Journal of Statistical Physics

, Volume 78, Issue 5–6, pp 1571–1589 | Cite as

Ergodicity, ensembles, irreversibility in Boltzmann and beyond

  • Giovanni Gallavotti
Articles

Abstract

The contents of a not too well-known paper by Boltzmann are critically examined. The etymology of the word ergodic and its implications are discussed. A connection with the modern theory of Ruelle is attempted.

Key Words

Boltzmann ergodicity irreversibility Ruelle principle SRB measures chaos nonequilibrium 

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References

  1. 1.
    L. Boltzmann, Über die mechanische Bedeutung des zweiten Haupsatzes der Wärmetheorie, inWissenschaftliche Abhandlungen, F. Hasenöhrl, ed. (reprinted Chelsea, New York), Vol. I, pp. 9–33.Google Scholar
  2. 2.
    L. Boltzmann, Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten, inWissenschaftliche Abhandlungen, F. Hasenöhrl, ed. (reprinted Chelsea, New York), Vol. I, pp. 49–96.Google Scholar
  3. 3.
    L. Boltzmann, Analytischer Beweis des zweiten Hauptsatzes der mechanischen Wärmetheorie aus den Sätzen über das Gleichgewicht des lebendigen Kraft, inWissenschaftliche Abhandlungen, F. Hasenöhrl, ed. (reprinted Chelsea, New York), Vol. I, pp. 288–308.Google Scholar
  4. 4.
    L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, inWissenschaftliche Abhandlungen, F. Hasenöhrl, ed. (reprinted Chelsea, New York), Vol. I, pp. 316–402 [English transl., in S. Brush, ed.,Kinetic Theory (Pergamon Press, Oxford), Vol. 2, p. 88].Google Scholar
  5. 5.
    L. Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respektive den Sätzen über das Wärmegleichgewicht, inWissenschaftliche Abhandlungen, F. Hasenöhrl, ed. (reprinted Chelsea, New York, 1968), Vol. II, pp. 164–223.Google Scholar
  6. 6.
    L. Boltzmann, Über die Eigenschaften monzyklischer und anderer damit verwandter Systeme, inWissenschaftliche Abhandlungen, F. P. Hasenöhrl, ed. (reprinted Chelsea, New York, 1968), Vol. III.Google Scholar
  7. 7.
    L. Boltzmann, Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, in S. Brush, ed.,Kinetic Theory (Pergamon Press, Oxford), Vol. 2, p. 218 [English transl.].Google Scholar
  8. 8.
    L. Boltzmann, Zu Hrn. Zermelo's Abhandlung “Ueber die mechanische Erklärung irreversibler Vorgänge,” in S. Brush, ed.,Kinetic Theory (Pergamon Press, Oxford), Vol. 2, p. 238 [English transl.].Google Scholar
  9. 9.
    L. Boltzmann,Lectures on Gas Theory [annotated by S. Brush] (University of California Press, Berkeley, 1964).Google Scholar
  10. 10.
    A. Bach, Boltzmann's probability distribution of 1877,Arch. History Exact Sci. 41:1–40 (1990).Google Scholar
  11. 11.
    S. Brush,The Kind of Motion We Call Heat (North-Holland, Amsterdam, 1976/Vol.II; 1986/Vol. I).Google Scholar
  12. 12.
    L. Bunimovitch, Y. Sinai, and N. Chernov, Statistical properties of two dimensional hyperbolic billiards,Russ. Math. Surv. 45(3):105–152 (1990).Google Scholar
  13. 13.
    R. Clausius, The nature of the motion which we call heat, inKinetic Theory, S. Brush, ed. (Pergamon Press, Oxford), pp. 111–147.Google Scholar
  14. 14.
    K. Chernov, G. Eyink, J. Lebowitz, and Y. Sinia, Steady state electric conductivity in the periodic Lorentz gas,Commun. Math. Phys. 154:569–601 (1993).Google Scholar
  15. 15.
    R. Dugas,La théorie physique au sens de Boltzmann (Griffon, Neuchâtel, 1959).Google Scholar
  16. 16.
    U. Dressler, Symmetry property of the Lyapunov exponents of a class of dissipative dynamical systems with viscous damping,Phys. Rev. 38A:2103–2109 (1988).Google Scholar
  17. 17.
    D. Evans, E. Cohen, and G. Morriss, Viscosity of a simple fluid from its maximal Lyapunov exponents,Phys. Rev. 42A:5990–5997 (1990).Google Scholar
  18. 18.
    D. Evans, E. Cohen, and G. Morris, Probability of second law violations in shearing steady flows,Phys. Rev. Lett. 71:2401–2404 (1993).Google Scholar
  19. 19.
    P. Ehrenfest and T. Ehrenfest,The Conceptual Foundations of the Statistical Approach in Mechanics (Dover, New York, 1990) [reprint].Google Scholar
  20. 20.
    J. Gibbs,Elementary Principles in Statistical Mechanics (Ox Bow Press, 1981) [reprint].Google Scholar
  21. 21.
    G. Gallavotti, Aspetti della teoria ergodica qualitativa e statistica del moto,Quaderni UMI (Pitagora, Bologna)21 (1982).Google Scholar
  22. 22.
    G. Gallavotti, L'hypothèse ergodique et Boltzmann, inDictionnaire Philosophique (Presses Universitaires de France, Paris, 1989), pp. 1081–1086.Google Scholar
  23. 23.
    G. Gallavotti, Meccanica Statistica, inEnciclopedia italiana delle scienze fisiche (Rome, 1994);Fisiche (Rome, 1994); Equipartizione e critica della Meccanica Statistica Classica, inEnciclopedia italiana delle scienze; Teoria Ergodica, inEnciclopedia del Novecento (in press).Google Scholar
  24. 24.
    G. Gallavotti, Insiemi statistici, inEnciclopedia italiana delle scienze fisiche (Rome, 1994).Google Scholar
  25. 25.
    H. Helmholtz, Principien der Statik monocyklischer Systeme, inWissenschaftliche Abhandlungen (Leipzig, 1895), Vol. III, pp. 142–162, 179–202.Google Scholar
  26. 26.
    H. Helmholtz, Studien zur Statik monocyklischer Systeme, inWissenschaftliche Abhandlungen (Leipzig, 1895), Vol. III, pp. 163–172, 173–178.Google Scholar
  27. 27.
    B. Holian, W. Hoover, and H. Posch, Resolution of Loschmidt's paradox: The origin of irreversible behaviour in reversible atomistic dynamics,Phys. Rev. Lett. 59:10–13 (1987).Google Scholar
  28. 28.
    K. Jacobs, Ergodic theory and combinatorics,Contemp. Math. 26:171–187 (1984).Google Scholar
  29. 29.
    T. Kuhn,Black Body Theory and the Quantum Discontinuity. 1814–1912 (University of Chicago Press, Chicago, 1987).Google Scholar
  30. 30.
    M. Klein, Maxwell and the beginning of the quantum theory,Arch. History Exact Sci. 1:459–479 (1962).Google Scholar
  31. 31.
    M. Klein, Mechanical explanations at the end of the nineteenth century,Centaurus 17:58–82 (1972).Google Scholar
  32. 32.
    M. Klein, The development of Boltzmann's statistical ideas, inThe Boltzmann Equation, E. Cohen and W. Thirring, eds.,Acta Physica Austriaca, Suppl. X, pp. 53–106.Google Scholar
  33. 33.
    J. Lebowitz, Boltzmann's entropy and time's arrow,Phys. Today 1993(September):32–38.Google Scholar
  34. 34.
    R. Livi, A. Politi, and S. Ruffo (1986). Distribution of characteristic exponents in the thermodynamic limit,J. Phys. 19A:2033–2040 (1986).Google Scholar
  35. 35.
    H. Liddell and R. Scott,Greek-English Lexicon (Oxford, University Press, Oxford, 1994).Google Scholar
  36. 36.
    J. Maxwell, On Boltzmann's theorem on the average distribution of energy in a system of material points, inThe Scientific Papers of J. C. Maxwell, W. Niven, ed. (Cambridge University Press, Cambridge, 1890), Vol. II, pp. 713–741.Google Scholar
  37. 37.
    M. Mathieu, On the origin of the notion “Ergodic Theory,”Expositiones Math. 6:373–377 (1988).Google Scholar
  38. 38.
    J. von Plato, Boltzmann's ergodic hypothesis,Arch. History Exact Sci. 44:71–89 (1992).Google Scholar
  39. 39.
    J. von Plato,Creating Modern Probability (Cambridge University Press, Cambridge, 1994).Google Scholar
  40. 40.
    H. Posch and W. Hoover, Nonequilibrium molecular dynamics of a classical fluid, inMolecular Liquids: New Perspectives in Physics and Chemistry, J. Teixeira-Dias, ed. (Kluwer, Dordrecht, 1992), pp. 527–547.Google Scholar
  41. 41.
    D. Ruelle, Measures describing a turbulent flow,Ann. N.Y. Acad. Sci. 357:1–9 (1980); see also J. Eckmann and D. Ruelle, Ergodic theory of strange attractors,Rev. Mod. Phys. 57:617–656 (1985); D. Ruelle, Ergodic theory of differentiable dynamical systems,Publ. Math. IHES 50:275–306 (1980).Google Scholar
  42. 42.
    Y. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billards,Russ. Math. Surv. 25:137–189 (1970).Google Scholar
  43. 43.
    J. Schwartz, The pernicious influence of mathematics on science, inDiscrete Thoughts: Essays in Mathematics, Science, and Philosophy, M. Kac, G. Rota, and J. Schwartz, eds. (Birkhauser, Boston, 1986), pp. 19–25.Google Scholar
  44. 44.
    S. Sarman, D. Evans and G. Morris, Conjugate pairing rule and thermal transport coefficients,Phys. Rev. 45A:2233–2242 (1992).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Giovanni Gallavotti
    • 1
  1. 1.Dipartimento di FisicaUniversità di RomaRomeItaly

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