Anomalous behavior of the Kramers rate at bifurcations in classical field theories

TL;DR

This paper analyzes the divergence of the Kramers rate at bifurcations in a stochastic Ginzburg-Landau PDE and derives a corrected, finite formula for the transition point, confirming previous conjectures.

Contribution

It provides an explicit corrected Kramers formula at bifurcation points for a stochastic PDE, resolving divergence issues and confirming prior theoretical conjectures.

Findings

Derived a finite, noise-dependent Kramers prefactor at bifurcations.

Explicit formulas involve Bessel and error functions for different boundary conditions.

Confirmed the conjecture by Maier and Stein regarding the divergence correction.

Abstract

We consider a Ginzburg-Landau partial differential equation in a bounded interval, perturbed by weak spatio-temporal noise. As the interval length increases, a transition between activation regimes occurs, in which the classical Kramers rate diverges [R.S. Maier and D.L. Stein, Phys. Rev. Lett. 87, 270601 (2001)]. We determine a corrected Kramers formula at the transition point, yielding a finite, though noise-dependent prefactor, confirming a conjecture by Maier and Stein [vol. 5114 of SPIE Proceeding (2003)]. For both periodic and Neumann boundary conditions, we obtain explicit expressions of the prefactor in terms of Bessel and error functions.