Developing time to frequency-domain descriptors for relaxation processes: Local trends

It is common practice while studying complex liquids to analyze their relaxations in time as well as in frequency. Unfortunately, there are not often at hand short and compact expressions corresponding simultaneously to the mathematical formulation of a same phenomenon in both spaces. Therefore, this work is focused towards the approximation of Fourier Transform of certain Weibull distributions (the time derivative of the Kohlrausch-Williams-Watts function) by Havriliak-Negami functions. In particular, it was found that a small interval of low frequencies are needed to recover the main traits of the relaxation for the stretched (β ≤ 1) and squeezed (β > 1) instances. However, it's easily recognizable that the weight of the low frequency part competes with the weight of the high frequency part, and the former distorts the power law behavior, diverging from  −β. In consequence, the tail's sturdiness influences the asymptotic trend of HN, suggesting a careful design of the approximant, the method of optimization, the absent of data errors, and of course the frequency domain. In this sense, we were able to explain how the asymptotic laws naturally emerge as a function ω, and validate the suitability-flexibility-instability of our local approximants.