Statistical properties of point vortex equilibria on the sphere

We describe a Brownian ratchet scheme which we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a linear algebra problem $A\Gamma = 0$ where $A$ is a $N \times N(N − 1)/2$ non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant, and $\Gamma \in R^N$ is the (unit) vector of vortex strengths which must lie in the nullspace of $A$. Existence of an equilibrium is expressed by the condition $det($A^TA) = 0$, while uniqueness follows if $Rank(A) = N−1$. The singular value decomposition of $A$ is used to calculate an optimal basis set for the nullspace, yielding all values of the vortex strengths for which the configuration is an equilibrium. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and after each random step we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the singular value drops below a predetermined convergence threshold, an equilibrium configuration is achieved and we find a basis set for the nullspace of A by calculating the right singular vectors corresponding to the singular values that are zero. For each $N = 4 \rightarrow 10$, we generate an ensemble of 1000 equilibrium configurations which we then use to calculate statistically averaged singular value distributions in order to obtain the averaged Shannon entropy and Frobenius norm of the collection. We show that the statistically averaged singular values produce an average Shannon entropy that closely follows a power-law scaling of the form $< S > \sim N^\beta$, where $\beta \sim 2/3$. We also show that the length of the conserved center-of-vorticity vector clusters at a value of one and the total vortex strength of the configurations cluster at the two extreme values ±1, indicating that the ensemble average produces a single vortex of unit strength which necessarily sits at the tip of the center-ofvorticity vector. The Hamiltonian energy averages to zero reflecting a relatively uniform distribution of points around the sphere, with vortex strengths of mixed sign