Single-molecule chemical reactions: Reexamination of the Kramers approach

Single-molecule chemical reactions yield insight into fluctuation phenomena that are obscured in the measurement of the ensemble of molecules. Kramers escape problem is investigated here in a framework suitable for single-molecule reactions. In particular we obtain distributions of escape times in simple limiting cases, rather than their mean, and investigate their sensitivity on initial conditions. Rich physical behaviors are observed: sub-Poissonian statistics when the dynamics is only slightly deviating from the Newtonian, super-Poissonian behavior when diffusion is dominating, and Poissonian behavior when Kramers original conditions hold. By varying initial conditions escape time distributions can follow a (usual) exponential or a τ^−3∕2 decay, due to regular diffusion. We briefly address experimental results that yield the τ^−3∕2 behavior (with cutoffs) and propose that this behavior is universal.