Metriplectic 4-bracket algorithm for constructing thermodynamically consistent dynamical systems

TL;DR

This paper introduces a thermodynamic algorithm based on a 4-bracket structure to construct dynamical systems that conserve energy and produce entropy, with applications to fluid and phase separation models.

Contribution

It presents a unified thermodynamic algorithm using a metriplectic 4-bracket for constructing consistent dynamical systems, extending classical models.

Findings

Successfully applied to Navier-Stokes-Fourier system

Extended to Cahn-Hilliard-Navier-Stokes system

Generalized Brenner-Navier-Stokes-Fourier equations

Abstract

A unified thermodynamic algorithm (UTA) is presented for constructing thermodynamically consistent dynamical systems, i.e., systems that have Hamiltonian and dissipative parts that conserve energy while producing entropy. The algorithm is based on the metriplectic 4-bracket given in Morrison and Updike [Phys.\ Rev.\ E 109, 045202 (2024)]. A feature of the UTA is the force-flux relation $\mathbf{J}^\alpha = - L^{\alpha\beta}\, \nabla(\delta H / \delta \xi^\beta)$ for phenomenological coefficients $L^{\alpha\beta}$, Hamiltonian $H$ and dynamical variables $\xi^\beta$. The algorithm is applied to the Navier-Stokes-Fourier, the Cahn-Hilliard-Navier-Stokes, and and Brenner-Navier-Stokes-Fourier systems, and significant generalizations of these systems are obtained.