The statistical foundation of entropy in extended irreversible thermodynamics

Abstract

Abstract In the theory of extended irreversible thermodynamics (EIT), the flux-dependent entropy function plays a key role; it is a fundamental distinction between EIT and the usual flux-independent entropy function adopted by classical irreversible thermodynamics (CIT). However, its existence, as a prerequisite for EIT, and its statistical origin have never been justified. In this work, by studying the macroscopic limit of an ϵ-dependent Langevin dynamics, which admits a large deviations (LD) principle, we show that the stationary LD rate functions of probability density p ϵ (x, t) and joint probability density p ϵ ( x , x ̇ , t ) actually turn out to be the desired flux-independent entropy function in CIT and flux-dependent entropy function in EIT respectively. The difference of the two entropy functions is determined by the time resolution for Brownian motions times a Lagrangian, the latter arises from the LD Hamilton–Jacobi equation and can be used for constructing conserved Lagrangian/Hamiltonian dynamics.