I'm given to understand that entropy in thermodynamics and entropy in information theory are functionally interchangeable. Informally, I can accept that the amount of work required to achieve a physical state that realizes a function of a random variable or process (or can be described by one) with a certain entropy is proportionate to the entropy of that state.
My problem is that thermodynamic entropy and info-theoretic entropy seem to point in opposite directions. So here's a thought experiment:
Imagine I have a pot with two fluids in it, on top of a burner. These two fluids don't chemically bond, but can produce a homogeneous mixture if you stir enough. They also settle out to different levels, being of different densities.
If I heat the fluids with the burner until one or both of them is near boiling, I know that I'll get a pretty good mixture after a while, like butter in chocolate. My notion is that this is a high-entropy state for the mixture, as it's as "scrambled" as it can be. If I were to try to describe this mixture with a string (a la Komolgorov), I would have to write a very long one to describe which particular state each of the fluid particles were in at a given instant.
On the other hand, though, this is a state I achieved by adding energy to the system through the burner. I obviously put the fluids into a state from which I could extract work. (At the most absurd level, I could put a little wheel in the fluid that would spin as they convected, and use it to power an LED light). Doesn't that mean that I've decreased the entropy? After all, isn't the Earth provided with energy to do work from the radiated energy of the Sun?
The reverse is just as contradictory to me. If I let the heat radiate back out of the pot and turn off the burner, the fluids will eventually settle out into layers. This is a much simpler arrangement to describe, and closer to that of an "unbroken egg" than the heated mixture. But now I have less energy to do work with, and it seems like this is a higher entropy state, since I've just "let it run down".
So what's wrong? My intuition, my definitions, or the experiment?