Phase diagram of a double-occupancy cell model of a fluid with Curie-Weiss interaction

Abstract: A double-occupancy cell model of a fluid with Curie-Weiss interaction is studied. First, we show that the model is isomorphic to the Blume-Capel model on a complete graph through a simple transformation from spin to occupancy variables. We then investigate its phase behavior within the grand-canonical ensemble using a combination of analytical and numerical methods. Despite its simplicity, the model exhibits a remarkably rich thermodynamic behavior depending on the ratio between the local repulsive and global attractive interactions. We identify regimes characterized by a single critical point, two distinct critical points, tricritical behavior, and triple-point formation. For sufficiently strong repulsion, the system possesses three fluid phases of different densities, leading to both gas-liquid and liquid-liquid coexistence. The locations of the critical, tricritical, and triple points are determined, and the corresponding phase diagrams are constructed. These results demonstrate that the competition between double-occupancy repulsion and long-range attraction is sufficient to generate complex phase behavior in a minimal multiple-occupancy lattice-gas model.

• The paper demonstrates that incorporating double occupancy with Curie-Weiss attraction yields rich phase behavior, including unique gas–liquid and liquid–liquid transitions.
• The analysis employs a saddle-point expansion and mapping to the Blume-Capel model, providing an exact mean-field framework for multifluid coexistence.
• The findings highlight how local repulsion and infinite-range attraction interact to stabilize triple points and integer-density phases in soft-matter systems.

Phase Diagram and Critical Properties of the Double-Occupancy Cell Model with Curie-Weiss Interaction

Introduction and Model Mapping

The double-occupancy cell model is a minimal lattice-gas system where each cell can host up to two particles. The Hamiltonian incorporates global (Curie-Weiss) attraction between all particles and a local repulsion penalizing double occupancy. This framework is directly mapped, via $n_l = S_l + 1$, to the Blume-Capel model on a complete graph, establishing isomorphism between mean-field spin-1 systems and multiple-occupancy lattice gases. The local repulsion strength $a = \Delta/J$ is the key parameter controlling the competition between bounded repulsion and unbounded attraction, directly affecting the emergent phase behavior.

Analytical Formalism and Thermodynamic Limit

The model is treated in the grand-canonical ensemble, with the partition function reduced, via a Gaussian integral representation and a saddle-point (Laplace) expansion, to a parametric mean-field equation of state governed by an effective single-cell partition function $K_0(T^*;z)$. This function accounts for the occupancy constraint ($n_l \leq 2$) and explicitly couples to the temperature and chemical potential. The thermodynamic observables, such as reduced density and pressure, are expressed in terms of ratios of $K_j$ special functions, analogous to moments of the occupancy distribution.

Distinct formulations are presented for distinguishable and indistinguishable particles, the latter enforcing the appropriate symmetry for direct comparison with the Blume-Capel model.

Phase Diagram Topology and Criticality

The interplay between $a$ and $T^*$ leads to a rich, nontrivial structure in the $(T^*,\rho^*,a)$ phase space. For distinguishable particles, three primary regimes are identified:

• Low repulsion ($a < a_{\mathrm{tc}}$): The model exhibits a single fluid critical point at $\rho^*_c = 1$, with the coexistence line resembling that of a standard mean-field lattice gas. The critical temperature $a = \Delta/J$0 increases as $a = \Delta/J$1 decreases.
• Intermediate regime ($a = \Delta/J$2): A tricritical point emerges at $a = \Delta/J$3. Here, two distinct critical points exist, corresponding to two first-order transitions—gas–liquid and liquid–liquid—terminating at their respective critical endpoints. Between these, a triple point is realized where three fluid phases (gas, low-density liquid, and high-density liquid) coexist.
• Strong repulsion ($a = \Delta/J$4): The coexistence region splits into two, with two critical points persisting at low $a = \Delta/J$5. The triple point vanishes; at low $a = \Delta/J$6, integer-density phases ($a = \Delta/J$7, $a = \Delta/J$8, $a = \Delta/J$9) are stabilized due to the occupancy cut-off.

  Figure 1: Phase diagram of the double-occupancy cell model for distinguishable particles, illustrating the critical line ($K_0(T^*;z)$0), two critical lines ($K_0(T^*;z)$1), and the tricritical/triple-point lines in $K_0(T^*;z)$2 space.

  

Figure 2: Example $K_0(T^*;z)$3 phase diagram at $K_0(T^*;z)$4, displaying the single critical point and the corresponding coexistence region.

Figure 3: Combined 3D $K_0(T^*;z)$5 representation showing coexistence lines, critical points, and the triple-point line.

For indistinguishable particles, the phenomenology is topologically similar, but the location of the tricritical point and critical temperatures shift, with $K_0(T^*;z)$6 and $K_0(T^*;z)$7 values matching those of the mean-field Blume-Capel model. This confirms particle indistinguishability as the proper prescription for alignment with the spin-1 canonical ensemble.

The absolute values for critical densities and temperatures, as well as the boundaries of the triple-point regime, are provided in detail in the paper; e.g., for large $K_0(T^*;z)$8, the critical densities converge to $K_0(T^*;z)$9 and $n_l \leq 2$0 at $n_l \leq 2$1.

Triple Points and Multiphase Coexistence

The interval $n_l \leq 2$2 (distinguishable particles) is characterized by the presence of a triple point where three distinct phases coexist at a common $n_l \leq 2$3 and $n_l \leq 2$4. As $n_l \leq 2$5 increases, $n_l \leq 2$6 rapidly decreases and vanishes for $n_l \leq 2$7. The triple-point densities smoothly interpolate between $n_l \leq 2$8 (at $n_l \leq 2$9) and $K_j$0 (as $K_j$1). This behavior is absent in single-occupancy lattice gases, underlining the essential role of multiple occupancy in producing gas–liquid–liquid transitions.

Physical and Theoretical Implications

The model demonstrates that a minimal extension of the lattice-gas paradigm (allowing $K_j$2) suffices to generate not only a gas-liquid critical point, but also a distinct liquid–liquid phase transition and associated criticality, as well as genuine triple points—a hallmark of core-softened and ultrasoft matter systems. The results reveal that mean-field competition between finite local repulsion and infinite-range attraction is a sufficient mechanism for stabilizing multiphase coexistence phenomena previously observed in models of soft-matter clustering, penetrable spheres, and the Gaussian core fluid.

These findings supply a tractable analytical benchmark for more realistic simulations and for validating approximate treatments of liquid–liquid criticality.

Structural Interpretation

The analysis of the pair distribution function $K_j$3 further corroborates the interpretation of the high-density phase as a fluid and not a crystalline state. In all phases, $K_j$4 exhibits mean-field character—stepwise behavior determined by the occupancy constraint—with no indication of periodic spatial order typical of crystalline phases.

Figure 4: The pair distribution function $K_j$5 as a function of separation, displaying stepwise mean-field fluid behavior in all thermodynamic phases.

Conclusions

The double-occupancy cell model with Curie-Weiss attraction and local bounded repulsion admits an exact mean-field treatment exhibiting canonical features of ultrasoft matter: multiple critical points, a liquid–liquid transition line, triple points, and coexistence of three distinct integer-density fluid phases in certain regions of parameter space. Mapping to the Blume-Capel model on the complete graph guarantees thermodynamic consistency and places this lattice gas in the context of well-studied spin systems. The explicit analytical results establish this model as a minimal, analytically tractable template for theoretical studies of multiphase soft matter systems, and a reference point for more complex or microscopic modeling approaches. The demonstrated phenomenology motivates further study into extensions (unrestricted occupancy, finite-range interactions, off-lattice generalization) to quantitatively capture the full range of density anomalies and clustering inherent in soft-matter systems.

Reference:

"Phase diagram of a double-occupancy cell model of a fluid with Curie-Weiss interaction" (2607.01009).