## 4 The Current Understanding of Dielectric Relaxation Processes
Later, [PERSON]/[PERSON] [12] expanded this approach by introducing a correlation factor \(g_{K}\) that indicates the degree of correlation between the permanent dipole moments of the molecules, resulting in the relation  
\[\Delta\epsilon=\frac{\mu^{2}}{3\epsilon_{o}k_{B}T}Fg_{K}\frac{N}{V} \tag{2}\]  
where technically, the correlation factor can be calculated as a sum of the pairwise angular correlations of the test dipole with its neighbors  
\[g_{K}=\frac{\langle\sum_{i}\mu_{i}\sum_{j}\mu_{j}\rangle}{N\mu^{2}}=1+\frac{ \langle\sum_{i=1}^{N}\sum_{j>i}\mu_{i}\mu_{j}\rangle}{N\mu^{2}}=1+z\langle \cos(\psi)\rangle \tag{3}\]  
which is often simplified using the number of nearest neighbor molecules \(z\) and their angle to the test dipole \(\psi\). This correlation factor is supposed to describe the ratio of the macroscopically observable or effective average dipole moment \(\langle\mu_{eff}\rangle\) and the average molecular dipole \(\langle\mu\rangle\). Theoretically, one could calculate \(g_{K}\) if the coordinates of the molecular dipole moments were known. Practically, the reverse is attempted; \(g_{K}\) is calculated from the dielectric relaxation strength as a measure of \(\langle\mu_{eff}\rangle\), an assumed molecular dipole moment, to gain insight into molecular correlations [33,44,45,46]. These modifications of [PERSON]'s original relation appear necessary to overcome its limitation to dilute systems. Nevertheless, [PERSON]'s approach still considers only a single species of dipoles--an assumption which is certainly questionable for molecules with internal degrees of freedom that permit librations of distinct subunits with different dipole moments. This is already demonstrated by the presence of a \(\beta\)-relaxation, commonly understood as localized motion of a small submolecular unit, in addition to the structural or \(\alpha\)-relaxation in the polyalcohols (Figure 3a,b). Consequently, this correction might not be specific enough to accurately estimate the impact on a single relaxation in a system of various types of fluctuating species. In contrast, the theoretical foundation of the Kirkwood correlation factor seems to capture interactions with neighboring dipoles regardless of their species or even the type of interaction (besides electrostatic interactions, one might also expect steric interactions). However, the concept covers angular (e.g., structural) correlations only, while dynamic correlations are not included, even though these must be highly relevant in a fluctuating system. These problems seem to become relevant in more recent studies, which have taken into account intra-and inter-molecular interactions, as well as the conformations of the molecules [47], and have already obtained pronounced differences between theory and experiment for the low molecular weight liquid methanol, which cannot be explained on the basis of current theories. Further inconsistencies are found in the published correlation factors of glycerol: early studies find values of \(g_{K}\)\(\sim\)5-5.5 (based on a molecular dipole moment of 2.67 D), which surprisingly dropped to a value of \(\sim\)1.5 at a temperature of 263 K [45]. Later studies found (using the same value for the molecular dipole moment) \(g_{K}\)\(\sim\)2.5-3 [44]. Considering these results, we analyze the polyalcohol series glycerol, xylitol and sorbitol, and specifically calculate an effective dipole moment using their molar mass \(M\) and their mass density \(\rho\) according to  
\[\langle u_{eff}\rangle_{mol}=\sqrt{\frac{3\varepsilon_{o}k_{B}T\Delta\varepsilon M }{\rho N_{A}}}=\langle u\rangle\sqrt{Fg_{K}} \tag{4}\]  
which exhibits a large variation among these similarly composed molecules (Figure 4). The resulting values are on the order of 6-10 D which exceed the interatomic dipole moments obtained from simulation studies [41] for the O-H and the C-O group of 4.5 D (\(15.1\times 10^{-30}\) cm) and 5.3 D (\(17.5\times 10^{-30}\) cm), respectively (Table 2).