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2024 | OriginalPaper | Chapter

Mean Field Limit for the Kac Model and Grand Canonical Formalism

Authors : Thierry Paul, Mario Pulvirenti, Sergio Simonella

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

We consider the classical Kac’s model for the approximation of the Boltzmann equation, and study the correlation error measuring the defect of propagation of chaos in the mean field limit. This contribution is inspired by a recent paper of the same authors [23] where a large class of models, including quantum systems, are considered. Here we outline the main ideas in the context of grand canonical measures, for which both the evolution equations and the proof simplify.

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Aoki, K., Pulvirenti, M., Simonella, S., Tsuji, T.: Backward clusters, hierarchy and wild sums for a hard sphere system in a low-density regime. Math. Mod. and Meth. in App. Sci. 25(5), 995–1010 (2015)

Bogolyubov, N.N.: Problems of a Dynamical Theory in Statistical Physics. Moscow: State Technical Press (1946) in Russian; English translation in Studies in Statistical Mechanics I, edited by J. de Boer and G. E. Uhlenbeck, part A, Amsterdam: North-Holland (1962)

Boldrighini, C., De Masi, A., Pellegrinotti, A.: Nonequilibrium fluctuations in particle systems modelling reaction-diffusion equations. Stoch. Process. Appl. 42, 1–30 (1992)

Caprino, S., Pulvirenti, M.: A cluster expansion approach to a one-dimensional Boltzmann equation: a validity result. Commun. Math. Phys. 166(3), 603–631 (1995)MathSciNetCrossRef

Caprino, S., De Masi, A., Presutti, E., Pulvirenti, M.: A derivation of the Broadwell equation. Commun. Math. Phys. 135(3), 443–465 (1991)MathSciNetCrossRef

Caprino, S., Pulvirenti, M., Wagner, W.: A particle systems approximating stationary solutions to the Boltzmann equation SIAM. J. Math. Anal. 4, 913–934 (1998)

Carlen, E., Carvalho, M.-C., Gabetta, E.: Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. CPAM 53(3), 370–397 (2000)MathSciNet

Carlen, E., Carvalho, M.-C., Gabetta, E.: On the relation between rates of relaxation and convergence of wild sums for solutions of the Kac equation. J. Funct. Anal. 220(2), 362–387 (2005)MathSciNetCrossRef

De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity 7, 633–696 (1994)

10.

De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamical Limits. In: Lecture Notes in Mathematics, vol. 1501. Springer-Verlag (1991)

11.

Graham, C., Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25, 115–132 (1997)MathSciNetCrossRef

12.

Grünbaum, F.A.: Propagation of chaos for the Boltzmann equation. Arch. Rat. Mech. Anal. 42, 323–345 (1971)MathSciNetCrossRef

13.

Hauray, M., Mischler, S.: On Ka’s chaos and related problems. J. Funct. Anal. 266, 6055–6157 (2014)MathSciNetCrossRef

14.

Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley and Los Angeles (1956)

15.

Kac, M.: Probability and Related Topics in Physical Sciences. Interscience, London-New York (1959)

16.

Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with Kac potentials. II. Fluctuations. Nonlinearity 9, 27–51 (1996)

17.

Masi, A., Presutti, E., Tsagkarogiannis, D., Vares, M.E.: Truncated correlations in the stirring process with births and deaths. Electron. J. Probab. 17, 1–35 (2012)MathSciNet

18.

Mischler, S., Mouhot, C.: Kac’s program in kinetic theory. Inven. Math. 193, 1–147 (2013)MathSciNetCrossRef

19.

Mischler, S., Mouhot, C., Wennberg, B.: A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. Probab. Theory Relat. Fields 161(1–2), 1–59 (2015)MathSciNetCrossRef

20.

Nota, A., Velázquez, J.J.L., Winter, R.: Interacting particle systems with long-range interactions: scaling limits and kinetic equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 32(2), 335–377 (2021)

21.

A. Nota, J.J.L. Velázquez, Winter, R.: Interacting particle systems with long-range interactions: approximation by tagged particles in random fields. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33(2), 439–506 (2022)

22.

Paul, T., Pulvirenti, M.: Asymptotic expansion of the mean-field approximation. Discret. Contin. Dyn. Syst. A 39, 1891–1921 (2019)MathSciNetCrossRef

23.

Paul, T., Pulvirenti, M., Simonella, S.: On the size of chaos in the mean field dynamics. Arch. Rat. Mech. Anal. 231, 285–317 (2019)MathSciNetCrossRef

24.

Pulvirenti, M., Simonella, S.: A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory. In: Albi, G., Merino-Aceituno, S., Nota, A., Zanella, M. (eds.) Trails in Kinetic Theory. SEMA SIMAI Springer Series, vol. 25, pp. 183–208 (2021)

25.

Pulvirenti, M., Simonella, S.: The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error. Inven. Math. 207(3), 1135–1237 (2017)MathSciNetCrossRef

26.

Spohn, H.: Large Scale Dynamics of Interacting Particles. In: Texts and Monographs in Physics. Springer-Verlag, Heidelberg (1991)

27.

Sznitman, A.-S.: Topics in propagation of chaos. In: École d’été de Probabilités de Saint-Flour XIX 1989. Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991)

Title: Mean Field Limit for the Kac Model and Grand Canonical Formalism
Authors: Thierry Paul
Mario Pulvirenti
Sergio Simonella
Publisher: Springer International Publishing
Book: From Particle Systems to Partial Differential Equations
Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-65195-3_12

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