## 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.
- 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.

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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.
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- 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.

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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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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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### Keywords

- Boltzmann Equation
- Thermodynamic Behavior
- Typicality Account
- Liouville Measure
- Molecular Chaos