Entropy and Chaos in a Lattice Gas Cellular Automata

  • Franco Bagnoli
  • Raúl Rechtman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5191)

Abstract

We find a simple linear relation between the thermodynamic entropy and the largest Lyapunov exponent (LLE) of an discrete hydrodynamical system, a deterministic, two-dimensional lattice gas automaton (LGCA). This relation can be extended to irreversible processes considering the Boltzmann’s H function and the expansion factor of the LLE. The definition of LLE for cellular automata is based on the concept of Boolean derivatives and is formally equivalent to that of continuous dynamical systems.

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References

  1. 1.
    Lebowitz, J.: Phys.Today, p. 32 (September 1973); Zaslavsky, G. M.: Phys. Today, p. 39 (August 2004)Google Scholar
  2. 2.
    Dzugutov, M., Aurell, E., Vulpiani, A.: Phys. Rev. Lett. 81, 1762 (1998)Google Scholar
  3. 3.
    Gaspard, P., Nicolis, G.: Phys. Rev. Lett. 65, 1693 (1990);  Gaspard, P., Dorfman, J. R.: Phys. Rev. E 52, 3525 (1995);  Evans, D.J., Morris, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Academic Press, London (1990); Hoover, W.G.: Computational Statistical Mechanics. Elsevier, Amsterdam (1991); Dorfman, J.R.,  van Beijeren, H.: Physica A 240, 12 (1997)Google Scholar
  4. 4.
    Falcioni, M., Palatella, L., Vulpiani, A.: Phys. Rev. E 71, 016118 (2005);  Latora, V., Baranger, M.: Phys. Rev. Lett. 82, 520 (1999)Google Scholar
  5. 5.
    d’Humiéres, D., Lallemand, P., Frisch, U.: Europhys. Lett. 2, 291 (1986);  Chopard, B., Droz, M.: Phys. Lett. 126A, 476 (1988);  Chen, S.,  Lee, M.,  Zhao, K., Doolen, G.D.: Physica B 37, 42 (1989)Google Scholar
  6. 6.
    Hardy, J., Pomeau, Y., de Pazzis, O.: J. Math. Phys. 14, 1746, (1973); Frisch, U., Hasslacher, B., Pomeau, Y.: Phys. Rev. Lett. 56, 1505 (1986); Frisch, U., d’Humiéres, D., Hasslacher, B., Lallemand, P., Pomeau, Y., Rivet, J.P.: Complex Systems 1, 649 (1987). Reprinted in Lecture Notes on Turbulence, Herring, J.R., McWilliams, J.C.(eds.): p. 649. World Scientific, Singapore (1989) .Google Scholar
  7. 7.
    Bagnoli, F., Rechtman, R., Zanette, D.: Rev. Mex. Fís. 39, 763 (1993)Google Scholar
  8. 8.
    Bagnoli, F., Rechtman, R., Ruffo, S.: Phys. Lett. A, 172, 34 (1992)Google Scholar
  9. 9.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  10. 10.
    Ehrenfest, P., Ehrenfest, T.: Enzyclopädie der Mathematischen Wissenschaften, vol. 4 (Teubner, Leipzig, 1911) English translation:  Moravcik, M. J.: Conceptual Foundations of Statistical Approach in Mechanics. Cornell University Press, Ithaca (1959).  Ernst, M. H.,  van Velzen, G. A.: J. Phys. A: Math. Gen. 22, 4611 (1989)Google Scholar
  11. 11.
    Bagnoli, F.: Int. J. Mod. Phys. 3, 307 (1992)Google Scholar
  12. 12.
    Bagnoli, F., Rechtman, R., Ruffo, S.: Lyapunov Exponents for Cellular Automata. In: López de Haro, M., Varea, C. (eds.) Lectures on Thermodynamics and Statistical Mechanics, p. 200. World Scientific, Singapore (1994)Google Scholar
  13. 13.
    Salcido, A., Rechtman, R.: Equilibrium Properties of a Cellular Automaton for Thermofluid Dynamics. In: Cordero, P., Nachtergaele, B. (eds.) Nonlinear Phenomena in Fluids, Solids and Other Complex Systems, p. 217. Elsevier, Amsterdam (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Franco Bagnoli
    • 1
  • Raúl Rechtman
    • 2
  1. 1.Dipartimento di EnergeticaUniversità di Firenze, Also CSDC and INFN, sez. FirenzeFirenzeItaly
  2. 2.Centro de Investigacíon en EnergíaUNAMTemixcoMexico

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