Theory of the Rotational Brownian Motion of a Free Rigid Body

Abstract

The orientation of a rigid body is specified by the Cayley-Klein parameters. A system of such bodies subject to small random changes in orientation but not subject to any externally applied torque is then considered in some detail. A diffusion equation is derived with certain linear combinations of the Cayley-Klein parameters as independent variables. This equation is expressed in terms of quantum-mechanical angular momentum operators and a Green's function for the equation is obtained as an expansion in angular momentum eigenfunctions. This expansion can be used to calculate averages of various physical quantities in a nonequilibrium distribution of orientations. It may also be used to calculate the spectral density of fluctuating quantities in an equilibrium distribution. Illustrative examples of both of these applications are given.