Skip to main content
SpringerLink
Log in

Ergodic Problems

  • Chapter
Statistical Physics I

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 30))

  • 1090 Accesses

Abstract

In the preceding chapters, we have described the general methods of statistical mechanics mostly on the basis of quantum mechanics. But in this chapter, we shall describe the ergodic problems based on classical and quantum mechanics, the reason for which will first be made clear.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L.D. Landau, E.M. Lifshitz: Statistical Physics, transl. by D. Shoenberg (Clarendon, Oxford 1938, Pergamon 1958)

    MATH  Google Scholar 

  2. A.I. Khinchin: Mathematical Foundations of Statistical Mechanics (Dover, New York 1949)

    MATH  Google Scholar 

  3. D. ter Haar: Rev. Mod. Phys. 27, 289 (1955)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Proceedings of the International School of Physics “Enrico Fermi” XIV, Ergodic Theories (Academic, New York 1961)

    Google Scholar 

  5. I.E. Farquhar: Ergodic Theory in Statistical Mechanics (Interscience, New York 1964)

    Google Scholar 

  6. R. Jancel: Foundations of Classical and Quantum Statistical Mechanics (Pergamon, Oxford 1963)

    Google Scholar 

  7. E. Hopf: Ergodentheorie (Chelsea, New York 1948)

    Google Scholar 

  8. V.I. Arnold, A. Avez: Ergodic Problems of Classical Mechanics (Benjamin, New York 1968)

    Google Scholar 

  9. P. and T. Ehrenfest: The Conceptual Foundations of the Statistical Approach in Mechanics [Transl. by M.J. Moravcsik] (Cornell University Press, Ithaca 1959)

    MATH  Google Scholar 

  10. R. Abraham, J.E. Marsden: Foundation of Mechanics, 2nd ed. (Benjamin, New York 1978)

    Google Scholar 

  11. G.D. Birkhoff: Proc. Nat. Acad. Sci. 17, 656 (1931)

    Article  ADS  Google Scholar 

  12. B.O. Koopman: Proc. Nat. Acad. Sci. 17, 315 (1931)

    Article  ADS  Google Scholar 

  13. G.D. Birkhoff, B.O. Koopman: Proc. Nat. Acad. Sci. 18, 279 (1932)

    Article  ADS  Google Scholar 

  14. P.A. Smith: J. Math. 7, 365 (1928)

    Google Scholar 

  15. L. Van Hove: Math. Rev. 17, 812 (1956)

    Google Scholar 

  16. D.S. Ornstein: Adv. Math. 4, 337 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Aizawa: J. Phys. Soc. Jpn. 33, 1693 (1972)

    Article  ADS  Google Scholar 

  18. J.G. Sinai: Ergodicity of Boltzmann’s gas model. In Statistical Mechanics, Foundations and Applications, Proc. of the I.U.P.A.P. Meeting, Copenhagen, 1966, ed. by T.A. Bak (Benjamin, New York 1967) p. 559

    Google Scholar 

  19. S. Jorna (ed.): Topics in Nonlinear Dynamics, AIP Conf. Proc, No. 46 (American Institute of Physics, New York 1978). Especially, see papers by J. Moser, M.V. Berry, J. Ford, Y.M. Treve

    Google Scholar 

  20. J. Ford: Fundamental Problems in Statistical Mechanics 1, ed. by E.D.G. Cohen (North-Holland, Amsterdam 1975)

    Google Scholar 

  21. G.E.O. Giacaglia: Perturbation Methods in Nonlinear Systems (Springer, Berlin, Heidelberg 1972)

    Book  Google Scholar 

  22. C.L. Siegel, J.K. Moser: Lectures in Celestial Mechanics (Springer, Berlin, Heidelberg 1971)

    Book  Google Scholar 

  23. J. Moser: Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton 1973)

    MATH  Google Scholar 

  24. Y. Hagiwara: Celestial Mechanics, Vol. 4, Part 2 (Japan Soc. for the Promotion of Science, Tokyo 1975)

    Google Scholar 

  25. G. Casati, J. Ford (eds.): Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Como, 1977, Lecture Notes Phys., Vol. 93 (Springer, Berlin, Heidelberg 1979)

    Google Scholar 

  26. I.C. Percival: Semiclassical Theory of Bound States, ed. by I. Prigogine, S.A. Rice, Advances in Chemical Physics, Vol. 36 (Wiley, New York 1977)

    Google Scholar 

  27. B.V. Chirikov: Phys. Rep. 52, 263 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  28. R.H.G. Helleman: In Fundamental Problems in Statistical Mechanics V, ed. by E.G.D. Cohen (North-Holland, Amsterdam 1980)

    Google Scholar 

  29. H. Bruns: Acta Math. 11, 25 (1887)

    Article  MathSciNet  Google Scholar 

  30. H. Poincaré: Les Methodes Nouvelles de la Mécanique Céleste (Dover, New York 1957)

    MATH  Google Scholar 

  31. E. Fermi: Phys. Z. 24, 261 (1923)

    MATH  Google Scholar 

  32. E. Fermi, J. Pasta, S. Ulam: Los Alamos Rpt. LA-1940 (1955); Collected Papers of Enrico Fermi, Vol. II, University of Chicago press, Chicago 1965) p. 977

    Google Scholar 

  33. P.C. Hemmer: Dynamic and Stochastic Types of Motion in the Linear Chain, Thesis (Trondheim 1959)

    Google Scholar 

  34. J. Ford, J. Waters: J. Math. Phys. 4, 1293 (1963)

    Article  ADS  Google Scholar 

  35. N. Saitô, N. Hirotomi, A. Ichimura: J. Phys. Soc. Jpn. 39, 1931 (1975)

    Article  Google Scholar 

  36. M. Hénon, C. Heiles: Astron. J. 69, 73 (1964)

    Article  ADS  Google Scholar 

  37. J.N. Greene: J. Math. Phys. 9, 760 (1968)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. A.N. Kolmogorov: Intern. Congress Mathematician, Amsterdam 1, 315 (1954)

    Google Scholar 

  39. J. Moser: Commun. Pure Appl. Math. 9, 673 (1956)

    Article  MATH  Google Scholar 

  40. N. Saitô, H. Hirooka, J. Ford, F. Vivaldi, G.H. Walker: Physica 5D, 273 (1983);

    ADS  Google Scholar 

  41. H. Hirooka, J. Ford, N. Saitô: J. Phys. Soc. Jpn. 53, 385 (1984)

    Article  Google Scholar 

  42. N. Saitô: J. Phys. Soc. Jpn. 51, 374 (1982)

    Article  ADS  Google Scholar 

  43. N. Saitô, Y. Matsunaga: J. Phys. Soc. Jpn. 58, 3089 (1989)

    Article  ADS  Google Scholar 

  44. G. Ludwig: In Ref. [5.4] and Die Grundlagen der Quantenmechanik (Springer, Berlin, Heidelberg 1954)

    Google Scholar 

  45. N.G. van Kampen: Physica 20, 603 (1954)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. M. Fierz: Helv. Phys. Acta 28, 705 (1955)

    MathSciNet  MATH  Google Scholar 

  47. J.E. Farquhar, P.T. Landsberg: Proc. R. Soc. London 239, 134 (1957)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. P. Bocchieri, A. Loinger: Phys. Rev. 111, 668 (1958);

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. P. Bocchieri, A. Loinger: Phys. Rev. 114, 948 (1959);

    Article  MathSciNet  ADS  Google Scholar 

  50. G.M. Prosperi, A. Scotti: Nuovo Cimento 13, 1007 (1959);

    Article  Google Scholar 

  51. G.M. Prosperi, A. Scotti: Nuovo Cimento. 17, 267 (1960);

    Article  MATH  Google Scholar 

  52. P. Bocchieri, G.M. Prosperi: J. Phys. Soc. Jpn. 33, (1972) [5.17, p. 17]

    Google Scholar 

  53. S. Golden, H.C. Longuet-Higgins: J. Chem. Phys. 33, 1479 (1960)

    Article  MathSciNet  ADS  Google Scholar 

  54. B. Eckhardt: Phys. Rep. 163, 205 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  55. B. Eckhardt:Les Houches Lectures 34, 205 (1988).

    MathSciNet  Google Scholar 

  56. B. Eckhardt:Chaotic Behaviors of Deterministic Systems 1983, 205 52

    Google Scholar 

  57. B. Eckhardt Chaos and Quantum Physics (1991) (North Holland Pub. Co.)

    Google Scholar 

  58. M.V. Berry, M. Tabor: Proc. R. Soc. London A 356, 375 (1977)

    Google Scholar 

  59. M.L. Mehta: Random Matrices and the Statistical Theory of Energy Levels (Academic, New York 1967) Revised ed. (1991)

    MATH  Google Scholar 

  60. T.A. Brody, J.B. Flores, P.A. Mello, A. Pandey, S.S.M. Wong: Rev. Mod. Phys. 53, 385 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  61. A. Shudo, N. Saitô: J. Phys. Soc. Jpn. 56, 2641 (1987)

    Article  ADS  Google Scholar 

  62. L.A. Bunimovich: Funct. Anal. Appl. 8, 254 (1974)

    Article  MATH  Google Scholar 

  63. S.W. McDonald, A.N. Kaufman: Phys. Rev. Lett. 42, 1189 (1979)

    Article  ADS  Google Scholar 

  64. E.J. Heller: Phys. Rev. Lett. 53, 1515 (1984);

    Article  MathSciNet  ADS  Google Scholar 

  65. E.J. Heller: In Quantum Chaos and Statistical Nuclear Physics. Proc. Cuernavaca, Mexico 1986, ed. by T.H. Seligman, H. Nishioka, Lect. Notes Phys., Vol. 263 (Springer, Berlin, Heidelberg 1986)

    Google Scholar 

  66. E.B. Bogomolny: Physica D 31, 405 (1988)

    Article  MathSciNet  Google Scholar 

  67. M.V. Berry: Proc. R. Soc. London A 423, 219 (1989)

    Article  ADS  Google Scholar 

  68. A. Shudo: Prog. Theor. Phys. Suppl. 98, 173 (1989)

    Article  ADS  Google Scholar 

  69. P. Pechukas: Phys. Rev. Lett. 51, 943 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  70. T. Yukawa, T. Ishikawa: Prog. Theor. Phys. Suppl. 98, 157 (1989)

    Article  ADS  Google Scholar 

  71. N. Saitô: Prog. Theor. Phys. Suppl. 98, 376 (1989)

    Article  ADS  Google Scholar 

  72. K. Takahashi: Prog. Theor. Phys. Suppl. 98, 1109 (1989)

    Article  Google Scholar 

  73. P.W. O’Connor, E.J. Heller: Phys. Rev. Lett. 61, 2288 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  74. N. Saitô and A. Shudo: J. Phys. Soc. Jpn. 62, 53 (1993)

    Article  ADS  MATH  Google Scholar 

  75. N. Saitô: to be published

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 5-29-8-108 Yoyogi, Shibuja-ku, 151, Tokyo, Japan

    Professor Dr. Morikazu Toda

  2. Department of Applied Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, 169, Tokyo, Japan

    Professor Dr. Nobuhiko Saitô

Authors
  1. Professor Dr. Morikazu Toda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Professor Dr. Ryogo Kubo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Professor Dr. Nobuhiko Saitô
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Toda, M., Kubo, R., Saitô, N. (1992). Ergodic Problems. In: Statistical Physics I. Springer Series in Solid-State Sciences, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58134-2_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-58134-2_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53662-8

  • Online ISBN: 978-3-642-58134-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation