The Markov Process Admits a Consistent Steady-State Thermodynamic Formalism

TL;DR

This paper develops a universal steady-state thermodynamic formalism for Markov processes, unifying various non-equilibrium models and establishing rigorous connections between Chapman-Kolmogorov, master, and Fokker-Planck equations.

Contribution

It introduces a novel, mathematically rigorous steady-state thermodynamic framework applicable to general Markov processes, bridging different equations in non-equilibrium thermodynamics.

Findings

Formalism consistent across Chapman-Kolmogorov, master, and Fokker-Planck equations

Rigorous derivation of formalism limits for continuous time and large system size

Unified thermodynamic description for Markov processes at steady state

Abstract

The seek for a new universal formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that: 1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; 2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; 3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that, with respect to the steady state, general Markov processes admit a unified and self-consistent non-equilibrium thermodynamic formulation, regardless of underlying detailed models.