On the Range of Validity of Integral Transform Methods in Tsallis Statistical Mechanics

TL;DR

This paper investigates the limitations of integral transform methods in Tsallis statistical mechanics, identifying valid parameter ranges and deriving new formulas with broader applicability for one-dimensional gases.

Contribution

It determines the limited validity of existing formulas and introduces new integral representations that extend the range of valid solutions in Tsallis statistical mechanics.

Findings

Existing formulas are valid only within specific q ranges

New integral formulas have wider validity for 1D gases

Applied formulas to ideal and Tonks gases successfully

Abstract

We show that if we require positive definite probabilities, then frequently cited results of Prato, of Lenzi and Hilhorst on the nonextensive equilibrium statistical mechanics of gases are valid only for a limited range of the Tsallis parameter, $q$. We determine the range of validity of the Hilhorst and the Lenzi formulae. We then use various integral representations of the Gamma function to derive new formulae for one-dimensional gases whose range of validity is wider than that of the Hilhorst and the Prato formulae. We then apply the new formulae to the classical ideal gas and the Tonks gas.