Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-23T16:42:24.441Z Has data issue: false hasContentIssue false
  • English
  • Français

Why Equilibrium Statistical Mechanics Works: Universality and the Renormalization Group

Published online by Cambridge University Press:  01 April 2022

Robert W. Batterman
Robert W. Batterman*
Affiliation:
Department of Philosophy, Ohio State University
Get access Rights & Permissions [Opens in a new window]

Abstract

Discussions of the foundations of Classical Equilibrium Statistical Mechanics (SM) typically focus on the problem of justifying the use of a certain probability measure (the microcanonical measure) to compute average values of certain functions. One would like to be able to explain why the equilibrium behavior of a wide variety of distinct systems (different sorts of molecules interacting with different potentials) can be described by the same averaging procedure. A standard approach is to appeal to ergodic theory to justify this choice of measure. A different approach, eschewing ergodicity, was initiated by A. I. Khinchin. Both explanatory programs have been subjected to severe criticisms. This paper argues that the Khinchin type program deserves further attention in light of relatively recent results in understanding the physics of universal behavior.

Type
Research Article
Information
Philosophy of Science , Volume 65 , Issue 2 , June 1998 , pp. 183 - 208
Copyright
Copyright © Philosophy of Science Association 1998

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

†

Send reprint requests to the author. Department of Philosophy, 350 University Hall, Ohio State University, Columbus, OH 43210.

‡

This material is based upon work supported by the National Science Foundation under Award No. SBR-9529052. I would like to thank Roger Jones, David Malament, and Abner Shimony for helpful comments and encouragement. A version of this paper was read at the 1997 APA Central division meetings in Pittsburgh. I would especially like to thank Yuri Balashov for his insightful criticisms as commentator there. I hope I have been able to address some of his worries.

References

Barenblatt, Grigory. I. (1996), Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Batterman, Robert. W. (1998), “Universality, Unification, and Understanding”, Preprint.Google Scholar
Bleher, P. M. and Sinai, Ya. G. (1973), “Investigation of the Critical Point in Models of the Type of Dyson's Hierarchical Models”, Communications in Mathematical Physics 33: 23–42.10.1007/BF01645604CrossRefGoogle Scholar
Cassandra, M. and Jona-Lasinio, G. (1978), “Critical Point Behaviour and Probability Theory”, Advances in Physics 27: 913–941.CrossRefGoogle Scholar
Earman, John, and Rédei, M. (1996), “Why Ergodic Theory Does Not Explain the Success of Equilibrium Statistical Mechanics”, British Journal of the Philosophy of Science 47: 63–78.CrossRefGoogle Scholar
Gallavotti, Giovanni, and Martinlöf, A. (1975), “Block-spin Distributions for Short-range Attractive Ising Models”, Il Nuovo Cimento 25 B: 425–441.CrossRefGoogle Scholar
Gnedenko, Boris. V. and Kolmogorov, A. N. ([1949] 1968), Limit Distributions for Sums of Independent Random Variables. Translated by K. L. Chung. Reading MA: Addison-Wesley.Google Scholar
Goldenfeld, Nigel, Martin, O., and Oono, Y. (1989), “Intermediate Asymptotics and Renormalization Group Theory”, Journal of Scientific Computing 4: 355–372.CrossRefGoogle Scholar
Ibraghimov, Ildar. A. and Linnik, Yu. V. (1969), Independent and Stationary Sequences of Random Variables. Groeningen: Walter Noordhoff Publishing Company.Google Scholar
Jona-Lasinio, G. (1975), “The Renormalization Group: A Probabilistic View”, Il Nuovo Cimento 26 B: 99–119.CrossRefGoogle Scholar
Khinchin, Alexander. I. (1949), Mathematical Foundations of Statistical Mechanics. Translated by G Gamow. New York: Dover Publications.Google Scholar
Lanford, Oscar E. III (1973), “Entropy and Equilibrium States in Classical Statistical Mechanics”, in Lenard, A. (ed.), Statistical Mechanics and Mathematical Problems Berlin: Springer-Verlag, pp. 1–113.Google Scholar
Malament, David and Zabell, S. (1980), “Why Gibbs Phase Averages Work-The Role of Ergodic Theory”, Philosophy of Science 47: 339–349.10.1086/288941CrossRefGoogle Scholar
Mazur, Peter and van der Linden, J. (1963), “Asymptotic Form of the Structure Function for Real Systems”, Journal of Mathematical Physics 4: 271–277.CrossRefGoogle Scholar
Ruelle, David (1969), Statistical Mechanics: Rigorous Results. New York: W. A. Benjamin.Google Scholar
Sinai, Yakov G. (1978), “Mathematical Foundations of the Renormalization Group Method in Statistical Physics”, in Dell'Antonio, G., Doplicher, S., and Jona-Lasinio, G. (eds.), Mathematical Problems in Theoretical Physics. Berlin: Springer-Verlag, pp. 303–311.CrossRefGoogle Scholar
Sinai, Yakov G. (1982), Theory of Phase Transitions: Rigorous Results. Oxford: Pergamon Press.Google Scholar
Sinai, Yakov G. (1992), Probability Theory: An Introductory Course. Translated by D. Haughton. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Sklar, Lawrence (1973), “Statistical Explanation and Ergodic Theory”, Philosophy of Science 40: 194–212.CrossRefGoogle Scholar
Sklar, Lawrence (1993), Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: Cambridge University Press.10.1017/CBO9780511624933CrossRefGoogle Scholar
Truesdell, Clifford (1961), “Ergodic Theory in Classical Statistical Mechanics”, in P. Caldirola (ed.), Ergodic Theories, volume 14 of Proceedings of the International School of Physics “Enrico Fermi”. New York: Academic Press, pp. 21–56.Google Scholar
Wightman, Arthur S. (1983), “Regular and Chaotic Motions in Dynamical Systems: Introduction to the Problems”, in Velo, G. and Wightman, A. S., (eds.), G. Velo and A. S. Wightman, New York: Plenum Press, pp. 1–26.Google Scholar
Wilson, Kenneth G. and Kogut, J. (1974), “The Renormalization Group and the ∊ Expansion”, Physics Reports 12: 75–200.CrossRefGoogle Scholar