Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function
Abstract
We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B ->C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t−1 to the anomalous t−1/4 behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.