反常扩散与分数阶对流-扩散方程

反常扩散现象在自然界和社会系统中广泛存在.考虑了扩散过程的时间相关和时空相关性，用非局域性的处理方法，在传统的二阶对流 扩散方程基础上，得到了分数阶对流 扩散方程，以此方程来描述反常扩散.在此方程中，弥散项和对时间的导数为分数阶导数所代替.由此分数阶对流 扩散方程，对传统的费克扩散定律进行推广，得到了广义的分数费克扩散定律，分数费克扩散定律说明某时刻空间中某点的流量不仅与其领域内的浓度梯度有关，而且与整个空间中其他不同点的粒子浓度、浓度变化的历史，甚至初始时刻的浓度有关.讨论了方程的解——分数稳定分布，并由此说明了扩散运动的平均平方位移是运移时间的非线性函数. 关键词： 扩散 分数阶微积分 稳定分布（Lévy分布） 费克扩散定律     

Fick's law and Fokker-Planck equation in inhomogeneous environments

In inhomogeneous environments, the correct expression of the diffusive flux is not always given by the Fick's law Γ=−D∇n. The most general hydrodynamic equation modelling diffusion is indeed the Fokker-Planck equation (FPE). The microscopic dynamics of each specific system may affect the form of the FPE, either establishing connections between the diffusion and the convection term, as well as providing supplementary terms. In particular, the Fick's form for the diffusion equation may arise only in consequence of a specific kind of microscopic dynamics. It is also shown how, in the presence of sharp inhomogeneities, even the hydrodynamic FPE limit may becomes inaccurate and mask some features of the true solution, as computed from the Master equation.     

Thermal interdiffusion in InGaAs/GaAs and GaAsSb/GaAs strained quantum wells as a function of doping density

We studied the thermal stability and interdiffusion of InGaAs/GaAs and GaAsSb/GaAs single quantum wells as a function of temperature for both Be and Si doping at various doping concentrations. The interdiffusion was monitored using the photoluminescence from the ground states of the valence- and conduction-band quantum wells. Using a Green's function method to solve the diffusion equation, assuming Fick's law behaviour, the evolution of the well shape during annealing was determined, and Schrödinger's equation was solved for this well shape to provide the ground-state energy levels of the system using the diffusion constant as the only fitting parameter. The validity of this model as applied to both systems is discussed.Strained Layer Structures Research Group.     

多孔材料分形扩散模型的Fourier-Bessel级数算法及其应用

将Fick扩散定律的Fourier三角级数算法推广成多孔材料分形扩散模型的Fourier-Bessel级数算法,并把它应用于化学工程中吸附问题涉及的浓度分布与相对吸附量的计算中,取得一些规律性认识.由于分形扩散模型是在Fick扩散定律的基础上增加了表征微观结构的参数d_f和θ,研究多孔材料中的浓度分布与相对吸附量时,与Fick扩散定律的研究结果相比,定性上基本一致,在定量上有差别,d_f和θ对扩散传质过程的影响各有侧重,用它们可更好地描述多孔材料中的扩散过程.     

Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.Dedicated to Res Jost and Arthur WightmanSupported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft     

On the process of filtration of fractional viscoelastic liquid food

In the process of filtration, fluid impurities precipitate/accumulate; this results in an uneven inner wall of the filter, consequently leading to non-uniform suction/injection. The Riemannian–Liouville fractional derivative model is used to investigate viscoelastic incompressible liquid food flowing through a permeable plate and to generalize Fick’s law. Moreover, we consider steady-state mass balance during ultrafiltration on a plate surface, and a fractional-order concentration boundary condition is established, thereby rendering the problem real and complex. The governing equation is numerically solved using the finite difference algorithm. The effects of the fractional constitutive models, generalized Reynolds number, generalized Schmidt number, and permeability parameter on the velocity and concentration fields are compared. The results show that an increase in fractionalorderαin the momentum equation leads to a decrease in the horizontal velocity. Anomalous diffusion described by the fractional derivative model weakens the mass transfer; therefore, the concentration decreases with increasing fractional derivativeγin the concentration equation.     

Phenomenological diffusion laws in solids

A method is considered for study of diffusion in the solid phase, free of the shortcomings of Fick's equations. For the stationary case analytical expressions are obtained for the probability of transmission, reflection, and absorption of diffusing particles by a layer of specified thickness. Principles are formulated for reduction of the nonstationary problem to the stationary case. The results obtained are applied to a study of the kinetics of oxidation processes. A generalization of Fick's first law to the nonstationary case is presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 31–36, April, 1979.     

Vertical Sediment Concentration Distribution in High-Concentrated Flows: An Analytical Solution Using Homotopy Analysis Method

Transport of suspended sediment in open channel flow has an enormous impact on real life situations, viz. control and management of reservoir sedimentation, geomorphic evolution such as dunes, rivers, and coastlines etc. Transport entails advection and diffusion. Turbulent diffusion is governed by the concept of Fick's law, which is based on the molecular diffusion theory, and the equation that represents the distribution of sediment concentration is the advection-diffusion equation. The study uses the existing governing equation which considers different phases for solid and fluid, and then couples the two phases. To deal with high-concentrated flow, sediment and turbulent diffusion coefficients are taken to be different from each other. The effect of hindered settling on sediment particles is incorporated in the governing equation, which makes the equation highly non-linear. This study derives an explicit closed-form analytical solution to the generalized one-dimensional diffusion equation representing the vertical sediment concentration distribution with an arbitrary turbulent diffusion coefficient profile. The solution is obtained by Homotopy Analysis Method, which does not rely on the small parameters present in the equation. Finally, the solution is validated by comparing it with the implicit solution and the numerical solution. A relevant set of laboratory data is selected to check the applicability of the model, and a close agreement shows the potential of the model in the context of application to high-concentrated sediment-laden open channel flow.     

Model for evolution of periodic layered structure in the SiO2/Mg system

A mathematical model for the growth of MgO/Mg₂Si periodic layered structure (PLS) is presented based on Fick's diffusion law and conservation of matter. The model explains the temporal evolution of the width of periodic layers at different temperatures, for layers distant enough from the Mg source. It is demonstrated that the difference in thickness of a pair of any two adjacent layers decreases with increasing temperature.     

Hard Sphere Diffusion Behaviour of Polymer Translocating through Interacting Pores

The translocation of polymer chain through a small pore from a high concentration side (cis side) to a low concentration side (trans side) is simulated by using Monte Carlo technique. The effect of the polymer--pore interaction on the translocation is studied. We find a special interaction at which the decay of the number of polymer chain, N, at the cis side obeys Fick's law, i.e. N decreases exponentially with time. The behaviour is analogous to the diffusion of hard sphere.     

Microscopic basis for Fick's law for self-diffusion

We investigate self-diffusion in a classical fluid composed of two species which are distinguished through the color of their particles, either black or white, but are identical as regards their mechanical properties. Disregarding color the fluid is in thermal equilibrium. We show that if a single test particle in the one-component fluid moves asymptotically as Brownian motion, then the color density and current in certain classes of nonequilibrium states are related, on the appropriate macroscopic scale, through Fick's law, and the former is governed by the diffusion equation. If in addition several test particles move asymptotically as independent Brownian motions, then the colored fluid is, on a macroscopic scale, in local equilibrium with parameters governed by the solution of the diffusion equation.Part of this work was done while both authors were at IHES, Bures-sur-Yvette, France.Supported in part by NSF Grant No. PHY 78-15920-02.Supported by a Heisenberg fellowship of the Deutsche Forschungsgemeinschaft.     

Exact fixed points in discrete hydrodynamics

Recently a discrete formulation of hydrodynamics was introduced, which was shown to be exactly renormalizable in a certain sense: a procedure was given for computing the equations of motion on a coarse space and time scale from those on a finer scale. In this paper we carry out this coarsening procedure explicitly, giving exact numerical results for a one-dimensional diffusive system. The coarsening transformation is found to have a one-parameter family of nontrivial fixed points, parameterized by a diffusion parameterD. This result gives a new way of understanding why so many systems obey Fick's lawj = – D'dn/dx. Any of an extremely broad class of microscopic equations of motion, when viewed on a coarse enough scale, obey the fixed-point equations (which are equivalent to Fick's law). The methods used here are equally applicable to higher-dimensionality systems such as fluids.Research partially supported by the Chemistry Division of NSF through Grant No. CHE-7906649.     

Degradation formula and working lifetime prediction for high-temperature coating

A model for predicting the degradation mode of high-temperature coating is developed in this paper based on Fick's law. The degradation formula is deduced according to this model and then used for predicting the working lifetime of the coating on the basis of the dependent relation among the effective thickness of the coating, the oxidation time and the thickness of the degradation layer resulting from oxidation and coating-substrate interdiffusion. Finally combination of the dominant degradation mode and formula gives the possible measures taken to improve the oxidation-resistant property of the coating.     

锗硅/硅异质结材料的化学气相淀积生长动力学模型

基于化学气相淀积(CVD)的Grove理论和Fick第一定律,提出并建立了锗硅(SiGe)/硅(Si)异质结材料减压化学气相淀积(RPCVD)生长动力学模型.与以前锗硅/硅异质结材料生长动力学模型仅考虑表面反应控制不同,本模型同时考虑了表面反应和气相传输两种控制机理,并给出了两种控制机理极限情况下的模型.本模型不仅适用于低温锗硅/硅应变异质结材料生长的表征,也适用于表征高温锗硅/硅弛豫异质结材料生长的表征.将模型计算值与实验结果进行了对比,无论是625℃低温下的应变SiGe的生长,还是900℃高温下的弛豫 关键词： SiGe/Si异质结材料 化学气相淀积生长动力学模型 Grove理论 Fick第一定律     

Spin diffusion in trapped clouds of cold atoms with resonant interactions

We show that puzzling recent experimental results on spin diffusion in a strongly interacting atomic gas may be understood in terms of the predicted spin diffusion coefficient for a generic strongly interacting system. Three important features play a central role: (a)?Fick's law for diffusion must be modified to allow for the trapping potential; (b)?the diffusion coefficient is inhomogeneous, due to the density variations in the cloud; and (c)?the diffusion approximation fails in the outer parts of the cloud, where the mean free path is long.     

Collective diffusion in carbon nanotubes: Crossover between one dimension and three dimensions

Using non-equilibrium molecular dynamics and Monte Carlo methods, we study the collective diffusion of helium in carbon nanotubes. The results show that the collective diffusion coefficient(CDC) increases with the dimension of the channel. The collective diffusion coefficient has a linear relationship with the temperature and the concentration. There exist a ballistic transport in short carbon nanotubes and a diffusive transport in long carbon nanotubes. Fick's law has an invalid region in the nanoscale channel.     

A growth kinetics model of rate decomposition for Si1-xGex alloy based on dimer theory

According to the dimer theory on semiconductor surface and chemical vapor deposition(CVD) growth characteristics of Si1-xGex, two mechanisms of rate decomposition and discrete flow density are proposed. Based on these two mechanisms, the Grove theory and Fick’s first law, a CVD growth kinetics model of Si1-xGex alloy is established. In order to make the model more accurate, two growth control mechanisms of vapor transport and surface reaction are taken into account. The paper also considers the influence of the dimer structure on the growth rate. The results show that the model calculated value is consistent with the experimental values at different temperatures.     

Modeling of Self-Diffusion and Relaxation Time NMR in Multi-Compartment Systems

The theory of pulsed field gradient (pfg) NMR applied to molecules in cellular systems which contain different subcellular compartments separated by permeable membranes, acting as diffusion barriers, has been extended. A numerical model of restricted diffusion and magnetization relaxation behavior in pfg-CPMG NMR experiments, based on the Fick's second law of diffusion, is presented. This model is applicable to a wide range of systems and allows the exploration of temporal and spatial behavior of the magnetization with and without the influence of gradient pulses. Results of the numerical experiments show their correspondence to the previously observed ones and demonstrate the importance of the inclusion of the time domain data in analyzing diffusion measurements.     

Dynamical aspects of the adsorption hysteresis phenomenon

Equilibrium and nonequilibrium transport properties of adsorbates in mesoporous Vycor porous glass have been experimentally studied using nuclear magnetic resonance techniques. With the known geometrical characteristics of porous glass and with measured self-diffusivities, transient sorption curves have been quantitatively compared to those predicted within a Fick's law model. This model correctly describes data outside a hysteresis region. In contrast, in the hysteresis region, a two-step mechanism of density relaxation is required to explain the behavior. These two mechanisms are identified as diffusion at early stages and activated density redistribution at later stages of adsorption. The latter mechanism, being intrinsically slow in nature, is anticipated to prevent the system from reaching equilibrium.