Renormalization-group approach to an Abelian sandpile model on planar lattices.
Renormalization-group approach to an Abelian sandpile model on planar lattices.
复制标题
DOI：
10.1103/physreve.66.021307
复制
发表时间：
2002-04
期刊：
影响因子：
--
通讯作者：
Chai-Yu Lin;Chin-Kun Hu
中科院分区：
文献类型：
--
作者：
Chai-Yu Lin;Chin-Kun Hu
在晶格沙堆模型的重整化群（RG）方法中，一个重要的步骤是精确枚举在RG变换的单元内沙堆动力学的所有可能的倾倒过程。在此，我们提出了一种计算机算法，用于在Bak-Tang-Wiesenfeld沙堆模型的RG方法中对平面晶格的细胞进行这种精确枚举。并考虑由Pietronero， Vespignani和Zapperi （PVZ）提出的约化高RG方程[物理学。与Ivashkevich提出的实高度RG方程[物理学报，2002,19(1994)]。教化。76,3368(1996)]。使用该算法，我们可以在较大的单元格尺寸下更快地进行RG变换，例如PVZ RG方程中的SQ晶格为3x3单元格，这是目前最大的单元格尺寸，并且发现了先前论文[Phys. 5]中的一些错误。Rev. 51, 1711(1995)]。对于SQ和PT晶格，我们得到了每个晶格唯一吸引的不动点，并计算了雪崩指数tau和动态指数z。我们的结果表明，PVZ - RG变换中增加细胞大小并不能得到更准确的结果。讨论了这一结果的意义。
One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.