1 Introduction

[page 399, §1]   
[399.1.1.1] A most remarkable chemical and physical universality is known from relaxation experiments near the glass transition of superooled liquids and other glass formers [1]. [399.1.1.2] Dielectric spectroscopy, viscoelastic modulus measurements, quasielastic light scattering, shear modulus and shear compliance as well as specific heat measurements for glass formers of different chemical composition all show “strange” or “anomalous” dynamics with anasymmetrically broadened relaxation peak that deviates strongly from exponential Debye relaxation [2].

[399.2.1.1] My objective in the present paper is to reinterpret the slow anomalous dynamics observed in broad band dielectric spectroscopy data as evidence for the physical reality of fractional time evolutions [3]. [399.2.1.2] Although well known in mathematics fractional semi-groups were first introduced on general grounds into physics in Refs. [4, 5]. [399.2.1.3] In [6](see also [7, 3] for later references) specific examples of [page 400, §0]    fractional time evolution, namely fractional diffusion and master equations, were for the first time identified as special cases of the well developed theory of continuous time random walks [8, 9, 10, 11, 12, 13, 14, 15, 16] thereby giving a solid and intuitive physical interpretation of the new concept that inspired many subsequent workers (see e.g. [17]). [400.1.0.1] Of course, fractional diffusion equations had been investigated long before as a purely mathematical exercise that generalizes ordinary diffusion [18, 19], but the profound implications for the foundations of physics were not discussed or investigated in these papers. [400.1.0.2] Replacing an ordinary time derivative with a fractional derivative is a profound change in the foundations of physics if the replacement is accompanied with the explicit or tacit claim that the fractional derivative is the generator of the physical time evolution. [400.1.0.3] Experimental evidence is necessary to justify such a dramatic change in the foundations of physics. [400.1.0.4] My motivation for the work presented here was to extend the experimental evidence for the physical reality of fractional time evolutions beyond the well known examples of anomalous diffusion and idealized fractional relaxation.

[400.1.1.1] Despite many years of work glassy dynamics remains an active research topic (see [20] for a recent review). [400.1.1.2] Excess wing and asymmetry of the so called α-peak are considered to be characteristics of glassy dynamics that have eluded theoretical understanding. [400.1.1.3] It is the purpose of this paper to show that both features, asymmetry and excess wing, appear simultaneously if the time evolution becomes fractional.

[400.1.2.1] Given the objectives the paper is organized as follows. [400.1.2.2] Let me begin by repeating the definition of fractional time evolutions, fractional derivatives and dynamical susceptibilities measured in experiment. [400.1.2.3] On the basis of these concepts it is shown in Section 5 how fractional time evolution gives rise to Cole-Cole susceptibilities. [400.1.2.4] Reconsidering the micro-macro transition it is argued in Section 6 that composite fractional time evolutions are more realistic. [400.1.2.5] In Section 7 the composite fractional relaxation equation is introduced and novel composite fractional susceptibilities are derived. [400.1.2.6] As an application the composite susceptibilities are used to fit broadband dielectric spectra of glycerol over up to 13 decades in frequency. [400.2.0.1] More important than the quantitative agreement however is the result that not only an asymmetric α-peak but also the excess wing region can result from a single stretching exponent.