su-chan park absorbing phase transitions 1.complex patterns in nature. 2.universality of complex...
TRANSCRIPT
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.