Spectral Content of a Single Non-Brownian Trajectory
Spectral Content of a Single Non-Brownian Trajectory
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DOI：
10.1103/physrevx.9.011019
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发表时间：
2019-01-31
影响因子：
12.5
通讯作者：
Xu, Xinran
中科院分区：
文献类型：
--
作者：
Krapf, Diego;Lukat, Nils;Xu, Xinran
时间相关过程通常使用功率谱密度（PSD）进行分析，功率谱密度通过对单个轨迹进行适当的傅里叶变换并找到相关的集合平均值来计算。通常情况下，可用的实验数据集对于这样的集合平均值来说太小，因此，理解在何种程度上可以从S（f，T）（单个轨迹的PSD）获得相关信息具有重要的概念和实际意义。在这里，我们专注于这个随机的行为，实现依赖于变量参数化的频率f和观察时间T，为一个广泛的家庭的异常扩散分数布朗运动与赫斯特指数H，并准确地推导出它的概率密度函数。我们表明，S（f，T）是成比例的随机数值因子，其普遍分布，我们确定的整体平均PSD。对于亚扩散（H < 1/2），我们发现S（f，T）类似于A/f（2 H +1），具有随机振幅A.与此形成鲜明对比的是，对于超扩散（H > 1/2），S（f，T）类似于具有随机振幅B的BT 2 H-1/f（2）。值得注意的是，当H > 1/2时，PSD表现出与布朗运动相同的频率依赖性，这是一种欺骗性的性质，在解释实验数据时可能导致错误的结论。值得注意的是，对于H > 1/2，PSD是老化的并且依赖于T。在活细胞和琼脂糖水凝胶中的实验以及广泛的模拟证实了我们对亚扩散和超扩散的预测。
Time-dependent processes are often analyzed using the power spectral density (PSD) calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble average. Frequently, the available experimental datasets are too small for such ensemble averages, and hence, it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from S(f, T), the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable parametrized by frequency f and observation time T, for a broad family of anomalous diffusions-fractional Brownian motion with Hurst index H-and derive exactly its probability density function. We show that S(f, T) is proportional-up to a random numerical factor whose universal distribution we determine-to the ensemble-averaged PSD. For subdiffusion (H < 1/2), we find that S(f, T) similar to A/f(2H+1) with random amplitude A. In sharp contrast, for superdiffusion (H > 1/2) S(f, T) similar to BT2H-1/f(2) with random amplitude B. Remarkably, for H > 1/2 the PSD exhibits the same frequency dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for H > 1/2 the PSD is ageing and is dependent on T. Our predictions for both sub-and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels and by extensive simulations.