Demystifying the second law of thermodynamics

Entropy is not a law. It is a derivation. You can derive entropy mathematically from other axioms we assume are true in reality.

Let me state it plainly in a sentence that is not entirely correct but will help elucidate the meaning of entropy:

Systems tend toward disorder because there are more possible disordered configurations then there are ordered configurations.

This is entropy (sort of).

If your system is a bunch of 6 sided dice one ordered configuration is rolling a 1 on all the dice. Another ordered configuration is rolling a 5 on all the dice. A more likely but ordered configuration is rolling either a 1 or a 2.

There are numerous ordered configurations of a system but the number of unordered outcomes is by far greater then ordered outcomes. If you have a system of dice all facing up with 1s the system is in a low entropy ordered state. If you shake all the dice, it is more likely for the dice to come out in a random state all with random numbers. Shaking the dice is increasing entropy. Entropy increases. But you see here because you relate the dice to probability it's easy to see how by probability the dice should trend toward more disordered states.

The same phenomena goes for particles in a box. Instead of the numbers on the side of the dice replace it with the position of each particle. Shaking the dice is exactly isomorphic to particles traveling in random directions. By probability, dice will roll random disordered numbers just like how by probability randomly moving particles will move into randomly disordered positions.

The same probabilistic intuition that you have for why throwing a bunch of bricks at the ground won't be accident construct a house is exactly the intuition behind probability AND as a result entropy.

The real phenomena here is probability. Why does it apply to reality? We don't know. But entropy is just a consequence of it. Entropy is simply saying (again sort of) systems tend toward more disordered states because disordered states are more numerous and therefore more probable.

The axiom or "law" as you call it is probability and entropy is a theorem or derivation that is a consequence of probability.

This is just a really simple explanation of entropy. Entropy is actually more general then this. It applies even to systems where "ordered" states outnumber "unordered" states. You can see my other explanations in this thread I get into it deeply.

In that sense there is an underlying reality to entropy, but it's impossible to be sure, an ordered state could seem disordered if we lack the data that would describe it. I think we understand entropy to the same degree we understand data, and as data science develops I hope we'll develop a better understanding of entropy as a consequence.

Heat makes for a good proxy though.

Entropy must be described relative to a system and an arbitrary definition of macrostates and microstates. It actually has nothing to do with disorder or order, I used the term previously because it helps with intuition but it is actually categorically wrong.

For example for a system of 5 loaded dice that roll six 99% of the time. With a microstate defined as the value of each dice after a roll and a macrostate defined as the number of 6s.

With this system entropy goes up as you roll more sixes. Order also goes up with entropy. Entropy is an arbitrary concept that is defined relative to your choice of a "system", "macrostates" and "microstates." You can choose systems that have higher probabilities of being ordered like say magnetic cubes in a box versus regular cubes. The magnetic cubes are more likely to be stacked perfectly and that is defined as a higher entropic state even when there is "less" information needed to "describe" it.

I never read about information theory but I'm assuming that the choice of "system" in information theory is usually pretty simple as in you can have numbers like the dice, but you usually don't work with "loaded" dice. So under the case where the microstate of each entity has equal probability of occuring. In this case there is a direct relationship between whether data describing something can be compressed and the entropy of of that something. The higher the entropy the less it can be compressed.

Information places idealistic restrictions on microstates... However this is not the case in nature. If the microstate of our universe of atoms is defined as the cartesian coordinates of each atom. Then atoms have a tendency to coalesce into spheres (planets, starts, black holes) due to gravity and as a result each microstate does not have equal probability of occuring.

Planets are in fact a higher entropic state then the cloud of dust that the planet initially started out as. I can now describe all those atoms with in compressed form (a macrostate): "Planet." Thereby leading to an opposite relationship between compressibility and entropy. In this case the higher the entropy of a system the less information is needed to describe it.

The popular intuition of entropy is often wrong and incomplete. The most general definition is here: https://en.wikipedia.org/wiki/Entropy_(information_theory)

You will see it is inline with what I'm talking about. Entropy is an arbitrary concept defined relative to your choice of a system (entities axioms and theorems) and arbitrary macrostates and arbitary microstates defined for the system.

Nobody has this down, they all talk about entropy without defining what kind of entropy they're talking about. Usually it's a half baked definition involving temperature and conservation of energy.

In general when you add conservation of energy into your system, microstates and macrostates then it becomes more inline with the popular notion of entropy. Because for atoms to self organize into planets they must stop moving, but because of conservation of energy that motion must be transferred into something else. So in general self organization in one part of the universe must mean another part of the universe gets hotter.

If I define a macrostate and microstate to account for conservation of energy meaning that my microstate must make sure that if a particle stops moving another one must start moving then things usually fits the notion of getting more disordered over time.

If I define the microstate to be just the position of atoms then things appear to become ordered over time as the position of atoms coalesce into spheres.

So there's different perspectives on it and all perspectives are true. It's just one perspective is examining the phenomenon of conservation of energy the other is not.