Malarz Krzysztof, Galam Serge
AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, PL-30059 Kraków, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jan;71(1 Pt 2):016125. doi: 10.1103/PhysRevE.71.016125. Epub 2005 Jan 18.
We report site percolation thresholds for square lattice with neighbor bonds at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (NN), next-nearest neighbors (NNN), next-next-nearest neighbors (4N), and fifth-nearest neighbors (6N) yield the same pc = 0.592... . The fourth-nearest neighbors (5N) give pc = 0.298... . This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with (NN+NNN), (NN+4N), (NN+NNN+4N), and (NN+5N). The calculated associated thresholds are respectively pc = 0.407..., 0.337..., 0.288..., and 0.234... . The existing Galam-Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.
我们报告了在不同增加范围内具有相邻键的方形晶格的位点渗流阈值。使用蒙特卡罗技术，我们发现最近邻（NN）、次近邻（NNN）、次次近邻（4N）和第五近邻（6N）产生相同的渗流阈值(p_c = 0.592...)。第四近邻（5N）给出的渗流阈值(p_c = 0.298...)。利用对称性论证证明了这种相等性在数学上是精确的。然后我们考虑各种邻域的组合，如（NN + NNN）、（NN + 4N）、（NN + NNN + 4N）和（NN + 5N）。计算得到的相关阈值分别为(p_c = 0.407...)、(0.337...)、(0.288...)和(0.234...)。现有的加拉姆 - 莫热渗流阈值通用公式无法重现这些数据，这表明维度和配位数不足以构建一个扩展到复杂晶格的通用定律。
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