Approximate and numerically exact solutions of the Fokker-Planck equation with bistable potentials

The kinetic rate for a symmetric bistable potential is calculated from the Fokker-Planck operator on both position and momentum. Numerical results are obtained by applying the Lanczos algorithm to the matrix representation of the time evolution operator. Both the continued fraction representation of the spectral density and the first positive eigenvalue which determines the transition rate, are obtained from the computed tridiagonal matrix. The numerical results are compared with the available analytical approximations of the kinetic rate for intermediate potential barriers. In particular, the localized functions method leads to an approximation which accurately describes the approach to the intermediate friction regime from the diffusion limit. Comparison is made also with the available approximations covering all the friction range, and derived under the asymptotic condition of very large potential barriers. A satisfactory agreement is found between the Mel'nikov-Meshkov equation and the numerical results for moderately large potential barriers.