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a/0NFJT4oBgHgl3EQfjiwF/content/tmp_files/2301.11574v1.pdf.txt b/0NFJT4oBgHgl3EQfjiwF/content/tmp_files/2301.11574v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..74fcb597652da4421fb2f806608617b40abb4cbf --- /dev/null +++ b/0NFJT4oBgHgl3EQfjiwF/content/tmp_files/2301.11574v1.pdf.txt @@ -0,0 +1,1837 @@ +Damage Preserving Transformation for Materials with Microstructure +Philip P. Müllera, Falk K. Wittela, David S. Kammera,∗ +aInstitute for Building Materials (IfB), ETH Zuerich, Laura-Hezner-Weg 7, 8093, Zuerich, Switzerland +Abstract +The failure of heterogeneous materials with microstructures is a complex process of damage nucleation, growth and +localisation. This process spans multiple length scales and is challenging to simulate numerically due to its high com- +putational cost. One option is to use domain decomposed multi-scale methods with dynamical refinement. If needed, +these methods refine coarse regions into a fine-scale representation to explicitly model the damage in the microstructure. +However, damage evolution is commonly restricted to fine-scale regions only. Thus, they are unable to capture the full +complexity and breath of the degradation process in the material. In this contribution, a generic procedure that allows +to account for damage in all representations is proposed. The approach combines a specially designed damage law, +with a scheme to generate pre-damaged fine-scale microstructures. Results indicate that the damage approximation for +the coarse representation works well. Furthermore, the generated fine-scale damage patterns are overall consistent with +explicitly simulated damage patterns. Minor discrepancies occur in the generation but subsequently vanish when explicit +damage evolution continuous; for instance under increased load. The presented approach provides a methodological basis +for adaptive multi-scale simulation schemes with consistent damage evolution. +Keywords: +Lattice, Continuum damage mechanics, Microstrutured disordered material, Anisotropic damage, +Multi-scale simulation, Harmonic decomposition, Damage modelling +∗Corresponding author +Email addresses: phimuell@ethz.ch (Philip P. Müller), +fwittel@ethz.ch (Falk K. Wittel), dkammer@ethz.ch (David S. +Kammer) + +Contents +1 +Introduction +3 +2 +Materials and Methods +4 +2.1 +Generic Damage Transforming Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Continuum Representation of 2D Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.3 +Exemplary Material Motive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.4 +Determining the Damage Law for the Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.5 +Process for the Construction of a Damaged Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.6 +Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.6.1 +The UniformSim Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.6.2 +The MultiLoadSim Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.6.3 +The ReconstrSim Simulation Setup +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3 +Results +8 +3.1 +Details of a Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Estimation of the Damage Law �D(#–κ) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.3 +Test of the Damage Evolution Law �D(#–κ) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3.4 +Estimation of the Transfer Function #–�r (#–κ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3.5 +Tests of the Reconstruction Process +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +4 +Summary and Conclusion +15 +5 +CRediT +15 +6 +Declaration of Competing Interest +15 +7 +Data Availability +15 +Appendix A +Parameters of �rx(κx) and �ry(κy) +16 +References +17 +2 + +1. Introduction +At a certain scale even heterogeneous materials will +appear homogeneous and some can even be considered +isotropic. +Among others, this is true for concrete, one +of the most widely used commodity on earth, a mixture +made of sand, aggregates, cement, water and chemical ad- +mixtures. The growth of damage inside concrete is highly +affected by the particular microstructure, where, depend- +ing on the scale, aggregates or even sand grains act either +as focal points for stresses or obstacles for damage. +Damage initiates at very small scales, long before the +macroscopic structure itself will fail or crack. Instead, the +damage leads to a reduction of the material’s stiffness. +Nevertheless, at one point the accumulated damage be- +comes so widespread, that even its smallest increase, will +trigger the previously isolated nuclei to merge. This leads +to a cascade of increasingly larger defects, culminating in +the emergence of a macroscopic crack. +Continuum based methods are the methods of choice +if large structures should be simulated, due to their com- +putational efficiency. For taking into account intrinsic de- +generative processes, constitutive laws are used. One of +the earliest, but still widely used laws for modelling dam- +age in concrete was proposed by Mazars (Lemaître, 2001; +Mazars and Lemaître, 1985). +It employs a scalar dam- +age variable to degrade the material’s stiffness. However, +even if the material was initially isotropic, damage will +induce anisotropy into the material’s behaviour. Clearly +any scalar damage variable is inherently unable to capture +this. During the years, a variety of anisotropic damage +models were proposed to address this issue (Brancherie +and Ibrahimbegovic, 2009; Braun et al., 2021; H. Chen et +al., 2016; Delaplace and Desmorat, 2008; Desmorat et al., +2007; Gaede et al., 2013). All of them consider the ac- +cumulated effects of the damage’s growth, represented by +internal state variables at the material points and by that +disregard the actual microstructure, whose degeneration is +the actual cause for the emerging damage. +To overcome this deficiency, the entire microstructure +could be explicitly represented and simulated. Unfortu- +nately, even with today’s fast computers, this is only pos- +sible for small sizes. A way to overcome this barrier are +multi-scale methods. They allow to invest computational +power exactly where it is needed, by combining different +representations. Although, many different methods were +proposed over the past years, they can be classified to be +either of hierarchical or of concurrent nature (Liu, 2018; +Zhang et al., 2012). +Hierarchical methods are characterised by a full sepa- +ration of scales, which allows to treat every level indepen- +dently from each other. Thus, the information is passed +between the different levels as one serves as input for the +hierarchically higher level. +Opposed to this, concurrent methods lack the full sepa- +ration of scales and typically decompose the computational +domain into different regions. Imagine a typical setting +where high accuracy is only needed inside a small part of +the computational domain, for example around a crack tip. +Ideally, one limits methods with high accuracy but large +computational burden to these small regions, while the +remaining part of the computational domain is described +by much more efficient methods. The flow of information +between the different regions must be handled by a cou- +pling scheme such as the Arlequin method (Anciaux et al., +2008; Bauman et al., 2008; Guidault and Belytschko, 2007; +Unger and Eckardt, 2011; Wellmann and Wriggers, 2012). +Whenever the decomposition is not available in advance, +one must resort to adaptive methods to refine regions on +demand (e.g., P. Y. Chen et al., 2021; Evangelista, Alves, +et al., 2020; Evangelista and Moreira, 2020; Rodrigues et +al., 2018). However, important questions are (i) how are +the regions that need to be refined identified, and (ii) how +is the loading history of the coarse connected to the initial +state of the newly created fine scale representation? Espe- +cially (ii) does not seem to be addressed well in literature. +Most authors assume that the coarse representation does +not accumulate any damage before being refined (L. Chen +et al., 2021; P. Y. Chen et al., 2021; Rodrigues et al., +2018; Unger, Eckardt, and Konke, 2011). Consequently, +damage is only allowed to evolve inside of the fine scale +representations that start off as undamaged. This is in- +consistent because it essentially disregards the entire load +history, including the damage, that would have degraded +a real material. +In this paper we propose, to the best of our knowledge, +a generic approach for the refinement step in adaptive con- +current multi-scale simulations, that is able to account for +the preceding damage evolution inside the coarse repre- +sentation. Thus, the created fine scale representation con- +tains an initial damage that is mechanically consistent to +the damage that has evolved inside the coarse represen- +tation. Our solution is to equip both, the fine and coarse +scale representations, with their own damage measure. We +analyse these damage measures and establish a connection +between them. Our approach is actually able to address +both questions raised above. By interpreting the coarse +damage measure as a “measure of suitability”, critical re- +gions that require refinement are regions whose damage +measure surpassed some predetermined threshold value. +Further, the coarse damage is used to initialise the fine +scale damage. +While the approach is generic and rather simple, its +practical details highly depend on the selected represen- +tations. Thus, we demonstrate it by applying it to one +particular test case. The reminder of this paper is organ- +ised as follows: In Sec. 2, we explain our method in more +detail and present the proposed techniques. In Sec. 3 we +determine the parameters of our method and asses its ap- +plicability, before we draw final conclusions in Sec. 4. +3 + +2. Materials and Methods +The particular choice of the material’s microstructure, +also called motive, is in general arbitrary, but should fol- +low principles of representative volume elements (RVE) +(Lemaître and Desmorat, 2005). The state of a discrete +representation with its inherent characteristic structure is +fully given by r, that describes every single discrete ele- +ment (right side of Fig. 1). In this representation, damage +D(r) is given by the irreversible degeneration of the con- +stituting elements. On the left side of Fig. 1, the smeared +continuum representation is shown, which lacks such an +explicit microstructure and only considers cumulative ef- +fects of the damage through internal state variables added +to the constitutive law. Here, damage is given by D, which +depends on the state #–κ at a particular location and is em- +bedded in the constitutive law. +Figure 1: The continuum damage, at a certain point, is given by D, which +depends on the respective state #– +κ . The discrete damage D(r) depends on +r and hence on the state of all discrete elements of the lattice. The two +representations are interconnected to each other by homogenisation and +refinement processes. Scale of the lattice is exaggerated. +Since the continuum representation loses its validity +once cracks localise, one must refine the continuum to its +discrete twin in such way that all important aspects of the +fracture will be captured accurately on the fine scale. The +key for a meaningful adaptive modelling of the damage +evolution lies in the transformation of the continuum to +the discrete representation, that conserves the degraded +mechanical behaviour found inside the continuum. One +focus of this work is an approach to construct a discrete +representation that respects the preceding damage present +in the continuum representation. +Even though the procedure is generic and in principal +not restricted to specific numerical material representa- +tions, this paper focuses on one particular choice. How- +ever, we will outline the generic way of working with the +method (see Sec. 2.1), before we start with our specific +choice. +We exemplarily chose a two-dimensional plane +stress, isotropic material (see Sec. 2.2) with an under- +lying material heterogeneity represented by a triangular +network of beam-truss elements with linear-elastic, brittle +behaviour with quenched disorder of breaking thresholds +(see Sec. 2.3). We then discuss the particular choice of +the damage law as well as the reconstruction step (see +Secs. 2.4, 2.5). To determine and test them, we use data +obtained from numerical simulations (see Sec. 2.6). +2.1. Generic Damage Transforming Method +Initially, the domain is described as a continuum with- +out any internal structure, whose state is fully described +by the continuum state variable #–κ. In the continuum, the +damage evolution is fully govern by the damage function +�D(#–κ). Therefore, �D(#–κ) can be interpreted as the macro- +scopic damage, that is expected for a hypothetical dis- +crete representation with identical loading. Thus we can +determine the function describing the macroscopic dam- +age by homogenising the discrete damage D(r). This leads +to a perspective on the damage law that is different from +the conventional one, where the damage law is calibrated +against a physical material. Instead, here the law is cali- +brated against a particular numerical representation of the +material. +When the continuum model experiences a certain dam- +age limit, it is no longer suitable and has to be refined to +a discrete representation. However, this discrete state has +to be consistent with the previous continuum representa- +tion. This includes stiffness and damage, which have to +be preserved as much as possible by the transformation. +Determining this reconstruction process challenging, since +it is by its very nature not unique. +2.2. Continuum Representation of 2D Isotropic Continua +To represent a two-dimensional isotropic material un- +der plane stress, the Finite Element Method (FEM) and +as damage measure continuum damage mechanics (CDM) +is used (Lemaître, 2001; Lemaître, 1996; Lemaître and +Desmorat, 2005). We use the well known material law: +σ = (I − D) Cε, +(1) +where σ and ε denote the continuum stress and strain ten- +sors, respectively, and C is the continuum stiffness tensor +of the undamaged material. Due to the choice of CDM, as +continuum damage measure, the damage variable D can +directly be identified with the damage function �D(#–κ). Fur- +ther, we identify #–κ as the continuum state variable given +as +#–κ := +�κx +κy +� +. +(2) +A zoning approach is used to divide the principal strain +space along an angle χ, known as zone boundary, into an +x- (shaded red parts) and y-zone (shaded yellow parts in +Fig. 2). The two components κx and κy represent the max- +imal reached principal tensile strain in x and y direction, +respectively, i.e. +κx := max +� +κx, ⟨ε1⟩+ +� +, +κy := max +� +κy, ⟨ε2⟩+ +� +. +While ε1 and ε2 are the eigenvalues of the strain tensor +ε, its eigenvectors form a Givens rotation matrix of angle +Γ, which is sometimes called eigenangle. +The angle Γ, +together with the boundary χ, determines which zone the +eigenvalues are associated with, see Fig. 2. +4 + +Figure 2: Interpretation of the zone boundary parameter χ. While ε1 +and ε2 are the eigenvalues of ε, its eigenvectors are described by the +value Γ. The eigenangle Γ and the zone boundary χ determines which +eigenvalue acts in which direction. +Since the continuum damage is only used during the +initial phase with low damage, two assumptions are made: +(i) It is assumed that the damage is orthotropic which re- +duces D and �D(#–κ) to diagonal matrices. (ii) It is assumed +that κx only acts on the x-damage while κy only affects +the y-damage, which means that we assume no correlation +between the directions. +2.3. Exemplary Material Motive +The example material motive chosen here is based on +models proposed in Refs. (Herrmann et al., 1989; Mier, +2017), namely a regular triangular lattice but formed by +3rd order Reddy truss-beam elements with characteristic +lattice size ℓ (Reddy, 1997; Reddy et al., 1997). +Using +beams allows to include bending properties and the re- +sulting lattice is able to represent a Cosserat continuum +(Ostoja-Starzewski, 2008; Vardoulakis, 2019). +The mi- +croscopical beams consist of an isotropic material with +Young’s modulus Eb and Poisson’s ratio νb. A list of all +used material parameters is given in Tab. I. In a multi- +scale simulation, Eb has to be chosen such that the result- +ing behaviour of the discrete structure matches the one of +the continuum, i.e. its stiffness tensor C. However, since +this paper studies the refinement step in isolation, with- +out having an actual continuum phase, the choice of Eb is +actually irrelevant. +Lattice Geometry. +The motive is defined by the number of +nodes (Nx, Ny) in x- and y-direction, the spatial extension +in x-direction Lx, with resulting characteristic lattice size +ℓ := Lx/(Nx −1) and spatial y-extension Ly := Nyℓ +√ +3/2. +An out-of-plane height of H is assumed. To remove the +symmetries of the lattice, topological disorder is intro- +duced (Moukarzel and Herrmann, 1992; Wittel, 2006) by +adding the random displacement +#–x ∆ +i := a ℓ +2 +#–x ∗ +i +(3) +to every internal node of the grid, where #–x ∗ +i is a random +vector sampled uniformly from the unit circle (see Fig. 3a). +Table I: Parameters of the discrete material motive. +Property +Value +Unit +Nx, Ny +300, 346 +[−] +Lx, Ly +2, 1.998 +m +H +1 +m +Eb +1 × 106 +Pa +νb +0.3 +[−] +kε +3 +[−] +λε +0.02 +[−] +kΦ +3 +◦ +λΦ +0.02 +[−] +The distortion is controlled by parameter a ∈ [0, 1[, known +as distortion level. +Figure 3: (a) Distortion of the central node, ignoring the distortion of the +surrounding nodes. The location of the distorted node (yellow circle), is +randomly selected within the blue circle of radius aℓ/2. Afterwards, the +length of the beams are adjusted to match the new node location (black +lines). (b) The thickness of beam i is given as ti := A(O) +i +/ℓi, where ℓi is +its length and A(O) +i +is the area the beam is representing. Points zL and +zK are centres of the adjacent triangles’ incircles. +Geometrical Properties of Beam-Truss Elements. +The +thickness of beam i, denoted as ti, depends on the lat- +tice’s geometry. It is given as ti := A(O) +i +/ℓi, where A(O) +i +is the area represented by the beam and ℓi its length, see +Fig. 3b. The area A(O) +i +is formally defined as the set of +points that are closer to beam i than any other beam, but +are inside the lattice. It can be determined by finding the +intersection of the angle’s bisectors, i.e. centre of the incir- +cle, of the two adjacent triangles denoted as zK and zL in +Fig. 3b. In case the beam is part of the boundary A(O) +i +is +artificially doubled. This ensures that in a regular lattice +all beams have the same axial rigidity. +Damage Criterion Applied to the Beam-Truss Lattice. +In +the discrete representation, damage is the irreversible fail- +ure of elements, namely the reduction of their contributing +stiffness to an insignificant level. To determine if a beam +has surpassed its loading capacity, the elliptical criterion +� εi +εi; th +�2 ++ +max +����Φ(r) +i +��� , +���Φ(l) +i +��� +� +Φi; th +=: Ψi ≥ 1 +(4) +is used, where εi;th and Φi;th are the beam’s elongation +and bending thresholds, respectively (Herrmann et al., +5 + +KicKy +E2a +(b) +ZK +a +ZL +Φ +(r)1989). Both thresholds are sampled independently from +the Weibull distributions εi;th +iid +∼ Weib (kε, λε) and Φi;th +iid +∼ +Weib (kΦ, λΦ). +The Discrete State Variable #–r . +The +discrete +state +is +uniquely described by r. However, for the context of this +paper the surrogate discrete state variable +#–r := +� +rx +ry +� +(5) +is introduced and termed “discrete state variable”. Since +#–r has only two components it does not uniquely describe +the damaged state. +This ambiguity will be resolved by +the reconstruction process (see Sec. 2.5). +#–r is a purely +mathematical quantity designed to have certain properties. +First, its 1-norm �r := ∥#–r ∥1 := |rx| + |rx| equals to Nf/NT, +where Nf is the number of failed beams and NT the total +number of beams in the lattice. �r is also called the ratio of +failed beams (rfb). Second, its components are defined by +associating them to the x- and y-zone, respectively, similar +to #–κ (see Sec. 2.2). But while κx is connected to strains +in the x-zone, rx is related to the amount of beams that +have failed due to κx. +2.4. Determining the Damage Law for the Continuum +The damage function �D(#–κ) will take the role of the +damage variable D inside the constitutive equation (1). +Thus, �D(#–κ) has to be designed such that its evolution +mimics the expected behaviour of D (see Sec. 3.2). For +the extraction, which involves two steps, the Uniform- +Sim simulation data of fully discrete lattices is used (see +Sec. 2.6.1). +Step 1: Effective Material Stiffness Tensor C. +First, the +effective stiffness tensor C is calculated by homogenisation. +After the convergence of each loading step, the following +seven strain-states +� +� +� +� +� +1 +2 +3 +� +�, +� +� +4 +5 +0 +� +�, +� +� +6 +0 +0 +� +�, +� +� +0 +7 +0 +� +�, +� +� +0 +0 +8 +� +�, +� +� +9 +0 +10 +� +�, +� +� +0 +9 +8 +� +� +� +� +� , +(6) +denoted as (εxx, εyy, 2εxy)T × 10−3, were applied to the +lattice, while blocking further damage to measure the re- +sulting stresses. This results in an overdetermined system +of 21 equations for the 6 unknown coefficients of C, which +is solved by a least-square approach. +Step 2: Determining the Damage Variable D. +Second, +the damage variable D is extracted from the effective stiff- +ness tensors of the lattice. For this, a technique originally +presented by Oliver-Leblond et al. (2021) is used. For com- +pleteness, the relevant equations are replicated to be +d(T ) := tr1,2[T ] = Tkkij, +(7a) +K := 1 +4 tr +� +d(C) +� +, +(7b) +D := D +� +C, �C +� +:= +1 +2K +� +d(C) − d +� +�C +� � +. +(7c) +The tensor defined by Eq. (7a) is also known as dilatation +second order tensor, while scalar K of Eq. (7b) is the bulk +modulus. Eq. (7c) combines the effective C and undam- +aged stiffness tensor �C to the damage variable D. D is +by construction a real symmetric 2 × 2 matrix, thus fully +characterised by its two eigenvalues d(x) and d(y) as well as +a single scalar Γ, describing the rotation of its eigenbasis +(see Fig. 2). +2.5. Process for the Construction of a Damaged Lattice +The reconstruction process, i.e. the creation of a dis- +crete lattice with a particular damage, involves two compo- +nents: (i) The transfer function #–�r (#–κ), which transforms +the continuum state #–κ into the discrete surrogate state +variable #–r . (ii) A scheme which transforms the surrogate +state #–r into the full discrete state r. Hence, the scheme +must be able to resolve the inherently present ambiguity in +#–r . As direct consequence of the definition of the discrete +state #–r (see Eq. (5)), the transfer function is given as +#–�r (#–κ) := +��rx(κx) +�ry(κy) +� +. +(8) +As for the damage function �D(#–κ), we are using data ob- +tained from the UniformSim simulations (see Sec. 2.6.1) +to empirically determine the function #–�r (#–κ) that approxi- +mates #–r . Due to the nature of #–r it is impossible to mea- +sure its components and thus to fit them directly. How- +ever, it is easy to measure and fit the quantity �r := ∥#–r ∥1. +Because of the specific design of the simulations and as- +sumptions, it is possible to associate the value �r to the +components of #–r , see Sec. 2.6.1. +For reconstructing the full discrete state, a probabilis- +tic scheme was devised. It starts by constructing an un- +damaged lattice from which certain beams are removed, +such that the resulting damage matches in a statistical +sense the one given by #–κ. Due to the assumed decoupling +between the x- and y-zone, it is possible to handle the two +directions independently. For each direction α, i.e. x and +y, the following steps must be done: +1. From the continuum state κα the corresponding dis- +crete state +variable, +rα = �rα(κα) +is +computed. +Through the relationship Nα := rα·NT , it is possible +to determine how many failed beams are associated +to this direction. +2. Each beam is assigned a probability defined as +pi ∝ +1 +εi; th +��� +�#–b i, #–t α +���� +k +, +(9) +where εi; th is the elongation threshold and #–b i the di- +rection of the beam. The vector #–t α, called “damage +basis”, represents the main damage direction. In our +motive, it is either #–t x := (1, 0)T or #–t y := (0, 1)T. +Finally, parameter k, called “directional weight”, is +6 + +a tuning parameter that balances the relative impor- +tance of the two terms and needs to be determined +(see Sec. 3.4). +3. The Nα many beams to fail are drawn from the prob- +ability distribution defined by Eq. (9) without re- +placement. +4. The selected beams are marked as failed. +2.6. Numerical Simulations +For estimating and testing the damage function �D(#–κ) +and the transfer function #–�r (#–κ), a series of different numer- +ical simulations are carried out on fully discrete lattices. +We employ for this a customised version of the Akantu +FEM library (Richart and Molinari, 2015). +Due to the +randomness of the lattice, 30 realisations were made for +each case. +2.6.1. The UniformSim Simulation Setup +The first type of simulation, called UniformSim, is used +for estimating the transfer function #–�r (#–κ) and the damage +law �D(#–κ). These simulations realise an uni-axial strain +Figure 4: Boundary conditions used by the UniformSim series, shown for +the case of ϕ = 0°. In general, the boundary conditions given by Eq. (10) +are applied to the whole boundary. Scale of the lattice is exaggerated. +state of the lattice that is also rotated by an arbitrary but +constant angle ϕ, called the pull direction. (see Fig. 4). +Thus +εϕ := Rϕ +T +��ε1 +0 +0 +0 +� +Rϕ, +(10) +where Rϕ is the Givens rotation matrix for angle ϕ that +is applied to the lattice’s boundary. In each loading step, +�ε1 is increased by 0.0001 until 0.005 is reached. The limit +is chosen to ensure that no localisation will occur and that +damage maintains its diffuse character. +The particular setup of the UniformSim simulations +together with the previous assumptions on D and #–�r (#–κ) +allows the following conclusions and simplifications: +(i) A pull direction is either associated to the x- or y- +zone (see Sec. 2.2). +This allows to probe the be- +haviour of a single zone. Which zone is probed de- +pends on ϕ and the yet unknown zone boundary +value χ. +(ii) A simulation, i.e. a particular value of ϕ, will only +affect the state of either the x- or the y-zone. Thus, +an increase of �ε1 will only affect one eigenvalue of D +and a single component of #–κ as well as #–r . Which +component is affected depends on ϕ and χ. +(iii) For the continuum state variable the relation �κ +!= +∥#–κ∥1 +!= |κα| +!= �ε1 holds. +Thus, one component +equals the applied uni-axial strain �ε1, while the other +is zero. +(iv) For the discrete state variable, the relation �r +!= ∥#–r ∥1 +!= +|rα| holds. Thus, only one component is non zero and +equals �r. This can be used to determine the compo- +nents of #–r from �r, once the x- and y-zones are known +(see Sec. 3.4). +2.6.2. The MultiLoadSim Simulation Setup +For testing the damage function as well as the recon- +struction procedure, a second type of simulation is used, +called MultiLoadSim. +It realises a bi-axial strain state, +imposed along the x- and y-axes (see Fig. 5). Both strains +are increased until εxx = εyy = εfin is reached, where εfin +is the control parameter. For each simulation, the loading +is imposed in three different ways, but each time the same +initial lattice is used: +XThenYSim: εxx is increased in steps of 0.0001 until it +reaches εfin and then maintained. Then εyy is in- +creased by the same increment until εfin is reached. +YThenXSim: The same as XThenYSim, however, the order +of loading the axes is switched. +BothXYSim: Both strains εxx and εyy are increased simul- +taneously, in steps of 0.0001 until εfin is reached. +All three paths reach the same final state, εxx = εyy = +κx = κy = εfin, but via different paths. As a consequence, +the special relation �κ := ∥#–κ∥1 = |εxx| + |εyy| holds in +these simulations. Both, the XThenYSim and the YThenX- +Sim loading path impose in the first half of the loading an +uni-axial strain state and then switch to a bi-axial strain +state for the second half, while the BothXYSim loading path +imposes a bi-axial strain state from the beginning. +Figure 5: Boundary conditions used in the verification simulations. Scale +of the lattice is exaggerated. +7 + +2.6.3. The ReconstrSim Simulation Setup +For testing the reconstruction process (see Sec. 2.5), a +third type of simulation is used, called ReconstrSim. The +basic setup is equivalent to UniformSim, but for ϕ = 0°. +Further, a lattice with a certain initial damage is used. +This damage was constructed to match the continuum +state #–κ = (�ε, 0)T, i.e. +the damage created by an uni- +axial strain of �ε applied along the x-direction. Another +important difference is, that the loading does not start at +zero but at �ε. In case of a perfectly working reconstruc- +tion process, one would expect no additional damage for +strains ≤ �ε. +Unlike the MultiLoadSim tests, which focuses on the +value of the reconstructed damage, these tests focus on +how the reconstructed lattices behave after their recon- +struction. In essence, this test simulates the exchange of +the continuum representation with the discrete represen- +tation, i.e. the refinement process, which is the core ap- +plication of the proposed method. +3. Results +Our proposed method relies on the damage law �D(#–κ) +used inside the continuum and the reconstruction process. +First, we discuss how we will use the techniques introduced +in Sec. 2 to process the data that we have collected from +the numerical simulations. Then, we determine the dam- +age law �D(#–κ) and the zone boundary value χ (see Sec. 3.2) +followed by an assessment of its accuracy (see Sec. 3.3). +Thereafter, we repeat the process to determine the trans- +fer function #–�r (#–κ) and the directional weight parameter k +(see Sec. 3.4). Finally, we demonstrate the applicability of +our method (see Sec. 3.5). +3.1. Details of a Numerical Simulation +Let’s consider a setting similar to UniformSim but with +ϕ = 0° (see Fig. 6a). Only one realisation is simulated and +the loading goes beyond �ε1 = 0.005. +We measure the normalised stresses (solid lines) and +compare them with the expected ones in case of suppressed +damage, i.e. undamaged case (dashed lines) (Fig. 6b). As +expected, initially the stresses behave predominantly lin- +ear. However, once a strain of about 0.003 is reached, we +observe that �σxx starts to deviate from the undamaged +case. +While this deviation increases with further load- +ing, we cannot observe it for �σyy, that is much less af- +fected by the loading. At one point, we observe that both +stresses suddenly drop. This is caused by the emergence +of a macroscopic crack, which is the expected behaviour +for a brittle disordered material, such as concrete. +Using the techniques presented in Sec. 2.4, it is possible to +extract the macroscopic damage variable D for the lattice, +at any loading step. In Fig. 6c, we show the eigenvalues of +the extracted damage variables where we can see that d(x) +exceeds d(y). This also explains why �σxx deviates much +more from the undamaged behaviour when compared to +�σyy. The underlying reason of this difference are the hor- +izontal beams. They experience much larger strains than +inclined beams, since they align with the loading and thus +fail at much larger number. From Fig. 6c, we can also see +that d(y) drops for a strain at around 0.002. This is a non- +physical behaviour as damage should always increase. It is +caused by some numerical issues during the determination +of the stiffness tensor (see Sec. 2.4) and the sensitivity of +damage extraction process. +Fig. 6d shows the ratio of failed beams (rfb), �r := ∥#–r ∥1 := +Nf/NT, where Nf is the total number of number of failed +beams and NT the total beams in the lattice. We will use +it to indirectly estimate the transfer function #–�r (#–κ). +Figs. 6e-h show snapshots of the lattice’s underlying +microstructure. While taken at different loading steps (see +Fig. 6d), they always show the same set of nodes, located +roughly at the lattice’s centre. While the exact damage +pattern depends on the realisation of the lattice and lo- +cations where the snapshot was taken, statistically they +all look the same. +It is this statistical damage pattern +that we want to capture by the transfer function #–�r (#–κ) +and recreate by the reconstruction process. Whereas the +damage law �D(#–κ) captures the accumulated effects on the +lattice’s macroscopical stiffness. +3.2. Estimation of the Damage Law �D(#–κ) +We now study the behaviour of the damage variable D +that we have extracted from the data of the UniformSim +simulations. From these observations, we will determine +the damage function �D(#–κ) as well as the zone boundary +value χ (see Fig. 2). +Functional Form of �dx(κx) and �dy(κy). +Since we have +assumed an orthotropic damage variable (see Sec. 2.2), we +have to assume the same for the damage function. Thus +arriving the tentative form of the damage function is given +by +�D(#–κ) := +�dxx(#–κ) +0 +0 +dyy(#–κ) +� +. +To account for deviations from this assumption, we will +connect the two diagonal elements of the damage func- +tion with the eigenvalues of the measured damage vari- +able. Thus, we have only two functions that we need to +determine. +Fig. 7 shows the evolution of the eigenvalues d(x) and +d(x) from the extracted damage variable for the pull di- +rections ϕ ∈ { 0°, 60° } at various distortion levels. +For +ϕ = 0°, the eigenvalue d(x) is much larger than d(y), while +for ϕ = 60° the opposite is observed. Later, we will use +this to determine the zone boundary value χ. Most im- +portantly, the figures show that both eigenvalues follow a +power law, irrespective of the pull direction and distortion. +Thus we approximate the diagonal elements/eigenvalues of +8 + +Figure 6: Representative simulation example. +(a) Schematics of the model configuration, scale of the lattice is exaggerated. +(b-d) Evolution of +continuum and microstructure properties of the lattice. Dotted lines denote strains beyond the limit of 0.005 used in UniformSim. (b) Normalised +measured stresses. Dashed lines represent the behaviour in case of suppressed damage. (c) Eigenvalues of the extracted damage variable D, see Eq. (1). +(d) Ratio of failed beams �r in the specimen. (e-h) Snapshots of a small section of the microstructure. Colours indicate the remaining load carrying +capacity of the beams � +Ψi := 1 − Ψi, where Ψi is defined by Eq. (4). Associated states are indicated in (d) by markers. Bending of beams is not shown. +�D(#–κ) as: +dxx ≈ �dx(κx; a, ϕ) := α(x) +a,ϕ · κx +β(x) +a,ϕ, +(11a) +dyy ≈ �dy(κy; a, ϕ) := α(y) +a,ϕ · κy +β(y) +a,ϕ. +(11b) +The parameters of these approximations depend on the +distortion level a and the pull direction ϕ. Later, we will +eliminate their dependency on ϕ and obtain the final pa- +rameters that only depend on a, which is constant. Fur- +ther, this choice guarantees that the damage is strictly +increasing. +Because of our previous assumption about the indepen- +dence of the directions, the approximations of the eigenval- +ues only depend on a single component of the continuum +state #–κ (Sec. 2.2). While this could be justified due to their +large differences, that we can see in Fig. 7, we clearly see +that even for ϕ = 0°, there is a certain coupling between +d(x) and d(y). To handle this, we use a simple coupling +scheme, which leads to the final damage function +�D(#–κ) := +(12) +� +� +� +� +max +� +�dx(κx), � +dy(κy) +η +� +0 +0 +max +� +�dy(κy), � +dx(κx) +η +� +� +� +� +�, +where �dx(κx) and �dy(κy) are the approximations of the +eigenvalues defined by Eq. (11) but without the depen- +dence on ϕ. The coupling ensures that the eigenvalues of +the damage function �D(#–κ) will at most differ by a factor +of η, which is exactly what we see in the case of uni-axial +loading (see Fig. 7). Here, we will assume that the em- +pirical parameter η equals 10 in all cases. We will later +give a justification of the form and value of the proposed +coupling. It is important to notice that this coupling is +designed for the uni-axial case. However, a more elabo- +rated coupling might be needed, depending on the details +of other material motives. +Parameters of �dx(κx) and �dy(κy). +Since the data, espe- +cially for the non-dominant eigenvalue shows strong vari- +ation for small strains, only data points corresponding to +9 + +a) +ouy +0.005 +<6 +0.000 +( +d(c) +d(y) +-4 +-6 +d +0.02 +(h) + 0.01 +(g) +(f) +(e) +0.00 +0.000 +0.002 +0.004 +0.006 +0.008 +remaining load carrying capacity := 1 - +Err [-] +>0Figure 7: Eigenvalues of the extracted damage variable D for pull direc- +tions ϕ = 0° (a) and 60° (b). Solid lines correspond to d(x), while dashed +lines correspond to d(y). Colours indicate different distortion levels a of +the underlying lattice. +strains above 0.002 were used for the parameter estima- +tion. +In Figs. 8a,b, we see that for small values of ϕ, +the α(x) +a,ϕ-parameters are very close to each other, while for +larger values of ϕ one observes a much larger scattering. +Interestingly, α(y) +a,ϕ-parameters behave inversely. Further- +more, on Fig. 8b we can clearly observe the α(y) +a,ϕ depen- +dence on ϕ. We see that α(y) +a,ϕ is small if ϕ is small too, but +above a certain value of ϕ, the parameters become much +larger and their scattering increases. The same, but in an +opposite way, holds for the α(x) +a,ϕ-parameters but in a less +pronounced fashion. +The estimates for the β-parameters (see Figs. 8c,d) +show a similar behaviour with respect to ϕ. +However, +while we observed a significant change in the behaviour +of the α-parameters’ values, from a particular value of ϕ +on we just observe an increase of the variability of β. +In summary, from Fig. 8 we can conclude that the β- and +especially the α(y)-parameters have different regimes de- +pending on ϕ. Further, inside such a regime, their partic- +Figure 8: Values of the α- (a,b) and β-parameters (c,d) with respect to +the pull direction ϕ. Colours indicate different distortion levels. Solid +lines correspond to lg α(x) +a,ϕ and β(x) +a,ϕ, while dashed lines to lg α(y) +a,ϕ and +β(y) +a,ϕ. Error bars indicate the 95% confidence interval. +ular value does not depend much on ϕ. +We also saw that the values for the β(x)-parameters for +small values of ϕ and β(y)-parameters for large values of +ϕ are both close to three. This means that the growth +behaviour of �dx(κx) and �dy(κy) are very similar. This jus- +tifies the form of the coupling used in the damage function +in Eq. (12). +Zone Boundary χ. +In Figs. 7 and 8, we have observed +that depending on the pull direction either d(x) or d(y) is +dominant. We now exploit this fact to define χ. To this +end, we define the dominance function ζ as: +ζ(a, ϕ) := lg +� +d(x) +a,ϕ; ˜κ=0.005 +d(y) +a,ϕ; ˜κ=0.005 +� +, +(13) +with d(α) +a,ϕ; ˜κ=0.005 as the damage eigenvalue associated to +direction α, once the uni-axial strain has reached 0.005. +10 + +a + = 0.0° +10-2 +d(r) a = 0.0 +d(r) aα = 0.1 +d(r) a = 0.2 +10-3 +d(r) a = 0.3 +d(r) α = 0.5 +10-4 +~ d(y) +(α)p +10-5 +10-6 +10-7 +L +(b) +Φ = 60.0° +10-2 +d(y) a = 0.1 +d(y) a = 0.2 +10-3 +d(y) a = 0.3 +d(y) α = 0.5 +10-4 +~ d(r) +(6)p +10-5 +10-6 +10-7 +10-4 +10-3 + [-](a) +5.0 +4.0 +(b) +5.5 +[-] ° +5.0 +4.5 +4.0 +(c) ++ +a= 0.0 +α= 0.3 +4.0 +α = 0.1 +a = 0.5 +a = 0.2 +a = 0.7 +3.5 +3.0 +(d) +4.0 +二 +3.5 +3.0 +0 +20 +40 +60 +80 +[] The most important aspects of this function are its sign +and root, to a lesser extend its value. ζ > 0 means that +d(x) is dominant, while ζ < 0 indicates that d(y) is domi- +nant. Thus, χ, which might depend on the distortion a, is +defined as ζ(a, χ) +!= 0. +Figure 9: Dominance function ζ(a, ϕ), Eq. 13, for different distortion +parameters a. The x-dominated region, i.e. d(x) ≫ d(y), is defined by +ζ > 0, while the y-dominated (grey shaded) region, i.e. d(x) ≪ d(y), is +defined by ζ < 0. The UniformSim data was used. +Examining Fig. 9, we see that, irrespective of the distor- +tion, χ must lie between 30° and 45°. After some exper- +imentation, we decided to use 40° as zone boundary, ir- +respective of the distortion level. A closer analysis might +yield different estimations. +ζ can be seen as a measure of the coupling between +d(x) and d(y). Thus, we can used it to determine the value +of the empirical coupling parameter η, see Eq. (12). Our +value η = 10 was selected because it is roughly the mean +value for ϕ = 0°. +Final Parameters of �dx(κx) and �dy(κy). +Eliminating the +dependency of the α- and β-parameters on the pull di- +rection ϕ will results in parameters that are valid inside +the entire x- or y-zone. For this, we combine the different +estimates as: +lg α(x) +a +:= 1 +|X| +� +ϕ∈X +lg α(x) +a,ϕ, +lg α(y) +a := 1 +|Y| +� +ϕ∈Y +lg α(y) +a,ϕ, (14a) +β(x) +a +:= 1 +|X| +� +ϕ∈X +β(x) +a,ϕ, +β(y) +a +:= 1 +|Y| +� +ϕ∈Y +β(y) +a,ϕ, +(14b) +where X contains all the pull directions associated to the +x- and Y the ones associated to the y-zone. Parameters +associated to the transversal directions are simply ignored, +e.g. lg α(y) +a,ϕ=0°. Further, the functional form of �dx(κx) and +�dy(κy) is still given by Eq. (12), just without the depen- +dency on ϕ. Note that Eq. (14) weights the different pull +directions equally. +3.3. Test of the Damage Evolution Law �D(#–κ) +We now evaluate how well the damage function is able +to predict the damage of a fully discrete simulation. For +this purpose, the MultiLoadSim simulations are used. +Figure 10: Damage eigenvalues for the three different loading paths, de- +scribed in Sec. 2.6.2, with final strain εfin = 0.002, plotted against +τ := ˜κ/2 εfin. Using a fully discrete simulation (solid) as reference and +the CDM damage law � +D(#– +κ ) (dash-dotted). +The colours indicates the +three different loading paths. The distortion of the lattices was a = 0.3. +Fig. 10 shows the results of such an experiment for +εfin = 0.002. We can see the eigenvalues, once computed +for the reference (solid), i.e. a fully discrete simulation, +and alternatively computed by the damage function �D(#–κ) +(dash-dotted), i.e. +CDM. They are plotted against the +normalised total strain τ := ˜κ/2 εfin. Thus, both the X- +ThenYSim (orange) and the YThenXSim (green) load paths +switch from an uni-axial to a bi-axial strain state at τ = +0.5. We observe that irrespective of the loading path the +same final damage values are reached. The value depends +on the used method, since the final value of the CDM is +different from the reference value. Note that this is not +problematic since the CDM is only used during the initial +phase. +11 + +2.0 +a +=0.0 +c-dominated +a = 0.1 +a = 0.2 +1.0 +a = 0.3 +a = 0.5 +[-] ( +a = 0.7 +a += 0.8 +0.0 +‘p)S +y-dominated +-1.0 +X +-2.0 +0 +20 +40 +60 +80 +6e +10-3 +10-4 +10-5 +(a)p +10-6 +10-7 +(b) +10-3 +10-4 +I +10-5 +(r)p +10-6 +ref. BothXY +10-7 +ref. XThenY +ref. YThenX +CDM +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T := K/2efin [-]Fig. 10 also demonstrates that the damage for the X- +ThenYSim and YThenXSim are very similar to each other. +However, the eigenvalues are flipped, which is the expected +behaviour. During the first half of the loading (i.e. τ < +0.5), the prediction of the dominant eigenvalue matches +well with the reference value for both loading paths. At +the same time, the non-dominant eigenvalue, i.e. the one +belonging to the transverse direction, is captured with less +but still acceptable accuracy. +The mismatch is entirely +due to the rather crude choice of the η coupling parameter +(see Eq. (12)). However, it indicates that the proposed +coupling is indeed working. +Nevertheless, for the second half of the loading (i.e. τ > +0.5) the CDM is unable to capture the evolution to a satis- +factory degree. In case of XThenYSim (orange lines), we see +that the CDM approximation of the x-eigenvalue d(x) re- +mains constant, since κx is not affected by a loading along +the y-axis. However, we see that in the reference system +d(x) continuously increase (see Fig. 11 for more). The y- +eigenvalue d(y) predicted by the CDM remains initially +constant due to the coupling. Once �dy(κy) has become +larger than � +dx(εfin)/η �dy(κy) starts to increase. However, +as it can be seen form Fig. 10b, the reference d(y) starts +to increase almost immediately. +A different case is the BothXYSim loading path. From +Fig. 10, it seems that for τ < 0.5 its damage grows slower +than the dominant damage observed for the other two +paths. This is because BothXYSim only has half the num- +bers of loading steps the other two have. If this is corrected +for then it would actually grow faster. This indicates that +there is some form of coupling between the two directions +that is not considered correctly. +In Fig. 11, we can see how the final damage, i.e. val- +ues of d(x) and d(y) at εxx = εyy = εfin, depend on the +control parameter εfin, using either the reference (solid +lines), the CDM (dash-dotted lines) or the reconstruction +(dashed lines). The colours distinguish the three different +load paths that were tested (see Sec. 2.6.2). The collapse +of the lines indicate that the damage is indeed path in- +dependent, regardless of the final strain εfin. However, +the final value depends on the particular method that was +used. In in Fig. 10, we observe a gap between the final +damage attained by the reference and the one predicted +by the CDM. We can now see that this gap is systematic +and actually increases with larger εfin. This is again in- +dicating that there is some form of coupling between the +directions that is not take into account yet. +3.4. Estimation of the Transfer Function #–�r (#–κ) +Our procedure to reconstruct a discrete lattice repre- +sentation based on a damaged continuum state (presented +in Sec. 2.5) requires two unknown quantities: First, the +transfer function #–�r (#–κ), which maps the continuum state +#–κ to the corresponding discrete surrogate state #–r . Sec- +ond, the directional weight parameter k, which balances +the orientation and the strength of a beam during the re- +Figure 11: Final value of the d(x) and d(y) damage eigenvalues, com- +puted using the reference (solid), CDM (dash doted) and reconstruction +(dashed) method, plotted against εfin. The colours indicate the three +different loading cases from Sec. 2.6.2. All lattices have a distortion of +a = 0.3. +construction process (see Eq. (9)). +Analogously to the +determination of the damage function, the data from the +UniformSim is used. +Functional Form of �rx(κx) and �ry(κy). +As mentioned be- +fore, it is impossible to measure the components of #–r di- +rectly. However, as outlined in Sec. 2.6.1 #–r is connected +to the ratio of failed beam as �r = ∥#–r ∥1 = Nf/NT +!= rα. +Thus, we can estimate #–r indirectly. Fig. 12 shows �r for +the pull direction ϕ = 0° at various distortion levels. As we +can see, the distortion level has only minor influence. Dif- +ferent pull directions do not lead to a qualitative change +(data not shown). For that reason, we approximate the +mean rfb as +�r ≈ +���#–�r (#–κ) +��� +1 := �r(κ; a, ϕ) := α(r) +a,ϕ · κβ(r) +a,ϕ, +(15) +with the two fit parameters α(r) +a,ϕ and β(r) +a,ϕ. Both depend +on the distortion a and the pull direction ϕ. Due to our +12 + +0.007 +0.006 +0.005 +0.004 +(c)p +0.003 +0.002 +0.001 +0.000 +b +0.007 +k = 6, a= 0.3 +王 +ref. BothXY +0.006 +ref. XThenY +ref. YThenX +0.005 +CDM +-王- +PD, k = 6 +I +0.004 +(r)p +0.003 +0.002 +0.001 +0.000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 + fin [-]Figure 12: ⟨˜r⟩ := � +∥#–r ∥1 +� +for pull direction ϕ = 0°, at different values +of the distortion parameter a. +No qualitative change is observed for +different pull directions ϕ. +previous assumptions, we can identify its argument κ di- +rectly with �κ. To eliminate the dependency on ϕ we use +the same method as for the damage function (see Sec. 3.2). +However, parameters associated to pull directions in X are +used to determine �rx(κx), while the ones belonging to Y +determine �ry(κy). This will transform the approximation +of the scalar quantity +���#–�r (#–κ) +��� +1 into the one for #–�r (#–κ). +For the discussion about the estimated α- and β-para- +meters please see Appendix A. +Directional Weight Parameter k. +The empirical tuning +parameter k influences the selection of beams during the +reconstruction process. It balances a beam’s strength, i.e. +its elongation threshold εth, against how well it aligns with +the damage basis #–t α (see Sec. 2.5). We determine k such +that the reconstructed damage variable D resembles the +reference damage D most closely. To this end, we define +Υk:= +��D11−D11 +�� +ℓ2+ +��D22−D22 +�� +ℓ2+2 +��D12−D12 +�� +ℓ2 (16) +as a measure of separation between the two damages. For +minimising Υk, we select a heuristic approach, in which +the reconstruction process (see Sec. 2.5) is run for differ- +ent values of k. +The k that minimises Υk will then be +used for the remaining part of this paper. However, for +this particular reconstruction process, the used α(r)- and +β(r)-parameters still depended on ϕ. Further, zoning was +ignored and as damage basis the pull direction ϕ was used. +The underlying lattice was not distorted. From Fig. 13 we +see that k = 6 minimises Υk independent of the pull di- +rection. We also see that pull directions { 30°, 90° } seem +to be almost unaffected by k, however, their values match +Υk=6. This is an artefact caused by the regular structure +of the underlying lattice and the scalar product used in the +definition of the selection probability (see Eq. (9)). How- +ever, this artefact is indicating that k = 6 is indeed a good +Figure 13: Υk, Eq. (16), for some values of k and various pull directions. +The reconstruction process was done on regular grids, without zoning. +Further the pull direction and its orthogonal was used as damage basis. +choice. +3.5. Tests of the Reconstruction Process +Now we evaluate the performance of the proposed +reconstruction scheme to create a mechanically equiva- +lent lattice, based solely on the continuum state #–κ (see +Sec. 2.5). For verification, we use the MultiLoadSim sim- +ulations (see Sec. 2.6.2). +In addition, we use the Re- +constrSim simulations to simulate a refinement step (see +Sec. 2.6.3). +The MultiLoadSim Results. +In Fig. 14, we see the re- +sults from the MultiLoadSim setup with εfin = 0.002 +(see Sec. 2.6.2). +They impose the bi-axial strain state +εxx = εyy = εfin, but with loading applied via three dif- +ferent paths. We used this setup before to assess the CDM +(see Sec. 3.3). An important note concerning the recon- +structed states is, that in each loading step the lattice and +hence the damage is constructed anew. Thus, although it +looks like a damage evolution, the damage at any loading +step has no connection to the previous one. However, each +time the same undamaged but distorted lattice was used. +If we now compare the damage from the references +(solid lines) with the one from the reconstructed lattices +(dashed lines) in Fig. 14, we see that the overall dam- +age values are very similar to the ones obtained by the +CDM. As before, we observe that for τ < 0.5 the dom- +inant eigenvalues, i.e. +d(x) for XThenYSim and d(y) for +YThenXSim, are captured well. Then, for τ > 0.5, these +reconstructed eigenvalues stop growing and thus deviate +from the references (solid lines). +An effect we observed +for CDM (dashed lines), too. But if we look at the other +eigenvalues, i.e. d(y) for XThenYSim (orange) and d(x) for +the YThenXSim (green), we see that they start to increase +almost immediately like the reference. This was not the +case for the CDM (dash-dotted lines shown in Fig. 10). +13 + +10-2 +0=0.0° +00=D +10-3 +a = 0.1 +a = 0.2 +a = 0.3 +10-4 +a = 0.5 +10-5 +10-6 +10-7 +10-4 +10-3 +i [-]a = 0.0 +β = 0.0° +§ = 15.0° +§ = 30.0° +§ = 45.0° +P = 60.0° +Φ = 75.0° += 90.0° +6 +10-4 +1 +4 +6 +8 +k [-]Figure 14: Damage eigenvalues for the three different loading paths, de- +scribed in Sec. 2.6.2, with final strain εfin = 0.002, plotted against +τ := ˜κ/2 εfin. Using fully discrete simulations (solid) as reference and the +reconstructed damage (dashed). The colours indicate the three different +loading paths that were taken. The distortion of the lattices was a = 0.3. +See Fig. 10 for the damage evolution predicted by the CDM. +The reconstruction process is affected by the ignored cou- +pling between the directions as well. However, the damage +eigenvalues generated by it follow the reference much bet- +ter than the ones computed by the CDM. +The ReconstrSim Results. +Now we are using the Recon- +strSim simulation setup, described in Sec. 2.6.3. The lat- +tices that are used here were reconstructed for the con- +tinuum state #–κ = (�ε, 0)T. The system is loaded under +uni-axial strain along the x-axis, starting at �ε. This setup +simulates how a discrete region that was loaded up to �ε, as +continuum and then refined behaves upon further loading. +In Fig. 15, we see how the damage eigenvalues d(x) and +d(y) (dashed and dotted lines, respectively) and the rfb �r +(solid lines), evolve for different reconstruction strains �ε, +indicated by different colours. The grey lines correspond +to the reference without any reconstruction. +The circles in Fig. 15 indicate the values for d(x), d(y) +and �r have in the reconstructed lattice before any loading +was applied to them. The circles associated to ∥#–r ∥1 show, +Figure 15: Behaviour of the d(x) and d(y) damage eigenvalues and �r. +Colours indicate different reconstruction parameters �ε. Grey is the ref- +erence, i.e. no reconstruction. Circles indicate damage/rfb values of the +lattices directly after reconstruction. Squares indicate damage/rfb values +of the lattices for an applied strain of �ε. +that the reconstructed lattices have a matching rfb value �r +which is a consequence of its construction. It is, however, +much more important, that the reconstructed d(x) eigen- +value (circles), matches the one predicted by the reference. +Thus the process is able to reconstruct the dominant eigen- +value. +We are also observing that d(y) are not as well re- +constructed. +This is a consequence of the assumptions +that the components of #–r are independent. +Since Re- +constrSim only impose strains along the x-axis, we have +κy ≡ 0 ⇒ ry ≡ 0. Thus, the reconstructed y-eigenvalues +we are seeing are caused by a directional sampling effect +created during the reconstruction of the damage. How- +ever, since d(y) is the non-dominant eigenvalue, we expect +and allow that it is less well reconstructed. +The squares in Fig. 15 indicate the state of the lat- +tices after an uni-axial strain of �ε along the x-axis was +applied to them. The difference between a square and its +associated circle proves that this strain causes the failure +of additional beams. If the reconstruction process would +work perfectly, any strain below or equal �ε should not lead +to the failure of any beam. Therefore, we might have re- +moved the right number of beams and these were more or +less correctly oriented, the selection of some of them was +not fully optimal. +Furthermore, we see that for subsequent loading steps, +the damage and rfb continue to increase (Fig. 15). While +the observed values for the restored systems remain above +the reference, the restored lattices slowly converge towards +them. This is because the damage created by the subse- +quent loading steps starts to dominate the artificial one, +14 + +e +10-3 +10-4 +(a)p +10-5 +10-6 +10-7 +(b) +10-3 +10-4 +工 +(6)p +10-5 +a = 0.3, fin = 0.002 +10-6 +ref. BothXY +ref. XThenY +ref. YThenX +PD. k = 6 +10-7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T := K/2efin [-]= 0.003 += 0.002 += 0.001 +=0.0 +d(c) +10-2 +d(y) +10-3 +10-4 +k = 6, a = 0.3 +10-5 += 0.001 += 0.003 +10-6 += 0.002 +restored: after loading +restored: before loading +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +i [-]that we created through the restoration process. +4. Summary and Conclusion +In this study, we presented a generic approach for the +creation of a discrete twin of a continuum representation +containing an initial damage. The discrete twin’s damage +is created in such a way, that it is mechanically consistent +to the original’s continuum damage. This is a step towards +adaptive multi-scale simulations, which take the state of +the coarse description of a region into account upon its +refinement. +While the method is general and has no restrictions +concerning the used numerical representations, we pre- +sented it in form of a concrete example. As continuum +representation, we have used FEM with CDM as damage +measure. +For the discrete representation, we have used +a lattice based on a triangular grid consisting of brittle +beam-truss elements. +One part of our method is the damage measure used +inside the continuum representation. This measure is used +during the initial continuum phase to track the evolving +continuum damage. Unlike classical CDM, that are cali- +brated to match the degradation of a particular material, +we calibrated the CDM against the degeneration of the +discrete numerical representation. Thus, it measures the +degeneration that would occur on a hypothetical fine scale +representation. +We saw that the determined CDM is indeed able to +capture the damage caused by uni-axial strains to a sat- +isfying degree. However, for bi-axial loading, the CDM is +unable to achieve the same. This is explained by the as- +sumption that the directions are independent. Directional +coupling must be used to further improve the CDM’s ac- +curacy +The second part of our method is the ability to con- +struct a discrete damage that is mechanically consistent +to a given continuum state #–κ. Since this problem is obvi- +ously not unique, we devised a stochastic scheme to gen- +erate representations containing such a particular discrete +damage. +We have seen, that the reconstruction process is indeed +able to create discrete lattices, whose initial degeneration +is consistent with the given continuum state #–κ. A draw- +back is, that imposing a strain that corresponds to #–κ it- +self, leads to the failing of some additional beams. This +is indicating that the selection process needs to be further +refined. Furthermore, as for the CDM, we observed prob- +lems for bi-axial strains, which are again caused by the +independence assumed between the directions. +Nevertheless, our data is indicating that our approach +works well for the case of uni-axial loading and is in prin- +cipal able to work for bi-axial loading. The next step is to +integrate our method into an adaptive multi-scale simula- +tion scheme. +5. CRediT +Philip Müller: Conceptualisation, Methodology, Soft- +ware, Validation, Writing - Original Draft, Visualisation; +Falk Wittel: Conceptualisation, Writing - Review & Edit- +ing, Supervision; David Kammer: Conceptualisation, Writ- +ing - Review & Editing, Supervision. +6. Declaration of Competing Interest +The authors declare that they have no known compet- +ing financial interests or personal relationships that could +have appeared to influence the work reported in this paper. +7. Data Availability +The simulation data generated in this study have been +deposited in the ETH Research collection database avail- +able at TBA. +15 + +Appendix A. Parameters of �rx(κx) and �ry(κy) +The fitting parameters (see Eq. (15)) of the ratio of +failed beams (rfb) �r, denoted as α(r) +a,ϕ and β(r) +a,ϕ were esti- +mated in the same way as the ones for the two damage +functions �dx(κx) and �dy(κy). However, they behave much +more stable and thus show less variations. +Figs. A.16a +Figure A.16: +Dependence of the two fitting parameters of Eq. (15), +lg α(r) +a, ϕ (a) and β(r) +a, ϕ (b), on the pull direction ϕ. +Colours indicating +different distortion levels a. The error bars is given by the 95% confi- +dence interval. +show the values for the α- and A.16b for the β-parameters. +Compared with the parameters we obtained for the dam- +age law (see Fig. 8), we see much less variability here. 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In: Philosophical Magazine 92.28-30. doi: https://doi.org/10.1080/14786435.2012. +731087. +18 + diff --git a/0NFJT4oBgHgl3EQfjiwF/content/tmp_files/load_file.txt b/0NFJT4oBgHgl3EQfjiwF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54309babfb945627928bb5e9ee8d323e9540c992 --- /dev/null +++ b/0NFJT4oBgHgl3EQfjiwF/content/tmp_files/load_file.txt @@ -0,0 +1,1887 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf,len=1886 +page_content='Damage Preserving Transformation for Materials with Microstructure Philip P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Müllera, Falk K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Wittela, David S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Kammera,∗ aInstitute for Building Materials (IfB), ETH Zuerich, Laura-Hezner-Weg 7, 8093, Zuerich, Switzerland Abstract The failure of heterogeneous materials with microstructures is a complex process of damage nucleation, growth and localisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This process spans multiple length scales and is challenging to simulate numerically due to its high com- putational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' One option is to use domain decomposed multi-scale methods with dynamical refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' If needed, these methods refine coarse regions into a fine-scale representation to explicitly model the damage in the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, damage evolution is commonly restricted to fine-scale regions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, they are unable to capture the full complexity and breath of the degradation process in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In this contribution, a generic procedure that allows to account for damage in all representations is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The approach combines a specially designed damage law, with a scheme to generate pre-damaged fine-scale microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Results indicate that the damage approximation for the coarse representation works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Furthermore, the generated fine-scale damage patterns are overall consistent with explicitly simulated damage patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Minor discrepancies occur in the generation but subsequently vanish when explicit damage evolution continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' for instance under increased load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The presented approach provides a methodological basis for adaptive multi-scale simulation schemes with consistent damage evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Keywords: Lattice, Continuum damage mechanics, Microstrutured disordered material, Anisotropic damage, Multi-scale simulation, Harmonic decomposition, Damage modelling ∗Corresponding author Email addresses: phimuell@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='ch (Philip P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Müller), fwittel@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='ch (Falk K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Wittel), dkammer@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='ch (David S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Kammer) Contents 1 Introduction 3 2 Materials and Methods 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 Generic Damage Transforming Method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 13 4 Summary and Conclusion 15 5 CRediT 15 6 Declaration of Competing Interest 15 7 Data Availability 15 Appendix A Parameters of �rx(κx) and �ry(κy) 16 References 17 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Introduction At a certain scale even heterogeneous materials will appear homogeneous and some can even be considered isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Among others, this is true for concrete, one of the most widely used commodity on earth, a mixture made of sand, aggregates, cement, water and chemical ad- mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The growth of damage inside concrete is highly affected by the particular microstructure, where, depend- ing on the scale, aggregates or even sand grains act either as focal points for stresses or obstacles for damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Damage initiates at very small scales, long before the macroscopic structure itself will fail or crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Instead, the damage leads to a reduction of the material’s stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Nevertheless, at one point the accumulated damage be- comes so widespread, that even its smallest increase, will trigger the previously isolated nuclei to merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This leads to a cascade of increasingly larger defects, culminating in the emergence of a macroscopic crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Continuum based methods are the methods of choice if large structures should be simulated, due to their com- putational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For taking into account intrinsic de- generative processes, constitutive laws are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' One of the earliest, but still widely used laws for modelling dam- age in concrete was proposed by Mazars (Lemaître, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Mazars and Lemaître, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It employs a scalar dam- age variable to degrade the material’s stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, even if the material was initially isotropic, damage will induce anisotropy into the material’s behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Clearly any scalar damage variable is inherently unable to capture this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' During the years, a variety of anisotropic damage models were proposed to address this issue (Brancherie and Ibrahimbegovic, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Braun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Delaplace and Desmorat, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Desmorat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Gaede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' All of them consider the ac- cumulated effects of the damage’s growth, represented by internal state variables at the material points and by that disregard the actual microstructure, whose degeneration is the actual cause for the emerging damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To overcome this deficiency, the entire microstructure could be explicitly represented and simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Unfortu- nately, even with today’s fast computers, this is only pos- sible for small sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' A way to overcome this barrier are multi-scale methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' They allow to invest computational power exactly where it is needed, by combining different representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Although, many different methods were proposed over the past years, they can be classified to be either of hierarchical or of concurrent nature (Liu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Hierarchical methods are characterised by a full sepa- ration of scales, which allows to treat every level indepen- dently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, the information is passed between the different levels as one serves as input for the hierarchically higher level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Opposed to this, concurrent methods lack the full sepa- ration of scales and typically decompose the computational domain into different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Imagine a typical setting where high accuracy is only needed inside a small part of the computational domain, for example around a crack tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Ideally, one limits methods with high accuracy but large computational burden to these small regions, while the remaining part of the computational domain is described by much more efficient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The flow of information between the different regions must be handled by a cou- pling scheme such as the Arlequin method (Anciaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Bauman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Guidault and Belytschko, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Unger and Eckardt, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Wellmann and Wriggers, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Whenever the decomposition is not available in advance, one must resort to adaptive methods to refine regions on demand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Evangelista, Alves, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Evangelista and Moreira, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Rodrigues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, important questions are (i) how are the regions that need to be refined identified, and (ii) how is the loading history of the coarse connected to the initial state of the newly created fine scale representation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Espe- cially (ii) does not seem to be addressed well in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Most authors assume that the coarse representation does not accumulate any damage before being refined (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Rodrigues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Unger, Eckardt, and Konke, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Consequently, damage is only allowed to evolve inside of the fine scale representations that start off as undamaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is in- consistent because it essentially disregards the entire load history, including the damage, that would have degraded a real material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In this paper we propose, to the best of our knowledge, a generic approach for the refinement step in adaptive con- current multi-scale simulations, that is able to account for the preceding damage evolution inside the coarse repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, the created fine scale representation con- tains an initial damage that is mechanically consistent to the damage that has evolved inside the coarse represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Our solution is to equip both, the fine and coarse scale representations, with their own damage measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We analyse these damage measures and establish a connection between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Our approach is actually able to address both questions raised above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' By interpreting the coarse damage measure as a “measure of suitability”, critical re- gions that require refinement are regions whose damage measure surpassed some predetermined threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further, the coarse damage is used to initialise the fine scale damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While the approach is generic and rather simple, its practical details highly depend on the selected represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, we demonstrate it by applying it to one particular test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The reminder of this paper is organ- ised as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2, we explain our method in more detail and present the proposed techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3 we determine the parameters of our method and asses its ap- plicability, before we draw final conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Materials and Methods The particular choice of the material’s microstructure, also called motive, is in general arbitrary, but should fol- low principles of representative volume elements (RVE) (Lemaître and Desmorat, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The state of a discrete representation with its inherent characteristic structure is fully given by r, that describes every single discrete ele- ment (right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In this representation, damage D(r) is given by the irreversible degeneration of the con- stituting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' On the left side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 1, the smeared continuum representation is shown, which lacks such an explicit microstructure and only considers cumulative ef- fects of the damage through internal state variables added to the constitutive law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Here, damage is given by D, which depends on the state #–κ at a particular location and is em- bedded in the constitutive law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figure 1: The continuum damage, at a certain point, is given by D, which depends on the respective state #– κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The discrete damage D(r) depends on r and hence on the state of all discrete elements of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The two representations are interconnected to each other by homogenisation and refinement processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Scale of the lattice is exaggerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since the continuum representation loses its validity once cracks localise, one must refine the continuum to its discrete twin in such way that all important aspects of the fracture will be captured accurately on the fine scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The key for a meaningful adaptive modelling of the damage evolution lies in the transformation of the continuum to the discrete representation, that conserves the degraded mechanical behaviour found inside the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' One focus of this work is an approach to construct a discrete representation that respects the preceding damage present in the continuum representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Even though the procedure is generic and in principal not restricted to specific numerical material representa- tions, this paper focuses on one particular choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' How- ever, we will outline the generic way of working with the method (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1), before we start with our specific choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We exemplarily chose a two-dimensional plane stress, isotropic material (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2) with an under- lying material heterogeneity represented by a triangular network of beam-truss elements with linear-elastic, brittle behaviour with quenched disorder of breaking thresholds (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We then discuss the particular choice of the damage law as well as the reconstruction step (see Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To determine and test them, we use data obtained from numerical simulations (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Generic Damage Transforming Method Initially, the domain is described as a continuum with- out any internal structure, whose state is fully described by the continuum state variable #–κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In the continuum, the damage evolution is fully govern by the damage function �D(#–κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Therefore, �D(#–κ) can be interpreted as the macro- scopic damage, that is expected for a hypothetical dis- crete representation with identical loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus we can determine the function describing the macroscopic dam- age by homogenising the discrete damage D(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This leads to a perspective on the damage law that is different from the conventional one, where the damage law is calibrated against a physical material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Instead, here the law is cali- brated against a particular numerical representation of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' When the continuum model experiences a certain dam- age limit, it is no longer suitable and has to be refined to a discrete representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, this discrete state has to be consistent with the previous continuum representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This includes stiffness and damage, which have to be preserved as much as possible by the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Determining this reconstruction process challenging, since it is by its very nature not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Continuum Representation of 2D Isotropic Continua To represent a two-dimensional isotropic material un- der plane stress, the Finite Element Method (FEM) and as damage measure continuum damage mechanics (CDM) is used (Lemaître, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Lemaître, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Lemaître and Desmorat, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We use the well known material law: σ = (I − D) Cε, (1) where σ and ε denote the continuum stress and strain ten- sors, respectively, and C is the continuum stiffness tensor of the undamaged material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Due to the choice of CDM, as continuum damage measure, the damage variable D can directly be identified with the damage function �D(#–κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Fur- ther, we identify #–κ as the continuum state variable given as #–κ := �κx κy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (2) A zoning approach is used to divide the principal strain space along an angle χ, known as zone boundary, into an x- (shaded red parts) and y-zone (shaded yellow parts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The two components κx and κy represent the max- imal reached principal tensile strain in x and y direction, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' κx := max � κx, ⟨ε1⟩+ � , κy := max � κy, ⟨ε2⟩+ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While ε1 and ε2 are the eigenvalues of the strain tensor ε, its eigenvectors form a Givens rotation matrix of angle Γ, which is sometimes called eigenangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The angle Γ, together with the boundary χ, determines which zone the eigenvalues are associated with, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 4 Figure 2: Interpretation of the zone boundary parameter χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While ε1 and ε2 are the eigenvalues of ε, its eigenvectors are described by the value Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The eigenangle Γ and the zone boundary χ determines which eigenvalue acts in which direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since the continuum damage is only used during the initial phase with low damage, two assumptions are made: (i) It is assumed that the damage is orthotropic which re- duces D and �D(#–κ) to diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (ii) It is assumed that κx only acts on the x-damage while κy only affects the y-damage, which means that we assume no correlation between the directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Exemplary Material Motive The example material motive chosen here is based on models proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (Herrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Mier, 2017), namely a regular triangular lattice but formed by 3rd order Reddy truss-beam elements with characteristic lattice size ℓ (Reddy, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Using beams allows to include bending properties and the re- sulting lattice is able to represent a Cosserat continuum (Ostoja-Starzewski, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Vardoulakis, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The mi- croscopical beams consist of an isotropic material with Young’s modulus Eb and Poisson’s ratio νb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' A list of all used material parameters is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In a multi- scale simulation, Eb has to be chosen such that the result- ing behaviour of the discrete structure matches the one of the continuum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' its stiffness tensor C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, since this paper studies the refinement step in isolation, with- out having an actual continuum phase, the choice of Eb is actually irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Lattice Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The motive is defined by the number of nodes (Nx, Ny) in x- and y-direction, the spatial extension in x-direction Lx, with resulting characteristic lattice size ℓ := Lx/(Nx −1) and spatial y-extension Ly := Nyℓ √ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' An out-of-plane height of H is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To remove the symmetries of the lattice, topological disorder is intro- duced (Moukarzel and Herrmann, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Wittel, 2006) by adding the random displacement #–x ∆ i := a ℓ 2 #–x ∗ i (3) to every internal node of the grid, where #–x ∗ i is a random vector sampled uniformly from the unit circle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Table I: Parameters of the discrete material motive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Property Value Unit Nx, Ny 300, 346 [−] Lx, Ly 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='998 m H 1 m Eb 1 × 106 Pa νb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 [−] kε 3 [−] λε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='02 [−] kΦ 3 λΦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='02 [−] The distortion is controlled by parameter a ∈ [0, 1[, known as distortion level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figure 3: (a) Distortion of the central node, ignoring the distortion of the surrounding nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The location of the distorted node (yellow circle), is randomly selected within the blue circle of radius aℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Afterwards, the length of the beams are adjusted to match the new node location (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (b) The thickness of beam i is given as ti := A(O) i /ℓi, where ℓi is its length and A(O) i is the area the beam is representing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Points zL and zK are centres of the adjacent triangles’ incircles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Geometrical Properties of Beam-Truss Elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The thickness of beam i, denoted as ti, depends on the lat- tice’s geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It is given as ti := A(O) i /ℓi, where A(O) i is the area represented by the beam and ℓi its length, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The area A(O) i is formally defined as the set of points that are closer to beam i than any other beam, but are inside the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It can be determined by finding the intersection of the angle’s bisectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' centre of the incir- cle, of the two adjacent triangles denoted as zK and zL in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In case the beam is part of the boundary A(O) i is artificially doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This ensures that in a regular lattice all beams have the same axial rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Damage Criterion Applied to the Beam-Truss Lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In the discrete representation, damage is the irreversible fail- ure of elements, namely the reduction of their contributing stiffness to an insignificant level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To determine if a beam has surpassed its loading capacity, the elliptical criterion � εi εi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' th �2 + max ����Φ(r) i ��� , ���Φ(l) i ��� � Φi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' th =: Ψi ≥ 1 (4) is used, where εi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='th and Φi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='th are the beam’s elongation and bending thresholds, respectively (Herrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', 5 KicKy E2a (b) ZK a ZL Φ (r)1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Both thresholds are sampled independently from the Weibull distributions εi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='th iid ∼ Weib (kε, λε) and Φi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='th iid ∼ Weib (kΦ, λΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The Discrete State Variable #–r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The discrete state is uniquely described by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, for the context of this paper the surrogate discrete state variable #–r := � rx ry � (5) is introduced and termed “discrete state variable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since #–r has only two components it does not uniquely describe the damaged state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This ambiguity will be resolved by the reconstruction process (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' #–r is a purely mathematical quantity designed to have certain properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' First, its 1-norm �r := ∥#–r ∥1 := |rx| + |rx| equals to Nf/NT, where Nf is the number of failed beams and NT the total number of beams in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' �r is also called the ratio of failed beams (rfb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Second, its components are defined by associating them to the x- and y-zone, respectively, similar to #–κ (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' But while κx is connected to strains in the x-zone, rx is related to the amount of beams that have failed due to κx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Determining the Damage Law for the Continuum The damage function �D(#–κ) will take the role of the damage variable D inside the constitutive equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, �D(#–κ) has to be designed such that its evolution mimics the expected behaviour of D (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For the extraction, which involves two steps, the Uniform- Sim simulation data of fully discrete lattices is used (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Step 1: Effective Material Stiffness Tensor C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' First, the effective stiffness tensor C is calculated by homogenisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' After the convergence of each loading step, the following seven strain-states � � � � � 1 2 3 � �, � � 4 5 0 � �, � � 6 0 0 � �, � � 0 7 0 � �, � � 0 0 8 � �, � � 9 0 10 � �, � � 0 9 8 � � � � � , (6) denoted as (εxx, εyy, 2εxy)T × 10−3, were applied to the lattice, while blocking further damage to measure the re- sulting stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This results in an overdetermined system of 21 equations for the 6 unknown coefficients of C, which is solved by a least-square approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Step 2: Determining the Damage Variable D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Second, the damage variable D is extracted from the effective stiff- ness tensors of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For this, a technique originally presented by Oliver-Leblond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (2021) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For com- pleteness, the relevant equations are replicated to be d(T ) := tr1,2[T ] = Tkkij, (7a) K := 1 4 tr � d(C) � , (7b) D := D � C, �C � := 1 2K � d(C) − d � �C � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (7c) The tensor defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (7a) is also known as dilatation second order tensor, while scalar K of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (7b) is the bulk modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (7c) combines the effective C and undam- aged stiffness tensor �C to the damage variable D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' D is by construction a real symmetric 2 × 2 matrix, thus fully characterised by its two eigenvalues d(x) and d(y) as well as a single scalar Γ, describing the rotation of its eigenbasis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Process for the Construction of a Damaged Lattice The reconstruction process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' the creation of a dis- crete lattice with a particular damage, involves two compo- nents: (i) The transfer function #–�r (#–κ), which transforms the continuum state #–κ into the discrete surrogate state variable #–r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (ii) A scheme which transforms the surrogate state #–r into the full discrete state r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Hence, the scheme must be able to resolve the inherently present ambiguity in #–r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As direct consequence of the definition of the discrete state #–r (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (5)), the transfer function is given as #–�r (#–κ) := ��rx(κx) �ry(κy) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (8) As for the damage function �D(#–κ), we are using data ob- tained from the UniformSim simulations (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1) to empirically determine the function #–�r (#–κ) that approxi- mates #–r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Due to the nature of #–r it is impossible to mea- sure its components and thus to fit them directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' How- ever, it is easy to measure and fit the quantity �r := ∥#–r ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Because of the specific design of the simulations and as- sumptions, it is possible to associate the value �r to the components of #–r , see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For reconstructing the full discrete state, a probabilis- tic scheme was devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It starts by constructing an un- damaged lattice from which certain beams are removed, such that the resulting damage matches in a statistical sense the one given by #–κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Due to the assumed decoupling between the x- and y-zone, it is possible to handle the two directions independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For each direction α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' x and y, the following steps must be done: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' From the continuum state κα the corresponding dis- crete state variable, rα = �rα(κα) is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Through the relationship Nα := rα·NT , it is possible to determine how many failed beams are associated to this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Each beam is assigned a probability defined as pi ∝ 1 εi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' th ��� �#–b i, #–t α ���� k , (9) where εi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' th is the elongation threshold and #–b i the di- rection of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The vector #–t α, called “damage basis”, represents the main damage direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In our motive, it is either #–t x := (1, 0)T or #–t y := (0, 1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Finally, parameter k, called “directional weight”, is 6 a tuning parameter that balances the relative impor- tance of the two terms and needs to be determined (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The Nα many beams to fail are drawn from the prob- ability distribution defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (9) without re- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The selected beams are marked as failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Numerical Simulations For estimating and testing the damage function �D(#–κ) and the transfer function #–�r (#–κ), a series of different numer- ical simulations are carried out on fully discrete lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We employ for this a customised version of the Akantu FEM library (Richart and Molinari, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Due to the randomness of the lattice, 30 realisations were made for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The UniformSim Simulation Setup The first type of simulation, called UniformSim, is used for estimating the transfer function #–�r (#–κ) and the damage law �D(#–κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' These simulations realise an uni-axial strain Figure 4: Boundary conditions used by the UniformSim series, shown for the case of ϕ = 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In general, the boundary conditions given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (10) are applied to the whole boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Scale of the lattice is exaggerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' state of the lattice that is also rotated by an arbitrary but constant angle ϕ, called the pull direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus εϕ := Rϕ T ��ε1 0 0 0 � Rϕ, (10) where Rϕ is the Givens rotation matrix for angle ϕ that is applied to the lattice’s boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In each loading step, �ε1 is increased by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0001 until 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The limit is chosen to ensure that no localisation will occur and that damage maintains its diffuse character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The particular setup of the UniformSim simulations together with the previous assumptions on D and #–�r (#–κ) allows the following conclusions and simplifications: (i) A pull direction is either associated to the x- or y- zone (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This allows to probe the be- haviour of a single zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Which zone is probed de- pends on ϕ and the yet unknown zone boundary value χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (ii) A simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' a particular value of ϕ, will only affect the state of either the x- or the y-zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, an increase of �ε1 will only affect one eigenvalue of D and a single component of #–κ as well as #–r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Which component is affected depends on ϕ and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (iii) For the continuum state variable the relation �κ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= ∥#–κ∥1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= |κα| !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= �ε1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, one component equals the applied uni-axial strain �ε1, while the other is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (iv) For the discrete state variable, the relation �r !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= ∥#–r ∥1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= |rα| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, only one component is non zero and equals �r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This can be used to determine the compo- nents of #–r from �r, once the x- and y-zones are known (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The MultiLoadSim Simulation Setup For testing the damage function as well as the recon- struction procedure, a second type of simulation is used, called MultiLoadSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It realises a bi-axial strain state, imposed along the x- and y-axes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Both strains are increased until εxx = εyy = εfin is reached, where εfin is the control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For each simulation, the loading is imposed in three different ways, but each time the same initial lattice is used: XThenYSim: εxx is increased in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0001 until it reaches εfin and then maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Then εyy is in- creased by the same increment until εfin is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' YThenXSim: The same as XThenYSim, however, the order of loading the axes is switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' BothXYSim: Both strains εxx and εyy are increased simul- taneously, in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0001 until εfin is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' All three paths reach the same final state, εxx = εyy = κx = κy = εfin, but via different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As a consequence, the special relation �κ := ∥#–κ∥1 = |εxx| + |εyy| holds in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Both, the XThenYSim and the YThenX- Sim loading path impose in the first half of the loading an uni-axial strain state and then switch to a bi-axial strain state for the second half, while the BothXYSim loading path imposes a bi-axial strain state from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figure 5: Boundary conditions used in the verification simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Scale of the lattice is exaggerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The ReconstrSim Simulation Setup For testing the reconstruction process (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5), a third type of simulation is used, called ReconstrSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The basic setup is equivalent to UniformSim, but for ϕ = 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further, a lattice with a certain initial damage is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This damage was constructed to match the continuum state #–κ = (�ε, 0)T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' the damage created by an uni- axial strain of �ε applied along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Another important difference is, that the loading does not start at zero but at �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In case of a perfectly working reconstruc- tion process, one would expect no additional damage for strains ≤ �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Unlike the MultiLoadSim tests, which focuses on the value of the reconstructed damage, these tests focus on how the reconstructed lattices behave after their recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In essence, this test simulates the exchange of the continuum representation with the discrete represen- tation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' the refinement process, which is the core ap- plication of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Results Our proposed method relies on the damage law �D(#–κ) used inside the continuum and the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' First, we discuss how we will use the techniques introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2 to process the data that we have collected from the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Then, we determine the dam- age law �D(#–κ) and the zone boundary value χ (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2) followed by an assessment of its accuracy (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thereafter, we repeat the process to determine the trans- fer function #–�r (#–κ) and the directional weight parameter k (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Finally, we demonstrate the applicability of our method (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Details of a Numerical Simulation Let’s consider a setting similar to UniformSim but with ϕ = 0° (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Only one realisation is simulated and the loading goes beyond �ε1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We measure the normalised stresses (solid lines) and compare them with the expected ones in case of suppressed damage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' undamaged case (dashed lines) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As expected, initially the stresses behave predominantly lin- ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, once a strain of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='003 is reached, we observe that �σxx starts to deviate from the undamaged case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While this deviation increases with further load- ing, we cannot observe it for �σyy, that is much less af- fected by the loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' At one point, we observe that both stresses suddenly drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is caused by the emergence of a macroscopic crack, which is the expected behaviour for a brittle disordered material, such as concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Using the techniques presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4, it is possible to extract the macroscopic damage variable D for the lattice, at any loading step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6c, we show the eigenvalues of the extracted damage variables where we can see that d(x) exceeds d(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This also explains why �σxx deviates much more from the undamaged behaviour when compared to �σyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The underlying reason of this difference are the hor- izontal beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' They experience much larger strains than inclined beams, since they align with the loading and thus fail at much larger number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6c, we can also see that d(y) drops for a strain at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is a non- physical behaviour as damage should always increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It is caused by some numerical issues during the determination of the stiffness tensor (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4) and the sensitivity of damage extraction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6d shows the ratio of failed beams (rfb), �r := ∥#–r ∥1 := Nf/NT, where Nf is the total number of number of failed beams and NT the total beams in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We will use it to indirectly estimate the transfer function #–�r (#–κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6e-h show snapshots of the lattice’s underlying microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While taken at different loading steps (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6d), they always show the same set of nodes, located roughly at the lattice’s centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While the exact damage pattern depends on the realisation of the lattice and lo- cations where the snapshot was taken, statistically they all look the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It is this statistical damage pattern that we want to capture by the transfer function #–�r (#–κ) and recreate by the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Whereas the damage law �D(#–κ) captures the accumulated effects on the lattice’s macroscopical stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Estimation of the Damage Law �D(#–κ) We now study the behaviour of the damage variable D that we have extracted from the data of the UniformSim simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' From these observations, we will determine the damage function �D(#–κ) as well as the zone boundary value χ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Functional Form of �dx(κx) and �dy(κy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since we have assumed an orthotropic damage variable (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2), we have to assume the same for the damage function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus arriving the tentative form of the damage function is given by �D(#–κ) := �dxx(#–κ) 0 0 dyy(#–κ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To account for deviations from this assumption, we will connect the two diagonal elements of the damage func- tion with the eigenvalues of the measured damage vari- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, we have only two functions that we need to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 7 shows the evolution of the eigenvalues d(x) and d(x) from the extracted damage variable for the pull di- rections ϕ ∈ { 0°, 60° } at various distortion levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For ϕ = 0°, the eigenvalue d(x) is much larger than d(y), while for ϕ = 60° the opposite is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Later, we will use this to determine the zone boundary value χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Most im- portantly, the figures show that both eigenvalues follow a power law, irrespective of the pull direction and distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus we approximate the diagonal elements/eigenvalues of 8 Figure 6: Representative simulation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (a) Schematics of the model configuration, scale of the lattice is exaggerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (b-d) Evolution of continuum and microstructure properties of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Dotted lines denote strains beyond the limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 used in UniformSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (b) Normalised measured stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Dashed lines represent the behaviour in case of suppressed damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (c) Eigenvalues of the extracted damage variable D, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (d) Ratio of failed beams �r in the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (e-h) Snapshots of a small section of the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Colours indicate the remaining load carrying capacity of the beams � Ψi := 1 − Ψi, where Ψi is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Associated states are indicated in (d) by markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Bending of beams is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' �D(#–κ) as: dxx ≈ �dx(κx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' a, ϕ) := α(x) a,ϕ · κx β(x) a,ϕ, (11a) dyy ≈ �dy(κy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' a, ϕ) := α(y) a,ϕ · κy β(y) a,ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (11b) The parameters of these approximations depend on the distortion level a and the pull direction ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Later, we will eliminate their dependency on ϕ and obtain the final pa- rameters that only depend on a, which is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Fur- ther, this choice guarantees that the damage is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Because of our previous assumption about the indepen- dence of the directions, the approximations of the eigenval- ues only depend on a single component of the continuum state #–κ (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While this could be justified due to their large differences, that we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 7, we clearly see that even for ϕ = 0°, there is a certain coupling between d(x) and d(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To handle this, we use a simple coupling scheme, which leads to the final damage function �D(#–κ) := (12) � � � � max � �dx(κx), � dy(κy) η � 0 0 max � �dy(κy), � dx(κx) η � � � � �, where �dx(κx) and �dy(κy) are the approximations of the eigenvalues defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (11) but without the depen- dence on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The coupling ensures that the eigenvalues of the damage function �D(#–κ) will at most differ by a factor of η, which is exactly what we see in the case of uni-axial loading (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Here, we will assume that the em- pirical parameter η equals 10 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We will later give a justification of the form and value of the proposed coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It is important to notice that this coupling is designed for the uni-axial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, a more elabo- rated coupling might be needed, depending on the details of other material motives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Parameters of �dx(κx) and �dy(κy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since the data, espe- cially for the non-dominant eigenvalue shows strong vari- ation for small strains, only data points corresponding to 9 a) ouy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 <6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='000 ( d(c) d(y) 4 6 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='02 (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='01 (g) (f) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='008 remaining load carrying capacity := 1 - Err [-] >0Figure 7: Eigenvalues of the extracted damage variable D for pull direc- tions ϕ = 0° (a) and 60° (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Solid lines correspond to d(x), while dashed lines correspond to d(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Colours indicate different distortion levels a of the underlying lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' strains above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 were used for the parameter estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 8a,b, we see that for small values of ϕ, the α(x) a,ϕ-parameters are very close to each other, while for larger values of ϕ one observes a much larger scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Interestingly, α(y) a,ϕ-parameters behave inversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further- more, on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 8b we can clearly observe the α(y) a,ϕ depen- dence on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We see that α(y) a,ϕ is small if ϕ is small too, but above a certain value of ϕ, the parameters become much larger and their scattering increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The same, but in an opposite way, holds for the α(x) a,ϕ-parameters but in a less pronounced fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The estimates for the β-parameters (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 8c,d) show a similar behaviour with respect to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, while we observed a significant change in the behaviour of the α-parameters’ values, from a particular value of ϕ on we just observe an increase of the variability of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In summary, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 8 we can conclude that the β- and especially the α(y)-parameters have different regimes de- pending on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further, inside such a regime, their partic- Figure 8: Values of the α- (a,b) and β-parameters (c,d) with respect to the pull direction ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Colours indicate different distortion levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Solid lines correspond to lg α(x) a,ϕ and β(x) a,ϕ, while dashed lines to lg α(y) a,ϕ and β(y) a,ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Error bars indicate the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' ular value does not depend much on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We also saw that the values for the β(x)-parameters for small values of ϕ and β(y)-parameters for large values of ϕ are both close to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This means that the growth behaviour of �dx(κx) and �dy(κy) are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This jus- tifies the form of the coupling used in the damage function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Zone Boundary χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 7 and 8, we have observed that depending on the pull direction either d(x) or d(y) is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We now exploit this fact to define χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To this end, we define the dominance function ζ as: ζ(a, ϕ) := lg � d(x) a,ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' ˜κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 d(y) a,ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' ˜κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 � , (13) with d(α) a,ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' ˜κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 as the damage eigenvalue associated to direction α, once the uni-axial strain has reached 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° 10-2 d(r) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 d(r) aα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 d(r) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 10-3 d(r) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 d(r) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 10-4 ~ d(y) (α)p 10-5 10-6 10-7 L (b) Φ = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° 10-2 d(y) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 d(y) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 10-3 d(y) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 d(y) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 10-4 ~ d(r) (6)p 10-5 10-6 10-7 10-4 10-3 [-](a) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 (b) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 [-] ° 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 (c) + a= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 二 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 0 20 40 60 80 [] The most important aspects of this function are its sign and root, to a lesser extend its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' ζ > 0 means that d(x) is dominant, while ζ < 0 indicates that d(y) is domi- nant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, χ, which might depend on the distortion a, is defined as ζ(a, χ) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figure 9: Dominance function ζ(a, ϕ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 13, for different distortion parameters a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The x-dominated region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' d(x) ≫ d(y), is defined by ζ > 0, while the y-dominated (grey shaded) region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' d(x) ≪ d(y), is defined by ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The UniformSim data was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Examining Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 9, we see that, irrespective of the distor- tion, χ must lie between 30° and 45°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' After some exper- imentation, we decided to use 40° as zone boundary, ir- respective of the distortion level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' A closer analysis might yield different estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' ζ can be seen as a measure of the coupling between d(x) and d(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, we can used it to determine the value of the empirical coupling parameter η, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Our value η = 10 was selected because it is roughly the mean value for ϕ = 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Final Parameters of �dx(κx) and �dy(κy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Eliminating the dependency of the α- and β-parameters on the pull di- rection ϕ will results in parameters that are valid inside the entire x- or y-zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For this, we combine the different estimates as: lg α(x) a := 1 |X| � ϕ∈X lg α(x) a,ϕ, lg α(y) a := 1 |Y| � ϕ∈Y lg α(y) a,ϕ, (14a) β(x) a := 1 |X| � ϕ∈X β(x) a,ϕ, β(y) a := 1 |Y| � ϕ∈Y β(y) a,ϕ, (14b) where X contains all the pull directions associated to the x- and Y the ones associated to the y-zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Parameters associated to the transversal directions are simply ignored, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' lg α(y) a,ϕ=0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further, the functional form of �dx(κx) and �dy(κy) is still given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (12), just without the depen- dency on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (14) weights the different pull directions equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Test of the Damage Evolution Law �D(#–κ) We now evaluate how well the damage function is able to predict the damage of a fully discrete simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For this purpose, the MultiLoadSim simulations are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figure 10: Damage eigenvalues for the three different loading paths, de- scribed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2, with final strain εfin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002, plotted against τ := ˜κ/2 εfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Using a fully discrete simulation (solid) as reference and the CDM damage law � D(#– κ ) (dash-dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The colours indicates the three different loading paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The distortion of the lattices was a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10 shows the results of such an experiment for εfin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We can see the eigenvalues, once computed for the reference (solid), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' a fully discrete simulation, and alternatively computed by the damage function �D(#–κ) (dash-dotted), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' They are plotted against the normalised total strain τ := ˜κ/2 εfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, both the X- ThenYSim (orange) and the YThenXSim (green) load paths switch from an uni-axial to a bi-axial strain state at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We observe that irrespective of the loading path the same final damage values are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The value depends on the used method, since the final value of the CDM is different from the reference value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Note that this is not problematic since the CDM is only used during the initial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 a =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 c-dominated a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 [-] ( a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='7 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 ‘p)S y-dominated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 0 20 40 60 80 6e 10-3 10-4 10-5 (a)p 10-6 10-7 (b) 10-3 10-4 I 10-5 (r)p 10-6 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' BothXY 10-7 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' XThenY ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' YThenX CDM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 T := K/2efin [-]Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10 also demonstrates that the damage for the X- ThenYSim and YThenXSim are very similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, the eigenvalues are flipped, which is the expected behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' During the first half of the loading (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' τ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5), the prediction of the dominant eigenvalue matches well with the reference value for both loading paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' At the same time, the non-dominant eigenvalue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' the one belonging to the transverse direction, is captured with less but still acceptable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The mismatch is entirely due to the rather crude choice of the η coupling parameter (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, it indicates that the proposed coupling is indeed working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Nevertheless, for the second half of the loading (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5) the CDM is unable to capture the evolution to a satis- factory degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In case of XThenYSim (orange lines), we see that the CDM approximation of the x-eigenvalue d(x) re- mains constant, since κx is not affected by a loading along the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, we see that in the reference system d(x) continuously increase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 11 for more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The y- eigenvalue d(y) predicted by the CDM remains initially constant due to the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Once �dy(κy) has become larger than � dx(εfin)/η �dy(κy) starts to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, as it can be seen form Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10b, the reference d(y) starts to increase almost immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' A different case is the BothXYSim loading path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10, it seems that for τ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 its damage grows slower than the dominant damage observed for the other two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is because BothXYSim only has half the num- bers of loading steps the other two have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' If this is corrected for then it would actually grow faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This indicates that there is some form of coupling between the two directions that is not considered correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 11, we can see how the final damage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' val- ues of d(x) and d(y) at εxx = εyy = εfin, depend on the control parameter εfin, using either the reference (solid lines), the CDM (dash-dotted lines) or the reconstruction (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The colours distinguish the three different load paths that were tested (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The collapse of the lines indicate that the damage is indeed path in- dependent, regardless of the final strain εfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, the final value depends on the particular method that was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10, we observe a gap between the final damage attained by the reference and the one predicted by the CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We can now see that this gap is systematic and actually increases with larger εfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is again in- dicating that there is some form of coupling between the directions that is not take into account yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Estimation of the Transfer Function #–�r (#–κ) Our procedure to reconstruct a discrete lattice repre- sentation based on a damaged continuum state (presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5) requires two unknown quantities: First, the transfer function #–�r (#–κ), which maps the continuum state #–κ to the corresponding discrete surrogate state #–r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Sec- ond, the directional weight parameter k, which balances the orientation and the strength of a beam during the re- Figure 11: Final value of the d(x) and d(y) damage eigenvalues, com- puted using the reference (solid), CDM (dash doted) and reconstruction (dashed) method, plotted against εfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The colours indicate the three different loading cases from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' All lattices have a distortion of a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' construction process (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Analogously to the determination of the damage function, the data from the UniformSim is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Functional Form of �rx(κx) and �ry(κy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As mentioned be- fore, it is impossible to measure the components of #–r di- rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, as outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 #–r is connected to the ratio of failed beam as �r = ∥#–r ∥1 = Nf/NT !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='= rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, we can estimate #–r indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 12 shows �r for the pull direction ϕ = 0° at various distortion levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As we can see, the distortion level has only minor influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Dif- ferent pull directions do not lead to a qualitative change (data not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For that reason, we approximate the mean rfb as �r ≈ ���#–�r (#–κ) ��� 1 := �r(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' a, ϕ) := α(r) a,ϕ · κβ(r) a,ϕ, (15) with the two fit parameters α(r) a,ϕ and β(r) a,ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Both depend on the distortion a and the pull direction ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Due to our 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='004 (c)p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='000 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='007 k = 6, a= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 王 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' BothXY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='006 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' XThenY ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' YThenX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 CDM 王- PD, k = 6 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='004 (r)p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0025 fin [-]Figure 12: ⟨˜r⟩ := � ∥#–r ∥1 � for pull direction ϕ = 0°, at different values of the distortion parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' No qualitative change is observed for different pull directions ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' previous assumptions, we can identify its argument κ di- rectly with �κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To eliminate the dependency on ϕ we use the same method as for the damage function (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, parameters associated to pull directions in X are used to determine �rx(κx), while the ones belonging to Y determine �ry(κy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This will transform the approximation of the scalar quantity ���#–�r (#–κ) ��� 1 into the one for #–�r (#–κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For the discussion about the estimated α- and β-para- meters please see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Directional Weight Parameter k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The empirical tuning parameter k influences the selection of beams during the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It balances a beam’s strength, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' its elongation threshold εth, against how well it aligns with the damage basis #–t α (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We determine k such that the reconstructed damage variable D resembles the reference damage D most closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' To this end, we define Υk:= ��D11−D11 �� ℓ2+ ��D22−D22 �� ℓ2+2 ��D12−D12 �� ℓ2 (16) as a measure of separation between the two damages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For minimising Υk, we select a heuristic approach, in which the reconstruction process (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5) is run for differ- ent values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The k that minimises Υk will then be used for the remaining part of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, for this particular reconstruction process, the used α(r)- and β(r)-parameters still depended on ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further, zoning was ignored and as damage basis the pull direction ϕ was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The underlying lattice was not distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 13 we see that k = 6 minimises Υk independent of the pull di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We also see that pull directions { 30°, 90° } seem to be almost unaffected by k, however, their values match Υk=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is an artefact caused by the regular structure of the underlying lattice and the scalar product used in the definition of the selection probability (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' How- ever, this artefact is indicating that k = 6 is indeed a good Figure 13: Υk, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (16), for some values of k and various pull directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The reconstruction process was done on regular grids, without zoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Further the pull direction and its orthogonal was used as damage basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Tests of the Reconstruction Process Now we evaluate the performance of the proposed reconstruction scheme to create a mechanically equiva- lent lattice, based solely on the continuum state #–κ (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For verification, we use the MultiLoadSim sim- ulations (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In addition, we use the Re- constrSim simulations to simulate a refinement step (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The MultiLoadSim Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 14, we see the re- sults from the MultiLoadSim setup with εfin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' They impose the bi-axial strain state εxx = εyy = εfin, but with loading applied via three dif- ferent paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We used this setup before to assess the CDM (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' An important note concerning the recon- structed states is, that in each loading step the lattice and hence the damage is constructed anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, although it looks like a damage evolution, the damage at any loading step has no connection to the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, each time the same undamaged but distorted lattice was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' If we now compare the damage from the references (solid lines) with the one from the reconstructed lattices (dashed lines) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 14, we see that the overall dam- age values are very similar to the ones obtained by the CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As before, we observe that for τ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 the dom- inant eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' d(x) for XThenYSim and d(y) for YThenXSim, are captured well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Then, for τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5, these reconstructed eigenvalues stop growing and thus deviate from the references (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' An effect we observed for CDM (dashed lines), too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' But if we look at the other eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' d(y) for XThenYSim (orange) and d(x) for the YThenXSim (green), we see that they start to increase almost immediately like the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This was not the case for the CDM (dash-dotted lines shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 13 10-2 0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° 00=D 10-3 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 10-4 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 10-5 10-6 10-7 10-4 10-3 i [-]a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° § = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° § = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° § = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° P = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° Φ = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0° 6 10-4 1 4 6 8 k [-]Figure 14: Damage eigenvalues for the three different loading paths, de- scribed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2, with final strain εfin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002, plotted against τ := ˜κ/2 εfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Using fully discrete simulations (solid) as reference and the reconstructed damage (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The colours indicate the three different loading paths that were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The distortion of the lattices was a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 10 for the damage evolution predicted by the CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The reconstruction process is affected by the ignored cou- pling between the directions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, the damage eigenvalues generated by it follow the reference much bet- ter than the ones computed by the CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The ReconstrSim Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Now we are using the Recon- strSim simulation setup, described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The lat- tices that are used here were reconstructed for the con- tinuum state #–κ = (�ε, 0)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The system is loaded under uni-axial strain along the x-axis, starting at �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This setup simulates how a discrete region that was loaded up to �ε, as continuum and then refined behaves upon further loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 15, we see how the damage eigenvalues d(x) and d(y) (dashed and dotted lines, respectively) and the rfb �r (solid lines), evolve for different reconstruction strains �ε, indicated by different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The grey lines correspond to the reference without any reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 15 indicate the values for d(x), d(y) and �r have in the reconstructed lattice before any loading was applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The circles associated to ∥#–r ∥1 show, Figure 15: Behaviour of the d(x) and d(y) damage eigenvalues and �r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Colours indicate different reconstruction parameters �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Grey is the ref- erence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' no reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Circles indicate damage/rfb values of the lattices directly after reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Squares indicate damage/rfb values of the lattices for an applied strain of �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' that the reconstructed lattices have a matching rfb value �r which is a consequence of its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' It is, however, much more important, that the reconstructed d(x) eigen- value (circles), matches the one predicted by the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus the process is able to reconstruct the dominant eigen- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We are also observing that d(y) are not as well re- constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is a consequence of the assumptions that the components of #–r are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since Re- constrSim only impose strains along the x-axis, we have κy ≡ 0 ⇒ ry ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, the reconstructed y-eigenvalues we are seeing are caused by a directional sampling effect created during the reconstruction of the damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' How- ever, since d(y) is the non-dominant eigenvalue, we expect and allow that it is less well reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 15 indicate the state of the lat- tices after an uni-axial strain of �ε along the x-axis was applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The difference between a square and its associated circle proves that this strain causes the failure of additional beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' If the reconstruction process would work perfectly, any strain below or equal �ε should not lead to the failure of any beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Therefore, we might have re- moved the right number of beams and these were more or less correctly oriented, the selection of some of them was not fully optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Furthermore, we see that for subsequent loading steps, the damage and rfb continue to increase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While the observed values for the restored systems remain above the reference, the restored lattices slowly converge towards them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is because the damage created by the subse- quent loading steps starts to dominate the artificial one, 14 e 10-3 10-4 (a)p 10-5 10-6 10-7 (b) 10-3 10-4 工 (6)p 10-5 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3, fin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 10-6 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' BothXY ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' XThenY ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' YThenX PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' k = 6 10-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 T := K/2efin [-]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='003 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='001 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 d(c) 10-2 d(y) 10-3 10-4 k = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 10-5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='001 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='003 10-6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 restored: after loading restored: before loading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='005 i [-]that we created through the restoration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Summary and Conclusion In this study, we presented a generic approach for the creation of a discrete twin of a continuum representation containing an initial damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The discrete twin’s damage is created in such a way, that it is mechanically consistent to the original’s continuum damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is a step towards adaptive multi-scale simulations, which take the state of the coarse description of a region into account upon its refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' While the method is general and has no restrictions concerning the used numerical representations, we pre- sented it in form of a concrete example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' As continuum representation, we have used FEM with CDM as damage measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' For the discrete representation, we have used a lattice based on a triangular grid consisting of brittle beam-truss elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' One part of our method is the damage measure used inside the continuum representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This measure is used during the initial continuum phase to track the evolving continuum damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Unlike classical CDM, that are cali- brated to match the degradation of a particular material, we calibrated the CDM against the degeneration of the discrete numerical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Thus, it measures the degeneration that would occur on a hypothetical fine scale representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We saw that the determined CDM is indeed able to capture the damage caused by uni-axial strains to a sat- isfying degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, for bi-axial loading, the CDM is unable to achieve the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is explained by the as- sumption that the directions are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Directional coupling must be used to further improve the CDM’s ac- curacy The second part of our method is the ability to con- struct a discrete damage that is mechanically consistent to a given continuum state #–κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Since this problem is obvi- ously not unique, we devised a stochastic scheme to gen- erate representations containing such a particular discrete damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' We have seen, that the reconstruction process is indeed able to create discrete lattices, whose initial degeneration is consistent with the given continuum state #–κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' A draw- back is, that imposing a strain that corresponds to #–κ it- self, leads to the failing of some additional beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is indicating that the selection process needs to be further refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Furthermore, as for the CDM, we observed prob- lems for bi-axial strains, which are again caused by the independence assumed between the directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Nevertheless, our data is indicating that our approach works well for the case of uni-axial loading and is in prin- cipal able to work for bi-axial loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The next step is to integrate our method into an adaptive multi-scale simula- tion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' CRediT Philip Müller: Conceptualisation, Methodology, Soft- ware, Validation, Writing - Original Draft, Visualisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Falk Wittel: Conceptualisation, Writing - Review & Edit- ing, Supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' David Kammer: Conceptualisation, Writ- ing - Review & Editing, Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Declaration of Competing Interest The authors declare that they have no known compet- ing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Data Availability The simulation data generated in this study have been deposited in the ETH Research collection database avail- able at TBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 15 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Parameters of �rx(κx) and �ry(κy) The fitting parameters (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (15)) of the ratio of failed beams (rfb) �r, denoted as α(r) a,ϕ and β(r) a,ϕ were esti- mated in the same way as the ones for the two damage functions �dx(κx) and �dy(κy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' However, they behave much more stable and thus show less variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='16a Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='16: Dependence of the two fitting parameters of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' (15), lg α(r) a, ϕ (a) and β(r) a, ϕ (b), on the pull direction ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Colours indicating different distortion levels a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' The error bars is given by the 95% confi- dence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' show the values for the α- and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='16b for the β-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Compared with the parameters we obtained for the dam- age law (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 8), we see much less variability here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' This is because it is far easier to measure this quantity than the damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='075 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='050 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='025 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='975 a= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='950 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='1 a= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='925 a= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='3 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content='900 0 20 40 60 80 [oReferences Anciaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Coulaud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Roman, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Zerah (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Universität Stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' isbn: 3-930683-59-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Wu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' Zheng (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 731087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFJT4oBgHgl3EQfjiwF/content/2301.11574v1.pdf'} diff --git a/0dFQT4oBgHgl3EQfDjUj/vector_store/index.faiss b/0dFQT4oBgHgl3EQfDjUj/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b1510b95e080494ab4251e1084f21c9111eefa76 --- /dev/null +++ b/0dFQT4oBgHgl3EQfDjUj/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8b49c52cee60bab97dd5392732161cc2ae0a08194235ab8a990537cc8d5f1e81 +size 31981613 diff --git a/0tE3T4oBgHgl3EQfnAp3/content/tmp_files/2301.04621v1.pdf.txt b/0tE3T4oBgHgl3EQfnAp3/content/tmp_files/2301.04621v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a8e7f6fcaae11eb00066507eef778522edcb9f6 --- /dev/null +++ b/0tE3T4oBgHgl3EQfnAp3/content/tmp_files/2301.04621v1.pdf.txt @@ -0,0 +1,1870 @@ +Entropy of different phases formed by soft rods +Jayeeta Chattopadhyay,1 Shiang-Tai Lin,2 and Prabal K. Maiti1, a) +1)Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, +India +2)Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan +(Dated: 12 January 2023) +Computation of entropy in liquids and liquid crystal phases is a big challenge in statistical physics. In this work, we +extend the two-phase thermodynamic model (2PT) to shape anisotropic soft repulsive spherocylinders (SRSs) and +report the absolute values of entropy for different liquid crystal (LC) phases for a range of aspect ratios L/D = 2 − 5. +We calculate the density of states (DoS) for different LC phases and decompose it into contributions arising from +translational and rotational degrees of freedom. +The translational and rotational modes are further partitioned into +diffusive, gas-like, and non-diffusive, solid-like components using a fluidicity factor. In the dilute limit, the entropy +values obtained from the 2PT method match exactly those of an ideal rigid rotor. We find that, for a given packing +fraction, the magnitude of the total entropy is roughly equal regardless of the different LC phases associated with +different aspect ratios. We also compute the excess entropy (for L/D = 5) and compare those with the values obtained +using the standard integration approach of molecular dynamics (MD) or Monte Carlo (MC) equation of state (EOS) of +SRS. The values obtained using both approaches match very well. The rotational and translational fluidicity factors are +further used to determine the phase boundaries of different liquid crystal phases for the respective aspect ratios. +I. +INTRODUCTION +The phase behavior of shape anisotropic particles is an +emerging field of research that gives rise to various liquid +crystal (LC) phases1–3. Examples span from living organisms +like tobacco mosaic virus4–6, fd virus7 to synthetic systems +of rod-like particles like boehmite8, silica9 etc. Different liq- +uid crystal phases can be identified based on their microscopic +arrangements, as well as positional and orientational order. +Onsager, in his seminal work10, showed that a system of +thin and hard rods could undergo a phase transition from +disordered isotropic to orientationally ordered nematic phase +above a critical aspect ratio (L/D > 3.7) that is mainly driven +by entropy. The loss of orientational entropy in the nematic +phase is compensated by the increase of translational entropy +due to the ordered structure. +Similarly, for the other LC +phases, entropy plays an important role in studying the stabil- +ity of the phases. Entropy of a fluid can be expressed as a mul- +tiparticle correlation expansion of statistical entropy devel- +oped by Green and Nettleton11,12 and generalized by Lazaridis +and co-workers13,14 for the non-spherical bodies. Costa et al. +first used this method to calculate the entropy of a system of +hard spherocylinders (HSCs)15,16 and later, by Cuetos et al.17 +in a system of soft repulsive spherocylinders (SRSs). It is also +worth mentioning several interesting works by Dhar et al.18,19 +where they have calculated entropy of hard rods and rigid rect- +angles in 3D and 2D using analytically solvable lattice model +and MC simulations. +In 2003, Lin et al.20 developed the two-phase thermody- +namic (2PT) model to calculate the entropy, free energy, and +other thermodynamic properties of liquids from a short MD +trajectory (20 picoseconds (ps)). +2PT model has emerged +as an efficient and accurate method in calculating various +a)Electronic mail: maiti@iisc.ac.in +thermodynamic properties of Lennard-Jones fluids for the di- +verse setting of state points both in 2D21 and 3D20, water +in bulk22 and under different confinement, carbon dioxide23 +and other organic and inorganic molecules24. +The results +match very well with those of the experimental studies. In +the 2PT method, the density of state (DoS) of a liquid, which +is calculated from the Fourier transform of the velocity auto- +correlation function (VACF), is decomposed into vibrational +(solid) and diffusive (gas) components. The thermodynamic +quantities, including entropy, are then calculated using har- +monic oscillator approximation to the solid component and +hard sphere approximation to the gas component. For the ro- +tational mode, the diffusive part is calculated from the rigid +rotor approximation20,22. In 2PT method, the entropy of a +definite state point is calculated from a single MD trajectory. +Thus, it is far more efficient than the conventional integration +approach of MD or MC equation of state of the SRS, which +entails several discrete MD/MC trajectories along the integra- +tion path. This is advantageous for the systems for which the +analytical form of the equation of state is unknown (such as +SRS). +In this work, we extend the 2PT method to calculate en- +tropy of various liquid crystal phases formed by a system +of soft repulsive spherocylinders of different aspect ratios +(length/diameter) L/D = 2,3,3.5,4 and 5. We validate our +method by comparing the entropy values obtained using the +standard integration approach of equation of state of the SRS +of L/D = 5 at T ∗ = 516,17. +We find that the entropy val- +ues do not have any strong dependence on the aspect ratio +but strongly depend on the packing fraction (η)of the system. +We also find that LC phase transitions are governed by the +change of pair entropy. The loss of orientational pair entropy +in the nematic phase is compensated by the increase of trans- +lational pair entropy. Similarly, in case of the smectic phase, +the loss of translational pair entropy is compensated by the +residual entropy arising from the multi-particle contribution. +In addition, we present an alternative way to identify the phase +boundaries of different liquid crystal phases from the fluidic- +arXiv:2301.04621v1 [cond-mat.soft] 11 Jan 2023 + +2 +ity factor that is directly related to the diffusivity of the sys- +tem: the packing fraction at which the translational fluidicity +ftrans saturates but rotational fluidicity frot decreases sharply +indicates the phase boundary of the isotropic to nematic (I- +N) phase transition. Similarly, the nematic to smectic (N-Sm) +transition is located where frot saturates but ftrans keeps de- +creasing. +The rest of the paper is organized as follows: In section +II, we briefly describe the theoretical background of the 2PT +method and summarize the multiparticle correlation expan- +sions method and the integration approach of equation of state +to calculate the entropy of SRS; in section III, we describe the +SRS model and the simulation protocol. We present the results +and analysis in section IV. Finally, in section V, we conclude +with the discussion on the major benefits of the 2PT method +and possible applications. +II. +MODEL AND COMPUTATIONAL DETAILS +We model the system as a collection of spherocylinders +(cylinder with hemispherical caps) of aspect ratios L/D = +2,3,3.5,4,5. The interacting potential is only due to the ex- +cluded volume interaction described by the Weeks-Chandler- +Andersen (WCA) potential given as follows25: +USRS = 4ε[( D +dm +)12 −( D +dm +)6]+ε +if +dm < 2 +1 +6 D += 0 +if +dm ≥ 2 +1 +6 D +(1) +Here, dm is the shortest distance between two SRS that +determines their relative orientation2,17,26,27,34. +For conve- +nience, thermodynamic quantities are expressed in terms of +interaction strength ε, diameter of the SRS D and mass m: +temperature T ∗ = kBT +ε , pressure P∗ = Pvhsc +kBT , packing fraction +η = vhscρ, where ρ is the number density of the system +defined as, ρ = N +V and vhsc = πD2( D +6 + L +4) is the volume of the +spherocylinder; energy E∗ = E +ε , entropy S∗ = S +kB , Helmholtz +free energy A∗ = A +ε , Gibbs free energy G∗ = G +ε , diffusivity +d∗ = d( m +ε )1/2/D and the time t∗ = t +� +ε/m/D. To compute +entropy using 2PT method, we convert all the thermody- +namic quantities in real units using the parameters of argon +(ε = 0.238 kcal/mol, σ = 3.405Å and mass m = 39.948 +g/mol) and then again convert them into the reduced units. +We build the system in a hexagonal-closed-packed (HCP) +crystal structure. As the particles are inherently anisotropic +in shape, we choose the number of particles in the x, y and +z directions such that the simulation box can be built in a +near-cubic geometry. If nx, ny, nz are the number of particles +in the x, y, z direction respectively and nu is the number of +particles in one unit cell, then the total number of particles +in one simulation box N = nu × nx × ny × nz. In our case, +number of SRSs is chosen to be N = 1024. +The periodic +boundary condition in all three directions are used. +We have carried out a series of MD simulations for a wide +range of state points spanning the melting transition from solid +(crystal) to gas (isotropic) for all the aspect ratios. We melt +the initial crystal structure slowly by reducing the pressure in +NPT ensemble (Constant particle number, pressure and tem- +perature) at T ∗ = 5 for each L/D. The positions and velocities +of the SRSs are updated using Verlet algorithm28 and the rota- +tional motion by quaternion-based rigid-body dynamics29–33. +The temperature and pressure of the system are controlled +using Berendsen thermostat and barostat35 with a tempera- +ture relaxation time τT = 0.05 and pressure relaxation time +τP = 2 respectively. We perform 1×105 to 2×105 MD steps +(with an integration time step δt = 0.001 in reduced unit) to +reach equilibrium condition and another 2−5×103 steps (5- +30 ps in real unit using the above-mentioned parameters) with +δt = 5×10−4(1 fs)in real unit for the 2PT method. +III. +THEORY +A. +Two phase thermodynamic method +1. +Density of State Function +The density of state (DoS) function G(ν) is defined as the +mass weighted sum of the atomic spectral densities. +This +can be obtained from Fourier transform of velocity auto- +correlation function (VACF) obtained from MD trajectory20. +G(ν) = +1 +kBT +Natom +∑ +l=1 +3 +∑ +k=1 +lim +τ→−∞ +ml +τ +���� +� τ +−τ vk +l (t)e−i2πνtdt +���� +2 +(2) +Here, Natom is the total number of atoms in the system. ml is +mass of the lth atom and vk +l is the velocity of the lth atom in kth +direction ( k indicates spatial coordinates x,y,z respectively). +G(ν) represents distribution of normal modes in the system i.e +G(ν) dν represents number of normal modes in the frequency +range ν to ν + dν. So, total number of modes in the system +i.e degrees of freedom of the system 3N +� ∞ +0 G(ν)dν = 3N +(3) +The diffusion constant (D) of the system is directly related +to the zero-frequency density of state of the system G(0): +D = kBT +12mN G(0) +(4) +For a rigid SRS, there is no vibrational motion. So, total +number of degrees of freedom for a rigid SRS is 5 compris- +ing 3 translational and 2 rotational motion. Therefore, total +number of modes in the system is: +� ∞ +0 G(ν)dν = 5N +(5) +Density of state G(ν) is decomposed into translational and +rotational part: +G(ν) = Gtrans(ν)+Grot(ν) +(6) + +3 +where, Gtrans(ν) is obtained from the translational component +of the center of mass velocity of the SRS: +Gtrans(ν) = +1 +kBT +N +∑ +j=1 +3 +∑ +k=1 +lim +τ→−∞ +m j +τ +���� +� τ +−τ vktrans +j +(t)e−i2πνtdt +���� +2 +(7) +here, N is the total number of SRS in the system and m j is +the mass of the SRS. vktrans +j +is the translational velocity of jth +SRS in kth direction. +Grot(ν) = +1 +kBT +N +∑ +j=1 +2 +∑ +k=1 +lim +τ→−∞ +Ik +j +τ +���� +� τ +−τ ωk +j (t)e−i2πνtdt +���� +2 +(8) +here, Ik +j is the moment of inertia of jth SRS along kth the +principal axis. As SRS is linear, the moment of inertia along +its director is 0. Therefore, k runs from 1 to 2. ωk +j represents +the angular velocity. +2. +Thermodynamic properties from 2PT method +Various thermodynamic quantities like energy, entropy of +a system can be expressed as a summation over the contribu- +tions from translational and rotational motion of SRS)22,23: +E = E0 +Etrans +Erot, +(9) +S = Strans +Srot. +(10) +Here, E0 is the reference energy. In 2PT method, the density +of states corresponding to translational or rotational motion is +partitioned as: +Gk(ν) = Gs +k(ν)+Gg +k(ν) +(11) +where, the subscript k stands for translational, or rotational +motion. The 1st term in Eq. 11 refers to the solid-like and the +2nd term in Eq. 11 refers to the gas-like contributions. For a +solid-like system, the DoS can be exactly determined by that +of harmonic oscillator. But for a liquid, harmonic approxima- +tion is no longer valid at the low frequency regime due to the +strong effect of anharmonicity. Also, the diffusive model at +the zero frequency can lead to singularity. In the 2PT model, +the anharmonicity effect at the low frequency is treated by de- +composing the DoS into gas-like and solid-like components +as mentioned in Eq. 11. The gas-like component is evaluated +from the DoS at the zero frequency and the fluidicity factor fk +using the following equation : +Gg +k(ν) = +Gk(0) +1+ +� +πνGk(0) +6fkN +�2 . +(12) +The fluidicity factor fk is calculated using the equation below: +2∆−9/2 +k +f 15/2 +k +−6∆−3 +k +f 5 +k −∆−3/2 +k +f 7/2 +k ++6∆−3/2 +k +f 5/2 +k ++2fk−2 = 0, +(13) +where, ∆k is the diffusivity constant in reduced unit that is +defined as: +∆k(T,V,N,k,Gk(0)) = 2Gk(0) +9N +�πkBT +k +�1/2 �N +V +�1/3 � 6 +π +�2/3 +. +(14) +The above equation Eq. +14 indicates ∆k only depends +on the thermodynamic state points (T,V,N) and Gk(0) that +can uniquely determines the fluidicity factor fk for different +modes. Once we calculate Gg +k(ν) from Eq. 12, the solid-like +component can be determined by subtracting it from the total +DoS Gk(ν) (Eq. 11) obtained from velocity auto-correlation. +Once we calculate Gg +k(ν) and Gs +k(ν), each component +(translational, rotational) of the thermodynamic quantities +(energy from Eq. 9 and entropy from Eq. 10) can be de- +termined by integrating the DoS using appropriate weighting +functions for the respective thermodynamic quantities: +Ek = β −1 +�� ∞ +0 dνGs +k(ν)W s +E,k(ν)+ +� ∞ +0 dνGg +k(ν)W g +E,k(ν) +� +, +(15) +Sk = kB +�� ∞ +0 dνGs +k(ν)W s +S,k(ν)+ +� ∞ +0 dνGg +k(ν)W g +S,k(ν) +� +, +(16) +Ak = β −1 +�� ∞ +0 dνGs +k(ν)W s +A,k(ν)+ +� ∞ +0 dνGg +k(ν)W g +A,k(ν) +� +, +(17) +where, β = (kBT)−1 and W g/s +l,k +is the weighting function for +thermodynamic quantity l (E/S/A) for each mode k (transla- +tion/rotation) partitioned into gas-like (g) or solid-like (s) con- +tribution. Here, +W s +E = βhν +2 ++ +βhν +exp(βhν)−1, +(18) +W s +S = +βhν +exp(βhν)−1 −ln[1−exp(−βhν)], +(19) +W g +E,trans(ν) = W g +E,rot(ν) = 0.5, +(20) +W g +S,trans(ν) = 1 +3 +SHS +kB +, +(21) +W g +S,rot(ν) = 1 +3 +SR +kB +(22) +where, SHS is the hard-sphere entropy and SR is the rotational +entropy of ideal gas modelled as rigid rotor: +SHS +kB += 5 +2 +ln +��2πmkBT +h2 +�3/2 V +ftrN z(y) +� ++ y(3y−4) +(1−y)2 , (23) +SR +kB += 1+ln +� T +σΘr +� +, +(24) + +4 +here, y = f 5/2 +trans/∆3/2 +trans and z(y) is the compressibility factor of +hard sphere from the Carnahan-Starling equation of state36. +Θr is the rotational temperature defined as Θr = +h2 +8π2IrkB and +σ is the rotational symmetry. The reference energy now be- +comes, +E0 = EMD −β −13N(1−0.5 ftrans −0.5 frot), +(25) +where, EMD is the total energy calculated from the MD simu- +lation. +B. +Entropy using multiparticle correlation expansion method +and integration approach on the SRS equation of state +The configurational entropy Scon is defined as:13–15,17. +Scon +tot = Sid + +∞ +∑ +n=2 +Sn, +(26) +where, Sid denotes the entropy of an ideal gas and Sn denotes +the entropy due to n-particle spatial correlation. Therefore, +the excess entropy can be calculated from well-known multi- +particle correlation expansion of the configurational entropy +(ME) Sex can be written as: +Sex = +∞ +∑ +n=2 +Sn = Scon +tot −Sid, +(27) +If S2 represents the entropy due to pair interaction, then the +residual entropy ∆s that includes the spatial correlation for n ≥ +3 becomes: +∆s = Sex −S2. +(28) +Pair entropy S2 can be expressed as: +S2 = Strans +2 ++Srot +2 , +(29) +Strans +2 += −2πρ +� +[g(r)lng(r)−g(r)+1]r2dr, +(30) +Srot +2 = 4πρ +� +g(r)qrot(r)r2dr, +(31) +qrot(r) = −1 +4 +� π +0 g(θ|r)sinθdθ. +(32) +In a system of linear molecules, the probability distribution +function g(r,θ) can be factorized as15, g(r,θ) = g(r)g(θ|r), +where, g(r) denotes the radial distribution and g(θ|r) denotes +the conditional probability distribution function between two +rods at a r distance with a relative angle between θ to θ +dθ. +The excess entropy can be exactly calculated using the +equation of state (EOS) of the SRS defined below17: +SEOS +ex +(ρ) = Uex +T − +� ρ +0 +� +P +kBTρ′ −1 +� dρ′ +ρ′ , +(33) +where, Uex represents the excess energy, which is the potential +energy per particle in the units of kB. +IV. +RESULTS AND DISCUSSION +We present equilibrium phase diagram of SRS of aspect +ratios L/D = 2 − 5 at the temperature T ∗ = 5 (Fig. +1 +and Fig.7(a)). +The magnitude of the pressures and densi- +ties corresponding to different phases for different aspect ra- +tios are listed in table IV. We obtain 4 stable phases for +L/D ≥ 3.5 :17,37–40 crystal (K), smectic (Sm), nematic (N) and +isotropic (I); 3 stable phases for L/D = 3: crystal, smectic, +and isotropic and two stable phases for L/D = 2: crystal and +isotropic. For further details of these phases and their charac- +terization, we refer the reader to our earlier work39,40. Here +we are interested in entropy computations of these phases. +I +I +I +N +Sm +N +N +Sm +Sm +K +K +K +(a) +(c) +(b) +FIG. 1. (a) Equation of state (b) nematic order parameter S (c) po- +tential energy per particle U∗/N are plotted with packing fraction +η for the system of soft repulsive spherocylinders of aspect ratio +L/D = 5. Thermodynamic quantities are defined in the reduced unit: +pressure P∗ = Pvhsc/kBT and packing fraction, η = ρvhsc where vhsc +is the volume of the spherocylinder. We observe four stable phases: +isotropic (I), nematic (N), smectic (Sm) and crystal (K). The vertical +gray lines indicate boundaries between two phases. +A. +Validation of 2PT method +In the dilute limit, the entropy and Helmholtz free energy of +SRS, calculated using 2PT method, can be compared with the +values obtained for an ideal diatomic gas modeled as a rigid +rotor. The analytical expressions for the partition function Z, + +15 +10 +N/n +5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +n20 +15 +10 +5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +n0.8 +0.6 +S +0.4 +0.2 +0 +0.1 +0.2 +0.3 +0.40.5 +0.6 +0.7 +0.8 +0.95 +entropy S, and Helmholtz free energy A of an ideal rigid rotor +are as follows: +Z(V,T) = +�2πmkBT +h2 +�3/2 +V 8π2IkBT +σh2 +, +(34) +S +NkB += ln +�2π(m1 +m2)kBT +h2 +�3/2 Ve5/2 +N ++ln8π2IkBTe +σh2 +. +(35) +A +NkBT = − +� +ln +�2π(m1 +m2)kBT +h2 +�3/2 V +N +ln8π2IkBT +σh2 ++1 +� +. +(36) +The 1st term in Eq. 35 is due to the translational motion, +and the 2nd term is due to the rotational motion (for an ideal +rigid rotor, there is no vibrational motion). In Table I and +II, we compare the entropy of the SRS system in a dilute limit +calculated from the 2pt method with that of an ideal rigid rotor +at the same state point calculated using the above equations for +different aspect ratios which are found to be in a very good +agreement. +TABLE I. Comparison of the total Stot, translational Strans and ro- +tational Srot entropy of SRS of different aspect ratios from the 2PT +method at the temperature T ∗ = 5 and number density ρ∗ = 0.01 +with that of a rigid rotor at the same state points calculated using Eq. +35. Here, entropy is calculated in kB/particle unit. +L/D +ρ∗ +Sid +trans +Sidrot +Sid +tot +S2PT +trans S2PT +rot +S2PT +tot +5 +0.01 18.36 12.18 30.54 18.26 12.30 30.56 +3 +0.01 18.36 11.15 29.51 18.36 11.25 29.61 +2 +0.01 18.36 10.34 28.70 18.36 10.44 28.80 +TABLE II. Comparison of the Helmholtz free energy of SRS of dif- +ferent aspect ratios from the 2PT method with that of the ideal rigid +rotor using Eq.36 at the dilute limit, temperature T ∗ = 5 and number +density ρ∗ = 0.01. A∗tot designates the total Helmholtz free energy +and A∗trans, A∗rot designate the translational and rotational components +respectively. +L/D +ρ∗ +Aid +trans +Aidrot +Aid +tot +A2PT +trans +A2PT +rot +A2PT +tot +5 +0.01 -84.24 -55.81 -139.95 -84.91 -55.95 -140.86 +3 +0.01 -84.24 -50.71 -134.98 -85.63 -50.65 -136.28 +2 +0.01 -84.24 -46.66 -130.88 -83.74 -48.33 -132.07 +B. +Density of states of liquid crystal phases +We calculate the density of state G(ν) of different liquid +crystal phases using 2PT method as shown in Fig. 2. For +each phase, we show the total DoS and its decomposition +into translational, rotational modes. +The translational and +rotational modes are further decomposed into gas-like and +solid-like components, as mentioned in the 2PT method +section. +In Fig.2(a), we plot DoS for the state point P∗ = 1.78,η = +0.29 which corresponds to the isotropic phase as shown in +the equilibrium phase diagram [Fig.1]. We find that both the +translational Gtrans and rotational Grot DoS are dominated by +the gas like contribution and decay exponentially. At the zero +frequency ν = 0, both of Gtrans and Grot have large finite val- +ues, indicating that the system possesses high translational +and rotational diffusivity. Similarly, in Fig.2(b), we plot DoS +of nematic phase for the state point P∗ = 6.23,η = 0.50. We +see that Gtrans decays exponentially and have a fine value at +ν = 0 indicating gas-like behaviour. However, Grot is domi- +nated by solid-like behaviour with a low rotational diffusivity. +In the case of the smectic phase ( P∗ = 8,η = 0.6), both of +the Gtrans and Grot are dominated by solid-like contribution. +However, Gtrans has a very low value at zero-frequency indi- +cating a low-diffusivity which is due to the in-layer fluid-like +motion. In crystal phase (state point P∗ = 15.13,η = 0.78), +G(ν) is roughly zero at ν = 0 indicating absence of diffusive +mode in the system. Both translational and rotational DoS +exhibit solid-like behaviour. +C. +Fluidicity factor of liquid crystal phases +The decomposition of the translational and rotational DoS +into gas-like and solid-like components is carried out by cal- +culating the fluidicity factor f as discussed in Section III-A. +We find that both of translational and rotational fluidicity fac- +tors are very high in the isotropic phase, very low in the crystal +phase and intermediate in the LC phases as mentioned in the +Table III and in Fig. 3. We also calculate the phase bound- +aries of different LC phases from the change of ftrans and frot. +In Fig.3, we find that, both of the ftrans and frot decrease +with packing fraction η in the isotropic phase. In the ne- +matic phase, ftrans remains almost constant at its value in the +isotropic phase, while frot keeps decreasing. This is also con- +sistent with the DoS calculation showing that rotational dif- +fusivity is much lower in the nematic phase than translational +diffusivity. The I-N phase boundary is therefore defined as +the packing fraction where frot keeps decreasing but ftrans be- +comes constant (η∗ +I−N ≈ 0.41−0.44 for L/D = 5). Similarly, +in the Smectic phase, frot remains nearly constant at its value +in the nematic phase while ftrans drops sharply. Hence, the +N-Sm phase boundary can be located at the packing fraction +where ftrans continues to decrease but frot remains almost con- +stant (η∗ +N−Sm ≈ 0.54 − 0.57 for L/D = 5). Both of ftrans and +frot acquire a very low value in the crystal phase. These anal- +yses suggest another method of quantifying the phase bound- + +6 +aries using the fluidicity factor. +TABLE III. Translational and rotational fluidicity factors for different +liquid crystal phases for the aspect ratio L/D = 5. +P∗ +η +ftrans +frot Phase +1.78 0.29 0.62 0.53 +I +6.23 0.50 0.41 0.18 +N +8.01 0.60 0.26 0.07 +Sm +15.13 0.78 0.09 0.06 +K +D. +Entropy calculation from 2PT method +In Table IV and in Fig.4, Fig.7, we mention the total en- +tropy Stot and its decomposition into the translational Strans +and rotational Srot modes for different liquid crystal phases +associated to different aspect ratios. We find that entropy de- +creases as a function of packing fraction for the given aspect +ratios. We also find that, at a certain packing fraction, to- +tal entropy is close by for the given aspect ratios, irrespec- +tive of the different liquid crystal phases they exhibit. As for +example, at η = 0.60, the magnitude of the total entropy is +Stot = 18.54 − 19.03 kB/particle; however, it shows smectic +structure for L/D ≥ 3 and isotropic structure for L/D = 2. +Similarly, at η = 0.54, Stot = 19.40−19.92 kB/particle while +it shows nematic structure for L/D ≥ 3.5 and isotropic struc- +ture for L/D = 3,2. These results indicate that, total entropy +depends on the thermodynamic state points only, not on the +different liquid crystal phases corresponding to different L/D +s. However, the entropy of different L/D s differs at the higher +packing fractions, as mentioned in Fig. 7(b). +In Fig.6, we calculate the pair entropy S2 of different LC +phases using Eq. 29 and its decomposition into translational +Str +2 and rotational Srot +2 +parts. We observe that, Srot +2 +decreases +sharply at the I-N phase boundary, while Str +2 decreases slowly. +For N-Sm transition, Str +2 decreases more rapidly than that of +I-N phase boundary. Our results are consistent with those of +Cuetos et al.17. These analyses indicate that the change of +entropy at the LC phase transition points are mainly driven by +the translational or rotational pair entropy. The sharp decrease +of rotational pair entropy at the I-N phase boundary is com- +pensated by residual entropy ∆s arising from the multi particle +correlation (Eq. 28). Similarly, the N-Sm phase transition is +driven by the sharp decrease of translational pair entropy that +is also compensated by residual entropy. +E. +Comparison of excess entropy from 2PT method and +integrating on the SRS equation of state +We calculate the excess entropy, Sex which is defined as the +amount of entropy arises due to the particles’ interaction us- +ing Eq.27. It is calculated from the difference between the +absolute entropy calculated from the 2PT method or integrat- +ing over MD/MC equation of state and the entropy of an ideal +rigid rotor at the same state point. We mention the magni- +tude of Sex for different liquid crystal phases in Table V for +L/D = 5 at T ∗ = 5. In Fig. 8, we compare the excess entropy +of SRS at different packing fractions from the 2PT method +with those of the standard integration approach on the (a) MD +equation of state of SRS from our simulation and (b) MC +equation of state of SRS employed by Cuetos et al.17 We ob- +serve that the magnitude of Sex are in good agreement at the +lower densities for the given methods. At the higher densi- +ties, Sex calculated from 2PT method matches well with the +MD equation of state, but it differs from the MC equation of +state17. +V. +CONCLUSION AND OUTLOOK +We describe a technique based on the two-phase thermody- +namic model (2PT) for computing the entropy of liquid crystal +phases of SRS with a range of aspect ratios L/D = 2−5. For +various liquid crystal phases, we compute the density of state +(DoS) functions and its decomposition into translational and +rotational motions. In the dilute limit, the entropy calculated +using the 2PT method matches exactly with that of an ideal +rigid rotor. We find that, at a definite packing fraction, the +magnitude of the total entropy is roughly equal regardless of +the different LC phases associated to different aspect ratios. +We compare the excess entropy with that of the conventional +integration approach on equation of state of SRS, that matches +well. The phase boundaries of different liquid crystal phases +are also calculated using the rotational and translational flu- +idicity factors. Our future study will involve to utilise this +method in calculating absolute value of entropy and other ther- +modynamic quantities of various liquid crystal molecules and +compare it with experiments. +ACKNOWLEDGMENTS +We thank SERB, India for financial support through provid- +ing computational facility. JC acknowledges support through +an INSPIRE fellowship. JC thanks S. Siva Nasarayya Chari +for insightful discussions. +1P.-G. De Gennes and J. Prost, The physics of liquid crystals, Vol. 83 (Oxford +university press, 1993). +2P. Bolhuis and D. Frenkel, The Journal of chemical physics 106, 666 (1997). +3S. C. McGrother, D. C. Williamson, and G. Jackson, The Journal of Chem- +ical Physics 104, 6755 (1996). +4Z. Dogic and S. Fraden, Phys. Rev. Lett. 78, 2417 (1997). +5H. Graf and H. Löwen, Phys. Rev. E 59, 1932 (1999). +6S. Fraden, G. Maret, D. L. D. Caspar, and R. B. Meyer, Phys. Rev. Lett. +63, 2068 (1989). +7Z. Dogic and S. Fraden, Phys. Rev. Lett. 78, 2417 (1997). +8P. Buining and H. Lekkerkerker, The Journal of Physical Chemistry 97, +11510 (1993). +9A. Kuijk, D. V. Byelov, A. V. Petukhov, A. Van Blaaderen, and A. Imhof, +Faraday discussions 159, 181 (2012). +10L. Onsager, Annals of the New York Academy of Sciences 51, 627 (1949). + +7 +0 +100 +200 +300 +400 +ν +0 +20 +40 +60 +G( ν) +η = 0.78 (Crystal) +0 +100 +200 +300 +400 +ν +0 +20 +40 +60 +80 +100 +G( ν) +η = 0.60 (SmecticA) +0 +100 +200 +300 +400 +ν +0 +25 +50 +75 +100 +125 +G( ν) +η = 0.50 (Nematic) +0 +25 +50 +75 +100 +125 +150 +ν +0 +100 +200 +300 +400 +G( ν) +η = 0.29 (Isotropic) +Gtrans +g +Gtrans +s +Grot +g +Grot +s +Gtrans +Grot +Gor +Total +(a) +(b) +(c) +(d) +FIG. 2. Density of state (DoS) G(ν) for (a) isotropic (b) nematic (c) smectic and (d) crystal phase. The components of the entropy are +mentioned in the legend. The snapshots of the configurations are shown for the respective phases. Here, we see that DoS of nematic phase +[Fig. (b)] comprises both solid and gas like components, whereas for smectic phase [Fig.(c)], it is dominated by solid like components only. +11H. Green, “The molecular theory of fluids. amsterdam: North holland publ,” +(1952). +12R. Nettleton and M. 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Pascal, W. A. Goddard III, P. K. Maiti, and S.-T. Lin, +Journal of chemical theory and computation 7, 1893 (2011). +24B. J. Borah, P. K. Maiti, C. Chakravarty, and S. Yashonath, The Journal of +Chemical Physics 136, 174510 (2012). +25J. D. Weeks, D. Chandler, and H. C. Andersen, The Journal of chemical +physics 54, 5237 (1971). +26M. P. Allen, G. T. Evans, D. Frenkel, and B. Mulder, Advances in chemical +physics 86, 1 (1993). +27C. Vega and S. Lago, Computers & chemistry 18, 55 (1994). +28L. Verlet, Physical review 159, 98 (1967). +29I. P. Omelyan, Computers in Physics 12, 97 (1998). +30N. S. Martys and R. D. Mountain, Physical Review E 59, 3733 (1999). +31M. Rotunno, T. Bellini, Y. Lansac, and M. A. Glaser, The Journal of chem- +ical physics 121, 5541 (2004). +32Y. Lansac, P. K. Maiti, N. A. Clark, and M. A. Glaser, Physical Review E +67, 011703 (2003). +33P. K. Maiti, Y. Lansac, M. A. Glaser, and N. A. Clark, Physical review +letters 88, 065504 (2002). +34D. Rajendra, J. Mandal, Y. Hatwalne, and P. K. Maiti, Soft Matter (2022). +35H. J. Berendsen, J. v. Postma, W. F. van Gunsteren, A. DiNola, and J. R. +Haak, The Journal of chemical physics 81, 3684 (1984). +36N. F. Carnahan and K. E. Starling, The Journal of Chemical Physics 53, 600 +(1970). + +8 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +η +0 +0.1 +0.2 +0.3 +0.4 +0.5 +fluidicity (f) +ftrans +frot +I +N +Sm +K +FIG. 3. Phase diagram of the SRS with aspect ratio L/D = 5 at +T ∗ = 5 in fluidicity, packing fraction (f − η) space. ftrans and frot +represent the translational and rotational components respectively. +The black dotted lines denote phase boundaries of different phases. +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +η +0 +5 +10 +15 +20 +25 +30 +S(kB/particle) +Strans +Srot +Stot +I +N +Sm +K +FIG. 4. Total entropy and its translational and rotational components +of different liquid crystal phases for the aspect ratio L/D = 5 at T ∗ = +5. The black dotted lines denote the phase boundaries. +37A. Cuetos and B. Martínez-Haya, Molecular Physics 113, 1137 (2015). +38D. J. Earl, J. Ilnytskyi, and M. R. Wilson, Molecular physics 99, 1719 +(2001). +39J. Chattopadhyay, S. Pannir-Sivajothi, K. Varma, S. Ramaswamy, C. Das- +gupta, and P. K. Maiti, Phys. Rev. E 104, 054610 (2021). +40J. Chattopadhyay, S. Ramaswamy, C. Dasgupta, and P. K. Maiti, arXiv +preprint arXiv:2205.00667 (2022). + +9 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +η +-200 +-150 +-100 +-50 +0 +A* +A*trans +A*rot +A*tot +I +N +Sm +K +FIG. 5. Helmholtz free energy A∗tot and its translational A∗trans and +rotational A∗rot components of different liquid crystal phases for the +aspect ratio L/D = 5 at T ∗ = 5. The black dotted lines denote the +phase boundaries. +-6 +-5 +-4 +-3 +-2 +-1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +η +S2tr (kB/particle) +I +N +Sm +FIG. 6. Translational pair entropy per particle of different liquid crys- +tal phases for the aspect ratio L/D = 5 at T ∗ = 5. The black dotted +lines denote the phase boundaries. + +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +-- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +-- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +-10 +2 +4 +6 +8 +10 +12 +14 +16 +18 +0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 +P* +� +L/D = 2.0 +L/D = 3.0 +L/D = 3.5 +L/D = 4.0 +L/D = 5.0 +15 +16 +17 +18 +19 +20 +21 +22 +23 +0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 +Stot (kB/particle) +� +L/D = 2.0 +L/D = 3.0 +L/D = 3.5 +L/D = 4.0 +L/D = 5.0 +9.5 +10 +10.5 +11 +11.5 +12 +12.5 +13 +0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 +Strans (kB/particle) +� +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +9.5 +10 +0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 +Srot (kB/particle) +� +(a) +(c) +(b) +(d) +FIG. 7. (a) Equation of state, (b) total entropy Stot and its decomposition into (c) translational Strans and (d) rotational Srot motion as a function +of packing fraction η for different L/Ds at the temperature T ∗ = 5. Here we see that, at a certain packing fraction, total entropy is roughly +same irrespective of the different LC phases corresponding to different L/Ds. + +11 +0 +0.2 +0.4 +0.6 +0.8 +η +-10 +-8 +-6 +-4 +-2 +0 +Sex(kB/particle) +MD_EOS +MC EOS [Ref. 17] +2PT +I +N +Sm +FIG. 8. Excess entropy Sex = S2PT/EOS +tot +− Sid +tot vs packing fraction η for L/D = 5 calculated using 2PT method and equation of state (EOS) +of SRS. Here, we compare the excess entropy calculated using MD equation of state and 2PT method from our simulation with those of the +Monte Carlo (MC) EOS from Cuetos et al.17. The black dotted lines denote the phase boundaries. + +12 +TABLE IV. Total entropy S2PT +tot +and its decomposition into translational S2PT +trans and rotational S2PT +rot +degrees of freedom for different liquid +crystal phases associated to different aspect ratios L/D s at T ∗ = 5. Here, P∗,ρ∗ and η indicate pressure, number density and packing fraction +respectively. +P∗ +ρ∗ +η +S2PT +trans +S2PT +rot +S2PT +tot +Phase +L/D = 5 +4.45 +0.093 +0.413 +12.574 +9.230 +21.804 +I +4.90 +0.098 +0.434 +12.509 +8.985 +21.494 +N +5.34 +0.103 +0.457 +12.483 +8.667 +21.150 +N +5.79 +0.107 +0.477 +12.435 +8.391 +20.827 +N +6.23 +0.111 +0.496 +12.320 +8.354 +20.674 +N +6.68 +0.116 +0.517 +12.228 +7.928 +20.157 +N +7.12 +0.121 +0.539 +12.170 +7.751 +19.921 +N +7.57 +0.130 +0.580 +11.924 +7.344 +19.268 +SmA +8.01 +0.135 +0.599 +11.780 +7.231 +19.011 +SmA +8.46 +0.137 +0.611 +11.730 +7.039 +18.768 +SmA +9.35 +0.144 +0.639 +11.355 +6.841 +18.197 +SmA +9.79 +0.147 +0.652 +11.298 +6.761 +18.059 +SmA +10.68 +0.151 +0.674 +11.221 +6.951 +18.172 +SmA +12.46 +0.160 +0.710 +10.909 +6.553 +17.462 +SmA +14.24 +0.171 +0.762 +10.161 +5.950 +16.111 +SmA +16.02 +0.177 +0.788 +9.879 +5.944 +15.823 +K +17.80 +0.184 +0.818 +9.656 +5.563 +15.219 +K +L/D = 4 +5.13 +0.121 +0.444 +12.287 +9.081 +21.368 +I +5.86 +0.127 +0.466 +11.999 +8.783 +20.782 +I +6.60 +0.134 +0.490 +11.845 +8.493 +20.338 +I +6.96 +0.137 +0.502 +11.812 +8.438 +20.251 +N +7.33 +0.141 +0.516 +11.706 +8.290 +19.996 +N +7.70 +0.146 +0.537 +11.740 +8.008 +19.748 +N +8.06 +0.155 +0.568 +11.720 +7.787 +19.507 +SmA +8.43 +0.165 +0.606 +11.630 +7.401 +19.031 +SmA +8.80 +0.169 +0.620 +11.446 +7.346 +18.792 +SmA +11.00 +0.187 +0.685 +11.019 +6.625 +17.644 +K + +13 +P∗ +ρ∗ +η +S2PT +trans +S2PT +rot +S2PT +tot +Phase +L/D = 3.5 +7.20 +0.155 +0.508 +11.708 +8.369 +20.077 +I +7.85 +0.160 +0.523 +11.608 +8.203 +19.810 +I +8.18 +0.162 +0.531 +11.549 +8.135 +19.684 +N +8.51 +0.166 +0.543 +11.385 +8.041 +19.427 +N +8.84 +0.171 +0.559 +11.412 +7.873 +19.285 +N +9.16 +0.191 +0.624 +11.410 +7.134 +18.544 +SmA +9.49 +0.196 +0.643 +11.233 +6.931 +18.164 +SmA +9.82 +0.198 +0.647 +11.306 +6.870 +18.176 +SmA +10.14 +0.203 +0.663 +11.130 +6.714 +17.844 +K +10.47 +0.205 +0.672 +11.082 +6.607 +17.688 +K +13.09 +0.224 +0.732 +10.282 +6.142 +16.423 +K +L/D = 3 +2.30 +0.122 +0.352 +13.438 +9.802 +23.239 +I +6.91 +0.176 +0.506 +11.708 +8.453 +20.161 +I +8.06 +0.185 +0.534 +11.387 +8.116 +19.504 +I +8.35 +0.187 +0.539 +11.345 +8.058 +19.403 +I +9.50 +0.195 +0.562 +11.142 +7.878 +19.020 +I +9.79 +0.198 +0.570 +11.026 +7.736 +18.762 +SmA +10.08 +0.211 +0.608 +11.161 +7.460 +18.621 +SmA +10.37 +0.227 +0.653 +11.052 +6.920 +17.972 +SmA +10.66 +0.233 +0.670 +10.963 +6.614 +17.577 +K +12.67 +0.249 +0.717 +10.507 +6.486 +16.992 +K +L/D = 2 +10.47 +0.282 +0.590 +10.955 +7.584 +18.539 +I +11.10 +0.287 +0.600 +10.776 +7.510 +18.286 +I +11.94 +0.293 +0.613 +10.614 +7.404 +18.017 +I +12.15 +0.326 +0.683 +10.457 +6.351 +16.808 +K +12.57 +0.344 +0.721 +9.830 +5.992 +15.822 +K +12.99 +0.348 +0.729 +9.786 +5.862 +15.647 +K + +14 +TABLE V. Total entropy S2PT +tot +and the excess entropy S2PT +ex +calculated from the 2PT method, entropy of ideal rigid rotor Sid +tot calculated using +Eq. 35 and excess entropy using Monte Carlo equation of state from Cuetos et al.17 SEOS +ex +for different liquid crystal phases of L/D = 5 at +T ∗ = 5: +P∗ +ρ∗ +S2PT +tot +Sid +tot +S2PT +ex +SEOS +ex +Ref17 +Phase +4.45 +0.093 +21.803 +28.316 +-6.512 +-4.023 +I +4.90 +0.098 +21.494 +28.267 +-6.772 +-4.454 +N +5.34 +0.103 +21.150 +28.215 +-7.064 +-4.598 +N +5.79 +0.107 +20.827 +28.172 +-7.345 +-4.885 +N +6.23 +0.112 +20.674 +28.134 +-7.459 +-5.316 +N +6.68 +0.116 +20.156 +28.093 +-7.936 +-5.891 +N +7.12 +0.121 +19.921 +28.051 +-8.131 +-6.322 +N +7.57 +0.130 +19.268 +27.977 +-8.709 +-6.753 +SmA +8.01 +0.135 +19.011 +27.946 +-8.935 +-7.328 +SmA +8.46 +0.137 +18.768 +27.926 +-9.158 +-7.615 +SmA +8.90 +0.141 +18.425 +27.898 +-9.473 +- +SmA +9.35 +0.144 +18.197 +27.880 +-9.683 +-8.046 +SmA +9.79 +0.147 +18.059 +27.860 +-9.801 +-8.333 +SmA + diff --git a/0tE3T4oBgHgl3EQfnAp3/content/tmp_files/load_file.txt b/0tE3T4oBgHgl3EQfnAp3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..237550d0e3db52da3ffb6651508d21f002d7e55d --- /dev/null +++ b/0tE3T4oBgHgl3EQfnAp3/content/tmp_files/load_file.txt @@ -0,0 +1,1126 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf,len=1125 +page_content='Entropy of different phases formed by soft rods Jayeeta Chattopadhyay,1 Shiang-Tai Lin,2 and Prabal K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Maiti1, a) 1)Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India 2)Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan (Dated: 12 January 2023) Computation of entropy in liquids and liquid crystal phases is a big challenge in statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In this work, we extend the two-phase thermodynamic model (2PT) to shape anisotropic soft repulsive spherocylinders (SRSs) and report the absolute values of entropy for different liquid crystal (LC) phases for a range of aspect ratios L/D = 2 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We calculate the density of states (DoS) for different LC phases and decompose it into contributions arising from translational and rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The translational and rotational modes are further partitioned into diffusive, gas-like, and non-diffusive, solid-like components using a fluidicity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In the dilute limit, the entropy values obtained from the 2PT method match exactly those of an ideal rigid rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We find that, for a given packing fraction, the magnitude of the total entropy is roughly equal regardless of the different LC phases associated with different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We also compute the excess entropy (for L/D = 5) and compare those with the values obtained using the standard integration approach of molecular dynamics (MD) or Monte Carlo (MC) equation of state (EOS) of SRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The values obtained using both approaches match very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The rotational and translational fluidicity factors are further used to determine the phase boundaries of different liquid crystal phases for the respective aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' INTRODUCTION The phase behavior of shape anisotropic particles is an emerging field of research that gives rise to various liquid crystal (LC) phases1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Examples span from living organisms like tobacco mosaic virus4–6, fd virus7 to synthetic systems of rod-like particles like boehmite8, silica9 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Different liq- uid crystal phases can be identified based on their microscopic arrangements, as well as positional and orientational order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Onsager, in his seminal work10, showed that a system of thin and hard rods could undergo a phase transition from disordered isotropic to orientationally ordered nematic phase above a critical aspect ratio (L/D > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7) that is mainly driven by entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The loss of orientational entropy in the nematic phase is compensated by the increase of translational entropy due to the ordered structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, for the other LC phases, entropy plays an important role in studying the stabil- ity of the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Entropy of a fluid can be expressed as a mul- tiparticle correlation expansion of statistical entropy devel- oped by Green and Nettleton11,12 and generalized by Lazaridis and co-workers13,14 for the non-spherical bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' first used this method to calculate the entropy of a system of hard spherocylinders (HSCs)15,16 and later, by Cuetos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='17 in a system of soft repulsive spherocylinders (SRSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' It is also worth mentioning several interesting works by Dhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='18,19 where they have calculated entropy of hard rods and rigid rect- angles in 3D and 2D using analytically solvable lattice model and MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In 2003, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='20 developed the two-phase thermody- namic (2PT) model to calculate the entropy, free energy, and other thermodynamic properties of liquids from a short MD trajectory (20 picoseconds (ps)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 2PT model has emerged as an efficient and accurate method in calculating various a)Electronic mail: maiti@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='in thermodynamic properties of Lennard-Jones fluids for the di- verse setting of state points both in 2D21 and 3D20, water in bulk22 and under different confinement, carbon dioxide23 and other organic and inorganic molecules24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The results match very well with those of the experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In the 2PT method, the density of state (DoS) of a liquid, which is calculated from the Fourier transform of the velocity auto- correlation function (VACF), is decomposed into vibrational (solid) and diffusive (gas) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The thermodynamic quantities, including entropy, are then calculated using har- monic oscillator approximation to the solid component and hard sphere approximation to the gas component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For the ro- tational mode, the diffusive part is calculated from the rigid rotor approximation20,22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In 2PT method, the entropy of a definite state point is calculated from a single MD trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Thus, it is far more efficient than the conventional integration approach of MD or MC equation of state of the SRS, which entails several discrete MD/MC trajectories along the integra- tion path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' This is advantageous for the systems for which the analytical form of the equation of state is unknown (such as SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In this work, we extend the 2PT method to calculate en- tropy of various liquid crystal phases formed by a system of soft repulsive spherocylinders of different aspect ratios (length/diameter) L/D = 2,3,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5,4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We validate our method by comparing the entropy values obtained using the standard integration approach of equation of state of the SRS of L/D = 5 at T ∗ = 516,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We find that the entropy val- ues do not have any strong dependence on the aspect ratio but strongly depend on the packing fraction (η)of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We also find that LC phase transitions are governed by the change of pair entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The loss of orientational pair entropy in the nematic phase is compensated by the increase of trans- lational pair entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, in case of the smectic phase, the loss of translational pair entropy is compensated by the residual entropy arising from the multi-particle contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In addition, we present an alternative way to identify the phase boundaries of different liquid crystal phases from the fluidic- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='04621v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='soft] 11 Jan 2023 2 ity factor that is directly related to the diffusivity of the sys- tem: the packing fraction at which the translational fluidicity ftrans saturates but rotational fluidicity frot decreases sharply indicates the phase boundary of the isotropic to nematic (I- N) phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, the nematic to smectic (N-Sm) transition is located where frot saturates but ftrans keeps de- creasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The rest of the paper is organized as follows: In section II, we briefly describe the theoretical background of the 2PT method and summarize the multiparticle correlation expan- sions method and the integration approach of equation of state to calculate the entropy of SRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' in section III, we describe the SRS model and the simulation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We present the results and analysis in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Finally, in section V, we conclude with the discussion on the major benefits of the 2PT method and possible applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' MODEL AND COMPUTATIONAL DETAILS We model the system as a collection of spherocylinders (cylinder with hemispherical caps) of aspect ratios L/D = 2,3,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5,4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The interacting potential is only due to the ex- cluded volume interaction described by the Weeks-Chandler- Andersen (WCA) potential given as follows25: USRS = 4ε[( D dm )12 −( D dm )6]+ε if dm < 2 1 6 D = 0 if dm ≥ 2 1 6 D (1) Here, dm is the shortest distance between two SRS that determines their relative orientation2,17,26,27,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For conve- nience, thermodynamic quantities are expressed in terms of interaction strength ε, diameter of the SRS D and mass m: temperature T ∗ = kBT ε , pressure P∗ = Pvhsc kBT , packing fraction η = vhscρ, where ρ is the number density of the system defined as, ρ = N V and vhsc = πD2( D 6 + L 4) is the volume of the spherocylinder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' energy E∗ = E ε , entropy S∗ = S kB , Helmholtz free energy A∗ = A ε , Gibbs free energy G∗ = G ε , diffusivity d∗ = d( m ε )1/2/D and the time t∗ = t � ε/m/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' To compute entropy using 2PT method, we convert all the thermody- namic quantities in real units using the parameters of argon (ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='238 kcal/mol, σ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='405Å and mass m = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='948 g/mol) and then again convert them into the reduced units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We build the system in a hexagonal-closed-packed (HCP) crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' As the particles are inherently anisotropic in shape, we choose the number of particles in the x, y and z directions such that the simulation box can be built in a near-cubic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' If nx, ny, nz are the number of particles in the x, y, z direction respectively and nu is the number of particles in one unit cell, then the total number of particles in one simulation box N = nu × nx × ny × nz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In our case, number of SRSs is chosen to be N = 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The periodic boundary condition in all three directions are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We have carried out a series of MD simulations for a wide range of state points spanning the melting transition from solid (crystal) to gas (isotropic) for all the aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We melt the initial crystal structure slowly by reducing the pressure in NPT ensemble (Constant particle number, pressure and tem- perature) at T ∗ = 5 for each L/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The positions and velocities of the SRSs are updated using Verlet algorithm28 and the rota- tional motion by quaternion-based rigid-body dynamics29–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The temperature and pressure of the system are controlled using Berendsen thermostat and barostat35 with a tempera- ture relaxation time τT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='05 and pressure relaxation time τP = 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We perform 1×105 to 2×105 MD steps (with an integration time step δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='001 in reduced unit) to reach equilibrium condition and another 2−5×103 steps (5- 30 ps in real unit using the above-mentioned parameters) with δt = 5×10−4(1 fs)in real unit for the 2PT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Two phase thermodynamic method 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Density of State Function The density of state (DoS) function G(ν) is defined as the mass weighted sum of the atomic spectral densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' This can be obtained from Fourier transform of velocity auto- correlation function (VACF) obtained from MD trajectory20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' G(ν) = 1 kBT Natom ∑ l=1 3 ∑ k=1 lim τ→−∞ ml τ ���� � τ −τ vk l (t)e−i2πνtdt ���� 2 (2) Here, Natom is the total number of atoms in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' ml is mass of the lth atom and vk l is the velocity of the lth atom in kth direction ( k indicates spatial coordinates x,y,z respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' G(ν) represents distribution of normal modes in the system i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='e G(ν) dν represents number of normal modes in the frequency range ν to ν + dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' So, total number of modes in the system i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='e degrees of freedom of the system 3N � ∞ 0 G(ν)dν = 3N (3) The diffusion constant (D) of the system is directly related to the zero-frequency density of state of the system G(0): D = kBT 12mN G(0) (4) For a rigid SRS, there is no vibrational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' So, total number of degrees of freedom for a rigid SRS is 5 compris- ing 3 translational and 2 rotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Therefore, total number of modes in the system is: � ∞ 0 G(ν)dν = 5N (5) Density of state G(ν) is decomposed into translational and rotational part: G(ν) = Gtrans(ν)+Grot(ν) (6) 3 where, Gtrans(ν) is obtained from the translational component of the center of mass velocity of the SRS: Gtrans(ν) = 1 kBT N ∑ j=1 3 ∑ k=1 lim τ→−∞ m j τ ���� � τ −τ vktrans j (t)e−i2πνtdt ���� 2 (7) here, N is the total number of SRS in the system and m j is the mass of the SRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' vktrans j is the translational velocity of jth SRS in kth direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Grot(ν) = 1 kBT N ∑ j=1 2 ∑ k=1 lim τ→−∞ Ik j τ ���� � τ −τ ωk j (t)e−i2πνtdt ���� 2 (8) here, Ik j is the moment of inertia of jth SRS along kth the principal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' As SRS is linear, the moment of inertia along its director is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Therefore, k runs from 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' ωk j represents the angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Thermodynamic properties from 2PT method Various thermodynamic quantities like energy, entropy of a system can be expressed as a summation over the contribu- tions from translational and rotational motion of SRS)22,23: E = E0 +Etrans +Erot, (9) S = Strans +Srot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (10) Here, E0 is the reference energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In 2PT method, the density of states corresponding to translational or rotational motion is partitioned as: Gk(ν) = Gs k(ν)+Gg k(ν) (11) where, the subscript k stands for translational, or rotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The 1st term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 11 refers to the solid-like and the 2nd term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 11 refers to the gas-like contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For a solid-like system, the DoS can be exactly determined by that of harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' But for a liquid, harmonic approxima- tion is no longer valid at the low frequency regime due to the strong effect of anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Also, the diffusive model at the zero frequency can lead to singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In the 2PT model, the anharmonicity effect at the low frequency is treated by de- composing the DoS into gas-like and solid-like components as mentioned in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The gas-like component is evaluated from the DoS at the zero frequency and the fluidicity factor fk using the following equation : Gg k(ν) = Gk(0) 1+ � πνGk(0) 6fkN �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (12) The fluidicity factor fk is calculated using the equation below: 2∆−9/2 k f 15/2 k −6∆−3 k f 5 k −∆−3/2 k f 7/2 k +6∆−3/2 k f 5/2 k +2fk−2 = 0, (13) where, ∆k is the diffusivity constant in reduced unit that is defined as: ∆k(T,V,N,k,Gk(0)) = 2Gk(0) 9N �πkBT k �1/2 �N V �1/3 � 6 π �2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (14) The above equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 14 indicates ∆k only depends on the thermodynamic state points (T,V,N) and Gk(0) that can uniquely determines the fluidicity factor fk for different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Once we calculate Gg k(ν) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 12, the solid-like component can be determined by subtracting it from the total DoS Gk(ν) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 11) obtained from velocity auto-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Once we calculate Gg k(ν) and Gs k(ν), each component (translational, rotational) of the thermodynamic quantities (energy from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 9 and entropy from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 10) can be de- termined by integrating the DoS using appropriate weighting functions for the respective thermodynamic quantities: Ek = β −1 �� ∞ 0 dνGs k(ν)W s E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k(ν)+ � ∞ 0 dνGg k(ν)W g E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k(ν) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (15) Sk = kB �� ∞ 0 dνGs k(ν)W s S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k(ν)+ � ∞ 0 dνGg k(ν)W g S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k(ν) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (16) Ak = β −1 �� ∞ 0 dνGs k(ν)W s A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k(ν)+ � ∞ 0 dνGg k(ν)W g A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k(ν) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (17) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' β = (kBT)−1 and W g/s l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='k is the weighting function for thermodynamic quantity l (E/S/A) for each mode k (transla- tion/rotation) partitioned into gas-like (g) or solid-like (s) con- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here, W s E = βhν 2 + βhν exp(βhν)−1, (18) W s S = βhν exp(βhν)−1 −ln[1−exp(−βhν)], (19) W g E,trans(ν) = W g E,rot(ν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5, (20) W g S,trans(ν) = 1 3 SHS kB , (21) W g S,rot(ν) = 1 3 SR kB (22) where, SHS is the hard-sphere entropy and SR is the rotational entropy of ideal gas modelled as rigid rotor: SHS kB = 5 2 +ln ��2πmkBT h2 �3/2 V ftrN z(y) � + y(3y−4) (1−y)2 , (23) SR kB = 1+ln � T σΘr � , (24) 4 here, y = f 5/2 trans/∆3/2 trans and z(y) is the compressibility factor of hard sphere from the Carnahan-Starling equation of state36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Θr is the rotational temperature defined as Θr = h2 8π2IrkB and σ is the rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The reference energy now be- comes, E0 = EMD −β −13N(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 ftrans −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 frot), (25) where, EMD is the total energy calculated from the MD simu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Entropy using multiparticle correlation expansion method and integration approach on the SRS equation of state The configurational entropy Scon is defined as:13–15,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Scon tot = Sid + ∞ ∑ n=2 Sn, (26) where, Sid denotes the entropy of an ideal gas and Sn denotes the entropy due to n-particle spatial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Therefore, the excess entropy can be calculated from well-known multi- particle correlation expansion of the configurational entropy (ME) Sex can be written as: Sex = ∞ ∑ n=2 Sn = Scon tot −Sid, (27) If S2 represents the entropy due to pair interaction, then the residual entropy ∆s that includes the spatial correlation for n ≥ 3 becomes: ∆s = Sex −S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (28) Pair entropy S2 can be expressed as: S2 = Strans 2 +Srot 2 , (29) Strans 2 = −2πρ � [g(r)lng(r)−g(r)+1]r2dr, (30) Srot 2 = 4πρ � g(r)qrot(r)r2dr, (31) qrot(r) = −1 4 � π 0 g(θ|r)sinθdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (32) In a system of linear molecules, the probability distribution function g(r,θ) can be factorized as15, g(r,θ) = g(r)g(θ|r), where, g(r) denotes the radial distribution and g(θ|r) denotes the conditional probability distribution function between two rods at a r distance with a relative angle between θ to θ +dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The excess entropy can be exactly calculated using the equation of state (EOS) of the SRS defined below17: SEOS ex (ρ) = Uex T − � ρ 0 � P kBTρ′ −1 � dρ′ ρ′ , (33) where, Uex represents the excess energy, which is the potential energy per particle in the units of kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' RESULTS AND DISCUSSION We present equilibrium phase diagram of SRS of aspect ratios L/D = 2 − 5 at the temperature T ∗ = 5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The magnitude of the pressures and densi- ties corresponding to different phases for different aspect ra- tios are listed in table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We obtain 4 stable phases for L/D ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 :17,37–40 crystal (K), smectic (Sm), nematic (N) and isotropic (I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 3 stable phases for L/D = 3: crystal, smectic, and isotropic and two stable phases for L/D = 2: crystal and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For further details of these phases and their charac- terization, we refer the reader to our earlier work39,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here we are interested in entropy computations of these phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' I I I N Sm N N Sm Sm K K K (a) (c) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (a) Equation of state (b) nematic order parameter S (c) po- tential energy per particle U∗/N are plotted with packing fraction η for the system of soft repulsive spherocylinders of aspect ratio L/D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Thermodynamic quantities are defined in the reduced unit: pressure P∗ = Pvhsc/kBT and packing fraction, η = ρvhsc where vhsc is the volume of the spherocylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We observe four stable phases: isotropic (I), nematic (N), smectic (Sm) and crystal (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The vertical gray lines indicate boundaries between two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Validation of 2PT method In the dilute limit, the entropy and Helmholtz free energy of SRS, calculated using 2PT method, can be compared with the values obtained for an ideal diatomic gas modeled as a rigid rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The analytical expressions for the partition function Z, 15 10 N/n 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='9 n20 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='9 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='95 entropy S, and Helmholtz free energy A of an ideal rigid rotor are as follows: Z(V,T) = �2πmkBT h2 �3/2 V 8π2IkBT σh2 , (34) S NkB = ln �2π(m1 +m2)kBT h2 �3/2 Ve5/2 N +ln8π2IkBTe σh2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (35) A NkBT = − � ln �2π(m1 +m2)kBT h2 �3/2 V N +ln8π2IkBT σh2 +1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (36) The 1st term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 35 is due to the translational motion, and the 2nd term is due to the rotational motion (for an ideal rigid rotor, there is no vibrational motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In Table I and II, we compare the entropy of the SRS system in a dilute limit calculated from the 2pt method with that of an ideal rigid rotor at the same state point calculated using the above equations for different aspect ratios which are found to be in a very good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Comparison of the total Stot, translational Strans and ro- tational Srot entropy of SRS of different aspect ratios from the 2PT method at the temperature T ∗ = 5 and number density ρ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 with that of a rigid rotor at the same state points calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here, entropy is calculated in kB/particle unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' L/D ρ∗ Sid trans Sidrot Sid tot S2PT trans S2PT rot S2PT tot 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='36 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='18 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='54 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='26 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='56 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='15 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='51 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='25 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='61 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='36 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='34 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='70 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='36 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='44 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='80 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Comparison of the Helmholtz free energy of SRS of dif- ferent aspect ratios from the 2PT method with that of the ideal rigid rotor using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='36 at the dilute limit, temperature T ∗ = 5 and number density ρ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' A∗tot designates the total Helmholtz free energy and A∗trans, A∗rot designate the translational and rotational components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' L/D ρ∗ Aid trans Aidrot Aid tot A2PT trans A2PT rot A2PT tot 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 -84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='24 -55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='81 -139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='95 -84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='91 -55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='95 -140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='86 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 -84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='24 -50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='71 -134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='98 -85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='63 -50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='65 -136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='28 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 -84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='24 -46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='66 -130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='88 -83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='74 -48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='33 -132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='07 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Density of states of liquid crystal phases We calculate the density of state G(ν) of different liquid crystal phases using 2PT method as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For each phase, we show the total DoS and its decomposition into translational, rotational modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The translational and rotational modes are further decomposed into gas-like and solid-like components, as mentioned in the 2PT method section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2(a), we plot DoS for the state point P∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='78,η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='29 which corresponds to the isotropic phase as shown in the equilibrium phase diagram [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We find that both the translational Gtrans and rotational Grot DoS are dominated by the gas like contribution and decay exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' At the zero frequency ν = 0, both of Gtrans and Grot have large finite val- ues, indicating that the system possesses high translational and rotational diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2(b), we plot DoS of nematic phase for the state point P∗ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='23,η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We see that Gtrans decays exponentially and have a fine value at ν = 0 indicating gas-like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' However, Grot is domi- nated by solid-like behaviour with a low rotational diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In the case of the smectic phase ( P∗ = 8,η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6), both of the Gtrans and Grot are dominated by solid-like contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' However, Gtrans has a very low value at zero-frequency indi- cating a low-diffusivity which is due to the in-layer fluid-like motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In crystal phase (state point P∗ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='13,η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='78), G(ν) is roughly zero at ν = 0 indicating absence of diffusive mode in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Both translational and rotational DoS exhibit solid-like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Fluidicity factor of liquid crystal phases The decomposition of the translational and rotational DoS into gas-like and solid-like components is carried out by cal- culating the fluidicity factor f as discussed in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We find that both of translational and rotational fluidicity fac- tors are very high in the isotropic phase, very low in the crystal phase and intermediate in the LC phases as mentioned in the Table III and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We also calculate the phase bound- aries of different LC phases from the change of ftrans and frot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3, we find that, both of the ftrans and frot decrease with packing fraction η in the isotropic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In the ne- matic phase, ftrans remains almost constant at its value in the isotropic phase, while frot keeps decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' This is also con- sistent with the DoS calculation showing that rotational dif- fusivity is much lower in the nematic phase than translational diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The I-N phase boundary is therefore defined as the packing fraction where frot keeps decreasing but ftrans be- comes constant (η∗ I−N ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='41−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='44 for L/D = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, in the Smectic phase, frot remains nearly constant at its value in the nematic phase while ftrans drops sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Hence, the N-Sm phase boundary can be located at the packing fraction where ftrans continues to decrease but frot remains almost con- stant (η∗ N−Sm ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='54 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='57 for L/D = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Both of ftrans and frot acquire a very low value in the crystal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' These anal- yses suggest another method of quantifying the phase bound- 6 aries using the fluidicity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Translational and rotational fluidicity factors for different liquid crystal phases for the aspect ratio L/D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' P∗ η ftrans frot Phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='53 I 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='18 N 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='07 Sm 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='06 K D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Entropy calculation from 2PT method In Table IV and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7, we mention the total en- tropy Stot and its decomposition into the translational Strans and rotational Srot modes for different liquid crystal phases associated to different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We find that entropy de- creases as a function of packing fraction for the given aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We also find that, at a certain packing fraction, to- tal entropy is close by for the given aspect ratios, irrespec- tive of the different liquid crystal phases they exhibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' As for example, at η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='60, the magnitude of the total entropy is Stot = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='54 − 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='03 kB/particle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' however, it shows smectic structure for L/D ≥ 3 and isotropic structure for L/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, at η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='54, Stot = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='40−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='92 kB/particle while it shows nematic structure for L/D ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 and isotropic struc- ture for L/D = 3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' These results indicate that, total entropy depends on the thermodynamic state points only, not on the different liquid crystal phases corresponding to different L/D s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' However, the entropy of different L/D s differs at the higher packing fractions, as mentioned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6, we calculate the pair entropy S2 of different LC phases using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 29 and its decomposition into translational Str 2 and rotational Srot 2 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We observe that, Srot 2 decreases sharply at the I-N phase boundary, while Str 2 decreases slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For N-Sm transition, Str 2 decreases more rapidly than that of I-N phase boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Our results are consistent with those of Cuetos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' These analyses indicate that the change of entropy at the LC phase transition points are mainly driven by the translational or rotational pair entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The sharp decrease of rotational pair entropy at the I-N phase boundary is com- pensated by residual entropy ∆s arising from the multi particle correlation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Similarly, the N-Sm phase transition is driven by the sharp decrease of translational pair entropy that is also compensated by residual entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Comparison of excess entropy from 2PT method and integrating on the SRS equation of state We calculate the excess entropy, Sex which is defined as the amount of entropy arises due to the particles’ interaction us- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' It is calculated from the difference between the absolute entropy calculated from the 2PT method or integrat- ing over MD/MC equation of state and the entropy of an ideal rigid rotor at the same state point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We mention the magni- tude of Sex for different liquid crystal phases in Table V for L/D = 5 at T ∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 8, we compare the excess entropy of SRS at different packing fractions from the 2PT method with those of the standard integration approach on the (a) MD equation of state of SRS from our simulation and (b) MC equation of state of SRS employed by Cuetos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='17 We ob- serve that the magnitude of Sex are in good agreement at the lower densities for the given methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' At the higher densi- ties, Sex calculated from 2PT method matches well with the MD equation of state, but it differs from the MC equation of state17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK We describe a technique based on the two-phase thermody- namic model (2PT) for computing the entropy of liquid crystal phases of SRS with a range of aspect ratios L/D = 2−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' For various liquid crystal phases, we compute the density of state (DoS) functions and its decomposition into translational and rotational motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' In the dilute limit, the entropy calculated using the 2PT method matches exactly with that of an ideal rigid rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We find that, at a definite packing fraction, the magnitude of the total entropy is roughly equal regardless of the different LC phases associated to different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' We compare the excess entropy with that of the conventional integration approach on equation of state of SRS, that matches well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The phase boundaries of different liquid crystal phases are also calculated using the rotational and translational flu- idicity factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Our future study will involve to utilise this method in calculating absolute value of entropy and other ther- modynamic quantities of various liquid crystal molecules and compare it with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank SERB, India for financial support through provid- ing computational facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' JC acknowledges support through an INSPIRE fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' JC thanks S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Siva Nasarayya Chari for insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' De Gennes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Prost, The physics of liquid crystals, Vol.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='78 (Crystal) 0 100 200 300 400 ν 0 20 40 60 80 100 G( ν) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='60 (SmecticA) 0 100 200 300 400 ν 0 25 50 75 100 125 G( ν) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='50 (Nematic) 0 25 50 75 100 125 150 ν 0 100 200 300 400 G( ν) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='29 (Isotropic) Gtrans g Gtrans s Grot g Grot s Gtrans Grot Gor Total (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Density of state (DoS) G(ν) for (a) isotropic (b) nematic (c) smectic and (d) crystal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The components of the entropy are mentioned in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The snapshots of the configurations are shown for the respective phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here, we see that DoS of nematic phase [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (b)] comprises both solid and gas like components, whereas for smectic phase 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' van Gunsteren, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' DiNola, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Haak, The Journal of chemical physics 81, 3684 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 36N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Carnahan and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Starling, The Journal of Chemical Physics 53, 600 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 η 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 fluidicity (f) ftrans frot I N Sm K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Phase diagram of the SRS with aspect ratio L/D = 5 at T ∗ = 5 in fluidicity, packing fraction (f − η) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' ftrans and frot represent the translational and rotational components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The black dotted lines denote phase boundaries of different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='9 η 0 5 10 15 20 25 30 S(kB/particle) Strans Srot Stot I N Sm K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Total entropy and its translational and rotational components of different liquid crystal phases for the aspect ratio L/D = 5 at T ∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The black dotted lines denote the phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 37A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Cuetos and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Martínez-Haya, Molecular Physics 113, 1137 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 38D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' J.' metadata={'source': 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+page_content=' Pannir-Sivajothi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Varma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Ramaswamy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Das- gupta, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Maiti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' E 104, 054610 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 40J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Chattopadhyay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Ramaswamy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Dasgupta, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Maiti, arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='00667 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='9 η 200 150 100 50 0 A* A*trans A*rot A*tot I N Sm K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Helmholtz free energy A∗tot and its translational A∗trans and rotational A∗rot components of different liquid crystal phases for the aspect ratio L/D = 5 at T ∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The black dotted lines denote the phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 6 5 4 3 2 1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 η S2tr (kB/particle) I N Sm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Translational pair entropy per particle of different liquid crys- tal phases for the aspect ratio L/D = 5 at T ∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The black dotted lines denote the phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' -- -- 10 2 4 6 8 10 12 14 16 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='85 P* � L/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 L/D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 L/D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 L/D = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 L/D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 15 16 17 18 19 20 21 22 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='85 Stot (kB/particle) � L/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 L/D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 L/D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 L/D = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 L/D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='85 Strans (kB/particle) � 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='85 Srot (kB/particle) � (a) (c) (b) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' (a) Equation of state, (b) total entropy Stot and its decomposition into (c) translational Strans and (d) rotational Srot motion as a function of packing fraction η for different L/Ds at the temperature T ∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here we see that, at a certain packing fraction, total entropy is roughly same irrespective of the different LC phases corresponding to different L/Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='8 η 10 8 6 4 2 0 Sex(kB/particle) MD_EOS MC EOS [Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 17] 2PT I N Sm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Excess entropy Sex = S2PT/EOS tot − Sid tot vs packing fraction η for L/D = 5 calculated using 2PT method and equation of state (EOS) of SRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here, we compare the excess entropy calculated using MD equation of state and 2PT method from our simulation with those of the Monte Carlo (MC) EOS from Cuetos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' The black dotted lines denote the phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 12 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Total entropy S2PT tot and its decomposition into translational S2PT trans and rotational S2PT rot degrees of freedom for different liquid crystal phases associated to different aspect ratios L/D s at T ∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' Here, P∗,ρ∗ and η indicate pressure, number density and packing fraction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' P∗ ρ∗ η S2PT trans S2PT rot S2PT tot Phase L/D = 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='413 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='574 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='230 21.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content=' 35 and excess entropy using Monte Carlo equation of state from Cuetos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='17 SEOS ex for different liquid crystal phases of L/D = 5 at T ∗ = 5: P∗ ρ∗ S2PT tot Sid tot S2PT ex SEOS ex Ref17 Phase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='093 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE3T4oBgHgl3EQfnAp3/content/2301.04621v1.pdf'} +page_content='803 28.' metadata={'source': 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Mitra†, Sudip Mittal‡ +Department of Computer Science & Engineering, Mississippi State University +{tc2006∗, sm3843†}@msstate.edu, mittal‡@cse.msstate.edu +Abstract—Critical servers can be secured against distributed +denial of service (DDoS) attacks using proof of work (PoW) +systems assisted by an Artificial Intelligence (AI) that learns +contextual network request patterns. In this work, we introduce +CAPOW, a context-aware anti-DDoS framework that injects la- +tency adaptively during communication by utilizing context-aware +PoW puzzles. In CAPOW, a security professional can define +relevant request context attributes which can be learned by the AI +system. These contextual attributes can include information about +the user request, such as IP address, time, flow-level information, +etc., and are utilized to generate a contextual score for incoming +requests that influence the hardness of a PoW puzzle. These +puzzles need to be solved by a user before the server begins +to process their request. Solving puzzles slow down the volume +of incoming adversarial requests. Additionally, the framework +compels the adversary to incur a cost per request, hence making +it expensive for an adversary to prolong a DDoS attack. We +include the theoretical foundations of the CAPOW framework +along with a description of its implementation and evaluation. +I. INTRODUCTION +An organization protects its critical servers from distributed +denial of service (DDoS), which may contain valuable infor- +mation, such as intellectual property, trade secrets, employee +personally identifiable information (PII), etc. To launch a +DDoS attack, the malicious users send a flood of requests to +these servers. As a result, requests from legitimate users either +experience delays or their requests are dropped. For more than +two decades, DDoS attacks have been a prominent issue and +even today it is far from being solved as these attacks are +cheaper to launch than to defend, especially with the rise of +DoS-as-a-Service [25]. +PoW system works by requiring incoming requests to ex- +pend resources solving an computational puzzles to prove ones +legitimacy. The general system consists of two parts: prover +and verifier. The prover finds the solution to the computational +puzzles, when solved, sends the solution to the verifier. In +a simple networked client-server environment, the user-side +contains the prover component, and the server-side contains +the verifier components. Researchers have proposed PoW- +based solutions for DDoS which makes the attack expensive to +launch [4], [21], [34]. Although, these solutions suffer from a +lack of intuition on how to set puzzle difficulty and adaptability +in different settings. +In this paper, we develop a defensive tool that emphasizes +on learning the normal activity patterns of legitimate users. +The idea behind the tool is to penalize the users that deviates +from normal activity patterns by issuing them hard puzzles +and at the same time issuing easy puzzles to users who +follow the pattern. We leverage a context-aware AI model +that can learn these normal activity patterns by contextual +information. The term context within the scope of legitimate +activity patterns can be defined as request attributes, such +as, IP address, time, flow-level information, etc. When the +context is IP address, network activity is considered deviated +if the source IP address is part of a known blocked IP +list. Whereas, when the context is time, network activity is +considered deviated if it arrives at an unusual time compared +to the normal activity pattern. Security professionals can select +relevant request context attributes which can be learned by the +AI models. The concept of context-aware AI models is derived +from context-aware computing introduced by Dey et. al [9]. +We introduce CAPOW tool, a context-aware AI-assisted +PoW system that helps to secure critical servers against DDoS +attacks. Our framework utilizes context-aware AI models that +learn the expected context pattern from server-side activity- +logs. The activity-logs are stored and managed by the server +which contains user activity (IP address, timestamp, flow- +level data, etc). The deviation from the learned pattern is then +leveraged to generate a contextual score for incoming requests +which tunes the difficulty level of the PoW puzzle to be solved. +The underlying defensive strategy curtails the ability of a +malicious user to prolong the attack by adaptively introducing +latency through PoW puzzles and compelling malicious users +to expend more resources to complete an attack. The main +contributions of this paper are as follows. +Contribution 1: We introduce CAPOW, an anti-DDoS frame- +work that injects latency adaptively, i.e., the framework en- +sures that malicious users incur higher latency than legitimate +users based on the deviation in context pattern. We discuss +the process of context score calculation from deviation in +Section III-B. +Contribution 2: We propose a policy component that is +created by security personnel to incorporate server-specific +security demands. We provide intuition for policy construction +in Section III-C. +Contribution 3: We discuss an instance of CAPOW imple- +mentation and perform evaluation to illustrate the effective- +ness of CAPOW. The implementation details are discussed +Section IV. The code is released on GitHuB [3]. +The rest of the paper is structured as follows. In Section II +we discuss the threat model and attack definitions. We discuss +the theoretical foundation of CAPOW in Section III and +arXiv:2301.11767v1 [cs.CR] 27 Jan 2023 + +CAPOW implementation in Section IV. We discuss related +works of the PoW system and DoS defense in Section V, +followed by the conclusion in Section VI. +II. THREAT MODEL +In this section, we present a series of assumptions associated +with the adversary’s abilities. An adversary A initiates a DDoS +attack by sending a flood of requests to the server. The ad- +versary’s intention is to overwhelm the server’s computational +resources and disrupt legitimate user communication with the +server. Although the attack described is a variant of DDoS, +the usefulness of CAPOW can be extended to other variants. +These assumptions described below are similar to previous +literature on DDoS defense using proof of work [17] and in +some sense, we consider a stronger adversary. +Assumption 1. Adversary A can eavesdrop on the communi- +cation channel of the server. A cannot modify any user request +and cannot read any request payload data. +Assume a secure network communication channel is used +by the user to send request packets to the server. The user +performs encryption on the payload data, including the puzzle +solution, and sends the packet to the server. When an adversary +eavesdrops on the channel, they can read the source and +destination IP of the packet, but they cannot read the encrypted +payload consisting of the puzzle parameters. Additionally, the +adversary cannot flip bits of the packet and pollute the puzzle +solution included in the payload. Hence, we assume that the +adversary has no knowledge of the puzzle parameters solved +by a user nor can it deny service to a user who has correctly +solved the puzzle. In Section IV, we utilize assumption 1 to +claim that the adversary cannot reverse engineer the base AI +models to receive easier PoW puzzles. +Assumption 2 Adversary A can spoof user identifiers, such as +IP addresses, and deceive a subset of underlying AI models. +CAPOW uses AI models to learn legitimate network ac- +tivity patterns and the deviation from the pattern is directly +proportional to the difficulty of PoW puzzles to be solved by +the user. A can spoof a legitimate user IP address and send +requests to the server. An intelligent adversary would send +probe packets to the server using a set of spoofed IP addresses +and only utilize IPs that require puzzles to be solved. This way, +the adversary is able to deceive the AI model and reduce the +latency introduced. In Section IV, we discuss that sending +probe packets becomes costly for an adversary to deceive +multiple base AI models. +Assumption 3 Adversary A cannot pollute the training data +of the AI models. +The AI model used by CAPOW learns normal activity +patterns and calculates a deviation which directly influences +the hardness of the puzzle. Hence, it is essential that the AI +learns normal activity patterns from an unpolluted activity-log +to maximize the effectiveness of CAPOW. In Section IV-B, +we describe the training process of a context-aware AI model +where a security professional is deployed to select secure data +to train the base AI models. +III. CAPOW ARCHITECTURAL DESIGN AND +THEORETICAL FOUNDATIONS +In this section, we describe the high-level architecture of +the core components and their inner workings that molds the +CAPOW framework. As shown in Figure 1, CAPOW consists +of four core components: request context extractor, context- +aware AI models, policy, and proof-of-work. +The AI models learn the normal activity pattern from +previous activity-logs. When an incoming request packet is +seen, first the context attributes are extracted from the new +request packet (see Section III-A). Then, the deviation between +the learned normal context pattern and new request contexts is +computed to calculate context score. We elaborate on AI model +training and score calculation in Section III-B. The policy +component of CAPOW provides security professionals with +certain abilities that strengthen the effectiveness of CAPOW +in various security settings (see Section III-C). The context +score influences the difficulty of PoW puzzle. In Section III-D, +we discuss the proof-of-work component and how the PoW +puzzles can curtails the ability of a malicious user to prolong +the attack by adaptively introducing latency. +Data Flow. From Figure 1, the flow of data between different +components of CAPOW is described below. (1) When a new +incoming packet is seen, the request packet is forwarded to +the request context extractor. (2) The extracted request context +attributes are passed to context-aware AI models which learned +expected context patterns from activity logs. The context +score generated by individual AI models is combined using +a function f to produce the final context score (Φ). (3) The +context score is forwarded to the policy component which sets +certain parameters, such as, it maps the context score to a +puzzle difficulty level. (4) The difficulty level is passed to the +puzzle solver which solves a puzzle of the defined difficulty +level using a function func. (5) The computed solution is sent +to the verifier. (6) When the solution is correct, the request +packet is placed on the server queue for processing. +A. Context Extraction from Request Packet +The concept of context-aware computing was introduced +by Dey et. al [9], where the proposed mechanism improved +human-to-computer interaction by delivering contextually rel- +evant data. In the paper, the author proposed an abstract defini- +tion of context, which is a piece of information that clarifies the +characteristics of an entity. When a system contains contextual +data about a situation or entity, the system can take context- +aware decisions which improve the overall quality of any +general decision-making. +In a security setting, a certain request is deemed suspicious +if the associated request attributes deviate from the usual +network activity pattern. For instance, a request packet of +payload size 65500 bytes is considered suspicious due to +deviation when the expected normal payload size pattern +is in the order of a few hundred bytes. To this end, we +2 + +Src IP +Dst IP +Payload +Context-Aware AI Model +f +Incoming Packet +Activity +Logs +Context Score +Range +Difficulty +Range +Policy File(s) +Solution = func (puzzle parameter) +Security +Professional +Policy +d-constrained +solution found? +func (solution parameter) +Proof-of-work +Server Queue +Packet n +Packet n-1 +Packet n-2 +Context C1 +Model +Context Score (Φ) +W1 +W2 +Wn +d1 +d2 +d3 +d4 +… +Φ1 +Φ2 +Φ3 +Φ4 +… +Calculated +context +score +forwarded +to policy +module +Puzzle Verifier +Puzzle Solver +Mapped +puzzle +difficulty +level +forwarded +to puzzle +solver +Packets with correct puzzle +solution placed in server +queue to process. +Send solution +1 +2 +3 +4 +5 +Request Context Extraction +C1 +C2 +Context C2 +Model +Context Ck +Model +Puzzle Parameters +Model Parameters +… +Ck +6 +Fig. 1. The figure illustrates the architecture of CAPOW framework. CAPOW consists of four core components: request context extractor, context-aware AI model, policy, and +proof of work. The AI model learns context patterns from previous activity-logs selected by security personnel and calculates a context score based on the deviation of the incoming +packet. The calculated score is mapped to the PoW puzzle difficulty level as defined by the security professional in policy files. The proof of work component performs evaluations +to find the constrained solution. The request with a correct solution is placed on the server queue to process. +define context of a request packet as request attributes, such +as source IP address, time of arrival, port address, time +to live (TTL), and other flow-level attributes. The contexts +attributes to be extracted are selected by security personnel +via policy component. The list of selected context attributes +are reformed periodically to update the defensive posture of +the organization deployed. When a new request packet is seen, +the request context extractor component extracts the selected +context attributes from the request packet and feeds it to the +context-aware AI models. +B. Context-Aware AI Model +The framework component consumes activity-logs supplied +by security personnel as input to generate a context-aware AI +model. The model is generated by considering a set of request +packets from the activity-log λ = {λ0, λ1, λ2, ..., λi}. Each +request packet λi consists of a set of request context attributes, +Cλi = {C0λi, C1λi, C2λi, ..., Ckλi} +(1) +where k is the number of request context attributes. Ck is +represented as n-dimensional vector. When an n-dimensional +vector of a single context for λ requests is projected in +Euclidean space, such relative positioning produces a cluster. +For k context attributes, k clusters are generated. The clusters +represent the normal activity pattern. To evaluate a new incom- +ing request, request context extractor from Section III-A, feeds +the context attributes which are then projected in Euclidean +space. The deviation ∆(p, q) of context Ck is calculated as the +Euclidean distance between the corresponding normal activity +cluster and the new request projection, +∆(p, q) = +� +� +� +� +n +� +j=1 +(qj − pj)2 +(2) +where p is projected single context attribute of the new request +and q is center of a normal cluster of the same context. +Consequently, the context score Φ for Ck is calculated as, +Φ(Ck) = +�∆(p, q) +∆max +� +× I +(3) +where ∆max is the maximum possible deviation for Ck. The +score is in the range of [0, I], where I ∈ Z+. In Section IV-B, +we discuss the implementation of context-aware AI models. +C. Policy +The policy component is a rule-based strategy that facilitates +the adaptive security guarantees of CAPOW. The rules are set +in policy files that determine certain CAPOW characteristics. +These characteristics include context-aware AI model specifi- +cations, such as, which activity-logs are supplied to train the +AI models, which context attributes hold more significance +over the others, etc. Additionally, these parameters include +proof-of-work components specifications, such as, what is +the rule to translate context score to puzzle difficulty, which +variant of PoW puzzle to be used, etc. Hence, it is evident +that policy construction is a non-trivial task and requires +consideration of various facets of the deployed server to bolster +the effectiveness of CAPOW in different security settings. +To perform the convoluted task of policy designing, security +professionals are deployed to design server-specific policies. +Intuition for AI model parameters. From Section III-A, +a request packet consists of several context attributes. The +significance of some contexts holds more importance over +others depending on the type of attack defense. For instance, +payload size is an important context attribute to protect against +large payload DDoS attacks [37], but less important to de- +fend volumetric DDoS attacks. Policy includes the weight +associated with context attributes to provide an attack-specific +defense. Additionally, a policy includes the source of data +3 + +to train the AI models to avoid model data pollution attacks +(Assumption 3). +Intuition for proof-of-work parameters. The context score +produced by the context-aware AI model is translated to +the PoW difficulty level. The policy includes the rules to +translate context scores to puzzle difficulty. In Section IV-C, +we implemented three rules to show that the translation leads +to adaptive latency injected. As stated by Green et. al [13], +amongst groups of users, the CPU capacity of each device can +vary 10x times, whereas memory capacity may only vary 4x +times. Hence, when a memory-bound PoW puzzle is used, it is +less likely for the adversary to have an edge over a legitimate +user as the discrepancy in memory power as the resource is +less compared to CPU-bound puzzles. The policy includes the +means to set variants of puzzles depending on the expected +user base. +D. Proof of Work +Classical proof of work systems [4], [10], [34] consists +of two main components – prover and verifier. The prover +provides verifiable evidence of expanding computational re- +sources by solving puzzles as assigned by the server. On the +other hand, the verifier validates whether the solved puzzle +yielded the desired solution. When PoW systems are used as +DoS defense [4], [26], [35], a user commits some computation +resources (CPU cycle, bandwidth, etc.) and burns one of these +resources for solving the PoW puzzle to prove their legitimacy. +In CAPOW, when a user deviates from a normal activity +pattern, the PoW component issues a PoW puzzle to request +proof of legitimacy. The difficulty level of PoW puzzle is +a function of context score. The rule to translate to context +score to difficulty level is defined under policy component +(Section III-C). PoW solver uses a function func to solve the +assigned difficulty puzzle (see Figure 1). In general terms, this +function injects two types of cost: (1) direct cost of resource +burning [14], and (2) indirect cost of latency. The notion +of resource burning cost represents the resource consumption +of a user, where the resource could be computational power, +memory, network bandwidth, or human capital [14]. This cost +directly impacts the ability of the adversary to conduct a DDoS +attack as every request requires the adversary to spend real- +life resources. The notion of latency cost captures the delay +in time introduced in communication due to the act of puzzle +solving. This cost indirectly impacts the adversarial intent by +throttling the rate of adversarial requests reaching the server +queue. Both costs ultimately cripple the adversarial capability +to prolong an ongoing DDoS attack. +IV. CAPOW IMPLEMENTATION, TOOL INSTANCE +DEPLOYMENT, AND EVALUATION +In this section, we present a deployment of CAPOW frame- +work by implementing a single instance of each core compo- +nent: context extractor, context-aware AI models, policy, and +proof-of-work. First, the context extractor instance extracts se- +lected request context attributes. Second, the extracted contexts +are relayed to context-aware AI model instances where each +base AI model is generated using server-side activity-logs. +Then, the trained AI models calculate the deviation of selected +contexts to produce a context score. Third, we provide three +policy designs that maps context score to difficulty of PoW +puzzle. Finally, we implemented a hash-based PoW puzzle +instance which, over repeated trials, finds the constrained +solution of assigned difficulty level. The costs inflicted due +to the our puzzle instance are CPU-cycles (resource burning) +and time spent (latency). For the purposes of validating our +contribution via evaluation, we consider that the main cost +injected is latency which, when injected, throttles the rate of +adversarial requests. +Now, we will describe our evaluation setup. We split the +CIC-IDS2017 dataset [24] into test and train files where day +1 to day 5 (Monday - Thursday) is used to train the models +and day 6 (Friday) is used to evaluate CAPOW. From day +1 to day 5, we deleted the attack traffic to learn normal +activity pattern. Consider five users sending requests to the +server U1, U2, U3, U4, and U5. We fixed four user identifiers +from day 5 to map the four above-mentioned users. Let the +fifth user U5, be mapped to the user identifier that performs +DoS on day 6. Since, the user identifier in CIC-IDS2017 +is IP address, let the mapped IP of user U1, U2, U3, U4, and +U5 is represented by 104.20.30.120, 83.66.160.22, +37.59.195.0, 104.16.84.55, and 205.174.165.73 +respectively. Through our evaluation scenario, we provided +evidence that CAPOW injects latency adaptively based on +the calculated context score of user U5 which throttles the +adversarial requests and make it expensive for an adversary to +prolong a DDoS attack. +A. Context Extraction Instance +The context extraction instance consumes the request packet +and extracts context attributes from the request packet. For +our implementation, we select three context attributes: (1) IP +address, (2) temporal activity, and (3) flow-level data. For +evaluation, we used feature attributes of CIC-IDS2017 dataset +to serve as context attributes. The source IP feature becomes +the IP address context, the timestamp feature becomes the +temporal activity context, and the remaining features become +flow-level context. +B. Context-Aware AI Model Instance +We propose an ensemble learner that consists of dedicated +base AI models to learn individual contextual patterns. The +base AI model receives the context attributes from the context +extractor as inputs. The model that (1) learns the IP address +pattern is called dynamic attribute-based reputation (DAbR), +(2) learns the temporal activity pattern is called temporal ac- +tivity model (TAM), and (3) learns the flow-level data pattern +is called flow-level model (FLOW). Each model computes a +context score in the range between [0, 10]. Context scores +from three AI models are combined using the argmax function. +Next, we discuss three base models where the subsections are +divided into model generation, context score calculation, and +evaluation. +4 + +User +Temporal Activity +User 1 +[[500, 501, …,600], [800, 801, +…,900]] +User 2 +[[770, 771, …,800], [850, 851, +…,860], …] +User 3 +[[100, 101, …,110], [300, 301, …, +315]] +User 4 +[[550, 551, …,560]] +User +Temporal Activity +User 1 +[[100, 102, …,220], [500,501,…,510]] +User 2 +[[200, 201,…, 230],] +User 3 +[[190, 191, …, 200], [630, …690]] +User 4 +[[100, 101, …,250], [260, 261, … 410]] +User 1 +Time (seconds) +100 +200 +400 +300 +500 +600 +700 +User 2 +User 3 +User 4 +User +Temporal Activity +User 1 +[[650, 651, …, 700], [760, 761, …, 800]] +User 2 +[[175, 176, …,190], [790, 791, …,800]] +User 3 +[[530, 531, …,602], [740, 741, …, 750]] +User 4 +[[350, 351, …,440], [690, 691, …, 701]] +User +Temporal Activity +User 1 +[[300, 301, …,405], [500,501,…,510]] +User 2 +[[505, 540], [640, 641, …680]] +User 3 +[[190,…200], [410, 530]] +User 4 +[[100, 101, …, 250], [260, 261, …, +410], [500, 501, …, 515]] +Activity log on t-3 day +Activity log t-1 day +Activity log t day +Activity log on t-2 day +Aged Activity Logs +User Activity Cluster +Current Activity Logs +Fig. 2. The figure shows that selected activity-logs (left) are used to generate a temporal activity model (TAM) (right). The illustration shows that out of four activity logs, currently +only two activity logs are used to form the model (blue box). The remaining activity-logs are aged in an attempt to keep the model up-to-date. +Dynamic Attribute-based Reputation (DAbR): We utilize +DAbR [29] as the base AI model that learns context patterns +for IP attributes. The AI model is generated by projecting +malicious IP attributes from Cisco Talos dataset [31] into +Euclidean space. The dataset contains a list of malicious +IP addresses and IP-related attributes [29]. The red dots in +Figure 3(A) represent the projected malicious IP attributes that +form a cluster in Euclidean space. When a new request is +evaluated, the IP attributes of the new request are projected +in Euclidean space and a deviation is calculated as Euclidean +distance to the malicious cluster center. The distance calculated +produces the context score for DAbR (α). The multi-colored +stars represent U1, U2, U3, U4, and U5. User U1, U2, U3, U4, and +U5 receives 2.87, 1.16, 3.15, 2.18, and 2.98 reputation score +respectively. +Temporal Activity Model (TAM): We propose a temporal +activity model (TAM) that learns the pattern of user request +activity based on time of arrival from activity-logs. The model +is generated using previous t-days server activity-logs. The +selected activity-logs can be either previous t consecutive days, +or t specific days (as defined in the policy). The temporal +model can be updated by aging the older activity models +(see Figure 2). The red rectangular blocks in Figure 3(B) +represent an activity cluster per user. The term active in +practice can represent a user session or concurrent requests. +When a user request U arrives at the server, the server finds +the corresponding user activity cluster (UCLS) formed by the +temporal activity model. The user activity cluster (UCLS) is a +list of time intervals that represents the user’s historical activity +times. The deviation in time is calculated as the distance +between the two nearest clusters. From CIC-IDS2017 dataset, +the cluster formed for user U1 shows that the user was active +between 130 − 140 minutes, 160 − 170 minutes, 600 − 670 +minutes, and 720−760 minutes. When user U1 arrived at time +700 minutes on day 6, the two nearest clusters are 600 − 670 +and 720−760 (see Figure 3(B)). This deviation is called ∆local +which is the distance between the two nearest clusters. Finally, +the context score for TAM is calculated as, +β = ∆local +∆max +× 10 +(4) +where, ∆max represents the maximum deviation possible +which in our implementation is 720 minutes. Note that no +cluster is found for U5, hence the context score calculates is +the highest in range. +Flow-level Model (FLOW): Flow-level Model (FLOW) learns +network flow context patterns from activity-logs. The network +flow attributes of a request packet are flow-related data, such as +TTL, flow duration, payload size, protocol, etc. To generate the +model, the n-dimensional flow attribute vectors are projected +in Euclidean space. In Figure 3(C), the green dots represent +projected network flow attributes of legitimate requests, and +the red dots represent projected network flow attributes of +malicious requests. When a new request is seen, its flow- +level attributes are projected and the Euclidean distance to +malicious and legitimate clusters are computed. The context +score is calculated as, +γ = ∆l,m +∆max +× 10 +(5) +where, ∆l,m is the deviation from malicious and legitimate +clusters and ∆max is the maximum deviation possible in flow- +level context. +C. Policy Component Instance +We constructed three policy instances, policy 1, policy 2, +and policy 3. These policies only set the mapping function +between context scores to the PoW puzzle difficulty level. +Context score is directly proportional to the difficulty of the +PoW puzzle, such as the increase in contextual deviation leads +to a higher difficulty puzzle and more latency injected. +Policies 1 and 2: Linear mapping. Assume a linear map +function. Policy 1 maps f(Φ) → d, where Φ ∈ [0, 10] is the +range of context score and d ∈ [0, 10] is the difficulty levels of +the PoW puzzle. Similar to policy 1, policy 2 maps f(Φ) → d, +where Φ ∈ [0, 10] and d ∈ [10, 20]. Note that, the error bar +in Figure 4 shows the discrepancy in time to solve d-level +PoW puzzle. As discussed in Section III-C, this discrepancy +in time to solve can be avoided by using memory-bound PoW +puzzles. +Policy 3: Error range mapping For policy 3, we incorporated +the error ϵ of the context-aware AI model. Assume a linear +map function. Policy 3 maps f(Φ) → d, where Φ ∈ [0, 10] and +d ∈ [0, 10]. The final difficulty level assigned is a difficulty +value chosen at random in the interval [⌈di − ϵ⌉, ⌈di + ϵ⌉], +where ϵ = 0.2. Figure 4 shows that as contextual deviation +increases, the amount of injected latency increases. +5 + +0.2 +0.4 +0.6 +0.8 +1.0 +Malicious cluster centre +User 1 +User 2 +User 3 +User 4 +User 5 +Malicious IP +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +4 +6 +8 +10 +Time (minutes) +Context Score +100 +200 +400 +300 +500 +600 +700 +800 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +Iu87 +Malicious +User 1 +User 2 +User 3 +User 4 +User 5 +Benign +User 1 +Context Score +User 4 +User 3 +User 2 +User 5 +Model B +Model C +Model A +2 +4 +6 +10 +8 +(A) +(B) +(C) +(D) +Fig. 3. The figure contains four sub-figures. (A) Representation of trained DAbR in the 2-D plot. The red dot cluster represents malicious IP attributes. (B) Representation of +trained TAM. The stars represent the current time of arrival. (C) Representation of FLOW. The green cluster represents legitimate flow-level attributes and the red cluster represents +malicious ones. (D) Represents the calculated context score after combining scores from Model A is DAbR, Model B is TAM, and Model C is FLOW. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Context Score ( +) +0 +200 +400 +600 +800 +Latency (millisecond) +Policy 1 +Policy 2 +Policy 3 +Fig. 4. An evaluation of our three implemented policies. The median of 30 trials is +reported for each reputation score. +D. PoW Instance – Hash Function +We discuss two sub-components of CAPOW that mimic +proof-of-work system: puzzle solver, and puzzle verifier. +Puzzle Solver. The puzzle solver takes user identifiers as input, +such as the timestamp of the arrival of the request packet (t), +and the user IP address (u). Additionally, the solver takes +a server seed value (ρ) to protect against pre-computational +attacks. To this, a n-bit string is added, which the client +modifies upon each hash function evaluation. We call this +string nonce denoted by η. +The user evaluates this input until it finds an output string +Y where Y = H(u||t||ρ||η) with d leading zeroes, where d is +the difficulty level assigned to the request packet. The puzzle +solver is a user-end component that is installed either in the +browser [19] or kernel-level. After solving, the user sends the +nonce back to the server for verification. +Puzzle Verifier. Puzzle verification is a server-side compo- +nent that performs straightforward verification of the puz- +zle solution by performing one hash evaluation, i.e., Y ′ = +H(u||t||ρ||η). If the sent η value leads to desired number of +leading 0’s, then the solution is verified. +Summary of CAPOW implementation evaluation. The con- +text scores produced by DAbR, TAM, and FLOW models are +combined to produce the final context score (Φ). As discussed +in Section III-C, some contexts might be more relevant than +others to provide attack specific defense. We denote weight w +as the significance of each context in the final context score. +The weights for each AI model are fixed through the policy +instance as discussed in Section IV-C. +Φ = arg max(w1α, w2β, w3γ) +(6) +where w1, w2, and w3 represent weights associated with +DAbR, TAM, and FLOW respectively. Figure 3(D) illustrates +the combined context score where w1, w2, and w3 is set to 1. +User U1 and U2 show that the final context score is decided +by FLOW model. Similarly, U3, U4, and U5 the final score +is decided by TAM model. Using policy 2, user U5 incurs +≈ 300ms latency for a context score of 8, which is the highest +latency amongst other users introduced by CAPOW. +Notably, the evaluation performed using a simulated dataset +might not reflect the worst case efficiency of CAPOW as in +practice, user U5 might not be deviate in a temporal activity +context. In this section, we discuss that the cost of deceiving +multiple AI models is expensive for the adversary. In our +implementation, user U5 has to deceive three AI models to +receive an easy PoW puzzle by receiving lower context scores. +User U5 can receive a lower context score for DAbR by +trivially spoofing the IP address (Assumption 2). To deceive +TAM, the user can engineer the requests around the same time +as noticed during eavesdropping (Assumption 1). As reading +or tracking flow-level data embedded in request payload data +while eavesdropping is not possible (Assumption 1), the only +way to deceive FLOW is by sending multiple probe packets to +land on a low context score. This is an extensive approach as +a security personnel may select new contexts to improve the +defensive posture of the organization periodically. Therefore, +deceiving all AI models becomes expensive for the adversary. +To validate contribution 3, we designed and evaluated an +implementation instance on CAPOW and provided policy +designs to validate contribution 2. Finally, CAPOW ensures +that malicious users incur higher latency than legitimate users +based on the deviation in context pattern that prevents DDOS. +Hence, we validate contribution 1 (see Section I). +V. RELATED WORKS +In this section, we discuss the overview of proof-of-work +(PoW) literature in DDoS. Relevant to our work, we will also +discuss the current advances in AI-assisted cybersecurity. +6 + +A. Classical Proof-of-Work +Dwork et. al [10] coined the term proof-of-work (PoW) +when they proposed the use of cryptographic hash functions +(also known as client puzzles) to combat unsolicited bulk +emails (junk emails). Following that, Franklin et. al [11] +proposed a lightweight website metering scheme in 1997 to +prevent fraudulent web server owners from inflating their +website’s popularity. In 1999, Jakobsson et. al [16] proposed +MicroMinting (originally proposed by Rivest et. al [30] as a +digital payment scheme) as a candidate problem that can reuse +the computational effort of solving the POW puzzle. Later that +year, Laurie et. al [18] proposed that proof of work does not +work in a spam setting. +B. Proof-of-Work as DoS defense +Similar to spam emails, in DDoS, it is significantly cheaper +for the attacking party to launch a DDoS attack than to defend +an infrastructure with the defending party. According to Arbor +network, launching a DoS attack costs an average of $66 per +attack and can cause damage to the victim of around $500 per +minute [20]. Aura et. al [4] proposed the first client puzzle +authentication protocol for a DoS resilient system. Mankins +et. al [21] investigated methods for tuning the amount of re- +source consumption to access server resources based on client +behavior, where the costs imposed can be either monetary +or computational. In a similar vein, Wang and Reiter [33] +investigate how clients can bid on puzzles through auctions. +Ndibwile et. al [22] proposed web traffic authentication as a +replacement for CAPTCHA-based defenses. Wu et. al [36] +proposed a software puzzle framework that disqualifies the +adversary’s ability to gain an advantage by using a GPU to +solve puzzles. A framework was put forth by Dean et. al [8] to +reduce DoS in TLS servers. A DoS variant was introduced by +Wood et. al [35]. Certain PoW defenses against DoS are layer- +specific. The network layer of the proof-of-work system used +by Parno et. al [26] prioritizes users who use more CPU time to +solve puzzles. The Heimdall architecture, which can detect any +change in network flow in routers, was introduced by Chen et. +al [7]. When a change in network flow is identified for any new +connection, a puzzle is generated and sent to the new user. The +difficulty of the computational challenges used in the context +of DoS attacks on the transport layer was recently assessed +using game theory by Noureddine et. al [23]. Walfish et. +al [32] propose an alternative resource called communication +capacity as a defense against application-layer flood attacks. +Other research has concentrated on incorporating PoW puzzles +into practical browsing experiences [5], [6], [19]. +C. Automated DoS defense +In this section, we revisit the literature on ensemble learning +techniques for network traffic classification problems. En- +semble learning is a branch of supervised machine learning +technique that aggregates the learning of multiple base learners +to improve overall prediction accuracy [28]. Like network +traffic classification problems, each base learner is trained to +become an expert in the local area of the total feature space. +Gaikwad et. al [12] proposed a bagging ensemble approach +using REPTree base learners to improve classification over +weaker AI models. Gupta et. al [2] suggested an IDS system +that uses ensemble learning to address a class imbalance +problem. The ensemble learner uses three base learners. First, +the deep neural network classifies normal and suspicious +traffic. Second, eXtreme Gradient Boosting is used to identify +major attacks. Third, random forest is used to classify minor +attacks. Zhou et. al [1] proposed feature selection process +using ensemble learning in two stages. The first stage involves +feature reduction using the heuristic method CFS and the +Bat Algorithm (BA). The second stage involves aggregating +C4.5 and Random Forest (RF) algorithms. Jabbar et. al [15] +suggested an ensemble classifier that uses Alternating Decision +Tree (ADTree) and the k-Nearest Neighbor algorithm (kNN) +as base AI models. Paulauskas and Auskalnis [27] proposed an +ensemble learner that employs four base classifiers: J48, C5.0, +Naive Bayes, and Partial Decision List (PART) to improve +classification results over individual AI models. +VI. CONCLUSION AND FUTURE WORK +In this paper, we design and evaluate CAPOW a context- +aware AI-assisted PoW framework that protects critical servers +against DDoS. The underlying defensive strategy involves +adaptively introducing latency on malicious users. To achieve +this functionality, our framework employs an AI model that +takes the context attributes from the incoming user request +packet as input. The AI model computes deviation from +normal activity patterns to output a context score. This score +influences the difficulty level of a PoW puzzle that injects +latency adaptively during communication. 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In Proceedings +of the 11th ACM Conference on Computer and Communications Security +(CCS), pages 246–256, 2004. +[35] Paul Wood, Christopher Gutierrez, and Saurabh Bagchi. +Denial of +service elusion (dose): Keeping clients connected for less. +In 2015 +IEEE 34th Symposium on Reliable Distributed Systems (SRDS), pages +94–103, 2015. +[36] Yongdong Wu, Zhigang Zhao, Feng Bao, and Robert H. Deng. Software +puzzle: A countermeasure to resource-inflated denial-of-service attacks. +IEEE Transactions on Information Forensics and Security, 10(1):168– +177, 2015. +[37] S¨uleyman ¨Ozdel, Pelin Damla Ates¸, C¸ a˘gatay Ates¸, Mutlu Koca, and +Emin Anarım. Network anomaly detection with payload-based analysis. +In 2022 30th Signal Processing and Communications Applications +Conference (SIU), pages 1–4, 2022. +8 + diff --git a/3NFKT4oBgHgl3EQfQi0f/content/tmp_files/load_file.txt b/3NFKT4oBgHgl3EQfQi0f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4374459a70fb3164799df8ef40655490ae7ebe8 --- /dev/null +++ b/3NFKT4oBgHgl3EQfQi0f/content/tmp_files/load_file.txt @@ -0,0 +1,675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf,len=674 +page_content='CAPoW: Context-Aware AI-Assisted Proof of Work based DDoS Defense Trisha Chakraborty∗, Shaswata Mitra†, Sudip Mittal‡ Department of Computer Science & Engineering, Mississippi State University {tc2006∗, sm3843†}@msstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='edu, mittal‡@cse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='msstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='edu Abstract—Critical servers can be secured against distributed denial of service (DDoS) attacks using proof of work (PoW) systems assisted by an Artificial Intelligence (AI) that learns contextual network request patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In this work, we introduce CAPOW, a context-aware anti-DDoS framework that injects la- tency adaptively during communication by utilizing context-aware PoW puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In CAPOW, a security professional can define relevant request context attributes which can be learned by the AI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' These contextual attributes can include information about the user request, such as IP address, time, flow-level information, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=', and are utilized to generate a contextual score for incoming requests that influence the hardness of a PoW puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' These puzzles need to be solved by a user before the server begins to process their request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Solving puzzles slow down the volume of incoming adversarial requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Additionally, the framework compels the adversary to incur a cost per request, hence making it expensive for an adversary to prolong a DDoS attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We include the theoretical foundations of the CAPOW framework along with a description of its implementation and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' INTRODUCTION An organization protects its critical servers from distributed denial of service (DDoS), which may contain valuable infor- mation, such as intellectual property, trade secrets, employee personally identifiable information (PII), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To launch a DDoS attack, the malicious users send a flood of requests to these servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' As a result, requests from legitimate users either experience delays or their requests are dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For more than two decades, DDoS attacks have been a prominent issue and even today it is far from being solved as these attacks are cheaper to launch than to defend, especially with the rise of DoS-as-a-Service [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' PoW system works by requiring incoming requests to ex- pend resources solving an computational puzzles to prove ones legitimacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The general system consists of two parts: prover and verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The prover finds the solution to the computational puzzles, when solved, sends the solution to the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In a simple networked client-server environment, the user-side contains the prover component, and the server-side contains the verifier components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Researchers have proposed PoW- based solutions for DDoS which makes the attack expensive to launch [4], [21], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Although, these solutions suffer from a lack of intuition on how to set puzzle difficulty and adaptability in different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In this paper, we develop a defensive tool that emphasizes on learning the normal activity patterns of legitimate users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The idea behind the tool is to penalize the users that deviates from normal activity patterns by issuing them hard puzzles and at the same time issuing easy puzzles to users who follow the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We leverage a context-aware AI model that can learn these normal activity patterns by contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The term context within the scope of legitimate activity patterns can be defined as request attributes, such as, IP address, time, flow-level information, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When the context is IP address, network activity is considered deviated if the source IP address is part of a known blocked IP list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Whereas, when the context is time, network activity is considered deviated if it arrives at an unusual time compared to the normal activity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Security professionals can select relevant request context attributes which can be learned by the AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The concept of context-aware AI models is derived from context-aware computing introduced by Dey et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We introduce CAPOW tool, a context-aware AI-assisted PoW system that helps to secure critical servers against DDoS attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Our framework utilizes context-aware AI models that learn the expected context pattern from server-side activity- logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The activity-logs are stored and managed by the server which contains user activity (IP address, timestamp, flow- level data, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The deviation from the learned pattern is then leveraged to generate a contextual score for incoming requests which tunes the difficulty level of the PoW puzzle to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The underlying defensive strategy curtails the ability of a malicious user to prolong the attack by adaptively introducing latency through PoW puzzles and compelling malicious users to expend more resources to complete an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The main contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Contribution 1: We introduce CAPOW, an anti-DDoS frame- work that injects latency adaptively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=', the framework en- sures that malicious users incur higher latency than legitimate users based on the deviation in context pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We discuss the process of context score calculation from deviation in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Contribution 2: We propose a policy component that is created by security personnel to incorporate server-specific security demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We provide intuition for policy construction in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Contribution 3: We discuss an instance of CAPOW imple- mentation and perform evaluation to illustrate the effective- ness of CAPOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The implementation details are discussed Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The code is released on GitHuB [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section II we discuss the threat model and attack definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We discuss the theoretical foundation of CAPOW in Section III and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='11767v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='CR] 27 Jan 2023 CAPOW implementation in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We discuss related works of the PoW system and DoS defense in Section V, followed by the conclusion in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' THREAT MODEL In this section, we present a series of assumptions associated with the adversary’s abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' An adversary A initiates a DDoS attack by sending a flood of requests to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The ad- versary’s intention is to overwhelm the server’s computational resources and disrupt legitimate user communication with the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Although the attack described is a variant of DDoS, the usefulness of CAPOW can be extended to other variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' These assumptions described below are similar to previous literature on DDoS defense using proof of work [17] and in some sense, we consider a stronger adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Adversary A can eavesdrop on the communi- cation channel of the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' A cannot modify any user request and cannot read any request payload data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Assume a secure network communication channel is used by the user to send request packets to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The user performs encryption on the payload data, including the puzzle solution, and sends the packet to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When an adversary eavesdrops on the channel, they can read the source and destination IP of the packet, but they cannot read the encrypted payload consisting of the puzzle parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Additionally, the adversary cannot flip bits of the packet and pollute the puzzle solution included in the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Hence, we assume that the adversary has no knowledge of the puzzle parameters solved by a user nor can it deny service to a user who has correctly solved the puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section IV, we utilize assumption 1 to claim that the adversary cannot reverse engineer the base AI models to receive easier PoW puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Assumption 2 Adversary A can spoof user identifiers, such as IP addresses, and deceive a subset of underlying AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' CAPOW uses AI models to learn legitimate network ac- tivity patterns and the deviation from the pattern is directly proportional to the difficulty of PoW puzzles to be solved by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' A can spoof a legitimate user IP address and send requests to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' An intelligent adversary would send probe packets to the server using a set of spoofed IP addresses and only utilize IPs that require puzzles to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' This way, the adversary is able to deceive the AI model and reduce the latency introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section IV, we discuss that sending probe packets becomes costly for an adversary to deceive multiple base AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Assumption 3 Adversary A cannot pollute the training data of the AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The AI model used by CAPOW learns normal activity patterns and calculates a deviation which directly influences the hardness of the puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Hence, it is essential that the AI learns normal activity patterns from an unpolluted activity-log to maximize the effectiveness of CAPOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section IV-B, we describe the training process of a context-aware AI model where a security professional is deployed to select secure data to train the base AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' CAPOW ARCHITECTURAL DESIGN AND THEORETICAL FOUNDATIONS In this section, we describe the high-level architecture of the core components and their inner workings that molds the CAPOW framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' As shown in Figure 1, CAPOW consists of four core components: request context extractor, context- aware AI models, policy, and proof-of-work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The AI models learn the normal activity pattern from previous activity-logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When an incoming request packet is seen, first the context attributes are extracted from the new request packet (see Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Then, the deviation between the learned normal context pattern and new request contexts is computed to calculate context score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We elaborate on AI model training and score calculation in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The policy component of CAPOW provides security professionals with certain abilities that strengthen the effectiveness of CAPOW in various security settings (see Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The context score influences the difficulty of PoW puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section III-D, we discuss the proof-of-work component and how the PoW puzzles can curtails the ability of a malicious user to prolong the attack by adaptively introducing latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Data Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' From Figure 1, the flow of data between different components of CAPOW is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (1) When a new incoming packet is seen, the request packet is forwarded to the request context extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (2) The extracted request context attributes are passed to context-aware AI models which learned expected context patterns from activity logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The context score generated by individual AI models is combined using a function f to produce the final context score (Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (3) The context score is forwarded to the policy component which sets certain parameters, such as, it maps the context score to a puzzle difficulty level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (4) The difficulty level is passed to the puzzle solver which solves a puzzle of the defined difficulty level using a function func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (5) The computed solution is sent to the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (6) When the solution is correct, the request packet is placed on the server queue for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Context Extraction from Request Packet The concept of context-aware computing was introduced by Dey et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [9], where the proposed mechanism improved human-to-computer interaction by delivering contextually rel- evant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In the paper, the author proposed an abstract defini- tion of context, which is a piece of information that clarifies the characteristics of an entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When a system contains contextual data about a situation or entity, the system can take context- aware decisions which improve the overall quality of any general decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In a security setting, a certain request is deemed suspicious if the associated request attributes deviate from the usual network activity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For instance, a request packet of payload size 65500 bytes is considered suspicious due to deviation when the expected normal payload size pattern is in the order of a few hundred bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To this end, we 2 Src IP Dst IP Payload Context-Aware AI Model f Incoming Packet Activity Logs Context Score Range Difficulty Range Policy File(s) Solution = func (puzzle parameter) Security Professional Policy d-constrained solution found?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' func (solution parameter) Proof-of-work Server Queue Packet n Packet n-1 Packet n-2 Context C1 Model Context Score (Φ) W1 W2 Wn d1 d2 d3 d4 … Φ1 Φ2 Φ3 Φ4 … Calculated context score forwarded to policy module Puzzle Verifier Puzzle Solver Mapped puzzle difficulty level forwarded to puzzle solver Packets with correct puzzle solution placed in server queue to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Send solution 1 2 3 4 5 Request Context Extraction C1 C2 Context C2 Model Context Ck Model Puzzle Parameters Model Parameters … Ck 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The figure illustrates the architecture of CAPOW framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' CAPOW consists of four core components: request context extractor, context-aware AI model, policy, and proof of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The AI model learns context patterns from previous activity-logs selected by security personnel and calculates a context score based on the deviation of the incoming packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The calculated score is mapped to the PoW puzzle difficulty level as defined by the security professional in policy files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The proof of work component performs evaluations to find the constrained solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The request with a correct solution is placed on the server queue to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' define context of a request packet as request attributes, such as source IP address, time of arrival, port address, time to live (TTL), and other flow-level attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The contexts attributes to be extracted are selected by security personnel via policy component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The list of selected context attributes are reformed periodically to update the defensive posture of the organization deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When a new request packet is seen, the request context extractor component extracts the selected context attributes from the request packet and feeds it to the context-aware AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Context-Aware AI Model The framework component consumes activity-logs supplied by security personnel as input to generate a context-aware AI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The model is generated by considering a set of request packets from the activity-log λ = {λ0, λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=', λi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Each request packet λi consists of a set of request context attributes, Cλi = {C0λi, C1λi, C2λi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=', Ckλi} (1) where k is the number of request context attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Ck is represented as n-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When an n-dimensional vector of a single context for λ requests is projected in Euclidean space, such relative positioning produces a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For k context attributes, k clusters are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The clusters represent the normal activity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To evaluate a new incom- ing request, request context extractor from Section III-A, feeds the context attributes which are then projected in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The deviation ∆(p, q) of context Ck is calculated as the Euclidean distance between the corresponding normal activity cluster and the new request projection, ∆(p, q) = � � � � n � j=1 (qj − pj)2 (2) where p is projected single context attribute of the new request and q is center of a normal cluster of the same context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Consequently, the context score Φ for Ck is calculated as, Φ(Ck) = �∆(p, q) ∆max � × I (3) where ∆max is the maximum possible deviation for Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The score is in the range of [0, I], where I ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section IV-B, we discuss the implementation of context-aware AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policy The policy component is a rule-based strategy that facilitates the adaptive security guarantees of CAPOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The rules are set in policy files that determine certain CAPOW characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' These characteristics include context-aware AI model specifi- cations, such as, which activity-logs are supplied to train the AI models, which context attributes hold more significance over the others, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Additionally, these parameters include proof-of-work components specifications, such as, what is the rule to translate context score to puzzle difficulty, which variant of PoW puzzle to be used, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Hence, it is evident that policy construction is a non-trivial task and requires consideration of various facets of the deployed server to bolster the effectiveness of CAPOW in different security settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To perform the convoluted task of policy designing, security professionals are deployed to design server-specific policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Intuition for AI model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' From Section III-A, a request packet consists of several context attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The significance of some contexts holds more importance over others depending on the type of attack defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For instance, payload size is an important context attribute to protect against large payload DDoS attacks [37], but less important to de- fend volumetric DDoS attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policy includes the weight associated with context attributes to provide an attack-specific defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Additionally, a policy includes the source of data 3 to train the AI models to avoid model data pollution attacks (Assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Intuition for proof-of-work parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The context score produced by the context-aware AI model is translated to the PoW difficulty level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The policy includes the rules to translate context scores to puzzle difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Section IV-C, we implemented three rules to show that the translation leads to adaptive latency injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' As stated by Green et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [13], amongst groups of users, the CPU capacity of each device can vary 10x times, whereas memory capacity may only vary 4x times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Hence, when a memory-bound PoW puzzle is used, it is less likely for the adversary to have an edge over a legitimate user as the discrepancy in memory power as the resource is less compared to CPU-bound puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The policy includes the means to set variants of puzzles depending on the expected user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Proof of Work Classical proof of work systems [4], [10], [34] consists of two main components – prover and verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The prover provides verifiable evidence of expanding computational re- sources by solving puzzles as assigned by the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' On the other hand, the verifier validates whether the solved puzzle yielded the desired solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When PoW systems are used as DoS defense [4], [26], [35], a user commits some computation resources (CPU cycle, bandwidth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=') and burns one of these resources for solving the PoW puzzle to prove their legitimacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In CAPOW, when a user deviates from a normal activity pattern, the PoW component issues a PoW puzzle to request proof of legitimacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The difficulty level of PoW puzzle is a function of context score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The rule to translate to context score to difficulty level is defined under policy component (Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' PoW solver uses a function func to solve the assigned difficulty puzzle (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In general terms, this function injects two types of cost: (1) direct cost of resource burning [14], and (2) indirect cost of latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The notion of resource burning cost represents the resource consumption of a user, where the resource could be computational power, memory, network bandwidth, or human capital [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' This cost directly impacts the ability of the adversary to conduct a DDoS attack as every request requires the adversary to spend real- life resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The notion of latency cost captures the delay in time introduced in communication due to the act of puzzle solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' This cost indirectly impacts the adversarial intent by throttling the rate of adversarial requests reaching the server queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Both costs ultimately cripple the adversarial capability to prolong an ongoing DDoS attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' CAPOW IMPLEMENTATION, TOOL INSTANCE DEPLOYMENT, AND EVALUATION In this section, we present a deployment of CAPOW frame- work by implementing a single instance of each core compo- nent: context extractor, context-aware AI models, policy, and proof-of-work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' First, the context extractor instance extracts se- lected request context attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Second, the extracted contexts are relayed to context-aware AI model instances where each base AI model is generated using server-side activity-logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Then, the trained AI models calculate the deviation of selected contexts to produce a context score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Third, we provide three policy designs that maps context score to difficulty of PoW puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Finally, we implemented a hash-based PoW puzzle instance which, over repeated trials, finds the constrained solution of assigned difficulty level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The costs inflicted due to the our puzzle instance are CPU-cycles (resource burning) and time spent (latency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For the purposes of validating our contribution via evaluation, we consider that the main cost injected is latency which, when injected, throttles the rate of adversarial requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Now, we will describe our evaluation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We split the CIC-IDS2017 dataset [24] into test and train files where day 1 to day 5 (Monday - Thursday) is used to train the models and day 6 (Friday) is used to evaluate CAPOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' From day 1 to day 5, we deleted the attack traffic to learn normal activity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Consider five users sending requests to the server U1, U2, U3, U4, and U5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We fixed four user identifiers from day 5 to map the four above-mentioned users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Let the fifth user U5, be mapped to the user identifier that performs DoS on day 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Since, the user identifier in CIC-IDS2017 is IP address, let the mapped IP of user U1, U2, U3, U4, and U5 is represented by 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='120, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='22, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='0, 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='55, and 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='73 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Through our evaluation scenario, we provided evidence that CAPOW injects latency adaptively based on the calculated context score of user U5 which throttles the adversarial requests and make it expensive for an adversary to prolong a DDoS attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Context Extraction Instance The context extraction instance consumes the request packet and extracts context attributes from the request packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For our implementation, we select three context attributes: (1) IP address, (2) temporal activity, and (3) flow-level data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For evaluation, we used feature attributes of CIC-IDS2017 dataset to serve as context attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The source IP feature becomes the IP address context, the timestamp feature becomes the temporal activity context, and the remaining features become flow-level context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Context-Aware AI Model Instance We propose an ensemble learner that consists of dedicated base AI models to learn individual contextual patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The base AI model receives the context attributes from the context extractor as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The model that (1) learns the IP address pattern is called dynamic attribute-based reputation (DAbR), (2) learns the temporal activity pattern is called temporal ac- tivity model (TAM), and (3) learns the flow-level data pattern is called flow-level model (FLOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Each model computes a context score in the range between [0, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Context scores from three AI models are combined using the argmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Next, we discuss three base models where the subsections are divided into model generation, context score calculation, and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 4 User Temporal Activity User 1 [[500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='600],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='900]] User 2 [[770,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 771,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='800],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [850,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 851,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='860],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …] User 3 [[100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='110],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 301,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 315]] User 4 [[550,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 551,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='560]] User Temporal Activity User 1 [[100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 102,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='220],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='510]] User 2 [[200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 230],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='] User 3 [[190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 200],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [630,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …690]] User 4 [[100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='250],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [260,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 261,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' … 410]] User 1 Time (seconds) 100 200 400 300 500 600 700 User 2 User 3 User 4 User Temporal Activity User 1 [[650,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 651,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 700],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [760,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 761,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 800]] User 2 [[175,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 176,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='190],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [790,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 791,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='800]] User 3 [[530,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 531,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='602],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [740,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 741,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 750]] User 4 [[350,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 351,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='440],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [690,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 691,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 701]] User Temporal Activity User 1 [[300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 301,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='405],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='510]] User 2 [[505,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 540],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [640,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 641,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …680]] User 3 [[190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='…200],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [410,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 530]] User 4 [[100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 250],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [260,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 261,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 410],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' [500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 515]] Activity log on t-3 day Activity log t-1 day Activity log t day Activity log on t-2 day Aged Activity Logs User Activity Cluster Current Activity Logs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The figure shows that selected activity-logs (left) are used to generate a temporal activity model (TAM) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The illustration shows that out of four activity logs, currently only two activity logs are used to form the model (blue box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The remaining activity-logs are aged in an attempt to keep the model up-to-date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Dynamic Attribute-based Reputation (DAbR): We utilize DAbR [29] as the base AI model that learns context patterns for IP attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The AI model is generated by projecting malicious IP attributes from Cisco Talos dataset [31] into Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The dataset contains a list of malicious IP addresses and IP-related attributes [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The red dots in Figure 3(A) represent the projected malicious IP attributes that form a cluster in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When a new request is evaluated, the IP attributes of the new request are projected in Euclidean space and a deviation is calculated as Euclidean distance to the malicious cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The distance calculated produces the context score for DAbR (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The multi-colored stars represent U1, U2, U3, U4, and U5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' User U1, U2, U3, U4, and U5 receives 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='87, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='16, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='18, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='98 reputation score respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Temporal Activity Model (TAM): We propose a temporal activity model (TAM) that learns the pattern of user request activity based on time of arrival from activity-logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The model is generated using previous t-days server activity-logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The selected activity-logs can be either previous t consecutive days, or t specific days (as defined in the policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The temporal model can be updated by aging the older activity models (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The red rectangular blocks in Figure 3(B) represent an activity cluster per user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The term active in practice can represent a user session or concurrent requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When a user request U arrives at the server, the server finds the corresponding user activity cluster (UCLS) formed by the temporal activity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The user activity cluster (UCLS) is a list of time intervals that represents the user’s historical activity times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The deviation in time is calculated as the distance between the two nearest clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' From CIC-IDS2017 dataset, the cluster formed for user U1 shows that the user was active between 130 − 140 minutes, 160 − 170 minutes, 600 − 670 minutes, and 720−760 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When user U1 arrived at time 700 minutes on day 6, the two nearest clusters are 600 − 670 and 720−760 (see Figure 3(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' This deviation is called ∆local which is the distance between the two nearest clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Finally, the context score for TAM is calculated as, β = ∆local ∆max × 10 (4) where, ∆max represents the maximum deviation possible which in our implementation is 720 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Note that no cluster is found for U5, hence the context score calculates is the highest in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Flow-level Model (FLOW): Flow-level Model (FLOW) learns network flow context patterns from activity-logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The network flow attributes of a request packet are flow-related data, such as TTL, flow duration, payload size, protocol, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To generate the model, the n-dimensional flow attribute vectors are projected in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In Figure 3(C), the green dots represent projected network flow attributes of legitimate requests, and the red dots represent projected network flow attributes of malicious requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When a new request is seen, its flow- level attributes are projected and the Euclidean distance to malicious and legitimate clusters are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The context score is calculated as, γ = ∆l,m ∆max × 10 (5) where, ∆l,m is the deviation from malicious and legitimate clusters and ∆max is the maximum deviation possible in flow- level context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policy Component Instance We constructed three policy instances, policy 1, policy 2, and policy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' These policies only set the mapping function between context scores to the PoW puzzle difficulty level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Context score is directly proportional to the difficulty of the PoW puzzle, such as the increase in contextual deviation leads to a higher difficulty puzzle and more latency injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policies 1 and 2: Linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Assume a linear map function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policy 1 maps f(Φ) → d, where Φ ∈ [0, 10] is the range of context score and d ∈ [0, 10] is the difficulty levels of the PoW puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Similar to policy 1, policy 2 maps f(Φ) → d, where Φ ∈ [0, 10] and d ∈ [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Note that, the error bar in Figure 4 shows the discrepancy in time to solve d-level PoW puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' As discussed in Section III-C, this discrepancy in time to solve can be avoided by using memory-bound PoW puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policy 3: Error range mapping For policy 3, we incorporated the error ϵ of the context-aware AI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Assume a linear map function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Policy 3 maps f(Φ) → d, where Φ ∈ [0, 10] and d ∈ [0, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The final difficulty level assigned is a difficulty value chosen at random in the interval [⌈di − ϵ⌉, ⌈di + ϵ⌉], where ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Figure 4 shows that as contextual deviation increases, the amount of injected latency increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='0 Malicious cluster centre User 1 User 2 User 3 User 4 User 5 Malicious IP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='0 0 2 4 6 8 10 Time (minutes) Context Score 100 200 400 300 500 600 700 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='0 Iu87 Malicious User 1 User 2 User 3 User 4 User 5 Benign User 1 Context Score User 4 User 3 User 2 User 5 Model B Model C Model A 2 4 6 10 8 (A) (B) (C) (D) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The figure contains four sub-figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (A) Representation of trained DAbR in the 2-D plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The red dot cluster represents malicious IP attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (B) Representation of trained TAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The stars represent the current time of arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (C) Representation of FLOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The green cluster represents legitimate flow-level attributes and the red cluster represents malicious ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' (D) Represents the calculated context score after combining scores from Model A is DAbR, Model B is TAM, and Model C is FLOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 10 Context Score ( ) 0 200 400 600 800 Latency (millisecond) Policy 1 Policy 2 Policy 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' An evaluation of our three implemented policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The median of 30 trials is reported for each reputation score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' PoW Instance – Hash Function We discuss two sub-components of CAPOW that mimic proof-of-work system: puzzle solver, and puzzle verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Puzzle Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The puzzle solver takes user identifiers as input, such as the timestamp of the arrival of the request packet (t), and the user IP address (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Additionally, the solver takes a server seed value (ρ) to protect against pre-computational attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To this, a n-bit string is added, which the client modifies upon each hash function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We call this string nonce denoted by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The user evaluates this input until it finds an output string Y where Y = H(u||t||ρ||η) with d leading zeroes, where d is the difficulty level assigned to the request packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The puzzle solver is a user-end component that is installed either in the browser [19] or kernel-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' After solving, the user sends the nonce back to the server for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Puzzle Verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Puzzle verification is a server-side compo- nent that performs straightforward verification of the puz- zle solution by performing one hash evaluation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=', Y ′ = H(u||t||ρ||η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' If the sent η value leads to desired number of leading 0’s, then the solution is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Summary of CAPOW implementation evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The con- text scores produced by DAbR, TAM, and FLOW models are combined to produce the final context score (Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' As discussed in Section III-C, some contexts might be more relevant than others to provide attack specific defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' We denote weight w as the significance of each context in the final context score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The weights for each AI model are fixed through the policy instance as discussed in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Φ = arg max(w1α, w2β, w3γ) (6) where w1, w2, and w3 represent weights associated with DAbR, TAM, and FLOW respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Figure 3(D) illustrates the combined context score where w1, w2, and w3 is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' User U1 and U2 show that the final context score is decided by FLOW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Similarly, U3, U4, and U5 the final score is decided by TAM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Using policy 2, user U5 incurs ≈ 300ms latency for a context score of 8, which is the highest latency amongst other users introduced by CAPOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Notably, the evaluation performed using a simulated dataset might not reflect the worst case efficiency of CAPOW as in practice, user U5 might not be deviate in a temporal activity context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In this section, we discuss that the cost of deceiving multiple AI models is expensive for the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In our implementation, user U5 has to deceive three AI models to receive an easy PoW puzzle by receiving lower context scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' User U5 can receive a lower context score for DAbR by trivially spoofing the IP address (Assumption 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To deceive TAM, the user can engineer the requests around the same time as noticed during eavesdropping (Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' As reading or tracking flow-level data embedded in request payload data while eavesdropping is not possible (Assumption 1), the only way to deceive FLOW is by sending multiple probe packets to land on a low context score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' This is an extensive approach as a security personnel may select new contexts to improve the defensive posture of the organization periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Therefore, deceiving all AI models becomes expensive for the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To validate contribution 3, we designed and evaluated an implementation instance on CAPOW and provided policy designs to validate contribution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Finally, CAPOW ensures that malicious users incur higher latency than legitimate users based on the deviation in context pattern that prevents DDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Hence, we validate contribution 1 (see Section I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' RELATED WORKS In this section, we discuss the overview of proof-of-work (PoW) literature in DDoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Relevant to our work, we will also discuss the current advances in AI-assisted cybersecurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Classical Proof-of-Work Dwork et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [10] coined the term proof-of-work (PoW) when they proposed the use of cryptographic hash functions (also known as client puzzles) to combat unsolicited bulk emails (junk emails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Following that, Franklin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [11] proposed a lightweight website metering scheme in 1997 to prevent fraudulent web server owners from inflating their website’s popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In 1999, Jakobsson et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [16] proposed MicroMinting (originally proposed by Rivest et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [30] as a digital payment scheme) as a candidate problem that can reuse the computational effort of solving the POW puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Later that year, Laurie et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [18] proposed that proof of work does not work in a spam setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Proof-of-Work as DoS defense Similar to spam emails, in DDoS, it is significantly cheaper for the attacking party to launch a DDoS attack than to defend an infrastructure with the defending party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' According to Arbor network, launching a DoS attack costs an average of $66 per attack and can cause damage to the victim of around $500 per minute [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Aura et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [4] proposed the first client puzzle authentication protocol for a DoS resilient system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Mankins et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [21] investigated methods for tuning the amount of re- source consumption to access server resources based on client behavior, where the costs imposed can be either monetary or computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' In a similar vein, Wang and Reiter [33] investigate how clients can bid on puzzles through auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Ndibwile et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [22] proposed web traffic authentication as a replacement for CAPTCHA-based defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Wu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [36] proposed a software puzzle framework that disqualifies the adversary’s ability to gain an advantage by using a GPU to solve puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' A framework was put forth by Dean et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [8] to reduce DoS in TLS servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' A DoS variant was introduced by Wood et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Certain PoW defenses against DoS are layer- specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The network layer of the proof-of-work system used by Parno et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [26] prioritizes users who use more CPU time to solve puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The Heimdall architecture, which can detect any change in network flow in routers, was introduced by Chen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' When a change in network flow is identified for any new connection, a puzzle is generated and sent to the new user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The difficulty of the computational challenges used in the context of DoS attacks on the transport layer was recently assessed using game theory by Noureddine et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Walfish et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [32] propose an alternative resource called communication capacity as a defense against application-layer flood attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Other research has concentrated on incorporating PoW puzzles into practical browsing experiences [5], [6], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Automated DoS defense In this section, we revisit the literature on ensemble learning techniques for network traffic classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' En- semble learning is a branch of supervised machine learning technique that aggregates the learning of multiple base learners to improve overall prediction accuracy [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Like network traffic classification problems, each base learner is trained to become an expert in the local area of the total feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Gaikwad et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [12] proposed a bagging ensemble approach using REPTree base learners to improve classification over weaker AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Gupta et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [2] suggested an IDS system that uses ensemble learning to address a class imbalance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The ensemble learner uses three base learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' First, the deep neural network classifies normal and suspicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Second, eXtreme Gradient Boosting is used to identify major attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Third, random forest is used to classify minor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Zhou et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [1] proposed feature selection process using ensemble learning in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The first stage involves feature reduction using the heuristic method CFS and the Bat Algorithm (BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The second stage involves aggregating C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='5 and Random Forest (RF) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Jabbar et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' al [15] suggested an ensemble classifier that uses Alternating Decision Tree (ADTree) and the k-Nearest Neighbor algorithm (kNN) as base AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' Paulauskas and Auskalnis [27] proposed an ensemble learner that employs four base classifiers: J48, C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content='0, Naive Bayes, and Partial Decision List (PART) to improve classification results over individual AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this paper, we design and evaluate CAPOW a context- aware AI-assisted PoW framework that protects critical servers against DDoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The underlying defensive strategy involves adaptively introducing latency on malicious users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' To achieve this functionality, our framework employs an AI model that takes the context attributes from the incoming user request packet as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' The AI model computes deviation from normal activity patterns to output a context score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' This score influences the difficulty level of a PoW puzzle that injects latency adaptively during communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' CAPOW ensures that the ability of a malicious user to prolong the attack is constrained by adaptively introducing latency through PoW puzzles and compelling malicious users to expend more re- sources to complete an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' For future work, different design variants of CAPOW can be configured to combat different DDoS attack types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' PoW systems suffer from inherent pitfalls of resource wastage which can be circumvented by replacing the model with proof of stake (PoS) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFKT4oBgHgl3EQfQi0f/content/2301.11767v1.pdf'} +page_content=' 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sha256:5e260e8d2552dbfce50b63fdcad9760a0550d0685febde21cc515ff410fc533f +size 4194349 diff --git a/69E4T4oBgHgl3EQf2A2F/content/tmp_files/2301.05295v1.pdf.txt b/69E4T4oBgHgl3EQf2A2F/content/tmp_files/2301.05295v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4edeb9db42a9279a58162527b15df2c1abe21cbe --- /dev/null +++ b/69E4T4oBgHgl3EQf2A2F/content/tmp_files/2301.05295v1.pdf.txt @@ -0,0 +1,416 @@ +Rock Guitar Tablature Generation via +Natural Language Processing +Josue Casco-Rodriguez +Rice University +Houston, TX, USA +jc135@rice.edu +Abstract +Deep learning has recently empowered and democratized generative modeling of images and +text [1, 2], with additional concurrent works exploring the possibility of generating more complex +forms of data, such as audio [3, 4]. However, the high dimensionality, long-range dependencies, +and lack of standardized datasets currently makes generative modeling of audio and music very +challenging. +We propose to model music as a series of discrete notes upon which we can use +autoregressive natural language processing techniques for successful generative modeling. While +previous works used similar pipelines on data such as sheet music and MIDI [5, 6], we aim to +extend such approaches to the under-studied medium of guitar tablature. Specifically, we develop +the first work to our knowledge that models one specific genre—heavy rock—as guitar tablature. +Unlike other works in guitar tablature generation, we have a freely available public demo at +https://huggingface.co/spaces/josuelmet/Metal Music Interpolator.1 +1 +Introduction +Music, like images and language, is a fundamental form of art and a quintessential piece of the hu- +man experience. Despite the fact that recent works—such as Stable Diffusion and OpenAI’s DALL-E +[7, 1]—have produced explosive breakthroughs in the generation and modeling of visual art, such +breakthroughs for music production have not yet been realized; however, recent works, such as Ope- +nAI’s Jukebox [4], have made progress towards advanced music generation. A large factor in why such +breakthroughs have yet to be is that music is challenging to model, requiring sequence-modeling of +data mediums that are not as well-understood or intuitive as images or words. +When investigating how a machine can learn to understand or generate music, one can begin by +understanding how people learn to work with music. Although music exists as a continuous-time au- +dio signal, people most efficiently understand and analyze music as a pattern of discrete frequencies +(for example, the note “A” = 440 Hz) that are played for discretely quantized intervals of time (e.g., +quarter-, half-, and whole-notes). As such, sequence-modeling techniques for understanding sequences +of discrete data can be leveraged towards music modeling, given a sufficient dataset of music samples +represented in discrete forms. +While sheet music and Musical Instrument Digital Interface (MIDI) files are conventional forms of +discretely compressed representations of music, one prominent form of music representation that has +been studied less is guitar tablature (see fig. 1 and fig. 2). While appearing similar to traditional +sheet music, guitar tablature differs by representing notes as fret and string indices upon which the +instrument-players must place their fingers so as to produce a specific note, since stringed instruments +are unique in that most notes that can be played on them have more than one fret and string combi- +nation that can produce them. +We develop a new dataset of compressed representations of guitar tablature from one specific genre of +1Our source code is used to train final demo model is available at https://github.com/Josuelmet/Metal-Music- +Interpolator. +1 +arXiv:2301.05295v1 [eess.AS] 12 Jan 2023 + +Figure 1: +A guitar tablature snippet of two measures written in 4/4 time. Each note is represented +not as its pitch, but rather as the specific fret index upon which a player should press upon a certain +string so as to produce the note. The fret index of a note on the string it is played on is equivalent to +the number of semitones between the note’s pitch and the lowest pitch that the string can play. Note +that certain notes contain information about their dynamics: for example, the P.M. symbol indicates +that certain notes should be played in a semi-muted fashion. +Figure 2: +Another guitar tablature snippet in 4/4 time, this time consisting of the iconic first measure +of Sweet Child O’ Mine by Guns N’ Roses. The six pitches arranged vertically on the left are the lowest +pitches that each of the six guitar strings can play. +music (heavy rock), as well as a neural network architecture that can leverage sequence modeling (such +as in long short-term memory networks or natural language processing models) to produce new guitar +tablature sequences when conditioned on a brief snippet of an existing guitar tablature. Specifically, +the proposed autoregressive model aims to estimate the most likely new tablature token xN+1 when +given the previous tokens x1, x2, ..., xN (i.e., estimate the conditional probability p(xN+1|x1, ..., xN)), +thus enabling an iterative procedure through which an M-token sequence can be generated from an +N-token sequence, for M > N. +2 +Background +2.1 +Sequence Models +Recurrent networks. Sequence modeling is a long-standing problem in machine learning and statis- +tics, with one of its earliest prominent efforts being recurrent neural networks [8]. While recurrent +neural networks are able to leverage a form of memory to model sequences of theoretically unbounded +contexts, in practice they and their recent variants [9] struggle to do so, in part due to gradient prop- +agation issues [10]. +Transformers. +Meanwhile, transformers [11] circumvent the problems with recurrent neural net- +works by replacing recurrent operations with one feedforward attention operation that compares every +element of a sequence with every other element of the sequence; such an approach could initially +seem disadvantaged due to the memoryless nature, inherently finite bounded context [12], and O(N 2) +runtime of an attention mechanism on a sequence of length N [13]. However, when combined with +additional innovations such as positional embeddings and token dimensionality reduction via vector- +ization, transformer architectures have yielded enormous advances in sequence and image modeling +[2, 1, 14]. +Self-attention. The key behind any transformer architecture is the self-attention mechanism. Given +a vector-valued input sequence X = [x1, x2, ..., xN] ∈ RN×Dx such that each element is Dx-dimensional +and the transformer feedforward dimension is D, a self-attention head transforms X into an output +sequence ˆV through the following: [13] +2 + +P. M. +P. M. +P. M. +4 +5 +18 +仆9 +10 +0 +0 +12 +0 +0 +11 +0 +81 +D# +15 +14 +4 +A# +15 +F# +14 +—12 +14 +14 +C# +4 +12 +G# +D#1. Using the weights WQ, WK ∈ RD×Dx and WV ∈ RDv×Dx, project X into three distinct +matrices—the query, key, and value matrices Q, K, and V—via these linear transformations: +Q = XWT +Q +K = XWT +K +V = XWT +V +2. Let us express the query, key, and value matrices as Q = [q1, ..., qN]T , K = [k1, ..., kN]T , and +V = [v1, ..., vN]T , where the vectors qi, ki, vi for i ∈ {1, 2, ..., N} are the query, key, and value +vectors, respectively. +Each output sequence vector ˆvi is calculated by multiplying each value vector vj by a score +determined as the similarity between the query vector qi the key vector kj : +ˆvi = +N +� +j=1 +softmax +�qT +i kj +√ +D +� +vj +Calculation of ˆV = [ˆv1, ..., ˆvN]T can thus be simply expressed as: +ˆV = softmax +� +QKT +√ +D +� +V = AV, +where the attention matrix A is computed by applying the softmax operation to each row of the +matrix QKT / +√ +D [13]. +2.2 +Related Works +Music/audio generation. Our key contribution to musical sequence modeling is publicly available +guitar tablature modeling of heavy rock. Various previous and ongoing works have approached music +generation both continuous and discrete data modalities. For example, the recent SaShiMi [3] and +Jukebox [4] architectures approach audio and music generation in the spaces of continuous waveforms +and discrete notes, respectively. The advent of diffusion models [15] has also found influence in a new +model combining spectrogram and MIDI music generation [16]. +Guitar tablature literature. +The field of guitar tablature analysis is small but growing, with +various works tackling challenges such as graph-based solo analysis [17], transcription [18], dataset +collection, and sequence modeling [19, 20, 21]. Of particular importance to our work are AnimeTab +[19] and DadaGP [20], since they also opt for a transformer-based approach to statistically generate +sequences of guitar tablature. Unlike DadaGP, our model has token representations that are much +more simple and easy to understand, is trained on one specific genre, and has a publicly available +demo. While our model may share some similarities with AnimeTab, which was published during the +development of this work and is supposed to have a demo released soon, our model has a demo already +available and is trained on the genre of heavy rock music instead of anime/video game music. +3 +Methods +3.1 +Data Processing +Initial preprocessing. The success of statistical inference methods often reflects the quality of data +used for training—data preprocessing is just as important to a successful model as the model itself. +Our data preprocessing pipeline begins by first collecting a sizeable volume of songs, in guitar tabla- +ture format, that accurately represent one subgenre of music2. For each tablature file, every song is +first converted into 4/4 time for ease of processing, and is then converted into a Python object via +PyGuitarPro3 for ease of querying. Each track of each song (i.e., each instrument or voice of each +2Complete list of songs: https://github.com/Josuelmet/Metal-Music-Interpolator/blob/main/songs/README.md +3https://github.com/Perlence/PyGuitarPro +3 + +Figure 3: +Note embedding scheme illustrated for the example note of a whole note on fret 0 of the +lowest string on a guitar/bass. The fret value is one-hot encoded as 0, the note length is one-hot +encoded as a whole note, and none of the flags are set to 1 because the note is neither dotted nor +palm-muted nor a dead/rest/tied note. +song, not including drums) is then converted into a one-dimensional list containing each note in the +song; each note is represented as a tuple containing the note’s pitch (with special designations for tied, +dead4, and rest notes) , duration, the chordal nature if applicable (with the represented chords being +4th, diminished 5th, and perfect 5th chords), and two flags indicating whether the note is dotted and +whether the note is muted. Note that the pitch of each note is represented not as the musical pitch +of each note (e.g., “A4” or “C3”), but rather as the fret on the guitar (or bass) upon which a player +should place their finger so as to generate the note. Once all songs’ notes have been represented as +tuples, each tuple is converted to an integer via an invertible dictionary map. +Embedding initialization. After initial pre-processing, each song exists as a set of sequences, where +each sequence represents one voice or instrument and contains integers that represent each note. While +a na¨ıve sequence model could attempt inference upon these scalar sequences, modern sequence models +have found success in instead representing the individual tokens or elements of a sequence as vectors, +allowing for more expressive and informative representations token modalities. Unlike previous works +[20], we opt for a simple, but effective, initial token vectorization illustrated in fig. 3. Each initial +vectorized token embedding, before training, has 72 dimensions: the first 59 are reserved for one-hot +encoding the number of semitones (equivalent to the number of frets on a guitar or bass) between the +pitch value and the lowest pitch playable by the given instrument; the next 3 dimensions are flags indi- +cating if the note is a dead, rest, or tied note; the next 8 dimensions one-hot encode the note’s duration +(e.g., whole-, half-, and quarter-notes); and the last two dimensions are flags indicating if the note’s +duration is dotted and if the note is played in a muted fashion. While the vectorized token embeddings +are trained, and thus iteravely refined, during the training process, we found that our hand-crafted +initialization scheme performed better than default random token initialization. As in any other suc- +cessful transformer architecture, each token also has a positional embedding that accompanies the +vectorized token embedding before going into the transformer model. +3.2 +Model Architecture +In addition to implementing the entire data preprocessing pipeline with the only starting point being +PyGuitarPro for querying tablature files as Python objects, we also had to manually implement the +transformer architecture, since we opted for a mini-GPT model due to the relatively small dataset at +our disposal compared to the number of sequences and number of parameters used to train conven- +tional language models [2]. Since we were not using a pre-written transformer model, we also had to +implement a causal masking mechanism to ensure that the transformer cannot use information from +any tokens after the token it is trying to predict. After extensive hyperparameter tuning on a 90-10 +testing-validation, split, our final hyperparameter values are located in table 1. The final mini-GPT ar- +chitecture consists of an embedding layer (as described earlier), three transformer blocks in sequence, +4Dead notes are noted that are heavily muted such that they lose a distinct sense of pitch. +4 + +0 +0 +1 +2 +57 +58 +Extra +Dead +Rest +Tied +Fret: +Note +10 +0 +0 +0 +0 +0 +0 +0 +Types: +cat +Dotted +Palm +Whole +Half +64th +128th +Duration +Mute +Note +1 +0 +Length: +0 +0 +0 +0 +0Hyperparameter +Value +N (sequence length) +100 +Output dimensionality (number of unique tokens) +1629 +Initial embedding dimensionality +72 +Number of transformer blocks +3 +Transformer feedforward dimension +512 +Transformer dropout rate +30% +Attention heads per transformer +10 +Initial learning rate +0.003 +β1 (Adam parameter) +0.96 +Batch size +512 +Training epochs (determined by early stopping) +∼ 120 +Table 1: +Hyperparameters of the final mini-GPT model. +Figure 4: +Training and validation accuracy and loss of the final mini-GPT model using a 90-10 +training-validation split. +and one final feedforward layer that returns a 1629-dimensional vector representing the conditional +probability of the next token’s value p (xN+1|x1, ..., xN). +4 +Results +Performance. Using the hyperparameters specified in table 1 and early stopping based on the valida- +tion loss, our mini-GPT model achieved over 70% training accuracy and over 60% validation accuracy, +as shown in fig. 4. While it was possible to improve the training accuracy by reducing the transformer +dropout, we empirically found that doing so worsened validation generalization due to overfitting. +Qualititative evaluation of our model is made possible through the interactive demo provided5. Not +only is our work the first, to our knowledge, that provides a publicly accessible generation demo, but +our model’s success also further validates the usage of and need for natural language processing tech- +niques in the realm of audio and music generation. +Future works. Straightforward and interesting extensions of our model include more explicity incor- +porating rhythmic information [21] in the embeddings or as a separate embedding; such an approach +would be better able to capture and understand the differences between notes that land on downbeats, +upbeats, and backbeats. Another extension would be to model multiple tracks or voices at a time, +allowing for modeling of rich chords or drum sequences. For note-based chord or drum modeling, some +form of specialized translational invariance could be key to ensuring that the autoregressive model +understands that, when multiple notes are placed at the same time, their order in a sequence does not +matter. Using a more novel attention mechanism [13] could further improve performance. Diffusion +5https://huggingface.co/spaces/josuelmet/Metal Music Interpolator +5 + +Accuracy +Loss +0.7 +Train Accuracy +6 +Train LosS +Val Accuracy +Val Los5 +0.6 +5 +0.5 +4 +0.4 +3 +0.3 +0.2 +2 - +0.1 +1 +0 +20 +40 +60 +80 +100 +120 +20 +40 +60 +80 +100 +120models could also play a role in generating sequences of guitar tablature, but whether they can out- +perform autoregressive language models has yet to be shown; recent work has shown that transformers +and diffusion models can be combined to produce state-of-the-art results in music generation [16]. +References +[1] A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen, “Hierarchical text-conditional image +generation with CLIP latents,” arXiv preprint arXiv:2204.06125, 2022. +[2] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, +G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, +D. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. 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Engel, “Multi- +instrument music synthesis with spectrogram diffusion,” in International Society for Music Infor- +mation Retrieval Conference, vol. 23, 2022. +[17] S. Ferretti, “Guitar solos as networks,” in IEEE International Conference on Multimedia and +Expo, 2016. +[18] Y.-H. Chen, W.-Y. Hsiao, T.-K. Hsieh, J.-S. R. Jang, and Y.-H. Yang, “Towards automatic +transcription of polyphonic electric guitar music: A new dataset and a multi-loss transformer +model,” in IEEE International Conference on Acoustics, Speech and Signal Processing, 2022. +[19] Y. Zhou, Y. Ju, and L. Xie, “Animetab: A new guitar tablature dataset of anime and game +music,” 2022. +[20] P. Sarmento, A. Kumar, C. Carr, Z. Zukowski, M. Barthet, and Y.-H. Yang, “DadaGP: a dataset +of tokenized GuitarPro songs for sequence models,” in International Society for Music Information +Retrieval Conference, vol. 22, 2021. +[21] Y.-H. Chen, Y.-H. Huang, W.-Y. Hsiao, and Y.-H. Yang, “Automatic composition of guitar tabs +by transformers and groove modeling,” in International Society for Music Information Retrieval +Conference, vol. 21, 2020. +7 + diff --git a/69E4T4oBgHgl3EQf2A2F/content/tmp_files/load_file.txt b/69E4T4oBgHgl3EQf2A2F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30881c6143067c66888589cb023b0dc74186a762 --- /dev/null +++ b/69E4T4oBgHgl3EQf2A2F/content/tmp_files/load_file.txt @@ -0,0 +1,323 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf,len=322 +page_content='Rock Guitar Tablature Generation via Natural Language Processing Josue Casco-Rodriguez Rice University Houston, TX, USA jc135@rice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='edu Abstract Deep learning has recently empowered and democratized generative modeling of images and text [1, 2], with additional concurrent works exploring the possibility of generating more complex forms of data, such as audio [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' However, the high dimensionality, long-range dependencies, and lack of standardized datasets currently makes generative modeling of audio and music very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' We propose to model music as a series of discrete notes upon which we can use autoregressive natural language processing techniques for successful generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While previous works used similar pipelines on data such as sheet music and MIDI [5, 6], we aim to extend such approaches to the under-studied medium of guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Specifically, we develop the first work to our knowledge that models one specific genre—heavy rock—as guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Unlike other works in guitar tablature generation, we have a freely available public demo at https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='co/spaces/josuelmet/Metal Music Interpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='1 1 Introduction Music, like images and language, is a fundamental form of art and a quintessential piece of the hu- man experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Despite the fact that recent works—such as Stable Diffusion and OpenAI’s DALL-E [7, 1]—have produced explosive breakthroughs in the generation and modeling of visual art, such breakthroughs for music production have not yet been realized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' however, recent works, such as Ope- nAI’s Jukebox [4], have made progress towards advanced music generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' A large factor in why such breakthroughs have yet to be is that music is challenging to model, requiring sequence-modeling of data mediums that are not as well-understood or intuitive as images or words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' When investigating how a machine can learn to understand or generate music, one can begin by understanding how people learn to work with music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Although music exists as a continuous-time au- dio signal, people most efficiently understand and analyze music as a pattern of discrete frequencies (for example, the note “A” = 440 Hz) that are played for discretely quantized intervals of time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', quarter-, half-, and whole-notes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' As such, sequence-modeling techniques for understanding sequences of discrete data can be leveraged towards music modeling, given a sufficient dataset of music samples represented in discrete forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While sheet music and Musical Instrument Digital Interface (MIDI) files are conventional forms of discretely compressed representations of music, one prominent form of music representation that has been studied less is guitar tablature (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While appearing similar to traditional sheet music, guitar tablature differs by representing notes as fret and string indices upon which the instrument-players must place their fingers so as to produce a specific note, since stringed instruments are unique in that most notes that can be played on them have more than one fret and string combi- nation that can produce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' We develop a new dataset of compressed representations of guitar tablature from one specific genre of 1Our source code is used to train final demo model is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='com/Josuelmet/Metal-Music- Interpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='05295v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='AS] 12 Jan 2023 Figure 1: A guitar tablature snippet of two measures written in 4/4 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Each note is represented not as its pitch, but rather as the specific fret index upon which a player should press upon a certain string so as to produce the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The fret index of a note on the string it is played on is equivalent to the number of semitones between the note’s pitch and the lowest pitch that the string can play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Note that certain notes contain information about their dynamics: for example, the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' symbol indicates that certain notes should be played in a semi-muted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Figure 2: Another guitar tablature snippet in 4/4 time, this time consisting of the iconic first measure of Sweet Child O’ Mine by Guns N’ Roses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The six pitches arranged vertically on the left are the lowest pitches that each of the six guitar strings can play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' music (heavy rock), as well as a neural network architecture that can leverage sequence modeling (such as in long short-term memory networks or natural language processing models) to produce new guitar tablature sequences when conditioned on a brief snippet of an existing guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Specifically, the proposed autoregressive model aims to estimate the most likely new tablature token xN+1 when given the previous tokens x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', xN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', estimate the conditional probability p(xN+1|x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', xN)), thus enabling an iterative procedure through which an M-token sequence can be generated from an N-token sequence, for M > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='1 Sequence Models Recurrent networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Sequence modeling is a long-standing problem in machine learning and statis- tics, with one of its earliest prominent efforts being recurrent neural networks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While recurrent neural networks are able to leverage a form of memory to model sequences of theoretically unbounded contexts, in practice they and their recent variants [9] struggle to do so, in part due to gradient prop- agation issues [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Meanwhile, transformers [11] circumvent the problems with recurrent neural net- works by replacing recurrent operations with one feedforward attention operation that compares every element of a sequence with every other element of the sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' such an approach could initially seem disadvantaged due to the memoryless nature, inherently finite bounded context [12], and O(N 2) runtime of an attention mechanism on a sequence of length N [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' However, when combined with additional innovations such as positional embeddings and token dimensionality reduction via vector- ization, transformer architectures have yielded enormous advances in sequence and image modeling [2, 1, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The key behind any transformer architecture is the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Given a vector-valued input sequence X = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', xN] ∈ RN×Dx such that each element is Dx-dimensional and the transformer feedforward dimension is D, a self-attention head transforms X into an output sequence ˆV through the following: [13] 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 4 5 18 仆9 10 0 0 12 0 0 11 0 81 D# 15 14 4 A# 15 F# 14 —12 14 14 C# 4 12 G# D#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Using the weights WQ, WK ∈ RD×Dx and WV ∈ RDv×Dx, project X into three distinct matrices—the query, key, and value matrices Q, K, and V—via these linear transformations: Q = XWT Q K = XWT K V = XWT V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Let us express the query, key, and value matrices as Q = [q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', qN]T , K = [k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', kN]T , and V = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', vN]T , where the vectors qi, ki, vi for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', N} are the query, key, and value vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Each output sequence vector ˆvi is calculated by multiplying each value vector vj by a score determined as the similarity between the query vector qi the key vector kj : ˆvi = N � j=1 softmax �qT i kj √ D � vj Calculation of ˆV = [ˆv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', ˆvN]T can thus be simply expressed as: ˆV = softmax � QKT √ D � V = AV, where the attention matrix A is computed by applying the softmax operation to each row of the matrix QKT / √ D [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='2 Related Works Music/audio generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Our key contribution to musical sequence modeling is publicly available guitar tablature modeling of heavy rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Various previous and ongoing works have approached music generation both continuous and discrete data modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' For example, the recent SaShiMi [3] and Jukebox [4] architectures approach audio and music generation in the spaces of continuous waveforms and discrete notes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The advent of diffusion models [15] has also found influence in a new model combining spectrogram and MIDI music generation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Guitar tablature literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The field of guitar tablature analysis is small but growing, with various works tackling challenges such as graph-based solo analysis [17], transcription [18], dataset collection, and sequence modeling [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Of particular importance to our work are AnimeTab [19] and DadaGP [20], since they also opt for a transformer-based approach to statistically generate sequences of guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Unlike DadaGP, our model has token representations that are much more simple and easy to understand, is trained on one specific genre, and has a publicly available demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While our model may share some similarities with AnimeTab, which was published during the development of this work and is supposed to have a demo released soon, our model has a demo already available and is trained on the genre of heavy rock music instead of anime/video game music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 3 Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='1 Data Processing Initial preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The success of statistical inference methods often reflects the quality of data used for training—data preprocessing is just as important to a successful model as the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Our data preprocessing pipeline begins by first collecting a sizeable volume of songs, in guitar tabla- ture format, that accurately represent one subgenre of music2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' For each tablature file, every song is first converted into 4/4 time for ease of processing, and is then converted into a Python object via PyGuitarPro3 for ease of querying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Each track of each song (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', each instrument or voice of each 2Complete list of songs: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='com/Josuelmet/Metal-Music-Interpolator/blob/main/songs/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='md 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='com/Perlence/PyGuitarPro 3 Figure 3: Note embedding scheme illustrated for the example note of a whole note on fret 0 of the lowest string on a guitar/bass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The fret value is one-hot encoded as 0, the note length is one-hot encoded as a whole note, and none of the flags are set to 1 because the note is neither dotted nor palm-muted nor a dead/rest/tied note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' song, not including drums) is then converted into a one-dimensional list containing each note in the song;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' each note is represented as a tuple containing the note’s pitch (with special designations for tied, dead4, and rest notes) , duration, the chordal nature if applicable (with the represented chords being 4th, diminished 5th, and perfect 5th chords), and two flags indicating whether the note is dotted and whether the note is muted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Note that the pitch of each note is represented not as the musical pitch of each note (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', “A4” or “C3”), but rather as the fret on the guitar (or bass) upon which a player should place their finger so as to generate the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Once all songs’ notes have been represented as tuples, each tuple is converted to an integer via an invertible dictionary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Embedding initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' After initial pre-processing, each song exists as a set of sequences, where each sequence represents one voice or instrument and contains integers that represent each note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While a na¨ıve sequence model could attempt inference upon these scalar sequences, modern sequence models have found success in instead representing the individual tokens or elements of a sequence as vectors, allowing for more expressive and informative representations token modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Unlike previous works [20], we opt for a simple, but effective, initial token vectorization illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Each initial vectorized token embedding, before training, has 72 dimensions: the first 59 are reserved for one-hot encoding the number of semitones (equivalent to the number of frets on a guitar or bass) between the pitch value and the lowest pitch playable by the given instrument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' the next 3 dimensions are flags indi- cating if the note is a dead, rest, or tied note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' the next 8 dimensions one-hot encode the note’s duration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', whole-, half-, and quarter-notes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' and the last two dimensions are flags indicating if the note’s duration is dotted and if the note is played in a muted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While the vectorized token embeddings are trained, and thus iteravely refined, during the training process, we found that our hand-crafted initialization scheme performed better than default random token initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' As in any other suc- cessful transformer architecture, each token also has a positional embedding that accompanies the vectorized token embedding before going into the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='2 Model Architecture In addition to implementing the entire data preprocessing pipeline with the only starting point being PyGuitarPro for querying tablature files as Python objects, we also had to manually implement the transformer architecture, since we opted for a mini-GPT model due to the relatively small dataset at our disposal compared to the number of sequences and number of parameters used to train conven- tional language models [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Since we were not using a pre-written transformer model, we also had to implement a causal masking mechanism to ensure that the transformer cannot use information from any tokens after the token it is trying to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' After extensive hyperparameter tuning on a 90-10 testing-validation, split, our final hyperparameter values are located in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' The final mini-GPT ar- chitecture consists of an embedding layer (as described earlier), three transformer blocks in sequence, 4Dead notes are noted that are heavily muted such that they lose a distinct sense of pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 4 0 0 1 2 57 58 Extra Dead Rest Tied Fret: Note 10 0 0 0 0 0 0 0 Types: cat Dotted Palm Whole Half 64th 128th Duration Mute Note 1 0 Length: 0 0 0 0 0Hyperparameter Value N (sequence length) 100 Output dimensionality (number of unique tokens) 1629 Initial embedding dimensionality 72 Number of transformer blocks 3 Transformer feedforward dimension 512 Transformer dropout rate 30% Attention heads per transformer 10 Initial learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='003 β1 (Adam parameter) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='96 Batch size 512 Training epochs (determined by early stopping) ∼ 120 Table 1: Hyperparameters of the final mini-GPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Figure 4: Training and validation accuracy and loss of the final mini-GPT model using a 90-10 training-validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' and one final feedforward layer that returns a 1629-dimensional vector representing the conditional probability of the next token’s value p (xN+1|x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=', xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 4 Results Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Using the hyperparameters specified in table 1 and early stopping based on the valida- tion loss, our mini-GPT model achieved over 70% training accuracy and over 60% validation accuracy, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' While it was possible to improve the training accuracy by reducing the transformer dropout, we empirically found that doing so worsened validation generalization due to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Qualititative evaluation of our model is made possible through the interactive demo provided5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Not only is our work the first, to our knowledge, that provides a publicly accessible generation demo, but our model’s success also further validates the usage of and need for natural language processing tech- niques in the realm of audio and music generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Straightforward and interesting extensions of our model include more explicity incor- porating rhythmic information [21] in the embeddings or as a separate embedding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' such an approach would be better able to capture and understand the differences between notes that land on downbeats, upbeats, and backbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Another extension would be to model multiple tracks or voices at a time, allowing for modeling of rich chords or drum sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' For note-based chord or drum modeling, some form of specialized translational invariance could be key to ensuring that the autoregressive model understands that, when multiple notes are placed at the same time, their order in a sequence does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Using a more novel attention mechanism [13] could further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Diffusion 5https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='co/spaces/josuelmet/Metal Music Interpolator 5 Accuracy Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='7 Train Accuracy 6 Train LosS Val Accuracy Val Los5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='2 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content='1 1 0 20 40 60 80 100 120 20 40 60 80 100 120models could also play a role in generating sequences of guitar tablature, but whether they can out- perform autoregressive language models has yet to be shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' recent work has shown that transformers and diffusion models can be combined to produce state-of-the-art results in music generation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Ramesh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'} +page_content=' Dhariwal, A.' metadata={'source': 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b/7NE4T4oBgHgl3EQfCQsk/content/tmp_files/2301.04858v1.pdf.txt @@ -0,0 +1,1675 @@ +Advancing carrier transport models for InAs/GaSb type-II superlattice MWIR +photodetectors +Rohit Kumar, Anup Kumar Mandia, Anuja Singh and Bhaskaran Muralidharan∗ +Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India +(Dated: January 13, 2023) +In order to provide the best possible performance, modern infrared photodetector designs necessitate +extremely precise modeling of the superlattice absorber region. We advance the Rode’s method for +the Boltzmann transport equation in conjunction with the k.p band structure and the envelope +function approximation for a detailed computation of the carrier mobility and conductivity of layered +type-II superlattice structures, using which, we unravel two crucial insights. First, the significance of +both elastic and inelastic scattering mechanisms, particularly the influence of the interface roughness +and polar optical phonon scattering mechanisms in technologically relevant superlattice structures. +Second, that the structure-specific Hall mobility and Hall scattering factor reveals that temperature +and carrier concentrations significantly affect the Hall scattering factor, which deviates significantly +from unity even for small magnetic fields. +This reinforces the caution that should be exercised +when employing the Hall scattering factor in experimental estimations of drift mobilities and carrier +concentrations. +Our research hence offers a comprehensive microscopic understanding of carrier +dynamics in such technologically relevant superlattices. Our models also provide highly accurate +and precise transport parameters beyond the relaxation time approximation and thereby paving the +way to develop physics-based device modules for mid-wavelength infrared photodetectors. +I. +INTRODUCTION +Modeling state-of-the-art infrared (IR) photodetectors +[1–6] require highly accurate transport parameters for de- +veloping dark and photocurrent performance projections +[5, 7–10]. Current technologically relevant IR photode- +tectors use III-V materials such as InAs/GaSb [11, 12] +due to numerous advantages [13, 14]. Type-II superlat- +tices (T2SLs) based on stacks of InAs/GaSb [1, 2, 14] are +thus extensively used to design high-performance third- +generation IR detectors [15, 16]. Despite the fact that the +mobility of the photogenerated minority carriers has a +significant impact on the performance of IR photodetec- +tors, carrier transport in technologically relevant T2SL +structures has not as extensively been explored. Recent +explorations in this context [17–23] which include carrier +mobility calculations [24], do not conclusively bring to +the fore structure-specific impact of important scatter- +ing mechanisms such as Piezoelectric (PZ), polar opti- +cal phonon (POP), acoustic deformation potential (ADP) +scattering mechanisms and most importantly the inter- +face roughness scattering (IRS). +With the necessity to develop a deeper understand- +ing of carrier transport in technologically relevant T2SLs, +this work advances an accurate model for transport calcu- +lations, wherein, we investigate different scattering lim- +ited transport under low-field in InAs/GaSb superlattices +(SLs) as a function of free electron carrier concentration, +temperature, and SL structural parameters. In our cal- +culations, five primary scattering mechanisms that limit +carrier mobility are the ionized impurity (II) [25], the PZ +[26], the ADP [27], the POP and the IRS [28–31]. +∗ corresponding author: bm@ee.iitb.ac.in +We advance the Rode’s method [32–34] which goes be- +yond the relaxation time approximation (RTA) [35, 36], +coupled with band structure calculations via the k.p [37– +41] technique that also includes the strain effect due to +lattice mismatch between InAs and GaSb materials [42]. +We demonstrate the effect of both the elastic and the +inelastic scattering mechanisms [43] on the electron mo- +bility of the composite structure for a wide range of tem- +peratures and doping concentrations. Our studies reveal +that the low-temperature mobility of T2SLs is limited +by the II, PZ and IRS scattering mechanisms. In con- +trast, the mobility at higher temperatures is mainly lim- +ited by the POP scattering mechanism, an inelastic and +anisotropic process. At intermediate temperatures, how- +ever, the mobility decreases due to a combined effect of +ADP and IRS mechanisms. The effects of several struc- +tural parameters including layer thicknesses, interface +roughness heights, correlation lengths, and ion densities +are thoroughly investigated. Our calculations thereby re- +inforce the superiority of the Rode’s method [32, 34] over +the conventionally employed RTA, wherein, the former is +applicable over a wide temperature range in the presence +of inelastic and anisotropic scattering mechanism. +In order to experimentally obtain the carrier concen- +tration and drift mobility in a SL structure, it is also im- +portant to ascertain the Hall scattering factor, which is +frequently thought of as being equal to unity, indicating +that the Hall mobility and the drift mobility are equal. +However, in many heterostructures, it differs significantly +from unity, which results in inaccurate estimates of the +carrier density and drift mobility. We clearly show that +the temperature and carrier concentrations significantly +affect the Hall scattering factor, and that it ranges from +0.3 to about 1.48 even for weak magnetic fields, thereby +reinforcing that caution should be exercised when em- +ploying this factor in calculations involving drift mobil- +arXiv:2301.04858v1 [cond-mat.mtrl-sci] 12 Jan 2023 + +2 +ity and carrier concentration. The models developed here +pave the way to develop physics-based device modules for +mid-wavelength IR (MWIR) photodetectors. +This paper is structured as follows. In Sec. II we de- +scribe the k.p model to compute the band structure, elec- +tron distribution function, Boltzmann transport formal- +ism, Rode’s approach and various scattering processes. +In Sec. III we illustrate the simulation methodology. In +Sec. IV, we explain the findings and finally, in Sec. V, +we summarize our results. +II. +ANALYTICAL FORMALISM +A. +Electronic band structure +The energy band structure of T2SLs can be calcu- +lated using various theoretical approaches like the den- +sity functional theory (DFT) [44], the empirical tight- +binding method [35, 45, 46], the empirical pseudopoten- +tial method [47, 48], many-body perturbation theory [49] +and the k.p perturbation method [37]. For this study, we +use the k.p technique with the envelope function approx- +imation (EFA) [24, 50, 51] since it overcomes the compu- +tational limitations of first-principles methods. The k.p +model is extensively used because of its superiority in +computing the energy band gap. Unlike ab − initio and +tight binding methods, the k.p technique requires fewer +input parameters I, with the related calculation proce- +dure being straightforward. +In this work, we solve the 8-band Kane Hamilto- +nian [52], by perturbatively extending the wave function +around high-symmetry points of the reciprocal space, em- +ploying the Lowdin’s perturbation approach [52]. +We +also consider the spin-orbit coupling [53] in our com- +putation, which provides additional contributions to the +spin splitting of the energy bands [3]. The SL wavefunc- +tions (Φn(z)) in the orbital basis states (u0(z)) along the +growth direction (z) are articulated in terms of the slowly +varying envelope functions (F(z)), which are given as +Φn(z) = +� +j +Fj(z)uj0(z). +(1) +Such envelope functions under the periodic boundary +conditions can be rewritten as +F i +j(k, z0) = e−iadF i +j(k, zM), +F i +j(k, zM+1) = eiadF i +j(k, z1), +(2) +where, d denotes the thickness of a period, M represents +the number of grid points, a denotes the Bloch vector of +the envelope function that spans the Brillouin zone (BZ) +and k represents the momentum along the transverse di- +rection. The final Hamiltonian of the SL in the basis set +comprises three matrices (H0, HI and HII), given by +H(k, kz) = H0 + HI � +−i ∂ +∂z +� ++ +� +−i ∂ +∂z +� +HII � +−i ∂ +∂z +� +. The +entire coupled differential equation is then solved using +a numerical finite difference method [54], as described in +earlier work [3]. +The interface between the InAs and the GaSb layers +is very abrupt as depicted in Fig. 1(a). The energy dif- +ference between the conduction band minimum (CBM) +and the first heavy hole (HH) maximum at the center of +the BZ determines the band gap in an InAs/GaSb-based +T2SL, as shown in Fig. 1(b). Figure 1(b) also demon- +strates that the InAs conduction band (CB) is lower than +the GaSb valence band (VB), indicating that the band +structure is a staggered T2SL [55]. +B. +Carrier transport model +1. +Boltzmann transport equation and its solution +In order to characterize the behavior of the T2SL sys- +tem, we solve the Boltzmann transport equation (BTE) +and compute the probability of finding a carrier with a +crystal momentum k at a location r at a time t as indi- +cated by the distribution function f(r, k, t). Solving the +BTE (3) yields the average distribution of the carriers in +both the position and the momentum space. The BTE +can be written as [56–58] +∂f +∂t − ∂f +∂t +��� +diff +− ∂f +∂t +��� +forces += ∂f +∂t +��� +coll ++ s(r, p, t) . +(3) +The term s(r, p, t), in Eq. +(3), represents generation- +recombination processes [59], where p is the classical mo- +mentum. The term (∂f/∂t)forces, represents the change +in the distribution function due to applied electric and +magnetic fields. +The term (∂f/∂t)forces = −F · ∇pf, +where, F = (dp/dt) = ℏ(dk/dt) = −e(E + v × B), repre- +sents the total force equal to the sum of the electric-force +and the Lorentz-force owing to the magnetic flux density +B, where e is the electron charge, E is the applied elec- +tric field and v denotes the group velocity of the carriers. +The term (∂f/∂t)diff = −v.∇rf, refers to the spatial +change in the distribution function caused by tempera- +ture or concentration gradients, which results in carrier +diffusion in the coordinate space. Here, (∂f/∂t)coll is the +collision term, which indicates how the distribution func- +tion changes over time due to collision events, and can +be described as the difference between the in- and the +out-scattering processes, i.e., +�∂f +∂t +� +coll += +� +k1 +� +S(k1, k) f (k1) +� +1 − f (k) +� +− S(k, k1) f (k) +� +1 − f (k1) +�� +, +(4) +where, S(k, k1) and S(k1, k) are the transition rates +for an electron moving between states k and k1. +Un- +der steady-state, ∂f +∂t = 0, in case of spatial homogeneity, +∇rf = 0, and assuming that there is no recombination- + +3 +(a) +(b) +FIG. 1. Preliminaries. (a) Schematic of InAs/GaSb based T2SL structure. The electron wave function in the InAs layer +extends beyond the interface into the GaSb layer and overlaps with the heavy hole wave function. Here, nML and mML are +the numbers of monolayers of InAs and GaSb respectively, in a single period d. (b) Band alignment of InAs/GaSb based +T2SL system showing the optical transition between the heavy-hole valence miniband and the electrons from lowest conduction +minibands that is employed to detect IR radiation. The periodic potential of the d period emerges in the material due to the +modulation of the semiconductor layers. The creation of hole (electron) minibands in the valence (conduction) band is caused +by the overlap of hole (electron) wave functions between adjacent GaSb (InAs) layers. The difference between the first electron +miniband in the CB and the first heavy hole miniband in the valence band is used to compute the effective bandgap energy Eg +of the T2SL (highlighted in black). +generation term, the BTE (3) can be rewritten as +−eE +ℏ · ∇kf = +� +k1 +� +S(k1, k) f (k1) +� +1 − f (k) +� +− S(k, k1) f (k) +� +1 − f (k1) +�� +. +(5) +In the low-electric field regime, the distribution func- +tion can be represented as [60] +f(k) = f0 +� +ε(k) +� ++ g(k) cos θ , +(6) +where, k=|k|, f denotes the actual electron distribution +function, which includes both the elastic and the inelas- +tic scattering mechanisms, g(k) is the perturbation term +to f0[ε(k)] produced by the electric field, θ is the angle +between applied electric field (along the symmetry axis) +and the electron wave vector k, and f0 represents the dis- +tribution function under equilibrium conditions, which is +taken according to Fermi-Dirac statistics [35, 59]. +By +solving Eqs. +(5) and (6), the perturbation term g(k), +can be calculated as [32, 34, 61, 62] +gi(k) = +�Si +� +gi(k) +� +− (−e)E +ℏ +� +∂f0 +∂k +� +So(k) + +1 +τel(k) +� +, +(7) +FIG. 2. +The various dominant scattering mechanisms in- +volved in a T2SL structure. +where E = |E|, and gi(k) appears on both sides of Eq. +(7). +Hence, we solve Eq. +(7) iteratively and the con- +vergence is exponentially fast which takes a few itera- +tions. Once gi(k) is obtained, we calculate the mobil- +ity. In Eq. (7), the term i indicates the iteration index, +and the terms, Si & So are the in-scattering and the +out-scattering operators, respectively, for inelastic scat- +tering mechanisms, as explained in Sec. II B. The term +1 +τel(k), represents the total momentum relaxation rate of +all the elastic scattering mechanisms, which is calculated +according to the Matthiessen’s rule (8), and can be writ- +ten as + +Interface scattering +GaSb +Growth direction +Extended Phonons +Interface +nMI +KeD +InAs +Temperature gradientInAs. +GaSb +InAs +GaSb +InAs +GaSb +InAs +z +GaSb +GaSb +CB +GaSb +VB +Ec +Eg +HH +LH +M +InAs +InAs +InAs +InAs +Spatial coordinateDominant scattering mechanisms in T2SL +Defect scattering +Lattice scattering +IRS +Impurity +Intravalley +4 +Ionized +Acoustic +Optical +Deformation potential +Piezoelectric +Polar4 +1 +τel(k) = +1 +τII(k) + +1 +τP Z(k) + +1 +τADP (k) + +1 +τIRS(k) . +(8) +The various dominant scattering mechanisms involved +in an InAs/GaSb-based T2SL structure are shown in Fig. +2. +2. +Ionized impurity scattering +The II scattering mechanism [25] arises due to the +Coulomb interactions between electrons and ions, when +a charged center is introduced inside the bulk material. +The II scattering mechanism is entirely elastic and dom- +inates usually at high doping concentrations and low +temperatures. The II scattering mechanism dominates +near the CB edge but reduces drastically as the energy +increases [63]. +The scattering rate for the II increases +rapidly with decreasing temperature. Here, we use the +Brooks-Herring approach [64] for the calculation of II +scattering rate [34, 65], which is given by +1 +τII(k) = +e4N +8π ν(k) (ϵ0 ϵs)2 (ℏ k)2 +� +P(k) ln +� +1 + 4 +� k +β +�2� +− Q(k) +� +, +(9) +where, ϵ0 is the permittivity of the free space, ϵs is the +static dielectric constant, ℏ is the reduced Planck’s con- +stant and N is the ionized impurity concentration, which +is the sum of the acceptor and donor impurity concen- +tration i.e., N = NA + ND. Here, β indicates the inverse +screening length, which is given as +β = +� +e2 +ϵ0 ϵs kB T +� +DS(ε)f0(1 − f0)dε , +(10) +where, DS(ε) is the density of states (DOS) at energy ε +and kB is the Boltzmann constant. P(k) and Q(k) can +be expressed as follows [34, 62] +P(k) = +� 3 +4 +�β c(k) +k +�4 ++ 2 +�β c(k) +k +�2 ++ 1 +� +, +(11) +Q(k) = +�3 β4 + 6 β2 k2 − 8 k4 +(β2 + 4 k2) k2 +� +c4(k) ++ 8 +� β2 + 2k2 +β2 + 4 k2 +� +c2(k) + +� +4 +� +k/β +�2 +1 + 4 +� +k/β +�2 +� +. +(12) +The detailed explanation of the P and Q parameters are +given in the literature [34]. Here, the wave function ad- +mixture c(k) represents the contribution of the p-orbital +to the wave function of the band. +3. +Piezoelectric scattering +The PZ effect arises due to the acoustic phonon scat- +tering in polar semiconductors. Being a weak effect, the +PZ scattering is elastic and significant only at low doping +concentrations and low temperatures, where other scat- +tering mechanisms are weak. The momentum relaxation +rate for the PZ scattering is given by [32, 65] +1 +τP Z(k) = +(eP)2 kB T +6π ϵ0 ϵs ν(k) ℏ2 +� +4c4(k) − 6c2(k) + 3 +� +, +(13) +where, P is a piezoelectric coefficient, which is a dimen- +sionless quantity. For the zincblende structure, it is given +as [34, 62] +P 2 = h2 +14 ϵ0 ϵs +35 +��12 +cl +� ++ +�16 +ct +�� +, +(14) +where, h14 is an element of the PZ stress tensor, and ct +and cl represents the spherically averaged elastic con- +stants for transverse and longitudinal modes, respec- +tively, and are given by [26, 32, 34] +cl = 3 +5c11 + 1 +5 +� +2c12 + 4c44 +� +, +ct = 1 +5 +� +c11 − c12 +� ++ 3 +5c44 , +(15) +where c11, c12, and c44 are three independent elastic con- +stants. +4. +Acoustic deformation potential scattering +The ADP scattering mechanism is caused by the inter- +action of electrons with non-polar acoustic phonons. It +is approximately elastic near room temperature For the +ADP scattering mechanism, the momentum relaxation +rate is given by [34, 65] +1 +τADP (k) = +kB T +� +e ΞD k +�2 +3π cel ν(k) ℏ2 +� +6c4(k)−8c2(k)+3 +� +, (16) + +5 +where, cel denotes the spherically averaged elastic con- +stant and ΞD represents the acoustic deformation poten- +tial, which is obtained by the CB shift (in eV) per unit +strain, owing to the acoustic waves(17). To calculate the +acoustic deformation potential (ΞD), we use the following +relation (17) +ΞD = −V × +� +∂ECBM +∂V +������ +V =V0 +, +(17) +where, V denotes the volume, ECBM represents the en- +ergy of the CBM and V0 is the zero pressure volume of +the structure. +5. +Interface roughness scattering +The existence of the interface roughness in a T2SL +[17, 18, 23, 29, 66] structure leads to endemic variations +in InAs well widths, causes modulation of the associated +energy levels and introduces an unstable potential for the +motion of the confined electrons. The IRS mechanism +can occur due to the imperfections that arise during the +growth of the material. The earlier related works [67, 68] +show that the degree of scattering decreases in propor- +tion to the well width hence it is important in MWIR +detectors. The IRS mechanism is an elastic process and +dominates at low temperatures in thin-film systems for +a short period of T2SL, and it is significant at high elec- +tron density. The momentum relaxation rate for the IRS +mechanism is given as [57, 69, 70] +1 +τIRS(k) = +� +e2 ∆ Λ +ϵ0 ϵ∞ +�2 +k +ℏ2 ν(k) +� +Nd + Ns +2 +�2 +× +1 +� +1 + (kΛ)2 ε +� +kΛ +� +1 + (kΛ)2 +� +, +(18) +where, Λ is the lateral correlation length, ∆ is the rough- +ness height, Ns is the sheet carrier concentration, and Nd +is the doping carrier density. +6. +Polar optical phonon scattering +The POP scattering results from the interaction of op- +tical phonons with electrons. The POP scattering mech- +anism is inelastic and anisotropic, which occurs via the +emission or the absorption of a phonon hence, RTA is in- +applicable in such SL structures. The scattering rate due +to the POP scattering mechanism is approximately con- +stant at very high energies, and it depends on the POP +frequencies. The POP scattering dominates in the higher +temperature domain. Hence, it is significant at both near +and beyond room temperature. The out-scattering oper- +ator is given by [34] +So = +� +Npop + 1 − f −� +λ− +o + +� +Npop + f +� +λ+ +o , +(19) +λ± +o = L±� +(A±)2ln +���k± + k +k± − k +��� − A±cc± − aca±c±� +, (20) +L± = +e2 ωpop k± +4π ℏ k ν(k±) +�ϵs − ϵ∞ +ϵs ϵ∞ +� +, +(21) +where, ϵ∞ and ϵs are high and low-frequency dielectric +constants, respectively. +A± = aa± + [(k±)2 + k2] cc±/ 2 k±k , +(22) +where c, c±, a and a± are the wave function coefficients, +k± is the solution of Eq. ε(k) ± ℏωpop. Any quantity +superfixed by plus/minus is to be evaluated at the en- +ergy corresponding to k+ or k−. The superscript plus +denotes scattering by the absorption and is evaluated at +an energy ε(k) + ℏωpop. Similarly, superscript minus de- +notes scattering by the emission and is evaluated at en- +ergy ε(k)−ℏωpop. Emission of phonons is possible only if +the phonons’ energy is greater than ℏωpop energy. There- +fore, if the phonon energy is less than ℏωpop, the term λ− +o +has to be considered as zero. The term Npop, indicates +the number of optical phonons and is given by the Bose +distribution as [32, 34] +Npop = +1 +exp (ℏ ωpop / kB T) − 1 . +(23) +The in-scattering operator Si, is given by +Si = (Npop + 1 − f)λ+ +i g+ + (Npop + f)λ− +i g− , +(24) +where, plus and minus superscripts indicate the absorp- +tion and emission processes, respectively. +The term +λ± +i (k) can be expressed as +λ± +i (k) = L± �(k±)2 + k2 +2 k± k +(A±)2 ln +���k± + k +k± − k +��� +− (A±)2 − c2(k) (c±(k))2 +3 +� +. +(25) +The mobility can be calculated after calculating the +rates of all the elastic scattering mechanisms +1 +τel(k) (8) +and the influence of inelastic scattering mechanisms on g +(7) through the terms Si(g) (24) and So (19). The rates +of various elastic scattering mechanisms are calculated by +using the expressions given in Eqs. (9), (13), (16), (18). +C. +Mobility and conductivity +The RTA [56] cannot be used if the scattering process +is inelastic and anisotropic because there is no way to +define the relaxation time that is independent of the dis- +tribution function. +In such instances, Rode’s iterative +approach can be applied to compute the real distribu- +tion function under low-field conditions. After calculat- +ing the perturbation distribution by using Rode’s algo- +rithm, we finally calculate the low-field carrier mobility, + +6 +FIG. 3. Flowchart for the calculation of electronic transport +parameters. +µ [32, 34, 65] +µ = +1 +3E +� +ν(ε) DS(ε) g(ε) dε +� +DS(ε) f0(ε) dε +. +(26) +The term g(ε), can be obtained from Eq. (7) and the +carrier velocity ν(k) can be calculated from the band +structure as +ν(k) = 1 +ℏ +∂ε +∂k . +(27) +Once the mobility is determined, it is pretty easy to +calculate the electrical conductivity by using +σ = n e µ , +(28) +where, µ is the electron drift mobility, and n is the elec- +tron carrier concentration. The entire sequence for cal- +culating the transport coefficients using Rode’s approach +is shown in Fig. 3. +Similarly, in the presence of an arbitrary magnetic +field, the BTE can be solved. The distribution function +in such cases can be written as [33, 71] +f(k) = f0[ε(k)] + xg(k) + yh(k) , +(29) +where, y is the direction, cosine from B × E to k, and +h(k) is the perturbation distribution function due to the +magnetic field. Substituting Eq. (29) in (3) gives a pair +of coupled equations that can be solved iteratively [33] +gi+1(k) = Si(gi(k) − (−e)E +ℏ +( ∂f0 +∂k ) + βSi(hi(k)) +So(k) (1 + β2) +, +(30) +hi+1(k) = Si(hi(k) + β (−e)E +ℏ +( ∂f0 +∂k ) − βSi(gi(k)) +So(k) (1 + β2) +, (31) +where, β = +(−e)ν(k)B +ℏkSo(k) , and B is the applied magnetic +field. The expression for the Hall mobility and the Hall +scattering factor can be written as [60] +µH = 1 +B +� +ν(ε) DS(ε) h(ε) dε +� +ν(ε) DS(ε) g(ε) dε , +(32) +rH = µH +µ , +(33) +where, µH and µ are the Hall and the drift mobility, +respectively, and rH is the Hall scattering factor. This +solution gives a more accurate result for the Hall scatter- +ing factor compared with the other expressions based on +the RTA [71]. +III. +SIMULATION APPROACH +First, we calculate the band structure using the k.p +technique as discussed in Sec. II A and then analytically +fit it to produce a smooth curve for the calculation of +group velocity [62]. By using Eq. (34), the Fermi level is +determined with a smooth band structure obtained after +the analytical fitting, where V0 represents the volume of +the cell and εc represents the energy at the bottom of the +CB. +n = 1 +V0 +� ∞ +εc +DS(ε)f(ε)dε . +(34) +Equations (9), (13), (16), (18), (19), (24) are used to +calculate the various scattering rates, and the perturba- +tion in the distribution function is determined using Eq. +(7) with Si(k) = 0. The term g(k), is calculated itera- +tively until g(k) converges and it gives results beyond the +RTA. +IV. +RESULTS AND DISCUSSION +A. +Dispersion relation for T2SL +We calculate the band structure of an InAs/GaSb- +based T2SL, with layer widths nML/mML, where n, m += 8, 8 correspondingly, using the 8 × 8 k.p technique +as described in Sec. II A, at a temperature of T=77 K, +and the results are shown in Fig. 4. In a single period +of 8ML/8ML InAs/GaSb configuration, the thickness of + +Start +Calculation of Input Parameters and +Band Structure +Analytical Fitting of Band Structure +Calculation of Fermi-Level +Calculation of Various Scattering +Rates +Si (g(k))=0 +Perturbation g(k) of the +distribution function +(-e)Ecfo +Si(gi(k) - +h +Cok +gi(k) +7 +So(k) + +Tel(k) +No +g(k) Converged ? +Si (g(k))=0 +Yes +Transport Coefficients +Stop7 +TABLE I. +Material parameters required to calculate the electronic band structure using the k.p technique at T = 77 K +[37, 72–74] +Quantity +Unit +InAs +GaSb +Lattice constant +˚A +6.0584 +6.0959 +Effective mass of electron (m∗ +e) +- +0.022 +0.0412 +Energy band gap at 0 K +eV +0.418 +0.814 +Luttinger parameter γ1 +- +19.4 +11.84 +Luttinger parameter γ2 +- +8.545 +4.25 +Luttinger parameter γ3 +- +9.17 +5.01 +Varshini Parameter α +meV/K +0.276 +0.417 +Varshini Parameter β +K +93 +140 +Interband mixing parameter Ep +eV +21.5 +22.4 +Spin-orbit splitting (SO) +eV +0.38 +0.76 +Valence band offset (VBO) +eV +-0.56 +0 +(a) (110) +(b) (001) +FIG. 4. Calculated band structure in the first BZ using the +periodic boundary condition of a T2SL based on 8 ML InAs +/ 8 ML GaSb at T = 77 K using the k.p method (a) The +in-plane dispersion and (b) the out-of-plane dispersion. +FIG. 5. +DOS calculated using the k.p method in an +InAs/GaSb SL as a function of energy. +The inset clearly +shows how the DOS for the carriers in the VB varies as a +function of energy. +each layer is roughly 24 ˚A. The dispersion curve along the +in-plane and the out-of-plane directions are presented in +Figs. 4(a) and 4(b), respectively and the calculated band +gap is 270 meV. The band gap of 270 meV corresponds +to a cut-off wavelength of 4.59 µm which confirms that +our model is best suited for the MWIR spectrum. In Fig. +5 we show the DOS of an SL as a function of energy, cal- +culated using the k.p method. Table I summarizes the +values of the parameters, utilized in the k.p calculations. +B. +Scattering rates +In Fig. 6, we show the dependence of scattering rates +with energy for the temperatures of 77 K, 300 K, and +500 K at doping densities of ND = 1 × 1013 cm−3 +and ND = 2 × 1017 cm−3. +Here, we show the rela- +tive importance of each of the scattering mechanisms in +a T2SL. The IRS mechanism is the strongest scatter- +ing mechanism for low as well as high doping densities +at a temperature of 77 K and 300 K as shown in Fig. +6. At a temperature of 77 K and a doping density of +ND = 1 × 1013 cm−3, the most dominant contributions +are due to the IRS followed by the ADP and the POP +scattering mechanisms. The II scattering mechanism is +the least significant scattering mechanism at this partic- +ular temperature and doping density, whereas it has a +significant contribution at higher doping densities. +At room temperature, the average energy of the carri- +ers is 3/2kBT = 0.0388 eV , indicating that the majority +of the carriers are in the low-energy region. Hence, it is +clear from Fig. 6(e) that at room temperature, the signif- +icant contribution comes from the IRS mechanism as well +as the POP scattering mechanism. Both scattering mech- +anisms are dominant at this temperature, and the dom- +inance of the POP scattering mechanism changes with +respect to temperature and the average energy of the +carriers, which signifies that the POP scattering mech- +anism plays a significant role in such a T2SL structure. +As a result, it is important to note that the POP scatter- +ing mechanism is the primary factor limiting the carrier’s +mobility from room temperature to higher temperatures. +At a temperature of 500 K, the average energy of +the carriers is 0.0646 eV and, most of the carrier con- +tributes to the POP scattering mechanism hence, this + +Energy (eV) +Ec +Eg +0 +HH +.1 +-0.5 +0 +0.5 +r /a +TEnergy (eV) +0.4 +Ec +0.2 +HH +0 +-0.2 +-0.4 +-1 +0 +1 +T /L1020 +(ev-1cm* +1018 +1020 +DOS +1019 +@1018 +sO +-0.02405 +-0.02340 +Energy (eV) +-0.05 0.25 +0.35 +0.45 +Energy (eV)8 +(a) T=77 K +(b) T=300 K +(c) T=500 K +(d) T=77 K +(e) T=300 K +(f) T=500 K +FIG. 6. +Scattering rates for 8ML/8ML InAs/GaSb based T2SL with roughness parameters Λ = 3 nm and ∆ = 0.3 nm as +a function of electron energy at +(a) T = 77 K and ND = 1 × 1013 cm−3 +(b) T = 300 K and ND = 1 × 1013 cm−3 +(c) +T = 500 K and ND = 1 × 1013 cm−3 (d) T = 77 K and ND = 2 × 1017 cm−3 (e) T = 300 K and ND = 2 × 1017 cm−3 and (f) +T = 500 K and ND = 2 × 1017 cm−3 . +TABLE II. Material parameters required to compute the various scattering rates [32, 73, 75–78]. +Parameter +Unit +InAs +GaSb +Elastic constant c11 +GPa +832.9 +884.2 +Elastic constant c12 +GPa +452.6 +402.6 +Elastic constant c44 +GPa +395.9 +432.2 +Acoustic deformation potential +eV +4.90 +6.70 +Low freq. dielectric constant +- +14.55 +15.00 +High freq. dielectric constant +- +11.78 +13.80 +Piezoelectric coefficient +C/m2 +0.045 +0.126 +Optical phonon frequency +1/cm +240 (LO)a, 218 (TO)b +193 (LO)a, 215 (TO)b +a LO : Longitudinal Optical Phonon Frequency. +b TO : Transverse Optical Phonon Frequency. +again demonstrates that the POP scattering mechanism +is the most dominant scattering mechanism for T2SL at +and beyond the ambient temperature for both doping +densities, as shown in Figs. 6(c) and 6(f). Figure 6 shows +a sudden change in the POP scattering rate after partic- +ular energy, which is because if the electron energy is +less than the POP energy, the electron can only scat- +ter by the absorption of the optical phonons, whereas if +the energy is greater than the phonon energy, the elec- +tron can scatter by both the absorption and the emission +of phonons, where the optical phonon energy is deter- +mined using ℏωP OP . The PZ scattering is the least domi- +nant scattering mechanism at higher doping densities, as +shown in Figs. 6(d), 6(e), 6(f). Table II lists the ma- +terial parameters that are used to compute the various +scattering rates. +It is generally known that the ADP scattering mech- +anism becomes substantial at temperatures of 77 K and +above, reducing electron mobility. Therefore, it is also +important to include the effect of the ADP scattering +mechanism, which is significant near the room tempera- +ture for low as well as high doping densities, which was + +PZ +(sec) +POP +IRS +ADP +Total +rate ( +Scattering +1011 +107 +0.03 +0.055 +0.1 +Energy (eV)1015 +PZ +(sec +POP +IRS +ADP +Total +rate +Scattering +1011 +107 +0.03 +0.055 +0.1 +Energy (eV)1015 +PZ +(sec +POP +IRS +ADP +Total +rate ( +Scattering +1011 +107 +0.03 +0.055 +0.1 +Energy (eV)PZ +(sec) +POP +IRS +ADP +Total +rate ( +Scattering +1011 +107 +0.03 +0.055 +0.1 +Energy (eV)1015 +PZ +(sec' +POP +IRS +ADP +Total +rate +M +Scattering +1011 +107 +0.03 +0.055 +0.1 +Energy (eV)1015 +PZ +(sec +POP +IRS +ADP +Total +rate +Scattering +1011 +107 +0.03 +0.055 +0.1 +Energy (eV)9 +FIG. 7. +Calculated mobility contribution for electrons due +to the various scattering mechanism involved in (8ML/8ML) +InAs/GaSb T2SL as a function of temperature for ND = 9 × +1016 cm−3. +FIG. 8. +Calculated low-field electron drift mobility in +8ML/8ML InAs/GaSb SL as a function of doping concen- +tration for temperatures of 77 K, 120 K and 150 K. +not highlighted in the earlier works for such SL struc- +tures. At lower temperatures and in the thin-film sys- +tems, the IRS scattering is considerable, and to compute +the roughness scattering rate, we utilize a sheet carrier +density Ns, of 4.6 × 1012 cm−2 and a doping carrier den- +sity Nd, of 1 × 1011 cm−2 with the roughness height ∆, +fixed at 0.3 nm, and the correlation length of the fluctu- +ations Λ kept at 3 nm. The IRS mechanism is temper- +ature independent, but the carrier distribution function +depends on the temperature. Therefore, the electron mo- +bility through the IRS mechanism is somewhat tempera- +ture sensitive. Except for the IRS scattering rate, which +is temperature independent, we see that all the scattering +rates increase as the temperature rises as shown in Figs. +6(a), 6(b), 6(c). When the temperature is either low or +intermediate, the II scattering rate increases with an in- +crease in the doping concentration, which suppress the +contribution from the PZ scattering, as shown in Figs. +6(a), 6(d), 6(b), 6(e). +FIG. 9. +Comparison of conductivity in a T2SL as a func- +tion of temperature, calculated using the Rode’s and the RTA +method for various doping concentrations. +FIG. 10. +Calculated temperature dependence of electronic +mobility with IRS heights for a correlation length of 3 nm +& ND = 9 × 1016 cm−3. The mobility due to only the IRS +mechanism is shown. +C. +Electron transport parameters +We calculate the mobility and the conductivity for +a T2SL at various temperatures and doping concentra- +tions. +Figure 7 shows the contribution to the mobil- +ity due to various scattering mechanisms calculated for +ND = 9 × 1016 cm−3. +To the best of our knowledge, +the combined effect of these scattering mechanisms in a +T2SL structure has never been shown in earlier works. +These five types of scattering mechanisms show their sig- +nificant contribution to the overall mobility calculation. +From Fig. 7 it turns out that the scattering mechanism +with the lowest mobility values is the dominant one in +that temperature range. Therefore, starting at a tem- +perature of 150 K, the POP scattering mechanism is the +most dominant scattering mechanism until 700 K; below +77 K, a significant contribution to the mobility comes +from the II scattering and the IRS mechanisms as shown + +1010 +Overall +RTAO +pOp +ADP +PZO +IRS + Overall + RTA + Il+ POP + ADP + PZ + IRS +103 +102 +300 +500 +700 +Temperature (K) +106 +102 +20 +150 +300 +500 +700 +Temperature (K)6 +10 +△77K口120K*150K +104 +10° +1012 +1014 +1016 +1018 +Doping Concentration (cm-3104 +Conductivity (S/cm) +-. 1×1016 cm-3-*. 1×1016 cm-3 +- - 9×1016 cm-3...*.. 9×1016 cm-3 +101 +....0..0....0 +::: +10-2 +Rode +RTA +10-5 +20 +150 +300 +500 +700 +Temperature (K)Mobility (cm? / V-sec) +A=0.1nm-0-A=0.3nm-0A=0.5nm +△=0.7nm +106 +105 +104 +103 +20 +150 +300 +Temperature (K)10 +FIG. 11. +Calculated mobility for electrons in an 8ML +InAs/8ML GaSb SL as a function of temperature and cor- +relation length for an IRS height of 0.3 nm with ND = +9 × 1016 cm−3. Here, the mobility due to only the IRS mech- +anism is shown. +FIG. 12. Temperature dependence of electron Hall mobility +in a T2SL calculated using the Rode’s and the RTA method +at B = 0.69 T for various doping concentrations. +in Fig. 7. +In case of II scattering mechanism, with increasing +temperature, the electron density increases exponentially +and causes growth in the screening length. +As a re- +sult, the mobility at low temperatures increases sharply +with rising temperatures because the scattering rates are +inversely related to the square of the screening length. +Since the POP scattering mechanism is more prominent +above 150 K; hence the overall mobility is reduced as +shown in Fig. 7. In Fig. 7, we also compare the mo- +bility computed using the RTA approach to the overall +mobility calculated using Rode’s method and it is found +that in the RTA approach, the mobility is underestimated +because the POP scattering mechanism is inelastic and +nonrandomizing, making it impossible to characterize the +perturbation in the distribution function using the relax- +FIG. 13. Hall scattering factor versus temperature at B= 0.69 +T for ND = 9 × 1017cm−3. +FIG. 14. Hall scattering factor as a function of temperature +and carrier concentration at B= 0.69 T. +ation time. The POP scattering mechanism becomes in- +significant at low temperatures, resulting in nearly com- +parable mobilities determined using the RTA and Rode’s +iterative technique. +In Fig. 8, we demonstrate the overall mobility versus +doping concentration at different temperatures and em- +phasize on the mobility at 77K, which is the usual operat- +ing temperature of most high-performance IR detectors. +The graph illustrates a decrease in mobility as the doping +concentration increases due to a rise in the number of ion- +ized centers. As we raise the temperature, the mobility +diminishes as expected because at higher temperatures +the phonon scattering increases. The mobility values do +not differ significantly for low carrier concentrations be- +cause the II scattering mechanism is less significant at +this range and the primary contributions for lower doping +concentration at low temperatures come from the PZ and +the ADP scattering mechanisms, while at greater doping +concentrations, the II scattering mechanism is compara- +ble to the ADP and the PZ scattering mechanisms. The +mobility owing to the II scattering mechanism is a de- + +A=30nm +A=20nm +^=12nm +Correlation lengthN +106 +A=8nm +Λ=6nm +Λ=3nm +Λ=2nm +103 +A=1nm +20 +150 +300 +Temperature +(K)Hall mobility (cm? / V-sec) +105 +D—1×1013 cm-3——1×1013 cm-3 +-0- 1×1016 cm-3-★- 1×1016 cm-3 +.O..9×1016 cm-3...*...9×1016 cm-3 +104 +Rode +RTA +20 +150 +300 +Temperature (K)Hall scattering factor r +.o..Rode ..o..RTA +1.00 +0.75 +20 +100 +200 +Temperature (K)3 +Carrier conc. (cm* +1016 +H +1014 +2 +0.5 +12 +10 +30 +77 +120 +150 +200 +Temperature(K)11 +creasing function of ND, the mobility begins to decrease +as ND exceeds 1 × 1016 cm−3. +In Fig. 9, we show the conductivity versus temperature +for the doping concentrations of ND = 1 × 1013 cm−3, +ND = 1 × 1016 cm−3 and ND = 9 × 1016 cm−3, re- +spectively, and to demonstrate the supremacy of our +approach, we compare the results obtained using both +the Rode’s and the RTA method. At higher tempera- +tures, the difference in the result of Rode’s method and +the RTA is due to the POP scattering mechanism, the +POP scattering is weaker at lower temperatures hence +both the RTA and the Rode exhibit the same conduc- +tivity. We demonstrate that the conductivity in a T2SL +increases with an increase in the carrier concentration +but decreases as we increase the temperature. +In Figs. 10 and 11, we show the mobility due to only +the IRS mechanism. +The calculated mobilities are vi- +tal functions of the roughness parameters and the carrier +scattering. The existing mobility calculations reveal that, +up to temperatures where the POP scattering mechanism +takes over, the IRS is the dominating scattering mecha- +nism in T2SL. The screening is included in our calcu- +lation using Thomas-Fermi screening which lowers the +scattering rates and increases the mobility. As illustrated +in Fig. 10, the mobility is shown to be strongly reliant +on the roughness height ∆, and decreases monotonically +with increasing ∆, and is proportional to ∆−2. +Figures 10 and 11 show that at low temperatures, the +mobility rises since the value of ∂f +∂ε is an ascending func- +tion of temperature and the denominator of Eq. (26) is +virtually constant at lower temperatures. Also, the elec- +tron density increases at higher temperatures and hence +the mobility drop smoothly. Figure 11 shows that the +mobility is high for smaller values of correlation length +Λ, and drops rapidly as the correlation length of rough- +ness increases until it reaches a saturation point. The +mobility reaches its maximum value at roughly 50 K for +smaller values of Λ, and this maximum point moves to- +ward the higher temperatures for greater values of Λ. +The Hall mobility in InAs/GaSb T2SLs is depicted in +Fig. 12. At temperatures above 50 K, the mobility re- +duces as expected from a combination of the ADP and +the POP scattering mechanisms. In T2SL, the mobility +increases with decreasing temperature, preferable to the +T −3/2 dependency associated with the phonon scattering. +The greater temperature dependency of the electron mo- +bility in InAs/GaSb-based T2SL may indicate stronger +electron-phonon coupling than in the bulk material. The +increased mobility near 50 K could be attributed to a +longer scattering time or a lower electron-effective mass +at the CB edge. +When the Hall scattering factor rH, deviates signifi- +cantly from unity, it indicates that to derive the elec- +tron drift mobility from the experimentally calculated +Hall mobility data, the Hall scattering factor must be +precisely determined. Figure 13 shows the predicted val- +ues of the Hall scattering factor against the temperature +at B = 0.69 T for ND = 9 × 1017 cm−3, while Fig. 14 +depicts the Hall scattering factor as a function of tem- +perature and the carrier concentration at B = 0.69 T. +To the best of our knowledge, calculations of the Hall +scattering factor in such SLs have not been performed +yet in earlier works. The contribution of various scat- +tering mechanisms decides the Hall scattering factor’s +value. Figures 13 and 14 indicate that the value of rH at +low temperatures deviates significantly from unity, while +many researchers use one as an ideal value for a variety +of calculations and studies, which is not accurate. The +carrier concentration and the drift mobility may both be +overestimated and underestimated when the Hall scatter- +ing factor is used as unity. The Hall scattering factor, in +our calculation, fluctuates between the values as low as +0.3 at low temperature and electron concentration, and +as high as 1.48 and even more at high temperature and +electron concentration as shown in Fig. 14. Therefore, +it is worth pointing out that, while evaluating the car- +rier concentration and the drift mobility in such SLs, one +must use caution. +In this work, we calculate the precise values of the +Hall scattering factor and show that for a doping value +of ND = 9 × 1017 cm−3, the computed values of rH are +0.914, 0.952 and 1.01 at temperatures of 77 K, 150 K +and 190 K, respectively, as also depicted in Fig. +13. +At higher temperatures, the value of the Hall scatter- +ing factor is more than unity, indicating that the drift +mobility is lower than the Hall mobility, implying that +the phonon-assisted scattering mechanisms are substan- +tial and diminish the drift mobility. As shown in Fig. +14, at temperatures of 30 K and 77 K, the Hall scat- +tering factor is equal to 0.335 & 0.638 for lower doping +concentrations of ND = 1 × 1012 cm−3 and it is equal +to 0.369 & 0.691 with slightly higher doping concentra- +tions of ND = 5×1015 cm−3 which signifies that the Hall +scattering factor increases as the temperature and elec- +tron concentrations rise, but as we increase the carrier +concentration beyond 3 × 1017 cm−3, the Hall scattering +factor starts decreasing. The higher electron concentra- +tion causes a rapid variation in the Hall factor. +V. +CONCLUSION +In this paper, we developed the Rode algorithm on the +BTE in conjunction with the k.p band structure and the +EFA for a detailed computation of the carrier mobility +and conductivity, in order to primarily unravel two cru- +cial insights. First, the significance of both elastic and in- +elastic scattering mechanisms, particularly the influence +of the IRS and POP scattering mechanisms in techno- +logically relevant SL structures. Second, the structure +specific Hall mobility and Hall scattering factor, which +reveals that temperature and carrier concentrations sig- +nificantly affect the Hall scattering factor, which devi- +ates significantly from unity, i.e., from 0.3 to about 1.48, +even for small magnetic fields. This reinforces the cau- +tion that should be exercised when employing the Hall + +12 +scattering factor in experimental estimations of drift mo- +bilities and carrier concentrations. Our research offers a +comprehensive microscopic understanding of carrier dy- +namics in such technologically relevant SLs. 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Christol, AIP Advances 9, 055012 (2019). + diff --git a/7NE4T4oBgHgl3EQfCQsk/content/tmp_files/load_file.txt b/7NE4T4oBgHgl3EQfCQsk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e599557b8f870770e5ba73ac197bbc19581cacac --- /dev/null +++ b/7NE4T4oBgHgl3EQfCQsk/content/tmp_files/load_file.txt @@ -0,0 +1,860 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf,len=859 +page_content='Advancing carrier transport models for InAs/GaSb type-II superlattice MWIR photodetectors Rohit Kumar, Anup Kumar Mandia, Anuja Singh and Bhaskaran Muralidharan∗ Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India (Dated: January 13, 2023) In order to provide the best possible performance, modern infrared photodetector designs necessitate extremely precise modeling of the superlattice absorber region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' We advance the Rode’s method for the Boltzmann transport equation in conjunction with the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p band structure and the envelope function approximation for a detailed computation of the carrier mobility and conductivity of layered type-II superlattice structures, using which, we unravel two crucial insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' First, the significance of both elastic and inelastic scattering mechanisms, particularly the influence of the interface roughness and polar optical phonon scattering mechanisms in technologically relevant superlattice structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Second, that the structure-specific Hall mobility and Hall scattering factor reveals that temperature and carrier concentrations significantly affect the Hall scattering factor, which deviates significantly from unity even for small magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' This reinforces the caution that should be exercised when employing the Hall scattering factor in experimental estimations of drift mobilities and carrier concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Our research hence offers a comprehensive microscopic understanding of carrier dynamics in such technologically relevant superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Our models also provide highly accurate and precise transport parameters beyond the relaxation time approximation and thereby paving the way to develop physics-based device modules for mid-wavelength infrared photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' INTRODUCTION Modeling state-of-the-art infrared (IR) photodetectors [1–6] require highly accurate transport parameters for de- veloping dark and photocurrent performance projections [5, 7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Current technologically relevant IR photode- tectors use III-V materials such as InAs/GaSb [11, 12] due to numerous advantages [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Type-II superlat- tices (T2SLs) based on stacks of InAs/GaSb [1, 2, 14] are thus extensively used to design high-performance third- generation IR detectors [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Despite the fact that the mobility of the photogenerated minority carriers has a significant impact on the performance of IR photodetec- tors, carrier transport in technologically relevant T2SL structures has not as extensively been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Recent explorations in this context [17–23] which include carrier mobility calculations [24], do not conclusively bring to the fore structure-specific impact of important scatter- ing mechanisms such as Piezoelectric (PZ), polar opti- cal phonon (POP), acoustic deformation potential (ADP) scattering mechanisms and most importantly the inter- face roughness scattering (IRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' With the necessity to develop a deeper understand- ing of carrier transport in technologically relevant T2SLs, this work advances an accurate model for transport calcu- lations, wherein, we investigate different scattering lim- ited transport under low-field in InAs/GaSb superlattices (SLs) as a function of free electron carrier concentration, temperature, and SL structural parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In our cal- culations, five primary scattering mechanisms that limit carrier mobility are the ionized impurity (II) [25], the PZ [26], the ADP [27], the POP and the IRS [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' ∗ corresponding author: bm@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='in We advance the Rode’s method [32–34] which goes be- yond the relaxation time approximation (RTA) [35, 36], coupled with band structure calculations via the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p [37– 41] technique that also includes the strain effect due to lattice mismatch between InAs and GaSb materials [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' We demonstrate the effect of both the elastic and the inelastic scattering mechanisms [43] on the electron mo- bility of the composite structure for a wide range of tem- peratures and doping concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Our studies reveal that the low-temperature mobility of T2SLs is limited by the II, PZ and IRS scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In con- trast, the mobility at higher temperatures is mainly lim- ited by the POP scattering mechanism, an inelastic and anisotropic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At intermediate temperatures, how- ever, the mobility decreases due to a combined effect of ADP and IRS mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The effects of several struc- tural parameters including layer thicknesses, interface roughness heights, correlation lengths, and ion densities are thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Our calculations thereby re- inforce the superiority of the Rode’s method [32, 34] over the conventionally employed RTA, wherein, the former is applicable over a wide temperature range in the presence of inelastic and anisotropic scattering mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In order to experimentally obtain the carrier concen- tration and drift mobility in a SL structure, it is also im- portant to ascertain the Hall scattering factor, which is frequently thought of as being equal to unity, indicating that the Hall mobility and the drift mobility are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' However, in many heterostructures, it differs significantly from unity, which results in inaccurate estimates of the carrier density and drift mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' We clearly show that the temperature and carrier concentrations significantly affect the Hall scattering factor, and that it ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3 to about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='48 even for weak magnetic fields, thereby reinforcing that caution should be exercised when em- ploying this factor in calculations involving drift mobil- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='04858v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 2 ity and carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The models developed here pave the way to develop physics-based device modules for mid-wavelength IR (MWIR) photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' II we de- scribe the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p model to compute the band structure, elec- tron distribution function, Boltzmann transport formal- ism, Rode’s approach and various scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' III we illustrate the simulation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' IV, we explain the findings and finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' V, we summarize our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' ANALYTICAL FORMALISM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Electronic band structure The energy band structure of T2SLs can be calcu- lated using various theoretical approaches like the den- sity functional theory (DFT) [44], the empirical tight- binding method [35, 45, 46], the empirical pseudopoten- tial method [47, 48], many-body perturbation theory [49] and the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p perturbation method [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' For this study, we use the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p technique with the envelope function approx- imation (EFA) [24, 50, 51] since it overcomes the compu- tational limitations of first-principles methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p model is extensively used because of its superiority in computing the energy band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Unlike ab − initio and tight binding methods, the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p technique requires fewer input parameters I, with the related calculation proce- dure being straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In this work, we solve the 8-band Kane Hamilto- nian [52], by perturbatively extending the wave function around high-symmetry points of the reciprocal space, em- ploying the Lowdin’s perturbation approach [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' We also consider the spin-orbit coupling [53] in our com- putation, which provides additional contributions to the spin splitting of the energy bands [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The SL wavefunc- tions (Φn(z)) in the orbital basis states (u0(z)) along the growth direction (z) are articulated in terms of the slowly varying envelope functions (F(z)), which are given as Φn(z) = � j Fj(z)uj0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (1) Such envelope functions under the periodic boundary conditions can be rewritten as F i j(k, z0) = e−iadF i j(k, zM), F i j(k, zM+1) = eiadF i j(k, z1), (2) where, d denotes the thickness of a period, M represents the number of grid points, a denotes the Bloch vector of the envelope function that spans the Brillouin zone (BZ) and k represents the momentum along the transverse di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The final Hamiltonian of the SL in the basis set comprises three matrices (H0, HI and HII), given by H(k, kz) = H0 + HI � −i ∂ ∂z � + � −i ∂ ∂z � HII � −i ∂ ∂z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The entire coupled differential equation is then solved using a numerical finite difference method [54], as described in earlier work [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The interface between the InAs and the GaSb layers is very abrupt as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The energy dif- ference between the conduction band minimum (CBM) and the first heavy hole (HH) maximum at the center of the BZ determines the band gap in an InAs/GaSb-based T2SL, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figure 1(b) also demon- strates that the InAs conduction band (CB) is lower than the GaSb valence band (VB), indicating that the band structure is a staggered T2SL [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Carrier transport model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Boltzmann transport equation and its solution In order to characterize the behavior of the T2SL sys- tem, we solve the Boltzmann transport equation (BTE) and compute the probability of finding a carrier with a crystal momentum k at a location r at a time t as indi- cated by the distribution function f(r, k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Solving the BTE (3) yields the average distribution of the carriers in both the position and the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The BTE can be written as [56–58] ∂f ∂t − ∂f ∂t ��� diff − ∂f ∂t ��� forces = ∂f ∂t ��� coll + s(r, p, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (3) The term s(r, p, t), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (3), represents generation- recombination processes [59], where p is the classical mo- mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term (∂f/∂t)forces, represents the change in the distribution function due to applied electric and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term (∂f/∂t)forces = −F · ∇pf, where, F = (dp/dt) = ℏ(dk/dt) = −e(E + v × B), repre- sents the total force equal to the sum of the electric-force and the Lorentz-force owing to the magnetic flux density B, where e is the electron charge, E is the applied elec- tric field and v denotes the group velocity of the carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term (∂f/∂t)diff = −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='∇rf, refers to the spatial change in the distribution function caused by tempera- ture or concentration gradients, which results in carrier diffusion in the coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, (∂f/∂t)coll is the collision term, which indicates how the distribution func- tion changes over time due to collision events, and can be described as the difference between the in- and the out-scattering processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=', �∂f ∂t � coll = � k1 � S(k1, k) f (k1) � 1 − f (k) � − S(k, k1) f (k) � 1 − f (k1) �� , (4) where, S(k, k1) and S(k1, k) are the transition rates for an electron moving between states k and k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Un- der steady-state, ∂f ∂t = 0, in case of spatial homogeneity, ∇rf = 0, and assuming that there is no recombination- 3 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (a) Schematic of InAs/GaSb based T2SL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The electron wave function in the InAs layer extends beyond the interface into the GaSb layer and overlaps with the heavy hole wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, nML and mML are the numbers of monolayers of InAs and GaSb respectively, in a single period d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (b) Band alignment of InAs/GaSb based T2SL system showing the optical transition between the heavy-hole valence miniband and the electrons from lowest conduction minibands that is employed to detect IR radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The periodic potential of the d period emerges in the material due to the modulation of the semiconductor layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The creation of hole (electron) minibands in the valence (conduction) band is caused by the overlap of hole (electron) wave functions between adjacent GaSb (InAs) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The difference between the first electron miniband in the CB and the first heavy hole miniband in the valence band is used to compute the effective bandgap energy Eg of the T2SL (highlighted in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' generation term, the BTE (3) can be rewritten as −eE ℏ · ∇kf = � k1 � S(k1, k) f (k1) � 1 − f (k) � − S(k, k1) f (k) � 1 − f (k1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (5) In the low-electric field regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' the distribution func- tion can be represented as [60] f(k) = f0 � ε(k) � + g(k) cos θ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (6) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' k=|k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' f denotes the actual electron distribution function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' which includes both the elastic and the inelas- tic scattering mechanisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' g(k) is the perturbation term to f0[ε(k)] produced by the electric field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' θ is the angle between applied electric field (along the symmetry axis) and the electron wave vector k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' and f0 represents the dis- tribution function under equilibrium conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' which is taken according to Fermi-Dirac statistics [35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' By solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (5) and (6), the perturbation term g(k), can be calculated as [32, 34, 61, 62] gi(k) = �Si � gi(k) � − (−e)E ℏ � ∂f0 ∂k � So(k) + 1 τel(k) � , (7) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The various dominant scattering mechanisms in- volved in a T2SL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' where E = |E|, and gi(k) appears on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Hence, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (7) iteratively and the con- vergence is exponentially fast which takes a few itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Once gi(k) is obtained, we calculate the mobil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (7), the term i indicates the iteration index, and the terms, Si & So are the in-scattering and the out-scattering operators, respectively, for inelastic scat- tering mechanisms, as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term 1 τel(k), represents the total momentum relaxation rate of all the elastic scattering mechanisms, which is calculated according to the Matthiessen’s rule (8), and can be writ- ten as Interface scattering GaSb Growth direction Extended Phonons Interface nMI KeD InAs Temperature gradientInAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' GaSb InAs GaSb InAs GaSb InAs z GaSb GaSb CB GaSb VB Ec Eg HH LH M InAs InAs InAs InAs Spatial coordinateDominant scattering mechanisms in T2SL Defect scattering Lattice scattering IRS Impurity Intravalley 4 Ionized Acoustic Optical Deformation potential Piezoelectric Polar4 1 τel(k) = 1 τII(k) + 1 τP Z(k) + 1 τADP (k) + 1 τIRS(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (8) The various dominant scattering mechanisms involved in an InAs/GaSb-based T2SL structure are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Ionized impurity scattering The II scattering mechanism [25] arises due to the Coulomb interactions between electrons and ions, when a charged center is introduced inside the bulk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The II scattering mechanism is entirely elastic and dom- inates usually at high doping concentrations and low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The II scattering mechanism dominates near the CB edge but reduces drastically as the energy increases [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The scattering rate for the II increases rapidly with decreasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, we use the Brooks-Herring approach [64] for the calculation of II scattering rate [34, 65], which is given by 1 τII(k) = e4N 8π ν(k) (ϵ0 ϵs)2 (ℏ k)2 � P(k) ln � 1 + 4 � k β �2� − Q(k) � , (9) where, ϵ0 is the permittivity of the free space, ϵs is the static dielectric constant, ℏ is the reduced Planck’s con- stant and N is the ionized impurity concentration, which is the sum of the acceptor and donor impurity concen- tration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=', N = NA + ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, β indicates the inverse screening length, which is given as β = � e2 ϵ0 ϵs kB T � DS(ε)f0(1 − f0)dε , (10) where, DS(ε) is the density of states (DOS) at energy ε and kB is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' P(k) and Q(k) can be expressed as follows [34, 62] P(k) = � 3 4 �β c(k) k �4 + 2 �β c(k) k �2 + 1 � , (11) Q(k) = �3 β4 + 6 β2 k2 − 8 k4 (β2 + 4 k2) k2 � c4(k) + 8 � β2 + 2k2 β2 + 4 k2 � c2(k) + � 4 � k/β �2 1 + 4 � k/β �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (12) The detailed explanation of the P and Q parameters are given in the literature [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, the wave function ad- mixture c(k) represents the contribution of the p-orbital to the wave function of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Piezoelectric scattering The PZ effect arises due to the acoustic phonon scat- tering in polar semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Being a weak effect, the PZ scattering is elastic and significant only at low doping concentrations and low temperatures, where other scat- tering mechanisms are weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The momentum relaxation rate for the PZ scattering is given by [32, 65] 1 τP Z(k) = (eP)2 kB T 6π ϵ0 ϵs ν(k) ℏ2 � 4c4(k) − 6c2(k) + 3 � , (13) where, P is a piezoelectric coefficient, which is a dimen- sionless quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' For the zincblende structure, it is given as [34, 62] P 2 = h2 14 ϵ0 ϵs 35 ��12 cl � + �16 ct �� , (14) where, h14 is an element of the PZ stress tensor, and ct and cl represents the spherically averaged elastic con- stants for transverse and longitudinal modes, respec- tively, and are given by [26, 32, 34] cl = 3 5c11 + 1 5 � 2c12 + 4c44 � , ct = 1 5 � c11 − c12 � + 3 5c44 , (15) where c11, c12, and c44 are three independent elastic con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Acoustic deformation potential scattering The ADP scattering mechanism is caused by the inter- action of electrons with non-polar acoustic phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' It is approximately elastic near room temperature For the ADP scattering mechanism, the momentum relaxation rate is given by [34, 65] 1 τADP (k) = kB T � e ΞD k �2 3π cel ν(k) ℏ2 � 6c4(k)−8c2(k)+3 � , (16) 5 where, cel denotes the spherically averaged elastic con- stant and ΞD represents the acoustic deformation poten- tial, which is obtained by the CB shift (in eV) per unit strain, owing to the acoustic waves(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' To calculate the acoustic deformation potential (ΞD), we use the following relation (17) ΞD = −V × � ∂ECBM ∂V ������ V =V0 , (17) where, V denotes the volume, ECBM represents the en- ergy of the CBM and V0 is the zero pressure volume of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Interface roughness scattering The existence of the interface roughness in a T2SL [17, 18, 23, 29, 66] structure leads to endemic variations in InAs well widths, causes modulation of the associated energy levels and introduces an unstable potential for the motion of the confined electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The IRS mechanism can occur due to the imperfections that arise during the growth of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The earlier related works [67, 68] show that the degree of scattering decreases in propor- tion to the well width hence it is important in MWIR detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The IRS mechanism is an elastic process and dominates at low temperatures in thin-film systems for a short period of T2SL, and it is significant at high elec- tron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The momentum relaxation rate for the IRS mechanism is given as [57, 69, 70] 1 τIRS(k) = � e2 ∆ Λ ϵ0 ϵ∞ �2 k ℏ2 ν(k) � Nd + Ns 2 �2 × 1 � 1 + (kΛ)2 ε � kΛ � 1 + (kΛ)2 � , (18) where, Λ is the lateral correlation length, ∆ is the rough- ness height, Ns is the sheet carrier concentration, and Nd is the doping carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Polar optical phonon scattering The POP scattering results from the interaction of op- tical phonons with electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The POP scattering mech- anism is inelastic and anisotropic, which occurs via the emission or the absorption of a phonon hence, RTA is in- applicable in such SL structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The scattering rate due to the POP scattering mechanism is approximately con- stant at very high energies, and it depends on the POP frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The POP scattering dominates in the higher temperature domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Hence, it is significant at both near and beyond room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The out-scattering oper- ator is given by [34] So = � Npop + 1 − f −� λ− o + � Npop + f +� λ+ o , (19) λ± o = L±� (A±)2ln ���k± + k k± − k ��� − A±cc± − aca±c±� , (20) L± = e2 ωpop k± 4π ℏ k ν(k±) �ϵs − ϵ∞ ϵs ϵ∞ � , (21) where, ϵ∞ and ϵs are high and low-frequency dielectric constants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' A± = aa± + [(k±)2 + k2] cc±/ 2 k±k , (22) where c, c±, a and a± are the wave function coefficients, k± is the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' ε(k) ± ℏωpop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Any quantity superfixed by plus/minus is to be evaluated at the en- ergy corresponding to k+ or k−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The superscript plus denotes scattering by the absorption and is evaluated at an energy ε(k) + ℏωpop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Similarly, superscript minus de- notes scattering by the emission and is evaluated at en- ergy ε(k)−ℏωpop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Emission of phonons is possible only if the phonons’ energy is greater than ℏωpop energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' There- fore, if the phonon energy is less than ℏωpop, the term λ− o has to be considered as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term Npop, indicates the number of optical phonons and is given by the Bose distribution as [32, 34] Npop = 1 exp (ℏ ωpop / kB T) − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (23) The in-scattering operator Si, is given by Si = (Npop + 1 − f)λ+ i g+ + (Npop + f)λ− i g− , (24) where, plus and minus superscripts indicate the absorp- tion and emission processes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term λ± i (k) can be expressed as λ± i (k) = L± �(k±)2 + k2 2 k± k (A±)2 ln ���k± + k k± − k ��� − (A±)2 − c2(k) (c±(k))2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (25) The mobility can be calculated after calculating the rates of all the elastic scattering mechanisms 1 τel(k) (8) and the influence of inelastic scattering mechanisms on g (7) through the terms Si(g) (24) and So (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The rates of various elastic scattering mechanisms are calculated by using the expressions given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (9), (13), (16), (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Mobility and conductivity The RTA [56] cannot be used if the scattering process is inelastic and anisotropic because there is no way to define the relaxation time that is independent of the dis- tribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In such instances, Rode’s iterative approach can be applied to compute the real distribu- tion function under low-field conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' After calculat- ing the perturbation distribution by using Rode’s algo- rithm, we finally calculate the low-field carrier mobility, 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Flowchart for the calculation of electronic transport parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' µ [32, 34, 65] µ = 1 3E � ν(ε) DS(ε) g(ε) dε � DS(ε) f0(ε) dε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (26) The term g(ε), can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (7) and the carrier velocity ν(k) can be calculated from the band structure as ν(k) = 1 ℏ ∂ε ∂k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (27) Once the mobility is determined, it is pretty easy to calculate the electrical conductivity by using σ = n e µ , (28) where, µ is the electron drift mobility, and n is the elec- tron carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The entire sequence for cal- culating the transport coefficients using Rode’s approach is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Similarly, in the presence of an arbitrary magnetic field, the BTE can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The distribution function in such cases can be written as [33, 71] f(k) = f0[ε(k)] + xg(k) + yh(k) , (29) where, y is the direction, cosine from B × E to k, and h(k) is the perturbation distribution function due to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (29) in (3) gives a pair of coupled equations that can be solved iteratively [33] gi+1(k) = Si(gi(k) − (−e)E ℏ ( ∂f0 ∂k ) + βSi(hi(k)) So(k) (1 + β2) , (30) hi+1(k) = Si(hi(k) + β (−e)E ℏ ( ∂f0 ∂k ) − βSi(gi(k)) So(k) (1 + β2) , (31) where, β = (−e)ν(k)B ℏkSo(k) , and B is the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The expression for the Hall mobility and the Hall scattering factor can be written as [60] µH = 1 B � ν(ε) DS(ε) h(ε) dε � ν(ε) DS(ε) g(ε) dε , (32) rH = µH µ , (33) where, µH and µ are the Hall and the drift mobility, respectively, and rH is the Hall scattering factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' This solution gives a more accurate result for the Hall scatter- ing factor compared with the other expressions based on the RTA [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' SIMULATION APPROACH First, we calculate the band structure using the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p technique as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' II A and then analytically fit it to produce a smooth curve for the calculation of group velocity [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (34), the Fermi level is determined with a smooth band structure obtained after the analytical fitting, where V0 represents the volume of the cell and εc represents the energy at the bottom of the CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' n = 1 V0 � ∞ εc DS(ε)f(ε)dε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (34) Equations (9), (13), (16), (18), (19), (24) are used to calculate the various scattering rates, and the perturba- tion in the distribution function is determined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (7) with Si(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The term g(k), is calculated itera- tively until g(k) converges and it gives results beyond the RTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Dispersion relation for T2SL We calculate the band structure of an InAs/GaSb- based T2SL, with layer widths nML/mML, where n, m = 8, 8 correspondingly, using the 8 × 8 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p technique as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' II A, at a temperature of T=77 K, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In a single period of 8ML/8ML InAs/GaSb configuration, the thickness of Start Calculation of Input Parameters and Band Structure Analytical Fitting of Band Structure Calculation of Fermi-Level Calculation of Various Scattering Rates Si (g(k))=0 Perturbation g(k) of the distribution function (-e)Ecfo Si(gi(k) - h Cok gi(k) 7 So(k) + Tel(k) No g(k) Converged ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Si (g(k))=0 Yes Transport Coefficients Stop7 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Material parameters required to calculate the electronic band structure using the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p technique at T = 77 K [37, 72–74] Quantity Unit InAs GaSb Lattice constant ˚A 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='0584 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='0959 Effective mass of electron (m∗ e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='0412 Energy band gap at 0 K eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='814 Luttinger parameter γ1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='84 Luttinger parameter γ2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='545 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='25 Luttinger parameter γ3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='01 Varshini Parameter α meV/K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='417 Varshini Parameter β K 93 140 Interband mixing parameter Ep eV 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='4 Spin-orbit splitting (SO) eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='76 Valence band offset (VBO) eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='56 0 (a) (110) (b) (001) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Calculated band structure in the first BZ using the periodic boundary condition of a T2SL based on 8 ML InAs / 8 ML GaSb at T = 77 K using the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p method (a) The in-plane dispersion and (b) the out-of-plane dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' DOS calculated using the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p method in an InAs/GaSb SL as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The inset clearly shows how the DOS for the carriers in the VB varies as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' each layer is roughly 24 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The dispersion curve along the in-plane and the out-of-plane directions are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 4(a) and 4(b), respectively and the calculated band gap is 270 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The band gap of 270 meV corresponds to a cut-off wavelength of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='59 µm which confirms that our model is best suited for the MWIR spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 5 we show the DOS of an SL as a function of energy, cal- culated using the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Table I summarizes the values of the parameters, utilized in the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Scattering rates In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6, we show the dependence of scattering rates with energy for the temperatures of 77 K, 300 K, and 500 K at doping densities of ND = 1 × 1013 cm−3 and ND = 2 × 1017 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, we show the rela- tive importance of each of the scattering mechanisms in a T2SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The IRS mechanism is the strongest scatter- ing mechanism for low as well as high doping densities at a temperature of 77 K and 300 K as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At a temperature of 77 K and a doping density of ND = 1 × 1013 cm−3, the most dominant contributions are due to the IRS followed by the ADP and the POP scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The II scattering mechanism is the least significant scattering mechanism at this partic- ular temperature and doping density, whereas it has a significant contribution at higher doping densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At room temperature, the average energy of the carri- ers is 3/2kBT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='0388 eV , indicating that the majority of the carriers are in the low-energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Hence, it is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6(e) that at room temperature, the signif- icant contribution comes from the IRS mechanism as well as the POP scattering mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Both scattering mech- anisms are dominant at this temperature, and the dom- inance of the POP scattering mechanism changes with respect to temperature and the average energy of the carriers, which signifies that the POP scattering mech- anism plays a significant role in such a T2SL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' As a result, it is important to note that the POP scatter- ing mechanism is the primary factor limiting the carrier’s mobility from room temperature to higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At a temperature of 500 K, the average energy of the carriers is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='0646 eV and, most of the carrier con- tributes to the POP scattering mechanism hence, this Energy (eV) Ec Eg 0 HH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='5 r /a TEnergy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='4 Ec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='2 HH 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='4 1 0 1 T /L1020 (ev-1cm* 1018 1020 DOS 1019 @1018 sO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='02405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='02340 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='45 Energy (eV)8 (a) T=77 K (b) T=300 K (c) T=500 K (d) T=77 K (e) T=300 K (f) T=500 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Scattering rates for 8ML/8ML InAs/GaSb based T2SL with roughness parameters Λ = 3 nm and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3 nm as a function of electron energy at (a) T = 77 K and ND = 1 × 1013 cm−3 (b) T = 300 K and ND = 1 × 1013 cm−3 (c) T = 500 K and ND = 1 × 1013 cm−3 (d) T = 77 K and ND = 2 × 1017 cm−3 (e) T = 300 K and ND = 2 × 1017 cm−3 and (f) T = 500 K and ND = 2 × 1017 cm−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Material parameters required to compute the various scattering rates [32, 73, 75–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Parameter Unit InAs GaSb Elastic constant c11 GPa 832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='9 884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='2 Elastic constant c12 GPa 452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='6 402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='6 Elastic constant c44 GPa 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='9 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='2 Acoustic deformation potential eV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='90 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='70 Low freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' dielectric constant 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='55 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='00 High freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' dielectric constant 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='78 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='80 Piezoelectric coefficient C/m2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='126 Optical phonon frequency 1/cm 240 (LO)a, 218 (TO)b 193 (LO)a, 215 (TO)b a LO : Longitudinal Optical Phonon Frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' b TO : Transverse Optical Phonon Frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' again demonstrates that the POP scattering mechanism is the most dominant scattering mechanism for T2SL at and beyond the ambient temperature for both doping densities, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6(c) and 6(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figure 6 shows a sudden change in the POP scattering rate after partic- ular energy, which is because if the electron energy is less than the POP energy, the electron can only scat- ter by the absorption of the optical phonons, whereas if the energy is greater than the phonon energy, the elec- tron can scatter by both the absorption and the emission of phonons, where the optical phonon energy is deter- mined using ℏωP OP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The PZ scattering is the least domi- nant scattering mechanism at higher doping densities, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6(d), 6(e), 6(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Table II lists the ma- terial parameters that are used to compute the various scattering rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' It is generally known that the ADP scattering mech- anism becomes substantial at temperatures of 77 K and above, reducing electron mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Therefore, it is also important to include the effect of the ADP scattering mechanism, which is significant near the room tempera- ture for low as well as high doping densities, which was PZ (sec) POP IRS ADP Total rate ( Scattering 1011 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1 Energy (eV)1015 PZ (sec POP IRS ADP Total rate Scattering 1011 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1 Energy (eV)1015 PZ (sec POP IRS ADP Total rate ( Scattering 1011 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1 Energy (eV)PZ (sec) POP IRS ADP Total rate ( Scattering 1011 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content="1 Energy (eV)1015 PZ (sec' POP IRS ADP Total rate M Scattering 1011 107 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1 Energy (eV)1015 PZ (sec POP IRS ADP Total rate Scattering 1011 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1 Energy (eV)9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Calculated mobility contribution for electrons due to the various scattering mechanism involved in (8ML/8ML) InAs/GaSb T2SL as a function of temperature for ND = 9 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Calculated low-field electron drift mobility in 8ML/8ML InAs/GaSb SL as a function of doping concen- tration for temperatures of 77 K, 120 K and 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' not highlighted in the earlier works for such SL struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At lower temperatures and in the thin-film sys- tems, the IRS scattering is considerable, and to compute the roughness scattering rate, we utilize a sheet carrier density Ns, of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='6 × 1012 cm−2 and a doping carrier den- sity Nd, of 1 × 1011 cm−2 with the roughness height ∆, fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3 nm, and the correlation length of the fluctu- ations Λ kept at 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The IRS mechanism is temper- ature independent, but the carrier distribution function depends on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Therefore, the electron mo- bility through the IRS mechanism is somewhat tempera- ture sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Except for the IRS scattering rate, which is temperature independent, we see that all the scattering rates increase as the temperature rises as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6(a), 6(b), 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' When the temperature is either low or intermediate, the II scattering rate increases with an in- crease in the doping concentration, which suppress the contribution from the PZ scattering, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 6(a), 6(d), 6(b), 6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Comparison of conductivity in a T2SL as a func- tion of temperature, calculated using the Rode’s and the RTA method for various doping concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Calculated temperature dependence of electronic mobility with IRS heights for a correlation length of 3 nm & ND = 9 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The mobility due to only the IRS mechanism is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Electron transport parameters We calculate the mobility and the conductivity for a T2SL at various temperatures and doping concentra- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figure 7 shows the contribution to the mobil- ity due to various scattering mechanisms calculated for ND = 9 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' To the best of our knowledge, the combined effect of these scattering mechanisms in a T2SL structure has never been shown in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' These five types of scattering mechanisms show their sig- nificant contribution to the overall mobility calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 7 it turns out that the scattering mechanism with the lowest mobility values is the dominant one in that temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Therefore, starting at a tem- perature of 150 K, the POP scattering mechanism is the most dominant scattering mechanism until 700 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' below 77 K, a significant contribution to the mobility comes from the II scattering and the IRS mechanisms as shown 1010 Overall RTAO pOp ADP PZO IRS Overall + RTA + Il+ POP + ADP + PZ + IRS 103 102 300 500 700 Temperature (K) 106 102 20 150 300 500 700 Temperature (K)6 10 △77K口120K*150K 104 10° 1012 1014 1016 1018 Doping Concentration (cm-3104 Conductivity (S/cm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 1×1016 cm-3-*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 1×1016 cm-3 - 9×1016 cm-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='. 9×1016 cm-3 101 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.0 ::: 10-2 Rode RTA 10-5 20 150 300 500 700 Temperature (K)Mobility (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' / V-sec) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='1nm-0-A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3nm-0A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='5nm △=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='7nm 106 105 104 103 20 150 300 Temperature (K)10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Calculated mobility for electrons in an 8ML InAs/8ML GaSb SL as a function of temperature and cor- relation length for an IRS height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3 nm with ND = 9 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Here, the mobility due to only the IRS mech- anism is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Temperature dependence of electron Hall mobility in a T2SL calculated using the Rode’s and the RTA method at B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='69 T for various doping concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In case of II scattering mechanism, with increasing temperature, the electron density increases exponentially and causes growth in the screening length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' As a re- sult, the mobility at low temperatures increases sharply with rising temperatures because the scattering rates are inversely related to the square of the screening length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Since the POP scattering mechanism is more prominent above 150 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' hence the overall mobility is reduced as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 7, we also compare the mo- bility computed using the RTA approach to the overall mobility calculated using Rode’s method and it is found that in the RTA approach, the mobility is underestimated because the POP scattering mechanism is inelastic and nonrandomizing, making it impossible to characterize the perturbation in the distribution function using the relax- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Hall scattering factor versus temperature at B= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='69 T for ND = 9 × 1017cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Hall scattering factor as a function of temperature and carrier concentration at B= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='69 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' ation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The POP scattering mechanism becomes in- significant at low temperatures, resulting in nearly com- parable mobilities determined using the RTA and Rode’s iterative technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 8, we demonstrate the overall mobility versus doping concentration at different temperatures and em- phasize on the mobility at 77K, which is the usual operat- ing temperature of most high-performance IR detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The graph illustrates a decrease in mobility as the doping concentration increases due to a rise in the number of ion- ized centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' As we raise the temperature, the mobility diminishes as expected because at higher temperatures the phonon scattering increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The mobility values do not differ significantly for low carrier concentrations be- cause the II scattering mechanism is less significant at this range and the primary contributions for lower doping concentration at low temperatures come from the PZ and the ADP scattering mechanisms, while at greater doping concentrations, the II scattering mechanism is compara- ble to the ADP and the PZ scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The mobility owing to the II scattering mechanism is a de- A=30nm A=20nm ^=12nm Correlation lengthN 106 A=8nm Λ=6nm Λ=3nm Λ=2nm 103 A=1nm 20 150 300 Temperature (K)Hall mobility (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' / V-sec) 105 D—1×1013 cm-3——1×1013 cm-3 0- 1×1016 cm-3-★- 1×1016 cm-3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.9×1016 cm-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='9×1016 cm-3 104 Rode RTA 20 150 300 Temperature (K)Hall scattering factor r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.Rode .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='.RTA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='75 20 100 200 Temperature (K)3 Carrier conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (cm* 1016 H 1014 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='5 12 10 30 77 120 150 200 Temperature(K)11 creasing function of ND, the mobility begins to decrease as ND exceeds 1 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 9, we show the conductivity versus temperature for the doping concentrations of ND = 1 × 1013 cm−3, ND = 1 × 1016 cm−3 and ND = 9 × 1016 cm−3, re- spectively, and to demonstrate the supremacy of our approach, we compare the results obtained using both the Rode’s and the RTA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At higher tempera- tures, the difference in the result of Rode’s method and the RTA is due to the POP scattering mechanism, the POP scattering is weaker at lower temperatures hence both the RTA and the Rode exhibit the same conduc- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' We demonstrate that the conductivity in a T2SL increases with an increase in the carrier concentration but decreases as we increase the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 10 and 11, we show the mobility due to only the IRS mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The calculated mobilities are vi- tal functions of the roughness parameters and the carrier scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The existing mobility calculations reveal that, up to temperatures where the POP scattering mechanism takes over, the IRS is the dominating scattering mecha- nism in T2SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The screening is included in our calcu- lation using Thomas-Fermi screening which lowers the scattering rates and increases the mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 10, the mobility is shown to be strongly reliant on the roughness height ∆, and decreases monotonically with increasing ∆, and is proportional to ∆−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figures 10 and 11 show that at low temperatures, the mobility rises since the value of ∂f ∂ε is an ascending func- tion of temperature and the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' (26) is virtually constant at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Also, the elec- tron density increases at higher temperatures and hence the mobility drop smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figure 11 shows that the mobility is high for smaller values of correlation length Λ, and drops rapidly as the correlation length of rough- ness increases until it reaches a saturation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The mobility reaches its maximum value at roughly 50 K for smaller values of Λ, and this maximum point moves to- ward the higher temperatures for greater values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The Hall mobility in InAs/GaSb T2SLs is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At temperatures above 50 K, the mobility re- duces as expected from a combination of the ADP and the POP scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In T2SL, the mobility increases with decreasing temperature, preferable to the T −3/2 dependency associated with the phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The greater temperature dependency of the electron mo- bility in InAs/GaSb-based T2SL may indicate stronger electron-phonon coupling than in the bulk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The increased mobility near 50 K could be attributed to a longer scattering time or a lower electron-effective mass at the CB edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' When the Hall scattering factor rH, deviates signifi- cantly from unity, it indicates that to derive the elec- tron drift mobility from the experimentally calculated Hall mobility data, the Hall scattering factor must be precisely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figure 13 shows the predicted val- ues of the Hall scattering factor against the temperature at B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='69 T for ND = 9 × 1017 cm−3, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 14 depicts the Hall scattering factor as a function of tem- perature and the carrier concentration at B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='69 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' To the best of our knowledge, calculations of the Hall scattering factor in such SLs have not been performed yet in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The contribution of various scat- tering mechanisms decides the Hall scattering factor’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Figures 13 and 14 indicate that the value of rH at low temperatures deviates significantly from unity, while many researchers use one as an ideal value for a variety of calculations and studies, which is not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The carrier concentration and the drift mobility may both be overestimated and underestimated when the Hall scatter- ing factor is used as unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The Hall scattering factor, in our calculation, fluctuates between the values as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3 at low temperature and electron concentration, and as high as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='48 and even more at high temperature and electron concentration as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Therefore, it is worth pointing out that, while evaluating the car- rier concentration and the drift mobility in such SLs, one must use caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' In this work, we calculate the precise values of the Hall scattering factor and show that for a doping value of ND = 9 × 1017 cm−3, the computed values of rH are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='914, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='952 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='01 at temperatures of 77 K, 150 K and 190 K, respectively, as also depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' At higher temperatures, the value of the Hall scatter- ing factor is more than unity, indicating that the drift mobility is lower than the Hall mobility, implying that the phonon-assisted scattering mechanisms are substan- tial and diminish the drift mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' 14, at temperatures of 30 K and 77 K, the Hall scat- tering factor is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='335 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='638 for lower doping concentrations of ND = 1 × 1012 cm−3 and it is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='369 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='691 with slightly higher doping concentra- tions of ND = 5×1015 cm−3 which signifies that the Hall scattering factor increases as the temperature and elec- tron concentrations rise, but as we increase the carrier concentration beyond 3 × 1017 cm−3, the Hall scattering factor starts decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' The higher electron concentra- tion causes a rapid variation in the Hall factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' CONCLUSION In this paper, we developed the Rode algorithm on the BTE in conjunction with the k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='p band structure and the EFA for a detailed computation of the carrier mobility and conductivity, in order to primarily unravel two cru- cial insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' First, the significance of both elastic and in- elastic scattering mechanisms, particularly the influence of the IRS and POP scattering mechanisms in techno- logically relevant SL structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Second, the structure specific Hall mobility and Hall scattering factor, which reveals that temperature and carrier concentrations sig- nificantly affect the Hall scattering factor, which devi- ates significantly from unity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=', from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='3 to about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content='48, even for small magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' This reinforces the cau- tion that should be exercised when employing the Hall 12 scattering factor in experimental estimations of drift mo- bilities and carrier concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Our research offers a comprehensive microscopic understanding of carrier dy- namics in such technologically relevant SLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Our model also provides highly accurate and precise transport pa- rameters beyond the RTA and hence paves the way to develop physics based device modules for MWIR pho- todetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors acknowledge funding from ISRO under the ISRO-IIT Bombay Space Technology Cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Smith and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' Mailhiot, Journal of Applied Physics 62, 2545 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfCQsk/content/2301.04858v1.pdf'} +page_content=' [2] A.' metadata={'source': 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+University of Florence +Via di Santa Marta, 3, 50139, Florence, Italy +pierluigi.mansueto@unifi.it +ABSTRACT +In this paper, we deal with the Front Steepest Descent algorithm for multi-objective optimization. +We point out that the algorithm from the literature is often incapable, by design, of spanning large +portions of the Pareto front. We thus introduce some modifications within the algorithm aimed to +overcome this significant limitation. We prove that the asymptotic convergence properties of the +algorithm are preserved and numerically show that the proposed method significantly outperforms +the original one. +Keywords Multi-objective optimization · Steepest descent · Pareto front +Mathematics Subject Classification (2020) 90C29 · 90C30 +1 +Introduction +In this paper, we are interested in optimization problems of the form +min +x∈Rn F(x) = (f1(x), . . . , fm(x))T , +(1) +where F : Rn → Rm is a vector-valued continuously differentiable function. We are thus dealing with smooth, +unconstrained multi-objective optimization problems, where many functions have to be simultaneously minimized and +Pareto’s efficiency concepts have to be considered to establish optimality. We refer the reader to [8] for an introduction +to multi-objective optimization. +Multi-objective descent methods [9–11, 16] constitute a class of algorithmic approaches designed to tackle these +problems; these approaches basically extend classical iterative optimization algorithms for scalar optimization to the +multi-objective setting. Descent methods are receiving increasing attention and have consistently become significant +alternatives to scalarization methods [6, 7, 15] and evolutionary algorithms [4]. This is particularly true for recent +versions of descent approaches that are specifically designed to handle sets of points and to construct an approximation +of the entire Pareto front, rather than a single solution. +In this short manuscript, we focus on the Front Steepest Descent (FSD) algorithm proposed in [2]. In particular, we +argue that, although being far superior than the original single point steepest descent algorithm for multi-objective +optimization [10], FSD as defined in [2] has limited exploration capabilities and it is quite frequently unable to span +large portions of the Pareto front. +We thus propose small but crucial modifications to the algorithm, that allow to turn it tremendously effective at spanning +the entire Pareto front, regardless of the starting set of points. We show that the proposed approach still enjoys the nice +convergence guarantees of the original FSD. +The rest of the paper is organized as follows: in Section 2, we summarize the FSD algorithm, recalling its convergence +properties; we then point out in Section 2.1 that in certain, common situations the algorithm is unable to span large +arXiv:2301.03310v1 [math.OC] 9 Jan 2023 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +portions of the Pareto front. In Section 3 we introduce the novel strategy for generating nondominated solutions within +FSD and we provide the convergence analysis for the resulting algorithm in Section 3.1. In Section 4, we present the +results of numerical experiments showing that the proposed modification significantly improves effectiveness and +consistency of the FSD algorithm. We finally give some concluding remarks in Section 5. +2 +The Front Steepest Descent algorithm +The Front Steepest Descent algorithm [2] was designed to solve problem (1) according to Pareto’s optimality concepts. +Given the standard partial ordering in Rm, i.e., +u ≤ v ⇐⇒ uj ≤ vj, ∀ j = 1, . . . , m, +u < v ⇐⇒ uj < vj, ∀ j = 1, . . . , m, +u ≨ v ⇐⇒ u ≤ v ∧ u ̸= v, +the aim is to find solutions ¯x ∈ Rn that satisfy the following properties, listed in decreasing order of strength: +• Pareto optimality: ∄ y ∈ Rn s.t. F(y) ≨ F(¯x); +• Weak Pareto optimality: ∄ y ∈ Rn s.t. F(y) < F(¯x); +• Pareto stationarity: min +d∈Rn +max +j=1,...,m ∇fj(¯x)T d = 0. +In fact, there typically exist many Pareto optimal solutions (the Pareto set) that account for different trade-offs between +the contrasting objectives; these trade-offs, that constitute in the objectives space the Pareto front, can a posteriori be +evaluated by the decision makers, who are thus willing to have the broadest possible range of available options. +FSD method specifically aims to construct an approximation of the entire Pareto front; the algorithm works in an iterative +fashion, maintaining at each iteration a set Xk of solutions that are mutually nondominated, i.e., for any x ∈ Xk there +is no y ∈ Xk such that F(y) ≨ F(x). +The points for the set Xk+1 are computed carrying out search steps starting from the points ˆx ∈ Xk along: +• the steepest common descent direction [10]: +v(ˆx) = arg min +d∈Rn +max +j=1,...,m ∇fj(ˆx)T d + 1 +2∥d∥2; +(2) +• the steepest partial descent directions [1,2]: given I ⊂ {1, . . . , m}, +vI(ˆx) = arg min +d∈Rn +max +j∈I ∇fj(ˆx)T d + 1 +2∥d∥2. +(3) +The use of equality notation in the definition of steepest descent directions is justified by the uniqueness of the solution +set for the above optimization problems (the objective is strongly convex and continuous). Given any subset of objectives +I, a partial descent direction exists if +θI(ˆx) = min +d∈Rn max +j∈I ∇fj(ˆx)T d + 1 +2∥d∥2 < 0; +of course, the steepest common descent direction v(ˆx) and the corresponding θ (ˆx) are considered when I = {1, . . . , m}. +Both mappings vI(ˆx) and θI(ˆx) are continuous [10]. +The instructions of the FSD procedure are summarized in Algorithm 1. In brief, at each iteration k, all points in the +current set of nondominated solutions, Xk, are considered; for each one of these points, xc, a line search along the +steepest partial descent direction is carried out for any subset of objectives I ⊆ {1, . . . , m} such that θI(xc) < 0; in +addition, a subset I is only considered for xc if the point is nondominated with respect to that subset of objectives. +The line search is an Armijo-type procedure whose scheme is reported in Algorithm 2. Given a nondominated point and +a search direction w.r.t. the objectives in I, the algorithm returns a new point such that it is “sufficiently nondominated”. +The obtained point is added to the set of nondominated points, while all the points that are now dominated by it are +filtered out. +Algorithm 2 enjoys the following finite termination properties. +2 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +Algorithm 1: FrontSteepestDescent +1 Input: F : Rn → Rm, X0 set of mutually nondominated points w.r.t. F. +2 k = 0 +3 while a stopping criterion is not satisfied do +4 +ˆXk = Xk +5 +forall xc ∈ Xk do +6 +forall I ⊆ {1, . . . , m} such that +• ∄y ∈ ˆXk s.t. FI(y) ≨ FI(xc) and +• θI(xc) < 0 +7 +do +8 +α = ArmijoLS(F(·), I, ˆXk, xc, vI(xc), θI(xc)) +9 +ˆXk = ˆXk \ {y ∈ ˆXk | F(xc + αvI(xc)) ≨ F(y)} ∪ {xc + αvI(xc)} +10 +Xk+1 = ˆXk +11 +k = k + 1 +12 return Xk +Algorithm 2: ArmijoLS +1 Input: F : Rn → Rm, I ⊆ {1, . . . , m}, ˆX set of mutually nondominated points w.r.t. F, xc ∈ ˆX, vI(xc) ∈ Rn, +θI(xc) ∈ R, α0 > 0, δ ∈ (0, 1), γ ∈ (0, 1). +2 α = α0 +3 Let ˆXI be the set of points in ˆX that are mutually nondominated w.r.t. FI +4 while ∃ y ∈ ˆXI s.t. FI(y) + 1γαθI(xc) < FI(xc + αvI(xc)) do +5 +α = δα +6 return α +Proposition 1 ( [2, Proposition 4]). Let I ⊆ {1, . . . , m}, ˆX be a set of mutually nondominated solutions containing +xc; xc is also nondominated w.r.t. FI and it is such that θI(xc) < 0. Then, ∃ α > 0, sufficiently small, such that +FI(y) + 1γαθI(xc) ≮ FI(xc + αvI(xc)), +∀ y ∈ ˆXI, +with ˆXI being the set of points in ˆX that are mutually nondominated w.r.t. FI. Furthermore, the produced point +xc + αvI(xc) is nondominated by any point in ˆX with respect to F. +Remark 1. An improved version of Algorithm 2 was also proposed in [2], which is based on an extrapolation strategy +and allows to possibly obtain many nondominated solutions along the search direction. When used within Algorithm 1, +the extrapolation technique does not alter theoretical convergence results, but the resulting algorithm is reported to be +significantly more effective. +Now, we shall recall the convergence properties of Algorithm 1, which are based on the concept of linked sequence [14]. +Definition 1. Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 1. We define a linked +sequence as a sequence {xjk} such that, for any k = 1, 2, . . ., the point xjk ∈ Xk is generated at iteration k − 1 of +Algorithm 1 by the point xjk−1 ∈ Xk−1. +Proposition 2 ( [2, Proposition 5]). Let us assume that there exists x0 ∈ X0 such that +• x0 is not Pareto stationary; +• the set L(x0) = �m +j=1{x ∈ Rn | fj(x) ≤ fj(x0)} is compact. +Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 1. Let {xjk} be a linked sequence, +then it admits limit points and every limit point is Pareto-stationary for problem (1). +3 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(d) +Figure 1: Pareto fronts obtained by the FSD algorithm on the convex JOS problem (n = 5). (a) FSD starts from 1 Pareto +point; (b) FSD starts from 2 Pareto points; (c) 3 independent FSD runs, started from 3 different random points; (d) 3 +independent runs of FSD with the extrapolation strategy, started from the same 3 random points as in (c). +2.1 +FSD may not span the Pareto front +The FSD algorithm constitutes, in practice, a significant improvement w.r.t. the simple multi-start steepest descent +strategy for multi-objective optimization. However, in experimental settings, it is not uncommon to observe situations +where FSD is unable to retrieve large portions of the Pareto front. +Here, we highlight this shortcoming and argue that it is the direct result of algorithmic design. In particular, the first +condition at step 6 of Algorithm 1 makes the outcome of the algorithm very strongly dependent on the starting point(s). +When a point xc is considered for exploration in Algorithm 1, a partial descent direction obtained according to the +subset of objectives I ⊆ {1, . . . , m} is only considered if xc is nondominated within Xk w.r.t. FI; in other words, there +is no y ∈ Xk such that FI(y) ≨ FI(xc). This condition was required by the authors of [2] in order to establish finite +termination properties for the line search (Algorithm 2). +Unfortunately, that same condition results in a limited fraction of points in Xk to be used for starting a partial descent +search. This fact can be visualized, with very extreme outcomes, in the bi-objective case; indeed, when m = 2, for +each of the two proper subsets of indices, I1 = {1} and I2 = {2} there is only one point that satisfies the (partial) +nondominance condition: xI1 = arg minx∈Xk f1(x) and xI2 = arg minx∈Xk f2(x). +Thus, partial descent is only carried out starting from the two current extreme points in the Pareto front. Moreover, +these partial descent steps will only allow to explore, outwards, the extreme parts of the current front approximation, +whereas the other descent step will mainly drive points to Pareto stationarity; as a result, even large holes within the +current solutions set cannot be filled. +Taking the reasoning to the extreme, let us assume that the starting set of solutions already lies on the Pareto front; if the +set contains only one point, then by repeated partial descent w.r.t. I1 and I2 the entire Pareto front can be spanned quite +uniformly; this situation is depicted in Figure 1a. If, on the other hand, there are two starting solutions, possibly far +away from each other in the objective space, then only the extreme parts of the front will be spanned, while the gap +4 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +between the two points is not tackled (Figure 1b). Of course, the same reasoning applies with more than two starting +points. +The paradoxical behavior of the algorithm is such that it might be convenient to start far away from the Pareto front. +In this way, FSD may have many iterations at its disposal to increase the size of the set Xk and uniformly span the +objectives space; points are then driven to Pareto stationarity thanks to steps carried out considering I = {1, 2}. +Anyhow, the results are still influenced, somewhat randomly, by the starting solutions, as shown in Figure 1c. Moreover, +the extreme parts of the front are always spanned much more densely than the central one. We shall remark that, as the +intermediate regions of the front often provide the most interesting trade-offs to users, this is a very significant issue in +practice. +The extrapolation technique proposed in [2] might allow to partly alleviate the issue discussed here, as much more +nondominated solutions are obtained at each iteration; however, it is again the exploration of the extreme regions that is +mainly enhanced and sped up, with possibly overall counterproductive results (Figure 1d). +3 +Improved Front Steepest Descent +In Algorithm 3, we report the scheme of a modified Front Steepest Descent (IFSD) algorithm that overcomes the +limitations of Algorithm 1 discussed in Section 2.1. +Algorithm 3: ImprovedFrontSteepestDescent +1 Input: F : Rn → Rm, X0 set of mutually nondominated points w.r.t. F, α0 > 0, δ ∈ (0, 1), γ ∈ (0, 1). +2 k = 0 +3 while a stopping criterion is not satisfied do +4 +ˆXk = Xk +5 +forall xc ∈ Xk do +6 +if xc ∈ ˆXk then +7 +if θ(xc) < 0 then +8 +αk +c = maxh=0,1,...{α0δh | F(xc + α0δhv(xc)) ≤ F(xc) + 1γα0δhθ(xc)} +9 +else +10 +αk +c = 0 +11 +zk +c = xc + αk +cv(xc) +12 +ˆXk = ˆXk \ {y ∈ ˆXk | F(zk +c ) ≨ F(y)} ∪ {zk +c } +13 +forall I ⊆ {1, . . . , m} s.t. θI(zk +c ) < 0 do +14 +if zk +c ∈ ˆXk then +15 +αI +c = maxh=0,1,...{α0δh | ∀ y ∈ ˆXk ∃j ∈ {1, . . . , m} s.t. fj(zk +c + α0δhvI(zk +c )) < fj(y)} +16 +ˆXk = ˆXk \ {y ∈ ˆXk | F(zk +c + αI +cvI(zk +c )) ≨ F(y)} ∪ {zk +c + αI +cvI(zk +c )} +17 +Xk+1 = ˆXk +18 +k = k + 1 +19 return Xk +Algorithm 3 includes a bunch of modifications w.r.t. the original FSD approach: +• for any point in Xk that is still nondominated when it is considered for exploration, a preliminary steepest +descent step is carried out; this step exploits a classical single point Armijo line search [10]; +• further searches w.r.t. subsets of objectives start at the obtained point, as long as it is not dominated; +• for partial descent searches, we require the obtained point to be nondominated by all other points in ˆXk. +The idea is that, with these modifications, all points may be used to start exploration based on partial descent; +convergence of all the produced points towards stationarity is then forced by means of the “preliminary” steepest +descent step, that ensures the sufficient decrease. In the next section we prove that the algorithm is well defined and +actually produces convergent sequences of points. +5 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +3.1 +Convergence analysis +In this section, we provide the formal convergence analysis for Algorithm 3. +Proposition 3. The line search at step 8 of Algorithm 3 is well defined. +Proof. The result follows from [10, Lemma 4] and by the if condition at step 7 that ensures that θ(xc) < 0. +Proposition 4. Step 15 of Algorithm 3 is well defined if zk +c is nondominated with respect to points in ˆXk. +Proof. Let y be any point in ˆXk; if F(y) = F(zk +c ), then by [10, Lemma 4] and the condition θI(zk +c ) < 0, there exists +¯α > 0 such that FI(zk +c + αvI(zk +c )) < FI(zk +c ) = FI(y) for all α < ¯α; thus there exists h sufficiently large such that +fj(zk +c +α0δhvI(zk +c )) < fj(y) for all j ∈ I. If, on the other hand, there exists j ∈ {1, . . . , m} such that fj(zk +c ) < fj(y), +then by the continuity of F there exists α = α0δh sufficiently small such that fj(zk +c + αvI(zk +c )) < fj(y). Thus, the +condition can be satisfied for all y ∈ ˆXk and αI +c is the minimum of the corresponding values of α0δh. +Proposition 5. If Xk contains mutually nondominated points with respect to F, then ˆXk contains nondominated points +at any time during iteration k; thus step 15 is always well defined and Xk+1 is finally a set of nondominated solutions. +Proof. At iteration k, the set ˆXk is initialized with the nondominated points Xk; then, it is only updated at steps 12 +and 16. At step 12, either zk +c = xc, and the set is not modified, or, by the definition of αk +c, zk +c dominates xc, which in +turn was nondominated. Thus, the added point zk +c is nondominated, while all the newly dominated points are removed. +At step 16, the added point zk +c + αI +cvI(zk +c ) is nondominated by the definition of αI +c; all the newly dominated points are +removed. Thus, ˆXk always contains mutually nondominated solutions. By Proposition 4 step 15 is therefore always +well defined. Moreover, since Xk+1 = ˆXk at the end of the iteration, Xk+1 inherits the nondominance property from +ˆXk. +Lemma 1. After step 12 of Algorithm 3, zk +c belongs to ˆXk. Moreover, for all ˜k > k, there exists y ∈ X˜k such that +F(y) ≤ F(zk +c ). +Proof. The first assertion of the proposition trivially follows from the update rule of ˆXk, at step 12. Now, either +zk +c ∈ X˜k or zk +c /∈ X˜k; in the former case, we trivially have y = zk +c ; otherwise, we can notice that, by the instructions of +the algorithm, any set X˜k, ˜k > k, is the result of repeated application of steps 12 and 16, starting from ˆXk at some point +when zk +c ∈ ˆXk. When zk +c was removed from the set, a point y1 was certainly inserted such that F(y1) ≤ F(zk +c ). Then, +either y1 ∈ X˜k, or y1 was removed when a point y2 such that F(y2) ≤ F(y1) was added. By recursively applying +the reasoning, we have that there is certainly a point yt ∈ X˜k such that F(yt) ≤ F(yt−1) ≤ . . . ≤ F(y2) ≤ F(y1) ≤ +F(zk +c ). This completes the proof. +Proposition 6. Let X0 be a set of mutually nondominated points and x0 ∈ X0 be a point such that the set L(x0) = +�m +j=1{x ∈ Rn | fj(x) ≤ fj(x0)} is compact. Let {Xk} be the sequence of sets of nondominated points produced by +Algorithm 3. Let {xjk} be a linked sequence, then it admits limit points and every limit point is Pareto-stationary for +problem (1). +Proof. For any k, either x0 ∈ Xk or x0 /∈ Xk. In the former case, since all points in Xk are mutually nondominated, +we certainly have xjk ∈ L(x0). Otherwise, by a similar reasoning as in the proof of Lemma 1, we have that there is a +point yk ∈ Xk such that F(yk) ≤ F(x0); since yk does not dominate xjk, we have that there exists h ∈ {1, . . . , m} +such that fh(xjk) ≤ fh(yk) ≤ fh(x0); thus, again, xjk ∈ L(x0). Therefore the entire sequence {xjk} belongs to the +compact set L(x0), and thus admits limit points. +Now, let us consider a limit point ¯x of a linked sequence {xjk}, i.e., there exists K ⊆ {1, 2, . . .} such that +lim +k→∞ +k∈K +xjk = ¯x. +We assume by contradiction that θ(¯x) < 0 and thus there exists ε > 0 such that for all k ∈ K sufficiently large we have +θ(xjk) ≤ −ε < 0. Let zjk = xjk + αjkv(xjk) the point obtained at step 11 of the algorithm starting from xjk. Now, +αjk ∈ [0, α0], which is a compact set, thus there exists a further subsequence K1 ⊆ K such that αjk → ¯α ∈ [0, α0]. +6 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +Moreover, function v(·) is continuous, thus v(xjk) → v(¯x) for k → ∞, k ∈ K1. Hence, taking the limits along K1 we +also get that zjk → ¯x + ¯αv(¯x) = ¯z. +By the definition of αjk and zjk (steps 8-11) we have that +F(zjk) ≤ F(xjk) + 1γαjkθ(xjk). +Taking the limits for k ∈ K1, k → ∞, recalling the continuity of θ(·), we get +F(¯z) ≤ F(¯x) + 1γ ¯αθ(¯x) ≤ F(¯x) − 1γ ¯αε. +(4) +Now, given k ∈ K1, let k1(k) be the smallest index in K1 such that k1(k) > k. By Lemma 1, there exists yjk1(k) ∈ +Xk1(k) such that F(yjk1(k)) ≤ F(zjk); moreover, xjk1(k) ∈ Xk1(k); by Proposition 5, the points in Xk1(k) are mutually +nondominated, hence there exists h(k) ∈ {1, . . . , m} such that +fh(k)(xjk1(k)) ≤ fh(k)(yjk1(k)) ≤ fh(k)(zjk). +Considering a further subsequence K2 ⊆ K1 such that h(k) = h for all k ∈ K2 and taking the limits, we obtain +fh(¯x) ≤ fh(¯z). +Putting this last result together with (4), we get +fh(¯x) ≤ fh(¯z) ≤ fh(¯x) − γ ¯αε. +Since ¯α ∈ [0, α0], ε > 0 and γ > 0, the above chain of inequalities can only hold if ¯α = limk→∞,k∈K2 αjk = 0. +For all k ∈ K2 sufficiently large, we have θ(xjk) < 0 and, thus, αjk is defined at step 8. Since αjk → 0, for any +q ∈ N, for all k ∈ K2 large enough we certainly have αjk < α0δq; thus, the Armijo condition F(xjk + αv(xjk)) ≤ +F(xjk) + 1γαθ(xjk) is not satisfied by α = α0δq, i.e., there exists ˜h(k) such that +f˜h(k)(xjk + α0δqv(xjk)) > f˜h(k)(xjk) + γα0δqθ(xjk). +Taking the limits along a suitable subsequence such that ˜h(k) = ˜h, we get +f˜h(¯x + α0δqv(¯x)) ≥ f˜h(¯x) + γα0δqθ(¯x). +Now, since q is arbitrary and θ(¯x) < 0, this is absurd by [10, Lemma 4]. The proof is thus complete. +4 +Numerical results +In this section, we show the results of computational experiments, supporting the discussion in Sections 2-3. The code, +which was written in Python3, was executed on a computer with the following characteristics: Ubuntu 22.04, Intel +Xeon Processor E5-2430 v2 6 cores 2.50 GHz, 16 GB RAM. In order to solve instances of problems (2)-(3), the Gurobi +optimizer (version 9.5) was employed. +We compared our approach (IFSD) to the original FSD, Algorithm 1, equipped with the base line search (Algorithm 2) +or the extrapolation strategy (EFSD). The following parameters setting was used for line searches: α0 = 1, δ = 0.5, γ = +10−4. +With respect to the conceptual scheme in Algorithm 3, we employed within IFSD a strategy to limit the number of points +used for partial descent searches, in order to improve the efficiency of the overall procedure and avoid the production +of too many, very close solutions. In particular, we added a condition based on the crowding distance [4] to decide +whether a point should be considered for further exploration after the steepest descent step or not. +The benchmark used for the comparisons consists of the following unconstrained problems: CEC09_2, CEC09_3 [17], +JOS_1 [12], MAN [13] (m = 2) and CEC09_10 (m = 3) [17]. For all the problems, we considered instances with +values of n in {5, 10, 20, 30, 40, 50, 100, 200}. Moreover, each problem was tested twice, with different strategies for +the initial points: a) n points are uniformly sampled from the hyper-diagonal of a suitable box; b) only the midpoint of +the hyper-diagonal is selected. The hyper-diagonal refers to the box constituting the constraints in the bounded version +of CEC and MAN problems, whereas it is [−100, 100]n for the JOS problem. +In order to appreciate the relative performance and robustness of the approaches, we employed the performance +profiles [5]. In brief, this tool shows the probability that a metric value achieved by a method in a problem is within +a factor τ ∈ R of the best value obtained by any of the algorithms in that problem. We employed classical metrics +for multi-objective optimization: purity, Γ–spread, ∆–spread [3] and hyper-volume [18]. Purity and hyper-volume +7 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +f2 +(c) +Figure 2: Pareto fronts obtained by the IFSD algorithm on the convex JOS problem (n = 5) starting from different +initial points: (a) 1 Pareto point as in Figure 1a; (b) 2 Pareto points as in Figure 1b; (c) 3 independent runs from the +same random points as those of Figure 1(c)-(d). +have increasing values for better solutions: then, the corresponding profiles are produced considering the inverse of the +obtained values. +In Figure 2, the behavior of the proposed approach in the same setting as in Figure 1 is shown. In this example we can +observe that now, regardless, of the starting point(s), the entire Pareto front is effectively spanned, with not even tiny +holes. +For a more consistent assessment of algorithms performance, we report in Figure 3 the performance profiles for the +IFSD, FSD and EFSD algorithms on the entire benchmark of 80 problem instances. +We observe a remarkable superiority of the proposed approach w.r.t. the original variants of the algorithm, especially in +terms of the spread metrics, which points out that the Pareto front is indeed spanned more widely and uniformly. The +strong hypervolume performance also supports this result. As for purity metric, the three algorithms appear to be closer, +but we still observe a slight advantage of IFSD. +5 +Conclusions +In this paper, we introduced an improved Front Steepest Descent algorithm with asymptotic convergence guarantees +similar as those of the original method. The novel algorithm is designed so as to overcome some empirically evident +limitation of FSD, that is often unable to span large portions of the Pareto front. Numerical evidence suggests that the +proposed procedure effectively achieves this goal. +Future work should be focused on the integration of the proposed approach and the extrapolation strategy proposed +in [2]. Moreover, the employment of the proposed approach within memetic procedures for global multi-objective +optimization [13] might be considered. Finally, the algorithm defined in this work could be extended to deal with +constrained optimization problems. +8 + +Improved Front Steepest Descent for MOO +MATTEO LAPUCCI AND PIERLUIGI MANSUETO +1 +2 +3 +4 +5 +6 +7 +8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative +Purity +IFSD +FSD +EFSD +(a) Purity profile +1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative +Hypervolume +(b) Hypervolume profile +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative +-spread +(c) Γ-spread profile +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative +-spread +(d) ∆-spread profile +Figure 3: Performance profiles for the IFSD, FSD and EFSD algorithms on a benchmark of 80 multi-objective problems. +Conflict of interest +The authors declare that they have no conflict of interest. +References +[1] G. Cocchi, M. Lapucci, and P. Mansueto. Pareto front approximation through a multi-objective augmented +lagrangian method. EURO Journal on Computational Optimization, page 100008, 2021. +[2] G. Cocchi, G. Liuzzi, S. Lucidi, and M. Sciandrone. On the convergence of steepest descent methods for +multiobjective optimization. Computational Optimization and Applications, pages 1–27, 2020. +[3] A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente. Direct multisearch for multiobjective optimization. +SIAM Journal on Optimization, 21(3):1109–1140, 2011. +[4] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. +IEEE Transactions on Evolutionary Computation, 6(2):182–197, 2002. +[5] E. D. Dolan and J. J. Moré. Benchmarking optimization software with performance profiles. Mathematical +Programming, 91(2):201–213, 2002. +[6] L. G. Drummond, N. Maculan, and B. F. Svaiter. On the choice of parameters for the weighting method in vector +optimization. Mathematical Programming, 111(1-2):201–216, 2008. +[7] G. Eichfelder. 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Springer Berlin Heidelberg. +10 + diff --git a/8dE1T4oBgHgl3EQfngSj/content/tmp_files/load_file.txt b/8dE1T4oBgHgl3EQfngSj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e2d57a78ce2398e19d078e2bdee61b4c34f0bc0 --- /dev/null +++ b/8dE1T4oBgHgl3EQfngSj/content/tmp_files/load_file.txt @@ -0,0 +1,638 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf,len=637 +page_content='IMPROVED FRONT STEEPEST DESCENT FOR MULTI-OBJECTIVE OPTIMIZATION Matteo Lapucci Global Optimization Laboratory (GOL) Department of Information Engineering University of Florence Via di Santa Marta, 3, 50139, Florence, Italy matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='lapucci@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='it Pierluigi Mansueto Global Optimization Laboratory (GOL) Department of Information Engineering University of Florence Via di Santa Marta, 3, 50139, Florence, Italy pierluigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='mansueto@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='it ABSTRACT In this paper, we deal with the Front Steepest Descent algorithm for multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We point out that the algorithm from the literature is often incapable, by design, of spanning large portions of the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We thus introduce some modifications within the algorithm aimed to overcome this significant limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We prove that the asymptotic convergence properties of the algorithm are preserved and numerically show that the proposed method significantly outperforms the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Keywords Multi-objective optimization · Steepest descent · Pareto front Mathematics Subject Classification (2020) 90C29 · 90C30 1 Introduction In this paper, we are interested in optimization problems of the form min x∈Rn F(x) = (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , fm(x))T , (1) where F : Rn → Rm is a vector-valued continuously differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We are thus dealing with smooth, unconstrained multi-objective optimization problems, where many functions have to be simultaneously minimized and Pareto’s efficiency concepts have to be considered to establish optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We refer the reader to [8] for an introduction to multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Multi-objective descent methods [9–11, 16] constitute a class of algorithmic approaches designed to tackle these problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' these approaches basically extend classical iterative optimization algorithms for scalar optimization to the multi-objective setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Descent methods are receiving increasing attention and have consistently become significant alternatives to scalarization methods [6, 7, 15] and evolutionary algorithms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' This is particularly true for recent versions of descent approaches that are specifically designed to handle sets of points and to construct an approximation of the entire Pareto front, rather than a single solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In this short manuscript, we focus on the Front Steepest Descent (FSD) algorithm proposed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In particular, we argue that, although being far superior than the original single point steepest descent algorithm for multi-objective optimization [10], FSD as defined in [2] has limited exploration capabilities and it is quite frequently unable to span large portions of the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We thus propose small but crucial modifications to the algorithm, that allow to turn it tremendously effective at spanning the entire Pareto front, regardless of the starting set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We show that the proposed approach still enjoys the nice convergence guarantees of the original FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The rest of the paper is organized as follows: in Section 2, we summarize the FSD algorithm, recalling its convergence properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' we then point out in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='1 that in certain, common situations the algorithm is unable to span large arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='03310v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='OC] 9 Jan 2023 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO portions of the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In Section 3 we introduce the novel strategy for generating nondominated solutions within FSD and we provide the convergence analysis for the resulting algorithm in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In Section 4, we present the results of numerical experiments showing that the proposed modification significantly improves effectiveness and consistency of the FSD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We finally give some concluding remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 2 The Front Steepest Descent algorithm The Front Steepest Descent algorithm [2] was designed to solve problem (1) according to Pareto’s optimality concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Given the standard partial ordering in Rm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=', u ≤ v ⇐⇒ uj ≤ vj, ∀ j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m, u < v ⇐⇒ uj < vj, ∀ j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m, u ≨ v ⇐⇒ u ≤ v ∧ u ̸= v, the aim is to find solutions ¯x ∈ Rn that satisfy the following properties, listed in decreasing order of strength: Pareto optimality: ∄ y ∈ Rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' F(y) ≨ F(¯x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Weak Pareto optimality: ∄ y ∈ Rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' F(y) < F(¯x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Pareto stationarity: min d∈Rn max j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=',m ∇fj(¯x)T d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In fact, there typically exist many Pareto optimal solutions (the Pareto set) that account for different trade-offs between the contrasting objectives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' these trade-offs, that constitute in the objectives space the Pareto front, can a posteriori be evaluated by the decision makers, who are thus willing to have the broadest possible range of available options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FSD method specifically aims to construct an approximation of the entire Pareto front;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' the algorithm works in an iterative fashion, maintaining at each iteration a set Xk of solutions that are mutually nondominated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=', for any x ∈ Xk there is no y ∈ Xk such that F(y) ≨ F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The points for the set Xk+1 are computed carrying out search steps starting from the points ˆx ∈ Xk along: the steepest common descent direction [10]: v(ˆx) = arg min d∈Rn max j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=',m ∇fj(ˆx)T d + 1 2∥d∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (2) the steepest partial descent directions [1,2]: given I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m}, vI(ˆx) = arg min d∈Rn max j∈I ∇fj(ˆx)T d + 1 2∥d∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (3) The use of equality notation in the definition of steepest descent directions is justified by the uniqueness of the solution set for the above optimization problems (the objective is strongly convex and continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Given any subset of objectives I, a partial descent direction exists if θI(ˆx) = min d∈Rn max j∈I ∇fj(ˆx)T d + 1 2∥d∥2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' of course, the steepest common descent direction v(ˆx) and the corresponding θ (ˆx) are considered when I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Both mappings vI(ˆx) and θI(ˆx) are continuous [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The instructions of the FSD procedure are summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In brief, at each iteration k, all points in the current set of nondominated solutions, Xk, are considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' for each one of these points, xc, a line search along the steepest partial descent direction is carried out for any subset of objectives I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} such that θI(xc) < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' in addition, a subset I is only considered for xc if the point is nondominated with respect to that subset of objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The line search is an Armijo-type procedure whose scheme is reported in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Given a nondominated point and a search direction w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' the objectives in I, the algorithm returns a new point such that it is “sufficiently nondominated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The obtained point is added to the set of nondominated points, while all the points that are now dominated by it are filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Algorithm 2 enjoys the following finite termination properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 2 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO Algorithm 1: FrontSteepestDescent 1 Input: F : Rn → Rm, X0 set of mutually nondominated points w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 2 k = 0 3 while a stopping criterion is not satisfied do 4 ˆXk = Xk 5 forall xc ∈ Xk do 6 forall I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} such that ∄y ∈ ˆXk s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FI(y) ≨ FI(xc) and θI(xc) < 0 7 do 8 α = ArmijoLS(F(·), I, ˆXk, xc, vI(xc), θI(xc)) 9 ˆXk = ˆXk \\ {y ∈ ˆXk | F(xc + αvI(xc)) ≨ F(y)} ∪ {xc + αvI(xc)} 10 Xk+1 = ˆXk 11 k = k + 1 12 return Xk Algorithm 2: ArmijoLS 1 Input: F : Rn → Rm, I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m}, ˆX set of mutually nondominated points w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' F, xc ∈ ˆX, vI(xc) ∈ Rn, θI(xc) ∈ R, α0 > 0, δ ∈ (0, 1), γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 2 α = α0 3 Let ˆXI be the set of points in ˆX that are mutually nondominated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FI 4 while ∃ y ∈ ˆXI s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FI(y) + 1γαθI(xc) < FI(xc + αvI(xc)) do 5 α = δα 6 return α Proposition 1 ( [2, Proposition 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m}, ˆX be a set of mutually nondominated solutions containing xc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' xc is also nondominated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FI and it is such that θI(xc) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Then, ∃ α > 0, sufficiently small, such that FI(y) + 1γαθI(xc) ≮ FI(xc + αvI(xc)), ∀ y ∈ ˆXI, with ˆXI being the set of points in ˆX that are mutually nondominated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Furthermore, the produced point xc + αvI(xc) is nondominated by any point in ˆX with respect to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' An improved version of Algorithm 2 was also proposed in [2], which is based on an extrapolation strategy and allows to possibly obtain many nondominated solutions along the search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' When used within Algorithm 1, the extrapolation technique does not alter theoretical convergence results, but the resulting algorithm is reported to be significantly more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Now, we shall recall the convergence properties of Algorithm 1, which are based on the concept of linked sequence [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We define a linked sequence as a sequence {xjk} such that, for any k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=', the point xjk ∈ Xk is generated at iteration k − 1 of Algorithm 1 by the point xjk−1 ∈ Xk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proposition 2 ( [2, Proposition 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let us assume that there exists x0 ∈ X0 such that x0 is not Pareto stationary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' the set L(x0) = �m j=1{x ∈ Rn | fj(x) ≤ fj(x0)} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let {xjk} be a linked sequence, then it admits limit points and every limit point is Pareto-stationary for problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 3 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 f2 (d) Figure 1: Pareto fronts obtained by the FSD algorithm on the convex JOS problem (n = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (a) FSD starts from 1 Pareto point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (b) FSD starts from 2 Pareto points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (c) 3 independent FSD runs, started from 3 different random points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (d) 3 independent runs of FSD with the extrapolation strategy, started from the same 3 random points as in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='1 FSD may not span the Pareto front The FSD algorithm constitutes, in practice, a significant improvement w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' the simple multi-start steepest descent strategy for multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' However, in experimental settings, it is not uncommon to observe situations where FSD is unable to retrieve large portions of the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Here, we highlight this shortcoming and argue that it is the direct result of algorithmic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In particular, the first condition at step 6 of Algorithm 1 makes the outcome of the algorithm very strongly dependent on the starting point(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' When a point xc is considered for exploration in Algorithm 1, a partial descent direction obtained according to the subset of objectives I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} is only considered if xc is nondominated within Xk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' FI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' in other words, there is no y ∈ Xk such that FI(y) ≨ FI(xc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' This condition was required by the authors of [2] in order to establish finite termination properties for the line search (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Unfortunately, that same condition results in a limited fraction of points in Xk to be used for starting a partial descent search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' This fact can be visualized, with very extreme outcomes, in the bi-objective case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' indeed, when m = 2, for each of the two proper subsets of indices, I1 = {1} and I2 = {2} there is only one point that satisfies the (partial) nondominance condition: xI1 = arg minx∈Xk f1(x) and xI2 = arg minx∈Xk f2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Thus, partial descent is only carried out starting from the two current extreme points in the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Moreover, these partial descent steps will only allow to explore, outwards, the extreme parts of the current front approximation, whereas the other descent step will mainly drive points to Pareto stationarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' as a result, even large holes within the current solutions set cannot be filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Taking the reasoning to the extreme, let us assume that the starting set of solutions already lies on the Pareto front;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' if the set contains only one point, then by repeated partial descent w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' I1 and I2 the entire Pareto front can be spanned quite uniformly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' this situation is depicted in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' If, on the other hand, there are two starting solutions, possibly far away from each other in the objective space, then only the extreme parts of the front will be spanned, while the gap 4 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO between the two points is not tackled (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Of course, the same reasoning applies with more than two starting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The paradoxical behavior of the algorithm is such that it might be convenient to start far away from the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In this way, FSD may have many iterations at its disposal to increase the size of the set Xk and uniformly span the objectives space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' points are then driven to Pareto stationarity thanks to steps carried out considering I = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Anyhow, the results are still influenced, somewhat randomly, by the starting solutions, as shown in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Moreover, the extreme parts of the front are always spanned much more densely than the central one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We shall remark that, as the intermediate regions of the front often provide the most interesting trade-offs to users, this is a very significant issue in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The extrapolation technique proposed in [2] might allow to partly alleviate the issue discussed here, as much more nondominated solutions are obtained at each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' however, it is again the exploration of the extreme regions that is mainly enhanced and sped up, with possibly overall counterproductive results (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 3 Improved Front Steepest Descent In Algorithm 3, we report the scheme of a modified Front Steepest Descent (IFSD) algorithm that overcomes the limitations of Algorithm 1 discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Algorithm 3: ImprovedFrontSteepestDescent 1 Input: F : Rn → Rm, X0 set of mutually nondominated points w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' F, α0 > 0, δ ∈ (0, 1), γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 2 k = 0 3 while a stopping criterion is not satisfied do 4 ˆXk = Xk 5 forall xc ∈ Xk do 6 if xc ∈ ˆXk then 7 if θ(xc) < 0 then 8 αk c = maxh=0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='{α0δh | F(xc + α0δhv(xc)) ≤ F(xc) + 1γα0δhθ(xc)} 9 else 10 αk c = 0 11 zk c = xc + αk cv(xc) 12 ˆXk = ˆXk \\ {y ∈ ˆXk | F(zk c ) ≨ F(y)} ∪ {zk c } 13 forall I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' θI(zk c ) < 0 do 14 if zk c ∈ ˆXk then 15 αI c = maxh=0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='{α0δh | ∀ y ∈ ˆXk ∃j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' fj(zk c + α0δhvI(zk c )) < fj(y)} 16 ˆXk = ˆXk \\ {y ∈ ˆXk | F(zk c + αI cvI(zk c )) ≨ F(y)} ∪ {zk c + αI cvI(zk c )} 17 Xk+1 = ˆXk 18 k = k + 1 19 return Xk Algorithm 3 includes a bunch of modifications w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' the original FSD approach: for any point in Xk that is still nondominated when it is considered for exploration, a preliminary steepest descent step is carried out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' this step exploits a classical single point Armijo line search [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' further searches w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' subsets of objectives start at the obtained point, as long as it is not dominated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' for partial descent searches, we require the obtained point to be nondominated by all other points in ˆXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The idea is that, with these modifications, all points may be used to start exploration based on partial descent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' convergence of all the produced points towards stationarity is then forced by means of the “preliminary” steepest descent step, that ensures the sufficient decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In the next section we prove that the algorithm is well defined and actually produces convergent sequences of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 5 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='1 Convergence analysis In this section, we provide the formal convergence analysis for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The line search at step 8 of Algorithm 3 is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The result follows from [10, Lemma 4] and by the if condition at step 7 that ensures that θ(xc) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Step 15 of Algorithm 3 is well defined if zk c is nondominated with respect to points in ˆXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let y be any point in ˆXk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' if F(y) = F(zk c ), then by [10, Lemma 4] and the condition θI(zk c ) < 0, there exists ¯α > 0 such that FI(zk c + αvI(zk c )) < FI(zk c ) = FI(y) for all α < ¯α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' thus there exists h sufficiently large such that fj(zk c +α0δhvI(zk c )) < fj(y) for all j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' If, on the other hand, there exists j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} such that fj(zk c ) < fj(y), then by the continuity of F there exists α = α0δh sufficiently small such that fj(zk c + αvI(zk c )) < fj(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Thus, the condition can be satisfied for all y ∈ ˆXk and αI c is the minimum of the corresponding values of α0δh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' If Xk contains mutually nondominated points with respect to F, then ˆXk contains nondominated points at any time during iteration k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' thus step 15 is always well defined and Xk+1 is finally a set of nondominated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' At iteration k, the set ˆXk is initialized with the nondominated points Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' then, it is only updated at steps 12 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' At step 12, either zk c = xc, and the set is not modified, or, by the definition of αk c, zk c dominates xc, which in turn was nondominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Thus, the added point zk c is nondominated, while all the newly dominated points are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' At step 16, the added point zk c + αI cvI(zk c ) is nondominated by the definition of αI c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' all the newly dominated points are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Thus, ˆXk always contains mutually nondominated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' By Proposition 4 step 15 is therefore always well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Moreover, since Xk+1 = ˆXk at the end of the iteration, Xk+1 inherits the nondominance property from ˆXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' After step 12 of Algorithm 3, zk c belongs to ˆXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Moreover, for all ˜k > k, there exists y ∈ X˜k such that F(y) ≤ F(zk c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The first assertion of the proposition trivially follows from the update rule of ˆXk, at step 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Now, either zk c ∈ X˜k or zk c /∈ X˜k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' in the former case, we trivially have y = zk c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' otherwise, we can notice that, by the instructions of the algorithm, any set X˜k, ˜k > k, is the result of repeated application of steps 12 and 16, starting from ˆXk at some point when zk c ∈ ˆXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' When zk c was removed from the set, a point y1 was certainly inserted such that F(y1) ≤ F(zk c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Then, either y1 ∈ X˜k, or y1 was removed when a point y2 such that F(y2) ≤ F(y1) was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' By recursively applying the reasoning, we have that there is certainly a point yt ∈ X˜k such that F(yt) ≤ F(yt−1) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' ≤ F(y2) ≤ F(y1) ≤ F(zk c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let X0 be a set of mutually nondominated points and x0 ∈ X0 be a point such that the set L(x0) = �m j=1{x ∈ Rn | fj(x) ≤ fj(x0)} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let {xjk} be a linked sequence, then it admits limit points and every limit point is Pareto-stationary for problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' For any k, either x0 ∈ Xk or x0 /∈ Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In the former case, since all points in Xk are mutually nondominated, we certainly have xjk ∈ L(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Otherwise, by a similar reasoning as in the proof of Lemma 1, we have that there is a point yk ∈ Xk such that F(yk) ≤ F(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' since yk does not dominate xjk, we have that there exists h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} such that fh(xjk) ≤ fh(yk) ≤ fh(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' thus, again, xjk ∈ L(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Therefore the entire sequence {xjk} belongs to the compact set L(x0), and thus admits limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Now, let us consider a limit point ¯x of a linked sequence {xjk}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=', there exists K ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='} such that lim k→∞ k∈K xjk = ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We assume by contradiction that θ(¯x) < 0 and thus there exists ε > 0 such that for all k ∈ K sufficiently large we have θ(xjk) ≤ −ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Let zjk = xjk + αjkv(xjk) the point obtained at step 11 of the algorithm starting from xjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Now, αjk ∈ [0, α0], which is a compact set, thus there exists a further subsequence K1 ⊆ K such that αjk → ¯α ∈ [0, α0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 6 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO Moreover, function v(·) is continuous, thus v(xjk) → v(¯x) for k → ∞, k ∈ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Hence, taking the limits along K1 we also get that zjk → ¯x + ¯αv(¯x) = ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' By the definition of αjk and zjk (steps 8-11) we have that F(zjk) ≤ F(xjk) + 1γαjkθ(xjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Taking the limits for k ∈ K1, k → ∞, recalling the continuity of θ(·), we get F(¯z) ≤ F(¯x) + 1γ ¯αθ(¯x) ≤ F(¯x) − 1γ ¯αε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (4) Now, given k ∈ K1, let k1(k) be the smallest index in K1 such that k1(k) > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' By Lemma 1, there exists yjk1(k) ∈ Xk1(k) such that F(yjk1(k)) ≤ F(zjk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' moreover, xjk1(k) ∈ Xk1(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' by Proposition 5, the points in Xk1(k) are mutually nondominated, hence there exists h(k) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' , m} such that fh(k)(xjk1(k)) ≤ fh(k)(yjk1(k)) ≤ fh(k)(zjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Considering a further subsequence K2 ⊆ K1 such that h(k) = h for all k ∈ K2 and taking the limits, we obtain fh(¯x) ≤ fh(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Putting this last result together with (4), we get fh(¯x) ≤ fh(¯z) ≤ fh(¯x) − γ ¯αε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Since ¯α ∈ [0, α0], ε > 0 and γ > 0, the above chain of inequalities can only hold if ¯α = limk→∞,k∈K2 αjk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' For all k ∈ K2 sufficiently large, we have θ(xjk) < 0 and, thus, αjk is defined at step 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Since αjk → 0, for any q ∈ N, for all k ∈ K2 large enough we certainly have αjk < α0δq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' thus, the Armijo condition F(xjk + αv(xjk)) ≤ F(xjk) + 1γαθ(xjk) is not satisfied by α = α0δq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=', there exists ˜h(k) such that f˜h(k)(xjk + α0δqv(xjk)) > f˜h(k)(xjk) + γα0δqθ(xjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Taking the limits along a suitable subsequence such that ˜h(k) = ˜h, we get f˜h(¯x + α0δqv(¯x)) ≥ f˜h(¯x) + γα0δqθ(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Now, since q is arbitrary and θ(¯x) < 0, this is absurd by [10, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 4 Numerical results In this section, we show the results of computational experiments, supporting the discussion in Sections 2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The code, which was written in Python3, was executed on a computer with the following characteristics: Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='04, Intel Xeon Processor E5-2430 v2 6 cores 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='50 GHz, 16 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In order to solve instances of problems (2)-(3), the Gurobi optimizer (version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5) was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We compared our approach (IFSD) to the original FSD, Algorithm 1, equipped with the base line search (Algorithm 2) or the extrapolation strategy (EFSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The following parameters setting was used for line searches: α0 = 1, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5, γ = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' With respect to the conceptual scheme in Algorithm 3, we employed within IFSD a strategy to limit the number of points used for partial descent searches, in order to improve the efficiency of the overall procedure and avoid the production of too many, very close solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In particular, we added a condition based on the crowding distance [4] to decide whether a point should be considered for further exploration after the steepest descent step or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The benchmark used for the comparisons consists of the following unconstrained problems: CEC09_2, CEC09_3 [17], JOS_1 [12], MAN [13] (m = 2) and CEC09_10 (m = 3) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' For all the problems, we considered instances with values of n in {5, 10, 20, 30, 40, 50, 100, 200}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Moreover, each problem was tested twice, with different strategies for the initial points: a) n points are uniformly sampled from the hyper-diagonal of a suitable box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' b) only the midpoint of the hyper-diagonal is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The hyper-diagonal refers to the box constituting the constraints in the bounded version of CEC and MAN problems, whereas it is [−100, 100]n for the JOS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In order to appreciate the relative performance and robustness of the approaches, we employed the performance profiles [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In brief, this tool shows the probability that a metric value achieved by a method in a problem is within a factor τ ∈ R of the best value obtained by any of the algorithms in that problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We employed classical metrics for multi-objective optimization: purity, Γ–spread, ∆–spread [3] and hyper-volume [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Purity and hyper-volume 7 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 f1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 f2 (c) Figure 2: Pareto fronts obtained by the IFSD algorithm on the convex JOS problem (n = 5) starting from different initial points: (a) 1 Pareto point as in Figure 1a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (b) 2 Pareto points as in Figure 1b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' (c) 3 independent runs from the same random points as those of Figure 1(c)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' have increasing values for better solutions: then, the corresponding profiles are produced considering the inverse of the obtained values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In Figure 2, the behavior of the proposed approach in the same setting as in Figure 1 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' In this example we can observe that now, regardless, of the starting point(s), the entire Pareto front is effectively spanned, with not even tiny holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' For a more consistent assessment of algorithms performance, we report in Figure 3 the performance profiles for the IFSD, FSD and EFSD algorithms on the entire benchmark of 80 problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' We observe a remarkable superiority of the proposed approach w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' the original variants of the algorithm, especially in terms of the spread metrics, which points out that the Pareto front is indeed spanned more widely and uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The strong hypervolume performance also supports this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' As for purity metric, the three algorithms appear to be closer, but we still observe a slight advantage of IFSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 5 Conclusions In this paper, we introduced an improved Front Steepest Descent algorithm with asymptotic convergence guarantees similar as those of the original method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' The novel algorithm is designed so as to overcome some empirically evident limitation of FSD, that is often unable to span large portions of the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Numerical evidence suggests that the proposed procedure effectively achieves this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Future work should be focused on the integration of the proposed approach and the extrapolation strategy proposed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Moreover, the employment of the proposed approach within memetic procedures for global multi-objective optimization [13] might be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' Finally, the algorithm defined in this work could be extended to deal with constrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 8 Improved Front Steepest Descent for MOO MATTEO LAPUCCI AND PIERLUIGI MANSUETO 1 2 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 Cumulative Purity IFSD FSD EFSD (a) Purity profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='50 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 Cumulative Hypervolume (b) Hypervolume profile 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 Cumulative spread (c) Γ-spread profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content='0 Cumulative spread (d) ∆-spread profile Figure 3: Performance profiles for the IFSD, FSD and EFSD algorithms on a benchmark of 80 multi-objective problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfngSj/content/2301.03310v1.pdf'} diff --git a/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf b/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a488e75a91b7fb8b2f021df8167bd3c16eb829bc --- /dev/null +++ b/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0a48cb595468a4712b1971bb44d162c0289044cd35a7dfcf91a4b391f845f897 +size 325307 diff --git a/8dE4T4oBgHgl3EQfCwtu/vector_store/index.faiss b/8dE4T4oBgHgl3EQfCwtu/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..303b8925b67c7d21030a3d62b642931833bd2e04 --- /dev/null +++ b/8dE4T4oBgHgl3EQfCwtu/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ba16a8c0aeda513b2671c5a5eb75ac176bed51bd96a19ba5fa1b0d15f8855748 +size 4194349 diff --git 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TURNER,1, 2 BHAL CHANDRA JOSHI,3 MAURA A. MCLAUGHLIN,1, 2 AND DANIEL R. STINEBRING4 +1Department of Physics and Astronomy, West Virginia University, P.O. Box 6315, Morgantown, WV 26506, USA +2Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA +3National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Post Bag 3, Ganeshkhind, Pune - 411007, India +4Department of Physics and Astronomy, Oberlin College, Oberlin, OH 44074, USA +ABSTRACT +We use the Upgraded Giant Metrewave Radio Telescope to measure scintillation arc properties in six bright +canonical pulsars with simultaneous dual frequency coverage. These observations at frequencies from 300 to +750 MHz allowed for detailed analysis of arc evolution across frequency and epoch. We perform more robust +determinations of arc curvature and scattering delay frequency-dependence than allowed by single-frequency- +band-per-epoch measurements, which we find to agree with theory and previous literature. We report the dis- +covery of a strong correlation between arc asymmetry and arc curvature, potentially indicating a link between +scattering screen distance and refraction strength or the effect of asymmetric distribution of scattering material +on a scattering screen. The inclusion of a 155 minute observation allowed us to resolve the scale of scintilla- +tion variations on short timescales, which we find to be directly tied to the amount of ISM sampled over the +observation. Some of our pulsars showed either consistent or emerging asymmetries in arc curvature, indicating +instances of refraction across their lines of sight. The presence of significant features in various pulsars, such +as multiple scintillation arcs in PSR J1136+1551 and flat arclets in PSR J1509+5531, that have been found in +previous works, were also sufficiently detected. Possible evidence for a timescale over which a given scatter- +ing screen dominates signal propagation was found by tracking visible scintillation arcs in each epoch in PSR +J1136+1551. The interesting pulsar science accomplished with this upgraded telescope shows strong promise +for important future work in pulsar astronomy. +Keywords: methods: data analysis – stars: pulsars – ISM: general – ISM: structure +1. INTRODUCTION +The scintillation of pulsar emission occurs as the result of +the propagation of this emission through non-uniform distri- +butions of free electrons in the ionized interstellar medium +(IISM). This interaction results in frequency-dependent and +time-evolving variations in the flux density of the pulsar sig- +nal as measured at a detector. +When these variations are +examined across observing frequency and time in so-called +dynamic spectra, representations of the change in the pulsar +signal’s intensity across frequency and time, for a given ob- +servation, they can provide valuable insight into the structure +of these electron density variations along our line of sight +(LOS) to a given pulsar. For observations where the scin- +tles, bright patches in a dynamic spectrum resulting from +constructive interference between photons as the result of +propagation through the free electrons in the ISM, are fully +resolved in frequency, have sufficient coverage in time, and +exhibit structures over the course of observations including, +but not limited to, non-zero slopes across frequency and time, +known as scintillation drift, as well as “crisscrossing” scin- +tillation patterns, then additional information can be gained +about the ISM structure along the LOS by examining the +parabolic arcs, known as scintillation arcs, that can emerge +by examining the Fourier transform of the dynamic spec- +trum (Stinebring et al. 2001). Some current hypotheses on +the physical origins of these arcs postulate that they originate +from compressed plasma along the boundaries of 50−100 pc +size bubbles in the ISM (Stinebring et al. 2022). +Traditional measurements of scintillation arcs have typ- +ically been limited to either one observing band over all +epochs (i.e., Trang & Rickett (2007)), or alternated between +observing bands from epoch to epoch (i.e., Stinebring et al. +(2019)). While generally sufficient for most analyses, this +band limit results in a bottleneck for examining the evo- +lution of various frequency-dependent effects over shorter +timescales, including scintillation arc curvature, structures +within individual arcs, and asymmetries in both arc bright- +ness and power as a function of differential time delay. By +making use of the subarray capabilities of the Upgraded Gi- +ant Metrewave Radio Telescope (uGMRT), we can effec- +arXiv:2301.05306v1 [astro-ph.HE] 12 Jan 2023 + +2 +J. E. TURNER ET AL. +tively create an ultra wideband receiver by setting multiple +groups of dishes to simultaneously observe at different fre- +quencies. This work is primarily data-focused and aims to +highlight the results of some multi-frequency analyses per- +formed on a small survey of six strong canonical pulsars us- +ing this approach. In Section 2 we discuss the data taken +as part of our survey. Section 3 describes the analyses per- +formed and the physical parameters extracted. Section 4 de- +tails the results of these analyses. Finally, Section 5 summa- +rizes our results and discusses possible next steps. +2. DATA +Our data were taken across across eight epochs span- +ning MJD 58987−59497 using 22 dishes split into subarrays +for simultaneous multi-frequency observations at uGMRT’s +Band 3 and Band 4, centered at 400 MHz and 650 MHz, +respectively, each with 200 MHz of bandwidth. This simul- +taneous low-frequency accessibility is comparable to instru- +ments like CHIME that can observe continuously between +400-800 MHz and better than instruments such as the Green +Bank Telescope, which, while having a wide range of low +frequency coverage, can only observe below one GHz with +at most 240 MHz of bandwidth at frequencies close to one +GHz and less than 200 MHz of bandwidth in lower frequency +ranges. Observations were also made at Band 5 centered +at 1360 MHz, although due to a combination of RFI and +low S/N no scintles were detectable in the dynamic spec- +tra. The observing bands were split into 4096 49 kHz wide +frequency channels and observed with 10 second subintegra- +tions. These data were flux calibrated using observations of +either 3C147 or 3C286 taken at the beginning of every ob- +serving session and every pulsar was phase calibrated with +a nearby source for five minutes once every 40 minutes of +observing time on the pulsar. Two to three pulsars were ob- +served at each epoch for 40 minutes each, except for MJD +59497, where three pulsars were observed for 155 minutes +each. As a result of the phase calibration, each of those obser- +vations were comprised of three 40 minute sub-observations +plus an additional 20 minute sub-observation. +3. ANALYSIS +All observations were processed to extract their dynamic +spectra by calculating the intensity, S, of the pulsar’s signal +at each observing frequency, ν, and time, t, via +S(ν, t) = Pon(ν, t) − Poff(ν, t) +Pbandpass(ν, t) +, +(1) +where Pbandpass is the total power of the observation as a +function of observing frequency and time, and Pon and Poff +are the power in all on- and off-pulse components, respec- +tively, as a function of frequency and time. Each dynamic +spectrum was then broken up into four 50 MHz spectra to +allow for more in-depth frequency-dependent analyses and +manually zapped by examining dynamic spectra data arrays +and removing pixels that were brighter than the brightest +scintle maxima. +Secondary spectra were then created by +taking the absolute square of the Fourier transform (i.e., the +power spectrum) of the corresponding dynamic spectrum and +converting it to units of dB. The primary (brightest) scintilla- +tion arcs on the positive and negative side of each secondary +spectrum’s fringe frequency axis were then found via sepa- +rate fν = ηf 2 +t fits, where fν is the differential time delay, η +is the arc curvature, and ft is the fringe frequency. +We also determined scintillation parameters by using the +python package Pypulse (Lam 2017) to create the 2D au- +tocorrelation functions (ACFs) of each dynamic spectrum +and fit 2D Gaussians to these ACFs to determine their scin- +tillation bandwidth, ∆νd, defined as the half-width at half- +maximum (HWHM) along the frequency axis of the ACF at +lag 0, scintillation timescale, ∆td, defined as the half-width +at e−1 along the time axis at ACF lag 0, and scintillation +drift rate, dν/dt, defined as the rotation of the 2D Gaussian +fit to the 2D ACF in the plane of the frequency and time lags. +For our scattering delay scaling index analysis, our measured +scintillation bandwidths were converted to scattering delays +using +2π∆νdτ = C1, +(2) +where C1 is a dimensionless quantity between 0.6 − 1.5 that +depends on the spectrum of the electron density fluctuations +and geometry of the medium (Cordes & Rickett 1998). For +this work we use C1 = 1. +4. RESULTS & DISCUSSION +Measured arc curvatures and their corresponding dynamic +spectrum scintillation drift rates can be found in Table 1. All +curvatures and their uncertainties have been scaled to their +corresponding value at 1 GHz assuming a ν−2 frequency de- +pendence (Hill et al. 2003). Here errors on the arc curva- +tures represent fitting errors from the linear least squares fit. +Some pulsars on MJD 59497 have multiple curvature mea- +surements at a given frequency, which is the result of this +epoch being 155 minutes instead of the 40 minutes of the +other observations. As a result, a new η was measured after +every 40 minutes since these observations were broken up +into 40 minute chunks separated by five minute phase cali- +brations. On days where measurements are given for the left +or right arm only, arcs on the other side of the fringe fre- +quency axis may have been present, but were unmeasureable +due to either having insufficient extension along the differ- +ential delay axis, insufficient flux relative to the background +noise, being too diffuse, being too close to central spike in +flux that commonly occurs around a fringe frequency of 0 +mHz, or some combination of these factors. + +SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY +3 +Table 1. Pulsar Scintillation Arc Curvatures and Drift Rates +Pulsar +MJD +Frequency +ηL +σηL +ηR +σηR +dν/dt +σdν/dt +(MHz) +(s3) +(s3) +(s3) +(s3) +(MHz/min) +(MHz/min) +J0630–2834 +58987 +325 +— +— +0.2071 +0.0069 +–0.0164 +0.0009 +J0630–2834 +58987 +375 +— +— +0.2331 +0.0160 +–0.0676 +0.0097 +J0630–2834 +58987 +425 +— +— +0.2932 +0.0223 +–0.0343 +0.0027 +J0630–2834 +58987 +475 +— +— +0.2037 +0.0207 +–0.0574 +0.0059 +J0630–2834 +58987 +575 +— +— +0.1220 +0.0061 +0.0076 +0.0025 +J0630–2834 +58987 +625 +— +— +0.1233 +0.0049 +–0.0779 +0.0097 +J0630–2834 +58987 +675 +— +— +0.1553 +0.0058 +0.0818 +0.0118 +J0630–2834 +58987 +725 +— +— +0.2566 +0.0162 +0.2685 +0.0196 +J1136+1551 +58987 +325 +0.0154 +0.0007 +0.0238 +0.0006 +— +— +J1136+1551 +58987 +375 +0.0198 +0.0009 +0.0231 +0.0005 +0.0202 +0.0067 +J1136+1551 +58987 +425 +0.0195 +0.0010 +0.0225 +0.0004 +–0.0287 +0.0074 +J1136+1551 +58987 +575 +0.0214 +0.0005 +0.0218 +0.0003 +–0.0173 +0.0067 +J1136+1551 +58987 +625 +0.0202 +0.0007 +0.0229 +0.0002 +— +— +J1136+1551 +58987 +675 +0.0199 +0.0005 +0.0249 +0.0003 +–0.0322 +0.0250 +J1136+1551 +58987 +725 +0.0203 +0.0007 +0.0282 +0.0006 +0.00004 +0.7446 +J1136+1551 +58991 +325 +0.0126 +0.0004 +0.0234 +0.0005 +0.0048 +0.0093 +J1136+1551 +58991 +375 +0.0149 +0.0007 +0.0253 +0.0007 +0.2686 +0.0128 +J1136+1551 +58991 +425 +0.0167 +0.0009 +0.0250 +0.0006 +— +— +J1136+1551 +58991 +475 +0.0183 +0.0007 +0.0225 +0.0005 +–0.0402 +0.0123 +J1136+1551 +58991 +575 +0.0165 +0.0005 +0.0254 +0.0005 +0.0097 +0.0026 +J1136+1551 +58991 +625 +0.0178 +0.0008 +0.0271 +0.0005 +–0.0088 +0.0034 +J1136+1551 +58991 +675 +0.0173 +0.0007 +0.0261 +0.0004 +— +— +J1136+1551 +58991 +725 +0.0171 +0.0007 +0.0239 +0.0003 +0.00003 +0.4211 +J1136+1551 +59115 +325 +0.0084 +0.0002 +0.0099 +0.0002 +0.2219 +0.0095 +J1136+1551 +59115 +375 +0.0085 +0.0003 +0.0116 +0.0003 +0.1232 +0.0115 +J1136+1551 +59115 +425 +0.0095 +0.0005 +0.0139 +0.0004 +— +— +J1136+1551 +59115 +475 +0.0092 +0.0005 +0.0129 +0.0004 +0.0869 +0.0079 +J1136+1551 +59115 +575 +0.0095 +0.0003 +0.0111 +0.0002 +0.0057 +0.0014 +J1136+1551 +59115 +625 +0.0135 +0.0007 +0.0141 +0.0006 +— +— +J1136+1551 +59115 +675 +0.0143 +0.0004 +0.0154 +0.0003 +0.0159 +0.0057 +J1136+1551 +59115 +725 +0.0130 +0.0005 +0.0160 +0.0005 +0.0109 +0.0036 +J1136+1551 +59497 +325 +0.0065 +0.0002 +0.0070 +0.0003 +–0.1251 +0.0104 +J1136+1551 +59497 +375 +0.0078 +0.0004 +0.0077 +0.0005 +–0.0554 +0.0069 +J1136+1551 +59497 +425 +0.0079 +0.0002 +0.0072 +0.0002 +–0.0119 +0.0019 +J1136+1551 +59497 +475 +0.0070 +0.0003 +0.0068 +0.0004 +–0.0293 +0.0044 +J1136+1551 +59497 +325 +0.0070 +0.0001 +0.0067 +0.0002 +0.0076 +0.0030 +J1136+1551 +59497 +375 +0.0077 +0.0002 +0.0068 +0.0002 +0.2637 +0.0498 +J1136+1551 +59497 +425 +0.0072 +0.0002 +0.0076 +0.0002 +–0.0544 +0.0066 +Table 1 continued + +4 +J. E. TURNER ET AL. +Table 1 (continued) +Pulsar +MJD +Frequency +ηL +σηL +ηR +σηR +dν/dt +σdν/dt +(MHz) +(s3) +(s3) +(s3) +(s3) +(MHz/min) +(MHz/min) +J1136+1551 +59497 +475 +0.0069 +0.0001 +0.0067 +0.0002 +–0.0302 +0.0045 +J1136+1551 +59497 +325 +0.0075 +0.0002 +0.0075 +0.0002 +–0.0651 +0.0058 +J1136+1551 +59497 +375 +0.0076 +0.0002 +0.0076 +0.0003 +–0.0877 +0.0175 +J1136+1551 +59497 +425 +0.0082 +0.0002 +0.0075 +0.0002 +–0.0656 +0.0064 +J1136+1551 +59497 +475 +0.0067 +0.0002 +0.0074 +0.0003 +–0.0417 +0.0053 +J1136+1551 +59497 +575 +0.0063 +0.0004 +0.0079 +0.0013 +–0.0977 +0.0507 +J1136+1551 +59497 +675 +0.0059 +0.0004 +0.0075 +0.0007 +— +— +J1136+1551 +59497 +725 +0.0057 +0.0005 +0.0068 +0.0008 +— +— +J1509+5531 +58987 +575 +0.3155 +0.0191 +0.1798 +0.0063 +— +— +J1509+5531 +58987 +625 +0.3642 +0.0184 +0.2021 +0.0060 +–0.0284 +0.0224 +J1509+5531 +58987 +675 +0.3410 +0.0207 +0.1978 +0.0078 +–1.5130 +0.0117 +J1509+5531 +59064 +575 +0.2611 +0.0147 +0.2495 +0.0116 +0.0015 +0.0030 +J1509+5531 +59064 +625 +0.2890 +0.0170 +0.2767 +0.0118 +— +— +J1509+5531 +59064 +675 +0.2736 +0.0181 +0.2714 +0.0128 +0.0039 +0.0016 +J1509+5531 +59064 +725 +0.2921 +0.0186 +0.2930 +0.0142 +0.1192 +0.0728 +J1509+5531 +59115 +575 +0.0852 +0.0029 +0.0928 +0.0037 +— +— +J1509+5531 +59115 +625 +0.0868 +0.0029 +0.0964 +0.0044 +0.0173 +0.0066 +J1509+5531 +59115 +675 +0.0958 +0.0031 +0.0908 +0.0044 +–0.0970 +0.0628 +J1509+5531 +59115 +725 +0.1074 +0.0026 +0.0872 +0.0030 +— +— +J1509+5531 +59497 +575 +0.0992 +0.0034 +0.1139 +0.0045 +–0.9922 +0.0226 +J1509+5531 +59497 +625 +0.1057 +0.0032 +0.1257 +0.0043 +0.0186 +0.0140 +J1509+5531 +59497 +675 +0.1088 +0.0033 +0.1160 +0.0035 +— +— +J1509+5531 +59497 +725 +0.0973 +0.0026 +0.1337 +0.0036 +0.0517 +0.0395 +J1509+5531 +59497 +575 +0.0706 +0.0028 +0.1054 +0.0047 +— +— +J1509+5531 +59497 +625 +0.0741 +0.0026 +0.1101 +0.0046 +0.0059 +0.0047 +J1509+5531 +59497 +675 +0.0782 +0.0022 +0.1200 +0.0047 +0.0109 +0.0086 +J1509+5531 +59497 +725 +0.0808 +0.0021 +0.1265 +0.0040 +— +— +J1509+5531 +59497 +675 +0.0606 +0.0020 +0.1194 +0.0042 +— +— +J1645–0317 +59074 +575 +0.0832 +0.0051 +— +— +0.2015 +0.0091 +J1645–0317 +59074 +625 +0.0885 +0.0059 +— +— +–0.1688 +0.0098 +J1645–0317 +59074 +675 +0.0806 +0.0063 +— +— +–0.1812 +0.0091 +J1645–0317 +59074 +725 +0.0832 +0.0061 +— +— +–0.2235 +0.0105 +J1932+1059 +58997 +325 +0.0382 +0.0030 +0.0331 +0.0018 +–0.1092 +0.0227 +J1932+1059 +58997 +375 +0.0358 +0.0016 +0.0364 +0.0020 +— +— +J1932+1059 +58997 +425 +0.0346 +0.0006 +0.0335 +0.0007 +–0.0224 +0.0086 +J1932+1059 +58997 +475 +0.0365 +0.0007 +0.0335 +0.0005 +0.1987 +0.0427 +J1932+1059 +58997 +575 +0.0351 +0.0005 +0.0360 +0.0004 +0.0181 +0.0093 +J1932+1059 +58997 +625 +0.0366 +0.0005 +0.0335 +0.0004 +0.0614 +0.0352 +J1932+1059 +58997 +675 +0.0359 +0.0010 +0.0358 +0.0010 +–0.0062 +0.0016 +J1932+1059 +58997 +725 +0.0373 +0.0007 +0.0348 +0.0012 +–0.0074 +0.0022 +Table 1 continued + +SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY +5 +Table 1 (continued) +Pulsar +MJD +Frequency +ηL +σηL +ηR +σηR +dν/dt +σdν/dt +(MHz) +(s3) +(s3) +(s3) +(s3) +(MHz/min) +(MHz/min) +J1932+1059 +59062 +325 +0.0330 +0.0006 +0.0271 +0.0004 +0.0669 +0.0111 +J1932+1059 +59062 +375 +0.0352 +0.0007 +0.0285 +0.0008 +0.1331 +0.0219 +J1932+1059 +59062 +425 +0.0347 +0.0005 +0.0293 +0.0004 +–0.0326 +0.0256 +J1932+1059 +59062 +475 +0.0345 +0.0007 +0.0278 +0.0006 +–0.1200 +0.0297 +J1932+1059 +59062 +575 +0.0337 +0.0007 +0.0303 +0.0004 +— +— +J1932+1059 +59062 +625 +0.0332 +0.0004 +0.0323 +0.0004 +–0.0221 +0.0095 +J1932+1059 +59062 +675 +0.0350 +0.0006 +0.0356 +0.0006 +–0.0221 +0.0095 +J1932+1059 +59062 +725 +0.0297 +0.0005 +0.0304 +0.0006 +–0.0192 +0.0152 +J1932+1059 +59497 +325 +0.0365 +0.0008 +0.0279 +0.0009 +–0.8215 +0.0090 +J1932+1059 +59497 +375 +0.0351 +0.0007 +0.0294 +0.0004 +0.1062 +0.0302 +J1932+1059 +59497 +425 +0.0399 +0.0004 +0.0320 +0.0005 +0.1479 +0.0287 +J1932+1059 +59497 +475 +0.0389 +0.0007 +0.0341 +0.0008 +0.2495 +0.3407 +J1932+1059 +59497 +575 +0.0405 +0.0016 +0.0363 +0.0012 +— +— +J1932+1059 +59497 +625 +0.0393 +0.0014 +0.0358 +0.0013 +–0.0815 +0.0392 +J1932+1059 +59497 +675 +0.0433 +0.0017 +0.0385 +0.0012 +–0.0222 +0.0103 +J1932+1059 +59497 +725 +0.0461 +0.0022 +0.0416 +0.0013 +— +— +J2048–1616 +59062 +325 +0.0221 +0.0006 +0.0108 +0.0009 +–0.4158 +0.0081 +J2048–1616 +59062 +375 +0.0198 +0.0007 +0.0113 +0.0023 +–0.0211 +0.0077 +J2048–1616 +59062 +425 +0.0247 +0.0020 +0.0108 +0.0012 +–0.1529 +0.0139 +J2048–1616 +59062 +625 +0.0152 +0.0004 +0.0132 +0.0005 +–0.0058 +0.0012 +J2048–1616 +59062 +675 +0.0152 +0.0004 +0.0138 +0.0005 +0.0208 +0.0090 +J2048–1616 +59062 +725 +0.0148 +0.0004 +0.0135 +0.0005 +0.00005 +0.0001 +NOTE—Scintillation arc measurements and drift rates in the left and right primary arms of each epoch at all +frequencies where measurable. ηL and ηR are the the arc curvature measurements for the left and right arms, +respectively, and dν/dt is the measured scintillation drift rate, with the matching σ’s representing the corre- +sponding uncertainties. All curvatures and their errors have been scaled to 1 GHz and errors on curvature here +are fit uncertainties. Some pulsars on MJD 59497 have multiple curvature measurements at a given frequency, +due to this epoch being 155 minutes instead of the 40 minutes of the other observations, and so a new η was +measured after every 40 minutes. +4.1. Scintillation Arc Curvature Scaling Behavior +As mentioned earlier, Hill et al. (2003) demonstrated +through both theoretical and observational means that the +arc curvature η should follow a ν−2 dependence, implying +that scattering is dominated by one or several thin screens +along the LOS. While over 2 GHz of bandwidth was used in +those observations (10-12.5 MHz of bandwidth centered at +430 MHz and either 50 or 100 MHz of bandwidth centered at +1175 MHz, 1400 MHz, and 2250 MHz), the frequency cov- +erage was discontinuous and all η measurements used in their +corresponding fits were from different epochs. Generally the +latter point should not be an issue as long as the observations +were taken within a period shorter than the pulsar’s refrac- +tive timescale. Indeed, for the data used in their fits, their +measured arc curvatures at a given frequency did not vary +significantly on day or week timescales, making them suit- +able for this type of analysis. However, the ideal situation +would be to obtain many measurements at many frequencies +during the same observation, preferably at the same time for +optimal consistency. With our high resolution and sufficient +observing time, we have the ability to make up to eight con- +current arc measurements over 450 MHz of bandwidth at low +frequency and can consequentially provide a more definitive +examination of the theory. +Following the methodology of Hill et al. (2003), for a scal- +ing index α, we performed a weighted linear least-squares fit +of the form +log10 η = α log10 ν + β +(3) +on the unscaled curvatures for each pulsar at each MJD. Ex- +ample fits can be seen in Figure 1, with all measured in- + +6 +J. E. TURNER ET AL. +dices listed in Table 2. We find that, overall, our scaling +indices are consistent with a theoretical index of −2, which +assumes thin screen scattering (Stinebring et al. 2001), with +PSRs J1136+1551 and J1932+1059 being especially consis- +tent. This effect can also be seen in Table 1, where arc cur- +vatures at all frequencies from a given pulsar at the same +MJD are generally in strong agreement after being scaled to +1 GHz. Interestingly, a weighted average of all curvature +fits shows that our left arm fits are overall more consistent +with an index of −2 than our right arms, with a weighted av- +erage of −1.99±0.03 across all left arm fits compared with +−1.69±0.02 across all right arm fits, indicating that refrac- +tion may play a role in how closely arc curvature scales as +expected with frequency. +4.2. Scattering Delay Scaling Behavior +Our wide frequency coverage also allowed us to examine +the scaling index of scattering delays. Under the assumption +that ISM fluctuations follow behaviors consistent with a Kol- +mogorov medium and that scattering can be modeled as the +result of interactions of pulsar emission with an infinite, thin, +scattering screen during its propagation, we should expect +that scattering delays scale with frequency as τd ∝ ν−4.4 +(Romani et al. 1986; Cordes & Rickett 1998). +Previous +studies examining the scattering indices of various pulsars +have done so using a number of methods, including simul- +taneous multi-frequency measurements (Bhat et al. 2004; +Bansal et al. 2019), splitting up measurements from a single +frequency band into multiple subbands (Levin et al. 2016; +Turner et al. 2021), and using measurements from many +epochs taken at two observing bands non-simultaneously +(Turner et al. 2021). Since more measurements and more fre- +quency coverage in a single epoch is ideal, the method used +in Bhat et al. (2004) and Bansal et al. (2019) is the most pre- +ferred of the three. The method in this paper utilizes a com- +bination of this approach and the subband approach to max- +imize the number of delay measurements per epoch, which +can be done thanks to our high frequency resolution and sen- +sitivity in both observing bands. +Similar to Equation 3 used to determine the arc curva- +ture scaling index, our scattering delay scaling indices ξ at +each epoch were determined by performing a weighted lin- +ear least-squares fit of the form +log10 τd = ξ log10 ν + b. +(4) +Example fits can be seen in Figure 2, with all measured in- +dices listed in Table 3. We find that half of our measured +indices are consistent with a Kolmogorov medium, while the +other half are consistent with a shallower medium. This be- +havior agrees well with previous studies, as both Bhat et al. +(2004) and Bansal et al. (2019) found indices either consis- +tent with a Kolmogorov medium or shallower than a Kol- +mogorov medium, while Levin et al. (2016) and Turner et al. +(2021) only found indices that were shallower than a Kol- +mogorov medium. +Many explanations have been given for why shallower- +than-Kolmogorov medium behavior has been observed so +frequently. Physical arguments have called into question the +validity of the simple infinite, thin screen model, demonstrat- +ing that shallower scaling indices are more consistent with +finite, thin screens (Rickett et al. 2009). This is expected to +be much more common among low DM pulsars (Cordes & +Lazio 2001), which agrees with our results, as all of the pul- +sars have dispersion measures below 40 pc cm−3. Shallower +indices have also been attributed to the existence of multi- +ple finite screens along the LOS (Lewandowski et al. 2013). +This hypothesis agrees well with our measured indices for +PSR B1133+16, as its indices are consistently shallower than +that of a Kolmogorov medium and it is also known to have at +least six distinct scattering screens (McKee et al. 2022). +Quality-of-data arguments have also been proposed. +Turner et al. (2021) suggested their shallower indices may +be at least partially attributable to an imbalance of lower fre- +quency data to higher frequency data for their multiple epoch +approach as well as a lack of sufficient frequency resolution +in their lower frequency band in some epochs. However, nei- +ther of these issues should affect our results, as our observa- +tions have a consistently even balance of low and high fre- +quency measurements at all epochs and all of our measure- +ments are well-resolved in frequency. +4.3. 155 Minute Observation +The inclusion of a 155 minute observation in our survey +on MJD 59497 allowed for an analysis of short-term arc cur- +vature variation in some pulsars, as observations had to be +paused every 40 minutes for a five minute phase calibration, +resulting in multiple 40 minute sub-observations. For pulsars +with at least two measurements in a given scintillation arc at a +given frequency, we examined overall variation in that arc at +that frequency by looking at the percent difference between a +given curvature measurement and the weighted average cur- +vature for that arm and frequency over the entire epoch. +For PSR J1136+1551, all observing frequencies centered +at or below 475 MHz had three measurements in each pri- +mary arm (the brightest arm, overwhelmingly often the arm +with the lowest curvature) at each frequency, with the accu- +mulation of all percent differences yielding a bimodal distri- +bution with peaks around percent differences of 2% and 7%. +The largest percent difference away from a weighted mean +was 7.9±0.2% and the smallest was 0.14±1.91%, although +the majority of all percent differences was below 3%. All of +this strongly indicates the ISM underwent very little change +along the LOS to this pulsar over the course of a given ob- +servation. This result is supported by this pulsar’s incredibly +low dispersion measure, meaning it does not sample a size- + +SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY +7 +(a) α fit for PSR J1932+1059 on MJD 58997 +(b) α fit for PSR J1136+1551 on MJD 58997. The inclusion of multiple points at +certain frequencies is the result of this epoch containing a 155 minute +observation instead of the 40 minutes of the other observations, and so a new η +was measured after every 40 minutes. +Figure 1. Example fits for the arc curvature scaling index +Table 2. Fitted Pulsar Scintillation Arc Curvature Scaling Indices +Pulsar +MJD +Scaling Index Left Arc +Scaling Index Error Left Arc +Nη +Scaling Index Right Arc +Scaling Index Error Right Arc +Nη +J0630–2834 +58987 +— +— +— +–2.48 +0.31 +8 +J1136+1551 +58987 +–1.79 +0.11 +7 +–1.89 +0.12 +7 +J1136+1551 +58991 +–1.65 +0.12 +8 +–1.94 +0.08 +8 +J1136+1551 +59115 +–1.36 +0.13 +8 +–1.52 +0.15 +8 +J1136+1551 +59497 +–2.12 +0.12 +15 +–1.99 +0.10 +15 +J1509+5531 +58987 +–1.49 +0.83 +3 +–1.34 +0.55 +3 +J1509+5531 +59064 +–1.62 +0.26 +4 +–1.41 +0.22 +4 +J1509+5531 +59115 +–0.93 +0.21 +4 +–2.31 +0.18 +4 +J1509+5531 +59497 +–2.01 +0.87 +9 +–1.34 +0.21 +9 +J1645–0317 +59074 +–2.09 +0.26 +4 +— +— +— +J1932+1059 +58997 +–1.93 +0.05 +8 +–1.93 +0.09 +8 +J1932+1059 +59062 +–2.08 +0.07 +8 +–1.75 +0.07 +8 +J1932+1059 +59497 +–1.77 +0.09 +8 +–1.53 +0.04 +8 +J2048–1616 +59062 +–2.52 +0.07 +6 +–1.68 +0.05 +6 +NOTE—Fitted arc curvature scaling indices for both left and right primary arcs. Nη indicates the number of arc curvature measurements used +in each fit. Measurements on MJD 59497 may have Nη > 8 due to this epoch being 155 minutes rather than the 40 minutes of the other +observations, and so a new η was measured after every 40 minutes. Arc curvature measurements used in these fits were left unscaled. +able portion of the ISM along its LOS relative to many pul- +sars that are observed (Bilous et al. 2016; Manchester et al. +2005; Pilkington et al. 1968). +For PSR J1509+5531, all observing frequencies centered +at or above 575 MHz had at least two measurements in each +arm at each frequency, with the accumulation of all per- +cent differences resulting in a one-sided distribution peaked + +J1932+1059MJD 5899T +4 × 10-1 +nL; Scaling Index= -1.93 ± 0.05 +nR; Scaling Index = -1.93 ± 0.09 +3 × 10-1. +2 × 10-1 +n +10-1 +6 × 10-2 +400 +500 +600 +700 +Frequency [MHz]J1136+1551MJD 5949T +nL; Scaling Index= -2.12 ± 0.12 +6 × 10-2 +nr; Scaling Index = -1.99 ± 0.10 +4 × 10-2 +3 × 10-2 +n +2 × 10-2 +10-1 +400 +500 +600 +700 +Frequency [MHz]8 +J. E. TURNER ET AL. +(a) Scattering delay scaling index fit for PSR J1932+1059 on MJD 59497 +(b) Scattering delay scaling index fit for PSR J1136+1551 on MJD 58991. +Figure 2. Example fits for the scattering delay scaling index +Table 3. Fitted Pulsar Scattering Delay Scaling Indices +Pulsar +MJD +Scaling Index +Index Error +Nτd +J0630–2834 +58987 +–4.19 +1.43 +8 +J1136+1551 +58987 +–1.44 +0.71 +6 +J1136+1551 +58991 +–3.78 +0.62 +7 +J1136+1551 +59115 +–2.72 +1.05 +6 +J1136+1551 +59497 +–1.71 +0.57 +13 +J1645–0317 +59074 +–4.60 +0.75 +4 +J1932+1059 +58997 +–1.83 +0.31 +7 +J1932+1059 +59062 +–1.74 +0.31 +6 +J1932+1059 +59497 +–4.14 +0.39 +6 +J2048–1616 +59062 +–3.77 +1.39 +6 +NOTE—Fitted scattering delay scaling indices, with a minimum +of four delay measurements (Nτd) required in a given epoch to +obtain a scaling index. Errors are the parameter uncertainties +from parameter fits. Half of our measured indices were con- +sistent with a Kolmogorov medium, while the other half were +consistent with a shallower medium. Measurements on MJD +59497 may have Nτd > 8 due to this epoch being 155 minutes +rather than the 40 minutes of the other observations, and so a +new η was measured after every 40 minutes. +around 6%. +The smallest percent difference away from +a weighted mean was 1.2±2.1%, while the largest was +36±0.1%, although the next largest after that was only +22±0.1%, meaning this maximum was an extreme outlier. +The majority of all percent differences was below 7%. As +with the previous pulsar, this also strongly indicates the ISM +underwent very little change along the LOS to this pul- +sar over the course of a given observation, a result again +supported by this pulsar’s fairly low dispersion measure +(Huguenin et al. 1968; Manchester et al. 2005). The fact that +this pulsar shows higher variation of this observation com- +pared to PSR J1136+1551 is likely due to PSR J1509+5531 +having a dispersion measure four times higher and a trans- +verse velocity 45% larger (Bilous et al. 2016; Huguenin et al. +1968; Manchester et al. 2005; Pilkington et al. 1968; Stovall +et al. 2015), so a significantly larger fraction of the ISM was +sampled during its observation, increasing the likelihood of +larger scintillation-based variations. +The next few subsections will be dedicated to highlighting +the features of a few pulsars in the survey. +4.4. J0630-2834 +In the one epoch for which we were able to resolve a scin- +tillation arc, only the right arm was resolvable across all fre- +quencies, with its relative brightness relative to the left side +of the fringe frequency axis consistently decreasing as fre- +quency increased. An example of this asymmetry can be seen +in Figure 3.This strong asymmetry is known to be the result +of refraction leading to scintillation drifting in the dynamic +spectra (Cordes et al. 2006). Interestingly, despite our asym- +metry appearing to decrease with frequency, the magnitude +of our measured scintillation drift rates seem to mildly favor +an increase with frequency, whereas one would expect an in- +crease in scintillation drift to coincide with an increase in the +asymmetry. +4.5. J1136+1551 + +J1932+1059 MJD 59497 +Scaling Index= -4.14 ± 0.39 +102 +(ns) +101 +400 +500 +600 +700 +Frequency [MHz]J1136+1551MJD 58991 +Scaling Index= -3.78 ± 0.62 +102 +(ns) +101 +400 +500 +600 +700 +Frequency [MHz]SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY +9 +Figure 3. An example dynamic (top) and secondary (bottom) spec- +trum from PSR J0630-2834 on MJD 58987 centered at 425 MHz. +There is a clear asymmetry in the secondary spectrum, with the right +arm being the dominant feature. This is likely the result of refraction +along the line of sight. The green line represents the arc curvature +fit and the arc curvature measurement quoted is unscaled. Scaled +uncertainties of the arc curvature can be found in Table 1. +This pulsar is well known for having the uncommon fea- +ture of multiple scintillation arcs, implying multiple scatter- +ing screens along its LOS (Hill et al. 2003; Stinebring et al. +2019). In the literature six distinct sets of arcs have been +found over a ∼34 year span of observations (McKee et al. +2022). In three of the four epochs in we which observed +this pulsar, we observed multiple arcs, an example of which +can be seen in Figure 4. After scaling our measurements +to 1400 MHz and using the convention from McKee et al. +(2022), we can conclude that we detected arcs E, C, and B +on MJDs 58987 and 58991 and arcs D and C on MJD 59115, +with arc C being the only detectable arc on MJD 59497. All +multiple-arc detections were made only in the observations +using uGMRT’s band 4, which was centered at 650 MHz. +The fact that the two epochs closest to each other in our sur- +vey (58987 and 58991) both detected the same sets of arcs +may hint at a timescale over which certain screens have a +larger influence over the pulsar signal propagation. +An examination of the power in each of the arms show +notable levels of asymmetry along the delay axis, and conse- +quently a notable amount of refraction, in all detectable arms +and across all frequencies in the first two epochs, with the +right arm having more power and extending further out on the +delay axis. This asymmetry clearly decreases over the course +of our observations across all frequencies until our final ob- +servation, where the arcs have approximately even levels of +power or the left arc starts to dominate in the asymmetry. +This trend is generally supported by the measured scintilla- +tion drift rates as well, especially for data taken at band 4 +(650 MHz), i.e., the same band where the multiple arcs were +visible, as measured drifts are generally positive during the +first three epochs and then considerably negative during the +final epoch. +Perhaps the most interesting finding from our observations +of this pulsar is the discovery of a strong correlation between +the measured arc curvatures and the arc asymmetry index, +which is a metric that describes the relative power between +the left and right arcs and is found by comparing the average +power along each arc via +A = PR(fν) − PL(fν) +PR(fν) + PL(fν) +, +(5) +with a larger index magnitude indicating greater asymmetry. +We believe this phenomenon has never before been reported +and is therefore worth further examination in future observa- +tions. As briefly mentioned earlier, asymmetry in arcs has +long been attributed to either the refraction of pulsar emis- +sion at the scattering screen or as the result of an asymmetric +distribution of the material within the screen (Cordes et al. +2006), while arc curvature is known to indicate the distance +between a given scattering screen and the observer (Stine- +bring et al. 2001). The correlation between the two suggests +further study of this effect may result in a better understand- +ing of how screen asymmetry and/or refraction affects pulsar +emission depending on the scattering screen’s proximity to +the pulsar. +An example dynamic and secondary spectrum pair is +shown in Figure 5, with its corresponding normalized sec- +ondary spectrum power profile, which is used to determine +the asymmetry index, shown in Figure 6, while the scatter +plot showing the relation between measured arc curvature +and arc asymmetry index across all measurements taken in +the 650 MHz band is shown in Figure 7. Of particular note in +Figure 7 are the three distinct clumps, which we believe are +the result of our observations being dominated by a differ- +ent scattering screen at each epoch (two of our observations +were taken four days apart, and so are dominated by the same +screen). It is likely that this pulsar’s at least six known scat- +tering screens are the main reason why we were able to see +this correlation in our data in the first place, as individual +scattering screens likely do not vary enough in distance over +time for this trend to become apparent. Indeed, the limited +number of pulsars with multiple known screens is probably +the main reason why this trend has not been reported in ear- +lier studies. +4.6. J1509+5531 +In the observations of this pulsar in the 650 MHz band, +all secondary spectra featured patchy rather than continuous + +PSR J0630-2834 MJD 58987 +450 +Flux Density (Arbitrary Units) +0.15 +[MHz] +440 +0.10 +430 +Frequency [ +0.05 +420 +0.00 +410 +-0.05 +400 +0 +10 +20 +30 +40 +Time [Min] +10 +nL =1.534 s3 +40 +nR =1.623 s3 +Log Power (dB) +5 +Delay [μs] +20 +0 +0 +-5 +20 +-40 +-10 +-40 +-20 +0 +20 +40 +Fringe Frequency [10-3 Hz]10 +J. E. TURNER ET AL. +(a) Scintillation arcs without overlaid fits +(b) Scintillation arcs with overlaid fits +Figure 4. Secondary spectrum of PSR J1136+1551 at 650 MHz on MJD 58987 showing the detection of three distinct scintillation arcs. +Figure 5. Dynamic (top) and secondary (bottom) spectra of PSR +J1136+1551 centered at 650 MHz on MJD 58987. The top half of +the secondary spectrum shows the overlaid arc fits in green. Scaled +uncertainties of the arc curvature can be found in Table 1. +arcs, particularly in the left arm. This patchiness indicates +a detection of this pulsar’s arclets, which result from sub- +structures in the ISM thought to arise from scattering inter- +ference between an inhomogeniously scattered distribution +of material and some distinct offset region (Walker & Stine- +bring 2005; Cordes et al. 2006). In the particular case of this +Figure 6. Normalized secondary spectrum power profile of PSR +J1136+1551 centered at 650 MHz on MJD 58987. The vertical +dashed lines indicate where the arcs fall on the normalized delay +axis. +pulsar these substructures are roughly AU in scale. Unique +to these arclets is their distinctly flat nature, which has been +attributed to its exceptionally high transverse velocity of over +960 km s−1 (Manchester et al. 2005). Interestingly, the arc +curvatures measured in the last two epochs (MJDs 59115 and +59497) are a factor of two to three times smaller than the first + +PSR J1136+1551MJD 5898T +14000 +70 +12000 +60 +10000 +50 +8000 +({_w) f +40 +6000 +30 +4000 +20 +2000 +10 +0 +-20 +0 +20 +-40 +40 +ft (mHz)PSR J1136+1551MJD5898T +14000 +70 +12000 +60 +10000 +50 +8000 +({_w) f +40 +6000 +30 +4000 +20 +2000 +10 +-20 +0 +20 +-40 +40 +ft (mHz)PSR J1136+1551MJD 5898T +750 +( )s +0.35 +0.30 +Frequency [MHz] +700 +0.25 +650 +0.15 +0.10 +600 +0.05 +550 +0.00 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Time [Min] +10 +nL =0.056 s3 +60 +nR =0.060 s3 +5 + Power (dB) +40 +Delay [μs] +20 +0 +0 +Log +20 +-10 +-40 +-20 +0 +20 +40 +Fringe Frequency [10-3 Hz]18 +16 +12 +10 +8 +-2 +0 +-1 +2 +Normalized ftSIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY +11 +Figure 7. Scatter plot showing measured arc curvatures and the +corresponding asymmetry indices for all measurements of PSR +J1136+1551 taken with Band 4. All arc curvatures have been scaled +to their corresponding 1 GHz equivalent. The three distinct clumps +are the result of the observations being dominated by three different +scattering screens. +two epochs (MJDs 59064 and 58987), possibly indicating a +detection of multiple scattering screens along the LOS to this +pulsar. This result augments the results of Sprenger et al. +(2022), who also found significant variability along the LOS +to this pulsar during the same period of time. An example +observation from the earlier two epochs is shown in Figure +8, while an example from the later two epochs is shown in +Figure 9. +4.7. J1932+1059 +Due to having the lowest DM in our survey, this pulsar +showed the least variation in arc curvature from epoch to +epoch across all frequencies. Its close proximity to Earth also +resulted in wide scintles in frequency, leading to high scintle +resolution and, consequently, very bright, narrow, and well +defined arcs. The sharpness of these arcs may also indicate +scattering that is highly anisotropic along the LOS (Walker +et al. 2004; Cordes et al. 2006), as well as originating from +a discrete, localized source (Stinebring et al. 2001). Overall +this was our most consistent pulsar in all aspects of scintilla- +tion. +This consistency lines up with its other astrophysical pa- +rameters, as its dispersion measure of 3.18 pc cm−3 (Large +et al. 1968; Manchester et al. 2005) was the lowest in our +survey and its transverse velocity of 152 km s−1 (Bilous +et al. 2016; Manchester et al. 2005) was the second low- +est. While its transverse velocity is a bit larger than PSR +Figure 8. Dynamic (top) and secondary (bottom) spectra of PSR +J1509+5531 centered at 625 MHz on MJD 58987. The top half of +the secondary spectrum shows the overlaid arc fits in green. Scaled +uncertainties of the arc curvature can be found in Table 1. During +this period of observations, visible arcs were considerably narrower +than later observations. +Figure 9. Dynamic (top) and secondary (bottom) spectra of PSR +J1509+5531 centered at 650 MHz on MJD 59115. The top half of +the secondary spectrum shows the overlaid arc fits in green. Scaled +uncertainties of the arc curvature can be found in Table 1. During +this period of observations, visible arcs were considerably wider +than later observations. +J0630−2834 and their distances are almost equivalent, PSR +J0630−2834 has a dispersion measure 10 times higher than +PSR J1932+1059 (Large et al. 1968, 1969; Manchester et al. + +J1136+1551 High Frequencies PL =0.92; PR =0.88 +Referenced at 1 GHz +nL +0.12 +NR +0.10 +0.08 +0.06 +0.04 +0.02 +0.00 +-0.02 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +n (s3)PSR J1509+5531MJD 58987 +650 +Flux Density (Arbitrary Units) +0.125 +[MHz] +640 +0.100 +0.075 +630 +Frequency +0.050 +620 +0.025 +610 +0.000 +-0.025 +600 +0 +10 +20 +30 +40 +Time [Min] +10 +nL =0.932 s3 +40 +NR =0.518 s3 +5 +Log Power (dB) +20 +Delay [μs] +0 +0 +20 +-5 +一 +-40 +-10 +-40 +-20 +0 +20 +40 +Fringe Frequency [10-3 Hz]PSR J1509+5531 MJD 59115 +750 +(n ) s +0.35 +0.30 +Frequency [MHz] +700 +0.25 +0.20 +650 +0.15 +0.10 +600 +0.05 +0.00 +550 +0 +5 +10 +15 +20 +25 +30 +35 +Time [Min] +10 +60 +nL =0.224 s3 +NR =0.187 s3 +40 +5 + Power (dB) +Delay [μs] +20 +0 +0 +Logl +-5 +-20 +-10 +-40 +-20 +0 +20 +40 +Fringe Frequency [10-3 Hz]12 +J. E. TURNER ET AL. +2005). This means that a much denser ISM was sampled +in PSR J0630−2834 than in PSR J1932+1059, meaning that +PSR J1932+1059 had decisively the least amount of ISM +sampled over our survey, making it the least likely to ex- +perience large scintillation-related variations. An example +observation is shown in Figure 10. +Figure 10. Dynamic (top) and secondary (bottom) spectra of PSR +J1932+1059 centered at 725 MHz on MJD 58987. The top half of +the secondary spectrum shows the overlaid arc fits in green. Scaled +uncertainties of the arc curvature can be found in Table 1. +5. CONCLUSIONS & FUTURE WORK +We performed simultaneous dual-frequency observations +of six bright canonical pulsars using the uGMRT. We ex- +tracted scintillation arc, bandwidth, and drift rate measure- +ments for each of these pulsars to examine a variety of sci- +ence. We examined how arc curvature scaled with frequency +and found our observations to be consistent with the index +predicted by theory, while at the same time using a more as- +tronomically ideal setup to perform these measurements. We +also measured scattering delay scaling indices for five of our +six pulsars and found indices consistent with or shallower +than what is expected from a Kolmogorov medium, agreeing +with previous literature. Finally, we find an interesting and +strong correlation between arc curvature and arc asymmetry +in PSR J1136+1551, demonstrating a potential connection +between screen asymmetry and/or refraction and scattering +screen location along the LOS, and the which we intend to +follow up with additional observations. +This study demonstrates the value of array-based tele- +scopes such as uGMRT to the pulsar astronomy community, +as well as the strengths of simultaneous multiband studies of +pulsars and the wide variety of science that can be done with +such an approach. This also shows strong promise for the +future observations using ultrawideband (UWB) receivers, +which are coming online at instruments such as the Green +Bank Telescope. +We thank the staff at the uGMRT who have made these +observations possible. The uGMRT is run by the National +Centre for Radio Astrophysics of the Tata Institute of Funda- +mental Research. We gratefully acknowledge support of this +effort from the NSF Physics Frontiers Center grants 1430284 +and 2020265 to NANOGrav. Some of the data processing +in this work utilized the resources of the Bowser computing +cluster at West Virginia University. +Software: +SCINTOOLS Reardon et al. (2020), PYPULSE +Lam (2017), SCIPY Virtanen et al. (2020), NUMPY van der +Walt et al. (2011), and MATPLOTLIB Hunter (2007). +REFERENCES +Bansal, K., Taylor, G. B., Stovall, K., & Dowell, J. 2019, The +Astrophysical Journal, 875, 146, doi: 10.3847/1538-4357/ab0d8f +Bhat, N. D. R., Cordes, J. M., Camilo, F., Nice, D. J., & Lorimer, +D. R. 2004, The Astrophysical Journal, 605, 759, +doi: 10.1086/382680 +Bilous, A. V., Kondratiev, V. I., Kramer, M., et al. 2016, A&A, +591, A134, doi: 10.1051/0004-6361/201527702 +Cordes, J. M., & Lazio, T. J. W. 2001, The Astrophysical Journal, +549, 997, doi: 10.1086/319442 +Cordes, J. M., & Rickett, B. 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R. 2005, MNRAS, 362, 1279, +doi: 10.1111/j.1365-2966.2005.09396.x + diff --git a/9tE4T4oBgHgl3EQf3g1J/content/tmp_files/load_file.txt b/9tE4T4oBgHgl3EQf3g1J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..915e2b0e288e9156a7005164b9b3688a51205d5a --- /dev/null +++ b/9tE4T4oBgHgl3EQf3g1J/content/tmp_files/load_file.txt @@ -0,0 +1,1269 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf,len=1268 +page_content='DRAFT VERSION JANUARY 16, 2023 Typeset using LATEX twocolumn style in AASTeX63 A Simultaneous Dual-Frequency Scintillation Arc Survey of Six Bright Canonical Pulsars Using the Upgraded Giant Metrewave Radio Telescope JACOB E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER,1, 2 BHAL CHANDRA JOSHI,3 MAURA A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Tata Institute of Fundamental Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Post Bag 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Ganeshkhind,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Pune - 411007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' India 4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Oberlin College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Oberlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' OH 44074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' USA ABSTRACT We use the Upgraded Giant Metrewave Radio Telescope to measure scintillation arc properties in six bright canonical pulsars with simultaneous dual frequency coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' These observations at frequencies from 300 to 750 MHz allowed for detailed analysis of arc evolution across frequency and epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We perform more robust determinations of arc curvature and scattering delay frequency-dependence than allowed by single-frequency- band-per-epoch measurements, which we find to agree with theory and previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We report the dis- covery of a strong correlation between arc asymmetry and arc curvature, potentially indicating a link between scattering screen distance and refraction strength or the effect of asymmetric distribution of scattering material on a scattering screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The inclusion of a 155 minute observation allowed us to resolve the scale of scintilla- tion variations on short timescales, which we find to be directly tied to the amount of ISM sampled over the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Some of our pulsars showed either consistent or emerging asymmetries in arc curvature, indicating instances of refraction across their lines of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The presence of significant features in various pulsars, such as multiple scintillation arcs in PSR J1136+1551 and flat arclets in PSR J1509+5531, that have been found in previous works, were also sufficiently detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Possible evidence for a timescale over which a given scatter- ing screen dominates signal propagation was found by tracking visible scintillation arcs in each epoch in PSR J1136+1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The interesting pulsar science accomplished with this upgraded telescope shows strong promise for important future work in pulsar astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Keywords: methods: data analysis – stars: pulsars – ISM: general – ISM: structure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' INTRODUCTION The scintillation of pulsar emission occurs as the result of the propagation of this emission through non-uniform distri- butions of free electrons in the ionized interstellar medium (IISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This interaction results in frequency-dependent and time-evolving variations in the flux density of the pulsar sig- nal as measured at a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' When these variations are examined across observing frequency and time in so-called dynamic spectra, representations of the change in the pulsar signal’s intensity across frequency and time, for a given ob- servation, they can provide valuable insight into the structure of these electron density variations along our line of sight (LOS) to a given pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' For observations where the scin- tles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' bright patches in a dynamic spectrum resulting from constructive interference between photons as the result of propagation through the free electrons in the ISM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' are fully resolved in frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' have sufficient coverage in time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' and exhibit structures over the course of observations including,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' but not limited to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' non-zero slopes across frequency and time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' known as scintillation drift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' as well as “crisscrossing” scin- tillation patterns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' then additional information can be gained about the ISM structure along the LOS by examining the parabolic arcs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' known as scintillation arcs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' that can emerge by examining the Fourier transform of the dynamic spec- trum (Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Some current hypotheses on the physical origins of these arcs postulate that they originate from compressed plasma along the boundaries of 50−100 pc size bubbles in the ISM (Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Traditional measurements of scintillation arcs have typ- ically been limited to either one observing band over all epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=', Trang & Rickett (2007)), or alternated between observing bands from epoch to epoch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=', Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' While generally sufficient for most analyses, this band limit results in a bottleneck for examining the evo- lution of various frequency-dependent effects over shorter timescales, including scintillation arc curvature, structures within individual arcs, and asymmetries in both arc bright- ness and power as a function of differential time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' By making use of the subarray capabilities of the Upgraded Gi- ant Metrewave Radio Telescope (uGMRT), we can effec- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05306v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='HE] 12 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' tively create an ultra wideband receiver by setting multiple groups of dishes to simultaneously observe at different fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This work is primarily data-focused and aims to highlight the results of some multi-frequency analyses per- formed on a small survey of six strong canonical pulsars us- ing this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' In Section 2 we discuss the data taken as part of our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Section 3 describes the analyses per- formed and the physical parameters extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Section 4 de- tails the results of these analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Finally, Section 5 summa- rizes our results and discusses possible next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' DATA Our data were taken across across eight epochs span- ning MJD 58987−59497 using 22 dishes split into subarrays for simultaneous multi-frequency observations at uGMRT’s Band 3 and Band 4, centered at 400 MHz and 650 MHz, respectively, each with 200 MHz of bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This simul- taneous low-frequency accessibility is comparable to instru- ments like CHIME that can observe continuously between 400-800 MHz and better than instruments such as the Green Bank Telescope, which, while having a wide range of low frequency coverage, can only observe below one GHz with at most 240 MHz of bandwidth at frequencies close to one GHz and less than 200 MHz of bandwidth in lower frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Observations were also made at Band 5 centered at 1360 MHz, although due to a combination of RFI and low S/N no scintles were detectable in the dynamic spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The observing bands were split into 4096 49 kHz wide frequency channels and observed with 10 second subintegra- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' These data were flux calibrated using observations of either 3C147 or 3C286 taken at the beginning of every ob- serving session and every pulsar was phase calibrated with a nearby source for five minutes once every 40 minutes of observing time on the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Two to three pulsars were ob- served at each epoch for 40 minutes each, except for MJD 59497, where three pulsars were observed for 155 minutes each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' As a result of the phase calibration, each of those obser- vations were comprised of three 40 minute sub-observations plus an additional 20 minute sub-observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' ANALYSIS All observations were processed to extract their dynamic spectra by calculating the intensity, S, of the pulsar’s signal at each observing frequency, ν, and time, t, via S(ν, t) = Pon(ν, t) − Poff(ν, t) Pbandpass(ν, t) , (1) where Pbandpass is the total power of the observation as a function of observing frequency and time, and Pon and Poff are the power in all on- and off-pulse components, respec- tively, as a function of frequency and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Each dynamic spectrum was then broken up into four 50 MHz spectra to allow for more in-depth frequency-dependent analyses and manually zapped by examining dynamic spectra data arrays and removing pixels that were brighter than the brightest scintle maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Secondary spectra were then created by taking the absolute square of the Fourier transform (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=', the power spectrum) of the corresponding dynamic spectrum and converting it to units of dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The primary (brightest) scintilla- tion arcs on the positive and negative side of each secondary spectrum’s fringe frequency axis were then found via sepa- rate fν = ηf 2 t fits, where fν is the differential time delay, η is the arc curvature, and ft is the fringe frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We also determined scintillation parameters by using the python package Pypulse (Lam 2017) to create the 2D au- tocorrelation functions (ACFs) of each dynamic spectrum and fit 2D Gaussians to these ACFs to determine their scin- tillation bandwidth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' ∆νd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' defined as the half-width at half- maximum (HWHM) along the frequency axis of the ACF at lag 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' scintillation timescale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' ∆td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' defined as the half-width at e−1 along the time axis at ACF lag 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' and scintillation drift rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' dν/dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' defined as the rotation of the 2D Gaussian fit to the 2D ACF in the plane of the frequency and time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' For our scattering delay scaling index analysis, our measured scintillation bandwidths were converted to scattering delays using 2π∆νdτ = C1, (2) where C1 is a dimensionless quantity between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='6 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='5 that depends on the spectrum of the electron density fluctuations and geometry of the medium (Cordes & Rickett 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' For this work we use C1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' RESULTS & DISCUSSION Measured arc curvatures and their corresponding dynamic spectrum scintillation drift rates can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' All curvatures and their uncertainties have been scaled to their corresponding value at 1 GHz assuming a ν−2 frequency de- pendence (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Here errors on the arc curva- tures represent fitting errors from the linear least squares fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Some pulsars on MJD 59497 have multiple curvature mea- surements at a given frequency, which is the result of this epoch being 155 minutes instead of the 40 minutes of the other observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' As a result, a new η was measured after every 40 minutes since these observations were broken up into 40 minute chunks separated by five minute phase cali- brations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' On days where measurements are given for the left or right arm only, arcs on the other side of the fringe fre- quency axis may have been present, but were unmeasureable due to either having insufficient extension along the differ- ential delay axis, insufficient flux relative to the background noise, being too diffuse, being too close to central spike in flux that commonly occurs around a fringe frequency of 0 mHz, or some combination of these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Pulsar Scintillation Arc Curvatures and Drift Rates Pulsar MJD Frequency ηL σηL ηR σηR dν/dt σdν/dt (MHz) (s3) (s3) (s3) (s3) (MHz/min) (MHz/min) J0630–2834 58987 325 — — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='2071 0.' metadata={'source': 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+page_content='0066 Table 1 continued 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Table 1 (continued) Pulsar MJD Frequency ηL σηL ηR σηR dν/dt σdν/dt (MHz) (s3) (s3) (s3) (s3) (MHz/min) (MHz/min) J1136+1551 59497 475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='00005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='0001 NOTE—Scintillation arc measurements and drift rates in the left and right primary arms of each epoch at all frequencies where measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' ηL and ηR are the the arc curvature measurements for the left and right arms, respectively, and dν/dt is the measured scintillation drift rate, with the matching σ’s representing the corre- sponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' All curvatures and their errors have been scaled to 1 GHz and errors on curvature here are fit uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Some pulsars on MJD 59497 have multiple curvature measurements at a given frequency, due to this epoch being 155 minutes instead of the 40 minutes of the other observations, and so a new η was measured after every 40 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scintillation Arc Curvature Scaling Behavior As mentioned earlier, Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2003) demonstrated through both theoretical and observational means that the arc curvature η should follow a ν−2 dependence, implying that scattering is dominated by one or several thin screens along the LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' While over 2 GHz of bandwidth was used in those observations (10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='5 MHz of bandwidth centered at 430 MHz and either 50 or 100 MHz of bandwidth centered at 1175 MHz, 1400 MHz, and 2250 MHz), the frequency cov- erage was discontinuous and all η measurements used in their corresponding fits were from different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Generally the latter point should not be an issue as long as the observations were taken within a period shorter than the pulsar’s refrac- tive timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Indeed, for the data used in their fits, their measured arc curvatures at a given frequency did not vary significantly on day or week timescales, making them suit- able for this type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' However, the ideal situation would be to obtain many measurements at many frequencies during the same observation, preferably at the same time for optimal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' With our high resolution and sufficient observing time, we have the ability to make up to eight con- current arc measurements over 450 MHz of bandwidth at low frequency and can consequentially provide a more definitive examination of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Following the methodology of Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2003), for a scal- ing index α, we performed a weighted linear least-squares fit of the form log10 η = α log10 ν + β (3) on the unscaled curvatures for each pulsar at each MJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Ex- ample fits can be seen in Figure 1, with all measured in- 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' dices listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We find that, overall, our scaling indices are consistent with a theoretical index of −2, which assumes thin screen scattering (Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2001), with PSRs J1136+1551 and J1932+1059 being especially consis- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This effect can also be seen in Table 1, where arc cur- vatures at all frequencies from a given pulsar at the same MJD are generally in strong agreement after being scaled to 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Interestingly, a weighted average of all curvature fits shows that our left arm fits are overall more consistent with an index of −2 than our right arms, with a weighted av- erage of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='03 across all left arm fits compared with −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='02 across all right arm fits, indicating that refrac- tion may play a role in how closely arc curvature scales as expected with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scattering Delay Scaling Behavior Our wide frequency coverage also allowed us to examine the scaling index of scattering delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Under the assumption that ISM fluctuations follow behaviors consistent with a Kol- mogorov medium and that scattering can be modeled as the result of interactions of pulsar emission with an infinite, thin, scattering screen during its propagation, we should expect that scattering delays scale with frequency as τd ∝ ν−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='4 (Romani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Cordes & Rickett 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Previous studies examining the scattering indices of various pulsars have done so using a number of methods, including simul- taneous multi-frequency measurements (Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2019), splitting up measurements from a single frequency band into multiple subbands (Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2021), and using measurements from many epochs taken at two observing bands non-simultaneously (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Since more measurements and more fre- quency coverage in a single epoch is ideal, the method used in Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2004) and Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2019) is the most pre- ferred of the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The method in this paper utilizes a com- bination of this approach and the subband approach to max- imize the number of delay measurements per epoch, which can be done thanks to our high frequency resolution and sen- sitivity in both observing bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Similar to Equation 3 used to determine the arc curva- ture scaling index, our scattering delay scaling indices ξ at each epoch were determined by performing a weighted lin- ear least-squares fit of the form log10 τd = ξ log10 ν + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (4) Example fits can be seen in Figure 2, with all measured in- dices listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We find that half of our measured indices are consistent with a Kolmogorov medium, while the other half are consistent with a shallower medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This be- havior agrees well with previous studies, as both Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2004) and Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2019) found indices either consis- tent with a Kolmogorov medium or shallower than a Kol- mogorov medium, while Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2016) and Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2021) only found indices that were shallower than a Kol- mogorov medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Many explanations have been given for why shallower- than-Kolmogorov medium behavior has been observed so frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Physical arguments have called into question the validity of the simple infinite, thin screen model, demonstrat- ing that shallower scaling indices are more consistent with finite, thin screens (Rickett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This is expected to be much more common among low DM pulsars (Cordes & Lazio 2001), which agrees with our results, as all of the pul- sars have dispersion measures below 40 pc cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Shallower indices have also been attributed to the existence of multi- ple finite screens along the LOS (Lewandowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This hypothesis agrees well with our measured indices for PSR B1133+16, as its indices are consistently shallower than that of a Kolmogorov medium and it is also known to have at least six distinct scattering screens (McKee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Quality-of-data arguments have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2021) suggested their shallower indices may be at least partially attributable to an imbalance of lower fre- quency data to higher frequency data for their multiple epoch approach as well as a lack of sufficient frequency resolution in their lower frequency band in some epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' However, nei- ther of these issues should affect our results, as our observa- tions have a consistently even balance of low and high fre- quency measurements at all epochs and all of our measure- ments are well-resolved in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 155 Minute Observation The inclusion of a 155 minute observation in our survey on MJD 59497 allowed for an analysis of short-term arc cur- vature variation in some pulsars, as observations had to be paused every 40 minutes for a five minute phase calibration, resulting in multiple 40 minute sub-observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' For pulsars with at least two measurements in a given scintillation arc at a given frequency, we examined overall variation in that arc at that frequency by looking at the percent difference between a given curvature measurement and the weighted average cur- vature for that arm and frequency over the entire epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' For PSR J1136+1551, all observing frequencies centered at or below 475 MHz had three measurements in each pri- mary arm (the brightest arm, overwhelmingly often the arm with the lowest curvature) at each frequency, with the accu- mulation of all percent differences yielding a bimodal distri- bution with peaks around percent differences of 2% and 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The largest percent difference away from a weighted mean was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='2% and the smallest was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='14±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='91%, although the majority of all percent differences was below 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' All of this strongly indicates the ISM underwent very little change along the LOS to this pulsar over the course of a given ob- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This result is supported by this pulsar’s incredibly low dispersion measure, meaning it does not sample a size- SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY 7 (a) α fit for PSR J1932+1059 on MJD 58997 (b) α fit for PSR J1136+1551 on MJD 58997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The inclusion of multiple points at certain frequencies is the result of this epoch containing a 155 minute observation instead of the 40 minutes of the other observations, and so a new η was measured after every 40 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Example fits for the arc curvature scaling index Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Fitted Pulsar Scintillation Arc Curvature Scaling Indices Pulsar MJD Scaling Index Left Arc Scaling Index Error Left Arc Nη Scaling Index Right Arc Scaling Index Error Right Arc Nη J0630–2834 58987 — — — –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='31 8 J1136+1551 58987 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='11 7 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 7 J1136+1551 58991 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 8 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='08 8 J1136+1551 59115 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='13 8 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='15 8 J1136+1551 59497 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 15 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='10 15 J1509+5531 58987 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='83 3 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='55 3 J1509+5531 59064 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='26 4 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='22 4 J1509+5531 59115 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='21 4 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='18 4 J1509+5531 59497 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='87 9 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='21 9 J1645–0317 59074 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='26 4 — — — J1932+1059 58997 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 8 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='09 8 J1932+1059 59062 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='07 8 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='07 8 J1932+1059 59497 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='09 8 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='04 8 J2048–1616 59062 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='07 6 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 6 NOTE—Fitted arc curvature scaling indices for both left and right primary arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Nη indicates the number of arc curvature measurements used in each fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Measurements on MJD 59497 may have Nη > 8 due to this epoch being 155 minutes rather than the 40 minutes of the other observations, and so a new η was measured after every 40 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Arc curvature measurements used in these fits were left unscaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' able portion of the ISM along its LOS relative to many pul- sars that are observed (Bilous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Pilkington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' For PSR J1509+5531, all observing frequencies centered at or above 575 MHz had at least two measurements in each arm at each frequency, with the accumulation of all per- cent differences resulting in a one-sided distribution peaked J1932+1059MJD 5899T 4 × 10-1 nL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaling Index= -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 nR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaling Index = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='09 3 × 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2 × 10-1 n 10-1 6 × 10-2 400 500 600 700 Frequency [MHz]J1136+1551MJD 5949T nL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaling Index= -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 6 × 10-2 nr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaling Index = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='10 4 × 10-2 3 × 10-2 n 2 × 10-2 10-1 400 500 600 700 Frequency [MHz]8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (a) Scattering delay scaling index fit for PSR J1932+1059 on MJD 59497 (b) Scattering delay scaling index fit for PSR J1136+1551 on MJD 58991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Example fits for the scattering delay scaling index Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Fitted Pulsar Scattering Delay Scaling Indices Pulsar MJD Scaling Index Index Error Nτd J0630–2834 58987 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='43 8 J1136+1551 58987 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='71 6 J1136+1551 58991 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='62 7 J1136+1551 59115 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 6 J1136+1551 59497 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='57 13 J1645–0317 59074 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='75 4 J1932+1059 58997 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='31 7 J1932+1059 59062 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='31 6 J1932+1059 59497 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='39 6 J2048–1616 59062 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='39 6 NOTE—Fitted scattering delay scaling indices, with a minimum of four delay measurements (Nτd) required in a given epoch to obtain a scaling index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Errors are the parameter uncertainties from parameter fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Half of our measured indices were con- sistent with a Kolmogorov medium, while the other half were consistent with a shallower medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Measurements on MJD 59497 may have Nτd > 8 due to this epoch being 155 minutes rather than the 40 minutes of the other observations, and so a new η was measured after every 40 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' around 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The smallest percent difference away from a weighted mean was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='1%, while the largest was 36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='1%, although the next largest after that was only 22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='1%, meaning this maximum was an extreme outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The majority of all percent differences was below 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' As with the previous pulsar, this also strongly indicates the ISM underwent very little change along the LOS to this pul- sar over the course of a given observation, a result again supported by this pulsar’s fairly low dispersion measure (Huguenin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The fact that this pulsar shows higher variation of this observation com- pared to PSR J1136+1551 is likely due to PSR J1509+5531 having a dispersion measure four times higher and a trans- verse velocity 45% larger (Bilous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Huguenin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Pilkington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Stovall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2015), so a significantly larger fraction of the ISM was sampled during its observation, increasing the likelihood of larger scintillation-based variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The next few subsections will be dedicated to highlighting the features of a few pulsars in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' J0630-2834 In the one epoch for which we were able to resolve a scin- tillation arc, only the right arm was resolvable across all fre- quencies, with its relative brightness relative to the left side of the fringe frequency axis consistently decreasing as fre- quency increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' An example of this asymmetry can be seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='This strong asymmetry is known to be the result of refraction leading to scintillation drifting in the dynamic spectra (Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Interestingly, despite our asym- metry appearing to decrease with frequency, the magnitude of our measured scintillation drift rates seem to mildly favor an increase with frequency, whereas one would expect an in- crease in scintillation drift to coincide with an increase in the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' J1136+1551 J1932+1059 MJD 59497 Scaling Index= -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='39 102 (ns) 101 400 500 600 700 Frequency [MHz]J1136+1551MJD 58991 Scaling Index= -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='62 102 (ns) 101 400 500 600 700 Frequency [MHz]SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' An example dynamic (top) and secondary (bottom) spec- trum from PSR J0630-2834 on MJD 58987 centered at 425 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' There is a clear asymmetry in the secondary spectrum, with the right arm being the dominant feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This is likely the result of refraction along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The green line represents the arc curvature fit and the arc curvature measurement quoted is unscaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaled uncertainties of the arc curvature can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This pulsar is well known for having the uncommon fea- ture of multiple scintillation arcs, implying multiple scatter- ing screens along its LOS (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' In the literature six distinct sets of arcs have been found over a ∼34 year span of observations (McKee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' In three of the four epochs in we which observed this pulsar, we observed multiple arcs, an example of which can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' After scaling our measurements to 1400 MHz and using the convention from McKee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2022), we can conclude that we detected arcs E, C, and B on MJDs 58987 and 58991 and arcs D and C on MJD 59115, with arc C being the only detectable arc on MJD 59497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' All multiple-arc detections were made only in the observations using uGMRT’s band 4, which was centered at 650 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The fact that the two epochs closest to each other in our sur- vey (58987 and 58991) both detected the same sets of arcs may hint at a timescale over which certain screens have a larger influence over the pulsar signal propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' An examination of the power in each of the arms show notable levels of asymmetry along the delay axis, and conse- quently a notable amount of refraction, in all detectable arms and across all frequencies in the first two epochs, with the right arm having more power and extending further out on the delay axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This asymmetry clearly decreases over the course of our observations across all frequencies until our final ob- servation, where the arcs have approximately even levels of power or the left arc starts to dominate in the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This trend is generally supported by the measured scintilla- tion drift rates as well, especially for data taken at band 4 (650 MHz), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=', the same band where the multiple arcs were visible, as measured drifts are generally positive during the first three epochs and then considerably negative during the final epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Perhaps the most interesting finding from our observations of this pulsar is the discovery of a strong correlation between the measured arc curvatures and the arc asymmetry index, which is a metric that describes the relative power between the left and right arcs and is found by comparing the average power along each arc via A = PR(fν) − PL(fν) PR(fν) + PL(fν) , (5) with a larger index magnitude indicating greater asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We believe this phenomenon has never before been reported and is therefore worth further examination in future observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' As briefly mentioned earlier, asymmetry in arcs has long been attributed to either the refraction of pulsar emis- sion at the scattering screen or as the result of an asymmetric distribution of the material within the screen (Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2006), while arc curvature is known to indicate the distance between a given scattering screen and the observer (Stine- bring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The correlation between the two suggests further study of this effect may result in a better understand- ing of how screen asymmetry and/or refraction affects pulsar emission depending on the scattering screen’s proximity to the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' An example dynamic and secondary spectrum pair is shown in Figure 5, with its corresponding normalized sec- ondary spectrum power profile, which is used to determine the asymmetry index, shown in Figure 6, while the scatter plot showing the relation between measured arc curvature and arc asymmetry index across all measurements taken in the 650 MHz band is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Of particular note in Figure 7 are the three distinct clumps, which we believe are the result of our observations being dominated by a differ- ent scattering screen at each epoch (two of our observations were taken four days apart, and so are dominated by the same screen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' It is likely that this pulsar’s at least six known scat- tering screens are the main reason why we were able to see this correlation in our data in the first place, as individual scattering screens likely do not vary enough in distance over time for this trend to become apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Indeed, the limited number of pulsars with multiple known screens is probably the main reason why this trend has not been reported in ear- lier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' J1509+5531 In the observations of this pulsar in the 650 MHz band, all secondary spectra featured patchy rather than continuous PSR J0630-2834 MJD 58987 450 Flux Density (Arbitrary Units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='15 [MHz] 440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='10 430 Frequency [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='00 410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 400 0 10 20 30 40 Time [Min] 10 nL =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='534 s3 40 nR =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='623 s3 Log Power (dB) 5 Delay [μs] 20 0 0 5 20 40 10 40 20 0 20 40 Fringe Frequency [10-3 Hz]10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (a) Scintillation arcs without overlaid fits (b) Scintillation arcs with overlaid fits Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Secondary spectrum of PSR J1136+1551 at 650 MHz on MJD 58987 showing the detection of three distinct scintillation arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Dynamic (top) and secondary (bottom) spectra of PSR J1136+1551 centered at 650 MHz on MJD 58987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The top half of the secondary spectrum shows the overlaid arc fits in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaled uncertainties of the arc curvature can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' arcs, particularly in the left arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This patchiness indicates a detection of this pulsar’s arclets, which result from sub- structures in the ISM thought to arise from scattering inter- ference between an inhomogeniously scattered distribution of material and some distinct offset region (Walker & Stine- bring 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' In the particular case of this Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Normalized secondary spectrum power profile of PSR J1136+1551 centered at 650 MHz on MJD 58987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The vertical dashed lines indicate where the arcs fall on the normalized delay axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' pulsar these substructures are roughly AU in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Unique to these arclets is their distinctly flat nature, which has been attributed to its exceptionally high transverse velocity of over 960 km s−1 (Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Interestingly, the arc curvatures measured in the last two epochs (MJDs 59115 and 59497) are a factor of two to three times smaller than the first PSR J1136+1551MJD 5898T 14000 70 12000 60 10000 50 8000 ({_w) f 40 6000 30 4000 20 2000 10 0 20 0 20 40 40 ft (mHz)PSR J1136+1551MJD5898T 14000 70 12000 60 10000 50 8000 ({_w) f 40 6000 30 4000 20 2000 10 20 0 20 40 40 ft (mHz)PSR J1136+1551MJD 5898T 750 ( )s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='30 Frequency [MHz] 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='25 650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='10 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='00 0 5 10 15 20 25 30 35 40 Time [Min] 10 nL =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='056 s3 60 nR =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='060 s3 5 Power (dB) 40 Delay [μs] 20 0 0 Log 20 10 40 20 0 20 40 Fringe Frequency [10-3 Hz]18 16 12 10 8 2 0 1 2 Normalized ftSIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY 11 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scatter plot showing measured arc curvatures and the corresponding asymmetry indices for all measurements of PSR J1136+1551 taken with Band 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' All arc curvatures have been scaled to their corresponding 1 GHz equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The three distinct clumps are the result of the observations being dominated by three different scattering screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' two epochs (MJDs 59064 and 58987), possibly indicating a detection of multiple scattering screens along the LOS to this pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This result augments the results of Sprenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' (2022), who also found significant variability along the LOS to this pulsar during the same period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' An example observation from the earlier two epochs is shown in Figure 8, while an example from the later two epochs is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' J1932+1059 Due to having the lowest DM in our survey, this pulsar showed the least variation in arc curvature from epoch to epoch across all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Its close proximity to Earth also resulted in wide scintles in frequency, leading to high scintle resolution and, consequently, very bright, narrow, and well defined arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The sharpness of these arcs may also indicate scattering that is highly anisotropic along the LOS (Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2006), as well as originating from a discrete, localized source (Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Overall this was our most consistent pulsar in all aspects of scintilla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This consistency lines up with its other astrophysical pa- rameters, as its dispersion measure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='18 pc cm−3 (Large et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005) was the lowest in our survey and its transverse velocity of 152 km s−1 (Bilous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005) was the second low- est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' While its transverse velocity is a bit larger than PSR Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Dynamic (top) and secondary (bottom) spectra of PSR J1509+5531 centered at 625 MHz on MJD 58987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The top half of the secondary spectrum shows the overlaid arc fits in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaled uncertainties of the arc curvature can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' During this period of observations, visible arcs were considerably narrower than later observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Dynamic (top) and secondary (bottom) spectra of PSR J1509+5531 centered at 650 MHz on MJD 59115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The top half of the secondary spectrum shows the overlaid arc fits in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaled uncertainties of the arc curvature can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' During this period of observations, visible arcs were considerably wider than later observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' J0630−2834 and their distances are almost equivalent, PSR J0630−2834 has a dispersion measure 10 times higher than PSR J1932+1059 (Large et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 1968, 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' J1136+1551 High Frequencies PL =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='92;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' PR =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='88 Referenced at 1 GHz nL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='12 NR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='030 n (s3)PSR J1509+5531MJD 58987 650 Flux Density (Arbitrary Units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='125 [MHz] 640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='075 630 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='050 620 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='025 610 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='025 600 0 10 20 30 40 Time [Min] 10 nL =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='932 s3 40 NR =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='518 s3 5 Log Power (dB) 20 Delay [μs] 0 0 20 5 一 40 10 40 20 0 20 40 Fringe Frequency [10-3 Hz]PSR J1509+5531 MJD 59115 750 (n ) s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='30 Frequency [MHz] 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='20 650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='10 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='00 550 0 5 10 15 20 25 30 35 Time [Min] 10 60 nL =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='224 s3 NR =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content='187 s3 40 5 Power (dB) Delay [μs] 20 0 0 Logl 5 20 10 40 20 0 20 40 Fringe Frequency [10-3 Hz]12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' TURNER ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This means that a much denser ISM was sampled in PSR J0630−2834 than in PSR J1932+1059, meaning that PSR J1932+1059 had decisively the least amount of ISM sampled over our survey, making it the least likely to ex- perience large scintillation-related variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' An example observation is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Dynamic (top) and secondary (bottom) spectra of PSR J1932+1059 centered at 725 MHz on MJD 58987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The top half of the secondary spectrum shows the overlaid arc fits in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Scaled uncertainties of the arc curvature can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' CONCLUSIONS & FUTURE WORK We performed simultaneous dual-frequency observations of six bright canonical pulsars using the uGMRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We ex- tracted scintillation arc, bandwidth, and drift rate measure- ments for each of these pulsars to examine a variety of sci- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We examined how arc curvature scaled with frequency and found our observations to be consistent with the index predicted by theory, while at the same time using a more as- tronomically ideal setup to perform these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We also measured scattering delay scaling indices for five of our six pulsars and found indices consistent with or shallower than what is expected from a Kolmogorov medium, agreeing with previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Finally, we find an interesting and strong correlation between arc curvature and arc asymmetry in PSR J1136+1551, demonstrating a potential connection between screen asymmetry and/or refraction and scattering screen location along the LOS, and the which we intend to follow up with additional observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This study demonstrates the value of array-based tele- scopes such as uGMRT to the pulsar astronomy community, as well as the strengths of simultaneous multiband studies of pulsars and the wide variety of science that can be done with such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' This also shows strong promise for the future observations using ultrawideband (UWB) receivers, which are coming online at instruments such as the Green Bank Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We thank the staff at the uGMRT who have made these observations possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' The uGMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Funda- mental Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' We gratefully acknowledge support of this effort from the NSF Physics Frontiers Center grants 1430284 and 2020265 to NANOGrav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Some of the data processing in this work utilized the resources of the Bowser computing cluster at West Virginia University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQf3g1J/content/2301.05306v1.pdf'} +page_content=' Software: SCINTOOLS Reardon et al.' metadata={'source': 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a/B9E4T4oBgHgl3EQfeA2w/content/tmp_files/2301.05096v1.pdf.txt b/B9E4T4oBgHgl3EQfeA2w/content/tmp_files/2301.05096v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..400bf7bfe5b2ce7205df643ad4cb8f99301e4405 --- /dev/null +++ b/B9E4T4oBgHgl3EQfeA2w/content/tmp_files/2301.05096v1.pdf.txt @@ -0,0 +1,734 @@ +Asynchronous training of quantum reinforcement learning +Samuel Yen-Chi Chen1 +1Wells Fargo +(Dated: January 13, 2023) +Abstract +The development of quantum machine learning (QML) has received a lot of interest recently +thanks to developments in both quantum computing (QC) and machine learning (ML). One of +the ML paradigms that can be utilized to address challenging sequential decision-making issues is +reinforcement learning (RL). It has been demonstrated that classical RL can successfully complete +many difficult tasks. A leading method of building quantum RL agents relies on the variational +quantum circuits (VQC). However, training QRL algorithms with VQCs requires significant amount +of computational resources. This issue hurdles the exploration of various QRL applications. In +this paper, we approach this challenge through asynchronous training QRL agents. Specifically, +we choose the asynchronous training of advantage actor-critic variational quantum policies. We +demonstrate the results via numerical simulations that within the tasks considered, the asyn- +chronous training of QRL agents can reach performance comparable to or superior than classical +agents with similar model sizes and architectures. +1 +arXiv:2301.05096v1 [quant-ph] 12 Jan 2023 + +I. +INTRODUCTION +Quantum computing (QC) has been posited as a means of achieving computational supe- +riority for certain tasks that classical computers struggle to solve [1]. Despite this potential, +the lack of error-correction in current quantum computers has made it challenging to ef- +fectively implement complex quantum circuits on these ”noisy intermediate-scale quantum” +(NISQ) devices [2]. To harness the quantum advantages offered by NISQ devices, the devel- +opment of specialized quantum circuit architectures is necessary. +Recent advances in the hybrid quantum-classical computing framework [3] that utilizes +both classical and quantum computing. Under this approach, certain computational tasks +that are expected to benefit from quantum processing are executed on a quantum computer, +while others, such as gradient calculations, are performed on classical computers. This hybrid +approach aims to take advantage of the strengths of both types of computing to address a +wide range of tasks. Hybrid algorithms that utilize variational quantum circuits (VQC) +have proven to be effective in a variety of machine learning tasks. VQCs are a subclass of +quantum circuits that possess tunable parameters, and their incorporation into QML models +has demonstrated success in a wide range of tasks [3, 4]. +Reinforcement learning (RL) is a branch of machine learning that deals with sequential +decision making tasks. Deep neural network-based RL has achieved remarkable results in +complicated tasks with human-level [5] or super-human performance [6]. However, quantum +RL is a developing field with many unresolved issues and challenges. The majority of existing +quantum RL models are based on VQC [7–11]. Although these models have been shown to +perform well in a variety of benchmark tasks, training them requires a significant amount +of computational resources. The long training time limits the exploration of quantum RL’s +broad application possibilities. We propose an asynchronous training framework for quantum +RL agents in this paper. We focus on the asynchronous training of advantage actor-critic +quantum policies using multiple instances of agents running in parallel. +We show, using numerical simulations, that quantum models may outperform or be sim- +ilar to classical models in the various benchmark tasks considered. Furthermore, the sug- +gested training approach has the practical advantage of requiring significantly less time for +training, allowing for more quantum RL applications. +The structure of this paper is as follows: In SectionII, we provide an overview of relevant +2 + +prior work and compare our proposal to these approaches. In SectionIII, we provide a brief +overview of the necessary background in reinforcement learning. In SectionIV, we introduce +the concept of variational quantum circuits (VQCs), which serve as the building blocks of +our quantum reinforcement learning agents. In SectionV, we present our proposed quantum +A3C framework. In Section VI, we describe our experimental setup and present our results. +Finally, in Section VII, we offer some concluding remarks. +II. +RELEVANT WORKS +The work that gave rise to quantum reinforcement learning (QRL) [12] may be traced +back to [13]. However, the framework demands a quantum environment, which may not +be met in most real-world situations. Further studies utilizing Grover-like methods include +[14, 15]. Quantum linear system solvers are also used to implement quantum policy iteration +[16]. We will concentrate on recent advancements in VQC-based QRL dealing with classical +environments. +The first VQC-based QRL [7], which is the quantum version of deep Q- +learning (DQN), considers discrete observation and action spaces in the testing environments +such as Frozen-Lake and Cognitive-Radio. Later, more sophisticated efforts in the area of +quantum DQN take into account continuous observation spaces like Cart-Pole [8, 9]. A +further development along this direction includes the using of quantum recurrent neural +networks such as QLSTM as the value function approximator [17] to tackle challenges such +as partial observability or environments requiring longer memory of previous steps. Various +methods such as hybrid quantum-classical linear solver are developed to find value functions +[18]. A further improvement of DQN which can improve the agent convergence such as +Double DQN (DDQN) are also implemented within VQC framework in the work [19], in +which the authors apply QRL to solve robot navigation task. Recent advances in QRL +have led to the development of frameworks that aim to learn policy functions, denoted as +π, directly. These frameworks are able to learn the optimal policy for a given problem, +in addition to learning value functions such as the Q-function. +For example, the paper +[10] describes the quantum policy gradient RL through the use of REINFORCE algorithm. +Then, the work [11] consider an improved policy gradient algorithm called PPO with VQCs +and show that even with a small number of parameters, quantum models can outperform +their classical counterparts. Provable quantum advantages of policy gradient are shown in +3 + +the work [20]. Additional research, such as the work in [21], has explored the impact of +various post-processing methods for VQC on the performance of quantum policy gradients. +Several improved quantum policy gradient algorithms have been proposed in recent years, +including actor-critic [22] and soft actor-critic (SAC) [23, 24]. These modifications seek to +further improve the efficiency and effectiveness of QRL methods. QRL has also been applied +to the field of quantum control [25] and has been extended to the multi-agent setting [26– +28]. The work [29] were the first to explore the use of evolutionary optimization for QRL. +In their work, multiple agents were initialized and run in parallel, with the top performing +agents being selected as parents to generate the next generation of agents. In the work [30], +the authors studied the use of advanced quantum policy gradient methods, such as the deep +deterministic policy gradient (DDPG) algorithm, for QRL in continuous action spaces. +In this work, we extend upon previous research on quantum policy gradient [10, 11, 22] by +introducing an asynchronous training method for quantum policy learning. While previous +approaches have employed single-threaded training, our method utilizes an asynchronous +approach, which may offer practical benefits such as reduced training time through the use +of multi-core CPU computing resources and the potential for utilizing multiple quantum +processing units (QPUs) in the future. +Our approach shares some similarities with the +evolutionary QRL method presented in [29], which also utilizes parallel computing resources. +However, our approach differs in that individual agents can share their gradients directly +with the shared global gradient asynchronously, rather than waiting for all agents to finish +before calculating fitness and creating the next generation of agents. This characteristic +may further improve the efficiency of the training process. These contributions represent a +novel advancement in the field of quantum reinforcement learning. +III. +REINFORCEMENT LEARNING +Reinforcement learning (RL) is a machine learning framework in which an agent learns to +accomplish a given goal by interacting with an environment E in discrete time steps [31]. The +agent observes a state st at each time step t and then chooses an action at from the action +space A based on its current policy π. The policy is a mapping from a specific state st to the +probabilities of choosing one of the actions in A. After performing the action at, the agent +gets a scalar reward rt and the state of the following time step st+1 from the environment. +4 + +For episodic tasks, the procedure is repeated across a number of time steps until the agent +reaches the terminal state or the maximum number of steps permitted. Seeing the state +st along the training process, the agent aims to maximize the expected return, which can +be expressed as the value function at state s under policy π, V π(s) = E [Rt|st = s], where +Rt = �T +t′=t γt′−trt′ is the return, the total discounted reward from time step t. The value +function can be further expressed as V π(s) = � +a∈A Qπ(s, a)π(a|s), where the action-value +function or Q-value function Qπ(s, a) = E[Rt|st = s, a] is the expected return of choosing +an action a ∈ A in state s according to the policy π. The Q-learning is RL algorithm to +optimize the Qπ(s, a) via the following formula +Q (st, at) ← Q (st, at) ++ α +� +rt + γ max +a +Q (st+1, a) − Q (st, at) +� +. +(1) +In contrast to value-based reinforcement learning techniques, such as Q-learning, which +rely on learning a value function and using it to guide decision-making at each time step, +policy gradient methods focus on directly optimizing a policy function, denoted as π(a|s; θ), +parametrized by θ. The parameters θ are updated through a gradient ascent procedure +on the expected total return, E[Rt]. A notable example of a policy gradient algorithm is +the REINFORCE algorithm, introduced in [32]. In the standard REINFORCE algorithm, +the parameters θ are updated along the direction ∇θ log π (at|st; θ) Rt, which is an unbiased +estimate of ∇θE [Rt]. However, this policy gradient estimate often suffers from high variance, +making training difficult. +To reduce the variance of this estimate while maintaining its +unbiasedness, a term known as the baseline can be subtracted from the return. This baseline, +denoted as bt(st), is a learned function of the state st. +The resulting update becomes +∇θ log π (at|st; θ) (Rt − bt (st)). A common choice for the baseline bt(st) in RL is an estimate +of the value function V π(st). +Using this choice for the baseline often results in a lower +variance estimate of the policy gradient [31]. The quantity Rt − bt = Q(st, at) − V (st) can +be interpreted as the advantage A(st, at) of action at at state st. Intuitively, the advantage +can be thought of as the ”goodness or badness” of action at relative to the average value +at state st. This approach is known as the advantage actor-critic (A2C) method, where the +policy π is the actor and the baseline, which is the value function V , is the critic [31]. +The asynchronous advantage actor-critic (A3C) algorithm [33] is a variant of the A2C +method that employs multiple concurrent actors to learn the policy through parallelization. +5 + +Asynchronous training of RL agents involves executing multiple agents on multiple instances +of the environment, allowing the agents to encounter diverse states at any given time step. +This diminished correlation between states or observations enhances the numerical stability +of on-policy RL algorithms such as actor-critic [33]. Furthermore, asynchronous training does +not require the maintenance of a large replay memory, thus reducing memory requirements +[33]. By harnessing the advantages and gradients computed by a pool of actors, A3C exhibits +impressive sample efficiency and robust learning performance, making it a prevalent choice +in the realm of reinforcement learning. +IV. +VARIATIONAL QUANTUM CIRCUIT +Variational quantum circuits (VQCs), also referred to as parameterized quantum circuits +(PQCs), are a class of quantum circuits that contain tunable parameters. These parame- +ters can be optimized using various techniques from classical machine learning, including +gradient-based and non-gradient-based methods. A generic illustration of a VQC is in the +central part of Figure 1. +The three primary components of a VQC are the encoding circuit, the variational circuit, +and the quantum measurement layer. The encoding circuit, denoted as U(x), transforms +classical values into a quantum state, while the variational circuit, denoted as V (θ), serves +as the learnable part of the VQC. The quantum measurement layer, on the other hand, +is utilized to extract information from the circuit. It is a common practice to repeatedly +execute the circuit, also known as ”shots,” in order to obtain the expectation values of +each qubit. A common choice is to use the Pauli-Z expectation values. Instead of being +binary integers, the values are received as floats. Additionally, other components, such as +additional VQCs or classical components such as DNN, can process the values obtained from +the circuit. +The VQC can operate with other classical components such as tensor networks (TN) [29, +34, 35] and deep neural networks (NN) to perform data pre-processing such as dimensional +reduction or post-processing such as scaling. We call such VQCs as dressed VQC, as shown +in Figure 1. The whole model can be trained in an end-to-end manner via gradient-based +[34, 35] or gradient-free methods [29]. For the gradient-based methods, the whole model +can be represented as a directed acyclic graph (DAG) and then back-propagation can be +6 + +applied. The success of such end-to-end optimization relies on the capabilities of calculating +the quantum gradients such as parameter-shift rule [36]. VQC-based QML models have +shown success in areas such as classification [34–38], natural language processing [39–41] +and sequence modeling [42, 43]. +Hybrid VQC +U(x) +V(θ) +|0⟩ +|0⟩ +|0⟩ +|0⟩ +NN +NN +FIG. 1. Hybrid variational quantum circuit (VQC) architecture. The hybrid VQC archi- +tecture includes a VQC and classical neural networks (NN) before and after the VQC. NN can be +used to reduce the dimensionality of the input data and refine the outputs from the VQC. +V. +QUANTUM A3C +The proposed quantum asynchronous advantage actor-critic (QA3C) framework consists +of two main components: a global shared memory and process-specific memories for each +agent. The global shared memory maintains the dressed VQC policy and value parameters, +which are modified when an individual process uploads its own gradients for parameter up- +dates. Each agent has its own process-specific memory that maintains local dressed VQC +policy and value parameters. These local models are used to generate actions during an +episode within individual processes. When certain criteria are met, the gradients of the +local model parameters are uploaded to the global shared memory, and the global model +parameters are modified accordingly. The updated global model parameters are then im- +mediately downloaded to the local agent that just uploaded its own gradients. The overall +concept of QA3C is depicted in Figure 2. +We construct the quantum policy π (at | st; θ) and value V (st; θv) function with the +dressed VQC as shown in Figure 1, in which the VQC follows the architecture shown in +7 + +⋯ +Worker 1 +Worker 2 +Worker 3 +Worker n +Environment 1 +Environment 2 +Environment 3 +Environment n +Global Parameter +st +π(at|st) +V(st) +FIG. 2. +Quantum asynchronous advantage actor-critic (A3C) learner. +The proposed +quantum A3C includes a global shared parameters and multiple parallel workers. +The action +generation process within each local agent is performed using the dressed VQC policy and value +functions stored in the process-specific memories. Upon meeting certain criteria, the gradients of +the local model parameters are uploaded to the global shared memory, where the global model +parameters are updated. The updated global model parameters are then immediately downloaded +to the local agent that just uploaded its own gradients. +Figure 3. This VQC architecture has been studied in the work such as quantum recurrent +neural networks (QRNN) [42], quantum recurrent RL [17], quantum convolutional neural +networks [44], federated quantum classification [38] and has demonstrated superior perfor- +mance over their classical counterparts under certain conditions. In addition, we employ +the classical DNN before and after the VQC to dimensionally reduce the data and fine-tune +the outputs from the VQC, respectively. The neural network components in this hybrid +architecture consist of single-layer networks for dimensionality conversion. Specifically, the +network preceding the VQC is a linear layer with an input dimension equal to the size of the +observation vector and an output dimension equal to the number of qubits in the VQC. The +networks following the VQC are linear layers with input dimensions equal to the number +of qubits in the VQC and output dimensions equal to the number of actions (for the actor +function π (at | st; θ)) or 1 (for the critic function V (st; θv)). These layers serve to convert +8 + +the output of the VQC for use in the actor-critic algorithm. The policy and value function +are updated after every S steps or when the agent reaches the terminal state. The details +of the algorithm such as the gradient update formulas are presented in Algorithm 1. +|0⟩ +H +Ry(arctan(x1)) +Rz(arctan(x2 +1)) +• +• +R(α1, β1, γ1) +|0⟩ +H +Ry(arctan(x2)) +Rz(arctan(x2 +2)) +• +• +R(α2, β2, γ2) +|0⟩ +H +Ry(arctan(x3)) +Rz(arctan(x2 +3)) +• +• +R(α3, β3, γ3) +|0⟩ +H +Ry(arctan(x4)) +Rz(arctan(x2 +4)) +• +• +R(α4, β4, γ4) +FIG. 3. VQC architecture for quantum A3C. The VQC used here includes Ry and Rz for +encoding classical values x, multiple CNOT gates to entangle qubits, general unitary rotations R +and the final measurement. The output of the VQC consists of Pauli-Z expectation values, which +are obtained through multiple runs (shots) of the circuit. +These values are then processed by +classical neural networks for further use. We use a 4-qubit system as an example here, however, it +can be enlarge or shrink based on the problem of interest. In this work, the number of qubit is 8. +VI. +EXPERIMENTS AND RESULTS +A. +Testing Environments +1. +Acrobot +The Acrobot environment from OpenAI Gym [45] consists of a system with two linearly +connected links, with one end fixed. The joint connecting the two links can be actuated by +applying torques. The goal is to swing the free end of the chain over a predetermined height, +starting from a downward hanging position, using as few steps as possible. The observation +in this environment is a six-dimensional vector comprising the sine and cosine values of the +two rotational joint angles, as well as their angular velocities. The agents are able to take +one of three actions: applying −1, 0, or +1 torque to the actuated joint. An action resulting +in the free end reaching the target height receives a reward of 0 and terminates the episode. +Any action that does not lead to the desired height receives a reward of −1. The reward +threshold is −100. +9 + +FIG. 4. The Acrobat environment from OpenAI Gym. +2. +Cart-Pole +Cart-Pole is a commonly used evaluation environment for simple RL models that has +been utilized as a standard example with in OpenAI Gym [45] (see Figure 5). +A fixed +junction connects a pole to a cart traveling horizontally over a frictionless track in this +environment. The pendulum initially stands upright, and the aim is to keep it as near to its +starting position as possible by moving the cart left and right. Each time step, the RL agent +learns to produce the right action according on the observation it gets. The observation in +this environment is a four dimensional vector st containing values of the cart position, cart +velocity, pole angle, and pole velocity at the tip. Every time step where the pole is near to +being upright results in a +1 award. An episode ends if the pole is inclined more than 15 +degrees from vertical or the cart moves more than 2.4 units away from the center. +FIG. 5. The Cart-Pole environment from OpenAI Gym. +10 + +3. +MiniGrid-SimpleCrossing +The MiniGrid-SimpleCrossing environment [46] is more sophisticated, with a lot bigger +observation input for the RL agent. In this scenario, the RL agent receives a 7 × 7 × 3 = +147 dimensional vector through observation and must choose an action from the action +space A, which offers six options. It is important to note that the 147-dimensional vector +is a compact and efficient representation of the environment rather than the real pixels. +There are six actions 0,· · · ,5 in the action space A for the agent to choose. +They are +turn left, turn right, move forward, pick up an object, drop the object being carried and +toggle. Only the first three of them are having actual effects in this case. The agent is +expected to learn this fact. In this environment, the agent receives a reward of 1 upon +reaching the goal. +A penalty is subtracted from this reward based on the formula 1 − +0.9 × (number of steps/max steps allowed), where the maximum number of steps allowed is +defined as 4 × n × n, and n is the grid size [46]. In this work, n is set to 9. This reward +scheme presents a challenge because it is sparse, meaning that the agent does not receive +rewards until it reaches the goal. As shown in Figure 6, the agent (shown in red triangle) +is expected to find the shortest path from the starting point to the goal (shown in green). +We consider three cases in this environment: MiniGrid-SimpleCrossingS9N1-v0, MiniGrid- +SimpleCrossingS9N2-v0 and MiniGrid-SimpleCrossingS9N3-v0. Here the N represents the +number of valid crossings across walls from the starting position to the goal. +B. +Hyperparameters and Model Size +In the proposed QA3C, we use the Adam optimizer with learning rate 1×10−4, β1 = 0.92 +and β2 = 0.999. The local agents will update the parameters with the global shared memory +every S = 5 steps. The discount factor γ is set to be 0.9. For the VQC, we set the number +of qubits to be 8 and two variational layers are used. Therefore, for each VQC, there are +8 × 3 × 2 = 48 quantum parameters. Actor and critic both have their own VQC, thus +the total number of quantum parameters is 96. The VQC architecture are the same across +various testing environments considered in this work. As we described in the Section V, +single layer networks are used before and after the VQC to convert the dimensions of data. +The networks preceding the VQC have input dimensions based on the environments that +11 + +(a) +(b) +(c) +FIG. 6. The SimpleCrossing environment from MiniGrid. The three environments from +MiniGrid-SimpleCrossing we consider in this work. +In each environment, there are also walls +which span 1 unit on each side (not shown in the figure). +(a), (b) and (c) represent exam- +ples from the MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-SimpleCrossingS9N2-v0 and MiniGrid- +SimpleCrossingS9N3-v0 environments, respectively. +the agent is to solve. For the classical benchmarks, we consider the model which are very +similar to the dressed VQC model. Specifically, we keep the architecture of classical model +similar to the one presented in Figure 1 while we replace the 8-qubit VQC with a single +layer with input and output dimensions equal to 8. This makes the architecture very similar +to the quantum model and the number of parameters are also very close. We summarize +the number of parameters in Table I. We utilize the open-source PennyLane package [47] +QA3C +Classical +Classical Quantum Total +Total +Acrobot +148 +96 +244 +292 +Cart-Pole +107 +96 +203 +251 +SimpleCrossing +2431 +96 +2527 +2575 +TABLE I. Number of parameters. We provide details on the number of parameters in the +proposed QA3C model, which includes both quantum and classical components. +The classical +benchmarks were designed with architectures similar to the quantum model, resulting in similar +model sizes. +to construct the quantum circuit models and the PyTorch as a overall machine learning +12 + +framework. The number of CPU cores and hence the number of parallel agents is 80 in +this work. We present simulation results in which the scores from the past 100 episodes are +averaged. +C. +Results +1. +Acrobot +We begin by evaluating the performance of our models on the Acrobot environment. The +simulation results of this experiment are presented in Figure7. The total number of episodes +was 100,000. As shown in the figure, the quantum model exhibits a gradual improvement +during the early training episodes, while the classical model struggles to improve its policy. +In terms of average score, the quantum model demonstrates superior performance compared +to the classical model. Furthermore, the quantum model exhibits a more stable convergence +pattern, without significant fluctuations or collapses after reaching optimal scores. These +results suggest that the quantum model may be more robust and reliable in this environment. +0 +20000 +40000 +60000 +80000 +100000 +Episode # +500 +400 +300 +200 +100 +0 +Average Score +Quantum +Classical +FIG. 7. Results: Quantum A3C in the Acrobot environment. +13 + +2. +Cart-Pole +The next experiment was conducted in the Cart-Pole environment. The total number of +episodes was 100,000. As illustrated in Figure 8, the quantum model achieved significantly +higher scores than the classical model. While the classical model demonstrated faster learn- +ing in the early training episodes, the quantum model eventually surpassed it and reached +superior scores. These results suggest that the quantum model may be more effective in this +environment. +0 +20000 +40000 +60000 +80000 +100000 +Episode # +0 +100 +200 +300 +400 +500 +Average Score +Quantum +Classical +FIG. 8. Results: Quantum A3C in the CartPole environment. +3. +MiniGrid-SimpleCrossing +The final experiment was conducted in the MiniGrid-SimpleCrossing environment, com- +prising a total of 100,000 episodes. +As depicted in Figure 9, among the three scenar- +ios, MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-SimpleCrossingS9N2-v0, and MiniGrid- +SimpleCrossingS9N3-v0, the quantum model outperformed the classical model in two of +the three scenarios, MiniGrid-SimpleCrossingS9N2-v0 and MiniGrid-SimpleCrossingS9N3- +v0, demonstrating faster convergence and higher scores. Even in the remaining scenario, +MiniGrid-SimpleCrossingS9N1-v0, the difference in performance between the two models +14 + +was minor. +0 +20000 +40000 +60000 +80000 +100000 +Episode # +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Average Score +S9N1-Quantum +S9N1-Classical +0 +20000 +40000 +60000 +80000 +100000 +Episode # +Average Score +S9N2-Quantum +S9N2-Classical +0 +20000 +40000 +60000 +80000 +100000 +Episode # +Average Score +S9N3-Quantum +S9N3-Classical +FIG. 9. Results: Quantum A3C in the MiniGrid-SimpleCrossing environment. +VII. +CONCLUSION +In this study, we demonstrate the effectiveness of an asynchronous training framework for +quantum RL agents. Through numerical simulations, we show that in the benchmark tasks +considered, advantage actor-critic quantum policies trained asynchronously can outperform +or match the performance of classical models with similar architecture and sizes. +This +technique affords a strategy for expediting the training of quantum RL agents through +parallelization, and may have potential applications in various real-world scenarios. +ACKNOWLEDGMENTS +The views expressed in this article are those of the authors and do not represent the views +of Wells Fargo. This article is for informational purposes only. Nothing contained in this +article should be construed as investment advice. Wells Fargo makes no express or implied +warranties and expressly disclaims all legal, tax, and accounting implications related to this +article. +15 + +Appendix A: Algorithms +1. +Quantum-A3C +Algorithm 1 Quantum asynchronous advantage actor-critic learning (algorithm for each +actor-learner process) +Define the global update parameter S +Assume global shared hybrid VQC policy parameter θ +Assume global shared hybrid VQC value parameter θv +Assume global shared episode counter T = 0 +Assume process-specific hybrid VQC policy parameter θ′ +Assume process-specific hybrid VQC value parameter θ′ +v +Initialize process-specific counter t = 1 +while T < Tmax do +Reset gradients dθ ← 0 and dθv ← 0 +Set tstart = t +Reset the environment and get state st +while st non-terminal or t − tstart < tmax do +Perform at according to policy π(at|st; θ′) +Receive reward rt and the new state st+1 +Update process-specific counter t ← t + 1 +if t mod S = 0 or reach terminal state then +Set R = +� +� +� +0 +for terminal st +V (st, θ′ +v) +for non-terminal st +for i ∈ {t − 1, . . . , tstart } do +R ← ri + γR +Accumulate gradients wrt θ′: dθ ← dθ + ∇θ′ log π (ai | si; θ′) (R − V (si; θ′ +v)) +Accumulate gradients wrt θ′ +v: dθv ← dθv + ∂ (R − V (si; θ′ +v))2 /∂θ′ +v +end for +Perform asynchronous update of θ using dθ and of θv using dθv +Update process-specific parameters from global parameters: θ′ ← θ and θ′ +v ← θv +end if +end while +end while +16 + +[1] M. 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Killoran, “Pen- +nylane: Automatic differentiation of hybrid quantum-classical computations,” arXiv preprint +arXiv:1811.04968, 2018. +20 + diff --git a/B9E4T4oBgHgl3EQfeA2w/content/tmp_files/load_file.txt b/B9E4T4oBgHgl3EQfeA2w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3518e96247f8a57579b5e317419613dd72b7873f --- /dev/null +++ b/B9E4T4oBgHgl3EQfeA2w/content/tmp_files/load_file.txt @@ -0,0 +1,721 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf,len=720 +page_content='Asynchronous training of quantum reinforcement learning Samuel Yen-Chi Chen1 1Wells Fargo (Dated: January 13, 2023) Abstract The development of quantum machine learning (QML) has received a lot of interest recently thanks to developments in both quantum computing (QC) and machine learning (ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' One of the ML paradigms that can be utilized to address challenging sequential decision-making issues is reinforcement learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' It has been demonstrated that classical RL can successfully complete many difficult tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A leading method of building quantum RL agents relies on the variational quantum circuits (VQC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' However, training QRL algorithms with VQCs requires significant amount of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This issue hurdles the exploration of various QRL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In this paper, we approach this challenge through asynchronous training QRL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Specifically, we choose the asynchronous training of advantage actor-critic variational quantum policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We demonstrate the results via numerical simulations that within the tasks considered, the asyn- chronous training of QRL agents can reach performance comparable to or superior than classical agents with similar model sizes and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='05096v1 [quant-ph] 12 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' INTRODUCTION Quantum computing (QC) has been posited as a means of achieving computational supe- riority for certain tasks that classical computers struggle to solve [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Despite this potential, the lack of error-correction in current quantum computers has made it challenging to ef- fectively implement complex quantum circuits on these ”noisy intermediate-scale quantum” (NISQ) devices [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' To harness the quantum advantages offered by NISQ devices, the devel- opment of specialized quantum circuit architectures is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Recent advances in the hybrid quantum-classical computing framework [3] that utilizes both classical and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Under this approach, certain computational tasks that are expected to benefit from quantum processing are executed on a quantum computer, while others, such as gradient calculations, are performed on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This hybrid approach aims to take advantage of the strengths of both types of computing to address a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Hybrid algorithms that utilize variational quantum circuits (VQC) have proven to be effective in a variety of machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' VQCs are a subclass of quantum circuits that possess tunable parameters, and their incorporation into QML models has demonstrated success in a wide range of tasks [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Reinforcement learning (RL) is a branch of machine learning that deals with sequential decision making tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Deep neural network-based RL has achieved remarkable results in complicated tasks with human-level [5] or super-human performance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' However, quantum RL is a developing field with many unresolved issues and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The majority of existing quantum RL models are based on VQC [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Although these models have been shown to perform well in a variety of benchmark tasks, training them requires a significant amount of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The long training time limits the exploration of quantum RL’s broad application possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We propose an asynchronous training framework for quantum RL agents in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We focus on the asynchronous training of advantage actor-critic quantum policies using multiple instances of agents running in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We show, using numerical simulations, that quantum models may outperform or be sim- ilar to classical models in the various benchmark tasks considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Furthermore, the sug- gested training approach has the practical advantage of requiring significantly less time for training, allowing for more quantum RL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The structure of this paper is as follows: In SectionII, we provide an overview of relevant 2 prior work and compare our proposal to these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In SectionIII, we provide a brief overview of the necessary background in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In SectionIV, we introduce the concept of variational quantum circuits (VQCs), which serve as the building blocks of our quantum reinforcement learning agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In SectionV, we present our proposed quantum A3C framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In Section VI, we describe our experimental setup and present our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Finally, in Section VII, we offer some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' RELEVANT WORKS The work that gave rise to quantum reinforcement learning (QRL) [12] may be traced back to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' However, the framework demands a quantum environment, which may not be met in most real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Further studies utilizing Grover-like methods include [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Quantum linear system solvers are also used to implement quantum policy iteration [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We will concentrate on recent advancements in VQC-based QRL dealing with classical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The first VQC-based QRL [7], which is the quantum version of deep Q- learning (DQN), considers discrete observation and action spaces in the testing environments such as Frozen-Lake and Cognitive-Radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Later, more sophisticated efforts in the area of quantum DQN take into account continuous observation spaces like Cart-Pole [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A further development along this direction includes the using of quantum recurrent neural networks such as QLSTM as the value function approximator [17] to tackle challenges such as partial observability or environments requiring longer memory of previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Various methods such as hybrid quantum-classical linear solver are developed to find value functions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A further improvement of DQN which can improve the agent convergence such as Double DQN (DDQN) are also implemented within VQC framework in the work [19], in which the authors apply QRL to solve robot navigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Recent advances in QRL have led to the development of frameworks that aim to learn policy functions, denoted as π, directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These frameworks are able to learn the optimal policy for a given problem, in addition to learning value functions such as the Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' For example, the paper [10] describes the quantum policy gradient RL through the use of REINFORCE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Then, the work [11] consider an improved policy gradient algorithm called PPO with VQCs and show that even with a small number of parameters, quantum models can outperform their classical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Provable quantum advantages of policy gradient are shown in 3 the work [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Additional research, such as the work in [21], has explored the impact of various post-processing methods for VQC on the performance of quantum policy gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Several improved quantum policy gradient algorithms have been proposed in recent years, including actor-critic [22] and soft actor-critic (SAC) [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These modifications seek to further improve the efficiency and effectiveness of QRL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' QRL has also been applied to the field of quantum control [25] and has been extended to the multi-agent setting [26– 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The work [29] were the first to explore the use of evolutionary optimization for QRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In their work, multiple agents were initialized and run in parallel, with the top performing agents being selected as parents to generate the next generation of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In the work [30], the authors studied the use of advanced quantum policy gradient methods, such as the deep deterministic policy gradient (DDPG) algorithm, for QRL in continuous action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In this work, we extend upon previous research on quantum policy gradient [10, 11, 22] by introducing an asynchronous training method for quantum policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' While previous approaches have employed single-threaded training, our method utilizes an asynchronous approach, which may offer practical benefits such as reduced training time through the use of multi-core CPU computing resources and the potential for utilizing multiple quantum processing units (QPUs) in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Our approach shares some similarities with the evolutionary QRL method presented in [29], which also utilizes parallel computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' However, our approach differs in that individual agents can share their gradients directly with the shared global gradient asynchronously, rather than waiting for all agents to finish before calculating fitness and creating the next generation of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This characteristic may further improve the efficiency of the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These contributions represent a novel advancement in the field of quantum reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' REINFORCEMENT LEARNING Reinforcement learning (RL) is a machine learning framework in which an agent learns to accomplish a given goal by interacting with an environment E in discrete time steps [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The agent observes a state st at each time step t and then chooses an action at from the action space A based on its current policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The policy is a mapping from a specific state st to the probabilities of choosing one of the actions in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' After performing the action at, the agent gets a scalar reward rt and the state of the following time step st+1 from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 4 For episodic tasks, the procedure is repeated across a number of time steps until the agent reaches the terminal state or the maximum number of steps permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Seeing the state st along the training process, the agent aims to maximize the expected return, which can be expressed as the value function at state s under policy π, V π(s) = E [Rt|st = s], where Rt = �T t′=t γt′−trt′ is the return, the total discounted reward from time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The value function can be further expressed as V π(s) = � a∈A Qπ(s, a)π(a|s), where the action-value function or Q-value function Qπ(s, a) = E[Rt|st = s, a] is the expected return of choosing an action a ∈ A in state s according to the policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The Q-learning is RL algorithm to optimize the Qπ(s, a) via the following formula Q (st, at) ← Q (st, at) + α � rt + γ max a Q (st+1, a) − Q (st, at) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' (1) In contrast to value-based reinforcement learning techniques, such as Q-learning, which rely on learning a value function and using it to guide decision-making at each time step, policy gradient methods focus on directly optimizing a policy function, denoted as π(a|s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ), parametrized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The parameters θ are updated through a gradient ascent procedure on the expected total return, E[Rt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A notable example of a policy gradient algorithm is the REINFORCE algorithm, introduced in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In the standard REINFORCE algorithm, the parameters θ are updated along the direction ∇θ log π (at|st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ) Rt, which is an unbiased estimate of ∇θE [Rt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' However, this policy gradient estimate often suffers from high variance, making training difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' To reduce the variance of this estimate while maintaining its unbiasedness, a term known as the baseline can be subtracted from the return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This baseline, denoted as bt(st), is a learned function of the state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The resulting update becomes ∇θ log π (at|st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ) (Rt − bt (st)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A common choice for the baseline bt(st) in RL is an estimate of the value function V π(st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Using this choice for the baseline often results in a lower variance estimate of the policy gradient [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The quantity Rt − bt = Q(st, at) − V (st) can be interpreted as the advantage A(st, at) of action at at state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Intuitively, the advantage can be thought of as the ”goodness or badness” of action at relative to the average value at state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This approach is known as the advantage actor-critic (A2C) method, where the policy π is the actor and the baseline, which is the value function V , is the critic [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The asynchronous advantage actor-critic (A3C) algorithm [33] is a variant of the A2C method that employs multiple concurrent actors to learn the policy through parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 5 Asynchronous training of RL agents involves executing multiple agents on multiple instances of the environment, allowing the agents to encounter diverse states at any given time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This diminished correlation between states or observations enhances the numerical stability of on-policy RL algorithms such as actor-critic [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Furthermore, asynchronous training does not require the maintenance of a large replay memory, thus reducing memory requirements [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' By harnessing the advantages and gradients computed by a pool of actors, A3C exhibits impressive sample efficiency and robust learning performance, making it a prevalent choice in the realm of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' VARIATIONAL QUANTUM CIRCUIT Variational quantum circuits (VQCs), also referred to as parameterized quantum circuits (PQCs), are a class of quantum circuits that contain tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These parame- ters can be optimized using various techniques from classical machine learning, including gradient-based and non-gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A generic illustration of a VQC is in the central part of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The three primary components of a VQC are the encoding circuit, the variational circuit, and the quantum measurement layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The encoding circuit, denoted as U(x), transforms classical values into a quantum state, while the variational circuit, denoted as V (θ), serves as the learnable part of the VQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The quantum measurement layer, on the other hand, is utilized to extract information from the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' It is a common practice to repeatedly execute the circuit, also known as ”shots,” in order to obtain the expectation values of each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A common choice is to use the Pauli-Z expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Instead of being binary integers, the values are received as floats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Additionally, other components, such as additional VQCs or classical components such as DNN, can process the values obtained from the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The VQC can operate with other classical components such as tensor networks (TN) [29, 34, 35] and deep neural networks (NN) to perform data pre-processing such as dimensional reduction or post-processing such as scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We call such VQCs as dressed VQC, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The whole model can be trained in an end-to-end manner via gradient-based [34, 35] or gradient-free methods [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' For the gradient-based methods, the whole model can be represented as a directed acyclic graph (DAG) and then back-propagation can be 6 applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The success of such end-to-end optimization relies on the capabilities of calculating the quantum gradients such as parameter-shift rule [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' VQC-based QML models have shown success in areas such as classification [34–38], natural language processing [39–41] and sequence modeling [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Hybrid VQC U(x) V(θ) |0⟩ |0⟩ |0⟩ |0⟩ NN NN FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Hybrid variational quantum circuit (VQC) architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The hybrid VQC archi- tecture includes a VQC and classical neural networks (NN) before and after the VQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' NN can be used to reduce the dimensionality of the input data and refine the outputs from the VQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' QUANTUM A3C The proposed quantum asynchronous advantage actor-critic (QA3C) framework consists of two main components: a global shared memory and process-specific memories for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The global shared memory maintains the dressed VQC policy and value parameters, which are modified when an individual process uploads its own gradients for parameter up- dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Each agent has its own process-specific memory that maintains local dressed VQC policy and value parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These local models are used to generate actions during an episode within individual processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' When certain criteria are met, the gradients of the local model parameters are uploaded to the global shared memory, and the global model parameters are modified accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The updated global model parameters are then im- mediately downloaded to the local agent that just uploaded its own gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The overall concept of QA3C is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We construct the quantum policy π (at | st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ) and value V (st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θv) function with the dressed VQC as shown in Figure 1, in which the VQC follows the architecture shown in 7 ⋯ Worker 1 Worker 2 Worker 3 Worker n Environment 1 Environment 2 Environment 3 Environment n Global Parameter st π(at|st) V(st) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Quantum asynchronous advantage actor-critic (A3C) learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The proposed quantum A3C includes a global shared parameters and multiple parallel workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The action generation process within each local agent is performed using the dressed VQC policy and value functions stored in the process-specific memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Upon meeting certain criteria, the gradients of the local model parameters are uploaded to the global shared memory, where the global model parameters are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The updated global model parameters are then immediately downloaded to the local agent that just uploaded its own gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This VQC architecture has been studied in the work such as quantum recurrent neural networks (QRNN) [42], quantum recurrent RL [17], quantum convolutional neural networks [44], federated quantum classification [38] and has demonstrated superior perfor- mance over their classical counterparts under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In addition, we employ the classical DNN before and after the VQC to dimensionally reduce the data and fine-tune the outputs from the VQC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The neural network components in this hybrid architecture consist of single-layer networks for dimensionality conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Specifically, the network preceding the VQC is a linear layer with an input dimension equal to the size of the observation vector and an output dimension equal to the number of qubits in the VQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The networks following the VQC are linear layers with input dimensions equal to the number of qubits in the VQC and output dimensions equal to the number of actions (for the actor function π (at | st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ)) or 1 (for the critic function V (st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These layers serve to convert 8 the output of the VQC for use in the actor-critic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The policy and value function are updated after every S steps or when the agent reaches the terminal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The details of the algorithm such as the gradient update formulas are presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' |0⟩ H Ry(arctan(x1)) Rz(arctan(x2 1)) R(α1, β1, γ1) |0⟩ H Ry(arctan(x2)) Rz(arctan(x2 2)) R(α2, β2, γ2) |0⟩ H Ry(arctan(x3)) Rz(arctan(x2 3)) R(α3, β3, γ3) |0⟩ H Ry(arctan(x4)) Rz(arctan(x2 4)) R(α4, β4, γ4) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' VQC architecture for quantum A3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The VQC used here includes Ry and Rz for encoding classical values x, multiple CNOT gates to entangle qubits, general unitary rotations R and the final measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The output of the VQC consists of Pauli-Z expectation values, which are obtained through multiple runs (shots) of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These values are then processed by classical neural networks for further use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We use a 4-qubit system as an example here, however, it can be enlarge or shrink based on the problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In this work, the number of qubit is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Testing Environments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Acrobot The Acrobot environment from OpenAI Gym [45] consists of a system with two linearly connected links, with one end fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The joint connecting the two links can be actuated by applying torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The goal is to swing the free end of the chain over a predetermined height, starting from a downward hanging position, using as few steps as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The observation in this environment is a six-dimensional vector comprising the sine and cosine values of the two rotational joint angles, as well as their angular velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The agents are able to take one of three actions: applying −1, 0, or +1 torque to the actuated joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' An action resulting in the free end reaching the target height receives a reward of 0 and terminates the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Any action that does not lead to the desired height receives a reward of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The reward threshold is −100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The Acrobat environment from OpenAI Gym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Cart-Pole Cart-Pole is a commonly used evaluation environment for simple RL models that has been utilized as a standard example with in OpenAI Gym [45] (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A fixed junction connects a pole to a cart traveling horizontally over a frictionless track in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The pendulum initially stands upright, and the aim is to keep it as near to its starting position as possible by moving the cart left and right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Each time step, the RL agent learns to produce the right action according on the observation it gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The observation in this environment is a four dimensional vector st containing values of the cart position, cart velocity, pole angle, and pole velocity at the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Every time step where the pole is near to being upright results in a +1 award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' An episode ends if the pole is inclined more than 15 degrees from vertical or the cart moves more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='4 units away from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The Cart-Pole environment from OpenAI Gym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' MiniGrid-SimpleCrossing The MiniGrid-SimpleCrossing environment [46] is more sophisticated, with a lot bigger observation input for the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In this scenario, the RL agent receives a 7 × 7 × 3 = 147 dimensional vector through observation and must choose an action from the action space A, which offers six options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' It is important to note that the 147-dimensional vector is a compact and efficient representation of the environment rather than the real pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' There are six actions 0,· · · ,5 in the action space A for the agent to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' They are turn left, turn right, move forward, pick up an object, drop the object being carried and toggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Only the first three of them are having actual effects in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The agent is expected to learn this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In this environment, the agent receives a reward of 1 upon reaching the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' A penalty is subtracted from this reward based on the formula 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='9 × (number of steps/max steps allowed), where the maximum number of steps allowed is defined as 4 × n × n, and n is the grid size [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In this work, n is set to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This reward scheme presents a challenge because it is sparse, meaning that the agent does not receive rewards until it reaches the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' As shown in Figure 6, the agent (shown in red triangle) is expected to find the shortest path from the starting point to the goal (shown in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We consider three cases in this environment: MiniGrid-SimpleCrossingS9N1-v0, MiniGrid- SimpleCrossingS9N2-v0 and MiniGrid-SimpleCrossingS9N3-v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Here the N represents the number of valid crossings across walls from the starting position to the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Hyperparameters and Model Size In the proposed QA3C, we use the Adam optimizer with learning rate 1×10−4, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='92 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The local agents will update the parameters with the global shared memory every S = 5 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The discount factor γ is set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' For the VQC, we set the number of qubits to be 8 and two variational layers are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Therefore, for each VQC, there are 8 × 3 × 2 = 48 quantum parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Actor and critic both have their own VQC, thus the total number of quantum parameters is 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The VQC architecture are the same across various testing environments considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' As we described in the Section V, single layer networks are used before and after the VQC to convert the dimensions of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The networks preceding the VQC have input dimensions based on the environments that 11 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The SimpleCrossing environment from MiniGrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The three environments from MiniGrid-SimpleCrossing we consider in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In each environment, there are also walls which span 1 unit on each side (not shown in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' (a), (b) and (c) represent exam- ples from the MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-SimpleCrossingS9N2-v0 and MiniGrid- SimpleCrossingS9N3-v0 environments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' the agent is to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' For the classical benchmarks, we consider the model which are very similar to the dressed VQC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Specifically, we keep the architecture of classical model similar to the one presented in Figure 1 while we replace the 8-qubit VQC with a single layer with input and output dimensions equal to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This makes the architecture very similar to the quantum model and the number of parameters are also very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We summarize the number of parameters in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We utilize the open-source PennyLane package [47] QA3C Classical Classical Quantum Total Total Acrobot 148 96 244 292 Cart-Pole 107 96 203 251 SimpleCrossing 2431 96 2527 2575 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We provide details on the number of parameters in the proposed QA3C model, which includes both quantum and classical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The classical benchmarks were designed with architectures similar to the quantum model, resulting in similar model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' to construct the quantum circuit models and the PyTorch as a overall machine learning 12 framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The number of CPU cores and hence the number of parallel agents is 80 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' We present simulation results in which the scores from the past 100 episodes are averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Acrobot We begin by evaluating the performance of our models on the Acrobot environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The simulation results of this experiment are presented in Figure7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The total number of episodes was 100,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' As shown in the figure, the quantum model exhibits a gradual improvement during the early training episodes, while the classical model struggles to improve its policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' In terms of average score, the quantum model demonstrates superior performance compared to the classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Furthermore, the quantum model exhibits a more stable convergence pattern, without significant fluctuations or collapses after reaching optimal scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These results suggest that the quantum model may be more robust and reliable in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 0 20000 40000 60000 80000 100000 Episode # 500 400 300 200 100 0 Average Score Quantum Classical FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Results: Quantum A3C in the Acrobot environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Cart-Pole The next experiment was conducted in the Cart-Pole environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' The total number of episodes was 100,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' As illustrated in Figure 8, the quantum model achieved significantly higher scores than the classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' While the classical model demonstrated faster learn- ing in the early training episodes, the quantum model eventually surpassed it and reached superior scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' These results suggest that the quantum model may be more effective in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 0 20000 40000 60000 80000 100000 Episode # 0 100 200 300 400 500 Average Score Quantum Classical FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Results: Quantum A3C in the CartPole environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' MiniGrid-SimpleCrossing The final experiment was conducted in the MiniGrid-SimpleCrossing environment, com- prising a total of 100,000 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' As depicted in Figure 9, among the three scenar- ios, MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-SimpleCrossingS9N2-v0, and MiniGrid- SimpleCrossingS9N3-v0, the quantum model outperformed the classical model in two of the three scenarios, MiniGrid-SimpleCrossingS9N2-v0 and MiniGrid-SimpleCrossingS9N3- v0, demonstrating faster convergence and higher scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Even in the remaining scenario, MiniGrid-SimpleCrossingS9N1-v0, the difference in performance between the two models 14 was minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 0 20000 40000 60000 80000 100000 Episode # 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='2 Average Score S9N1-Quantum S9N1-Classical 0 20000 40000 60000 80000 100000 Episode # Average Score S9N2-Quantum S9N2-Classical 0 20000 40000 60000 80000 100000 Episode # Average Score S9N3-Quantum S9N3-Classical FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Results: Quantum A3C in the MiniGrid-SimpleCrossing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' CONCLUSION In this study, we demonstrate the effectiveness of an asynchronous training framework for quantum RL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Through numerical simulations, we show that in the benchmark tasks considered, advantage actor-critic quantum policies trained asynchronously can outperform or match the performance of classical models with similar architecture and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This technique affords a strategy for expediting the training of quantum RL agents through parallelization, and may have potential applications in various real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' ACKNOWLEDGMENTS The views expressed in this article are those of the authors and do not represent the views of Wells Fargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' This article is for informational purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Nothing contained in this article should be construed as investment advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' Wells Fargo makes no express or implied warranties and expressly disclaims all legal, tax, and accounting implications related to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 15 Appendix A: Algorithms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Quantum-A3C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Algorithm 1 Quantum asynchronous advantage actor-critic learning (algorithm for each ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='actor-learner process) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Define the global update parameter S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Assume global shared hybrid VQC policy parameter θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Assume global shared hybrid VQC value parameter θv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Assume global shared episode counter T = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Assume process-specific hybrid VQC policy parameter θ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Assume process-specific hybrid VQC value parameter θ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Initialize process-specific counter t = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='while T < Tmax do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Reset gradients dθ ← 0 and dθv ← 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Set tstart = t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Reset the environment and get state st ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='while st non-terminal or t − tstart < tmax do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='Perform at according to policy π(at|st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ′) Receive reward rt and the new state st+1 Update process-specific counter t ← t + 1 if t mod S = 0 or reach terminal state then Set R = � � � 0 for terminal st V (st, θ′ v) for non-terminal st for i ∈ {t − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' , tstart } do R ← ri + γR Accumulate gradients wrt θ′: dθ ← dθ + ∇θ′ log π (ai | si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ′) (R − V (si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ′ v)) Accumulate gradients wrt θ′ v: dθv ← dθv + ∂ (R − V (si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' θ′ v))2 /∂θ′ v end for Perform asynchronous update of θ using dθ and of θv using dθv Update process-specific parameters from global parameters: θ′ ← θ and θ′ v ← θv 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“Pen- nylane: Automatic differentiation of hybrid quantum-classical computations,” arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content='04968, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfeA2w/content/2301.05096v1.pdf'} diff --git a/C9E0T4oBgHgl3EQfQQB9/content/2301.02190v1.pdf b/C9E0T4oBgHgl3EQfQQB9/content/2301.02190v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1f7aaf2cba18870bcf335933f830ec73c6da01ce --- /dev/null +++ b/C9E0T4oBgHgl3EQfQQB9/content/2301.02190v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:74997de1f465b85232df4d694855acbb614c063b0502ed876800c2fb5b34f7a3 +size 298396 diff --git 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100644 index 0000000000000000000000000000000000000000..fe321d94c153edea4a66a7cd4ff363e4b5963dc7 --- /dev/null +++ b/C9E1T4oBgHgl3EQf-AYx/content/tmp_files/2301.03562v1.pdf.txt @@ -0,0 +1,898 @@ +arXiv:2301.03562v1 [math.FA] 9 Jan 2023 +(Non-)amenability of B(E) and Banach space geometry +Matthew Daws and Matthias Neufang +Abstract +Let E be a Banach space, and B(E) the algebra of all bounded linear operators on E. The question +of amenability of B(E) goes back to Johnson’s seminal memoir [39] from 1972. We present the first +general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space +geometry of E, which imply that B(E) is non-amenable. We cover all spaces for which this is known so +far (with the exception of one particular example), with much shorter proofs, such as ℓp for p ∈ [1, ∞] +and c0, but also many new spaces: the numerous classes of spaces covered range from all Lp-spaces +for p ∈ (1, ∞) to Lorentz sequence spaces and reflexive Orlicz sequence spaces, to the Schatten classes +Sp for p ∈ [1, ∞], and to the James space J, the Schlumprecht space S, and the Tsirelson space T , +among others. Our approach also highlights the geometric difference to the only space for which B(E) +is known to be amenable, the Argyros–Haydon space, which solved the famous scalar-plus-compact +problem. +1 +Introduction +Amenability of Banach algebras is a central notion in functional analysis; cf., e.g., [56, 59]. We mention +Johnson’s fundamental result [39] that a locally compact group G is amenable if and only if its group +algebra L1(G) is amenable. Amenability is also a key concept in operator algebra theory. We note the +deep classical result, due to Connes [21], Bunce–Paschke [14] and Haagerup [38], that for C∗-algebras, +amenability is equivalent to nuclearity (the forward implication is due to Connes and Bunce–Paschke). +Together with work of Wassermann [66], it follows that the algebra B(H) of all bounded linear operators +on a Hilbert space H is non-amenable. However, it was a long-standing open problem whether this holds +for B(ℓp) for all p ∈ [1, ∞]. Indeed, Johnson already asked in 1972 in [39], where he introduced the very +notion of amenability, whether B(E) is ever amenable for an infinite-dimensional Banach space E. Since +then, determining if B(E) is amenable or not for a given infinite-dimensional Banach space E, has proven +a very difficult open problem; cf. the survey article [58]. +Let us briefly recall the history of this problem. For many years, ℓ2 remained the only space for +which the problem was solved. Then Read showed in [54] that B(ℓ1) is non-amenable. His proof was +simplified by Pisier in [51], using expanders, and further simplified by Ozawa in [47], where also a proof +of non-amenability of B(ℓ2) is given avoiding the use of the above results of Connes, Bunce–Paschke and +Wassermann. Moreover, it is shown by Ozawa in [47] that B(ℓ∞) and (implicitly) B(c0) are not amenable. +However, the case of B(ℓp) for p ∈ (1, ∞)\{2} remained open, as noted by Ozawa in [47], by Pisier in [51], +and by Read in [54]. In [57], Runde finally established the celebrated result that B(ℓp) is not amenable +for any p ∈ (1, ∞), through a technically involved proof, building in particular on his earlier work with +Daws ([26], [27]) and on Ozawa’s work. More generally, it is shown in [57] that B(ℓp(E)) is non-amenable +for all Lp-spaces E, where p ∈ (1, ∞). Moreover, Choi shows in [19] that B(L1[0, 1]) is non-amenable; +this is also proven by Aldabbas in [2]. Further, Choi gives a criterion, applying to E = T (ℓ2), the trace +class operators on ℓ2, showing that B(E) is not amenable; see Remark 4.19 below. However, as noted by +Dales (cf. [57, Corollary 2.5]), the Argyros–Haydon space, which solved the famous scalar-plus-compact +problem, provides an example of an infinite-dimensional space E such that B(E) is amenable. Note that +this space is a hereditarily indecomposable L∞-space. +1 + +In this paper, we develop entirely different methods from the ones used so far and establish, employing +our new techniques, the first general principles which encode non-amenability of B(E), for very wide classes +of spaces E, in terms of the Banach space geometry of E. For instance, we show that if E is reflexive +with the approximation property and isomorphic to its square, then B(E) is non-amenable. We also show +that if E is infinite-dimensional and reflexive with a subsymmetric basis, then B(E) is non-amenable. +Our results cover all spaces known so far (with the exception of one particular example, cf. Remark 4.19 +below), with much shorter proofs, and many new (classes of) spaces. The following is a list of spaces E +– obviously always assumed infinite-dimensional – for which we show that B(E) is non-amenable, as a +result of our general approach (we write “∼=” to denote a Banach space isomorphism): +1. all Lp(Ω, B, µ)-spaces and, more generally, all Lp spaces for p ∈ (1, ∞) – this generalises the main +result of [57]; +2. the Lorentz sequence spaces d(w, p), the Garling sequence spaces g(w, p), and the Baernstein spaces +Bp, for all p ∈ (1, ∞); +3. all reflexive Orlicz sequence spaces whose Orlicz function satisfies the ∆2 condition at 0; +4. all separable reflexive rearrangement invariant (r.i.) function spaces; +5. the Schatten classes Sp for all p ∈ [1, ∞]; +6. the non-commutative Lp-spaces Lp(M) for all p ∈ [2, ∞), whenever M is an infinite-dimensional +von Neumann algebra such that Lp(M) has the approximation property; +7. c0, ℓ1, ℓ∞; +8. the non-commutative counterparts of the above, i.e., the spaces K(H), T (H) and B(H) of compact, +trace class and all bounded linear operators on a separable Hilbert space H, and, more generally, +K(ℓp), N(ℓp) and B(ℓp) for all p ∈ (1, ∞); +9. C(K) for any infinite compact metric space K (that is, all separable C(K) spaces), and all separable +L1(Ω, B, µ)-spaces (note that L∞[0, 1] ∼= ℓ∞, so this space is covered as well); +10. the Hardy spaces Hp for all p ∈ [1, ∞) (note that Hp ∼= Lp[0, 1] for p ∈ (1, ∞)); +11. the vector-valued spaces Lp(µ, X) for all p ∈ (1, ∞), for σ-finite µ with Lp(µ) infinite-dimensional, +whenever X∗ has the bounded approximation property and the Radon-Nikodym Property (for ex- +ample, X is reflexive with the approximation property); +12. the vector-valued spaces C(K, X) for any infinite compact metric space K, whenever X∗ has the +bounded approximation property and the Radon-Nikodym Property; +13. particular spaces such as the James space J, the Schlumprecht space S, and the Tsirelson space T. +The paper is organized as follows. In Section 2, we first establish a result on a new hereditary property +of amenability which generalises a well-known theorem of Gourdeau and Ghahramani–Loy–Willis. From +this we shall derive, in Section 3, the general criteria for non-amenability of B(E) in the case of reflexive +spaces E. We treat the non-reflexive situation in Section 4. In the last Section, we present an alternative, +short proof of the non-amenability of B(ℓp) for all p ∈ (1, ∞] which uses operator algebra techniques +and harmonic analysis, instead of Banach space geometry. We also present an “elementary” proof of the +non-amenability of B(ℓ2), which has long been sought after: it is even shorter and of course avoids the +use of nuclearity for C∗-algebras. +2 + +2 +A new hereditary property of amenability +The central idea of this paper is the following. +For a Banach space E, write A(E) for the space of +approximable operators, the norm closure of the finite-rank operators in B(E). +Write K(E) for the +compact operators on E. Often A(E) = K(E); this is true when E has the approximation property +(AP), [60, Chapter 4]. When E is reflexive with the AP, we can identify the bidual K(E)∗∗ with B(E), +where K(E)∗∗ is given the (first) Arens product. +Indeed, K(E) is Arens regular, and so both Arens +products agree. This result goes back to [48, Theorem 2], see also [24, Section 6], and in more generality, +[25]. Consequently, to study B(E), one might study the bidual of A = K(E). A well-known result in +this direction if that when A∗∗ is amenable, also A is amenable, a result shown by Gourdeau ([34], [35, +Theorem 2.3]) and, independently, Ghahramani–Loy–Willis [33, Theorem 1.8]. This result is not directly +useful to us, as K(E) is often amenable (cf. [11, 36]). We will thus substantially generalise this result +below, and this generalisation will be central to our new approach. +With this motivation outlined, we start with some generality, and consider a Banach algebra A and +its bidual A∗∗ equipped with the first Arens product. Let us recall the Arens products. We turn A∗, A∗∗ +into A-bimodules in the usual way. We then define bilinear maps A∗∗ × A∗ → A∗, A∗ × A∗∗ → A∗ by +⟨F · µ, a⟩ = ⟨F, µ · a⟩, +⟨µ · F, a⟩ = ⟨F, a · µ⟩ +(a ∈ A, µ ∈ A∗, F ∈ A∗∗). +We then define bilinear maps ✷, ✸ : A∗∗ × A∗∗ → A∗∗ by +⟨F✷G, µ⟩ = ⟨F, G · µ⟩, +⟨F✸G, µ⟩ = ⟨G, µ · F⟩ +(F, G ∈ A∗∗, µ ∈ A∗). +Direct calculations show that these are Banach algebra products, the first and second Arens products, +respectively. The canonical map κA : A → A∗∗ is a homomorphism, and κA(a)✷F = a·F, F✷κA(a) = F·a, +and similarly for ✸. Henceforth, we always equip A∗∗ with ✷ unless otherwise stated. +Our first step, following [33], is to link A∗∗ �⊗A∗∗ with (A�⊗A)∗∗, where �⊗ denotes the completed +projective Banach space tensor product. We wish to use slightly more concrete identifications than [33], +and we shall hence identify (A�⊗A)∗ with B(A, A∗), say B(A, A∗) ∋ T ↔ ϕT ∈ (A�⊗A)∗ by +⟨ϕT , a ⊗ b⟩ = ⟨T(a), b⟩ +(a, b ∈ A). +(1) +Compare with [60, Section 2.2], for example. Define Ψ : A∗∗ �⊗A∗∗ → (A�⊗A)∗∗ by +⟨Ψ(F ⊗ G), ϕT ⟩ = ⟨F, T ∗(G)⟩ +(F, G ∈ A∗∗). +(2) +Clearly Ψ extends by bi-linearity and continuity to a contraction. Let us just check that this agrees with +[33, Lemma 1.7], so choose bounded nets (ai), (bj) in A converging weak∗ to F, G ∈ A∗∗, respectively. +Then +⟨Ψ(F ⊗ G), ϕT ⟩ = ⟨F, T ∗(G)⟩ = lim +i ⟨T ∗(G), ai⟩ = lim +i lim +j ⟨T(ai), bj⟩ = lim +i lim +j ⟨ϕT , ai ⊗ bj⟩, +which is the same extension given by [33, Lemma 1.6] (cf. [5, §3]). +We now record some useful facts about this map. +Proposition 2.1 ([33, Lemma 1.7]). We have the following commutative diagram +A∗∗ �⊗A∗∗ +Ψ +� (A�⊗A)∗∗ +A�⊗A +�� +κA⊗κA +� +�� +κA � +⊗A +�q +q +q +q +q +q +q +q +q +q +With πA : A�⊗A → A the product map, and similarly for πA∗∗, we have that (πA)∗∗ ◦ Ψ = πA∗∗. Further- +more, Ψ is an A-bimodule map. +3 + +An obvious, but key, property is that A(E) is always an ideal in B(E). We abstract this idea as follows. +Firstly identify A with κA(A) ⊆ A∗∗. Suppose that B ⊆ A∗∗ is some closed subalgebra containing A as +an ideal; we write A ✂ B ⊆ A∗∗. As A is an ideal in B, naturally A is a B-bimodule, and hence also A�⊗A +is a B-bimodule. In the standard way, hence also A∗ and A∗∗, so also A∗∗ �⊗A∗∗, and (A�⊗A)∗∗, become +B-bimodules. However, as B ⊆ A∗∗, also A∗∗ and A∗ have B-actions for the restriction of the A∗∗ actions. +Lemma 2.2. The two B actions on A∗ agree, while the left action of B on A∗∗ agrees with ✸, and the +right action of B on A∗∗ agrees with ✷. +Proof. In this proof, to avoid confusion, let us write b ⊲ a and a ⊳ b for a ∈ A, b ∈ B, to denote the +B-bimodule actions arising from viewing A as an ideal in B, and similarly for the actions of B on A∗ and +A∗∗. Given b ∈ B, a ∈ A, µ ∈ A∗, we find that +⟨b ⊲ µ, a⟩ = ⟨µ, a ⊳ b⟩ = ⟨a · b, µ⟩ = ⟨b, µ · a⟩ = ⟨b · µ, a⟩, +and so b ⊲ µ = b · µ. To be precise, when we write “a · b” we are considering b ∈ A∗∗ and the natural A +action on A∗∗; similarly b · µ is the A∗∗ action on A∗ where we view B as a subalgebra of A∗∗. +Similarly +⟨µ ⊳ b, a⟩ = ⟨µ, b ⊲ a⟩ = ⟨b · a, µ⟩ = ⟨b, a · µ⟩ = ⟨µ · b, a⟩, +so that µ ⊳ b = µ · b. +Then, given b ∈ B, F ∈ A∗∗, µ ∈ A∗, we have +⟨b ⊲ F, µ⟩ = ⟨F, µ ⊳ b⟩ = ⟨F, µ · b⟩ = ⟨b✸F, µ⟩, +⟨F ⊳ b, µ⟩ = ⟨F, b ⊲ µ⟩ = ⟨F, b · µ⟩ = ⟨F✷b, µ⟩, +as claimed. +As both A∗∗ �⊗A∗∗ and (A�⊗A)∗∗ are B-bimodules, we might ask if Ψ is a B-bimodule map. Again, the +situation is slightly complicated as both Arens products arise. +Lemma 2.3. Given b ∈ B and F, G ∈ A∗∗ we have +b · Ψ(F ⊗ G) = Ψ(b✸F ⊗ G), +Ψ(F ⊗ G) · b = Ψ(F ⊗ G✷b). +Proof. Again, we identify T ∈ B(A, A∗) with ϕT ∈ (A�⊗A)∗ as in (1), and in the proof we continue to +write ⊲, ⊳ for the B-bimodule actions. For b ∈ B, a1, a2 ∈ A, +⟨b ⊲ ϕT , a1 ⊗ a2⟩ = ⟨ϕT , a1 ⊗ a2 ⊳ b⟩ = ⟨T(a1), a2 ⊳ b⟩ = ⟨b ⊲ T(a1), a2⟩, +⟨ϕT ⊳ b, a1 ⊗ a2⟩ = ⟨ϕT , b ⊲ a1 ⊗ a2⟩ = ⟨T(b ⊲ a1), a2⟩. +Set b ⊲ ϕT = ϕT1 and ϕT ⊳ b = ϕT2, so that T1(a1) = b ⊲ T(a1) and T2(a1) = T(b ⊲ a1). Then, for G ∈ A∗∗, +⟨T ∗ +1 (G), a1⟩ = ⟨G, b ⊲ T(a1)⟩ = ⟨T ∗(G ⊳ b), a1⟩, +⟨T ∗ +2 (G), a1⟩ = ⟨G, T(b ⊲ a1)⟩ = ⟨T ∗(G), b ⊲ a1⟩ = ⟨T ∗(G) ⊳ b, a1⟩. +Then, for F, G ∈ A∗∗, from (2), and using Lemma 2.2, +⟨b · Ψ(F ⊗ G), ϕT ⟩ = ⟨Ψ(F ⊗ G), ϕT2⟩ = ⟨F, T ∗ +2 (G)⟩ = ⟨F, T ∗(G) ⊳ b⟩ = ⟨Ψ(b✸F ⊗ G), ϕT ⟩, +⟨Ψ(F ⊗ G) · b, ϕT ⟩ = ⟨Ψ(F ⊗ G), ϕT1⟩ = ⟨F, T ∗ +1 (G)⟩ = ⟨F, T ∗(G ⊳ b)⟩ = ⟨Ψ(F ⊗ G✷b), ϕT ⟩, +as claimed. +4 + +We write Zt(A∗∗) for the (first) topological centre (denoted by Z(1) +t +(A∗∗) in [24, 25]), that is, +Zt(A∗∗) = {F ∈ A∗∗ : F✷G = F✸G (G ∈ A∗∗)}. +The following is now immediate from Lemmas 2.2 and 2.3. +Corollary 2.4. Let A ✂ B ⊆ A∗∗ and suppose that B ⊆ Zt(A∗∗). Then the B-bimodule actions on A∗∗ +agree with the product ✷, and Ψ is a B-bimodule map. +We now state and prove the main result of this Section, which shows that amenability of A∗∗ (or more +generally a subalgebra) passes to B when A ✂ B. This generalises [35, Theorem 2.3] and [33, Theorem +1.8], where the statement is shown for the special case B = A and C = A∗∗. +Theorem 2.5. Let A✂B ⊆ A∗∗ and suppose that B ⊆ Zt(A∗∗). Let C ⊆ A∗∗ be a closed subalgebra which +is amenable, with B ⊆ C. Then B is amenable. +Proof. The canonical maps +A�⊗A → B�⊗B → C �⊗C → A∗∗ �⊗A∗∗ +are all contractions, but the overall composition is an isometry, [60, Corollary 2.14], and so each individual +map must also be an isometry, and we identify each tensor product as a closed subspace of A∗∗ �⊗A∗∗. As +B ⊆ Zt(A∗∗), Lemma 2.2 shows that each inclusion is also a B-bimodule map. +As C is amenable, [56, Theorem 2.2.4] or [59, Theorem 2.2.5], it has a bounded approximate diagonal +(di) ⊆ C �⊗C ⊆ A∗∗ �⊗A∗∗, that is, +∥c · di − di · c∥ → 0, +∥πC(di)✷c − c∥ → 0 +(c ∈ C). +(3) +For each i let ni = Ψ(di) ∈ (A�⊗A)∗∗. As B ⊆ C, (3) holds for each member of B, and so it follows from +Corollary 2.4 that for each b ∈ B, +∥b · ni − ni · b∥ = ∥Ψ(b · di) − Ψ(di · b)∥ ≤ ∥b · di − di · b∥ → 0, +(4) +∥(πA)∗∗(ni)✷b − b∥ = ∥(πA)∗∗(Ψ(di))✷b − b∥ = ∥πA∗∗(di)✷b − b∥ = ∥πC(di)✷b − b∥ → 0, +(5) +the second claim using Proposition 2.1. +The bi-adjoint of the inclusion A�⊗A → B�⊗B gives an (isometric) inclusion (A�⊗A)∗∗ → (B�⊗B)∗∗. +For each i let mi be the image of ni in (B�⊗B)∗∗, and let m ∈ (B�⊗B)∗∗ be a weak∗-cluster point of the +bounded net (mi). As the B-bimodule actions on (B�⊗B)∗∗ are weak∗-continuous, it follows from (4) that +b · m = m · b for each b ∈ B. As A is a subalgebra of B, it follows that (πB)∗∗(mi) = (πA)∗∗(ni) for each +i, and so (5) shows that (πB)∗∗(m)✷b = b for each b ∈ B. That is, m is a virtual diagonal for B, showing +that B is amenable, [56, Theorem 2.2.4] or [59, Theorem 2.2.5]. +We immediately obtain the following +Corollary 2.6. Let A ✂ B ⊆ A∗∗ and suppose that A is Arens regular. If A∗∗ is amenable, then so is B. +3 +The general criteria in the reflexive case, and applications +In this Section, we apply Theorem 2.5 in the classical situation when E is reflexive with the AP. As +explained above, it follows that if we set A = A(E) then A = K(E) and A∗∗ is isomorphic to B(E). As +A is Arens regular in this case, the condition on Zt(A∗∗) is vacuous; see Corollary 2.6. Our aim is to use +contradiction to show that B(E) cannot be amenable, by applying Corollary 2.6 to a suitable algebra B +which is “obviously” not amenable. Our technique for finding such B is the following idea which, combined +with Theorem 2.5 or Corollary 2.6, is key to our simplified approach. +5 + +Proposition 3.1. Let E be a Banach space, and suppose there exists T ∈ B(E) which is not compact, +but with T 2 ∈ K(E). Let B be the Banach algebra generated by K(E) and T. Then B is the linear span +of K(E) and T, and B is not amenable. +Proof. As T 2 ∈ K(E) it is easy to see that B is the linear span of K(E) and T. Consider the quotient +algebra C = B/K(E), and let x be the image of T in this quotient. As T is non-compact, x ̸= 0, but x2 = 0 +as T 2 is compact. Thus C is the one-dimensional algebra spanned by x. As C is obviously not unital, it +cannot be amenable. Thus B also cannot be amenable, as amenability passes to quotient algebras. +Corollary 3.2. Let E be a reflexive Banach space with the AP such that there exists T ∈ B(E) which is +not compact, but with T 2 ∈ K(E). Then B(E) is not amenable. +Proof. This follows from Corollary 2.6 applied with A = A(E), so that A∗∗ = B(E), and with B as in +Proposition 3.1. +Theorem 3.3. Let E be a reflexive Banach space with the AP such that E ∼= E0 ⊕ E1 with E0, E1 +isomorphic as Banach spaces. When E is infinite-dimensional, B(E) is not amenable. +Proof. Let T ∈ B(E) be the composition of the projection from E onto E0, the isomorphism from E0 +to E1, and the inclusion E1 → E. Then T 2 = 0. As E is infinite dimensional, so are E0, E1, and hence +T is not compact. Indeed, by the Riesz Lemma, we can find a sequence of unit vectors (xn) in E0 with +∥xn − xm∥ ≥ 1/2 for n ̸= m. Treating E0 ⊆ E, we have that (T(xn)) is a bounded sequence of vectors, +with ∥T(xn) − T(xm)∥ ≥ c/2 for each n ̸= m, where c > 0 is a constant depending on the isomorphisms +E ∼= E0 ⊕ E1 and E0 ∼= E1. Thus T cannot be compact. The result then follows from Corollary 3.2. +Corollary 3.4. Let E be a reflexive Banach space with the AP. If E ∼= E ⊕E then B(E) is not amenable. +For the definition of tree translation equivalent Banach spaces used below, we refer the reader to [10, +Section 4]. This technical definition is useful to us precisely because it allows us to prove the following. +Corollary 3.5. Let E be a reflexive, tree translation equivalent Banach space. Then B(E) is not amenable. +Proof. By assumption, E has a tree translation equivalent basis, so in particular has a basis and hence +has the AP, and by [10, Theorem 4.6], satisfies E ∼= E ⊕ E. Corollary 3.4 now yields the claim. +Generalising the above idea, we obtain the following. +Theorem 3.6. Let E be a reflexive Banach space with the AP, such that there exist closed, infinite- +dimensional, isomorphic subspaces E0 and E1, and a projection P from E onto E0 with P(E1) finite- +dimensional. Then B(E) is not amenable. +Proof. Let T0 : E0 → E1 be an isomorphism, let P : E → E0 be a projection, and set T = T0P : E → +E1 ⊆ E. As PT0P(x) ∈ P(E1) for any x ∈ E, and as P(E1) is finite-dimensional, PT0P is compact, +and so T 2 is compact. As T(x) = T0(x) for each x ∈ E0, and as E0 is infinite-dimensional, and T0 an +isomorphism, it follows that T is not compact. The result now follows from Corollary 3.2. +For the next result, recall the notion of a subsymmetric basis, [46, Definition 3.a.2], which is an +unconditional basis (en) such that (eni) is equivalent to (en) for all increasing sequences (ni). +Corollary 3.7. Let E be a reflexive Banach space with the AP. If E has an infinite-dimensional com- +plemented subspace with a subsymmetric basis, then B(E) is not amenable. +In particular, if E is an +infinite-dimensional reflexive Banach space with a subsymmetric basis, then B(E) is not amenable. +6 + +Proof. Let (ei) be the subsymmetric basis of the infinite-dimensional complemented subspace X. Put +E0 := lin{e2i | i ∈ N} and E1 := lin{e2i−1 | i ∈ N}. As (ei) is subsymmetric, by definition the map +ei �→ e2i extends linearly and continuously to an isomorphism between E and E0; similarly E and E1 are +isomorphic, whence also E0 ∼= E1. As X is complemented, we have a projection P from E onto X. As (ei) +is unconditional, we have a projection Q from X onto E0. Obviously, QP(E) = E0 and QP(E1) = {0}. +By Theorem 3.6, we obtain that B(E) is non-amenable. +We now apply the above results to various classes of Banach spaces. We start with Lp-spaces; for +details on this very important class of spaces, we refer the reader to [3, Section 5] or [44, 45]. We use [28, +Section 23] below, which takes a different definition, but [28, Section 23.3] shows that the latter covers +the Lp-spaces. +Corollary 3.8. Let E be any infinite-dimensional Lp-space, where p ∈ (1, ∞); for instance, E is any +infinite-dimensional Lp(Ω, B, µ) space. Then B(E) is not amenable. +Proof. Let E be an Lp-space, for p ∈ (1, ∞). By [44, Theorem 7.1], E is isomorphic to a complemented +subspace of an Lp space, so certainly reflexive. Further, by [28, Section 21.6, Corollary 1], taking account +of the aforementioned [28, Section 23.3], E has the bounded approximation property. Finally, by [44, +Proposition 7.3], E contains a complemented subspace isomorphic to ℓp. As ℓp has a symmetric basis, +Corollary 3.7 yields the claim. +Remark 3.9. The above corollary generalises [57, Theorem 4.4], the main result of [57], and provides +a much shorter proof. More precisely, it is shown in [57, Theorem 4.4] that B(ℓp(E)) is non-amenable +for any Lp-space E. We note that ℓp(E) is again an Lp- or an L2-space. Indeed, E is isomorphic to +a complemented subspace of some Lp-space, by [45, Corollary 1]. Hence, ℓp(E) is also isomorphic to a +complemented subspace of some Lp-space. Thus, ℓp(E) is an Lp- or an L2-space, by [45, Corollary 1]. +(See also the introduction to [3, Section 5].) +The Baernstein space B2 was introduced by Baernstein in [8], and the p generalisations Bp by Seifert +in his dissertation [63]. They are now viewed as being strongly related to Tsirelson’s space, see [16]. +Corollary 3.10. The Baernstein spaces Bp satisfy that B(Bp) is non-amenable for all p ∈ (1, ∞). +Proof. Each Bp is reflexive and has a basis, hence the AP, and contains a complemented subspace iso- +morphic to ℓp; cf. [16, Theorem 0.15]. As above, the claim now follows from Corollary 3.7. +We now consider the most fundamental examples of non-commutative Lp spaces, namely the Schatten +classes Sp. Recall that Sp is the collection of operators u in B(ℓ2) with Tr(|u|p) < ∞, and norm ∥u∥p = +Tr(|u|p)1/p. For p ∈ (1, ∞), Sp is reflexive (having canonically dual Sq for 1 +p + 1 +q = 1). Letting Pn ∈ B(ℓ2) +be the projection onto the first n coordinates, we have that PnuPn ∈ Sp for each u ∈ Sp, with ∥PnuPn∥p ≤ +∥u∥p. Further, PnuPn → u is norm, hence showing that Sp has the (metric) approximation property. +Corollary 3.11. For p ∈ (1, ∞) the Schatten class Sp satisfies that B(Sp) is not amenable. +Proof. The operation of projecting an operator in B(ℓ2) onto its diagonal restricts to Sp and gives a +projection of Sp onto a subspace isomorphic to ℓp, see the discussion in [4, pages 84–85] for example. The +claim now follows from Corollary 3.7. +More generally, given a von Neumann algebra M, one can consider the non-commutative Lp-spaces +over M, denoted by Lp(M). Again, for p ∈ (1, ∞), Lp(M) is reflexive (having canonically dual Lq(M) for +1 +p + 1 +q = 1). To our knowledge, it is not in general known when Lp(M) has the (Banach space) AP, but +there has been some study of when Lp(M) possesses various Operator Space approximation properties, +7 + +all of which imply the AP; see for example [41] which shows in particular that for a discrete group Γ with +the (group) approximation property, and with M = V N(Γ) the group von Neumann algebra, Lp(M) has +the Operator Space Approximation Property, [41, Theorem 1.1]. +Corollary 3.12. Let p ∈ [2, ∞), and let M be an infinite-dimensional von Neumann algebra such that +Lp(M) has the AP. Then B(Lp(M)) is not amenable. +Proof. By [53, Theorem 0.2], Lp(M) contains a complemented subspace isomorphic to ℓ2 or ℓp. Under +our hypothesis, Corollary 3.7 now applies to give the result. +In the following, we consider various classes of “classical” Banach spaces; in the statement of the result +we give references regarding the properties of the spaces needed for Corollary 3.7 to apply. +Corollary 3.13. For the following infinite-dimensional Banach spaces E we have that B(E) is non- +amenable: +(i) the Lorentz sequence spaces d(w, p) for all p ∈ (1, ∞), see [46, Section 3.a, Section 4.e]; +(ii) the reflexive Orlicz sequence spaces whose Orlicz function satisfies the ∆2 condition at 0, see [46, +Section 4.a, Proposition 4.a.4], and for when such a space is reflexive, [46, Proposition 4.b.2]; +(iii) the Garling sequence spaces g(w, p) for all p ∈ (1, ∞), see [1, Proposition 2.4, Theorem 3.1]; +(iv) the Schlumprecht space S, [62], which is reflexive, [18, Theorem 2.1] or [6, Corollary 8.3]. +Proof. Each Banach space above is infinite-dimensional and reflexive with a subsymmetric basis, which +the given references show. Corollary 3.7 hence implies the claims. +For the Tsirelson space T, we follow the modern convention and view T as the dual of the original +construction of Tsirelson, so T is as defined in [31, Section 2]. +Corollary 3.14. For the following Banach spaces E we have that B(E) is non-amenable: +(i) all infinite-dimensional separable reflexive rearrangement invariant (r.i.) function spaces; +(ii) the Tsirelson space T. +Proof. Each Banach space above is infinite-dimensional, reflexive (cf. [31, Section 2] for T) and, by +[10, Examples 4.3 and 4.4], tree translation equivalent (note that the Haar system is a tree translation +equivalent basis in case (i)). Hence Corollary 3.5 yields the claims. +Remark 3.15. The isomorphism T ∼= T ⊕T, used by Corollary 3.5, also appears in [9]. There, the author +defines a generalised way of constructing “Tsirelson-like” spaces, which are denoted by S. The space T +follows as a special case, see [9, page 209]. The remarks after [9, Corollary 4.7] show that S ∼= S ⊕ T and +so in particular, T ∼= T ⊕ T. +In this direction, we should also remark that A(T) is known to be non-amenable, [11, Corollary 5.8], +and so [33, 35] shows also that B(T) cannot be amenable. +Finally, we note that the uniformly convex Banach space E (of Tsirelson type) with symmetric basis +which contains no isomorphic copy of any ℓp for p ∈ (1, ∞), constructed by Figiel–Johnson in [31, Section +4], also satisfies that B(E) is non-amenable, by Corollary 3.7. +We finish this Section by comparing these constructions with the one known example of an infinite- +dimensional space E with B(E) amenable. +8 + +Remark 3.16. Note that the Argyros–Haydon space X, [7], for which B(X) is amenable, [57, Corol- +lary 2.5], is a hereditarily indecomposable L∞-space. Its dual X∗ is isomorphic to ℓ1 and hence has the +AP, so X has the AP, [60, Corollary 4.7]. Yet, besides being non-reflexive, the key geometric property +of X of being hereditarily indecomposable is in stark contrast to the Banach space properties which we +employ above, all of which say that the space is “very decomposable”. +4 +The case of non-reflexive Banach spaces +In this Section, we treat the case when E is not reflexive. Then A(E) is not Arens regular, and so we +need to consider the (first) topological centre when applying Theorem 2.5. +We follow [25], which is a dense article, so we recall some of the definitions. Given a Banach space E +we write N(E) for the nuclear operators on E, the image of E∗ �⊗E in B(E) equipped with the quotient +norm. Write I(E) for the integral operators; there is always a norm-decreasing inclusion N(E) ⊆ I(E). +Given x∗ ∈ E∗ and x ∈ E we write θx,x∗ for the rank-one operator y �→ ⟨x∗, y⟩x. Then A(E) is by +definition the closed linear span of such operators in B(E). Trace duality gives a natural pairing between +A(E) and I(E∗) which extends the pairing +⟨J, θx,x∗⟩ = Tr(Jθ∗ +x,x∗) = ⟨J(x∗), x⟩ +(J ∈ I(E∗), θx,x∗ ∈ A(E)). +With respect to this pairing, we have that A(E)∗ = I(E∗). +For T ∈ B(E∗∗) define +η(T) = κ∗ +E ◦ T ∗ ◦ κE∗ ∈ B(E∗), +and let Q(T) = η(T)∗. Direct calculation shows that η(T ∗) = T for T ∈ B(E∗), as κ∗ +ET ∗∗κE∗ = κ∗ +EκE∗T = +T. Similarly, given S ∈ B(E∗∗) we see that η(T ∗S) = κ∗ +ES∗T ∗∗κE∗ = η(S)T and so Q(T ∗S) = T ∗Q(S). +Define a bilinear map ⋆ on B(E∗∗) by T ⋆ S = Q(T) ◦ S. Then ⋆ is a Banach algebra product, see [25, +Proposition 2.5]. +We write W(E) for the ideal of weakly compact operators in B(E). +Proposition 4.1. Let E be a Banach space. Suppose that A(E) admits a bounded approximate identity +(eα). This holds if and only if E∗ has the bounded approximation property (BAP). Let Φ0 ∈ A(E)∗∗ be a +weak∗-cluster point of the net (eα). There are bounded maps ψ1, ψ2 : B(E∗∗) → A(E)∗∗ = I(E∗)∗ given +by +⟨ψ1(T), J⟩ = ⟨Φ0, η(TJ∗)⟩, +⟨ψ2(T), J⟩ = ⟨Φ0, η(T)J⟩ +(T ∈ B(E∗∗), J ∈ I(E∗)). +Then ψ1 is an isomorphism onto its range, and a homomorphism B(E∗∗) → (A(E)∗∗, ✷), and ψ2 is +a homomorphism (B(E∗∗), ⋆) → (A(E)∗∗, ✸). +The map ψ2, restricted to {T ∗ : T ∈ B(E∗)}, is an +isomorphism onto its range. For a ∈ A(E) we have that ψ1(a∗∗) = ψ2(a∗∗) = κA(E)(a). For T ∈ W(E) +we have that ψ1(T ∗∗) = ψ2(T ∗∗). +We have the identification +Zt(A(E)∗∗) = +� +ψ2(T ∗) : T ∈ B(E∗), TI(E∗) ⊆ N(E∗) +� +. +Proof. The equivalence of A(E) having a bai and E∗ having the BAP is shown in [37], building on the +work of many authors. [25, Theorem 5.17] shows the claims about ψ1, ψ2, while that ψ1(T ∗∗) = ψ2(T ∗∗) +for T ∈ W(E) is observed at the end of the proof of [25, Corollary 5.22]. Finally [25, Corollary 5.22] +shows the claim about Zt(A(E)∗∗), which is denoted by Z(1) +t +(A(E)∗∗) in [25]. +To obtain a class of operators T ∈ B(E∗) with TI(E∗) ⊆ N(E∗), we use [25, Theorem 3.31] or [60, +Theorem 5.47], which shows that when T ∈ W(E∗), we have TI(E∗) ⊆ N(E∗). +9 + +Theorem 4.2. Let E be a Banach space such that E∗ has the BAP. Suppose there is S ∈ W(E) \ K(E) +with S2 ∈ K(E). Then B(E) is not amenable. +Proof. That S is weakly compact means that S∗ is weakly compact by Gantmacher’s Theorem, [22, +Theorem 5.5] for example, and so ψ2(S∗∗) is in Zt(A(E)∗∗). Further, ψ1(S∗∗) = ψ2(S∗∗). +Set A = A(E) and let C = {ψ1(T ∗∗) : T ∈ B(E)} ⊆ A∗∗, an algebra isomorphic to B(E), as +B(E) → B(E∗∗), T �→ T ∗∗ is a homomorphism. Let B0 be the algebra generated by A(E) = K(E) and S, +so as S2 ∈ K(E), B0 is the linear span of A(E) and S. Let B be the image of B0 in C. As ψ1(S∗∗) = ψ2(S∗∗) +and ψ1, ψ2 agree on A, we see that B ⊆ Zt(A∗∗). +If B(E) is amenable, so is C, so by Theorem 2.5, B is amenable, hence B0 is amenable. Thus B0/A +is amenable, which as in the proof of Proposition 3.1 leads to the required contradiction. So B(E) is not +amenable. +Corollary 4.3. Let K be an uncountable compact metric space. Then B(C(K)) is not amenable. +Proof. By Milutin’s Theorem, [55, Theorem 2.1], we have C(K) ∼= C[0, 1], so it is enough to consider +E = C[0, 1]. We apply Theorem 4.2, so we seek S ∈ W(E) \ K(E) with S2 ∈ K(E). Firstly, suppose we +can find any T ∈ W(E) \ K(E). Define linear maps +T0 : C[0, 1] → C[0, 1], +T0(f)(s) = f(2s) +(f ∈ C[0, 1], s ∈ [0, 1]), +and also T1 : C[0, 1] → C[0, 1] in the following way. +Pick a small δ > 0 and a linear bijection ϕ : +[1/2+δ, 1−δ] → [0, 1]. For f ∈ C[0, 1] define T1(f) as follows. For 1/2+δ ≤ t ≤ 1−δ let T1(f)(t) = f(ϕ(t)), +and for t < 1/2 let T1(f)(t) = 0. +Set T1(f)(1/2) = T1(f)(1) = 0, and linearly interpolate on the +intervals [1/2, 1/2 + δ] and [1 − δ, 1]. +Then T0 is a metric surjection, T1 is an isometry, and hence +S = T1TT0 ∈ W(E)\K(E). As T0T1 = 0 also S2 = 0. In fact, the square of any weakly compact operator +on any C(K) is always compact by a result of Grothendieck, [55, Corollary 4.2], and so already T works +in Theorem 4.2, but we prefer to give this explicit construction. +It hence remains to find a suitable T. We use [52, Example 6] which gives an example of an absolutely +summing but non-compact map R : C[0, 1] → c0; for instance, with rn : [0, 1] → {±1} the Rademacher +functions, we may define +R(f) = +� � 1 +0 +rn(t)f(t) dt +�∞ +n=1. +By [60, Corollary 6.20], R is weakly compact as it is absolutely summing. It remains to find an isometry +c0 → C[0, 1] which when composed with R will give us our required T. This is well-known (cf., e.g., +[55, Lemma 2.5 (d)]) but we give a construction. +Let f1 ∈ C[0, 1] be the piece-wise linear function +with f1(0) = f1(1/2) = f1(1) = 0, f1(3/4) = 1, let f2 ∈ C[0, 1] be the piece-wise linear function with +f2(0) = f2(1/4) = f2(1/2) = f2(1) = 0, f2(3/8) = 1, and so forth. Then (fn) is a copy of the standard +unit vector basis of c0, and the map c0 → C[0, 1], (an) �→ � +n anfn is our isometry. Here the sum is to be +interpreted pointwise, but as (an) ∈ c0, it actually converges absolutely. +Corollary 4.4. B(L1[0, 1]) is not amenable. +Proof. We apply Theorem 4.2 to E = L1[0, 1]. As E∗ = L∞[0, 1] has the BAP, we need only find a +suitable S ∈ W(E). As E ∼= E ⊕ E, we can obtain S2 = 0 so long as we can find any R ∈ W(E) \ K(E). +To find such an R, we can follow [19, Example 3.4], for instance. +Corollary 4.5. Let E be a Banach space containing a complemented subspace isomorphic to ℓp for some +p ∈ (1, ∞). If E∗ has the BAP then B(E) is not amenable. +10 + +Proof. We can easily find R ∈ B(ℓp) with R non-compact but R2 = 0. Indeed, if (en) is the standard +unit vector basis of ℓp, define R by R(e2n) = e2n+1 and R(e2n−1) = 0 for each n ∈ N. Let P from E +onto ℓp be a projection, so as ℓp is reflexive, RP is weakly compact. As R2 = 0 also (RP)2 = 0. As P +is a projection onto ℓp and by the construction of R, we see that RP is not compact. Now Theorem 4.2 +implies the result. +Remark 4.6. Note that the above yields another proof of the non-amenability of B(E) for any Lp-space +E, where p ∈ (1, ∞). +Corollary 4.7. The Hardy spaces Hp for all p ∈ [1, ∞) satisfy that B(Hp) is not amenable. +Proof. For p ∈ (1, ∞) we have Hp ∼= Lp[0, 1] by the classical result [12], so the claim follows from +Corollary 3.8. Now consider H1. By [42, Section 3], H1 contains a complemented subspace isomorphic +to ℓ2 (this is due to Paley, cf. the references in [42]). Also, by [40, Corollary 1], the dual H∗ +1 ∼= BMO +has the uniform approximation property, hence the BAP. Now Corollary 4.5 entails that B(H1) is not +amenable. +Next we derive a result concerning vector-valued Lp spaces over σ-finite measure spaces. So let µ be a +σ-finite measure, E a Banach space, and consider Lp(µ, E). When E∗ has the Radon-Nikodym Property +(RNP), then Lp(µ, E)∗ = Lq(µ, E∗), where 1 +p + 1 +q = 1, cf. [30, Section IV, Theorem 1]. For the RNP, see, +e.g., [30, Chapter III] or [25, Definition 3.16]. All reflexive spaces, and all separable dual spaces have the +RNP. More generally, E∗ has the RNP if and only if every separable subspace of E has separable dual, +[30, Section VII.2, Corollary 8]. We note that [30] works only with finite measures, but the σ-finite case +is a routine generalisation from this. +To apply our result, we wish to know when Lq(µ, E∗) has the BAP. The following result is surely +known, but as we have not found a suitable reference, we give a proof. +Lemma 4.8. Let E have the BAP, and let p ∈ [1, ∞). Let µ be a σ-finite measure. Then Lp(µ, E) has +the BAP. +Proof. We shall use the ∆p norm from [28, Chapter 7] which is not quite a “tensor norm” on Lp(µ) ⊗ E; +in particular, it fails the usual mapping property. Nevertheless, Lp(µ) ⊗ E is dense in Lp(µ, E) for the +norm ∆p. +It is easy to see that we can witness that Lp(µ) has the metric approximation property by finite-rank, +positive operators (Ti), cf. [30, Section VIII.3, Example 11] for instance. For any positive operator T on +Lp(µ), we do have that T ⊗ idE is bounded for ∆p, with bound ∥T∥, and so extends to Lp(µ, E), see [28, +Theorem 7.3]. +Any S ∈ B(E) extends to idLp(µ) ⊗ S on Lp(µ, E) with norm ∥S∥. Thus if (Sj) is a bounded net of +finite-rank operators on E witnessing that E has the BAP, then (Ti ⊗ Sj) is a bounded net of finite-rank +operators on Lp(µ, E) which tends in the point-norm topology to the identity on Lp(µ) ⊗ E, and thus by +boundedness and density, on all of Lp(µ, E). Hence Lp(µ, E) has the BAP. +Corollary 4.9. Let E be a Banach space such that E∗ has the BAP and the RNP (for example, E is +reflexive with the AP), and let p ∈ (1, ∞). Let µ be a σ-finite measure with Lp(µ) infinite-dimensional. +Then B(Lp(µ, E)) is not amenable. +Proof. The discussion above shows that under these hypotheses, Lp(µ, E) has dual Lq(µ, E∗). By Lemma +4.8, also Lq(µ, E∗) has the BAP, as E∗ does. Furthermore, Lp(µ, E) contains a complemented subspace +isomorphic to ℓp, by [17, Proposition 1.4.1]. Now Corollary 4.5 implies the result. +For the definition of the James space J, we refer the reader to [46, Example 1.d.2]. +11 + +Corollary 4.10. The James space J satisfies that B(J) is not amenable. +Proof. By combining [15, Theorem 5] with the remark after [15, Corollary 4] and [15, Corollary 3], we +find that J admits a complemented subspace isomorphic to ℓ2. Also, as J has a shrinking basis, J∗ has +a basis by [46, Proposition 1.b.1], and so the BAP. Again, Corollary 4.5 yields the claim. +4.1 +When the nuclear and integral operators on E∗ agree +If we assume that N(E∗) = I(E∗) then we can say more. +Recall the discussion of the RNP after +Corollary 4.7 above. A useful result is that when E∗ has the RNP, then N(E∗) = I(E∗) with equal +norms, [25, Theorem 3.18] and [60, Section 5.3]. +We continue with the notation from Proposition 4.1. +Lemma 4.11. Let E be a Banach space such that E∗ has the BAP, and N(E∗) = I(E∗). Then ψ1(T ∗) = +ψ2(T ∗) for each T ∈ B(E∗). +Proof. Firstly, for general E, given T ∈ B(E∗) and J ∈ I(E∗), we always have +⟨ψ1(T ∗), J⟩ = ⟨Φ0, η(T ∗J∗)⟩ = ⟨Φ0, JT ⟩, +⟨ψ2(T ∗), J⟩ = ⟨Φ0, η(T ∗)J⟩ = ⟨Φ0, TJ⟩. +(6) +As in Proposition 4.1, here Φ0 is a weak∗-cluster point of a bai (eα) for A(E). Now let J ∈ N(E∗), say +J = θx∗,x∗∗. Then +⟨Φ0, JT ⟩ = ⟨Φ0, θx∗,T ∗(x∗∗)⟩ = lim +α ⟨θx∗,T ∗(x∗∗), eα⟩ = lim +α ⟨T ∗(x∗∗), e∗ +α(x∗)⟩ = ⟨T ∗(x∗∗), x∗⟩, +here using that e∗ +α(x∗) → x∗ for each x∗ ∈ E∗. Indeed, as θx,x∗eα = θx,e∗α(x∗) and ∥θx,x∗ − θx,x∗eα∥ → 0, +it follows that ∥x∥∥x∗ − e∗ +α(x∗)∥ → 0. We similarly see that +⟨Φ0, TJ⟩ = ⟨Φ0, θT(x∗),x∗∗⟩ = lim +α ⟨θT(x∗),x∗∗, eα⟩ = lim +α ⟨x∗∗, e∗ +α(T(x∗))⟩ = ⟨x∗∗, T(x∗)⟩. +It follows that ⟨Φ0, TJ⟩ = ⟨Φ0, JT⟩, and by linearity and continuity, this holds for all J ∈ N(E∗). The +result now follows from (6). +Theorem 4.12. Let E be a Banach space such that E∗ has the BAP, and N(E∗) = I(E∗). Let B be +a closed subalgebra with K(E) ✂ B ⊆ B(E). If any of B(E), B(E∗) or B(E∗∗) is amenable, then B is +amenable. +Proof. Set A = A(E), and let C = {ψ1(T ∗∗) : T ∈ B(E)} ⊆ A∗∗, an algebra isomorphic to B(E). +As I(E∗) = N(E∗), we know from Proposition 4.1 that Zt(A∗∗) = {ψ2(T ∗) : T ∈ B(E∗)}, and by +Lemma 4.11, this equals {ψ1(T ∗) : T ∈ B(E∗)}. Thus C ⊆ Zt(A∗∗), and so when B(E) is amenable, also +C is amenable, and hence Theorem 2.5 yields the result. +When B(E∗) is amenable, we instead set C = {ψ1(T ∗) : T ∈ B(E∗)} ⊆ A∗∗, an algebra anti-isomorphic +to B(E∗), so that C is amenable. Now C = Zt(A∗∗). We identify B with {ψ1(T ∗∗) : T ∈ B}, so that B ⊆ C, +and A ✂ B. Again, Theorem 2.5 yields the claim. +Finally, suppose that B(E∗∗) is amenable. As A∗ = I(E∗) = N(E∗) by hypothesis, and as E∗ has the +BAP so that N(E∗)∗ = B(E∗∗), it follows that A∗∗ = B(E∗∗) with ψ1 being an isomorphism. For this, +see [25, Section 5.2] and [24, Section 6]. Now set C = A∗∗ which is thus amenable, and again identify B +with {ψ1(T ∗∗) : T ∈ B}, so that B ⊆ Zt(A∗∗). Again Theorem 2.5 implies the result. +Corollary 4.13. Let K be an infinite countable compact metric space. Then B(C(K)) is not amenable. +12 + +Proof. By [55, Lemma 2.5(d)], we know that C(K) contains an isometric copy of c0. Using Sobczyk’s +theorem, see [65], as C(K) is separable, there is a projection P from C(K) onto c0. As K is countable, +we have C(K)∗ = M(K) ∼= ℓ1(K) which is a separable dual space, and hence has the RNP; also it has +the BAP. We then apply Theorem 4.12 with B = K(C(K)) ⊕ CS for a suitable operator S. Again, we use +Proposition 3.1, so we seek S non-compact with S2 compact. It is easy to find T ∈ B(c0) non-compact +with T 2 = 0, compare the proof of Corollary 4.16 below. Set S = TP, so that S is non-compact as P is +a projection onto c0, while PT = T so S2 = TPTP = T 2P = 0. Theorem 4.12 now yields that B(C(K)) +is not amenable. +Corollary 4.14. Let K be an infinite compact metric space, equivalently, let K be an infinite compact +space with C(K) separable. Then B(C(K)) is not amenable. +Proof. This follows immediately from Corollaries 4.3 and 4.13. +We now consider the vector-valued spaces C(K, E) for a Banach space E. These can be realised as the +injective tensor product C(K)ˇ⊗E, see [60, Section 3.2]. The following generalises the previous corollary, +in that we can take E = C. +Corollary 4.15. Let K be an infinite compact metric space, and let E be a Banach space such that E∗ +has the BAP and the RNP. Then B(C(K, E)) is not amenable. +Proof. We may identify C(K, E)∗ with the space of regular vector measures of bounded variation, defined +on the Borel subsets of K, with values in E∗, see for example [60, page 112]. As E∗ has the RNP, [60, +Corollary 5.23] shows that this space coincides with M(K)�⊗E∗, paired against C(K)ˇ⊗E = C(K, E) in +the canonical way. As both M(K) and E∗ have the BAP, the proof of Lemma 4.8 is readily adapted to +show that M(K)�⊗E∗ = C(K, E)∗ has the BAP; cf. also [29, Corollary 1.18]. +As C(K, E) = C(K)ˇ⊗E, fixing x ∈ E and µ ∈ E∗ with ∥x∥ = ∥µ∥ = ⟨µ, x⟩ = 1, the maps +C(K)ˇ⊗E → C(K), f ⊗ y �→ ⟨µ, y⟩f +and +C(K) → C(K)ˇ⊗E, f �→ f ⊗ x +establish that C(K) is a complemented subspace of C(K, E). When K is uncountable, we may use the +complemented copy of C(K) together with the proof of Corollary 4.3 to show that there is T ∈ B(C(K, E)) +which is weakly compact but not compact, and with T 2 compact. Thus Theorem 4.2 yields the result. +Now suppose that K is countable. As E∗ has the RNP, we know that separable subspaces of E have +separable duals. Let X ⊆ C(K, E) be separable, and let (fn) be a dense subset. Then {fn(k) : n ∈ N, k ∈ +K} is a countable subset of E and so its closed linear span is a separable subspace of E, say E0. Then +X ⊆ C(K, E0), and as K is countable, it follows easily that C(K, E0) is separable. We know that E∗ +0 is +separable, and so C(K, E0)∗ = ℓ1(K)�⊗E∗ +0 is separable, as ℓ1(K) is separable. We have hence shown that +separable subspaces of C(K, E) have separable dual. So C(K, E)∗ has the RNP and the BAP in this case. +As in the proof of Corollary 4.13, C(K) contains a complemented copy of c0, and hence so does C(K, E). +We can now argue exactly as in the proof of Corollary 4.13 to see that B(C(K, E)) is not amenable. +Corollary 4.16. B(c0), B(ℓ1) and B(ℓ∞) are not amenable. +Proof. Set E = c0. Then E∗ = ℓ1 has the BAP, and as a separable dual space, has the RNP, so that +N(E∗) = I(E∗). We can find S ∈ B(c0) \ K(c0) with S2 = 0. Indeed, if (en) is the standard unit vector +basis of c0 then define S by S(e2n) = e2n+1 and S(e2n−1) = 0 for each n ∈ N. Choosing B = K(c0) ⊕ CS +gives a non-amenable Banach algebra by Proposition 3.1, and then Theorem 4.12 shows that B(c0), B(ℓ1) +and B(ℓ∞) are not amenable. +Corollary 4.17. Let E be an infinite-dimensional separable L1 space. Then B(E) is not amenable. +13 + +Proof. There is a classification of such E, [67, p. 83]: indeed, either E ∼= L1[0, 1] or E ∼= ℓ1, so the result +follows from Corollaries 4.4 and 4.16. +We now establish the non-commutative version of Corollary 4.16. +Corollary 4.18. Let H be an infinite-dimensional separable Hilbert space. Then B(K(H)), B(T (H)) and +B(B(H)) are not amenable. +Proof. Set E = K(H), so that E∗ = T (H), the trace class operators, and E∗∗ = B(H). Then E∗ has +the BAP, indeed, even a (Schauder) basis, which follows from [60, Proposition 4.25] for example. Also +T (H) is a separable dual space, and so has the RNP, hence N(E∗) = I(E∗). As in the proof of the +next corollary, we can find an operator S ∈ B(E) which is non-compact with S2 = 0, thus showing that +B = K(E) ⊕ CS is not amenable, by Proposition 3.1. Now Theorem 4.12 yields that B(K(H)), B(T (H)) +and B(B(H)) are not amenable. +Remark 4.19. In [19, Section 4], Choi shows that when E is a Banach space with the BAP such that +E∗ does not have the BAP, then B(E) is not amenable. This criterion is rather restrictive; the canonical +example, thanks to Szankowski’s result [64], is E = T (H). +Choi’s argument uses again [37] which shows that, under these hypotheses, A(E) has a one-sided but +no two-sided bounded approximate identity. As A(E) is an ideal in B(E), this contradicts B(E) being +amenable, cf. [19, Lemma 2.2]. Our proof above that B(T (H)) is non-amenable avoids the use of the very +deep result of Szankowski. +There is to our knowledge one way to construct spaces which Choi’s result covers, giving non-amenability +of B(E), and where our methods do not apply. By [43, Corollary 3], see also [46, Theorem 1.e.7(b)], start- +ing with any Banach space F which fails to have the AP, one can show that there exists a Banach space +E with the BAP (indeed, a Schauder basis) such that E∗ does not have the AP. +In fact, more generally, we obtain the following. +Corollary 4.20. For all p ∈ (1, ∞) we have that B(K(ℓp)), B(N(ℓp)) and B(B(ℓp)) are not amenable. +Proof. We proceed at first with some generality. Let F be a Banach space such that F ∗∗ is separable +with the BAP. Also F ∗ is separable, and a dual space, and so has the RNP. Further, F ∗ has the BAP, +see [25, Corollary 3.22] for example, and so there are bounded nets (ti), (sj) of finite-rank operators in +B(F ∗), B(F ∗∗), respectively, converging in the point-norm topology to the identity. Set E = K(F) = A(F), +so that E∗ = I(F ∗) = N(F ∗) = F ∗ �⊗F ∗∗ by the hypotheses on F. Then (ti ⊗ sj) is a bounded net of +finite-rank operators converging in the point-norm topology to the identity on F ∗ �⊗F ∗∗, showing that E∗ +has the BAP. As F ∗∗ and F ∗ are separable, also E∗ is separable. Thus we can apply Theorem 4.12 to +find that B(E), B(E∗) and B(E∗∗) are not amenable, provided a suitable B can be constructed. +Now specialise to the case F = ℓp for p ∈ (1, ∞). Let (en) be the usual unit vector basis of ℓp, and +set F0, F1 to be the closed span of (e2n), (e2n−1), respectively. For i = 0, 1 there is a natural projection +Pi : F → Fi, and an inclusion ιi : Fi → F. Further, there is an isometry j : F0 → F1. For x ∈ A(ℓp) +define S(x) = jP0xP0 ∈ A(ℓp). As P0j = 0, we see that S2(x) = jP0S(x)P0 = jP0jP0xP0 = 0 for each +x. For y ∈ A(F0) we can set x = ι0yP0 ∈ A(ℓp), and then S(x) = jP0ι0yP0 = jyP0, so in particular, +∥S(x)∥ = ∥yP0∥ = ∥y∥ = ∥x∥. This shows that S is non-compact, and so B = A(E) ⊕ CS is a suitable +non-amenable algebra. +Remark 4.21. Note that the Argyros–Haydon space X, for which B(X) is amenable, cf. Remark 3.16, +satisfies that X∗ is isomorphic to ℓ1, so X∗ has the BAP and the RNP, whence N(X∗) = I(X∗). However, +as B(X) = K(X) ⊕ CidX, there is obviously no operator S ∈ B(X) \ K(X) with S2 compact. +14 + +5 +Alternative proofs of the non-amenability of B(ℓp) for p ∈ (1, ∞] +In this Section, we present alternative, quick proofs that B(ℓp) is non-amenable for all p ∈ (1, ∞] using +operator algebra methods and harmonic analysis, rather than Banach space geometry. We first give a +short proof of the non-amenability of B(ℓ2), avoiding the use of nuclearity for C∗-algebras (we remark +that [13] quoted below was written when there was no relationship known between amenability and +nuclearity for C∗-algebras, as noted therein). Given a discrete group G, we write C∗ +r (G) for its reduced +group C∗-algebra. Recall that K(ℓ2(G)) is Arens regular being a C∗-algebra, and the Arens product on +K(ℓ2(G))∗∗ = B(ℓ2(G)) is the usual composition of operators. +Theorem 5.1. B(ℓ2) is not amenable. +Proof. Realize ℓ2 as ℓ2(G) for some countable discrete non-amenable group G, such as F2. +Suppose +that B(ℓ2(G)) is amenable. Put A := K(ℓ2(G)), and consider the C∗-subalgebra B := A ⊕ C∗ +r (G) of +B(ℓ2(G)) = A∗∗ (note that K(ℓ2(G)) ∩ C∗ +r (G) = {0} by [20, Proposition 3.2]). As in Section 3 we can +apply Corollary 2.6 to see that amenability of B(ℓ2(G)) passes to B, and hence to the quotient C∗ +r (G). +Thus G is amenable by [13, Proposition 2] – a contradiction. +For the case of B(ℓp), p ∈ (1, ∞), we will argue similarly. We will consider the p-analogue of C∗ +r (G), +i.e., the algebra PFp(G) of p-pseudofunctions on G, defined as the Banach algebra generated in B(ℓp(G)) +by λp(ℓ1(G)), where λp is the representation of ℓ1(G) on ℓp(G) given by left convolution. +We are grateful to N.C. Phillips for pointing out the following +Lemma 5.2. Let G be a countable discrete group, and p ∈ (1, ∞). Then the canonical quotient map +q : B(ℓp(G)) → B(ℓp(G))/K(ℓp(G)) is isometric on PFp(G). +Proof. This is shown for p = 2 in [49, Proposition 4.5], and inspection of the proof shows that the +argument carries over, mutatis mutandis, to the case of general p. +For the following, note that given a discrete group G, K(ℓp(G)) is Arens regular by [23, Theorem +2.6.23], and the product on K(ℓp(G))∗∗ = B(ℓp(G)) is the usual composition of operators. +Theorem 5.3. B(ℓp) is not amenable for any p ∈ (1, ∞). +Proof. Let p ∈ (1, ∞). Realize ℓp as ℓp(G) for some countable discrete non-amenable group G, such as +F2. Suppose that B(ℓp(G)) is amenable. Put A := K(ℓp(G)), and consider the space B := A ⊕ PFp(G) +(note that K(ℓp(G)) ∩ PFp(G) = {0} as the proof of [20, Proposition 3.2] for p = 2 carries over to +the case of general p). +Let q : B(ℓp(G)) → B(ℓp(G))/K(ℓp(G)) be the canonical quotient map. +By +Lemma 5.2, q(PFp(G)) ⊆ B(ℓp(G))/K(ℓp(G)) is closed. Hence B = q−1(q(PFp(G))) is a closed subalgebra +of B(ℓp(G)) = A∗∗. Again, by Corollary 2.6, B is amenable. So the quotient PFp(G) is amenable, whence +G is amenable (see the proof of [32, Theorem 6.4], which uses work of Phillips [50]) – a contradiction. +We shall now give an alternative proof of the non-amenability of B(ℓ∞). To this end, let G be a +countable discrete group. We recall that K(c0(G))∗∗ = B(ℓ∞(G)), with the first Arens product being the +usual composition of operators, and Zt(K(c0(G))∗∗) = Bσ(ℓ∞(G)), where the latter denotes the maps in +B(ℓ∞(G)) which are weak∗-weak∗-continuous; see pages 59–61, in particular Example 6.2, in [24]. (This +also follows from Proposition 4.1 in this special case when E = c0(G), as then E∗ = ℓ1(G) has the RNP +and so the ideas of Section 4.1 apply.) We also recall that Φ : ℓ1(G) → B(c0(G)), where +Φ(f)(g) = f ∗ g for all f ∈ ℓ1(G), g ∈ c0(G), +is an isometric representation. We see B(c0(G)) as a subalgebra of B(ℓ∞(G)) (by taking second adjoints). +So we have B(c0(G)) ⊆ Bσ(ℓ∞(G)). +We have the following +15 + +Lemma 5.4. Let G be a countable discrete group. Then the canonical quotient map q : B(c0(G)) → +B(c0(G))/K(c0(G)) is isometric on Φ(ℓ1(G)). +Proof. Again, this follows, mutatis mutandis, as in the proof of [49, Proposition 4.5], replacing ℓ2(I) by +c0(I). Note that all elements of Φ(ℓ1(G)) commute with right translations in B(c0(G)). +We shall now prove +Theorem 5.5. B(ℓ∞) is not amenable. +Proof. Realize ℓ∞ as ℓ∞(G) for some countable discrete non-amenable group G, such as F2. Suppose that +B(ℓ∞(G)) is amenable. Put A := K(c0(G)), and consider the space B := A ⊕ Φ(ℓ1(G)) ⊆ B(c0(G)) ⊆ +Bσ(ℓ∞(G)); note that K(c0(G))∩Φ(ℓ1(G)) = {0} follows from [61, proof of Theorem 1]. Let q : B(c0(G)) → +B(c0(G))/K(c0(G)) be the canonical quotient map. By Lemma 5.4, q(Φ(ℓ1(G))) ⊆ B(c0(G))/K(c0(G)) is +closed. Hence B = q−1(q(Φ(ℓ1(G)))) is a closed subalgebra of Bσ(ℓ∞(G)) = Zt(A∗∗). By Theorem 2.5, B +is amenable. So the quotient Φ(ℓ1(G)) is amenable. Thus, ℓ1(G) is amenable, whence G is amenable by +Johnson’s classical result, [59, Theorem 2.1.10] or [56, Theorem 2.1.8] – a contradiction. +Acknowledgements +The first named author is partially supported by EPSRC grant EP/T030992/1. For the purpose of open +access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript +version arising. No data were created or analysed in this study. The second named author is partially +supported by NSERC; this support is gratefully acknowledged. +References +[1] F. Albiac, J.L. 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A panorama, Springer Monographs in Mathematics, Springer- +Verlag, New York, 2020. +[60] R.A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics. +Springer-Verlag London, Ltd., London, 2002. +[61] S. Sakai, Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664. +[62] T. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. +[63] C.J. Seifert, Averaging in Banach spaces, Ph.D. thesis, Kent State University, 1977. +[64] A. Szankowski, B(H) does not have the approximation property, Acta Math. 147 (1981), no. 1-2, +89–108. +[65] W.A. Veech, Short proof of Sobczyk’s theorem, Proc. Amer. Math. Soc. 28 (1971), 627–628. +[66] S. Wassermann, On tensor products of certain group C∗-algebras, J. Funct. Anal. 23 (1976), no. 3, +239–254. +[67] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25, Cam- +bridge University Press, Cambridge, 1991. +19 + +Authors’ affiliations +Matthew Daws +Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, United Kingdom +Email: matt.daws@cantab.net; mdaws@uclan.ac.uk +Matthias Neufang +School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S +5B6, Canada +and +Laboratoire de Math´ematiques Paul Painlev´e (UMR CNRS 8524), Universit´e de Lille, D´epartement de +Math´ematiques, 59655 Villeneuve d’Ascq Cedex, France +Email: mneufang@math.carleton.ca; matthias.neufang@univ-lille.fr +20 + diff --git a/C9E1T4oBgHgl3EQf-AYx/content/tmp_files/load_file.txt b/C9E1T4oBgHgl3EQf-AYx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9234b425061028886b7444da380f94bd36e1605 --- /dev/null +++ b/C9E1T4oBgHgl3EQf-AYx/content/tmp_files/load_file.txt @@ -0,0 +1,1289 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf,len=1288 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='03562v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='FA] 9 Jan 2023 (Non-)amenability of B(E) and Banach space geometry Matthew Daws and Matthias Neufang Abstract Let E be a Banach space, and B(E) the algebra of all bounded linear operators on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The question of amenability of B(E) goes back to Johnson’s seminal memoir [39] from 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We present the first general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space geometry of E, which imply that B(E) is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We cover all spaces for which this is known so far (with the exception of one particular example), with much shorter proofs, such as ℓp for p ∈ [1, ∞] and c0, but also many new spaces: the numerous classes of spaces covered range from all Lp-spaces for p ∈ (1, ∞) to Lorentz sequence spaces and reflexive Orlicz sequence spaces, to the Schatten classes Sp for p ∈ [1, ∞], and to the James space J, the Schlumprecht space S, and the Tsirelson space T , among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Our approach also highlights the geometric difference to the only space for which B(E) is known to be amenable, the Argyros–Haydon space, which solved the famous scalar-plus-compact problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 1 Introduction Amenability of Banach algebras is a central notion in functional analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', [56, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We mention Johnson’s fundamental result [39] that a locally compact group G is amenable if and only if its group algebra L1(G) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Amenability is also a key concept in operator algebra theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We note the deep classical result, due to Connes [21], Bunce–Paschke [14] and Haagerup [38], that for C∗-algebras, amenability is equivalent to nuclearity (the forward implication is due to Connes and Bunce–Paschke).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Together with work of Wassermann [66], it follows that the algebra B(H) of all bounded linear operators on a Hilbert space H is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' However, it was a long-standing open problem whether this holds for B(ℓp) for all p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, Johnson already asked in 1972 in [39], where he introduced the very notion of amenability, whether B(E) is ever amenable for an infinite-dimensional Banach space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Since then, determining if B(E) is amenable or not for a given infinite-dimensional Banach space E, has proven a very difficult open problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the survey article [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let us briefly recall the history of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For many years, ℓ2 remained the only space for which the problem was solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then Read showed in [54] that B(ℓ1) is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' His proof was simplified by Pisier in [51], using expanders, and further simplified by Ozawa in [47], where also a proof of non-amenability of B(ℓ2) is given avoiding the use of the above results of Connes, Bunce–Paschke and Wassermann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Moreover, it is shown by Ozawa in [47] that B(ℓ∞) and (implicitly) B(c0) are not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' However, the case of B(ℓp) for p ∈ (1, ∞)\\{2} remained open, as noted by Ozawa in [47], by Pisier in [51], and by Read in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In [57], Runde finally established the celebrated result that B(ℓp) is not amenable for any p ∈ (1, ∞), through a technically involved proof, building in particular on his earlier work with Daws ([26], [27]) and on Ozawa’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' More generally, it is shown in [57] that B(ℓp(E)) is non-amenable for all Lp-spaces E, where p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Moreover, Choi shows in [19] that B(L1[0, 1]) is non-amenable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' this is also proven by Aldabbas in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further, Choi gives a criterion, applying to E = T (ℓ2), the trace class operators on ℓ2, showing that B(E) is not amenable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='19 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' However, as noted by Dales (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [57, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5]), the Argyros–Haydon space, which solved the famous scalar-plus-compact problem, provides an example of an infinite-dimensional space E such that B(E) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Note that this space is a hereditarily indecomposable L∞-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 1 In this paper, we develop entirely different methods from the ones used so far and establish, employing our new techniques, the first general principles which encode non-amenability of B(E), for very wide classes of spaces E, in terms of the Banach space geometry of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For instance, we show that if E is reflexive with the approximation property and isomorphic to its square, then B(E) is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We also show that if E is infinite-dimensional and reflexive with a subsymmetric basis, then B(E) is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Our results cover all spaces known so far (with the exception of one particular example, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='19 below), with much shorter proofs, and many new (classes of) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The following is a list of spaces E – obviously always assumed infinite-dimensional – for which we show that B(E) is non-amenable, as a result of our general approach (we write “∼=” to denote a Banach space isomorphism): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' all Lp(Ω, B, µ)-spaces and, more generally, all Lp spaces for p ∈ (1, ∞) – this generalises the main result of [57];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the Lorentz sequence spaces d(w, p), the Garling sequence spaces g(w, p), and the Baernstein spaces Bp, for all p ∈ (1, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' all reflexive Orlicz sequence spaces whose Orlicz function satisfies the ∆2 condition at 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' all separable reflexive rearrangement invariant (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=') function spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the Schatten classes Sp for all p ∈ [1, ∞];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the non-commutative Lp-spaces Lp(M) for all p ∈ [2, ∞), whenever M is an infinite-dimensional von Neumann algebra such that Lp(M) has the approximation property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' c0, ℓ1, ℓ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the non-commutative counterparts of the above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', the spaces K(H), T (H) and B(H) of compact, trace class and all bounded linear operators on a separable Hilbert space H, and, more generally, K(ℓp), N(ℓp) and B(ℓp) for all p ∈ (1, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' C(K) for any infinite compact metric space K (that is, all separable C(K) spaces), and all separable L1(Ω, B, µ)-spaces (note that L∞[0, 1] ∼= ℓ∞, so this space is covered as well);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the Hardy spaces Hp for all p ∈ [1, ∞) (note that Hp ∼= Lp[0, 1] for p ∈ (1, ∞));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the vector-valued spaces Lp(µ, X) for all p ∈ (1, ∞), for σ-finite µ with Lp(µ) infinite-dimensional, whenever X∗ has the bounded approximation property and the Radon-Nikodym Property (for ex- ample, X is reflexive with the approximation property);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the vector-valued spaces C(K, X) for any infinite compact metric space K, whenever X∗ has the bounded approximation property and the Radon-Nikodym Property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' particular spaces such as the James space J, the Schlumprecht space S, and the Tsirelson space T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In Section 2, we first establish a result on a new hereditary property of amenability which generalises a well-known theorem of Gourdeau and Ghahramani–Loy–Willis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' From this we shall derive, in Section 3, the general criteria for non-amenability of B(E) in the case of reflexive spaces E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We treat the non-reflexive situation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In the last Section, we present an alternative, short proof of the non-amenability of B(ℓp) for all p ∈ (1, ∞] which uses operator algebra techniques and harmonic analysis, instead of Banach space geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We also present an “elementary” proof of the non-amenability of B(ℓ2), which has long been sought after: it is even shorter and of course avoids the use of nuclearity for C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 2 2 A new hereditary property of amenability The central idea of this paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For a Banach space E, write A(E) for the space of approximable operators, the norm closure of the finite-rank operators in B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Write K(E) for the compact operators on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Often A(E) = K(E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' this is true when E has the approximation property (AP), [60, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' When E is reflexive with the AP, we can identify the bidual K(E)∗∗ with B(E), where K(E)∗∗ is given the (first) Arens product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, K(E) is Arens regular, and so both Arens products agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This result goes back to [48, Theorem 2], see also [24, Section 6], and in more generality, [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Consequently, to study B(E), one might study the bidual of A = K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' A well-known result in this direction if that when A∗∗ is amenable, also A is amenable, a result shown by Gourdeau ([34], [35, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3]) and, independently, Ghahramani–Loy–Willis [33, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This result is not directly useful to us, as K(E) is often amenable (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [11, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We will thus substantially generalise this result below, and this generalisation will be central to our new approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' With this motivation outlined, we start with some generality, and consider a Banach algebra A and its bidual A∗∗ equipped with the first Arens product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let us recall the Arens products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We turn A∗, A∗∗ into A-bimodules in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We then define bilinear maps A∗∗ × A∗ → A∗, A∗ × A∗∗ → A∗ by ⟨F · µ, a⟩ = ⟨F, µ · a⟩, ⟨µ · F, a⟩ = ⟨F, a · µ⟩ (a ∈ A, µ ∈ A∗, F ∈ A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We then define bilinear maps ✷, ✸ : A∗∗ × A∗∗ → A∗∗ by ⟨F✷G, µ⟩ = ⟨F, G · µ⟩, ⟨F✸G, µ⟩ = ⟨G, µ · F⟩ (F, G ∈ A∗∗, µ ∈ A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Direct calculations show that these are Banach algebra products, the first and second Arens products, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The canonical map κA : A → A∗∗ is a homomorphism, and κA(a)✷F = a·F, F✷κA(a) = F·a, and similarly for ✸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Henceforth, we always equip A∗∗ with ✷ unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Our first step, following [33], is to link A∗∗ �⊗A∗∗ with (A�⊗A)∗∗, where �⊗ denotes the completed projective Banach space tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We wish to use slightly more concrete identifications than [33], and we shall hence identify (A�⊗A)∗ with B(A, A∗), say B(A, A∗) ∋ T ↔ ϕT ∈ (A�⊗A)∗ by ⟨ϕT , a ⊗ b⟩ = ⟨T(a), b⟩ (a, b ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (1) Compare with [60, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Define Ψ : A∗∗ �⊗A∗∗ → (A�⊗A)∗∗ by ⟨Ψ(F ⊗ G), ϕT ⟩ = ⟨F, T ∗(G)⟩ (F, G ∈ A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (2) Clearly Ψ extends by bi-linearity and continuity to a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let us just check that this agrees with [33, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7], so choose bounded nets (ai), (bj) in A converging weak∗ to F, G ∈ A∗∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then ⟨Ψ(F ⊗ G), ϕT ⟩ = ⟨F, T ∗(G)⟩ = lim i ⟨T ∗(G), ai⟩ = lim i lim j ⟨T(ai), bj⟩ = lim i lim j ⟨ϕT , ai ⊗ bj⟩, which is the same extension given by [33, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [5, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We now record some useful facts about this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1 ([33, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We have the following commutative diagram A∗∗ �⊗A∗∗ Ψ � (A�⊗A)∗∗ A�⊗A �� κA⊗κA � �� κA � ⊗A �q q q q q q q q q q With πA : A�⊗A → A the product map, and similarly for πA∗∗, we have that (πA)∗∗ ◦ Ψ = πA∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further- more, Ψ is an A-bimodule map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 3 An obvious, but key, property is that A(E) is always an ideal in B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We abstract this idea as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Firstly identify A with κA(A) ⊆ A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Suppose that B ⊆ A∗∗ is some closed subalgebra containing A as an ideal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' we write A ✂ B ⊆ A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As A is an ideal in B, naturally A is a B-bimodule, and hence also A�⊗A is a B-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In the standard way, hence also A∗ and A∗∗, so also A∗∗ �⊗A∗∗, and (A�⊗A)∗∗, become B-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' However, as B ⊆ A∗∗, also A∗∗ and A∗ have B-actions for the restriction of the A∗∗ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The two B actions on A∗ agree, while the left action of B on A∗∗ agrees with ✸, and the right action of B on A∗∗ agrees with ✷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In this proof, to avoid confusion, let us write b ⊲ a and a ⊳ b for a ∈ A, b ∈ B, to denote the B-bimodule actions arising from viewing A as an ideal in B, and similarly for the actions of B on A∗ and A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Given b ∈ B, a ∈ A, µ ∈ A∗, we find that ⟨b ⊲ µ, a⟩ = ⟨µ, a ⊳ b⟩ = ⟨a · b, µ⟩ = ⟨b, µ · a⟩ = ⟨b · µ, a⟩, and so b ⊲ µ = b · µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' To be precise, when we write “a · b” we are considering b ∈ A∗∗ and the natural A action on A∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' similarly b · µ is the A∗∗ action on A∗ where we view B as a subalgebra of A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Similarly ⟨µ ⊳ b, a⟩ = ⟨µ, b ⊲ a⟩ = ⟨b · a, µ⟩ = ⟨b, a · µ⟩ = ⟨µ · b, a⟩, so that µ ⊳ b = µ · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then, given b ∈ B, F ∈ A∗∗, µ ∈ A∗, we have ⟨b ⊲ F, µ⟩ = ⟨F, µ ⊳ b⟩ = ⟨F, µ · b⟩ = ⟨b✸F, µ⟩, ⟨F ⊳ b, µ⟩ = ⟨F, b ⊲ µ⟩ = ⟨F, b · µ⟩ = ⟨F✷b, µ⟩, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As both A∗∗ �⊗A∗∗ and (A�⊗A)∗∗ are B-bimodules, we might ask if Ψ is a B-bimodule map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, the situation is slightly complicated as both Arens products arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Given b ∈ B and F, G ∈ A∗∗ we have b · Ψ(F ⊗ G) = Ψ(b✸F ⊗ G), Ψ(F ⊗ G) · b = Ψ(F ⊗ G✷b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, we identify T ∈ B(A, A∗) with ϕT ∈ (A�⊗A)∗ as in (1), and in the proof we continue to write ⊲, ⊳ for the B-bimodule actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For b ∈ B, a1, a2 ∈ A, ⟨b ⊲ ϕT , a1 ⊗ a2⟩ = ⟨ϕT , a1 ⊗ a2 ⊳ b⟩ = ⟨T(a1), a2 ⊳ b⟩ = ⟨b ⊲ T(a1), a2⟩, ⟨ϕT ⊳ b, a1 ⊗ a2⟩ = ⟨ϕT , b ⊲ a1 ⊗ a2⟩ = ⟨T(b ⊲ a1), a2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set b ⊲ ϕT = ϕT1 and ϕT ⊳ b = ϕT2, so that T1(a1) = b ⊲ T(a1) and T2(a1) = T(b ⊲ a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then, for G ∈ A∗∗, ⟨T ∗ 1 (G), a1⟩ = ⟨G, b ⊲ T(a1)⟩ = ⟨T ∗(G ⊳ b), a1⟩, ⟨T ∗ 2 (G), a1⟩ = ⟨G, T(b ⊲ a1)⟩ = ⟨T ∗(G), b ⊲ a1⟩ = ⟨T ∗(G) ⊳ b, a1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then, for F, G ∈ A∗∗, from (2), and using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2, ⟨b · Ψ(F ⊗ G), ϕT ⟩ = ⟨Ψ(F ⊗ G), ϕT2⟩ = ⟨F, T ∗ 2 (G)⟩ = ⟨F, T ∗(G) ⊳ b⟩ = ⟨Ψ(b✸F ⊗ G), ϕT ⟩, ⟨Ψ(F ⊗ G) · b, ϕT ⟩ = ⟨Ψ(F ⊗ G), ϕT1⟩ = ⟨F, T ∗ 1 (G)⟩ = ⟨F, T ∗(G ⊳ b)⟩ = ⟨Ψ(F ⊗ G✷b), ϕT ⟩, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 4 We write Zt(A∗∗) for the (first) topological centre (denoted by Z(1) t (A∗∗) in [24, 25]), that is, Zt(A∗∗) = {F ∈ A∗∗ : F✷G = F✸G (G ∈ A∗∗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The following is now immediate from Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let A ✂ B ⊆ A∗∗ and suppose that B ⊆ Zt(A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then the B-bimodule actions on A∗∗ agree with the product ✷, and Ψ is a B-bimodule map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We now state and prove the main result of this Section, which shows that amenability of A∗∗ (or more generally a subalgebra) passes to B when A ✂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This generalises [35, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3] and [33, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8], where the statement is shown for the special case B = A and C = A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let A✂B ⊆ A∗∗ and suppose that B ⊆ Zt(A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let C ⊆ A∗∗ be a closed subalgebra which is amenable, with B ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The canonical maps A�⊗A → B�⊗B → C �⊗C → A∗∗ �⊗A∗∗ are all contractions, but the overall composition is an isometry, [60, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='14], and so each individual map must also be an isometry, and we identify each tensor product as a closed subspace of A∗∗ �⊗A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As B ⊆ Zt(A∗∗), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2 shows that each inclusion is also a B-bimodule map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As C is amenable, [56, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4] or [59, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5], it has a bounded approximate diagonal (di) ⊆ C �⊗C ⊆ A∗∗ �⊗A∗∗, that is, ∥c · di − di · c∥ → 0, ∥πC(di)✷c − c∥ → 0 (c ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (3) For each i let ni = Ψ(di) ∈ (A�⊗A)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As B ⊆ C, (3) holds for each member of B, and so it follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4 that for each b ∈ B, ∥b · ni − ni · b∥ = ∥Ψ(b · di) − Ψ(di · b)∥ ≤ ∥b · di − di · b∥ → 0, (4) ∥(πA)∗∗(ni)✷b − b∥ = ∥(πA)∗∗(Ψ(di))✷b − b∥ = ∥πA∗∗(di)✷b − b∥ = ∥πC(di)✷b − b∥ → 0, (5) the second claim using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The bi-adjoint of the inclusion A�⊗A → B�⊗B gives an (isometric) inclusion (A�⊗A)∗∗ → (B�⊗B)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For each i let mi be the image of ni in (B�⊗B)∗∗, and let m ∈ (B�⊗B)∗∗ be a weak∗-cluster point of the bounded net (mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As the B-bimodule actions on (B�⊗B)∗∗ are weak∗-continuous, it follows from (4) that b · m = m · b for each b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As A is a subalgebra of B, it follows that (πB)∗∗(mi) = (πA)∗∗(ni) for each i, and so (5) shows that (πB)∗∗(m)✷b = b for each b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' That is, m is a virtual diagonal for B, showing that B is amenable, [56, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4] or [59, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We immediately obtain the following Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let A ✂ B ⊆ A∗∗ and suppose that A is Arens regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' If A∗∗ is amenable, then so is B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 3 The general criteria in the reflexive case, and applications In this Section, we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 in the classical situation when E is reflexive with the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As explained above, it follows that if we set A = A(E) then A = K(E) and A∗∗ is isomorphic to B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As A is Arens regular in this case, the condition on Zt(A∗∗) is vacuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Our aim is to use contradiction to show that B(E) cannot be amenable, by applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6 to a suitable algebra B which is “obviously” not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Our technique for finding such B is the following idea which, combined with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 or Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6, is key to our simplified approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 5 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space, and suppose there exists T ∈ B(E) which is not compact, but with T 2 ∈ K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let B be the Banach algebra generated by K(E) and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B is the linear span of K(E) and T, and B is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As T 2 ∈ K(E) it is easy to see that B is the linear span of K(E) and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Consider the quotient algebra C = B/K(E), and let x be the image of T in this quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As T is non-compact, x ̸= 0, but x2 = 0 as T 2 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus C is the one-dimensional algebra spanned by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As C is obviously not unital, it cannot be amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus B also cannot be amenable, as amenability passes to quotient algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a reflexive Banach space with the AP such that there exists T ∈ B(E) which is not compact, but with T 2 ∈ K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6 applied with A = A(E), so that A∗∗ = B(E), and with B as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a reflexive Banach space with the AP such that E ∼= E0 ⊕ E1 with E0, E1 isomorphic as Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' When E is infinite-dimensional, B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let T ∈ B(E) be the composition of the projection from E onto E0, the isomorphism from E0 to E1, and the inclusion E1 → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then T 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As E is infinite dimensional, so are E0, E1, and hence T is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, by the Riesz Lemma, we can find a sequence of unit vectors (xn) in E0 with ∥xn − xm∥ ≥ 1/2 for n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Treating E0 ⊆ E, we have that (T(xn)) is a bounded sequence of vectors, with ∥T(xn) − T(xm)∥ ≥ c/2 for each n ̸= m, where c > 0 is a constant depending on the isomorphisms E ∼= E0 ⊕ E1 and E0 ∼= E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus T cannot be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The result then follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a reflexive Banach space with the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' If E ∼= E ⊕E then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the definition of tree translation equivalent Banach spaces used below, we refer the reader to [10, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This technical definition is useful to us precisely because it allows us to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a reflexive, tree translation equivalent Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By assumption, E has a tree translation equivalent basis, so in particular has a basis and hence has the AP, and by [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6], satisfies E ∼= E ⊕ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4 now yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Generalising the above idea, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a reflexive Banach space with the AP, such that there exist closed, infinite- dimensional, isomorphic subspaces E0 and E1, and a projection P from E onto E0 with P(E1) finite- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let T0 : E0 → E1 be an isomorphism, let P : E → E0 be a projection, and set T = T0P : E → E1 ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As PT0P(x) ∈ P(E1) for any x ∈ E, and as P(E1) is finite-dimensional, PT0P is compact, and so T 2 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As T(x) = T0(x) for each x ∈ E0, and as E0 is infinite-dimensional, and T0 an isomorphism, it follows that T is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The result now follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the next result, recall the notion of a subsymmetric basis, [46, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2], which is an unconditional basis (en) such that (eni) is equivalent to (en) for all increasing sequences (ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a reflexive Banach space with the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' If E has an infinite-dimensional com- plemented subspace with a subsymmetric basis, then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In particular, if E is an infinite-dimensional reflexive Banach space with a subsymmetric basis, then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let (ei) be the subsymmetric basis of the infinite-dimensional complemented subspace X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Put E0 := lin{e2i | i ∈ N} and E1 := lin{e2i−1 | i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As (ei) is subsymmetric, by definition the map ei �→ e2i extends linearly and continuously to an isomorphism between E and E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' similarly E and E1 are isomorphic, whence also E0 ∼= E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As X is complemented, we have a projection P from E onto X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As (ei) is unconditional, we have a projection Q from X onto E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Obviously, QP(E) = E0 and QP(E1) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6, we obtain that B(E) is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We now apply the above results to various classes of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We start with Lp-spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' for details on this very important class of spaces, we refer the reader to [3, Section 5] or [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We use [28, Section 23] below, which takes a different definition, but [28, Section 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3] shows that the latter covers the Lp-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be any infinite-dimensional Lp-space, where p ∈ (1, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' for instance, E is any infinite-dimensional Lp(Ω, B, µ) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be an Lp-space, for p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By [44, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1], E is isomorphic to a complemented subspace of an Lp space, so certainly reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further, by [28, Section 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6, Corollary 1], taking account of the aforementioned [28, Section 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3], E has the bounded approximation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Finally, by [44, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3], E contains a complemented subspace isomorphic to ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As ℓp has a symmetric basis, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7 yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The above corollary generalises [57, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4], the main result of [57], and provides a much shorter proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' More precisely, it is shown in [57, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4] that B(ℓp(E)) is non-amenable for any Lp-space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We note that ℓp(E) is again an Lp- or an L2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, E is isomorphic to a complemented subspace of some Lp-space, by [45, Corollary 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Hence, ℓp(E) is also isomorphic to a complemented subspace of some Lp-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus, ℓp(E) is an Lp- or an L2-space, by [45, Corollary 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (See also the introduction to [3, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=') The Baernstein space B2 was introduced by Baernstein in [8], and the p generalisations Bp by Seifert in his dissertation [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' They are now viewed as being strongly related to Tsirelson’s space, see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The Baernstein spaces Bp satisfy that B(Bp) is non-amenable for all p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Each Bp is reflexive and has a basis, hence the AP, and contains a complemented subspace iso- morphic to ℓp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [16, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As above, the claim now follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We now consider the most fundamental examples of non-commutative Lp spaces, namely the Schatten classes Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Recall that Sp is the collection of operators u in B(ℓ2) with Tr(|u|p) < ∞, and norm ∥u∥p = Tr(|u|p)1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For p ∈ (1, ∞), Sp is reflexive (having canonically dual Sq for 1 p + 1 q = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Letting Pn ∈ B(ℓ2) be the projection onto the first n coordinates, we have that PnuPn ∈ Sp for each u ∈ Sp, with ∥PnuPn∥p ≤ ∥u∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further, PnuPn → u is norm, hence showing that Sp has the (metric) approximation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For p ∈ (1, ∞) the Schatten class Sp satisfies that B(Sp) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The operation of projecting an operator in B(ℓ2) onto its diagonal restricts to Sp and gives a projection of Sp onto a subspace isomorphic to ℓp, see the discussion in [4, pages 84–85] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The claim now follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' More generally, given a von Neumann algebra M, one can consider the non-commutative Lp-spaces over M, denoted by Lp(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, for p ∈ (1, ∞), Lp(M) is reflexive (having canonically dual Lq(M) for 1 p + 1 q = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' To our knowledge, it is not in general known when Lp(M) has the (Banach space) AP, but there has been some study of when Lp(M) possesses various Operator Space approximation properties, 7 all of which imply the AP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' see for example [41] which shows in particular that for a discrete group Γ with the (group) approximation property, and with M = V N(Γ) the group von Neumann algebra, Lp(M) has the Operator Space Approximation Property, [41, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let p ∈ [2, ∞), and let M be an infinite-dimensional von Neumann algebra such that Lp(M) has the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(Lp(M)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By [53, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2], Lp(M) contains a complemented subspace isomorphic to ℓ2 or ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Under our hypothesis, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7 now applies to give the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In the following, we consider various classes of “classical” Banach spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' in the statement of the result we give references regarding the properties of the spaces needed for Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7 to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the following infinite-dimensional Banach spaces E we have that B(E) is non- amenable: (i) the Lorentz sequence spaces d(w, p) for all p ∈ (1, ∞), see [46, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='a, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='e];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (ii) the reflexive Orlicz sequence spaces whose Orlicz function satisfies the ∆2 condition at 0, see [46, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='a, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4], and for when such a space is reflexive, [46, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (iii) the Garling sequence spaces g(w, p) for all p ∈ (1, ∞), see [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (iv) the Schlumprecht space S, [62], which is reflexive, [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1] or [6, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Each Banach space above is infinite-dimensional and reflexive with a subsymmetric basis, which the given references show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7 hence implies the claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the Tsirelson space T, we follow the modern convention and view T as the dual of the original construction of Tsirelson, so T is as defined in [31, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the following Banach spaces E we have that B(E) is non-amenable: (i) all infinite-dimensional separable reflexive rearrangement invariant (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=') function spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (ii) the Tsirelson space T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Each Banach space above is infinite-dimensional, reflexive (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [31, Section 2] for T) and, by [10, Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4], tree translation equivalent (note that the Haar system is a tree translation equivalent basis in case (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Hence Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 yields the claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The isomorphism T ∼= T ⊕T, used by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5, also appears in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' There, the author defines a generalised way of constructing “Tsirelson-like” spaces, which are denoted by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The space T follows as a special case, see [9, page 209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The remarks after [9, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7] show that S ∼= S ⊕ T and so in particular, T ∼= T ⊕ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In this direction, we should also remark that A(T) is known to be non-amenable, [11, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8], and so [33, 35] shows also that B(T) cannot be amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Finally, we note that the uniformly convex Banach space E (of Tsirelson type) with symmetric basis which contains no isomorphic copy of any ℓp for p ∈ (1, ∞), constructed by Figiel–Johnson in [31, Section 4], also satisfies that B(E) is non-amenable, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We finish this Section by comparing these constructions with the one known example of an infinite- dimensional space E with B(E) amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 8 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Note that the Argyros–Haydon space X, [7], for which B(X) is amenable, [57, Corol- lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5], is a hereditarily indecomposable L∞-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Its dual X∗ is isomorphic to ℓ1 and hence has the AP, so X has the AP, [60, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Yet, besides being non-reflexive, the key geometric property of X of being hereditarily indecomposable is in stark contrast to the Banach space properties which we employ above, all of which say that the space is “very decomposable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 4 The case of non-reflexive Banach spaces In this Section, we treat the case when E is not reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then A(E) is not Arens regular, and so we need to consider the (first) topological centre when applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We follow [25], which is a dense article, so we recall some of the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Given a Banach space E we write N(E) for the nuclear operators on E, the image of E∗ �⊗E in B(E) equipped with the quotient norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Write I(E) for the integral operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' there is always a norm-decreasing inclusion N(E) ⊆ I(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Given x∗ ∈ E∗ and x ∈ E we write θx,x∗ for the rank-one operator y �→ ⟨x∗, y⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then A(E) is by definition the closed linear span of such operators in B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Trace duality gives a natural pairing between A(E) and I(E∗) which extends the pairing ⟨J, θx,x∗⟩ = Tr(Jθ∗ x,x∗) = ⟨J(x∗), x⟩ (J ∈ I(E∗), θx,x∗ ∈ A(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' With respect to this pairing, we have that A(E)∗ = I(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For T ∈ B(E∗∗) define η(T) = κ∗ E ◦ T ∗ ◦ κE∗ ∈ B(E∗), and let Q(T) = η(T)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Direct calculation shows that η(T ∗) = T for T ∈ B(E∗), as κ∗ ET ∗∗κE∗ = κ∗ EκE∗T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Similarly, given S ∈ B(E∗∗) we see that η(T ∗S) = κ∗ ES∗T ∗∗κE∗ = η(S)T and so Q(T ∗S) = T ∗Q(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Define a bilinear map ⋆ on B(E∗∗) by T ⋆ S = Q(T) ◦ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then ⋆ is a Banach algebra product, see [25, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We write W(E) for the ideal of weakly compact operators in B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Suppose that A(E) admits a bounded approximate identity (eα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This holds if and only if E∗ has the bounded approximation property (BAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let Φ0 ∈ A(E)∗∗ be a weak∗-cluster point of the net (eα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' There are bounded maps ψ1, ψ2 : B(E∗∗) → A(E)∗∗ = I(E∗)∗ given by ⟨ψ1(T), J⟩ = ⟨Φ0, η(TJ∗)⟩, ⟨ψ2(T), J⟩ = ⟨Φ0, η(T)J⟩ (T ∈ B(E∗∗), J ∈ I(E∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then ψ1 is an isomorphism onto its range, and a homomorphism B(E∗∗) → (A(E)∗∗, ✷), and ψ2 is a homomorphism (B(E∗∗), ⋆) → (A(E)∗∗, ✸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The map ψ2, restricted to {T ∗ : T ∈ B(E∗)}, is an isomorphism onto its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For a ∈ A(E) we have that ψ1(a∗∗) = ψ2(a∗∗) = κA(E)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For T ∈ W(E) we have that ψ1(T ∗∗) = ψ2(T ∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We have the identification Zt(A(E)∗∗) = � ψ2(T ∗) : T ∈ B(E∗), TI(E∗) ⊆ N(E∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The equivalence of A(E) having a bai and E∗ having the BAP is shown in [37], building on the work of many authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [25, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='17] shows the claims about ψ1, ψ2, while that ψ1(T ∗∗) = ψ2(T ∗∗) for T ∈ W(E) is observed at the end of the proof of [25, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Finally [25, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='22] shows the claim about Zt(A(E)∗∗), which is denoted by Z(1) t (A(E)∗∗) in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' To obtain a class of operators T ∈ B(E∗) with TI(E∗) ⊆ N(E∗), we use [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='31] or [60, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='47], which shows that when T ∈ W(E∗), we have TI(E∗) ⊆ N(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 9 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space such that E∗ has the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Suppose there is S ∈ W(E) \\ K(E) with S2 ∈ K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' That S is weakly compact means that S∗ is weakly compact by Gantmacher’s Theorem, [22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5] for example, and so ψ2(S∗∗) is in Zt(A(E)∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further, ψ1(S∗∗) = ψ2(S∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set A = A(E) and let C = {ψ1(T ∗∗) : T ∈ B(E)} ⊆ A∗∗, an algebra isomorphic to B(E), as B(E) → B(E∗∗), T �→ T ∗∗ is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let B0 be the algebra generated by A(E) = K(E) and S, so as S2 ∈ K(E), B0 is the linear span of A(E) and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let B be the image of B0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As ψ1(S∗∗) = ψ2(S∗∗) and ψ1, ψ2 agree on A, we see that B ⊆ Zt(A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' If B(E) is amenable, so is C, so by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5, B is amenable, hence B0 is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus B0/A is amenable, which as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1 leads to the required contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' So B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let K be an uncountable compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(C(K)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By Milutin’s Theorem, [55, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1], we have C(K) ∼= C[0, 1], so it is enough to consider E = C[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2, so we seek S ∈ W(E) \\ K(E) with S2 ∈ K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Firstly, suppose we can find any T ∈ W(E) \\ K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Define linear maps T0 : C[0, 1] → C[0, 1], T0(f)(s) = f(2s) (f ∈ C[0, 1], s ∈ [0, 1]), and also T1 : C[0, 1] → C[0, 1] in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Pick a small δ > 0 and a linear bijection ϕ : [1/2+δ, 1−δ] → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For f ∈ C[0, 1] define T1(f) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For 1/2+δ ≤ t ≤ 1−δ let T1(f)(t) = f(ϕ(t)), and for t < 1/2 let T1(f)(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set T1(f)(1/2) = T1(f)(1) = 0, and linearly interpolate on the intervals [1/2, 1/2 + δ] and [1 − δ, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then T0 is a metric surjection, T1 is an isometry, and hence S = T1TT0 ∈ W(E)\\K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As T0T1 = 0 also S2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In fact, the square of any weakly compact operator on any C(K) is always compact by a result of Grothendieck, [55, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2], and so already T works in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2, but we prefer to give this explicit construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' It hence remains to find a suitable T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We use [52, Example 6] which gives an example of an absolutely summing but non-compact map R : C[0, 1] → c0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' for instance, with rn : [0, 1] → {±1} the Rademacher functions, we may define R(f) = � � 1 0 rn(t)f(t) dt �∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By [60, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='20], R is weakly compact as it is absolutely summing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' It remains to find an isometry c0 → C[0, 1] which when composed with R will give us our required T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This is well-known (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', [55, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 (d)]) but we give a construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let f1 ∈ C[0, 1] be the piece-wise linear function with f1(0) = f1(1/2) = f1(1) = 0, f1(3/4) = 1, let f2 ∈ C[0, 1] be the piece-wise linear function with f2(0) = f2(1/4) = f2(1/2) = f2(1) = 0, f2(3/8) = 1, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then (fn) is a copy of the standard unit vector basis of c0, and the map c0 → C[0, 1], (an) �→ � n anfn is our isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Here the sum is to be interpreted pointwise, but as (an) ∈ c0, it actually converges absolutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' B(L1[0, 1]) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2 to E = L1[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As E∗ = L∞[0, 1] has the BAP, we need only find a suitable S ∈ W(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As E ∼= E ⊕ E, we can obtain S2 = 0 so long as we can find any R ∈ W(E) \\ K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' To find such an R, we can follow [19, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space containing a complemented subspace isomorphic to ℓp for some p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' If E∗ has the BAP then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We can easily find R ∈ B(ℓp) with R non-compact but R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, if (en) is the standard unit vector basis of ℓp, define R by R(e2n) = e2n+1 and R(e2n−1) = 0 for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let P from E onto ℓp be a projection, so as ℓp is reflexive, RP is weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As R2 = 0 also (RP)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As P is a projection onto ℓp and by the construction of R, we see that RP is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2 implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Note that the above yields another proof of the non-amenability of B(E) for any Lp-space E, where p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The Hardy spaces Hp for all p ∈ [1, ∞) satisfy that B(Hp) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For p ∈ (1, ∞) we have Hp ∼= Lp[0, 1] by the classical result [12], so the claim follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now consider H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By [42, Section 3], H1 contains a complemented subspace isomorphic to ℓ2 (this is due to Paley, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the references in [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Also, by [40, Corollary 1], the dual H∗ 1 ∼= BMO has the uniform approximation property, hence the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 entails that B(H1) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Next we derive a result concerning vector-valued Lp spaces over σ-finite measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' So let µ be a σ-finite measure, E a Banach space, and consider Lp(µ, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' When E∗ has the Radon-Nikodym Property (RNP), then Lp(µ, E)∗ = Lq(µ, E∗), where 1 p + 1 q = 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [30, Section IV, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the RNP, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', [30, Chapter III] or [25, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' All reflexive spaces, and all separable dual spaces have the RNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' More generally, E∗ has the RNP if and only if every separable subspace of E has separable dual, [30, Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2, Corollary 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We note that [30] works only with finite measures, but the σ-finite case is a routine generalisation from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' To apply our result, we wish to know when Lq(µ, E∗) has the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The following result is surely known, but as we have not found a suitable reference, we give a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E have the BAP, and let p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let µ be a σ-finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then Lp(µ, E) has the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We shall use the ∆p norm from [28, Chapter 7] which is not quite a “tensor norm” on Lp(µ) ⊗ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' in particular, it fails the usual mapping property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Nevertheless, Lp(µ) ⊗ E is dense in Lp(µ, E) for the norm ∆p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' It is easy to see that we can witness that Lp(µ) has the metric approximation property by finite-rank, positive operators (Ti), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [30, Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3, Example 11] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For any positive operator T on Lp(µ), we do have that T ⊗ idE is bounded for ∆p, with bound ∥T∥, and so extends to Lp(µ, E), see [28, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Any S ∈ B(E) extends to idLp(µ) ⊗ S on Lp(µ, E) with norm ∥S∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus if (Sj) is a bounded net of finite-rank operators on E witnessing that E has the BAP, then (Ti ⊗ Sj) is a bounded net of finite-rank operators on Lp(µ, E) which tends in the point-norm topology to the identity on Lp(µ) ⊗ E, and thus by boundedness and density, on all of Lp(µ, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Hence Lp(µ, E) has the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space such that E∗ has the BAP and the RNP (for example, E is reflexive with the AP), and let p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let µ be a σ-finite measure with Lp(µ) infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(Lp(µ, E)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The discussion above shows that under these hypotheses, Lp(µ, E) has dual Lq(µ, E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8, also Lq(µ, E∗) has the BAP, as E∗ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Furthermore, Lp(µ, E) contains a complemented subspace isomorphic to ℓp, by [17, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the definition of the James space J, we refer the reader to [46, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 11 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The James space J satisfies that B(J) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By combining [15, Theorem 5] with the remark after [15, Corollary 4] and [15, Corollary 3], we find that J admits a complemented subspace isomorphic to ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Also, as J has a shrinking basis, J∗ has a basis by [46, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1], and so the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1 When the nuclear and integral operators on E∗ agree If we assume that N(E∗) = I(E∗) then we can say more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Recall the discussion of the RNP after Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' A useful result is that when E∗ has the RNP, then N(E∗) = I(E∗) with equal norms, [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='18] and [60, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We continue with the notation from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space such that E∗ has the BAP, and N(E∗) = I(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then ψ1(T ∗) = ψ2(T ∗) for each T ∈ B(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Firstly, for general E, given T ∈ B(E∗) and J ∈ I(E∗), we always have ⟨ψ1(T ∗), J⟩ = ⟨Φ0, η(T ∗J∗)⟩ = ⟨Φ0, JT ⟩, ⟨ψ2(T ∗), J⟩ = ⟨Φ0, η(T ∗)J⟩ = ⟨Φ0, TJ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (6) As in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1, here Φ0 is a weak∗-cluster point of a bai (eα) for A(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now let J ∈ N(E∗), say J = θx∗,x∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then ⟨Φ0, JT ⟩ = ⟨Φ0, θx∗,T ∗(x∗∗)⟩ = lim α ⟨θx∗,T ∗(x∗∗), eα⟩ = lim α ⟨T ∗(x∗∗), e∗ α(x∗)⟩ = ⟨T ∗(x∗∗), x∗⟩, here using that e∗ α(x∗) → x∗ for each x∗ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, as θx,x∗eα = θx,e∗α(x∗) and ∥θx,x∗ − θx,x∗eα∥ → 0, it follows that ∥x∥∥x∗ − e∗ α(x∗)∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We similarly see that ⟨Φ0, TJ⟩ = ⟨Φ0, θT(x∗),x∗∗⟩ = lim α ⟨θT(x∗),x∗∗, eα⟩ = lim α ⟨x∗∗, e∗ α(T(x∗))⟩ = ⟨x∗∗, T(x∗)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' It follows that ⟨Φ0, TJ⟩ = ⟨Φ0, JT⟩, and by linearity and continuity, this holds for all J ∈ N(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The result now follows from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be a Banach space such that E∗ has the BAP, and N(E∗) = I(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let B be a closed subalgebra with K(E) ✂ B ⊆ B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' If any of B(E), B(E∗) or B(E∗∗) is amenable, then B is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set A = A(E), and let C = {ψ1(T ∗∗) : T ∈ B(E)} ⊆ A∗∗, an algebra isomorphic to B(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As I(E∗) = N(E∗), we know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1 that Zt(A∗∗) = {ψ2(T ∗) : T ∈ B(E∗)}, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='11, this equals {ψ1(T ∗) : T ∈ B(E∗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus C ⊆ Zt(A∗∗), and so when B(E) is amenable, also C is amenable, and hence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' When B(E∗) is amenable, we instead set C = {ψ1(T ∗) : T ∈ B(E∗)} ⊆ A∗∗, an algebra anti-isomorphic to B(E∗), so that C is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now C = Zt(A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We identify B with {ψ1(T ∗∗) : T ∈ B}, so that B ⊆ C, and A ✂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Finally, suppose that B(E∗∗) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As A∗ = I(E∗) = N(E∗) by hypothesis, and as E∗ has the BAP so that N(E∗)∗ = B(E∗∗), it follows that A∗∗ = B(E∗∗) with ψ1 being an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For this, see [25, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2] and [24, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now set C = A∗∗ which is thus amenable, and again identify B with {ψ1(T ∗∗) : T ∈ B}, so that B ⊆ Zt(A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5 implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let K be an infinite countable compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(C(K)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By [55, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5(d)], we know that C(K) contains an isometric copy of c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Using Sobczyk’s theorem, see [65], as C(K) is separable, there is a projection P from C(K) onto c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As K is countable, we have C(K)∗ = M(K) ∼= ℓ1(K) which is a separable dual space, and hence has the RNP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' also it has the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We then apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12 with B = K(C(K)) ⊕ CS for a suitable operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, we use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1, so we seek S non-compact with S2 compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' It is easy to find T ∈ B(c0) non-compact with T 2 = 0, compare the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set S = TP, so that S is non-compact as P is a projection onto c0, while PT = T so S2 = TPTP = T 2P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12 now yields that B(C(K)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let K be an infinite compact metric space, equivalently, let K be an infinite compact space with C(K) separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(C(K)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This follows immediately from Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We now consider the vector-valued spaces C(K, E) for a Banach space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' These can be realised as the injective tensor product C(K)ˇ⊗E, see [60, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The following generalises the previous corollary, in that we can take E = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let K be an infinite compact metric space, and let E be a Banach space such that E∗ has the BAP and the RNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(C(K, E)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We may identify C(K, E)∗ with the space of regular vector measures of bounded variation, defined on the Borel subsets of K, with values in E∗, see for example [60, page 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As E∗ has the RNP, [60, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='23] shows that this space coincides with M(K)�⊗E∗, paired against C(K)ˇ⊗E = C(K, E) in the canonical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As both M(K) and E∗ have the BAP, the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8 is readily adapted to show that M(K)�⊗E∗ = C(K, E)∗ has the BAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' also [29, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As C(K, E) = C(K)ˇ⊗E, fixing x ∈ E and µ ∈ E∗ with ∥x∥ = ∥µ∥ = ⟨µ, x⟩ = 1, the maps C(K)ˇ⊗E → C(K), f ⊗ y �→ ⟨µ, y⟩f and C(K) → C(K)ˇ⊗E, f �→ f ⊗ x establish that C(K) is a complemented subspace of C(K, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' When K is uncountable, we may use the complemented copy of C(K) together with the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3 to show that there is T ∈ B(C(K, E)) which is weakly compact but not compact, and with T 2 compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2 yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now suppose that K is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As E∗ has the RNP, we know that separable subspaces of E have separable duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let X ⊆ C(K, E) be separable, and let (fn) be a dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then {fn(k) : n ∈ N, k ∈ K} is a countable subset of E and so its closed linear span is a separable subspace of E, say E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then X ⊆ C(K, E0), and as K is countable, it follows easily that C(K, E0) is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We know that E∗ 0 is separable, and so C(K, E0)∗ = ℓ1(K)�⊗E∗ 0 is separable, as ℓ1(K) is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We have hence shown that separable subspaces of C(K, E) have separable dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' So C(K, E)∗ has the RNP and the BAP in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As in the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='13, C(K) contains a complemented copy of c0, and hence so does C(K, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We can now argue exactly as in the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='13 to see that B(C(K, E)) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' B(c0), B(ℓ1) and B(ℓ∞) are not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set E = c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then E∗ = ℓ1 has the BAP, and as a separable dual space, has the RNP, so that N(E∗) = I(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We can find S ∈ B(c0) \\ K(c0) with S2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Indeed, if (en) is the standard unit vector basis of c0 then define S by S(e2n) = e2n+1 and S(e2n−1) = 0 for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Choosing B = K(c0) ⊕ CS gives a non-amenable Banach algebra by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1, and then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12 shows that B(c0), B(ℓ1) and B(ℓ∞) are not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let E be an infinite-dimensional separable L1 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' There is a classification of such E, [67, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 83]: indeed, either E ∼= L1[0, 1] or E ∼= ℓ1, so the result follows from Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We now establish the non-commutative version of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let H be an infinite-dimensional separable Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then B(K(H)), B(T (H)) and B(B(H)) are not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set E = K(H), so that E∗ = T (H), the trace class operators, and E∗∗ = B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then E∗ has the BAP, indeed, even a (Schauder) basis, which follows from [60, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='25] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Also T (H) is a separable dual space, and so has the RNP, hence N(E∗) = I(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As in the proof of the next corollary, we can find an operator S ∈ B(E) which is non-compact with S2 = 0, thus showing that B = K(E) ⊕ CS is not amenable, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12 yields that B(K(H)), B(T (H)) and B(B(H)) are not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In [19, Section 4], Choi shows that when E is a Banach space with the BAP such that E∗ does not have the BAP, then B(E) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This criterion is rather restrictive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' the canonical example, thanks to Szankowski’s result [64], is E = T (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Choi’s argument uses again [37] which shows that, under these hypotheses, A(E) has a one-sided but no two-sided bounded approximate identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As A(E) is an ideal in B(E), this contradicts B(E) being amenable, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Our proof above that B(T (H)) is non-amenable avoids the use of the very deep result of Szankowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' There is to our knowledge one way to construct spaces which Choi’s result covers, giving non-amenability of B(E), and where our methods do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By [43, Corollary 3], see also [46, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='7(b)], start- ing with any Banach space F which fails to have the AP, one can show that there exists a Banach space E with the BAP (indeed, a Schauder basis) such that E∗ does not have the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' In fact, more generally, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For all p ∈ (1, ∞) we have that B(K(ℓp)), B(N(ℓp)) and B(B(ℓp)) are not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We proceed at first with some generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let F be a Banach space such that F ∗∗ is separable with the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Also F ∗ is separable, and a dual space, and so has the RNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further, F ∗ has the BAP, see [25, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='22] for example, and so there are bounded nets (ti), (sj) of finite-rank operators in B(F ∗), B(F ∗∗), respectively, converging in the point-norm topology to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Set E = K(F) = A(F), so that E∗ = I(F ∗) = N(F ∗) = F ∗ �⊗F ∗∗ by the hypotheses on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then (ti ⊗ sj) is a bounded net of finite-rank operators converging in the point-norm topology to the identity on F ∗ �⊗F ∗∗, showing that E∗ has the BAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As F ∗∗ and F ∗ are separable, also E∗ is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='12 to find that B(E), B(E∗) and B(E∗∗) are not amenable, provided a suitable B can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Now specialise to the case F = ℓp for p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let (en) be the usual unit vector basis of ℓp, and set F0, F1 to be the closed span of (e2n), (e2n−1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For i = 0, 1 there is a natural projection Pi : F → Fi, and an inclusion ιi : Fi → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Further, there is an isometry j : F0 → F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For x ∈ A(ℓp) define S(x) = jP0xP0 ∈ A(ℓp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As P0j = 0, we see that S2(x) = jP0S(x)P0 = jP0jP0xP0 = 0 for each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For y ∈ A(F0) we can set x = ι0yP0 ∈ A(ℓp), and then S(x) = jP0ι0yP0 = jyP0, so in particular, ∥S(x)∥ = ∥yP0∥ = ∥y∥ = ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This shows that S is non-compact, and so B = A(E) ⊕ CS is a suitable non-amenable algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Note that the Argyros–Haydon space X, for which B(X) is amenable, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='16, satisfies that X∗ is isomorphic to ℓ1, so X∗ has the BAP and the RNP, whence N(X∗) = I(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' However, as B(X) = K(X) ⊕ CidX, there is obviously no operator S ∈ B(X) \\ K(X) with S2 compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 14 5 Alternative proofs of the non-amenability of B(ℓp) for p ∈ (1, ∞] In this Section, we present alternative, quick proofs that B(ℓp) is non-amenable for all p ∈ (1, ∞] using operator algebra methods and harmonic analysis, rather than Banach space geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We first give a short proof of the non-amenability of B(ℓ2), avoiding the use of nuclearity for C∗-algebras (we remark that [13] quoted below was written when there was no relationship known between amenability and nuclearity for C∗-algebras, as noted therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Given a discrete group G, we write C∗ r (G) for its reduced group C∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Recall that K(ℓ2(G)) is Arens regular being a C∗-algebra, and the Arens product on K(ℓ2(G))∗∗ = B(ℓ2(G)) is the usual composition of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' B(ℓ2) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Realize ℓ2 as ℓ2(G) for some countable discrete non-amenable group G, such as F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Suppose that B(ℓ2(G)) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Put A := K(ℓ2(G)), and consider the C∗-subalgebra B := A ⊕ C∗ r (G) of B(ℓ2(G)) = A∗∗ (note that K(ℓ2(G)) ∩ C∗ r (G) = {0} by [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' As in Section 3 we can apply Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6 to see that amenability of B(ℓ2(G)) passes to B, and hence to the quotient C∗ r (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus G is amenable by [13, Proposition 2] – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the case of B(ℓp), p ∈ (1, ∞), we will argue similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We will consider the p-analogue of C∗ r (G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=', the algebra PFp(G) of p-pseudofunctions on G, defined as the Banach algebra generated in B(ℓp(G)) by λp(ℓ1(G)), where λp is the representation of ℓ1(G) on ℓp(G) given by left convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We are grateful to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Phillips for pointing out the following Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let G be a countable discrete group, and p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then the canonical quotient map q : B(ℓp(G)) → B(ℓp(G))/K(ℓp(G)) is isometric on PFp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' This is shown for p = 2 in [49, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5], and inspection of the proof shows that the argument carries over, mutatis mutandis, to the case of general p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the following, note that given a discrete group G, K(ℓp(G)) is Arens regular by [23, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='23], and the product on K(ℓp(G))∗∗ = B(ℓp(G)) is the usual composition of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' B(ℓp) is not amenable for any p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Realize ℓp as ℓp(G) for some countable discrete non-amenable group G, such as F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Suppose that B(ℓp(G)) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Put A := K(ℓp(G)), and consider the space B := A ⊕ PFp(G) (note that K(ℓp(G)) ∩ PFp(G) = {0} as the proof of [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2] for p = 2 carries over to the case of general p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let q : B(ℓp(G)) → B(ℓp(G))/K(ℓp(G)) be the canonical quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2, q(PFp(G)) ⊆ B(ℓp(G))/K(ℓp(G)) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Hence B = q−1(q(PFp(G))) is a closed subalgebra of B(ℓp(G)) = A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='6, B is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' So the quotient PFp(G) is amenable, whence G is amenable (see the proof of [32, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4], which uses work of Phillips [50]) – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We shall now give an alternative proof of the non-amenability of B(ℓ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' To this end, let G be a countable discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We recall that K(c0(G))∗∗ = B(ℓ∞(G)), with the first Arens product being the usual composition of operators, and Zt(K(c0(G))∗∗) = Bσ(ℓ∞(G)), where the latter denotes the maps in B(ℓ∞(G)) which are weak∗-weak∗-continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' see pages 59–61, in particular Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='2, in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' (This also follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1 in this special case when E = c0(G), as then E∗ = ℓ1(G) has the RNP and so the ideas of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1 apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=') We also recall that Φ : ℓ1(G) → B(c0(G)), where Φ(f)(g) = f ∗ g for all f ∈ ℓ1(G), g ∈ c0(G), is an isometric representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We see B(c0(G)) as a subalgebra of B(ℓ∞(G)) (by taking second adjoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' So we have B(c0(G)) ⊆ Bσ(ℓ∞(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We have the following 15 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let G be a countable discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Then the canonical quotient map q : B(c0(G)) → B(c0(G))/K(c0(G)) is isometric on Φ(ℓ1(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Again, this follows, mutatis mutandis, as in the proof of [49, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5], replacing ℓ2(I) by c0(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Note that all elements of Φ(ℓ1(G)) commute with right translations in B(c0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' We shall now prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' B(ℓ∞) is not amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Realize ℓ∞ as ℓ∞(G) for some countable discrete non-amenable group G, such as F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Suppose that B(ℓ∞(G)) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Put A := K(c0(G)), and consider the space B := A ⊕ Φ(ℓ1(G)) ⊆ B(c0(G)) ⊆ Bσ(ℓ∞(G));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' note that K(c0(G))∩Φ(ℓ1(G)) = {0} follows from [61, proof of Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Let q : B(c0(G)) → B(c0(G))/K(c0(G)) be the canonical quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='4, q(Φ(ℓ1(G))) ⊆ B(c0(G))/K(c0(G)) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Hence B = q−1(q(Φ(ℓ1(G)))) is a closed subalgebra of Bσ(ℓ∞(G)) = Zt(A∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='5, B is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' So the quotient Φ(ℓ1(G)) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Thus, ℓ1(G) is amenable, whence G is amenable by Johnson’s classical result, [59, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='10] or [56, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='8] – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Acknowledgements The first named author is partially supported by EPSRC grant EP/T030992/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' No data were created or analysed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' The second named author is partially supported by NSERC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 3, 239–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' [67] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25, Cam- bridge University Press, Cambridge, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' 19 Authors’ affiliations Matthew Daws Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, United Kingdom Email: matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='daws@cantab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='net;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' mdaws@uclan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='uk Matthias Neufang School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada and Laboratoire de Math´ematiques Paul Painlev´e (UMR CNRS 8524), Universit´e de Lille, D´epartement de Math´ematiques, 59655 Villeneuve d’Ascq Cedex, France Email: mneufang@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content=' matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='neufang@univ-lille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} +page_content='fr 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E1T4oBgHgl3EQf-AYx/content/2301.03562v1.pdf'} diff --git a/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf b/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e21d5d5fe15e4dd53eaabd7a403c8084e00fe2ad --- /dev/null +++ b/CdAyT4oBgHgl3EQf4fpA/content/2301.00786v1.pdf @@ -0,0 +1,3 @@ +version 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Peets10, Weiwei Zhao4,5, Yanfeng Guo6, +Shiyan Li9,11,12,ǂ, Wenge Yang2, ¶ and Bernd Büchner1,13,§ + +1Leibniz Institute for Solid State and Materials Research (IFW-Dresden), 01069 Dresden, Germany +2Center for High Pressure Science and Technology Advanced Research, 201203 Shanghai, China +3State Key Laboratory of Superhard Materials, Department of Physics, Jilin University, 130012 +Changchun, China +4State Key Laboratory of Advanced Welding & Joining, Harbin Institute of Technology, 150001 Harbin, +China +5Flexible Printed Electronics Technology Center, Harbin Institute of Technology (Shenzhen), 518055 +Shenzhen, China +6School of Physical Science and Technology, ShanghaiTech University, 200031 Shanghai, China +7Institute for Applied Physics, University of Science and Technology Beijing, 100083 Beijing, China +8Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, +East China Normal University, 200241 Shanghai, China +9State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, 200438 +Shanghai, China +10Institute of Solid State and Materials Physics, Technische Universität Dresden, 01069 Dresden, +Germany +11Collaborative Innovation Center of Advanced Microstructures, 210093 Nanjing, China +12Shanghai Research Center for Quantum Sciences, 201315 Shanghai, China +13Institute of Solid State and Materials Physics and Würzburg-Dresden Cluster of Excellence—ct.qmat, +Technische Universität Dresden, 01062 Dresden, Germany + + + +2 + +Abstract +The noncentrosymmetric RAlPn (R = rare earth elements, Pn = Si or Ge) with space- +inversion (SI) and/or time-reversal (TR) symmetry breaking host multiple types of +Weyl fermions, providing a fertile platform for the exploration of novel topological +states. In particular, when the magnetic configuration is coupled to electronic +wavefunctions, exotic anomalous transverse transport phenomena emerge. Here, by +employing electrical and thermoelectrical transport, we systematically study the +ferromagnetic Weyl semimetal CeAlSi and its nonmagnetic analog LaAlSi. For +LaAlSi, an anomalous Nernst effect (ANE) with an anomalous Nernst angle of ~ +15.4 at 15.3 K is revealed. In addition, quantum oscillations reveal five frequencies, +some of which possess nontrivial Berry phase. Moreover, a possible temperature- +induced Lifshitz transition is also unveiled. For CeAlSi, in addition to the anomalous +Hall effect (AHE), an ANE is also discovered. The AHE and ANE arise in the +paramagnetic state, and then are enhanced when temperature approaches the +ferromagnetic ordering temperature, evidencing the interplay between magnetism +and topology in CeAlSi. High-pressure electrical transport and x-ray diffraction +measurements demonstrate multiple phase transitions, i.e., a pressure-induced +Lifshitz transition at ~ 10 GPa, a magnetic transition from the ferromagnetic state +to a paramagnetic state beyond ~ 20 GPa, and a structural phase transition at ~ 40 +GPa. Under pressure, a sign change in anomalous Hall resistivity takes place at ~ 5.6 +GPa with an enhancement of anomalous Hall angle. These findings indicate that +LaAlSi and CeAlSi provide unique and tunable platforms to explore exotic +topological physics, phase transitions, and potential platforms for an array of +promising applications. + +Introduction +In the past decade, the success of theoretical predictions and experimental +confirmations of topological semimetals has propelled the development of research on +topological states of matter and topotronics1–5. In topological materials, particularly +significant efforts have been devoted to searching for and characterizing novel topological +states due to their exotic properties, such as the presence of low-energy excitations, +extremely large magnetoresistance (MR), topological surface states, Fermi arcs, chiral +anomaly1–5. The addition of magnetic elements in magnetic topological materials (MTMs) + +3 + +breaks the time-reversal (TR) symmetry, leading to various intriguing phenomena6–9, such +as intrinsic anomalous Hall/Nernst (AHE/ANE)1–4,10 and topological Hall/Nernst effects +(THE/TNE)11–17, and topological magnetic textures (for example, skyrmions11–18, +hedgehogs19,20, merons21, magnetic bubbles13, hopfions22). The interplay between +magnetic configuration and topology remains mysterious but is expected to be promising +for the realization of novel topological states. Hitherto research on this interplay is limited +to a few cases, for example, a magnetic-field-induced ideal type-II Weyl state in Mn(Bi1- +xSbx)2Te4 23,24, a magnetic-exchange-induced Weyl state in EuCd2Sb2 25, a spin- +fluctuation-induced Weyl semimetal state in EuCd2As2 17,26, a magnetism-induced +topological transition in EuAs3 27, and magnetization-tunable Weyl states in EuB6 28, etc. +To exploit more novel phenomena and elaborate the relationship, more systems are called +for. +Recently, the noncentrosymmetric RAlPn series (R = rare earth elements, Pn = Si, +Ge) were proposed and demonstrated to be Weyl semimetals21,29–49. Weyl semimetals host +low-energy excitations, namely Weyl fermions which can be described by the Weyl +equation with 2  2 complex Pauli matrices1–4. Weyl fermions arise in the vicinity of +doubly degenerate electronic band crossing point (the Weyl node), and a pair of Weyl +nodes possess opposite chirality1–4. In general, to attain Weyl states, space-inversion (SI) +or TR symmetry should be broken, and there are few cases in which SI and TR +symmetries are simultaneously broken1–4. For the nonmagnetic LaAlPn, SI symmetry is +naturally broken31,32, and the systems host two types of Weyl states (type-I and type-II), +evidenced by angle-resolved photoemission spectroscopy (ARPES) measurements in +LaAlGe31, and Shubnikov–de Haas (SdH) oscillations together with band calculations in +LaAlSi32. Moreover, for LaAlPn, a spin Hall angle that is comparable to MTe2 (M = W, +Mo) has been predicted50–54. More intriguingly, pressure-induced superconductivity and +robust topology against pressure up to 80.4 GPa have been uncovered in LaAlPn, making +LaAlPn potential candidates for the realization of topological superconductivity55. +In contrast to LaAlPn, both SI and TR symmetries are broken in the magnetic +siblings, i.e., RAlPn (R = Ce, Pr, Nd, Sm, Pn = Si, Ge), rendering them rare cases for +studying novel topological properties with the simultaneous breaking of SI and TR +symmetries. Indeed, various phenomena have been discovered, such as AHE/ANE and +possible axial gauge fields in PrAlGe36, a singular angular MR in CeAlGe49, Weyl- +mediated magnetism in NdAlSi29, and Weyl-mediated spiral magnetism and Kramers + +4 + +nodal lines in SmAlSi39,56. CeAlSi is a ferromagnetic Weyl semimetal with noncollinear +magnetic ordering41. Electrical transport measurements for CeAlSi revealed an +anisotropic AHE and a loop-shaped Hall effect (LHE) in the ferromagnetic state, and a +nontrivial Berry phase41. ARPES experiments unveiled surface Fermi arcs and bulk Weyl +cones, further demonstrating the existence of Weyl fermions42. The flat band stemming +from Ce 4f electrons was also detected, indicating that electronic correlations may also +play a role42. More interestingly, scanning superconducting quantum interference device +(sSQUID) and magneto-optical Kerr effect (MOKE) microscopy on CeAlSi found the +presence of nontrivial chiral domain walls that contributed to the topological properties43- +45. In previous studies, the AHE was only found in the ferromagnetic state41,45, but the +anomalous transverse transport in the paramagnetic state remains less explored. On top +of that, the interplay between magnetism and topology in CeAlSi has not been elaborated. +Moreover, pressure could serve as a useful knob to tune the crystal structure and +consequently the band structure and topological properties of CeAlSi45. However, the +evolution of the band structure and topological properties under higher pressure has not +been investigated40,45. +In this work, we study the electrical and thermoelectrical transport properties of +CeAlSi and its nonmagnetic counterpart LaAlSi at ambient pressure, and the pressure +evolution of the resistivity, the crystal structure, and the electronic band structure of +CeAlSi. Given that LaAlSi has a very similar electronic band structure to CeAlSi, thus +scrutinization on them would help to shed light on how the topology and magnetism affect +the anomalous transverse transport. For LaAlSi, an ANE with a giant anomalous Nernst +angle (ANA) of ~ 15.4% at 15.3 K is discovered. Distinct de Haas-van Alphen (dHvA) +and Nernst oscillations reveal five oscillation frequencies, and demonstrate the presence +of nontrivial Berry phase. In addition, a possible temperature-induced Lifshitz transition +is discovered. For CeAlSi, both ANE and AHE are unveiled. The ANE and AHE arise in +the paramagnetic state and then increase when temperature approaches the ferromagnetic +transition temperature (TC), which suggests that magnetism interacts with topology and +then facilitates the anomalous transverse transport. Under pressure, multiple phase +transitions are found. At P ~ 8.5 GPa, a pressure-induced Lifshitz transition occurs. The +Hall resistivity changes its sign from positive to negative, indicating the system changes +from hole- to electron-dominated transport, which has been verified by our DFT +calculations. With pressure, the TC increases till 13.2 GPa, beyond which pressure +gradually drives this system from the ferromagnetic state to a paramagnetic state, as also + +5 + +verified by our DFT calculations. Above 40 GPa, pressure induces a new structural phase. +These results suggest the ferromagnetic Weyl semimetal CeAlSi together with its non- +magnetic analog LaAlSi provide a fertile platform for studying the novel topological +states arising from the interplay among magnetism, topology, and electronic correlations. + +Results +Anomalous transverse transport and quantum oscillations in LaAlSi. RAlSi (R = La, +Ce) crystallize in the same tetragonal structure with the space group I41md (No. 109), as +shown in the inset of Fig. 1(a)32,41. High-quality single crystals of RAlSi (R = La, Ce) +have been synthesized through a flux method32,41. The largest natural surface is the ab +plane [Supplementary Fig. 1]. For all transport measurements relevant to this work, the +current or heat flow is applied in the ab plane with the magnetic field parallel to the c axis. +LaAlSi is a nonmagnetic semimetal, while CeAlSi possesses in-plane (ab plane) +noncollinear ferromagnetic ordering below ~ 10 K [Fig. 1(a)]32,41. The temperature +dependence of resistivity for LaAlSi and CeAlSi is displayed in Fig. 1(b). The +ferromagnetic ordering in CeAlSi is evident, while LaAlSi shows semimetallic behavior. +Figures 1(c) and 1(d) display the calculated electronic band structure and associated Berry +curvature for LaAlSi and CeAlSi, respectively. +LaAlSi possesses a nonlinear-field dependence in the Hall resistivity (yx) [Fig. 1(e)]. +In general, a nonlinear Hall resistivity profile signifies the coexistence of two types of +carriers, i.e., electrons and holes, which can be described by a two-carrier model. In this +fashion, we fit the Hall resistivity at several selected temperatures [Supplementary Fig. +S2(e)] and extract the carrier density and mobility [Supplementary Fig. S2(f)]. From the +fit, the mobility of hole carriers is two orders of magnitude larger than of electrons, but +the density is much lower. Strikingly, we found there is an anomaly in the temperature +dependence of density and mobility around ~ 12 K for both carriers, indicating a possible +temperature-induced Lifshitz transition, which will be discussed later. For nonmagnetic +topological systems, in addition to AHE/ANE, orbital magnetization is also proposed to +be relevant to Berry curvature for it is the integral of the product of anomalous Hall +conductivity and Fermi-Dirac function in energy space57. For LaAlSi, a ferromagnetic- +like hysteresis has been uncovered, the origin of which is unclear (see Supplementary +Note 3 for more details). +According to the Mott relation, the thermoelectric signals are proportional to the + +6 + +derivative of the conductivities with respect to Fermi energy at EF 10. This also applies to +the anomalous Hall conductivity (𝜎𝑥𝑦 +𝐴 ) and anomalous Nernst conductivity (𝑥𝑦 +𝐴 )10. +Therefore, thermoelectric transport is exquisitely sensitive to the band structure and the +anomalous contributions near EF. Figures 2(a) and 2(b) show the magneto-Seebeck and +Nernst signals at selected temperatures, respectively. Here, we plot 𝑆𝑥𝑥(𝐻)/𝑆𝑥𝑥(0) +[ 𝑆𝑥𝑥(𝐻) = 𝑆𝑥𝑥(𝐻) − 𝑆𝑥𝑥(0) ] and 𝑆𝑥𝑦/𝑇 for better comparison. The field- +independent anomalous components in Nernst are evident at high fields and low +temperatures. To further analyze the ANE, an empirical approach is adopted57: + +𝑆𝑥𝑦 = 𝑆𝑥𝑦 +𝑁 + 𝑆𝑥𝑦 +𝐴 , +(1) + +𝑆𝑥𝑦 +𝑁 = 𝑆0 +𝑁 +𝜇𝐵 +1+(𝜇𝐵)2, +(2) + +𝑆𝑥𝑦 +𝐴 = 𝑆𝑥𝑦 +𝐴 tanh (𝐵/𝐵0), +(3) +Here, 𝑆𝑥𝑦 +𝑁 and 𝑆𝑥𝑦 +𝐴 represent conventional and anomalous contributions, respectively. +𝑆0 +𝑁, 𝑆𝑥𝑦 +𝐴 , , and B0 denote the amplitude of the conventional semiclassical contribution, +the amplitude of the anomalous contribution, the carrier mobility, and the saturation field +above which the plateau appears. From the fit, the amplitude of the anomalous Nernst +signal (|𝑆𝑥𝑦 +𝐴 |/𝑇) is extracted for low temperatures, as shown in the inset of Fig. 2(b). +With increasing temperature, |𝑆𝑥𝑦 +𝐴 |/𝑇 peaks at ~ 10.3 K, and then decreases +monotonically, which is consistent with the temperature evolution of the density and +mobility. The temperature dependence of the anomalous Nernst angle ( ANA ≡ +arctan (𝑆𝑥𝑦 +𝐴 /𝑆𝑥𝑥) ~ 𝑆𝑥𝑦 +𝐴 /𝑆𝑥𝑥) is also plotted. The ANA also increases abruptly at ~ +10.3 K, and then reaches a maximum of ~ 15.4% at 15.3 K. Such a giant ANA found in +the nonmagnetic material LaAlSi is comparable to that found in the famous Heusler +ferromagnet Co2MnGa58-62. +Now, we focus on the quantum oscillations in LaAlSi. In addition to SdH oscillations +[Fig. 1(e)], quantum oscillations in thermoelectric signals and magnetization are also +evident [Figs. 2]. We analyzed the oscillatory components of different physical quantities +via fast Fourier transform (FFT) and compared them in Supplementary Fig. 4(b). In the +previous SdH oscillation study, two oscillation frequencies of 7.96 T and 47.78 T were +revealed32. However, five oscillation frequencies (4.2, 20.2, 28.5, 34.7, and 42.4 T +assigned to α, , , , and γ bands, respectively) are identified in this work [Fig. 2(c), +Supplementary Fig. S4]. To further verify these frequencies, electronic band calculations +(EF = -0.0015Ry) have been conducted, revealing 6, 21.5, 29.6, 35, 206.8, 357.4 T for hole + +7 + +pockets, and 24.1, 30.65, 46.85 T for electron pockets. These results coincide with each +other very well. Therefore, the α and  frequencies are denoted to hole pockets, and the γ +frequency to electron pocket. It is hard to assign the  and  frequencies as either electron +or hole pockets due to the broad peak. The pockets with the oscillation frequencies of +206.8 and 357.4 T cannot be resolved in our experiments nor in previous SdH studies in +magnetic fields up to 32 T32. +The quantum oscillations can be described by the Lifshitz-Kosevich (LK) +equation63–65. For dHvA, M follows the expression, + +𝑀 ∝ −𝐵𝜆𝑅𝑇𝑅𝐷𝑅𝑆sin [2𝜋( +𝐹 +𝐵 − 𝛾 − 𝛿)], +(4) +where 𝑅𝑇 = +2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵 +sinh(2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵), 𝑅𝐷 = exp (− +2𝜋2𝑘𝐵𝑚∗𝑇𝐷 +𝑒ℏ𝐵 +), and 𝑅𝑆 = cos ( +𝜋𝑔𝑎 +2 ). TD and +a are the Dingle temperature and the ratio of cyclotron effective mass (m*) to free electron +mass (m0). B is the average of the field range of the oscillations, 1/B = (1/B1 +1/B2)/2 with +B1 and B2 the minimum and maximum values, respectively. The phase factor −  −  in +Eq. (4) describes the oscillation of M, in which γ = 1/2 − 𝜙𝐵/2𝜋, and 𝜙𝐵 is the +Berry phase63–65. The phase shift  is 0 or  1/8 for a quasi-2-dimensional (quasi-2D) or +a corrugated 3-dimensional (3D) Fermi surface, respectively. m* can be extracted by +fitting the temperature dependence of the amplitude of the oscillations (the thermal +damping factor RT). The fit gives 0.028(2)m0, 0.042(2)m0, 0.0042(1)m0, and 0.038(2)m0 +for α, , γ, and , respectively [Fig. 2(e)]. Due to the weak amplitude of the  band in M +[Supplementary Fig. 4(a)], its m* cannot be obtained. The thermoelectric signal also +follows a similar expression, viz.57, + +𝐴 +𝑇 ∝ +2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵 +sinh (2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵), +(5) +where A is the amplitude of Sxy. The fit yields 0.026(3)m0, 0.042(2)m0, 0.0054(3)m0, +0.032(3)m0 and 0.069(2)m0 for α, , γ,  and , respectively [Fig. 2(e)], which are +consistent with those from the dHvA analysis. The Landau level index diagram is also +plotted, as shown in Fig. 2(f). Here, we assign integer indices to the peak positions, and +half-integer indices to valley positions in M (Sxy). Intercepts between −1/8 and 1/8 +indicate the existence of Berry phase, while those between 3/8 and 5/8 are trivial. Note +that because Sxx and Sxy are the diagonal and off-diagonal terms of tensor S, respectively, +the maxima in Sxy have a phase shift with a quarter of a period66,67. Therefore, we shift +the Sxy first, and then take the positions of peaks or valleys for the Landau level index + +8 + +diagram. From the linear fits to the data, we obtained the intercepts as seen in the inset to +Fig. 2(f), and we argue that α and  are trivial, while , γ, and  are nontrivial. + +Anomalous transverse transport in CeAlSi. Figure 3(a) shows the Hall resistivity of +CeAlSi at different temperatures. For better comparison, the data at 2 K [Fig. 1(f)] is +replotted. There is a turning point at ~ 2.5 T, above which the Hall resistivity profile with +a positive slope displays a linear dependence. The turning point persists up to ~ 10 K, and +then broadens and shifts to higher fields. Above ~ 100 K, the Hall resistivity displays +linear behavior. Figure 3(b) depicts the magnetization of CeAlSi at various temperatures +with magnetic field applied along the c axis. Above 15 K, the magnetization displays a +linear dependence, evidencing that CeAlSi is paramagnetic. When temperature decreases +below 15 K, the system approaches the regime of magnetic fluctuations, and nonlinear +components start to contribute. To obtain the anomalous contributions, we subtract the +linear background by adopting the expression, 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝜌𝑦𝑥 +𝐴 41,45. The anomalous +Hall resistivity is plotted in Figs. 1(f) and 3(c) for 2 K and higher temperatures, +respectively. A loop-shaped Hall effect (LHE), a hysteresis produced during the upward +and downward scan of fields, is also verified in our sample [right inset of Fig. 1(f)], as +reported in previous studies41,45. The LHE found in CeAlSi is rather unusual, and it was +proposed that the LHE may derive from topological surface states41. Note that both +magnetism and topology will contribute to Berry curvature, leading to the AHE10. To +further verify that the AHE at low temperature arises from magnetic ordering, we fit the +data at 2 K by using the equation 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝑅𝑠𝑀. As may be seen in the inset of Fig. +3(c), the turning point can be well fitted, implying that the AHE in the ferromagnetic state +has a close relation to magnetic ordering. The anomalous Hall conductivity [𝑥𝑦 +𝐴 = +𝜌𝑦𝑥 +𝐴 /(𝜌𝑦𝑥 +𝐴 2 + 𝜌𝑥𝑥 +2)] and anomalous Hall angle [AHA ≡ arctan (𝑥𝑦 +𝐴 /𝜎𝑥𝑥) ~ 𝑥𝑦 +𝐴 /𝜎𝑥𝑥, +𝜎𝑥𝑥 = 𝜌𝑥𝑥/(𝜌𝑥𝑥 +2 + 𝜌𝑦𝑥 +2) shown in Supplementary Fig. 5(b)] are also calculated. As +displayed in Fig. 3(d), the AHE (AHA) arises below ~ 100 K and then ascends with +temperature approaching the regime of magnetic fluctuations (below 15 K) that is defined +according to the dM/dH [see the lower panel in Fig. 3(h)]. When the system enters into +the ferromagnetic state, the anomalous 𝑥𝑦 +𝐴 and AHA not vary much. +To further address the anomalous transverse transport in CeAlSi, we performed +thermoelectrical transport measurements. Figures 3(e) and 3(f) show the magneto- +Seebeck and Nernst signals, respectively. ANE is clearly evident at low temperatures. We + +9 + +fit the data to Eq. (1), as shown in Supplementary Fig. 5(c), and the anomalous Nernst +signal (|𝑆𝑥𝑦 +𝐴 |/𝑇) is extracted [the inset to Fig. 3(f)]. Upon decreasing temperature below +~ 31 K, the ANE appears and attains a plateau below 15.3 K. Similarly, there is a sudden +enhancement of the ANA when the system enters into regime of magnetic fluctuations, +and then the ANA reaches ~ 9.5% at 7.8 K. The ANA of CeAlSi is smaller than LaAlSi, +although the Berry curvature of the former is nearly one order of magnitude larger than +the latter. The Nernst conductivity (𝑥𝑦 = 𝜎𝑥𝑥𝑆𝑥𝑦 + 𝜎𝑥𝑦𝑆𝑥𝑥) is shown in Supplementary +Fig. 5(d). The data for 31 K and 51.5 K nearly overlap, and therefore we take the data of +51.5 K as the contribution from ordinary Nernst signal. The anomalous Nernst +conductivity −xy is obtained by subtracting the ordinary contribution, i.e., −𝑥𝑦 = +−[𝑥𝑦(𝑇) − 𝑥𝑦(51.5 K)], as displayed in Fig. 3(g), which shows similar behavior. We +plot the contour plots of 𝑥𝑦 +𝐴 , −xy and dM/dH in Fig. 3(h). As mentioned above, +magnetization starts to display a nonlinear dependence below 15 K, indicating the onset +of magnetic fluctuations, and this is more evident in the dM/dH plot. Above 15 K, the +magnetization is linear. However, 𝑥𝑦 +𝐴 and −xy develop above 15 K, which is in sharp +contrast to the linear dependence in magnetization. This means that the AHE and ANE +do not scale with the magnetization, they arise from topology rather than magnetism. +When the system is in the vicinity of the temperature where magnetic fluctuations start to +play a role, 𝑥𝑦 +𝐴 and −xy are significantly enhanced, implying that magnetism interacts +with the topology, and the interplay between them facilitates the anomalous transverse +transport in CeAlSi. According to DFT calculations, the Weyl nodes arise from the SI +symmetry breaking, and the TR symmetry breaking does not change the classification of +topology but just shifts the positions of Weyl nodes in the BZ as the ferromagnetism acts +as a simple Zeeman coupling30,41. In MTMs, the coupling between magnetic +configuration and external magnetic field could produce various intermediate magnetic +or topological states, and hence the variation of AHE may root in these states23-28. +However, for CeAlSi, it was proposed that the angle between the noncollinear spins does +not change with applied magnetic field up to 8 T, which distinguishes the AHE in CeAlSi +from the THE41. Therefore, the enhancement of anomalous transverse transport in CeAlSi +possibly arises from the shift of the positions of Weyl nodes. + +Pressure-induced phase transitions in CeAlSi. Previously, high-pressure studies on +CeAlSi revealed a monotonic enhancement of the TC with increasing pressure up to 21.4 + +10 + +GPa40,45. The AHE and the LHE are suppressed with pressure up to 2.7 GPa, while the +negligible pressure effect on the magnetic structure and electronic band structure under +low pressure implies the importance of domain walls for the topological behavior in +CeAlSi43,45. Such tunable chiral magnetic domain walls were also reported in the +antiferromagnetic sibling CeAlGe47. Figure 4(a) displays the resistivity profiles at various +pressures, and the inset shows the device for electrical transport measurements. Under +pressure, the TC increases monotonically with pressure up to 13.2 GPa [Figs. 4(a) and +4(b)], beyond which it cannot be resolved. The pressure evolution of TC is roughly +consistent with previous reports40,45. Above 15.9 GPa, the resistivity shows metallic +behavior. Figure 4(c) shows the Hall resistivity at 2 K at various pressures. With +increasing pressure to 3.2 GPa, the Hall resistivity decreases slightly, followed by a slight +enhancement at 5.6 GPa. Surprisingly, when the pressure reaches 8.5 GPa, the slope of +the Hall resistivity changes sign abruptly, indicating a possible pressure-induced Lifshitz +transition. The contour profiles of the derivative of the normalized resistivity with respect +to temperature, the pressure evolutions of the Hall coefficient (RH), and the resistivity at +2 K are plotted in Figs. 4(e-f). As may be seen, the pressure-induced Lifshitz transition +seems to correspond to the evolution of magnetism. To further shed light on the transition, +we calculated the magnetic moments under pressure via DFT calculations, yielding +0.8364B, 0.9657B, 0.6811B, and 0.00256B for 0, 10, 20, and 40 GPa, respectively, +which is overall consistent with the experimental data. Thus, the enhancement of TC under +low pressure derives from the pressure-driven enhancement of magnetic moments. Under +higher pressure the magnetic moments decrease gradually, and then disappear, leading to +a magnetic phase transition from the ferromagnetic to a paramagnetic state. The pressure +evolution of magnetic moments provides a strong hint that the Lifshitz transition has a +close relation to the coupling between the electronic band structure and magnetic +configurations, although the evolution of the magnetic structure itself is hitherto unclear. +Now, we turn to the AHE under pressure. Figure 4(c) displays the Hall resistivity +profiles under various pressures at 2 K. We obtain the anomalous Hall resistivity by +subtracting the ordinary contribution, via 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝜌𝑦𝑥 +𝐴 . Upon increasing pressure, +𝜌𝑦𝑥 +𝐴 initially decreases, which is consistent with the previous study45. However, beyond +1.8 GPa, 𝜌𝑦𝑥 +𝐴 increases, reaching a maximum at 5.6 GPa. At 8.5 GPa, a sign change from +positive to negative accompanied by a slight reduction in 𝜌𝑦𝑥 +𝐴 implies that the dominant +carriers change from hole to electron. Upon further compression, 𝜌𝑦𝑥 +𝐴 decreases + +11 + +monotonically and then cannot be resolved above 32.5 GPa. The pressure-dependent +AHA and the absolute value of anomalous Hall conductivity |𝑥𝑦 +𝐴 | are also calculated, +as plotted in Fig. 4(g), which have a similar evolution as 𝜌𝑦𝑥 +𝐴 in Fig. 4(d). Supplementary +Figs. 6(c) and 6(d) show the longitudinal resistivity and conductivity, respectively. The +anomalous Hall angles are ~ 9.0% and ~ 9.7% for 0.6 GPa and 5.6 GPa, respectively. To +further shed light on the intrinsic AHE for pressurized CeAlSi, the anomalous Hall +conductivity as a function of the longitudinal conductivity is summarized in +Supplementary Fig. 7. For the intrinsic AHE, the anomalous Hall conductivity is +independent of the longitudinal conductivity (|𝑥𝑦 +𝐴 |vs.𝜎𝑥𝑥 ~ constant) 10,58–62,68. Clearly, +the data adhere to the universal law both at high and ambient pressures, verifying the +intrinsic nature of the AHE in CeAlSi. +Finally, to obtain more information about the pressure-induced phase transitions, we +investigate the pressure evolution of the crystal structures and electronic band structures. +Figure 5(a) displays the high-pressure XRD profiles. Under pressure, the crystal structure +with the space group of I41md persists up to 39.3 GPa. Upon further compression, a new +diffraction peak situated at ~ 10.9 arises, indicative of a structural phase transition. The +determination of the high-pressure phase is beyond the scope of this paper. The emerging +high-pressure phase coexists with the I41md phase up 60 GPa. The lattice constants are +extracted from Rietveld refinements, and the relative changes with respect to 1 GPa are +displayed in the upper panel of Fig. 5(c). The ratio of a/c is also plotted in the lower panel +of Fig. 5(c). As can be seen, in addition to the structural phase transition, there are two +anomalies at ~ 10 GPa and ~ 20 GPa, which correspond with the pressures where the +Lifshitz transition and the transition from the magnetic state to a paramagnetic state in +resistivity appear, respectively. The band structures at several selected pressures are +calculated, which remain overall unchanged [Supplementary Fig. 8], except that the hole +pockets along the –X line become smaller with pressure and then transform to electron +pockets at ~ 10 GPa [Figs. 5(d-i)], which confirms the pressure-induced Lifshitz transition +in CeAlSi. This also implies that the pockets along the –X line dominate the transport +behavior (the Hall coefficient under pressure changes from positive to negative) in +CeAlSi. At 0 GPa, the Weyl nodes along the –X line are located 74 meV above EF, +whereas they shift to −57 meV and −78 meV below EF for 10 GPa and 20 GPa, +respectively. This indicates that pressure serves as an ideal parameter to tune the crystal +structure of CeAlSi, which consequently has an effect on the evolution of topology + +12 + +accompanied by the changes of AHE. Since there is no distinct anomaly in the calculated +band structures for 10 GPa and 20 GPa, the structural anomalies probably arise from +magnetostriction/magnetoelastic effects that are altered by pressure45,47. + +Discussion +First, we discuss the possibility of the temperature-induced Lifshitz transition in +LaAlSi. Temperature-dependent carrier density and mobility show anomalies around ~ +12 K, indicating the topological change of Fermi surfaces, i.e., the Lifshitz transition. +Among topological materials, a temperature-induced Lifshitz transition has been reported +in a few cases, such as WTe2 69, ZrTe5 70, HfTe5 71, ZrSiSe 72, EuAs3 27, Bi4Br4 73, in which +it usually has a close relationship with the transport anomalies. To this end, we check and +compare the low-temperature data of LaAlSi, as shown in Supplementary Fig. 9. In the +heat capacity, no distinct anomaly can be distinguished. However, there is a weak +anomaly in the slope of the resistivity at ~ 15 K. In contrast to the resistivity, the anomaly +in the Seebeck signal is more evident, displaying a sudden ascent from 10 K to 15 K, akin +to WTe2 69. In addition, from the FFT results in quantum oscillations [Fig. 2(c) and +Supplementary Fig. 4(a)], the  band can be easily identified below 12 K but hardly +distinguished at higher temperatures. Therefore, if the topologically nontrivial  band is +relevant to the Lifshitz transition, it will be very interesting to investigate if or to what +extent the Weyl nodes are involved. Together, these findings provide a strong hint for a +temperature-induced Lifshitz transition in LaAlSi. Pressure induces superconductivity in +LaAlSi55, and there are two possible scenarios for its origin. First, the superconductivity +in LaAlSi may stem from a pressure-induced new structural phase. However, due to the +weak intensity of the new XRD peaks as also observed here in CeAlSi, the new phase +could not be resolved in previous study55. If this is the case, the qualification of LaAlSi +for +topological +superconductivity +should +be +scrutinized. Alternatively, +the +superconductivity in LaAlSi may arise from other effects, for example, the pressure- +enhanced electron-phonon coupling. +For CeAlSi, the local 4f-moments of Ce3+ interact within the lattice, leading to a +noncollinear ferromagnetic ordering41. Differing from LaAlSi that hosts both type-I and +type-II Weyl nodes, CeAlSi in the paramagnetic state possesses only type-I Weyl nodes， +and the TR breaking in the ferromagnetic state does not change the classification of Weyl +nodes41. Thus, to a certain extent, the band structure of CeAlSi is quite different from + +13 + +LaAlSi. For CeAlSi, electron or hole doping can easily alter the Hall resistivity, as verified +by Yang et al41. They measured five samples with the same residual resistivity ratios, but +the Hall resistivity profiles were very different41. All samples in their study exhibited +overall negative slopes of the Hall resistivity, indicative of the domination of electron +carriers, even when EF crossed both the electron and hole pockets41. By comparing the EF +values with the energy positions of the Weyl nodes, they proposed that the AHE and the +LHE arise from a set of Weyl nodes that lie 24 meV above EF 41. However, in this work, +both Sample 1 and Sample 2 of CeAlSi show positive slopes of the Hall resistivity, +implying the domination of hole carriers. The densities of hole carriers for Sample 1, and +Sample 2 at 0.6 GPa, are calculated through a linear fit to the high-field data, yielding +4.051021 cm-3 and 5.431020 cm-3, respectively. Therefore, our samples are hole-doped, +and the observation of the AHE/ANE (Fig. 3) and the LHE [Fig. 1(f)] may arise in the +vicinity of a set of Weyl nodes that are located 9 meV below EF. +Note that the Hall resistivity profiles of Sample 1 of CeAlSi are different from +Sample 2, and the carrier density of the former is about 7.5 times larger than the latter. +For CeAlSi, ARPES experiments revealed the presence of the band deriving from Ce 4f +electrons below EF 42. Therefore, Sample 1 is closer to the Ce 4f-electron band than +Sample 2, which may account for the discrepancies in Hall resistivity. This indicates that +the electronic correlation from Ce 4f electrons also plays a crucial role. As mentioned +above, nontrivial domain walls are also relevant to the topological properties45,47. Under +pressure, the anomalous Hall resistivity of CeAlSi is significantly enhanced. Considering +that the dimensions of the single crystal we used are comparable to the size of one single +domain41,44, the magnetic texture as well as the topological properties can be easily altered +by +the +domain-wall +landscapes +or +other +effects, +for +example, +magnetostriction/magnetoelastic effects43. As a consequence, pressure serves as an +efficient +route +to +tune +the +landscapes +of +domain +walls +and +the +magnetostriction/magnetoelastic effects, and then in turn affect the AHE. +In summary, by employing electrical, thermoelectrical transport, and high-pressure +techniques, we systematically studied the ferromagnetic Weyl semimetal CeAlSi and the +nonmagnetic LaAlSi. For LaAlSi, quantum oscillations reveal five oscillation frequencies +and the existence of nontrivial Berry phase. An ANE with a giant Nernst angle of 15.4% +has been unveiled. In addition, a possible temperature-induced Lifshitz transition is +uncovered. For CeAlSi at ambient pressure, we found that both AHE and ANE arise from + +14 + +the paramagnetic state and are then strengthened when temperature approaches the +ferromagnetic transition, implying that magnetism interacts with topology, and then their +interplay promotes the anomalous transverse transport. Under pressure, multiple phase +transitions are discovered, i.e., a Lifshitz transition at ~ 10 GPa, a magnetic transition +from the ferromagnetic state to a paramagnetic state beyond ~ 20 GPa, and a structural +phase transition above ~ 40 GPa. These results suggest that magnetic CeAlSi and LaAlSi +could serve as fertile and tunable platforms to explore novel topological states with +anomalous transverse transport, and the interplay among magnetism, topology, and +electronic correlations. + +During the preparation of this manuscript, we noticed that the anomalous Nernst effect +with different results from ours in CeAlSi has been reported by other workers74. + +Methods +Sample synthesis. For the growth of LaAlSi and CeAlSi single crystals, a self-flux +method was adopted, as described in the literature32. The as-grown single crystals were +characterized by x-ray diffraction (XRD) measurements, as shown in Supplementary Fig. +1. + +Electrical, thermoelectrical transport, and thermodynamic measurements. For +transport measurements, a single crystal was cut into a bar shape. A standard six-probe +method was used for the longitudinal resistivity and transverse Hall measurements. For +thermoelectrical transport measurements, the Seebeck and Nernst signal were measured +simultaneously, and the temperature gradient (T) was determined by a differential +AuFe/chromel-P thermocouple which had been calibrated carefully in magnetic fields. +The cold end of the thermocouple was directly connected to the heat sink, and the +temperature of the cold end was the same as the base temperature (TB) which was +measured by a Cernox thermometer. The temperature of the sample (Ts), i.e., the T used +in Figs. 2 and 3, was determined to be the average of the cold and the hot ends, i.e., Ts = +TB + T/2. Electrical transport data were collected in a physical property measurement +system (PPMS, Quantum Design), and thermoelectrical transport data were collected in +a home-built 4He cryostat. Magnetic susceptibility and specific heat measurements were +performed in a magnetic property measurement system (MPMS, Quantum Design) and a + +15 + +PPMS, respectively. +For electrical transport measurements under high pressure, a van der Pauw method +was used, as shown in the inset of Fig. 4(a). The single crystal with a dimension of ~ 70 +m  70 m  15 m was cut by a focused ion beam (FIB) along the c axis, and the +surface of the single crystal is the ab plane. Magnetic field was applied perpendicular to +the plane. For the calculation of resistivity, the following equation is adopted75, + +𝜌 = +𝜋𝑑 +ln2 ( +𝑅𝐴𝐵+𝑅𝐴𝐷 +2 +) ∙ 𝑓( +𝑅𝐴𝐵 +𝑅𝐴𝐷), +where the function 𝑓(𝑥) satisfies the equation, + +exp (− +ln2 +𝑓(𝑥)) ∙ cosh [( +𝑥−1 +𝑥+1) +ln2 +𝑓(𝑥)] = 1/2, +d is the thickness of the sample, RAB and RAD are the resistance of the sample along +different directions, as shown in the inset of Fig. 4(a). Given the square shape of the +sample and the tetragonal structure of CeAlSi, RAB/RAD is assumed to be ~ 1. For x < 2.2, +𝑓(𝑥) ≈ 1/cosh (ln (𝑥)/2.403) with an error of less than 0.1%. + +Synchrotron XRD measurements under pressure. High-pressure angle-dispersive +XRD (wavelength: 0.434 Å) measurements of ground CeAlSi powder were performed at +beamline 13-BMC of the Advanced Photon Source, Argonne National Laboratory. The +powder of CeAlSi was loaded into a sample chamber sealed by a rhenium gasket. A +symmetric diamond anvil cell (DAC) was used to generate quasi-hydrostatic pressure +using silicone oil as the pressure-transmitting medium. The pressure inside the sample +chamber was determined by the shift of ruby fluorescence76. Experimental parameters +between the sample and detector were calibrated using the standard LaB6. All two- +dimensional XRD images were analyzed using Dioptas77, yielding one-dimensional +intensity versus diffraction angle patterns. Rietveld analyses were performed by using the +general structure and analysis system (GSAS) software78. + +Density functional theory (DFT) calculations. First-principles calculations were carried +out by using the Vienna ab initio Simulation Package (VASP)79,80. Exchange-correlation +effects were treated by using a Perdew-Burke-Ernzerhof (PBE)-type generalized gradient +approximation (GGA)81,82 with the projector-augmented-wave (PAW) potential83,84. An +on-site Coulomb interaction was added for Ce f-electrons within the GGA+U scheme +with Ueff = 6 eV. The cutoff energy of the plane-wave basis was fixed at 500 eV. A + +16 + +151515 -centered k mesh based on the Monkhorst-Pack method was selected to +sample the Brillouin zone. The energy and force difference criteria were defined as 10-6 +eV and 0.01 eV/Å for self-consistent convergence. To simulate paramagnetic CeAlSi, we +treated the 4f electrons on Ce as core electrons. 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E.J.C. acknowledges the financial support from the Alexander +von Humboldt Foundation. W.G.Y. acknowledges the National Natural Science +Foundation of China (Grant No. U1930401). S.Y.L. acknowledges the National Natural +Science Foundations of China (Grant No. 12174064), and the Ministry of Science and +Technology of China (Grant No. 2022YFA1402203). W.W.Z. is supported by the +Shenzhen Peacock Team Plan (KQTD20170809110344233) and the Bureau of Industry +and Information Technology of Shenzhen through the Graphene Manufacturing +Innovation Center (201901161514). Y.F.G. acknowledges research funds from the State +Key Laboratory of Surface Physics and Department of Physics, Fudan University (Grant +No. KF2019 06). Y.J.X. was supported by the National Natural Science Foundation of +China (Grant No. 12204033). + +Author Contributions +E.J.C. conceived the idea and designed the experiments. E.J.C., J.Y., W.X., and Y.F.G. +prepared the single crystals. E.J.C. and L.M.Y. were responsible for electrical transport +experiments under pressure. L.M.Y. and W.G.Y. conducted the high-pressure XRD +measurements and analysis. X.B.S. and W.W.Z. performed the DFT calculations for the +pressure evolution of the electronic band structure. E.J.C. and M.B. conducted +thermoelectric measurements. Y.M.W. helped with the data collection. Y.J.X. performed +the DFT calculations for the pressure evolution of the magnetic moments of CeAlSi. N.P. +and D.C.P. helped with the orientation of the single crystals for thermoelectric +measurements. E.J.C., L.M.Y., X.B.S., and Y.X. analyzed the data. E.J.C. wrote the paper. +E.J.C., S.Y.L., W.G.Y., and B.B. supervised the project. E.J.C., L.M.Y., X.B.S., and M.B. +contributed equally to this work. All authors discussed the results and commented on the +manuscript. + +Competing interests +The authors declare no competing interests. + +23 + + +Additional Information +Supplementary information is available for this paper at the URL inserted when +published. +Correspondence and requests for materials should be addressed to E.J.C. (e.cheng@ifw- +dresden.de). + +Figure captions +Figure 1 | Basic properties and Berry curvature of RAlSi (R = La, Ce). a Zero-field- +cooling (ZFC) and field-cooling (FC) magnetization as a function of temperature for +RAlSi with the magnetic field applied along the c axis. Inset shows the schematic structure +of RAlSi. RAlSi possesses a noncentrosymmetric structure with the space group of I41md. +b Longitudinal resistivity (xx) of LaAlSi and CeAlSi. Inset displays the derivative of +resistivity with respect to temperature. There is no distinct anomaly in LaAlSi, while a +peak situated at TC ~ 10.1 K corresponds to the ferromagnetic transition in CeAlSi. c and +d Band structure and associated Berry curvature for LaAlSi and CeAlSi, respectively. e +and f Transverse Hall resistivity (yx) and magnetoresistance [MR = xx(H)/xx(0)100%, +xx(H) = xx(H) − xx(0)] at 2 K with the magnetic field applied along the c axis for +LaAlSi and CeAlSi, respectively. The left and right insets in (f) display the anomalous +Hall resistivity (𝜌𝑦𝑥 +A ) after subtracting the ordinary contribution through 𝜌𝑦𝑥 = 𝑅0𝐵 + +𝜌𝑦𝑥 +𝐴 , and the loop-shaped Hall effect (LHE), respectively. + +Figure 2 | Anomalous Nernst effect (ANE) and quantum oscillations analysis of +LaAlSi. a Normalized magneto-Seebeck signal [Sxx(H)/Sxx(0), Sxx(H) = Sxx(H) - Sxx(0)] +at different temperatures with the magnetic field applied along the c axis. b Nernst signal +normalized to the temperature at different temperatures. Inset shows the temperature +dependence of the amplitude of anomalous Nernst signal normalized to the temperature +(|𝑆𝑥𝑦 +A |/𝑇 ) and anomalous Nernst angle (ANA). The bold dashed line represents the +empirical expression fit to the data of 7.1 K. c The fast Fourier transform (FFT) results at +various temperatures, derived from the oscillations in the Nernst signal. Inset displays the +oscillatory component, Sxy/T. Five oscillation frequencies (4.2, 20.2, 28.5, 34.7, and 42.4 +T assigned to α, , , , and γ bands, respectively) have been distinguished. d The de +Haas-van Alphen (dHvA) oscillations at different temperatures. e FFT amplitude (Amp.) + +24 + +as a function of temperature. The solid lines represent the fits to the Lifshitz-Kosevich +formula to obtain the cyclotron effective mass (m*). f Landau index n plotted against +1/0H for the Nernst and magnetization oscillations. Lines represent linear fits. The left +panel shows the extrapolation of 1/0H to zero. The right panel displays the intercepts for +α, β, γ,  and  pockets. + +Figure 3 | Anomalous Hall effect (AHE) and anomalous Nernst effect (ANE) of +CeAlSi. a Hall resistivity of CeAlSi at different temperatures with the magnetic field +applied along the c axis. As a comparison, the data of 2 K is replotted. b Field dependence +of magnetization at various temperatures with the magnetic field applied along the c axis. +Inset shows the low-field data. c Anomalous Hall resistivity (𝜌𝑦𝑥 +A ) at various temperatures. +Inset shows the representative fit to the Hall resistivity at 2 K through 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝑅𝑠𝑀. +d Anomalous Hall conductivity ( 𝑥𝑦 +A ) at various temperatures. Inset displays the +anomalous Hall angle (AHA). e Normalized magneto-Seebeck signal at different +temperatures with the magnetic field applied along the c axis. f Nernst signal normalized +to the temperature at different temperatures. Inset shows the temperature dependence of +the amplitude of anomalous Nernst signal normalized to the temperature (|𝑆𝑥𝑦 +A |/𝑇), and +the anomalous Nernst angle (ANA). g Anomalous Nernst conductivity (−xy) as a +function of field at several selected temperatures, −𝑥𝑦 = −[𝑥𝑦(𝑇) − 𝑥𝑦(51.5 K)]. +h Contour plots of the 𝑥𝑦 +A , −xy and the derivative of magnetization (dM/dH). The +background color represents the magnitude of their values. + +Figure 4 | Pressure-induced phase transitions in CeAlSi. a Temperature dependence +of longitudinal resistivity at different pressures. Inset shows a picture of the sample +chamber. The single crystal is ~ 70  70  15 m3. b Low-temperature resistivity +normalized to the data at 50 K. With increasing pressure, the ferromagnetic transition +temperature (TC) initially increases. c Hall resistivity at various pressures. Above 5.6 GPa, +the slope of Hall resistivity changes sign, indicating that the dominant carriers change +from holes to electrons. Inset shows the high-pressure data above 15.9 GPa. d Anomalous +Hall resistivity (𝜌𝑦𝑥 +A ) at various pressures. Inset shows the high-pressure data. e Contour +plot of the derivative of normalized resistivity at different pressures. The background +color represents the d(xx/𝜌𝑥𝑥 +50K )/dT value. The pressure evolution of TC is added. f + +25 + +Pressure-dependent Hall coefficient (RH) and the resistivity at 2 K (𝜌𝑥𝑥 +2K). RH is obtained +through linear fits to the high-field data. The shaded area represents the pressure region +where RH changes sign, suggesting the existence of a pressure-induced Lifshitz transition. +g Pressure dependence of anomalous Hall angle (AHA) and absolute value of anomalous +Hall conductivity (|𝑥𝑦 +A |). + +Figure 5 | Pressure evolution of the crystal structure and band structure of CeAlSi. +a X-ray diffraction (XRD) pattern of CeAlSi at room temperature up to 60 GPa. The +ambient-pressure structure with the space group of I41md persists to ~ 39.3 GPa, beyond +which a new diffraction peak emerges (marked with a dashed line and asterisk), indicating +that a pressure-induced structural phase transition occurs. 0 represents that the pressure +inside the sample chamber is released to zero, indicating that the emerging new structural +phase is unstable at ambient pressure. b The Rietveld refinement of the XRD pattern at +1.0 GPa. The refined value is RP = 2.23% with weighted profile RWP = 1.60%. The upper +panel in (c) shows the pressure-dependent normalized parameters a/a0, c/c0 and V/V0 +extracted from powder diffraction GSAS refinements. The lower panel in (c) shows the +pressure evolution of the a/c ratio. d-f Band structures of CeAlSi along the -W-X line +for 0 GPa, 10 GPa, and 20 GPa, respectively. g-i Calculated 3-dimensional (3D) Fermi +surfaces for 0 GPa, 10 GPa, and 20 GPa, respectively. The violet and dark yellow color +represent electron pockets and hole pockets, respectively. At 10 GPa, pressure drives hole +pockets (the red dashed circle as marked in g) into electron pockets, demonstrating the +pressure-induced Lifshitz transition observed in Hall resistivity under pressure. + + + +26 + +Figure 1 + + +E (eV) +-ΩZ(k) (10-4 Å-2) +-1 +0 +1 +0 +4 +2 +6 +Χ +Γ +Σ +Ζ Γ +ΝΣ1 +LaAlSi +-1 +0 +1 +0 +2 +3 +-1 +1 +Χ +Γ +Σ +Ζ Γ +ΝΣ1 +CeAlSi +a +b +e +c +d +f +E (eV) +-ΩZ(k) (10-3 Å-2) +c +a +b +Si +Al +R + +27 + +Figure 2 + +a +b +c +d +e +f + +28 + +Figure 3 + + +a +b +c +d +e +h +f +g + +29 + +Figure 4 + +a +c +d +b +f +g +Ruby +Sample +100 µm +A +B +C +D +e + +30 + +Figure 5 + + + + +pockect +pockect +- ++ +a +b +c +d +e +f +g +h +i +0 GPa +10 GPa +20 GPa +E (eV) +-0.6 +-0.3 +0 +0.3 +0.6 +W +0 GPa +-0.6 +-0.3 +0 +0.3 +0.6 +E (eV) +10 GPa +W +-0.6 +-0.3 +0 +0.3 +0.6 +E (eV) +20 GPa +W + +品031 + +Supplementary Information for +Anomalous transverse transport and phase transitions in Weyl +semimetals RAlSi (R = La, Ce) +Erjian Cheng1,*,⸙, Limin Yan2,3,*, Xianbiao Shi4,5,*, Mahdi Behnami1,*, Jian Yuan6, Yuanji Xu7,Yang +Xu8, Yimin Wan9, Wei Xia6, Nikolai Pavlovskii10, Darren C. Peets10, Weiwei Zhao4,5, Yanfeng Guo6, +Shiyan Li9,11,12,ǂ, Wenge Yang2, ¶ and Bernd Büchner1,13,§ + +1Leibniz Institute for Solid State and Materials Research (IFW-Dresden), 01069 Dresden, Germany +2Center for High Pressure Science and Technology Advanced Research, 201203 Shanghai, China +3State Key Laboratory of Superhard Materials, Department of Physics, Jilin University, 130012 +Changchun, China +4State Key Laboratory of Advanced Welding & Joining, Harbin Institute of Technology, 150001 Harbin, +China +5Flexible Printed Electronics Technology Center, Harbin Institute of Technology (Shenzhen), 518055 +Shenzhen, China +6School of Physical Science and Technology, ShanghaiTech University, 200031 Shanghai, China +7Institute for Applied Physics, University of Science and Technology Beijing, 100083 Beijing, China +8Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, +East China Normal University, 200241 Shanghai, China +9State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, 200438 +Shanghai, China +10Institute of Solid State and Materials Physics, Technische Universität Dresden, 01069 Dresden, +Germany +11Collaborative Innovation Center of Advanced Microstructures, 210093 Nanjing, China +12Shanghai Research Center for Quantum Sciences, 201315 Shanghai, China +13Institute of Solid State and Materials Physics and Würzburg-Dresden Cluster of Excellence—ct.qmat, +Technische Universität Dresden, 01062 Dresden, Germany + + +32 + +Supplementary Note 1: X-ray diffraction pattern of the as-grown RAlSi +(R = La, Ce) single crystals + + +Supplementary Figure 1 | X-ray diffraction (XRD) pattern of the as-grown RAlSi (R += La, Ce) single crystals. a XRD pattern from the largest natural surface of a RAlSi +single crystal. The largest natural surface is the ab plane. b Rocking curves of the (004) +peaks for RAlSi. Sample 1 and Sample 2 of CeAlSi came from different batches. The full +width at half maximum (FWHM) for LaAlSi is 0.12. The FWHMs for Sample 1 and +Sample 2 of CeAlSi are 0.02 and 0.04, respectively. The small values of FWHMs +indicate the high quality of the as-grown single crystals. Sample 1 of CeAlSi was used +for electrical and thermoelectrical transport measurements at ambient pressure, while +Sample 2 comes from another batch from which it was used for high-pressure electrical +transport measurements. For transport measurements, the magnetic field is applied +perpendicular to the ab plane. For thermoelectrical transport measurements, the +orientation of the samples was further verified by a Laue camera. + +Supplementary Note 2: Electrical transport behavior of LaAlSi + +Supplementary Figs. 2(a-d) show the resistivity, Hall resistivity, conductivity, and +Hall conductivity profiles at different temperatures, respectively. Supplementary Fig. 2(e) +displays representative fits to the Hall resitivity using a two-carrier model, i.e., 𝜌𝑦𝑥 = +− ( +𝐵 +𝑒) [(𝑛𝑒𝜇𝑒 +2 − 𝑛ℎ𝜇ℎ +2) + 𝜇𝑒 +2𝜇ℎ +2(𝑛ℎ − 𝑛𝑒)𝐵2]/[(𝑛𝑒𝜇𝑒 + 𝑛ℎ𝜇ℎ)2 + 𝜇𝑒 +2𝜇ℎ +2(𝑛ℎ − 𝑛𝑒)2𝐵2] , +where ne, nh, e, and h denote electron density, hole density, electron mobility, and hole +a +b + +33 + +mobility, respectively. The fitting results are displayed in Supplementary Fig. 2(f). The +density and mobility values of electrons are on the order of 1020 cm-3 and 10 cm2V-1s-1, +respectively, which are 4 orders of magnitude larger and 2 orders of magnitude smaller +than holes. The density of electrons we obtain is consistent with that previously reported1,2, +however the mobility is much lower. We also fit the data using a single-band model, and +the density is nearly the same (~ 1020 cm-3), but the mobility is ~ 2280 cm2V-1s-1 at 2 K, +comparable to the previous report1,2. These results suggest that the low-density hole +carriers with high mobility also play crucial roles. Strikingly, the temperature +dependences of density and mobility show an anomaly around ~ 12 K for both carriers, +indicating the change of Fermi surfaces, suggesting a possible temperature-induced +Lifshitz transition. + + +Supplementary Figure 2 | Electrical transport behavior of LaAlSi. a-d Longitudinal +resistivity, transverse Hall resistivity, longitudinal conductivity, and Hall conductivity of +LaAlSi at different temperatures, respectively. e The fits to Hall resistivity at different +temperatures using a two-carrier model. f Temperature dependence of carrier density and +mobility from the fitting. + + + +a +b +c +d +e +f + +34 + +Supplementary Note 3: Anomalous magnetization in LaAlSi + + +Supplementary Figure 3 | Anomalous magnetization in LaAlSi. a and b The +magnetization with the magnetic field applied along the c axis at selected temperatures +and an expanded view at low field of LaAlSi (Sample 1), respectively. c and d The +magnetization and an expanded view at low field with the magnetic field applied in plane +at selected temperatures, respectively. A ferromagnetic-like hysteresis up to room +temperature has been observed for both in-plane and out-of-plane fields. e Heat capacity +of Sample 1. No distinct anomaly that can be resolved, besides the phonon peak around +50 K, excluding a ferromagnetic transition in LaAlSi. f Comparison of the magnetization +for different samples at 2 K. Sample 2 with the residual resistivity ratio (RRR  +𝜌300K/𝜌2K) of ~ 3.7 comes from the same batch as Sample 1. Sample 3 (RRR ~ 3) and +Sample 4 (RRR ~ 1.4) come from two different batches. + +The magnetization behavior of LaAlSi does not resemble that of a paramagnet. A +ferromagnetic-like hysteresis at low fields for magnetic fields applied both in plane and +out of plane is very unusual [Supplementary Fig. 3(a-d)]. The temperature dependence of +zero-field-cooling (ZFC) and field-cooling (FC) magnetization [Fig. 1(a)], heat capacity +[Supplementary Fig. 3(e)], and the derivative of resistivity [the inset of Fig. 1(b)] do not +a +b +d +e +c +f + +(T) Ho +0'S +7.0- +0.0 +r.0 +S.0 +20 +300 +500 +100 +-0'4 +1 (K) +rB tn"-J) +0.0 +0'4 +Hsp +8.0(T) Hoμ +0. +2.0- +0.0 +2.0 +0.下 +-0'4 +0.0 +300 +0'4 +500 +100 +20 +1 (K) +HIIc +8.035 + +show any distinct anomalies, which implies the intrinsic nature of the anomaly in +magnetization. To further check the anomaly, we measure another three samples +[Supplementary Fig. 3(f)]. Sample 2 with the residual resistivity ratio of ~ 3.7 comes from +the same batch as Sample 1. Compared with Sample 1, the magnetization of Sample 2 +overall displays similar behavior, but the ferromagnetic-like hysteresis is weakened [the +inset of Supplementary Fig. 3(f)]. Sample 3 and Sample 4 (RRR ~ 1.4) come from another +two batches. Sample 3 (RRR ~ 3) and Sample 4 (RRR ~ 1.4) host similar magnetization +behavior. In contrast to Sample 1 and Sample 2, the magnitude of magnetization for +Sample 3 and Sample 4 is weakened. Quantum oscillations can be distinguished in +Sample 3, but cannot be resolved in Sample 4. Nevertheless, Sample 3 and Sample 4 +show a weak ferromagnetic-like hysteresis [the inset of Supplementary Fig. 3(f)]. These +results suggest that sample quality plays a significant role in the magnetization. For +Sample 3 and Sample 4 above ~ 2.5 T, the derivative of magnetization with respect to the +field (dM/dH) is negative, and such a negative value may originate from diamagnetism +of conduction carriers, as observed in the Mott insulator Ca2RuO43. For Cd3As2, a +ferromagnetic-like hysteresis at low fields was also reported, which was proposed to be +possibly relevant to the orbital magnetization4. The orbital magnetization follows the +equation 𝑀 = ∫ 𝑓(𝑥)𝜎𝑥𝑦 +𝐴 (𝜀)𝑑𝜀, where 𝜎𝑥𝑦 +𝐴 is the anomalous Hall conductivity at zero +temperature with Fermi energy , and f () is the Fermi-Dirac function [see Supplementary +Ref. 4 for more details]. Therefore, we argue that the anomaly in magnetization observed +in LaAlSi may be nontrivial, and probably relevant to Berry curvature. + + + + + +36 + +Supplementary Note 4: Fast Fourier transform (FFT) frequencies in +LaAlSi + + +Supplementary Figure 4 | The analysis of de Haas-van Alphen (dHvA) quantum +oscillations and the comparison of FFT frequencies from different physical +quantities. a FFT results of magnetization oscillations at various temperatures. Magnetic +field is applied along the c axis. Inset displays the oscillatory component M. b +Comparison of FFT results from the oscillations in different physical quantities, i.e., +Seebeck signal, resistivity, magnetization, and Nernst signal. Compared with Seebeck and +resistivity, the quantum oscillations in Nernst and magnetization were stronger, and more +FFT frequencies can be obtained. + + + + +a +b + +37 + +Supplementary Note 5: Basic preoperties of CeAlSi at ambient pressure + + +Supplementary Figure 5 | Magnetoresistance, conductivity, the fit to Nernst, and +Nernst conductivity of CeAlSi. a Magnetoresistance (MR) of CeAlSi at different +temperatures. b Longitudinal conductivity at different temperatures. c Fits to the Nernst +signal normalized to the temperature at selected temperatures. d Nernst conductivity +(−xy) as a function of field at several selected temperatures. The Nernst conductivities +for 31 and 51.5 K nearly overlap. To obtain the anomalous contributions, the data at 51.5 +K is taken as the ordinary contribution to be deducted. + + +a +b +c +d + +38 + +Supplementary Note 6: Magnetization (Sample 2) and electrical +transport results under pressure of CeAlSi + + +Supplementary Figure 6 | Magnetization (Sample 2) and electrical transport results +under pressure of CeAlSi. a Magnetization at 2 K with the magnetic field applied along +the c axis. The upper inset shows the low-field data. The lower inset displays the +temperature dependence of the magnetization with magnetic field of 0.1 T, revealing a +ferromagnetic transition temperature consistent with Sample 1 of CeAlSi and previous +studies5–8. b Hall conductivity at 0.6 GPa with the magnetic field applied along the c axis. +c and d Magnetoresistance (MR) and conductivity as a function of field at 2 K for different +pressures. + +a +b +c +d + +39 + +Supplementary Note 7: Universal scaling relation between the +anomalous Hall conductivity and the longitudinal conductivity in +CeAlSi. + + +Supplementary Figure 7 | Absolute value of anomalous Hall conductivity |𝐱𝐲 +𝐀 | as a +function of longitudinal conductivity xx of CeAlSi under ambient and high pressure. +For better comparison, several pure metals (Fe, Co, Ni, Gd)9, oxides [Nd2(MoNb)2O7, +La1-x(Sr,Ca)xMnO3, SrRuO3]10, chalcogenide spinels (Cu1-xZnxCr2Se4)11, magnetic +semiconductors (GaMnAs, anatase–Co–TiO2, rutile–Co–TiO2) 10, Co3Sn2S212, MnSi13, +Fe1-xCoxSi13, Mn3Ge14, and Mn3Sn14 have been plotted together. The solid lines in three +regimes represent |xy +A | µxx +1.6, |xy +A | ~ const., and |xy +A | µxx, for the dirty, intermediate, +and clean regimes, respectively9,15. For CeAlSi at both ambient and high pressures, the +anomalous Hall conductivity is located in the intermediate regime, suggesting an intrinsic +origin of the anomalous Hall effect. + + + + +40 + +Supplementary Note 8: The pressure evolution of band structure of +CeAlSi + + +Supplementary Figure 8 | The evolution of band structure of CeAlSi under pressure +with spin-orbit coupling (SOC) included. For all pressures, CeAlSi remains a Weyl +semimetal, and the band structure does not change much. With increasing pressure, the +hole pockets along the –X line become smaller, and then turn into electron pockets at +8.5 ~ 10 GPa, evidencing a pressure-induced Lifshitz transition. + + +-1 +0 +1 +Energy (eV) +-1 +0 +1 +Energy (eV) +0 GPa +5 GPa +8.5 GPa +10 GPa +15 GPa +20 GPa +30 GPa +24 GPa +40 GPa +-1 +0 +1 +Energy (eV) + +41 + +Supplementary Note 9: Possible evidence for the temperature-induced +Lifshitz transition in LaAlSi + + +Supplementary Figure 9 | Possible evidence for the temperature-induced Lifshitz +transition in LaAlSi. In the heat capacity, no distinct anomaly can be resolved below 30 +K. In the resistivity, the slops of the profile display a weak anomaly at ~ 15 K. The +temperature dependence of the Seebeck signal (−Sxx) ascends from ~ 10 K to ~ 15 K, +consistent with the anomalies in the temperature evolution of carrier density and mobility. +These results provide a strong hint that a temperature-induced Lifshitz transition exists in +LaAlSi. + +13.0 +13.2 +13.4 +13.6 +-0.1 +0.0 +0.1 +0 +2 +4 +6 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +2 +4 +6 +xx (W cm) +dxx/dT (a.u.) +LaAlSi +Cp (J mol-1K-1) +dCp/dT (a.u.) +0 T +-Sxx (VK-1) +T (K) + +42 + +Supplementary References +1. Su, H. et al. Multiple Weyl fermions in the noncentrosymmetric semimetal LaAlSi. +Phys. Rev. B 103, 165128 (2021). +2. Cao, W. et al. Pressure-induced superconductivity in the noncentrosymmetric Weyl +semimetals LaAlX (X = Si, Ge). Phys. Rev. B 105, 174502 (2022). +3. Mattoni, G., Yonezawa, S. & Maeno, Y. Diamagnetic-like response from localized +heating of a paramagnetic material. Appl. Phys. Lett. 116, 172405 (2020). +4. Liang, T. et al. Anomalous Nernst effect in the Dirac semimetal Cd3As2. Phys. Rev. +Lett. 118, 136601 (2017). +5. Xu, B. et al. Picoscale magnetoelasticity governs heterogeneous magnetic domains +in a noncentrosymmetric ferromagnetic Weyl semimetal. Adv. Quantum Technol. 4, +2000101 (2021). +6. Piva, M. M. et al. Tuning the nontrivial topological properties of the Weyl semimetal +CeAlSi. Preprint at http://arxiv.org/abs/2111.05742 (2021). +7. He, X. et al. Pressure tuning domain-wall chirality in noncentrosymmetric magnetic +Weyl semimetal CeAlGe. Preprint at http://arxiv.org/abs/2207.08442 (2022). +8. Yang, H.-Y. et al. Noncollinear ferromagnetic Weyl semimetal with anisotropic +anomalous Hall effect. Phys. Rev. B 103, 115143 (2021). +9. Miyasato, T. et al. Crossover behavior of the anomalous Hall effect and anomalous +Nernst effect in itinerant ferromagnets. Phys. Rev. Lett. 99, 086602 (2007). +10. Onoda, S., Sugimoto, N. & Nagaosa, N. Quantum transport theory of anomalous +electric, thermoelectric, and thermal Hall effects in ferromagnets. Phys. Rev. B 77, 165103 +(2008). +11. Lee, W.-L., Watauchi, S., Miller, V. L., Cava, R. J. & Ong, N. P. Anomalous Hall heat +current and Nernst effect in the CuCr2Se4−xBrx ferromagnet. Phys. Rev. Lett. 93, 226601 +(2004). +12. Liu, E. et al. Giant anomalous Hall effect in a ferromagnetic kagome-lattice +semimetal. Nat. Phys. 14, 1125–1131 (2018). +13. Manyala, N. et al. Large anomalous Hall effect in a silicon-based magnetic + +43 + +semiconductor. Nat. Mater. 3, 255–262 (2004). +14. Chen, T. et al. Anomalous transport due to Weyl fermions in the chiral +antiferromagnets Mn3X, X = Sn, Ge. Nat. Commun. 12, 572 (2021). +15. Onoda, S., Sugimoto, N. & Nagaosa, N. Intrinsic versus extrinsic anomalous Hall +effect in ferromagnets. Phys. Rev. Lett. 97, 126602 (2006). + diff --git a/F9E2T4oBgHgl3EQfTQcm/content/tmp_files/load_file.txt b/F9E2T4oBgHgl3EQfTQcm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa33d52ea5f5d6916d6df66df2c2ac30ad9b3419 --- /dev/null +++ b/F9E2T4oBgHgl3EQfTQcm/content/tmp_files/load_file.txt @@ -0,0 +1,1698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf,len=1697 +page_content='1 Anomalous transverse transport and phase transitions in Weyl semimetals RAlSi (R = La, Ce) Erjian Cheng1,*,⸙, Limin Yan2,3,*, Xianbiao Shi4,5,*, Mahdi Behnami1,*, Jian Yuan6, Yuanji Xu7,Yang Xu8, Yimin Wan9, Wei Xia6, Nikolai Pavlovskii10, Darren C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Peets10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Weiwei Zhao4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Yanfeng Guo6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Shiyan Li9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='ǂ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Wenge Yang2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' ¶ and Bernd Büchner1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='§ 1Leibniz Institute for Solid State and Materials Research (IFW-Dresden),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 01069 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Germany 2Center for High Pressure Science and Technology Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 201203 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 3State Key Laboratory of Superhard Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Jilin University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 130012 Changchun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 4State Key Laboratory of Advanced Welding & Joining,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Harbin Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 150001 Harbin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 5Flexible Printed Electronics Technology Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Harbin Institute of Technology (Shenzhen),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 518055 Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 6School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' ShanghaiTech University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 200031 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 7Institute for Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 100083 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 8Key Laboratory of Polar Materials and Devices (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' School of Physics and Electronic Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' East China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 200241 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 9State Key Laboratory of Surface Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 200438 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 10Institute of Solid State and Materials Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Technische Universität Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 01069 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Germany 11Collaborative Innovation Center of Advanced Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 210093 Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 12Shanghai Research Center for Quantum Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 201315 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 13Institute of Solid State and Materials Physics and Würzburg-Dresden Cluster of Excellence—ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='qmat, Technische Universität Dresden, 01062 Dresden, Germany 2 Abstract The noncentrosymmetric RAlPn (R = rare earth elements, Pn = Si or Ge) with space- inversion (SI) and/or time-reversal (TR) symmetry breaking host multiple types of Weyl fermions, providing a fertile platform for the exploration of novel topological states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In particular, when the magnetic configuration is coupled to electronic wavefunctions, exotic anomalous transverse transport phenomena emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Here, by employing electrical and thermoelectrical transport, we systematically study the ferromagnetic Weyl semimetal CeAlSi and its nonmagnetic analog LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For LaAlSi, an anomalous Nernst effect (ANE) with an anomalous Nernst angle of ~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4\uf025 at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 K is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In addition, quantum oscillations reveal five frequencies, some of which possess nontrivial Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Moreover, a possible temperature- induced Lifshitz transition is also unveiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi, in addition to the anomalous Hall effect (AHE), an ANE is also discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The AHE and ANE arise in the paramagnetic state, and then are enhanced when temperature approaches the ferromagnetic ordering temperature, evidencing the interplay between magnetism and topology in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' High-pressure electrical transport and x-ray diffraction measurements demonstrate multiple phase transitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', a pressure-induced Lifshitz transition at ~ 10 GPa, a magnetic transition from the ferromagnetic state to a paramagnetic state beyond ~ 20 GPa, and a structural phase transition at ~ 40 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under pressure, a sign change in anomalous Hall resistivity takes place at ~ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa with an enhancement of anomalous Hall angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These findings indicate that LaAlSi and CeAlSi provide unique and tunable platforms to explore exotic topological physics, phase transitions, and potential platforms for an array of promising applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Introduction In the past decade, the success of theoretical predictions and experimental confirmations of topological semimetals has propelled the development of research on topological states of matter and topotronics1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In topological materials, particularly significant efforts have been devoted to searching for and characterizing novel topological states due to their exotic properties, such as the presence of low-energy excitations, extremely large magnetoresistance (MR), topological surface states, Fermi arcs, chiral anomaly1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The addition of magnetic elements in magnetic topological materials (MTMs) 3 breaks the time-reversal (TR) symmetry, leading to various intriguing phenomena6–9, such as intrinsic anomalous Hall/Nernst (AHE/ANE)1–4,10 and topological Hall/Nernst effects (THE/TNE)11–17, and topological magnetic textures (for example, skyrmions11–18, hedgehogs19,20, merons21, magnetic bubbles13, hopfions22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The interplay between magnetic configuration and topology remains mysterious but is expected to be promising for the realization of novel topological states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Hitherto research on this interplay is limited to a few cases, for example, a magnetic-field-induced ideal type-II Weyl state in Mn(Bi1- xSbx)2Te4 23,24, a magnetic-exchange-induced Weyl state in EuCd2Sb2 25, a spin- fluctuation-induced Weyl semimetal state in EuCd2As2 17,26, a magnetism-induced topological transition in EuAs3 27, and magnetization-tunable Weyl states in EuB6 28, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To exploit more novel phenomena and elaborate the relationship, more systems are called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Recently, the noncentrosymmetric RAlPn series (R = rare earth elements, Pn = Si, Ge) were proposed and demonstrated to be Weyl semimetals21,29–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Weyl semimetals host low-energy excitations, namely Weyl fermions which can be described by the Weyl equation with 2 \uf0b4 2 complex Pauli matrices1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Weyl fermions arise in the vicinity of doubly degenerate electronic band crossing point (the Weyl node), and a pair of Weyl nodes possess opposite chirality1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In general, to attain Weyl states, space-inversion (SI) or TR symmetry should be broken, and there are few cases in which SI and TR symmetries are simultaneously broken1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For the nonmagnetic LaAlPn, SI symmetry is naturally broken31,32, and the systems host two types of Weyl states (type-I and type-II), evidenced by angle-resolved photoemission spectroscopy (ARPES) measurements in LaAlGe31, and Shubnikov–de Haas (SdH) oscillations together with band calculations in LaAlSi32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Moreover, for LaAlPn, a spin Hall angle that is comparable to MTe2 (M = W, Mo) has been predicted50–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' More intriguingly, pressure-induced superconductivity and robust topology against pressure up to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 GPa have been uncovered in LaAlPn, making LaAlPn potential candidates for the realization of topological superconductivity55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In contrast to LaAlPn, both SI and TR symmetries are broken in the magnetic siblings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', RAlPn (R = Ce, Pr, Nd, Sm, Pn = Si, Ge), rendering them rare cases for studying novel topological properties with the simultaneous breaking of SI and TR symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Indeed, various phenomena have been discovered, such as AHE/ANE and possible axial gauge fields in PrAlGe36, a singular angular MR in CeAlGe49, Weyl- mediated magnetism in NdAlSi29, and Weyl-mediated spiral magnetism and Kramers 4 nodal lines in SmAlSi39,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' CeAlSi is a ferromagnetic Weyl semimetal with noncollinear magnetic ordering41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Electrical transport measurements for CeAlSi revealed an anisotropic AHE and a loop-shaped Hall effect (LHE) in the ferromagnetic state, and a nontrivial Berry phase41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' ARPES experiments unveiled surface Fermi arcs and bulk Weyl cones, further demonstrating the existence of Weyl fermions42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The flat band stemming from Ce 4f electrons was also detected, indicating that electronic correlations may also play a role42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' More interestingly, scanning superconducting quantum interference device (sSQUID) and magneto-optical Kerr effect (MOKE) microscopy on CeAlSi found the presence of nontrivial chiral domain walls that contributed to the topological properties43- 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In previous studies, the AHE was only found in the ferromagnetic state41,45, but the anomalous transverse transport in the paramagnetic state remains less explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' On top of that, the interplay between magnetism and topology in CeAlSi has not been elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Moreover, pressure could serve as a useful knob to tune the crystal structure and consequently the band structure and topological properties of CeAlSi45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, the evolution of the band structure and topological properties under higher pressure has not been investigated40,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In this work, we study the electrical and thermoelectrical transport properties of CeAlSi and its nonmagnetic counterpart LaAlSi at ambient pressure, and the pressure evolution of the resistivity, the crystal structure, and the electronic band structure of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Given that LaAlSi has a very similar electronic band structure to CeAlSi, thus scrutinization on them would help to shed light on how the topology and magnetism affect the anomalous transverse transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For LaAlSi, an ANE with a giant anomalous Nernst angle (ANA) of ~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4% at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 K is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Distinct de Haas-van Alphen (dHvA) and Nernst oscillations reveal five oscillation frequencies, and demonstrate the presence of nontrivial Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In addition, a possible temperature-induced Lifshitz transition is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi, both ANE and AHE are unveiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The ANE and AHE arise in the paramagnetic state and then increase when temperature approaches the ferromagnetic transition temperature (TC), which suggests that magnetism interacts with topology and then facilitates the anomalous transverse transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under pressure, multiple phase transitions are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' At P ~ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 GPa, a pressure-induced Lifshitz transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The Hall resistivity changes its sign from positive to negative, indicating the system changes from hole- to electron-dominated transport, which has been verified by our DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' With pressure, the TC increases till 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2 GPa, beyond which pressure gradually drives this system from the ferromagnetic state to a paramagnetic state, as also 5 verified by our DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Above 40 GPa, pressure induces a new structural phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These results suggest the ferromagnetic Weyl semimetal CeAlSi together with its non- magnetic analog LaAlSi provide a fertile platform for studying the novel topological states arising from the interplay among magnetism, topology, and electronic correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Results Anomalous transverse transport and quantum oscillations in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' RAlSi (R = La, Ce) crystallize in the same tetragonal structure with the space group I41md (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 109), as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(a)32,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' High-quality single crystals of RAlSi (R = La, Ce) have been synthesized through a flux method32,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The largest natural surface is the ab plane [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For all transport measurements relevant to this work, the current or heat flow is applied in the ab plane with the magnetic field parallel to the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' LaAlSi is a nonmagnetic semimetal, while CeAlSi possesses in-plane (ab plane) noncollinear ferromagnetic ordering below ~ 10 K [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(a)]32,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The temperature dependence of resistivity for LaAlSi and CeAlSi is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The ferromagnetic ordering in CeAlSi is evident, while LaAlSi shows semimetallic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figures 1(c) and 1(d) display the calculated electronic band structure and associated Berry curvature for LaAlSi and CeAlSi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' LaAlSi possesses a nonlinear-field dependence in the Hall resistivity (\uf072yx) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In general, a nonlinear Hall resistivity profile signifies the coexistence of two types of carriers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', electrons and holes, which can be described by a two-carrier model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In this fashion, we fit the Hall resistivity at several selected temperatures [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' S2(e)] and extract the carrier density and mobility [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' S2(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' From the fit, the mobility of hole carriers is two orders of magnitude larger than of electrons, but the density is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Strikingly, we found there is an anomaly in the temperature dependence of density and mobility around ~ 12 K for both carriers, indicating a possible temperature-induced Lifshitz transition, which will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For nonmagnetic topological systems, in addition to AHE/ANE, orbital magnetization is also proposed to be relevant to Berry curvature for it is the integral of the product of anomalous Hall conductivity and Fermi-Dirac function in energy space57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For LaAlSi, a ferromagnetic- like hysteresis has been uncovered, the origin of which is unclear (see Supplementary Note 3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' According to the Mott relation, the thermoelectric signals are proportional to the 6 derivative of the conductivities with respect to Fermi energy at EF 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' This also applies to the anomalous Hall conductivity (𝜎𝑥𝑦 𝐴 ) and anomalous Nernst conductivity (\uf061𝑥𝑦 𝐴 )10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, thermoelectric transport is exquisitely sensitive to the band structure and the anomalous contributions near EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figures 2(a) and 2(b) show the magneto-Seebeck and Nernst signals at selected temperatures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Here, we plot \uf044𝑆𝑥𝑥(𝐻)/𝑆𝑥𝑥(0) [ \uf044𝑆𝑥𝑥(𝐻) = 𝑆𝑥𝑥(𝐻) − 𝑆𝑥𝑥(0) ] and 𝑆𝑥𝑦/𝑇 for better comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The field- independent anomalous components in Nernst are evident at high fields and low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further analyze the ANE, an empirical approach is adopted57: 𝑆𝑥𝑦 = 𝑆𝑥𝑦 𝑁 + 𝑆𝑥𝑦 𝐴 , (1) 𝑆𝑥𝑦 𝑁 = 𝑆0 𝑁 𝜇𝐵 1+(𝜇𝐵)2, (2) 𝑆𝑥𝑦 𝐴 = \uf044𝑆𝑥𝑦 𝐴 tanh (𝐵/𝐵0), (3) Here, 𝑆𝑥𝑦 𝑁 and 𝑆𝑥𝑦 𝐴 represent conventional and anomalous contributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 𝑆0 𝑁, \uf044𝑆𝑥𝑦 𝐴 , \uf06d, and B0 denote the amplitude of the conventional semiclassical contribution, the amplitude of the anomalous contribution, the carrier mobility, and the saturation field above which the plateau appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' From the fit, the amplitude of the anomalous Nernst signal (|\uf044𝑆𝑥𝑦 𝐴 |/𝑇) is extracted for low temperatures, as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' With increasing temperature, |\uf044𝑆𝑥𝑦 𝐴 |/𝑇 peaks at ~ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 K, and then decreases monotonically, which is consistent with the temperature evolution of the density and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The temperature dependence of the anomalous Nernst angle ( ANA ≡ arctan (\uf044𝑆𝑥𝑦 𝐴 /𝑆𝑥𝑥) ~ \uf044𝑆𝑥𝑦 𝐴 /𝑆𝑥𝑥) is also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The ANA also increases abruptly at ~ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 K, and then reaches a maximum of ~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4% at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Such a giant ANA found in the nonmagnetic material LaAlSi is comparable to that found in the famous Heusler ferromagnet Co2MnGa58-62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Now, we focus on the quantum oscillations in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In addition to SdH oscillations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(e)], quantum oscillations in thermoelectric signals and magnetization are also evident [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' We analyzed the oscillatory components of different physical quantities via fast Fourier transform (FFT) and compared them in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In the previous SdH oscillation study, two oscillation frequencies of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='96 T and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='78 T were revealed32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, five oscillation frequencies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='7, and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 T assigned to α, \uf068, \uf064, \uf062, and γ bands, respectively) are identified in this work [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(c), Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further verify these frequencies, electronic band calculations (EF = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0015Ry) have been conducted, revealing 6, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6, 35, 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='8, 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 T for hole 7 pockets, and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='65, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='85 T for electron pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These results coincide with each other very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, the α and \uf062 frequencies are denoted to hole pockets, and the γ frequency to electron pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' It is hard to assign the \uf068 and \uf064 frequencies as either electron or hole pockets due to the broad peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The pockets with the oscillation frequencies of 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='8 and 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 T cannot be resolved in our experiments nor in previous SdH studies in magnetic fields up to 32 T32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The quantum oscillations can be described by the Lifshitz-Kosevich (LK) equation63–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For dHvA, \uf044M follows the expression, \uf044𝑀 ∝ −𝐵𝜆𝑅𝑇𝑅𝐷𝑅𝑆sin [2𝜋( 𝐹 𝐵 − 𝛾 − 𝛿)], (4) where 𝑅𝑇 = 2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵 sinh(2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵), 𝑅𝐷 = exp (− 2𝜋2𝑘𝐵𝑚∗𝑇𝐷 𝑒ℏ𝐵 ), and 𝑅𝑆 = cos ( 𝜋𝑔𝑎 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' TD and a are the Dingle temperature and the ratio of cyclotron effective mass (m*) to free electron mass (m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' B is the average of the field range of the oscillations, 1/B = (1/B1 +1/B2)/2 with B1 and B2 the minimum and maximum values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The phase factor − \uf067 − \uf064 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' (4) describes the oscillation of \uf044M, in which γ = 1/2 − 𝜙𝐵/2𝜋, and 𝜙𝐵 is the Berry phase63–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The phase shift \uf064 is 0 or \uf0b1 1/8 for a quasi-2-dimensional (quasi-2D) or a corrugated 3-dimensional (3D) Fermi surface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' m* can be extracted by fitting the temperature dependence of the amplitude of the oscillations (the thermal damping factor RT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The fit gives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='028(2)m0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='042(2)m0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0042(1)m0, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='038(2)m0 for α, \uf062, γ, and \uf064, respectively [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Due to the weak amplitude of the \uf068 band in \uf044M [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(a)], its m* cannot be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The thermoelectric signal also follows a similar expression, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='57, 𝐴 𝑇 ∝ 2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵 sinh (2𝜋2𝑘𝐵𝑚∗𝑇/𝑒ℏ𝐵), (5) where A is the amplitude of \uf044Sxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The fit yields 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='026(3)m0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='042(2)m0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0054(3)m0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='032(3)m0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='069(2)m0 for α, \uf062, γ, \uf064 and \uf068, respectively [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(e)], which are consistent with those from the dHvA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The Landau level index diagram is also plotted, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Here, we assign integer indices to the peak positions, and half-integer indices to valley positions in \uf044M (\uf044Sxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Intercepts between −1/8 and 1/8 indicate the existence of Berry phase, while those between 3/8 and 5/8 are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Note that because Sxx and Sxy are the diagonal and off-diagonal terms of tensor S, respectively, the maxima in \uf044Sxy have a phase shift with a quarter of a period66,67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, we shift the \uf044Sxy first, and then take the positions of peaks or valleys for the Landau level index 8 diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' From the linear fits to the data, we obtained the intercepts as seen in the inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(f), and we argue that α and \uf064 are trivial, while \uf062, γ, and \uf068 are nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Anomalous transverse transport in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 3(a) shows the Hall resistivity of CeAlSi at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For better comparison, the data at 2 K [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(f)] is replotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' There is a turning point at ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 T, above which the Hall resistivity profile with a positive slope displays a linear dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The turning point persists up to ~ 10 K, and then broadens and shifts to higher fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Above ~ 100 K, the Hall resistivity displays linear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 3(b) depicts the magnetization of CeAlSi at various temperatures with magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Above 15 K, the magnetization displays a linear dependence, evidencing that CeAlSi is paramagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' When temperature decreases below 15 K, the system approaches the regime of magnetic fluctuations, and nonlinear components start to contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To obtain the anomalous contributions, we subtract the linear background by adopting the expression, 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝜌𝑦𝑥 𝐴 41,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The anomalous Hall resistivity is plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(f) and 3(c) for 2 K and higher temperatures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A loop-shaped Hall effect (LHE), a hysteresis produced during the upward and downward scan of fields, is also verified in our sample [right inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(f)], as reported in previous studies41,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The LHE found in CeAlSi is rather unusual, and it was proposed that the LHE may derive from topological surface states41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Note that both magnetism and topology will contribute to Berry curvature, leading to the AHE10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further verify that the AHE at low temperature arises from magnetic ordering, we fit the data at 2 K by using the equation 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝑅𝑠𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As may be seen in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(c), the turning point can be well fitted, implying that the AHE in the ferromagnetic state has a close relation to magnetic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The anomalous Hall conductivity [\uf073𝑥𝑦 𝐴 = 𝜌𝑦𝑥 𝐴 /(𝜌𝑦𝑥 𝐴 2 + 𝜌𝑥𝑥 2)] and anomalous Hall angle [AHA ≡ arctan (\uf073𝑥𝑦 𝐴 /𝜎𝑥𝑥) ~ \uf073𝑥𝑦 𝐴 /𝜎𝑥𝑥, 𝜎𝑥𝑥 = 𝜌𝑥𝑥/(𝜌𝑥𝑥 2 + 𝜌𝑦𝑥 2) shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 5(b)] are also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(d), the AHE (AHA) arises below ~ 100 K and then ascends with temperature approaching the regime of magnetic fluctuations (below 15 K) that is defined according to the dM/dH [see the lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' When the system enters into the ferromagnetic state, the anomalous \uf073𝑥𝑦 𝐴 and AHA not vary much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further address the anomalous transverse transport in CeAlSi, we performed thermoelectrical transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figures 3(e) and 3(f) show the magneto- Seebeck and Nernst signals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' ANE is clearly evident at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' We 9 fit the data to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' (1), as shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 5(c), and the anomalous Nernst signal (|𝑆𝑥𝑦 𝐴 |/𝑇) is extracted [the inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Upon decreasing temperature below ~ 31 K, the ANE appears and attains a plateau below 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Similarly, there is a sudden enhancement of the ANA when the system enters into regime of magnetic fluctuations, and then the ANA reaches ~ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5% at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The ANA of CeAlSi is smaller than LaAlSi, although the Berry curvature of the former is nearly one order of magnitude larger than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The Nernst conductivity (\uf061𝑥𝑦 = 𝜎𝑥𝑥𝑆𝑥𝑦 + 𝜎𝑥𝑦𝑆𝑥𝑥) is shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The data for 31 K and 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 K nearly overlap, and therefore we take the data of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 K as the contribution from ordinary Nernst signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The anomalous Nernst conductivity −\uf044\uf061xy is obtained by subtracting the ordinary contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', −\uf044\uf061𝑥𝑦 = −[\uf061𝑥𝑦(𝑇) − \uf061𝑥𝑦(51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 K)], as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(g), which shows similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' We plot the contour plots of \uf073𝑥𝑦 𝐴 , −\uf044\uf061xy and dM/dH in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As mentioned above, magnetization starts to display a nonlinear dependence below 15 K, indicating the onset of magnetic fluctuations, and this is more evident in the dM/dH plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Above 15 K, the magnetization is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, \uf073𝑥𝑦 𝐴 and −\uf044\uf061xy develop above 15 K, which is in sharp contrast to the linear dependence in magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' This means that the AHE and ANE do not scale with the magnetization, they arise from topology rather than magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' When the system is in the vicinity of the temperature where magnetic fluctuations start to play a role, \uf073𝑥𝑦 𝐴 and −\uf044\uf061xy are significantly enhanced, implying that magnetism interacts with the topology, and the interplay between them facilitates the anomalous transverse transport in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' According to DFT calculations, the Weyl nodes arise from the SI symmetry breaking, and the TR symmetry breaking does not change the classification of topology but just shifts the positions of Weyl nodes in the BZ as the ferromagnetism acts as a simple Zeeman coupling30,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In MTMs, the coupling between magnetic configuration and external magnetic field could produce various intermediate magnetic or topological states, and hence the variation of AHE may root in these states23-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, for CeAlSi, it was proposed that the angle between the noncollinear spins does not change with applied magnetic field up to 8 T, which distinguishes the AHE in CeAlSi from the THE41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, the enhancement of anomalous transverse transport in CeAlSi possibly arises from the shift of the positions of Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Pressure-induced phase transitions in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Previously, high-pressure studies on CeAlSi revealed a monotonic enhancement of the TC with increasing pressure up to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 10 GPa40,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The AHE and the LHE are suppressed with pressure up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='7 GPa, while the negligible pressure effect on the magnetic structure and electronic band structure under low pressure implies the importance of domain walls for the topological behavior in CeAlSi43,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Such tunable chiral magnetic domain walls were also reported in the antiferromagnetic sibling CeAlGe47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 4(a) displays the resistivity profiles at various pressures, and the inset shows the device for electrical transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under pressure, the TC increases monotonically with pressure up to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2 GPa [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(a) and 4(b)], beyond which it cannot be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The pressure evolution of TC is roughly consistent with previous reports40,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Above 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='9 GPa, the resistivity shows metallic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 4(c) shows the Hall resistivity at 2 K at various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' With increasing pressure to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2 GPa, the Hall resistivity decreases slightly, followed by a slight enhancement at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Surprisingly, when the pressure reaches 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 GPa, the slope of the Hall resistivity changes sign abruptly, indicating a possible pressure-induced Lifshitz transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The contour profiles of the derivative of the normalized resistivity with respect to temperature, the pressure evolutions of the Hall coefficient (RH), and the resistivity at 2 K are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(e-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As may be seen, the pressure-induced Lifshitz transition seems to correspond to the evolution of magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further shed light on the transition, we calculated the magnetic moments under pressure via DFT calculations, yielding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='8364\uf06dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='9657\uf06dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6811\uf06dB, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='00256\uf06dB for 0, 10, 20, and 40 GPa, respectively, which is overall consistent with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Thus, the enhancement of TC under low pressure derives from the pressure-driven enhancement of magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under higher pressure the magnetic moments decrease gradually, and then disappear, leading to a magnetic phase transition from the ferromagnetic to a paramagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The pressure evolution of magnetic moments provides a strong hint that the Lifshitz transition has a close relation to the coupling between the electronic band structure and magnetic configurations, although the evolution of the magnetic structure itself is hitherto unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Now, we turn to the AHE under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 4(c) displays the Hall resistivity profiles under various pressures at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' We obtain the anomalous Hall resistivity by subtracting the ordinary contribution, via 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝜌𝑦𝑥 𝐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Upon increasing pressure, 𝜌𝑦𝑥 𝐴 initially decreases, which is consistent with the previous study45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, beyond 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='8 GPa, 𝜌𝑦𝑥 𝐴 increases, reaching a maximum at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' At 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 GPa, a sign change from positive to negative accompanied by a slight reduction in 𝜌𝑦𝑥 𝐴 implies that the dominant carriers change from hole to electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Upon further compression, 𝜌𝑦𝑥 𝐴 decreases 11 monotonically and then cannot be resolved above 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The pressure-dependent AHA and the absolute value of anomalous Hall conductivity |\uf073𝑥𝑦 𝐴 | are also calculated, as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(g), which have a similar evolution as 𝜌𝑦𝑥 𝐴 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Supplementary Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 6(c) and 6(d) show the longitudinal resistivity and conductivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The anomalous Hall angles are ~ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0% and ~ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='7% for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further shed light on the intrinsic AHE for pressurized CeAlSi, the anomalous Hall conductivity as a function of the longitudinal conductivity is summarized in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For the intrinsic AHE, the anomalous Hall conductivity is independent of the longitudinal conductivity (|\uf073𝑥𝑦 𝐴 |vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='𝜎𝑥𝑥 ~ constant) 10,58–62,68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Clearly, the data adhere to the universal law both at high and ambient pressures, verifying the intrinsic nature of the AHE in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Finally, to obtain more information about the pressure-induced phase transitions, we investigate the pressure evolution of the crystal structures and electronic band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 5(a) displays the high-pressure XRD profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under pressure, the crystal structure with the space group of I41md persists up to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Upon further compression, a new diffraction peak situated at ~ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='9\uf0b0 arises, indicative of a structural phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The determination of the high-pressure phase is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The emerging high-pressure phase coexists with the I41md phase up 60 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The lattice constants are extracted from Rietveld refinements, and the relative changes with respect to 1 GPa are displayed in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The ratio of a/c is also plotted in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As can be seen, in addition to the structural phase transition, there are two anomalies at ~ 10 GPa and ~ 20 GPa, which correspond with the pressures where the Lifshitz transition and the transition from the magnetic state to a paramagnetic state in resistivity appear, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The band structures at several selected pressures are calculated, which remain overall unchanged [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 8], except that the hole pockets along the \uf047–X line become smaller with pressure and then transform to electron pockets at ~ 10 GPa [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 5(d-i)], which confirms the pressure-induced Lifshitz transition in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' This also implies that the pockets along the \uf047–X line dominate the transport behavior (the Hall coefficient under pressure changes from positive to negative) in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' At 0 GPa, the Weyl nodes along the \uf047–X line are located 74 meV above EF, whereas they shift to −57 meV and −78 meV below EF for 10 GPa and 20 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' This indicates that pressure serves as an ideal parameter to tune the crystal structure of CeAlSi, which consequently has an effect on the evolution of topology 12 accompanied by the changes of AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Since there is no distinct anomaly in the calculated band structures for 10 GPa and 20 GPa, the structural anomalies probably arise from magnetostriction/magnetoelastic effects that are altered by pressure45,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Discussion First, we discuss the possibility of the temperature-induced Lifshitz transition in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Temperature-dependent carrier density and mobility show anomalies around ~ 12 K, indicating the topological change of Fermi surfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', the Lifshitz transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Among topological materials, a temperature-induced Lifshitz transition has been reported in a few cases, such as WTe2 69, ZrTe5 70, HfTe5 71, ZrSiSe 72, EuAs3 27, Bi4Br4 73, in which it usually has a close relationship with the transport anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To this end, we check and compare the low-temperature data of LaAlSi, as shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In the heat capacity, no distinct anomaly can be distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, there is a weak anomaly in the slope of the resistivity at ~ 15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In contrast to the resistivity, the anomaly in the Seebeck signal is more evident, displaying a sudden ascent from 10 K to 15 K, akin to WTe2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In addition, from the FFT results in quantum oscillations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(c) and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(a)], the \uf068 band can be easily identified below 12 K but hardly distinguished at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, if the topologically nontrivial \uf068 band is relevant to the Lifshitz transition, it will be very interesting to investigate if or to what extent the Weyl nodes are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Together, these findings provide a strong hint for a temperature-induced Lifshitz transition in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Pressure induces superconductivity in LaAlSi55, and there are two possible scenarios for its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' First, the superconductivity in LaAlSi may stem from a pressure-induced new structural phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, due to the weak intensity of the new XRD peaks as also observed here in CeAlSi, the new phase could not be resolved in previous study55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' If this is the case, the qualification of LaAlSi for topological superconductivity should be scrutinized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Alternatively, the superconductivity in LaAlSi may arise from other effects, for example, the pressure- enhanced electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi, the local 4f-moments of Ce3+ interact within the lattice, leading to a noncollinear ferromagnetic ordering41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Differing from LaAlSi that hosts both type-I and type-II Weyl nodes, CeAlSi in the paramagnetic state possesses only type-I Weyl nodes， and the TR breaking in the ferromagnetic state does not change the classification of Weyl nodes41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Thus, to a certain extent, the band structure of CeAlSi is quite different from 13 LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi, electron or hole doping can easily alter the Hall resistivity, as verified by Yang et al41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' They measured five samples with the same residual resistivity ratios, but the Hall resistivity profiles were very different41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' All samples in their study exhibited overall negative slopes of the Hall resistivity, indicative of the domination of electron carriers, even when EF crossed both the electron and hole pockets41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' By comparing the EF values with the energy positions of the Weyl nodes, they proposed that the AHE and the LHE arise from a set of Weyl nodes that lie 24 meV above EF 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' However, in this work, both Sample 1 and Sample 2 of CeAlSi show positive slopes of the Hall resistivity, implying the domination of hole carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The densities of hole carriers for Sample 1, and Sample 2 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa, are calculated through a linear fit to the high-field data, yielding 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='05\uf0b41021 cm-3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='43\uf0b41020 cm-3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, our samples are hole-doped, and the observation of the AHE/ANE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3) and the LHE [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(f)] may arise in the vicinity of a set of Weyl nodes that are located 9 meV below EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Note that the Hall resistivity profiles of Sample 1 of CeAlSi are different from Sample 2, and the carrier density of the former is about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 times larger than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi, ARPES experiments revealed the presence of the band deriving from Ce 4f electrons below EF 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, Sample 1 is closer to the Ce 4f-electron band than Sample 2, which may account for the discrepancies in Hall resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' This indicates that the electronic correlation from Ce 4f electrons also plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As mentioned above, nontrivial domain walls are also relevant to the topological properties45,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under pressure, the anomalous Hall resistivity of CeAlSi is significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Considering that the dimensions of the single crystal we used are comparable to the size of one single domain41,44, the magnetic texture as well as the topological properties can be easily altered by the domain-wall landscapes or other effects, for example, magnetostriction/magnetoelastic effects43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As a consequence, pressure serves as an efficient route to tune the landscapes of domain walls and the magnetostriction/magnetoelastic effects, and then in turn affect the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In summary, by employing electrical, thermoelectrical transport, and high-pressure techniques, we systematically studied the ferromagnetic Weyl semimetal CeAlSi and the nonmagnetic LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For LaAlSi, quantum oscillations reveal five oscillation frequencies and the existence of nontrivial Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' An ANE with a giant Nernst angle of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4% has been unveiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In addition, a possible temperature-induced Lifshitz transition is uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi at ambient pressure, we found that both AHE and ANE arise from 14 the paramagnetic state and are then strengthened when temperature approaches the ferromagnetic transition, implying that magnetism interacts with topology, and then their interplay promotes the anomalous transverse transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Under pressure, multiple phase transitions are discovered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', a Lifshitz transition at ~ 10 GPa, a magnetic transition from the ferromagnetic state to a paramagnetic state beyond ~ 20 GPa, and a structural phase transition above ~ 40 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These results suggest that magnetic CeAlSi and LaAlSi could serve as fertile and tunable platforms to explore novel topological states with anomalous transverse transport, and the interplay among magnetism, topology, and electronic correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' During the preparation of this manuscript, we noticed that the anomalous Nernst effect with different results from ours in CeAlSi has been reported by other workers74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Methods Sample synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For the growth of LaAlSi and CeAlSi single crystals, a self-flux method was adopted, as described in the literature32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The as-grown single crystals were characterized by x-ray diffraction (XRD) measurements, as shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Electrical, thermoelectrical transport, and thermodynamic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For transport measurements, a single crystal was cut into a bar shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A standard six-probe method was used for the longitudinal resistivity and transverse Hall measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For thermoelectrical transport measurements, the Seebeck and Nernst signal were measured simultaneously, and the temperature gradient (\uf044T) was determined by a differential AuFe/chromel-P thermocouple which had been calibrated carefully in magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The cold end of the thermocouple was directly connected to the heat sink, and the temperature of the cold end was the same as the base temperature (TB) which was measured by a Cernox thermometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The temperature of the sample (Ts), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', the T used in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2 and 3, was determined to be the average of the cold and the hot ends, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Ts = TB + \uf044T/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Electrical transport data were collected in a physical property measurement system (PPMS, Quantum Design), and thermoelectrical transport data were collected in a home-built 4He cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Magnetic susceptibility and specific heat measurements were performed in a magnetic property measurement system (MPMS, Quantum Design) and a 15 PPMS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For electrical transport measurements under high pressure, a van der Pauw method was used, as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The single crystal with a dimension of ~ 70 \uf06dm \uf0b4 70 \uf06dm \uf0b4 15 \uf06dm was cut by a focused ion beam (FIB) along the c axis, and the surface of the single crystal is the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Magnetic field was applied perpendicular to the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For the calculation of resistivity, the following equation is adopted75, 𝜌 = 𝜋𝑑 ln2 ( 𝑅𝐴𝐵+𝑅𝐴𝐷 2 ) ∙ 𝑓( 𝑅𝐴𝐵 𝑅𝐴𝐷), where the function 𝑓(𝑥) satisfies the equation, exp (− ln2 𝑓(𝑥)) ∙ cosh [( 𝑥−1 𝑥+1) ln2 𝑓(𝑥)] = 1/2, d is the thickness of the sample, RAB and RAD are the resistance of the sample along different directions, as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Given the square shape of the sample and the tetragonal structure of CeAlSi, RAB/RAD is assumed to be ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For x < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2, 𝑓(𝑥) ≈ 1/cosh (ln (𝑥)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='403) with an error of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Synchrotron XRD measurements under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' High-pressure angle-dispersive XRD (wavelength: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='434 Å) measurements of ground CeAlSi powder were performed at beamline 13-BMC of the Advanced Photon Source, Argonne National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The powder of CeAlSi was loaded into a sample chamber sealed by a rhenium gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A symmetric diamond anvil cell (DAC) was used to generate quasi-hydrostatic pressure using silicone oil as the pressure-transmitting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The pressure inside the sample chamber was determined by the shift of ruby fluorescence76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Experimental parameters between the sample and detector were calibrated using the standard LaB6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' All two- dimensional XRD images were analyzed using Dioptas77, yielding one-dimensional intensity versus diffraction angle patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Rietveld analyses were performed by using the general structure and analysis system (GSAS) software78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Density functional theory (DFT) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' First-principles calculations were carried out by using the Vienna ab initio Simulation Package (VASP)79,80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Exchange-correlation effects were treated by using a Perdew-Burke-Ernzerhof (PBE)-type generalized gradient approximation (GGA)81,82 with the projector-augmented-wave (PAW) potential83,84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' An on-site Coulomb interaction was added for Ce f-electrons within the GGA+U scheme with Ueff = 6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The cutoff energy of the plane-wave basis was fixed at 500 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A 16 15\uf0b415\uf0b415 \uf047-centered k mesh based on the Monkhorst-Pack method was selected to sample the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The energy and force difference criteria were defined as 10-6 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='01 eV/Å for self-consistent convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To simulate paramagnetic CeAlSi, we treated the 4f electrons on Ce as core electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Spin-orbit coupling (SOC) was considered in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The WANNIER90 package85,86 was adopted to construct Wannier functions from the first-principles results without an iterative maximal- localization procedure87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The WANNIERTOOLS code88 was used to find Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For the calculations for the pressure evolution of magnetic moments, the full-potential augmented plane-wave and local orbital methods, as implemented in the WIEN2k code, was adopted89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The PBE-type GGA was used for the exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The RMTKMAX were set to be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 and we used 1000 k-point meshes for the whole Brillouin zone with Ueff = 6 eV for Ce with turning on SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Data availability The data that support the findings of this study are available from the corresponding authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Yan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' & Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Prog.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 8, 337–354 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Jia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' & Hasan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Weyl semimetals, Fermi arcs and chiral anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Troyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' & Soluyanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' WannierTools: An open-source software package for novel topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 224, 405–416 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Blaha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Schwarz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Madsen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Kvasnicka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' & Luitz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Wien2k: An augmented plane wave plus local orbital program for calculating the crystal properties (Technical University of Wien in Austria, ISBN39501031-1-2) (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 22 Acknowledgments We thank Simin Nie for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the project C03 and C07 of the Collaborative Research Center SFB 1143 (project-ID 247310070), the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='qmat (EXC 2147, Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 390858490).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' All authors discussed the results and commented on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 23 Additional Information Supplementary information is available for this paper at the URL inserted when published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Correspondence and requests for materials should be addressed to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='cheng@ifw- dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure captions Figure 1 | Basic properties and Berry curvature of RAlSi (R = La, Ce).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a Zero-field- cooling (ZFC) and field-cooling (FC) magnetization as a function of temperature for RAlSi with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the schematic structure of RAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' RAlSi possesses a noncentrosymmetric structure with the space group of I41md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Longitudinal resistivity (\uf072xx) of LaAlSi and CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset displays the derivative of resistivity with respect to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' There is no distinct anomaly in LaAlSi, while a peak situated at TC ~ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1 K corresponds to the ferromagnetic transition in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c and d Band structure and associated Berry curvature for LaAlSi and CeAlSi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' e and f Transverse Hall resistivity (\uf072yx) and magnetoresistance [MR = \uf044\uf072xx(H)/\uf072xx(0)\uf0b4100%, \uf044\uf072xx(H) = \uf072xx(H) − \uf072xx(0)] at 2 K with the magnetic field applied along the c axis for LaAlSi and CeAlSi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The left and right insets in (f) display the anomalous Hall resistivity (𝜌𝑦𝑥 A ) after subtracting the ordinary contribution through 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝜌𝑦𝑥 𝐴 , and the loop-shaped Hall effect (LHE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 2 | Anomalous Nernst effect (ANE) and quantum oscillations analysis of LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a Normalized magneto-Seebeck signal [\uf044Sxx(H)/Sxx(0), \uf044Sxx(H) = Sxx(H) - Sxx(0)] at different temperatures with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Nernst signal normalized to the temperature at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the temperature dependence of the amplitude of anomalous Nernst signal normalized to the temperature (|\uf044𝑆𝑥𝑦 A |/𝑇 ) and anomalous Nernst angle (ANA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The bold dashed line represents the empirical expression fit to the data of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c The fast Fourier transform (FFT) results at various temperatures, derived from the oscillations in the Nernst signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset displays the oscillatory component, \uf044Sxy/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Five oscillation frequencies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='7, and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 T assigned to α, \uf068, \uf064, \uf062, and γ bands, respectively) have been distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' d The de Haas-van Alphen (dHvA) oscillations at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' e FFT amplitude (Amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=') 24 as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The solid lines represent the fits to the Lifshitz-Kosevich formula to obtain the cyclotron effective mass (m*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' f Landau index n plotted against 1/\uf06d0H for the Nernst and magnetization oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Lines represent linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The left panel shows the extrapolation of 1/\uf06d0H to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The right panel displays the intercepts for α, β, γ, \uf064 and \uf068 pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 3 | Anomalous Hall effect (AHE) and anomalous Nernst effect (ANE) of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a Hall resistivity of CeAlSi at different temperatures with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' As a comparison, the data of 2 K is replotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Field dependence of magnetization at various temperatures with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the low-field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c Anomalous Hall resistivity (𝜌𝑦𝑥 A ) at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the representative fit to the Hall resistivity at 2 K through 𝜌𝑦𝑥 = 𝑅0𝐵 + 𝑅𝑠𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' d Anomalous Hall conductivity ( \uf073𝑥𝑦 A ) at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset displays the anomalous Hall angle (AHA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' e Normalized magneto-Seebeck signal at different temperatures with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' f Nernst signal normalized to the temperature at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the temperature dependence of the amplitude of anomalous Nernst signal normalized to the temperature (|𝑆𝑥𝑦 A |/𝑇), and the anomalous Nernst angle (ANA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' g Anomalous Nernst conductivity (−\uf044\uf061xy) as a function of field at several selected temperatures, −\uf044\uf061𝑥𝑦 = −[\uf061𝑥𝑦(𝑇) − \uf061𝑥𝑦(51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 K)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' h Contour plots of the \uf073𝑥𝑦 A , −\uf044\uf061xy and the derivative of magnetization (dM/dH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The background color represents the magnitude of their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 4 | Pressure-induced phase transitions in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a Temperature dependence of longitudinal resistivity at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows a picture of the sample chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The single crystal is ~ 70 \uf0b4 70 \uf0b4 15 \uf06dm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Low-temperature resistivity normalized to the data at 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' With increasing pressure, the ferromagnetic transition temperature (TC) initially increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c Hall resistivity at various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Above 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa, the slope of Hall resistivity changes sign, indicating that the dominant carriers change from holes to electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the high-pressure data above 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='9 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' d Anomalous Hall resistivity (𝜌𝑦𝑥 A ) at various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset shows the high-pressure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' e Contour plot of the derivative of normalized resistivity at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The background color represents the d(\uf072xx/𝜌𝑥𝑥 50K )/dT value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The pressure evolution of TC is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' f 25 Pressure-dependent Hall coefficient (RH) and the resistivity at 2 K (𝜌𝑥𝑥 2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' RH is obtained through linear fits to the high-field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The shaded area represents the pressure region where RH changes sign, suggesting the existence of a pressure-induced Lifshitz transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' g Pressure dependence of anomalous Hall angle (AHA) and absolute value of anomalous Hall conductivity (|\uf073𝑥𝑦 A |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Figure 5 | Pressure evolution of the crystal structure and band structure of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a X-ray diffraction (XRD) pattern of CeAlSi at room temperature up to 60 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The ambient-pressure structure with the space group of I41md persists to ~ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 GPa, beyond which a new diffraction peak emerges (marked with a dashed line and asterisk), indicating that a pressure-induced structural phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 0\uf0a2 represents that the pressure inside the sample chamber is released to zero, indicating that the emerging new structural phase is unstable at ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b The Rietveld refinement of the XRD pattern at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The refined value is RP = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='23% with weighted profile RWP = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The upper panel in (c) shows the pressure-dependent normalized parameters a/a0, c/c0 and V/V0 extracted from powder diffraction GSAS refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The lower panel in (c) shows the pressure evolution of the a/c ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' d-f Band structures of CeAlSi along the \uf047-W-X line for 0 GPa, 10 GPa, and 20 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' g-i Calculated 3-dimensional (3D) Fermi surfaces for 0 GPa, 10 GPa, and 20 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The violet and dark yellow color represent electron pockets and hole pockets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' At 10 GPa, pressure drives hole pockets (the red dashed circle as marked in g) into electron pockets, demonstrating the pressure-induced Lifshitz transition observed in Hall resistivity under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 26 Figure 1 E (eV) ΩZ(k) (10 4 Å 2) 1 0 1 0 4 2 6 Χ Γ Σ Ζ Γ ΝΣ1 LaAlSi 1 0 1 0 2 3 1 1 Χ Γ Σ Ζ Γ ΝΣ1 CeAlSi a b e c d f E (eV) ΩZ(k) (10 3 Å 2) c a b Si Al R 27 Figure 2 a b c d e f 28 Figure 3 a b c d e h f g 29 Figure 4 a c d b f g Ruby Sample 100 µm A B C D e 30 Figure 5 pockect pockect + a b c d e f g h i 0 GPa 10 GPa 20 GPa E (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 W 0 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 E (eV) 10 GPa W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 E (eV) 20 GPa W 品031 Supplementary Information for Anomalous transverse transport and phase transitions in Weyl semimetals RAlSi (R = La, Ce) Erjian Cheng1,*,⸙, Limin Yan2,3,*, Xianbiao Shi4,5,*, Mahdi Behnami1,*, Jian Yuan6, Yuanji Xu7,Yang Xu8, Yimin Wan9, Wei Xia6, Nikolai Pavlovskii10, Darren C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Peets10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Weiwei Zhao4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Yanfeng Guo6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Shiyan Li9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='ǂ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Wenge Yang2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' ¶ and Bernd Büchner1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='§ 1Leibniz Institute for Solid State and Materials Research (IFW-Dresden),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 01069 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Germany 2Center for High Pressure Science and Technology Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 201203 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 3State Key Laboratory of Superhard Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Jilin University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 130012 Changchun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 4State Key Laboratory of Advanced Welding & Joining,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Harbin Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 150001 Harbin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 5Flexible Printed Electronics Technology Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Harbin Institute of Technology (Shenzhen),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 518055 Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 6School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' ShanghaiTech University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 200031 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 7Institute for Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 100083 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 8Key Laboratory of Polar Materials and Devices (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' School of Physics and Electronic Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' East China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 200241 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 9State Key Laboratory of Surface Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 200438 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 10Institute of Solid State and Materials Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Technische Universität Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 01069 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Germany 11Collaborative Innovation Center of Advanced Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 210093 Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 12Shanghai Research Center for Quantum Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 201315 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' China 13Institute of Solid State and Materials Physics and Würzburg-Dresden Cluster of Excellence—ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='qmat, Technische Universität Dresden, 01062 Dresden, Germany 32 Supplementary Note 1: X-ray diffraction pattern of the as-grown RAlSi (R = La, Ce) single crystals Supplementary Figure 1 | X-ray diffraction (XRD) pattern of the as-grown RAlSi (R = La, Ce) single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a XRD pattern from the largest natural surface of a RAlSi single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The largest natural surface is the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Rocking curves of the (004) peaks for RAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 1 and Sample 2 of CeAlSi came from different batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The full width at half maximum (FWHM) for LaAlSi is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='12\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The FWHMs for Sample 1 and Sample 2 of CeAlSi are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='02\uf0b0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='04\uf0b0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The small values of FWHMs indicate the high quality of the as-grown single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 1 of CeAlSi was used for electrical and thermoelectrical transport measurements at ambient pressure, while Sample 2 comes from another batch from which it was used for high-pressure electrical transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For transport measurements, the magnetic field is applied perpendicular to the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For thermoelectrical transport measurements, the orientation of the samples was further verified by a Laue camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Supplementary Note 2: Electrical transport behavior of LaAlSi Supplementary Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(a-d) show the resistivity, Hall resistivity, conductivity, and Hall conductivity profiles at different temperatures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(e) displays representative fits to the Hall resitivity using a two-carrier model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', 𝜌𝑦𝑥 = − ( 𝐵 𝑒) [(𝑛𝑒𝜇𝑒 2 − 𝑛ℎ𝜇ℎ 2) + 𝜇𝑒 2𝜇ℎ 2(𝑛ℎ − 𝑛𝑒)𝐵2]/[(𝑛𝑒𝜇𝑒 + 𝑛ℎ𝜇ℎ)2 + 𝜇𝑒 2𝜇ℎ 2(𝑛ℎ − 𝑛𝑒)2𝐵2] , where ne, nh, \uf06de, and \uf06dh denote electron density, hole density, electron mobility, and hole a b 33 mobility, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The fitting results are displayed in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The density and mobility values of electrons are on the order of 1020 cm-3 and 10 cm2V-1s-1, respectively, which are 4 orders of magnitude larger and 2 orders of magnitude smaller than holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The density of electrons we obtain is consistent with that previously reported1,2, however the mobility is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' We also fit the data using a single-band model, and the density is nearly the same (~ 1020 cm-3), but the mobility is ~ 2280 cm2V-1s-1 at 2 K, comparable to the previous report1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These results suggest that the low-density hole carriers with high mobility also play crucial roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Strikingly, the temperature dependences of density and mobility show an anomaly around ~ 12 K for both carriers, indicating the change of Fermi surfaces, suggesting a possible temperature-induced Lifshitz transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Supplementary Figure 2 | Electrical transport behavior of LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a-d Longitudinal resistivity, transverse Hall resistivity, longitudinal conductivity, and Hall conductivity of LaAlSi at different temperatures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' e The fits to Hall resistivity at different temperatures using a two-carrier model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' f Temperature dependence of carrier density and mobility from the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a b c d e f 34 Supplementary Note 3: Anomalous magnetization in LaAlSi Supplementary Figure 3 | Anomalous magnetization in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a and b The magnetization with the magnetic field applied along the c axis at selected temperatures and an expanded view at low field of LaAlSi (Sample 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c and d The magnetization and an expanded view at low field with the magnetic field applied in plane at selected temperatures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A ferromagnetic-like hysteresis up to room temperature has been observed for both in-plane and out-of-plane fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' e Heat capacity of Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' No distinct anomaly that can be resolved, besides the phonon peak around 50 K, excluding a ferromagnetic transition in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' f Comparison of the magnetization for different samples at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 2 with the residual resistivity ratio (RRR \uf0ba 𝜌300K/𝜌2K) of ~ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='7 comes from the same batch as Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 3 (RRR ~ 3) and Sample 4 (RRR ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4) come from two different batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The magnetization behavior of LaAlSi does not resemble that of a paramagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' A ferromagnetic-like hysteresis at low fields for magnetic fields applied both in plane and out of plane is very unusual [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(a-d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The temperature dependence of zero-field-cooling (ZFC) and field-cooling (FC) magnetization [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1(a)], heat capacity [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(e)], and the derivative of resistivity [the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=" 1(b)] do not a b d e c f (T) Ho 0'S 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 20 300 500 100 0\'4 1 (K) rB tn" J) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content="0 0'4 Hsp 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0(T) Hoμ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content="下 0'4 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content="0 300 0'4 500 100 20 1 (K) HIIc 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='035 show any distinct anomalies, which implies the intrinsic nature of the anomaly in magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To further check the anomaly, we measure another three samples [Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 2 with the residual resistivity ratio of ~ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='7 comes from the same batch as Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Compared with Sample 1, the magnetization of Sample 2 overall displays similar behavior, but the ferromagnetic-like hysteresis is weakened [the inset of Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 3 and Sample 4 (RRR ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4) come from another two batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Sample 3 (RRR ~ 3) and Sample 4 (RRR ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4) host similar magnetization behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In contrast to Sample 1 and Sample 2, the magnitude of magnetization for Sample 3 and Sample 4 is weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Quantum oscillations can be distinguished in Sample 3, but cannot be resolved in Sample 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Nevertheless, Sample 3 and Sample 4 show a weak ferromagnetic-like hysteresis [the inset of Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 3(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These results suggest that sample quality plays a significant role in the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For Sample 3 and Sample 4 above ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 T, the derivative of magnetization with respect to the field (dM/dH) is negative, and such a negative value may originate from diamagnetism of conduction carriers, as observed in the Mott insulator Ca2RuO43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For Cd3As2, a ferromagnetic-like hysteresis at low fields was also reported, which was proposed to be possibly relevant to the orbital magnetization4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The orbital magnetization follows the equation 𝑀 = ∫ 𝑓(𝑥)𝜎𝑥𝑦 𝐴 (𝜀)𝑑𝜀, where 𝜎𝑥𝑦 𝐴 is the anomalous Hall conductivity at zero temperature with Fermi energy \uf065, and f (\uf065) is the Fermi-Dirac function [see Supplementary Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 4 for more details].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Therefore, we argue that the anomaly in magnetization observed in LaAlSi may be nontrivial, and probably relevant to Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 36 Supplementary Note 4: Fast Fourier transform (FFT) frequencies in LaAlSi Supplementary Figure 4 | The analysis of de Haas-van Alphen (dHvA) quantum oscillations and the comparison of FFT frequencies from different physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a FFT results of magnetization oscillations at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Magnetic field is applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Inset displays the oscillatory component \uf044M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Comparison of FFT results from the oscillations in different physical quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', Seebeck signal, resistivity, magnetization, and Nernst signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Compared with Seebeck and resistivity, the quantum oscillations in Nernst and magnetization were stronger, and more FFT frequencies can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a b 37 Supplementary Note 5: Basic preoperties of CeAlSi at ambient pressure Supplementary Figure 5 | Magnetoresistance, conductivity, the fit to Nernst, and Nernst conductivity of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a Magnetoresistance (MR) of CeAlSi at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Longitudinal conductivity at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c Fits to the Nernst signal normalized to the temperature at selected temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' d Nernst conductivity (−\uf044\uf061xy) as a function of field at several selected temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The Nernst conductivities for 31 and 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 K nearly overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' To obtain the anomalous contributions, the data at 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 K is taken as the ordinary contribution to be deducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a b c d 38 Supplementary Note 6: Magnetization (Sample 2) and electrical transport results under pressure of CeAlSi Supplementary Figure 6 | Magnetization (Sample 2) and electrical transport results under pressure of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a Magnetization at 2 K with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The upper inset shows the low-field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The lower inset displays the temperature dependence of the magnetization with magnetic field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1 T, revealing a ferromagnetic transition temperature consistent with Sample 1 of CeAlSi and previous studies5–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' b Hall conductivity at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 GPa with the magnetic field applied along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' c and d Magnetoresistance (MR) and conductivity as a function of field at 2 K for different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' a b c d 39 Supplementary Note 7: Universal scaling relation between the anomalous Hall conductivity and the longitudinal conductivity in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Supplementary Figure 7 | Absolute value of anomalous Hall conductivity |\uf073𝐱𝐲 𝐀 | as a function of longitudinal conductivity \uf073xx of CeAlSi under ambient and high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For better comparison, several pure metals (Fe, Co, Ni, Gd)9, oxides [Nd2(MoNb)2O7, La1-x(Sr,Ca)xMnO3, SrRuO3]10, chalcogenide spinels (Cu1-xZnxCr2Se4)11, magnetic semiconductors (GaMnAs, anatase–Co–TiO2, rutile–Co–TiO2) 10, Co3Sn2S212, MnSi13, Fe1-xCoxSi13, Mn3Ge14, and Mn3Sn14 have been plotted together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The solid lines in three regimes represent |\uf073xy A | µ\uf073xx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6, |\uf073xy A | ~ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=', and |\uf073xy A | µ\uf073xx, for the dirty, intermediate, and clean regimes, respectively9,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For CeAlSi at both ambient and high pressures, the anomalous Hall conductivity is located in the intermediate regime, suggesting an intrinsic origin of the anomalous Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 40 Supplementary Note 8: The pressure evolution of band structure of CeAlSi Supplementary Figure 8 | The evolution of band structure of CeAlSi under pressure with spin-orbit coupling (SOC) included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' For all pressures, CeAlSi remains a Weyl semimetal, and the band structure does not change much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' With increasing pressure, the hole pockets along the \uf047–X line become smaller, and then turn into electron pockets at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 ~ 10 GPa, evidencing a pressure-induced Lifshitz transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 1 0 1 Energy (eV) 1 0 1 Energy (eV) 0 GPa 5 GPa 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='5 GPa 10 GPa 15 GPa 20 GPa 30 GPa 24 GPa 40 GPa 1 0 1 Energy (eV) 41 Supplementary Note 9: Possible evidence for the temperature-induced Lifshitz transition in LaAlSi Supplementary Figure 9 | Possible evidence for the temperature-induced Lifshitz transition in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In the heat capacity, no distinct anomaly can be resolved below 30 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' In the resistivity, the slops of the profile display a weak anomaly at ~ 15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' The temperature dependence of the Seebeck signal (−Sxx) ascends from ~ 10 K to ~ 15 K, consistent with the anomalies in the temperature evolution of carrier density and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' These results provide a strong hint that a temperature-induced Lifshitz transition exists in LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='1 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='6 0 10 20 30 2 4 6 \uf072xx (\uf06dW cm) d\uf072xx/dT (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=') LaAlSi Cp (J mol 1K 1) dCp/dT (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=') 0 T Sxx (\uf06dVK 1) T (K) 42 Supplementary References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Multiple Weyl fermions in the noncentrosymmetric semimetal LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQfTQcm/content/2301.03800v1.pdf'} +page_content=' Phys.' 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a/KtAzT4oBgHgl3EQfkP1s/content/tmp_files/2301.01528v1.pdf.txt b/KtAzT4oBgHgl3EQfkP1s/content/tmp_files/2301.01528v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c40db7282659c2899fcce37e3d3fe0cfbeb0b9f --- /dev/null +++ b/KtAzT4oBgHgl3EQfkP1s/content/tmp_files/2301.01528v1.pdf.txt @@ -0,0 +1,621 @@ +Condensed Matter Physics, 2022, Vol. 25, No. 4, 43705: 1–10 +DOI: 10.5488/CMP.25.43705 +http://www.icmp.lviv.ua/journal +Dielectric relaxation induced by oxygen vacancies in +Na0.5Bi0.5TiO3 ceramics +V. M. Sidak +1∗, M. P. Trubitsyn +2, T. V. Panchenko +2 +1 Dnipro State Medical University, 9 Vernadsky St., 49044 Dnipro, Ukraine +2 Oles Honchar Dnipro National University, 72 Gagarina Ave., 49045 Dnipro, Ukraine +Received June 30, 2022, in final form August 31, 2022 +Dielectric permittivity was studied in ceramics of relaxor ferroelectric bismuth-sodium titanate Na0.5Bi0.5TiO3. +The measurements were performed on as sintered and heat treated in vacuum samples. The diffuse dielectric +anomalies associated with the structural phase transitions were observed in as sintered samples. The intense +peak of permittivity (𝜀max ∼ 104) appeared after heat treating in vacuum. The anomaly of 𝜀(𝑇) was contributed +by slow polarization processes ( 𝑓 < 10 kHz) and was non-stable, vanishing on heating in air up to ∼ 800 K. Tem- +perature and frequency dependencies of 𝜀 were described by using Cole-Cole model with accounting thermally +stimulated decay of the non-stable polarization. It is supposed that the dielectric anomaly is determined by +space charge polarization mechanism. Oxygen vacancies V•• +O and electrons localized on titanium ions Ti′ +Ti are +assumed to be responsible for the phenomenon observed. +Key words: dielectric properties, permettivity, perovsikes, defects +1. Introduction +High sensitivity for external fields is the most valuable requirement for functional materials used to +transform energy from certain kind to another one. An increased susceptibility often results from lattice +instability in the range of structural phase transition. That is why crystalline compounds undergoing +structural transformations are intensively investigated by researchers and technologists involved in cre- +ation of new functional materials for piezoelectric, thermoelectric, photovoltaic and other converters. The +crystals with perovskite ABO3 structure have found wide range of applications in modern electronics. +Consequently, the compounds of the perovskite family are among the most popular objects for studies in +materials sciences. Variations of chemical composition, formation of the structure on nano- and micro- +meter levels, control on the lattice defects make it possible to create the materials with a broad variety of +physical properties. Thus, ceramics based on Pb-ZrTiO3 show extremely high electro-mechanical para- +meters and are used in piezoelectric devices [1]. Introducing the transition groups ions into the structural +ABO3 unit leads to the appearance of magneto-electrical coupling in multiferroic materials (BiMnO3, +BiFeO3, TbMnO3) [2]. Some crystals with complex perovskite structure like ACu3Ti4O12 (A = Ca, Ba, +Sr) possess extremely high dielectric constants (∼ 104 − 105) which opens new prospects to be used as +the materials with high permittivity in memory and microwave devices [3, 4]. +It is well known that structural imperfections can strongly affect the properties of crystals and +even are capable of inducing new phenomena which are not observed in a perfect lattice. That is why +comprehensive information on typical intrinsic and extrinsic lattice defects becomes of high importance. +At the present time, numerous works are aimed at studying the mechanisms of the influence of defects on +the properties of crystals. Based on the knowledge gained, the technological approaches are developed +that allow to control qualitatively and quantitatively the defectiveness of the crystal structure. Doping with +iso- or heterovalent impurities, heat treatment in various atmospheres, applying external fields make it +∗Corresponding author: vasylsidak@gmail.com. +This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution +of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. +43705-1 +arXiv:2301.01528v1 [cond-mat.mtrl-sci] 4 Jan 2023 + +V. M. Sidak, M. P. Trubitsyn, T. V. Panchenko +Figure 1. The crystal structure of NBT in tetragonal phase [6]. +possible to stimulate the appearance of the defects that improve the targeted characteristics or, conversely, +to reduce the content of undesirable defects that degrade the useful parameters. Intensive experimental +and technological studies aimed at controlling the subsystem of defects, have led to the appearance of the +“defect engineering” concept [5]. +Modern requirements in the field of environmental protection considerably changed the situation in +the production of functional materials and urge the search for new compositions free from health harmful +chemical elements. Lead-free bismuth-sodium titanate Na0.5Bi0.5TiO3 (NBT) meets these requirements +and shows a number of attractive physical properties. NBT crystal belongs to a group of complex +perovskites with the structure of A’A”BO3 type, where sodium and bismuth atoms are randomly distributed +through the A-site (figure 1) [6]. The extremely high electro-mechanical coupling is the most prominent +physical property of NBT crystal and solid solutions based on it [7]. The specific properties are directly +related to the structural phase states observed in NBT. On cooling from high-temperature side NBT +undergoes the following sequence of phase transitions: from cubic to tetragonal ferroelastic phase at +𝑇𝐶 ≈ 810 K, and further to rhombohedral ferroelectric phase at 𝑇𝑅 ≈ 490 K [6]. In the range of T𝑅, NBT +demonstrates high permittivity and specific dielectric dispersion peculiar to relaxor ferroelectrics [8]. +Besides, the properties of NBT can be substantially modified by doping and technological treatments [9– +12]. +Recently, the strong dielectric anomaly (∼ 104) was observed near 670–690 K in NBT single crystal [9, +13–15] and Na0.5Bi0.5TiO3 – BaTiO3 (NBT–BT) solid solutions [16]. The 𝜀(𝑇) dependence showed an +anomalous temperature behaviour and unusual frequency dispersion (here and below symbol 𝜀 without +prime means real part of permittivity). In addition, permittivity peak disappeared after heat treatment +in air (∼ 800 K) and could be restored by heat treating in vacuum (∼ 1070 K). The authors of [15] +supposed that dielectric anomaly was contributed by the dipole defects formed by oxygen vacancies +(V•• +O ) and electrons localized on the nearest titanium ions Ti′ +Ti. The associated dipole defects (Ti′ +Ti-V•• +O )• +were considered as unstable and decomposing upon heating. +These results were obtained for NBT single crystals. Of course, for practical applications, NBT +ceramics can be expected as more commercially and technologically acceptable. In this paper anomalous +dielectric relaxation mentioned above is studied in NBT ceramics. By accounting the permittivity value +in maximum (∼ 104), the previous interpretation based on the dipole defects [15, 16] is considered +critically. It is supposed that a strong dielectric peak can be associated with space charge polarization +phenomenon. The possible microscopic mechanisms of the dielectric relaxation are briefly discussed. +43705-2 + +NaBi +TiDielectric relaxation induced by oxygen vacancies in Na0.5Bi0.5TiO3 ceramics +2. Experimental results +The NBT ceramics were prepared by usual sintering technique. The samples for electrical properties +measurements were cut off as the plane-parallel plates with the edges of about 5 × 5 × 0.8 mm3. The Pt +electrodes were deposited on the main planes of the samples by cathode sputtering method. Electrical +properties were measured using AC bridge P 5083 in the temperature interval 300–800 K for the frequency +range 0.5–100 kHz. Two types of the samples were used: i) prepared from as sintered ceramics and ii) +heat treated in vacuum. The regimes of heat treating were the same as those previously used for single +crystals (𝑇 = 1070 K, 𝑡 = 2 h, 𝑝 ≈ 1 Pa) [16]. +400 +600 +800 +0,5 +1,0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +e, 10 +4 +T, K +1 +2 +3 +10 +-8 +10 +-7 +10 +-6 +s, Ohm +-1 +cm +-1 +10 +3 +/T, K +-1 +Figure 2. The permittivity dependencies 𝜀(𝑇) in as sintered NBT ceramics. The AC field frequency was +f = 0.5 (1); 0.8 (2); 1 (3); 2 (4); 5 (5); 10 (6); 50 (7); 100 (8) kHz. The inset shows Arrhenius plot of +conductivity 𝜎(1/𝑇) dependencies. +The temperature dependencies of dielectric permittivity 𝜀 and electrical conductivity 𝜎 measured on +heating for as sintered NBT ceramics are shown in figure 2. In contrast to the data obtained for NBT and +NBT-BT single crystals [15, 16], 𝜀(𝑇) dependence does not show intense relaxation anomaly and reflects +the structural transformations in the range of T𝐶 and T𝑅 only (figure 2). The 𝜀(𝑇) dependencies measured +on the next cooling run and on the subsequent heating-cooling cycles coincide with each other. The inset +to figure 2 shows the temperature dependencies of conductivity 𝜎 plotted in Arrhenius scale. One can +see that 𝜎 increases with AC field frequency 𝑓 and weakly depends on temperature. This behaviour is +typical of dielectrics at relatively low temperatures. Only at frequencies 𝑓 < 2 kHz and for 𝑇 ⩾ 500 K +conductivity starts to grow exponentially on heating that gives nearly linear regions in the Arrhenius plot. +Such a behaviour reflects a growing contribution of thermally activated charge transfer. +Next, the sample of NBT ceramics was heat treated in vacuum, cooled to room temperature and +after that its electrical properties were measured. The data obtained are shown in figure 3. One can see +that after heat treating, in the range 700 – 780 K, 𝜀(𝑇) demonstrates intense maximum (𝜀max ∼ 5 · 104, +𝑓 = 0.5 kHz) which is strongly dependent on frequency 𝑓 . As 𝑓 increases, the peak of 𝜀(𝑇) sharply +decreases in magnitude and shifts to higher temperatures. Similarly to the data observed for NBT and +NBT-BT single crystals [15, 16], intense 𝜀(𝑇) maximum (figure 3) could be detected for the first +heating run only and disappeared for the next cooling and heating runs. Corresponding dependencies of +conductivity 𝜎(1/𝑇) are shown in the inset to figure 3. In the low-temperature interval (𝑇 < 500 K), 𝜎 +demonstrates nearly the same behaviour as in the untreated sample (the inset to figure 2), but for higher +temperatures conductivity shows an intense peak corresponding to the relaxation maximum of 𝜀(𝑇). +In subsequent temperature runs, the 𝜎(1/𝑇) dependencies did not show contribution from dielectric +relaxation and were the same as shown in the inset to figure 2. +43705-3 + +V. M. Sidak, M. P. Trubitsyn, T. V. Panchenko +400 +600 +800 +0 +1 +3 +5 + 1 + 5 + 2 + 6 + 3 + 7 + 4 + 8 +e, 10 +4 +T, K +1 +2 +3 +10 +-8 +10 +-6 +10 +-4 +s, Ohm +-1 +cm +-1 +10 +3 +/T, K +-1 +Figure 3. The dependencies 𝜀(𝑇) measured in the first heating run of NBT ceramics previously heat +treated in vacuum (1070 K, 2 h). The AC field frequencies are indicated in the caption to figure 2. The +solid lines were calculated by using (3.1), (3.4). The inset shows corresponding 𝜎(1/𝑇) dependencies. +3. The model +As mentioned in section 2, the dielectric anomaly 𝜀(𝑇) similar to the one shown in figure 3 was +earlier detected in single crystals of NBT and NBT-BT [15, 16]. Special attention was paid to the nearly +symmetrical shape of permittivity peak, that was quite different from the asymmetrical 𝜀(𝑇) anomaly +of Debye relaxator. It was proposed that dipoles or associated complexes responsible for the dielectric +anomaly were thermally destroyed on heating. Such decomposition was noticeable in the temperature +range where the dielectric relaxation was detected. Consequently, the high-temperature wing of the 𝜀(𝑇) +anomaly decreased more sharply. +The first attempt to explain the specific character of the dielectric anomaly (figure 3) was made in [15] +where a decrease of the dipoles or consentration of mobile defects was described as simple exponential +temperature decay. The possible role of configurational and vibrational entropy of the dipole defects was +considered somewhat later in [14]. Nevertheless, these approaches allowed to interpret the data only at +the qualitative level and not provide a correct quantitative description of the experimental results. More +accurately, the 𝜀(𝑇, 𝑓 ) behavior in NBT-BT single crystal was described in [16], where Debye relaxator +model was combined with the kinetic equation that determins the decay of the polarizing entities with +temperature. +Dielectric response of real structures in an external AC field can be described by Cole-Cole, Davidson- +Cole and other models [17]. These models predict different types of dielectric spectra, symmetrical or +non-symmetrical diagrams in complex (𝜀′–𝜀′′) plane, where 𝜀′ and 𝜀′′ represent real and imaginary parts +of permittivity. The dielectric anomaly shown in figure 3 is observed practically in the same temperature- +frequency range (𝑇 > 500 K, 𝑓 < 10 kHz), where charge transfer processes notably contribute to +conductivity (the nearly linear regions in the 𝜎(1/𝑇) dependencies, the insert to figure 2). That is why +the experimental diagrams plotted in (𝜀′–𝜀′′) for the used frequency region do not permit to make a +reliable choice between the models mentioned. Hence, the experimental data are described by Cole-Cole +model [17] +𝜀∗(𝑇, 𝜔) = 𝜀∞ + +𝐶/𝑇 +1 + (i𝜔𝜏𝑅)1−𝛼 , +(3.1) +where 𝜔 = 2π 𝑓 is an AC field frequency; 𝑘 is Boltzmann constant. Expression (3.1) includes a minimum +number of the fitting parameters: 𝜀∞ – permittivity at high-frequency; Curie constant 𝐶 ∼ 𝑛 which is +directly proportional to the concentration 𝑛 of the dipoles; 𝜏𝑅(𝑇) = 𝜏0 +𝑅 exp(𝐸/𝑘𝑇) is the time of the +43705-4 + +Dielectric relaxation induced by oxygen vacancies in Na0.5Bi0.5TiO3 ceramics +relaxation of dipole moments in an external field; energy parameter 𝐸 estimates the height of the potential +barrier which is overcome at the reorientation dipole moments; phenomenological parameter 0 ⩽ 𝛼 < 1 +describes the distribution of relaxation times 𝜏𝑅 in disordered structures. +It should be noted that for the samples heat treated in vacuum (figure 3), the anomalies of permittivity +imaginary part 𝜀′′(𝑇, 𝑓 ) contain contributions from dielectric relaxation and charge transfer in the same +temperature-frequency range (see the comments above). Hence, the analysis of 𝜀′′(𝑇, 𝑓 ) dependences +needs to separate these contributions with apriori unknown parameters. The anomalies of permittivity +real part 𝜀′(𝑇, 𝑓 ) (figure 3) are mainly contributed by the polarization processes. Thus, the parameters +of the discussed dielectric relaxation can be determined more directly from 𝜀′(𝑇, 𝑓 ) dependencies. In +addition, non-stable nature of polarization causes a specific type of anomalous behaviour which is more +evident just for dependencies 𝜀′(𝑇). That is why the following analysis is focused on permittivity real +part dependencies. +Further, it should be considered that polarizing entities (dipoles, associated complexes) are non- +equilibrium and undergo thermal decomposition. One can assume that a decrease of their concentration +𝑛 can be described by the simple kinetic equation [18] +d𝑛 +d𝑡 = − 𝑛 +𝜏𝐷 +. +(3.2) +Here, 𝜏𝐷(𝑇) = 𝜏0 +𝐷 exp(𝑈/𝑘𝑇) and 𝑈 are the time and energy parameters determining the thermal decay +of non-stable polarization. In (3.2), one can go from differentiation in time to a derivative in temperature +by considering that during the experiments, the samples were heated and cooled with a constant rate. +Thus, the temperature of the samples can be written as𝑇(𝑡) = 𝑇0+𝛾𝑡 , where T0 is an initial temperature; 𝛾 +is the rate of temperature changes; 𝑡 is the current time. Thus, remembering that 𝐶 ∼ 𝑛, from equation 3.2 +one can rewrite Curie constant as +𝐶(𝑇) = 𝐶0 · exp +������ +− 1 +𝛾𝜏0 +𝐷 +· +𝑇∫ +𝑇0 +exp +� +− 𝑈 +𝑘𝑇 +� +d𝑇 +������ +. +(3.3) +Direct fitting of expression (3.1), with accounting (3.3), to the experimental data was complicated and +did not give reliable results. Nevertheless, an approximate integration in (3.3) performed in [19] yielded +the following expression +𝐶(𝑇) = 𝐶0 · exp +� +− +𝑘𝑇2 +𝛾𝜏0 +𝐷(𝑈 + 𝑘𝑇) exp +� +− 𝑈 +𝑘𝑇 +�� +. +(3.4) +Thus, the experimental data shown in figure 3 can be described using Cole-Cole formulae 3.1 +combined with the approximate solution 3.4 of the kinetic equation 3.2. +4. Discussion +It is well known that oxygen vacancies V•• +O are the typical defect for the crystals of complex oxides. +In tightly packed structures like perovskites ABO3, the excess positive charge (+2e) associated with V•• +O +more probably can be compensated by the necessary number of cationic vacancies, which gives rise to the +appearance of Schottky-type defects. If the concentration of V•• +O is too high with respect to the number +of cation vacancies, the additional electronic defects appear and the valence of the cations neighboring +the V•• +O can decrease. In ABO3 structures, the weakly bound electrons can be localized on titanium ions +and as a result, Ti′ +Ti centers can arise [20–22]. The energy levels of Ti′ +Ti centers are shallow enough, and +the electrons that hop via regular titanium ions can participate in the charge transfer. The presence of the +nearest neighboring V•• +O stabilizes the localized electrons and as a result, associated pairs (Ti′ +Ti-V•• +O )• can +arise [20]. +One can expect that thermal treatment of NBT ceramics in vacuum (T = 1070 K) should increase +mainly the concentration of V•• +O . Each V•• +O that arose in the treated ceramics could cause the emergence of +43705-5 + +V. M. Sidak, M. P. Trubitsyn, T. V. Panchenko +two Ti′ +Ti centers. Correspondingly, the appearance of intense 𝜀(𝑇) anomaly after heat treatment (figure 3) +can be just associated with the defects formed by V•• +O . That is why in the previous works [9, 14–16] a +slow dielectric relaxation in NBT single crystals was attributed to re-orientations of (Ti′ +Ti-V•• +O )• dipoles +resulting from hopping of V•• +O through oxygen octahedra vertices. Thermal decay of polarization (3.2) +was interpreted as a result of disassociation of (Ti′ +Ti-V•• +O )• centers occurring on heating. Nevertheless, a +great value of permittivity in the peak (∼ 5 · 104 at 𝑓 = 0.5 kHz, see figure 3) can be hardly attributed +to the dipole defects, the concentration of which is assumed to be low enough. More probably, such a +high value of 𝜀 can be the result of space charge polarization which is often observed in inhomogeneous +media. Usually, permittivity of such substances is defined as an effective one. +Let us consider more in detail the assumption that intense 𝜀(𝑇) peak in figure 3 is contributed by +mobile charge defects which in an external electric field can accumulate near certain inhomogeneities. +Earlier in [9, 14–16] we supposed that reorientation of the dipole complexes (Ti′ +Ti-V•• +O )• in an external +field and their thermal dissociation on heating occurred through V•• +O hopping. That is why calculating +the 𝜀(𝑇, 𝑓 ) dependencies by using (3.1), (3.4), we associated the pre-exponential factors 𝜏0 +𝑅, 𝜏0 +𝐷 for +the relaxation times with inverse Debye frequency. Correspondingly, the values of 𝜏0 +𝑅, 𝜏0 +𝐷 were fixed +(≈ 2 · 10−13 s) and estimated from Debye temperatures (𝜃 ≈ 260–350 K) typical of perovskites [23]. +1,5 +2,0 +2,5 +10 +-5 +10 +0 +10 +5 +t +R +, t +D +, s +10 +3 +/T, K +t +D +t +R +Figure 4. The dependencies of the dielectric relaxation time 𝜏𝑅(1/𝑇) and polarization decay time 𝜏𝐷(1/𝑇) +calculated from the data given in table 1. +Assuming the space charge polarization to be the main effect, we have no apriori information on the +values of 𝜏0 +𝑅, 𝜏0 +𝐷 and that is why we set them free. We should add that considering 𝜏0 +𝑅, 𝜏0 +𝐷 as the fitting +parameters allowed to reduce by about an order of magnitude the mean square deviation of the calculated +data from the experimental ones. The curves calculated with the help of the model discussed in section 3, +are drawn in figure 3 with the solid lines. It should be noted that the background contribution to 𝜀(𝑇) due +to the structural phase transitions was taken into account as it was described earlier in [9, 14–16]. In the +scale chosen, this contribution shows only a weak dependency on temperature and in pure form can be +seen in figure 2 where intense 𝜀(𝑇) peak is absent. The values of the parameters, used in (3.1), (3.4) and +averaged for all studied frequencies, are presented in table 1. One can see that fitting the calculated data +to the experimental ones, in contrast to the previous assumption in [16], gives strongly different values +for the pre-exponential factors 𝜏0 +𝑅 and 𝜏0 +𝐷. The value of 𝜏0 +𝑅 corresponds to the order of typical lattice +frequencies. By contrast, the factor 𝜏0 +𝐷 for polarization decay is found to be twelve orders longer which +corresponds to the infra-low frequency range. Seemingly, such extremely high value of 𝜏0 +𝐷 indirectly +evidence in favor of space charge polarization mechanism. +The dependencies of dielectric relaxation time 𝜏𝑅(1/𝑇) and polarization decay time 𝜏𝐷(1/𝑇), calcu- +43705-6 + +Dielectric relaxation induced by oxygen vacancies in Na0.5Bi0.5TiO3 ceramics +Table 1. The values of the parameters in (3.1), (3.4) obtained from the 𝜀(𝑇, 𝑓 ) dependencies (figure 3). +C0, K +𝛼 +𝜏0 +𝑅, s +𝐸, eV +𝜏0 +𝐷, s +𝑈, eV +3.4(8) · 107 +0.09(1) +1.2(2) · 10−13 +1.21(1) +1.1(2) · 10−1 +0.67(2) +400 +800 +0 +2 +4 +De, 10 +4 +T, K +1 +2 +3 +4 +5 +400 +800 +0 +4 +C, 10 +7 + K +T, K +Figure 5. The contribution from non-stable polarization to dielectric anomaly Δ𝜀(𝑇). The dashed curves +are calculated by using the following heating rates 𝛾 = 0.1 (1); 1 (2); 10 (3); 102 (4); 106 (5) K/min. The +parameters in (3.1), (3.4) are taken from processing the experimental data measured at 𝑓 = 1 kHz. The +solid line is calculated for the experimental data in figure 3 (𝛾 = 1.7 K/min, 𝑓 = 1 kHz). The inset shows +the corresponding dependencies of Curie constant 𝐶(𝑇). +lated with the help of the data in table 1, are shown in figure 4. One can see that for the whole studied +interval, 𝜏𝐷 values considerably exceed the typical time (∼ 1 s) of a single measurement at certain 𝑇 +and 𝑓 . On the other hand, 𝜏𝐷 values are comparable with the time (∼ 3 − 4 h) of a single measuring run. +One can show that the model (section 3) combining Cole-Cole formulae (3.1) with kinetic equa- +tion (3.2) allows one to describe the specific features of the dielectric relaxation discussed. Thus, (3.4) +predicts that the form of the dielectric anomaly 𝜀(𝑇, 𝑓 ) should depend on time and on heating rate +𝛾. Obviously, such effects can be tested in experiment. Besides, one can expect that the behaviour of +𝜀(𝑇, 𝑓 ) should depend on the ratio between the rates of dielectric relaxation 𝜏−1 +𝑅 and polarization decay +𝜏−1 +𝐷 . Really, the anomaly 𝜀(𝑇, 𝑓 ) takes a specific form (figure 3) since the decay of non-equilibrium +polarization becomes notable in the same temperature range where dielectric peak is detected. Thus, +one can expect that the type of dielectric anomaly for certain 𝜏0 +𝑅, 𝜏0 +𝐷 values should depend on the ratio +between activation energies 𝑈/𝐸. Let us consider the effects mentioned. +Figure 5 shows the 𝜀(𝑇) anomaly calculated for different 𝛾 values. One can see how considerably +the variations of 𝛾 can change the 𝜀(𝑇) behaviour. For high values of 𝛾 during the whole measuring +cycle, the decay of non-equilibrium polarization remains practically negligible. Correspondingly, in +the limit 𝛾 → ∞, the behaviour of 𝜀(𝑇) approaches a classic behavior of Debye relaxator (figure 5). +For the intermediate values of 𝛾, the anomaly of 𝜀(𝑇) takes the form of nearly symmetrical peak. At +infinitely low heating rate (𝛾 → 0), non-equilibrium polarization has enough time to decay totally +before the permittivity peak can be detected. As a result, on lowering 𝛾, the permittivity peak decreases +in amplitude and finally disappears (figure 5). The inset to figure 5 shows the calculated temperature +dependence of Curie constant. On heating, 𝐶(𝑇) shows a step-like decrease manifesting a decay of non- +equilibrium polarization. For high rates of 𝛾, Curie constant possesses a maximum possible value and is +practically temperature independent in the whole interval studied. For low values of 𝛾, Curie constant on +heating decreases to zero before the dielectric relaxation can be detected. +43705-7 + +V. M. Sidak, M. P. Trubitsyn, T. V. Panchenko +400 +800 +0 +2 +4 +De, 10 +4 +T, K +1 +2 +3 +4 +5 +6 +400 +800 +0 +4 +C, 10 +7 + K +T, K +Figure 6. The dielectric anomaly Δ𝜀(𝑇) (dashed lines) calculated for the following ratios 𝑈/𝐸 = 0.4 (1); +0.5 (2); 0.6 (3); 0.7 (4); 0.8 (5); 1.2 (6). The value of 𝐸 = 1.21 eV is fixed, and the value of 𝑈 is varied. +The solid line corresponds to the experimental data in figure 3 (𝑈/𝐸 = 0.55, 𝑓 = 1 kHz). The 𝐶(𝑇) +dependencies are shown in the inset. +Figure 6 illustrates how 𝜀(𝑇) anomaly changes its form when the ratio between activation energies +𝑈/𝐸 varies. For higher values of 𝑈/𝐸, one has a Debye-type behavior of 𝜀. On lowering the ratio 𝑈/𝐸, +permittivity 𝜀(𝑇) takes the intermediate peak-like form and finally it disappears when the ratio 𝑈/𝐸 +decreases. The inset to figure 6 shows the corresponding dependencies of Curie constant 𝐶(𝑇). +Assuming that an intense 𝜀(𝑇) peak (figure 3) is determined by space charge polarization effects, +and for NBT ceramics one can consider the same typical defects such as oxygen vacancies V•• +O , electrons +localized on titanium Ti′ +Ti and probably associated complexes based on them. V•• +O can be assumed +to be more heavy defects, whereas localized electrons Ti′ +Ti can be supposed to be more light ones. +The mobile charge defects can accumulate near the following inhomogeneities in NBT ceramics: i) +intergrain boundaries; ii) ferroelectric or ferroelastic domains boundaries; iii) near-electrode regions. +Presumably, one can expect that during long enough period the oxygen vacancies can accumulate near +certain inhomogeneities and form the regions with higher V•• +O concentration. In the applied electric field, +electrons Ti′ +Ti move between the regions with increased V•• +O concentration. On heating, due to diffusion, +the regions with high V•• +O concentration dissolve. More information on the nature of the inhomogeneities +which can cause space charge polarization in NBT can be obtained from comparison of the experimental +data measured for ceramic and single crystalline NBT. This work is in progress at the moment. +5. Summary +An intense low frequency anomaly of dielectric permittivity appeared in NBT ceramics after heat +treating in vacuum (𝑇 = 1070 K, 𝑡 = 2 h). The corresponding polarization was found to be non-stable +and disappeared after heating in air up to ∼ 800 K. The results of the thermal treatment evidenced that +the observed dielectric relaxation was contributed by the defects including oxygen vacancies. Dielectric +maxima were detected in the same temperature-frequency range where charge transfer processes gave +notable contribution to conductivity in AC field. That is why we could not examine reliably the type of +the experimental diagrams plotted in the complex plane of permittivity. The temperature and frequency +dependencies of 𝜀 were described on the basis of Cole-Cole model which could be used to describe +dielectric relaxation in partially disordered structures. Thermal decay of the non-equilibrium polarization +was described using the simple kinetic equation. The analysis was focused on the behaviour of permittivity +real part which was mainly contributed by the polarization processes. Combination of Cole-Cole model +with kinetic equation allowed us to describe the experimental data with a good accuracy and to predict +43705-8 + +Dielectric relaxation induced by oxygen vacancies in Na0.5Bi0.5TiO3 ceramics +the evolution of dielectric anomaly under variations of the experimental conditions and characteristics of +the phenomena observed. The great value of permittivity in the maximum (𝜀max ∼ 5 · 104, 𝑓 = 0.5 kHz), +observed for NBT ceramics, made doubtful the assumption that the dielectric anomaly could be due to +the dipole defects the concentration of which was assumed to be not extremely high. That is why it was +supposed that the observed dielectric relaxation was determined by space charge polarization mechanism. +Oxygen vacancies V•• +O and electrons localized on titanium ions Ti′ +Ti were assumed to be responsible for +the phenomena studied. One can hope that more details on the microscopic mechanism of the thermally +non-stable dielectric relaxation can be derived from comparative studies of the electrical properties of +NBT single crystals and ceramics treated in atmospheres enriched and depleted in oxygen. +Acknowledgements +The study was funded by Ministry of Education and Science of Ukraine according to the research +projects No. 0119U100694, No. 0120U102239 and No. 0122U001228. +References +1. Bobic J. D., Review of the Most Common Relaxor Ferroelectrics and their Applications. In: Magnetic, Ferro- +electric, and Multiferroic Metal Oxides, Stojanovic B. D. (Ed.), Elsevier, 2018, 233–249, +doi:10.1016/B978-0-12-811180-2.00011-6. +2. Glinchuk M. D., Ragulya A. 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V., Sidak V. M., Trubitsyn M. P., Popov S. A., Tuluk A. Yu., Suchanicz J., Acta Phys. Pol. A, 2018, +133, 816–818, doi:10.12693/APhysPolA.133.816. +11. Suchanicz J., Kluczewska K., Czaja P., Kania A., Konieczny K., Handke B., Sokolowski M., Trubitsyn M.P., +Kruzina T.V., Ceram. Int., 2017, 43, 17194–17201, doi:10.1016/j.ceramint.2017.09.144. +12. Suchanicz J., Wąs M., Nowakowska-Malczyk M., Konieczny K., Czaja P., Kluczewska-Chmielarz K., +Marchewka J., Wcisło D., Wolański R., Stanuch K., Trubitsyn M. P., Sokolowski M., Phase Transitions, 2021, +94, No. 3–4, 210–218, doi:10.1080/01411594.2021.1931204. +13. Sidak V., Trubitsyn M., In: 2015 International Young Scientists Forum on Applied Physics (YSF), IEEE, +Dnipropetrovsk, Ukraine, 1–2, doi:10.1109/YSF.2015.7333268. +14. Sidak V., Trubitsyn M., J. Phys. Electron., 2020, 28, 87–90, doi:10.15421/332026. +15. Sidak V. M., Trubitsyn M. P., Appl. Nanosci., 2022, 12, 775–780, doi:10.1007/s13204-021-01712-y. +16. Sidak V. M., Trubitsyn M. P., Appl. Nanosci., (in press). +17. Poplavko Yu., Yakymenko Yu., Functional Dielectrics for Electronics Fundamentals of Conversion Properties, +Woodhead Publishing, Duxford, 2020. +18. Chen R., Kirsh Y., The Analysis of Thermally Stimulated Processes. In: International series in the science of +the solid state, Vol. 15, Pamplin B. (Ed.), Elsevier, 1981, 1–16, doi:10.1016/C2009-0-14588-9. +19. Simmons J. G., Taylor G. W., Phys. Rev., 1972, 5, No. 4, 1619–1629, doi:10.1103/physrevb.5.1619. +20. Erdem E., Jakes P., Eichel R.-A., Sinclair D. C., Pasha M., Reaney I. M., Funct. Mater. Lett., 2010, 03, No. 01, +65–68, doi:10.1142/s1793604710000956. +21. Yang F., Li M., Li L., Wu P., Pradal-Velázquez E., Sinclair D. C., J. Mater. Chem. A, 2018, 6, No. 13, 5243–5254, +doi:10.1039/c7ta09245h. +43705-9 + +V. M. Sidak, M. P. Trubitsyn, T. V. Panchenko +22. Scharfschwerdt R., Mazur A., Schirmer O. F., Hesse H., Mendricks S., Phys. Rev. B, 1996, 54, No. 21, 15284– +15290, doi:10.1103/PhysRevB.54.15284. +23. Shebanovs L., Ferroelectrics, 2002, 269, No. 1, 87–92, doi:10.1080/00150190211137. +Дiелектрична релаксацiя, iндукована кисневими +вакансiями в керамiцi Na0.5Bi0.5TiO3 +В. М. Сiдак1, М. П. Трубiцин2, Т. В. Панченко2 +1 Днiпровський державний медичний унiверситет, Україна, 49044 Днiпро, вул. Вернадського, 9 +2 Днiпровський нацiональний унiверситет iменi Олеся Гончара, Україна, 49045 Днiпро, пр. Гагарiна, 72 +Дослiджено дiелектричну проникнiсть керамiки релаксорного сегнетоелектрика натрiй-вiсмутового тита- +нату Na0.5Bi0.5TiO3. Вимiрювання проводили на необроблених i термiчно оброблених у вакуумi зразках. +В необроблених зразках спостерiгалися розмитi дiелектричнi аномалiї, пов’язанi зi структурними фазови- +ми переходами. Iнтенсивний пiк дiелектричної проникностi (𝜀max ∼ 104) з’явився пiсля термiчної оброб- +ки у вакуумi. Аномалiя 𝜀(𝑇) була спричинена повiльними процесами поляризацiї ( 𝑓 < 10 кГц) i була +нестабiльною, зникаючи при нагрiваннi в повiтрi до 800 K. Температурнi та частотнi залежностi 𝜀(𝑇) опи- +сано за допомогою моделi Коула-Коула з урахуванням термостимульованого затухання нестабiльної поля- +ризацiї. Передбачається, що дiелектрична аномалiя визначається механiзмом поляризацiї просторового +заряду. Кисневi вакансiї V•• +O та електрони, локалiзованi на iонах титану Ti′ +Ti, вважаються вiдповiдальними +за спостережуване явище. +Ключовi слова: дiелектричнi властивостi, дiелектрична проникнiсть, перовскiти, дефекти +43705-10 + diff --git a/KtAzT4oBgHgl3EQfkP1s/content/tmp_files/load_file.txt b/KtAzT4oBgHgl3EQfkP1s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38e4fe87a03af1415bbac9ba81947fef4d231a4e --- /dev/null +++ b/KtAzT4oBgHgl3EQfkP1s/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf,len=632 +page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 4, 43705: 1–10 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='43705 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='ua/journal Dielectric relaxation induced by oxygen vacancies in Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 ceramics V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Sidak 1∗, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Trubitsyn 2, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Panchenko 2 1 Dnipro State Medical University, 9 Vernadsky St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=', 49044 Dnipro, Ukraine 2 Oles Honchar Dnipro National University, 72 Gagarina Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=', 49045 Dnipro, Ukraine Received June 30, 2022, in final form August 31, 2022 Dielectric permittivity was studied in ceramics of relaxor ferroelectric bismuth-sodium titanate Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The measurements were performed on as sintered and heat treated in vacuum samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The diffuse dielectric anomalies associated with the structural phase transitions were observed in as sintered samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The intense peak of permittivity (𝜀max ∼ 104) appeared after heat treating in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The anomaly of 𝜀(𝑇) was contributed by slow polarization processes ( 𝑓 < 10 kHz) and was non-stable, vanishing on heating in air up to ∼ 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Tem- perature and frequency dependencies of 𝜀 were described by using Cole-Cole model with accounting thermally stimulated decay of the non-stable polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' It is supposed that the dielectric anomaly is determined by space charge polarization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Oxygen vacancies V•• O and electrons localized on titanium ions Ti′ Ti are assumed to be responsible for the phenomenon observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Key words: dielectric properties, permettivity, perovsikes, defects 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Introduction High sensitivity for external fields is the most valuable requirement for functional materials used to transform energy from certain kind to another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' An increased susceptibility often results from lattice instability in the range of structural phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why crystalline compounds undergoing structural transformations are intensively investigated by researchers and technologists involved in cre- ation of new functional materials for piezoelectric, thermoelectric, photovoltaic and other converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The crystals with perovskite ABO3 structure have found wide range of applications in modern electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Consequently, the compounds of the perovskite family are among the most popular objects for studies in materials sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Variations of chemical composition, formation of the structure on nano- and micro- meter levels, control on the lattice defects make it possible to create the materials with a broad variety of physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thus, ceramics based on Pb-ZrTiO3 show extremely high electro-mechanical para- meters and are used in piezoelectric devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Introducing the transition groups ions into the structural ABO3 unit leads to the appearance of magneto-electrical coupling in multiferroic materials (BiMnO3, BiFeO3, TbMnO3) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Some crystals with complex perovskite structure like ACu3Ti4O12 (A = Ca, Ba, Sr) possess extremely high dielectric constants (∼ 104 − 105) which opens new prospects to be used as the materials with high permittivity in memory and microwave devices [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' It is well known that structural imperfections can strongly affect the properties of crystals and even are capable of inducing new phenomena which are not observed in a perfect lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why comprehensive information on typical intrinsic and extrinsic lattice defects becomes of high importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' At the present time, numerous works are aimed at studying the mechanisms of the influence of defects on the properties of crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Based on the knowledge gained, the technological approaches are developed that allow to control qualitatively and quantitatively the defectiveness of the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Doping with iso- or heterovalent impurities, heat treatment in various atmospheres, applying external fields make it ∗Corresponding author: vasylsidak@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 43705-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='01528v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='mtrl-sci] 4 Jan 2023 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Sidak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Trubitsyn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Panchenko Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The crystal structure of NBT in tetragonal phase [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' possible to stimulate the appearance of the defects that improve the targeted characteristics or, conversely, to reduce the content of undesirable defects that degrade the useful parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Intensive experimental and technological studies aimed at controlling the subsystem of defects, have led to the appearance of the “defect engineering” concept [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Modern requirements in the field of environmental protection considerably changed the situation in the production of functional materials and urge the search for new compositions free from health harmful chemical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Lead-free bismuth-sodium titanate Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 (NBT) meets these requirements and shows a number of attractive physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' NBT crystal belongs to a group of complex perovskites with the structure of A’A”BO3 type, where sodium and bismuth atoms are randomly distributed through the A-site (figure 1) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The extremely high electro-mechanical coupling is the most prominent physical property of NBT crystal and solid solutions based on it [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The specific properties are directly related to the structural phase states observed in NBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' On cooling from high-temperature side NBT undergoes the following sequence of phase transitions: from cubic to tetragonal ferroelastic phase at 𝑇𝐶 ≈ 810 K, and further to rhombohedral ferroelectric phase at 𝑇𝑅 ≈ 490 K [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In the range of T𝑅, NBT demonstrates high permittivity and specific dielectric dispersion peculiar to relaxor ferroelectrics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Besides, the properties of NBT can be substantially modified by doping and technological treatments [9– 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Recently, the strong dielectric anomaly (∼ 104) was observed near 670–690 K in NBT single crystal [9, 13–15] and Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 – BaTiO3 (NBT–BT) solid solutions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The 𝜀(𝑇) dependence showed an anomalous temperature behaviour and unusual frequency dispersion (here and below symbol 𝜀 without prime means real part of permittivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In addition, permittivity peak disappeared after heat treatment in air (∼ 800 K) and could be restored by heat treating in vacuum (∼ 1070 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The authors of [15] supposed that dielectric anomaly was contributed by the dipole defects formed by oxygen vacancies (V•• O ) and electrons localized on the nearest titanium ions Ti′ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The associated dipole defects (Ti′ Ti-V•• O )• were considered as unstable and decomposing upon heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' These results were obtained for NBT single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Of course, for practical applications, NBT ceramics can be expected as more commercially and technologically acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In this paper anomalous dielectric relaxation mentioned above is studied in NBT ceramics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' By accounting the permittivity value in maximum (∼ 104), the previous interpretation based on the dipole defects [15, 16] is considered critically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' It is supposed that a strong dielectric peak can be associated with space charge polarization phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The possible microscopic mechanisms of the dielectric relaxation are briefly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 43705-2 NaBi TiDielectric relaxation induced by oxygen vacancies in Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 ceramics 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Experimental results The NBT ceramics were prepared by usual sintering technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The samples for electrical properties measurements were cut off as the plane-parallel plates with the edges of about 5 × 5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='8 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The Pt electrodes were deposited on the main planes of the samples by cathode sputtering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Electrical properties were measured using AC bridge P 5083 in the temperature interval 300–800 K for the frequency range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5–100 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Two types of the samples were used: i) prepared from as sintered ceramics and ii) heat treated in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The regimes of heat treating were the same as those previously used for single crystals (𝑇 = 1070 K, 𝑡 = 2 h, 𝑝 ≈ 1 Pa) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 400 600 800 0,5 1,0 1 2 3 4 5 6 7 8 e, 10 4 T, K 1 2 3 10 8 10 7 10 6 s, Ohm 1 cm 1 10 3 /T, K 1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The permittivity dependencies 𝜀(𝑇) in as sintered NBT ceramics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The AC field frequency was f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='8 (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 1 (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 2 (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 5 (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 10 (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 50 (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 100 (8) kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The inset shows Arrhenius plot of conductivity 𝜎(1/𝑇) dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The temperature dependencies of dielectric permittivity 𝜀 and electrical conductivity 𝜎 measured on heating for as sintered NBT ceramics are shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In contrast to the data obtained for NBT and NBT-BT single crystals [15, 16], 𝜀(𝑇) dependence does not show intense relaxation anomaly and reflects the structural transformations in the range of T𝐶 and T𝑅 only (figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The 𝜀(𝑇) dependencies measured on the next cooling run and on the subsequent heating-cooling cycles coincide with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The inset to figure 2 shows the temperature dependencies of conductivity 𝜎 plotted in Arrhenius scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can see that 𝜎 increases with AC field frequency 𝑓 and weakly depends on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' This behaviour is typical of dielectrics at relatively low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Only at frequencies 𝑓 < 2 kHz and for 𝑇 ⩾ 500 K conductivity starts to grow exponentially on heating that gives nearly linear regions in the Arrhenius plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Such a behaviour reflects a growing contribution of thermally activated charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Next, the sample of NBT ceramics was heat treated in vacuum, cooled to room temperature and after that its electrical properties were measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The data obtained are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can see that after heat treating, in the range 700 – 780 K, 𝜀(𝑇) demonstrates intense maximum (𝜀max ∼ 5 · 104, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5 kHz) which is strongly dependent on frequency 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' As 𝑓 increases, the peak of 𝜀(𝑇) sharply decreases in magnitude and shifts to higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Similarly to the data observed for NBT and NBT-BT single crystals [15, 16], intense 𝜀(𝑇) maximum (figure 3) could be detected for the first heating run only and disappeared for the next cooling and heating runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Corresponding dependencies of conductivity 𝜎(1/𝑇) are shown in the inset to figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In the low-temperature interval (𝑇 < 500 K), 𝜎 demonstrates nearly the same behaviour as in the untreated sample (the inset to figure 2), but for higher temperatures conductivity shows an intense peak corresponding to the relaxation maximum of 𝜀(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In subsequent temperature runs, the 𝜎(1/𝑇) dependencies did not show contribution from dielectric relaxation and were the same as shown in the inset to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 43705-3 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Sidak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Trubitsyn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Panchenko 400 600 800 0 1 3 5 1 5 2 6 3 7 4 8 e, 10 4 T, K 1 2 3 10 8 10 6 10 4 s, Ohm 1 cm 1 10 3 /T, K 1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The dependencies 𝜀(𝑇) measured in the first heating run of NBT ceramics previously heat treated in vacuum (1070 K, 2 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The AC field frequencies are indicated in the caption to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The solid lines were calculated by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The inset shows corresponding 𝜎(1/𝑇) dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The model As mentioned in section 2, the dielectric anomaly 𝜀(𝑇) similar to the one shown in figure 3 was earlier detected in single crystals of NBT and NBT-BT [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Special attention was paid to the nearly symmetrical shape of permittivity peak, that was quite different from the asymmetrical 𝜀(𝑇) anomaly of Debye relaxator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' It was proposed that dipoles or associated complexes responsible for the dielectric anomaly were thermally destroyed on heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Such decomposition was noticeable in the temperature range where the dielectric relaxation was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Consequently, the high-temperature wing of the 𝜀(𝑇) anomaly decreased more sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The first attempt to explain the specific character of the dielectric anomaly (figure 3) was made in [15] where a decrease of the dipoles or consentration of mobile defects was described as simple exponential temperature decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The possible role of configurational and vibrational entropy of the dipole defects was considered somewhat later in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Nevertheless, these approaches allowed to interpret the data only at the qualitative level and not provide a correct quantitative description of the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' More accurately, the 𝜀(𝑇, 𝑓 ) behavior in NBT-BT single crystal was described in [16], where Debye relaxator model was combined with the kinetic equation that determins the decay of the polarizing entities with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Dielectric response of real structures in an external AC field can be described by Cole-Cole, Davidson- Cole and other models [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' These models predict different types of dielectric spectra, symmetrical or non-symmetrical diagrams in complex (𝜀′–𝜀′′) plane, where 𝜀′ and 𝜀′′ represent real and imaginary parts of permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The dielectric anomaly shown in figure 3 is observed practically in the same temperature- frequency range (𝑇 > 500 K, 𝑓 < 10 kHz), where charge transfer processes notably contribute to conductivity (the nearly linear regions in the 𝜎(1/𝑇) dependencies, the insert to figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why the experimental diagrams plotted in (𝜀′–𝜀′′) for the used frequency region do not permit to make a reliable choice between the models mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Hence, the experimental data are described by Cole-Cole model [17] 𝜀∗(𝑇, 𝜔) = 𝜀∞ + 𝐶/𝑇 1 + (i𝜔𝜏𝑅)1−𝛼 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1) where 𝜔 = 2π 𝑓 is an AC field frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 𝑘 is Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1) includes a minimum number of the fitting parameters: 𝜀∞ – permittivity at high-frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Curie constant 𝐶 ∼ 𝑛 which is directly proportional to the concentration 𝑛 of the dipoles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 𝜏𝑅(𝑇) = 𝜏0 𝑅 exp(𝐸/𝑘𝑇) is the time of the 43705-4 Dielectric relaxation induced by oxygen vacancies in Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 ceramics relaxation of dipole moments in an external field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' energy parameter 𝐸 estimates the height of the potential barrier which is overcome at the reorientation dipole moments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' phenomenological parameter 0 ⩽ 𝛼 < 1 describes the distribution of relaxation times 𝜏𝑅 in disordered structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' It should be noted that for the samples heat treated in vacuum (figure 3), the anomalies of permittivity imaginary part 𝜀′′(𝑇, 𝑓 ) contain contributions from dielectric relaxation and charge transfer in the same temperature-frequency range (see the comments above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Hence, the analysis of 𝜀′′(𝑇, 𝑓 ) dependences needs to separate these contributions with apriori unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The anomalies of permittivity real part 𝜀′(𝑇, 𝑓 ) (figure 3) are mainly contributed by the polarization processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thus, the parameters of the discussed dielectric relaxation can be determined more directly from 𝜀′(𝑇, 𝑓 ) dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In addition, non-stable nature of polarization causes a specific type of anomalous behaviour which is more evident just for dependencies 𝜀′(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why the following analysis is focused on permittivity real part dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Further, it should be considered that polarizing entities (dipoles, associated complexes) are non- equilibrium and undergo thermal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can assume that a decrease of their concentration 𝑛 can be described by the simple kinetic equation [18] d𝑛 d𝑡 = − 𝑛 𝜏𝐷 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2) Here, 𝜏𝐷(𝑇) = 𝜏0 𝐷 exp(𝑈/𝑘𝑇) and 𝑈 are the time and energy parameters determining the thermal decay of non-stable polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2), one can go from differentiation in time to a derivative in temperature by considering that during the experiments, the samples were heated and cooled with a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thus, the temperature of the samples can be written as𝑇(𝑡) = 𝑇0+𝛾𝑡 , where T0 is an initial temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 𝛾 is the rate of temperature changes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 𝑡 is the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thus, remembering that 𝐶 ∼ 𝑛, from equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2 one can rewrite Curie constant as 𝐶(𝑇) = 𝐶0 · exp ������ − 1 𝛾𝜏0 𝐷 𝑇∫ 𝑇0 exp � − 𝑈 𝑘𝑇 � d𝑇 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='3) Direct fitting of expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1), with accounting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='3), to the experimental data was complicated and did not give reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Nevertheless, an approximate integration in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='3) performed in [19] yielded the following expression 𝐶(𝑇) = 𝐶0 · exp � − 𝑘𝑇2 𝛾𝜏0 𝐷(𝑈 + 𝑘𝑇) exp � − 𝑈 𝑘𝑇 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4) Thus, the experimental data shown in figure 3 can be described using Cole-Cole formulae 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1 combined with the approximate solution 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4 of the kinetic equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Discussion It is well known that oxygen vacancies V•• O are the typical defect for the crystals of complex oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In tightly packed structures like perovskites ABO3, the excess positive charge (+2e) associated with V•• O more probably can be compensated by the necessary number of cationic vacancies, which gives rise to the appearance of Schottky-type defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' If the concentration of V•• O is too high with respect to the number of cation vacancies, the additional electronic defects appear and the valence of the cations neighboring the V•• O can decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In ABO3 structures, the weakly bound electrons can be localized on titanium ions and as a result, Ti′ Ti centers can arise [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The energy levels of Ti′ Ti centers are shallow enough, and the electrons that hop via regular titanium ions can participate in the charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The presence of the nearest neighboring V•• O stabilizes the localized electrons and as a result, associated pairs (Ti′ Ti-V•• O )• can arise [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can expect that thermal treatment of NBT ceramics in vacuum (T = 1070 K) should increase mainly the concentration of V•• O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Each V•• O that arose in the treated ceramics could cause the emergence of 43705-5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Sidak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Trubitsyn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Panchenko two Ti′ Ti centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Correspondingly, the appearance of intense 𝜀(𝑇) anomaly after heat treatment (figure 3) can be just associated with the defects formed by V•• O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why in the previous works [9, 14–16] a slow dielectric relaxation in NBT single crystals was attributed to re-orientations of (Ti′ Ti-V•• O )• dipoles resulting from hopping of V•• O through oxygen octahedra vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thermal decay of polarization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2) was interpreted as a result of disassociation of (Ti′ Ti-V•• O )• centers occurring on heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Nevertheless, a great value of permittivity in the peak (∼ 5 · 104 at 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5 kHz, see figure 3) can be hardly attributed to the dipole defects, the concentration of which is assumed to be low enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' More probably, such a high value of 𝜀 can be the result of space charge polarization which is often observed in inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Usually, permittivity of such substances is defined as an effective one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Let us consider more in detail the assumption that intense 𝜀(𝑇) peak in figure 3 is contributed by mobile charge defects which in an external electric field can accumulate near certain inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Earlier in [9, 14–16] we supposed that reorientation of the dipole complexes (Ti′ Ti-V•• O )• in an external field and their thermal dissociation on heating occurred through V•• O hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why calculating the 𝜀(𝑇, 𝑓 ) dependencies by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4), we associated the pre-exponential factors 𝜏0 𝑅, 𝜏0 𝐷 for the relaxation times with inverse Debye frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Correspondingly, the values of 𝜏0 𝑅, 𝜏0 𝐷 were fixed (≈ 2 · 10−13 s) and estimated from Debye temperatures (𝜃 ≈ 260–350 K) typical of perovskites [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 1,5 2,0 2,5 10 5 10 0 10 5 t R , t D , s 10 3 /T, K t D t R Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The dependencies of the dielectric relaxation time 𝜏𝑅(1/𝑇) and polarization decay time 𝜏𝐷(1/𝑇) calculated from the data given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Assuming the space charge polarization to be the main effect, we have no apriori information on the values of 𝜏0 𝑅, 𝜏0 𝐷 and that is why we set them free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' We should add that considering 𝜏0 𝑅, 𝜏0 𝐷 as the fitting parameters allowed to reduce by about an order of magnitude the mean square deviation of the calculated data from the experimental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The curves calculated with the help of the model discussed in section 3, are drawn in figure 3 with the solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' It should be noted that the background contribution to 𝜀(𝑇) due to the structural phase transitions was taken into account as it was described earlier in [9, 14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In the scale chosen, this contribution shows only a weak dependency on temperature and in pure form can be seen in figure 2 where intense 𝜀(𝑇) peak is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The values of the parameters, used in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4) and averaged for all studied frequencies, are presented in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can see that fitting the calculated data to the experimental ones, in contrast to the previous assumption in [16], gives strongly different values for the pre-exponential factors 𝜏0 𝑅 and 𝜏0 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The value of 𝜏0 𝑅 corresponds to the order of typical lattice frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' By contrast, the factor 𝜏0 𝐷 for polarization decay is found to be twelve orders longer which corresponds to the infra-low frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Seemingly, such extremely high value of 𝜏0 𝐷 indirectly evidence in favor of space charge polarization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The dependencies of dielectric relaxation time 𝜏𝑅(1/𝑇) and polarization decay time 𝜏𝐷(1/𝑇), calcu- 43705-6 Dielectric relaxation induced by oxygen vacancies in Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 ceramics Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The values of the parameters in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4) obtained from the 𝜀(𝑇, 𝑓 ) dependencies (figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' C0, K 𝛼 𝜏0 𝑅, s 𝐸, eV 𝜏0 𝐷, s 𝑈, eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4(8) · 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='09(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2(2) · 10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='21(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1(2) · 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='67(2) 400 800 0 2 4 De, 10 4 T, K 1 2 3 4 5 400 800 0 4 C, 10 7 K T, K Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The contribution from non-stable polarization to dielectric anomaly Δ𝜀(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The dashed curves are calculated by using the following heating rates 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 1 (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 10 (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 102 (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 106 (5) K/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The parameters in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4) are taken from processing the experimental data measured at 𝑓 = 1 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The solid line is calculated for the experimental data in figure 3 (𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='7 K/min, 𝑓 = 1 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The inset shows the corresponding dependencies of Curie constant 𝐶(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' lated with the help of the data in table 1, are shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can see that for the whole studied interval, 𝜏𝐷 values considerably exceed the typical time (∼ 1 s) of a single measurement at certain 𝑇 and 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' On the other hand, 𝜏𝐷 values are comparable with the time (∼ 3 − 4 h) of a single measuring run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can show that the model (section 3) combining Cole-Cole formulae (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='1) with kinetic equa- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2) allows one to describe the specific features of the dielectric relaxation discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4) predicts that the form of the dielectric anomaly 𝜀(𝑇, 𝑓 ) should depend on time and on heating rate 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Obviously, such effects can be tested in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Besides, one can expect that the behaviour of 𝜀(𝑇, 𝑓 ) should depend on the ratio between the rates of dielectric relaxation 𝜏−1 𝑅 and polarization decay 𝜏−1 𝐷 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Really, the anomaly 𝜀(𝑇, 𝑓 ) takes a specific form (figure 3) since the decay of non-equilibrium polarization becomes notable in the same temperature range where dielectric peak is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thus, one can expect that the type of dielectric anomaly for certain 𝜏0 𝑅, 𝜏0 𝐷 values should depend on the ratio between activation energies 𝑈/𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Let us consider the effects mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Figure 5 shows the 𝜀(𝑇) anomaly calculated for different 𝛾 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can see how considerably the variations of 𝛾 can change the 𝜀(𝑇) behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' For high values of 𝛾 during the whole measuring cycle, the decay of non-equilibrium polarization remains practically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Correspondingly, in the limit 𝛾 → ∞, the behaviour of 𝜀(𝑇) approaches a classic behavior of Debye relaxator (figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' For the intermediate values of 𝛾, the anomaly of 𝜀(𝑇) takes the form of nearly symmetrical peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' At infinitely low heating rate (𝛾 → 0), non-equilibrium polarization has enough time to decay totally before the permittivity peak can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' As a result, on lowering 𝛾, the permittivity peak decreases in amplitude and finally disappears (figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The inset to figure 5 shows the calculated temperature dependence of Curie constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' On heating, 𝐶(𝑇) shows a step-like decrease manifesting a decay of non- equilibrium polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' For high rates of 𝛾, Curie constant possesses a maximum possible value and is practically temperature independent in the whole interval studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' For low values of 𝛾, Curie constant on heating decreases to zero before the dielectric relaxation can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 43705-7 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Sidak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Trubitsyn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Panchenko 400 800 0 2 4 De, 10 4 T, K 1 2 3 4 5 6 400 800 0 4 C, 10 7 K T, K Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The dielectric anomaly Δ𝜀(𝑇) (dashed lines) calculated for the following ratios 𝑈/𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='4 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5 (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='6 (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='7 (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='8 (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='2 (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The value of 𝐸 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='21 eV is fixed, and the value of 𝑈 is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The solid line corresponds to the experimental data in figure 3 (𝑈/𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='55, 𝑓 = 1 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The 𝐶(𝑇) dependencies are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Figure 6 illustrates how 𝜀(𝑇) anomaly changes its form when the ratio between activation energies 𝑈/𝐸 varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' For higher values of 𝑈/𝐸, one has a Debye-type behavior of 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' On lowering the ratio 𝑈/𝐸, permittivity 𝜀(𝑇) takes the intermediate peak-like form and finally it disappears when the ratio 𝑈/𝐸 decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The inset to figure 6 shows the corresponding dependencies of Curie constant 𝐶(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Assuming that an intense 𝜀(𝑇) peak (figure 3) is determined by space charge polarization effects, and for NBT ceramics one can consider the same typical defects such as oxygen vacancies V•• O , electrons localized on titanium Ti′ Ti and probably associated complexes based on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' V•• O can be assumed to be more heavy defects, whereas localized electrons Ti′ Ti can be supposed to be more light ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The mobile charge defects can accumulate near the following inhomogeneities in NBT ceramics: i) intergrain boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' ii) ferroelectric or ferroelastic domains boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' iii) near-electrode regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Presumably, one can expect that during long enough period the oxygen vacancies can accumulate near certain inhomogeneities and form the regions with higher V•• O concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' In the applied electric field, electrons Ti′ Ti move between the regions with increased V•• O concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' On heating, due to diffusion, the regions with high V•• O concentration dissolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' More information on the nature of the inhomogeneities which can cause space charge polarization in NBT can be obtained from comparison of the experimental data measured for ceramic and single crystalline NBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' This work is in progress at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Summary An intense low frequency anomaly of dielectric permittivity appeared in NBT ceramics after heat treating in vacuum (𝑇 = 1070 K, 𝑡 = 2 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The corresponding polarization was found to be non-stable and disappeared after heating in air up to ∼ 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The results of the thermal treatment evidenced that the observed dielectric relaxation was contributed by the defects including oxygen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Dielectric maxima were detected in the same temperature-frequency range where charge transfer processes gave notable contribution to conductivity in AC field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why we could not examine reliably the type of the experimental diagrams plotted in the complex plane of permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The temperature and frequency dependencies of 𝜀 were described on the basis of Cole-Cole model which could be used to describe dielectric relaxation in partially disordered structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Thermal decay of the non-equilibrium polarization was described using the simple kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The analysis was focused on the behaviour of permittivity real part which was mainly contributed by the polarization processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Combination of Cole-Cole model with kinetic equation allowed us to describe the experimental data with a good accuracy and to predict 43705-8 Dielectric relaxation induced by oxygen vacancies in Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 ceramics the evolution of dielectric anomaly under variations of the experimental conditions and characteristics of the phenomena observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' The great value of permittivity in the maximum (𝜀max ∼ 5 · 104, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5 kHz), observed for NBT ceramics, made doubtful the assumption that the dielectric anomaly could be due to the dipole defects the concentration of which was assumed to be not extremely high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' That is why it was supposed that the observed dielectric relaxation was determined by space charge polarization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Oxygen vacancies V•• O and electrons localized on titanium ions Ti′ Ti were assumed to be responsible for the phenomena studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' One can hope that more details on the microscopic mechanism of the thermally non-stable dielectric relaxation can be derived from comparative studies of the electrical properties of NBT single crystals and ceramics treated in atmospheres enriched and depleted in oxygen.' metadata={'source': 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+page_content=' Дiелектрична релаксацiя, iндукована кисневими вакансiями в керамiцi Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3 В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Сiдак1, М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Трубiцин2, Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Панченко2 1 Днiпровський державний медичний унiверситет, Україна, 49044 Днiпро, вул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Вернадського, 9 2 Днiпровський нацiональний унiверситет iменi Олеся Гончара, Україна, 49045 Днiпро, пр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Гагарiна, 72 Дослiджено дiелектричну проникнiсть керамiки релаксорного сегнетоелектрика натрiй-вiсмутового тита- нату Na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content='5TiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Вимiрювання проводили на необроблених i термiчно оброблених у вакуумi зразках.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' В необроблених зразках спостерiгалися розмитi дiелектричнi аномалiї, пов’язанi зi структурними фазови- ми переходами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Iнтенсивний пiк дiелектричної проникностi (𝜀max ∼ 104) з’явився пiсля термiчної оброб- ки у вакуумi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Аномалiя 𝜀(𝑇) була спричинена повiльними процесами поляризацiї ( 𝑓 < 10 кГц) i була нестабiльною, зникаючи при нагрiваннi в повiтрi до 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Температурнi та частотнi залежностi 𝜀(𝑇) опи- сано за допомогою моделi Коула-Коула з урахуванням термостимульованого затухання нестабiльної поля- ризацiї.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Передбачається, що дiелектрична аномалiя визначається механiзмом поляризацiї просторового заряду.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Кисневi вакансiї V•• O та електрони, локалiзованi на iонах титану Ti′ Ti, вважаються вiдповiдальними за спостережуване явище.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} +page_content=' Ключовi слова: дiелектричнi властивостi, дiелектрична проникнiсть, перовскiти, дефекти 43705-10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfkP1s/content/2301.01528v1.pdf'} diff --git a/KtFJT4oBgHgl3EQfxS3S/content/2301.11634v1.pdf b/KtFJT4oBgHgl3EQfxS3S/content/2301.11634v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..70ad6c66683880bcec839312d75e23367b425aa8 --- /dev/null +++ b/KtFJT4oBgHgl3EQfxS3S/content/2301.11634v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7038f29de384c258451b1c67170697d4d722725a3678693c2032860a0a09e3c1 +size 716257 diff --git a/KtFJT4oBgHgl3EQfxS3S/vector_store/index.faiss b/KtFJT4oBgHgl3EQfxS3S/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..bc7d9d8d010a907af675f59ac4c6bec6a67f420c --- /dev/null +++ b/KtFJT4oBgHgl3EQfxS3S/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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a/MNE3T4oBgHgl3EQfYgqz/content/tmp_files/2301.04489v1.pdf.txt b/MNE3T4oBgHgl3EQfYgqz/content/tmp_files/2301.04489v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a500f5847fb8aef8821466825b4f9353ac6d16ec --- /dev/null +++ b/MNE3T4oBgHgl3EQfYgqz/content/tmp_files/2301.04489v1.pdf.txt @@ -0,0 +1,2088 @@ +arXiv:2301.04489v1 [math.AP] 11 Jan 2023 +Pressure, Intermittency, Singularity +Peter Constantin +ABSTRACT. We give conditions for regularity of solutions of three dimensional incompressible Navier-Stokes +equations based on the pressure and on structure functions. +On the occasion of the centennial anniversary of O. A. Ladyzhenskaya +1. Introduction +We consider solutions of incompressible Navier-Stokes equations in R3, with smooth and localized +initial data, and discuss conditions in terms of pressure and structure functions that are easily accessible +and guarantee that solutions which are smooth on a time interval [0, T) have smooth (and hence unique) +extensions beyond T. The literature on regularity issues for Navier-Stokes equations is so extensive that we +are not able to give here even the beginning of a survey. We mention just some minimal references in this +short paper, with apologies to the many authors and works we knowingly or unknowingly leave out. +We discuss unforced Navier-Stokes equations +∂tu + u · ∇u − ν∆u + ∇p = 0, +(1) +with +∇ · u = 0, +(2) +and +u(x, 0) = u0. +(3) +The kinematic viscosity ν is a strictly positive constant, u is the velocity, p is the pressure. Most of this +paper is concerned with solutions in the whole space, but there will be a few instances in which we refer to +the bounded domain case. In that case the assumed boundary conditions are homogeneous Dirichlet, +u| ∂Ω = 0. +(4) +We maintain a sparing notation throughout the paper, omitting arguments and indices as often as we can. +We recall the local existence result for initial data (at time T0) in V . The spaces H (mentioned below) +and V are spaces of divergence-free vector fields which are completions of smooth compactly supported +divergence free fields in the topologies of L2 and H1. The norm in V is called the enstrophy. In the whole +space it corresponds to the ˙H1 norm. Initial data with finite enstrophy lead to local strong solutions, that is +unique solutions belonging to L∞(T0, T0+τ; V )∩L2(T0, T0+τ; H2∩V ) for some τ > 0. Strong solutions +are C∞ smooth for t > T0 in smooth domains [5]. By ”conditions for regularity” for smooth solutions on +a time interval [0, T) we mean conditions which guarantee u(T) ∈ V . These are global regularity condi- +tions. We note here that we are not talking about ǫ- regularity concepts ([11]) which are conditions on weak +solutions in space-time cylinders, which imply pointwise local regularity inside a smaller cylinder. When +assembled over space time, these conditions lead to partial regularity, and may lead to global regularity if +additional assumptions are in place, (for instance a single potential first singularity at one point). In this +paper we consider conditions which lead directly to persistence of regularity. +Key words and phrases. Navier-Stokes, pressure, intermittency, singularity. +MSC Classification: 35Q35, 35Q86. +1 + +2 +PETER CONSTANTIN +There are several well-known conditions for regularity. One of the simplest is +ˆ T +0 +∥u(t)∥4 +V dt ≤ MV < ∞. +(5) +From it, we have in a straightforward manner [5] that +∥u(t)∥2 +V ≤ ∥u(0)∥2 +V exp +� +Cν−3MV +� +(6) +for all 0 ≤ t ≤ T. We adhere to the good practice that arguments of exponentials or logarithms should be +nondimensional. Another easy to prove explicit condition is based on ∥∇u∥L3 (see below, Theorem 5). +The celebrated Ladyzhenskaya-Prodi-Serrin conditions [11] are +ˆ T +0 +∥u(t)∥p +Lqdt ≤ Mp,q < ∞, +(7) +with +2 +p + 3 +q = 1, +(8) +and 3 < q ≤ ∞. When q = 3 the condition is +∥u(t)∥L3 ≤ M3 < ∞, +t − a.e. +on +[0, T]. +(9) +As it is very well-known, the Ladyzhenskaya-Prodi-Serrin conditions imply regularity. The following is the +explicit bound on the enstrophy. +THEOREM 1. Let Ω be a bounded open domain in R3 with smooth boundary, let q > 3 and let u be a +strong solution of the Navier-Stokes equations in Ω on the interval [0, T]. There exists an absolute constant +C such that +∥u(t)∥2 +V ≤ ∥u(0)∥2 +V exp +� +Cν− q+3 +q−3 +ˆ t +0 +∥u(s)∥ +2q +q−3 +Lq ds +� +. +(10) +holds for 0 ≤ t ≤ T. In particular, if (7) holds then +∥u(t)∥2 +V ≤ ∥u(0)∥2 +V exp +� +Cν− q+3 +q−3 Mp,q +� +. +(11) +PROOF. Here is a brief proof. We recall that the Stokes operator is defined as +Au = −P∆u, +(12) +where P is the Leray projector on divergence-free vector fields. We recall +∥u∥H2(Ω) ≤ C|Au|H, +(13) +the fact that +∥u∥V = ∥u∥H1(Ω), +(14) +and the notation +B(u, v) = P(u · ∇v). +(15) +We start with the enstrophy evolution +1 +2 +d +dt∥u∥2 +V + ν|Au|2 +H = −(B(u, u), Au)H ≤ |B(u, u)|H|Au|H +(16) +Now, because P is a projector, it follows that +|B(u, u)|H ≤ ∥u · ∇u∥L2. +(17) +A H¨older inequality with exponents q, 2q +q−2, 2 yields +|B(u, u)|H ≤ ∥u∥Lq∥∇u∥ +L +2q +q−2 , +(18) + +3 +and because 2 < +2q +q−2 < 6, interpolation yields +∥u∥Lq∥∇u∥ +L +2q +q−2 ≤ ∥u∥Lq∥∇u∥ +1− 3 +q +L2 ∥∇u∥ +3 +q +L6 +(19) +Using the embedding H2(Ω) ⊂ W 1,6(Ω), (13) and (14), we have +|B(u, u)|H ≤ C∥u∥Lq∥u∥ +1− 3 +q +V +∥Au∥ +3 +q +H +(20) +Thus, from(16) we have +1 +2 +d +dt∥u∥2 +V + ν|Au|2 +H ≤ ∥u∥Lq∥u∥ +1− 3 +q +V +|Au| +1+ 3 +q +H +(21) +and Young’s inequality with exponents (1 +2(1 + 3 +q))−1, (1 +2(1 − 3 +q)))−1 yields +1 +2 +d +dt∥u∥2 +V + ν|Au|2 +H ≤ ν +2|Au|2 +H + Cν− q+3 +q−3∥u∥ +2q +q−3 +Lq ∥u∥2 +V . +(22) +The claimed inequality (10) follows by integrating the ODE inequality (22). +□ +REMARK 1. The same result holds in R3 or T3 with the same proof. +If q > 3, the bound on the enstrophy is precise and quantitative. In the case q = 3, in order to have +a good quantitative control it is useful to have a form of finite uniform integrability of |u(x, t)|3. This +condition is +∃δ > 0, ∀t, ∀A, +|A| ≤ δ ⇒ +ˆ +A +|u(x, t)|3dx ≤ +� ν +2C +�3 +. +(23) +In the left hand side, |A| is the Lebesgue measure of A. In the right hand side, ν is the kinematic viscosity +and C is the constant in Morrey’s inequality, +∥u∥L6(R3) ≤ C∥u∥ ˙H1(R3). +(24) +REMARK 2. The condition (23) is uniform in time, but it is much weaker than uniform integrability, +because +ν +2C is fixed. +THEOREM 2. Let u be a strong solution of the NSE in R3 on [0, T]. Assume (23). Then +∥u(t)∥2 +˙H1 ≤ min + + + +∥u0∥2 +˙H1 exp {t +� +∥u0∥2 +L2 +δν +� +}, +∥u0∥2 +˙ +H1 + +2 +δν2 ∥u0∥4 +L2, +(25) +where δ is the constant in (23). +REMARK 3. The time exponential bound is better than the time independent bound for times shorter +than +δν +∥u0∥2 +L2 log +� +1 + +2∥u0∥4 +L2 +δν2∥u∥2 +˙H1 +� +. After that time, the time independent bound is smaller. In either case, the +bound (25) implies that the enstrophy is bounded on [0, T], which in turn implies that the solution has a +unique strong extension beyond T. +PROOF. The proof (based on ([7]) follows from the enstrophy equation (16) using the fact that +|{x; |u(x, t)| ≥ U}| ≤ U −2∥u0∥2 +L2 +(26) +with the choice of +U = δ− 1 +2 ∥u0∥L2, +(27) +and estimating the nonlinear term separately in the region where |u(x, t)| ≥ U and where |u(x, t)| ≤ U by +∥u · ∇u∥L2 ≤ +�ˆ +|u||≥U +|u|3dx +� 1 +3 +∥∇u∥L6 + U∥∇u∥L2. +(28) + +4 +PETER CONSTANTIN +Then, using the assumption (23) we obtain +∥u · ∇u∥L2 ≤ ν +2∥∆u∥L2 + U∥∇u∥L2, +(29) +we absorb the first term in half the dissipation and use a Young inequality in the second term. We end up +with ODE inequality +˙y ≤ U 2 +ν y +(30) +for the quantity y = ∥u∥2 +˙H1 with U given by (27). The exponential bound in inequality (25) follows from +Gronwall, and the time independent bound follows by using ν +´ t +0 y(s)ds ≤ 2∥u0∥2 +L2. +□ +REMARK 4. The same result holds in bounded domains Ω or the periodic case T3, with the same proof. +We use the enstrophy equation (16), and the inequality (17). Then we estimate like in (28) and note that +∥∇u∥L6 ≤ C|Au|H. +As the reader may have already noticed, we are interested in explicit conditions, involving constants +known a priori, without the need to sample solutions, and which yield explicit enstrophy bounds. These +conditions are useful if additionally it is true that if they are satisfied uniformly on solutions of approxima- +tions that converge only almost everywhere, then the solutions are smooth. We refer to such conditions as +”easily accessible”. The conditions (7), (23) are easily accessible. +THEOREM 3. Let un(x, t) be an approximation of the NSE solution u(x, t) on the interval [0, T]. As- +sume that +un(x, t) → u(x, t) +(x, t) − a.e. +on +Ω × [0, T]. +(31) +(i) If there exists Mp,q < ∞, such that (7) holds for un uniformly for all n, then u obeys (7) with the same +constant Mp,q. +(ii) If there exists M3 such that (9) holds for un uniformly for all n, then u obeys (9) withe the same constant +M3. +(iii) If there exists a constant δ > 0 such that (23) holds for un uniformly for all n, then u obeys (23) withe +the same constant δ. +REMARK 5. In the case (i), the solution u obeys the quantitative bound (11). In the case (iii), the +solution u obeys the quantitative bound (25). In these cases the H1 bound on u(T) is explicit. +PROOF. The proof of (i), (ii), (iii) follows from applications of Fatou’s lemma. +□ +This paper is devoted to conditions based on pressure and on structure functions. There is a good motiva- +tion to seek conditions in terms of the pressure. In the absence of the pressure, the Navier-Stokes equations +are Burgers equations in 3D and obey a maximum principle. This implies that the velocity is bounded, (if +initially so), and the solutions are smooth for all time. The pressure in the Navier-Stokes equations is the +only reason the equations are not local and the velocity magnitude is not a priori controlled. Conditions of +regularity in terms of the pressure are known in L2 [1] and one sided in L∞ [12]. We present in this paper +an L +3 +2 condition, the analogue of the q = 3 condition (23) expressed in terms of only the pressure (Theo- +rem 8, Theorem 9). We also give the analogues of the Ladyzhenskaya-Prodi-Serrin conditions (Theorem 10). +The motivation to express conditions for regularity in terms of structure functions comes from experi- +mental, numerical and theoretical turbulence studies. Structure functions are averages of moments of veloc- +ity increments. They obey remarkable and robust statistical relations. The relations need interpretation and +then the may serve as reasonable hypothesis for the solutions of Navier-Stokes equations. One of the more +widely verified relations is the ”four-fifths” law [8] +⟨(δ∥ +ℓ u)3⟩ = −4 +5ǫ|ℓ| +(32) + +5 +where δ∥ +ℓ (u) = (u(x + ℓ) − u(x)) · +ℓ +|ℓ| is the longitudinal velocity increment and ǫ = −⟨dE +dt ⟩ is the rate +of dissipation of energy, which in the case of unforced NSE equals ν⟨|∇u|2⟩. The four-fifths law is shown +to hold for homogeneous and isotropic turbulence in the limit of time to infinity, followed by Reynolds +number to infinity, followed by ℓ → 0, in this order. The Navier-Stokes solutions are assumed to be smooth. +The braces ⟨·⟩ are expectations (ensemble average). Long time and space averages are usually assumed to +realize them, and in numerical experiments these averages are used. The assumption of finite positive ǫ +is also made, in the limit of time to infinity, followed by Reynolds number to infinity, in this order. The +Reynolds number is defined as +Re = UL +ν +(33) +where U is a velocity scale, and L is a length scale. The classical K’41 Kolmogorov theory proposes scaling +exponents ζp = p +3 for structure functions +⟨(δ∥ +ℓ u)p⟩ = Cp(ǫ|ℓ|)ζp. +(34) +These relations are expected to hold in a range of scales, |ℓ| ∈ (η, L) where η is the Kolmogorov dissipation +scale, +η = +�ν3 +ǫ +� 1 +4 +(35) +which is determined by the kinematic viscosity and energy dissipation rate, alone. Below the Kolmogorov +dissipation scale, it is assumed that viscous effects dominate, with smooth behavior. The length scale L +is the integral scale of turbulence. Turbulence findings are average statements, they refer to typical long +time behavior, and are asymptotic in Reynolds number. Interpreting them for the initial value problem for +Navier-Stokes equations is challenging. It is however reasonable to expect that there are many solutions +which give statistical weight to the turbulence laws and have properties that are consistent with them. +We give quantitative conditions in Theorem 11, Theorem 12. They involve a cutoff scale r = r(t). We +modify the structure function S2(x, r) (see (54) below) to take into account a possibly non-universal viscous +regularization below r. The regularity condition (101) requires +´ T +0 r(t)−4dt < ∞, a condition satisfied by +the Kolmogorov length r = η. The condition requires in addition the smallness of +´ +A S +3 +2 +2 (x, r)dx ≤ +� ν +C +�3 +on sets of small enough measure, |A| ≤ δ. +Modern theories modify the scaling ζp = p +3 in (34) of the K’41 theory, reflecting experimental and nu- +merical observation of intermittency. The turbulent signal is intermittent, that is, regions of high gradients of +velocity are found to be sparse in both time and space. The connection between intermittency and regularity +was explored in several mathematical works, (see for instance [10] and references therein) where assump- +tions of sparse behavior in physical space are used to deduce improved conditional regularity. In a different +setting [2], multifractal scaling exponents were connected to ratios of volume averages, and conditions for +regularity were given on the basis of intermittency dimension. +In terms of the exponents, it is found numerically (see for instance [9]) that ζ2 > 2 +3 and, while ζ3 remains +close to 1, ζp become smaller than p/3 for large p, and perhaps even tends asymptotically to a constant, +suggesting depletion of regularity, significantly below H¨older. We express conditions of regularity in terms +of a Dini modulus of continuity which give regularity if logarithmic scaling is assumed (Theorem 13). In +Section 4.3 we consider a multifractal scenario where regularity still persists. In Section 4.4 we give a +condition for regularity which requires small increments of velocity only in time dependent regions of high +velocity and high gradients. +The proofs are based on observations concerning the pressure. +2. The pressure +We consider solutions of +− ∆p = ∇ · (u · ∇u) +(36) + +6 +PETER CONSTANTIN +in Ω ⊂ R3, where u is divergence-free and sufficiently regular. We recall representation results from [4]. +We use the notation +f(x, r) = +1 +4πr2 +ˆ +|x−y|=r +f(y)dS(y) = + +|ξ|=1 +f(x + rξ)dS(ξ) +(37) +where dS is surface area and +ffl +denotes the integral normalized by the area of the region of integration, +which in the above case is 4π. We denote +σij( � +y − x) = 3(yi − xi)(yj − xj) +|y − x|2 +− δij +(38) +where +� +y − x = y − x +|y − x|. +(39) +The following lemma was proved in [4]. +LEMMA 1. Let x ∈ Ω ⊂ R3, let 0 < r < dist(x, ∂Ω), and let p solve (36) with divergence-free +u ∈ C2(Ω)3. Let v ∈ R3. Then +p(x) = p(x, r) − 1 +3|u(x) − v|2 + +ffl +|ξ|=1 |ξ · (u(x + rξ) − v)|2 dS(ξ) ++P.V. +´ r +0 +dρ +ρ +ffl +|ξ|=1 σij(ξ)(ui(x + ρξ) − vi)(uj(x + ρξ) − vj)dS(ξ). +(40) +All terms in the right hand side of (40) are determined solely by information in the ball of radius r about +x. We denote the singular integral +K(x, r) = P.V. 1 +4π +ˆ +|y−x|≤r +σij +� +� +y − x +� +|y − x|3 +(ui(y) − vi)(uj(y) − vj)dy. +(41) +Thus, (40) reads +p(x) = p(x, r) − 1 +3|u(x) − v)|2 + + +|ξ|=1 +|ξ · (u(x + rξ) − v)|2 dS(ξ) + K(x, r). +(42) +Evidently, K depends on the choice of the vector v. In applications we want to be able to choose v appro- +priately. For divergence free functions u, it holds that + +|ξ|=1 +ξi (ξ · u(x + rξ)) dS(ξ) + 1 +4π PV +ˆ +B(x,r) +σij( � +x − y) +|x − y|3 uj(y)dy = 1 +3ui(x). +(43) +This follows from +1 +4π +ˆ +B(x,r) +yi − xi +|y − x|3 (∇ · u)(y)dy = 0 +by integration by parts. We also note that + +|ξ|=1 +(ξ · v)2dS(ξ) = 1 +3|v|2. +(44) +This is true for any v that does not depends on ξ. Therefore, the representation (40) is valid even if v is a +function of x and r (but not ξ). Indeed, this follows by opening brackets in the right hand side of (40), using +(43) and (44), and identifying what remains as (40) for v = 0, which was proved independently in [4]. +We average (42) 1 +R +´ 2R +R +dr and obtain the representation [4] +THEOREM 4. Let p solve (36) with divergence-free u ∈ C2(Ω)3. Let x ∈ Ω ⊂ R3, v ∈ R3 and let +0 < r < 1 +2dist(x, ∂Ω). Then, +p(x) = β(x, r) + π(x, r) − 1 +3r +ˆ 2r +r +|u(x) − v|2dρ +(45) + +7 +with +β(x, r) = 1 +r +ˆ 2r +r +p(x, ρ)dρ +(46) +and +π(x, r) = 1 +r +ˆ 2r +r +� +K(x, ρ) + + +|ξ|=1 +|ξ · (u(x + ρξ) − v)|2dS(ξ) +� +dρ +(47) +The explicit expression for π(x, r) is +π(x, r) = P.V. 1 +4π +´ +|x−y|≤2r w +� +|y−x| +r +� σij(� +y−x) +|y−x|3 (ui(y) − vi)(uj(y) − vj)dy ++ 1 +4πr +´ +r≤|y−x|≤2r +1 +|y−x|2 +� +y−x +|y−x| · (u(y) − v) +�2 +dy, +(48) +where the weight w is given by +w(λ) = + + + +1, +if 0 ≤ λ ≤ 1, +2 − λ +if 1 ≤ λ ≤ 2, +0 +if λ ≥ 2 +(49) +We recall bounds on β and π (Propositions 2 and 3, [4]) +PROPOSITION 1. There exists an absolute constant C such that, for any r > 0, +∥∇β(·, r)∥L2(R3) ≤ Cr−1∥u∥2 +L4(R3) +(50) +and +∥β(·, r)∥L∞ ≤ Cr−2∥∇u∥L2∥u∥L2 +(51) +hold. Moreover, for any 1 < q < ∞, there exists Cq independent of r, such that +∥β(·, r)∥Lq(R3) ≤ Cq∥u∥2 +L2q(R3) +(52) +holds. +We choose now v = u(x). Then from (52) and the corresponding bound for p it follows also that +∥π(·, r)∥Lq(R3) ≤ Cq∥u∥2 +L2q(R3). +(53) +We denote by +S2(x, r) = 1 +4π +ˆ +|y|≤2r +1 +|y|3 |u(x + y) − u(x)|2dy +(54) +. We note that +|π(x, r)| ≤ 2S2(x, r) +(55) +follows from (48). For any measurable set A ⊂ Ω with dist(A, ∂Ω) > 2r we have +ˆ +A +S2(x, r)q ≤ Cqr2q +ˆ +A+rB(0,1) +|∇u(x)|2qdx +(56) +for any q ≥ 1 where B(0, 1) is the unit ball in R3. This follows in straightforward manner by writing +the integral in (54) in polar coordinates with y = ρξ, ρ = |y|, ξ = �y, expressing u(x + y) − u(x) = +´ 1 +0 +d +dλu(x + λρξ)dλ, and using Schwarz and H¨older inequalities. +REMARK 6. The bounds (52) and (53) for β and π are valid in bounded domains with smooth boundary +if we add bounds for ∥p∥Lq(∂Ω), see Lemma 2 in [6]. Once these are obtained the rest of the bounds which +are local bounds are valid. + +8 +PETER CONSTANTIN +3. Conditional regularity +THEOREM 5. Assume +ˆ T +0 +∥∇u(t)∥2 +L3dt = N < ∞. +(57) +There exists an absolute constant C such that +sup +0≤t≤T +∥∇u(t)∥2 +L2 ≤ ∥∇u0∥2 +L2 exp CN +ν +(58) +holds. +PROOF. +1 +2 +d +dt +ˆ +R3 |∇u|2dx + ν +ˆ +|∆u|2dx = +ˆ +R3(u · ∇u) · ∆udx ≤ ∥u∥L6∥∇u∥L3∥∆u∥L2 +followed by Schwartz and Morrey inerqualities results in +1 +2 +d +dt∥∇u(t)∥2 +L2 ≤ C +ν ∥∇u(t)∥2 +L3∥∇u(t)∥2 +L2, +and the proof is finished by Gronwall. +□ +Let +Λ = (−∆) +1 +2 . +(59) +THEOREM 6. Assume +ˆ T +0 +∥Λ +3 +2u(t)∥2 +L2dt = M < ∞. +(60) +There exists an absolute constant C such that +sup +0≤t≤T +∥∇u(t)∥2 +L2 ≤ ∥∇u0∥2 +L2 exp CM +ν +(61) +holds. +PROOF. This is a consequence of (58) and of the inequality N ≤ CM which follows from the familiar +[13] Riesz potential inequality +∥f∥L3 ≤ C∥Λ +1 +2 f∥L2 +(62) +with f = ∂iuj, for i, j = 1, . . . 3. +□ +REMARK 7. The finiteness (58) is a stronger regularity condition (i.e., more general) than the finiteness +(61). Both conditions are analogues of the Ladyzhenskaya-Prodi-Serrin condition (7) for q = ∞. The +finiteness (58) and (7) for q = ∞ are logically independent of each other. +THEOREM 7. There exists an absolute constant C such that +sup +0≤t≤T +∥Λ +1 +2 u(t)∥L2 ≤ ∥Λ +1 +2 u0∥L2 + C +ˆ T +0 +∥Λ +3 +2u(t)∥2 +L2dt +(63) +holds for smooth solutions of Navier-Stokes equations. +PROOF. We take the scalar product of the Navier-Stokes equation with Λu(t) and integrate. We obtain +1 +2 +d +dt∥Λ +1 +2u∥2 +L2 + ν∥Λ +3 +2 u∥2 +L2 ≤ +���� +ˆ +(u · ∇u) · Λudx +���� ≤ C∥u∥L3∥∇u∥L3∥Λu∥L3 ≤ C∥Λ +1 +2u∥L2∥Λ +3 +2 u∥2 +L3, +(64) +where we used (62). We divide by ∥Λ +1 +2u∥L2 and integrate in time. +□ +REMARK 8. In view of (62), the inequality (63) implies the boundedness of the L3 norm. Hence, via +the Navier-Stokes equation, the finiteness (60) implies (9). The main virtue of (63) is that it is viscosity- +independent, and is valid for Euler equations as well. + +9 +The analogue of the L3-based condition (9) in terms of only the pressure is the following. +THEOREM 8. There exists an absolute constant C, such that, if p = RiRj(uiuj) satisfies the finite +uniform integrability condition +∃δ > 0, ∀t, ∀A +|A| ≤ δ ⇒ +ˆ +A +|p(x, t)| +3 +2 dx ≤ +� ν +C +�3 +(65) +on [0, T], then u ∈ L∞(0, T; L3(R3)) with explicit bounds depending only on δ, ν, T, ∥u0∥L3, ∥u0∥L2. +PROOF. We take the evolution of the L3 norm of u: +1 +3 +d +dt +ˆ +R3 |u|3dx + ν +ˆ +R3 |u|[|∇u|2 + |∇|u||2]dx = − +ˆ +R3 |u|(u · ∇p)dx = +ˆ +R3 p(u · ∇|u|)dx +(66) +Let U be a large positive number and φ a positive smooth function of one variable which is compactly +supported in [0, 2], satisfies 0 ≤ φ ≤ 1 and identically equals 1 on [0, 1]. We split the RHS of (66) , +ˆ +R3 p(u · ∇|u|)dx = +ˆ +R3 φ +�|u| +U +� +p(u · ∇|u|)dx + +ˆ +R3 +� +1 − φ +�|u| +U +�� +p(u · ∇|u|)dx. +We estimate the first term, using that on the support of φ we have |u| ≤ 2U, +���� +ˆ +R3 φ +�|u| +U +� +p(u · ∇|u|)dx +���� ≤ (2U) +1 +2 √ +D +�ˆ +R3 p2dx +(67) +where +D = +ˆ +R3 |u||∇u|2dx +(68) +and then using the boundedness of Riesz transforms in Lp spaces and then interpolating L4 between L3 and +L9, we have +ˆ +R3 p2 ≤ C +ˆ +R3 |u|4dx ≤ C∥u∥ +5 +2 +L3∥u∥ +3 +2 +L9. +(69) +Now we use the fact that there exists a constant C such that +D ≥ C∥u∥3 +L9. +(70) +This fact follows from Morrey’s inequality ∥∇f∥L2 ≥ C∥f∥L6 applied for with f = |u| +3 +2 . Thus, from (67) +and (70) we have +���� +ˆ +R3 φ +�|u| +U +� +p(u · ∇|u|)dx +���� ≤ CU +1 +2D +3 +4 ∥u∥ +5 +4 +L3. +(71) +With Young’s inequality we have +���� +ˆ +R3 φ +�|u| +U +� +p(u · ∇|u|)dx +���� ≤ ν +2D + CU 2ν−3∥u∥5 +L3. +(72) +We know from the energy inequality, interpolation L2 − L6 and Morrey’s inequality that +ˆ T +0 +∥u∥4 +L3dt ≤ C∥u0∥2 +L2 +ˆ T +0 +∥∇u∥2 +L2dt ≤ Cν−1∥u0∥4 +L2, +(73) +and thus the factor CU 2ν−3∥u∥2 +L3 multiplying the ∥u∥3 +L3 is time integrable. The second term is estimated +as +���� +ˆ +R3 +� +1 − φ +�|u| +U +�� +p(u · ∇|u|)dx +���� ≤ C +√ +D∥u∥ +1 +2 +L9∥((1 − φ)p∥L +9 +4 +(74) +by using H¨older with exponents 9 +4, 18, 2. One more interpolation, +∥(1 − φ)p∥L +9 +4 ≤ ∥(1 − φ)p∥ +1 +2 +L +9 +2 ∥(1 − φ)p∥ +1 +2 +L +3 +2 + +10 +PETER CONSTANTIN +and the inequality based on the fact that Riesz transforms are bounded in Lp +∥(1 − φ)p∥L +9 +2 ≤ C∥u∥2 +L9, +together with (70) yield from (74) +ˆ +R3 +� +1 − φ +�|u| +U +�� +p(u · ∇|u|)dx ≤ CD∥((1 − φ)p∥ +1 +2 +L +3 +2 . +(75) +Now the support of 1 − φ +� +|u(x,t)| +U +� +is included in the set +BU(t) = {x | |u(x, t)| ≥ U} +(76) +which has uniformly small Lebesgue measure +|BU(t)| ≤ U −2∥u0∥2 +L2 +(77) +and, by assumption, the function x �→ |p(x, t)| +3 +2 satisfies (65), so that +∥((1 − φ)p∥ +1 +2 +L +3 +2 ≤ ν +2C +(78) +holds uniformly for t ∈ [0, T], if U is chosen large enough. +□ +REMARK 9. The condition (65) is weaker than uniform integrability, because ν +C is fixed. The condition +holds if |p(x, t)| +3 +2 is uniformly integrable in x on [0, T]. In particular, it holds if +p ∈ C(0, T; L +3 +2 (R3)), +(79) +or if p is piece-wise continuous on [0, T] with values in L +3 +2, because in these cases the curve t �→ p belongs +to a compact subset of L +3 +2 and therefore |p(x, t)| +3 +2 is uniformly integrable in x on [0, T]. The condition also +holds if |p(x, t)| ≤ f(x), x-a.e., with f ∈ L +3 +2 (R3) time independent, because this again implies uniform +integrabilty. +REMARK 10. As it is seen in the proof above, the condition (65) is not applied on just any set, but rather +on a set of interest, namely the set where the absolute magnitude of velocity exceeds a fixed large threshhold, +(76). It is easy to see that this set can be replaced by a smaller, and even more interesting set +BU,G(t) = {x | |u(x, t)| ≥ U, and |∇u(x, t)| ≥ G}. +(80) +The proof of this fact is similar to the proof above, using two cutoffs, and showing that the regions |u| ≤ U +and, separately |∇u| ≤ G lead each to a priori bounds on the size of the L3 norm of u, leaving only the +contributions from BU,G(t) to require control. +The following result shows that the condition (65) leads to easily accessible bounds. +THEOREM 9. Let r ≥ 4. There exists a constant C = Cr such that if (65) holds on [0, T] then +u ∈ L∞(0, T; Lr(R3)) +(81) +holds. More precisely, we have the single exponential bound +∥u(·, t)∥Lr ≤ ∥u0∥Lr exp +�Ct∥u0∥2 +L2 +νδ +� +(82) +where δ is the constant in (65). +PROOF. The evolution of the Lr norm is given by +1 +r +d +dt +ˆ +R3 |u|rdx + ν +ˆ +R3[|∇u|2|u|r−2 + (r − 2)|∇|u||2|u|r−2]dx = +ˆ +R3 pu · ∇|u|r−2dx. +(83) + +11 +We let U be a large positive number and φ a positive smooth function of one variable which is compactly +supported in [0, 2], satisfies 0 ≤ φ ≤ 1 and identically equals 1 on [0, 1]. We split the RHS of (66) . We split +the right hand side of (83), +ˆ +R3 p(u · ∇|u|r−2)dx = +ˆ +R3 φ +�|u| +U +� +p(u · ∇|u|r−2)dx + +ˆ +R3 +� +1 − φ +�|u| +U +�� +p(u · ∇|u|r−2)dx. +We estimate the first term using H¨older inequalities with exponents r +2 for p, 2 for a term involving the +gradient, |u| +r−2 +2 |∇u|, and +2r +r−4 for the term |u| +r−2 +2 , taking advantage of the fact that on the support of φ we +have |u| ≤ 2U, and using the boundedness of Riesz transforms in Lp spaces. We deduce that the first term +is bounded by +���� +ˆ +R3 φ +�|u| +U +� +p(u · ∇|u|r−2)dx +���� ≤ CU∥u∥ +r +2 +LrD +1 +2, +(84) +where +D = +ˆ +R3 |u|r−2|∇u|2dx. +(85) +We used, in view of r(r−2) +r−4 += r + +2r +r−4, that +�ˆ +|u|≤U +|u| +r(r−2) +r−4 dx +� r−4 +2r +≤ U∥u∥ +r +2 −2 +Lr . +(86) +The bound (84) is valid for r = 4 as well, we just take |u| ≤ U outside the integral and use L2 −L2 bounds. +Hiding +√ +D in 1 +2νD, we see that the inequality (84) leads to an exponential growth +∥u(·, t)∥Lr ≤ ∥u0∥LreCν−1 ´ t +0 U2ds +(87) +if the second term does not contribute to growth, The second term is bounded using +D ≥ C∥u∥r +L3r. +(88) +We first bound +��� +´ +R3 +� +1 − φ +� +|u| +U +�� +p(u · ∇|u|r−2)dx +��� ≤ ∥ +� +1 − φ +� +|u| +U +�� +p∥ +L +3r +r+1 ∥u∥ +r−2 +2 +L3r D +1 +2 +≤ C∥ +� +1 − φ +� +|u| +U +�� +p∥ +L +3r +r+1 D1− 1 +r , +(89) +and then use +∥ +� +1 − φ +� +|u| +U +�� +p∥ +L +3r +r+1 ≤ ∥ +� +1 − φ +� +|u| +U +�� +p∥ +1 +2 +L +3 +2 ∥ +� +1 − φ +� +|u| +U +�� +p∥ +1 +2 +L +3r +2 +≤ C∥ +� +1 − φ +� +|u| +U +�� +p∥ +1 +2 +L +3 +2 D +1 +r +(90) +to deduce +���� +ˆ +R3 +� +1 − φ +�|u| +U +�� +p(u · ∇|u|r−2)dx +���� ≤ C∥ +� +1 − φ +�|u| +U +�� +p∥ +1 +2 +L +3 +2 D. +(91) +The proof is completed by the assumption of finite uniform integrability (65) which shows this term to be +absorbed in the remaining dissipative term 1 +2νD. +□ +REMARK 11. The result above can be used together with Theorem (1) to give a quantitative bound +depending on δ on the supremum in time of the enstrophy. +The following result is the analogue of the Ladyzhenskaya-Prodi-Serrin condition in terms of the pres- +sure. + +12 +PETER CONSTANTIN +THEOREM 10. Let p = RiRj(uiuj). Assume that there exists q > 3 +2 such that +ˆ T +0 +∥p(t)∥ +2q +2q−3 +Lq(R3)dt < ∞ +(92) +Then u ∈ L∞(0, T; L3(R3)) obey u ∈ L∞(0, T; L3(R3)) with explicit bounds depending in addition to +(92) only on ν, T, ∥u0∥L3, ∥u0∥L2. +PROOF. The proof follows from the evolution of the L3 norm (66) by estimating as in (74) using H¨older +with exponents 9/4, 18, 2, +���� +ˆ +R3 p(u · ∇|u|)dx +���� ≤ C +√ +D∥u∥ +1 +2 +L9∥p∥L +9 +4 ≤ CD +2 +3∥p∥L +9 +4 +(93) +where we used also (70). We distinguish three ranges of q. +When q ≤ 9 +4 we interpolate +∥p∥L +9 +4 ≤ ∥p∥ +2q +9−2q +Lq +∥p∥ +9−4q +9−2q +L +9 +2 +. +(94) +We use ∥p∥L +9 +2 ≤ CD +2 +3 which folllows from (70) and the boundedness of Riesz transforms in Lp spaces, to +deduce +���� +ˆ +R3 p(u · ∇|u|)dx +���� ≤ CD1−α∥p∥ +2q +9−2q +Lq +(95) +with α = +2q−3 +(9−2q). From Young’s inequality, the boundedness of the L3 norm of u follows if we know that +´ T +0 ∥p∥ +2q +2q−3 +Lq +dt is finite, which was assumed in (92). +When q ∈ [9 +4, 9 +2], we use Young’s inequality in (93) and deduce that we need to estimate +´ T +0 ∥p∥3 +L +9 +4 dt. +We interpolate +∥p∥L +9 +4 ≤ ∥q∥ +2q +3(2q−3) +Lq +∥p∥ +4q−9 +3(2q−3) +L +3 +2 +≤ C∥q∥ +2q +3(2q−3) +Lq +∥u∥ +8q−18 +3(2q−3) +L3 +(96) +and thus +∥q∥3 +L +9 +4 ≤ ∥p∥ +2q +2q−3 +Lq +∥u∥α +L3 +(97) +with α = 8q−18 +2q−3 . If q ≤ 9 +2 we have α ≤ 3, and the condition (92) ensures that ∥u∥L3 remains bounded. +When q ≥ 9 +2 we estimate +���� +ˆ +R3 p(u · ∇|u|)dx +���� ≤ C +√ +D∥u∥ +1 +2 +L3∥p∥L3 +(98) +using H¨older with exponents 2, 3, 6. Using Young’s inequality we need to consider the effect of the quantity +∥u∥L3∥p∥2 +L3. We interpolate +∥p∥L3 ≤ ∥p∥ +q +2q−3 +Lq +∥p∥ +q−3 +2q−3 +L +3 +2 +≤ ∥p∥ +q +2q−3 +Lq +∥u∥ +2(q−3) +2q−3 +L3 +, +(99) +and it follows that +∥u∥L3∥p∥2 +L3 ≤ C∥p∥ +2q +2q−3 +Lq +∥u∥ +6q−15 +2q−3 +L3 +(100) +Because 6q−15 +2q−3 < 3, the condition (92) implies a uniform bound on ∥u∥L3 on [0, T] in this last case. +□ + +13 +4. Structure function +We assume +∃r(t), +´ T +0 r(t)−4dt < ∞, ∃δ > 0, ∀A, ∀t ∈ [0, T] +|A| ≤ δ ⇒ +´ +A S2(x, 2r(t)) +3 +2 dx ≤ +� ν +C +�3 +(101) +holds where S2(x, r) is given in (54). +THEOREM 11. There exists an absolute constant C, such that, for any T > 0, if a strong solution u of +NSE satisfies (101) for all 0 ≤ t < T, then u ∈ L∞(0, T; L3(R3)) with explicit bounds depending only on +ν, T, ∥u0∥L3, ∥u0∥L2 and the assumed δ > 0, +´ T +0 r(t)−4dt. +PROOF. We use (66) and decompose the pressure p = π + β as in (45), at each time t, with the choice +r = r(t). We bound the term +���� +ˆ +R3 |u|(u · ∇β)dx +���� ≤ Cr−1∥u∥4 +L4 +(102) +using (50), and then by interpolation (used also in (69)) we obtain +���� +ˆ +R3 |u|(u · ∇β)dx +���� ≤ Cr−1∥u∥ +5 +2 +L3∥u∥ +3 +2 +L9. +(103) +The dissipation D (68) obeys (70) and, so +���� +ˆ +R3 |u|(u · ∇β)dx +���� ≤ ν +6D + Cν−1r(t)−2∥u∥5 +L3. +(104) +In view of (73) and the assumption +´ T +0 r(t)−4dt < ∞, it follows that the factor r−2∥u∥2 +L3 multiplying +∥u∥3 +L3 is time integrable a priori, +r(t)−2∥u(t)∥2 +L3 ∈ L1([0, T]), +(105) +and thus this term leads to an explicit uniform bound on ∥u∥L3, in terms of the initial data and +´ T +0 r−4dt. +We integrate by parts in the term +���� +ˆ +R3 |u|(u · ∇π)dx +���� = +���� +ˆ +R3 π(u · ∇|u|)dx +���� ≤ +ˆ +|u|≤U +|π||u||∇|u||dx + +ˆ +|u|≥U +|π||u||∇|u||dx +(106) +where U is a large time independent constant, at our disposal. We estimate the first term using (52): +ˆ +|u|≤U +|π||u||∇|u||dx ≤ C2U +1 +2√ +D∥u∥2 +L4 ≤ ν +6D + CU 2ν−3∥u∥5 +L3. +(107) +where we used interpolation (used also in (69)) and (70). We argue like in the proof of Theorem (8), +invoking (73), which implies that the term U 2∥u∥2 +L3 multiplying ∥u∥3 +L3 in the right hand side of (107) is +time integrable with an explicit a priori bound, and as such it leads via Grownwalll to an explicit bound on +∥u∥L3. +Finally, taking U large enough so that the set BU(t) of (76) has small measure as in (77), using (55), +proceeding in the same manner as for (75), and using the assumption (101), we have +ˆ +|u|≥U +|π||u||∇|u||dx ≤ CD +�ˆ +BU +S2(·, r(t)) +3 +2 dx +� 1 +3 +≤ ν +6D. +(108) +This term is absorbed in the remaining dissipative term, ending the proof. +□ +As in the case of the condition regarding the finite uniform integrability of the pressure (65), the structure +function finite integrability condition (101) leads to easily accessible bounds. + +14 +PETER CONSTANTIN +THEOREM 12. Let q ≥ 4. There exists a constant C depending on q such that if (101) holds on [0, T] +then +u ∈ L∞(0, T; Lq(R3)) +(109) +holds. More precisely, we have the single exponential bound +∥u(·, t)∥Lq ≤ ∥u0∥Lq exp +�Ct∥u0∥2 +L2 +νδ ++ ν− 3 +2 ∥u0∥2 +L2Γ(t) +� +(110) +where δ is the constant in (101) and +Γ(t) = +�ˆ t +0 +r−4(s)ds +(111) +is bounded on [0, T] by assumption. +PROOF. The proof follows closely the proof of Theorem 9. We use the evolution of the Lq norm (83) +where we changed r to q because r has now a different meaning. We split at each time p = β(·, r) + π(·, r) +using r = r(t). We bound the term +ˆ +R3 βu · ∇|u|q−2dx ≤ ∥β∥Lq∥u∥ +q−2 +2 +Lq +√ +D ≤ C∥β∥ +1 +2 +L∞∥u∥ +q +2 +Lq +√ +D +(112) +where we used H¨older with exponents q, 2 and +2q +q−2, interpolated ∥β∥Lq ≤ ∥β∥ +1 +2 +L∞∥β∥ +1 +2 +L +q +2 and used the +bound ∥β∥L +q +2 ≤ C∥u∥2 +Lq. Hiding +√ +D in 1 +3νD, this term leads to a growth factor +ν−1 +ˆ t +0 +∥β(·, r(s))∥L∞ds ≤ Cν− 3 +2∥u0∥2 +L2Γ(t) +(113) +where we used (51) and the Navier-Stokes energy inequality. We treat the terms involving π in exactly the +same manner as we treated p in the proof of Theorem 9. We omit further details. +□ +REMARK 12. Theorem 12 is stronger than Theorem 11 for strong solutions, which have initial data in +H1. +REMARK 13. A particularly significant small scale r = η is given by classical turbulence theory, where +the Kolmogorov dissipation wave number kd, inverse of the viscous dissipation scale η is given by +kd = η−1 = ν− 3 +4 ǫ +1 +4 = ν− 1 +2 (⟨|∇u(t)|2⟩) +1 +4 . +(114) +We note that it is a priori time integrable to power 4. +4.1. A nearly selfsimilar example. Theorem 11 (or rather, its proof) applies to functions which have +small translation increments in L3. We consider +u(x, t) = V + smooth +(115) +where the leading term V satisfies +∥V (y + ·) − V (·)∥L3 ≤ W(t)|y|s +(116) +for some s > 0. In order to have nondimensional quantities, we write +W = UL1−s. +(117) +The typical example is of the form +V (x, t) = U(t)P +� x +L(t) +� +(118) +where P is time independent and ∥P(· + z) − P(·)∥L3 ≤ C|z|s. We note that this condition is satisfied by +many functions with slow decay which are not in L3(R3) or even in L2(R3), such as P(z) = (1 + |z|)−β, +β > 0. Of course, the condition is also satisfied on Bs +3,∞(R3). + +15 +We have that (116) reads +∥V (y + ·) − V (·)∥L3 ≤ UL +�|y| +L +�s +, +(119) +and define +UL +ν += Re(V ). +(120) +We take ǫ > 0 such that s > 1 +2ǫ, and, writing |y|−3 = |y|−1+ǫ|y|−2−ǫ we use a H¨older inequality with +exponents 3, 3 +2 to bound +S2(x, r) +3 +2 ≤ Cǫ− 1 +2 r +3ǫ +2 +ˆ +|y|≤r +|y|−3− 3ǫ +2 |V (y + x) − V (x)|3 dy +(121) +Integrating dx on A and switching the order of integration we deduce +´ +A S2(x, r) +3 +2dx ≤ Cǫ− 1 +2 r +3ǫ +2 ´ +|y|≤r |y|−3− 3ǫ +2 ´ +A |V (y + x) − V (x)|3 dxdy +≤ Cǫ− 1 +2 W 3r3s = Cǫ− 1 +2(UL)3 � r +L +�3s . +(122) +Assuming a bound on the Reynolds number of the profile, +Re(V ) ≤ R +(123) +and fixing ǫ < 2s, we have +ˆ +A +S2(x, r) +3 +2 dx ≤ Cs +� r +L +�3s +(Re(V )3ν3 ≤ Cs +� r +L +�3s +R3ν3 +(124) +The condition (101) is satisfied if +� r +L +�s +R ≤ C +− 1 +3 +s +(2C)−1. +(125) +REMARK 14. The condition (125) shows that for self-similar profiles with time dependent collapsing +inner scale L, the condition is satisfied choosing r small compared to the collapsing scale L. Regularity +follows if L−4(t) is time integrable. In particular, if the leading term V is given by (118) and U(t)L(t) ≤ +Rν, then regularity follows. The proof of this fact follows verbatim the proof of Theorem (11) including the +estimate (107). In that estimate now U is time dependent and it is bounded above by RL(t)−1. The term +U −2∥u∥2 +L3 is still time integrable and that is why the result continues to hold. +4.2. A Dini Condition. +THEOREM 13. Assume that u satisfies +∥δyu∥L3 ≤ m(|y|) +(126) +where δyu(x, t) = u(x + y, t) − u(x, t), and where 0 ≤ m is a time independent function satisfying +ˆ 1 +0 +m2(ρ)dρ +ρ < ∞. +(127) +Then u satisfies (101) with r time independent, and consequently, smooth solutions of Navier-Stokes equa- +tions obeying (126) with (127) on [0, T) obey u ∈ L∞(0, T; L3(R3)) with explicit bounds depending only +on m, ν, T, ∥u0∥L3, ∥u0∥L2. +PROOF. The proof follows from the fact that +∥S2(·, r)∥L +3 +2 ≤ C +ˆ +|y|≤2r +∥δyu∥2 +L3 dy +|y|3 . +(128) +This inequality is proved by duality, integrating S2 against a test function in L3 +ˆ +R3 S2φdx = 1 +4π +ˆ +|y|≤2r +dy +|y|3 +ˆ +R3 φ(x)|δyu(x)|2dx ≤ 1 +4π∥φ∥L3 +ˆ +|y|≤2r +∥δyu∥2 +L3 dy +|y|3 . +(129) + +16 +PETER CONSTANTIN +From (128) and the assumed Dini condition, we deduce that the ∥S2(·, r)∥L +3 +2 ≤ +� ν +C +�2 if r is chosen small +enough so that +ˆ 2r +0 +m2(ρ)dρ +ρ ≤ +� ν +C +�2 +. +(130) +□ +REMARK 15. Clearly m(r) ∼ log−α(r−1) with α > +1 +2 is sufficient. As we remarked before, the +smallness of the L3 increment does not imply that the function needs to be in L3. We also remark that m can +be allowed to depend on time, if m(r)r−1 is uniformly integrable on [0, 1], or more generally, if denoting +Im(t)(r) = +ˆ 2r +0 +m2(ρ, t)dρ +ρ +(131) +we have that the preimage of +� ν +C +�2 under Im(t), that is r(t) = I−1 +m(t) +�� ν +C +�2� +, obeys +´ T +0 r(t)−4dt < ∞. +4.3. Multifractal intermittent scenario. We consider the region BU(t) = {x | |u(x, t)| ≥ U} de- +fined before in (76). We take U time independent. We introduce a time independent length scale L > 0, and +we require +U 2 ≥ L−3 +ˆ +R3 |u|2dx +(132) +so that +|BU| ≤ L3. +(133) +We assume that the velocity increments +s2(x, r) = + +|y|=r +|u(x + y) − u(x)|2dS(y) +(134) +obey bounds +s2(x, r) ≤ G2 � r +L +�2α(x) +(135) +with G > 0 constant (with units of velocity), L > 0 as above, constant, (with units of length) and with +0 < α(x) ≤ 1. This upper bound is assumed to hold a.e. in x ∈ BU(t) and for all 0 < r < r0, where +0 < r0 < L is a fixed positive constant. Because +S2(x, r) = +ˆ 2r +0 +s2(x, ρ)dρ +ρ , +(136) +we have that +S2(x, r0) ≤ CG2 +1 +α(x) +�r0 +L +�2α(x) +(137) +holds a.e in x ∈ BU. In multifractal turbulent intermittent scenarios, it is assumed that there is a spectrum +of near-singularities of H¨older exponent h and that these are achieved on sets Σh of dimension d(h) ≤ 3 +which occur randomly with probability dµ(h). +The dimension d(h) is implemented in the following manner. We take a region Vh around Σh and +partition it in small disjoint cubes of size ρ with ρ ≤ r0. This region is a ”collar”of cross-section size ρ +around the set Σh ∩ BU. The multifractal assumption is that the number of such cubes of Vh is of the order +Nh(ρ) = +� ρ +L +�−d(h). Assuming α(x) ≥ h to hold on each such cube, we have from (137), on each cube +S2(x, r0) ≤ CG2h−1 �r0 +L +�2h +. +(138) +Writing the volume of the cube as L3( ρ +L)3, we have +ˆ +BU∩Vh +S2(x, r0) +3 +2dx ≤ C(GL)3 1 +h +3 +2 +�r0 +L +�3h � ρ +L +�3−d(h) +≤ C(GL)3h− 3 +2 +�r0 +L +�3−d(h)+3h +. +(139) + +17 +Above we used ρ ≤ r0. Summing in h, remembering the frequency, we obtain +ˆ +BU ∩(∪hVh) +S +3 +2 +2 (x, r0)dx ≤ C(GL)3 +ˆ 1 +0 +h− 3 +2 +�r0 +L +�3−d(h)+3h +dµ(h) +(140) +In the multifractal formalism, the structure function exponents are defined by +ζp = inf +h (3 − d(h) + ph). +(141) +The inequality (140) above implies +ˆ +BU ∩(∪hVh) +S2(x, r0) +3 +2 dx ≤ Cµ(GL)3 �r0 +L +�ζ3 +(142) +where +Cµ = C +ˆ 1 +0 +h− 3 +2 dµ(h) +(143) +is assumed to be finite. Introducing the Reynolds number based on G, +RG = GL +ν +(144) +and recalling that BU \ ∪hVh was assumed to have measure zero, we have +ˆ +BU +S2(x, r0) +3 +2 dx ≤ +� +CµR3 +G +�r0 +L +�ζ3� +ν3 +(145) +we see that the condition (101) is satisfied for BU if +R3 +G +�r0 +L +�ζ3 ≤ (C3Cµ)−1. +(146) +In classical turbulence theory ζ3 = 1. If ζ3 > 0, under the above scenario, it is enough to have r0 +L small +enough in order to deduce that no singularities in finite time can occur. +4.4. Time dependent regions of interest. As me noted before, the finite uniform integrability of con- +dition (101) is not needed, all we need is control of S2 on certain small sets of interest. We consider the set +BU,G(t) = {x | |u(x, t)| ≥ U, and |∇u(x, t)| ≥ G} defined in (80). We note that +|BU,G(t)| ≤ C min{U −2∥u0∥2 +L2; G−1∥ω0∥L1}. +(147) +where ω = ∇ × u. The first term in the inequality follows from the Markov-Chebyshev inequality and the +fact that the L2 norms of solutions of Navier-Stokes equations are non-increasing in time. The second term +follows from the fact that the map ω �→ ∇u is weak type 1, that is from G|{x | |∇u| ≥ G} ≤ C∥ω∥L1, and +the fact that the L1 norm of vorticity of solutions of Navier-Stokes equations is non-increasing in time [3]. +THEOREM 14. Let U(t), G(t) and r(t) be positive numbers such that +ˆ T +0 +(r(t)−4 + U(t)4 + G(t))dt < ∞. +(148) +Consider the set +B(t) = {x | |u(x, t)| ≥ U and |∇u(x, t)| ≥ G}. +(149) +There exists an absolute constant C such that, if +ˆ +|y|≤r(t) +�ˆ +B(t) +|δyu(x, t)|3dx +� 2 +3 dy +|y|3 ≤ +� ν +C +�2 +(150) +then the smooth solution of Navier-Stokes equations obeys u ∈ L∞(0, T; L3(R3)) with explicit bounds +depending only on ν, T, ∥u0∥L3, ∥u0∥L2. + +18 +PETER CONSTANTIN +PROOF. We follow the proof of Theorem 11. The β term is estimated as in (104). The contribution of +the term involving π from the region |u| ≤ U is estimated as in (107), noting that the term U 2∥u∥2 +L3 is time +integrable in view of (148). A new term is +ˆ +|u|≥U,|∇u|≤G +|π||u||∇|u||dx ≤ CG∥u∥3 +L3, +(151) +and, in view of (148) it leads via Grownwalll to an explicit bound on ∥u∥L3. We are left with +ˆ +B(t) +|π||u||∇|u||dx ≤ CD +�ˆ +B(t) +S +3 +2 +2 (x, r(t))dx +� 1 +3 +(152) +Now we use +�ˆ +B +S +3 +2 +2 dx +� 2 +3 +≤ 1 +4π +ˆ +|y|≤r +dy +|y|3 +ˆ +B +|δyu(x)|2dx +(153) +proved by duality, testing against arbitrary L3(B) functions. The assumption (150) implies +ˆ +B(t) +|π||u||∇|u||dx ≤ νD +6 +(154) +and concludes the proof. +□ +THEOREM 15. Let q ≥ 4, and assume (150) where the functions U(t), r(t) and G(t) obey +ˆ T +0 +(U 2(t) + r−4(t) + G(t))dt < ∞. +(155) +Then we have the single exponential bound +∥u(·, t)∥Lq ≤ ∥u0∥Lq exp + +Cν−1 +ˆ t +0 +U 2ds + C +ˆ t +0 +G(s)ds + Cν− 3 +2 ∥u0∥2 +L2 +�ˆ t +0 +r−4(s)ds + + +(156) +PROOF. We start as in the proof of Theorem 12 by splitting p = β + π in the estimate the evolution of +the Lq norm of u, and deduce the bound (112) leading to the exponential growth factor (113). We are left o +estimate the contribution of π, that is +I = +ˆ +R3 πu · ∇|u|q−2dx. +(157) +We bound the integral +|I| ≤ I1 + I2 + IB += (q − 2) +�´ +|u|≤U |π||u|q−2|∇u|dx + +´ +|∇u|≤G |π||u|q−2|∇u|dx + +´ +B(t) |π||u|q−2|∇u|dx +� +. +(158) +We bound I1 like in (84), +I1 ≤ CU∥u∥ +q +2 +Lq +√ +D +(159) +where we use the fact that ∥π(·, r)∥L +q +2 ≤ C∥u∥2 +Lq holds with C an absolute constant, independent of r, and +�ˆ +|u|≤U +|u| +q(q−2) +q−4 dx +� q−4 +2q +≤ U∥u∥ +q +2 −2 +Lq . +(160) +The bound (159) is valid for q = 4 as well, we just take |u| ≤ U outside the integral and use L2 − L2 +bounds. The term I2 is bound directly +|I2| ≤ CG∥u∥q +Lq +(161) +The last term is smaller than the dissipation, using the arguments similar to the ones leading to (91). We +omit further details. +□ + +19 +REMARK 16. The condition U ∈ L2(0, T) appearing in (155) is better than the condition U ∈ L4(0, T) +of (148) of Theorem 14. That is just because in that theorem the desire was to bound the L3 norm in terms +solely of itself. Theorem 15 is strictly stronger that Theorem 14 (it implies it for strong solutions), by +bounding first the L4 norm of the solution, and then returning to the proof of the bound of the L3 norm. +Acknowledgment. Research partially supported by NSF grant DMS-2106528. +References +[1] L. Berselli, G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Pro- +ceedings of the AMS 130 (12) (2002), 3585-3595. +[2] A. Cheskidov, R. Shvydkoy, Volumetric theory of intermittency in fully developed turbulence arXiv:2203.11060, (2022). +[3] P. Constantin, Navier-Stokes equations and area of interfaces, Commun. Math. Phys. 129 (1990), 241 - 266. +[4] P. Constantin, Local formulas for the hydrodynamic pressure and applications, Russian Mathematical Surveys, 69 (2014), +395-418. +[5] P. Constantin, C. Foias. Navier-Stokes equations, University of Chicago Press, Chicago (1988). +[6] L. Escauriaza, S. Montaner, Some remarks on the Lp regularity of second derivatives of solutions to non-divergence ellip- +tic equations and the Dini condition. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 1, pp. 49–63. DOI +10.4171/RLM/751 +[7] C. Foias, personal communication, unpublished. +[8] U. Frisch, Turbulence, the legacy of A.N. Kolmogorov, Cambridge University Press Cambridge (1995). +[9] K.P. Iyer, K.R. Sreenivasan, P.K. Yeung, Scaling exponents saturate in three-dimensional isotropic turbulence, Phys. Rev. +Fluids 5, (5), (2020), 054605. +[10] Z. Grujic, L. Xu, Asymptotic criticality of the Navier-Stokes regularity problem, arXiv:1911.00974v4, (2023). +[11] G. Seregin, Lecture notes on regularity theory for Navier-Stokes equations, World Scientific Publishing Singapore, (2015). +[12] G. Seregin and V. Sverak, Regularity criteria for Navier–Stokes solutions, in Y. Giga and A. Novotny, editors, Handbook of +Mathematical Analysis in Mechanics of Viscous Fluids, pp. 829–867. Springer International Publishing, Berlin (2018). +[13] E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, (1970) +DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ 08544 +Email address: const@math.princeton.edu + diff --git a/MNE3T4oBgHgl3EQfYgqz/content/tmp_files/load_file.txt b/MNE3T4oBgHgl3EQfYgqz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fa2b88e0989990e825593b73513847a48b5fef0 --- /dev/null +++ b/MNE3T4oBgHgl3EQfYgqz/content/tmp_files/load_file.txt @@ -0,0 +1,489 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf,len=488 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='04489v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='AP] 11 Jan 2023 Pressure, Intermittency, Singularity Peter Constantin ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We give conditions for regularity of solutions of three dimensional incompressible Navier-Stokes equations based on the pressure and on structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' On the occasion of the centennial anniversary of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Ladyzhenskaya 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Introduction We consider solutions of incompressible Navier-Stokes equations in R3, with smooth and localized initial data, and discuss conditions in terms of pressure and structure functions that are easily accessible and guarantee that solutions which are smooth on a time interval [0, T) have smooth (and hence unique) extensions beyond T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The literature on regularity issues for Navier-Stokes equations is so extensive that we are not able to give here even the beginning of a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We mention just some minimal references in this short paper, with apologies to the many authors and works we knowingly or unknowingly leave out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We discuss unforced Navier-Stokes equations ∂tu + u · ∇u − ν∆u + ∇p = 0, (1) with ∇ · u = 0, (2) and u(x, 0) = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (3) The kinematic viscosity ν is a strictly positive constant, u is the velocity, p is the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Most of this paper is concerned with solutions in the whole space, but there will be a few instances in which we refer to the bounded domain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In that case the assumed boundary conditions are homogeneous Dirichlet, u| ∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (4) We maintain a sparing notation throughout the paper, omitting arguments and indices as often as we can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We recall the local existence result for initial data (at time T0) in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The spaces H (mentioned below) and V are spaces of divergence-free vector fields which are completions of smooth compactly supported divergence free fields in the topologies of L2 and H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The norm in V is called the enstrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In the whole space it corresponds to the ˙H1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Initial data with finite enstrophy lead to local strong solutions, that is unique solutions belonging to L∞(T0, T0+τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' V )∩L2(T0, T0+τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' H2∩V ) for some τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Strong solutions are C∞ smooth for t > T0 in smooth domains [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' By ”conditions for regularity” for smooth solutions on a time interval [0, T) we mean conditions which guarantee u(T) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' These are global regularity condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We note here that we are not talking about ǫ- regularity concepts ([11]) which are conditions on weak solutions in space-time cylinders, which imply pointwise local regularity inside a smaller cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' When assembled over space time, these conditions lead to partial regularity, and may lead to global regularity if additional assumptions are in place, (for instance a single potential first singularity at one point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In this paper we consider conditions which lead directly to persistence of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Navier-Stokes, pressure, intermittency, singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' MSC Classification: 35Q35, 35Q86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 1 2 PETER CONSTANTIN There are several well-known conditions for regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' One of the simplest is ˆ T 0 ∥u(t)∥4 V dt ≤ MV < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (5) From it, we have in a straightforward manner [5] that ∥u(t)∥2 V ≤ ∥u(0)∥2 V exp � Cν−3MV � (6) for all 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We adhere to the good practice that arguments of exponentials or logarithms should be nondimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Another easy to prove explicit condition is based on ∥∇u∥L3 (see below, Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The celebrated Ladyzhenskaya-Prodi-Serrin conditions [11] are ˆ T 0 ∥u(t)∥p Lqdt ≤ Mp,q < ∞, (7) with 2 p + 3 q = 1, (8) and 3 < q ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' When q = 3 the condition is ∥u(t)∥L3 ≤ M3 < ∞, t − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (9) As it is very well-known, the Ladyzhenskaya-Prodi-Serrin conditions imply regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The following is the explicit bound on the enstrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let Ω be a bounded open domain in R3 with smooth boundary, let q > 3 and let u be a strong solution of the Navier-Stokes equations in Ω on the interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists an absolute constant C such that ∥u(t)∥2 V ≤ ∥u(0)∥2 V exp � Cν− q+3 q−3 ˆ t 0 ∥u(s)∥ 2q q−3 Lq ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (10) holds for 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In particular, if (7) holds then ∥u(t)∥2 V ≤ ∥u(0)∥2 V exp � Cν− q+3 q−3 Mp,q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (11) PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Here is a brief proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We recall that the Stokes operator is defined as Au = −P∆u, (12) where P is the Leray projector on divergence-free vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We recall ∥u∥H2(Ω) ≤ C|Au|H, (13) the fact that ∥u∥V = ∥u∥H1(Ω), (14) and the notation B(u, v) = P(u · ∇v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (15) We start with the enstrophy evolution 1 2 d dt∥u∥2 V + ν|Au|2 H = −(B(u, u), Au)H ≤ |B(u, u)|H|Au|H (16) Now, because P is a projector, it follows that |B(u, u)|H ≤ ∥u · ∇u∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (17) A H¨older inequality with exponents q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 2q q−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 2 yields |B(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' u)|H ≤ ∥u∥Lq∥∇u∥ L 2q q−2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (18) 3 and because 2 < 2q q−2 < 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' interpolation yields ∥u∥Lq∥∇u∥ L 2q q−2 ≤ ∥u∥Lq∥∇u∥ 1− 3 q L2 ∥∇u∥ 3 q L6 (19) Using the embedding H2(Ω) ⊂ W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='6(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (13) and (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' we have |B(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' u)|H ≤ C∥u∥Lq∥u∥ 1− 3 q V ∥Au∥ 3 q H (20) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' from(16) we have 1 2 d dt∥u∥2 V + ν|Au|2 H ≤ ∥u∥Lq∥u∥ 1− 3 q V |Au| 1+ 3 q H (21) and Young’s inequality with exponents (1 2(1 + 3 q))−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (1 2(1 − 3 q)))−1 yields 1 2 d dt∥u∥2 V + ν|Au|2 H ≤ ν 2|Au|2 H + Cν− q+3 q−3∥u∥ 2q q−3 Lq ∥u∥2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (22) The claimed inequality (10) follows by integrating the ODE inequality (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The same result holds in R3 or T3 with the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' If q > 3, the bound on the enstrophy is precise and quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In the case q = 3, in order to have a good quantitative control it is useful to have a form of finite uniform integrability of |u(x, t)|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' This condition is ∃δ > 0, ∀t, ∀A, |A| ≤ δ ⇒ ˆ A |u(x, t)|3dx ≤ � ν 2C �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (23) In the left hand side, |A| is the Lebesgue measure of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In the right hand side, ν is the kinematic viscosity and C is the constant in Morrey’s inequality, ∥u∥L6(R3) ≤ C∥u∥ ˙H1(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (24) REMARK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition (23) is uniform in time, but it is much weaker than uniform integrability, because ν 2C is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let u be a strong solution of the NSE in R3 on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Assume (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Then ∥u(t)∥2 ˙H1 ≤ min \uf8f1 \uf8f2 \uf8f3 ∥u0∥2 ˙H1 exp {t � ∥u0∥2 L2 δν � }, ∥u0∥2 ˙ H1 + 2 δν2 ∥u0∥4 L2, (25) where δ is the constant in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' REMARK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The time exponential bound is better than the time independent bound for times shorter than δν ∥u0∥2 L2 log � 1 + 2∥u0∥4 L2 δν2∥u∥2 ˙H1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' After that time, the time independent bound is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In either case, the bound (25) implies that the enstrophy is bounded on [0, T], which in turn implies that the solution has a unique strong extension beyond T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proof (based on ([7]) follows from the enstrophy equation (16) using the fact that |{x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' |u(x, t)| ≥ U}| ≤ U −2∥u0∥2 L2 (26) with the choice of U = δ− 1 2 ∥u0∥L2, (27) and estimating the nonlinear term separately in the region where |u(x, t)| ≥ U and where |u(x, t)| ≤ U by ∥u · ∇u∥L2 ≤ �ˆ |u||≥U |u|3dx � 1 3 ∥∇u∥L6 + U∥∇u∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (28) 4 PETER CONSTANTIN Then, using the assumption (23) we obtain ∥u · ∇u∥L2 ≤ ν 2∥∆u∥L2 + U∥∇u∥L2, (29) we absorb the first term in half the dissipation and use a Young inequality in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We end up with ODE inequality ˙y ≤ U 2 ν y (30) for the quantity y = ∥u∥2 ˙H1 with U given by (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The exponential bound in inequality (25) follows from Gronwall, and the time independent bound follows by using ν ´ t 0 y(s)ds ≤ 2∥u0∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The same result holds in bounded domains Ω or the periodic case T3, with the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We use the enstrophy equation (16), and the inequality (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Then we estimate like in (28) and note that ∥∇u∥L6 ≤ C|Au|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' As the reader may have already noticed, we are interested in explicit conditions, involving constants known a priori, without the need to sample solutions, and which yield explicit enstrophy bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' These conditions are useful if additionally it is true that if they are satisfied uniformly on solutions of approxima- tions that converge only almost everywhere, then the solutions are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We refer to such conditions as ”easily accessible”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The conditions (7), (23) are easily accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let un(x, t) be an approximation of the NSE solution u(x, t) on the interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' As- sume that un(x, t) → u(x, t) (x, t) − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' on Ω × [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (31) (i) If there exists Mp,q < ∞, such that (7) holds for un uniformly for all n, then u obeys (7) with the same constant Mp,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (ii) If there exists M3 such that (9) holds for un uniformly for all n, then u obeys (9) withe the same constant M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (iii) If there exists a constant δ > 0 such that (23) holds for un uniformly for all n, then u obeys (23) withe the same constant δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' REMARK 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In the case (i), the solution u obeys the quantitative bound (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In the case (iii), the solution u obeys the quantitative bound (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In these cases the H1 bound on u(T) is explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proof of (i), (ii), (iii) follows from applications of Fatou’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ This paper is devoted to conditions based on pressure and on structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There is a good motiva- tion to seek conditions in terms of the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In the absence of the pressure, the Navier-Stokes equations are Burgers equations in 3D and obey a maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' This implies that the velocity is bounded, (if initially so), and the solutions are smooth for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The pressure in the Navier-Stokes equations is the only reason the equations are not local and the velocity magnitude is not a priori controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Conditions of regularity in terms of the pressure are known in L2 [1] and one sided in L∞ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We present in this paper an L 3 2 condition, the analogue of the q = 3 condition (23) expressed in terms of only the pressure (Theo- rem 8, Theorem 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We also give the analogues of the Ladyzhenskaya-Prodi-Serrin conditions (Theorem 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The motivation to express conditions for regularity in terms of structure functions comes from experi- mental, numerical and theoretical turbulence studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Structure functions are averages of moments of veloc- ity increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' They obey remarkable and robust statistical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The relations need interpretation and then the may serve as reasonable hypothesis for the solutions of Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' One of the more widely verified relations is the ”four-fifths” law [8] ⟨(δ∥ ℓ u)3⟩ = −4 5ǫ|ℓ| (32) 5 where δ∥ ℓ (u) = (u(x + ℓ) − u(x)) · ℓ |ℓ| is the longitudinal velocity increment and ǫ = −⟨dE dt ⟩ is the rate of dissipation of energy, which in the case of unforced NSE equals ν⟨|∇u|2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The four-fifths law is shown to hold for homogeneous and isotropic turbulence in the limit of time to infinity, followed by Reynolds number to infinity, followed by ℓ → 0, in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The Navier-Stokes solutions are assumed to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The braces ⟨·⟩ are expectations (ensemble average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Long time and space averages are usually assumed to realize them, and in numerical experiments these averages are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The assumption of finite positive ǫ is also made, in the limit of time to infinity, followed by Reynolds number to infinity, in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The Reynolds number is defined as Re = UL ν (33) where U is a velocity scale, and L is a length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The classical K’41 Kolmogorov theory proposes scaling exponents ζp = p 3 for structure functions ⟨(δ∥ ℓ u)p⟩ = Cp(ǫ|ℓ|)ζp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (34) These relations are expected to hold in a range of scales, |ℓ| ∈ (η, L) where η is the Kolmogorov dissipation scale, η = �ν3 ǫ � 1 4 (35) which is determined by the kinematic viscosity and energy dissipation rate, alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Below the Kolmogorov dissipation scale, it is assumed that viscous effects dominate, with smooth behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The length scale L is the integral scale of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Turbulence findings are average statements, they refer to typical long time behavior, and are asymptotic in Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Interpreting them for the initial value problem for Navier-Stokes equations is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' It is however reasonable to expect that there are many solutions which give statistical weight to the turbulence laws and have properties that are consistent with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We give quantitative conditions in Theorem 11, Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' They involve a cutoff scale r = r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We modify the structure function S2(x, r) (see (54) below) to take into account a possibly non-universal viscous regularization below r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The regularity condition (101) requires ´ T 0 r(t)−4dt < ∞, a condition satisfied by the Kolmogorov length r = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition requires in addition the smallness of ´ A S 3 2 2 (x, r)dx ≤ � ν C �3 on sets of small enough measure, |A| ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Modern theories modify the scaling ζp = p 3 in (34) of the K’41 theory, reflecting experimental and nu- merical observation of intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The turbulent signal is intermittent, that is, regions of high gradients of velocity are found to be sparse in both time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The connection between intermittency and regularity was explored in several mathematical works, (see for instance [10] and references therein) where assump- tions of sparse behavior in physical space are used to deduce improved conditional regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In a different setting [2], multifractal scaling exponents were connected to ratios of volume averages, and conditions for regularity were given on the basis of intermittency dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In terms of the exponents, it is found numerically (see for instance [9]) that ζ2 > 2 3 and, while ζ3 remains close to 1, ζp become smaller than p/3 for large p, and perhaps even tends asymptotically to a constant, suggesting depletion of regularity, significantly below H¨older.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We express conditions of regularity in terms of a Dini modulus of continuity which give regularity if logarithmic scaling is assumed (Theorem 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='3 we consider a multifractal scenario where regularity still persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='4 we give a condition for regularity which requires small increments of velocity only in time dependent regions of high velocity and high gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proofs are based on observations concerning the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The pressure We consider solutions of − ∆p = ∇ · (u · ∇u) (36) 6 PETER CONSTANTIN in Ω ⊂ R3, where u is divergence-free and sufficiently regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We recall representation results from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We use the notation f(x, r) = 1 4πr2 ˆ |x−y|=r f(y)dS(y) = |ξ|=1 f(x + rξ)dS(ξ) (37) where dS is surface area and ffl denotes the integral normalized by the area of the region of integration, which in the above case is 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We denote σij( � y − x) = 3(yi − xi)(yj − xj) |y − x|2 − δij (38) where � y − x = y − x |y − x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (39) The following lemma was proved in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' LEMMA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let x ∈ Ω ⊂ R3, let 0 < r < dist(x, ∂Ω), and let p solve (36) with divergence-free u ∈ C2(Ω)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let v ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Then p(x) = p(x, r) − 1 3|u(x) − v|2 + ffl |ξ|=1 |ξ · (u(x + rξ) − v)|2 dS(ξ) +P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' ´ r 0 dρ ρ ffl |ξ|=1 σij(ξ)(ui(x + ρξ) − vi)(uj(x + ρξ) − vj)dS(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (40) All terms in the right hand side of (40) are determined solely by information in the ball of radius r about x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We denote the singular integral K(x, r) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 1 4π ˆ |y−x|≤r σij � � y − x � |y − x|3 (ui(y) − vi)(uj(y) − vj)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (41) Thus, (40) reads p(x) = p(x, r) − 1 3|u(x) − v)|2 + |ξ|=1 |ξ · (u(x + rξ) − v)|2 dS(ξ) + K(x, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (42) Evidently, K depends on the choice of the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In applications we want to be able to choose v appro- priately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' For divergence free functions u, it holds that |ξ|=1 ξi (ξ · u(x + rξ)) dS(ξ) + 1 4π PV ˆ B(x,r) σij( � x − y) |x − y|3 uj(y)dy = 1 3ui(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (43) This follows from 1 4π ˆ B(x,r) yi − xi |y − x|3 (∇ · u)(y)dy = 0 by integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We also note that |ξ|=1 (ξ · v)2dS(ξ) = 1 3|v|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (44) This is true for any v that does not depends on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Therefore, the representation (40) is valid even if v is a function of x and r (but not ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Indeed, this follows by opening brackets in the right hand side of (40), using (43) and (44), and identifying what remains as (40) for v = 0, which was proved independently in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We average (42) 1 R ´ 2R R dr and obtain the representation [4] THEOREM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let p solve (36) with divergence-free u ∈ C2(Ω)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let x ∈ Ω ⊂ R3, v ∈ R3 and let 0 < r < 1 2dist(x, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Then, p(x) = β(x, r) + π(x, r) − 1 3r ˆ 2r r |u(x) − v|2dρ (45) 7 with β(x, r) = 1 r ˆ 2r r p(x, ρ)dρ (46) and π(x, r) = 1 r ˆ 2r r � K(x, ρ) + |ξ|=1 |ξ · (u(x + ρξ) − v)|2dS(ξ) � dρ (47) The explicit expression for π(x, r) is π(x, r) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 1 4π ´ |x−y|≤2r w � |y−x| r � σij(� y−x) |y−x|3 (ui(y) − vi)(uj(y) − vj)dy + 1 4πr ´ r≤|y−x|≤2r 1 |y−x|2 � y−x |y−x| · (u(y) − v) �2 dy, (48) where the weight w is given by w(λ) = \uf8f1 \uf8f2 \uf8f3 1, if 0 ≤ λ ≤ 1, 2 − λ if 1 ≤ λ ≤ 2, 0 if λ ≥ 2 (49) We recall bounds on β and π (Propositions 2 and 3, [4]) PROPOSITION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists an absolute constant C such that, for any r > 0, ∥∇β(·, r)∥L2(R3) ≤ Cr−1∥u∥2 L4(R3) (50) and ∥β(·, r)∥L∞ ≤ Cr−2∥∇u∥L2∥u∥L2 (51) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Moreover, for any 1 < q < ∞, there exists Cq independent of r, such that ∥β(·, r)∥Lq(R3) ≤ Cq∥u∥2 L2q(R3) (52) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We choose now v = u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Then from (52) and the corresponding bound for p it follows also that ∥π(·, r)∥Lq(R3) ≤ Cq∥u∥2 L2q(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (53) We denote by S2(x, r) = 1 4π ˆ |y|≤2r 1 |y|3 |u(x + y) − u(x)|2dy (54) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We note that |π(x, r)| ≤ 2S2(x, r) (55) follows from (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' For any measurable set A ⊂ Ω with dist(A, ∂Ω) > 2r we have ˆ A S2(x, r)q ≤ Cqr2q ˆ A+rB(0,1) |∇u(x)|2qdx (56) for any q ≥ 1 where B(0, 1) is the unit ball in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' This follows in straightforward manner by writing the integral in (54) in polar coordinates with y = ρξ, ρ = |y|, ξ = �y, expressing u(x + y) − u(x) = ´ 1 0 d dλu(x + λρξ)dλ, and using Schwarz and H¨older inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' REMARK 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The bounds (52) and (53) for β and π are valid in bounded domains with smooth boundary if we add bounds for ∥p∥Lq(∂Ω), see Lemma 2 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Once these are obtained the rest of the bounds which are local bounds are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 8 PETER CONSTANTIN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Conditional regularity THEOREM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Assume ˆ T 0 ∥∇u(t)∥2 L3dt = N < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (57) There exists an absolute constant C such that sup 0≤t≤T ∥∇u(t)∥2 L2 ≤ ∥∇u0∥2 L2 exp CN ν (58) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 1 2 d dt ˆ R3 |∇u|2dx + ν ˆ |∆u|2dx = ˆ R3(u · ∇u) · ∆udx ≤ ∥u∥L6∥∇u∥L3∥∆u∥L2 followed by Schwartz and Morrey inerqualities results in 1 2 d dt∥∇u(t)∥2 L2 ≤ C ν ∥∇u(t)∥2 L3∥∇u(t)∥2 L2, and the proof is finished by Gronwall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ Let Λ = (−∆) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (59) THEOREM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Assume ˆ T 0 ∥Λ 3 2u(t)∥2 L2dt = M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (60) There exists an absolute constant C such that sup 0≤t≤T ∥∇u(t)∥2 L2 ≤ ∥∇u0∥2 L2 exp CM ν (61) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' This is a consequence of (58) and of the inequality N ≤ CM which follows from the familiar [13] Riesz potential inequality ∥f∥L3 ≤ C∥Λ 1 2 f∥L2 (62) with f = ∂iuj, for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The finiteness (58) is a stronger regularity condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=', more general) than the finiteness (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Both conditions are analogues of the Ladyzhenskaya-Prodi-Serrin condition (7) for q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The finiteness (58) and (7) for q = ∞ are logically independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists an absolute constant C such that sup 0≤t≤T ∥Λ 1 2 u(t)∥L2 ≤ ∥Λ 1 2 u0∥L2 + C ˆ T 0 ∥Λ 3 2u(t)∥2 L2dt (63) holds for smooth solutions of Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We take the scalar product of the Navier-Stokes equation with Λu(t) and integrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We obtain 1 2 d dt∥Λ 1 2u∥2 L2 + ν∥Λ 3 2 u∥2 L2 ≤ ���� ˆ (u · ∇u) · Λudx ���� ≤ C∥u∥L3∥∇u∥L3∥Λu∥L3 ≤ C∥Λ 1 2u∥L2∥Λ 3 2 u∥2 L3, (64) where we used (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We divide by ∥Λ 1 2u∥L2 and integrate in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In view of (62), the inequality (63) implies the boundedness of the L3 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Hence, via the Navier-Stokes equation, the finiteness (60) implies (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The main virtue of (63) is that it is viscosity- independent, and is valid for Euler equations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 9 The analogue of the L3-based condition (9) in terms of only the pressure is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists an absolute constant C, such that, if p = RiRj(uiuj) satisfies the finite uniform integrability condition ∃δ > 0, ∀t, ∀A |A| ≤ δ ⇒ ˆ A |p(x, t)| 3 2 dx ≤ � ν C �3 (65) on [0, T], then u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L3(R3)) with explicit bounds depending only on δ, ν, T, ∥u0∥L3, ∥u0∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We take the evolution of the L3 norm of u: 1 3 d dt ˆ R3 |u|3dx + ν ˆ R3 |u|[|∇u|2 + |∇|u||2]dx = − ˆ R3 |u|(u · ∇p)dx = ˆ R3 p(u · ∇|u|)dx (66) Let U be a large positive number and φ a positive smooth function of one variable which is compactly supported in [0, 2], satisfies 0 ≤ φ ≤ 1 and identically equals 1 on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We split the RHS of (66) , ˆ R3 p(u · ∇|u|)dx = ˆ R3 φ �|u| U � p(u · ∇|u|)dx + ˆ R3 � 1 − φ �|u| U �� p(u · ∇|u|)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We estimate the first term, using that on the support of φ we have |u| ≤ 2U, ���� ˆ R3 φ �|u| U � p(u · ∇|u|)dx ���� ≤ (2U) 1 2 √ D �ˆ R3 p2dx (67) where D = ˆ R3 |u||∇u|2dx (68) and then using the boundedness of Riesz transforms in Lp spaces and then interpolating L4 between L3 and L9, we have ˆ R3 p2 ≤ C ˆ R3 |u|4dx ≤ C∥u∥ 5 2 L3∥u∥ 3 2 L9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (69) Now we use the fact that there exists a constant C such that D ≥ C∥u∥3 L9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (70) This fact follows from Morrey’s inequality ∥∇f∥L2 ≥ C∥f∥L6 applied for with f = |u| 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Thus, from (67) and (70) we have ���� ˆ R3 φ �|u| U � p(u · ∇|u|)dx ���� ≤ CU 1 2D 3 4 ∥u∥ 5 4 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (71) With Young’s inequality we have ���� ˆ R3 φ �|u| U � p(u · ∇|u|)dx ���� ≤ ν 2D + CU 2ν−3∥u∥5 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (72) We know from the energy inequality, interpolation L2 − L6 and Morrey’s inequality that ˆ T 0 ∥u∥4 L3dt ≤ C∥u0∥2 L2 ˆ T 0 ∥∇u∥2 L2dt ≤ Cν−1∥u0∥4 L2, (73) and thus the factor CU 2ν−3∥u∥2 L3 multiplying the ∥u∥3 L3 is time integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The second term is estimated as ���� ˆ R3 � 1 − φ �|u| U �� p(u · ∇|u|)dx ���� ≤ C √ D∥u∥ 1 2 L9∥((1 − φ)p∥L 9 4 (74) by using H¨older with exponents 9 4, 18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' One more interpolation, ∥(1 − φ)p∥L 9 4 ≤ ∥(1 − φ)p∥ 1 2 L 9 2 ∥(1 − φ)p∥ 1 2 L 3 2 10 PETER CONSTANTIN and the inequality based on the fact that Riesz transforms are bounded in Lp ∥(1 − φ)p∥L 9 2 ≤ C∥u∥2 L9, together with (70) yield from (74) ˆ R3 � 1 − φ �|u| U �� p(u · ∇|u|)dx ≤ CD∥((1 − φ)p∥ 1 2 L 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (75) Now the support of 1 − φ � |u(x,t)| U � is included in the set BU(t) = {x | |u(x, t)| ≥ U} (76) which has uniformly small Lebesgue measure |BU(t)| ≤ U −2∥u0∥2 L2 (77) and, by assumption, the function x �→ |p(x, t)| 3 2 satisfies (65), so that ∥((1 − φ)p∥ 1 2 L 3 2 ≤ ν 2C (78) holds uniformly for t ∈ [0, T], if U is chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition (65) is weaker than uniform integrability, because ν C is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition holds if |p(x, t)| 3 2 is uniformly integrable in x on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In particular, it holds if p ∈ C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L 3 2 (R3)), (79) or if p is piece-wise continuous on [0, T] with values in L 3 2, because in these cases the curve t �→ p belongs to a compact subset of L 3 2 and therefore |p(x, t)| 3 2 is uniformly integrable in x on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition also holds if |p(x, t)| ≤ f(x), x-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=', with f ∈ L 3 2 (R3) time independent, because this again implies uniform integrabilty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' REMARK 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' As it is seen in the proof above, the condition (65) is not applied on just any set, but rather on a set of interest, namely the set where the absolute magnitude of velocity exceeds a fixed large threshhold, (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' It is easy to see that this set can be replaced by a smaller, and even more interesting set BU,G(t) = {x | |u(x, t)| ≥ U, and |∇u(x, t)| ≥ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (80) The proof of this fact is similar to the proof above, using two cutoffs, and showing that the regions |u| ≤ U and, separately |∇u| ≤ G lead each to a priori bounds on the size of the L3 norm of u, leaving only the contributions from BU,G(t) to require control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The following result shows that the condition (65) leads to easily accessible bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let r ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists a constant C = Cr such that if (65) holds on [0, T] then u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Lr(R3)) (81) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' More precisely, we have the single exponential bound ∥u(·, t)∥Lr ≤ ∥u0∥Lr exp �Ct∥u0∥2 L2 νδ � (82) where δ is the constant in (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The evolution of the Lr norm is given by 1 r d dt ˆ R3 |u|rdx + ν ˆ R3[|∇u|2|u|r−2 + (r − 2)|∇|u||2|u|r−2]dx = ˆ R3 pu · ∇|u|r−2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (83) 11 We let U be a large positive number and φ a positive smooth function of one variable which is compactly supported in [0, 2], satisfies 0 ≤ φ ≤ 1 and identically equals 1 on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We split the RHS of (66) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We split the right hand side of (83), ˆ R3 p(u · ∇|u|r−2)dx = ˆ R3 φ �|u| U � p(u · ∇|u|r−2)dx + ˆ R3 � 1 − φ �|u| U �� p(u · ∇|u|r−2)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We estimate the first term using H¨older inequalities with exponents r 2 for p, 2 for a term involving the gradient, |u| r−2 2 |∇u|, and 2r r−4 for the term |u| r−2 2 , taking advantage of the fact that on the support of φ we have |u| ≤ 2U, and using the boundedness of Riesz transforms in Lp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We deduce that the first term is bounded by ���� ˆ R3 φ �|u| U � p(u · ∇|u|r−2)dx ���� ≤ CU∥u∥ r 2 LrD 1 2, (84) where D = ˆ R3 |u|r−2|∇u|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (85) We used, in view of r(r−2) r−4 = r + 2r r−4, that �ˆ |u|≤U |u| r(r−2) r−4 dx � r−4 2r ≤ U∥u∥ r 2 −2 Lr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (86) The bound (84) is valid for r = 4 as well, we just take |u| ≤ U outside the integral and use L2 −L2 bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Hiding √ D in 1 2νD, we see that the inequality (84) leads to an exponential growth ∥u(·, t)∥Lr ≤ ∥u0∥LreCν−1 ´ t 0 U2ds (87) if the second term does not contribute to growth, The second term is bounded using D ≥ C∥u∥r L3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (88) We first bound ��� ´ R3 � 1 − φ � |u| U �� p(u · ∇|u|r−2)dx ��� ≤ ∥ � 1 − φ � |u| U �� p∥ L 3r r+1 ∥u∥ r−2 2 L3r D 1 2 ≤ C∥ � 1 − φ � |u| U �� p∥ L 3r r+1 D1− 1 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (89) and then use ∥ � 1 − φ � |u| U �� p∥ L 3r r+1 ≤ ∥ � 1 − φ � |u| U �� p∥ 1 2 L 3 2 ∥ � 1 − φ � |u| U �� p∥ 1 2 L 3r 2 ≤ C∥ � 1 − φ � |u| U �� p∥ 1 2 L 3 2 D 1 r (90) to deduce ���� ˆ R3 � 1 − φ �|u| U �� p(u · ∇|u|r−2)dx ���� ≤ C∥ � 1 − φ �|u| U �� p∥ 1 2 L 3 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (91) The proof is completed by the assumption of finite uniform integrability (65) which shows this term to be absorbed in the remaining dissipative term 1 2νD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The result above can be used together with Theorem (1) to give a quantitative bound depending on δ on the supremum in time of the enstrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The following result is the analogue of the Ladyzhenskaya-Prodi-Serrin condition in terms of the pres- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 12 PETER CONSTANTIN THEOREM 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let p = RiRj(uiuj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Assume that there exists q > 3 2 such that ˆ T 0 ∥p(t)∥ 2q 2q−3 Lq(R3)dt < ∞ (92) Then u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L3(R3)) obey u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L3(R3)) with explicit bounds depending in addition to (92) only on ν, T, ∥u0∥L3, ∥u0∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proof follows from the evolution of the L3 norm (66) by estimating as in (74) using H¨older with exponents 9/4, 18, 2, ���� ˆ R3 p(u · ∇|u|)dx ���� ≤ C √ D∥u∥ 1 2 L9∥p∥L 9 4 ≤ CD 2 3∥p∥L 9 4 (93) where we used also (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We distinguish three ranges of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' When q ≤ 9 4 we interpolate ∥p∥L 9 4 ≤ ∥p∥ 2q 9−2q Lq ∥p∥ 9−4q 9−2q L 9 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (94) We use ∥p∥L 9 2 ≤ CD 2 3 which folllows from (70) and the boundedness of Riesz transforms in Lp spaces, to deduce ���� ˆ R3 p(u · ∇|u|)dx ���� ≤ CD1−α∥p∥ 2q 9−2q Lq (95) with α = 2q−3 (9−2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' From Young’s inequality, the boundedness of the L3 norm of u follows if we know that ´ T 0 ∥p∥ 2q 2q−3 Lq dt is finite, which was assumed in (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' When q ∈ [9 4, 9 2], we use Young’s inequality in (93) and deduce that we need to estimate ´ T 0 ∥p∥3 L 9 4 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We interpolate ∥p∥L 9 4 ≤ ∥q∥ 2q 3(2q−3) Lq ∥p∥ 4q−9 3(2q−3) L 3 2 ≤ C∥q∥ 2q 3(2q−3) Lq ∥u∥ 8q−18 3(2q−3) L3 (96) and thus ∥q∥3 L 9 4 ≤ ∥p∥ 2q 2q−3 Lq ∥u∥α L3 (97) with α = 8q−18 2q−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' If q ≤ 9 2 we have α ≤ 3, and the condition (92) ensures that ∥u∥L3 remains bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' When q ≥ 9 2 we estimate ���� ˆ R3 p(u · ∇|u|)dx ���� ≤ C √ D∥u∥ 1 2 L3∥p∥L3 (98) using H¨older with exponents 2, 3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Using Young’s inequality we need to consider the effect of the quantity ∥u∥L3∥p∥2 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We interpolate ∥p∥L3 ≤ ∥p∥ q 2q−3 Lq ∥p∥ q−3 2q−3 L 3 2 ≤ ∥p∥ q 2q−3 Lq ∥u∥ 2(q−3) 2q−3 L3 , (99) and it follows that ∥u∥L3∥p∥2 L3 ≤ C∥p∥ 2q 2q−3 Lq ∥u∥ 6q−15 2q−3 L3 (100) Because 6q−15 2q−3 < 3, the condition (92) implies a uniform bound on ∥u∥L3 on [0, T] in this last case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Structure function We assume ∃r(t), ´ T 0 r(t)−4dt < ∞, ∃δ > 0, ∀A, ∀t ∈ [0, T] |A| ≤ δ ⇒ ´ A S2(x, 2r(t)) 3 2 dx ≤ � ν C �3 (101) holds where S2(x, r) is given in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists an absolute constant C, such that, for any T > 0, if a strong solution u of NSE satisfies (101) for all 0 ≤ t < T, then u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L3(R3)) with explicit bounds depending only on ν, T, ∥u0∥L3, ∥u0∥L2 and the assumed δ > 0, ´ T 0 r(t)−4dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We use (66) and decompose the pressure p = π + β as in (45), at each time t, with the choice r = r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We bound the term ���� ˆ R3 |u|(u · ∇β)dx ���� ≤ Cr−1∥u∥4 L4 (102) using (50), and then by interpolation (used also in (69)) we obtain ���� ˆ R3 |u|(u · ∇β)dx ���� ≤ Cr−1∥u∥ 5 2 L3∥u∥ 3 2 L9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (103) The dissipation D (68) obeys (70) and, so ���� ˆ R3 |u|(u · ∇β)dx ���� ≤ ν 6D + Cν−1r(t)−2∥u∥5 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (104) In view of (73) and the assumption ´ T 0 r(t)−4dt < ∞, it follows that the factor r−2∥u∥2 L3 multiplying ∥u∥3 L3 is time integrable a priori, r(t)−2∥u(t)∥2 L3 ∈ L1([0, T]), (105) and thus this term leads to an explicit uniform bound on ∥u∥L3, in terms of the initial data and ´ T 0 r−4dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We integrate by parts in the term ���� ˆ R3 |u|(u · ∇π)dx ���� = ���� ˆ R3 π(u · ∇|u|)dx ���� ≤ ˆ |u|≤U |π||u||∇|u||dx + ˆ |u|≥U |π||u||∇|u||dx (106) where U is a large time independent constant, at our disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We estimate the first term using (52): ˆ |u|≤U |π||u||∇|u||dx ≤ C2U 1 2√ D∥u∥2 L4 ≤ ν 6D + CU 2ν−3∥u∥5 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (107) where we used interpolation (used also in (69)) and (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We argue like in the proof of Theorem (8), invoking (73), which implies that the term U 2∥u∥2 L3 multiplying ∥u∥3 L3 in the right hand side of (107) is time integrable with an explicit a priori bound, and as such it leads via Grownwalll to an explicit bound on ∥u∥L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Finally, taking U large enough so that the set BU(t) of (76) has small measure as in (77), using (55), proceeding in the same manner as for (75), and using the assumption (101), we have ˆ |u|≥U |π||u||∇|u||dx ≤ CD �ˆ BU S2(·, r(t)) 3 2 dx � 1 3 ≤ ν 6D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (108) This term is absorbed in the remaining dissipative term, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ As in the case of the condition regarding the finite uniform integrability of the pressure (65), the structure function finite integrability condition (101) leads to easily accessible bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 14 PETER CONSTANTIN THEOREM 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let q ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' There exists a constant C depending on q such that if (101) holds on [0, T] then u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Lq(R3)) (109) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' More precisely, we have the single exponential bound ∥u(·, t)∥Lq ≤ ∥u0∥Lq exp �Ct∥u0∥2 L2 νδ + ν− 3 2 ∥u0∥2 L2Γ(t) � (110) where δ is the constant in (101) and Γ(t) = �ˆ t 0 r−4(s)ds (111) is bounded on [0, T] by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proof follows closely the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We use the evolution of the Lq norm (83) where we changed r to q because r has now a different meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We split at each time p = β(·, r) + π(·, r) using r = r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We bound the term ˆ R3 βu · ∇|u|q−2dx ≤ ∥β∥Lq∥u∥ q−2 2 Lq √ D ≤ C∥β∥ 1 2 L∞∥u∥ q 2 Lq √ D (112) where we used H¨older with exponents q, 2 and 2q q−2, interpolated ∥β∥Lq ≤ ∥β∥ 1 2 L∞∥β∥ 1 2 L q 2 and used the bound ∥β∥L q 2 ≤ C∥u∥2 Lq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Hiding √ D in 1 3νD, this term leads to a growth factor ν−1 ˆ t 0 ∥β(·, r(s))∥L∞ds ≤ Cν− 3 2∥u0∥2 L2Γ(t) (113) where we used (51) and the Navier-Stokes energy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We treat the terms involving π in exactly the same manner as we treated p in the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We omit further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ REMARK 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Theorem 12 is stronger than Theorem 11 for strong solutions, which have initial data in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' REMARK 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' A particularly significant small scale r = η is given by classical turbulence theory, where the Kolmogorov dissipation wave number kd, inverse of the viscous dissipation scale η is given by kd = η−1 = ν− 3 4 ǫ 1 4 = ν− 1 2 (⟨|∇u(t)|2⟩) 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (114) We note that it is a priori time integrable to power 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' A nearly selfsimilar example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Theorem 11 (or rather, its proof) applies to functions which have small translation increments in L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We consider u(x, t) = V + smooth (115) where the leading term V satisfies ∥V (y + ·) − V (·)∥L3 ≤ W(t)|y|s (116) for some s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In order to have nondimensional quantities, we write W = UL1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (117) The typical example is of the form V (x, t) = U(t)P � x L(t) � (118) where P is time independent and ∥P(· + z) − P(·)∥L3 ≤ C|z|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We note that this condition is satisfied by many functions with slow decay which are not in L3(R3) or even in L2(R3), such as P(z) = (1 + |z|)−β, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Of course, the condition is also satisfied on Bs 3,∞(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 15 We have that (116) reads ∥V (y + ·) − V (·)∥L3 ≤ UL �|y| L �s , (119) and define UL ν = Re(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (120) We take ǫ > 0 such that s > 1 2ǫ, and, writing |y|−3 = |y|−1+ǫ|y|−2−ǫ we use a H¨older inequality with exponents 3, 3 2 to bound S2(x, r) 3 2 ≤ Cǫ− 1 2 r 3ǫ 2 ˆ |y|≤r |y|−3− 3ǫ 2 |V (y + x) − V (x)|3 dy (121) Integrating dx on A and switching the order of integration we deduce ´ A S2(x, r) 3 2dx ≤ Cǫ− 1 2 r 3ǫ 2 ´ |y|≤r |y|−3− 3ǫ 2 ´ A |V (y + x) − V (x)|3 dxdy ≤ Cǫ− 1 2 W 3r3s = Cǫ− 1 2(UL)3 � r L �3s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (122) Assuming a bound on the Reynolds number of the profile, Re(V ) ≤ R (123) and fixing ǫ < 2s, we have ˆ A S2(x, r) 3 2 dx ≤ Cs � r L �3s (Re(V )3ν3 ≤ Cs � r L �3s R3ν3 (124) The condition (101) is satisfied if � r L �s R ≤ C − 1 3 s (2C)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (125) REMARK 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition (125) shows that for self-similar profiles with time dependent collapsing inner scale L, the condition is satisfied choosing r small compared to the collapsing scale L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Regularity follows if L−4(t) is time integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In particular, if the leading term V is given by (118) and U(t)L(t) ≤ Rν, then regularity follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proof of this fact follows verbatim the proof of Theorem (11) including the estimate (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In that estimate now U is time dependent and it is bounded above by RL(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The term U −2∥u∥2 L3 is still time integrable and that is why the result continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' A Dini Condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Assume that u satisfies ∥δyu∥L3 ≤ m(|y|) (126) where δyu(x, t) = u(x + y, t) − u(x, t), and where 0 ≤ m is a time independent function satisfying ˆ 1 0 m2(ρ)dρ ρ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (127) Then u satisfies (101) with r time independent, and consequently, smooth solutions of Navier-Stokes equa- tions obeying (126) with (127) on [0, T) obey u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L3(R3)) with explicit bounds depending only on m, ν, T, ∥u0∥L3, ∥u0∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The proof follows from the fact that ∥S2(·, r)∥L 3 2 ≤ C ˆ |y|≤2r ∥δyu∥2 L3 dy |y|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (128) This inequality is proved by duality, integrating S2 against a test function in L3 ˆ R3 S2φdx = 1 4π ˆ |y|≤2r dy |y|3 ˆ R3 φ(x)|δyu(x)|2dx ≤ 1 4π∥φ∥L3 ˆ |y|≤2r ∥δyu∥2 L3 dy |y|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (129) 16 PETER CONSTANTIN From (128) and the assumed Dini condition, we deduce that the ∥S2(·, r)∥L 3 2 ≤ � ν C �2 if r is chosen small enough so that ˆ 2r 0 m2(ρ)dρ ρ ≤ � ν C �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (130) □ REMARK 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Clearly m(r) ∼ log−α(r−1) with α > 1 2 is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' As we remarked before, the smallness of the L3 increment does not imply that the function needs to be in L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We also remark that m can be allowed to depend on time, if m(r)r−1 is uniformly integrable on [0, 1], or more generally, if denoting Im(t)(r) = ˆ 2r 0 m2(ρ, t)dρ ρ (131) we have that the preimage of � ν C �2 under Im(t), that is r(t) = I−1 m(t) �� ν C �2� , obeys ´ T 0 r(t)−4dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Multifractal intermittent scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We consider the region BU(t) = {x | |u(x, t)| ≥ U} de- fined before in (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We take U time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We introduce a time independent length scale L > 0, and we require U 2 ≥ L−3 ˆ R3 |u|2dx (132) so that |BU| ≤ L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (133) We assume that the velocity increments s2(x, r) = |y|=r |u(x + y) − u(x)|2dS(y) (134) obey bounds s2(x, r) ≤ G2 � r L �2α(x) (135) with G > 0 constant (with units of velocity), L > 0 as above, constant, (with units of length) and with 0 < α(x) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' This upper bound is assumed to hold a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' in x ∈ BU(t) and for all 0 < r < r0, where 0 < r0 < L is a fixed positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Because S2(x, r) = ˆ 2r 0 s2(x, ρ)dρ ρ , (136) we have that S2(x, r0) ≤ CG2 1 α(x) �r0 L �2α(x) (137) holds a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='e in x ∈ BU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' In multifractal turbulent intermittent scenarios, it is assumed that there is a spectrum of near-singularities of H¨older exponent h and that these are achieved on sets Σh of dimension d(h) ≤ 3 which occur randomly with probability dµ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The dimension d(h) is implemented in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We take a region Vh around Σh and partition it in small disjoint cubes of size ρ with ρ ≤ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' This region is a ”collar”of cross-section size ρ around the set Σh ∩ BU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The multifractal assumption is that the number of such cubes of Vh is of the order Nh(ρ) = � ρ L �−d(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Assuming α(x) ≥ h to hold on each such cube, we have from (137), on each cube S2(x, r0) ≤ CG2h−1 �r0 L �2h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (138) Writing the volume of the cube as L3( ρ L)3, we have ˆ BU∩Vh S2(x, r0) 3 2dx ≤ C(GL)3 1 h 3 2 �r0 L �3h � ρ L �3−d(h) ≤ C(GL)3h− 3 2 �r0 L �3−d(h)+3h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (139) 17 Above we used ρ ≤ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Summing in h, remembering the frequency, we obtain ˆ BU ∩(∪hVh) S 3 2 2 (x, r0)dx ≤ C(GL)3 ˆ 1 0 h− 3 2 �r0 L �3−d(h)+3h dµ(h) (140) In the multifractal formalism, the structure function exponents are defined by ζp = inf h (3 − d(h) + ph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (141) The inequality (140) above implies ˆ BU ∩(∪hVh) S2(x, r0) 3 2 dx ≤ Cµ(GL)3 �r0 L �ζ3 (142) where Cµ = C ˆ 1 0 h− 3 2 dµ(h) (143) is assumed to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Introducing the Reynolds number based on G, RG = GL ν (144) and recalling that BU \\ ∪hVh was assumed to have measure zero, we have ˆ BU S2(x, r0) 3 2 dx ≤ � CµR3 G �r0 L �ζ3� ν3 (145) we see that the condition (101) is satisfied for BU if R3 G �r0 L �ζ3 ≤ (C3Cµ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (146) In classical turbulence theory ζ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' If ζ3 > 0, under the above scenario, it is enough to have r0 L small enough in order to deduce that no singularities in finite time can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Time dependent regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' As me noted before, the finite uniform integrability of con- dition (101) is not needed, all we need is control of S2 on certain small sets of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We consider the set BU,G(t) = {x | |u(x, t)| ≥ U, and |∇u(x, t)| ≥ G} defined in (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We note that |BU,G(t)| ≤ C min{U −2∥u0∥2 L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' G−1∥ω0∥L1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (147) where ω = ∇ × u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The first term in the inequality follows from the Markov-Chebyshev inequality and the fact that the L2 norms of solutions of Navier-Stokes equations are non-increasing in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The second term follows from the fact that the map ω �→ ∇u is weak type 1, that is from G|{x | |∇u| ≥ G} ≤ C∥ω∥L1, and the fact that the L1 norm of vorticity of solutions of Navier-Stokes equations is non-increasing in time [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' THEOREM 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let U(t), G(t) and r(t) be positive numbers such that ˆ T 0 (r(t)−4 + U(t)4 + G(t))dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (148) Consider the set B(t) = {x | |u(x, t)| ≥ U and |∇u(x, t)| ≥ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (149) There exists an absolute constant C such that, if ˆ |y|≤r(t) �ˆ B(t) |δyu(x, t)|3dx � 2 3 dy |y|3 ≤ � ν C �2 (150) then the smooth solution of Navier-Stokes equations obeys u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' L3(R3)) with explicit bounds depending only on ν, T, ∥u0∥L3, ∥u0∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' 18 PETER CONSTANTIN PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We follow the proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The β term is estimated as in (104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The contribution of the term involving π from the region |u| ≤ U is estimated as in (107), noting that the term U 2∥u∥2 L3 is time integrable in view of (148).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' A new term is ˆ |u|≥U,|∇u|≤G |π||u||∇|u||dx ≤ CG∥u∥3 L3, (151) and, in view of (148) it leads via Grownwalll to an explicit bound on ∥u∥L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We are left with ˆ B(t) |π||u||∇|u||dx ≤ CD �ˆ B(t) S 3 2 2 (x, r(t))dx � 1 3 (152) Now we use �ˆ B S 3 2 2 dx � 2 3 ≤ 1 4π ˆ |y|≤r dy |y|3 ˆ B |δyu(x)|2dx (153) proved by duality, testing against arbitrary L3(B) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The assumption (150) implies ˆ B(t) |π||u||∇|u||dx ≤ νD 6 (154) and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ THEOREM 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Let q ≥ 4, and assume (150) where the functions U(t), r(t) and G(t) obey ˆ T 0 (U 2(t) + r−4(t) + G(t))dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (155) Then we have the single exponential bound ∥u(·, t)∥Lq ≤ ∥u0∥Lq exp \uf8eb \uf8edCν−1 ˆ t 0 U 2ds + C ˆ t 0 G(s)ds + Cν− 3 2 ∥u0∥2 L2 �ˆ t 0 r−4(s)ds \uf8f6 \uf8f8 (156) PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We start as in the proof of Theorem 12 by splitting p = β + π in the estimate the evolution of the Lq norm of u, and deduce the bound (112) leading to the exponential growth factor (113).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We are left o estimate the contribution of π, that is I = ˆ R3 πu · ∇|u|q−2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (157) We bound the integral |I| ≤ I1 + I2 + IB = (q − 2) �´ |u|≤U |π||u|q−2|∇u|dx + ´ |∇u|≤G |π||u|q−2|∇u|dx + ´ B(t) |π||u|q−2|∇u|dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (158) We bound I1 like in (84), I1 ≤ CU∥u∥ q 2 Lq √ D (159) where we use the fact that ∥π(·, r)∥L q 2 ≤ C∥u∥2 Lq holds with C an absolute constant, independent of r, and �ˆ |u|≤U |u| q(q−2) q−4 dx � q−4 2q ≤ U∥u∥ q 2 −2 Lq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' (160) The bound (159) is valid for q = 4 as well, we just take |u| ≤ U outside the integral and use L2 − L2 bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The term I2 is bound directly |I2| ≤ CG∥u∥q Lq (161) The last term is smaller than the dissipation, using the arguments similar to the ones leading to (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' We omit further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' □ 19 REMARK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' The condition U ∈ L2(0, T) appearing in (155) is better than the condition U ∈ L4(0, T) of (148) of Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' That is just because in that theorem the desire was to bound the L3 norm in terms solely of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Theorem 15 is strictly stronger that Theorem 14 (it implies it for strong solutions), by bounding first the L4 norm of the solution, and then returning to the proof of the bound of the L3 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE3T4oBgHgl3EQfYgqz/content/2301.04489v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': 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Frandsen,1 Raju Baral,1 Barry Winn,2 and V. Ovidiu Garlea2 +1)Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, +USA +2)Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, +USA +(*Electronic mail: benfrandsen@byu.edu) +(Dated: 3 January 2023) +We report the first example of magnetic pair distribution function (mPDF) data obtained through use of neutron po- +larization analysis. Using the antiferromagnetic semiconductor MnTe as a test case, we present high-quality mPDF +data collected on the HYSPEC instrument at the Spallation Neutron Source using longitudinal polarization analysis to +isolate the magnetic scattering cross section. Clean mPDF patterns are obtained for MnTe in both the magnetically +ordered state and the correlated paramagnet state, where only short-range magnetic order is present. We also demon- +strate significant improvement in the quality of high-resolution mPDF data through application of ad hoc corrections +that require only minimal human input, minimizing potential sources of error in the data processing procedure. We +briefly discuss the current limitations and future outlook of mPDF analysis using polarized neutrons. Overall, this work +provides a useful benchmark for mPDF analysis using polarized neutrons and provides an encouraging picture of the +potential for routine collection of high-quality mPDF data. +I. +INTRODUCTION +Magnetic pair distribution function (mPDF) analysis of +neutron total scattering data has recently emerged as a valu- +able tool for investigating local magnetic correlations in mag- +netic materials1,2. +In analogy to the more familiar atomic +pair distribution function (PDF) method3, the mPDF is ob- +tained by Fourier transforming the total magnetic scattering, +which refers to the scattering arising both from long-range +magnetic correlations (resulting in magnetic Bragg peaks) +and from short-range magnetic correlations (resulting in dif- +fuse magnetic scattering). This yields the real-space, pair- +wise magnetic correlation function. The mPDF technique is +most useful for the study of short-range magnetic correlations +such as those in a correlated paramagnet or a quantum disor- +dered magnet, for which the real-space mPDF can be easier +to interpret and model than the corresponding diffuse scatter- +ing pattern in reciprocal space. On the other hand, conven- +tional magnetic Bragg diffraction analysis will typically re- +main the preferred choice for the determination of long-range +ordered magnetic crystalline states. Since the introduction of +the mPDF technique in 2014, it has been applied to numerous +systems with short-range magnetic correlations ranging from +quantum magnets to functional magnetic materials4–14. +For an isotropic powder sample of a typical magnetic mate- +rial possessing localized spins that belong to a single magnetic +species, the mPDF is given by1,7 +Gmag(r) = 2 +π +� ∞ +Qmin +Q +� +(dσ/dΩ)mag +2 +3NsS(S+1)(γr0)2[f(Q)]2 −1 +� +sin(Qr)dQ +(1) += +3 +2S(S+1) +� +1 +Ns ∑ +i̸=j +� +Aij +r δ(r −rij)+Bij +r +r3 +ij +Θ(rij −r) +� +−4πrρ0 +2 +3m2 +� +. +(2) +The first equation defines the experimental mPDF, while the +second equation shows how to calculate the mPDF for a given +magnetic structure. Here, Q is the magnitude of the scatter- +ing vector, Qmin is the minimum measured scattering vec- +tor (assumed to exclude the small-angle scattering regime), +(dσ/dΩ)mag is the magnetic differential scattering cross sec- +tion, r is real-space distance, r0 = µ0 +4π +e2 +me is the classical elec- +tron radius, γ = 1.913 is the neutron magnetic moment in units +of nuclear magnetons, S is the spin quantum number in units +of ¯h, f(Q) is the magnetic form factor, Ns is the number of +spins in the system, i and j label individual spins Si and Sj sep- +arated by the distance rij, Aij = ⟨Sy +i Sy +j⟩, Bij = 2⟨Sx +i Sx +j⟩−⟨Sy +i Sy +j⟩, +Θ is the Heaviside step function, m is the average magnetic +moment in Bohr magnetons (which is zero for anything with +no net magnetization, such as antiferromagnets), and ρ0 is +the number of spins per unit volume. +A local coordinate +system is used for each spin pair to compute Aij and Bij, +given by ˆx = +rj−ri +|rj−ri| and ˆy = Si− ˆx(Si· ˆx) +|Si− ˆx(Si· ˆx)|. The generalization +arXiv:2301.00137v1 [cond-mat.mtrl-sci] 31 Dec 2022 + +Magnetic PDF Data from Polarized Neutrons +2 +to magnetic materials with multiple types of magnetic atoms +and/or nonzero orbital contributions to the magnetic moment +is straightforward and given elsewhere1. Generally speaking, +a positive (negative) peak at a given distance corresponds to +net parallel (antiparallel) orientation between spins separated +by that distance, lending an intuitive interpretation to mPDF +data. We note that in Eq. 1, the subtraction of unity from +the term in parentheses amounts to the removal of the self- +scattering contribution to the magnetic scattering cross sec- +tion, which is desirable because the self-scattering contains +no information about the correlations between distinct mag- +netic moments. Eq. 1 is fully equivalent to the corresponding +integral expression used in the definition of the atomic PDF. +One of the biggest experimental challenges for obtaining +high-quality mPDF data is accurately measuring the magnetic +scattering to a sufficiently large momentum transfer and sep- +arating it from the (typically much larger) nuclear scattering. +For a typical experiment using unpolarized neutrons, this can +be done either by fitting a structural model to the nuclear +Bragg peaks and subtracting them out to leave just the mag- +netic scattering, or by subtracting a reference measurement +taken at a temperature where no coherent magnetic scatter- +ing is present (e.g. at high temperature well above a mag- +netic transition) from a diffraction pattern collected at a lower +temperature with nonzero magnetic scattering. If the atomic +structure is identical at both temperatures, then the difference +between the scattering patterns gives the magnetic scattering. +However, both of these strategies can result in significant ar- +tifacts due to imperfect modeling of the nuclear Bragg peaks +(or nuclear diffuse scattering if short-range structural correla- +tions exist) or imperfect temperature subtraction, where even +simple thermal expansion causes shifts in the nuclear Bragg +peak positions that can be difficult to treat accurately. An- +other source of error in this strategy is the over-subtraction of +magnetic scattering which could still be present in the form of +incoherent paramagnetic scattering that follows a form factor +decay as a function of Q. +An alternative approach is to make no attempt to separate +the magnetic scattering from the nuclear scattering and in- +stead simply generate the PDF from the combined diffraction +signal following standard protocols. The atomic and mag- +netic structures can then be modeled together in real space2. +However, this only works if the total scattering data are suit- +able for conventional PDF analysis, meaning that this ap- +proach is typically suitable only for data collected on dedi- +cated PDF diffractometers with access to large values of mo- +mentum transfer (typically more than 20 Å−1). Even then, +the mPDF signal can often be one or more orders of magni- +tude smaller than the nuclear PDF signal, making it difficult +detect above the noise. Furthermore, this approach treats mag- +netic scattering the same as nuclear scattering in the PDF data +processing protocol, meaning that no attempt is made to nor- +malize the magnetic scattering by the squared magnetic form +factor, as should normally be done according to Eq. 1. This +has the effect of twice convoluting Gmag(r) with the Fourier +transform of the magnetic form factor, broadening the mPDF +by approximately +√ +2 times the real-space width of the elec- +tronic wavefunction giving rise to the magnetic moment. This +“non-deconvoluted mPDF” (previously called the “unnormal- +ized mPDF”2) takes the form2 +dmag(r) = 2 +π +� ∞ +Qmin +Q +�dσ +dΩ +� +mag +sin(Qr)dQ +(3) += C1 ×Gmag(r)∗S(r)+C2 × dS +dr , +(4) +where +C1 += +Ns +2π +� γr0 +2 +�2 2 +3⟨g +� +J(J +1)⟩2 +and +C2 += +Ns +2π +� γr0 +2 +�2 2 +3⟨g2J(J + 1)⟩ are constants, +∗ represents the +convolution operation, and S(r) = F { fm(Q)} ∗ F { fm(Q)} +(where F denotes the Fourier transform). The second term +on the right of Eq. 4, which arises from the self-scattering +contribution to the magnetic differential scattering cross +section, results in a peak at low r (below approximately 1 Å). +The strategy of co-modeling the atomic and magnetic PDF +data in real space requires working with this lower-resolution, +non-deconvoluted mPDF signal. To keep the distinction clear, +we will refer to dmag(r) as the non-deconvoluted mPDF and +Gmag(r) as the proper mPDF, deconvoluted mPDF, or just the +mPDF. +Another challenge exists even in cases where the magnetic +scattering has been cleanly separated from the nuclear scat- +tering: accurately normalizing the magnetic scattering by the +squared magnetic form factor prior to performing the Fourier +transform. +Because the squared magnetic form factor for +most magnetic ions approaches zero between 5 and 8 Å−1, +any noise or leftover nuclear scattering in this range will be +greatly amplified during the normalization step, inevitably +leading to high-frequency artifacts in the mPDF data. Often, +these artifacts are sufficiently severe that it is preferable just to +skip the normalization step altogether and settle for the non- +deconvoluted mPDF2; however, this comes with the cost of +significantly reduced real-space resolution, so it is also not +ideal. +Many useful mPDF studies have been carried out in spite +of these experimental challenges. Nevertheless, the technique +will have a greater impact if the data quality improves. Here, +we report two developments in mPDF data collection and pro- +cessing that help address the dual challenges of accurately +isolating the magnetic scattering and reliably mitigating the +artifacts introduced during the magnetic form factor normal- +ization step. The first is using longitudinal polarization anal- +ysis with a polarized neutron beam to separate the magnetic, +nuclear, and nuclear spin-incoherent cross sections directly15, +with no need for temperature subtraction or modeling of nu- +clear Bragg peaks. The second is using a polynomial func- +tion to perform ad hoc data corrections that minimize the +real-space artifacts resulting from the form factor normaliza- +tion16,17. These are both well-established methods, but they +are applied here to mPDF analysis for the first time. In the fol- +lowing, we briefly describe longitudinal polarization analysis +and the procedure for performing the ad hoc data corrections, +and then provide examples of their application to mPDF data +on the antiferromagnetic semiconductor manganese telluride +(MnTe). Code for performing the data corrections is freely +available as part of the diffpy.mpdf python package18 and +in the Supplementary Information of this article. These devel- + +Magnetic PDF Data from Polarized Neutrons +3 +opments represent significant progress toward routine collec- +tion of high-resolution mPDF data on a variety of magnetic +materials of fundamental and technological interest. +II. +LONGITUDINAL POLARIZATION ANALYSIS +The method of XYZ longitudinal polarization analysis of +neutron scattering data collected with large solid-angle detec- +tor banks (also called 6-pt polarization analysis, or 10-pt po- +larization analysis in its more general form including out-of- +plane scattering) allows unambiguous separation of nuclear, +magnetic, and nuclear spin-incoherent scattering cross sec- +tions. This is accomplished by measuring the spin-flip and +non-spin-flip cross sections for various directions of polariza- +tion of the incident neutron beam; detailed descriptions of the +technique are found elsewhere15,19,20. A small handful of neu- +tron diffractometers and spectrometers have the capability to +perform longitudinal polarization analysis, such as D7 at the +Institut Laue Langevin19, DNS at FRM-II in Munich21, and +HYSPEC at the Spallation Neutron Source (SNS) and Oak +Ridge National Laboratory (ORNL)22. The former two in- +struments can access a maximum momentum transfer of ∼4.0 +and 4.84 Å−1, respectively, while HYSPEC can reach just +over 6 Å−1 for certain choices of incident energy and detec- +tor positions. Given the importance of maximizing the range +of accessible momentum transfer for the Fourier transform, +HYSPEC is currently the instrument with polarization analy- +sis that is most suitable for mPDF analysis. +III. +AD HOC DATA CORRECTIONS +In any attempt to isolate the magnetic scattering inten- +sity, normalize it by the squared magnetic form factor, and +perform the Fourier transform to generate the experimental +mPDF, errors will inevitably be introduced due to imperfect +time-independent instrumental background corrections, inter- +ference from nuclear and nuclear spin incoherent scattering, +inaccuracies in the measured or approximated magnetic form +factor, statistical noise, etc. Because typical squared magnetic +form factors approach zero between 5 and 8 Å−1, any errors +present in this range of the data become particularly magnified +if included in the Fourier transform, introducing nonphysi- +cal artifacts into the mPDF. These challenges are similar to +those encountered when generating x-ray PDF data, where +the x-ray scattering atomic form factor plays an analogous +role as the magnetic form factor. The traditional approach +in x-ray PDF analysis has been to apply corrections for each +source of error manually3, but this can be an onerous task +and is itself prone to errors16, particularly for researchers new +to the technique. As has been discussed in detail for x-ray +PDF16,17, ad hoc corrections using a polynomial correction +function can largely mitigate the errors with minimal human +input. This ad hoc approach is also highly automatable, allow- +ing for rapid throughput PDF data processing using tools such +as PDFgetX317. Here, we have implemented the PDFgetX3 +algorithm in a custom-built python program and applied it to +magnetic scattering data to obtain high-resolution mPDF data +with no manual corrections necessary. The python code has +been made available as part of the diffpy.mpdf package18 +and as a standalone program in the Supplementary Informa- +tion. +One technical note merits discussion here. As explained +in Ref. 17, the polynomial correction function results in non- +physical contributions to the real-space PDF for r values less +than rpoly = πn/Qmaxinst, where n is the degree of the poly- +nomial and Qmaxinst is the maximum value of Q to which the +correction polynomial is fit. As long as rpoly is shorter than the +nearest-neighbor distance, then the artifacts resulting from the +ad hoc corrections will be restricted to the r-region below the +first peak in the PDF data. In practice, an appropriate value of +rpoly is selected (typically below 1 Å for x-ray PDF data), and +a weighted average of the PDFs obtained using the nearest +integer values of n corresponding to the selected rpoly value +is generated. For synchrotron x-ray PDF experiments with +Qmaxinst values of 25 Å−1 or greater, typical values for n range +from 7 to 9. When applied to mPDF, Qmaxinst will often be +much smaller, anywhere from 5 to 8 Å−1, so n should likewise +be smaller to keep rpoly acceptably short. For HYSPEC, where +the largest accessible value for Qmaxinst is about 6 Å−1, typi- +cal values of n are 3 and 4, resulting in rpoly values of 1.57 Å +and 2.09 Å respectively. Although these values would be too +large for typical atomic PDF data, they are acceptable for typ- +ical magnetic structures, where the nearest-neighbor distance +between magnetic atoms is unlikely to be as short as 2.09 Å. +IV. +APPLICATION TO MANGANESE TELLURIDE +We demonstrate the use of polarized neutrons and the ad +hoc data correction procedure to generate mPDF data for the +antiferromagnetic semiconductor MnTe. This material has a +hexagonal crystal structure (space group P63/mmc) with a +magnetic ordering temperature of TN = 307 K, below which +the Mn2+ spins order with parallel alignment within the ab +planes and antiparallel alignment between neighboring ab +planes, as has been established by previous neutron diffrac- +tion studies23–25. Short-range antiferromagnetic correlations +are known to survive well into the paramagnetic phase above +TN, likely up to 900 K or higher26. The short-range mag- +netism in MnTe was the subject of a recent mPDF study14 +using data collected with unpolarized neutrons, together with +three-dimensional 3D-∆mPDF27 data collected from a single +crystal. +A. +Experimental Details +A large powder sample of MnTe (∼8 g) was synthesized +according to the procedure in Ref. 14, pressed into a pellet +of diameter 6 mm, and loaded into an aluminum sample can. +A closed cycle refrigerator was used to control the temper- +ature. The incident neutron energy used for the experiment +was Ei = 28 meV, and the Fermi chopper was operated at + +Magnetic PDF Data from Polarized Neutrons +4 +a frequency of 60 Hz. This configuration provided an en- +ergy resolution at the elastic position of 5.7 meV full width +at half maximum (FWHM). The data were integrated over en- +ergy transfers between −5 and 5 meV. The incident beam at +HYSPEC is polarized by a vertically focusing array of mag- +netically saturated Heusler (Cu2MnAl) crystals. A Mezei flip- +per was used to reverse the polarization direction of the inci- +dent beam, and the polarization analysis of the scattered beam +was accomplished with a 60◦-wide-angle multi-channel ra- +dial supermirror-polarizer array22. Spin-flip (SF) non-spin- +flip (NSF) data were collected with the neutron polarization +oriented vertically, Pz, for two different positions of the the +60◦ detector bank to cover the 2θ scattering range 3 - 112◦. In +this configuration, the SF setting measures the magnetic scat- +tering, and the NSF gives the combined nuclear and magnetic +contributions. The nuclear spin-incoherent scattering contri- +bution from MnTe is expected to be insignificant. A flipping +ratio of 13 was determined by evaluating the scattering from +nuclear Bragg peaks in SF and NSF configurations. +Post- +processing data corrections for the angle-dependent supermir- +ror transmission and for the flipping ratio efficiency were car- +ried out using the algorithms implemented in MANTID soft- +ware, as discussed in Refs. 22 and 28. The magnetic scattering +used in this study corresponds to the SF intensity corrected for +the flipping ratio according to Eq. 2 in Ref. 22. +B. +Results +1. +Magnetic Scattering +In Fig. 1, we display the magnetic scattering at 50 K (a) and +330 K (b). Sharp antiferromagnetic Bragg peaks are present +at 50 K. At 330 K, the magnetic scattering is diffuse but still +quite structured, indicating the persistence of short-range cor- +relations into the paramagnetic phase, as expected from earlier +results. The nuclear scattering cross section at 50 K, obtained +by subtracting the SF scattering from the NSF scattering, is +shown in Fig. 1(c). A slight oversubtraction of the strong mag- +netic scattering is visible around 0.9 Å−1, but we see minimal +interference effects elsewhere, so we conclude that the separa- +tion of the different cross sections is sufficiently accurate for +our purposes. +2. +mPDF Data and Fits +Fig. 2 shows key results from the ad hoc data correction +process to produce the mPDF from the scattering data col- +lected at 50 K. In panel (a), we display the magnetic scattering +data and the squared magnetic form factor for Mn2+, scaled +to match the scattering profile between 3 and 6 Å−1 as closely +as possible. We used the analytical approximation of the form +factor as reported in the International Tables for Crystallog- +raphy29. Panel (b) shows the uncorrected form of the reduced +magnetic structure function Fmag(Q) (not to be confused with +the magnetic structure factor used in conventional magnetic +0 +1 +2 +3 +4 +5 +6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +a +Magnetic 50 K +0 +1 +2 +3 +4 +5 +6 +0.0 +0.1 +0.2 +0.3 +Intensity (arb. units) +b +Magnetic 330 K +0 +1 +2 +3 +4 +5 +6 +Q (Å +1) +0 +1 +2 +3 +4 +c +Nuclear 50 K +FIG. 1. Magnetic scattering intensity for MnTe at (a) 50 K and (b) +330 K. Sharp Bragg peaks are present at 50 K due to the long-range +ordered antiferromagnetic phase, whereas only diffuse peaks remain +at 330 K in the correlated paramagnetic regime. (c) Nuclear scatter- +ing at 50 K. +diffraction analysis), given by +Fmag,uc(Q) = Q +� +(dσ/dΩ)mag +A[f(Q)]2 +−1 +� +, +(5) +where A is the scaling constant used in panel (a), replacing the +coefficient on [f(Q)]2 shown in Eq. 1 that would be appropri- +ate if the data were on an absolute scale. The “uc” in the +subscript of the vertical axis label stands for “uncorrected”. +Note that the increasing behavior of Fmag(Q) at high Q is non- +physical, indicating an imperfect normalization by the squared +magnetic form factor. Indeed, the correct form of Fmag(Q) +should oscillate around zero at high Q, since the correct nor- +malization of the magnetic scattering in Eq. 5 puts it on the +scale of unity. To correct this error, we used least-squares min- +imization to fit a polynomial to Fmag,uc, in accordance with the + +Magnetic PDF Data from Polarized Neutrons +5 +a +b +c +d +FIG. 2. Procedure for obtaining the mPDF from magnetic scattering +data from MnTe at 50 K. (a) Magnetic scattering intensity (black +symbols) overlaid by the square of the magnetic form factor of Mn2+ +(orange curve; scaled to match the magnitude of the data at high Q). +(b) Uncorrected reduced magnetic structure function Fmag,uc (black +symbols) with the best-fit degree-3 polynomial correction function +(orange curve). (c) Corrected Fmag,c after subtracting the polynomial +fit. (d) Resulting mPDF using Qmax = 5.5 Å−1. +ad hoc approach used for x-ray PDF in PDFgetX317. Fig. 2(b) +shows the best-fit cubic polynomial as the orange curve [note +that this is not the squared magnetic form factor, which is in- +stead displayed in Fig. 2(a)]. With a polynomial of degree +3 and Qmaxinst = 6.0 Å−1, the corresponding value of rpoly is +1.57 Å, safely below the nearest-neighbor Mn-Mn distance of +∼3.37 Å. Due to the relatively low degree of the polynomial, +it effectively captures the slowly modulating, nonphysical be- +havior of Fmag,uc without fitting to the physically meaningful +features in the data, which have characteristically faster mod- +ulations in Q. This polynomial is sometimes referred to as +a background function, which is accurate in the sense that it +captures the slowly modulating errors in Fmag,uc due to imper- +fect normalization by the squared magnetic form factor, but +it should not be interpreted as an attempt to correct for the +background scattering from the instrumental setup. We as- +sume the instrumental background has already been removed. +In panel (c), we display the corrected form of Fmag(Q), la- +beled Fmag,c, resulting from subtracting the fitted polynomial. +We note that Fmag,c now oscillates around zero as Q becomes +large, as expected when all errors have been removed. Finally, +panel (d) shows the resulting mPDF using Qmax = 5.5 Å−1, +Qmaxinst = 6.0 Å−1, and rpoly = 1.57 Å. +In Fig. 3, we display fits to the mPDF data at 50 K us- +ing the published magnetic structure of MnTe. +We plot +the non-deconvoluted mPDF data (given by Eq. 3) and fit +in panel (a) in blue symbols and a solid red curve, respec- +tively. The fit agrees closely with the data and is free from +0 +5 +10 +15 +20 +25 +1 +0 +1 +2 +3 +dmag(r) (arb. units) +MnTe 50 K a +0 +5 +10 +15 +20 +25 +r (Å) +2 +1 +0 +1 +Gmag(r) (arb. units) +b +FIG. 3. Magnetic PDF data for MnTe at 50 K, including (a) the non- +deconvoluted mPDF dmag(r) and (b) the proper mPDF Gmag(r). The +blue circles and red curve show the experimental data and best-fit +mPDF, respectively. In panel (b), the vertical dashed line indicates +the selected value of rpoly = 1.55 Å, below which artifacts from the +polynomial correction function are expected to be significant. +the high-frequency noise that accompanies non-deconvoluted +mPDF data obtained together with standard atomic PDF data, +highlighting the advantage of using polarized neutrons. In +Fig. 3(b), we plot the deconvoluted mPDF data and fit, where +the greatly improved real-space resolution is immediately ap- +parent. The fit successfully captures the most prominent fea- +tures of the mPDF data, demonstrating the success of our ap- +proach. As fitting parameters, we included an arbitrary scale +factor, a real-space damping term to account for the finite +Q-space resolution of the data (known as Qdamp in conven- +tional PDF analysis), and an r-dependent real-space broaden- +ing term (Qbroad for conventional PDF) to account for statis- +tical noise in the scattering data30. The fitted values of Qdamp +and Qbroad were 0.037 Å−1 and 1.68 Å−1, respectively. We +note the presence of artifacts for r ≲ 1.5 Å (corresponding +closely to rpoly = 1.57 Å) and minor misfits throughout the +full data range, but considering that this experimental mPDF +curve was generated without any human intervention or fine +tuning (in contrast to earlier efforts2,31), the overall level of +agreement is highly encouraging. We further note that, ignor- +ing any distortions or errors in the data, the mPDF pattern in + +Magnetic PDF Data from Polarized Neutrons +6 +0 +5 +10 +15 +20 +0.0 +0.5 +1.0 +1.5 +2.0 +dmag(r) (arb. units) +MnTe 330 K a +0 +5 +10 +15 +20 +r (Å) +0.2 +0.1 +0.0 +0.1 +0.2 +Gmag(r) (arb. units) +b +FIG. 4. Magnetic PDF data for MnTe at 330 K, including (a) the non- +deconvoluted mPDF dmag(r) and (b) the proper mPDF Gmag(r). The +blue circles and red curve show the experimental data and best-fit +mPDF with an anisotropic correlation model, respectively. In panel +(b), the vertical dashed line indicates the selected value of rpoly = +3.0 Å. +the ordered state is equivalent to the mPDF that would be ob- +tained if one were to perform a perfect nuclear and magnetic +Rietveld refinement to an unpolarized neutron diffraction pat- +tern over a sufficiently large Q-range and then apply the nor- +malization and Fourier transform procedure to the magnetic +part of the Rietveld fit. +In many cases, short-range magnetic correlations will give +rise to the most physically interesting mPDF data. In that +spirit, we plot in Fig. 4 the mPDF data and fits for MnTe at +330 K, somewhat above TN = 307 K and in the “correlated +paramagnet” regime where short-range correlations persist in +the absence of long-range magnetic order. Once again, the +non-deconvoluted and deconvoluted mPDF patterns are dis- +played in panels (a) and (b), respectively. Because the mag- +netic scattering is significantly weaker in the paramagnetic +phase, the Q-range of usable scattering data for the Fourier +transform was more limited, so the data shown were gener- +ated with Qmax = 4.5 Å−1, together with rpoly = 3.0 Å and +Qmaxinst = 6.0 Å−1. We also used a Fermi-Dirac modifica- +tion function with a width of 1.0 Å−1 to generate the decon- +voluted mPDF (see next section for details about the modi- +fication function). The non-deconvoluted fit in panel (a) is +reasonably good, although truncation effects are apparent in +the data. The deconvoluted mPDF data in Fig. 4(b) exhibit +somewhat larger artifacts that the ad hoc corrections could not +remedy, but the near-neighbor antiferromagnetic correlations +are clearly evident in the data, e.g. as the alternating negative +and positive peaks around 3.4 Å (nearest neighbor, antipar- +allel), 4.2 Å (second nearest neighbor, parallel), and 5.4 Å +(third nearest neighbor, antiparallel). The mPDF fit captures +the general features of the data. We attribute the reduced data +quality in the correlated paramagnetic state to the more re- +stricted Q-range and greater statistical noise in the scattering +data compared to the data collected at 50 K. This underscores +the need for high-statistics measurements over a sufficient Q- +range to produce optimal high-resolution mPDF data. +Despite the more limited data quality, we find it encourag- +ing that the fitted magnetic model produced from the prop- +erly deconvoluted mPDF data agrees well with the model pro- +duced by the fit to the non-deconvoluted data and to our ear- +lier mPDF fits performed on a different instrument with unpo- +larized neutrons14. In particular, the short-range correlations +in MnTe were shown to be spatially anisotropic in our ear- +lier study, with a longer correlation length along the crystallo- +graphic c axis than within the hexagonal ab plane. The present +fits included two free parameters capturing this anisotropic ef- +fect in the magnetic model. The fit naturally converged to +correlation lengths of 7.1(5) and 4.6(6) Å along c and in the +ab plane, respectively, in good agreement to those reported +in Ref. 14 at a similar temperature. The consistency of these +results validates the earlier report and the use of polarized neu- +trons for obtaining useful mPDF data in a correlated param- +agnet. +3. +Effect of Modification Functions +Finally, we discuss the effect of applying a Q-space mod- +ification function32, also known as a window function, to +Fmag,c(Q) prior to the Fourier transform. In the context of +atomic PDF analysis, various modification functions have +been used to suppress the effect of statistical or systematic er- +rors in the scattering data at high Q, reducing high-frequency +ripples in the PDF (but with the cost of reduced real-space +resolution and possible loss of physically meaningful infor- +mation)32. +A modification function w(Q) operates simply +through multiplication with the corrected form of Fmag(Q), +such that w(Q) × Fmag(Q) is the quantity to be Fourier trans- +formed. We consider here three modification functions: the +default step function +ws(Q) = +� +1 +if Q ≤ Qmax +0 +if Q > Qmax +, +(6) +a modified Fermi-Dirac function +wFD(Q) = +� +2 +e(Q−Qmax)/∆+1 −1 +if Q ≤ Qmax +0 +if Q > Qmax +, +(7) + +Magnetic PDF Data from Polarized Neutrons +7 +and the conventional Lorch function33 +wL(Q) = +� Qmax +πQ sin +� +πQ +Qmax +� +if Q ≤ Qmax +0 +if Q > Qmax +. +(8) +We note that ws is equivalent to applying no modification +function at all, since it is unity up until the hard cutoff of the +Fourier transform at Qmax. These three modification functions +are displayed in Fig. 5(a), together with the squared mag- +netic form factor for Mn2+ for reference. In all cases, we +use Qmax = 5.5 Å−1, and for wFD, we set ∆ to 0.5 Å−1. The +mPDF curves at 50 K generated with the application of each +of these modification functions are given in Fig. 5(b). Both +wFD and wL suppress high-frequency features in the mPDF +relative to the default mPDF using ws. The effect is particu- +larly pronounced for wL. However, we note that for the avail- +able Q-range of 5.5 Å−1 included in the Fourier transform, +wL results in a significant suppression of Fmag(Q) over nearly +the entire range similar in scale to the squared magnetic form +factor shown in Fig. 5(a). In effect, then, application of wL +undoes the normalization by the squared form factor, yield- +ing a quantity that is similar in real-space resolution to the +non-deconvoluted mPDF that we were originally hoping to +improve upon. We therefore suggest that a Lorch window is +of limited usefulness for producing deconvoluted mPDF pat- +terns when the available Q-range does not significantly exceed +the extent of the squared magnetic form factor. On the other +hand, wFD with a modestly chosen value of ∆ may provide +value by suppressing artifacts from high-Q noise without sac- +rificing meaningful mPDF data due to excessive broadening. +Fig. 5(c) shows the mPDF curves at 330 K obtained for each +type of modification function, this time with Qmax = 4.5 Å−1 +and ∆ = 1.0 Å−1. The Lorch and Fermi-Dirac functions again +result in greatly suppressed Fourier ripples, with the Fermi- +Dirac function preserving somewhat more real-space resolu- +tion than the Lorch function. +In the mPDF fits presented in the previous section, we +found no need to use a modification function for the data col- +lected at 50 K where the magnetic scattering is strong. How- +ever, given the weaker magnetic scattering at 330 K, we found +it beneficial to apply a Fermi-Dirac modification function with +a width of 1.0 Å−1. We chose this width because, after test- +ing several different values, we found that 1.0 Å−1 offered a +good balance between minimizing artifacts in the mPDF data +and preserving the real-space resolution to a reasonable de- +gree. Ideally, the calculated mPDF used for modeling the data +should be convoluted with the Fourier transform of the mod- +ification function to account correctly for the change in real- +space resolution caused by the modification function. +V. +DISCUSSION AND CONCLUSION +We have reported the first use of polarized neutrons to gen- +erate mPDF data and the first application of PDFgetX3-style +ad hoc corrections to produce high-resolution, properly de- +convoluted Gmag(r) curves with minimal human input. Po- +larization analysis allowed us to isolate the magnetic scat- +tering cross section and produce the resulting mPDF data +0 +2 +4 +6 +Q (Å +1) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Window Amplitude +a +Step +Fermi-Dirac +Lorch +Mn2 + +0 +5 +10 +15 +20 +r (Å) +2 +1 +0 +1 +Gmag(r) (arb. units) +b +MnTe 50 K +0 +5 +10 +15 +20 +r (Å) +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +Gmag(r) (arb. units) +c +MnTe 330 K +FIG. 5. (a) modification functions as defined in the main text with +Qmax = 5.5 Å−1. The Fermi-Dirac modification function uses ∆ = +0.5 Å−1. The squared magnetic form factor of Mn2+ is also shown +for reference. (b) The mPDF patterns for MnTe at 50 K resulting +from application of the modification functions in (a), using the same +legend. (c) Same as (b), but for 330 K. The transform and mask +parameters are Qmax = 4.5 Å−1 and ∆ = 1.0 Å−1 . + +Magnetic PDF Data from Polarized Neutrons +8 +with little to no interference from nuclear scattering. +On +HYSPEC, we achieved a Qmax value of 5.5 Å−1 in the mag- +netically ordered state of MnTe. This was sufficient for an ex- +ceptionally clean non-deconvoluted mPDF curve and a high- +quality proper mPDF curve produced with the help of the ad +hoc corrections, demonstrating the promise of this method +for producing mPDF data. +In the paramagnetic state, we +obtained good-quality non-deconvoluted and deconvoluted +mPDF curves suitable for quantitative mPDF refinements that +confirmed a prior result. However, the weaker magnetic scat- +tering at 330 K necessitated a more limited Q-range and in- +troduced significantly more noise and artifacts into the data. +These issues could be ameliorated in large part simply by col- +lecting more data to improve the statistics over a large enough +range of momentum transfer (ideally up to 5.5 Å−1 or greater). +Finally, our results suggest that if a modification function is to +be used to reduce high-frequency noise in the mPDF data, a +modified Fermi Dirac function or similar is preferred over the +traditional Lorch function. +Overall, the results reported here represent a significant +step forward in the development of mPDF methodologies +by highlighting the promise of polarized neutrons and mod- +ern data processing techniques to produce high-quality, high- +resoultion mPDF data. We have provided a realistic view of +both the capabilities and the limitations posed by current ex- +perimental resources when generating high-resolution mPDF +data from weak and diffuse magnetic scattering. With ma- +jor developments in neutron capabilities on the near horizon +at the European Spallation Source, the Second Target Station +(STS) of the Spallation Neutron Source, and elsewhere, in- +strumental limitations are expected to be less of a concern +in the future. We mention particularly the planned STS in- +strument VERDI34, for which the combination of high beam +intensity, large Q coverage, and polarization analysis should +enable routine collection of high-resolution mPDF data of ex- +cellent quality, which we expect to transform the landscape of +mPDF studies in condensed matter and materials physics. +VI. +SUPPLEMENTARY MATERIAL +The supplemental materials include the magnetic scattering +cross section at 50 K and 330 K, the nuclear scattering cross +section at 50 K and 330 K, python code for generating the +mPDF with the ad hoc corrections, and an example python +script showing how to use the code. +ACKNOWLEDGMENTS +We thank Melissa Graves-Brook for assistance with the +HYSPEC experiment. Work by B.A.F. and R.B. was sup- +ported by the U.S. Department of Energy, Office of Science, +Office of Basic Energy Sciences (DOE-BES) through Award +No. DE-SC0021134. 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Instrum. 93, 065103 (2022). + diff --git a/RNAyT4oBgHgl3EQfU_fg/content/tmp_files/load_file.txt b/RNAyT4oBgHgl3EQfU_fg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..78d0b8417d3b2b3392b4c979933fd59b0fa9b9f8 --- /dev/null +++ b/RNAyT4oBgHgl3EQfU_fg/content/tmp_files/load_file.txt @@ -0,0 +1,769 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf,len=768 +page_content='Magnetic PDF Data from Polarized Neutrons Magnetic Pair Distribution Function Data Using Polarized Neutrons and ad hoc Corrections Benjamin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Frandsen,1 Raju Baral,1 Barry Winn,2 and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Ovidiu Garlea2 1)Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, USA 2)Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (*Electronic mail: benfrandsen@byu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='edu) (Dated: 3 January 2023) We report the first example of magnetic pair distribution function (mPDF) data obtained through use of neutron po- larization analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Using the antiferromagnetic semiconductor MnTe as a test case, we present high-quality mPDF data collected on the HYSPEC instrument at the Spallation Neutron Source using longitudinal polarization analysis to isolate the magnetic scattering cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Clean mPDF patterns are obtained for MnTe in both the magnetically ordered state and the correlated paramagnet state, where only short-range magnetic order is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We also demon- strate significant improvement in the quality of high-resolution mPDF data through application of ad hoc corrections that require only minimal human input, minimizing potential sources of error in the data processing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We briefly discuss the current limitations and future outlook of mPDF analysis using polarized neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Overall, this work provides a useful benchmark for mPDF analysis using polarized neutrons and provides an encouraging picture of the potential for routine collection of high-quality mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' INTRODUCTION Magnetic pair distribution function (mPDF) analysis of neutron total scattering data has recently emerged as a valu- able tool for investigating local magnetic correlations in mag- netic materials1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In analogy to the more familiar atomic pair distribution function (PDF) method3, the mPDF is ob- tained by Fourier transforming the total magnetic scattering, which refers to the scattering arising both from long-range magnetic correlations (resulting in magnetic Bragg peaks) and from short-range magnetic correlations (resulting in dif- fuse magnetic scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This yields the real-space, pair- wise magnetic correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The mPDF technique is most useful for the study of short-range magnetic correlations such as those in a correlated paramagnet or a quantum disor- dered magnet, for which the real-space mPDF can be easier to interpret and model than the corresponding diffuse scatter- ing pattern in reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' On the other hand, conven- tional magnetic Bragg diffraction analysis will typically re- main the preferred choice for the determination of long-range ordered magnetic crystalline states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Since the introduction of the mPDF technique in 2014, it has been applied to numerous systems with short-range magnetic correlations ranging from quantum magnets to functional magnetic materials4–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' For an isotropic powder sample of a typical magnetic mate- rial possessing localized spins that belong to a single magnetic species, the mPDF is given by1,7 Gmag(r) = 2 π � ∞ Qmin Q � (dσ/dΩ)mag 2 3NsS(S+1)(γr0)2[f(Q)]2 −1 � sin(Qr)dQ (1) = 3 2S(S+1) � 1 Ns ∑ i̸=j � Aij r δ(r −rij)+Bij r r3 ij Θ(rij −r) � −4πrρ0 2 3m2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (2) The first equation defines the experimental mPDF, while the second equation shows how to calculate the mPDF for a given magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Here, Q is the magnitude of the scatter- ing vector, Qmin is the minimum measured scattering vec- tor (assumed to exclude the small-angle scattering regime), (dσ/dΩ)mag is the magnetic differential scattering cross sec- tion, r is real-space distance, r0 = µ0 4π e2 me is the classical elec- tron radius, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='913 is the neutron magnetic moment in units of nuclear magnetons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' S is the spin quantum number in units of ¯h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' f(Q) is the magnetic form factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Ns is the number of spins in the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' i and j label individual spins Si and Sj sep- arated by the distance rij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Aij = ⟨Sy i Sy j⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Bij = 2⟨Sx i Sx j⟩−⟨Sy i Sy j⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Θ is the Heaviside step function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' m is the average magnetic moment in Bohr magnetons (which is zero for anything with no net magnetization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' such as antiferromagnets),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' and ρ0 is the number of spins per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A local coordinate system is used for each spin pair to compute Aij and Bij, given by ˆx = rj−ri |rj−ri| and ˆy = Si− ˆx(Si· ˆx) |Si− ˆx(Si· ˆx)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The generalization arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='00137v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='mtrl-sci] 31 Dec 2022 Magnetic PDF Data from Polarized Neutrons 2 to magnetic materials with multiple types of magnetic atoms and/or nonzero orbital contributions to the magnetic moment is straightforward and given elsewhere1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Generally speaking, a positive (negative) peak at a given distance corresponds to net parallel (antiparallel) orientation between spins separated by that distance, lending an intuitive interpretation to mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1, the subtraction of unity from the term in parentheses amounts to the removal of the self- scattering contribution to the magnetic scattering cross sec- tion, which is desirable because the self-scattering contains no information about the correlations between distinct mag- netic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1 is fully equivalent to the corresponding integral expression used in the definition of the atomic PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' One of the biggest experimental challenges for obtaining high-quality mPDF data is accurately measuring the magnetic scattering to a sufficiently large momentum transfer and sep- arating it from the (typically much larger) nuclear scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' For a typical experiment using unpolarized neutrons, this can be done either by fitting a structural model to the nuclear Bragg peaks and subtracting them out to leave just the mag- netic scattering, or by subtracting a reference measurement taken at a temperature where no coherent magnetic scatter- ing is present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' at high temperature well above a mag- netic transition) from a diffraction pattern collected at a lower temperature with nonzero magnetic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' If the atomic structure is identical at both temperatures, then the difference between the scattering patterns gives the magnetic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' However, both of these strategies can result in significant ar- tifacts due to imperfect modeling of the nuclear Bragg peaks (or nuclear diffuse scattering if short-range structural correla- tions exist) or imperfect temperature subtraction, where even simple thermal expansion causes shifts in the nuclear Bragg peak positions that can be difficult to treat accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' An- other source of error in this strategy is the over-subtraction of magnetic scattering which could still be present in the form of incoherent paramagnetic scattering that follows a form factor decay as a function of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' An alternative approach is to make no attempt to separate the magnetic scattering from the nuclear scattering and in- stead simply generate the PDF from the combined diffraction signal following standard protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The atomic and mag- netic structures can then be modeled together in real space2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' However, this only works if the total scattering data are suit- able for conventional PDF analysis, meaning that this ap- proach is typically suitable only for data collected on dedi- cated PDF diffractometers with access to large values of mo- mentum transfer (typically more than 20 Å−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Even then, the mPDF signal can often be one or more orders of magni- tude smaller than the nuclear PDF signal, making it difficult detect above the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Furthermore, this approach treats mag- netic scattering the same as nuclear scattering in the PDF data processing protocol, meaning that no attempt is made to nor- malize the magnetic scattering by the squared magnetic form factor, as should normally be done according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This has the effect of twice convoluting Gmag(r) with the Fourier transform of the magnetic form factor, broadening the mPDF by approximately √ 2 times the real-space width of the elec- tronic wavefunction giving rise to the magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This “non-deconvoluted mPDF” (previously called the “unnormal- ized mPDF”2) takes the form2 dmag(r) = 2 π � ∞ Qmin Q �dσ dΩ � mag sin(Qr)dQ (3) = C1 ×Gmag(r)∗S(r)+C2 × dS dr , (4) where C1 = Ns 2π � γr0 2 �2 2 3⟨g � J(J +1)⟩2 and C2 = Ns 2π � γr0 2 �2 2 3⟨g2J(J + 1)⟩ are constants, ∗ represents the convolution operation, and S(r) = F { fm(Q)} ∗ F { fm(Q)} (where F denotes the Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The second term on the right of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 4, which arises from the self-scattering contribution to the magnetic differential scattering cross section, results in a peak at low r (below approximately 1 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The strategy of co-modeling the atomic and magnetic PDF data in real space requires working with this lower-resolution, non-deconvoluted mPDF signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' To keep the distinction clear, we will refer to dmag(r) as the non-deconvoluted mPDF and Gmag(r) as the proper mPDF, deconvoluted mPDF, or just the mPDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Another challenge exists even in cases where the magnetic scattering has been cleanly separated from the nuclear scat- tering: accurately normalizing the magnetic scattering by the squared magnetic form factor prior to performing the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Because the squared magnetic form factor for most magnetic ions approaches zero between 5 and 8 Å−1, any noise or leftover nuclear scattering in this range will be greatly amplified during the normalization step, inevitably leading to high-frequency artifacts in the mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Often, these artifacts are sufficiently severe that it is preferable just to skip the normalization step altogether and settle for the non- deconvoluted mPDF2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' however, this comes with the cost of significantly reduced real-space resolution, so it is also not ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Many useful mPDF studies have been carried out in spite of these experimental challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Nevertheless, the technique will have a greater impact if the data quality improves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Here, we report two developments in mPDF data collection and pro- cessing that help address the dual challenges of accurately isolating the magnetic scattering and reliably mitigating the artifacts introduced during the magnetic form factor normal- ization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The first is using longitudinal polarization anal- ysis with a polarized neutron beam to separate the magnetic, nuclear, and nuclear spin-incoherent cross sections directly15, with no need for temperature subtraction or modeling of nu- clear Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The second is using a polynomial func- tion to perform ad hoc data corrections that minimize the real-space artifacts resulting from the form factor normaliza- tion16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' These are both well-established methods, but they are applied here to mPDF analysis for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In the fol- lowing, we briefly describe longitudinal polarization analysis and the procedure for performing the ad hoc data corrections, and then provide examples of their application to mPDF data on the antiferromagnetic semiconductor manganese telluride (MnTe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Code for performing the data corrections is freely available as part of the diffpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='mpdf python package18 and in the Supplementary Information of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' These devel- Magnetic PDF Data from Polarized Neutrons 3 opments represent significant progress toward routine collec- tion of high-resolution mPDF data on a variety of magnetic materials of fundamental and technological interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' LONGITUDINAL POLARIZATION ANALYSIS The method of XYZ longitudinal polarization analysis of neutron scattering data collected with large solid-angle detec- tor banks (also called 6-pt polarization analysis, or 10-pt po- larization analysis in its more general form including out-of- plane scattering) allows unambiguous separation of nuclear, magnetic, and nuclear spin-incoherent scattering cross sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This is accomplished by measuring the spin-flip and non-spin-flip cross sections for various directions of polariza- tion of the incident neutron beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' detailed descriptions of the technique are found elsewhere15,19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A small handful of neu- tron diffractometers and spectrometers have the capability to perform longitudinal polarization analysis, such as D7 at the Institut Laue Langevin19, DNS at FRM-II in Munich21, and HYSPEC at the Spallation Neutron Source (SNS) and Oak Ridge National Laboratory (ORNL)22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The former two in- struments can access a maximum momentum transfer of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='84 Å−1, respectively, while HYSPEC can reach just over 6 Å−1 for certain choices of incident energy and detec- tor positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Given the importance of maximizing the range of accessible momentum transfer for the Fourier transform, HYSPEC is currently the instrument with polarization analy- sis that is most suitable for mPDF analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' AD HOC DATA CORRECTIONS In any attempt to isolate the magnetic scattering inten- sity, normalize it by the squared magnetic form factor, and perform the Fourier transform to generate the experimental mPDF, errors will inevitably be introduced due to imperfect time-independent instrumental background corrections, inter- ference from nuclear and nuclear spin incoherent scattering, inaccuracies in the measured or approximated magnetic form factor, statistical noise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Because typical squared magnetic form factors approach zero between 5 and 8 Å−1, any errors present in this range of the data become particularly magnified if included in the Fourier transform, introducing nonphysi- cal artifacts into the mPDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' These challenges are similar to those encountered when generating x-ray PDF data, where the x-ray scattering atomic form factor plays an analogous role as the magnetic form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The traditional approach in x-ray PDF analysis has been to apply corrections for each source of error manually3, but this can be an onerous task and is itself prone to errors16, particularly for researchers new to the technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' As has been discussed in detail for x-ray PDF16,17, ad hoc corrections using a polynomial correction function can largely mitigate the errors with minimal human input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This ad hoc approach is also highly automatable, allow- ing for rapid throughput PDF data processing using tools such as PDFgetX317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Here, we have implemented the PDFgetX3 algorithm in a custom-built python program and applied it to magnetic scattering data to obtain high-resolution mPDF data with no manual corrections necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The python code has been made available as part of the diffpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='mpdf package18 and as a standalone program in the Supplementary Informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' One technical note merits discussion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' As explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 17, the polynomial correction function results in non- physical contributions to the real-space PDF for r values less than rpoly = πn/Qmaxinst, where n is the degree of the poly- nomial and Qmaxinst is the maximum value of Q to which the correction polynomial is fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' As long as rpoly is shorter than the nearest-neighbor distance, then the artifacts resulting from the ad hoc corrections will be restricted to the r-region below the first peak in the PDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In practice, an appropriate value of rpoly is selected (typically below 1 Å for x-ray PDF data), and a weighted average of the PDFs obtained using the nearest integer values of n corresponding to the selected rpoly value is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' For synchrotron x-ray PDF experiments with Qmaxinst values of 25 Å−1 or greater, typical values for n range from 7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' When applied to mPDF, Qmaxinst will often be much smaller, anywhere from 5 to 8 Å−1, so n should likewise be smaller to keep rpoly acceptably short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' For HYSPEC, where the largest accessible value for Qmaxinst is about 6 Å−1, typi- cal values of n are 3 and 4, resulting in rpoly values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='57 Å and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='09 Å respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Although these values would be too large for typical atomic PDF data, they are acceptable for typ- ical magnetic structures, where the nearest-neighbor distance between magnetic atoms is unlikely to be as short as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='09 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' APPLICATION TO MANGANESE TELLURIDE We demonstrate the use of polarized neutrons and the ad hoc data correction procedure to generate mPDF data for the antiferromagnetic semiconductor MnTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This material has a hexagonal crystal structure (space group P63/mmc) with a magnetic ordering temperature of TN = 307 K, below which the Mn2+ spins order with parallel alignment within the ab planes and antiparallel alignment between neighboring ab planes, as has been established by previous neutron diffrac- tion studies23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Short-range antiferromagnetic correlations are known to survive well into the paramagnetic phase above TN, likely up to 900 K or higher26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The short-range mag- netism in MnTe was the subject of a recent mPDF study14 using data collected with unpolarized neutrons, together with three-dimensional 3D-∆mPDF27 data collected from a single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Experimental Details A large powder sample of MnTe (∼8 g) was synthesized according to the procedure in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 14, pressed into a pellet of diameter 6 mm, and loaded into an aluminum sample can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A closed cycle refrigerator was used to control the temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The incident neutron energy used for the experiment was Ei = 28 meV, and the Fermi chopper was operated at Magnetic PDF Data from Polarized Neutrons 4 a frequency of 60 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This configuration provided an en- ergy resolution at the elastic position of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='7 meV full width at half maximum (FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The data were integrated over en- ergy transfers between −5 and 5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The incident beam at HYSPEC is polarized by a vertically focusing array of mag- netically saturated Heusler (Cu2MnAl) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A Mezei flip- per was used to reverse the polarization direction of the inci- dent beam, and the polarization analysis of the scattered beam was accomplished with a 60◦-wide-angle multi-channel ra- dial supermirror-polarizer array22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Spin-flip (SF) non-spin- flip (NSF) data were collected with the neutron polarization oriented vertically, Pz, for two different positions of the the 60◦ detector bank to cover the 2θ scattering range 3 - 112◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In this configuration, the SF setting measures the magnetic scat- tering, and the NSF gives the combined nuclear and magnetic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The nuclear spin-incoherent scattering contri- bution from MnTe is expected to be insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A flipping ratio of 13 was determined by evaluating the scattering from nuclear Bragg peaks in SF and NSF configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Post- processing data corrections for the angle-dependent supermir- ror transmission and for the flipping ratio efficiency were car- ried out using the algorithms implemented in MANTID soft- ware, as discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 22 and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The magnetic scattering used in this study corresponds to the SF intensity corrected for the flipping ratio according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Magnetic Scattering In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1, we display the magnetic scattering at 50 K (a) and 330 K (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Sharp antiferromagnetic Bragg peaks are present at 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' At 330 K, the magnetic scattering is diffuse but still quite structured, indicating the persistence of short-range cor- relations into the paramagnetic phase, as expected from earlier results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The nuclear scattering cross section at 50 K, obtained by subtracting the SF scattering from the NSF scattering, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A slight oversubtraction of the strong mag- netic scattering is visible around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='9 Å−1, but we see minimal interference effects elsewhere, so we conclude that the separa- tion of the different cross sections is sufficiently accurate for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' mPDF Data and Fits Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 2 shows key results from the ad hoc data correction process to produce the mPDF from the scattering data col- lected at 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In panel (a), we display the magnetic scattering data and the squared magnetic form factor for Mn2+, scaled to match the scattering profile between 3 and 6 Å−1 as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We used the analytical approximation of the form factor as reported in the International Tables for Crystallog- raphy29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Panel (b) shows the uncorrected form of the reduced magnetic structure function Fmag(Q) (not to be confused with the magnetic structure factor used in conventional magnetic 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 a Magnetic 50 K 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='3 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) b Magnetic 330 K 0 1 2 3 4 5 6 Q (Å 1) 0 1 2 3 4 c Nuclear 50 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Magnetic scattering intensity for MnTe at (a) 50 K and (b) 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Sharp Bragg peaks are present at 50 K due to the long-range ordered antiferromagnetic phase, whereas only diffuse peaks remain at 330 K in the correlated paramagnetic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (c) Nuclear scatter- ing at 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' diffraction analysis), given by Fmag,uc(Q) = Q � (dσ/dΩ)mag A[f(Q)]2 −1 � , (5) where A is the scaling constant used in panel (a), replacing the coefficient on [f(Q)]2 shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1 that would be appropri- ate if the data were on an absolute scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The “uc” in the subscript of the vertical axis label stands for “uncorrected”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Note that the increasing behavior of Fmag(Q) at high Q is non- physical, indicating an imperfect normalization by the squared magnetic form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Indeed, the correct form of Fmag(Q) should oscillate around zero at high Q, since the correct nor- malization of the magnetic scattering in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 5 puts it on the scale of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' To correct this error, we used least-squares min- imization to fit a polynomial to Fmag,uc, in accordance with the Magnetic PDF Data from Polarized Neutrons 5 a b c d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Procedure for obtaining the mPDF from magnetic scattering data from MnTe at 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (a) Magnetic scattering intensity (black symbols) overlaid by the square of the magnetic form factor of Mn2+ (orange curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' scaled to match the magnitude of the data at high Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (b) Uncorrected reduced magnetic structure function Fmag,uc (black symbols) with the best-fit degree-3 polynomial correction function (orange curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (c) Corrected Fmag,c after subtracting the polynomial fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (d) Resulting mPDF using Qmax = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' ad hoc approach used for x-ray PDF in PDFgetX317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 2(b) shows the best-fit cubic polynomial as the orange curve [note that this is not the squared magnetic form factor, which is in- stead displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' With a polynomial of degree 3 and Qmaxinst = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1, the corresponding value of rpoly is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='57 Å, safely below the nearest-neighbor Mn-Mn distance of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='37 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Due to the relatively low degree of the polynomial, it effectively captures the slowly modulating, nonphysical be- havior of Fmag,uc without fitting to the physically meaningful features in the data, which have characteristically faster mod- ulations in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This polynomial is sometimes referred to as a background function, which is accurate in the sense that it captures the slowly modulating errors in Fmag,uc due to imper- fect normalization by the squared magnetic form factor, but it should not be interpreted as an attempt to correct for the background scattering from the instrumental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We as- sume the instrumental background has already been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In panel (c), we display the corrected form of Fmag(Q), la- beled Fmag,c, resulting from subtracting the fitted polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We note that Fmag,c now oscillates around zero as Q becomes large, as expected when all errors have been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Finally, panel (d) shows the resulting mPDF using Qmax = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1, Qmaxinst = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1, and rpoly = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='57 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 3, we display fits to the mPDF data at 50 K us- ing the published magnetic structure of MnTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We plot the non-deconvoluted mPDF data (given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 3) and fit in panel (a) in blue symbols and a solid red curve, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The fit agrees closely with the data and is free from 0 5 10 15 20 25 1 0 1 2 3 dmag(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) MnTe 50 K a 0 5 10 15 20 25 r (Å) 2 1 0 1 Gmag(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Magnetic PDF data for MnTe at 50 K, including (a) the non- deconvoluted mPDF dmag(r) and (b) the proper mPDF Gmag(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The blue circles and red curve show the experimental data and best-fit mPDF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In panel (b), the vertical dashed line indicates the selected value of rpoly = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='55 Å, below which artifacts from the polynomial correction function are expected to be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' the high-frequency noise that accompanies non-deconvoluted mPDF data obtained together with standard atomic PDF data, highlighting the advantage of using polarized neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 3(b), we plot the deconvoluted mPDF data and fit, where the greatly improved real-space resolution is immediately ap- parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The fit successfully captures the most prominent fea- tures of the mPDF data, demonstrating the success of our ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' As fitting parameters, we included an arbitrary scale factor, a real-space damping term to account for the finite Q-space resolution of the data (known as Qdamp in conven- tional PDF analysis), and an r-dependent real-space broaden- ing term (Qbroad for conventional PDF) to account for statis- tical noise in the scattering data30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The fitted values of Qdamp and Qbroad were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='037 Å−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='68 Å−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We note the presence of artifacts for r ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å (corresponding closely to rpoly = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='57 Å) and minor misfits throughout the full data range, but considering that this experimental mPDF curve was generated without any human intervention or fine tuning (in contrast to earlier efforts2,31), the overall level of agreement is highly encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We further note that, ignor- ing any distortions or errors in the data, the mPDF pattern in Magnetic PDF Data from Polarized Neutrons 6 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 dmag(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) MnTe 330 K a 0 5 10 15 20 r (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 Gmag(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Magnetic PDF data for MnTe at 330 K, including (a) the non- deconvoluted mPDF dmag(r) and (b) the proper mPDF Gmag(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The blue circles and red curve show the experimental data and best-fit mPDF with an anisotropic correlation model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In panel (b), the vertical dashed line indicates the selected value of rpoly = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' the ordered state is equivalent to the mPDF that would be ob- tained if one were to perform a perfect nuclear and magnetic Rietveld refinement to an unpolarized neutron diffraction pat- tern over a sufficiently large Q-range and then apply the nor- malization and Fourier transform procedure to the magnetic part of the Rietveld fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In many cases, short-range magnetic correlations will give rise to the most physically interesting mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In that spirit, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 4 the mPDF data and fits for MnTe at 330 K, somewhat above TN = 307 K and in the “correlated paramagnet” regime where short-range correlations persist in the absence of long-range magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Once again, the non-deconvoluted and deconvoluted mPDF patterns are dis- played in panels (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Because the mag- netic scattering is significantly weaker in the paramagnetic phase, the Q-range of usable scattering data for the Fourier transform was more limited, so the data shown were gener- ated with Qmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1, together with rpoly = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å and Qmaxinst = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We also used a Fermi-Dirac modifica- tion function with a width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1 to generate the decon- voluted mPDF (see next section for details about the modi- fication function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The non-deconvoluted fit in panel (a) is reasonably good, although truncation effects are apparent in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The deconvoluted mPDF data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 4(b) exhibit somewhat larger artifacts that the ad hoc corrections could not remedy, but the near-neighbor antiferromagnetic correlations are clearly evident in the data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' as the alternating negative and positive peaks around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='4 Å (nearest neighbor, antipar- allel), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 Å (second nearest neighbor, parallel), and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='4 Å (third nearest neighbor, antiparallel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The mPDF fit captures the general features of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We attribute the reduced data quality in the correlated paramagnetic state to the more re- stricted Q-range and greater statistical noise in the scattering data compared to the data collected at 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This underscores the need for high-statistics measurements over a sufficient Q- range to produce optimal high-resolution mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Despite the more limited data quality, we find it encourag- ing that the fitted magnetic model produced from the prop- erly deconvoluted mPDF data agrees well with the model pro- duced by the fit to the non-deconvoluted data and to our ear- lier mPDF fits performed on a different instrument with unpo- larized neutrons14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In particular, the short-range correlations in MnTe were shown to be spatially anisotropic in our ear- lier study, with a longer correlation length along the crystallo- graphic c axis than within the hexagonal ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The present fits included two free parameters capturing this anisotropic ef- fect in the magnetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The fit naturally converged to correlation lengths of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='1(5) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='6(6) Å along c and in the ab plane, respectively, in good agreement to those reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 14 at a similar temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The consistency of these results validates the earlier report and the use of polarized neu- trons for obtaining useful mPDF data in a correlated param- agnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Effect of Modification Functions Finally, we discuss the effect of applying a Q-space mod- ification function32, also known as a window function, to Fmag,c(Q) prior to the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In the context of atomic PDF analysis, various modification functions have been used to suppress the effect of statistical or systematic er- rors in the scattering data at high Q, reducing high-frequency ripples in the PDF (but with the cost of reduced real-space resolution and possible loss of physically meaningful infor- mation)32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A modification function w(Q) operates simply through multiplication with the corrected form of Fmag(Q), such that w(Q) × Fmag(Q) is the quantity to be Fourier trans- formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We consider here three modification functions: the default step function ws(Q) = � 1 if Q ≤ Qmax 0 if Q > Qmax , (6) a modified Fermi-Dirac function wFD(Q) = � 2 e(Q−Qmax)/∆+1 −1 if Q ≤ Qmax 0 if Q > Qmax , (7) Magnetic PDF Data from Polarized Neutrons 7 and the conventional Lorch function33 wL(Q) = � Qmax πQ sin � πQ Qmax � if Q ≤ Qmax 0 if Q > Qmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (8) We note that ws is equivalent to applying no modification function at all, since it is unity up until the hard cutoff of the Fourier transform at Qmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' These three modification functions are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 5(a), together with the squared mag- netic form factor for Mn2+ for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In all cases, we use Qmax = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1, and for wFD, we set ∆ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The mPDF curves at 50 K generated with the application of each of these modification functions are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Both wFD and wL suppress high-frequency features in the mPDF relative to the default mPDF using ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The effect is particu- larly pronounced for wL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' However, we note that for the avail- able Q-range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1 included in the Fourier transform, wL results in a significant suppression of Fmag(Q) over nearly the entire range similar in scale to the squared magnetic form factor shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In effect, then, application of wL undoes the normalization by the squared form factor, yield- ing a quantity that is similar in real-space resolution to the non-deconvoluted mPDF that we were originally hoping to improve upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We therefore suggest that a Lorch window is of limited usefulness for producing deconvoluted mPDF pat- terns when the available Q-range does not significantly exceed the extent of the squared magnetic form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' On the other hand, wFD with a modestly chosen value of ∆ may provide value by suppressing artifacts from high-Q noise without sac- rificing meaningful mPDF data due to excessive broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 5(c) shows the mPDF curves at 330 K obtained for each type of modification function, this time with Qmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1 and ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The Lorch and Fermi-Dirac functions again result in greatly suppressed Fourier ripples, with the Fermi- Dirac function preserving somewhat more real-space resolu- tion than the Lorch function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In the mPDF fits presented in the previous section, we found no need to use a modification function for the data col- lected at 50 K where the magnetic scattering is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' How- ever, given the weaker magnetic scattering at 330 K, we found it beneficial to apply a Fermi-Dirac modification function with a width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We chose this width because, after test- ing several different values, we found that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1 offered a good balance between minimizing artifacts in the mPDF data and preserving the real-space resolution to a reasonable de- gree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Ideally, the calculated mPDF used for modeling the data should be convoluted with the Fourier transform of the mod- ification function to account correctly for the change in real- space resolution caused by the modification function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION We have reported the first use of polarized neutrons to gen- erate mPDF data and the first application of PDFgetX3-style ad hoc corrections to produce high-resolution, properly de- convoluted Gmag(r) curves with minimal human input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Po- larization analysis allowed us to isolate the magnetic scat- tering cross section and produce the resulting mPDF data 0 2 4 6 Q (Å 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Window Amplitude a Step Fermi-Dirac Lorch Mn2 + 0 5 10 15 20 r (Å) 2 1 0 1 Gmag(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) b MnTe 50 K 0 5 10 15 20 r (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='4 Gmag(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' units) c MnTe 330 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (a) modification functions as defined in the main text with Qmax = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The Fermi-Dirac modification function uses ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The squared magnetic form factor of Mn2+ is also shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (b) The mPDF patterns for MnTe at 50 K resulting from application of the modification functions in (a), using the same legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' (c) Same as (b), but for 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' The transform and mask parameters are Qmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1 and ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='0 Å−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Magnetic PDF Data from Polarized Neutrons 8 with little to no interference from nuclear scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' On HYSPEC, we achieved a Qmax value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1 in the mag- netically ordered state of MnTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This was sufficient for an ex- ceptionally clean non-deconvoluted mPDF curve and a high- quality proper mPDF curve produced with the help of the ad hoc corrections, demonstrating the promise of this method for producing mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' In the paramagnetic state, we obtained good-quality non-deconvoluted and deconvoluted mPDF curves suitable for quantitative mPDF refinements that confirmed a prior result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' However, the weaker magnetic scat- tering at 330 K necessitated a more limited Q-range and in- troduced significantly more noise and artifacts into the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' These issues could be ameliorated in large part simply by col- lecting more data to improve the statistics over a large enough range of momentum transfer (ideally up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='5 Å−1 or greater).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Finally, our results suggest that if a modification function is to be used to reduce high-frequency noise in the mPDF data, a modified Fermi Dirac function or similar is preferred over the traditional Lorch function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Overall, the results reported here represent a significant step forward in the development of mPDF methodologies by highlighting the promise of polarized neutrons and mod- ern data processing techniques to produce high-quality, high- resoultion mPDF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We have provided a realistic view of both the capabilities and the limitations posed by current ex- perimental resources when generating high-resolution mPDF data from weak and diffuse magnetic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' With ma- jor developments in neutron capabilities on the near horizon at the European Spallation Source, the Second Target Station (STS) of the Spallation Neutron Source, and elsewhere, in- strumental limitations are expected to be less of a concern in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' We mention particularly the planned STS in- strument VERDI34, for which the combination of high beam intensity, large Q coverage, and polarization analysis should enable routine collection of high-resolution mPDF data of ex- cellent quality, which we expect to transform the landscape of mPDF studies in condensed matter and materials physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL The supplemental materials include the magnetic scattering cross section at 50 K and 330 K, the nuclear scattering cross section at 50 K and 330 K, python code for generating the mPDF with the ad hoc corrections, and an example python script showing how to use the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Melissa Graves-Brook for assistance with the HYSPEC experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Work by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' was sup- ported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Basic Energy Sciences (DOE-BES) through Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' DE-SC0021134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' This study used resources at the Spal- lation Neutron Source (SNS), a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The data that support the findings of this study are available within the article and its supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' AUTHOR DECLARATIONS The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Frandsen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Yang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQfU_fg/content/2301.00137v1.pdf'} +page_content=' Billinge, “Magnetic pair distribution 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+We use the probabilistic iteration to determine sufficient conditions +for maximizing the weighted domination probability. In the second +part, we study a weighted generalization of the chromatic number +and estimate the minimum number of colours needed to satisfy the +constraints when the weights themselves are random. We show that +the “extra” cost incurred for weighted colouring is small if the weights +have sufficiently large moments. We also consider inhomogenous ran- +dom graphs where the edge probabilities are not necessarily all same +and obtain bounds for the chromatic number in terms of its averaged +edge probabilities. +Key words: Weighted Domination; Maximum Domination Proba- +bility; Weighted Colouring; Inhomogenous Random graphs, +AMS 2020 Subject Classification: Primary: 60C05. +1 +Introduction +In this section, we describe briefly the two problems studied in this paper. +∗E-Mail: gganesan82@gmail.com +1 + +Weighted Domination +Records in a sequence of independent and identically distributed (i.i.d.) ran- +dom variables have been extensively studied in the context of statistical es- +timation. The probability that the kth random variable is a record (as com- +pared to the previous k − 1 values) grows inversely in the index k and the +corresponding record events are mutually independent [15]. Various aspects +of record properties with differing assumptions have been studied since and +as examples, [4] study large deviation properties for weak records and [11] +studies characterizations of geometric distributions based on kth record val- +ues. Also, [7] study records in a sequence of maximally dependent random +variables and obtain expressions for corresponding record moments and dis- +tributions. +In this paper, we study records from a graph theoretic view point and use +probabilistic iteration to estimate the probability that a given set of vertices +form a weighted dominating set. +Weighted Colouring +The chromatic number χ(G) of a graph G is the smallest number of colours +needed for a proper colouring of G and many upper and lower bounds +for χ(G) exist in terms of various graph parameters like maximum vertex +degree, independence number, clique number etc. (see Chapter 5, [19]). For +homogenous random graphs where each edge is present with the same prob- +ability pn, concentration of the chromatic number around its mean has been +well-studied [18, 6, 17]. Many bounds for the expected chromatic number +have also been derived using clique sizes in the complement graph and a com- +bination of second moment and martingale methods [5, 2]. The paper [16] +uses counting arguments to obtain sharp lower bounds on the chromatic +number. In a related direction, [3] obtain two point concentration of the +chromatic number for edge probability pn = n−1/2−δ with δ > 0. The anal- +ysis proceeds through k−choosable graphs. Extending this to pn = +d +n, [1] +use analytical techniques to obtain the two possible values of the chromatic +number for d > 0 a constant. +Recently, in [10] we have studied the chromatic number of homogenous +random graphs whose edge probabilities do not necessarily form a convergent +sequence. In the first part of this paper, we obtain estimates for the chromatic +number of an inhomogenous graph in terms of averaged edge probabilities. +2 + +We use an auxiliary bound for chromatic number of a deterministic graph, +in terms of its maximum averaged degree, that is of independent interest. +In the second part of this paper, we study weighted colouring in ran- +dom graphs. +Generalizing radio labelling of graphs [8, 12] which we call +as weighted colouring number, we use the probabilistic method to estimate +the weighted colouring number when the edge weights themselves are ran- +dom. Such situations arise frequently in applications and we show that if the +weights have sufficiently large moments, then the “cost” incurred due to the +unboundedness of weights is small. +The paper is organized as follows: In the following section, we state and +prove our first main result regarding maximizing the domination probability +in weighted random graphs. Next, in Section 3 we state and prove our two +main results regarding weighted and constrained colouring in random graphs +and finally, in Section 4, we collect the auxiliary combinatorial lemmas used +in the proof of the main theorems. +2 +Weighted Domination +In this section, we study domination in weighted random graphs obtained as +follows. Let Kn be the complete graph on n vertices and equip vertex v with a +random weight Sv. The random variables {Sv}1≤v≤n are i.i.d. continuous with +a common cumulative distribution function (cdf) F. For each vertex v, we now +associate a deterministic domination neighbourhood D(v) ⊂ {1, 2, . . . , n} \ +{v} and say that v is a weighted dominating vertex if Sv > maxu∈D(v) Su. +Similarly, we say that a finite set of vertices V := {v1, . . . , vk} is a weighted +dominating set if each vertex in V is a weighted dominating vertex. +Let Edom = Edom(V) be the event that V is a weighted dominating set +and also let Rtot := � +1≤l≤k 11(Avl) be the total number of weighted dominat- +ing vertices in V. For any vertex v, we have by symmetry that P(Av) = +1 +d(v), +where d(v) = 1 + #D(v) and #D(v) is the size of the domination neighbour- +hood of v. Therefore if the vertices in V have disjoint domination neighbour- +hoods, then the events Av, v ∈ V are independent and so +P(Edom) = +k +� +l=1 +1 +d(vl) =: pdom. +In general, if the domination neighbourhoods are not disjoint, the events Av, v ∈ +V are correlated and so we expect that Edom occurs with lesser probability, as +3 + +described in the following result. Recalling that V = {v1, . . . , vk} with v1 < +v2 < . . . < vk, we define C(vl) := �l +j=1 D(vj) and set c(vl) := 1 + #C(vl) +for 1 ≤ l ≤ k. +Theorem 1. If +� x +0 F k−1(y)dF(y) = F k(x) +k +for each integer k ≥ 1 and x > 0 +and +vk /∈ C(vk) and vl ∈ D(vl+1) \ C(vl) for each 1 ≤ l ≤ k − 1, +(2.1) +then +P (Edom) = +� +1≤l≤k +1 +c(vl) ≤ pdom and var(Rtot) ≤ ERtot = +k +� +l=1 +1 +d(vl). +(2.2) +Moreover, the events {Avl}1≤l≤k are mutually independent if and only if +D(vl) ⊂ D(vl+1) strictly, for each 1 ≤ l ≤ k − 1. +In words, the above result says that if the domination neighbourhoods +satisfy the “weak-nested” property (2.1), then pdom is the maximum possible +weighted domination probability and moreover, this value is achieved only if +the neighbourhoods satisfy a strong-nested property. +From a statistical view point, we could also interpret Theorem 1 as a +“graph-theoretic” version of records. For context, we recall that “time-based” +record events in a sequence of i.i.d. random variables are known to be mu- +tually independent when the comparison set consists of the entire past [15]. +From Theorem 1, we see that graph-theoretic record events are negatively +correlated for weakly nested neighbourhoods and are mutually independent +for strongly nested neighbourhoods. +Proof of Theorem 1: Let Bk := �k +j=1 Avj and for x > 0 define +Avk(x) := Avk +� +{Svk < x} = +� +max +j∈D(vk) Sj < Svk < x +� +(2.3) +and Bk(x) := Bk−1 +� Avk(x), with the notation that the maximum of the +empty set is zero and B0 = Ω. We first show by induction that for every x > 0, +P(Bk(x)) = +F c(vk)(x) +c(v1) · · ·c(vk) = P(Bk) · F c(vk)(x). +(2.4) +4 + +For the basis step, we see that the random variable X with cdf F is continuous +and so F(x) = P(X ≤ x) = P(X < x). Conditioning on Sv1 = y, we therefore +get that P (B1(x)) = +� x +0 P +� +maxj∈D(v1) Sj < y +� +dF(y) equals +� x +0 +P + + � +j∈D(v1) +{Sj < y} + + dF(y) = +� x +0 +F d(v1)−1(y)dF(y) = F d(v1)(x) +d(v1) +. (2.5) +Since d(v1) = c(v1) this proves the basis step. +To prove the induction step, we now assume that the relation (2.4) is +true for the event Bk−1(x) and consider the event Bk(x). If Bk−1 = �k−1 +j=1 Avj +occurs, then using the weak nested property in the statement of the theorem, +we already have Svk−1 > maxl∈C(vk−1) Svl and so Svk is a record if and only +if Svk > maxj∈Ek Sj and Svk > Svk−1, where Ek := D(vk) \ (C(vk−1) ∪ vk−1) . +Letting 11(.) denote the indicator function, we then have that +11 (Bk(x)) += +11 +� +Avk(x) +� +Bk−1 +� += +11 +� +x > Svk > max +j∈Ek Sj +� +11(x > Svk > Svk−1)11(Bk−1) (2.6) +and since the event Bk−1 depends only on the values of {Sj}j∈C(vk−1)∪{vk−1}, +we condition on Svk = z and get from (2.6) that P (Bk(x)) = +� x +z=0 tk(z)dF(z) +where +tk(z) +:= +P +� +max +j∈Ek Sj < z +� +P +� +Bk−1 +� � +Svk−1 < z +�� += +� +j∈Ek +P(Sj < z)P +� +Bk−1 +� � +Svk−1 < z +�� +. +(2.7) +To evaluate (2.7), we use the weak nested property vk−1 ∈ D(vk)\C(vk−1) +to get that Ek = D(vk) \ (C(vk−1) ∪ vk−1) has cardinality c(vk) − c(vk−1) − 1. +Consequently +tk(z) += +F c(vk)−c(vk−1)−1(z)P +� +Bk−1 +� � +Svk−1 < z +�� += +F c(vk)−c(vk−1)−1(z)P (Bk−1(z)) +and substituting this into the expression for P(Bk(x)) determined prior to (2.7), +we get +P (Bk(x)) = +� x +0 +F c(vk)−c(vk−1)−1(z)P (Bk−1(z)) dF(z). +5 + +By induction assumption we have P (Bk−1(z)) = +F c(vk−1)(z) +c(v1)·c(v2)···c(vk−1) and so +P (Bk(x)) += +1 +c(v1) · c(v2) · · ·c(vk−1) +� x +0 +F c(vk)−1(z)dF(z) += +1 +c(v1) · c(v2) · · ·c(vk)F c(vk)(x). +(2.8) +This proves the induction step and therefore completes the proof of (2.4). +The first relation in (2.2) follows directly from (2.4) by setting x = ∞. +Moreover, var(Rtot) = V1 + 2V2 where +V1 = +k +� +l=1 +P(Avl) − (P(Avl))2 = +k +� +l=1 +� +1 +d(vl) − +1 +d2(vl) +� +≤ +k +� +l=1 +1 +d(vl) = ERtot +and V2 = � +1≤l1 0 such that +P +�C1n1−β +log n +≤ χ(G) ≤ C2n1−β +log n +� += 1 − o(1), +(3.2) +where o(1) −→ 0 as n → ∞. +The following result obtains upper and lower bounds for the chromatic +number of an inhomogenous random graph G, in terms of “averaged” edge +probabilities. +Theorem 2. Suppose there are constants 0 ≤ βup ≤ βlow ≤ α and positive +constants C1, C2 such that +C1 +nβlow ≤ 1 +�l +2 +� +� +u,v∈V +p(u, v) ≤ C2 +nβup +(3.3) +7 + +for any set V ⊆ V containing l ≥ nα(log n)3 vertices. There are positive +constants Di, 1 ≤ i ≤ 4 such that +P +� +χ(G) ≥ +n1−α +(log n)3 +� +≥ 1 − exp +� +−D2 · n2α−βlow · (log n)6� +(3.4) +and +P +� +χ(G) ≤ D3nmax(α,1−βup) · (log n)4� +≥ 1 − exp +� +−D4n2α−βlow · (log n)5� +(3.5) +for all n large. +We prove the lower bound in (3.5) by obtaining an upper bound for the +maximum size of a stable set and prove the upper bound in (3.5) using a +maximum average degree estimate for the chromatic number, obtained in +Lemma 4 of Section 4. We provide the details below. Throughout, we use +the following standard Deviation Estimate. Let Zi, 1 ≤ i ≤ t be independent +Bernoulli random variables satisfying +P(Zi = 1) = pi = 1 − P(Zi = 0). +If Wt = �t +i=1 Zi and µt = EWt, then for any 0 < ǫ < 1 +2 we have that +P (|Wt − µt| ≥ ǫµt) ≤ 2 exp +� +−ǫ2 +4 µt +� +. +(3.6) +For a proof of (3.6), we refer to Corollary A.1.14, pp. 312, Alon and Spencer +(2008). +Proof of (3.4) in Theorem 2: For a set of vertices V of size #V = l, let +pav(V) := 1 +�l +2 +� +� +u,v∈V +p(u, v) +(3.7) +be the average edge probability of edges formed by the vertices of V. For a +constant 0 < α < 1 set +plow := min +V pav(V) ≤ max +V +pav(V) =: pup +(3.8) +where the minimum and maximum are taken over all sets V of cardinality l +satisfying nα · (log n)3 ≤ l ≤ n. +8 + +If α(G) is the largest size of a stable set in the random graph G, then we +know that (Proposition 5.1.7, pp. 193, West (2000)) +χ(G) ≥ +n +α(G). +(3.9) +To estimate the probability of the event {α(G) ≥ t}, we let V be any set of t +vertices and let F(V) be the event that V is a stable set. We have that +P (F(V)) = +� +u̸=v∈V +(1 − p(u, v)) ≤ e−(t +2)·pav(V), +where pav(V) is as defined in (3.7). Using (3.3), we get that +P (F(V)) ≤ +� +− +�t +2 +� +· C1 +nβlow +� +≤ e−D1t2·n−βlow +for some constant D1 > 0. Therefore letting Ftot := � +V F(V) where the union +is over all sets of size t, we get by the union bound that +P (Ftot) ≤ +�n +t +� +· e−D1t2·n−βlow ≤ nt · e−D1t2·n−βlow. +(3.10) +For t = nα · (log n)3, we have that +nt · e−D1t2·n−βlow += +exp +� +−t log n +� +D1nα−βlow(log n)2 − 1 +�� +≤ +exp +� +−2D2 · tnα−βlow · (log n)3� += +exp +� +−2D2 · n2α−βlow · (log n)6� +, +(3.11) +for some constant D2 > 0. +Substituting (3.11) into (3.10) we get that +P(Ftot) ≤ exp +� +−2D2 · n2α−βlow · (log n)6� +for all n large. If the complement event F c +tot occurs, then the largest size of +the stable set α(H) ≤ nα · (log n)3. Plugging this into (3.9), we get that +P +� +χ(H) ≥ +n1−α +(log n)3 +� +≥ 1 − exp +� +−D2 · n2α−βlow · (log n)6� +and this proves (3.4). +9 + +Proof of (3.5) in Theorem 2: We prove (3.5) using the estimate (4.1) +of Lemma 4(b) for χ(G) and the estimate (4.4) for the maximum average +degree hav(G), in the proof of Lemma 4(a). +For a set V ⊆ {1, 2, . . . , n} of size #V = l, let m(V) be the random +number of edges in the induced subgraph HV of H formed by the vertices +of V. The expected number of edges in HV is +Em(V) = +�l +2 +� +pav(V) +and so by the standard deviation estimate (3.6) we have for 0 < ǫ < 1 +2 that +P +� +m(V) ≥ +�l +2 +� +pav(V)(1 + ǫ) +� +≤ exp +� +−ǫ2 +4 +�l +2 +� +pav(V) +� +. +We henceforth set ǫ = +1 +√log n. If nα · (log n)3 ≤ l ≤ n then we have from (3.8) +that +P(m(V) ≥ +�l +2 +� +pup(1 + ǫ)) ≤ exp +� +−ǫ2 +4 +�l +2 +� +plow +� +≤ exp +� +−l2ǫ2 +16 plow +� +. +(3.12) +Defining the event +Ebad := +� +nα≤#V=l≤n +� +m(V) ≥ +�l +2 +� +pup(1 + ǫ) +� +(3.13) +we have from (3.12) and the union bound that +P(Ebad) ≤ +� +nα≤l≤n +�n +l +� +· exp +� +−l2ǫ2 +16 plow +� +(3.14) +Using +�n +l +� +≤ nl we have that +�n +l +� +· exp +� +−l2ǫ2 +16 plow +� +≤ nl · exp +� +−l2ǫ2 +16 plow +� += exp(−Al), +(3.15) +where Al = Al(n) := l +� +lǫ2·plow +16 +− log n +� +. Since ǫ = +1 +√log n and l ≥ nα·(log n)3, +we have that +lǫ2 · plow ≥ nαplow · (log n)2 ≥ C1 · nα−βlow · (log n)2, +10 + +by (3.3) and so using α ≥ βlow and we get that +Al ≥ l · 2C · nα−βlow · (log n)2 ≥ 2C · n2α−βlow · (log n)5 +(3.16) +for all n large and some constant C > 0. Plugging (3.16) into (3.15) and +using (3.14) we get that +P(Ebad) +≤ +� +nα≤l≤n +exp +� +−2Cn2α−βlow · (log n)5� +≤ +n · exp +� +−2Cn2α−βlow · (log n)5� +≤ +exp +� +−Cn2α−βlow · (log n)5� +(3.17) +for all n large. +If the complement event Ec +bad occurs, then in any induced subgraph Hl +of H containing T := nα · (log n)3 ≤ l ≤ n vertices, the number of edges is +at most m(Hl) ≤ +�l +2 +� +pup(1 + ǫ) and so +max +T+1≤l≤n +2m(Hl) +l +≤ (n − 1) · pup(1 + ǫ) ≤ C2 · n1−βup(1 + ǫ) +(3.18) +where C2 > 0 is the constant in (3.3). Substituting (3.18) into (4.4) and +using the fact that ǫ = +1 +√log n, we get that the maximum average degree +hav ≤ nα · (log n)3 + C2n1−βup(1 + ǫ) ≤ D · nmax(α,1−βup) · (log n)3 +(3.19) +with probability at least 1 − ωn, for some constant D > 0. Here ωn is the +final expression in (3.17). Plugging (3.19) into (4.1) of Lemma 4(b), we get +the upper bound in (3.5). This completes the proof of Theorem 2. +Weighted Colouring +In this subsection, we study a weighted generalization of proper colouring +defined as follows. +Equip each edge h of the random graph G obtained +in (3.1) with a deterministic weight w(h) ≥ 1. Let f : V → {1, 2, . . .} be any +map from the vertex set of H to the set of positive integers. +Definition 1. We say that f is a proper w−weighted colouring or simply +proper weighted colouring of H if for each edge h = (u, v) ∈ E with endver- +tices u and v we have +|f(u) − f(v)| ≥ w(u, v). +(3.20) +11 + +We define the weighted colouring number of H as +χw = χw(H) := min +f +max +(u,v)∈E |f(u) − f(v)|, +(3.21) +where the minimum is over all proper weighted colourings of H. +We could interpret χw(H) as a measure of the effect of weights on a proper +colouring of H. Indeed if w(u, v) = 1 for all edges, then the weighted colouring +number χw(H) = χ(H), the chromatic number. In general, χw(H) ≥ χ(H) +and if w(u, v) ≤ K for some K and all edges (u, v), then χw(H) ≤ Kχ(H). +Henceforth we consider weighted colouring number of G when the edge +weights are random and unbounded. Formally, we equip the +�n +2 +� +edges of Kn +with independent and identically distributed (i.i.d.) +weights {w(h)}h∈Kn +having a constant mean 1 ≤ µ0 < ∞ and satisfying w(h) ≥ 1 a.s. for all +edges h ∈ Kn. The edge weights are also independent of G. +For a realization ω of the random graph G, let Pω be the probability +measure associated with the edge weights of ω. For constants +C, ǫ1, ǫ2, ǫ3 > 0 we define the event +Egood(C, ǫ1, ǫ2) := +� +ω : Pω (χw(ω) ≤ Cnǫ1) ≥ 1 − C +nǫ2 +� +and have the following result regarding the weighted colouring number χw(G). +For a vertex u, we define of the average edge probability pav(u) := +1 +n−1 +� +v̸=u p(u, v). +Theorem 3. Suppose Ew2s(u, v) < ∞ for some integer s ≥ 1. +(a) Suppose for each vertex u the average edge probability satisfies +C1 +nβ ≤ +pav(u) ≤ C2 +nβ for some positive constants C1, C2 and 0 < β < 1. For every s > +1 +1−β there is a constant C > 0 such that +P (Egood(C, 1 − β, s(1 − β) − 1)) ≥ 1 − e−C−1n1−β. +(3.22) +(b) Suppose the average edge probability pav := +�n +2 +�−1 � +h p(h) satisfies C1 +nβ ≤ +pav ≤ C2 +nβ for some 2 +s < β < 1. For every 0 < γ < β +2 − 1 +s, there is a con- +stant D > 0 such that +P(Egood(D, 1 − γ, β)) ≥ 1 − e−D−1n2−β +(3.23) +Part (a) of the above result essentially says that for nearly all realiza- +tions, the weighted colouring number is at most of the order of n1−β with +12 + +high probability. From part (b) of Theorem 3, we see that even under the +weaker condition on the overall edge probability average, the weighted colour- +ing number is still o(n) with high probability, provided the weights have a +bounded sth moment for s sufficiently large. Here and henceforth o(n) de- +notes a sequence satisfying o(n) +n +−→ 0 as n → ∞. +Proof of Theorem 3(a): For 1 ≤ i ≤ n let Ni be the number of neigh- +bours of the vertex i in the graph G. From the bounds on the average edge +probability pav(i), we see that the expected number of neighbours for ver- +tex i lies C1n1−β and C2n1−β. Therefore by the concentration estimate (3.6) +with ǫ = 1 +2, we have that +P +�C1n1−β +2 +≤ Ni ≤ 2C2n1−β +� +≥ 1 − e−Dn1−β, +for some positive constant D. Letting Enei := � +1≤i≤n{ C1n1−β +2 +≤ Ni ≤ 2C2n1−β}, +we have by the union bound that +P(Enei) ≥ 1 − ne−Dn1−β. +(3.24) +Let ω ∈ Enei be a realization of G. To find an upper bound for χw(ω), +we use the locally averaged bound (4.7) in Proposition 5. For 1 ≤ i ≤ n +let Ji := � +v∼i w(i, v) be the sum of the weights of edges containing i as an +endvertex. Using the fact that there are at most 2C2n1−β nodes adjacent +to i, we have that Ji is stochastically dominated by �2C2n1−β +i=1 +Zi where Zi +are i.i.d. with the same distribution as the edge weights. Using the fact that +the edge weights have bounded 2sth moment, we now show that there are +constants D1, D2 > 0 such that +P + + +2C2n1−β +� +i=1 +Zi ≥ D1n1−β + + ≤ +D2 +ns(1−β). +(3.25) +Indeed, denoting the expectation of Zi as µZ and using the Markov inequality +we have for any integer t ≥ 2s that +P +� +t +� +i=1 +Zi ≥ 2tµZ +� +≤ E +��t +i=1(Zi − µZ) +�2s +(tµZ)2s +. +(3.26) +Expanding +��t +i=1(Zi − µZ) +�2s , we see that any term of the form +� +j(Zij −µZ)bj has expectation zero, unless each bj is even. This implies that +13 + +the total number of terms with non-zero expectation is �s +l=1 +�t +l +� +≤ s +�t +s +� +≤ sts +by the monotonicity of the binomial coefficient for t ≥ 2s. Thus +E +� +t +� +i=1 +(Zi − µZ) +�2s +≤ C1sts +for some constant C1 > 0 and plugging this into (3.26), we get that +P +��t +i=1 Zi ≥ 2tµZ +� +≤ C2 +ts . Setting t = 2C2n1−β then gives us (3.25). +Letting Fwt := � +1≤i≤n{Ji ≤ D1n1−β} and using the union bound, we get +from (3.25) that +Pω (Fwt) ≥ 1 − +D2 +ns(1−β)−1. +(3.27) +If Fwt occurs, then using (4.7), we get that χw ≤ D3n1−β for some con- +stant D3 > 0. Thus we get (3.22) from (3.24) and (3.27). +Proof of Theorem 3(b): The expected number of edges in G equals +�n +2 +� +pav +and so if Eedge is the event that the number of edges in the random graph G +lies between +�n +2 +� pav +2 +and +�n +2 +� 3pav +2 , then by the concentration estimate (3.6) +with ǫ = 1 +2, we have that +P(Ec +edge) ≤ exp +� +−C1 +�n +2 +� +pav +� +≤ exp +� +−C2n2−β� +(3.28) +for some constants C1, C2 > 0. +Let ω ∈ Eedge be any realization of G and let q = q(ω) be the number +of edges in ω. By definition q is at least of the order of n2−β and so if Wtot +denotes the total weight of edges in ω, then by Chebychev’s inequality, we +have for constant ǫ > 0 that +Pω(|Wtot − qµ0| ≥ ǫqµ0) ≤ var(Wtot) +ǫ2q2µ2 +0 +≤ C3 +q ≤ +C4 +n2−β +(3.29) +for some constants C3, C4 > 0. Next let Ewt be the event that the maximum +edge weight in ω is at most n(2+δ)/s for some constant δ > 0 to be determined +later. By the bounded sth moment assumption of the edge weights and the +Markov inequality we have for any edge h of ω that +Pω +� +w(h) ≥ n(2+δ)/s� +≤ C5 +n2+δ +14 + +for some constant C5 > 0. Since the number of edges of ω is at most of the +order of n2−β, we get by the union bound that +Pω(Ec +wt) ≤ C6 +nβ+δ +(3.30) +for some constant C6 > 0. +Set ǫ = 1 and suppose that Etot := Ewt∩{Wtot ≤ 2qµ0} occurs. From (3.29) +and (3.30), we get that +Pω(Etot) ≥ 1 − C7 +n2−β − C7 +nβ+δ +(3.31) +for some constant C7 > 0. We choose δ > 0 small so that 2 − β > β + δ and +this is possible since β < 1. If the event Etot occurs, then setting K = n(2+δ)/s +and mµ = Wtot in the bound (4.8), we get that +χw(ω) ≤ C8n +2+δ +2s · √q ≤ C9n +2+δ +2s n1− β +2 = C9n1−( β +2 − 1 +s− δ +2s) +for some constants C8, C9 > 0. From the statement of the theorem we know +that β > 2 +s and so we choose δ > 0 smaller if necessary so that β +2 − 1 +s − δ +2s > 0 +as well. With this choice of δ and setting Egood = Eedge, we obtain (3.23) +from (3.28) and (3.31). +4 +Combinatorial Lemmas +In this section, we state and prove the combinatorial lemmas used in the +proof of the Theorems in the previous section. Throughout H = (V, E) is a +deterministic graph with vertex set V = {1, 2, . . ., n} and edge set E. +Our first result in concerns an upper bound for the chromatic number +of a graph in terms of its maximum average degree, used in the proof of +Theorem 2. We begin with a couple of definitions. For a graph H = (V, E), +we define the maximum average degree as hav = hav(H) := maxΓ dav(Γ), +where the maximum is over all subgraphs Γ ⊆ G and dav(Γ) is the average +vertex degree in Γ. By definition, we have that dav ≤ hav ≤ ∆, the maximum +degree of a vertex in H. The following result obtains bounds for the chromatic +number of H in terms of its maximum average degree. +15 + +Lemma 4. For any graph H containing n vertices and m edges, we have +that the chromatic number +χ(H) ≤ 2 max(1, hav) · log +� +ne +max(1, hav) +� +, +(4.1) +where hav satisfies dav ≤ hav ≤ min +�√ +2m, 6(m∆)1/3� +. +From the above Lemma, we see that if H has bounded maximum average +degree, then χ(H) = O(log n). For context we recall from the edge count +bound that if the average degree of H is bounded, then χ(H) ≤ 2√m = +� +2dav(G) · √n = O(√n). This is the best possible since if H contains a +complete subgraph formed by √n vertices, then the chromatic number is at +least √n. On the other hand, if H has bounded maximum average degree, +then every subgraph of H is also sparse and so we expect H to have low +chromatic number. This is reflected in the estimate (4.1). +Proof of Lemma 4: A stable set in H is a set of vertices no two of which are +adjacent in H. Let I1 = {v1, . . . , vt} be a maximum stable set, i.e. a stable +set of maximum size, in G1 := H. We assign the colour 1 to all vertices +in I1. We remove all the vertices of I1 from G1 to obtain a graph G2. We +now repeat the above procedure with G2. Letting I2 be the maximum stable +set in G2, we assign the colour 2 to all vertices in I2. Continuing the above +procedure, let Gk+1 be the graph obtained at the end of k iterations. The +partial colouring of H obtained so far uses k colours and by construction is +proper. Therefore if Gk+1 has nk+1 vertices, then χ(Gk+1) ≤ nk+1 and so +χ(G) ≤ k + nk+1. +(4.2) +To estimate nk+1, we estimate the size of Ii for each 1 ≤ i ≤ k. Recalling +that dav and hav ≥ dav are the average degree and maximum average degree +of H, respectively, we get from Theorem 3.2.1, pp. 29, Alon and Spencer +(2008), that I1 has cardinality +#I1 ≥ +n +2 max(1, dav) ≥ +n +2 max(1, hav) := nα. +(4.3) +From (4.3), we see that the graph G2 has n2 ≤ n(1−α) vertices. The graph G2 +has a maximum average degree of at most hav and so arguing as in (4.3) we get +that #I2 ≥ n2α. This implies that G3 has n3 ≤ n2(1−α) ≤ n(1−α)2 vertices. +Continuing this way, we get that Gk+1 has nk+1 ≤ n(1 − α)k vertices and so +from (4.2), we get that χ(G) ≤ k + ne−θk =: g(k) where θ := | log(1 − α)|. +16 + +We see that g(k) is minimized if k satisfies 1 − nθ · e−θk = 0 and for +this value of k, we get that χ(G) ≤ log(nθ)+1 +θ +. Using | log(1 − x)| > x we get +that θ > α and since α = +1 +2 max(1,hav) ≤ 1 +2, we also get that +θ = | log(1 − α)| ≤ +� +j≥1 +αj = +α +1 − α ≤ 2α. +In effect α ≤ θ ≤ 2α and so +χ(G) ≤ log(2αθ) + 1 +α += 2 max(1, hav) · +� +log +� +n +max(1, hav) +� ++ 1 +� +. +This proves (4.1). +To obtain the bounds for hav, we use the fact that for any integer 1 ≤ +T ≤ n, the maximum average degree satisfies +hav += +max +� +max +1≤l≤T max +Hl dav(Hl), +max +T+1≤l≤n max +Hl dav(Hl) +� +≤ +max +� +T, +max +T+1≤l≤n max +Hl dav(Hl) +� += +max +� +T, +max +T+1≤l≤n max +Hl +2m(Hl) +l +� +(4.4) +where Hl is an induced subgraph of G containing l vertices and m(Hl) is the +number of edges in Hl. For any subgraph Hl we have that m(Hl) ≤ m, the +total number of edges in H and so from (4.4) we see that +hav(G) ≤ max +� +T, +max +T+1≤l≤n +2m +l +� +≤ max +� +T, 2m +T +� +(4.5) +and the final expression in (4.5) attains its maximum at T = +√ +2m. Thus we +get that hav(G) ≤ +√ +2m. +For the final estimate hav ≤ (m∆)1/3, we again use (4.4) as follows. +For x > 0, the probability that a randomly chosen vertex has degree larger +than x is bounded above by the Markov inequality as dav +x and so the number +of vertices in H with degree larger than x is at most ndav +x += m +x . For any sub- +graph Hl on l vertices, we therefore have that the number of edges m(Hl) ≤ +17 + +m +x · ∆ + lx and plugging this into (4.4), we get that +hav +≤ +max +� +T, +max +T+1≤l≤n +2m∆ +lx ++ 2x +� +≤ +max +� +T, 2m∆ +Tx + 2x +� +≤ +T + 2m∆ +Tx + 2x. +(4.6) +The final expression in (4.6) is minimized at T = +� +2m∆ +x +and so +hav ≤ 2 +� +2m∆ +x ++ 2x. +Setting x = (m∆)1/3, we then get the desired bound hav ≤ (2 +√ +2+2)(m∆)1/3 +and this completes the proof of the Lemma. +To prove Theorem 3, we use the following deterministic result that obtains +bounds for the weighted colouring number χw(H) in terms of averaged and +locally averaged edge weights. +Lemma 5. Suppose H is a graph with m edges and let w(u, v) be the weight +associated with edge (u, v). +(a) We have that +χw(H) ≤ 1 + max +v∈H +� +u∼v +(2w(u, v) − 1). +(4.7) +(b) If w(h) ≤ K for some K > 0 and all edges h ∈ H, then +χw(H) ≤ 2 +� +2mKµ +(4.8) +where µ := 1 +m +� +h w(h) is the average edge weight in H. +The bound in part (a) of Lemma 5 is a generalization of the maximum +degree bound χ(H) ≤ ∆ + 1. In fact if w(u, v) = 1 for all edges (u, v) ∈ H, +then the right side of (4.7) reduces to ∆+1. As in the proof of the maximum +degree bound, we use a greedy colouring procedure to obtain (4.7). Part (b) +of Proposition 5 establishes bounds for the weighted colouring number in +18 + +terms of the average edge weight. The bound in (4.8) is essentially the best +possible since if H contains a clique of size c√m for some c > 0 then using +the fact that each edge weight is at least one, we get χw(H) ≥ c√m. +We also remark here that a weaker upper bound of χw(H) ≤ 2K√m +can be obtained using a direct counting argument as follows: Let C := +{K, 2K, . . ., rK}, where r = χ(H) is the smallest possible integer that allows +a proper colouring of G using only colours from C. Let Vi be the set of all +vertices with colour iK for 1 ≤ i ≤ r. By the minimality of r, there must +exist at least one edge between a vertex in Vi and a vertex in Vj, for any i ̸= j +and so +�r +2 +� +≤ m. Consequently χw ≤ rK ≤ 2K√m. For µ ≤ K +3 the bound +in (4.8) is stronger and is obtained using the probabilistic method below. +We now prove Lemma 5 and Theorem 3 in that order below. +Proof of Lemma 5(a): Let L + 1 := 1 + maxv∈H +� +u∼v(2w(u, v) − 1), the +right hand side of (4.7). We iteratively colour the vertices of H from the +set {1, 2, . . ., L + 1}. Pick a vertex u1 and assign the colour l(u1) = 1. For +the ith iteration, i ≥ 2, we pick an uncoloured vertex ui and let l(v1), . . . , l(vt) +be the colours of its coloured neighbours. Let I(vj) be the set of all integers l +satisfying |l − l(vj)| ≤ w(ui, vj) − 1. There are 2w(ui, vj) − 1 integers in I(vj) +and so the number of integers in � +1≤j≤t I(vj) is at most L. We therefore assign +a colour l(ui) ∈ {1, 2, . . . , L + 1} \ � +1≤j≤t I(vj) to the vertex ui. Continuing +this process we obtain a proper weighted colouring of H and this proves +Lemma 5(a). +Proof of Lemma 5(b): For integer θ ≥ 1 to be determined later, let Xi, 1 ≤ +i ≤ n be independent and identically distributed (i.i.d.) random variables +uniformly randomly chosen from {1, 2, . . . , θ}. Say that edge h = (u, v) is bad +if (3.20) does not hold. Let hi = (ui, vi), 1 ≤ i ≤ q be the set of all bad edges. +We now iteratively change the labels assigned to the vertices of bad edges +until no bad edge remains. We begin by setting w1 := u1 and f(w1) = θ + K +and marking u1. We then pick an unmarked endvertex w2 of some bad edge +and set f(w2) = θ + 2K. We proceed this way until each endvertex of a bad +edge is marked. Finally, for the rest of the vertices we set f(u) = Xu. +Since w(h) ≤ K for all h, we have that f is a proper weighted colouring +of H. Moreover, if the number of bad edges is Nbad, then 1 ≤ f(u) ≤ θ+NbadK +and so +χw ≤ θ + NbadK. +(4.9) +To bound Nbad, we first estimate ENbad as follows. For any two vertices u +19 + +and v and any integer 1 ≤ j ≤ θ − 1, we have that |Xu − Xv| = j if and +only if either Xu = l and Xv = l + j for some 1 ≤ l ≤ θ − j or Xv = l +and Xu = l + j for some 1 ≤ l ≤ θ − j. Thus P(|Xu − Xv| = j) = +2 +θ2(θ − j) +and so for any t ≤ θ, we have that +P(|Xu − Xv| ≤ t) = 2 +θ2 +� +θt − t(t + 1) +2 +� +≤ 2t +θ . +For an edge h with weight w(h), we have that the probability that h is bad +is at most 2w(h) +θ +and so +ENbad ≤ +� +h +2w(h) +θ += 2mµ +θ +(4.10) +where µ = 1 +m +� +h w(h) is the average edge weight. +From (4.10), we see that there exists a realization such that Nbad ≤ 2mµ +θ +and plugging this into (4.9), we get that +χw ≤ θ + 2mKµ +θ +(4.11) +and the right side of (4.11) is minimized at θ = √2mKµ. Thus χw(H) ≤ +2√2mKµ and this completes the proof of the Lemma. +Acknowledgement +I thank Professors Rahul Roy, Alberto Gandolfi, C. R. Subramanian and K. +Adhikari for crucial comments and also thank IMSc and IISER Bhopal for +my fellowships. +References +[1] D. Achlioptas and A. Naor. (2005). The Two Possible Values of the +Chromatic Number of a Random Graph. Annals of Mathematics, 162, +pp. 1335–1351. +[2] N. Alon and M. Krivelevich. (1997). The Concentration of the Chromatic +Number of Random Graphs. Combinatorica, 17(3), pp. 303–313. +20 + +[3] N. Alon and J. Spencer. (2008). The Probabilistic Method. Wiley Inter- +science. +[4] Bairamov, I. and Stepanov, A. (2006). A note on large deviations of +weak records. Statistics and Probability Letters, 76, pp. 1449–1453. +[5] B. Bollob´as. (1988). The Chromatic Number of Random Graphs. Com- +binatorica, 8, pp. 49–55. +[6] B. Bollob´as. (2001). Random Graphs. Cambridge University Press. +[7] Charalambides, C. A. and Rychlik, T. (2008). Distributions and mo- +ments of record values in a sequence of maximally dependent random +variables. Journal of Statistical Planning and Inference, 138, pp. 2253– +2266. +[8] G. Chartrand, D. Erwin, P. Zhang, F. Harary. (2001). Radio labelings +of graphs. Bull. Inst. Combin. Appl., 33 pp. 77-85. +[9] A. Frieze. (1990). On the Independence Number of Random Graphs. +Discrete Mathematics, 81, pp. 171–176. +[10] G. Ganesan. (2020). Cliques and Chromatic Number in Multiregime +Random Graphs. Sankhya, doi : https : //doi.org/10.1007/s13171 − +020 − 00205 − 4. +[11] Jasi´nski, K. (2018). Characterizations of geometric distributions based +on kth record values. Statistics, 52, pp. 1116–1127. +[12] D. Korze, Z. Shao and A. Vesel. (2021). New results on radio k−labelings +of distance graphs. +Discrete Applied Mathematics, Available online, +September 2021. +[13] T. Luczak. (1991). The Chromatic Number of Random Graphs. Com- +binatorica, 11, pp. 45–54. +[14] M. Molloy and B. Reed. (2002). Graph Colouring and the Probabilistic +Method. Springer. +[15] Nevzorov, V. B. (1986). Records. Theory of Probability and Applications, +32, pp. 201–223. +21 + +[16] K. Panagiotou and A. Steger. (2009). A Note on the Chromatic Number +of a Dense Random Graph. Discrete Mathematics, 309, pp. 3420–3423. +[17] A. Scott. (2008). On the Concentration of the Chromatic Number of +Random Graphs. Arxiv Link: https://arxiv.org/abs/0806.0178. +[18] E. Shamir and J. Spencer. (1987). Sharp Concentration of the Chromatic +Number on Random Graphs Gn,p. Combinatorica, 7, pp. 121–129. +[19] D. West. (2000). Introduction to Graph Theory. Prentice Hall. +22 + diff --git a/U9E5T4oBgHgl3EQfBQ5V/content/tmp_files/load_file.txt b/U9E5T4oBgHgl3EQfBQ5V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dce3e96ff415ee0d223d53ee543f25ed837801c1 --- /dev/null +++ b/U9E5T4oBgHgl3EQfBQ5V/content/tmp_files/load_file.txt @@ -0,0 +1,590 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf,len=589 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='05385v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='PR] 13 Jan 2023 Weighted Domination and Colouring in Random Graphs Ghurumuruhan Ganesan ∗ IISER, Bhopal Abstract In the first part of this paper, we consider weighted domination in the case where the vertices of the complete graph on n vertices are equipped with independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=') weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' We use the probabilistic iteration to determine sufficient conditions for maximizing the weighted domination probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In the second part, we study a weighted generalization of the chromatic number and estimate the minimum number of colours needed to satisfy the constraints when the weights themselves are random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' We show that the “extra” cost incurred for weighted colouring is small if the weights have sufficiently large moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' We also consider inhomogenous ran- dom graphs where the edge probabilities are not necessarily all same and obtain bounds for the chromatic number in terms of its averaged edge probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Key words: Weighted Domination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Maximum Domination Proba- bility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Weighted Colouring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Inhomogenous Random graphs, AMS 2020 Subject Classification: Primary: 60C05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' 1 Introduction In this section, we describe briefly the two problems studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' ∗E-Mail: gganesan82@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='com 1 Weighted Domination Records in a sequence of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=') ran- dom variables have been extensively studied in the context of statistical es- timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' The probability that the kth random variable is a record (as com- pared to the previous k − 1 values) grows inversely in the index k and the corresponding record events are mutually independent [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Various aspects of record properties with differing assumptions have been studied since and as examples, [4] study large deviation properties for weak records and [11] studies characterizations of geometric distributions based on kth record val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Also, [7] study records in a sequence of maximally dependent random variables and obtain expressions for corresponding record moments and dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In this paper, we study records from a graph theoretic view point and use probabilistic iteration to estimate the probability that a given set of vertices form a weighted dominating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Weighted Colouring The chromatic number χ(G) of a graph G is the smallest number of colours needed for a proper colouring of G and many upper and lower bounds for χ(G) exist in terms of various graph parameters like maximum vertex degree, independence number, clique number etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' (see Chapter 5, [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' For homogenous random graphs where each edge is present with the same prob- ability pn, concentration of the chromatic number around its mean has been well-studied [18, 6, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Many bounds for the expected chromatic number have also been derived using clique sizes in the complement graph and a com- bination of second moment and martingale methods [5, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' The paper [16] uses counting arguments to obtain sharp lower bounds on the chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In a related direction, [3] obtain two point concentration of the chromatic number for edge probability pn = n−1/2−δ with δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' The anal- ysis proceeds through k−choosable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Extending this to pn = d n, [1] use analytical techniques to obtain the two possible values of the chromatic number for d > 0 a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Recently, in [10] we have studied the chromatic number of homogenous random graphs whose edge probabilities do not necessarily form a convergent sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In the first part of this paper, we obtain estimates for the chromatic number of an inhomogenous graph in terms of averaged edge probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' 2 We use an auxiliary bound for chromatic number of a deterministic graph, in terms of its maximum averaged degree, that is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In the second part of this paper, we study weighted colouring in ran- dom graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Generalizing radio labelling of graphs [8, 12] which we call as weighted colouring number, we use the probabilistic method to estimate the weighted colouring number when the edge weights themselves are ran- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Such situations arise frequently in applications and we show that if the weights have sufficiently large moments, then the “cost” incurred due to the unboundedness of weights is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' The paper is organized as follows: In the following section, we state and prove our first main result regarding maximizing the domination probability in weighted random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Next, in Section 3 we state and prove our two main results regarding weighted and constrained colouring in random graphs and finally, in Section 4, we collect the auxiliary combinatorial lemmas used in the proof of the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' 2 Weighted Domination In this section, we study domination in weighted random graphs obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Let Kn be the complete graph on n vertices and equip vertex v with a random weight Sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' The random variables {Sv}1≤v≤n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' continuous with a common cumulative distribution function (cdf) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' For each vertex v, we now associate a deterministic domination neighbourhood D(v) ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' , n} \\ {v} and say that v is a weighted dominating vertex if Sv > maxu∈D(v) Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Similarly, we say that a finite set of vertices V := {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' , vk} is a weighted dominating set if each vertex in V is a weighted dominating vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Let Edom = Edom(V) be the event that V is a weighted dominating set and also let Rtot := � 1≤l≤k 11(Avl) be the total number of weighted dominat- ing vertices in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' For any vertex v, we have by symmetry that P(Av) = 1 d(v), where d(v) = 1 + #D(v) and #D(v) is the size of the domination neighbour- hood of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Therefore if the vertices in V have disjoint domination neighbour- hoods, then the events Av, v ∈ V are independent and so P(Edom) = k � l=1 1 d(vl) =: pdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In general, if the domination neighbourhoods are not disjoint, the events Av, v ∈ V are correlated and so we expect that Edom occurs with lesser probability, as 3 described in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Recalling that V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' , vk} with v1 < v2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' < vk, we define C(vl) := �l j=1 D(vj) and set c(vl) := 1 + #C(vl) for 1 ≤ l ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' If � x 0 F k−1(y)dF(y) = F k(x) k for each integer k ≥ 1 and x > 0 and vk /∈ C(vk) and vl ∈ D(vl+1) \\ C(vl) for each 1 ≤ l ≤ k − 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='1) then P (Edom) = � 1≤l≤k 1 c(vl) ≤ pdom and var(Rtot) ≤ ERtot = k � l=1 1 d(vl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='2) Moreover, the events {Avl}1≤l≤k are mutually independent if and only if D(vl) ⊂ D(vl+1) strictly, for each 1 ≤ l ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' In words, the above result says that if the domination neighbourhoods satisfy the “weak-nested” property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='1), then pdom is the maximum possible weighted domination probability and moreover, this value is achieved only if the neighbourhoods satisfy a strong-nested property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' From a statistical view point, we could also interpret Theorem 1 as a “graph-theoretic” version of records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' For context, we recall that “time-based” record events in a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' random variables are known to be mu- tually independent when the comparison set consists of the entire past [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' From Theorem 1, we see that graph-theoretic record events are negatively correlated for weakly nested neighbourhoods and are mutually independent for strongly nested neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Proof of Theorem 1: Let Bk := �k j=1 Avj and for x > 0 define Avk(x) := Avk � {Svk < x} = � max j∈D(vk) Sj < Svk < x � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='3) and Bk(x) := Bk−1 � Avk(x), with the notation that the maximum of the empty set is zero and B0 = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' We first show by induction that for every x > 0, P(Bk(x)) = F c(vk)(x) c(v1) · · ·c(vk) = P(Bk) · F c(vk)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='4) 4 For the basis step, we see that the random variable X with cdf F is continuous and so F(x) = P(X ≤ x) = P(X < x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Conditioning on Sv1 = y, we therefore get that P (B1(x)) = � x 0 P � maxj∈D(v1) Sj < y � dF(y) equals � x 0 P \uf8eb \uf8ed � j∈D(v1) {Sj < y} \uf8f6 \uf8f8 dF(y) = � x 0 F d(v1)−1(y)dF(y) = F d(v1)(x) d(v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='5) Since d(v1) = c(v1) this proves the basis step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' To prove the induction step, we now assume that the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='4) is true for the event Bk−1(x) and consider the event Bk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' If Bk−1 = �k−1 j=1 Avj occurs, then using the weak nested property in the statement of the theorem, we already have Svk−1 > maxl∈C(vk−1) Svl and so Svk is a record if and only if Svk > maxj∈Ek Sj and Svk > Svk−1, where Ek := D(vk) \\ (C(vk−1) ∪ vk−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Letting 11(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=') denote the indicator function, we then have that 11 (Bk(x)) = 11 � Avk(x) � Bk−1 � = 11 � x > Svk > max j∈Ek Sj � 11(x > Svk > Svk−1)11(Bk−1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='6) and since the event Bk−1 depends only on the values of {Sj}j∈C(vk−1)∪{vk−1}, we condition on Svk = z and get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='6) that P (Bk(x)) = � x z=0 tk(z)dF(z) where tk(z) := P � max j∈Ek Sj < z � P � Bk−1 � � Svk−1 < z �� = � j∈Ek P(Sj < z)P � Bk−1 � � Svk−1 < z �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='7) To evaluate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='7), we use the weak nested property vk−1 ∈ D(vk)\\C(vk−1) to get that Ek = D(vk) \\ (C(vk−1) ∪ vk−1) has cardinality c(vk) − c(vk−1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Consequently tk(z) = F c(vk)−c(vk−1)−1(z)P � Bk−1 � � Svk−1 < z �� = F c(vk)−c(vk−1)−1(z)P (Bk−1(z)) and substituting this into the expression for P(Bk(x)) determined prior to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='7), we get P (Bk(x)) = � x 0 F c(vk)−c(vk−1)−1(z)P (Bk−1(z)) dF(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' 5 By induction assumption we have P (Bk−1(z)) = F c(vk−1)(z) c(v1)·c(v2)···c(vk−1) and so P (Bk(x)) = 1 c(v1) · c(v2) · · ·c(vk−1) � x 0 F c(vk)−1(z)dF(z) = 1 c(v1) · c(v2) · · ·c(vk)F c(vk)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='8) This proves the induction step and therefore completes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' The first relation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='2) follows directly from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content='4) by setting x = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E5T4oBgHgl3EQfBQ5V/content/2301.05385v1.pdf'} +page_content=' Moreover, var(Rtot) = V1 + 2V2 where V1 = k � l=1 P(Avl) − (P(Avl))2 = k � l=1 � 1 d(vl) − 1 d2(vl) � ≤ k � l=1 1 d(vl) = ERtot and V2 = � 1≤l1 0.25 Jy/pixel, P ′/σP ′ > 2 +beam FWHM +p′ = 10% +HAWC+ 154 µm Polarised Flux image +I contours: 0.125( +√ +2)16 Jy/pixel +Polo. data selection: +I > 0.25 Jy/pixel, P ′/σP ′ > 2 +beam FWHM +p′ = 10% +Figure 1. (Top) SOFIA HAWC+ band D (154µm) total intensity (Stokes I) image of BYF 73 on a logarithmic scale, overlaid by white +contours as labelled. (In all figures, we use the notation x(y)z for contours running from level x in steps of y to level z.) All HAWC+ band +D images have 2′′.75 pixels, or 0.2× the 13′′.6 beam. At every 2nd pixel (0.4 beam) satisfying the indicated selection criteria, we also display +black “vectors” showing the measured polarisation percentage (p′) and position angle (with the usual ±π degeneracy) of the plane-of-sky +B field component (i.e., rotated 90◦ from the observed polarisation direction). The peak I intensity is 17.58 Jy/pixel with a typical rms +error in the interior of the image 2–3 mJy/pixel, rising to 4–6 mJy/pixel around the image boundary due to the dither pattern of the +observations; the peak S/N in the I image is >5000. Inside the I = 0.5 Jy/pixel contour, nearly all p′ vectors have S/N ranging from ∼5 +to >30; for 0.25 2.5 +Figure 2. +(Top) ALMA 13-pointing mosaic of 3 mm continuum emission from BYF 73, in a 3.5 GHz-wide band centred at an effective +frequency of 102.1346 GHz. The contrast is enhanced to bring out the fainter structures, in particular the E-W Streamer, as indicated by +the colour bar to the right. The point sources MIR 2 and 11 (see Fig. 3) peak at 21 and 7 mJy/beam, respectively. The synthesised beam +(2′′.93×2′′.74 @ –38◦.0) is shown in the bottom-left corner, and the noise σrms = 0.13 mJy/beam where the primary beam correction is small, +away from the map edge, which is at a primary beam cutoff of 0.2. This gives a peak S/N of 170. For reference, magenta contours are +overlaid from the HAWC+ 154µm Stokes I data, at levels 0.44(0.10)0.84, 1.5, 3, 5, and 9 Jy/pixel (as in Fig. 3). +(Bottom) Zoom in to all detectable 3 mm continuum polarised emission within a deeper, single ALMA pointing of BYF 73’s central +structures, framed by the yellow box in the top panel. The image is the debiased polarised flux on the colour scale to the right, peaking at +0.55 mJy/beam for MIR 2 (S/N = 24, σrms = 23 µJy/beam). The debiased percent polarisation vectors are overlaid in magenta, rotated +by 90◦ to show the B field orientation at every second pixel in l and b (as in Fig. 1). Away from MIR 2, most vectors shown have S/N > 5 +with typical noise σrms = 4% in amplitude and 5◦ in angle. The grey contours here (at 0.2, 0.6, 1.1, 1.6, 2.5, 5, 10 mJy/beam) show the +ALMA Stokes I from the mosaic in the top panel. The single-field I map has noise σrms = 85 µJy/beam for a peak S/N =240 at MIR 2, +slightly deeper than the mosaic. The noisy polarisation features near the N-S ionisation front west of MIR 2 are probably real, but are not +accurately calibrated outside the roughly 1/3 FWHM primary beam limit (20′′, large yellow circle) of ALMA’s polarisation mode in Cycle +7. The synthesised beam (2′′.61×2′′.52 @ 21◦.0) is shown in the top-left corner with a 30% polarisation scale bar. + +mJy/beam +3 +22 +286.19 +286.20 +Longitude +286.21 +Galactic +286.22 +286.23 +0.19 +0.18 +0.16 +0.15 +Galactic Latitudebeam +Kru +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +286.205 + Longitude +286.210 +Galactic I +286.215 +172 +.74 +0.168 +166 +0.1 +0 +Galactic Latitude6 +Barnes et al. +MIR 12 is not detected shortward of 10 µm, like MIR 11. +Together, however, MIR 12+13 are marginally detected +in the HAWC+ I image as distinct extensions to the FIR +emission; in combination with their 3mm continuum de- +tections at S/N∼10, we consider them to also be probable +(lower-mass) protostars. In the spectral-line data (§4), +while MIR 11–13 are all outside the single 12CO field of +view, the mosaics show interesting features near MIR 11 +consistent with its protostar status (§4.5). In contrast, +the mosaics near MIR 12 & 13 are completely unremark- +able. Based on the spectroscopy, none of MIR 11–13 seem +to have any impact on the wider cloud’s evolution. +This extensive multi-wavelength data, showing a rela- +tive paucity of mm-wave point sources and almost-as- +scarce mid-IR (i.e., 8–18 µm) point sources, supports +P18’s inference that most of the plethora of near-IR (i.e., +1–5 µm) stars are likely to be in the foreground of the +BYF 73 cloud. That is, while scores of stars within the T- +ReCS field show embedded near-IR colours (Andersen et +al. 2017), most of these cannot be deeply embedded, since +P18 only directly detected 8 of them with T-ReCS, sug- +gesting a lack of embedding envelopes. Based on compar- +isons with their near-IR visibility, T-ReCS would likely +only have detected 2 more sources outside the observed +mosaic, MIR 9 and 10, at P18’s sensitivity level. +Even among these 10 mid-IR bright sources, only MIR +2 is detectable at all in the 3 mm continuum; specifically, +even the very mid-IR-bright stars MIR 1, 3, 9, and 10 are +not detected with ALMA. By comparison with MIR 2, +this suggests that these other four mid-IR bright stars +have very minimal (if any) protostellar dust envelopes, +of mass <3–4 M⊙ (ALMA’s 3σ detection limits in the +two observing modes). Therefore, it is reasonable to con- +clude that, among all these point sources, only MIR 11– +13 have similarly “cold” spectral energy distribution to +MIR 2, and are in a similarly early stage of protostellar +evolution. Scaling MIR 11’s 3mm flux density to MIR 2, +which is 3× as bright, suggests that its mass may very +approximately be 80 M⊙, still large by protostellar stan- +dards. Similarly scaling MIR 12+13’s 3mm flux densities +yields dust masses ∼7 M⊙ and 10 M⊙, respectively. +For the extended emission, both the polarised and +unpolarised 154 µm structures simultaneously trace two +very different dust populations, each in their own way: +(1) the warm dust permeating and surrounding the H ii +region, arcing out to the west and northwest from the +molecular clump, and seen well in Herschel and Spitzer +images at 70 µm and shorter wavelengths, and (2) the +cold dust in the massive (2×104 M⊙) molecular clump +to the east, traced well by the usual mm-wave molecular +lines and longer wavelength (≥250 µm) continuum. +Similarly, the 3 mm emission mostly traces the cold +molecular structures, but apparently also some high- +density warm dust associated with the N-S ionisation +front (IF) between the molecular cloud and H ii region. +Circumstantially, MIR 3 appears to be the main driver of +the IF in Fig. 3a–d; MIR 4 also lies close to the southern +end of the IF, but seems not to have as much impact +on its surroundings. The main extended features in the +3 mm continuum are the rather striking arcs of emission +running mainly east and west of MIR 2, which for lack of +a better term, we call the “Streamer”.12 There is also a +12 We resist calling it a filament since that term has a specific +notable ∼5′′×10′′ gap (the “Hole”) in the 3 mm emission +within the Streamer, immediately adjacent to MIR 2 on +its eastern side. It is unclear from its continuum prop- +erties whether this is a true lack of emission due to an +absence of material in the Streamer, or whether it is the +shadow of an extremely cold, high-optical-depth compo- +nent in the foreground of the Streamer, completely ab- +sorbing the 3 mm emission beyond it. The spectral-line +maps, however, resolve this question; see §§4.1,4.4. +3.3. Magnetic Field Structures in the Molecular Core: +HAWC+ +We begin our exploration of the molecular core’s B +field as revealed by HAWC+. Zooming in to the inner +portion of the molecular cloud near the massive proto- +star MIR 2 (P18), a very striking feature of the polarised +emission stands out immediately – see Figure 4. There is +a strong, narrow null in P ′ curving around the western +side of the molecular peak, between it and the H ii region, +in particular the darker blue colours indicating very low +P ′. +The EW width of this null is quite small, appar- +ently only 1 pixel or ≲7000 AU (i.e., much less than the +angular resolution of HAWC+, but we show this is not +unphysical below). Further, this null can be traced most +of the way around the molecular peak, although it broad- +ens out somewhat to the N, S, and E, to an approximate +width of 3–6 pixels, or 0.1–0.2 pc. +The area enclosed by this boundary layer, signified by +where P ′ rises above 0 on the inside, is an approximately +elliptical zone of size 0.55×0.40 pc at PA ≈ 80◦ (labelled +IBL for inner boundary layer in Fig. 4). The outer bound- +ary layer of the null is a vaguely elliptical polygon of +approximate height 0.91 pc and width 0.60 pc, at PA ∼ +20◦ (labelled OBL in Fig. 4). The space between these +boundary layers has essentially zero (i.e., S/N<3) FIR +polarisation and B⊥. +To see why the narrow western null is real, we show in +Figure 5 the Stokes Q and U maps in the same zoomed +area as Figure 4. Carefully comparing the three images +in Figures 4-5 on the western side of the MIR 2 core, one +sees that where P ′ ≈ 0 (e.g., close to the I = 5.6 Jy/pixel +contour), Q drops from positive to negative values going +into the centre, and U behaves similarly (Q and U are +both 0 where the images are orange). +Therefore, the +P ′ null is very narrow here because both Q and U are +changing rapidly through 0 at the same locations (but +in a manner consistent with HAWC+’s angular resolu- +tion), from larger positive to larger negative values as +one traverses into the centre of the core. +In other words, each of these components of P ′ is +strongly reversing sign on this boundary, correspond- +ing to a null in P ′ and 90◦ change in direction (due to +the definition of the Stokes parameters) for both the ob- +served polarisation angle and inferred B field direction +θB⊥ across this boundary. West of MIR 2, this change +is very sharp, much less than a beamwidth. +Indeed, +this change of direction is visible in the p′ vectors them- +selves, as shown in Figure 4. The vectors just outside the +null, i.e., in the H ii region and also east of the MIR 2 +meaning in SF studies, which we do not wish to pre-judge. For +example, the IF also looks filamentary, but as is common with +such interfaces, it is likely only prominent in Fig. 2 due to its flat +geometry being viewed edge-on. + +Magnetic Fields and Gas Structures in BYF 73 +7 +HAWC+ I contours: 0.44(0.10)0.84, 1.5, 3, 5, 9 Jy/pixel +HAWC+ P ′ contours: 50(16)98, 140 mJy/pixel +IRAC band 4 +IRAC band 2 +IRAC band 3 +IRAC band 1 +IRAC band 2 +AAT K +(a) +(b) +ALMA I contours: 0.4, 1, 2, 4, 8, 16 mJy/beam +HAWC+ I, HAWC+ P ′, ALMA I contours +IRAC band 4 +IRAC band 2 +IRAC band 3 +IRAC band 1 +IRAC band 2 +AAT K +(c) +(d) +ALMA I, HAWC+ I contours +ALMA I, HAWC+ I contours +IRAC band 4 +IRAC band 2 +IRAC band 3 +IRAC band 1 +IRAC band 2 +AAT K +(e) +(f) +Figure 3. +(All panels) Composite RGB images of Spitzer and Anglo-Australian Telescope data (Barnes et al. 2013, P18) as labelled in +each panel, plus locations of MIR sources 1–8 from P18 and new designations MIR 9–13 in this work. Contours are also colour-coded and +labelled at the top of each panel. Panels in the left column are IRAC 4-3-2 composites, with c,e being more saturated than a in order to +bring out fainter features. Panels in the right column are IRAC 2-1 + AAT K composites. Panels on the same row are on the same scale +to facilitate comparisons; the top row is a wider view, other rows are successive zooms. + +286.205 +286.210 +Galactic +286.215 +286.220 +.80 +75 +.170 +6 +1 +1 +0 +0 +.0 +0 +Latitude +Galactic286.218 +286.220 +286.222 +286.224 +286.226 +286.228 +286.230 +.68 +0.166 +0.162 +.60 +0.158 +0.1 +0.16 +0.1 +Latitude +Galactic286.218 +286.220 +286.222 + Longitude +286.224 +Galactic I +286.226 +286.228 +286.230 +.68 +.66 +.62 +09 +0.158 +0.1 +0.1 +0.1 +0.1 +0.1 +Latitude +Galactic286.18 +286.20 +Galactic +286.22 +2 +3 +0.20 +8 +6 +1 +0 +Galactic Latitude286.18 +286.20 +8 +286.22 +2 +0.20 +0.18 +0.16 +Latitude +Galactic286.205 +Longitude +286.210 +Galactic I +286.215 +286.220 +0.180 +.175 +0.170 +0.165 +0 +Latitude +Galactic8 +Barnes et al. +Figure 4. +Zoom in to P ′ image from Fig. 1’s lower panel on the +indicated colour scale, for all pixels with I > 0.25 Jy/pixel, plus +white I contours at 0.25( +√ +2)16 Jy/pixel, blue HAWC+ beam, and +red 10% p′ vector scale. We also show the p′ vectors at every pixel +(0.2 beam) unmasked by S/N, since here low-S/N p′ vectors are +very small anyway. Also shown are positions of the outer & inner +boundary layers (OBL, black; IBL, red), described in the text. +core, are all oriented roughly east-west, while the vec- +tors inside the null, especially close to the MIR 2 core, +are mostly oriented roughly north-south. The change in +PA is sharpest where Q+U are reversing most sharply, +just west of MIR 2. For the small-P ′ locations between +the two boundary layers to the N, E, and S of the MIR 2 +core, the changes in sign for Q and U are more gradual. +But they do change sign in each case, when looking from +the outer areas of the cloud to the centre. +In the more gradually changing boundary N, E, & S of +the core, the P ′ nulls seem to indicate an area where the +2D plane-of-sky B field component (B⊥) is dropping to +zero, since the P ′-null boundary’s width is roughly beam- +sized and there are few pixels with P ′>3σ. In contrast, +the sharp P ′-null boundary west of MIR 2 is much thin- +ner. Rather than merely dropping to zero, this seems to +be due to a 90◦ change in direction alone, i.e., that B⊥ +is sharply changing in direction at the boundary layer +around MIR 2, producing a null in P ′ without a corre- +sponding null in B⊥. +An alternative situation at the IBL is that the 3D B +vector is aligned very close to our line of sight there, i.e., +that B consists only of B||. In other words, looking pro- +gressively from the H ii region (west) through the IBL +and towards MIR 2 and its immediate east, the 3D B +field orientation points one way (∼EW) in the H ii re- +gion, twists through 90◦ at the null so B points directly +towards or away from us at the IBL, and then twists an- +other 90◦ to point in another 3D direction (∼NS) nearer +to MIR 2 within the IBL, with Q & U changing sign +in the process. We consider this a less likely proposition, +however, since the P ′ is so narrow, any B|| would put ad- +ditional unresolved structure into the IBL, and it would +require a rather large flux annulus to be pointed right at +us, a fairly significant “finger of God” effect in our view. +On the other hand, a pure 90◦ change to B⊥ alone might +require a discontinuity in B⊥. To resolve this, higher- +resolution polarisation data could reveal more details to +this narrow change in B⊥ (see §3.4), while Zeeman mea- +HAWC+ Q image +I cntrs: 0.25( +√ +2)16 Jy/pix +HAWC+ U image +I cntrs: 0.25( +√ +2)16 Jy/pix +Figure 5. (Top) Same area of the inner molecular clump as Fig. 4, +except showing only Stokes Q data with I contours. (Bottom) Same +as top panel but for Stokes U. Both Q and U images are displayed +on the same scale, with zero as orange. +surements could measure B|| directly, potentially distin- +guishing between a single 90◦ B⊥ twist or additional B|| +in the IBL. +A third possibility for the IBL’s apparent null is that +the polarisation signal comes from dust emission on the +low-opacity (west, H ii region) side, but dust absorption +on the high-opacity (east, molecular core) side, making +the null more of a polarisation radiative transfer effect, +without necessarily implying anything significant for the +cloud’s inherent B field. This would be similar to the +FIR polarisation pattern in Sgr B2, although observed at +a much coarser physical resolution of 1.5 pc (35′′ beam) +with the KAO (Novak et al. 1997). In order to be rel- +evant, the dust opacity in the core would need to be +> +∼1; however, based on P18’s SED fitting to Herschel +and other data, we compute an effective peak τ ≈ 0.001 +for BYF 73 at 154 µm and 37′′ resolution. Allowing for +HAWC+’s 7.4× smaller beam area, it is unlikely that +the peak τ within the IBL is more than 7× higher than +this. Even in ALMA’s beam, ∼30× smaller in area than +HAWC+’s, τ could rise to ∼0.2 at 154 µm if all the flux +were coming from MIR 2, but this is still less than 1 and +we know MIR 2 contains less than half the flux there, +§3.1. +Therefore, we can discount this possibility. +For + +Jy/pixel +5 +0.10 +0.05 +0.00 +0 +286.20 +Longitude +286.21 +IBL +Galactic I +286.22 +286.23 +8 +6 +0.1 +Galactic LatitudeJy/pixel +0.05 +0.10 +5 +0.10 +0.05 +0.1 +0.00 +286.20 +Galactic Longitude +286.21 +OBL +286.22 +286.23 +8 +6 +0.16 +0 +0 +Galactic LatitudeJy/pixel +0.05 +-0.10 +5 +0.10 +0.05 +0.1 +0.00 +286.20 +Galactic Longitude +286.21 +OBL +286.22 +286.23 +6 +1 +0 +0 +0 +Galactic LatitudeMagnetic Fields and Gas Structures in BYF 73 +9 +now, we prefer the pure B⊥-twist interpretation since it +is the simplest. +Upon further inspection of the inner 0.5 pc ∼ 0◦.01, one +can see another significant feature around MIR 2 in the +polarisation maps, even where the central I structure is +very smooth. While the I and P ′ emission peak exactly +on MIR 2’s position, the I morphology is slightly more +extended to the east, compared to the sharper decline +towards the west/H ii region. This morphology is mim- +icked in the inner P ′ distribution, i.e., where P ′ > 0 in- +side the IBL, except that in P ′ the point-source nature of +MIR 2 is much more distinct, while the extension to the +east is revealed as a semi-circular ring structure adjacent +to MIR 2. We dub this the “eastern polarisation lobe” +(hereafter EPL). Morphologically, it seems unlikely that +this lobe is associated with any of MIR 1–8, as can be +seen in Figure 6, which overlays their positions on both +the P ′ and p′ images. This is underscored by indications +from P18 that MIR 1 and 3–8 are possibly on the near +side of the BYF 73 cloud, and not as deeply associated +with the MIR 2 core. +Intriguingly, the EPL shows a similar polarisation sig- +nature to the MIR 2 core proper (see top panel of Fig. 5) +but inverted in Q, going from negative values outside +the EPL to positive values across it. +This is equiv- +alent to a sharp rotation of the inferred B field be- +tween each structure as we will see in the next sec- +tion, further suggesting that the EPL is distinct from +the MIR 2 core/protostar, each having its own physics, +despite the much more amorphous appearance around +MIR 2 in Stokes I (Fig. 1). Identification of these fea- +tures was based solely on the HAWC+ data, and before +the ALMA maps (next) were in hand. +3.4. Magnetic Field Structures in the Molecular Core: +ALMA +Turning now to the ALMA data, the only high-column- +density dust structures seen in the 3 mm continuum +(Fig. 2) are (i) MIR 2; (ii) the 1–3 mJy/beam structures +east and west of it which we call the “Streamer” and +“Streamer-west,” respectively; (iii) a ∼0.5 mJy/beam lin- +ear feature aligned almost exactly N-S with the H ii re- +gion’s ionisation front (IF); (iv) another ∼0.5 mJy/beam +patch NE of MIR 2 aligned with the EPL; (v) some even +weaker diffuse features ∼1′ to the east of MIR 2; plus +(vi) three other eastern point sources which we have des- +ignated MIR 11-13 in Figure 3. The larger-scale features +in Stokes I at the two different wavelengths have a very +nice overall correspondence, despite the different resolu- +tions: the EW extension of the brightest core emission, +the weaker emission extending north along the IF, and +the new point sources MIR 11–13 and diffuse emission ex- +tending to the east all look mutually consistent. Even the +EPL’s structure, inferred from the HAWC+ data alone, +is easily and gratifyingly verified in the ALMA I images. +The detectable ALMA polarisation, however, is limited +to a subset of these features, namely MIR 2, both sides of +the Streamer, the EPL, and possibly the southernmost +parts of the IF (near MIR 4). +The ALMA P ′ map is therefore much more spatially +compact than the HAWC+ P ′ map. +However, the +ALMA p′ values in the molecular core are quite large, +typically 5–20% or more in high S/N areas, as opposed +to the more typical HAWC+ p′ values around 1% within +HAWC+ P ′ image +I contours +P ′ contours +HAWC+ p′ image +I contours +P ′ contours +Figure 6. +Further zoom-in (compared with Figs. 4–5) to MIR 2 +area in P ′ (top) and p′ (bottom), with the same contours as in Fig. 3 +(i.e., I = yellow at 0.44(0.10)0.84, 1.5, 3, 5, 9 Jy/pixel, P ′ = grey +at 50(16)98, 140 mJy/pixel). Here we also label the locations of +MIR 1–8 from P18 (numbers with magenta circles), the MIR 2 core +(black), the eastern polarisation lobe (EPL, blue), and the inner +P ′ = 0 boundary layer (IBL, red). +the IBL/molecular core (HAWC+ p′ is 10% or more only +in the H ii region, but does rise to ∼5% in the diffuse, +eastern extremes of the molecular cloud). This lower per- +centage polarisation at shorter wavelengths could be due +to two effects: +(A) The polarisation signal is being diluted in the +larger HAWC+ beam due to its origin in small structures, +such as those found in the ALMA maps, but which to +some extent cancel each other out in the HAWC+ beam. +For example in the MIR 2 core, the correspondence be- +tween HAWC+ and ALMA vectors is modest, and the +ALMA vector PAs vary more strongly than the HAWC+ +PAs. However, due to the correspondence between both +HAWC+ and ALMA inferred B field morphologies de- +scribed below, we discount this effect. +(B) More probably, in the denser parts of the cloud, +the 3 mm emission is more efficiently polarised by the +cold dust than the 154 µm emission: the “polarisation +spectral index” (PSI) is >1. This would run counter to + +ixel +0.15 +0.10 +0.05 +0.00 +286.200 +286.205 +2 +Galactic +286.210 +286.215 +180 +165 +0.1 +Galactic Latitudepercent +6 +4 +2 +286.200 +286.205 +Cor +2 +MIR +Galactic +286.210 +IBL +286.215 +081 +0.170 +.65 +0.1' +0 +Galactic Latitude10 +Barnes et al. +HAWC+ ALMA P ′ images +2% 30% +Figure 7. +Zoom in to all high-S/N polarisation data among the central structures of the BYF 73 molecular core, from HAWC+ (green P ′ +image with displayed range 0–190 mJy/pixel, and cyan rotated p′ vectors with maximum 1.47%) and ALMA (red P ′ image with displayed +range 0–32 mJy/beam, and pink rotated p′ vectors with maximum 65%). These are like Fig. 4 for HAWC+ and the bottom panel of Fig. 2 +for ALMA, with coloured p′ vector scale bars in the bottom-left corner and coloured beam sizes in the bottom-right. For HAWC+, the +typical uncertainty above 50 mJy/pixel is ∆P ′ = 5–6 mJy/pixel and ∆θ = 2◦, for a S/N = 10–30. For ALMA, the uncertainty is ∆P ′ = +23 µJy/beam, giving a typical ∆θ = 5◦and S/N = 5–10. Six of the 8 MIR point sources from P18 are also shown for reference, as are labels +for the MIR 2 core, and in orange, the ALMA Streamer and the arc of the EPL. +the situation in the ρ Oph cloud, where radiative torques +from external illumination are thought to more efficiently +align grains in the less dense parts of that cloud, giving a +PSI < 1 (Santos et al. 2019). Here, we argue that MIR 2’s +radiation could be aligning grains more efficiently in the +cloud core, if radiative torques from internal illumination +are the cause (Lazarian 2007). +We overlay both instruments’ polarisation maps of the +IBL, the peak column density area in all maps, in Figure +7. Within the cold, high column density dust preferen- +tially traced by the 3 mm maps, we note two distinct +magnetic domains comprised of five sub-structures, each +with its own orientation and character: +(1a) Close in to MIR 2, the field is oriented mostly +N-S, which is very similar to that inferred from the +HAWC+ data, but with amplitudes p′∼1% for HAWC+ +and p′> +∼3% for ALMA. We call this the “MIR 2 core.” +(1b) Just to the SE of MIR 2, at the western end of +the main Streamer, there is a small patch of polarisation +with a similar N-S orientation, which we call the “MIR 2 +extension.” +These two structures comprise the predominantly N- +S magnetic domain inside the IBL. The following three +structures comprise a different magnetic domain, ori- +ented mostly E-W or somewhat NE-SW. +(2a) Across the EPL, the uniformity is almost as good +as in the MIR 2 core, with most HAWC+ vectors p′∼1% +@ N60◦E, while the ALMA vectors range over p′=10– +20% and run mostly E-W, although some vectors turn +towards ∼N60◦E at the more distant fringe from MIR 2. +(2b) Along the main Streamer east of MIR 2, ALMA +vectors are p′∼5–15% while running mostly E-W nearer +to MIR 2, but again turning more towards ∼N60◦E as +they move away from MIR 2. HAWC+ does not detect +high S/N polarised emission from the Streamer; thus, its +vectors are somewhat jumbled in orientation there, but +their alignment with the ALMA vectors is still reasonably +good. +(3) In the Streamer-west, while the HAWC+ vectors +continue to align N-S, the ALMA vectors turn E-W, but +this is beyond the reliably-calibrated polarisation radius +in the ALMA field, so the divergence may not be signifi- +cant. It is also possible that, because of the larger beam, +the HAWC+ data are dominated by the bright emission +from MIR 2 (polarised N-S) further into the Streamer- +W, IF, and H ii region than are the ALMA data, before +HAWC+ finally picks up the E-W B field orientation in +the H ii region itself. If true, this would make the po- +larisation signal from both instruments more consistent +with each other here too, as per effect B above. +In terms of the P ′ null and sharp 90◦ B⊥ twist seen in +the HAWC+ data, Figure 7 seems to suggest that it is +mostly an artifact of resolution and sensitivity, as per the +pure B⊥-twist explanation (§3.3). In other words, we can +see the inferred B⊥ direction change quickly between the +MIR 2 core and the Streamer-west, right under the edge +of the N-S B⊥ vectors in Figure 7, even if the ALMA +Streamer-west vectors are less reliable there. + +286.204 +286.206 +4 +MIR 2core +Longitude +2 +286.208 +5 +Galactic +286.210 +EPI +6 +286.212 +streamer +286.214 +0.174 +0.172 +0.170 +0.168 +Latitude +GalacticMagnetic Fields and Gas Structures in BYF 73 +11 +Figure 8. +12CO outflow features overlaid on the 3 mm Stokes I continuum mosaic from Fig. 2; the latter is displayed as a greyscale + +grey contours at 0.4, 1, 2, & 5 mJy/beam. The blue and red wings of the 12CO Stokes I emission are shown with blue and red contours at +levels of 30(40)450 K km s−1 in each case, integrated from –60 to –22 km s−1 and –17.5 to +18 km s−1 respectively, for all voxels above 5σ +using the smooth-and-mask (SAM) algorithm (Rots et al. 1990). The lowest red and blue contours also approximately indicate the circular +boundary of the single ALMA polarisation field at the 20% primary beam cutoff. The average rms noise in the 12CO line wing maps is 0.14 +(blue) and 0.25 (red) K km s−1, giving S/N peaks ∼2400 and 1850, respectively. Green labels show various continuum features as discussed +in the text, along with selected MIR point sources (Fig. 3) in magenta with white labels, except for MIR 2 which is shown in black. The +synthesised beam is 2.′′83×2.′′66 @–35.4◦, shown in the top-left corner as a gold ellipse. +In summary, the significant B field structures in the +molecular core of BYF 73 are MIR 2, the EPL, and the +Streamer, where both the HAWC+ and ALMA inferred +B fields are broadly consistent. The B field structures +seen by both facilities in the H ii region may also be con- +sistent with each other. We reserve discussion of the B +field structures in the H ii region for §5.1. +4. FEATURES OF THE SPECTRAL LINE EMISSION +4.1. A Strong Bipolar Outflow in 12CO +The continuum structures seen in the molecular core at +both the SOFIA/HAWC+ and ALMA wavelengths are +intriguing, both in total intensity and polarised emission. +However, while apparently related to the dominance of +MIR 2, from their structure alone their physical signifi- +cance is not entirely obvious, nor is how they are con- +nected to BYF 73’s star-forming activity. +Not surprisingly, the ALMA spectroscopy provides im- +portant insights; perhaps more surprising is that it is the +12CO data that provide the key. +Although the 12CO +spectral polarisation cubes only cover a single ALMA +field as in the bottom panel of Figure 2, the information +they reveal about the nature of the continuum emission +in BYF 73 is pivotal. First, the brightest 12CO emission +by far lies in the highly Doppler-shifted line wings, ex- +tending up to ±35–40 km s−1 from the cloud’s systemic +VLSR, as illustrated in Figure 8. +Spatially, these line +wings delineate a massive bipolar outflow clearly ema- +nating from MIR 2. +The opening angle appears small +near MIR 2, θopen< +∼10◦, and the outflow appears to im- +pact the Streamer: the flow directions are apparently +strongly affected by the large inertia of ambient cloud +material in the Streamer, and deviate from their initial +vectors. Based on the small θopen and lack of overlap +between the red & blue wings, we estimate the outflow’s +inclination to the line of sight lies in 40◦< +∼θincl < +∼80◦. +From detailed inspection of the 12CO Stokes I cube, +the intrinsic outflow direction from MIR 2 is along the +magenta line in Figure 8 at a Galactic PA = 120◦, but +this terminates at the magenta boxes at each end of that +line. The red-shifted outflow then deviates around both +sides of the Streamer-West along the paths indicated by +the gold arrows in Figure 8. The blue-shifted outflow is +apparently deflected by the highest-density portion of the +Streamer into the direction shown by the cyan arrow of +Figure 8. Spectrally, the highest relative speeds appear +to lie near MIR 2 in the red wing, but are displaced from +MIR 2 by ∼14′′ = 0.17 pc downstream in the blue wing. +Apart from this offset, the outflow speeds are generally +more modest as one looks further downstream. +In the case of the blue wing, the outflow direction along +Galactic PA = 120◦ close to MIR 2 is seemingly deflected + +beam +0.004 +0.003 +0.002 +0.001 +0.000 +3 +286.205 +286.210 +Longitude +EPI +IBL +Galactic j +286.215 +9 +D +286.220 +Q +0.180 +0.175 +0.170 +0.165 +0.160 +Latitude +Galactic12 +Barnes et al. +by a clear 47◦ “bounce” into a single new direction along +PA = 73◦. The flow then continues to at least the east- +ern edge of the single polarisation field, 0.6 pc away from +MIR 2. This deflection is clearly seen in the individual +12CO Stokes I cube’s channel maps, and is not an arti- +fact, for example, of opacity-masking of a more southerly +portion of the blue wing by the Streamer, hiding a con- +tinued blue outflow along PA = 120◦. This is because (1) +the Streamer and outflow are well separated in VLSR, so +there is no opportunity for the Streamer to mask some +parts of the blue wing (see also below); and (2) the 13CO +line, which is much lower opacity than 12CO, shows the +same structural features coincident with the deflection +9′′ east of MIR 2. +In the case of the red wing, the initial outflow from +MIR 2 along PA = –60◦ appears to be somewhat blocked +by the Streamer-west, such that the flow deviates to ei- +ther side of this obstruction, before continuing to flow at +the same PA to the western edge of the field, 0.3 pc away +from MIR 2. These deviations are similarly easy to see +in the channel maps. +Note that the outflow widths, at ∼10′′–15′′, are well +under the ALMA MRS in the single 12CO field, so we +believe we recover essentially all the outflow structure in +the line wings. It is difficult to say, however, if the out- +flow continues beyond these boundaries (e.g., into the +H ii region), since the 12CO data are limited by the field +of view. +But it is fairly obvious that the outflow and +Streamer interact strongly, one sculpting the other, in- +cluding the appearance of the EPL and Hole. +The second noticeable feature of the 12CO data, apart +from where the outflow can be specifically traced in to +MIR 2, is that the rest of the cloud is much fainter in +the line core between roughly –22 and –17 km s−1, with +any non-outflow features being <10% as bright. +This +confirms that the cloud is extremely optically thick ev- +erywhere, and that except for the large outflow-driven +Doppler shifts, virtually no 12CO emission can escape +from the cloud’s interior. +Third, and even more interestingly, we detect the +linearly-polarised Goldreich-Kylafis effect almost every- +where in the outflow, and at high S/N in Pd in almost all +channels which trace the outflow in I. This is presented +and analysed in §5.3. +4.2. Cloud Architecture from Spectral Line Mosaics +The ALMA 13CO, C18O, & CN mosaics provide fur- +ther details for analysis of the BYF 73 cloud emission, +but we focus here on kinematic features associated with +the Streamer and MIR 2, in order to shed further light +on the B field structures described above, and their dy- +namics. +In contrast to the very compact 3 mm continuum emis- +sion (compared to the mosaic size, Fig. 2), the 13CO, +C18O, & CN mosaics illustrated in Figure 9 all show +much more extended structure, although this emission +is brightest near the continuum features, and for 13CO, +across much of the IF visible in the Spitzer images as well +(Fig. 3). The 13CO and C18O emission fills most of the +mosaic, seemingly even extending beyond it, in all direc- +tions for 13CO, and to the north and east for C18O. Even +the CN is somewhat extended, although less so than the +iso-CO lines. The 13CO+C18O extents include parts of +the H ii region (the area west of the IF), presumably due +to residual molecular gas on its near and far sides that +has not yet been ionised by the UV field or swept up in +the general H ii region expansion. Some of this effect is +more easily visible in the LV diagram of Figure 10, where +the 13CO is brightest at velocities slightly redward and +blueward of the C18O across the cold cloud. +In fact, in the data cubes this is very widespread: the +effect can be seen at positions and velocities of nearly all +structures, even deep within the molecular cloud. There +are many extended, often filamentary features with shal- +low velocity gradients and a distinct 13CO layer lying just +westward of, and slightly red- or blue-shifted from, each +bright C18O structure. Evidently, the cloud is actually +somewhat porous to the UV field emanating from the H ii +region, despite the cloud’s more opaque appearance in +the near-IR. The C18O structures then seem to delineate +the colder, more shielded parts of the cloud’s interior, +while the cocooning 13CO around each feature may define +its more excited side, facing the H ii region. Curiously, +the brightest CN emission seems to track better with +bright 13CO in position and velocity rather than with +C18O, although widespread fainter CN does lie across +the mosaic and various C18O features. The variation of +line ratios with position and velocity is difficult to por- +tray here, but Figures 9 and 10 give some idea of the +complexity. +The most noticeable kinematic feature in the spec- +tral line cubes is an EW velocity gradient across the +Streamer, one remarkable in several respects. Near the +bright continuum emission (and thus the brightest parts +of each emission line), the gradient is consistent across +all three species, reaching a maximum blueshift of –22.0 +±0.1 km s−1 about 20′′ = 0.25 pc east of MIR 2, and a +maximum redshift of –17.0±0.2 km s−1 around 11′′±2′′ += 0.13±0.02 pc northwest of MIR 2: see Figure 11. The +gradient is also at its sharpest exactly across the middle +of MIR 2 itself, ∼2–3 km s−1 across only 1 ALMA beam +(∼6000 AU = 0.03 pc), or ∼75 km s−1 pc−1. The steep- +est part of the gradient, defining a symmetry axis, lies +on a nearly N-S curve, just like the B field orientation +in Figure 7. +Moreover, this symmetry axis across the +middle of MIR 2 (straddling the width of the Streamer) +looks identical in all 3 lines, underscoring its dynami- +cal importance and strongly suggesting a flat NS feature +within MIR 2 as the origin for the outflow (more on this +next). Meanwhile, the full extent of the EW blue-to-red +velocity gradient lies along a curved line across l∼286◦ +.23–286◦.20, roughly 1.3 pc. +These details suggest the possibility that the main and +western extensions of the Streamer form part of a rather +large structure (perhaps including a disk), in which in- +ward flow to MIR 2 must occur and there generate the +outflow. While the case is somewhat circumstantial so +far, the evidence becomes much stronger upon closer in- +spection. +4.3. High Velocity 13CO Emission: +A Massive Keplerian Disk or Freefalling Accretion? +For BYF 73 as a whole, we estimate that its systemic +velocity is Vsys = –19.6±0.2 km s−1 on the LSR scale, +based on inspection of the C18O data cube. +As seen +in Figures 9+10, nearly all of the mosaics’ line emis- +sion lies between –24 and –16 km s−1. However, there + +Magnetic Fields and Gas Structures in BYF 73 +13 +Figure 9. +Slightly cropped composite RGB image of ALMA spectral line mosaics’ total intensities (species as labelled, integrated from +–24.5 to –15.5 km s−1 for all voxels above 5σ using SAM; Rots et al. 1990) and overlaid by contours (at 0.4, 1, 2, and 5 mJy/beam) of +the 3 mm continuum (Fig. 2) plus selected MIR point sources (Fig. 3). The IBL, EPL, and IF (Figs. 6–8) are also labelled in yellow. The +brightness scales in each colour channel run from dark at 0 to saturation at 38.15, 11.71, & 11.65 K km s−1, respectively; the respective +overall peak levels, and error levels in areas away from the map edges, are 53.28±0.11, 13.42±0.08, and 16.66±0.21 K km s−1. Thus, the +C18O and CN moment maps have typical S/N ≈ 20–60, while that of 13CO is 100–300 across much of the mosaic. +Figure 10. +Longitude-velocity diagram of the same data presented in Fig. 9, i.e., integrated over the same latitude range as displayed +therein. The brightness scales are dark for 0 K arcmin in each colour channel, and the saturation/peak ± uncertainty levels (in areas away +from the map edges) are 13CO 10.42/12.77±0.04 (red), C18O 4.14/5.08±0.04 (green), and CN 1.73/2.69±0.10 K arcmin (blue). +is clear evidence in the 13CO cube of high-velocity line +wings close to the position of MIR 2: up to ∆Vblue = +–11.5 km s−1 and ∆Vred = +16 km s−1, for a total VLSR +range of 27.5 km s−1 (i.e., from –31 to –3.5 km s−1). This +emission is quite small in spatial extent, with length ∼ +20′′ = 0.25 pc and width 10′′–15′′ = 0.12–0.18 pc for each +lobe: see Figure 11. +This compact configuration is completely different to +the massive, more extended bipolar outflow clearly vis- +ible in 12CO (§4.1), which is indicated in the bottom +panel of Figure 12 to help distinguish the LV patterns of +the two species. In particular, the 12CO outflow emis- +sion that extends beyond an area ±10′′ around MIR 2 +must be relatively low opacity τ and high excitation Tex, +since it is much brighter than the superimposed 13CO +emission, where the latter is even detectable at the same +(l,V ) coordinates. +In contrast, the high-velocity emis- +sion coincident with MIR 2 has 13CO almost as bright as +12CO, suggesting a much higher τ and lower Tex. The +respective excitation conditions are consistent with gas +being entrained by a powerful mechanical outflow, and +gas responding to the local gravitational potential. +Finally, the extended, low-velocity 13CO wings are not +visible in 12CO due to the latter’s high τ, although much +of the 13CO line wing emission (Fig. 11) is spatially ori- +ented similarly to the inner 12CO outflow, including the + +286.20 +3 +Galactic Longitude +286.21 +286.22 +2 +3 +9 +13CO +C180 +CN +286.23 +0.18 +0.17 +0.16 +Latitude +Galactic286.20 +286.21 +Galactic j +286.22 +286.23 +-16 +-18 +24 +1 +1 +1 +1 +(km/s) +Velocity14 +Barnes et al. +Figure 11. +Velocity fields (first moments with SAM) of 13CO line wings integrated from –31 to –24.5 km s−1 (left, blue wing) and from +–15 to –3.5 km s−1 (right, red wing). The colour scales match the corresponding truncated velocity wedges. Overlaid are the same 3mm +continuum contours as in Fig. 9, plus selected labels. +43◦ bend in the blue wing and the deviations around the +Streamer-west in the red wing (Fig. 8). Line wings simi- +lar to either the 12CO or 13CO high-velocity patterns are +not detectable in either the C18O or CN cubes. +Thus, while the brightest 13CO emission is probably +also tracing the outflow, the small extent of the high- +∆V emission is more peculiar. It becomes progressively +tinier as the velocity channel being viewed moves further +from the cloud’s systemic value, contracting to within a +beamwidth of MIR 2 at the highest velocities. This is +the opposite of what is typically seen in protostellar jets, +where the highest velocities are usually at the most distal +parts of the outflow (Lee et al. 2000). +Instead, the LV-moment diagrams in Figure 12 suggest +this pattern might arise from a Keplerian disk. Sample +rotation curves Vrot = +� +GM/R are overlaid for 3 differ- +ent central masses in Figure 12 as well. The only free +parameter in fitting such curves is the central mass:13 +the position of MIR 2 is well-constrained, as is the veloc- +ity extent of the emission. Only the highest mass curve +of the 3 examples fits the high-∆V 13CO emission enve- +lope adequately. The lower mass curves and P18’s mass +estimates are all much too small, and are strongly ruled +out under this interpretation. +Indeed, a 1350±50 M⊙ curve fits the data remarkably +well over a longitude extent of 0◦.008 = 0.35 pc, or a +radius of 36,000 AU, which is about half the longitude +extent (286◦.203–216) of the IBL as measured in the +HAWC+ data (Fig. 6). Such a disk would also be ∼half +the size of the Keplerian disk seen in another massive +cloud, K3-50 (Howard et al. 1997), and about half or less +of K3-50’s disk mass (Barnes et al. 2015), so these num- +bers are not unheard of in high-mass SF. But it means +that the central mass of MIR 2 (presumably comprising +a massive protostar and its envelope) is 5–6× larger than +P18’s preferred estimate, and that it dominates the dy- +13 Of course, the distance (2.50±0.27 kpc) also matters. If this +is changed, the linear scale and mass will change proportionately. +However, with an 11% uncertainty, the implied mass of MIR 2 re- +mains above 1200 M⊙ for rotation, or above 850 M⊙for infall. +namics of the gas over that span. +Alternatively, the kinematics can also be explained by +gas in free fall towards a 950±35 M⊙ object, since Vff = +√ +2Vrot decreases the central mass required to produce +the curves seen in Figure 12 by +√ +2. Reference to either +scenario hereafter is meant to include both as feasible +physical settings near MIR 2. +In our view, such rotation/freefall curves are so distinc- +tive that there is no other reasonable explanation for the +motions of the gas, at least within the IBL (i.e., excluding +a much smaller amount of apparently counter-rotating +gas in the same window). Outside the IBL, the devia- +tions from Keplerian rotation are stronger, and may be +due to more typical, modestly turbulent cloud motions +and/or internal structures. +4.4. The Eastern Polarisation Lobe (EPL) +Even if we have constructed a plausible picture of +BYF 73’s internal structure and mass flows, the EPL is a +distinctive feature in both the SOFIA and ALMA polar- +isation maps (Fig. 7) with an as-yet undetermined role or +import. It lies north of the plane of the Streamer/disk +around MIR 2, yet the inferred B field directions through +it still point towards/away from MIR 2. It is bright in +line emission as well (Fig. 9), particularly C18O, but also +13CO and CN in turn around its arc. The velocity fields +(Fig. 10) reveal little except that it is close to systemic +(–20 km s−1) in C18O at its northern apex, and slightly +blueshifted (–21.5 km s−1) in both CN and C18O at its +base near the Streamer, but blending smoothly with the +Streamer’s rotational pattern. +Scanning through the +channels in the 13CO cube, the changing pattern gives +the impression of a splash effect or prominence-like ten- +dril, driven by the outflow off ambient Streamer material +downstream of the 47◦ bend in the blue wing, with a +relatively gentle relative blueshift of 1.5 km s−1. +If true this would be quite interesting: does it signify +the expulsion of surface material from the Streamer? +Or is it a wisp from the wider cloud, falling in at a +slightly higher speed, unconstrained by the Streamer due + +2 +0 +286.205 +Longitude +286.210 +Galactic l +C +286.215 +286.220 + Latitude +Galactic9 +22 +2 +286.205 +Longitude +286.210 +IBI +Galactic +5 +286.215 +286.220 +8 +5 +5 +6 +0 +0 +0 +0 +Latitude +GalacticMagnetic Fields and Gas Structures in BYF 73 +15 +Streamer +12CO Outflow + + + + + }Rotation +} +Figure 12. (Top Left) Integrated longitude-velocity diagram (ze- +roth moment) of 13CO emission across the Streamer, latitudes 0◦ +.1677–0◦.1733. +The log10 brightness scale (needed to display the +image’s dynamic range of ∼200) peaks at 4.92±0.02 K.arcmin. The +emission at –8.5 km s−1 is presumed to arise in a diffuse foreground +cloud unrelated to BYF 73. Overlaid are coloured curves represent- +ing Keplerian rotation for 3 sample masses contained within 1.′′8 of +MIR 2’s position, each half joined by a straight line inside that ra- +dius. Dotted lines indicate MIR 2’s longitude (l = 286◦.20745 from +T-ReCS astrometry; P18) and BYF 73’s Vsys = –19.6 km s−1. +(Top Right) Longitude-velocity first moment map of the same data +as in the top panel. The colours represent the intensity-weighted +mean latitude of the integrated emission, e.g., the highest-velocity +emission lies at the same latitude as MIR 2 (the cyan colour, b = +0◦.16975±0◦.00004 in T-ReCS’ 0◦.00007 = 0.′′25 PSF/beam). +(Bottom Right) Longitude-velocity second moment (intensity- +weighted latitude dispersion, or latitude width of the emission) of +the same data as in the above panels, with additional labels to dis- +tinguish between the 12CO and 13CO LV patterns. On this scale, +unresolved features smaller than the ALMA beam (∼2.′′6) are a +medium blue or darker, such as the highest-velocity emission, all +velocities along the inner edge of the disk, and along a good por- +tion of the 1350 M⊙ curve. Most of the interior of the disk, i.e., the +portions between the 1350 M⊙ curve and VLSR = –19.6 km s−1, +have widths < +∼ 5′′, while the super-rotating and solid-body portions +of the Streamer (the brighter features in the top panel) have widths +up to ∼7′′. +to its northerly approach? While at a lower surface den- +sity than the disk, perhaps 1027 m−2 or a tenth of the +Streamer (§5.2), the continuum data show that its mass +is not insignificant, perhaps 15 M⊙ in total. Higher reso- +lution polarisation data would be desirable to determine +exactly what the EPL signals. +4.5. A Second Massive Protostar +Around MIR 11 there appears to be a second strong ve- +locity gradient, similar to that straddling MIR 2 (§4.1). +This gradient is somewhat vague in 13CO, clearer in +C18O, but most obvious in CN (see Fig. 13), and is +oriented with the strongest ∆V along a similar axis +(∼N40◦E) as the biconical mid-IR nebula (Figs. 3e, f): +blue-shifted emission on the more IR-visible side to the +NE, and red-shifted emission to the SW. The various line +profiles show only a narrow velocity range, however, ±2– +5 km s−1, rather than a strong outflow signature, and +for 13CO and C18O, the red- and blue-shifted emission +overlaps somewhat on the sky. Both lines have strong +self-absorption around the line centre of at –19.8 km s−1. +The CN is better separated on the sky into blue and red +lobes, with no self-absorption. These biconical character- +istics suggest the possibility of MIR 11 being in an even +earlier, pre-outflow stage of evolution than MIR 2. +5. MAGNETIC FIELD ANALYSIS +5.1. Davis-Chandrasekhar-Fermi (DCF) Method +5.1.1. Preamble +We start with the method of Barnes et al. (2015) +to make some reasonable estimates of B field strength, +based on the dispersion s in inferred field directions from +the polarisation data. The basic DCF approach (Davis +1951; Chandrasekhar & Fermi 1953) assumes a statisti- +cal connection between turbulent motions in the gas and +the dispersion in B field direction in the presence of a +transverse MHD wave. Such analysis is necessarily ap- +proximate, since other thermal, rotational, gravitational, + +log(K.arcmin) +5 +0 +5 +5 +0.0 +286.20 +286.21 +Galactic +286.22 +G +286.23 +5 +5 +25 +30 +Velocity (km/Latitude(degrees) +0.172 +0.171 +0.170 +0.169 +3 +1 +286.20 +286.21 +Galactic +286.22 +286.23 +5 +5 +5 +30arcsec +8 +2 +286.20 +286.21 +Galactic +286.22 +286.23 +5 +5 +5 +22 +Velocity (km/s16 +Barnes et al. +Figure 13. +ALMA mosaic velocity field (first moment) of main +hyperfine component of CN around MIR 11 (black), overlaid by +white ALMA 3mm I continuum contours at 0.4, 1, 2, 4 mJy/beam +and orange HAWC+ I contours at 0.64(0.10)0.84 Jy/pix (compare +with Figs. 3e, f). +or even magnetic effects may affect the two processes +treated by DCF. On the other hand, even for supersonic +(M ∼ 5–9) MHD gases (as in H ii regions), Ostriker et +al. (2001)’s numerical simulations showed that the DCF +method can give some useful information, despite not +being developed for such a setting. +One approach to evaluating the behaviour of s is that +of Myers & Goodman (1991). +In their language, the +goal is to identify a “correlation length” in the implied +B field orientation, within which the B field directions +are correlated and aligned with each other, and outside +of which they are not. +Using the formalism of Myers & Goodman (1991), we +fit the distributions of polarisation position angle θ with a +simple gaussian e−θ2 +B/2s2 to obtain a best-fit value for the +dispersion s in θB (measured in radians). Our method is +a simplified version of Myers & Goodman (1991)’s anal- +ysis, since they showed that this approach gives very +reliable results even for their comprehensive data (hun- +dreds of stellar polarisation measurements) on the Tau- +rus molecular clouds. We have a smaller data set of θB, +so will not need the full Myers & Goodman (1991) treat- +ment. +5.1.2. HAWC+ Data Analysis +In the HAWC+ data, the measured θB⊥ in any given +region with S/N > +∼5 will have an uncertainty in orienta- +tion dominated by instrumental noise, ∆θB⊥ +< +∼3◦. This +occurs at a level slightly less than the P ′ = 50 mJy/pixel +contour in Figures 1 and 3, and we show the correspond- +ing θB⊥ maps in Figure 14. There are three regions of in- +terest (hereafter ROI) satisfying these criteria: two arcs +of polarised emission in the H ii region (labelled north +& south), and the area within the IBL containing the +MIR 2 core (but excluding the EPL due to the paucity +of statistics, ∼20 pixels). To begin the DCF analysis, +we show in Figure 15 the θB⊥ distribution in all pixels +within each ROI. None of these is really gaussian, but we +overlay such fits in order to compute effective dispersions +in θB⊥ for reasons which will become clear shortly. +We next construct histograms for subsets (comprised of +square boxes of area A) of each ROI, computing a disper- +sion s(A) in θB⊥ for each subset. The smaller the boxes, +the more choice we have of where to fit them inside each +ROI. We then compute a mean dispersion +(± a standard deviation) in the field orientation for all +small boxes of a given area A, no matter where they are +placed within the ROI. Finally, in Figure 16 we plot all +such results as a function of box size A1/2, ranging from +a minimal useful size of A = 3×3 pixels (roughly one +Nyquist sample given the HAWC+ band D beam) to the +full size of each ROI. +In each ROI, the mean dispersion within all boxes +of area A rises as the box size increases, meaning that +θB⊥ is more correlated on small scales (e.g., s < 5◦ within +the H ii region over spans < 0.5 pc), and becomes less cor- +related over longer distances (s∼10◦across > +∼1 pc within +the H ii region). The scale at which s seems to plateau +is where we identify the correlation length as per My- +ers & Goodman (1991). Thus, the HAWC+ polarisation +data suggest that, as far as the B field is concerned, the +H ii region consists of one or perhaps two coherent struc- +tures, with a correlation length ∼0.5 pc as evidenced by +the plateauing of s at 6◦–7◦ in the smaller H ii-North +(Fig. 16), and the slow rise of s across H ii-South as the +B field orientation slowly changes across the 1 pc arc of +the H ii region. +In contrast, the interior of the IBL (already much +smaller than the H ii region) shows two closely-spaced +and distinct B field configurations, namely the EPL and +MIR 2 core. +While no useful statistics could be com- +piled for the EPL, even the MIR 2 core has insufficient +resolution to discern more than a strongly rising s at all +measured scales, and no correlation length can be defined +beyond the ∼0.2 pc extent of the core itself, as in Figure +7. +5.1.3. ALMA Data Analysis +The same approach can be used with the ALMA polar- +isation data, which has high enough resolution to resolve +the DCF analysis for the MIR 2 core and its environment, +unlike its minimal representation in Figures 14–16. We +therefore define the ALMA ROIs in Figure 17 where θB⊥ +has S/N > 2.5 but is typically 5–10, as in Figures 2 and 7 +and conforming to the description in §3.4. Then in Fig- +ure 18 we show the ALMA θB⊥ distributions, compiling +the ALMA DCF statistics in Figure 19. +We first notice that, in the overlapping range of scales +(0.1–0.3 pc), the dispersion s in the IBL from both +ALMA and HAWC+ data are in agreement, rising ap- +proximately from 10◦ to 20◦ when averaging broadly over +the 5 structures in Figure 19. Focusing next on the small- +est ALMA scales (0.02 pc), s in the IBL also starts out +at a few degrees, then gradually rises. In each of the 5 +ROIs, Figure 19 hints at an s plateau for each structure, +before rising further as other uncorrelated structures are +included. In Table 1 we compile the (A1/2 +corr, scorr) pairs +that can be read off the trends in Figure 19, and for com- +pleteness also include the values for the H ii region ROIs +from Figure 16. The EPL seems to have the fastest-rising +s and the least well-defined plateau in Figure 19, suggest- +ing that it has not mapped out a single correlation length +in its structure. +Therefore, the measurable correlation lengths in the +IBL are perhaps only a tenth those in the H ii region, +0.05–0.15 pc compared to ∼1 pc. The dispersions in B +field direction at those lengths are respectively ∼6◦–20◦, + +8 +9 +0 +2 +2 +2 +2 +286.222 +286.224 +e +286.226 +Galactic +286.228 +286.230 +2 +g +9 +1 +1 +1 +0 +0 +Galactic LatitudeMagnetic Fields and Gas Structures in BYF 73 +17 +H iinorth +EPL + +MIR 2 core +H iisouth +Figure 14. +Cutouts of the θB⊥ distribution in HAWC+ ROIs with S/N(P ′) > +∼5, overlaid by P ′ contours 50(16)98,140 mJy/pixel (the +same as in Figs. 3, 6). These ROIs correspond to arcs of polarised emission in the H ii region, and to the EPL+MIR 2 core within the IBL. +In the analysis shown in Figs. 15 and 16, however, the ROI enclosing MIR 2 excludes the pixels covering the EPL. +compared to 7◦–12◦for the H ii region. Such lengths and +dispersions are the scales above which B field directions +are not correlated with each other between neighbouring +areas, and are therefore the scales we should explore for +other physical thresholds, and particularly for any con- +straints on B field strength. +5.1.4. Numerical Results +Figure 15. +Histograms of θB⊥ at all pixels within each of the +ROI cutouts from Fig. 14, as labelled. Also shown are gaussian fits +to, and the dispersions in, each θB⊥ distribution. +As described by Ostriker et al. (2001), Crutcher et al. +(2004), Barnes et al. (2015) and many other workers, the +standard DCF analysis (Davis 1951; Chandrasekhar & +Fermi 1953) directly links the dispersions in polarisation +angle δθ = s (tracing variations in the B field orien- +tation) to three other physical parameters that simply +and naturally describe the propagation of a transverse +Figure 16. +Mean dispersion of polarization position angles θB⊥, +with the dispersion in the dispersion shown as error bars, as a +function of box size within each ROI shown in Figs. 14, 15. + +deg +-50 +50 +0 +286.1 +286.18 +286.19 +Galactic +286.20 +286.21 +9 +8 +6 +5 +0 +0 +0 +0 +Galactic LatitudeBYF73-HIIsouth +disp = 12.3° +BYF73-HIInorth +disp = 7.6° +BYF73-MIR2core +100 +disp = 16.70 +Incidence +50 +0 +-50 +0 +50 + [degrees]15 +[degrees] +10 +5 +BYF73-HIIsouth +BYF73-HInorth +BYF73-MIR2core +0 +0.5 +1 +1.5 +(Area)1/2 +pc18 +Barnes et al. +Figure 17. +Cutouts of the θB⊥ distribution in ALMA ROIs) with S/N(P ′) > +∼3, overlaid by P ′ contours 50(16)98,140 mJy/pixel (the +same as in Figs. 2, 7). These ROIs correspond to polarised emission from the Streamer, EPL, and parts of the MIR 2 core within the IBL. +MHD wave in a turbulent plasma: the gas density n, the +line-of-sight velocity dispersion δV , and the plane-of-sky +magnetic field strength B⊥. That is, s will increase as +(1) B decreases, since then the magnetic restoring forces +are reduced; (2) n increases, since then the medium’s in- +ertia to the MHD wave is greater; or (3) δV increases, +since that describes the strength of the MHD wave. +According to Crutcher et al. (2004), with appropriate +SI unit conversions (1 nT = 10 µG) the projected B field +Figure 18. +Histograms of θB⊥ at all pixels within each of the +ROI cutouts from Fig. 17, as labelled. Also shown are gaussian fits +to, and the dispersions in, each θB⊥ distribution. +strength +B⊥,DCF = 0.543 pT √µn (∆V/s) , +(1) +where n (m−3) is the gas density with mean molecular +weight µ=2.35, ∆V = +√ +8ln2 δV is the velocity FWHM +( km s−1) in the cloud, and s is measured in degrees. In- +cluded in the constant is a numerical factor Q = 0.5 from +Crutcher et al. (2004) to correct for various smoothing +Figure 19. +Mean dispersion of polarization position angles θB⊥, +with the dispersion in the dispersion shown as error bars, as a +function of box size within each ROI shown in Figs. 17, 18. + +deg +0 +5 +0 +286.204 +286.206 +286.208 +n +-ext' +286.210 +Galactic +IBL +286.212 +286.214 +286.216 +.74 +02 +0.1 +0.1' +0.1 +Galactic Latitude80 +MIR2west +disp = 20.8° +EPL +disp = 24.1° +Streamer +60 +disp = 17.1° +MIR2core +disp = 21.1° +MIR2ext +disp = 6.70 +Incidence +40 +20 +0 +-50 +0 +50 +, [degrees]20 +[degrees] +s +Dispersion +10 +MIR2west +EPL +Streamer +MIR2core +MIR2ext +0.1 +0.2 +0.3 +(Area)/2 [pc]Magnetic Fields and Gas Structures in BYF 73 +19 +effects (e.g., see Ostriker et al. 2001). +For an illustrative example, consider the region of dens- +est gas inside the IBL. From modelling of HCO+ emis- +sion by Barnes et al. (2010) at the 40′′ Mopra resolution +(roughly 3× the HAWC+ beam), we take δV ∼ 1 km s−1 +as a median intrinsic value and the estimated peak n ∼ +5×1011 m−3 ostensibly near MIR 2, to connect scorr in +our polarisation maps to the field strength B⊥. Eq. (1) +then becomes +B⊥,DCF(MIR 2, Mopra) ∼ 92 nT (s/15◦)−1 +(2) +at these values of n and δV around MIR 2 (or 0.9 mG in +cgs units). Indeed, the value for n may well be higher in +the smaller HAWC+ beam, and is certainly much higher +in the 0.03 pc structures revealed by ALMA (see §5.2), +peaking at 3.6×1013 m−3. Then, +B⊥,DCF(MIR 2, ALMA) ∼ 1.18 µT (s/10◦)−1 +(3) +(12 mG), a value which has not been observed in any star- +forming region outside of maser spots. Even the smaller +value in Eq. (2) is among the highest non-maser B field +strengths in similar massive star-forming clouds, accord- +ing to the compilation of (Crutcher 2012, his Fig. 7). +Despite the possibly record-setting value for B⊥ near +MIR 2, it is commensurate with MIR 2’s high gas den- +sity, i.e., together they indicate a mass-to-flux ratio that +is very close to critical (see below). +In other words, +any field strength much less than this (or density much +greater) would probably not provide sufficient support +against gravitational collapse (allowing, of course, for the +likelihood of |B| > B⊥). However, this density is derived +from SED fitting which, as we have already noted, may +be significantly underestimating the gravitational poten- +tial near MIR 2, based on the apparently Keplerian or +infalling motions seen in the 13CO data (§4.3). In that +case, even this high B field cannot avoid criticality. +As a contrasting example, we also consider the H ii re- +gion ROIs. In such regions, bulk expansion speeds are +typically 2–3× the velocity dispersion (= sound speed) +in the roughly 8000 K ionised gas, ∼12 km s−1 (Habing & +Israel 1979; Franco et al. 1990). Such flows are thought +to dominate the energetics in the gas. +For BYF 73, +the H ii region has a total flux density at 843 MHz of +only 60 mJy (Green et al. 1999). This corresponds to a +small emission measure EM = n2D = 9.5×1015 pc m−6 +(Mezger & Henderson 1967). With an apparent diameter +2R=D∼0.5 pc, this yields a much lower electron density +ne ≈ 1.4×108 m−3 than in the molecular cloud,14 but we +also have a somewhat larger dust-based estimate of nd ≈ +7×109 m−3 from SED fitting (§5.2) which may average +in material from outside the H ii region. We bracket this +uncertainty by combining these values in Eq. (1) with two +estimates (e.g., using a lower µ=1.28 in the ionised gas), +B⊥,DCF(H II, ions) ∼ 21 nT (s/5◦)−1 +and +B⊥,DCF(H II, dust) ∼ 195 nT (s/5◦)−1 , +(4) +for the H ii region ROIs, depending on which parts of the +line of sight through the H ii region are being sampled by +14 From these parameters, one can also derive an excitation pa- +rameter U = Rn2/3 = 6.7×104 pc m−2 (Mezger et al. 1967) for the +H ii region, needing only a single ∼B1 star (Panagia 1973) to power +it, and confirming its modest impact on the molecular cloud. +Table 1 +Davis-Chandrasekhar-Fermi correlation statistics for +BYF 73 polarisation structures from SOFIA & ALMA data +Structure +Correlation +B Field Angular +σB/ ¯B⊥ +Scale A1/2 +corr +Dispersion scorr +H ii-North +0.5 pc +6◦ +0.1 +H ii-South +1.5 pc +12◦ +0.3 +Streamer-W +0.14 pc +18◦ +0.3 +MIR 2 core +0.08 pc +16◦ +0.4 +EPL +0.12 pc +22◦ +0.6 +Streamer +0.14 pc +13◦ +0.3 +MIR 2 extn. +0.05 pc +6◦ +0.1 +the HAWC+ polarisation data. +Even the lower (pure-H ii) value seems somewhat high +compared to a more typical 1 nT in other H ii region +studies (Crutcher 2012; Barnes et al. 2015); whether this +value is reasonable is unknown, but B field measurements +could also be obtained, for example, via high-resolution +HI Zeeman observations. +The higher dust-based esti- +mate would apply if the polarisation contribution is pre- +dominantly from outside the H ii region, but then the B +field configuration still suggests a connection to the H ii +expansion. This could be reconciled with an origin in +a sheathing, higher-density PDR layer rather than the +ionised cavity. +5.1.5. Energetic Considerations +For our purposes, though, the point is whether the en- +ergy density M in these somewhat strong B fields exceeds +or is less than the kinetic energy density K in the ionised +outflow. Borrowing from Eq. (15) in §5.3, we can write +this ratio as M +K = 5.2% +� +∆V/Vrel +s/6◦ +�2 +. From the flow in the +H ii region, we have that ∆V/Vrel ∼ 1, while from Fig- +ure 16 we have s = 6◦, 12◦ in the H ii region-north and +-south, respectively. Thus, the B field energy density in +the H ii region is probably still small compared to the +kinetic energy in the ionised flow. +This approach is only valid, however, if the situation of +the DCF analysis holds, namely the presence of an MHD +wave with turbulent motions. If other processes operate, +then the B field strength may be indeterminate without +direct measurements, either smaller or larger than the +DCF value. For example, if other motions enhance vari- +ations in θB⊥, s may be larger than the DCF-only value, +underestimating B⊥. +In expanding H ii regions, the kinetic energy density of +the expansion K often exceeds the thermal energy density +T by a wide factor (i.e., in addition to exceeding M): +K/T = 2 +3M2 (Barnes et al. 2015), where M (= 2–3 in +the example above) is the Mach number of the flow. For +star formation in cold molecular gas, the most interesting +question is the relationship between B fields and gravity. +If M ≪ G, gravity dominates and the gas is considered +“supercritical;” if M ≫ G, it is “subcritical.” We discuss +this question further in §§5.2, 5.3, and 6.3. +However, there is an additional factor in criticality: we +need to understand not only the value of the dispersion +s, but also its behaviour on different length scales. As +explained by Myers & Goodman (1991), where the DCF +method applies, these dispersions are related to the ratio + +20 +Barnes et al. +Figure 20. +Relative alignment between polarization position an- +gle θB⊥ and the tangent to iso-column density N contours, as a +function of N across the HAWC+ field. +An angle of 0◦ means +the B field is oriented along the iso-N contours, while at 90◦ the +field is perpendicular to the contours and aligned with the gradi- +ent ∇N. Also shown as red lines and labelled in N, in units of +1026m−2, are the boundaries of the separate bands in N for which +each histogram in Figure 21 was computed. The boundaries were +chosen to ensure histogram equalisation, i.e., to divide all N data +into 10 equally-populated bins with comparable statistical noise in +each N-bin. +of the disordered vs. ordered B field strengths via +scorr = +σB +L1/2 ¯B⊥ +. +(5) +Here scorr is measured in radians, L is the number of B +field correlation lengths (assumed ≈ Acorr) in the line of +sight, ¯B⊥ is the strength of the ordered component of +the projected B field, and σB is the dispersion in the +strength of the random component of the projected B +field. For now, we estimate L from the behaviour of s +as seen in the DCF structure functions (Figs. 16, 19), ie., +where s plateaus in each structure at some size scale A, +and so constrain somewhat the ratio σB/ ¯B⊥. +In the H ii region, L = 1–2, scorr ≈ 0.1, so we estimate +(σB/ ¯B⊥)HII ≈ 0.1–0.3. This is another way of describing +the highly ordered appearance of the B field vectors over +large scales (Fig. 1). Likewise, for the 5 structures in the +IBL, effectively L = (1–3) × A1/2 +corr by construction, and +we estimate the random:ordered B field strength ratios +for all 7 structures described here in the same way, and +list them in Table 1. +5.2. Histogram of Relative Orientations (HRO) +The HRO method of analysing B field orientations +in star-forming gas is by now a fairly standard tech- +nique (e.g., Soler et al. 2017, and references therein). +In the Vela C molecular complex, for example, Soler et +al. (2017) used BLASTPol data with a resolution 3.′0 = +0.6 pc at Vela to examine how the B field orientation +changes with column density. They found that at lower +molecular gas column densities ∼1026 m−2, the B field +direction tends to be mostly parallel to or not show any +Figure 21. +HRO plots in each of the N-bins shown in Figure +20. Each window is labelled by the range of column density N in +that bin and its fitted HRO shape parameter ξ ± uncertainty, as +defined by Soler et al. (2017). +Figure 22. +HRO shape parameter ξ as a function of column +density N as fitted in Figure 21. The black labels and dashed line +are solutions to the parameters C (the slope) and X (log of the +N-axis intercept) of a linear regression to all the ξ data. +preferred direction relative to gas structures, whereas the +B field is mostly perpendicular to higher column density +structures ∼1027 m−2. This generally confirms a series of + +15.0 +x +x +x +x ++ +x +美 +XK +x ++ +XxX +x +x +x +x +x +X +xx +x +++ +x +炎 +xx +x +X +x +x +x +x +x +Xx +X +×× +X +x +x +X +X +x +x +++ +XX +x +x +x +X +++ +x +++ +x +x +X +x +X +_ +x +x +XX +x +x +x +x ++ +++ +x +* +x +x +x +[molecules/m +交 +X +x +x +x +x +* +x +X +x +x +×× +xx +X +x +x +×x +X: +X +x +X +X +xx +X +Xat +++ +xx +X +X +xx +X +X +x +6.0 +X +x +x +x +X +XX +x +xX +X +xx +X +X +X ++ +. +xxx +X. +x +x +x +X +X +. +XX +X +x +* +X +x +X. +Xx ++ +X +xx +X +X +3. + x +x +x +X +茶 +x +++ +XX +X +x +X +xx +x +X +2.5 +X +2.0 +1.8 +Column +X +x +1.2 +X +x +C +1. +. +0.8 +? +XX +XX +X +x +0 +20 +40 +60 +80 +Relative +Alignment. +[degrees1ogN = 26.78-27.18 +F= +-0.371±0.057 +60 +40 +20 +60 +logN +26.55-26.78 +0.084±0.074 +40 +20 +60 +logN = 26.40-26.55 += 0.266±0.068 +40 +20 +logN +26.91 +26.40 +0.311±0.065 +60 +26.24 +# = 0.238±0.065 +40 +89 +logw +26. +0.260±0.065 +40 +20 +60 +OgN +26.03-26.14 +=0.371±0.059 +40 +20 +60 +Nol +25.96-26.03 +=0.200±0.062 +40 +LogN +25.89 +5796 +=0.645±0.049 +logN +25.81- +-25.89 +=0.284±0.079 +0 +20 +40 +60 +80 +Relative Alignment [degrees0.6 +0.4 +parameter +0.2 +shape +0 +HRO +-0.2 +-0.4 +1026 +Column Density N(H,) [molecules/m?Magnetic Fields and Gas Structures in BYF 73 +21 +Figure 23. +Overlay of the HAWC+ B field vectors (similar to Fig. 1 but including all vectors with P ′/σP ′ > 1.4, with a 20% p′ scale in +the bottom-right corner) and a column density map (contours 0.8, 1.1, 1.58, 2.27, 3.5, 5.1, 7, 10, 13×1026 m−2) derived from the SED-fit +NH2 map from Herschel data (Pitts et al. 2019) (hence the two beams). +results by the lower resolution (∼10′) Planck Collabora- +tion (e.g., Planck Collaboration 2016) over a wider range +of molecular gas column densities. +In the various structural components of Vela C, the +transition from mostly parallel to mostly perpendicular +can be rather sharp at a certain column density for each +structure, but this transition density is different in each +structure. This is widely attributed to a transition from +subcritical gas at lower densities, where the flow is at +least guided to some extent by the B field, to near-critical +or slightly supercritical gas at higher densities, where +gravity is capable of overwhelming the magnetic pressure, +allowing stars to form. +5.2.1. HAWC+ Data +With our substantially higher resolution ALMA and +SOFIA/HAWC+ data, it should be instructive to con- +duct a similar HRO analysis from several pc down to +<0.1 pc scales in the massive star formation environment +of BYF 73. We show first in Figure 20 the relative align- +ment of B field vectors with the SED-fit column density +N map (Pitts et al. 2019) as a proxy for “structure” in +the molecular gas, in all the HAWC+ data as shown in +Figure 1. That is, where the rotated polarisation vectors +θB⊥ are aligned with the tangent to the iso-N contours, +the relative angle is close to 0◦ and the field is considered +to be “parallel” to the gas structures. Where θB⊥ is per- +pendicular to the contours and aligned with the column +density gradient ∇N, the relative angle is close to 90◦ +and the field is considered to be “perpendicular” to the +gas structures. +This approach has the advantage of not imposing any +preconceived interpretation of whether the gas structures +represent “clumps,” “cores,” “filaments,” or any other po- +tentially subjective term (see, e.g., Planck Collaboration +2016; Soler et al. 2017). The distribution is quantified +by computing histograms on each N-bin separately as in +Figure 21, including the HRO shape parameter –1 < ξ +< 1 computed on each N bin’s HRO, as described by +Soler et al. (2017). This parameter objectively indicates +whether there is a preponderance of parallel (ξ > 0) or +perpendicular (ξ < 0) alignments in the data, and can be +plotted as a function of N (Fig. 22) to reveal any trends +via linear regression, +ξ = CHRO (logN − XHRO) . +(6) +Already in Figure 20 we can see that the distribution +of relative alignments has definite patterns in various col- +umn density ranges. +These observations are reflected +numerically in Figures 21 and 22. Thus, in the lower- +N ranges, there is an overabundance of parallel align- +ments (relative PA < 20◦) between the inferred B field +orientation θB⊥ and the iso-N contours, and ξ > 0 at +high significance, 3–13σ each across 8 N-bins. +In the +top two N ranges, however, there is a sudden transition +to clearly more perpendicular alignments, PA ∼ 40◦– +70◦ in the penultimate N bin (ξ ≈ 0), and 60◦–90◦ in +the top bin (ξ < 0 at 6σ). Indeed, compared to Vela +C (Soler et al. 2017), the slope CHRO is substantially +closer to –1 in the HAWC+ data for BYF 73, indicating +an even stronger alignment trend with increasing N and +a sharper transition (XHRO intercept) than in the Vela +cloud, at Ncrit = (3.9±1.0 +0.8)×1026 m−2. Interestingly, the +steepness of CHRO as seen in Planck+BLASTPol large +scale maps may be correlated with the inclination angle +of the mean B field (e.g., Sullivan et al. 2021). One sees a +shallower slope in clouds where the polarisation fraction +levels indicate that the B field is inclined closer to the +line of sight. Thus, the steeper slope in BYF 73 may be +related to its B field lying close to the plane of the sky +(see §§4.1, 5.4, 6). +The distribution of points in Figure 20 can be more + +286.16 +286.18 +286.2 +286.22 +22 +286.24 +22 +0.2 +0.18 +0.1622 +Barnes et al. +Figure 24. +Similarly to Fig. 20, this shows the relative alignment +between polarization position angle θB⊥ and the tangent to iso- +column density N contours, as a function of N, except here across +the ALMA field. The red N-bin boundaries for each histogram in +Figure 25 are labelled here in units of 1027m−2. +intuitively understood in Figure 23, which overlays the +HAWC+ p′ vectors and N contours from the Herschel- +based SED fits (Pitts et al. 2019). This map shows that +the large number of points with N < +∼ 1026 m−2 and pref- +erentially parallel alignments arises in the H ii region, +while the other large concentration of points with N ≈ +2×1026 m−2 arises mostly from the extended IF to the +north and the similar arc bounding the H ii region to the +southwest of MIR 2. The transition between these two +column density levels contains relatively few points due +to the sharp density gradient across the IF. For N ≥ +3×1026 m−2 and progressively closer to MIR 2, the align- +ments become preferentially more perpendicular. +5.2.2. ALMA Data +We can repeat the HRO analysis on the smaller scale of +the ALMA field. Figures 24–26 similarly show the over- +all relative alignment distribution as a function of N, the +HROs in separate N-bins, and the ξ vs. N plot for all +ALMA data. The relative alignment distribution shows +similarly striking changes with N as in the HAWC+ data. +In the three lowest-N bins, the B field shows no partic- +ular preference for parallel or perpendicular alignments +in the ALMA maps (ξ ≈ 0 within the uncertainties). In +the middle six N bins, though, the distribution changes +to show a substantial preference for perpendicular struc- +tures (ξ < 0 at a S/N of 2.5–5σ in each bin). These 9 +bins together behave similarly to the BLASTPol results +in Vela C, again including the existence of a sharp transi- +tion from positive to negative ξ, but now at a 4× higher +N ≈ 1.6×1027 m−2 than in the HAWC+ data. +However, +in the highest-N +bin, +the distribution +changes back to a very strong (7σ) parallel signature, ξ += 0.60±0.08. This produces a very atypical non-negative +result in Figure 26 for the fitted HRO parameter CHRO +(black labels and dashed line). In Vela C and elsewhere, +a negative CHRO means that ξ changes systematically +Figure 25. +HRO plots in each of the N-bins shown in Figure +24. Each window is labelled by the range of column density N in +that bin and its fitted HRO shape parameter ξ ± uncertainty, as +defined by Soler et al. (2017). +Figure 26. +HRO shape parameter ξ as a function of column +density N as fitted in Figure 25. The black labels and dashed line +are solutions to the parameters C and X of a linear regression to +all the ξ data, while the red labels and dashed line are for a fit to +all data except the highest column density bin with logN > 28. +from weakly positive to definitely negative values as N + +43.4 +x +X +x +Xx +x +x +x +美 +x +x +x +x +x +x +x +x +xx +x ++ +x +x +x +x +x +x +x +x +X +x +x +×× +[molecules/m +x +x +x +X +X +xx +x +x +x +x ++ +x +x +x +x +x +× ××× +x +X +x +x +x +x +x +x +x +x +x +X +X +xX +x +x +X +x +++ +7.8 +. +x +xx +x +X +x +X +X +x +X +X +X +xx +Density N(H,) +x +xx +X +X +X +X +5.0 +xX +X +3.9 +3. +2.9 +XK +2.3 +x +X +X +XX +x +x +X +Column +X +X x +X. +X +x +16 +X ++ +x +x +xx + ++ +++ +x +xX +X +x +x +++ +x +& +x +x +x ++ +X +X +x x +×× +XXX +0 +20 +40 +60 +80 +Relative +Alignment [degreesloN +27.90 28.64 + = 0.604±0.079 +30 +logN +27.70- +-27.89 +0.250±0.099 +20 +10 +30 +1ogN = 27.60-27.70 +0.576±0.082 +20 +10 +30 +1ogN = 27.52-27.60 +0.600±0.080 +20 +10 +30 +logN = 27.46-27.52 +360±0.093 +20 +logN = 27.37-27.46 +0.360±0.093 +20 +logN +27.21 +-27.37 +-0.248±0.096 +20 +08 +27.06-27.21 + = 0.119±0.095 +20 +10 +30 +logN + 26.94 +27.04 += 0.143±0.104 +20 +10 +30 +1ogN = 26.72-26.94 +0.099±0.118 +20 +10 +0 +20 +40 +60 +80 +Relative Alignment [degreesCHRo=0.29±0.33 +CHRo=-0.69±0.23 +0.5 +0 +-0.5 +1027 +1028 +Column Density N(H,) [molecules/m"]Magnetic Fields and Gas Structures in BYF 73 +23 +Figure 27. +Overlay of ALMA B field vectors (similar to Figs. 2, 7, 17 but including all vectors with P ′/σP ′ > 1.4; 20% p′ scale in +bottom-left corner) with a column density map (contours 0.3, 0.6, 1, 1.6, 2.5, 5, 10×1027 m−2) derived from scaling the ALMA I mosaic +(Fig. 2) to an SED-fit Tdust map from Herschel data (Pitts et al. 2019). +rises, meaning a transition from parallel or random B +field alignments to perpendicular ones, often with a sharp +transition across ξ = 0 at a particular N. In this context, +the last data point in Figure 26 may be anomalous, but +as it turns out, this may not be that significant. +To see this, consider the θB⊥ and N maps together +(Fig. 27), where one can see where each of the ten N +bins are located. The three lowest-N bins with ξ ≈ 0 +arise in the weaker emission features of the Streamer to +the west and farthest east, and southern parts of the IF. +The middle six N bins with ξ < 0 arise in the brighter +emission of the EPL, MIR 2-ext, and the main part of the +Streamer. The highest-N bin arises exclusively from the +brightest parts of the MIR 2 peak, where the structure is +actually not well-resolved in the 2.′′6 ALMA beam. This +would not only preclude accurate θB⊥ measurements at +MIR 2, but also might include Q and U cancellation +within the ALMA beam, underestimating P ′. Resolu- +tion alone would make any alignment inferences ques- +tionable, but in addition we recall that MIR 2 is near the +limit of the reliably-calibrated window of the ALMA po- +larisation field. Therefore, we cannot accurately quantify +the alignment measurements or their uncertainties right +at the MIR 2 peak, and conclude that the ξ value in this +N bin should be discounted. +As an exercise, therefore, we also computed the regres- +sion parameters C and X for the nine lower-N bins in +Figure 26, and show these as red labels and a dashed +line. In this case C is definitely negative (3σ) and more +in line with the Vela C results, while the XHRO intercept +gives Ncrit = (9.5±5.3 +3.4)×1026 m−2. Based on this alone, +it seems desirable to obtain an even higher-resolution po- +larisation map of MIR 2 and its immediate surroundings. +Such a map would allow us to explore the massive proto- +stellar core’s B field in much finer detail and track the ξ +trend to even higher N, not to mention better resolving +Figure 28. +Combined HRO ξ vs. N plot from both HAWC+ +and ALMA data (Figs. 22,26), where the two data sets overlap in +the N-bins from 0.5 to 1.5×1027 m−2, and we have increased the +number of N-bins to 25 because of the roughly doubled number of +points. The overall results of the fitting, however, are very similar +for any number of N-bins between 10 and 30. As in Fig. 26, the +black labels and dashed line are solutions to the parameters C +and X of a linear regression to all the ξ data, while the red labels +and dashed line are for a fit to all data except the highest column +density bin with logN > 28. Overlaid in green and cyan are the +respective fits from Figs. 22 and 26 for comparison. +the core itself (e.g., 250 AU at 0.′′1). +5.2.3. Combined Data + +205 +286. +2 +286. +5 +286.21 +3.22 +286. +0.175 +0.17 +0.165X HRo =26.74±0.09 +0.5 +parameter + shape +0 +HRO +-0.5 +1026 +1027 +1028 +Column Density N(H,) [molecules/m?24 +Barnes et al. +The ξ–N trends in Figures 22 & 26 overlap nicely in +column density, and we present a combined plot in Fig- +ure 28. +There, the slope C is slightly shallower com- +pared to either the HAWC+ or ALMA-only results, but +the overall trend is firmer (the uncertainties in C & X +are smaller) due to the wider range of N being sam- +pled. In combination, the data suggest that the transi- +tion to perpendicularity in BYF 73’s Streamer and MIR 2 +core occurs (from the red intercept X in Fig. 28) at Ncrit += (6.6±1.2 +0.9)×1026 m−2, near the geometric mean of the +transitions from the individual instruments. +This can +also be converted into an equivalent critical gas density if +we assume a line-of-sight depth to the Streamer approxi- +mately equal to its projected width, n ∼ N/D, where D +≈ 0.087 pc. Then, ncrit = (2.0±0.5 +0.4)×1011 m−3. +This is actually a rather suggestive threshold: in Fig- +ure 27, it is the column density of the second-lowest con- +tour, and includes the MIR 2 core, ∼all of the Streamer- +main and -west, and much of the IF. It suggests that +the Streamer’s width may be related, locally at least, +to the transition between MHD forces governing the gas +dynamics and self-gravity, as seen in §4. +It is instructive to compare this result with other HRO +studies. For example, Planck Collaboration (2016) used +10′ resolution Planck data to study B field orientations +in 10 nearby (150–450 pc) Gould Belt clouds, with a +finest physical resolution similar to our HAWC+ data +and ranging up: ∼0.4–40 pc. At this scale the median gas +densities are n ≈ 5×108 m−3, substantially less than is +typical in the Streamer as estimated above. The thresh- +old column densities in these 10 clouds are also lower +than that for BYF 73, by a factor of 10 on average, X +≈ 6×1025 m−2. Similarly in Vela-C, a massive but rela- +tively unevolved cloud at 700–900 pc, Soler et al. (2017); +Zucker et al. (2020) used Herschel and BLASTpol data at +3′ resolution for their HRO analysis (i.e., with a similar +physical resolution to Planck Collaboration 2016), and +found a typical X ≈ 3×1026 m−3 or about half BYF 73’s +value. Finally, in two portions of L1688 in Ophiuchus at +140 pc, Lee et al. (2021) combined HAWC+ and Planck +data (giving similar physical resolutions to our ALMA +data) to confirm a column density threshold similar to +Planck Collaboration (2016)’s 10 clouds, and a volume +density threshold n ≈ 1010 m−3. +We can relate this column density threshold to the +equivalent B field threshold if gravity and the magnetic +pressure were critically balanced. Using the mass-to-flux +ratio approach of Crutcher et al. (2004) as adapted by +Barnes et al. (2015), we have +λ = (M/Φ)obs +(M/Φ)crit += 0.064 NH2/1024m−2 +BTOT/nT +, +(7) +or +Bcrit (nT) = NH2/(1.57 × 1025m−2) , +(8) +when λ = 1. +With the above threshold, we obtain +Bcrit = 25±6 nT for the HAWC+ measurements, Bcrit += 61±27 nT for the ALMA data, and Bcrit = 42±7 nT +averaged over the mapped HAWC+ and ALMA emission +in BYF 73, based on the combined HRO analysis. +Again, we can compare this result to equivalent Bcrit +for the nearby clouds of ∼4 nT (Planck Collaboration +2016), ∼20 nT for Vela-C (Soler et al. 2017), and ∼4 nT +for L1688 (Lee et al. 2021), placing BYF 73 at a higher- +Ncrit and -Bcrit level than these other clouds. +Our +Bcrit result for BYF 73 generally is also about half the +DCF value at the peak of MIR 2 with Mopra’s reso- +lution (Eq. 2), which is not unreasonable given the re- +spective density levels. Thus, both the DCF and HRO +analyses give us mutually consistent clues about the B +field strengths in BYF 73, which seem to be significantly +stronger than in other clouds. +5.3. The Goldreich-Kylafis (GK) effect in 12CO +5.3.1. Widespread Polarisation in the Outflow +The GK effect can arise when B fields (even weak ones) +and velocity & excitation gradients in molecular gas com- +bine to produce imbalances from thermal equilibrium in +populations of magnetic sublevels M of spectral lines +with opacity ∼1 (Goldreich & Kylafis 1981; Girart et +al. 2004; Crutcher 2012). This can produce linearly po- +larised spectral line emission that is either aligned with +(π transitions) or perpendicular to (σ transitions) the lo- +cal B field, depending on the radiative transfer circum- +stances, namely the unknown angles between the radia- +tion anisotropy, the line of sight, and the B field direc- +tion. In addition, the classical Zeeman effect can give +rise to circularly polarised σ transitions parallel to B, +observable for that component of B oriented along the +line of sight. +We report here the widespread detection of strong, +linearly polarised emission in the 12CO line wings from +BYF 73 (i.e., its bipolar outflow) that is consistent with +the GK effect. The native Stokes data have high S/N, +up to 21 for P ′ and p′, in individual 0.16 km s−1-wide +channels and across much of the outflow visible in both +line wings. However, where the polarisation signal weak- +ens, the p′ values tend to rise and, with the larger un- +certainties, the resulting vector maps become somewhat +confusing to look at. Therefore, we averaged (binned) +the native Stokes data into ∼3 km s−1-wide channels for +display purposes only, and formed the polarisation prod- +ucts on the binned cubes with proper noise weighting: +the significant polarisation features are then more easily +visualised in the binned data. Figure 29 shows overlays +of the blue- and red-shifted I and P ′ emission in these +binned data, together with all observed polarisation vec- +tors above 4σrms (and up to 72σ) across the full velocity +range of the line wings. +Interpreting GK polarisation vectors requires some res- +olution of the 90◦ ambiguity described above, depending +on whether σ transitions from the M=±1 magnetic sub- +levels (polarised perpendicularly to the B field) will be +stronger or weaker than the π transitions from M=0 sub- +levels (parallel to B). In general, whether outflows are +driven by magnetocentrifugal forces anchored in proto- +stellar disks, or by collimated protostellar jets carrying +their own B fields, the nominal expectation is that in- +ferred B fields should be aligned along outflows. In Fig- +ure 29 we have rotated the vectors by 90◦ from those +observed, and it is this orientation which, remarkably +clearly, shows an overwhelming orientation along the out- +flow direction in each wing, especially for the higher-S/N +pixels. Equivalent plots of the observed vectors show a +near-universal circumferential alignment around MIR 2, +which would seem to be unphysical based on the above + +Magnetic Fields and Gas Structures in BYF 73 +25 +Figure 29. +BYF 73 12CO outflow polarisation maps shown in 3 km s−1-wide panels, each labelled by their centre velocities in the top-left +corner and a 5% polarisation vector scale (yellow bar) in the bottom-left corner. The panels overlay several components, averaged over +the same velocity ranges: polarised flux P ′ images scaled to the colour bar on the right in K; I contours in red at 2,4,8,16 K, dashed for +negative values from missing short-spacing information; and orange percentage p′ vectors at every second pixel above 4σ, with PAs rotated +by 90◦ to indicate θB⊥. For the P ′, p′, and θB⊥ data in each panel, they were constructed by first binning the native Stokes data by 19 +channels, and then forming the products P ′, p′, and θB⊥ on the new binned cubes. In order to better display the high-S/N features, the +top 12/bottom 8 panels respectively show the blue-/red-shifted line wings vignetted to the east/west of MIR 2. +understanding. +Indeed, the high-S/N vectors track both the bend in +the blue wing and the fork in the red wing inferred solely +from the I emission pattern (Fig. 8). +There are some +low-S/N vectors which don’t align with this general pat- +tern, however, typically near the p′ threshold. This is +most notable in the –24 km s−1 panel, both north and +south of the outflow itself. In the northern portion of +this polarised emission, the alignment is instead approx- +imately across the EPL as mapped by HAWC+ (Figs. 6– +8). South of the outflow, the emission appears to be an +artifact of missing short spacings in the field of view, so +we discount it. +An arrangement with outflows oriented along the B +field direction is typical of Crutcher (2012)’s summary of +outflow studies via GK mapping. The data for BYF 73 +show that the observed polarisation is preferentially ori- +ented 90◦ from the presumed B field direction down the + +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +286.21 +GLON (degre +286.215 +286.21 +GLON (degree +286.215 +286.21 +GLON (degre +286.215 +286.21 +GLON (degre +286.215 +-57.09 +45.03 +32.96 +0.172 +0.17 +0.168 +0.166 +0.172 +0.17 +0.168 +0.166 +0.172 +0.17 +0.168 +0.166 +GLAT (degrees) +() +(se) 286.205 +GLON (degrees) +286.21 +286.205 +GLON (degrees) +286.21 +3.21 +286.205 +286.21 +11.85 +2.0 +286.205 +286.21 +-14.87 +82- +0.172 +0.17 +0.168 +0.166 +0.172 +0.17 +0.168 +0.166 +GLAT (degrees) +GLAT (degrees)26 +Barnes et al. +Figure 30. +Simplified DCF analysis of polarisation orientations in the ALMA 12CO data (left, blue shifted emission; right, red-shifted +emission) as a function of velocity, treated as single boxes encompassing all polarised emission in each channel. All pixels of θB⊥ above +4σ are shown as black dots; their mean values in each channel are connected by a red line, while the dispersions in each channel’s θB⊥ +distributions are drawn as green error bars. The dark blue line shows χ2 values (on the right ordinate axis of each panel) of gaussian fits to +the θB⊥ distributions in each channel. For reference, the horizontal magenta and cyan lines respectively show the orientation of the red- +and blue-shifted outflow axes, as illustrated in Fig. 8, and each left ordinate is additionally labelled with compass directions. +outflow axes, and so supports an excitation condition in +which the σ M=±1 transitions are robustly overpopu- +lated in the outflow relative to the π M=0 transitions. +5.3.2. Simplified DCF Analysis +Quantifying this description, however, is challenging +due to the sheer volume and effectively 4D nature of the +data. As a first attempt, we perform a simplified DCF +analysis per unbinned 0.16 km s−1-wide channel in the +data. We argue that this is reasonable, even though the +original DCF method was not developed for outflows. +Indeed, we believe that DCF analysis of spectral-line lin- +ear polarisation will give a better result than for dust +polarisation, in the following sense. +A GK-imbued spectral line directly samples the tur- +bulence, density, and polarisation dispersion in the same +region. A problem with application of DCF to dust po- +larisation is that one needs to estimate the density sam- +pled by the dust polarisation, plus a turbulent linewidth. +Although we have dust-based column density maps of +BYF 73 (Figs. 23, 27) from which density estimates can +be simply inferred, densities and linewidths are more +generally inferred from observations of spectral lines: +excitation analysis for density, and directly measured +linewidth. However, different spectral lines sample dif- +ferent density regimes and different lines have different +linewidths, so just what is appropriate for the dust po- +larisation analysis is never clear. One often ends up with +some sort of ill-defined average along the line of sight. +On the other hand, for spectral-line polarisation things +are self-consistent. One can infer the polarisation disper- +sion, density, and measure the turbulence directly in the +same parcel of gas, namely that sampled by the spectral +line being observed. DCF then gives an estimate of the +B field strength in that spatial and density parcel, not +for some ill-defined and possibly different regions along +the line of sight. +If GK polarisation can be detected in multiple spectral +lines that sample different density regimes, one can in +principle build up a 3D picture of the B field. So far, +however, detections of the GK effect in species other than +CO have been rare, but presumably that will improve as +time goes on. +For this channel-DCF analysis of BYF 73, we do not +include sub-ROIs of each channel at smaller scales, as in +Figs. 16 and 19, and assume instead for simplicity that +the whole-channel-ROI gives an approximate measure of +the θB⊥ correlation length for that channel, since the +polarised emission is dominated by the outflow structure, +as seen in Figure 29. The results are shown in Figure 30 +for both 12CO line wings. +Several features are immediately evident. +The most +significant are the clear trends in θB⊥, for the blue wing +from VLSR = –22 to –36 km s−1, and for the red wing +from VLSR = –17 to –2 km s−1, showing a B field orien- +tation that changes gradually, in both cases, from EW to +more along the outflow axis and then back to EW, as one +looks from the lower to higher outflow speeds. This in- +ternal consistency is not so surprising since the statistics +in these channels (a few hundred pixels each) are quite +robust. Observationally, however, there is no reason to +expect the polarisation to line up so reliably, channel by +channel, unless the polarisation signal in all channels is +strongly governed by the intrinsic physics of the outflow. +Thus, over these velocity ranges, the dispersion in θB⊥ +for each channel is quite small, s = 11◦±4◦, even where +a few pixels appear as outliers in the θB⊥ distribution of +some channels. This is not much larger than the aver- +age noise-derived ∆θrms ≈ 7◦, giving an intrinsic mean +dispersion s ≈ ±8◦.6, or as little as ±7◦in some places. +Overall, the polarised emission at these velocities appears +very well organised. + +200 +40 +NS +[degrees] +150 +30 +PA +in +Dispersion +100 +20 +EW +and +Mean +10 +-50 +-40 +-30 +Visr [km/s]NS +[degrees] +150 +30 +PA +in +100 +20 +EW + and +50 +10 +Mean +-15 +-10 +0 +[km/s] +LSRMagnetic Fields and Gas Structures in BYF 73 +27 +At velocities from VLSR = –36 to –55 km s−1 and >– +2 km s−1, however, the mean θB⊥ direction in each chan- +nel becomes more erratic as the outflow speed increases, +on average still lying near the outflow directions but +with a dispersion among the channels s ≈ ±40◦ for the +blue wing. This wider variation probably reflects poorer +statistics, with typically only a few, or a few dozen, pixels +per channel. +At velocities closer to the line core, VLSR = –22 to – +17 km s−1, aliasing of extended line emission throughout +the field of view introduces many polarisation features +with probably unreliable θB⊥, indicated in Figure 30 by +both the increasing density of dots at all θB⊥ and higher +χ2 from the non-gaussian θB⊥ distribution, all becom- +ing more noticeable as the velocity approaches Vsys = +–19.6 km s−1. +5.3.3. Magnetic Field Strength Calculations +As with the continuum data (§5.1), we can use this ba- +sic DCF information on the dispersion in θB⊥ per chan- +nel from the GK effect, to make estimates of the B field +strength in the outflow. For Eq. (1) from §5.1, we first es- +timate the 12CO column density in the line wing emission +I12CO via the velocity-resolved, opacity-corrected con- +version law from Barnes et al. (2018, their Fig. 5b, not +their integrated law in Fig. 9b). Next, we convert that +to an H2 column density with Pitts & Barnes (2021)’s +dust temperature-dependent abundance law. Finally, we +turn this into a volume density assuming a line-of-sight +depth D ≈ 0.087 pc through the outflow (and correlation +length, later) equal to the outflow’s average projected +width: +nch = N0 +D +(I12COdV/K km s−1)p +10[−10 log2(Td/T0)+logX0] . +(9) +From Barnes et al. (2018), N0 = 1.27×1020 m−2 and p += 1.92; the ALMA channel width dV = 0.159 km s−1 +converts I to the proper units; and from Pitts & Barnes +(2021), T0 = 20 K is the dust temperature at which the +gas phase CO abundance relative to H2 peaks, at a value +X0 = 0.74×10−4. +Eq. (9) thus converts the I12CO data cube into a cube +of H2 density per channel at the observed velocity. To +combine this with Eq. (1), we need a turbulent veloc- +ity dispersion. We can choose a velocity FWHM ∆V in +the gas corresponding to 1 ALMA channel to be consis- +tent with the above formulation, but in reality it may +be several times larger, since the outflow is likely to be +turbulent at some level related to the ±25 km s−1 range +of flow speeds. In that case, the true velocity FWHM +in the gas would re-scale the single-channel B⊥,DCF esti- +mated via Eq. (1) by ψ = (∆V /dV )1+p/2, since we would +need to evaluate I in Eq. (9) over the same ∆V -wide bins +(the column density inferred from I, and hence the den- +sity, is additive across channels). +With p as above, ψ +∝ (∆V /dV )2 approximately, or as the ∼square of the +number of channels in a ∆V bin. +So for molecular gas with µ as before and s = 7◦ in the +inner part of the flow, where minimal gas-phase densities +inferred from Eq. (9) are nch ∼ 109 m−3 ch−1, we obtain +B⊥,DCF,outflow = 0.61 nT +� +nch +109m−3 +� ψ +s/7◦ +� +or +(10) +≈ 4.5 nT +� I12COdV +10 K km s−1 +�p/2� ψ +s/7◦ +� +(11) +as a minimum for B in the molecular outflow, when ap- +proximating the dust temperature at 20 K throughout to +compute a minimal H2 density at the peak CO abun- +dance. +As described above, 1 channel probably slices the tur- +bulent structure in the outflow rather finely: for ∆V += 3 km s−1, for example, the scaling would increase the +single-channel B⊥,DCF coefficient by ψ = 320×, e.g., to +1.4 µT for the same value of I in Eq. (11). Indeed, the +brightness of the mapped 12CO outflow emission in Fig- +ure 29 ranges up to 90 K km s−1 in 3 km s−1-wide bins, +suggesting that B fields might be stronger still in some lo- +cations. Of course, this scaling may not actually be valid: +while simulations of turbulent plasmas suggest that DCF +estimates are reasonable up to Mach numbers M ∼ 5–9 +(Ostriker et al. 2001), such values may be far exceeded +(M>100) in the outflow. +5.3.4. Energy Densities +Despite these somewhat large uncertainties, we at least +have rough estimates for B⊥,DCF field strengths in the +outflow. As a final exercise, we compare the energy den- +sity M that would exist in such B fields with the kinetic +energy density K of the outflowing gas. M follows di- +rectly from Eq. (10), +M= B2/2µ0 = 1.49 × 10−13Jm−3 n9 +� ψ +s/7◦ +�2 +, (12) +where µ0 is the permeability of free space (or the +magnetic constant in preferred SI usage) and n9 = +nch/109m−3. +Thus, smaller values for M will obtain +at lower gas densities (and approximately I, through +Eq. (9)), either at the edges of the outflow or at higher +velocities, whereas larger M will lie closer to the outflow +origin where the density or I is higher. +For K we can assume a cylindrical outflow geometry +(diameter D, length L) to approximately compute +K = +1 +2MV 2 +rel +1 +4πD2L = 1 +2ρV 2 +rel += 3.9 × 10−12 Jm−3 n9 (Vrel/km s−1)2 +(13) +where L ≈ 0.6 pc is the physical length of the 50′′-long +blue lobe of the outflow. The second expression is much +simpler to use, since the mass density ρ = 2µ mHnch, +with the number density nch from Eq. (9). This can ac- +tually be done separately for each channel if we use its +relative outflow speed Vrel as measured from Vsys = – +19.6 km s−1. Thus, the value of K will be larger at higher +ρ ∝ I (approximately) but especially at higher Vrel, or +smaller at lower I or especially lower Vrel. +We can now compute a datacube of the M/K ratio, +M +K = 3.8% +n9 +� +ψ +s/7◦ +�2 +n9(Vrel/km s−1)2 , +(14) + +28 +Barnes et al. +Figure 31. +Logarithm of the ratio of magnetic M and kinetic K energy densities (colour bar on the right) in the 12CO outflow wings of +BYF 73. Each panel is a binned average of 3 channels (0.47 km s−1 wide) labelled by their mean VLSR in the top-left corner, and covering +both the red & blue vignettes of Fig. 29. Red contours in each panel are of the respective binned Stokes I of 12CO at 2,4,8,16,32 K. +which turns out to be independent of the density as long +as we measure both over the same channels or velocity +bins. Since ψ is part of the density scaling, this simplifies +to +M +K = 3.8% +�∆V/Vrel +s/7◦ +�2 +, +(15) +which we can evaluate per native channel of width dV +(or any other binning), even while using a larger ∆V +to represent the turbulence in the flow. In other words, +the M/K ratio can reasonably be estimated with some +knowledge of only the gas turbulence ∆V and polarisa- +tion dispersion s in the B field directions in each channel +at Vrel, which is ultimately just a restatement of the DCF +method, per unit volume. +For purposes of illustration, we take ∆V = 3 km s−1 +and a more conservative s = 13◦, and present the M/K +ratio results in Figure 31 at some representative channels +Vrel. Different ∆V or s values would obviously scale the +ratios as (∆V /s)2. Despite the larger s and smaller M in +Figure 31 than discussed above, M/K peaks at 37, i.e., +≫1. We discuss this further in §6. +5.4. The Zeeman effect in CN +As the only observational technique capable of directly +measuring B field strengths, the Zeeman effect has been +widely utilised over five decades (Crutcher 2012). How- +ever, successful detections are notoriously difficult: for +extended thermal emission from molecular clouds, only +HI, OH, and CN have yielded B field detections, and +among these, only CN can provide information on field +strengths in dense (> +∼1011m−3) gas. Despite considerable +effort, there still exist only 14 individual CN Zeeman +measurements from a heterogeneous sample of clouds +(Falgarone et al. 2008). But with the advent of full po- +larisation capability in Cycle 7, anticipation has been +high that ALMA might fundamentally change the state +of play in this field. +Unfortunately, despite the very high S/N (∼200) in the +ALMA Stokes I data for BYF 73, covering 8 of the 9 hy- +perfine transitions of the CN J=1→0 line, the V cube +shows nothing discernible above the noise. Computing +the ratio of Stokes V to dI/dV and scaling this to the +Zeeman splitting coefficient of any of the brightest hyper- +fine transitions (as in Table 1 of Falgarone et al. 2008) +yields only 3σ limits ∼1 µT (10 mG), as seen in Figure 32. +This is near the upper end of the range of field strengths +seen before in dense gas (Crutcher 2012), but the noise +level would need to be at least halved to obtain reliable +measurements even at those levels. A further issue was +the Cycle 7 limit for accurately-calibrated V data lying +within the inner 10% of the primary beam. +This also means we can’t use Zeeman data to dis- +tinguish between the scenarios (a pure B⊥-twist or an +additional B|| component) put forward to explain the + +1.5 +0.5 +-1.5 +2- +90 +-2.5 +286.205 +286.21 +GLON +-15.03 +286.215 +286.205 +286.21 +-15.50 +286.215 +286.205 +286.21 +GLON ( +5.98 +286.215 +286.205 +286.21 +-16.46 +286.215 +0.166 0.168 0.17 0.172 +0.166 0.168 0.17 0.172 +0.172 +0.166 0.168 0.17 +0.166 0.168 0.17 0.172 +0.166 0.168 0.17 0.172 +GLAT (degrees) +GLAT (degrees) +GLAT (degrees) +GLAT (degrees) +GLAT (degrees)Magnetic Fields and Gas Structures in BYF 73 +29 +HAWC+ P ′ null on the western edge of the IBL (§3.3). +While somewhat discouraging, the non-detection may +partly be due to the cloud’s orientation. +That is, the +Zeeman effect can only measure the line-of-sight compo- +nent B||. The fact that the outflow is viewed close to +side-on (§4.1) suggests that most organised structures in +the molecular cloud, such as an accretion disk around +MIR 2, would also probably be presented edge-on to us, +as might any structures being accelerated away from it, +thus possibly maximising B⊥ and minimising B||. +6. DISCUSSION +6.1. Dynamics: ALMA Reveals the Outflow +and Isolates the Inflow +Based on 40′′ resolution Mopra HCO+ maps, Barnes +et al. (2010) first described a massive infall of dense, cold +material within the wider BYF 73 cloud, without any ev- +idence of an outflow characteristic of lower-mass YSOs. +This suggested an extremely early evolutionary state for +a very massive protostar, which seemed to be confirmed +by the mid- and far-IR data of P18. With the higher- +resolution SOFIA and ALMA data presented here, par- +ticularly the strong bipolar 12CO outflow, we see that +the original appearance of outflow-free, extremely young +massive star formation may have been something of a +masquerade.15 Nevertheless, through the ALMA 13CO +data, we are able to discern more specific clues to the +configuration of the inflow originally seen in HCO+. +However, the 3D relationship between the Streamer, +outflow, and disk or infall as described in §4 remains +puzzling. The disk can be traced from an outer radius +of 0.18 pc = 36,000 AU to an inner radius no larger than +the limit of the ALMA resolution, 1.′′8 = 4500 AU. Ap- +parently, this disk is close to edge-on based on the sharp +velocity gradient across MIR 2, so the filamentary im- +pression of the Streamer may be an illusion. For exam- +ple, we see that both the outflow and rotational/infall +patterns are oriented EW, but this arrangement would +seem physically counterintuitive. Undoubtedly, there is +some depth to these features in the line of sight, and +it is possible that any inflow to the disk might be ap- +proaching MIR 2 from behind its eastern side, even while +the blue jet is receding from MIR 2 on the same side. +Likewise, inflow overlying the western Streamer may be +from the front, while the red jet recedes from MIR 2 as +it encounters the H ii region. Separating these features +in the line of sight requires only a few ×104 AU ∼ 0.1 pc, +so we consider this scenario reasonable. Moreover, the +Streamer/disk is clearly not flat; there is a measurable +width and curvature to its structure. +What are the dimensions of the disk/infall zone cen- +tred on MIR 2? The modal value of the latitude across all +the disk emission, from the LV-1st moment map in Fig- +ure 12 (middle panel), is b = 0◦.16997±0◦.00005, only 0.′′8 +15 Inspired by the ALMA results, we re-examined the Mopra +12CO data (Barnes et al. 2018) to see if we could tease out hints of +the outflow, but still found no clear evidence of the strong red- and +blue-shifted emission so easily visible in the ALMA maps. However, +convolving the ALMA data to the Mopra resolution, and adding in +the missing short-spacing 12CO information plus the higher Mopra +noise per 40′′ beam, we found that the outflow became invisible to +Mopra at that sensitivity. So the two instruments’ results are con- +sistent, and provide an object lesson against similar masquerades +in other sources. +Figure 32. +(Top) Sample Zeeman calculation of B|| at the lati- +tude labelled in the top-left corner, using the Miriad task zeemap +for the brightest CN hyperfine component at 113.490982 GHz, pre- +sented as an LV diagram. (Bottom) S/N ratio of the data in the top +panel. Note that for Cycle 7, the reliably-calibrated Stokes V data +are limited to the inner 10% of the ALMA primary beam, which +encompasses only the area within 286◦.214 > +∼ l > +∼ 286◦.212 (∼6′′). +Also, to the west (right) of this area, zeemap underestimates the +noise due to the ALMA primary beam correction, and so the few +pixels with apparently larger S/N are actually not. +Effectively, +there are no pixels with B|| measurements > 3σ. += 0.3 ALMA beamwidths north of MIR 2 itself. Three- +quarters of this emission lies within 0◦.169 < b < 0◦.172, +a span of only 11′′ or ∼4 ALMA beams, strongly sug- +gesting a somewhat narrow structure for the high-∆V +material. +We also computed an LV-2nd moment map +(Fig. 12, bottom panel) to examine the latitude width, +confirming that it is indeed thin, from only ∼3 ALMA +beams = 7′′ thick to <1 beam. From an inspection of +all three LV moment maps, we see that the eastern side +of the disk at the higher velocities (VLSR < –25 km s−1) +seems to lie mostly at one latitude close to that of MIR 2, +b = 0◦.1698±0◦.0005, and so is indeed quite flat in the EW +plane to within 1 ALMA beamwidth. This is across an +extent of 0◦.01 = 36′′, for an aspect ratio of 15–20:1 ori- +ented EW. +Given this, it is hard to imagine a disk oriented in the + +Ln +2 +4 +2 +0 +5 +286.205 +2 +286.210 +e +5 +Galactic +1 +36.2 +8 +2 +G +286.220 +2 +9 +1688759 +6 +8 +0 +2 +4 +0 +1 +2 +1 +2 +2 +Glat: +(km/s) +Velocityratio +2 +4 +2 +0 +5 +286.205 +2 +286.210 +Uor +5 +Galactic +286.21 +G +3.220 +86. +2 +9 +1688759 +6 +8 +0 +2 +0 +4 +1 +1 +2 +2 +2 +Glat: +Velocity +(km/s)30 +Barnes et al. +same direction as the outflow it is supposed to be driv- +ing. This favours the 13CO data tracing free-falling ma- +terial onto a 950 M⊙ MIR 2 within a 36′′×2′′ structure, +rather than a Keplerian disk, since such a disk ought to +be oriented close to NS, parallel to the sharpest velocity +gradient across MIR 2. If this is indicative, the gradi- +ent suggests a disk thickness perhaps < +∼2′′ or 5000 AU, +but possibly even narrower. +On the other hand, even +if we separate the infall from the outflow along our line +of sight, it is equally hard to see how a predominantly +EW infall (i.e., along a polar direction) produces a disk +oriented NS. So the puzzle persists. +Additionally, if both the MIR absorption and FIR +emission mass estimates at MIR 2’s peak position are 5– +12× too small (P18), this is possible evidence for signif- +icant grain growth in MIR 2’s protostellar envelope; any +free-free emission from MIR 2 (§3.1) would make this dis- +crepancy worse. P18’s gravitational energy release lumi- +nosity also scales with the mass, raising it to perhaps +20–33% of MIR 2’s total luminosity. +Indeed, if future +higher-resolution observations revealed an impact radius +for the inflow only 5× smaller at 1000 AU, this could not +only account completely for MIR 2’s brightness via grav- +itational energy release, but also possibly reveal it to be +the first example of a massive “first hydrostatic core.” +At velocities closer to systemic, the brightest LV emis- +sion in the eastern disk, corresponding to the Streamer +at 286◦.22 > l > 286◦.21, also lies very close to this EW +plane, although slightly north of it. However, it is not +distributed along any of the Kerplerian curves: instead, +its velocity drops linearly from –23 km s−1 to systemic +over its 36′′ length, with kinematics mimicking that of +solid-body rotation. +This portion of the eastern disk +is thicker, 5′′–7′′, than the high-velocity emission there, +<3′′, giving it an aspect ratio around 6:1. Continuing +the non-Keplerian behaviour, east of l = 286◦.22 or out- +side the Streamer’s distance from MIR 2, there appears +to be some material in “super-rotation” in the bottom-left +quadrant of the curves, i.e., with VLSR exceeding the ro- +tational curve for 1350 M⊙. This lies at b = 0◦.172 (red in +the middle panel of Fig. 12), or 7′′ north of MIR 2, but is +again about as thin (1 ALMA beam) as the high-velocity +disk material. However, the envelope of this material’s +super-rotation is moving at close to +√ +2× this curve, sug- +gesting either free-fall of ambient material towards the +Streamer/disk from the rear of the cloud, or that the +enclosed mass at this radius has ∼doubled. +In contrast, the western side of the disk seems to curve +somewhat north of the EW plane of the eastern disk, to +a latitude as far as 12′′ north of MIR 2 at 0◦.173, and +at a moderately high velocity (VLSR = –10 km s−1) from +systemic. The rest of the western disk ranges in latitude +from MIR 2’s value up to this limit, and the line of maxi- +mum velocity in the red-shifted wing map (Fig. 11, right +panel) is clearly curved to the north-west from MIR 2. +The western disk’s thickness (Fig. 12) is also broader +than for the eastern disk overall, up to 7′′, but is also +thin (<3′′) in many places, even where it curves to the +NW. It is worth noticing that the solid-body portion of +the Streamer/eastern disk seems to continue part-way +(0◦.006 = 22′′) into the western disk in both the 1st- and +2nd-LV-moment maps, but then seems to reverse bluntly +back to MIR 2’s longitude at VLSR = –16 km s−1, while +thickening to a width of almost 10′′ just west of MIR 2, +almost as if the Streamer’s infall (if that’s what it is) were +being deflected from the EW plane by some obstacle west +of MIR 2. +What of the counter-rotating parts of the LV diagrams +(i.e., the “empty” top-left and bottom-right quadrants +of the rotation curves)? +Much of this emission, espe- +cially the brighter portions thereof, lie north (b > 0◦.172, +magenta) or south (b < 0◦.168, black) of the disk, and +outside the longitude range of the IBL (286◦.216 > l > +286◦.203): they appear to be associated with other in- +ternal structures of the cloud, supporting the rotational +interpretation for the inner parts of the Streamer. +While MIR 2’s mass seems dynamically dominant +within the IBL, the mass in the Streamer/disk must nev- +ertheless also be significant. +In §5.2 we find a mean +column density 3×1027 m−2 ≈ 6000 M⊙/pc2 along the +Streamer, or roughly 15 M⊙ per 4′′ box, assuming there +is not the same mass deficit/degree of grain growth as in +the MIR 2 core. A rough total along the full 72′′ length +and 8′′ width of the streamer is then perhaps 500 M⊙. +This would explain why MIR 2’s gravitational influence +seems to drop beyond the outer Keplerian radius deter- +mined above, since there the gas mass of the broader +cloud starts rivalling MIR 2’s effects. +In such a disk, the 0.034 M⊙/yr mass accretion rate +determined by Barnes et al. (2010), still a record as far +as we know, can be supported by a merely 0.01%/yr +“leakage” of mass through the disk onto MIR 2’s core, or +alternatively, that the Streamer can supply this accre- +tion rate for another 104 yr. On the other hand, the time +required to build up the more massive MIR 2 core at this +accretion rate is closer to 40,000 yr instead of the 7,000 yr +estimated by P18, assuming that Barnes et al. (2010)’s +accretion rate is correct. +Without detailed modelling, +we cannot refine the accretion rate value here beyond an +approximate calculation below, but the true rate seems +unlikely to be too much less than this, with such a mas- +sive reservoir available for accumulation. +As one example, consider the velocity difference be- +tween the brightest disk material within the bottom-left +quadrant of the rotation curves, and the 1350 M⊙ curve +itself, as seen in Figure 12 between 286◦.217 > +∼l > +∼286◦ +.208 and at latitudes ∼4′′ north of MIR 2. +This dif- +ference runs from 0 km s−1 at the eastern end of this +window to ∼+10 km s−1 at MIR 2, in the sense of be- +ing “sub-rotational” in our line of sight. If we suppose +that the rotational motion here is being translated into +proper motions inward to MIR 2, the effective accretion +speed can be taken (very approximately!) +as Vaccr ∼ +5 km s−1 = 5 pc/Myr. The emission along this feature +(which covers perhaps half of the main part of the 8′′- +wide Streamer) averages ∼10 K/ch in brightness, or con- +servatively, ∼20 K km s−1 integrated. +Using a simple conversion factor X(13CO) = 1026 m−2 +(K km s−1)−1 (probably an underestimate; Barnes et al. +2018), the column density in this feature alone runs +around Naccr ∼ 2×1027 m−2 = 4000 M⊙/pc2, or perhaps +2 +3 of the Streamer’s total column density as seen in Figure +27. This translates to a linear density Λaccr ∼ 400 M⊙/pc +within the 0.1 pc width of the presumed accretion stream. +Therefore we have a mass flux in this accretion stream +of +˙Maccr = ΛaccrVaccr ∼ 2×10−3 M⊙/yr. + +Magnetic Fields and Gas Structures in BYF 73 +31 +This very rough estimate is still on the large side com- +pared to other massive protostars (Rygl et al. 2013), but +smaller than Barnes et al. (2010)’s rate of 0.034 M⊙/yr. +The true value is likely a multiple of the above example, +however, due to several factors: our conservative starting +column density, other accretion flows such as the western +side of the Streamer, the super-rotating material, higher +density streams traced better by C18O or CN, 2× faster +flow closer to MIR 2, and a probably 5× larger effective +X(13CO). Thus, the Barnes et al. (2010) rate may still +be a reasonable global estimate. +In summary, the complex yet potentially understand- +able structure of the Streamer near MIR 2, and Keplerian +disk/freefalling infall zone around it, may have much to +tell us about heavy mass accretion onto a massive proto- +star. Clearly, MIR 2 is an exceptional and exciting object +that demands further study. +6.2. Magnetic Fields: Driving the Outflow? +The 12CO outflow from MIR 2 is fairly massive: with +a typical line brightness I12CO ∼ 10–30 K per 0.16 km s−1 +ALMA channel or average integrated intensity ∼200 K +km s−1, we can use Eq. (9) or its ilk (Barnes et al. 2018) +to estimate gas column densities of about 8×1025 m−2 +or 160 M⊙ pc−2 in the outflow. Inside the flow dimen- +sions of roughly 1×0.1 pc, this gives a total outflow mass +of perhaps 16 M⊙. This mass is being driven to speeds +of 10s of km s−1, so the kinetic energy of the flow is +similarly large, about 1.6×1039 J. If this emerges over +timescales of 10s of kyr, then the mass outflow rate is +˙Mout = ΛoutVout ∼ 5×10−4 M⊙/yr (or ∼10% of the in- +fall rate as estimated above, similar to other outflows; +Pudritz & Ray 2019) and the mechanical luminosity of +the outflow Lout ∼ 4 L⊙. Could this mechanical power +be imparted by B fields? +From Eq. (15) and Figure 31, we see that the energy +density in the B field is typically well below the kinetic +energy density in the higher-velocity and lower-density +gas: the B field is therefore likely a passenger in the flow +at these points. On the other hand, M may rival or even +exceed K where Vrel is small. This result, however, should +be taken as merely suggestive, since M/K > 1 only in the +20 lowest-Vrel channels, where the 12CO opacity is still +high and we may not be mapping much of the outflow via +12CO. But B fields do seem to be detected throughout +the outflow, at a few × 10 nT. It seems reasonable to +suppose that similar B fields (at least!) should exist close +to BYF 73’s Vsys, and specifically close to the base of the +flow at MIR 2. If this were true, the B field would at +least have the potential of being energetically important. +As such, this is circumstantial yet valuable evidence +that the B field may be intimately involved in driving, +or at least shaping, the outflow. Pudritz & Ray (2019) +showcase some other recent observational results mak- +ing this B field connection to the outflow, typically on +∼100 AU (i.e., disk) scales. As far as we are aware, this is +the first instance where the structure of the whole molec- +ular outflow might at least partially be attributable to +the B field at its origin. The connection is not always +clear, however: a rare case where the B field seems to +play an important role in massive star formation is the +compact H ii region K3-50, where a strong ionised out- +flow emanates from a high-mass protostellar object, sur- +rounded by a Keplerian disk extending over radii 0.1– +0.7 pc (Howard et al. 1997). There, the B field inferred +at the inner edge of the disk seems to be strong enough +to provide support against gravity; however, even in K3- +50, the B field does not seem strong enough to influence +the outflow (Barnes et al. 2015). +Our results for BYF 73 seem to provide additional +observational +support +for +the +picture +of +magneto- +centrifugally powered protostellar outflows (Shu et al. +1994; Ouyed & Pudritz 1997; Tomisaka 2011), as opposed +to the main competing model of turbulent entrainment +of gas from a bipolar jet (e.g., Raga et al. 1993). Recent +numerical work on solar coronal mass ejections (Jiang et +al. 2021) may even supply a specific mechanism for the +high speeds in such outflows, namely sudden magnetic re- +connection in bipolar loops, presumably anchored in the +inner parts of a protostellar accretion disk. It is tempt- +ing to suppose such reconnecting loops drive the vigorous +outflows widely seen in other star-forming clouds, as may +be happening here with MIR 2/BYF 73. +Future work +in this area, supported by ALMA+SOFIA observations, +should be very interesting. +6.3. Magnetic Criticality +We briefly also note that the critical N, n, and B⊥ val- +ues derived from HRO analysis of the SOFIA+ALMA +data (§5.2) place BYF 73 just below the maximum B|| +trend line in Crutcher (2012)’s n vs B summary plot +(his Fig. 6), nicely among other dense clouds’ CN-Zeeman +measurements. In contrast, BYF 73 is slightly above the +line of criticality in Crutcher (2012)’s N vs B|| plot (his +Fig. 7), on the side of being subcritical and a little above +its supercritical counterparts in this regard. Given the +uncertainties in our results, however, this may not be +terribly significant. Further, BYF 73 is not the most ex- +treme strong-B outlier compared to the line of critical- +ity, but it may be the highest column density subcritical +cloud. As Crutcher (2012) points out, this would be un- +usual in the sense of at least supporting the possibility +of ambipolar diffusion playing an important role in cloud +stability (Mouschovias & Ciolek 1999). +However, our +HRO result is not the same as a direct Zeeman strength +measurement, and it should probably not yet be overin- +terpreted without some confirming evidence. +Nevertheless, it is tempting to wonder what higher- +resolution and -sensitivity observations might reveal +about the material closer in to MIR 2, where the infall +might be more clearly imaged, the outflow may be driven, +and its originating disk might be discerned. +7. CONCLUSIONS +We have presented a range of new observational data +exploring details of the massive molecular clump BYF 73, +previously thought to harbour a massive (240 M⊙), +very young (7,000 yr), Class 0 protostar (MIR 2) with +the largest mass inflow rate (0.034 M⊙/yr) observed to +date. The new data include far-IR (SOFIA/HAWC+) +and 3mm (ALMA) continuum emission, mm-wave spec- +troscopy of several molecular species, and polarisation +maps in both the continuum and spectral lines from both +facilities. The polarisation data in particular have been +analysed in order to learn about the structure, strength, +role, and significance of the B field in this cloud (as sum- +marised in Table 2), and the continuum and spectral line + +32 +Barnes et al. +data were analysed and interpreted in this context. Our +results include the following. +• The 14′′ resolution HAWC+ data show a centrally +concentrated cloud with generally low polarised emission +(a few percent) from the central 0.5 pc of the molecular +clump, but at a relatively high polarisation (10–20%) +extended across the adjacent, 2 pc wide, low-power H ii +region. The polarisation structure east of MIR 2 shows a +second, distinct feature in the form of an arc, the eastern +polarisation loop (EPL); there is also a clear, very-low +to zero-polarisation boundary layer (IBL) around MIR 2 +and the EPL. +• The 2.′′5 resolution ALMA continuum data show four +main features: a narrow, massive EW Streamer of cold, +dense gas; a fainter, NS line of emission coincident with +the ionisation front (IF) facing the H ii region; another +faint spur of emission aligned with the EPL; and a small +number (5) of 3mm point sources, of which MIR 2 is by +far the brightest. These 3mm point sources are far fewer +than the number seen at near- or mid-IR wavelengths, +suggesting that many of the latter may be relatively low- +extinction and/or more evolved objects. The polarised +3mm emission comes from parts of the Streamer, IF, and +EPL; it is somewhat patchy but also mainly oriented EW, +switching to a NS orientation across MIR 2, with very +high (20–40%) fractional polarisation in most locations. +• The ALMA 12CO Stokes I cube reveals a prominent, +powerful, bipolar outflow from MIR 2, extending over a +velocity range almost ±40 km s−1 from the cloud’s Vsys. +Both the 0.4 pc red and 0.6 pc blue wings of this out- +flow appear to be deflected from their starting vectors +by inertially significant parts of the Streamer. The EPL +may be a result of this deflection in the blue wing of the +outflow. +• The wider ALMA 13CO, C18O, and CN mosaics re- +veal much more extended emission across the cloud than +in the 3mm continuum, with only modest structural or +kinematic correspondence to the Streamer, IF, or point +sources. These suggest that the wider cloud is somewhat +porous to the UV radiation from the adjacent H ii region. +The outflow can, however, be traced closer in to MIR 2 +within the 13CO cube. +• In the same area, the 13CO also shows clear evi- +dence for material in either Keplerian rotation about, or +free-fall onto, MIR 2; the apparent axial geometry of this +material, however, is puzzling. If Keplerian, it implies a +gravitating mass 1350±50 M⊙ within 1.′′8 = 4500 AU of +MIR 2 and any envelope; if freely infalling, the implied +mass within that radius is 950±35 M⊙. +These masses +are > +∼5× larger than from SED fitting, suggesting possi- +bly significant grain growth has occurred in MIR 2. The +larger mass in a small radius also suggests up to 33% +of MIR 2’s luminosity could be powered by gravitational +energy release. In light of these higher-resolution and - +sensitivity data, the prior mass infall rate is found to be +reasonable; however with a 5× larger mass, MIR 2’s age +may be more like 40,000 yr. +• Davis-Chandrasekhar-Fermi (DCF) analysis of the +continuum polarisation data suggest relatively strong B +fields are present in the gas near MIR 2: 92 nT at the +Mopra scale ≈ 2×the HAWC+ scale, and 1.18 µT at the +ALMA scale, the latter a possible record in cold, non- +masering molecular gas. Despite these high values, they +Table 2 +Summary of Magnetic Field Results in BYF 73 +Structure +Facility +Method +B +n +(nT) +(m−3) +H ii-N,ionsa +HAWC+ +DCFc +17.5 +1.4e8 +H ii-N,dustb +HAWC+ +DCF +162 +7e9 +H ii-S,ionsa +HAWC+ +DCF +8.8 +1.4e8 +H ii-S,dustb +HAWC+ +DCF +81 +7e9 +MIR 2 core +HAWC+ +DCF +77 +4e11 +Streamer-W +ALMA +DCF +92 +8e11 +MIR 2 core +ALMA +DCF +740 +3.6e13 +MIR 2 extn. +ALMA +DCF +330 +1e12 +EPL +ALMA +DCF +49 +3e11 +Streamer +ALMA +DCF +106 +5e11 +Streamer +HAWC+ +HRO +25±6 +1e11 +Streamer +ALMA +HRO +61±27 +3e11 +Streamer +HAWC+ALMA +HRO +42±7 +2e11 +a Using a mean molecular weight µ = 1.28 for ionised gas. +b Using a mean molecular weight µ = 2.35 for molecular gas. +c DCF B field values (Eq. (1)) are scaled to the local dispersion +s (e.g., Table 1) with uncertainties ∼ ±30%. +are nominally consistent with critical balance between +B fields and gravity. With the higher central mass for +MIR 2 indicated by the Keplerian pattern in the 13CO +data, the gas is supercritical in these areas. In the H ii re- +gion, the DCF estimate is 21 nT, also somewhat stronger +than typical in such gas, but where the ionised flow still +dominates the energetics. +• Histogram of relative orientations (HRO) analysis +gives a sharper estimate of where the B field might reach +criticality in the gas. In the Streamer, we obtain thresh- +olds for criticality of Bcrit = 42±7 nT at log(Ncrit/m2) += 26.74±0.09 or log(ncrit/m3) = 11.31± 0.09, where B +likely dominates gravity and helps organise the gas struc- +tures below these thresholds, and gravity likely domi- +nates above them. +• The 12CO polarisation cubes reveal the presence of +the Goldreich-Kylafis effect almost everywhere in the +outflow. +The orientation of the B field is seen to lie +closely along the outflow direction, consistent with prior +studies but in a far more widespread manner than seen +before. A simplified DCF analysis of the 12CO emission +in each channel shows that, for most of the outflow, the +B field does not dominate the kinetic energy of the flow. +However, the two energy densities may be comparable at +the lowest outflow velocities, where the B field may even +drive the flow close to MIR 2. +• Despite a peak S/N of 200 in Stokes I, the ALMA +CN polarisation data detects no Zeeman effect above the +noise in the Stokes V cube, with a 3σ limit of 1 µT. This +may partly be attributable to the outflow’s predominant +orientation across our line of sight, perhaps organising +other cloud structures in a similar direction, and min- +imising B||. +These results suggest that even higher resolution +and/or sensitivity data on BYF 73 and MIR 2 would pro- +duce exciting constraints on early stages of massive star +formation. +We thank the SOFIA crew and scientific staff, and the +ALMA-North America staff, for outstanding support of +their respective facilities. We also thank the anonymous +referee for a careful reading of the manuscript and many +helpful suggestions which improved the presentation of + +Magnetic Fields and Gas Structures in BYF 73 +33 +the paper. PJB gratefully acknowledges financial sup- +port for this work provided by NASA through awards +SOF 07-0089 and 09-0048 issued by USRA. Based in +part on observations made with the NASA/DLR Strato- +spheric Observatory for Infrared Astronomy (SOFIA). +SOFIA is jointly operated by the Universities Space Re- +search Association, Inc. +(USRA), under NASA con- +tract NNA17BF53C, and the Deutsches SOFIA Insti- +tut (DSI) under DLR contract 50 OK 2002 to the Uni- +versity of Stuttgart. This paper makes use of the fol- +lowing ALMA data: ADS/JAO.ALMA.2019.1.01031.S. +ALMA is a partnership of ESO (representing its mem- +ber states), NSF (USA) and NINS (Japan), together with +NRC (Canada), MOST and ASIAA (Taiwan), and KASI +(Republic of Korea), in cooperation with the Republic +of Chile. 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P., Goodman, A., & Alves, J. 2021, A&A, 633, +51 + diff --git a/UtE2T4oBgHgl3EQfCwbV/content/tmp_files/load_file.txt b/UtE2T4oBgHgl3EQfCwbV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d8b54cf7251c9c98cfe7faa788b614ffa609297 --- /dev/null +++ b/UtE2T4oBgHgl3EQfCwbV/content/tmp_files/load_file.txt @@ -0,0 +1,2400 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf,len=2399 +page_content='Draft version January 11, 2023 Preprint typeset using LATEX style emulateapj v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 12/16/11 SOFIA AND ALMA INVESTIGATE MAGNETIC FIELDS AND GAS STRUCTURES IN MASSIVE STAR FORMATION: THE CASE OF THE MASQUERADING MONSTER IN BYF 73 Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Barnes1,2, Stuart D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Ryder3,4, Giles Novak5,6, Richard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Crutcher7, Laura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Fissel8, Rebecca L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Pitts9, and William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Schap III10 Draft version January 11, 2023 ABSTRACT We present SOFIA+ALMA continuum and spectral-line polarisation data on the massive molec- ular cloud BYF 73, revealing important details about the magnetic field morphology, gas structures, and energetics in this unusual massive star formation laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The 154µm HAWC+ polarisation map finds a highly organised magnetic field in the densest, inner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='55×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='40 pc portion of the cloud, compared to an unremarkable morphology in the cloud’s outer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The 3mm continuum ALMA polarisation data reveal several more structures in the inner domain, including a pc-long, ∼500 M⊙ “Streamer” around the central massive protostellar object MIR 2, with magnetic fields mostly paral- lel to the east-west Streamer but oriented north-south across MIR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The magnetic field orientation changes from mostly parallel to the column density structures to mostly perpendicular, at thresholds Ncrit = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='6×1026 m−2, ncrit = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='5×1011 m−3, and Bcrit = 42±7 nT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' ALMA also mapped Goldreich- Kylafis polarisation in 12CO across the cloud, which traces in both total intensity and polarised flux, a powerful bipolar outflow from MIR 2 that interacts strongly with the Streamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The magnetic field is also strongly aligned along the outflow direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' energetically, it may dominate the outflow near MIR 2, comprising rare evidence for a magnetocentrifugal origin to such outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' A portion of the Streamer may be in Keplerian rotation around MIR 2, implying a gravitating mass 1350±50 M⊙ for the protostar+disk+envelope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' alternatively, these kinematics can be explained by gas in free fall towards a 950±35 M⊙ object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The high accretion rate onto MIR 2 apparently occurs through the Streamer/disk, and could account for ∼33% of MIR 2’s total luminosity via gravitational energy release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Keywords: ISM: magnetic fields — stars: formation — ISM: kinematics and dynamics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' INTRODUCTION Magnetic fields (hereafter B fields) in astrophysical set- tings are very widespread and may play an important role in the evolution of the interstellar medium (ISM), stars, galaxies, and the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Yet, they are techni- cally challenging to measure, limiting our ability to un- derstand the full physics within these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' This is because B field measurements depend on accurate values for the polarised contributions to emission or absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', the Stokes parameters Q, U, V ), which are usually much weaker than the total intensity I, and then inter- preting the data in terms of particular physical polarisa- pbarnes@spacescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='org 1 Space Science Institute, 4765 Walnut St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Suite B, Boulder, CO 80301, USA 2 School of Science and Technology, University of New Eng- land, Armidale NSW 2351, Australia 3 School of Mathematical & Physical Sciences, Macquarie Uni- versity, NSW 2109, Australia 4 Astronomy, Astrophysics and Astrophotonics Research Cen- tre, Macquarie University, NSW 2109, Australia 5 Center for Interdisciplinary Exploration & Research in As- trophysics (CIERA), 1800 Sherman Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', Evanston, IL 60201, USA 6 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' of Physics and Astronomy, Northwestern University, 2145 Sheridan Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', Evanston, IL 60208, USA 7 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' of Astronomy, University of Illinois, 1010 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Green St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', Urbana, IL 60801, USA 8 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' of Physics, Engineering Physics & Astronomy, Queen’s University, 64 Bader Lane, Kingston, ON, K7L 3N6, Canada 9 Niels Bohr Institute, Centre for Star & Planet Formation, University of Copenhagen, Øster Voldgade 5-7, 1350 Copen- hagen K, Denmark 10 Astronomy Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', University of Florida, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Box 112055, Gainesville, FL 32611, USA tion mechanisms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', as explained by Crutcher (2012) or Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' In star formation (SF), the role and importance of B fields is a long-standing problem (McKee & Ostriker 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Crutcher 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' This is largely due to observa- tional challenges of high-quality B field measurements in large cloud samples at high spatial dynamic range (SDR), and relating these to the clouds’ other physi- cal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Prior work on the Zeeman effect shows that, below a threshold density n0 ≈ 300 cm−3, B fields can support gas against gravity and have fairly uniform strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Above this level, studies suggest the line-of- sight component increases with density, B|| ∝ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='65, and the ratio of magnetic to gravitational forces is close to critical (Crutcher 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Confirming the higher-density behaviour is important to SF theory, since SF is not observed in low-density gas (Lada 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Tracking local variations in the transi- tion density n0 is also significant, since this could change the SF efficiency and/or initial mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Catching massive protostars, especially, in the act of formation is even more difficult compared to low-mass protostars, be- cause of their greater distances, accelerated timescales, and rapid alteration of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The plane-of-sky component B⊥ has recently begun to be mapped at high SDR via linear polarisation of mm–µm continuum emission or absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', Planck Collaboration 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' This probably arises from non- spherical dust grains aligned by radiative torques to the B field: while not all alignment mechanisms are mag- netic, non-magnetic mechanisms are not thought to be arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='03618v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='GA] 9 Jan 2023 2 Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' dominant (Lazarian 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' If the alignment is magnetic, statistical methods can convert turbulent variations in field orientation θB⊥ to estimates of |B⊥| (Davis 1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Chandrasekhar & Fermi 1953, hereafter DCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Although approximate, DCF methods have been effectively used from cloud (10 pc) to core (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='1 pc) scales (Myers & Good- man 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2015) to meaningfully constrain the importance of B fields in different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Large-scale maps of FIR/submm polarisation from Planck and other missions coupled with new analysis methods and high-quality molecular gas data (Fissel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Soler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2018) permit new insights into the role of B fields in SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' In Vela C, for example, the alignment of θB⊥ with dense structures changes from parallel to perpendicular near the same threshold n0 as in the Zeeman data (Fissel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' However, data on massive cluster-scale clumps, where most massive protostars likely also form, are very sparse: we need to precisely measure both |B| and n in a wider variety of clouds and environments to test these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' As part of a long-term project to systematically inves- tigate the physics of B fields and dense gas in a uniform sample of CN-bright, massive molecular clumps that are likely sites of high-mass star formation (Sharpe 2019), we obtained observing time with both the Stratospheric Observatory For Infrared Astronomy (SOFIA) and Ata- cama Large Millimeter/submillimeter Array (ALMA) to map the first few targets in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' We used the polarimetric far-infrared (FIR) High-resolution Airborne Wideband Camera-plus (HAWC+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2018) aboard SOFIA and ALMA’s full-polarisation mode in both the 3 mm continuum and spectral line observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' We report here the first results for this project, an analysis of the B field properties in the molecular cloud BYF 73 with the most massive protostellar inflow rate known (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2010), and following up recent multi-wavelength work on the same cloud (Pitts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2018, hereafter P18) from Gemini with T-ReCS, SOFIA with FIFI-LS, and ATCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' P18 found that, of the 8 mid- IR point sources imaged with T-ReCS, MIR 2 seems to be the overwhelmingly dominant protostellar source in terms of mass (240 M⊙) and luminosity (4700 L⊙), yet comprises only ∼1% of the cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' After ruling out gravitational energy release from the inflow and other forms of mechanical or thermal energy, it was not clear what MIR 2’s energy source is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' MIR 2 also seems re- markably young, perhaps only 7000 yr old at the very high mass accretion rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='034 M⊙ yr−1) in the cloud (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2010), making it potentially the most mas- sive and youngest Class 0 protostar known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Our intent was to map the global (5′) B field structure and gas kinematics across this exceptional cloud exhibit- ing such large-scale mass motions, at a high enough res- olution (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='′′6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='′′5 for SOFIA and ALMA, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=') to potentially constrain the role of the B field, gas dynam- ics, and energy balance in this very unusual context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' In §2 we describe the observational and data reduction approach, briefly overview the continuum data, and compare their calibra- tion with prior studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' In §3 we explore features of the FIR and 3 mm continuum emission globally and in de- tail, including the polarisation data and inferred B field morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' In §4 we present the velocity-resolved 3 mm spectral-line mosaics and polarisation products, includ- ing key insights into the significance of the continuum features based on the lines’ kinematic and dynamical in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' In §5 we use two standard statistical methods, one in a new way for the spectroscopy, to analyse our po- larisation data and obtain constraints on the role of B fields in this cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' We discuss all these results in §6 in order to highlight new insights from the data as well as their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' We present our conclusions in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' SOFIA/HAWC+ We mapped BYF 73 on 2019 July 17 at 0832–0905 UT with HAWC+’s band D (λ154 µm) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='11 Chopping and nodding were done asymmetrically due to the nearby FIR emission to the Galactic west and south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The to- tal on-source integration time was 784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Pipeline pro- cessing with HAWC-DRP produced final Level 4 qual- ity image products which were downloaded from the SOFIA archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' This processing produces data that has all known instrumental and atmospheric effects removed, giving an absolute Stokes I calibration uncertainty of 20%, a relative polarisation uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='3% in flux and 3◦ in angle, and astrometry which should be accu- rate to better than 3′′ (Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' However, we found the HAWC+ L4 astrometry was still consistently offset ∼2′′ to the Galactic south compared to the Gemini 10 µm, Herschel 70 µm, and ALMA & ATCA 3mm maps, all of which are strongly and consistently peaked on the massive protostellar core MIR 2 (allowing for MIR 1’s proximity to MIR 2 in the Gemini data), so we inserted this correction by hand into the HAWC+ data files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' At a distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='27 kpc (near NGC 3324;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Samson 2021), the scale for BYF 73 is 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='01 = 36′′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='44 pc, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='1 pc = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='′′25 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Thus, HAWC+ band D gives us a useful spatial dynamic range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='16 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='6 pc, a linear factor of 22 and almost 500 resolution elements in area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The resulting full-field images in both total intensity Stokes I and the debiased polarised flux P ′ = � Q2 + U 2 − n2rms, where nrms is the combined in- strumental and sky noise, are shown in Figure 1, overlaid also with the inferred B field polarisation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' ALMA BYF 73 was observed with ALMA at 3 mm wavelength on 2020 January 1 in the C-43 array (baselines 15–314 m) and in two correlator setups and mapping modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' the to- tal on-source integration time was ∼7500 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The first mode mapped a standard, 13-pointing mosaic of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='′8 centered on the peak molecular line emission as measured in the Mopra maps (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' 2010), similar in extent to the ATCA mosaic reported by P18, in both the 3 mm cold dust continuum plus the J=1→0 lines of 13CO & C18O and N=1→0 line of CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The second mode was a single-pointing, full-polarisation, deeper integration at the peak emission position near MIR 2, to map the B field strength & structure with (1) the cold dust contin- uum, and potentially with (2) the Goldreich-Kylafis ef- fect in the line wings of 12CO (Goldreich & Kylafis 1981), 11 See the HAWC+ description at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='sofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='usra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='edu/ instruments/hawc, its Data Handbook at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='sofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='usra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' edu/sites/default/files/Instruments/HAWC_PLUS/Documents/ hawc_data_handbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='pdf, and the Cycle 7 Observer’s Handbook at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='sofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='usra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='edu/sites/default/files/Other/Documen ts/OH-Cycle7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='pdf for details of the observing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Magnetic Fields and Gas Structures in BYF 73 3 HAWC+ 154 µm Stokes I image I contours: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='125( √ 2)16 Jy/pixel Polo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' data selection: I > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='25 Jy/pixel, P ′/σP ′ > 2 beam FWHM p′ = 10% HAWC+ 154 µm Polarised Flux image I contours: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='125( √ 2)16 Jy/pixel Polo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' data selection: I > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='25 Jy/pixel, P ′/σP ′ > 2 beam FWHM p′ = 10% Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' (Top) SOFIA HAWC+ band D (154µm) total intensity (Stokes I) image of BYF 73 on a logarithmic scale, overlaid by white contours as labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' (In all figures, we use the notation x(y)z for contours running from level x in steps of y to level z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=') All HAWC+ band D images have 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='75 pixels, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='2× the 13′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='6 beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' At every 2nd pixel (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='4 beam) satisfying the indicated selection criteria, we also display black “vectors” showing the measured polarisation percentage (p′) and position angle (with the usual ±π degeneracy) of the plane-of-sky B field component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=', rotated 90◦ from the observed polarisation direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' The peak I intensity is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='58 Jy/pixel with a typical rms error in the interior of the image 2–3 mJy/pixel, rising to 4–6 mJy/pixel around the image boundary due to the dither pattern of the observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' the peak S/N in the I image is >5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' Inside the I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='5 Jy/pixel contour, nearly all p′ vectors have S/N ranging from ∼5 to >30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content=' for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQfCwbV/content/2301.03618v1.pdf'} +page_content='25 0. In the periodic lattice, we +let Dm +i = Di + m, for m ∈ Λ, and then denote the full lattice as +D = +� +m∈Λ +N +� +i=1 +Dm +i . +We will define a finite system of resonators resulting from truncation of the periodic lattice. Let +Ir ⊂ Λ be all lattice points within distance r from the origin +Ir = {m ∈ Λ | |m| < r}. +We define the finite collection of resonators Df = Df(r) as +Df(r) = +� +m∈Ir +D + m. +In this setting, Df is a finite lattice where D is the single, repeated unit. The goal is to clarify in which +sense the spectral properties of a finite, but large, lattice can be approximated by the corresponding +infinite one. +We let G be the Green’s function for Laplace’s equation in three dimensions: +G(x) = − +1 +4π|x|. +Given a bounded domain Ω ⊂ R3, we then define the single layer potential SΩ : L2(∂Ω) → H1(∂Ω) as +SΩ[ϕ](x) := +� +∂Ω +G(x − y)ϕ(y) dσ(y), +x ∈ ∂Ω. +Specifically, SΩ is known to be invertible [7]. For a finite lattice, we define the capacitance coefficients +as +(Cmn +f +)ij(r) = +� +∂Dm +i +S−1 +Df [χ∂Dn +j ] dσ, +(2.1) +for 1 ≤ i, j ≤ N and m, n ∈ Ir. Here, we explicitly indicate the dependence of the size r of the +truncated lattice. For m, n ∈ Ir, we observe that Cmn +f +(r) is a matrix of size N × N, while the block +matrix Cf = (Cmn +f +) is a matrix of size N|Ir| × N|Ir|. +We next define the capacitance coefficients for the infinite lattice. We begin by defining the dual +lattice Λ∗ of Λ as the lattice generated by α1, ..., αd satisfying αi · lj = 2πδij and P⊥αi = 0, for +i, j = 1, ..., d. We define the Brillouin zone Y ∗ as Y ∗ := �Rd × {0}�/Λ∗, where 0 is the zero-vector in +R3−d. We remark that Y ∗ can be written as Y ∗ = Y ∗ +d × {0}, where Y ∗ +d has the topology of a torus in +d dimensions. +When α /∈ Y \ {0}, we can define the quasi-periodic Green’s function Gα(x) as +Gα(x) := +� +m∈Λ +G(x − m)eiα·m. +(2.2) +The series in (2.2) converges uniformly for x and y in compact sets of Rd, with x ̸= y and α ̸= 0. Given +a bounded domain Ω ⊂ Y , we can then define the quasi-periodic single layer potential Sα +Ω : L2(∂Ω) → +H1(∂Ω) as +Sα +Ω[ϕ](x) := +� +∂Ω +Gα(x − y)ϕ(y) dσ(y), +x ∈ ∂Ω. +(2.3) +For α ∈ Y ∗ and for 1 ≤ i, j ≤ N, the quasi-periodic capacitance matrix (“dual-space” representation) +is the N × N-matrix defined as +�Cα +ij = +� +∂Di +(Sα +D)−1[χ∂Dj] dσ. +(2.4) +3 + +· · · +· · · +· · · +· · · +m = +�−2 +−1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�−2 +0 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�−2 +1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�−1 +−1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�−1 +0 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�−1 +1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +� 0 +−1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�0 +0 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�0 +1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +� 1 +−1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�1 +0 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�1 +1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +� 2 +−1 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�2 +0 +� +i=1 +i=3 +i=2 +· · · +· · · +· · · +· · · +m = +�2 +1 +� +i=1 +i=3 +i=2 +m = +�−1 +−1 +� +i=1 +i=3 +i=2 +m = +�−1 +0 +� +i=1 +i=3 +i=2 +m = +�−1 +1 +� +i=1 +i=3 +i=2 +m = +� 0 +−1 +� +i=1 +i=3 +i=2 +m = +�0 +0 +� +i=1 +i=3 +i=2 +m = +�0 +1 +� +i=1 +i=3 +i=2 +m = +� 1 +−1 +� +i=1 +i=3 +i=2 +m = +�1 +0 +� +i=1 +i=3 +i=2 +m = +�1 +1 +� +i=1 +i=3 +i=2 +r +r +(2.1) +Df(r) +(2.4) +D +�Cα, α ∈ Y ∗ +quasi-periodic capacitance +matrix +Inverse Floquet +transform (2.5) +C +real-space capacitance +matrix +Truncate +Ct(r), r ∈ (0, ∞) +truncated capacitance +matrix +Cf(r), r ∈ (0, ∞) +finite capacitance +matrix +Figure 1: This work studies the convergence of the eigenfrequencies of defect modes in a truncated periodic +material to the spectrum of the corresponding infinite material. We use capacitance matrices as a canonical +model for many-body scattering of time-harmonic waves. The aim of this work is to show how eigenvalues of +the finite capacitance matrix Cf(r) converge to those of the real-space capacitance matrix C. The calligraphic +font for C denotes the fact that this is an infinite matrix. Our strategy is to compare the spectrum of Cf(r) with +the truncated capacitance matrix Ct(r), which is obtained by truncating all but a finite O(r) number of rows +in C, before letting r → ∞. Throughout this work, we use the block matrix notation (Cmn)ij to refer to the +i, j ∈ {1, . . . , N} entry of the n, m ∈ Λ block in a matrix C. +4 + +Figure 2: An example of a localized defect mode for a system of 31 resonators. The eigenvalues of the finite ma- +trix BtCf are computed, where Cf is the generalized capacitance matrix for a system of evenly spaces resonators +and Bt is the identity matrix but with the central entry (Bt)0 +11 = 2. +For 1 ≤ i, j ≤ N, we can then define the “real-space” capacitance coefficients at the lattice point m by +Cm +ij = +1 +|Y ∗| +� +Y ∗ +�Cα +ije−iα·m dα. +(2.5) +Here, C0 +ij corresponds to the diagonal block which contains the capacitance coefficients of the resonators +within a single unit cell. We use the notation C to denote the infinite matrix that contains all the Cm +ij +coefficients, for all 1 ≤ i, j ≤ N and all m ∈ Λ. +A final, important quantity for the analysis in this work is the truncated capacitance matrix Ct. +This is obtained by keeping only N|Ir| × N|Ir| coefficients from C, to give a matrix that is the same +size as Cf. A schematic of the various pieces of notation used in this article and how they related to +each other is given in Figure 1. The proof strategy deployed in this work is to compare the spectra of +Cf with that of Ct, and then let r → ∞ in order to approximate the spectrum of C. In particular, the +modes that we will compare are defect modes, which are spatially localized modes that exist due to +the presence of defects in the otherwise periodic material, an example of which is shown in Figure 2. +We will model defect modes through pre-multiplication by a defect matrix B. For each m ∈ Λ, we +let Bm be an N × N diagonal matrix +Bm = +ábm +1 +0 +· · · +0 +0 +bm +2 +· · · +0 +... +... +... +... +0 +0 +· · · +bm +N +ë +, +(2.6) +where the diagonal entries bm +i +are real-valued parameters. In this work, we only consider compact +defects, where bm +i += 1 for all but finitely many i and m. For the infinite structure, we let B be the +infinite block-diagonal matrix that contains Bm for all m ∈ Λ. Under the assumption on the bm +i , B is +said to be a compact perturbation of the identity. The spectrum of the infinite structure is given by +the solutions to the spectral problem +BCu = λu. +(2.7) +For the finite structure of size r, we let Bt be the block-diagonal matrix (Bm), m ∈ Ir and consider +the spectral problem +BtCfu = λu. +An example of such a defect mode is shown in Figure 2. A system of 31 resonators is modelled, with +the finite defect matrix Bt chosen to be the identity, perturbed so that its central element is (Bt)0 +11 = 2. +The generalized capacitance matrix serves not only as a canonical model for coupled resonators +(whose interaction terms decay as r−1), but can also be derived from first principles in certain physical +settings. For example, in Appendix A we briefly explain how this model arises for a system of high- +contrast resonators in which case the eigenstates of the generalized capacitance matrix fully characterize +the subwavelength resonant spectrum of the system. +2.2 +Convergence of capacitance coefficients +Based on the layer-potential characterization of capacitance, we prove in this section that the capaci- +tance coefficients of a large but finite structure converge, as the size grows, to corresponding coefficients +5 + +of the infinite structure. We begin with the following result, which collects some well-known results +on the capacitance matrices [1, 8]. +Lemma 2.1. Let �Cα and Cf be the quasi-periodic and finite capacitance matrix, respectively. Then +(i) �Cα and Cf are symmetric, positive definite matrices; +(ii) �Cα and Cf are strictly diagonally dominant matrices; +(iii) We have ( �Cα)ii > 0 and (Cmm +f +)ii > 0. Moreover, for i ̸= j and m ̸= n we have ( �Cα)ij < 0 and +(Cmn +f +)ij < 0. +The next result shows that a fixed block of the infinite capacitance matrix is approximately equal +to corresponding block of the capacitance matrix of the finite structure. In other words, the finite- +structure capacitance coefficients can be approximated through the infinite structure as long as we are +sufficiently far away from the edges of the finite structure. +Theorem 2.2. For fixed m, n ∈ Λ, we have as r → ∞, +lim +r→∞ Cmn +f +(r) = Cm−n. +Proof. Firstly, observe that +SDf[ψ] = +� +m∈Ir +SD+m[ψm], +where ψm = ψ|∂D+m. Recall that the quasi-periodic single-layer potential is defined as +Sα +D[φ] = +� +∂D +� +m∈Λ +G(x − y − m)eiα·mφ(y) dσ. +Given φ ∈ L2(D), we define φα +m ∈ L2(D + m) as +φα +m(y) = φ(y − m)eiα·m. +Then it is clear that +Sα +D[φ] = +� +m∈Λ +SD+m[φα +m]. +We can then decompose +Sα +D[φ] = +� +m∈Ir +SD+m[φα +m] + +� +∂D +� +m∈Λ\Ir +G(x − y − m)eiα·mφ(y) dσ += SDf[φα] + Rα[φ], +where, in the operator norm, Rα = o(1) as r → ∞. From the Neumann series, we now have +(Sα +D)−1[χ∂Di] = S−1 +Df [χα +i ] + o(1), +(2.8) +where χα +i is defined as +χα +i = +� +m∈Ir +χ∂Dm +i eiα·m. +From Lemma B.2 in Appendix B, we know that the error term in (2.8) holds uniformly in α. +If +m, n ∈ Ir are fixed and i, j = 1, ..., N, we then have from (2.8) that +Cm−n +ij += +1 +|Y ∗| +� +Y ∗ +� +∂Di +e−iα·(m−n)S−1 +Df [χα +j ] dσ dα + o(1) += +� +∂Dm +i +S−1 +Df [χ∂Dn +j ] dσ + o(1) += (Cmn +f +)ij(r) + o(1). +This proves the claim. +The numerical results presented in Figure 3 demonstrate the convergence of the capacitance co- +efficients, as established by Theorem 2.2. We plot |(Cf)0 +11 − C0 +11| for both a one-dimensional and a +two-dimensional lattice and show that this error converges algebraically to zero as the size of the finite +lattice increases. +6 + +· · · +· · · +. +(a) One-dimensional lattice +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +. +(b) Two-dimensional square lattice +Figure 3: Convergence of the capacitance coefficient of large finite lattices. (a) A one-dimensional lattice with +a single resonator in the unit cell (N = 1). (b) A two-dimensional lattice with a single resonator in the unit +cell (N = 1). In both cases, we plot |(Cf)0 +11 − C0 +11| for increasing numbers of resonators r. Here, the error +scales as O(r−3.3) for sufficiently small r. +3 +Convergence to pure point spectrum +In this section, we study a problem where the infinite structure has a pure point spectrum, correspond- +ing to a localized mode. We introduce a defect to the model in order to create such a mode. For a +finite, truncated structure, there will be an eigenvalue arbitrarily close to the pure point spectrum. +3.1 +Example of a defect structure +Before developing any convergence theory, we present an example of a defect structure exhibiting a +pure point spectrum, corresponding to a localized mode. We take a lattice with a single resonator +N = 1 inside each unit cell. We take a single resonator with perturbed (“defect”) material parameter. +In other words, +bm +1 = +® +1, +m ̸= 0, +1 + x, +m = 0, +(3.1) +for some parameter x > −1. The eigenvalues of the (infinite-dimensional) generalized capacitance +matrix BC in this setting was studied in [2]. It was found that λ is an eigenvalue of BC if and only if +it is a root of the equation +x +|Y ∗| +� +Y ∗ +λα +1 +λ − λα +1 +dα = 1, +(3.2) +where λα +1 is the single eigenvalue of the quasi-periodic capacitance matrix �Cα of the unperturbed +periodic structure. This equation has a solution λ = λ0 precisely in the case x > 0. In other words, the +7 + +defect induces an eigenvalue λ0 in the pure point spectrum of BC, corresponding to an exponentially +localized eigenmode. An example of such a localized eigenmode was shown in Figure 2. +3.2 +Convergence of defect modes +In this section, we prove that, if the infinite structure has a localized mode, there will be an eigenvalue +of the truncated structure arbitrarily close to the localized frequency. +We let C denote the infinite capacitance matrix. As before, we let Cf denote the capacitance matrix +of a finite structure of size N|Ir| × N|Ir|. Furthermore, we let Ct denote the truncated matrix of C +of size N|Ir| × N|Ir|, and similarly let Bt be the truncation of B. At this point, we emphasize that +Ct is “nonphysical” in the sense that it does not correspond to a capacitance matrix associated to any +physical structure but, rather, to the finite matrix obtained by simply truncating the infinite matrix +C. +We assume that BC has a localized eigenmode u, and let ut be the truncation of u of size N|Ir|. +The first result follows only from the decay of the localized mode. +Lemma 3.1. Assume that B is a compact perturbation of the identity, such that BC has a localized +eigenmode u with corresponding eigenvalue λ. Then there is an eigenvalue ˜λ = ˜λ(r) of BtCt satisfying +lim +r→∞ +˜λ(r) = λ. +Proof. We let ut be the infinite vector obtained by padding ut with 0. Since u is in ℓ2(Λ), for any ε > 0 +we can choose large enough r so that +∥u − ut∥ℓ2 < ε. +Since BC is a bounded operator, we then have +∥λu − BCut∥ℓ2 < Kε, +for some K > 0. Restricting to the finite block of size r, we have +∥λut − BtCtut∥2 < Kε. +In other words, λ is in the Kε-pseudospectrum of BtCt, and since BtCt is normal, we have an eigenvalue +˜λ of BtCt satisfying +|˜λ(r) − λ| = Kε. +This proves the claim. +Next, we study the properties of Cf as the size of the finite structure increases. +Lemma 3.2. For i = 1, ..., N + 1, assume that Bi ⊂ R3 are disjoint, connected domains and let +B = +N +� +n=1 +Bi +�B = +N+1 +� +n=1 +Bi. +Let Cij, �Cij denote the capacitance coefficients associated to B and �B, respectively. Then +Cii ≤ �Cii +i = 1, ..., N. +Proof. We will use a variational characterization of the capacitance coefficients. +Let H = {v ∈ +H1 +loc(R3) | v(x) ∼ |x|−1 as x → ∞} and let +V = {v ∈ H | v|∂Bj = δij for j = 1, ..., N}, +�V = {v ∈ H | v|∂Bj = δij for j = 1, ..., N + 1}. +Observe that �V ⊂ V. It then follows that +Cii = min +v∈V +� +R3 |∇v|2 dx ≤ min +v∈�V +� +R3 |∇v|2 dx = �Cii. +8 + +Remark 3.3. Lemma 3.2 states that the diagonal capacitance coefficients will always increase when +adding additional resonators. In the physical situation of electrostatics this result is intuitive: the +self-capacitance of a conductor can only increase if additional conductors are introduced. +Lemma 3.4. As r → ∞, we have ∥Cf∥2 < K for some K independent of r. +Proof. We know that the capacitance matrix Cf is diagonally dominant: +(Cmm +f +)ii > +� +n∈Z,j̸=i +��(Cmn +f +)ij +��, +for any i, m. For fixed i and m, we know from Lemma 3.2 that (Cmm +f +)ii(r) is increasing in r, and for +all r we have +(Cmm +f +)ii(r) < C0 +ii, +where, as before, C0 +ii is the corresponding entry of the infinite capacitance matrix C. In particular, the +eigenvalues of Cf(r) are bounded as r → ∞, which shows the claim. +As discussed above, the matrix Ct appearing in Lemma 3.1 is nonphysical, as it is a truncation +of the matrix for the infinite system. Instead, we need to phrase the result for the matrix Cf, which +describes the finite system. The following theorem is the main result of this section. +Theorem 3.5. Assume that B is a compact perturbation of the identity, such that BC has a localized +eigenmode u with corresponding eigenvalue λ. Then there is an eigenvalue ˆλ = ˆλ(r) of BtCf satisfying +lim +r→∞ +ˆλ(r) = λ. +Proof. We let +K1 = sup +r>0 +∥Cf(r) − Ct∥2, +and observe from Lemma 3.4 that K < ∞. We also let +K2 = ∥Bf∥2. +Given ε > 0, we pick r0 > 0 such that the following four terms are small: +∥C0,f −C0,t∥2 < +ε +4K2 +, +∥ut−u0,t∥2 < +ε +4K1K2 +, +∥Bf(Ct−C0,t)u0,t∥2 < ε +4, +∥Bf(Cf −C0,f)u0,t∥2 < ε +4 +for all r large enough; the first inequality follows from Theorem 2.2 while the subsequent inequalities +follow from the ℓ2(Λ)-decay of u. Here, C0,t, u0,t, and C0,f are the truncations of Ct, ut, and Cf to the +smaller lattice of radius r0 (padded with zero where needed for the matrix operations). We know from +Lemma 3.1 that we can take r large enough so that BtCt has an eigenvalue ˜λ of distance ε from λ. +We then have +BfCfut = BfCtut + Bf(Cf − Ct)(ut − u0,t) + Bf(C0,f − C0,t)u0,t ++ Bf(Cf − C0,f)u0,t − Bf(Ct − C0,t)u0,t. +(3.3) +Then +∥(BtCf − BtCt)ut∥2 < ε, +which means that there is an eigenvalue ˆλ of distance ε from �λ, and hence |ˆλ − λ| < 2ε. +Remark 3.6. As an example, B and C as given in Section 3.1 satisfy the assumptions of Theorem 3.5. +9 + +· · · +1 +1 +1 +1+x +1 +1 +1 +· · · +(a) One-dimensional lattice +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +. +. +(b) Two-dimensional square lattice +Figure 4: Convergence of the frequency of the defect modes, for a defect on the central resonator (with x = 1) +created by perturbing a single entry of B. (a) A one-dimensional lattice with a single resonator in the unit +cell (N = 1). Here, the difference between the defect frequency computed for a finite structure and for the +corresponding infinite structure scales as O(r−1.4), where r is the length of the truncated structure. This is +shown in the upper plot. The lower plot shows the spectrum of successively larger lattices. The defect frequency +for the infinite structure is computed using (3.2). In the geometry sketch on the right, the corresponding entry +bm +1 +from the matrix B is shown above each resonator. +(b) A two-dimensional square lattice with a single +resonator in the unit cell (N = 1). Here, the error scales as O(r−3.3), where r is the width of the (square) +truncated structure. +10 + +25 +米 +米 +米 +米 +米 +米 +3 +Frequency +20 +来 +米 +15 +米 +米 +米米 +米 +米 +米 +米 +米 +10 +9 +19 +39 +59 +99 +199 +8 +Size of structure r· · · +· · · +. +Figure 5: Convergence of the frequency of the defect modes in a lattice with resonators arranged in pairs (N = 2) +and a defect corresponding to the two central resonators being removed. This gives two topologically protected +edge modes. Here, the error scales as O(r−1.7) for the even mode and O(r−3.8) for the odd mode, where r is +the length of the truncated structure. +3.3 +Numerical illustration +Figure 4 shows the convergence of the difference between the defect frequency computed for a finite +structure and for the corresponding infinite structure, computed analytically using e.g. +(3.2). +It +is evident that the frequency converges algebraically, unlike related work on one-dimensional systems +where the truncated frequency converges exponentially as a function of the length of the finite structure +[14]. This holds irrespective of the dimensionality d ∈ {1, 2, 3} of the lattice: the results in Figure 4a +are for a one-dimensional lattice while Figure 4b shows a two-dimensional square lattice. +The reason for the algebraic convergence observed in this matrix model is the presence of long- +range interactions between the coupled resonators (which scale inversely with the distance between +resonators) and the fact that the capacitance matrix has all non-zero entries. Conversely, the exponen- +tial convergence observed in one-dimensional models, e.g. by [13, 14, 16], is due to the fact that the +corresponding capacitance matrix is tridiagonal in this case [9]. It can be shown that the exponential +convergence is a general property of tridiagonal matrix systems, whereas the algebraic convergence +observed here is typical of three-dimensional scattering problems, where interactions scale inversely +with distances. +Remark 3.7. Comparing Figure 4a and Figure 3a, it appears that the error of the frequency of the +defect mode is inheriting the O(r−1.4) convergence of the capacitance coefficients. Similar behaviour +is observed for the square lattice, whereby O(r−3.3) convergence is observed both for the defect mode +in Figure 4b and the capacitance coefficient in Figure 3b. While this is unsurprising, it turns out not +to be the case for other types of compact defect. For example, in Figure 5 we show the convergence of +the defect modes in a dislocated Su-Schrieffer-Heeger (SSH) lattice, which is a one-dimensional lattice +of resonators arranged in pairs (so N = 2). This system supports two defect modes that are known +to be topologically protected and benefit from enhanced robustness properties (see [3] for details). The +even mode experiences O(r−1.7) convergence while the odd mode converges at a faster O(r−3.8) rate. +Understanding these different convergence rates is a valuable question for future study. +4 +Convergence to continuous spectrum +Through numerical illustrations, we can illustrate how the discrete spectrum of the truncated structure +approximates the Floquet-Bloch spectral bands of the infinite structure. Making analytic statements +relating these two quantities, however, is a challenging problem that is beyond the scope of the present +work. The two spectra have very different fundamental characteristics and a greater understanding +of the edge effects that occur at the ends of the finite structure would be needed in order to make +progress on this fiendish question. +We now outline the method used to compute the discrete band structure which, given the set of +eigenpairs (ωj, uj) of a truncated structure, approximates the band structure of the periodic struc- +ture. If we take the size r of the truncated structure to be reasonably large, the eigenmode uj will +approximately be a linear combination of Bloch modes with frequency ωj. To compare the discrete +11 + +eigenvalues of the truncated problem to the continuous spectrum of the periodic problem, we ’reverse +engineer’ the appropriate quasi-periodicities α corresponding to these Bloch modes. Observe that uj +is a vector of length N|Ir|. If we let (uj)m denote the vector of length N associated to cell m ∈ Λ, we +define the truncated Floquet transform of uj as +(ˆuj)α = +� +m∈Ir +(uj)meiα·m, +α ∈ Y ∗. +(4.1) +Observe that (ˆuj)α is a vector of length N. Looking at the 2-norm ∥(ˆuj)α∥2 as a function of α, this +function has distinct peaks at certain values of α. We then take the quasi-periodicitiy associated to +the mode uj as +argmax +α∈Y ∗ ∥(ˆuj)α∥2. +(4.2) +Note that the symmetry of the problem means that if α is an approximate quasi-periodicity then so +will −α be. In cases of additional symmetries of the lattice, we expect additional symmetries of the +quasi-periodicities. +· · · +· · · +. +(a) Single periodic resonators (N = 1) +· · · +· · · +. +(b) Periodic pairs of resonators (N = 2) +Figure 6: The continuous spectrum of the infinite structure and the discrete spectrum of the truncated structure +for a one-dimensional lattices. (a) Single periodic resonators (N = 1) with a truncated structure consisting of +50 resonators. (b) Periodic pairs of resonators (N = 2) with a truncated structure containing 100 resonators. +In both cases, the truncated Floquet transform (4.1) is used to approximate the quasi-periodicity of the truncated +modes. +Figure 6a shows the subwavelength continuous spectrum of an infinite array of resonators, which +takes the form of a single spectral band. It is plotted alongside the discrete spectrum of a truncated +array of 50 resonators, for which the quasi-periodicities have been approximated using the method +outlined above. +The discrete band structure mostly follows closely the infinite one, even for this +relatively small truncated array. The frequencies close to zero are not exhibited in the finite structure, +as the edge effects have the greatest effect on low-frequency modes. We would need to consider a much +larger truncated structure to capture the lowest frequency part of the spectrum. +This behaviour can also be observed in more complicated structures. In Figure 6b, we compare the +continuous and truncated spectra of an array of resonators arranged in pairs (dimers). The truncated +structure has 100 resonators arranged in 50 pairs. This geometry is an example of the famous SSH +12 + +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +· · · +· · · +... +... +. +(a) Square lattice +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +. +(b) Honeycomb lattice +Figure 7: Examples of continuous and discrete spectra of the infinite and truncated structures, respectively. (a) +A square lattice with two resonators per unit cell, resulting in two bands separated by a gap. (b) A honeycomb +lattice with Dirac cones at the vertices of the Brillouin zone. In both cases, the truncated structure have 800 +resonators and the truncated Floquet transform is used to approximate the quasi-periodicity of the truncated +modes. +chain [17] which has been shown to have fascinating topological properties [5]. This system has two +subwavelength spectral bands and the truncated modes are split between approximating the two bands. +Additionally, we can consider this method for lattices of higher dimension. Figure 7a shows the +case of a square lattice of resonator dimers. Similarly to Figure 6b, there is a band gap between the +first and the second bands, and we see a close agreement between the discrete and the continuous band +structure. Figure 7b shows a similar figure in the case of a honeycomb lattice, where the finite lattice +is truncated along zig-zag edges of the lattice. As shown in [6], there are Dirac cones on each corner +of the Brillouin zone. In the truncated structure, in addition to the “bulk modes” whose frequencies +closely agree with the continuous spectrum, there are “edge modes” which are localized around the +edges and whose points in the band structure lie away from the continuous bands. +5 +Concluding remarks +In this work, we have demonstrated the convergence of defect modes in large resonator arrays to the +corresponding modes in the infinite, periodic structure. We have studied this using the generalized +capacitance matrix, which is a canonical model for three-dimensional wave scattering by resonant +systems with long-range interactions. +Our conclusions could also be generalized to other models, +since the decay of the Helmholtz Green’s function is the key feature that underpins our results. In +13 + +0 +2 +α1-2Freo +10 +5米 +米 +米 +米 +米 +米53 +uency +15米 +米米米15 +*类 +1025 +20 +**20 +***0 +-20 +2 +α2一 +一 +- +0 +-2 +2 +α1-410 +e +5-米 +米53 +uency +15米 +米 +米米 +米 +米米* +米 +米 +** +米 +米 +米 +米 +米 +米 +米米 +** +米米 +米 +米 +米 +米 +* +米 +米 +米 +米1025 +2020 +1.5+0 +一 +-4 +-2一 +0 +2 +44 +α2particular, the long-range “1/r” decay is the cause of the algebraic convergence we observe, in contrast +to the exponential convergence observed in analogous one-dimensional settings. +A significant advantage of the model used in this work is that the Bloch modes, in addition to +the defect modes, are also concisely characterized. As detailed in Section 4, this provides a numerical +method for approximating the continuous spectrum. Importantly, this constructive approach presents +a possible avenue for proving statements about the convergence of eigenvalues to the continuous spec- +trum. We see developing this convergence theory as a challenging but important problem for future +investigation. Even in one-dimensional models, demonstrating convergence to the continuous spectrum +remains an open problem. +A +Asymptotic derivation of the model +In this brief appendix, we recall how the generalized capacitance matrix arises through an asymptotic +treatment of a system of coupled high-contrast resonators. In particular, it can be used to characterize +the subwavelength (i.e. asymptotically low-frequency) resonance of the system. For more details and +a review of extensions to other settings (such as non-Hermitian and time-modulated systems) see [1]. +We will present the results for a finite system of resonators. Analogous results hold for infinite +periodic systems, by modifying the Green’s function appropriately [1]. We suppose that the material +inclusions Di ⊂ R3, as considered already in this work, represent the material inclusions that will act +as our resonators. We consider the scattering of time-harmonic waves with frequency ω and will solve +a Helmholtz scattering problem in three dimensions. This Helmholtz problem, which can be used to +model acoustic, elastic and polarized electromagnetic waves, represents the simplest model for wave +propagation that still exhibits the rich phenomena associated to subwavelength physics. +We use vi denote the wave speed in each resonator Di. In which case, ki = ω/vi is the wave number +in Di. Similarly, the wave speed and wave number in the background medium are denoted by v and +k. Finally, we must introduce the material contrast parameters δ1, . . . , δN. These parameters describe +the contrast between the material inside Di and the background material. For example, in the case of +an acoustic system, δi is the density of the material inside Di divided by the density of the background +material. We will want these contrast parameters to be small (an air bubble in water is one famous +example in the setting of acoustics). Then for the domain +D = +� +m∈Ir +N +� +i=1 +(Di + m), +we consider the Helmholtz resonance problem +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∆u + k2u = 0 +in Rd \ D, +∆u + k2 +i u = 0 +in Di + m, for i = 1, . . . , N, m ∈ Ir, +u|+ − u|− = 0 +on ∂D, +δi +∂u +∂ν +���� ++ +− ∂u +∂ν +���� +− += 0 +on ∂Di + m for i = 1, . . . , N, m ∈ Ir, +u(x) satisfies the Sommerfeld radiation condition, +(A.1) +where the Sommerfeld radiation condition says that +lim +|x|→∞ |x| +d−1 +2 +Å ∂ +∂|x| − ik +ã +u = 0, +uniformly in all directions x/|x|, +(A.2) +and guarantees that energy is radiated outwards by the scattered solution. +The asymptotic regime we consider is that the material contrast parameters are all small while the +wave speeds are all of order one. That is, there exists some δ > 0 such that +δi = O(δ) +and +v, vi = O(1) +as +δ → 0, for i = 1, . . . , N. +(A.3) +Within this setting, we are interested in solutions to the resonance problem (A.1) that are subwavelength +in the sense that +ω → 0 +as +δ → 0. +(A.4) +14 + +To be able to characterize the subwavelength resonant modes of this system, we must define the +generalized capacitance coefficients. Recall the capacitance coefficients (Cmn +f +)ij from (2.1). Then, we +define the corresponding generalized capacitance coefficient as +(Cmn +f +)ij = δiv2 +i +|Dm +i |(Cmn +f +)ij, +(A.5) +where |Dm +i | is the volume of the bounded subset Dm +i . +Then, the eigenvalues of Cf determine the +subwavelength resonant frequencies of the system, as prescribed by the following theorem. +Theorem A.1. Consider a system of N|Ir| subwavelength resonators in R3. For sufficiently small +δ > 0, there exist N|Ir| subwavelength resonant frequencies ω1(δ), . . . , ωN|Ir|(δ) with non-negative real +parts. Further, the subwavelength resonant frequencies are given by +ωn = +� +λn + O(δ) +as +δ → 0, +where {λn : n = 1, . . . , N|Ir|} are the eigenvalues of the generalized capacitance matrix Cf, which satisfy +λn = O(δ) as δ → 0. +A similar result exists for an infinite periodic structure, in terms of the eigenvalues of the generalized +quasi-periodic capacitance matrix, as defined in (2.4), see [1] for details. +The definition (A.5) clarifies the motivation for pre-multiplying by the perturbation matrix B +to describe defects. +When B is a compact perturbation of the identity, it describes defects that +correspond to changing the material parameters on a finite number of resonators, so that the quantity +δiv2 +i corresponding to those resonators is altered. +B +Uniformity across the Brillouin zone +In this appendix, we provide additional details of the proof of Theorem 2.2. +The main result is +Lemma B.2, which shows that (Sα +D)−1 is in operator norm, uniformly bounded for α in a neighbourhood +of 0. The analysis is similar to [4, Section 3.3]. +From e.g. [7], we have a dual-space representation of Gα given by +Gα(x) = − 1 +|Y | +� +q∈Λ∗ +ei(α+q)·x +|α + q|2 = −eiα·x +|Y ||α|2 − 1 +|Y | +� +q∈Λ∗\{0} +ei(α+q)·x +|α + q|2 . +Define the periodic Green’s function G0 as +G0(x) = − 1 +|Y | +� +q∈Λ∗\{0} +eiq·x +|q|2 . +For α close to zero, we then have +Gα(x) = +−1 +|Y ||α|2 − iα · x +|Y ||α|2 + (α · x)2 +2|Y ||α|2 + G0(x) + O(|α|). +Consequently, for α close to zero, we have the an expansion of the single-layer potential Sα +D: +Sα +D[ψ](x) = − +1 +|Y ||α|2 +� +∂D +ψ(y) dσ − +i +|Y ||α|2 +� +∂D +α · (x − y)ψ(y) dσ ++ +1 +2|Y ||α|2 +� +∂D +�α · (x − y)�2ψ(y) dσ + S0 +D[ψ](x) + O(|α|). +(B.1) +Lemma B.1. If S0 +D[ϕ] = Kχ∂D for some constant K and some ϕ ∈ L2(∂D) satisfying +� +∂D ϕ dσ = 0, +then ϕ = 0. +15 + +Proof. For x ∈ R3 \ D, define V (x) := S0 +D[ϕ](x). Then V solves the following differential problem, +� +� +� +� +� +� +� +∆V = 0 +in R3 \ D, +V |+ = K +on ∂D, +V (x + m) = V (x) +for all m ∈ Λ. +(B.2) +Moreover, using the jump relations and integration by parts, we have that +� +∂D +ϕ dσ = K +� +Y \D +|∇V |2 dx = 0. +If K ̸= 0, it follows from (B.2) that +� +Y \D |∇V |2 dx ̸= 0 which is a contradiction. In other words we +must have K = 0, so that S0 +D[ϕ] = 0 and +� +∂D ϕ dσ = 0. From [4, Lemma 3.7], we have that ϕ = 0. +Lemma B.2. ∥(Sα +D)−1∥, in operator norm, is bounded for α in a neighbourhood of 0. +Proof. To reach a contradiction, we assume that Sα +D[φ] = O(|α|) for some φ, which can be written as +φ = φ0 + |α|φ1, where φ0 is nonzero, does not depend on α, and φ1 = O(1) as |α| → 0. Also define +v = +α +|α|. From (B.1) it follows that +� +∂D +φ0 dσ = 0, +� +∂D +φ1(y) dσ + i +� +∂D +v · (x − y)φ0(y) dσ = O(|α|), +K(v) − iv · x +� +∂D +φ1(y) dσ + 1 +2 +� +∂D +�v · (x − y)�2φ0(y) dσ + |Y |S0 +D[φ0] = O(|α|), +where K is constant as function of x. Simplifying, we have that +1 +2 +� +∂D +�v · (x − y)�2φ0(y) dσ = −(v · x) +� +∂D +(v · y)φ0(y) dσ + 1 +2 +� +∂D +(v · y)2φ0(y) dσ. +In total we get +S0 +D[φ0](x) = ˜K(v) + 2(v · x) +|Y | +� +∂D +(v · y)φ0(y) dσ, +where ˜K is constant in x. Observe that S0 +D[φ0](x) is independent of v. As a function of x, this function +is constant for x ∈ v⊥, and so this function is constant for all x. From Lemma B.1 we get that φ0 = 0 +which proves the claim. +References +[1] H. Ammari, B. Davies, and E. O. Hiltunen. Functional analytic methods for discrete approxima- +tions of subwavelength resonator systems. arXiv preprint arXiv:2106.12301, 2021. +[2] H. Ammari, B. Davies, and E. O. Hiltunen. Anderson localization in the subwavelength regime. +arXiv preprint arXiv:2205.13337, 2022. +[3] H. Ammari, B. Davies, and E. O. Hiltunen. Robust edge modes in dislocated systems of subwave- +length resonators. J. London Math. Soc., 106(3):2075–2135, 2022. +[4] H. Ammari, B. Davies, E. O. Hiltunen, H. Lee, and S. Yu. Exceptional points in parity–time- +symmetric subwavelength metamaterials. SIAM J. Math. Anal., 54(6):6223–6253, 2022. +[5] H. Ammari, B. Davies, E. O. Hiltunen, and S. Yu. Topologically protected edge modes in one- +dimensional chains of subwavelength resonators. J. Math. Pures Appl., 144:17–49, 2020. +[6] H. Ammari, B. Fitzpatrick, E. O. Hiltunen, H. Lee, and S. Yu. Honeycomb-lattice Minnaert +bubbles. SIAM J. Math. Anal., 52(6):5441–5466, 2020. +16 + +[7] H. Ammari, H. Kang, and H. Lee. Layer Potential Techniques in Spectral Analysis, volume 153 +of Mathematical Surveys and Monographs. American Mathematical Society, Providence, 2009. +[8] R. A. Diaz and W. J. Herrera. The positivity and other properties of the matrix of capacitance: +Physical and mathematical implications. J. Electrost., 69(6):587–595, 2011. +[9] F. Feppon, Z. Cheng, and H. Ammari. Subwavelength resonances in 1D high-contrast acoustic +media. HAL preprint hal-03697696, 2022. +[10] N. L. Hills and S. N. Karp. Semi-infinite diffraction gratings – I. Commun. Pure Appl. Math., +18:203–233, 1965. +[11] L. M. Joseph and R. V. Craster. Reflection from a semi-infinite stack of layers using homogeniza- +tion. Wave Motion, 54:145–156, 2015. +[12] P. A. Kuchment. Floquet Theory for Partial Differential Equations, volume 60 of Operator Theory: +Advances and Applications. Springer Science & Business Media, 1993. +[13] J. Lin. A perturbation approach for near bound-state resonances of photonic crystal with defect. +European J. Appl. Math., 27(1):66–86, 2016. +[14] J. Lin and F. Santosa. Resonances of a finite one-dimensional photonic crystal with a defect. +SIAM J. Appl. 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Wave Motion, 86:44–62, 2019. +17 + diff --git a/XNE1T4oBgHgl3EQfvwUc/content/tmp_files/load_file.txt b/XNE1T4oBgHgl3EQfvwUc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2e0fc2f1045700ae152572f19e5d730936fb9aa --- /dev/null +++ b/XNE1T4oBgHgl3EQfvwUc/content/tmp_files/load_file.txt @@ -0,0 +1,1190 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf,len=1189 +page_content='Spectral convergence of defect modes in large finite resonator arrays Habib Ammari∗ Bryn Davies† Erik Orvehed Hiltunen‡ Abstract We show that defect modes in infinite systems of resonators have corresponding modes in finite systems which converge as the size of the system increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We study the generalized capacitance matrix as a model for three-dimensional coupled resonators with long-range interactions and con- sider defect modes that are induced by compact perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' If such a mode exists, then there are elements of the discrete spectrum of the corresponding truncated, finite system converging algebraically to each element of the pure point spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This result, which concerns periodic lattices of arbitrary dimension in a three-dimensional differential system, is in contrast with the exponential convergence observed in one-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This is due to the presence of long- range interactions in the system, which gives a dense matrix model and shows that exponential convergence cannot be expected in physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Mathematics Subject Classification (MSC2010): 35J05, 35C20, 35P20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Keywords: finite crystals, metamaterials, edge effects, capacitance coefficients, subwavelength reso- nance 1 Introduction Much of the physical literature concerning wave propagation in periodic media relies on a believable but highly non-trivial piece of logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' That is, researchers want to be able to relate the spectral properties of infinite periodic structures with truncated, finite versions of the same material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The motivation for this is that infinite periodic structures can be described very concisely using Floquet-Bloch analysis [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' However, finite, truncated versions of the structure are often required when it comes to either numerical or physical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' It is perfectly plausible that the two structures should behave similarly, particularly away from the edges of the truncated structure and especially when the truncated structure is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' However, a precise convergence theory relating the spectra of these two quite different differential operators is, in general, yet to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The many interesting phenomena that occur at the edges of periodic arrays have been studied in some detail [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For example, there is a tendency for wave energy to be localized to the edges of the structure, taking the form of surface waves [15, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This is an example of an edge effect and highlights that there will always be fundamental differences between how infinite and truncated structures interact with waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Another important question that has been explored in this field, and is intimately related to the results presented in this work, is the extent to which waves incident on the edge of a truncated periodic structure can excite Bloch waves in the structure (thus, replicating the behaviour of its infinite counterpart) [11, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The central question of this work is the extent to which the resonant spectra of infinite and truncated structures can be related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will focus on localized modes which decay quickly outside of some compact region, meaning they are less severely affected by edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Additionally, localized modes are the eigenmodes of interest for many wave guiding applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Existing results have shown that in certain one-dimensional systems, any defect mode of the infinite structure will have a corresponding ∗Department of Mathematics, ETH Zurich, Zurich, Switzerland (habib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='ammari@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='ch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' †Department of Mathematics, Imperial College London, London, UK (bryn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='davies@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' ‡Department of Mathematics, Yale University, New Haven, USA (erik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='hiltunen@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='03402v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='CD] 9 Jan 2023 mode in the truncated structure converging to the defect mode as the size tends to infinity [13, 14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A defect mode is a mode that is created by making a perturbation to introduce a defect to the periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Such a mode is characterized by being spatially localized (in the sense that it decays quickly enough to be square integrable along the axis or axes of periodicity) and having an eigenfrequency that belongs to the pure point spectrum of the perturbed periodic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The terminology “pure point spectrum” and “defect mode eigenfrequency” are preferred by spectral analysts and wave physicists, respectively, and we will use them somewhat interchangeably here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The rest of the spectrum will typically be composed of the continuous spectrum, which corresponds to the Bloch modes that propagate through the material without decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In previous works, it was shown that the convergence of defect mode eigenfrequency to the pure point spectrum of the infinite periodic operator was exponential with respect to the size of the trun- cated array [13, 14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' These results concerned one-dimensional systems (or, equivalently, tridiagonal matrix models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In this work, we study the generalized capacitance matrix, which is a dense resonator model that includes long-range interactions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will recall how this model can be derived from a three-dimensional scattering problem with high-contrast resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' However, it can be viewed more generally as a canonical model for coupled resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In this setting, we will prove that any defect mode eigenfrequency of the infinite structure has a sequence of eigenvalues of the truncated structures converging to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' However, in contrast to the one-dimensional and tridiagonal models explored previ- ously, we observed that this convergence is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This occurs due to the structure of the problem and is an inherent consequence of the density of the matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We believe that similar behaviour will be observed in other multi-dimensional differential systems and dense matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This paper is split into three main parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In Section 2, we introduce the matrix model (the generalized capacitance matrix) that we will study and prove some elementary properties that lay the foundations for the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Section 3 contains the main results of this work, which show that the truncated structures have eigenfrequencies that converge to the pure point spectrum of the infinite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Finally, in Section 4, we present numerical evidence for the convergence of the truncated spectra to the continuous spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proving convergence to these Bloch modes remains an open problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' however, the constructive nature of the generalized capacitance matrix approach presented in this work provides a promising platform for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 2 The generalized capacitance matrix model In this section, we will introduce the generalized capacitance matrix model that will be the object of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Its definition uses layer potentials to capture the (potentially complex) shapes of the resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In Appendix A we briefly present asymptotic results showing how this model can be deduced from a subwavelength resonance problem with a system of high-contrast resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Finally, we will prove a convergence result for the capacitance coefficients that will be the basis of the theorems in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1 Definition We study a system of periodically repeated resonators in a lattice in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We take lattice vectors l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , ld ∈ R3, where 0 < d < 3, and let Λ denote the lattice generated by these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In other words, Λ := {m1l1 + · · · + mdld | mi ∈ Z} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' At this point, we remark that there are three possible cases: d = 1, corresponding to a chain of resonators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' d = 2, corresponding to a screen of resonators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' or d = 3, corresponding to a crystal of resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For simplicity, we assume that the lattice is aligned with the first d coordinate axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We take Y ⊂ R3 to be a single unit cell, Y = � � � � � {c1l1 + x2e2 + x3e3 | 0 ≤ c1 ≤ 1, x2, x3 ∈ R}, d = 1, {c1l1 + c2l2 + x3e3 | 0 ≤ c1, c2 ≤ 1, x3 ∈ R}, d = 2, {c1l1 + c2l2 + c3l3 | 0 ≤ c1, c2, c3 ≤ 1}, d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 2 We let D ⊂ Y be a collection of N resonators contained in Y D = N � i=1 Di, where Dn are disjoint domains in Y with boundary ∂Di ∈ C1,s for s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In the periodic lattice, we let Dm i = Di + m, for m ∈ Λ, and then denote the full lattice as D = � m∈Λ N � i=1 Dm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will define a finite system of resonators resulting from truncation of the periodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Let Ir ⊂ Λ be all lattice points within distance r from the origin Ir = {m ∈ Λ | |m| < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We define the finite collection of resonators Df = Df(r) as Df(r) = � m∈Ir D + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In this setting, Df is a finite lattice where D is the single, repeated unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The goal is to clarify in which sense the spectral properties of a finite, but large, lattice can be approximated by the corresponding infinite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We let G be the Green’s function for Laplace’s equation in three dimensions: G(x) = − 1 4π|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Given a bounded domain Ω ⊂ R3, we then define the single layer potential SΩ : L2(∂Ω) → H1(∂Ω) as SΩ[ϕ](x) := � ∂Ω G(x − y)ϕ(y) dσ(y), x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Specifically, SΩ is known to be invertible [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For a finite lattice, we define the capacitance coefficients as (Cmn f )ij(r) = � ∂Dm i S−1 Df [χ∂Dn j ] dσ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) for 1 ≤ i, j ≤ N and m, n ∈ Ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Here, we explicitly indicate the dependence of the size r of the truncated lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For m, n ∈ Ir, we observe that Cmn f (r) is a matrix of size N × N, while the block matrix Cf = (Cmn f ) is a matrix of size N|Ir| × N|Ir|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We next define the capacitance coefficients for the infinite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We begin by defining the dual lattice Λ∗ of Λ as the lattice generated by α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', αd satisfying αi · lj = 2πδij and P⊥αi = 0, for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We define the Brillouin zone Y ∗ as Y ∗ := �Rd × {0}�/Λ∗, where 0 is the zero-vector in R3−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We remark that Y ∗ can be written as Y ∗ = Y ∗ d × {0}, where Y ∗ d has the topology of a torus in d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' When α /∈ Y \\ {0}, we can define the quasi-periodic Green’s function Gα(x) as Gα(x) := � m∈Λ G(x − m)eiα·m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) The series in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) converges uniformly for x and y in compact sets of Rd, with x ̸= y and α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Given a bounded domain Ω ⊂ Y , we can then define the quasi-periodic single layer potential Sα Ω : L2(∂Ω) → H1(∂Ω) as Sα Ω[ϕ](x) := � ∂Ω Gα(x − y)ϕ(y) dσ(y), x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3) For α ∈ Y ∗ and for 1 ≤ i, j ≤ N, the quasi-periodic capacitance matrix (“dual-space” representation) is the N × N-matrix defined as �Cα ij = � ∂Di (Sα D)−1[χ∂Dj] dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='· · ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='i=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) Df(r) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4) D �Cα, α ∈ Y ∗ quasi-periodic capacitance matrix Inverse Floquet transform (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5) C real-space capacitance matrix Truncate Ct(r), r ∈ (0, ∞) truncated capacitance matrix Cf(r), r ∈ (0, ∞) finite capacitance matrix Figure 1: This work studies the convergence of the eigenfrequencies of defect modes in a truncated periodic material to the spectrum of the corresponding infinite material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We use capacitance matrices as a canonical model for many-body scattering of time-harmonic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The aim of this work is to show how eigenvalues of the finite capacitance matrix Cf(r) converge to those of the real-space capacitance matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The calligraphic font for C denotes the fact that this is an infinite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Our strategy is to compare the spectrum of Cf(r) with the truncated capacitance matrix Ct(r), which is obtained by truncating all but a finite O(r) number of rows in C, before letting r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Throughout this work, we use the block matrix notation (Cmn)ij to refer to the i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , N} entry of the n, m ∈ Λ block in a matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 4 Figure 2: An example of a localized defect mode for a system of 31 resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The eigenvalues of the finite ma- trix BtCf are computed, where Cf is the generalized capacitance matrix for a system of evenly spaces resonators and Bt is the identity matrix but with the central entry (Bt)0 11 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For 1 ≤ i, j ≤ N, we can then define the “real-space” capacitance coefficients at the lattice point m by Cm ij = 1 |Y ∗| � Y ∗ �Cα ije−iα·m dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5) Here, C0 ij corresponds to the diagonal block which contains the capacitance coefficients of the resonators within a single unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We use the notation C to denote the infinite matrix that contains all the Cm ij coefficients, for all 1 ≤ i, j ≤ N and all m ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A final, important quantity for the analysis in this work is the truncated capacitance matrix Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This is obtained by keeping only N|Ir| × N|Ir| coefficients from C, to give a matrix that is the same size as Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A schematic of the various pieces of notation used in this article and how they related to each other is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The proof strategy deployed in this work is to compare the spectra of Cf with that of Ct, and then let r → ∞ in order to approximate the spectrum of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In particular, the modes that we will compare are defect modes, which are spatially localized modes that exist due to the presence of defects in the otherwise periodic material, an example of which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will model defect modes through pre-multiplication by a defect matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For each m ∈ Λ, we let Bm be an N × N diagonal matrix Bm = ábm 1 0 · · 0 0 bm 2 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 0 0 · · bm N ë , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='6) where the diagonal entries bm i are real-valued parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In this work, we only consider compact defects, where bm i = 1 for all but finitely many i and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For the infinite structure, we let B be the infinite block-diagonal matrix that contains Bm for all m ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Under the assumption on the bm i , B is said to be a compact perturbation of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The spectrum of the infinite structure is given by the solutions to the spectral problem BCu = λu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='7) For the finite structure of size r, we let Bt be the block-diagonal matrix (Bm), m ∈ Ir and consider the spectral problem BtCfu = λu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' An example of such a defect mode is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A system of 31 resonators is modelled, with the finite defect matrix Bt chosen to be the identity, perturbed so that its central element is (Bt)0 11 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The generalized capacitance matrix serves not only as a canonical model for coupled resonators (whose interaction terms decay as r−1), but can also be derived from first principles in certain physical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For example, in Appendix A we briefly explain how this model arises for a system of high- contrast resonators in which case the eigenstates of the generalized capacitance matrix fully characterize the subwavelength resonant spectrum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2 Convergence of capacitance coefficients Based on the layer-potential characterization of capacitance, we prove in this section that the capaci- tance coefficients of a large but finite structure converge, as the size grows, to corresponding coefficients 5 of the infinite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We begin with the following result, which collects some well-known results on the capacitance matrices [1, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Let �Cα and Cf be the quasi-periodic and finite capacitance matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then (i) �Cα and Cf are symmetric, positive definite matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (ii) �Cα and Cf are strictly diagonally dominant matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (iii) We have ( �Cα)ii > 0 and (Cmm f )ii > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Moreover, for i ̸= j and m ̸= n we have ( �Cα)ij < 0 and (Cmn f )ij < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The next result shows that a fixed block of the infinite capacitance matrix is approximately equal to corresponding block of the capacitance matrix of the finite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In other words, the finite- structure capacitance coefficients can be approximated through the infinite structure as long as we are sufficiently far away from the edges of the finite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For fixed m, n ∈ Λ, we have as r → ∞, lim r→∞ Cmn f (r) = Cm−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Firstly, observe that SDf[ψ] = � m∈Ir SD+m[ψm], where ψm = ψ|∂D+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Recall that the quasi-periodic single-layer potential is defined as Sα D[φ] = � ∂D � m∈Λ G(x − y − m)eiα·mφ(y) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Given φ ∈ L2(D), we define φα m ∈ L2(D + m) as φα m(y) = φ(y − m)eiα·m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then it is clear that Sα D[φ] = � m∈Λ SD+m[φα m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We can then decompose Sα D[φ] = � m∈Ir SD+m[φα m] + � ∂D � m∈Λ\\Ir G(x − y − m)eiα·mφ(y) dσ = SDf[φα] + Rα[φ], where, in the operator norm, Rα = o(1) as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' From the Neumann series, we now have (Sα D)−1[χ∂Di] = S−1 Df [χα i ] + o(1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='8) where χα i is defined as χα i = � m∈Ir χ∂Dm i eiα·m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' From Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2 in Appendix B, we know that the error term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='8) holds uniformly in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' If m, n ∈ Ir are fixed and i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', N, we then have from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='8) that Cm−n ij = 1 |Y ∗| � Y ∗ � ∂Di e−iα·(m−n)S−1 Df [χα j ] dσ dα + o(1) = � ∂Dm i S−1 Df [χ∂Dn j ] dσ + o(1) = (Cmn f )ij(r) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The numerical results presented in Figure 3 demonstrate the convergence of the capacitance co- efficients, as established by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We plot |(Cf)0 11 − C0 11| for both a one-dimensional and a two-dimensional lattice and show that this error converges algebraically to zero as the size of the finite lattice increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 6 · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) One-dimensional lattice · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) Two-dimensional square lattice Figure 3: Convergence of the capacitance coefficient of large finite lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) A one-dimensional lattice with a single resonator in the unit cell (N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) A two-dimensional lattice with a single resonator in the unit cell (N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In both cases, we plot |(Cf)0 11 − C0 11| for increasing numbers of resonators r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Here, the error scales as O(r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3) for sufficiently small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 3 Convergence to pure point spectrum In this section, we study a problem where the infinite structure has a pure point spectrum, correspond- ing to a localized mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We introduce a defect to the model in order to create such a mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For a finite, truncated structure, there will be an eigenvalue arbitrarily close to the pure point spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1 Example of a defect structure Before developing any convergence theory, we present an example of a defect structure exhibiting a pure point spectrum, corresponding to a localized mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We take a lattice with a single resonator N = 1 inside each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We take a single resonator with perturbed (“defect”) material parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In other words, bm 1 = ® 1, m ̸= 0, 1 + x, m = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) for some parameter x > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The eigenvalues of the (infinite-dimensional) generalized capacitance matrix BC in this setting was studied in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' It was found that λ is an eigenvalue of BC if and only if it is a root of the equation x |Y ∗| � Y ∗ λα 1 λ − λα 1 dα = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) where λα 1 is the single eigenvalue of the quasi-periodic capacitance matrix �Cα of the unperturbed periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This equation has a solution λ = λ0 precisely in the case x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In other words, the 7 defect induces an eigenvalue λ0 in the pure point spectrum of BC, corresponding to an exponentially localized eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' An example of such a localized eigenmode was shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2 Convergence of defect modes In this section, we prove that, if the infinite structure has a localized mode, there will be an eigenvalue of the truncated structure arbitrarily close to the localized frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We let C denote the infinite capacitance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As before, we let Cf denote the capacitance matrix of a finite structure of size N|Ir| × N|Ir|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Furthermore, we let Ct denote the truncated matrix of C of size N|Ir| × N|Ir|, and similarly let Bt be the truncation of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' At this point, we emphasize that Ct is “nonphysical” in the sense that it does not correspond to a capacitance matrix associated to any physical structure but, rather, to the finite matrix obtained by simply truncating the infinite matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We assume that BC has a localized eigenmode u, and let ut be the truncation of u of size N|Ir|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The first result follows only from the decay of the localized mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Assume that B is a compact perturbation of the identity, such that BC has a localized eigenmode u with corresponding eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then there is an eigenvalue ˜λ = ˜λ(r) of BtCt satisfying lim r→∞ ˜λ(r) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We let ut be the infinite vector obtained by padding ut with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Since u is in ℓ2(Λ), for any ε > 0 we can choose large enough r so that ∥u − ut∥ℓ2 < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Since BC is a bounded operator, we then have ∥λu − BCut∥ℓ2 < Kε, for some K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Restricting to the finite block of size r, we have ∥λut − BtCtut∥2 < Kε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In other words, λ is in the Kε-pseudospectrum of BtCt, and since BtCt is normal, we have an eigenvalue ˜λ of BtCt satisfying |˜λ(r) − λ| = Kε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Next, we study the properties of Cf as the size of the finite structure increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', N + 1, assume that Bi ⊂ R3 are disjoint, connected domains and let B = N � n=1 Bi �B = N+1 � n=1 Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Let Cij, �Cij denote the capacitance coefficients associated to B and �B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then Cii ≤ �Cii i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will use a variational characterization of the capacitance coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Let H = {v ∈ H1 loc(R3) | v(x) ∼ |x|−1 as x → ∞} and let V = {v ∈ H | v|∂Bj = δij for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', N}, �V = {v ∈ H | v|∂Bj = δij for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=', N + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Observe that �V ⊂ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' It then follows that Cii = min v∈V � R3 |∇v|2 dx ≤ min v∈�V � R3 |∇v|2 dx = �Cii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 8 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2 states that the diagonal capacitance coefficients will always increase when adding additional resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In the physical situation of electrostatics this result is intuitive: the self-capacitance of a conductor can only increase if additional conductors are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As r → ∞, we have ∥Cf∥2 < K for some K independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We know that the capacitance matrix Cf is diagonally dominant: (Cmm f )ii > � n∈Z,j̸=i ��(Cmn f )ij ��, for any i, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For fixed i and m, we know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2 that (Cmm f )ii(r) is increasing in r, and for all r we have (Cmm f )ii(r) < C0 ii, where, as before, C0 ii is the corresponding entry of the infinite capacitance matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In particular, the eigenvalues of Cf(r) are bounded as r → ∞, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As discussed above, the matrix Ct appearing in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1 is nonphysical, as it is a truncation of the matrix for the infinite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Instead, we need to phrase the result for the matrix Cf, which describes the finite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The following theorem is the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Assume that B is a compact perturbation of the identity, such that BC has a localized eigenmode u with corresponding eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then there is an eigenvalue ˆλ = ˆλ(r) of BtCf satisfying lim r→∞ ˆλ(r) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We let K1 = sup r>0 ∥Cf(r) − Ct∥2, and observe from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4 that K < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We also let K2 = ∥Bf∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Given ε > 0, we pick r0 > 0 such that the following four terms are small: ∥C0,f −C0,t∥2 < ε 4K2 , ∥ut−u0,t∥2 < ε 4K1K2 , ∥Bf(Ct−C0,t)u0,t∥2 < ε 4, ∥Bf(Cf −C0,f)u0,t∥2 < ε 4 for all r large enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' the first inequality follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2 while the subsequent inequalities follow from the ℓ2(Λ)-decay of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Here, C0,t, u0,t, and C0,f are the truncations of Ct, ut, and Cf to the smaller lattice of radius r0 (padded with zero where needed for the matrix operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1 that we can take r large enough so that BtCt has an eigenvalue ˜λ of distance ε from λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We then have BfCfut = BfCtut + Bf(Cf − Ct)(ut − u0,t) + Bf(C0,f − C0,t)u0,t + Bf(Cf − C0,f)u0,t − Bf(Ct − C0,t)u0,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3) Then ∥(BtCf − BtCt)ut∥2 < ε, which means that there is an eigenvalue ˆλ of distance ε from �λ, and hence |ˆλ − λ| < 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As an example, B and C as given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1 satisfy the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 9 · · 1 1 1 1+x 1 1 1 · · (a) One-dimensional lattice · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) Two-dimensional square lattice Figure 4: Convergence of the frequency of the defect modes, for a defect on the central resonator (with x = 1) created by perturbing a single entry of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) A one-dimensional lattice with a single resonator in the unit cell (N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Here, the difference between the defect frequency computed for a finite structure and for the corresponding infinite structure scales as O(r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4), where r is the length of the truncated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This is shown in the upper plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The lower plot shows the spectrum of successively larger lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The defect frequency for the infinite structure is computed using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In the geometry sketch on the right, the corresponding entry bm 1 from the matrix B is shown above each resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) A two-dimensional square lattice with a single resonator in the unit cell (N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Here, the error scales as O(r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3), where r is the width of the (square) truncated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 10 25 米 米 米 米 米 米 3 Frequency 20 来 米 15 米 米 米米 米 米 米 米 米 10 9 19 39 59 99 199 8 Size of structure r· · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Figure 5: Convergence of the frequency of the defect modes in a lattice with resonators arranged in pairs (N = 2) and a defect corresponding to the two central resonators being removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This gives two topologically protected edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Here, the error scales as O(r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='7) for the even mode and O(r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='8) for the odd mode, where r is the length of the truncated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3 Numerical illustration Figure 4 shows the convergence of the difference between the defect frequency computed for a finite structure and for the corresponding infinite structure, computed analytically using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' It is evident that the frequency converges algebraically, unlike related work on one-dimensional systems where the truncated frequency converges exponentially as a function of the length of the finite structure [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This holds irrespective of the dimensionality d ∈ {1, 2, 3} of the lattice: the results in Figure 4a are for a one-dimensional lattice while Figure 4b shows a two-dimensional square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The reason for the algebraic convergence observed in this matrix model is the presence of long- range interactions between the coupled resonators (which scale inversely with the distance between resonators) and the fact that the capacitance matrix has all non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Conversely, the exponen- tial convergence observed in one-dimensional models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' by [13, 14, 16], is due to the fact that the corresponding capacitance matrix is tridiagonal in this case [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' It can be shown that the exponential convergence is a general property of tridiagonal matrix systems, whereas the algebraic convergence observed here is typical of three-dimensional scattering problems, where interactions scale inversely with distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Comparing Figure 4a and Figure 3a, it appears that the error of the frequency of the defect mode is inheriting the O(r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4) convergence of the capacitance coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Similar behaviour is observed for the square lattice, whereby O(r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3) convergence is observed both for the defect mode in Figure 4b and the capacitance coefficient in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' While this is unsurprising, it turns out not to be the case for other types of compact defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For example, in Figure 5 we show the convergence of the defect modes in a dislocated Su-Schrieffer-Heeger (SSH) lattice, which is a one-dimensional lattice of resonators arranged in pairs (so N = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This system supports two defect modes that are known to be topologically protected and benefit from enhanced robustness properties (see [3] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The even mode experiences O(r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='7) convergence while the odd mode converges at a faster O(r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='8) rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Understanding these different convergence rates is a valuable question for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 4 Convergence to continuous spectrum Through numerical illustrations, we can illustrate how the discrete spectrum of the truncated structure approximates the Floquet-Bloch spectral bands of the infinite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Making analytic statements relating these two quantities, however, is a challenging problem that is beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The two spectra have very different fundamental characteristics and a greater understanding of the edge effects that occur at the ends of the finite structure would be needed in order to make progress on this fiendish question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We now outline the method used to compute the discrete band structure which, given the set of eigenpairs (ωj, uj) of a truncated structure, approximates the band structure of the periodic struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' If we take the size r of the truncated structure to be reasonably large, the eigenmode uj will approximately be a linear combination of Bloch modes with frequency ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' To compare the discrete 11 eigenvalues of the truncated problem to the continuous spectrum of the periodic problem, we ’reverse engineer’ the appropriate quasi-periodicities α corresponding to these Bloch modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Observe that uj is a vector of length N|Ir|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' If we let (uj)m denote the vector of length N associated to cell m ∈ Λ, we define the truncated Floquet transform of uj as (ˆuj)α = � m∈Ir (uj)meiα·m, α ∈ Y ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) Observe that (ˆuj)α is a vector of length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Looking at the 2-norm ∥(ˆuj)α∥2 as a function of α, this function has distinct peaks at certain values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We then take the quasi-periodicitiy associated to the mode uj as argmax α∈Y ∗ ∥(ˆuj)α∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) Note that the symmetry of the problem means that if α is an approximate quasi-periodicity then so will −α be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In cases of additional symmetries of the lattice, we expect additional symmetries of the quasi-periodicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) Single periodic resonators (N = 1) · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) Periodic pairs of resonators (N = 2) Figure 6: The continuous spectrum of the infinite structure and the discrete spectrum of the truncated structure for a one-dimensional lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) Single periodic resonators (N = 1) with a truncated structure consisting of 50 resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) Periodic pairs of resonators (N = 2) with a truncated structure containing 100 resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In both cases, the truncated Floquet transform (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) is used to approximate the quasi-periodicity of the truncated modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Figure 6a shows the subwavelength continuous spectrum of an infinite array of resonators, which takes the form of a single spectral band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' It is plotted alongside the discrete spectrum of a truncated array of 50 resonators, for which the quasi-periodicities have been approximated using the method outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The discrete band structure mostly follows closely the infinite one, even for this relatively small truncated array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The frequencies close to zero are not exhibited in the finite structure, as the edge effects have the greatest effect on low-frequency modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We would need to consider a much larger truncated structure to capture the lowest frequency part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This behaviour can also be observed in more complicated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In Figure 6b, we compare the continuous and truncated spectra of an array of resonators arranged in pairs (dimers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The truncated structure has 100 resonators arranged in 50 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This geometry is an example of the famous SSH 12 · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) Square lattice · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) Honeycomb lattice Figure 7: Examples of continuous and discrete spectra of the infinite and truncated structures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (a) A square lattice with two resonators per unit cell, resulting in two bands separated by a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (b) A honeycomb lattice with Dirac cones at the vertices of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In both cases, the truncated structure have 800 resonators and the truncated Floquet transform is used to approximate the quasi-periodicity of the truncated modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' chain [17] which has been shown to have fascinating topological properties [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This system has two subwavelength spectral bands and the truncated modes are split between approximating the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Additionally, we can consider this method for lattices of higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Figure 7a shows the case of a square lattice of resonator dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Similarly to Figure 6b, there is a band gap between the first and the second bands, and we see a close agreement between the discrete and the continuous band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Figure 7b shows a similar figure in the case of a honeycomb lattice, where the finite lattice is truncated along zig-zag edges of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As shown in [6], there are Dirac cones on each corner of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In the truncated structure, in addition to the “bulk modes” whose frequencies closely agree with the continuous spectrum, there are “edge modes” which are localized around the edges and whose points in the band structure lie away from the continuous bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 5 Concluding remarks In this work, we have demonstrated the convergence of defect modes in large resonator arrays to the corresponding modes in the infinite, periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We have studied this using the generalized capacitance matrix, which is a canonical model for three-dimensional wave scattering by resonant systems with long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Our conclusions could also be generalized to other models, since the decay of the Helmholtz Green’s function is the key feature that underpins our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In 13 0 2 α1-2Freo 10 5米 米 米 米 米 米53 uency 15米 米米米15 类 1025 20 **20 ***0 20 2 α2一 一 0 2 2 α1-410 e 5-米 米53 uency 15米 米 米米 米 米米* 米 米 ** 米 米 米 米 米 米 米米 ** 米米 米 米 米 米 米 米 米 米1025 2020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5+0 一 4 2一 0 2 44 α2particular, the long-range “1/r” decay is the cause of the algebraic convergence we observe, in contrast to the exponential convergence observed in analogous one-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A significant advantage of the model used in this work is that the Bloch modes, in addition to the defect modes, are also concisely characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As detailed in Section 4, this provides a numerical method for approximating the continuous spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Importantly, this constructive approach presents a possible avenue for proving statements about the convergence of eigenvalues to the continuous spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We see developing this convergence theory as a challenging but important problem for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Even in one-dimensional models, demonstrating convergence to the continuous spectrum remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A Asymptotic derivation of the model In this brief appendix, we recall how the generalized capacitance matrix arises through an asymptotic treatment of a system of coupled high-contrast resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In particular, it can be used to characterize the subwavelength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' asymptotically low-frequency) resonance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For more details and a review of extensions to other settings (such as non-Hermitian and time-modulated systems) see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will present the results for a finite system of resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Analogous results hold for infinite periodic systems, by modifying the Green’s function appropriately [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We suppose that the material inclusions Di ⊂ R3, as considered already in this work, represent the material inclusions that will act as our resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We consider the scattering of time-harmonic waves with frequency ω and will solve a Helmholtz scattering problem in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' This Helmholtz problem, which can be used to model acoustic, elastic and polarized electromagnetic waves, represents the simplest model for wave propagation that still exhibits the rich phenomena associated to subwavelength physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We use vi denote the wave speed in each resonator Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In which case, ki = ω/vi is the wave number in Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Similarly, the wave speed and wave number in the background medium are denoted by v and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Finally, we must introduce the material contrast parameters δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , δN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' These parameters describe the contrast between the material inside Di and the background material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For example, in the case of an acoustic system, δi is the density of the material inside Di divided by the density of the background material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' We will want these contrast parameters to be small (an air bubble in water is one famous example in the setting of acoustics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then for the domain D = � m∈Ir N � i=1 (Di + m), we consider the Helmholtz resonance problem � � � � � � � � � � � � � � � � � � � ∆u + k2u = 0 in Rd \\ D, ∆u + k2 i u = 0 in Di + m, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , N, m ∈ Ir, u|+ − u|− = 0 on ∂D, δi ∂u ∂ν ���� + − ∂u ∂ν ���� − = 0 on ∂Di + m for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , N, m ∈ Ir, u(x) satisfies the Sommerfeld radiation condition, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) where the Sommerfeld radiation condition says that lim |x|→∞ |x| d−1 2 Å ∂ ∂|x| − ik ã u = 0, uniformly in all directions x/|x|, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) and guarantees that energy is radiated outwards by the scattered solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The asymptotic regime we consider is that the material contrast parameters are all small while the wave speeds are all of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' That is, there exists some δ > 0 such that δi = O(δ) and v, vi = O(1) as δ → 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3) Within this setting, we are interested in solutions to the resonance problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) that are subwavelength in the sense that ω → 0 as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4) 14 To be able to characterize the subwavelength resonant modes of this system, we must define the generalized capacitance coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Recall the capacitance coefficients (Cmn f )ij from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then, we define the corresponding generalized capacitance coefficient as (Cmn f )ij = δiv2 i |Dm i |(Cmn f )ij, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5) where |Dm i | is the volume of the bounded subset Dm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then, the eigenvalues of Cf determine the subwavelength resonant frequencies of the system, as prescribed by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Consider a system of N|Ir| subwavelength resonators in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For sufficiently small δ > 0, there exist N|Ir| subwavelength resonant frequencies ω1(δ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , ωN|Ir|(δ) with non-negative real parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Further, the subwavelength resonant frequencies are given by ωn = � λn + O(δ) as δ → 0, where {λn : n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' , N|Ir|} are the eigenvalues of the generalized capacitance matrix Cf, which satisfy λn = O(δ) as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' A similar result exists for an infinite periodic structure, in terms of the eigenvalues of the generalized quasi-periodic capacitance matrix, as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='4), see [1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='5) clarifies the motivation for pre-multiplying by the perturbation matrix B to describe defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' When B is a compact perturbation of the identity, it describes defects that correspond to changing the material parameters on a finite number of resonators, so that the quantity δiv2 i corresponding to those resonators is altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' B Uniformity across the Brillouin zone In this appendix, we provide additional details of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The main result is Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2, which shows that (Sα D)−1 is in operator norm, uniformly bounded for α in a neighbourhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' The analysis is similar to [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' From e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' [7], we have a dual-space representation of Gα given by Gα(x) = − 1 |Y | � q∈Λ∗ ei(α+q)·x |α + q|2 = −eiα·x |Y ||α|2 − 1 |Y | � q∈Λ∗\\{0} ei(α+q)·x |α + q|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Define the periodic Green’s function G0 as G0(x) = − 1 |Y | � q∈Λ∗\\{0} eiq·x |q|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For α close to zero, we then have Gα(x) = −1 |Y ||α|2 − iα · x |Y ||α|2 + (α · x)2 2|Y ||α|2 + G0(x) + O(|α|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Consequently, for α close to zero, we have the an expansion of the single-layer potential Sα D: Sα D[ψ](x) = − 1 |Y ||α|2 � ∂D ψ(y) dσ − i |Y ||α|2 � ∂D α · (x − y)ψ(y) dσ + 1 2|Y ||α|2 � ∂D �α · (x − y)�2ψ(y) dσ + S0 D[ψ](x) + O(|α|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' If S0 D[ϕ] = Kχ∂D for some constant K and some ϕ ∈ L2(∂D) satisfying � ∂D ϕ dσ = 0, then ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' For x ∈ R3 \\ D, define V (x) := S0 D[ϕ](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Then V solves the following differential problem, � � � � � � � ∆V = 0 in R3 \\ D, V |+ = K on ∂D, V (x + m) = V (x) for all m ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) Moreover, using the jump relations and integration by parts, we have that � ∂D ϕ dσ = K � Y \\D |∇V |2 dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' If K ̸= 0, it follows from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2) that � Y \\D |∇V |2 dx ̸= 0 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In other words we must have K = 0, so that S0 D[ϕ] = 0 and � ∂D ϕ dσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' From [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='7], we have that ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' ∥(Sα D)−1∥, in operator norm, is bounded for α in a neighbourhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' To reach a contradiction, we assume that Sα D[φ] = O(|α|) for some φ, which can be written as φ = φ0 + |α|φ1, where φ0 is nonzero, does not depend on α, and φ1 = O(1) as |α| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Also define v = α |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' From (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1) it follows that � ∂D φ0 dσ = 0, � ∂D φ1(y) dσ + i � ∂D v · (x − y)φ0(y) dσ = O(|α|), K(v) − iv · x � ∂D φ1(y) dσ + 1 2 � ∂D �v · (x − y)�2φ0(y) dσ + |Y |S0 D[φ0] = O(|α|), where K is constant as function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Simplifying, we have that 1 2 � ∂D �v · (x − y)�2φ0(y) dσ = −(v · x) � ∂D (v · y)φ0(y) dσ + 1 2 � ∂D (v · y)2φ0(y) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' In total we get S0 D[φ0](x) = ˜K(v) + 2(v · x) |Y | � ∂D (v · y)φ0(y) dσ, where ˜K is constant in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Observe that S0 D[φ0](x) is independent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' As a function of x, this function is constant for x ∈ v⊥, and so this function is constant for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' From Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='1 we get that φ0 = 0 which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Ammari, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Davies, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Hiltunen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Functional analytic methods for discrete approxima- tions of subwavelength resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='12301, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Ammari, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Davies, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Hiltunen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Anderson localization in the subwavelength regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content='13337, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Ammari, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' Davies, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE1T4oBgHgl3EQfvwUc/content/2301.03402v1.pdf'} +page_content=' 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Kleykamp,1, ∗ S. Akhter,2 Z. Ahmad Dar,3, 2 V. Ansari,2 M. V. Ascencio,4, † M. Sajjad Athar,2 A. Bashyal,5, ‡ +A. Bercellie,1 M. Betancourt,6 A. Bodek,1 J. L. Bonilla,7 A. Bravar,8 H. Budd,1 G. Caceres,9, § T. Cai,1, 10 +M.F. Carneiro,5, 9, ¶ G.A. D´ıaz,1 H. da Motta,9 S.A. Dytman,11 J. Felix,7 L. Fields,12 A. Filkins,3, ∗∗ R. Fine,1, †† +A.M. Gago,4 H. Gallagher,13 S.M. Gilligan,5 R. Gran,14 E.Granados,7 D.A. Harris,10, 6 S. Henry,1 D. Jena,6 +S. Jena,15 A. Klustov´a,16 M. Kordosky,3 D. Last,17 A. Lozano,9 X.-G. Lu,18, 19 E. Maher,20 S. Manly,1 +W.A. Mann,13 C. Mauger,17 K.S. McFarland,1 B. Messerly,11, ‡‡ J. Miller,21 O. Moreno,3, 7 J.G. Morf´ın,6 +D. Naples,11 J.K. Nelson,3 C. Nguyen,22 A. Olivier,1 V. Paolone,11 G.N. Perdue,6, 1 K.-J. Plows,19 M.A. Ram´ırez,17, 7 +R.D. Ransome,23 H. Ray,22 D. Ruterbories,1 H. Schellman,5 C.J. Solano Salinas,24 H. Su,11 M. Sultana,1 +V.S. Syrotenko,13 E. Valencia,3, 7 N.H. Vaughan,5 A.V. Waldron,25, 16 C. Wret,1 B. Yaeggy,21, §§ and L. Zazueta3 +(The MINERνA Collaboration) +1Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627 USA +2Department of Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India +3Department of Physics, William & Mary, Williamsburg, Virginia 23187, USA +4Secci´on F´ısica, Departamento de Ciencias, Pontificia Universidad Cat´olica del Per´u, Apartado 1761, Lima, Per´u +5Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA +6Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA +7Campus Le´on y Campus Guanajuato, Universidad de Guanajuato, Lascurain +de Retana No. 5, Colonia Centro, Guanajuato 36000, Guanajuato M´exico. +8University of Geneva, 1211 Geneva 4, Switzerland +9Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Rio de Janeiro, 22290-180, Brazil +10York University, Department of Physics and Astronomy, Toronto, Ontario, M3J 1P3 Canada +11Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA +12Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA +13Physics Department, Tufts University, Medford, Massachusetts 02155, USA +14Department of Physics, University of Minnesota – Duluth, Duluth, Minnesota 55812, USA +15Department of Physical Sciences, IISER Mohali, Knowledge City, SAS Nagar, Mohali - 140306, Punjab, India +16The Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom +17Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 +18Department of Physics, University of Warwick, Coventry, CV4 7AL, UK +19Oxford University, Department of Physics, Oxford, OX1 3PJ United Kingdom +20Massachusetts College of Liberal Arts, 375 Church Street, North Adams, MA 01247 +21Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Avenida Espa˜na 1680 Casilla 110-V, Valpara´ıso, Chile +22University of Florida, Department of Physics, Gainesville, FL 32611 +23Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA +24Facultad de Ciencias, Universidad Nacional de Ingenier´ıa, Apartado 31139, Lima, Per´u +25G O Jones Building, Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK +(Dated: January 9, 2023) +This paper presents the first simultaneous measurement of the quasielastic-like neutrino-nucleus +cross sections on C, water, Fe, Pb and scintillator (hydrocarbon or CH) as a function of longitudinal +and transverse muon momentum. The ratio of cross sections per nucleon between Pb and CH is +always above unity and has a characteristic shape as a function of transverse muon momentum +that evolves slowly as a function of longitudinal muon momentum. The ratio is constant versus +longitudinal momentum within uncertainties above a longitudinal momentum of 4.5GeV/c. The +cross section ratios to CH for C, water, and Fe remain roughly constant with increasing longitudinal +momentum, and the ratios between water or C to CH do not have any significant deviation from +unity. Both the overall cross section level and the shape for Pb and Fe as a function of transverse +muon momentum are not reproduced by current neutrino event generators. These measurements +provide a direct test of nuclear effects in quasielastic-like interactions, which are major contributors +to long-baseline neutrino oscillation data samples. +The charged-current quasielastic neutrino interaction +(i.e. νµn → µ−p) contributes the majority of selected +signal interactions in current accelerator-based neutrino +oscillation experiments [1–6]. Because the interaction’s +final state is simple, the lepton flavor is easily identi- +fied. The neutrino energy may be estimated assuming +arXiv:2301.02272v1 [hep-ex] 5 Jan 2023 + +2 +two-body kinematics where the target is assumed to be a +neutron at rest. However, for those nuclei in use in oscil- +lation experiments these assumptions can bias neutrino +energy reconstruction because of finite initial neutron +momentum inside the nucleus [7]. In addition, quasielas- +tic scattering can be mimicked by other processes; for +example, when final-state particles are absorbed in the +nucleus. These biases are already significant in current +experiments [1–4] and risk becoming dominant uncertain- +ties in the future, for example in DUNE [5] and Hyper- +Kamiokande [6]. Oscillation experiments may also use +different target nuclei for their near detector than for +their far detector [1], so direct measurements on different +nuclei provide insight on biases that might be introduced +by that choice. +The MINERvA experiment published a measurement +of quasielastic-like cross sections on a variety of nuclei at +a mean neutrino energy of 3 GeV [8], using interactions +where both a final-state muon and proton were identified. +This letter describes a measurement made with a data +set that is over twenty times as large due to the following +factors: the mean neutrino energy is higher by a factor of +two, the integrated number of protons on target is larger +by a factor of four, and the requirement for a final-state +proton has been removed. The increased statistics allow +a more detailed probe of this process. +MINERvA +recently +measured +charged-current +charged-pion production on different nuclei [9]. +Using +the same neutrino beam as used here, MINERvA found +that the ratio of pions produced on Fe or Pb compared +to scintillator is lower than predicted by current models. +This has implications not only for the background, but +also for the signal in this analysis. Neutrino interactions +in which pions are produced but absorbed in the nucleus +can be quasielastic-like, and are thus included as a +signal process. +In addition, due to the possibility of +interactions with nucleon pairs, +the quasielastic-like +definition allows any number of protons and neutrons in +the final state. +The MINERvA detector [10] consists of a nuclear tar- +get region of several thin passive targets interspersed +with 1.7 cm-thick active scintillator planes, followed by a +scintillator-only region followed by electromagnetic and +hadronic calorimetry. +The MINOS near detector [11], +located 2 m downstream of MINERvA, measures the +charge and momentum of final-state muons. MINERvA’s +targets include regions made up of C, Fe, Pb, and water. +The solid targets are configured in such a way that the +total amount of passive material a particle traverses be- +tween the start of the interaction and the scintillator-only +region (in g/cm2) is approximately the same. The wa- +ter target is a flattened circular neoprene balloon that +is between 17-24 cm thick in the beam direction. The +detector is modeled using a hit-level Geant4-based simu- +lation overlaid with random beam data to simulate beam- +related accidental activity. The simulation includes the +time dependence of both the proton beam intensity and +the configuration of the water target. +The NuMI beam is produced by a 120-GeV proton +beam incident on a two-interaction-length graphite tar- +get followed by two parabolic focusing horns and a 675- +m decay pipe and 200 m of earth to shield the tertiary +muons. For these data the horn currents are set to focus +positively-charged pions, creating a neutrino-dominated +broad-band beam with a peak energy of 6.5 GeV. +The beam line is modeled with a Geant4-based [12, 13] +simulation (g4numi [14] version 6, built against Geant +version v.9.4.p2). +There are known discrepancies be- +tween measurements and Geant4 predictions of pion pro- +duction from proton-on-carbon interactions relevant to +NuMI flux predictions [14]. +MINERvA corrects these +predictions using hadron-production data. In addition, +measurements of neutrino-electron (ν −e) scattering [15] +and interactions with low recoil energy [16] are used to +constrain the flux prediction. +This analysis uses data +that correspond to 10.61×1020 protons on target (POT), +where the first (second) half of the exposure was with the +water target empty (full) to allow the non-water back- +ground interactions to be measured directly and sub- +tracted from the full target sample. +The GENIE 2.12.6 event generator [17] is used to sim- +ulate neutrino interactions on nuclei. +For quasielastic +scattering on nucleons the Llewellyn-Smith formalism is +used [18]. +Nuclear effects are incorporated by using a +Bodek-Ritchie high momentum tail [19] in the Fermi mo- +mentum distribution of the initial-state nucleons. The +default GENIE interaction model is adjusted to match +previous MINERvA data via a GENIE tune v1 (Mn- +vGENIEv1), which includes three additional modifica- +tions. First, the Valencia Random Phase Approximation +(RPA) correction, considered as a “weak nuclear screen- +ing” [20, 21] for a Fermi gas [22, 23], is added as a func- +tion of neutrino energy and three-momentum transfer. +Second, the prediction for multi-nucleon scattering given +by the Valencia model [24–26] in GENIE 2.12.6 is added +and modified with an empirical fit [27] based on previ- +ous MINERvA data on CH. The modification increases +the integrated 2-particle 2-hole (2p2h) interaction rate by +49%. This same fractional increase per proton-neutron +pair is applied for all nuclei. Finally, non-resonant pion +production is reduced by 57% to agree with a fit to mea- +surements on deuterium [28]. +Interactions are selected by requiring a muon candidate +that originates in the MINERvA detector and is recon- +structed in the MINOS near detector. There is no min- +imum number requirement for proton tracks, and there +must be no electron candidates resulting from the pion +to muon to electron decay chain (“Michel”s) near the +interaction vertex or any track endpoint. Backgrounds +and efficiencies are determined separately for the sam- +ples with and without the identified proton tracks. To +further reject interactions containing charged pions, any + +3 +non-muon reconstructed track is required to satisfy pro- +ton identification cuts based on the energy deposition +pattern. +To remove interactions with neutral pions, a +cut is made requiring no more than one isolated cluster +of energy in the detector. +The muon momentum is found by the addition of the +momentum determined by range inside the MINERvA +detector plus the momentum determined by range or +curvature inside MINOS [29]. The muon angle is mea- +sured in the MINERvA detector. To address the MINOS +acceptance, only interactions with muons reconstructed +within 17o of the neutrino beam and with momenta above +1.5 GeV/c and below 40 GeV/c are retained. The cross +sections we report are defined as any interaction with +a muon in the final state, where the muon has an an- +gle of no more than 17o and a momentum between 2 and +20 GeV/c. For these interactions any number of nucleons +is allowed, but no photons above 10 MeV (to accommo- +date nuclear excitations) and no mesons. +FIG. 1: +Reconstructed vertex location in the upstream re- +gion of MINERvA along the detector axis shown in data and +simulation in the full water target configuration. Interactions +correspond to those with two reconstructed tracks. +There are two primary categories of backgrounds: in- +teractions that originated in the scintillator but whose +vertex +is mis-reconstructed in a target, +and +non- +quasielastic-like interactions that are correctly recon- +structed in a nuclear target but are incorrectly recon- +structed as quasielastic-like. Predictions for both back- +grounds are constrained by comparing the data to the +simulation in sidebands. The background between one +nuclear target and another at the same vertex z location +is small due to fiducial volume cuts and is constrained by +interactions in the other target. +The prediction for the scintillator background can be +constrained by the ratio between the data and the sim- +ulation for interactions reconstructed in the scintillator +surrounding each of the nuclear targets. Figure 1 shows +the reconstructed interaction vertex position as a func- +tion of distance along the detector axis for all interac- +tions that have a reconstructed muon in MINOS and an +additional reconstructed track: the interactions in the +denser nuclear target material show up as clear peaks in +this distribution, and the normalization of the scintilla- +tor background comes from the interactions at least one +scintillator plane away from each target. +FIG. 2: +Top: Data and prediction for the (left) single Michel +Electron sideband, (right) Extra Energy cluster sideband for +Pb. Bottom: signal region in Pb (left) and CH (right), all +after the backgrounds and the signal have been tuned, for +the peak longitudinal momentum (P||) bin. The scintillator +background to Pb has been constrained and subtracted. +The second category of backgrounds comes from in- +teractions that take place in the target of interest, but +are not quasielastic-like. In this case one or more neutral +or charged pions have been misidentified as a proton or +not seen at all. The single neutral pion background is +constrained by MINERvA’s earlier measurement of neu- +tral pion production [30]. To determine the backgrounds +from other neutrino interaction channels, two different +sidebands are used where the data is compared to the +simulation. The first sideband requires a Michel electron +to provide a sample enriched with charged pions; the sec- +ond requires at least two extra clusters of energy away +from the interaction vertex to provide a sample enriched +in neutral pions, as in Ref. [31]. +Figure 2 (top) shows the data and simulation in the +two sidebands for the Pb target as a function of trans- +verse momentum (PT ) in the peak longitudinal momen- +tum (P||) region (4.5