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

Source abstract

Abstract We prove the existence of a phase transition in dimension $$d>1$$ d > 1 in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range $$\gamma ^{-1}$$ γ - 1 , with $$\gamma >0$$ γ > 0 . This transition is “close”, for small positive $$\gamma$$ γ , to the one proved previously by Lebowitz and Penrose in the van der Waals limit $$\gamma \downarrow 0$$ γ ↓ 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 $$\gamma ^{-d}$$ γ - 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.