Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities ρ_A(t) and ρ_B(t), and the random spatial structure ξ(r) = (ξ_A(t)., ξ_B(t)), for the diffusion-controlled chemical reaction A+B→inert. For equal initial densities ρ_B(0) = ρ_b(0) there is a change in behavior from d≤ 4, where ρ_A(t) =ρ_B(t) ≈C/t^d/4, to d ≥ 4, where ρ_A(t) =ρ_b(t) ≈C/t as t → ∞; the term C depends on the initial densities and changes with d. There is a corresponding change in the spatial structure. In d < 4, the particle types separate with only one type present locally, and ξ, after suitable rescaling, tends to a random Gaussian process. In d >4, both particle types are, after large times, present locally in concentrations not depending on type or location. In d=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.