Power spectral density of a single Brownian trajectory: what one can and cannot learn from it
Abstract
The power spectral density (PSD) of any timedependent stochastic process X _{t} is a meaningful feature of its spectral content. In its textbook definition, the PSD is the Fourier transform of the covariance function of X _{t} over an infinitely large observation time T, that is, it is defined as an ensembleaveraged property taken in the limit T\to ∞ . A legitimate question is what information on the PSD can be reliably obtained from singletrajectory 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 ddimensional Brownian motion (BM) we calculate the probability density function of a singletrajectory 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 frequencydependence 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 ensembleaveraged and singletrajectory 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 singleparticle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discretetime lattice random walk. We highlight some differences in the behavior of a singletrajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.
 Publication:

New Journal of Physics
 Pub Date:
 February 2018
 DOI:
 10.1088/13672630/aaa67c
 arXiv:
 arXiv:1801.02986
 Bibcode:
 2018NJPh...20b3029K
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematics  Statistics Theory
 EPrint:
 24 pages, 13 figures