Random walk with a hop-over site: a novel approach to tagged diffusion and its applications (original) (raw)

One-dimensional random walks on a lattice with energetic disorder

Physical review. B, Condensed matter, 1994

results are obtained for random walks of excitation on a one-dimensional lattice with a Gaussian energy distribution of site energies. The distribution 4(t) of waiting times is studied for different degrees of energetic disorder. It is shown that at T=0, %'(t) is described by a biexponential dependence and at T@0 the distribution %(t) broadens due to the power-law "tail" t '~t hat corresponds to the description of %(t) in the framework of the continuous-time random walk model. The parameter y depends linearly on T for strong (T~O) and moderate disorder. For the case of T=O the number of new sites S(t) visited by a walker is calculated at t~~. The results are in accordance with Monte Carlo data. The survival probability 4(t) for strong disorder in the long-time limit is characterized by the power-law dependence 4(t)-t s with p=cy, where c is the trap concentration and for moderate disorder the decay 4(t) is faster than t

Effect of a forbidden site on a d -dimensional lattice random walk

Journal of Physics A: Mathematical and General, 2001

We study the effect of a single excluded site on the diffusion of a particle undergoing random walk in a d-dimensional lattice. The determination of the characteristic function allows to find explicitly the asymptotical behaviour of physical quantities such as the particle average position (drift) x (t) and the mean square deviation x 2 (t) − x 2 (t). Contrarily to the one-dimensional case, where x (t) diverges at infinite times ( x (t) ∼ t 1/2 ) and where the diffusion constant D is changed due to the impurity, the effects of the latter are shown to be much less important in higher dimensions: for d ≥ 2, x (t) is simply shifted by a constant and the diffusion constant remains unaltered although dynamical corrections (logarithmic for d = 2) still occur. Finally, the continuum space version of the model is analyzed; it is shown that d = 1, is the lower dimensionality above which all the effects of the forbidden site are irrelevant. *

Biased random walk in energetically disordered lattices

Physical Review E, 1998

We utilize our previously reported model of energetically disordered lattices to study diffusion properties, where we now add the effect of a directional bias in the motion. We show how this leads to ballistic motion at low temperatures, but crosses over to normal diffusion with increasing temperature. This effect is in addition to the previously observed subdiffusional motion at early times, which is also observed here, and also crosses over to normal diffusion at long times. The interplay between these factors of the two crossover points is examined here in detail. The pertinent scaling laws are given for the crossover times. Finally, we deal with the case of the frequency dependent bias, which alternates ͑switches͒ its direction with a given frequency, resulting in a different type of scaling. ͓S1063-651X͑98͒11008-5͔

Markov chain analysis of random walks in disordered media

Physical Review E, 1994

We study the dynamical exponents dwd_{w}dw and dsd_{s}ds for a particle diffusing in a disordered medium (modeled by a percolation cluster), from the regime of extreme disorder (i.e., when the percolation cluster is a fractal at p=pcp=p_{c}p=pc) to the Lorentz gas regime when the cluster has weak disorder at p>pcp>p_{c}p>pc and the leading behavior is standard diffusion. A new technique of relating the velocity autocorrelation function and the return to the starting point probability to the asymptotic spectral properties of the hopping transition probability matrix of the diffusing particle is used, and the latter is numerically analyzed using the Arnoldi-Saad algorithm. We also present evidence for a new scaling relation for the second largest eigenvalue in terms of the size of the cluster, ∣lnlambdamax∣simS−dw/df|\ln{\lambda}_{max}|\sim S^{-d_w/d_f}∣lnlambdamax∣simS−dw/df, which provides a very efficient and accurate method of extracting the spectral dimension dsd_sds where ds=2df/dwd_s=2d_f/d_wds=2df/dw.

Kinetic effects in diffusion on a disordered square lattice

In this work, the effect of fluctuations in a disordered square lattice on diffusion of a test particle is studied using kinetic Monte Carlo simulations. Diffusion is relevant to a wide variety of problems, both within physics and outside of physics. Kinetic effects in diffusion are often hidden in a thermodynamical description of the problem. In this work, no assumptions based on energy are made, and diffusion occurs purely based on the attempt rate of the test particle and the occupation and fluctuation rate of the lattice. Although the average transition rate of the particle is the same for a static or fluctuating lattice with specific occupation, the diffusion constant is kinetically affected in a fluctuating, disordered lattice. If the lattice fluctuates faster than the attempt rate of the particle, diffusion is controlled by the attempt rate of the particle. However, if the lattice fluctuates slower than the attempt rate of the particle, diffusion is affected by the fluctuations. The slower the lattice fluctuates, the lower the diffusion constant. Furthermore, it is found that for fast fluctuating lattices, diffusion is due to Brownian motion. If the lattice fluctuates slower than the particle, diffusion becomes anomalous depending on the occupation of the lattice.

Single random walker on disordered lattices

Journal of Statistical Physics, 1984

Random walks on square lattice percolating clusters were followed for up to 2 • 10 ~ steps. The mean number of distinct sites visited (SN) gives a spectral dimension of d s = 1.30 5:0.03 consistent with superuniversality (d S = 4/3) but closer to the alternative ds= 182/139, based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a "crossover" from fractal-to-Euclidean behavior. Walks on two-and three-dimensional lattices are similar, except that those in three dimensions are more efficient.

Distribution of the number of distinct sites visited by random walks in disordered lattices

Physical Review E, 1995

The distributions of the number of distinct sites S visited by random walks of n steps in the infinite cluster of two-dimensional lattices at the percolation threshold are studied. Different lattice sizes, different origins of the walks, and different realizations of the disorder are investigated by Monte Carlo simulations. The distribution of the mean values of (S") appears to have selfaveraging features. The probability distribution of the normalized values of (S") is investigated with respect to its multifractal behavior. The distributions of the probabilities p($) for fixed S" are presented and analyzed. These distributions are wide and their moments show behavior that cannot be characterized by multifractal scaling exponents.

Effect of temperature on biased random walks in disordered media

Physical Review E, 1997

We study diffusion on an energetically disordered lattice, where each bond between sites is characterized as a random energy barrier. In such a model it had previously been observed that the mean square displacement is sublinear with time at early times, but eventually reaches the classical linear behavior at long times, as a strong function of the temperature. In the current work we add the effect of directional bias in the random walk motion, in which along one axis only, motion in one direction is assigned a higher probability while along the opposite direction a reduced probability. We observe that for low temperatures a ballistic character dominates, as shown by a slope of 2 in the R 2 vs time plot, while at high temperatures the slope reverts to 1, manifesting that the effect of the bias parameter is obliterated. Thus, we show that for a biased random walk diffusion may proceed faster at lower temperatures. The details of how this crossover takes place, and the scaling law of the crossover temperature as a function of the bias are also given. ͓S1063-651X͑97͒51207-4͔