Some remarks on the effect of the Random Batch Method on phase transition

TL;DR

This paper investigates how the Random Batch Method affects phase transitions in particle systems, showing that it preserves the transition but lowers the critical temperature due to added noise.

Contribution

It provides a theoretical analysis of the impact of RBM-induced noise on phase transition behavior in mean-field models.

Findings

RBM preserves phase transition phenomena.

The critical temperature for phase transition is reduced by RBM.

RBM introduces noise that influences the effective dynamics of the system.

Abstract

In this article, we focus on two toy models : the Curie-Weiss model and the system of $N$ particles in linear interactions in a double well confining potential. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the $O(N^2)$ interactions per time step, the Random Batch Method (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size $p>1$, and computing the interactions only within each batch, thus reducing the numerical complexity to $O(Np)$ per time step. The convergence of this numerical method has been proved in other works. This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the effective dynamics of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.