Okay I've thought through matters again and I have boiled things down
to two questions:
1. The philosophical question. Is it possible to have an ideal gas
composed of particles that are identical but distinguishable?
Versteegh spends quite a bit of space in his paper arguing that the
answer is yes, but I'm beginning to have doubts. When I say "is it
possible" I do NOT mean "in theory" but "in reality". Specifically
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. ... The Gibbs paradox can be
resolved without invoking the indistinguishability of particles. We
see that quantum theory does not change the relevant characteristics
of the situation, so that quantum theory is irrelevant to the
resolution of the paradox. ... If the particles are (approximately)
localized, permutation produces a different microstate just as in
classical mechanics."
My problem with this description is that it seems to me that in a gas
composed of a large number of particles flying around at high speeds,
the particles WILL collide from time to time in the sense of overlap
their wavefunctions and hence it will NOT be possible to keep track
of the phase-space trajectories of individual particles indefinitely.
I think a simple calculation involving the thermal de Broglie
wavelength, Maxwell-Boltzmann average speed, and realistic parameters
of say helium-4 at STP should settle the matter. Someone encourage me
to do the calculation and I will.
2. The practical question. Let's ignore question 1 and assume an
equilibrated spinless monatomic ideal gas of identical but
distinguishable particles exists, at least in theory. Assume the
temperature is high enough that we can treat the translational states
as a continuum and replace the partition function sums by integrals;
but assume the temperature is low enough that no electronic or
nuclear excited states are populated. Room temperature easily fits
between these two limits. Assume 1 mole of gas so we can use
Stirling's approximation. Since we're only considering translations
in 3D, we have internal energy U = 1.5 N k T. Take the volume to be
say V = 1 L cube (10 cm on a side). Here comes the question: TRUE or
FALSE - the entropy of the gas is given by the standard
Sackur-Tetrode equation? Specifically, suppose we double V, N, and U
for the gas - does S double? (Operationally, put two copies of the
gas systems side by side, and then remove the partition separating
them.)
If you have Schroeder "Thermal Physics" handy, you may wish to look
at Eqs. (2.55) and (2.57) and the discussion surrounding them.
Schroeder's solution is to simply say that an ideal gas of identical
but distinguishable particles CANNOT exist and so this practical
question is moot. But other authors like Swendsen and Versteegh
disagree.
Does this posting make it clearer what my problems are?
--
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363 mailto:mungan@usna.eduhttp://usna.edu/Users/physics/mungan/