ACTIVATED DECAY-RATE - FINITE-BARRIER CORRECTIONS

The activated escape of an underdamped Brownian particle out of a deep potential well is characterized by weak friction gamma much less than omega (gamma is the coefficient of friction and omega is a typical fr equency of the intrawell motion) and by a large barrier height U0 much greater than T (U0 is the barrier height and T is the temperature). T he approach developed previously to calculate the decay rate is based on the derivation of an integral equation and enables one to sum up an infinite series in powers of the ratio gammaU0/Tomega approximately 1 contributing to the preexponential factor of the Arrhenius law. In th e present paper it is shown that the leading correction to the above r esult comes from the slowing down of the particle motion near the top of the barrier and is of the order of (T/U0)ln(U0/T). To calculate it explicitly, one needs to find a correction to the kernel of the above- mentioned integral equation. Beyond the leading-logarithmic approximat ion, two different factors contribute corrections of the order of T/U0 approximately gamma/omega. The noise-induced effects in the barrier c rossing-recrossing by particles in a narrow energy range epsilon appro ximately gammmaT/omega can be easily incorporated into the general sch eme of the calculations. On the other hand, a more accurate derivation of the kernel of the integral equation is required to take into accou nt small variations of the intrawell particle motion caused by variati ons of the particle energy on the scale T much less than U0 under the effects of friction and thermal noise. The proposed consistent expansi on in terms of the small parameters of the problem provides an effecti ve approach to a quantitative investigation of the turnover behavior i n the Kramers problem. For the regime of an intermediate-to-strong fri ction, the finite-barrier corrections can be neglected, since, for typ ical barrier shapes, they are always small.