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 \cite{GP} and \cite{KM}. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, $θ^{\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $θ< θ^{\sharp}$ and prove, abstractly, that a {\it critical} transition must occur at $θ= θ^{\sharp}$. However for this system we show that under generic conditions -- $L$ large, $d \geq 2$ and isotropic interactions -- the phase transition is in fact discontinuous and occurs at some $θ\t < θ^{\sharp}$. Finally, for H--stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the $θ\t(L)$ tend to a definitive non--trivial limit as $L\to\infty$.

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