  We theoretically investigate an electric-field-driven system of charged
spheres as a primitive model of concentrated electrolytes under an applied
electric field. First, we provide a unified formulation for the stochastic
charge and density dynamics of the electric-field-driven primitive model using
the stochastic density functional theory (DFT). The stochastic DFT integrates
various frameworks of the equilibrium and dynamic DFTs, the liquid state
theory, and the field-theoretic approach, which allows us to justify in a
unified manner various modifications previously made for the
Poisson-Nernst-Planck model. Next, we consider stationary density-density and
charge-charge correlation functions of the primitive model with a static
electric field. We focus on an electric-field-induced synchronization between
the emergence of density and charge oscillations, or the crossover from
monotonic to oscillatory decay of density-density and charge-charge
correlations. The correlation function analysis demonstrates the appearance of
stripe states formed by segregation bands perpendicular to the external field.
We also predict the following: (i) the electric-field-induced crossover occurs
prior to the conventional Kirkwood crossover without an applied electric field,
and (ii) the ion concentration dependence of the decay lengths at the
electric-field-induced crossovers bears a similarity to the underscreening
behavior found by simulation and theoretical studies on the oscillatory decay
length in equilibrium.
