Canonical distributions and phase transitions

K.B. Athreya; J.D.H. Smith

Discussiones Mathematicae Probability and Statistics (2000)

  • Volume: 20, Issue: 2, page 167-176
  • ISSN: 1509-9423

Abstract

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Entropy maximization subject to known expected values is extended to the case where the random variables involved may take on positive infinite values. As a result, an arbitrary probability distribution on a finite set may be realized as a canonical distribution. The Rényi entropy of the distribution arises as a natural by-product of this realization. Starting with the uniform distributionon a proper subset of a set, the canonical distribution of equilibriumstatistical mechanics may be used to exhibit an elementary phase transition, characterized by discontinuity of the partition function.

How to cite

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K.B. Athreya, and J.D.H. Smith. "Canonical distributions and phase transitions." Discussiones Mathematicae Probability and Statistics 20.2 (2000): 167-176. <http://eudml.org/doc/287676>.

@article{K2000,
abstract = {Entropy maximization subject to known expected values is extended to the case where the random variables involved may take on positive infinite values. As a result, an arbitrary probability distribution on a finite set may be realized as a canonical distribution. The Rényi entropy of the distribution arises as a natural by-product of this realization. Starting with the uniform distributionon a proper subset of a set, the canonical distribution of equilibriumstatistical mechanics may be used to exhibit an elementary phase transition, characterized by discontinuity of the partition function.},
author = {K.B. Athreya, J.D.H. Smith},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {canonical distribution; canonical ensemble; Gibbs state; phase transition; entropy maximization; Rényi entropy},
language = {eng},
number = {2},
pages = {167-176},
title = {Canonical distributions and phase transitions},
url = {http://eudml.org/doc/287676},
volume = {20},
year = {2000},
}

TY - JOUR
AU - K.B. Athreya
AU - J.D.H. Smith
TI - Canonical distributions and phase transitions
JO - Discussiones Mathematicae Probability and Statistics
PY - 2000
VL - 20
IS - 2
SP - 167
EP - 176
AB - Entropy maximization subject to known expected values is extended to the case where the random variables involved may take on positive infinite values. As a result, an arbitrary probability distribution on a finite set may be realized as a canonical distribution. The Rényi entropy of the distribution arises as a natural by-product of this realization. Starting with the uniform distributionon a proper subset of a set, the canonical distribution of equilibriumstatistical mechanics may be used to exhibit an elementary phase transition, characterized by discontinuity of the partition function.
LA - eng
KW - canonical distribution; canonical ensemble; Gibbs state; phase transition; entropy maximization; Rényi entropy
UR - http://eudml.org/doc/287676
ER -

References

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  1. [1] J.W. Gibbs, Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics, Yale University Press New Haven, CT 1902. Zbl33.0708.01
  2. [2] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (1957), 620. Zbl0084.43701
  3. [3] M. Tribus, Rational Descriptions, Decisions and Designs Pergamon, Elmsford, NY 1969. 
  4. [4] R.D. Levine and M. Tribus (eds.), The Maximum Entropy Formalism, MIT Press, Cambridge, MA 1979. 
  5. [5] J.H. Justice (ed.), Maximum Entropy and Bayesian Methods in Applied Statistics, Cambridge University Press, Cambridge 1986. 
  6. [6] W.T. Grandy, Jr, Foundations of Statistical Mechanics, Volume I, Reidel, Dordrecht 1987. Zbl0721.60119
  7. [7] A. Rényi, On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability 1960, Volume 1, University of California Press, Berkeley, CA 1961. Zbl0115.35502
  8. [8] J. Aczél, Measuring information beyond information theory - why some generalized information measures may be useful, others not, Aequationes Math. 27 (1984), 1. Zbl0547.94004
  9. [9] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, ``Handbuch der Physik, Bd. 24/1 Quantentheorie'', Springer, Berlin 1933. Zbl0007.13504
  10. [10] C. Shannon, A mathematical theory of communication, Bell Systems Technical Journal 23 (1948), 349. Zbl1154.94303
  11. [11] K.B. Athreya, Entropy maximization, IMA Preprint Series # 1231, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 1994. 

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