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Connecting N points pairwise in volume V, average density of lines?

  1. Aug 26, 2012 #1
    Say we take N random points in a volume V and connect the points pairwise with line-segments. I would like to estimate the number of segments that intersect some small volume v, and where N is large enough so that any small random sample volume v will have many intersections. Little volume v may or may not enclose any points.

    Does this get me close? First let us estimate the total length of the line segments, Ʃ. Use an average separation distance D between each pair of points. The distance D is of order one half the length of the volume V, V = L^3, D = L/2.

    The total number of segments is N(N-1)/2 so an estimate for the length of line-segments,

    Ʃ = D*N(N-1)/2 for large N this is about D*N^2/2
    Ʃ ≈ D*N^2/2

    Assume this total length is evenly divided into each small volume v. The length in volume v is the fraction [d^3/D^3] times Ʃ,

    [d^3/D^3]*Ʃ = [d^3/D^3]*D*N^2/2 = σ

    Assume the average length of the line-segments that intersect the little volume v is one-half the length the little volume v, d/2.

    Then the average number of line-segments in v, ω, is,

    ω = σ/[d/2] = {[d^3/D^3]*D*N^2/2}/[d/2] = d^2*N^2/D^2

    using Wolfram calculator,

    http://www.wolframalpha.com/

    Using our Universe as an example, let d = 1m, N = 10^80, D = [3.5*10^80m^3]^.3333 ≈ 7*10^26

    ω = 2*10^106 segments intersecting a volume of 1m^3.

    We can ask what must the size of the volume v above be so that on average there will be only one line-segment intersecting it.

    Set ω = 1 = d^2*N^2/D^2 now d is unknown and we use N and D above,

    d = D/N = 7*10^26/10^80 = 7*10^-54m. If we are too near a point this estimate is bad. If we enclose a point ω jumps by about N

    We can also ask how many points must a volume v have so that ω above changes significantly because of the additional line-segments from the enclosed points.

    I made many bad estimates but I think I'm within a factor of a billion above?

    Thanks for any help!
     
  2. jcsd
  3. Aug 29, 2012 #2
    This is an interesting problem! I wonder, are the endpoints important, or do they just serve as a way of defining some random lines? Would it serve your purpose equally well to have some number of lines (the infinite kind) randomly distributed throughout your space?
     
  4. Aug 29, 2012 #3
    I had something in mind for the endpoints. If you have N charged particles in some volume then the electrstatic potential energy is given by the sum of the potential energy of each pair of charged particles. And for N particles with mass the gravitational potential energy (Newtonian Gravity) is given by a similar pairwise sum. In either case the length of the line segments is important. So if there were some unseen connection between each pair of particles in our universe I was curious just how dense would those connections be. Just an idea. It seems for the numbers I used if there were some type of unseen connections they would have to be very dense. I also had the idea that disturbances of these connections might be gravitons and photons. Spinors would have to pop out of such an idea and I don't know if that is possible.

    After some thought I realized that for N random points in a box that the density of line segments would fall off near the inside boundary of the box. Things could be more uniform in some closed space like S^3 but then a pair of points in S^3 gives two (or more) geodesic paths between the points? Maybe you include both paths?

    So do you think I'm in the ballpark with those numbers?
     
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