Nonlinear chemical dynamics in low dimensions: An exactly soluble model

Restricting space to low dimensions can cause deviations from the mean-field behavior in certain statistical systems. We investigate, both numerically and analytically, the behavior of the chemical reaction A+2X3X in one and two dimensions. In one dimension, we produce exact results showing that the trimolecular reaction system stabilizes in a nonequilibrium, locally frozen, asymptotic state in which the ratior of A to X particles is a constant number,r=0.38, quite different from the mean-field ratio,r_MF=1. The same trimolecular model, however, reaches the mean-field limit in two dimensions. In contrast, the bimolecular chemical reaction A+X2X is shown to agree with the mean-field predictions in all dimensions. For both models, we show that the adoption of certain types of transition rules in the laws of evolution can lead to oscillatory steady states.