Heat equation and Brownian motion of an overdamped rotating sphere

The governing equations of Brownian rigid bodies that both translate and rotate are of interest in fields such as self-assembly of proteins, anisotropic colloids, dielectric theory, and liquid crystals. In this paper, the partial differential equation that describes the evolution of concentration is derived from the stochastic differential equation of a sphere experiencing Brownian motion in a viscous medium where a potential field may be present. The potential field may be either interactions between particles or applied externally. The derivation is performed once for particles whose orientation can be specified by a vector ($S^2$), and again for particles which require a rotation matrix ($SO(3)$). The derivation shows the important difference between probability density and concentration, the Ito and Stratonovich calculus, and a Piola-type identity is obtained to complete the derivation.