Mean first-passage time to a small absorbing target in three-dimensional elongated domains

TL;DR

This paper derives an approximate formula for the mean first-passage time to a small target within elongated, axisymmetric 3D domains, simplifying the problem to a 1D model and validating it with simulations.

Contribution

It introduces a novel approximation method reducing a 3D MFPT problem to a 1D problem for elongated domains with arbitrary-shaped targets.

Findings

The formula accurately predicts MFPT dependence on target distance and domain profile.

Validation confirms the approximation's accuracy through Monte Carlo simulations.

The method applies to arbitrary target shapes within axisymmetric domains.

Abstract

We derive an approximate formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape inside an elongated domain of a slowly varying axisymmetric profile. For this purpose, the original Poisson equation in three dimensions is reduced an effective one-dimensional problem on an interval with a semi-permeable semi-absorbing membrane. The approximate formula captures correctly the dependence of the MFPT on the distance to the target, the radial profile of the domain, and the size and the shape of the target. This approximation is validated by Monte Carlo simulations.