Next Article in Journal
Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and Support Vector Machine
Next Article in Special Issue
Infrared Cloaking, Stealth, and the Second Law of Thermodynamics
Previous Article in Journal
A New Entropy Optimization Model for Graduation of Data in Survival Analysis
Previous Article in Special Issue
Association of Finite-Dimension Thermodynamics and a Bond-Graph Approach for Modeling an Irreversible Heat Engine
Entropy 2012, 14(8), 1317-1342; doi:10.3390/e14081317
Article

Equivalence of Partition Functions Leads to Classification of Entropies and Means

1,†
 and 2,*
1 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 2 Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada Present address: Bell Mobility Inc., Mississauga, Ontario L4W 5N2, Canada.
* Author to whom correspondence should be addressed.
Received: 24 April 2012 / Revised: 26 June 2012 / Accepted: 9 July 2012 / Published: 27 July 2012
(This article belongs to the Special Issue Advances in Applied Thermodynamics)
Download PDF [385 KB, uploaded 24 February 2015]

Abstract

We derive a two-parameter family of generalized entropies, Spq, and means mpq. To this end, assume that we want to calculate an entropy and a mean for n non-negative real numbers {x1,,xn}. For comparison, we consider {m1,,mk} where mi = m for all i = 1,,k and where m and k are chosen such that the lp and lq norms of {x1,,xn} and {m1,,mk} coincide. We formally allow k to be real. Then, we define k, log k, and m to be a generalized cardinality kpq, a generalized entropy Spq, and a generalized mean mpq respectively. We show that this family of entropies includes the Shannon and Rényi entropies and that the family of generalized means includes the power means (such as arithmetic, harmonic, geometric, root-mean-square, maximum, and minimum) as well as novel means of Shannon-like and Rényi-like forms. A thermodynamic interpretation arises from the fact that the lp norm is closely related to the partition function at inverse temperature β = p. Namely, two systems possess the same generalized entropy and generalized mean energy if and only if their partition functions agree at two temperatures, which is also equivalent to the condition that their Helmholtz free energies agree at these two temperatures.
Keywords: cardinality; dimensionality; entropy; equivalence; free energy; information and thermodynamics; norm; mean; partition function; Shannon and Rényi axioms cardinality; dimensionality; entropy; equivalence; free energy; information and thermodynamics; norm; mean; partition function; Shannon and Rényi axioms
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Share & Cite This Article

Export to BibTeX |
EndNote


MDPI and ACS Style

Elnaggar, M.S.; Kempf, A. Equivalence of Partition Functions Leads to Classification of Entropies and Means. Entropy 2012, 14, 1317-1342.

View more citation formats

Article Metrics

Comments

Citing Articles

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert