The Uniformity of Jammed Soft Disk Packings

Rattler-free jammed packings were conjectured by Torquato \& Stillinger to be hyperuniform, such that volume-fraction fluctuations across a set of $L^d$ measuring windows is $\sigma_\phi^2(L)\sim 1/L^{d+1}$. For simulations of bidisperse soft disks of average area $\langle a\rangle$, we thus propose to quantify the uniformity of the packings by the value of a hyperuniformity disorder length, $h_e$, defined by $\sigma_\phi^2(L)/\phi = 4 \langle a\rangle h_e/L^{3}$ and equal to the distance from the window boundary over which density fluctuations occur. Independent of system size, preparation protocol, and fraction of rattlers, we find $h_e=0.084\sqrt{\langle a\rangle}$, which is only 14\% larger (i.e. only 14\% less uniform) than for a triangular lattice of close-packed disks. However, for windows larger than a certain size $L_e$ we find liquid-like Poissonian fluctuations of $h(L)=(h_e/L_e)L$, as defined by $\sigma_\phi^2(L)/\phi = 4 \langle a\rangle h(L)/L^{3}\sim 1/L^2$. For a rapid quench protocol, the value is $L_e=65\sqrt{\langle a\rangle}$, independent of system size and fraction of rattlers. For slower quenches, $L_e$ increases and is the subject of current study.