Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T21:08:38.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Typicality and the Approach to Equilibrium in Boltzmannian Statistical Mechanics

Published online by Cambridge University Press:  01 January 2022

Roman Frigg
Get access Rights & Permissions [Opens in a new window]

Abstract

An important contemporary version of Boltzmannian statistical mechanics explains the approach to equilibrium in terms of typicality. The problem with this approach is that it comes in different versions, which are, however, not recognized as such and not clearly distinguished. This article identifies three different versions of typicality-based explanations of thermodynamic-like behavior and evaluates their respective successes. The conclusion is that the first two are unsuccessful because they fail to take the system's dynamics into account. The third, however, is promising. I give a precise formulation of the proposal and present an argument in support of its central contention.

Type
Research Article
Information
Philosophy of Science , Volume 76 , Issue 5 , December 2009 , pp. 997 - 1008
Copyright
Copyright © The Philosophy of Science Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Special thanks goes to Charlotte Werndl for invaluable discussions about dynamical systems. I also would like to thank Craig Callender, Hartry Field, Shelly Goldstein, Stephan Hartmann, Carl Hoefer, Wolfgang Pietsch, and Nino Zanghì for valuable comments on earlier drafts and helpful discussions. Thanks to the audiences in Madrid, Oxford, Pittsburgh, and Jerusalem for stimulating discussions.

References

Albert, David Z. (2000), Time and Chance. Cambridge, MA: Harvard University Press.Google Scholar
Argyris, John, Faust, Gunter, and Haase, Maria (1994), An Exploration of Chaos. Amsterdam: Elsevier.Google Scholar
Bricmont, Jean (1996), “Science of Chaos or Chaos in Science?”, in Gross, Paul R., Levitt, Norman, and Lewis, Martin W. (eds.), The Flight from Science and Reason. Annals of the New York Academy of Sciences 775. New York: New York Academy of Sciences, 131–175.Google Scholar
Callender, Craig (2001), “Taking Thermodynamics Too Seriously”, Taking Thermodynamics Too Seriously 32:539–553.Google Scholar
Dürr, Detlef (1998), “Über den Zufall in der Physik”, paper given at the 1998 Leopoldina Meeting, Halle, http://www.mathematik.uni-muenchen.de/%7Eduerr/Zufall/zufall.html.Google Scholar
Earman, John, and Rédei, Miklós (1996), “Why Ergodic Theory Does Not Explain the Success of Equilibrium Statistical Mechanics”, Why Ergodic Theory Does Not Explain the Success of Equilibrium Statistical Mechanics 47:63–78.Google Scholar
Frigg, Roman (2009), “Probability in Boltzmannian Statistical Mechanics”, in Ernst, Gerhard and Hüttemann, Andreas (eds.), Time, Chance and Reduction: Philosophical Aspects of Statistical Mechanics. Cambridge: Cambridge University Press.Google Scholar
Goldstein, Sheldon (2001), “Boltzmann's Approach to Statistical Mechanics”, in Bricmont, Jean, Dürr, Detlef, Galavotti, Maria Carla, Ghirardi, Gian Carlo, Petruccione, Francesco, and Zanghì, Nino (eds.), Chance in Physics: Foundations and Perspectives. Berlin: Springer, 39–54.CrossRefGoogle Scholar
Goldstein, Sheldon, and Lebowitz, Joel L. (2004), “On the (Boltzmann) Entropy of Non-equilibrium Systems”, On the (Boltzmann) Entropy of Non-equilibrium Systems 193:53–66.Google Scholar
Lavis, David (2005), “Boltzmann and Gibbs: An Attempted Reconciliation”, Boltzmann and Gibbs: An Attempted Reconciliation 36:245–273.Google Scholar
Lavis, David (2008), “Boltzmann, Gibbs, and the Concept of Equilibrium”, Boltzmann, Gibbs, and the Concept of Equilibrium 75:682–696.Google Scholar
Lebowitz, Joel L. (1993a), “Boltzmann's Entropy and Time's Arrow”, Physics Today (September): 32–38.CrossRefGoogle Scholar
Lebowitz, Joel L. (1993b), “Macroscopic Laws, Microscopic Dynamics, Time's Arrow and Boltzmann's Entropy”, Macroscopic Laws, Microscopic Dynamics, Time's Arrow and Boltzmann's Entropy 194:1–27.Google Scholar
Lebowitz, Joel L. (1999), “Statistical Mechanics: A Selective Review of Two Central Issues”, Statistical Mechanics: A Selective Review of Two Central Issues 71:346–357.Google Scholar
Lichtenberg, Allan J., and Liebermann, M. A. (1992), Regular and Chaotic Dynamics. 2nd ed. Berlin: Springer.CrossRefGoogle Scholar
Markus, Lawrence, and Meyer, Kenneth R. (1974), “Generic Hamiltonian Dynamical Systems Are neither Integrable nor Ergodic”, Generic Hamiltonian Dynamical Systems Are neither Integrable nor Ergodic 144:1–52.Google Scholar
Maudlin, Tim (2007), “What Could Be Objective about Probabilities?”, What Could Be Objective about Probabilities? 38:275–291.Google Scholar
Ott, Edward (1993), Chaos in Dynamical Systems. Cambridge: Cambridge University Press.Google Scholar
Sklar, Lawrence (1993), Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Smith, Peter (1998), Explaining Chaos. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Uffink, Jos (2007), “Compendium of the Foundations of Classical Statistical Physics”, in Butterfield, Jeremy and Earman, John (eds.), Philosophy of Physics. Amsterdam: North-Holland, 923–1047.CrossRefGoogle Scholar
Volchan, Sérgio B. (2007), “Probability as Typicality”, Probability as Typicality 38:801–814.Google Scholar
Vranas, Peter B. M. (1998), “Epsilon-Ergodicity and the Success of Equilibrium Statistical Mechanics”, Epsilon-Ergodicity and the Success of Equilibrium Statistical Mechanics 65:688–708.Google Scholar
Zanghì, Nino (2005), “I fondamenti concettuali dell’approccio statistico in fisica”, in Allori, Valia, Dorato, Mauro, Laudisa, Federico, and Zanghì, Nino (eds.), La natura delle cose: Introduzione ai fundamenti e alla filosofia della fisica. Rome: Carocci.Google Scholar