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A graphical performance comparison of the Philleo, the Pleau and Pigeon, and
the Lu and Torquato estimates of the paste-void proximity probability density
function is shown in Fig. 7. The
sphere radii are lognormally distributed with a number density of
240 mm^{−3}. The Philleo estimate terminates at *s* equals zero since
it is already normalized for the fraction of paste within *s* of an
air void surface. The Lu and Torquato estimate is virtually exact
at the resolution of this experiment. The Philleo estimate is fairly accurate
for *s* greater than zero, while that of
Pleau and Pigeon is noticeably in error.
These qualitative differences are born out in the previously reported
results.

**Figure 7:** Estimates of the paste-void probability density function by
Philleo, Pleau and Pigeon, and Lu and Torquato for lognormally
distributed sphere radii with a number density of 240 mm^{−3}.
Measured values are shown as filled circles.