Chaotic Dynamics and the Origin of Statistical Laws

AUG 01, 1999

Chaotic dynamics in real systems does not provide finite relaxation time to equilibrium or fast decay of fluctuations, and chaotic systems are not completely random in the sense originally postulated for statistical systems. These properties may require rethinking some of the fundamental assumptions of thermodynamics.

DOI: 10.1063/1.882777

George M. Zaslavsky

The problem of the foundation of statistical physics emerged with the derivation by Ludwig Boltzmann of a kinetic equation for a gas of molecules that required monotonic growth of entropy. Boltzmann’s theory leads to modern thermodynamics, and, for example, to the impossibility of gas spontaneously gathering in one part of a container in the absence of external forces. This result, known as the H‐theorem, met with strong contemporary opposition, especially from mathematician Ernst Zermelo.

References

1. L. Boltzmann, Wien Ber. 66, 275 (1872).
2. E. Zermelo, Wied. Ann. 57, 485 (1896).
3. P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics, Cornell U. P., Ithaca, N.Y. (1959).
4. M. Kac, Probability and Related Topics in Physical Sciences, Interscience, New York (1957).
5. E. Fermi, J. Pasta. S. Ulam, Los Alamos Scientific Report LA‐1940 (1955),
reprinted in Collected Works of Enrico Fermi, vol. 2, p. 978, U. of Chicago P., Chicago (1965).
N. Zabusky, J. Comput. Physics 43, 195 (1981).https://doi.org/JCTPAH
6. B. V. Chirikov, Physics Reports 52, 263 (1981);
N. S. Krylov, Works on the Foundations of Statistical Physics, Princeton U. P., Princeton, N.J. (1979).
7. G. M. Zaslavsky, Chaos in Dynamic Systems, Harwood, New York (1984).
8. H. S. Leff and A. F. Rex, eds., Maxwell’s Demon, Princeton U. P., Princeton, N.J. (1990).
9. M. F. Shlesinger, G. M. Zaslavsky, J. Klafter, Nature 363, 31 (1993).https://doi.org/NATUAS
10. J. D. Meiss, Rev. Mod. Phys. 64, 795 (1992).https://doi.org/RMPHAT
11. L. Kuznetsov, G. M. Zaslavsky, Phys. Rev. E 58, 7330 (1998). https://doi.org/PLEEE8
A. Babiano, G. Boffetta, A. Provenzale, A. Vulpiani, Phys. Fluids 6, 2465 (1994).https://doi.org/PHFLE6
12. G. M. Zaslavsky, D. Stevens, H. Weitzner, Phys. Rev. E 48, 1683 (1993). https://doi.org/PLEEE8
M. F. Shlesinger, B. J. West, J. Klafter, Phys. Rev. Lett. 58, 1100 (1987). https://doi.org/PRLTAO
T. Solomon, E. Weeks, H. Swinney, Phys. Rev. Lett. 71, 3975 (1993). https://doi.org/PRLTAO
S. Benkadda, S. Kassibrakis, R. White, G. M. Zaslavsky, Phys. Rev. E 55, 4909 (1997).https://doi.org/PLEEE8
13. G. M. Zaslavsky, M. Edelman, B. Niyazov, Chaos 7, 159 (1997). https://doi.org/CHAOEH
V. Afraimovich, G. M. Zaslavsky, Phys. Rev. E 55, 5418 (1997).https://doi.org/PLEEE8
14. J. Machta, J. Stat. Phys. 32, 555 (1983).https://doi.org/JSTPBS
15. G. M. Zaslavsky, Chaos 5, 653 (1995). https://doi.org/CHAOEH
G. M. Zaslavsky, M. Edelman, Phys. Rev. E 56, 5310 (1997).https://doi.org/PLEEE8
16. E. W. Montroll, M. F. Shlesinger, in Studies in Statistical Mechanics, J. Lebowitz, E. Montroll, eds., North Holland, Amsterdam (1984), vol. 11, p. 1.
M. F. Shlesinger, Ann. Rev. Phys. Chem. 39, 269 (1988).https://doi.org/ARPLAP
17. P. Lévy, Theorie de l’Addition des Variables Aletoires, Gauthier‐Villiars, Paris (1937).
18. B. Sundaram, G. M. Zaslavsky, Phys. Rev. E 59, 7231 (1999).https://doi.org/PLEEE8

More about the Authors

George M. Zaslavsky. New York, University.