SPATIAL STRUCTURE IN DIFFUSION-LIMITED 2-PARTICLE REACTIONS

We analyze the limiting behavior of the densities rho(A)(t) and rho(B)(t), and the random spatial structure zeta(t) = (zeta(A)(t)), zeta(B)(t)), for the diffusion-controlled chemical reaction A + B --> inert. For equal initial densities rho(A)(0) = rho(B)(0) there is a change in behavior from d less-than-or-equal-to 4, where rho(A)(t) = rho(B)(t) almost-equal-to C/t(d/4), to d greater-than-or-equal-to 4, where rho(A)(t) = rho(B)(t) almost-equal-to C/t as t --> infinity; 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 zeta, 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.