Canonical distributions and phase transitions

• Autorzy: K.B. Athreya , J.D.H. Smith
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

canonical distribution; canonical ensemble; Gibbs state; phase transition; entropy maximization; Rényi entropy

Kategorie tematyczne

• 94A17: Measures of information, entropy
• 82B26: Phase transitions (general)

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae Probability and Statistics
• Rocznik: 2000
• Tom: 20
• Numer: 2
• Strony: 167-176

Daty

wydano

2000

otrzymano

1999-02-03

poprawiono

2000-03-15

Twórcy

autor

K.B. Athreya

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

autor

J.D.H. Smith

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

Bibliografia

• [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.
• [2] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (1957), 620.
• [3] M. Tribus, Rational Descriptions, Decisions and Designs Pergamon, Elmsford, NY 1969.
• [4] R.D. Levine and M. Tribus (eds.), The Maximum Entropy Formalism, MIT Press, Cambridge, MA 1979.
• [5] J.H. Justice (ed.), Maximum Entropy and Bayesian Methods in Applied Statistics, Cambridge University Press, Cambridge 1986.
• [6] W.T. Grandy, Jr, Foundations of Statistical Mechanics, Volume I, Reidel, Dordrecht 1987.
• [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.
• [8] J. Aczél, Measuring information beyond information theory - why some generalized information measures may be useful, others not, Aequationes Math. 27 (1984), 1.
• [9] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, ``Handbuch der Physik, Bd. 24/1 Quantentheorie'', Springer, Berlin 1933.
• [10] C. Shannon, A mathematical theory of communication, Bell Systems Technical Journal 23 (1948), 349.
• [11] K.B. Athreya, Entropy maximization, IMA Preprint Series # 1231, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 1994.

Identyfikatory

DOI

10.7151/dmps.1009