Microscopic models for dielectric relaxation in disordered systems

It is shown how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended to yield the empirical Havriliak-Negami (HN) equation of anomalous dielectric relaxation from a microscopic model based on a kinetic equation just as in the Debye model. This kinetic equation is obtained by means of a generalization of the noninertial Fokker-Planck equation of conventional Brownian motion (generally known as the Smoluchowski equation) to fractional kinetics governed by the HN relaxation mechanism. For the simple case of noninteracting dipoles it may be solved by Fourier transform techniques to yield the Green function and the complex dielectric susceptibility corresponding to the HN anomalous relaxation mechanism.

Figure 1

$f_1\left(t\right)∕f_1\left(0\right)$ as a function of $t∕\tau$ for $\sigma =0.5$ and different values of $\nu$ [Eq. (A4), solid line] with the short [Eq. (34), dotted lines] and long [Eq. (35), dashed lines] time asymptotes.

Figure 2

$f_1\left(t\right)∕f_1\left(0\right)$ as a function of $t∕\tau$ for $\nu ==0.5$ and different values of $\sigma$ [Eq. (A4), solid line] with the short [Eq. (34), dotted lines] and long [Eqs. (35) (for $\sigma =0.2$ and 0.5) and (36) (for $\sigma =1$), dashed lines] time asymptotes.