Metastability for the exclusion process with mean-field interaction

We consider an exclusion particle system with long-range, mean-field-type i nteractions at temperature 1/beta. The hydrodynamic limit of such a system is given by an integrodifferential equation with one conservation law on th e circle C: it is the gradient flux of the Kac free energy functional F-bet a. For beta less than or equal to 1, any constant function with value m is an element of [ -1, +1] is the global minimizer of F-beta in the space {u : integral?, u(x) dx = m}. For beta > 1, F-beta restricted to {u : integral c, u(x) dx = m} may have several local minima: in particular, the constant solution may not be the absolute minimizer of F-beta. We therefore study th e long-time behavior of the par tide system when the initial condition is c lose to a homogeneous stable state, giving results on the time of exit from (suitable) subsets of its domain of attraction. We follow the Freidlin-Wen tzell approach: first, we study in detail Fg together with the time asympto tics of the solution of the hydrodynamic equation; then we study the probab ility of rare events for thr particle system, i.e., large deviations From t he hydrodynamic limit.