Entropy and chaos in the Kac model

Kinetic and Related Models

Volumn 3, Issue 1, 2010, Pages 85-122

  Carlen, Eric A ^a   Carvalho, Maria C ^b   Le Roux, Jonathan ^c   Loss, Michael ^d   Villani, Cédric ^b  

^a RUTGERS UNIVERSITY   (United States)

^b UNIVERSITY OF LISBON   (Portugal)

^c UNIVERSITY OF TOKYO   (Japan)

^d GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Entropy; Propagation of chaos

Indexed keywords

EID: 79953283102     PISSN: 19375093     EISSN: 19375077     Source Type: Journal    
DOI: 10.3934/krm.2010.3.85     Document Type: Article

References (26)

1. C. Bardos, F. Golse and D. Levermore, Fluid dynamical limits of kinetic equations, I: Formal derivation, J. Statist. Phys., 63 (1991), 323-344.
2. F. Barthe, D. Cordero-Erausquin and B. Maurey, Entropy of spherical marginals and related inequalities, J. Math. Pures Appl., 86 (2006), 89-99.
3. G. Ben Arous and O. Zeitouni, Increasing propagation of chaos for mean field models, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 85-102.
4. A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Statist. Phys., 94 (1999), 603-618.
5. P. Caputo, On the spectral gap of the Kac walk and other binary collision processes, ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 205-222.
6. T. Carleman, Sur la solution de l'équation intégrodifférentielle de Boltzmann, Acta. Math., 60 (1933), 91-146.
7. E. A. Carlen, The rate of local equilibration in kinetic theory, Contemporary Mathematics, 437 (2007), 71-88.
8. E. A. Carlen, M. C. Carvalho and M. Loss, Many-body aspects of approach to equilibrium, In "Journées Équations aux Dérivées Partielles," N. D. et. al., Ed. Univ. of Nantes, Nantes, 2000, 165-185.
9. E. A. Carlen, M. Carvalho and M. Loss, Determination of the spectral gap for Kac's master equation and related stochastic evolution, Acta Math., 191 (2003), 1-54.
10. E. A. Carlen, E. Lieb and M. Loss, A sharp analog of Young's inequality on SN and related entropy inequalities, Jour. Geom. Analysis, 14 (2004), 487-520.
11. E. A. Carlen, J. Gernimo and M. Loss, Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions, To appear in SIAM Jour. on Math. Analysis, 2008
12. E. A. Carlen and X. Lu, Fast and slow convergence to equilibrium Maxwellian molecules via wild sums, Jour. Stat. Phys., 112 (2003), 59-134.
13. M. D. Donsker and S. R. S. Varadhan, Assymptotic evaluation of certain Markov processes for large time, I, Comm. Pure Appl. Math., 28 (1975), 1-47.
14. W. Feller, "An Introduction to Probability Theory and its Applications. Vol. II," Second edition. John Wiley & Sons Inc., New York, 1971.
15. D. Foata and A. Fuchs, "Calcul des Probabilités," Dunod, Paris, 1986.
16. F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.
17. E. Janvresse, Spectral gap for Kac's model of Boltzmann equation, Ann. Probab., 29 (2001), 288-304.
18. M. Kac, Foundations of kinetic theory, Proc. 3rd Berkeley Symp. Math. Stat. Prob., J. Ney-man, ed. Univ. of California, 3 (1956), 171-197.
19. E. Kosygina, The behavior of the specific entropy in the hydrodynamic scaling limit, Ann. Probab., 29 (2001), 1086-1110.
20. J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 169 (2009), 903-991.
21. P. Major, Series of problems in probability theory. The central limit theorem and Fourier analysis, I, Available online at www.renyi.hu/~major/probability/central.html.
22. D. Maslen, The eigenvalues of Kac's master equation, Math. Zeit., 243 (2003), 291-331.
23. F. G. Mehler, Ueber die Entwicklung einer function von beliebig vielen variablen nach Laplaschen functionen höherer Ordnungn, Crelle's Journal, 66 (1866), 161-176.
24. A.-S. Sznitman, Topics in propagation of chaos, In "École d'Été de Probabilités de Saint-Flour XIX-1989," Springer, Berlin, 1991, 165-251.
25. C. Villani, A review of mathematical topics in collisional kinetic theory, In "Handbook of Mathematical Fluid Dynamics, Vol. I," North-Holland, Amsterdam, 2002, 71-305.
26. C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.