Anomalous Diffusion Of Particles Research Articles

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

Laser Measurement of Anomalous Electron Diffusion in a Crossed-Field Plasma.

The anomalous diffusion of particles and energy in magnetized plasma systems is a widespread phenomenon that can adversely impact their operation and preclude predictive models. In this Letter, this diffusion is characterized noninvasively in a low-temperature, Hall-type plasma. Laser-induced fluorescence and incoherent Thomson scattering measurements are combined with a 1D generalized Ohm's law to infer the time-averaged inverse Hall parameter, a transport coefficient that governs cross-field diffusion. While the measured diffusion profile agrees with model-based estimates in magnitude, the measurements do not exhibit the steep "transport barrier" typically imposed in models. Instead, these results show that the electric field is primarily driven by a diamagnetic contribution due to the large peak electron temperature exceeding 75eV. This finding motivates a reconsideration of nonclassical energy transport across field lines in low-temperature plasmas.

Simulation study of the influence of the cross‐field anomalous diffusion on the tungsten impurity transport in the scrape‐off layer with activated drifts

AbstractTungsten (W) impurity eroded from the divertor target can degrade the confinement performance of future fusion reactors. The mechanism of the effect of E × B drift on the W transport in the scrape‐off layer (SOL) was investigated with a fixed cross‐field anomalous particle diffusion coefficient for W ions (D⊥,W) of 1 m2s−1 in our previous simulation work, which indicates that the W flux entering the confined region (Γenter) and mean W concentration in the confined region () can be increased by more than one order of magnitude by drifts. In this work, the influence of the cross‐field anomalous diffusion on the W transport in the SOL is further investigated. The basic flow pattern of the W ion flux (ΓW) keeps when D⊥,W varying from 0.5 to 3 m2s−1, and Γenter mainly comes from the hotter divertor region. Results show that D⊥,W value affects the W leakage from divertor region and entry into confined region. Therefore, Γenter and both decrease with increasing D⊥,W. Mechanisms for cases w/wo drifts are compared.

Temporal Asymptotic Form of the Survival Probability in the Effective Medium Approximation for Trapping of Particles in Media with Anomalous Diffusion

The capture of particles diffusing anomalously in accordance with a power law in absorbing traps is analyzed in the effective medium approximation. A new slower power-law dependence of the asymptotic form of the particle survival probability over long times has been established. This result is due to the anomalous diffusion of particles in strongly anisotropic media.

Estimation of near-bed sediment concentrations in turbulent flow beyond normality

Abstract Recent experiments have established that sediment particle motion, especially for particles near the bed, may not follow a normal (Fickian) diffusion behavior. To modify the diffusion equation when the fluctuation velocity is based on a normal distribution is a key research goal. This study represents an attempt to acquire some insight into the particle diffusion behavior by considering the turbulent fluctuation velocity based on bivariate probability distributions and by introducing the stochastic particle alignment into the particle–bed collision process in an open channel. The fluctuation velocity distribution is obtained using the Gram–Charlier expansion, which considers the first four statistical moments of turbulent fluctuation velocity. Fluctuation velocities in both streamwise and vertical directions are generated by conducting correlative Monte Carlo simulations. Moreover, in this study, the stochastic particle alignment composed of random particle sizes on the bed, as well as the particle collision process with random angles of incidence of particles is introduced to the particle collision mechanism. The associated probabilities of particle rebounding then would depend on the conditional probability distribution of the collision height. Particle trajectories simulated based on the saltation model and the random particle-bed collision process can be used to estimate the ensemble mean sediment concentration profile and the ensemble variance of sediment concentrations. In this study, the collision process, sediment concentration, and variance of particle motion are verified using available experimental data. Anomalous diffusion of particles can be observed in this study. In the proposed model, particle diffusion in the streamwise direction is found to be superdiffusive and that in the vertical direction is subdiffusive.

Continuous-time random walks and Fokker-Planck equation in expanding media

We consider a continuous random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates the probability distribution function (pdf) of finding a particle at a given position and time to the single-step jump length and waiting time pdfs is provided. The equation takes the form of a generalized Fokker-Planck equation when the jump length pdf of the particle has a finite variance. This generalized equation becomes a fractional Fokker-Planck equation in the case of a heavy-tailed waiting time pdf. These equations allow us to study the relationship between expansion, diffusion and external force. We establish the conditions under which the dominant contribution to transport stems from the diffusive transport rather than from the drift due to the medium expansion. We find that anomalous diffusion processes under a constant external force in an expanding medium described by means of our continuous random walk model are not Galilei invariant, violate the generalized Einstein relation, and lead to propagators that are qualitatively different from the ones found in a static medium. Our results are supported by numerical simulations.

Fractional Sturm–Liouville Equations: Self-Adjoint Extensions

In this report we study a fractional analogue of Sturm–Liouville equation. A class of self-adjoint fractional Sturm–Liouville operators is described. We give a biological interpretation of the fractional order equation and nonlocal boundary conditions that arise in describing the systems separated by a membrane. In particular, the connection with so called “fractional kinetic” equations is observed. Also, some spectral properties of the fractional kinetic equations are derived. An application to the anomalous diffusion of particles in a heterogeneous system of the fractional Sturm–Liouville equations is discussed.

Anomalous Diffusion of Particles Dispersed in Xanthan Solutions Subjected to Shear Flow

Xanthan gum exhibits viscoelastic and shear-thinning properties. We investigate the Brownian motion of particles dispersed in xanthan gum solutions that are subjected to simple shear flow. The mean...

Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion.

It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases.

Anomalous diffusion in a field of randomly distributed scatterers

We consider the motion of particles which are scattered by randomly distributed obstacles. In between scattering events the particles move uniformly. The governing master equation is obtained by mapping the problem onto a master equation which was previously devised for the description of anomalous diffusion of particles with inertia [R. Friedrich, Phys. Rev. Lett. 96, 230601 (2006)]. We show that for a scale-free distance distribution of scatterers a time-fractional master equation arises. The corresponding diffusion equation which exhibits a power-law diffusion coefficient is solved in d dimensions via the method of subordination.

Modelling of Lévy walk kinetics of charged particles in edge electrostatic turbulence in tokamaks

We model and discuss the possible types of motion that charged particles may undergo in a stationary and spatially periodic electrostatic potential and a homogeneous magnetic field. The model is considered to be the simplest approximation of more complex phenomena of plasma edge turbulence in tokamaks. Therein, low frequency turbulence appears in the plasma edge, resulting in a fluctuation of the electron density, and also in the generation of a turbulent electrostatic field. Typical parameters of this turbulent electrostatic field are an electrical potential amplitude of 10–100 V and wave numbers k≈103 m-1. In our model, we consider these regimes, together with a homogeneous magnetic field with a magnitude of 1 T. We investigate the dynamics of singly-ionized carbon ions – a typical plasma impurity – with kinetic energies on the order of 10 eV. Besides the obvious Larmor and drift motions, a motion of random-walk and of Levy walk character appear therein. All of these types of motion can play an important role in the modelling of the anomalous diffusion of particles from the plasma edge turbulence region. The dynamics mentioned will cause an inevitable escape of energetic particles and thus of power loss from the thermonuclear reactor. Moreover, Levy walk kinetics represents a very interesting kind of kinetics, currently of great interest, which was previously not so often discussed.

Anomalous Diffusion of Particles with Inertia in External Potentials

Recently a new type of Kramers-Fokker-Planck equation has been proposed [Friedrich, R.; Jenko, F.; Baule, A.; Eule, S. Phys. Rev. Lett. 2006, 96, 230601] describing anomalous diffusion in external potentials. In the present paper, the explicit cases of a harmonic potential and a velocity-dependent damping are incorporated. Exact relations for moments for these cases are presented, and the asymptotic behavior for long times is discussed. Interestingly, the bounding potential and the additional damping by itself lead to a subdiffusive behavior. While acting together, the particle becomes localized for long periods of time.

Liquid–liquid phase transition and anomalous diffusion in simulated liquid GeO 2

We perform molecular dynamics (MD) simulation of diffusion in liquid GeO 2 at the temperatures ranged from 3000 to 5000 K and densities ranged from 3.65 to 7.90 g/cm 3. Simulations were done in a model containing 3000 particles with the new interatomic potentials for liquid and amorphous GeO 2, which have weak Coulomb interaction and Morse-type short-range interaction. We found a liquid–liquid phase transition in simulated liquid GeO 2 from a tetrahedral to an octahedral network structure upon compression. Moreover, such phase transition accompanied with an anomalous diffusion of particles in liquid GeO 2 that the diffusion constant of both Ge and O particles strongly increases with increasing density (e.g. with increasing pressure) and it shows a maximum at the density around 4.95 g/cm 3. The possible relation between anomalous diffusion of particles and structural phase transition in the system has been discussed.

Anomalous particle diffusion and Lévy random walk of magnetic field lines in three-dimensional solar wind turbulence

Plasma transport in the presence of turbulence depends on a variety of parameters such as the fluctuation level, δB/B0, the ratio between the particle Larmor radius and the turbulence correlation length, and the turbulence anisotropy. In this paper, we present the results of numerical simulations of plasma and magnetic field line transport in the case of anisotropic magnetic turbulence, for parameter values close to those of the solar wind. We assume a uniform background magnetic field B0 = B0 ez and a Fourier representation for magnetic fluctuations, which includes wavectors oblique with respect to B0. The energy density spectrum is a power law, and in k space it is described by the correlation lengths lx, ly, lz, which quantify the anisotropy of turbulence. For magnetic field lines, transport perpendicular to the background field depends on the Kubo number R = (δB/B0) (lz/lx). For small Kubo numbers, R ≪ 1, anomalous, non-Gaussian transport regimes (both sub- and superdiffusive) are found, which can be described as a Lévy random walk. Increasing the Kubo number, i.e. the fluctuation level, δB/B0, or the ratio lz/lx, we find first a quasilinear regime and then a percolative regime, both corresponding to Gaussian diffusion. For particles, we find that transport parallel and perpendicular to the background magnetic field depends heavily on the turbulence anisotropy and on the particle Larmor radius. For turbulence levels typical of the solar wind, δB/B0≃ 0.5–1, when the ratio between the particle Larmor radius and the turbulence correlation lengths is small, anomalous regimes are found in the case lz/lx ⩽ 1, with a Lévy random walk (superdiffusion) along the magnetic field and subdiffusion in the perpendicular directions. Conversely, for lz/lx > 1 normal Gaussian diffusion is found. A possible expression for generalized double diffusion is discussed.

Anomalous diffusion of particles with internal degrees of freedom

Abstract Anomalous dynamics have been extensively investigated, especially in modelling charge transfer in photoconductivity and, more generally, for subballistic and superballistic transport, as found, say, in turbulent ionic motion in plasmas. In many such cases, work has focused on particles without internal degrees of freedom. Here we present models in which the anomalous dynamics is due to the internal modes of the moving particles, and less to the disorder of the medium or to the external fields.

Anomalous diffusion of particles with internal degrees of freedom

Anomalous dynamics have been extensively investigated, especially in modelling charge transfer in photoconductivity and, more generally, for subballistic and superballistic transport, as found, say, in turbulent ionic motion in plasmas. In many such cases, work has focused on particles without internal degrees of freedom. Here we present models in which the anomalous dynamics is due to the internal modes of the moving particles, and less to the disorder of the medium or to the external fields.

Anomalous diffusion of particles driven by correlated noise

We study the effect of an arbitrary stationary random force on the motion of damped particles. Using a Langevin description, we derive exact expressions for the dispersion of the particle position, of the particle velocity, and their cross dispersion. The particles can exhibit anomalous diffusion, and the connection between this behavior and the functional form of the noise correlations is investigated in detail. We also study anomalous diffusion for the special cases of overdamped and undamped particles.

Anomalous diffusion of particles with a finite free-motion velocity

The asymptotic behavior of the diffusion packet width is investigated in the anomalous diffusion of a particle with a finite free-motion velocity and a power distribution of free paths and waiting times in traps.

Anomalous self-diffusion of particles in a classically mesoscopic regime

An anomalous diffusion of particles has been found by a method of molecular dynamics calculation in the distribution p( x,t) of time-developing Cartesian component of particle displacement, x = x( t) − x(0), in a classically mesoscopic regime in some condensed matters. The distribution law is modified from that of the Gaussian process and is found to have the form p( x,t) ∞ exp{− ax β/ t α}, ( a > 0, 0⩽α⩽1, 0 < β⩽2), in thermal equilibrium state. Despite of this generalized distribution law, the mean square displacement 〈 x 2〉( t) ∞ t 2α/β seems to be t-linear, i.e. 2α/β ≅ 1, in some cases.

Trapping, percolation, and anomalous diffusion of particles in a two-dimensional random field

We analyze from first principles the advection of particles in a velocity field with HamiltonianH(x, y)=¯ V 1 y−¯ V 2 x+W 1 (y)-W 2 (x), whereW i , i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratioρ=max(|¯V1¦/Ū; ¦¯V2|/Ū), withŪ =〈|W 1 ′|2〉1/2=rms fluctuations. Remarkably, the model is exactly solvable forρ ⩾1 and well suited for Monte Carlo simulations for all ρ, providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines forρ=0, deriving power laws for pnc(L) and 〈λ(L)〉, which are, respectively, the escape probability and the length of escaping trajectories for a box of sizeL, L » 1. We also obtain a characterization of the “statistical topography” of the HamiltonianH. Second, we study the large-scale transport of advected particles withρ > 0. For 0 <ρ < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for ρ⩾ 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variablest − v/2[x(t)−〈x(t)〉] andt −v/2 [y(t)-〈(t)〉]. The large-scale motions are shown to be either Fickian (ν=1), or superdiffusive (ν=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for ρ ⩾ 1 and extended to the regime 0<ρ<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of $$(\bar V_1 ,\bar V_2 )$$ ) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.