A new set of Monte Carlo moves for lattice random-walk models of biased diffusion

Abstract

We recently demonstrated that standard fixed-time lattice random-walk models cannot be modified to properly represent biased diffusion processes in more than two dimensions. The origin of this fundamental limitation appears to be the fact that traditional Monte Carlo moves do not allow for simultaneous jumps along each spatial direction. We thus propose a new algorithm to transform biased diffusion problems into lattice random walks such that we recover the proper dynamics for any number of spatial dimensions and for arbitrary values of the external field. Using a hypercubic lattice, we redefine the basic Monte Carlo moves, including the transition probabilities and the corresponding time durations, in order to allow for simultaneous jumps along all Cartesian axes. We show that our new algorithm can be used both with computer simulations and with exact numerical methods to obtain the mean velocity and the diffusion coefficient of point-like particles in any dimensions and in the presence of obstacles.