Entropy of the cell fluid model with Curie-Weiss interaction

TL;DR

The paper derives explicit analytical formulas for the entropy of a cell-fluid lattice gas with Curie-Weiss attraction and unlimited occupancy, expressing the entropy per particle $S^*$ and per cell $S^*_v$ as functions of the reduced temperature $T^*$ and reduced chemical potential $\mu^*$ via parametric representations using $\bar{z}_{\max}$ and $\tilde{K}_j$. It reveals an infinite cascade of first-order phase transitions at sufficiently low $T^*$, with entropy exhibiting jumps across phases and minima near integer densities. The work draws qualitative parallels with Hubbard-type models and van der Waals fluids, highlighting a universal entropic signature of multiple occupancy in lattice gases and providing analytic benchmarks for mean-field theories of such systems. It also discusses artifacts such as negative entropy at high densities arising from the classical treatment, suggesting how these features relate to occupancy and interaction parameters.

Abstract

Entropy of the cell fluid model with Curie-Weiss interaction is obtained in analytical form as a function of temperature and chemical potential. A parametric equation is derived representing the entropy as a function of density. Features of both the entropy per particle and the entropy per cell are investigated at the entropy-density and entropy-chemical potential planes. The considered cell model is a multiple-occupancy model and possesses an infinite sequence of first-order phase transitions at sufficiently low temperatures. We find that the entropy exhibits pronounced minima at around integer-valued particle densities, which may be a generic feature of multiple-occupancy models.

Figures (2)

• Figure 1: Entropy: (a) Entropy per particle versus chemical potential. (b) Entropy per particle versus density. (c) Entropy per cell versus chemical potential. (d) Entropy per cell versus density. In all figures, color curves correspond to the following temperatures: Red - $T^*=0.40$; Green - $T^*=0.30$ Black - $T^*=0.25$; Blue - $T^* = 0.20$. Parameters are taken as $a=1.2$ and $v^*=5.0$.
• Figure 2: Entropy of the van der Waals fluid: (a) Entropy per particle versus chemical potential. (b) Entropy per particle versus density. (c) Entropy per cell versus chemical potential. (d) Entropy per cell versus density. In all figures, color curves correspond to the following temperatures: Red - $T^*_{\mathrm{vdW}}=0.30$; Green - $T^*_{\mathrm{vdW}}=0.25$ Black - $T^*_{\mathrm{vdW}}=0.225$; Blue - $T^*_{\mathrm{vdW}} = 0.20$. The value of the parameter: $v^*_{\mathrm{vdW}}=5.0$.