1. Condensation of Networks and Pair Exclusion Process Jae Dong Noh 노 재 동 盧 載 東 University of Seoul ( 서울市立大學校 ) Interdisciplinary Applications of Statistical Physics & Complex Networks (KITPC/ITP-CAS, Feb 28-Apr 1, 2011)

2. Condensation Condensation : macroscopic number of particles in a single microscopic state  condensate Bose-Einstein condensation of ideal boson gases in 3D  s = n 0 / N  n = 1 –  s http://www.colorado.edu/physics/2000/bec/three_peaks.html Rb gas

3. Condensation Condensation : macroscopic number of particles in a single microscopic state  condensate Balls in boxes [Eggers, PRL 83, 5322 (1999)]

4. Condensation Condensation : macroscopic number of particles in a single microscopic state  condensate Traffic jam condensate of empty sites slow carfast car

5. Condensation Condensation : macroscopic number of particles in a single microscopic state  condensate Percolation condensate = giant cluster

6. Lattice models : Conserved Mass Aggregation [Majumdar et al `98] Mean field theory (Rajesh et al `01) Bias (Rajesh et al `02) Networks (Kwon and Kim `06) Open boundary (Ha et al `08) hopping (1) chipping (  ) aggregate exponential

7. Lattice model: Zero Range Process N sites (i=1, ,N) Mass m i (=0,1,  ) at each site i Jumping rate u i (m) Stochastic matrix Dynamics A particle jumps out of site i at the rate u i (m i ), and then hops to a site j selected with probability T j à i. 1 2 3 i N [F. Spitzer, Adv. Math. 5, 246, (1970).] on-site (zero range) interaction quenched disorder graph structure

8. Zero Range Process Factorized stationary state (little spatial correlation) Condensation induced by on-site attractions, random pinning potential, structural disorder,...  random walk problem Evans&Hanney, JPA 38 R195 (2005) e.g., network (Noh et al `05)function form of u(l)site-dep. hopping function u i (l)

9. ZRP in D-dim. Lattice Interaction driven condensation Jumping rate function : u i (m) = u(m) = 1+b/m Phase Diagram b  2 condensed phase normal phase macroscopic condensates m s » N 1 ln m ln p(m) exponential power-law power-law + condensate

10. ZRP in D-dim. Lattice Jumping rate function a < 1 : condensation always a = 1 : marginal case a > 1 : no condensation n u(n)u(n)

11. ZRP on Scale-Free Networks On SF networks with degree distribution Hopping rate function n u(n)u(n) a < 1 a = 1 a >1 [Noh et al `05]

12. Condensation of Networks Coevolving networks – Information flow on networks : independent random walkers on networks – Flow pattern is determined by the underlying network structure : e.g., (visiting frequency to a site i) / (degree of i) – Edge rewiring dynamics : The busier, the robuster. condensates = hubs [Kim and Noh `08, `09] [Noh and Rieger `04]

13. Network vs. Driven diffusive system Mapping from a network to a driven diffusive system – nodes  lattice sites – edges  particles – degree k i  occupation number m i – edge rewiring  particle hopping

14. Mesoscopic condensation Not macroscopic but mesoscopic condensation Not a single condensate but many condensates k » N 1/2 N hub » N 1/2

15. Network vs. Driven diffusive system Mapping from a network to a driven diffusive system Self-loop constraint  pair exclusion

16. Pair Exclusion Process : model definition There are M/2 distinct pairs of particles distributed over N sites. Dynamics 1.A particle jumps out of a site i at the rate u i (m i ). 2.It tries to hop to a site j selected with the probability T j à i 3.If the hopping particle finds its partner at site j, then the hopping is rejected.  Pair Exclusion (weak) (  k,  k ) {k=1, ,M/2} Non-Zero-Range Process : not solvable on-site interaction hopping dynamics depending on underlying geometry

17. PEP : configuration Configuration : constraint : i(  l )  i(  l ) for all pairs l=1, ,M/2 A particle at site i can hop to j only if the target site is not occupied by its enemy/partner Assumption : the particle species distribution is uncorrelated and random so that a configuration is only specified with the occupation number distribution  Configuration : m = {m 1,m 2, ,m N }

18. Approximate particle hopping rate Hopping rate of a particle from site i to j : Accepting probability = 1 ( <1 to cover soft exclusion) For large M =  N, rejection probability ij

19. Solvability of the PEP Approximate hopping rate for the PEP Necessary condition for solvability – v(m) = c (zero range process) : factorized state for any hopping matrix T – u(m) = c (target process) [Luck&Godreche `07] : factorized state when T satisfies the detailed balance – Both u(m) and v(m) are not a constant function [Kim&Lee&Noh `10] : factorized state when T satisfies the detailed balance – General forms of W ji (m,m’) [Evans&Hanney, Luck&Godreche]

20. PEP in 1D with Symmetric Hopping u(ni)u(ni) 1/2 v(m i+1 ) v(m i-1 )

21. PEP in 1D with Symmetric Hopping Solvable with the approximate hopping rate Factorized state Solvable PEP with symmetric hopping ' ZRP with

22. PEP in 1D with symmetric hopping Canonical hopping rate u i (m) = 1+b/m Approximate rejection probability seems to be okay.

23. PEP in 1D with Symmetric Hopping Factorized state : where Pair exclusion : cutoff in the mass at m cutoff » N 1/2 macroscopic condensation  mesoscopic condensation cf) ZRP with [Schwarzkopf et al `08]

24. PEP in 1D with Symmetric Hopping Analytic results A single macroscopic condensate breaks into N con mesoscopic condensates of mass m con. b  2 condensed phase normal phase Multiple number of mesoscopic condensates m con » (N ln N) 1/2

25. PEP in 1D with Symmetric Hopping Numerical results

26. PEP in 1D with Asymmetric Hopping u(ni)u(ni) 1 v(m i+1 )

27. PEP in 1D with Asymmetric Hopping Non-solvable Factorization fails Numerical simulations mesoscopic condensation Same type of mesoscopic condensation m con ~ (NlnN) 1/2 N con ~ (N/lnN) 1/2

28. PEP in 1D with Asymmetric Hopping Factorization fails  Spatial correlation Space-time plot Clustering of condensates

29. Other systems with pair exclusion Cluster aggregation : two clusters merge into a larger one successively under the pair exclusion constraint Cluster merging probability : disorder-order transition  disorder-criticality transition giant cluster density mean cluster size

30. Summary Why are there hubs? : feedback between structure and dynamics Single macroscopic condensate vs. many mesoscopic condensate Self-loop constraint Pair Exclusion Process

31. Kim, Beom Jun 김 범 준