## 3 Results
### Numerical Study of the Loss Tangent of Dielectric Materials
Here, the parameter set \(\zeta_{i}\) corresponds to the set of experimental, numerical values for the \(i\)-th relaxator, \(\zeta_{i}=\zeta_{\text{exp},i}\), while \(\zeta_{i,\max}\) corresponds to the set of theoretical numerical values for the \(i\)-th relaxator, \(\zeta_{i,\max}=\zeta_{\text{th},i}\). As in [4], calculations were performed for two scenarios: (1) without considering quantum tunneling of protons (theoretical curves at experimental maxima are denoted with double indices without a \"prime\"), and (2) considering quantum tunneling of protons through the potential barrier during parameter calculations (121) and (122) in Formula (151). In this case, theoretical curves at experimental maxima are denoted with double indices including a \"prime\"--see Figures 6 and 7. Following a similar approach and using the method of minimizing the comparison function between theory and experiment, theoretical graphs of \(\text{Re}\big{(}\epsilon_{\text{th}}^{\text{(T)}}(\
u)\big{)}\) were constructed, and numerical calculations of relaxator parameters were performed based on the experimental graphs of \(\text{Re}\big{(}\epsilon_{\text{exp}}^{\text{(T)}}(\
u)\big{)}\). Parameter sets \(\zeta_{i}=\zeta_{\text{exp},i}\), \(\zeta_{i,\max}=\zeta_{\text{th},i}\) were calculated by matching the functions \(\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{exp},i}(\
u ),\,\,\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{th},i }(\
u)\) in the vicinity of frequencies \(\
u_{\text{exp},i,\max}\) corresponding to maxima on branches or graphs 1 and 2 of the experimental spectra (Figures 8 and 9), using the minimization condition:  
\[\left|\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{exp},i,\max}(\zeta_{i})-\,\,\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)} \right)_{\text{th},i,\max}(\zeta_{i,\max})\right|\rightarrow\min.\]  
The results show that across the entire domain of the measured functions \(\text{tg}\delta_{\text{exp}}^{\text{(T)}}(\
u)\), \(\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{exp}}(\
u)\), at the points \(\
u_{\text{exp},i,\min}\) the theoretical functions \(\text{tg}\delta_{\text{th}}^{\text{(T)}}(\
u)\), \(\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{th}}(\
u)\) yield parameter sets \(\zeta_{i}=\zeta_{\text{exp},i}\), \(\zeta_{i,\max}=\zeta_{\text{th},i}\) that are mathematically equivalent, with a high degree of precision, to those calculated and presented in Figures 6-9. Similarly, we investigated numerical minimization schemes for the points corresponding to minima \(\
u_{\text{exp},i,\min}\) on graphs 1 and 2 (Figures 8 and 9) for the functions \(\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{exp},i}(\
u ),\,\,\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{th},i }(\
u)\) using the condition:  
\[\left|\left(\text{Re}\big{(}\epsilon^{\text{(T)}}\big{)}\right)_{\text{exp},i,\min}\big{(}\zeta_{\text{exp},i}\big{)}-\,\,\left(\text{Re}\big{(}\epsilon^{ \text{(T)}}\big{)}\right)_{\text{th},i,\min}(\zeta_{\text{th},i})\right| \rightarrow\min.\]  
An equally important aspect of dielectric spectroscopy is the study of the temperature spectra of dielectric losses \(\text{tg}\delta_{\text{exp}}^{\text{(v)}}(\text{T})\), measured at a fixed frequency of the alternating field within a certain temperature range accessible under the given experimental conditions. Figure 10 presents the results of combined experimental and theoretical studies of this type. On the theoretical side, these were performed using the quasi-classical kinetic theory outlined in this article (derived from expression (151), with consideration of parameters (120), (121), (122)).