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, 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 \(\theta_{\text{T}}<\theta^{\sharp }\) . 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→∞.