Anomalous Diffusion Regime Research Articles

Effect of force and location of bottleneck for particle moving through window under encapsulation

One of the central assumptions to explain appearance of an anomalous diffusion regime in microporous materials is the cancellation of forces experienced by a moving particle at the bottleneck. In this work, we examine this issue of force cancellation inside a microporous material exploring the diffusion path and cross-checking bottleneck associated with diffusion. Conventional wisdom is that window of a microporous material is the bottleneck for diffusion. Our study reveals that bottleneck for diffusion is not at the window, it is always 2.6 A to 3.0 A away from the window plane and depending on guest size, it lies in different parts of the inner surface of $$\alpha $$ -cage. Window has significant effect in diffusion process only when diffusing particle size is comparable or larger than the window free diameter. Window does not play any role if particle size is smaller than window free diameter. We show that in the anomalous diffusion regime there is no signature of force cancellation at the window. Activation energy and total force along the diffusion path are responsible for the existence of anomalous diffusion regime in microporous materials. Correlation between diffusivity (D) and activation energy ( $$E_a$$ ) in faujasite zeolite is shown. Force cancellation at the window of microporous material does not play any role in the diffusion process unless diffusing particle size is larger than the window free diameter. The location of bottleneck for diffusion is not at the window, it is always 2.6 A to 3.0 A away from the window plane.

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

Divergent Diffusion Coefficients in Simulations of Fluids and Lipid Membranes.

We investigate the dependence of single-particle diffusion coefficients on the size and shape of the simulation box in molecular dynamics simulations of fluids and lipid membranes. We find that the diffusion coefficients of lipids and a carbon nanotube embedded in a lipid membrane diverge with the logarithm of the box width. For a neat Lennard-Jones fluid in flat rectangular boxes, diffusion becomes anisotropic, diverging logarithmically in all three directions with increasing box width. In elongated boxes, the diffusion coefficients normal to the long axis diverge linearly with the height-to-width ratio. For both lipid membranes and neat fluids, this behavior is predicted quantitatively by hydrodynamic theory. Mean-square displacements in the neat fluid exhibit intermediate regimes of anomalous diffusion, with t ln t and t(3/2) components in flat and elongated boxes, respectively. For membranes, the large finite-size effects, and the apparent inability to determine a well-defined lipid diffusion coefficient from simulation, rationalize difficulties in comparing simulation results to each other and to those from experiments.

Experimental Validation of a Nonextensive Scaling Law in Confined Granular Media.

In this Letter, we address the relationship between the statistical fluctuations of grain displacements for a full quasistatic plane shear experiment, and the corresponding anomalous diffusion exponent α. We experimentally validate a particular case of the Tsallis-Bukman scaling law, α=2/(3-q), where q is obtained by fitting the probability density function (PDF) of the displacement fluctuations with a q-Gaussian distribution, and the diffusion exponent is measured independently during the experiment. Applying an original technique, we are able to evince a transition from an anomalous diffusion regime to a Brownian behavior as a function of the length of the strain window used to calculate the displacements of the grains. The outstanding conformity of fitting curves to a massive amount of experimental data shows a clear broadening of the fluctuation PDFs as the length of the strain window decreases, and an increment in the value of the diffusion exponent-anomalous diffusion. Regardless of the size of the strain window considered in the measurements, we show that the Tsallis-Bukman scaling law remains valid, which is the first experimental verification of this relationship for a classical system at different diffusion regimes. We also note that the spatial correlations show marked similarities to the turbulence in fluids, a promising indication that this type of analysis can be used to explore the origins of the macroscopic friction in confined granular materials.

Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions

We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction ϕ of the crowders and the crowder–crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational—but not rotational—diffusivity of the particles D is a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law with the area fraction ϕ occupied by the crowders and the critical value Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.

Diffusion in Jammed Particle Packs.

Using random walk simulations we explore diffusive transport through monodisperse sphere packings over a range of packing fractions ϕ in the vicinity of the jamming transition at ϕ(c). Various diffusion properties are computed over several orders of magnitude in both time and packing pressure. Two well-separated regimes of normal "Fickian" diffusion, where the mean squared displacement is linear in time, are observed. The first corresponds to diffusion inside individual spheres, while the latter is the long-time bulk diffusion. The intermediate anomalous diffusion regime and the long-time value of the diffusion coefficient are both shown to be controlled by particle contacts, which in turn depend on proximity to ϕ(c). The time required to recover normal diffusion t* scales as (ϕ-ϕ(c))(-0.5) and the long-time diffusivity D(∞)∼(ϕ-ϕ(c))0.5, or D(∞)∼1/t*. It is shown that the distribution of mean first passage times associated with the escape of random walkers between neighboring particles controls both t* and D(∞) in the limit ϕ→ϕ(c).

Anomalous dispersion of sea ice in the Fram Strait region

AbstractThe single‐particle dispersion of sea ice in the Fram Strait region is investigated using ice drift buoys deployed from 2002 to 2009 within the Fram Strait Cyclones and the Arctic Climate System Study campaigns. A new method to estimate the direction of the mean flow, based on a satellite drift product, is introduced. As a result, the bias in the dispersion introduced by the mean flow is eliminated considering only the displacements of the buoys in the cross‐stream direction. Results show an absolute dispersion growing quadratically in time for the first 3 days and an anomalous dispersion regime exhibiting a strongly self‐similar scaling following a 5/4 power law for time scales larger than 6 days persisting over the whole time series of length 32 days. The non‐Gaussian distribution of the velocity fluctuations with a skewness of −0.15 and a kurtosis of 7.33 as well as the slope of the Lagrangian frequency spectrum between −2 and −1 are in agreement with the anomalous diffusion regime. Comparison with data from the International Arctic Buoy Program yields similar results with an anomalous dispersion starting after 10 days and persisting over the whole time series of length 50 days. The results suggest the presence of deformation and shear acting on the sea ice dispersion. The high correlation between the cross‐stream displacements and the cross‐stream wind velocities shows the important role of the wind as a source for the anomalous dispersion.

Anomalous diffusion process applied to magnetic resonance image enhancement

Diffusion process is widely applied to digital image enhancement both directly introducing diffusion equation as in anisotropic diffusion (AD) filter, and indirectly by convolution as in Gaussian filter. Anomalous diffusion process (ADP), given by a nonlinear relationship in diffusion equation and characterized by an anomalous parameters q, is supposed to be consistent with inhomogeneous media. Although classic diffusion process is widely studied and effective in various image settings, the effectiveness of ADP as an image enhancement is still unknown. In this paper we proposed the anomalous diffusion filters in both isotropic (IAD) and anisotropic (AAD) forms for magnetic resonance imaging (MRI) enhancement. Filters based on discrete implementation of anomalous diffusion were applied to noisy MRI T2w images (brain, chest and abdominal) in order to quantify SNR gains estimating the performance for the proposed anomalous filter when realistic noise is added to those images. Results show that for images containing complex structures, e.g. brain structures, anomalous diffusion presents the highest enhancements when compared to classical diffusion approach. Furthermore, ADP presented a more effective enhancement for images containing Rayleigh and Gaussian noise. Anomalous filters showed an ability to preserve anatomic edges and a SNR improvement of 26% for brain images, compared to classical filter. In addition, AAD and IAD filters showed optimum results for noise distributions that appear on extreme situations on MRI, i.e. in low SNR images with approximate Rayleigh noise distribution, and for high SNR images with Gaussian or non central χ noise distributions. AAD and IAD filter showed the best results for the parametric range 1.2 < q < 1.6, suggesting that the anomalous diffusion regime is more suitable for MRI. This study indicates the proposed anomalous filters as promising approaches in qualitative and quantitative MRI enhancement.

Fluctuation effects in bidirectional cargo transport

We discuss a theoretical model for bidirectional cargo transport in biological cells, which is driven by teams of molecular motors and subject to thermal fluctuations. The model describes explicitly the directed motion of the molecular motors on the filament. The motor-cargo coupling is implemented via linear springs. By means of extensive Monte Carlo simulations we show that the model describes the experimentally observed regimes of anomalous diffusion, i.e. subdiffusive behavior at short times followed by superdiffusion at intermediate times. The model results indicate that the subdiffusive regime is induced by thermal fluctuations while the superdiffusive motion is generated by correlations of the motors’ activity. We also tested the efficiency of bidirectional cargo transport in crowded areas by measuring its ability to pass barriers with increased viscosity. Our results show a remarkable gain of efficiency for high viscosities.

Diffusion tensor MRI phantom exhibits anomalous diffusion.

This paper reports diffusion weighted MRI measurements of cyclohexane in a novel diffusion tensor MRI phantom composed of hollow coaxial electrospun fibers (average diameter 10.2 μm). Recent studies of the phantom demonstrated its potential as a calibration standard at low b values (less than 1000 s/mm<;sup>2<;/sup>) for mean diffusivity and fractional anisotropy. In this paper, we extend the characterization of cyclohexane diffusion in this heterogeneous, anisotropic material to high b values (up to 5000 s/mm<;sup>2<;/sup>), where the apparent diffusive motion of the cyclohexane exhibits anomalous behavior (i.e., the molecular mean squared displacement increases with time raised to the fractional power 2α/β). Diffusion tensor MRI was performed at 9.4 T using an Agilent imaging scanner and the data fit to a fractional order Mittag-Leffler (generalized exponential) decay model. Diffusion along the fibers was found to be Gaussian (2α/β=l), while diffusion across the fibers was sub-diffusive (2α/β<;l). Fiber tract reconstruction of the data was consistent with scanning electron micrograph images of the material. These studies suggest that this phantom material may be used to calibrate MR systems in both the normal (Gaussian) and anomalous diffusion regimes.

Ultraslow diffusion in an exactly solvable non-Markovian random walk.

We study a one-dimensional discrete-time non-Markovian random walk with strong memory correlations subjected to pauses. Unlike the Scher-Montroll continuous-time random walk, which can be made Markovian by defining an operational time equal to the random-walk step number, the model we study keeps a record of the entire history of the walk. This new model is closely related to the one proposed recently by Kumar, Harbola, and Lindenberg [Phys. Rev. E 82, 021101 (2010)], with the difference that in our model the stochastic dynamics does not stop even in the extreme limit of subdiffusion. Surprisingly, this small difference leads to large consequences. The main results we report here are exact results showing ultraslow diffusion and a stationary diffusion regime (i.e., localization). Specifically, the equations of motion are solved analytically for the first two moments, allowing the determination of the Hurst exponent. Several anomalous diffusion regimes are apparent, ranging from superdiffusion to subdiffusion, as well as ultraslow and stationary regimes. We present the complete phase diffusion diagram, along with a study of the persistence and the statistics in the regions of interest.

Exact solution of an anisotropic 2D random walk model with strong memory correlations

Over the last decade, there has been progress in understanding one-dimensional non-Markovian processes via analytic, sometimes exact, solutions. The extension of these ideas and methods to two and higher dimensions is challenging. We report the first exactly solvable two-dimensional (2D) non-Markovian random walk model belonging to the family of the elephant random walk model. In contrast to Lévy walks or fractional Brownian motion, such models incorporate memory effects by keeping an explicit history of the random walk trajectory. We study a memory driven 2D random walk with correlated memory and stops, i.e. pauses in motion. The model has an inherent anisotropy with consequences for its diffusive properties, thereby mixing the dominant regime along one dimension with a subdiffusive walk along a perpendicular dimension. The anomalous diffusion regimes are fully characterized by an exact determination of the Hurst exponent. We discuss the remarkably rich phase diagram, as well as several possible combinations of the independent walks in both directions. The relationship between the exponents of the first and second moments is also unveiled.

Anomalous versus Slowed-Down Brownian Diffusion in the Ligand-Binding Equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.

Dynamical localization in chaotic systems: Spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems

We study the kicked rotator in the classically fully chaotic regime using Izrailev's N-dimensional model for various N≤4000, which in the limit N→∞ tends to the quantized kicked rotator. We do treat not only the case K=5, as studied previously, but also many different values of the classical kick parameter 5≤K≤35 and many different values of the quantum parameter kε[5,60]. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable Λ=l(∞)/N to the case of anomalous diffusion in the classical phase space by deriving the localization length l(∞) for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents β(BR). (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by β(loc) in the interval [0,1]. The level repulsion parameters β(BR) and β(loc) are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between β(loc) and the relative localization length Λ, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).

Admittance of diffusion limited adsorption coupled to reversible charge transfer on rough and finite fractal electrodes

A theoretical model is developed to quantify the effect of diffusion controlled adsorption of electroactive species on (reversible) charge transfer admittance at a rough electrode. Results are obtained for the linearized isotherm and applied on various electrode roughness models, viz. (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, viz. the power spectrum of roughness and (iii) random profile function with finite self-affine fractal properties. The general expressions for admittance density and total admittance have a systematic operator structure in Fourier transformed surface profile function. Realistic model of roughness requires inclusion of randomness in surface morphology, hence the quantity of interest, the mean admittance is obtained (by taking ensemble average over all possible random configurations). Mathematical expression for the mean admittance of random morphology is the functional of the power spectrum of roughness. A special case of finite fractal roughness is characterized as self-affine scaling property over limited length scales and hence possess two lateral cutoff lengths along with the fractal dimension and a topothesy length (ℓτ). Our analysis explicitly relates the problem of rough electrode admittance with diffusion length (D/ω), isotherm (length) parameter (K) and fractal morphological characteristics. The three characteristic frequencies in this theory are: (a) lower adsorption crossover frequency –ωad≈D/K2, (b) adsorption–roughness topothesy dependent weak pseudo-quasireversibility characteristic frequency –ωq≈D/(ℓτK) and (c) high frequency small cutoff length (ℓ) dependent Warburg – anomalous Warburg crossover frequency –ωw≈D/ℓ2. Finally, theory shows emergence of pseudo-quasireversibility in intermediate frequency region due to electrode roughness in adsorption–diffusion system and curtailment of high frequency anomalous diffusion regime.

Universality classes of transport in time-dependent random potentials

The growth of the average kinetic energy of classical particles is studied for potentials that are random both in space and time. Such potentials are relevant for recent experiments in optics and in atom optics. It is found that for small velocities uniform acceleration takes place, and at a later stage fluctuations of the potential are encountered, resulting in a regime of anomalous diffusion. This regime was studied in the framework of the Fokker-Planck approximation. The diffusion coefficient in velocity was expressed in terms of the average power spectral density, which is the Fourier transform of the potential correlation function. This enabled to establish a scaling form for the Fokker-Planck equation and to compute the large and small velocity limits of the diffusion coefficient. A classification of the random potentials into universality classes, characterized by the form of the diffusion coefficient in the limit of large and small velocity, was performed. It was shown that one-dimensional systems exhibit a large variety of universality classes, contrary to systems in higher dimensions, where only one universality class is possible. The relation to Chirikov resonances, which are central in the theory of chaos, was demonstrated. The general theory was applied and numerically tested for specific physically relevant examples.

Free induction decays in entangled polymer melts

The relaxation of the transverse magnetization components caused by both dipolar interactions between the spins of different polymer chains and the dipolar coupling between CH-protons on an isolated Kuhn segment along a single polymer chain have been calculated. Explicit expressions for the transverse relaxation function are given in terms of the absolute mean squared displacement of the Kuhn segment during melt gr(t), the tangent vector dynamical correlation function 〈bn(t)b(0)〉, the segmental relaxation time τs, the Kuhn segment length b, the bond length a0, the internuclear distance d, and the spin number density ρs. It is shown that the functional dependence of the intramolecular relaxation function on 〈bn(t)b(0)〉 is fairly weak. The time-dependence of the intramolecular contribution to the transverse relaxation function is dominated by the probability density distribution function of the end-to-end vector of the Kuhn segment. The long-time decay of the intramolecular contribution to the transverse relaxation function is found to scale as t−3/2 for τs<<t<<τmax, where τmaxis the maximum relaxation time of polymer chains in melts. For times much less than the spin–spin relaxation time, T2≈10−3−10−2s, we show that the intermolecular contribution to the relaxation function is given by the following expression: exp(−λ1(b, τs, ρs)t2/gr3/2(t)). Both the numerical coefficient and the functional dependence of λ1on b, τs and ρs reproduce the expression obtained from the frequently used second cumulant approximation. For longer times (T2≤t<<τmax), the intermolecular contribution is determined by the following relation: exp(−λ2(b, τs, ρs, t)gr(t)). We show that λ2 increases logarithmically with t. The molecular mass independence of λ1and λ2 shows that, in polymer melts with molecular masses Mw far above the critical value Mc, the relevant experimental window for the decay of the intermolecular relaxation function is connected with the anomalous diffusion regime. Comparison with the experimental data suggests that the intermolecular contribution plays a significant role in the NMR relaxation process in polymer systems close to the melting point.

ANOMALOUS DIFFUSION LIMIT FOR KINETIC EQUATIONS WITH DEGENERATE COLLISION FREQUENCY

This paper is devoted to hydrodynamic limits for collisional linear kinetic equations. It is a classical result that under certain conditions on the collision operator, the long time/small mean free path asymptotic behavior of the density of particles can be described by diffusion-type equations. We are interested in situations in which the degeneracy of the collision frequency for small velocities causes this limit to break down. We show that the appropriate asymptotic analysis leads to an anomalous diffusion regime.

Mathematical modeling of anomalous diffusion in porous media

Analysis of diffusion in a complex environment shows that the conventional diffusion equation based on Fick's law fails to model the anomalous character of the diffusive mass trans- port observed in the field and laboratory experiments. New mathematical models of diffusive transport, different from Fick's law, were proposed and validated in literature. In the present pa- per the examples of the equations that can be used for describing the anomalous mass transport are presented and some important properties of these equations are discussed. Two regimes of anomalous diffusion are identified. One regime, which is called sub-diffusion, is characterized by the slower propagation of the concentration front, so that the squared distance of the front pas- sage requires longer time than in the case of the classical Fickian diffusion. The second regime (called super-diffusion) is characterized by the higher diffusion rate, so that the particles will pass the specified distance faster than in the case of classical Fickian diffusion. Both regimes can be modeled by non-local diffusion equation with temporal and spatial fractional derivatives. It is shown that equation with spatially variable diffusivity proposed by O'Shaughnessy and Pro- caccia (1985), which provides a relatively good model of diffusion on a regular fractal, is less applicable for describing the effects of sub and super diffusion that may take place in a fractured porous medium or any other complex medium.

Three-dimensional Brownian diffusion of rod-like macromolecules in the presence of randomly distributed spherical obstacles: Molecular dynamics simulation

Brownian diffusion of rod-like polymers in the presence of randomly distributed spherical obstacles is studied using molecular dynamics simulations. It is observed that dependence of the reduced diffusion coefficient of these macromolecules on the available volume fraction can be described reasonably by a power law function. Despite the case of obstructed diffusion of flexible polymers in which reduced diffusion coefficient has a weak dependence on the polymer length, this dependence is noticeably strong in the case of rod-like polymers. Diffusion of these macromolecules in the presence of obstacles is observed that is anomalous at short time scales and normal at long times. Duration time of the anomalous diffusion regime is found that increases very rapidly with increasing both the polymer length and the obstructed volume fraction. Dynamics of diffusion of these polymers is observed that crosses over from Rouse to reptation type with increasing the density of obstacles.