492 research outputs found
Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
We have derived a fractional Fokker-Planck equation for subdiffusion in a
general space-and- time-dependent force field from power law waiting time
continuous time random walks biased by Boltzmann weights. The governing
equation is derived from a generalized master equation and is shown to be
equivalent to a subordinated stochastic Langevin equation.Comment: 5 page
The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations
In this work, we consider subordinated processes controlled by a family of
subordinators which consist of a power function of time variable and a negative
power function of stable random variable. The effect of parameters in
the subordinators on the subordinated process is discussed. By suitable
variable substitutions and Laplace transform technique, the corresponding
fractional Fokker-Planck-type equations are derived. We also compute their mean
square displacements in a free force field. By choosing suitable ranges of
parameters, the resulting subordinated processes may be subdiffusive, normal
diffusive or superdiffusive.Comment: 11 pages, accepted by J. Stat. Mech.: Theor. Ex
Fractional diffusion in periodic potentials
Fractional, anomalous diffusion in space-periodic potentials is investigated.
The analytical solution for the effective, fractional diffusion coefficient in
an arbitrary periodic potential is obtained in closed form in terms of two
quadratures. This theoretical result is corroborated by numerical simulations
for different shapes of the periodic potential. Normal and fractional spreading
processes are contrasted via their time evolution of the corresponding
probability densities in state space. While there are distinct differences
occurring at small evolution times, a re-scaling of time yields a mutual
matching between the long-time behaviors of normal and fractional diffusion
Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis
This paper is concerned with a non-homogeneous in space and non-local in time
random walk model for anomalous subdiffusive transport of cells. Starting with
a Markov model involving a structured probability density function, we derive
the non-local in time master equation and fractional equation for the
probability of cell position. We show the structural instability of fractional
subdiffusive equation with respect to the partial variations of anomalous
exponent. We find the criteria under which the anomalous aggregation of cells
takes place in the semi-infinite domain.Comment: 18 pages, accepted for publicatio
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
Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
A detailed study is presented for a large class of uncoupled continuous-time
random walks (CTRWs). The master equation is solved for the Mittag-Leffler
survival probability. The properly scaled diffusive limit of the master
equation is taken and its relation with the fractional diffusion equation is
discussed. Finally, some common objections found in the literature are
thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page
From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion
Einstein's explanation of Brownian motion provided one of the cornerstones
which underlie the modern approaches to stochastic processes. His approach is
based on a random walk picture and is valid for Markovian processes lacking
long-term memory. The coarse-grained behavior of such processes is described by
the diffusion equation. However, many natural processes do not possess the
Markovian property and exhibit to anomalous diffusion. We consider here the
case of subdiffusive processes, which are semi-Markovian and correspond to
continuous-time random walks in which the waiting time for a step is given by a
probability distribution with a diverging mean value. Such a process can be
considered as a process subordinated to normal diffusion under operational time
which depends on this pathological waiting-time distribution. We derive two
different but equivalent forms of kinetic equations, which reduce to know
fractional diffusion or Fokker-Planck equations for waiting-time distributions
following a power-law. For waiting time distributions which are not pure power
laws one or the other form of the kinetic equation is advantageous, depending
on whether the process slows down or accelerates in the course of time
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