A perturbation method for the solution of the Kramers equation for a particle moving under an alternating force ? application to the theory of dielectric and Kerr effect relaxation

Abstract

A perturbation method for the solution of the Kramers Eq. for a particle moving in a cosine potential where the amplitude of the potential is a cissoidal function of time is described. The solution is effected by expanding the distribution function in the usual Fourier-Hermite series yielding a set of ordinary differential-difference Eqs. giving the exact time dependence of the ensemble averages. These Eqs. are a double matrix set inn the order of the Hermite functions andp the order of the circular functions. The perturbation is applied by expanding the solution set in powers of the small parameter, amplitude of the potential/kT. This allows one to systematically uncouple then and thep dependencies in the original set, thus that set may be solved to any order inn by limiting the size of then matrix.