## 1 Introduction
### Comparative Analysis of Various Theoretical Methods for Describing Dielectric Relaxation in HBCs
In the experimental range of variation of the parameters \(\rm E_{0}\approx 10^{5}\)-\(10^{6}\frac{B}{M}\), \(\rm T\approx 70\div 450\) K--wherein the smallness condition of the dimensionless parameter \(\rm\varepsilon_{0}=\frac{dE_{0}t}{k_{B}T}\approx 0.001\)-\(0.01\) is set at any combination of the values \(\rm E_{0}\) and \(\rm T\) to describe the relaxation polarization in HBCs in the alternating electric field \(\rm E_{\rm pdf}(t)=E_{0}\cdot exp(iot)\)--is a sufficiently nonlinear system of Fokker-Planck and Poisson equations [1], constructed in _the first approximation_ using the small parameter \(\rm\zeta(x;t)=\frac{dE(xt)a}{2k_{B}T}<1\)[2; 3; 4]. The solution to this system was devised using a decomposition in a power series at degrees of other small parameters \(\rm\gamma=\frac{\mu_{max}^{(1)}\cdot dE_{0}}{D_{0\rm pdf}^{(0)}}=\varsigma_{0} \frac{W^{(1)}}{W^{(0)}}<1\)[2; 4; 69; 72; 135] (\(W^{(0)}\) and \(W^{(1)}\) from (10) in this article). The coefficients \(W^{(0)}\) and \(W^{(1)}\) for the decomposition of \(W^{(\pm)}_{ij}\)\(\approx\)\(W^{(0)}\)\(\mp\)\(W^{(1)}\)\(\cdot\)\(\zeta_{ij}\), where \(\rm\zeta_{ij}=\frac{|\Delta U_{ij}|}{k_{B}T}<<1\)[2; 4; 69; 72; 135], are calculated for models with a rectangular [1] and parabolic potential barrier [1; 71] and meet the following condition: \(\frac{W^{(1)}}{W^{(0)}}\leq 1\)[135]. So, according to Formulas (10) and (10a) as well as (10b) from this article,  
\[\rm\gamma(T)=\varsigma_{0}(T)\times\frac{W^{(1)}(T)}{W^{(0)}(T)}=\frac{q\rm aE _{0}}{k_{B}T}\cdot\left(\frac{exp\left(-\frac{U_{0}}{k_{B}T}\right)+D^{(1)}(T )}{exp\left(-\frac{U_{0}}{k_{B}T}\right)+D^{(0)}(T)}\right). \tag{11}\]  
Based on (11), and including (10b) and (10c),  
\[W^{(0)}(T)=\frac{\rm\
u_{0}}{2}\left(\frac{exp\ (-\Lambda)-\frac{\Lambda}{X(T)} \exp\ (-X(T))}{1-\frac{\Lambda}{X(T)}}\right),\]  
\[W^{(1)}(T)=\frac{\rm\
u_{0}}{2}\times\frac{\Lambda}{X(T)}\left(\frac{exp\ (-\Lambda)-exp\ (-X(T))}{1-\frac{\Lambda}{X(T)}}\right),\]  
and we arrive at the following expression:  
\[\rm\gamma(T)=\varsigma_{0}(T)\times\frac{\frac{\Lambda}{X(T)}\times(exp\ (-\Lambda)-exp\ (-X(T)))}{exp\ (-\Lambda)-\frac{\Lambda}{X(T)}\exp\ (-X(T))}=\varsigma_{0}(T)\times\frac{exp\ (-\Lambda)-exp\ (-X(T))}{\frac{X(T)}{\Lambda}\exp\ (-\Lambda)-exp\ (-X(T))}. \tag{12}\]  
At the same time, in the temperature range \(\rm T\approx\ (100\)-\(250)\) K, when the dielectric relaxation in the HBC is determined mainly by thermally activated (classical) proton transitions and, by virtue of (11) \(\rm\gamma_{therm}=\frac{q\rm aE_{0}}{k_{B}T}\), \(\rm W^{(1)}_{therm}=\frac{q\rm aE_{0}}{k_{B}T}\approx 10^{-3}-10^{-2}\), the results of linear approximation via the parameter \(\rm\gamma\)[1] are in good agreement with the experimental results [125; 133]. In the field of low temperatures (\(\rm T\approx 70\)-\(100\) K), the question of studying non-linearities caused by the influence of proton tunnel transitions on non-Debye patterns of behavior of the frequency-temperature spectra with complex dielectric permittivity (CDP) is relevant [135]. In this case, the values of  
\[\rm\gamma_{\it quant,\it{tunn}}=\frac{q\rm aW^{(1)}_{\it quant,\it{tunn}}(T)E _{0}}{k_{B}T\cdot W^{(0)}_{\it quant,\it{tunn}}(T)}=\frac{q\rm aE_{0}}{k_{B}T} \cdot\frac{D^{(1)}(T)}{D^{(0)}(T)}\]  
increase significantly, with \(\rm\gamma_{\it tunn}\)\(\approx 10^{-2}-10^{-1}\), and, when solving the kinetic equation [1], in continuation of the linear theory [1], members of a higher (starting from the second order) order of perturbation theory should be taken into account. This problem, in principle, can be solved in via the parameter \(\rm\gamma\)[1; 135]; however, in [1], the kinetic coefficients are calculated without taking into account the transparency of the potential barrier, and in [135], the theoretical spectra \(\epsilon^{/}(\omega;\mathrm{T})\), \(\epsilon^{/}/(\omega;\mathrm{T})\) are not studied. The physical model of proton relaxation was accepted according to the information presented in [1, 135].