Rotational Brownian Motion

Abstract

A Fokker-Planck equation for the joint probability density of the orientation and angular velocity of a body of general shape is derived by use of a rotational Langevin equation. Equations governing the seperate distributions of orientation and angular velocity are deduced from the equation for the joint probability density. For the special case of a spherical body, two expressions for the orientation distribution are calculated, one valid for small values of the frictional constant occurring in the rotational Langevin equation, and the other valid for large values of the frictional constant. The latter expression includes previously presented results of rotational-diffusion theory and Steele's modification of rotational-diffusion theory, and the calculation provides conditions of validity for these theories. Expressions are calculated for time-correlation functions of spherical tensors, such as spherical harmonics, which involve functions of the orientation of a body.