Generalized Diffusion Equation Research Articles

A generalized linear equation for non-linear diffusion in external fields and non-ideal systems

Based on macroscopic thermodynamic analysis, we have established a generalized linear theory for non-linear diffusion in external fields and non-ideal systems, which was classically described by the Fokker–Planck equation (or the Smoluchowski equation) and the non-linear Fickian equation, respectively. The new theory includes three basic equations expressed in ‘apparent variables’ as defined in this paper: (i) a generalized linear flux equation for non-linear diffusion; (ii) an apparent mass conservation equation and (iii) a generalized linear non-steady state equation for non-linear diffusion. Our analysis shows that (i) all of the existing linear and non-linear equations are the special cases of the new non-steady state general diffusion equation. It was also demonstrated that the general equation of the non-steady state is equivalent to the Fokker–Planck equation; (ii) coupling diffusion with multiple driving forces can be unified to a single force: the apparent concentration gradient; (iii) the exact relationship between diffusion coefficient and concentration in the non-linear Fickian equation under non-ideal conditions could be established and (iv) the potential energy is conservative in a diffusion process. An application of the generalized linear equation showed that the solution is simple. For the first time, an analytic solution of the Smoluchowski equation with a time-dependent potential in the algebraic form was obtained.

Master equation with two times: Diffusion in external field

Abstract The master equation for diffusion involving two times applies to the problem of diffusion in a time-dependent (in general inhomogeneous) external field. We consider the case of the quasi Fokker–Planck approximation, when the probability transition function for diffusion (PTD-function) does not possess a long tail in coordinate space and can be expanded as the function of instantaneous displacements. For relatively weak external field the linear expansion of the PTD function leads to a simple generalization of diffusion equation, containing the retardation factors.

Spin injection in quantum wells with spatially dependent rashba interaction

We consider Rashba spin–orbit effects on spin transport driven by an electric field in semiconductor quantum wells. We derive spin diffusion equations that are valid when the mean free path and the Rashba spin–orbit interaction vary on length scales larger than the mean free path in the weak spin–orbit coupling limit. From these general diffusion equations, we derive boundary conditions between regions of different spin–orbit couplings. We show that spin injection is feasible when the electric field is perpendicular to the boundary between two regions. When the electric field is parallel to the boundary, spin injection only occurs when the mean free path changes within the boundary, in agreement with the recent work by Tserkovnyak et al (Preprint cond-mat/0610190).

Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation

AbstractIn this paper, we consider a class of generalized diffusion equations which are of great interest in mathematical physics. For some of these equations that model fast diffusion, nonclassical and nonclassical potential symmetries are derived. These symmetries allow us to increase the number of solutions. These solutions are unobtainable neither from classical nor from classical potential symmetries. Copyright © 2007 John Wiley & Sons, Ltd.

Generalized diffusion and pretransitional fluctuations statistics

(a) For diffusion type processes, non-Gaussian distributions are obtained, in a generic manner, from a generalization of classical linear response theory; (b) Statistical properties of hydrodynamic fields reveal pretransitional fluctuations in fingering processes, and these precursors are found to exhibit power law distributions; (c) These power laws are shown to follow from q-Gaussian structures which are solutions to the generalized diffusion equation. The present analysis (i) offers a physical picture of the precursors dynamics, (ii) suggests a physical interpretation of nonextensivity from the structure of the precursors, and (iii) provides an illustration of the emergence of statistics from dynamics.

Fractional diffusion equation with an absorbent term and a linear external force: Exact solution

We investigate the solutions of a generalized diffusion equation which contains space and time fractional derivatives by taking an absorbent (or source) term and a linear external force into account. They are expressed in terms of the Fox H functions and, for some special cases, they are directly related to the Levy distributions. We also analyze the first passage time distribution for the case characterized by the absence of absorbent (or source) term and external force for a semi-infinite interval with absorbent boundary conditions. We also discuss the connection of the results presented here with the anomalous diffusion.

Quatro abordagens para o movimento browniano

A "dança" aleatória de pequenas partículas em suspensão num líquido, um fenômeno conhecido como movimento browniano, foi primeiramente explicado por Einstein na sua famosa tese de doutorado. Seguindo uma perspectiva histórica, mostramos como este fenômeno pode ser descrito de quatro maneiras distintas, a saber: o tratamento difusivo de Einstein, a variante estocástica ou de força flutuante proposta por Paul Langevin, a abordagem via equação de Fokker-Planck, e finalmente, as caminhadas aleatórias de Mark Kac. Discutiremos também as limitações presentes na abordagem difusiva. Em particular, mostramos que a equação parabólica na qual Einstein baseou sua explicação deve ser substituída por uma equação do tipo hiperbólica que também surge naturalmente no tratamento via caminhadas aleatórias. A solução geral dessa equação é obtida e comparada com o resultado padrão. Para tempos curtos, comparados com as escalas de tempo características do sistema, o movimento das partículas segue um comportamento ondulatório.

Approximate analytical solutions for a class of nonlinear differential equations

This paper presents a theoretical analysis for a class of generalized diffusion equations. An effective analytical decomposition and numerical technique is given by similarity transformations and the analytical decomposition technique. The approximate analytical solution may be expressed in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of approximate solutions are illustrated by good agreement with numerical results.

Generalized diffusion equation for anisotropic anomalous diffusion

Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.

The diffusional evolution of chemical composition in a solar model

We consider the basic physical processes resulting in a differential, microscopic redistribution of stellar matter, generally known as diffusion. The main effect of diffusion in the solar interior is a segregation of light and heavy elements in the gravitational field. As a result, the abundance of helium and heavy elements in the solar envelope is reduced, while it becomes enriched by hydrogen. We present estimates of the degree of settling for a sequence of evolutionary models via numerical solution of the generalized diffusion equation. The effect of the ion charge (in the approximation of full ionization) on the settling rate is studied in detail. Abundance variations are given for the centers of the models, as well as for the convective envelope of the modern Sun. We analyze the effects of the thermal and concentrational diffusion on the evolution of the chemical-composition profile. Quantitatively, the effect of thermal diffusion is not very large, but it leads to the appearance of new features in the hydrogen-abundance profile, namely, a discontinuity at the base of the convective zone. The effect of concentration diffusion is relatively small, and is appreciable only at the model center at late stages of the evolution, and also close to the base of the convective envelope. All the mechanisms studied are necessary components of a modern model for the internal structure and evolution of the Sun.

Spatial distribution of cutaneous microvasculature and local drug clearance after drug application on the skin

We analysed quantitatively blood microvessels distribution in normal skin. We conclude that the segmental area of blood vessels peaks approximately 0.1 mm below the skin surface, where the upper cutaneous blood plexus resides. Total blood vessels area then decreases quasi-exponentially to a depth of approx. − 0.75 mm, with a decay length of ∼ 0.1 mm, which is site and skin condition dependent, but at greater depths the decrease is approx. 6-times less steep. The corresponding permeability sink exhibits a similar, but superficially steeper, depth-profile. The lateral localisation of superficial blood vessels is such that ensures maximum diffusion from and into the capillaries, which affects transdermal drug delivery: each hairpin-like loop is in the centre of a papilla that corresponds to a cluster of corneocytes surrounded by main diffusion pathways. The aggregate area of blood vessels in the skin is ≥ 2.5-fold greater than total organ surface area under normal physiological conditions. The molecules diffusing through the skin barrier are thus largely cleared in outermost 20% of the organ, which may create a drug concentration maximum in the dermis, if clearance increases significantly with time. Skin microdialysis data are therefore extremely sensitive to cutaneous blood flow (distribution) and sampling. Skin microvasculature and its distribution must consequently be considered in all topical or transdermal drug transport studies, for example, by including suitably formulated clearance term into generalised diffusion equation.

Fractional nonlinear diffusion equation, solutions and anomalous diffusion

We devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms. The presence of external forces and absorbent terms is also considered. The solutions found here can have a compact or long tail behavior and, in particular, for the last case in the asymptotic limit, we relate these solutions to the Lévy or Tsallis distributions. In addition, from the results presented here a rich class of diffusive processes, including normal and anomalous ones, can be obtained.

Anomalous surfactant diffusion in a living polymer system

Random processes are generally described by Gaussian statistics as formulated by the central limit theorem. However, there exists a large number of exceptions to this rule that can be found in a variety of fields. Diffusion processes are often analyzed by the scaling law <r2> approximately t2beta, where the second moment of the diffusion propagator or molecular mean square displacement, <r2>, in the case of Gaussian diffusion is proportional to t, i.e., beta=1/2. A deviation from Gaussian behavior may be either superdiffusion (beta>1/2) or subdiffusion (beta<1/2). In this paper we demonstrate that all three diffusion regimes may be observed for the surfactant self-diffusion, on the length scale of 10(-6) m and the time scale of 0.02-0.8 s. in a system of wormlike micelles, depending on small variations in the sample composition. The self-diffusion is followed by pulsed gradient NMR where one not only measures the second moment of the diffusion propagator, but actually measures the Fourier transform of the full diffusion propagator itself. A generalized diffusion equation in terms of fractional time derivatives provides a general description of all the different diffusion regimes, and where 1beta can be interpreted as a dynamic fractal dimension. Experimentally, we find beta=1/4 and 3/4, in the regimes of sub- and superdiffusion, respectively. The physical interpretation of the subdiffusion behavior is that the dominating diffusion mechanism corresponds to a lateral diffusion along the contour of the wormlike micelles. Superdiffusion is obtained near the overlap concentration where the average micellar size is smaller so that the center of mass diffusion of the micelles contributes to the transport of surfactant molecules.

Nonclassical symmetry reductions for an inhomogeneous nonlinear diffusion equation

In this paper we consider a class of generalised diffusion equations which are of great interest in mathematical physics. For some of these equations model, fast diffusion nonclassical symmetries are derived. We find the connection between classes of nonclassical symmetries of the equation and of an associated system. These symmetries allow us to increase the number of solutions. Some of these solutions are unobtainable by classical symmetries and exhibit an interesting behaviour.

Dynamics of DNA Ejection from Bacteriophage

The ejection of DNA from a bacterial virus (i.e., phage) into its host cell is a biologically important example of the translocation of a macromolecular chain along its length through a membrane. The simplest mechanism for this motion is diffusion, but in the case of phage ejection a significant driving force derives from the high degree of stress to which the DNA is subjected in the viral capsid. The translocation is further sped up by the ratcheting and entropic forces associated with proteins that bind to the viral DNA in the host cell cytoplasm. We formulate a generalized diffusion equation that includes these various pushing and pulling effects and make estimates of the corresponding speedups in the overall translocation process. Stress in the capsid is the dominant factor throughout early ejection, with the pull due to binding particles taking over at later stages. Confinement effects are also investigated, in the case where the phage injects its DNA into a volume comparable to the capsid size. Our results suggest a series of in vitro experiments involving the ejection of DNA into vesicles filled with varying amounts of binding proteins from phage whose state of stress is controlled by ambient salt conditions or by tuning genome length.

Impact of Sulfate Ion on Dolomite Concrete Performance

Concrete deterioration is an important cause of sudden collapse of buildings. Sulfate ionattack is a prominent cause of concrete deterioration. The main cause of sulfate attackinduceddamage is ettringite formation in voids, cracks, and contact zones between aggregatesand hardened cement paste. Scanning electron microscopy (SEM) images are taken to detectdelayed ettringite crystal development in voids. General diffusion equations are utilized toinvestigate the sulfate ion ingress and the distribution of sulfate ion concentration in concreterelated to time. A numerical model demonstrates the relation between the diffusion coefficientand concentration. A computer program is developed to model the problem of ion diffusion inconcrete. The sulfate attack on concrete by one-dimensional diffusion process is simulated.The numerical model demonstrates the relation between the diffusion coefficient andconcentration.The computer program is developed to calculate the SO4 ion diffusion. Compared with theSO4 ion diffusion coefficient in free water, the high diffusion value and the low compressivestrength of concrete specimen are the values of concrete deterioration. The experimentalprogram included mix design of 6 concrete mixes. The parameters of the experimentalprogram were: cement type and content, presence of silica fume, sulfates concentration, andexposure period.The results show that, the concrete deterioration by sulfate attack is strongly influenced by thesulfates concentration in the aggressive medium and the exposure time. There is a linearrelation between strength loss and concentration for mixes containing silica fume. OrdinaryPortland cement (OPC) concrete limits the strength reduction about 10% after one yearexposure to 5,000 ppm sulfate solution. In case of 100,000 ppm sulfate solution, the reductionis (13.1%). Increasing the cement content increases the strength loss by sulfates attack even inthe presence of silica fume.

Equação da difusão fracionária não-linear: solução exata

We devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms. The presence of external forces and absorbent terms is also considered. The solutions found here can have a compact or long tail behavior. Particularly in the last case in the asymptotic limit, we relate these solutions to the Levy or Tsallis distributions. In addition, from the results presented here, a rich class of diffusive processes, including normal and anomalous ones, can be obtained.

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations u t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ u m at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.

A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes

This article reviewed anomalous diffusion phenomena coupled with facile and sluggish charge-transfer reactions at fractal interface. Firstly, the generalised diffusion equation (GDE) involving a fractional derivative which describes diffusion towards and from fractal interface was briefly introduced. And then, anomalous diffusion coupled with facile charge-transfer reaction at fractal interface, i.e., diffusion-controlled transfer process across fractal interface, was mathematically examined by the generalised Cottrell, Sand, Randles-Sevcik and Warburg equations theoretically derived from the analytical solutions to the GDE under the semi-infinite boundary condition. Finally, in order to provide a guideline in analysing anomalous diffusion coupled with sluggish charge-transfer reaction at fractal interface, i.e., non-diffusion-controlled transfer process across fractal interface, this review covered the recent researches into the effect of surface roughness on non-diffusion-controlled transfer process within the intercalation electrodes.