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

TL;DR

This paper derives coarse-grained fluctuating hydrodynamic equations for many Brownian particles under non-equilibrium conditions, revealing long-range correlations in certain interaction regimes.

Contribution

It systematically derives hydrodynamic equations for different interaction length scales and analyzes their correlation properties under non-equilibrium driving.

Findings

Long-range correlations of order r^{-2} for large interaction length regimes.

No long-range correlations observed for small interaction length regimes.

Provides a theoretical framework for non-equilibrium fluctuating hydrodynamics.

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, $\ell_{\rm int} \ll \ell$ 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 $\ell_{\rm int}\ll \ell$.