论文标题
三维狭窄捕获问题中目标通量的渐近分析
Asymptotic analysis of target fluxes in the three-dimensional narrow capture problem
论文作者
论文摘要
我们在三维（3D）狭窄的捕获问题中对目标通量进行了渐近分析。后者涉及一个扩散搜索过程，其中目标比搜索域的大小要小得多。小的目标假设使我们能够使用匹配的渐近扩展和Green的功能来求解拉普拉斯空间中的扩散方程。特别是，我们以非二米化目标尺寸$ε$的力量得出了拉普拉斯的渐近膨胀向每个目标转化为每个目标。直接使用通量工作的一个主要优点是，一个人可以生成统计数量，例如分裂概率和有条件的第一通道时间矩，而无需在每种情况下解决单独的边界值问题。但是，为了得出这些数量的渐近扩展，有必要消除格林在限制$ s \ rightarrow 0 $中出现的函数奇异性，其中$ s $是laplace变量。我们通过考虑$ε$，$ s $和$λ\simε/s $的三重扩展来实现这一目标。这使我们能够以$λ$的形式对无限功率系列进行部分求和，从而导致$λ^n/（1+λ）^n $的乘法因子。由于$λ^n/（1+λ）^n \ rightarrow 1 $ as $ s \ rightarrow 0 $，因此消除了$ s $中的奇异性。然后，我们展示如何以小$ $ s $限制得出分裂概率和条件MFPT的相应渐近扩展。最后，我们通过在球形搜索域中考虑一对目标来说明理论，可以明确计算绿色的功能。
We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than the size of the search domain. The small target assumption allows us to use matched asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in powers of the non-dimensionalized target size $ε$. One major advantage of working directly with fluxes is that one can generate statistical quantities such as splitting probabilities and conditional first passage time moments without having to solve a separate boundary value problem in each case. However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate Green's function singularities that arise in the limit $s\rightarrow 0$, where $s$ is the Laplace variable. We achieve this by considering a triple expansion in $ε$, $s$ and $Λ\sim ε/s$. This allows us to perform partial summations over infinite power series in $Λ$, which leads to multiplicative factors of the form $Λ^n/(1+Λ)^n $. Since $Λ^n/(1+Λ)^n \rightarrow 1$ as $s\rightarrow 0$, the singularities in $s$ are eliminated. We then show how corresponding asymptotic expansions of the splitting probabilities and conditional MFPTs can be derived in the small-$s$ limit. Finally, we illustrate the theory by considering a pair of targets in a spherical search domain, for which the Green's functions can be calculated explicitly.
