Diffusion equations from master equations -- A discrete geometric approach

Shin-itiro Goto, Hideitsu Hino

Published 2019-08-13Version 1

In this paper, 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 play roles, and master equations are described on graphs that consist of vertexes 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 Laplacian is defined as a self-adjoint operator with respect to an introduced inner product. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.