Physica A 392 (2013) 136–148 Contents lists available at SciVerse ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Log-periodic corrections to the Cole–Cole expression in dielectric relaxation A.A. Khamzin ∗ , R.R. Nigmatullin, I.I. Popov Institute of Physics, Kazan (Volga Region) Federal University, 420008, Kremlevskaya str. 18, Kazan, Tatarstan, Russia article info Article history: Received 10 May 2012 Received in revised form 13 August 2012 Available online 5 September 2012 Keywords: Cole–Cole expression Fractional derivation Fractals Dielectric permittivity Log-periodic oscillations abstract 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. 1. Introduction At the present time, detailed research in soft condensed physics matter [1] as a means of understanding the structure dynamics and relaxation phenomena in complex systems is observed. These complex systems represent a wide class of different materials with disordered structure. In particular, the investigation of (bio)polymers, colloids (emulsions and microemulsions), biologic cells, porous materials and liquid crystals can be also related to a complex system. In most methods used to investigate disordered materials, dielectric spectroscopy plays a dominant role on the mesoscopic scale [1]. However, the simple exponential relaxation law and classical model of Brownian diffusion cannot describe a wide class of relaxation phenomena and kinetics in these soft condensed materials. So, it is necessary to develop other approaches for the description of nonexponential relaxation behavior and anomalous diffusion processes (which nowadays are referred to as ‘‘strange kinetics’’ [2]). Usually, for detailed descriptions of the kinetic processes taking place in different complex systems, it is necessary to use many different experimental methods to cover the corresponding frequency range. From this point of view, dielectric spectroscopy has an undoubted advantage over all other methods, because modern dielectric spectrometers allow one to cover a very wide frequency range (from 10 −6 to 10 11 Hz) [3–5]. The total polarization P (t ) of a dielectric placed in an external electric field E (t ) contains two parts [6]: P (t ) = P 0 (t ) + P 1 (t ), (1) where P 0 (t ) defines the instantaneous component while P 1 (t ) determines its retarded component. In the classic theory of relaxation [6], it is supposed that the rate of change of the retarded component is proportional to the difference between its ∗ Corresponding author. E-mail address: airat.khamzin@rambler.ru (A.A. Khamzin). 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.08.011