From stretched exponential to inverse power-law: fractional dynamics, Cole-Cole relaxation processes, and beyond

We present a generalisation of the classical exponential relaxation based on the fractional Fokker-Planck equation framework. We show how fractional dynamics modifies the Brownian dynamics underlying standard relaxation processes, and gives rise to the Mittag-Leffler relaxation of modes and moments. The latter is characterised through a turnover from an initial stretched exponential to a final inverse power-law pattern and the associated complex susceptibility corresponds to the Cole-Cole pattern. (C) 2002 Published by Elsevier Science B.V.