Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T21:08:58.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Mind the Gap: Boltzmannian versus Gibbsian Equilibrium

Published online by Cambridge University Press:  01 January 2022

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

Abstract

There are two main theoretical frameworks in statistical mechanics, one associated with Boltzmann and the other with Gibbs. Despite their well-known differences, there is a prevailing view that equilibrium values calculated in both frameworks coincide. We show that this is wrong. There are important cases in which the Boltzmannian and Gibbsian equilibrium concepts yield different outcomes. Furthermore, the conditions under which equilibriums exists are different for Gibbsian and Boltzmannian statistical mechanics. There are, however, special circumstances under which it is true that the equilibrium values coincide. We prove a new theorem providing sufficient conditions for this to be the case.

Type
Physical Sciences
Information
Philosophy of Science , Volume 84 , Issue 5 , December 2017 , pp. 1289 - 1302
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.)

References

Baxter, Rodney J. 1982. Exactly Solved Models in Statistical Mechanics. London: Academic Press.Google Scholar
Berger, Arno. 2001. Chaos and Chance: An Introduction to Stochastic Aspects of Dynamics. New York: de Gruyter.CrossRefGoogle Scholar
Cipra, Barry. 1987. “An Introduction to the Ising Model.” American Mathematical Monthly 94:937–54.CrossRefGoogle Scholar
Davey, Kevin. 2009. “What Is Gibbs’s Canonical Distribution?” Philosophy of Science 76 (5): 970–83.CrossRefGoogle 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
Goldstein, Sheldon. 2001. “Boltzmanns Approach to Statistical Mechanics.” In Chance in Physics: Foundations and Perspectives, ed. Bricmont, Jean, Dürr, Detlef, Galavotti, Maria Carla, Ghirardi, Gian Carlo, Petruccione, Francesco, and Zangh, Nino, 39–54. Berlin: Springer.CrossRefGoogle Scholar
Khinchin, Alexandr I. 1949. Mathematical Foundations of Statistical Mechanics. Mineola, NY: Dover.Google Scholar
Lavis, David. 2005. “Boltzmann and Gibbs: An Attempted Reconciliation.” Studies in History and Philosophy of Modern Physics 36:245–73.CrossRefGoogle Scholar
Lebowitz, Joel L. 1993. “Boltzmann’s Entropy and Time’s Arrow.” Physics Today 46:32–38.CrossRefGoogle Scholar
Malament, David B., and Zabell, Sandy L. 1980. “Why Gibbs Phase Averages Work: The Role of Ergodic Theory.” Philosophy of Science 56:339–49.Google Scholar
Secular, Paul. 2015. “Monte-Carlo Simulation of Small 2D Ising Lattice with Metropolis Dynamics.” http://secular.me.uk/physics/ising-model.pdf.Google Scholar
Uffink, Jos. 2007. “Compendium of the Foundations of Classical Statistical Physics.” In Philosophy of Physics, ed. Butterfield, Jeremy and Earman, John, 923–1047. Amsterdam: North Holland.CrossRefGoogle Scholar
Werndl, Charlotte, and Frigg, Roman. 2015a. “Reconceptionalising Equilibrium in Boltzmannian Statistical Mechanics.” Studies in History and Philosophy of Modern Physics 49:19–31.CrossRefGoogle Scholar
Werndl, Charlotte, and Frigg, Roman 2015b. “Rethinking Boltzmannian Equilibrium.” Philosophy of Science 82:1224–35.CrossRefGoogle Scholar
Werndl, Charlotte, and Frigg, Roman 2017. “Boltzmannian Equilibrium in Stochastic Systems.” In EPSA15 Selected Papers: The 5th Conference of the European Philosophy of Science Association in Düsseldorf, ed. Massimi, Michela, Romeijn, Jan-Willem, and Schurz, Gerhard. Berlin: Springer.Google Scholar
Werndl, Charlotte, and Frigg, Roman Forthcominga. “Transcript of the Talk and Discussion at the Oxford Workshop in Honour of David Wallace.” In Proceedings of the Quantum Foundations of Statistical Mechanics Workshop in Oxford, ed. Daniel Bendingham, Owen Marony, and Christopher Timpson. Oxford: Oxford University Press.Google Scholar
Werndl, Charlotte, and Frigg, Roman Forthcomingb. “When Does a Boltzmannian Equilibrium Exist?” In Quantum Foundations of Statistical Mechanics, ed. Daniel Bedingham, Owen Maroney, and Christopher Timpson. Oxford: Oxford University Press.Google Scholar