Building reliable lattice Monte Carlo models for real drift and diffusion problems

Abstract

We revisit the well-known issue of representing an overdamped drift-and-diffusion system by an equivalent lattice random-walk model. We demonstrate that commonly used Monte Carlo algorithms do not conserve the diffusion coefficient when a driving field of arbitrary amplitude is present, and that such algorithms would actually require fluctuating jumping times and one clock per Cartesian direction to work properly. Although it is in principle possible to construct valid algorithms with fixed time steps, we show that no such algorithm can be used in more than two dimensions if the jumps are made along only one axis at each time step.

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

“…In this case, the mean jumping times are shorter along the field axis, but one can easily renormalize the jumping probabilities to use a single time step. In a recent paper [9], we demonstrated that although this method does give the correct drift velocity for arbitrary values of the driving field, it fails to give the correct diffusion coefficient. The problem is due to the often neglected fact that the variance of the jumping time affects the diffusion process in the presence of a net drift [10].…”

“…In the same article [9], we showed how to modify a one-dimensional LMC algorithm with the addition of a stochastic jumping time τ ± ∆τ , where the appropriate value of the standard-deviation ∆τ was again obtained from the resolution of the local FPP. For simulations in higher spatial dimensions d > 1, it is possible to use our one-dimensional algorithm with the proper method to alternate between the dimensions as long as the Monte Carlo clock advances only when the particle moves along the field direction [9]. LMC simulations of diffusion processes actually use stochastic methods to resolve a discrete problem that can be written in terms of coupled linear equations.…”

“…We recently solved the problem of defining a LMC algorithm with both a fixed time step and the proper temporal fluctuations [9]. This required the addition of a probability to stay put on the current lattice site during a given time step (of course, this change also implies a renormalization of the jumping probabilities).…”

“…Unfortunately, our LMC algorithm [9] has a fatal flaw: for dimensions d > 2, some of the jumping probabilities turn out to be negative. This failure suggests that there is a fundamental problem with this class of models, or more precisely with standard LMC moves (however, note that it is still possible to use computer simulations and fluctuating jumping times τ ± ∆τ , as explained above).…”

“…This failure is due to the lack of temporal fluctuations in such a LMC algorithm (at each step, the particle would jump either forward (p + ) or backward (p − ), and all jumps would take the same time τ ). As mentioned above, it is possible to fix this problem [9] with a stochastic time step like τ ± ∆τ where ∆τ can also be calculated exactly within the framework of FPP's [17]:…”