Phase Transitions in a Continuum Curie-Weiss System: A Quantitative Analysis

Abstract

Phase transitions in a continuum Curie-Weiss system of interacting particles are studied quantitatively. The interaction is determined by a division of the underlying space $\mathbb {R}^d$ into congruent cubic cells. For a region $V\subset \mathbb {R}^d$ consisting of $N\in \mathbb {N}$ cells, each two particles contained in V attract each other with intensity $J_1/N$. The particles contained in the same cell repel each other with intensity $J_2>J_1$. For fixed values of the intensities $J_1, J_2$, the temperature and the chemical potential, the thermodynamic phase is defined as a probability measure on the space of occupation numbers of cells. There is shown that the half-plane $J_1\,\times \,$  chemical potential contains phase coexistence points, and thus multiple thermodynamic phases of the system may exist at the same values of the temperature and chemical potential. The numerical calculations describing such phenomena are presented.