An encounter-based approach to the escape problem

TL;DR

This paper extends the encounter-based approach to diffusion reactions by incorporating escape regions and reflecting boundaries, deriving spectral expansions and probabilistic interpretations relevant to chemistry and biophysics.

Contribution

It introduces a generalized framework for encounter-based diffusion reactions with escape regions, providing spectral solutions and probabilistic insights beyond traditional models.

Findings

Derived spectral expansion for the full propagator.

Obtained joint probability density of escape time and encounters.

Discussed applications in chemistry and biophysics.

Abstract

We revise the encounter-based approach to imperfect diffusion-controlled reactions, which employs the statistics of encounters between a diffusing particle and the reactive region to implement surface reactions. We extend this approach to deal with a more general setting, in which the reactive region is surrounded by a reflecting boundary with an escape region. We derive a spectral expansion for the full propagator and investigate the behavior and probabilistic interpretations of the associated probability flux density. In particular, we obtain the joint probability density of the escape time and the number of encounters with the reactive region before escape, and the probability density of the first-crossing time of a prescribed number of encounters. We briefly discuss generalizations of the conventional Poissonian-type surface reaction mechanism described by Robin boundary condition and potential applications of this formalism in chemistry and biophysics.