The Annals of Probability

Stochastic particle approximations for generalized Boltzmann models and convergence estimates

Carl Graham and Sylvie Méléard

Full-text: Open access

Abstract

We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniques and a representation of the nonlinear process on a Boltzmann tree. No regularity nor uniqueness assumption is needed. We then consider a nonlinear equation with both Vlasov and Boltzmann terms and give a weak pathwise propagation of chaos result using a compactness-uniqueness method which necessitates some regularity. These results imply functional laws of large numbers and extend to multitype models. We give algorithms simulating or approximating the particle systems.

Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 115-132.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404281

Digital Object Identifier
doi:10.1214/aop/1024404281

Mathematical Reviews number (MathSciNet)
MR1428502

Zentralblatt MATH identifier
0873.60076

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F17: Functional limit theorems; invariance principles 47H15 65C05: Monte Carlo methods 76P05: Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05] 82C40: Kinetic theory of gases 82C80: Numerical methods (Monte Carlo, series resummation, etc.)

Keywords
Boltzmann equation nonlinear diffusion with jumps random graphs and tress coupling propagation of chaos Monte Carlo algorithms

Citation

Graham, Carl; Méléard, Sylvie. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25 (1997), no. 1, 115--132. doi:10.1214/aop/1024404281. https://projecteuclid.org/euclid.aop/1024404281


Export citation

References

  • 1 BABOVSKY, H. and ILLNER, R. 1994. A convergence proof for Nanbu's simulation method for the full Boltzmann equation. SIAM J. Numer. Anal. 26 45 65.
  • 2 CERCIGNANI, C. 1988. The Boltzmann Equation and Its Applications. Springer, New York.
  • 3 CHAUVIN, B. 1993. Branching processes, trees and the Boltzmann equation. Proceedings du Congres Probabilites Numeriques, INRIA. ´ ´
  • 4 DIPERNA, R. J. and LIONS, P. L. 1989. On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. Math. 130 321 366.
  • 5 GRAHAM, C. 1992. Nonlinear diffusion with jumps. Ann. Inst. H. Poincare 28 393 402. ´
  • 6 GRAHAM, C. 1992. McKean Vlasov Ito Skorohod equations, and nonlinear diffusions with discrete jump sets. Stochastic Process. Appl. 40 69 82.
  • 7 GRAHAM, C. and MELEARD, S. 1993. Propagation of chaos for a fully connected loss network ´ ´ with alternate routing. Stochastic Process. Appl. 44 159 180.
  • 8 GRAHAM, C. and MELEARD, S. 1994. Chaos hypothesis for a system interacting through ´ ´ shared resources. Probab. Theory Related Fields 100 157 173.
  • 9 ILLNER, R. and NEUNZERT, H. 1987. On simulation methods for the Boltzmann equation. Transport Theory Statist. Phys. 16 141 154.
  • 10 JOFFE, A. and METIVIER, M. 1986. Weak convergence of sequences of semimartingales with ´ applications to multitype branching processes. Adv. in Appl. Probab. 18 20 65.
  • 11 LANFORD, III, O. E. 1975. Time evolution of large classical systems. Lecture Notes in Phys. 38 1 111. Springer, Berlin.
  • 12 LEONARD, C. 1995. Large deviations for long range interacting particle systems with ´ jumps. Ann. Inst. H. Poincare.´
  • 13 NANBU, K. 1983. Interrelations between various direct simulation methods for solving the Boltzmann equation. J. Phys. Soc. Japan 52 3382 3388.
  • 14 NEUNZERT, H., GROPENGEISSER, F. and STRUCKMEIER, J. 1991. Computational methods forthe Boltzmann equation. In Applied and Industrial Mathematics R. Spigler, ed. 111 140. Kluwer, Dordrecht.
  • 15 PERTHAME, B. 1994. Introduction to the theory of random particle methods for Boltzmann equation. In Progresses on Kinetic Theory. World Scientific, Singapore.
  • 16 PERTHAME, B. and PULVIRENTI, M. 1996. On some large systems of random particles which approximate scalar conservation laws. Asympt. Anal. To appear.
  • 17 PULVIRENTI, M., WAGNER, W. and ZAVELANI ROSSI, M. B. 1993. Convergence of particle schemes for the Boltzmann equation. Preprint 49, Institut fur Angewandte Analysis und Stochastik, Berlin.
  • 18 SZNITMAN, A. S. 1984. Equations de type de Boltzmann, spatialement homogenes.Wahrsch. Verw. Gebeite 66 559 592. ´
  • 19 SZNITMAN, A. S. 1991. Propagation of chaos. Ecole d'Ete de Probabilites de Saint-Flour ´ ´1989 Lecture Notes in Math. 1464 165 251. Springer, Berlin.
  • 20 UCHIYAMA, K. 1988. Derivation of the Boltzmann equation from particle dynamics. Hiroshima Math. J. 18 245 297.
  • 21 WAGNER, W. 1992. A convergence proof for Bird's direct simulation method for the Boltzmann equation. J. Statist. Phys. 66 1011 1044.
  • 22 WAGNER, W. 1994. A functional law of large numbers for Boltzmann type stochastic particle systems. Preprint 93, Institut fur Angewandte Analysis und Stochastik, Berlin.
  • CMAP, ECOLE POLYTECHNIQUE UNIVERSITE PARIS 6 F-91128 PALAISEAU F-75231 PARIS FRANCE FRANCE E-MAIL: mata@cmapx.polytechnique.fr