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《Physica A》2007
Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process. 相似文献
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Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Le?vy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit. 相似文献
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用连续时间无规行走(CTRW)理论处理陷阱控制的无序点阵上的无规行走问题,首次导出行走者可有自发衰变及受陷态具有有限寿命情形下,行走者存活几率P(t)满足的方程。对一种广泛使用的等待时间分布密度ψ(t)=ααt-(1-α)exp(-αtα)0<α≤1,在受陷态寿命无限长情况下,给出适用于任意陷阱浓度和任意时间的P(t)的级数解。结合实验事实和Ngai的低能激发理论,指出同时考虑动力学关联和结构无序对解释实际过程的必要性。并提出包括可由Ngai低能激发理论描写的动力学关联在内的连续时间无规行走理论,其物理图象与目前的CTRW理论有根本不同。
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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. 相似文献
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We study financial distributions within the framework of the continuous time random walk (CTRW). We review earlier approaches and present new results related to overnight effects as well as the generalization of the formalism which embodies a non-Markovian formulation of the CTRW aimed to account for correlated increments of the return. 相似文献
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We determine the probability distribution of the first passage time for a class of non-Markovian processes. This class contains, amongst others, the well-known continuous time random walk (CTRW), which is able to account for many properties of anomalous diffusion processes. In particular, we obtain the mean first passage time for CTRW processes with truncated power-law transition time distribution. Our treatment is based on the fact that the solutions of the non-Markovian master equation can be obtained via an integral transform from a Markovian Langevin process. 相似文献
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Random walk particle tracking simulations of non-Fickian transport in heterogeneous media 总被引:1,自引:0,他引:1
G. Srinivasan D.M. Tartakovsky M. Dentz H. Viswanathan B. Berkowitz B.A. Robinson 《Journal of computational physics》2010,229(11):4304-4314
Derivations of continuum nonlocal models of non-Fickian (anomalous) transport require assumptions that might limit their applicability. We present a particle-based algorithm, which obviates the need for many of these assumptions by allowing stochastic processes that represent spatial and temporal random increments to be correlated in space and time, be stationary or non-stationary, and to have arbitrary distributions. The approach treats a particle trajectory as a subordinated stochastic process that is described by a set of Langevin equations, which represent a continuous time random walk (CTRW). Convolution-based particle tracking (CBPT) is used to increase the computational efficiency and accuracy of these particle-based simulations. The combined CTRW–CBPT approach enables one to convert any particle tracking legacy code into a simulator capable of handling non-Fickian transport. 相似文献
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本文采用近年来在解释凝聚物质中的低频涨落、耗散和弛豫现象时普遍适用的一个新的非纯指数形式的等待时间分布函数(以下称WTDF)ψ(t),讨论了连续时间无规行走(以下称CTRW)问题的渐近解。得到了许多有意义的和与实验一致的结果。它们是:平均位移、色散迁移率、平均平方位移、色散扩散系数、Nernst-Einstein关系、方差与标准方差、点阵统计学、初始位置占有几率、色散电导率、色散电输运和记忆函数。所有结果表明:由这个WTDF ψ(t)所描写的CTRW过程在短时区内表现为非Markov的,在长时区内则表现为Markov的,即所有结果都含有一个与介质的微观结构有关的且决定着色散程度的单参数——红外发散指数n(0≤n<1)。n越大,色散越大。当n=0时,色散消失,所有结果立即退化成Markov即经典形式,这与已有的实验事实一致。
关键词: 相似文献
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Barkai E Metzler R Klafter J 《Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics》2000,61(1):132-138
We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE. 相似文献
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基于连续时间随机行走(CTRW)理论,实现反常扩散条件下对跳跃步长和等待时间分布函数的抽样,改进Metropolis抽样判定方法以适用于存在非线性势的情况.数值研究布朗粒子在亚稳势下的逃逸速率.结果显示,稳定逃逸速率γst随反常指数α非单调变化,在超扩散条件下存在极大值和位垒相消现象. 相似文献
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We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks. 相似文献
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Helena Stage Sergei Fedotov 《The European Physical Journal B - Condensed Matter and Complex Systems》2017,90(11):225
A standard assumption of continuous time random walk (CTRW) processes is that there are no interactions between the random walkers, such that we obtain the celebrated linear fractional equation either for the probability density function of the walker at a certain position and time, or the mean number of walkers. The question arises how one can extend this equation to the non-linear case, where the random walkers interact. The aim of this work is to take into account this interaction under a mean-field approximation where the statistical properties of the random walker depend on the mean number of walkers. The implementation of these non-linear effects within the CTRW integral equations or fractional equations poses difficulties, leading to the alternative methodology we present in this work. We are concerned with non-linear effects which may either inhibit anomalous effects or induce them where they otherwise would not arise. Inhibition of these effects corresponds to a decrease in the waiting times of the random walkers, be this due to overcrowding, competition between walkers or an inherent carrying capacity of the system. Conversely, induced anomalous effects present longer waiting times and are consistent with symbiotic, collaborative or social walkers, or indirect pinpointing of favourable regions by their attractiveness. 相似文献
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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. 相似文献
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Bisquert J 《Physical review letters》2003,91(1):010602
We investigate the macroscopic diffusion of carriers in the multiple-trapping (MT) regime, in relation with electron transport in nanoscaled heterogeneous systems, and we describe the differences, as well as the similarities, between MT and the continuous-time random walk (CTRW). Diffusion of free carriers in MT can be expressed as a generalized continuity equation based on fractional time derivatives, while the CTRW model for diffusive transport generalizes the constitutive equation for the carrier flux. 相似文献
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J. Noolandi 《Solid State Communications》1977,24(7):477-480
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. 相似文献