# Thought experiment with opposite intuitions from thermo/info theory

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?

• How are you defining entropy in an information-theoretic sense? – probably_someone Dec 25 '16 at 7:37
• @probably_someone I was trying to stay super informal with this question, but I was going for "simple" meaning "least compressed", in the sense of the system's state having, say, a Komolgorov complexity shorter than that of the actual state itself. The minimum length of the description of the mixed state is much longer than the same of the separated state. – bright-star Dec 25 '16 at 7:46
• That definition might be what gets you into trouble. I would be very careful how you define information-theoretic entropy; it tends to be non-intuitive at best. – probably_someone Dec 25 '16 at 7:47

As far as extraction of work is concerned, what matters is change in entropy of universe. If the process that occurs increases entropy of universe, then you can in principle extract work out of it. During that process a particular sub-system's entropy may increase, decrease, or remain constant.

In your boiling pot example, you have indeed increased entropy of the contents of the pot by heating it. But this doesn't mean you cannot extract work out of it. In fact the convective motion inside the pot tells you there is still mechanical energy inside the pot that could be converted to heat, and therefore further increase entropy of universe. So it is not surprising that you can extract some work out of it using some contraption like a turbine wheel.

When you turn off heating and the pot cools down and returns to equilibrium, the universe has reached the state of maximum entropy (to avoid complications, think of boiling pot and heating mechanism to be located inside an isolated chamber). From there no other higher entropy state is accessible to the universe, so no work can be extracted from the cooled down pot.

In short, always think in terms of entropy change of the entire universe, and not of a particular system.

A closed system can change entropy, entropy is not related "definitively" to how much energy it has, rather to how ordered it isn't (to get the sign correct). Bear with me.

Any interaction on a closed system, including moving energy around will increase the entropy, which is why entropy is sometimes called the arrow of time.

If you add energy from outside the system, it is possible to achieve either increases or decrases in entropy of the system. The overall super-system including your energey source, though, will always increase in entropy (as per last para). If your function (the way you put energy in) is realising a random state, then all the energy is introducing disorder. It's actually a new rule to me (to qualify my answer) but I suggest this is when the measure of disorder, entropy, is proportional to the work done. (work done of course being all the energy introduced to the system, to close the point)