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Composition law for the Cole-Cole relaxation and ensuing evolution equations

By K. Górska, A. Horzela and A. Lattanzi

Abstract

Physically natural assumption says that the any relaxation process taking place in the time interval $[t_{0},t_{2}]$, $t_{2} > t_{0}\ge 0$ may be represented as a composition of processes taking place during time intervals $[t_{0}, t_{1}]$ and $[t_{1},t_{2}]$ where $t_{1}$ is an arbitrary instant of time such that $t_{0} \leq t_{1} \leq t_{2}$. For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation. The latter leads to differential equations involving fractional derivatives, either of the Caputo or the Riemann-Liouville senses, which are equivalent to the special case of the fractional Fokker-Planck equation satisfied by the Mittag-Leffler function known to describe the Cole-Cole relaxation in the time domain

Topics: Condensed Matter - Mesoscale and Nanoscale Physics, Mathematical Physics
Publisher: 'Elsevier BV'
Year: 2018
DOI identifier: 10.1016/j.physleta.2019.03.008
OAI identifier: oai:arXiv.org:1805.12013

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