Thomas-Fermi Method For Particles Obeying Generalized Exclusion Statistics

TL;DR

This paper applies the Thomas-Fermi method to analyze the thermodynamics of particles with generalized exclusion statistics, deriving exact density and equation of state for specific potentials and exploring Bose-Einstein condensation conditions.

Contribution

It provides new exact results for particle densities and equations of state in one-dimensional systems obeying Haldane exclusion statistics, including for harmonic and linear potentials.

Findings

Exact one-particle spatial density for harmonic potential

Closed-form equation of state at finite temperature

Bose-Einstein condensation only occurs for bosons

Abstract

We use the Thomas-Fermi method to examine the thermodynamics of particles obeying Haldane exclusion statistics. Specifically, we study Calogero-Sutherland particles placed in a given external potential in one dimension. For the case of a simple harmonic potential (constant density of states), we obtain the exact one-particle spatial density and a {\it closed} form for the equation of state at finite temperature, which are both new results. We then solve the problem of particles in a $x^{2/3} ~$ potential (linear density of states) and show that Bose-Einstein condensation does not occur for any statistics other than bosons.