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

[John Denker <jsd@av8n.com>]

On 06/23/2011 10:55 AM, Carl Mungan wrote:
> 1. The philosophical question. Is it possible to have an ideal gas 
> composed of particles that are identical but distinguishable? 

There is a notion of "distinguishable by position" ... 
within limits.

Without this notion:
 -- Classical physics would not exist.
 -- Specifically, we would not be able to talk -- ever --
  about "an electron".  We would have to antisymmetrize the
  wavefunction with respect to every electron in the universe.

This notion exists ... WITHIN LIMITS.

> Sec VII in Versteegh is beginning to bother me. I quote: "When the 
> individual one-particle wave packets in the many-particle state have 
> small spatial extensions and do not overlap, ... they approximately 
> follow the paths of classical particles. ... Narrow wave packets can 
> follow trajectories through space-time, and as long as they remain 
> separated from each other, these wave packets remain as 
> distinguishable as classical particles. 

That's crazy.  It is at best a statement about properties of the
null set, because in a gas the particles do not remain always
separated from each other.  In an ordinary sample of gas, each
particle collides with another a billion times per second, so
even if the particles are separated from each other on average,
they do not "remain separated" long enough to perform the relevant
experiments (such as a Gibbs-style mixing experiment).

To reach the same conclusion another way, identical-particle effects 
must always come into play at some point;  otherwise we would predict
the wrong answer to the Gibbs experiment.

One way to think about the real physics is to consider a gas of
particles that have a nonzero nuclear spin, such as 3He.  At 
ordinary (not too cold) temperatures, the nuclear spin has no 
effect on the equation of motion, i.e. on the particle/particle 
scattering events.  The spin is just a label.  Let's put up-labeled 
particles on one side and down-labeled particles on the other 
side and pull out the partition.  Mixing occurs.  In terms of 
macroscopic pressure and temperature and heat capacity this 
scenario is not appreciably different from the corresponding 
unlabeled scenario ... but the entropy is different!  There is 
entropy of mixing of the labeled particles.  For some purposes 
you may not care, but you will for sure care if you ever try to 
separate them.  You will have to expend energy on the order of 
T dS to unmix them.

  A concept that may be useful here is the notion of "spectator
  entropy".  You might decide that for purposes of temperature,
  pressure, and ordinary chemistry you don't care about the
  entropy of the nuclear spin labels.  You can call it 
  spectator entropy and ignore it.  That's your choice, and
  you can get away with it for some purposes ... but only for
  *some* purposes.  That entropy is real, and it is still there
  even if you choose not to account for it.  There are *some*
  things (such as heat capacity) that depend only on derivatives
  of the entropy, so that the zero of entropy is arbitrary ...
  but there remain other things (such as Bose condensation) that
  depend on the absolute and you simply must account for all of
  the entropy, including spin labels or any other kind of labels.

Forsooth, at low temperatures the equation of motion is affected
by the labeling, i.e. by the identical particle effects.  The
diffusion constant changes in magnitude, and will pick up an
imaginary part, possibly leading to spin waves.  There is what
chemists would call a "molecular field" created by the up-particles
such that a particle with some other spin-orientation will precess 
in this field.  This is observed even in dilute gases, where the 
particles remain quite far apart on average.  You could argue that 
the identical-particle effects only occur rarely, when one particle 
encounters another, but even so, the effects are cumulative ("always 
clockwise") and cannot be neglected.

The experiment has been done.  I don't offhand know if it has
been done in vapor-phase 3He, but I know for sure it has been
done in monatomic hydrogen.  Spectacular spin waves were observed
at low temperatures in a dilute gas (roughly atmospheric density).
  http://prl.aps.org/abstract/PRL/v52/i17/p1508_1

If anybody thinks such things cannot happen in a dilute gas, they're
just wrong.

Also there is a venerable cottage industry in observing spin waves 
in dilute /liquid/ 3He, although you might need to think for a 
moment to realize that that is equivalent.