Time-dependent reactivity for diffusion-controlled annihilation and coagulation in two dimensions.

Abstract

We present a method to obtain the concentration decay for coagulation, A+A\ensuremath{\rightarrow}A (\ensuremath{\epsilon}=1), and annihilation, A+A\ensuremath{\rightarrow}0 (\ensuremath{\epsilon}=2), of diffusing particles, on two-dimensional lattices. By mapping these reactions into processes that preserve the particle number, we find an approximate solution, which, as compared with numerical simulations for the square lattice, turns out to be exact for short and long times. The particle number along the whole course of the reaction is obtained in a closed form, and can be written as a function of only the mean number of distinct sites visited by a single particle. For a homogeneous initial particle distribution the particle number behaves at long times as N(t)\ensuremath{\sim}\ensuremath{\epsilon}N(0)(8aDt${)}^{\mathrm{\ensuremath{-}}1}$ln(8bDt), where D is the diffusivity and a and b depend on the lattice type. On the other hand, for a strongly inhomogeneous (fractal) initial distribution, the particle number decays at long times as N(t)\ensuremath{\sim}\ensuremath{\epsilon}N(0)(8aDt${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}/2}$ln(8bDt), where \ensuremath{\gamma} is the fractal dimension of the initial particle distribution (02).