62 research outputs found
Fractional Diffusion Equation for a Power-Law-Truncated Levy Process
Truncated Levy flights are stochastic processes which display a crossover
from a heavy-tailed Levy behavior to a faster decaying probability distribution
function (pdf). Putting less weight on long flights overcomes the divergence of
the Levy distribution second moment. We introduce a fractional generalization
of the diffusion equation, whose solution defines a process in which a Levy
flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A
closed form for the characteristic function of the process is derived. The pdf
of the displacement slowly converges to a Gaussian in its central part showing
however a power law far tail. Possible applications are discussed
Barrier Grossing Induced by Fractional Gaussian Noise
A problem of the rate of escape of a particle under the influence of the external fractional Gaussian noise is studied by using the method of numerical integration of an overdamped Langevin equation. Considering a truncated harmonic potential, the dependences of the mean escape time on the noise intensity and Hurst index are evaluated, together with the probability density functions for the escape times. It is found that, like the corresponding classical problem with white Gaussian noise, they both obey an exponential law.За допомогою чисельного iнтегрування передемпфованого рiвняння Ланжевена дослiджено задачу про швидкiсть вильоту частинки iз потенцiальної ями пiд дiєю дробового гаусового шуму. На прикладi обрiзаного гармонiчного потенцiалу отримано залежностi середнього часу вильоту вiд iнтенсивностi шуму та показника Херста, а також обчислено функцiї розподiлу часiв вильоту. Зроблено висновок, що, як i у випадку класичної задачi з бiлим гаусовим шумом, цi величини є експоненцiальними функцiями вiдповiдних параметрiв
Fluctuation-driven directed transport in the presence of Levy flights
Numerical evidence of directed transport driven by symmetric Levy noise in
time-independent ratchet potentials in the absence of an external tilting force
is presented. The results are based on the numerical solution of the fractional
Fokker-Planck equation in a periodic potential and the corresponding Langevin
equation with Levy noise. The Levy noise drives the system out of thermodynamic
equilibrium and an up-hill net current is generated. For small values of the
noise intensity there is an optimal value of the Levy noise index yielding the
maximum current. The direction and magnitude of the current can be manipulated
by changing the Levy noise asymmetry and the potential asymmetry
Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion
A considerable number of systems have recently been reported in which Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth in time of the mean squared displacement, yet the probability density function of the particle displacement is distinctly non-Gaussian, and often of exponential (Laplace) shape. This apparently ubiquitous behaviour observed in very different physical systems has been interpreted as resulting from diffusion in inhomogeneous environments and mathematically represented through a variable, stochastic diffusion coefficient. Indeed different models describing a fluctuating diffusivity have been studied. Here we present a new view of the stochastic basis describing time-dependent random diffusivities within a broad spectrum of distributions. Concretely, our study is based on the very generic class of the generalised Gamma distribution. Two models for the particle spreading in such random diffusivity settings are studied. The first belongs to the class of generalised grey Brownian motion while the second follows from the idea of diffusing diffusivities. The two processes exhibit significant characteristics which reproduce experimental results from different biological and physical systems. We promote these two physical models for the description of stochastic particle motion in complex environments.the DFG within project ME 1535/6-
Stable Equilibrium Based on L\'evy Statistics: Stochastic Collision Models Approach
We investigate equilibrium properties of two very different stochastic
collision models: (i) the Rayleigh particle and (ii) the driven Maxwell gas.
For both models the equilibrium velocity distribution is a L\'evy distribution,
the Maxwell distribution being a special case. We show how these models are
related to fractional kinetic equations. Our work demonstrates that a stable
power-law equilibrium, which is independent of details of the underlying
models, is a natural generalization of Maxwell's velocity distribution.Comment: PRE Rapid Communication (in press
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
Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers
The emerging diffusive dynamics in many complex systems shows a
characteristic crossover behaviour from anomalous to normal diffusion which
is otherwise fitted by two independent power-laws. A prominent example for
a subdiffusive-diffusive crossover are viscoelastic systems such as lipid
bilayer membranes, while superdiffusive-diffusive crossovers occur in systems
of actively moving biological cells. We here consider the general dynamics of
a stochastic particle driven by so-called tempered fractional Gaussian noise,
that is noise with Gaussian amplitude and power-law correlations, which are
cut off at some mesoscopic time scale. Concretely we consider such noise with
built-in exponential or power-law tempering, driving an overdamped Langevin
equation (fractional Brownian motion) and fractional Langevin equation motion.
We derive explicit
expressions for the mean squared displacement and correlation functions,
including different shapes of the crossover behaviour depending on the
concrete tempering, and discuss the physical meaning of the tempering. In the
case of power-law tempering we also find a crossover behaviour from faster to
slower superdiffusion and slower to faster subdiffusion. As a direct application
of our model we demonstrate that the obtained dynamics quantitatively described
the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer
systems. We also show that a model of tempered fractional Brownian motion recently
proposed by Sabzikar and Meerschaert leads to physically very different behaviour
with a seemingly paradoxical ballistic long time scaling
A Model for Persistent Levy Motion
We propose the model, which allows us to approximate fractional Levy noise
and fractional Levy motion. Our model is based (i) on the Gnedenko limit
theorem for an attraction basin of stable probability law, and (ii) on
regarding fractional noise as the result of fractional
integration/differentiation of a white Levy noise. We investigate self - affine
properties of the approximation and conclude that it is suitable for modeling
persistent Levy motion with the Levy index between 1 and 2.Comment: 14 pages, REVTeX, 5 figures PostScrip
Truncated Levy Random Walks and Generalized Cauchy Processes
A continuous Markovian model for truncated Levy random walks is proposed. It
generalizes the approach developed previously by Lubashevsky et al. Phys. Rev.
E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing
for nonlinear friction in wondering particle motion and saturation of the noise
intensity depending on the particle velocity. Both the effects have own reason
to be considered and individually give rise to truncated Levy random walks as
shown in the paper. The nonlinear Langevin equation governing the particle
motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta
method and the obtained numerical data were employed to calculate the geometric
mean of the particle displacement during a certain time interval and to
construct its distribution function. It is demonstrated that the time
dependence of the geometric mean comprises three fragments following one
another as the time scale increases that can be categorized as the ballistic
regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive
one), and the standard motion of Brownian particles. For the intermediate Levy
type part the distribution of the particle displacement is found to be of the
generalized Cauchy form with cutoff. Besides, the properties of the random
walks at hand are shown to be determined mainly by a certain ratio of the
friction coefficient and the noise intensity rather then their characteristics
individually.Comment: 7 pages, 3 figure
Density and potential fluctuations in the edge plasma of the Uragan-3m torsatron
Ion saturation current and floating potential fluctuations are recorded by movable array of 4 Langmuir probes near
the boundary of the confinement region in the l = 3/m = 9 Uragan-3M torsatron with an RF produced and heated
plasma. On the basis of these data main spectral and time characteristics of the low frequency electrostatic turbulence
have been derived. The existence of the radial electric field shear and reversal of poloidal phase velocity of the
fluctuations at the plasma boundary have been confirmed. The time function of the turbulent E×B particle flux contains
intermittent bursts with the amplitude multiply exceeding the average flux. Up to 70% of the total fluctuating flux is
carried in these bursts
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