## 1 Introduction
Taking proton relaxation polarization in HBCs as a special case, this work prioritizes the study of quasi-classical methods for the theoretical description of ion relaxation polarization and conductivity when the main mathematical apparatus comprises the schemes and methods of the classical kinetic theory (in the form of kinetic Boltzmann equations, the balance of the number of particles in various equilibrium states, the collision integral, etc.) and continuum electrodynamics (equations of the electric field in matter and its boundary conditions), with elements of non-relativistic quantum theory required to account for the effects of the quantum tunneling of basic charge carriers in constructing kinetic coefficients and other parameters of the kinetic equation. A direct study of the methods of attaining a quantum mechanical description of tunnel relaxation polarization in proton semiconductors and dielectrics (PSDs) is devoted to the work in [66; 68; 75; 76; 120; 121], which will be summarized in a future review article. The methodology of this review article, as described in Sections 23, 25, 26 and 28-2.10, is based on the analysis of various methods developed by the first author of this article, Kalytka V., independently (in the period 2012-2024) and with his supervisor M.P. Tonkonogov (in the period 2000-2005). The methods of analytical study relate to the kinetics of relaxation polarization, particularly proton relaxation polarization in the case of HBCs, due to the diffusion movement of the most mobile charge carriers (in the general case, ions; in HBCs, protons). Descriptions of mathematical models of dielectric relaxation in HBCs and in wide-class ion dielectrics in a temperature range of T = 1-1550 K and electric field strengths of 0.1-1000 MW/m [1; 6; 7; 8; 9; 10] are performed. In summary, this study considers methods of quasi-classical kinetic theory, which is most convenient when studying high-temperature relaxation polarization related to the temperature region T = 100-450 K, when the dielectric relaxation mechanism in HBCs is based on the movement of Bjerrum ionization, orientation defects, and water molecules due to the Maxwell relaxation transfer of (100-250 K) relaxers (protons) in the crystal structure dielectric when placed in an electric field. At higher temperatures (250-550 K), the processes are reduced to volume charge polarization associated with the accumulation over time of spatial charge in the space between the electrodes, the distribution of which over the volume of the dielectric occurs due to the movement of protons between ions of the anion sublattice and the interaction of protons with the oscillations of ions in the anion sublattice. Within the framework of classical statistical theory, this can be interpreted as some operator reflecting, as a function of spatial variables and time, the influence of the ion environment on the proton subsystem, where the main element of this operator is the potential of proton interactions with the potential field of the anion sublattice (the hydrogen bond field). The additional smaller component of the operator is described by a numerical constant reflecting the stationary state of the temperature of the phonon subsystem (250-550 K). That is, in quasi-classical theory, at the mathematical level, the influence of the phonon subsystem on relaxers is described using a simple numerical constant, which is reduced in the model equations. In this case, the theoretical description of volume charge polarization in HBCs is developed using the same methods as in the high-temperature region (250-550 K) but only takes into account nonlinear kinetic phenomena reflected in the form of the interaction operator of volumetric charge relaxation modes, differing in the order of perturbation theory k [1; 2] on the frequency harmonic of the established order r (as a rule, you need to start with the frequency harmonic of the first order (r = 1)).