The McKean-Vlasov equation in finite volume

Abstract

We study the McKean-Vlasov equation on the finite tori of length scale L in d-dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194-209, 1970) and Kirkwood and Monroe (J. Chem. Phys. 9:514-526, 1941). Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ# of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for θ