A σ-model for glassy dynamics

Leticia F. Cugliandolo

LPTHE Jussieu & LPT-ENS Paris France – IUF

leticia@lpt.ens.fr

In collaboration with C. Chamon and

J. Arenzon, S. Bustingorry, H. Castillo, P. Charbonneau, D. Domínguez,

S. Franz, M. P. Kennett, J. L. Iguain, D. Reichman, A. Sicilia, M. Sellitto,

H. Yoshino

LPS-ENS, 05/04/2006

Plan

• What is the glassy problem ? Overview.

• Some theoretical ideas coming from mean-field theory.

• Beyond .

The glassy phenomenon

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

)

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

0.14

0.12

0.10

0.08

0.06

0.04

0.02

|g1(

t w,t)

|20.01 0.1 1 10 100 1000

d t (sec)

twVarious shear histories

a)

b)

L-J mixture J-L Barrat & W. Kob (99) Colloidal suspension B. Viasnoff & F. Lequeux (03)

τmicro ≪ τexp ≪ τrelax that changes by ≈ 10 orders of magnitude !

Time-scale separation & slow non-equilibrium dynamics

Glassy dynamicsStructure factor : nothing special happens at Tg.

One-time quantities decay non-exponentially,

e.g. energy density in a relaxing magnet,

density in a compactifying granular system

radial distribution function in a particle system

Two-time quantities age, i.e. the stationary relaxion is lost

and there is a separation of time-scales, rapid-slow,

controlled by tw.

Many systems, many techniques

Simulation Confocal microscopy

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

Decoration Sketch

Vortex (Bi2Sr2CaCu2O8) Polymer melt

Questions

• Can one characterize the global/bulk dynamics ?

(Mean-field/large dimensional models)

• What about the fluctuations ? Local/mesoscopic dynamics

Idea : accept the glass without explaining how and why it appears

and describe its dynamics in detail.

(cfr. phonons in solids...)

Focus on two-time quantities.

• Which is the reason for the slowing down ?

• Is there some growing hidden order ?

Modelling

The system is coupled to its environment

~ri evolve according to some stochastic rule, e.g. Langevin dynamics

mrai (t) + γrai (t) = −δV ({~ri})

rai (t)+ ξai (t)

〈 ξai (t)ξbj(t

′) 〉 = 2γkBTδijδabδ(t− t′)

m is a mass, γ the friction coefficient, T is the temperature of the bath

and kB the Boltzmann constant

V ({~ri}) is the potential energy and − δV ({~ri})δra

i

the deterministic force

Key quantities

Much of the global dynamics can be described with

• the global correlation functions, e.g.

C(t, tw) = N−1∑N

i=1〈si(t)si(tw)〉 in spin systems,

Cs(q; t, tw) = N−1∑N

i=1〈 ei~q[~ri(t)−~ri(tw)] 〉 in particle systems.

• their associated linear response functions, e.g.

R(t, tw) = N−1∑N

i=1δ〈si(t)〉δhi(tw)

∣∣∣h=0

in spin systems.

Solvable modelsLarge N limit and/or large d limit.

Exact Schwinger-Dyson equations

∂tC(t, tw) =

∫dt′ Σ(t, t′)C(t′, tw) +

∫dt′ D(t, t′)R(tw, t

′) ,

∂tR(t, tw) =

∫dt′ Σ(t, t′)R(t′, tw) ,

where the self-energy and vertex are functions of C and R :

D(t, tw) = D[C(t, tw)] , Σ(t, tw) = D′[C(t, tw)]R(t, tw) .

Solvable numerically and analytically in the long tw limit.

LFC & J. Kurchan (93)

Separation of time-scales

In the long tw limit

Fast

1e-02

1e-01

1e+00

1e+01 1e+03 1e+05 1e+07

C

t-tw

q

tw1tw2tw3

Slow

Cs(t, tw) ≈ fc

(L(t)L(tw)

)

∂tCs(t, tw) ≪ Cs(t, tw)

Eqs. for the slow relaxation Cs ≡ C < q :

Approx. asymptotic time-reparametization invariance t→ h(t)

Time-reparametrizationExample : the equation ∂tR(t, tw) =

∫dt′ Σ(t, t′)R(t′, tw)

• Separation of time-scales (drop ∂tR, approximate the integral) :

µ∞Rs(t, tw) ∼

∫dt′ D′[Cs(t, t′)]Rs(t, t′)Rs(t′, tw) . (1)

• The transformation

t→ ht ≡ h(t) ,

Cs(t, tw) → Cs(ht, htw) ,

Rs(t, tw) → dhtw

dtwRs(ht, htw) .

with ht positive and monotonic leaves eq. (1) invariant :

µ∞Rs(ht, htw) ∼

∫dht′ D

′[Cs(ht, ht′)]Rs(ht, ht′) R

s(ht′ , htw) .

Time reparametrization invarianceA nuissance

Similar to the matching problem in non-linear diff. eqs.

0

1

2

3

0 1 2 3

y

λ

dy

dλ= g[y(λ)]

Many asymptotic solutions.

Time reparametrization invarianceOne can compute analytically fc and χs(Cs)

Cs(t, tw) ∼ fc

(L(t)

L(tw)

),

χs(t, tw) ≡

∫ t

tw

dt′R(t, t′) ∼1 − q

T+

1

TeffCs(t, tw)

but not the ‘clock’ L(t) .

Finite dimensions

• Slow dynamics : observed

• Separation of time-scales : observed, though less clear-cut.

Num. sol. MF model Sim. L-J mixture Exp. colloidal susp.

10−2

10−1

100

101

102

103

104

1050.0

0.2

0.4

0.6

0.8

1.0 (a)

tw=63100

tw=10

τ

Ck(

t w+

τ,t w

)

tw=0

k=7.23

0.14

0.12

0.10

0.08

0.06

0.04

0.02

|g1(

t w,t)

|2

0.01 0.1 1 10 100 1000d t (sec)

twVarious shear histories

a)

b)

B. Kim & A. Latz (00) J-L Barrat & W. Kob (99) B. Viasnoff & F. Leq ueux (03)

Finite dimensionsAnalytically

S = Sslow + Sfast + Sint

• RG argument based on separation of time-scales allows one to show

the approximate asymptotic invariance of Sslow under

global time-reparametrizations

t→ ht ≡ h(t) ,

Csr (t, tw) → Cs

r (ht, htw) ,

Rsr(t, tw) → dhtw

dtwRsr(ht, htw) .

Symmetry breaking terms become less important as tw, t− tw → ∞.

3d Edwards-Anderson spin-glass Chamon, Kennett, Castillo & LFC (02).

The Heisenberg ferromagnet

An analogy : a nuissance turned into a model

φi φj

φi -φj

Si Sj

Landau free-energy

F =

∫ddr

{[∇~m(~r)]2 + λ

[m2(~r) −m2

0

]2}.

Invariant under the global rotation ma(~r) = Rabmb(~r).

(Global time-reparametrization invariance)

Statics of the Heisenberg ferro

Ground state : ~m(~r) = ~m0 for all ~r.

Fluctuations : ~m(~r) = ~m0 + δ ~m(~r).

Longitudinal (easy) &

transverse (hard) fluctuations. Spin-waves

m0( r )

Longitudinal

TransverseLow energy excitation

(Time-reparametrization waves)

Leading fluctuations

Scaling of the slow part of the global correlation

Cs(t, tw) ≈ fc

(L(t)

L(tw)

).

The global time-reparametrization invariance ⇒ Csr (t, tw) ≈ fc

(hr(t)hr(tw)

).

Ex. hr1 = tt0

, hr2 = ln(tt0

), hr3 = e

lna“

tt0

”

on different regions

1e-02

1e-01

1e+00

1e+00 1e+02 1e+04 1e+06

C

t-tw

h1h2h3

Same tw, slower and faster decays

Castillo, Chamon, LFC, Iguain, Kennett (02,03).

Turn it useful : σ model

Easy fluctuations t→ hr(t).

• Ideally : derive the action S[hr(t)].

Doable in quasi-mean-field models, C. Chamon, LFC, S. Franz, in progress.

• In practice : propose the action S[hr(t)] ;

derive predictions from S[hr(t)],

e.g. ρ[Csr ; t, tw] ; ρ[Rs

r; t, tw] ; ρ[Csr , R

sr; t, tw]

that can be checked numerically & experimentally.

P. Chamon, Charbonneau, LFC, D. Reichman & M. Sellitto (04).

σ-model

Slow decay in terms of hr(t) ≡ e−ϕr(t)

Csr (t, tw) ≈ fc

(hr(t)

hr(tw)

)= fc

(e−

R t

twdt′∂t′ϕr(t′)

)

The simplest

(i) global time-reparametrization invariant ;

(ii) local in space ;

(iii) positive definite (∂thr(t) > 0 ⇒ ∂tϕr(t) > 0) ;

(iv) invariant under ϕr(t) → ϕr(t) + Φ(r) as Csr effective action is

A = K

∫ddr

∫dt

[∇∂tϕr(t)]2

∂tϕr(t)

σ-model

Using the ‘proper’ time τ(t) ≡ lnL(t)

with L(t) the “growth” law in the global corr.

& the change of variables ψ2r(τ) ≡ ∂τϕr(τ)

Csr (t, tw) ≈ fc

(e−

R ln L(t)ln L(tw)

dτ ′ψ2r (τ ′)

)

A = K

∫ddr

∫dτ [∇ψr(τ)]

2

Chamon, Charbonneau, LFC, Reichman & Sellitto (04)

cfr. Bramwell, Holdsworth & Pinton (98) xy-model – spin waves ;Antal & Rácz (94) Edwards-Wilkinson manifold.

Some consequences• Temporal scaling of the pdf of local correlations dictated by the global

correlation ρ(Cr; t, tw) = ρ[Cr;Cs(t, tw)] .

• Negatively-skewed, non-Gaussian ρ(Cr;Cs) for 0 < Cs < q.

• The two-time dependent correlation length ξ(t, tw),[∑

i

Csi (t, tw)Cs

j (t, tw)

]

c

≈ e−|~ri−~rj |/ξ(t,tw) ,

should diverge with t and tw.

• Constant of motion. ρ[Cr, χr; t, tw] should follow the global FDT rel. :

limtw→∞;C(t,tw)=C

χ(t, tw) = χ(C) .

All can be tested with simulations & experiments.

pdf of local correlationsKinetically constrained model ; four (t, tw)/ C(t, tw) = 0.8.

Similar results for the 3d spin-glass.

-6

-5

-4

-3

-2

-1

0

-6 -4 -2 0 2 4 6

log

( C - < C > )r r /σ

10(σ

ρ )

10e+0110e+0210e+0310e+05

GaussianGumbel a=12

Chamon, Charbonneau, LFC, Reichman & Sellitto (04)

cfr. E. Bertin (05)

pdf of correlations & responses

3d Edwards-Anderson spin-glass.

Cr(t, tw) ≡1

Vr

∑

i∈Vr

si(t)si(tw) , χr(t, tw) ≡1

Vr

∑

i∈Vr

∫ t

tw

dt′δsi(t)

δhi(t′)

∣∣∣∣h=0

0

0.5Cr 0

0.5

1

χr

5

15

25

ρ

(a)

0

0.5Cr 0

0.5

1

χr

5

15

25

ρ++++

+++++ Bulk

FDT(b)

0 0.5 1Cr

0

0.5

1

χr

+ Bulk : Parametric plot χ(t, tw) vs C(t, tw) for tw fixed and 7 t (> tw).

ρ corresponds to the maximum t yielding the smallest C (left-most +).

Castillo, Chamon, LFC, Kennett (02).

How general is this ?

• Critical dynamics

e.g. the 2d xy model, an elastic line in a random environment

0.7

0.8

0.9

1

1e+00 1e+02 1e+04 1e+06

C

t

1e+011e+021e+03 tx

(a)

∼(t−

tw)1/

2

t − tw〈w

2(t

,tw)〉

106105104103102101100

103

102

101

100

10−1

10−2

〈w2〉∞ −−−

t− tw

(b)

1018101610141012101010810610410210010−2

α = 0.145

t/tw − 1

〈w2 (

t/t w

)〉

10410210010−210−410−6

101

100

10−1

10−2

10−3

1

Berthier, Holdsworth, Sellitto (03) Yoshino (96), Busting orry, Iguain, LFC, Chamon, Domínguez (06)

Multiplicative scaling

C(t, tw) ≈ t−αw fc

(L(t)

L(tw)

)Take care of t−αw & saturation !

How general is this ?

• Coarsening – domain growth Just scale invariance

e.g. the d-dimensional O(N) model in the large N limit (continuous

space limit of the Heisenberg ferro with N → ∞)

φα(~r, t) = ∇2φα(~r, t) + λ|N−1φ2(~r, t) − 1|φα(~r, t) + ~ξα(~r, t)

Different mechanism, linked to extreme violation of the fluctuation-dissipation

equilibrium relation between correlations and responses (Teff → ∞).

Chamon, LFC, Yoshino (06)

Is it this way for all coarsening systems ? Arenzon, Chamon, LFC, Sicilia, in progress

ExperimentsTime fluctuations in Brownian particles

a micellar polycrystal

A. Duri, P. Ballesta, L. Cipelletti, H. Bissig, & V. Trappe (0 4)

Spatial fluctuations in polymer glasses (cantilevers) K. Sinnathamby, H. Oukris, N. Israeloff (06) ;

colloidal suspensions (confocal microscopy) P. Wang, C. Song, H. Makse et al, in progress

Summary

Theory for the nonequilibrium dynamics in the glassy phase.

dictated by (the assumption) of

Global time reparametrization invariance

cfr. Spin-waves in Heisenberg ferromagnets.

Predictions for the behaviour of local correlations and responses,

in rather good agreeement with simulations in disordered spin models

and kinetically constrained models ; experiments on colloidal systems on

their way

SummaryClassification of non-equilibrium systems ?

The theory suggests a strong link between Teff and the fluctuations.

• Structural and spin glasses – aging, Teff < +∞.

• Critical dynamics – interrupted aging, no asymptotic Teff .

•Domain growth – aging in the correlations but ‘no memory’, Teff → ∞

with different properties of the fluctuations.

to be confirmed !