Improvements to Kramers turnover theory

Abstract

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 1980s. Two formulations were given, one by Mel'nikov and Meshkov (MM) [V. I. Mel'nikov and S. V. Meshkov, 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 Hänggi (PGH) [E. Pollak, H. Grabert, and P. Hänggi, 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. Then, this does not seriously affect the theory. Analysis and application for a cubic potential and Ohmic friction are presented.