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

Abstract

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, ${\ell }_{\mathrm{int}}\ll \ell$ and ${\ell }_{\mathrm{int}}\gg \ell$, 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 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 ${\ell }_{\mathrm{int}}\gg \ell$, while no such behavior is observed for the case ${\ell }_{\mathrm{int}}\ll \ell$.

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