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

Abstract

Equations of motion for the microscopic number density $\hat{\rho }(\mathbf{x},t)$ and the momentum density $\hat{\mathbf{g}}(\mathbf{x},t)$ 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 $\rho (\mathbf{x},t)$, 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.

10 pages

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Cited by in corpus (7)

• Statistical Mechanics of Active Ornstein Uhlenbeck Particles
• Classical dynamical density functional theory: from fundamentals to applications
• Stochastic Kinetic Theory for Collective Behavior of Hydrodynamically Interacting Active Particles
• Mean-field microrheology of a very soft colloidal suspension: inertia induces shear-thickening
• Calculation of displacement correlation tensor indicating vortical cooperative motion in two-dimensional colloidal liquids
• Stochastic density functional theory for ions in a polar solvent
• Stochastic dynamics and thermodynamics around a metastable state based on the linear Dean-Kawasaki equation