有限体积中的McKean-Vlasov方程
The McKean-Vlasov Equation in Finite Volume
摘要：
研究了$d$ -维中长度尺度$L$有限环面上的McKean-Vlasov方程.我们推导出相变存在的充分必要条件，这是基于在\cite{GP}和\cite{KM}首次发现的标准。在其中和在随后的工作中，人们发现在特定的模型相关值，相互作用参数的$θ^{\sharp}$的临界转变的指示。我们表明，均匀的密度（这可能被解释为液相）是动态稳定的$θ< θ^{\sharp}$和证明，抽象地说，{ \it临界}转变必须发生在$θ= θ^{\sharp}$。然而，对于这个系统，我们表明，在一般条件下- $L$大，$d \geq 2$和各向同性的相互作用-相变实际上是不连续的，并发生在一些$θ\t < θ^{\sharp}$。最后，对于H-稳定的，有界的相互作用与不连续的过渡，我们表明，适当的缩放，$θ\t(L)$往往是一个明确的非平凡的极限为$L\to\infty$。
英文摘要：
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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