From: "JACK L. URETSKY (C)1998; HEP DIVISION, ARGONNE NATIONAL LAB ARGONNE, IL 60439" <JLU@HEP.ANL.GOV>
Date: Tue, 11 May 1999 06:53:49 -0500
Hi Jim-
I*m not qualified to comment on the last paragraph of your
posting, but I think that the rest of your question misunderstands the
meaning of a density function.
Let "Int" stand for the integral sign, and consider some distribution
of matter along a line. Then I can define a density function r(x), and a
"particle mass" m, such that the total mass on the line is
M = m*Int r(x)dx where the integral is over the entire
line. Then I can call Int r(x)dx, integrated over a segment of length delta,
"the number of particles" in the segment of length delta. Now take the limit,
remembering that dx is truly an infinitesimal, so that r(x)dx is, poetically
speaking, "the number of particles in an infinitesimal length dx (at x)". But
r(x) has the dimensions of particles per unit length, so we are entitled to
think of dx, again somewhat poetically, as a "length" assigned to one particle.
The real physics, such as it is, is in the density function r(x).
Now turn to thermo in one dimension, and generalize the preceding
argument. Instead of integrating over x, you must integrate over phase space
which is dxdp for each particle. To get a partition function, divide each
dxdp by an h that makes the units come out right, and take care of the correct
Boltzman counting of the particles. Just remember that all the physics is in
the density function.
Note that the argument is most easily made from an integral (or
macroscopic viewpoint) and the microscopic language is just fiction that leads
to the right answer.
As to texts, you might find Huang (1963) and Landau and Lifshitz clearer.
Regards,
Jack
*************************************************************
Ok, Jack: I have a tendance to ask generalized questions -- I think that I
learn more that way, but let me give a specific example:
Consider an ideal gas of N "particles" in a box (sides L) with total energy
E -- How do I count the number of available states?
Some texts (eg Stowe, big Rief, and I think Kittel -- but I can't find my
copy just now) do a Byzantine calculation based on Heisenberg or wave
packets -- something like each particle (or each state) occupies a volume
of DxDyDzDpxDpyDpz in 6-dimensional phase space. And get something like
Omega = (1/h^3)V(real) V(momentum)
I guess my problem is that I am uncomfortable saying that the particles are
restricted to the resulting 6-d grid -- Why can't a particle be at say
(1/3),(1/8), 1/7), etc in the cell? Why should I assume that the particles
are centered in their cells?
Am I making any sense -- or am I just old and senile and am having a mental
block?
"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography