Re: classical physics: Gibbs paradox

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

I second John Denker's nomination of the not well-appreciated
"Gibbs paradox" as a worthy "problem with classical physics. His
message prompted me to dig into my dusty boxes to find the report
of a library research project I did as an undergraduate on this
interesting topic.

John writes:

> 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.

It's interesting, however, to note that even in the light of
quantum physics, which does seem to offer a fundamental
explanation for the discontinuity between distinguishability and
indistiguishability, one can still imagine gases in mixtures of
states that vary continuously (in the sense of the degree of
orthogonality of the wave functions) from distinguishable to
indistinguishable and one obtains a corresponding fraction of the
usual entropy of mixing result. (See Von Neumann's "Mathematical
Foundations of Quantum Mechanics, 1955, pp.367-379).

More accessible treatments of related issues are also available in
(old) issues of AJP:

Boyer, AJP, V38, pp. 771-773, 1970
Hestenes, AJP, V38, pp. 840-845, 1970

(Looks at the entropy of mixing from a classical standpoint via
the construction of filters that will separate initially mixed
gases of spherical and cubical molecules with the cubical
molecules becoming increasingly spherical via shaved off edges.)

Klein, AJP, V26, pp. 80-81, 1958

(Looks at the entropy of mixing of gases which differ only in
excitation level and which subsequently decay to the same level
and inquires into the implications for the second law of thermo.)

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm