Some frameworks for dissipative evolution in multiscale non-equilibrium thermodynamics

TL;DR

This paper surveys geometric frameworks for dissipative evolution in multiscale non-equilibrium thermodynamics, unifying approaches such as gradient dynamics, GENERIC, metriplectic systems, and Rayleigh-type dissipation. It clarifies how entropy and energy constraints guide dissipation, showing connections among variational principles, d'Alembert formulations, and Hamiltonian-Poisson structures. Key contributions include establishing generalized variational frameworks for GENERIC, recasting Rayleigh dissipation within gradient and Hamiltonian formalisms, and presenting extended and Ehrenfest regularizations that produce energy decay or entropy production while preserving key invariants. The work highlights how these frameworks can be chosen or blended to model complex dissipative behavior in fluids, kinetic theory, and complex materials, with implications for robust multiscale modeling and numerical implementations.

Abstract

In this paper, we review and compare some frameworks for dissipation in non-equilibrium thermodynamics. We start with a brief overview of classical irreversible thermodynamics and gradient dynamics. Then we discuss several specific frameworks including Rayleigh dissipation potential and the dissipative d'Alembert framework, showing their relations with gradient dynamics. Finally, we discuss frameworks for dissipative evolution generated from Poisson brackets.