Quantum Techniques for Reaction Networks

Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum physics, such as second quantization, can be adapted to apply to such systems. Here we use these techniques to study how the “master equation” describing stochastic time evolution for a reaction network is related to the “rate equation” describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a “coherent state”, meaning that the numbers of entities of each kind are described by independent Poisson distributions. Remarkably, in this case the rate equation and master equation give the exact same formula for the time derivative of the expected number of particles of each species.