  We derive a set of modified Poisson-Nernst-Planck (mPNP) equations for ion
transport from the variation of the free energy functional which includes the
many-body Coulomb correlation in media of variable dielectric coefficient. The
correlation effects are considered through the Debye charging process in which
the self energy of an ion is governed by the generalized Debye-H\"uckel
equation. We develop the asymptotic expansions of the self energy taking the
ion radius as the small parameter such that the multiscale model can be solved
efficiently by numerical methods. It is shown that the variations of the energy
functional give the self-energy-modified PNP equations which satisfy a proper
weak formulation. We present numerical results from different asymptotic
expansions with a semi-implicit conservative numerical method and investigate
the effect of the Coulomb correlation.
