Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.     

Grain-boundary diffusion in nanocrystals with a time-dependent diffusion coefficient

A solution to the equation of grain-boundary diffusion is obtained under conditions where migration of the diffusant from the boundaries into the grains is absent and the diffusion coefficient decreases with time from an increased value to a value characteristic of equilibrium grain boundaries. The specific features of the grain-boundary diffusion in nanocrystals are qualitatively analyzed in terms of this solution.     

Comparison between fixed and Gaussian steplength in Monte Carlo simulations for diffusion processes

We analyze the different degrees of accuracy of two Monte Carlo methods for the simulation of one-dimensional diffusion processes with homogeneous or spatial dependent diffusion coefficient that we assume correctly described by a differential equation. The methods analyzed correspond to fixed and Gaussian steplengths. For a homogeneous diffusion coefficient it is known that the Gaussian steplength generates exact results at fixed time steps Δt. For spatial dependent diffusion coefficients the symmetric character of the Gaussian distribution introduces an error that increases with time. As an example, we consider a diffusion coefficient with constant gradient and show that the error is not present for fixed steplength with appropriate asymmetric jump probabilities.     

Solutions for a non-Markovian diffusion equation

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.     

Effective relaxation time of the impurity concentration in a medium with a slightly nonuniform diffusion coefficient

Based on the previously introduced integral criterion and a new approach to estimating the time characteristics of diffusion, the relaxation time of impurity concentration in a medium with a slightly nonuniform diffusion coefficient is determined. The diffusion coefficient profile that cuts the relaxation time is found.     

Positronium diffusion in crystals

The temperature dependence of the diffusion coefficient is analyzed for the two cases of localized and delocalized positronium in a crystal. The diffusion coefficient for the localized positronium is calculated from the Kubo formula, while that for the free positronium is calculated in the relaxation-time approximation. The temperature dependence of the tunneling diffusion coefficient for positronium is specified by analogy with impuritons in a quantum crystal.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 25–28, July, 1981.     

Dependence of chaotic diffusion on the size and position of holes

A particle driven by deterministic chaos and moving in a spatially extended environment can exhibit normal diffusion, with its mean square displacement growing proportional to the time. Here, we consider the dependence of the diffusion coefficient on the size and the position of areas of phase space linking spatial regions ('holes') in a class of simple one-dimensional, periodically lifted maps. The parameter dependent diffusion coefficient can be obtained analytically via a Taylor-Green-Kubo formula in terms of a functional recursion relation. We find that the diffusion coefficient varies non-monotonically with the size of a hole and its position, which implies that a diffusion coefficient can increase by making the hole smaller. We derive analytic formulas for small holes in terms of periodic orbits covered by the holes. The asymptotic regimes that we observe show deviations from the standard stochastic random walk approximation. The escape rate of the corresponding open system is also calculated. The resulting parameter dependencies are compared with the ones for the diffusion coefficient and explained in terms of periodic orbits.     

Anisotropic diffusion in media with absorbing traps (application to the Keller–Dykhne theorem)

Anisotropic diffusion in multidimensional media with absorbing traps is investigated. It is shown that the diffusion asymptotic forms of the survival probability over long time intervals are determined by a new effective diffusion coefficient. Exact expressions derived for the effective diffusion coefficient generalize the well-known Dykhne–Keller 2D results to the 3D case. In other words, the dynamic diffusion approach is used for obtaining a generalization of the Dykhne–Keller theorem to the 3D case.     

Measurement of dye diffusion in agar gel by use of low-coherence interferometry

We demonstrate low-coherence interferometry for diffusion measurements. We have measured the diffusion coefficient of a phthalocyanine dye in 1.5% agar gel with a two-wavelength interferometer; one wavelength was matched to the absorption peak of the dye at 675 nm, while the other, 805 nm, was not affected by the dye. The diffusion coefficient of the dye was found by fitting a mathematical model for the interferometer signal to the measured low-coherence interferometry amplitude. A 95% confidence interval for the diffusion coefficient was found to be D = (2.5 +/- 0.2) x 10(-10) m2/s. The influence of speckle averaging and experiment time on the determination of the diffusion coefficient has been studied. The presented technique allows in situ characterization of diffusion in semitransparent media.     

氢气在多层球壳中扩散过程的模拟计算

 从Fick第二定律出发,推导了氢气在多层球壳中,扩散过程的浓度分布表达式,即氢气浓度与扩散时间和扩散距离的关系。每层中,在一定的扩散时间内,一定量的氢气的扩散浓度与扩散深度之间近似为抛物线关系,说明渗透物沿着聚合物链段膨胀的方向进行扩散。在同一位置,一定量氢气的浓度,随着时间的延长而增加,说明渗透物使聚合物链段发生膨胀,自由体积增大,扩散路径增多,扩散系数随着浓度的增加而增大。     

A lattice-Boltzmann scheme for Cattaneo’s diffusion equation

A lattice-Boltzmann (LB) formulation is developed for the simulation of Cattaneo’s diffusion equation. To do this, the collision term is computed from a 1-step back relaxation dynamics which, in turn, induces a hyperbolic-type diffusion effect. The Fickian LB formulation is recovered in the limit as a relaxation time coefficient goes to zero.     

A theory of modulational diffusion

Modulational diffusion, a weak instability that frequently occurs in many-dimensional, nonlinear Hamiltonian systems, is studied both analytically and numerically. Modulational diffusion arises when an oscillation in one of the degrees of freedom is phase modulated at a slow driving frequency, producing a “modulational layer” of overlapping resonances in phase spaces. Because the motion within this layer is chaotic, any coupling to the oscillation of another degree of freedom produces a long-time diffusion of its associated action along the layer. The diffusion rate for this process is evaluated analytically, and is compared with numerical calculations for a model, two-degree-of-freedom, nonautonomous Hamiltonian. The diffusion coefficient is found to vary in a series of descending steps as the frequency difference between the two oscillators is increased. Good agreement between analytical and numerical results has been obtained over many orders of magnitude in the diffusion coefficient.     

Entropy as a measure of diffusion

The time variation of entropy, as an alternative to the variance, is proposed as a measure of the diffusion rate. It is shown that for linear and time-translationally invariant systems having a large-time limit for the density, at large times the entropy tends exponentially to a constant. For systems with no stationary density, at large times the entropy is logarithmic with a coefficient specifying the speed of the diffusion. As an example, the large-time behaviors of the entropy and the variance are compared for various types of fractional-derivative diffusions.     

Acquisition of pure-phase diffusion spectra using oscillating-gradient spin echo

Oscillating-gradient spin echo (OGSE) diffusion experiments have long been used to measure the short-time apparent diffusion coefficient, D(app)(t), in the presence of restricted diffusion, as well as the spectrum of the slow-motion velocity autocorrelation function. In this work, we focus on two previously unexplored aspects of OGSE experiments: convection compensation and acquisition of pure-phase diffusion spectra in the presence of homonuclear scalar couplings. We demonstrate that convection compensation afforded by single-echo OGSE compares well with that in double-echo convection-compensated PGSE experiments. We also show that, in the presence of homonuclear scalar couplings, setting the OGSE echo time to 1/2J enables acquisition of pure-phase diffusion spectra and yields more reliable D estimates than mixed-phase PGSE or OGSE spectra. Pure-phase OGSE acquisition is also compatible with measurements of the apparent diffusion coefficient at an arbitrary diffusion time. These features of OGSE can be valuable in diffusion measurements of scalar-coupled small-molecule probes in cellular and other heterogeneous systems.     

Spin diffusion in multiple pulse spin-locking in solids containing paramagnetic impurities

Spin diffusion and spin-lattice relaxation in solids containing paramagnetic impurities under influence of a multiple-pulse spin-locking radio-frequency sequence are studied theoretically and experimentally. The diffusion equation obtained provides a clue for determination of the time dependent magnetization. The spin-lattice relaxation time is calculated as a function of the correlation time and multiple-pulse field parameters. From the experimental data the spin diffusion coefficient, the radius of the spin diffusion barrier, and the correlation time for very slow molecular motion in polycrystalline (C(2)F)(n) system are estimated and found to be D～7.1×10(-12)cm(2)/s, r(c)～4.8×10(-10)m, and τ(c)～10.2μs, respectively.     

Generalized Fick law for anomalous diffusion in the multidimensional comb model

The problem of multidimensional diffusion is considered within the framework of the comb model. It is shown that the diffusion current for the case of anomalous subdiffusion random walks is described by the generalized Fick law containing the diffusion tensor instead of the usual coefficient. The form of the diffusion tensor components is an unusual form of operator as fractional time derivatives. The orders of the fractional exponents are different for different directions.     

Excitonic diffusion in thin molecular films

The analyses of exciton diffusion in thin molecular films have shown that the diagonal elements of the diffusion tensor, usually called diffusion coefficients, depend on the layer index labeling layers in the direction of disturbed symmetry. The particular analysis was done for a thin film having four layers. In this structure only two layers are occupied by optical excitations. It means that in the four layer film two films occur in which optical excitations can travel. The subfilm contains a boundary layer that noticeably differs from the subfilm with internal layers. If the subfilm contains the boundary layer, the diffusion coefficient of this layer differs from the diffusion coefficient of any internal layer. If the subfilm contains two internal layers, the diffusion coefficient of these layers are equal, expectably from the viewpoint of physics. The exciton diffusion is very low due to the high exciton energies. This work was supported by the Serbian Ministry of Science and Technology: Grant No 1895.     

用液芯柱透镜快速测量液相扩散系数-折射率空间分布瞬态测量法

本文提出了一种快速测量液相扩散系数的方法, 该方法以液芯柱透镜作为液相扩散池和成像元件, 利用柱透镜成像过程中特有的折射率空间分辨测量能力, 只需记录一幅瞬态扩散图像, 根据图像的像宽与折射率的对应关系, 基于扩散定律快速计算出液相扩散系数. 实验研究了室温(25℃)下乙二醇和纯水间的扩散过程, 用折射率空间分布法测量了扩散系数, 和其他测量方法得到的结果进行了分析对比, 结果表明:用折射率空间分布法测量液相扩散系数具有数据采集耗时短(～20 ms)、测量速度快(<1 s)、精度高(相对误差<3%)和操作简单的特点, 为快速测定液相扩散系数提供了一种有效的新方法.     

Active diffusion of motor particles

The movement of motor particles consisting of one or several molecular motors bound to a cargo particle is studied theoretically. The particles move on patterns of immobilized filaments. Several patterns are described for which the motor particles undergo nondirected but enhanced diffusion. Depending on the walking distance of the particles and the mesh size of the patterns, the active diffusion coefficient exhibits three different regimes. For micrometer-sized motor particles in water, e.g., this diffusion coefficient can be enhanced by 2 orders of magnitude.     

NMR studies of restricted diffusion in lyotropic systems

Tortuosity, 1/alpha, and surface-to-volume ratio, S/V, were determined in aqueous solutions of decylammonium, dodecylammonium and tetradecylammonium chlorides of various concentrations by measuring the apparent diffusion coefficient of water, D(app)(delta). This was found to be much smaller than in the bulk state. Such restricted diffusion is interpreted in terms of the Mitra model, where D(delta) depends on diffusion time and is controlled primarily by S/V. The samples exhibit lamellar (L), hexagonal (H) and isotropic (I) liquid crystalline phases. We observed changes in S/V upon phase transition. In the lamellar and hexagonal phases, the system is ordered, resulting in relatively small S/V ratios, compared to the micellar-isotropic phase. We did not observe a dependence on the diffusion time, delta, in the isotropic phase, because the duration of the experiment was not sufficiently short to observe the change from D(app)(delta) to D(eff). We observed the effective diffusion coefficient of water, which directly probes the tortuosity of the system. The S/V ratios were obtained by fitting the Mitra model, using known values of the bulk water diffusion coefficients, and the assumption that D(app) --> D0 for delta --> 0. S/V is correlated with the type of structure, increasing on transition to the isotropic phase and decreasing on transition to other phases. The change in tortuosity is small, but slightly larger for the isotropic phase.