Non-Markovian reversible diffusion-influenced reactions in two dimensions

TL;DR

This paper studies non-Markovian reversible diffusion-influenced reactions in two dimensions, deriving exact and approximate solutions for the unbinding probability, revealing different long-time decay behaviors compared to Markovian cases.

Contribution

It provides an exact expression for the unbinding probability in non-Markovian 2D diffusion reactions, extending previous Markovian models to include slower residence time decay.

Findings

Ultimate dissociation is complete, similar to Markovian case.

Long-time decay of unbinding probability follows a t^{- extsigma} log t pattern.

Exact solutions valid for all times and arbitrary .

Abstract

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 $1-S(t|\ast)\sim t^{-\sigma}\ln t$.