A study of Kramers’ turnover theory in the presence of exponential memory friction

Originally, the challenge of solving Kramers' turnover theory was limited to Ohmic friction, or equivalently, motion of the escaping particle governed by a Langevin equation. Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)] (MM) presented a solution valid for Ohmic friction. The turnover theory was derived more generally and for memory friction by Pollak, Grabert, and Hänggi [J. Chem. Phys. 91, 4073 (1989)] (PGH). Mel'nikov proceeded to also provide finite barrier corrections to his theory [Phys. Rev. E 48, 3271 (1993)]. Finite barrier corrections were derived only recently within the framework of PGH theory [E. Pollak and R. Ianconescu, J. Chem. Phys. 140, 154108 (2014)]. A comprehensive comparison between MM and PGH theories including finite barrier corrections and using Ohmic friction showed that the two methods gave quantitatively similar results and were in quantitative agreement with numerical simulation data. In the present paper, we extend the study of the turnover theories to exponential memory friction. By comparing with numerical simulation, we find that PGH theory is rather accurate, even in the strong friction long memory time limit, while MM theory fails. However, inclusion of finite barrier corrections to PGH theory leads to failure in this limit. The long memory time invalidates the fundamental assumption that consecutive traversals of the well are independent of each other. Why PGH theory without finite barrier corrections remains accurate is a puzzle.