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The spherical model of logarithmic potentials as examined by Monte Carlo methods

We examine Euler’s equations for inviscid fluid flow by a discretized version representing the fluid as a piecewise constant finite approximation based on Voronoi cells. The strengths of these cells are constrained to conserve circulation and enstrophy. With this model we examine by a Monte Carlo Metropolis–Hastings algorithm the dependence of the system on such parameters as the number of points, the statistical mechanics temperature, and the number of sweeps used in the simulation. Tools to examine the system include the mean nearest neighbor parity, energy, distance between extreme-valued sites, and statistical study of an individual site or of all mesh site values. In negative statistical mechanics temperatures a solid-body rotation state is found. The positive-temperature state is not as strongly organized. Numerical evidence supports our expectation of a single phase transition, between positive and negative temperatures.