Entropy and entropy production for the Kac's model

Abstract

Using a probabilistic model due to Kac and called by him a caricature of the dilute hard sphere gas, we compute the entropy and entropy production according to the generalized definitions proposed by the Brussels group. As the generalized ℋ-theorem describes the approach to equilibrium of the whole system, correlations play a role even if they are such that reduced distribution functions for a finite number of variables satisfy the molecular chaos condition. Nevertheless, close to equilibrium, one recovers Boltzmann's definitions. Using the results of another paper, we can give a precise meaning to what is meant by «close to equilibrium» : it is the asymptotic regime in which, to describe the evolution, one can restrict oneself to those contributions which correspond to the longest time scale in the system.