Pair distribution function of the cell fluid model with Curie-Weiss interaction

TL;DR

The paper addresses the equilibrium structure of a cell-fluid system with Curie–Weiss interaction by deriving an exact expression for the pair distribution function $g^{(2)}$ in the grand-canonical ensemble. It leverages an asymptotic analysis of the grand partition function and an occupancy-based measure $Q_{T^*,\\mu^*}(n)$, yielding representations of $g^{(2)}$ and the one- and two-particle densities via functions $K_j$ and tilde $K_j$, and enabling parametric descriptions with respect to $T^*,\\mu^*,\\rho^*$. The main results show that $g^{(2)}({\\bf r}_1,{\\bf r}_2)=1$ when particles occupy different cells, while $g^{(2)}=\\langle n(n-1) \\rangle_Q / \\langle n \\rangle_Q^2$ when they share a cell, with no intra-cell spatial structure and a discontinuity at cell boundaries; the low-density limit recovers the Boltzmann factor of the Curie–Weiss interaction. Overall, the work establishes that the cell-fluid with Curie–Weiss interaction behaves as a mean-field fluid and provides exact analytic tools for analyzing pair correlations in this lattice-like soft-potential system, including occupancy-driven density effects and phase-coexistence behavior.

Abstract

In this paper we present results for the pair distribution function for the cell fluid model with Curie-Weiss interaction. As a supplementary result, one- and two-particle densities are calculated.

Figures (4)

• Figure 1: Pair distribution function $g^{(2)}$ versus $\bar{z}$ at two temperatures. Curve 1 (Black): $T^* = 0.4$, the temperature higher than $T^*_c$; Curve 2 (Blue): $T^*=0.2$, the temperature lower than $T^*_c$. The critical temperature $T^*_c \approx 0.25$. At Curve 2, the bold solid sections correspond to the stable phases, the dashed sections correspond to the metastable and unstable regions.
• Figure 3: Pair distribution function $g^{(2)}$ versus $\rho^*$ at different temperatures. (a) Curve 1 (Black): $T^* = 0.4$ - temperature is higher than $T^*_c$; Curve 2 (Blue): $T^*=0.2$ - temperature is lower than $T^*_c$. (b) Blue curve: $T^*=0.1$ - temperature is much lower than $T^*_c$. The critical temperature $T^*_c \approx 0.25$. For undercritical temperatures, the bold solid sections correspond to the stable phases, the dashed sections correspond to the metastable and unstable regions. In Panel (b), the stable regions are too small to be noticeable in the plot.
• Figure 4: Pair distribution function $g^{(2)}$ versus $\mu^*$ at two different temperatures. Curve 1 (Black): $T^* = 0.4$, the temperature higher than $T^*_c$; Curve 2 (Blue): $T^*=0.2$, the temperature lower than $T^*_c$. The critical temperature $T^*_c \approx 0.25$. At Curve 2, the bold solid sections correspond to the stable phases, the dashed sections correspond to the metastable and unstable regions.
• Figure 5: Pair distribution function $g^{(2)}({\vb r}_1, {\vb r}_2)$ as a function of separation $r = \abs{{\vb r}_1 - {\vb r}_2}$. The quantity $r^*$ denotes distance from the point with coordinate ${\vb r}_1$ to the boundary of the cubic cell containing ${\vb r}_1$, in the direction of ${\vb r}_2$. (a) Phase I: $T^* = 0.2$, $\rho^*=0.1$; Phase II: $T^* = 0.2$, $\rho^* = 1.0$; Phase III: $T^*_c = 0.2$, $\rho^* = 2.0$; Supercritical: $T^* = 0.5$, $\rho^* = 0.5$. At $r > r^*$, $g^{(2)} = 1$. (b) Phase I: $T^* = 0.2$, $\rho^*=0.1$; Low-density limit: $g^{(2)} \approx \exp(-a/T^*)$, with $T^*=0.2$, $a=1.2$.