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

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.