Characterization of frequency dispersion in the impedance response of a distributed model from the mathematical properties of the distribution function of relaxation times

Heterogeneity and disorder univocally leads to distortion phenomena in the immittance response of a system. The analysis of this response in terms of a distribution of relaxation times (DRT) has become an important topic of basic and applied research. In this work we theoretically and numerically study the impact of the mathematical properties of a distribution function of relaxation times (DFRT) on the frequency dispersion displayed by the immittance response of a distributed model for an ideal dielectric system such as an ideally-polarized electrode, paying special attention to constant-phase-element (CPE) behavior. The analysis of the problem encompasses both explicit and implicit results reported in the literature, and a number of new findings. It is shown, for instance, that CPE exponent is upper bounded by 1, as found in experiments. Conditions to be fulfilled by the DFRT in order to give frequency dispersion are revealed, and CPE behavior is related to scale invariance. The extent of this capacitance dispersion on the frequency spectrum of the response is also addressed for both infinite and finite systems.