In this paper we compute and analyse the transition rates and duration of
reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the
Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude
white noise, as well as for small and large domain. We demonstrate that
extremely rare reactive trajectories corresponding to direct transitions
between two metastable states are efficiently computed using an algorithm
called adaptive multilevel splitting. This algorithm is dedicated to the
computation of rare events and is able to provide ensembles of reactive
trajectories in a very efficient way. In the small noise limit, our numerical
results are in agreement with large-deviation predictions such as
instanton-like solutions, mean first passages and escape probabilities. We show
that the duration of reactive trajectories follows a Gumbel distribution like
for one degree of freedom systems. Moreover, the mean duration growths
logarithmically with the inverse temperature. The prefactor given by the
potential curvature grows exponentially with size. The main novelty of our work
is that we also perform an analysis of reactive trajectories for large noises
and large domains. In this case, we show that the position of the reactive
front is essentially a random walk. This time, the mean duration grows linearly
with the inverse temperature and quadratically with the size. Using a
phenomenological description of the system, we are able to calculate the
transition rate, although the dynamics is described by neither
Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results
confirm our analysis
