Brownian motion in a periodic potential: Application to dielectric relaxation

Abstract

The relaxational dynamics of a planar rotator in anM-fold cosine potential subject to a random torque is investigated in detail. For the case of a periodic potential with large barrier height, the numerical results of the relaxation dynamics are in complete agreement with an approximate analytical solution. The latter, is derived on assuming a harmonic potential at the bottom of the potential minima and a large time-scale separation between the short-time libration inside each potential minima and a long-time hopping phenomenon over the potential barriers. ForM≧2, the hopping phenomenon is found to be the dominant feature of the orientational autocorrelation function. The average hopping time is explained satisfactorily in terms of the Kramers activation rate theory. In particular a complete agreement is found between the numerical results of the escape rate and those obtained from the modified Kramers' predictions valid for low friction coefficient. The cosine model is applied to the study of dielectric spectroscopy. The particle mobility and the complex permitivity, of a dielectric material are calculated by numerical solutions for rotational velocity and orientational auto-correlation functions, respectively. The main features of the experimental observables are determined analytically and compared to the corresponding numerical results. The applicability of the plane rotator model to dielectric spectroscopy is also discussed.