在吸收性纳米柱森林中的存活
Survival in a nanoforest of absorbing pillars
AI总结：
本文以含单吸收柱的圆柱管近似周期性纳米柱森林，通过模式匹配求解修正 Helmholtz 方程，精确刻画扩散粒子的存活概率与首次通过时间分布。
摘要：
我们研究了一个粒子在两个平行反射平面之间扩散并朝向周期性吸收柱阵列运动时的存活概率。我们用包含单个柱的圆柱管来近似该系统的周期性单胞。利用模式匹配方法，我们得到了该区域内修正 Helmholtz 方程的精确解，该解决定了存活概率的 Laplace 变换以及相关的首次通过时间分布。这一解揭示了若干几何参数各自的作用：柱的高度和半径、柱间距离以及约束平面之间的距离。该模型使我们能够探索首次通过时间概率密度中的不同渐近区域。在约束平面间距离较大这一具有实际意义的情形中，我们论证了平均首次通过时间远大于典型时间，因而不具信息量。我们还阐明了电容近似在 Laplace 算子主特征值上的失效。文中讨论了若干实际意义和未来展望。
英文摘要：
We investigate the survival probability of a particle diffusing between two parallel reflecting planes toward a periodic array of absorbing pillars. We approximate the periodic cell of this system by a cylindrical tube containing a single pillar. Using a mode matching method, we obtain an exact solution of the modified Helmholtz equation in this domain that determines the Laplace transform of the survival probability and the associated distribution of first-passage times. This solution reveals the respective roles of several geometric parameters: the height and radius of the pillar, the inter-pillar distance, and the distance between confining planes. This model allows us to explore different asymptotic regimes in the probability density of the first-passage time. In the practically relevant case of a large distance between confining planes, we argue that the mean first-passage time is much larger than the typical time and thus uninformative. We also illustrate the failure of the capacitance approximation for the principal eigenvalue of the Laplace operator. Some practical implications and future perspectives are discussed.
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