Journal of Statistical Physics

, Volume 49, Issue 1–2, pp 139–222 | Cite as

Cellular automata and statistical mechanical models

  • Pàl Rujàn


We elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata. Transient and stationary properties of probabilistic automata are linked to surface and bulk properties, respectively, of restricted statistical mechanical systems. It is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata. Some exact results are derived, including duality transformations, exact mappings, “disorder,” and “linear” solutions. Many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties. This approach is considered to be a first step toward the design of fully parallel,probabilistic systems whose computational abilities rely on the cooperative behavior of their components.

Key words

Cellular automata multispin Ising models static and dynamic critical phenomena duality relations exact solutions stochastic processes predictible behavior topological invariants microcanonical simulations 


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  1. 1.
    J. von Neumann,Theory of Self-Reproducing Automata (edited and completed by A. W. Burks) (University of Illinois Press, 1966).Google Scholar
  2. 2.
    E. F. Codd,Cellular Automata (Academic Press, 1968); A. W. Burks,Essays on Cellular Automata (University of Illinois Press, 1970).Google Scholar
  3. 3.
    L. Bienenstock, F. Fogelman, and G. Weisbuch (eds.),Disordered Systems and Biological Organization (Springer-Verlag, 1986); D. J. Depew and B. H. Weber (eds.),Evolution at a Crossroads (MIT Press, 1985); R. L. Dobrushin, V. I. Kryukov, and A. L. Toom (eds.),Locally Interacting Systems and their Application in Biology (Springer-Verlag, 1978); B. Batchelor (ed.),Pattern Recognition (Plenum Press, 1978); T. Poggio, V. Torre, and C. Koch,Nature 317:314 (1985), and references therein.Google Scholar
  4. 4.
    J. E. Hopcroft and J. D. Ullman,Introduction to Automata Theory, Languages, and Computation (Addison-Wesley, 1979); F. Fogelman-Soulie, Y. Robert, and M. Tschuente (eds.),Automata Networks in Computer Science, Theory and Applications (Manchester University Press, 1986); R. Vollmar,Algorithmen in Zellularautomaten (B. G. Teubner Verlag, Stuttgart, 1979).Google Scholar
  5. 5.
    S. Wolfram,Theory and Applications of Cellular Automata (World Scientific, 1986).Google Scholar
  6. 6.
    S. Wolfram,Rev. Mod. Phys. 55:601 (1983); G. Y. Vichniak, inDisordered Systems and Biological Organization, L. Bienenstock, F. Fogelman, and G. Weisbuch, eds. (Springer-Verlag, 1986).Google Scholar
  7. 7.
    H. J. Hilhorst, A. F. Bakker, C. Bruin, A. Compagner, and A. Hoogland,J. Stat. Phys. 34:987 (1984); H. Herrmann, inProceedings of the Statphys' 16 (North-Holland, 1986).Google Scholar
  8. 8.
    W. A. Little,Math. Biosci. 19:101 (1974); W. A. Little and G. L. Shaw,Math. Biosci. 39:281 (1978); J. J. Hopfield,Proc. Natl. Acad. Sci. USA 79:2554 (1982); P. Peretto,Biol. Cybern. 50:51 (1984); D. J. Amit, H. Gutfreund, and H. Sampolinsky,Phys. Rev. A 32:1007 (1985).Google Scholar
  9. 9.
    E. Domany, R. Meir, and W. Kinzel,Europhys. Lett. 2:175 (1986); E. Domany and R. Meir, to be published.Google Scholar
  10. 10.
    B. A. Huberman and T. Hogg,Phys. Rev. Lett. 52:1024 (1984);J. Stat. Phys. 41:115 (1985); U.S. Patent No. 4591980; D. d'Humieres and B. A. Huberman,J. Stat. Phys. 34:361 (1984).Google Scholar
  11. 11.
    T. G. Lewis and W. H. Payne,J. Assoc. Comput. Mach. 20:456 (1973); S. Kirkpatrick and E. P. Stoll,J. Comp. Phys. 40:517 (1981).Google Scholar
  12. 12.
    M. H. Kalos,Phys. Rev. 128:1781 (1962); D. M. Ceperley and M. H. Kalos, inMonte Carlo Methods in Statistical Physics, K. Binder, ed. (Springer, 1979); J. H. Hetherington,Phys. Rev. B 30:2713 (1984).Google Scholar
  13. 13.
    J. Hardy, O. de Pazzis, and Y. Pomeau,J. Math. Phys. 14:470 (1973); U. Frisch, B. Hasslacher, and Y. Pomeau,Phys. Rev. Lett. 56:1505 (1986); D. d'Humieres, P. Lallemand, and U. Frisch,Europhys. Lett. 2:291 (1986).Google Scholar
  14. 14.
    Y. Pomeau,J. Phys. A 17:L415 (1984).Google Scholar
  15. 15.
    G. Vichniac, inTheory and Applications of Cellular Automata, S. Wolfram, ed. (World Scientific, Singapore, 1986).Google Scholar
  16. 16.
    H. Herrmann,J. Stat. Phys. 45:154 (1986).Google Scholar
  17. 17.
    E. Goles and G. Y. Vichniac,J. Phys. A 19:L961 (1986).Google Scholar
  18. 18.
    D. A. Ackley, G. E. Hinton, and T. J. Sejnowski,Cognitive Sci. 9:147 (1985); G. E. Hinton, T. J. Sejnowski, and D. A. Ackley, Boltzmann machines: Constraint satisfaction networks that learn, Technical Report CMU-CS-84-119, Carnegie-Mellon University (May 1984).Google Scholar
  19. 19.
    R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, 1982).Google Scholar
  20. 20.
    O. Perron,Math. Ann. 64:248 (1907); S. B. Frobenius,Preuss. Akad. Wiss. 471 (1908); E. Seneta,Non-negative Matrices (Wiley, 1973); P. Ròzsa,Linear Algebra and Applications (Müszaki Kiadò, 1974) (in Hungarian).Google Scholar
  21. 21.
    H. A. Kramers and G. H. Wannier,Phys. Rev. 60:263 (1941); R. J. Baxter,J. Stat. Phys. 19:461 (1978); P. Rujàn,Physica A 96:379 (1979); R. J. Baxter and P. J. Forrester,J. Phys. A 17:2675 (1984).Google Scholar
  22. 22.
    F. Haake and K. Thol,Z. Phys. B 40:219 (1980).Google Scholar
  23. 23.
    C. Domb and M. S. Green (eds.),Phase Transitions and Critical Phenomena, Vol.6 (Academic Press, 1976).Google Scholar
  24. 24.
    P. Nightingale,J. Appl. Phys. 53:7927 (1982); H. H. Roomany and H. W. Wyld,Phys. Rev. D 21:3341 (1980).Google Scholar
  25. 25.
    M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz (eds.) (Academic Press, 1983).Google Scholar
  26. 26.
    F. Alcaraz, M. N. Barber, J. Kuti, and P. Rujàn, to be published.Google Scholar
  27. 27.
    P. Rujàn, inNon-linear Equations in Classical and Quantum Field Theory, N. Sanchez, ed. (Springer-Verlag, 1985), p. 286; J. M. Maillard, inProceedings of the 2nd UC Conference on Statistical Physics (March 26–29 1986, Davis, California).Google Scholar
  28. 28.
    C. H. Bennett and G. Grinstein,Phys. Rev. Lett. 55:657 (1985).Google Scholar
  29. 29.
    H. A. Kramers and G. H. Wannier,Phys. Rev. 60:252 (1941).Google Scholar
  30. 30.
    W. Kinzel,Z. Phys. B 60:205 (1985).Google Scholar
  31. 31.
    G. Grinstein, C. Jayaprakash, and Y. He,Phys. Rev. Lett. 55:2527 (1985).Google Scholar
  32. 32.
    S. Redner,J. Stat. Phys. 25:15 (1981).Google Scholar
  33. 33.
    Z. Ràcz,Phys. Rev. B 21:4012 (1980); R. J. Baxter, I. G. Enting, and S. K. Tsang,J. Stat. Phys. 22:465 (1980).Google Scholar
  34. 34.
    S. J. Chang and J. Wright,Phys. Rev. A 23:1419 (1981).Google Scholar
  35. 35.
    S. Wolfram,Commun. Math. Phys. 96:15 (1984).Google Scholar
  36. 36.
    M. Gardner,Sci. Am. 223(4):120 (1970); E. R. Berlekamp, J. H. Conway, and R. K. Guy,Winning Ways, Vol. 2 (Academic Press, 1982); M. Gardner,Wheels, Life and Other Mathematical Amusements (Freeman, 1982).Google Scholar
  37. 37.
    P. Grassberger,Physica D 10:52 (1984).Google Scholar
  38. 38.
    C. Fan and B. McCoy,Phys. Rev. 182:614 (1969); J. F. Fernandez,Phys. Rev. B 16:5125 (1977); R. Bruinsma and G. Aeppli,Phys. Rev. Lett. 50:1494 (1983); G. Györgyi and P. Rujàn,J. Phys. C 17:4207 (1984).Google Scholar
  39. 39.
    C. L. Henley,Phys. Rev. Lett. 54:2030 (1985); D. Kutasov, A. Aharony, E. Domany, and W. Kinzel,Phys. Rev. Lett. 56:2229 (1986); A. Maritan and A. L. Stella,J. Phys. A 19:L269 (1986).Google Scholar
  40. 40.
    Z. Ràcz,Phys. Rev. Lett. 55:1707 (1985); Z. Ràcz and M. Plischke,Acta Phys. Hung. (May 1987).Google Scholar
  41. 41.
    J. Marro, J. L. Lebowitz, H. Spohn, and M. H. Kalos,J. Stat. Phys. 38:725 (1985).Google Scholar
  42. 42.
    P. C. Hohenberg and B. I. Halperin,Rev. Mod. Phys. 49:435 (1977).Google Scholar
  43. 43.
    F. Y. Wu, Unpublished results.Google Scholar
  44. 44.
    R. Balian and G. Toulouse,Ann. Phys. (N.Y.) 83:28 (1974).Google Scholar
  45. 45.
    T. D. Lee and C. N. Yang,Phys. Rev. 87:410 (1952); M. E. Fisher,Phys. Rev. Lett. 40:1610 (1978); D. A. Kurtze,J. Stat. Phys. 30:15 (1983).Google Scholar
  46. 46.
    R. J. Glauber,J. Math. Phys. 4:294 (1963); B. U. Felderhof,Rep. Math. Phys. 1:215 (1972);2:151 (1972); E. Siggia,Phys. Rev. B 16:2319 (1979); see also H. Falk,J. Math. Phys. 26:692 (1985).Google Scholar
  47. 47.
    W. Kinzel and J. Yeomans,J. Phys. A 14:L163 (1981).Google Scholar
  48. 48.
    H. Yahata,Prog. Theor. Phys. 52:871 (1974); T. Schneider, M. Zannetti, and R. Badii,Phys. Rev. B 31:2941 (1985); T. Schneider and M. Schwartz,Phys. Rev. B 31:7484 (1985).Google Scholar
  49. 49.
    E. Domany and J. E. Gubernatis,Phys. Rev. B 32:3354 (1985); A. Aharony, E. Domany, R. M. Hornreich, T. Schneider, and M. Zannetti,Phys. Rev. B 32:3358 (1985).Google Scholar
  50. 50.
    E. Domany and W. Kinzel,Phys. Rev. Lett. 53:311 (1984).Google Scholar
  51. 51.
    C. J. Hamer,J. Phys. A 15:L675 (1982); H. Takano and Y. Saito,Prog. Theor. Phys. 73:1369 (1985).Google Scholar
  52. 52.
    B. J. Adler and T. E. Wainwright,Phys. Rev. 127:359 (1962); J. Orban and A. Bellemans,J. Chem. Phys. 49:363 (1968); K. Binder and L. P. Landau,Phys. Rev. B 21:1941 (1980).Google Scholar
  53. 53.
    J. Stephenson,J. Math. Phys. 11:420 (1970);Can. J. Phys. 48:2118 (1970);Phys. Rev. B 1:4405 (1970).Google Scholar
  54. 54.
    T. R. Welberry and G. H. Miller,J. Appl. Cryst. 6:87 (1973);8:636 (1982);Acta Cryst. A 34:120 (1978).Google Scholar
  55. 55.
    I. Peschel and V. J. Emery,Z. Phys. B 43:241 (1981); J. Kurmann, H. Thomas, and G. Müller,Physica A 112:235 (1982).Google Scholar
  56. 56.
    I. G. Enting,J. Phys. A 10:1023, 1737 (1977);11:555, 2001 (1978); A. M. W. Verhagen,J. Stat. Phys. 15:219 (1976); P. Rujàn,J. Stat. Phys. 29:231, 247 (1982);43:615 (1984); I. Peschel and F. Rys,Phys. Lett. A 91:187 (1982); R. J. Baxter,J. Phys. A 17:L911 (1984).Google Scholar
  57. 57.
    E. Domany,Phys. Rev. Lett. 52:871 (1984).Google Scholar
  58. 58.
    M. T. Jaeckel and J. M. Maillard,J. Phys. A 18:641, 2271 (1985); F. Y. Wu,J. Stat. Phys. 40:613 (1985).Google Scholar
  59. 59.
    W. Selke, inLecture Notes in Physics, Vol. 206, A. Pekalski and J. Sznajd, eds. (Springer-Verlag, 1984).Google Scholar
  60. 60.
    P. Rujàn,J. Stat. Phys. 29:231,247 (1982);43:615 (1984).Google Scholar
  61. 61.
    D. Dhar,Phys. Rev. Lett. 49:959 (1982);51:853 (1983).Google Scholar
  62. 62.
    V. Hakim and J. P. Nadal,J. Phys. A 16:L213.Google Scholar
  63. 63.
    B. Derrida and J. P. Nadal,J. Phys. (Paris) Lett. 45:L701 (1984).Google Scholar
  64. 64.
    J. M. Maillard and R. Rammal,J. Phys. A 16:353;18:833 (1985).Google Scholar
  65. 65.
    I. G. Enting,J. Phys. A 11:2001 (1978); A. Georges, D. Hansel, P. Le Doussal, and J. M. Maillard,J. Phys. A 19:L329, 1001 (1986).Google Scholar
  66. 66.
    D. Dhar and J. M. Maillard,J. Phys. A 18:L383 (1985).Google Scholar
  67. 67.
    M. Y. Choi and B. A. Huberman,J. Phys. A 17:L765 (1984).Google Scholar
  68. 68.
    C. Fan and F. Y. Wu,Phys. Rev. 179:560 (1969); E. Müller-Hartmann and J. Zittartz,Z. Phys. B 27:261 (1977); J. Villain and P. Bak,J. Phys. (Paris) 42:657 (1981).Google Scholar
  69. 69.
    J. L. Cardy,J. Phys. A 15:L593 (1982).Google Scholar
  70. 70.
    R. Savit,Rev. Mod. Phys. 52:453 (1980).Google Scholar
  71. 71.
    H. Giacomini,J. Phys. A 18:1499,1505,1579 (1985).Google Scholar
  72. 72.
    I. M. Krichever,Funct. Anal. Appl. Math. 15:92 (1981); K. Sogo, M. Uchimani, Y. Akutsu, and M. Wadati,Prog. Theor. Phys. 68:508 (1982); J. M. Maillard and T. Garel,J. Phys. A 17:1251,1257 (1984).Google Scholar
  73. 73.
    A. B. Zamolodchikov,Commun. Math. Phys. 69:165 (1979);79:489 (1981); M. T. Jaeckel and J. M. Maillard,J. Phys. A 15:1309 (1982).Google Scholar
  74. 74.
    W. Kinzel, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, Bristol, 1983).Google Scholar
  75. 75.
    L. S. Schulmann and P. E. Seiden,J. Stat. Phys. 27:83 (1982); H. Gutowitz, J.Victor, B. Knight,Physica D, to be published.Google Scholar
  76. 76.
    H. Falk,Physica A 119:580 (1983).Google Scholar
  77. 77.
    J. Kogut,Rev. Mod. Phys. 51:659 (1979).Google Scholar
  78. 78.
    A. Patkòs and P. Rujàn,J. Phys. A 18:1765 (1985).Google Scholar
  79. 79.
    F. Beleznay,J. Phys. A 19:551 (1986).Google Scholar
  80. 80.
    H. W. Blötte and H. J. Hilhorst,J. Phys. A 15:L631 (1982); B. Nienhuis, H. J. Hilhorst, and H. W. Blötte,J. Phys. A 17:3559 (1984).Google Scholar
  81. 81.
    P. Rujàn and A. Patkòs,Phys. Lett. B 129:437 (1983).Google Scholar
  82. 82.
    E. Jen,J. Stat. Phys. 43:243 (1986).Google Scholar
  83. 83.
    E. H. Lieb and F. Y. Wu, inPhase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds. (Academic Press, 1972); P. W. Kasteleyn, inFundamental Problems in Statistical Mechanics III, E. G. D. Cohe, ed. (North-Holland, 1975).Google Scholar
  84. 84.
    M. Creutz,Ann. Phys. (N.Y.) 167:62 (1986).Google Scholar
  85. 85.
    M. Novotny,J. Appl. Phys. 53:7997 (1982).Google Scholar
  86. 86.
    Y. Aharonov and M. Schwartz,Phys. Rev. Lett. 48:1137 (1982);Phys. Lett. B 157:57 (1985); inQuantum Concepts in Space and Time, O. Penrose, ed. (Oxford Press, 1986).Google Scholar
  87. 87.
    P. Rujàn and M. Schwartz, to be published.Google Scholar
  88. 88.
    J. W. Essam, inPhase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, eds. (Academic Press, 1972); D. Stauffer,Phys. Rep. 54:1 (1979).Google Scholar
  89. 89.
    M. Minsky and S. Papert,Perceptrons (MIT Press, 1969).Google Scholar
  90. 90.
    P. Bak,Rep. Prog. Phys. 45:587 (1982).Google Scholar
  91. 91.
    V. L. Pokrovsky and G. V. Uimin,J. Phys. C 11:3535 (1978); J. Hubbard,Phys. Rev. B 17:494 (1978); S. Aubry, inSolitons and Condensed Matter Physics, A. R. Bishop and T. Schneider, eds. (Springer-Verlag, 1979); P. Bak and R. Bruinsma,Phys. Rev. Lett. 49:249 (1982).Google Scholar
  92. 92.
    B. Derrida, Y. Pomeau, and J. Vannimenus,J. Phys. C 11:4749 (1978).Google Scholar
  93. 93.
    J. Kanamori,J. Phys. Soc. Japan 53:250 (1984).Google Scholar
  94. 94.
    J. Villain,Ferroelectrics 66:143 (1986).Google Scholar
  95. 95.
    S. Kirkpatrick and G. Toulouse,J. Phys. (Paris) 46:1277 (1985), and references therein.Google Scholar
  96. 96.
    M. S. Dresselhaus, G. Dresselhaus, J. E. Fischer, and M. J. Moran (eds.),Intercalated Graphite, Vol. 20 (Elsevier, 1983); M. S. Dresselhaus and G. Dresselhaus,Adv. Phys. 30:139 (1981).Google Scholar
  97. 97.
    S. Ostlund,Phys. Rev. B 24:398 (1981); D. A. Huse,Phys. Rev. B 24:5180 (1981).Google Scholar
  98. 98.
    D. A. Huse, M. E. Fisher, and J. Yeomans,Phys. Rev. B 23:180 (1981).Google Scholar
  99. 99.
    R. Rammal and G. Toulouse,J. Phys. (Paris) Lett. 44:L13 (1983); J. V. Josè,J. Phys. A 16:L205 (1983); S. Grossmann, inNon-Equilibrium Cooperative Phenomena in Physics and Related Fields, M. G. Velarde, ed. (ASI-Series, Plenum Press, 1983); R. B. Stinchcombe, inScaling Phenomena in Disordered Systems (NATO Advanced Study Institute, 1985).Google Scholar
  100. 100.
    M. Keane,Z. Wahr. Verw. Geb. 10:335 (1968); F. M. Dekking,Z. Wahr. Verw. Geb. 41:221 (1978).Google Scholar
  101. 101.
    F. Axel, J. P. Allouche, M. Klèman, M. Mendès-France, and J. Peyrière,J. Phys. Coll. C3:181 (1986).Google Scholar
  102. 102.
    J. M. Luck and D. Petritis,J. Stat. Phys. 42:289 (1986), and references therein.Google Scholar
  103. 103.
    J. Luck,J. Phys. A 20:1259 (1987); C. Godrèche, J. M. Luck, and H. Orland,J. Stat. Phys. 45:777 (1986).Google Scholar
  104. 104.
    B. Derrida, inChance and Matter (Les Houches Summer School, July 1986); B. Derrida and Y. Pomeau,Europhys. Lett. 1:45 (1986); B. Derrida and D. Stauffer, unpublished.Google Scholar
  105. 105.
    B. Derrida and H. Flybjerg,J. Phys. A 19:L1003 (1986).Google Scholar
  106. 106.
    M. Kerszberg and A. Bergman, inComputer Simulation in Brain Science (Copenhagen, 1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Pàl Rujàn
    • 1
  1. 1.Institut für Festkörperforschung der Kernforschungsanlage Jülich GmbHJülichFederal Republic of Germany

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