The Effects of Weak Spatiotemporal Noise on a Bistable One-Dimensional System

TL;DR

This paper analytically investigates how weak spatiotemporal noise influences the reversal dynamics of a one-dimensional bistable system, revealing a transition in activation regimes as the system size varies.

Contribution

It extends Kramers and Langer-Coleman theories to finite spatial domains with spatially dependent noise, identifying a new transition in reversal mechanisms.

Findings

Reversal rate prefactor diverges at a critical system size.

Transition from homogeneous to inhomogeneous reversal occurs as domain size increases.

Finite volume effects require higher transcendental functions for analysis.

Abstract

We treat analytically a model that captures several features of the phenomenon of spatially inhomogeneous reversal of an order parameter. The model is a classical Ginzburg-Landau field theory restricted to a bounded one-dimensional spatial domain, perturbed by weak spatiotemporal noise having a flat power spectrum in time and space. Our analysis extends the Kramers theory of noise-induced transitions to the case when the system acted on by the noise has nonzero spatial extent, and the noise itself is spatially dependent. By extending the Langer-Coleman theory of the noise-induced decay of a metastable state, we determine the dependence of the activation barrier and the Kramers reversal rate prefactor on the size of the spatial domain. As this is increased from zero and passes through a certain critical value, a transition between activation regimes occurs, at which the rate prefactor diverges. Beyond the transition, reversal preferentially takes place in a spatially inhomogeneous rather than in a homogeneous way. Transitions of this sort were not discovered by Langer or Coleman, since they treated only the infinite-volume limit. Our analysis uses higher transcendental functions to handle the case of finite volume. Similar transitions between activation regimes should occur in other models of metastable systems with nonzero spatial extent, perturbed by weak noise, as the size of the spatial domain is varied.