Journal of Statistical Physics

, Volume 45, Issue 3–4, pp 471–526 | Cite as

Cellular automaton fluids 1: Basic theory

  • Stephen Wolfram


Continuum equations are derived for the large-scale behavior of a class of cellular automaton models for fluids. The cellular automata are discrete analogues of molecular dynamics, in which particles with discrete velocities populate the links of a fixed array of sites. Kinetic equations for microscopic particle distributions are constructed. Hydrodynamic equations are then derived using the Chapman-Enskog expansion. Slightly modified Navier-Stokes equations are obtained in two and three dimensions with certain lattices. Viscosities and other transport coefficients are calculated using the Boltzmann transport equation approximation. Some corrections to the equations of motion for cellular automaton fluids beyond the Navier-Stokes order are given.

Key words

Cellular automata derivation of hydrodynamics molecular dynamics kinetic theory Navier-Stokes equations 


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  1. 1.
    S. Wolfram, ed.,Theory and Applications of Cellular Automata (World Scientific, 1986).Google Scholar
  2. 2.
    S. Wolfram, Cellular automata as models of complexity,Nature 311:419 (1984).Google Scholar
  3. 3.
    N. Packard and S. Wolfram, Two-dimensional cellular automata,J. Stat. Phys. 38:901 (1985).Google Scholar
  4. 4.
    D. J. Tritton,Physical Fluid Dynamics (Van Nostrand, 1977).Google Scholar
  5. 5.
    W. W. Wood, Computer studies on fluid systems of hard-core particles, inFundamental Problems in Statistical Mechanics 3 E. D. G. Cohen, ed. (North-Holland, 1975).Google Scholar
  6. 6.
    J. Hardy, Y. Pomeau, and O. de Pazzis, Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions,J. Math. Phys. 14:1746 (1973); J. Hardy, O. de Pazzis, and Y. Pomeau, Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions,Phys. Rev. A 13:1949 (1976).Google Scholar
  7. 7.
    U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice gas automata for the Navier-Stokes equation,Phys. Rev. Lett. 56:1505 (1986).Google Scholar
  8. 8.
    J. Salem and S. Wolfram, Thermodynamics and hydrodynamics with cellular automata, inTheory and Applications of Cellular Automata, S. Wolfram, ed. (World Scientific, 1986).Google Scholar
  9. 9.
    D. d'Humieres, P. Lallemand, and T. Shimomura, An experimental study of lattice gas hydrodynamics, Los Alamos preprint LA-UR-85-4051; D. d'Humieres, Y. Pomeau, and P. Lallemand, Simulation d'allees de Von Karman bidimensionnelles a l'aide d'un gaz sur reseau,C. R. Acad. Sci. Paris II 301:1391 (1985).Google Scholar
  10. 10.
    J. Broadwell, Shock structure in a simple discrete velocity gas,Phys. Fluids 7:1243 (1964).Google Scholar
  11. 11.
    H. Cabannes, The discrete Boltzmann equation, Lecture Notes, Berkeley (1980).Google Scholar
  12. 12.
    R. Gatignol,Theorie cinetique des gaz a repartition discrete de vitesse (Springer, 1975).Google Scholar
  13. 13.
    J. Hardy and Y. Pomeau, Thermodynamics and hydrodynamics for a modeled fluid,J. Math. Phys. 13:1042 (1972).Google Scholar
  14. 14.
    S. Harris,The Boltzmann Equation (Holt, Rinehart and Winston, 1971).Google Scholar
  15. 15.
    R. Caflisch and G. Papanicolaou, The fluid-dynamical limit of a nonlinear model Boltzmann equation,Commun. Pure Appl. Math. 32:589 (1979).Google Scholar
  16. 16.
    B. Nemnich and S. Wolfram, Cellular automaton fluids 2: Basic phenomenology, in preparation.Google Scholar
  17. 17.
    S. Wolfram,SMP Reference Manual (Inference Corporation, Los Angeles, 1983); S. Wolfram, Symbolic mathematical computation,Commun. ACM 28:390 (1985).Google Scholar
  18. 18.
    A. Sommerfeld,Thermodynamics and Statistical Mechanics (Academic Press, 1955).Google Scholar
  19. 19.
    S. Wolfram, Origins of randomness in physical systems,Phys. Rev. Lett. 55:449 (1985).Google Scholar
  20. 20.
    S. Wolfram, Random sequence generation by cellular automata,Adv. Appl. Math. 7:123 (1986).Google Scholar
  21. 21.
    J. P. Boon and S. Yip,Molecular Hydrodynamics (McGraw-Hill, 1980).Google Scholar
  22. 22.
    E. M. Lifshitz and L. P. Pitaevskii,Statistical Mechanics, Part 2 (Pergamon, 1980), Chapter 9.Google Scholar
  23. 23.
    R. Liboff,The Theory of Kinetic Equations (Wiley, 1969).Google Scholar
  24. 24.
    D. Levermore, Discretization effects in the macroscopic properties of cellular automaton fluids, in preparation.Google Scholar
  25. 25.
    E. M. Lifshitz and L. P. Pitaevskii,Physical Kinetics (Pergamon, 1981).Google Scholar
  26. 26.
    P. Resibois and M. De Leener,Classical Kinetic Theory of Fluids (Wiley, 1977).Google Scholar
  27. 27.
    L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon, 1959).Google Scholar
  28. 28.
    M. H. Ernst, B. Cichocki, J. R. Dorfman, J. Sharma, and H. van Beijeren, Kinetic theory of nonlinear viscous flow in two and three dimensions,J. Stat. Phys. 18:237 (1978).Google Scholar
  29. 29.
    J. R. Dorfman, Kinetic and hydrodynamic theory of time correlation functions, inFundamental Problems in Statistical Mechanics 3, E. D. G. Cohen, ed. (North-Holland, 1975).Google Scholar
  30. 30.
    R. Courant and K. O. Friedrichs,Supersonic Flows and Shock Waves (Interscience, 1948).Google Scholar
  31. 31.
    D. Levermore, private communication.Google Scholar
  32. 32.
    J. Milnor, private communication.Google Scholar
  33. 33.
    V. Yakhot, B. Bayley, and S. Orszag, Analogy between hyperscale transport and cellular automaton fluid dynamics, Princeton University preprint (February 1986).Google Scholar
  34. 34.
    H. S. M. Coxeter,Regular Polytopes (Macmillan, 1963).Google Scholar
  35. 35.
    M. Hammermesh,Group Theory (Addison-Wesley, 1962), Chapter 9.Google Scholar
  36. 36.
    L. D. Landau and E. M. Lifshitz,Quantum Mechanics (Pergamon, 1977), Chapter 12.Google Scholar
  37. 37.
    H. Boerner,Representations of Groups (North-Holland, 1970), Chapter 7.Google Scholar
  38. 38.
    L. D. Landau and E. M. Lifshitz,Theory of Elasticity (Pergamon, 1975), Section 10.Google Scholar
  39. 39.
    D. Levineet al, Elasticity and dislocations in pentagonal and icosahedral quasicrystals,Phys. Rev. Lett. 14:1520 (1985).Google Scholar
  40. 40.
    L. D. Landau and E. M. Lifshitz,Statistical Physics (Pergamon, 1978), Chapter 13.Google Scholar
  41. 41.
    B. K. Vainshtein,Modern Crystallography, (Springer, 1981), Chapter 2.Google Scholar
  42. 42.
    J. H. Conway and N. J. A. Sloane, to be published.Google Scholar
  43. 43.
    R. L. E. Schwarzenberger,N-Dimensional Crystallography (Pitman, 1980).Google Scholar
  44. 44.
    J. Milnor, Hubert's problem 18: On crystallographic groups, fundamental domains, and on sphere packing,Proc. Symp. Pure Math. 28:491 (1976).Google Scholar
  45. 45.
    B. G. Wybourne,Classical Groups for Physicists (Wiley, 1974), p. 78; R. Slansky, Group theory for unified model building,Phys. Rep. 79:1 (1981).Google Scholar
  46. 46.
    B. Grunbaum and G. C. Shephard,Tilings and Patterns (Freeman, in press); D. Levine and P. Steinhardt, Quasicrystals I: Definition and structure, Univ. of Pennsylvania preprint.Google Scholar
  47. 47.
    N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane,Nedl. Akad. Wetensch. Indag. Math. 43:39 (1981); J. Socolar, P. Steinhardt, and D. Levine, Quasicrystals with arbitrary orientational symmetry,Phys. Rev. B 32:5547 (1985).Google Scholar
  48. 48.
    R. Penrose, Pentaplexity: A class of nonperiodic tilings of the plane,Math. Intelligencer 2:32 (1979).Google Scholar
  49. 49.
    J. P. Rivet and U. Frisch, Automates sur gaz de reseau dans l'approximation de Boltzmann,C. R. Acad. Sci. Paris II 302:267 (1986).Google Scholar
  50. 50.
    P. J. Davis,Circulant Matrices (Wiley, 1979).Google Scholar
  51. 51.
    L. D. Landau and E. M. Lifshitz,Statistical Physics (Pergamon, 1978), Chapter 5.Google Scholar
  52. 52.
    E. Kolb and S. Wolfram, Baryon number generation in the early universe,Nucl. Phys. B 172:224 (1980), Appendix A.Google Scholar
  53. 53.
    I. S. Gradshteyn and I. M. Ryzhik,Table of integrals, Series and Products (Academic Press, 1965).Google Scholar
  54. 54.
    U. Frisch, private communication.Google Scholar
  55. 55.
    P. Roache,Computational Fluid Mechanics (Hermosa, Albuquerque, 1976).Google Scholar
  56. 56.
    S. Omohundro and S. Wolfram, unpublished (July 1985).Google Scholar
  57. 57.
    D. d'Humieres, private communication.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • Stephen Wolfram
    • 1
    • 2
  1. 1.The Institute for Advanced StudyPrinceton
  2. 2.Thinking Machines CorporationCambridge

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