Spatial structure in diffusion-limited two-particle reactions

Abstract

We analyze the limiting behavior of the densitiesρ _A(t) andρ _B(t), and the random spatial structureξ(r) = (ξ _A(t).,ξ _B(t)), for the diffusion-controlled chemical reaction A+B→inert. For equal initial densitiesρ _B(0) = ρ_b(0) there is a change in behavior fromd⩽ 4, where ρ_A(t) =ρ _B(t) ≈C/t^d/4, tod ⩾ 4, where ρ_A(t) =ρ _b(t) ≈C/t ast → ∞; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and ξ, after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.