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

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.