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


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.


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

Mathematics Subject Classification (2000)

82C40 76P05 82C22 60K35 


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  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

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