Fluctuations in Glassy Systems

Claudio Chamon

DMR 0305482DMR 0128922

PRL 89, 217201 (2002)PRL 88, 237201 (2002)PRB 68, 134442 (2003)J. Chem. Phys. 121, 10120 (2004)JSTAT P01006 (2006)JSTAT P05001 (2007)

In collaboration with Leticia Cugliandolo -Univ. Paris VI & IUF

and

Malcolm Kennett - Simon Fraser Univ.Horacio Castillo - Ohio Univ.

Jose Luis Iguain - Univ. MontrealPatrick Charbonneau - Harvard Univ.David Reichman - Columbia Univ.Mauro Sellitto - ICTP TriesteHajime Yoshino - Osaka Univ.Silvio Franz - ICTP TriesteMarco Picco - LPTHE JussieuLudovic Jaubert - ENS LyonGabriel Fabricius - UNLP

Review in JSTAT (2007)

So ... Why Glasses?

Do we really know?

Do we need to fully answer why glasses? before we really further our understanding of glassy dynamics?

Here we take the point of view that, by starting from the fact that glassy systems exist (as nature presents us with concrete examples), we can then attempt to characterize whatever possible universal properties there are in glassy dynamics.

Physical aging

Source: L.C.E. Struik, Physical aging in amorphous polymers and other materials, Elsevier, Amsterdam (1978)

Physical aging II

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Orlyanchik & Ovadyahu, PRL (2004)

2D electron glass

2D thin films of crystalline In2O3-x

Source: Orlyanchik & Ovadyahu, PRL (2004)

Conceptual approach:

Analogy:

Why glasses?

Aging dynamics, spatial heterogeneities, universal scalings,...

Even without the answer,

If glass exist, then one has

Why crystals?

Bloch states, quasimomentum, phonon spectrum,...

Even without the answer,

If crystals exist, then one has

Phil Anderson inConcepts in Solids

Guiding principle:symmetry

Invariance of an effective dynamical action under uniform reparametrizations of the time scales

S. L. Ginzburg, Zh. Eksp. Teor. Fiz. 90, 754 (1986) [Sov. Phys. JETP 63, 439 (1986)]. L. B. Ioffe, Phys. Rev. B 38, 5181 (1988). L. F. Cugliandolo and J. Kurchan, J. Phys. A 27, 5749-5772 (1994).

Invariance at the level of eqs. of motion was long noted

“Many solutions” of eq. of motion! This “annoyance”, we claim, has physical meaning

and consequences

Time reparametrization:

RAG(t1, t2) → R̃AG(t1, t2) =∂h

∂t2RAG(h(t1), h(t2))

CAG(t1, t2) → C̃AG(t1, t2) = CAG(h(t1), h(t2))

t → h(t)

FDT plots:

Susceptibility vs. Correlation

0 1

Aging

ST

Off-Equilibrium Fluctuation Dissipation Relation - OEFDR

0 1

Aging

ST

χ(t1, t2) =

∫t2

t1

dt′R(t, t′) C(t1, t2)vs.

Time reparametrization:

RAG(t1, t2) → R̃AG(t1, t2) =∂h

∂t2RAG(h(t1), h(t2))

CAG(t1, t2) → C̃AG(t1, t2) = CAG(h(t1), h(t2))

t → h(t)

RAG(t1, t2) = βeff(CAG)∂

∂t2CAG(t1, t2)

Reparametrization invariance holds at the level of the action, not only eqs. of motionTrue for both short range and infinite range models.

Chamon, Kennett, Castillo, and Cugliandolo - PRL 2002a

under t → h(t)

Qi(t1, t2) → Q̃i(t1, t2) =

(

∂h

∂t1

)∆A(

∂h

∂t2

)∆R

Qi(h(t1), h(t2))

Reparametrization Group (RpG)

SAG[QAG] → S̃AG[Q̃AG] = SAG[Q̃AG]

Kennett and Chamon - PRL 2001

Two assumptions: 1) unitarity and causality and 2) separation of time scales

Castillo, in preparationDetailed proof for soft spin version of Edwards-Anderson model

Consequences of Reparametrization Invariance ...

What are the consequences of reparametrization invariance?

Global symmetry Slow spatial fluctuationslocal reparametrizations

Transversefluctuations

0 1

Aging

ST

Transversefluctuations

basis for aDynamical theory of local aging in glasses

What are the consequences of reparametrization invariance?

1. A growing dynamical correlation length.

2. Scaling of the pdf of local two-time functions.

3. Functional form of the pdf of local two-time functions.

4. Triangular relations between two-time functions.

5. Scaling relations for general multi-time functions.

6. Local fluctuation-dissipation relations.

7. Infinite susceptibilities.

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

++

++++

+ Bulk

FDT

0 0.5 1Cr

0

0.5

1

χr

Surface plot of the joint pdf for t/tw = 16.Points within the shown contour accountfor 2/3 of the total probability.

Time evolution of ρ(Cr(t, tw),χr(t, tw)), fort/tw = 1.00005, 1.001, 1.06, 2, 8, 32(from right to left, 2/3-probability contours).The crosses are the bulk χ(C) for the samepairs of times.

L = 64, T = 0.72Tc, V = 133, tw = 4 × 10

4 MC steps.

Joint probability distribution function ρ(C!r,χ!r).

Random surfaces and local correlations

local correlations:

Csr (t, tw) ∼ f

(

hr(tw)

hr(t)

)

= f(

e−[φr(t)−φr(tw)])

= f(

e−∆φr|t

tw

)

ii) If our interest is in short-ranged problems, the action must be written usinglocal terms. The action can thus contain products evaluated at a single timeand point in space of terms such as φr(t), φ̇r(t), ∇φr(t), ∇φ̇r(t), and similarderivatives.

i) The action must be invariant under a global time reparametrization t → s(t).

iii) The scaling form of CSr is invariant under φr(t) → φr(t) + Φr, with Φrindependent of time. Thus, the action must also contain this symmetry.

iv) The action must be positive definite.

⇒ Seff = K

∫

ddr

∫

dt

(

∇φ̇r(t))2

φ̇r(t)

Random surfaces and local correlations

from which:

collapse of PDFs at same global correlation

Seff = K

∫ddr

∫ds

(∇ϕ̇r(s))2

ϕ̇r(s)= K

∫ddr

∫ds (∇ψr(s))

2

where ψr =

√

φ̇r

s ≡ lnh(t)

∆φr|t

tw=

∫s(t)

s(tw)ds φ̇r(s) =

∫s(t)

s(tw)ds ψ2

r(s)

〈∆φr1 |ttw

∆φr2 |ttw

· · ·∆φrN |ttw〉c = [s(t) − s(tw)] F(r1, r2, . . . , rN )

= ln

(

h(t)

h(tw)

)

F(r1, r2, . . . , rN )

Scaling and collapse of the distribution of local correlations

Bulk correlations: C(t, tw) ≈ f(t/tw)Local correlation distribution: Pt,tw[Clocal] ≈ Pt/tw[Clocal]

0

1

2

3

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

P[c]

c

t/tw=4tw=1e04tw=2e04tw=4e04tw=8e04

tw=1.6e05tw=3.2e05

0

1

2

3

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

P[c]

c

t/tw=8tw=1e04tw=2e04tw=4e04tw=8e04

tw=1.6e05

0

1

2

3

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

P[c]

c

t/tw=16tw=1e04tw=2e04tw=4e04tw=8e04

0

1

2

3

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

P[c]

c

t/tw=32tw=1e04tw=2e04tw=4e04

Correction from the growing correlation length

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

!(C r

)

Cr

tw=1ktw=10ktw=100k

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

!(C r

)

Cr

tw=1ktw=10ktw=100k

PDF of local coarse-grained correlations Cr at different times t and tw in the 3d Edwards-Anderson model with L = 100 at T = 0.6 < Tc. The waiting-times are given in the key and the global correlation is fixed to C = 0.4 < qEA.

(a) The coarse-graining boxes have linear size ℓ = 9 in all cases. The curves do not collapse, a slow drift with increasing tw is clear in the figure.

(b) Variable coarse-graining length ℓ chosen so as to held ℓ/ξ approximately constant. The collapse improves considerably with respect to panel (a).

(a) (b)

Triangular relations

orh(!x, t1)h(!x, t2)

≈ f−1 (C!x(t1, t2)/qEA)C!x(t1, t2) ≈ qEA × f(

h(!x, t1)h(!x, t2)

)

If f−1(z) ∼ zλ

1 =h(!x, t1)h(!x, t2)

× h(!x, t2)h(!x, t3)

× h(!x, t3)h(!x, t1)

≈ f−1 (C!x(t1, t2)/qEA)× f−1 (C!x(t2, t3)/qEA)

f−1 (C!x(t1, t3)/qEA)

C!x(t1, t2)× C!x(t2, t3)C!x(t1, t3)

≈ qEA

Take t1 < t2 < t3

Numerical check of triangular relations

0 50 100 150 200 250 300 350 400 450

Cr 23 (Cg 13/Cr 13)1/2

Cr 12 (Cg 13/Cr 13)1/2

0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600

Cr 23 (Cg 13/Cr 13)1/2

Cr 12 (Cg 13/Cr 13)1/2

0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Cr 12 (Cg 13/Cr 13)1/2

Cr 23 (Cg 13/Cr 13)1/2

25%50%75%

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Cr 12 (Cg 13/Cr 13)1/2

Cr 23 (Cg 13/Cr 13)1/2

25%50%75%

Colloidal Glasses

Data from Weeks et al.

!Rα(t)→ Si(t)

Particle position to spin mapping

Lagrange vs. Euler

Lack of positional order 〈Si(t)Sj(t)〉 → 0|i− j|exponentially fast in

Glass type order 〈Si(t)Si(tw)〉Edwards-Anderson type ordering

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000 7000

C

t-tw (s)

200s460s840s

1500s2000s2680s3580s

Global correlation and aging

1

10 100 1000

C

t-tw

various tw, evolution with t

010013017023031042056075100134179

FIG. 5: The global two-time correlation function in the super-cooled liquid. Raw data as a function

of t − tw; the decay is approximately described by a stretched exponential.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300

C

t-tw

site 1 Mxy=3site 2 Mxy=3

site 1 Mxy=11site 2 Mxy=11

FIG. 6: The local coarse-grained two-time correlation function on two random sites called 1 and 2.

The sample evolves in the glassy regime. The coarse-graining box has linear sizes !x = !y = 7 and

!z = 9 in one pair of curves and !x!y = 23 and !z = 9 in the other (! = 2M + 1 and these lengths

are measured in pixel units). The thick lines are a smoothing of the curves.

6

Global correlation and aging

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3 3.5 4 4.5 5

C

t/tw

840s1500s2000s2680s3580s

Correlation length

0.0001

0.001

0.01

0.1

1

10

0 5 10 15 20 25

distance (µm)

velocitymobility

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16

distance (µm)

disconnectedsubtracting only CC

connected

S4(tw, t; !∆r) =1

Nsites

∑

!ri−!rj= !∆r

Si(t) Si(tw) Sj(t) Sj(tw)

C2(tw, t; !∆r) =1

Nsites

∑

!ri−!rj= !∆r

Si(t) Sj(tw)

Sc4(tw, t; !∆r) = S4(tw, t; !∆r) −[C2(tw, t; 0)]2

−C2(tw, t; !∆r) C2(tw, t;− !∆r)−C2(tw, tw; !∆r) C2(t, t; !∆r)

Weeks, Crocker & Weitz, JPCM 07 -- Lagrange approach Euler approach

S!u(tw, t;∆r) =1N!u

∑

|!rα−!rβ |=∆r

!uα · !uβ

Sδu(tw, t;∆r) =1Nδu

∑

|"rα−"rβ |=∆r

δuα δuβ

δuα = |"uα|−1

Npart

∑

α

|"uα|

0.0001

0.001

0.01

0.1

1

10

0 5 10 15 20 25

distance (µm)

velocitymobility

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16

distance (µm)

disconnectedsubtracting only CC

connected

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cr 12 (Cg 13/Cr 13)1/2

Cr 23 (Cg 13/Cr 13)1/2

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cr 12 (Cg 13/Cr 13)1/2

Cr 23 (Cg 13/Cr 13)1/2

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cr 12 (Cg 13/Cr 13)1/2

Cr 23 (Cg 13/Cr 13)1/2

Triangular relations t1 = 1500s t3 = 6400st2 = 1600s, 3600s, 6300s

Conclusions:

• Approach: understand dynamics once glasses are presented by nature

• Reparametrization invariance as guiding symmetry

• Soft reparametrization modes: aging is spatially heterogeneous in glasses

• Distribution of ages shows universality - collapse of PDF of local correlations

• Connection between the distribution of local correlations and the theory of random dynamical manifolds

• Triangular relations

• Colloidal glasses vs. Spin glasses

The End