  We study the statistical properties of many Brownian particles under the We
study the statistical properties of many Brownian particles under the influence
of both a spatially homogeneous driving force and a periodic potential with
period $\ell$ in a two-dimensional space. In particular, we focus on two
asymptotic cases, $\ell_{\rm int} \ll \ell$ and $\ell_{\rm int} \gg \ell$,
where $\ell_{\rm int}$ represents the interaction length between two particles.
We derive fluctuating hydrodynamic equations describing the evolution of a
coarse-grained density field defined on scales much larger than $\ell$ for both
the cases. Using the obtained equations, we calculate the equal-time
correlation functions of the density field to the lowest order of the
interaction strength. We find that the system exhibits the long-range
correlation of the type $r^{-d}$ ($d=2$) for the case $\ell_{\rm int} \gg
\ell$, while no such behavior is observed for the case $\ell_{\rm int}\ll
\ell$.
