Reaction-diffusion on metric graphs: From 3D to 1D

We provide stochastic foundations for the analysis of a class of reaction-diffusion systems using as an example the known Temporal Analysis of Products (TAP) experiments, showing how to effectively obtain explicit solutions to the associated equations by approximating the 3-dimensional domain of diffusion U (the reactor) by 1-dimensional network models. In a typical TAP experiment a pulse of reactant gas of species A is injected into U , which is filled with chemically inert material, permeable to gas diffusion. Particles of catalyst are placed amid this inert medium, forming active sites where the reaction A→ B may occur. On part of the boundary of U designated as the exit, a mixture of A and B can escape. We study the problem of determining the (molar) fraction of product gas in the mixture after U is fully evacuated. This fraction is identified with the reaction probability—that is, the probability of a single diffusing molecule reacting before leaving U . Specifically, we are interested in how this probability depends on the reaction rate constant k. After giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability, we study a class of domains, called fat graphs, comprising a network of thin tubes with active sites at junctures. The main result of the paper is that in the limit, as the thin tubes approach curves in 3-dimensional space, reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k. By this 3D to 1D reduction, the simpler processes on metric graphs can be used as model systems for more realistic 3-dimensional configurations. This is illustrated with analytic and numerical examples. One example, in particular, illustrates an important application of our method: finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant. Reaction-diffusion, metric graphs.