On the differences between discrete and continuum diffusion–reaction systems in two and three dimensions

Abstract One of the long‐standing issues in discrete and continuum diffusion–reaction systems is that in two and three dimensions, there are non‐negligible differences between the two systems, in contrast to one dimension, where simulations with a finite jump distance can produce results identical to those of continuum diffusion–reaction systems. The differences are analyzed as a function of the lattice constant. It has been observed that these differences do not decrease monotonically as the lattice constant decreases, due to coincidental cancelation arising from the mathematical complexities of the discrete system. This non‐monotonic behavior indicates that the lattice constant should be chosen carefully when studying diffusion–reaction systems. In addition, a dynamic time step method is introduced in Monte Carlo simulations for computational efficiency, based on the relationship between the jump distance and the time step. The accuracy of the method is confirmed and this method is particularly efficient when reactions are rare at long times.