Journal of Statistical Physics

, Volume 51, Issue 5–6, pp 981–990 | Cite as

Lorentz lattice gases: Basic theory

  • M. H. Ernst
  • P. M. Binder
Articles

Abstract

We present several ballistic models of the Lorentz gas in two-dimensional lattices with deterministic and stochastic deflection rules, and their corresponding Liouville equations. Boltzmann-level-equation results are obtained for the diffusion coefficient and velocity autocorrelation function for models with stochastic deflection rules. The long-time behavior of the mean square displacement is briefly discussed and the possibility of abnormal diffusion indicated. Even if the diffusion coefficient exists, its low-density limit may not be given correctly by the Boltzmann equation.

Key words

Lorentz gas lattice gases Liouville equation Boltzmann equation diffusion coefficient 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. A. Lorentz,Proc. R. Acad. Amst. 7:438, 585, 684 (1905).Google Scholar
  2. 2.
    P. Ehrenfest,Collected Scientific Papers (North-Holland, Amsterdam, 1959).Google Scholar
  3. 3.
    J. M. J. van Leeuwen and A. Weyland,Physica 36:457 (1967); A. Weyland and J. M. J. van Leeuwen,Physica 38:35 (1968).Google Scholar
  4. 4.
    M. H. Ernst and A. Weyland,Phys. Lett. 34A:39 (1971).Google Scholar
  5. 5.
    E. H. Hauge and E. G. D. Cohen,J. Math. Phys. 10:397 (1969).Google Scholar
  6. 6.
    E. H. Hauge, inTransport Phenomena, G. Kirczenow and J. Marro, eds. (Springer-Verlag, Berlin, 1974).Google Scholar
  7. 7.
    D. J. Gates,J. Math. Phys. 13:1005, 1315 (1972).Google Scholar
  8. 8.
    M. H. Ernst, P. F. J. van Velthoven, and Th. M. Nieuwenhuizen,J. Phys. A 20:949 (1987).Google Scholar
  9. 9.
    H. van Beijeren,Rev. Mod. Phys. 54:195 (1982).Google Scholar
  10. 10.
    P. M. Binder,Complex Systems 1:559 (1987); Statistical properties of Lorentz lattice gases, Los Alamos Preprint LA-UR-87-3471 (1987).Google Scholar
  11. 11.
    U. Frisch, B. Hasslacher, and Y. Pomeau,Phys. Rev. Lett. 56:1505 (1986).Google Scholar
  12. 12.
    Complex Systems 1(4) (1987).Google Scholar
  13. 13.
    M. H. Ernstet al., Physica 45:127 (1969).Google Scholar
  14. 14.
    H. van Beijeren, private communication.Google Scholar
  15. 15.
    D. W. Jepsen,J. Math. Phys. 6:405 (1965); J. L. Lebowitz and J. K. Perçus,Phys. Rev. 155:122 (1967); J. L. Lebowitz, J. K. Perçus, and J. Sykes,Phys. Rev. 171:224 (1968).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • M. H. Ernst
    • 1
    • 2
  • P. M. Binder
    • 2
    • 3
    • 4
  1. 1.Institute for Theoretical PhysicsState UniversityTA UtrechtThe Netherlands
  2. 2.Los Alamos National LaboratoryCenter for Nonlinear Studies, MS-B258Los Alamos
  3. 3.Applied Physics SectionYale UniversityNew Haven
  4. 4.Groupe de Physique des SolidesÉcole Normale SupérieureParisFrance

Personalised recommendations