Underdamped, anomalous kinetics in double-well potentials

The noise-driven motion in a bistable potential acts as the archetypal model of various physical phenomena. Here, we contrast properties of the overdamped escape dynamics with the full (underdamped) dynamics. In the weak noise limit, for the overdamped particle driven by nonequilibrium, α-stable noise the ratio of forward to backward transition rates depends only on the width of a potential barrier separating both minima. Using analytical and numerical methods, we show that in the regime of full dynamics, contrary to the overdamped case, the ratio of transition rates depends on both the widths and the heights of the potential barrier separating minima of the double-well potential. The derived analytical formula for the ratio of transition rates is corroborated by extensive numerical simulations. Results of numerical simulations follow especially well the analytical predictions in the weak noise limit when the most probable escape scenario is via a single, strong, noise kick, which is sufficient to induce a quasideterministic transition over the potential barrier. Such an escape trajectory can be analyzed in terms of the instantaneous velocity, which is fully characterized by its density function, which is of the same type as the probability density underlying the noise distribution.