統(tǒng)計(jì)物理學(xué)習(xí)講義

1、統(tǒng)計(jì)物理學(xué)習(xí)講義中科院數(shù)學(xué)院復(fù)雜系統(tǒng)研究中心復(fù)雜系統(tǒng)學(xué)習(xí)班 (CSSGBJ)韓 靖2003年10月27日統(tǒng)計(jì)物理、自旋玻璃和復(fù)雜系統(tǒng)統(tǒng)計(jì)物理做什么？自旋玻璃(Spin Glasses)是什么？它們?cè)趶?fù)雜系統(tǒng)研究中有何應(yīng)用？它們的局限性？探討：對(duì)我們的研究有何啟發(fā)？學(xué)習(xí)提綱和計(jì)劃 (歡迎補(bǔ)充修改)基本概念介紹Entropy, Boltzmann分布(partition function)Example: K-SAT問(wèn)題的相變Dynamics and Landscapes各態(tài)歷盡, landscapes, Monte Carlo SimulationExample: Simulated Annea

2、ling(模擬退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于網(wǎng)絡(luò)動(dòng)力學(xué)的例子Replica Symmetry 用于組合問(wèn)題的例子Cavity Methods: Survey Propagation Critical Phenomena & Power-law相變SOC, HOT/COLD理論誰(shuí)報(bào)名來(lái)主講？統(tǒng)計(jì)物理Statistical physics is about systems composed of many parts. 集體行為 組合數(shù)學(xué)和概率理論Traditional examples:氣體、液體、固體 - 原

3、子或分子；金屬、半導(dǎo)體 - 電子;量子場(chǎng) - 量子，電磁場(chǎng) - 光子等Complex systems examples:生態(tài)系統(tǒng) - 物種社會(huì)系統(tǒng) - 人計(jì)算機(jī)網(wǎng)絡(luò) - 計(jì)算機(jī)市場(chǎng) - 經(jīng)紀(jì)人agent魚群 - 魚、鳥群 - 鳥、蟻群 - 螞蟻組合問(wèn)題 變量 研究復(fù)雜系統(tǒng)為什么要學(xué)習(xí)統(tǒng)計(jì)物理？Collective Behavior 群體行為集體行為：系統(tǒng)由大量相似的個(gè)體組成全局行為不依賴于個(gè)體的精確細(xì)節(jié)，而相互作用必須合理定義，并且不要太復(fù)雜；個(gè)體在單獨(dú)存在的行為與在整體中的行為很不一樣.(在整體中各個(gè)體行為變得相似)；相互作用的類型：吸引、抗拒、對(duì)齊主要的集體現(xiàn)象：相變、模式形成、群組運(yùn)動(dòng)、

4、同步 研究手段：統(tǒng)計(jì)物理、多主體計(jì)算機(jī)模擬“磁化”現(xiàn)象:go個(gè)體行為 鄰居動(dòng)作的平均方向同步掌聲恐慌現(xiàn)象http:/angel.elte.hu/vicsek/自旋玻璃(Spin Glasses)簡(jiǎn)單的理想模型，性質(zhì)豐富，易于研究個(gè)體:spin si; 系統(tǒng):多個(gè)spin局部相互作用以最簡(jiǎn)單的Ising模型為例：si=1 或者 1在lattice上排列，相鄰spin之間有相互作用能量(Hamiltonian)：E = - J(i-1)isi-1siJij0, 偏好相鄰?fù)?；Jij0, 偏好相鄰不同向；Jij=0,無(wú)相互作用考慮外部場(chǎng) E = - Jijsisj - hisi性質(zhì)：有序/無(wú)序、受挫

5、、相變、對(duì)稱破缺現(xiàn)實(shí)中的例子：組合問(wèn)題、恐慌人群、經(jīng)濟(jì)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問(wèn)題經(jīng)典NP-完全問(wèn)題N個(gè)布爾變量: xi=True/False, si=1/-1M個(gè)clauses: M個(gè)含k個(gè)變量的邏輯表達(dá)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,目標(biāo)：滿足所有M個(gè)clauses 的 N個(gè)布爾變量的一組賦值Spin glass 的能量 E =- a=1,M(Ca =T),Ground

7、 State E=-M 解狀態(tài)結(jié)果：當(dāng)K=3, M/N 4.25, 問(wèn)題求解困難 恐慌現(xiàn)象行人建模：期望移動(dòng)速度、與他人的排斥力、與墻壁的作用力、個(gè)人速度的擾動(dòng)恐慌（由于火災(zāi)或者大眾心理）：人們希望移動(dòng)更快人與人之間的物理沖突更厲害；出口處障礙、堵塞形成；危險(xiǎn)壓力出現(xiàn)；人群開(kāi)始出現(xiàn)大眾恐慌心理；看不到其它的出口；計(jì)算機(jī)模擬實(shí)驗(yàn)： (Go) 單出口房間：無(wú)恐慌、恐慌、驚跑、帶圓柱、火災(zāi)走廊：直走廊、中間加寬的走廊人群：個(gè)人主義、群體心理、兩者綜合Begin統(tǒng)計(jì)物理能做什么？怎么做？基本點(diǎn)：只關(guān)心狀態(tài)的概率，并不關(guān)心演化的過(guò)程（假設(shè)各態(tài)歷經(jīng)）熵最大核心： Boltzmann分布 (partitio

8、n function)學(xué)習(xí)提綱和計(jì)劃基本概念介紹Entropy, Boltzmann分布(partition function)Example: K-SAT問(wèn)題的相變Dynamics and Landscapes各態(tài)歷盡, landscapes, Monte Carlo SimulationExample: Simulated Annealing(模擬退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于網(wǎng)絡(luò)動(dòng)力學(xué)的例子Replica Symmetry 用于組合問(wèn)題的例子Cavity Methods: Survey Propagat

9、ion Critical Phenomena & Power-law相變SOC, HOT/COLD理論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.學(xué)習(xí)提綱和計(jì)劃基本概念介紹Entropy, Boltzmann分布(partition function)Example:

15、K-SAT問(wèn)題的相變Dynamics and Landscapes各態(tài)歷盡, landscapes, Monte Carlo SimulationExample: Simulated Annealing(模擬退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于網(wǎng)絡(luò)動(dòng)力學(xué)的例子Replica Symmetry 用于組合問(wèn)題的例子Cavity Methods: Survey Propagation Critical Phenomena & Power-law相變SOC, HOT/COLD理論各態(tài)歷盡對(duì)任意2個(gè)系統(tǒng)狀態(tài)r1和r2, r1

16、可以經(jīng)過(guò)有限部變換到r2. 00011011熵最大分布的三個(gè)條件 Rij=probability of ri changes to rj 方程的平衡狀態(tài)是熵最大分布,必須要滿足：p=Rp, R 有唯一的主特征向量(特征值為1)各態(tài)歷經(jīng)細(xì)致平衡：平衡態(tài)時(shí)，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設(shè)定狀態(tài)轉(zhuǎn)換矩陣，使得系統(tǒng)演化服從我們希望的狀態(tài)分布 P。如果各態(tài)歷盡和細(xì)致平衡，有 把P代入就可以得到Rij Simulated Annealing目標(biāo)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學(xué)習(xí)提綱和計(jì)劃基本概念介紹Entropy, Boltzmann分布(partition function)Example: K-SAT問(wèn)題的相變Dynamics and Landscapes各態(tài)歷盡, landscapes, Monte Carlo SimulationExample: Simulated Annealing(模擬退火)Meanfield, Replica Symmetry, Cavity MethodsMeanfield 用于網(wǎng)絡(luò)動(dòng)力學(xué)的例子Replica Symmetry 用于組合問(wèn)題的例子Cavity Methods: Survey Propagation Critical Phenomena & Power-law相變SOC, HOT/COLD理論Replica Approach and P(J)For a given J, free energy density:fJ=-1/(N