### 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 language | English (US) |
---|---|

Pages (from-to) | 291-357 |

Number of pages | 67 |

Journal | Journal of Statistical Physics |

Volume | 83 |

Issue number | 3-4 |

State | Published - May 1996 |

### Fingerprint

### 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

*Journal of Statistical Physics*,

*83*(3-4), 291-357.

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

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 83, no. 3-4, pp. 291-357.

}

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

VL - 83

SP - 291

EP - 357

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -