Accumulation time of diffusion in a 2D singularly perturbed domain

TL;DR

This paper develops a mathematical approach combining asymptotic analysis and Green's functions to compute the accumulation time for diffusion relaxation in 2D domains with small holes, relevant to cellular processes.

Contribution

It introduces a novel method to calculate accumulation time in singularly perturbed 2D domains, improving understanding of local relaxation dynamics beyond eigenvalue-based measures.

Findings

Derived explicit formulas for accumulation time in 2D domains with holes.

Showed accumulation time better captures local relaxation behavior.

Validated the approach with analytical and numerical examples.

Abstract

A general problem of current interest is the analysis of diffusion problems in singularly perturbed domains, within which small subdomains are removed from the domain interior and boundary conditions imposed on the resulting holes. One major application is to intracellular diffusion, where the holes could represent organelles or biochemical substrates. In this paper we use a combination of matched asymptotic analysis and Green's function methods to calculate the so-called accumulation time for relaxation to steady state in 2D domains. The standard measure of the relaxation rate is in terms of the principal nonzero eigenvalue of the negative Laplacian. However, this global measure does not account for possible differences in the relaxation rate at different spatial locations, is independent of the initial conditions, and relies on the assumption that the eigenvalues have sufficiently large spectral gaps. As previously established for diffusion-based morphogen gradient formation, the accumulation time provides a better measure of the relaxation process.