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

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}.