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The problem is simple: I have discs of diameter $d$ and I am placing them randomly on a plane, area A. I place them at $n$ discs per unit area.

What is the average separation of the discs if they are placed randomly? Assume they can't overlap, only touch on the edges.

My first thought was to say that the discs all have areas of $\pi r^2$ and the ratio of the area of the discs to the area of the plane will be $n \pi r^2$ divided by A.

So if the length of the pane on one side is L then the average distance you go without hitting a disc is related to that.

Then I looked up the mean free path calculations, and saw that a mean free path is $l = n\sigma$ where $\sigma$ is the area of the particle involved that's zooming through a gas. Applied to this problem that would tell me the distance a random disc entering the plane from the edge will go without hitting another disc, but it won't tell me the average separation between them.

So I was wondering if anyone could help out with this. In three dimensions I see with gas laws that if you assume a molecule is a hard sphere taking up a volume $d^3$ then the average separation is the cube root of the average volume occupied per molecule. So I wold assume something similar applies here, with the average separation being proportional to the square root of the average area occupied by a disc.

Is that intuition correct?

Jesse
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  • Does not look like an easy problem! To get a starting estimate I would work the problem backwards. Assume a uniform placement of disks in square or hexagonal packing and get the relationship between area and distances. Next I would do numerical testing with random numbers and see what seems to be a normal range of answers. Interesting problem though. – Maesumi May 13 '14 at 02:19
  • The tricky part is "Assume they can't overlap". As soon as you say that, the disks are no longer independently uniformly distributed on the plane, and it is not clear what the distribution actually is. But if $d$ is extremely small, we can get an approximation from the average separation between uniformly distributed points on a plane. –  May 13 '14 at 02:30
  • How would you define average distance of disks? – Maesumi May 13 '14 at 02:38
  • To answer my question: Find minimum distance of each disk from all others and then average these distances! – Maesumi May 13 '14 at 02:44
  • but wouldn't the minimum be the diameter of the disk? Then the average would just be that, which doesn't make a lot of sense to me. Unless you mean average the minimum and maximum... – Jesse May 13 '14 at 03:25

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