Application of moments to the general linear multicomponent reaction-diffusion equation

A sequence of spatial moments to the solution of the linear multicomponent reaction-diffusion equation in the infinite domain with general nondiagonal diffusion and rate matrices can be established without prior determination of the solution. This is significant for several reasons. First, while the solutions to full reaction-diffusion equations cannot generally be found in closed analytical form, a simple recursion relation can be established for the moments. Secondly the lower moments can be given a clear physical meaning and they can be explicitly represented in terms of their contribution from the kinetic and diffusion portions of the problem. These results suggest that conversion of raw experimental concentration profiles into moments would be useful for physical interpretation as well as analysis in terms of the underlying diffusion and rate matrices. As a particular point of illustration, it is shown that the lower moments can be combined to yield expressions for the mean-squared diffusion length for each component of the mixture. It is shown that the time behavior of the mean-squared diffusion length in a reaction-diffusion system deviates from that in simple Fick's law diffusion only if there is both chemical reaction and a nondiagonal diffusion matrix. As a specific illustration the mean-squared diffusion lengths are calculated for a two-solute system undergoing general diffusion and a reversible isomerziation reaction.