We present a new time-dependent density functional approach to study 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 approximated 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 in time of the density depend on the functional derivatives of the grand canonical free energy functional F[ρ] of the system. In particular the static solutions of the equation for the density correspond to the exact equilibrium profiles provided one is able to determine the exact form of F[ρ]. 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, with some interesting exceptions, which are discussed and explained. In addition, we consider the case where one is forced to use an approximate form of F[ρ]. Finally we compare the present method with the stochastic equation for the density proposed by other authors [Kawasaki, Kirkpatrick etc.] and discuss the role of the thermal fluctuations.

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A geometrically based fundamental-measure free-energy density functional unified the scaled-particle and Percus-Yevick theories for the hard-sphere fluid mixture. It has been successfully applied to the description of simple ͑''atomic''͒ three-dimensional ͑3D͒ fluids in the bulk and in slitlike pores, and has been extended to molecular fluids. However, this functional was unsuitable for fluids in narrow cylindrical pores, and was inadequate for describing the solid. In this work we analyze the reason for these deficiencies, and show that, in fact, the fundamental-measure geometrically based theory provides a free-energy functional for 3D hard spheres with the correct properties of dimensional crossover and freezing. After a simple modification of the functional, as we propose, it retains all the favorable Dϭ3 properties of the original functional, yet gives reliable results even for situations of extreme confinements that reduce the effective dimensionality D drastically. The modified functional is accurate for hard spheres between narrow plates (Dϭ2), and inside narrow cylindrical pores (Dϭ1), and it gives the exact excess free energy in the Dϭ0 limit ͑a cavity that cannot hold more than one particle͒. It predicts the ͑vanishingly small͒ vacancy concentration of the solid, provides the fcc hard-sphere solid equation of state from closest packing to melting, and predicts the hard-sphere fluid-solid transition, all in excellent agreement with the simulations.

We present a new time-dependent Density Functional approach to study 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 approximated 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 in time of the density depend on the functional derivatives of the grand canonical free energy functional F [ρ] of the system. In particular the static solutions of the equation for the density correspond to the exact equilibrium profiles provided one is able to determine the exact form of F [ρ]. 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, with some interesting exceptions, which are discussed and explained. In addition, we consider the case where one is forced to use an approximate form of F [ρ]. Finally we compare the present method with the stochastic equation for the density proposed by other authors [Kawasaki,Kirkpatrick etc.] and discuss the role of the thermal fluctuations.

A new free energy density functional for hard spheres is presented, along the lines of the fundamental measure theory, which reproduces the Percus-Yevick equation of state and direct correlation function for the fluid, with a tensor weighted density. The functional, based on the zero-dimension limit, is exact for any one-dimensional density distribution of the spheres. The application to the hard sphere crystals gives excellent results, solving all of the qualitative problems of previous density functional approximations, including the unit cell anisotropy in the fcc lattice and the description of the metastable bcc lattice.