Propagation of chaos and approximation error of random batch particle system in the mean field regime

TL;DR

This paper proves the propagation of chaos and provides sharp approximation error bounds for the random batch particle system, enhancing understanding of its convergence to the mean-field limit in particle simulations.

Contribution

It establishes the propagation of chaos for the random batch system and derives precise error bounds in terms of particle number and time step, using the BBGKY hierarchy.

Findings

Sharp bound of (k^2/N^2 + ktau^2) on the relative entropy

Demonstrates convergence of the random batch system to the mean-field limit

Provides insights into the system's behavior in the mean field regime

Abstract

The random batch method [J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulation of classical $N$-particle systems and their mean-field limit, but also a new model for interacting particle system that could be more physical in some applications. In this work, we establish the propagation of chaos for the random batch particle system and at the same time obtain its sharp approximation error to the classical mean field limit of $N$-particle systems. The proof leverages the BBGKY hierarchy and achieves a sharp bound both in the particle number $N$ and the time step $\tau$. In particular, by introducing a coupling of the division of the random batches to resolve the $N$-dependence, we derive an $\mathcal{O}(k^2/N^2 + k\tau^2)$ bound on the $k$-particle relative entropy between the law of the system and the tensorized law of the mean-field limit. This result provides a useful understanding of the convergence properties of the random batch system in the mean field regime.