Probability distribution of (Schw\"{a}mmle and Tsallis) two-parameter entropies and the Lambert W-function

TL;DR

This paper derives the probability distribution of a two-parameter entropy using the Lambert W-function and discusses its extensivity properties, contributing to the understanding of non-extensive entropies in complex systems.

Contribution

It introduces a new explicit form of the probability distribution for Schw"{a}mmle and Tsallis's two-parameter entropy using the Lambert W-function and analyzes its extensivity.

Findings

Probability distribution expressed via Lambert W-function.

Extensivity condition for the two-parameter entropy.

Relationship between subsystem and composite system probabilities.

Abstract

We investigate a two-parameter entropy introduced by Schw\"{a}mmle and Tsallis and obtain its probability distribution in the canonical ensemble. The probability distribution is given in terms of the Lambert W-function which has been used in many branches of physics, especially in fractal structures. Also, extensivity of $S_{q,q'}$ is discussed and a relationship is found to exist between the probabilities of a composite system and its subsystems so that the two-parameter entropy, $S_{q,q'}$, is extensive.