Power spectral density of a single Brownian trajectory: what one can and cannot learn from it
2018
The power spectral density (PSD) of any time-dependent stochastic processes
$X_t$ is a meaningful feature of its spectral content. In its text-book
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 ensemble-averaged property taken in the limit $T \to \infty$.
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 \textit{single} trajectory recorded for a \textit{finite} observation
time $T$. In quest for this answer, for a $d$-dimensional Brownian motion
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 Brownian motion 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 Brownian motion, and the case of a discrete-time
lattice random walk. We highlight some differences in the behavior of a
single-trajectory PSD for Brownian motion and for the two latter situations.
The framework developed herein will allow for meaningful physical analysis
of experimental stochastic trajectories.
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