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

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.

arXiv comment16 pages

References in corpus (5)

• Droplet Nucleation and Domain Wall Motion in a Bounded Interval
• Dynamics of Kinks: Nucleation, Diffusion and Annihilation
• Periodic Instanton Bifurcations and Thermal Transition Rate
• Group Theoretical Properties and Band Structure of the Lame Hamiltonian
• Critical Behavior of the Kramers Escape Rate in Asymmetric Classical Field Theories

Cited by in corpus (8)

• Rare Events Statistics in Reaction--Diffusion Systems
• Critical Behavior of the Kramers Escape Rate in Asymmetric Classical Field Theories
• Spatio-temporal Bounded Noises, and transitions induced by them in solutions of real Ginzburg-Landau model
• Anomalous behavior of the Kramers rate at bifurcations in classical field theories
• Exchange of stability as a function of system size in a nonequilibrium system
• Fluctuational Instabilities of Alkali and Noble Metal Nanowires
• The Escape Problem in a Classical Field Theory With Two Coupled Fields
• Noisy Classical Field Theories with Two Coupled Fields: Dependence of Escape Rates on Relative Field Stiffnesses