Next: Void-Void Proximity Distribution Up: Lu and Torquato Previous: Lu and Torquato
The approach of Lu and Torquato was to derive the probability that a point chosen at random throughout the entire system would have no part of an air void within a distance s from it. The region of thickness s about the point constitutes a test sphere of radius s. This test sphere of radius s constitutes the Lu and Torquato "void". This void exclusion probability is given by 
with s<0 corresponding to a sphere with radius (−s) being entirely inside an air void. The averaged quantity in Eqn. 27 has the same definition as before:
Again, the quantity Θ(r + s) is the Heaviside function , and insures that the argument (s+r) remains positive.
This result can be recast into the air void problem. Since eV(s) represents the probability of a random point not being within a distance s of an air void surface, the probability of finding the nearest void surface within a distance s of a randomly chosen point is the complement of the void exclusion probability:
The probability of finding the nearest air void surface a distance s from a random point in the paste portion only is
This gives the fraction of the paste volume within a distance s of an air void surface, which is equivalent to the definition of the paste-void proximity cumulative distribution function.