The Physical Origins of Entropy Production, Free Energy Dissipation and their Mathematical Representations

TL;DR

This paper develops a comprehensive mathematical framework for nonequilibrium thermodynamics of stochastic systems, linking entropy, free energy, and heat flows, and clarifying their roles in open and closed systems.

Contribution

It introduces a unified theory connecting entropy production, free energy dissipation, and heat flows in stochastic systems, including the concept of housekeeping heat.

Findings

Derived fundamental entropy balance equations.

Clarified the relationship between entropy production and heat dissipation.

Introduced the concept of housekeeping heat in steady states.

Abstract

A complete mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented. As generalizations of isothermal entropy and free energy, two functions of states play central roles: the Gibbs entropy $S$ and the relative entropy $F$, which are related via the stationary distribution of the stochastic dynamics. $S$ satisfies the fundamental entropy balance equation $dS/dt=e_p-h_d/T$ with entropy production rate $e_p\ge 0$ and heat dissipation rate $h_d$, while $dF/dt=-f_d\le 0$. For closed systems that satisfy detailed balance: $Te_p(t)=f_d(t)$. For open system one has $Te_p(t)=f_d(t)+Q_{hk}(t)$ where the housekeeping heat $Q_{hk}\ge 0$ was first introduced in the phenomenological nonequilibrium steady state thermodynamics. Entropy production $e_p$ consists of free energy dissipation associated with spontaneous relaxation, $f_d$, and active energy pumping that sustains the open system $Q_{hk}$. The amount of excess heat involved in the relaxation $Q_{ex}=h_d-Q_{hk} = f_d-T(dS/dt)$.