247 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
Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
Anomalous diffusion is frequently described by scaled Brownian motion (SBM),
a Gaussian process with a power-law time dependent diffusion coefficient. Its
mean squared displacement is with
for . SBM may provide a
seemingly adequate description in the case of unbounded diffusion, for which
its probability density function coincides with that of fractional Brownian
motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a
significant amplitude scatter of the time averaged mean squared displacement.
More severely, we demonstrate that under confinement, the dynamics encoded by
SBM is fundamentally different from both fractional Brownian motion and
continuous time random walks. SBM is highly non-stationary and cannot provide a
physical description for particles in a thermalised stationary system. Our
findings have direct impact on the modelling of single particle tracking
experiments, in particular, under confinement inside cellular compartments or
when optical tweezers tracking methods are used.Comment: 7 pages, 5 figure
Stationary States in Bistable System Driven by L\'evy Noise
We study the properties of the probability density function (PDF) of a
bistable system driven by heavy tailed white symmetric L\'evy noise. The shape
of the stationary PDF is found analytically for the particular case of the
L\'evy index \alpha = 1 (Cauchy noise). For an arbitrary L\'evy index we employ
numerical methods based on the solution of the stochastic Langevin equation and
space fractional kinetic equation. In contrast with the bistable system driven
by Gaussian noise, in the L\'evy case the positions of maxima of the stationary
PDF do not coincide with the positions of minima of the bistable potential. We
provide a detailed study of the distance between the maxima and the minima as a
function of the potential's depth and L\'evy noise parameters.Comment: Accepted to EPJS
The problem of analytical calculation of barrier crossing characteristics for Levy flights
By using the backward fractional Fokker-Planck equation we investigate the
barrier crossing event in the presence of Levy noise. After shortly review
recent results obtained with different approaches on the time characteristics
of the barrier crossing, we derive a general differential equation useful to
calculate the nonlinear relaxation time. We obtain analytically the nonlinear
relaxation time for free Levy flights and a closed expression in quadrature of
the same characteristics for cubic potential.Comment: 12 pages, 2 figures, presented at 5th International Conference on
Unsolved Problems on Noise, Lyon, France, 2008, to appear in J. Stat. Mech.:
Theory and Experimen
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