Cellular Automata and Condensed Matter Physics

  • Stephen Wolfram


Cellular automata are mathematical models for systems containing many identical components with local interactions. These notes describe some of their properties, and discuss applications to condensed matter physics.


Cellular Automaton Regular Language Cellular Automaton Model Pseudorandom Generator Rule Number 


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  1. 1.
    S. Wolfram, “Cellular automata as models of complexity”, Nature 311 (1984) 419.CrossRefGoogle Scholar
  2. 2.
    Cellular automata,edited by D. Farmer, T. Toffoli and S. Wolfram, Physica 10D (1984) nos. 1 and 2, and North-Holland Publishing Co. (1984).Google Scholar
  3. 3.
    S. Wolfram, “Statistical mechanics of cellular automata”, Rev. Mod. Phys. 55 (1983) 601.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    N. Packard and S. Wolfram, “Two-dimensional cellular automata”, J. Stat. Phys. 38 (1985) 901.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Wolfram, “Twenty problems in the theory of cellular automata”, Phys. Scripta TO (1985) 170.Google Scholar
  6. 6.
    B. Mandelbrot, The fractal geometry of nature, Freeman (1982).Google Scholar
  7. 7.
    R. Shaw, “Strange attractors, chaotic behaviour and information flow”, Z. Naturforsch. 36a (1981) 80.MathSciNetMATHGoogle Scholar
  8. 8.
    S. Wolfram, “Universality and complexity in cellular automata”, Physics bOD (1984) 1.Google Scholar
  9. 9.
    N. Packard, “Complexity of growing patterns in cellular automata”, Institute for Advanced Study preprint (October 1983).Google Scholar
  10. 10.
    Martin, A. Odlyzko and S. Wolfram, “Algebraic properties of cellular automata”, Commun. Math. Phys. 93 (1984) 219.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    S. Wolfram, “Computation theory of cellular automata”, Commun. Math. Phys. 96 (1984) 15.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    J. Hoperoft and J. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley (1979).Google Scholar
  13. 13.
    L. Ilurd, “Formal language characterizations of cellular automaton limit sets”, Princeton University preprint (May 1985).Google Scholar
  14. 14.
    N. Margolus, “Physics-like models of computation”, Physica 10D (1984) 81.MathSciNetGoogle Scholar
  15. 15.
    N. Packard, “Cellular automaton models for dendritic crystal growth”, Institute for Advanced Study preprint (May 1985).Google Scholar
  16. 16.
    J. M. Greenberg, B. D. Hassard and S. P. Hastings, “Pattern formation and periodic structures in systems modelled by reaction-diffusion equations”, Bull. Amer. Math. Soc. 84 (1975) 1296; B. Madore and W. Freedman, “Computer simulations of the Belousov-Zhabotinsky reaction”, Science 222 (1983) 615.Google Scholar
  17. 17.
    R. Stinchcombe, these proceedings.Google Scholar
  18. 18.
    S. Wolfram, “Origins of randomness in physical systems”, to be published.Google Scholar
  19. 19.
    D. Knuth, Seminurnerical algorithms, Addison-Wesley (1981).Google Scholar
  20. 20.
    L. Sander, these proceedings.Google Scholar
  21. 21.
    J. Milnor and S. Wolfram, “Cryptography with cellular automata”, in preparation.Google Scholar
  22. 22.
    P. Grassberger, “Chaos and diffusion in deterministic cellular automata”, Physica 10D (1984) 52.MathSciNetGoogle Scholar
  23. 23.
    P. Bak, these proceedings.Google Scholar
  24. 24.
    W. Kinzel, `Phase transitions of cellular automata“, Z. Phys. B58 (1985) 229; P. Grassherger, F. Krause and T. von der Twer, ”A new type of kinetic critical phenomena“, J. Phys. A17 (1984) L105.Google Scholar
  25. 25.
    E. Domany and W. Kinzel, “Equivalence of cellular automata to Ising models and directed percolation”, Phys. Rev. Lett. 53 (1984) 311.MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Creutz, “Deterministic Ising dynamics”, Ann. Phys., to be published.Google Scholar
  27. 27.
    S. Wolfram, “Computation theory and the Second Law of thermodynamics”, in preparation.Google Scholar
  28. 28.
    D. Millis, “The Connection Machine: A computer architecture based on cellular automata”, Physica 1OD (1984) 213; S. Wolfram, “Scientific computation with the Connection Machine”, unpublished report (1985).Google Scholar
  29. 29.
    N. Packard and R. Shaw, private communication.Google Scholar
  30. 30.
    S. Wolfram, “Undecidability and intractability in theoretical physics”, Phys. Rev. Lett. 51 (1985) 735; “Computer software in science and mathematics”, Sci. Amer. 251 (Sept. 1984) 188.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Stephen Wolfram
    • 1
  1. 1.The Institute for Advanced StudyPrincetonUSA

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