Coherence and Uniqueness Theorems for Averaging Processes in Statistical Mechanics

Statistical mechanics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental questions, what is thermodynamics and how it may be applied outside the realm of physical particles, have remained unanswered. We answer these questions here: Statistical mechanics in its most general form is variational calculus applied to probability distributions and by extension to stochastic processes in general; as a mathematical theory, it is independent of physical hypotheses but provides the means to incorporate our knowledge and model assumptions about the particular problem. The fundamental ensemble is a microcanonical space of probability distributions sampled via a bias functional that establishes a probability measure on this space. The maximization of this measure expresses the most probable distribution via a set of parameters (microcanonical partition function, canonical partition function and generalized temperature) that are connected through a set of mathematical relationships that we recognize as the familiar equations of thermodynamic. Any distribution in in this space maybe endowed with the status of the most probable distribution under an appropriately constructed bias functional. Entropy, Kullback-Leibler divergence and the second law have simple interpretations in this theory. We obtain statistical mechanics as a special application to molecular systems and make contact with Information Theory and Bayesian inference. We use numerical examples to demonstrate the thermodynamic treatment of generic probability distributions, present a thermodynamic algorithm (the cluster ensemble) to sample arbitrary distributions with positive argument by analogy to reacting particles and discuss the extension of statistical mechanics to stochastic processes in general.

The paper continues a long series of our research and considers a secondorder nonlinear evolutionary parabolic system. The system can be a model of various convective and diffusion processes in continuum mechanics, including mass transfer in a binary mixture. In hydrology, ecology, and mathematical biology, it describes the propagation of pollutants in water and air, as well as population dynamics, including the interaction of two different biological species. We construct solutions that have the type of diffusion (heat) wave propagating over a zero background with a finite velocity. Note that the system degenerates on the line where the perturbed and zero (unperturbed) solutions are continuously joined. A new existence and uniqueness theorem is proved in the class of analytical functions. In this case, the solution has the desired type and is constructed in the form of characteristic series, the convergence of which is proved by the majorant method. We also present two new classes of exact solutions, the construction of which, due to ansatzes of a specific form, reduces to integrating systems of ordinary differential equations that inherit a singularity from the original formulation. The obtained results are expected to be helpful in modeling the evolution of the Baikal biota and the propagation of pollutants in the water of Lake Baikal near settlements

The relationship between a description by means of classical statistical mechanics and by means of hydrodynamics is considered for a multicomponent nonrelativistic mixture. It is shown that in the class of weak sources of a hydrodynamic type, the transition from statistics to hydrodynamics is exact. The material equations of the hydrodynamic model turn out to be nonlocal in space and time. The corresponding kernels are determined by the Hamiltonian dynamics of particles of the mixture. The hydrodynamic model is dissipative, despite the equivalence of the defined class of processes in Hamiltonian mechanics.

A review is presented of some mathematical procedures for the derivation of evolution equations which describe irreversible Markoffian processes in statistical mechanics.

We present a new approach to the construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd of Ginibre's Fermi-Dirac gas of such loops. This approach is based on the cluster expansion method. We obtain the existence of Gibbs perturbations of a large class of point processes. Moreover, it is shown that certain “limiting Gibbs processes” are Gibbs in the sense of Dobrushin, Lanford, and Ruelle if the underlying potential is positive. Finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.

This work proposes an optimization algorithm in optical design based on the concepts of ergodic and stochastic processes in statistical mechanics. In mixed-variable optimization problems, pseudo-random number and discrete-to-continuous variable conversion dramatically increase the speed at which the system solves for the optimal solution. Pseudo-random numbers are mainly applied in two important steps in the optimization algorithm: determining the combination of glasses involved and the order in which the successive glass parameters are replaced by real glasses. After two series of stochastic processes, the merit function value decreases rapidly along the steepest descent path, and thus the optical system approaches the optimal solution within a very short duration of time. By using the method proposed in this paper, a plan apochromatic objective with a long working distance was optimized, and finally, a high-quality optical system was obtained.

The aim of this paper is to provide better understanding of a few approaches that have been proposed for treating nonequilibrium (time-dependent) processes in statistical mechanics with the emphasis on the interrelation between theories. The ensemble method, as it was formulated by Gibbs, has great generality and broad applicability to equilibrium statistical mechanics. Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. In the present work, the statistical theory of nonequilibrium processes which is based on nonequilibrium ensemble formalism is discussed. We also outline the reasoning leading to some other useful approaches to the description of the irreversible processes. The kinetic approach to dynamic many-body problems, which is important from the point of view of the fundamental theory of irreversibility, is alluded to. Appropriate references are made to papers dealing with similar problems arising in other fields. The emphasis is on the method of the nonequilibrium statistical operator (NSO) developed by Zubarev. The NSO method permits one to generalize the Gibbs ensemble method to the nonequilibrium case and to construct a nonequilibrium statistical operator which enables one to obtain the transport equations and calculate the transport coefficients in terms of correlation functions, and which, in the case of equilibrium, goes over to the Gibbs distribution. Although some space is devoted to the formal structure of the NSO method, the emphasis is on its utility. Applications to specific problems such as the generalized transport and kinetic equations, and a few examples of the relaxation and dissipative processes, which manifest the operational ability of the method, are considered.

CONTENTS Introduction § 1. Generalized functions and Gaussian integrals on finite-dimensional non-Archimedean spaces § 2. Non-Archimedean Hilbert spaces § 3. Spaces of functions square-integrable with respect to Gauss and Lebesgue distributions on non-Archimedean spaces § 4. The Schrödinger and Bargmann-Fock representations in non-Archimedean quantum mechanics § 5. Non-Archimedean statistical quantum mechanics § 6. Existence and uniqueness theorems for the solution of linear partial differential equations on non-Archimedean spaces § 7. The solubility of Schrödinger's, Heisenberg's and Liouville's equations in non-Archimedean mechanics § 8. Trotter's formula in non-Archimedean commutative Banach algebras, and the chronological representation of the solution of Schrödinger's equation with a potential § 9. Generalized functions and Gaussian integrals on infinite-dimensional non-Archimedean spaces § 10. Existence and uniqueness theorems for the solution of linear partial differential equations on infinite-dimensional non-Archimedean spaces § 11. The path integral for a non-Archimedean scalar bosonic field § 12. Non-Archimedean bosonic strings in a photonic-like gauge § 13. Non-Archimedean bosonic string fields in a photonic-like gauge § 14. Singular functions of a non-Archimedean scalar bosonic field References

On the statistical, quantum and practical mechanics of electrostatic atomization

The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions.

Noise, chaos, and (ε, τ)-entropy per unit time

In a recent work, the authors derived kinetic equations for the spin autocorrelation function for a paramagnetic spin system using the resummation procedure introduced by R\'esibois and De Leener in the framework of the statistical mechanics of irreversible processes due to Prigogine and co-workers. These equations are non-Markovian and nonlinear in the high-field, high-temperature, and Weiss-limit approximations. In the present paper, methods of approximation are given to solve such kinetic equations and are applied to the study of two important NMR problems, namely, free induction decay (FID) and spin diffusion. The general characteristics of the FID are obtained even in the lowest order of approximation owing to the resummation procedure, whereas the next higher-order correction leads to very good agreement with the experimental results given by Barnaal and Lowe. The following asymptotic form is also derived: $\ensuremath{\Gamma}(t)=(acos\ensuremath{\alpha}t+bsin\ensuremath{\alpha}t){e}^{\ensuremath{-}\ensuremath{\beta}t}$ A diffusion equation is obtained for the magnetization. From this the diffusion coefficient is computed and is found to be in agreement with that proposed by several authors. However, consideration of higher-order corrections does not seem to explain the strong dependence on the orientation of the external magnetic field which was observed experimentally by Leppelmeier.

Machine learning (ML) is transforming all areas of science. The complex and time-consuming calculations in molecular simulations are particularly suitable for an ML revolution and have already been profoundly affected by the application of existing ML ...Read More

Variants of fluctuation theorems recently discovered in the statistical mechanics of nonequilibrium processes may be used for the efficient determination of high-dimensional integrals as typically occurring in Bayesian data analysis. In particular for multimodal distributions, Monte Carlo procedures not relying on perfect equilibration are advantageous. We provide a comprehensive statistical error analysis for the determination of the prior-predictive value (the evidence) in a Bayes problem, building on a variant of the Jarzynski equation. Special care is devoted to the characterization of the bias intrinsic to the method and statistical errors arising from exponential averages. We also discuss the determination of averages over multimodal posterior distributions with the help of a consequence of the Crooks relation. All our findings are verified by extensive numerical simulations of two model systems with bimodal likelihoods.

The problem of adequately describing transport processes of fluids in confined conditions is solved using methods of nonequilibrium statistical mechanics.The 'fluid-channel wall' system is regarded as a two-fluid medium, in which each phase has a particular velocity and temperature.The obtained results show that the transport equations in confined spaces should contain not only the stress tensor and the heat flux vector, but also the interfacial forces responsible for the transfer of momentum and heat due to the interaction with the wall surfaces.The stress tensor and the heat flux vector fluid can be expressed in terms of the effective viscosity and thermal conductivity.However, the constitutive relations contain additive terms that correspond to fluid-surface interactions.Thus, not only do the fluid transport coefficients in nanochannels differ from the bulk transport coefficients, but they are also not only determined by the parameters of the fluid.