Metastability in a continuous mean-field model at low temperature and strong interaction

We consider a system of N∈N mean-field interacting stochastic differential equations that are driven by Brownian noise and a single-site potential of the form z↦z4∕4−z2∕2. The strength of the noise is measured by a small parameter ε>0 (which we interpret as the temperature), and we suppose that the strength of the interaction is given by J>0. Choosing the empirical mean (P:RN→R, Px=1∕N∑ixi) as the macroscopic order parameter for the system, we show that the resulting macroscopic Hamiltonian has two global minima, one at −mε⋆<0 and one at mε⋆>0. Following this observation, we are interested in the average transition time of the system to P−1(mε⋆), when the initial configuration is drawn according to a probability measure (the so-called last-exit distribution), which is supported around the hyperplane P−1(−mε⋆). Under the assumption of strong interaction, J>1, the main result is a formula for this transition time, which is reminiscent of the celebrated Eyring–Kramers formula (see Bovier et al. (2004)) up to a multiplicative error term that tends to 1 as N→∞ and ε↓0. The proof is based on the potential-theoretic approach to metastability.