Journal of Statistical Physics

, Volume 32, Issue 3, pp 477–501 | Cite as

On the Boltzmann equation for the Lorentz gas

  • C. Boldrighini
  • L. A. Bunimovich
  • Ya. G. Sinai
Articles

Abstract

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if ω is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter λ>0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.

Key words

Boltzmann-Grad limit Boltzmann equation Lorentzmodel 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Grad, Principles of the Kinetic Theory of Gases, inHandbunch der Physik, Vol. 12, S. Flügge, ed. (Springer, Berlin, 1958).Google Scholar
  2. 2.
    H. Spohn,Rev. Mod. Phys. 52:569–615 (1980).Google Scholar
  3. 3.
    G. Gallavotti, Rigorous Theory of the Boltzmann equation in the Lorentz Gas, Nota interna n∘ 358, Istituto di Fisica, Università di Roma, 1972.Google Scholar
  4. 4.
    G. Gallavotti,Phys. Rev. 185:308 (1969).Google Scholar
  5. 5.
    Ya. G. Sinai,Russ. Math. Surv. 25:132 (1970).Google Scholar
  6. 6.
    G. Gallavotti, Lectures on the Billiard, inDynamical Systems, Theory and Application, J. Moser, ed. (Springer, Berlin, 1975).Google Scholar
  7. 7.
    L. A. Bunimovich and Ya. G. Sinai,Commun. Math. Phys. 78:247–280 (1980).Google Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • C. Boldrighini
    • 1
  • L. A. Bunimovich
    • 2
  • Ya. G. Sinai
    • 2
  1. 1.Istituto Matematico dell'Università, CamerinoC.N.R., G.N.F.M.Italy
  2. 2.Landau Institute for Theoretical PhysicsAcademy of Sciences of the U.S.S.R.Moscow

Personalised recommendations