Chaos and entropic chaos in Kac's model without high moments

Kleber Carrapatoso (Université Paris-Dauphine)
Amit Einav (University of Cambridge)


In this paper we present a new local Lévy Central Limit Theorem, showing convergence to stable states that are not necessarily the Gaussian, and use it to find new and intuitive entropically chaotic families with underlying one-particle function that has moments of order $2\alpha$, with $1<\alpha<2$. We also discuss a lower semi continuity result for the relative entropy with respect to our specific family of functions, and use it to show a form of stability property for entropic chaos in our settings.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-38

Publication Date: August 27, 2013

DOI: 10.1214/EJP.v18-2683


  • Bassetti, Federico; Gabetta, Ester. Survey on probabilistic methods for the study of Kac-like equations. Boll. Unione Mat. Ital. (9) 4 (2011), no. 2, 187--212. MR2840602
  • Bassetti, Federico; Ladelli, Lucia; Matthes, Daniel. Central limit theorem for a class of one-dimensional kinetic equations. Probab. Theory Related Fields 150 (2011), no. 1-2, 77--109. MR2800905
  • Barthe, F.; Cordero-Erausquin, D.; Maurey, B. Entropy of spherical marginals and related inequalities. J. Math. Pures Appl. (9) 86 (2006), no. 2, 89--99. MR2247452
  • Ben Arous, G.; Zeitouni, O. Increasing propagation of chaos for mean field models. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), no. 1, 85--102. MR1669916
  • Carlen, E.; Carvalho, M. C.; Loss, M. Many-body aspects of approach to equilibrium. Séminaire: Équations aux Dérivées Partielles, 2000–2001, Exp. No. XIX, 12 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2001. MR1860691
  • Carlen, E. A.; Carvalho, M. C.; Loss, M. Determination of the spectral gap for Kac's master equation and related stochastic evolution. Acta Math. 191 (2003), no. 1, 1--54. MR2020418
  • Carlen, Eric A.; Carvalho, Maria C.; Le Roux, Jonathan; Loss, Michael; Villani, Cédric. Entropy and chaos in the Kac model. Kinet. Relat. Models 3 (2010), no. 1, 85--122. MR2580955
  • Carlen, Eric A.; Geronimo, Jeffrey S.; Loss, Michael. Determination of the spectral gap in the Kac model for physical momentum and energy-conserving collisions. SIAM J. Math. Anal. 40 (2008), no. 1, 327--364. MR2403324
  • Carrapatoso K.: Quantitative and Qualitative Kac's Chaos on the Boltzmann Sphere. ARXIV
  • Cercignani, C. $H$-theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech. (Arch. Mech. Stos.) 34 (1982), no. 3, 231--241 (1983). MR0715658
  • Dolera, Emanuele; Gabetta, Ester; Regazzini, Eugenio. Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem. Ann. Appl. Probab. 19 (2009), no. 1, 186--209. MR2498676
  • Einav, Amit. On Villani's conjecture concerning entropy production for the Kac master equation. Kinet. Relat. Models 4 (2011), no. 2, 479--497. MR2786394
  • Einav, Amit. A counter example to Cercignani's conjecture for the $d$ dimensional Kac model. J. Stat. Phys. 148 (2012), no. 6, 1076--1103. MR2975524
  • Einav A.: A Few Ways to Destroy Entropic Chaoticity on Kac's Sphere. phTo appear in Comm. Math. Sci. ARXIV
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp. MR0270403
  • Goudon, Thierry; Junca, Stéphane; Toscani, Giuseppe. Fourier-based distances and Berry-Esseen like inequalities for smooth densities. Monatsh. Math. 135 (2002), no. 2, 115--136. MR1894092
  • Hauray M. and Mischler S.: On Kac's Chaos and Related Problems. HAL:
  • Janvresse, Elise. Spectral gap for Kac's model of Boltzmann equation. Ann. Probab. 29 (2001), no. 1, 288--304. MR1825150
  • Kac, M. 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. MR0084985
  • Lanford, Oscar E., III. Time evolution of large classical systems. Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1--111. Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975. MR0479206
  • Lott, John; Villani, Cédric. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009), no. 3, 903--991. MR2480619
  • Maslen, David K. The eigenvalues of Kac's master equation. Math. Z. 243 (2003), no. 2, 291--331. MR1961868
  • Matthes, Daniel; Toscani, Giuseppe. Propagation of Sobolev regularity for a class of random kinetic models on the real line. Nonlinearity 23 (2010), no. 9, 2081--2100. MR2672637
  • McKean, H. P., Jr. An exponential formula for solving Boltmann's equation for a Maxwellian gas. J. Combinatorial Theory 2 1967 358--382. MR0224348
  • Mischler S. and Mouhot C.: Kac's Program in Kinetic Theory. phTo appear in Inventiones Mathematicae. ARXIV
  • Pinsker, M. S. Information and information stability of random variables and processes. Translated and edited by Amiel Feinstein Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1964 xii+243 pp. MR0213190
  • Sznitman, Alain-Sol. Topics in propagation of chaos. École d'Été de Probabilités de Saint-Flour XIX—1989, 165--251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. MR1108185
  • Villani, Cédric. A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, 71--305, North-Holland, Amsterdam, 2002. MR1942465
  • Villani, Cédric. Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234 (2003), no. 3, 455--490. MR1964379
  • Villani, Cédric. Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. ISBN: 978-3-540-71049-3 MR2459454

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.