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Full Canonical Information from Grand-Potential Density-Functional Theory week ending PRL 113, 238304 (2014) PHYSICAL REVIEW LETTERS 5 DECEMBER 2014 Full Canonical Information from Grand-Potential Density-Functional Theory Daniel de las Heras and Matthias Schmidt* Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany (Received 17 September 2014; published 5 December 2014) 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. DOI: 10.1103/PhysRevLett.113.238304 PACS numbers: 82.70.Dd, 05.70.-a, 71.15.Mb, 82.35.-x Classical density functional theory (DFT) [1–3] forms an Further interest in a canonical description originates essential tool for the investigation of a broad spectrum of from the highly successful dynamical DFT for Brownian simple [4] and complex systems [5] in soft condensed dynamics [1,10]. Here the time evolution via the underly- matter. While DFT was shown to describe a broad range ing Brownian many-body dynamics is intrinsically particle of the occurring phenomena, typically the investigated conserving, and should, hence, be more appropriately systems contain at least one large spatial dimension, such modeled canonically than, as is usually done, grand that a thermodynamic limit can be performed, where the canonically. volume V → ∞ and the number of particles N → ∞ upon Previous efforts to construct a canonical DFT were keeping the mean particle density N=V ¼ const. based, e.g., on adding a correction term to the grand Evans [1] and Mermin [3] originally formulated the potential functional [11] in order to suppress the density exact variational principle of DFT using the grand ensem- fluctuations. In Ref. [6], González et al. expressed the ble, where N fluctuates and the chemical potential μ, the canonical density distribution via a series expansion in volume V, and the absolute temperature T constitute the powers of the inverse average number of particles. By independent thermodynamic variables. The equilibrium truncating the series, they could obtain rather accurate thermodynamic grand potential Ω0ðμ;V;TÞ generalizes canonical profiles from grand canonical DFT in a system of to a grand potential functional Ωð½ρ; μ;V;TÞ of the one- confined hard spheres. The method was subsequently body density distribution ρðrÞ, where r indicates position. applied to a system of one-dimensional rods [12]. The Within the variational theory, ρðrÞ forms the trial field formal extension of the DFT formalism to the canonical which is a priori unknown. For a given thermodynamic ensemble can be based on the Mermin-Evans argument state at μ, V, and T it is the equilibrium grand canonical [13] or on Levy’s constrained search method [14]. density distribution ρ0ðrÞ which minimizes Ω [1,3]. Hence, In this Letter we present a general and formally exact at the minimum method to obtain the canonical one-body density distribu- tion and the canonical partition function using a given δΩð½ρ; μ;T;VÞ ¼ 0: ð Þ grand canonical density functional. The method is based δρðrÞ 1 μ ρ0ðrÞ on decomposing the smooth variation with of the grand canonical quantities into the underlying discrete, Evaluating the functional at ρ0ðrÞ yields the thermody- N-dependent canonical quantities. In order to demonstrate namic grand potential, Ω½ρ0¼Ω0. the validity of the method we consider the one-dimensional Both ensembles, grand canonical and canonical, are system of confined hard particles. As Percus’ (grand) free equivalent in the thermodynamic limit and the differences energy functional for this system is exact, we obtain the between them can be negligible if N ≳ 102. Nevertheless, exact canonical density profiles and the exact canonical there are many interesting systems where the number of partition functions for each discrete value of N. particles is small and the occurring structuring depends In detail, we consider N one-dimensional hard rods of nontrivially on N. Examples are the confinement of hard length σ confined between two parallel walls that are spheres in spherical cavities [6], isolated colloidal clusters separated by a distance h. The interaction between a with ∼10 particles [7], spherical colloids confined in particle and each wall is also hard core; i.e., the particles adaptive two-dimensional cavities [8], and the formation cannot penetrate the walls. We compute the grand canonical of domain walls in two-dimensional confined anisotropic density profiles ρ0ðx; μÞ, where x is the space variable, particles [9]. using the exact grand canonical density functional [15]. 0031-9007=14=113(23)=238304(5) 238304-1 © 2014 American Physical Society week ending PRL 113, 238304 (2014) PHYSICAL REVIEW LETTERS 5 DECEMBER 2014 The minimization is performed using a standard conjugate canonical profile for N ¼ 1, cf. Fig. 1(a), which is constant gradient method (see, e.g., Ref. [16] for details about the within the allowed region (0.5σ ≤ x ≤ h − 0.5σ). However, implementation). As a reference we use benchmark results the corresponding grand canonical profile with N¯ ¼ 1 for canonical density profiles ρNðxÞ from our canonical exhibits clear structuring due to the combination of under- Monte Carlo (MC) simulations, where we first equilibrate lying canonical profiles. the system by performing 106 MC steps and then run First we describe how to obtain the canonical partition 8 10 10 –10 steps to obtain the density profiles. functions ZNðV;TÞ from the grand canonical DFT. The In order to illustrate the differences between canonical grand partition function of a given system is and grand canonical ensembles we show, in Fig. 1, density profiles for a pore with h=σ ¼ 4.9. The values of μ in the X∞ Ξðμ;V;TÞ¼ eβμNZ ðV;TÞ; ð Þ grand ensemble are chosen such that the resulting average N 4 total number N¯ of particles equals their (integer) number N¼0 N in the corresponding canonical system. In both grand where β ¼ 1=kBT, with kB being the Boltzmann constant. canonical and canonical ensembles the number of particles Note that in a closed system of hard cores, such as the can be calculated via the spatial integrals of the corre- present one, the sum truncates at a maximal number of sponding density profiles: particles, N . The thermodynamic grand potential is Z max ¯ NðμÞ¼ dxρ0ðx; μÞ; ð2Þ Ω0ðμ;V;TÞ¼−kBT ln Ξðμ;V;TÞ: ð5Þ V Z Combining Eqs. (4) and (5), we find N ¼ dxρNðxÞ: ð3Þ V X∞ βμN exp½−βΩ0ðμ;V;TÞ − 1 ¼ e ZNðV;TÞ; ð6Þ Because of the hard-core nature of the confined system, the N¼1 N ¼ 4 maximal number of particles is max for the chosen value of h. The density profiles in both ensembles are where the partition function of the empty state is Z0 ¼ 1 strikingly different from each other, as is exemplified by the (which only affects the normalization of the grand canoni- cal partition function, and therefore an unimportant additive x / σ x / σ constant in the grand potential). Equation (6) can be 0 123 4 0 123 4 rewritten as (a) (b) 0.3 0.8 X∞ 0.6 1 ¼ cNðμ;T;Ω0ÞZN; ð7Þ σ 0.2 σ N¼1 (x) (x) ρ 0.4 ρ 0.1 0.2 with 0 0 exp½βμN þ βΩ0ðμÞ 2 (c) (d) cNðμ;T;Ω0Þ¼ ; ð8Þ 4 1 − exp½βΩ0ðμÞ 1.5 3 Ω Z σ σ where we have suppressed the dependence of 0 and N on (x) (x) 1 V T ρ ρ 2 and in the notation. Given a grand canonical DFT, minimizing the functional for given values of μ, V, and T 0.5 1 yields results for Ω0ðμÞ. This enables us to calculate the c N 0 0 coefficients N for each according to Eq. (8).By 0123401234 μ x / σ x / σ repeating the procedure for different values of we obtain a set of linear equations, cf. Eq. (7), where the only FIG. 1 (color online). Scaled density profiles ρðxÞσ as a unknown variables are the canonical partition functions x=σ function of in the canonical ensemble (black symbols) ZN. Numerically, solving this set of linear equations yields and in the grand canonical ensemble (red lines) obtained with results for the ZN. In Fig. 2, we show the numerically canonical MC simulation and the exact grand canonical DFT, obtained canonical partition functions for the one- respectively. The pore width is h=σ ¼ 4.9. The canonical number of particles N equals the average grand canonical number of dimensional system as a function of the wall separation N¯ N ¼ 1 N ¼ 2 N ¼ 3 h. We compare these results with the analytic expression for particles in cases (a) ; (b) ; (c) .In N (d) N ¼ 4 and N¯ ≈ 3.95. Note that N¯ ¼ 4 corresponds to the the partition function [17], ZN ¼ (ðh − NσÞ=Λ) =N!, limit μ → ∞ that cannot be achieved by numerical minimization where Λ is the (irrelevant) thermal wavelength that we of the functional. have set as Λ ¼ σ. The agreement is perfect. 238304-2 week ending PRL 113, 238304 (2014) PHYSICAL REVIEW LETTERS 5 DECEMBER 2014 3 (a) 4 3 Z1 2 N 2 Z2 N Z 1 Z3 1 0 -4 -2 0 2 4 6 Z4 βµ Z 5 (b) 1 0 12345 6 σ p p h / 0 3 p p p 4 1 2 N FIG.
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