6,698 research outputs found
L\'{e}vy flights as subordination process: first passage times
We obtain the first passage time density for a L\'{e}vy flight random process
from a subordination scheme. By this method, we infer the asymptotic behavior
directly from the Brownian solution and the Sparre Andersen theorem, avoiding
explicit reference to the fractional diffusion equation. Our results
corroborate recent findings for Markovian L\'{e}vy flights and generalize to
broad waiting times.Comment: 4 pages, RevTe
Towards deterministic equations for Levy walks: the fractional material derivative
Levy walks are random processes with an underlying spatiotemporal coupling.
This coupling penalizes long jumps, and therefore Levy walks give a proper
stochastic description for a particle's motion with broad jump length
distribution. We derive a generalized dynamical formulation for Levy walks in
which the fractional equivalent of the material derivative occurs. Our approach
will be useful for the dynamical formulation of Levy walks in an external force
field or in phase space for which the description in terms of the continuous
time random walk or its corresponding generalized master equation are less well
suited
On solutions of a class of non-Markovian Fokker-Planck equations
We show that a formal solution of a rather general non-Markovian
Fokker-Planck equation can be represented in a form of an integral
decomposition and thus can be expressed through the solution of the Markovian
equation with the same Fokker-Planck operator. This allows us to classify
memory kernels into safe ones, for which the solution is always a probability
density, and dangerous ones, when this is not guaranteed. The first situation
describes random processes subordinated to a Wiener process, while the second
one typically corresponds to random processes showing a strong ballistic
component. In this case the non-Markovian Fokker-Planck equation is only valid
in a restricted range of parameters, initial and boundary conditions.Comment: A new ref.12 is added and discusse
Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals
We show that the generalized diffusion coefficient of a subdiffusive
intermittent map is a fractal function of control parameters. A modified
continuous time random walk theory yields its coarse functional form and
correctly describes a dynamical phase transition from normal to anomalous
diffusion marked by strong suppression of diffusion. Similarly, the probability
density of moving particles is governed by a time-fractional diffusion equation
on coarse scales while exhibiting a specific fine structure. Approximations
beyond stochastic theory are derived from a generalized Taylor-Green-Kubo
formula.Comment: 4 pages, 3 eps 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
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