The rotational Brownian motion of a nonspherical body and its application to the theory of dielectric relaxation

The conditional probability density function in angular velocity space for a symmetrical body is obtained exactly both from the stochastic differential equations governing the rotational Brownian motion and from the Fokker–Planck equation. Furthermore, it is shown how the Laplace transform of the angular velocity correlation function for an asymmetrical body may be calculated from the Fokker–Planck equation. At the same time, the Laplace transform of the dipole correlation function for the symmetrical body is calculated both from a stochastic integrodifferential equation and Fokker–Planck–Kramers equation. Also, the Laplace transform of the dipole correlation function for the asymmetrical body is obtained using the Fokker–Planck–Kramers equation. Also, the dipole correlation function as an explicit function of time is calculated numerically for a symmetrical body based on the Fokker–Plank–Kramers equation, and is compared to that obtained from the free rotational model and to a single exponential decay function with the Debye relaxation time.