Visualize a large cubic room, say 100 feet on a side. 1,000 ping pong balls
will be removed from an urn and placed in the room via random drawings from
a uniform distribution for the x, y, and z position. The balls will be held
stationary in their assigned position by means of magic. Once all the balls
are in place compute the distance between each ball and its nearest
neighbor. Take the mean of all those 1,000 computations. *That* is what I
want to be able to derive analytically. That number may be called mean free
distance, it may not. But the proper name to affix to the number is not
what I am in search of. I can write a Monte Carlo simulation but an
analytic solution seems possible.
If the balls are small enough, their size is immaterial, the only rule is
that the only one ball can occupy one space. Assume infinitesimally small
balls if you wish. If the room is big enough boundary conditions at the
walls can safely be ignored.
This was triggered by a statement I read that, on average, if galaxies were
represented by a tennis ball, the nearest neighbor to a ball would be 50
feet away. I have tried to distill this down to get to what I see as the
essence of the problem. I have no interest in pursuing the *actual* method
used or the estimates *actually* used in formulating that statement. I am
for now only interested in the mathematics of the sterilized situation.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at
http://jse.stat.ncsu.edu/
=================================================================
