Journal of Statistical Physics

, Volume 78, Issue 5–6, pp 1521–1554 | Cite as

Generalized Boltzmann equation for lattice gas automata

  • H. J. Bussemaker
  • M. H. Ernst
  • J. W. Dufty


In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.

Key Words

Non-Gibbs states lack of detailed balance static pair correlations lattice gas automata BBGKY hierarchy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. R. Dorfman and H. van Beijeren, In “Statistical Mechanics, Part B: Time-Dependent Processes, B. J. Berne, ed. (Plenum Press, New York, 1977), p. 65.Google Scholar
  2. 2.
    E. G. D. Cohen, InFundamental Problems in Statistical Mechanis II, E. G. D. Cohen, ed. (North-Holland, Amsterdam, (1968), p. 228.Google Scholar
  3. 3.
    J. R. Dorfman, InFundamental Problems in Statistical Mechanics III, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1975), p. 277.Google Scholar
  4. 4.
    T. R. Kirkpatrick and M. H. Ernst,Phys. Rev. A 44:8051 (1991).Google Scholar
  5. 5.
    G. A. van Velzen, R. Brito, and M. H. Ernst,J. Stat. Phys. 70:811 (1993).Google Scholar
  6. 6.
    G. A. van Velzen and M. H. Ernst,J. Phys. A: Math. Gen. 22:4611 (1989).Google Scholar
  7. 7.
    A. J. H. Ossendrijver, A. Santos, and M. H. Ernst,J. Stat. Phys. 71:1015 (1993).Google Scholar
  8. 8.
    R. van Roij and M. H. Ernst,J. Stat. Phys. 73:47 (1993).Google Scholar
  9. 9.
    R. Brito and M. H. Ernst,Phys. Rev. A 46:875 (1992).Google Scholar
  10. 10.
    M. Gerits, M. H. Ernst, and D. Frenkel,Phys. Rev. E 48:988 (1993).Google Scholar
  11. 11.
    B. M. Boghosian and W. Taylor, Preprint (available from bulletin board Scholar
  12. 12.
    M. S. Green and R. A. Picirelli,Phys. Rev. 132:1388 (1963).Google Scholar
  13. 13.
    B. Dubrulle, U. Frisch, M. Hénon, and J.-P. Rivet,J. Stat. Phys. 59:1187 (1990).Google Scholar
  14. 14.
    J. A. Somers and P. C. Rem, InCellular Automata and Modeling of Complex Physical Systems, P. Manneville, ed. (Springer, Berlin, in 1989), p. 161.Google Scholar
  15. 15.
    M. Hénon,J. Stat. Phys. 68:409 (1992).Google Scholar
  16. 16.
    H. J. Bussemaker and M. H. Ernst,J. Stat. Phys. 68:431 (1992).Google Scholar
  17. 17.
    U. Frisch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet,Complex Systems 1:31 (1987) 31 [reprinted in G. Doolen, ed.,Lattice Gas Methods for Partial Differential Equations (Addison-Wesley, Singapore, 1990)].Google Scholar
  18. 18.
    E. C. G. Stueckelberg,Helv. Phys. Acta 25:577 (1952).Google Scholar
  19. 19.
    M. Hénon, InProceedings of NATO workshop, Waterloo, Canada, June 7–12 1993, A. Lawniczak and R. Kapral, eds. (Fields Institute Communications, American Mathematical Society).Google Scholar
  20. 20.
    G. Zanetti,Phys. Rev. A 40:1539 (1989).Google Scholar
  21. 21.
    M. Hénon,Complex System 1:763 (1987).Google Scholar
  22. 22.
    M. H. Ernst and E. G. D. Cohen,J. Stat. Phys. 25:153 (1981).Google Scholar
  23. 23.
    D. Frenkel and M. H. Ernst,Phys. Rev. Lett. 63:2165 (1989).Google Scholar
  24. 24.
    T. Naitoh, M. H. Ernst, and J. W. Dufty,Phys. Rev. A 42:7187 (1990).Google Scholar
  25. 25.
    R. Kapral, A. Lawniczak, and P. Masiar,J. Chem. Phys. 96:2762 (1992); X.-G. Wu and R. Kapral,Phys. Rev. Lett. 70: 1940 (1993).Google Scholar
  26. 26.
    O. Biham, A. A. Middleton, and D. Levine,Phys. Rev. A 46:6124 (1992); J. M. Molera, F. C. Martinez, J. A. Cuesta, and R. Brito, Preprint (available from bulltin board Scholar
  27. 27.
    B. Kamgar-Parsi, E. G. D. Cohen, and I. M. de Schepper,Phys. Rev. A. 35:4781 (1987).Google Scholar
  28. 28.
    S. P. Das, H. J. Bussemaker, and M. H. Ernst,Phys. Rev. E 48:245 (1993).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • H. J. Bussemaker
    • 1
  • M. H. Ernst
    • 1
  • J. W. Dufty
    • 2
  1. 1.Institute for Theoretical PhysicsUniversity of UtrechtThe Netherlands
  2. 2.Physics DepartmentUniversity of FloridaGainesville

Personalised recommendations