From Classical Critical Phenomena to Species Borders - An...

Classical Critical Phenomena and Phase TransitionsNon-Equilibrium Critical Phenomena

From Classical Critical Phenomenato Species Borders

An Overview

Gunnar Pruessner

Department of PhysicsImperial College London

Eötvös Loránd University, Budapest, March 2006

G. Pruessner (Imperial College London) Critical Phenomena and Species Borders Eötvös University, 03/2006 1 / 24

Outline

1 Classical Critical Phenomena and Phase TransitionsHistorical OverviewClassic Example: Bond PercolationScaling and Finite Size Scaling

2 Non-Equilibrium Critical PhenomenaAbsorbing State Phase TransitionSpecies BordersSummary

Historical OverviewClassic Example: Bond PercolationScaling and Finite Size Scaling

Outline

Classical Critical PhenomenaA Historical Overview I

Andrews (1869): The critical pointVan der Waals (1873): Equation of stateWeiss (1907): FerromagnetismLenz (1920): Ising modelIsing (1925): Solution of 1D-IsingEhrenfest (1933): Classification of transitionsLandau (1937): Unified theory, universalityKramers and Wannier (1941): Tc for 2D-IsingOnsager (1942): Solution of 2D-IsingYang (1952): M(T) for 2D-Ising

Classical Critical PhenomenaA Historical Overview II

Stueckelberg and Petermann (1953): Ultraviolet renormalisationGell-Mann and Low (1954): Renormalisation scheme in QEDDomb and Hunter, and Widom (1965): Scaling hypothesisKadanoff (1966): Generalised scaling and block spinsWilson (1971): Use RG in critical phenomenaWilson and Fisher (1972): Small parameter ε = dc − d

Gross, Politzer and Wilczek (1973): Asymptotic freedom

Outline

Classic Example: Bond PercolationThe Model

On a 2D grid, bonds are active with probability pCluster: set of sites connected through active bondsTemperature-like variable: p (drives transition)

On a 2D grid, bonds are activewith probability p

Cluster: set of sites connectedthrough active bonds

Temperature-like variable: p(drives transition)

Introduce site-site correlation function g(~r1,~r2)

Asymptotically: Exponential decay g(~r1,~r2) ∝ exp(−|~r1 −~r2|/ξ)

At p = pc the correlation length ξ diverges→ No characteristic scale

History:Three dimensional polymers: Flory 1941Mathematics: Hammersley and Broadbent 1954pc = 1/2 conjectured in 1955, proven (Kesten) 1980Renaissance: CFT 1992, SLE 2001

Observables. . .Order parameter θ, percolation probability (coverage by infinitecluster): vanishes in one phase, picks up in the otherCluster size distribution (cluster number density) P(s)Crossing probability Ep (cluster connects two opposite sides)

Observables:Order parameter θ, percolation probability (coverage by infinitecluster): vanishes in one phase, picks up in the otherCluster size distribution (cluster number density) P(s)Crossing probability Ep (cluster connects two opposite sides)

Scanning through the transition

Deep inside disordered phase: Mean Field TheoryCrossover region: Fluctuations start to take overCritical region: Non-trivial scalingCrossover towards ordered phaseWarning: Gaussian Theory (trivial theory) does not order

Outline

Scaling around the critical point (p = pc)The infinite system

Scaling

Divergent correlation length: ξ ∝ |p − pc|−ν

Observables and parameters are related by scaling relationsExample: 〈s〉 ∝ |p − pc|

−(2−τ)/σ

Observables “look the same under rescaling”: . . .Even in infinite systems: Scaling is asymptotic (lower cutoff)Infinite system away from critical point:

P(s) = as−τG(s/ξD)

Universal scaling function: G

Scaling

−(2−τ)/σ

Scaling

−(2−τ)/σ

Observables “look the same under rescaling”:Probability for cluster size s is P(s)Probability for cluster size s ′ = 2s is a multiple of P(s)This multiple is the same, independent of sP(s) does not possess an intrinsic scaleP(s) has the form of a power law: P(s) = as−τ

Even in infinite systems: Scaling is asymptotic (lower cutoff)Infinite system away from critical point:

Scaling

−(2−τ)/σ

Scaling

−(2−τ)/σ

What is so striking about scaling?Part I: Self-similarity!

Central idea: “Power law” means no intrinsic scale

The function ex − 1 on asmall scale

The function ex − 1 on a largescale

The function ex − 1 on asmall scale

The function ex − 1 on a largescale

The function x2 on a smallscale

The function x2 on a largescale

The function x2 on a smallscale

The function x2 on a largescale

What is so striking about scaling?Part I: Self-similarity!Central idea: “Power law” means no intrinsic scale

Trivial powerlaws (can) correspond to dimensional constraintsEven trivial powerlaws (can) have deep physical meaningNon-trivial powerlaws

Need finite (microscopic) scale for dimensional consistencyAre not result of dimensional constraints

Scaling in finite systems

Infinite systems: Original scenarioFinite system: Critical scaling where ξ � LCrossover into finite size scaling region

width: L−1/ν

What is so striking about scaling?Part II: Universality

Scale invariance and UniversalitySingle scale: correlation lengthUniversality classes – universal quantities

ExponentsAmplitude ratiosEffectively same physics everywhere in asymptotia

Many microscopic (interaction) details irrelevant for universalfeaturesHowever: Universality of long range behaviour onlyNon-universal: pc, amplitudes, lower cutoff, amplitude of uppercutoff

Absorbing State Phase TransitionSpecies BordersSummary

Outline

Non-Equilibrium Critical PhenomenaExample: Absorbing State Phase Transition

Absorbing State Phase Transition

System runs “until it hits an absorbing state”Two phases: Eventually in absorbing state or always activeContinuous transition between absorbing and activeProblem: Finite systemsParadigm: directed percolationGrassberger: Single absorbing state? DP!

Absorbing State Phase TransitionContact Process

CP is DP with asynchronous updatesActive site become inactive: rate ε

Inactive site become active: rate c per active neighbourRescale time, so that ε effectively disappears and c → λ = c/ε

Parameter that drives transition: λ → λc = 1.6488 . . .

Outline

Contact Process in Population DynamicsSpatially varying temperature-like variable

Biological systems: Spatially varying λ

(like scanning through all λ simultaneously)λ = λ(x) = λc + λ ′x

Continuum vs. lattice (vanishing vs. finite ξ)

Contact Process in Population DynamicsSpatially varying temperature-like variable

Biological systems: Spatially varying λ

(like scanning through all λ simultaneously)λ = λ(x) = λc + λ ′x

Continuum vs. lattice (vanishing vs. finite ξ)

The Contact Process with λ(x)Part I: The interface

Interface extends inside the disordered region“Autonomous” fluctuations only in ordered phaseAfter long times: Disordered phase fluctuates by invasion only

How to locate/define the interface?Properties? Wetting?

What is the effect of higher order interactions?G. Pruessner (Imperial College London) Critical Phenomena and Species Borders Eötvös University, 03/2006 19 / 24

The Contact Process with λ(x)Part II: Scaling regions

Spatial behaviour reflects critical regions of equilibrium systemWhere L � ξ: critical scalingFSS region: L ≈ ξ. . .

The Contact Process with λ(x)Part III: Identifying the FSS region ∆X

What is the size ∆x of the scaling region?“Naïve” FSS region: ξbulk ≈ L∆x ∝ λ′ −1L−1/ν

Problem: Width of correlated region underestimatedBetter guess(?): Correlated patches

Identify a patch around λc, within which every point can at least“reach back” to λc

∆x ∝ λ′ −ν/(1+ν)

General consideration: Anisotropy, physical space versus latticespace

∆x ∝ λ′ −ν/(1+ν)

What is the size ∆x of the scaling region?Better guess(?): Correlated patches

∆x ∝ λ′ −ν/(1+ν)

General consideration: Anisotropy, physical space versus latticespaceG. Pruessner (Imperial College London) Critical Phenomena and Species Borders Eötvös University, 03/2006 21 / 24

∆x ∝ λ′ −ν/(1+ν)

The Contact Process with λ(x)Part IV: Some Numerical Results

Numerics compatible to “scaling region argument”, but not perfect

Outline

SummaryPhase transition driven by spatial variation of temperature-likevariable: Theoretically very appealingIn different phases, standard methods should applyOpen problem: What is the size and the scaling of the scalingregion?Further numerics needed

Many thanks to Nicholas P. Moloney, Zoltán Rácz and Beáta Oborny!

From Classical Critical Phenomena to Species Borders - An...

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