The role of the Havriliak-Negami relaxation in the description of local structure of Kohlrausch's function in the frequency domain. Part II

TL;DR

This paper explores the use of double Havriliak-Negami functions with frequency-dependent parameters to accurately model the Fourier Transform of Kohlrausch's function, proposing new models that improve local structure representation.

Contribution

It introduces two novel models based on adiabatic parameter variation of Havriliak-Negami functions for better Fourier Transform approximation of Kohlrausch's function.

Findings

Double Havriliak-Negami approximant fits Kohlrausch's Fourier Transform well.

New models provide a systematic approach for local and global approximation.

Extended Havriliak-Negami functions improve frequency domain representation.

Abstract

The suitability of a double Havriliak-Negami (HN) approximant to represent the Fourier Transform of the time derivative of Kohlrausch-Williams-Watts function, -\psi_{\beta}, has been discussed in the first part of this work. There, it is established the local character of the approximation and how, with slight variation of the parameters \{\alpha_{1,2},\gamma_{1,2},\tau_{1,2},\lambda\} with frequency, Ap_{2}HN can describe a perfect fit with the objective function, \psi_{\beta}. Such adiabatic behavior is commonly misunderstood as an argument against the approximation by means of basic relaxation functions as Havriliak-Negami; this fact it is best interpreted as the need for a wider family of relaxations with a known local portrayal. Two new sets of models for describing compactly the Fourier Transform of Kohlrausch-Williams-Watts are proposed, both based on the adiabatic variation of parameters of a double Havriliak-Negami approximation along the whole interval of frequencies. The first one is relying, obviously, on the use of a well-behaved-pair of patches of the mentioned type of approximants, \mathcal{A}p_{2}HN(\omega). The second is obtained by altering the simple functions HN(\omega) and making dissimilar the couple. They are proposed the guidelines of a new and systematic approach with extended Havriliak-Negami functions which is global, (non local), and of constant parameters. The latter at the cost of a more complicated dependency with the low frequencies than 1+(i\omega\tau_{HN})^{\alpha}.