【作者】 李丽；

【导师】 何文辰；

【作者基本信息】 河北工业大学 ， 理论物理， 2003， 硕士

【摘要】 重整化群方法是研究相变的临界现象最有力的工具，它是一种由理论求得分形维数的方法。这种方法从概念上看它回避直接求配分函数，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。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。实空间(位置空间)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，晶格动物，无规行走等广泛地被应用。 本论文从石墨的空间结构特点出发，说明了其层状结构，并且把石墨一层内的晶格即六角格子作为研究对象。首先从分形的定义以及六角格子的形成过程两个方面说明了其分形特性。然后根据对称性不变的原则，从整体晶格中选取结构单元，把结构单元和生长模型分别作为重整化变换前、后的图形来进行重整化变换，选取热力学函数易逸度为参量，写出了重整化变换前后的配分函数和重整化变换关系式，求出了这一变换的不动点。最后给出了六角格子发生相变时关联长度的临界指数和分形维数。并利用标度律求出了比热临界指数。其中在计算配分函数时所采用的P-V路方法，它最初用于化学方面研究苯型烃系统的开库勒结构。我们首次引入此方法，并把粒子在P-V路上的行为看作一种受约束条件的无规行走，行走的轨迹是一种无规分形，从而把P-V路方法与无规分形联系起来。我们虽然利用P-V路方法写出了重整化变换前后配分函数的形式，但并没有直接研究配分函数本身的性质，而采用了重整化群的思想，研究使配分函数保持不变的变换。 另外，在本论文中，我们所研究的粒子是在石墨一层内的晶格即六角格子上的行走，这就使本文的研究具有一定的物理背景，从而对石墨和苯型烃的研究工作具有一定的理论指导作用和意义。更多还原

【Abstract】 Renormalization group method which calculates fractal dimension theoretically is the most useful tool when we study critical phenomena of phase transition. This way seems to avoid partition function conceptively, but study the transformation which makes partition 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 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. The real-space (or position-space) renormalization group method is close to fractal and is widely used in geometric phase transition systems without Hamilton, for example, seepage, lattice animal and random walk.In this paper, beginning with the characteristic of space structure of graphite, we explain its layer structure and choose lattice in a layer which is called hexangular lattice as our studied object. At first, the fractal property of hexangular lattice is illustrated from two aspects which are the definition of fractal and the form of hexangular lattice. Then the structure unit of hexangular lattice is found according to the principle of invariable symmetry. Renormalization transformation is processed when we regard the structure unit and the growth model as graphs before and after transformation respectively. After choosing the11thermodynamic function fugacity as parameter, we can write out the partition functions before and after transformation and the formula of renormalization transformation. At the same time, we also work out the fixed points of transformation. At last, the critical exponent and fractal dimension of correlative length are calculated when phase transition happens. Using the scaling law, we know the critical exponent of specific heat too. The peak-valley path method in computing partition function is originally used in chemistry to study Kekulean structure of benzenoid hydrocarbon. We introduce it firstly. The behavior of particles on peak-valley path can be regarded as a random walk in some restrictive conditions, the track of which is a kind of random fractal. Thus peak-valley path method is connected with fractal. Though we know the forms of partition function before and after transformation, we will not study directly the property of partition function itself but the transformation which makes it unchanged with the idea of renormalization group.Otherwise, in this paper, the random walk of particles is processed on lattices of graphite in a layer i.e. hexangular lattices, which makes our study have some physical background. Thus our results have some instructional function theoretically on the study of graphite and benzenoid hydrocarbon.更多还原