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 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.
Highlights
A “reaction network” describes how various kinds of classical particles can interact and turn into other kinds. They are commonly used in chemistry. This reaction network gives a very simplified picture of what is happening in a glass of water: H2O H+ + OH− (1)
The reactions here are all reversible, but this is not required by the reaction network formalism
We are seeing a similar phenomenon in stochastic mechanics: the deterministic dynamics given by the rate equation match the stochastic dynamics given by the master equation in the special case of coherent states
Summary
A “reaction network” describes how various kinds of classical particles can interact and turn into other kinds. This is called a “pure state”: it describes a completely definite situation where there are l1 particles of the first kind and l2 of the second kind This method of describing a probability distribution using a power series is far from new. In Theorem 9 we show that this equation gives the rate equation exactly when the numbers of particles of each species are described by independent Poisson distributions This relies on a nice fact: the generating function of a product of independent Poisson distributions is an eigenvector of all the annihilation operators. We are seeing a similar phenomenon in stochastic mechanics: the deterministic dynamics given by the rate equation match the stochastic dynamics given by the master equation in the special case of coherent states We explore this further in a companion paper [3]. See the review by Biamonte, Faccin, and De Domenico [13]
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