Coarse-grained forms for equations describing the microscopic motion of particles in a fluid

TL;DR

This paper derives coarse-grained stochastic equations for fluid particle densities by averaging microscopic equations, linking Dean's density equation to dynamic density functional theory through thermally averaged interactions.

Contribution

It introduces a method to obtain stochastic PDEs for coarse-grained densities from microscopic Langevin equations, connecting Dean's equation with density functional theory.

Findings

Derived stochastic PDEs for coarse-grained densities.

Linked Dean's microscopic density equation to DFT.

Provided a framework for averaging microscopic dynamics.

Abstract

Equations of motion for the microscopic number density $\hat{\rho}({\bf x},t)$ and the momentum density $\hat{\bf g}({\bf x},t)$ of a fluid have been obtained in the past from the corresponding Langevin equations representing the dynamics of the fluid particles. In the present work we average these exact equations of microscopic dynamics over the local equilibrium distribution to obtain stochastic partial differential equations for the coarse grained densities with smooth spatial and temporal dependence. In particular, we consider Dean's exact balance equation for the microscopic density of a system of interacting Brownian particles to obtain the basic equation of the dynamic density functional theory. In the thermally averaged equation for the coarse grained density ${\rho}({\bf x},t)$, the related dependence on the bare interaction potential in Dean's equation is converted to that on the corresponding direct correlation functions of the density functional theory.