>>12094089>EntropyEvery microstate maps to a unique macrostate.
Not every macrostate is mapped to from a unique microstate.
The number of microstates that map to the same macrostate is the "entropy" of that macrostate.
It quantifies the degree of "non injectiveness" of the map.
In that sense, it's similar to the concept of "degeneracy" in quantum mechanics.
Think of a game involving two dice, eg craps, monopoly, etc.
There are thirty six possible outcomes, six for each die - these are the microstates.
There are eleven possible outcomes for the sum of the two values - this is a macrostate.
If we only know that we rolled (6,6), we can calculate that the sum is 12.
If we only know that the sum is 12, we can invert the map and calculate that we rolled (6,6).
The important point here is that we can calculate the missing information in both directions.
If we only know that we rolled (2,2), we can calculate that the sum is 4.
If we only know that the sum is 4, we have no way to invert the map and calculate what we rolled.
It could be either (2,2), (1,3) or (3,1) - the inverse map is multi-valued.
The degree of "multi-valuedness" of the inverse map is what entropy is quantifying.
The most likely sum for a roll of two dice is 7, and it is no coincidence that this is the sum with the highest entropy.
You are six times more likely to get a sum of 7 (maximum entropy) than you are to get a sum of 1 or 12 (minimum entropy).
If you increase the number of dice, the difference between the min and max entropy sums gets larger and larger.
For a mole of dice, the likelihood of getting the maximum entropy sum is close to 100%.
In thermodynamics, the concept of an "irreversible process" is intimately connected to the non invertability of many-to-one functions.
In that sense, entropy is "real", as we can observe that many processes are not spontaneously reversible.
