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.

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. ͓S1063-651X͑97͒51207-4͔PACS number͑s͒: 05.40.ϩj, 05.60.ϩw Biased random walk is a prototype model for studies of particle diffusion in disorded solids ͓1,2͔ where kinetic problems are concerned, such as conduction, viscous flow, polymer dynamics, etc. The characteristic of the bias implies a preferential direction for the motion, as opposed to purely stochastic motion. The problem becomes much more complicated when the underlying space is not a simple lattice, but contains a certain degree of randomness itself, such as, for example, a rugged energy landscape model that we recently introduced ͓3-5͔. Every lattice site has its own energy ͑usually chosen randomly, from a random number distribution͒. Thus, the problem is not directly amenable to exact analytical theories, except mean field arguments. In the present study we continue the investigation of such systems, with the inclusion now of the bias characteristics.The model used here is the one used in our previous studies ͓3-5͔, in which we now incorporate an external field ͑bias͒. Briefly, a square lattice is generated. Each bond between any two sites is a barrier with a height that is randomly chosen from a given distribution. The height of the barriers depends on the mean value ͗E͘ and on the dispersion parameter of the E distribution in the following way:

We study the hull of the territory visited by N random walkers after t time steps. The walkers move on two-dimensional substrates, starting all from the same position. For the substrate, we consider (a). a square lattice and (b). a percolation cluster at criticality. On the square lattice, we (c). also allow for birth and death processes, where at every time step, alphaN walkers die and are removed from the substrate, and simultaneously the same number of walkers is added randomly at the positions of the remaining walkers, such that the total numbers of walkers is constant in time. We perform numerical simulations for the three processes and find that for all of them, the structure of the hull is self-similar and described by a fractal dimension d(H) that slowly approaches, with an increasing number of time steps, the value d(H)=4/3. For process (c), however, the time to approach the asymptotic value increases drastically with increasing fraction of N/alpha, and can be observed numerically only for sufficiently small values of N/alpha.

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Diffraction experiments can be used easily to measure the time evolution of a system under nonequilibrium conditions to attain a new equilibrium state and deduce ''nonequilibrium'' surface diffusion coefficients. It is not clear how the ''nonequilibrium'' diffusion coefficients extracted from such diffraction experiments should be interpreted. We study with Monte Carlo simulations the behavior of the ''nonequilibrium'' tracer and collective diffusion coefficients in a lattice-gas model with attractive nearest-neighbor interactions, as the system evolves in time from an initial random state to attain the (1ϫ1) ordered state for temperatures T/T c Ͻ1. We calculate the dependence of the mean-square displacement ͗R 2 ͘ on time and the collective diffusion from the relaxation of the nonequilibrium structure factor S(q ជ ,t) within time sub-intervals under the assumption that quasiequilibrium holds. We determine the time-dependent ''nonequilibrium'' diffusion coefficients and extract the time-dependent activation energies. For both diffusion coefficients the ''nonequilibrium'' values obtained at late times are compared to the corresponding values obtained at equilibrium.