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Entropy 2012, 14(8), 1317-1342; doi:10.3390/e14081317

Equivalence of Partition Functions Leads to Classification of Entropies and Means

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)
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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 (CC BY 3.0).

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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.

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