34 research outputs found
Random time averaged diffusivities for L\'evy walks
We investigate a L\'evy-Walk alternating between velocities with
opposite sign. The sojourn time probability distribution at large times is a
power law lacking its mean or second moment. The first case corresponds to a
ballistic regime where the ensemble averaged mean squared displacement (MSD) at
large times is , the latter to enhanced diffusion with , . The correlation function and the time averaged
MSD are calculated. In the ballistic case, the deviations of the time averaged
MSD from a purely ballistic behavior are shown to be distributed according to a
Mittag-Leffler density function. In the enhanced diffusion regime, the
fluctuations of the time averages MSD vanish at large times, yet very slowly.
In both cases we quantify the discrepancy between the time averaged and
ensemble averaged MSDs
Asymptotic front behavior in an reaction under subdiffusion
We discuss the front propagation in the reaction under
subdiffusion which is described by continuous time random walks with a
heavy-tailed power law waiting time probability density function. Using a
crossover argument, we discuss the two scaling regimes of the front
propagation: an intermediate asymptotic regime given by the front solution of
the corresponding continuous equation, and the final asymptotics, which is
fluctuation-dominated and therefore lays out of reach of the continuous scheme.
We moreover show that the continuous reaction subdiffusion equation indeed
possesses a front solution that decelerates and becomes narrow in the course of
time. This continuous description breaks down for larger times when the front
gets atomically sharp. We show that the velocity of such fronts decays in time
faster than in the continuous regime
Front propagation in A+B -> 2A reaction under subdiffusion
We consider an irreversible autocatalytic conversion reaction A+B -> 2A under
subdiffusion described by continuous time random walks. The reactants'
transformations take place independently on their motion and are described by
constant rates. The analog of this reaction in the case of normal diffusion is
described by the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) equation leading
to the existence of a nonzero minimal front propagation velocity which is
really attained by the front in its stable motion. We show that for
subdiffusion this minimal propagation velocity is zero, which suggests
propagation failure
Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking
Anomalous diffusion has been widely observed by single particle tracking
microscopy in complex systems such as biological cells. The resulting time
series are usually evaluated in terms of time averages. Often anomalous
diffusion is connected with non-ergodic behaviour. In such cases the time
averages remain random variables and hence irreproducible. Here we present a
detailed analysis of the time averaged mean squared displacement for systems
governed by anomalous diffusion, considering both unconfined and restricted
(corralled) motion. We discuss the behaviour of the time averaged mean squared
displacement for two prominent stochastic processes, namely, continuous time
random walks and fractional Brownian motion. We also study the distribution of
the time averaged mean squared displacement around its ensemble mean, and show
that this distribution preserves typical process characteristic even for short
time series. Recently, velocity correlation functions were suggested to
distinguish between these processes. We here present analytucal expressions for
the velocity correlation functions. Knowledge of the results presented here are
expected to be relevant for the correct interpretation of single particle
trajectory data in complex systems.Comment: 15 pages, 15 figures; References adde
Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method
In this work we present a new numerical method for the solution of the distributed order time fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed
Asymptotic densities of ballistic Lévy walks
We propose an analytical method to determine the shape of density profiles in
the asymptotic long time limit for a broad class of coupled continuous time
random walks which operate in the ballistic regime. In particular, we show that
different scenarios of performing a random walk step, via making an
instantaneous jump penalized by a proper waiting time or via moving with a
constant speed, dramatically effect the corresponding propagators, despite the
fact that the end points of the steps are identical. Furthermore, if the speed
during each step of the random walk is itself a random variable, its
distribution gets clearly reflected in the asymptotic density of random
walkers. These features are in contrast with more standard non-ballistic random
walks