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In this paper we compute and analyse the transition rates and duration ofreactive trajectories of the stochastic 1-D Allen-Cahn equations for both theFreidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitudewhite noise, as well as for small and large domain. We demonstrate thatextremely rare reactive trajectories corresponding to direct transitionsbetween two metastable states are efficiently computed using an algorithmcalled adaptive multilevel splitting. This algorithm is dedicated to thecomputation of rare events and is able to provide ensembles of reactivetrajectories in a very efficient way. In the small noise limit, our numericalresults are in agreement with large-deviation predictions such asinstanton-like solutions, mean first passages and escape probabilities. We showthat the duration of reactive trajectories follows a Gumbel distribution likefor one degree of freedom systems. Moreover, the mean duration growthslogarithmically with the inverse temperature. The prefactor given by thepotential curvature grows exponentially with size. The main novelty of our workis that we also perform an analysis of reactive trajectories for large noisesand large domains. In this case, we show that the position of the reactivefront is essentially a random walk. This time, the mean duration grows linearlywith the inverse temperature and quadratically with the size. Using aphenomenological description of the system, we are able to calculate thetransition rate, although the dynamics is described by neitherFreidlin--Wentzell or Eyring--Kramers type of results. Numerical resultsconfirm our analysis.

Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm