Exact results and self-averaging properties for random-random walks on a one-dimensional infinite lattice (original) (raw)

Velocity and diffusion coefficient of a random asymmetric one-dimensional hopping model

Journal de Physique, 1989

2014 La vitesse et le coefficient de diffusion d'une particule sur un réseau périodique unidimensionnel de période N avec des taux de transfert aléatoires et asymétriques sont calculés de manière simple grâce à une méthode basée sur une relation de récurrence, qui permet d'établir une analogie aux grands temps avec un modèle de marche strictement dirigée. Les résultats pour un système complètement aléatoire sont obtenus en prenant la limite N ~ ~. On montre qu'un calcul, reposant sur une hypothèse d'échelle dynamique, de la vitesse et du coefficient de diffusion dans un réseau désordonné infini conduit aux mêmes résultats.

Random walk on a disordered directed Bethe lattice

Journal of Statistical Physics, 1991

The random walk of a particle on a directed Bethe lattice of constant coordinance Z is examined in the case of random hopping rates. As a result, the higher the coordinance, the narrower the regions of anomalous drift and diffusion. The annealed and quenched mean square dispersions are calculated in all dynamical phases. In opposition to the one-dimensional (Z=2) case, the annealed and quenched mean quadratic dispersions are shown to be identical in all phases.

Two-dimensional random-random walks: Dynamical exponents in a quenched directed model

Journal of Statistical Physics, 1991

This paper presents a study of the dynamics of a particle undergoing a directed random walk in a two-dimensional disordered square lattice. We derive the asymptotical behaviors of the coordinate and of the mean square displacement. All the dynamical exponents are calculated both in the normal and the anomalous regimes. It is shown that, as contrasted to the one-dimensional case, the so-called quenched and annealed diffusion "constants" indeed coincide. KEY WORDS: Fluctuation phenomena; random processes and Brownian motion. 1Groupe de Physique des Solides, Laboratoire associ6 au CNRS (UA no. 17) et aux Universit6s Paris VII et Paris VI,

Lyapunov exponent of random walkers on a bond-disordered lattice

Physica A: Statistical Mechanics and its Applications, 1997

The chaotic properties of a random walker in a quenched random environment are studied analytically, following the work of Gaspard et al. on Lorentz gases, for systems with closed (periodic) or open (absorbing) boundaries. The model of interest describes random walkers hopping on a disordered lattice, on which the hopping probabilities across bonds are quenched random variables.

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

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

Hopping transport on site-disordered d -dimensional lattices

Physical Review A, 1987

We consider here the hopping of an electron among a band of localized electronic states on a ddimensional lattice. The hopping rates are assumed to be stochastic variables determined by some probability distribution. We restrict our attention to nearest-neighbor transport in the limit in which the fluctuations in the hopping rates are large. In this limit we construct an exact expansion for the frequency-dependent diffusion coefficient D(c) that is applicable to a wide range of transport phenomena (d =1 conductors, trapping phenomena, molecularly based electronic devices, etc.) in any spatial dimension. For the case of hopping transport with d =1, our method confirms earlier results that strong Auctuations in the hopping rates give rise to a non-Markovian c'-correction to normal diffusion. In two dimensions, we establish explicitly the existence of a non-Markovian logarithmic correction c, inc to D(c,). The formalism is then extended to d dimensions and the frequency corrections are discussed. For d =3, two frequency corrections must be retained. One is linear in c and the other proportional to c,. It is shown that only the c correction contributes to the longtime tail t ' in the time-dependent diffusion coefficient D(t). From these results we show that the long-time tail in the velocity autocorrelation function which is a consequence of the strong fluctuations in the hopping rates is of the form t "+" '. Comparison is made with earlier results.

Non-Markoffian diffusion in a one-dimensional disordered lattice

Journal of Statistical Physics, 1982

Recent treatments of diffusion in a one-dimensional disordered lattice by Machta using a renormalization-group approach, and by Alexander and Orbach using an effective medium approach, lead to a frequency-dependent (or non-Markoffian) diffusion coefficient. Their resUlts are confirmed by a direct calculation of the diffusion coefficient.