# 11 Experimental Evidence

[218.2.1] If fractional time evolutions from eq. (27)
with α<1 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 α-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 G with time scale τ>0
and fractional order βτ

GTβττt=GTβττ1tGTβττ-τ1t=GTβττ1tGTβτ1+τ2τ2t | | (36) |

holds generally with τ=τ1+τ2.
[218.3.2] A physical system typically shows different
physical phenomena on different time scales τi.
[218.3.3] In [53, 54] it was assumed that
the second factor in eq. (36) becomes
approximately fractional in the sense that

GTβτ1+τ2τ2t≈GTατ2τ2t | | (37) |

holds in the weak* or strong topology with

[218.3.4] The resulting composite time evolution
GTβττ1tGTατ2τ2t
was studied in [53, 54]
for the case βτ=1.
[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]

τ1dfdt+τ2αAαf=-f | | (39) |

[page 219, §0]
with Aα from eq. (33) and inital value f0=1.
[219.0.1] Its solution is

ft=E1,1-α,1-tτ1,-τ2ατ1t1-α | | (40) |

where

Ea1,a2,bz1,z2=∑k=0∞∑ℓ1≥0∑ℓ2≥0ℓ1+ℓ2=kk!ℓ1!ℓ2!z1ℓ1z2ℓ2Γb+a1ℓ1+a2ℓ2 | | (41) |

with a1,a2>0 and b,z1,z2∈C
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
χu=1-uf^u
where f^u is the Laplace transform
of ft and u=iω 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 T=193K 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 T=323K to T=184K.
[221.1.2] The fits are based on a trinomial fractional relaxation
function as detailed in [54, 61].