Polynomial dynamical systems, reaction systems, and the global attractor conjecture

Many dynamical systems with polynomial right-hand side can be regarded as “reaction systems”, i.e., mathematical models for the dynamics of concentrations in a network of reactions. We discuss the connection between special classes of reaction systems (such as detailed-balanced and vertex-balanced systems) and the Boltzmann equation. In particular, vertex-balanced systems are believed to have globally attracting states (this is the “global attractor conjecture"). We also describe some applications to quantum Boltzmann equations, acoustic wave turbulence, and the current state of the art for the proof of the global attractor conjecture.