# Elastic Slow Dynamics in Polycrystalline Metal Alloys
## 4 Multi-Relaxation Theory and Results
### Theory
#### 4.1.1 Discrete Relaxation Times Distribution in the Log-Space  
As a first step, to derive in a rough approximation the main properties of the function \(F(\tau)\), we approximate the solution as a finite sum of exponentials. To this purpose a basis of relaxation times should be defined. Since the basis must span a large time interval, it is more practical to work in a log space. Therefore the basis is defined as a vector \(B=\{B_{1},B_{2},...B_{N}\}\) where \(B_{i}=log_{10}(\tau_{i})\). The elements of the basis are equally spaced with spacing \(\Delta_{B}=(B_{N}-B_{1})/(N-1)\) and different basis could be chosen, varying its dimension \(N\) or its upper and lower limits \(B_{1}\) and \(B_{N}\).  
In this framework, the fitting function becomes:  
\[\delta c/c=\sum_{i=1}^{N}G_{i}e^{-t/10^{B_{i}}} \tag{5}\]  
where the coefficients \(G_{i}\) could be derived fitting the experimental data.  
Since the goal is to derive continuous distributions, the fitting procedure can be repeated \(K\) times changing the starting point of the basis \(B_{1}=-2+j*\Delta_{B}/K\) with \(j=0,1,\ldots,K-1\). The procedure was applied with \(K=30\) and results are shown in Figure 5. For both materials considered we obtain a pseudo-continuous distribution. Thus, the assumption that  
\[\delta c/c=\int_{B_{1}}^{B_{N}}G(B)e^{-t/10^{B}}dB \tag{6}\]  
is valid.  
From results of Figure 5 it could be deduced that:  
* in both cases a continuum distribution is found;
* in the case of the Aluminum sample (Alu-C20) shown in subplot(a) the distribution presents a well defined peak and a marked asymmetry with a steep increase for low values of the relaxation time \(\tau\). The same behaviour was also obtained for the other Al sample and for the Magnesium samples (not reported for brevity);
* in the case of the Titanium alloy sample the distribution presents two peaks for different values of \(\tau\) preserving the marked asymmetry in the distribution for both.  
To obtain the continuous distribution \(F(\tau)\) in Equation (4) one more step is needed. Substituting \(dB=d(log\tau)=1/\tau log(10)d\tau\), we obtain:  
\[\delta c/c=\int_{B_{1}}^{B_{N}}\frac{G(B)}{\tau log(10)}e^{-t/\tau}d\tau \tag{7}\]  
It follows  
\[\hat{F}(\tau)=G(B)/\tau log(10) \tag{8}\]  
where \(B=log(\tau)\). The resulting distribution could be reasonably well approximated by a \However, the use of the fitting procedure in the log-space to derive a continuous distribution in the lin-space is biased by several approximations which make it unreliable, as further proven in the Supplemental Material:  
* the distribution of the basis used for the discrete fit is not equally spaced in the linear space;
* the basis elements do not correspond to the central point of the interval in the linear space, e.g., assuming \(\Delta B=1\), the element \(B_{i}=0\) represents the interval \(-0.5<B_{i}<0.5\), which means that in the linear space \(\tau_{i}=1\) represents the interval \(1/\sqrt{10}<\tau<\sqrt{10}\), whose central point is not \(\tau=1\);
* the procedure of dividing by \(\tau\) when deriving the distribution in the lin space might amplify errors present in the estimation of the \(B_{i}\)s, especially for low \(\tau\) values, thus leading to distortions and/or to a limitation to the definition of details of the distribution \(\hat{F}(\tau)\).  
As a consequence, the procedure described so far could only be reliably used as a qualitative indication of the properties of the underlying continuous relaxation spectrum \(F(\tau)\). As such, as shown in the next Subsection, it could suggest a plausible analytical expression for \(F(\tau)\).