In this paper, we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in Duncan et al. (Brownian motion in an N-scale periodic potential, arXiv:1605.05854, 2016b). We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean–Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

中文翻译：

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两尺度电势中相互作用扩散的平均场极限。

在本文中，我们研究了一个弱相互作用的扩散系统的组合平均场和均化极限，该系统以邓肯（Duncan）等人所考虑的形式在两尺度局部周期性约束势中移动。（N尺度周期性势中的布朗运动，arXiv：1605.05854，2016b）。我们表明，尽管平均场和均质化极限通勤有限次，但它们通常不会在长时限内通勤。特别是，根据我们采用两个极限的顺序，稳态的分叉图可能会有所不同。此外，在传递到均化极限之前，我们以两级电位构造了平稳的McKean-Vlasov方程的分歧图。