Analyzing the equilibrium states of quasi-neutral spatially   inhomogeneous system of charges above liquid dielectric film basing on first   principles of quantum statistics
  The theory of quasi-neutral equilibrium states of charges above liquid dielectric surface is built. This theory is based on first principles of quantum statistics for systems, comprising many identical particles. The proposed approach is concerned with applying the variation principle, modified for the considered systems, and the Thomas-Fermi model. In terms of the developed theory a self-consistency equations are obtained. These equations provide the relation between the main parameters, describing the system: the potential of static electric field, the distribution function of charges and the surface profile of liquid dielectric. The equations are used to study the phase transition in the system to a spatially periodic state. The proposed method can be applied to analyzing the properties of the phase transition in the system to a spatially periodic states of wave type. Using the analytical and numerical methods, we make a detailed research of the dependence of critical parameters of such phase transition on the thickness of liquid dielectric film. Some stability criteria of the new asymmetric phase of the studied system are discussed. 
D.M. Lytvynenko and Yu.V. Slyusarenko
cond-mat.stat-mech