Non-Markovian reversible diffusion-influenced reactions in two dimensions

We investigate the reversible diffusion-influenced reaction of an isolated pair in the presence of a non-Markovian generalization of the backreaction boundary condition in two space dimensions. Following earlier work by Agmon and Weiss, we consider residence time probability densities that decay slower than an exponential and that are characterized by a parameter $0<\sigma\leq 1$. We calculate an exact expression for the probability $S(t|\ast)$ that the initially bound particle is unbound, which is valid for arbitrary $\sigma$ and for all times. Furthermore, we derive an approximate solution for long times. We show that the ultimate fate of the bound state is complete dissociation, as in the 2D Markovian case. However, the limiting value is approached quite differently: Instead of a $\sim t^{-1}$ decay, we obtain $1-S(t|\ast)\sim t^{-\sigma}\ln t$.