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.