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 prob ability on d-dimensional cubic lattices for d = 2,3, and 4, at low densitie s. Standard bimolecular kinetics are observed for d = 3 and 4. Small but si gnificant deviations from bimolecular kinetics are observed for d = 2, as e xpected from earlier studies. Although application of the Smoluchowski appr oach to diffusion-controlled reactions leads to the conclusion that the rat e coefficients should be proportional to the diffusion coefficients of the particles, themselves functions of the jump probability, in fact marked dev iations from linearity are found. A low-density mean-field theory is develo ped which agrees well with the simulation results in three dimensions and v ery well in four dimensions. In two dimensions the sequence which defines t he theoretical rate coefficient converges to zero, although intermediate va lues are in reasonable qualitative agreement with the simulation results. T he convergence to zero is due to the fact that random walks in two dimensio ns return with certainty to the origin; dominance of the kinetics of diffus ion-controlled reactions in low dimensions is ascribed to the same cause. ( C) 1999 Elsevier Science B.V. All rights reserved.