Microscopic model of a non-Debye dielectric relaxation: The Cole–Cole law and its generalization

Based on a self-similar spatial–temporal structure of the relaxation process, we construct a microscopic model for a non-Debye (nonexponential) dielectric relaxation in complex systems. In this model, we derive the Cole–Cole expression for the complex dielectric permittivity and show that the exponent $\alpha$ involved in that expression is equal to the fractal dimension of the spatial–temporal self-similar ensemble characterizing the structure of the medium and the relaxation process occurring in it. We find a relation between the macroscopic relaxation time and the micro- and mesoparameters of the system. We obtain a generalized Cole–Cole expression for the complex dielectric permittivity involving log-periodic corrections that occur because of a discrete scaling invariance of the fractal structure generating the relaxation process on the mesoscopic scale. The found expression for the dielectric permittivity can be used to interpret dielectric spectra in disordered dielectrics.