Statistical Mechanics Research Articles

Energy distributions of Frenkel–Kontorova-type atomic chains: Transition from conservative to dissipative dynamics

We investigate energy distributions of Frenkel–Kontorova-type atomic chains generated from large number of independent identically distributed (iid) random initial atomic positionings under two cases. In the first case, atoms at the free-end chains without dissipation (conservative case) are only coupled to one other atom, whereas each atom inside the bulk is coupled to its 2 nearest neighbors. Here, atoms located at the chain are all at the same type. Such kind of systems can be modelled by conservative standard map. We show that, when the coupling is non-linear (which leads chaotic arrangement of the atoms) for energy distribution, the Boltzmann–Gibbs statistical mechanics is constructed, namely, exponential form emerges as Boltzmann factor P(E)∝e−βE. However, when the coupling is linear (which leads linear arrangement of the atoms) the Boltzmann–Gibbs statistical mechanics fails and the exponential distribution is replaced by a q-exponential form, which generalizes the Boltzmann factor as P(E)∝eq−βqE=[1−(1−q)βqE]1/(1−q). We also show for each type of atom localization with N number of atoms, β (or βq) values can be given as a function of 1/N. In the second case, although the couplings among the atoms are exactly the same as the previous case, atoms located at the chain are now considered as being at different types. We show that, for energy distribution of such linear chains, each of the distributions corresponding to different dissipation parameters (γ) are in the q-exponential form. Moreover, we numerically verify that βq values can be given as a linear function of 1/∑n=1N(1−γ)(n−2). On the other hand, although energy distributions of the chaotic chains for different dissipation parameters are in exponential form, a linear scaling between β and γ values cannot be obtained. This scaling is possible if the energies of the chains are scaled with 1/(1−γ)−N. For both cases, clear data collapses among distributions are evident.

Concentration Scales and Solvation Thermodynamics: Some Theoretical and Experimental Observations Regarding Spontaneity and the Partition Ratio.

The solvation thermodynamics (ST) formalism proposed by A. Ben-Naim is a mathematically rigorous and physically grounded theory for describing properties related to solvation. It considers the solvation process as the transfer of a molecule ("solute") from a fixed position in the ideal gas phase to a fixed position within the solution. Thus, it removes any contribution to the solvation process that is not related to the interactions between this molecule and its environment in the solution. Because ST is based on statistical thermodynamics, the natural variable is number density, which leads to the amount (or "molar") concentration scale. However, this choice of concentration scale is not unique in classical thermodynamics and the solvation properties can be different for commonly used concentration scales. We proposed and performed experiments with diethylamine in a water/hexadecane heterogeneous mixture to confront the predictions of the ST, based on the amount (or "molar") concentration scale, and the Fowler-Guggenheim formalism, based on the mole fraction scale. By means of simple acid-base titration and 1H NMR measurements, it was established that the predictions of differences in the solvation Gibbs energy and the partition ratio (or "partition coefficient") of diethylamine between water and hexadecane are consistent with the ST formalism. Additionally, with current literature data, we have shown additional experimental support for the ST. However, due to the arbitrariness of the relative amount of solvents in the partition ratio, the choice of a single concentration scale within the classical thermodynamics is still not possible.

Tough Cortical Bone-Inspired Tubular Architected Cement-Based Material with Disorder.

Cortical bone is a tough biological material composed of tube-like osteons embedded in the organic matrix surrounded by weak interfaces known as cement lines. The cement lines provide a microstructurally preferable crack path, hence triggering in-plane crack deflection around osteons due to cement line-crack interaction. Inspired by this toughening mechanism and facilitated by a hybrid (3D-printing/casting) process, the study engineers architected tubular cement-based materials with the stepwise cracking toughening mechanism, that enables a non-brittle fracture. Using experimental and theoretical approaches, the study demonstrates the competition between tube size and shape on stress intensity factor from which engineering stepwise cracking can emerge. Two competing mechanisms, both positively and negatively affected by the growing tube size, arise to significantly enhance the overall fracture toughness by up to 5.6-fold compared to the monolithic brittle counterpart without sacrificing the specific strength. This is enabled by crack-tube interaction and engineering the tube size, shape, and orientation, which promotes rising resistance-curves (R-curve). "Disorder" curves and statistical mechanics parameters are proposed for the first time to quantitatively characterize the degree of disorder for describing the representation of the architected arrangement of materials in lieu of otherwise inadequate "periodicity" classification and misperceiveddisorder parameters (perturbation and Voronoi tessellation methods).

Early attempts to integrate kinetic theory into economics in Italy between the wars

Between the two world wars, a group of Italian mathematicians, statisticians and economists set out to use the tools of statistical mechanics to interpret the Pareto income distribution curve as a probability distribution. Models based on Boltzmann's thermal equilibrium were independently developed to find an economic variable, comparable to the kinetic energy of gas theory, that could explain the heavy tail of the Pareto curve. The various avenues explored eventually led them back to Pareto's much debated explanation, namely that the hyperbolic distribution of income was due to the unequal distribution of talent.

Free energy and extension of stiff polymer chains confined in nanotubes with diverse cross-sectional shapes

The statistical mechanics of stiff polymer chains confined within narrow tubes is a foundational topic in polymer physics, extensively analyzed in prior research. For cylindrical, rectangular, and slit-like confinements, the chains’ free energy and extension adhere to a scaling law consistent with the Odijk theory. While this scaling law may not apply to tubes with different cross-sectional geometries, there is a lack of research examining the behavior of stiff chains in tubes with intricate cross-sectional shapes. In this study, we investigate the partition function of a stiff chain confined within an elliptic tube using the path integral approach, deriving a deflection length in a concise closed form through dimensional analysis. This length scale facilitates straightforward expressions for the chain's free energy and extension. Notably, we discover a shape-independent property of these expressions applicable to tubes with a wide variety of cross-sectional geometries. Extensive numerical simulations are conducted using a biased chain-growth Monte Carlo method, incorporating the Pruned and Enriched Rosenbluth algorithm, to validate the theoretical predictions on the confinement free energy and extension of chains in tubes with differing shapes.

Statistical physical view of statistical inference in Bayesian linear regression model.

This paper considers similarities between statistical physics and Bayes inference through the Bayesian linear regression model. Some similarities have been discussed previously, such as the analogy between the marginal likelihood in Bayes inference and the partition function in statistical mechanics. In particular, this paper considers the proposal to associate discrete sample size with inverse temperature [C. H. LaMont and P. A. Wiggins, Phys. Rev. E 99, 052140 (2019)2470-004510.1103/PhysRevE.99.052140]. The previous study suggested that incorporating this similarity motivates the derivation of analogs of thermodynamic functions such as energy and entropy. The study also anticipated that those analogous functions have potential to describe Bayes estimation from physical points of view and to provide physical insights into mechanisms of estimation. This paper incorporates a macroscopic perspective as an asymptotics similar to the thermodynamic limit into the previous suggestion. Its motivation stems from the statistical mechanical concept of deriving thermodynamic functions that characterize macroscopic properties of macroscopic systems. This incorporation not only allows analogs of macroscopic thermodynamic functions to be considered but also suggests a candidate for an analog of inverse temperature with continuity, which is partly consistent with the previous proposal to associate the discrete sample size with inverse temperature. On the basis of this suggestion, we analyze analogs of macroscopic thermodynamic functions for a Bayesian linear regression model which is the basis of various machine learning models. We further investigate, through the behavior of these functions, how Bayes estimation is described from the perspective of physics and what kind of physical insight is obtained. As a result, the estimation of regression coefficients, which is the primary task of regression, appears to be described by the physical picture of balance between decreasing energy and increasing entropy as in equilibrium states of thermodynamic systems. More specifically we observe the physical view of Bayes inference as follows: the estimation succeeds where the effect of decreasing energy is dominant at low temperature. On the other hand, the estimation fails where the effect of increasing entropy is dominant at high temperature.

Non-equilibrium Statistical Mechanics in Biological Systems

Unlike equilibrium systems, which are characterized by a time-independent distribution of particles and energy, non-equilibrium systems experience constant fluxes of matter or energy, often due to external driving forces. These systems can exhibit a wide range of dynamic behaviors some of which can appear as self-organizing. Building on the framework of non-equilibrium statistical mechanics, Jeremy England in his paper "Statistical Physics of Adaptation" derives an equation which relates the relative forwards probability of transitioning between two possible macrostates with their respective reverse probability, entropy, and average energy dissipation across all possible micro trajectories. He showed that structures which are better at absorbing and dissipating energy into their surroundings during their formation have a higher likelihood of occurring, which can possibly explain how life-like behaviors emerge in non-equilibrium systems. In this project, we test England's equation on a toy model of self replicating populations, as well as attempt to apply it to a biological dataset. In the toy model under certain constraints we do find that our intuition of the systems dynamics does track with England's equation, that is we observe that the population with the greatest replication rate tends to have the highest energy dissipation when it's population makes up the greater fraction of the total population of particles. In the biological dataset we analyze protein-protein interactions and how the relative bonding energy and binding rates compare for different mutations. We explore the limitations of his equation when trying to make predictions about highly complex biological systems.

Random Transitions of a Binary Star in the Canonical Ensemble.

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a "binary star" consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a "dilute" state, where the particles are well separated, and a "condensed" state, where the particles are bound together. We show that the evolution of the system is given by a Fokker-Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.

Comparative study of multiple statistical damage mechanics models for rock behaviors under high temperature

Under the influence of external factors such as high temperature, chemical corrosion, and loading-unloading cycles, micro-cracks and pores may develop within rock structures. Continuum damage mechanics (CDM) characterizes these micro-defects arising within rocks, investigating the physico-mechanical behaviors of rocks in damaged states. This study, grounded in the theoretical framework of CDM and utilizing the Mohr-Coulomb failure criterion, integrates different probability distributions and damage coupling methods to develop three statistical damage models. The investigation primarily explores the effects of strength criteria, probability functions, and damage coupling mechanisms on the statistical damage models. In addressing the complex interplay between rock micro-defects and external stressors such as high temperatures and chemical corrosion, this study leverages the principles of CDM within the Mohr-Coulomb criterion framework. It advances the field by developing three innovative statistical damage models, enriched by diverse probability distributions and damage coupling methods. The research delineates the substantial impact of strength criteria, probability functions, and damage coupling mechanisms on the behavior of these models. Notably, it achieves a synthesis of theoretical constructs and experimental validation, demonstrating the models' capacity to reflect the nuanced behaviors of damaged rock accurately. Furthermore, this investigation contributes a methodological blueprint for statistical damage model construction, underscoring the critical selection of strength criteria and probability distributions. These efforts fortify the theoretical underpinnings necessary for the enhanced engineering application of statistics-driven damage analysis.

Arbitrage equilibria in active matter systems

The motility-induced phase separation (MIPS) phenomenon in active matter has been of great interest for the past decade or so. A central conceptual puzzle is that this behavior, which is generally characterized as a nonequilibrium phenomenon, can yet be explained using simple equilibrium models of thermodynamics. Here, we address this problem using a new theory, statistical teleodynamics, which is a conceptual synthesis of game theory and statistical mechanics. In this framework, active agents compete in their pursuit of maximum effective utility, and this self-organizing dynamics results in an arbitrage equilibrium in which all agents have the same effective utility. We show that MIPS is an example of arbitrage equilibrium and that it is mathematically equivalent to other phase-separation phenomena in entirely different domains, such as sociology and economics. As examples, we present the behavior of Janus particles in a potential trap and the effect of chemotaxis on MIPS.

Modern chemical graph theory

AbstractGraph theory has a long history in chemistry. Yet as the breadth and variety of chemical data is rapidly changing, so too do graph encoding methods and analyses that yield qualitative and quantitative insights. Using illustrative cases within a basic mathematical framework, we showcase modern chemical graph theory's utility in Chemists' analysis and model development toolkit. The encoding of both experimental and simulation data is discussed at various levels of granularity of information. This is followed by a discussion of the two major classes of graph theoretical analyses: identifying connectivity patterns and partitioning methods. Measures, metrics, descriptors, and topological indices are then introduced with an emphasis upon enhancing interpretability and incorporation into physical models. Challenging data cases are described that include strategies for studying time dependence. Throughout, we incorporate recent advancements in computer science and applied mathematics that are propelling chemical graph theory into new domains of chemical study.This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Structure and Mechanism > Computational Materials Science Structure and Mechanism > Molecular Structures

A Statistical Framework for a New two-parameter Unit Bilal Distribution with Application to Model Asymmetric Data

In the realm of statistical mechanics, unit models are commonly employed to elucidate tangible numerical values that fall within the range of 0 to 1, encompassing entities like proportions, percentages, and probabilities. Unit models are frequently created by adapting an existing model originally defined in a broader domain, while others are explicitly formulated independently. This study aims to develop a novel unit interval lifespan model with a small number of parameters that is capable of describing more complex failure patterns, such as growing and bathtub-shaped failure rates. The unit Bilal (UB) model, based on the inverse-exponential technique and the Bilal model, is explained in this research study. Different statistical features and aspects of reliability are investigated. The results have both theoretical and practical implications. We derive explicit expressions for primary functions from the UB distribution from a theoretical perspective. To evaluate parameter, estimate approaches, we demonstrate various numerical and graphical simulation findings. The applicability of the model to real-world data is also discussed.

Development of Air–Ti Gas-Surface Reaction Mechanisms and Rate Calculations

Gas-surface reaction mechanisms and rates were developed for an air–titanium surface to support finite-rate catalytic surface reaction models used in computational fluid dynamics. For the surface dissociation/recombination reactions, a two-step reaction method was applied based on the assumption that two atoms must be positioned in adjacent cells to recombine. Transition state theory was used to calculate the surface reactions for adsorbed diatomic molecules to dissociate to the adjacent unit cell or atoms located in the adjacent cells to recombine into diatomic molecules. For the second step, atoms located in the adjacent cells are allowed to further dissociate to another cell (total separation), or totally separated atoms recombine into the adjacent cells. A stochastic gas-surface kinetics simulation code, STOKIN, was developed to simulate gas-surface reactions and to calculate the equilibrium constants of second molecular dissociation ⇌ recombination processes. Thermodynamic properties of the adsorbates (enthalpies, entropies, and heat capacities) were calculated using density functional theory along with statistical mechanics assuming no rotational and translational contributions to the entropies and heat capacities due to the adsorbates’ inability to freely move or rotate on the surface.

Correlations in classical non-equilibrium systems and their connection with temperature

Abstract Correlations are essential for the description of the properties of complex systems, particularly from the point of view of statistical mechanics. In this work we discuss the existence of correlations between parts of a non-equilibrium, composite system in a steady state, and its relation with the concept of temperature fluctuations. For this purpose, we review a recently proposed definition of steady-state temperature, namely the fundamental inverse temperature, together with a descriptor of temperature uncertainty, the inverse temperature covariance, showing that both of these quantities are invariants upon the choice of subsystems.

Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy

We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.

An anisotropic damage visco-hyperelastic model for multiaxial stress-strain response and energy dissipation in filled rubber

In this article, we introduce a novel physically-based anisotropic damage visco-hyperelastic model designed to predict the history-dependent inelastic behavior of multiaxially stretched filled rubber. The model integrates both the anisotropic Mullins effect and intrinsic viscosity through the consideration of internal physics, represented by two distinct networks: an elastic ground network and a superimposed viscous network. The rupture of molecular bonds within the elastic network chain backbone is modeled using statistical mechanics, while the effects of anisotropy-induced chain orientation at the upper scale are addressed through a microsphere-based scale transition method. The intrinsic viscosity is represented by the viscous network, which is governed by time-dependent equations to account for the viscous overstress. The influence of fillers is captured through the concept of strain amplification, applied to the two networks within the rubber matrix. The effectiveness of the model in capturing the biaxial behavior of filled rubber is evaluated by comparing its outputs with experimental data from a filled rubber system. This assessment specifically considers the impact of pre-stretching under various loading conditions and across a wide range of filler concentrations. Notably, it successfully predicts anisotropic stress-strain response and energy dissipation, and the coupled effects of damage and viscosity.

Inequality in the Distribution of Wealth and Income as a Natural Consequence of the Equal Opportunity of All Members in the Economic System Represented by a Scale-Free Network

The purpose of this work is to examine the nature of the historically observed and empirically described by the Pareto law inequality in the distribution of wealth and income in an economic system. This inequality is presumed to be the result of unequal opportunity by its members. An analytical model of the economic system consisting of a large number of actors, all having equal access to its total wealth (or income) has been developed that is formally represented by a scale-free network comprised of nodes (actors) and links (states of wealth or income). The dynamic evolution of the complex network can be mapped in turn, as is known, into a system of quantum particles (links) distributed among various energy levels (nodes) in thermodynamic equilibrium. The distribution of quantum particles (photons) at different energy levels in the physical system is then derived based on statistical thermodynamics with the attainment of maximal entropy for the system to be in a dynamic equilibrium. The resulting Planck-type distribution of the physical system mapped into a scale-free network leads naturally into the Pareto law distribution of the economic system. The conclusions of the scale-free complex network model leading to the analytical derivation of the empirical Pareto law are multifold. First, any complex economic system behaves akin to a scale-free complex network. Second, equal access or opportunity leads to unequal outcomes. Third, the optimal value for the Pareto index is obtained that ensures the optimal, albeit unequal, outcome of wealth and income distribution. Fourth, the optimal value for the Gini coefficient can then be calculated and be compared to the empirical values of that coefficient for wealth and income to ascertain how close an economic system is to its optimal distribution of income and wealth among its members. Fifth, in an economic system with equal opportunity for all its members there should be no difference between the resulting income and wealth distributions. Examination of the wealth and income distributions described by the Gini coefficient of national economies suggests that income and particularly wealth are far off from their optimal value. We conclude that the equality of opportunity should be the fundamental guiding principle of any economic system for the optimal distribution of wealth and income. The practical application of this conclusion is that societies ought to shift focus from policies such as taxation and payment transfers purporting to produce equal outcomes for all, a goal which is unattainable and wasteful, to policies advancing among others education, health care, and affordable housing for all as well as the re-evaluation of rules and institutions such that all members in the economic system have equal opportunity for the optimal utilization of resources and the distribution of wealth and income. Future research efforts should develop the scale-free complex network model of the economy as a complement to the current standard models.

Statistical mechanics description of an empirical scenario in which gravity was discussed when coupled to a harmonic oscillator

Statistical mechanics description of an empirical scenario in which gravity was discussed when coupled to a harmonic oscillator

Statistical mechanics of passive Brownian particles in a fluctuating harmonic trap.

We consider passive Brownian particles trapped in an "imperfect" harmonic trap. The trap is imperfect because it is randomly turned off and on, and as a result particles fail to equilibrate. Another way to think about this is to say that a harmonic trap is time dependent on account of its strength evolving stochastically in time. Particles in such a system are passive and activity arises through external control of a trapping potential, thus, no internal energy is used to power particle motion. A stationary Fokker-Planck equationof this system can be represented as a third-order differential equation, and its solution, a stationary distribution, can be represented as a superposition of Gaussian distributions for different strengths of a harmonic trap. This permits us to interpret a stationary system as a system in equilibrium with quenched disorder.

Theory and Monte Carlo simulation of the ideal gas with shell particles in the canonical, isothermal-isobaric, grand canonical, and Gibbs ensembles.

Theories of small systems play an important role in the fundamental understanding of finite size effects in statistical mechanics, as well as the validation of molecular simulation results as no computer can simulate fluids in the thermodynamic limit. Previously, a shell particle was included in the isothermal-isobaric ensemble in order to resolve an ambiguity in the resulting partition function. The shell particle removed either redundant volume states or redundant translational degrees of freedom of the system and yielded quantitative differences from traditional simulations in this ensemble. In this work, we investigate the effect of including a shell particle in the canonical, grand canonical, and Gibbs ensembles. For systems comprised of a pure component ideal gas, analytical expressions for various thermodynamic properties are obtained. We also derive the Metropolis Monte Carlo simulation acceptance criteria for these ensembles with shell particles, and the results of the simulations of an ideal gas are in excellent agreement with the theoretical predictions. The system size dependence of various important ensemble averages is also analyzed.