球对称二维区域中的出口时间分布
Exit time distribution in spherically symmetric two-dimensional domains
摘要：
计算了布朗粒子在球对称二维区域（圆盘、角扇区、环）和边界上有出口的矩形中的出口时间分布。本文给出了Helmholtz型偏微分方程在混合Dirichlet- Neumann边界条件下的解析解.我们提出了一个精确的解决方案依赖于矩阵求逆，和一个近似的显式解决方案。近似的解决方案被证明是精确的出口消失的大小，即使是大出口是准确的。对于角扇区，我们也得到了精确的显式公式的时刻的出口时间。对于环形和矩形，平均出口时间的近似表达式示出是非常准确的，即使对于大出口。该分析还扩展到偏置扩散。由于在微流体，热传播，量子台球和声学中遇到了混合边界条件的亥姆霍兹方程，所开发的方法可以找到许多应用程序以外的出口过程。
英文摘要：
The distribution of exit times is computed for a Brownian particle in spherically symmetric two- dimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet- Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoustics, the developed method can find numerous applications beyond exit processes.
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