Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View

Regazzini, Eugenio

Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 175-198 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Résumé

Let $f(\cdot,t)$ be the probability density function representing the solution of Kac's Boltzmann-like equation at time $t$ , with initial data $f_{0}$ , and let $g_{\sigma}$ be the Gaussian density with zero mean and variance $\sigma^{2}$ , $\sigma^{2}$ being the value of the second moment of $f_{0}$ . Henry McKean Jr. put forward the conjecture that the total variation distance between $f(\cdot,t)$ and $g_{\sigma}$ goes to zero, as $t\to+\infty$ , with an exponential rate equal to $-1/4$ . This lecture aims at explaining the main efforts made to a view to validating this conjecture, and concludes with the theorem stating that this holds true whenever $f_{0}$ has finite fourth moment and its Fourier transform $\varphi_{0}$ satisfies $|\varphi_{0}(\xi)|=o(|\xi|^{-p})$ as $|\xi|\to+\infty$ , for some $p>0$ . The first part of the lecture expounds the derivation of the Kac Boltzmann-like equation from the Kac master equation. A detailed description of the probabilistic methods resorted to prove the above-mentioned theorem is then given. The final part mentions further applications of these methods to other kinetic models.

Publié le : 2009-02-01

MR 2493650 | Zbl 1177.82093

@article{BUMI_2009_9_2_1_175_0,
     author = {Eugenio Regazzini},
     title = {Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {2},
     year = {2009},
     pages = {175-198},
     zbl = {1177.82093},
     mrnumber = {2493650},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_1_175_0}
}

Regazzini, Eugenio. Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 175-198. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_1_175_0/

Bibliographie

[1] Bassetti, F. - Ladelli, L. - Regazzini, E., Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model. To appear in J. Stat Phys. | MR 2456941 | Zbl 1161.82337

[2] Beurling, A., Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle. In 9th Congr. Math: Scandinaves (Helsinki, 1938), 199-210. Tryekeri, Helsinki (1939). | Zbl 65.0483.02

[3] Billingsley, P., Probability and Measure, 3rd ed. Wiley, New York (1995). | MR 1324786

[4] Billingsley, P., Convergence of Probability Measures, 2nd ed. Wiley, New York (1999). | MR 1700749 | Zbl 0944.60003

[5] Carlen, E. A. - Carvalho, M. C., Probabilistic methods in kinetic theory. Riv. Mat. Univ. Parma, 7 (2003), 101-149. | MR 2052787 | Zbl 1140.82324

[6] Carlen, E. A. - Carvalho, M. C. - Gabetta, E., On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. Journal of Functional Analysis, 220 (2005), 362-387. | MR 2119283 | Zbl 1108.82036

[7] Carlen, E. A. - Carvalho, M. C. - Loss, M., Determination of the spectral gap for Kac's master equation and related stochastic evolution. Acta Mat., 191 (2003), 1-54. | MR 2020418 | Zbl 1080.60091

[8] Carlen, E. A. - Gabetta, E. - Regazzini, E., Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution. J. Appl. Prob. 45 (2008), 95-106. | MR 2409313 | Zbl 1142.60013

[9] Carlen, E. A. - Gabetta, E. - Toscani, G., Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm. Math. Phys. 305 (1999), 521-546. | MR 1669689 | Zbl 0927.76088

[10] Chen, L. H. Y. - Shao, Q., A non-uniform Berry-Esseen bound via Stein's method. Probab. Theory Related Fields, 120 (2001), 236-254. | MR 1841329 | Zbl 0996.60029

[11] Y. S CHOW - Teicher, H., Probability Theory. Independence, Interchangeability, Martingales, 3rd edition. Springer Verlag, New York (1997). | MR 1476912

[12] Comtet, L., Analyse Combinatoire. Presses universitaires de France, Paris (1970).

[13] Diaconis, P. - Saloff-Coste, L., Bounds for Kac's master equation. Comm. Math. Phys., 209 (2000), 729-755. | MR 1743614 | Zbl 0953.60098

[14] Dolera, E., Applicazione del Teorema Centrale del Limite all'Analisi della Velocità di Convergenza dell'Equilibrio della Soluzione dell'Equazione di Kac, nella Metrica della Variazione Totale. Degree thesis (2006). Università degli Studi di Pavia, Dipartimento di Matematica.

[15] Dolera, E. - Gabetta, E. - Regazzini, E., Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem. To appear in Ann. Applied Probability. | MR 2498676 | Zbl 1163.60007

[16] Dudley, R. M., Real Analysis and Probability, revised reprint. Cambridge University Press, Cambridge (2002). | MR 1932358

[17] Feller, W., On the Berry-Esseen theorem. Z. Wahrsch. Verw. Gebiete, 10 (1968), 261-268. | MR 239639 | Zbl 0167.17304

[18] Fortini, S. - Ladelli, L. - Regazzini, E., A central limit problem for partially exchangeable random variables. Theory Probab. Appl., 41 (1996), 224-246. | MR 1445757 | Zbl 0881.60019

[19] Gabetta, E. - Regazzini, E., Some new results for McKean's graphs with applications to Kac's equation. J. Statist. Phys., 125 (2006), 947-974. | MR 2283786 | Zbl 1107.82046

[20] Gabetta, E. - Regazzini, E., Central limit theorem for the solution of the Kac equation. To appear in Ann. Applied Probability. | MR 2474538 | Zbl 1161.82018

[21] Gabetta, E. - Regazzini, E., Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics. Revised for Probab. Theory Related Fields. | MR 2574735 | Zbl 1181.60030

[22] Gini, C., Five papers in Atti del R. Ist. Veneto di Sc. Lettere ed Arti, 74 (1915), 185-213, 583-610, 1903-1942, 75 (1916) 309-331, 1419-1461.

[23] Janvresse, E., Spectral gap for Kac's model of Boltzmann equation. The Annals of Probability, 29 (2001), 288-304. | MR 1825150 | Zbl 1034.82049

[24] Kac, M., Foundations of kinetic theory. Proc. 3rd Berkeley Sympos. (J. Neyman, ed.) 3 (1956), 171-197. | MR 84985 | Zbl 0072.42802

[25] Kac, M., M. Probability and Related Topics in Physical Science. Wiley, New York (1959). | Zbl 0087.33003

[26] Lévy, P., Calcul des Probabilités. Gauthier-Villars, Paris (1925).

[27] Linnik, Y. V., An information-theoretic proof of the central limit theorem with the Lindeberg condition. Theory Prob. Appl., 4 (1959), 288-299. | MR 124081 | Zbl 0097.13103

[28] Matthes, D. - Toscani, G., On steady distributions of kinetic models of conservative economies. J. Stat. Phys., 130 (2008), 1087-1117. | MR 2379241 | Zbl 1138.91020

[29] Mckean, H. P., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rational Mech. Anal., 21 (1966), 343-367. | MR 214112 | Zbl 1302.60049

[30] Petrov, V. V., Sums of Independent Random Variables. Springer-Verlag, Berlin (1975). | MR 388499 | Zbl 0322.60042

[31] Petrov, V. V., Limit Theorems of Probability Theory. Clarendon Press, Oxford (1995). | MR 1353441 | Zbl 0826.60001

[32] Rachev, S. T., Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester (1991). | MR 1105086 | Zbl 0744.60004

[33] Stroock, D. W., Probability Theory, an Analytic View. Cambridge University Press, Cambridge (1993). | MR 1267569 | Zbl 0925.60004

[34] Tanaka, S., An extension of Wild's sum for solving certain nonlinear equations of measures. Proc. Japan Acad., 44 (1968), 884-889. | MR 237927 | Zbl 0177.44902

[35] Villani, C., Mathematics of granular materials. J. Statist. Phys.124 (2006), 781-822. | MR 2264625 | Zbl 1134.82040

[36] Villani, C., Entropy production and convergence to equilibrium. (Lectures from a Special Semester at the Centre Emile Borel, Inst. H. Poincaré, Paris 2001). Lecture Notes in Math., 1916 (2008), 1-70. | MR 2409050

[37] Wild, E., On Boltzmann's equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc., 47 (1951), 602-609. | MR 42999 | Zbl 0043.43703