Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-12T10:09:45.316Z Has data issue: false hasContentIssue false

6 - Probability in Boltzmannian statistical mechanics

from Part II - Probability and chance

Published online by Cambridge University Press:  04 August 2010

By
Roman Frigg
Edited by
Gerhard Ernst  and
Andreas Hüttemann
Gerhard Ernst
Affiliation:
Universität Stuttgart
Andreas Hüttemann
Affiliation:
Westfälische Wilhelms-Universität Münster, Germany
Get access

Summary

Introduction

A cup of tea, left to its own, cools down while the surrounding air heats up until both have reached the same temperature, and a gas, confined to the left half of a room, uniformly spreads over the entire available space as soon as the confining wall is removed. Thermodynamics (TD) characterizes such processes in terms of an increase of thermodynamic entropy, which attains its maximum value at equilibrium, and the second law of thermodynamics posits that in an isolated system entropy cannot decrease. The aim of statistical mechanics (SM) is to explain the behaviour of these systems, in particular their conformity with the second law, in terms of the dynamical laws governing the individual molecules of which the systems are made up. In what follows these laws are assumed to be the ones of classical mechanics.

An influential suggestion of how this could be achieved was made by Ludwig Boltzmann (1877), and variants of it are currently regarded by many as the most promising option among the plethora of approaches to SM. Although these variants share a commitment to Boltzmann's basic ideas, they differ widely in how these ideas are implemented and used. These differences become most tangible when we look at how the different approaches deal with probabilities. There are two fundamentally different ways of introducing probabilities into SM, and even within these two groups there are important disparities as regards both technical and interpretational issues.

Type
Chapter
Information
Time, Chance, and Reduction
Philosophical Aspects of Statistical Mechanics
 , pp. 92 - 118
Publisher: Cambridge University Press
Print publication year: 2010

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

Abraham, R. and Marsden, J. E. (1980). Foundations of Mechanics, 2nd edn. London: Benjamin-Cummings.
Albert, D. (2000). Time and Chance. Cambridge, MA: Harvard University Press.
Arnold, V. and Avez, A. (1968). Ergodic Problems in Classical Mechanics. New York, Benjamin.
Boltzmann, L. (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. Reprinted in (1909). Wissenschaftliche Abhandlungen, vol. 2, ed. Hasenöhrl, F.. Leipzig: J. A. Barth, pp. 164–223.
Callender, C. (1999) Reducing thermodynamics to statistical mechanics: the case of entropy. Journal of Philosophy, 96, 348–373.CrossRefGoogle Scholar
Clark, P. (2001). Statistical mechanics and the propensity interpretation of probability. In Chance in Physics: Foundations and Perspectives. ed. Bricmont, J.et al. Berlin: Springer-Verlag, pp. 271–281.CrossRef
Earman, J. (1986). A Primer on Determinism. Dordrecht: Kluwer.CrossRef
Earman, J. (2006). The ‘past hypothesis’: not even false. Studies in History and Philosophy of Modern Physics, 37, 399–430.CrossRefGoogle Scholar
Earman, J. and Rédei, M. (1996). Why ergodic theory does not explain the success of equilibrium statistical mechanics. British Journal for the Philosophy of Science, 47, 63–78.CrossRefGoogle Scholar
Ehrenfest, P. and Ehrenfest, T. (1907). Über Zwei Bekannte Einwände gegen das Boltzmannsche H-Theorem. Phyikalische Zeitschrift, 8, 311–14.Google Scholar
Ehrenfest, P. and Ehrenfest, T. (1912/1959). The Conceptual Foundations of the Statistical Approach in Mechanics. Mineola, NY: Dover (Reprinted 2002. First published in German in 1912; first English Translation 1959.)
Elga, A. (2004). Infinitesimal chances and the laws of nature. Australasian Journal of Philosophy, 82, 67–76.CrossRefGoogle Scholar
Frigg, R. and Hoefer, C. (2007). Probability in GRW theory. Studies in the History and Philosophy of Modern Physics, 38, 371–389.CrossRefGoogle Scholar
Frigg, R. (2008). A field guide to recent work on the foundations of statistical mechanics. In: The Ashgate Companion to Contemporary Philosophy. ed. Rickles, D.. London: Ashgate, pp. 99–196.
Goldstein, H. (1981). Classical Mechanics. Reading, MA: Addison Wesley.
Goldstein, S. (2001). Boltzmann's approach to statistical mechanics. In Chance in Physics: Foundations and Perspectives, ed. Bricmont, J.et al. Berlin: Springer-Verlag.
Goldstein, S. and Lebowitz, J. L. (2004). On the (Boltzmann) entropy of non-equilibrium systems. Physica D: Nonlinear Phenomena, 193 (1–4), 53–66.CrossRefGoogle Scholar
Hitchcock, C. (ed.) (2004). Contemporary Debates in Philosophy of Science. Oxford: Blackwell.
Hoefer, C. (2007). The third way on objective probability: a skeptic's guide to objective chance. Mind, 116, 549–596.CrossRefGoogle Scholar
Khinchin, A. I. (1949). Mathematical Foundations of Statistical Mechanics. Mineola, NY: Dover.
Lavis, D. (2005). Boltzmann and Gibbs: an attempted reconciliation. Studies in History and Philosophy of Modern Physics, 36, 245–273.CrossRefGoogle Scholar
Lebowitz, J. L. (1993a). Boltzmann's entropy and time's arrow. Physics Today, September issue, 32–38.Google Scholar
Lebowitz, J. L. (1993b). Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropy. Physica A, 194, 1–27.Google Scholar
Lebowitz, J. L. (1999). Statistical mechanics: a selective review of two central issues. Reviews of Modern Physics, 71, 346–357.CrossRefGoogle Scholar
Lewis, D. (1986). A Subjectivist's Guide to Objective Chance and Postscripts to ‘A subjectivist's guide to objective Chance’. In Philosophical Papers, vol. 2, Oxford: Oxford University Press, pp. 83–132.
Lewis, D. (1994). Humean supervenience debugged. Mind, 103, 473–90.CrossRefGoogle Scholar
Loewer, B. (2001). Determinism and chance. Studies in History and Philosophy of Modern Physics, 32, 609–629.CrossRefGoogle Scholar
Loewer, B. (2004). David Lewis' Humean theory of objective chance. Philosophy of Science, 71, 1115–1125.CrossRefGoogle Scholar
Meacham, C. (2005). Three proposals regarding a theory of chance. Philosophical Perspectives, 19, 281–307.CrossRefGoogle Scholar
Redhead, M. (1995). From Physics to Metaphysics. Cambridge: Cambridge University Press.CrossRef
Sklar, L. (1993). Physics and Chance. Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: Cambridge University Press.CrossRef
Lith, J. (2001). Ergodic theory, interpretations of probability and the foundations of statistical mechanics. Studies in History and Philosophy of Modern Physics, 32, 581–594.CrossRefGoogle Scholar
Uffink, J. (2007). Compendium of the foundations of classical statistical physics. In Philosophy of Physics, ed. Butterfield, J. and Earman, J.. Amsterdam: North-Holland, pp. 923–1047.CrossRef
Plato, J. (1988). Ergodic theory and the foundations of probability. In Causation, Chance and Credence, vol. 1, ed. Skyrms, B. and Harper, W. L.. Dordrecht: Kluwer, pp. 257–277.
Plato, J. (1989). Probability in dynamical systems. In Logic, Methodology and Philosophy of Science, vol. VIII, ed. Fenstad, J. E., Frolov, I. T. and Hilpinen, R.. Amsterdam: North-Holland, pp. 427–443.
Plato, J. (1994). Creating Modern Probability. Cambridge: Cambridge University Press.CrossRef
Winsberg, E. (2004a). Can conditioning on the ‘past hypothesis’ militate against the reversibility objections? Philosophy of Science, 71, 489–504.Google Scholar
Winsberg, E. (2004b). Laws and statistical mechanics. Philosophy of Science, 71, 707–718.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×