> how is it then, that things are becoming more spread out over time?

This may help: https://www.researchgate.net/figure/Classical-evolution-in-p...

If we knew the present we could predict the future - at least in classical physics. We have a very approximative knowledge of the present though, based on a macroscopic description. We can still predict a range of outcomes and see what it means in macroscopic terms. The initial set of states consistent with what we know “spreads out” as time goes by. We “lose” information by keeping only a “coarse” macroscopic description.

> what is the property about the past, that it seems have a lot of uncommon states and not a lot of the common ones?

This is called as Past hypothesis.

https://en.wikipedia.org/wiki/Past_hypothesis

>how is it then, that things are becoming more spread out over time?

It's more likely for things to spread out then to concentrate in one corner when you randomly move all particles in a box. If all particles moved to one corner of a box you would assume there's an intelligence at work moving the particles because such movement is too low of a probability to happen without intelligent intervention.

>if common states are mathematically more likely to be common, why is it that the future has them and the past does not, in general?

Common states are called "common" because their are more of them in general. Think in terms of things with a few states like rolling dice. You have a machine that rolls rice continuously and checks the result. Use this as an analogy of particles of gas moving around in a box and then use that as an analogy of the universe.

Nobody knows why probability works this way. Probability is what differentiates the arrow of time.

>like, why hasn't heat death happened? the probability thing seems to be almost proof-like, you cannot really argue against it. but clearly, the universe did, to a great degree, at some point in the past. why?

Entropy is just a phenomenon of probability. Don't let the concept of entropy rule your brain and override what's going on. Heat death hasn't happened because entropy can be frozen. You freeze ice then the particles stop moving.

Additionally all of what I said above doesn't apply to things with gravity like black holes. With gravity things begin to automatically self organize into circular orbits or spherical planets. Why? It's because entropy isn't measuring disorder. That's a mistake. Entropy is just saying that systems drift towards high probability macrostates. In systems with gravity, spherical shapes and circular orbits ARE a higher probability macrostate then one where the particles disordered. In this system a higher entropic state is actually MORE ordered then a lower entropic macrostate.

I don't know if there's going to be a "heat" death, but for sure we are moving towards higher entropy as the law says. But this does not necessarily mean more disorder or things getting spread out.

That's it. I think the word entropy just confuses everyone. It's just someone observed these weird phenomenon with heat and called it entropy. Then we realized it's just a bunch of particles moving into high probability patterns.

But that's a semantics game. Sub probability for entropy. Why do we live in a world where low probability states were in the past and high probability ones are in the future? What intrinsic property of the universe causes this asymmetry? One can imagine a symmetric k-negative universe where high probability macrostates trend towards low probability macrostates. Or a k-zero universe where the dice never rolls.

None of such questions follow definitionally from the second law ^H^H^H probability.

Yes. It is a semantics game. I feel people understand probability but they don't understand entropy hence it's easier to just use the term probability state.

And yes the questions you pose don't follow from the 2nd law. But they are the big question.