How many particles must a two-dimensional dusty plasma have to appear infinite?

A complex (dusty) plasma disk (CPD) is a two-dimensional system of $n$ particles interacting through a shielded Coulomb potential with Debye length $\lambda$ and confined in an isotropic parabolic well. The emergence of macroscopic behavior in a CPD is studied by considering the dependence of the breathing frequency $\omega_{{\rm br}}$ on $n$, $\lambda$, the disk radius $R_{0}$, and the nearest neighbor distance $a$. An approximate analytical expression for $\omega_{{\rm br}}$ is derived for $R_{0}\gg\lambda$ with $a/\lambda$ finite. In the plasma regime $a < \lambda$, so that each particle interacts with many other particles, $\omega_{{\rm br}}^{2}\approx4$ independent of $n$. In the ``condensed-matter'' regime $a > \lambda$, nearest-neighbor interactions dominate and $\omega_{{\rm br}}^{2}\sim a/\lambda$. Exact solutions for $n=100$ to 3200 particles approach the unbounded-plasma limit as $n$ increases. Solutions with $n=3200$ and $a/\lambda$ between 0.25 and 0.5 are found to provide the best approximation to an infinite plasma.