Generalized Entropies Depending Only on the Probability and Their Quantum Statistics

Abstract

Modified entropies have been extensively considered by several authors in articles published almost anywhere. Among the most well known are the Rényi entropy and the Havdra-Charvtá and Tsallis entropy. All these depend on one or several parameters. By means of a modification to Superstatistics, one of the authors (Obregón) has proposed generalized entropies that depend only on the probability. There are three entropies: S I = k ∑ l = 1 Ω ( 1 - p l p l ) , S I I = k ∑ l = 1 Ω ( p l - p l - 1 ) and their linear combination S I I I = k ∑ l = 1 Ω p l - p l - p l p l 2 . It is interesting to notice that the expansion in series of these entropies having as a first term S = - k ∑ l = 1 Ω p l ln p l in the parameter x l ≡ p l ln p l ≤ 1 cover, up to the first terms, any other expansion of any other possible function in x l , one would want to propose as another entropy. The three proposed entropies by Obregón are then the only possible generalizations of the Boltzmann-Gibbs (BG) or Shannon entropies that depend only of the probability. One obtains a superposition of two statistics (that of β and that of p l ), hence the name superstatistics. One may define an averaged Boltzmann factor as B ( E ) = ∫ 0 ∞ f ( β ) e β E d β where f ( β ) is the distribution of β . This work will deal with the analysis of the first two generalized entropies and will propose and deduce their associated quantum statistics; namely Bose-Einstein and Fermi-Dirac. The results will be compared with the standard ones and those due to the entropies by Tsallis. It will be seen in both cases that the BEO (the Bose-Einstein statistics corresponding to the entropies proposed by Obregón) statistic differs slightly from the usual BE statistic and in the same way for FDO the difference is small from the usual FD.

Keywords

• entropy
• quantum statistic
• Bose-Einstein
• Fermi-Dirac
• nonextensive entropy