  Simulation and experimental studies have demonstrated non-equilibrium
ordering in driven colloidal suspensions: with increasing driving force, a
uniform colloidal mixture transforms into a locally demixed state characterized
by the lane formation or the emergence of strongly anisotropic stripe-like
domains. Theoretically, we have found that a linear stability analysis of
density dynamics can explain the non-equilibrium ordering by adding a
non-trivial advection term. This advection arises from fluctuating flows due to
non-Coulombic interactions associated with oppositely driven migrations. Recent
studies based on the dynamical density functional theory (DFT) without
multiplicative noise have introduced the flow kernel for providing a general
description of the fluctuating velocity. Here, we assess and extend the above
deterministic DFT by treating electric-field-driven binary ionic mixtures as
the primitive model. First, we develop the stochastic DFT with multiplicative
noise for the laning phenomena. The stochastic DFT considering the fluctuating
flows allows us to determine correlation functions in a steady state. In
particular, asymptotic analysis on the stationary charge-charge correlation
function reveals that the above dispersion relation for linear stability
analysis is equivalent to the pole equation for determining the oscillatory
wavelength of charge--charge correlations. Next, the appearance of stripe-like
domains is demonstrated not only by using the pole equation but also by
performing the 2D inverse Fourier transform of the charge--charge correlation
function without the premise of anisotropic homogeneity in the electric field
direction.
