In this note I present the main results about the quantitative and qualitative propagation of chaos for the Boltzmann-Kac system obtained in collaboration with C. Mouhot in [33] which gives a possible answer to some questions formulated by Kac in [25]. We also present some related recent results about Kac’s chaos and Kac’s program obtained in [34, 23, 13] by K. Carrapatoso, M. Hauray, C. Mouhot, B. Wennberg and myself.
@article{SLSEDP_2012-2013____A22_0, author = {St\'ephane Mischler}, title = {Kac{\textquoteright}s chaos and {Kac{\textquoteright}s} program}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:22}, pages = {1--17}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2012-2013}, doi = {10.5802/slsedp.48}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.48/} }
TY - JOUR AU - Stéphane Mischler TI - Kac’s chaos and Kac’s program JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:22 PY - 2012-2013 SP - 1 EP - 17 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.48/ DO - 10.5802/slsedp.48 LA - en ID - SLSEDP_2012-2013____A22_0 ER -
%0 Journal Article %A Stéphane Mischler %T Kac’s chaos and Kac’s program %J Séminaire Laurent Schwartz — EDP et applications %Z talk:22 %D 2012-2013 %P 1-17 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.48/ %R 10.5802/slsedp.48 %G en %F SLSEDP_2012-2013____A22_0
Stéphane Mischler. Kac’s chaos and Kac’s program. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Talk no. 22, 17 p. doi : 10.5802/slsedp.48. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.48/
[1] Arkeryd, L., Caprino, S., and Ianiro, N. The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation. J. Statist. Phys. 63, 1-2 (1991), 345–361. | MR
[2] Bodineau, T., Gallagher, I., and Saint-Raymond, L. The brownian motion as the limit of a deterministic system of hard-spheres. preprint (2013). | arXiv
[3] Boissard, E., and Le Gouic, T. On the mean speed of convergence of empirical and occupation measures in wassserstein distance. | arXiv
[4] Bolley, F., Guillin, A., and Malrieu, F. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. M2AN Math. Model. Numer. Anal. 44, 5 (2010), 867–884. | Numdam | MR | Zbl
[5] Boltzmann, L. Weitere studien über das wärmegleichgewicht unter gasmolekülen. Sitzungsberichte der Akademie der Wissenschaften 66 (1872), 275–370. Translation: Further studies on the thermal equilibrium of gas molecules, in Kinetic Theory 2, 88âÃì174, Ed. S.G. Brush, Pergamon, Oxford (1966).
[6] Carleman, T. Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60, 1 (1933), 91–146. | MR
[7] Carlen, E., Carvalho, M. C., and Loss, M. Spectral gap for the Kac model with hard collisions. | arXiv
[8] Carlen, E. A., Carvalho, M. C., Le Roux, J., Loss, M., and Villani, C. Entropy and chaos in the Kac model. Kinet. Relat. Models 3, 1 (2010), 85–122. | MR | Zbl
[9] Carlen, E. A., Carvalho, M. C., and Loss, M. Determination of the spectral gap for Kac’s master equation and related stochastic evolution. Acta Math. 191, 1 (2003), 1–54. | MR | Zbl
[10] Carlen, E. A., Gabetta, E., and Toscani, G. Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm. Math. Phys. 199, 3 (1999), 521–546. | MR | Zbl
[11] Carlen, E. A., Geronimo, J. S., and Loss, M. Determination of the spectral gap in the Kac model for physical momentum and energy-conserving collisions. SIAM J. Math. Anal. 40, 1 (2008), 327–364. | MR | Zbl
[12] Carrapatoso, K. Propagation of chaos for the spatially homogeneous Landau equation for maxwellian molecules. | HAL
[13] Carrapatoso, K. Quantitative and qualitative Kac’s chaos on the Boltzmann sphere. | HAL
[14] Fournier, N., Hauray, M., and Mischler, S. Propagation of chaos for the 2d viscous vortex model. To appear in J. Eur. Math. Soc.
[15] Fournier, N., and Méléard, S. Monte Carlo approximations and fluctuations for 2d Boltzmann equations without cutoff. Markov Process. Related Fields 7 (2001), 159–191. | MR | Zbl
[16] Fournier, N., and Méléard, S. A stochastic particle numerical method for 3d Boltzmann equation without cutoff. Math. Comp. 71 (2002), 583–604. | MR | Zbl
[17] Fournier, N., and Mischler, S. Rate of convergence of the Nanbu particle system for hard potentials. | HAL
[18] Fournier, N., and Mouhot, C. On the well-posedness of the spatially homogeneous boltzmann equation with a moderate angular singularity. Comm. Math. Phys. 283, 3 (2009), 803–824. | MR | Zbl
[19] Grad, H. Principles of the kinetic theory of gases. In Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase. Springer-Verlag, Berlin, 1958, pp. 205–294. | MR
[20] Graham, C., and Méléard, S. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. The Annals of Probability 25 (1997), 115–132. | MR | Zbl
[21] Grünbaum, F. A. Propagation of chaos for the Boltzmann equation. Arch. Rational Mech. Anal. 42 (1971), 323–345. | MR | Zbl
[22] Hauray, M. Fisher information decay for the Boltzman-Kac system associtaed to maxwell molecules. Personnal communication.
[23] Hauray, M., and Mischler, S. On Kac’s chaos and related problems, work in progress.
[24] Janvresse, E. Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29, 1 (2001), 288–304. | MR | Zbl
[25] Kac, M. Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III (Berkeley and Los Angeles, 1956), University of California Press, pp. 171–197. | MR | Zbl
[26] Kolokoltsov, V. N. Nonlinear Markov processes and kinetic equations, vol. 182 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2010. | MR | Zbl
[27] Lanford, III, O. E. Time evolution of large classical systems. In Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974). Springer, Berlin, 1975, pp. 1–111. Lecture Notes in Phys., Vol. 38. | MR | Zbl
[28] Lu, X., and Mouhot, C. On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium. | arXiv
[29] Maslen, D. K. The eigenvalues of Kac’s master equation. Math. Z. 243, 2 (2003), 291–331. | MR | Zbl
[30] Maxwell, J. C. On the dynamical theory of gases. Philos. Trans. Roy. Soc. London Ser. A 157 (1867), 49–88.
[31] McKean, H. P. The central limit theorem for Carleman’s equation. Israel J. Math. 21, 1 (1975), 54–92. | MR | Zbl
[32] McKean, Jr., H. P. An exponential formula for solving Boltmann’s equation for a Maxwellian gas. J. Combinatorial Theory 2 (1967), 358–382. | MR | Zbl
[33] Mischler, S., and Mouhot, C. Kac’s program in kinetic theory. Invent. Math. 193, 1 (2013), 1–147. | MR | Zbl
[34] Mischler, S., Mouhot, C., and Wennberg, B. A new approach to quantitative chaos propagation for drift, diffusion and jump processes. To appear in Probab. Theory Related Fields, . | HAL
[35] Otto, F., and Villani, C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 2 (2000), 361–400. | MR | Zbl
[36] Peyre, R. Some ideas about quantitative convergence of collision models to their mean field limit. J. Stat. Phys. 136, 6 (2009), 1105–1130. | MR | Zbl
[37] Sznitman, A.-S. Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66, 4 (1984), 559–592. | MR | Zbl
[38] Sznitman, A.-S. Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989, vol. 1464 of Lecture Notes in Math. Springer, Berlin, 1991, pp. 165–251. | MR | Zbl
[39] Tanaka, H. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46, 1 (1978/79), 67–105. | MR | Zbl
[40] Tanaka, H. Some probabilistic problems in the spatially homogeneous Boltzmann equation. In Theory and application of random fields (Bangalore, 1982), vol. 49 of Lecture Notes in Control and Inform. Sci. Springer, Berlin, 1983, pp. 258–267. | MR | Zbl
[41] Toscani, G., and Villani, C. Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys. 94, 3-4 (1999), 619–637. | MR | Zbl
[42] Villani, C. Fisher information estimates for Boltzmann’s collision operator. J. Math. Pures Appl. (9) 77, 8 (1998), 821–837. | MR | Zbl
[43] Villani, C. Cercignani’s conjecture is sometimes true and always almost true. Comm. Math. Phys. 234, 3 (2003), 455–490. | MR | Zbl
Cited by Sources: