Research On Phase Transition Behaviors Of Ising Model And Percolation Model

The studies of the phase transition in Ising model and percolation model are presented in this dissertation.On the basis of plentiful literatures about Ising model phase transition and network percolation transition,the physical background,practical significance and the research status and trend of these two models are summarized comprehensively.Then,the phase transition properties of three-dimensional Ising model with plaquette four-spin and tetrahedron four-spin on a simple cubic are studied by probability theory and numerical methods.The partition function of the system is calculated by using correlation approximation.Then the qualitative and quantitative properties of the phase transition of the three-dimensional Ising model in the thermodynamic limit are analyzed carefully.The error sources in the mean field calculation are analyzed strictly by using the correlation approximation on a finite lattice system.The Ising model on graphene structure with four-spins interaction in external fields is analyzed by correlation fucntions The field-induced and temperature-induced phase transitions in the model are discussed numerically.Secondly,a fast algorithm for fragmentation of tree network is designed,thus the evolution and fragmentation of Achlioptas type on random tree networks and specific tree networks are discussed successfully.Finally,the percolation properties of graphene structure with pentagon-heptagon defect at any concentration are studied numerically.A statistical method for determining the critical threshold is successfully proposed.Further,the effect of defect concentration on percolation phase transition is analyzed.The innovative research results on phase transition in Ising model,tree network and graphene structure system are obtained in present study.The paper has important theoretical value for error analysis of mean field calculation method,provides a powerful experimental method for scholars to study tree network by numerical analysis,reduces the influence of scale effect on the results,and completely solves the effect of defect concentration on percolation transition of graphene system.The research results have certain reference value for understanding and studying the physical properties of two dimensional materials.This study has important scientific significance and practical value for the progress of phase transition research in theory and practice1.In this investigation,the correlation analysis in probability theory is successfully applied to the calculation of partition function of Ising model with four-spins interactions. For the first time,the sources of errors in mean field calculation are systematically analyzed.It is found that the inappropriate limit approximation is the main error source of the mean field calculation,which provides important guidance for the accurate analysis in the future.The critical points in discontinuous phase transition of Ising model with plaquette and tetrahedron four-spins are analytically given.The influence of the four-spins interaction intensity on the phase transition behavior is discussed qualitatively and quantitatively,and the phase transition properties are obtained.The phase transition behavior of a four-spins Ising model on an ideal graphene structure system is also discussed in an external field.The effect of the change of the spin-to-spin interaction intensity in Ising model on the ferromagnetic properties of the system is quantitatively analyzed,and the phase transition types are discussed and classified successfully,which has significance for the further study of the properties of graphene materials2.By designing a labeling method of tree network,a fast algorithm for fragmentation of tree network is successfully presented.The algorithm minimizes the influence of finite scale effect on the results of research in numerical analysis method,and provides a good computer simulation tool for discussing the phase transition behavior of tree network in the thermodynamic limit.The system size calculated with this method is much larger than that calculated with depth-first search or width-first search.3.A simple method to determine the critical threshold according to the components size distribution is presented.Two phenomena of percolation transition,namely,the Achlioptas evolution process and the Achlioptas fragmentation process of tree,have very different percolaiton transition behavior,which is the first time that has been discovered The Achlioptas process on a specific tree evolution model is discontinuous phase transition while the fragmentation process of Achlioptas on a random tree is continuous.It is found that random selection,preference selection of edges and selection of evolutionary objects have significant effects on the robustness(stability)of tree networks4.The percolation characteristics of graphene structure with pentagon-heptagon defect at any concentration are studied theoretically for the first time.The correctness and applicability of the proposed model and method are verified by comparing with the existing analytical results.By using this method,the percolation threshold of the thermodynamic limit of graphene structure with arbitrary defect concentration can be successfully obtained by simulating graphene system with finite sizes.It is found that the defect concentration had little effect on the percolation properties.By using the statistical sampling technique proposed in this paper,the system size computed on it in Monte Carlo sampling can be greatly improved.This method can also be applied to other types of lattice system.