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

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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.