Problems In Statistical Mechanics Research Articles

Efficient optimization of variational autoregressive networks with natural gradient.

Estimating free energy is a fundamental problem in statistical mechanics. Recently, machine-learning-based methods, particularly the variational autoregressive networks (VANs) have been proposed to minimize variational free energy and to approximate the Boltzmann distribution. VAN enjoys notable advantages, including the exact computation of the normalized joint distribution and fast sampling, which are critical features often missing in Markov chain Monte Carlo algorithms. However, VAN also faces significant computational challenges. These include difficulties in the optimization of variational free energy in a complicated parameter space and slow convergence of learning. In this work, we introduce an optimization technique based on natural gradients to the VAN framework, namely ng-VAN, to enhance the learning efficiency and accuracy of the conventional VAN. The method has computational complexity cubic in the batch size rather than in the number of model parameters, hence it can be efficiently implemented for a large VAN model. We carried out extensive numerical experiments on the Sherrington-Kirkpatrick model, spin glasses on random graphs, and the two-dimensional Ising model. Our results indicate that compared with the conventional VAN, ng-VAN significantly improves the accuracy in estimating free energy and converges much faster with shorter learning time. This allows extending the VAN framework's applicability to challenging statistical mechanics problems that were previously not accessible.

Projected state ensemble of a generic model of many-body quantum chaos

Abstract The projected ensemble is based on the study of the quantum state of a subsystem A conditioned on projective measurements in its complement. Recent studies have observed that a more refined measure of the thermalization of a chaotic quantum system can be defined on the basis of convergence of the projected ensemble to a quantum state design, i.e. a system thermalizes when it becomes indistinguishable, up to the kth moment, from a Haar ensemble of uniformly distributed pure states. Here we consider a random unitary circuit with the brick-wall geometry and analyze its convergence to the Haar ensemble through the frame potential and its mapping to a statistical mechanical problem. This approach allows us to highlight a geometric interpretation of the frame potential based on the existence of a fluctuating membrane, similar to those appearing in the study of entanglement entropies. At large local Hilbert space dimension q, we find that all moments converge simultaneously with a time scaling linearly in the size of region A, a feature previously observed in dual unitary models. However, based on the geometric interpretation, we argue that the scaling at finite q on the basis of rare membrane fluctuations, finding the logarithmic scaling of design times t k = O ( log ⁡ k ) . Our results are supported with numerical simulations performed at q = 2.

Escaping Kinetic Traps Using Nonreciprocal Interactions.

Kinetic traps are a notorious problem in equilibrium statistical mechanics, where temperature quenches ultimately fail to bring the system to low energy configurations. Using multifarious self-assembly as a model system, we introduce a mechanism to escape kinetic traps by utilizing nonreciprocal interactions between components. Introducing nonequilibrium effects offered by broken action-reaction symmetry in the system pushes the trajectory of the system out of arrested dynamics. The dynamics of the model is studied using tools from the physics of interfaces and defects. Our proposal can find applications in self-assembly, glassy systems, and systems with arrested dynamics to facilitate escape from local minima in rough energy landscapes.

Interplay between an Absorbing Phase Transition and Synchronization in a Driven Granular System.

Absorbing phase transitions (APTs) are widespread in nonequilibrium systems, spanning condensed matter, epidemics, earthquakes, ecology, and chemical reactions. APTs feature an absorbing state in which the system becomes entrapped, along with a transition, either continuous or discontinuous, to an active state. Understanding which physical mechanisms determine the order of these transitions represents a challenging open problem in nonequilibrium statistical mechanics. Here, by numerical simulations and mean-field analysis, we show that a quasi-2D vibrofluidized granular system exhibits a novel form of APT. The absorbing phase is observed in the horizontal dynamics below a critical packing fraction, and can be continuous or discontinuous based on the emergent degree of synchronization in the vertical motion. Our results provide a direct representation of a feasible experimental scenario, showcasing a surprising interplay between dynamic phase transition and synchronization.

Gauge Field Marginal of an Abelian Higgs Model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon–Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler (Gauge theories as a problem of constructive quantum field theory and statistical mechanics, volume 159 of lecture notes in physics, Springer, Berlin, p v+192. https://doi.org/10.1007/3-540-11559-5, 1982) but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the Hölder–Besov-type norms introduced in Chevyrev (Commun Math Phys 372(3):1027–1058. https://doi.org/10.1007/s00220-019-03567-5, 2019) are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.

Magic in generalized Rokhsar-Kivelson wavefunctions

Magic is a property of a quantum state that characterizes its deviation from a stabilizer state, serving as a useful resource for achieving universal quantum computation e.g., within schemes that use Clifford operations. In this work, we study magic, as quantified by the stabilizer Renyi entropy, in a class of models known as generalized Rokhsar-Kivelson systems, i.e., Hamiltonians that allow a stochastic matrix form (SMF) decomposition. The ground state wavefunctions of these systems can be written explicitly throughout their phase diagram, and their properties can be related to associated classical statistical mechanics problems, thereby allowing powerful analytical and numerical approaches that are not usually available in conventional quantum many body settings. As a result, we are able to express the SRE in terms of wave function coefficients that can be understood as a free energy difference of related classical problems. We apply this insight to a range of quantum many body SMF Hamiltonians, which affords us to study numerically the SRE of large high-dimensional systems, and in some cases to obtain analytical results. We observe that the behaviour of the SRE is relatively featureless across quantum phase transitions in these systems, although it is indeed singular (in its first or higher order derivative, depending on the nature of the transition). On the contrary, we find that the maximum of the SRE generically occurs at a cusp away from the quantum critical point, where the derivative suddenly changes sign. Furthermore, we compare the SRE and the logarithm of overlaps with specific stabilizer states, asymptotically realised in the ground state phase diagrams of these systems. We find that they display strikingly similar behaviors, which in turn establish rigorous bounds on the min-relative entropy of magic.

A NEW CLASS OF GIBBS MEASURES FOR THREE-STATE SOS MODEL ON A CAYLEY TREE

The phase transition phenomenon is one of the central problems of statistical mechanics. It occurs when the model possesses multiple Gibbs measures. In this paper, we consider a three-state SOS (solid-on-solid) model on a Cayley tree. We reduce description of Gibbs measures to solving of a non-linear functional equation, each solution of which corresponds to a Gibbs measure.We give some sufficiency conditions on the existence of multiple Gibbs measures for the model. We give a review of some known (translation-invariant, periodic, non-periodic) Gibbs measures of the model and compare them with our new measures. We show that the Gibbs measures found in the paper differ from the known Gibbs measures, i.e, we show that these measures are new.

The set of basis functions generated by Pearson type IV distributions and its application to problems of statistical data analysis and quantum mechanics

На примере распределений Пирсона IV типа предложена процедура дополнения классического распределения вероятностей до квантового состояния. Получена волновая функция, отвечающая распределениям Пирсона IV типа, а также построен соответствующий набор базисных функций. Продемонстрировано применение разработанного метода к задачам статистического анализа данных и квантовой механики. Показана эффективность разработанного подхода при аппроксимации статистических распределений с тяжелыми хвостами.

Exploring student reasoning in statistical mechanics: Identifying challenges in problem-solving groups

Statistical mechanics has received limited attention in physics education research and remains a relatively underrepresented topic even in research on upper-division physics courses. The purpose of this study was to explore potential challenges that physics students encounter when they solve statistical mechanics problems in groups. Adopting a grounded approach, we video recorded and analyzed nine small student groups engaging in collaborative problem solving on the topic. The analysis involved iterative thematic coding, which gave rise to ten emergent categories of challenges. These were later divided into two broad groupings: and . In the first grouping, we list seven identified categories related to the concepts of macrostates and microstates, distinguishable and indistinguishable particles, temperature, entropy, energy, equilibrium, heat bath, the Boltzmann distribution, and the partition function. In the second grouping, we list three categories related to the inappropriate application of common relations, difficulty managing tensions between calculated results and qualitative reasoning, and coming up with definitions of new and inconsistent concepts. Some of our findings are supported by existing research on the topic, and others are previously unreported. Based on our findings, we propose that future research should investigate the relations between the identified challenges on one hand, and students’ epistemological framing, reasoning, and use of multiple representations on the other. Finally, we suggest that teachers should spend time engaging students in a conceptual discussion of the central ideas of statistical mechanics, motivating the choice and pointing out limitations of commonly used toy models, and linking course content to real-world phenomena. Published by the American Physical Society 2024

Whitham approach to certain large fluctuation problems in statistical mechanics

We show the relationship between the strongly non-linear limit (also termed the dispersionless or the Whitham limit) of the macroscopic fluctuation theory of certain statistical models and the inverse scattering method. We show that in the strongly non-linear limit the inverse scattering problem can be solved using the steepest descent method of the associated Riemann–Hilbert problem. The importance of establishing this connection, is that the equations in the strongly non-linear limit can often be solved exactly by simple means, the connection then provides a limit in which one can solve the inverse scattering problem, thus aiding potentially the exact solution of a particular large deviation problem.

Linear growth of circuit complexity from Brownian dynamics

How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ−1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ~ kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are analytically tractable and are directly applicable to practical applications calling for unitary k-designs.

Size of a 2D ring polymer topologically unentangled with a planar array of obstacles

We readdress the statistical mechanical problem of the size of a 2D ring polymer, topologically unentangled with a planar lattice array of regularly spaced obstacles. It is commonly assumed in the literature that such a polymer adopts a randomly branched type of configuration, in order to ostensibly maximise chain entropy, while minimising obstacle entanglement. Via an innovative analytic approach, valid in the condensed polymer region, we are able to provide a greater theoretical understanding, and justification, for this presumed polymer behaviour. Our theoretically derived results could also potentially have important implications for the structure of interphase chromosomes, as well as electrophoretic ring polymer dynamics.

Slow melting of a disordered quantum crystal

The melting of the corner of a crystal is a classical, real-world, non-equilibrium statistical mechanics problem which has shown several connections with other branches of physics and mathematics. For a perfect, classical crystal in two and three dimensions the solution is known: the crystal melts reaching a certain asymptotic shape, which keeps expanding ballistically. In this paper, we move onto the quantum realm and show that the presence of quenched disorder slows down severely the melting process. Nevertheless, we show that there is no many-body localization transition, which could impede the crystal to be completely eroded. We prove such claim both by a perturbative argument, using the forward approximation, and via numerical simulations. At the same time we show how, despite the lack of localization, the erosion dynamics is slowed from ballistic to logarithmic, therefore pushing the complete melting of the crystal to extremely long timescales.

О некоторых строгих результатах в теории конденсированного состояния

The work contains a critical analysis of the methods and results currently used and applied in the theory of condensed matter. A unified mathematical apparatus has been developed - the method of functional integration which is equally applicable to all main distributions of statistical physics: microcanonical, canonical, and grand canonical ensembles. Within the framework of this method, an exact factorization of the configuration integral with respect to atomic coordinates was performed and a connection between the microcanonical and canonical ensembles was established. It is shown that interatomic interactions can be eliminated by renormalizing external random fields, and random external fields can be eliminated by renormalizing interatomic potentials. A new formulation of problems of equilibrium statistical mechanics as a dynamic field theory with a Hamiltonian depending on temperature is proposed.

Realizability of iso-g2 processes via effective pair interactions.

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction ϕ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal" packing fraction ϕc = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through ϕc. For ϕ near but below ϕc, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[ - κ(ϕ)r] for d = 1, r-1/2exp[ - κ(ϕ)r] for d = 2, and r-1exp[ - κ(ϕ)r] (Yukawa form) for d = 3, where κ-1(ϕ) is a screening length, and for ϕ = ϕc, the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost" random sequential addition at the maximum packing fraction ϕc = 1/8 is much shorter ranged than that for the 3D unit-step function g2 at ϕc; thus, it does not constrain the realizability of the unit-step function g2. Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso-g2 process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution.

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function Z, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for Z to be minimally plausible. Specifically, we present six necessary-but not sufficient-conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken "solutions", unveiling their shortcomings.

The interpolation between random walk and self-avoiding walk by avoiding marked sites

The self-avoiding walk (SAW) on a regular lattice is one of the most important and classic problems in statistical mechanics with major applications in polymer chemistry. Random walk is an exactly solved problem while SAW is an open problem so far. We interpolate these two limits in 1D and 2D by considering a model in which the walker marks certain sites in time and does not visit them again. We study two variants: (a) the walker marks sites at every k time-steps, in this case, it is possible to enumerate all possible paths of a given length. (b) The walker marks sites with a certain probability p. For k = 1 or p = 1, the walk reduces to the usual SAW. We compute the average trapping time and distance covered by a walker as a function of the number of steps and radius of gyration in these cases. We observe that 1D deterministic, 1D probabilistic, and 2D deterministic cases are in the same universality class as SAW while 2D probabilistic case shows continuously varying exponents.

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium.

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g_{2}(r) or corresponding structure factor S(k) in d-dimensional Euclidean space R^{d} is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)2470-004510.1103/PhysRevE.101.032124] conjectured that any realizable g_{2}(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g_{2}(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L_{2}-norm errors that are an order of magnitude smaller than that of previous methods. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

Spinning hexagons

We reduce the computation of three point function of three spinning operators with arbitrary polarizations in \U0001d4a9 = 4 SYM to a statistical mechanics problem via the hexagon formalism. The central building block of these correlation functions is the hexagon partition function. We explore its analytic structure and use it to generate perturbative data for spinning three point functions. For certain polarizations and any coupling, we express the full asymptotic three point function in determinant form. With the integrability approach established we open the ground to study the large spin limit where dualities with null Wilson loops and integrable pentagons must appear.

Physics-informed graph neural networks enhance scalability of variational nonequilibrium optimal control.

When a physical system is driven away from equilibrium, the statistical distribution of its dynamical trajectories informs many of its physical properties. Characterizing the nature of the distribution of dynamical observables, such as a current or entropy production rate, has become a central problem in nonequilibrium statistical mechanics. Asymptotically, for a broad class of observables, the distribution of a given observable satisfies a large deviation principle when the dynamics is Markovian, meaning that fluctuations can be characterized in the long-time limit by computing a scaled cumulant generating function. Calculating this function is not tractable analytically (nor often numerically) for complex, interacting systems, so the development of robust numerical techniques to carry out this computation is needed to probe the properties of nonequilibrium materials. Here, we describe an algorithm that recasts this task as an optimal control problem that can be solved variationally. We solve for optimal control forces using neural network ansatz that are tailored to the physical systems to which the forces are applied. We demonstrate that this approach leads to transferable and accurate solutions in two systems featuring large numbers of interacting particles.