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

Abstract

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.

References in corpus (6)

• The large deviation approach to statistical mechanics
• Random changes of flow topology in two dimensional and geophysical turbulence
• Generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes
• The instanton method and its numerical implementation in fluid mechanics
• Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations
• Non-classical large deviations for a noisy system with non-isolated attractors

Cited by in corpus (22)

• Computation of extreme heat waves in climate models using a large deviation algorithm
• Population dynamics method with a multi-canonical feedback control
• A rare event algorithm links transitions in turbulent flows with activated nucleations
• Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation
• Computing return times or return periods with rare event algorithms
• Rare event computation in deterministic chaotic systems using genealogical particle analysis
• Applications of large deviation theory in geophysical fluid dynamics and climate science
• Extreme events in transitional turbulence
• An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two
• Multistability and rare spontaneous transitions in barotropic $\beta$-plane turbulence
• Application of Adaptive Multilevel Splitting to High-Dimensional Dynamical Systems
• Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain
• Numerical study of extreme mechanical force exerted by a turbulent flow on a bluff body by direct and rare-event sampling techniques
• Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems
• Collapse of transitional wall turbulence captured using a rare events algorithm
• Instantons for rare events in heavy-tailed distributions
• Data-driven methods to estimate the committor function in conceptual ocean models
• On a new class of score functions to estimate tail probabilities of some stochastic processes with Adaptive Multilevel Splitting
• A Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields
• Computing non-equilibrium trajectories by a deep learning approach
• Finite size analysis of a double crossover in transitional wall turbulence
• Does rare, noise-induced, bypass transition in plane Couette flow bypass instantons ?