Temporal fractal model for the anomalous dielectric relaxation of inhomogeneous media with chaotic structure.

Abstract

The potential of the fractional derivative technique is demonstrated on the example of derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimensionality of a temporal fractal ensemble (in a sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of a microstructure of inhomogeneous media exhibiting nonexponential dielectric relaxation is built by singling out groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level, the relaxation should be of a classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times.