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By suitable reformulations, we review the mathematical frameworks of sixdifferent approaches to the description of non-equilibrium dynamics with thepurpose to set up a unified formulation of the Maximum Entropy Production (MEP)principle valid in all these contexts. In this way, we extend to suchframeworks the concept of Steepest Entropy Ascent dynamics introduced by thepresent author in previous work on quantum thermodynamics. Actually, thepresent formulation constitutes a generalization also in the quantumthermodynamics framework. The analysis emphasizes that in the SEA-inspiredimplementation of the MEP principle, a key role is played by the geometricalmetric with respect to which to measure the length of a trajectory in statespace. The metric tensor turns out to be directly related to the inverse of theOnsager's generalized conductivity tensor. We conclude that in most of theexisting theories of non-equilibrium the time evolution of the staterepresentative can be seen to actually follow in state space the path of SEAwith respect to a suitable metric connected with the generalizedconductivities. The resulting unified family of SAE/MEP dynamical models areall intrinsically consistent with the second law of thermodynamics. Thenonnegativity of the entropy production is a general and readily proved featureof SEA dynamics. In several of the different approaches to non-equilibriumdescription we consider here, the SEA concept has not been investigated before.Therefore, it is hoped that the present unifying approach may prove useful inproviding a fresh basis for effective, thermodynamically consistent, numericalmodels and theoretical treatments of irreversible conservative relaxationtowards equilibrium from far non-equilibrium states.

Steepest-Entropy-Ascent and Maximal-Entropy-Production Dynamical Models  of Irreversible Relaxation to Stable Equilibrium from Any Non-Equilibrium  State. Unified Treatment for Six Non-Equilibrium Frameworks