27,573 research outputs found
Fractional Kinetics for Relaxation and Superdiffusion in Magnetic Field
We propose fractional Fokker-Planck equation for the kinetic description of
relaxation and superdiffusion processes in constant magnetic and random
electric fields. We assume that the random electric field acting on a test
charged particle is isotropic and possesses non-Gaussian Levy stable
statistics. These assumptions provide us with a straightforward possibility to
consider formation of anomalous stationary states and superdiffusion processes,
both properties are inherent to strongly non-equilibrium plasmas of solar
systems and thermonuclear devices. We solve fractional kinetic equations, study
the properties of the solution, and compare analytical results with those of
numerical simulation based on the solution of the Langevin equations with the
noise source having Levy stable probability density. We found, in particular,
that the stationary states are essentially non-Maxwellian ones and, at the
diffusion stage of relaxation, the characteristic displacement of a particle
grows superdiffusively with time and is inversely proportional to the magnetic
field.Comment: 15 pages, LaTeX, 5 figures PostScrip
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
Anomalous subdiffusion with multispecies linear reaction dynamics
We have introduced a set of coupled fractional reaction-diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics. The model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics. The effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms. One consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species. The general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail. We have carried out Monte Carlo simulations corresponding to these special cases and we find excellent agreement with theory
Fractional Chemotaxis Diffusion Equations
We introduce mesoscopic and macroscopic model equations of chemotaxis with
anomalous subdiffusion for modelling chemically directed transport of
biological organisms in changing chemical environments with diffusion hindered
by traps or macro-molecular crowding. The mesoscopic models are formulated
using Continuous Time Random Walk master equations and the macroscopic models
are formulated with fractional order differential equations. Different models
are proposed depending on the timing of the chemotactic forcing.
Generalizations of the models to include linear reaction dynamics are also
derived. Finally a Monte Carlo method for simulating anomalous subdiffusion
with chemotaxis is introduced and simulation results are compared with
numerical solutions of the model equations. The model equations developed here
could be used to replace Keller-Segel type equations in biological systems with
transport hindered by traps, macro-molecular crowding or other obstacles.Comment: 25page
Fractional Reaction-Diffusion Equation
A fractional reaction-diffusion equation is derived from a continuous time
random walk model when the transport is dispersive. The exit from the encounter
distance, which is described by the algebraic waiting time distribution of jump
motion, interferes with the reaction at the encounter distance. Therefore, the
reaction term has a memory effect. The derived equation is applied to the
geminate recombination problem. The recombination is shown to depend on the
intrinsic reaction rate, in contrast with the results of Sung et al. [J. Chem.
Phys. {\bf 116}, 2338 (2002)], which were obtained from the fractional
reaction-diffusion equation where the diffusion term has a memory effect but
the reaction term does not. The reactivity dependence of the recombination
probability is confirmed by numerical simulations.Comment: to appear in Journal of Chemical Physic
On distributions of functionals of anomalous diffusion paths
Functionals of Brownian motion have diverse applications in physics,
mathematics, and other fields. The probability density function (PDF) of
Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger
equation in imaginary time. In recent years there is a growing interest in
particular functionals of non-Brownian motion, or anomalous diffusion, but no
equation existed for their PDF. Here, we derive a fractional generalization of
the Feynman-Kac equation for functionals of anomalous paths based on
sub-diffusive continuous-time random walk. We also derive a backward equation
and a generalization to Levy flights. Solutions are presented for a wide number
of applications including the occupation time in half space and in an interval,
the first passage time, the maximal displacement, and the hitting probability.
We briefly discuss other fractional Schrodinger equations that recently
appeared in the literature.Comment: 25 pages, 4 figure
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