1. Memory Effect in Rotational Brownian Motion

Abstract

Dynamical properties of dielectrics are fully characterized by absorption and dispersion and thus many theoretical methods have been proposed: Debye obtained an expression for the dielectric function of the form (1+iωτ)^<-1> where ω is the angular frequency and τ is the relaxation time. Although the Debye theory is applicable to various materials, the theory cannot explain several important aspects of the polar molecules; for instance, the behavior in the short time region t≪τ of relaxing dlpole moment, because the inertial effect, the memory effect, and the interactions of molecules are thoroughly neglected. Several authors took into account the inertial effect to explain the short time behavior. McConnel, for Instance, obtained the dielectric function with the use of the Langevln-type equation which, however, does not contain any memory effect. On the other hand, Van Vleck-Welsskopf and Frohllch considered harmonically oscillating charges and obtained the dielectric function, but the memory effect was neglected. In this thesis, a review of physical aspect of dielectrics and a summary of the damping theory are given in the first half and subsequently formulations are given by taking into account the memory effect and the inertial effect which may affect the dynamical properties of the dlpole moment. In our treatment the dielectric is assumed to be a dilute solution of disc polar molecules. In a basic Langevln-type equation, a random force, which originates from the molecules of the non-polar solvent, Is assumed to be a Gaussian or a two-state-jump Markoff process. In the case of the Gaussian fluctuating force, the Fokker-Planck type equation Is exactly constructed by the TCL formula of the damping theoryn; in the case of the two-state-jump fluctuating force, an approximate equation is obtained with the use of the TC formula. On the basis of the above equations, the relaxation function, the dielectric function, and the distribution function are derived, and the fluctuation of the dipole moment Is also determined. In the white noise limit, McConnel's result is derived in the case of the Gaussian fluctuating force and Van Vleck-Welsskopf, Frohlichs1 result is obtained in the case of the two-state-jump fluctuating force. The Debye's result is found in the narrowing limit irrespective of the stochastic processes. Detailed numerical calculations are given and several new aspects are found. Especially for the dielectric function, deviation at high frequencies from the Debye's result Is found. This may corresponds to an experiment for polar liquid, though our model is so simple.