Indeed John, some very exciting ideas that are new to me in this paper! In particular, towards the end he has worked out a challenge example that Anne Shiu, Ezra Miller and I set out in our paper arXiv:1305.5303. I hope to read it thoroughly and write more about it in my next blog entry. Sorry it’s taking so long to get done.

]]>• Alexander N. Gorban, General H-theorem and entropies that violate the second law, *Entropy* **16** (2014), 2408–2432.

For more remarks comparing different paper on Lyapunov functions for chemical reaction networks and evolutionary games, go here. I will try to summarize all these ideas in a blog post at some point, but right now we’ve got two parallel conversations going on: one featuring chemical reaction network theorists and one featuring evolutionary game theorists… both talking about free energy as a Lyapunov function.

]]>That’s a very nice proof, I had not seen it before. Thanks!

]]>I’ve fixed the LaTeX in your comment, but it’s nice to have a PDF version too.

I’m glad to see you here! We’ve spent a lot of time here discussing your work on stationary solutions of the master equation for complex-balanced systems.

Thanks for posting this comment! It’s great to see another proof of this important result.

]]>I think it is worth noting that Martin Feinberg proved things in a different way than did Horn and Jackson. Marty posted some nice lecture notes (from a series of lectures in Wisconsin in 1979) that can be found at: http://crnt.engineering.osu.edu/LecturesOnReactionNetworks

In particular, his proof that the function being discussed here is, in fact, a Lyapunov function is quite nice. Let me redo it here as it is quite nice, though I will change the proof slightly by dropping the notion of “complex-space,” which Marty makes use of in his notes. I like this proof as it *really* shows (to me at least) where the complex balance condition comes in.

**********

Let be a deterministically modeled chemical reaction system with mass-action kinetics. Suppose that there are precisely species. We denote the th reaction by

and denote the span of the reaction vectors by

The ODE governing the dynamics of the system is

Assume that the system is complex-balanced with complex-balanced equilibrium . This means that for each ,

where the sum on the left is over all reactions for which is the source complex, and the sum on the right is over all reactions for which is the product complex.

Now define the function by

The fact that is a Lyapunov function for the system is captured in the following result.

**Theorem.** Suppose that with . Then

with equality if and only if .

**Proof.** Note that

Using that for any real numbers we have with equality if and only if (consider secant lines of ), we have

where the final equality holds by the condition above on complex-balancing.

Thus, we have a strict inequality unless

for all . That is, we have a strict inequality unless

Following precisely the argument on page 4–33 of Feinberg’s notes, we now note that if both

and

hold, then

which, by the monotonicity of the function, can only happen if for all .

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