The cell fluid model with Curie-Weiss interactions: special cases and analytical results

Inspired by previous extensive numerical studies of a cell fluid model with Curie-Weiss interactions, we concentrate on some analytically tractable special cases in its description. The key ingredient of the model is a competition between global attraction and local repulsion interactions between particles with coupling constants $J_1$ and $J_2$, respectively. We provide analytical results in several limiting cases, including the ideal-gas limit $J_1=J_2=0$ and the strong-repulsion limit $J_2\gg J_1$. For $J_2\gg J_1$, a detailed analytical study is presented. We derive explicit expressions for the critical point parameters, the equation of state, and the binodal and spinodal curves in closed form. The equation of state is found to be in full agreement with that of the van der Waals lattice gas, and the order parameter satisfies the standard Curie-Weiss equation. In a neighborhood of the critical point, a Landau expansion is shown to have the same form and symmetry as that of the classical lattice gas within the mean-field approximation. Moreover, based on the explicit knowledge of a few leading terms in the asymptotic expansion of the deformed exponential function governing the physics of the cell model, we extend its validity range to include the marginal case of thermodynamic stability, $J_1=J_2$. In particular, this extension makes a consideration of the ideal-gas limit $J_1=J_2=0$ formally legitimate. For the generic marginal case $J_1=J_2\ne0$ systematically avoided in previous works, we present numerical data and phase diagrams that augment their findings for $J_2>J_1$.