34 research outputs found
Infinite densities for L\'evy walks
Motion of particles in many systems exhibits a mixture between periods of
random diffusive like events and ballistic like motion. In many cases, such
systems exhibit strong anomalous diffusion, where low order moments with below a critical value exhibit diffusive scaling while for
a ballistic scaling emerges. The mixed dynamics constitutes a
theoretical challenge since it does not fall into a unique category of motion,
e.g., the known diffusion equations and central limit theorems fail to describe
both aspects. In this paper we resolve this problem by resorting to the concept
of infinite density. Using the widely applicable L\'evy walk model, we find a
general expression for the corresponding non-normalized density which is fully
determined by the particles velocity distribution, the anomalous diffusion
exponent and the diffusion coefficient . We explain how
infinite densities play a central role in the description of dynamics of a
large class of physical processes and discuss how they can be evaluated from
experimental or numerical data.Comment: Phys. Rev. E, in pres
Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
Strong anomalous diffusion, where with a nonlinear spectrum \nu(q) \neq \mbox{const}, is wide spread
and has been found in various nonlinear dynamical systems and experiments on
active transport in living cells. Using a stochastic approach we show how this
phenomena is related to infinite covariant densities, i.e., the asymptotic
states of these systems are described by non-normalizable distribution
functions. Our work shows that the concept of infinite covariant densities
plays an important role in the statistical description of open systems
exhibiting multi-fractal anomalous diffusion, as it is complementary to the
central limit theorem.Comment: PRL, in pres
Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
We find a general formula for the distribution of time-averaged observables
for systems modeled according to the sub-diffusive continuous time random walk.
For Gaussian random walks coupled to a thermal bath we recover ergodicity and
Boltzmann's statistics, while for the anomalous subdiffusive case a weakly
non-ergodic statistical mechanical framework is constructed, which is based on
L\'evy's generalized central limit theorem. As an example we calculate the
distribution of : the time average of the position of the particle,
for unbiased and uniformly biased particles, and show that exhibits
large fluctuations compared with the ensemble average .Comment: 5 pages, 2 figure
1/f^beta noise in a model for weak ergodicity breaking
In systems with weak ergodicity breaking, the equivalence of time averages
and ensemble averages is known to be broken. We study here the computation of
the power spectrum from realizations of a specific process exhibiting 1/f^beta
noise, the Rebenshtok-Barkai model. We show that even the binned power spectrum
does not converge in the limit of infinite time, but that instead the resulting
value is a random variable stemming from a distribution with finite variance.
However, due to the strong correlations in neighboring frequency bins of the
spectrum, the exponent beta can be safely estimated by time averages of this
type. Analytical calculations are illustrated by numerical simulations.Comment: 10 pages, 7 figures; extended references and summary, smaller
corrections; final versio
A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
We give a closer look at the Central Limit Theorem (CLT) behavior in
quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one
for long-range-interacting classical many-body systems. We present new
calculations which show that, following their time evolution, we can observe
and classify three kinds of long-standing quasi-stationary states (QSS) with
different correlations. The frequency of occurrence of each class depends on
the size of the system. The different microsocopic nature of the QSS leads to
different dynamical correlations and therefore to different results for the
observed CLT behavior.Comment: 11 pages, 8 figures. Text and figures added, Physica A in pres
On distributions of functionals of anomalous diffusion paths
Functionals of Brownian motion have diverse applications in physics,
mathematics, and other fields. The probability density function (PDF) of
Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger
equation in imaginary time. In recent years there is a growing interest in
particular functionals of non-Brownian motion, or anomalous diffusion, but no
equation existed for their PDF. Here, we derive a fractional generalization of
the Feynman-Kac equation for functionals of anomalous paths based on
sub-diffusive continuous-time random walk. We also derive a backward equation
and a generalization to Levy flights. Solutions are presented for a wide number
of applications including the occupation time in half space and in an interval,
the first passage time, the maximal displacement, and the hitting probability.
We briefly discuss other fractional Schrodinger equations that recently
appeared in the literature.Comment: 25 pages, 4 figure
Weakly non-ergodic Statistical Physics
We find a general formula for the distribution of time averaged observables
for weakly non-ergodic systems. Such type of ergodicity breaking is known to
describe certain systems which exhibit anomalous fluctuations, e.g. blinking
quantum dots and the sub-diffusive continuous time random walk model. When the
fluctuations become normal we recover usual ergodic statistical mechanics.
Examples of a particle undergoing fractional dynamics in a binding force field
are worked out in detail. We briefly discuss possible physical applications in
single particle experiments
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