Single-trajectory spectral analysis of scaled Brownian motion

Abstract

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  ¥. 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  ¥ 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 singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.

Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

“…The scaled Brownian motion (SBM) [71], admits a variable diffusion coefficient in time, i.e., D(t), which is considerable to modeling anomalous diffusion phenomena. Currently, the SBM has been applied in different methods of statistical physics, such as renewal resetting [72], ultra-slow diffusion [73], weak and non-ergodic systems [74,75] and spectral analyses of single trajectory [76]. Here, we consider a stochastic and deterministic diffusion coefficient in time as follows.…”

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.…”

Aging power spectrum of membrane protein transport and other subordinated random walks

“…The main interest is in the study of the diffusive motion of individual molecules, taking advantage of developments in single-particle tracking [25][26][27][28][29]. With the goal of understanding the properties of individual trajectories, the PSD of stochastic processes gained renewed interest, for example with respect to the fluctuations of PSD estimator from a single trajectory [12,[30][31][32][33][34].…”