Godec Aljaž, Metzler Ralf
Institute of Physics &Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany.
National Institute of Chemistry, 1000 Ljubljana, Slovenia.
Sci Rep. 2016 Feb 8;6:20349. doi: 10.1038/srep20349.
The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.
首次通过是一个通用概念，用于量化诸如扩散分子的位置或股票价值等随机量首次越过预设阈值（目标）的时间。在过去十年中，出现了一系列具有启发性的新结果，主要集中在这类过程的所谓平均首次通过时间和全局首次通过时间（分别为MFPT和GFPT）上。在这里，我们将对首次通过过程的理解进一步推进。对于一个简单的非均匀系统，我们严格推导了首次通过时间（FPT）的完整分布。我们的结果表明，典型的FPT与MFPT有显著差异，MFPT对应于FPT分布的长时间行为。相反，短时间行为被证明对应于直接从初始值连接到目标的轨迹。值得注意的是，我们揭示了首次通过动力学中一个先前被忽视的第三个特征时间尺度，它反映了短暂偏离目标的情况。
