Abstract
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 , 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 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.
- Received 29 September 2018
- Revised 22 December 2018
DOI:https://doi.org/10.1103/PhysRevX.9.011019
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
With single-molecule localization microscopy, which relies on a variety of techniques to precisely pinpoint the location of single particles, scientists can record individual particle trajectories across impressive distances. However, it is often challenging to reliably evaluate the physical information encoded in such data and to extract relevant parameters. Here, we present a simple yet powerful approach to retrieve that information with broad relevance to applications and theory.
We analyze the power spectral density of a single realization of fractional Brownian motion—one of the most widespread stochastic processes that can be subdiffusive, diffusive, or superdiffusive. We calculate exactly the full distribution of such a power spectral density and make several rigorous statements about the spectral content of a single trajectory. We confirm our theoretical predictions with simulations as well as experiments in live cells and agarose hydrogels. Furthermore, we offer a criterion that permits us to prove the anomalous character of random motion in situations where analysis of the mean-squared displacement leads to ambiguous conclusions.
This research paves the way for future experimental, theoretical, and numerical work dealing with the Fourier analysis of the broad array of stochastic processes encountered in biological and other complex systems and the exact influence of measurement-induced noise.