The McKean-Vlasov Equation in Finite Volume
The McKean-Vlasov Equation in Finite Volume
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DOI：
10.1007/s10955-009-9913-z
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发表时间：
2010-02-01
影响因子：
1.6
通讯作者：
Panferov, V.
中科院分区：
文献类型：
--
作者：
Chayes, L.;Panferov, V.
我们研究了 d 维长度尺度 L 的有限圆环上的 McKean-Vlasov 方程。我们推导了相变存在的充分必要条件，这些条件基于 Gates 和 Penrose (Commun. Math. Phys. 17: 194-209, 1970) 以及 Kirkwood 和 Monroe (J. Chem. Phys. 9: 514-526, 1941) 首次发现的标准。在其中和后续工作中，人们发现了指向特定模型相关值（交互参数的 theta(#)）处的关键转变的指示。我们证明，均匀密度（可以解释为液相）对于 theta < theta(#) 是动态稳定的，并抽象地证明，临界转变必须发生在 theta = theta(#) 处。然而，对于该系统，我们表明，在一般条件下（L 大、d >= 2 和各向同性相互作用），相变实际上是不连续的，并且发生在某些 theta(T) < theta(#) 处。最后，对于 H 稳定、具有不连续跃迁的有界相互作用，我们表明，通过适当的缩放，theta(T)(L) 趋于确定的非平凡极限，即 L -> 无穷大。
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, 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 critical transition must occur at theta = theta(#). 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(T) < theta(#). 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 -> infinity.