Asymptotic behavior of densities in diffusion dominated two-particle reactions

We analyze the limiting behavior of the densities ρA(t) and ρB(t) for the diffusion controlled chemical reaction A + B → inert. We prove that for equal initial densities ρA(0)=ρB(0) there is a change in behavior from d ⩽4, where ρA(t)=ρB(t)∼Ctd4, to d⩾4, where ρA(t)=ρB(t)∼Ct as t→∞. For unequal initial densities ρA(t)(0)<ρB(0), ρA(t)∼e -Ct in d=1, ρA(t)∼e−Ct/log t in d=2, ρA(t)∼e−Ct in d⩾3. The term C depends on the initial densities and changes with d.