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Explaining Thermodynamic-Like Behavior in Terms of Epsilon-Ergodicity

Published online by Cambridge University Press:  01 January 2022

Roman Frigg  and
Charlotte Werndl
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Abstract

Why do gases reach equilibrium when left to themselves? The canonical answer, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer is now widely regarded as flawed. We argue that some of the main objections in particular arguments based on the Kolmogorov-Arnold-Moser theorem and the Markus-Meyer theorem are beside the point. We then argue that something close to Boltzmann’s proposal is true: gases behave thermodynamic-like if they are epsilon-ergodic, that is, ergodic on the phase space except for a small region of measure epsilon. This answer is promising because there is evidence that relevant systems are epsilon-ergodic.

Type
Research Article
Information
Philosophy of Science , Volume 78 , Issue 4 , October 2011 , pp. 628 - 652
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

‡

Authors are listed alphabetically. This work is fully collaborative. Earlier versions of this article have been presented at the 2010 British Society for the Philosophy of Science conference and at the Universities of Utrecht and Oxford; we would like to thank the audiences for valuable discussions. We also want to thank Scott Dumas, David Lavis, Pierre Lochak, and David Wallace for helpful comments. Roman Frigg also wishes to acknowledge support from the Spanish government research project FFI2008-01580/consolider ingenio CSD2009-0056.

References

Albert, David. 2000. Time and Chance. Cambridge, MA: Harvard University Press.Google Scholar
Arnold, Vladimir I. 1963. “Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics.” Russian Mathematical Surveys 18:85–193.CrossRefGoogle Scholar
Arnold, Vladimir I.. 1964. “Instabilities in Dynamical Systems with Several Degrees of Freedom.” Soviet Mathematics Doklady 5:581–85.Google Scholar
Arnold, Vladimir I.. 1980. Mathematical Methods of Classical Mechanics. New York: Springer.Google Scholar
Arnold, Vladimir I.. 1994. “Mathematical Problems in Classical Physics.” In Trends and Perspectives in Applied Mathematics, ed. Sirovich, Lawrence, 1–20. Berlin: Springer.Google Scholar
Arnold, Vladimir I., Kozlov, Valery I., and Neishtat, Anatoly I.. 1985. Dynamical Systems. Vol. 3. Heidelberg: Springer.Google Scholar
Bennetin, Giancarlo, Livi, Roberto, and Ponno, Antonio. 2009. “The Fermi-Pasta-Ulam Problem: Scaling Laws vs. Initial Conditions.” Journal of Statistical Physics 135:873–93.Google Scholar
Bennetin, Giancarlo, Lo Vecchio, Guido, and Tenenbaum, Alexander. 1980. “Stochastic Transition in Two-Dimensional Lennard-Jones Systems.” Physics Review A 22:1709–19.Google Scholar
Berti, Massimiliano, Biasco, Luca, and Bolle, Philippe. 2003. “Drift in Phase Space: A New Variational Mechanism with Optimal Diffusion Time.” Journal de Mathématiques Pures et Appliqués 82:613–64.Google Scholar
Bocchieri, P., Scotti, Antonio, Bearzi, Bruno, and Loinger, A.. 1970. “Anharmonic Chain with Lennard-Jones Interaction.” Physical Review A 2:213–19.Google Scholar
Boltzmann, Ludwig. 1871. “Einige allgemeine Sätze über Wärmegleichgewicht.” Wiener Berichte 53:670–711.Google Scholar
Bricmont, Jean. 2001. “Bayes, Boltzmann and Bohm: Probabilities in Physics.” In Chance in Physics: Foundations and Perspectives, ed. Bricmont, Jean, Dürr, Detlef, Galavotti, Maria C., Ghirardi, Gian C., Pettrucione, Francesco, and Zanghi, Nino, 3–21. Berlin: Springer.Google Scholar
Callender, Craig. 1999. “Reducing Thermodynamics to Statistical Mechanics: The Case of Entropy.” Journal of Philosophy 96:348–73.Google Scholar
Callender, Craig. 2001. “Taking Thermodynamics Too Seriously.” Studies in History and Philosophy of Modern Physics 32:539–53.CrossRefGoogle Scholar
Chernov, Nikolai, and Markarian, Roberto. 2006. Chaotic Billiards. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Chernov, Nikolai, and Young, Lai-Sang. 2000. “Decay of Lorentz Gases and Hard Balls.” In Hard Ball Systems and the Lorentz Gas, ed. Szász, Domokos, 89–120. Berlin: Springer.CrossRefGoogle Scholar
Chirikov, Boris V. 1979. “A Universal Instability of Many-Dimensional Oscillator Systems.” Physics Reports 56:263–379.CrossRefGoogle Scholar
Chirikov, Boris V.. 1991. “Patterns in Chaos.” Chaos, Solitons and Fractals 1:79–103.CrossRefGoogle Scholar
Dellago, Christoph, and Posch, Harald A.. 1997. “Mixing, Lyapunov Instability, and the Approach to Equilibrium in a Hard-Sphere Gas.” Physical Review E 55:9–12.Google Scholar
Delshams, Amadeu, and Huguet, Gemma. 2009. “Geography of Resonances and Arnold Diffusion in A Priori Unstable Hamiltonian Systems.” Nonlinearity 22:1997–2077.CrossRefGoogle Scholar
De Souza, Vanessa K., and Wales, David J.. 2005. “Diagnosing Broken Ergodicity Using an Energy Fluctuation Metric.” Journal of Chemical Physics 123:134504.CrossRefGoogle ScholarPubMed
Diana, E., Galgani, Luigi, Casartelli, Mario, Casati, Giulio, and Scotti, Antonio. 1976. “Stochastic Transition in a Classical Nonlinear Dynamical System: A Lennard-Jones Chain.” Theoretical and Mathematical Physics 29:1022–27.CrossRefGoogle Scholar
Dizadji-Bahmani, Foad, Frigg, Roman, and Hartmann, Stephan. 2010. “Who’s Afraid of Nagelian Reduction?” Erkenntnis 73:393–412.CrossRefGoogle Scholar
Donnay, Victor J. 1999. “Non-ergodicity of Two Particles Interacting via a Smooth Potential.” Journal of Statistical Physics 96:1021–48.CrossRefGoogle Scholar
Donnay, Victor J., and Liverani, Carlangelo. 1991. “Potentials on the Two-Torus for Which the Hamiltonian Flow Is Ergodic.” Communications in Mathematical Physics 135:267–302.CrossRefGoogle Scholar
Earman, John, and Rédei, Miklós. 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
Emch, Gerard G., and Liu, Chuang. 2002. The Logic of Thermostatistical Physics. Berlin: Springer.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
Frigg, Roman. 2009a. “Probability in Boltzmannian Statistical Mechanics.” In Time, Chance and Reduction: Philosophical Aspects of Statistical Mechanics, ed. Ernst, Gerhard and Hüttemann, Andreas, 92–118. Cambridge: Cambridge University Press.Google Scholar
Frigg, Roman. 2009b. “Typicality and the Approach to Equilibrium in Boltzmannian Statistical Mechanics.” Philosophy of Science 76 (Proceedings): 997–1008.CrossRefGoogle Scholar
Frigg, Roman. 2010. “Why Typicality Does Not Explain the Approach to Equilibrium.” In Probabilities, Causes and Propensities in Physics, ed. Suárez, Mauricio, 77–93. Berlin: Springer.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, Gonzalez, Wesley, Harmann, Stephan, Weber, Marcel, Stadler, Friedrich, and Uebel, Thomas, 351–72. Berlin: Springer.CrossRefGoogle Scholar
Froeschlé, Claude, Guzzo, Massimiliano, and Lega, Elena. 2000. “Graphical Evolution of the Arnold Web: From Order to Chaos.” Science 289:2108–10.CrossRefGoogle ScholarPubMed
Froeschlé, Claude, and Schneidecker, Jean-Paul. 1975. “Stochasticity of Dynamical Systems with Increasing Degrees of Freedom.” Physical Review A 12:2137–43.Google Scholar
Goldstein, Sheldon. 2001. “Boltzmann’s Approach to Statistical Mechanics.” In Chance in Physics: Foundations and Perspectives, ed. Bricmont, Jean, Dürr, Detlef, Galavotti, Maria C., Ghirardi, Gian C., Pettrucione, Francesco, and Zanghi, Nino, 39–54. Berlin: Springer.Google Scholar
Guzzo, Massimiliano, Lega, Elena, and Froeschlé, Claude. 2005. “First Numerical Evidence of Global Arnold Diffusion in Quasi-Integrable Systems.” Discrete and Continuous Dynamical Systems B 5:687–98.Google Scholar
Hirsch, Morris W. 1976. Differential Topology. Berlin: Springer.CrossRefGoogle 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. 2011. “An Objectivist Account of Probabilities in Statistical Physics.” In Probabilities in Physics, ed. Beisbart, Claus and Hartmann, Stephan. Oxford: Oxford University Press, forthcoming.Google Scholar
Markus, Larry, and Meyer, Kenneth R.. 1969. “Generic Hamiltonians Are Not Ergodic.” In Proceedings of the 9th Conference on Nonlinear oscillations, ed. Mitropolsky, Yu A., 311–32. Kiev: Kiev Naukova Dumka.Google Scholar
Markus, Larry, and Meyer, Kenneth R.. 1974. “Generic Hamiltonian Dynamical Systems Are neither Integrable nor Ergodic.” Memoirs of the American Mathematical Society 144:1–52.Google Scholar
Mather, John N. 2004. “Arnold Diffusion I: Announcement of Results.” Journal of Mathematical Sciences 124:5275–89.CrossRefGoogle Scholar
McQuarrie, Donald A. 2000. Statistical Mechanics. Sausalito, CA: University Science.Google Scholar
Ott, Edward. 2002. Chaos in Dynamical Systems. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Oxtoby, John C. 1980. Measure and Category. New York: Springer.CrossRefGoogle Scholar
Penrose, Oliver. 1979. “Foundations of Statistical Physics.” Reports on Progress in Physics 42:1937–2006.CrossRefGoogle Scholar
Penrose, Oliver, and Lebowitz, Joel L.. 1973. “Modern Ergodic Theory.” Physics Today 26:23–29.Google Scholar
Percival, Ian. 1986. “Integrable and Nonintegrable Hamiltonian Systems.” In Nonlinear Dynamics Aspects of Particle Accelerators, ed. Jowett, John M., Month, Melvin, and Turner, Stuart, 12–36. Berlin: Springer.Google Scholar
Pettini, Marco. 2007. Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics. New York: Springer.CrossRefGoogle Scholar
Pettini, Marco, and Cerruti-Sola, Monica. 1991. “Strong Stochasticity Thresholds in Nonlinear Large Hamiltonian Systems: Effect on Mixing Times.” Physical Review A 44:975–87.Google Scholar
Reichl, Linda E. 1998. A Modern Course in Statistical Physics. New York: Wiley.Google Scholar
Reidl, Charles J., and Miller, Bruce N.. 1993. “Gravity in One Dimension: The Critical Population.” Physical Review E 48:4250–56.Google Scholar
Simányi, Nandor. 1992. “The K-Property of N Billiard Balls.” Inventiones Mathematicae 108:521–48.CrossRefGoogle Scholar
Simányi, Nandor. 1999. “Ergodicity of Hard Spheres in a Box.” Ergodic Theory and Dynamical Systems 19:741–66.CrossRefGoogle Scholar
Simányi, Nandor. 2003. “Proof of the Boltzmann-Sinai Ergodic Hypothesis for Typical Hard Disk Systems.” Inventiones Mathematicae 154:123–78.CrossRefGoogle Scholar
Simányi, Nandor. 2009. “Conditional Proof of the Boltzmann-Sinai Ergodic Hypothesis.” Inventiones Mathematicae 177:381–413.CrossRefGoogle Scholar
Sinai, Yakov G. 1963. “On the Foundations of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics.” Soviet Mathematics Doklady 4:1818–22.Google Scholar
Sinai, Yakov G.. 1970. “Dynamical Systems with Elastic Reflections: Ergodic Properties of Dispersing Billiards.” Uspekhi Matematicheskikh Nauk 25:141–92.Google Scholar
Sklar, Lawrence. 1993. Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Stoddard, Spotswood D., and Ford, Joseph. 1973. “Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System.” Physical Review A 8:1504–12.Google Scholar
Szász, Domokos. 1996. “Boltzmann’s Ergodic Hypothesis: A Conjecture for Centuries?” Studia Scientiarum Mathematicarum Hungarica 31:299–322.Google Scholar
Uffink, Jos. 1996. “Nought but Molecules in Motion.” Review of Physics and Chance, by Lawrence Sklar. Studies in History and Philosophy of Modern Physics 27:373–87.CrossRefGoogle Scholar
Uffink, Jos. 2007. “Compendium to the Foundations of Classical Statistical Physics.” In Philosophy of Physics, ed. Butterfield, Jeremy and Earman, John, 923–1074. Amsterdam: North-Holland.Google Scholar
Vivaldi, Franco. 1984. “Weak-Instabilities in Many-Dimensional Hamiltonian Systems.” Review of Modern Physics 56:737–53.CrossRefGoogle Scholar
Vranas, Peter B. M. 1998. “Epsilon-Ergodicity and the Success of Equilibrium Statistical Mechanics.” Philosophy of Science 65:688–708.Google Scholar
Werndl, Charlotte. 2009a. “Are Deterministic Descriptions and Indeterministic Descriptions Observationally Equivalent?” Studies in History and Philosophy of Modern Physics 40:232–42.CrossRefGoogle Scholar
Werndl, Charlotte. 2009b. “Justifying Definitions in Mathematics: Going Beyond Lakatos.” Philosophia Mathematica 17:313–40.CrossRefGoogle Scholar
Werndl, Charlotte. 2009c. “What Are the New Implications of Chaos for Unpredictability?” British Journal for the Philosophy of Science 60:195–220.CrossRefGoogle Scholar
Werndl, Charlotte. 2011. “On the Observational Equivalence of Continuous-Time Deterministic and Indeterministic Descriptions.” European Journal for the Philosophy of Science 1 (2): 193–225.CrossRefGoogle Scholar
Wright, Harold, and Miller, Bruce N.. 1984. “Gravity in One Dimension: A Dynamical and Statistical Study.” Physical Review A 29:1411–18.Google Scholar
Yoshimura, Kazuyoshi. 1997. “Strong Stochasticity Threshold in Some Anharmonic Lattices.” Physica D 104:148–62.Google Scholar
Zheng, Zhigang, Hu, Gang, and Zhang, Juyuan. 1996. “Ergodicity in Hard-Ball Systems and Boltzmann’s Entropy.” Physical Review E 53:3246–53.Google Scholar