Geometric Frustration and Interparticle Gap Size Distributions in Ordered Hexagonal Polydisperse Disk Packs
| dc.creator | Snowman, Daniel P. | |
| dc.date | 2007-11-27 | |
| dc.date.accessioned | 2026-07-07T08:45:45Z | |
| dc.date.available | 2026-07-07T08:45:45Z | |
| dc.description | 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 $Δr$. 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 $Γ$. 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 $Δr = 1x10^{-10}$ m corresponding to the scale of one atomic radius. In each case, probability density distributions of gap sizes have been calculated for those $Δr$ above, and also for 2$Δr$ and 0.5$Δr$, respectively. A general result is presented for the probability of a nonzero normalized ($\fracΓ{Δr}$) gap size arising, $P(Γ\geq αΔr)= 1-0.124α$, where $α$ 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. | |
| dc.description | 1 Manuscript File and 14 figures have been included in a *.zip file | |
| dc.identifier | https://arxiv.org/abs/0711.4381 | |
| dc.identifier | http://arxiv.org/abs/0711.4381 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/143062 | |
| dc.subject | Statistical Mechanics | |
| dc.title | Geometric Frustration and Interparticle Gap Size Distributions in Ordered Hexagonal Polydisperse Disk Packs | |
| dc.type | text |