[218.2.1] If fractional time evolutions from eq. (27) with must be expected on general grounds, then they should be observable in experiment. [218.2.2] Numerous experimental examples of anomalous dynamics or strange kinetics have been identified (see [17, 18, 19, 20] and the present volume for reviews). [218.2.3] Here the example of dielectric -relaxation in glasses is briefly discussed[53, 54], because it provides experimental data over up to 19 decades in time [55], and because the explanation of its excess wing has been a matter of debate.

[218.3.1] For every induced time evolution on with time scale and fractional order

(36) |

holds generally with . [218.3.2] A physical system typically shows different physical phenomena on different time scales . [218.3.3] In [53, 54] it was assumed that the second factor in eq. (36) becomes approximately fractional in the sense that

(37) |

holds in the weak* or strong topology with

(38) |

[218.3.4] The resulting composite time evolution was studied in [53, 54] for the case . [218.3.5] Rescaling this composite operator as in the case of Debye relaxation and computing the infinitesimal generator yields the fractional differential equation [53, 54]

(39) |

[page 219, §0] with from eq. (33) and inital value . [219.0.1] Its solution is

(40) |

where

(41) |

with and is the binomial Mittag-Leffler function [61].

[page 220, §1] [220.1.1] The complex frequency dependent susceptibility is obtained from the the normalized relaxation function as where is the Laplace transform of and is the imaginary circular frequency [54, p. 402, eq.(18)]. [220.1.2] The real part of the complex dielectric susceptibility for propylene carbonate at temperature K is plotted in Figure 1, its imaginary part in Figure 2. [220.1.3] These figures are taken from [53]. [220.1.4] Crosses represent experimental data. [220.1.5] Different fit functions are shifted by half a decade for better visibility. [220.1.6] The range over which the data were fitted is indicated by two dashed vertical lines. [page 221, §0] [221.0.1] The curve labelled FD (fractional dynamics) is the susceptibility corresponding to the relaxation function in eq. (40) It reproduces the high frequency wing even outside the range of its fit. [221.0.2] This is not the case for the other four curves curves, labelled Debye, KWW, CD and HN. They correspond to four popular fit functions for dielectric relaxation [55, 62]. [221.0.3] The curve Debye corresponds to a simple exponential function, KWW (Kohlrausch-Williams-Watts) is a stretched exponential relaxation function. [221.0.4] The relaxation functions for the two remaining cases, CD (Cole-Davison) and HN (Havriliak-Negami) were given for the first time in [58, 60].

[221.1.1] Figure 3 from [54] shows the real and imaginary part of the dielectric susceptibility for glycerol as its temperature varies over the glass transition range from K to K. [221.1.2] The fits are based on a trinomial fractional relaxation function as detailed in [54, 61].