Metriplectic 4-bracket algorithm for constructing thermodynamically consistent dynamical systems

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.