【作者】 石少波；

【导师】 何文辰；

【作者基本信息】 河北工业大学 ， 理论物理， 2004， 硕士

【摘要】 相变和临界现象是凝聚态物理学和统计物理学中十分活跃和重要的研究领域。 分形概念的引入揭开了相变研究的新篇章。分形是具有自相似对称性的几何图形，可用来模拟自然界中在一定尺度范围内具有自相似对称性的不规则结构，如Koch曲线可用来模拟海岸线，渗流模型可模拟金属绝缘体混合物，自回避无规行走模型可模拟线性聚合物，岩裂模型可模拟真实岩体的脆性破裂等。 重整化群方法已被证明是一种研究该类问题较为有效的方法。这种方法的依据是：系统发生相变时，对临界行为起作用的不是小尺度行为，而是大尺度行为(关联长度无穷大)。因而，系统的一些微观细节(小尺度行为)变得不重要。该方法假定在标度变换下系统的结构不变，讨论在长度标度l与λl下系统结构的关系。它回避了直接求配分函数，而代之以研究使配分函数保持不变的变换，这些变换构成所谓重整化群。然后找出重整化变换的不动点，在所有不动点中那些不稳定不动点是发生相变的临界点。实空间重整化群方法与分形有密切的关系，在不具有哈密顿的几何相变系统，如渗流，岩裂，自回避无规行走等模型广泛地被应用。 本文利用实空间重整化群方法对渗流、岩裂、飞蚁模型进行了研究。我们的主要工作如下： 1．重整化群方法处理渗流模型时，“导通”规则的选取至关重要，我们定义一个描述“导通”规则的系数(简称规则系数)。在此基础上，给出一个经验公式72/n+3/8(Z-6)=R，由此可通过格子的配位数来确定立方格子点渗流模型重整化渗流、岩裂、飞蚁模型的实空间重整化群方法研究群方法研究中“导通”规则选取方式。依据该原则选取“导通”规则，分析kadanoff元胞“导通”情况，得出三种格子(SC、BCC、户℃C)不动点方程，从而解出临界值p。和临界指数v。接着，我们就简立方格子点渗流模型引入“鬼”场，求得了全部的临界指数。由普适性理论知，这些指数应该是三维近邻点渗流模型的临界指数。另外，我们采用位置空间重整化群方法，对二维次近邻三角格子键渗流模型进行了研究。类似与渗流模型，就岩裂模型我们提出岩裂规则系数的概念。在此基础上，对该模型进行了研究。得到一些有趣的结论。我们提出了一种新的自回避行走模型(飞蚁模型)，用重整化群方法计算了该模型的临界值和分形维数分别为Kc=0.545069、D=0.814909。接着，和真实自回避行走(TSAw)模型的结果相比较说明了所得结果的合理性。最后，得出结论并就本文以后的工作做了一些展望。更多还原

【Abstract】 Phase transition and critical phenomena in condensed matter physics and statistical physics is a quite important field of inquiry.The introduction of fractal conception opens a new chapter of the study of phase transition. The fractal is a geometrical figure with self-similar symmetry, and it is an important tool for characterizing irregular structures in nature that are self-similar on certain length scales. For example, the Koch curves can be viewed as a mathematical model for coastlines, percolation model can be used to mimic mixture of mental and insulator, self-avoiding walks can serve as a model for linear polymers, and rock fracture mode can stimulate the process of brittle fracture of rock, and so on.In the study of these problems, the technique of real space renormalization group is proved to a comparatively powerful means. The gist of this method is that the behavior, which contributes to critical phenomena, is not small scaling one, but big scaling one (i.e. the correlative length is infinite) when phase transition happens. Thus some microscopic details (small scaling behavior) become unimportant. This theory is based on a model for how the structure on one length scale l is related to that on another scale M, with the implicit assumption that the structure is invariant under the renormalization group scale transformation T(X). This way seems to avoid partition function, but study the transformation that makespartition function unchanged. These transformations are made up of renormalization group. Then the fixed point of transformation can be found, among which those unstable ones are critical points of phase transition. The real-space renormalization group approach is close to fractal and is widely used in geometric phase transition systems without Hamilton, for example, percolation model, rock fracture model, flit ant model.The models of percolation, rock fracture and flit ant are studied on the real-space renormalization group approach. Some researches have been done as follows:1. In the study of percolation model, a coefficient, which is used to describe "conduction" rule, is defined. On base of a experience formula, fit "conduction" rules of percolation on cubic lattices are chosen, Using these rules, we analyze the states of Kadanoff cells and obtain fixed equations. The results (The threshold pc and the critical exponent v) basically agree with known results. By "ghost" field .The percolation threshold pc and the critical exponents α , β ,γ , ν, δ , η are obtained. These values are the critical exponents of three-dimension site-lattice. We study the two-dimension triangular-bond lattice percolation with next-nearest-neighbor interactions on the renormalization group approach as well.2. Similar to percolation, the critical rock fracture model is established by renormalization group theory approach, and the relation between the fracture rules and the critical probability, and the fractal dimension, and the critical exponents is studied.3. A new TSAW model are proposed, we use the real space renormalization group approach to treat the model on square lattice. The threshold Kc and the fractal dimension D are obtained respectively. Comparison of the new model with the TSAW, the results show that the model and the calculation are reasonable.In the end, the main points of this thesis and the prospects for this investigation were shown.更多还原