Superdiffusive Regime Research Articles

Magnetic flux conservation in an imploding plasma.

The theory of magnetic flux conservation is developed for a subsonic plasma implosion and used to describe the magnetic flux degradation in the MagLIF concept [S. A. Slutz etal., Phys. Plasmas 17, 056303 (2010)10.1063/1.3333505]. Depending on the initial magnetic Lewis and Péclet numbers and the electron Hall parameter, the implosion falls into either a superdiffusive regime in which the magnetization decreases or a magnetized regime in which the magnetization increases. Scaling laws for magnetic field, temperature, and magnetic flux losses in the hot spot of radius R are obtained for both regimes. The Nernst velocity convects the magnetic field outwards, pushing it against the liner and enhancing the magnetic field diffusion, thereby reducing the magnetic field compression and degrading the implosion performance. However, in the magnetized regime, the core of the hot spot becomes magnetically insulated and undergoes an ideal adiabatic compression (T∼R^{-4/3} compared to T∼R^{-2/3} without magnetic field), while the detrimental Nernst term is confined to the outer part of the hot spot. Its effect is drastically reduced, improving the magnetic flux conservation.

A martingale approach for the elephant random walk

The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter p which lies between zero and one. This behavior is totally different in the diffusive regime , the critical regime , and the superdiffusive regime . In the diffusive and critical regimes, we establish some new results on the almost sure asymptotic behavior of the ERW, such as the quadratic strong law and the law of the iterated logarithm. In the superdiffusive regime, we provide the first rigorous mathematical proof that the limiting distribution of the ERW is not Gaussian.

A random acceleration model of individual animal movement allowing for diffusive, superdiffusive and superballistic regimes

Patterns of individual animal movement attracted considerable attention over the last two decades. In particular, question as to whether animal movement is predominantly diffusive or superdiffusive has been a focus of discussion and controversy. We consider this problem using a theory of stochastic motion based on the Langevin equation with non-Wiener stochastic forcing that originates in animal’s response to environmental noise. We show that diffusive and superdiffusive types of motion are inherent parts of the same general movement process that arises as interplay between the force exerted by animals (essentially, by animal’s muscles) and the environmental drag. The movement is superballistic with the mean square displacement growing with time as langle {x}^{2}(t)rangle sim {t}^{4} at the beginning and eventually slowing down to the diffusive spread langle {x}^{2}(t)rangle sim t. We show that the duration of the superballistic and superdiffusive stages can be long depending on the properties of the environmental noise and the intensity of drag. Our findings demonstrate theoretically how the movement pattern that includes diffusive and superdiffusive/superballistic motion arises naturally as a result of the interplay between the dissipative properties of the environment and the animal’s biological traits such as the body mass, typical movement velocity and the typical duration of uninterrupted movement.

Modeling methanol transfer in the mesoporous catalyst for the methanol-to-olefins reaction by the time-fractional diffusion equation

The solutions of the time-fractional diffusion equation for the short and long times are obtained via an application of the asymptotic Green's functions. The derived solutions are applied to analysis of the methanol mass transfer through H-ZSM-5/alumina catalyst grain. It is demonstrated that the methanol transport in the catalysts pores may be described by the obtained solutions in a fairly good manner. The measured fractional exponent is equal to 1.20 ± 0.02 and reveals the super-diffusive regime of the methanol mass transfer. The presence of the anomalous transport may be caused by geometrical restrictions and the adsorption process on the internal surface of the catalyst grain's pores.

Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Method to characterize directional changes in Arctic sea ice drift and associated deformation due to synoptic atmospheric variations using Lagrangian dispersion statistics

Abstract. A framework is developed to assess the directional changes in sea ice drift paths and associated deformation processes in response to atmospheric forcing. The framework is based on Lagrangian statistical analyses leveraging particle dispersion theory which tells us whether ice drift is in a subdiffusive, diffusive, ballistic, or superdiffusive dynamical regime using single-particle (absolute) dispersion statistics. In terms of sea ice deformation, the framework uses two- and three-particle dispersion to characterize along- and across-shear transport as well as differential kinematic parameters. The approach is tested with GPS beacons deployed in triplets on sea ice in the southern Beaufort Sea at varying distances from the coastline in fall of 2009 with eight individual events characterized. One transition in particular follows the sea level pressure (SLP) high on 8 October in 2009 while the sea ice drift was in a superdiffusive dynamic regime. In this case, the dispersion scaling exponent (which is a slope between single-particle absolute dispersion of sea ice drift and elapsed time) changed from superdiffusive (α ∼ 3) to ballistic (α ∼ 2) as the SLP was rounding its maximum pressure value. Following this shift between regimes, there was a loss in synchronicity between sea ice drift and atmospheric motion patterns. While this is only one case study, the outcomes suggest similar studies be conducted on more buoy arrays to test momentum transfer linkages between storms and sea ice responses as a function of dispersion regime states using scaling exponents. The tools and framework developed in this study provide a unique characterization technique to evaluate these states with respect to sea ice processes in general. Application of these techniques can aid ice hazard assessments and weather forecasting in support of marine transportation and indigenous use of near-shore Arctic areas.

Dispersion of the solar magnetic flux in the undisturbed photosphere as derived from SDO/HMI data

To explore the magnetic flux dispersion in the undisturbed solar photosphere, magnetograms acquired by Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamic Observatory (SDO) were utilized. Two areas, a coronal hole area (CH) and an area of super-granulation pattern, SG, were analyzed. We explored the displacement and separation spectra and the behavior of the turbulent diffusion coefficient, $K$. The displacement and separation spectra are very similar to each other. Small magnetic elements (of size 3-100 squared pixels and the detection threshold of 20 Mx sm$^{-2}$) in both CH and SG areas disperse in the same way and they are more mobile than the large elements (of size 20-400 squared pixels and the detection threshold of 130 Mx sm$^{-2}$). The regime of super-diffusivity is found for small elements ($\gamma \approx 1.3 $ and $K$ growing from $\sim$100 to $\sim$ 300 km$^2$ s$^{-1}$). Large elements in the CH area are scanty and show super-diffusion with $\gamma \approx 1.2$ and $K$ = (62-96) km$^2$ s$^{-1}$ on rather narrow range of 500-2200 km. Large elements in the SG area demonstrate two ranges of linearity and two diffusivity regimes: sub-diffusivity on scales (900-2500) km with $\gamma=0.88$ and $K$ decreasing from $\sim$130 to $\sim$100 km$^2$ s$^{-1}$, and super-diffusivity on scales (2500-4800) km with $\gamma \approx 1.3$ and $K$ growing from $\sim$140 to $\sim$200 km$^2$ s$^{-1}$. Comparison of our results with the previously published shows that there is a tendency of saturation of the diffusion coefficient on large scales, i.e., the turbulent regime of super-diffusivity is gradually replaced by normal diffusion.

Penetration of probe field in heated plasma

Dependences of the effective penetration depth of probe pulse field in current-carrying plasma with time-dependent temperatures of particles are determined. It is shown that the field penetration in fully and weakly ionized plasmas takes place in the subdiffusion and superdiffusion regimes, respectively.

Central limit theorem and related results for the elephant random walk

We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on ℤ with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior. We prove a law of large numbers and a central limit theorem. Remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive. Inside the superdiffusive regime, the ERW converges to a non-degenerate random variable which is not normal. We also obtain explicit expressions for the correlations of increments of the ERW.

Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging

The quantification of anomalous diffusion is increasingly being recognised as an advanced modality of analysis for the evaluation of tissue microstructure in magnetic resonance imaging (MRI). One powerful framework to account for anomalous diffusion in biological and structurally heterogeneous tissues is the use of diffusion operators based on fractional calculus theory, which generalises the physical principles of standard diffusion in homogeneous media. However, their non-locality makes analytical solutions often unavailable, limiting the applicability of these modelling and analysis techniques. In this paper, we derive compact analytical signal decays for practical MRI sequences in the anisotropic fractional Bloch–Torrey setting, as described by the space fractional Laplacian and importantly the time Caputo derivative. The attained solutions convey relevant characteristics of MRI in biological tissues not replicated by standard diffusion, including super-diffusive and sub-diffusive regimes in signal decay and the diffusion-driven incomplete refocusing of spins at the end of the sequence. These results may therefore have significant implications for advancing the current interpretation of MRI, and for the estimation of tissue properties based on exact solutions to underlying diffusive processes.

Dynamics of Branched Polymers in Random Layered Flows with Intramolecular Hydrodynamic Coupling: Star and Dendrimer

The authors develop a theoretical formalism to incorporate the effect of intramolecular hydrodynamic interactions (HIs) on the dynamics of flexible branched polymer in the presence of random layered flows. The influence of HIs on the anomalous diffusive behavior of branched polymers is illustrated through the preaveraged Oseen tensor approach. Although the formalism is valid for polymer structures with arbitrary topology, particular attention is paid here to the study of stars and dendrimers. The macromolecular property that is evaluated is the average square displacement (ASD) of drift center of the polymer. Qualitatively, our analysis highlights two anomalous power‐law regimes, viz. subdiffusive (intermediate‐time polymer stretching and flow induced diffusion) and superdiffusive (long‐time flow induced diffusion). The time dependence of the ASD in the presence of HIs within the preaveraging approximation reveals the anomalous long‐time dynamics which is governed by scaling behavior, t2 − α/2. The introduction of HIs in random flows speeds up the dynamics resulting in the shorter crossover time (from subdiffusive to superdiffusive regime) with enhanced magnitude of ASD compared to the free‐draining limit. image

Non-Fickian diffusion of methanol in mesoporous media: Geometrical restrictions or adsorption-induced?

The methanol mass transfer in the mesoporous silica and alumina/zeolite H-ZSM-5 grains has been studied. We demonstrate that the methanol diffusion is characterized as a time-fractional for both solids. Methanol transport occurs in the super-diffusive regime, which is faster comparing to the Fickian diffusion. We show that the fractional exponents defining the regime of transport are different for each porous grain. The difference between the values of the fractional exponents is associated with a difference in the energetic strength of the active sites of the surface of the media of different chemical nature as well as the geometrical restrictions of the porous media. Increasing by six-fold, the pore diameter leads to a 1.1 fold increase of the fractional exponent. Decreasing by three-fold, the methanol desorption energy results into the same increasing the fractional exponent. Our findings support that mainly the adsorption process, which is defined by the energetic disorder of the corresponding surface active sites, is likely to be the driving force of the abnormality of the mass transfer in the porous media. Therefore, the fractional exponent is a fundamental characteristic which is individual for each combination of the porous solid and diffusing species.

Relaxation of charge in monolayer graphene: Fast nonlinear diffusion versus Coulomb effects

Pristine monolayer graphene exhibits very poor screening because the density of states vanishes at the Dirac point. As a result, charge relaxation is controlled by the effects of zero-point motion (rather than by the Coulomb interaction) over a wide range of parameters. Combined with the fact that graphene possesses finite intrinsic conductivity, this leads to a regime of relaxation described by a nonlinear diffusion equation with a diffusion coefficient that diverges at zero charge density. Some consequences of this fast diffusion are self-similar superdiffusive regimes of relaxation, the development of a charge depleted region at the interface between electron- and hole-rich regions, and finite extinction times for periodic charge profiles.

Solutions for a mass transfer process governed by fractional diffusion equations with reaction terms

We investigate the behavior of a mass transfer process governed by a set of fractional diffusion equations coupled by appropriate reaction terms. The presence of memory effects in the diffusive term is also considered. For this set of equations, we obtain solutions and analyze the influence of the reaction terms on the spreading of these solutions. Particularly, we observe that for reversible reaction processes the reaction terms play an important role for intermediate times and for long times the processes are essentially governed by the bulk equations. These results show a rich class of behaviors which can be connected to sub- or superdiffusive regime.

Anomalous diffusion and diffusion anomaly in confined Janus dumbbells

Self-assembly and dynamical properties of Janus nanoparticles have been studied by molecular dynamic simulations. The nanoparticles are modeled as dimers and they are confined between two flat parallel plates to simulate a thin film. One monomer from the dumbbells interacts by a standard Lennard-Jones potential and the other by a two-length scales shoulder potential, typically used for anomalous fluids. Here, we study the effects of removing the Brownian effects, typical from colloidal systems immersed in aqueous solution, and consider a molecular system, without the drag force and the random collisions from the Brownian motion. Self-assembly and diffusion anomaly are preserved in relation to the Brownian system. Additionally, a superdiffusive regime associated to a collective reorientation in a highly structured phase is observed. Diffusion anomaly and anomalous diffusion are explained in the two length scale framework.

Dynamics of comb-of-comb-network polymers in random layered flows.

We analyze the dynamics of comb-of-comb-network polymers in the presence of external random flows. The dynamics of such structures is evaluated through relevant physical quantities, viz., average square displacement (ASD) and the velocity autocorrelation function (VACF). We focus on comparing the dynamics of the comb-of-comb network with the linear polymer. The present work displays an anomalous diffusive behavior of this flexible network in the random layered flows. The effect of the polymer topology on the dynamics is analyzed by varying the number of generations and branch lengths in these networks. In addition, we investigate the influence of external flow on the dynamics by varying flow parameters, like the flow exponent α and flow strength W_{α}. Our analysis highlights two anomalous power-law regimes, viz., subdiffusive (intermediate-time polymer stretching and flow-induced diffusion) and superdiffusive (long-time flow-induced diffusion). The anomalous long-time dynamics is governed by the temporal exponent ν of ASD, viz., ν=2-α/2. Compared to a linear polymer, the comb-of-comb network shows a shorter crossover time (from the subdiffusive to superdiffusive regime) but a reduced magnitude of ASD. Our theory displays an anomalous VACF in the random layered flows that scales as t^{-α/2}. We show that the network with greater total mass moves faster.

Anisotropic macroturbulence and diffusion associated with a westward zonal jet: From laboratory to planetary atmospheres and oceans.

Turbulence with inverse energy cascade and its transport properties are investigated experimentally in a flow associated with a westward propagating jet. Turbulence and the jet were produced by an electromagnetic force in a rotating tank filled with an electrolytic saline solution. The parabolic free surface emulated the topographic β effect which evoked the zonation. The spectral and transport flow characteristics were highly anisotropic. Turbulence is diagnosed by exploring the analogy between vertical and horizontal turbulent overturns in, respectively, stably stratified and quasigeostrophic flows which gives rise to a method of potential vorticity (PV) monotonizing. The anisotropization of transport properties of the flow is investigated using the finite scale Lyapunov exponent technique. After initial exponential particle separation, radial (meridional in geophysical and planetary applications) diffusion attains a short-ranged Richardson regime which transitions to the Taylor (scale-independent diffusivity) one. The azimuthal (zonal) diffusion exhibits a double-plateau structure which attains a superdiffusive regime on large scales. The transition to the Taylor regime for the radial diffusion takes place at a scale of turbulence anisotropization. The radial eddy diffusivity in both regimes as well as the transition scale are all determined by the rate of the inverse energy cascade, ε, that can be diagnosed by the PV monotonizing. Conversely, ε can be deduced from the scale of the Richardson-Taylor regime transition in the radial eddy diffusivity which, thus, provides an additional tool of diagnosing anisotropic macroturbulence with inverse energy cascade.

Diagnostics of turbulent and fractal properties of photospheric plasma outside active regions of the Sun

Results of analysis of multi-scale and turbulent properties of observed photospheric granulation patterns in undisturbed solar photosphere are presented. Data were obtained with the New Solar Telescope at Big Bear Solar observatory. Different types of magnetic environment were explored: a coronal hole (CH) area, a quiet sun (QS) intranetwork area, a QS/network area, and an area with small pores. The property of multifractality was revealed for granulation patterns in all environments on scales below 600 km. The degree of multifractality tends to be stronger as the magnetic environment becomes weaker. Analysis of turbulent diffusion on scales less than 1000–2000 km revealed the regime of super-diffusivity for all data sets. Super-diffusion becomes stronger from the QS/network to the QS/intranetwork to the CH. Both multifractality and super-diffusivity on very small scales are associated with the fast turbulent dynamo action. The results show that the most favorable conditions for the fast turbulent dynamo are met outside the network, inside vast areas of weakest magnetic fields, which supports the idea of nonlocal, deep turbulent dynamo.

Transport of relativistic electrons at shocks in shell-type supernova remnants: diffusive and superdiffusive regimes

Context. Understanding the transport properties of energetic particles in the presence of magnetic turbulence is crucial for interpreting observations of supernova remnants (SNR) and for assessing the cosmic-ray acceleration mechanism.Aims. We aim at obtaining information on the transport regimes of energetic electrons upstream and downstream of SNR blast waves by studying the X-ray rims.Methods. We considered emission profiles expected when synchrotron energy losses dominate, normal diffusion (typically causing an exponential decay), and superdiffusion (causing a power-law decay) for a spherically symmetric model. Then we compared the model profiles, projected on the plane of the sky, with Chandra observations of supernova (SN)1006 and Tycho’s SN.Results. Our study shows that downstream of the blast wave the observed profile is exponentially cut-off due to synchrotron energy losses in the amplified and shock compressed magnetic field present. Upstream of the blast wave and close to the shock, the observed profile of SN1006 is well reproduced by electron diffusion in the Bohm regime. However, the long X-ray tail far upstream of the shock is better explained by considering a change in the energetic electrons’ transport regime, that appears to become superdiffusive and gives rise to a power law intensity profile on scales over which the magnetic field strength can be assumed to be constant. Similar results are obtained for Tycho, although in this case the nonlinear effects associated with particle acceleration appear to be stronger close to the shock.Conclusions. The analysis of Chandra observations of the X-ray thin rims of SN1006 and of Tycho’s SN, SN 1572, can be well explained assuming normal diffusion in the vicinity of the blast wave and superdiffusion far upstream, similarly to what has been found for particles accelerated at interplanetary shocks. This suggests that anomalous transport is common in both interplanetary space and interstellar medium.

An Investigation of Methanol Anomalous Diffusion in Mesoporous Silica

Methanol transport in mesoporous silica is investigated. It is demonstrated that usual approach based on the second Fick’s law fails describing the experimental kinetic data. Contrary, the solution of the time-fractional diffusion equation fits the experimental data in a fairly good manner. Obtained value of the fractional order reveals the presence of fast super-diffusive regime of transport.