Steepest-Entropy-Ascent Irreversible Relaxation Towards Thermodynamic Equilibrium: The Dynamical Ansatz that Completes the Gyftopoulos-Hatsopoulos Unified Theory with a General Quantal Law of Causal Evolution

Abstract

We overview the main features of the general equation of motion that completes the Gyftopoulos-Hatsopoulos unified theory of mechanics and thermodynamics with a quantal law of causal evolution that entails relaxation towards stable equilibrium for any non-equilibrium state, no matter how far from thermodynamic equilibrium. We illustrate with numerical examples the behavior of the equation of motion by discussing spontaneous energy redistribution within an isolated, closed system composed of non-interacting identical particles with energy levels ei and i = 1, 2,…, N. For this system the time-dependent occupation probabilities pi(t) obey the nonlinear rate equations which include functions of the pi(t)’s that maintain invariant the mean energy and the normalization condition. The entropy is a non-decreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions of the rate equations are unique and well-defined for arbitrary initial conditions pi(0) and for all times, -∞<t<+∞. Existence and uniqueness both forward and backward in time allows the reconstruction of the ancestral or primordial lowest entropy state. We also illustrate the structure and main properties of the nonlinear dynamics for a composite system. An initial version of this paper was published in July of 2006 in the proceedings of ECOS’06, Aghia Pelagia, Crete, Greece.