Sure. Actually there are two main things we call states. The first one (macroscopic state, or macrostate) is what we think as characteristic properties of a bit of matter, e.g. "1 kg of liquid water at 300 K under atmospheric pressure" compared to "1 kg of solid water at 250 K under atmospheric pressure".

The second one (microscopic states, or microstate) is the way the particles that constitute this bit of matter are arranged. The naming is a bit unfortunate and it can get technical quite quickly, but the distinction between macroscopic and microscopic states is crucial.

So, a more precise version of my previous post would be something like that. In the liquid example, there are many, many ways of distributing the H2O molecules that would result in the same macroscopic description. This means that there are many microscopic states that are consistent with the macroscopic state we observe. And, looking at a glass of water, we cannot say where the molecules are.

On the other hand, in a perfect ice crystal, the positions of all the atoms constituting al the H2O molecules are uniquely determined by the crystal structure. So, looking at a perfect ice cube we can say where every molecule is. It gets a bit more complicated in reality because no crystal is ever perfect. There are defects that introduce some disorder, so there are more than one microscopic state, but much fewer than in the liquid.

Entropy is larger for things that have more microscopic states consistent with their macroscopic state. You can also see here a hint of the link with information entropy in CS, if you think about the number of microscopic states as our knowledge of the molecules' positions.

> Doesn't the entropy change when I change my definition of the state? If I go to the extreme and there is only one kind of state, an actual arrangement of particles, fields etc., the entropy would be the same of each and everything (one possible arrangement per state).

It does sound very subjective. The conventional naming is a bit unfortunate and a consequence of the historical roots of statistical Physics in 19th-century thermodynamics. After all, there is no clear boundary between macroscopic and microscopic.

If this bothers you, you can say that "macroscopic" refers to the thing you are looking at as a whole, and "microscopic" to its constituents. This framework works as long as the stuff you are studying is made up of smaller things. We commonly use atoms and molecules in examples because it is somewhat intuitive, but you could consider an atom itself as a macroscopic system and its quarks as its constituents. Or the universe and its galaxies clusters.

The caveat is that the mathematical formalism is exact in the limit where the number of constituents is infinite, but might break down if their number is too small. That's why the distinction between macroscopic and microscopic is helpful.

> Does that make entropy an entirely subjective measure?

In a way, a bit. You can define the constituents seemingly arbitrarily (like considering molecules, or atoms as separate entities or not, adding electrons, etc). Adding more details gives a more accurate answer, but at some point it becomes irrelevant. So it is actually less subjective than relative.

To keep (ab)using the water example, a molecule is made up of 3 atoms, each one having a position (3 positions, each one being a 3-dimensional vector, so 9 parameters in total). But when grouped in a molecule these positions are not independent, and need to be consistent with the O-H bond length and the H-O-H angle. A molecule is characterised by a position (3-d vector), an orientation along the axes of the reference frame (3 other parameters), an angle and two bond lengths (again 9 parameters in total). So there is no more information if you describe the ice cube as a collection of atoms than as a collection of molecules, even though the choice seems arbitrary.

Take a deck of cards, say, a pre-ordered one (technically, you know what microstate it's in) - so let's say that its "disorder" is "zero", and your entropy about it is zero.

Now shuffle it face down really well, now both its "disorder" and your entropy about it is pretty high (it has a specific microstate that you don't know, and the ensemble of all potential microstates form a macrostate).

Now look at this shuffled deck face up again - it's "disorder" is still high, but now your entropy about it is zero again, because you know what microstate it's in.