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基于高效的单团簇生长算法, 采用蒙特卡罗方法模拟了考虑最近邻、次近邻, 直至第五近邻格点的二维正方格子的键渗流. 计算得到了二十余种格点模型高精度的键渗流阈值, 并深入探讨了渗流阈值
$p_{\rm c}$ 与格点结构之间的关联. 通过引入参数$\xi = \displaystyle\sum\nolimits_{i} z_{i} r_{i}^{2} / i$ (其中$z_{i}$ 和$r_{i}$ 分别为第i近邻格点的配位数及到中心格点的距离)来消除“简并”, 研究发现$p_{\rm c}$ 随ξ的变化较好地满足幂律关系$p_{\rm c} \propto \xi^{-\gamma}$ , 数值拟合得$\gamma \approx 1$ .Based on an effective single cluster growth algorithm, bond percolation on square lattice with the nearest neighbors, the next nearest neighbors, up to the 5th nearest neighbors are investigated by Monte Carlo simulations. The bond percolation thresholds for more than 20 lattices are deduced, and the correlations between percolation threshold$p_{\rm c}$ and lattice structures are discussed in depth. By introducing the index$\xi = \displaystyle\sum\nolimits_{i} z_{i} r_{i}^{2} / i$ to remove the degeneracy, it is found that the thresholds follow a power law$p_{\rm c} \propto \xi^{-\gamma}$ , with$\gamma \approx 1$ , where$z_{i}$ is the ith neighborhood coordination number, and$r_{i}$ is the distance between sites in the i-th coordination zone and the central site.-
Keywords:
- percolation /
- single cluster growth algorithm /
- Monte Carlo simulation
[1] Broadbent S R, Hammersley J M 1957 Math. Proc. Cambridge Phil. Soc. 53 629Google Scholar
[2] Stauffer D, Aharony A 1994 Introduction to Percolation Theory (Boca Raton: CRC Press)
[3] 韩伟涛, 伊鹏 2019 物理学报 68 078902Google Scholar
Han W T, Yi P 2019 Acta Phys. Sin. 68 078902Google Scholar
[4] 李乐, 李克非 2015 物理学报 64 136402Google Scholar
Li L, Li K F 2015 Acta Phys. Sin. 64 136402Google Scholar
[5] 王小娟, 宋梅, 郭世泽, 杨子龙 2015 物理学报 64 044502Google Scholar
Wang X J, Song M, Guo S Z, Yang Z L 2015 Acta Phys. Sin. 64 044502Google Scholar
[6] 李炎, 唐刚, 宋丽建, 寻之朋, 夏辉, 郝大鹏 2013 物理学报 62 046401Google Scholar
Li Y, Tang G, Song L J, Xun Z P, Xia H, Hao D P 2013 Acta Phys. Sin. 62 046401Google Scholar
[7] Koza Z, Kondrat G, Suszczynski K 2014 J. Stat. Mech.: Th. Exp. 2014 P11005Google Scholar
[8] Koza Z, Pola J 2016 J. Stat. Mech.: Th. Exp. 2016 103206Google Scholar
[9] Kleinberg J M 2000 Nature 406 845Google Scholar
[10] Sander L M, Warren C P, Sokolov I M 2003 Physica A 325 1Google Scholar
[11] Ziff R M 2021 Physica A 568 125723Google Scholar
[12] Domb C 1972 Biometrika 59 209Google Scholar
[13] Dalton N W, Domb C, Sykes M F 1964 Proc. Phys. Soc. 83 496Google Scholar
[14] Domb C, Dalton N W 1966 Proc. Phys. Soc. 89 859Google Scholar
[15] Gouker M, Family F 1983 Phys. Rev. B 28 1449Google Scholar
[16] Jerauld G R, Scriven L E, Davis H T 1984 J. Phys. C: Solid State 17 3429Google Scholar
[17] Gawron T R, Cieplak M 1991 Acta Phys. Pol. A 80 461Google Scholar
[18] d'Iribarne C, Rasigni G, Rasigni M 1995 Phys. Lett. A 209 95Google Scholar
[19] d'Iribarne C, Rasigni M, Rasigni G 1999 J. Phys. A: Math. Gen. 32 2611Google Scholar
[20] d'Iribarne C, Rasigni M, Rasigni G 1999 Phys. Lett. A 263 65Google Scholar
[21] Malarz K, Galam S 2005 Phys. Rev. E 71 016125Google Scholar
[22] Majewski M, Malarz K 2007 Acta Phys. Pol. B 38 2191
[23] Kurzawski K, Malarz K 2012 Rep. Math. Phys. 70 163Google Scholar
[24] Malarz K 2015 Phys. Rev. E 91 043301Google Scholar
[25] Kotwica M, Gronek P, Malarz K 2019 Int. J. Mod. Phys. C 30 1950055Google Scholar
[26] Malarz K 2020 Chaos 30 123123Google Scholar
[27] Ouyang Y, Deng Y J, Blote H W J 2018 Phys. Rev. E 98 062101Google Scholar
[28] Deng Y J, Ouyang Y, Blote H W J 2019 J. Phys.: Conf. Ser. 1163 012001Google Scholar
[29] Xun Z P, Ziff R M 2020 Phys. Rev. Research 2 013067Google Scholar
[30] Xun Z P, Ziff R M 2020 Phys. Rev. E 102 012102Google Scholar
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[32] van der Marck S C 1998 Int. J. Mod. Phys. C 9 529Google Scholar
[33] Xun Z P, Hao D P, Ziff R M 2021 Phys. Rev. E 103 022126Google Scholar
[34] Frei S, Perkins E 2016 Electron. J. Probab. 21 1Google Scholar
[35] Xu W H, Wang J F, Hu H, Deng Y J 2021 Phys. Rev. E 103 022127Google Scholar
[36] Piec S, Malarz K, Kulakowski K 2005 Int. J. Mod. Phys. C 16 1527Google Scholar
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图 2 SQ-1, 2格子键渗流在不同占据概率p下
$s^{\tau-2}P_{\geqslant s}$ 随$ s^{\sigma} $ 的变化曲线, 其中$ \tau = 187/91 $ ,$ \sigma = 36/91 $ . 插图表示主图中所示结果线性部分的斜率随占据概率p的变化关系Fig. 2. Plot of
$s^{\tau-2}P_{\geqslant s}$ vs.$ s^{\sigma} $ with$ \tau = 187/91 $ and$ \sigma = 36/91 $ for the bond percolation of the$\text{SQ-}1, 2$ lattice under different values of p. The inset indicates the slope of the linear portions of the curves shown in the main figure as a function of p.图 3 SQ-1, 2格子键渗流在不同占据概率p下
$s^{\tau-2}P_{\geqslant s}$ 随$s^{-\varOmega}$ 的变化曲线, 其中$ \tau = 187/91 $ ,$\varOmega = 72/91$ Fig. 3. Plot of
$s^{\tau-2}P_{\geqslant s}$ vs.$s^{-\varOmega}$ with$ \tau = 187/91 $ and$ \varOmega = 72/91 $ for the bond percolation of the$\text{SQ-}1, 2$ lattice under different values of p.表 1 含复杂近邻格点的二维正方格子的键渗流阈值
Table 1. Bond percolation thresholds of square lattice with complex neighborhoods.
格点模型 总配
位数 z标量
参数ξ键渗流阈值 $p_{\rm c}$ SQ-1, 2,
SQ-2, 58 8 0.2503683(7) $\text{SQ-}1, 3$ 8 9.33 0.2214989(9) $\text{SQ-}1, 5$ 8 10.4 0.1972557(13) $\text{SQ-}4$ 8 10 0.1937380(10) SQ-1, 2, 3,
SQ-2, 3, 512 13.33 0.1522201(9) $\text{SQ-}1, 2, 5$ 12 14.4 0.1380527(7) $\text{SQ-}1, 4$ 12 14 0.1362105(5) $\text{SQ-}2, 4$ 12 14 0.1345500(10) $\text{SQ-}1, 3, 5$ 12 15.73 0.1342972(8) $\text{SQ-}3, 4$ 12 15.33 0.1309686(14) $\text{SQ-}4, 5$ 12 16.4 0.1247135(15) $\text{SQ-}1, 2, 4$ 16 18 0.1059928(8) $\text{SQ-}1, 2, 3, 5$ 16 19.73 0.1032173(7) $\text{SQ-}1, 3, 4$ 16 19.33 0.1027026(6) $\text{SQ-}2, 3, 4$ 16 19.33 0.1011488(8) $\text{SQ-}1, 4, 5$ 16 20.4 0.0978026(14) $\text{SQ-}2, 4, 5$ 16 20.4 0.0967349(11) $\text{SQ-}3, 4, 5$ 16 21.73 0.0954613(7) $\text{SQ-}1, 2, 3, 4$ 20 23.33 0.0841507(7) $\text{SQ-}1, 2, 4, 5$ 20 24.4 0.0804649(9) $\text{SQ-}1, 3, 4, 5$ 20 25.73 0.0790839(9) $\text{SQ-}2, 3, 4, 5$ 20 25.73 0.0780764(6) $\text{SQ-}1, 2, 3, 4, 5$ 24 29.73 0.0671855(5) 表 2 正方格子不同近邻格点的相关参数
Table 2. Parameters of different nearest neighbors on square lattice.
第 i 近邻 距中心格点
距离的平方 $r_{i}^{2}$第i近邻
格点数 $z_{i}$总配位数 z 1 1 4 4 2 2 4 8 3 4 4 12 4 5 8 20 5 8 4 24 -
[1] Broadbent S R, Hammersley J M 1957 Math. Proc. Cambridge Phil. Soc. 53 629Google Scholar
[2] Stauffer D, Aharony A 1994 Introduction to Percolation Theory (Boca Raton: CRC Press)
[3] 韩伟涛, 伊鹏 2019 物理学报 68 078902Google Scholar
Han W T, Yi P 2019 Acta Phys. Sin. 68 078902Google Scholar
[4] 李乐, 李克非 2015 物理学报 64 136402Google Scholar
Li L, Li K F 2015 Acta Phys. Sin. 64 136402Google Scholar
[5] 王小娟, 宋梅, 郭世泽, 杨子龙 2015 物理学报 64 044502Google Scholar
Wang X J, Song M, Guo S Z, Yang Z L 2015 Acta Phys. Sin. 64 044502Google Scholar
[6] 李炎, 唐刚, 宋丽建, 寻之朋, 夏辉, 郝大鹏 2013 物理学报 62 046401Google Scholar
Li Y, Tang G, Song L J, Xun Z P, Xia H, Hao D P 2013 Acta Phys. Sin. 62 046401Google Scholar
[7] Koza Z, Kondrat G, Suszczynski K 2014 J. Stat. Mech.: Th. Exp. 2014 P11005Google Scholar
[8] Koza Z, Pola J 2016 J. Stat. Mech.: Th. Exp. 2016 103206Google Scholar
[9] Kleinberg J M 2000 Nature 406 845Google Scholar
[10] Sander L M, Warren C P, Sokolov I M 2003 Physica A 325 1Google Scholar
[11] Ziff R M 2021 Physica A 568 125723Google Scholar
[12] Domb C 1972 Biometrika 59 209Google Scholar
[13] Dalton N W, Domb C, Sykes M F 1964 Proc. Phys. Soc. 83 496Google Scholar
[14] Domb C, Dalton N W 1966 Proc. Phys. Soc. 89 859Google Scholar
[15] Gouker M, Family F 1983 Phys. Rev. B 28 1449Google Scholar
[16] Jerauld G R, Scriven L E, Davis H T 1984 J. Phys. C: Solid State 17 3429Google Scholar
[17] Gawron T R, Cieplak M 1991 Acta Phys. Pol. A 80 461Google Scholar
[18] d'Iribarne C, Rasigni G, Rasigni M 1995 Phys. Lett. A 209 95Google Scholar
[19] d'Iribarne C, Rasigni M, Rasigni G 1999 J. Phys. A: Math. Gen. 32 2611Google Scholar
[20] d'Iribarne C, Rasigni M, Rasigni G 1999 Phys. Lett. A 263 65Google Scholar
[21] Malarz K, Galam S 2005 Phys. Rev. E 71 016125Google Scholar
[22] Majewski M, Malarz K 2007 Acta Phys. Pol. B 38 2191
[23] Kurzawski K, Malarz K 2012 Rep. Math. Phys. 70 163Google Scholar
[24] Malarz K 2015 Phys. Rev. E 91 043301Google Scholar
[25] Kotwica M, Gronek P, Malarz K 2019 Int. J. Mod. Phys. C 30 1950055Google Scholar
[26] Malarz K 2020 Chaos 30 123123Google Scholar
[27] Ouyang Y, Deng Y J, Blote H W J 2018 Phys. Rev. E 98 062101Google Scholar
[28] Deng Y J, Ouyang Y, Blote H W J 2019 J. Phys.: Conf. Ser. 1163 012001Google Scholar
[29] Xun Z P, Ziff R M 2020 Phys. Rev. Research 2 013067Google Scholar
[30] Xun Z P, Ziff R M 2020 Phys. Rev. E 102 012102Google Scholar
[31] Galam S, Mauger A 1996 Phys. Rev. E 53 2177Google Scholar
[32] van der Marck S C 1998 Int. J. Mod. Phys. C 9 529Google Scholar
[33] Xun Z P, Hao D P, Ziff R M 2021 Phys. Rev. E 103 022126Google Scholar
[34] Frei S, Perkins E 2016 Electron. J. Probab. 21 1Google Scholar
[35] Xu W H, Wang J F, Hu H, Deng Y J 2021 Phys. Rev. E 103 022127Google Scholar
[36] Piec S, Malarz K, Kulakowski K 2005 Int. J. Mod. Phys. C 16 1527Google Scholar
[37] Gutman I 1994 J. Chem. Inf. Comp. Sci. 34 1087Google Scholar
[38] Schultz H P 1989 J. Chem. Inf. Comp. Sci. 29 227Google Scholar
[39] Malarz K 2021 Phys. Rev. E 103 052107Google Scholar
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