Mean first-passage time to a small absorbing target in an elongated planar domain

TL;DR

This paper presents an explicit formula for estimating the mean first-passage time to a small target in elongated planar domains, combining conformal mapping and homogenisation techniques for practical applications.

Contribution

The authors develop a fully explicit approximation for MFPT in elongated domains, integrating conformal mapping, boundary homogenisation, and Fick-Jacobs equation, validated against numerical solutions.

Findings

Approximate formula accurately predicts MFPT away from the target.

Method is validated through comparison with numerical solutions.

Provides a practical tool for rapid MFPT estimation in physical and biological systems.

Abstract

We derive an approximate but fully explicit formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape in a general elongated domain in the plane. Our approximation combines conformal mapping, boundary homogenisation, and Fick-Jacobs equation to express the MFPT in terms of diffusivity and geometric parameters. A systematic comparison with a numerical solution of the original problem validates its accuracy when the starting point is not too close to the target. This is a practical tool for a rapid estimation of the MFPT for various applications in chemical physics and biology.