Condensation in Pair Exclusion Process

Condensation of Networks and Pair Exclusion ProcessJae Dong Noh University of Seoul ()Interdisciplinary Applications of Statistical Physics & Complex Networks(KITPC/ITP-CAS, Feb 28-Apr 1, 2011)CondensationCondensation : 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.htmlRb gasCondensationCondensation : macroscopic number of particles in a single microscopic state condensate

Balls in boxes

[Eggers, PRL 83, 5322 (1999)]

CondensationCondensation : macroscopic number of particles in a single microscopic state condensate

Traffic jam condensate of empty sitesslow carfast carCondensationCondensation : macroscopic number of particles in a single microscopic state condensate

Percolation

condensate = giant clusterLattice 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 ()

aggregateexponentialLattice model: Zero Range ProcessN 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 Tji .123iN

[F. Spitzer, Adv. Math. 5, 246, (1970).] on-site (zero range) interaction quenched disorder graph structureZero Range ProcessFactorized stationary state (little spatial correlation)

Condensation induced byon-site attractions, random pinning potential, structural disorder,...

random walk problemEvans&Hanney, JPA 38 R195 (2005)e.g., network (Noh et al `05)function form of u(l)site-dep. hopping function ui(l)ZRP in D-dim. LatticeInteraction driven condensationJumping rate function : ui(m) = u(m) = 1+b/mPhase Diagramb2

condensedphasenormalphasemacroscopic condensatesms N1ln mln mln mln p(m)ln p(m)ln p(m)

exponentialpower-lawpower-law +condensateZRP in D-dim. Lattice Jumping rate function

a < 1 : condensation alwaysa = 1 : marginal casea > 1 : no condensationnu(n)

ZRP on Scale-Free NetworksOn SF networks with degree distribution

Hopping rate function

nu(n)a < 1a = 1a >1

[Noh et al `05]Condensation of NetworksCoevolving networksInformation flow on networks: independent random walkers on networksFlow 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]Network vs. Driven diffusive systemMapping from a network to a driven diffusive system nodes lattice sitesedges particlesdegree ki occupation number mi edge rewiring particle hopping

Mesoscopic condensationNot macroscopic but mesoscopic condensation

Not a single condensate but many condensatesk N1/2Nhub N1/2

Network vs. Driven diffusive systemMapping from a network to a driven diffusive system

Self-loop constraint pair exclusionPair Exclusion Process : model definitionThere 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 Tji

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 interactionhopping dynamics depending on underlying geometryPEP : configurationConfiguration :

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}

Approximate particle hopping rateHopping rate of a particle from site i to j :

Accepting probability

= 1 (