Dynamical density functional theory for molecular and colloidal fluids: a microscopic approach to fluid mechanics

TL;DR

This paper develops a microscopic dynamical density functional theory (DDFT) that connects fluid mechanics with particle-scale dynamics, capable of describing both colloidal and atomic fluids, and extends to glass transition phenomena.

Contribution

It extends existing DDFTs by deriving a generalized Euler equation from the Kramers equation, linking microscopic particle dynamics to continuum fluid mechanics.

Findings

Derivation of a DDFT from the Kramers equation that generalizes the Euler equation.

The DDFT accurately predicts equilibrium density profiles consistent with classical DFT.

The theory can be approximated to recover mode coupling theory for glass transitions.

Abstract

In recent years, a number of dynamical density functional theories (DDFTs) have been developed for describing the dynamics of the one-body density of both colloidal and atomic fluids. In the colloidal case, the particles are assumed to have stochastic equations of motion and theories exist for both the case when the particle motion is over-damped and also in the regime where inertial effects are relevant. In this paper we extend the theory and explore the connections between the microscopic DDFT and the equations of motion from continuum fluid mechanics. In particular, starting from the Kramers equation which governs the dynamics of the phase space probability distribution function for the system, we show that one may obtain an approximate DDFT that is a generalisation of the Euler equation. This DDFT is capable of describing the dynamics of the fluid density profile down to the scale of the individual particles. As with previous DDFTs, the dynamical equations require as input the Helmholtz free energy functional from equilibrium density functional theory (DFT). For an equilibrium system, the theory predicts the same fluid one-body density profile as one would obtain from DFT. Making further approximations, we show that the theory may be used to obtain the mode coupling theory that is widely used for describing the transition from a liquid to a glassy state.