Dynamical Density Functional Theory for Brownian Dynamics of Colloidal Particles

Abstract

Variational methods play a key role in physics. Density functional theory (DFT) is a special and important example of such a variational formulation: There is a functional of the one-particle density which gives access to the equilibrium thermodynamics when it is minimized with respect to the density. This important theory can be both applied to quantum-mechanical electrons and to classical systems. In this book chapter we shall consider a nonequilibrium situation where an explicit time-dependence comes into play. This addresses the point of nonequilibrium dynamics and is therefore much more complicated than the traditional DFT. In the special case of completely overdamped Brownian dynamics of classical colloidal particles typically described by the Langevin or Smoluchowski equations, a dynamical version of DFT, the so-called dynamical density functional theory (DDFT), is available and makes dynamical predictions which are in good agreement with computer simulations. Here we shall derive DDFT for Brownian dynamics in a tutorial way from the Smoluchowski equation and mention some applications. The theory will then be generalized towards hydrodynamic interactions between the particles and to orientational degrees of freedom describing e.g. rod-like colloids. Finally some recent developments will be discussed including active Brownian particles and the derivation of DDFT from projection operator techniques which can be viewed as a variational problem.