Journal of Statistical Physics

, Volume 67, Issue 1–2, pp 289–302 | Cite as

Recurrence properties of Lorentz lattice gas cellular automata

  • L. A. Bunimovich
  • S. E. Troubetzkoy


Recurrence properties of a point particle moving on a regular lattice randomly occupied with scatterers are studied for strictly deterministic, nondeterministic, and purely random scattering rules.

Key words

Recurrence percolation cellular automata tiling lattice gas Lorentz gas wind-tree model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. A. Lorentz, The motion of electrons in metallic bodies,Proc. Roy. Acad. Amst. 7:438, 585, 684 (1905).Google Scholar
  2. 2.
    P. Ehrenfest,Collected Scientific Papers (North-Holland, Amsterdam, 1959), p. 229.Google Scholar
  3. 3.
    Th. W. Ruijgrok and E. G. D. Cohen, Deterministic lattice gas models,Phys. Lett. A 133:415 (1988).Google Scholar
  4. 4.
    G. Grimmett,Percolation (Springer-Verlag, New York, 1989), p. 240.Google Scholar
  5. 5.
    X. P. Kong and E. G. D. Cohen, Anomalous diffusion in a lattice-gas wind-tree model,Phys. Rev. B 40:4838 (1989).Google Scholar
  6. 6.
    X. P. Kong and E. G. D. Cohen, Diffusion and propagation in triangular Lorentz lattice gas cellular automata,J. Stat. Phys. 62:737 (1991).Google Scholar
  7. 7.
    X. P. Kong and E. G. D. Cohen, A kinetic theorist's look at lattice gas cellular automata,Physica D 47:9 (1991).Google Scholar
  8. 8.
    E. G. D. Cohen, New types of diffusion in lattice gas cellular automata, inMicroscopic Simulations of Complex Hydrodynamic Phenomena (Plenum Press, New York, to appear).Google Scholar
  9. 9.
    R. M. Ziff, X. P. Kong, and E. G. D. Cohen, A Lorentz lattice gas and kinetic walk model,Phys. Rev. A 44:2410 (1991).Google Scholar
  10. 10.
    C. Pickover, Picturing randomness with Truchet tiles,J. Recreational Math. 21:256 (1989).Google Scholar
  11. 11.
    C. Pickover, Several short classroom experiments,AMS Notices 38:192 (1991).Google Scholar
  12. 12.
    S. Roux, E. Guyon, and D. Sornette, Hull percolation,J. Phys. A 21:L475 (1988).Google Scholar
  13. 13.
    R. Bradley, Exactθ point and exponents for polymer chains on an oriented two-dimensional lattice,Phys. Rev. A 39:3738 (1989).Google Scholar
  14. 14.
    B. Duplantier, Hull percolation and standard percolation,J. Phys. A 21:3969 (1988).Google Scholar
  15. 15.
    S. Manna and A. Guttmann, Kinetic growth walks and trails on oriented square lattices: Hull percolation and percolation hulls,J. Phys. A 22:3113 (1989).Google Scholar
  16. 16.
    R. Ziff, P. Cummings, and G. Stell, Generation of percolation cluster perimeters by a random walk,J. Phys. A 17:3009 (1984).Google Scholar
  17. 17.
    J. M. F. Gunn and M. Ortuno, Percolation and motion in a simple random environment,J. Phys. A 18:L1095 (1985).Google Scholar
  18. 18.
    B. Tóth, Persistent random walks in random environment,Prob. Theory Related Fields 71:615 (1986).Google Scholar
  19. 19.
    H. Kesten,Percolation Theory for Mathematicians (Birkhäuser, Boston, 1982).Google Scholar
  20. 20.
    M. V. Menshikov, S. A. Molchanov, and A. F. Sidorenko, Percolation theory and some applications,J. Sov. Math. 42:1766 (1988).Google Scholar
  21. 21.
    F. Spitzer,Principles of Random Walk (Springer-Verlag, New York, 1964), p. 83.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • L. A. Bunimovich
    • 1
  • S. E. Troubetzkoy
    • 1
  1. 1.Forschungszentrum BiBoSUniversität BielefeldGermany

Personalised recommendations