854 research outputs found
Levy Anomalous Diffusion and Fractional Fokker--Planck Equation
We demonstrate that the Fokker-Planck equation can be generalized into a
'Fractional Fokker-Planck' equation, i.e. an equation which includes fractional
space differentiations, in order to encompass the wide class of anomalous
diffusions due to a Levy stable stochastic forcing. A precise determination of
this equation is obtained by substituting a Levy stable source to the classical
gaussian one in the Langevin equation. This yields not only the anomalous
diffusion coefficient, but a non trivial fractional operator which corresponds
to the possible asymmetry of the Levy stable source. Both of them cannot be
obtained by scaling arguments. The (mono-) scaling behaviors of the Fractional
Fokker-Planck equation and of its solutions are analysed and a generalization
of the Einstein relation for the anomalous diffusion coefficient is obtained.
This generalization yields a straightforward physical interpretation of the
parameters of Levy stable distributions. Furthermore, with the help of
important examples, we show the applicability of the Fractional Fokker-Planck
equation in physics.Comment: 22 pages; To Appear in Physica
Levy ratchets with dichotomic random flashing
Additive symmetric L\'evy noise can induce directed transport of overdamped
particles in a static asymmetric potential. We study, numerically and
analytically, the effect of an additional dichotomous random flashing in such
L\'evy ratchet system. For this purpose we analyze and solve the corresponding
fractional Fokker-Planck equations and we check the results with Langevin
simulations. We study the behavior of the current as function of the stability
index of the L\'evy noise, the noise intensity and the flashing parameters. We
find that flashing allows both to enhance and diminish in a broad range the
static L\'evy ratchet current, depending on the frequencies and asymmetry of
the multiplicative dichotomous noise, and on the additive L\'evy noise
parameters. Our results thus extend those for dichotomous flashing ratchets
with Gaussian noise to the case of broadly distributed noises.Comment: 15 pages, 6 figure
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 chemotaxis diffusion equations
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk 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, macromolecular crowding or other obstacles
A Fractional Fokker-Planck Model for Anomalous Diffusion
In this paper we present a study of anomalous diffusion using a Fokker-Planck
description with fractional velocity derivatives. The distribution functions
are found using numerical means for varying degree of fractionality observing
the transition from a Gaussian distribution to a L\'evy distribution. The
statistical properties of the distribution functions are assessed by a
generalized expectation measure and entropy in terms of Tsallis statistical
mechanics. We find that the ratio of the generalized entropy and expectation is
increasing with decreasing fractionality towards the well known so-called
sub-diffusive domain, indicating a self-organising behavior.Comment: 22 pages, 14 figure
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