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Here's a draft of an article by Manoj Gopalkrishnan:
It needs work, which he will probably do. Below are my comments to him. Do any of you have further suggestions?
Your post looks good! It's technical, but we've gotten so deep into reaction networks on this blog that there will definitely be plenty of readers who will be interested.
It compiles fine. There are some typos that I'll fix, and also deviations from my preferred style, but you don't need to worry about those.
Here's my one big suggestion. Near the start you say:
"Today we will see that for complex-balanced systems, the pseudo-Helmholtz free energy function is what mathematicians call a Lyapunov function, and physicists call an H-function: it is monotonically decreasing along trajectories of the rate equation. I'm going to explain this statement, and give you most of the proof!"
Before you proceed, it would be great to explain why we should care about this fact. You explain at the very end:
"The whole point of having a Lyapunov function is Lyapunov's theorem which implies stability for the dynamics."
But many readers won't make it that far, and fewer will make it if they don't know why they should bother.
I can expand on that. When writing, I always have a mental model of how a certain fraction of the readers get bored and quit reading each time something hard comes along. The unexplained phrases complex-balanced system, pseudo-Helmholtz free energy function, Lyapunov function, and H-function, all packed into one paragraph, will kill off about 30% of your readers, since many will assume that to understand what's coming next, they should know those phrases... even though you explain two of them right after you invoke them! (People aren't very persistent when reading blog articles.) So, if I were writing this, my first step would be to eliminate Lyapunov function and H-function from this passage, and say
"Today we will see that for complex-balanced systems, the pseudo-Helmholtz free energy function is monotonically decreasing along trajectories of the rate equation. I'm going to explain this statement, and give you most of the proof!"
I would later introduce the jargon Lyapunov function and H-function, after explaining why it's interesting to have a function that decreases along trajectories.
In my second pass, I would also eliminate the adverb monotonically, since it's not strictly necessary in order to get the idea, and it makes the paragraph more scary:
"Today we will see that for complex-balanced systems, the pseudo-Helmholtz free energy function decreases along trajectories of the rate equation. I'm going to explain this statement, and give you most of the proof!"
The scary phrases complex-balanced system and rate equation can stay, since they're necessary to quickly convey the idea, and you can probably afford to kill off readers who don't know these phrases, since I've explained them many times.
Personally, I would probably either add a joke about how scary the phrase pseudo-Helmholtz free energy function is, or make up a shorter term just for the purposes of this blog article. Anything "pseudo" or "quasi" sounds technical and vaguely repulsive.
Anyway, once the hurdle presented by this paragraph is low enough, a lot of people will survive it and be very interested t read, in the very next paragraph, why it's interesting to have a function that decreases along trajectories. An intuitive explanation of Lyapunov's theorem, with a bare minimum of jargon, would be nice.
One last comment: near the very end, you never define conservation class. You could probably get the idea across pretty quick.
Anyway, apart from these changes I guess all that's needed are the figures! I need to finish editing and post Marc Harper's article on relative entropy as a Lyapunov function in evolutionary game theory before I post yours, so there's no need to get this done in less than a week.
It's great how we're getting more guest posts on related issues! If you care, you can see Marc's post in draft form here.