Generalized Measure of Entropy, Mathai's Distributional Pathway Model, and Tsallis Statistics

TL;DR

This paper explores the scalar pathway model of Mathai, linking it to various probability distributions in physics through a pathway parameter, and demonstrates its connection to Tsallis statistics and superstatistics via entropy maximization.

Contribution

It extends Mathai's pathway model to scalar cases, unifying many physics-related distributions and connecting them through a generalized entropy framework.

Findings

Pathway parameter 'alpha' generates a continuum of distributions.

The model encompasses Tsallis statistics and superstatistics.

Different families of densities are connected via entropy maximization.

Abstract

The pathway model of Mathai (2005) mainly deals with the rectangular matrix-variate case. In this paper the scalar version is shown to be associated with a large number of probability models used in physics. Different families of densities are listed here, which are all connected through the pathway parameter 'alpha', generating a distributional pathway. The idea is to switch from one functional form to another through this parameter and it is shown that basically one can proceed from the generalized type-1 beta family to generalized type-2 beta family to generalized gamma family when the real variable is positive and a wider set of families when the variable can take negative values also. For simplicity, only the real scalar case is discussed here but corresponding families are available when the variable is in the complex domain. A large number of densities used in physics are shown to be special cases of or associated with the pathway model. It is also shown that the pathway model is available by maximizing a generalized measure of entropy, leading to an entropic pathway. Particular cases of the pathway model are shown to cover Tsallis statistics (Tsallis, 1988) and the superstatistics introduced by Beck and Cohen (2003).