Metastability for the Exclusion Process with Mean-Field Interaction

Abstract

We consider an exclusion particle system with long-range, mean-field-type interactions at temperature 1/β. The hydrodynamic limit of such a system is given by an integrodifferential equation with one conservation law on the circle $C$: it is the gradient flux of the Kac free energy functional F _β. For β≤1, any constant function with value m ∈ [−1, +1] is the global minimizer of F _β in the space $\{ u:\int_C {u(x)} \,dx = m\}$. For β>1, F _β restricted to $\{ u:\int_C {u(x)} \,dx = m\}$ may have several local minima: in particular, the constant solution may not be the absolute minimizer of F _β. We therefore study the long-time behavior of the particle system when the initial condition is close to a homogeneous stable state, giving results on the time of exit from (suitable) subsets of its domain of attraction. We follow the Freidlin–Wentzell approach: first, we study in detail F _β together with the time asymptotics of the solution of the hydrodynamic equation; then we study the probability of rare events for the particle system, i.e., large deviations from the hydrodynamic limit.