Equivalence of Partition Functions Leads to Classification of Entropies and Means

Entropy

Michel S. Elnaggar¹ 

[1] Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; E-Mail:

关键词: cardinality;    dimensionality;    entropy;    equivalence;    free energy;    information and thermodynamics;    ℓ^p norm;    mean;    partition function;    Shannon and Rényi axioms;   

DOI  :  10.3390/e14081317

来源: mdpi

【 摘 要 】

We derive a two-parameter family of generalized entropies, S_pq, and means m_pq. To this end, assume that we want to calculate an entropy and a mean for n non-negative real numbers {x₁,…,x_n}. For comparison, we consider {m₁,…,m_k} where m_i = m for all i = 1,…,k and where m and k are chosen such that the l^p and l^q norms of {x₁,…,x_n} and {m₁,…,m_k} coincide. We formally allow k to be real. Then, we define k, log k, and m to be a generalized cardinality k_pq, a generalized entropy S_pq, and a generalized mean m_pq 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 l^p 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.

【 授权许可】

CC BY   
© 2012 by the authors; licensee MDPI, Basel, Switzerland.