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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 T_{\alpha}(t) with 0<\alpha\leq 1. [403.1.1.2] Hence the transition from a microscopic time scale to a macroscopic time scale amounts to the replacement T(t)\to T_{\alpha}(t). [403.1.1.3] As a consequence the infinitesimal generator A_{1}=-\mathrm{d}/\mathrm{d}t has to be replaced with the infinitesimal generator A_{\alpha}=-D^{{\alpha}}.

[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

f(0+)=\lim _{{t\to 0}}f(t)=f_{0} (22)

with 0<f_{0}<\infty 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 A_{\alpha}=-D^{{\alpha}} further as a derivative -D^{{\alpha,1}}_{{0+}} of order \alpha and type \mu=1 with lower limit 0 [3]. [403.1.2.4] Thus one arrives at the fractional relaxation equation

\tau^{\alpha}D^{{\alpha,1}}_{{0+}}\hat{f}(t)+\hat{f}(t)=0 (23)

of type 1 with the initial condition \hat{f}(0+)=1 from eq. (22). [403.1.2.5] The relaxation time \tau 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 \alpha and type 1. [403.1.3.3] The solution of the idealized fractional relaxation equation (of type 1) (23) reads

\hat{f}(t)=E_{\alpha}\left(-\left(\frac{t}{\tau}\right)^{\alpha}\right) (24)

where

E_{\alpha}(z)=\sum _{{k=0}}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+1)} (25)

is the Mittag-Leffler function [37]. [403.1.3.4] For idealized fractional relaxation of type \mu\neq 1 see [3]. [403.1.3.5] For \alpha=1 one has E_{1}(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

\hat{\chi}(u)=\frac{1}{1+(u\tau)^{\alpha}} (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 \beta-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 \beta-peak with the so called Havriliak-Negami susceptibility [40] for the \alpha-peak. [403.2.2.3] The full expression for the traditional fit function is then

\hat{\chi}(u)=\frac{1}{\left(1+(u\tau _{1})^{{\alpha _{1}}}\right)^{{\alpha _{2}}}}+\frac{C}{1+(u\tau _{2})^{{\alpha _{3}}}} (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 \alpha-peak including the excess wing at high frequencies or a possible slow \beta- 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.