  It is shown that in adiabatic approximation for nuclei the many-component
Coulomb system cannot be described on the basis of the grand canonical
ensemble. Using the variational Bogolyubov's procedure for the free energy, the
Hohenberg-Kohn theorem is proved in the canonical ensemble for inhomogeneous
electron gas at finite temperature. The principal difference between
consideration in the framework of quantum statistics in the canonical ensemble
and quantum-mechanical consideration of a finite number of particles in
infinite volume is established. The problem of universality of the density
functional for describing the inhomogeneous electron density in a disordered
nuclei field is considered.
