Fundamental measure theory for predicting many-body correlation functions

Abstract

We study many-body correlation functions within various Fundamental Measure Theory (FMT) formulations and compare their predictions to Monte Carlo simulations of hard-sphere fluids. FMT accurately captures the qualitative behavior of three- and four-body structure, particularly at low and intermediate wavevectors. At higher wavevectors, the predictions of FMT vary in quantitative accuracy. We show that the dominant contributions to the four-point structure factor arise from direct triplet correlations, allowing the evaluation of four-point correlations to be greatly simplified. In glass-forming liquids at high volume fractions, FMT correctly reproduces deviations from the convolution approximation, highlighting FMT's ability to capture growing structural multipoint correlations upon supercooling.

Figures (7)

• Figure 1: (a) Fourier-space two-body direct correlation function $\rho c_2(k)$ for a hard-sphere fluid at volume fraction $\eta = 0.49$, comparing various theoretical models to Monte Carlo (MC) simulation data (black symbols). (b) Magnification of the low-$k$ region. FMT-based predictions are shown for the original Rosenfeld (R), White Bear I (WBI), White Bear II (WBII), Malijevský (M) Lutsko (L) and Gül (G) functionals, alongside results obtained from Ornstein–Zernike closures, specifically the Verlet (V) and Modified Hypernetted Chain (MHNC) theories. For this and all following figures, the error bars denote the standard deviation of the mean from 100 different simulation runs. Here, the error bars are typically smaller than the markers.
• Figure 2: (a) Static structure factor $S_2(k)$ of the hard-sphere fluid at $\eta = 0.49$ as a function of the wavenumber $k$, comparing Monte Carlo data (black circles) to the same theoretical models as in Fig. \ref{['fig:c2']}. Panels (b) and (c) magnify the first and second peak regions, respectively.
• Figure 3: Triplet structure factor $S_3(k_1,k_2,\theta)$ at $\eta = 0.49$ from Monte Carlo (MC) simulations (black circles) compared to predictions from the different FMTs. The convolution approximation ($c_3=0$) is also shown, for which we have used the Rosenfeld prediction of $S_2$. Top row: $S_3$ as a function of $k_1$ at fixed $\theta$ and $k_2$. Bottom row: $S_3$ as a function of $\cos\theta$ at fixed $k_1 = k_2$.
• Figure 4: Triplet direct correlation function $\rho^2 c_3(k_1,k_2,\theta)$ for a hard-sphere fluid at $\eta = 0.49$. Black symbols show Monte Carlo (MC) results. (a) Direct triplet correlations $c_3$ against $k$ for $k_1 = k_2 = k$ and $\theta = 0.25\pi$. (b–d) $c_3$ as a function of $\cos\theta$ for various $k_1 = k_2$ values.
• Figure 5: Four-body static structure factor $S_4(k_1,k_2,k_3,\theta_{12},\theta_{13},\phi_{23})$ for a hard-sphere fluid at $\eta = 0.49$. Monte Carlo data (black symbols) are compared to various FMT predictions and the convolution approximation obtained with the Rosenfeld $S_2$ (black, dashed). The wavenumbers are fixed at $k_1 = k_2 = 7.0$, while $k_3 \in \{2.0, 7.0\}$ as labeled. Each panel corresponds to a different angular configuration.