Liquid-Vapor Transition in a Model of a Continuum Particle System with Finite-Range Modified Kac Pair Potential

We prove the existence of a phase transition in dimension $d>1$ in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range ${\gamma }^{-1}$ , with $\gamma >0$ . This transition is "close", for small positive $\gamma$ , to the one proved previously by Lebowitz and Penrose in the van der Waals limit $\gamma \downarrow 0$ . It is of the type of the liquid-vapor transition observed when a fluid, like water, heated at constant pressure, boils at a given temperature. Previous results on phase transitions in continuum systems with stable potentials required the use of unphysical four-body interactions or special symmetries between the liquid and vapor. The pair interaction we consider is obtained by partitioning space into cubes of volume ${\gamma }^{-d}$ , and letting the Kac part of the pair potential be uniform in each cube and act only between adjacent cubes. The "short-range" part of the pair potential is quite general (in particular, it may or may not include a hard core), but restricted to act only between particles in the same cube. Our setup, the "boxed particle model", is a special case of a general "spin" system, for which we establish a first-order phase transition using reflection positivity and the Dobrushin-Shlosman criterion.