Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics

By reformulating the steepest-entropy-ascent (SEA) dynamical model for nonequilibrium thermodynamics
in the mathematical language of differential geometry, we compare it with the primitive formulation of the
general equation for the nonequilibrium reversible-irreversible coupling (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 on the dissipative, i.e., entropy generating, 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 dissipative and nondissipative 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 dissipative
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.