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)?
b) What is the mean nearest neighbor interparticle distance for this gas?
What is the mean volume of the largest sphere centered on a randomly
chosen gas particle that contains only that particle in its interior?
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?
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?
e) If this problem was generalized to a generic D-dimensional volume
container how would your answers to parts c) and d) be different?
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.