Condensation of Networks and Pair Exclusion Process

1. Condensation of Networks and Pair Exclusion Process JaeDongNoh 노재 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 Rbgas Condensation : macroscopic number of particles in a single microscopic state  condensate Bose-Einstein condensation of ideal boson gases in 3D s = n0 / N n = 1 – s http://www.colorado.edu/physics/2000/bec/three_peaks.html

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

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

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

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

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

8. ZRP in D-dim. Lattice lnp(m) exponential lnm macroscopic condensates ms»N1 lnp(m) power-law  condensed phase lnm normal phase lnp(m) power-law + condensate lnm b 2 Interaction driven condensation Jumping rate function : ui(m) = u(m) = 1+b/m Phase Diagram

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

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

11. Condensation of Networks [Noh and Rieger `04] condensates = hubs [Kim and Noh `08, `09] • 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.

12. Network vs. Driven diffusive system • Mapping from a network to a driven diffusive system • nodes  lattice sites • edges  particles • degree ki  occupation number mi • edge rewiring  particle hopping

13. Mesoscopic condensation k»N1/2 Nhub»N1/2 Not macroscopic but mesoscopic condensation Not a single condensate but many condensates

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

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

16. 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 = {m1,m2,,mN}

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

18. 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 Wji(m,m’) [Evans&Hanney, Luck&Godreche]

19. PEP in 1D with Symmetric Hopping 1/2 1/2 u(ni) v(mi-1) v(mi+1)

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

21. PEP in 1D with Symmetric Hopping [Schwarzkopf et al `08] Factorized state : where Pair exclusion : cutoff in the mass at mcutoff»N1/2 macroscopic condensation  mesoscopic condensation cf) ZRP with

22. PEP in 1D with Symmetric Hopping Multiple number of mesoscopic condensates mcon» (NlnN)1/2 condensed phase  normal phase b Analytic results A single macroscopic condensate breaks into Nconmesoscopic condensates of mass mcon. 2

23. PEP in 1D with Symmetric Hopping Numerical results

24. PEP in 1D with Asymmetric Hopping 1 u(ni) v(mi+1)

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

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

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