A scaling theory of bifurcations in the symmetric weak-noise escape problem

Robert S. Maier, D. L. Stein

    Research output: Contribution to journalArticle

    Abstract

    We consider two-dimensional overdamped double-well systems perturbed by white noise. In the weak-noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double-well system are varied, a unique MPEP may bifurcate into two equally likely MPEPs. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. We quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our construction relies on a new scaling theory, which yields "critical exponents" describing weak-noise behavior at the bifurcation point, near the saddle.

    Original languageEnglish (US)
    Pages (from-to)291-357
    Number of pages67
    JournalJournal of Statistical Physics
    Volume83
    Issue number3-4
    StatePublished - May 1996

    Fingerprint

    Scaling Theory
    saddles
    escape
    Bifurcation Point
    Bifurcation
    scaling
    Separatrix
    Saddle
    white noise
    Probable
    approximation
    boundary layers
    exponents
    activation
    Quasi-stationary Distribution
    WKB Method
    Equally likely
    Smoluchowski Equation
    Path
    Perturbed System

    Keywords

    • Boundary layer
    • Caustics
    • Double well
    • Fokker-Planck equation
    • Lagrangian manifold
    • Large deviation theory
    • Large fluctuations
    • Maslov-WKB method
    • Non-Arrhenius behavior
    • Nongeneric caustics
    • Pearcey function
    • Singular perturbation theory
    • Smoluchowski equation

    ASJC Scopus subject areas

    • Mathematical Physics
    • Physics and Astronomy(all)
    • Statistical and Nonlinear Physics

    Cite this

    A scaling theory of bifurcations in the symmetric weak-noise escape problem. / Maier, Robert S.; Stein, D. L.

    In: Journal of Statistical Physics, Vol. 83, No. 3-4, 05.1996, p. 291-357.

    Research output: Contribution to journalArticle

    Maier, Robert S. ; Stein, D. L. / A scaling theory of bifurcations in the symmetric weak-noise escape problem. In: Journal of Statistical Physics. 1996 ; Vol. 83, No. 3-4. pp. 291-357.
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