Anomalous Diffusion Behavior Research Articles

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

Variable-order derivative time fractional diffusion model for heterogeneous porous media

Constant-order derivative (COD) time fractional diffusion models have been successfully employed to describe transport processes where the rate of diffusion or the diffusion behavior differs from the classical Brownian motion (i.e. anomalous diffusion). More so, the Variable-order derivative (VOD) time fractional diffusion models have been recently acknowledged as an alternative and more rigorous approach to handle time-dependent anomalous diffusion behavior observed in highly heterogeneous fractured porous media.This study exhibits two finite difference approximations based on control volume finite difference approximations to handle a VOD time fractional diffusion mathematical model describing fluid flow in a heterogeneous porous medium. Subsequently, the numerical models are validated against the exact solution of the simplified COD time fractional diffusion mathematical model with constant rock and fluid properties. The well-established COD time fractional diffusion mathematical model is a special case of the herein presented VOD time fractional diffusion mathematical model. Furthermore, a modified incremental balance check and a modified wellbore flow model are presented to account for the variable order fractional exponent. The results from the numerical simulations are presented to reveal the peculiar features of the VOD time fractional diffusion model. This study would herald more applications of the VOD time fractional diffusion models in numerical modelling of fluid flow in porous media of fractal geometry or heterogeneous media.

Noise, delocalization, and quantum diffusion in one-dimensional tight-binding models.

As an unusual type of anomalous diffusion behavior, namely (transient) superballistic transport, has been experimentally observed recently, but it is not yet well understood. In this paper, we investigate the white noise effect (in the Markov approximation) on quantum diffusion in one-dimensional tight-binding models with a periodic, disordered, and quasiperiodic region of size L attached to two perfect lattices at both ends in which the wave packet is initially located at the center of the sublattice. We find that in a completely localized system, inducing noise could delocalize the system to a desirable diffusion phase. This controllable system may be used to investigate the interplay of disorder and white noise, as well as to explore an exotic quantum phase.

Anomalously high Na+ and low Li+ mobility in intercalated Na2Ti6O13.

We report an anomalous diffusion behavior in intercalated Na2Ti6O13. Using first-principles calculations, the direct migration of inserted Na+ along the tunnel direction is predicted to have a barrier of 0.24-0.44 eV, while the migration of inserted Li+ along the tunnel direction has a barrier of 0.86-1.15 eV. Although Li+ can also diffuse along a zig-zag path in the tunnel, the barrier of 0.86-0.99 eV is still much higher than that for Na+. Our results surprisingly lead to the conclusion that the diffusion of larger Na+ is 4-8 orders of magnitude faster than Li+ in the same host lattice, and explain the experimentally observed exceptional rate capability of Na2Ti6O13 as the Na-ion battery anode. The anomalous diffusion behavior is attributed to the geometric features of Na2Ti6O13. For migration of Li+ it is necessary to weaken Li-O bonds and to overcome the repulsion between Li and host Na ions simultaneously, while for Na+ diffusion the improved Na-O bonding at the transition state partially compensates for the energy penalty from the repulsion of host Na ions.

Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

Grown-in beryllium diffusion in indium gallium arsenide: An ab initio, continuum theory and kinetic Monte Carlo study

A roadblock in utilizing InGaAs for scaled-down electronic devices is its anomalous dopant diffusion behavior; specifically, existing models are not able to explain available experimental data on beryllium diffusion consistently. In this paper, we propose a more comprehensive model, taking self-interstitial migration and Be interaction with Ga and In into account. Density functional theory (DFT) calculations are first used to calculate the energy parameters and charge states of possible diffusion mechanisms. Based on the DFT results, continuum modeling and kinetic Monte Carlo simulations are then performed. The model is able to reproduce experimental Be concentration profiles. Our results suggest that the Frank-Turnbull mechanism is not likely, instead, kick-out reactions are the dominant mechanism. Due to a large reaction energy difference, the Ga interstitial and the In interstitial play different roles in the kick-out reactions, contrary to what is usually assumed. The DFT calculations also suggest that the influence of As on Be diffusion may not be negligible.

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.

Dynamics of single semiflexible polymers in dilute solution.

We study the dynamics of a single semiflexible chain in solution using computer simulations, where we systematically investigate the effect of excluded volume, chain stiffness, and hydrodynamic interactions. We achieve excellent agreement with previous theoretical considerations, but find that the crossover from the time τb, up to which free ballistic motion of the monomers describes the chain dynamics, to the times W-1 or τ0, where anomalous monomer diffusion described by Rouse-type and Zimm-type models sets in, requires two decades of time. While in the limit of fully flexible chains the visibility of the anomalous diffusion behavior is thus rather restricted, the t3/4 power law predicted for stiff chains without hydrodynamic interactions is verified. Including hydrodynamics, evidence for the predicted [tln(t)]3/4 behavior is obtained. Similar good agreement with previous theoretical predictions is found for the decay of the bond autocorrelation functions and the end-to-end vector correlation. Finally, several predictions on the variation of characteristic relaxation times with persistence length describing the chain stiffness are tested.

Modified cumulative distribution function in application to waiting time analysis in the continuous time random walk scenario

The continuous time random walk model plays an important role in modelling of the so-called anomalous diffusion behaviour. One of the specific properties of such model is the appearance of constant time periods in the trajectory. In the continuous time random walk approach they are realizations of the sequence called waiting times. In this work we focus on the analysis of waiting time distribution by introducing novel methods of parameter estimation and statistical investigation of such a distribution. These methods are based on the modified cumulative distribution function. In this paper we consider three special cases of waiting time distributions, namely α-stable, tempered stable and gamma. However, the proposed methodology can be applied to broad set of distributions—in general it may serve as a method of fitting any distribution function if the observations are rounded. The new statistical techniques are applied to the simulated data as well as to the real data of concentration in indoor air.

A modified memory-based mathematical model describing fluid flow in porous media

This study presents a modified memory-based mathematical model describing fluid flow in porous media. For the first time such model is derived employing the Grünwald–Letnikov (G–L) definition of the Riemann–Liouville (R–L) time fractional operator via a generalized Darcy’s equation. The proposed mathematical model is suitable to describe the anomalous diffusion behavior observed in a medium of fractal geometry, as well as in disordered and highly heterogeneous porous media. A numerical scheme based on existing discretization method is employed to handle the modified memory-based mathematical model. The accuracy of the numerical model is validated through the analytical solution for a simplified problem. In addition, numerical experiments are presented to demonstrate the effect of the anomalous diffusion exponent on the predicted reservoir pressure, wellbore pressure, and volumetric flux considering the flow of an under-saturated oil in a hydrocarbon reservoir as an example. Results show that decrease in magnitude of the anomalous diffusion exponent results in larger pore and wellbore pressure. This study would open a door to apply the G–L interpretation of the time fractional operator in numerical modeling of fluid flow through porous media.

Efficient implementation to numerically solve the nonlinear time fractional parabolic problems on unbounded spatial domain

Anomalous diffusion behavior in many practical problems can be described by the nonlinear time-fractional parabolic problems on unbounded domain. The numerical simulation is a challenging problem due to the dependence of global information from time fractional operators, the nonlinearity of the problem and the unboundedness of the spacial domain. To overcome the unboundedness, conventional computational methods lead to extremely expensive costs, especially in high dimensions with a simple treatment of boundary conditions by making the computational domain large enough. In this paper, based on unified approach proposed in [25], we derive the efficient nonlinear absorbing boundary conditions (ABCs), which reformulates the problem on unbounded domain to an initial boundary value problem on bounded domain. To overcome nonlinearity, we construct a linearized finite difference scheme to solve the reduced nonlinear problem such that iterative methods become dispensable. And the stability and convergence of our linearized scheme are proved. Most important, we prove that the numerical solutions are bounded by the initial values with a constant coefficient, i.e., the constant coefficient is independent of the time. Overall, the computational cost can be significantly reduced comparing with the usual implicit schemes and a simple treatment of boundary conditions. Finally, numerical examples are given to demonstrate the efficiency of the artificial boundary conditions and theoretical results of the schemes.

Infiltration experiments demonstrate an explicit connection between heterogeneity and anomalous diffusion behavior

AbstractTransport in systems containing heterogeneity distributed over multiple length scales can exhibit anomalous diffusion behaviors, where the time exponent, determining the spreading length scale of the transported scalar, differs from the expected value of . Here we present experimental measurements of the infiltration of glycerin, under a fixed pressure head, into a Hele‐Shaw cell containing a 3‐D printed distribution of flow obstacles; a system that is an analog for infiltration into a porous medium. In support of previously presented direct simulation results, we experimentally demonstrate that, when the obstacles are distributed as a fractal carpet with fractal dimension H < 2, the averaged progress of infiltration exhibits a subdiffusive behavior . We further show that observed values of the subdiffusion time exponent appear to be quadratically related to the fractal dimension of the carpet.

Computations of anomalous phase change

Purpose – The purpose of this paper is to demonstrate how anomalous diffusion behaviors can be manifest in physically realizable phase change systems. Design/methodology/approach – In the presence of heterogeneity the exponent in the diffusion time scale can become anomalous, exhibiting values that differ from the expected value of 1/2. Here the author investigates, through directed numerical simulation, the two-dimensional melting of a phase change material (PCM) contained in a pattern of cavities separated by a non-PCM matrix. Under normal circumstances we would expect that the progress of melting F(t) would exhibit the normal diffusion time exponent, i.e., F∼t1/2. The author’s intention is to investigate what features of the PCM cavity pattern might induce anomalous phase change, where the progress of melting has a time exponent different from n=1/2. Findings – When the PCM cavity pattern has an internal length scale, i.e., when there is a sub-domain pattern which, when reproduced, gives us the full domain pattern, the direct simulation recovers the normal ∼t1/2 phase change behavior. When, however, there is no internal length scale, e.g., the pattern is a truncated fractal, an anomalous super diffusive behavior results with melting going as t n; n > 1/2. By studying a range of related fractal patterns, the author is able to relate the observed sub-diffusive exponent to the cavity pattern’s fractal dimension. The author also shows, how the observed behavior can be modeled with a non-local fractional diffusion treatment and how sub-diffusion phase change behavior (F∼t n; n < 1/2) results when the phase change nature of the materials in the cavity and matrix are inverted. Research limitations/implications – Although the results clearly demonstrate under what circumstances anomalous phase change behavior can be practically produced, the question of an exact theoretical relationship between the cavity pattern geometry and the observed anomalous time exponent is not known. Practical implications – The clear role of the influence of heterogeneity on heat flow behavior is illustrated. Suggesting that modeling heat and fluid flow in heterogeneous systems requires careful consideration. Originality/value – The novel direct simulation of melting in a two-dimensional PCM cavity pattern provides a clear illustration of anomalous behavior in a classic heat and fluid flow system and by extension provides motivation to continue the numerical investigation of anomalous and non-local behaviors and fractional calculus tools.

Applications of the Fractional Sturm-Liouville Problem to the Space-Time Fractional Diffusion in a Finite Domain

Abstract The space–time fractional diffusion equations on finite domain model anomalous diffusion behavior with large particle jumps combined with long waiting times. In this work we prove existence of strong solutions for such equations. Our proofs strongly depend on the fractional Sturm–Liouville theory, precisely on the problem of finding eigenvalues and corresponding eigenfunctions to the certain fractional differential equation. Using the method of separating variables and applying theorem ensuring existence of solutions to the fractional Sturm–Liouville problem we solve several types of fractional diffusion equations.

Diffusion of Ellipsoids in Bacterial Suspensions.

Active fluids such as swarming bacteria and motile colloids exhibit exotic properties different from conventional equilibrium materials. As a peculiar example, a spherical tracer immersed inside active fluids shows an enhanced translational diffusion, orders of magnitude stronger than its intrinsic Brownian motion. Here, rather than spherical tracers, we investigate the diffusion of isolated ellipsoids in a quasi-two-dimensional bacterial bath. Our study shows a nonlinear enhancement of both translational and rotational diffusions of ellipsoids. More importantly, we uncover an anomalous coupling between particles' translation and rotation that is strictly prohibited in Brownian diffusion. The coupling reveals a counterintuitive anisotropic particle diffusion, where an ellipsoid diffuses fastest along its minor axis in its body frame. Combining experiments with theoretical modeling, we show that such an anomalous diffusive behavior arises from the generic straining flow of swimming bacteria. Our work illustrates an unexpected feature of active fluids and deepens our understanding of transport processes in microbiological systems.

Parameter estimation for the fractional fractal diffusion model based on its numerical solution

In this paper, we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media. The Bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution. Firstly, the center Box difference method is employed to solve the fractional fractal diffusion equation subject to the initial–boundary conditions. Then we apply the Bayesian method to estimate three parameters for the model simultaneously, that is the fractional order α, the fractal dimension df and the structure parameter θ. To testify the validity of the results for the model’s parameter estimation, experimental data from the fast desorption process of methane in coal are used. It is shown that the numerical results modeled with parameters given by the Bayesian method coincide well with the desorption experimental data, indicating that the Bayesian method is efficient and valid in identifying multi-parameter for the fractional fractal diffusion model. Besides, the fractional fractal diffusion model is demonstrated to be more capable than the classical Fick’s model to describe the anomalous diffusion behavior of gas in coal. To clarify the parameters’ effect on the relative diffusion amount of methane for the fractional fractal diffusion model, parameter sensitivity analysis is also carried out. This paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model.

Scaling forms of particle densities for Lévy walks and strong anomalous diffusion.

We study the scaling behavior of particle densities for Lévy walks whose transition length r is coupled with the transition time t as |r|∝t^{α} with an exponent α>0. The transition-time distribution behaves as ψ(t)∝t^{-1-β} with β>0. For 1<β<2α and α≥1, particle displacements are characterized by a finite transition time and confinement to |r|<t^{α} while the marginal distribution of transition lengths is heavy tailed. These characteristics give rise to the existence of two scaling forms for the particle density, one that is valid at particle displacements |r|≪t^{α} and one at |r|≲t^{α}. As a consequence, the Lévy walk displays strong anomalous diffusion in the sense that the average absolute moments 〈|r|^{q}〉 scale as t^{qν(q)} with ν(q) piecewise linear above and below a critical value q_{c}. We derive explicit expressions for the scaling forms of the particle densities and determine the scaling of the average absolute moments. We find that 〈|r|^{q}〉∝t^{qα/β} for q<q_{c}=β/α and 〈|r|^{q}〉∝t^{1+αq-β} for q>q_{c}. These results give insight into the possible origins of strong anomalous diffusion and anomalous behaviors in disordered systems in general.

Anomalous diffusion in fast cellular flows at intermediate time scales

It is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a Brownian motion. This paper studies the behaviour on “intermediate” time scales before diffusion sets in. Various heuristics suggest that an anomalous diffusive behaviour should be observed. We prove that the variance on intermediate time scales grows like \(O(\sqrt{t})\). Hence, on these time scales the effective behaviour can not be purely diffusive, and is consistent with an anomalous diffusive behaviour.

Evaluation of pH-sensitive poly(2-hydroxyethyl methacrylate-co-2-(diisopropylamino)ethyl methacrylate) copolymers as drug delivery systems for potential applications in ophthalmic therapies/ocular delivery of drugs

Smart polymers like pH sensitive systems can improve different pharmacological treatment. In this work the behavior of copolymers containing 2-hydroxyethyl methacrylate (HEMA) with different proportions of 2-(diisopropy- lamino)ethyl methacrylate (DPA) and different amounts of cross-linker agent, ethylene glycol dimethacrylate (EGDMA) are evaluated as pH-sensitive drug delivery systems for potential application in ophthalmic therapies. A detailed character- ization of the pH-responsive behavior was performed by swelling studies and scanning electron microscopy (SEM) analy- sis. Drug loading and release studies at different pH values were evaluated using Rhodamine 6G (Rh6G) as a model drug. The interaction between Rh6G and hydrogels was studied by Fourier Transform Infrared (FTIR) spectroscopy and scanning electron microscopy (SEM). The results show that the presence of DPA in the copolymers confers pH-responsive properties to the polymer, as noted in swelling and SEM studies, when the pH decreases below 7.40 the swelling degree increases and a porous morphology is observed. The apparent pKa of copolymers was estimated between 6.80 and 7.17 depending on the composition. The amount of Rh6G loaded depends mainly on the medium pH and the interaction between the drug and the copolymers, observed by SEM and FTIR spectrum. The release of Rh6G of copolymers p(HEMA/DPA) show a normal Fickian or anomalous diffusion behavior at different pH values, depending on the HEMA/DPA ratio.

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.