On the role of the energy loss in turnover theories of activated rate processes

Abstract

The diffusion theory of chemical reactions established by Kramers models a chemical reaction as the escape of a Brownian particle from a potential well. Kramers studied the dependence of the escape rate of the particle on the frictional damping in two limits, namely, when the damping is weak so that the rate limiting step is the transfer of energy from the bath to the particle, and in the spatial diffusion regime when the transfer of energy is fast enough to maintain thermal equilibrium of escaping particles. Mel'nikov and Meshkov extended the Kramers theory to the full damping range by using the Wiener-Hopf method. The systematic solution of the non-Markovian turnover problem was given by Pollak, Grabert and Hänggi who proposed a theory that combines the normal mode technique, as well as the approach by Mel'nikov and Meshkov. The key quantity appearing in both turnover theories is the loss of energy of the particle per oscillation. The theories, however, are asymptotic in the energy loss in the sense that their approximations for this quantity are correct only in the weak damping limit. In this paper, we present an alternative to the existing approximations for the energy loss, which approaches the correct limiting behavior for both weak and strong friction. The basic idea is to employ a properly defined energy loss of the deterministic particle dynamics. Its use in the overall rate expression is shown to considerably improve the agreement between analytical calculations and exact numerical results for the escape rate.