Discontinuous percolation transitions in real physical systems

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Discontinuous percolation transitions in real physical systems

Y. S. Cho and B. Kahng

Phys. Rev. E 84, 050102(R) – Published 23 November 2011

Abstract

We study discontinuous percolation transitions (PTs) in the diffusion-limited cluster aggregation model of the sol-gel transition as an example of real physical systems, in which the number of aggregation events is regarded as the number of bonds occupied in the system. When particles are Brownian, in which cluster velocity depends on cluster size as $v_{s} \sim s^{η}$ with $η = - 0.5$ , a larger cluster has less probability to collide with other clusters because of its smaller mobility. Thus, the cluster is effectively more suppressed in growth of its size. Then the giant cluster size increases drastically by merging those suppressed clusters near the percolation threshold, exhibiting a discontinuous PT. We also study the tricritical behavior by controlling the parameter $η$ , and the tricritical point is determined by introducing an asymmetric Smoluchowski equation.

Received 21 March 2011

DOI:https://doi.org/10.1103/PhysRevE.84.050102

Authors & Affiliations

Y. S. Cho and B. Kahng

Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea

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Vol. 84, Iss. 5 — November 2011

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Images

Figure 1
(a) Plot of the giant cluster size $G$ vs $t$ when particles are Brownian. The giant cluster grows continuously from $t = 0$ . (b) Plot of $G$ vs $p$ . $G$ grows drastically near $p_{f} = 1 - 1 / N$ . Inset: Plot of the relationship between $t$ and $p$ . $t$ increases drastically but in a power-law manner as $p$ approaches $p_{f}$ . Simulations are carried out with $N = 8000$ monoparticles at $t = 0$ on $400 \times 400$ square lattices.Reuse & Permissions
Figure 2
(a) Plot of $G_{N}$ vs $p$ for different particle numbers, $N = 50 000$ ( $□$ ), $N = 98 000$ ( $\circ$ ), $N = 162 000$ ( $▵$ ), $N = 242 000$ ( $▽$ ), and $N = 338 000$ ( $⋄$ ) at a fixed density $ρ = 0.05$ . $G_{N}$ begins to increase drastically near $p \approx 1$ and thus the data are not distinguishable for different sizes in a region that is far smaller than $p = 1$ . (b) Zoomed-in plot of $G_{N}$ vs $p$ near $p_{c}$ . As the system size grows, the giant cluster grows more drastically. (c) Plot of $d G_{N} (p) / d p$ calculated at $p_{c} (N)$ vs $N$ . The slope $d G_{N} (p_{c}) / d p$ increases as $N^{0.86 \pm 0.02}$ , indicating that it diverges in the thermodynamic limit. (d) Plot of $G$ vs $\bar{p} \equiv (p - p_{d}) d G_{N} (p_{c}) / d p$ for different $N$ . $p_{d} (N)$ is the $p$ intercept of the tangent of the curve $G_{N} (p)$ at $p_{c}$ . (e) Plot of $χ_{1} / N^{0.95}$ vs $\bar{p}$ . (f) Plot of $χ_{2} / N$ vs $\bar{p}$ . For (d), (e), and (f), the data collapse well onto a single curve.Reuse & Permissions
Figure 3
Snapshots of the system for various values of $η$ at $p = 0.99$ . The velocity of each cluster is given as $v_{s} \propto s^{η}$ , where $s$ is the cluster size. Numerical simulations are carried out for $N = 8000$ particles on $L \times L = 400 \times 400$ square lattices. Since $p$ is fixed, the number of clusters for each case is equal. The giant cluster is represented in a different color (gray/orange). The cluster-size distribution becomes more heterogeneous as $η$ increases.Reuse & Permissions
Figure 4
(a) Numerical estimations of $k_{i}^{'} / C^{'}$ at $p_{d}$ for $η = - 0.5$ $(□)$ , $η = 0$ $(\circ)$ , $η = 0.4$ $(▵)$ , and $η = 0.8$ ( $\nabla$ ). Simulations are carried out for $N = 8000$ and $L = 400$ . Slopes of the guidelines are $- 0.23 \pm 0.02$ for ( $η = - 0.5$ ), $0.32 \pm 0.04$ ( $η = 0$ ), $0.62 \pm 0.01$ ( $η = 0.4$ ), and $0.88 \pm 0.01$ ( $η = 0.8$ ). (b) Numerical estimations of the $k_{i} / C$ at $p_{d}$ . The same symbols are used as in (a). The system size is $N = 8000$ and $L = 400$ . Slopes of the guidelines are $0.35 \pm 0.04$ ( $η = - 0.5$ ), $0.32 \pm 0.04$ ( $η = 0$ ), $0.2 \pm 0.01$ ( $η = 0.4$ ), and $0.05 \pm 0.005$ ( $η = 0.8$ ).Reuse & Permissions
Figure 5
Plot of $G_{N}$ vs $p$ for the Smoluchowski equation with $k_{i} \sim 1$ and $k_{j}^{'} \sim j^{0.8}$ (a), and $k_{i} \sim 1$ and $k_{j}^{'} \sim j^{1.5}$ (b). As the system size grows, $G_{N}$ grows more drastically and $p_{c} (N)$ approaches one in (a), but it decreases to $p_{c} (\infty) > 0$ in (b). For (b), the transition is continuous. A similar plot for the DLCA with $η = 0.8$ (c) and $η = 1.5$ (d). Similar behaviors are shown. Simulations are performed for $N / 10^{3} = 8$ , 32, 72, 128, and 200. Inset of (d): Finite-size scaling of $G_{N} (p)$ for different system sizes with $1 / ν = 0.28$ , $β / ν = 0.5$ , and $p_{c} (\infty) = 0.42$ . These numerical values depend on $η$ . Thus, for $η > η_{c}$ , the critical points for different $η$ form a critical line.Reuse & Permissions

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