The McKean-Vlasov Equation in Finite Volume

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, theta-sharp of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for theta<theta-sharp and prove, abstractly, that a critical transition must occur at theta=theta-sharp. However for this system we show that under generic conditions -- L large, d>=2 and isotropic interactions -- the phase transition is in fact discontinuous and occurs at some theta r < theta shapr . Finally, for H-stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the theta r (L) tend to a definitive non-trivial limit as L--> infinite.