Models of dielectric relaxation based on completely monotone functions

TL;DR

This paper surveys various dielectric relaxation models, emphasizing their representation through completely monotone functions in the time domain and characterizing them using fractional differential operators.

Contribution

It provides a comprehensive overview of dielectric models with a focus on their mathematical properties and introduces fractional calculus for their characterization.

Findings

All models exhibit completely monotone relaxation functions.

Time-domain functions are characterized by fractional differential operators.

The approach unifies different models under a common mathematical framework.

Abstract

The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.