  The Kramers turnover problem, that is obtaining a uniform expression for the
rate of escape of a particle over a barrier for any value of the external
friction was solved in the eighties. Two formulations were given, one by
Melnikov and Meshkov (MM) (J. Chem. Phys. 85, 1018 (1986)), which was based on
a perturbation expansion for the motion of the particle in the presence of
friction. The other, by Pollak, Grabert and Haenggi (PGH) (J. Chem. Phys. 91,
4073 (1989)), valid also for memory friction, was based on a perturbation
expansion for the motion along the collective unstable normal mode of the
particle. Both theories did not take into account the temperature dependence of
the average energy loss to the bath. Increasing the bath temperature will
reduce the average energy loss. In this paper, we analyse this effect, using a
novel perturbation theory. We find that within the MM approach, the thermal
energy gained from the bath diverges, the average energy gain becomes infinite,
implying an essential failure of the theory. Within the PGH approach increasing
the bath temperature reduces the average energy loss but only by a finite small
amount, of the order of the inverse of the reduced barrier height. This then
does not seriously affect the theory. Analysis and application for a cubic
potential and Ohmic friction are presented.
