Two-state random walk model of diffusion. 2. Oscillatory diffusion

Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast, we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk, we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement, dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem (the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive regimes) is demonstrated.     

A myopic random walk on a finite chain

We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,N) in discrete time, with ‘myopic boundary conditions’—a walker at 1 (orN) at timen moves to 2 (orN − 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of the walker oscillate about certain mean values asn → ∞, with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out.     

First passage time and escape time distributions for continuous time random walks

We consider an arbitrary continuous time random walk (ctrw)via unbiased nearest-neighbour jumps on a linear lattice. Solutions are presented for the distributions of the first passage time and the time of escape from a bounded region. A simple relation between the conditional probability function and the first passage time distribution is analysed. So is the structure of the relation between the characteristic functions of the first passage time and escape time distributions. The mean first passage time is shown to diverge for all (unbiased)ctrw’s. The divergence of the mean escape time is related to that of the mean time between jumps. A class ofctrw’s displaying a self-similar clustering behaviour in time is considered. The exponent characterising the divergence of the mean escape time is shown to be (1−H), whereH(0<H<1) is the fractal dimensionality of thectrw.     

基于连续时间无规行走模型研究反常扩散

基于连续时间无规行走(CTRW)理论，数值研究了布朗粒子的欠扩散、正常扩散和超扩散三种扩散行为.解决了CTRW模型的跳跃步长和等待时间分布函数的可实现化问题，对Metropolis抽样方法进行了改进以适用于周期势.探讨了布朗马达依靠闪烁棘轮和摇摆棘轮整流反常扩散所获得的定向速度，结果显示，闪烁布朗马达定向流极大值出现在超扩散条件下；摇摆布朗马达定向流最大值出现在弹道扩散条件下.     

基于连续时间无规行走模型研究反常扩散

基于连续时间无规行走(CTRW)理论,数值研究了布朗粒子的欠扩散、正常扩散和超扩散三种扩散行为.解决了CTRW模型的跳跃步长和等待时间分布函数的可实现化问题,对Metropolis抽样方法进行了改进以适用于周期势.探讨了布朗马达依靠闪烁棘轮和摇摆棘轮整流反常扩散所获得的定向速度,结果显示,闪烁布朗马达定向流极大值出现在超扩散条件下;摇摆布朗马达定向流最大值出现在弹道扩散条件下. 关键词： 无规行走 反常扩散 Metropolis抽样 棘轮势     

运用CTRW-Metropolis模型数值研究亚稳势中粒子逃逸问题

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γ_st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.     

On the Wang–Uhlenbeck Problem in Discrete Velocity Space

The arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values, ±v with rate constant k. For this model, which is the continuous-space version of a persistent random walk, analytic expressions are found for the first passage time distributions to the origin. Since the evolution equation of this model can be regarded as the two-state finite-difference approximation in velocity space of the Kramers–Klein equation, this work constitutes a solution of the simplest version of the Wang–Uhlenbeck problem. Formal solution (in Laplace space) of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the Maxwell distribution is also provided.     

On an interpolation model for the transition operator for Markov and non-Markov processes

A phenomenological interpolation model for the transition operator of a stationary Markov process is shown to be equivalent to the simplest difference approximation in the master equation for the conditional density. Comparison with the formal solution of the Fokker-Planck equation yields a criterion for the choice of the correlation time in the approximate solution. The interpolation model is shown to be form-invariant under an iteration-cum-rescaling scheme. Next, we go beyond Markov processes to find the effective time-development operator (the counterpart of the conditional density) in the following very general situation: the stochastic interruption of the systematic evolution of a variable by an arbitrary stationary sequence of randomizing pulses. Continuous-time random walk theory with a distinct first-waiting-time distribution is used, along with the interpolation model for the transition operator, to obtain the solution. Convenient closed-form expressions for the ‘averaged’ time-development operator and the autocorrelation function are presented in various special cases. These include (i) no systematic evolution, but correlated pulses; (ii) systematic evolution interrupted by an uncorrelated (Poisson) sequence of pulses.     

Random walk representations of the Heisenberg model

We develop random walk representations for the spin-S Heisenberg ferromagnet with nearest neighbor interactions. We show that the spin-S Heisenberg model is a diffusion with local times controlled by the spin-S Ising model. As a consequence, expectations for the Heisenberg model conditioned on zero diffusion are shown to be Ising expectations.     

Space–time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model

In this paper, we consider a type of continuous time random walk model where the jump length is correlated with the waiting time. The asymptotic behaviors of the coupled jump probability density function in the Fourier–Laplace domain are discussed. The corresponding fractional diffusion equations are derived from the given asymptotic behaviors. Corresponding to the asymptotic behaviors of the joint probability density function in the Fourier–Laplace space, the asymptotic behaviors of the waiting time probability density and the conditional probability density for jump length are also discussed.     

Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Reflection principles for biased random walks and application to escape time distributions

We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such frozen processes.     

Asymptotic properties of multistate random walks. I. Theory

A calculation is presented of the long-time behavior of various random walk properties (moments, probability of return to the origin, expected number of distinct sites visited) formultistate random walks on periodic lattices. In particular, we consider inhomogeneous periodic lattices, consisting of a periodically repeated unit cell which contains a finite number of internal states (sites). The results are identical to those for perfect lattices except for a renormalization of coefficients. For walks without drift, it is found that all the asymptotic random walk properties are determined by the diffusion coefficients for the multistate random walk. The diffusion coefficients can be obtained by a simple matrix algorithm presented here. Both discrete and continuous time random walks are considered. The results are not restricted to nearest-neighbor random walks but apply as long as the single-step probability distributions associated with each of the internal states have finite means and variances.     

A computer simulation study of ionic conductivity in polymer electrolytes

In this paper we present a computer simulation study of ionic conductivity in solid polymeric electrolytes. The multiphase nature of the material is taken into account. The polymer is represented by a regular lattice whose sites represent either crystalline or amorphous regions with the charge carrier performing a random walk. Different waiting times are assigned to sites corresponding to the different phases. A random walk (RW) is used to calculate the conductivity through the Nernst-Einstein relation. Our walk algorithm takes into account the reorganization of the different phases over time scales comparable to time scales for the conduction process. This is a characteristic feature of the polymer network. The qualitative nature of the variation of conductivity with salt concentration agrees with the experimental values for PEO-NH₄I and PEO-NH₄SCN. The average jump distance estimated from our work is consistent with the reported bond lengths for such polymers.     

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 Walks and Polymers in the Presence of Quenched Disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both ‘random walk models’, where the random walk is a dynamical process generated by local transition rules, and on ‘polymer models’, where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points: thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures T _c(i,L) over the ensemble of samples (i) of size L. We describe the results of this analysis for the bidimensional wetting and for the Poland–Scheraga model of DNA denaturation.Conference Proceedings “Mathematics and Physics”, I.H.E.S., France, November 2005     

Analytical representations of non-Gaussian laws of random walks

An analytical representation of a random process with independent increments in some space (random walks introduced by Pearson) is considered. The law of random walk distribution in space is derived from the general representation of stochastic elementary hops (distribution law of hop probability) using Kadanoff’s concept of the unit increment as one hop. For limited hop laws and laws of hop distributions with all moments there naturally arises Chandrasekhar’s result that describes ordinary physical diffusion. For laws of hop distributions without the second and highest moments there also arise known Lévy walks (flights) sometimes treated as superdiffusion. For the intermediate case, where the distributions of hops have at least the second moment and not all finite moments (these hops are sometimes called truncated Lévy walks), the asymptotic form of the random walk distribution was obtained for the first time. The results obtained are compared with the experimental laws known in econophysics. Satisfactory agreement is observed between the developed theory and the empirical data for insufficiently studied truncated Lévy walks.     

Directional quantum random walk induced by coherence

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker's position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker's position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker's position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker's position distribution.     

Laplace operator and random walk on one-dimensional nonhomogeneous lattice

A classical result of probability theory states that under suitable space and time renormalization, a random walk converges to Brownian motion. We prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice. Under suitable conditions on the nonhomogeneous medium, we prove convergence to Brownian motion and explicitly compute the diffusion coefficient. The proofs are based on the study of the spectrum of random matrices of increasing dimension.