A σ -model for glassy dynamics Leticia F. Cugliandolo LPTHE Jussieu & LPT-ENS Paris France – IUF
[email protected]
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 ! a)
tw Various shear histories
b)
4.0
Tf=0.1 Tf=0.3 Tf=0.4
gAA(r)
gAA(r)
6.0
4.0
0.14 0.12
|g1(tw,t)|
2
3.0 2.0
2.0
Tf=0.435 0.0 0.9
1.0
1.1
1.2
1.3 r 1.4
1.0 t=0
0.0
t=10
1.0
2.0
L-J mixture
0.10 0.08 0.06 0.04 0.02
Tf=0.4
3.0
r 4.0
J-L Barrat & W. Kob (99)
0.01
0.1
1
10
100
1000
d t (sec)
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 dynamics Structure 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 (Bi2 Sr2 CaCu2 O8 )
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
m¨ ria (t)
+
γ r˙ia (t)
δV ({~ri }) =− a + ξia (t) ri (t)
h ξia (t)ξjb (t′ ) i = 2γkB T δij δ ab δ(t − t′ ) m is a mass, γ the friction coefficient, T is the temperature of the bath and kB the Boltzmann constant ({~ ri }) V ({~ri }) is the potential energy and − δV δr the deterministic force a i
Key quantities Much of the global dynamics can be described with
• the global correlation functions, e.g. C(t, tw ) = N
−1
C s (q; t, tw ) =
PN
i=1 hsi (t)si (tw )i PN i~ q [~ ri (t)−~ ri (tw )] −1 h e N i=1
in spin systems,
i
• their associated linear response functions, e.g. P δhsi (t)i R(t, tw ) = N −1 N i=1 δhi (tw ) h=0
in particle systems.
in spin systems.
Solvable models Large N limit and/or large d limit. Exact Schwinger-Dyson equations
∂t C(t, tw ) = ∂t R(t, tw ) =
Z
Z
dt′ Σ(t, t′ )C(t′ , tw ) +
Z
dt′ D(t, t′ )R(tw , t′ ) ,
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+00
Slow
C
q
C s (t, tw ) ≈ fc
1e-01
1e-02
L(t) L(tw )
∂t C s (t, tw ) ≪ C s (t, tw )
tw1 tw2 tw3 1e+01 1e+03 1e+05 1e+07 t-tw
Eqs. for the slow relaxation C s
≡C 0 ⇒ ∂t ϕr (t) > 0) ; (iv) invariant under ϕr (t) → ϕr (t) + Φ(r) as Crs effective action is A=K
Z
dd r
Z
[∇∂t ϕr (t)]2 dt ∂t ϕr (t)
σ -model Using the ‘proper’ time τ (t)
≡ ln L(t)
with L(t) the “growth” law in the global corr. & the change of variables ψr2 (τ )
≡ ∂τ ϕr (τ ) R ln L(t) ′ 2 ′ − ln L(t ) dτ ψr (τ ) s w Cr (t, tw ) ≈ fc e Z Z A = K dd r dτ [∇ψr (τ )]2 Chamon, Charbonneau, LFC, Reichman & Sellitto (04)
xy-model – spin waves ; Antal & Rácz (94) Edwards-Wilkinson manifold.
cfr. Bramwell, Holdsworth & Pinton (98)
Some consequences • Temporal scaling of the pdf of local correlations dictated by the global correlation ρ(Cr ; t, tw ) = ρ[Cr ; C s (t, tw )] . • Negatively-skewed, non-Gaussian ρ(Cr ; C s ) for 0 < C s < q . • The two-time dependent correlation length ξ(t, tw ), "
X
#
Cis (t, tw )Cjs (t, tw )
i
≈ e−|~ri −~rj |/ξ(t,tw ) , c
should diverge with t and tw .
• Constant of motion. ρ[Cr , χr ; t, tw ] should follow the global FDT rel. : lim
tw →∞;C(t,tw )=C
χ(t, tw ) = χ(C) .
All can be tested with simulations & experiments.
pdf of local correlations Kinetically constrained model ; four (t, tw )/ C(t, tw )
= 0.8.
Similar results for the 3d spin-glass. 0
-2
log
10
(σ ρ )
-1
-3
10e+01 10e+02 10e+03 10e+05 Gaussian Gumbel a=12
-4 -5 -6 -6
-4
-2
0
2
4
6
( C r - < C r > ) /σ Chamon, Charbonneau, LFC, Reichman & Sellitto (04) cfr. E. Bertin (05)
pdf of correlations & responses 3d Edwards-Anderson spin-glass. 1 X Cr (t, tw ) ≡ si (t)si (tw ) , Vr i∈Vr
25
Z t X 1 ′ δsi (t) dt χr (t, tw ) ≡ Vr δhi (t′ ) h=0 tw i∈Vr
1
(a)
(b)
FDT
ρ 15
χr 0.5
++ + +
5
+ Bulk
0
1 0.5
Cr
0.5 0
χr
+++ +
0 0
0.5
1
Cr
+ 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
1
103
hw 2(t, tw )i
(a)
C
0.9
hw2 i∞ − −− 2 1/
101 ∼
− (t
) tw
100
1e+01 1e+02 1e+03
101
g2 hw (t/tw )i
0.8
10
(b)
2
100
tx
10−1
α = 0.145
10−1
10−2
10−3 10−6
0.7 1e+00
10
1e+02
1e+04
1e+06
100
101
102
103
104
105
Berthier, Holdsworth, Sellitto (03)
10−2
100
t/tw − 1
102
104
106 10−2 100 102 104 106 108 10101012101410161018
t − tw
t − tw
t
10−4
−2
Yoshino (96), Bustingorry, Iguain, LFC, Chamon, Domínguez (06)
Multiplicative scaling
C(t, tw ) ≈
t−α w fc
L(t) L(tw )
Take care of
t−α & saturation ! w 1
How general is this ? • Coarsening – domain growth
Just scale invariance
d-dimensional O(N ) model in the large N limit (continuous space limit of the Heisenberg ferro with N → ∞) e.g. the
φ˙ α (~r, t) = ∇2 φα (~r, t) + λ|N −1 φ2 (~r, t) − 1| φα (~r, t) + ξ~α (~r, t) Different mechanism, linked to extreme violation of the fluctuation-dissipation equilibrium relation between correlations and responses (Tef f
→ ∞).
Chamon, LFC, Yoshino (06)
Is it this way for all coarsening systems ?
Arenzon, Chamon, LFC, Sicilia, in progress
Experiments Time fluctuations in Brownian particles
a micellar polycrystal
A. Duri, P. Ballesta, L. Cipelletti, H. Bissig, & V. Trappe (04)
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
Summary Classification of non-equilibrium systems ? The theory suggests a strong link between Tef f and the fluctuations.
• Structural and spin glasses – aging, Tef f < +∞. • Critical dynamics – interrupted aging, no asymptotic Tef f . • Domain growth – aging in the correlations but ‘no memory’, Tef f → ∞ with different properties of the fluctuations. to be confirmed !