Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

Journal of Nonlinear Science
S N Gomes, G A Pavliotis

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.

References

Dec 1, 1966·Proceedings of the National Academy of Sciences of the United States of America·H P McKean
Sep 26, 2012·Physical Review Letters·Benjamin D GoddardSerafim Kalliadasis
Oct 16, 2016·Physical Review. E·A B DuncanM Pradas
Dec 9, 2017·Physical Review Letters·Farzad Farkhooi, Wilhelm Stannat

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Citations

Apr 20, 2019·Physical Review. E·Susana N GomesPetr Yatsyshin
Dec 18, 2020·Physical Review. E·Bertrand Ottino-Löffler, Mehran Kardar
Jan 7, 2021·Proceedings. Mathematical, Physical, and Engineering Sciences·Valerio LucariniNiccolò Zagli

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