Non-Equilibrium Phase Transitions an Introduction

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Non-equilibrium phase transitions An Introduction

Lecture III

Haye Hinrichsen

University of Würzburg, Germany

March 2006 Directed Percolation Other Universality Classes

Third Lecture: Outline

1 Directed Percolation Scaling Theory Langevin Equation

2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes

Outline

1 Directed Percolation Scaling Theory Langevin Equation

2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes

Phenomenological Scaling Theory – Scale Invariance

Scaling hypothesis: In the scaling regime the large-scale properties of absorbing phase transitions are invariant under scale transformations (zoom in – zoom out)

∆ → λ ∆ (∆ = p − pc) ρ → λβρ 0 P → λβ P

−ν⊥ ξ⊥ → λ ξ⊥ −νk ξk → λ ξk

0 with the four exponents β, β , ν⊥, νk.

This is called simple scaling (opposed to multiscaling). Directed Percolation Other Universality Classes

Using Scale Invariance

If you have an expression...

multiply the distance from criticality p − pc by λ. multiply all quantities that measure local activity by λβ 0 multiply all quantities that activate sites locally by λβ multiply all lengths by λ−ν⊥ multiply all times by λ−νk

...then this expression should be asymptotically invariant. Directed Percolation Other Universality Classes

Using Scale Invariance

1st Example: Decay of the density at criticality:

−α β −νk −α ρ ∼ t ⇒ λ ρ ∼ (λ t) ⇒ α = β/νk

The same more formally:

ρ(t) = λ−βρ(λ−νk t)

Set λ = t1/νk .

⇒ ρ(t) = t−β/ν⊥ ρ(1) Directed Percolation Other Universality Classes

Using Scale Invariance

2nd Example: Decay of the density for ∆ 6= 0:

ρ(∆, t) = λ−βρ(λ∆, λ−νk t)

Set again λ = t1/νk .

⇒ ρ(∆, t) = t−β/ν⊥ ρ(∆t1/νk , 1)

ρ(∆, t) = t−β/ν⊥ f (∆t1/νk ) This is a scaling form, f (ξ) is called scaling function. Directed Percolation Other Universality Classes

Using Scale Invariance – Scaling function

ρ(∆, t) = t−β/ν⊥ f (∆t1/νk )

The scaling function f (ξ) has certain asymptotic properties:

f (ξ) → const for ξ → 0 f (ξ) ∼ ξβ for ξ → ∞, ∆ > 0 f (ξ) → 0 for ξ → ∞, ∆ < 0

Check scaling and visualize f (ξ) by a data collapse, plotting ρ(∆, t)tβ/ν⊥ versus ∆t1/νk . Directed Percolation Other Universality Classes

Using Scale Invariance – Data Collapse

ρ(∆, t) = t−β/ν⊥ f (∆t1/νk )

0 10 0 10

-1 10 -1 10 α -2 ρ(t) 10 ρ(t) t -2 10 -3 10

-3 10 -4 10 0 1 2 3 4 -4 -3 -2 -1 0 1 10 10 10 10 10 10 10 10 10 10 10 ν t t τ || Directed Percolation Other Universality Classes

Using Scale Invariance

3rd Example: Two-point correlation function:

c(r, τ, ∆) = hs(r1, t1)s(r2, t2)i

Here r = |r2 − r1| and τ = t2 − t1

c(λ−ν⊥ r, λ−νk τ, λ∆) ' λ−2βc(r, τ, ∆)

Setting λ = τ 1/νk we obtain:

c(r, ∆t, ∆) = τ −2β/νk g(r/τ ν⊥/νk , ∆τ 1/νk ) , Directed Percolation Other Universality Classes

Universality of Scaling Functions

The form of scaling functions (apart from metric factors) is universal, e.g., the same in all models for DP.

Scaling functions are as universal as critical exponents.

Like critical exponents scaling functions cannot always be expressed exactly in a closed form.

A universality class is characterized by values of the critical exponents and speciﬁc scaling functions. Directed Percolation Other Universality Classes

Using Scale Invariance

There is much more about scaling:

Empty-interval probability distribution External ﬁeld and ﬂuctuations Finite-size scaling Pair connectedness function Spreading from a seed Early-time regime / critical initial slip

...see lecture notes. Directed Percolation Other Universality Classes

Outline

1 Directed Percolation Scaling Theory Langevin Equation

2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes

Phenomenological Langevin Equation

Let ρ(x, t) be a mesoscopic density of active sites. The Langevin equation for DP reads:

2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t)

Here ξ(x, t) is a density-dependent Gaussian noise with the correlations

hξ(x, t)i = 0 , hξ(x, t)ξ(x0, t0)i = Γ ρ(x, t) δd (x − x0) δ(t − t0)

The amplitude of ξ(x, t) is proportional to pρ(x, t), Directed Percolation Other Universality Classes

DP Langevin Equation: Form of the Noise

Why ξ(x, t) ∝ pρ(x, t)?

The noise describes local ﬂuctuations of the coarse-grained density ρ(x, t) Directed Percolation Other Universality Classes

DP Langevin Equation: Form of the Noise

Each active site creates offspring (+1) or dies (-1), hence generates bounded noise.

Central limit theorem:

Total noise amplitude is proportional to the square-root of the number of noise sources.

⇒ amplitude[ξ(x, t)] ∝ pρ(x, t) Directed Percolation Other Universality Classes

Langevin Equation – Static Mean Field Approximation

2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t)

Stationary state: ∂t ρ(x, t) = 0

0 absorbing state ⇒ ρstat = a/b active state

The parameter a plays the role of ∆ = p − pc. The critical point is ac = 0. The density exponent is βMF = 1. Directed Percolation Other Universality Classes

Langevin Equation – Scale Invariance

Apply scale transformation

ρ → λβρ

−ν⊥ ξ⊥ → λ ξ⊥ −νk ξk → λ ξk

to the Langevin equation

2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t)

The result reads:

β+ν β 2β 2 β+2ν⊥ 2 λ k ∂t ρ(x, t) = λ aρ(x, t) − λ bρ (x, t) + λ D∇ ρ(x, t) 1 (β+dν +ν ) + λ 2 ⊥ k ξ(x, t) Directed Percolation Other Universality Classes

Langevin Equation – Mean Field Critical Exponents

To determine mean ﬁeld exponents require scale invariance:

β+ν β 2β 2 β+2ν⊥ 2 λ k ∂t ρ(x, t) = λ aρ(x, t) − λ bρ (x, t) + λ D∇ ρ(x, t) 1 (β+dν +ν ) + λ 2 ⊥ k ξ(x, t)

The scaling factors drop out if:

a = 0 βMF = 1 MF ν⊥ = 1/2 MF νk = 1

d = 4 Directed Percolation Other Universality Classes

Langevin Equation – Upper Critical Dimension

β+ν β 2β 2 β+2ν⊥ 2 λ k ∂t ρ(x, t) = λ aρ(x, t) − λ bρ (x, t) + λ D∇ ρ(x, t) 1 (β+dν +ν ) + λ 2 ⊥ k ξ(x, t)

⇑ The dimension enters only here.

d noise d>4 irrelevant mean field correct d=4 marginal mean field + log corrections d<4 relevant full field theory needed

dc = 4 is called upper critical dimension. Directed Percolation Other Universality Classes

Langevin Equation – Higher Order Terms

2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t)

Higher-order terms such as

ρ3, ∇4ρ, (∇ρ)2,...

would be irrelevant under rescaling.

This is the origin of universality. Directed Percolation Other Universality Classes

Directed Percolation – Field theory

Backbone of a two-point function:

⇒ Lecture by U. C. Täuber (Monday 11:30) Directed Percolation Other Universality Classes

DP conjecture

Janssen / Grassberger

Any stochastic model with the properties... short-range interactions two-state model single absorbing state spontaneous removal and offspring production no quenched disorder no unconventional symmetries

...belongs to the DP universality class. Directed Percolation Other Universality Classes

Outline

1 Directed Percolation Scaling Theory Langevin Equation

2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes

Parity-conserving Class

Reaction-diffusion schemes:

DP-class A → 0 A → 2A 2A → A + diffusion PC-class 2A → 0 A → 3A + diffusion

PC-processes conserve the number of particles modulo 2.

Example: Branching Annihilating Random Walk with even number of offspring (BARW): Directed Percolation Other Universality Classes

PC Transitions belong to a Different Class

Parity-conserving class : 2A → 0 , A → 3A and diffusion

Two different dynamical sectors.

Absorbing (subcritical) phase governed√ by 2A → 0 decays algebraically, in 1d as 1/ t. Field theory (Cardy, Täuber) with two critical dimensions: dc = 2 for annihilation, dc = 4/3 for branching. Different set of critical exponents in 1d: 0 β = β νk ν⊥ DP ≈ 0.276 ≈ 1.734 ≈ 1.097 PC 0.92(2) 3.22(6) 1.83(3) Directed Percolation Other Universality Classes

Visualizing Scale Invariance:

1/z Plot x/t versus ln t, where z = νk/ν⊥ is the expected dynamical exponent: Directed Percolation Other Universality Classes

Visualizing Scale Invariance in the Critical BARW:

x → λ−ν⊥ x t → λ−νk t ρ → λβρ Directed Percolation Other Universality Classes

Parity-Conserving Class

2A → 0 ,A → 3A and diffusion

Is it only parity conservation that drives the transition away from DP? Directed Percolation Other Universality Classes

Parity-Conserving Class Directed Percolation Other Universality Classes

Outline

1 Directed Percolation Scaling Theory Langevin Equation

2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes

Voter Universality Class

Voter model:

If you can’t make up your mind...

with two states: Z2-symmetry.

simply adopt the opinion of a randomly chosen nearest neighbour. Directed Percolation Other Universality Classes

Voter Model in One Dimension

In 1d the voter model is like Glauber-Ising dynamics at T = 0:

In 1d the kinks between Z2-symmetric domains can be interpreted as particles:

2A → 0 Directed Percolation Other Universality Classes

Voter Model in Two Dimensions

Glauber−Ising model at T=0 Classical voter model Directed Percolation Other Universality Classes

Voter Phase Transitions (Z2-symmetric transitions)

Take the voter model and add interfacial noise: Directed Percolation Other Universality Classes

Voter Phase Transitions

In 1d voter phase transitions belong to the parity-conserving class.

In 2d voter transitions form a new class.

A Langevin equation has been proposed by Hammal et al.:

∂ p ρ = (aρ − bρ3)(1 − ρ2) + D∇2ρ + σ 1 − ρ2ξ ∂t

Z2-symmetric Directed Percolation Other Universality Classes

Outline

1 Directed Percolation Scaling Theory Langevin Equation

2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes

Contact Process by Lévy Flights

motivation:

lattice model: Directed Percolation Other Universality Classes

Lévy Flights in Space

Lévy Flights are random moves over distances r distributed as P(r) ∼ r −d−σ

σ=1.5 Directed Percolation Other Universality Classes

Lévy Flights in Space

Ordinary Diffusion: d ρ(x, t) = D∇2ρ(x, t) dt ∇2eikx = −k 2eikx 1 x2 ⇒ ρ(x, t) = exp − (4πDt)d/2 4Dt Anomalous Diffusion by Lévy ﬂights: d ρ(x, t) = D∇σρ(x, t) dt ∇σeikx = −|k|σeikx 1 Z ⇒ ρ(x, t) = d d k expikx − D|k|σt (2π)d Directed Percolation Other Universality Classes

Lévy Flights and Ordinary Diffusion: Fixed Points

σ > 2 :

1 x2 ρ(x, t) = exp − (4πDt)d/2 4Dt

σ < 2 :

1 Z ρ(x, t) = d d k expikx − D|k|σt (2π)d

Levy−stable distributions Gaussian fixed point σ 0 1 2 3 Directed Percolation Other Universality Classes

Contact Process by Lévy Flights – How it looks like Directed Percolation Other Universality Classes

Contact Process by Lévy Flights – Langevin Equation

Take DP Langevin equation:

2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t)

hξ(x, t)ξ(x0, t0)i = Γ ρ(x, t) δd (x − x0) δ(t − t0)

Add fractional operator for Lévy Flights in space:

2 2 σ ∂t ρ(x, t) = aρ(x, t)−bρ (x, t)+D∇ ρ(x, t)+D∇ ρ(x, t)+ξ(x, t) Directed Percolation Other Universality Classes

Contact Process by Lévy Flights – Mean Field

2 2 σ ∂t ρ(x, t) = aρ(x, t)−bρ (x, t)+D∇ ρ(x, t)+D∇ ρ(x, t)+ξ(x, t)

Power counting yields mean ﬁeld exponents

β = 1, ν⊥ = 1/σ, νk = 1

and the upper critical dimension

dc = 2σ.

Interesting because by choosing σ we can be close to the upper critical dimension, even in a simulation of a one-dimensional system. Directed Percolation Other Universality Classes

Contact Process by Lévy Flights – Field Theory

The Lévy term is not renormalized by loop diagrams.

Exact scaling relation:

νk − ν⊥(σ − d) − 2β = 0

Prediction of the threshold σc, above which DP is recovered.

νk − 2β σc = d + > 2 ν⊥ DP Directed Percolation Other Universality Classes

Contact Process by Lévy Flights – Results ν T 2

ν 1 ||

0 0 0.5 1 1.5 2 2.5 σ Directed Percolation Other Universality Classes

Contact Process by temporal Lévy Flights Directed Percolation Other Universality Classes

Spatio-Temporal Lévy Flights

log t Directed Percolation Other Universality Classes

Contact Process by temporal Lévy Flights

P(τ) ∼ τ −1−κ

Temporal ﬂights controlled by the exponent κ > 0

Guessed Langevin equation:

κ 2 2 ∂t ρ(x, t) + ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t)

κ iωt κ iωt ∂t e = (iω) e

...or we may even combine both. Directed Percolation Other Universality Classes

Spatio-Temporal Lévy Flights – Phase Diagram

L DP 1,5 MFL dominated by spatial Levy flights C

κ 1 A LI I mixed phase dominated by incubation times 0,5 MF B MFI 0 0 0,5 1 1,5 2 2,5 3 σ Directed Percolation Other Universality Classes

The End

Thank you !