Density of states for a low-energy point vortex gas on the sphere

Abstract

Point vortex systems are useful models of two-dimensional fluids. The equilibrium statistical mechanical properties of point vortices in a planar region include some unusual features, such as negative temperature states, that depend on the details of the limiting process used. The limiting form of the density of states as the vortex number increases is of central importance. Esler et al. (2013, 2015) expressed the limiting density of states for point vortices modeling low-energy fluid systems in a planar region as the inverse Fourier transform of a function determined by the spectrum of a certain differential operator. In this paper random variables related to the density of states are found for systems of point vortices on the sphere using the theory of weighted U-statistics. The limiting distributions of these statistics were determined by limits of the Fourier transforms of the probability density function; the results take a form analogous to that found in those earlier works. Two systems are investigated. The first is a neutral or weakly non-neutral point vortex gas on the sphere, using the same low-energy limiting process. The second is a lattice model where the vortex positions are fixed and the circulations are random. In both cases the limiting form of the moment generating function is found in closed form, from which the density of states can be found by inverse Fourier transform.