Phase diagram of Ising systems with additional long-range forces

Abstract. We consider ferromagnetic Ising systems where the interaction is given by the sum of a fixed reference potential and a Kac potential of intensity λ ≥ 0 and scaling parameter γ> 0. In the Lebowitz Penrose limit γ → 0+ the phase diagram in the (T, λ) positive quadrant is described by a critical curve λmf(T) which separates the regions with one and two phases, respectively below and above the curve. We prove that if λ> λmf(T), i.e. above the curve, there are at least two Gibbs states for small values of γ. If instead λ < λmf(T) and if the reference Gibbs state (i.e. without the Kac potential) satisfies a mixing condition at the temperature T, then, at the same temperature the full interaction (i.e. with also the Kac potential) satisfies the Dobrushin Shlosman uniqueness condition for small values of γ so that there is a unique Gibbs state. 1. Introduction. In the original van der Waals theory the occurrence of phase transitions is due to long attractive forces between molecules. In its Statistical Mechanics formulation, [11], these forces are described by Kac potentials that depend on a scaling parameter γ> 0 which controls the strength and the range of the potential. In a large variety of systems th