## 1 Introduction
### Comparative Analysis of Different Ion-Relaxation Polarization Models
Strictly speaking, in comparison with the quasiclassical approach, the quantum mechanical calculation parameter \(\gamma\) also depends on a configuration (like the symmetry and parameters of structure of a quasidiscrete power range of protons \(E_{n}^{(0)}(U_{0},\delta_{0},\gamma_{0})\) and on the distribution law of protons at the levels \(E_{n}^{(0)}\)[66]. The parameter \(\varepsilon_{0}=\frac{\mathrm{d}E_{0}}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\)[1] does not provide such information. 2. At temperatures of T \(\ll\) T\({}_{\mathit{cr},\;move}\)[65; 70], assuming \(\frac{\Delta}{\mathrm{T}}=\frac{\mathrm{T}}{\Gamma_{\mathit{cr},\;move}}<<1\), from (10a) and (11), we have \(W^{(0)}(\mathrm{T})\to W^{(0)}_{\mathit{quant},\mathit{t}un}(\mathrm{T}) \approx\frac{\gamma_{0}}{2}\cdot\frac{\exp(-\Lambda)}{1-\frac{\Gamma}{\Gamma_{ \mathit{cr},\;move}}}\), \(W^{(1)}(\mathrm{T})\to W^{(1)}_{\mathit{quant},\mathit{t}un}(\mathrm{T}) \approx\frac{\gamma_{0}}{2}\cdot\frac{\frac{\mathrm{T}}{\Gamma_{\mathit{cr}, \;move}}\exp(-\Lambda)}{1-\frac{\Gamma}{\Gamma_{\mathit{cr},\;move}}}\). Then, \(\gamma(\mathrm{T})\to\gamma_{\mathit{quant},\mathit{t}un}(\mathrm{T}) \approx\varepsilon_{0}(\mathrm{T})\frac{\mathrm{T}}{\Gamma_{\mathit{cr},\;move }}=\frac{\mathrm{d}E_{0}}{\mathrm{k}_{\mathrm{B}}\Gamma_{\mathit{cr},\;move}}\). At temperatures of T \(\gg\) T\({}_{\mathit{cr},\;move}\), owing to \(W^{(0)}(\mathrm{T})\to W^{(0)}_{\mathit{therm},\mathit{active}}(\mathrm{T}) =\frac{\gamma_{0}}{2}\cdot\exp(-\mathrm{X})\), \(W^{(1)}(\mathrm{T})\to W^{(1)}_{\mathit{therm},\mathit{active}}(\mathrm{T}) =\frac{\gamma_{0}}{2}\cdot\exp(-\mathrm{X})\), we have \(\gamma(\mathrm{T})\to\gamma_{\mathit{therm},\mathit{active}}=\varepsilon_{0}= \frac{\mathrm{d}E_{0}}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\). The relation \(\frac{\gamma_{0}}{\gamma_{\mathit{CL}}}\approx\frac{\mathrm{T}}{\Gamma_{ \mathit{cr}}}\) for low-temperature relaxers, when T \(>\)T\({}_{\mathit{cr}}\), T\({}_{\mathit{cr}}\approx\) 100 K (in HBCs), indicates that the small parameter \(\gamma\) in the tunnel relaxation area (T\(<<\)T\({}_{\mathit{cr}}\)) is 1 to 2 orders of magnitude higher than in the classical relaxation domain (T\(>>\)T\({}_{\mathit{cr}}\)), which requires accounting for higher approximations of perturbation theory than those used in [1; 125; 133] when solving the Fokker-Planck equation in the T \(<\) 100 K region. At a classical relaxation temperature of T\(>>\)T\({}_{\mathit{cr}}\), in strong fields, the parameter \(\gamma_{\mathit{CL}}=\varepsilon_{0}\) is 1-2 orders higher than T at tensions of E\({}_{0}\approx\left(10^{5}\div 10^{6}\right)\frac{\mathrm{V}}{\mathrm{m}}\), and the non-linear influence across the field of members upon solving the Fokker-Planck equation significantly increases. Section 2.8 of this scientific work is devoted to a detailed analytical study of the effects of nonlinearities of the quasi-classical kinetic equation of the original phenomenological model--in the form of a linearized generalized kinetic equation (see Equation (28)) reduced to the form of (40a) that satisfies the Fokker-Planck equation [1; 65; 125; 135]--on the mechanism of formation of volume-charge distributions in ion dielectrics of various classes (as a special case, in proton semiconductors and HBC class dielectrics). The universality of the nonlinear model of ion-relaxation polarization developed in Section 2.8 is justified by the applicability of the equations of this model to the description of polarization kinetic phenomena in solid-state structures, characterized in a wide range of fields and temperatures by similar physical mechanisms of ionic relaxation and conductivity in various dielectrics with high ionic conductivity, wherein the physical mechanism of the diffusive proton transfer characteristic of HBCs is a special case of a more generalized kind of process associated with ion-relaxation polarization.