because the classic result assumes zero correlation between where any two molecules are, and once any kind of significant density difference exists between the two sides of the volume this will break.
The assumption of zero correlation is valid for ideal gases. It will not break if there is a density difference. We're talking about statistical correlation here.
Statistical independence means the chance that a molecule is at a particular spot depends not at all on where the other molecules are. Certainly if the molecules never hit each other, they only bounce off the walls of the volume, then this would be true as the molecules don't interract with each other so their probability of being one place or another is not changed by putting the other molecules anywhere, as long as they don't interract.
But molecules in a gas do interact they bounce off each other. Even an ideal gas. There is an average distance they travel before bouncing off another molecule called a mean free path. A situation where the mean free path is << size of volume is typical at STP.
Does this interaction break non-correlation? My intuition is that it does. But the thing I know for sure is that the only derivation I have ever seen for calculating the probability that all the gas is in 1/2 the volume was done with the assumptions of zero correlations, which we only know is the case for zero interaction, which is NOT an assumption required in the ideal gas models. And is certainly not true of any real gases.
"Entropy is in the mind" doesn't mean that you need consciousness for entropy to exist. All you need is a model of the world.
This is as true for Entropy as it is for Energy. By this standard, Entropy and Energy are both in the mind, neither one is "realer" than the other.
Entropy is in the mind in exactly the same sense that probability is in the mind. See the relevant Sequence post if you don't know what that means.
The usual ideal gas model is that collisions are perfectly elastic, so even if you do factor in collisions they don't actually change anything. Interactions such as van der Waals have been factored in. The ideal gas approximation should be quite close to the actual value for gases like Helium.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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