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arXiv · 2607.15896

On the diffusive-mean field limit of a kinetic weakly interacting particle system

Raphaël Gastaldello·Grigorios A. Pavliotis·Gabriel Stoltz·Urbain Vaes
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Abstract

We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics, extending the results of Delgadino, Gvalani and Pavliotis (Arch. Ration. Mech. Anal., 2021) to the hypoelliptic/hypocoercive case. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the $O(2)$ model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean-Vlasov-Fokker-Planck partial differential equation.

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Raphaël Gastaldello, Grigorios A. Pavliotis, Gabriel Stoltz, Urbain Vaes. 2026-07-17. On the diffusive-mean field limit of a kinetic weakly interacting particle system. https://arxiv.org/abs/2607.15896

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