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Foundations of Physics

, Volume 17, Issue 9, pp 939–963 | Cite as

A fundamental link between system theory and statistical mechanics

  • H. Atmanspacher
  • H. Scheingraber
Article

Abstract

A fundamental link between system theory and statistical mechanics has been found to be established by the Kolmogorov entropy K. By this quantity the temporal evolution of dynamical systems can be classified into regular, chaotic, and stochastic processes. Since K represents a measure for the internal information creation rate of dynamical systems, it provides an approach to irreversibility. The formal relationship to statistical mechanics is derived by means of an operator formalism originally introduced by Prigogine. For a Liouville operator L and an information operator\(\tilde M\) acting on a distribution in phase space, it is shown that i[L,\(\tilde M\)]≡KI (I=identity operator). As a first consequence of this equivalence, a relation is obtained between the chaotic correlation time of a system and Prigogine's concept of a “finite duration of presence.” Finally, the existence of chaos in quantum systems is discussed with respect to the existence of a quantum mechanical time operator.

Keywords

Entropy Phase Space System Theory Quantum System Temporal Evolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. Haken,Synergetics—An Introduction, 3rd edn. (Springer, Berlin, 1983).Google Scholar
  2. 2.
    G. Nicolis and I. Prigogine,Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).Google Scholar
  3. 3.
    J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57, 617 (1985).Google Scholar
  4. 4.
    I. Prigogine,Vom Sein zum Werden, 2nd edn. (Piper, München, 1985), and references therein; I. Prigogine,From Being to Becoming, 2nd edn. (Freeman, New York, forthcoming).Google Scholar
  5. 5.
    I. Prigogine,Non-Equilibrium Statistical Mechanics (Interscience, New York, 1962).Google Scholar
  6. 6.
    V. I. Oseledec,Trans. Mosc. Math. Soc. 19, 197 (1968) [Tr. Mosk. Mat. Ova. 19, 179 (1968)].Google Scholar
  7. 7.
    P. Grassberger, “Information aspects of strange attractors,” inChaos in Astrophysics, J. R. Buchler, J. M. Perdang, and E. A. Spiegel, eds. (Reidel, Dordrecht, 1985).Google Scholar
  8. 8.
    M. W. Hirsch and S. Smale,Differential Equations, Dynamic Systems, and Linear Algebra (Academic Press, New York, 1965).Google Scholar
  9. 9.
    J. B. Pesin,Russ. Math. Surv. 32, 455 (1977) [Usp. Mat. Nauk 32, 55 (1977)].Google Scholar
  10. 10.
    I. Procaccia,Phys. Scr. T 9, 40 (1985).Google Scholar
  11. 11.
    J. Balatoni and A. Renyi, inSelected Papers of A. Renyi, Vol. 1 (Akademie Budapest, 1976), p. 588.Google Scholar
  12. 12.
    C. F. v. Weizsäcker,Aufbau der Physik (Hanser, München, 1985).Google Scholar
  13. 13.
    C. E. Shannon and C. Weaver,The Mathematical Theory of Communication (University of Illinois Press, Urbana, Illinois, 1962).Google Scholar
  14. 14.
    L. Arnold and W. Kliemann, “Qualitative theory of stochastic systems,” inProbability Analysis and Related Topics, Vol. 3 (Academic Press, New York, 1983).Google Scholar
  15. 15.
    F. Takens, inLecture Notes in Mathematics 898, D. A. Rand and L. S. Young, eds. (Springer, Berlin, 1981).Google Scholar
  16. 16.
    N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw,Phys. Rev. Lett. 45, 712 (1980).Google Scholar
  17. 17.
    P. Grassberger and I. Procaccia,Phys. Rev. A 28, 2591 (1983).Google Scholar
  18. 18.
    A. M. Albano, J. Abounadi, T. H. Chyba, C. E. Searle, S. Yong, R. S. Gioggia, and N. B. Abraham,J. Opt. Soc. Am. B 2, 47 (1985).Google Scholar
  19. 19.
    H. Atmanspacher and H. Scheingraber,Phys. Rev. A 34, 253 (1986).Google Scholar
  20. 20.
    G. Mayer-Kress, ed.,Dimensions and Entropies in Chaotic Systems (Springer, Berlin, 1986).Google Scholar
  21. 21.
    P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).Google Scholar
  22. 22.
    H. Haken,Phys. Scr. 32, 274 (1985).Google Scholar
  23. 23.
    A. Ben Mizrachi, I. Procaccia, and P. Grassberger,Phys. Rev. A 29, 975 (1984).Google Scholar
  24. 24.
    S. Grossmann and H. Horner,Z. Phys. B 60, 79 (1985).Google Scholar
  25. 25.
    F. T. Arecchi,Phys. Scr. T 9, 85 (1985).Google Scholar
  26. 26.
    G. Birkhoff,Lattice Theory, 3rd edn. (AMS Coll. Publ., Vol. 25, Providence, Rhode Island, 979).Google Scholar
  27. 27.
    Y. Elskens and I. Prigogine,Proc. Natl. Acad. Sci. USA 83, 5756 (1986); M. Courbage,Physica A 122, 459 (1983); S. Goldstein and O. Penrose,J. Stat. Phys. 24, 325 (1981).Google Scholar
  28. 28.
    S. Goldstein,Israel J. Math. 38, 241 (1981).Google Scholar
  29. 29.
    B. Misra,Proc. Natl. Acad. Sci. USA 75, 1627 (1978).Google Scholar
  30. 30.
    B. V. Chirikov,Found. Phys. 16, 39 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • H. Atmanspacher
    • 1
  • H. Scheingraber
    • 1
  1. 1.Max-Planck-Institut für Physik und Astrophysik, Institut für Extraterrestrische PhysikGarchingFRG

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