Generalized information entropies depending only on the probability distribution

TL;DR

This paper introduces generalized entropies based solely on probability distributions, deriving explicit forms for specific fluctuation models and connecting them to known entropy frameworks without relying on additional parameters.

Contribution

It develops new entropy functions depending only on probabilities, calculates their forms for certain fluctuation distributions, and links them to existing entropy concepts like Kaniadakis and Sharma-Mittal.

Findings

Closed-form entropy for Gamma distribution derived

Boltzmann factors coincide for small fluctuations

Entropy validity confirmed via saddle-point approximation

Abstract

Systems with a long-term stationary state that possess as a spatio-temporally fluctuation quantity $\beta$ can be described by a superposition of several statistics, a "super statistics". We consider first, the Gamma, log-normal and $F$-distributions of $\beta$. It is assumed that they depend only on $p_l$, the probability associated with the microscopic configuration of the system. For each of the three $\beta-$distributions we calculate the Boltzmann factors and show that they coincide for small variance of the fluctuations. For the Gamma distribution it is possible to calculate the entropy in a closed form, depending on $p_l$, and to obtain then an equation relating $p_l$ with $\beta E_l$. We also propose, as other examples, new entropies close related with the Kaniadakis and two possible Sharma-Mittal entropies. The entropies presented in this work do not depend on a constant parameter $q$ but on $p_l$. For the $p_l$-Gamma distribution and its corresponding $B_{p_l}(E)$ Boltzmann factor and the associated entropy, we show the validity of the saddle-point approximation. We also briefly discuss the generalization of one of the four Khinchin axioms to get this proposed entropy.