Systematic derivation of coarse-grained fluctuating hydrodynamic equations for many Brownian particles under non-equilibrium condition

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 $\mathrm{\ell }$ in a two-dimensional space. In particular, we focus on two asymptotic cases, ${\mathrm{\ell }}_{\mathrm{i}\mathrm{n}\mathrm{t}}\ll \mathrm{\ell }$ and ${\mathrm{\ell }}_{\mathrm{i}\mathrm{n}\mathrm{t}}\gg \mathrm{\ell }$, where ${\mathrm{\ell }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ 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 $\mathrm{\ell }$ 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 $\ell_{\rm int} \gg \ell$,whilenosuchbehaviorisobservedforthecase$\ell_{\rm int}\ll \ell$.