在非均匀介质中用多个小陷阱最小化捕获时间

中文摘要 AI 辅助

我们研究了在具有空间依赖性扩散系数$D(x)$和给定初始粒子分布$\omega(x)$的非均匀介质中，最优放置大量小吸收陷阱以最小化粒子平均首次通过时间（MFPT）的问题。在二维和三维中，我们根据陷阱系综消耗粒子的强度识别出两种不同的机制。在弱捕获机制（陷阱较少）中，最优陷阱密度$\rho$是归一化逆扩散系数$\mu \propto 1/D$和初始粒子分布$\omega$的算术平均值，即$\rho = \frac12(\mu+\omega)$。在强捕获机制（陷阱较多）中，最优陷阱密度与其几何平均值成正比，即$\rho \propto \sqrt{\mu \omega}$。我们重新审视了经典的格林函数方法，并表明它仅适用于弱捕获；在二维中，这要求陷阱尺寸在陷阱数量上呈指数级小，因此大多数感兴趣的应用反而属于强捕获机制。因此，我们开发了一种能够同时捕获两种机制的均匀化方法。初始粒子分布的一个特殊选择是$\omega \propto 1/D$，这对应于根据Itô解释在无陷阱情况下的平衡粒子分布。在这种情况下，我们表明最优陷阱分布本身就是平稳粒子分布，无论捕获强度如何，并且这使得MFPT在整个域内保持恒定；相同的陷阱分布也最小化最坏情况下的捕获时间。最后，在一维中，我们发现$\rho\propto\left(\mu \omega\right)^{1/3}$，类似的分析涵盖了空间依赖漂移和变截面薄域。直接数值优化证实了分析结果。

英文摘要

We study the problem of optimally placing a large number of small absorbing traps to minimize the mean first-passage time (MFPT) of particles diffusing in a heterogeneous medium with space-dependent diffusivity $D(x)$ and prescribed initial particle distribution $ω(x)$. In two and three dimensions we identify two distinct regimes, depending on how strongly the trap ensemble depletes the particles. In the weak trapping regime (fewer traps) the optimal trap density $ρ$ is the arithmetic average $ρ= \frac12(μ+ω)$ of the normalized inverse diffusivity $μ\propto 1/D$ and the initial particle distribution $ω$. In the strong trapping regime (more traps), it is proportional to their geometric average, $ρ\propto \sqrt{μω}$. We revisit the classical Green's function approach and show that it applies to weak trapping only; in 2D this requires the trap size to be exponentially small in the number of traps, so that most applications of interest fall in the strong trapping regime instead. We therefore develop a homogenization approach that captures both regimes. A special choice of initial particle distribution is $ω\propto 1/D$, which corresponds to the equilibrium particle distribution in the absence of traps according to the Itô interpretation. In this case we show that the optimal trap distribution is the stationary particle distribution itself, regardless of the trapping strength, and this makes the MFPT constant throughout the domain; the same trap distribution also minimizes the worst-case capture time. Finally, in one dimension we find instead $ρ\propto\left(μω\right) ^{1/3}$, and a similar analysis covers space-dependent drift and thin domains of variable cross-section. Direct numerical optimization confirms the analytical results.

发表机构

• Laboratoire de Physique de la Matière Condensée, CNRS – Ecole Polytechnique, Institut Polytechnique de Paris（凝聚态物质物理实验室，法国国家科学研究中心-巴黎综合理工学院，巴黎理工学院）
• Dalhousie University（达尔豪斯大学）

机构由 AI 辅助整理，请以论文原文为准。