2,471 research outputs found
Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
Anomalous diffusion is frequently described by scaled Brownian motion (SBM),
a Gaussian process with a power-law time dependent diffusion coefficient. Its
mean squared displacement is with
for . SBM may provide a
seemingly adequate description in the case of unbounded diffusion, for which
its probability density function coincides with that of fractional Brownian
motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a
significant amplitude scatter of the time averaged mean squared displacement.
More severely, we demonstrate that under confinement, the dynamics encoded by
SBM is fundamentally different from both fractional Brownian motion and
continuous time random walks. SBM is highly non-stationary and cannot provide a
physical description for particles in a thermalised stationary system. Our
findings have direct impact on the modelling of single particle tracking
experiments, in particular, under confinement inside cellular compartments or
when optical tweezers tracking methods are used.Comment: 7 pages, 5 figure
Residual mean first-passage time for jump processes: theory and applications to L\'evy flights and fractional Brownian motion
We derive a functional equation for the mean first-passage time (MFPT) of a
generic self-similar Markovian continuous process to a target in a
one-dimensional domain and obtain its exact solution. We show that the obtained
expression of the MFPT for continuous processes is actually different from the
large system size limit of the MFPT for discrete jump processes allowing
leapovers. In the case considered here, the asymptotic MFPT admits
non-vanishing corrections, which we call residual MFPT. The case of L/'evy
flights with diverging variance of jump lengths is investigated in detail, in
particular, with respect to the associated leapover behaviour. We also show
numerically that our results apply with good accuracy to fractional Brownian
motion, despite its non-Markovian nature.Comment: 13 pages, 8 figure
Comparison of pure and combined search strategies for single and multiple targets
We address the generic problem of random search for a point-like target on a
line. Using the measures of search reliability and efficiency to quantify the
random search quality, we compare Brownian search with L\'evy search based on
long-tailed jump length distributions. We then compare these results with a
search process combined of two different long-tailed jump length distributions.
Moreover, we study the case of multiple targets located by a L\'evy searcher.Comment: 16 pages, 12 figure
Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
A growing number of biological, soft, and active matter systems are observed
to exhibit normal diffusive dynamics with a linear growth of the mean squared
displacement, yet with a non-Gaussian distribution of increments. Based on the
Chubinsky-Slater idea of a diffusing diffusivity we here establish and analyze
a minimal model framework of diffusion processes with fluctuating diffusivity.
In particular, we demonstrate the equivalence of the diffusing diffusivity
process with a superstatistical approach with a distribution of diffusivities,
at times shorter than the diffusivity correlation time. At longer times a
crossover to a Gaussian distribution with an effective diffusivity emerges.
Specifically, we establish a subordination picture of Brownian but non-Gaussian
diffusion processes, that can be used for a wide class of diffusivity
fluctuation statistics. Our results are shown to be in excellent agreement with
simulations and numerical evaluations.Comment: 19 pages, 6 figures, RevTeX. Physical Review X, at pres
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
Leapover lengths and first passage time statistics for L\'evy flights
Exact results for the first passage time and leapover statistics of symmetric
and one-sided Levy flights (LFs) are derived. LFs with stable index alpha are
shown to have leapover lengths, that are asymptotically power-law distributed
with index alpha for one-sided LFs and, surprisingly, with index alpha/2 for
symmetric LFs. The first passage time distribution scales like a power-law with
index 1/2 as required by the Sparre Andersen theorem for symmetric LFs, whereas
one-sided LFs have a narrow distribution of first passage times. The exact
analytic results are confirmed by extensive simulations.Comment: 4 pages, 5 figures, REVTe
First passage behaviour of fractional Brownian motion in two-dimensional wedge domains
We study the survival probability and the corresponding first passage time
density of fractional Brownian motion confined to a two-dimensional open wedge
domain with absorbing boundaries. By analytical arguments and numerical
simulation we show that in the long time limit the first passage time density
scales as t**{-1+pi*(2H-2)/(2*Theta)} in terms of the Hurst exponent H and the
wedge angle Theta. We discuss this scaling behaviour in connection with the
reaction kinetics of FBM particles in a one-dimensional domain.Comment: 6 pages, 4 figure
Bulk-mediated diffusion on a planar surface: full solution
We consider the effective surface motion of a particle that intermittently
unbinds from a planar surface and performs bulk excursions. Based on a random
walk approach we derive the diffusion equations for surface and bulk diffusion
including the surface-bulk coupling. From these exact dynamic equations we
analytically obtain the propagator of the effective surface motion. This
approach allows us to deduce a superdiffusive, Cauchy-type behavior on the
surface, together with exact cutoffs limiting the Cauchy form. Moreover we
study the long-time dynamics for the surface motion.Comment: 12 pages, 1 figur
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