Abstract
This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform
Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail
In the regime of smallα, ifα¦logz¦≪1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ α (z) for variousα in the range 0<α<1. The functionQ α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as
with
It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ n (ω) is shown to be directly related toQ α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions
the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).
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Montroll, E.W., Bendler, J.T. On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation. J Stat Phys 34, 129–162 (1984). https://doi.org/10.1007/BF01770352
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DOI: https://doi.org/10.1007/BF01770352