Field-induced transport in random media

Abstract

We review the problem of particle transport in random media in the presence of an external field. The random medium is modeled by the infinite cluster above the percolation threshold. The field imposes a preferred direction of motion along which diffusing particles (random walkers are more likely to move than against. Two kinds of traps occur - branches pointing in the direction of the field, and backbends, in which particles must move against the field. For noninteracting particles, the drift velocity is a nonmonotonic function of the biasing field, and the two kinds of traps make the current vanish above a threshold value of the bias. If there is hard-core repulsion between the particles, branches get filled up and eventually cease to be traps. Below the directed percolation threshold, transport is rate-limited by backbends, and the particle current flows predominantly along those paths on the percolation backbone on which the length of every backbend is bounded. The current is a nonmonotonic function of the biasing field. We also consider a different sort of interparticle interaction which leads to levels of particles equalising near backbend bottoms. The motion along a typical path is then described by 'drop-push' dynamics: between backbends, particles drop down, assisted by the field, and push those on the next backbend, possibly leading to a cascade of overflows. Drop-push dynamics has interesting connections with other lattice gas automata, and Monte Carlo simulations show that the model supports kinematic waves and exhibits interesting behaviour of time-dependent correlations.