非均匀圆盘和球体中的扩散：出射时间和均匀化公式的新的封闭形式表达式
Diffusion in heterogeneous discs and spheres: new closed-form expressions for exit times and homogenization formulae
摘要：
扩散传输的数学模型支持我们对化学、生物化学和生物传输现象的理解。这种模型的分析往往集中在相对简单的几何形状和处理通过高度理想化的均匀介质的扩散。相反，扩散输运理论的实际应用不可避免地涉及处理更复杂的几何形状以及处理非均匀介质。扩散输运的最基本性质之一是平均粒子寿命或平均离开时间的概念，其是首次通过时间概念的特定应用，并提供扩散粒子到达吸收边界所需的平均时间。平均粒子寿命的大多数形式分析适用于相对简单的几何形状，通常具有均匀（空间不变）的材料属性。在这项工作中，我们提出了一个一般的框架，提供精确的数学洞察平均粒子寿命，粒子寿命的较高时刻，点粒子扩散在非均匀的磁盘和球体与径向对称。我们的分析适用于具有任意数量和排列的不同层的几何形状，其中每层中的传输的特征在于不同的扩散率。我们获得了在任意位置释放的扩散粒子的平均粒子寿命的精确闭合形式表达式，并且我们概括了这些结果，以针对一系列不同的边界条件给出粒子寿命的任何高阶矩的精确闭合形式表达式。最后，利用这些结果，我们构建了新的均匀化公式，提供了一个准确的简化描述扩散通过异质盘和球。
英文摘要：
Mathematical models of diffusive transport underpin our understanding of chemical, biochemical and biological transport phenomena. Analysis of such models often focusses on relatively simple geometries and deals with diffusion through highly idealised homogeneous media. In contrast, practical applications of diffusive transport theory inevitably involve dealing with more complicated geometries as well as dealing with heterogeneous media. One of the most fundamental properties of diffusive transport is the concept of mean particle lifetime or mean exit time, which are particular applications of the concept of first passage time, and provide the mean time required for a diffusing particle to reach an absorbing boundary. Most formal analysis of mean particle lifetime applies to relatively simple geometries, often with homogeneous (spatially-invariant) material properties. In this work, we present a general framework that provides exact mathematical insight into the mean particle lifetime, and higher moments of particle lifetime, for point particles diffusing in heterogeneous discs and spheres with radial symmetry. Our analysis applies to geometries with an arbitrary number and arrangement of distinct layers, where transport in each layer is characterised by a distinct diffusivity. We obtain exact closed-form expressions for the mean particle lifetime for a diffusing particle released at an arbitrary location and we generalise these results to give exact, closed-form expressions for any higher-order moment of particle lifetime for a range of different boundary conditions. Finally, using these results we construct new homogenization formulae that provide an accurate simplified description of diffusion through heterogeneous discs and spheres.
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