Dynamic Density Functional Theory (DDFT)

Abstract

Synonyms Time-dependent density functional theory (TDFT) Definition Dynamic density functional theory (DDFT) is, on the one hand, a time-dependent (dynamic) extension of the static density functional theory (DFT) and, on the other hand, the generalization of Fick's law to the diffusion of interacting particles. The time evolution of the ensemble-averaged density of Brownian particles is given as an integrodifferential equation in terms of the equilibrium Helmholtz free energy functional (or the grand canonical functional). DDFT resolves density variations on length scales down to the particle size but only works for slow relaxing dynamics close to equilibrium. Overview One can prove that in thermal equilibrium, in a grand canonical ensemble (i.e., volume, chemical potential, and temperature are fixed), the grand canonical free energy O(r(r)) of a system can be written as a functional of the one-body density r(r) alone, which will depend on the position r in inhomogeneous systems. The density distribution r eq (r) which minimizes the grand potential functional is the equilibrium density distribution. This statement is the basis of the equilibrium density functional theory (DFT) for classical fluids which has been used with great success to describe a variety of inhomogeneous fluid phenomena, in particular the structure of confined liquids, wetting, anisotropic fluids, and fluid-fluid interfaces. For a historical overview and further references , see [1, 2]. Out of equilibrium there is no such rigorous principle. However, macroscopically one can find a large variety of phenomenological equations for the time evolution which are based on macro-scopic quantities alone, e.g., the diffusion equation, the heat transport equation, and the Navier-Stokes equations for hydrodynamics. A microscopic dynamical theory for the time evolution of slow variables such as the momentum density or the particle density with molecular spatial resolution is highly desirable. In perhaps one of the simplest microscopic cases, a system of interacting Brownian (i.e., diffusing) particles, and in a local equilibrium approximation, one can write the time evolution of the ensemble-averaged one-body density as a functional of the density [3, 4]