presumably coming \nfrom the small regions of pure CH$_4$ which have been detected\n(see above). \n\nQuaoar, though having significantly less coverage of CH$_4$ on its\nsurface than Makemake, also appears to have detectable C$_2$H$_6$ \nembedded within its surface, which would be expected for an object\nwith CH$_4$ and little to no N$_2$. Quaoar is also unusual for\nhaving both deep water ice absorption features and a very \nred color. The radiation processing of CH$_4$ appears\nlikely to be the cause of this coloration; long-term irradiation\nof CH$_4$ will cause the ice to turn red \\citep{2006ApJ...644..646B}.\n2007 OR10, which appears in a similar regime\nof volatile retention as Quaoar, has a similar very red surface\ncombined with deep water ice absorptions, leading to the hypothesis\nthat it, too, contains CH$_4$, and, presumably, its radiation \nproducts.\n\nAfter the creation of C$_2$H$_6$, continued irradiation leads to\nC$_2$H$_2$ (acetylene) and C$_2$H$_4$ (ethylene).\nThese molecules have increasingly difficult to detect spectral\nfeatures and have not been detected on Makemake -- or any other\nobject in the solar system -- to date, but it is likely that they\nare present as temporary products which eventually lead to\nthe red coloration of the long-chain hydrocarbons.\n\nOrcus is similar in size to Quaoar but warmer, and thus is not\nexpected to retain volatiles. \\citet{2010A&A...520A..40D} nonetheless suggest\nthe detection of C$_2$H$_6$, an involatile irradiation \nproduct on this surface. \nWhile it is possible to imagine a scenario in which CH$_4$ irradiation\nleads to irradiation products which then stay behind as a lag deposit\nafter the CH$_4$ has been lost, it is difficult to construct a scenario\nin which the surface does not also retain the red coloration of\nmore complex irradiation products. Orcus is instead amongst the most\nneutrally colored KBOs. We suspect the higher signal-to-noise spectra\nof Orcus will confirm a lack of methane irradiation products on its surface.\n\n\\subsection{Haumea and its family}\nHaumea is unique in the solar system. It is the fastest rotating and\nmost elongated\ngravitationally bound object in the solar system, has a density\nnearly that of rock \\citep{2006ApJ...639.1238R}, and is surrounded by two\nmoons \\citep{2005ApJ...632L..45B, 2006ApJ...639L..43B}. Though it is one of the largest\nobjects in the Kuiper belt and would be predicted by the simple\nmodel above to contain abundant volatiles, its surface is instead\ndominated by what appears to be nearly pure water ice \\citep{2007ApJ...655.1172T}.\nThe moons likewise appear to have surfaces of \npure water ice \\citep{2006ApJ...640L..87B, 2009ApJ...695L...1F},\nand a dynamical family of KBOs with pure water ice surfaces\nexists in orbits separated from Haumea by only a few hundred meters per\nsecond (Fig 4).\n\\begin{figure}\n\\includegraphics[scale=.75]{fig4.eps}\n\\caption{\nNear infrared reflectance spectra of Haumea, its satellite Hi'iaka, \nand some members of its collisional\nfamily, compared to a model of a laboratory spectrum of pure ice.\nNo objects with comparably deep water ice absorption features are\nfound anywhere in the Kuiper belt other than the dynamical\nvicinity of Haumea. The spectra are consistent with nearly pure\nwater ice. In all cases with sufficient signal-to-noise, the\nspectrum shows the 1.65 $\\mu$m absorption feature of crystalline ice.\nThis feature is ubiquitous in the outer solar system on objects\nlarge and small and appears to be caused by exogenic rather than\nendogenic processes.}\n\\end{figure}\n\nWhile Haumea is large enough to be differentiated, so that we can imagine\nan icy outer layer surrounding a predominantly rocky body, the\nknown dynamical family members and the satellites range in size from $\\sim$70\nto $\\sim$ 370 km (assuming 0.7 albedos), likely too small to be differentiated. For these objects,\nthe fact that the surfaces appear to be nearly pure water ice\nstrongly suggests that these objects are nearly pure water ice\nin their interiors, also. In particular, the mutual orbital\nperturbations of the two Haumea satellites show that their masses are\nconsistent with them having bulk densities even lower than 1 g cm$^{-3}$\n\\citep{2009AJ....137.4766R}.\n\nAll of these characteristics are likely the product of a single\noblique giant impact onto a differentiated proto-Haumea earlier \nin the history of the solar system \\citep{2007Natur.446..294B}. Prior to impact, the proto-Haumea\nwould have been differentiated into a rocky core, a relatively pure water\nice mantle, and a crust of volatiles that had been driven to the surface\nand their irradiation products.\nThe oblique impact\nthen gave Haumea its fast rotation, causing its\nextreme elongation. In addition, the impact must have blasted off\na significant fraction of the proto-Haumea's icy mantle, which then\nbecame the satellites and the collisional family.\n\nIn this scenario, the nearly pure water ice surface of Haumea (along with\nits high density) occurs because we are viewing the exposed layers of the\ninterior pure icy mantle. The nearly pure water ice surfaces of\nthe dynamical family members and the satellites occur because they are\nfragments of the interior of the mantle. Long term irradiation\nof these pure ice fragments does not cause any of the reddening or\ndarkening that would be expected if any hydrocarbons were present.\n Fragments of the crust, which\nshould have contained more irradiated hydrocarbons, presumably were\ncreated also, but the volume of crust is much smaller than that of\nmantle, and there is currently no method for\npositive identification of non-ice components of\nHaumea, so to date only mantle fragments are known.\n\nEven if the post-impact\nHaumea and its collisional family originally began with\nnearly pure water ice surfaces, the fact that these surfaces\nremain relatively uncontaminated with dust, fragments of dark\nimpactors or irradiated hydrocarbons is a surprise. It has even\nbeen suggested that the collision must have been relatively recent\nfor the surfaces to appear so pure \\citep{2008AJ....136.1502R}. Dynamically \nmodeling of the diffusion of Haumea with the 12:7 mean motion resonance\nwith Neptune suggests, however, that the collision occurred billions of\nyears ago \\citep{2007AJ....134.2160R}. Modeling of collision probabilities suggests\nthat the collision must have occurred near the beginning of the solar \nsystem \\citep{2008AJ....136.1079L}. Pure water ice surfaces in the Kuiper must be\ncapable of staying relative pristine for billions of years. While\nthe reasons for this longevity are not clear, we will use this empirically\ndetermined fact below when discussing water ice on the mid-sized objects.\n\nIn addition to being relatively pristine, the spectrum of water ice\non Haumea and its family members always contains the 1.65 $\\mu$m\nabsorption feature due to crystalline water ice. While this 1.65 $\\mu$m\nfeature appear ubiquitous throughout the outer solar system, its\npresence is perhaps surprising. Irradiation studies suggest that\ncrystalline water ice should turn to amorphous water ice on\nrelatively short time scales unless some heating is applied to\nrecrystallize the water ice. This 1.65 $\\mu$m feature is thus\noften taken as evidence of some type of internal heating, from\ncryovolcanism \\citep{2004Natur.432..731J} \nto radioactivity and tidal forces \\citep{2011A&A...528A.105D}. The fact\nthat the 1.65 $\\mu$m absorption can be seen \neven on Haumea family members as small as 1995 SM55, with\nan estimated size of $\\sim$180 km in diameter (assuming\nan identical albedo to Haumea) \ndemonstrates, however, that no\nsuch internal mechanism is required for the appearance of this feature.\nThese objects are far to small to maintain the liquid water beneath\ntheir surface that would be necessary to support any current\ncryovolcanism  and as fragments of the icy mantle of \nHaumea they are thoroughly lacking in the rocks that would give rise\nto radioactive heating. It seems clear, even if not understood, that \nthe appearance of the 1.65 $\\mu$m absorption feature of crystalline\nwater ice does not contain any information about internal processes,\nbut rather contains new information about the physics of crystallization\nunder these conditions.\n\n\\section{Mid-sized objects}\nWhile the surface compositions of the largest object appear well \nunderstood and even predictable, less attention has been paid to the\nmid-size $\\sim$500 - 1000 km diameter KBOs which are just a little\ntoo small to be able to have retained surface volatiles. In some\nways, these objects form a more interesting class than the \nlarger objects, as these objects do not have a frosty veneer\nhiding the inherent surface composition. Unfortunately, however, while\nhigh quality spectroscopy of the largest and brightest objects\ncan be readily obtained, these mid-sized objects are considerably more\ndifficult to observe even with the largest telescopes in the world. \nMajor observing campaigns with the Keck telescope \\citep{2008AJ....135...55B,\nspecsub} and\nthe VLT \\citep{2009Icar..201..272G,2011Icar..214..297B} have provided much of what we know of\nthe surface compositions of these objects.\n\\subsection{Water ice}\n\\citet{2011Icar..214..297B} and \\citet{specsub} both analyze extensive\ncollections of high quality spectra of KBOs and their progeny.\nIn \\citet{2011Icar..214..297B} the possibility of water ice in VLT spectra\nis first assessed by\nsimply calculating the depth of a potential 2 $\\mu$m absorption feature\nand then followed by detailed modeling including many different components.\n\\citet{specsub} concentrate on a examining the possibility of water\nice in Keck spectra by fitting a simple water ice plus sloped continuum\nmodel to the near infrared spectrum. \nThe VLT and Keck results are in agreement both on the broad\nresults and on most of the individual objects. \n\\begin{figure}\n\\plotone{fig5.eps}\n\\caption{\nThe water ice spectra fraction -- a measure of the amount of\nwater ice absorption in an object spectrum -- as a function of\nabsolute magnitude. Absolute magnitude is a measure of the intrinsic brightness\nof the object, which is related to size. The largest objects -- on the left of the plot --\nhave the smallest absolute magnitudes. A clear trend is seen for the largest\nobjects to have the deepest water ice absorption spectra. The\ntransition around $H\\sim4$ from moderate to deep water ice\nabsorption is a potential indicator of the size at which\ninterior oceans caused surface ice flows in the past.\nThe water ice fraction on smaller objects is indistinguishable\nfrom that on the even smaller centaurs.}\n\\end{figure}\n\n\nIn Figure 5, we show the water ice spectral fraction -- a measure\nof the depth of the water ice absorption in the spectrum -- as \na function of the absolute magnitude (a proxy for diameter) \nof KBOs and also of centaurs (see below),\nfrom the \\citet{specsub} sample.\nA clear trend is evident:\nbrighter than an absolute magnitude of $H=3$, all KBOs have deep\nwater ice absorption (water ice spectral fraction $>0.2$). Fainter than an absolute magnitude of $H=5$, deep water ice absorption is never seen.\n\nWhile such a correlation of absolute magnitude and presence of\nwater ice might be expected simply from the increased albedo\nof objects with more water ice on their surfaces, Spitzer radiometry\nhas shown that the $H>3$ KBOs are indeed smaller than the $H<3$ KBOs\n\\citep{2008ssbn.book..161S}.\nSomewhere between the $\\sim$650 km diameter of Ixion and \nthe $\\sim$900 km diameters\nof Quaoar and Orcus, KBOs appear\nto dramatically change in surface composition.\n\nThe change in surface composition is clearly not monotonic with\nsize. Ixion, 2002 UX25, 2002 AW197, 2004 GV9,\nand  2003 AZ84 have $\\sim$650 km diameters\nwithin their uncertainties \\citep{2008ssbn.book..161S}, but\nonly 2003 AZ84 has a large abundance of water ice on the surface. \nLikewise, the smaller 2005 RM43, which would have a diameter of \n$\\sim$520 km if it has the same albedo as 2003 AZ84, has water\nice absorption as deep as the larger objects Quaoar and 2007 OR10,\nwhile the similarly-sized objects Varuna and Huya have much\nsmaller absorption depths. \n\nSmaller than the $\\sim$520 km size of 2005 RM43, however, no (non-Haumea\nfamily member) KBO has been found with strong water ice absorption.\nFew objects in this range are bright enough for\nhigh quality spectroscopy (1996 TP66, with a diameter of $\\sim$180 km\nfor an assumed 0.1 albedo is the smallest KBO with a well measured\nspectrum), however. Nonetheless, spectroscopic evidence from\ncentaurs and photometric evidence from smaller KBOs (see below)\nsuggests that this trend continues, and that between 500 and 700 km\nin diameter KBOs transition from typical surfaces of small KBOs\nto those dominated by absorption due to water ice.\n\nAdditional evidence for a surface change in this diameter\nrange comes from measurements\nof albedos. \nThough the size and\nalbedo data available from Spitzer photometry have large\nuncertainties, a jump in albedo at this size range \nis apparent.\nThe same effect is see in the\ncumulative absolute magnitude distribution of the large\nobjects. While this distribution follows a power law from \nabsolute magnitudes of 3 until at least 4.5, for objects \nbrighter than $H=3$, the cumulative number distribution is\nlarger than would be expected from this power law, a clear\nsign of the increased albedos of these mid-sized\nobjects \\citep{2008ssbn.book..335B}. Implications for this surface\nchange are discussed below.\n\n\\subsection{Ammonia}\nOf these mid-sized objects with strong water ice absorption\nfeatures, two have evidence for a 2.25 $\\mu$m absorption\nfeature due to ammonia. Ammonia was first detected on Charon\n\\citep{2000Sci...287..107B,2007ApJ...663.1406C,2010Icar..210..930M} and later suggested to be present also on\nOrcus \\citep{2008A&A...479L..13B}. In both cases, the presence of ammonia\nwas postulated to be due to the flow of ammonia-rich interior\nliquid water on to the surface of the object at some point in\nthe past. Detailed models of the interior structure and evolution\nof these two bodies have suggested that such a scenario \nappears reasonable \\citep{2007ApJ...663.1406C, 2010A&A...520A..40D}.\n\nAmmonia has not been detected on the other water ice-rich objects,\nbut for Quaoar a bright object where high quality\nspectra can be obtained, the 2.3 $\\mu$m region of the spectrum\nwhere ammonia has its only detectable near-infrared feature is\ninstead dominated by the strongest absorption feature due to CH$_4$.\nAmmonia has additional absorption features beyond 3 $\\mu$m which\ncould be distinguished from CH$_4$ but spectral measurement\nat these wavelengths awaits larger ground or space-based telescopes.\nThe smaller water-ice-rich objects 2003 AZ84 and 2005 RM43 have spectra with\ninsufficient signal-to-noise to detect ammonia. Larger telescopes will\nagain be required.\n\nBased on the likely presence of ammonia on Charon and Orcus,\nand the sharp increase in water ice absorption with larger\nsize for these objects, \nwe hypothesize that on these largest objects the presence\nof water ice -- whether ammonia is detectable or not -- is a\nremnant of past liquid flows on the surface. We predict that\nfor these objects ammonia will always be detected when sufficient\nsignal-to-noise is available. The liquid flows\nneed not be recent -- the purity of the water ice on the surface of\nHaumea and its family shows that water ice can remain pristine through\nthe age of the solar system -- but the liquid flows must have \noccurred after the volatiles whose irradiation would cause coloration\n(see below) have all escaped. The increase in water ice absorption\nwith size would be a natural consequence of the larger \ninterior liquid reservoirs of larger objects. The sharp increase\nin water ice absorption starting at diameters around $\\sim$650 km\ngives an important clue into the physics of liquid interiors\nand surface water flows.\n\n\\section{Small objects: spectroscopic constraints}\n\\subsection{Kuiper belt objects}\n\nMany objects smaller than the size where strong water ice\nabsorption is present are statistically\nconsistent with having no water ice on their surface, but there is\na significant prevalence in the data for positive detection of water ice.\nRandom and known systematic errors would not produce a bias towards\nwater ice detection. Indeed, we regard the nearly complete lack of \nobjects with negative water ice fraction as an indication of the\nrobustness of our method. We conclude, therefore, that even the low\nlevel of water ice fraction detected in the majority of the\nobjects is a real indication that water ice at a low level\nis common even on the objects smaller than Ixion.\n\n2002 VE95, the smallest Kuiper belt object with a very high quality \nspectrum ($\\sim$330 km diameter assuming a 0.1 albedo), shows\nclear evidence of crystalline water ice on its surface\n\\citep{2008AJ....135...55B, 2006A&A...455..725B}.\nIndeed, the 1.65 $\\mu$m\nabsorption feature of crystalline water ice can always be detected when\nthe signal-to-noise is sufficiently high. \n\nBased on their smaller sizes, it\nseems likely that the water ice on these smaller objects is\nnot caused by liquid flows on the exterior, but rather by the\nexposure of crustal water ice. An important test of this expectation\nwould be that these smaller KBOs should not show the presence\nof ammonia on their surfaces. Currently feasible spectroscopy\ncannot achieve the signal-to-noise required to make this test, however.\n\n\n\\subsection{Centaurs}\nKBOs smaller than those discussed above are too faint for high quality\nspectroscopy even with the largest telescopes in the world. To understand\nthe compositions of these objects, we have to resort to\nobservational proxies. One important\nproxy has been spectroscopic observations of Centaurs. Centaurs are\nformer KBOs which have been perturbed onto short-lived planet-crossing\norbits. Being much closer to the sun than typical KBOs, they are \nbrighter and easier to study in detail.\n\nSpectroscopically, the Centaurs appear indistinguishable\nfrom the smallest KBOs whose spectra can be measured: objects contain\neither no detectable absorption features, a small amount of absorption \ndue to water ice, or (in one case) absorption due to water ice and \nmethanol. Figure 5 includes water ice absorption depth for a large\nsample of Centaurs, compared to KBOs. No discernible difference\ncan be seen between the largest of the centaurs and the smallest of\nthe measured KBOs. \nThe amount of water ice absorption seen on the surface of a centaur\ndoes not appear to correlate with anything, including\nperihelion, semimajor axis, optical color, dynamical lifetime, or \nactivity. Interestingly, while much speculation has occurred about \nCentaur surface evolution as objects move closer to the sun and \nbegin heating, infrared spectroscopy shows no such evidence of any\nchange in the distribution of water ice absorption depth.\n\n\\subsection{Methanol}\nAn absorption band at $\\sim$2.27 $\\mu$m\nwas first detected on the bright\ncentaur Pholus \\citep{1993Icar..102..166D}, and \\citet{1998Icar..135..389C} present \nthe case that this band is plausibly due to the presence of\nmethanol, though they point out that the identification is\nnot unique and that other low molecular weight hydrocarbons\nor photolytic products of methanol might fit the spectrum equally well.\nPholus is one of the reddest objects in the solar system, again\nfitting our picture of optical colors of irradiated hydrocarbons well.\n\nBased on lower signal-to-noise spectra with properties\nwhich resemble the water-ice-plus-methanol spectrum of Pholus, the presence of\nmethanol was suggested on the KBOs 1996 GQ21 and 2002 VE95\n\\citep{2008ssbn.book..143B}. \nTo examine this possibility more closely, we combine the Keck spectra\nof these two objects to increase signal-to-noise and consider\nthe presence of methanol. While the signal-to-noise\nremains low, the presence of absorption features similar\nto those on Pholus is certainly plausible.\nBoth objects, like Pholus, are red.\n\nA handful of other KBOs have recently been reported to also have\nabsorption features near 2.27 $\\mu$m, but, unlike\nPholus, 1996 GQ21 and 2002 VE95, to not have absorption\ndue to water ice \\citep{2011Icar..214..297B}. The signal-to-noise in\nthe spectral region are low,\nso it is difficult to determine\nif the absorption features are real. \nTo examine the possibility that\na $\\sim$2.27 $\\mu$m absorption feature occurs on faint KBOs,\nwe first examine all of the KBOs and centaurs in the Keck sample\n(which includes none of the potential methanol objects from\nthe VLT sample).\nWe find that a small number of objects have absorption features\nat or near the 2.27 $\\mu$m methanol absorption line, but\nno single spectrum is sufficiently reliable in this region to\nassert a positive detection. To increase the signal-to-noise,\nwe sum the spectra of all of the KBOs and centaurs in the Keck\nsample\nexcept for those with water ice absorption at the level of\nthat seen on 2003 AZ84 or deeper and those already suspected\nto contain methanol-like features (1996 GQ21 and 2002 VE95). This combined spectrum shows\nresidual absorptions due to water, as would be inferred from\nthe positive detections on most objects. The only major deviation\nfrom the water ice spectrum occurs at precisely the wavelength\nof the feature seen on Pholus and suspected on 1996 GQ21 and 2002 VE95.\nWe conclude that\nthe methanol-like feature is indeed present at a low-level on\nKBOs even though the feature cannot be reliably identified in\nindividual spectra\n(Figure 6).\n\\begin{figure}\n\\includegraphics[scale=.75]{fig6.eps}\n\\caption{The region of the near-infrared spectrum containing \nthe 2.27 $\\mu$m absorption feature attributed to methanol.\nThe feature can clearly be seen on Pholus, superimposed on a \nsmall amount of water ice absorption. The combined\n(to increase signal-to-noise) spectrum of 1996 GQ21 and 2002 VE95\nshows hints of a feature at the same location. The sum of 38\nKBO and centaur spectra, none of which individually show clear evidence\nfor the feature, clearly shows a feature at the same spot.\n}\n\\end{figure}\n\n\nNo hypothesis has ever been formulated for the \nsporadic presence of methanol on KBOs or centaurs, other\nthan to point out that methanol is common in cometary comae,\nso it is expected to be present in the interior of KBOs.\nIts presence is less expected on the surface of KBOs, however,\nas the absorption features of hydrocarbon ices quickly degrade\nunder irradiation while the remnants turn red \\citep{2006ApJ...644..646B}.\nVisible methanol absorption features suggest that the\nmethanol has only recently been exposed at the surface, perhaps\nas a result of a collision exposing the subsurface. \nOne prediction of this suggestion would be that the amount of\nmethanol observed would vary as different faces of the object were\nobserved. While such a test is possible in principle, in practice\nspectroscopy of these faint objects \nis sufficiently difficult that variation would be difficult to\nprove. If, however, irradiation preferentially turned the regions\nwith exposed methanol red, these objects should perhaps show color\nvariation with rotation, something which is otherwise rarely \nobserved in objects of this size \\citep{2008ssbn.book..129S}.\n\n\\subsection{Silicates}\nIn addition to ices and their irradiation products, we should\nexpect that KBOs that are small enough to be\nundifferentiated should expose some of their\nrock component on the surface. While silicates such as\nolivine or pyroxene\nare often included in detailed models of KBO spectra \n\\citep[i.e.][]{2010Icar..208..945M,2011Icar..214..297B}\n no\nspecific absorptions are easily observable: olivine, for example, has a \nbroad absorption centered at around 1 $\\mu$m, where observations\nare usually poor. These silicates are thus included in models\nto fit the overall spectral shapes over the $\\sim$0.6 - 1.2 $\\mu$m\nrange.\n\nMid-infrared observations have the possibility of positively\nidentifying silicate emission in the 10 and 20 $\\mu$m region \non KBOs. To date only the centaur Asbolus has had a \nspectrum measured with sufficient signal-to-noise to even\ndetect these spectral regions, and, in this case, emissivity\npeaks around 10 and 20 $\\mu$m are indeed seen, similar to\nthat observed in Trojan asteroids \\citep{2006Icar..182..496E,2008ssbn.book..143B},\nand interpreted to be due to fine-grained silicates.\nFuture observations will require improved space-based \nmid-infrared facilities, but characterization of \nactual silicate composition is indeed possible.\n\nWhile olivines or pyroxenes cannot be specifically identified\nin the visible to near-infrared range,\nthere have been a few reports of shallow broad absorption\nfeatures in the visible wavelength range\nsimilar to absorptions seen on some asteroids.\nOn asteroids these are generally attributed to aqueously \naltered silicates.\nConfirmation of these features has been difficult; the absorptions\nare subtle and have frequently appeared changed or absent upon reobservation\nof the same object \\citep{2004A&A...421..353F,2004A&A...416..791D, 2009A&A...508..457F, 2008A&A...487..741A, 2003AJ....125.1554L}\nWhile these changes are usually attributed to rotational\nvariability, it is worth noting that this speculation has never\nbeen verified. Indeed, for one object observations over half \nof a rotational period showed no signs of the visible absorption \nfeatures.\n\nWhile it is possible that these difficult-to-confirm features \nare a product of sporadic systematic error, the possibility\nof the existence of aqueously altered silicates is \nan interesting one to consider. The presence of\nliquid water on surfaces in the Kuiper belt might seem surprising, but\nhydrous materials seem to be present in small comets, interplanetary\ndust particles, and debris disks \\citep{2004A&A...416..791D}. \nIf these detections are indeed real, the most surprising thing\nabout them, perhaps, is that they are uncommon. \n\n\n\n\\section{Small objects: photometric constraints}\nSmall objects in the Kuiper belt are too faint to detect spectroscopically,\nbut photometric measurements can still give information -- albeit limited --\nabout the surfaces of these objects. To date, the single most robust\nconclusion based on photometry is that Kuiper belt surfaces are\ndiverse.\nLarge surveys of KBO optical\ncolors have found a wider range of\nsurface colors\nin the Kuiper belt than in any other small body population in the \nsolar system. The colors are uncorrelated with\nmost dynamical or physical properties \\citep[see review in][]{2008ssbn.book...91D}.\nA few systematic patterns\nhave been found, however, which are important clues to understanding the\nsurface compositions of these outer solar system objects.\n\n\\subsection{The bifurcated colors of centaurs} \nThe range of centaur optical colors generally covers the same wide\n range of colors\nfound in the Kuiper belt, but the centaurs are deficient in colors in\nthe middle part of the range, giving the distribution of centaur optical\ncolors\na bimodal appearance \\citep{2008ssbn.book..105T} with a neutral\nand a red clump of objects. Interestingly, this bifurcation is not\nseen only in the centaurs, but it appears to extend to all objects with\nlow perihelion distance whether the objects are dynamically unstable or\nnot. This result immediately suggests that the bifurcation in the \noptical colors is somehow formed through the increased heating or\nirradiation experienced by lower perihelion objects.\n\nThe H\/WTSOSS (Hubble\/WFC3 Test of Surfaces in the Outer Solar System) survey which used the Hubble Space Telescope to extend\n optical photometry of centaurs (and other low\nperihelion objects) into the near-infrared\n\\citep{hwtsoss} found that the neutral and red clumps do not consist\nof two groups with identical surfaces, \nbut rather are best described by two groups that fall along two\nseparate \nmixing lines.\nThe neutral clump of objects consists of a mixture of\na nearly neutrally reflecting material and a slightly red material,\nwhile the red clump of objects consists of a mixture of the same\nneutrally reflecting material and a much red material\n(Fig 7). \n\\begin{figure}\n\\includegraphics[scale=.65]{fig7.eps}\n\\caption{\nThree-color HST photometry\nof objects with perihelia inside of 30 AU from\nthe H\/WTSOSS survey. The filters are at 0.606, 0.814 1.39, and\n1.54 $\\mu$m, corresponding roughly to $R$, $I$, $H$-continuum, \nand a filter sampling the 1.6 $\\mu$m water ice absorption \nfeature. \nThe yellow point shows the\ncolors of the sun. \nThe objects appear to bifurcate into two clumps, one with near-solar optical colors\n(the ''neutral'' clump) and one with significantly redder optical colors (the ''red clump'').\nThe neutral clump (on the left in [F606W]-[F814W]\ncolors) shows a clear spread in the other colors, while the\nred clump shows a spread in all three colors. The color variation in\nboth clumps can be described by a mixing line (purple lines) where\nthe neutral and red clumps both have a single \ncommon end member but are mixed with either a more neutral or more \nred component, respectively.}\n\n\n\\end{figure}\n\n\nWhile three color photometry is generally incapable of identifying specific \nices or minerals, \\citet{hwtsoss} find that the neutral component \ncommon to all centaurs is consistent with some of the same\nhydrated silicates suggested from the optical spectroscopy\ndiscussed above. Significantly more spectral work is required,\nhowever, to further explore this possibility.\n\n \\subsection{The uniform colors of the cold classical KBOS.}\nWhile most of the rest of the \nKuiper belt appears to be composed of essentially the\nsame distribution\n of neutral to red objects \\citep{morbandbrown,2008ssbn.book...91D}\none dynamical region stands out for its\nhomogeneous composition. The cold classical Kuiper belt was first identified\nas a dynamically unique region of the Kuiper belt -- a difficult-to-explain \noverabundance of low inclination, dynamically cold objects beyond about\n41 AU \\citep{2001AJ....121.2804B}. \nSubsequent observations revealed that these objects\nshared a common red coloring \\citep{2002ApJ...566L.125T}. \nIn the H\/WTSOSS survey these objects do not fall along mixing\nlines, as the neutral and red clumps of centaurs do, but instead appear\nnearly uniform in color space \\citep{hwtsoss}.\nThese cold classical KBOs are\nalso know to be unique in their lack of large bodies \n\\citep{2001AJ....121.1730L}, their\nhigher abundance of satellites \\citep{2008Icar..194..758N}, \nand their different size distribution \\citep{2010Icar..210..944F}.\nAnd though our understanding of albedos in the Kuiper belt is\nstill poor, preliminary results suggest that the cold classical KBOs also appear to have higher albedos than\nthose of the remaining population\n\\citep{2009Icar..201..284B}. \nAll of these properties appear to signify a population with a \ndifferent -- and perhaps unique -- formation location or history. \nUnderstanding the surface compositions of these objects will be \na challenge given their distance and small sizes. In addition, these\nobjects are dynamically stable, so we likely never get samples\nof this population as centaurs or comets.\n\n\n\\subsection{The diverse colors of the remainder of the Kuiper belt}\nOther than the unique color properties of the centaurs and cold classical\nKBOs, the bulk of KBOs have no discernible pattern to their colors.\nThis finding in itself is significant for understanding the causes\nof the colors of KBOs.\nThis lack of\na connection between color and any dynamical property, \nparticularly with semi-major axis or perihelion\/aphelion \ndistance argues strongly that local heating, UV irradiation, \nand solar wind and cosmic ray\nbombardment \\citep{2003EM&P...92..261C} cannot be responsible for\nthe varying colors of the Kuiper belt. Local conditions appear to\nhave no primary influence on the colors of KBOs. \nFurthermore, careful measurement of the colors of the separate \ncomponents of binary KBOs has shown a tight correlation over the full\nrange of Kuiper belt colors \\citep{2009Icar..200..292B}. \nThe colors of two KBOs in orbit around\neach other are almost always nearly identical.\nThis fact immediately rules out any of the stochastic process such as\ncollisions for the causes of these Kuiper belt colors. Indeed, \ngiven the lack of correlation of color with local conditions, the\nnearly identical colors of binary KBOs argues that colors are simply primordial.\nIf binary KBOs were formed by early mutual capture in a \nquiescent disk \\citep{2002Natur.420..643G}, \nthe two component would likely have formed in very similar locations. If,\nalternatively, binary KBOs were formed in an initial gravitational\ncollapse \\citep{2010AJ....140..785N}, the objects would of necessity have formed at the \nsame location and of the same materials. \n\n\\subsection{The transition from KBOs to centaurs}\nThe manner in which primordial KBO surfaces evolve to become\nthe color-bifurcated centaur population could provide important \nclues to the compositions of both surfaces. While it appears\nthat a transition from a unimodal to bimodal color distribution\nmust occur as objects move to lower perihelia, the actual\nevidence for change is, in fact, weak.\nThe KBOs and centaurs with measured colors\nhave very different ranges of sizes, \nand, as demonstrated above, large KBOs have their\nsurfaces modified by the presence of volatiles and water ice. We\nthus must only compare like-sized objects. In addition, the cold\nclassical KBOs, with their unique surfaces, likely never enter the\ncentaur population, so these should be excluded from the \ncomparison. Finally,\nthe KBOs, being more distant, are likely to have higher uncertainty in\ntheir color measurements. \n\n\\begin{figure}\n\\plotone{fig8.eps}\n\\caption{\nA comparison of the colors of (non-cold classical)\nKBOs (thin line) and centaurs (thick line) in the $6<H<9$ range.\nA K-S test cannot distinguish any significant difference between\nthe distributions. While it is commonly assumed that the surfaces\nof KBOs evolve to have the color distribution of the centaur\npopulation, we find no statistical evidence that such evidence\noccurs. A similar conclusion can be drawn from the spectra\nof Figure 5 and from the three-color data of the H\/WTSOSS survey.}\n\\end{figure}\n\nWhen the centaurs are compared to the appropriate KBOs, little evidence\nof surface evolution can be found. \nIn Figure 8 we compare well-measured optical colors of centaurs with absolute\nmagnitudes between 6 and 9\n to those of non-cold classical KBOs with the same range\nof absolute magnitudes. While colors of the Kuiper belt as a whole \nappear significantly different from those of the centaurs, the \ndifference is much less apparent when we compare the correct samples.\nIn fact,\na Kolmogorov-Smirnov test of the two distributions cannot distinguish the two at\ngreater than a 48\\% confidence limit. \nUsing all of the available color data, we find\nno statistical evidence that any color evolution has occurred\nas objects move inward from the Kuiper belt to become centaurs.\n\\citet{hwtsoss} come to the same conclusion from the optical and near-infrared\nphotometry of the H\/WTSOSS survey. In addition, the albedos of KBOs measured\nwith Spitzer appear to be similarly distributed to those of the\nsmaller KBOs \\citep{2008ssbn.book..161S}.\n\nWhile there is no evidence in the spectroscopic or photometric data\nfor surface evolution in centaurs, in at least one way centaurs and\nKBOs are clearly different. A small number of centaurs is known to\nshow cometary activity, at least sporadically \\citep{2009AJ....137.4296J}. \nTo date,\nsuch activity has appeared confined to objects in the neutral \nclump, though the numbers remain small and the color measurements themselves\ncould be compromised by the presence of a coma \\citep{2009AJ....137.4296J}. \nIn addition, all but 2 of the 13 of the known active centaurs have perihelia\ninside of 10 AU.\nIt has been speculated that outgassing could lead to surface \nmodification and even to the bifurcated optical colors of the centaurs\n\\citep{2008ssbn.book..105T}.\nInterestingly, an examination of the 16 centaurs observed in the\nH\/WTSOSS survey\nshows no evidence for activity in any of them. Only\n4 of the 16 have perihelia inside of 10 AU, though. Nonetheless,\nthe colors bifurcation seen in this sample of currently inactive\ncentaurs suggests that color bifurcation is not directly\ncaused by centaur activity.\n\nRobustly determining whether or not centaur surfaces are evolved\nis difficult. The numbers of centaurs available is small, the\ncomparison KBOs are faint, and the expected distributions are \nunknown. It is possible that a complete understanding of this\nquestion will require large scale surveys like that of \nthe currently-proposed\nLarge Scale Synoptic Telescope (LSST)\nto find and characterize\nmany more objects and acquire sufficient statistics to understand \nthese populations.\n\n\n\\subsection{The causes of colors}\nTo date, only three hypotheses have been advanced to explain the colors\nof KBOs. \nThe earliest hypotheses suggested\nrandomized collisional excavation\n \\citep{1996AJ....112.2310L} or velocity dependent impact resurfacing \n\\citep{2002AJ....124.2297S}, but the correlated colors\nof KBO binaries \\citep{2009Icar..200..292B} effectively rules out these or any other stochastic\nprocesses.  \\citet{2003EM&P...92..261C} have suggested that the uniform red\ncolors of the cold classical KBOs are caused by their existence\nin an irradiation minimum environment, with irradiation increasing interior\ndue to increasing fluxes from solar energetic ions and increasing\nexterior by increased flux of energetic ions from the termination shock,\nbut this hypothesis has no general explanation for the remaining colors,\nnor would it lead to the sizable observed population\nneutrally colored higher\ninclination objects in the same environment. \n\nA new hypothesis has suggested that the colors of KBOs are set by\nearly evaporation and irradiation of volatiles followed by dynamical\nmixing \\citep{2011ApJ...739L..60B}.\nIn this hypothesis, objects are formed outside of Neptune\nin the $\\sim$15-30 AU region in the early solar system out\nof a diverse and variable mix of materials. When the nebula disperses\nand objects are first exposed to sunlight, their surfaces heat, and\nvolatiles are driven off. For widely variable starting compositions,\nthe surface compositions of objects at the same distance from the sun\nwill become nearly identical. Figure 9 shows a model from\n\\citet{2011ApJ...739L..60B} which shows the distance at which different volatiles would be\ncompletely removed from the surface of an object as a function of size\nof the object. Regardless of the diversity of initial compositions, \nthe surfaces of the objects will quickly have strong gradients in \nvolatile composition. These volatiles are then irradiated and \nbegin to develop the colors expected from their particular mix of\nremaining surface volatiles.\n\\begin{figure}\n\\plotone{fig9.eps}\n\\caption{\nA hypothesis for the surface colors of small KBOs. In the early solar\nsystem KBOs form with a variety of composition, but as the nebula\ndisappears and the sun begins to heat their surfaces, strong\ngradients in their surface compositions form. Leftward of each\nlabeled line, each species would be fully depleted from the surface.\nKBOs formed interior to about 20 AU would only have H$_2$O and CO$_2$\non their surfaces, \nwhile KBOs which formed outside of $\\sim$20 AU\nwould retain methanol on their surfaces. Irradiation of the methanol-free\nsurfaces could lead to the dark neutral objects, while irradiation of\nmethanol-containing surfaces could lead to the brighter red objects.\nOutside of $\\sim$30 AU objects can also retain NH$_3$, which could\nperhaps explain some of the unique surface characteristics of the\ncold classical KBOs if they formed {\\it in situ}.\n}\n\\end{figure}\n\n\nIn this model, methanol is the most important \ncoloring agent. Methanol is depleted on all surfaces inside of\nabout 20 AU, and present on all of those exterior.\nWhile hydrocarbons are generally expected to carbonize and \nturn dark and neutral upon prolonged irradiation, \nlaboratory experiments have suggested that irradiation of\nmethanol to dosages expected on KBOs over the age of\nthe solar system leads to bright red surfaces.\nIn this hypothesis, then,  the objects interior to the methanol line\nbecome the neutral population, while those outside become the red\npopulation. A solar system scale instability such as that envisioned by\nthe Nice model then scatters the objects onto the orbits where they\ncurrently reside, mixing the neutral and red populations. The cold classical\nKBOs form in place \\citep{2012ApJ...744L...3B}, and their unique colors form by the\nadditional presence of ammonia on their surfaces. \nAs KBOs scatter inward and become centaurs, their surfaces remain unchanged.\n\nWhile the model of \\citet{2011ApJ...739L..60B} is the only quantitative model to attempt to\nexplain all of the current observations of colors of KBOs, it is\nclearly speculative. Much of the laboratory data required to trace\nthe specific irradiation chemistry has not been performed, and the question\nof the evolution of the surfaces of the centaurs remains open. \nExtending this model to the Hubble Space Telescope survey of \\citet{hwtsoss}, which suggests mixing\non the surfaces of KBOs, we would conclude that the irradiated volatiles\nare the neutral and red materials that are mixed in with the hydrated\nsilicate-like material. No obvious explanation exists for the\ndifferent amounts of mixing between the irradiated volatiles\nand the silicates.\n\n\\section{Bulk composition}\nWhile most work on the composition of the Kuiper belt has focused on\nthe surface composition, all of the spectral and photometric results \ndiscussed probe an insignificantly small depth into the surface. To\nunderstand the true composition of KBOs requires an understanding\nof the bulk composition. Detailed measurement of the bulk\ncomposition is of course impossible, but one important proxy --\nthe ice-to-rock fraction -- is available for some KBOs. Measurement\nof the ice-to-rock ratio requires measurement of the density which, in\nturn, requires measurement of the mass and radius of the objects. \nWhile measurement of the size of an object is possible through multiple \nmeans (to date far infrared radiometry from the Spitzer Space Telescope\nhas been the dominant method), a measurement of the mass is only possible\nif the object has a satellite whose orbit is known. \n\n\nKBOs\nwere expected to be a relatively homogeneous group. \nThey all are thought to have\ngrown gradually through accretion, \nsampling similar regions of the solar nebula, so their compositions\nshould have been nearly identical. Indeed, when Pluto\nwas the only known large Kuiper belt object, its density of \n$\\sim$2 g cm$^{-3}$ was taken to indicate a $\\sim$30-70 ice-rock \nmix in the outer solar nebula as a whole \\citep{1988Natur.335..240M}.\n\n\\begin{figure}\n\\plotone{fig10.eps}\n\\caption{Measured densities of KBOs. In all cases the uncertainty\nin the size dominates the density measurement, so error bars follow\nlines of constant mass with varying diameter. The green lines show\nthe density as a function of size for different rock fractions for\nfully differentiated body; the ice phase change at higher\npressures causes ice to get denser. The red line shows the maximum\ndensity increase that could be expected from a simple model where\nsmall amounts of ice are preferentially removed in the collisions\nrequired to build larger bodies from smaller ones. \nData for sizes and densities come from \\citet{2006ApJ...639.1238R,\n2007Sci...316.1585B, \n2009DPS....41.6204M,\n2010AJ....139.2700B, \n2010ApJ...714.1547F,2010Icar..207..978B}.\n}\n\\end{figure}\nOne of the biggest outer solar system surprises of the past few years,\ntherefore, has been the discovery that the ice fraction\nmeasured in KBOs varies from essentially 0 to 1 (Figure 10). Objects\nhave been found with densities significantly less than 1 g cm$^{-3}$\n\\citep{2009DPS....41.6204M,2010Icar..207..978B}\nindicating both a near-unity ice fraction and significant porosity,\nwhile other objects have been found with densities of nearly\npure rock \\citep{2010ApJ...714.1547F}. \n\nExamination of the measured densities \ndiameter reveals \nthat the density measurements have extremely large\nuncertainties. The large error bars are a function of the \nuncertainty in the measured sizes and the 3-times-higher uncertainties\nin the associated volumes. Detailed understanding of the trends and\ncauses of ice-rock fractions in the Kuiper belt clearly requires \nsignificantly higher quality size measurements.\n\nEven with these large uncertainties, however, two trends are apparent.\nFirst, there is a general trend for an increase in density as a function \nof size. While an object with a fixed ice fraction will \nundergo a density increase with increased size owing to\nthe  small density increase that occurs due to the change \nof ice to higher density phase as pressure is increased, \nthe general trend of increased density with size seen in the Kuiper\nbelt is significantly larger than expected unless\nthe rock fraction itself is increasing in larger objects (Fig. 10). \n\nThe second general trend is the difference in densities between objects\nwhich exclusively have small satellites and those which have larger satellites.\nThe objects which exclusively have\nsmall satellites -- Haumea, Quaoar, and Eris -- have been hypothesized\nto have undergone giant impacts which led to these satellite \\citep{2006ApJ...639L..43B}.\nInterestingly, these objects have higher densities than every other\nmeasured object.\n\nSuch a wide range of ice-rock ratios is astounding.\nNo dynamical evidence exists that the large KBOs came from dramatically\ndifferent regions of the solar nebula, and\nno reasonable formation mechanism appears capable of delivering\nsuch variability among large accreted objects. \nWe suggest three scenarios for explaining the extreme variability of\nthe bulk composition of KBOs below, but we do not find any \nof the explanations completely satisfying.\n\n\\subsection{Single collision densification}\nThe extreme densities of all of the KBOs which exclusively have\nsmall satellites immediately suggests a connection between the\nsatellite-forming giant impact and the high density. Indeed, in\nthe case of Haumea, the creation of an icy collisional family\ndynamically surrounding the object demonstrates that these\nimpacts do indeed remove ice from a differentiated mantle. \nIt appears reasonable to suggest that the three known objects\nwith the highest densities all achieved these densities \nthrough single catastrophic collisions which removed \nsubstantial amounts of overlying ice in a process reminiscent of\nthat envisioned for Mercury.\n\nUnfortunately, the types of collisions that would be required\nare infeasible in the solar system. Using scalings from numerical\nsimulations of \\citet{2009ApJ...691L.133S}, we find that to \nremove enough ice mass to create \nan object with the $\\sim$2.6 g cm$^{-3}$ of Eris from an object that initially had\nthe $\\sim$2.0 g cm$^{-3}$ of Pluto requires hitting Triton with\na $\\sim$ 500 km KBO traveling 40 km s$^{-1}$! \n\nThough the connection between high densities and small satellites\nis striking, we can envision no physical mechanism by which the\ntwo can be related.\n\n\\subsection{Water ice loss during accretion}\nWhile single impacts cannot lead to significant ice loss, \nmultiple impacts during accretion can.\nThe red line in Figure 10 shows, for example,\na scenario where large bodies are preferentially grown by\naccreting other like-sized bodies. For two bodies which collide\nat their mutual escape velocities, as such bodies would, \nthe \\citet{2009ApJ...691L.133S} scalings suggest that $\\sim$10\\%\nof the total mass will be removed from the system. If for \ndifferentiated objects the ice is exclusively removed, these objects\nwill grow in density as they grow in size. \n\nThe model shown here\nthen represent the maximum density \ngrowth that could happen from an assumed starting\nsize and starting density (300 km, 1.3 g cm$^{-3}$)\nfor differentiated objects which grow to \nbecome the larger dwarf planets. While a general increase such as that\nseen through Charon, Pluto, and Triton can perhaps be explained,\nthe three very high density objects are well away from this line.\nTo create Quaoar, for example, would require starting with 300 km \ndifferentiated objects\nwith initial densities of 1.9 g cm$^{-3}$, which would then be\ninconsistent with the other observed densities.\n\nWhile accretional loss of water ice may play some role in the \ngeneral trend of increasing densities with size, that process\nclearly cannot explain all of the observations.\n\n\\subsection{Extreme inhomogeneity in the disk}\nOne of the only alternatives to invoking collisions as a means to \nmodify densities is to suggest that the densities are primordial.\nThe general trend of densities increasing with size could be caused\nby an increase in the ice-to-rock ratio as a function of heliocentric\ndistance, combined by a tendency of the largest objects to form\ncloser to the sun where formation timescales are shorter and\nlarger objects can grow. However, this scenario would also \nsuggest the existence of a population of smaller objects\nwhich also formed at these closer distances and should therefore\nalso have the high densities of the largest objects. No such\npopulation appears. \n\nThe alternative is that the ice-to-rock ratio did not vary\nsmoothly with distance, but rather the disk was chemically \ninhomogeneous. It is difficult to reconcile this possibility\nwith the assumption that the large objects were accreted out\nof large numbers of small ones sampling multiple portions of the disk.\n\n\n\\bigskip\nNone of the three proposed scenarios provides a satisfactory \nexplanation for the trends and variability seen in densities.\nThe study of the bulk composition of KBOs is, however,\n in its infancy. Radii of KBOs are still poorly determined, and\nthe number of objects with measured satellite orbits remains small.\nWe anticipate that continued work in this areas will eventually \nyield the insights that will allow the bulk compositions\nof these objects to be understood.\n\n\n\\section{Implications, speculations, and open questions}\nWe have attempted to discern general trends and formulate\noverlying principles for understanding the compositions of the\nobjects in the \nKuiper belt. We are at a stage in our understanding of the \nKuiper belt that, for the first time, we have at least a first-order\nunderstanding of the compositions of the surfaces\nof KBOs and valid working hypotheses for understanding these compositions.\nA comparable understanding of the bulk composition is -- however --\nfar away.\n\nUsing all of the spectroscopic and photometric data as well as \nthe current best understanding of the physics and chemistry of these\nbodies, \nwe conceptually divide Kuiper belt surfaces into major groups, \nwith differences being caused by size, formation location, and history.\nThe six broad types of KBO surfaces are as follow:\n\\begin{itemize}\n\\item{Volatile rich (Triton, Eris, Pluto): objects large enough to \nretain significant reservoirs of N$_2$, CH$_4$, and CO. Their spectra\nare dominated by CH$_4$ absorption, but their surfaces \nare likely dominated by N$_2$ with CH$_4$ a moderate contributor.}\n\\item{Volatile transition (Makemake, Quaoar, 2007 OR10): objects which \nare on the verge of losing their volatiles but still contain (at least)\nCH$_4$. With CH$_4$ the dominant molecule on the surface, radiation\nprocessing occurs much more quickly, so ethane and other methane irradiation\nproducts are present on the surface.}\n\\item{Water ice plus ammonia rich surfaces (Charon, Orcus, Quaoar, \n2007 OR10, AZ84, other): Slightly smaller ($\\sim 500$ km $< D < \\sim$1200 km)\nobjects on which volatiles have been mostly\nor fully depleted and the spectra are dominated by significant crystalline\nwater ice absorption. Where high S\/N is available and methane doesn't\nhide the spectral signature, we expect that all of these will show\nthe presence of ammonia. We suggest that these are due to water\nice flowing on the surface after volatile irradiation has otherwise\nset the surface color and composition.}\n\\item{Neutral surfaces of small objects:} Objects which are too small to\nretain volatiles or to have had water flows and formed inside $\\sim$20 AU\nwill have their surfaces depleted in all major ices except H$_2$O and CO$_2$.\nIrradiation will then cause dark neutrally colored surfaces to develop.\nThese objects show a range of optical-near IR colors based on the\namount of silicate mixed with the irradiated ices. No hypothesis has\nbeen suggested for why objects differ in their surface rock-ice ratio.\n\\item{Red surfaces of small objects:} Small objects which formed outside\nof $\\sim$20 AU would have been able to retain CH$_3$OH on their surface. Upon\nirradiation these surfaces would turn red and retain moderate albedos.\nThese objects show a range of optical and optical-near IR colors based\non the amount of silicate mixed with the irradiated ices. \n\\item{Pure water ice (Haumea and its family and satellites): \npure water ice surfaces\nform when the nearly-pristine water ice mantle of a differentiated KBO\nis exposed in an impact.}\n\\end{itemize}\n\n\nWhile these six classes of Kuiper belt surfaces provide an overall framework\nfor understanding the the composition of KBO surfaces, there are \nopen questions about multiple aspects of this framework that must\nbe answered before we can be certain that our first-order understanding\nof these surfaces is correct. In addition, some overall questions still\nremain. We summarize some of the most important questions to be\nanswered by future research:\n\n\\begin{itemize}\n\\item{Is water ice on medium-sized KBOs a sign of previous water flow on\nthe surface? Evidence that this hypothesis is true would include\nthe presence of ammonia on every medium-sized objects on which substantial\nwater ice absorption is present. Images of Charon from the New Horizon\nspacecraft may also shed light on this question.}\n\\item{Is the sporadic methanol seen on small objects a transient sign\nof a recent impact? Direct evidence for this hypothesis is difficult\nwithout resolved imaging, for which there are no future plans. However,\nrotationally revolved spectroscopy could help begin\nto answer this question. Such spectroscopy will require the next\ngeneration of large telescopes, but rotationally resolved color measurement\nmay provide a currently feasible initial step.}\n\\item{Do KBO surfaces become modified as they enter the centaur region? \nExtremely high precision color measurements of large numbers of \nhigher perihelion non-cold classical KBOs\ncould answer this question. Such precise measurements are possible\nwith current telescopes. Understanding the conditions that lead to\ncentaur activity and that effect of activity on the surface -- if any --\nwill also be important.}\n\\item{Can the volatile loss hypothesis for the colors of KBOs be\nsupported by new laboratory data? The hypothesis suggests specific \nchemical combinations that give the colors seen in the Kuiper belt;\nirradiation of these combinations would show if the predictions\nare valid.}\n\\item{Are silicates visible at the surface? Reliably reproducible \nresults on the presence of aqueously altered silicates would\nbe strong evidence. Midinfrared spectroscopy is difficult but\nshould eventually prove feasible. The silicate-colored\nmaterial identified in the mixtures from the purely\nphotometric H\/WTSOSS surface could perhaps be isolated \nspectroscopically with carefully targeted measurements.}\n\n\\item{What causes the extreme bulk\ncompositional variability of the Kuiper belt? Three hypotheses have\nbeen suggested, but none of them is satisfactory. Significant more\nwork is needed in this area.}\n\\end{itemize}\n\nThe last decade has seen studies of the composition\nof the Kuiper belt blossom from simple cataloging to \nthe development of actual conceptual frameworks which\ncan explain many of the observed properties. Much work remains, however,\nto both verify and extend this framework, and to use these\ndistant bodies to continue to provide new insights into the formation\nand evolution of the solar system.\n\nThe preparation of this review has been supported by grant\nNNX09AB49G from the NASA Planetary Astronomy program. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe interference of single photons formed the basis of many applications in quantum optics, from Young's double slit experiment to ascertain the wave-particle duality \\cite{Young:04}, to Michelson-type interferometers as a cosmic probe \\cite{Michelson:87}, to Bell's inequality experiments via two-photon correlations \\cite{Shih:88}.\nMore recently, the interference of photons has proved to be an indispensable tool in quantum communications and quantum information processing, and notably in the pursuit of scalable quantum computing via quantum networked architectures of small processors (nodes) interacting via quantum channels\\cite{Kimble:08,Reiserer:15}. \nEntanglement across the nodes is generated via deterministic direct entanglement transfer \\cite{Cirac:97,Ritter:12} (where a mediating photon's entanglement with its emitter is transferred to an entanglement between the emitter and the recipient) or via measurement based entanglement \\cite{Cabrillo:99,Feng:03,Duan:03, Simon:03,Campbell:08}.\nWith the former, successful entanglement is detected \\it{a posteriori}\\rm{} (--accompanied with its collapse). The latter, albeit an inherently probabilist protocol, is favoured for scaling as successful entanglement is heralded upon detection of readout photons \\cite{Reiserer:15}.\nIt has led to successful elementary implementations with ions \\cite{Moehring:07}, neutral atoms \\cite{Hofmann:12}, defect centres in diamonds \\cite{Hensen:15} and micromechanical oscillators \\cite{Riedinger:18}. \nUsually, the polarisation of the photons in such processes is well defined and controlled. The rate of entanglement generation in these experiments has been primarily limited by the collection efficiency of photons from the emitters. The use of optical cavities has further improved our abilities to control the temporal shape of the photon \\cite{Vasilev:10} and has significantly improved collection efficiencies through the Purcell effect \\cite{Purcell:46}, both of which play a key role in enhancing entanglement fidelities and rates \\cite{Rohde:05, Goto:19}.\nTo this end, recent experiments have ventured toward the integration of miniature optical cavities \\cite{Hunger:10,NisbetJones:11,Bransdstatter:13,Steiner:13, Gulati:17,Takahashi:20, Brekenfeld:20} with more emerging\nSuch cavities with small radii of curvatures are likely to exhibit birefringence effects owing to limited precision in manufacturing and coating capabilities for the required small radii of curvatures \\cite{Jacob:95, Moriwaki:97,Takahashi:14, Ott:16}.\n\nRecently, it was demonstrated that the polarisation of photons from birefringent emitter-cavity systems undergoes polarisation oscillation \\cite{Barrett:19} - the produced photon's polarisation changes on the timescale of its length. The effects have been harnessed as a means of enhancing extraction rates\\cite{Barrett:20}.   \n \n In this paper, we point out that polarisation oscillation in measurement based entanglement schemes, where the qubit is often encoded in the photons' polarisation degree of freedom, can cause irreversible loss in the entanglement fidelity and therefore must be carefully considered in such implementations.\n\n\n\n\n \n\n \n \n \t\n\n\n\n\n\n\\begin{figure}\n\t\\includegraphics[height=10cm, , trim=0.0cm 0.2cm 0.0cm 0.0cm, clip]{.\/Fig1_ab.pdf}\n\t\\includegraphics[height=10cm]{.\/Fig1_c2_column.pdf}\n\t\\caption{\\label{fig:fig1}Measurement based entanglement setup for polarisation encoded photonic qubits.\\textbf{(a)} Two nodes composed of an emitter and a cavity emit into the input modes (a) and (b) of a non-polarising beam splitter (NPBS). At each node, the emitter is in superposition of the $\\ket{ \\uparrow}$ and $\\ket{ \\downarrow}$ states, entangled with orthogonal photon polarisations (see \\textbf{(c)}). The cavities at nodes 1 and 2 are birefringent-free but aren't necessarily identical, hence emit temporally distinguishable photons. For simplicity we only depict the temporal profile of one of the polarisations ($\\alpha_H$ and $\\beta_H$) of the superposed states. \nThe output at modes (c) and (d) are directed to polarising beam splitters for detecting the orthogonal components at the photodetector (PD). A quarter wave plate (QWP) allows for the mapping of the cavity emission onto a desired polarisation basis. In setup \\textbf{(b)}, node 2 is birefringent, hence its emission's polarisation changes along its pulse length with $\\beta_H^H$($\\beta_V^H$) showing the $H$ ($V$) polarised component (where the superscript denotes the expected polarisation in the absence of birefringence). \\textbf{(c)} Energy level scheme of the emitter-cavity system at different node types. A bichromatic pump (black arrows) with detunings $\\delta_1$ and $\\delta_2$ transfers the emitter population from the ground state $\\ket{g}$ to the qubit state $\\ket{ \\downarrow}$ ($\\ket{ \\uparrow}$) with the emission of a $\\sigma^-$($\\sigma^+$) photon mediated by a cavity, with frequency $\\omega_{cav}$, coupled to the $\\ket{e} \\leftrightarrow \\ket{ \\downarrow}$ and $\\ket{e} \\leftrightarrow \\ket{ \\uparrow}$ transitions. The qubit states are shown to have a Zeeman splitting of $\\Delta$. In the birefringent node, the cavity is shown to exhibit linear birefringence with energy $\\omega_H$($\\omega_V$) for the horizontal(vertical) polarisation eigenmode, and with splitting $\\Omega_B$ (with units setting $\\hbar=1$).\n}\n\\end{figure}\n\n\n\\section{Method}\n\n\\subsection{Non-birefringent system}\t\t\nWe consider the entangler depicted in Fig. \\ref{fig:fig1}. An emitter-cavity system at node (a) generates the balanced\\footnote{For simplicity, we consider the naturally desired balanced entangled state (which maximises entanglement generation rates) without loss of generality on the effects of birefringence. The formulation can simply be extended to an unbalanced entangled sate.} emitter-photon entangled state\n\n\\begin{align}\n\t\\ket{\\psi^{(\\textrm{a})}} &= \\frac{1}{\\sqrt{2}}\\ket{\\uparrow, H^{(\\textrm{a})}} +  \\ket{\\downarrow, V^{(\\textrm{a})}} \\\\\n\t& =  \\frac{1}{\\sqrt{2}}\\left( \\adag{}{H} \\ket{\\uparrow} + \\adag{}{V} \\ket{\\downarrow} \\right) \\ket{0}\n\\end{align}\nwhere $\\adag{}{x}$ is the $x$-polarised photon creation operator at mode (a) of the non-polarising beam splitter, with $x = \\{H \\textrm{(horizontal)}, V \\textrm{(vertical)} \\}$.\nThe photon's temporal waveform, which has so far not been considered, plays a key role in role in interference based processes, and consequently on entanglement fidelities as we shall later see.\nWe incorporate the photons waveform to the emitter-photon state by introducing the \\it{photon-wavepacket creation operator} \\rm \\cite{Loudon:00},\n\\begin{equation}\n\t\\adag{}{x} \\rightarrow \\Adag{\\alpha}{}{x} : = \\int \\alpha^{*}_x(t) \\adag{}{x}(t) \\dd{t},\n\\end{equation} \nwhich holds under normalization conditions $\\int |\\alpha_x(t)|^2 \\dd{t} = 1$ and where $ \\alpha_x(t)$ is the temporal wavepacket amplitude of the $x$-polarised emission as depicted in Fig.\\ref{fig:fig1}(a).\nWith the beam splitter transformation\n\\begin{equation}\n\\mqty(a_x^{}(t) \\\\ b_x^{}(t)) = \\frac{1}{\\sqrt{2}} \\mqty(1 & -1\\\\ 1 & 1)\\mqty(c_x^{}(t) \\\\ d_x^{}(t)),\n\\end{equation}\nwhere $b(t), c(t)$ and $d(t)$ are annihilation operators for modes (b), (c) and (d) respectively, the wave packet creation operators transform as\n\\begin{equation}\n\\Adag{\\alpha}{}{x}\n= \\frac{1}{\\sqrt{2}}  \\left(\\Cdag{\\alpha}{}{x}- \\Ddag{\\alpha}{}{x}\\right).\n\\end{equation}\nfor the wavepacket from node (a), where $\\Cdag{\\alpha}{}{x} := \\int \\alpha^{*}_x(t) \\cdag{}{x}(t) \\dd{t}$ and similarly, $\\Ddag{\\alpha}{}{x} := \\int \\alpha^{*}_x(t) \\ddag{}{x}(t) \\dd{t}$.  We define the wavepacket creation operator for node (b), $\\Bdag{\\beta}{}{x}$, which transforms as $\\Bdag{\\beta}{}{x} = \\frac{1}{\\sqrt{2}}  \\left(\\Cdag{\\beta}{}{x} + \\Ddag{\\beta}{}{x}\\right)$.  \nThe output state thus becomes\n\\begin{equation}\\label{eq:phit}\n\\begin{aligned}\n\\ket{\\phi_{\\textrm{out}}} \n\t& = \\frac{1}{4} \\Big[ \\left(\\Cdag{\\alpha}{}{H} - \\Ddag{\\alpha}{}{H}\\right)\\left(\\Cdag{\\beta}{}{H} + \\Ddag{\\beta}{}{H}\\right) \\ket{\\uparrow\\uparrow} \\\\\n\t& \\quad + \\left(\\Cdag{\\alpha}{}{V} - \\Ddag{\\alpha}{}{V}\t\\right)  \\left(\\Cdag{\\beta}{}{V} + \\Ddag{\\beta}{}{V}\t\\right) \\ket{\\downarrow\\downarrow} \\\\\n& \\quad + \\Cdag{\\alpha}{}{H} \\Cdag{\\beta}{}{V} \\ket{\\uparrow \\downarrow} +   \\Cdag{\\alpha}{}{V} \\Cdag{\\beta}{}{H} \\ket{\\downarrow\\uparrow}  \\\\\n& \\quad + \\Cdag{\\alpha}{}{H} \\Ddag{\\beta}{}{V}  \\ket{\\uparrow \\downarrow} -  \\Ddag{\\alpha}{}{V} \\Cdag{\\beta}{}{H} \\ket{\\downarrow\\uparrow}  \\\\\n& \\quad - \\Ddag{\\alpha}{}{H} \\Cdag{\\beta}{}{V}    \\ket{\\uparrow \\downarrow} +   \\Ddag{\\alpha}{}{V} \\Cdag{\\beta}{}{H} \\ket{\\downarrow\\uparrow}   \\\\\n& \\quad - \\Ddag{\\alpha}{}{H} \\Ddag{\\beta}{}{V}    \\ket{\\uparrow \\downarrow} -    \\Ddag{\\alpha}{}{V} \\Ddag{\\beta}{}{H} \\ket{\\downarrow\\uparrow}  \\Big] \\ket{0}\\\\\n\\end{aligned}\n\\end{equation}\nwhere the last 4 lines correspond to relevant pair of terms for detection of a $H$ and $V$ photons leading to successful heralds. The projected state upon the detection event of a $H$ and $V$ photons at node (c) at respective times $t_H$ and $t_V$ is\n\\begin{equation}\n\\begin{aligned}\n\\ket{\\phi_{H_cV_c} (t_H,t_V)} &=\\cann{H}{t_H} \\cann{V}{t_V} \\ket{\\phi_{\\textrm{out}}} \\\\ &= \\frac{1}{4} \\cann{H}{t_H} \\cann{V}{t_V}  \\Big[\\Cdag{\\alpha}{}{H} \\Cdag{\\beta}{}{V} \\ket{\\uparrow \\downarrow} +  \\Cdag{\\alpha}{}{V} \\Cdag{\\beta}{}{H} \\ket{\\downarrow \\uparrow } \\Big] \\ket{0} \\\\\n& = \\frac{1}{4} \\Big[ \\iint \\alpha^*_H(s_H) \\delta(t_H-s_H) \\beta^*_V(s_V) \\delta(t_V-s_V) \\dd{s_H}\\dd{s_V} \\ket{\\uparrow \\downarrow} \\quad \\\\\n& + \\iint \\alpha^*_V(s_V) \\delta(t_V-s_V) \\beta^*_H(s_H) \\delta(t_H-s_H) \\dd{s_H}\\dd{s_V}\\ket{\\downarrow \\uparrow} \\Big]\\ket{0} \\\\\n& = \\frac{1}{4} \\left(\\alpha^*_H(t_H) \\beta^*_V(t_V) \\ket{\\uparrow\\downarrow }+ \\alpha^*_V(t_V) \\beta^*_H(t_H) \\ket{\\downarrow \\uparrow}\\right) \\ket{0}, \n\\end{aligned}\n\\end{equation}\nwhere we have used $[\\cann{x}{t_x},\\cdag{}{x}(s_x)] = \\delta(t_x-s_x)$. \nThe fidelity of the conditional atomic state for the above detection event, $\\ket{\\psi^+} = \\bra{0}\\cann{H}{t_H} \\cann{V}{t_V} \\ket{\\psi_{\\textrm{out}}}$, with the Bell state\n$\\ket{\\Psi_{\\textrm{Bell}}^+} = \\frac{1}{\\sqrt{2}}(\\ket{\\uparrow \\downarrow} +\\ket{\\downarrow\\uparrow})$ is given by\n\\begin{equation}\\label{eq:FidelityNoBiref}\n\t\\begin{aligned}\n\t\\mathcal{F}&= \\frac{1}{\\Tr{\\ket{\\psi^+}\\bra{\\psi^+}}}|\\braket{\\Psi_{\\textrm{Bell}}^+}{\\psi^+}|^2 \\\\\n\t&= \\frac{1}{2}\\left( 1+\\frac{\\alpha^*_H(t_H) \\beta_H(t_H)\\beta^*_V(t_V) \\alpha_V(t_V) + \\text{c.c.}}{|\\alpha_H(t_H) \\alpha_V(t_V)|^2 + |\\alpha_V(t_V)\\alpha_H(t_H)|^2}\\right). \n\t\\end{aligned}\n\\end{equation}\n\nIntegrating over all detection events, we find the fidelity to be\n\\begin{equation*}\n\t\\mathcal{F} = \\frac{1}{2} \\left(1 + \\Re{\\int \\dd{t_H}\\alpha^*_H(t_H) \\beta_H(t_H) \\int \\dd{t_V}\\beta^*_V(t_V) \\alpha_V(t_V)}\\right).\n\\end{equation*}\n\nWhen the emission wavepackets from the emitter-cavity nodes are identical, the fidelity with the Bell state becomes unity.\n\n\\subsection{Introducing birefringence}\n\nWe now consider the case where the second emitter-cavity node suffers from birefringence. It has been shown in \\cite{Barrett:19} that the polarisation of a photon emitted from a birefringent cavity will generally exhibit a time dependence. The different path length for each eigenmode of the cavity leads to a relative path difference, which accumulates during the roundtrips in the cavity, and results in an effective rotation of the polarisation states. If the path difference accumulates faster than the cavity decay rate, a time-dependent polarisation will be observed as depicted in Fig.\\ref{fig:fig1}(b). The birefringence transforms the waveform $\\beta_H(t)$  to a superposition of a $H$ component with waveform $\\beta^H_H(t)$, and a $V$ component,  $\\beta^H_V(t)$. The photon-wavepacket creation operator from the non-birefringent system can thus be substituted as the following for the birefringent node:\n\\begin{align}\n\t\\Bdag{\\beta}{}{H} &\\rightarrow \\Bdag{\\beta}{,H}{H} + \\Bdag{\\beta}{,H}{V},\\\\\n\t\\Bdag{\\beta}{}{V} &\\rightarrow  \\Bdag{\\beta}{,V}{V} + \\Bdag{\\beta}{,V}{H},\n\\end{align}\nwhere $\\Bdag{j,x}{}{x}$  are wavepacket creation operators, with $\\Bdag{\\beta}{,H}{H} = h^{\\beta,H} \\int \\beta^{H*}_H(t)\\adag{2}{H}(t) \\dd{t}$ and $\\Bdag{\\beta}{,H}{V} = v^{\\beta,H} \\int \\beta^{H*}_V(t)\\adag{2}{V}(t) \\dd{t}$ where the coefficients $h^{\\beta,H}$ and $v^{\\beta,H}$, are introduced for normalisation while retaining the $\\int |\\beta^H_x(t)|^2 \\dd{t} = 1$ conditions required for the photon-wavepacket creation operator formalism. This transformation makes no assumptions about the relative decay rate for each cavity eigenmode nor about potential back action of the emission onto the emitter-cavity dynamics.\n\nThe resulting state at node (b) thus becomes\n\\begin{align*}\n\t\\ket{\\psi^{(b)}} = \\frac{1}{\\sqrt{2}}\\Big(\\left(\\Bdag{\\beta}{,H}{H} + \\Bdag{\\beta}{,H}{V}\\right)\\ket{\\uparrow}  + \\left(\\Bdag{\\beta}{,V}{H} + \\Bdag{\\beta}{,V}{V}\\right)\\ket{\\downarrow}\\Big).\n\\end{align*}\n\n$\\Bdag{\\beta}{,H}{x}$ and  $\\Bdag{\\beta}{,V}{x}$ transform similarly to $\\Bdag{\\beta}{}{x}$ under the beam splitter.\nIf we consider again the detection event of a $H$- and a $V$-polarised photons at node (c), the relevant terms in the expansion of $\\ket{\\phi_{\\textrm{out}}} = \\ket{\\psi^{(1)}}\\otimes\\ket{\\psi^{(2)}}$ are\n\\begin{align*}\n\t\\Cdag{\\alpha}{}{H}\\Cdag{\\beta}{,H}{V}\\ket{\\uparrow\\uparrow}, \\Cdag{\\alpha}{}{H}\\Cdag{\\beta}{,V}{V}\\ket{\\uparrow\\downarrow}, \\Cdag{\\alpha}{}{V}\\Cdag{\\beta}{,H}{H}\\ket{\\downarrow\\uparrow}, \\textrm{and } \\Cdag{\\alpha}{}{V}\\Cdag{\\beta}{,V}{H}\\ket{\\downarrow\\downarrow}.\n\\end{align*}\nThe projected atomic state for this detection event at respective times $t_H$ and $t_V$ for the $H$ and $V$ polarised photons is\n\\begin{equation}\\label{eq:BirefOut1}\n\t\\begin{aligned}\n\t\\ket{\\Psi^{+}{'}}&= \\bra{0} \\cann{H}{t_H}\\cann{V}{t_V}\\ket{\\phi_{out}}\n\t\\\\&= \\frac{1}{4} \\cann{H}{t_H}\\cann{V}{t_V} \\Big[ \\Cdag{\\alpha}{}{H}\\ket{\\uparrow}\\Big(\\Cdag{\\beta}{,H}{V}\\ket{\\uparrow}  +  \\Cdag{\\beta}{,V}{V}\\ket{\\downarrow}\\Big) + \\Cdag{\\alpha}{}{V}\\ket{\\downarrow}\\left(\\Cdag{\\beta}{,H}{H}\\ket{\\uparrow} + \\Cdag{\\beta}{,V}{H}\\ket{\\downarrow}\\right) \\Big].\n\t\\end{aligned}\n\\end{equation}\n\nWe can clearly see from this that the entangled state is not in the desired Bell basis \\{$ \\ket{\\uparrow\\downarrow},\\ket{\\downarrow\\uparrow}$\\}, but rather in the \\it basis \\rm \\{$ \\ket{\\uparrow\\downarrow'},\\ket{\\downarrow\\uparrow'}$\\} where $\\ket{\\downarrow'} = \\left(\\Cdag{\\beta}{,H}{V}\\ket{\\uparrow} + \\Cdag{\\beta}{,V}{V}\\ket{\\downarrow}\\right)$ and $ \\ket{\\uparrow'}=\\left(\\Cdag{\\beta}{,H}{H}\\ket{\\uparrow} + \\Cdag{\\beta}{,V}{H}\\ket{\\downarrow}\\right)$. We will see that this is more than a trivial rotation in bases and that it necessitates a careful evaluation owing to complications introduced by the arbitrary detection times, $t_H$ and $t_V$.\nSimplifying the terms in Eq.\\eqref{eq:BirefOut1},\nwe find the conditional atomic state for the detection of $H$ and $V$ photons at node (c) at respective times $t_H$ and $t_V$ to be:\n\\begin{equation}\n\\begin{aligned}\\label{eq:HeraldedStateWBiref}\n\\ket{\\Psi^{+}{'}} &= \\bra{0}\\cann{H}{t_H}\\cann{V}{t_V} \\ket{\\phi_{\\textrm{out}}} \\\\\n&= \\frac{1}{4} \\Big[ \\alpha^{*}_H(t_H) \\ket{\\uparrow} \\left( v^{\\beta,H} \\beta^{H*}_V(t_V) \\ket{\\uparrow} + v^{\\beta,V}\\beta^{V*}_V(t_V) \\ket{ \\downarrow}  \\right) + \\alpha^{*}_V(t_V) \\ket{\\downarrow} \\left( h^{\\beta,H} \\beta^{H*}_H(t_H) \\ket{\\uparrow} + h^{\\beta,V}\\beta^{V*}_H(t_H) \\ket{ \\downarrow}  \\right) \\Big] \n\\end{aligned}\n\\end{equation}\nThe fidelity with the Bell state $\\ket{\\Psi_{\\textrm{Bell}}^{+''}} = \\frac{1}{\\sqrt{2}}(\\ket{\\uparrow \\downarrow''} +\\ket{\\downarrow\\uparrow''})$ where $ \\{\\ket{ \\uparrow''},\\ket{ \\downarrow''}\\}$ is a measurement basis of node 2 that can be suitably chosen to maximise the fidelity, is given by\n\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}&= \\frac{1}{\\Tr{\\rho}}\\abs{\\braket{\\Psi_{\\textrm{Bell}}^{+''}}{\\Psi^{+}{'}}}^2 \\\\\n\\end{aligned}\n\\end{equation}\nwhere $\\rho = \\ket{\\Psi^{+}{'}}\\bra{\\Psi^{+}{'}}$. For the traditional choice of the basis$ \\{\\ket{ \\uparrow''} \\equiv \\ket{ \\uparrow}  ,\\ket{ \\downarrow''} \\equiv \\ket{ \\downarrow} \\}$, the fidelity becomes \n\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\mathcal{F}= \\frac{1}{2}\\Big[1 + \\frac{ \\Re{\\alpha^{*}_H(t) \\beta^H_H(t_H)\\alpha^{*}_V(t_V) \\beta^{V*}_V(t_V)v^{\\beta,V}h^{\\beta,H*}}}{8\\Tr{\\rho}}\n\t\t-\\frac{\\abs{\\alpha_H(t_H) v^{\\beta,H} \\beta^H_V(t_V)}^2 + \\abs{\\alpha_V(t_V) h^{\\beta,V} \\beta^V_H(t_H)}^2}{16\\Tr{\\rho}}  \\Big]\n\t\\end{aligned}\n\\end{equation}\nwhich closely resembles the form  for the non-birefringent case \\eqref{eq:FidelityNoBiref} with an extra reduction term. In addition to the condition of identical $H$ and $V$ waveforms from the nodes, the maximum fidelity in the presence of birefringence requires $\\abs{h^{\\beta,H}} = \\abs{v^{\\beta,V}} = 1$, as is expected.This traditional choice of $ \\{\\ket{ \\uparrow''},\\ket{ \\downarrow''}\\}$ doesn't maximise the fidelity in general. For instance, one could select $\\ket{ \\downarrow''} = \\left( v^{\\beta,H} \\beta^{H*}_V(t_V) \\ket{\\uparrow} + v^{\\beta,V}\\beta^{V*}_V(t_V) \\ket{ \\downarrow}  \\right) $; in this scenario, the fidelity will be higher than for the traditional choice of $\\ket{ \\uparrow''}$, provided the detection times are close, $t_H \\simeq t_V$ (in addition to the cavity decay rates for both eigenmodes being similar \\footnote{When the cavity decay rates are not polarisation dependent, we have $\\beta^H_H(t) = \\beta^V_V(t)$, $\\beta^H_V(t) = \\beta^V_H(t)$, $h^{\\beta,H} = v^{\\beta,V}$ and $h^{\\beta,V} = v^{\\beta,H}$, assuming all other parameters for the production of $H$ and $V$ photons are identical.}).\nTo show this, we simplify Eq.\\eqref{eq:HeraldedStateWBiref} to the normalised state\n\\begin{equation}\n\\begin{aligned}\n\t\\ket{\\phi_{H_cV_c}} = c_1 \\ket{ \\uparrow}\\ket{ \\downarrow''} + c_2 \\ket{ \\downarrow}\\left(\\sqrt{1 - p(\\delta, \\Delta)}\\ket{ \\uparrow''}+ \\sqrt{p(\\delta, \\Delta)}\\ket{ \\downarrow''}\\right)\n\\end{aligned}\n\\end{equation}\nand find fidelity\n\\begin{equation}\n\t\\mathcal{F} = \\frac{1}{2}\\left(1 +  2 \\Re{c_1^*c_2 \\sqrt{1 - p}}  - \\abs{c_2}p\\right)\n\\end{equation}\nwhere $p$ is a function of $\\delta = t_V-t_H$ and $\\Delta$,  a function of the decay rates and the relative coupling strength to the cavity eigenmodes,  governing the overlap between the states $\\ket{ \\uparrow''}$ and $\\ket{ \\uparrow'}$. For identical cavity mode decay rates and effective coupling strengths, the fidelity is generally maximised when $t_H=t_V$, a condition where $p$ is minimised. This is a crude assessment which assumes a driving pulse which produces photons with no `ringing' on their waveforms.\n\nThe approach of choosing an appropriate basis $\\{\\ket{ \\uparrow''},\\ket{ \\downarrow''}\\}$ naturally assumes post-selection of detection time-stamps to filter desired fidelities. An alternative, and complementary, approach to restore the loss in fidelity from the birefringence effects is to make local corrections to the stationary qubits, conditioned on the detection time-stamps. We elaborate on this technique later. \n\nFirst, we consider if the effects of birefringence, namely the distortion of the photonic waveforms, can be restored before the interference based measurement. The mentioned methods require strict time-stamping of photon detection events. In an ideal scenario where there is no birefringence, the heralding of fully entangled states is independent of the relative detection times. This is a tremendous advantage which  removes the necessity of extra, and often noise-injecting, local operations. Naturally, one wonders if one can introduce a serial device to restore time-independence of the photon's polarisation. Phase retarders are routinely employed in laboratories to manipulate the polarisation of an incoming beam. However, typically, the input beam has a fixed polarisation (compared to the timescale required to adjust the phase retarder's settings). In the considered case, the length of photons\nare desired to be short \\cite{Keller:04, Takahashi:20} to maximise repetition rate (- note, fidelity can be traded for rate and \\it vice versa \\rm) and are typically on the order of ~500 ns. The polarisation of the produced photons can oscillate on shorter timescales as demonstrated in \\cite{Barrett:19}. To the best of the authors' knowledge, there are no commercially available linear devices capable of performing a time-dependent rotation of such photons' polarisation to produce a time independent polarisation \\footnote{The fastest identified optics were Pockels cells, typically employed in photon routing and Q-switching,  with $\\sim$nanosecond optical rise times \\cite{Leysop} and modulation bandwidths generally limited to hundreds of kHz.}.\nFor the sake of completeness, we point out that such a device could also be another emitter-cavity system with complementary properties. However, if such process could be performed efficiently, the need for a heralding entangler would be defeated. \nIt thus becomes necessary to seek other options in order to recover the desired dynamics. \n\n\n\n\\section{Results}\n\\subsection{Raw fidelity}\nWe numerically study the fidelity of the entangled states produced from the system depicted in Fig.\\ref{fig:fig1}(b) with the corresponding Bell states in the case of degenerate and non-degenerate qubit states. The emitter is modelled as a 4 level atom ( Fig.\\ref{fig:fig1}(c)). A v-stirap (vacuum-stimulated Raman adiabatic passage) scheme is employed to transfer the population from the ground-state, $\\ket{g}$, to the stationary qubit states, $\\ket{ \\uparrow}$ and $\\ket{\\downarrow}$ via the intermediate state $\\ket{e}$. We consider only adiabatically produced photons - that is, spontaneous emission does not repopulate the ground state\\footnote{The coherently produced photons are usually the dominant contributors to the temporally mixed photonic output. In cavity-assisted Raman transition schemes the effective spontaneous emission rate falls with the square of the pump detuning whilst the effective coupling falls linearly with the detuning. Hence, temporal mode mixing could be sufficiently eliminated.}.  The system at one node exhibits no birefringence whilst the system at the second node exhibits birefringence (Fig. \\ref{fig:fig1}(b)) characterised by $\\Omega_B$, the frequency splitting of the eigenmodes.\nFig. \\ref{fig:fig2} summarises the results. Here, we set $g = \\kappa = \\gamma\/0.6 $, where $g$ is the emitter-cavity coupling strength (set to be the same for each of the v-stirap branches), $\\kappa$ is the cavity decay rate (assumed to be identical for both modes), and $\\gamma$ is the spontaneous emission rate of the $\\ket{e}$ state. The qubit states are degenerate and a transition to either qubit state from the $\\ket{e}$ state is equally probable. The relation $g\\sim\\kappa$ is shown to be near ideal for optimal extraction efficiency \\cite{Goto:19}, and the choice $\\gamma = 0.6g$ is loosely based on currently achievable desired coupling regimes \\cite{Takahashi:20}.\n\\begin{figure*}\n\n\t\\includegraphics[width=\\textwidth, trim=0.5cm 0.5cm 0.5cm 0.5cm, clip]{.\/Fig2mega_2.pdf}\n\t\\caption{\\label{fig:fig2} Entanglement dynamics for degenerate qubit states when the birefringence at the birefringent node is set to 1\/10, 1\/3, 2\/3 and 1 in units of $\\kappa$.\n\t\tPopulations in the emitter-photon state space \\textbf{(a)}, raw fidelities for detections of $H$ and $V$ photons at respective times $t_H$ and $t_V$ \\textbf{(b)}, fidelities post-correction for the same detection events \\textbf{(c)}, corresponding detection probability densities \\textbf{(d)}, magnitude of local qubit rotation required for fidelity correction at the birefringent node, with arrows indicating the azimuthal rotation direction \\textbf{(e)} and fidelity-success rate trade-offs \\textbf{(f)}, all for the degree of birefringence quoted at the head of the corresponding column. The blue contours in \\textbf{(b)} show the detection event that need to be retained for an average raw fidelity of 99.9\\% (no post-selection would be required for $\\Omega_B=0.1\\kappa$). Likewise, the red contours in \\textbf{(c)}show the post-selectable region for an average fidelity of 99.9\\% (no post-selection is required for $\\Omega_B \\leq \\kappa\/3$). The respective success probabilities for these contours are marked by the dashed lines in \\textbf{(f)}. }\n\\end{figure*}\n\\begin{figure*}\n\n\t\\includegraphics[width=\\textwidth,trim=0.5cm 0.5cm 0.4cm 0.5cm, clip]{.\/Fig3mega_2.pdf}\n\t\\caption{\\label{fig:fig3} Entanglement dynamics for non-degenerate qubit states when the birefringence at the birefringent node is set to 1\/10, 1\/3, 2\/3 and 1 in units of $\\kappa$.\n\t\tPopulations in the emitter-photon state space \\textbf{(a)}, raw fidelities for detections of $H$ and $V$ photons at respective times $t_H$ and $t_V$ \\textbf{(b)}, fidelities post-correction for the same detection events \\textbf{(c)}, corresponding detection probability densities \\textbf{(d)}, magnitude of local qubit rotation required for fidelity correction at the birefringent node, with arrows indicating the azimuthal rotation direction \\textbf{(e)} and fidelity-rate trade-offs \\textbf{(f)}, all for the degree of birefringence quoted at the head of the corresponding column. The blue contours in \\textbf{(b)} show the detection event that need to be retained for an average raw fidelity of 99.9\\% (no post-selection would be required for $\\Omega_B=0.1\\kappa$). Likewise, the red contours in \\textbf{(c)}show the post-selectable region for an average fidelity of 99.9\\% (no post-selection is required for $\\Omega_B \\leq \\kappa\/10$). The respective success probabilities for these contours are marked by the dashed lines in \\textbf{(f)}. }\n\\end{figure*}\n\nFig. \\ref{fig:fig2}(a) show the photons produced from the birefringent system for\nthe $\\Omega_B$ setting given at the top of the columns,\nand when the qubits states are degenerate ($\\Delta = 0$).\nIf there were no birefringence, the production of a $H$($V$) photon would only be associated with the atomic $\\ket{\\uparrow}$($\\ket{\\downarrow}$) state. However, polarisation oscillations create a non-zero population in the $\\ket{\\uparrow V}$ and $\\ket{\\downarrow H}$ states. We look at the fidelity landscape for all detection times \n(Fig.\\ref{fig:fig2}(b)).\nWe note that the fidelity is highest when detection events occur at times corresponding to the onset of the photonic pulse. This is intuitive as these timescales are shorter than those required for polarisation oscillations to take effect. However these sets of events happen with very low probability (see Fig.\\ref{fig:fig2}(d)), which further restricts the fidelity versus entanglement generation rate trade-off.\nWe also note a symmetry about the diagonal $t_H = t_V$; this is the case when the cavity decay rate has no polarisation dependency and the effective couplings to the stationary qubit states are identical; the fidelity generally falls away from this diagonal.\n \nWe study the fidelity versus entanglement rate trade-off attainable by post selection of detection events. The blue contours in Fig. \\ref{fig:fig2}(b) show the set of detection events yielding an {average} fidelity equal to or greater than 99\\%. Note that this average fidelity is obtained by weighing the raw fidelities (Fig. \\ref{fig:fig2}(b)) with the corresponding probability densities for each event  (Fig. \\ref{fig:fig2}(c)). In a similar way, we produce  Fig. \\ref{fig:fig2}(f)\nwhich show the probability of success associated with a minimum raw fidelity (blue solid lines). The blue dashed vertical lines are analogous to the blue contours in Fig. \\ref{fig:fig2}(b), marking the 99\\% raw average fidelity boundary. It is immediately noticeable that birefringence causes great detriment to achievable minimum fidelity for a given success rate. When the birefringence is $\\Omega_B = 0.1\\kappa$, post selection is not required to retain a high average fidelity. On the other hand, for larger birefringence, acute post selection becomes necessary reducing the probability of successful high-fidelity heralding close to nil. \n\n The theoretical maximum probability of heralding from $H$ and $V$ photon detections at different nodes is 25\\% when there is no birefringence. However, when placing restrictions on the maximum emission length (with the aim of enhancing emission rates), stronger driving pumps are required which in turn reduces photon extraction efficiencies due to spontaneous emission scattering. This results  in an overall lower success rate.\n In the absence of birefringence, all detection events can, in principle, give unit fidelity if one is not concerned with the length of the emitted photons. On the other hand, when birefringence enters the picture (with all other parameters, notably the diving powers and detunings, kept the same), the maximum attainable success rate can drop due to reduced photon emission probabilities.\n\n\n\nFig. \\ref{fig:fig3} is a reproduction of the dynamics in Fig. \\ref{fig:fig2} where the degeneracy of the qubit states is lifted by $\\Delta = 5\\kappa$ and the transition\n$\\ket{e} \\leftrightarrow \\ket{\\uparrow}$ state is 5\/4 times more probable than the $\\ket{e} \\leftrightarrow \\ket{\\downarrow}$ transition. The Rabi frequencies of the drives have been adjusted to produce photons of the similar length as in Fig. \\ref{fig:fig2}(a), and with their detunings appropriately adjusted for resonant Raman transitions. The resulting photonic profiles Fig. \\ref{fig:fig3}(a) are significantly distorted due to a beating effect emanating from the bichromatic drive.  This leads to a worsened fidelity-rate relationship in contrast to when $\\Delta \\simeq 0$. The probability density landscapes (Fig. \\ref{fig:fig3}(d)) become patterned, and so do the  high fidelity regions in Fig. \\ref{fig:fig3}(b) adding a layer of complexity for post-selection of high-fidelity detection events. \n\n\n\n\n\nThe results presented in Fig. \\ref{fig:fig2}(f) and Fig. \\ref{fig:fig3}(f) help deduce the tolerable degree of birefringence for desired fidelities (optionally conditioned on a minimum success rates), and provide specifications for the machining of cavities.\n\n\nWe have seen that the uniform distribution of the fidelity against success probability for the non-birefringent pair of nodes is significantly impaired when one node suffers from birefringence. In the following we assess if and how the fidelities can be recovered.  \n\n\\subsection{Restoring fidelity} Heralded entanglement schemes are employed because efficient deterministic global gate operations are presently not available for remote systems.\nAs such, whilst it would be theoretically possible to restore the loss in fidelities via global operations, it would defeat the practical need of employing heralded entanglers. However, local operations are well within the remit of experiments \\cite{Brown:11,Harty:14, Sheng:18}. We therefore find, for each detection time pairs $t_H$ and $t_V$, the local qubit rotation that would maximise the fidelity of the resulting state with the Bell state.\nThe resulting fidelities, required corrections and success probability-fidelity trade-offs are presented in Fig.\\ref{fig:fig2}(c), Fig.\\ref{fig:fig2}(e) and Fig.\\ref{fig:fig2}(f) (red solid lines), respectively, when the qubits are degenerate. The corresponding results for the non-degenerate case are presented in Fig.\\ref{fig:fig3}(c), Fig.\\ref{fig:fig3}(e) and Fig.\\ref{fig:fig3}(f) (red solid lines). Fig.\\ref{fig:fig2}(e) and Fig. \\ref{fig:fig3}(e) show the magnitude of the qubit rotation required to optimise fidelity for each detection event. The overlaid black arrows show the azimuthal direction of the required qubit rotation\\footnote{The altitude contributions aren't generally negligible but the azimuthal contribution were deemed sufficient to gauge a sense of the variability in the required correction for different $\\Omega_B$ settings.}. Fig.\\ref{fig:fig2}(c) and Fig.\\ref{fig:fig3}(c) show the fidelity landscapes for each detection time pairs following the qubit corrections. A sizeable difference is noticed in the region with average fidelity of 99\\% or greater (red contours) when compared to the raw fidelities (Fig.\\ref{fig:fig2}(b) and Fig.\\ref{fig:fig3}(b) ).\n The fidelity is near-fully restored in the absence of energy splitting between the qubit states (Fig.\\ref{fig:fig2}(f)). When the degeneracy between the qubit states is lifted however, restoration to fidelities greater than or equal to $ 99\\% $ is limited to a small fraction of the detection events (see red dashed lines in Fig. \\ref{fig:fig3}(f)).  \n\n When the degeneracy between the qubits is lifted, the photonic temporal profiles are heavily distorted from the preferred Gaussian-like profile \\cite{Rohde:05}. Furthermore, the population profile in the $\\ket{ \\uparrow}\\ket{H}$ and $\\ket{ \\downarrow}\\ket{V}$ states are no longer identical. These result in a practically non-trivial complexity in the qubit corrections landscape required to optimise fidelity (Fig. \\ref{fig:fig3}(e)).  \n \n \\begin{figure}[h!]\n \n \t\\includegraphics[width=0.9\\columnwidth]{.\/Fig4_v_onecolumntext.pdf}\n \t\\caption{\\label{fig:fig5} (a) In addition to $\\Omega_B$, a large birefringence, $\\delta_B$, is deliberately induced in the cavity's $H-V$ basis. \\textbf{Top}: Here the atom is emitting circularly polarised photons. Polarisation oscillations would result as the $\\sigma^\\pm$ emission couple to both cavity eigenmodes. \\textbf{Bottom}: The emission is mapped to $H$ and $V$ polarised photons by an appropriate choice of quantisation axis as in \\cite{Stute:12}. The drive detunings and\/or the Zeeman splitting are adjusted for resonant emission into the $H$ and $V$ modes. Note that the states $\\ket{ \\downarrow}$ and $\\ket{ \\uparrow}$ here do not necessarily correspond to the same Zeeman sublevel states as those in the $\\sigma^\\pm$ emission scheme. (b-c) Fidelity vs deliberate birefringence, $\\delta_B$, superposed on an undesired birefringence of $\\Omega_B=1.0\\kappa$ with (red) and without (blue) local qubit corrections to restore fidelity. The green traces show the associated success probability. In (b), $\\Delta = 0$ whilst in (c), $\\Delta = 5\\kappa$. }\n \\end{figure}\n\n\\subsection{Deliberate birefringence} We have seen that birefringence can lead to severe loss in fidelity in measurement based entanglement, some of which can be recovered via local operations. Extra local qubit operations are often themselves a source of noise; experimentalists may therefore naturally prefer to avoid them by opting for non-birefringent cavities. However, due to machining tolerances, it is impossible to eliminate birefringence fully. In the case of non-negligible birefringence, we propose an alternative to the local qubit corrections presented above. This entails matching-up the atomic emissions' basis as well as frequencies to those of the cavity's birefringence (Fig \\ref{fig:fig5}). We show that by selecting appropriate Zeeman splitting and quantisation axis together with a large birefringence, one can reduce polarisation oscillations and therefore be able to obtain high fidelities without the need for local qubit corrections. We extend this approach to demonstrate that even in the presence of off-axis \\it{undesired }\\rm birefringence (emerging from machining tolerances, for example), adding large birefringence can be used to suppress polarisation oscillation from the undesired birefringence.\n\nWe consider the case where there's an off-axis\/undesired birefringence of $\\Omega = 1.0\\kappa$ as in the last columns of Figs. \\ref{fig:fig2} and \\ref{fig:fig3}. We impose an additional linear birefringence, $\\delta_B$, up to $\\delta_B = 10 \\kappa$,  achievable by engineering deliberate ellipticity onto the cavity mirrors as in \\cite{Uphoff:15} and implemented in \\cite{Brekenfeld:20}. As $\\delta_B$ increases, the eigenmodes in the $H-V$ basis becomes more and more spectrally resolved, and the degree of polarisation oscillation between these eigenmodes falls. In the limit of large birefringence, this becomes equivalent to coupling the atomic emissions to 2 independent cavities.  We thus find that (Fig. \\ref{fig:fig5}b-c) high ion-ion entanglement fidelity does not require local qubit corrections for large $\\delta_B$. In the case of degenerate qubit states (Fig. \\ref{fig:fig5}b), we note a marginal drop in success probability, likely due  a \\it{beating } \\rm effect in the photonic outputs due to the bichromatic drive.  A reduced degree of adiabaticity leads to reduced emission probability. In the case of non-generate qubits, the drive frequencies need not change greatly for different $\\delta_B$; the dynamics in success probability is thus not dominated by frequency beating effects, but likely due to changes in effective cooperativity.\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nBirefringent cavities are known to impose time-dependence on the polarisation of the emitted photons,\nbut the resulting effect on the fidelity of interference-based entanglement-swapping experiments had not previously been investigated. In this paper, we have provided a detailed study of the impact on the performance of heralded, two-photon schemes used to generate entanglement remotely between emitters at distant nodes.\nWe have shown that, in the presence of birefringence, the detection of orthogonal photons no longer heralds fully entangled states, resulting in significantly lower entanglement fidelity.\nAs such, we conclude that the best approach for high entanglement fidelity and rate is the use of non-birefringent cavities. Where cavity birefringence cannot be made negligible a remedy is the use of local qubit operations to attempt to restore orthogonality in the  collapse state subspace of the emitters. However, applying the appropriate correction requires sampling from a continuum of unitary correction pulses areas and angles on the basis of the photon arrival times. This approach can near-fully recover fidelities when the qubit states are degenerate, and in this regime the pulse parameters vary slowly with photon time. However, when a large splitting separates the qubit energies and the system is driven with a bichromatic Raman field, recovery of fidelities greater than 99\\% is possible only for a fraction of the detection space with the optimal pulse parameters varying rapidly with the arrival times, making reliable correction onerous. Considering this situation, we presented an alternative approach to suppress polarisation oscillations. By inducing a large, deliberate birefringence co-linear with the emission bases, the coupling between orthogonal emissions due to stray birefringence can be reduced and hence high entanglement fidelities retained. In summary, uncontrolled birefringence can lead to significant loss in fidelity and success rates in measurement-based entanglement and these effects should be carefully considered and mitigated when designing such experiments.\n\n\\section*{Acknowledgements}\n\nThe authors are grateful for helpful discussions with A. Kuhn and T. Barrett. This work was funded by the United\nKingdom Engineering and Physical Sciences Research\nCouncil ``Networked Quantum Information Technology'' and ``Quantum Computing and Simulation'' Hubs.  \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA few years ago we have proposed\na quark-parton model of cumulative phenomena\nin the interactions with nuclei \\cite{NPB94,YF97}\nbased on\nperturbative QCD calculations of the corresponding quark\ndiagrams near the thresholds, at which other quarks\n(\"donors\") in the nuclear flucton transfer all their\nlongitudinal momenta to the distinguished active quark\nand become soft.\n\nConsider the scattering of\na hadronic projectile off a nucleus with the momentum $P$\nin the c.m. system.\nAt high energies\nthe momentum $K$ of the produced pion belongs to\nthe cumulative region if $K_z>P_z\/A$.\nAs a reasonable first approximation, we treat the nucleus as a\ncollection of $N=3A$ valence quarks, which, on the average, carry each\nlongitudinal momentum $x_{0}P_{z}\/A$ with $x_{0}=1\/3$.\nIn our approach the cumulative pion production proceeds in two steps.\nFirst a valence quark with a scaling variable $x>1$ is created.\nAfterwards\nit decays into the observed hadron with its scaling variable $x$ close\nto the initial cumulative quark's one. This second step is described by\nthe well-known quark fragmentation functions\n\\cite{CapellaT81} and will not be\ndiscussed here.\n\nThe produced cumulative (\"active\") quark acquires\nthe momentum much greater than $x_{0}P_{z}\/A$ only if this\nquark has interacted by means of gluon exchanges\nwith other $p$ quarks of flucton (\"donors\")\nand has taken some of their longitudinal momenta (see Fig.1).\nIf this active quark accumulates all\nlongitudinal momentum of these $p$ quarks then\n$K_z=(p+1)x_0P_z\/A$ and the donors  become soft.\nIt is well-known that interactions which make the longitudinal momentum\nof one of the quark equal to zero may be treated by perturbation theory\n\\cite{Brodsky92}. This allows to calculate the part of Fig. 1\nresponsible for the creation of a cumulative quark explicitly.\nThis was done in \\cite{NPB94,YF97}, to which papers we refer  the\nreader for all the details. As a result we were able to explain the\nexponential fall-off of the production rate in the cumulative region.\n\nSince with  the rise of $x$ the active quark has to interact with\na greater number of donors, one expects that its average transverse momentum\nalso grows with $x$. Roughly one expects that $\\langle K_{\\perp}^2 \\rangle$\nis proportional to the number of interactions, that is, to $x$.\nIn \\cite{NPB94,YF97}\nthis point was not studied:\nwe have limited ourselves with the inclusive cross-section\nintegrated over the transverse momenta,\nwhich lead to some simplifications.\nThe aim of the present paper is to find\nthe pion production rate dependence on\nthe transverse momentum\nand the mean value of the latter\nas a function of $x$ in the cumulative region.\nThis dependence and also the magnitude of $\\langle K_{\\perp}^2 \\rangle$\nhave been studied experimentally. The comparison of our predictions\nwith the data allows to obtain further support for our model and\nfix one of the two its parameters (the infrared cutoff).\n\n\n\\section{The $K_{\\perp}$ dependence}\n\nRepeating the calculations of the diagram in Fig.1 described in\n\\cite{NPB94,YF97} but not\nlimiting  ourselves with the inclusive cross-section\nintegrated over the transverse momentum,\nwe readily find that all dependence upon\nthe transverse momentum $K_{\\perp}$ of the produced particle\nis concentrated in a factor:\n\\begin{equation}\nJ (K_{\\perp})=\n\\int\n\\rho_A(\\underbrace{r,...,r}_{p+1}|\\underbrace{\\ol{r},...,\\ol{r}}_{p+1})\nG(c_1,...,c_p)\n\\prod_{i=1}^{p}\n\\lambda(c_i-r)\\lambda(c_i-\\ol{r})d^2 c_i\ne^{i(\\ol{r}-r)K_{\\perp}}\nd^2 r d^2 \\ol{r}\n\\end{equation}\nHere $\\rho_A$ is the (translationally invariant)\nquark density matrix of the nucleus:\n\\begin{equation}\n\\rho_A(r_i|\\ol{r}_i)\\equiv\n\\int\n\\psi_{\\perp A}(r_i,r_m)\n\\psi^*_{\\perp A}(\\ol{r}_i,r_m)\n\\prod_{m=p+2}^{N} d^2 r_m\n\\end{equation}\nwhere\n$\\psi_{\\perp A}$ is the transverse part of\nthe nuclear quark wave function. The propgation of soft donor quarks\nis decribed by\n\\begin{equation}\n\\lambda(c)=\\frac{K_0(m|c|)}{2\\pi}\n\\label{lam}\n\\end{equation}\nwhere $m$ is the constituent quark mass and  $K_0$\nis the modified Bessel function (the Mac-Donald function).\nThe interaction with the projectile contributes a factor\n\\begin{equation}\nG(c_1,...,c_p)=\n\\int\n\\prod_{i=1}^{p}\n\\sigma_{qq}(c_i-b_i)\n\\eta_H(b_1,...,b_p) d^2 b_i\n\\end{equation}\nwhere $\\sigma_{qq}(c)$ is the quark-quark cross-section\nat a given value of impact parameter $c$ and\n\\begin{equation}\n\\eta_H(b_1,...,b_p)=\n\\sum_{L\\geq p}\\frac{L!}{(L-p)!}\n\\int|\\psi_{\\perp H}(b_i)|^2\n\\delta^{(2)}(\\frac{1}{L}\\sum_{i=1}^{L} b_i)\n\\ d^2 b_{p+1}...d^2 b_L\n\\end{equation}\nis a multiparton distribution in the projectile, expressed via\nthe transverse part of\nits partonic wave function $\\psi_{\\perp H}$ .\nIf we integrate $J (K_{\\perp})$ over $K_{\\perp}$ we come back\nto our old result (Eq. (33) in \\cite{NPB94}):\n$$\n\\int J (K_{\\perp})\n\\frac{d^2 K_{\\perp}}{(2\\pi)^2}=\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n\\int\nG(c_1,...,c_p)\n\\prod_{i=1}^{p} \\lambda^2(c_i-r)\nd^2 c_i d^2 r\n$$\n\nIf one assumes factorization of the multiparton distribution\n$\\eta_H(b_1,...,b_p)$ then\\\\\n$G(c_1,...,c_p)$ also factorizes:\n\\begin{equation}\nG(c_1,...,c_p)=\n\\prod_{i=1}^{p}\nG_0(c_i)\n\\label{facG}\n\\end{equation}\nFollowing \\cite{YF97} we use the quasi-eikonal approximation\nfor $\\eta_H$:\n$$\n\\eta_H(b_1,...,b_p)=\n\\xi^{(p-1)\/2}\\nu_H^{p}\\prod_{i=1}^{p}\\eta_H(b_i)\n$$\nwhere $\\xi$ is the quasi-eikonal diffraction factor,\n$\\nu_H^{}$ is the mean number of partons in the projectile hadron\nand the single parton distribution $\\eta_H(b)$\nis normalized to unity.\nIn a Gaussian approximation\nfor $\\sigma(c)$ and $\\eta_H(b)$ we find:\n$$\nG_0(c)=\n\\xi^{\\frac{1}{2}-\\frac{1}{2p}}\n\\frac{\\nu_H\\sigma_{qq}}{\\pi r_{0H}^2}\ne^{-\\frac{c^2}{r_{0H}^2}}\n$$\nwhere $\\sigma_{qq}$ is the total  quark-quark cross-section,\n$r_{0H}^2=r_0^2+r_H^2$, $r_0$ and $r_H$ are the widths of\n$\\sigma(c)$ and $\\eta_H(b)$ respectively.\n\nWith the factorised $G(c_1,...,c_p)$ (\\ref{facG}) we have\n$$\nJ (K_{\\perp})=\n\\int\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{\\ol{r}-r,...,\\ol{r}-r}_{p+1})\nj^p(r,\\ol{r})\ne^{i(\\ol{r}-r)K_{\\perp}}\nd^2 r d^2 \\ol{r}\n$$\nwhere\n$$\nj(r,\\ol{r})=\n\\int d^2 c\nG_0(c)\n\\lambda(c-r)\n\\lambda(c-\\ol{r})\n$$\nWe also have used the translational invariance of the $\\rho$-matrix.\nNote that near the real threshold we have no spectators and\n$$\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{\\ol{r}-r,...,\\ol{r}-r}_{p+1})\n=\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n$$\nIn any case large $K_{\\perp}$ corresponds to small $\\ol{r}-r$ so\nwe factor $\\rho_A$ out of the integral in zero point.\nIn the rest integral we pass to the variables\n$$\nB=\\frac{r+\\ol{r}}{2}, \\hs 1 b=\\ol{r}-r\n$$\nand shift the integration variable $c$, then\n\\begin{equation}\nJ (K_{\\perp})=\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n\\int\nj^p(B,b)\ne^{ibK_{\\perp}}\nd^2 b d^2 B\n\\end{equation}\nwhere\n\\begin{equation}\nj(B,b)=\n\\int\nG_0(B+c)\n\\lambda(\\frac{b}{2}-c)\n\\lambda(\\frac{b}{2}+c)\nd^2 c\n\\label{j}\n\\end{equation}\n\n\n\\section{The calculation of $\\langle |K_{\\perp}|\\rangle $}\n\nNow we would like to find the width of the distribution on $K_{\\perp}$\nas a function of $p$ or what is the same of the cumulative\nnumber $x=(p+1)\/3$.\nFrom the mathematical point of view it is simpler to calculate\nthe mean squared width of the distribution $\\langle K_{\\perp}^2\\rangle $.\nUnfortunately in our case this quantity is logarithmically\ndivergent at large $K_{\\perp}$.\nThis divergency results from the behavior of $j(B,b)$\nat small $b$. This behavior is determined by the behavior of the\n$ \\lambda(b)=K_0(m|b|)\/(2\\pi) $ (\\ref{lam}),\nwhich has a logarithmical singularity\nat $|b|=0$. Smooth $G_0(B+c)$ does not affect this behavior.\n\nFor this reason we shall rather calculate $\\langle |K_{\\perp}|\\rangle $:\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\n=\\frac{1}{J_N}\n\\int\nj^p(B,b)\n|K_{\\perp}|e^{ibK_{\\perp}}\nd^2 b d^2 B\n\\frac{d^2 K_{\\perp}}{(2\\pi)^2}\n\\end{equation}\nwhere $J_N$ is the same integral as in the numerator\nbut without $|K_{\\perp}|$.\nPresenting $|K_{\\perp}|$ as $K_{\\perp}^2\/|K_{\\perp}|$ and\n$K_{\\perp}^2$ as the Laplacian $\\Delta_b$ applied to\nthe exponent we find\n$$\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{1}{J_N}\n\\int\nj^p(B,b)\n\\Delta_b e^{ibK_{\\perp}}\nd^2 b d^2 B\n\\frac{d^2 K_{\\perp}}{|K_{\\perp}|(2\\pi)^2}\n$$\nTwice integrating by parts\nand using the formula\n$$\n\\int\n\\frac{d^2 K_{\\perp}}{|K_{\\perp}|}e^{ibK_{\\perp}} =\n\\frac{2\\pi}{|b|}\n$$\nwe find\n$$\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{1}{2\\pi J_N}\n\\int\n\\frac{1}{|b|}\n\\Delta_b j^p(B,b)\nd^2 b d^2 B\n$$\nNow we again integrate by parts once to find\n$$\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{1}{2\\pi J_N}\n\\int\nd^2 B\n\\frac{d^2 b}{|b|^2}\n(n_b \\nabla_b) j^p(B,b)\n$$\nwhere  $n_b=b\/|b|$. This leads to our final formula\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{p}{2\\pi J_N}\n\\int\nd^2 B\n\\frac{d^2 b}{|b|^2}\nj^{p-1}(B,b)\n(n_b \\nabla_b) j(B,b)\n\\label{Kperp}\n\\end{equation}\nwhere $j(B,b)$ is given by (\\ref{j}),\n$ \\lambda(b) $ is given by (\\ref{lam}) and\n$$\nJ_N=\\int d^2 B j^{p}(B,b=0)\n$$\n\n\n\\section{Approximations}\nTo simplify numerical calculations we make some additional\napproximations, which are not very essential but are well supported\nby the comparison with exact calulations at a few sample points.\n\nAs follows from the\nthe asymptotics of  $K_0(z)$ at large $z$\n$$\nK_0(z)\\simeq\\sqrt{\\frac{\\pi}{2z}}e^{-z}\n$$\nthe width of $\\lambda(b)$ (\\ref{lam}) is of the order $m^{-1}$.\nThe function $G_0$ is smooth in the vicinity of the origin\nand its width $r_{0H}=\\sqrt{r_0^2+r_H^2}$  is substancially\nlarger than the width of $\\lambda$.\nFor this reason we factor $G_0(B+c)$ out of the integral (\\ref{j})\nover $c$ at the point $B$:\n\\begin{equation}\nj(B,b)=\nG_0(B)\\Lambda(b),\n\\hs 1\n\\Lambda(b) \\equiv\n\\int\n\\lambda(c)\n\\lambda(c-b)\nd^2 c=\n\\frac{|b|}{4\\pi m}K_1(m|b|)\n\\label{Lam}\n\\end{equation}\nThen we find that the integrals over $B$ and $b$ decouple\n\\begin{equation}\nJ (K_{\\perp})=\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n\\int G^p_0(B) d^2 B\n\\int \\Lambda^p(b)\ne^{ibK_{\\perp}}\nd^2 b\n\\end{equation}\n\nIn this approximation we find that $\\langle |K_{\\perp}|\\rangle $\ndepends only\non one parameter - the constituent quark mass $m$,\nwhich  in our approach plays the role of an infrared cutoff:\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\n=pm\n\\int_0^{\\infty} dz K_0(z) (zK_1(z))^{p-1}\n\\label{apprKperp}\n\\end{equation}\nThis allows to relate $m$ directly to the experimental data on\nthe transverse momentum dependence.\n\n\\section{Comparison with the data and discussion}\n\nThe integral in (\\ref{apprKperp}) can be easy calculated\nnumerically. For values of $p=1,...,12$ it is very well\napproximated by a power dependence (see Fig.2), so that we obtain\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\/m\n=1.594\\, p^{0.625}\n\\end{equation}\nAs we observe, the rise of $\\langle |K_{\\perp}|\\rangle$ turns out to be\neven faster than\nexpected on naive physical grounds mentioned in the Introduction\n($\\sim\\sqrt{p}$).\nThe resulting plots for $\\langle |K_{\\perp}|\\rangle^2$\nas a function of the cumulative number $x=(p+1)\/3$\nat different values of parameter $m$ are shown in Fig.3\ntogether with avaiable experimental data from \\cite{Boyarinov94}\non $\\langle K_{\\perp}^2\\rangle$ for pion production obtained in\nexperiments\n\\cite{Boyarinov94}-\\cite{Boyarinov87}\nwith 10 $GeV$ protons\nand \\cite{Baldin82, Baldin83} with 8.94 $GeV$ protons.\n\nNote that earlier publications of the first group\n\\cite{Boyarinov92,Boyarinov87}\nreported a much stronger increase of $\\langle K_{\\perp}^2\\rangle$\nwith $x$, up to  value 2 $(GeV\/c)^2$ at $x=3$ for pion production.\nIn our approach such an increase would require the quark mass to be as high\nas $m \\simeq 225 MeV$.\nIn a more recent publication \\cite{Boyarinov94}\nthe rise of $\\langle K_{\\perp}^2\\rangle$ is substancially weaker\n(it corresponds to $m \\simeq 175 MeV$ in our approach).\nThe authors of \\cite{Boyarinov94} explain this\nby new experimental data obtained and by a cutoff $K_{\\perp max}$\nintroduced in calculations of $\\langle K_{\\perp}^2\\rangle$ in \\cite{Boyarinov94}.\nThe introduction of this cutoff considerably\n(approximatly two times)\ndecreases the experimental value of $\\langle K_{\\perp}^2\\rangle$ at $x=3$.\nIn our opinion this is a confirmation that the\ncumulative pion production rate only weakly decreases with $K_{\\perp}$\nin the cumulative region so that the\nthe integral over $K_{\\perp}^2$ which enters the definition of\n$\\langle K_{\\perp}^2\\rangle$ is weakly convergent or even divergent,\nas in our approach.\nUndoubtedly presentation of the experimental data in terms of\nthe mean value\n$\\langle |K_{\\perp}| \\rangle^2$,\nrather than $\\langle K_{\\perp}^2\\rangle$ should reduce\nthe dependence on the cutoff\n$K_{\\perp max}$ and make the results more informative.\n\nOne of the ideas behind the investigations of the cumulative\nphenomena is that they may be a manifestation\nof a cold quark-gluon plasma formed when several nucleons overlap in the\nnuclear matter. In \\cite{NPB94} we pointed out  that our model does not\ncorrespond to this picture. It implies  coherent interactions of the\nactive quark with donors and, as a result, strong correlations between the\nlongitudinal and transverse motion. Predictions for the dependence of\n$\\langle |K_{\\perp}| \\rangle$ on $x$ are also  different. From the cold\nquark-gluon plasma model one expects $\\langle |K_{\\perp}| \\rangle$ to behave\nas $x^{1\/3}$, since the Fermi momentum of the quarks inside the overlap\nvolume is proportional to the cubic root of the quark density. Our model\npredicts a much faster increase, with a power twice larger. The\nexperimental data seem to support our predictions.\n\n\\section{Acknowledgments}\n\nThe authors are greatly thankful to Prof. P.Hoyer who attracted\ntheir attention to the problem.\n\nThis work is supported by the Russian Foundation for Fundamental\nResearch, Grant No. 97-02-18123.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acknowledgments}\nWe would like to thank De Meng, Maryam Fazel and Mehran Mesbahi for their help.\n\n\\section{Conclusions}\\label{conclusion}\nWe derived a class of $N$ player, weighted potential games under heterogeneous MDP dynamics and with application to multi-agent path coordination. For these games, we show equivalence between the unique Nash equilibrium and the global solution of a potential minimization problem, which we solve via gradient descent and single-player dynamic programming. Future work include deriving learning-based solutions for the games and \\change{integrating partially observable scenarios in which players have local observations only.} \n\n\n\n\n\\section{Multi-agent path coordination}\\label{sec:mapf}\n\\noindent We apply our game model to a multi-agent pick up and delivery scenario \\change{with stochastic package arrival times}. As shown in Figure~\\ref{fig:grid_world}, $N$ players navigate a 2D \\change{space}. \\change{Each player's goal is} to  transport packages from the pick up chutes to the drop off chutes while \\change{avoiding collision with others}. Code for the simulation is available at \\url{https:\/\/github.com\/lisarah\/mdp_path_coordination}.\n\\begin{figure}\n    \\centering\n    \\vspace*{0.3cm} \n    \\includegraphics[width=0.7\\columnwidth]{Figure\/plzwork.png}\n    \\caption{Operation environment for multi-robot warehouse scenario.}\n    \\label{fig:grid_world}\n\\end{figure}\n\\subsection{Stationary MDP Model}\n\\change{Players operate in a} two dimensional grid world with $5$ rows and $10$ columns. In addition to capturing location, each state also dictates whether the robot is in pick up or delivery mode. The state space is given by\n\\[[S] = \\Big\\{ (\\change{v, w}, m) \\ | \\ 1\\leq \\change{v} \\leq 5, \\ 1\\leq \\change{w} \\leq 10, \\ m \\in \\change{\\{1,2\\}}\\Big\\}.\\]\nAt each state, available actions are $[A] = \\{u, d, r, l, s\\}$, corresponding to up, down, right, left, stay. Player transition dynamics and rewards are \\emph{stationary} in time.\nThe transition probability of each state $(\\change{v, w}, m)$ extends the location-based transition probabilities $P^0$. \n\n\\noindent\\textbf{Location-based transition}. Let $u =\\change{(v, w)}$ denote the location component of the state. At each location, each action either points to a feasible target $u_{targ}(a)$ or is infeasible. The set of all feasible targets from $u$ is $\\mathcal{N}(u)$. \nWhen a target exists, players have $1 > q > 0$ chance of reaching it and $1 - q$ chance of reaching other states in $\\mathcal{N}(u)$. \n\\begin{equation}\\label{eqn:feasible_location_transitions}\nP^0_{u'ua} = \\begin{cases}\nq & u' = u_{targ}(a), \\\\\n\\frac{1 - q}{|\\mathcal{N}(u)|} & u' \\in \\mathcal{N}(u)\/\\{u_{targ}(a)\\},\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nWhen the target location is infeasible, the player transitions into a neighboring state $u' \\in \\mathcal{N}(u)$ at random.\n\\begin{equation}\\label{eqn:infeasible_location_transitions}\nP^0_{u'ua} = \\begin{cases}\n\\frac{1}{|\\mathcal{N}(u)|} & u' \\in \\mathcal{N}(u),\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\\textbf{Full transition dynamics}. \\change{Within the same mode}, players transition between locations via dynamics $P^0$. Player modes \\change{transition at pick up chutes $\\mathcal{P}$ and drop off chutes $\\mathcal{D}$}.\n\\change{\n\\begin{enumerate}\n    \\item When player $i$ is in mode $1$ (pick up) and about to transition into pick chute $p^i \\in \\mathcal{P}$, player $i$'s mode has $r^i$ probability of switching to mode $2$ (drop off). \n    \\[\\begin{cases}\n     P^i_{t(p^i, 2)sa} = r^iP^0_{tp^iua},\\\\\n     P^i_{t(p^i, 1)sa} = (1 - r^i)P^0_{tp^iua},\n    \\end{cases}\\  \\forall s = (u, 1), \\ s \\in [S].\\]\n    \\item When player $i$ is in mode $2$ (drop off) and about to transition into drop chute $d^i \\in \\mathcal{D}$, player $i$ transitions to mode $1$ with probability $1$.\n    \\[\\textstyle \\begin{cases}\n     \\textstyle P^i_{t(d^i, 1)sa} = P^0_{td^iua},\\\\\n     \\textstyle P^i_{t(p^i, 2)sa} = 0,\n    \\end{cases}\\  \\forall s = (u, 2), \\ s \\in [S].\\]\n\\end{enumerate}\n}\nHere, $r^i \\in {\\mathbb{R}}$ denotes the probability of package arrival when player $i$ is in \\change{$p^i$}. Modeled as an independent Poisson process with rate $\\lambda_i$ and interval $\\Delta t = 1s$, $r^i = \\exp(-\\lambda_i \\Delta_t)$.\n\n\\subsection{Player Costs}\n\\noindent For all $(t,s, a) \\in \\mathcal{T}\\times[S]\\times[A]$ and congestion distribution $y$~\\eqref{eqn:congestion_distribution}, player $i$'s \\change{cost is given by\n\\[\\textstyle\\ell_{tsa}^i(y, x^i) = \\epsilon x^i_{tsa} - c^i_{tsa} + \\alpha_i f_{ts}(y).\\]}\n\\noindent The player-specific objective $c^i_{tsa}$ is defined as \n\\begin{equation}\nc^i_{t(\\change{v, w}, m) a} = \\begin{cases}\n1 & \\change{(v, w)  = p^i}, \\ m = \\change{1}, \\\\\n1 & \\change{(v, w)  = d^i}, \\ m = \\change{2}, \\\\\n0 & \\text{otherwise.}\n\\end{cases}\n\\end{equation}\nThe congestion function is strictly state-based and is an exponential function given by\n\\begin{equation}\\label{eqn:simulation_congestion_function}\nf_{t\\change{(v, w, m)}}(y) = -\\beta\\exp\\big(\\beta(\\sum_{m' \\in \\change{\\{1,2\\}}}\\sum_{a' \\in [A]}y_{t(\\change{v, w}, m')a'} - 1)\\big),\n\\end{equation}\nwhere $\\alpha_i> 0$ for all $(t,s,a) \\in \\mathcal{T}\\times[S]\\times[A]$. \\change{As opposed to~\\eqref{eqn:individual_costs}, function~\\eqref{eqn:simulation_congestion_function} calculates the congestion in $(v,w, \\cdot)$ using both $(v,w, 1)$'s and $(v, w, 2)$'s congestion level. }\n\\begin{figure}\n    \\centering\n    \\vspace*{0.3cm} \n    \\includegraphics[trim={0 0 0 0.5cm},width=0.85\\columnwidth]{Figure\/polished_figures\/distribution_convergence.png}\n    \\caption{$\\norm{\\cdot}_2$ of player $i$'s state-action distribution over Algorithm~\\ref{alg:frank_wolfe} iterations.}\n    \\label{fig:frank_wolfe}\n\\end{figure}\n\\subsection{Simulation Results}\n\\noindent We simulate the path coordination game using parameters from Table~\\ref{tab:sim_params}. Player $i$'s pick up locations is the $i^{th}$ element of $ \\mathcal{P}= \\{(4, \\change{w^i}) \\ | \\ \\change{w^i} \\in [8, 7, 2]\\}$, \\change{and its drop-off location is the $i^{th}$ element of $\\mathcal{D} = \\{(0, w^i) \\ | \\ w^i \\in [4,5,8]\\}$}. At $t = 0$, players are initialized \\change{at their drop off location. }\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|ccccccccc|}\n\\hline\n$N$  & $q$ & $\\gamma_i$ & $\\lambda_i$ & \\change{$\\alpha_i$} &  $\\Delta t$ & $T$ & \\change{$\\epsilon$} & \\change{$\\beta$} \\\\\n\\hline\n3 & 0.98 & 0.99& 0.5 & \\change{\\{0.5, 1, 1.5\\}} & 1s & \\change{120s} & \\change{1e-3} & \\change{40}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for simulation environment.}\\label{tab:sim_params}\n\\end{table}\n\n\\noindent We run Algorithm~\\ref{alg:frank_wolfe} for $100$ iterations, where line~\\ref{alg:mdp} is solved via value iteration~\\eqref{eqn:value_iteration}. \\change{The two norm of $x^i$ is shown in Figure~\\ref{fig:frank_wolfe} as a function of the algorithm iterations. We see that the state-action densities stabilize in about $20$ steps.} Performance is evaluated by: 1) expected number of collisions, 2) expected packages delivery time, 3) worst package delivery time. The results over $100$ random trials are visualized in Figures~\\ref{fig:collision_results} and~\\ref{fig:waiting_time_results}. \n\\begin{figure}\n    \\centering\n    \\vspace*{0.3cm} \n    \\includegraphics[trim={0 0 0 0.5cm},width=0.8\\columnwidth]{Figure\/polished_figures\/collision_time_line.png}\n    \\caption{Collisions per player as a function of MDP time step $t$.}\n    \\label{fig:collision_results}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{Figure\/polished_figures\/collision_stats.png}\n    \\caption{Average waiting time per package, worst case waiting time per package, and average number of collisions in $T$ for each player.}\n    \\label{fig:waiting_time_results}\n\\end{figure}\n\n\\noindent\\change{\nWe compare the \\emph{jointly optimal congestion-free wait time} computed using Algorithm~\\ref{alg:frank_wolfe}, and compare them to the shortest \\change{wait time} available in the absence of opponents.\n}\nEach path is the number of steps to complete the drop off-pick up-drop off cycle. \\change{Based on players' pick-up and drop-off locations, their shortest \\change{wait time} in the absence of opponents is $16$, $12$, $20$ respectively.} We note that this matches well with the average wait time shown in Figure~\\ref{fig:waiting_time_results}.\n\nWe set the player impact factors as $\\{0.5, 1, 1.5\\}$ as in Table~\\ref{tab:sim_params}. From Figure~\\ref{fig:waiting_time_results}, the impact factors directly correlate with rate of collision players experience. Player $0$ impacts congestion the least and is the least sensitive to congestion. As a result, it encountered more collisions. Player $2$ impacts congestion the most and is the most sensitive to congestion. As a result, it encountered the least collisions. The collision rate is spread out evenly over $\\mathcal{T}$ (Figure~\\ref{fig:collision_results}).\n\\section{Individual Players}\n\nTo analyze how the penalty functions affect the individual player's decision making, we develop their perspective through the value function formulation, make connections to Q-Functions, and show that the penalties directly affect the value functions of individual players.\n\n\\subsection{Dual Formulation: Value Iteration}\nIt is well know that the dual of the linear programming MDP optimization problem is the dynamic programming formulation of the same problem. Here we show that in the non-linear, state dependent case, the dual constraints forms the dynamic programming problem new players joining the game would solve at time $t$.\n\nIndividual players try to optimize their value functions through solving the dual of the primal MDP problem. Given the MDP game primal problem, Eqn. \\ref{MDP game}, the dual can be written as:  \n\\begin{equation}\n\\begin{aligned}\n\\max_{V_{ts}}  & \\quad \\sum_s V_{0s}p_s \\\\\n\\text{s.t } & V_{ts} \\leq \\sum_{s'} P(s'| s, a)V_{t+1,s'} + \\ell_{tsa}(y_{tsa}) \\quad \\\\\n& \\forall \\, s \\in \\mathcal{S}, a\\in \\mathcal{A}, t \\in [T-1] \\\\\n& V_{Ts} \\leq \\ell_{tsa}(y_{tsa}) \\quad \\forall \\, s \\in \\mathcal{S},\\,\\, a\\in \\mathcal{A}\n\\end{aligned} \\label{unconstrained dual}\n\\end{equation}\nFrom this we can derive the value iteration method shown for latency functions $\\ell_{tsa}(y_{tsa}) $. Let the Q function be defined as following: \n\\begin{equation}\\label{q function}\nQ_{tsa}(y_{tsa}) = \\sum_{s'} P(s'| s, a)V_{t+1,s'} + \\ell_{tsa}(y_{tsa}) \\\\\n\\end{equation}\nThe difference $V^*_{ts} -  Q_{tsa}(y^*_{tsa}) = \\mu^*_{tsa} \\geq 0$ is the dual variable of constraint $y_{tsa} \\geq 0 $, and is also the inefficiency of the action $a$. Any action $\\bar{a} $ whose inefficiency is zero is the optimal action at that particular state and time. \n\n\nWhich leads to the value iteration method Alg. \\ref{value iteration}. \n\\begin{algorithm}\n\\caption{Value iteration}\n\\begin{algorithmic}[h]\n\\Require \\(R\\), \\(P\\).\n\\Ensure \\(V^\\star\\), \\(\\pi^\\star\\).\n\\ForEach{\\(s,\\in\\mathcal{S}\\)}\n\\State{\\(V^\\star_{Ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{min}} \\,\\,\\ell_{Tsa}(y_{Tsa})\\)}\n\\State{\\(\\pi^\\star_{Ts}= \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}} \\,\\,\\ell_{Tsa}(y_{Tsa})\\)}\n\\EndFor\n\\For{\\(t=T-1, T-2, \\ldots, 0\\)}\n\\ForEach{\\(s\\in\\mathcal{S}\\)}\n\\State{\\(\\displaystyle V^\\star_{ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{min}}\\,\\, Q_{tsa}(y_{tsa})\\)}\n\\State{\\(\\displaystyle \\pi^\\star_{ts} = \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}}\\,\\, Q_{tsa}(y_{tsa})\\)}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\label{value iteration}\n\\end{algorithm}\n\n\\begin{theorem}\nAssuming existing players have reached nash equilibirum, and the rewards are state dependent $R_{tsa}(y_{tsa}) $, algorithm \\ref{value iteration} with optimal  $y^*_{tsa} $ is the dynamic programming problem that any new player solves entering the game at time $t $.\n\nFurthermore, the best reward that the new player can gain is defined by \n\\[\\min_s V^*_{ts}\\]\n\\end{theorem}\n\n\\begin{proof}\nSince the algorithm depends on current state-action distribution, $y_{tsa}$. At Nash Equilibrium the constraints would be: \n\\[V_{ts} \\leq \\sum_{s'} P(s'| s, a)V_{t+1,s'} + \\ell_{tsa}(y^*_{tsa}) \\]\n\\[V_{Ts} \\leq \\ell_{tsa}(y^*_{tsa}) \\]\nAny player entering at time t would need to maximize its value function according to $V_{ts} $. Since the game is non-atomic, the new player would not affect the state depedent reward functions. Therefore, the new player's optimal policy can be found from algorithm \\ref{value iteration}.\n\n\\end{proof}\n\n\n\nSuppose we impose constraint $y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}\\bar{a}} $ for one state $\\bar{s} $ and all times t, and suppose it was violated during time step $\\bar{t}$. From an individual game player's perspective, the exact penalty function coefficients $k_t(\\bar{s},\\bar{a}) $,  directly offsets the value functions at constrained optimal $y_c^*(s,a)$.  \n\nUsing the exact penalty theorem, we find the minimum coefficient $k_{\\bar{t}\\bar{s}\\bar{a}} \\geq \\tau^*_{\\bar{t}\\bar{s}\\bar{a}} $ to ensure that the optimal solutions are between the penalized problem and the unconstrained problem are the same. However in order to ensure that the next player will also avoid the constraints, additional \"toll\" need to be charged. \n\n\n\\begin{theorem}\nIn addition to $\\tau^*_{\\bar{t}\\bar{s}}$ the dual variable's optimal value corresponding to the constraint $y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}} $ in problem \\eqref{MDP game}. Let \n\\[k_{\\bar{t}\\bar{s}\\bar{a}} > \\max\\{\\tau^*_{\\bar{t}\\bar{s}\\bar{a}}, \\mu_{\\bar{t}\\bar{s}a_{new}}   \\}\\]\n\\[a_{new} = \\text{argmin}_{a \\neq \\bar{a}} Q_{\\bar{t}\\bar{s}a} \\]\nThen we can guarantee that the next player will not choose to violate the constraint $y_{\\bar{t}\\bar{s}\\bar{a}} \\leq d_{\\bar{t}\\bar{s}\\bar{a}} $\n\n\\end{theorem}\n\n\\begin{proof}\nFrom the exact penalty theorem, the solution to the unconstrained problem \\eqref{MDP Game} and the constrained problem \\eqref{cMDP game} share the same value and argument. \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\sum\\limits_{a\\in\\mathcal{A}} \\int_0^{y_{tsa}}\\ell_{tsa}(x)dx\\\\\n\\mbox{s.t.} &\\sum\\limits_{a\\in\\mathcal{A}} y_{t+1, sa} = \\sum\\limits_{s'\\in\\mathcal{S}}\\sum\\limits_{a\\in\\mathcal{A}}P_{s'as}y_{ts'a}, \\quad t\\in[T-1],\\\\\n&\\sum\\limits_{a\\in\\mathcal{A}}y_{0sa}=p_s,\\\\\n& y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}\\bar{a}}, \\\\\n&y_{tsa}\\geq 0,\\quad \\forall s\\in\\mathcal{S}, a\\in\\mathcal{A}, t\\in[T]\n\\end{aligned}\\label{cMDP game}\n\\end{equation}\n\nFor at the optimal solution of this constrained MDP game, the dual value iteration method occurs according to algorithm \\ref{constrained value iteration}. Here we see the value of $k_{t\\bar{s}\\bar{a}} $ directly offsets the Q-functions at the times when the unconstrained solution violated the state constraints.\n\n\\begin{algorithm}\n\\caption{Constrained Value iteration at constrained optimal $y_c^* $}\n\\begin{algorithmic}[h]\n\\Require \\(R\\), \\(P\\).\n\\Ensure \\(V^\\star\\), \\(\\pi^\\star\\).\n\\ForEach{\\(s,\\in\\mathcal{S}\\)}\n\\State{\\(\nV^\\star_{Ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{min}} \\,\\,Q_{Tsa} + k_T(s,a)\\)}\n\\State{\\(\n\\pi^\\star_{T}(s)= \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}} \\,\\,Q_{Tsa} + k_T(s,a)\\)}\n\\EndFor\n\\For{\\(t=T-1, T-2, \\ldots, 0\\)}\n\\ForEach{\\(s\\in\\mathcal{S}\\)}\n\\State{\\(\\displaystyle V^\\star_{t}(s)= \\underset{a\\in\\mathcal{A}}{\\mbox{min}}\\,\\, Q_{tsa} + k_t(s,a)\\)}\n\\State{\\(\\displaystyle \\pi^\\star_{t}(s) = \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}}\\,\\, Q_{tsa}+ k_t(s,a)\\)}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\label{constrained value iteration}\n\\end{algorithm}\n\nSuppose the violated constraint occurs at $\\bar{t}, \\bar{s}, \\bar{a} $. Let $\\tilde{\\mu} $ be the smallest inefficiency of the set of actions available at $\\bar{s} $, excluding optimal action $\\bar{a} $. If $k_{\\bar{t}\\bar{s}\\bar{a}} > \\tilde{\\mu} $, then $\\bar{a} $ is no longer the optimal, and so the player will not choose $\\bar{a} $ as optimal policy. Therefore, the original constraint $y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}\\bar{a}} $ cannot be violated at $\\bar{t} $ if the original solution was feasible. \n\nSince $k_{\\bar{t}\\bar{s}\\bar{a}} > \\tau^*_{\\bar{t}\\bar{s}\\bar{a}} $, then we still satisfy the exact penalty theorem, and so the optimal solution and arguments of the penalized problem still correspond to the constrained problem's solution and arguments. \n\\end{proof}\n\n\\begin{theorem}\nFor a new player entering the game at time $t $, their best expected reward is upperbounded by the optimal value function at time $t $. \n\\[\\max_{s} \\tilde{V}_{ts} \\]\nWhere $\\tilde{V}_{ts} $ is the result of solving algorithm \\ref{constrained value iteration}. Furthermore, the player can achieve this value by starting at state $s = \\text{argmax} V_{ts} $ and follow optimal policy $\\pi_{ts}^* $from algorithm \\ref{constrained value iteration}.\n\\end{theorem}\n\\begin{proof}\nThis follows from the fact that the value functions is equivalent to the expected sum of rewards over time.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\\label{sec:introduction}\nAs autonomous \\change{path} planning algorithms become widely-adapted by aeronautical, robotics, and operational sectors~\\cite{yun2020multi,ota2006multi}, the standard underlying assumption that the operating environment is stationary is no longer sufficient. More likely, autonomous players \\emph{share} the operating environment with other players who may have conflicting objectives. While the possibility for multi-agent conflicts has pushed single-agent \\change{path} planning towards greater emphasis on \\change{robust planning} and collision avoidance, we believe that the overarching goal should be to consider other players' trajectories and achieve optimality with respect to the \\emph{multi-agent dynamics}. \n\nWe focus on the scenario where a group of heterogeneous players collectively perform path planning in response to stochastic demands. We are inspired by fleets of robo-taxis fulfilling ride demands while avoiding congestion in traffic~\\cite{vosooghi2019robo} and warehouse robots retrieving packages under dynamic arrival rates~\\cite{kumar2018development,li2020mechanism} while avoiding collisions. The common feature in these applications is that the players must plan with respect to a forecasted demand distribution rather than a deterministic demand. We assume that the desirable outcome is a competitive equilibrium. Beyond competitive settings, a competitive equilibrium can be used in cooperative settings to ensure that each player achieves identical costs and each demand is \\emph{optimally} fulfilled with respect to other demands, thus ensuring a degree of fairness.\n\nWe propose MDP congestion games as \\change{a theoretical framework for analyzing the resulting path coordination problem}. By leveraging common congestion features in multi-agent \\change{path} planning, our key contribution is \\emph{reducing} the $N$-player coupled MDP problem to a \\emph{single} potential minimization problem. As a result, we can use optimization techniques to analyze the Nash equilibrium as well as apply gradient descent methods to compute it.\n\n\n\n\\textbf{Contributions}. To address the lack of \\change{game-theoretical models for} path \\change{coordination under} MDP dynamics, we propose an MDP congestion game with finite players and heterogeneous \\change{player} costs and dynamics. We define Bellman equation-type conditions for the Nash equilibrium, formulate a \\emph{potential function} and provide a necessary and sufficient condition for its existence. \\change{Under certain assumptions on the player costs}, we show equivalence between the \\change{Nash equilibrium} and the \\change{global solution of} the potential minimization problem, and provide sufficient conditions for a unique Nash equilibrium. Specifically \\change{for multi-player path coordination}, we \\change{formulate a class of cost functions that allows players to have different sensitivities to the total congestion and to find congestion-free paths that optimally achieve their individual objectives}. Finally, we provide a \\change{distributed} algorithm that converges to the Nash equilibrium and give rates of \\change{its} convergence. We demonstrate our model and algorithm on a 2D autonomous warehouse problem where robots retrieve and deliver packages \\change{with stochastic arrival times} while sharing a common navigation space. \n\n \n\n\n\n\n\n\n\n\n\n\\section{Unknown Initial Distribution}\n\n(I believe this is the the most general way to constrain the policy to maintain safety in the MDP.)\n\nI think we want to solve the following problem for all $t' = 0,\\dots,T$.\n\n\\begin{align}\n\\max_{y,p_{t'}} & \\quad \\sum_{s} \\sum_{a} \\Gamma_{t'}(s'|s,a) K_{t'}(s,a)p_{t'}(s) \\\\\n\\text{s.t.} & \\quad \\sum_s \\sum_a y_0(s,a) = 1 \\notag \\\\\n& \\quad \\sum_a y_{t+1}(s',a) = \\sum_{s \\in \\mathcal{S}} \\sum_{a \\in \\mathcal{A}} \\Gamma_t(s'|s,a) y_t(s,a) \\notag \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad t = 0,\\dots,t'-1  \\notag \\\\\n& \\quad \\sum_a y_{t'}(s,a) = p_{t'}(s) \\notag \\\\\n& \\quad 0 \\leq y_t(s,a) \\leq d_t(s,a) \\qquad t=0,\\dots,t' \\notag\n\\end{align}\n\nWe want to get a bound for the maximum amount of mass in state $s^*$.  \n\\begin{align}\n\\min & \\quad \\lambda + \\sum_t\\sum_s\\sum_a d_t(s,a)\\tau_t(s,a) \\\\\n\\text{s.t.} & \\quad \\lambda = V_0(s)+\\tau_0(s,a) \\qquad \\qquad \\qquad \\forall s \\in \\mathcal{S} \\notag \\\\\n& \\quad V_t(s') = \\sum_s \\Gamma_t(s'|s,a)\\Big(V_{t+1}(s)+\\mu_t(s,a)\\Big)+\\tau_t(s,a) \\notag \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\forall a \\in \\mathcal{A} \\notag \\\\\n& \\quad V_{t'} = \\sum_a \\Gamma_{t'}(s^*|s,a)K_{t'}(s,a) + \\tau_{t'}(s,a) \\notag\n\\end{align}\n\n\n\\section{Useless Stuff}\n\nWe now consider the case where the initial distribution is not known a-priori but rather only constrained to be within some set.  We assume $p_0 \\in \\mathcal{P}_0$.  We first consider box capacity constraints on $p_0 \\leq d_0$ and defining the least conservative constant constraints on the policy that keep the set invariant.  Set $\\bar{d}_0 = d_0$. Let $\\bar{d}_t$ be upper bounds on the state distribution at time $t$.  We now derive bounds on the entries of $K_t(s,a)$.  If the constraint at the next time step \n\\begin{align}\nd_{t+1}(s')= \\Gamma_t(s'|s,a)K_t(s,a)\\bar{d}_t(s)\n\\end{align}\n\\begin{align}\nK_t(s,a) = \n\\frac{d_{t+1}(s')}{\\Gamma_t(s'|s,a)\\bar{d}_t(s)}\n\\end{align}\n\\begin{align}\nD_t(s,a) = \\min \\left\\{\\min_{s'}\n\\frac{d_{t+1}(s')}{\\Gamma_t(s'|s,a)\\bar{d}_t(s)},1\n\\right\\}\n\\end{align}\n\\begin{align}\n\\bar{d}_{t+1}(s') = \\max_{y_t} & \\quad \\sum_s \\sum_a \\Gamma_t(s'|s,a)y_t(s,a) \\\\ \n\\text{s.t.}& \\quad \\sum_a y_t(s,a) \\leq \\bar{d}_t(s) \\notag \\\\\n& \\quad \\sum_{s,a} y_t(s,a) = 1 \\notag \\\\\n& \\quad y_t(s,a) \\leq \\text{diag}(y_t\\mathbf{1})D_t(s,a) \\notag\n\\end{align}\n\\section{Heterogeneous MDP Congestion Game}\\label{sec:notation}\nConsider a finite number of players $[N] = \\{1, \\ldots, N\\}$ with a \\emph{shared finite state-action space} given by $([S], [A])$ and common time interval $\\mathcal{T} = \\{0, 1, \\ldots, T\\}$. Each player $i$ has \\emph{individual} time-dependent transition probabilities given by $P^i\\in {\\mathbb{R}}_+^{TSSA}$, where at time $t$, $P^i_{ts'sa}$ is the transition probability from state $s$ to state $s'$ using action $a$ \\change{satisfying} the simplex constraints:\n\\begin{equation}\n    \\textstyle\\sum_{s'} P^i_{t s' s a} = 1, \\ \\forall (i, t, s,  a) \\in [N]\\times \\change{[T]}\\times [S] \\times [A].\n\\end{equation}\n\\textbf{State-action distribution}. \\change{At time $t$, let player $i$'s state be $s^i(t) \\in [S]$ and action taken be $a^i(t) \\in [A]$, then $x^i_{tsa} = \\mathbb{P}[s^i(t) = s, a^i(t) = a]$ is  player $i$'s probability of being in state $s$ taking action $a$ at time $t$.}\n\\change{Player $i$'s state-action probability trajectory over time period $\\mathcal{T}$ is $x^i \\in{\\mathbb{R}}^{(T+1)SA}$, its \\emph{state-action distribution}.} \\change{We use $\\mathcal{X}(P^i, z^i_0)$ to denote the set of all feasible state-action distributions under transition dynamics $P^i$ and initial condition $z^i_0 \\in{\\mathbb{R}}^S_+$, where $z^i_{0s}= \\mathbb{P}[s^i(0) = s]$ is player $i$'s probability of starting in state $s$.}\n\\begin{multline}\\label{eqn:feasible_mdp_flows}\n\\textstyle\\mathcal{X}(P^i, \\change{z^i_{0}}):= \\Big\\{\\change{x^i} \\in {\\mathbb{R}}_+^{(T+1)SA} \\Bigg| \\sum_{a} \\change{x^i_{0sa} = z^i_{0s}}, \\forall s \\in [S], \\Big. \\\\ \n\\textstyle\\Big.\\sum_{s', a} P^i_{tss'a} \\change{x^i_{(t-1)s'a}} = \\sum_{a} \\change{x^i_{tsa}}, \\ \\forall (t,s) \\in  \\change{[T]}\\times [S] \\Big\\}.\n\\end{multline}\nThe \\emph{joint state-action distribution} of all players is given by\n\\begin{equation}\\label{eqn:joint_state_action}\n    x = (x^1, \\ldots, x^N) \\in {\\mathbb{R}}_+^{N(T+1)SA}.\n\\end{equation}\n\\change{We assume that $x$ is fully observable and may denote it as $x = (x^i, x^{-i})$ where $x^{-i} = (x^j)_{j \\in [N]\/\\{i\\}}$.}\n\n\\noindent\\textbf{\\change{Player} costs}. \\change{Similar to stochastic games, the player} costs are continuously differentiable \\emph{functions} of $x$: player $i$ incurs a cost $\\ell^i_{tsa}(x)$ for taking action $a$ at state $s$ and time $t$.\n\\begin{equation}\\label{eqn:player_cost_def}\n    \\ell^i_{tsa}: {\\mathbb{R}}_+^{N(T+1)SA} \\mapsto {\\mathbb{R}}, \\ \\forall (i,t,s,a) \\in [N]\\times \\mathcal{T} \\times [S] \\times [A].\n\\end{equation}\n\\change{Compared to stochastic games where player costs are coupled to the opponent policies,~\\eqref{eqn:player_cost_def} is better suited to model collision events. For example, the expectation of the log-barrier function for players $i$ and $j$ at time $t$ can be modeled as $\\sum_{s, s' \\in [S]} (\\sum_a x^i_{tsa})(\\sum_a x^j_{ts'a})\\log(d_{s,s'})$, in which $d_{s,s'}$ denotes the distance between states $s, s'\\in [S]$. }\n\n\\noindent The \\emph{cost vector} of $(\\ell^1,\\ldots \\ell^N)$~\\eqref{eqn:player_cost_def} is given by $\\xi:{\\mathbb{R}}_+^{N(T+1)SA}\\mapsto {\\mathbb{R}}_+^{N(T+1)SA}$,\n\\begin{equation}\\label{eqn:cost_vector}\n    \\xi(x) = [\\ell^1_{011}(x), \\ell^1_{012}(x), \\ldots, \\ell^N_{TSA}(x)] \\in {\\mathbb{R}}_+^{N(T+1)SA}.\n\\end{equation}\nWe assume that $\\xi$ has a positive definite gradient in $x$. \n\\begin{assumption}\\label{assum:positive_definite} The player cost vector $\\xi$~\\eqref{eqn:cost_vector} satisfies $\\nabla \\xi(x) \\succ 0$ for all $x$~\\eqref{eqn:joint_state_action} where $x^i \\in \\mathcal{X}(P^i, x_0^i), \\ \\forall i \\in [N]$.\n\\end{assumption}\n\\noindent For the class of player costs considered in Section~\\ref{sec:path_coordination_costs}, Assumption~\\ref{assum:positive_definite} implies that the player costs strictly increase as the number of players increases.\n\n\\noindent\\textbf{Coupled MDPs}. Given an initial distribution $z_0^i \\in {\\mathbb{R}}^S_+$ and fixed state-action distributions \\change{$x^{-i}$}~\\eqref{eqn:joint_state_action}, player $i$ solves the following optimization problem under MDP dynamics.\n\\begin{equation}\n\\begin{aligned}\\label{eqn:individual_player_mdp}\n     \\underset{x^i}{\\mbox{min}} & \\change{\\sum_{t, s, a}\\int_0^{x^i_{tsa}}\\ell^i_{tsa}(u^i, x^{-i}) \\partial u^i_{tsa}} \\ \\mbox{s.t. } & x^i \\in \\mathcal{X}(P^i, z^i_{0}).\n\\end{aligned}\n\\end{equation}\n\\change{In~\\eqref{eqn:individual_player_mdp}, we note that each integral is taken over $u^i_{tsa}$, the $(t, s, a)^{th}$ element of $u^i$.} \nWhen $\\ell^i_{tsa}(x)$ is constant for all $(t,s,a) \\in \\mathcal{T} \\times [S]\\times [A]$, player $i$ solves a standard \\emph{linear program} MDP.\n\n\\noindent\\textbf{Dynamic programming}. At a joint state-action distribution $x$~\\eqref{eqn:joint_state_action}, player $i$'s cost-to-go in~\\eqref{eqn:individual_player_mdp} can be recursively defined via Q-value functions~\\cite{puterman2014markov} as\n\\begin{multline}\n   Q^i_{Tsa}(x) := \\ell^i_{Tsa}(x), \\label{eqn:q_value} \\\\ \n   Q^i_{(t-1)sa}(x) := \\ell^i_{(t-1)sa}(x) + \\textstyle\\sum_{s'} P^i_{ts'sa}\\underset{a'}{\\min}\\, Q^i_{t,s'a'}(x),\\\\\n\\forall \\ t \\in [T]\n\\end{multline}\n\\noindent\\change{\nThe optimal solution of~\\eqref{eqn:individual_player_mdp} can be stated using~\\eqref{eqn:q_value}.}\n\n\\begin{theorem}\n\\change{Under Assumption~\\ref{assum:positive_definite}, ${x}^i$~\\eqref{eqn:feasible_mdp_flows} uniquely minimizes~\\eqref{eqn:individual_player_mdp} with respect to the state-action distribution $x^{-i}$ if and only if its associated $Q^i({x}^i, x^{-i})$~\\eqref{eqn:q_value} satisfies\n\\begin{equation}\\label{eqn:individual_optimality}\n    {x}^i_{tsa} > 0 \\Rightarrow \\textstyle Q^i_{tsa}({x}^i, x^{-i})= \\min_{a'} Q^i_{tsa'}({x}^i, x^{-i}),\n\\end{equation}\nfor all $(t, s, a) \\in \\mathcal{T}\\times[S]\\times[A]$.} I.e., ${x}^i$ is optimal for~\\eqref{eqn:individual_player_mdp} if and only if every action played with nonzero probability achieves the minimum cost-to-go~\\eqref{eqn:q_value} among available actions. \n\\end{theorem}\n\\begin{proof}\n\\change{\nLet $F({x}^i, x^{-i}) = \\sum_{t, s, a}\\int_0^{{x}^i_{tsa}}\\ell^i_{tsa}(u^i, x^{-i}) \\partial u^i_{tsa}$, then $\\partial F({x}^i, x^{-i}) \/\\partial x^i = \\ell({x}^i, x^{-i})$. We then apply Proposition A\\ref{prop:dp_to_minimizer} to~\\eqref{eqn:individual_player_mdp} and the theorem's results follow directly.}\n\\end{proof}\n\\noindent\\change{\nWhen all players jointly achieve the optimal cost-to-go~\\eqref{eqn:individual_optimality}, a stable equilibrium for unilateral optimality is achieved.\n}\n\\begin{definition}[Nash Equilibrium] \\label{def:wardrop} \nThe joint state-action distribution $\\hat{x}= [\\hat{x}^1, \\ldots, \\hat{x}^N]$~\\eqref{eqn:joint_state_action}\nis a \\emph{Nash equilibrium} if \\change{$\\big(\\hat{x}^i, Q^i(\\hat{x})\\big)$} satisfies~\\eqref{eqn:individual_optimality} for all $i \\in [N]$.\n\\end{definition}\n\\subsection{Potential optimization form}\n\\noindent We are interested in MDP congestion games that can be reduced from the coupled MDPs~\\eqref{eqn:individual_player_mdp} to a single minimization problem given by\n\\begin{equation}\n    \\begin{aligned}\\label{eqn:convex_opt_eqn}\n        \\min_{x^1, \\ldots, x^N}  F(x), \\ \n    \\text{s.t. } x^i \\in \\mathcal{X}(P^i, z^i_0) , \\ \\forall \\ i \\in [N],\n   \n    \\end{aligned}\n\\end{equation}\nwhere $F$ is the \\emph{potential function} of the corresponding game.\n\n\\begin{definition}[Potential Function]\\label{def:potential}\nWe say an MDP congestion game with {player costs} $\\{\\ell^{i}\\}_{i \\in [N]}$~\\eqref{eqn:player_cost_def} has a \\emph{potential function} $F: {\\mathbb{R}}^{N(T+1) S A}\\mapsto {\\mathbb{R}}$ if $F$ satisfies\n\\begin{equation}\\label{eqn:potential_first_order}\n        \\frac{\\partial F(x)}{\\partial x^i_{tsa}} = \\ell^i_{tsa}(x), \\ \\forall \\ (i, t, s, a) \\in [N]\\times\\mathcal{T}\\times[S]\\times [A].\n    \\end{equation}\n\\end{definition}\n\\noindent The following assumption on $\\{\\ell^{i}\\}_{i \\in [N]}$ is necessary and sufficient for the existence of $F$~\\cite[Eqn.2.44]{patriksson2015traffic}.\n\\begin{assumption}\\label{ass:potential_existence}\nFor all $(i, t, s, a) , (i', t', s', a') \\in [N]\\times\\mathcal{T}\\times[S]\\times [A]$, the player costs $\\{\\ell^{i}\\}_{i \\in [N]}$ satisfy\n    \\begin{equation}\\label{eqn:potential_second_order}\n   \n        \\frac{\\partial \\ell^i_{tsa}(x)}{\\partial x^{i'}_{t's'a'}} = \\frac{\\partial \\ell^{i'}_{t's'a'}(x)}{\\partial x^i_{tsa}}.\n    \\end{equation}\n\\end{assumption}\n\\begin{remark}\nAssumption~\\ref{ass:potential_existence} \\change{is equivalent to} $F$ \\change{being} conservative: $\\forall \\ x_1, x_2 \\in \\{x^i_{tsa}\\ | \\ (i, t, s, a) \\in [N] \\times \\mathcal{T} \\times [S] \\times [A]\\}$,   \n\\begin{equation}\\label{eqn:potential_conservative}\n    {\\partial^2 F(x)}\/{\\partial x_1\\partial x_2} = {\\partial^2 F(x)}\/{\\partial x_2\\partial x_1}.\n\\end{equation}\nIn other words, the Jacobian of $\\xi$~\\eqref{eqn:cost_vector}, ${\\partial\\xi(x)}\/{\\partial x}$, is symmetrical.\n\\end{remark}\n\\noindent Verifying the existence of $F$~\\eqref{eqn:potential_first_order} is non-trivial. However, if $F$ exists, the \\change{solution} of~\\eqref{eqn:convex_opt_eqn} is the Nash equilibrium~\\cite{calderone_finite}.  \n\\begin{theorem}\\label{thm:kkt_pts} \n\\change{If the player costs $\\{\\ell^{i}\\}_{i \\in [N]}$~\\eqref{eqn:player_cost_def} satisfy Assumption~\\ref{assum:positive_definite},  \n\\begin{enumerate}\n    \\item the potential function (Definition~\\ref{def:potential}) exists,\n    \\item $\\hat{x}$~\\eqref{eqn:joint_state_action} is the global optimal solution of~\\eqref{eqn:convex_opt_eqn} if and only if $\\hat{x}$ is a Nash equilibrium (Definition~\\ref{def:wardrop}). \n\\end{enumerate}\n}\n\\end{theorem}\n\\begin{proof}\n\\change{We prove statement $1$ by showing that Assumption~\\ref{assum:positive_definite} implies Assumption~\\ref{ass:potential_existence}: if $\\nabla \\xi(x)\\succ 0$ for all feasible joint state-action distributions $x$~\\eqref{eqn:joint_state_action}, then $\\nabla \\xi(x)$ is symmetrical and satisfies~\\eqref{eqn:potential_second_order}.}\n\\change{Next, we show the forward direction of the statement $2$. If $(\\hat{x}^1,\\ldots \\hat{x}^N)$ minimizes~\\eqref{eqn:convex_opt_eqn}, then for each $i\\in [N]$, $\\hat{x}^i$ minimizes~\\eqref{eqn:mdp_variant} at $\\hat{x}^{-i}$. From Proposition A\\ref{prop:dp_to_minimizer}, $\\hat{x}^i$ satisfies~\\eqref{eqn:individual_optimality} for all $i\\in [N]$, therefore $\\hat{x}$ is a Nash equilibrium. To show the reverse direction of $2$, if~\\eqref{eqn:individual_optimality} is satisfied for all $i \\in [N]$, $\\hat{x}^i$ is coordinate-wise optimal for coordinate $i$ (Proposition A\\ref{prop:dp_to_minimizer}). Under Assumption~\\ref{assum:positive_definite},~\\eqref{eqn:convex_opt_eqn} has a strictly convex differentiable objective with separable convex constraints $\\mathcal{X}(P^i, z^i_0)$---each $x^i$ is constrained independently of $x^j$, $\\forall j \\in[N]\/\\{i\\}$, then the jointly coordinate-wise optimal $\\hat{x}$ is the global optimal solution of~\\eqref{eqn:convex_opt_eqn}~\\cite[Thm 4.1]{tseng2001convergence}. }\n\\end{proof}\n\\subsection{Path Coordination as an MDP Congestion Game}\\label{sec:path_coordination_costs}\n\\noindent We now model the path coordination problem as an MDP congestion game and demonstrate how players can achieve individual objectives while avoiding each other.\n\n\\noindent To reflect the congestion level of each state-action, we first define a \\textbf{congestion distribution} as the weighted sum of individual state-action distributions.\n\\begin{equation}\\label{eqn:congestion_distribution}\n    \\textstyle y := \\sum_{i\\in[N]} \\alpha_ix^i \\in {\\mathbb{R}}^{(T+1)SA}, \\ \\alpha_i > 0, \\  \\forall i \\in [N],\n\\end{equation}\n\\noindent where $\\alpha_i$ is player $i$'s \\emph{impact factor}. If all players contribute to congestion equally, $\\alpha_i  = 1 \\ \\forall i \\in [N]$.  \n\n\\noindent\\textbf{Player costs}. \\change{We derive a class of player costs that satisfy Assumption~\\ref{assum:positive_definite}, incorporate congestion-based penalties, and enable players to pursue individual objectives. For all $(i,t,s,a) \\in [N]\\times\\mathcal{T}\\times[S]\\times[A]$, the player cost is given by }\n\\begin{equation}\\label{eqn:individual_costs}\n  \\textstyle\\ell^i_{tsa}(y, x^i) = \\alpha_i f_{ts}\\big(\\sum_{a'} y_{tsa'}\\big) + \\alpha_i g_{tsa}\\big(y_{tsa}\\big) + h^i_{tsa}(x^i_{tsa}),\n\\end{equation}\nwhere \\change{$\\alpha_i$ is the same as in~\\eqref{eqn:congestion_distribution}, $f_{ts}:{\\mathbb{R}} \\mapsto{\\mathbb{R}}$ is the state-dependent congestion and takes the congestion level of $(t,s)$ as input, $g_{tsa}:{\\mathbb{R}} \\mapsto{\\mathbb{R}}$ is the state-action-dependent congestion and takes the congestion level of $(t,s,a)$ as input, and $h^i_{tsa}:{\\mathbb{R}}\\mapsto{\\mathbb{R}}$ is the player-specific objective and takes player $i$'s probability of being in $(t,s,a)$ as input. Player-specific objectives such as obstacle avoidance and target reachability can be incorporated as constant offsets in $h^i$}. \n\\begin{remark}[Effect of $\\alpha_i$]\nThe impact factor $\\alpha_i$ scales player $i$'s \\change{relative impact on the total congestion and the total congestion's impact on player $i$. When $\\alpha_i < \\alpha_j$, player $i$ impacts congestion less and cares about the congestion less than player $j$. When $\\alpha_i > \\alpha_j$, player $i$ impacts congestion more and cares about the congestion more than player $j$. }\n\\end{remark}\n\\noindent The potential function~\\eqref{eqn:potential_first_order} of the game with costs~\\eqref{eqn:individual_costs} is\n\\begin{equation}\n \\begin{aligned}\\label{eqn:state_state_action_potential}\n    F(x)= & \n\\textstyle   \\sum_{t,s} \\int_0^{\\sum_{a'}y_{tsa'}}f_{ts}(u) \\partial u + \\sum_{t,s,a} \\int_0^{y_{tsa}}g_{tsa}(u) \\partial u   \\\\\n    & \\textstyle \n    + \\sum_{i, t, s, a} \\int_{0}^{x^i_{tsa}} h^i_{tsa} (u)\\partial u.\n\\end{aligned}   \n\\end{equation}\n\\begin{remark}\n\\change{Congestion costs $f$ and $g$ must be identical for all players in order for a potential (Definition~\\ref{def:potential}) to exist.}\n\\end{remark}\n\\begin{example}[Road-sharing Vehicles]\n\\change{\nConsider a sedan (player $1$, $\\alpha_1=1$) and a trailer (player $2$, $\\alpha_2=2$) sharing a road network modeled by $[S]\\times [A]$. Player $i$ wants to reach state $s_i \\in [S]$. The player-specific objective is $h^i_{tsa}(x^i_{tsa}) = -\\mathbb{1}[s = s_i] + \\epsilon_i x^i_{tsa}$, where $\\mathbb{1}[w]$ is $1$ when $w$ is true and $0$ otherwise. The term $\\epsilon_i x^i_{tsa}$ where $\\epsilon_i > 0$ encourages player $i$ to randomize its policy over all optimal actions. Players experience state-based congestion as $f_{ts}(w) =\\exp(w)$. The player cost~\\eqref{eqn:individual_costs} is $\\ell^i_{tsa}(y, x^i) = \\alpha_i\\exp(\\sum_{a'} y_{tsa'}) + \\epsilon_ix^i_{tsa} -\\mathbb{1}[s = s_i]$.  }\n\\end{example}\n\n\n\n\n\n\n\n\n\n\\begin{corollary}\\label{cor:unique_ne}\nPlayer costs of form~\\eqref{eqn:individual_costs} \\change{satisfy Assumption~\\ref{assum:positive_definite} if  $h^i_{tsa}(\\cdot)$ is strictly increasing and $f_{ts}(\\cdot)$, $g_{tsa}(\\cdot)$ are non-decreasing} $\\forall (i,t,s,a)\\in [N]\\times\\mathcal{T}\\times[S]\\times[A]$.\n\\end{corollary}\n\\begin{proof}\n \\change{Let $I_Z$ be an identity matrix of size $Z\\times Z$, $\\mathbf 1_Z$ be a ones vector of size $Z\\times 1$, $\\vec{\\alpha} = [\\alpha_1,\\ldots,\\alpha_N] \\in {\\mathbb{R}}^{N\\times 1}$, $h(x) = [h^1(x),\\ldots,h^N(x)] \\in {\\mathbb{R}}^{N(T+1)SA}$, and $\\otimes$ be a kronecker product. We define the matrices $M = \\vec{\\alpha} \\otimes I_{(T+1)SA}$ and $J = (I_{(T+1)S}\\otimes \\mathbf 1_A^\\top)M$, and verify that $Mx = y$, $[Jx]_{ts} = \\sum_{a'}y_{tsa'}$ $\\forall (t,s) \\in \\mathcal{T}\\times[S]$, and $\\xi(x) = J^\\top f(Jx) + M^\\top g(Mx) + h(x)$. Let $w = Jx$, we can take $\\xi$'s gradient as\n$\n\\textstyle \\nabla \\xi(x) = \n J^\\top \\nabla f(w) J + \n M^\\top \\nabla g(y) M + \\nabla h(x).\n $\nUnder Corollary assumptions,\n$\\nabla f(w)$ and $\\nabla g(y)$ are non-negative diagonal matrices and $\\nabla h(x) $ is a strictly positive diagonal matrix. Therefore}, $\\nabla \\xi(x) \\succ 0$. \n\\end{proof}\n\\begin{remark}\nCorollary~\\ref{cor:unique_ne} implies that a strictly increasing $h^i$ is crucial to ensuring a unique Nash equilibrium. \nTherefore, $h^i$ can be interpreted as a regularization term.\n\\end{remark}\n\\subsection{Frank-Wolfe Learning Dynamics}\n\\noindent We find the Nash equilibrium of MDP congestion games by {leveraging} single-agent dynamic programming. \n\\begin{algorithm}[ht!]\n\\caption{Frank-Wolfe with dynamic programming}\n\\begin{algorithmic}[1]\n\\Require \\(\\{\\ell^i\\}_{i\\in[N]}\\), \\(\\{P^i\\}_{i\\in[N]}\\), \\(\\{z^i_0\\}_{i\\in[N]}\\), \\(N\\), \\([S], [A], \\mathcal{T}\\).\n\\Ensure \\(\\change{\\{\\hat{x}^{i}_{tsa}\\}_{t \\in \\mathcal{T}, s\\in[S], a \\in [A]}}\\).\n\\State{\\( x^{i0}\\in\\mathcal{X}(P^i, z^i_0) \\in {\\mathbb{R}}^{(T+1)SA}, \\quad \\forall \\ i \\in [N]. \\)}\n\n\\For{\\(k = 1, 2, \\ldots, \\)}\n\\For{\\(i = 1,\\ldots,N\\)}\n    \\State{\\(\\change{C^{ik}} = \\ell^i([x^{1k},\\ldots, x^{Nk}])\\)}\\label{alg:cost_retrieval}\n\t\\State{\\(\\pi^i\\) = \\text{MDP}(\\change{\\(C^{ik}\\)}, \\(P^i\\), \\([S]\\), \\([A]\\), \\(T\\), \\change{\\(z^i_0\\)})}\\label{alg:mdp}\n\t\\State{\\(b^{ik} = \\) \\Call{RetrieveDensity}{\\(P\\), \\(z^i_0\\), \\(\\pi^i\\)}}\\Comment{Alg.~\\ref{alg:density}}\n\t\\State{\\(x^{i(k+1)}= (1 -  \\frac{2}{k+1})x^{ik} +  \\frac{2}{k+1} b^{ik}\\)}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\label{alg:frank_wolfe}\n\\end{algorithm}\n\n\\noindent In Algorithm~\\ref{alg:frank_wolfe}, each player can access an \\emph{oracle} \\change{that} returns the cost for a given joint state-action distribution. \n\\change{In line~\\ref{alg:mdp}, $\\pi^i \\in [A]^{(T+1)S}$ is any deterministic policy that solves the finite time MDP with cost $C^{ik}$, transition probability $P^i$, and initial distribution $z^i_0$. We use value iteration to recursively find $\\pi^i$ as}\n\\change{\\begin{equation}\n\\begin{aligned}\\label{eqn:value_iteration}\n    V^i_{Ts} & = \\textstyle\\min_{a}C^{ik}_{Tsa}, \\ \\pi^i_{Ts} \\in  \\mathop{\\rm argmin}_{a}C^{ik}_{Tsa}, \\\\\n    V^i_{(t-1)s} &  \\textstyle = \\min_{a} C^{ik}_{(t-1)sa} + {\\sum_{s'}}P^i_{ts'sa}V^i_{ts'} \\ \\forall t \\in [T] \\\\\n    \\pi^i_{(t-1)s} & \\textstyle \\in \\mathop{\\rm argmin}_{a} C^{ik}_{(t-1)sa} + {\\sum_{s'}}P^i_{ts'sa}V^i_{ts'} \\ \\forall t \\in [T]\n\\end{aligned}\n\\end{equation}}\n\\noindent \\change{Algorithm~\\ref{alg:frank_wolfe}} then retrieves the corresponding state-action density $b^{ik}$ via Algorithm~\\ref{alg:density} and \\change{combines it} with the current state-action density $x^{ik}$ to derive the next joint state-action density. All steps within lines 4 to 7 are parallelizable.\n\n\\begin{algorithm}[ht!]\n\\caption{Retrieving \\change{state-action distribution} from \\change{$\\pi$} }\n\\begin{algorithmic}[1]\n\\Require \\(P\\), \\(z\\), \\(\\pi\\).\n\\Ensure \\(\\{d_{tsa}\\}_{t \\in\\mathcal{T}, s\\in[S], a\\in[A]}\\)\n\\State{\\(d_{0s\\pi_{0s}} = z_s, \\ \\forall s \\in [S]\\)}\n\\For{\\(t=1, \\ldots, T\\)}\n\t  \n\t  \n\t  \n\t\t    \\State{\\(d_{ts(\\pi_{ts})} = \\textstyle\\sum_a\\sum_{s'} P_{tss'a}d_{(t-1)s'a}, \\ \\forall\\ s \\in [S]\\)}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:density}\n\\end{algorithm}\n\n\\begin{theorem}\nUnder Assumption~\\ref{assum:positive_definite}, \nAlgorithm~\\ref{alg:frank_wolfe} converges towards the Nash equilibrium \\change{$\\hat{x} = (\\hat{x}^1,\\ldots, \\hat{x}^N)$} as \n\\begin{equation}\\label{eqn:alg_convergence}\n   \\textstyle \\frac{\\alpha}{2}\\sum_{i\\in[N]} \\norm{x^{ik} - \\change{\\hat{x}^{i}}}^2_2 \\leq \\frac{2C_F}{k+2} \n\\end{equation}\nwhere $C_F$ is the potential function $F$'s~\\eqref{eqn:potential_first_order} \\emph{curvature constant} given by\n\\[ C_F = \\underset{\\substack{x^i, s^i \\in \\mathcal{X}(P^i, z^i_0) \\\\  \\gamma \\in [0, 1]\\\\\nw^i  = x^i + \\gamma(s^i - x^i)}}{\\sup} \\frac{2}{\\gamma^2} \\Big(F(\\change{s}) - F(x) - \\sum_{i \\in [N]}(x^i - w^i)^\\top \\ell^i(x)\\change{\\Big)}.\\]\n\\end{theorem}\n\\begin{proof}\nAlgorithm~\\ref{alg:frank_wolfe} is a straight-forward implementation of~\\cite[Alg.2]{jaggi2013revisiting}. \n    \\change{From Assumption~\\ref{assum:positive_definite},  $\\nabla \\xi(\\hat{x}) \\succ 0$. Therefore}, the potential function $F$ is \\change{strongly} convex and satisfies $ \\frac{\\alpha}{2}\\sum_{i\\in[N]} \\norm{x^{ik} - \\change{\\hat{x}^{i}}}^2_2 \\leq F(x^{k}) - F(\\hat{x})$. Equation~\\eqref{eqn:alg_convergence} then follows directly from~\\cite[Thm.1]{jaggi2013revisiting}.\n\\end{proof}\n\\begin{remark}[Scalability]\nAlgorithm~\\ref{alg:frank_wolfe} has linear complexity in the number of players.\n\\end{remark}\n\\subsection{Population Perspective: Numerical Method}\\label{sec:dualFormulation}\n\nAfter the social planner has offered incentives, the population plays the Wardrop equilibrium defined by modified rewards~\\eqref{eq:generalReward}; this equilibrium can be computed using the Frank Wolfe (FW) method~\\cite{freund2016new}, given in Algorithm~\\ref{alg:MSA}, with known optimal variables $\\{\\tau_i^{\\star}\\}$. \n\nFW is a numerical method for convex optimization problems with continuously differentiable objectives and compact feasible sets \\cite{powell1982convergence}, including routing games.   One advantage of this learning paradigm is that the population does not need to know the function $r(\\cdot)$. Instead, they simply react to the realized rewards of previous game at each iteration. It also provides an interpretation for how a Wardrop equilibrium might be asymptotically reached by agents in MDPCG in an online fashion.\n\nAssume that we have a repeated game play, where players execute a fixed strategy determined at the start of each game. At the end of each game $k$, rewards of game $k$ based on $y^k_{tsa}$ are revealed to all players. FW models the population as having two sub-types: \\emph{adventurous} and \\emph{conservative}. Upon receiving reward information $\\ell_{tsa}(y^{k}_{tsa})$, the adventurous population decides to change its strategy while the conservative population does not. To determine its new strategy, the adventurous population uses value iteration on the latest reward information---i.e.~Algorithm \\ref{alg:valueIteration}---to compute a new optimal policy. Their resultant density trajectory is then computed using Algorithm \\ref{alg:policy iteration}. The step size at each iteration is equivalent to the fraction of total population who switches strategy.  \nThe stopping criteria for the FW algorithm is determined by the Wardrop equilibrium notion---that is, as the population iteratively gets closer to an optimal strategy, the marginal increase in potential decreases to zero.\n\\begin{algorithm}\n\\caption{Value Iteration Method}\n\\begin{algorithmic}[h]\n\\Require \\(r\\), \\(P\\).\n\\Ensure \\( \\{\\pi^{\\star}_{ts}\\}_{ t \\in [T], \\, s\\in \\mathcal{S}} \\)\n\\For{\\(t=T, \\ldots, 1\\)}\n\t\t\\State{\\(\\displaystyle V_{ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{max}}\\, Q_{tsa}\\), \\(\\displaystyle \\pi^{\\star}_{ts} = \\underset{a\\in\\mathcal{A}}{\\mbox{argmax}}\\, Q_{tsa},\\ \\forall \\ s \\in \\mathcal{S}\\)} \\Comment{Eqn.~\\eqref{eq:qvalue}}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:valueIteration}\n\\end{algorithm}\n\\begin{algorithm}\n\\caption{Retrieving density trajectory from a policy }\n\\begin{algorithmic}[h]\n\\Require \\(P\\), \\(p\\), \\(\\pi\\).\n\\Ensure \\(\\{d_{tsa}\\}_{t \\in[T], s\\in\\mathcal{S}, a\\in\\mathcal{A}}\\)\n\\State{\\(d_{tsa} = 0, \\ \\forall \\ t \\in [T], s \\in \\mathcal{S}, a \\in \\mathcal{A}\\)}\n\\State{\\(d_{1s\\pi_{1s}} = p_s, \\ \\forall s \\in \\mathcal{S}\\)}\n\\For{\\(t=2, \\ldots, T\\)}\n\t  \n\t  \n\t  \n\t\t    \\State{\\(d_{ts(\\pi_{ts})} = \\sum\\limits_{a \\in \\mathcal{A}}\\sum\\limits_{s' \\in \\mathcal{S}} P_{t-1,ss'a}d_{t-1,s'a}, \\ \\forall\\ s \\in \\mathcal{S} \\)}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:policy iteration}\n\\end{algorithm}\n\\begin{algorithm}\n\\caption{Frank Wolfe Method with Value Iteration}\n\\begin{algorithmic}[h]\n\\Require \\(\\bar{\\ell}\\), \\(P\\), \\(p_s\\), \\(N\\), \\(\\epsilon\\).\n\\Ensure \\(\\{y^{\\star}_{tsa}\\}_{t \\in [T], s\\in\\mathcal{S}, a \\in \\mathcal{A}}\\).\n\\State{\\( y^0 = 0 \\in {\\mathbb{R}}^{T \\times|\\mathcal{S}| \\times |\\mathcal{A}|} \\)}\n\\For{\\(k = 1, 2, \\ldots, N\\)}\n    \\State{\\(c^k_{tsa} = \\bar{\\ell}_{tsa}(y^k)\\), \\(\\quad \\forall \\,\\, t \\in [T], s \\in \\mathcal{S}, a \\in \\mathcal{A}\\)}\n\t\\State{\\(\\pi_{ts} = \\) \\Call{ValueIteration}{\\(c^k\\), \\(P\\)}}\n\t\\Comment{Alg.~\\ref{alg:valueIteration}}\n\t\\State{\\(d^k\\) = \\Call{RetrieveDensity}{\\(P, p_s, \\pi_{ts}\\)}}\\Comment{Alg.~\\ref{alg:policy iteration}}\n\n\t\\State{\\( y^k = (1 -  \\frac{2}{k+1})y^{k-1} +  \\frac{2}{k+1} d^k\\)}\n\t\\State{Stop if }\n\t\\State{\\(\\quad \\sum\\limits_{t\\in[T]} \\sum\\limits_{s\\in\\mathcal{S}} \\sum\\limits_{a\\in\\mathcal{A}}\\Big(c^k_{tsa} - c^{k-1}_{tsa}\\Big)^2 \\leq \\epsilon\\)}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:MSA}\n\\end{algorithm}\n\nIn contrast to implementations of FW in routing game literature, Algorithm \\ref{alg:MSA}'s descent direction is determined by solving an MDP \\cite[Section~4.5]{puterman2014markov} as opposed to a shortest path problem from origin to destination \\cite[Sec.4.1.3]{patriksson2015traffic}.\nAlgorithm~\\ref{alg:MSA} is guaranteed to converge to a Wardrop equilibirum if the predetermined step sizes decrease to zero as a harmonic series \\cite{freund2016new}---e.g., $\\frac{2}{k+1}$.\nFW with predetermined step sizes has been shown to have sub-linear worst case convergence in routing games \\cite{powell1982convergence}. \nOn the other hand, replacing fixed step sizes with optimal step sizes found by a line search method leads to a much better convergence rate.\n \n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Problem Formulation}\\label{problemFormulation}\n\n\\section{Problem Formulation}\\label{problemFormulation}\nConsider a finite horizon, Markov Decision Process (MDP) congestion game, where a group of players individually solves an MDP to maximize expected self gain while competing for global resources. We assume that all players have the same set of possible actions and stochastic transitions. This type of finite horizon, Markov Decision Process congestion game (MDPCG) is formulated as a convex optimization problem in \\cite{MDPCongestionGame}. \n\nIn this paper, we develop a dual interpretation for the role that state constraints play in MDPCG: through a social planner's perspective and individual players' perspective. Players will attempt to maximize their individual objectives by finding an optimal sequence of actions, while a social planner strives for game-level objectives, such as maximizing total gain or enforcing state constraints. To achieve these objectives, the planner cannot impose explicit constraints, but instead can adjust individual objectives to incentivize\/discourage certain actions. A toll\/reward method is developed for implicitly realizing global state dependent constraints .\n\nThe rest of the paper is organized as follows: in Section \\ref{notation}, we introduce the convex optimization model of state-constrained MDPCG. Section \\ref{convergence} connects the constrained optimal solution to Wardrop equilibrium with respect to a generalized cost function. In Section \\ref{dualFormulation}, we perform simulations of repeated game play using a fixed step-sized Frank Wolfe Method. The simulations show that when the cost functions are adjusted for tolls, players indeed converge at the optimal strategy that does not violate corresponding state constraints.\n\n\n\\section{Related work}\n\\label{sec:related work}\n\n\n\nAn MDP congestion game~\\cite{calderone2017infinite} is a stochastic population game and is related to potential mean field games \\cite{lasry2007mean,gueant2011infinity} in the discrete time and state-action space \\cite{gomes2010discrete} and mean field games on graphs \\cite{gueant2015existence}. \n\\change{In this paper, we extend our previous framework from continuous populations of identical MDP decision makers~\\cite{calderone2017infinite} to a finite number of heterogenous MDP decision makers.} In the continuous population case, MDP congestion games have been analyzed for constraint satisfaction in~\\cite{li2019tolling} and sensitivity to \\change{hyper}parameters in~\\cite{li2019sensitivity}. \n\nModel-based multi-agent \\change{path} planning is typically solved via graph-based searches~\\cite{cohen2019optimal} and mixed integer linear programming~\\cite{chen2020scalable}. Recently, reinforcement learning has been introduced as a viable method for solving multi-agent \\change{path} planning~\\cite{semnani2020multi,yun2020multi}. In most scenarios, the \\change{path} planning problem is modeled as an MDP~\\cite{bayerlein2021multi,lo2021towards}. In particular, \\cite{lo2021towards} adopts a stochastic game model for human-robot collision avoidance, but focuses more on algorithm development rather than game structure analysis. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reward Uncertainty}\nHere we introduce slightly different notation: \n\\[y_{ts} = \\begin{pmatrix}\ny_{tsa_1} \\\\ y_{tsa_2}\\\\\\vdots\\\\y_{tsa_m}\\\\\n\\end{pmatrix} \\in {\\mathbb{R}}^{|\\mathcal{A}|} \\]\n\\[\n\\ell_{ts}(y_{ts}) = \\begin{pmatrix}\n\\ell_{tsa_1}(y_{tsa_1}) \\\\ \\ell_{tsa_2}(y_{tsa_2})\\\\\\vdots\\\\\\ell_{tsa_m}(y_{tsa_m})\\\\\n\\end{pmatrix} \\in {\\mathbb{R}}^{|\\mathcal{A}|}\n\\]\nThe optimization problem can be rewritten in the new vector variable form: \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\int_0^{y_{ts}}\\ell_{ts}(x)dx\\\\\n\\mbox{s.t.} & \\mathbf 1^T y_{t+1, s} = \\sum\\limits_{s'\\in\\mathcal{S}}P_{s\\cdot s'}^Ty_{ts'}, \\quad t\\in[T-1],\\\\\n&\\mathbf 1^Ty_{0s}=p_s,\\\\\n&y_{ts}\\geq 0,\\quad \\forall s\\in\\mathcal{S}, t\\in[T]\n\\end{aligned}\\label{mdp game vector}\n\\end{equation}\n\n\\subsection{Uncertainty Model}\nFor each individual solving the problem, their model of the reward may be entirely experience based and therefore prone to uncertainty. \n\nHere, we model their uncertainty as the following: for $(t,s,a)$, the reward can be anywhere within the following ellipse, which is also state dependent in its mean value\n\\begin{equation}\n\\begin{aligned}\n\\ell_{ts}(x) &\\in \\mathcal{E}_{ts} \\quad \\forall \\,\\, x \\in [0,1] \\\\\n& \\in \\{R_{ts}x + Q_{ts}u_{ts}\\, | \\quad \\norm{u}_2 \\leq 1\\} \\\\\n\\end{aligned}\n\\end{equation}\nWhere $R_{ts} \\in {\\mathbb{R}}^{|S| \\times |S| } $ is a time and state dependent matrix. \n\nSuppose we are considering worstcase scenario, the objective can be rewritten as: \n\\begin{equation}\n\\begin{aligned}\n&\\sup\\{\\mathbf 1^T\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\int_0^{y_{ts}}\\ell_{ts}(x)dx\\} \\\\\n= &\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}  \\mathbf 1^T\\int_0^{y_{ts}}R_{ts}xdx + \\sup\\{\\mathbf 1^T\\int_0^{y_{ts}}Q_{ts}u dx | \\norm{u}_2 \\leq 1\\} \\\\\n\\leq &\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 + \\sup\\{\\norm{u_{ts}^TQ_{ts}^Ty_{ts}}_2| \\norm{u}_2 \\leq 1\\} \\\\\n\\leq &\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 +\\norm{Q_{ts}^Ty_{ts}}_2 \\\\\n\\end{aligned}\n\\end{equation} \nThen the optimization problem becomes: \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 +\\norm{Q_{ts}^Ty_{ts}}_2 \\\\\n\\mbox{s.t.} & \\mathbf 1^T y_{t+1, s} = \\sum\\limits_{s'\\in\\mathcal{S}}P_{s\\cdot s'}^Ty_{ts'}, \\quad t\\in[T-1],\\\\\n&\\mathbf 1^Ty_{0s}=p_s,\\\\\n&y_{ts}\\geq 0,\\quad \\forall s\\in\\mathcal{S}, t\\in[T]\n\\end{aligned}\\label{mdp game vector}\n\\end{equation}\n\nThe Lagangian becomes\n\\begin{equation}\n\\begin{aligned}\nL(y_{ts}, V_{ts}, \\mu_{ts}) & = \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}y_{ts}}^2_2 +\\norm{Q_{ts}^Ty_{ts}}_2 \\\\\n& + \\sum\\limits_{t \\in [1,T]}\\sum\\limits_{s\\in\\mathcal{S}} V_{ts}\\Big(\\sum\\limits_{s' \\in \\mathcal{S}}P^T_{s\\cdot s'} y_{t-1,s'} - \\mathbf 1^Ty_{ts}\\Big) \\\\\n& + \\sum\\limits_{s \\in \\mathcal{S}}V_{0s}\\Big(p_s - \\mathbf 1^Ty_{0s}\\Big) \\\\\n& - \\sum\\limits_{s \\in \\mathcal{S}}\\sum\\limits_{t \\in[T]}\\mu_{ts}^Ty_{ts} \\\\\n\\end{aligned}\n\\end{equation}\n\nThen the corresponding KKT conditions for the value iteration is: \n\n\\begin{equation}\n\\begin{aligned}\n&t = 0 \\cdots T-1: \\\\\n&\\quad V_{ts}\\mathbf 1 = -\\mu_{ts} + R_{ts}y_{ts} + \\frac{1}{\\norm{Q_{ts}^Ty_{ts}}_2}Q_{ts}Q_{ts}^Ty_{ts}\\\\\n& \\quad\\quad + \\sum_{s'}P_{s'\\cdot s}V_{t+1,s'} \\\\\n&t = T: \\\\\n&\\quad V_{Ts}\\mathbf 1 = -\\mu_{tsa}  + R_{Ts}y_{Ts} + \\frac{1}{\\norm{Q_{Ts}^Ty_{Ts}}_2}Q_{Ts}Q_{Ts}^Ty_{Ts}\\\\\n\\end{aligned} \\label{robust kkt condition}\n\\end{equation} \n\nWe can estimate the worst case of $\\frac{1}{\\norm{Q_{ts}^Ty_{ts}}_2}Q_{ts}Q_{ts}^Ty_{ts} $ by the following inequality based on Cauchy-Schwarz inequality: \n\\[\\frac{1}{\\norm{Q_{ts}^Ty_{ts}}_2}Q_{ts}Q_{ts}^Ty_{ts} \\geq -\\norm{Q_{ts}}_2 \\]\n\nThen the worst case dual problem is the following: \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2  - y^T_{ts}R_{ts}y_{ts}+ \\sum\\limits_{s \\in \\mathcal{S}}p_sV_{0s} \\\\\n\\mbox{s.t.} & V_{ts}\\mathbf 1 \\leq R_{ts}y_{ts} - \\norm{Q_{ts}}_2  + \\sum_{s'}P_{s'\\cdot s}V_{t+1,s'} \\\\\n&V_{Ts}\\mathbf 1 \\leq R_{Ts}y_{Ts} - \\norm{Q_{Ts}}_2 \\\\\n\\end{aligned}\\label{mdp game vector}\n\\end{equation}\nNotice that since $\\norm{Q_{ts}}_2  = \\lambda_{max} $, this is equivalent to offsetting the reward function by the worst possible variation, to obtain a robust reward function $R_{ts} - \\lambda_{max}(Q_{ts}) $\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}