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Selection of an Attractor in a Continuum of Stable Solutions: Descriptions of a Wave Front at Different Scales

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Abstract

Details of the particle dynamics are shown to modify the mean speed of a chemical wave front propagating into an unstable stationary state. The comparison of several description methods at different scales allows us to discriminate between different sources of departure from the macroscopic deterministic prediction and to give a quantitative expression of the speed corrections induced by discretization of the variables, internal fluctuations in small systems, and departure from local equilibrium in the presence of a fast reaction.

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REFERENCES

  1. R. A. Fisher, Ann. Eugenics 7:335 (1937).

    Google Scholar 

  2. A. Kolmogorov, I. Petrovsky, and N. Piskunov, Bull. Univ. Moscow. Ser. Int. Sec. A 1:1 (1937).

    Google Scholar 

  3. D. G. Aronson and H. F. Weinberger, Adv. Math. 30:33 (1978); W. van Saarloos, Phys. Rev. Lett. 58:2571 (1987); W. van Saarloos, Phys. Rev. A 37:211 (1988); W. van Saarloos, Phys. Rev. A 39:6367 (1989).

    Google Scholar 

  4. S. Cornell, M. Droz, and B. Chopard, Phys. Rev. A 44:4826 (1991); B. Chopard, M. Droz, T. Karapiperis, and Z. Racz, Phys. Rev. E 47:R40 (1993).

    Google Scholar 

  5. J. Riordan, C. R. Doering, and D. ben-Avraham, Phys. Rev. Lett. 75:565 (1995).

    Google Scholar 

  6. H.-P. Breuer, W. Huber, and F. Petruccione, Physica D 73:259 (1994); ibid., Europhys. Lett. 30:69 (1995).

    Google Scholar 

  7. A. Lemarchand, A. Lesne, and M. Mareschal, Phys. Rev. E 51:4457 (1995).

    Google Scholar 

  8. M. Karzazi, A. Lemarchand, and M. Mareschal, Phys. Rev. E 54:4888 (1996).

    Google Scholar 

  9. Ch. Dellago and H. A. Posch, Physica A 240:68 (1997).

    Google Scholar 

  10. R. van Zon, H. van Beijeren, and Ch. Dellago, Phys. Rev. Lett. 80:2035 (1998).

    Google Scholar 

  11. J. Mai, I. M. Sokolov, and A. Blumen, Europhys. Lett. 44:7 (1998).

    Google Scholar 

  12. E. Brunet and B. Derrida, Phys. Rev. E 56:2597 (1997).

    Google Scholar 

  13. D. A. Kessler, Z. Ner, and L. M. Sander, Phys. Rev. E 58:107 (1998).

    Google Scholar 

  14. A. Lemarchand and B. Nowakowski, Europhys. Lett. 41:455 (1998).

    Google Scholar 

  15. A. Lemarchand and B. Nowakowski, J. Chem. Phys. 109:7028 (1998).

    Google Scholar 

  16. A. Lemarchand and B. Nowakowski, J. Chem. Phys. 111:6190 (1999).

    Google Scholar 

  17. M. Velikanov and R. Kapral, J. Chem. Phys. 110:109 (1999).

    Google Scholar 

  18. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).

    Google Scholar 

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

    Google Scholar 

  20. J. D. Murray, Mathematical Biology (Springer, Berlin, 1989).

    Google Scholar 

  21. A. J. Koch and H. Meinhardt, Rev. Mod. Phys. 66:1481 (1994).

    Google Scholar 

  22. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1977).

  23. C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985).

    Google Scholar 

  24. D. T. Gillespie, J. Comput. Phys. 22:403 (1976).

    Google Scholar 

  25. G. A. Bird, Molecular Gas Dynamics (Clarendon, Oxford, 1976).

    Google Scholar 

  26. I. Prigogine and E. Xhrouet, Physica 15:913 (1949).

    Google Scholar 

  27. B. Shizgal and M. Karplus, J. Chem. Phys. 52:4262 (1970); ibid. 54:4345 and 4357 (1971).

    Google Scholar 

  28. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, London, 1953).

    Google Scholar 

  29. D. Napier and B. Shizgal, Phys. Rev. E 52:3797 (1995); B. Shizgal and D. Napier, Physica A 223:50 (1996).

    Google Scholar 

  30. B. Nowakowski and J. Popielawski, J. Chem. Phys. 100:7602 (1994).

    Google Scholar 

  31. B. V. Alexeev, A. Chikhaoui, and I. T. Grushin, Phys. Rev. E 49:2809 (1994).

    Google Scholar 

  32. B. Nowakowski and A. Lemarchand, J. Chem. Phys. 106:3965 (1997).

    Google Scholar 

  33. A. Lemarchand, I. Nainville, and M. Mareschal, Europhys. Lett. 36:227 (1996).

    Google Scholar 

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

    Google Scholar 

  35. M. Droz and A. McKane, J. Phys. A 27:L467 (1994).

    Google Scholar 

  36. P.-A. Rey and M. Droz, J. Phys. A 30:1101 (1997).

    Google Scholar 

  37. A. Lemarchand and B. Nowakowski, Physica A 271:87 (1999).

    Google Scholar 

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Lemarchand, A. Selection of an Attractor in a Continuum of Stable Solutions: Descriptions of a Wave Front at Different Scales. Journal of Statistical Physics 101, 579–598 (2000). https://doi.org/10.1023/A:1026430625726

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  • DOI: https://doi.org/10.1023/A:1026430625726

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