统计物理学习讲义

1、统计物理学习讲义中科院数学院复杂系统研究中心复杂系统学习班 (CSSGBJ)韩 靖2003年10月27日统计物理、自旋玻璃和复杂系统统计物理做什么？自旋玻璃(Spin Glasses)是什么？它们在复杂系统研究中有何应用？它们的局限性？探讨：对我们的研究有何启发？学习提纲和计划 (欢迎补充修改)基本概念介绍Entropy, Boltzmann分布(partition function)Example: K-SAT问题的相变Dynamics and Landscapes各态历尽, landscapes, Monte Carlo SimulationExample: Simulated Annea

2、ling(模拟退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于网络动力学的例子Replica Symmetry 用于组合问题的例子Cavity Methods: Survey Propagation Critical Phenomena & Power-law相变SOC, HOT/COLD理论谁报名来主讲？统计物理Statistical physics is about systems composed of many parts. 集体行为 组合数学和概率理论Traditional examples:气体、液体、固体 - 原

3、子或分子；金属、半导体 - 电子;量子场 - 量子，电磁场 - 光子等Complex systems examples:生态系统 - 物种社会系统 - 人计算机网络 - 计算机市场 - 经纪人agent鱼群 - 鱼、鸟群 - 鸟、蚁群 - 蚂蚁组合问题 变量 研究复杂系统为什么要学习统计物理？Collective Behavior 群体行为集体行为：系统由大量相似的个体组成全局行为不依赖于个体的精确细节，而相互作用必须合理定义，并且不要太复杂；个体在单独存在的行为与在整体中的行为很不一样.(在整体中各个体行为变得相似)；相互作用的类型：吸引、抗拒、对齐主要的集体现象：相变、模式形成、群组运动、

4、同步 研究手段：统计物理、多主体计算机模拟“磁化”现象:go个体行为 邻居动作的平均方向同步掌声恐慌现象http:/angel.elte.hu/vicsek/自旋玻璃(Spin Glasses)简单的理想模型，性质丰富，易于研究个体:spin si; 系统:多个spin局部相互作用以最简单的Ising模型为例：si=1 或者 1在lattice上排列，相邻spin之间有相互作用能量(Hamiltonian)：E = - J(i-1)isi-1siJij0, 偏好相邻同向；Jij0, 偏好相邻不同向；Jij=0,无相互作用考虑外部场 E = - Jijsisj - hisi性质：有序/无序、受挫

5、、相变、对称破缺现实中的例子：组合问题、恐慌人群、经济模型(-)(+)(+) ?sisi+1si-1J(i-1)iJi(i+1)E=- JijsisjSpin GlassConfiguration r = s1,s2,snHamiltonian (E, Cost function): E(r)J=HJ(r) = -JiksiskQuenched variable: J, random variable a probability distribution P(J)Different Spin model: different P(J)Notation:=PJ(s)g(s)So-called D

6、isorder: Structural parameter J is random and have large complexity自旋玻璃例子- K-SAT问题经典NP-完全问题N个布尔变量: xi=True/False, si=1/-1M个clauses: M个含k个变量的逻辑表达式K=3, 3-SAT: c1:x1 or (not x3) or x8, c2:(not x2) or x3 or (not x4), c3:x3 or x7 or x9,目标：满足所有M个clauses 的 N个布尔变量的一组赋值Spin glass 的能量 E =- a=1,M(Ca =T),Ground

7、 State E=-M 解状态结果：当K=3, M/N 4.25, 问题求解困难 恐慌现象行人建模：期望移动速度、与他人的排斥力、与墙壁的作用力、个人速度的扰动恐慌（由于火灾或者大众心理）：人们希望移动更快人与人之间的物理冲突更厉害；出口处障碍、堵塞形成；危险压力出现；人群开始出现大众恐慌心理；看不到其它的出口；计算机模拟实验： (Go) 单出口房间：无恐慌、恐慌、惊跑、带圆柱、火灾走廊：直走廊、中间加宽的走廊人群：个人主义、群体心理、两者综合Begin统计物理能做什么？怎么做？基本点：只关心状态的概率，并不关心演化的过程（假设各态历经）熵最大核心： Boltzmann分布 (partitio

8、n function)学习提纲和计划基本概念介绍Entropy, Boltzmann分布(partition function)Example: K-SAT问题的相变Dynamics and Landscapes各态历尽, landscapes, Monte Carlo SimulationExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于网络动力学的例子Replica Symmetry 用于组合问题的例子Cavity Methods: Survey Propagat

9、ion Critical Phenomena & Power-law相变SOC, HOT/COLD理论EntropyMicrostate r: a specific configuration of systemMacrostate R: an evaluation value(R): number of microstates related to a macrostateMicro-canonical entropy: S(R)=k log (R) More General forms:A macrostate R: pi for system be found in a microsta

10、te i A distribution of microstates.Gibbs Entropy: S(R) =-k pi logpi Maximum the most possible distribution of microstates Without constraint on pi, pi=1/N S is maximized (ni)=M!/n1!n2!.nN!, pi=ni/MWith Constraint on pi: Partition Function ZObservable quantity E (Hamiltonian)Ergodic Hypothesis (time

11、average=ensemble average)We know: From experiments: , Ei for all ri, and = = piEi, pi=1.We want to know the most probable distribution of microstates Maximize S=-kpilogpi and we get: pi=e-Ei/Z, Z=ie-Ei (=(kT)-1)So, pi and is decided by Ei and Knowing or T and Ei, we can define the most possible dist

12、ribution of microstates pi and Z T Z distribution is less symmetricalToy ExampleThree microstates: E1=0, E2=2, E3=3We have p1E1+p2E2+p3E3= e.g. 2p2+3p3=, and p1+p2+p3=1 3 temperatures: decreasing order of TZ p1p2p311.50.1052.5400.3930.3190.287210.4201.7160.5830.2520.16530.31.0831.1540.8670.0990.034I

13、mportant conceptsPartition function: Z(T,E)=re- E(r)/T Knowing this, we can do a lot of things!Variance of E, #sol, Free Energy: F = -k T lnZ (?)Entropy S=- (F/ T)E=-k pilnpi Z and #sol (ground state)Z (T)=re-E(r)/T = H=1,2,r|E(r)=H e-H/T When T0, system are most likely in the ground state. e-E(r)/T

14、 0 except E(r)=0Z(0)= r|E(r)=0 e-0 = r|E(r)=0So, number of ground states = Z(0).In T0, Z also counts other r that E(r)0. But the lower T, the r with lower E(r) Z counts. Z is decreasing when T is decreasing.The K-SAT result considers T=0.学习提纲和计划基本概念介绍Entropy, Boltzmann分布(partition function)Example:

15、K-SAT问题的相变Dynamics and Landscapes各态历尽, landscapes, Monte Carlo SimulationExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于网络动力学的例子Replica Symmetry 用于组合问题的例子Cavity Methods: Survey Propagation Critical Phenomena & Power-law相变SOC, HOT/COLD理论各态历尽对任意2个系统状态r1和r2, r1

16、可以经过有限部变换到r2. 00011011熵最大分布的三个条件 Rij=probability of ri changes to rj 方程的平衡状态是熵最大分布,必须要满足：p=Rp, R 有唯一的主特征向量(特征值为1)各态历经细致平衡：平衡态时，piRij=pjRjiErgodicity breaking and LandscapeMapping of microstates onto energiesbarrierr1r2r3rnVery high, unlikely to cross, when system size is large,T is low:pi/pj=e-(Ei-E

17、j)/TMonte Carlo Simulation设定状态转换矩阵，使得系统演化服从我们希望的状态分布 P。如果各态历尽和细致平衡，有 把P代入就可以得到Rij Simulated Annealing目标P是Boltzmann分布：pie-Ei/T。Rij/Rji=e-(Ej-Ei)/T Rij= 1if EjEi e-(Ej-Ei)/T if EjEiSimulated Annealing:We want to minimize ET=0, ergodicity breaking, favors minimal ET0, barriers can be crossed, favors mo

18、re states Most problems have many metastable states (local optima), various scales of barriers heights学习提纲和计划基本概念介绍Entropy, Boltzmann分布(partition function)Example: K-SAT问题的相变Dynamics and Landscapes各态历尽, landscapes, Monte Carlo SimulationExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry,

19、Cavity MethodsMeanfield 用于网络动力学的例子Replica Symmetry 用于组合问题的例子Cavity Methods: Survey Propagation Critical Phenomena & Power-law相变SOC, HOT/COLD理论Replica Approach and P(J)For a given J, free energy density:fJ=-1/(N) ln ZJFor a P(J), we want to know: =P(J)fJFor n replicas: Zn=JP(J)(ZJ)n (ZJ)n=s1s2sn exp-a=1nHJ(sa)si is the i th replica. fn=-1/(nN) ln Zn, ln Z= Lim n0 (Zn-1/n)We get: = Lim n0 fn f0参考教材 /group/CSSGBJ/Mark Newman 2001 复杂系统暑期学校教材