Random Walks on Fractals

Abstract

We study the pseudounimolecular reaction A+B→P, with A(0)<<B(0) and we model it through a random-walk picture. Two distinct cases are considered: the trapping problem, in which the minority, A-particles, move while the B-molecules are stationary, and the target problem, in which the A molecules act as fixed targets and only the B-particles move. As underlying structures we consider both translationally-symmetric lattices and also fractals (Sierpinski gaskets). Fractals are a large class of geometrical objects; apart from being interesting by themselves, fractals also provide a way to mimic disorder. Although both the target and the trapping problem can be described by the same reaction-diffusion equation, their decays are different: It is the trapping which shows at longer times the slower decay and the larger deviations from standard kinetic decay laws. Both, for the trapping and target problems, the deviations get more pronounced if the dimensionality of the underlying structure decreases; the decay laws found for Sierpinski gaskets are intermediate between the decays for one- and two-dimensional regular lattices.