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, ${\theta }^{\mathrm{♯}}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\theta <{\theta }^{\mathrm{♯}}$ and prove, abstractly, that a {\it critical} transition must occur at $\theta ={\theta }^{\mathrm{♯}}$. However for this system we show that under generic conditions – $L$ large, $d\ge 2$ and isotropic interactions – the phase transition is in fact discontinuous and occurs at some $\theta \text{t}<{\theta }^{\mathrm{♯}}$. Finally, for H–stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the $\theta \text{t}(L)$ tend to a definitive non–trivial limit as $L\to \mathrm{\infty }$.