Abstract
This article highlights and discusses the Ehrenfests’ use of toy models to explore irreversibility in statistical mechanics. In particular, we explore their urn and P–Q models and highlight that, while the former was primarily used to provide a simple counter-example to Zermelo’s objection to Boltzmann’s statistical mechanical underpinning of the Second Law of Thermodynamics, the latter was intended to highlight the role and importance of the Stoßzahlansatz as a cause of the tendency of systems to exhibit entropy increase. We also explain the sense in which these models are toy models and why agents can use them, as the Ehrenfests’ did, to carry out this important work, despite the fact that they do not represent any real system.
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Notes
- 1.
- 2.
Historically, however, intentionality conditions have usually been added to analyses of scientific representation only after they have been rejected for failing to account of the logical properties of scientific representations.
- 3.
See Bartha (2013) for a comprehensive discussion of analogies and analogical reasoning.
- 4.
For a discussion of other important functions performed by toy models see Luczak (2017).
- 5.
The urn model was introduced in Ehrenfest and Ehrenfest (1907).
- 6.
unvorstellbar.
- 7.
in der Regel.
- 8.
See Ehrenfest and Ehrenfest (1907, p. 2).
- 9.
For the original description of the set-up, see Ehrenfest and Ehrenfest (1907, pp. 2–4).
- 10.
hüpft.
- 11.
See Ehrenfest and Ehrenfest (1907, p. 2–3).
- 12.
Ehrenfest and Ehrenfest (1907, p. 2): Es ist immer wahrscheinlicher, daß die jeweils aufgerufene Kugel in der volleren, als daß sie in der leereren Urne angetroffen wird. Solange also die Urne A noch viel voller ist als die Urne B, wird sich bei den folgenden Ziehungen in der Regel die Urne A in B entleeren und nur ausnahmsweise eine Kugel aus B erhalten.
- 13.
See Ehrenfest and Ehrenfest (1907, p. 4).
- 14.
molekulare Unordnung.
- 15.
Ehrenfest and Ehrenfest (1907, p. 4, footnote 1): Deshalb lassen wir hier die Frage durchaus unberührt, inwieweit der Nachweis des H-Theorems etwa als lückenlos angesehen werden kann; welchen Sinn man im speziellen der Hypothese einer dauernden “molekularen Unordnung” geben soll.
- 16.
See e.g. Kac (1956).
- 17.
See Luczak (2017).
- 18.
The P–Q model was introduced in Ehrenfest and Ehrenfest (1909).
- 19.
Zwischenstück.
- 20.
Ehrenfest and Ehrenfest (1909, p. 19): ... welche Stellung der Stoßzahlansatz in den zuletzt erwähnten Maxwell-Boltzmannschen Untersuchungen einnimmt.
- 21.
For a description of the model’s set-up, see Ehrenfest and Ehrenfest (1909, pp. 19–20).
- 22.
Ehrenfest and Ehrenfest (1909, p. 19): ... auf jedes grösseres Gebiet sollen nahe gleichviel entfallen ...
- 23.
Ehrenfest and Ehrenfest (1909, p. 20): Das Analogon zu dem mehrfach genannten Stoßzahlansatz besteht nun in der folgenden Behauptung: Von den P-Molekülen jeder einzelnen Bewegungsrichtung entfällt auf die Streifen S ein solcher Bruchteil, als dem Verhältnis der Gesamtfläche aller S zur totalen freien Fläche entspricht.
- 24.
Ehrenfest and Ehrenfest (1909, p. 20): Wenn bei der Berechnung der Zahlen \(N_{12}\), \(N_{21}\), \(N_{23}\), \(N_{32}\) etc. für jedes Zeitelement \(\Delta t\) immer wieder der Stoßzahlansatz (7) zugrunde gelegt wird, so erhält man eine monotone Abnahme der Unterschiede der Zahlen \(f_1\), \(f_2\), \(f_3\), \(f_4\).
- 25.
See Ehrenfest and Ehrenfest (1909, p. 19).
- 26.
- 27.
See Boltzmann (1872).
- 28.
The quantity we call H was originally denoted E in Boltzmann’s early work. See Boltzmann (1872).
- 29.
See Brown et al. (2009) for more on Boltzmann’s H-theorem.
- 30.
- 31.
- 32.
This is easiest to see if we consider the extreme disequilibrium state. Suppose that, at \(t = 0\), \(k_{-x,j,\Delta _t} = 1\), and that the Stoßzahlansatz holds. A time \(\Delta _t\) later, a fraction of these P-molecules have been scattered, half into the \(-y\)-direction, half into the y-direction. Suppose, now, we reverse the velocities, and ask what fraction of, say, the y-direction P-molecules will collide with a Q-molecule in time \(\Delta _t\). Answer: all of them! The P-molecules that are not travelling in the x-direction are all on collision courses that will turn them into x-travelling molecules.
- 33.
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Luczak, J., Zuchowski, L. (2021). The Ehrenfests’ Use of Toy Models to Explore Irreversibility in Statistical Mechanics. In: Uffink, J., Valente, G., Werndl, C., Zuchowski, L. (eds) The Legacy of Tatjana Afanassjewa. Women in the History of Philosophy and Sciences, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-47971-8_6
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