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.     

Random walk in a random medium in one dimension

Applying scaling and universality arguments, the long-time behavior of the probability distribution for a random walk in a one-dimensional random medium satisfying Sinai's constraint is obtained analytically. The convergence to this asymptotic limit and the fluctuations of this distribution are evaluated by solving numerically the stochastic equations for this walk.     

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.     

Two models of quantum random walk

We present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.     

Discrete Green's Function for Lattice Random Walk

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Discrete Green's Function for Lattice Random Walk

A lattice random walk model based on particles scattering on discrete lattice of homogenous space is introduced. The discrete Green's function (DFG) for two-dimensional and three-dimensional lattice random walk of photon is found and proved by mathematical induction. The convolution theorem of photon lattice random walk is presented. They can be used with the method of images to calculate the photon density distribution in semi-infinite and finite slab homogenous turbid media such as tissue.     

度关联无标度网络上的有倾向随机行走

有倾向随机行走是研究网络上数据包路由策略的有效方法. 由于许多真实技术网络包括互联网都具有负的度关联特征, 因此本文研究这种网络上的有倾向随机行走性质. 研究表明: 在负关联网络上粒子可以在连接度较大的节点上均匀分布, 而连接度小的节点上粒子较少; 负关联网络上随机行走的速度比非关联网络更快; 找到了负关联网络上的最佳倾向性系数, 在此情况下负关联网络上随机行走的速度远快于非关联网络. 负关联网络既可以利用度小的节点容纳粒子, 又可以利用度大的节点快速传输, 这是负关联网络上高行走效率产生的机制.     

Area Distribution for Directed Random Walks

We study the probability distribution for the area under a directed random walk in the plane. The walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed. This model captures some of the qualitative features of earthquakes, avalanches, and other self-organized critical phenomena in one dimension. By finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4/3.     

A study on correlated exponential random walks

Based on the linear Boltzmann transport formulation, we investigate the statistics of correlated exponential random walks that are continuous in space and discrete in time. We show that asymptotically, the correlated random walk process is diffusive and derive an effective diffusion constant. We investigate the power spectral characteristics of the associated random forces. We also present some results on the first passage time distribution and establish that asymptotically it reduces to that associated with simple Gaussian walks.     

CTRW模型等待时间分布的随机抽样方法

研究等待时间分布为截断幂律分布的连续时间随机行走(CTRW)模型,分析比较截断幂律分布的可能随机抽样方法,并提出一套可以对其进行精确抽样的乘分布舍选算法.数值结果表明:当幂律指数α > 0.2时,该算法的抽样效率在80%以上.利用该算法可以成功得出CTRW模型描述的三种反常输运过程的标度关系.     

Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions

We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher’s equations—as well as their fractional generalizations—from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations.     

Diffusion of oriented particles in porous media

Diffusion of particles in porous media often shows subdiffusive behavior. Here, we analyze the dynamics of particles exhibiting an orientation. The features we focus on are geometrical restrictions and the dynamical consequences of the interactions between the local surrounding structure and the particle orientation. This interaction can lead to particles getting temporarily stuck in parts of the structure. Modeling this interaction by a particular random walk dynamics on fractal structures we find that the random walk dimension is not affected while the diffusion constant shows a variety of interesting and surprising features.     

Theory of anomalous dispersion in a-Se

A theory of multiple trapping expressed in terms of generalized first-order transport equations is used to explain the change in dispersion with temperature of the photocurrent transients in a-Se. The theory is shown to be equivalent to the continuous-time random walk (CTRW) model of Scher and Montroll, and the hopping-time distribution function is computed for the CTRW model in terms of the trap parameters.     

Asymptotic properties of multistate random walks. II. Applications to inhomogeneous periodic and random lattices

The previously developed formalism for the calculation of asymptotic properties of multistate random walks is used to study random walks on several inhomogeneous periodic lattices, where the periodically repeated unit cell contains a number of inequivalent sites, as well as on lattices with a random distribution of inequivalent sites. We concentrate on the question whether the random walk properties depend on the spatial arrangement of the sites in the unit cell, or only on the number density of the different types of sites. Specifically we consider lattices with periodic and random arrangements of columns and lattices with periodic and random arrangements of anisotropic scatterers.     

Biased random walk with restart for essential proteins prediction

Predicting essential proteins is crucial for discovering the process of cellular organization and viability. We propose biased random walk with restart algorithm for essential proteins prediction, called BRWR. Firstly, the common process of practice walk often sets the probability of particles transferring to adjacent nodes to be equal, neglecting the influence of the similarity structure on the transition probability. To address this problem, we redefine a novel transition probability matrix by integrating the gene express similarity and subcellular location similarity. The particles can obtain biased transferring probabilities to perform random walk so as to further exploit biological properties embedded in the network structure. Secondly, we use gene ontology (GO) terms score and subcellular score to calculate the initial probability vector of the random walk with restart. Finally, when the biased random walk with restart process reaches steady state, the protein importance score is obtained. In order to demonstrate superiority of BRWR, we conduct experiments on the YHQ, BioGRID, Krogan and Gavin PPI networks. The results show that the method BRWR is superior to other state-of-the-art methods in essential proteins recognition performance. Especially, compared with the contrast methods, the improvements of BRWR in terms of the ACC results range in 1.4%-5.7%, 1.3%-11.9%, 2.4%-8.8%, and 0.8%-14.2%, respectively. Therefore, BRWR is effective and reasonable.     

A Note on Transience Versus Recurrence for a Branching Random Walk in Random Environment

We consider a branching random walk in random environment on ^d where particles perform independent simple random walks and branch, according to a given offspring distribution, at a random subset of sites whose density tends to zero at infinity. Given that initially one particle starts at the origin, we identify the critical rate of decay of the density of the branching sites separating transience from recurrence, i.e., the progeny hits the origin with probability <1 resp. =1. We show that for d3 there is a dichotomy in the critical rate of decay, depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk. The dichotomy marks a transition from local to global behavior in the progeny that hits the origin. We also consider the situation where the branching sites occur in two or more types, with different offspring distributions, and show that the classification is more subtle due to a possible interplay between the types. This note is part of a series of papers by the second author and various co-authors investigating the problem of transience versus recurrence for random motions in random media.     

Fractional Feynman-Kac Equation with Space-Dependent Anomalous Exponent

To describe the distribution of functionals for inhomogeneous subdiffusion in space- and time-dependent force field, we derive forward and backward time-fractional Feynman-Kac equations with space-dependent anomalous exponent based on the space-jump random walk model. In our examples, we get the statistic of occupation fraction and first passage time for anomalous infiltration of free particles in disordered systems by applying the backward version.     

Migration and proliferation dichotomy in tumor-cell invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW), we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting-time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.     

First passage times for correlated random walks and some generalizations

It is generally difficult to solve Fokker-Planck equations in the presence of absorbing boundaries when both spatial and momentum coordinates appear in the boundary conditions. In this note we analyze a simple, exactly solvable model of the correlated random walk and its continuum analogue. It is shown that one can solve for the moments recursively in one dimension in exact analogy with first passage problems for the Fokker-Planck equation, although the boundary conditions are somewhat more complicated. Further generalizations are suggested to multistate random walks.