Bimolecular reaction A + B --> 0 at steady state on fractals: Anomalous rate law and reactant self-organization

Abstract

This paper is a theoretical investigation of the diffusion limited reaction A + B --> 0 at steady state and on fractal structures. We propose an extension of a scheme previously applied to Euclidean spaces, using the so-called fractal diffusion operator defined by O'Shaughnessy and Procaccia. We show that for a particular type of source, we obtain distributions of reactants and macroscopic reaction laws that interpolate the results previously found in Euclidean dimensions. More specifically, we show that the relevant dimension of the problem is the spectral dimension ds and for ds < 2 we have a mesoscopic segregation in the medium that may imply anomalous orders of reaction. Some of the predictions are tested via Monte Carlo simulations. More generally, we find that these results can be viewed as examples of a more general property common to other elementary diffusion limited bimolecular reactions.