Jamming transition of kinetically constrained models in rectangular systems

We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: For long and wide channels it decreases algebraically $v_c\sim W^{-1/2}$ with the system's width $W$, while in square systems it decreases logarithmically $v_c\sim 1/\ln L$ with length $L$. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as $W\approx 10$.

Figure 1

The difference between frozen and unfrozen particles in the Kob-Andersen model: White square are vacancies. Light-gray particles can move in this initial configuration. Dark-gray particles cannot move now, but after some other particle(s) move, they too are mobile. Black particles are permanently frozen and will never move.

Figure 2

Division into sections, represented by different colors: Columns $1–2$, $3–9$, and $10–15$. The strips are columns $1–2$, $7–9$, and $14–15$.

Figure 3

Average fraction $\frac{〈N〉}{n}$ of frozen particles in sections of length $l$ in the KA model with hard-wall boundary conditions for different widths and densities: $W=4$, $\rho =0.8$ (blue squares), $W=10$, $\rho =0.89$ (purple circles), $W=20$, $\rho =0.93$ (yellow triangles), $W=40$, $\rho =0.95$ (green stars). Continuous lines are approximations (6), with $\lambda$ taken from simulations of long rectangles ($L=200W$).

Figure 4

The effective $\lambda$ as a function of $W/L$ for hard-wall (a) and periodic (b) boundary conditions. In this range, ${\lambda }_{\text{eff}}$ does not change much with the length $L$, and converges as $L\to \infty$ to a finite value for each width, $W$. For each system size, ${\lambda }_{\text{eff}}$ for the KA model is higher than in the FA model, and it is higher with hard-wall boundary conditions than with periodic boundary conditions, which means that $n_{\text{PF}}^{\text{KA}}>n_{\text{PF}}^{\text{FA}}$ and $n_{\text{PF}}^{\text{hard}\text{-wall}}>n_{\text{PF}}^{\text{periodic}}$, as expected. At $L\to \infty$, $\lambda$ is almost the same for $W=20$ and 100 but different for $W=10$.

Figure 5

The effective $\lambda$ as a function of the system's width $W$ for hard-wall (a) and periodic boundary conditions (b). The plots show the value of ${\lambda }_{\text{eff}}$ at $L=200W$ (for $W\le 100$), $L=500W$ (for $W=120$), $L=2000W$ (for $W=150$), and $L=5000W$ (for $W=200$), which is large enough to be considered infinite in this range of widths, and at $L=W$. For $L\gg W$ the effective $\lambda$ has a minimum at $W\approx 50$, but there is no drastic change in its value in the range $20\le W\le 100$. Even at $W=200$ the relative difference from the minimum is $0.08$. ${\lambda }_{\text{eff}}$ for long tunnels ($L\gg W$) and squares ($L=W$) is almost the same at $60\le W\le 100$. We expect that at $W\to \infty$, $\lambda$ will converge to ${\pi }^2/18\approx 0.54$.

Figure 6

Fraction of frozen particles $n_{\text{PF}}$ vs density $\rho$ for hard-wall (left) and periodic (right) boundary conditions and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of ${\lambda }_{\text{eff}}$. As the system's size increases, the approximation becomes better and the value of ${\lambda }_{\text{eff}}$ becomes dependent mostly on the model, and not on the system's size. $\lambda$ was set to ${\lambda }_{\text{eff}}(W,L)$, as shown in Fig. 4.

Figure 7

Ratio between critical vacancy density in rectangles of dimension $W\times L$ and critical vacancy density in squares of dimension $L\times L$ in the KA model with hard-wall boundary conditions. Symbols are results of numerical simulations and continuous line is the approximations (17) and (19). The ratio between the critical densities significantly differs from 1 only for $W<W_c\left(L\right)$. $W_c/L$ decreases with $L$, and the ratio between the critical densities increases with $L$.

Figure 8

The critical vacancy density, $v_c$, as a function of the width for very long systems [$L=200W\gg L_c\left(W\right)$] with hard-wall boundaries. The symbols are the results from the numerical simulations and the continuous lines are the approximation [Eq. (19)] with $\lambda ={\lambda }_{\text{eff}}(100,100)$.

Figure 9

Fraction of frozen particles vs aspect ratio $r=W/L$ at constant area $WL={10}^4$ and particle density $\rho =0.92$ with hard-wall boundaries. Symbols are results of numerical simulations for KA (blue squares) and FA (purple circles) models. Continuous lines are analytical approximations with ${\lambda }_{\text{eff}}(100,100)\approx 0.271$ (KA), $0.252$ (FA).

Figure 10

Critical vacancy density (a) and transition width (b) with hard-wall boundaries. Symbols are the same in both panels. In panel (a), full lines are the approximations (17) and (19) for large $W$. In panel (b), full lines are the results from the full analytical expression and the dashed lines are the large-$W$ approximations.

Figure 11

Fraction of frozen particles vs. density for the FA (full symbols) and KA (empty symbols) models, for hard-wall (squares) and periodic (circles) boundary conditions, for $W=1,2$. Symbols are results of numerical simulations with $L=200W$, continuous line is analytical expression. For $W=2$ the results for KA with periodic boundaries and FA with hard-wall boundaries are almost the same.

Figure 12

Fraction of frozen particles vs density for $W=3,4,5,6$ with hard-wall boundaries. Dashed lines are result of the approximation of Sec. 2 for the KA (higher curve) and FA (lower curve) models. Continuous lines are results from the approximation of Sec. 4c for the KA (higher curve) and FA (lower curve) models. Full (KA) and empty (FA) squares are numerical results. Numerical simulations were done with $L=200W$. Analytical results are for $L=\infty$. As $W$ increases, larger sections should be included. For $W=3,4$, the current approximation is better than the approximation of Sec. 2, but for $W\ge 5$, longer sections are required.

Figure 13

(a) Density of sections which are completely frozen or completely unfrozen, hard-wall boundaries. The minimum for $W=40$ is much lower than for $W=7$ because there is a larger chance of a small, inconsequential unfrozen island. (b) The distribution of the fraction of frozen particles per section for $W=7$ and $\rho =0.9$ and for $W=40$ and $\rho =0.95$ [near the minima in panel (a)]. Most sections are either completely unfrozen or almost completely frozen.

Figure 14

Snapshots of (a) $100\times 2000$ system at $\rho =0.957$, (b) $7\times 140$ system at $\rho =0.85$. Both with hard-wall boundaries. Black squares are particles which are frozen in both models, dark gray are frozen only in the KA model, light gray are unfrozen in both models, and white squares are vacancies. Panel (a) contains two sections, which are both almost completely frozen for KA, but only one of them is frozen for FA. Small scattered islands are also visible. In panel (b) there are 11 sections, showing many different behaviors. Zooming in on the pictures allows to see the details. In print in the $100\times 2000$ system, what appears brightest are the light-gray areas, since the individual white sites are too small to be seen.

Figure 15

(a) Probability that a frozen particle belongs to a certain row at $W=3$. At some width-dependent density, the probability of a frozen particle to be in the extreme rows is maximal. (b) The ratio between the maximum for the extreme rows and the minimum for the middle row(s) as a function of $W$. This is maximal at $W=12$.