## 1 Introduction
### Comparative Analysis of Various Theoretical Methods for Describing Dielectric Relaxation in HBCs
In order to more strictly coordinate the results of measurements made in [8] and the calculated results in comparison to that achieved in [125; 133; 134], when studying the density spectra of TSTDs and dielectric loss spectra, it is necessary to use, taking into account the methods in [73; 74], the higher-quality and theoretically stricter methods set forth in [2; 3]. In Sections 2.5, 2.6 and 2.8-2.10, these methods will be described in considerable detail. They comprise a nonlinear phenomenological theory of dielectric relaxation, which allows, within the framework of the quasi-classical kinetic theory, the study of kinetic phenomena under ion-relaxation (in particular, proton-relaxation) polarization in dielectrics for a wide range of field parameters (0.1-1000 MV/m) and temperatures (0-1550 K). This theory [2; 3] is distinguished by universal mathematical expressions presented in the form of recurrent formulas for calculating the relaxation modes of the volume-charge distribution generated in the dielectric under the influence of an external electric field. The scientific significance and novelty of this mathematical model stem from its high degree of originality and adaptability to the numerical calculations of the theoretical spectra of the tangent of the dielectric loss angle in dielectrics with a complex crystal structure. The general results of numerical calculations from _theoretical graphs_\(\rm tg\delta_{th}(T)\) will be given in Sections 3.1 and 3.2. The application of nonlinear phenomenological theory [2; 3] to the study of thermostimulated depolarization currents in solid dielectrics is a separate and rather cumbersome task that requires the development of a number of specific methods related to the peculiarities of the physical and mathematical model of thermostimulated polarization and, above all, to the boundary conditions imposed on the kinetic equation, an important facet when studying the electret effect. Within the framework of this article, the solution to this problem, due to its incompleteness and mathematical bulkiness, will not yet be given, but we are counting on the possibility of addressing this problem in future publications. Within the framework of this article, before studying the phenomenological kinetic equation important for practical applications in Fokker-Planck form [2; 3], we will consider the justification and conclusion of the generalized kinetic equation of the quasi-classical kinetic theory of dielectric relaxation that is relevant from a fundamental point of view. Section 2.5 of this article is devoted to the study of the nonlinear properties and kinetic coefficients of this equation, a simplified version of which is the Fokker-Planck equation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\rm Mg_{3}\) (\(\rm Si_{4}\)\(\rm O_{10}\))(OH)\({}_{2}\)} & \multicolumn{4}{c}{**CaSO\({}_{4}\) -0.5\(\rm H_{2}\) O**} \\ \hline \multirow{2}{*}{\(\rm T_{max}\)\(\rm K\)} & \multicolumn{2}{c}{**Activation Energy, eV**} & \multicolumn{2}{c}{**Activation Energy, eV**} \\  & \(\rm U_{0,ep}\)[8] & \(\rm U_{0,th,2}\)[1] & \(\rm U_{0,th,2}\)[1] & \(\rm T_{max}\)\(\rm K\) & \(\rm U_{0,exp}\)[8] & \(\rm U_{0,th,1}\)[1] & \(\rm U_{0,th,2}\)[1] & \(\rm U_{0,th,2}\)[2] \\ \hline
160 & 0.9 \(\pm\) 0.02 & 0.87 & 0.89 & 145 & 1.1 \(\pm\) 0.02 & 0.95 & 0.97 \\
220 & 0.18 \(\pm\) 0.03 & 0.15 & 0.18 & 210 & 0.2 \(\pm\) 0.05 & 0.13 & 0.25 \\
265 & 0.36 \(\pm\) 0.04 & 0.33 & 0.39 & 270 & 0.45 \(\pm\) 0.07 & 0.43 & 0.51 \\
310 & 0.4 \(\pm\) 0.08 & 0.35 & 0.46 & 320 & 0.6 \(\pm\) 0.2 & 0.45 & 0.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Activation energy of relaxers in Onot talc and gypsum, calculated using the kinetic theory \(\rm U_{0,th,1}\)[1] and finite differences \(\rm U_{0,th,2}\)[1]. Here, we will make numerical and qualitative estimates of some parameters of nonlinear kinetic theory.