Asymptotic analysis of extended two-dimensional narrow capture problems

TL;DR

This paper develops an asymptotic analysis for 2D narrow capture problems, providing new insights into capture times, probabilities, and optimal search strategies in complex diffusive systems with small targets.

Contribution

It introduces a novel asymptotic expansion in $ u=-1/ ext{ln} \, ext{epsilon}$ for 2D narrow capture problems, enabling efficient analysis of capture probabilities and moments.

Findings

Derived asymptotic expansions for splitting probabilities and FPT moments.

Provided new results for stochastic resetting in search strategies.

Analyzed resource accumulation in multiple search-and-capture rounds.

Abstract

In this paper we extend our recent work on two-dimensional (2D) diffusive search-and-capture processes with multiple small targets (narrow capture problems) by considering an asymptotic expansion of the Laplace transformed probability flux into each target. The latter determines the distribution of arrival or capture times into an individual target, conditioned on the set of events that result in capture by that target. A characteristic feature of strongly localized perturbations in 2D is that matched asymptotics generates a series expansion in $\nu=-1/\ln \epsilon$ rather than $\epsilon$, $0<\epsilon\ll 1$, where $\epsilon$ specifies the size of each target relative to the size of the search domain. Moreover, it is possible to sum over all logarithmic terms non-perturbatively. We exploit this fact to show how a Taylor expansion in the Laplace variable $s$ for fixed $\nu$ provides an efficient method for obtaining corresponding asymptotic expansions of the splitting probabilities and moments of the conditional FPT densities. We then use our asymptotic analysis to derive new results for two major extensions of the classical narrow capture problem: optimal search strategies under stochastic resetting, and the accumulation of target resources under multiple rounds of search-and-capture.