  By suitable reformulations, we review the mathematical frameworks of six
different approaches to the description of non-equilibrium dynamics with the
purpose to set up a unified formulation of the Maximum Entropy Production (MEP)
principle valid in all these contexts. In this way, we extend to such
frameworks the concept of Steepest Entropy Ascent dynamics introduced by the
present author in previous work on quantum thermodynamics. Actually, the
present formulation constitutes a generalization also in the quantum
thermodynamics framework. The analysis emphasizes that in the SEA-inspired
implementation of the MEP principle, a key role is played by the geometrical
metric with respect to which to measure the length of a trajectory in state
space. The metric tensor turns out to be directly related to the inverse of the
Onsager's generalized conductivity tensor. We conclude that in most of the
existing theories of non-equilibrium the time evolution of the state
representative can be seen to actually follow in state space the path of SEA
with respect to a suitable metric connected with the generalized
conductivities. The resulting unified family of SAE/MEP dynamical models are
all intrinsically consistent with the second law of thermodynamics. The
nonnegativity of the entropy production is a general and readily proved feature
of SEA dynamics. In several of the different approaches to non-equilibrium
description we consider here, the SEA concept has not been investigated before.
Therefore, it is hoped that the present unifying approach may prove useful in
providing a fresh basis for effective, thermodynamically consistent, numerical
models and theoretical treatments of irreversible conservative relaxation
towards equilibrium from far non-equilibrium states.
