Reaction Networks as a Many-Body Problem

States or species and the reactions among them make up a reaction network. In the network, the abundances of species or the probabilities of states can be tracked as the reactions evolve in time by standard numerical techniques. In this work we look at a reaction network as a many body problem. The states or species make up this multi-level system and the energies are calculated from the branchings of a directed graph. The edges of the graph are reaction rates and the vertices are the states or species. We think of this problem in terms of the Matrix Forest Theorem where the constraints on the rate matrix are interpreted in terms of matrix minors. This approach allows us to compute an energy spectrum for the network, and we are able to understand the evolution of the network as transitions among different levels in the spectrum. We apply this approach to understand network equilibrium, dynamic equilibrium, quasi-static equilibrium, and reaction freeze out, especially in nucleosynthetic systems in stellar environments. Our approach can help clarify the governing role of particular reactions in element formation processes.