Boltzmann-Gibbs Distribution Research Articles

Relaxation and fluctuations of a mass- and dipole-conserving stochastic lattice gas

Han [] have recently introduced a classical stochastic lattice gas model which, in addition to particle conservation, also conserves the particles' dipole moment. Because of its intrinsic nonlinearity this model exhibits unusual macroscopic scaling behaviors, different from those of lattice gases that conserve only the number of particles. Here we investigate some basic relaxation and fluctuation properties of this model at large scales and at long times. These properties crucially depend on whether the total number of particles is infinite or finite. We find similarity solutions, describing relaxation of the dipole-conserving gas (DCG) in several standard settings. A major part of our effort is an extension to this model of the macroscopic fluctuation theory (MFT), previously developed for lattice gases where only the number of particles is conserved. We apply the MFT to the calculation of the variance of nonequilibrium fluctuations of the excess number of particles on the positive semi-axis when starting from an (either deterministic, or random) constant density at t=0. Using the MFT, we also identify the equilibrium Boltzmann-Gibbs distribution for the DCG. Finally, based on these results, we determine the probability distribution of, and the most probable density history leading to, a large deviation in the form of a macroscopic void of a given size in an initially uniform DCG at equilibrium. Published by the American Physical Society 2024

Dynamic fluctuations of current and mass in nonequilibrium mass transport processes

We study steady-state dynamic fluctuations of current and mass in several variants of random average processes on a ring of L sites. These processes violate detailed balance in the bulk and have nontrivial spatial structures: their steady states are not described by the Boltzmann–Gibbs distribution and can have nonzero spatial correlations. Using a microscopic approach, we exactly calculate the second cumulants, or the variance, ⟨Qi2(T)⟩c and ⟨Qsub2(l,T)⟩c , of cumulative (time-integrated) currents up to time T across the ith bond and across a subsystem of size l (summed over bonds in the subsystem), respectively. We also calculate the (two-point) dynamic correlation function of the subsystem mass. In particular, we show that, for large L≫1 , the second cumulant ⟨Qi2(T)⟩c of the cumulative current up to time T across the ith bond grows linearly as ⟨Qi2⟩c∼T for initial times T∼O(1) , subdiffusively as ⟨Qi2⟩c∼T1/2 for intermediate times 1≪T≪L2 , and then again linearly as ⟨Qi2⟩c∼T for long times T≫L2 . The scaled cumulant liml→∞,T→∞⟨Qsub2(l,T)⟩c/2lT of current across the subsystem of size l and up to time T converges to the density-dependent particle mobility χ(ρ) when the large subsystem-size limit is taken first, followed by the large-time limit; when the limits are reversed, it simply vanishes. Remarkably, regardless of the dynamical rules, the scaled current cumulant D⟨Qi2(T)⟩c/2χL≡W(y) as a function of scaled time y=DT/L2 can be expressed in terms of a universal scaling function W(y) , where D is the bulk-diffusion coefficient; interestingly, the intermediate-time subdiffusive and long-time diffusive growths can be connected through a single scaling function W(y) . The power spectra for current and mass are also exactly characterized by the respective scaling functions. Furthermore, we provide a microscopic derivation of equilibrium-like Green–Kubo and Einstein relations that connect the steady-state current fluctuations to an ‘operational’ mobility (i.e. the response to an external force field) and mass fluctuation, respectively.

Power-law relaxation of a confined diffusing particle subject to resetting with memory

We study the relaxation of a Brownian particle with long range memory under confinement in one dimension. The particle diffuses in an arbitrary confining potential and resets at random times to previously visited positions, chosen with a probability proportional to the local time spent there by the particle since the initial time. This model mimics an animal which moves erratically in its home range and returns preferentially to familiar places from time to time, as observed in nature. The steady state density of the position is given by the equilibrium Gibbs–Boltzmann distribution, as in standard diffusion, while the transient part of the density can be obtained through a mapping of the Fokker–Planck equation of the process to a Schrödinger eigenvalue problem. Due to memory, the approach at late times toward the steady state is critically self-organised, in the sense that it always follows a sluggish power-law form, in contrast to the exponential decay that characterises Markov processes. The exponent of this power-law depends in a simple way on the resetting rate and on the leading relaxation rate of the Brownian particle in the absence of resetting. We apply these findings to several exactly solvable examples, such as the harmonic, V-shaped and box potentials.

Control oriented modeling for particle size distributions in a spray drying process

This paper addresses the problem of predicting the Particle Size Distrubution (PSD) in spray drying processes in function of the liquid phase volumetric flow rate. For this purpose, based on a Boltzmann-Gibbs distribution representation the stochastic droplet dynamics is identified using experimental data and the Fokker-Planck Equation (FPE). The model is extended to account for piecewise constant flow rates in order to enable future studies on automatic control of the PSD. The results highlight the effectiveness of the stochastic modeling approach for predicting the PSD in spray drying processes, and its potentials for real-time control and process optimization.

Extensive and nonextensive statistical mechanics: Exp and log distribution functions

The dominance of Boltzmann–Gibbs distribution (BG) in statistical physics appears to endow an impression that this would be the only statistical approach available. In fact, a large class of statistical approaches already exists. This is generalizing BG statistics referring to various types of nonextensivity, nonadditivity, nonequilibrium, nonlinearity, etc. For instance, [Formula: see text] and [Formula: see text] functions play crucial roles in both extensive and nonextensive domains of statistical mechanics. Emerging in a physical system, BG statistics defines extensive entropy which relates the number of microstates to thermodynamic quantities or macroscopic states. In this regard, the Boltzmann distribution, extensive statistics, refers to a well-defined probability distribution, while the various types of nonextensive statistics categorically violate the fourth Shannon–Khinchen additivity axiom. The [Formula: see text] and [Formula: see text] distribution functions in BG, Tsallis, and generic statistics are systematically compared. We focus on the mathematical properties of both distribution functions and conclude that their compatibility exclusively depends on the nonextensive parameters, i.e. the mathematical properties of both distribution functions depend on the nonextensive parameters either that of Tsallis- or that of the generic-type of nonextensivity. We also conclude that the statistical nature of the underlying ensemble should be taken into consideration when applying the statistical approach.

Bending fluctuations in semiflexible, inextensible, slender filaments in Stokes flow: Toward a spectral discretization.

Semiflexible slender filaments are ubiquitous in nature and cell biology, including in the cytoskeleton, where reorganization of actin filaments allows the cell to move and divide. Most methods for simulating semiflexible inextensible fibers/polymers are based on discrete (bead-link or blob-link) models, which become prohibitively expensive in the slender limit when hydrodynamics is accounted for. In this paper, we develop a novel coarse-grained approach for simulating fluctuating slender filaments with hydrodynamic interactions. Our approach is tailored to relatively stiff fibers whose persistence length is comparable to or larger than their length and is based on three major contributions. First, we discretize the filament centerline using a coarse non-uniform Chebyshev grid, on which we formulate a discrete constrained Gibbs-Boltzmann (GB) equilibrium distribution and overdamped Langevin equation for the evolution of unit-length tangent vectors. Second, we define the hydrodynamic mobility at each point on the filament as an integral of the Rotne-Prager-Yamakawa kernel along the centerline and apply a spectrally accurate "slender-body" quadrature to accurately resolve the hydrodynamics. Third, we propose a novel midpoint temporal integrator, which can correctly capture the Ito drift terms that arise in the overdamped Langevin equation. For two separate examples, we verify that the equilibrium distribution for the Chebyshev grid is a good approximation of the blob-link one and that our temporal integrator for overdamped Langevin dynamics samples the equilibrium GB distribution for sufficiently small time step sizes. We also study the dynamics of relaxation of an initially straight filament and find that as few as 12 Chebyshev nodes provide a good approximation to the dynamics while allowing a time step size two orders of magnitude larger than a resolved blob-link simulation. We conclude by applying our approach to a suspension of cross-linked semiflexible fibers (neglecting hydrodynamic interactions between fibers), where we study how semiflexible fluctuations affect bundling dynamics. We find that semiflexible filaments bundle faster than rigid filaments even when the persistence length is large, but show that semiflexible bending fluctuations only further accelerate agglomeration when the persistence length and fiber length are of the same order.

A hybrid data-driven-physics-constrained Gaussian process regression framework with deep kernel for uncertainty quantification

Gaussian process regression (GPR) is a well-known machine learning method employed for various applications such as uncertainty quantifications. However, GPR is an inherently data-driven method that requires a sufficiently large dataset. If appropriate physics constraints (e.g. those expressed in partial differential equations) can be incorporated, the amount of data can be greatly reduced, and the accuracy can be further improved. In this study, we propose a hybrid data-driven-physics-constrained Gaussian process regression framework. We encode the physics knowledge with a Boltzmann-Gibbs distribution and derive our model using the maximum likelihood (ML) approach. We apply the deep kernel learning method. Through the training of a deep neural network, the proposed model learns an adaptive covariance function from both data and physics constraints. The proposed model achieves good results in high-dimensional problems, and correctly propagates the uncertainty while requiring very limited labelled data.

Modeling income distribution: An econophysics approach.

This study aims to develop appropriate models for income distribution in Iran using the econophysics approach for the 2006-2018 period. For this purpose, the three improved distributions of the Pareto, Lognormal, and Gibbs-Boltzmann distributions are analyzed with the data extracted from the target household income expansion plan of the statistical centers in Iran. The research results indicate that the income distribution in Iran does not follow the Pareto and Lognormal distributions in most of the study years but follows the generalized Gibbs-Boltzmann distribution function in all study years. According to the results, the generalized Gibbs-Boltzmann distribution also properly fits the actual data distribution and could clearly explain the income distribution in Iran. The generalized Gibbs-Boltzmann distribution also fits the actual income data better than both Pareto and Lognormal distributions.

Optimal income crossover for a two-class model using particle swarm optimization.

Personal income distribution may exhibit a two-class structure, such that the lower income class of the population (85-98%) is described by exponential Boltzmann-Gibbs distribution, whereas the upper income class (2-15%) has a Pareto power-law distribution. We propose a method, based on a theoretical and numerical optimization scheme, which allows us to determine the crossover income between the distributions, the temperature of the Boltzmann-Gibbs distribution, and the Pareto index. Using this method, the Brazilian income distribution data provided by the National Household Sample Survey was studied. The data was stratified into two dichotomies (sex/gender and color/race), so the model was tested using different subsets along with accessing the economic differences between these groups. Last, we analyze the temporal evolution of the parameters of our model and the Gini coefficient discussing the implication on the Brazilian income inequality. In this paper, we propose an optimization method to find a continuous two-class income distribution, which is able to delimit the boundaries of the two distributions. It also gives a measure of inequality which is a function that depends only on the Pareto index and the percentage of people in the high-income region. We found a temporal dynamics relation, that may be general, between the Pareto and the percentage of people described by the Pareto tail.

Per capita wealth in cities and regions fitted to Pareto, stretched exponential and econophysics Boltzmann–Gibbs distributions

From the years 2001 to 2017, per capita nominal and real (adjusted to inflation) GDP at purchasing power parity (PCGDP-PPP) distributions for cities and regions are fitted to various functions. For most years and regions, real PCGDP-PPP data are very well adjusted to the one-parameter Boltzmann–Gibbs distribution (BGD), in accordance with the exponential behavior predicted by the simple econophysics analogy between conserved money in economic trade and energy in elastic collisions in gases. Overall, fittings are better for large regions in recent years, which may reflect an increasing economic globalization in time. Cities, small regions and large regions values are well fitted by stretched exponential distributions.

Two-class structure of Forbes Global 2000 firms sales per employee similar to countries income distributions including some COVID-19 pandemic effects

Sales per capita (per number of employees) of the two thousand Forbes Magazine top publicly-traded companies (G-2000) for years 2015, 2020 and 2021 are statistically analyzed. Employing an econophysical model, the sales distributions per capita for the three years exhibit a two-class structure: a Pareto power law in the higher part and exponential in the lower part resembling a Boltzmann-Gibbs distribution. This distribution is consistent with income distributions around the world as if a fraction of a firms' wealth goes to its employees in the form of wages and salaries. We highlight some changes in sales between 2020 and 2021 on selected industries that had the biggest negative and positive impacts partially due to the COVID-19 pandemic.

Physics-inspired analysis of the two-class income distribution in the USA in 1983-2018.

The first part of this paper is a brief survey of the approaches to economic inequality based on ideas from statistical physics and kinetic theory. These include the Boltzmann kinetic equation, the time-reversal symmetry, the ergodicity hypothesis, entropy maximization and the Fokker-Planck equation. The origins of the exponential Boltzmann-Gibbs distribution and the Pareto power law are discussed in relation to additive and multiplicative stochastic processes. The second part of the paper analyses income distribution data in the USA for the time period 1983-2018 using a two-class decomposition. We present overwhelming evidence that the lower class (more than 90% of the population) is described by the exponential distribution, whereas the upper class (about 4% of the population in 2018) by the power law. We show that the significant growth of inequality during this time period is due to the sharp increase in the upper-class income share, whereas relative inequality within the lower class remains constant. We speculate that the expansion of the upper-class population and income shares may be due to increasing digitization and non-locality of the economy in the last 40 years. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Sales per employee of the world’s top combined publicly traded and state-owned companies

Sales per capita (per number of employees) of the 2000 Forbes Magazine top publicly-traded companies (G-2000) and some of the world’s leading state-owned enterprises (SOE) are statistically analyzed. This hybrid or combined cumulative probability sales distribution per capita exhibits a two-class structure: a Pareto power law in the higher part and exponential in the lower part resembling a Boltzmann–Gibbs distribution, where money is conserved in economic trade like energy is conserved in elastic collisions. This global per capita sales two-class distribution is qualitatively similar to income and wealth distributions in many countries around the world.

Design of biased random walks on a graph with application to collaborative recommendation

This work investigates a paths-based statistical physics formalism, inspired from the bag-of-paths framework, for the design of random walks on a graph in which the transition probabilities (the policy) are biased in favor of some node features. More precisely, given a weighted directed graph G and a non-negative cost assigned to each edge, the biased random walk is defined as the policy minimizing the expected cost rate along the walks while maintaining a constant relative entropy rate. As for the standard bag-of-paths and the randomized shortest paths frameworks, the model assigns a Gibbs–Boltzmann distribution to the set of infinite walks and allows to recover known results from the literature, derived here from a different perspective. Examples of quantities of interest are the partition function of the system, the cost rate, the optimal transition probabilities, etc. In addition, the same formalism allows the introduction of capacity constraints on the expected node visit rates and an algorithm for computing the optimal policy subject to such capacity constraints is developed. Simulation results indicate that the proposed procedure can be effectively used in order to define a Markov chain driving the walk towards nodes having some specific properties, like seniority, education level or low node degree (hub-avoiding walk). An application relying on this last property is proposed as a tool for improving serendipity in collaborative recommendation, and is tested on the MovieLens data.

Complexity Analysis of a Sampling-Based Interior Point Method for Convex Optimization

We develop a short-step interior point method to optimize a linear function over a convex body assuming that one only knows a membership oracle for this body. The approach is based a sketch of a universal interior point method using the so-called entropic barrier. It is well known that the gradient and Hessian of the entropic barrier can be approximated by sampling from Boltzmann-Gibbs distributions and the entropic barrier was shown to be self-concordant. The analysis of our algorithm uses properties of the entropic barrier, mixing times for hit-and-run random walks, approximation quality guarantees for the mean and covariance of a log-concave distribution, and results on inexact Newton-type methods.

Quantum Machine-Learning for Eigenstate Filtration in Two-Dimensional Materials.

Quantum machine-learning algorithms have emerged to be a promising alternative to their classical counterparts as they leverage the power of quantum computers. Such algorithms have been developed to solve problems like electronic structure calculations of molecular systems and spin models in magnetic systems. However, the discussion in all these recipes focuses specifically on targeting the ground state. Herein we demonstrate a quantum algorithm that can filter any energy eigenstate of the system based on either symmetry properties or a predefined choice of the user. The workhorse of our technique is a shallow neural network encoding the desired state of the system with the amplitude computed by sampling the Gibbs-Boltzmann distribution using a quantum circuit and the phase information obtained classically from the nonlinear activation of a separate set of neurons. We show that the resource requirements of our algorithm are strictly quadratic. To demonstrate its efficacy, we use state filtration in monolayer transition metal dichalcogenides which are hitherto unexplored in any flavor of quantum simulations. We implement our algorithm not only on quantum simulators but also on actual IBM-Q quantum devices and show good agreement with the results procured from conventional electronic structure calculations. We thus expect our protocol to provide a new alternative in exploring the band structures of exquisite materials to usual electronic structure methods or machine-learning techniques that are implementable solely on a classical computer.

Generalization of the maximum entropy principle for curved statistical manifolds

The maximum entropy principle (MEP) is one of the most prominent methods to investigate and model complex systems. Despite its popularity, the standard form of the MEP can only generate Boltzmann-Gibbs distributions, which are ill-suited for many scenarios of interest. As a principled approach to extend the reach of the MEP, this paper revisits its foundations in information geometry and shows how the geometry of curved statistical manifolds naturally leads to a generalization of the MEP based on the R\'enyi entropy. By establishing a bridge between non-Euclidean geometry and the MEP, our proposal sets a solid foundation for the numerous applications of the R\'enyi entropy, and enables a range of novel methods for complex systems analysis.

Phenomenological Tsallis distribution from thermal field theory

Classical and quantum Tsallis distributions have been widely used in many branches of natural and social sciences. But, the quantum field theory of the Tsallis distributions is relatively a less explored arena. In this paper, we derive the expression for the thermal two-point functions in the Tsallis statistics with the help of the corresponding statistical mechanical formulations. We show that the quantum Tsallis distributions used in the literature appear in the thermal part of the propagator much in the same way the Boltzmann–Gibbs distributions appear in the conventional thermal field theory. As an application of our findings, we calculate the thermal mass in the [Formula: see text] scalar field theory within the realm of the Tsallis statistics.

Stationarization and Multithermalization in spin glasses

We develop further the study of a system in contact with a multibath having different temperatures at widely separated timescales. We consider those systems that do not thermalize in finite times when in contact with an ordinary bath but may do so in contact with a multibath. Thermodynamic integration is possible, thus allowing one to recover the stationary distribution on the basis of measurements performed in a `multi-reversible' transformation. We show that following such a protocol the system is at each step described by a generalization of the Boltzmann-Gibbs distribution, that has been studied in the past. Guerra's bound interpolation scheme for spin-glasses is closely related to this: by translating it into a dynamical setting, we show how it may actually be implemented in practice. The phase diagram plane of temperature vs ``number of replicas", long studied in spin- glasses, in our approach becomes simply that of the two temperatures the system is in contact with. We suggest that this representation may be used to directly compare phenomenological and mean-field inspired models. Finally, we show how an approximate out of equilibrium probability distribution may be inferred experimentally on the basis of measurements along an almost reversible transformation.

Phenomenological determinism in Hamiltonian dynamical systems

Classical equations that govern the information in a Boltzmann–Gibbs Hamiltonian dynamical system are shown to entail the evolution of a phenomenological fraction of the information. The evolution of this fraction of information can be predicted along a preferential time direction provided a structural condition on the dynamics time scale is fulfilled. This preferential direction points towards the future or the past, starting from an initial time, provided an appropriate initial condition is fulfilled by the Boltzmann–Gibbs distribution at this time. Choosing one direction defines an arrow of time. That fraction of information is thus submitted to a phenomenological determinism. It permanently generates non-phenomenological components of the information, which accumulate, and experience an exponentially fast complexification. This complexification makes the accumulated non-phenomenological information unable to influence the evolution of the phenomenological information. That evolution is thus autonomous and deterministic.