zbMATH — the first resource for mathematics

Entropy decay for the Kac evolution. (English) Zbl 1405.35126
Summary: We consider solutions to the Kac master equation for initial conditions where \(N\) particles are in a thermal equilibrium and \(M \leq N\) particles are out of equilibrium. We show that such solutions have exponential decay in entropy relative to the thermal state. More precisely, the decay is exponential in time with an explicit rate that is essentially independent on the particle number. This is in marked contrast to previous results which show that the entropy production for arbitrary initial conditions is inversely proportional to the particle number. The proof relies on Nelson’s hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections.
35Q20 Boltzmann equations
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
Full Text: DOI
[1] Ball, K.: Volumes of sections of cubes and related problems. In: Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 1987-88, Volume 1376 of the Series Lecture Notes in Mathematics, pp 251-260. Springer, Berlin (1989)
[2] Ball, K., Volume ratios and a reverse isoperimetric inequality, J. Lond. Math. Soc. (Second Series), 44, 351-359, (1991) · Zbl 0694.46010
[3] Barthe, F., On a reverse form of the Brascamp-Lieb inequality, Inventiones Mathematicae, 134, 335-361, (1998) · Zbl 0901.26010
[4] Barthe, F., A Continuous Version of the Brascamp-Lieb Inequalities, 53-63, (2004), Berlin, Heidelberg · Zbl 1084.26011
[5] Bennett, J.; Carbery, A.; Christ, M.; Tao, T., The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal., 17, 1343-1415, (2008) · Zbl 1132.26006
[6] Bonetto, F.; Loss, M.; Tossounian, H.; Vaidyanathan, R., Uniform approximation of a Maxwellian thermostat by finite reservoirs, Commun. Math. Phys., 351, 311-339, (2017) · Zbl 1371.82064
[7] Bonetto, F.; Loss, M.; Vaidyanathan, R., The Kac model coupled to a thermostat, J. Stat. Phys., 156, 647-667, (2014) · Zbl 1302.35290
[8] Brascamp, H. J.; Lieb, E. H., Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math., 20, 151-173, (1976) · Zbl 0339.26020
[9] Carlen, E.A., Carvalho, M.C., Loss, M.: Many-body aspects of approach to equilibrium. In: Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), Exp. No. XI, 12 pp., Université de Nantes, Nantes, (2000)
[10] Carlen, E. A.; Carvalho, M. C.; Loss, M., Determination of the spectral gap for Kac’s master equation and related stochastic evolution, Acta Mathematica, 191, 1-54, (2003) · Zbl 1080.60091
[11] Carlen, E. A.; Cordero-Erausquin, D., Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities, Geom. Funct. Anal., 19, 373-405, (2009) · Zbl 1231.26015
[12] Carlen, E. A.; Loss, M., Extremals of functionals with competing symmetries, J. Funct. Anal., 88, 437-456, (1990) · Zbl 0705.46016
[13] Carlen, E. A.; Lieb, E. H.; Loss, M., A sharp analog of Young’s inequality on \(S\)\(N\) and related entropy inequalities, J. Geom. Anal., 14, 487-520, (2004) · Zbl 1056.43002
[14] Einav, A., On Villani’s conjecture concerning entropy production for the Kac master equation, Kinet. Relat. Models, 4, 479-497, (2011) · Zbl 1335.82023
[15] Federbush, P., Partially alternate derivation of a result of Nelson, J. Math. Phys., 10, 50-52, (1969) · Zbl 0165.58301
[16] Gabetta, E.; Toscani, G.; Wennberg, B., Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Stat. Phys., 81, 901-934, (1995) · Zbl 1081.82616
[17] Gross, L., Logarithmic Sobolev inequalities, Am. J. Math., 97, 1061-1083, (1975) · Zbl 0318.46049
[18] Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. In Dirichlet Forms (Varenna, 1992), Volume 1563 of the series Lecture Notes in Mathematics, pp 54-88. Springer Berlin Heidelberg, (1993) · Zbl 0812.47037
[19] Han, T. S., Nonnegative entropy measures of multivariate symmetric correlations, Inf. Control., 36, 133-156, (1978) · Zbl 0367.94041
[20] Janvresse, E., Spectral gap for Kac’s model of Boltzmann equation, Ann. Probab., 29, 288-304, (2001) · Zbl 1034.82049
[21] Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, pp. 171-197. University of California Press, Berkeley (1956)
[22] Lieb, E. H., Gaussian kernels have only Gaussian maximizers, Inventiones Mathematicae, 102, 179-208, (1990) · Zbl 0726.42005
[23] Loomis, L. H.; Whitney, H., An inequality related to the isoperimetric inequality, Bull. Am. Math. Soc., 55, 961-962, (1949) · Zbl 0035.38302
[24] Maslen, D. K., The eigenvalues of Kac’s master equation., Mathematische Zeitschrift, 243, 291-331, (2003) · Zbl 1016.82016
[25] Mischler, S.; Mouhot, C., About Kac’s program in kinetic theory, Comptes Rendus Mathématique (Académie Des Sciences Paris), 349, 1245-1250, (2011) · Zbl 1233.35153
[26] Mischler, S.; Mouhot, C., Kac’s program in kinetic theory, Inventiones Mathematicae, 193, 1-147, (2013) · Zbl 1274.82048
[27] Nelson, E., The free Markoff field, J. Funct. Anal., 12, 211-227, (1973) · Zbl 0273.60079
[28] Tossounian, H.; Vaidyanathan, R., Partially thermostated Kac model, J. Math. Phys., 56, 083301, (2015) · Zbl 1330.82036
[29] Villani, C., Cercignani’s conjecture is sometimes true and always almost true, Commun. Math. Phys., 234, 455-490, (2003) · Zbl 1041.82018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.