OK so this, like my baseball thing, is a puzzler that I’ve been chewing on for many years. (I am not high.)
In high school my physics teacher, in explaining entropy, pointed out that if you push a porcelain teacup off of a table and it smashes, that requires far less energy than making the teacup. And that’s entropy, as I was taught it - the universe wants things to be in a chaotic state, and it’s easier in terms of energy to get things into a chaotic state than to get them into an ordered state. Then there was some math business. Now, before anyone goes impugning my physics teacher, I should note that I failed this class, and indeed failed seven out of my eight semesters of science classes in high school. (A feat I matched in math, and I’m fairly sure I only passed both my second semester sophomore year because my dad died.) So there is every chance in the world that I misunderstood her.
But my question persists: I intuitively see the teacup as more ordered than the bunch of broken shards on the ground because I’m a human and I like to drink tea and so teacups are useful for me. But the universe isn’t a human and doesn’t care about drinking tea. So my question is, what makes the teacup more ordered than the shards, in a non anthropo-chauvinistic way? Like, we can imagine some alien race that finds the shards more ordered than the cup, right? That doesn’t sound too crazy to me for an alien intelligence; they’ll have massively different values and observations and perceptions of what ordered means. After all, it’s still the same material at the same mass. So what’s the non-human version of order that the non-smashed teacup satisfies better than the smashed one?
I feel like this is especially important because YouTube teaches me that entropy is the key to the arrow of time. Although perhaps “teaches” is not the best word.`
The quick answer is that there's only one way for the teacup to be a teacup, but there are many many (many many many) ways for the shards to be shards. Jumble up the teacup, and you get shards. Jumble up the shards, and you also get shards.
If you imagine any system (like the atoms in the teacup) as being described by some phase space, some configurations have larger shares of the phase space than others. The unbroken teacup is one particular arrangement -- a tiny island in the phase space. Because there are many many ways of the teacup being broken, most of the phase space will describe some variant of 'broken teacup'. So as the system evolves over time, and wanders around the phase space, it's going to spend a lot more time in states which dominate that phase space.
This is what entropy is -- the statistical tendency for systems to exist in states which have the largest share of the phase space.
Entropy is fundamentally not about disorder, or chaos, or whatever. It's about statistics, and about the distribution of energy, and it frankly only makes sense when you get down to the truly microscopic level, so I dislike when people present mostly macroscopic examples like the teacup.
Systems in the universe have energy. For example, a bunch of molecules has a bunch of energy, by virtue of those molecules moving their mass through space. Entropy is about the number of possible ways that energy can be distributed. For example, if I have a single molecule carrying 6 units of energy, there's only one way it can be distributed: that molecule has to carry all six. Conversely, if I have two molecules carrying the same amount of energy, suddenly there are a huge number of options. They could both have three units. One could have four and the other one two. Or decimal numbers. The options aren't infinite, due to quantum stuff (don't ask) but they are more numerous. So The two-molecule system has the same energy, but higher entropy, because the energy has more options for how it can be distributed.
Besides having more molecules, how else can we have higher entropy? Well, those molecules can have greater or lesser freedom of movement. Molecules in a solid are locked into a particular shape and orientation, and the only possible motion vibrational (i.e. sitting still and buzzing excitedly). Molecules in a liquid are able to not just vibrate but to spin, so there's a greater range of ways liquid molecules can carry their energy. And in a gas, they can not just vibrate or spin, but move through space.
The arrow of time tends towards higher entropy not because of any kind of preference about order or disorder, but out of statistical inevitability. Each possible configuration of a system (called a microstate) represents a kind of lottery ticket. The more tickets you buy, the greater your chance of winning, so the universe tends inexorably towards sets of conditions (macrostates) with a greater number of potential microstates.
*I am a chemist, but not a physical chemist and definitely not a physicist, so I'm not an expert on entropy, but I have had to learn about it a lot, and I've taught its rudiments many times. What I'm articulating here is a somewhat-less-than-rigorous explanation aimed at lay comprehensibility.*