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A cell dynamical system model of chemical turbulence

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Abstract

A cellular-automaton-like caricature of chemical turbulence on an infinite one-dimensional lattice is studied. The model exhibits apparently “turbulent” space-time patterns. To make this statement precise, the following problems or points are discussed: (1) The infinite-system-size limit of such cell-dynamical systems and its observability is defined. (2) It is proved that the invariant state in the large-system-size limit of the “turbulent” phase exhibits spatial patterns governed by a Gibbs random field. (3) Potential characteristics of “turbulent” space-time patterns are critically surveyed and a working definition of (weak) turbulence is proposed. (4) It is proved that the invariant state of the ‘turbulent” phase is actually (weak) turbulent. Furthermore, we conjecture that the turbulent phase of our model is an example of a K system that is not Bernoulli.

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References

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

    Google Scholar 

  2. J. Demongeot, E. Cole, and M. Tchuente, eds.,Dynamical Systems and Cellular Automata (Academic Press, New York, 1985).

    Google Scholar 

  3. K. Kaneko,Prog. Theor. Phys. 72:480 (1984); R. J. Deissler,Phys. Lett. 100A:451 (1984); R. Kapral,Phys. Rev. A 32:1076 (1985).

    Google Scholar 

  4. Y. Oono and M. Kohmoto,Phys. Rev. Lett. 55:2927 (1985).

    Google Scholar 

  5. D. Ruelle,Thermodynamic Formalism (Benjamin, New York, 1978).

    Google Scholar 

  6. D. Ruelle and F. Takens,Commun. Math. Phys. 20:167 (1971);23:343 (1971); S. Newhouse, D. Ruelle, and F. Takens,Commun. Math. Phys. 64:35 (1978).

    Google Scholar 

  7. J. B. McLaughlin and P. C. Martin,Phys. Rev. A 12:186 (1975).

    Google Scholar 

  8. E. N. Lorenz,J. Amos. Sci. 20:130 (1963).

    Google Scholar 

  9. T. Y. Li and J. A. Yorke,Am. Math. Month. 82:985 (1975).

    Google Scholar 

  10. R. M. May,Nature 261:459 (1976).

    Google Scholar 

  11. D. Ruelle,Commun. Math. Phys. 93:285 (1984); P. Constantine and C. Foias,Commun. Pure Appl. Math. 38:1 (1985).

    Google Scholar 

  12. A. R. Bishop, K. Fesser, P. S. Lomdahl, W. C. Kerr, M. B. William, D. F. Dubois, H. A. Rose, and B. Hafizi,Phys. Rev. Lett. 51:335 (1983); A. R. Bishop, K. Fesser, and D. S. Lomdahl,Physica 7D:259 (1983).

    Google Scholar 

  13. Y. Oono,Prog. Theor. Phys. 60:1583 (1979).

    Google Scholar 

  14. Y. Oono and M. Osikawa,Prog. Theor. Phys. 64:1804 (1980); M. Osikawa and Y. Oono,Publ. RIMS (Kyoto Univ.) 17:165 (1981).

    Google Scholar 

  15. J. Guckenheimer,Commun. Math. Phys. 70:133 (1979).

    Google Scholar 

  16. A. K. Zvonkin and L. A. Levin,Russ. Math. Surv. 25(6):83 (1970).

    Google Scholar 

  17. J. Ford, inLong-Time Prediction in Dynamics, C. W. Horton, Jr., L. E. Reichl, and V. G. Szebehely, eds. (Wiley, New York, 1982).

    Google Scholar 

  18. A. A. Brudno,Trans. Moscow Math. Soc. 44:127 (1983).

    Google Scholar 

  19. S. Wolfram,Physica 10D:35 (1984); inTheory and Applications of Cellular Automata, S. Wolfram, ed. (World Scientific, Singapore, 1986).

    Google Scholar 

  20. N. Packard, inDynamical Systems and Cellular Automata, J. Demongeot, E. Cole, and M. Tchuente, eds. (Academic Press, New York, 1985).

    Google Scholar 

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

    Google Scholar 

  22. Y. Oono and S. Puri,Phys. Rev. Lett. 58:836 (1987); R. Kapral and G. L. Oppo, preprint (1986).

    Google Scholar 

  23. Y. Takahashi and Y. Oono,Prog. Theor. Phys. 71:851 (1984).

    Google Scholar 

  24. S. Willson,Math. Systems Theor. 9:132 (1975).

    Google Scholar 

  25. M. Miyamoto,J. Math. Kyoto Univ. 19:525 (1979).

    Google Scholar 

  26. P. Walters,An Introduction to Ergodic Theory (Springer, New York, 1982).

    Google Scholar 

  27. Y. Kuramoto,Chemical Oscillation, Waves and Turbulence (Springer, New York, 1984).

    Google Scholar 

  28. N. Koppel, inProceedings of the International Congress of Mathematicians, Vol. 2, Section 18, Z. Ciesielski and C. Olech, eds. (Polish Scientific Publishers, Warsaw, 1984).

    Google Scholar 

  29. Y. Kuramoto and T. Yamada,Prog. Theor. Phys. 56:679 (1976); T. Yamada and Y. Kuramoto,Prog. Theor. Phys. 56:681 (1976); Y. Kuramoto,Prog. Theor. Phys. Suppl. 64:346 (1978); see also Y. Kuramoto,Prog. Theor. Phys. 63:1885 (1980).

    Google Scholar 

  30. G. J. Sivashinsky,Acta Astronaut. 4:1177 (1977).

    Google Scholar 

  31. H. Yamazaki, Y. Oono, and K. Hirakawa,J. Phys. Soc. Jpn. 44:335 (1978); H. Yamazaki, Y. Oono, and K. Hirakawa,J. Phys. Soc. Jpn. 46:721 (1979).

    Google Scholar 

  32. J. J. Tyson, inThe Belousov-Zhabotinsky Reaction, S. Levine, ed. (Springer, Berlin, 1976).

    Google Scholar 

  33. M. Fujisaka and T. Yamada,Z. Phys. 28B:239 (1977).

    Google Scholar 

  34. T. Niwa,J. Stat. Phys. 18:3 (1978).

    Google Scholar 

  35. Y. Oono, T. Kohda, and H. Yamazaki,J. Phys. Soc. Jpn. 48:738 (1980).

    Google Scholar 

  36. T. Winfree,Sci. Am. 230:82 (1974).

    Google Scholar 

  37. F. Madore and W. L. Freeman,Science 222:615 (1983).

    Google Scholar 

  38. P. Grassberger, F. Krause, and T. von der Twen,J. Phys. A 17:L105 (1984).

    Google Scholar 

  39. S. Smale,Bull. Am. Math. Soc. 73:747 (1967); R. Bowen,On Axiom A Diffeomorphisms (AMS, Providence, Rhode Island, 1977).

    Google Scholar 

  40. J. Milnor and W. Thurston, On Iterated Maps of the Interval I, II, mimeographed note Princeton (1977); Y. Takahashi,Osaka J. Math. 20:599 (1983).

  41. M. Loeve,Probability Theory I (Springer, New York, 1977), Section 4.3.

    Google Scholar 

  42. D. S. Ornstein,Ergodic Theory, Randomness and Dynamical Systems (Yale University, New Haven, 1984).

    Google Scholar 

  43. M. Aizenmann, S. Goldstein, and J. L. Lebowitz,Commun. Math. Phys. 39:289 (1975).

    Google Scholar 

  44. Ya. Sinai,Sov. Math. Dokl. 4:1818 (1963).

    Google Scholar 

  45. S. Kalikow,Ann. Math. 115:398 (1982).

    Google Scholar 

  46. M. Davis,Computabilily and Unsolvability (Dover, New York, 1982).

    Google Scholar 

  47. P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).

    Google Scholar 

  48. W. Krieger,Trans. Am. Math. Soc. 149:453 (1970); Erratum168:519 (1972).

    Google Scholar 

  49. O. Martin, A. Odlyzko, and S. Wolfram,Physica 10D:219 (1984).

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics and Material Research Laboratory, University of Illinois at Urbana-Champaign, 61801, Urbana, Illinois

    Y. Oono & C. Yeung

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  1. Y. Oono
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  2. C. Yeung
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Oono, Y., Yeung, C. A cell dynamical system model of chemical turbulence. J Stat Phys 48, 593–644 (1987). https://doi.org/10.1007/BF01019690

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