Leticia F. Cugliandolo Univ. Pierre et Marie Curie – Paris ...

Glassy dynamics

Leticia F. Cugliandolo

Univ. Pierre et Marie Curie – Paris VI

[email protected]

www.lpthe.jussieu.fr/ leticia

Slides & useful notes in front page

Kolkata, January 2012

Plan

• First lesson .

Introduction. Overview of glassy systems and methods.

• Second lesson .

Coarsening.

• Third lesson.

Back to glasses : models and free-energy landscapes.

• Fourth lesson .

Glassy dynamics.

Isolated systems

Dynamics of a classical isolated system

Foundations of statistical physics .

Question : does the dynamics of a particular system reach a flat distri-

bution over the constant energy surface in phase space ?

Ergodic theory , ∈ mathematical physics at present .

Dynamics of a quantum isolated system

a problem of current interest, recently boosted by cold atom experiments.

Question : after a quantum quench, i.e. a rapid variation of a parameter

in the system, are at least some observables described by thermal ones ?

When, how, which ? but we shall not discuss these issues here .

Dissipative systemsAim

Our interest is to describe the statics and dynamics of a classical or

quantum system coupled to a classical or quantum environment .

The Hamiltonian of the ensemble is

H = Hsyst +Henv +HintE∆

env

syst

The dynamics of all variables are given by Newton or Heisenberg rules, depen-

ding on the variables being classical or quantum.

The total energy is conserved, E = ct but each contribution is not, in particular,

Esyst 6= ct, and we’ll take Esyst ≪ Eenv

Reduced systemModel the environment and the interaction

E.g., an esemble of harmonic oscillators and a bi-linear coupling :

Henv +Hint =

N∑

α=1

[

p2α2mα

+mαω

2α

2

(cα

mαω2α

x− qα

)2]

Classically (coupled Newton equations) and quantum mechanically (easier in

a path-integral formalism) one can integrate out the oscillator variables.

Assuming the environment is coupled to the sample at the initial time and that

its variables are characterized by a Gibbs-Boltzmann density function ρ ∝

e−β(Henv+Hint) at inverse temperature β one finds :

a colored Langevin equation (classically) or

a reduced dynamic generating functional Zred (quantum mechanically).

See notes for explicit calculations.

General Langevin equation

The system, {rai }, with i = 1, . . . , N and a = 1, . . . , d, coupled to an

equilibrium environment evolves according to the Langevin eq.

Mrai (t)︸ ︷︷ ︸

+

∫ ∞

t0

dt′γ(t− t′)rai (t′)

︸ ︷︷ ︸

= −δV ({~ri})

rai (t)︸ ︷︷ ︸

+ ξai (t)︸︷︷︸

.

Inertia friction deterministic force noise

Coloured noise with correlation 〈 ξai (t)ξbj(t

′) 〉 = kBTδijδabΓ(t − t′)

and zero mean.

The friction kernel is γ(t− t′) = Γ(t− t′)θ(t− t′) with T the tempe-

rature of the bath and kB the Boltzmann constant.

Proof : see, e.g., U. Weiss 99 and notes .

Separation of time-scalesThe shortest : the bath ; white noise limit

In classical systems one usually takes a bath with the shortest relaxation

time τenv ≪ τsyst

The bath is approximated by the white form Γ(t− t′) = 2γδ(t− t′)

The Langevin equation becomes

Mvai (t) + γvai (t) = −δV ({~ri})

δrai (t)+ ξai (t) and rai = vai

with 〈ξai (t)ξbj(t

′)〉 = 2kBTγδijδabδ(t− t′).

γ is the friction coefficient that measures the strength of the coupling to

the bath.

Separation of time-scalesVelocities and coordinates

For t ≫ τv = M/γ one expects the velocities to equilibrate to the

Maxwell distribution P ({~v}) =∏

i

P (~vi) ∝∏

i

e−βMv2i /2

In this limit, one can drop Mvai and work with the

overdamped equation γrai = −V ({~ri})

δrai+ ξai .

The positions can have highly non-trivial dynamics , see examples.

Message : be very careful when trying to prove equilibration .

Different variables could behave very differently.

The potential energyA simple example : a harmonic oscillator

Hsyst =p2

2M+ V (x) =

p2

2M+

Mω2

2x2 V

Relaxation dynamics : Effective Langevin equation .

The momentum reaches the distribution ∝ e−βp2

2M for t ≫ τv =Mγ

.

The position reaches the distribution ∝ e−βV (x) for t ≫ τx =γ

Mω2 .

Exercise : prove these results, see notes.

Equilibrium : For times longer thanmax(τv, τx) the configurations (p, x)

are sampled from the canonical measure :

P (p, x) =e−βHsyst(p,x)

Z(β)

Out of equilibriumHow can a system stay out of equilibrium ?

• The equilibration time goes beyond the experimentally accessible times.

τsyst ≫ t

No confining potential, e.g. harmonic oscillator in the ω → 0 limit :

τx = γ/(Mω2) → ∞. E.g., Diffusion processes .

Macroscopic systems in which the equilibration time grows with

the system size, limN≫1 τsyst(N) ≫ t

E.g., Critical dynamics, coarsening, glassy physics .

• Driven systems ~F 6= ~∇V (~r)

E.g., Sheared liquids, vibrated powders, active matter .

What we knowabout the macroscopic systems

E0 ≪ Esyst : collective phenomena lead to equilibrium phase transitions .

E.g., thermal PM - FM transitions in classical magnetic systems .

We understand the nature of the equilibrium phases and the phase transi-

tions . We can describe the phases with mean-field theory and the critical be-

havior with scaling arguments and the renormalization group .

Quantum and thermal fluctuations conspire against the ordered phases.

We understand the equilibrium and out of equilibrium relaxation at the cri-

tical point or within the phases . We describe it with the dynamic RG at the

critical point or the dynamic scaling hypothesis in the ordered phase.

E.g., growth of critical structures or ordered domains .

What we do not knowIn systems with competing interactions there is no consensus upon :

• whether there are phase transitions,

• which is the nature of the putative ordered phases,

• which is the dynamic mechanism.

Examples are :

• systems with quenched disorder ;

• systems with geometric frustration ;

• all kinds of glasses.

Static and dynamic mean-field theory has been developed – both classically

and quantum mechanically – and it yield new concepts and predictions.

Extensions of the RG have been proposed and are currently being explored.

D. S. Fisher ; Le Doussal & Wiese

The rest of the 1st lecture

Discussion of several macroscopic systems with slow dynamics due to

limN≫1 τsyst(N) ≫ t

Examples :

Systems with quenched disorder.

Frustration, heterogeneity, self-averageness

Systems with frustrated interactions.

Geometric frustration.

List of references : books and review papers.

Quenched disorder

Quenched variables are frozen during time-scales over which other va-

riables fluctuate.

Time scales τmicro ≪ t ≪ τq

τq could be the diffusion time-scale for magnetic impurities, the magnetic

moments of which will fluctuate in a magnetic system or ;

the flipping time of impurities that create random fields acting on

other magnetic variables.

Weak disorder (modifies the critical properties but not the phases) vs.

strong disorder (modifies both).

E.g., random ferromagnets (Jij > 0) vs. spin-glasses (Jij>< 0).

The case of spin-glassesMagnetic impurities (spins) randomly placed in an inert hos t

Quenched random interactions

Magnetic impurities in a metal hostRKKY potential

V (rij) ∝cos 2kF rij

r3ijsisj

very rapid oscillations about 0

and slow power law decay.

Standard lore : there is a 2nd order static phase transition at Tsseparating a paramagnetic from a spin-glass phase.

No dynamic precursor above Ts. Glassy dynamics below Ts with aging,

memory effects, etc.

Measurements

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Mydosh et al. 81 Hérisson & Ocio 02Transition Dynamics

Pinning by impuritiesCompetition between elasticity and quenched randomness

d-dimensional elastic manifold in a transverse N -dimensional quenched

random potential.

OilWater

Interface between two phases ;vortex line in type-II supercond ;stretched polymer.

Distorted Abrikosov lattice

Goa et al. 01

Frustration

HJ [{s}] = −∑

〈ij〉 Jijsisj Ising model

+

++ + +

+

+

+ +

+

Disordered Geometric

Efrustgs > EFM

gs and Sfrustgs > SFMgs

Frustration enhances the ground-state enegy and entropy

Disordered example

Efrustgs = −2J > EFM

gs = −4J

Sfrustgs = ln 4 > SFM

gs = ln 2

Geometric case

Efrustgs = 3J > EFM

gs = −3J

Sfrustgs = ln 3 > SFM

gs = ln 2

Frustration

HJ [{s}] = −∑

〈ij〉 Jijsisj Ising model

One cannot satisfy all couplings simultaneously if∏

loop Jij < 0 .

One can expect to have metastable states too.

Trick : Avoid frustra-

tion by defining a spin

model on a tree, with

no loops, or in a di-

lute graph, with loops of

length lnN .

But it could be an over-simplication. Cayley, Bethe, Peierls, ...

Heterogeneity

Each variable, spin or other, feels a different local field,hi =∑z

j=1 Jijsj ,

contrary to what happens in a ferromagnetic sample, for instance.

Homogeneous Heterogeneous

hi = 4J ∀ i. hj = −2J hk = 0 hl = 2J .

Each sample is a priori different but,

do they all have a different thermodynamic and dynamic behav ior ?

Self-averagenessThe disorder-induced free-energy density distribution approaches a Gaus-

sian with vanishing dispersion in the thermodynamic limit :

limN→∞fN (β, J) = f∞(β) independently of disorder

– Experiments : all typical samples behave in the same way.

– Theory : one can perform a (hard) average of disorder, [. . . ],

−βNf∞(β) = limN→∞[lnZN(β, J)]

Exercise : Prove it for the 1d Ising chain ; argument for finite d systems.

Intensive quantities are also self-averaging.

Replica theory

−βf∞(β) = limN→∞ limn→0[Zn

N(β, J)]− 1

Nn

Everyday-life glasses

• 3000 BC Glass discovered in the Middle East. LUXURIOUS OBJECTS.

• 1st century BC Blowpipe discovered on the Phoenician coast. Glass

manufacturing flourished in the Roman empire. EVERYDAY-LIFE USE.

• By the time of the Crusades glass manufacture had been revived in

Venice. CRISTALLO

• After 1890, the engineering of glass as a material developed very fast

everywhere.

What do glasses look like ?

Simulation Confocal microscopy

Molecular (Sodium Silicate) Colloids (e.g. d ∼ 162 nm in water)

Experiment Simulation

Granular matter Polymer melt

Structural glassesCharacteristics

• Selected variables (molecules, colloidal particles, vortices or polymers

in the pictures) are coupled to their surroundings (other kinds of mol-

ecules, water, etc.) that act as thermal baths in equilibrium .

• There is no quenched disorder.

• The interactions each variable feels are still in competition , e.g. Lenard-

Jones potential, frustration .

• Each variable feels a different set of forces, time-dependent hetero-

geneity .

Sometimes one talks about self-generated disorder .

Correlation functionsStructure and dynamics

The two-time dependent density-density correlation :

g(r; t, tw) ≡ 〈 δρ(~x, t)δρ(~y, tw) 〉 with r = |~x− ~y|

The average over different dynamical histories (simulation/experiment)

〈. . .〉 implies isotropy (all directions are equivalent)

invariance under translations of the reference point ~x.

Its Fourier transform, F (q; t, tw) = N−1∑N

i,j=1〈 ei~q(~ri(t)−~rj(tw)) 〉

The incoherent intermediate or self correlation :

Fs(q; t, tw) = N−1∑N

i=1〈 ei~q(~ri(t)−~ri(tw)) 〉

Hansen & McDonald 06

Structural glasses

No obvious structural change but slowing down !

1.0 2.0 3.0 4.00.0

1.0

2.0

3.0

4.0

r

g AA(r

;t,t)

g AA(r

)

r

t=0

t=10

Tf=0.1Tf=0.3Tf=0.4

Tf=0.435

Tf=0.4

0.9 1.0 1.1 1.2 1.3 1.40.0

2.0

4.0

6.0

10−1

100

101

102

103

104

105

0.0

0.2

0.4

0.6

0.8

1.0

t−tw

Fs(

q,t−

tw)

T=5.0

T=0.466

q=7.25

A particles

LJ mixture Vαβ(r) = 4ǫαβ

[(σαβ

r

)12−(σαβ

r

)6]

τ0 ≪ τsyst < t but τsyst changes by 10 orders of magnitude !

Time-scale separation & slow (metastable) equilibrium dynamics

Molecular dynamics – J-L Barrat & Kob 99

Still lower temperatureOut of equilibrium relaxation

10−1

100

101

102

103

104

1050.0

0.2

0.4

0.6

0.8

1.0

tw=63100

tw=10

t−tw

Fs(

q;t,t

w)

tw=0

q=7.23

Tf=0.4

0.14

0.10

0.06

0.02

|g1(

t w,t)

|2

0.01 0.1 1 10 100 1000t-tw (sec)

twVarious shear histories

L-J mixture J-L Barrat & Kob 99 Colloids Viasnoff & Lequeux 03

τ0 ≪ t ≪ τsyst

The relaxation time goes beyond the experimentally accessible times

The same is observed in all other glasses.

Time-scalesCalorimetric measurement of entropy

What is making the relaxation so

slow ?

Is there growth of static order ?

Which one ?

Phase space picture ?

Below Tg ?

FluctuationsEnergy scales

Thermal fluctuations

– irrelevant for granular matter since mgd ≫ kBT ; dynamics is

induced by macroscopic external forces.

– important for magnets, colloidal suspensions, etc.

Quantum fluctuations

– when ~ω>∼ kBT quantum fluctuations are important.

– one can even set T → 0 and keep just quantum fluctuations.

Examples : quantum magnets, Wigner crystals, etc.

Why are they all ‘glasses’ ?What do they have in common ?

– No obvious spatial order : disorder (differently from crystals).

– Many metastable states

Rugged free-energy landscape

(independently of quenched disorder)

– Slow non-equilibrium relaxation

τ0 ≪ t ≪ τsyst

Time-scale separation experimental – system

– Hard to make them flow under external forces.

Pinning, creep, slow non-linear rheology.

Methodsfor classical and quantum disordered systems

Statics

TAP Thouless-Anderson-Palmer

Replica theory

fully-connected (complete graph)

Gaussian approx. to field-theories

Cavity or Peierls approx.

}

dilute (random graph)

Bubbles & droplet arguments

functional RG

finite dimensions

Dynamics

Generating functional for classical field theories (MSRJD).

Schwinger-Keldysh closed-time path-integral for quantum dissipative models

(the previous is recovered in the ~ → 0 limit).

Perturbation theory, renormalization group techniques, self-consistent approx.

Active matterNon-potential forces ; energy injection

Reviews : Fletcher & Geissler 09 ; Menon 10 ; Ramaswamy 10

References

– Liquids & glass transitionP. G. Debenedetti, Metastable liquids (Princeton Univ. Press, 1997).E. J. Donth, The glass transition : relaxation dynamics in liquids and disordered materials (Springer,2001).K. Binder and W. Kob, Glassy materials and disordered solids : an introduction to their statisticalmechanics (World Scientific, 2005).A. Cavagna, Supercooled liquids for pedestrians, Phys. Rep. 476, 51 (2009).L. Berthier & G. Biroli, A theoretical perspective on the glass transition and nonequilibrium pheno-mena in disordered materials, arXiv :1011.2578

– Spin-glassesK. H. Fischer and J. A. Hertz, Spin glasses (Cambridge Univ. Press, 1991).M. Mézard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond (World Scientific, 1986).T. Castellani & A. Cavagna, Spin-glass theory for pedestrians, J. Stat. Mech. (2005) P05012.F. Zamponi, Mean field theory of spin glasses, arXiv :1008.4844.N. Kawashima & H. Rieger, Recent progress in spin glasses in Frustrated spin systems, H. T. Dieped. (World Scientific, 2004).M. Talagrand, Spin glasses, a challenge for mathematicians (Springer-Verlag, 2003).

References

– Disorder elastic systemsT. Giamarchi & P. Le Doussal, Statics and dynamics of disordered elastic systems, arXiv :cond-mat/9705096.T. Giamarchi, A. B. Kolton, A. Rosso, Dynamics of disordered elastic systems, arXiv :cond-mat/0503437.

– Phase ordering kineticsA. J. Bray, Theory of phase ordering kinetics, Adv. Phys. 43, 357 (1994).S. Puri, Kinetics of Phase Transitions, (Vinod Wadhawan, 2009).L. F. Cugliandolo, Topics in coarsening phenomena, arXiv :0911.0771 , Physica A 389, 4360 (2010).

– GlassesE. J. Donth, The glass transition : relaxation dynamics in liquids and disordered materials (Springer,2001).K. Binder and W. Kob, Glassy materials and disordered solids : an introduction to their statisticalmechanics (World Scientific, 2005).L. F. Cugliandolo, Dynamics of glassy systems, Les Houches Session 77, arXiv :cond-mat/0210312.L. Berthier & G. Biroli, A theoretical perspective on the glass transition and nonequilibrium pheno-mena in disordered materials, arXiv :1011.2578.

– Active matter

D. A. Fletcher and P. L. Geissler, Active biological materials, Annu. Rev. Phys. Chem. 60, 469 (2009).G. I. Menon, Active matter, arXiv :1003.2032 in Rheology of complex fluids.S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Cond. Matt. Phys. 1, 323(2010).

End of 1st lecture

Plan

• First lesson .

Introduction. Overview of glassy systems and methods.

• Second lesson .

Coarsening.

• Third lesson.

Back to glasses : models and free-energy landscapes.

• Fourth lesson .

Glassy dynamics.

Phase ordering kineticsDynamics across a 2nd order phase transition

• Equilibrium phases are known on both sides of the transition.

• The dynamic mechanism can be understood.

• Interesting as a theoretical problem , beyond perturbation theory.

• To cf. the observables to similar ones in problems for which we do not

know the equilibrium phases nor the dynamic mechanisms.

• To investigate whether growth phenomena exist in problems with unk-

nown dynamic mechanisms. e.g. glasses

• To unveil “generic” features of macroscopic systems out of equilibrium

(classical or quantum).

out of equilibrium statistical mechanics or thermodynamic s

Phase ordering kineticsPlan of the lecture

• E.g. the PM-FM spontaneous symmetry breaking transition, competi-

tion between two equilibrium states related by symmetry, non-con-

served order parameter dynamics.

• Discussion of τsyst(L).

• Critical and sub-critical dynamics ; growing lengths.

• Space-time correlation. Separation of time-scales and dynamic scaling.

• Two-time correlation. Separation of time-scales and dynamic scaling.

• Universality classes : conserved scalar order parameter, vector order

parameter, etc.

• Linear susceptibility. First hint on effective temperatures

PM-FM transitione.g., up & down spins in a 2d Ising model (IM)

〈φ〉 = 0 〈φ〉 = 0 〈φ〉 6= 0

T → ∞ T = Tc T < Tc

Equilibrium configurations

In a canonical setting the control parameter is T/J .

2nd order phase-transitionbi-valued equilibrium states related by symmetry, e.g. Ising magnets

lowercriticalupper

φ

f

T

〈φ〉

Ginzburg-Landau free-energy function Scalar order parameter

Detailed proof in notes, sketch of it in 3rd lecture

EvolutionThe system is in contact with a thermal bath

Thermal agitation

Non-conserved order parameter (NCOP) 〈φ〉(t, T ) 6= ct

e.g. single spin flips with Glauber or Monte Carlo stochastic rules.

Development of magnetization in a ferromagnet.

Conserved order parameter (COP) 〈φ〉(t, T ) = 〈φ〉(0, T ) = ct

e.g. pair of antiparallel spin flips with Kawasaki stochastic rules.

Phase separation in binary fluids.

NCOP EvolutionA quench or an annealing across a phase transition

T (t) = Tc(1− t/τa)

t0

Tc

T

〈φ〉

Non-conserved order parameter 〈φ〉(t, T ) 6= ct

Development of magnetization in a ferromagnet after a quenc h.

NCOP EvolutionDevelopment of |m(t)| after a quench to Tf < Tc

e.g. single spin flips with Glauber or Monte Carlo stochastic rules.

-0.25

0

0.25

0.5

0.75

1

0 250 500 750 1000

|m|

t

L=20 40 85

100

Equilibrium

Out of equilibrium

System size dependence : L is the linear size of the sample.

τsyst(L, T )

The probleme.g. up & down spins in a 2d Ising model (IM)

0

50

100

150

200

0 50 100 150 200

’data’

0

50

100

150

200

0 50 100 150 200

’data’

0

50

100

150

200

0 50 100 150 200

’data’

Tf = Tc

0

50

100

150

200

0 50 100 150 200

’data’

0

50

100

150

200

0 50 100 150 200

’data’

0

50

100

150

200

0 50 100 150 200

’data’

Tf < Tc

Question : starting from equilibrium at T0 → ∞

how is equilibrium at Tf = Tc or Tf < Tc attained ?

Growth kinetics

• At Tf = Tc the system needs to grow critical structures

Critical coarsening.

At Tf < Tc : the system tries to order locally in one of the two com-

peting equilibrium states under the new conditions.

Sub-critical coarsening.

In both cases the linear size of the equilibrated patches increases

in time .

One extracts a growing linear size of equilibrated patches

R(t, T )

from a point-to-point correlationC(r, t) =1

N

N∑

i,j=1

〈δsi(t)δsj(t)〉|~ri−~rj |=r

Space-time correlationE.g., critical quench in the 2d Ising model

C(r, t) = 1N

∑Ni,j=1〈δsi(t)δsj(t)〉|~ri−~rj |=r

0

0.2

0.4

0.6

0.8

1

0 10 20 30

C(r

,t)

r

t=163264

128t=256

Equilibrium Tc

Ceq ≃ r−η

All dynamic data are far from the equilibrium curve.

For longer times, one needs longer distances to de-correlate by the same amount ;

e.g. reach C = 1/e growing typical length Rc(t).

Space-time correlatione.g., sub-critical quench in the 2d Ising model

C(r, t) = 1N

∑Ni,j=1〈δsi(t)δsj(t)〉|~ri−~rj |=r

0

0.2

0.4

0.6

0.8

1

1 2 3

C(r

,t)

r/R(t,T)

100

101

102

103

100 101 102 103

R2

t

Dynamic scaling : C(r, t) ≃ m2eq fc

(r

R(t, T )

)

Space-time correlationSeparation of fime-scales & dynamic scaling

Critical quench C(r, t) ≃ Ceq(r) f

(r

Rc(t)

)

Ceq(r) ≃ r2−d−η , limx→0 f(x) = 1 and limx→∞ f(x) = 0.

Sub-critical quench C(r, t) ≃ Ceq(r) +m2eq f

(r

R(t, T )

)

C(0, t) = 1 ∀t, limr→0Ceq(r) = 1−m2eq, limr→∞ Ceq(r) ∝ 〈δsi〉

2eq =

0, limx→0 f(x) = 1 and limx→∞ f(x) = 0.

Dynamic scalingCritical growth

Proven by dynamic RG in ǫ = du − d expansion.

Hohenberg & Halperin ; Janssen ; Calabrese & Gambassi

Rc(t) ≃ t1/zc

with zc = 2.0538 in the 2d Ising model with NCOP dynamics

zc = 2.0134 in the 3d Ising model with NCOP dynamics

zc = 2 in the 4d Ising model with NCOP dynamics

The scaling functions can also be computed with this method.

Very interesting geometric and statistical properties of critical growing struc-

tures. Blanchard, LFC & Picco, in preparation

Dynamic scalingSub-critical growth

Hypothesis : at late times there is a single length-scale, the typical radius

of the domains R(T, t), such that the domain structure is (in statistical

sense) independent of time when lengths are scaled by R(T, t), e.g.

C(r, t) ≡ 〈 si(t)sj(t) 〉||~ri−~rj |=r ∼ m2eq f

(r

R(t, T )

)

,

etc. when r ≫ ξ(T ), t ≫ t0 and C < m2eq(T ).

Suggested by experiments and numerical simulations. Proved for

• Ising chain with Glauber dynamics.

• Langevin dynamics of the O(N) model with N → ∞, and the

spherical ferromagnet. Review Bray, 1994.

• Distribution of hull-enclosed areas in 2d curvature driven coarsening.

Two-time self-correlatione.g., MC simulation of the 2dIM at T < Tc

C(t, tw) = N−1∑N

i=1〈si(t)si(tw)〉

0.1

1

1 10 100 1000

C(t

,t w)

t-tw

tw=248

163264

128256512

Stationary relaxation

Aging decay

Separation of time-scales : stationary – aging

C(t, tw) = Ceq(t− tw) +m2eq f

(R(t, T )

R(tw, T )

)

Ceq(0) = 1−m2eq, limx→∞ Ceq(x) = 0, f(1) = 1 ; limx→∞ f(x) = 0.

Two-time self-correlationComparison

Critical coarsening (Tf = Tc) Sub-critical coarsening (Tf < Tc)

C

t− tw

10510310110−1

100

10−1

10−2

tα

qea

Cagaging

stationary Cst

t− tw

10510310110−1

Separation of time-scales

Multiplicative Additive

Phase separationSpinodal decomposition in binary mixtures

A species ≡ spin up ; B species ≡ spin down

2d Ising model with Kawasaki dynamics at T

locally conserved order parameter

50 : 50 composition ; Rounder boundaries

R(t, T ) ≃ λ(T )t1/3 Huse, early 90s.

Dynamics in the 2d XY modelSchrielen pattern : gray scale according to sin2 2θi(t)

Defects are vortices (planar spins turn around these points)

After a quench vortices annihilate and tend to bind in pairs

R(t, T ) ≃ λ(T ){t/ ln[t/τ(T )]}1/2

Yurke et al 93, Bray & Rutenberg 94

Universality classesas classified by the growing length

R(t, T ) ≃

λ(T ) t1/2 scalar NCOP zd = 2

λ(T ) t1/3 scalar COP zd = 3

λ(T )

(t

ln t

)1/2

planar NCOP in d = 2

etc.

Defined by the time-dependence of R(t, T ).

Temperature and other parameters appear in the prefactor.

Super-universality ?

Are scaling functions independent of temperature and other parameters ?

Review Bray 94 ; Corberi, Lippiello, Mukherjee, Puri, Zanne tti 11

Frustrated magnetse.g., 2d spin ice or vertex models

Stripe growth in the FM phase

Anisotropic growth, R⊥(t, T ) and R‖(t, T )

Levis & LFC 11

Weak disordere.g., random ferromagnets

At short time scales the dynamics is relatively fast and independent of

the quenched disorder ; thus

R(t, T ) ≃ λ(T )t1/zd

At longer time scales domain-wall pinning by disorder dominates.

Assume there is a length-dependent barrier B(R) ≃ ΥRψ to over-

come

The Arrhenius time needed to go over such a barrier is tA ≃ t0 eB(R)kBT

This implies

R(t, T ) ≃

(kBT

Υln t/t0

)1/ψ

Weak disorderStill two ferromagnetic states related by symmetry

R(t, T ) ≃

λ(T )t1/zd R ≪ Lc(T ) curvature-driven

Lc(T )(ln t/t0)1/ψ R ≫ Lc(T ) activated

with Lc(T ) a growing function of T .

Inverting times as a function of length t ≃ [R/λ(T )]zd eR/Lc(T )

At intermediate times this equation can be approximated by an effective

power law with a T -dependent exponent :

t ≃ Rzd(T ) with zd(T ) ≃ zd [1 + ct/Lc(T )]

Bustingorry et al. 09

Commonly used in numerical studies but theoretically incorrect.

Linear responseTo a kick and to a step

− δ δ+

h

t t2 2

w w0 t����

��������

����

����

r(0)

r(tw)

tr( )

r( )th

The perturbation couples linearly to the observable H → H−hB({~ri})

The linear instantaneous response of another observable A({~ri}) is

RAB(t, tw) ≡δ〈A({~ri})(t)〉h

δh(tw)

∣∣∣∣h=0

The linear integrated response or dc susceptibility is

χAB(t, tw) ≡

∫ t

tw

dt′RAB(t, t′)

Linear responseCritical and sub-critical coarsening

Critical coarsening

χ(t, tw) = β − χeq(t− tw)g

(R(t, T )

R(tw, T )

)

Sub-critical coarsening

χ(t, tw) = χeq(t− tw) + [R(tw, T )]−aχ g

(R(t, T )

R(tw, T )

)

In both cases : χeq(t− tw) = −T−1dCeq(t− tw)/d(t− tw).

Interesting consequences that we shall discuss in the 4th lecture.

Reviews Crisanti & Ritort 03 ; Calabrese & Gambassi 05 ; Corberi et al. 07,

LFC 11

AgingOlder samples relax more slowly

Older samples need more time to :

relax spontaneously (e.g. correlation function) ;

relax after a change in conditions (e.g. response function).

tw is a reference time that measures the age of the sample.

The state of the sample at tw is compared to the one at one (or more)

later time(s).

Huge literature on this phenomenology. Some reviews were written by

Struick on polymer glasses. Vincent et al. & Nordblad et al. on spin-glasses.

McKenna et al. on all kinds of glasses.

Summary

• At and below Tc growth of equilibrium structures.

• The linear size of the equilibrium patches is measured by R(t, T )

• At Tc vanishing order parameter

Multiplicative scaling C ≃ CeqCag ; χ ≃ χeqχag

• Below Tc non-vanishing order parameter

Additive scaling C ≃ Ceq + Cag ; χ ≃ χeq + χag

Cag is finite while χag vanishes asymptotically.

We shall discuss χ and how it compares to C later.

Phase ordering kineticsThe lecture was about

• Growth of equilibrium patches at Tc and below Tc.

• Divergence of τsyst(L) with the system size.

• Existence of a single growing length R(t, T )

• Separation of time-scales and dynamic scaling, e.g. C = Ceq + Cag.

• Two kinds of correlations : Space-time and two-time ones.

• Dynamic universality classes at and below Tc.

• The more tricky/rich linear susceptibility.

Is there a static growing length in all systems with slow dyna mics ?

Which one ?

End of 2nd lecture