Metriplectic formulations of variational thermodynamics
Valentin Carlier
TL;DR
This work develops a metriplectic reformulation of variational principles for non-equilibrium thermodynamics, showing that constrained Lagrangian dynamics can be written as the sum of a Poisson bracket and a symmetric 4-bracket driven by the Hamiltonian $H$ and the entropy $S$. Starting from a Legendre transform of a Lagrangian $L(q,\dot{q},S)$ (with invertible mapping to $(q,p,S)$), the authors derive two compatible metriplectic constructions, one that remains valid when the dissipative rate $K$ is nonzero and another that remains well-defined when $K$ vanishes by assuming a symmetric friction tensor $\Lambda$. The framework is then extended to discrete systems, Euler–Poincaré reduced systems, and systems lacking a symplectic part, with explicit formulations of the brackets and illustrative examples including the one-piston and two-piston problems, viscous/heat-conducting fluids, and simple chemical reaction networks. A key outcome is the explicit Kulkarni–Nomizu product structure of the dissipative 4-brackets and their reduction to 2-brackets, which aids geometric interpretation and potential structure-preserving discretizations. The approach provides a unified way to derive dissipative dynamics from variational principles and offers tools for stability analysis and numerics in continuum mechanics and thermodynamics.
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-Poincaré reduced systems and systems with no symplectic part. Several example are shown, including infinite dimensional problems arising from continuum mechanics.