On the mean field limit of Random Batch Method for interacting particle systems

Shi Jin, Lei Li

Published 2020-05-24Version 1

The Random Batch Method proposed in our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] is not only a numerical method for interacting particle systems and its mean-field limit, but also can be viewed as a model of particle system in which particles interact, at discrete time, with randomly selected mini-batch of particles. In this paper we investigate the mean-field limit of this model as the number of particles $N \to \infty$. Unlike the propagation of chaos in the classical mean field limit for interacting particle systems, the chaos here is imposed at every discrete time. Despite this, we will not only justify this mean-field limit (discrete in time) but will also show that the limit, as the discrete time interval $\tau \to 0$, approaches to the solution of a nonlinear Fokker-Planck equation arising as the mean-field limit of the original interacting particle system, in Wasserstein-2 distance; namely, the two limits $\lim_{N\to\infty}$ and $\lim_{\tau \to 0}$ commute.