Thermodynamic fluctuations in
model glasses
Ludovic Berthier
Laboratoire Charles Coulomb
Université Montpellier 2 & CNRS
Glassy Systems and Constrained Stochastic Dynamics – Warwick, June 11, 2014
D4PARTICLES
title – p.1
=1
Coworkers
=1
12
(b) f = 0.18
(c)
• With:
D. Coslovich (Montpellier)
R. Jack (Bath)
W. Kob (Montpellier)
(e)
(f)
title – p.2
Temperature crossovers
• Glass formation characterized by several “accepted” crossovers. Onset,
mode-coupling & glass temperatures: directly studied at equilibrium.
14
12
Tg
10
log10 ( η / poise)
8
6
Tm
4
Tb
2
0
TK
T0
Tc T T
x
B
-2
TA
T*
-4
0.0020
[G. Tarjus]
0.0025
0.0030
0.0035
1 / T ( K-1)
0.0040
0.0045
0.0050
[Debenedetti & Stillinger]
• Extrapolated temperatures for dynamic and thermodynamic singularities:
T0 , TK . Existence and nature of “ideal glass transition” at Kauzmann
temperature is controversial.
title – p.3
Dynamic heterogeneity
• When density is large, particles must move in a correlated way. New
transport mechanisms revealed over the last decade: fluctuations matter.
90
• Spatial fluctuations grow (modestly)
near Tg .
80
70
60
• Clear indication that some kind of
phase transition is not far – which?
50
40
30
• Structural origin not clearly established: point-to-set lengthscales,
other structural indicators?
20
10
0
0
10
20
30
40
50
60
70
80
90
Dynamical heterogeneities in glasses, colloids and granular materials
Eds.: Berthier, Biroli, Bouchaud, Cipelletti, van Saarloos (Oxford Univ. Press, 2011).
title – p.4
Dynamic heterogeneity
• When density is large, particles must move in a correlated way. New
transport mechanisms revealed over the last decade: fluctuations matter.
• Spatial fluctuations grow (modestly)
near Tg .
• Clear indication that some kind of
phase transition is not far – which?
• Structural origin not clearly established: point-to-set lengthscales and
other structural indicators?
• Do “convincing” thermodynamic fluctuations even exist?
title – p.5
Dynamical view: Large deviations
P (m) = hδ(m − mt )i ∼ e−tN ψ(m) .
• Exponential tail: direct signature of
phase coexistence in (d + 1) dimensions: High and low activity phases.
Direct connection to dynamic heterogeneity.
10
P(m)
• Large deviations of fluctuations of
the (time integrated) local activity
R
Rt ′
mt = dx 0 dt m(x; t′ , t′ + ∆t):
10
10
-3
-6
10
10
0
tobs= 320
tobs= 640
tobs= 960
tobs= 1280
-9
-12
0.0
0.1
0.2
m
0.3
[Jack et al., JCP ’06]
• Equivalently, a field coupled to local dynamics induces a nonequilibrium
[Garrahan et al., PRL ’07]
first-order phase transition in the “s-ensemble”.
• Metastability controls this physics. “Complex” free energy landscape
gives rise to same transition, but the transition exists without multiplicity of
[Jack & Garrahan, PRE ’10]
glassy states: KCM, plaquette models.
title – p.6
Thermodynamic view: RFOT
• Random First Order Transition (RFOT) theory is a theoretical framework
constructed over the last 30 years (Parisi, Wolynes, Götze...) using a large
set of analytical techniques.
[Structural glasses and supercooled liquids, Wolynes & Lubchenko, ’12]
• Some results become exact for simple “mean-field” models, such as the
X
fully connected p-spin glass model: H = −
Ji1 ···ip si1 · · · sip .
i1 ···ip
• Recently demonstrated for hard spheres as d → ∞ [Kurchan, Zamponi et al.]
• Complex free energy landscape → sharp transitions: Onset (apparition
of metastable states), mode-coupling singularity (metastable states
dominate), and entropy crisis (metastable states become sub-extensive).
• Ideal glass = zero configurational entropy, replica symmetry breaking.
• Extension to finite dimensions (‘mosaic picture’) remains ambiguous.
title – p.7
‘Landau’ free energy
• Relevant thermodynamic fluctuations encoded in “effective potential”
V (Q). Free energy cost, i.e. configurational entropy, for 2 configurations to
[Franz & Parisi, PRL ’97]
have overlap Q:
Z
Z
Vq (Q) = −(T /N ) dr2 e−βH(r2 ) log
PN
1
where: Q12 = N i,j=1 θ(a − |r1,i − r2,j |).
dr1 e−βH(r1 ) δ(Q − Q12 )
0.08
V (Q)
• V (Q) is a ‘large deviation’ function, mainly studied in mean-field
RFOT limit.
Simple liquid
0.06
Tonset
0.04
Tmct
0.02
TK
0
0
0.2
0.4
Q
•
Σ(T )
0.6
P (Q)
=
hδ(Q − Q12 )i
∼
exp[−βN V (Q)]
0.8
• Overlap fluctuations reveal evolution of multiple metastable states. Finite
d requires ‘mosaic state’ because V (Q) must be convex: exponential tail.
title – p.8
Direct measurement?
• Principle: Take two equilibrated configurations 1 and 2, measure their
overlap Q12 , record the histogram of Q12 .
• (Obvious) problem: Two equilibrium configurations are typically
uncorrelated, with mutual overlap ≪ 1 and small (nearly Gaussian)
fluctuations.
102
P (Q)
10
T = 9, N = 256
0
10−2
10−4
10−6
0
0.1
0.2
0.3
Q
• Solution: Seek “large deviations” using umbrella sampling techniques.
[Berthier, PRE ’13]
title – p.9
Overlap fluctuations in 3d liquid
10
Va (Q)
• Idea: bias the dynamics using Wi (Q) = ki (Q − Qi )2 to explore of Q ≈ Qi .
T = 18, 15, 13, 11, 10, 9, 7
• Reconstruct P (Q) using
reweighting techniques.
• Static fluctuations may control fluctuations and phase
transitions in trajectory space.
5
0
T = 13, 11, 10, 8, 7
10
Vq (Q)
• Exponential tail below Tonset
→ phase coexistence between multiple metastable
states in 3d bulk liquid.
(a) Annealed
(b) Quenched
5
0
0
0.2
0.4
0.6
Q
0.8
1
title – p.10
Equilibrium phase transitions
• Non-convex V (Q) implies that an equilibrium phase transition can be
induced by a field conjugated to Q.
[Kurchan, Franz, Mézard, Cammarota, Biroli...]
• Annealed: 2 coupled copies.
T
Critical point
Tonset
εa
TK
ε
H = H1 + H2 − ǫa Q12
• Quenched: copy 2 is frozen.
εq
H = H1 − ǫq Q12
• Within RFOT: Some differences between quenched and annealed cases.
• First order transition emerges from TK ,
ending at a critical point near Tonset .
• Direct consequence of, but different
nature from, ideal glass transition.
title – p.11
Spin plaquette models
• Spin plaquette models are intermediate spin models between KCM and
spin glass RFOT models: statics not fully trivial, localized defects and
X
s1 s2 s3 s4 .
facilitated dynamics. E.g. in d = 2 on square lattice: E = −
• Plausible scenario for emergence of facilitated dynamics out of
interacting Hamiltonian with glassy dynamics. [Garrahan, JPCM ’03]
• Dynamic heterogeneity similar to
[Jack et al., PRE ’05]
standard KCM.
2
self-dual
line
1.5
• “High-order” or “multi-point” static
correlations develop without finite T
phase transitions.
T
1
0.5
• For triangular plaquette model, annealed transition occurs [Garrahan, PRE
’14]. Quenched? -> NO (Rob).
0
0
low
overlap
critical
endpoint
high
overlap
0.5
1
!
1.5
2
title – p.12
Numerical evidence in 3d liquid
20
(a)
• Investigate (T, ǫ) phase diagram using umbrella sampling.
ǫ
• Sharp jump of the overlap below Tonset ≈ 10.
T = 13, · · · , 7
• Suggests coexistence region
ending at a critical point.
10
N = 256
0
0
0.2
0.4
0.6
0.8
1
Q
10
8
P (Q)
• P (Q) bimodal for finite N .
(b)
T = 8, N = 108
• Bimodality and static susceptibility enhanced at larger N for
T . Tc ≈ 9.8.
6
ǫ=0
4.17
5.09
5.98
6.17
6.33
6.43
4
2
0
0
0.2
0.4
0.6
Q
→ Equilibrium first-order phase
transition.
0.8
1
[see also Parisi & Seoane PRE ’14]
title – p.13
Configurational entropy Σ(T )
• Σ = kB log N signals entropy crisis: Σ(T → TK ) = 0. Problematic when
d < ∞, because metastable states cannot be rigorously defined.
• Experiments and simulations require approximations: Σ ≈ Stot − Svib .
1
βV (Q)
T = 13
0.6
7
0.4
0.2
Σ
0
P (Q)
• Sensible estimate:
Σ = β[V (Qhigh ) − V (Qlow )]
Harmonic spheres N = 108
0.8
3
1.5
0
0
0.2
0.4
0.6
0.8
• Free energy cost to localize the system ‘near’ a given
configuration: Well-defined
even in finite d.
• Definition of ‘states’, exploration of energy land1 scape not needed.
Q
[Berthier and Coslovich, arXiv:1401.5260]
title – p.14
Ideal glass transition?
• ǫ perturbs the Hamiltonian: Affects the competition energy /
configurational entropy (possibly) controlling the ideal glass transition.
• Random pinning of a fraction c of particles: unperturbed Hamiltonian.
• Slowing down observed numerically.
[Kim, Scheidler...]
T
Critical point
Tonset
• Within RFOT, ideal glass transition line extends up to critical point.
[Cammarota & Biroli, PNAS ’12]
TK
Pinning
• Pinning reduces multiplicity of states, i.e.
decreases configurational entropy: Σ(c, T ) ≃
Σ(0, T ) − cY (T ). Equivalent of T → TK .
title – p.15
Pinning in plaquette models
• Random pinning studies in spin plaquette models offer an alternative
[Jack & Berthier, PRE ’12]
scenario to RFOT.
• Crossover f ⋆ (T ) from competition between bulk correlations and random
pinning: directly reveals growing static correlation lengthscale.
Light blue: mobile. Deep blue: frozen. Black: pinned.
title – p.16
Smooth crossover
=3
• Static overlap q increases rapidly with fraction f of pinned spins,
crossover f ⋆ = f ⋆ (T ), but no phase transition.
=
• Overlap fluctuations reveal growing static correlation length scale, but
susceptibility remains finite as N → ∞.
• Dynamics barely slows down with10f , unlike atomistic models.
t
1
=3
(b)
0.8
β=4
=3
q
β=3
(e)
1000
0.6
τ
β = 2.5
0.4
500
0.2
0
10
0
0.1
0.2
f
t
0.3
0.45
10
0
0
0.05 0.1 0.15 0.2 0.25
f
title – p.17
Random pinning in 3d liquid
• Challenge: fully exploring equilibrium configuration space in the
presence of random pinning: parallel tempering. Limited (for now) to small
system sizes: N = 64, 128.
[Kob & Berthier, PRL ’13]
Low-c fluid
High-c glass
• From liquid to equilibrium glass: freezing of amorphous density profile.
• We performed a detailed investigation of the nature of this phase
change, in fully equilibrium conditions.
title – p.18
Order parameter
• We detect “glass formation” using an equilibrium, microscopic order
parameter: The global overlap Q = hQ12 i.
0.7
0.6
overlap
0.5
0.4
T=4.8
T=5.0
T=5.5
T=6.0
T=7.0
T=8.0
T=13
T=20
0.3
0.2
0.1
0.0
0.00
0.10
0.20
c
0.30
0.40
N = 64
• Gradual increase at high T to more abrupt emergence of amorphous
order at low T at well-defined c value. First-order phase transition or
smooth crossover?
title – p.19
Fluctuations: Phase coexistence
• Probability distribution function of the overlap: P (Q) = hδ(Q − Qαβ )i .
c=0
c=0.063
c=0.125
c=0.188
c=0.250
c=0.281
c=0.313
c=0.375
P(q)
6
6
4
2
0
0.0
c=0
c=0.031
c=0.078
c=0.109
c=0.125
c=0.156
c=0.188
8
T=13
P(q)
8
T=4.8
4
2
0.1
0.2
0.3
0.4
q
0.5
0.6
0.7
0.8
0
0.0
0.1
0.2
0.3
0.4
q
0.5
0.6
0.7
0.8
0.9
N = 64
• Distributions remain nearly Gaussian at high T .
• Bimodal distributions appear at low enough T , suggestive of phase
coexistence at first-order transition, rounded by finite N effects.
title – p.20
Thermodynamic limit?
• Phase transition can only be proven using finite-size scaling techniques
to extrapolate toward N → ∞.
• Limited data support enhanced bimodality and larger susceptibility for
larger N . Encouraging, but not quite good enough: More work needed.
title – p.21
Equilibrium phase diagram
• Location of the transition from liquid-to-glass determined from
equilibrium measurements of microscopic order parameter on both sides.
10
τ=10
3
τ=10
4
τ=10
τ=10
8
Ton
5
6
T
τ=10
7
fluid
6
4
TK from γ
2
0
0
Tmct
glass
TK from χ
0.04
0.08
0.12
c
(c)
0.16
0.2
0.24
• Glass formation induced by random pinning has clear thermodynamic
signatures which can be studied directly.
• Results compatible with Kauzmann transition – this can now be decided.
title – p.22
Summary
• Non-trivial static fluctuations of the overlap in 3d bulk supercooled
liquids: non-Gaussian V (Q) losing convexity below ≈ Tonset .
• Adding a thermodynamic field can induce equilibrium phase transitions.
T
• Annealed coupling: first-order transition
ending at simple critical point. Universal?
Critical point
• Quenched coupling: first-order transition
ending at random critical point. Specific to
RFOT?
Tonset
TK
ε
• Random pinning: random first order transition ending at random critical point. Specific
to RFOT.
• Direct probes of peculiar thermodynamic underpinnings of RFOT theory.
• A Kauzmann phase transition may exist, and its existence be decided.
title – p.23