The Partition Function of a Multi-Component Coulomb Gas on a Circle

TL;DR

This paper analyzes the partition function of a multi-component Coulomb gas on a circle, deriving explicit formulas at special temperature using advanced mathematical tools, extending understanding beyond single-component systems.

Contribution

It provides analytical expressions for the partition function of a multi-component Coulomb gas on a circle at specific temperature, utilizing Toeplitz and Vandermonde determinants.

Findings

Partition functions simplify at temperature β=2.

Explicit formulas derived for canonical and grand canonical ensembles.

Extension of known results from single-component to multi-component systems.

Abstract

We study a two-dimensional Coulomb gas consisting of a mixture of particles carrying various positive multiple integer charges, confined on a unit circle. We consider the system in the canonical and grand canonical ensembles, and attempt to calculate the partition functions analytically, using Toeplitz and confluent Vandermonde determinants. Just like in the simple one-component system (Dyson gas), the partition functions simplify at special temperature $\beta=2$, allowing us to find compact expressions for them.