Brownian Motion on a Sphere: Distribution of Solid Angles

We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two state quantum systems (or polarised light) undergoing random evolution. Our results are also relevant to recent experiments which observe the Brownian motion of molecules on curved surfaces like micelles and biological membranes. Our theoretical analysis agrees well with the results of computer experiments.