A Monte Carlo algorithm for free and coaxial ring extremal states of the vortex N-body problem on a sphere

We show that the search for statistical equilibria at very low positive temperatures, using a Monte Carlo algorithm, can successfully locate dynamical equilibria of the N-vortex problem on a sphere. Numerical results are collected to show that for a wide range of particle numbers, this algorithm accurately and efficiently locates the ground state or lowest energy equilibrium. The extremal states found numerically are carefully compared with well-known exact configurations such as the regular polyhedra. Using an essential tool called the radial distribution function, we state a theorem that is useful for comparing N-vortex configurations which are related to one another by elements of the group O(3). It is found that by constraining the system to equally spaced latitudinal rings of vortices the computational cost may be reduced by an order of magnitude. Many of the results reported here apply directly to other N-body problems on a sphere, such as the distribution of N charges on a sphere