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.     

Long-time asymptotics in the one-dimensional trapping problem with large bias

The survival probability of a particle which moves according to a biased random walk in a one-dimensional lattice containing randomly distributed deep traps is studied at large times. Exact asymptotic expansions are deduced for fields exceeding a certain threshold, using the method of images. In order to cover the whole range of fields, we also derive the behavior of the survival probability below this threshold, using the eigenvalue expansion method. The connection with the continuous diffusion model is discussed.     

Survival probability in one dimension for theA+B→B reaction with hard-core repulsion

We study the effect of hard-core repulsion (known as the bus effect) betweenB particles on the reaction-diffusion systemA+BB in the continuous-time random walk model in one dimension with theA particles stationary. We show rigorously that the survival probability of theA particles is asymptotically bounded asC ₁lim _t{[–logS(t)]/t ^0.5}C ₂, whereC ₁ andC ₂ are constants. We also do simulations to confirm our results.     

Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs （complete cycle, complete Kn, charter and n-cube）. Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.     

Loop-erased self-avoiding random walk in two and three dimensions

If(n) is the position of the self-avoiding random walk in ^d obtained by erasing loops from simple random walk, then it is proved that the mean square displacementE(¦(n)¦²) grows at least as fast as the Flory predictions for the usual SAW, i.e., at least as fast asn ^3/2 ford=2 andn ^6/5 ford=3. In particular, if the mean square displacement of the usual SAW grows liken ^1.18... ind=3, as expected, then the loop-erased process is in a different universality class.     

混合物介质中声波尾波成因的随机过程分析*

该文基于声波在混合物介质中传播时反射及散射的随机特性,把混合物介质抽象为三维各向同性的马尔科夫链,把声波在混合物介质中传播过程抽象为声波在三维马尔科夫链中以声速进行“随机游走”的随机过程。用空间内某点接收到声波的概率类比该点接收波振幅,以声波到达该点所走过的步数类比接收波时域曲线的时间。此理论模型可较好解释声波在混合物介质中传播时“峰波延后”及“尾波”等现象。     

Solution of the persistent, biased random walk

The solution of the one-dimensional persistent, biased random walk is found. Its finite differences equation is derived and shown to be satisfied by the said solution.     

On the non-equivalence of two standard random walks

We focus on two models of nearest-neighbour random walks on d$d$-dimensional regular hyper-cubic lattices that are usually assumed to be identical—the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll–Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1$1$. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour.     

Sub-diffusive scaling with power-law trapping times

Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive scaling at long times.The statistical mechanical treatment of diffusion in a disordered environment is often quite involved. In this short review,we present a simple and unified view of the many quantitative results on anomalous diffusion in the literature, including the scaling of the diffusion front and the mean first-passage time. Various analytic calculations and physical arguments are examined to highlight the role of dimensionality, energy landscape, and rare events in affecting the particle trajectory statistics. The general understanding that emerges will aid the interpretation of relevant experimental and simulation results.     

Survival probabilities for random walks on lattices with randomly distributed traps

I present here a numerical procedure to compute survival probabilities for random walks on lattices with randomly distributed traps. The procedure has some advantages over existing methods, and its performance is evaluated for the 1D simple random walk, for which some exact results are known. Thereafter, I apply the procedure to 1D random walks with variable step length and to 3D simple random walks.     

Velocity and Dispersion for a Two-Dimensional Random Walk

In the paper, we consider the transport of a two-dimensional random walk. The velocity and the dispersion of this two-dimensional random walk are derived. It mainly show that： （i） by controlling the values of the transition rates, the direction of the random walk can be reversed; （ii） for some suitably selected transition rates, our two-dimensional random walk can be efficient in comparison with the one-dimensional random walk. Our work is motivated in part by the challenge to explain the unidirectional transport of motor proteins. When the motor proteins move at the turn points of their tracks （i.e., the cytoskeleton filaments and the DNA molecular tubes）, some of our results in this paper can be used to deal with the problem.     

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.     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Sharp Phase Boundaries for a Lattice Flux Line Model

We consider a model of nonintersecting flux lines in a rectangular region on the lattice ^d , where each flux line is a non-isotropic self-avoiding random walk constrained to begin and end on the boundary of the region. The thermodynamic limit is reached through an increasing sequence of such regions. We prove the existence of several distinct phases for this model, corresponding to different regimes for the flux line density—a phase with zero density, a collection of phases with maximal density, and at least one intermediate phase. The locations of the boundaries of these phases are determined exactly for a wide range of parameters. Our results interpolate continuously between previous results on oriented and standard nonoriented self-avoiding random walks.