Entropy, Dynamics, and Molecular Chaos

Abstract

With the help of simple probabilistic models of Kac and McKean, we discuss the meaning of the generalized expression for entropy that was recently introduced by our group and compare it with Boltzmann's expression. We emphasize the fact that Boltzmann's formulation in terms of the single particle distribution function, f(1), requires very restricted assumptions about the preparation of the system (chaos) and the nature of the collision mechanism (Markov processes).Our generalized [unk]-theorem, however, refers to the complete system; in general, it does not lead to an [unk]-theorem for the single particle distribution function, f(1). It is valid whatever the preparation of the system. In McKean's model, situations exist where it gives the correct behavior while the Boltzmann's expression for entropy becomes meaningless. In addition, in Kac's model, we show that correlations reach equilibrium more rapidly than f(1) and that there is an asymptotic regime where both formulations give the same result.