本文關(guān)鍵詞： 復(fù)雜網(wǎng)絡(luò) 混合自旋Ising模型 蒙特卡羅模擬 相變 損傷擴(kuò)散　出處：《陜西師范大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：近年來(lái),復(fù)雜網(wǎng)絡(luò)引起了科學(xué)家的廣泛關(guān)注,已經(jīng)成為包括數(shù)學(xué)、力學(xué)、物理學(xué)、計(jì)算機(jī)、生命科學(xué)、管理科學(xué)、系統(tǒng)科學(xué)、社會(huì)學(xué)、金融和經(jīng)濟(jì)學(xué)等許多科學(xué)領(lǐng)域的研究熱點(diǎn)。復(fù)雜網(wǎng)絡(luò)上的動(dòng)力學(xué)或物理狀態(tài)的演化是一個(gè)重要研究領(lǐng)域,而復(fù)雜網(wǎng)絡(luò)上自旋系統(tǒng)的相變行為研究是一個(gè)具有重要意義的方向。如果給復(fù)雜網(wǎng)絡(luò)的節(jié)點(diǎn)賦予某種自旋狀態(tài),給連邊賦予某種耦合或相互作用就可以建立復(fù)雜網(wǎng)絡(luò)上的自旋系統(tǒng),這類自旋模型可以用于刻畫諸如復(fù)雜網(wǎng)絡(luò)上疾病的傳播、謠言的擴(kuò)散等動(dòng)力學(xué)、以及社會(huì)學(xué)的問(wèn)題。因此,復(fù)雜網(wǎng)絡(luò)上自旋系統(tǒng)相變行為的研究具有重要意義,是服務(wù)于上述應(yīng)用的基礎(chǔ)。本論文采用蒙特卡羅Metropolis抽樣方法,研究了小世界網(wǎng)絡(luò)、小世界Sierpinski墊片(small world Sierpinski gasket-SWSG)網(wǎng)絡(luò)上的混合自旋Ising模型的相變特性,并探究了 Sierpinski墊片型晶格上損傷擴(kuò)散的動(dòng)力學(xué)行為。取得的主要成果如下:1、采用數(shù)值方法研究了一維NW小世界網(wǎng)絡(luò)上混合自旋Ising模型相變的行為,結(jié)果表明,對(duì)任意隨機(jī)加邊概率,該網(wǎng)絡(luò)上的混合自旋Ising系統(tǒng)存在連續(xù)相變。隨機(jī)加邊概率影響系統(tǒng)的臨界溫度,系統(tǒng)的相變溫度和隨機(jī)加邊概率之間呈冪律關(guān)系。小世界網(wǎng)絡(luò)上的混合自旋Ising模型具有平均場(chǎng)特性,其相變的臨界指數(shù)為α = 0,β= 1/2,γ = 1,v = 2和小世界網(wǎng)絡(luò)上的Ising模型屬于同一普適類。系統(tǒng)的相變溫度受晶格場(chǎng)影響,隨著晶格場(chǎng)的逐漸增強(qiáng),系統(tǒng)相變溫度會(huì)連續(xù)減小到零。2、發(fā)現(xiàn)Sierpinski墊片上混合自旋Ising模型存在損傷擴(kuò)散的相變現(xiàn)象。在轉(zhuǎn)變溫度以下,系統(tǒng)損傷不擴(kuò)散,當(dāng)溫度高于轉(zhuǎn)變溫度時(shí),系統(tǒng)損傷愈合。晶格場(chǎng)改變系統(tǒng)的轉(zhuǎn)變溫度,隨著晶格場(chǎng)的增強(qiáng),系統(tǒng)中自旋取0的概率增加,從而改變系統(tǒng)的關(guān)聯(lián)程度,進(jìn)而使系統(tǒng)損傷的轉(zhuǎn)變溫度降低。當(dāng)晶格場(chǎng)足夠強(qiáng)時(shí),系統(tǒng)的損傷消失。數(shù)值模擬損傷擴(kuò)散的弛豫時(shí)間揭示出系統(tǒng)靜態(tài)臨界指數(shù)Z不再是常量,Z的值是系統(tǒng)溫度和晶格場(chǎng)的函數(shù)。對(duì)于給定晶格場(chǎng),Z的值隨溫度的增加而逐漸減小;而當(dāng)系統(tǒng)溫度給定時(shí),Z的值隨晶格場(chǎng)的增強(qiáng)而減小,二者之間成線性關(guān)系。3、小世界Sierpinski墊片網(wǎng)絡(luò)是在傳統(tǒng)Sierpinski墊片網(wǎng)格上隨機(jī)加邊而構(gòu)建,具有小世界性、自相似性、無(wú)標(biāo)度性的拓?fù)涮匦�。基于該網(wǎng)絡(luò)上混合自旋Ising系統(tǒng)的蒙特卡羅模擬,我們計(jì)算了磁化率、比熱和四階矩。結(jié)果顯示:加邊條數(shù)直接影響SWSG網(wǎng)絡(luò)上系統(tǒng)熱力學(xué)量的特性,但不能導(dǎo)致系統(tǒng)發(fā)生有限溫度相變。對(duì)系統(tǒng)損傷擴(kuò)散的動(dòng)力學(xué)研究表明,系統(tǒng)弛豫時(shí)間和系統(tǒng)尺寸間不再是簡(jiǎn)單的冪律關(guān)系,而是指數(shù)關(guān)系。系統(tǒng)靜態(tài)動(dòng)力學(xué)指數(shù)Z的值受加邊的影響而不再是常量。
[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 偽 = 0, 尾 = 1 / 2, 緯 = 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.
【學(xué)位授予單位】：陜西師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O157.5

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