Fractional power exponential relaxation

In the nineteenth century, the Maxwell equations appeared, based upon electromagnetic theory and juxtaposed with light. In the twentieth century came the Debye tradition of regarding biomolecules as polar materials with exponential relaxation. However, experimental findings showed that most dielectrics do not show exponential relaxation, but fractional power law relaxation. Fractional power law was the universal property. Then came the Cole models based upon a new component, the constant phase element (CPE). 

But Jonscher was apparently not aware that Cole already in 1928 used the circle segment analysis in the Wessel plane and found many circular arcs with suppressed circle centers. The concept of CPE was introduced, and in Cole and Cole (1941), the idea was introduced that a dielectric could have a distribution of many relaxation time constants. The Debye model with ideal components presupposed one single relaxation time constant and therefore a complete semicircle. However, the Cole—Cole model implied that the distributed time constants do not correspond to one exponential, but a fractional power law. It seems that Jonscher (1996) did not accept the flieory of distributed... 

Figure 12 displays the Mittag-Leffler function as well as the two asymptotes, the dashed curve being the stretched exponential and the dotted curve the inverse power law. What is apparent from this discussion is the long-time memory associated with the fractional relaxation process, being an inverse power law rather than the exponential of ordinary relaxation. It is apparent that the Mittag-Leffler function smoothly joins these two empirically determined asymptotic distributions. 

Figure 12. The solid curve is the Mittag-Leffler function, the solution to the fractional relaxation equation. The dashed curve is the stretched exponential (Kohlrausch-Williams-Watts Law), and the dotted curve is the inverse power law (Nutting Law).

FIG. 8 (a) The schematic density profile for the case of adsorption from a semidilute solution we distinguish a layer of molecular thickness Z a where the pol5nner density depends on details of the interaction with the substrate and the monomer size, the proximal region a < z < D where the decay of the density is governed hy a universal power law (which cannot he obtained within mean-field theory), the central region for D < z < with a self-similar profile, and the distal region for < z, where the polymer concentration relaxes exponentially to the bulk volume fraction (b) The density profile for the case of depletion, where the concentration decrease close to the wall (j>g relaxes to its hulk value at a distance of the order of the bulk correlation length... 

---