Dynamic density functional theory of fluids

Abstract

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.

Dynamical Density Functional Theory with Hydrodynamic Interactions and Colloids in Unstable Traps

“…The theory is explicitly worked out for hydrodynamic interactions on the Rotne-Prager level and generalizes earlier formulations [16,17,18] where hydrodynamic interactions were neglected. The theory makes predictions for an arbitrary time-dependent external potential, i.e., for a general inhomogeneous nonequilibrium situation.…”

“…δρ(r,t) and persists when hydrodynamic interactions are neglected [17,18]. j 2 and j 3 are additional current densities which occur due to the solvent mediated hydrodynamic interactions.…”

Generalized Stochastic Fokker-Planck Equations

“…It can also be derived from a kinetic theory in which the transition probability from one site to another depends on the occupancy of these sites (see [24] and Section 2.11 of [46]). In this Appendix, we show that the generalized Smoluchowski equation (71) can also be derived from the dynamic density functional theory (DDFT) used in the theory of simple liquids when the particles experience short-range interactions [92]. In that case, the nonlinear pressure is due to the correlations induced by the short-range interactions, and the drift is due to the long-range interactions.…”

“…To evaluate the integral corresponding to the short-range interactions, we use an approximation that has become standard in the DDFT of fluids [92] and take ρ 2 (r, r , t)∇u SR (|r − r |) dr = ρ(r, t)∇ δF ex δρ [ρ(r, t)],…”

Hard rectangles near curved hard walls: Tuning the sign of the Tolman length

“…Reducing the threshold value 10 −8 to 10 −9 does not affect the results. Alternatively, the functional Ω[ρ( r, φ)] could also be minimized using dynamical density functional theory [87][88][89][90][91].…”