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

TL;DR

This paper derives a composition law for Cole-Cole relaxation processes, leading to fractional differential equations involving Caputo or Riemann-Liouville derivatives, generalizing the classical Debye model.

Contribution

It establishes a new composition law for Cole-Cole relaxation and derives associated evolution equations involving fractional derivatives.

Findings

The composition law for Cole-Cole relaxation is expressed as an integro-differential relation.

The evolution equations are equivalent to fractional Fokker-Planck equations.

The results generalize the classical Debye relaxation to more complex models.

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.