Single species diffusion-influenced reaction A+A→αA:: Validity of the Smoluchowski approach

We investigate the single species diffusion-influenced reaction. A + A --> alpha A with a finite reactivity in all dimensions. The reaction model includes a purr coagulation (alpha = 1) or a purl annihilation (alpha = 0) model. We apply the hierarchical Smoluchowski approach to study the dimensional aspects of the fluctuation, reactivity, particle size, and alpha(0 less than or equal to alpha less than or equal to 1). The theoretical results an compared with these of the Monte Carlo simulations in one, two, and three regular dimensions. The simulation results reveal that the classical Smoluchowski approach is exact in the short time limit in all dimensions and in the long time limit in three dimensions. The hierarchical Smoluchowski approach is found to be numerically exact at all times in two and three dimensions. A numerical method to obtain the exact result of the annihilation for a finite reactivity in one dimension is presented. We also propose a quite accurate analytic solution for an arbitrary alpha for the infinite reactivity in one dimension.