Diffusion equations from master equations—A discrete geometric approach

Hideitsu Hino,Shin-Itiro Goto

doi:10.1063/5.0003656

Hideitsu Hino, Shin-Itiro Goto

Open Access

https://doi.org/10.1063/5.0003656

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Abstract

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master equations are described on graphs that consist of vertices representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.

Highlights

Master equations are vital in the study of nonequilibrium statistical mechanics1–3 since they are mathematically simple and show relaxation processes toward equilibrium states under some conditions
Under the assumption that the detailed balance conditions hold, an equivalence between master equations and discrete diffusion equations is shown, where the Laplacians are constructed by choosing appropriate measures for inner products: Claim
Master equations under the detailed balance conditions are equivalent to discrete diffusion equations

Summary

INTRODUCTION

Master equations are vital in the study of nonequilibrium statistical mechanics since they are mathematically simple and show relaxation processes toward equilibrium states under some conditions. These equations describe the time evolution of probabilities, and they are first order differential or difference equations. Master equations are vital in the study of nonequilibrium statistical mechanics since they are mathematically simple and show relaxation processes toward equilibrium states under some conditions.4 These equations describe the time evolution of probabilities, and they are first order differential or difference equations. Under the assumption that the detailed balance conditions hold, an equivalence between master equations and discrete diffusion equations is shown, where the Laplacians are constructed by choosing appropriate measures for inner products: Claim. Master equations under the detailed balance conditions are equivalent to discrete diffusion equations (see Theorem 3.2 for details) By applying this statement, it is shown that probability distribution functions relax to the equilibrium state (see Corollary 3.1 and Proposition 3.2). II, some preliminaries are provided in order to keep this paper self-contained They include boundary operators, inner products, and Laplacians.

PRELIMINARIES

Standard operators

Expectation values

CONCLUSIONS