Variational theory of activated rate processes for an arbitrary barrier

The thermally activated escape of a Brownian particle over a smooth barrier of arbitrary shape and height is considered as an eigenproblem of the Fokker-Planck equation. For the case of moderate and large friction, the least nonzero eigenvalue of this equation is found via a Rayleigh-quotient-based perturbation method. A comparison with existing variational approaches and from numerical simulations for bistable potentials with parabolic and quartic barriers shows that the proposed expression gives unprecedentedly accurate results at all barrier heights including the limit of vanishingly low barrier.