Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-11-27
Physics
Condensed Matter
Statistical Mechanics
1 Manuscript File and 14 figures have been included in a *.zip file
Scientific paper
This work analyzes the distribution and size of interparticle gaps arising in an ensemble of hexagonal unit structures in the xy plane when packing disks with a Gaussian distribution of radii with mean (r) and standard deviation $\Delta r$. During the course of this investigation an equivalency is established between gaps arising in hexagonal unit structure packs and nine-ball billiard rack patterns. An analytic expression is derived for the probability distribution and location of interparticle gaps of magnitude $\Gamma$. Due to the number of variables and large number of possible arrangements, a Monte Carlo simulation has been conducted to complement and probe the analytic form for three very different systems: i) billiard balls with Billiard Congress of America (BCA) specifications, ii) US pennies with specifications of the US Mint, and iii) a hypothetical system with $r = 1.0 m$ and $\Delta r = 1x10^{-10}$ m corresponding to the scale of one atomic radius. In each case, probability density distributions of gap sizes have been calculated for those $\Delta r$ above, and also for 2$\Delta r$ and 0.5$\Delta r$, respectively. A general result is presented for the probability of a nonzero normalized ($\frac{\Gamma}{\Delta r}$) gap size arising, $P(\Gamma \geq \alpha \Delta r)= 1-0.124\alpha$, where $\alpha$ is a constant $\leq 5.0$. This curious result reflects the phenomenon of geometric frustration; the inability of the system to simultaneously satisfy all geometric constraints required by a perfect-rack sans interparticle gaps.
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