Exact Energy Computation of the One Component Plasma on a Sphere for Even Values of the Coupling Parameter

TL;DR

This paper presents an exact method to compute the energy and entropy of the two-dimensional one component plasma on a sphere for even coupling parameters, revealing universal finite-size corrections and crystallization features.

Contribution

It introduces a formalism using monomial basis expansion to exactly calculate energy and entropy for the 2dOCP on a sphere at even coupling values, including large system behavior.

Findings

Universal finite-size correction term in entropy and free energy

Crystallization features observed for small particle numbers at large coupling

Good agreement between exact and Monte Carlo energies for even coupling values

Abstract

The two dimensional one component plasma 2dOCP is a classical system consisting of $N$ identical particles with the same charge $q$ confined in a two dimensional surface with a neutralizing background. The Boltzmann factor at temperature $T$ may be expressed as a Vandermonde determinant to the power $\Gamma=q^2/(k_B T)$. Several statistical properties of the 2dOCP have been studied by expanding the Boltzmann factor in the monomial basis for even values of $\Gamma$. In this work, we use this formalism to compute the energy of the 2dOCP on a sphere. Using the same approach the entropy is computed. The entropy as well as the free energy in the thermodynamic limit have a universal finite-size correction term $\frac{\chi}{12}\log N$, where $\chi=2$ is the Euler characteristic of the sphere. A non-recursive formula for coefficients of monomial functions expansion is used for exploring the energy as well as structural properties for sufficiently large values of $\Gamma$ to appreciate the crystallization features for $N=2,3,\ldots,9$ particles. Finally, we make a brief comparison between the exact and numerical energies obtained with the Metropolis method for even values of $\Gamma$.