Family of closed-form solutions for two-dimensional correlated diffusion processes.
Family of closed-form solutions for two-dimensional correlated diffusion processes.
复制标题
二维相关扩散过程的封闭式解系列。
DOI：
--
复制
发表时间：
2019
影响因子：
2.4
通讯作者：
Jan Drugowitsch
中科院分区：
文献类型：
--
作者：
H. Shan;R. Moreno;Jan Drugowitsch
带边界的扩散过程是输运现象的模型，在许多领域具有广泛的适用性。这些过程由它们的概率密度函数（PDF）描述，其通常服从福克-普朗克方程（FPE）。虽然在没有边界的情况下获得解析解通常是可能的，但是一旦存在吸收边界，获得FPE的封闭形式解就更具挑战性。因此，对这些过程的分析在很大程度上依赖于近似或直接模拟。在本文中，我们研究了二维的，时间均匀的，空间相关的线性扩散，轴对齐，吸收边界。我们的主要结果是显式构造的一个完整的家庭的封闭形式的解决方案，他们的PDF使用图像的方法。我们发现，当且仅当两个扩散过程之间的相关系数ρ取一组任意值时，这样的解才能成立。利用几何论证，我们导出了能找到这类解的ρ的完备集，可解的ρ由ρ=-cos（π/k）给出，其中k∈Z^{+}<${+∞}.在模拟中验证了解决方案。过程的定性行为似乎在ρ上平滑地变化，允许从我们的解外推到ρ不可解的情况。
Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs). While obtaining analytical solutions is often possible in the absence of boundaries, obtaining closed-form solutions to the FPE is more challenging once absorbing boundaries are present. As a result, analyses of these processes have largely relied on approximations or direct simulations. In this paper, we studied two-dimensional, time-homogeneous, spatially correlated diffusion with linear, axis-aligned, absorbing boundaries. Our main result is the explicit construction of a full family of closed-form solutions for their PDFs using the method of images. We found that such solutions can be built if and only if the correlation coefficient ρ between the two diffusing processes takes one of a numerable set of values. Using a geometric argument, we derived the complete set of ρ's where such solutions can be found. Solvable ρ's are given by ρ=-cos(π/k), where k∈Z^{+}∪{+∞}. Solutions were validated in simulations. Qualitative behaviors of the process appear to vary smoothly over ρ, allowing extrapolation from our solutions to cases with unsolvable ρ's.