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

TL;DR

This paper presents an improved Havriliak-Negami approximation for the Fourier Transform of Weibull distributions related to Kohlrausch's function, offering high accuracy over a broad frequency range and insights into parameter sensitivities.

Contribution

It introduces a numerically stable double Havriliak-Negami model that accurately approximates the Fourier Transform of Weibull distributions over extensive frequency intervals.

Findings

The model is free from typical numerical distortions.

The double HN approximation is highly accurate locally.

Parameter models are sensitive to data and sampling errors.

Abstract

An improved approximation via Havriliak-Negami (HN) functions to the Fourier Transform (FT) of certain Weibull distributions, -\psi_{\beta}, (the time derivative of the Kohlrausch-Williams-Watts function), is given for a large interval of frequencies: \omega/2\pi\in[0,10^{12}] if 0<\beta\leq1 and \omega/2\pi\in[0,10^{7}] if 1<\beta\leq2. The model is free from the usual numerical distortions, or restrictions associated to sampling step and finite size, present in similar adjustments to complex relaxation functions. Further indicates that the identification of (FT) Weibull data with a double HN approximant, \psi_{\beta}\simeq\mathcal{A}p_{2}HN, is quite exact locally even though the parameters involved should vary adiabatically with the frequency, i.e. \{\alpha_{1,2},\gamma_{1,2},\tau_{1,2},\lambda\}(\omega). This fact is the base for the high sensibility of the parameter models, as functions of \beta, to the available data and sampling associated errors. Presumably is also the root of the different proposals for asymptotic traits of \alpha\cdot\gamma(\beta)_{1,2}|_{\omega\rightarrow\infty} functions found in scientific literature.