Robert Maier and Daniel Stein (University of Arizona)
A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem
(2800K, PostScript, 60 pp.; available via ftp from platinum.math.arizona.edu)
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 a new scaling theory,
which yields `critical exponents' describing weak-noise behavior near
the saddle, at the bifurcation point.