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.