Conservation-dissipation formalism of irreversible thermodynamics

TL;DR

This paper introduces a conservation-dissipation formalism (CDF) for modeling irreversible thermodynamic processes, ensuring stability and applicability far from equilibrium, with examples in heat conduction and fluid flow.

Contribution

The paper develops a systematic, nonlinear formalism for irreversible thermodynamics that unifies and generalizes existing laws like Cattaneo's and Maxwell's laws.

Findings

CDF equations are globally hyperbolic and suitable for numerical solutions.

The formalism applies to systems far from equilibrium.

New thermodynamics theories for heat conduction and fluid flow are formulated.

Abstract

We propose a conservation-dissipation formalism (CDF) for coarse-grained descriptions of irreversible processes. This formalism is based on a stability criterion for non-equilibrium thermodynamics. The criterion ensures that non-equilibrium states tend to equilibrium in long time. As a systematic methodology, CDF provides a feasible procedure in choosing non-equilibrium state variables and determining their evolution equations. The equations derived in CDF have a unified elegant form. They are globally hyperbolic, allow a convenient definition of weak solutions, and are amenable to existing numerics. More importantly, CDF is a genuinely nonlinear formalism and works for systems far away from equilibrium. With this formalism, we formulate novel thermodynamics theories for heat conduction in rigid bodies and non-isothermal compressible Maxwell fluid flows as two typical examples. In these examples, the non-equilibrium variables are exactly the conjugate variables of the heat fluxes or stress tensors. The new theory generalizes Cattaneo's law or Maxwell's law in a regularized and nonlinear fashion.