Random Close Packing as a Dynamical Phase Transition

Sphere packing is an ancient problem. The densest packing is known to be a face-centered cubic (FCC) crystal, with space-filling fraction ${\varphi }_{FCC}=\pi /\sqrt{18}\approx 0.74$. The densest “random packing,” random close packing (RCP), is yet ill defined, although many experiments and simulations agree on a value ${\varphi }_{RCP}\approx 0.64$. We introduce a simple absorbing-state model, biased random organization (BRO), which exhibits a Manna class dynamical phase transition between absorbing and active states that has as its densest critical point ${\varphi }_{c_{\max }}\approx 0.64\approx {\varphi }_{RCP}$ and, like other Manna class models, is hyperuniform at criticality. The configurations we obtain from BRO appear to be structurally identical to RCP configurations from other protocols. This leads us to conjecture that the highest-density absorbing state for an isotropic biased random organization model produces an ensemble of configurations that characterizes the state conventionally known as RCP.

Figure 1

The critical volume fraction ${\varphi }_c$ for 3D biased random organization (BRO) as a function of the displacement magnitude $\epsilon$. Traces correspond to different ratios of variance of repulsive to random displacements $\delta$, where $\delta =1$ (blue) consists of purely repulsive displacements, whereas $\delta =0$ (green) is random organization (RO) with all random displacements. For $\delta >0$, the small-$\epsilon$ limit for any fixed ratio of repulsive displacements ${\varphi }_c(\epsilon \to 0)\approx 0.640\pm 0.001\approx {\varphi }_{RCP}$.

Figure 2

Manna universal scaling of biased random organization. (a) The steady-state activity $f_a^{\infty }$ is plotted as a function of the reduced control parameter $(\varphi -{\varphi }_c)/{\varphi }_c$, showing a clear transition between absorbing $f_a^{\infty }=0$ and active $f_a^{\infty }>0$ states. The data for decreasing $\epsilon$ values of the BRO model are collapsed with the RO model $\delta =0$ (blue squares) and the Manna model (light blue diamonds). Collapse requires rescaling the reduced control parameter by ${\varphi }_c^{(-1/\beta )}$ and rescaling the activity by $A/\sqrt{\epsilon }$. (b) The same data from (a) are displayed using log-log plotting to show power-law scaling of $f_a^{\infty }$ above the critical point. All models show Manna/RO class scaling $f_a^{\infty }\sim (\varphi -{\varphi }_c{)}^{\beta }$, where $\beta =0.84$ in 3D. (c) For $\delta =1$ and epsilon corresponding to parts (a) and (b), the characteristic relaxation times $\tau$ from a Poisson random initial state to an absorbing state are plotted to show power-law divergence at ${\varphi }_c$.

Figure 3

Random close packing structure at ${\varphi }_c(\epsilon \to 0)$ for $\delta =1$. (a) The pair correlation function $g(r)$ is plotted for critical structures of BRO. All show an expected excluded volume region which corresponds to the particle size and a strong diverging peak at the nearest neighbor separation in the $\epsilon \to 0$ limit. Inset: The split in the second-nearest neighbor peak occurs at small epsilon ($\epsilon =0.025d$ is shown). The cusps at $r=\sqrt{3}d$ and $r=2d$ ($-$) are consistent with previous RCP studies. (b) A log-log plot of $g(r)$ at ${\varphi }_c$ as a function of $r/d-1$ emphasizes the power-law decay of $g(r)$ away from $r=d$. Small-$\epsilon$ structures at ${\varphi }_c$ show $g(r)\sim (r/d-1{)}^{-0.5}$ ($--$). (c) The average contact number $Z$ is counted for critical structures where $Z$ is the number of neighbors at a distance less than $d+x_{\mathrm{cut}}$ from a target particle. For each structure, $Z$ is undercounted if $x_{\mathrm{cut}}$ is too small, but as $\epsilon \to 0$, $Z$ approaches a plateau at the isostatic limit $Z=6$. The overcounted structures fit a form close to previous RCP calculations [33].

Figure 4

Random close packing is Manna class hyperuniform. (a) The angularly averaged structure factor $S(q)$ is plotted for BRO structures at ${\varphi }_c$. Different traces correspond to decreasing $\epsilon$ values where all critical structures are hyperuniform with Manna universal scaling $S(q\to 0)\sim q^{0.25\pm 0.01}$ for all $\epsilon$ values. (b) The critical structure factor $S(q)$ for three models of RCP: Lubachevsky-Stillinger (data from Donev et al.) [30], soft spheres, and BRO. The structure factors calculated from LS and soft spheres agree remarkably well with the $\epsilon \to 0$ limit of BRO, all three showing $S(q\to 0)\sim q^{\alpha }$. Manna class hyperuniform scaling is according to ${\alpha }_{LS}=0.24\pm 0.02$, ${\alpha }_{Soft}=0.27\pm 0.03$, and ${\alpha }_{BRO}=0.26\pm 0.02$. Inset: The relative difference structure factor $\mathrm{\Delta }S(q)={\int }_0^{5(2\pi /d)}|S(q,\varphi )/S(q,{\varphi }_c)-1|dq$ is a measure of $S(q)$ away from ${\varphi }_c$. Though all models agree at the critical point $\mathrm{\Delta }S\approx 0$, they differ on both sides of the transition.