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

Finite Larmor radius effects on non-diffusive tracer transport in a zonal flow

Finite Larmor radius (FLR) effects on non-diffusive transport in a prototypical zonal flow with drift waves are studied in the context of a simplified chaotic transport model. The model consists of a superposition of drift waves of the linearized Hasegawa-Mima equation and a zonal shear flow perpendicular to the density gradient. High frequency FLR effects are incorporated by gyroaveraging the ExB velocity. Transport in the direction of the density gradient is negligible and we therefore focus on transport parallel to the zonal flows. A prescribed asymmetry produces strongly asymmetric non- Gaussian PDFs of particle displacements, with L\'evy flights in one direction but not the other. For zero Larmor radius, a transition is observed in the scaling of the second moment of particle displacements. However, FLR effects seem to eliminate this transition. The PDFs of trapping and flight events show clear evidence of algebraic scaling with decay exponents depending on the value of the Larmor radii. The shape and spatio-temporal self-similar anomalous scaling of the PDFs of particle displacements are reproduced accurately with a neutral, asymmetric effective fractional diffusion model.Comment: 14 pages, 13 figures, submitted to Physics of Plasma

Tails of probability density for sums of random independent variables

The exact expression for the probability density $p_{_N}(x)$ for sums of a finite number $N$ of random independent terms is obtained. It is shown that the very tail of $p_{_N}(x)$ has a Gaussian form if and only if all the random terms are distributed according to the Gauss Law. In all other cases the tail for $p_{_N}(x)$ differs from the Gaussian. If the variances of random terms diverge the non-Gaussian tail is related to a Levy distribution for $p_{_N}(x)$. However, the tail is not Gaussian even if the variances are finite. In the latter case $p_{_N}(x)$ has two different asymptotics. At small and moderate values of $x$ the distribution is Gaussian. At large $x$ the non-Gaussian tail arises. The crossover between the two asymptotics occurs at $x$ proportional to $N$. For this reason the non-Gaussian tail exists at finite $N$ only. In the limit $N$ tends to infinity the origin of the tail is shifted to infinity, i. e., the tail vanishes. Depending on the particular type of the distribution of the random terms the non-Gaussian tail may decay either slower than the Gaussian, or faster than it. A number of particular examples is discussed in detail.Comment: 6 pages, 4 figure

Statistical Properties of the Interbeat Interval Cascade in Human Subjects

Statistical properties of interbeat intervals cascade are evaluated by considering the joint probability distribution $P(\Delta x_2,\tau_2;\Delta x_1,\tau_1)$ for two interbeat increments $\Delta x_1$ and $\Delta x_2$ of different time scales $\tau_1$ and $\tau_2$. We present evidence that the conditional probability distribution $P(\Delta x_2,\tau_2|\Delta x_1,\tau_1)$ may obey a Chapman-Kolmogorov equation. The corresponding Kramers-Moyal (KM) coefficients are evaluated. It is shown that while the first and second KM coefficients, i.e., the drift and diffusion coefficients, take on well-defined and significant values, the higher-order coefficients in the KM expansion are very small. As a result, the joint probability distributions of the increments in the interbeat intervals obey a Fokker-Planck equation. The method provides a novel technique for distinguishing the two classes of subjects in terms of the drift and diffusion coefficients, which behave differently for two classes of the subjects, namely, healthy subjects and those with congestive heart failure.Comment: 5 pages, 6 figure

Levy flights in quenched random force fields

Levy flights, characterized by the microscopic step index f, are for f<2 (the case of rare events) considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f-2 in the short range case and the critical fall-off exponent 2f-2 in the long range case. By means of a dynamic renormalization group analysis based on the momentum shell integration method, we determine flows, fixed point, and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient. Unlike the case of ordinary Brownian motion in a quenched force field characterized by a single critical dimension or fall-off exponent d=2, two critical dimensions appear in the Levy case. A critical dimension (or fall-off exponent) d=f below which the diffusion coefficient exhibits anomalous scaling behavior, i.e, algebraic spatial behavior and long time tails, and a critical dimension (or fall-off exponent) d=2f-2 below which the force correlations characterized by a non trivial fixed point become relevant. As a general result we find in all cases that the dynamic exponent z, characterizing the mean square displacement, locks onto the Levy index f, independent of dimension and independent of the presence of weak quenched disorder.Comment: 27 pages, Revtex file, 17 figures in ps format attached, submitted to Phys. Rev.

Biased diffusion in a piecewise linear random potential

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.Comment: 11 pages, 3 figure

Statistics of quantum transmission in one dimension with broad disorder

We study the statistics of quantum transmission through a one-dimensional disordered system modelled by a sequence of independent scattering units. Each unit is characterized by its length and by its action, which is proportional to the logarithm of the transmission probability through this unit. Unit actions and lengths are independent random variables, with a common distribution that is either narrow or broad. This investigation is motivated by results on disordered systems with non-stationary random potentials whose fluctuations grow with distance. In the statistical ensemble at fixed total sample length four phases can be distinguished, according to the values of the indices characterizing the distribution of the unit actions and lengths. The sample action, which is proportional to the logarithm of the conductance across the sample, is found to obey a fluctuating scaling law, and therefore to be non-self-averaging, in three of the four phases. According to the values of the two above mentioned indices, the sample action may typically grow less rapidly than linearly with the sample length (underlocalization), more rapidly than linearly (superlocalization), or linearly but with non-trivial sample-to-sample fluctuations (fluctuating localization).Comment: 26 pages, 4 figures, 1 tabl

Rates of convergence of nonextensive statistical distributions to Levy distributions in full and half spaces

The Levy-type distributions are derived using the principle of maximum Tsallis nonextensive entropy both in the full and half spaces. The rates of convergence to the exact Levy stable distributions are determined by taking the N-fold convolutions of these distributions. The marked difference between the problems in the full and half spaces is elucidated analytically. It is found that the rates of convergence depend on the ranges of the Levy indices. An important result emerging from the present analysis is deduced if interpreted in terms of random walks, implying the dependence of the asymptotic long-time behaviors of the walks on the ranges of the Levy indices if N is identified with the total time of the walks.Comment: 20 page

Diffusion, super-diffusion and coalescence from single step

From the exact single step evolution equation of the two-point correlation function of a particle distribution subjected to a stochastic displacement field \bu(\bx), we derive different dynamical regimes when \bu(\bx) is iterated to build a velocity field. First we show that spatially uncorrelated fields \bu(\bx) lead to both standard and anomalous diffusion equation. When the field \bu(\bx) is spatially correlated each particle performs a simple free Brownian motion, but the trajectories of different particles result to be mutually correlated. The two-point statistical properties of the field \bu(\bx) induce two-point spatial correlations in the particle distribution satisfying a simple but non-trivial diffusion-like equation. These displacement-displacement correlations lead the system to three possible regimes: coalescence, simple clustering and a combination of the two. The existence of these different regimes, in the one-dimensional system, is shown through computer simulations and a simple theoretical argument.Comment: RevTeX (iopstyle) 19 pages, 5 eps-figure