Exact solutions for the 2d one component plasma

TL;DR

This paper provides an exact solution for the classical equilibrium statistics of the 2D one component plasma, a model relevant for plasmas and related to random matrix theory, for all even non-negative temperatures.

Contribution

It presents the first exact integration of the 2D one component plasma ensemble for all even non-negative temperatures, solving a longstanding open problem.

Findings

Exact integration of the plasma ensemble achieved

Connection established with the 2D Selberg integral

Applicable for all even non-negative temperatures

Abstract

The 2d one component gas of pointlike charges in a uniform neutralizing background interacting with a logarithmic potential is a common model for plasmas. In its classical equilibrium statistics at fixed temperature (canonical ensemble) it is formally related to certain types of random matrices with Gaussian distribution and complex eigenvalues. In this paper, I present an exact integration of this ensemble for $N$ such particles (or alternatively $N\times N$ matrices) for all even non-negative temperatures, a significant open problem in statistical physics for several decades. I achieve this exact integration via an exact integration of a related ensemble, the two-dimensional Selberg integral.