Full Canonical Information from Grand-Potential Density-Functional Theory

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.

Figure 1

Scaled density profiles $\rho (x)\sigma$ as a function of $x/\sigma$ in the canonical ensemble (black symbols) and in the grand canonical ensemble (red lines) obtained with canonical MC simulation and the exact grand canonical DFT, respectively. The pore width is $h/\sigma =4.9$. The canonical number of particles $N$ equals the average grand canonical number of particles $\overline{N}$ in cases (a) $N=1$; (b) $N=2$; (c) $N=3$. In (d) $N=4$ and $\overline{N}\approx 3.95$. Note that $\overline{N}=4$ corresponds to the limit $\mu \to \infty$ that cannot be achieved by numerical minimization of the functional.

Figure 2

Canonical partition functions $Z_N$ for a range of particle numbers $N=1–5$ (as indicated) of a system of one-dimensional hard particles confined between hard walls as a function of the scaled wall separation distance $h/\sigma$. The lines represent the analytic solution. The symbols indicate the numerical results for the partition functions obtained from decomposition of the exact grand canonical DFT results.

Figure 3

(a) Average number of particles $\overline{N}$ as a function of the scaled chemical potential $\beta \mu$ in a pore with $h/\sigma =4.9$ calculated with three different methods. The lines lie on top of each other. (b) Probability $p_N$ of finding $N=0–4$ (as indicated) particles in a pore with $h/\sigma =4.9$ as a function of $\beta \mu$. (c) $p_N$ as a function of the average number of particles $\overline{N}$ inside the pore. (d) $p_N$, with $N=0–25$, as a function of $\overline{N}$ in a pore with $h/\sigma =25.9$ and a parabolic external potential.

Figure 4

Scaled canonical density profiles ${\rho }_N(x)\sigma$ obtained by MC simulation (black symbols) and using the grand canonical decomposition (red lines) of the exact grand canonical DFT for: (a) $N=1–4$ in a pore with $h/\sigma =4.9$, and (b) $N=20$ in a pore with $h/\sigma =25.9$ with a parabolic external potential $\beta V_{\text{ext}}(x)=0.05(x-h/2{)}^2{\sigma }^{-2}$. The dashed green line in (b) is the grand canonical profile for $\overline{N}=20$. Only the left half of the profiles are shown in (b).