## Appendix A Description of Relaxation in Time and Frequency  
The use of the Fourier transform to describe dynamic susceptibilities and scattering experiments has its foundations in linear response theory. In this appendix, the relations between response functions, relaxation functions, susceptibilities, correlation functions, and scattering laws shall be briefly summarized.  
The linear response \(B(t)\) to a perturbation \(A(t)\) can be written as  
\[B(t)=\int_{-\infty}^{t}\mathrm{d}t^{\prime}\,R(t-t^{\prime})\,A(t^{\prime}) \tag{73}\]  
Consider first the momentary perturbation \(A(t)=\delta(t)\). The response is \(B(t)=R(t)\). Therefore, the memory kernel \(R\) is identified as the _response function_.  
Consider next a perturbation \(A(t)=\mathrm{e}^{\eta t}\Theta(-t)\) that is slowly switched on and suddenly switched off (\(\Theta\) is the Heaviside step function, \(\eta\) is sent to \(0^{+}\) at the end of the calculation). For \(t>0\), one obtains \(B(t)=\Phi(t)\) where \(\Phi\) is the negative primitive of the response function  
\[R(t)=-\partial_{t}\Phi(t) \tag{74}\]  
Since \(\Phi\) describes the time evolution after an external perturbation has been switched off, it is called the _relaxation function_. [PERSON]'s stretched exponential function is a frequently used approximation for \(\Phi(t)\).  
Consider finally a periodic perturbation that is switched on adiabatically, \(A(t)=\exp(-i\omega t+\eta t)\), implying again the limit \(\eta\to 0^{+}\). Introducing the _dynamic susceptibility_  
\[\chi(\omega):=\int_{0}^{\infty}\mathrm{d}t\,\mathrm{e}^{i(\omega+i\eta)t}\,R(t) \tag{75}\]  
the response can be written \(B(t)=\chi(\omega)A(t)\). To avoid the differentiation (74) in the integrand, it is more convenient to transform the relaxation function,  
\[F(\omega):=\int_{0}^{\infty}\mathrm{d}t\,\mathrm{e}^{i\omega t}\,\Phi(t) \tag{76}\]  
This is Equation (4), the starting point of the present work.  
Partial integration yields a simple relation between \(\chi\) and \(F\):  
\[\chi(\omega)=\Phi(0)+i\omega F(\omega) \tag{77}\]  
In consequence, the _imaginary_ part of the susceptibility, which typically describes the loss peak in a spectroscopic experiment, is given by the _real_ part of the Fourier transform of the relaxation function, Im \(\eta=\omega\)Re \(F(\omega)\).  
Up to this point, the only physical input has been Equation (73). To make a connection with _correlation functions_, more substantial input is needed. Using the full apparatus of statistical mechanics(Poisson brackets, Liouville equation, Boltzmann distribution, Yvon's theorem), it is found [40] that for classical systems  
\[\langle A(t)B(0)\rangle=k_{\rm B}T\Phi(t) \tag{78}\]  
Pair correlation functions are typically measured in _scattering_ experiments. For instance, inelastic neutron scattering at wavenumber \(q\) measures the scattering law \(S(q,\omega)\), which is the Fourier transform of the density correlation function,  
\[S(q,\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\!\!{\rm d}t\,{\rm e}^{{\rm i} \omega t}\langle\rho(q,t)^{*}\rho(q,0)\rangle \tag{79}\]  
In contrast to Equations (4) and (75), this is a normal, two-sided Fourier transform. If we let \(\langle\rho(q,t)^{*}\rho(q,0)\rangle=\Phi_{q}(t)\), then the scattering law \(S(q,\omega)\) is  
\[S(q,\omega)=\frac{1}{\pi}\mbox{Re }F_{q}(\omega) \tag{80}\]