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Phase transitions of cellular automata

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Zeitschrift für Physik B Condensed Matter

Abstract

Cellular automata (CA) are simple mathematical models of the dynamics of discrete variables in discrete space and time, with applications in nonequilibrium physics, chemical reactions, population dynamics and parallel computers. Phase transitions of stochastic CA with absorbing states are investigated. Using transfermatrix scaling the phase diagrams, critical properties and the entropy of one-dimensional CA are calculated. The corners of the phase diagrams reduce to deterministic CA discussed by Wolfram (Rev. Mod. Phys.55, 601 (1983)). Three-state models are introduced and, for special cases, exactly mapped onto two-state CA. The critical behaviour of other threestate models with one or two absorbing states and with immunization is investigated. Finally CA with competing reactions and/or with disorder are studied.

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References

  1. Wu, F.Y.: Rev. Mod. Phys.54, 235 (1982)

    Google Scholar 

  2. Glauber, R.: J. Math. Phys.4, 234 (1963)

    Google Scholar 

  3. Binder, K.: Monte Carlo methods in statistical physics. In: Topics in Current Physics. Binder, K. (ed.), Vol. 7. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  4. Physica10 D. 1–247 (1984)

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

    Google Scholar 

  6. Wolfram, S.: in Ref. [4] Physica10 , p. 1 (1984)

  7. For a review see Kinzel, W.: In: Percolation structures and processes. Deutsch, G., Zallen, R., Adler, J. (eds.). Bristol: Adam Hilger 1983

    Google Scholar 

  8. Grassberger, P., Torre, A. de la: Ann. Phys.122, 373 (1979); Grassberger, P.: Z. Phys. B-Condensed Matter47, 365 (1982)

    Google Scholar 

  9. Grassberger, P.: Nucl. Phys. B125, 91 (1977); Cardy, J.L., Sugar, R.: J. Phys. A13, L423 (1980)

    Google Scholar 

  10. Haken, H.: Synergetics. Berlin, Heidelberg, New York: Springer 1978; Nicolis, G., Prigogine, I.: Self organization in non-equilibrium. New York: Wiley 1977

    Google Scholar 

  11. Verhagen, A.M.W.: J. Stat. Phys.15, 213 (1976); Enting, I.G.: J. Phys. C10, 1379 (1977); A11, 2001 (1978); Rujan, P.: J. Stat. Phys.29, 247 (1982);34, 615 (1984)

    Google Scholar 

  12. Domany, E., Kinzel, W.: Phys. Rev. Lett.53, 311 (1984)

    Google Scholar 

  13. Choi, M.Y., Huberman, B.A.: J. Phys. A17, L765 (1984)

    Google Scholar 

  14. Kinzel, W., Yeomans, J.: J. Phys. A14, L163 (1981)

    Google Scholar 

  15. Janssen, H.K.: A multistate CA belongs to the same universality class as a twostate CA if it has a single absorbing state and if all its components can diffuse (unpublished)

  16. Cardy, J.: J. Phys. A16, L709 (1983)

    Google Scholar 

  17. Grassberger, P., Krause, F., Twer, T.v.d.: J. Phys. A17, L105 (1984)

    Google Scholar 

  18. Domany, E., Kinzel, W.: Phys. Rev. Lett.47, 5 (1981)

    Google Scholar 

  19. Some of the following results have already been presented in Ref. [12]

    Google Scholar 

  20. Frobenius, S.B.: Preuss. Akad. Wiss. 471 (1908)

  21. Grassberger, P.: in Ref. [4] Physica10 , p. 52 (1984)

  22. Griffiths, R.B.: J. Math. Phys.8, 484 (1967)

    Google Scholar 

  23. As in Fig. 2 the determination of the phase boundaries close to class 30CA is not possible due to strong finite size effects

  24. see e.g. Ma, S.K.: Modern theory of critical phenomena. London: Benjamin 1976

    Google Scholar 

  25. Janssen, H.K.: Z. Phys. B-Condensed Matter42, 151 (1981)

    Google Scholar 

  26. Hamer, C.J.: J. Phys. A15, L675 (1982)

    Google Scholar 

  27. Barber, M.N.: In: Phase transitions and critical phenomena. Domb, C., Lebowitz, J.L. (eds.), Vol. 8. New York: Academic Press 1983

    Google Scholar 

  28. see e.g. Ott, E.: Rev. Mod. Phys.53, 655 (1981)

    Google Scholar 

  29. Stephenson, J.: J. Math. Phys.11, 420 (1970); Phys. Rev. B1, 4405 (1970)

    Google Scholar 

  30. Kinzel, W., Selke, W., Binder, K.: Surf. Sci.121, 13 (1982)

    Google Scholar 

  31. Caroll, C.E.: Surf. Sci.32, 119 (1972)

    Google Scholar 

  32. Harris, A.B.: J. Phys. C7, 1671 (1974)

    Google Scholar 

  33. Edwards, S.F., Anderson, P.W.: J. Phys. F5, 965 (1975) for a recent review see: Fischer, K.H.: Phys. Status Solidi B116, 357 (1983)

    Google Scholar 

  34. Sherrington, D., Kirkpatrick, S.: Phys. Rev. Lett.35, 1792 (1975)

    Google Scholar 

  35. Tanaka, E., Edwards, S.F.: J. Phys. F10, 2471 (1980); Bray, A.J., Moore, M.A.; J. Phys. C13, L469 (1980)

    Google Scholar 

  36. Mezard, M., Parisi, G., Sourlas, N., Toulouse, G., Virasoro, M.: Phys. Rev. Lett.52, 1156 (1984)

    Google Scholar 

  37. Hopfield, J.J.: Proc. Natl. Acad. Sci. USA79, 2554 (1982)

    Google Scholar 

  38. Anderson, P.W.: Proc. Natl. Acad. Sci. USA80, 3386 (1983)

    Google Scholar 

  39. Morgenstern, I., Binder, K.: Phys. Rev. Lett.43, 1615 (1979); Phys. Rev. B22, 288 (1980) see also: Binder, K., Kinzel, W.: Lectures Notes in Physics. Vol. 192, p. 279. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  40. Binder, K., Schröder, K.: Phys. Rev. B14, 2142 (1976); Kinzel, W.: Phys. Rev. B19, 4594 (1979)

    Google Scholar 

  41. Kinzel, W., Binder, K.: Phys. Rev. B29, 1300 (1984)

    Google Scholar 

  42. Young, A.P.: J. Phys. C18, L517 (1984)

    Google Scholar 

  43. Domb, C.: Proc. R. Soc. London Ser. A196, 36 (1949)

    Google Scholar 

  44. Wilkinson, J.H.: Algebraic eigenvalue problems. Oxford: Claredon Press 1965

    Google Scholar 

  45. Nightingale, P.: J. Appl. Phys.53, 7927 (1982)

    Google Scholar 

  46. Privman, V., Fisher, M.E.: J. Phys. A16, L295 (1983)

    Google Scholar 

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  1. Institut für Festkörperforschung der Kernforschungsanlage Jülich, Postfach 1913, D-5170, Jülich 1, Federal Republic of Germany

    W. Kinzel

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Kinzel, W. Phase transitions of cellular automata. Z. Physik B - Condensed Matter 58, 229–244 (1985). https://doi.org/10.1007/BF01309255

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  • DOI: https://doi.org/10.1007/BF01309255

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