  We consider the Dean-Kawasaki (DK) equation of overdamped Brownian particles
that forms the basis of the stochastic density functional theory. Recently, the
linearized DK equation has successfully reproduced the full Onsager theory of
symmetric electrolyte conductivity. In this paper, the linear DK equation is
applied to investigate density fluctuations around the ground state
distribution of strongly coupled counterions near a charged plate, focusing
especially on the transverse dynamics along the plate surface. Consequently, we
find a crossover scale above which the transverse density dynamics appears
frozen and below which diffusive behavior of counterions can be observed on the
charged plate. The linear DK equation provides a characteristic length of the
dynamical crossover that is similar to the Wigner-Seitz radius used in
equilibrium theory for the 2D one-component plasma, which is our main result.
Incidentally, general representations of longitudinal dynamics vertical to the
plate further suggest the existence of advective and electrically reverse
flows; these effects remain to be quantitatively investigated.
