Thermally activated escape processes in a double well coupled to a slow harmonic mode

Abstract

We present accurate calculations of thermally activated rates for a symmetric double well system coupled to a dissipative harmonic mode. Diffusive barrier crossing is treated by solving the time-independent two-dimensional Smoluchowski equation as a function of a coupling and a diffusion anisotropy parameter. The original problem is transformed to a Schrödinger equation with a Hamiltonian describing a reactive system coupled to a one-dimensional harmonic bath. The calculations are performed using a matrix representation of the Hamiltonian operator in a set of orthonormal basis functions. An effective system-specific basis is introduced which consists of adiabatically displaced eigenfunctions of the coupled harmonic oscillator and those of the uncoupled reactive subsystem. This representation provides a very rapid convergence rate. Just a few basis functions are sufficient to obtain highly accurate eigenvalues with a small computational effort. The presented results demonstrate the applicability of the method in all regimes of interest, reaching from inter-well thermal activation (fast harmonic mode) to deep intra-well relaxation (slow harmonic mode). Our calculations reveal the inapplicability of the Kramers–Langer theory in certain regions of parameter space not only when the anisotropy parameter is exponentially small, but even in the isotropic diffusion case when the coupling is weak. The calculations show also that even for large barrier heights there is a region in the parameter space with multiexponential relaxation towards equilibrium. An asymptotic theory of barrier crossing in the strongly anisotropic case is presented, which agrees well with the numerically exact results.