  We investigate non-Markovian barrier-crossing kinetics of a massive particle
in one dimension in the presence of a memory function that is the sum of two
exponentials with different memory times $\tau_1$ and $\tau_2$. Our Langevin
simulations for the special case where both exponentials contribute equally to
the total friction show that the barrier crossing time becomes independent of
the longer memory time if at least one of the two memory times is larger than
the intrinsic diffusion time. When we associate memory effects with coupled
degrees of freedom that are orthogonal to a one-dimensional reaction
coordinate, this counterintuitive result shows that the faster orthogonal
degrees of freedom dominate barrier-crossing kinetics in the non-Markovian
limit and that the slower orthogonal degrees become negligible, quite contrary
to the standard time-scale separation assumption and with important
consequences for the proper setup of coarse-graining procedures in the
non-Markovian case. By asymptotic matching and symmetry arguments, we construct
a crossover formula for the barrier crossing time that is valid for general
multi-exponential memory kernels. This formula can be used to estimate
barrier-crossing times for general memory functions for high friction, i.e. in
the overdamped regime, as well as for low friction, i.e. in the inertial
regime. Typical examples where our results are important include protein
folding in the high-friction limit and chemical reactions such as
proton-transfer reactions in the low-friction limit.
