  We consider the generalization of the Kramers escape over a barrier problem
to the case of a long chain molecule. The problem involves the motion of a
chain molecule of $N$ segments across a region where the free energy per
segment is higher, so that it has to cross a barrier. We consider the limit
where the length of the molecule is much larger than the width of the barrier.
The width is taken to be sufficiently wide that a coninuum description is
applicable to even to the portion over the barrier. We use the Rouse model and
analyze the mechanism of crossing a barrier. There can be two dominant
mechanisms. They are: end crossing and hairpin crossing. We find the free
energy of activation for the hairpin crossing is two times that for end
crossing. In both cases, the activation energy has a square root dependence on
the temperature $T$, leading to a non-Arrhenius form for the rate. We also show
that there is a special time dependent solution of the model, which corresponds
to a kink in the chain, confined to the region of the barrier. The movement of
the polymer from one side to the other is equivalent to the motion of the kink
on the chain in the reverse direction. We also consider the translocation of
hydrophilic polypeptides across hydrophobic pores, a process that is quite
common in biological systems. Biological systems accomplish this by having a
hydrophobic signal sequence at the end that goes in first. We find that for
such a molecule, the transition state resembles a hook, and this is in
agreement with presently accepted view in cell biology.
