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

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.