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

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