Temporal fractal model for the anomalous dielectric relaxation of inhomogeneous media with chaotic structure - art. no. 031504

The potential of the fractional derivative technique is demonstrated on the example of derivation of all three known patterns of anomalous, nonexponen tial dielectric relaxation of an inhomogeneous medium in the time domain. I t is explicitly assumed that the fractional derivative is related to the di mensionality of a temporal fractal ensemble (in a sense that the relaxation times are distributed over a self-similar fractal system). The proposed fr actal model of a microstructure of inhomogeneous media exhibiting nonexpone ntial dielectric relaxation is built by singling out groups of hierarchical ly subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are d erived assuming that the real (physical) ensemble of relaxation times is co nfined between the upper and lower limits of self-similarity. It is predict ed that at times, shorter than the relaxation time at the lowest (primitive ) selfsimilarity level, the relaxation should be of a classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times.