Abstract
The relation between the Boltzmannian and the Gibbsian formulations of statistical mechanics is one of the major conceptual issues in the foundations of the discipline. In their celebrated review of statistical mechanics, Paul Ehrenfest and Tatiana Ehrenfest-Afanassjewa discuss this issue and offer an argument for the conclusion that Boltzmannian equilibrium values agree with Gibbsian phase averages. In this paper, we analyse their argument, which is still important today, and point out that its scope is limited to dilute gases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The original paper was published in German under the title ‘Begriffiche Grundlagen der Statistischen Auffassung in der Mechanik’ in 1911. Throughout this paper, we quote the English translation that came out in 1959 under the title ‘The conceptual foundations of the statistical approach in mechanics’.
- 2.
- 3.
In this paper, we mostly follow Ehrenfest and Ehrenfest-Afanassjewa and consider deterministic systems. In our (2017) we discuss stochastic systems and show that the main results carry over to the stochastic context. We consider an explicitly stochastic system below in Sect. 4.6.
- 4.
For ease of notation, we suppress the subscript ‘i’ from now.
- 5.
Strictly speaking, this is true only under an additional assumption that we discuss in the next section.
- 6.
- 7.
- 8.
A variant of the Ehrenfest-Afanassjewa Condition requires that the observable f is constant nearly everywhere on phase space and does not take extreme values on the rest of the phase space:
There is an constant \(C\in \mathbb {R}\) and a \(\bar{X}\subseteq X\) with \(\mu _{X}(\bar{X})= 1-\delta \) (for a small \(\delta \ge 0\)) such that (i) \(|f(x)-C|\le \varepsilon \) for all \(x\in \bar{X}\) for a very small \(\varepsilon \ge 0\) and (ii) \(|\int _{X\setminus \bar{X}} f(x)d\mu _{X}-C\delta )| \le \gamma \) (for a very small \(\gamma \ge 0\)).
Because the Boltzmannian equilibrium macro-value \(F_{equ}\) takes up more than \(\delta \) of phase space, it follows that \(F_{equ}\) is very close to C. Therefore, \(|f(x)-F_{equ}|\le \epsilon _{1}\) for a small \(\varepsilon _{1}\ge 0\) for all \(x\in \bar{X}\) and \(|\int _{X\setminus \bar{X}} f(x)d\mu _{X}-F_{equ}\delta )| \le \gamma _{1}\) (for a very small \(\gamma _{1}\ge 0\)). This is in fact the original Ehrenfest-Afanassjewa Condition and so the variant is in fact equivalent to the original Ehrenfest-Afanassjewa Condition.
- 9.
See Uffink’s (2007, 1020–1028) for a discussion.
- 10.
Given a certain macro-variable f and an allowable difference between the Gibbsian phase average and the Boltzmannian equilibrium macro-value, one could precisely quantify what notion of ‘approximately the same macro-value as the Maxwell–Boltzmann distribution’ would be needed in order for the Khinchin theorem to go through by making use of the calculations in Jeans (1904, Sects. 46–56).
- 11.
Alternatively, ‘most’ can also be understood as referring to the fact that the system spends at least \(\alpha >1/2\) of its time in equilibrium, leading to the different notion of an \(\alpha \)-\(\varepsilon \)-equilibrium. Nothing in what follows hinges on which notion of equilibrium is adopted (cf. Werndl and Frigg 2015b and forthcoming references).
- 12.
As we have seen, for sufficiently low temperatures \(\{\xi ^*, \xi ^{+}\}\) is the largest macro-region. The higher the temperature, the more uniform is the probability measure. Hence, for sufficiently high temperatures, the largest macro-region will differ from \(\{\xi ^*,\xi ^{+}\}\). Because the canonical distribution is continuous in T, there exists a T such that \(\{\xi ^*, \xi ^{+}\}\) is the largest macro-region but its probability is \(\le 0.5\).
- 13.
Such micro-state corresponds to the smallest possible departure from the macro-state with zero energy because the number of downward pointing arrows is the same for all rows. From this, then follows that there has to be a perturbation in each row and that \(\sqrt{N}\) has to be the second lowest value of the internal energy (Lavis and Bell 1999).
- 14.
- 15.
It is assumed that N ia a multiple of k, i.e. \(N=k*s\) for some \(s\in \mathbb {N}\).
References
Baxter, R. J. (1982). Exactly solved models in statistical mechanics. London: Academic Press.
Boltzmann, L. (1877). Über die Beziehung zwischen dem zweiten Hauptsatze der mecha-nischen Wärmetheorie und der Wahrscheinlichkeitsrechnung resp. den Sätzen über das Wärmegleichgewicht. Wiener Berichte, 76, 373–435.
Davey, K. (2009). What is Gibbs’s canonical distribution? Philosophy of Science, 76, 970–983.
Ehrenfest, P., & Ehrenfest-Afanassjewa, T. (1959). The conceptual foundations of the statistical approach in mechanics. Ithaca, N.Y.: Cornell University Press.
Frigg, R. (2008). A field guide to recent work on the foundations of statistical mechanics. In D. Rickles (Ed.), The ashgate companion to contemporary philosophy of physics (pp. 99–196). London: Ashgate.
Frigg, R., & Werndl, C. (2019). Can somebody please say what Gibbsian statistical mechanics says? The British Journal for the Philosophy of Science, online first, https://doi.org/10.1093/bjps/axy057.
Jeans, J. H. (1904). The dynamical theory of gases. Cambridge: Cambridge University Press.
Khinchin, A. I. (1960) [1949]. Mathematical foundations of statistical mechanics. Mineola/NY: Dover Publications.
Lavis, D. (2005). Boltzmann and Gibbs: An attempted reconciliation. Studies in History and Philosophy of Modern Physics, 36, 245–73.
Lavis, D. (2008). Boltzmann, Gibbs and the concept of equilibrium. Philosophy of Science, 75, 682–96.
Lavis, D., & Bell, G. M. (1999). Statistical mechanics of lattice systems, Volume 1: Closed form and exact solutions. Berlin and Heidelberg: Springer.
Malament, D., & Zabell, S. L. (1980). Why Gibbs phase averages work. Philosophy of Science 47, 339–49.
Myrvold, W. C. (2016). Probabilities in statistical mechanics. In C. Hitchcock & A. Hájek (Eds.), The Oxford handbook of probability and philosophy (pp. 573–600). Oxford: Oxford University Press.
Slater, J. C. (1941). Theory of the transition in KH\(^2\)PO\(^4\). Journal of Chemical Physics, 9, 16–33.
Uffink, J. (2007). Compendium of the foundations of classical statistical physics. In J. Butterfield & J. Earman (Eds.), Philosophy of physics (pp. 923–1047). Amsterdam: North Holland.
Werndl, C., & Frigg, R. (2015a). Rethinking Boltzmannian equilibrium. Philosophy of Science, 82, 1224–35.
Werndl, C., & Frigg, R. (2015b). Reconceptionalising equilibrium in Boltzmannian statistical mechanics. Studies in History and Philosophy of Modern Physics, 49, 19–31.
Werndl, C., & Frigg, R. (2017a). Boltzmannian equilibrium in stochastic systems. In M. Michela & R. Jan-Willem (Eds.), Proceedings of the EPSA15 conference (pp. 243–254). Berlin and New York: Springer.
Werndl, C., & Frigg, R. (2017b). Mind the gap: Boltzmannian versus Gibbsian equilibrium. Philosophy of Science, 84, 1289–1302.
Werndl, C., & Frigg, R. Forthcoming. When does a Boltzmannian equilibrium exist?. In D. Bedingham, O. Maroney, C. Timpson (Eds.), Quantum foundations of statistical mechanics. Oxford: Oxford University Press.
Werndl, C., & Frigg, R. (2020). When do Gibbsian phase averages and Boltzmannian equilibrium values agree? Studies in History and Philosophy of Modern Physics, DOI:https://www.sciencedirect.com/science/article/abs/pii/S1355219820300903.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Werndl, C., Frigg, R. (2021). Ehrenfest and Ehrenfest-Afanassjewa on Why Boltzmannian and Gibbsian Calculations Agree. 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_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-47971-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-47970-1
Online ISBN: 978-3-030-47971-8
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)