Systematic derivation of coarse-grained fluctuating hydrodynamic equations for many Brownian particles under nonequilibrium conditions

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 ${\ell }_{\mathrm{int}}⪡\ell$ and ${\ell }_{\mathrm{int}}⪢\ell$, where ${\ell }_{\mathrm{int}}$ 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 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 long-range correlation of the type $r^{-d}$ $(d=2)$ for the case ${\ell }_{\mathrm{int}}⪢\ell$, while no such behavior is observed for the case ${\ell }_{\mathrm{int}}⪡\ell$.