Typicality, Irreversibility and the Status of Macroscopic Laws


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.

This is a preview of subscription content, log in to check access.

Fig. 1


  1. 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. 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. 3.

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

  4. 4.

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

  5. 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. 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. 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. 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. 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. 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. 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. 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).


  1. Albert, D. (2000). Time and chance. Cambridge: Harvard University Press.

    Google Scholar 

  2. Bernoulli, J. (1713). Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis. (Basel: Thurneysen Brothers). Reprinted: Bernoulli, J. (2006). The art of conjecturing. Baltimore: The John Hopkins University Press.

  3. Boltzmann, L. (1896). Vorlesungen über Gastheorie. (Leipzig: Verlag v. J. A. Barth, Leipzig). Nabu Public Domain Reprints.

  4. Boltzmann, L. (1896). Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo. Wiedemann’s Annalen, 57, 773–784.

    Google Scholar 

  5. Bricmont, J. (1995). Science of chaos or chaos in science? Physicalia Magazine, 17, 159–208.

    Google Scholar 

  6. Callender, C. (2004). There is no Puzzle about the low entropy past. In C. Hitchcock (Ed.), Contemporary debates in philosophy of science (pp. 240–255). London: Blackwell.

    Google Scholar 

  7. Carroll, S. (2010). From eternity to here. The quest for the ultimate theory of time. Dutton: Penguin Group.

    Google Scholar 

  8. Carroll, S., & Chen, J. (2004). Spontaneous Inflation and the origin of the arrow of time. ArXiv: hep-th/0410270.

  9. Davies, P. C. W. (1977). The physics of time asymmetry. Berkeley: University of California Press.

    Google Scholar 

  10. Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2010). Who’s Afraid of Nagelian reduction? Erkenntnis, 73, 393–412.

    Article  Google Scholar 

  11. Dürr, D. (2009). Bohmian mechanics. Berlin: Springer.

    Google Scholar 

  12. Einstein, A. (1949). Autobiographical notes. In: P. A. Schilpp (Ed.), Albert Einstein: Philosopher scientist, the library of living philosophers (p. 43), sixth printing 1995. La Salle, IL: Open Court.

  13. Feynman, R. (1967). The character of physical law. Cambridge: The MIT Press.

    Google Scholar 

  14. Frigg, R. (2009). Typicality and the approach to equilibrium in Boltzmannian statistical mechanics. Philosophy of Science, 76, 997–1008.

    Article  Google Scholar 

  15. Frigg, R. (2011). Why typicality does not explain the approach to equilibrium. In M. Suárez (Ed.), Probabilities, causes and propensities in physics (pp. 77–93). Dordrecht: Springer.

    Google Scholar 

  16. Frigg, R., & Werndl, C. (2011). Explaining thermodynamic-like behaviour in terms of epsilon-ergodicity. Philosophy of Science, 78, 628–652.

    Article  Google Scholar 

  17. Frigg, R., & Werndl, C. (2012). Demystifying typicality. Philosophy of Science, 79, 917–929.

    Article  Google Scholar 

  18. Gallagher, I., Saint Raymond, L., & Texier, B. (2012). From Newton to Boltzmann: The case of short-range potentials. Preprint: ArXiv: 1208.5753v1 [math.AP].

  19. Goldstein, S. (2001). Boltzmann’s approach to statistical mechanics. In J. Bricmont, D. Dürr, et al. (Eds.), Chance in physics. Foundations and perspectives (pp. 39–54). Berlin: Springer.

    Google Scholar 

  20. Goldstein, S. (2012). Typicality and notions of probability in physics. In Y. Ben-Menahem & M. Hemmo (Eds.), Probability in physics. The frontiers collection (pp. 59–71). Berlin: Springer.

    Google Scholar 

  21. Goldstein, S., & Lebowitz, J. (2004). On the (Boltzmann) entropy of non-equilibrium systems. Physica D: Nonlinear phenomena, 193(1–4), 53–66.

    Article  Google Scholar 

  22. King, F. (1975). BBGKY hierarchy for positive potentials. Dissertation, University of California at Berkeley.

  23. Kripke, S. (1980). Naming and necessity. Oxford: Blackwell.

    Google Scholar 

  24. Lanford, O. E. (1975). Time evolution of large classical systems. In J. Moser (Ed.), Lecture notes in physics (Vol. 38 pp. 1–111), Berlin: Springer.

  25. Lavis, D. (2005). Boltzmann and Gibbs: An attempted reconciliation. Studies in History and Philosophy of Modern Physics, 36, 245–273.

    Article  Google Scholar 

  26. Lebowitz, J. (1981). Microscopic dynamics and macroscopic laws. Annals New York Academy of Sciences, pp. 220–233.

  27. Lebowitz, J. (1993). Macroscopic laws, microscopic dynamics, time’s arrow and Boltzmann’s entropy. Physica A, 194, 1–27.

    Article  Google Scholar 

  28. Maudlin, T. (2007). What could be objective about probabilities? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38(2), 275–291.

    Article  Google Scholar 

  29. Penrose, R. (1999). The emperor’s new mind. Oxford: Oxford University Press.

    Google Scholar 

  30. Price, H. (1996). Time’s arrow & archimedes’ point. New directions for the physics of time. New York: Oxford University Press.

    Google Scholar 

  31. Price, H. (2002). Burbury’s last case: The Mystery of the entropic arrow. In C. Callender (Ed.), Time, reality & experience (pp. 19–56). Cambridge: Cambridge University Press.

    Google Scholar 

  32. Pulvirenti, M., Saffiro, C., & Simonella, S. (2013). On the validity of the Boltzmann equation for short range potentials. Preprint: ArXiv: 1301.2514v1 [math-ph].

  33. Schwartz, J. (1992). The pernicious influence of mathematics on science. In M. Kac, G.-C. Rota, & J. Schwartz (Eds.), Discrete thoughts (pp. 19–25). Boston: Birkhäuser.

    Google Scholar 

  34. Shafer, G., & Volk, V. (2006). The sources of Kolmogorov’s Grundbegriffe. Statistical Science, 21(1), 70–98.

    Article  Google Scholar 

  35. Sklar, L. (1973). Statistical explanation and ergodic theory. Philosophy of Science, 40(2), 194–212.

    Article  Google Scholar 

  36. Spohn, H. (1991). Large scale dynamics of interacting particles. Berlin: Springer.

    Google Scholar 

  37. Uffink, J. (2007). Compendium of the foundations of classical statistical physics. In J. Butterfield & J. Earman (Eds.), Handbook for the philosophy of physics (pp. 923–1047). Amsterdam: Elsevier.

    Google Scholar 

  38. Uffink, J. (2008). Boltzmann’s work in statistical pysics. The Stanford Encyclopedia of Philosophy.

  39. Villani, C. (2002). A review of mathematical topics in collisional kinetic theory. In S. Friedlander & D. Serre (Eds.), Handbook of mathematical fluid dynamics (Vol. 1, pp. 71–305). Amsterdam: Elsevier.

    Google Scholar 

  40. Zanghì, N. (2005). I fondamenti concettuali dell’approccio statistico in fisica. In V. Allori, M. Dorato, F. Laudisa, & N. Zanghì (Eds.), La Natura Delle Cose. Introduzione ai Fundamenti e alla Filosofia della Fisica (pp. 139–228). Roma: Carocci.

    Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Dustin Lazarovici.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


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