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An extended ensemble Monte Carlo study for a simple glass model . - - PowerPoint PPT Presentation

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Outline . . An extended ensemble Monte Carlo study for a simple glass model . . . . . Koji Hukushima mailto:hukusima@phys.c.u-tokyo.ac.jp The University of Tokyo, Komaba, Graduate school of Arts and Sciences 3 September, 2009 This

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Outline

. . . . . . .

An extended ensemble Monte Carlo study for a simple glass model Koji Hukushima

mailto:hukusima@phys.c.u-tokyo.ac.jp

The University of Tokyo, Komaba, Graduate school of Arts and Sciences

3 September, 2009 This work has been done in collaboration with Prof. S. Sasa

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 1 / 29

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Outline

Outline

. ..

1

A simple lattice glass model : Biroli-M´ ezard (BM) model . ..

2

Extended ensemble MC . ..

3

Test in BM model on a random graph . ..

4

BM model on a regular graph in two dimensions . ..

5

Summary

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 1 / 29

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Outline

General question

Glassy materials are ubiquitous, but the nature of glass still remains to be unclear. One of the most fundamental questions is that existence of thermodynamical glass transition? The spin-glass transitions are found both in experiments and in theoretical models. Does the lattice model in finite dimensions exhibit a thermodynamic transition?

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 2 / 29

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A simple lattice glass model : Biroli-M´ ezard (BM) model

A simple lattice glass model: Biroli-M´ ezard model (2002)

A given graph G(V,E)

Random graph Regular graph ...

An occupation variable σi is defined on each site σi = { 1 for occupied, for empty Particle configuration: σ = (σi)i=1,··· ,N . BM model with l = 1 . . . . . . . . Energy : H(σ) = ∞ # of particles in the neighbor sites is greater than a parameter l. H(σ) = 0 Otherwise

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 4 / 29

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A simple lattice glass model : Biroli-M´ ezard (BM) model

Equilibrium statisitcal mechanics of BM model

Grand partition function Z(µ) = ∑

σ

C(σ)eµ PN

i=1 σi

where C(σ) is a indicator function, C(σ) = { 1 for a possible configuration σ

  • therwise.

C(σ) is generally expressed as multi-body interactions. Probability P(σ): P(σ) = 1 Z(µ)C(σ) exp ( µ

N

∑

i=1

σi ) Expectation value A(µ) = ˆ Aµ = ∑

σ ˆ

A(σ)P(σ).

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 5 / 29

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A simple lattice glass model : Biroli-M´ ezard (BM) model

Phase transition of the BM model

. A typical snapshot of BM model with l = 1 . . . . . . . . . A typical snapshot of BM model with l = 1 . . . . . . . . . . . . . . . . .

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 6 / 29

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A simple lattice glass model : Biroli-M´ ezard (BM) model

Theoretical analyses: Rivoire et al(2004) , Krzakala-Tarzia-Zdeborov´

a(2008)

The model in a random graph by cavity method. One-step replica-symmetry-breaking transition occurs at ρc. Aaverage density ρ(µ) is continuous, but smooth at some density ρc. Dynamical transition occurs at ρd < ρ. For ρ > ρd local-update algorithm, like simulated annealing, cannot reach equilibrium. . ρ vs µ curve . . . . . . . .

ρ µ

Liquid Glass

µc thermodynamic glass transition dynamical transition ρc µd ρd

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 7 / 29

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A simple lattice glass model : Biroli-M´ ezard (BM) model

Questions

Physics: Does there exits thermodynamic glass transition in a statistical-mechnical model beyond mean-field analysis? the BM model defined on a honeycomb lattice where connectivity is 3. Algorithm: How can we go across “dynamical transition” to reveal thermodynamic transition? Florent provided a promising strategy in his talk, but... we study in a straightforward way by using an extended ensemble MC method

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 8 / 29

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Extended ensemble MC

Slow relaxation in MCMC

Under some circumstances, transitions in Markov chain are strongly

  • suppressed. Configurations get trapped into some small area of

configuration space.

✞ ✝ ☎ ✆

ergodicity breaking≃ slow relaxation/mixing

One may often face this difficulty in some physically interesting problems. .

.

.

1 slowing down of phase transition

critical slowing down of 2nd

  • rder transition

nucleation of 1st order transition

.

.

.

2 rugged free energy

(Spin) Glasses, proteins,

  • ptimizations...

T

many metastable states = multi-modal dist.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 10 / 29

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Extended ensemble MC

Some development of MCMC algorithm

.

.

.

1 non-local update (cluster algorithm)

Swendsen-Wang (1987), Wolf (88): based on Fortuin-Kasteleyn representation. pivot update algorithm for polymer simulation

.

.

.

2 Extended ensemble MC

Original probability distribution to be solved is modified or extended.

Multicanonical MC : Berg-Neuhaus, (1991)

entropic sampling : Lee Broad histogram MC : Oliveira (1998) Flat histogram MC，Transition Matrix MC : Wang (1999) Wang-Landau method....

Simulated tempering : Marinari-Parisi(1992),

Expanded ensemble method : Lyubartsev et. al. (1992)

Exchange MC method : Hukushima-Nemoto(1996),

Metropolis coupled Markov chain MC(Geyer) , Parallel tempering

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 11 / 29

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Extended ensemble MC

From SA to exchange MC

.

.

.

1 Temperature schedule

Practically important. Temperature is always lowered, but heating may be useful for escaping from local minima.

.

.

.

2 Sampling from distribution

Detailed balance conditions are not satisfied when temperature is lowered. When temperature is fixed, this corresponds to MCMC. At low temperatures, the difficulty of slow relaxation(mixing) is faced.

A goal is to construct an MCMC algorithm in which temperature keeps changing back and forth with preserving detailed balance con- ditions. Simulated tempering and exchange MC(parallel tempering). A dual method is multicanonical MC, where energy value changes in a wide range.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 12 / 29

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Extended ensemble MC

Exchange Monte Carlo Method (1)

Replicated System：For a given model H(X), M replicas of the system is introduced: Heff({X}) =

M

∑

m=1

βmH(Xm), “extended state” : {X} = {X1, X2, · · · , XM}. “Extended” probability distribution Peq({X}; {β}) =

M

∏

m=1

Peq(Xm; βm) =

M

∏

m=1

1 Z(βm) exp(−βmH(Xm)) Monte Carlo steps .

.

.

1 Update each replica configuration (Local update)

Xm = ⇒ X

′

m

.

.

.

2 Exchange of two configurations of two replicas Xm and Xn

{Xm, Xn} = ⇒ {Xn, Xm}

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 13 / 29

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Extended ensemble MC

Exchange Monte Carlo Method (2)

Detailed Balance conditions P({· · · , X, · · · , X ′, · · · }; {· · · , βm, · · · , βn, · · · }) × W (X, X ′; βm, βn) = P({· · · , X ′, · · · , X, · · · }; {· · · , βm, · · · , βn, · · · })W (X ′, X; βm, βn). W (X, X ′; βm, βn) W (X ′, X; βm, βn) = exp(−∆), where ∆(X, X ′; βm, βn) = (βn − βm)(H(X) − H(X ′)). Transition Probability for exchange process W (X, X ′; βm, βn) =    min[1, exp(−∆)], for Metropolis type,

1 2

( 1 + tanh(−∆

2 )

) , for heat-bath type.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 14 / 29

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Extended ensemble MC

Exchange Monte Carlo Method (3)

Monte Carlo Procedure

20 40 60 80 100 120 140 160 180 200

Temperature

.

.

.

1 Each replica is updated

simultaneously and independently as a canonical ensemble for a few MC steps {Xm} = ⇒ {X

′

m}

.

.

.

2 An exchange process between Xm

and Xm+1 is tried and accepted with probability W (Xm, Xm+1). {Xm, Xm+1} = ⇒ {Xm+1, Xm} Each replica wonders in the parameter space like random walker.

Self-organized annealing and heating

  • ne may have a chance to escape from meta-stable state at high temp.

Sampling from equilibrium distribution at each temperature.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 15 / 29

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Extended ensemble MC

Strategy of extended ensemble MCs

Acceleration of relaxation ... (reduction of mixing time)

Target Source

Low temperature ⇐ temperature ⇒ High temperature exchange MC High entropy

Mixing !!

Multi-canonical MC Low energy ⇐ Energy ⇒ High energy = ⇒ Slow relaxation = ⇒fast relaxation Slow relaxation found in the “TARGET” might be modified with the help of fast relaxation in the “SOURCE”.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 16 / 29

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Extended ensemble MC

Possible extensions of Extended ensemble MC

Multi-canonical MC Simulated tempering exchange MC (Parallel tempering) Z(β) = ∑ exp(−βE) Energy E Temperature β Z(µ) = ∑ exp(µn) Particle number n Chemical Potential µ magnetic interaction E ′ = −HM Magnetization M Magnetic field H E = E (1) + kE (2) E (2) k free particle+interaction An easy system with high entropy or at high temperature must be included in the parameter space. An extension to multiple parameters is OK.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 17 / 29

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Test in BM model on a random graph

Comparison between SA and EMC

. parameter schedule of SA . . . . . . . .

Monte Carlo time

1/ µ MCS

. EMC(parallel tempering) . . . . . . . .

Monte Carlo time

1/ µ MCS

The same set of µs is used in SA and EMC. A constant number of Monte Carlo steps(MCS) is performed at each µ. The annealing rate in SA is proportional to 1/MCS. Total MCS of SA and EMC is in common.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 19 / 29

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Test in BM model on a random graph

MCS dependence of ρ(µ)

. ρ-µ curve of BM model on random graph with N = 512 . . . . . . . . regular random graph: C = 3 model parameter : l = 1 (allowed occupation number in neighbor sites)

0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 3 4 5 6 7 8 9 10 ρ µ Exchange MC MCS= 256 MCS= 2048 MCS= 65536 MCS= 1048576 MCS=10240000 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 3 4 5 6 7 8 9 10 ρ µ Simulated Annealing MCS= 256 MCS= 2048 MCS= 65536 MCS= 1048576 MCS=16777216

. ρ-µ curve of BM model on random graph with N = 512 . . . . . . . .

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 20 / 29

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Test in BM model on a random graph

Annealing limit and equilibrium limit

. ρeq(N) − ρ vs MC step at µ = 10 . . . . . . . . N = 128, 256, 512 and 1024.

0.002 0.004 0.006 0.008 0.01 0.012 0.014 102 103 104 105 106 107 108 ρeq(N)-ρ MCS N=128(EMC) N=256(EMC) N=512(EMC) N=1024(EMC) N=128(SA) N=256(SA) N=512(SA) N=1024(SA)

. Size dep. of the densest packing ρRCP(N) . . . . . . . .

0.564 0.566 0.568 0.57 0.572 0.574 0.576 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 ρRCP(N) 1/N 1RSB cavity

.

.

.

1 For large N limit, SA data may saturate to a finite gap predicted by

the cavity method, but no tendency is found in EMC data up to MCS and sizes observed at least. .

.

.

2 Extrapolated value of the densest packing obtained by EMC with

finite N is consistent with that of the cavity method.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 21 / 29

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Test in BM model on a random graph

Glassy property of BM model with C = 3 and l = 1

Density ˆ ρ = 1

N

∑

i σi.

Overlap of density fluctuation ˆ q = 1 N

N

∑

i=1

(σi − ˆ ρ) ( σ′

i − ˆ

ρ′ ) where σ and σ′ are configurations of two independent systems. Overlap distribution P(q) P(q) = δ(q − ˆ q) . Overlap dist. P(q) with N = 512 . . . . . . . .

  • 0.1
  • 0.05

0.05 0.1 0.15 0.2 0.25 Q 5 6 7 8 9 10 µ 2 4 6 8 10 12 14 16 18

This supports a trasition from liquid to some glassy states. Two-peak structure of P(q) is consistent with 1RSB picture.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 22 / 29

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Test in BM model on a random graph

Glassy transition of BM model with C = 3 and l = 1

. Binder parameter . . . . . . . . gµ = 1 2 ( 3 − ˆ q4 ˆ q22 ) A negative dip appears at 1RSB transition.

  • 4
  • 3.5
  • 3
  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

0.5 2 3 4 5 6 7 8 9 10 g µ N=64 N=128 N=256 N=384 N=512

. Extrapolation of glass transition . . . . . . . .

4.5 5 5.5 6 6.5 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 µdip(N) 1/N 1RSB cavity

µK ≃ 6.7 (thanks to Florent). This is consistent with the 1RSB cavity analysis.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 23 / 29

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BM model on a regular graph in two dimensions

BM model with on a honeycomb lattice

C = 3: Honeycomb (HC) lattice l = 0: exactly solved model

Baxter’s hard hexagon model continuous phase transition from liquid to trigonal crystal.

l = 1 : not yet. . densest packing for l = 1 . . . . . . . . Complex mixtures are expected to be realized for dense region. . . . . . . . . . . hard-trigons crystal for l = 0 . . . . . . . .

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 25 / 29

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SLIDE 23

BM model on a regular graph in two dimensions

ρ(µ) curve and density fluctuation of BM in HC

density fluctuation χ(µ) = ∂ρ(µ) ∂µ = N(ˆ ρ − ρ(µ))2 Anomalous behavior in χ is found around µ ∼ 7.0 χ has a finite jump at transition?! . density fluctuation χ . . . . . . . .

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 4 5 6 7 8 9 10 χ µ

L=24 L=26 L=32 L=34 L=40 L=42 L=50 0.53 0.56 4 5 6 7 8 9 10 ρ µ

linear size L: 24 to 50.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 26 / 29

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SLIDE 24

BM model on a regular graph in two dimensions

Evidence of thermodynamic transition at finite µ

. Overlap distribution . . . . . . . .

  • 0.05

0.05 0.1 0.15 0.2 Q 5 6 7 8 9 10 µ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

. Binder parameter . . . . . . . .

  • 0.4
  • 0.2

0.2 0.4 0.6 2 3 4 5 6 7 8 9 10

gµ

µ

L=24 L=26 L=32 L=34 L=40 L=42 L=48 L=50 6.2 6.4 6.6 6.8 7 7.2 0.01 0.02 0.03 0.04 0.05

µdip 1/L

24 32 34 26 42 50 40 48

At high dense region, P(q) has a double-peak structure, often found in 1RSB phase. The transition point is estimated as µs ≃ 6.8 from dips of g. Surprisingly, the value is very close to the cavity estimate for random graph model µK ≃ 6.7.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 27 / 29

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SLIDE 25

Summary

Summary

A simple lattice glass model, Biroli-M´ ezard (BM) model has been studied by using an exchange Monte Carlo (EMC) method. Efficiency of EMC method has been found in applications to the BM model defined on a regular random graph.

Time needed for equilibrium for EMC is shorter than that for the simulated annealing (SA). Presumably, dynamical arrest also appears in EMC for large size, but no tendency is found up to 103. Results obtained by EMC with relatively small sizes are consistent with those predicted by the cavity analysis.

A thermodynamic phase transition is found in the BM model in a two-dimensional honeycomb lattice.

1RSB feature µhoneycomb

K

∼ 6.8 is very close to µC=3

K

= 6.7... Z(µ) = Z(BP)(1 + ...non-singular part)???

Thank you for your attention.

  • K. Hukushima (U. of Tokyo)

MCMC for glassy physics Santa Fe, September 2009 29 / 29