Random walks on the Comb model and its generalizations

Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to "effective reducing" of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed.     

Non-linear relativistic diffusions

We obtain a non-linear generalization of the relativistic diffusion. We discuss diffusion equations whose non-linearity is a consequence of quantum statistics. We show that the assumptions of the relativistic invariance and an interpretation of the solution as a probability distribution substantially restrict the class of admissible non-linear diffusion equations. We consider relativistic invariant as well as covariant frame-dependent diffusion equations with a drift. In the latter case we show that there can exist stationary solutions of the diffusion equation besides the equilibrium solution corresponding to the quantum or Tsallis distributions. We define the relative entropy as a function of the diffusion probability and prove that it is monotonically decreasing in time when the diffusion tends to equilibrium. We discuss its relation to the thermodynamic behavior of diffusing particles.     

Stabilization of the Eulerian model for incompressible multiphase flow by artificial diffusion

The commonly used Eulerian or continuum model for incompressible multiphase flow is known to be unstable to perturbations for all wavenumbers, even if viscosity terms are used in the momentum equations. In the present work the model is stabilized by adding explicit artificial diffusion to the mass equations. The artificial diffusion terms lead to improved stability properties: uniform flow becomes linearly stable for large wavenumbers, and above an analytically derived threshold for the artificial diffusivity, stability for all wavenumbers is achieved. The artificial diffusivity reappears in the momentum equations, in such a way that fundamental properties of the standard equations remain valid: Galilean invariance is maintained, total mass and momentum are conserved, decay of total kinetic energy is ensured in the absence of external forces, and a flow initially at rest at hydrostatic pressure remains unchanged, even if the spatial distribution of volume fractions is nonuniform. A staggered finite volume pressure correction method using central differencing (leading to energy conserving discretization of convective and pressure terms) is presented. Application of the method to one-dimensional two-phase flow of falling particles particles confirms that the equations are stable with and unstable without artificial diffusion in the volume fraction equation.     

Ionic diffusion on a lattice: Effects of the order-disorder transition on the dynamics of non-equilibrium systems

We examine the behaviour of the concentration profiles of particles with repulsive interactions diffusing on a host lattice. At low temperature, the diffusion process is strongly influenced by the presence of ordered domains. We use mean field equations and Monte-Carlo simulations to describe the various effects which influence the kinetic behaviour. An effective diffusion coefficient is determined analytically and is compared with the simulations. Finite gradient effects on the ordered domains and on the diffusion are discussed. The kinetics studied is relevant for superionic conductors, for intercalation and also for the diffusion of particles adsorbed on a substrate. Received: 26 June 1997 / Revised: 18 September 1997 / Accepted: 10 November 1997     

A stochastic model of Maxwell's equations in 1+1 dimensions

We show that the random walk model due to Mark Kac which underlies the telegraph equations may be modified to produce Maxwell's field equations in 1+1 dimensions. This provides the field equations with a representation in terms of classical particles. It also establishes the Kac model as a strong conceptual link between the diffusion, telegraph, and Maxwell equations, and suggests that recent simulations of the Schrödinger and Dirac equations are analogous to Maxwell's equation in terms of interpretation.     

Short-time correlations of many-body systems described by nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations

Analytical expressions for short-time correlation functions, diffusion coefficients, mean square displacement, and second order statistics of many-body systems are derived using a mean field approach in terms of nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations. The results are illustrated for the Desai–Zwanzig model, the nonlinear diffusion equation related to the Tsallis statistics, and a Vlasov–Fokker–Planck equation describing bunch particles in particle accelerator storage rings.     

Continuum description of anomalous diffusion on a comb structure

Anomalous diffusion on a comb structure consisting of a one-dimensional backbone and lateral branches (teeth) of random length is considered. A well-defined classification of the trajectories of random walks reduces the original problem to an analysis of classical diffusion on the backbone, where, however, the time of this process is a random quantity. Its distribution is dictated by the properties of the random walks of the diffusing particles on the teeth. The feasibility of applying mean-field theory in such a model is demonstrated, and the equation for the Green’s function with a partial derivative of fractional order is obtained. The characteristic features of the propagation of particles on a comb structure are analyzed. We obtain a model of an effective homogeneous medium in which diffusion is described by an equation with a fractional derivative with respect to time and an initial condition that is an integral of fractional order. Zh. éksp. Teor. Fiz. 114, 1284–1312 (October 1998)     

The renewal equation for persistent diffusion

Persistent diffusion in one dimension, in which the velocity of the diffusing particle is a dichotomic Markov process, is considered. The flow is non-Markovian, but the position and the velocity together constitute a Markovian diffusion process. We solve the coupled forward Kolmogorov equations and the coupled backward Kolmogorov equations with appropriate initial conditions, to establish a generalized (matrix) form of the renewal equation connecting the probability densities and first passage time distributions for persistent diffusion.     

Sum rules for Sαβ(k,ω) for two diffusing species

Starting from the Smoluchowski equation without hydrodynamic interactions for two species of spherical diffusing particles, sum rules are derived here for the first three moments of S_αβ(k,ω), i.e., for the initial value of the first, second and third time-derivatives of F_αβ(k, t) (the time-dependent correlations between the fluctuations in the local concentration of diffusing particles of species α and β). These sum rules are written in terms of the potential of interaction u_αβ(r) between the diffusing particles and the two- and three-particles distribution functions. This derivation is motivated by its potential use in the study of counterion effects on the diffusion of highly charged colloidal particles. Thus, we propose to approximate the memory function involved in the time evolution equation for F_αβ(k, t by a two-parameter model, with its (k-dependent) parameters being determined by the sum rules derived here. This procedure, along with Kirkwood's superposition approximation, reduces the dynamical problem to the knowledge of the radial distribution functions g_αβ(r).     

Diffusion phenomenon in the hyperbolic and parabolic regimes

We discuss the diffusion phenomenon in the parabolic and hyperbolic regimes. New effects related to the finite velocity of the diffusion process are predicted, that can partially explain the strange behavior associated to adsorption phenomenon. For sake of simplicity, the analysis is performed by considering a sample in the shape of a slab limited by two perfectly blocking surfaces, in such a manner that the problem is one-dimensional in the space. Two cases are investigated. In the former, the initial distribution of the diffusing particles is assumed of gaussian type, centered around the symmetry surface in the middle of the sample. In the latter, the initial distribution is localized close to the limiting surfaces. In both cases, we show that the evolution toward to the equilibrium distribution is not monotonic. In particular, close to the limiting surfaces the bulk density of diffusing particles present maxima and minima related to the finite velocity of the diffusion process connected to the second order time derivative in the partial differential equation describing the evolution of the bulk density in the sample.     

Scaling limits for interacting diffusions

We consider a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance. We derive a nonlinear bulk diffusion equation for the density of the particle distribution on the circle. The diffusion coefficient is determined as a function of density in terms of standard thermodynamical objects.This research was supported by a grant from the National Science Foundation DMS-89-01682     

Chaotic diffusion for particles moving in a time dependent potential well

The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and; (ii) by the solution of the diffusion equation. The dynamic of the diffusing particles is made by the use of a two dimensional, nonlinear area preserving map for the variables energy and time. The phase space of the system is mixed containing both chaos, periodic regions and invariant spanning curves limiting the diffusion of the chaotic particles. The chaotic evolution for an ensemble of particles is treated as random particles motion and hence described by the diffusion equation. The boundary conditions impose that the particles can not cross the invariant spanning curves, serving as upper boundary for the diffusion, nor the lowest energy domain that is the energy the particles escape from the time moving potential well. The diffusion coefficient is determined via the equation of the mapping while the analytical solution of the diffusion equation gives the probability to find a given particle with a certain energy at a specific time. The momenta of the probability describe qualitatively the behavior of the average energy obtained by numerical simulation, which is investigated either as a function of the time as well as some of the control parameters of the problem.     

On the Theory of Electric-Current Flow through a Mixture of Charged Particles

Current conduction through a mixture made of two species of positively charged particles is considered where one of the latter species participates in the exchange with the surrounding medium. A solution to the electrodiffusion equations together with Poisson's equation is obtained in the first approximation in terms of the small parameter. A condition is determined where the distribution of charged particles involved in the exchange with the surrounding medium is derived using the diffusion equation for neutral particles. It is shown that the solution to the electrodiffusion equations contains a component decaying with time.     

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.     

The Kinetics of the Coordination Transformation for Preparation of Nanosized CdS in a PVA Film

Research on the preparation kinetics of nanosized materials is considered. The layered diffusion model (LDM) for S^2? diffusing in a polymer–metal complex film, used for preparation of nanosized semiconductor metal sulfide by a coordination transformation method, is briefly introduced. An indirect measurement was employed to determine the concentration of residual S^2? in the solution during the diffusion in a PVA–Cd²⁺ complex film. The relationship between the diffusion amount and the diffusion time shows that the diffusion curves are independent of the amount of Cd²⁺ in the film when the total amount of S^2? is less than that of Cd²⁺. The experimental phenomena coincide basically with the theoretical curves deduced from LDM; thus they provide a proof for the LDM and the diffusion equations. The diffusion constant of S^2? diffusing in PVA–Cd²⁺ complex film is about 1.7×10^?3 cm²/s.     

莱维噪声诱发的速度双模分布

研究了非线性阻尼驱动的惯性莱维飞行在自由势场中的反常输运。通过引入依赖于速度的非线性阻尼,长程跳跃的莱维粒子被束缚,粒子的动力学行为由发散收敛为与时间呈正比的正常扩散。观察到在自由势场中莱维飞行的速度的定态概率分布呈双模分布形式,更为重要的是,观察到双模形式与莱维系数及非线性阻尼系数相关。     

The effect of surface contact conditions on grain boundary interdiffusion in a semi-infinite bicrystal

The Kirkendall effect is conditioned by active diffusion as well as by active sources and sinks for vacancies. In the case of grain boundaries under the condition of negligible bulk diffusion, the Kirkendall effect is highly localized and responsible for the formation of an extra material wedge in the grain boundary, which may lead to high stress concentrations. The Kirkendall effect in grain boundaries of a binary system is described by a set of partial differential equations for the mole fraction of one of the diffusing components and for the stress component normal to the grain boundary completed with the respective initial and boundary conditions. The contact conditions of the grain boundary with the surface layer acting as source of one of the diffusing components can be considered as equilibrium ones ensuring the continuity of generalized chemical potentials of both diffusing components. Thus, the boundary conditions are determined by the difference in chemistry (i.e. how the thermodynamic parameters depend on chemical composition) of the grain boundaries and of the surface layer. The simulations based on the present model indicate a drastic influence of the chemistry on the grain boundary interdiffusion and Kirkendall effect.     

Anomalous transport in fluid fleld with random waiting time depending on the preceding jump length

Anomalous(or non-Fickian) transport behaviors of particles have been widely observed in complex porous media.To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields,in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced,and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived.As examples,two generalized advection-dispersion equations for Gaussian distribution and levy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Anomalous transport in fluid field with random waiting time depending on the preceding jump length

Anomalous (or non-Fickian) transport behaviors of particles have been widely observed in complex porous media. To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields, in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced, and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived. As examples, two generalized advection-dispersion equations for Gaussian distribution and lévy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Influence of anisotropy of the diffusion coefficient on the asymptotic behavior of a spatial atom distribution in structures with complex geometry

Obtained by the method of continual integration in the optimal trajectory approximation is the asymptotic behavior of the Green's function of the Fick equation with anisotropic diffusion coefficient in structures with complex geometry when the domain boundary is impermeable to the diffusing particles. Using the expression found for the Green's function, topological properties of the equal-concentration surfaces, particularly the topological singularities of a diffusion p-n-junction, are determined by means of the given initial distribution of the diffusing atoms.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 9–12, May, 1988.