A non self-referential expression of Tsallis' probability distribution function

TL;DR

This paper derives a non self-referential expression for Tsallis' probability distribution, simplifying the mathematical form and unifying different formalisms in non-extensive statistical mechanics.

Contribution

It introduces an alternative non self-referential form of Tsallis distributions applicable to both first and third formalisms, improving mathematical clarity.

Findings

Derived a non self-referential expression for Tsallis distributions

Unified the first and third formalisms under a common non-referential framework

Simplified the mathematical structure of Tsallis probability distributions

Abstract

The canonical probability distribution function (pdf) obtained by optimizing the Tsallis entropy under the linear mean energy constraint (first formalism) or the escort mean energy constraint (third formalism) suffer self-referentiality. In a recent paper [Phys. Lett. A {\bf335} (2005) 351-362] the authors have shown that the pdfs obtained in the two formalisms are equivalent to the pdf in non self-referential form. Based on this result we derive an alternative expression, which is non self-referential, for the Tsallis distributions in both first and third formalisms.