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

Robert S Maier, D. L. Stein

Research output: Contribution to journalArticle

92 Citations (Scopus)

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

@article{da41a804898543b2ad7048a9414d6624,
title = "A scaling theory of bifurcations in the symmetric weak-noise escape problem",
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.",
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",
author = "Maier, {Robert S} and Stein, {D. L.}",
year = "1996",
month = "5",
language = "English (US)",
volume = "83",
pages = "291--357",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
number = "3-4",

}

TY - JOUR

T1 - A scaling theory of bifurcations in the symmetric weak-noise escape problem

AU - Maier, Robert S

AU - Stein, D. L.

PY - 1996/5

Y1 - 1996/5

N2 - 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.

AB - 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.

KW - Boundary layer

KW - Caustics

KW - Double well

KW - Fokker-Planck equation

KW - Lagrangian manifold

KW - Large deviation theory

KW - Large fluctuations

KW - Maslov-WKB method

KW - Non-Arrhenius behavior

KW - Nongeneric caustics

KW - Pearcey function

KW - Singular perturbation theory

KW - Smoluchowski equation

UR - http://www.scopus.com/inward/record.url?scp=0030526108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030526108&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030526108

VL - 83

SP - 291

EP - 357

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -