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.