?????????????? ??????? (CSSGBJ)

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????????

• ????????????????????? (CSSGBJ)
• ? ?
• 2003?10?27?

2
??????????????

• ????????
• ????(Spin Glasses)????
• ??????????????????????
• ?????????????

3
??????? (??????)

• ??????
• Entropy, Boltzmann??(partition function)
• Example K-SAT?????
• Dynamics and Landscapes
• ????, landscapes, Monte Carlo Simulation
• Example Simulated Annealing(????)
• Meanfield, Replica Symmetry, Cavity Methods
• Meanfield ??????????
• Replica Symmetry ?????????
• Cavity Methods Survey Propagation
• Critical Phenomena Power-law
• ??
• SOC, HOT/COLD??

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????

• Statistical physics is about systems composed of
  many parts. ???? ?????????
• Traditional examples
• ???????? - ?????
• ?????? - ??
• ??? - ??,??? - ???
• Complex systems examples
• ???? - ??
• ???? - ?
• ????? - ???
• ?? - ???agent
• ?? - ???? - ???? - ??
• ???? ?? ?????????????????

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Collective Behavior ????
http//angel.elte.hu/vicsek/

• ????
• ????????????
• ???????????????,???????????,???????
• ??????????????????????.(?????????????)
• ???????????????
• ??????????????????????
• ?????????????????
• ????go???? ? ?????????
• ????
• ????

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????(Spin Glasses)

• ???????,????,????
• ??spin si ????spin??????
• ?????Ising????
• si1 ?? 1
• ?lattice???,??spin???????
• ??(Hamiltonian)E - ?J(i-1)isi-1si
• Jijgt0, ??????Jijlt0, ???????Jij0,?????
• ????? E - ?Jijsisj - ?hisi
• ????/?????????????
• ????????????????????

E- ?Jijsisj
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Spin Glass

• Configuration r s1,s2,,sn
• Hamiltonian (E, Cost function) E(r)JHJ(r)
  -?Jiksisk
• Quenched variable J, random variable a
  probability distribution P(J)Different Spin
  model different P(J)
• Notationltggt?PJ(s)g(s)
• So-called Disorder Structural parameter J is
  random and have large complexity

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??????- K-SAT??

• ??NP-????
• N????? xiTrue/False, si1/-1
• M?clauses M??k?????????K3, 3-SAT c1x1 or
  (not x3) or x8, c2(not x2) or x3 or (not x4),
  c3x3 or x7 or x9,
• ??????M?clauses ? N??????????
• Spin glass ??? E - ?a1,M?(Ca T),Ground State
  E-M ????
• ???K3, M/N 4.25, ??????

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????

• ??????????????????????????????????
• ??(??????????)
• ????????
• ?????????????
• ??????????
• ??????
• ????????????
• ????????
• ??????? (Go)
• ?????????????????????
• ?????????????
• ????????????????

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Begin

• ?????????????
• ???
• ????????,?????????(??????)
• ???
• ?? Boltzmann?? (partition function)

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???????

• ??????
• Entropy, Boltzmann??(partition function)
• Example K-SAT?????
• Dynamics and Landscapes
• ????, landscapes, Monte Carlo Simulation
• Example Simulated Annealing(????)
• Meanfield, Replica Symmetry, Cavity Methods
• Meanfield ??????????
• Replica Symmetry ?????????
• Cavity Methods Survey Propagation
• Critical Phenomena Power-law
• ??
• SOC, HOT/COLD??

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Entropy

• Microstate r a specific configuration of system
• Macrostate R an evaluation value
• ?(R) number of microstates related to a
  macrostate
• Micro-canonical entropy S(R)k log ?(R)
• More General forms
• A macrostate R pi for system be found in a
  microstate i
• A distribution of
  microstates.
• Gibbs Entropy S(R) -k ?pi logpi Maximum ? the
  most possible distribution of microstates
• Without constraint on pi, pi1/N ? S is
  maximized

?(ni)M!/n1!n2!...nN!, pini/M
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With Constraint on pi Partition Function Z

• Observable quantity E (Hamiltonian)
• Ergodic Hypothesis (time averageensemble
  average)
• We know
• From experiments ltEgt,
• Ei for all ri, and ltEgt ltEgt ?piEi, ?pi1.
• We want to know the most probable distribution of
  microstates
• Maximize S-k?pilogpi and we get
• pie-ßEi/Z, Z?ie-ßEi (ß(kT)-1)
• So, pi and ß is decided by Ei and ltEgt
• Knowing ßor T and Ei, we can define the most
  possible distribution of microstates pi and Z
• ß? ? T? ? ltEgt? ? Z? distribution is less
  symmetrical

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Toy Example

• Three microstates E10, E22, E33
• We have p1E1p2E2p3E3ltEgt
• e.g. 2p23p3ltEgt, and p1p2p31
• 3 temperatures decreasing order of T

ltEgt ß Z p1 p2 p3
1 1.5 0.105 2.540 0.393 0.319 0.287
2 1 0.420 1.716 0.583 0.252 0.165
3 0.3 1.083 1.154 0.867 0.099 0.034
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Important concepts

• Partition 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

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Z and sol (ground state)

• Z (T)?re-E(r)/T ?H1,2,?rE(r)H e-H/T
• When T?0, system are most likely in the ground
  state. e-E(r)/T ?0 except E(r)0
• Z(0) ? rE(r)0 e-0 ? rE(r)0
• So, number of ground states Z(0).
• In Tgt0, Z also counts other r that E(r)gt0. But
  the lower T, the r with lower E(r) Z counts. Z
  is decreasing when T is decreasing.
• The K-SAT result considers T0.

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???????

• ??????
• Entropy, Boltzmann??(partition function)
• Example K-SAT?????
• Dynamics and Landscapes
• ????, landscapes, Monte Carlo Simulation
• Example Simulated Annealing(????)
• Meanfield, Replica Symmetry, Cavity Methods
• Meanfield ??????????
• Replica Symmetry ?????????
• Cavity Methods Survey Propagation
• Critical Phenomena Power-law
• ??
• SOC, HOT/COLD??

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????

• ???2?????r1?r2, r1??????????r2.

00
01
10
11
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??????????

• Rijprobability of ri changes to rj
• ?????????????,?????
• pRp, R ?????????(????1)
• ????
• ????????,piRijpjRji

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Ergodicity breaking and Landscape

• Mapping of microstates onto energies

barrier
Very high, unlikely to cross, when system size
is large,T is low pi/pje-(Ei-Ej)/T
r1
r2
r3
rn

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Monte Carlo Simulation

• ????????,????????????????? P?
• ???????????,?
• ?P???????Rij

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Simulated Annealing

• ??P?Boltzmann??pi?e-Ei/T?
• Rij/Rjie-(Ej-Ei)/T
• Rij 1 if Ej?Ei
• e-(Ej-Ei)/T if EjgtEi
• Simulated Annealing
• We want to minimize E
• T0, ergodicity breaking, favors minimal E
• Tgt0, barriers can be crossed, favors more states
  
• Most problems have many metastable states (local
  optima), various scales of barriers heights

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???????

• ??????
• Entropy, Boltzmann??(partition function)
• Example K-SAT?????
• Dynamics and Landscapes
• ????, landscapes, Monte Carlo Simulation
• Example Simulated Annealing(????)
• Meanfield, Replica Symmetry, Cavity Methods
• Meanfield ??????????
• Replica Symmetry ?????????
• Cavity Methods Survey Propagation
• Critical Phenomena Power-law
• ??
• SOC, HOT/COLD??

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Replica Approach and P(J)

• For a given J, free energy densityfJ-1/(ßN) ln
  ZJ
• For a P(J), we want to know ltfgt?P(J)fJ
• For n replicas Zn?JP(J)(ZJ)nlt(ZJ)ngt
• (ZJ)n?s1?s2?sn exp-?a1nßHJ(sa)si is
  the i th replica.
• fn-1/(ßnN) ln Zn, ln Z Lim n?0 (Zn-1/n)
• We get ltfgt Lim n?0 fn f0

25
????

• http//groups.yahoo.com/group/CSSGBJ/
• Mark Newman 2001 ?????????? http//www.santafe.edu
  /mark/budapest01/
• K-SAT?? Nature, Vol 400, July 1999, p133-137
• Survey Propagation Science, Vol 297, Aug. 2002,
  p812-815, p784-785.
• SOC ???????, Per Bak.
• HOT/COLD
• HOT Highly Optimized Tolerance A Mechanism for
  Power Laws in Designed Systems. J. M. Carlson,
  John Doyle. (April 27, 1999)
• COLD Optimal design, robustness, and risk
  aversion. M. E. J. Newman, Michelle Girvan and J.
  Doyne Farmer