A two-scale approach for sampling local free energy minimisers

When a free energy admits several local minimisers, the associated mean-field particle system exhibits a metastable behavior, undergoing random transitions between these minimisers. However, when the number of particles is large, these transitions are rare events and typically occur at a timescale which is out of reach for simulations. In this work, we introduce a system of particles coupled with a slow macroscopic external variable which approximately follows a Langevin dynamics associated to a coarse-grained free energy. The empirical distribution of the particle system thus samples all local minimisers of the initial free energy, with transition times which are now independent from the number of particles. The construction is inspired by a theoretical tool: the two-scale approach for proving log-Sobolev inequalities for Gibbs measures.