Hidden dynamical symmetries in ageing phenomena Malte Henkel - - PowerPoint PPT Presentation

Hidden dynamical symmetries in ageing phenomena

Malte Henkel

Laboratoire de Physique de Mat´ eriaux Universit´ e Henri Poincar´ e Nancy I, France

collaborators: M. Pleimling

• F. Baumann, X. Durang, S.B. Dutta,
• M. Ebbinghaus, H. Grandclaude

reviews: J. Phys. Cond. Matt. 19, 065101 (2007); J. Stat. Mech. P07015 (2007)

KIAS Seoul, 1st of July 2008

Contents :

• I. Ageing phenomena

physical ageing ; scaling behaviour and exponents

• II. Hidden dynamical symmetries

Local scaling with z = 2 ; stochastic field-theory ; computation

• f response and correlation functions
• III. Local scale-invariance for z = 2

Mass terms ; integrability ; test through responses and correlators in several models

• IV. Test case : 2D disordered Ising model
• V. Conclusions

• I. Ageing phenomena

why do materials ‘look old’ after some time ? which (reversible) microscopic processes lead to such macroscopic effects ? physical ageing known since historical (or prehistorical) times systematic studies first in glassy systems

Struik 78

a priori behaviour should depend on entire prehistory but evidence for reproducible and universal behaviour

for better conceptual understanding : study ageing first in simpler systems (i.e. disordered ferromagnets) ageing : defining characteristics and symmetry properties :

1 slow dynamics (i.e. non-exponential relaxation) 2 breaking of time-translation invariance 3 dynamical scaling

new evidence for larger, local scaling symmetries

Struik 78

• 1. observe slow relaxation after quenching PVC from melt to low T
• 2. creep curves depend on waiting time te and creep time t
• 3. find master curve for all (t, te) −

→ dynamical scaling → three defining properties of physical ageing

master curves of distinct materials are identical − → Universality ! good for theorists . . .

Struik 78

conceptual confirmation in phase-ordering : Allen-Cahn equation

easier to study : ageing in simple systems without disorder consider a simple magnet (ferromagnet, i.e. Ising model)

1 prepare system initially at high temperature T ≫ Tc > 0 2 quench to temperature T < Tc (or T = Tc)

→ non-equilibrium state

3 fix T and observe dynamics

Bray 94

competition : at least 2 equivalent ground states local fields lead to rapid local ordering no global order, relaxation time ∞ formation of ordered domains, of linear size L = L(t) ∼ t1/z dynamical exponent z universal Allen-Cahn equation v = −(d − 1)K for domain walls

Snapshots of spin configurations in several 2D/3D Ising models quenched to T < Tc, for three different times t = 25, 100, 225. Left : pure Middle : disordered Right : 3D spin glass

Scaling behaviour & exponents

single relevant time-dependent length scale L(t) ∼ t1/z

Bray 94, Janssen et al. 92, Cugliandolo & Kurchan 90s, Godr` eche & Luck 00, . . .

correlator C(t, s; r) := φ(t, r)φ(s, 0) = s−bfC t s , r (t − s)1/z

• response R(t, s; r)

:= δφ(t, r) δh(s, 0)

• h=0

= s−1−afR t s , r (t − s)1/z

• Surprise : scaling behaviour far away from the critical point,

in the entire phase T < Tc ?

How to understand these scaling forms → mean-field

Langevin eq. for order parameter m(t) dm(t) dt = 3λ2m(t) − m(t)3 + η(t) , η(t)η(s) = 2Tδ(t − s) contrˆ

• le parameter λ2 :

(1) λ2 > 0 : T < Tc, (2) λ2 = 0 : T = Tc, (3) λ2 < 0 : T > Tc two-time observables : response R(t, s), correlation C(t, s) R(t, s) = δm(t) δh(s)

• h=0

= 1 2T m(t)η(s) , C(t, s) = m(t)m(s) mean-field equation of motion : ∂tR(t, s) = 3

• λ2 − v(t)
• R(t, s) + δ(t − s)

∂sC(t, s) = 3

• λ2 − v(s)
• C(t, s) + 2TR(t, s)

with variance v(t) = m(t)2, ˙ v(t) = 6

• λ2 − v(t)
• v(t)

se λ2 ≥ 0 : fluctuations persist se λ2 < 0 : fluctuations disappear in the long-time limit t, s → ∞ : (t > s) R(t, s) ≃    1

• s/t

e−3|λ2|(t−s) ; C(t, s) ≃ T      2 min(t, s) ; λ2 > 0 s

• s/t

; λ2 = 0

1 (3|λ2|)e−3|λ2| |t−s|

; λ2 < 0 fluctuation-dissipation ratio measures distance from equilibrium X(t, s) = TR(t, s) ∂sC(t, s) ≃    1/2 + O(e−6λ2s) ; λ2 > 0 2/3 ; λ2 = 0 1 + O(e−|λ2| |t−s|) ; λ2 < 0 relaxation far from equilibrium, when X = 1, if λ2 ≥ 0 (T ≤ Tc)

Consequences : If λ2 > 0 : free random walk, the system never reaches equilibrium ! If λ2 = 0 : slow relaxation, because of critical fluctuations In both situations : observe

1 slow dynamics (non-exponential relaxation) 2 time-translation-invariance broken 3 dynamical scaling behaviour

− → the conditions for physical ageing are all satisfied if T ≤ Tc − → the system remains out of equilibrium If λ2 < 0 : rapid relaxation, with finite relaxation time τrel ∼ 1/|λ2|, towards unique equilibrium state

Return to scaling forms :

correlator C(t, s; r) = s−bfC t s , r (t − s)1/z

• response R(t, s; r)

= s−1−afR t s , r (t − s)1/z

• values of exponents : equilibrium correlator → classes S and L

Ceq(r) ∼ exp(−|r|/ξ) |r|−(d−2+η) = ⇒ class S class L = ⇒ a = 1/z a = (d − 2 + η)/z if T < Tc : z = 2 and b = 0 if T = Tc : z = zc and b = a for y → ∞ : fC,R(y, 0) ∼ y−λC,R/z, λC,R independent exponents Question : general arguments to find form of scaling functions ?

Tests of dynamical scaling : 3D Ising model, T < Tc

no time-translation invariance dynamical scaling C(t, s) : autocorrelation function, quenched to T < Tc scaling regime : t, s ≫ τmicro and t − s ≫ τmicro

Fluctuation-dissipation theorem

Ageing goes on far from equilibrium ! No fluctuation-dissipation theorem : R(t, s; r)=T∂C(t, s; r)/∂s rather use fluctuation-dissipation ratio to measure distance from equilibrium

Cugliandolo, Kurchan, Parisi 94

X(t, s) := TR(t, s) ∂sC(t, s) At equilibrium : X(t, s) = 1. Otherwise, X(t, s) = 1. Experimentalists often use effective temperature Teff := T/X(t, s) Teff is not a thermodynamic ensemble quantity, since it may depend on the observable

Calabrese & Gambassi 04

Experimental examples for the breaking of the FDT I

spin glass CdCr1.7In0.3S4, quenched to T/Tc = 0.8

Herisson & Ocio 02

trace susceptibility χZFC(t, s) = t

s du R(t, u) over against C(t, s)

sub-ageing corretions to scaling ? for C ≈ 1, straight line with slope −1/T in χ − C plot

Experimental examples for the breaking of the FDT II

mechanical response of a colloidal suspension of PMMA measure diffusive motion and drift (under an external perpendicular field – 2D sample) find subageing with truly small µ ∼ 0.48

Makse et al 06

• II. Hidden dynamical symmetries

A) Langevin equation (model A of Hohenberg & Halperin 77) 2M∂φ ∂t = ∆φ − δV[φ] δφ + η

• rder-parameter φ(t, r) non-conserved

M : kinetic co´ efficient V : Landau-Ginsbourg potential η : gaussian noise, cantered and with variance η(t, r)η(t′, r′) = 2Tδ(t − t′)δ(r − r′) fully disordered initial conditions (centred gaussian noise) B) master equation

e.g. Glauber 63

i.e. kinetic Ising model with heat-bath dynamics random initial state → relaxation towards equilibrium stationary states

Local scaling with z = 2 → LSI

Question : extended dynamical scaling for given z = 1 ? MH 92, 94, 02

motivation :

• 1. conformal invariance in equilibrium critical phenomena, z = 1
• 2. Schr¨
• dinger-invariance of simple diffusion, z = 2

Lie 1881, Niederer 72, Hagen 71, Kastrup 68

t → αt γt + δ , r → Rr + vt + a γt + δ , αδ = 1 Lie algebra age1 := X1,0, Y±1/2, M0 generators : (no TTI !) Xn = −tn+1∂t − n + 1 2 tnr∂r − n(n + 1) 4 Mtn−1r2 − x 2(n + 1) + nξ

• tn

Ym = −tm+1/2∂r −

• m + 1

2

• Mtm−1/2r

Mn = −tnM also contains ‘phase changes’ in the wave function ! (projective)

commutators in root diagramme Schr¨

• dinger operator : S = 2M∂t − ∂2

r

Schr¨

• dinger equation : Sφ = 0

Schr¨

• dinger-invariance :

[S, X0] = −S, [S, Y−1/2] = 0 and [S, X1] = −2tS + 2M(x + ξ − 1

2)

= ⇒ if x + ξ = 1/2, solutions of the 1D Schr¨

• dinger/diffusion equation mapped

by age1 onto another solution → local dynamical symmetry co-variant two-point function R12 := φ1(t, r)φ2(s, 0) : XR12 = 0 with X ∈ aged ⊂ confd+2

MH 94, MH & Unterberger 03, MH et al 06

R12 ∼ s−1−a t s 1+a′−λR/2 t s − 1 −1−a′ exp

• −M1

2 r2 t − s

• with 1 + a = x1+x2

2

, a′ − a = ξ1 + ξ2, λR = 2(x1 + ξ1), M1 + M2 = 0 causality condition t > s : R12 is a response function ! reproduced in some ageing systems with z = 2 WHY ? ?

choice of the quasi-primary operators ?

Finite transformation calculated from aged : t = β(t′), r = r′

• dβ(t′)

dt′

and β(0) = 0 φ(t, r) = ˙ β(t′)−x/2 d ln β(t′) d ln t′ −ξ

• extra transformation

exp

• −Mr′2

4 d ln ˙ β(t′) dt′

• mass term

φ′(t′, r′) reduce to usual age-quasiprimary operator Φ(t, r) := t−2ξ/zφ(t, r). Then Φ(t) = ˙ β(t′)−(x+2ξ)/zΦ′(t′) , transforms as a quasiprimary. a) mean-field equation ∂tm = ∆m + 3(λ2 − v(t))m reduces to diffusion equation ∂tΦ = ∆Φ via m(t, r) = Φ(t, r) exp t dτ 3

• λ2 − v(τ)
• two cases :
• if T = Tc ⇔ λ2 = 0 :

Φ(t) ∼ t1/2m(t) if T < Tc ⇔ λ2 > 0 : Φ(t) ∼ 1 · m(t)

⇒ magnetisation m(t) and quasiprimary operator Φ(t) distinct b) kinetic spherical model equation ∂tφ(t) = ∆φ(t) − v(t)φ(t) + noise , v(t) ∼ t−1 gauge transformation Φ(t) = φ(t) exp − t

0 dτ v(τ), gives diff. eq.

c) kinetic Glauber-Ising model T = Tc

MH, Enß, Pleimling 06

1D a′ − a = −1

2

2D a′ − a ≃ −0.17(2) 3D a′ − a ≃ −0.022(5) ∗ 2nd-order ǫ-expansion disagrees with lattice data Pleimling & Gambassi 05 ∗ a′ − a < 0 required to match LSI with lattice data, but still disagrees with FT ⇒ resum ǫ-expansion to be able to compare with lattice data ?

Stochastic field-theory

Langevin equations do not have non-trivial dynamical symmetries ! compare results of deterministic symmetries to stochastic models ? go to stochastic field-theory, action

Janssen, de Dominicis,. . . 70s-80s

J [φ, φ] =

• φ(2M∂t − ∆)φ +

φV′[φ]

• J0[φ,e

φ] : deterministic

−T

• φ2 −
• φt=0Cinit

φt=0

• + Jb[e

φ] : noise

• φ : response field ;

C(t, s) = φ(t)φ(s), R(t, s) = φ(t) φ(s) averages : A0 :=

• DφD

φ A[φ, φ] exp(−J0[φ, φ]) masses : Mφ = −Me

φ

Theorem : IF J0 is Galilei- and spatially translation-invariant,

then Bargman superselection rules

Bargman 54

• φ1 · · · φn

φ1 · · · φm

• 0 ∼ δn,m

(1)

computation of a response function

Picone & MH 04

R(t, s) =

• φ(t)

φ(s)

• =
• φ(t)

φ(s)e−Jb[e

φ]

=

• φ(t)

φ(s)

• 0 = R0(t, s)

Bargman eq. (1) = ⇒ response function does not depend on noise ! left side : computed in stochastic models right side : local scale-symmetry of deterministic equation application to ageing : aged-covariant two-point response function R(t, s; r) = r0s−1−a t s 1+a′−λR/zt s − 1 −1−a′ exp

• −M

2 r2 t − s

• (2)

confirmed in many phase-ordering systems

reviews : MH, J. Phys. Cond Matt. 19, 065101 (’07) MH & Baumann, J. Stat. Mech. P07015 (’07)

Correlation functions for z = 2

find C(t, s) = φ(t)φ(s) = φ(t)φ(s)e−Jb[e

φ]0 from Bargman rule

C(t, s) = a0 2

• RddR R(4)

0 (t, s, 0; R)

initial + T 2M ∞ du

• RddR R(4)

0 (t, s, u; R) thermal

R(4)

0 (t, s, u; r)

=

• φ(t, y)φ(s, y)

φ2(u, r + y)

• Four-point R(4)

function must be found from local scaling

Theorem : LSI with z = 2 =

⇒ λC = λR

Picone & MH 04

agrees with a different argument of Bray 94 – and explicit models test C(t, s) explicitly in Ising/Potts models

MH et al. 04, Lorenz & Janke 07

Tests of eq. (2) in exactly solvable models

A) 1D Glauber-Ising model, at T = 0 exact two-time response function of the order-parameter

Lippiello & Zannetti 00, Godr` eche & Luck 00, MH & Sch¨ utz 04

R(t, s; r) = R(t, s) exp

• −1

4 r2 t − s

• , R(t, s) = 1

π

• 1

2s(t − s) read off : a = 0, a′ = −1/2, λR = 1, z = 2, M = 1/2. B) Spherical model, d > 2 dimensions, with equation of motion ∂tφ = ∆φ − v(t)φ + η potential v(t) comes from spherical constraint

i S2 i = N. Write

g(t) = exp(2 t

0 dτ v(τ)) and g(t) ∼ t̥ solves a Volterra integral

• equation. ̥ = ̥(d) known. Reduce to (2) via a gauge

transformation.

Godr` eche & Luck 00, Picone & MH 04

what appear to be essential features of local scaling ?

no local scaling in full Langevin equation

noise terms only compatible with translation-invariance

local scaling in deterministic part → reduction formulæ hidden local scaling symmetry, at least when z = 2 Galilei-invariance of deterministic part

together with scaling, implies full ageing-invariance

MH & Unterberger 03

physical origin of Galilei-invariance ? testable predictions for responses and correlators

Tests of eq. (2) in 2D/3D Glauber-Ising models

χTRM(t, s) = s du R(t, u) = s−afM(t/s) integrated response (thermoremanent susceptibility) MTRM(t, s) for the Glauber-Ising model compared to LSI (a) 2D, T = 1.5, (b) 3D, T = 3 T < Tc, hence z = 2 compare data from master equation with local scale-symmetry MH & M. Pleimling, Phys. Rev. E68, 065101(R) (2003)

spatio-temporally integrated response Ising model T < Tc (a,b) 2D ; µ = 1, 2, 4 (c,d) 3D ; µ = 1, 2, s

0 du

√µs dr rd−1R(t, u; r) = sd/2−aρ(2)(t/s, µ) MH & M. Pleimling, Phys. Rev. E68, 065101(R) (2003)

Autocorrelation in the q-states Potts model, T < Tc

• E. Lorenz & W. Janke, Europhys. Lett. 77, 10003 (2007).

Test of C(t, s) in the 1D Glauber-Ising model

Quench from disordered initial state, to T = Tc = 0 : C(t, s) = 2 π arctan

• 2

t/s − 1 exact L& Z 00, G& L 00, H& S 04

!

= C0 1 dv v2µ t s − 1

• (1 − v)

−2µ−1/2 t s + 1 − 2v 2µ choose µ = −1/4 and C0 = √ 2 /π. require all extensions beyond standard equilbrium scaling

Retour to the non-disordered Ising model, at T < Tc : z = 2 requires a more precise form of the ‘initial’ correlator :

Ohta, Jasnow, Kawasaki ’82

C(t, t; r) = 2 π arcsin

• exp
• − r2

L(t)2

• hence autocorrelator in the scaling limit

C(ys, s) = C0yρ(y − 1)−ρ−λC /z ∞ dx e−xfν

• x

y − 1

• fν(√u )

= ∞ dv arcsin

• e−νv

J0(√uv )

• f practical importance :

‘good’ choice of ‘initial correlations Cini(r) = c0δ(r) not sufficient

• III. Local scale-invariance for z = 2

Extend known cases z = 1, 2 = ⇒ axioms of LSI :

MH 02, Baumann & MH 07

1 M¨

• bius transformations in time (generator Xn)

t → t′ = αt γt + δ ; αδ = 1 require commutator : [Xn, Xn′] = (n − n′)Xn+n′

2 Dilatation generator : X0 = −t∂t − 1 z r · ∂r − x z

Implies simple power-law scaling L(t) ∼ t1/z (no glasses !).

3 Spatial translation-invariance → 2e family Ym of generators. 4 Xn contain phase terms from the scaling dimension x = xφ 5 Xn, Ym contain further ‘mass terms’ (Galilei !) 6 finite number of independent conditions for n-point functions.

Extend to z = 1, 2 by generators with mass terms, for d = 1 : Y1−1/z := −t∂r − µzr∇2−z

r

− γz(2 − z)∂r∇−z

r

Galilei X1 := −t2∂t − 2 z tr∂r − 2(x + ξ) z t − µr2∇2−z

r

special −2γ(2 − z)r∂r∇−z

r

− γ(2 − z)(1 − z)∇−z

r

depend on two parameters γ, µ and on two dimensions x, ξ contains fractional derivative ( f : Fourier transform) ∇α

r f (r) := iα

• Rd

dk (2π)d |k|αeir·k f (k) some properties : ∇α

r ∇β r = ∇α+β r

, [∇α

r , ri] = α∂ri∇α−2 r

∇α

r exp(iq · r) = iα|q|α exp(iq · r)

Fact 1 : simple algebraic structure : [Xn, Xn′] = (n − n′)Xn+n′ , [Xn, Ym] = n z − m

• Yn+m

→ Generate Ym from Y−1/z = −∂r. Fact 2 : LSI-invariant Schr¨

• dinger operator :

S := −µ∂t + z−2∇z

r

Let x0 + ξ = 1 − 2/z + (2 − z)γ/µ. Then [S, Ym] = 0 and [S, X0] = −S , [S, X1] = −2tS + 2µ z (x − x0) = ⇒ Sφ = 0 is lsi-invariant equation, if xφ = x0. Physical assumption (hidden) : equations of motion remain of first order in ∂t, even after renormalisation.

Fact 3 : non-trivial conservation laws : iterated commutator with G := Y1−1/z, ad G. = [., G] Mℓ := (adG)2ℓ+1 Y−1/z = aℓµ2ℓ+1∇(2ℓ+1)(1−z)+1

r

For z = 2, aℓ = 0 if ℓ ≥ 1. For a n-point function F (n) = φ1 . . . φn, MℓF (n) = 0 gives in momentum space n

• i=1

µ2ℓ−1

i

|ki|2ℓ−(2ℓ−1)z

• F (n)({ti, ki})

= n

• i=1

ki

• F (n)({ti, ki})

= = ⇒ momentum conservation & conservation of |k|α ! analogous to relativistic factorisable scattering

Zamolodchikov2 79, 89

• equil. analogy : 2D Ising model at T = Tc in magnetic field

Consequence : a lsi-covariant 2n-point function F (2n) is only

non-zero, if the ‘masses’ µi can be arranged in pairs (µi, µσ(i)) with i = 1, . . . , n such that µi = −µσ(i) . generalised Galilei-invariance with z = 2 = ⇒ integrability Corollary 1 : Bargman rule : φ1 . . . φn φ1 . . . φm0 ∼ δn,m Corollary 2 : treat (linear) stochastic equations with lsi-invariant deterministic part, reduction formulæ Corollary 3 : response function noise-independent R(t, s; r) = R(t, s)F(µ1,γ1)(|r|(t − s)−1/z) R(t, s) = r0 s−a t s 1+a′−λR/z t s − 1 −1−a′ F(µ,γ)(u) =

• Rd

dk (2π)d |k|γ exp (iu · k − µ|k|z) Corollary 4 : Correlators obtained from factorised 4-point responses.

How to test the foundations of LSI

theory is built on : a) simple scaling – domain sizes L(t) ∼ t1/z b) invariance under M¨

• bius transformation t → t/(γt + δ)

c) Galilei-invariance generalised to z = 2 together with spatial translation-invariance = ⇒ extended Bargman rules = ⇒ factorisation of 2n-point functions M¨

• bius transformation

autoresponse R(t, s) generalised Galilei-invariance space-time response R(t, s; r) factorisation two-time correlation function

Tests of LSI for z = 2 :

spherical model with conserved order-parameter, T = Tc, z = 4

Baumann & MH 06

Mullins-Herring model for surface growth, z = 4

R¨

• thlein, Baumann, Pleimling 06

spherical model with long-ranged interactions, T ≤ Tc, 0 < z = σ < 2

Cannas et al. 01 ; Baumann, Dutta, MH 07

ferromagnets at their critical point (Ising, XY), z ≈ 2.0 − 2.2

MH, Enss, Pleimling 06 ; Abriet & Karevski 04

critical particle-reaction models (DP ?) z ≈ 1.6 − 2

´ Odor 06

particle-reaction models with L´ evy-flight transport Durang & MH 08 important : consideration of invariant differential equation

• IV. Phase-ordering in disordered Ising models

pure systems : dynamics through moving domains walls universal description through Allen-Cahn eq. : ˙ L = v = (1 − d)K∼ L−1 = ⇒ L(t) ∼ t1/2 disorder : pins domain walls, need thermal activation

Henley & Huse 85

dL(t) dt = D(T, L)L(t)−1 , D(T, L) = D0 exp(−EB(L)/T) a) power-law EB(L) = E0Lψ : = ⇒ L(t) ∼ (ln t)1/ψ

Henley & Huse 85

ψ = 4 in 2D b) log law EB(L) = ǫ ln(1 + L) : = ⇒ L(t) ∼ t1/z

Paul, Puri, Rieger 04

z = 2 + ǫ/T, depends on temperature and disorder

Confirmations of algebraic scaling of the domain sizes :

1 direct simulations of L(t) in bond- and site-disordered 2D

Ising models gives empirical identification ǫ = ε for bond-disorder

Paul, Puri, Rieger 04/05

2 analytical studies of SOS-model on disordered substrate

H =

• (i,j)

(hi − hj)2 , hi = ni

• ∈Z

+ di

• disorder,∈[0,1]

continuum limit described by Cardy-Ostlund model H =

• dr
• (∇φ(r))2 − g cos (2π[φ(r) − ξ(r)])
• such that z = 2 + (2π2/9)(Tg/T) if T ≪ Tg

Schehr & Le Doussal 04/05

= ⇒change contrˆ

• le parameters to modify z

2D Ising model with bond disorder

H = −

• (i,j)

Ji,jσiσj , σi = ±1 , Ji,j uniformly in [1 − ε/2, 1 + ε/2] quench to T < Tc(ε) ≈ Tc(0) ≃ 2.269 . . ., heat-bath dynamics Properties : algebraic scaling, L(t) ∼ t1/z, z = z(ε, T) superuniversality : scaling function C(t, r) = σr(t)σ0(t) = F(r/L(t)) independent of disorder ; L(t) is domain size

Fisher & Huse 88

Questions : really simple ageing ? finite-time corrections ?

• r rather ‘super-ageing’ ?

Paul, Schehr, Rieger 07

is the simple relation z = 2 + ε/T true ? how general is superuniversality ?

Sicilia et al 07, Aron et al 08

Tests of simple ageing behaviour, I

two-time autocorrelator C(t, s) (a) ε = 0.5, T = 1 (b) ε = 1, T = 0.4 (c) ε = 2, T = 0.4 two-time autoresponse MTRM(t, s) (a) ε = 0.4, T = 0.4 (b) ε = 1, T = 0.6 (c) ε = 2, T = 0.8

Tests of simple ageing behaviour, II

simple ageing in space-dependence (a) C(t, r) ε = 1, T = 0.8 (b) C(t, s; r) ε = T = 0.5 (c) MTRM(t, s; r) ε = 2, T = 0.6 ⇒ dynamical exponent z = z(ε/T) is non-linear ⇒ simple ageing, no evidence for ‘super-ageing’

Tests of simple ageing behaviour, III

non-equilibrium response exponent a = a(ε, T)= a(ε/T) relation a = 1/z, well- known from pure systems, is no longer valid rather find az < 1 Proposed explanation : disorder should turn domain walls into fractals with dimension df (pure systems : df = d − 1) standard scaling arguments then lead to az = d − df = ⇒ further tests needed need next domain size L(t) : find from C(t, L(t)) ! = 1

2

Tests of superuniversality, I

single-time correlator C(t, r) (a) 0 ≤ ε < 2 (b) ε = 2 green curve : case (a) = ⇒ superuniversality confirmed for 0 ≤ ε < 2

agrees with earlier tests of superuniversality Puri 91, Biswal 96, . . .

= ⇒ two distinct regimes for the form of the scaling function for 0 ≤ ε < 2 : bonds are weakend, but not cut for ε = 2 : bonds may be cut

Tests of superuniversality, II

two-time correlator C(t, s; r) (a) 0 ≤ ε < 2 (b) ε = 2 green curve : case (a) superuniversality also needs distances |r|/L(t) 0.5 in order to hold true

Tests of superuniversality, III

two-time response MTRM(t, s; r) (a) 0 ≤ ε < 2 (b) ε = 2 green curve : case (a) need distances |r|/L(t) 0.5 y = 2 y = 4 y = 10

Scaling and universality in the 2D disordered Ising model

continuously varying dynamical exponent z = z(ε/T) but not of a simple linear form simple ageing confirmed b = 0, but some exponents, such as a = a(ε/T), more complicated superuniversality in general confirmed but two important conditions for its validity

1 not too strong disorder 0 ≤ ε<2 2 spatial distances large enough, |r|/L(t) 0.5

looked like an ideal case to test LSI how to include aspects such as (partial) superuniversality into LSI ?

• V. Conclusions

1 look for extensions of dynamical scaling in ageing systems

recently, scaling derived for phase-ordering Arenzon et al. 07

2 here : hypothesis of generalised Galilei-invariance 3 leads to Bargman rule if z = 2

and further to ‘integrability’ if z = 1, 2.

4 hidden dynamical symmetry of deterministic part of (linear)

Langevin equations

5 Tests : derive two-time response and correlation functions 6 LSI exactly proven for linear Langevin equations

good numerical evidence for some non-linear systems Some questions (the list could/should be extended) : how to physically justify Galilei-invariance ? how to extend to non-linear equations ? choice of the type of fractional derivative ? what is the algebraic (non-Lie !) structure of LSI ? treatment of master equations with LSI ?