The McKean-Vlasov Equation in Finite Volume

TL;DR

This paper analyzes the phase transition behavior of the McKean-Vlasov equation on finite tori, establishing conditions for stability and discontinuity of phase transitions, and examining their limits as system size grows.

Contribution

It provides a detailed analysis of phase transition conditions, showing that under generic conditions the transition is discontinuous and identifying the limiting behavior as system size increases.

Findings

Uniform density is stable for  < heta^{\u2212}

Phase transition is discontinuous and occurs at  < heta^{\u2212} under generic conditions

Transition points (L) tend to a non-trivial limit as L  

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, $\theta^{\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\theta < \theta^{\sharp}$ and prove, abstractly, that a {\it critical} transition must occur at $\theta = \theta^{\sharp}$. 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 $\theta\t < \theta^{\sharp}$. Finally, for H--stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the $\theta\t(L)$ tend to a definitive non--trivial limit as $L\to\infty$.