Quantum techniques for studying equilibrium in reaction networks

Anderson, Craciun, and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits a stationary distribution when the deterministic model of the same system with mass-action kinetics admits an equilibrium solution obeying a certain ‘complex balance’ condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators, and coherent states. This is an example of ‘stochastic mechanics’, where we take techniques from quantum mechanics and replace amplitudes by probabilities. We explain how the systems studied here can be described using either of two equivalent formalisms: reaction networks, as in chemistry, or stochastic Petri nets, as in some other disciplines.