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# Coupling Through Emergent Conservation Laws

edited July 2 in Chat

I'm sorry I've been slower about cranking out lectures these days! I've been struggling to finish a paper with some students at Applied Category Theory 2018, and got mired in some complicated calculations. But it's almost done, and I'm starting to release it as a series of blog articles. The first is here:

There's no category theory visible in here, though if I ever get around to creating a publishable version it'll be a lot more mathematical and probably will have category theory in it. Open reaction networks are morphisms in a category!

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edited July 2

Here's the next part - a quick introduction to chemistry and thermodynamics:

Comment Source:Here's the next part - a quick introduction to chemistry and thermodynamics: * [Coupling through emergent conservation laws (part 2)](https://johncarlosbaez.wordpress.com/2018/06/27/coupling-through-emergent-conservation-laws-part-2/) - review of reaction networks and equilibrium thermodynamics. 
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Is it possible all conservation laws are really emergent conservation laws?

Comment Source:Is it possible all conservation laws are really emergent conservation laws?
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I apologize if you guys already know about this, but wanted to mention a practical resource in this area from systems biology. Genome scale models (e.g. downloadable from http://bigg.ucsd.edu) describe the coupling/stoichiometry for every known reaction in a living organism. These models describe a convex polytope of potentially accessible fluxes for every reaction in the organism.

Comment Source:It will be interesting to read more about this idea of emergent conservation laws. I apologize if you guys already know about this, but wanted to mention a practical resource in this area from systems biology. Genome scale models (e.g. downloadable from http://bigg.ucsd.edu) describe the coupling/stoichiometry for every known reaction in a living organism. These models describe a convex polytope of potentially accessible fluxes for every reaction in the organism.
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Tyler - that's cool! If we ever get some ideas that could use a workout on industrial-strenth data, that would be fun! Right now the biggest example we studied using our techniques is the citric acid cycle. But in fact our techniques wouldn't really be suitable for considering every reaction in the organism at once.

Comment Source:Tyler - that's cool! If we ever get some ideas that could use a workout on industrial-strenth data, that would be fun! Right now the biggest example we studied using our techniques is the citric acid cycle. But in fact our techniques wouldn't really be suitable for considering every reaction in the organism at once.
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edited July 2

Here's the next part: a basic statement of the problem concerning coupling:

Comment Source:Here's the next part: a basic statement of the problem concerning coupling: * [Coupling through emergent conservation laws (part 3)](https://johncarlosbaez.wordpress.com/2018/06/28/coupling-through-emergent-conservation-laws-part-3/) - what is coupling?
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Here are some more parts:

Comment Source:Here are some more parts: * <a href = "http://math.ucr.edu/home/baez/networks/coupling/coupling_4.html">Coupling through emergent conservation laws (part 4)</a> - interactions. * <a href = "https://johncarlosbaez.wordpress.com/2018/06/30/coupling-through-emergent-conservation-laws-part-5.html">Coupling through emergent conservation laws (part 5)</a> - coupling in quasiequilibrium states. * <a href = "https://johncarlosbaez.wordpress.com/2018/07/01/coupling-through-emergent-conservation-laws-part-6/">Coupling through emergent conservation laws (part 6)</a> - emergent conservation laws. 
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This is cool. Is there a symmetry related to the conservation laws? Or is that "symmetry" the "coupling" reactions them self? Or nothing at all.

Is there a "free-energy" <-> time scale transformation? Where looking at a slower time is like looking a system with higher needed energy?

Comment Source:This is cool. Is there a symmetry related to the conservation laws? Or is that "symmetry" the "coupling" reactions them self? Or nothing at all. Is there a "free-energy" <-> time scale transformation? Where looking at a slower time is like looking a system with higher needed energy? 
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edited July 5

Now that you mention it, I believe there probably are symmetries connected to these conserved quantities. They don't follow from the usual Noether's theorem because the "rate equation" describing the dynamics of chemical reactions is not derived from a Lagrangian. I proved a generalization of Noether's theorem to Markov processes, but it doesn't cover the rate equation, which is nonlinear. So, I may need to generalize it further! Thanks for suggesting it.

Is there a "free-energy" <-> time scale transformation? Where looking at a slower time is like looking a system with higher needed energy?

When you study chemistry quantum-mechanically you get the usual reciprocal relation between times and energies, related to the fact that $$\hbar$$ has units of time $$\times$$ energy. Here we are studying them phenomenologically, just writing down a rate equation involving some rate constants that come "out of the blue". There's more to say, but no time to say it!

Comment Source:Now that you mention it, I believe there probably are symmetries connected to these conserved quantities. They don't follow from the usual Noether's theorem because the "rate equation" describing the dynamics of chemical reactions is not derived from a Lagrangian. I proved a [generalization of Noether's theorem to Markov processes](https://arxiv.org/abs/1203.2035), but it doesn't cover the rate equation, which is nonlinear. So, I may need to generalize it further! Thanks for suggesting it. > Is there a "free-energy" <-> time scale transformation? Where looking at a slower time is like looking a system with higher needed energy? When you study chemistry quantum-mechanically you get the usual reciprocal relation between times and energies, related to the fact that \$$\hbar\$$ has units of time \$$\times\$$ energy. Here we are studying them phenomenologically, just writing down a rate equation involving some rate constants that come "out of the blue". There's more to say, but no time to say it!
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We're studying free and forgetful functors right now. Is there a free-forgetful adjunction between emergent conservation laws and regular conservation laws?

Comment Source:We're studying free and forgetful functors right now. Is there a free-forgetful adjunction between emergent conservation laws and regular conservation laws? 
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I haven't a clue - I don't even know what categories those laws would be objects of! As far as I know, they are just elements of sets. Maybe you can think of these sets as categories in a useful way, but I don't know how.

Comment Source:I haven't a clue - I don't even know what categories those laws would be objects of! As far as I know, they are just elements of sets. Maybe you can think of these sets as categories in a useful way, but I don't know how.
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Well a true conservation law can be thought of as a partition of the phase space. But we want to look at equations that are mostly true, that are true up to a possible fuzz, and we probably want morphisms to be some sort of "thickened" implication?

Comment Source:Well a true conservation law can be thought of as a partition of the phase space. But we want to look at equations that are mostly true, that are true up to a possible fuzz, and we probably want morphisms to be some sort of "thickened" implication?
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edited July 4

Following..very intriguing..thanks all for the posts

Comment Source:Following..very intriguing..thanks all for the posts
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Christopher wrote:

But we want to look at equations that are mostly true, that are true up to a possible fuzz, and we probably want morphisms to be some sort of "thickened" implication?

Too difficult for me!

Comment Source:Christopher wrote: > But we want to look at equations that are mostly true, that are true up to a possible fuzz, and we probably want morphisms to be some sort of "thickened" implication? Too difficult for me!
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I'm getting some nice comments on this followup post:

Comment Source:I'm getting some nice comments on this followup post: * [Toric geometry in reaction networks](https://johncarlosbaez.wordpress.com/2018/07/03/toric-geometry-in-reaction-networks/).
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Its not horribly complicated of a beast..okay its a little horrible...errr..,maybe a lot horrible. I mean I am getting like 4 different kinds of category or 2 category to really fit all the puzzle pieces together. None of the pieces individually are very complicated, but how they reflect each other is..tricky?

Comment Source:Its not horribly complicated of a beast..okay its a little horrible...errr..,maybe a lot horrible. I mean I am getting like 4 different kinds of category or 2 category to really fit all the puzzle pieces together. None of the pieces individually are very complicated, but how they reflect each other is..tricky?