On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics

TL;DR

This paper investigates the conditions for complete monotonicity of the Prabhakar function, extending the Havriliak--Negami model for non-Debye dielectric relaxation, and compares numerical methods for evaluating this function.

Contribution

It establishes new parameter conditions for the Prabhakar function's monotonicity and extends the classical model to broader parameter ranges, with a focus on numerical evaluation techniques.

Findings

Derived conditions for complete monotonicity of the Prabhakar function.

Extended the Havriliak--Negami model to wider parameter ranges.

Compared and validated three numerical evaluation methods.

Abstract

The three parameters Mittag--Leffler function (often referred as the Prabhakar function) has important applications, mainly in physics of dielectrics, in describing anomalous relaxation of non--Debye type. This paper concerns with the investigation of the conditions, on the characteristic parameters, under which the function is locally integrable and completely monotonic; these properties are essential for the physical feasibility of the corresponding models. In particular the classical Havriliak--Negami model is extended to a wider range of the parameters. The problem of the numerical evaluation of the three parameters Mittag--Leffler function is also addressed and three different approaches are discussed and compared. Numerical simulations are hence used to validate the theoretical findings and present some graphs of the function under investigation.