Modeling and asymptotic analysis of the concentration difference in a nanoregion between an influx and outflux diffusion across narrow windows

TL;DR

This paper develops an asymptotic model to analyze concentration differences in nanoregions with multiple narrow windows, providing insights into diffusion behavior relevant to biophysical applications.

Contribution

The study introduces a second-order asymptotic analysis for steady-state diffusion in domains with multiple narrow windows, incorporating geometric factors and validating with numerical simulations.

Findings

Derived explicit formulas for concentration distribution between windows.

Validated asymptotic results with numerical simulations.

Introduced a length scale for diffusion depth estimation.

Abstract

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady-state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace's equation using Green's function techniques and second order asymptotic analysis, and depends on the influx amplitude, the diffusion properties as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread. We discuss some applications in biophysics.