  The Kramers problem in the energy-diffusion limited regime of very low
friction is difficult to deal with analytically becasue of the repeated
recrossings of the barrier that typically occur before an asymptotic rate
constant is achieved. Thus, the transmission coefficient of particles over the
potential barrier undergoes oscillatory behavior in time before settling into a
steady state. Recently Kohen and Tannor (JCP Vol. 103, Pg. 6013, 1995)
developed a method based on the phase space distribution function to calculate
the transmission coefficient as a function of time in the high-friction regime.
Here we formulate a parallel method for the low-friction regime. We find
analytic results for the full time and temperature dependence of the rate
coefficient in this regime. Our low-friction result at long times reproduces
the equilibrium result of Kramers at very low friction and extends it to higher
friction and lower temperatures below the turn-over region. Our results
indicate that the single most important quantity in determining the entire time
evolution of the transmission coefficient is the rate of energy loss of a
particle that starts above the barrier. We test our resutls, as well as those
of Kohen and Tannor for the Kramers problem, against detailed numerical
simulations.
