论文标题
在弱相互作用扩散的扩散均值场上极限
On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
论文作者
论文摘要
本文的目的是分析在周期性相互作用潜力影响下发生的大量弱相互作用扩散过程的统计行为。我们将注意力集中在均值的平均场和扩散（均匀化）限制上。特别是，我们表明，如果将平均场系统限制为圆环进行相变，则这两个限制不会通勤，也就是说，如果它承认多个稳态。圆环上这种系统的典型例子是由平均野外旋转器的嘈杂的库拉莫托模型给出的。作为我们主要结果的副产品，我们还分析了中央限制定理的能量后果，以围绕平均场限制的波动，并在吉布斯的相对熵中得出最佳的收敛速率，以测量平均场能的（唯一）平均场能的（唯一）低于临界温度的极限。
The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive(homogenisation)limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.
