So after doing the Maxwell’s Demon thing, you say that mutual information decreases, the entropy of Y decreases, so we are left with the same amount of total entropy:
M1,Y1 → M1,Y1
M2,Y2 → M2,Y1
M3,Y3 → M3,Y1
M4,Y4 → M4,Y1
However, I don’t see why the mutual information would be lost; would the Demon know where he “put” the molecule, thus making the transition look more like:
M1,Y1 → M1,Y1
M2,Y2 → M1,Y1
M3,Y3 → M1,Y1
M4,Y4 → M1,Y1
This would of course shrink the phase space, violate the second law, etc. I just do not see how M would stay the same when Y changed (i.e. lose the mutual information).
I guess it would seem to me that what gets “overwritten” is the (now invalid) knowledge of where Y is, and what it is overwritten with is the new, valid position of it. I’ll have to chew on it for a while.
By the way, sort of unrelated, but I’ve always wondered why gravity acting on things is not considered a loss of entropy. For example I can drop a bowling ball from multiple distances, but it will always end up 0 feet from the ground:
B4 → B0
B3 → B0
B2 → B0
etc.
The only thing I can think of is that, when the ball hits the ground the collision creates enough heat (i.e. entropy) to balance everything out. Is that correct?