Lyapunov function for non-equilibrium transport processes

TL;DR

This paper establishes that for parabolic-like non-equilibrium transport processes, the dot products of fluxes and forces serve as Lyapunov functions, with entransy dissipation being more appropriate than entropy production in heat conduction.

Contribution

It introduces a theorem showing flux-force dot products as Lyapunov functions for certain transport processes, challenging the traditional entropy production principle.

Findings

Flux-force dot products serve as Lyapunov functions.

Entransy dissipation is a better Lyapunov function than entropy production in heat conduction.

Theoretical and numerical analyses support the new Lyapunov function concept.

Abstract

Irreversibility is a critical property of non-equilibrium transport processes. An opinion has long been insisted that the entropy production rate is a Lyapunov function for all kinds of processes, that is, the principle of minimum entropy production. However, such principle is based on some strong assumptions that are rarely valid in practice. Here, the common features of parabolic-like transport processes are discussed. A theorem is then put forward that the dot products of fluxes and corresponding forces serve as Lyapunov function for parabolic-like transport processes. Such fluxes and forces are defined by their actual constitutive relations (e.g., the Fourier's law, the Fick's law, etc.). Then, some typical transport processes are analyzed. Particularly for heat conduction, both the theoretical and numerical analyses demonstrate that its Lyapunov function is the entransy dissipation rather that the entropy production, when the Fourier's law is valid. The present work could be helpful for further understanding on the irreversibility and the mathematical interpretation of non-equilibrium processes.