Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions

TL;DR

This paper introduces a new theoretical framework for analyzing the behavior of diffusing particles in confined geometries through the surface hopping propagator, impacting the understanding of diffusion-influenced reactions.

Contribution

We derive explicit formulas for the surface hopping propagator in various Euclidean domains, providing a new theoretical tool for studying diffusion-mediated surface phenomena.

Findings

Explicit formulas for half-space, annuli, cylinders, and spherical shells.

Analysis of propagator behavior for immortal and mortal particles.

Provides a foundation for future studies of diffusion-influenced reactions.

Abstract

Dynamics of a particle diffusing in a confinement can be seen a sequence of bulk-diffusion-mediated hops on the confinement surface. Here, we investigate the surface hopping propagator that describes the position of the diffusing particle after a prescribed number of encounters with that surface. This quantity plays the central role in diffusion-influenced reactions and determines their most common characteristics such as the propagator, the first-passage time distribution, and the reaction rate. We derive explicit formulas for the surface hopping propagator and related quantities for several Euclidean domains: half-space, circular annuli, circular cylinders, and spherical shells. These results provide the theoretical ground for studying diffusion-mediated surface phenomena. The behavior of the surface hopping propagator is investigated for both "immortal" and "mortal" particles.