Big Jump Principle in Physical Models and Data Sets: From Levy Walk to Porous Media

该项目研究物理系统中的大跳跃原理，以及如何利用它来处理极端事件，特别是该项目研究污染传播中的损害减少。在简化的假设下，该原理是应用数学极值理论中一个众所周知的概念。它涉及一组独立的同分布的随机变量的和和最大值，这些随机变量是厚尾的。该原理指出，当总和变大时，它与最大值成正比。因此，一个单一的随机变量，即“大跳跃”，控制了总和的统计量。根据系统的不同，随机变量的总和代表不同的物理量，例如，在随机漫步中，步长加起来就是粒子的位置。然而，在大多数物理系统中，随机变量是相关的，因此刚才描述的大跳跃原理是无效的。该项目由两个相连的步骤组成。首先，提出了关联系统的极值理论。这为描述不同物理模型（如弹道Levy步行）中的大跳跃原理提供了必要的工具。其次，与魏茨曼科学研究所Brian Berkowitz教授的实验水文学小组合作，研究了多孔介质中污染扩散的大跳跃原理。在这次合作中，该项目进一步关注大跳跃原则如何帮助改善污染传播的风险管理。这里的大跳跃原理是指污染颗粒在某些热点（如死角孔）的最大捕获时间导致在某些边界（如地下水位）的突破时间延长。该项目研究了一个很有前景的减少伤害的想法：移除大跳跃。这可以通过在多孔介质中切除一个关键的死角孔来实现，也可以通过减去理论模型中一组随机变量的最大值来实现。去除大跳变后，长时间污染的极端事件的统计特性发生了显著变化。污染物被更快地冲走，环境健康得到恢复。另一个目标是开发数据分析工具，这些工具稍后将应用于降水等数据集。在这里，该项目是与马克斯普朗克复杂系统物理研究所的Holger Kantz教授的时间序列分析小组合作进行的。

The project studies the big jump principle in physical systems and how it can be used to deal with extreme events, in particular the project studies damage reduction in contamination spreading. Under simplified assumptions the principle is a well-known concept in extreme value theory which is part of applied mathematics. It relates the sum and the maximum of a set of independent and identically distributed random variables which are fat-tailed. The principle states that when the sum becomes large it is proportional to the maximum. Thus, a single random variable, the "big jump", controls the statistics of the sum. Depending on the system the random variables add up to sums which represent different physical quantities, e.g. in random walks the step sizes add up to the particle position. However, in most physical systems the random variables are correlated and therefore the just described big jump principle is not valid. The project consists of two connected steps. First, extreme value theory is advanced for correlated systems. This provides necessary tools to describe the big jump principle in different physical models such as the ballistic Levy walk. Secondly, the big jump principle in the context of contamination spreading in porous media is studied in collaboration with the experimental hydrology group of Prof. Brian Berkowitz of the Weizmann Institute of Science. Within this collaboration the project further focuses on how the big jump principle helps to improve risk management in contamination spreading. The big jump principle here means that the maximum trapping time of the contamination particles in some hot spot like a dead-end pore leads to extended breakthrough times at some boundary like the groundwater level. The project investigates a promising idea for damage reduction: the removal of the big jump. This can either be realized by cutting out a critical dead-end pore in the porous medium or by subtraction of the maximum in a set of random variables for a theoretical model. The statistical properties of the extreme events of long periods of contamination change dramatically upon removal of the big jump. The contaminant is washed out much quicker and the environmental health is restored. An additional objective is the development of data analysis tools which are later applied on data sets such as precipitation. Here the project is in collaboration with the time series analysis group of Prof. Holger Kantz of the Max Planck Institute for the Physics of Complex Systems.