62 research outputs found
Power spectral density of a single Brownian trajectory: what one can and cannot learn from it
The power spectral density (PSD) of any time-dependent stochastic processXt is ameaningful 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 timeT, that is, it is defined as an ensemble-averaged property taken
in the limitT ¥.Alegitimate question iswhat information on the PSDcan be reliably obtained from
single-trajectory experiments, if one goes beyond the standard definition and analyzes thePSD of a single
trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensionalBrownian
motion (BM) we calculate the probability density function of a single-trajectory PSDfor arbitrary
frequency f, finite observation timeTand arbitrary number k of projections of the trajectory on different
axes.We show analytically that the scaling exponent for the frequency-dependence of the PSDspecific to
an ensemble ofBMtrajectories can be already obtained from a single trajectory, while the numerical
amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating
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 truncatedWiener
representation ofBM, and the case of a discrete-time lattice randomwalk.Wehighlight some differences in
the behavior of a single-trajectory PSDforBMand for the two latter situations.The framework developed
herein will allow formeaningful physical analysis of experimental stochastic trajectories
1/f noise for intermittent quantum dots exhibits non-stationarity and critical exponents
The power spectrum of quantum dot fluorescence exhibits noise,
related to the intermittency of these nanosystems. As in other systems
exhibiting noise, this power spectrum is not integrable at low
frequencies, which appears to imply infinite total power. We report
measurements of individual quantum dots that address this long-standing
paradox. We find that the level of noise decays with the
observation time. The change of the spectrum with time places a bound on the
total power. These observations are in stark contrast with most measurements of
noise in macroscopic systems which do not exhibit any evidence for
non-stationarity. We show that the traditional description of the power
spectrum with a single exponent is incomplete and three additional
critical exponents characterize the dependence on experimental time.Comment: 16 pages, 4 figure
Modelling intermittent anomalous diffusion with switching fractional Brownian motion
The stochastic trajectories of molecules in living cells, as well as the
dynamics in many other complex systems, often exhibit memory in their path over
long periods of time. In addition, these systems can show dynamic
heterogeneities due to which the motion changes along the trajectories. Such
effects manifest themselves as spatiotemporal correlations. Despite the broad
occurrence of heterogeneous complex systems in nature, their analysis is still
quite poorly understood and tools to model them are largely missing. We
contribute to tackling this problem by employing an integral representation of
Mandelbrot's fractional Brownian motion that is compliant with varying motion
parameters while maintaining long memory. Two types of switching fractional
Brownian motion are analysed, with transitions arising from a Markovian
stochastic process and scale-free intermittent processes. We obtain simple
formulas for classical statistics of the processes, namely the mean squared
displacement and the power spectral density. Further, a method to identify
switching fractional Brownian motion based on the distribution of displacements
is described. A validation of the model is given for experimental measurements
of the motion of quantum dots in the cytoplasm of live mammalian cells that
were obtained by single-particle tracking
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