Abstract
The power spectral density (PSD) of any time-dependent stochastic process Xt is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.
Highlights
The power spectral density (PSD) of a stochastic process Xt, which is formally defined as μS (f, ∞) = lim T →∞ T E eiftXtdt, (1)provides important insights into the spectral content of Xt
The power spectral density (PSD) of any time-dependent stochastic processes Xt is a meaningful feature of its spectral content
A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T
Summary
The power spectral density (PSD) of a stochastic process Xt, which is formally defined as (see, e.g., Ref.[1]) μS (f, ∞) = lim T →∞ T E eiftXtdt , (1). Often the PSD as defined in Eq (1) has the form μS(f, ∞) = A/f β, where A is an amplitude and β, (typically, one has 0 < β ≤ 2 [2]), is the exponent characteristic of the statistical properties of Xt. In experiments and numerical modeling, the PSD has been determined using a periodogram estimate for a wide variety of systems in physics, biophysics, geology etc. The PSD has been analysed, as well, for the trajectories of tracers in artificial crowded fluids [11], for active micro-rheology of colloidal suspensions [12], Kardar-Parisi-Zhang interface fluctuations [13], for sequences of earthquakes [14], weather data [15,16,17], biological evolution [18], human cognition [19], network traffic [20] and even for the loudness of music recordings (see, e.g., Refs. [21, 22])
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