Non-Debye relaxations: The ups and downs of the stretched exponential vs Mittag-Leffler's matchings

TL;DR

This paper compares stretched exponential and Mittag-Leffler functions in describing dielectric relaxation, emphasizing the advantages of Mittag-Leffler functions for time domain analysis and their connection to stochastic processes.

Contribution

It introduces a stochastic process perspective to relate non-Debye relaxation functions, demonstrating the utility of Mittag-Leffler functions over traditional models.

Findings

Mittag-Leffler functions provide a more comprehensive description of dielectric relaxation.

Characteristic exponents can be expressed with elementary functions, simplifying analysis.

Comparison confirms the effectiveness of Mittag-Leffler functions in modeling non-Debye relaxations.

Abstract

Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are got according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of data are usually fitted by time or frequency dependent elementary functions which in turn may be analytically transformed among themselves using the Laplace transform and compared each other. This leads to the question on comparability of results got using just mentioned experimental procedures. If we would like to do that in the time domain we have to go beyond widely accepted Kohlrausch-Williams-Watts approximation and get acquainted with description using the Mittag-Leffler functions. To convince the reader that the latter is not difficult to understand we propose to look at the problem from the point of view of objects sitting in the heart of stochastic processes approach to relaxation. These are the characteristic exponents which are read out from the standard non-Debye frequency dependent patterns. Characteristic functions appear to be expressed in terms of elementary functions which asymptotic analysis is simple. This opens new possibility to compare behavior of functions used to describe non-Debye relaxations. Results of such done comparison are fully confirmed by calculations which use the powerful apparatus of the Mittag-Leffler functions.