Manuscript submitted to Website: http: AIMsciences org AIMS' Journals Volume X Number 0X XX 200X pp X–XX

Manuscript submitted to Website: AIMS' Journals Volume X, Number 0X, XX 200X pp. X–XX ENTROPY AND CHAOS IN THE KAC MODEL Dedicated to Professor Giuseppe Toscani on the occasion of his sixtieth birthday Eric A. Carlen Department of Mathematics, Hill Center Rutgers University Piscataway, NJ 08854,U.S.A. Maria C. Carvalho Department of Mathematics and CMAF University of Lisbon 1649-003 Lisbon,PORTUGAL Jonathan Le Roux Department of Information Physics and Computing The University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN Michael Loss School of Mathematics Georgia Institute of Technology Atlanta GA, 30332, U.S.A. Cedric Villani UMPA, ENS Lyon University of Lisbon 46 allee d'Italie, 69364 Lyon Cedex 07, FRANCE (Communicated by the associate editor name) Abstract. We investigate the behavior in N of the N–particle entropy func- tional for Kac's stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove results that bring together the notion of prop- agation 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.

• equation satisfies


• probability measures


• kac walk


• dimensional lebesgue measure


• mark kac


• vj


• continu- ous time


• boltzmann equation


• convergence against