Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings

Abstract

Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are obtained 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 functions which, in turn, may be analytically transformed among themselves using the Laplace transform. This leads to the question on comparability of results obtained 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 become 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 which appear in the 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 whose asymptotics is simple. This opens new possibility to compare behavior of functions used to describe non-Debye relaxations. It turnes out that the use of Mittag-Leffler function proves very convenient for such a comparison.