Journal of Statistical Physics

, Volume 124, Issue 2–4, pp 823–842 | Cite as

Systematic Density Expansion of the Lyapunov Exponents for a Two-Dimensional Random Lorentz Gas

  • H. V. Kruis
  • Debabrata Panja
  • Henk van Beijeren


We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form A 0ñln ñ+B 0ñ, where A 0 and B 0 are known constants and ñ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order (ñ2), the positive Lyapunov exponent is of the form A 0ñln ñ+B 0ñ+A 1ñ2ln ñ +B 1ñ2. Explicit numerical values of the new constants A 1 and B 1 are obtained by means of a systematic analysis. This takes into account, up to O2), the effects of all possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.


Lyapunov exponent Lorentz gas density expansion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65:1693 (1990).MATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London (1990).Google Scholar
  3. 3.
    J. R. Dorfman, An Introduction to Chaos in Non-Equilibrium Statistical Mechanics, Cambridge Univ. Press (1999).Google Scholar
  4. 4.
    W. G. Hoover, Computational Statistical Mechanics Elsevier Publ. Co., Amsterdam (1991).Google Scholar
  5. 5.
    Ya. G. Sinai, Russ. Math. Surv. 25:137 (1970).MathSciNetCrossRefGoogle Scholar
  6. 6.
    N. S. Krylov, Works on the Foundations of Statistical Mechanics, Princeton University Press, Princeton (1979).Google Scholar
  7. 7.
    H. van Beijeren and J. R. Dorfman, Phys. Rev. Lett. 74:4412 (1995).CrossRefADSGoogle Scholar
  8. 8.
    H. van Beijeren and J. R. Dorfman, Phys. Rev. Lett. 76:3238 (1996).CrossRefADSGoogle Scholar
  9. 9.
    H. van Beijeren, A. Latz and J. R. Dorfman, Phys. Rev. E 57:4077 (1998).MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Since we use the full collision frequency in our equations this should perhaps rather be called a Lorentz-Enskog equation. Substituting νc = 2nav one indeed recovers the Lorentz Boltzmann equation of [7].Google Scholar
  11. 11.
    H. van Beijeren, A. Latz, and J. R. Dorfman, Phys. Rev. E 63: 016312 (2001).Google Scholar
  12. 12.
    H. V. Kruis, Masters thesis, University of Utrecht, The Netherlands (1997).Google Scholar
  13. 13.
    D. Panja, Ph.D. thesis, University of Maryland, College Park, USA (2000).Google Scholar
  14. 14.
    D. Panja (unpublished).Google Scholar
  15. 15.
    J. M. J. van Leeuwen and A. Weyland, Physica 36:457 (1967); J. M. J. van Leeuwen and A. Weyland, Physica 39:35 (1968).Google Scholar
  16. 16.
    M. H. Ernst, J. R. Dorfman, W. R. Hoegy, and J. M. J. van Leeuwen, Physica 45:127 (1969).CrossRefADSGoogle Scholar
  17. 17.
    H. van Beijeren and M. H. Ernst, J. Stat. Phys. 21:125 (1979).CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • H. V. Kruis
    • 1
  • Debabrata Panja
    • 1
  • Henk van Beijeren
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of UtrechtUtrechtThe Netherlands

Personalised recommendations