Density functional for hard hyperspheres from a tensorial-diagrammatic series

We represent the free energy functional by a diagrammatic series with tensorial coefficients indexed by powers of the length scale. For hard cores, we obtain Percus’s exact functional in one dimension and the Kierlik-Rosinberg form of fundamental measures theory in three dimensions. In five dimensions, the functional describes bulk fluids better than does the Percus-Yevick theory. At planar walls, density profiles oscillate with smaller periods than in lower dimensions. Our findings open up avenues for treating both more general high-dimensional systems and three-dimensional mixtures via dimensional reduction.

Figure 1

Compressibility factor $\beta P/\rho$ as a function of packing fraction $\eta$ for 5D hard hyperspheres, as obtained from the DFT bulk free energy (thick line), and compared to the PY compressibility (PYc) and virial (PYv) results (dashed lines) and the LM (thin line) equation of state [9]. Also shown are DFT results from the test particle limit (TPL, symbols) using the virial theorem. The inset shows $\beta P/\rho$ as a function of the total packing fraction $\eta \equiv n_5$ for binary mixtures of 5D hard hyperspheres as obtained from DFT (solid lines), along with the equation of state of Ref. [10] (dashed lines). The upper pair of curves is for size ratio 4 and composition 0.75 of the larger species; the lower pair of curves is for an equimolar mixture with size ratio 2.5.

Figure 2

Bulk two-body direct correlation function $c(r)$ as a function of the scaled distance $r/\sigma$, for 5D hard hyperspheres, as obtained from DFT via the second functional derivative (solid lines), along with the PY result (dashed lines), for packing fractions $\eta =0.05,0.1,0.15,0.2$ (from top to bottom). Also shown is the result of the theory by Rohrmann and Santos (RS) [11] (symbols), which is based on the LM equation of state [9], for $\eta =0.2$ (we omit the small, positive Yukawa tail outside of the core). The inset shows DFT results for the partial pair direct correlation functions $c_{\mathit{ij}}(r)$ as a function of $r/{\sigma }_1$, where ${\sigma }_1=2R_1$ for a binary mixture of 5D hard hyperspheres with size ratio 4, composition of 0.75 for the (larger) species 2, and total density $\rho {\sigma }_1^5=1.4$, corresponding to $\eta =0.173$ (cf. inset of Fig. 1).

Figure 3

The radial distribution function $g(r)$ as a function of $r/\sigma$, for 5D hard hyperspheres, as obtained from DFT using the OZ equation (solid line) and the test particle method (symbols), and compared with the PY result (dashed line), for packing fraction $\eta =0.2$. The OZ result reaches $g(0)=-2.82$ at zero separation. The inset shows the partial pair distribution functions $g_{\mathit{ij}}(r)$ as obtained from the OZ method (lines) for the binary mixture described in the inset of Fig. 2, along with simulation results [12] (symbols).

Figure 4

Scaled density profiles $\rho (z){\sigma }^D$ as a function of the scaled distance $z/\sigma$ for packing fraction $\eta =0.7$ in 1D (dotted line), $\eta =0.367$ in 3D (dashed line), and $\eta =0.115$ in 5D (solid line). (a) Hard hypersphere fluids adsorbed at a hard wall; $z$ is the distance from the wall. The inset shows the surface tension $\beta \gamma {\sigma }^{D-1}$ due to the presence of the wall as a function of $\eta$. (b) Density distributions for hard hypersphere fluids in a planar slit with width $L=4.5\sigma$ in chemical equilibrium with a bulk having the same value of $\eta$ as given above. The inset shows the scaled solvation force $\beta f_S{\sigma }^D$ as a function of $L/\sigma$.