Thermal decay of a metastable state: the influence of the re-scattering on the quasistationary dynamical rate

When a Brownian particle, initially being in the potential well, overcomes the barrier and moves to the absorptive border, it still has a chance to be scattered back to the well by thermal fluctuations. We study this phenomenon carefully modeling numerically the motion of the particle with the Langevin equations. Four potentials which coincide near the well and the barrier but differ in the tail (i.e. beyond the barrier) are considered. It is shown that the potential for which the well and the barrier are described by two smoothly joined parabolas ("the parabolic potential") plays a role of a dividing range for the mutual layout of the quasistationary dynamical rate and the widely used in the literature Kramers rate. Namely, for the potentials with a steeper tails, the Kramers rate R_K0 underestimates the true quasistationary dynamical rate R_D, whereas for the less steep tails opposite holds (inversion of R_D/R_K0). It is proved that the mutual layout of the values of the R_D for different potentials is explained by the re-scattering of the particles from the potential tail.