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.

本文的目的是分析在周期性相互作用势的影响下演化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平均场和扩散（均质化）极限的组合上。尤其是，我们表明，如果约束到圆环的平均场系统经历相变（也就是说，如果它允许多个稳态），则这两个极限不会互换。这种在圆环上的系统的典型示例由平均场平面旋转器的嘈杂的Kuramoto模型给出。作为我们主要结果的副产品，