Solution of Kramers' problem for a moderately to heavily damped elastic string

We obtain the nucleation rate of critical droplets for an elastic string moving in a double-well potential and subject to noise and damping forces. We obtain this rate for a class of potentials that includes both the asymmetric φ4 and the φ6 potentials. The frequencies of small oscillations about the critical droplet are obtained from a Heun equation. We solve the Fokker-Planck equation for the phase-space probability density by projecting onto the eigenfunction basis. We present a comparison with simulations for the case of the asymmetric φ4 potential