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June  2011, 4(2): 479-497. doi: 10.3934/krm.2011.4.479

## On Villani's conjecture concerning entropy production for the Kac Master equation

 1 Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States

Received  September 2010 Revised  February 2011 Published  April 2011

In this paper we take an idea presented in recent paper by Carlen, Carvalho, Le Roux, Loss, and Villani ([3]) and push it one step forward to find an exact estimation on the entropy production. The new estimation essentially proves that Villani's conjecture is correct, or more precisely that a much worse bound to the entropy production is impossible in the general case.
Citation: Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic and Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479
##### References:
 [1] A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Statist. Phys., 94, (1999), 603-618. doi: 10.1023/A:1004537522686. [2] E. A. Carlen, M. C. Carvalho and M. Loss, Many-body aspects of approach to equilibrium, "Séminaire Equations aux Dérivées Partielles'' (La Chapelle sur Erdre, 2000), [3] E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122. doi: 10.3934/krm.2010.3.85. [4] E. Janvresse, Spectral gap for Kac's model of Boltzmann equation, Ann. Probab., 29 (2001), 288-304. doi: 10.1214/aop/1008956330. [5] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, pp. 171-197. University of California Press, Berkeley and Los Angeles, 1956. [6] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490. doi: 10.1007/s00220-002-0777-1.

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##### References:
 [1] A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Statist. Phys., 94, (1999), 603-618. doi: 10.1023/A:1004537522686. [2] E. A. Carlen, M. C. Carvalho and M. Loss, Many-body aspects of approach to equilibrium, "Séminaire Equations aux Dérivées Partielles'' (La Chapelle sur Erdre, 2000), [3] E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122. doi: 10.3934/krm.2010.3.85. [4] E. Janvresse, Spectral gap for Kac's model of Boltzmann equation, Ann. Probab., 29 (2001), 288-304. doi: 10.1214/aop/1008956330. [5] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, pp. 171-197. University of California Press, Berkeley and Los Angeles, 1956. [6] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490. doi: 10.1007/s00220-002-0777-1.
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