Random distribution of spheres

clamtrox

Hey guys, I need to fill up a box with uniformly distributed set of non-overlapping spheres. This is quite easy to do numerically. I was wondering what is expectation value for the asymptotic volume fraction of the spheres.

Suppose I have a big box with side L, and spheres with radius R<<L. I pick a random point x inside the box, and add it to my collection of spheres if |x-x_n|>R for all spheres already in the collection. I can keep on doing this until there's no room in the box to add another sphere; suppose that leaves me with N spheres. What is [itex] \frac{4\pi R^3}{3 L^3} E(N) [/itex] ?

0xDEADBEEF

This problem is called the "parking lot test" for random numbers. I am sure you will find the answer by googeling for it.