A SCALING THEORY OF BIFURCATIONS IN THE SYMMETRICAL WEAK-NOISE ESCAPEPROBLEM

We consider two-dimensional overdamped double-well systems perturbed b y 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 th e two wells. However, as the parameters of a symmetric double-well sys tem are varied, a unique MPEP may bifurcate into two equally likely MP EPs. At the bifurcation point in parameter space, the activation kinet ics 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 probabili ty distribution of the system that is valid in a boundary layer near t he separatrix. The approximation is a Formal asymptotic solution of th e Smoluchowski equation. Our construction relies on a new scaling theo ry, which yields ''critical exponents'' describing weak-noise behavior at the bifurcation point, near the saddle.