The Havriliak-Negami relaxation and its relatives: The response, relaxation and probability density functions

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

Research output: Contribution to journalArticle

6 Citations (Scopus)


We study functions related to the experimentally observed Havriliak- Negami dielectric relaxation pattern proportional in the frequency domain to [1 + (iωτ0)α]-βwith τ0 > 0 being some characteristic time. For α = l/k 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 β = (2 - q)/(q - 1) and τ0 = (q - 1)1/α(1 < q < 2) we show that for 0 < α < 1 the response functions fα,β(t/τ0) go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities gα,β (u), we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.
Original languageEnglish
Article number135202
Pages (from-to)-
JournalJournal of Physics A: Mathematical and Theoretical
Issue number13
Publication statusPublished - 5 Mar 2018
Externally publishedYes


All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

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