  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.
