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# 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 with . [403.1.1.2] Hence the transition from a microscopic time scale to a macroscopic time scale amounts to the replacement . [403.1.1.3] As a consequence the infinitesimal generator has to be replaced with the infinitesimal generator .

[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

 (22)

with 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 further as a derivative of order and type with lower limit [3]. [403.1.2.4] Thus one arrives at the fractional relaxation equation

 (23)

of type 1 with the initial condition from eq. (22). [403.1.2.5] The relaxation time 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 and type 1. [403.1.3.3] The solution of the idealized fractional relaxation equation (of type 1) (23) reads

 (24)

where

 (25)

is the Mittag-Leffler function [37]. [403.1.3.4] For idealized fractional relaxation of type see [3]. [403.1.3.5] For one has 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

 (26)

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 -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 -peak with the so called Havriliak-Negami susceptibility [40] for the -peak. [403.2.2.3] The full expression for the traditional fit function is then

 (27)

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 -peak including the excess wing at high frequencies or a possible slow - 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.