  Statistically studied are the equilibrium characteristics of a subsystem of
mobile charges of one sort, taking into account the subsystem of fixed charges
of the opposite sign creating a compensating electric background. The
distribution of these charges under the influence of the external field is
invariable. To represent free energy of the subsystem of mobile charges in the
form of a functional of their density and to calculate cell potentials of the
mean forces by the method of conditional distributions, a cumulant expansion
with respect to the renormalized Mayer functions is used. To take into account
the screening effects, the results of the collective variables method are used.
A system of integral equations for the potentials of mean forces is obtained
that accounts for the effects of near- and long-range interactions. The
calculations are made in the lattice approximation. The correlation component
distinguished in the expression for the binary distribution function makes it
possible to calculate the correlated and uncorrelated parts of the electric
potential using the Poisson equation. In the case of sufficiently small
electric fields, a linear expansion of the chemical potential in terms of
deviation of the charge concentration from the homogeneous distribution is
considered. In final calculations the correlation between particles is taken
into account in the approximation of the first neighbors. In this approximation
the potential and charge concentration distribution is described by a linear
differential equation of the fourth order. The results of its analytical
solution and subsequent numerical calculations for the characteristics of solid
electrolyte are analyzed.
