Non-Debye relaxations: smeared time evolution, memory effects, and the Laplace exponents

TL;DR

This paper links stochastic process theory with fractional dynamics to better understand non-Debye dielectric relaxation, deriving evolution equations for models like Havriliak-Negami and Jurlewicz-Weron-Stanislavsky.

Contribution

It introduces a novel approach connecting memory functions with Laplace exponents of infinitely divisible distributions, unifying stochastic and fractional dynamics in relaxation analysis.

Findings

Derived evolution equations for relaxation models

Linked spectral functions with Laplace exponents

Provided conditions for consistent multi-pattern relaxation descriptions

Abstract

The non-Debye, \textit{i.e.,} non-exponential, behavior characterizes a large plethora of dielectric relaxation phenomena. Attempts to find their theoretical explanation are dominated either by considerations rooted in the stochastic processes methodology or by the so-called \textsl{fractional dynamics} based on equations involving fractional derivatives which mimic the non-local time evolution and as such may be interpreted as describing memory effects. Using the recent results coming from the stochastic approach we link memory functions with the Laplace (characteristic) exponents of infinitely divisible probability distributions and show how to relate the latter with experimentally measurable spectral functions characterizing relaxation in the frequency domain. This enables us to incorporate phenomenological knowledge into the evolution laws. To illustrate our approach we consider the standard Havriliak-Negami and Jurlewicz-Weron-Stanislavsky models for which we derive well-defined evolution equations. Merging stochastic and fractional dynamics approaches sheds also new light on the analysis of relaxation phenomena which description needs going beyond using the single evolution pattern. We determine sufficient conditions under which such description is consistent with general requirements of our approach.