Metriplectic formulations of variational thermodynamics

TL;DR

This paper introduces a new metriplectic framework for non-equilibrium thermodynamics, combining Hamiltonian and dissipative dynamics through a novel bracket structure, applicable to various physical systems.

Contribution

It develops a metriplectic reformulation of variational principles for thermodynamics, unifying Hamiltonian and dissipative effects using a combined bracket structure.

Findings

Solutions can be generated by a sum of Poisson and metriplectic brackets.

Applicable to simple, discrete, and continuum systems.

Includes examples from continuum mechanics.

Abstract

We propose a metriplectic reformulation of Lagrangian variational formulations for non-equilibrium thermodynamics. We prove that solutions to these constrained variational principles can be generated by the sum of a classic Poisson bracket and a metriplectic 4-bracket, that takes the Hamiltonian and the entropy as generators. We study different cases: simple systems, discrete systems, Euler-Poincar\'e reduced systems and systems with no symplectic part. Several example are shown, including infinite dimensional problems arising from continuum mechanics.