Discussiones Mathematicae Probability and Statistics 20(2) (2000) 167-176

CANONICAL DISTRIBUTIONS AND PHASE TRANSITIONS

K.B. Athreya

School of Operations Research
and Industrial Engineering Cornell University
Ithaca, NY 14853, USA

J.D.H. Smith

Department of Mathematics, Iowa State University
Ames, IA 50011, USA

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.

Keywords: canonical distribution, canonical ensemble, Gibbs state, phase transition, entropy maximization, Rényi entropy.

1991 Mathematics Subject Classification: 82B26, 94A17.

References

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Received 3 February 1999
Revised 15 March 2000