The Havriliak–Negami relaxation and its relatives: the response, relaxation and probability density functions
Authors:
- K. Górska,
- A. Horzela,
- Ł. Bratek,
- G. Dattoli,
- K. A. Penson
Abstract
We study functions related to the experimentally observed Havriliak–Negami dielectric relaxation pattern proportional in the frequency domain to $[1+({\rm i}\omega\tau_{0}){\hspace{0pt}}^{\alpha}]^{-\beta}$ with $\tau_{0} > 0$ being some characteristic time. For $\alpha = l/k< 1$ (l and k being positive and relatively prime integers) and $\beta > 0$ we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their domain dual in the sense of the inverse Laplace transform. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization $\beta = (2-q)/(q-1)$ and $\tau_{0} = (q-1){\hspace{0pt}}^{1/\alpha}$ $(1 < q < 2)$ we show that for $0 < \alpha < 1$ the response functions $f_{\alpha, \beta}(t/\tau_{0})$ go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities $g_{\alpha, \beta}(u)$ , we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.
- Record ID
- CUT52029d02b54f4c2ead3b5fa4c81d60a6
- Publication categories
- ;
- Author
- Journal series
- Journal of Physics A-Mathematical and Theoretical, ISSN 1751-8113, e-ISSN 1751-8121
- Issue year
- 2018
- Vol
- 51
- No
- 13
- Pages
- [1-15]
- Article number
- 135202
- Other elements of collation
- wykr.; Bibliografia (na s.) - 14-15; Bibliografia (liczba pozycji) - 64; Oznaczenie streszczenia - Abstr.; Numeracja w czasopiśmie - Vol. 51, No 13
- Keywords in English
- the Havriliak-Negami relaxation, the Prabhakar function, the (three-parameter) generalized Mittag-Leffler functions
- DOI
- DOI:10.1088/1751-8121/aaafc0 Opening in a new tab
- URL
- http://iopscience.iop.org/article/10.1088/1751-8121/aaafc0/meta Opening in a new tab
- Language
- eng (en) English
- Score (nominal)
- 30
- Additional fields
- Indeksowana w: Web of Science, Scopus
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUT52029d02b54f4c2ead3b5fa4c81d60a6/
- URN
urn:pkr-prod:CUT52029d02b54f4c2ead3b5fa4c81d60a6
* presented citation count is obtained through Internet information analysis, and it is close to the number calculated by the Publish or PerishOpening in a new tab system.