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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
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.
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].
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.
L. Boltzmann, Über die Eigenschaften monzyklischer und anderer damit verwandter Systeme, inWissenschaftliche Abhandlungen, F. P. Hasenöhrl, ed. (reprinted Chelsea, New York, 1968), Vol. III.
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.].
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.].
L. Boltzmann,Lectures on Gas Theory [annotated by S. Brush] (University of California Press, Berkeley, 1964).
A. Bach, Boltzmann's probability distribution of 1877,Arch. History Exact Sci. 41:1–40 (1990).
S. Brush,The Kind of Motion We Call Heat (North-Holland, Amsterdam, 1976/Vol.II; 1986/Vol. I).
L. Bunimovitch, Y. Sinai, and N. Chernov, Statistical properties of two dimensional hyperbolic billiards,Russ. Math. Surv. 45(3):105–152 (1990).
R. Clausius, The nature of the motion which we call heat, inKinetic Theory, S. Brush, ed. (Pergamon Press, Oxford), pp. 111–147.
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).
R. Dugas,La théorie physique au sens de Boltzmann (Griffon, Neuchâtel, 1959).
U. Dressler, Symmetry property of the Lyapunov exponents of a class of dissipative dynamical systems with viscous damping,Phys. Rev. 38A:2103–2109 (1988).
D. Evans, E. Cohen, and G. Morriss, Viscosity of a simple fluid from its maximal Lyapunov exponents,Phys. Rev. 42A:5990–5997 (1990).
D. Evans, E. Cohen, and G. Morris, Probability of second law violations in shearing steady flows,Phys. Rev. Lett. 71:2401–2404 (1993).
P. Ehrenfest and T. Ehrenfest,The Conceptual Foundations of the Statistical Approach in Mechanics (Dover, New York, 1990) [reprint].
J. Gibbs,Elementary Principles in Statistical Mechanics (Ox Bow Press, 1981) [reprint].
G. Gallavotti, Aspetti della teoria ergodica qualitativa e statistica del moto,Quaderni UMI (Pitagora, Bologna)21 (1982).
G. Gallavotti, L'hypothèse ergodique et Boltzmann, inDictionnaire Philosophique (Presses Universitaires de France, Paris, 1989), pp. 1081–1086.
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).
G. Gallavotti, Insiemi statistici, inEnciclopedia italiana delle scienze fisiche (Rome, 1994).
H. Helmholtz, Principien der Statik monocyklischer Systeme, inWissenschaftliche Abhandlungen (Leipzig, 1895), Vol. III, pp. 142–162, 179–202.
H. Helmholtz, Studien zur Statik monocyklischer Systeme, inWissenschaftliche Abhandlungen (Leipzig, 1895), Vol. III, pp. 163–172, 173–178.
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).
K. Jacobs, Ergodic theory and combinatorics,Contemp. Math. 26:171–187 (1984).
T. Kuhn,Black Body Theory and the Quantum Discontinuity. 1814–1912 (University of Chicago Press, Chicago, 1987).
M. Klein, Maxwell and the beginning of the quantum theory,Arch. History Exact Sci. 1:459–479 (1962).
M. Klein, Mechanical explanations at the end of the nineteenth century,Centaurus 17:58–82 (1972).
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.
J. Lebowitz, Boltzmann's entropy and time's arrow,Phys. Today 1993(September):32–38.
R. Livi, A. Politi, and S. Ruffo (1986). Distribution of characteristic exponents in the thermodynamic limit,J. Phys. 19A:2033–2040 (1986).
H. Liddell and R. Scott,Greek-English Lexicon (Oxford, University Press, Oxford, 1994).
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.
M. Mathieu, On the origin of the notion “Ergodic Theory,”Expositiones Math. 6:373–377 (1988).
J. von Plato, Boltzmann's ergodic hypothesis,Arch. History Exact Sci. 44:71–89 (1992).
J. von Plato,Creating Modern Probability (Cambridge University Press, Cambridge, 1994).
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.
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).
Y. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billards,Russ. Math. Surv. 25:137–189 (1970).
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.
S. Sarman, D. Evans and G. Morris, Conjugate pairing rule and thermal transport coefficients,Phys. Rev. 45A:2233–2242 (1992).
Author information
Authors and Affiliations
Additional information
This is an expanded and revised version of a paper read at a conference celebrating the 150th anniversary of the birth of Boltzmann, Vienna, 24 February 1994. This paper is archived inmp_arc@math.utexas.edu, #94-66; updated copies (in Postscript) can also be obtained by sending a request to the author by E-mail.
Rights and permissions
About this article
Cite this article
Gallavotti, G. Ergodicity, ensembles, irreversibility in Boltzmann and beyond. J Stat Phys 78, 1571–1589 (1995). https://doi.org/10.1007/BF02180143
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02180143