Drift in a random walk along self-similar clusters

This paper is a study of the relationship between diffusion and conductivity when the random walks of particles occur via Lévy hops. It shows that because of the unusual nature of Lévy hops the particle mobility is a nonlinear function of the electric field in arbitrarily weak fields. The crossover to ordinary diffusion by introduction of a finite displacement in each step is also discussed. Zh. éksp. Teor. Fiz. 115, 1016–1023 (March 1999)     

On the relation between a fluctuating diffusion equation and long time tails in stationary random media

Ernst, Machta, Dorfman, and van Beijeren [J. Stat. Phys. 34:477 (1984);35:413 (1984)] have proposed that diffusion in a stationary random medium is described by a fluctuating diffusion equation involving a coarse-grained local diffusion coefficient K(r) and free volume fraction(r). We show that for a particular class of models [lattice diffusion with random transition rates and constant(r)], their prediction for the long time tail in the velocity autocorrelation function is the correct asymptotic limit.     

Subordinated diffusion and continuous time random walk asymptotics

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.     

The diffusion of self-avoiding random walk in high dimensions

We use the Brydges-Spencer lace expansion to prove that the mean square displacement of aT step strictly self-avoiding random walk in thed dimensional square lattice is asymptotically of the formDT asT approaches infinity, ifd is sufficiently large. The diffusion constantD is greater than one.     

Asymptotic solutions of the continuous-time random walk model of diffusion

Asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times. It is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these asymptotic distributions are of stable form. This indicates that the realm of applicability of the diffusion equation is limited. The Montroll-Weiss equation is rederived to include the influence of the initial waiting interval and the role of the stable distributions in physical problems is briefly discussed.     

On the asymmetry of a random walk in the presence of a field

Any ensemble of random walks with symmetric transition probabilities will have symmetric properties. However, any single realization of such a random walk may be asymmetric. In an earlier paper, Weiss and Weissman developed a measure of asymmetry and applied it to random walks in the absence of a field, showing that the degree of asymmetry (in the diffusion limit) is independent of time and that the most probable degree of asymmetry corresponds to the maximum possible. We show in the present paper how the presence of a symmetric field can change this result, both in making the degree of asymmetry depend on time, and driving the random walk toward a more symmetric state.     

On a coupled random walk process specified by a Hamiltonian

A coupled random walk process specified by an effective Hamiltonian in a potential field is proposed. The Hamiltonian is expressed in terms of a set of jumping probabilities which characterize the random walk processes. The steady state is expressed by the Hamiltonian. Conditions for the Hamiltonian to be reduced to the Ginzburg-Landau type are discussed.     

On the critical exponent for random walk intersections

The exponent _d for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2<₂3/4. Monte Carlo simulations are done to suggest ₂=0.61 and ₃0.29.     

Weak convergence of a random walk in a random environment

Let _i (x),i=1,...,d,xZ ^d , satisfy _i (x)>0, and ₁(x)+...+ _d (x)=1. Define a Markov chain onZ ^d by specifying that a particle atx takes a jump of +1 in thei ^th direction with probability 1/2 _i (x) and a jump of –1 in thei ^th direction with probability 1/2 _i (x). If the _i (x) are chosen from a stationary, ergodic distribution, then for almost all the corresponding chain converges weakly to a Brownian motion.     

Fractional diffusion in the multiple-trapping regime and revision of the equivalence with the continuous-time random walk

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.     

Generalised master equation of random walk in a random medium

A generalised formulation of continuous-time random-walks is introduced to study excitation transport in disordered systems containing permanent traps (the localised states). Its exact equivalence with the generalised master equation is established. The exact generalised transport equation obtained has been shown to reduce under special conditions to other random walk equations known in the literature.     

Random walk in a random multiplicative environment

We investigate the dynamics of a random walk in a random multiplicative medium. This results in a random, but correlated, multiplicative process for the spatial distribution of random walkers. We show how the details of these correlations determine the asymptotic properties of the walk, i.e., the central limit theorem does not apply to these multiplicative processes. We also study a periodic source-trap medium in which a unit cell contains one source, followed byL–1 traps. We calculate the asymptotic behavior of the number of particles, and determine the conditions for which there is growth or decay in this average number. Finally, we discuss the asymptotic behavior of a random walk in the presence of randomly distributed, partially-absoprbing traps. For this case, a temporal regime of purely exponential decay of the density can occur, before the asymptotic stretched exponential decay, exp(–at ^1/3), sets in.     

Einstein's relation between diffusion constant and mobility for a diffusion model

An Ornstein-Uhlenbeck process in a periodic potential inR ^d is considered. It has been shown previously that this process satisfies a central limit theorem in the sense that, by rescaling space and time in a suitable way, the distribution of the process converges to that of a Wiener process with nonsingular diffusion matrix. Here a rigorous proof is given of a version of Einstein's formula for this model, relating the diffusion constant to the mobility of the system.     

On the proximity relation between two surface-melted clusters involved in inter-cluster mass transfer

We explore the way free particles produced by dissociating “particle-hole pairs” on a surface-melted cluster can be transferred to a second, nearby surface-melted cluster. This mass transport is based on an inter-cluster direct transfer mechanism of the particles. We found that in this particular case one cluster may grow at the expense of another, obeying a temporal power law with the exponent 1/2 for the average radius (R∼t ^1/2). The change from the expected universal power law (R∼t ^1/3) is a consequence of the proximity relation between these two clusters which lead to enhance the effective transport rates. Received 4 December 2000     

On the survival probability of a random walk in a finite lattice with a single trap

We consider the survival of a random walker in a finite lattice with periodic boundary conditions. The initial position of the random walker is uniformly distributed on the lattice with respect to the trap. We show that the survival of a random walker, U _n>, can be exactly related to the expected number of distinct sites visted on a trap-free lattice by U _n=1–S _n/N ^D (*) whereN ^D is the number of lattice points inD dimensions. We then analyze the behavior of S_n in any number of dimensions by using Tauberian methods. We find that at sufficiently long times S _n decays exponentially withn in all numbers of dimensions. InD = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forN ²N 1 whenD = 1 andN lnN n 1 whenD = 2. No such crossover exists when Z3. The form of (*) suggests that the single trap approximation is indeed a valid low-concentration limit for survival on an infinite lattice with a finite concentration of traps.     

Application of random walk concept to the cyclic diffusion mechanisms for self-diffusion in intermetallic compounds

Huntington–Elcock–McCombie (HEM) mechanism involving six consecutive and correlated jumps, a triple-defect mechanism (TDM) involving three correlated jumps and an anti-structure bridge (ASB) mechanism invoking the migration of an anti-structure atom are the three mechanisms currently in vogue to explain the self- and solute diffusion in intermetallic compounds. Among them, HEM and TDM are cyclic in nature. The HEM and TDM constitute the theme of the present article. The concept of random walk is applied to them and appropriate expressions for the diffusion coefficient are derived. These equations are then employed to estimate activation energies for self-diffusion via HEM and TDM processes and compared with the available experimental data on activation energy for self-diffusion in intermetallic compounds. The resulting activation energies do not favour HEM and TDM for the self-diffusion in intermetallic compounds. A comparison of the sum of experimentally determined activation energies for vacancy formation and migration with the activation energies for self-diffusion determined from radioactive tracer method favours the conventional monovacancy-mediated process for self-diffusion in intermetallic compounds.