Spectral content of a single non-Brownian trajectory
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Time-dependent processes are often analysed 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 data sets 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 (fBm) 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)∼ A/f^2H+1 with random-amplitude A. In sharp contrast, for superdiffusion (H>1/2) S(f,T)∼ BT^2H-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.
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