Role of fluctuations for inhomogeneous reaction-diffusion phenomena

Although fluctuations have been known to change the long-time behavior of homogeneous diffusion-reaction phenomena dramatically in dimensions d≤4, simulations of reaction fronts in two-dimensional A+B→C inhomogeneous systems have only shown marginal departure from mean-field behavior. We perform cellular-automata simulations of the one-dimensional case and find that the width W(t) of the reaction front behaves as ${\mathit{t}}^{0.293\pm 0.005}$, in contrast to mean-field behavior ${\mathit{t}}^{1/6}$. We develop a scaling theory to obtain inequalities for the exponents in the more general mechanism nA+mB→C. Heuristic arguments about the range of fluctuations imply that the mean-field behavior should be correct in dimensions larger than an upper critical dimension ${\mathit{d}}_{\mathrm{u}\mathrm{p}=2\mathrm{}}$, irrespective of the values of n and m. This leads us to reinterpret the two-dimensional data obtained previously in terms of a logarithmic correction to mean-field behavior.