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Solution: gas particle distribution problem



I gather that since no one (other than bc with his comment about the
scattering of sunlight by atmospheric density fluctuations) has responded
yet to my problem involving the distribution of a macroscopic number of
ideal gas particles in a container, none of the PHYS-L readership may
have wanted to concern themselves with it for whatever reasons.

So as a matter of closure on an unsuccessful attempt at provoking
interest in a bus arrival problem-inspired generalization to 3-D physics,
I now pull the plug on this problem, and give the solution to it on the
outside chance that someone might care to see it.

In order to relate John D.'s Poisson process bus arrival problem to an
analogous problem in a couple of more dimensions but with with a more
physics-oriented flavor, consider the case of (a macroscopic number) N
particles of an ideal gas in thermal equilibrium in a container of
(macroscopic) volume V so the average volume per particle is v = V/N. We
assume that the smallest dimensional distance across the container is
very much larger than v^(1/3).

a) Pick a point in the container at random and consider the sphere
centered on the point picked whose radius is equal to the distance
from that point to the nearest gas particle. This sphere is the
largest sphere centered on the picked point that contains no particles
in its interior. What is the mean volume of this sphere in units of
v? What is this sphere's mean radius in units of v^(1/3)?

Let random variable X == sphere volume, and let random variable
R == sphere radius. Then the requested means are:

<X> = v
<R> = 0.5539602784*v^(1/3)

(BTW, the exact numerical coefficient above for <R> is really
GAMMA(4/3)*(3/(4*[pi]))^(1/3) .)

b) What is the mean nearest neighbor interparticle distance for this gas?

<R> = 0.5539602784 v^(1/3) (the same as in part a))

What is the mean volume of the largest sphere centered on a randomly
chosen gas particle that contains only that particle in its interior?

<X> = v (the same as in part a))

c) Suppose the constraint in part a) that there be no particles
(strictly) inside the sphere was changed to the case that there are
now exactly n particles in the interior of the sphere. Now what is
the distribution of the volume of the largest sphere with this
condition?

Probability density p(X) = ((X/v)^n)*exp(-X/v)/(n!*v)

d) Suppose part c) was changed so that the sphere was centered on a
randomly chosen gas particle. What is the distribution of volume for
the largest sphere that contains exactly n other particles in its
interior other than the one at the center?

Same answer as part c)

e) If this problem was generalized to a generic D-dimensional volume
container how would your answers to parts c) and d) be different?

No difference. Same answer as in parts c) & d) in *any*
dimensionality.

The specific value D of the problem's dimensionality does not affect the
(hyper)sphere's "volume" distribution. And it only affects the the
distribution of the (hyper)sphere's radius distribution merely
*indirectly* through a relatively trivial change-of-variable
transformation involving the geometric R^D dependence of the general
formula for a (hyper)sphere's "volume" in terms of its radius (i.e.
"V" = (([pi]^(D/2))/(D/2)!)*R^D ).

Note that since we have an ideal gas the particle volume is entirely
negligible compared to v, and the distribution of particle locations in
space is uniform with each particle's location independent of the
locations of all the other particles present.

Actually, the volume distribution for this problem is not only
independent of the container's dimensionality, it is also independent of
the container's "shape" *and* is independent of the "shape" of the region
of that volume (formerly a sphere above) chosen that is to have exactly n
particles in it--as long as that chosen region's location is picked at
random (but being fully contained inside the container).

David Bowman
David_Bowman@georgetowncollege.edu