Log-periodic corrections to the Cole-Cole expression in dielectric relaxation

A model of the self-similar process of relaxation is given, and a method of derivation of the kinetic equations for the total polarization based on the ideas of fractional kinetics is suggested. The derived kinetic equations contain integro-differential operators having non-integer order. They lead to the Cole-Cole expression for the complex dielectric permittivity. It is shown rigorously that the power-law exponent α in the Cole-Cole expression coincides with the dimension of the mixed space-temporal fractal ensemble. If the discrete scale invariance for the temporal-space structure of the dielectric medium considered becomes important, then the expression for the complex dielectric permittivity contains log-periodic corrections (oscillations) and, hence, it generalizes the conventional Cole-Cole expression. The corrections obtained in this model suggest another way of interpretation and analysis of dielectric spectra for different complex materials. © 2012 Elsevier B.V. All rights reserved