Drift and Trapping of Particles Under Biased Motion on Disordered Lattices

Abstract

This project deals with the properties of systems of biased random walkers on disordered lattices. We use a percolation cluster to model the disordered lattice and study how the motion of the walkers is affected by the disorder in the lattice. I developed an algorithm to find the backbone and branches of a cluster. We find the steady state for a system of non-interacting biased walkers on a percolation cluster theoretically. Using this, we plot the average velocity of the walkers as a function of the bias. We simulate interacting biased random walkers with hard-core interactions on a percolation cluster to understand the current of the walkers as a function of bias and walker density. The long-time velocity-velocity autocorrelation function is a slowly varying function of the bias for interacting random walkers on a regular comb. This slow decay is due to the dynamic heterogeneity in the random walkers’ motion, which means there are different regions where the walkers’ motion differs. In the region inside the branches, the average velocity of the walkers is low as most are trapped beneath other walkers. Meanwhile, closer to the backbone, the walkers are free to move and have a larger average velocity. The velocity-velocity autocorrelation function also shows bumps corresponding to the walkers trapped at different depths. We find the occupation probabilities in a regular comb using the partition function and compare it to the occupation probabilities in our simulations and the theoretical occupation probability for an infinite comb. A walker deep inside the trap stays there for a long time, which can be observed by plotting the probability that the trapping time is greater than τ vs τ , which is a slowly decaying function of τ and shows steps corresponding to the walkers being trapped at different depths.