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Rethinking Boltzmannian Equilibrium

Published online by Cambridge University Press:  01 January 2022

Charlotte Werndl  and
Roman Frigg
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Abstract

Boltzmannian statistical mechanics partitions the phase space of a system into macroregions, and the largest of these is identified with equilibrium. What justifies this identification? Common answers focus on Boltzmann’s combinatorial argument, the Maxwell-Boltzmann distribution, and maximum entropy considerations. We argue that they fail and present a new answer. We characterize equilibrium as the macrostate in which a system spends most of its time and prove a new theorem establishing that equilibrium thus defined corresponds to the largest macroregion. Our derivation is completely general and does not rely on assumptions about the dynamics or interparticle interactions.

Type
Classical Physics
Information
Philosophy of Science , Volume 82 , Issue 5 , December 2015 , pp. 1224 - 1235
Copyright
Copyright © The Philosophy of Science Association

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References

Ainsworth, Peter Mark. 2012. “Entropy in Statistical Mechanics.” Philosophy of Science 79:542–60.CrossRefGoogle Scholar
Albert, David. 2000. Time and Chance. Cambridge, MA: Harvard University Press.Google Scholar
Baxter, Rodney. 1982. Exactly Solved Models in Statistical Mechanics. San Diego, CA: Academic Press.Google Scholar
Boltzmann, Ludwig. 1877. “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung resp. den Sätzen über das Wärmegleichgewicht.” Wiener Berichte 76:373–435.Google Scholar
Brown, Harvey, and Uffink, Jos. 2001. “The Origins of Time-Asymmetry in Thermodynamics: The Minus First Law.” Studies in History and Philosophy of Modern Physics 32:525–38.CrossRefGoogle Scholar
Callender, Craig. 2001. “Taking Thermodynamics Too Seriously.” Studies in History and Philosophy of Modern Physics 32:539–53.CrossRefGoogle Scholar
Dizadji-Bahmani, Foad, Frigg, Roman, and Hartmann, Stephan. 2010. “Who Is Afraid of Nagelian Reduction?” Erkenntnis 73:393–412.CrossRefGoogle Scholar
Ehrenfest, Paul, and Ehrenfest, Tatiana. 1959. The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press.Google Scholar
Frigg, Roman. 2008. “A Field Guide to Recent Work on the Foundations of Statistical Mechanics.” In The Ashgate Companion to Contemporary Philosophy of Physics, ed. Rickles, Dean, 99–196. London: Ashgate.Google Scholar
Frigg, Roman, and Hoefer, Carl. 2010. “Determinism and Chance from a Humean Perspective.” In The Present Situation in the Philosophy of Science, ed. Dieks, Dennis, Gonzales, Wencelao, Hartmann, Stephan, Weber, Marcel, Stadler, Friedrich, and Übel, Thomas, 351–72. Berlin: Springer.Google Scholar
Frigg, Roman, and Werndl, Charlotte. 2011. “Explaining Thermodynamic-Like Behavior in Terms of Epsilon-Ergodicity.” Philosophy of Science 78:628–52.CrossRefGoogle Scholar
Frigg, Roman, and Werndl, Charlotte 2012. “Demystifying Typicality.” Philosophy of Science 79:917–29.CrossRefGoogle Scholar
Goldstein, Sheldon, and Lebowitz, Joel L.. 2004. “On the (Boltzmann) Entropy of Nonequilibrium Systems.” Physica D 193:53–66.Google Scholar
Gupta, Mool C. 2003. Statistical Thermodynamics. New Delhi: New Age.Google Scholar
Lavis, David. 2005. “Boltzmann and Gibbs: An Attempted Reconciliation.” Studies in History and Philosophy of Modern Physics 36:245–73.CrossRefGoogle Scholar
Lavis, David 2008. “Boltzmann, Gibbs and the Concept of Equilibrium.” Philosophy of Science 75:682–96.CrossRefGoogle Scholar
Penrose, Roger. 1989. The Emperor’s New Mind. Oxford: Oxford University Press.CrossRefGoogle Scholar
Petersen, Karl. 1983. Ergodic Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Reiss, Howard. 1996. Methods of Thermodynamics. Mineola, NY: Dover.Google Scholar
Uffink, Jos. 2001. “Bluff Your Way in the Second Law of Thermodynamics.” Studies in History and Philosophy of Modern Physics 32:305–94.CrossRefGoogle Scholar
Uffink, Jos 2007. “Compendium of the Foundations of Classical Statistical Physics.” In Philosophy of Physics, ed. Buttereld, Jeremy and Earman, John, 923–1047. Amsterdam: North-Holland.CrossRefGoogle Scholar
Wang, Genmiao, Sevinck, Edith M., Mittag, Emil, Searles, Debra J., and Evans, Denis J.. 2002. “Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales.” Physical Review Letters 89:050601.CrossRefGoogle ScholarPubMed
Werndl, Charlotte. 2013. “Justifying Typicality Measures in Boltzmannian Statistical Mechanics.” Studies in History and Philosophy of Modern Physics 44:470–79.CrossRefGoogle Scholar