The McKean-Vlasov Equation in Finite Volume

Abstract

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, ${\theta }^{♯}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\theta <{\theta }^{♯}$ and prove, abstractly, that a {\it critical} transition must occur at $\theta ={\theta }^{♯}$. 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 $θ\t < θ^{\sharp}$. 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 $L\to \infty$.

References in corpus (2)

• State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System
• Phase Transition in a Vlasov-Boltzmann Binary Mixture

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