Journal
NEW JOURNAL OF PHYSICS
Volume 21, Issue -, Pages -Publisher
IOP Publishing Ltd
DOI: 10.1088/1367-2630/ab2f52
Keywords
diffusion; anomalous diffusion; power spectral analysis; single trajectory analysis
Categories
Funding
- Deutsche Forschungsgemeinschaft (DFG) [ME1535/6-1, ME1535/7-1]
- Alexander von Humboldt Polish Honorary Research Scholarship from the Polish Science Foundation
- Deutsche Forschungsgemeinschaft (German Research Foundation)
- Potsdam University
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A standard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T -> infinity. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T -> infinity can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion. We demonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent. We also compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.
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