Graham: you might check out my edited version of Marc Harper's post on [[Blog - relative entropy in evolutionary dynamics|relative entropy in evolutionary dynamics]]. You'll be glad to know that even before reading your comment, I greatly expanded the discussion of Lyapunov functions and started talking about balls rolling down hills.

I did not talk about the need for syrup, nor did I mention that free energy or entropy are good places to start looking for such functions.

Before, in parts 9-13 of the [information geometry series](, I explained how relative entropy being a Lyapunov function for evolutionary dynamics is connected to the 2nd law of thermodynamics. However, it's probably worth another mention!

The issue of needing lots of friction to get our "rolling ball" intuition to apply to _first-order_ differential equations is extremely important, but I'm not sure this blog article is the place for it.