Statistical equilibrium of the Coulomb/vortex gas on the unbounded 2-dimensional plane
This paper presents the statistical equilibrium distributions of single-species vortex gases and cylindrical electron plasmas on the unbounded plane obtained by Monte Carlo simulations. We present detailed numerical evidence that at high values of \(\beta>0\) and \(\mu>0\), where \(\beta\) is the inverse temperature and \(mu\) is the Lagrange multiplier associated with the conservation of the moment of vorticity, the equilibrium vortex gas distribution is centered about a regular crystalline distribution with very low variance. This equilibrium crystalline structure has the form of several concentric nearly regular polygons within a bounding circle of radius \(R\). When \(\beta\sim O(1)\), the mean vortex distributions have nearly uniform vortex density inside a circular disk of radius \(R\). In all the simulations, the radius \(R=\sqrt{\beta\Omega/2\mu}\) where \(\Omega\) is the total vorticity of the point vortex gas or number of identical point charges. Using a continuous vorticity density model and assuming that the equilibrium distribution is a uniform one within a bounding circle of radius \(R\), we show that the most probable value of \(R\) scales with inverse temperature \(\beta>0\) and chemical potential \(\mu>0\) as in \(R=\sqrt{\beta\Omega/2\mu}\).
scientific article; zbMATH DE number 2157837