1. 自旋玻璃与消息传递算法 Spin Glass and Message-Passing Algorithms 周海军 http://www.itp.ac.cn/~zhouhj/ 中国科学院理论物理研究所

2. 2 提纲 1 。自旋玻璃理论 基本图像与平衡自 由能分布 空腔方法 2 。消息传递算法 Vertex-Cover 问题, 3- SAT 问题 Survey Propagation 算 法

3. 3 部分参考文献 1.Mezard, Parisi, Virasoro, “Spin Glass Theory and Beyond” (World Scientific, 1987 ） 2.Mezard, Parisi, “The Bethe lattice spin glass revisited”, European Physics Journal B 20: 217-233 (2001) 3.Mezard, Parisi, Zecchina, “Analytic and algorithmic solution of random satisfiability problems”, Science 297: 812-815 (2002) 如果对报告中所涉及的具体模型的计算细节感兴趣，请参考 http://www.itp.ac.cn/~zhouhj/mainen.html

4. 自旋玻璃理论： 自由能分

5. 5 Statistical mechanics of a (simple) system in equilibrium is well-established. Partition function, free energy, …. Mean-field treatment. Phase transitions. Correlation length, scaling exponents, …. Renormalization flow.

6. 6 …, but non-equilibrium dynamics of even a simple system may be difficult to understand Formal framework. Connection with equilibrium. Glassy dynamics. Why relaxation becomes so low and non-exponential? ……

7. 7 Equilibrium (static) and dynamical properties of complex systems are both difficult and interesting Quenched randomness, frustration, non- self averaging, …, broken ergodicity. NP-complete combinatorial optimizations, message-passing algorithms for information science (CDMA, for example!), econo-physics, …, biological systems.

8. 8 自旋玻璃：无序与阻错系统的简单模型 3D regular lattice (Edwards-Anderson, 1975) Complete graph (Sherrington-Kirkpatrick 1975) Random Poisson graph (Viana-Bray, 1985)

9. 9 What we learned from an equilibrium statistical mechanics course?

10. 10 What we learned from an equilibrium statistical mechanics course? (contl.)

11. 11 What we learned from an equilibrium statistical mechanics course? (contl.)

12. 12 ergodic vs non-ergodic ?

13. 13 repeated heating—annealing and the equilibrium Gibbs measure Complexity （复杂度）

14. 14 distribution of equilibrium free-energies 1

15. 15 distribution of equilibrium free-energies (contl.) 2

16. 16

17. 17 Which thermodynamic states contribute to the equilibrium properties? If Excited macrostates matter! Macrostates of minimal free energy density matter!

18. 18 3-spin-Interaction Ising model on a complete graph

19. 19 the mean free energy density

20. 20 Overlap Distribution

21. 自旋玻璃理论： 空腔方法（ cavity method)

22. 22 Let’s define an artificial system!

23. 23 Some examples of the grand free energy ： 2-body interactions Beta=+infinity The max-2-SAT problem Beta=1.25 The +/- J spin-glass model on a random regular graph of degree K=6

24. 24 How to calculate the grand free energy? The cavity approach

25. 25

26. 26 NN+2

27. 27 Population Dynamics Simulation

28. 28

29. Message-Passing Algorithms

30. 30 3-SAT 问题

31. 31 顶点覆盖问题

32. 32 This graph is covered, but not optimally covered.

33. 33 Minimal Vertex Cover Problem A vertex cover of the global minimal size. Is a NP-hard optimization problem. Efficient algorithms for constructing near- optimal solutions for a given graph.

34. 34 There are many optimal solutions for a given graph

35. 35 Three types of vertices ： (1) vertices that are always covered (frozen vertices, ) (2) vertices that are always uncovered (frozen vertices, ) (3) vertices that are covered in some solutions and uncovered in the remaining solutions (unfrozen vertices, )

36. 36 Mean-field analysis of the minimal vertex cover problem on a random graph

37. 37 自洽的空穴场方法 覆盖还是不覆盖 ? The vertex cover problem

38. 38 = always uncovered always covered unfrozen Weigt, Hartmann, PRL (2000), PRE (2001)

39. 39 New vertex un-covered New vertex partially covered New vertex always covered

40. 40 Mean-field theory result is lower than experimental values for c > e=2.7183 2.7183

41. 41 假定的相空间结构

42. 42 引入参数 y

43. 43 neighborsvertex iprobabilityVC size increase re-weighted probability all unfrozen or always covered always uncovered 0 at least one always uncovered unfrozen or always covered +1

44. 44

45. 45

46. 46 同样的消息传递的算法可以用于解决 神经网络，信息系统，满足性问题， … ， 中的许多计算困难

47. 47 Program and School in Beijing 2008 ICTP-ITP Spring School on “Statistical Physics and Interdisciplinary Applications” March 03-14, 2008 KITPC Program “Collective Dynamics in Information Systems” March 01-April 15, 2008