1. Absorbing Phase Transitions
Su-Chan Park
Absorbing Phase Transitions
Complex Patterns in Nature.
Universality of Complex Patterns
How to treat problems
Summary and Future
Talk at KAIST (May 7, 2003)

2. Complex Patterns in Nature
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 …

3. Spatiotemporal critical patterns in dynamic systems
time
1d BAW(1) model (DP class)
space
1d NKI model (DI class)
space

4. Gary Larson

5. Universality of Complex Patterns
Pattern classifications
Fractal dimensions
- Fractal dimensions do not depend on details of
system Hamiltonians or dynamic evolution rules.
Universality classes
Equilibrium critical systems:
symmetry, embedding dimensions, …
[symmetry between ground states in the ordered phase]
2d equilibrium critical patterns:
almost complete list is known by conformal field theory.

6. 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 ?

7. Absorbing Phase Transitions
Nonequilibrium phase transitions between
dead state and live state.
Configurational phase space
trapped state
Survival probability :
external parameter
dead (absorbing) phase
Live (active) phase
- 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 ?

8. Example : Contact process (epidemic spreading)
[1d version]
occupied state: infected person
vacant state : healthy person
Rule: a particle is annihilated with probability p
or creates another particle at a neighboring site with prob. 1-p
absorbing state: all lattice sites are empty. [epidemics are over.] p
absorbing phase
active phase pc

9. Directed Percolation Universality class
space
time
space
time
Correlation length :
Relaxation time :
dynamic exponent :
A complete set of relevant scaling exponents
Directed Percolation Universality class

10. 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, …..

11. Multiple absorbing state models
Nonequilibrium Kinetic Ising (NKI) model
[1d version]
Rule: T=0 single spin-flip dynamics with prob. p
or T= Kawasaki (pair spin-flip) dynamics with prob. 1-p
update
p/2 p
1-p
absorbing states : all spins up state, all spins down states.
(two symmetric absorbing states: Z2 symmetry) p
absorbing phase
active phase pc

12. 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

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-p
Infinitely 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

14. - Recent Issues
Pair Contact Process with Diffusion (PCPD)
- New universality class?
Absorbing state : no particle or single particle
Coupled to non-diffusive conserved field
- Related to the sand pile models
- SOC and absorbing phase transitions

15. How to Treat Problems
- Master Equation
- Quantum Hamiltonian ,

16. Monte Carlo Simulation
- Numerical Method
Monte Carlo Simulation
- Analytic Method
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.

17. 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.