计算机模拟 ppt课件

1、,1,计算机模拟在铁电物理研究中的应用,Monte Carlo Molecular Dynamics Cellular Automaton Phase Field Method,2,Monte Carlo Simulations,3,OUTLINES,Random number Sampling, Metropolis, Wolff algorithm Averaging Quantum Monte Carlo Micro-canonical Monte Carlo,4,Monte Carlo Method,Monaco Monte Carlo Casino Stochastic Comput

2、er Simulations Monte Carlo Simulations in other areas: Stock Market, City Traffic, Forest Fire, (epidemic), Nuclei Reaction, Crystal Growth etc.,5,6,TWO Theorems,Central limit,Large number,7,General Procedure of Monte Carlo Simulations,Generate random numbers, or pseudo-random numbers Sampling, algo

3、rithms Doing averaging, get physical quantities,8,Random number generator,The system supplied random number functions in Fortran or C is not good enough for MC simulations write random number generation routine takes times, and normally does not work properly copy routine from publications; download

4、 it from computer web-sites,9,Application in Statistical Physics,An observable A(x),macroscopic system N 1022 atoms; Monte Carlo accessible N 103 - 106 atoms, sampling,10,Metropolis sampling,Importance sampling,Weight from x to x, Markov chain,11,Ising model Hamiltonian,12,Single spin flip scheme,Ca

5、lculate Eold select a spin, flip it calculate Etrial E=Etrial-Eold if E0, take the state if E0, get W=exp(-E), If r W, accept the new state, otherwise retain the previous state, r is a random number,old,trial,13,General procedureof Metropolis sampling for Ising model,1. initial state: (a) random dis

6、tribution of spin-up or spin-down; (b) alignment up or down 2. chose a site: (a) randomly or (b) sequentially 3. flip the spin at the site, (single spin flip) 4. calculate the energy Etrial, and compute E= Etrial Eold, the energy difference,14,5. If 0E, accept the state, and go to (9), compute physi

7、cal quantities 6. If E is positive, compute W=exp(-E), 7. generate a random number r in unit interval 0,1 8 .If r W, accept the new state, otherwise retain the previous state 9. determine the value of the desired physical quantities, e.g. spin average etc,15,10. repeat (2)-(9) to obtain a sufficient

8、 number of states 11. compute the average over microstates, get the global average of spin M and energy U, also susceptibility and specific heat 12. calculate other properties, such as critical exponents etc,16,Susceptibility and specific heat,17,Wolff algorithm,identify clusters as sets of sites co

9、nnected by bonds create a bond, probability p=1-e-K, K=J/kBT if i=j. No bond otherwise,18,Quantum Monte Carlo,Ising model in a transverse field,Suzuki-Trotter formula,19,Partition function,20,Effective Hamiltonian,Mapping a d-dimension Hamitonian into a d+1 dimension Hamiltonian,where,21,Microcanoni

10、cal Monte Carlo,Canonical ensemble (正则系综) Fixed temperature Micro-canonical ensemble (微正则系综) Fixed total energy coexistence of two phases, first order phase transition annealing, heat conductivity,22,MC Summary,Metropolis sampling: still popular Wolff algorithm: fast, only for NN Quantum MC for IMTF

11、: Micro-canonical MCSimulations,23,Main References,BOOKs 1K Binder, Monte Carlo Methods in statistical physics, Springer-Verlag, 1979 2M Suzuki, Quantum Monte Carlo Methods in equilibrium and non-equilibrium systems, Springer Verlag, 1987 3M Toda, R Kubo and N Saito, Statistical physics, Springer Ve

12、rlag, 1985 PAPERs 4 M Suzuki, Prog Theo Phys 56 (1976) 1454 5 U Wollf, Phys Lett B228 (1989) 379 6 J S Wang and R H Swendsen, Physica A167 (1990) 565 7 I Vattulainen, T Ala-Nissila, K Kankaala, Phys Rev Lett, 73 (1994) 2513 8 D P Landau, Phys Rev, B13 (1976) 2997 9 N Metropolis, A W Rosenbluth, M N

13、Rosenbluth, A H Teller and E Teller, J Chem Phys, 21 (1953) 1087,24,Fatigue of ferroelectric thin films:Simulation with Monte Carlo Method,Ferroelectric thin films: due to the advancement of experimental preparation technology, thin films down to few hundreds nanometers can be achieved both in labor

14、atory and in industry.,An application of Monte Carlo Method,25,These films have practical applications in memory device and micro-electro-mechanic device as well as pyroelectric devices. However, some problems still exists in present thin films, one of them is the fatigue property. Fatigue means the

15、 polarization is reduced after the polarization is switched back and forth for many times.,26,Monte Carlo Simulation of fatigue in ferroelectric films,Models: modified diffusion-limited aggregation (DLA) model, Ising model, both in two-dimension space. Procedure: first use modified DLA to generate a

16、 dendrites to represent the defects formed near the electrodes; Then Ising model is implemented to calculate the polarization etc.,27,Diffused Limited Aggregation Model,DLA model,28,Diffused Limited Aggregation Model, DLA model, 扩散置限凝聚模型，扩散限制聚集模型 在1981年由Witten和Sander首次用计算机模拟建立的。,29,DLA模型的基本思想是： 在二维方

17、形点阵的中央放一静止的种子颗粒，然后在距离很远的边界处随机地释放一颗粒，让它作无规行走。 当颗粒运动到种子颗粒的近邻时就与种子颗粒粘结成一个集团（若颗粒到边界上则利用周期性边界条件重新引入）。,30,同时在边界处又释放一个新的颗粒继续作无规运动，直到粘结再集团表面为止。 如此反复进行就能在点阵的中央产生一个（树枝状图案的）分形集团。 下图是由3000个颗粒凝聚成的DLA集团。,31,32,改进: 假设二维方形点阵的两边(电极)都是静止的种子颗粒，然后在二维方形点阵中间处随机地释放一颗粒，让它作无规行走。 当颗粒运动到种子颗粒的近邻时就与种子颗粒粘结成一个集团（若颗粒到边界上则利用周期性边界条件

18、重新引入）。 这样就在两个电极之间形成了树枝状缺陷。,33,理论模拟:铁电薄膜的疲劳,铁电薄膜的疲劳特性是指在极化强度在电场作用下经过多次反转后，极化强度下降的性质。疲劳的产生可能有多种原因，实验上观察到多次反转后的样品中有树枝状图案从表面延伸到内部。根据这一现象，人们认为是电化学和空间电荷在电场作用下产生漂移造成，上图是根据这一思想进行的一个简单模拟的结果。,-电极 铁电 薄膜 -电极,34,A few more words,Monte Carlo is a very general and useful approach to deal with stochastic problems:

19、Nuclear reactions Statistical physics problems ,35,原胞自动机方法Cellular Automaton,36,What is cellular automata?,A quite useful tool in study dynamic and/or non-equilibrium, and/or spatially inhomogeneous systems.,元胞自动机是一种时间、空间、状态都离散的动力学模型，是非线性科学的一种重要研究方法，特别适合于复杂系统时空演化过程的动态模拟研究。,37,元胞自动机的构建没有固定的数学公式，构成方式繁

20、杂，变种很多，行为复杂。故其分类难度也较大，自元胞自动机产生以来，对于元胞自动机分类的研究就是元胞自动机的一个重要的研究课题和核心理论，在基于不同的出发点，元胞自动机可有多种分类。而基于维数的元胞自动机分类也是最简单和最常用的划分。,38,Applications,more realistic, especially when applied to: Biological problem Social problems: voting, epidemic etc,应用： 晶粒粗化和再结晶;生命的演化过程， 城市交通问题; 流行病趋势(Epidemic);森林火势蔓延,39,由一系列更新规则（u

21、pdating rules）组成。例：杨辉三角形（Pascal）,40,von Neumann and Moore neighboring,41,晶粒生长(应用),42,一个例子,郑容森,谭惠丽,秦继民(广西师范大学), 大学物理,24卷11期,10-13页,2005年 ”二维伊辛模型相变临界现象的元胞自动机模拟”,基本更新过程与Monte Carlo非常类似; 每一时步各元胞状态同时更新(与Monte Carlo不同); 能量不守恒问题,需要引入Q2R规则;,43,Main References,E Ahmed and A S Elgazzar, Physica A 296 (2001) 5

22、29-538 E Domany and W Kinzel, Phys Rev Lett, 53 (1984) 447 F Bagnoli, N Boccara and R Rechtman, 2000,Cond-mat 0002361 K P Hadeler, in O Diekman et al. (Eds.). Mathematics Inspired by biolgy, springer, Berlin, 1998 Bastien Chopard, Michel Droz, 物理系统的元胞自动机模拟,祝玉学,赵学龙 译,北京 清华大学出版社,2003,44,Molecular Dyna

23、mics Simulations,分子动力学模拟,45,Molecular Dynamic Simulation 用有限差分法求解多粒子体系的牛顿（经典力学）运动方程， Newtons law Potential function Physical Observable,46,基本过程： （1） 运算条件参数（温度、粒子数、密度、时间等） （2）体系初始化（初试位置和速度） （3）计算作用在所有粒子上的力 （4）解牛顿方程 （5）计算平均值，输出结果,47,初始条件：ri(t=0), vi(t=0),运动方程：,48,速度：,运动方程：,Initial conditions：,t=t:,49,

24、Verlet algorithm：,50,Nf体系的自由度数,温度：平均动能,51,力的计算： 势函数的选择 Hard sphere Potential(硬球势):,52,The Lennard-Jones potential,The Lennard-Jones potential is mildly attractive as two uncharged molecules or atoms approach one another from a distance, but strongly repulsive when they approach too close.,53,54,The

25、Morse Potential,where x is the bond length, re is the equilibrium bond length (the bond length at the potential minimum) and De is the equilibrium dissociation energy of the molecule (measured from the potential minimum).,55,56,Although this is an approximate potential, L-J potential has the feature

26、s needed to describe the interactions between closed shell atoms: namely, a strong repulsive short range interaction, a long range van der Waals attraction, and a potential well. Its mathematical and computational form is simpler, and actually more realistic in its treatment of long range interactio

27、ns, than the Morse function.,57,However, the Morse function is the more useful in analysis of the vibration spectroscopy of bound diatoms, and so does pop up quite often.,58,位置、速度： - 总能量，温度 - 表面的吸附、脱附，稳定结构，界面、位错等,59,60,势函数形式的选取 势函数参量的调整 密度泛函-分子动力学(DFT-MD),61,Phase Field Model,相场动力学模型,62,What can we

28、do with PFM,Grain growth in metal and ceramic Condensation from liquid state Domain pattern evolution,63,An arbitrary polycrystalline microstructure is described by a set of continuous field variables:,Where p is the number of possible orientations in the space, and i (i=1,2,p) is called orientation

29、 field variables which distinguish the different orientations of grains and are continuous in space. Their value continuously vary from -1.0 to 1.0.,64,With the diffused-interface theory, the total energy of an inhomogeneous system can be written as,Where f0 is the local free energy density which is

30、 a function of field variables i, and i are the gradient energy coefficients.,65,The spatial and temporal evolution of orientation field variables is described by the Ginzburg-Landau equation,Where Li is the kinetic coefficients related to the grain boundary mobility.,66,A simple form of local free

31、energy density,Where , and are the phenomenological parameters,67,Numerical Simulation Procedure,The Laplacian is discretized as following,Where x is the grid size, j represents the first-nearest neighbors of site i, and k represents the second-nearest neighbors of site i.,68,For discretization with

32、 respect to time, the explicit Euler equation is used,Where t is time step for integration.,69,To visualize the microstructure evolution using the orientation field variables, the following function is defined,Function (r) has the value of 1.0, within grains and significantly small values at grain boundaries.,70,71,References,LQ Chen and W Yang, Phys Rev B50 (1994) 15752 D Fan and L Q Chen, Computer simulation of grain growth using a continuum field model, Acta Mater, 45 (1997) 611 D Fan, C G Geng and L Q Chen, Computer simulation of topological evolution in