5 Idealized fractional relaxation

[page 403, §1]   
[403.1.1.1] It was shown in [21, 22, 3, 4, 5] that coarse graining a microscopic time evolution may lead to a fractional time evolution Tα⁢t with 0<α≤1. [403.1.1.2] Hence the transition from a microscopic time scale to a macroscopic time scale amounts to the replacement T⁢t→Tα⁢t. [403.1.1.3] As a consequence the infinitesimal generator A1=-d/d⁢t has to be replaced with the infinitesimal generator Aα=-Dα.

[403.1.2.1] To establish fractional differential equations of motion one also needs initial (and/or boundary) conditions and domains of definition. [403.1.2.2] In the rest of the paper the initial condition

with 0<f0<∞ will be used, and the functions will be assumed to be continuous and bounded unless larger or smaller spaces are needed. [403.1.2.3] The choice of initial condition suggests to specify the fractional derivative Aα=-Dα further as a derivative -D0+α,1 of order α and type μ=1 with lower limit 0 [3]. [403.1.2.4] Thus one arrives at the fractional relaxation equation

of type 1 with the initial condition f⌃⁢0+=1 from eq. (22). [403.1.2.5] The relaxation time τ serves to make the equation dimensionally correct.

[403.1.3.1] The fractional relaxation equation is the natural generalization of the Debye relaxation equation (19). [403.1.3.2] Its solutions are the eigenfunctions of fractional derivative operators of order α and type 1. [403.1.3.3] The solution of the idealized fractional relaxation equation (of type 1) (23) reads

where

is the Mittag-Leffler function [37]. [403.1.3.4] For idealized fractional relaxation of type μ≠1 see [3]. [403.1.3.5] For α=1 one has E1⁢x=exp⁡x and the solution reduces to the exponential Debye function given in eq. (20).

[403.2.1.1] Inserting the Laplace transform of (24) into eq. (18) yields the normalized susceptibility of idealized fractional relaxation as

which is recognized as the Cole-Cole expression employed in [38].

[403.2.2.1] Experimentally this susceptibility is often used to fit the so called slow β-relaxation peak of many glass-formers [39]. [403.2.2.2] In such fits one often uses a linear combination of the Cole-Cole susceptibility (26) for the β-peak with the so called Havriliak-Negami susceptibility [40] for the α-peak. [403.2.2.3] The full expression for the traditional fit function is then

where the first term represents the Havriliak-Negami susceptibility [40]. [403.2.2.4] This linear combination contains six fit parameters and allows to fit the asymmetric α-peak including the excess wing at high frequencies or a possible slow β- peak, but excluding the boson peak. [403.2.2.5] Next it will be shown that a fit function of similar quality but with fewer parameters can be obtained from composite fractional time evolutions.