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

1. 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

2. Our Research Interest Statistical Mechanics of Mesoscopic Complex Chemical Systems • Fluctuation Induced Phenomena • MultiscaleSimulation Methods with Application • Phase Transition Kinetics: Nucleation • Dynamical Self-Assembly of Self-Propelled Particles • Stochastic Thermodynamics and Fluctuation Theorems

3. Outline • Introduction - Physics of network • CGMethod: d-CG - Merging nodes with similar degrees - Using LMF scheme for the CG map - Statistically consistency • Nucleation kinetics - Scale-free network: Size effect - Modular network: Optimal modularity • Summary

4. Physics of Networks Nodes + Links

5. Scale Free Networks • Power-Law Distribution • Preferential Growth • Hub-Leaf: Heterogeneity • Ubiquitous Importance: Social, Physical, Biological, Chemical systems Metabolic Yeast Protein Modularity

6. Critical Phenomena on Networks • Equilibrium System: Ising Model Hamiltonian MC: Metropolis Dynamics Order-Disorder Phase Transition(P.T.) +1 disorder d=1：No P.T. d=2: Theory d=3: MC d≥4: MFT order h=0 -1 Critical Point Critical Exponent Finite Size Effects Network Topology

7. Critical Phenomena on Networks No spreading spreading • Non-Equilibrium System: SIS Model Epidemic Spreading Non-Equilibrium Phase Transition (KMC simulation) Susceptible Infection Rate Recovery Rate Infected Threshold Value N.E. Fluctuation Size Effects Network Topology

8. Part 1: CG method • Question: Promising Method? Microscopic Methods: MC, KMC Macroscopic Methods: MFT Expensive: Limit to small network size and time scale Phenomenological: Lack micro-details and fluctuations ? Coarse-Grained Method: Both Accurate and Efficient

9. CG Scheme：Local Mean Field • Merge micro-nodes into a CG-node Merge CG CG-Network Micro-Network • CG Connectivity: LMF Scheme 0 or 1 (0,1) 1

10. CG Ising Model • CG State Variable: • CG Hamiltonian(Closed at CG level) Intra-cell Inter-cell CG-MC Simulation: Spin-flip

11. Merging: Muitiple ways 1 1 I 4 4 5 2 5 III 2 3 3 II 6 6 1 4 5 2 3 6 II III I Which way is right (better)?

12. CG and Micro Configurations 1 2 3 4 • Degeneracy

13. 1 4 5 2 3 6 Statistical Consistency II 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 III I

14. d-CG: Satisfy CSC • IfMerging nodes with SIMILAR degrees • And using Annealed Network Approximation (ANA) for ensemble-averaged behavior • We can prove Error level ~

15. Details: CG-Ising Model • Insert ANA into H: • Split into Intra- and Inter- parts:

16. Details: CG-Ising Model • For exact d-CG , we have • Using ANA, we have and then • We can prove, for ,

17. 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

18. 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 Metastable Stable Network Topology Nucleation Rate Nucleation Pathway Size Effects ?

19. Nucleation  Rare event Chemical Reaction Nucleation Protein Folding Translocation Path sampling methods

20. Forward Flux Sampling(FFS) A B • 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(ii+1) using racket-like methods See：Enhanced Sampling of Nonequilibrium Steady States, Annu. Rev. Phys. Chem. 2010. 61:441–59

21. Homogeneous Nucleation

22. Homogeneous Nucleation Average degree of the nodes in the nucleus Probability distribution of the cluster size • New phase starts from nodes with smaller degrees • Cluster size of new phase follows power law distribution

23. 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

24. 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

25. Classical Nucleation Theory (CNT) Bulk term: Driven force Surface term: Penalty • Critical point

26. 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 CNT

27. Modular network: 2-steps • Question: How the overall nucleation rate depends on the modularity ?

28. Modified FFS R1 • Usual FFS: Trap to A’ • Modified FFS: Determine A’ adaptively • For two-step: R2 A A’ R B • 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

29. Optimal Modularity Nucleation Rate Free energy barrier and critical nucleus

30. Summary • CG-method Phys.Rev.E 82,011107(2010); 83, 066109(2011) • 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 • Nucleation Phys.Rev.E 83,031110(2010); 83, 046124(2011)

31. Acknowledgements Dr. Hanshuang Chen Dr. Chuansheng Shen Funding: National Science Foundation of China