Geodesic
Microscopic model of a~non-Debye dielectric relaxation: The~Cole--Cole law and its generalization
Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 2, pp. 314-332

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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.
Keywords: dielectric relaxation, complex dielectric permittivity, non-Debye dielectric spectrum, discrete scaling invariance, log-periodic oscillation.
Mots-clés : fractal 
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     author = {A. A. Khamzin and R. R. Nigmatullin and I. I. Popov},
     title = {Microscopic model of {a~non-Debye} dielectric relaxation: {The~Cole--Cole} law and its generalization},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {314--332},
     publisher = {mathdoc},
     volume = {173},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_173_2_a7/}
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A. A. Khamzin; R. R. Nigmatullin; I. I. Popov. Microscopic model of a~non-Debye dielectric relaxation: The~Cole--Cole law and its generalization. Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 2, pp. 314-332. http://geodesic.mathdoc.fr/item/TMF_2012_173_2_a7/