The effect of target orientation on the mean first passage time of a Brownian particle to a small elliptical absorber

TL;DR

This paper derives a high order asymptotic expansion for the mean first passage time of Brownian particles to a small elliptical trap, revealing how trap orientation influences capture efficiency in various 2D domains.

Contribution

It extends existing models by incorporating trap orientation effects into the asymptotic analysis of MFPT, validated through numerical simulations.

Findings

Optimal trap orientation depends on trap position within the domain.

In a disk, radial orientation minimizes MFPT when the trap is near the center.

Angular orientation minimizes MFPT when the trap is farther from the center.

Abstract

We develop a high order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations which confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centered at $0<r<r_c :=\sqrt{2-\sqrt{2}}$. When centered at position $r_c<r<1$, the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green's function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.