The effects of weak spatiotemporal noise on a bistable one-dimensional system

Robert S. Maier, D. L. Stein

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    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.

    Original languageEnglish (US)
    Title of host publicationProceedings of SPIE - The International Society for Optical Engineering
    EditorsL. Schimanski - Geier, D. Abbott, A. Neiman, C. Broeck
    Pages67-78
    Number of pages12
    Volume5114
    DOIs
    StatePublished - 2003
    EventNoise in Complex Systems and Stochastic Dynamics - Santa Fe, NM, United States
    Duration: Jun 2 2003Jun 4 2003

    Other

    OtherNoise in Complex Systems and Stochastic Dynamics
    CountryUnited States
    CitySanta Fe, NM
    Period6/2/036/4/03

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    Keywords

    • Activation barrier
    • Kramers prefactor
    • Kramers theory
    • Nonzero spatial extent
    • Rate prefactor
    • Spatially extended system
    • Spatiotemporal noise
    • Transition rate theory
    • Weak noise

    ASJC Scopus subject areas

    • Electrical and Electronic Engineering
    • Condensed Matter Physics

    Cite this

    Maier, R. S., & Stein, D. L. (2003). The effects of weak spatiotemporal noise on a bistable one-dimensional system. In L. Schimanski - Geier, D. Abbott, A. Neiman, & C. Broeck (Eds.), Proceedings of SPIE - The International Society for Optical Engineering (Vol. 5114, pp. 67-78) https://doi.org/10.1117/12.497641