Exact solutions for the 2d one component plasma

Timothy D. Andersen

Published 2011-11-02, updated 2013-02-13Version 3

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.