  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.
