Entropy of irreversible cooling and the “discrimination” of model stochastic processes

Abstract

Entropy changes are calculated for the irreversible cooling of a homogeneousN-particle system. The execution of an appropriate model stochastic process enables one to calculate the “discrimination”D (from the transition probabilities of the actual steps) and < − D> is shown to be equal to the external entropy change ΔS _ext. This is trivially true for the “Metropolislike” processes, where the individual particles maintain a direct heat exchange with a reservoir. “Cooperative” processes, which attribute the heat exchange to the mass ofN particlesin toto, are also considered; for these ΔS _ext is still equal to < − D>. Hence, knowing <D> and the entropy of the initial and final states of the system, one can calculate the net entropy production and study its minimization. Alternatively, a consistently probabilistic approach (independent of thermodynamic equivalents) postulates that statistical mechanical processes proceed with the least discrimination, Min<D>, for given conditions. The postulate is supported by its conformance with the second law of thermodynamics. Min<D> reduces to the Jaynes principle both at equilibrium and for isolated systems. Computer experiments illustrating the calculation ofD are presented. These describe the cooling of a square Ising lattice, with the help of the Metropolis and of the cooperative model processes; the latter, optimized for least entropy production, rapidly converge toward equilibrium.