Exact Finite Size Study of the 2dOCP at Gamma=4 and Gamma=6

TL;DR

This paper provides an exact numerical analysis of the 2dOCP at specific couplings, confirming theoretical predictions about free energy scaling and deriving new sum rules for correlation functions in finite systems.

Contribution

It offers the first exact finite N calculations of free energy and distribution functions for 2dOCP on disk and sphere geometries at Gamma=4 and 6, extending contact theorem ideas.

Findings

Free energy data supports universal logarithmic term with Euler characteristic.

Finite N density profile poorly matches contact theorem predictions.

Derived a sum rule for the second moment of pair correlation in finite disks.

Abstract

An exact numerical study is undertaken into the finite $N$ calculation of the free energy and distribution functions for the two-dimensional one-component plasma. Both disk and sphere geometries are considered, with the coupling $\Gamma$ set equal to 4 and 6. Extrapolation of our data for the free energy is consistent with the existence of a universal term ${\chi \over 12} \log N$, where $\chi$ denotes the Euler characteristic of the surface, as predicted theoretically. The exact finite $N$ density profile is shown to give poor agreement with the contact theorem relating the density at contact and potential drop to the pressure in the thermodynamic limit. This is understood theoretically via a known finite $N$ version of the contact theorem. Furthermore, the ideas behind the derivation of the latter result are extended to give a sum rule for the second moment of the pair correlation in the finite disk, which in the thermodynamic limit converges to the Stillinger-Lovett result.