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The early universe is always described to have begun in a state of extremely low entropy and it's been increasing ever since.

But the same amount of stuff exists now as it did back then. Only thing that's changed is how big the universe is now vs then.

So it seems to me entropy in a given system is a function of its volume.

Take the classic gas in a closed box example. It has 2 compartments, with a barrier in between. Take the barrier out, gas spreads out to both compartments until maximum entropy is achieved. But the gas was in its maximum entropy for the smaller compartment it occupied before, we just added more space. Add even more space to the box, gas will spread out even more. You essentially increase maximum entropy.

And the universe is expanding - i.e. we get more space all the time.

So the early universe is only low entropy, because space expanded to allow for more entropy.

And if it keeps expanding, it makes me conclude the "heat death" is (kinda) nonsensical. Perhaps things would slow down until interactions happen on absurdly long timescales, but they'd still keep happening.

What am I missing here?

(One thing I could think of, don't know how logically sound, is where, for every particle in the universe, each other particle finds itself outside of its cosmological horizon, removing it from the pool of particles that could interact with anything ever again.)

There's also some other philosophical arguments I can think of based on these ideas. Like for example the idea that space has to be quantized. Because of space was continuous, well, a finite region of space would have infinite entropy, because it'd have an infinite number of possible microstates.