标题： 二维狭窄捕获问题中部分可访问目标的渐近分析 显示英文标题

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

摘要： 本文中，我们利用奇异摄动理论解决二维空间中一组部分可访问的目标 $\calU_k$, $k=1,\ldots,N$ 在有界域 $\Omega\subset \R^2$ 内的狭窄捕获问题。与之前的狭窄捕获模型不同，我们假设当搜索者通过附着到部分吸附表面 $\partial \calU_k$ 找到目标时，并不能立即获得目标内部资源的访问权。相反，搜索者会停留在表面上一段随机等待时间 $\tau$，之后它要么进入目标内部获取资源（{\em 表面吸收}），要么脱离表面继续搜索过程（{\em 表面脱附)}）。此外，我们还考虑了两种不同的解吸情景——粒子要么从解吸点继续搜索，要么迅速返回初始搜索位置。 我们将狭窄捕获问题表述为一组更新方程，这些方程将吸收的概率密度和目标通量密度与不可逆吸附的相关量联系起来。这些更新方程有效地将吸附和解吸的连续轮次连接起来，在最终吸收事件之前提供了一个通用的概率框架，用于整合非马尔可夫解吸/吸收模型以及解吸后的不同搜索场景。 我们分两阶段求解一般的更新方程。首先，我们通过匹配渐近分析法，在小目标极限下，求解罗宾边值问题（BVP）以计算不可逆吸附的目标通量的拉普拉斯变换。然后，我们使用BVP的内解来求解相应的非马尔可夫解吸/吸收的拉普拉斯变换更新方程，从而得到对应目标通量的显式内曼级数展开式。 显示英文摘要

摘要： 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.