Soshichi Uchii, Kyoto University


Boltzmann, L.

Wissenschaftliche Abhandlungen, 3 vols. Edited by F Hasenoerl. J. A. Barth, Leipzig, 1909. Reprinted by Chelsea Pub. Co., New York, 1968.

[1868] Studien ueber das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten. Wien. Ber. 58, 517-560. Abhandlungen 1, 49-105.

[1871] Einige allgemeine Saetze ueber Waermegleichgewicht. Wien. Ber. 63, 679-711. Abhandlungen 1, 259-287.

[1872] Weitere Studien ueber das Waemegleichgewicht unter Gasmolekuelen. Wien. Ber. 66, 275-370. Abhandlungen 1, 316-402.

[1877] Bemerkungen ueber einige Probleme der mechanischen Waermetheorie. Wien. Ber. 75, 62-100. Abhandlungen 2, 112-148.

[1877a] Ueber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Waermetheorie und der Wahrshceinlichkeitsrechnung respektive den Saetzen ueber das Waermegleichgewicht. Wien. Ber. 76, 373-435. Abhandlungen 2, 164-223.

[1881] Referat ueber die Abhandlung von J. C. Maxwell "Ueber Boltzmanns Theorem betreffend die mittlere Verteilung der lebendigen Kraft in einem System materieller Punkte." Wied. Ann. Beiblaetter 5, 403-417. Abhandlungen 2, 582-595.

[1884] Ueber die Eigenschaften monocyclischer und anderer damit verwandter Systeme. J. r. ang. Math. (Crelles Journal) 98, 68-94. Abhandlungen 3, 122-152.

[1887] Ueber die mechanischen Analogien des zweiten Hauptsatzes der Thermodynamik. J. r. ang. Math. (Crelles Journal) 100, 201-212. Abhandlungen 3, 258-271.

[1896] Vorlesungen ueber Gastheorie I. J. A. Barth, Leipzig, 1896. Reprinted, Akademische Druck-u. Verlagsanstalt, Graz, 1981.

[1896a] Entgegnung auf die Waermetheoretischen Betrachtungen des Hrn. E. Zermelo. Ann. Phys. [3] 57. Abhandlungen 3, 567-578.

[1898] Vorlesungen ueber Gastheorie II. J. A. Barth, Leipzig, 1898. Reprinted, Akademische Druck-u. Verlagsanstalt, Graz, 1981.

Brush, S. G.

[1966] Kinetic Theory, Vol. 2. Pergamon Press.

[1976] The Kind of Motion we call Heat, 2 vols. North-Holland Publ. Co.

[1983] Statistical Physics and the Atomic Theory of Matter from Boyle and Newton to Landau and Onsager. Princeton University Press.

Ehrenfest, Paul and Tatiana

[1959] The Conceptual Foundations of the Statistical Approach in Mechanics. Dover Publications, Inc., 1990. [Translated by M. J. Moravcsik, from "Begriffliche Grundlagen der statistischen Auffassung in der Mechanik." Encyklopaedie der mathematische Wissenschaften, Vol. 4, 1912]

Krueger, Lorenz, G. Gigerenzer, and M. S. Morgan, eds.

[1987] The Probabilistic Revolution, 2 vols. MIT Press.

Loschmidt, Josef

[1876] Ueber den Zustand des Waermegleichgewichtes eines Systems von Koerpern mit Ruecksicht auf die Schwerkraft. Wien. Ber. 73, 75, 76.

Maxwell, James Clerk

The Scientific Papers of James Clerk Maxwell. Reprinted, Dover, 1965.

Maxwell on Molecules and Gases, ed. by E. Garber, S. Brush, and C. W. F. Everitt. MIT Press, 1986.

[1860] Illustrations of the Dynamical Theory of Gases. Phil. Mag. [4]19, 20.

[1867] On the Dynamical Theory of Gases. Phil. Trans. 157.

[1871] Theory of Heat. Longmans, Green, and Co.

[1879] On Boltzmann's Theorem on the Average Distribution of Energy in a System of Material Points. Trans. Camb. Phil. Soc. 12.

Tomonaga, S.

[1979] What is Physics? 2 vols. [In Japanese] Iwanami, Tokyo.

Uchii, S.

[1990] Is Theory Reduction Possible? [in Japanese] Frontiers of the Contemporary Philosophy (ed. by K. Kamino). Keiso Shobo, Tokyo.

von Plato, J.

[1987] Probabilistic Physics the Classical Way. In Krueger et al. 1987, vol. 2, pp. 379-407.

11. Appendix: Irreversibility still Unexplained?

One of the most perplexing problems for Boltzmann's explanation of irreversibility in terms of probability is this: Given a gas with an initial condition with small time-average, we can predict, with high probability, that its entropy will increase in the future; but we can extend the same reasoning to the past, and tell, with high probability, that its entropy must have been high in the distant past.

This point was already noticed by Boltzmann himself (1877).

I will mention here a peculiar consequence of Loschmidt's theorem, namely that when we follow the state of the world into the infinitely distant past, we are actually just as correct in taking it to be very probable that we would reach a state in which all temperature differences have disappeared, as we would be in the following the state of the world into the distant future. [Brush 1966, 193]

And this makes Boltzmann dubious about the applicability of the kinetic theory to the entire universe (ibid.).

Now, how should we analyze the problem? Maybe we should first distinguish (1) ordinary cases of thermodynamic phenomena from (2) the consideration of the entire universe.

Let us recall that the second law applies to an isolated system. And the preceding consideration applies only to such systems as isolated and having a long or infinite history in the past. Now, how does this affect the two cases?

(1) In the ordinary case, the asssumption that a given gas is an isolatd system is either idealization or approximation. Moreover, it is very hard to imagine that the given gas is literally an isolated system and has an infinite history in the past. Indeed, if we know that this gas has a low entropy state, it must be because we have somehow prepared that state only a short time ago. This means, of course, we cannot apply probability consideration to the past. We already know that the gas had an initial state of low entropy; and we also know that it can have only one-way history into the future. Thus, given this setting, Boltzmann's explanation in terms of probability seems to be all right.

(2) But if we consider the entire universe, it seems that both conditions of isolation and long past history are satisfied. However, is it then probable to conclude that the bulk of the histrory was occupied by high entropy states? Not really; because (a) if the universe has the beginning, its history up to now may occupy only a small proportion to its entire history including the future. On the other hand, (b) if the past history is infinite, the conclusion seems unavoidable. However, aside from the argument that this does not harmonize with our best available theory of the universe, it seems to go far beyond our experience of thermodynamic processes. Thus it seems to me that, in this context, we may not be justified in taking the irreversibility for granted as applicable to the entire universe. If we may take seriously Dr. Tomonaga's condition of insufficient knowledge for combining time-average with probability, Boltzmann's doubt should work in either of the following two ways: Either we should doubt the applicability of the kinetic theory to the entire universe with two-way infinite history, or we should doubt our belief in the irreversibility on such a grand scale.

[For some discussions of irreversibility by philosophers, see the following: L. Sklar, "The Elusive Object of Desire: In pursuit of the Kinetic Equations and the Second Law"; John Earman, "The Problem of Irreversibility," both in PSA 1986, Vol. 2, ed. by A. Fine and P. Machamer, Philosophy of Science Association, 1987. And, of course, Sklar's impressive book, Physics and Chance, Cambridge Univ. Press, 1993, appeared, after the present paper was finished.]

ĒŠ 10. For or Against Reductionism?

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March 1, 1999; last modified March 4. (c) Soshichi Uchii