  We revisit the Kibble-Zurek mechanism by analyzing the dynamics of phase
ordering systems during an infinitely slow annealing across a second order
phase transition. We elucidate the time and cooling rate dependence of the
typical growing length and we use it to predict the number of topological
defects left over in the symmetry broken phase as a function of time, both
close and far from the critical region. Our results extend the Kibble-Zurek
mechanism and reveal its limitations.
