Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic?

Abstract

We aim to clarify confusions in the literature as to whether or not dynamical density functional theories for the one-body density of a classical Brownian fluid should contain a stochastic noise term. We point out that a stochastic as well as a deterministic equation of motion for the density distribution can be justified, depending on how the fluid one-body density is defined -i.e. whether it is an ensemble averaged density distribution or a spatially and/or temporally coarse grained density distribution.

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

“…The question of whether to incorporate stochastic noise into DDFT's as a means of modeling fluctuations has been discussed in the literature. 31,35 The present theory, as formulated 21 and implemented here, gives a deterministic dynamics for the evolution of ensemble average density distribution in the system. In principle, one can incorporate a stochastic noise term into the present approach and the results would then have more of the character of individual Kawasaki dynamics trajectories.…”

Phase separation dynamics in a two-dimensional magnetic mixture

“…The integration over orientation in Eq. (19) then becomes 23,27 dω h B (r , ω )s −→ 2π (20) where L(r) = coth B(r) − 1/B(r) is the Langevin function. The latter also defines the local magnetization m(r) = dωh B (r, ω) cos θ = L(r).…”

“…To this end, we employ the DDFT approach, [16][17][18][19] in which the time evolution of the one-particle densities is governed by a generalized continuity equation. The latter may be derived by integrating the Smoluchowski equation, that is, the Fokker-Planck equation for a system of (colloidal) particles with overdamped stochastic equations of motion (i.e., the inertial terms in the microscopic Langevin equations of motion are neglected).…”

“…The key approximation of the DDFT approach is that the non-equilibrium two-body density distribution functions at time t are set equal to those of an equilibrium system with the same one-body density profile. [16][17][18][19] As a consequence, the currents entering the DDFT equations are determined by (functional derivatives of) the equilibrium Helmholtz free energy functional. As in most DDFT applications so far, we neglect here the effect of hydrodynamic (solvent-induced) interactions between the particles.…”

“…15 The advantage of addressing these topics using DFT is that the latter allows one to establish the link between the macroscopic behaviour of the system to the microscopic Hamiltonian, which is naturally incorporated via the excess Helmholtz free energy functional. In the last few years, the first steps in these directions have already been made on the basis of the so-called dynamical density functional theory [16][17][18][19] (DDFT), which consists of a generalized continuity equation for the one-body density distribution of a many-particle systems of overdamped (Brownian) colloidal particles. Recent applications of the DDFT to phase separation kinetics include spinodal decomposition in spherical fluids 18 and heterogeneous nucleation at solid surfaces.…”