Connecting the random organization transition and jamming within a unifying model system

While the random organization transition describes the change from reversible to irreversible dynamics in a nonequilibrium system, the athermal jamming transition at zero shear rate occurs when particles can no longer avoid overlaps. Despite the obvious differences between these two transitions, we show that they both occur within the same model packing problem. In this unifying model system the particles are first randomly distributed and then displaced in each step if they overlap. For random displacements we obtain a random organization transition, while jamming occurs in the case of deterministic shifts. We also analyze the critical behavior of random organization. Our results show that random organization and jamming are opposite limits of random sphere packings, and we expect that various equilibrium and nonequilibrium transitions can be formulated as related intermediate packing problems.

Figure 1

(a,b) In our model system overlapping particles are displaced either (a) in a random direction or (b) deterministically along their connecting line. In both cases the length of the displacement is chosen such that the particles are touching but no longer overlapping after being displaced. (c) Sketch of the original random organization model where repeatedly a shear with shear amplitude $A$ and subsequently a reverse shear is applied. Particles are displaced if they collide. (d) Collisions occur if the center of mass of one particle lies within a collision region of another particle (area shaded in light red). (e) A similar region (area shaded in light red) denoting overlaps, which can be constructed in our model. Here we have drawn elliptical particles that lead to the same transition as circles, because the system can be stretched in any direction without changing the transition properties.

Figure 2

Fraction of active particles $N_a/N$ as a function of the number of steps $t$ for various packing fractions $\varphi$. Particles are displaced either (a,c) in random directions or (b,d) deterministically when they overlap. In the main figure of (a,b) ten runs with $N=50000$ particles are considered in (b,d) ten runs with $N=10000$ each. (a,b) Results for a two-dimensional system leading to (a) a random organization transition or (b) showing the approach to a hexagonally close packed state. (c,d) Three-dimensional system with (c) a random organization transition and (d) jamming. The black solid lines in (a,c) mark the power law behavior at the transition packing fraction. The insets in (a,c) address the typical finite size behavior for selected packing fractions. The insets in (b,d) show the step numbers $\tau$ needed until the system becomes inactive as a function of $\varphi$. A Vogel-Fulcher-Tammann and an empirical fit are depicted (see text for details).

Figure 3

(a,c) Fraction of active particles left in the long time limit $N_a/N$ as a function of packing fraction $\varphi$ for the random organization system in (a) two dimensions and (c) three dimensions. For packing fractions above the random organization transition packing fractions ${\varphi }_{\text{RO}}$ a power law $N_a/N\propto {(\varphi -{\varphi }_{\text{RO}})}^{\beta }$ is fitted. The insets show log-log plots of the power law fits. (b,d) Relaxation times $\tau$ obtained from the results of Figs. 2 and 2 as a function of the packing fraction $\varphi$ in (b) two dimensions and (d) three dimensions. The dotted blue line is a fit according to $\tau =A_j{|\varphi -{\varphi }_{\text{RO}}|}^{-\nu }$, where $A_j=A_1$ for $\varphi <{\varphi }_{\text{RO}}$ and $A_j=A_2$ for $\varphi >{\varphi }_{\text{RO}}$. The insets show log-log plots of the relaxation time $\tau$ divided by the fitting prefactors $A_j$ depending on $|\varphi -{\varphi }_{\text{RO}}|$. The data for $\varphi <{\varphi }_{\text{RO}}$ are depicted by magenta circles, and the data for $\varphi >{\varphi }_{\text{RO}}$ are depicted by red squares.

Figure 4

Analysis of the critical behavior of the random organization transition of square particles with fixed orientation in two dimensions. (a) Fraction of active particles left in the long time limit as function of the packing fraction. For packing fractions above the random organization transition packing fractions ${\varphi }_{\text{RO}}^{\left(sq\right)}$ a power law $N_a/N\propto {(\varphi -{\varphi }_{\text{RO}}^{\left(sq\right)})}^{\beta }$ is fitted. (b) Relaxation times $\tau$ as a function of the packing fraction $\varphi$. The line denotes a fit according to $\tau =A|\varphi -{\varphi }_{\text{RO}}^{\left(sq\right)}{|}^{-\nu }$. We employed $N=1000$ squares. Due to finite size effects, no relaxation times for $\varphi <{\varphi }_{\text{RO}}^{\left(sq\right)}$ were determined. The insets show log-log plots. For further details see the corresponding results for circular particles [Figs. 2, 3, and 3].