Annihilation reactions: Crossover from mean-field to anomalous behaviours

Abstract

We study the coagulation reaction A + A → A with diffusion and probability p of reaction in a one dimensional lattice and the annihilation reaction A + A → 0 with diffusion and interaction in the two-dimensional fractal percolation cluster. Due to reaction the particle density ρ decreases as a function of time t. The crossover from mean-field (ρ ~ t −1) to anomalous behaviors (ρ ~ t −1/2 in one dimension and ρ ~ t −2/3 in the percolation cluster) is analyzed. From this analysis, in one dimension an analytical approximation of the density is found which agrees very well with Monte Carlo results of ρ(t) for all times and for small values of p. For the percolation cluster and large values of the nearest-neighbor repulsive interaction U between particles, we find that a simple mean-field approximation works at short times. The length of the time interval where this approximation holds increases as U increases.