Finite temperature analytical results for a harmonically confined gas obeying exclusion statistics in $d$-dimensions

TL;DR

This paper derives analytical expressions for the finite-temperature properties of a harmonically confined gas obeying exclusion statistics in any dimension, comparing Gentile and Haldane distributions and highlighting Gentile's simplicity and accuracy at low temperatures.

Contribution

It provides the first closed-form analytical results for the one-body density matrix and Wigner function of such gases, and compares different exclusion statistics.

Findings

Gentile distribution accurately describes the system at low temperatures.

Deviations between Gentile and Haldane distributions occur outside the degenerate regime.

Gentile's simple functional form is a practical alternative to Haldane's distribution.

Abstract

Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a $d$-dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile distribution continues to provide a good description of the system, with deviations only arising well outside of the degenerate regime. Our results illustrate that the exceedingly simple functional form of the Gentile distribution is an excellent alternative to the generally only implicit form of the Haldane distribution at low temperatures.