复杂网络上混合自旋Ising模型动力学特性的研究

发布时间：2018-02-16 05:31

本文关键词： 复杂网络混合自旋Ising模型蒙特卡罗模拟相变损伤扩散　出处：《陕西师范大学》2016年博士论文　论文类型：学位论文

【摘要】：近年来,复杂网络引起了科学家的广泛关注,已经成为包括数学、力学、物理学、计算机、生命科学、管理科学、系统科学、社会学、金融和经济学等许多科学领域的研究热点。复杂网络上的动力学或物理状态的演化是一个重要研究领域,而复杂网络上自旋系统的相变行为研究是一个具有重要意义的方向。如果给复杂网络的节点赋予某种自旋状态,给连边赋予某种耦合或相互作用就可以建立复杂网络上的自旋系统,这类自旋模型可以用于刻画诸如复杂网络上疾病的传播、谣言的扩散等动力学、以及社会学的问题。因此,复杂网络上自旋系统相变行为的研究具有重要意义,是服务于上述应用的基础。本论文采用蒙特卡罗Metropolis抽样方法,研究了小世界网络、小世界Sierpinski垫片(small world Sierpinski gasket-SWSG)网络上的混合自旋Ising模型的相变特性,并探究了 Sierpinski垫片型晶格上损伤扩散的动力学行为。取得的主要成果如下:1、采用数值方法研究了一维NW小世界网络上混合自旋Ising模型相变的行为,结果表明,对任意随机加边概率,该网络上的混合自旋Ising系统存在连续相变。随机加边概率影响系统的临界温度,系统的相变温度和随机加边概率之间呈幂律关系。小世界网络上的混合自旋Ising模型具有平均场特性,其相变的临界指数为α = 0,β= 1/2,γ = 1,v = 2和小世界网络上的Ising模型属于同一普适类。系统的相变温度受晶格场影响,随着晶格场的逐渐增强,系统相变温度会连续减小到零。2、发现Sierpinski垫片上混合自旋Ising模型存在损伤扩散的相变现象。在转变温度以下,系统损伤不扩散,当温度高于转变温度时,系统损伤愈合。晶格场改变系统的转变温度,随着晶格场的增强,系统中自旋取0的概率增加,从而改变系统的关联程度,进而使系统损伤的转变温度降低。当晶格场足够强时,系统的损伤消失。数值模拟损伤扩散的弛豫时间揭示出系统静态临界指数Z不再是常量,Z的值是系统温度和晶格场的函数。对于给定晶格场,Z的值随温度的增加而逐渐减小;而当系统温度给定时,Z的值随晶格场的增强而减小,二者之间成线性关系。3、小世界Sierpinski垫片网络是在传统Sierpinski垫片网格上随机加边而构建,具有小世界性、自相似性、无标度性的拓扑特性。基于该网络上混合自旋Ising系统的蒙特卡罗模拟,我们计算了磁化率、比热和四阶矩。结果显示:加边条数直接影响SWSG网络上系统热力学量的特性,但不能导致系统发生有限温度相变。对系统损伤扩散的动力学研究表明,系统弛豫时间和系统尺寸间不再是简单的幂律关系,而是指数关系。系统静态动力学指数Z的值受加边的影响而不再是常量。
[Abstract]:In recent years, complex networks have attracted the attention of scientists. They include mathematics, mechanics, physics, computer, life science, management science, systems science, sociology, etc. The evolution of dynamics or physical states on complex networks is an important research field. The study of the phase transition behavior of spin systems on complex networks is an important direction. If the nodes of complex networks are given some spin states, A spin system on a complex network can be established by giving some coupling or interaction to the connected edges, which can be used to characterize dynamics such as disease spread on complex networks, rumor diffusion, and sociological problems. It is of great significance to study the phase transition behavior of spin systems on complex networks, which is the basis of the above applications. In this paper, Monte Carlo Metropolis sampling method is used to study small-world networks. The phase transition characteristics of the mixed spin Ising model on small world Sierpinski gasket-SWSGs networks with small world Sierpinski gasket-SWSGs, The dynamic behavior of damage diffusion on Sierpinski gasket lattice is investigated. The main results are as follows: 1. The phase transition behavior of mixed spin Ising model on one-dimensional NW small-world network is studied by numerical method. The results show that, For arbitrary random edge addition probability, the mixed spin Ising system on the network has a continuous phase transition. The random edge addition probability affects the critical temperature of the system. There is a power law relationship between the temperature of phase transition and the probability of random edge addition. The mixed spin Ising model on a small-world network has an average field characteristic. The critical exponents of phase transition are 伪 = 0, 尾 = 1 / 2, 纬 = 1V = 2 and the Ising model on small-world networks belong to the same universal class. The temperature of phase transition of the system is affected by the lattice field and increases gradually with the lattice field. The phase transition temperature of the system decreases continuously to zero. 2. It is found that there is a phase transition phenomenon in the mixed spin Ising model on the Sierpinski gasket. Below the transition temperature, the system damage does not diffuse, and when the temperature is higher than the transition temperature, the system damage does not diffuse. The system damage heals. The lattice field changes the transition temperature of the system. With the increase of the lattice field, the probability of spin 0 in the system increases, thus changing the correlation degree of the system. Then the transition temperature of system damage is reduced. When the lattice field is strong enough, The relaxation time of numerical simulation shows that the static critical exponent Z is no longer a constant value, which is a function of system temperature and lattice field. For a given lattice field, the value of Z decreases with the increase of temperature. However, when the system temperature is given, the value of Z decreases with the increase of lattice field, and the relationship between them is linear. The small-world Sierpinski gasket network is built on the traditional Sierpinski gasket mesh with random edge addition, which is small worldwide and self-similar. Based on the Monte Carlo simulation of the mixed spin Ising system on the network, the magnetic susceptibility, specific heat and fourth order moments are calculated. The results show that the number of edge-added bars directly affects the thermodynamic properties of the system on the SWSG network. But it can not lead to the finite temperature phase transition of the system. The dynamic study of system damage diffusion shows that the system relaxation time and system size are no longer a simple power law relationship. The value of the index Z of system static dynamics is influenced by the addition of edges and is no longer a constant.
【学位授予单位】：陕西师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.5

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