American Institute of Mathematical Sciences

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March  2010, 3(1): 85-122. doi: 10.3934/krm.2010.3.85

Entropy and chaos in the Kac model

Eric A. Carlen 1, , Maria C. Carvalho 2, , Jonathan Le Roux 3, , Michael Loss 4, and  Cédric Villani 5,

1. 

Department of Mathematics, Hill Center, Rutgers University, Piscataway, NJ 08854, United States
2. 

Department of Mathematics and CMAF, University of Lisbon, 1649-003 Lisbon, Portugal
3. 

Department of Information Physics and Computing, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
4. 

School of Mathematics, Georgia Institute of Technology, Atlanta GA, 30332, United States
5. 

UMPA, ENS Lyon, University of Lisbon, 46 allée d’Italie, 69364 Lyon Cedex 07, France

Received  April 2009 Revised  November 2009 Published  January 2010

We investigate the behavior in $N$ of the $N$--particle entropy functional 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 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 here show 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.
Keywords: Entropy, propagation of chaos..
Mathematics Subject Classification: Primary: 76P05, 60G50; Secondary: 54C7.
Citation: Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic & Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
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