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

top
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

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.