Skip to main content
SpringerLink
Log in

Cellular Automata

  • Chapter
Fractals and Disordered Systems

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Wolfram: Rev. Mod. Phys. 55, 601 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. N.H. Packard, S. Wolfram: J. Stat. Phys. 38, 901 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  3. S. Wolfram: Theory and Applications of Cellular Automata (World Scientific, Singapore 1986)

    MATH  Google 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 preparation

    Google Scholar 

  5. G.Y. Vichniac: Physica D 10, 96 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  6. T. Toffoli, M. Margolus: Cellular Automata Machines (MIT Press, Cambridge MA 1987)

    Google Scholar 

  7. P. Bak, C. Tang, K. Wiesenfeld: Phys. Rev. Lett. 59, 381 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  8. P. Bak: Physica D 38, 5 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  9. P. Grassberger, S.S. Manna: J. Physique 51, 1077 (1990), with further references

    Article  Google Scholar 

  10. S.A. Kauffman: J. Theor. Biol. 22, 437 (1969)

    Article  MathSciNet  Google Scholar 

  11. U. Keller, B. Thomas, H.J. Pohley: J. Stat. Phys. 52, 1129 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. B. Derrida, Y. Pomeau: Europhys. Lett. 1, 59 (1986)

    Article  Google Scholar 

  13. B. Derrida, G. Weisbuch: J. Physique 47, 1297 (1986)

    Article  Google Scholar 

  14. J.G. Zabolitzky: Physica A 163, 447 (1990)

    Article  ADS  MATH  Google Scholar 

  15. B. Derrida, D. Stauffer: Europhys. Lett. 2, 739 (1986)

    Article  ADS  Google Scholar 

  16. D. Stauffer: Phil. Mag. B 56, 510 (1987)

    Google Scholar 

  17. U.M.S. Costa: J. Phys. A 20, L 583 (1987)

    Article  ADS  Google Scholar 

  18. B. Derrida, G. Weisbuch: Europhys. Lett. 4, 657 (1987)

    Article  ADS  Google Scholar 

  19. H.E. Stanley, D. Stauffer, J. Kertesz, H.J. Herrmann: Phys. Rev. Lett. 59, 2326 (1987)

    Article  ADS  Google Scholar 

  20. A. Coniglio, L. de Arcangelis, H.J. Herrmann, N. Jan: Europhys. Lett. 8, 315 (1989)

    Article  ADS  Google Scholar 

  21. G. Le Caer: Physica A 159, 329 (1989)

    Article  ADS  Google Scholar 

  22. L. de Arcangelis, H.J. Herrmann, A. Coniglio: J. Phys. A 23, L265 (1990)

    Article  Google Scholar 

  23. A. Coniglio, N. Jan, D. Stauffer: J. Phys. A 20, L1103 (1987)

    Article  ADS  Google Scholar 

  24. L.R. da Silva, H.J. Herrmann: J. Stat. Phys. 52, 463 (1988)

    Article  ADS  Google Scholar 

  25. D. Stauffer: Physica D 38, 341 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  26. 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. A. Hansen, S. Roux: Physica A 160, 275 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  28. L. de Arcangelis: J. Phys. A 20, L369 (1987)

    Article  Google Scholar 

  29. A. Hansen: J. Phys. A 21, 2481 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. S.P. Obukhov, D. Stauffer: J. Phys. A 22, 1715 (1989)

    Article  ADS  Google Scholar 

  31. 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)

    Article  ADS  Google Scholar 

  33. K.E. Kürten, in: Models of Brain Functions (Cambridge University Press, Cambridge 1990)

    Google Scholar 

  34. D. Stauffer: Physica A 157, 645 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. S.S. Manna, D. Stauffer: Physica A 162, 176 (1990)

    Article  ADS  Google Scholar 

  36. R.W. Gerling: Physica A 162, 187 and 196 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  37. D. Stauffer: J. Phys. A 23, 5933 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  38. H.A. Gutowitz, J.D. Victor, B.W. Knight: Physica D 28, 18 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. P.M. Binder: J. Phys. A 24, L31 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  40. J. Adler: Physica A 171, 453 (1991)

    Article  ADS  Google Scholar 

  41. S.A. Kauffman, The Origin of Order, Oxford University Press, New York 1993

    Google 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)

    Article  Google Scholar 

  43. P. Bak, M. Creutz, in: Fractals in Science, ed. by A. Bunde and S. Havlin (Springer, Heidelberg, New York 1994)

    Google Scholar 

  44. P. Bak, K. Sneppen: Phys. Rev. Lett. 71, 4087 (1993)

    Article  ADS  Google Scholar 

  45. T.S. Ray, N. Jan: Phys. Rev. Lett. 72, 4048 (1994)

    Article  ADS  Google Scholar 

  46. for game of life see P. Bak: Physica A 191, 41 (1992)

    Article  ADS  Google Scholar 

  47. R.J. de Boer, L.A. Segel, A.S. Perelson: J. Theor. Biol. 155, 295 (1992)

    Article  Google 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. J. Stewart, F.J. Varela: J. Theor. Biol. 153, 477 (1991)

    Article  Google Scholar 

  50. D. Stauffer, G. Weisbuch: Physica A 180, 42 (1992)

    Article  ADS  Google Scholar 

  51. S. Dasgupta: Physica A 189, 403 (1992)

    Article  ADS  Google Scholar 

  52. M. Deffner: Physica A 195, 279 (1993)

    Article  ADS  Google Scholar 

  53. R.M. Zorzenon dos Santos: Physica A 196, 12 (1993)

    Article  ADS  MATH  Google Scholar 

  54. M. Sahimi, D.Stauffer: Phys. Rev. Lett. 71, 4271 (1993)

    Article  ADS  Google Scholar 

  55. D. Chowdhury, V. Deshpande, D. Stauffer: Int. J. Mod. Phys. C 5, 1049 (1994)

    Article  ADS  Google Scholar 

  56. B. Barral, H. Chate, P. Manneville: Phys. Lett. A 163, 279 (1992), citing earlier papers of this group

    Article  ADS  Google Scholar 

  57. J. Hemmingson, G.W. Peng: J. Phys. A 27, 2735 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  58. G.A. Kohring: Physica A 186, 97 (1992)

    Article  ADS  Google 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)

    Article  ADS  Google Scholar 

  60. R. Benzi, S. Succi, M. Vergassola: Phys. Rep. 222, 3 (1992)

    Article  Google Scholar 

  61. K. Nagel, M. Schreckenberg: J. Physique I 2, 2221 (1992)

    Article  ADS  Google Scholar 

  62. T. Nagatani: J. Phys. Soc. Jpn. 63, 1228 (1994) and his earlier papers

    Article  ADS  Google Scholar 

  63. J.B. Rundle, W. Klein: J. Stat. Phys. 72, 405 (1993)

    Article  ADS  MATH  Google Scholar 

  64. B. Chopard, P. Luthi, M. Droz: Phys. Rev. Lett. 72, 1384 (1994)

    Article  ADS  Google Scholar 

  65. J.R. Weimar, J.P. Boon: Phys. Rev. E 49, 1749 (1994)

    Article  ADS  Google Scholar 

  66. C.F. Baillie, D.A. Johnston: Phys. Lett. B 326, 51 (1994)

    Article  ADS  Google Scholar 

  67. F. Bagnoli, R. Rechtman, S. Ruffo: Phys.Lett. A 172, 34 (1992)

    Article  ADS  Google Scholar 

  68. Ph. Blanchard, D. Gandolfo: J. Stat. Phys. 73, 399 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  69. G.A. Kohring: Physica A 182, 320 (1992)

    Article  ADS  Google Scholar 

  70. B. Voorhees, S. Bradshaw: Physica D 27, 152 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  71. D. Makowiec: Z. Physik B 95, 519 (1994)

    Article  ADS  Google Scholar 

  72. U. Frisch, B. Hasslacher, Y. Pomeau: Phys. Rev. Lett. 56, 1505 (1986)

    Article  ADS  Google Scholar 

  73. H.J. Herrmann: J. Stat. Phys. 45, 145 (1986)

    Article  ADS  Google Scholar 

  74. J.G. Zabolitzky, H.J. Herrmann: J. Comp. Phys. 76, 426 (1988)

    Article  ADS  MATH  Google Scholar 

  75. D. Stauffer, F.W. Hehl, V. Winkelmann, J.G. Zabolitzky: Computer Simulation and Computer Algebra, 2nd ed. (Springer, Berlin, Heidelberg 1989)

    Book  Google Scholar 

  76. M. Creutz: Phys. Rev. Lett. 50, 1411 (1983); Ann. Physics 167 62 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  77. M. Schulte, W. Stiefelhagen, E.S. Demme: J. Phys. A 20, L1023 (1987)

    Article  ADS  Google Scholar 

  78. C. Moukarzel, N. Parga: J. Phys. A 22, 943 (1989)

    Article  ADS  Google Scholar 

  79. C. Moukarzel: J. Phys. A 22, 4487 (1989)

    Article  ADS  Google Scholar 

  80. H.J. Herrmann, H.O. Carmesin, D. Stauffer: J. Phys. A 20, 4939 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  81. S.C. Glotzer, D. Stauffer, S. Sastry: Physica A 164, 1 (1990), and Ref. [9.9]

    Article  ADS  Google Scholar 

  82. D. Stauffer: J. Phys. A 23, 1847 (1990); J. Phys. A 24, 909 (1991), (review, see in particular Fig. 4 there)

    Article  MathSciNet  ADS  Google Scholar 

  83. I.R. Cohen, in: Theories of Immune Networks, ed. by H. Atlan, I.R. Cohen, Springer Ser. Syn., Vol.46 (Springer, Berlin, Heidelberg 1989)

    Chapter  Google 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. H. Atlan: Bull. Math. Biol. 51, 247 (1989)

    Google Scholar 

  87. M. Kauffman, J. Urbain, R. Thomas: J. Theor. Biol. 114, 527 (1985)

    Article  Google Scholar 

  88. G.W. Hoffman: J. Theor. Biol. 129, 355 (1987)

    Article  Google Scholar 

  89. for a recent differential equation approach see U. Behn, J.L. van Hemmen: J. Stat. Phys. 56, 533 (1989)

    Article  ADS  MATH  Google Scholar 

  90. D. Chowdhury, D. Stauffer: J. Stat. Phys. 59, 1019 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  91. I. Dayan, D. Stauffer, S. Havlin: J. Phys. A 21, 2473 (1988)

    Article  ADS  MATH  Google Scholar 

  92. U. Wiesner: J. Undergrad. Res. Phys. 7, 15 (1988)

    Google Scholar 

  93. A.U. Neumann: Physica A 162, 1 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  94. R.B. Pandey: J. Stat. Phys. 54, 997 (1989)

    Article  ADS  MATH  Google Scholar 

  95. ibid. 61, 235 (1990), and J. Phys. A 23, 3421 (1990)

    Article  ADS  Google Scholar 

  96. Ch.F. Kougias, J. Schulte: J. Stat. Phys. 60, 263

    Google Scholar 

  97. for a differential equation approach see A.R. McLean, T.B.L. Kirkwood: J. Theor. Biol. 147, 177 (1990)

    Article  Google Scholar 

  98. G. Weisbuch, H. Atlan: J. Phys. A 21, L 189 (1988)

    Article  ADS  Google Scholar 

  99. K. E. Kürten: J. Stat. Phys. 52, 489 (1988)

    Article  ADS  MATH  Google Scholar 

  100. I.R. Cohen, H. Atlan: J. Autoimmunity 2, 613 (1989)

    Article  Google Scholar 

  101. D. Chowdhury, D. Stauffer, P.V. Choudary: J. Theor. Biol. 145, 207 (1990)

    Article  Google Scholar 

  102. D. Chowdhury, M. Sahimi, D. Stauffer: J. Theor. Biol. 152, 263 (1991)

    Article  Google 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. J. Hardy, O. de Pazzis, Y. Pomeau: Phys. Rev A 13, 1949 (1976)

    Article  ADS  Google Scholar 

  105. S. Succi: Comp. Phys. Comm. 47, 173 (1987)

    Article  ADS  MATH  Google Scholar 

  106. 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)

    Article  Google Scholar 

  108. J.A.M.S. Duarte, U. Brosa: J. Stat. Phys. 59, 501 (1990)

    Article  ADS  Google Scholar 

  109. S. Chen, K. Diemer, G.D. Dooien, K. Eggert, C. Fu, S. Gutman, B. Travis, in Ref. [9.64]

    Google Scholar 

  110. G.A. Kohring: J. Stat. Phys. 63, 411 (1991); J. Physique I 1, 87 (1991) and preprints

    Article  ADS  Google Scholar 

  111. B.M. Bhogosian: Computers in Physics (APS), Jan/Feb. 1990

    Google Scholar 

  112. D.H. Rothman: Geophys. 53, 509 (1988)

    Article  Google 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. 18

    Google Scholar 

Download references

Authors
  1. Dietrich Stauffer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Theoretische Physik, Universität Giessen, Heinrich-Buff-Ring 16, D-35392, Giessen, Germany

    Armin Bunde

  2. Department of Physics, Bar-Ilan University, Ramat Gan, Israel

    Shlomo Havlin

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Stauffer, D. (1996). Cellular Automata. In: Bunde, A., Havlin, S. (eds) Fractals and Disordered Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84868-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-84868-1_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-84870-4

  • Online ISBN: 978-3-642-84868-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation