Re: [Phys-l] Gibbs paradox

[brian whatcott <betwys1@sbcglobal.net>]

On 4/1/2011 1:15 PM, John Denker wrote:
> On 04/01/2011 09:33 AM, Carl Mungan wrote:
> > I think the question of additivity is if you *don't* mix them. Take
> > two identical copies of the system (say your two sets of decks, each
> > individually shuffled), put one on top of the other but don't
> > reshuffle (in the gas case, that would mean keep a partition in
> > place), and now ask: what is the entropy of this new "doubled" system?
>
> If that's the question, I'm more confused than ever as to what
> this thread is about.
>
> a) There is a theorem that says that if the two subsystems have
>   independent probabilities, the entropy is additive.  This isn't
>   even physics;  it's just mathematics.  You can prove it on the
>   back of a post-it note.
>
> b) If we don't mix them, it cannot possibly matter whether they
>   are identical or distinguishable.
>
> What am I missing?
I THOUGHT the issue was this - rendered in comic form:

In a  rectangular cardboard box, with a central removable partition, 
place a dozen ping pong balls each side of the partition.
 Shake well.    1)
Remove the partition. Shake well.      2)
Replace the partition with equal numbers on each side again.    3)

THEN

In this cardboard box place the dozen balls in one side,
 and place  the same quantity of ping pong balls dyed red
in the other side of the central partition.    Shake Well.  4)
Remove the partition. Shake well.      5)
Replace the partition and rearrange the balls with equal numbers on each 
side again segregated by color.    6)

The questions: are the entropy values of arrangement 1) , 2) and 3) 
significantly different?
Are the entropy values of arrangement 4),  5)  and 6) significantly 
different?
Are the two sets of  Ss for 1,2,3, and 4,5,6 significantly different? Is 
this paradoxical?
Compare and contrast.    :-)

Brian W

Brian W