The narrow capture problem with partially absorbing targets and stochastic resetting

TL;DR

This paper analyzes the mean first-passage time for a diffusing particle with stochastic resetting in a bounded domain containing small, partially absorbing targets, using asymptotic methods to determine how resetting and reaction rates influence absorption times.

Contribution

It develops an asymptotic framework to quantify the flux into small, partially absorbing targets under stochastic resetting, incorporating the effects of reaction and resetting rates.

Findings

MFPT is often unimodal in resetting rate, with an optimal value.

Asymptotic expansions for fluxes depend on target size and absorption rate.

The model provides insights into optimizing search strategies with resetting.

Abstract

We consider a particle undergoing diffusion with stochastic resetting in a bounded domain $\calU\subset \R^d$ for $d=2,3$. The domain is perforated by a set of partially absorbing targets within which the particle may be absorbed at a rate $\kappa$. Each target is assumed to be much smaller than $|\calU|$, which allows us to use asymptotic and Green's function methods to solve the diffusion equation in Laplace space. In particular, we construct an inner solution within the interior and local exterior of each target, and match it with an outer solution in the bulk of $\calU$. This yields an asymptotic expansion of the Laplace transformed flux into each target in powers of $\nu=-1/\ln \epsilon$ ($d=2$) and $\epsilon$ ($d=3$), respectively, where $\epsilon$ is the non-dimensionalized target size. The fluxes determine how the mean first-passage time to absorption depends on the reaction rate $\kappa$ and the resetting rate $r$. For a range of parameter values, the MFPT is a unimodal function of $r$, with a minimum at an optimal resetting rate $r_{\rm opt}$ that depends on $\kappa$ and the target configuration.