320 research outputs found
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
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
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
Feynman-Kac equation for anomalous processes with space-and time-dependent forces
Invited contribution to the J. Phys. A special issue Emerging Talent
Pre-asymptotic corrections to fractional diffusion equations
The motion of contaminant particles through complex environments such as
fractured rocks or porous sediments is often characterized by anomalous
diffusion: the spread of the transported quantity is found to grow sublinearly
in time due to the presence of obstacles which hinder particle migration. The
asymptotic behavior of these systems is usually well described by fractional
diffusion, which provides an elegant and unified framework for modeling
anomalous transport. We show that pre-asymptotic corrections to fractional
diffusion might become relevant, depending on the microscopic dynamics of the
particles. To incorporate these effects, we derive a modified transport
equation and validate its effectiveness by a Monte Carlo simulation.Comment: 6 pages, 3 figure
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