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

Euler's equations for inviscid uid o w are examined by a dis- cretized version which represents the uid in a Hamiltonian point-vortex prob- lem on the surface of the unit sphere. The vorticities { the site strengths { of these points are constrained to satisfy a spherical model, constraining the circulation to a hyperplane and the enstrophy to the surface of a hypersphere. The distribution of site strengths is allowed to vary according to a Monte Carlo Metropolis-Hastings algorithm. With this model the dependence of the system { as measured by a tool called the mean nearest neighbor parity, by considering the energy of the sys- tem over a number of Monte Carlo sweeps, by considering the distance between the greatest and the least sites, and by the statistics of both the site values at a given number of sweeps and of a single site over a number of sweeps { on such parameters as the number of points, the statistical mechanics temperature, and the number of sweeps used in the simulation are examined. It is consistently found that in negative statistical mechanics temperatures a solid-body rotational state is found. These negative temperature states correspond to the expected distribution of site values, to the mean nearest neighbor parity, and to the energy of the solid-body rotational state. The positive-temperature state is not as strongly organized a state, and some of the diculties in distinguishing between chaotic and organized states in pos- itive temperatures are outlined. Nevertheless it is established that the Monte Carlo algorithm is an eectiv e and fast method to model this point-vortex system, and numerical evidence suggests that our expectation of nding a sin- gle phase transition, between positive and negative inverse temperatures, is supported experimentally.