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 ) ∼ 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.

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

“…As modern experiments such as single particle tracking routinely produce large amounts of data for the thermally and actively driven motion of test particles, the extraction of physical information from the garnered data becomes ever more important. On the one hand, this is met by the analysis of a growing number of complementary observables such as time averaged MSD [85,86], higher order moments or mean maximal statistics [87], p-variation methods [88], first-passage methods [89], or single trajectory power spectra [8,90]. On the other hand objective methods such as maximum likelihood approaches [91][92][93][94] or machine learning [95] are being recognised as useful tools.…”

Single-trajectory spectral analysis of scaled Brownian motion

“…The PSD is classically calculated by first performing a Fourier transform of an individual trajectory X(t) over the finite observation time T where the angular brackets denote the statistical averaging. In the second line of (2), taking into account that X(t) is real-valued, we took the absolute square and used the summation relation for trigonometric functions [2] to obtain the cosine function with the difference of the two times and the autocorrelation function X t X t 1 2 á ñ ( ) ( ) of the process X(t); see [1,3,4] for more details.…”

Tracer diffusion on a crowded random Manhattan lattice

“…For the former C v approaches for sufficiently large T and f a universal (i.e. regardless of the actual value of H>1/2), time T-independent constant value 2 [72], while for the latter-a universal time T-independent constant value 5 2 [73], the same which is observed for a standard Brownian motion [68].…”

“…Further on, the law μ(T, f )∼T 1/3 /f 2 was observed for other superdiffusive processes, such as a fractional Brownian motion with the Hurst index H=2/3 (i.e. γ=4/3) [72] or a super-diffusive scaled Brownian motion Z t described by the Langevin equation  z = Z t t t 1 6 [73], with ζ t being a Gaussian white-noise with zero mean. This latter process also produces a super-diffusive motion with γ=4/3, suggesting that the law μ(T, f )∼T 1/3 /f 2 might be a generic feature of processes with γ=4/3.…”

“…Therefore, fixing T and focussing only on the fdependence of μ(T, f ) garnered from numerical simulations, one can be led to an erroneous conclusion that the observed process is a Brownian motion. As an actual fact, this is precisely the T-dependence of μ(T, f ) which helps to realise that this is not the case (see also [72]).…”