Random channel kinetics for reaction–diffusion systems

Abstract

A random channel approach is developed for reaction–diffusion processes in disordered systems. Although the starting point of our research is the kinetic study of the decay and preservation of marine organic carbon, our approach can be used for describing other disordered kinetic catalytic processes with random pathways. We consider a generic catalytic mechanism with two species: (a) a catalyst, which is continuously produced by a variable number of independent sources randomly distributed in space; this catalyst diffuses from the sources and is degrading according to a first order kinetic law; the generation, the degradation and the diffusion of the catalyst balance each other out and a stationary concentration field is generated; (b) an active species, which decays according to a second order kinetic law; the decay rate is proportional to the product of the concentrations of the catalyst and the concentration of the active species. We show that the catalyst concentration field can be represented by the sum of a random number of Yukawa-like potentials. The average value of the survival function of the active species can be expressed as a grand canonical average of a nonlinear functional of the catalyst field and can be evaluated exactly. We show that a good approximation is given by a nearest neighbor approach, where only the contribution of the closest source is taken into account for the computation of the random concentration field of the catalyst. We discuss the application of the model to the problem of decay and preservation of marine organic carbon. With minor adaptation the model can be applied to other problems of disordered kinetics, such as spatially distributed heterogeneous catalytic processes.