Models of quantum and classical particles on the d-dimensional lattice ZZ d with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the particle m = µ/h 2 tends to infinity. For these models, it is proposed to define the convergence of the Euclidean Gibbs states, when m → +∞, by the weak convergence of the corresponding local Gibbs specifications, determined by conditional Gibbs measures. In fact it is proved that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with pair interactions possessing the translation invariance, has also been proven.

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A single-sort continuum Curie-Weiss system of interacting particles is studied. The particles are placed in the space R d divided into congruent cubic cells. For a region V ⊂ R d consisting of N ∈ N cells, every two particles contained in V attract each other with intensity J 1 /N. The particles contained in the same cell are subjected to binary repulsion with intensity J 2 > J 1 . For fixed values of the temperature, the interaction intensities, and the chemical potential the thermodynamic phase is defined as a probability measure on the space of occupation numbers of cells, determined by a condition typical of Curie-Weiss theories. It is proved that the half-plane J 1 × chemical potential contains phase coexistence points at which there exist two thermodynamic phases of the system. An equation of state for this system is obtained.