https://arxiv.org/abs/1009.5966 is on similar theme i think.

]]>• John Baez, Quantum techniques for reaction networks, *Advances in Mathematical Physics* **2018** (2018), 7676309.

It’s a long and comitragic tale which I feel like blogging about. Soon.

]]>The version on the arXiv is now a bit out of date. The latest version, with all known corrections made, is on my website. But I’ll eventually update the arXiv version.

]]>In the proof of Lemma 12, I think should be .

In the penultimate sentence, “…as show by…” should be “…as shown by…”

Hope that helps.

]]>as indeed they say at the beginning of section 3.1.

]]>It would be interesting to have some general results about, say, when coarse-graining a stochastic Petri net makes an equilibrium unviable, or makes a new equilibrium appear.

That sounds interesting but a bit tough. Since I’m pretty good at abstract nonsense, I’d be tempted to start out by building some general infrastructure: namely, working out the correction notion(s) of *morphism* between stochastic Petri nets, and figuring out how the rate equations and master equations are related, when we have two stochastic Petri nets related by a morphism.

Since a Petri net is a free symmetric monoidal category, one obvious notion of morphism, which you mentioned, is a symmetric monoidal functor. (If we want to intimidate people, we can note that a category of ‘free’ gadgets and the obvious morphisms between them is a Kleisli category. Knowing this can even be useful for things other than intimidation.)

A stochastic Petri net is a free symmetric monoidal category equipped with a symmetric monoidal functor to the multiplicative monoid , viewed as a symmetric monoidal category. This again suggests an obvious notion of morphism between stochastic Petri nets, at least to us category geeks.

But the interesting part is whether these morphisms give rise to some sort of map sending solutions of the master equation of one stochastic Petri net to solutions of the other. This would probably be ‘too perfect’ for anyone interested in realistic coarse-grainings… but I suspect that if you follow the tao of mathematics, it will quickly lead you to consider these ‘perfect’ coarse-grainings.

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