The Kirkwood–Salsburg equations for a bounded stable Kac potential. II. Instability and phase transitions

We prove that systems interacting via potentials of the form φ (x1,x2) =γsψ (γx12) where ψ is bounded stable and defined on bounded support are unstable to fluctuations of wavenumber k′min≠0 at a particular value v0 of v≡nβ, where n is the density and β=1/kBT in the limit γ→0 (VdW1). We also prove (in the VdW1) that the solution to the equation for the single particle distribution function bifurcates at this same value v0, that the nonconstant solution is periodic and has a reciprocal lattice vector with a magnitude k′min, and that there exists a type of long range order at v0. These results are interpreted to indicate the existence of a spinodal point on the liquid isotherm, and similarities between this system and the known properties of the hard sphere fluid are discussed. A theorem is also proven about the range of activity where one has a unique fluid phase, and it is shown that this system has no coexistence region in the usual sense.