Non-equilibrium phase transitions

Transcription

1 Non-equilibrium phase transitions An Introduction Lecture III Haye Hinrichsen University of Würzburg, Germany March 2006

2 Third Lecture: Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Parity-conserving Class Voter Universality Class DP with long-range interactions

3 Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Parity-conserving Class Voter Universality Class DP with long-range interactions

4 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 p c ) ρ λ β ρ P λ β P ξ λ ν ξ ξ λ ν ξ with the four exponents β, β, ν, ν. This is called simple scaling (opposed to multiscaling).

5 Using Scale Invariance If you have an expression... multiply the distance from criticality p p c by λ. multiply all quantities that measure local activity by λ β multiply all quantities that activate sites locally by λ β multiply all lengths by λ ν multiply all times by λ ν...then this expression should be asymptotically invariant.

6 Using Scale Invariance 1st Example: Decay of the density at criticality: ρ t α λ β ρ (λ ν t) α α = β/ν The same more formally: ρ(t) = λ β ρ(λ ν t) Set λ = t 1/ν. ρ(t) = t β/ν ρ(1)

7 Using Scale Invariance 2nd Example: Decay of the density for 0: ρ(, t) = λ β ρ(λ, λ ν t) Set again λ = t 1/ν. ρ(, t) = t β/ν ρ( t 1/ν, 1) ρ(, t) = t β/ν f ( t 1/ν ) This is a scaling form, f (ξ) is called scaling function.

8 Using Scale Invariance Scaling function ρ(, t) = t β/ν f ( t 1/ν ) The scaling function f (ξ) has certain asymptotic properties: f (ξ) const for ξ 0 f (ξ) ξ β for ξ, > 0 f (ξ) 0 for ξ, < 0 Check scaling and visualize f (ξ) by a data collapse, plotting ρ(, t)t β/ν versus t 1/ν.

9 Using Scale Invariance Data Collapse ρ(, t) = t β/ν f ( t 1/ν ) ρ(t) t ρ(t) t α t τ ν

10 Using Scale Invariance 3rd Example: Two-point correlation function: c(r, τ, ) = s(r 1, t 1 )s(r 2, t 2 ) Here r = r 2 r 1 and τ = t 2 t 1 c(λ ν r, λ ν τ, λ ) λ 2β c(r, τ, ) Setting λ = τ 1/ν we obtain: c(r, t, ) = τ 2β/ν g(r/τ ν /ν, τ 1/ν ),

11 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.

12 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.

13 Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Parity-conserving Class Voter Universality Class DP with long-range interactions

14 Phenomenological Langevin Equation Let ρ(x, t) be a mesoscopic density of active sites. The Langevin equation for DP reads: t ρ(x, t) = aρ(x, t) bρ 2 (x, t) + D 2 ρ(x, t) + ξ(x, t) Here ξ(x, t) is a density-dependent Gaussian noise with the correlations ξ(x, t) = 0, ξ(x, t)ξ(x, t ) = Γ ρ(x, t) δ d (x x ) δ(t t ) The amplitude of ξ(x, t) is proportional to ρ(x, t),

15 DP Langevin Equation: Form of the Noise Why ξ(x, t) ρ(x, t)? The noise describes local fluctuations of the coarse-grained density ρ(x, t)

16 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)] ρ(x, t)

17 Langevin Equation Static Mean Field Approximation t ρ(x, t) = aρ(x, t) bρ 2 (x, t) + D 2 ρ(x, t) + ξ(x, t) Stationary state: t ρ(x, t) = 0 { 0 absorbing state ρ stat = a/b active state The parameter a plays the role of = p p c. The critical point is a c = 0. The density exponent is β MF = 1.

18 Langevin Equation Scale Invariance Apply scale transformation ρ λ β ρ ξ λ ν ξ ξ λ ν ξ to the Langevin equation t ρ(x, t) = aρ(x, t) bρ 2 (x, t) + D 2 ρ(x, t) + ξ(x, t) The result reads: λ β+ν t ρ(x, t) = λ β aρ(x, t) λ 2β bρ 2 (x, t) + λ β+2ν D 2 ρ(x, t) + λ 1 2 (β+dν +ν ) ξ(x, t)

19 Langevin Equation Mean Field Critical Exponents To determine mean field exponents require scale invariance: λ β+ν t ρ(x, t) = λ β aρ(x, t) λ 2β bρ 2 (x, t) + λ β+2ν D 2 ρ(x, t) + λ 1 2 (β+dν +ν ) ξ(x, t) The scaling factors drop out if: a = 0 β MF = 1 ν MF = 1/2 ν MF = 1 d = 4

20 Langevin Equation Upper Critical Dimension λ β+ν t ρ(x, t) = λ β aρ(x, t) λ 2β bρ 2 (x, t) + λ β+2ν D 2 ρ(x, t) + λ 1 2 (β+dν +ν ) 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 d c = 4 is called upper critical dimension.

21 Langevin Equation Higher Order Terms t ρ(x, t) = aρ(x, t) bρ 2 (x, t) + D 2 ρ(x, t) + ξ(x, t) Higher-order terms such as ρ 3, 4 ρ, ( ρ) 2,... would be irrelevant under rescaling. This is the origin of universality.

22 Directed Percolation Field theory Backbone of a two-point function: Lecture by U. C. Täuber (Monday 11:30)

23 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.

24 Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Parity-conserving Class Voter Universality Class DP with long-range interactions

25 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):

26 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: d c = 2 for annihilation, d c = 4/3 for branching. Different set of critical exponents in 1d: β = β ν ν DP PC 0.92(2) 3.22(6) 1.83(3)

27 Visualizing Scale Invariance: Plot x/t 1/z versus ln t, where z = ν /ν is the expected dynamical exponent:

28 Visualizing Scale Invariance in the Critical BARW: x λ ν x t λ ν t ρ λ β ρ

29 Parity-Conserving Class 2A 0, A 3A and diffusion Is it only parity conservation that drives the transition away from DP?

30 Parity-Conserving Class

31 Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Parity-conserving Class Voter Universality Class DP with long-range interactions

32 Voter Universality Class Voter model: If you can t make up your mind... with two states: Z 2 -symmetry. simply adopt the opinion of a randomly chosen nearest neighbour.

33 Voter Model in One Dimension In 1d the voter model is like Glauber-Ising dynamics at T = 0: In 1d the kinks between Z 2 -symmetric domains can be interpreted as particles: 2A 0

34 Voter Model in Two Dimensions Glauber Ising model at T=0 Classical voter model

35 Voter Phase Transitions (Z 2 -symmetric transitions) Take the voter model and add interfacial noise:

36 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.: t ρ = (aρ bρ3 )(1 ρ 2 ) + D 2 ρ + σ 1 ρ 2 ξ Z 2 -symmetric

37 Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Parity-conserving Class Voter Universality Class DP with long-range interactions

38 Contact Process by Lévy Flights motivation: d lattice model:

39 Lévy Flights in Space Lévy Flights are random moves over distances r distributed as P(r) r d σ σ=1.5

40 Lévy Flights in Space Ordinary Diffusion: d dt ρ(x, t) = D 2 ρ(x, t) 2 e ikx = k 2 e ikx ρ(x, t) = 1 ( (4πDt) d/2 exp x 2 4Dt Anomalous Diffusion by Lévy flights: d dt ρ(x, t) = D σ ρ(x, t) σ e ikx = k σ e ikx ρ(x, t) = 1 (2π) d d d k exp ( ikx D k σ t ) )

41 Lévy Flights and Ordinary Diffusion: Fixed Points σ > 2 : ρ(x, t) = 1 ( (4πDt) d/2 exp x 2 ) 4Dt σ < 2 : ρ(x, t) = 1 (2π) d d d k exp ( ikx D k σ t ) Levy stable distributions Gaussian fixed point σ

42 Contact Process by Lévy Flights How it looks like

43 Contact Process by Lévy Flights How it looks like

44 Contact Process by Lévy Flights Langevin Equation Take DP Langevin equation: t ρ(x, t) = aρ(x, t) bρ 2 (x, t) + D 2 ρ(x, t) + ξ(x, t) ξ(x, t)ξ(x, t ) = Γ ρ(x, t) δ d (x x ) δ(t t ) Add fractional operator for Lévy Flights in space: t ρ(x, t) = aρ(x, t) bρ 2 (x, t)+d 2 ρ(x, t)+d σ ρ(x, t)+ξ(x, t)

45 Contact Process by Lévy Flights Mean Field t ρ(x, t) = aρ(x, t) bρ 2 (x, t)+d 2 ρ(x, t)+d σ ρ(x, t)+ξ(x, t) Power counting yields mean field exponents β = 1, ν = 1/σ, ν = 1 and the upper critical dimension d c = 2σ. Interesting because by choosing σ we can be close to the upper critical dimension, even in a simulation of a one-dimensional system.

46 Contact Process by Lévy Flights Field Theory The Lévy term is not renormalized by loop diagrams. Exact scaling relation: ν ν (σ d) 2β = 0 Prediction of the threshold σ c, above which DP is recovered. σ c = d + ν 2β ν > 2 DP

47 Contact Process by Lévy Flights Results 2 ν T 1 ν β

48 Contact Process by temporal Lévy Flights

49 Spatio-Temporal Lévy Flights log t

50 Contact Process by temporal Lévy Flights P(τ) τ 1 κ Temporal flights controlled by the exponent κ > 0 Guessed Langevin equation: κ t ρ(x, t) + t ρ(x, t) = aρ(x, t) bρ 2 (x, t) + D 2 ρ(x, t) + ξ(x, t) κ t e iωt = (iω) κ e iωt...or we may even combine both.

51 Spatio-Temporal Lévy Flights Phase Diagram 2 1,5 MFL L dominated by spatial Levy flights C DP κ 1 0,5 A MF LI mixed phase 0 0 0,5 1 1,5 2 2,5 3 B I dominated by incubation times MFI

52 The End Thank you!