Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

TL;DR

This paper explores Mittag-Leffler functions with multiple parameters as models for non-Debye dielectric relaxation, analyzing their properties and demonstrating their suitability for representing complex relaxation phenomena.

Contribution

It introduces a comprehensive analysis of multi-parameter Mittag-Leffler functions, establishing their relevance for modeling anomalous dielectric relaxation and connecting them to classical models.

Findings

Mittag-Leffler functions are suitable for non-Debye relaxation modeling.

The functions exhibit complete monotonicity under certain conditions.

Visualizations illustrate the response functions and spectral distributions.

Abstract

We revisit the Mittag-Leffler functions of a real variable $t$, with one, two and three order-parameters $\{\alpha, \beta, \gamma\}$, as far as their Laplace transform pairs and complete monotonicty properties are concerned. These functions, subjected to the requirement to be completely monotone for $t>0$, are shown to be suitable models for non--Debye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole-Cole, Davidson-Cole and Havriliak-Negami. We show 3D plots of the response functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters.