Essential equivalence of the GENERIC and Steepest Entropy Ascent models of dissipation for non-equilibrium thermodynamics

TL;DR

This paper demonstrates the fundamental equivalence between the GENERIC and Steepest Entropy Ascent models in non-equilibrium thermodynamics by reformulating them in differential geometric terms and comparing their structures.

Contribution

It provides a formal proof that, under certain conditions, GENERIC and SEA models are interchangeable in describing dissipation in thermodynamics.

Findings

Reformulation of SEA using differential geometry.

Comparison of GENERIC and SEA structures.

Proof of equivalence under conservation law degeneracies.

Abstract

By reformulating the Steepest-Entropy-Ascent (SEA) dynamical model for non-equilibrium thermodynamics in the mathematical language of Differential Geometry, we compare it with the primitive formulation of the GENERIC model and discuss the main technical differences of the two approaches. In both dynamical models the description of dissipation is of the "entropy-gradient" type. SEA focuses only onto the irreversible component of the time evolution, chooses a sub-Riemannian metric tensor as dissipative structure, and uses the local entropy density field as potential. GENERIC emphasizes the coupling between the reversible and irreversible components of the time evolution, chooses two compatible degenerate structures (Poisson and degenerate co-Riemannian), and uses the global energy and entropy functionals as potentials. As an illustration, we rewrite the known GENERIC formulation of the Boltzmann Equation in terms of the square-root of the distribution function adopted by the SEA formulation. We then provide a formal proof that in more general frameworks, whenever all degeneracies in the GENERIC framework are related to conservation laws, the SEA and GENERIC models of the irreversible component of the dynamics are essentially interchangeable, provided of course they assume the same kinematics. As part of the discussion, we note that equipping the dissipative structure of GENERIC with the Leibniz identity makes it automatically SEA on metric leaves.