Diffusion toward non-overlapping partially reactive spherical traps: fresh insights onto classic problems

TL;DR

This paper develops a semi-analytical method to analyze diffusion and reaction processes around multiple non-overlapping spherical traps, providing new tools for understanding complex diffusion-reaction systems in physics and biology.

Contribution

It introduces a generalized separation of variables approach and derives a semi-analytical Green function solution for diffusion-reaction problems involving multiple spherical traps.

Findings

Derived a semi-analytical Green function for the problem

Provided methods to compute eigenvalues and eigenfunctions in perforated domains

Discussed applications to diffusion-controlled reactions in physics and biology

Abstract

Several classic problems for particles diffusing outside an arbitrary configuration of non-overlapping partially reactive spherical traps in three dimensions are revisited. For this purpose, we describe the generalized method of separation of variables for solving boundary value problems of the associated modified Helmholtz equation. In particular, we derive a semi-analytical solution for the Green function that is the key ingredient to determine various diffusion-reaction characteristics such as the survival probability, the first-passage time distribution, and the reaction rate. We also present modifications of the method to determine numerically or asymptotically the eigenvalues and eigenfunctions of the Laplace operator and of the Dirichlet-to-Neumann operator in such perforated domains. Some potential applications in chemical physics and biophysics are discussed, including diffusion-controlled reactions for mortal particles.