Re: [Phys-l] Gibbs paradox (redux)

[Carl Mungan <mungan@usna.edu>]

> > By dividing the glass of milk into 2 equal portions, I seem to have
> > decreased the entropy of the system. This seems like a strange
> > result: where is corresponding gain of entropy in the surroundings to
> > ensure the 2nd law is not violated?
>
> Perhaps it would help to start with chocolate milk on one side and
> white milk laced with radioactive cesium on the other.  Remove the
> partition.  Observe the mixing.  Re-insert the partition.  Unmixing
> does not occur.

I accept what you say for chocolate and white milk. Also for the cards.

But I'm still stuck on what happens with just a glass of milk. My problem is that *macroscopically* I can't tell any difference between two half glasses of milk and one full glass of milk, in contrast to the case of chocolate and white where I can see the color gradations. Shouldn't there be some *macroscopically measurable* evidence of a change in entropy?

> Just because two subsystems are
physically separated at the moment does not mean their
> probabilities are statistically independent.

Does this mean my calculation of 2S_new - S = -Nkln2 is correct or not? I understand I should be ready to throw extensivity out, but I don't want to throw the 2nd law out also. Can you give more details of how S of the universe as a whole does not decrease when I pour a glass of milk into two half glasses? Can you actually show me how to calculate delta_S for the universe in this process? I'm not even sure if the answer is zero or positive, although I'm guessing it's zero.

If pouring is too complicated, I will accept a conceptually simpler experiment of your choice, as long as we end up with two half-volumes of the milk.

Oh and one more thing: Bernard asked about how identical the milk globules are allowed to be. Would anything change if we could somehow manufacture milk with *exactly* identical globules? (Okay, we surely couldn't do this with milk. So maybe I instead manufacture macroscopically small but microscopically large - mesoscopically sized? - metal balls and suspend them in some suitable - preferably dense - solvent.) I'm thinking that as long as the particles are large compared to their thermal de Broglie wavelengths then they are classical, whether exactly identical or not. Would exact identicality suddenly make the particles indistinguishable instead of distinguishable? I'm thinking not for large enough particles, but am willing to be open minded if you can give me a good reason.

Thanks for bearing with me and for the discussion so far; it has helped clarify my puzzlement about Gibbs paradox. -Carl