  A many-particle theory is presented for the effective quasistatic
permittivity of macroscopically homogeneous and isotropic systems of
inhomogeneous dielectric particles with different degrees of penetrability. The
theory is based upon our original compact-group approach, complemented by the
Hashin-Shtrikman variational principle. The governing equation is obtained by
summing up the statistical moments for the deviations of the local permittivity
in the system from the desired effective permittivity. The latter is, in
principle, recoverable from the governing equation as a functional of the
constituents' volume concentrations (expressed through statistical averages of
certain products of the particles' characteristic functions) and permittivity
profiles. Under the suggestion that the local permittivity is determined by the
shortest distance from the point of interest to the nearest sphere, a complete
analysis is carried out for hard and fully penetrable spheres with
piecewise-continuous radial permittivities. The results are contrasted with
other authors' analytical theories and simulation data. This comparison
validates our theory and also sheds light on possible computational errors
caused by the use of rectangular lattices to simulate dispersions of spherical
particles.
