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.     

Spatial Markov model of anomalous transport through random lattice networks

Flow through lattice networks with quenched disorder exhibits a strong correlation in the velocity field, even if the link transmissivities are uncorrelated. This feature, which is a consequence of the divergence-free constraint, induces anomalous transport of passive particles carried by the flow. We propose a Lagrangian statistical model that takes the form of a continuous time random walk with correlated velocities derived from a genuinely multidimensional Markov process in space. The model captures the anomalous (non-Fickian) longitudinal and transverse spreading, and the tail of the mean first-passage time observed in the Monte Carlo simulations of particle transport. We show that reproducing these fundamental aspects of transport in disordered systems requires honoring the correlation in the Lagrangian velocity.     

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.     

An approximate method for solving radiation transport problems in temporally and spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. It is assumed that the medium consists of randomly distributed lumps of material embedded in a background matrix and in each lump the properties may vary randomly with time. The coefficients for scattering and absorption are represented mathematically by members of a random characteristic set function, which depend on space and time. Different physical situations can be described by different forms and combinations of these set functions. In order to effect a solution of the resulting stochastic transport equation, which may be for photons or neutrons, we make the, a priori, assumption that the functional form for the solution of the transport equation, i.e. the stochastic flux, can be represented by the same mathematical form as the scattering and absorption coefficients (or cross sections), i.e. we introduce a stochastic ansatz. This procedure leads to a set of deterministic equations from which the mean and variance of the flux in space and time can be obtained. For the case of a two-phase medium, either two or four coupled integro-differential equations are obtained for the deterministic functions that arise (depending on the problem) and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those from the Levermore-Pomraning (LP) theory, but the new equations offer an opportunity to deal with more general forms of stochastic processes and combine simultaneously time and space fluctuations. The stochastic characteristics of the medium are defined by the correlation functions which appear in the equations and, by making plausible assumptions about the functional form of these autocorrelation functions, different physical situations can be simulated, according to the structure of the medium. The main contribution of the present work is to include space and time fluctuations simultaneously as a pseudo-dichotomic Markov process.     

Transport Equation Evaluation of Coupled Continuous Time Random Walks

The transport behavior of a migrating particle in a disordered medium is exhibited in the solution of a transport equation derived from a coupled continuous time random walk (CTRW). A core aspect of CTRW is the spectrum of transitions in displacement s and time t, ψ(s,t), that characterizes the disordered system, which determine the transport. In many applications the CTRW approach has successfully accounted for the anomalous or non-Fickian nature of the particle plume propagation based on a power-law dependence ψ(t) in a decoupled p(s)ψ(t) approximation to ψ(s,t). For example, this power-law dependence in t derives from the complex Darcy flow fields in geological formations. Recently, the fully coupled CTRW was analyzed using a particle tracking approach, demonstrating that the decoupled approximation is valid only for a compact distribution of s. In this paper we solve the nonlocal-in-time transport equation with a ψ(s,t) containing a power-law dependence in both s (a Lévy-like distribution) and t, which necessitates the strong s,t coupling. We show enhanced transport behavior (relative to the plume propagation behavior reported in the literature) that derives from the rare large displacements in s (limited by the transition t). The interplay between the two coupled power laws is clearly shown in the changes in the breakthrough curves in the arrival times, dispersion and dependence on the velocity (v=s/t) distribution. Similar enhancements are exhibited in the particle tracking results.     

Turbulent transport processes in a plasma as a diffusion process with random time

It is proposed to apply the statistical analysis of the increments of fluctuating particle fluxes to analyzing the probability characteristics of turbulent transport processes in plasma. Such an approach makes it possible to analyze the dynamic probability characteristics of the process under study. It is shown that, in the plasmas of the L-2M stellarator and the TAU-1 linear device, the increments of local fluctuating particle fluxes are of stochastic character and that their distributions are scale mixtures of Gaussians. In particular, for TAU-1, the increments have the Laplacian distribution (which is a scale mixture of Gaussians with an exponential mixing distribution). This implies that the rate of flux variations is a diffusion process with random time. It is shown that the characteristic growth (damping) time of fluctuations is one order of magnitude shorter than their characteristic correlation time. Physical mechanisms that may be responsible for the random character of the growth (damping) of fluctuations are discussed.     

Improved particle filters for multi-target tracking

We present a novel approach for improving particle filters for multi-target tracking. The suggested approach is based on drift homotopy for stochastic differential equations. Drift homotopy is used to design a Markov Chain Monte Carlo step which is appended to the particle filter and aims to bring the particle filter samples closer to the observations while at the same time respecting the target dynamics. We have used the proposed approach on the problem of multi-target tracking with a nonlinear observation model. The numerical results show that the suggested approach can improve significantly the performance of a particle filter.     

Hybrid three-dimensional variation and particle filtering for nonlinear systems

This work addresses the problem of estimating the states of nonlinear dynamic systems with sparse observations.We present a hybrid three-dimensional variation(3DVar) and particle piltering(PF) method,which combines the advantages of 3DVar and particle-based filters.By minimizing the cost function,this approach will produce a better proposal distribution of the state.Afterwards the stochastic resampling step in standard PF can be avoided through a deterministic scheme.The simulation results show that the performance of the new method is superior to the traditional ensemble Kalman filtering(EnKF) and the standard PF,especially in highly nonlinear systems.     

A PDF projection method: A pressure algorithm for stand-alone transported PDFs

In this paper, a new formulation of the projection approach is introduced for stand-alone probability density function (PDF) methods. The method is suitable for applications in low-Mach number transient turbulent reacting flows. The method is based on a fractional step method in which first the advection–diffusion–reaction equations are modelled and solved within a particle-based PDF method to predict an intermediate velocity field. Then the mean velocity field is projected onto a space where the continuity for the mean velocity is satisfied. In this approach, a Poisson equation is solved on the Eulerian grid to obtain the mean pressure field. Then the mean pressure is interpolated at the location of each stochastic Lagrangian particle. The formulation of the Poisson equation avoids the time derivatives of the density (due to convection) as well as second-order spatial derivatives. This in turn eliminates the major sources of instability in the presence of stochastic noise that are inherent in particle-based PDF methods. The convergence of the algorithm (in the non-turbulent case) is investigated first by the method of manufactured solutions. Then the algorithm is applied to a one-dimensional turbulent premixed flame in order to assess the accuracy and convergence of the method in the case of turbulent combustion. As a part of this work, we also apply the algorithm to a more realistic flow, namely a transient turbulent reacting jet, in order to assess the performance of the method.     

随机对流扩散方程的数值仿真

本文针对对流一扩散随机过程在随机输入（即随机输运和源项）,作用下进行数值仿真。我们先将对流扩散随机微分方程中的随机函数采用有限项截断的多项式浑沌展开（Polynomial Chaos Expansion）展开,再由Galerkin映射法得到求解浑沌展开系数的确定性方程组。这是一个在物理空间包含多尺度解的大方程组。为此我...     

A stochastic approach to hopping transport in semiconductors

Generalized charge carrier equations for hopping transport in semiconductors are derived which include also the widely used Van Roosbroeck equations. The approach is based on a microscopic stochastic interacting particle system which models the hopping of electrons on a random set of states.     

Dynamics and spatial organization of endosomes in mammalian cells

We combine particle tracking and stochastic simulations to analyze the dynamics and organization of early endocytic vesicles in mammalian cells. At short time scales (<10(1) sec) vesicles exhibit 1D symmetric bidirectional motor-driven transport on microtubules such that the mean squared displacement (MSD) scales as t3/2, but the MSD shows a crossover to facilitated diffusion at longer times (>10(1) sec). Facilitated diffusion results in rapid equilibration of vesicles on microtubules. The asterlike organization of microtubules causes perinuclear accumulation of vesicles despite symmetric transport.     

Channeling and percolation in two-dimensional chaotic dynamics

The Hamiltonian dynamics of a particle moving in a nearly periodic two-dimensional (2-D) potential of square symmetry is analyzed. The particle undergoes two types of unbounded stochastic or random walks in such a system: a quasi-1-D motion (a "stochastic channeling") and a 2-D motion which results from a sort of stochastic percolation. A scenario for the onset of this stochastic percolation is analyzed. The threshold energy for percolation is found as a function of the perturbation parameter. Each type of random walk has the property of intermittency. The particle transport is anomalous in certain energy intervals.     

Knudsen Gas in a Finite Random Tube: Transport Diffusion and First Passage Properties

We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and ergodic, non-interacting particles move straight with constant speed. Upon hitting the tube walls, they are reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. Steady state transport is studied by introducing an open tube segment as follows: We cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis. Particles which leave this piece through the segment boundaries disappear from the system. Through stationary injection of particles at one boundary of the segment a steady state with non-vanishing stationary particle current is maintained. We prove (i) that in the thermodynamic limit of an infinite open piece the coarse-grained density profile inside the segment is linear, and (ii) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube. Thus we prove Fick’s law and equality of transport diffusion and self-diffusion coefficients for quite generic rough (random) tubes. We also study some properties of the crossing time and compute the Milne extrapolation length in dependence on the shape of the random tube.     

Extremal behavior of a coupled continuous time random walk

Coupled continuous time random walks (CTRWs) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density governing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.     

The impact approximation in the theory of bimolecular quasi-resonant processes

The kinetic equation for a density matrix, which describes the relaxation of the internal states of encountering particles dissolved in an inert medium, has been derived under the following assumptions: a) the random motion of reacting particles in a liquid is considered to be a classical Markoffian process; b) the concentration of reacting particles is small enough. The equation obtained is shown to be the generalization of that of the familiar impact theory of pressure broadening for the case of any type of encountering particle motion. Our general formulae are concretized in accordance with the physical situations of rectilinear, diffusion, and stochastic jump motion of the encountering particles.     

Kernel principal component analysis for stochastic input model generation

Stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used. These input models are often constructed from a set of experimental samples of the underlying random field. To this end, the Karhunen–Loève (K–L) expansion, also known as principal component analysis (PCA), is the most popular model reduction method due to its uniform mean-square convergence. However, it only projects the samples onto an optimal linear subspace, which results in an unreasonable representation of the original data if they are non-linearly related to each other. In other words, it only preserves the first-order (mean) and second-order statistics (covariance) of a random field, which is insufficient for reproducing complex structures. This paper applies kernel principal component analysis (KPCA) to construct a reduced-order stochastic input model for the material property variation in heterogeneous media. KPCA can be considered as a nonlinear version of PCA. Through use of kernel functions, KPCA further enables the preservation of higher-order statistics of the random field, instead of just two-point statistics as in the standard Karhunen–Loève (K–L) expansion. Thus, this method can model non-Gaussian, non-stationary random fields. In this work, we also propose a new approach to solve the pre-image problem involved in KPCA. In addition, polynomial chaos (PC) expansion is used to represent the random coefficients in KPCA which provides a parametric stochastic input model. Thus, realizations, which are statistically consistent with the experimental data, can be generated in an efficient way. We showcase the methodology by constructing a low-dimensional stochastic input model to represent channelized permeability in porous media.     

An age-dependent approach to random walks on disordered lattices

We derive a new approach for the stochastic transport in random systems, starting from a phenomenological master equation with random transition rates. Our method combines the effective medium approximation with age-dependent dynamics. Within the framework of our approximation, the static disorder may be described by means of a system of age-dependent master equations. For translationally invariant systems which obey certain separability conditions, the approach is equivalent with the continuous time random walk theory. Moreover, for self-avoiding random walks our effective medium approximation is exact. For non self-avoiding random walks, the approximation neglects the correlations between successive transitions leading to closed paths on the lattice.     

Anomalous transport in heterogeneous media

The diffusion dynamics of particles in heterogeneous media is studied using particle-based simulation techniques. A special focus is placed on systems where the transport of particles at long times exhibits anomalies such as subdiffusive or superdiffusive behavior. First, a two-dimensional model system is considered containing gas particles (tracers) that diffuse through a random arrangement of pinned, disk-shaped particles. This system is similar to a classical Lorentz gas. However, different from the original Lorentz model, soft instead of hard interactions are considered and we also discuss the case where the tracer particles interact with each other. We show that the modification from hard to soft interactions strongly affects anomalous-diffusive transport at high obstacle densities. Second, non-linear active micro-rheology in a glass-forming binary Yukawa mixture is investigated, pulling single particles through a deeply supercooled state by applying a constant force. Here, we observe superdiffusion in force direction and analyze its origin. Finally, we consider the Brownian dynamics of a particle which is pulled through a two-dimensional random force field. We discuss the similarities of this model with the Lorentz gas as well as active micro-rheology in glass-forming systems.