Abstract
We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.
Highlights
Metastability is a common physical phenomenon, in which a system quickly moved across a first-order phase transition line takes a long time to settle in its equilibrium state
The second class of models consists of stochastic differential equations driven by weak
In the particular case where the drift term is given by minus the gradient of a potential, the attractors are local minima of the potential, and the mean transition time between local minima is governed by Kramers’ law [Eyr[35], Kra40]: In the small-noise limit, the transition time is exponentially large in the potential barrier height between the minima, with a multiplicative prefactor depending on the curvature of the potential at the local minimum the process starts in and at the highest saddle crossed during the transition
Summary
Metastability is a common physical phenomenon, in which a system quickly moved across a first-order phase transition line takes a long time to settle in its equilibrium state. One of the main ingredients of the proof is a result by Blomker and Jentzen on spectral Galerkin approximations [BJ13], which allows us to reduce the system to a finite-dimensional one This reduction requires some a priori bounds on moments of transition times, which we obtain by large-deviation techniques (though it might be possible to obtain them by other methods). Transition times for the finite-dimensional equation can be accurately estimated by the potential-theoretic approach of [BEGK04, BGK05], provided one can control capacities uniformly in the dimension. Such a control has been achieved in [BBM10] in a particular case, the so-called synchronised regime of a chain of coupled bistable particles introduced in [BFG07a, BFG07b].
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