Noh Jae Dong, Lee Hyun Keun, Park Hyunggyu
Department of Physics, University of Seoul, Seoul 130-743, Republic of Korea and School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jul;84(1 Pt 1):010101. doi: 10.1103/PhysRevE.84.010101. Epub 2011 Jul 20.
We investigate a critical scaling law for the cluster heterogeneity H in site and bond percolations in d-dimensional lattices with d = 2,...,6. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability p increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that H diverges algebraically, approaching the percolation critical point p(c) as H |p-p(c)|(-1/σ) with the critical exponent σ associated with the characteristic cluster size. Interestingly, its finite-size-scaling behavior is governed by a new exponent ν H = 1+d (f)/(d)ν, where d(f) is the fractal dimension of the critical percolating cluster and ν is the correlation length exponent. The corresponding scaling variable defines a singular path to the critical point. All results are confirmed by numerical simulations.
我们研究了二维到六维晶格中，位点渗流和键渗流中簇异质性(H)的临界标度律。簇异质性定义为不同簇大小的数量。随着占据概率(p)增加，在亚临界相中，簇大小分布从单分散分布演变为多分散分布，而在超临界相中又回到单分散分布。我们通过分析表明，(H)呈代数发散，当(H\sim|p - p(c)|^{-\frac{1}{\sigma}})时趋近于渗流临界点(p(c))，其中临界指数(\sigma)与特征簇大小相关。有趣的是，其有限尺寸标度行为由新指数(\nu_H = 1 + \frac{d_f}{d}\nu)控制，其中(d_f)是临界渗流簇的分形维数，(\nu)是关联长度指数。相应的标度变量定义了一条通向临界点的奇异路径。所有结果均通过数值模拟得到证实。
