Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps

TL;DR

This paper develops a hybrid asymptotic-numerical method to optimize the placement of small absorbing traps in 2D near-disk and elliptical domains, minimizing the mean first passage time for Brownian particles with applications in biophysics.

Contribution

It introduces a new explicit formula for the Neumann Green's function in ellipses and applies a hybrid approach to determine optimal trap configurations, including collinear patterns in elongated ellipses.

Findings

Optimal trap patterns depend on domain shape and aspect ratio.

For long, thin ellipses, traps align along the semi-major axis.

The hybrid theory accurately predicts MFPT and trap locations.

Abstract

The determination of the mean first passage time (MFPT) for a Brownian particle in a bounded 2-D domain containing small absorbing traps is a fundamental problem with biophysical applications. The average MFPT is the expected capture time assuming a uniform distribution of starting points for the random walk. We develop a hybrid asymptotic-numerical approach to predict optimal configurations of $m$ small stationary circular absorbing traps that minimize the average MFPT in near-disk and elliptical domains. For a general class of near-disk domains, we illustrate through several specific examples how simple, but yet highly accurate, numerical methods can be used to implement the asymptotic theory. From the derivation of a new explicit formula for the Neumann Green's function and its regular part for the ellipse, a numerical approach based on our asymptotic theory is used to investigate how the spatial distribution of the optimal trap locations changes as the aspect ratio of an ellipse of fixed area is varied. The results from the hybrid theory for the ellipse are compared with full PDE numerical results computed from the closest point method \cite{IWWC2019}. For long and thin ellipses, it is shown that the optimal trap pattern for $m=2,\ldots,5$ identical traps is collinear along the semi-major axis of the ellipse. For such essentially 1-D patterns, a thin-domain asymptotic analysis is formulated and implemented to accurately predict the optimal locations of collinear trap patterns and the corresponding optimal average MFPT.