Generalized Entropies Depending only on the Probability and their Quantum Statistics

Modfied entropies have been extensively considered in the literature [1]. Among the most well known are the Rényi entropy [2] and the Havdra-Charvát [3] and Tsallis entropy [4]. All these depend on one or several parameters.

 By means of a modification to Superstatistics [5], one of the authors (O. Obregón) has proposed generalized entropies that depend only on the probability [6, 7] and by generalizing the Replica trick [8] the entropies that correspond to the von Neumann entropy can also be found [7]. There are three entropies  S_I = k Σ^Ω_l=1 (1−p_l ^pl ), S_II = kΣ ^Ω_l=1 (p_l^−pl − 1),  and their linear combination    S_III = kΣ ^Ω_l=1 (p_l ^−pl −p_l ^pl ) / 2..

It is interesting to notice that the expansion in series of these entropies having as a first term S= −kΣ ^Ω_l=1 p_l ln p_l   in the parameter  X_l ≡ p_l ln pl ≤ 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 diference will be on the constants that multiply each of the terms in these expansions, but these small correction terms to the usual Boltzmann-Gibbs or Shannon entropies will be of the same order of magnitude for any possible expanded function. Then the three proposed entropies in [6,7] are the only possible generalizations of the Boltzmann-Gibbs or Shannon entropies that depend only of the probability.

 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 in [3, 4].

[1] M.D. Esteban, D. Morales, A summary on entropy statistics, Kybernetika V 31, No. 4, 337-347 (1995).

[2] A. Rényi, Probability Theory, North Holland, Amsterdam, (1970).

[3] J.Havdra, F. Charvát, Quantification method of classification processes. Concept entropy, Kybernetika V. 3, No. 1, 30-35, (1967).

[4] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistics Physiscs. 52, 479-487 (1988).

[5] C. Beck and E. G. D. Cohen, Physica A 322, 267 (2003).

[6] O. Obregón, Entropy 12 (9), 2067 (2010).

[7] O. Obregón, Generalized information and entanglement entropy, gravitation and holography, Int. J. Mod. Phys. A. Vol. 30, No. 16 (2015).

[8] P. Calebrese and J. Cardy, J. Stat. Mech. Theory Exp. 2004, 06002 (2014).