Numerical Simulation And Application Of Critical Phenomena In Nature

Phase transition and critical phenomenon are the important fields in sophisticated science. Phase transition is a mutation change, which indicates the system is changed from disorder state into order state and the other way round. It is mainly classified into equilibrium phase transition and nonequilibrium phase transition. The former includes gas-liquid phase transition, paramagnetic-ferromagnetic phase transition. The latter includes laser and Benard phenomenon. A lot of remarkable research findings have been accomplished since Andrews discovered the critical point. Based on the previous researches, this paper presents the Ising Model and its strict solution applied to the research of paramagnetic and ferromagnetic phase transition in equilibrium phase transition, the three methods dealt with phase transition and critical phenomenon, the theory of average field proposed by L. Landau, and the combined method of renormalization group and scale conversion pioneered by K. Wilson as well as the method of numerical simulation. Firstly, this paper uses the method of renormalization group to deal with the Ising Model of equilibrium phase transition and calculate and compare the critical index of the model of square cell with different lattices, which leads to the finding that the size of Kadanoff cell have greater influence on the approximate treatment of interactions between cells. If the smaller cell is chosen, the effect of primary cumulation expansion is not perfect. If the bigger cell is chosen, the interactions between cells relatively decrease. However, with the increase of lattice, internal intrinsic coordinates increase, thus the amount of calculation will be greatly increased. The method of renormalization group is also used to deal with the nonequilibrium phase transition of the instability and failure of rocks, which results in the theoretical critical point of instability and failure of rock material and the evolve direction of rock stability. Compared with the experimental result, it is proved that the model of renormalization model can accurately simulate the instability and failure of rocks and effectively infer the failure critical point, establishing the standard for determining rock stability. Besides, the two numerical methods are adopted to simulate equilibrium and nonequilibrum phase transition. The Monte-Carlo method is used to simulate Ising model, which results in the law of magnetization change with temperature and the critical temperature. The laws of different initial spin system under different temperatures are also investigated. The domain distribution is obtained under low temperature. Finally, the cellular automaton is used to simulate the equilibrium phase transition of rock instability and failure, thus, to construct a "complete"cellular automaton model which may reflect comprehensively and effectively the various characteristics such as heterogeneity, anisotropy, energy transmission and dissipation, and stress jump during the evolution process of failure of rock material. The power law of energy transmission and occurrence number have been obtained, which prove that rock damage and failure have self-organized criticality.