Research On The Models Of Percolation,Rock Fracture And Flit On The Real-Space Renormalization Group Approach

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.