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We present a new time-dependent density functional approach for studying the relaxational dynamics of an assembly of interacting particles, subject to thermal noise. Starting from the Langevin stochastic equations of motion for the velocities of the particles, we are able by means of an approximate closure to derive a self-consistent deterministic equation for the temporal evolution of the average particle density. The closure is equivalent to assuming that the equal-time two-point correlation function out of equilibrium has the same properties as its equilibrium version. The changes over time of the density depend on the functional derivatives of the grand canonical free-energy functional F[rho] of the system. In order to assess the validity of our approach, we performed a comparison between the Langevin dynamics and the dynamic density functional method for a one-dimensional hard-rod system in three relevant cases and found remarkable agreement. In addition, we consider the case where one is forced to use an approximate form of F[rho]

Dynamic density functional theory of fluids