TY  - JOUR
A1  - Krapf, Diego
A1  - Marinari, Enzo
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
A1  - Xu, Xinran
A1  - Squarcini, Alessio
T1  - Power spectral density of a single Brownian trajectory
BT  - what one can and cannot learn from it
JF  - New journal of physics : the open-access journal for physics
N2  - The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful 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 timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate 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 timeT. 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 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 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.
KW  - power spectral density
KW  - single-trajectory analysis
KW  - probability density function
KW  - exact results
Y1  - 2018
U6  - https://doi.org/10.1088/1367-2630/aaa67c
SN  - 1367-2630
VL  - 20
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Sandev, Trifce
A1  - Iomin, Alexander
A1  - Kantz, Holger
A1  - Metzler, Ralf
A1  - Chechkin, Aleksei V.
T1  - Comb Model with Slow and Ultraslow Diffusion
JF  - Mathematical modelling of natural phenomena
N2  - We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.
KW  - comb-like model
KW  - anomalous diffusion
KW  - mean squared displacement
KW  - probability density function
Y1  - 2016
U6  - https://doi.org/10.1051/mmnp/201611302
SN  - 0973-5348
SN  - 1760-6101
VL  - 11
SP  - 18
EP  - 33
PB  - EDP Sciences
CY  - Les Ulis
ER  -