Su-Chan Park

Absorbing Phase Transitions

1. Complex Patterns in Nature.

2. Universality of Complex Patterns

3. Absorbing Phase Transitions

4. How to treat problems

5. Summary and Future

Talk at KAIST (May 7, 2003)

Self-organized critical (SOC) patterns in open systems

Forest fire models

Diffusion limited aggregation

Sand pile avalanches, crystal growth, biological evolutions, social structuring, stock market fluctuations, earthquakes, landscapes, coastlines, galaxy distributions …

Complex Patterns in Nature

Spatiotemporal critical patterns in dynamic systems

1d NKI model (DI class)

space

time

1d BAW(1) model (DP class)

space

Gary Larson

Universality of Complex Patterns

Pattern classifications

- Fractal dimensions do not depend on details of system Hamiltonians or dynamic evolution

rules.

Fractal dimensions

Universality classes

- Equilibrium critical systems: symmetry, embedding dimensions,

…

2d equilibrium critical patterns: almost complete list is known by conformal field

theory.

[symmetry between ground states in the ordered phase]

Nonequilibrium critical systems ?

- Nonequilibrium phase transition models- Self-organized criticality models

Universality classes are not well established yet.

- symmetry between ground states (?),

- conservation laws (?), - embedding dimensions, ….

Simplest nonequilibrium phase transition models ?

Absorbing Phase Transitions

- Nonequilibrium phase transitions between dead state and live state.

- Configurational phase space

trapped stateSurvival probability : sP

external parameter

dead (absorbing) phase

Live (active) phase

0sP 0sP

- Absorbing state: nonequillibrium steady state farthest from equilibrium (zero measure entropy) : simplest one?- Probabilistic accessibility to each absorbing state determines the symmetry of the system ?

- Example : Contact process (epidemic spreading)

Rule: a particle is annihilated with probability p or creates another particle at a neighboring site with

prob. 1-p

occupied state: infected person

vacant state : healthy person

absorbing state: all lattice sites are empty. [epidemics are over.]

[1d version]

p

absorbing phaseactive phase

sP

pc

)(~ ppP cs

d)3(81.0

d)2(58.0

)d1(277.0

space

time

space

time

Correlation length :

)(~ ppc ]09.1[

||)(~ ppc

Relaxation time :

]73.1[ ||

dynamic exponent :

58.1||

z

),,( || A complete set of relevant scaling exponents

Directed Percolation Universality class

- Directed Percolation (DP)

DP conjecture

Continuous transitions from an active phase into

a single absorbing state should belong to DP class.- Various chemical reaction models,

- Branching annihilation random walk models with one offspring : BAW(1),- Pinning-depinning transitions,- SOC evolution model (Bak+Sneppen),- Roughening, wetting transitions, …..

- Multiple absorbing state models

Nonequilibrium Kinetic Ising (NKI) model

absorbing states : all spins up state, all spins down states.

(two symmetric absorbing states: Z2 symmetry)

Rule: T=0 single spin-flip dynamics with prob. p or T= Kawasaki (pair spin-flip) dynamics with

prob. 1-p

[1d version]

up

da

te

p/2 0p 1-p1-p 1-p

p

absorbing phaseactive phase

pc

Directed Ising (DI) Universality class

- Branching annihilation random walk models with two offspring : BAW(2), [A+A 0, A 3A : mod(2) conservation (parity-conserving class)]- Interacting monomer-dimer model, PCA models, …..

symmetry-breaking field : DI DP

7

13,

4

13,

14

13||

- Infinitely many absorbing states models

Pair Contact Process [1d version]

Rule: a pair of particles is annihilated with probability p or creates another particle at a neighboring site with

prob. 1-pInfinitely many absorbing states: mixture of isolated particles

and vacancies

DP class !!!

[Dimer-dimer models, dimer-trimer models, TTP, DR,…]

Probabilistic accessibility to the absorbing states does not have any explicit symmetry properties, separated by infinite dynamical barriers.

Global Z2 symmetry built-in models DI

- Recent Issues

Pair Contact Process with Diffusion (PCPD)

Coupled to non-diffusive conserved field

- Related to the sand pile models

- New universality class?

Absorbing state : no particle or single particle

- SOC and absorbing phase transitions

How to Treat Problems

- Master Equation

- Quantum Hamiltonian

,

- Numerical Method

- Analytic Method

Monte Carlo Simulation

Langevin Equation and Field Theory of DP

where

DP conjecture

No other symmetry, No conservation, No quenched disorder, No long range interaction : DP universality class.

Summary and Future

- Complex patterns can be characterized by a set of fractal dimensions.

- Fractal dimensions describe singular behaviors of critical systems.

- Absorbing transition models are the simplest models to study and find the most fundamental complex patterns in systems far from equilibrium.

- Wide applicability to various systems in nature.- Need to quest for new type complex patterns and

link to SOC, interface growth, etc.- Need to establish a firm classification scheme of

universality classes.