## 6 Conclusions  
1. From the comparative analysis of existing methods for the theoretical description of relaxation polarization and conductivity in crystals with ion-molecular chemical bonds (CIMB), it was established that the phenomenological model [1; 155; 156]was constructed via linear approximation according to a small dimensionless parameter \(\zeta(x;t)=\frac{\mathrm{q}E(x;t)a}{2\mathrm{k}_{\mathrm{B}}\mathrm{T}}<1\)[1,135], which was bounded by a polarizing field (E\({}_{0}\approx\) 0.1-1 MV/m) and different temperatures (T \(\approx\) 70-250 K). For comparisons with the experiment, the value range of the parameter is as follows: \(\zeta_{0}=\frac{\mathrm{q}E_{0}a}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\approx 0.001 \text{--}0.01\)[70]. At temperatures of T \(\approx\) 100-250 K, linear approximation via a small dimensionless parameter \(\gamma=\zeta_{0}\frac{W^{(1)}}{W^{(0)}}\approx\) 0.001 \text{--}0.01\)[1,73] is in good agreement with the experiment [125,135]. Outside this temperature range, the role of nonlinear polarization effects increases, which requires the consideration of subsequent (at least from the third) approximations of the perturbation theory via a small dimensionless parameter \(\gamma=\zeta_{0}\frac{W^{(1)}}{W^{(0)}}\) (Formulas (11) and (12)) [1, 2, 3, 4]. The nature of these nonlinearities is explained by tunnel (quantum) proton transitions (T \(\approx\) 250-450 K) [65,67,69,70,72,73] and volume charge relaxation (T \(\approx\) 250-450 K) [70,72,73]; these result in a deviation from the Debye laws with respect to the frequency-temperature of complex dielectric permittivity (CDP) [135]. 2. The output circuit is described, and the parameters and properties of the generalized quasi-classical kinetic Equation (17) are analyzed, which allows, on the basis of a single analytical circuit and, via the successive approximation method (31), the investigation of nonlinear volume-charge polarization mechanisms in the HBC within the temperature range of T \(\approx\) 1-1550 K and the field E\({}_{0}\approx\) 10\({}^{5}\)-10\({}^{8}\frac{\mathrm{V}}{\mathrm{M}}\)[69,73]. The Fokker-Planck equation [1,135] is obtained from Equation (17) via linear approximation with parameter \(\zeta(x;t)=\frac{\mathrm{q}E(x;t)a}{2\mathrm{k}_{\mathrm{B}}\mathrm{T}}\). The higher degrees of parameter \(\zeta_{0}=\frac{\mathrm{q}E_{0}a}{2\mathrm{k}_{\mathrm{B}}\mathrm{T}}\) in (17) enhances the influence of quantum effects on a small parameter of the perturbation theory (12). Formally, the generalized kinetic Equation (17) applies to other crystals with ion conductivity values similar to the HBC with respect to crystal lattice structure and properties. The applicability of the developed analytical methods to the study of superionic conductivity and quasi-sieve electric effects (1250 K, 1 kHz) in corundum-zirconium ceramics is not excluded [5]. 3. The effect of nonlinearities on relaxation times for the microscopic processes of ion transitions (in the HBC, protons) through a potential barrier was investigated (expressions (46) and (51)). The weak dependence of relaxation times on temperatures at the quantum transitions of ions (48) and (49) is established. Near the absolute zero temperature, relaxation times do not depend on temperature (50). 4. A comparative analysis of various methods and approximations in the theoretical description of dielectric relaxation in the HBC was performed. It was observed that quantum proton transitions enhance the role of nonlinear effects in the polarization of the HBC in the low-temperature region (T \(<\) 100 K). In strong fields, nonlinear effects are also manifested at high temperatures (T \(>\) 100 K). In this regard, a generalized solution of the nonlinear equations of the phenomenological ion relaxation polarization model (expressions (57)-(61)) should, within the framework of quasi-classical kinetic theory [2,3], be carried out more strictly than that in [1] via the methods of perturbation theory (power series (62)).