From variational to bracket formulations in nonequilibrium thermodynamics of simple systems

TL;DR

This paper demonstrates how a variational approach to nonequilibrium thermodynamics systematically derives various bracket formulations, unifying different methods and connecting them through symmetry reduction and geometric structures.

Contribution

It shows that a variational formulation can generate multiple bracket formulations in nonequilibrium thermodynamics, linking them through symmetry reduction and geometric insights.

Findings

Derivation of single and double generator brackets from a variational principle

Recovery of the metriplectic or GENERIC bracket in linear cases

Application of symmetry reduction and connection to double bracket dissipation

Abstract

A variational formulation for nonequilibrium thermodynamics was recently proposed in \cite{GBYo2017a,GBYo2017b} for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systematic way to derive from a unified perspective several bracket formulations for nonequilibrium thermodynamics proposed earlier in the literature, such as the single generator bracket and the double generator bracket. In the case of a linear relation between the thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC bracket is recovered. We also show how the processes of reduction by symmetry can be applied to these brackets. In the reduced setting, we also consider the case in which the coadjoint orbits are preserved and explain the link with double bracket dissipation. A similar development has been presented for continuum systems in \cite{ElGB2019} and applied to multicomponent fluids.