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

TL;DR

This paper unifies six non-equilibrium frameworks using Steepest Entropy Ascent and Maximal-Entropy-Production principles, extending the approach to quantum thermodynamics and emphasizing the geometric metric's role in thermodynamic consistency.

Contribution

It introduces a unified, geometrically grounded formulation of MEP and SEA dynamics applicable across multiple non-equilibrium theories, including quantum thermodynamics.

Findings

The metric tensor relates to Onsager's conductivity, guiding the trajectory in state space.

Most non-equilibrium theories follow SEA paths respecting the second law.

The approach provides a basis for thermodynamically consistent numerical models.

Abstract

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.