Abstract
Why do systems prepared in a non-equilibrium state approach, and eventually reach, equilibrium? An important contemporary version of the Boltzmannian approach to statistical mechanics answers this question by an appeal to the notion of typicality. The problem with this approach is that it comes in different versions, which are, however, not recognised as such, much less clearly distinguished, and we often find different arguments pursued side by side. The aim of this paper is to disentangle different versions of typicality-based explanations of thermodynamic behaviour and evaluate their respective success. My conclusion will be that the boldest version fails for technical reasons, while more prudent versions leave unanswered essential questions.
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
Notes
- 1.
- 2.
- 3.
If one takes the past state to be the state at the beginning of the universe, there is the further question of whether or not one needs to explain why the world came into being in such a special state. For opposite views on that matter see the contributions of Callender and Price to Hitchcock (2004).
- 4.
Tyicality measures often are, but need not be, probability measures (Zanghì, 2005, 188).
- 5.
This definition of typicality is adapted from Dürr (1998, Section 2), Lavis (2005, 258), Zanghì (2005, 185), and Volchan (2007, 805). Strictly speaking one should refer to this notion as ‘ɛ-typicality’ because the definition depends on the choice of ɛ and elements that are typical with respect to one choice of ɛ need not be typical with respect to another. However, nothing in what follows depends on a particular choice of ɛ and so there is no need to make this dependence explicit. Furthermore, there is an alternative definition of typicality which is stricter than the one adopted here in that it requires \(\nu(\Pi)/\nu(\Sigma) = 1\). This definition is unsuitable in the present context because it classifies as atypical certain elements that, from a physics point of view, clearly are typical.
- 6.
Square brackets indicate that Goldstein’s notation has been replaced by the notion used in this paper. I will use this convention throughout.
- 7.
Albert takes a similar stance and dismisses approaches to the foundations of SM that appeal to ergodicity as ‘sheer madness’ (2000, 70) and ergodic theory as an enterprise that has ‘produced beautiful mathematics’ but is ultimately, if we are interested in the foundation of SM, ‘nothing more nor less [...] than a waste of time’ (ibid.).
- 8.
- 9.
I neglect the possibility that there maybe x that come from or move into microstates of the same entropy. These cases could be accounted for by introducing the subsets \(\Gamma_{M_{i}}^{(0+)}\), etc., and rephrasing the argument accordingly. One can easily see that this would not alter the conclusions that I reach and I therefore neglect them in the interest of ease of discussion and notion.
- 10.
A point to this effect was first made by Ehrenfest and Ehrenfest-Afanassjewa (1912, 32–34). However, their argument is based on an explicitly probabilistic model and so its relevance to deterministic dynamical system is tenuous.
- 11.
A measure μ ′ is absolutely continuous with μ iff for any measurable region \(A \subseteq \Gamma_{E}\): if \(\mu(A)=0\) then \(\mu'(A)=0\). More colloquially, a measure μ ′ is absolutely continuous with another measure μ if it assigns measure zero to all sets that are assigned measure zero by μ, while, possibly, assigning different values to the sets to which μ assigns non-zero measure.
- 12.
Maybe an defence along the lines of Malement and Zabell (1980) would fit the bill, but this would need to be argued in detail.
- 13.
See Callender (2010) for a further discussion of the problems that arise in connection with gravity.
References
Albert, D. (2000), Time and Chance, Cambridge, MA: Harvard University Press.
Arnold, V. I. (2006), Ordinary Differential Equations. Heidelberg: Springer.
Boltzmann, L. (1877), ‘Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung resp. den Sätzen über das Wärmegleichgewicht. Wiener Berichte 76, 373-435. Reprinted in F. Hasenöhrl (ed.): Wissenschaftliche Abhandlungen. Leipzig: J. A. Barth 1909, Vol. 2, 164-223.
Bricmont, J. (1996), Science of Chaos or Chaos in science?, In P. R. Gross, N. Levitt, and M. W. Lewis (eds.), The Flight from Science and Reason, Annals of the New York Academy of Sciences, New York, NY Vol. 775, 131-175.
Bricmont, J. (2001), Bayes, Boltzmann and Bohm: Probabilities in physics, in Bricmont et al., 3–21.
Callender, C. (1999), Reducing thermodynamics to statistical mechanics: The case of entropy, Journal of Philosophy 96, 348–373.
Callender, C. (2001), Taking thermodynamics too seriously, Studies in the History and Philosophy of Modern Physics 32, 539–53.
Callender, C. (2010), The past hypothesis meets gravity, forthcoming In G. Ernst and A. Hüttemann (eds.), Time, Chance and Reduction. Philosophical Aspects of Statistical Mechanics. Cambridge: Cambridge University Press 2010, pp. 34–58.
Dürr, D. (1998), Über den Zufall in der Physik, manuscript presented at the 1998 Leopoldina Meeting in Halle. Available at http://www.mathematik.uni-muenchen.de/%7Eduerr/Zufall/zufall.html
Dürr, D. (2001), Bohmsche Mechanik als Grundlage der Quantenmechanik. Berlin: Springer.
Dürr, D., S. Goldsein and N. Zanghì (1992), Quantum Equilibrium and the Origin of Absolute Uncertainty, Journal of Statistical Physics 67, 843–907.
Earman, J. (2006), The “past hypothesis”: Not even False, Studies in History and Philosophy of Modern Physics 37, 399–430.
Earman, J. and Rédei, M. (1996), Why Ergodic Theory Does Not Explain the Success of Equilibrium Statistical Mechanics, British Journal for the Philosophy of Science 47, 63–78.
Ehrenfest, P. and Ehrenfest-Afanassjewa, T. (1912/1959), The Conceptual Foundations of the Statistical Approach in Mechanics. New York NY: Dover 2002. (First published in German in 1912; first English Translation 1959.)
Frigg, R. (2008), A Field Guide to Recent Work on the Foundations of Statistical Mechanics, In Rickles, D. (ed.), The Ashgate Companion to Contemporary Philosophy of Physics. London: Ashgate, 99–196.
Galvan, B. (2006), Typicality vs. probability in trajectory-based formulations of quantum mechanics, Foundations of Physics 37, 1540–1562.
Goldstein, S. (2001), Boltzmann’s Approach to Statistical Mechanics, in: Bricmont et al. 2001, 39–54.
Goldstein, S. and Lebowitz, J. L. (2004), On the (Boltzmann) entropy of non-equilibrium systems, Physica D 193, 53–66.
Goldstein, S., Lebowitz, J. L. Tomulka, R. and Zanghì, N. (2006), Canonical Typicality, Physical Review Letters 96, Issue 5.
Hitchcock, C. (ed.) (2004) Contemporary Debates in Philosophy of Science. Malden, MA: Blackwell.
Lavis, D. (2005), Boltzmann and Gibbs: An attempted reconciliation, Studies in History and Philosophy of Modern Physics 36, 245–273.
Lavis, D. (2008), Boltzmann, Gibbs and the concept of equilibrium, forthcoming in Philosophy of Science, 75 (December 2008), pp. 682–696.
Lebowitz, J. L. (1993a), Boltzmann’s entropy and time’s arrow, Physics Today, September Issue, 32–38.
Lebowitz, J. L. (1993b), Macroscopic laws, microscopic dynamics, time’s arrow and Boltzmann’s entropy, Physica A 194, 1–27.
Lebowitz, J. L. (1999), Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics 71, 346–357.
Loewer, B. (2001), Determinism and chance, Studies in History and Philosophy of Modern Physics 32, 609–629.
Malament, D. B. and Zabell, S. L. (1980), Why Gibbs phase averages work, Philosophy of Science 47, 339–349.
Maudlin, T. (2007), What could be objective about probabilities?, Studies in History and Philosophy of Modern Physics 38, 275–291.
Penrose, R. (1989), The Emperor’s New Mind. Oxford: Oxford University Press.
Salmon, W. (1992), Scientific Explanation, In M. Salmon, et al. (eds.), Introduction to the Philosophy of Science. Cambridge: Hackett, 7–23.
Sklar, L. (1973), Statistical explanation and ergodic theory, Philosophy of Science 40, 194–212.
Sklar, L. (1993), Physics and Chance. Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: Cambridge University Press.
Uffink, J. (2007), Compendium of the foundations of classical statistical physics, In J. Butterfield and J. Earman (eds.), Philosophy of Physics. Amsterdam: North Holland, 923–1047.
van Lith, J. (2001), Ergodic theory, interpretations of probability and the foundations of statistical mechanics, Studies in History and Philosophy of Modern Physics 32, 581–594.
Volchan, S. B. (2007), Probability as Typicality, Studies in History and Philosophy of Modern Physics 38, 801–814.
Zanghì, N. (2005), I Fondamenti concettuali dell’approccio statistico in Fisica, In V. Allori, M. Dorato, F. Laudisa and N. Zanghì (eds.), La Natura Delle Cose. Introduzione ai Fundamenti e alla Filosofia della Fisica. Roma: Carocci.
Acknowledgements
Special thanks goes to David Lavis for many illuminating discussions on SM in general, and the Boltzmannian approach in particular. I also would like to thank Craig Callender, Stephan Hartmann, Carl Hoefer, Wolfgang Pietsch, Charlotte Werndl, and two anonymous referees for valuable comments on earlier drafts. Thanks to Jean Bricmont for a helpful email conversation on his mixing condition discussed in Section 4.3, and to Detlef Dürr for drawing my attention to omissions in my first bibliography. Many thanks to Flavia Padovani for helping me with those passages in Zanghì’s chapter that were beyond the reach of my ‘FAPP Italian’. Thanks to Mauricio Suárez for organising the workshop at which this paper has first been presented, and thanks to the audiences in Madrid and Oxford for stimulating discussions. Finally, I would like to acknowledge financial support from two project grants of the Spanish Ministry of Science and Education (SB2005-0167 and HUM2005-04369).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Netherlands
About this chapter
Cite this chapter
Frigg, R. (2011). Why Typicality Does Not Explain the Approach to Equilibrium. In: Suárez, M. (eds) Probabilities, Causes and Propensities in Physics. Synthese Library, vol 347. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9904-5_4
Download citation
DOI: https://doi.org/10.1007/978-1-4020-9904-5_4
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-9903-8
Online ISBN: 978-1-4020-9904-5
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)