The McKean-Vlasov Equation in Finite Volume

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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 GP and KM. 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< θ and prove, abstractly, that a critical transition must occur at θ= θ. 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 θ < θ. Finally, for H--stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the θ(L) tend to a definitive non--trivial limit as L→∞.