Improvements to Kramers Turnover Theory

TL;DR

This paper improves Kramers turnover theory by analyzing temperature effects on energy loss, revealing limitations of previous models and proposing a novel perturbation approach for better accuracy.

Contribution

It introduces a new perturbation theory that accounts for temperature dependence, addressing the shortcomings of earlier models in Kramers turnover theory.

Findings

MM approach diverges at high temperatures

PGH approach shows limited temperature effect on energy loss

Application to cubic potential confirms theoretical predictions

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 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.