Anomalous diffusion in branched elliptical structure

Diffusion in narrow curved channels with dead-ends as in extracellular space in the biological tissues,e.g.,brain,tumors,muscles,etc.is a geometrically induced complex diffusion and is relevant to different kinds of biological,physical,and chemical systems.In this paper,we study the effects of geometry and confinement on the diffusion process in an elliptical comb-like structure and analyze its statistical properties.The ellipse domain whose boundary has the polar equationρ(θ)=b/√1−e^(2)cos^(2)θ with 0<e<1,θ∈[0,2π],and b as a constant,can be obtained through stretched radius r such that Υ=rρ(θ)with r∈[0,1].We suppose that,for fixed radius r=R,an elliptical motion takes place and is interspersed with a radial motion inward and outward of the ellipse.The probability distribution function(PDF)in the structure and the marginal PDF and mean square displacement(MSD)along the backbone are obtained numerically.The results show that a transient sub-diffusion behavior dominates the process for a time followed by a saturating state.The sub-diffusion regime and saturation threshold are affected by the length of the elliptical channel lateral branch and its curvature.

CONCENTRATION DISTRIBUTION OF FRACTIONAL ANOMALOUS DIFFUSION CAUSED BY AN INSTANTANEOUS POINT SOURCE

The Fox function expression and the analytic expression for the concentration distribution of fractional anomalous diffusion caused by an instantaneous point source in n-dimensional space (n= 1, 2 or 3) are derived by means of the condition of mass conservation , the time-space similarity of the solution , Mellin transform and the properties of the Fox function . And the asymptotic behaviors for the solutions are also given .

Computational analysis of the roles of biochemical reactions in anomalous diffusion dynamics

Most biochemical processes in cells are usually modeled by reaction-diffusion （RD） equations. In these RD models, the diffusive process is assumed to be Gaussian. However, a growing number of studies have noted that intracellular diffusion is anomalous at some or all times, which may result from a crowded environment and chemical kinetics. This work aims to computationally study the effects of chemical reactions on the diffusive dynamics of RD systems by using both stochastic and deterministic algorithms. Numerical method to estimate the mean-square displacement （MSD） from a deterministic algorithm is also investigated. Our computational results show that anomalous diffusion can be solely due to chemical reactions. The chemical reactions alone can cause anomalous sub-diffusion in the RD system at some or all times. The time-dependent anomalous diffusion exponent is found to depend on many parameters, including chemical reaction rates, reaction orders, and chemical concentrations.

ANOMALOUS DIFFUSION IN FRACTAL POROUS MEDIUM

Fractal media has many characteristics different from those of homogeneous media, it has a correlated self-similar structure. The particle diffusion in pore fractal is different from Fick' s diffusion, and its mean-squared displacement follows fractal scaling law. The model of particle diffusion in pore fractal by means of the statistics method of stochastic process is structured and some fractal characteristics and non-Markov property are proved.

SCALE-INVARIANT SOLUTION FOR FRACTIONAL ANOMALOUS DIFFUSION EOUATION

Fractional diffusion equation for transport phenomena in fractal media is an integro-partial differential equation. For solving the problem of this kind of equation the invariants of the group of scaling transformations are given and then used for deriving the integro-ordinary differential equation for the scale-invariant solution. With the help of Mellin transform the scale-invariant solution is obtained in terms of Fox functions. And the series and asymptotic representations for the solution are considered.

ANOMALOUS DIFFUSION IN FINITE LENGTH FINGERS COMB FRAME WITH THE EFFECTS OF TIME AND SPACE RIESZ FRACTIONAL CATTANEO-CHRISTOV FLUX AND POISEUILLE FLOW

This paper presents an investigation on the anomalous diffusion in finite length fingers comb frame, the time and space Riesz fractional Cattaneo-Christov flux is introduced with the Oldroyds＇ upper convective derivative and the effect of Poiseuille flow is also taken into account. Formulated governing equation possesses the coexisting characteristics of parabolicity and hyperbolicity. Numerical solution is obtained by the Ll-scheme and shifted Griinwald formulae, which is verified by introducing a source item to construct an exact solution. The effects, such as time and space fractional parameters, relaxation parameter and the ratio of the pressure gradient and viscosity coefficient, on the spatial and temporal evolution of particles distribution and dynamic characteristics are shown graphically and analyzed in detail.

Memory dependent anomalous diffusion in comb structure under distributed order time fractional dual-phase-lag model

This paper considers a novel distributed order time fractional dual-phase-lag model to analyze the anomalous diffusion in a comb structure,which has a widespread application in medicine and biology.The newly proposed constitution model is a generalization of the dual-phase-lag model,in which a spectrum of the time fractional derivatives with the memory characteristic governed by the weight coefficient is considered and the formulated governing equation contains both the diffusion and wave characteristics.With the L1-formula to discrete the time Caputo fractional derivatives,the finite difference method is used to discretize the model and the related numerical results are plotted graphically.By adding a source term,an exact solution is defined to verify the correctness of the numerical scheme and the convergence order of the error in spatial direction is presented.Finally,the dynamic characteristics of the particle distributions and the effects of involved parameters on the total number of particles in the x-direction are analyzed in detail.

Fractional single-phase-lagging heat conduction model for describing anomalous diffusion

The fractional single-phase-lagging(FSPL)heat conduction model is obtained by combining scalar time fractional conservation equation to the single-phase-lagging(SPL)heat conduction model.Based on the FSPL heat conduction model,anomalous diffusion within a finite thin film is investigated.The effect of different parameters on solution has been observed and studied the asymptotic behavior of the FSPL model.The analytical solution is obtained using Laplace transform method.The whole analysis is presented in dimensionless form.Numerical examples of particular interest have been studied and discussed in details.

Pattern formation in superdiffusion Oregonator model

Pattern formations in an Oregonator model with superdiffusion are studied in two-dimensional(2D) numerical simulations. Stability analyses are performed by applying Fourier and Laplace transforms to the space fractional reaction–diffusion systems. Antispiral, stable turing patterns, and travelling patterns are observed by changing the diffusion index of the activator. Analyses of Floquet multipliers show that the limit cycle solution loses stability at the wave number of the primitive vector of the travelling hexagonal pattern. We also observed a transition between antispiral and spiral by changing the diffusion index of the inhibitor.

Uphill anomalous transport in a deterministic system with speed-dependent friction coefficient

We investigate the transport of a deterministic Brownian particle theoretically, which moves in simple onedimensional, symmetric periodic potentials under the influence of both a time periodic and a static biasing force. The physical system employed contains a friction coefficient that is speed-dependent. Within the tailored parameter regime, the absolute negative mobility, in which a particle can travel in the direction opposite to a constant applied force, is observed.This behavior is robust and can be maximized at two regimes upon variation of the characteristic factor of friction coefficient. Further analysis reveals that this uphill motion is subdiffusion in terms of localization（diffusion coefficient with the form D（t） -t-（-1） at long times）. We also have observed the non-trivially anomalous subdiffusion which is significantly deviated from the localization; whereas most of the downhill motion evolves chaotically, with the normal diffusion.

Diffusion dynamics in branched spherical structure

Diffusion on a spherical surface with trapping is a common phenomenon in cell biology and porous systems.In this paper,we study the diffusion dynamics in a branched spherical structure and explore the influence of the geometry of the structure on the diffusion process.The process is a spherical movement that occurs only for a fixed radius and is interspersed with a radial motion inward and outward the sphere.Two scenarios govern the transport process in the spherical cavity:free diffusion and diffusion under external velocity.The diffusion dynamics is described by using the concepts of probability density function(PDF)and mean square displacement(MSD)by Fokker–Planck equation in a spherical coordinate system.The effects of dead ends,sphere curvature,and velocity on PDF and MSD are analyzed numerically in detail.We find a transient non-Gaussian distribution and sub-diffusion regime governing the angular dynamics.The results show that the diffusion dynamics strengthens as the curvature of the spherical surface increases and an external force is exerted in the same direction of the motion.

Ergodicity recovery of random walk in heterogeneous disordered media

Significant and persistent trajectory-to-trajectory variance are commonly observed in particle tracking experiments,which have become a major challenge for the experimental data analysis.In this theoretical paper we investigate the ergodicity recovery behavior,which helps clarify the origin and the convergence of trajectory-to-trajectory fluctuation in various heterogeneous disordered media.The concepts of self-averaging and ergodicity are revisited in the context of trajectory analysis.The slow ergodicity recovery and the non-Gaussian diffusion in the annealed disordered media are shown as the consequences of the central limit theorem in different situations.The strange ergodicity recovery behavior is reported in the quenched disordered case,which arises from a localization mechanism.The first-passage approach is introduced to the ergodicity analysis for this case,of which the central limit theorem can be employed and the ergodicity is recovered in the length scale of diffusivity correlation.

Analysis of Anisotropic Nonlocal Diffusion Models:Well-Posedness of Fractional Problems for Anomalous Transport

We analyze the well-posedness of an anisotropic,nonlocal diffusion equation.Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus,we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation.Furthermore,we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s∈[0.5,1).We also present an application of the advection-diffusion equation to anomalous transport of solutes.

Sub-diffusive scaling with power-law trapping times

Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive scaling at long times. The statistical mechanical treatment of diffusion in a disordered environment is often quite involved. In this short review, we present a simple and unified view of the many quantitative results on anomalous diffusion in the literature, including the scaling of the diffusion front and the mean first-passage time. Varioust analytic calculations and physical arguments are examined to highlight the role of dimensionality, energy landscape, and rare events in affecting the particle trajectory statistics. The general understanding that emerges will aid the interpretation of relevant experimental and simulation results.

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory： the jumping distance and the waiting time, are replaced by two new ones： the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement （x^2（t）） - t^a is realized numerically and analysed theoretically, where the value of the power index a is in a region of 0 〈 a 〈 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.

Fractional backward Kolmogorov equations

This paper derives the fractional backward Kolmogorov equations in fractal space-time based on the construction of a model for dynamic trajectories. It shows that for the type of fractional backward Kolmogorov equation in the fractal time whose coefficient functions are independent of time, its solution is equal to the transfer probability density function of the subordinated process X（Sα （t））, the subordinator Sα （t） is termed as the inverse-time a-stable subordinator and the process X（τ） satisfies the corresponding time homogeneous Ito stochastic differential equation.

Molecular diffusion in ternary poly(vinyl alcohol)solutions

The diffusion kinetics of a molecular probe-rhodamine B-in ternary aqueous solutions containing poly(vinyl alcohol),glycerol,and surfactants was investigated using fluorescence correlation spectroscopy and dynamic light scattering.We show that the diffusion characteristics of rhodamine B in such complex systems is determined by a synergistic effect of molecular crowding and intermolecular interactions between chemical species.The presence of glycerol has no noticeable impact on rhodamine B diffusion at low concentration,but significantly slows down the diffiision of rhodamine B above 3.9%(w/v)due to a dominating steric inhibition effect.Furthermore,introducing surfactants(cationic/nonionic/anionic)to the system results in a decreased diffusion coefficient of the molecular probe.In solutions containing nonionic surfactant,this can be explained by an increased crowding effect.For ternary poly(vinyl alcohol)solutions containing cationic or anionic surfactant,surfactant-polymer and surfactant-rhodamine B interactions alongside the crowding effect of the molecules slow down the overall diffiisivity of rhodamine B.The results advance our insight of molecular migration in a broad range of industrial complex formulations that incorporate multiple compounds,and highlight the importance of selecting the appropriate additives and surfactants in formulated products.

Pressure transient analysis of multiple fractured horizontal wells in naturally fractured unconventional reservoirs based on fractal theory and fractional calculus

Currently,most models for multiple fractured horizontal wells(MFHWs)in naturally fractured unconventional reservoirs(NFURs)are based on classical Euclidean models which implicitly assume a uniform distribution of natural fractures and that all fractures are homogeneous.While fractal theory provides a powerful method to describe the disorder,heterogeneity,uncertainty and complexity of the NFURs.In this paper,a fractally fractional diffusion model(FFDM)for MFHWs in NFURs is established based on fractal theory and fractional calculus.Particularly,fractal theory is used to describe the heterogeneous,complex fracture network,with consideration of anomalous behavior of diffusion process in NFURs by employing fractional calculus.The Laplace transformation,line source function,dispersion method,and superposition principle are used to solve this new model.The pressure responses in the real time domain are obtained with Stehfest numerical inversion algorithms.The type curves of MFHW with three different outer boundaries are plotted.Sensitivity analysis of some related parameters are discussed as well.This new model provides the relatively more accurate and appropriate evaluation results for pressure transient analysis for MFHWs in NFURs,which could be applied to accurately interpret the real pressure data of an MFHW in field.