Deterministic Cellular Automata with Diffusive Behavior

  • C. D. Levermore
  • B. M. Boghosian
Part of the Springer Proceedings in Physics book series (SPPHY, volume 46)


It is a classical result [1] that an ensemble of independent unbiased random walks on the one dimensional lattice, Z, and moving at discrete times, Z +, has a continuum limit given by a diffusion equation. More recently, systems of randomly walking particles interacting via an exclusion principle have been studied [2] [3]. Another interesting problem is that of using deterministic dynamical systems for the same purpose. Of course, to the extent that the underlying microscopic dynamics of atoms in real diffusing media are deterministic, we know that this should be possible. In this work we describe two completely deterministic cellular automata that exhibit diffusive behavior in one dimension, possibly with spatial inhomogeneity. We analyze these automata both theoretically and experimentally to investigate their continuum limits. In the first of these, we experimentally find significant deviations from the Chapman-Enskog theory; these deviations are due to a buildup of correlations that invalidates the Boltzmann molecular chaos assumption. In the second, the relevant correlations have been suppressed, and good agreement with the Chapman-Enskog theory is obtained.


Boltzmann Equation Diffusion Equation Cellular Automaton Continuum Limit Diffusive Behavior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W. Feller, An Introduction to Probability Theory and its Applications, Volume I, Chapter 14, Section 6 (J. Wiley, 1970).Google Scholar
  2. [2]
    See, for example, E. Presutti, Proceedings of the First Bernoulli Congress, September, 1986;Google Scholar
  3. J. L. Lebowitz, Sixteenth IUPAP Conference on Statistical Mechanics, Boston University, August 11–16, 1986.Google Scholar
  4. [3]
    B. M. Boghosian, C. D. Levermore, Complex Systems, 1 (1987) 17.MathSciNetMATHGoogle Scholar
  5. [4]
    See, for example, S. Wolfram, “Minimal Cellular Automaton Approximations to Continuum Systems”, in the proceedings of Cellular Automata’86, M.I.T., June, 1986; andGoogle Scholar
  6. S. Wolfram, Phys. Rev. Lett., 55, 5 (1985) 449.MathSciNetADSCrossRefGoogle Scholar
  7. [5]
    B. M. Boghosian, C. D. Levermore, “A Deterministic Cellular Automaton with Diffusive Behavior”, in the Proceedings of the International Workshop on Discrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrodynamics, Torino, World Scientific, September, 1988.Google Scholar
  8. [6]
    Hardy, J., de Pazzis, O., Pomeau, Y., Phys. Rev. A, 13, 5 (1976).CrossRefGoogle Scholar
  9. [7]
    D. E. Knuth, “The Art of Computer Programming,” 2, “Seminumerical Algorithms,” Addison-Wesley (1981).Google Scholar
  10. [8]
    U. Frisch, D. D’Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, J.-P. Rivet, Complex Systems, 1 (1987) 648.MathSciNetGoogle Scholar
  11. Connection Machine is a registered trademark of Thinking Machines Corporation.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • C. D. Levermore
    • 1
  • B. M. Boghosian
    • 2
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Thinking Machines CorporationCambridgeUSA

Personalised recommendations