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

TL;DR

This paper analyzes the mean-field limit of the Random Batch Method for interacting particle systems, showing convergence to a nonlinear Fokker-Planck equation as the number of particles grows and the time step shrinks.

Contribution

It establishes the mean-field limit of the Random Batch Method without relying on the law of large numbers, connecting it to a nonlinear Fokker-Planck equation.

Findings

The mean-field limit is justified as particle number N approaches infinity.

The discrete-time limit converges to a nonlinear Fokker-Planck equation.

The approach does not depend on classical chaos propagation methods.

Abstract

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 classical mean field limit for interacting particle systems where the law of large numbers plays the role and the chaos is propagated to later times, the mean field limit now does not rely on the law of large numbers and chaos 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 distance.