Adiabatic thermostatistics of the two parameter entropy and the role of Lambert's W-function in its applications

TL;DR

This paper develops a unified framework for adiabatic ensembles in generalized statistical mechanics using two-parameter entropy, deriving key theorems and illustrating applications with classical gases, highlighting the role of Lambert's W-function.

Contribution

It introduces a generalized adiabatic ensemble framework based on Schwammle-Tsallis two-parameter entropy and explores its implications with classical systems, including the use of Lambert's W-function.

Findings

Heat functions expressed via Lambert's W-function in large N limit.

Specific heat can be positive or negative depending on deformation parameters.

Gravitational effects influence internal energy and specific heat in microcanonical ensemble.

Abstract

A unified framework to describe the adiabatic class of ensembles in the generalized statistical mechanics based on Schwammle-Tsallis two parameter (q, q') entropy is proposed. The generalized form of the equipartition theorem, virial theorem and the adiabatic theorem are derived. Each member of the class of ensembles is illustrated using the classical nonrelativistic ideal gas and we observe that the heat functions could be written in terms of the Lambert's W-function in the large N limit. In the microcanonical ensemble we study the effect of gravitational field on classical nonrelativistic ideal gas and a system of hard rods in one dimension and compute their respective internal energy and specific heat. We found that the specific heat can take both positive and negative values depending on the range of the deformation parameters, unlike the case of one parameter Tsallis entropy.