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)

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 …

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

Gary Larson

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.

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 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 ?

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

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

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 [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

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

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

- 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

How to Treat Problems - Master Equation - Quantum Hamiltonian ,

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.

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.