Spectral content of a single non-Brownian trajectory

Abstract

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 $S(f,T)\sim A/f^{2H+1}$ with random-amplitude $A$. In sharp contrast, for superdiffusion $(H>1/2)$ $S(f,T)\sim B{T}^{2H-1}/f^2$ 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.

13 pages, 5 figures, Supplemental Material can be found at https://journals.aps.org/prx/supplemental/10.1103/PhysRevX.9.011019/prx_SM_final.pdf

References in corpus (5)

• Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
• The Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork
• Anomalous Diffusion of Self-Propelled Particles in Directed Random Environments
• Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model
• Superdiffusion dominates intracellular particle motion in the supercrowded space of pathogenic Acanthamoeba castellanii

Cited by in corpus (33)

• Measurement of Anomalous Diffusion Using Recurrent Neural Networks
• Bayesian deep learning for error estimation in the analysis of anomalous diffusion
• Full distribution of first exit times in the narrow escape problem
• Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries
• Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories
• Aging power spectrum of membrane protein transport and other subordinated random walks
• Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems
• Inferring pointwise diffusion properties of single trajectories with deep learning
• Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions
• Codifference can detect ergodicity breaking and non-Gaussianity
• Modelling intermittent anomalous diffusion with switching fractional Brownian motion
• Non-Gaussian displacement distributions in models of heterogeneous active particle dynamics
• Gramian Angular Fields for leveraging pretrained computer vision models with anomalous diffusion trajectories
• Variational inference of fractional Brownian motion with linear computational complexity
• Multifractional Brownian motion with telegraphic, stochastically varying exponent
• Path integrals for fractional Brownian motion and fractional Gaussian noise
• Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift
• Preface: Characterisation of Physical Processes from Anomalous Diffusion Data
• Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
• Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates
• First-passage area distribution and optimal fluctuations of fractional Brownian motion
• Estimation of the lowest limit of 1/f noise in semiconductor materials
• Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes
• Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes
• Efficient recurrent neural network methods for anomalously diffusing single particle short and noisy trajectories
• Two-dimensional Brownian motion with dependent components: turning angle analysis
• Unified approach to reset processes and application to coupling between process and reset
• Ergodic properties of functionals of Gaussian processes
• Recurrent neural network analysis of single trajectories switching between anomalous diffusion states
• Evaluating Gaussianity of heterogeneous fractional Brownian motion
• Classification of anomalous diffusion in animal movement data using power spectral analysis
• Scaling theory for the $1/f$ noise
• Two-dimensional fractional Brownian motion: Analysis in time and frequency domains