• Kenny Courser, *Open Systems: A Double Categorical Perspective*, Ph.D. thesis, U. C. Riverside, 2020.

• Maru Sarazola, Dynamical systems and their steady states, 2 April 2018.

She compares two papers:

• David Spivak, The steady states of coupled dynamical systems compose according to matrix arithmetic.

• John Baez and Blake Pollard, A compositional framework for reaction networks, *Reviews in Mathematical Physics* **29** (2017), 1750028.

(Blog article here.)

The Petri nets are familiar to the Chemist, though maybe not necessarily under that name :-). I see that you arrived at that by what we would call “reaction kinetic” considerations. When I started to think about it I had a different approach: just regard the thermodynamic equilibria.

So the basic question is about the types of the products and not how long it takes to get them. That would be something like a “reaction mechanism network”. (A side note here: what you write about the term “species” in Chemistry is entirely true, but there is another thing that can be mentioned, chemists are well trained to strictly separate between “particle/molecule,atom” and “compound” hierarchy. There is usually a set of terms which forms a partition with this respect and “species” strictly speaking belongs to the particle/molecule subset). So I thought about compounds (anything what you can put into any type of container) as the elements of the set . And the binary operation (lets write it multiplicative for the moment) being a perfect mixture followed by infinite time. Then reactions read for example like . Its clear that , you mix the compound with itself and nothing happens. (The nice feature of multiplicativity here I find is that total amounts do not appear, only concentrations.) Also is clear. But except closed-ness, there is nothing more. So my conclusion was its a CI-magma. What is missing are different reaction conditions at which we wait for the thermodynamic equilibrium. So I thought you get that by a set of indexed morphisms, the indices are the reaction conditions, I think its basically temperature and pressure, maybe some EM radiation. This morphism conserves idempotency and commutativity (besides closedness), so I thought one might call it “homomorphism” (but I am not sure).

An interesting feature of this formalization is, that one can for example model that there is a difference between pouring into and into , which seems to contradict commuativity at first glance, but thats not true: pouring a into b means essentially starting with and going via and to , which is in a CI-magma. So in that approach “kinetics” would enter from an more algebraic side the buisnes.

Would it make sense to you?

]]>Interesting, thank you very much!

]]>A common approach these days is to use a strict symmetric monoidal category where different substances (usually called ‘species’) are objects, the tensor product is denoted and a typical morphism would a reaction like this:

This is a tiny bit like a commutative idempotent magma, but different. You can read more in Section 25.2 here:

• John Baez and Jacob Biamonte *Quantum Techniques for Stochastic Mechanics*, World Scientific Press, Singapore, 2018. (Draft available

on the arXiv.)

That’s a very interesting question, which many people have thought about.

One answer people sometimes give is that analysis makes heavy use of alternating universal and existential quantifiers (“for every there exists a such that for all …”), while algebra tends to avoid these, making the reasoning “cleaner”. However, this neglects the fact that universal properties involve strings of alternating universal and existential quantifiers.

Another answer is more historical: category theory first arose at the borderline of algebra and topology when people realized they were using functors to reduce topology problems to algebra problems, and it hasn’t percolated into analysis as deeply.

I suspect the real answer will only be known when we *do* better unify analysis and other branches of math; this may require new ideas that we don’t have yet… and not having these ideas is what’s holding us back.

Thanks! I’m curious as to why algebra and analysis have resisted unification (unlike say, algebra and topology), but I suspect that’s way beyond addressable here.

]]>There’s a tradition of studying chemical reactions in which concentrations of chemicals of obey a set of differential equations called the ‘rate equation’, based on the **law of mass action**: the rate of a chemical reaction is proportional to the concentrations of the reacting substances.

In real life all chemical reactions are reversible and a closed system tends to settle down into an equilibrium obeying detailed balance: the rate of any reaction is equal to that of the reverse reaction. However, in some cases the rate of some reverse reactions are so slow that it’s enlightening to consider what happens when they’re zero. Then we can have interesting phenomena, like solutions of the rate equation where the concentrations of chemicals are periodic as a function of time, or even chaotic. A famous example is the Belousov-Zhabotinsky reaction.

But the Belousov-Zhabotinsky reaction actually involves about 18 reactions, so it’s interesting to look for simpler model systems that display similar behavior. A famous one is the Brusselator. The reactions here are not reversible (they’re shown in the Wikipedia article I linked to), and the analysis is simplest when 2 of the 5 chemicals involved are treated as having concentrations so much higher than the rest that we can approximate these concentrations as fixed… even though these chemicals are getting used up.

This sort of thing led mathematical chemists to study chemical reaction networks where not all reactions are reversible and some concentrations are held fixed, or **chemostatted**.

The big revolution in chemical reaction network theory came in the 1960s, with the work of Aris, Feinberg, Jackson, Horn and others. They proved many theorems about the behavior of the rate equation for different classes of chemical reaction networks: existence, uniqueness or nonuniqueness of equilibrium solutions, periodic solutions, etc.

Around the 1970s, Prigogine focused attention on the nonequilibrium thermodynamics of ‘open’ systems, where for example chemicals are constantly being added and/or removed from a system. Then we can have ‘nonequilibrium steady states’, where there’s no change in chemical concentrations, but nonzero net flows: that is, a lack of detailed balance. Sometimes there exists a unique nonequilibrium steady state that is stable, but sometimes it becomes unstable and one sees oscillatory behavior. Prigogine developed a formalism to study this.

My work with Blake Pollard formalizes the notion of ‘open chemical reaction network’. We describe the processes that let you build a big open chemical reaction network from smaller pieces. We describe the laws obeyed by these ‘building up’ processes. We describe the ‘open rate equation’, a generalization of the rate equation for open chemical reaction networks. We describe how the open rate equation of a big open chemical reaction network is built up from the open rate equations of the pieces. We also describe how nonequilibrium steady states of a big open chemical reaction network are built up from nonequilibrium steady states of the pieces. The right language for describing all this is category theory, since that provides techniques for studying how open systems can be connected end-to-end (‘composition’) or in parallel (‘tensoring’).

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