A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem

TL;DR

This paper develops a new scaling theory to analyze bifurcations in the most probable escape paths of symmetric double well systems under weak noise, revealing non-Arrhenius activation behavior at bifurcation points.

Contribution

It introduces a novel scaling theory and uses the Maslov-WKB method to quantify non-Arrhenius behavior at bifurcation points in weak-noise escape problems.

Findings

Quantifies non-Arrhenius activation kinetics at bifurcation points.

Develops critical exponents describing weak-noise behavior near the saddle.

Provides an asymptotic approximation of the quasistationary distribution near the separatrix.

Abstract

We consider the overdamped limit of two-dimensional double well systems perturbed by weak 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 MPEP's. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. In this paper 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 analysis relies on the development of a new scaling theory, which yields `critical exponents' describing weak-noise behavior near the saddle, at the bifurcation point.