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Typicality, Irreversibility and the Status of Macroscopic Laws

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Abstract

We discuss Boltzmann’s probabilistic explanation of the second law of thermodynamics providing a comprehensive presentation of what is called today the typicality account. Countering its misconception as an alternative explanation, we examine the relation between Boltzmann’s H-theorem and the general typicality argument demonstrating the conceptual continuity between the two. We then discuss the philosophical dimensions of the concept of typicality and its relevance for scientific reasoning in general, in particular for understanding the reduction of macroscopic laws to microscopic laws. Finally, we reply to various common criticisms of the typicality account.

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Notes

  1. The term “Past Hypothesis” is due to Albert (2000), though the necessity of such an assumption was already noted by Boltzmann (1896a, pp. 252–53). See also Feynman (1967) and Carroll (2010) for a very nice discussion.

  2. Similarly, the pertinent entry in the Stanford Encyclopedia of Philosophy Uffink (2008) presents Boltzmann’s work as a series of rather incoherent (and ultimately wanting) attempts to explain the second law.

  3. For a good introduction, see, for instance, Davies (1977). For a detailed mathematical treatment, see Spohn (1991); Villani (2002); Lebowitz (1981).

  4. While the “true” microscopic \(H(f_{X(t)}(q,v))\) fluctuates and only decreases “on average”.

  5. Assumption, unfortunately, is not a perfectly accurate translation of the German word Ansatz. Whereas the first is sometimes used synonymously with a logical premise, the later has a distinctly pragmatic element and can refer to something more akin to an “approximation” or a “working hypothesis”.

  6. See Lanford (1975) and King (1975) for the landmark results and Gallagher et al. (2012); Pulvirenti et al. (2013) for recent extensions to more general potentials.

  7. See Dizadji-Bahmani et al. (2010) for a recent defense of Nagelian reduction. On typicality, see, e.g., Maudlin (2007); Bricmont (1995); Dürr (2009); Goldstein (2012), and Zanghì (2005).

  8. On the other hand, many measures would yield a different notion of typicality. One can think, for instance, of singular measures, concentrated on a single point in phase space. Such a measure may even turn out to be stationary, in case that this particular microstate happens to be a stationary point of the dynamics. So why not take such a measure to define “typicality”, meaning that a property is typical if and only if it is instantiated by this one particular configuration? We trust the reader to answer this question for himself.

  9. See Bernoulli (1713). Such typicality statements can be understood in the sense of Cournot’s principle, which is one of the basic principles underlying the philosophy of Kolmogorov’s Grundbegriffe, but also stands in the philosophical tradition of great mathematicians such as Emile Borel, Maurice Fréchet or Paul Lévy. See Shafer and Volk (2006) for a beautiful essay on this topic.

  10. Of course, among all possible Newtonian universes there will be some with no thermodynamic arrow and no interesting structures at all. But here, to make a point, we consider universes that are hospitable to intelligent life, while the second law of thermodynamics fails to hold in branching systems just so often as to make a fool out of physicists.

  11. See, for instance, Penrose (1999) and his “Weyl curvature hypothesis” as a proposal for an additional law restricting the initial state of the universe, but also Callender (2004) arguing from a Humean perspective against the need for further explanation of the Past Hypothesis. See Carroll (2010) for a very nice discussion of the problem and ibid. as well as Carroll and Chen (2004) for an attempt to dispose of the Past Hypothesis altogether.

  12. Ergodicity is probably true for the hard-sphere gas, and almost certainly failing for any more realistic model. For why ergodicity is largely irrelevant in the first place, see e.g. Schwartz (1992); Bricmont (1995) and Goldstein (2001).

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Acknowledgments

We are grateful to Detlef Dürr, Sheldon Goldstein, Tim Maudlin and Nino Zanghì for teaching us almost everything we know about the subject of this paper. Thanks to Jean Bricmont, Mathias Frisch and Jenann Ismael for insightful remarks on various occasions.

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  1. Ludwig-Maximilians-Universität München, Mathematisches Institut, Theresienstr. 39, 80333, Munich, Germany

    Dustin Lazarovici & Paula Reichert

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  1. Dustin Lazarovici
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Correspondence to Dustin Lazarovici.

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Lazarovici, D., Reichert, P. Typicality, Irreversibility and the Status of Macroscopic Laws. Erkenn 80, 689–716 (2015). https://doi.org/10.1007/s10670-014-9668-z

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