Re: classical physics: Gibbs paradox

[John Denker <jsd@MONMOUTH.COM>]

At 10:12 AM 1/23/01 -0700, Larry Smith wrote:
> What are some of the problems (or unresolved issues) with classical physics
> that provided impetus for modern physics (relativity or the quantum
> revolution)?
>
> The blackbody ultraviolet catastrophe is always near the top of everybody's
> list; what else would you add to make the list complete?

What about the Gibbs paradox?  I'm not a historian, and I'm not sure
exactly what role it played, but one could imagine a scenario under which
it was a very big deal indeed.

As you recall, the Gibbs paradox has to do with the entropy of mixing.  It
teaches us something fundamental about the nature of entropy.  It stands in
contrast to expressions such as dS=dQ/T which I consider a non-fundamental
special case.

Gibbs asks us to consider a box with a partition.  On one side of the
partition we have one type of gas, say the helium-3 isotope.  On the other
side we have another gas, say the helium-4 isotope.  When we pull out the
partition, we observe macroscopic cooling.  This is irreversible because of
the entropy of mixing.

Now the interesting thing is that if we have helium-4 on both sides,
pulling the partition has no effect.  For some reason, helium-4 cannot mix
with itself, but it can mix with helium-3.

This is a tremendously deep result.  We have no choice but to conclude that
some things are identical, and some things are not -- and that it really
matters!

There is no helium-3.9999 isotope, so things cannot be "almost"
identical.  This Gedankenexperiment is one of the great turning points in
the history of science.  It tells us an incredible amount about what an
_atom_ is, and what a _state_ is.  Without a good understanding of
identical particle states, the folks who invented quantum mechanics would
have had a very hard time.

Somebody should check on this, but I have found no evidence that Prof.
Gibbs ever referred to this a paradox.
  http://asuwlink.uwyo.edu/~wtg/Issues/Issues6.html

My guess is that he saw the resolution of this paradox fairly quickly.  But
there must have been at least SOME time between starting to think about
such issues and fully understanding them.  During this time, however long
or short, one could argue that the Gibbs paradox was an unresolved issue
that provided the impetus for a huuuuge step toward modern physics (so this
might contribute to an answer to Larry's question).

==============
To take this to the next level:

It turns out that the helium-3 nucleus has a magnetic dipole moment.  It is
so weak (and so well sheltered deep inside the atom) that it has a
negligible effect on things like collisions in the gas.  However the
alignment of the magnetic vectors gives us a way, in classical theory, to
create two samples of gas that are "almost" identical -- just make their
moments "almost" aligned.  The only way out of this enhanced Gibbs paradox
is to conclude that the alignment-states are quantized also.  I have no
reason to believe that anybody worried about the entropy of mixing of
polarization states until long after quantum-mechanics was invented, but if
they had, it would have been a paradox of the highest order.