130 research outputs found
Spectral content of a single non-Brownian trajectory
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 , the PSD of a
single trajectory. Here we focus on the behavior of this random,
realization-dependent variable, parametrized by frequency and
observation-time , for a broad family of anomalous diffusions---fractional
Brownian motion (fBm) with Hurst-index ---and derive exactly its probability
density function. We show that is proportional---up to a random
numerical factor whose universal distribution we determine---to the
ensemble-averaged PSD. For subdiffusion () we find that with random-amplitude . In sharp contrast, for superdiffusion
with random amplitude . Remarkably, for
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 the PSD is ageing and is dependent on
. Our predictions for both sub- and superdiffusion are confirmed by
experiments in live cells and in agarose hydrogels, and by extensive
simulations.Comment: 13 pages, 5 figures, Supplemental Material can be found at
https://journals.aps.org/prx/supplemental/10.1103/PhysRevX.9.011019/prx_SM_final.pd
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