Anomalous Diffusion Model Research Articles

Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative

In this paper, we consider a fractional nonlinear problem for anomalous diffusion. The diffusion coefficient we use is of power type, and hence the investigated problem generalizes the porous-medium equation. A generalization is made by introducing a fractional time derivative. We look for self-similar solutions for which the fractional setting introduces other than classical space–time scaling. The resulting similarity equations are of nonlinear integro-differential type. We approximate these equations by an expansion of the integral operator and by looking for solutions in a power function form. Our method can be easily adapted to solve various problems in self-similar diffusion. The approximations obtained give very good results in numerical analysis. Their simplicity allows for easy use in applications, as our fitting with experimental data shows. Moreover, our derivation justifies theoretically some previously used empirical models for anomalous diffusion.

Distinguishing free and anomalous diffusion by rectangular fluorescence recovery after photobleaching: a Monte Carlo study

Fluorescence recovery after photobleaching (FRAP) is a common technique to probe mobility of fluorescently labeled proteins in biological membranes by monitoring the time-dependence of the spatially integrated fluorescence signals after a bleaching pulse. Discrimination by FRAP between free diffusion with an immobile fraction (FDIM) and the phenomenological model for anomalous diffusion based on the time-dependent diffusion coefficient (TDDC) is a challenging problem, requiring extremely long observation times for differentiation. Recently, rectangular FRAP (rFRAP) has been introduced for normal diffusion by considering not only the temporal but also spatial information, taking the effective point spread function of the optical system into account. In this work we provide an extension of rFRAP toward anomalous diffusion according to the continuous time random walk (CTRW). We explore whether the spatial information in rFRAP allows for enhanced discrimination between FDIM, TDDC, and CTRW in a single experiment within a feasible time window. Simulations indicate that rFRAP can indeed differentiate the different models by evaluating the spatial autocorrelation of the differences between the measured and fitted pixel values. Hence, rFRAP offers a tool that is capable of discriminating different types of diffusion at shorter time scales than in the case where spatial information is discarded.

Cosmic ray anisotropy in fractional differential models of anomalous diffusion

The problem of galactic cosmic ray anisotropy is considered in two versions of the fractional differential model for anomalous diffusion. The simplest problem of cosmic ray propagation from a point instantaneous source in an unbounded medium is used as an example to show that the transition from the standard diffusion model to the Lagutin-Uchaikin fractional differential model (with characteristic exponent α = 3/5 and a finite velocity of free particle motion), which gives rise to a knee in the energy spectrum at 106 GeV, increases the anisotropy coefficient only by 20%, while the anisotropy coefficient in the Lagutin-Tyumentsev model (with exponents α = 0.3 and β = 0.8, a long stay of particles in traps, and an infinite velocity of their jumps) is close to one. This is because the parameters of the Lagutin-Tyumentsev model have been chosen improperly.

Water, salt water, and alkaline solution uptake in epoxy thin films

ABSTRACTAs a means of characterizing the diffusion parameters of fiber reinforced polymer (FRP) composites within a relatively short time frame, the potential use of short term tests on epoxy films to predict the long‐term behavior is investigated. Reference is made to the literature to assess the effectiveness of Fickian and anomalous diffusion models to describe solution uptake in epoxies. The influence of differing exposure conditions on the diffusion in epoxies, in particular the effect of solution type and temperature, are explored. Experimental results, where the solution uptake in desiccated (D) or undesiccated (U) thin films of a commercially available epoxy matrix subjected to water (W), salt water (SW), or alkali concrete pore solution (CPS) at either 20 or 60°C, are also presented. It was found that the type of solution did not significantly influence the diffusion behavior at 20°C and that the mass uptake profile was anomalous. Exposure to 60°C accelerated the initial diffusion behavior and appeared to raise the level of saturation. In spite of the accelerated approach, conclusive values of uptake at saturation remained elusive even at an exposure period of 5 years. This finding questions the viability of using short‐term thin film results to predict the long‐term mechanical performance of FRP materials. © 2013 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 130: 1898–1908, 2013

Escape time of cosmic rays from the galaxy in the anomalous diffusion model

Two versions of the anomalous diffusion model (Lagutin and Uchaikin’s and Lagutin and Tyumentsev’s) based on fractional transport equations are considered within the leaky-box approximation with respect to cosmic ray problems. The distributions of the first passage and escape times are computed via the Monte Carlo method. The observed difference between the results for the two versions is found to be a result of incorrectly choosing the Lagutin-Tyumentsev version’s parameters.

Fickian and non-Fickian diffusion with bimolecular reactions

Mixing zone dynamics of a reaction product $C$ during diffusion of two species ($A$ and $B$) are examined, using a two-dimensional particle tracking model (PT) for the reaction $A+B\ensuremath{\rightarrow}C$, allowing for both Fickian and non-Fickian transitions. The form of PT we use is equivalent to a continuous time random walk, which is a widely used model for anomalous transport and diffusion. It is shown that the basic patterns of the $C$ dynamics---the temporal evolution of the spatial profile and the temporal $C$ production---are similar for both modes of diffusion. However, the distinctive time scale for the non-Fickian case is very much larger even when the median transition steps are matched with the Fickian case. For immobile $C$, the spatial profile pattern is a broadening (Gaussian) reaction front evolving to a concentration-fluctuation dominated (Lorentzian) shape. The temporal $C$ production is fit well by a stretched exponential for both diffusion types. In analyzing experiments, the appearance of a Gaussian $C$ profile does not prove that the diffusion process is Fickian.

Complex patterns of non-Gaussian diffusion in artificial anisotropic tissue models

Motional barriers due to geometrical restrictions are ubiquitous in heterogeneous media such as porous rocks or biological tissues. As a consequence, molecular propagation deviates from the patterns of Gaussian diffusion. In this paper, we report on the results of the application of several non-Gaussian and anomalous diffusion models used to describe experimental data in the artificial anisotropic system with well-defined properties. In particular, we focus on the influence of fibre packing density on the quantitative metrics of these models and compare their sensitivity to this parameter. The results are discussed in the context of the importance for better understanding the governing diffusion mechanisms in complex tissue microstructures.

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

Abstract The leading role of a special function of the Wright-type, referred to as M-Wright or Mainardi function, within a parametric class of self-similar stochastic processes with stationary increments, is surveyed. This class of processes, known as generalized grey Brownian motion, provides models for both fast and slow anomalous diffusion. In view of a subordination-type formula involving M-Wright functions, these processes emerge to have all finite moments and be uniquely defined by their mean and auto-covariance structure like Gaussian processes. The corresponding master equation is shown to be a fractional differential equation in the Erdélyi-Kober sense and the diffusive process is named Erdélyi-Kober fractional diffusion. In Appendix, an historical overview on the M-Wright function is reported.

Proton to helium ratio in cosmic rays: analysis of the PAMELA data

In our previous papers we proposed an anomalous diffusion model for solution of the knee problem in the primary cosmic-rays spectrum and explanation of different values of spectral exponent of protons and other nuclei at wide energy region. The anomaly results from large free paths (Levy flights) of particles between magnetic domains. The physical arguments and the calculations indicate that the bulk of observed cosmic rays with energy 108 – 1010 eV is formed by numerous distant (r > 1 kpc) sources. It means that the contribution of these sources to the observed flux may be evaluated in the framework of the steady-state approach (JG). The contribution of the nearby (r < 1 kpc) relatively young (t < 105 yrs) sources defines the spectrum in the high energy region and provides the knee (JL). Consequently the cosmic rays intensity from all galactic sources for some group of nuclei consists of local JL and global JG components.We show that in the framework of our anomalous diffusion model with two-component representation of CR intensity the behavior of proton to helium ratio observed by PAMELA can be reproduced.

Monte Carlo simulations of anomalous diffusion of cosmic rays

The anomalous diffusion model for cosmic rays transport in fractal-like galactic medium developed by the authors allows to describe main features of the observed spectrum of charged particles arriving in the Solar system. Anomalous diffusion equation underlying the model contains fractional derivative operators and do not take into account the finite speed of particles transport. In this regard, we conducted a Monte Carlo simulation of uncoupled continuous-time random walks of particles in this model. The particles can perform an anomalously large jumps between scattering centers (Levy flights) and also can stay a long time in the inhomogeneities of the medium (Levy traps). The probability of such behavior of particles is described by power-law distributions for the jumps length and for the residence time in the inhomogeneity. In this paper, we present the spatial distribution of particles obtained from the direct simulations of particle trajectories in the framework of our model for two versions: considering finite particle speed and the case of an instantaneous jump.

Fronts in anomalous diffusion–reaction systems

A review of recent developments in the field of front dynamics in anomalous diffusion-reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

Approximation of a semigroup model of anomalous diffusion in a bounded set

The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.

A Fractional Anomalous Diffusion Model and Numerical Simulation for Sodium Ion Transport in the Intestinal Wall

The authors present a fractional anomalous diffusion model to describe the uptake of sodium ions across the epithelium of gastrointestinal mucosa and their subsequent diffusion in the underlying blood capillaries using fractional Fick’s law. A heterogeneous two-phase model of the gastrointestinal mucosa is considered, consisting of a continuous extracellular phase and a dispersed cellular phase. The main mode of uptake is considered to be a fractional anomalous diffusion under concentration gradient and potential gradient. Appropriate partial differential equations describing the variation with time of concentrations of sodium ions in both the two phases across the intestinal wall are obtained using Riemann-Liouville space-fractional derivative and are solved by finite difference methods. The concentrations of sodium ions in the interstitial space and in the cells have been studied as a function of time, and the mean concentration of sodium ions available for absorption by the blood capillaries has also been studied. Finally, numerical results are presented graphically for various values of different parameters. This study demonstrates that fractional anomalous diffusion model is appropriate for describing the uptake of sodium ions across the epithelium of gastrointestinal mucosa.

Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

Swelling characteristics of konjac glucomannan superabsobent synthesized by radiation-induced graft copolymerization

Graft copolymerization of konjac glucomannan (KGM) and acrylic acid was induced by 60Co-γ irradiation at room temperature. The effects of radiation dose and monomer-to-KGM ratio on grafting yield and equilibrium water absorbency were investigated. The KGM-based superabsorbent polymer (KSAP) could absorb water 625 times of its dry weight when the radiation dose was 5.0kGy and monomer-to-KGM ratio was 5. The structure of KSAP was characterized by FTIR, XRD, and SEM. KSAP showed a lower crystallinity than KGM. The porous microstructure of KSAP was revealed by SEM. The diffusion mechanism of water in the hydrogel is consistent with the anomalous diffusion model. Cations, especially multivalent cations, greatly reduced water absorbency of KSAP. Rising temperature, acidic or basic solutions are not favorable for the swelling of KSAP.

On the wavelet-based simulation of anomalous diffusion

The field of microrheology is based on experiments involving particle diffusion. Microscopic tracer beads are placed into a non-Newtonian fluid and tracked using high speed video capture and light microscopy. The modelling of the behaviour of these beads is now an active scientific area which demands multiple stochastic and statistical methods. We propose an approximate wavelet-based simulation technique for two classes of continuous time anomalous diffusion models, the fractional Ornstein–Uhlenbeck process and the fractional generalized Langevin equation. The proposed algorithm is an iterative method that provides approximate discretizations that converge quickly and in an appropriate sense to the continuous time target process. As compared to previous works, it covers cases where the natural discretization of the target process does not have closed form in the time domain. Moreover, we propose to minimize the border effect via smoothing.

Spatiotemporal dynamics of the biological interface between cancer and the microenvironment: a fractal anomalous diffusion model with microenvironment plasticity

BackgroundThe invasion-metastasis cascade of cancer involves a process of parallel progression. A biological interface (module) in which cells is linked with ECM (extracellular matrix) by CAMs (cell adhesion molecules) has been proposed as a tool for tracing cancer spatiotemporal dynamics.MethodsA mathematical model was established to simulate cancer cell migration. Human uterine leiomyoma specimens, in vitro cell migration assay, quantitative real-time PCR, western blotting, dynamic viscosity, and an in vivo C57BL6 mouse model were used to verify the predictive findings of our model.ResultsThe return to origin probability (RTOP) and its related CAM expression ratio in tumors, so-called "tumor self-seeding", gradually decreased with increased tumor size, and approached the 3D Pólya random walk constant (0.340537) in a periodic structure. The biphasic pattern of cancer cell migration revealed that cancer cells initially grew together and subsequently began spreading. A higher viscosity of fillers applied to the cancer surface was associated with a significantly greater inhibitory effect on cancer migration, in accordance with the Stokes-Einstein equation.ConclusionThe positional probability and cell-CAM-ECM interface (module) in the fractal framework helped us decipher cancer spatiotemporal dynamics; in addition we modeled the methods of cancer control by manipulating the microenvironment plasticity or inhibiting the CAM expression to the Pólya random walk, Pólya constant.

Entropy and Information in a Fractional Order Model of Anomalous Diffusion

Fractional order dynamic models (e.g., systems of ordinary and partial differential equations of non-integer order in time and space) are becoming more popular for characterizing the behavior of complex systems. Justification for such models is typically based on improved fits to experimental data or a reduced mean squared error for models with the same number of fitting parameters. This rationale, however, is relative to the form of the selected fitting function, and is dependent on the order of the derivatives. Nevertheless, there seems to be a recognition that fractional order models work better than integer order models in describing the electrical and mechanical properties of multi-scale, heterogeneous materials. In order to address this issue and to offer a new approach for establishing the utility of fractional order models, we calculate the total Shannon spectral entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation of the continuous time, random walk model of diffusion gives a spectral entropy minimum for normal (i.e., Gaussian) diffusion with surrounding values leading to greater values of spectral entropy.

Dual stage modeling of moisture absorption and desorption in epoxy mold compounds

A dual stage diffusion model is developed in this paper for both absorption and desorption processes. Both stages in moisture absorption and desorption, i.e., Fickian and non-Fickian process, are described mathematically using a combination of Fickian terms. Absorption and desorption tests are also conducted on six distinct commercial epoxy mold compounds (EMCs) used in electronic packaging. For absorption, the samples are subjected to 85°C/85% relative humidity and 60°C/85% relative humidity soaking, respectively. Desorption conditions are above glass transition temperature at 140°C and 160°C. The dual stage models generate reasonable results for the diffusive properties and display outstanding experimental fits. All six compounds show strong non-Fickian diffusion behaviors, which are further verified by the experiments with different thicknesses. For absorption, while Fickian diffusion is dominant in the beginning of process, non-Fickian mechanism plays a large role with time increasing. Saturated moisture concentration associated with Fickian-stage diffusion appears to be independent of temperature under the tested conditions. For desorption, higher temperature leads to less percentage of the permanent residual moisture content in most compounds. At 160°C, 90% of the initial moisture for all samples is diffused out within 24h, following an approximate modified Fickian diffusion process. The dual stage model developed in this paper provides a mathematical formulation for modeling anomalous moisture diffusion behavior using commercial finite element analysis software.

Space–time fractional diffusion on bounded domains

Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space–time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.