Study ofA+A→0 with probability of reaction and diffusion in one dimension and in fractal substrata

Abstract

We present Monte Carlo simulations of annihilation reactionA+A→0 in one dimensional lattice and in three different fractal substrata. In the model, the particles diffuse independently and when two of them attempt to occupy the same substratum site, they react with a probabilityp. For different kinds of initial distributions and in the short an intermediate time regimes, the results for 0<p≪1 show that the density ofA particles approximately behaves as ρ(t)=ρ(t=0)f(t/t ₀), with the scaling functionf(x)≃1 forx≪1,f(x)∼x ^−y forx≫1. The crossover timet ₀, behaves ast ₀∼ρ ^−1y_0eff where theeffective initial densityρ _0eff depends on ρ(t=0) and on the kind of initial distribution. For a given substratum of spreading dimensiond _s, the exponenty(d _s/2<y<1) depends only onp and its value increases asp decreases (y→1 whenp→0). In the very long time regime it is expected thatp(t)∼t ^−ds/2 independently ofp.