Generalized Ensemble Theory with Non-extensive Statistics

TL;DR

This paper revisits non-extensive statistical mechanics using Tsallis entropy, deriving consistent thermodynamic relationships and generalized quantum distributions, and applying them to phenomena like blackbody radiation.

Contribution

It introduces a new formulation of non-extensive ensemble theory that avoids self-referential issues and extends to generalized quantum distributions and Planck law applications.

Findings

Derived consistent thermodynamic relations in non-extensive ensembles

Obtained q-deformed Bose-Einstein and Fermi-Dirac distributions

Applied theory to generalized Planck law showing distinct behaviors

Abstract

The non-extensive canonical ensemble theory is reconsidered with the method of Lagrange multipliers by maximizing Tsallis entropy, with the constraint that the normalized term of Tsallis' $q-$average of physical quantities, the sum $\sum p_j^q$, is independent of the probability $p_i$ for Tsallis parameter $q$. The self-referential problem in the deduced probability and thermal quantities in non-extensive statistics is thus avoided, and thermodynamical relationships are obtained in a consistent and natural way. We also extend the study to the non-extensive grand canonical ensemble theory and obtain the $q$-deformed Bose-Einstein distribution as well as the $q$-deformed Fermi-Dirac distribution. The theory is further applied to the generalized Planck law to demonstrate the distinct behaviors of the various generalized $q$-distribution functions discussed in literature.