Institute for Physics & Astronomy, University of Potsdam, 14476, Potsdam-Golm, Germany.
National Institute of Chemistry, 1000, Ljubljana, Slovenia.
Sci Rep. 2017 Jun 20;7(1):3878. doi: 10.1038/s41598-017-03712-x.
Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.
异常扩散在越来越多的系统中被发现。这种异常扩散的确切性质为研究系统的物理规律提供了重要信息。对于有限粒子运动时间序列的分析，其中一个核心性质是表观扩散率的固有可变性，通常用遍历破坏参数 EB 来量化。在这里，我们证明 EB 通常不足以对数据的可变性提供有意义的度量。相反，由偏度和峰度引入的更高阶矩提供了重要的附加信息。我们对三种流行的异常扩散模型分析了这些量。特别是，我们发现即使对于高斯分数布朗运动，物理测量结果中也会出现显著的偏度，需要考虑到这一点。有趣的是，峰度和偏度也可能为数据所隐含的异常扩散指数提供敏感的估计。我们还为分数布朗运动的 EB 参数推导了一个新的结果，该结果在整个异常扩散参数范围内有效。我们的结果对于异常扩散的分析很重要，但也为异常随机过程的理论提供了新的见解。
