## 1 Introduction
### Basic Theoretical Provisions for Physical and Mathematical Models of Relaxation Polarization  
Section 2 of this work is devoted to the description and comparative analysis of existing theoretical methods developed by the authors of this article. These theoretical methods were developed to describe the kinetics of relaxation processes occurring during the formation of a polarized state and during conductivity (in general, ions, and in HBCs, protons) in dielectric structures of a class of crystals with ionic-molecular chemical bonds in an alternating electric field. The existing directions in this scientific field and, accordingly, the models can be divided according to the specifics of the physical description of polarization processes in dielectrics into quasi-classical and quantum mechanical equations based on solutions of the nonlinear quasi-classical generalized nonlinear kinetic equation [70] (a special case is the Fokker-Planck equation solved in conjunction with the Poisson equation [67; 69; 71]) and rather strict solutions of the quantum kinetic Liouville equation for the ensemble of the most mobile charged particles in the crystal (main charge carriers), respectively [66; 68; 75; 76]. The quasi-classical kinetic equation is based, as a rule, on a model of the transfer of the main charge carriers or relaxers (ions) in a multi-pit potential crystalline field perturbed by a polarizing electric field [70; 122; 123]. This model is semi-classical and relies on classical statistical and kinetic theories using elements of quantum theories (the stationary Schrodinger equation, the Gibbs quantum canonical distribution, and different models of the energy spectrum of particles distributed to the energy levels of the continuous or, in the more stringent case, the discrete spectrum [75; 76]). The simplest version of the quasi-classical model of dielectric relaxation is the model of ion transport in a symmetric double potential well with a potential barrier of the same type of symmetry [1; 2; 72]. Thus, in a quasi-classical model, kinetic coefficients are written within the framework of a quasi-classical kinetic equation but take into account both classical (thermally activated) and quantum tunneling transitions of ions between potential wells, making it possible to study polarization kinetic phenomena in dielectrics with a complex crystal lattice structure (layered silicates, crystalline hydrates, ceramics, vermiculites, etc.) with sufficiently accurate theoretical results (when compared with experimental results) in relation to a wide range of temperatures (50-550 K) and field parameters (0.1-1 MV/m) [70; 71; 72; 122; 124]. In this case, due to the low transparency values of the potential barrier for heavy relaxers (with a mass much larger than that of the hydrogen ion), the quasi-classical model is insufficient in the region of low (50-100 K) and ultra-low temperatures (4-25 K), when the main contribution to dielectric relaxation in a number of ionic crystals related to crystals with hydrogen bonds (HBCs) introduce quantum tunnel proton transitions along the hydrogen bonds (the protons form a hydrogen sublattice and move in both classical and quantum ways within and between ions of the anionic sublattice) [1,8]. In an HBC, the hydrogen sublattice is geometrically integrated into the anionic sublattice and mobile, light (in comparison to massive anions) hydrogen ions (protons) actively move (perform a relaxation movement) against the background of the inactive anionic sublattice ions [1,8]. Thus, a mathematical description of the low-temperature dielectric relaxation in the HBC, which is reduced to quantum tunneling proton relaxation (the quantum diffusion transitions of protons over hydrogen bonds), is needed.