Activated decay of a metastable state: transient times for small and large dissipation

Abstract

The time evolution of the thermally activated decay rates is considered. This evolution is of particular importance for the recent nanoscale experiments discussed in the literature, where the potential barrier is relatively low (or the temperature is relatively high). The single-molecule pulling is one example of such experiments. The decay process is modeled in the present work through computer solving the stochastic (Langevin) equations. Altogether about a hundred of high precision rates have been obtained and analyzed. The rates are registered at the absorption point located far beyond the barrier to exclude the influence of the backscattering on the value of the quasistationary decay rate. The transient time, i.e., the time lapse during which the rate attains half of its quasistationary value, has been extracted. The dependence of the transient times upon a damping parameter is compared with that of the inverse quasistationary decay rate. Two analytical formulae approximating the time-dependences of the numerical rates are proposed and analyzed.

Appendix

The time step $\tau$ of the numerical modeling is a technical parameter which, being chosen correctly, should not influence the physical results. To demonstrate the accuracy of our numerical calculations, we show in Table

Table 1 The dependence of the quasistationary numerical decay rate $R_{D}$ upon the time step $\tau$ for different values of the damping ($\varphi$) and governing ($G$) parameters

1 how the calculated quasistationary decay rate $R_{{\text{D}}}$ depends upon $\tau$ for several values of the damping and governing parameters. While the time step is relatively large and decreases, the same happens to $R_{{\text{D}}}$: this means that $\tau$ is too big.

At the smaller values of the time step, the decay rate ceases to depend regularly upon $\tau$ as it is expected. Finally, when the time step becomes small enough the quasistationary decay rate $R_{{\text{D}}}$ stays constant within the statistical errors. One sees this behavior of $R_{{\text{D}}} \left( \tau \right)$ for any set of the damping and governing parameters in Table 1.