Nonlinear Dynamical Equation for Irreversible, Steepest-Entropy-Ascent Relaxation to Stable Equilibrium

Abstract

Abstract. We discuss the structure and main features of the nonlinear evolution equation proposed by this author as the fundamental dynamical law within the framework of Quantum Thermodynamics. The nonlinear equation generates a dynamical group providing a unique deterministic description of irreversible, conservative relaxation towards equilibrium from any non-equilibrium state, and satisfies a very restrictive stability requirement equivalent to Hatsopoulos-Keenan statement of the second law of thermodynamics. Here, we emphasize its mathematical structure and its applicability also within other contexts, such as Classical and Quantum Statistical Mechanics, and Information Theory.

Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations

“…It was for such pioneering QT framework that the present author designed the nonlinear master equation [ 58 , 59 , 60 ] and shortly thereafter [ 61 ] proved it admits a steepest-entropy-ascent (SEA) variational formulation that embodies at the local microscopic level the principle of maximal entropy production [ 43 , 44 , 62 , 90 , 94 , 100 , 134 ]. Starting in the mid eighties, during times when quantum thermodynamics was considered interesting only by a handful of pioneers, the present author has addressed both the quantum-foundations and mathematical–physics communities [ 61 , 77 , 78 , 80 , 100 , 107 , 129 , 130 , 131 , 135 , 136 , 137 , 138 ] and the engineering-thermodynamics and non-equilibrium-thermodynamics communities [ 79 , 139 , 140 , 141 , 142 , 143 , 144 ] to raise awareness about the requirement that fundamental and phenomenological models of dissipative and irreversible processes must incorporate, i.e., must not violate, general thermodynamic principles. More recently, the SEA principle has been shown to encompass all the major levels of description of nonequilibrium dynamics and irreversible processes [ 94 ], including the general modeling structure known as metriplectic dynamics [ 145 , 146 , 147 ] or GENERIC (see [ 35 ] and references therein), which even more recently has been shown to bear deep connections also with the mathematical theories of gradient flows [ 148 , 149 , 150...…”

“…Starting in the mid eighties, during times when quantum thermodynamics was considered interesting only by a handful of pioneers, the present author has addressed both the quantum-foundations and mathematical–physics communities [ 61 , 77 , 78 , 80 , 100 , 107 , 129 , 130 , 131 , 135 , 136 , 137 , 138 ] and the engineering-thermodynamics and non-equilibrium-thermodynamics communities [ 79 , 139 , 140 , 141 , 142 , 143 , 144 ] to raise awareness about the requirement that fundamental and phenomenological models of dissipative and irreversible processes must incorporate, i.e., must not violate, general thermodynamic principles.…”