Analytic results for the Tsallis thermodynamic variables

TL;DR

This paper provides exact analytical formulas for thermodynamic variables in Tsallis non-extensive statistics, simplifying calculations and enabling better understanding of hot dense systems across various scientific fields.

Contribution

It presents the first exact massive results for thermodynamic variables in Tsallis statistics, including a simplified method for complex momentum integrals.

Findings

Analytical formulas for pressure, temperature, chemical potential, and mass parameters.

Numerical results match standard numerical integrations.

Simplified computational method reduces complexity of calculations.

Abstract

We analytically investigate the thermodynamic variables of a hot and dense system, in the framework of the Tsallis non-extensive classical statistics. After a brief review, we start by recalling the corresponding massless limits for all the thermodynamic variables. We then present the detail of calculation for the exact massive result regarding the pressure -- valid for all values of the $q$-parameter -- as well as the Tsallis $T$-, $\mu$- and $m$- parameters, the former characterizing the non-extensivity of the system. The results for other thermodynamic variables, in the massive case, readily follow from appropriate differentiations of the pressure, for which we provide the necessary formulas. For the convenience of the reader, we tabulate all of our results. A special emphasis is put on the method used in order to perform these computations, which happens to reduce cumbersome momentum integrals into simpler ones. Numerical consistency between our analytic results and the corresponding usual numerical integrals are found to be perfectly consistent. Finally, it should be noted that our findings substantially simplify calculations within the Tsallis framework. The latter being extensively used in various different fields of science as for example, but not limited to, high-energy nucleus collisions, we hope to enlighten a number of possible applications.