Advertisement

Journal of Nonlinear Science

, Volume 20, Issue 1, pp 1–46 | Cite as

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

  • Karsten Matthies
  • Florian Theil
Article

Abstract

This paper introduces a new method to show the validity of a continuum description for the deterministic dynamics of many interacting particles. Here the many-particle evolution is analyzed for a hard sphere flow with the addition that after a collision the collided particles are removed from the system. We consider random initial configurations which are drawn from a Poisson point process with spatially homogeneous velocity density f 0(v). Assuming that the moments of order less than three of f 0 are finite and no mass is concentrated on lines, the homogeneous Boltzmann equation without gain term is derived for arbitrary long times in the Boltzmann–Grad scaling. A key element is a characterization of the many-particle flow by a hierarchy of trees which encode the possible collisions. The occurring trees are shown to have favorable properties with a high probability, allowing us to restrict the analysis to a finite number of interacting particles and enabling us to extract a single-body distribution. A counter-example is given for a concentrated initial density f 0 even to short-term validity.

Keywords

Boltzmann equation Boltzmann–Grad limit Validity Kinetic annihilation Deterministic dynamics Random initial data 

Mathematics Subject Classification (2000)

82C40 76P05 82C22 60K35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boldrighini, C., Bunimovich, L.A., Sinai, Y.G.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32, 477–501 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  2. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, Berlin (1994) zbMATHGoogle Scholar
  3. Coppex, F., Droz, M., Piasecki, J., Trizac, E., Wittwer, P.: Some exact results for Boltzmann’s annihilation dynamics. Phys. Rev. E 79, 21103 (2003) Google Scholar
  4. DiPerna, R., Lions, P.L.: On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130, 321–366 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  5. Droz, M., Frachebourg, L., Piasecki, J., Rey, P.-A.: Ballistic annihilation kinetics for a multivelocity one-dimensional ideal gas. Phys. Rev. E 51, 5541–5548 (1995) Google Scholar
  6. Durrett, R.: Probability: Theory and Examples, 3rd edn. Duxbury, N. Scituate (2004) Google Scholar
  7. Elskens, Y., Frisch, H.: Annihilation kinetics in the one-dimensional ideal gas. Phys. Rev. A 31, 3812–3816 (1985) CrossRefGoogle Scholar
  8. Gallavotti, G.: Rigorous theory of Boltzmann equation in the Lorentz gas. Preprint Nota interna, vol. 358, Univ. di Roma (1970) Google Scholar
  9. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. (N.S.) 37, 407–436 (2000). Reprinted from Bull. Am. Math. Soc. 8, 437–479 (1902) MathSciNetzbMATHCrossRefGoogle Scholar
  10. Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for two- and three-dimensional gas in vacuum. Erratum and improved result. Commun. Math. Phys. 121, 143–146 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  11. Illner, R., Shinbrot, M.: Blow-up of solutions of the gain-term only Boltzmann equation. Math. Methods Appl. Sci. 9, 251–259 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  12. Kallenberg, O.: Probabilistic Symmetries and Invariance Principles. Probability and its Applications. Springer, Berlin (2005) zbMATHGoogle Scholar
  13. Krug, J., Spohn, H.: Universality classes for deterministic surface growth. Phys. Rev. A 38, 4271–4283 (1988) MathSciNetCrossRefGoogle Scholar
  14. Lanford, O.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol. 38, pp. 1–111. Springer, Berlin (1975) CrossRefGoogle Scholar
  15. Lang, R., Nguyen, X.: Smoluchowski’s theory of coagulation holds rigorously in the Boltzmann–Grad limit. Z. Wahrs. Verw. Geb. 54, 227–280 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  16. Matthies, K., Theil, F.: Validity and non-validity of propagation of chaos. In: Mörters, P., et al. (eds.) Analysis and Stochastics of Growth Processes, pp. 101–119. Oxford University Press, London (2008) CrossRefGoogle Scholar
  17. Piasecki, J.: Ballistic annihilation in a one-dimensional fluid. Phys. Rev. E 51, 5535–5540 (1995) CrossRefGoogle Scholar
  18. Piasecki, J., Trizac, E., Droz, M.: Dynamics of ballistic annihilation. Phys. Rev. E 65, 66111 (2002) CrossRefGoogle Scholar
  19. Spohn, H.: The Lorentz process converges to a random flight process. Commun. Math. Phys. 60, 277–290 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Berlin (1991) zbMATHGoogle Scholar
  21. Sznitman, A.: Topics in the propagation of chaos. In: Hennequin, P. (ed.) Ecole d’Eté de Probabilités de Saint-Flour 1989. Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991) CrossRefGoogle Scholar
  22. Ushiyama, K.: On the Boltzmann–Grad limit for the Broadwell model of the Boltzmann equation. J. Stat. Phys. 52, 331–355 (1988) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations