Diffusion-controlled reactions on simple lattices: dependence of the rate coefficients on jump probability and dimension

In this work, rates of the diffusion-controlled annihilation reaction A+A→nothing are studied by stochastic simulation as functions of jump probability on d-dimensional cubic lattices for d=2,3, and 4, at low densities. Standard bimolecular kinetics are observed for d=3 and 4. Small but significant deviations from bimolecular kinetics are observed for d=2, as expected from earlier studies. Although application of the Smoluchowski approach to diffusion-controlled reactions leads to the conclusion that the rate coefficients should be proportional to the diffusion coefficients of the particles, themselves functions of the jump probability, in fact marked deviations from linearity are found. A low-density mean-field theory is developed which agrees well with the simulation results in three dimensions and very well in four dimensions. In two dimensions the sequence which defines the theoretical rate coefficient converges to zero, although intermediate values are in reasonable qualitative agreement with the simulation results. The convergence to zero is due to the fact that random walks in two dimensions return with certainty to the origin; dominance of the kinetics of diffusion-controlled reactions in low dimensions is ascribed to the same cause.