Anomalous Diffusion Model Research Articles

Features of Charging–Discharging of Supercapacitors

We present the results of experimental investigations of supercapacitors produced by Panasonic using several methods, namely, measurements of temporal dependences of charging–discharging currents, cyclic dc charging, and cyclic voltammetry. The values of the internal resistance, static and dynamic capacitance, as well as their dependences on the bias voltage of the capacitors, have been determined. Measurements have shown that the initial stage of relaxation of current is well approximated by the exponential time dependence, and then transition to a powerlike dependence occurs. This well-known effect is explained on the basis of the allowance for specific transport properties of a porous fractal-type medium. These processes are adequately described by the fractional-differential model of anomalous diffusion. A weak dependence of relaxation curves on the voltage at low values of the latter ( 3 V) is explained by the appearance of new percolation paths blocked at low charging voltages due to the presence of high-potential barriers. The internal resistance and the static capacitance have been determined by measuring the voltage across the supercapacitor in the mode of dc charging. These parameters have been shown to depend on the voltage applied to the capacitor. The dependence of the dynamic capacitance on the voltage has been determined using cyclic voltammetry. It has been shown that the capacitance depends not only on the voltage, but also on the prehistory of charging and discharging of the capacitor. Comparison of the experimental results and the published data on the models and equivalent circuits with passive R, L, and C elements allows one to conclude that such models and equivalent circuits can be applied only when explaining a limited number of phenomena, in particular, behavior at small relaxation times.

Conformable derivative approach to anomalous diffusion

By using a new derivative with fractional order, referred to conformable derivative, an alternative representation of the diffusion equation is proposed to improve the modeling of anomalous diffusion. The analytical solutions of the conformable derivative model in terms of Gauss kernel and Error function are presented. The power law of the mean square displacement for the conformable diffusion model is studied invoking the time-dependent Gauss kernel. The parameters related to the conformable derivative model are determined by Levenberg–Marquardt method on the basis of the experimental data of chloride ions transportation in reinforced concrete. The data fitting results showed that the conformable derivative model agrees better with the experimental data than the normal diffusion equation. Furthermore, the potential application of the proposed conformable derivative model of water flow in low-permeability media is discussed.

Overdamped Oscillator Model with a Complex Viscosity To Interpret Impedance Spectroscopy Data

We propose a new framework to interpret impedance spectroscopy data by taking into account the ionic motion near to the electrodes limiting the electrolytic sample. This ionic motion is described by an overdamped harmonic oscillator model (ODO). We introduce a complex viscosity approach that has some equivalence with an anomalous diffusion model. The results show that this complex viscosity model predicts an impedance behavior that is very similar to the one predicted by an equivalent circuit having a constant phase element (CPE). We demonstrate that this mechanical model based on the overdamped oscillator has always an equivalent circuit, and both approaches may be applied to analyze impedance spectroscopy data, especially in the low-frequency region.

Ergodic properties of Lévy flights coexisting with subdiffusion and related models

We analyze ergodic properties of two different stationary processes resulting from a combination of Lévy flights and subdiffusion. The first one comes from the Lamperti transformation of subordinated γ-stable process. The second one is a sequence of increments of subordinated Lévy process. We prove that both processes are mixing and ergodic. We also study increments and the Lamperti transformation of time-changed fractional Brownian motion. It appears that these models of anomalous diffusion are also ergodic and mixing. We believe that the derived results will help to verify ergodicity in single particle tracking experiments.

Formation of a relation of nonlocalities in the anomalous diffusion model

Построена модель случайного блуждания, в которой соотношение пространственно-временных нелокальностей определяется структурой потока памяти и стохастической моделью сил. Предлагаемая модель позволяет вычислять параметры, характеризующие нелокальность воздействия среды и памяти частицы.

Formation of a relation of nonlocalities in the anomalous diffusion model

We construct a model of a random walk in which the relation of space–time nonlocalities is defined by the structure of memory flow and a stochastic force model. The proposed model allows computing the parameters that characterize the nonlocality of the medium exposure and the particle memory.

Chung-type law of the iterated logarithm for continuous time random walk

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time random walk with jumps and waiting times in the domains of attraction of stable laws.

Apparent Anomalous Diffusion in the Cytoplasm of Human Cells: The Effect of Probes' Polydispersity.

This work, based on in vivo and in vitro measurements, as well as in silico simulations, provides a consistent analysis of diffusion of polydisperse nanoparticles in the cytoplasm of living cells. Using the example of fluorescence correlation spectroscopy (FCS), we show the effect of polydispersity of probes on the experimental results. Although individual probes undergo normal diffusion, in the ensemble of probes, an effective broadening of the distribution of diffusion times occurs-similar to anomalous diffusion. We introduced fluorescently labeled dextrans into the cytoplasm of HeLa cells and found that cytoplasmic hydrodynamic drag, exponentially dependent on probe size, extraordinarily broadens the distribution of diffusion times across the focal volume. As a result, the in vivo FCS data were effectively fitted with the anomalous subdiffusion model while for a monodisperse probe the normal diffusion model was most suitable. Diffusion time obtained from the anomalous diffusion model corresponds to a probe whose size is determined by the weight-average molecular weight of the polymer. The apparent anomaly exponent decreases with increasing polydispersity of the probes. Our results and methodology can be applied in intracellular studies of the mobility of nanoparticles, polymers, or oligomerizing proteins.

Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion.

In this Letter, we present a measurement of the phase-space density distribution (PSDD) of ultracold ^{87}Rb atoms performing 1D anomalous diffusion. The PSDD is imaged using a direct tomographic method based on Raman velocity selection. It reveals that the position-velocity correlation function C_{xv}(t) builds up on a time scale related to the initial conditions of the ensemble and then decays asymptotically as a power law. We show that the decay follows a simple scaling theory involving the power-law asymptotic dynamics of position and velocity. The generality of this scaling theory is confirmed using MonteCarlo simulations of two distinct models of anomalous diffusion.

Extended diffusion weighted magnetic resonance imaging with two-compartment and anomalous diffusion models for differentiation of low-grade and high-grade brain tumors in pediatric patients.

The purpose of this study was to examine advanced diffusion-weighted magnetic resonance imaging (DW-MRI) models for differentiation of low- and high-grade tumors in the diagnosis of pediatric brain neoplasms. Sixty-two pediatric patients with various types and grades of brain tumors were evaluated in a retrospective study. Tumor type and grade were classified using the World Health Organization classification (WHO I-IV) and confirmed by pathological analysis. Patients underwent DW-MRI before treatment. Diffusion-weighted images with 16 b-values (0-3500s/mm2) were acquired. Averaged signal intensity decay within solid tumor regions was fitted using two-compartment and anomalous diffusion models. Intracellular and extracellular diffusion coefficients (Dslow and Dfast), fractional volumes (Vslow and Vfast), generalized diffusion coefficient (D), spatial constant (μ), heterogeneity index (β), and a diffusion index (index_diff=μ×Vslow/β) were calculated. Multivariate logistic regression models with stepwise model selection algorithm and receiver operating characteristic (ROC) analyses were performed to evaluate the ability of each diffusion parameter to distinguish tumor grade. Among all parameter combinations, D and index_diff jointly provided the best predictor for tumor grades, where lower D (p=0.03) and higher index_diff (p=0.009) were significantly associated with higher tumor grades. In ROC analyses of differentiating low-grade (I-II) and high-grade (III-IV) tumors, index_diff provided the highest specificity of 0.97 and D provided the highest sensitivity of 0.96. Multi-parametric diffusion measurements using two-compartment and anomalous diffusion models were found to be significant discriminants of tumor grading in pediatric brain neoplasms.

Anomalous diffusion of ions in electrochromic tungsten oxide films

Amorphous tungsten oxide thin films were deposited by sputtering at different O2/Ar ratios onto conducting substrates. Ion intercalation and diffusion in the films was studied by electrochemical impedance spectroscopy measurements in the frequency range 10 mHz–100kHz and for potentials between 1.0 and 3.2V vs. Li/Li+, using the film as working electrode in a Li+ containing electrolyte. The impedance data were in very good agreement with anomalous diffusion models. Different models were found to be applicable at potentials >1.8V and <1.8V. At high potentials ion intercalation was found to be reversible and an anomalous diffusion model describing ion hopping was favored. At low potentials ion intercalation was found to be irreversible and ion trapping takes place. In this latter range an anomalous diffusion model for the case of non-conserved number of charge carriers gave the best fit to experimental data. We obtained potential dependent diffusion coefficients in the range from 10−9 to 10−11cm2/s, and anomalous diffusion exponents in the range 0.1 to 0.4, with the films deposited at lower O2/Ar ratios exhibiting the higher values.

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.

Directional sensitivity of anomalous diffusion in human brain assessed by tensorial fractional motion model

Anisotropic diffusion in the nervous system is most commonly modeled by apparent diffusion tensor, which is based on regular diffusion theory. However, the departure of diffusion-induced signal attenuation from a mono-exponential form implies that there is anomalous diffusion. Recently, a novel diffusion NMR theory based on the fractional motion (FM) model, which is an anomalous diffusion model, has been proposed. While the FM model has been applied to both healthy subjects and tumor patients, its anisotropy in the nervous system remains elusive. In this study, this issue was addressed by measuring the FM-related parameters in 12 non-collinear directions. A metric to quantify the directional deviation was derived. Furthermore, the FM-related parameters were modeled as tensors and analyzed in analogy with the conventional diffusion tensor imaging (DTI). Experimental results, which were obtained for 15 healthy subjects at 3T, exhibited pronounced anisotropy of the FM-related parameters, although the effects were smaller than the apparent diffusion coefficient (ADC). The tensorial nature for α, which is the Noah exponent in the FM model, showed behavior similar to the ADC, especially the principal eigenvector for α aligned with the dominant white matter fiber directions. The Hurst exponent H in the FM model, however, showed no correlation with the major fiber directions. The anisotropy of the FM model may provide complementary information to DTI and may have potential for tractography and detecting brain abnormalities.

Cusping, transport and variance of solutions to generalized Fokker–Planck equations

We study properties of solutions to generalized Fokker–Planck equations through the lens of the probability density functions of anomalous diffusion processes. In particular, we examine solutions in terms of their cusping, travelling wave behaviours, and variance, within the framework of stochastic representations of generalized Fokker–Planck equations. We give our analysis in the cases of anomalous diffusion driven by the inverses of the stable, tempered stable and gamma subordinators, demonstrating the impact of changing the distribution of waiting times in the underlying anomalous diffusion model. We also analyse the cases where the underlying anomalous diffusion contains a Lévy jump component in the parent process, and when a diffusion process is time changed by an uninverted Lévy subordinator. On the whole, we present a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems. We discuss possible applications in physical experiments, including, with reference to specific examples, the potential for model misclassification and how combinations of our four criteria may be used to overcome this issue.

Fractional and fractal derivative models for transient anomalous diffusion: Model comparison

Transient anomalous diffusion characterized by transition between diffusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon in real-world geologic media, due to the spatiotemporal variation of multiple physical, hydrologic, and chemical factors that can trigger non-Fickian diffusion. There are four fractional and fractal derivative models that can describe transient diffusion, including the distributed-order fractional diffusion equation (D-FDE), the tempered fractional diffusion equation (T-FDE), the variable-order fractional diffusion equation (V-FDE), and the variable-order fractal derivative diffusion equation (H-FDE). This study evaluates these models for transient sub-diffusion by comparing their mean squared displacement (which is the criteria for diffusion state), breakthrough curves (exhibiting nuance in diffusive state transition), and possible hydrogeologic origin (to build a potential link to medium properties). Results show that the T-FDE captures the slowest transition from sub-diffusion to normal-diffusion, and the D-FDE model only captures transient diffusion ending with sub-diffusion. The other two models, V-FDE and H-FDE, define a time-dependent scaling index to characterize complex transition states and rates. Preliminary field application shows that the V-FDE model, which provides a flexible transition rate, is appropriate to capture the fast transition from sub-diffusion to normal-diffusion for transport of a fluorescent water tracer dye (uranine) through a small-scale fractured aquifer. Further evaluations are needed using field measurements, so that practitioners can select the most reliable model for real-world applications.

Anomalous diffusion based on fractional calculus approach applied to drying analysis of apple slices: The effects of relative humidity and temperature

AbstractThe objective of this research was to evaluate the diffusion mechanism based on Fick's second law and anomalous diffusion modifying the drying operating conditions temperature (T) and relative humidity (RH) on apple slices (cv. Granny smith). Drying was performed at 30 °C, 40 °C, 50 °C and 30%, 50%, 70%RH based on a 32 experimental design. The drying curves were analyzed to determine the effective diffusion (Deff) using two methods: Fick's second law and anomalous diffusion model based on fractional calculus approach. Our results showed that the anomalous diffusion mechanism fit the experimental data better and revealed a super‐diffusive behavior (α &gt; 1). Therefore, Deff values were estimated using the anomalous diffusion model with α = 1.735. With respect to the use of anomalous diffusion model solution based on fractional calculus at different operations conditions allow to get better fitting of drying data than second Fick's law model, even keeping the Arrhenius behavior of Deff with temperature.Practical applicationsDiffusion process is typically analyzed using models based on Fick's second law. However, several assumptions inherent in Fick's second law are not fulfilled in food materials (e.g., structural heterogeneity). In addition, Fick's second law does not consider structural variations during the drying process, and it is well known that fruits such as apples experience significant structural changes during drying. Fractional calculus is a tool for mathematically representing the anomalous diffusion of solutes whose movements can be faster or slower than postulated in Fick's second law due to food structure. In general, with second Fick's law model not good fitting to data are obtained, therefore the prediction capacity of the model is limited. Based on this, anomalous diffusion solution based on fractional calculus was applied. The results showed good fit the data to the model, and also by the time exponent (α) is possible to identify the kind of diffusion process.

Influence of the finiteness of particle velocity on the energy spectrum of cosmic rays in an anomalous diffusion model with Lévy flights

An estimate of the influence the finiteness of particle velocity has on the results of a fractional differential (anomalous) model of cosmic ray propagation in the Galaxy with Levy flights developed by the authors is considered. The results from Monte Carlo simulations of particle diffusion in random walk models with finite and infinite velocities are presented. It is shown that considering particle velocity finiteness has almost no effect on the cosmic ray energy spectrum obtained for E > 1 GeV in the anomalous diffusion model with Levy flights for nearby young sources.

Zooming in on Anomalous Diffusion with Variable-Lengthscale Fluorescence Correlation Spectroscopy

The diffusion of macromolecules in cells, and in general in crowded or complex fluids, is often found to be anomalous. However, given that each different experimental technique usually probes diffusion at a single lengthscale, it remains difficult to determine which anomalous diffusion model is the most appropriate for each particular system. Here we show that variable-lengthscale fluorescence correlation spectroscopy, where the volume of observation is varied over several orders of magnitude, combined with a numerical inversion procedure of the correlation data, allows retrieving the mean-squared displacement of molecular diffusion processes for up to five decades in time.

Resonant behavior of a harmonic oscillator with fluctuating mass driven by a Mittag-Leffler noise

The resonant behavior of a generalized Langevin equation (GLE) in the presence of a Mittag-Leffler noise is studied analytically in this paper. Considering that a GLE with a Mittag-Leffler friction kernel is very useful for modeling anomalous diffusion processes with long-memory and long-range dependence, and the surrounding molecules do not only collide with the Brownian particle but also adhere to the Brownian particle for random time. Thus, we consider the Brownian particle with fluctuating mass, and the fluctuations of the mass are modelled as a dichotomous noise. Applying the stochastic averaging method, we obtain the exact expression of the output amplitude gain of the system. By studying the impact of the driving frequency and the noise parameters, we find the non-monotonic behaviors of the output amplitude gain. The results indicate that the bona fide SR, the wide sense SR and the conventional SR phenomena occur in the proposed harmonic oscillator with fluctuating mass driven by Mittag-Leffler noise. It is found that when we consider the output amplitude gain versus the driving frequency, the phenomena of stochastic multi-resonance (SMR) with two, three and four peaks are observed, and the quadruple-peaks SR phenomenon had never been observed in previous literature. Besides, when we investigate the dependence of output amplitude gain on the memory exponent, the inverse stochastic resonance (ISR) phenomenon takes place, in contrast to the well-known phenomenon of stochastic resonance. Furthermore, we compare the corresponding ordinary harmonic oscillator without memory to our generalized model, and found that the properties of long-memory and long-range dependence endows our generalized model with more abundant dynamic behaviors than the ordinary harmonic oscillator without memory.

Using spectral and cumulative spectral entropy to classify anomalous diffusion in Sephadex™ gels

Sephadex™ gel beads are commonly used to separate mixtures of similar molecules based on trapping and size exclusion from internal submicron diameter cavities. Water, as it freely moves through the porous gel and enclosed chambers of Sephadex™ beads, exhibits both normal (Gaussian) and anomalous (non-Gaussian) water diffusion. The apparent diffusion coefficient (ADC) of water in Sephadex™ gels can be measured using magnetic resonance imaging (MRI) by applying diffusion-weighted pulse sequences. This study investigates the relationship between the ADC of water and the complexity (i.e., size and number of cavities) of a series of Sephadex™ beads. We first classified the stochastic movement of water by using the solution to the space and time fractional diffusion equation to extract the ADC and the fractional time and space parameters (α, β), which are essentially the order of the respective fractional derivatives in Fick’s second law. From the perspective of the continuous time random walk (CTRW) model of anomalous diffusion, these parameters reflect waiting times (trapping) and jump increments (nano-flow) of the water in the gels. The observed MRI diffusion signal decay represents the Fourier transform of the diffusion propagator (i.e., the characteristic function of the stochastic process). In two series of Sephadex™ gel beads, we observed a strong inverse correlation between bead porosity (which is also responsible for molecular size exclusion) and the fractional order parameters; as the gels become more heterogeneous, the ADC decreases, both α and β are reduced and the diffusion exhibits anomalous (sub-diffusion) behavior. In addition, as a new measure for the structural complexity in Sephadex™ gel beads, we propose using the spectral and the cumulative spectral entropy that are derived from the observed characteristic function. We find that both measures of entropy increase with the porosity and tortuosity of the gel in a manner consistent with fractional order diffusional dynamics.