Closure schemes for joint density functions in diffusion-limited reactions

We study the macroscopic particle concentrations both in a diffusion-limited coagulation (A+A→A) and in a two-species diffusion-limited annihilation reaction (A+B→scrI), where scrI represents an inert result, in spatial dimension d, using a general closure scheme for truncating the hierarchies of the kinetic equations of the joint density functions. In the coagulation reaction, the concentration C(t) goes to zero asymptotically as ${\mathit{t}}^{\mathrm{-}\mathit{d}/2\mathrm{}}$ for dt/t for d=2, and as ${\mathit{t}}^{\mathrm{-}1}$ for d>2, while in the annihilation reaction with equal initial density ${\mathit{C}}_{\mathit{A}}$(0)=${\mathit{C}}_{\mathit{B}}$(0), the density ${\mathit{C}}_{\mathit{A}}$(t)=${\mathit{C}}_{\mathit{B}}$(t) goes to zero as ${\mathit{t}}^{\mathrm{-}\mathit{d}/4\mathrm{}}$ for d≤4 and ${\mathit{t}}^{\mathrm{-}1}$ for d>4. Our approximation reproduces correctly all these asymptotic concentrations. An alternative approximation is also applied to the two-species annihilation reaction in one dimension, and its result is the same as that obtained from the general closure scheme.