Lateral diffusion of the topological charge density in stochastic optical fields

Stochastic (i.e. random and quasi-random) optical fields may contain distributions of optical vortices that are represented by non-uniform topological charge densities. Numerical simulations are used to investigate the evolution under free-space propagation of topological charge densities that are inhomogeneous along one dimension. It is shown that this evolution is described by a diffusion process that has a diffusion parameter which depends on the propagation distance.     

Generalized Fick law for anomalous diffusion in the multidimensional comb model

The problem of multidimensional diffusion is considered within the framework of the comb model. It is shown that the diffusion current for the case of anomalous subdiffusion random walks is described by the generalized Fick law containing the diffusion tensor instead of the usual coefficient. The form of the diffusion tensor components is an unusual form of operator as fractional time derivatives. The orders of the fractional exponents are different for different directions.     

Spectral Analysis of Fractional Kinetic Equations with Random Data

We present a spectral representation of the mean-square solution of the fractional kinetic equation (also known as fractional diffusion equation) with random initial condition. Gaussian and non-Gaussian limiting distributions of the renormalized solution of the fractional-in-time and in-space kinetic equation are described in terms of multiple stochastic integral representations.     

Diffusion in a heterogeneous medium for low concentrations

The equations describing diffusion on a heterogeneous lattice for low concentrations are considered taking into account lattice site blocking. It is shown that lattice site blocking cannot be disregarded in the case of a strongly heterogeneous lattice even for low concentrations. It is established that the equation with a fractional time derivative holds only in a bounded time interval. Anomalous diffusion, which is described by the equation with a fractional time derivative at the initial stage, must be described over long time periods by an ordinary diffusion equation with a concentration-dependent diffusion coefficient.     

Model of superdiffusion

A locally nonequilibrium model of superdiffusion is proposed that is based on the partition of the set of diffusing particles into groups according to the flight length of these particles. The process of diffusion is described in terms of partial concentrations of particles belonging to different groups. As special limit cases, the model yields equations with fractional time derivative and the so-called porous medium equation. The basic equations of the model are Markov equations; therefore, they easily include reaction terms. The model can be applied to describing the types of diffusion in which the diffusing particles are in free flight most of the time.     

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.     

Lateral phase drift of the topological charge density in stochastic optical fields

The statistical distributions of optical vortices or topological charge in stochastic optical fields can be inhomogeneous in both transverse directions. Such two-dimensional inhomogeneous vortex or topological charge distributions evolve in a complex way during free-space propagation. While the evolution of one-dimensional topological charge densities can be described by a linear diffusion process, the evolution of two-dimensional topological charge densities exhibits some additional nonlinear dynamics. Here we propose a phase drift mechanism as a partial explanation for this additional nonlinear dynamics. Numerical results are presented in support of this proposal.     

Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

We examine a numerical method to approximate to a fractional diffusion equation with the Riesz fractional derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz fractional derivative, we use the “fractional centered derivative” approach. We determine the error of the Riesz fractional derivative to the fractional centered difference. We apply the Crank–Nicolson method to a fractional diffusion equation which has the Riesz fractional derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank–Nicolson method for the fractional diffusion equation with using fractional centered difference approach.     

The Influence of Fractional Diffusion in Fisher-KPP Equations

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.     

Matrix approach to discrete fractional calculus II: Partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny’s matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359–386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.     

One- and Two-Dimensional Pulse Sequences for Diffusion Experiments in the Fringe Field of Superconducting Magnets

The fringe field of superconducting magnets provides extremely strong and stable field gradients which can be used for diffusion experiments. A number of corresponding "supercon fringe-field" variants of the "pulsed-gradient spin-echo" method are presented. These include procedures and pulse sequences for the record of relaxation-independent diffusion decays, multislice excitation, and one- or two-dimensional Fourier-transform evaluation from the k to the z domain. Several test experiments are described. It is demonstrated that reliable diffusion decays are obtained. The one-dimensional z-domain evaluation produces grid patterns analogous to the optical "forced Rayleigh scattering" method. The two-dimensional experiments are suitable for the detection of spatially varying probability densities of diffusion or incoherent flow in heterogeneous media. As the resolution is particularly high, fluids can be studied close to solid surfaces.     

A nonlinear Fokker–Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics

We investigate the solutions for a set of coupled nonlinear Fokker–Planck equations coupled by the diffusion coefficient in presence of external forces. The coupling by the diffusion coefficient implies that the diffusion of each species is influenced by the other and vice versa due to this term, which represents an interaction among them. The solutions for the stationary case are given in terms of the Tsallis distributions, when arbitrary external forces are considered. We also use the Tsallis distributions to obtain a time dependent solution for a linear external force. The results obtained from this analysis show a rich class of behavior related to anomalous diffusion, which can be characterized by compact or long-tailed distributions.     

Finite difference approximations for the fractional advection-diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection-diffusion equation, the fractional Crank-Nicholson method (FCN) is given, which is based on shifted Grünwald-Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order ²(Δt). A numerical example is presented to confirm the conclusions.     

Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach

The use of reaction-diffusion models rests on the key assumption that the diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand reactive systems in the presence of this type of non-Gaussian diffusion. Here we study front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay, x(-(1+alpha)). Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, x(-alpha), of the front's tail.     

Anomalous diffusion for a correlated process with long jumps

We discuss diffusion properties of a dynamical system, which is characterised by long-tail distributions and finite correlations. The particle velocity has the stable Lévy distribution; it is assumed as a jumping process (the kangaroo process) with a variable jumping rate. Both the exponential and the algebraic form of the covariance–defined for the truncated distribution–are considered. It is demonstrated by numerical calculations that the stationary solution of the master equation for the case of power-law correlations decays with time, but a simple modification of the process makes the tails stable. The main result of the paper is a finding that–in contrast to the velocity fluctuations–the position variance may be finite. It rises with time faster than linearly: the diffusion is anomalously enhanced. On the other hand, a process which follows from a superposition of the Ornstein–Uhlenbeck–Lévy processes always leads to position distributions with a divergent variance which means accelerated diffusion.     

二不能压缩气体的分片合流运动

本文求出规定二不能压缩气体的分片合流运动时,速度与质量密度分布的微分方程式。假定两种气体的密度不同但温度则相等。不能压缩的定义是每单位体积中的两种气体分子数的和不变。本文只讨论一平面守恒注中所需要的微分方程式。粘滞流体运动中之边界近似法仍可应用。同样方法亦可用到守恒圆柱体注,半注及气体中温度不同诸问题。     

Untersuchung des Nichtgleichgewichtszustandes an stationären Edelgasplasmen unter Normaldruck

It is shown that in wall stabilized rare gas arcs under normal pressure deviations from Saha-equilibrium occur. These deviations are very strong for Helium and Neon, smaller for Argon and Krypton discharges and are caused by diffusion of neutral and charged particles. A numerical method is described for the evaluation of temperature and density distributions from measured line intensities in the case of non-equilibrium, based on the balance equations of the arc plasma. Results are given for Neon, Argon and Krypton arcs of 0.15 and 0.3 cm radius. A simple validity condition for local thermal equilibrium in a plasma with diffusion effects is derived. The influence of non-equilibrium on the determination of transition probabilities is discussed.     

Asymptotic solutions of the continuous-time random walk model of diffusion

Asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times. It is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these asymptotic distributions are of stable form. This indicates that the realm of applicability of the diffusion equation is limited. The Montroll-Weiss equation is rederived to include the influence of the initial waiting interval and the role of the stable distributions in physical problems is briefly discussed.     

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.