Statistical Thermodynamics of General Minimal Diffusion Processes: Constuction, Invariant Density, Reversibility and Entropy Production

TL;DR

This paper develops a thermodynamic framework for minimal diffusion processes by constructing their invariant measures, analyzing reversibility, and linking entropy production to the properties of the associated elliptic operators.

Contribution

It introduces a functional analytical method to construct solutions, establishes invariant measures, and connects reversibility with entropy production in diffusion processes.

Findings

Existence of positive invariant measure with density

Equivalence of self-adjointness, reversibility, and zero entropy production

Thermodynamic interpretation of diffusion processes

Abstract

The solution to nonlinear Fokker-Planck equation is constructed in terms of the minimal Markov semigroup generated by the equation. The semigroup is obtained by a purely functional analytical method via Hille-Yosida theorem. The existence of the positive invariant measure with density is established and a weak form of Foguel alternative proven. We show the equivalence among self-adjoint of the elliptic operator, time-reversibility, and zero entropy production rate of the stationary diffusion process. A thermodynamic theory for diffusion processes emerges.