If one is changing while the other is not, even though they both represent the same thing when it comes down to it, would this not be evidence of a limitation of the method of measurement as applied to logical entropy? How would that impact upon Thorne's formulation?
But, OTOH, if one of the gasses was also a lot hotter than the other one, you can tell after the fact how much white mixed with the black and vice-versa with Boltzmann's formulas. (It corresponds nicely with the amount of temperature change observed in both compartments.)
What Thorne describes--actually a historical narrative in which key insights were made by Bekenstein, Zel'dovich, and Hawking--not only involves such equivalence as there is between logical and thermodynamic entropy, but an equivalence between black hole dynamics and thermodynamics in general.
Thus it was that Hawking, in 1974, having proved firmly that a black hole radiates as though it had a temperature proportional to its surface gravity, went on to assert, without real proof, that all of the other similarities between the laws of black hole mechanics and the laws of thermodynamics were more than a coincidence. The black-hole laws are the same thing as the thermodynamic laws, but in disguise ...Sounds shaky, but Hawking's postulate is so far holding up. Thorne cites some of his own work as an example.
Throw into a black hole's atmosphere a small amount of material containing some small amount of energy (or, equivalently, mass) angular momentum (spin), and electric charge. From the atmosphere this material will continue on down through the horizon and into the hole. Once the material has entered the hole, it is impossible by examining the hole from outside to learn the nature of the injected material (whether it consisted of matter or of antimatter, of photons and heavy atoms, or of electrons and positrons), and it is impossible to learn just where the material was injected. Because a black hole has no "hair," all one can discover, by examining the hole from outside, are the total amounts of mass, angular momentum, and charge that entered the atmosphere.He goes on a bit later,... [T]he logarithm of the number of ways to inject must be the increase in the atmosphere's entropy, as described by the standard laws of thermodynamics. By a fairly simple calculation, Zurek and I were able to show that this increase in thermodynamic entropy is precisely equal to 1/4 times the increase in the horizon's area, divided by the Planck-Wheeler area; this is, it is precisely the increase in the horizon's area in disguise, the same disguise that Hawking inferred, in 1974, from the mathematical similarity of the laws of black-hole mechanics and the laws of thermodynamics.
The thought experiment also shows the second law of thermodynamics in action. The energy, angular momentum, and charge that one throws into the hole's atmosphere can have any form at all ... When the bag is thrown into the the hole's atmosphere, the entropy of the external universe is reduced by the amount of the entropy (randomness) in the bag. However, the entropy of the hole's atmosphere, and thence of the hole, goes up by more than the bag's entropy, so the total entropy of hole plus external Universe goes up. The second law of thermodynamics is obeyed.Similarly, it turns out, when the black hole evaporates some particles, its own surface area and entropy typically go down; but the particles get distributed randomly in the external Universe, increasing its entropy by more than the hole's entropy loss. Again, the second law is obeyed.