Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma

TL;DR

This paper uses advanced mathematical tools to expand Vandermonde powers in the 2dOCP, enabling the calculation of thermodynamic properties and correlation functions for small particle numbers.

Contribution

It introduces a novel application of Jack polynomial expansions to analyze the 2dOCP, facilitating the computation of moments and distributions for specific parameters.

Findings

Computed moments of the pair correlation function on a sphere.

Analyzed the distribution of radial linear statistics in the plane.

Demonstrated the effectiveness of Jack polynomial expansion for small N.

Abstract

The two-dimensional one-component plasma (2dOCP) is a system of $N$ mobile particles of the same charge $q$ on a surface with a neutralising background. The Boltzmann factor of the 2dOCP at temperature $T$ can be expressed as a Vandermonde determinant to the power $\Gamma=q^{2}/(k_B T)$. Recent advances in the theory of symmetric and anti-symmetric Jack polymonials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for $N$ values up to 14 and $\Gamma$ equal to 4, 6 and 8. In this work, we explore two applications of this formalism to study the moments of the pair correlation function of the 2dOCP on a sphere, and the distribution of radial linear statistics of the 2dOCP in the plane.