(Crossposted from my personal website)

Thermodynamics is really weird. Most people have probably encountered a bad explanation of the basics at some point in school, but probably don't remember more than

Energy is conserved
Entropy increases
There's something called the ideal gas law/ideal gas equation.

Energy conservation is not very mysterious. Apart from some weirdness around defining energy in general, it's just a thing you can prove from whatever laws of motion you're using.

But entropy is very weird. You've heard that it measures "disorder" in some vague sense. Maybe you've heard that it's connected to the Shannon entropy of a probability distribution $H (p) = \sum_x - p (x) ln p (x)$ . Probably the weirdest thing about it is the law it obeys: It's not conserved, but rather it increases with time. This is more or less the only law like that in physics.

It gets even weirder when you consider that at least classical Newtonian physics is time-symmetric. Roughly speaking, this means if you have a movie of things interacting under the laws of Newton, and you play it backwards, they're still obeying the laws of Newton. An orbiting moon just looks like it's orbiting in the other direction, which is perfectly consistent. A stone which is falling towards earth and accelerating looks like it's flying away from earth and decelerating - exactly as gravity is supposed to do.

But if there's some "entropy" quality out there that only increases, then that's obviously impossible! When you played the movie backwards, you'd be able to tell that entropy was decreasing, and if entropy always increases, some law is being violated. So what, is entropy some artefact of quantum mechanics? No, as it turns out. Entropy is an artefact of the fact that you can't measure all the particles in the universe at once. And the fact that it seems to always increase is a consequence of the fact that matter is stable at large scales.

The points in this post are largely from E.T. Jaynes' Macroscopic Prediction.

A proof that entropy doesn't always increase

Let $X$ be the set of states of some physical system. Here I will assume that there is a finite number of states and time advances in discrete steps - there is some function $T : X \to X$ which steps time forward one step. We assume that these dynamics are time-reversible in the weak sense that $T$ is a bijection - every state is the future of exactly one "past" state. ...

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