, Volume 80, Issue 4, pp 689–716 | Cite as

Typicality, Irreversibility and the Status of Macroscopic Laws

  • Dustin LazaroviciEmail author
  • Paula Reichert
Original Article


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.


Boltzmann Equation Thermodynamic Behavior Typicality Account Liouville Measure Molecular Chaos 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  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.Google Scholar
  3. Boltzmann, L. (1896). Vorlesungen über Gastheorie. (Leipzig: Verlag v. J. A. Barth, Leipzig). Nabu Public Domain Reprints.Google Scholar
  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.CrossRefGoogle 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.Google Scholar
  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.CrossRefGoogle 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.CrossRefGoogle Scholar
  16. Frigg, R., & Werndl, C. (2011). Explaining thermodynamic-like behaviour in terms of epsilon-ergodicity. Philosophy of Science, 78, 628–652.CrossRefGoogle Scholar
  17. Frigg, R., & Werndl, C. (2012). Demystifying typicality. Philosophy of Science, 79, 917–929.CrossRefGoogle 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].Google Scholar
  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.CrossRefGoogle 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.CrossRefGoogle Scholar
  22. King, F. (1975). BBGKY hierarchy for positive potentials. Dissertation, University of California at Berkeley.Google Scholar
  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.Google Scholar
  25. Lavis, D. (2005). Boltzmann and Gibbs: An attempted reconciliation. Studies in History and Philosophy of Modern Physics, 36, 245–273.CrossRefGoogle Scholar
  26. Lebowitz, J. (1981). Microscopic dynamics and macroscopic laws. Annals New York Academy of Sciences, pp. 220–233.Google Scholar
  27. Lebowitz, J. (1993). Macroscopic laws, microscopic dynamics, time’s arrow and Boltzmann’s entropy. Physica A, 194, 1–27.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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].Google Scholar
  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.CrossRefGoogle Scholar
  35. Sklar, L. (1973). Statistical explanation and ergodic theory. Philosophy of Science, 40(2), 194–212.CrossRefGoogle Scholar
  36. Spohn, H. (1991). Large scale dynamics of interacting particles. Berlin: Springer.CrossRefGoogle 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.CrossRefGoogle Scholar
  38. Uffink, J. (2008). Boltzmann’s work in statistical pysics. The Stanford Encyclopedia of Philosophy.Google Scholar
  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.CrossRefGoogle 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

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Ludwig-Maximilians-Universität München, Mathematisches InstitutMunichGermany

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