Quantum Techniques for Reaction Networks

TL;DR

This paper explores how quantum field theory techniques can be applied to reaction networks, linking stochastic master equations with deterministic rate equations, especially when the system is in a coherent state.

Contribution

It introduces a novel application of quantum field theory methods to analyze the relationship between stochastic and deterministic descriptions of reaction networks.

Findings

Strong relation between master and rate equations in coherent states

Quantum techniques reveal conditions for deterministic behavior in reaction networks

Poisson distributions characterize the state of entities in the system

Abstract

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 field theory 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.