Entropy and chaos in the Kac model

Abstract

We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochasticmodel of Boltzmann dynamics, and its relation to the entropy function forsolutions of Kac's one dimensional nonlinear model Boltzmann equation. We proveresults that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved hereshow that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open and the upper bounds obtained here show that this problem is somewhat subtle.