Anomalous diffusion in disordered lattices: Effect of bias

Abstract

We model a system of disorder by utilizing a lattice on which the sites are characterized by energies, which are highly disordered, as they are chosen from a given distribution of energy values (say, from a uniform distribution in a given range), and thus every single site is characterized by a different energy value. Diffusion of single isolated particles is thermally activated, and follows Boltzmann statistics. The behavior of the mean-square displacement is followed as a function of time. It has been observed in the past that in such a system there is a strong anomaly at early times, in that the system is strongly sub-linear, but crosses over to the classical expected linear behavior at long times.

In this system we now consider an additional effect, that of directional bias in the motion. A higher probability is assigned in a prescribed direction, which naturally leads to ballistic motion. An interesting effect is observed as the temperature is raised to very high values: motion crosses over to normal diffusion with increasing temperature, even in the presence of the bias. This effect is in addition to the previously observed sub-diffusional 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 of considerable interest, and it is examined here in detail. The pertinent scaling laws are given for the crossover times.

Furthermore, we consider the case of the dependence of the bias on some frequency, which makes it to alternate (switch) the direction of motion with the given frequency parameter, resulting in a rachet-type of picture, and in a different type of scaling.