200 research outputs found
Aging Scaled Brownian Motion
Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of
passive tracers in complex and biological systems. It is a highly
non-stationary process governed by the Langevin equation for Brownian motion,
however, with a power-law time dependence of the noise strength. Here we study
the aging properties of SBM for both unconfined and confined motion.
Specifically, we derive the ensemble and time averaged mean squared
displacements and analyze their behavior in the regimes of weak, intermediate,
and strong aging. A very rich behavior is revealed for confined aging SBM
depending on different aging times and whether the process is sub- or
superdiffusive. We demonstrate that the information on the aging factorizes
with respect to the lag time and exhibits a functional form, that is identical
to the aging behavior of scale free continuous time random walk processes.
While SBM exhibits a disparity between ensemble and time averaged observables
and is thus weakly non-ergodic, strong aging is shown to effect a convergence
of the ensemble and time averaged mean squared displacement. Finally, we derive
the density of first passage times in the semi-infinite domain that features a
crossover defined by the aging time.Comment: 10 pages, 8 figures, REVTe
Asymptotics of Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with Oscillating Boundary. Multiple Eigenvalue Case
We study the asymptotic behavior of the solutions of a spectral problem for
the Laplacian in a domain with rapidly oscillating boundary. We consider the
case where the eigenvalue of the limit problem is multiple. We construct the
leading terms of the asymptotic expansions for the eigenelements and verify the
asymptotics
On anomalous diffusion and the out of equilibrium response function in one-dimensional models
We study how the Einstein relation between spontaneous fluctuations and the
response to an external perturbation holds in the absence of currents, for the
comb model and the elastic single-file, which are examples of systems with
subdiffusive transport properties. The relevance of non-equilibrium conditions
is investigated: when a stationary current (in the form of a drift or an energy
flux) is present, the Einstein relation breaks down, as is known to happen in
systems with standard diffusion. In the case of the comb model, a general
relation, which has appeared in the recent literature, between the response
function and an unperturbed suitable correlation function, allows us to explain
the observed results. This suggests that a relevant ingredient in breaking the
Einstein formula, for stationary regimes, is not the anomalous diffusion but
the presence of currents driving the system out of equilibrium.Comment: 10 pages, 4 figure
Probabilistic properties of detrended fluctuation analysis for Gaussian processes
Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter
Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations
We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski
kinetic equations, which describe evolution of the systems influenced by
stochastic forces distributed with stable probability laws. These equations
generalize known kinetic equations of the Brownian motion theory and contain
symmetric fractional derivatives over velocity and space, respectively. With
the help of these equations we study analytically the processes of linear
relaxation in a force - free case and for linear oscillator. For a weakly
damped oscillator we also get kinetic equation for the distribution in slow
variables. Linear relaxation processes are also studied numerically by solving
corresponding Langevin equations with the source which is a discrete - time
approximation to a white Levy noise. Numerical and analytical results agree
quantitatively.Comment: 30 pages, LaTeX, 13 figures PostScrip
Effective surface motion on a reactive cylinder of particles that perform intermittent bulk diffusion
In many biological and small scale technological applications particles may
transiently bind to a cylindrical surface. In between two binding events the
particles diffuse in the bulk, thus producing an effective translation on the
cylinder surface. We here derive the effective motion on the surface, allowing
for additional diffusion on the cylinder surface itself. We find explicit
solutions for the number of adsorbed particles at one given instant, the
effective surface displacement, as well as the surface propagator. In
particular sub- and superdiffusive regimes are found, as well as an effective
stalling of diffusion visible as a plateau in the mean squared displacement. We
also investigate the corresponding first passage and first return problems.Comment: 26 pages, 5 figure
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