The probability density function for the Havriliak-Negami relaxation

K. Górska, A. Horzela, Ł. Bratek, K. A. Penson, G. Dattoli

Published 2016-11-19Version 1

We study the functions related to the Havriliak-Negami frequency relaxation $\sim [1+(i\omega\tau_{0})^{\alpha}]^{-\beta}$ with $\tau_{0}$ characteristic time, measured in many experiments. We furnish exact and explicit expression for the response function $f_{\alpha, \beta}(t/\tau_{0})$ in time domain and a probability density $g_{\alpha, \beta}(u)$ in space domain for $\alpha = l/k < 1$ and $\beta < k/l$, with $k$ and $l$ positive integers. For $0 < \alpha < 1$ and $\beta=1$ we reproduce the functions related to the Cole-Cole relaxation. We use the method of integral transforms. We show that $f_{\alpha, \beta}(t/\tau_{0})$ with $\beta = (2-q)/(q-1)$ and $\tau_{0} = (q-1)^{1/\alpha}$, $1 < q < 2$, goes over to the one-sided L\'{e}vy stable distribution when $q$ tends to one. Moreover, applying the self-similar property of $g_{\alpha, \beta}(u)$ we introduce two-variable density which satisfies the integral form of evolution equation.