Abstract
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.
Highlights
Systems of interacting particles, possibly subject to thermal noise, arise in several applications, ranging from standard ones such as plasma physics and galactic dynam-Communicated by Charles R
The combined mean field and homogenization limit for a system of interacting diffusions in a two-scale confining potential was studied in this paper
It was shown that the bifurcation diagrams can be completely different for small but finite and for the homogenized McKean–Vlasov equation
Summary
Possibly subject to thermal noise, arise in several applications, ranging from standard ones such as plasma physics and galactic dynam-. It should be clear from these two figures that the homogenization and mean field limits, when combined with the long time limit, do not necessarily commute. The homogenization process tends to smooth out local minima and to even “convexify” the confining potential—think of a quadratic potential perturbed by fast periodic fluctuations This implies, in particular, that even though many additional stationary solutions, i.e., branches in the bifurcation diagram may appear for all finite values of , most, if not all, of them may not be present in the bifurcation diagram for the homogenized dynamics.
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