# The probability distribution for the number of pairwise distances $\leq$ some threshold for points uniformly placed in a sphere

If I place place $N$ particles in a sphere of radius $R$, selecting positions across the sphere's volume with uniform probability, what is the exact probability distribution for the number of pairwise distances between points less than or equal to some distance $d$? Is an exact distribution known for rectangular or cylindrical boundary conditions?

Please note that I am looking for an exact probability distribution provided spherical boundary conditions. One can of course come up with a "good enough" approximation for some physical system, perhaps with the aid of simulation. But I'd like to understand how, if possible, one can arrived at a closed form expression for the distribution.

Update: To ask the question in a different way, what's the probability distribution for the number of particles that have a lowerbound nearest-neighbor distance $\geq d$?

• To get an exact distribution for large $N$, or even for which $d$ the probability that no points are within distance $d$ is greater than $0$, you would need to solve messy sphere packing problems. I don't see any hope of solving all of these sphere packing problems at once. I think $N=2$ is known, though. – Douglas Zare May 13 '13 at 2:50
• @Douglas Zare I appreciate that the problem would be intractable if there was an exclusion radius around each particle, but it wasn't so clear to me that it would be impossible without this restriction. However, I think I see your point... Are there any relevant papers you know of for this system? – LHoward May 13 '13 at 3:32
• I don't know of such a paper offhand, but I suggest searching for "two-point correlation function." – Douglas Zare May 13 '13 at 3:54
• @Douglas Zare Thanks, I'll look into it! – LHoward May 13 '13 at 4:03