[405.1.2.1] In this section the composite fractional susceptibility (of type ) given in eq. (36) is used to fit broad band dielectric data of glycerol [43, 20]. [405.1.2.2] For more discussion of the experimental data see the contribution of P. Lunkenheimer and A. Loidl in this special issue.

[405.1.3.1] Figure 1 shows a fit to the experimental data of glycerol with the composite fractional susceptibility function given in eq. (36). [405.1.3.2] The upper figure displays the real part, the lower figure the imaginary part of the frequency dependent susceptibility . [405.1.3.3] The different curves belong to different temperatures ranging from K down to . [405.1.3.4] The normalized composite fractional susceptibility contains three fit parameters, while the traditionally used linear combination from eq. (27) contains six (resp. five when ) fit parameters. [405.1.3.5] Because the experimental data are not normalized one additional parameter is needed in all cases to fit the data. [405.1.3.6] This extra paramter is the dielectric strength defined as

(39) |

[405.1.3.7] Figure 1 shows that not only the asymmetric -peak but also the excess wing at high frequencies can be fitted quantitatively at all except the three lowest temperatures (K) using the composite fractional susceptibility function (36) with only three essential fit parameters. [405.2.0.1] Note that for K the fit extends over almost 9 decades in frequency.

[405.2.1.1] If an iterated composite fractional time evolution [page 406, §0] with four parameters is introduced an even better quantitative agreement can be obtained at all availaible temperatures. [406.1.0.1] In Figure 2 the composite fractional susceptibility

(40) |

with four parameters was used to fit the same data as in Figure 1. [406.1.0.2] This fit function has still two (resp. one) parameter less than the conventional fit function of eq. (27). [406.1.0.3] Note that in this case for K the agreement extends over 13 decades in frequency including the full range of the excess wing.

[406.1.1.1] The values of the fit parameters were found to depend sensitively on the frequency range that was included in the fit. [406.1.1.2] For this reason real and imaginary part were fitted separately. [406.1.1.3] The variation of the fit parameters for real and imaginary part gives an impression of the quality of the fit. [406.1.1.4] One source for parameter variations might be that the experimental data are patched together from differentmeasurements. [406.2.0.1] The matching of different data sets leads to visible breakpoints in the experimental data sets.

[406.2.1.1] In Figure 3 and 4 the fit parameters for real and imaginary parts corresponding to the fits shown in Figure 1 are plotted against temperature. [406.2.1.2] Figure 4 shows the relaxation times in an Arrhenius plot. [406.2.1.3] Clear deviations from Arrhenius behaviour are found. [406.2.1.4] Figure 3 shows the exponent and dielectric strength from the normalized composite fractional susceptibility. [page 407, §0] [407.1.0.1] Note that the dependence of on temperature shows a qualitative different behaviour than for fits using Havriliak-Negami or Cole-Davidson functions. [407.1.0.2] In those cases the exponent decreases slowly with temperature from values around 0.8 to values around 0.5. [407.1.0.3] Here the values of seem to remain flat for a temperature window between 200-300K where they fall into the range between and . [407.1.0.4] The values seem to increase with lowering the temperature, but this could be an artefact because the low temperature fits are only qualitatively accurate. [407.1.0.5] On the other hand the increase at low could also suggest a return to an effective non-fractional time evolution at low temperatures in the glassy phase. [407.1.0.6] For the excess wing in the composite fractional susceptibility function becomes increasingly flat.

[407.1.1.1] In summary the present paper has derived a novel three parameter susceptibility function from the theory of fractional time evolutions [3]. [407.1.1.2] The new function contains only a single stretching exponent. [407.1.1.3] It shows two widespread characteristics of relaxation spectra in glass forming materials: i) an asymmetry of the -peak and ii) an excess wing at high frequencies. [407.1.1.4] The excess wingis not shown by the popular Cole-Cole, Cole-Davidson, Havriliak-Negami or Kohlrausch-Williams-Watts functions. [407.1.1.5] The new fit function with only three parameters yields agreement with broad band dielectric data over up to 9 decades in time. [407.1.1.6] A four parameter generalization gives good agreement over up to 13 decades in frequency. [407.1.1.7] Nevertheless the large uncertainty in the fit parameters indicates that smoother experimental data are needed to establish conclusively whether composite fractional time evolutions exist in experiment.