3 Fractional relaxation models

[1284.2.1] In this work we use a generalized form of the Debye relaxation model from eq. (1). [1284.2.2] It is based on the theory of fractional time evolutions for macroscopic states of many body systems first proposed in equation (5.5) in [24] and subsequently elaborated in [25, 26, 27, 28, 29, 30, 17, 31, 32, 33, 34]. [1284.2.3] As discussed in [30, 17] composite fractional time evolutions are expected near the glass transition. [1284.2.4] Such time evolutions give rise to generalized Debye laws of the form of model A:

or model B:

where the parameters obey 0<α,α1,α2<1, α1>α2 and the relaxation times τ1,τ2>0 are positive. [1284.2.5] Here the symbols τ1D+τ2αDα, respectively τ1D+τ1α1D+α1τ2α2Dα2 are the infinitesimal generators of composite fractional semigroups [page 1285, §0]    with Dα being a generalized fractional Riemann-Liouville derivative of order α and almost any type [29, 35]. [1285.0.1] If Dν represents a classical fractional Riemann-Liouville derivative of order ν then its definition reads (with ν∈R+)

where ν is the smallest integer greater or equal ν, Γ the gamma function and D=νd/νdtν.

[1285.1.1] The Laplace transform of the fractional Riemann-Liouville derivative is [36]

[page 1286, §1]    [1286.1.1] With these definitions the Laplace transformation of equations (6) and (7) gives with relation (2) the normalized dielectric susceptibilities of model A

and model B

[1286.1.2] These results apply also for other types of generalized Riemann-Liouville fractional derivatives introduced in [30, 17].

[1286.2.1] The functions from equations (11) and (12) are used to fit the dielectric spectroscopy data of 5-methyl-2-hexanol, glycerol and methyl-m-toluate. [1286.2.2] Real and imaginary part are fitted simultaneously with the parameters α, α1, α2, τ1 and τ2.

[1286.3.1] Additionally we fit the temperature dependent relaxation times τ1 and τ2 with the Vogel-Tammann-Fulcher function

where T is the absolute temperature, τ0 a material parameter, D the fragility and TVF the Vogel-Fulcher temperature. [1286.3.2] The fit parameters are τ0, D and TVF.