Asymptotic analysis of the 2D narrow-capture problem for partially accessible targets

Abstract

In this paper we use singular perturbation theory to solve the 2D narrow capture problem for a set of partially accessible targets $\calU_k$, $k=1,\ldots,N$, in a bounded domain $\Omega\subset \R^2$. In contrast to previous models of narrow capture, we assume that when a searcher finds a target by attaching to the partially adsorbing surface $\partial \calU_k$ it does not have immediate access to the resources within the target interior. Instead, the searcher remains attached to the surface for a random waiting time $\tau$, after which it either gains access to the resources within ({\em surface absorption}) or detaches and continues its search process ({\em surface desorption)}. We also consider two distinct desorption scenarios -- either the particle continues its search from the point of desorption or rapidly returns to its initial search position. We formulate the narrow capture problem in terms of a set of renewal equations that relate the probability density and target flux densities for absorption to the corresponding quantities for irreversible adsorption. The renewal equations, which effectively sew together successive rounds of adsorption and desorption prior to the final absorption event, provide a general probabilistic framework for incorporating non-Markovian models of desorption/absorption and different search scenarios following desorption. We solve the general renewal equations in two stages. First, we calculate the Laplace transformed target fluxes for irreversible adsorption by solving a Robbin boundary value problem (BVP) in the small-target limit using matched asymptotic analysis. We then use the inner solution of the BVP to solve the corresponding Laplace transformed renewal equations for non-Markovian desorption/absorption, which leads to explicit Neumann series expansions of the corresponding target fluxes.