  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 formulas approximating the
time-dependences of the numerical rates are proposed and analyzed.
