Phase transition on complex networks: Coarse-Grained simulation methods and nucleation kinetics

Zhonghuai Hou (侯中怀 )

Department of Chemical Physics &Hefei National Lab for Physical Science at Microscale,

University of Science & Technology of China

2012.8.12 Dalian

Our Research Interest

Statistical Mechanics of Mesoscopic Complex Chemical Systems

Fluctuation Induced Phenomena Multiscale Simulation Methods with Application Phase Transition Kinetics: Nucleation Dynamical Self-Assembly of Self-Propelled Particles Stochastic Thermodynamics and Fluctuation

Theorems

Outline1. Introduction

- Physics of network2. CG Method: d-CG

- Merging nodes with similar degrees- Using LMF scheme for the CG map- Statistically consistency

3. Nucleation kinetics- Scale-free network: Size effect- Modular network: Optimal modularity

4. Summary

1998 Nature

1999 Science

2002 RMP

2006 Phys. Rep. 2008 RMP

Physics of Networks

Nodes + Links

Scale Free Networks Power-Law Distribution Preferential Growth Hub-Leaf: Heterogeneity Ubiquitous Importance: Social,

Physical, Biological, Chemical systems

Metabolic Yeast Protein Modularity

Critical Phenomena on Networks Equilibrium System: Ising Model

Hamiltonianij i j i

i j i

H J A s s h s

+1

-1 h=0

MC: Metropolis Dynamics ' 1,expW s s Min E

Order-Disorder Phase Transition(P.T.)

order

disorder d=1： No P.T.d=2: Theoryd=3: MCd≥4: MFT

Critical PointCritical Exponent Finite Size Effects

Network Topology

Critical Phenomena on Networks Non-Equilibrium System: SIS Model

0.0 0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

0.4

0.5

dens

ity o

f sic

k po

pula

tion

No spreading

spreading

Non-Equilibrium Phase Transition(KMC simulation)

Threshold ValueN.E. Fluctuation Size Effects

Network Topology

Epidemic Spreading

S I I I

I S

Susceptible Infection Rate

Recovery RateInfected

Part 1: CG method Question: Promising Method?

Microscopic Methods: MC, KMC

Expensive: Limit to small network size and time scale

Macroscopic Methods: MFT

Phenomenological: Lack micro-details and fluctuations

Coarse-Grained Method:Both Accurate and Efficient

?

Merge CG

cij vA A

, ;

,

2 ,1

2 .

iji j C i jc

iji C j C

A ifq q

AA if

q q

Merge micro-nodes into a CG-node q C

0 or 1 (0,1)

CG Connectivity: LMF SchemeMicro-Network CG-Network

1

ijAcA

① CG Scheme： Local Mean Field

② CG Ising Model CG State Variable: CG Hamiltonian(Closed at CG level)

ii Cs

2

2c c cJH A q J A h

Intra-cell Inter-cell

( )prob 2 Min ,1cEcW n e

( 2) ( ) 2 (2 2) 2c c c c cE H H J A JA h

CG-MC Simulation: Spin-flip

Merging: Muitiple ways1

23

4

5

61

23

5

6

4

II

I

III

1

23

5

6

4

II

I III

0 1 2 01 2 1 1 60 1 6 1

cA

0 1 2 1 31 2 1 1 31 3 1 3 2 3

cA

Which way is right (better)?

CG and Micro Configurations

1 2

3 4

{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}

{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}{ 1, 1, 1, 1}

{ 2, 2}{ 2, 0}{ 2, 2}{0, 2}{0, 0}{0, 2}{ 2, 2}{ 2, 0}{ 2, 2}

! ! !ug q n n

Degeneracy

is is

③ Statistical Consistency

Condition of statistical consistency (CSC) The probability to find any given CG configuration

in equilibrium should be the same when calculated from the CG- or the Micro-model

II

I III

1

23

4

5

6

is

exp c

c

H

Z exp iH s

Z

! ! !ug q n n

g is

④ d-CG: Satisfy CSC If Merging nodes with SIMILAR degrees

2 2 1d D 22 1=

i C i

ii C

d D d

d dq

And using Annealed Network Approximation (ANA) for ensemble-averaged behavior

ij i jA d d DN We can prove

,ci iH s H s ~

2

2 1

d

D q

Error level

expexp= i

cis

c

H sg H

Z Z

Details: CG-Ising Model Insert ANA into H:

i j i j ii j i

JH d d s s h sDN

Split into Intra- and Inter- parts:

, ,

intra inter ex

c cN N

i j i j i j i j ii j i j i

i j C i C j C

JH d d s s d d s s h sDN

H H H

2 2intra

, ; ,

inter,

2

v

i j i ji j C i j i j C

i ji C j C

J JH D s s D s s qDN DN

JH D D s sDN

2ii Cs

Details: CG-Ising Model For exact d-CG , we have ,i C i Cd D d D

Using ANA, we have and then cA D D DN

2

2 2

2

2

c c cJH A q J A h

J D q J D D hDN

exH h

We can prove, for ,

H

1 cH H

Results d-CG shows excellent agreements

with the micro-MC results even with rather small CG-network size

CG model with random-merging scheme (dotted lines) fails

d-CG reproduce both the phase transition point and the fluctuation properties

d-CG can study the size effect very efficiently

The phase transition point diverges in the thermodynamic limit

Also apply to Non-Equi. SIS model Can be extended to general

weighted networks: s-CG

Part 2: Nucleation Kinetics Ising model on SF network: Tc diverges Hint: Large spin-cluster hard to change state Question: Phase transition kinetics ? We consider: Nucleation Process

Initial state: h>0, most spins up At t=0, suddenly change to h<0 Up-state becomes metastable Up Down: Nucleation

Nucleation RateNucleation PathwaySize Effects

Network Topology ?

Metastable

Stable

Nucleation Rare event

Chemical Reaction Nucleation Protein Folding Translocation

 

Path sampling methods

Forward Flux Sampling(FFS)

1

10

|n

AB A i ii

k P

AB

See： Enhanced Sampling of Nonequilibrium Steady States, Annu. Rev. Phys. Chem. 2010. 61:441–59

Stage 1: Calculate flux out of A state by dividing the number of crossings N0 by the total simulation time

Stage 2: Calculate the transition prob. P(ii+1) using racket-like methods

Homogeneous Nucleation

3

0

~

10006

2.590.7130880m

p k k

Nk

Th

Homogeneous Nucleation

New phase starts from nodes with smaller degrees Cluster size of new phase follows power law distribution

Average degree of the nodes in the nucleus:newk :cp N Probability distribution

of the cluster size

Critical Nucleus Committor probability : The average probability to

reach B before returning to A Critical nucleus : Committor distribution: Peak at 0.5 good RxC One may also use umbrella sampling (US) to get

( )B iP

c 0.5B cp

1

1( ) |n

B i j jj i

P P

c

474FFSc 451US

c

hom ~ exp cR F

Size effects

Both critical nucleus size and free energy barrier increase linearly with network size N

Homogeneous nucleation rate decreases exponentially with N Harder to nucleate in heterogeneous networks Nucleation is only relevant in finite size system

  

 

Classical Nucleation Theory (CNT) 

Bulk term: Driven force Surface term: Penalty 

2 NJ kN

Critical point

0

F

hom ~ exp cR F

Heterogeneous Nucleation Setup: Fix w seeds with down spins Two different ways: 1) Choose w nodes randomly 2) Choose w target nodes with the largest degrees

homln ~hetR R w 2 1homln ~hetR R w CNT

Modular network: 2-steps 

Question: How the overall nucleation rate depends on the modularity ?

Modified FFS

A

B

A’

R1

R2 Usual FFS: Trap to A’ Modified FFS: Determine A’

adaptively For two-step: 1 1 1

1 2R R R R

Monitor the sampling time for the probability p(i->i+1) between neighboring interfaces

If 1) and 2) , consider the interface i as the intermediate state A’

If such condition cannot be satisfied in the whole route to B, then the nucleation follows one-step process

it

1i it t 1 1p i i

Optimal ModularityNucleation Rate Free energy barrier

and critical nucleus

Summary CG-method

Nucleation

Condition of Statistical Consistency The d-CG approach: 1) CG-Map: LMF scheme 2) Merging: SIMILAR degree Size effect: TC diverges in the

thermodynamic limit on SF network

Phys.Rev.E 82,011107(2010); 83, 066109(2011)

Phys.Rev.E 83,031110(2010); 83, 046124(2011)

Acknowledgements

Funding: National Science Foundation of China

Dr. Hanshuang Chen Dr. Chuansheng Shen