Why is it said that entropy of a closed system may increase in classical physics?

Question

Why is it said that entropy of a closed system may increase in classical physics?

A classic thought experiment to explain this claim is that of a closed box with some moving billiard balls initially grouped at one side of the box.

After some time we expect the billiard balls to be uniformly randomly spread about the box. Such a state it is said reflects said increase in entropy.

However, if the balls and the box are perfectly elastic and the dynamics are reversible, then while the balls may end up uniformly randomly spread about the box, the entropy of the system should be just the same as that of its initial state. doesn't it?

Please answer in simple terms :)

Two notes:

1. This question popped into my head while reading a book on information theory, and recalling Susskind's lectures on classical mechanics where he asserts classical mechanics is deterministic and reversible.

2. If in this particular example entropy stays the same, then why is it used as a classic example?

Answer 1

Entropy is one of those words that can have different (related) meanings depending on who's using it and when they're using it, even among physicists. But for the sake of answering the question in simple terms, we can use this definition:

• Entropy is the number of microstates compatible with a given macrostate. (It's actually the natural log of that number, but that's not important here. Same idea.)

• A microstate specifies every detail, like specifying the position and velocity of every ball in the box.

• A macrostate specifies only a few overall features, like specifying only the total kinetic energy of all of the balls in the box. Any given macrostate is consistent with lots of different microstates.

One way to answer the question is to realize that different authors might be using different overall features to define their macrostates. If we define a macrostate by the total kinetic energy of all the balls in the box, then the entropy does not change (assuming the system is closed). But if we define a macrostate by the number of balls in one half of the box, then the entropy may change even if the system is closed, because the system may start in one macrostate (all the balls on one side) and evolve to a different macrostate (roughly equal numbers of balls on both sides).

Altogether, if two different people are talking about two different macroscopic features, they can make different statements about what happens to the entropy over time, and both statements can be correct. The definition of entropy depends on which features we're tracking, because entropy is the (log of the) number of microstates compatible with those features.