Mattos Thiago G, Mejía-Monasterio Carlos, Metzler Ralf, Oshanin Gleb
Max Planck Institute for Intelligent Systems, Heisenbergstrasse 3, 70569 Stuttgart, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Sep;86(3 Pt 1):031143. doi: 10.1103/PhysRevE.86.031143. Epub 2012 Sep 27.
We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ(1)/(τ(1)+τ(2)), which is a measure for how similar the first passage times τ(1) and τ(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.
我们研究了在具有不同形状以及吸附和反射边界配置的有界二维区域中，布朗运动到达吸附边界的首次通过统计量。通过广泛的数值分析，我们得到了随机变量ω=τ(1)/(τ(1)+τ(2))的概率P(ω)分布，其中τ(1)和τ(2)是从同一位置开始的布朗运动的两次独立实现的首次通过时间，ω衡量了它们的相似程度。我们为每个区域构建了一个图表，以确定P(ω)是呈现单峰、钟形形式，还是双峰、M形行为。在前一种情况下，平均首次通过时间（MFPT）是首次通过行为的有效特征，而在后一种情况下，它对于该过程来说是一个不充分的度量。引人注目的是，我们发现P(ω)的两种模式之间存在明显的转变，这对于区域形状以及吸附和反射边界的各自位置具有特征性。我们的结果表明，在二维区域中，首次通过时间可能经常出现大幅波动，这使得即使在所有首次通过分布的矩都存在的有界区域中，将MFPT作为实际行为的稳健度量的普遍应用也变得相当模糊。
