Cellular Automata

  • Dietrich Stauffer

Abstract

Cellular automata [9.1] offer a very wide field of choices, and within the space given to me by the editors it will not be possible to write, say, 4 294 967 296 different chapters, each dealing with one particular automaton separately. Instead, I will concentrate on two more systematic approaches. The first approach mixes all possible cellular automata together by letting each site select at the beginning which cellular automata rule it wants to obey. This is the Kauff-man model for genetics, and its fractal dimensions at the phase transition have been studied in detail by many people since 1986. The second approach tries to classify all possible cellular automata according to some unified simple criteria; such criteria could be the fractal dimensions, but since they are only beginning to be known we concentrate on the simpler Wolfram classification. Some readers may wish to look only at some restricted class of cellular automata, and not at all of them; for these readers I offer some appendices.

Keywords

Permeability Porosity Stein Defend Wolfram 

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References

  1. 9.1
    S. Wolfram: Rev. Mod. Phys. 55, 601 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  2. N.H. Packard, S. Wolfram: J. Stat. Phys. 38, 901 (1986)MathSciNetADSCrossRefGoogle Scholar
  3. S. Wolfram: Theory and Applications of Cellular Automata (World Scientific, Singapore 1986)MATHGoogle Scholar
  4. G. Weisbuch: Dynamique des systèmes complexes (Editions du CNRS, Paris 1989); English translation: Complex System Dynamics (Addison-Wesley, New York 1991); for programming techniques see also D. Stauffer: Computers in Physics, Jan/Feb 1991, p. 62, Ref. [9.51b], and R.W. Gerling, D. Stauffer: in preparationGoogle Scholar
  5. 9.3
    G.Y. Vichniac: Physica D 10, 96 (1984)MathSciNetADSCrossRefGoogle Scholar
  6. T. Toffoli, M. Margolus: Cellular Automata Machines (MIT Press, Cambridge MA 1987)Google Scholar
  7. 9.4
    P. Bak, C. Tang, K. Wiesenfeld: Phys. Rev. Lett. 59, 381 (1987)MathSciNetADSCrossRefGoogle Scholar
  8. P. Bak: Physica D 38, 5 (1989)MathSciNetADSCrossRefGoogle Scholar
  9. 9.5
    P. Grassberger, S.S. Manna: J. Physique 51, 1077 (1990), with further referencesCrossRefGoogle Scholar
  10. 9.6
    S.A. Kauffman: J. Theor. Biol. 22, 437 (1969)MathSciNetCrossRefGoogle Scholar
  11. U. Keller, B. Thomas, H.J. Pohley: J. Stat. Phys. 52, 1129 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  12. 9.7
    B. Derrida, Y. Pomeau: Europhys. Lett. 1, 59 (1986)CrossRefGoogle Scholar
  13. B. Derrida, G. Weisbuch: J. Physique 47, 1297 (1986)CrossRefGoogle Scholar
  14. 9.8
    J.G. Zabolitzky: Physica A 163, 447 (1990)ADSMATHCrossRefGoogle Scholar
  15. 9.9
    B. Derrida, D. Stauffer: Europhys. Lett. 2, 739 (1986)ADSCrossRefGoogle Scholar
  16. D. Stauffer: Phil. Mag. B 56, 510 (1987)Google Scholar
  17. U.M.S. Costa: J. Phys. A 20, L 583 (1987)ADSCrossRefGoogle Scholar
  18. B. Derrida, G. Weisbuch: Europhys. Lett. 4, 657 (1987)ADSCrossRefGoogle Scholar
  19. H.E. Stanley, D. Stauffer, J. Kertesz, H.J. Herrmann: Phys. Rev. Lett. 59, 2326 (1987)ADSCrossRefGoogle Scholar
  20. A. Coniglio, L. de Arcangelis, H.J. Herrmann, N. Jan: Europhys. Lett. 8, 315 (1989)ADSCrossRefGoogle Scholar
  21. G. Le Caer: Physica A 159, 329 (1989)ADSCrossRefGoogle Scholar
  22. L. de Arcangelis, H.J. Herrmann, A. Coniglio: J. Phys. A 23, L265 (1990)CrossRefGoogle Scholar
  23. 9.10
    A. Coniglio, N. Jan, D. Stauffer: J. Phys. A 20, L1103 (1987)ADSCrossRefGoogle Scholar
  24. 9.11
    L.R. da Silva, H.J. Herrmann: J. Stat. Phys. 52, 463 (1988)ADSCrossRefGoogle Scholar
  25. 9.12
    D. Stauffer: Physica D 38, 341 (1989)MathSciNetADSCrossRefGoogle Scholar
  26. 9.13
    D. Stauffer, in: Computer Simulation Studies in Condensed Matter Physics II, ed. by D.P. Landau, K.K. Mon, H.B. Schüttler (Springer, Berlin, Heidelberg 1989)Google Scholar
  27. 9.14
    A. Hansen, S. Roux: Physica A 160, 275 (1989)MathSciNetADSCrossRefGoogle Scholar
  28. 9.15
    L. de Arcangelis: J. Phys. A 20, L369 (1987)CrossRefGoogle Scholar
  29. 9.16
    A. Hansen: J. Phys. A 21, 2481 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 9.17
    S.P. Obukhov, D. Stauffer: J. Phys. A 22, 1715 (1989)ADSCrossRefGoogle Scholar
  31. 9.18
    G. Y.Vichniac, H. Hartmann, in: Disordered Systems and Biological Organization, ed. by E. Bienenstock, F. Fogelman-Soulie, G. Weisbuch (Springer, Berlin, Heidelberg 1986)Google Scholar
  32. L.R. da Silva: J. Stat. Phys. 53, 985 (1988)ADSCrossRefGoogle Scholar
  33. K.E. Kürten, in: Models of Brain Functions (Cambridge University Press, Cambridge 1990)Google Scholar
  34. 9.19
    D. Stauffer: Physica A 157, 645 (1989)MathSciNetADSMATHCrossRefGoogle Scholar
  35. S.S. Manna, D. Stauffer: Physica A 162, 176 (1990)ADSCrossRefGoogle Scholar
  36. R.W. Gerling: Physica A 162, 187 and 196 (1990)MathSciNetADSCrossRefGoogle Scholar
  37. D. Stauffer: J. Phys. A 23, 5933 (1990)MathSciNetADSCrossRefGoogle Scholar
  38. H.A. Gutowitz, J.D. Victor, B.W. Knight: Physica D 28, 18 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  39. 9.20
    P.M. Binder: J. Phys. A 24, L31 (1991)MathSciNetADSCrossRefGoogle Scholar
  40. 9.21
    J. Adler: Physica A 171, 453 (1991)ADSCrossRefGoogle Scholar
  41. 9.22
    S.A. Kauffman, The Origin of Order, Oxford University Press, New York 1993Google Scholar
  42. For later evolution simulations of Kauffman models see D. Stauffer, N. Jan: J. Theor. Biol. 168, 211 (1994); J. Stat. Phys. 74, 1293 (1994)CrossRefGoogle Scholar
  43. 9.23
    P. Bak, M. Creutz, in: Fractals in Science, ed. by A. Bunde and S. Havlin (Springer, Heidelberg, New York 1994)Google Scholar
  44. 9.24
    P. Bak, K. Sneppen: Phys. Rev. Lett. 71, 4087 (1993)ADSCrossRefGoogle Scholar
  45. T.S. Ray, N. Jan: Phys. Rev. Lett. 72, 4048 (1994)ADSCrossRefGoogle Scholar
  46. for game of life see P. Bak: Physica A 191, 41 (1992)ADSCrossRefGoogle Scholar
  47. 9.25
    R.J. de Boer, L.A. Segel, A.S. Perelson: J. Theor. Biol. 155, 295 (1992)CrossRefGoogle Scholar
  48. R.J. de Boer, J.D. van der Laan, P. Hogeweg, in: Thinking about Biology, ed. by F.J. Varela and W.D. Stein (Addison-Wesley, New York 1992)Google Scholar
  49. 9.26
    J. Stewart, F.J. Varela: J. Theor. Biol. 153, 477 (1991)CrossRefGoogle Scholar
  50. 9.27
    D. Stauffer, G. Weisbuch: Physica A 180, 42 (1992)ADSCrossRefGoogle Scholar
  51. 9.28
    S. Dasgupta: Physica A 189, 403 (1992)ADSCrossRefGoogle Scholar
  52. 9.29
    M. Deffner: Physica A 195, 279 (1993)ADSCrossRefGoogle Scholar
  53. 9.30
    R.M. Zorzenon dos Santos: Physica A 196, 12 (1993)ADSMATHCrossRefGoogle Scholar
  54. 9.31
    M. Sahimi, D.Stauffer: Phys. Rev. Lett. 71, 4271 (1993)ADSCrossRefGoogle Scholar
  55. 9.32
    D. Chowdhury, V. Deshpande, D. Stauffer: Int. J. Mod. Phys. C 5, 1049 (1994)ADSCrossRefGoogle Scholar
  56. 9.33
    B. Barral, H. Chate, P. Manneville: Phys. Lett. A 163, 279 (1992), citing earlier papers of this groupADSCrossRefGoogle Scholar
  57. J. Hemmingson, G.W. Peng: J. Phys. A 27, 2735 (1994)MathSciNetADSCrossRefGoogle Scholar
  58. 9.34
    G.A. Kohring: Physica A 186, 97 (1992)ADSCrossRefGoogle Scholar
  59. see also D.W. Grünau, T. Lookman, S.Y. Chen, A.S. Lapedes: Phys. Rev. Lett. 71, 4198 (1993); J.F. McCarthy: Phys. Fluids 6, 435 (1994)ADSCrossRefGoogle Scholar
  60. 9.35
    R. Benzi, S. Succi, M. Vergassola: Phys. Rep. 222, 3 (1992)CrossRefGoogle Scholar
  61. 9.36
    K. Nagel, M. Schreckenberg: J. Physique I 2, 2221 (1992)ADSCrossRefGoogle Scholar
  62. 9.37
    T. Nagatani: J. Phys. Soc. Jpn. 63, 1228 (1994) and his earlier papersADSCrossRefGoogle Scholar
  63. 9.38
    J.B. Rundle, W. Klein: J. Stat. Phys. 72, 405 (1993)ADSMATHCrossRefGoogle Scholar
  64. 9.39
    B. Chopard, P. Luthi, M. Droz: Phys. Rev. Lett. 72, 1384 (1994)ADSCrossRefGoogle Scholar
  65. J.R. Weimar, J.P. Boon: Phys. Rev. E 49, 1749 (1994)ADSCrossRefGoogle Scholar
  66. 9.40
    C.F. Baillie, D.A. Johnston: Phys. Lett. B 326, 51 (1994)ADSCrossRefGoogle Scholar
  67. 9.41
    F. Bagnoli, R. Rechtman, S. Ruffo: Phys.Lett. A 172, 34 (1992)ADSCrossRefGoogle Scholar
  68. 9.42
    Ph. Blanchard, D. Gandolfo: J. Stat. Phys. 73, 399 (1993)MathSciNetADSMATHCrossRefGoogle Scholar
  69. 9.43
    G.A. Kohring: Physica A 182, 320 (1992)ADSCrossRefGoogle Scholar
  70. B. Voorhees, S. Bradshaw: Physica D 27, 152 (1994)MathSciNetADSCrossRefGoogle Scholar
  71. 9.44
    D. Makowiec: Z. Physik B 95, 519 (1994)ADSCrossRefGoogle Scholar
  72. 9.45
    U. Frisch, B. Hasslacher, Y. Pomeau: Phys. Rev. Lett. 56, 1505 (1986)ADSCrossRefGoogle Scholar
  73. 9.46
    H.J. Herrmann: J. Stat. Phys. 45, 145 (1986)ADSCrossRefGoogle Scholar
  74. J.G. Zabolitzky, H.J. Herrmann: J. Comp. Phys. 76, 426 (1988)ADSMATHCrossRefGoogle Scholar
  75. 9.47
    D. Stauffer, F.W. Hehl, V. Winkelmann, J.G. Zabolitzky: Computer Simulation and Computer Algebra, 2nd ed. (Springer, Berlin, Heidelberg 1989)CrossRefGoogle Scholar
  76. 9.48
    M. Creutz: Phys. Rev. Lett. 50, 1411 (1983); Ann. Physics 167 62 (1986)MathSciNetADSCrossRefGoogle Scholar
  77. 9.49
    M. Schulte, W. Stiefelhagen, E.S. Demme: J. Phys. A 20, L1023 (1987)ADSCrossRefGoogle Scholar
  78. 9.50
    C. Moukarzel, N. Parga: J. Phys. A 22, 943 (1989)ADSCrossRefGoogle Scholar
  79. C. Moukarzel: J. Phys. A 22, 4487 (1989)ADSCrossRefGoogle Scholar
  80. 9.51
    H.J. Herrmann, H.O. Carmesin, D. Stauffer: J. Phys. A 20, 4939 (1987)MathSciNetADSCrossRefGoogle Scholar
  81. 9.52
    S.C. Glotzer, D. Stauffer, S. Sastry: Physica A 164, 1 (1990), and Ref. [9.9]ADSCrossRefGoogle Scholar
  82. 9.53
    D. Stauffer: J. Phys. A 23, 1847 (1990); J. Phys. A 24, 909 (1991), (review, see in particular Fig. 4 there)MathSciNetADSCrossRefGoogle Scholar
  83. 9.54
    I.R. Cohen, in: Theories of Immune Networks, ed. by H. Atlan, I.R. Cohen, Springer Ser. Syn., Vol.46 (Springer, Berlin, Heidelberg 1989)CrossRefGoogle Scholar
  84. W.E. Paul, in: Fundamental Immunology, ed. by W.E. Paul (Raven, New York 1989)Google Scholar
  85. A.S. Perelson, ed.: Theoretical Immunology (Addison Wesley, Reading, MA 1988)Google Scholar
  86. 9.55
    H. Atlan: Bull. Math. Biol. 51, 247 (1989)Google Scholar
  87. 9.56
    M. Kauffman, J. Urbain, R. Thomas: J. Theor. Biol. 114, 527 (1985)CrossRefGoogle Scholar
  88. G.W. Hoffman: J. Theor. Biol. 129, 355 (1987)CrossRefGoogle Scholar
  89. for a recent differential equation approach see U. Behn, J.L. van Hemmen: J. Stat. Phys. 56, 533 (1989)ADSMATHCrossRefGoogle Scholar
  90. 9.57
    D. Chowdhury, D. Stauffer: J. Stat. Phys. 59, 1019 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  91. 9.58
    I. Dayan, D. Stauffer, S. Havlin: J. Phys. A 21, 2473 (1988)ADSMATHCrossRefGoogle Scholar
  92. U. Wiesner: J. Undergrad. Res. Phys. 7, 15 (1988)Google Scholar
  93. 9.59
    A.U. Neumann: Physica A 162, 1 (1989)MathSciNetADSCrossRefGoogle Scholar
  94. 9.60
    R.B. Pandey: J. Stat. Phys. 54, 997 (1989)ADSMATHCrossRefGoogle Scholar
  95. ibid. 61, 235 (1990), and J. Phys. A 23, 3421 (1990)ADSCrossRefGoogle Scholar
  96. Ch.F. Kougias, J. Schulte: J. Stat. Phys. 60, 263Google Scholar
  97. for a differential equation approach see A.R. McLean, T.B.L. Kirkwood: J. Theor. Biol. 147, 177 (1990)CrossRefGoogle Scholar
  98. 9.61
    G. Weisbuch, H. Atlan: J. Phys. A 21, L 189 (1988)ADSCrossRefGoogle Scholar
  99. K. E. Kürten: J. Stat. Phys. 52, 489 (1988)ADSMATHCrossRefGoogle Scholar
  100. 9.62
    I.R. Cohen, H. Atlan: J. Autoimmunity 2, 613 (1989)CrossRefGoogle Scholar
  101. 9.63
    D. Chowdhury, D. Stauffer, P.V. Choudary: J. Theor. Biol. 145, 207 (1990)CrossRefGoogle Scholar
  102. D. Chowdhury, M. Sahimi, D. Stauffer: J. Theor. Biol. 152, 263 (1991)CrossRefGoogle Scholar
  103. See e.g., the Proceedings of the NATO Advanced Workshop on Lattice Gas Methods for PDE’s, ed. by G.D. Dooien, Physica D 47, 1 (1991)Google Scholar
  104. 9.65
    J. Hardy, O. de Pazzis, Y. Pomeau: Phys. Rev A 13, 1949 (1976)ADSCrossRefGoogle Scholar
  105. S. Succi: Comp. Phys. Comm. 47, 173 (1987)ADSMATHCrossRefGoogle Scholar
  106. 9.66
    G.D. Dooien: First Topical Conference on Computational Physics, APS, Boston 1989 (unpublished invited talk)Google Scholar
  107. U. Brosa: J. Physique 51, 1051 (1990)CrossRefGoogle Scholar
  108. 9.67
    J.A.M.S. Duarte, U. Brosa: J. Stat. Phys. 59, 501 (1990)ADSCrossRefGoogle Scholar
  109. 9.68
    S. Chen, K. Diemer, G.D. Dooien, K. Eggert, C. Fu, S. Gutman, B. Travis, in Ref. [9.64]Google Scholar
  110. 9.69
    G.A. Kohring: J. Stat. Phys. 63, 411 (1991); J. Physique I 1, 87 (1991) and preprintsADSCrossRefGoogle Scholar
  111. 9.70
    B.M. Bhogosian: Computers in Physics (APS), Jan/Feb. 1990Google Scholar
  112. 9.71
    D.H. Rothman: Geophys. 53, 509 (1988)CrossRefGoogle Scholar
  113. S. Succi, A. Cancelliere, C. Chang, E. Foti, M. Gramignani, D. Rothman, in: Computational Methods in Subsurface Hydrology, ed. by G. Gambolati, A. Rinaldo, C.A. Brebbia, W.G. Gray, G.F. Pinder (Springer, Berlin, Heidelberg 1990)Google Scholar
  114. J.P. Boon, in: Correlations and Connectivity — Geometric Aspects of Physics, Chemistry and Biology ed. by H.E. Stanley, N. Ostrowsky (Kluwer, Dordrecht 1990), p. 56 and D. Stauffer, p. 18Google Scholar

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  • Dietrich Stauffer

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