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.
Formulae for the dielectric and Kerr effect responses are given for any size of then×n matrix and the solution is carried out explicitly forn=2 for the linear dielectric response, the Kerr effect response of an assemply of dipoles having induced moments only (which is also a linear response as far as the solution of the Kramers Eq. is concerned), and the Kerr effect response of an assembly having permanent moment only; a nonlinear response. They reduce to the known results for <cosθ> and <cos2θ> when the inertia tends to zero. They agree forn=2 with the results obtained from the modified Smoluchowski Eq. In the case of the linear response with small intertial effects that Eq. provides an adequate description of the ensemble averages but not for the nonlinear response when higher powers ofn (at least equal to three) must be used.
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Coffey, W.T., Mc Goldrick, S.G. 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. Z. Physik B - Condensed Matter 72, 123–129 (1988). https://doi.org/10.1007/BF01313119
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DOI: https://doi.org/10.1007/BF01313119