Full Canonical Information from Grand-Potential Density-Functional Theory

Abstract

We present a general and formally exact method to obtain the canonical one-body density distribution and the canonical free energy from direct decomposition of classical density functional results in the grand ensemble. We test the method for confined one-dimensional hard-core particles for which the exact grand potential density functional is explicitly known. The results agree to within high accuracy with those from exact methods and our Monte Carlo many-body simulations. The method is relevant for treating finite systems and for dynamical density functional theory.

Bulk dynamics of Brownian hard disks: Dynamical density functional theory versus experiments on two-dimensional colloidal hard spheres

“…In previous theoretical studies addressing Brownian diffusion of three-dimensional colloidal hard spheres [12,14,24], it turned out that the quality of the results compared to those from simulations crucially depends on (i) the choice of the underlying functional characterizing hard-body correlations, and (ii) the precise treatment of the self component s, which represents one single particle. Generally, the free-energy functionals in DFT are of grand-canonical character, and fluctuations are known to yield unphysical contributions in systems where particles are treated explicitly [41,42]. Importantly, a method has been derived which removes possible self-interactions within the free-energy functional by considering the zerodimensional crossover and a proper modeling of the grand potential in that situation.…”

Particle-conserving dynamics on the single-particle level

“…As detailed in Ref. [14], we could reduce the number of equations by one upon removing the trivial case of zero particles in each species, which has been omitted in the present study due to the negligible effect on computation time. The above methods can be easily generalized to systems containing any number κ of different components, where the number of coupled linear equations to be solved grows exponentially with κ.…”

“…Note that the canonical partition function changes in each iteration step and generally differs from Z N entering in Eq. (14), which is calculated for the true external fields V . Dotted lines: speciesresolved density profiles for a mixture in which the particle order is strictly maintained, as given by Eq.…”

“…N − 1) with F N given by Eq. (14). In any case, it is necessary to determine at each time step the adiabatic potentials V N (x, t) in equilibrium, as described in Sec.…”

“…This issue was addressed in the late 1990's by González et al [12,13], who employed an expansion in inverse powers of the particle number to systematically approximate canonical density profiles using grand-canonical information as an input. The problem of calculating equilibrium density profiles in the canonical ensemble was revisited in 2014 by de las Heras and Schmidt [14] who showed how to obtain exact canonical information from grand-canonical DFT given an exact functional, performing explicit calculations for the one-dimensional hard-rod system. The method of de las Heras and Schmidt was further generalized in Ref.…”