具有Lévy行走和布朗运动的两态老化过程
Aging two-state process with Lévy walk and Brownian motion
摘要：
随着单态过程动力学研究的不断深入，双态过程由于在复杂系统中的广泛应用以及在动物觅食行为等诸多领域的有效应用，越来越受到人们的关注。该报告建立了两个状态的过程的理论基础：Lévy行走和布朗运动，已被证明是一个有效的间歇搜索过程。假设两个态的逗留时间分布都是指数为$α_\pm\in(0,2)$的重尾分布。通过分析弱老化和强老化情况下的整体平均和时间平均均方位移，得到了该两态过程的动力学行为。发现$α_\pm$的大小关系在很长一段时间内决定了两个态的分数，在这些MSD中起着至关重要的作用。根据MSDs的一般表达式，发现了两态过程的一些内在特征。在六种不同的情况下，详细地介绍了分数对这些可观量的影响。得到这些结果的关键是计算两态过程的速度关联函数，其方法可以推广到其它多态过程。
英文摘要：
With the rich dynamics studies of single-state processes, the two-state processes attract more and more interests of people, since they are widely observed in complex system and have effective applications in diverse fields, say, foraging behavior of animals. This report builds the theoretical foundation of the process with two states: Lévy walk and Brownian motion, having been proved to be an efficient intermittent search process. The sojourn time distributions in two states are both assumed to be heavy-tailed with exponents $α_\pm\in(0,2)$. The dynamical behaviors of this two-state process are obtained through analyzing the ensemble-averaged and time-averaged mean squared displacements (MSDs) in weak and strong aging cases. It is discovered that the magnitude relationship of $α_\pm$ decides the fraction of two states for long times, playing a crucial role in these MSDs. According to the generic expressions of MSDs, some inherent characteristics of the two-state process are detected. The effects of the fraction on these observables are detailedly presented in six different cases. The key of getting these results is to calculate the velocity correlation function of the two-state process, the techniques of which can be generalized to other multi-state processes.
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