Thermodynamic fluctuations inmodel glasses

Ludovic Berthier

Laboratoire Charles CoulombUniversite Montpellier 2 & CNRS

Glassy Systems and Constrained Stochastic Dynamics – Warwick, June 11, 2014

D4PARTICLES

title – p.1

Coworkers

• With:

D. Coslovich (Montpellier)

R. Jack (Bath)

W. Kob (Montpellier)

= 1= 1

12 (b) (c)

(e) (f)

f = 0.18

title – p.2

Temperature crossovers

• Glass formation characterized by several “accepted” crossovers. Onset,mode-coupling & glass temperatures: directly studied at equilibrium.

1 / T ( K-1)

0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050

log 10

( η

/ poi

se)

-4

-2

0

2

4

6

8

10

12

14

T0TK

TxTBTc

T*TA

Tg

Tb

Tm

[G. Tarjus] [Debenedetti & Stillinger]

• Extrapolated temperatures for dynamic and thermodynamic singularities:T0, TK . Existence and nature of “ideal glass transition” at Kauzmanntemperature is controversial.

title – p.3

Dynamic heterogeneity

• When density is large, particles must move in a correlated way. Newtransport mechanisms revealed over the last decade: fluctuations matter.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90

• Spatial fluctuations grow (modestly)near Tg.

• Clear indication that some kind ofphase transition is not far – which?

• Structural origin not clearly es-tablished: point-to-set lengthscales,other structural indicators?

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. Newtransport mechanisms revealed over the last decade: fluctuations matter.

• Spatial fluctuations grow (modestly)near Tg.

• Clear indication that some kind ofphase transition is not far – which?

• Structural origin not clearly estab-lished: point-to-set lengthscales andother structural indicators?

• Do “convincing” thermodynamic fluctuations even exist?title – p.5

Dynamical view: Large deviations

• Large deviations of fluctuations ofthe (time integrated) local activity

mt =∫dx

∫ t0dt′m(x; t′, t′ +∆t):

P (m) = 〈δ(m−mt)〉 ∼ e−tNψ(m).

• Exponential tail: direct signature ofphase coexistence in (d + 1) dimen-sions: High and low activity phases.Direct connection to dynamic hetero-geneity.

0.0 0.1 0.2 0.3m

10-12

10-9

10-6

10-3

100

P(m

)

tobs

= 320tobs

= 640tobs

= 960tobs

= 1280

[Jack et al., JCP ’06]

• Equivalently, a field coupled to local dynamics induces a nonequilibriumfirst-order phase transition in the “s-ensemble”. [Garrahan et al., PRL ’07]

• Metastability controls this physics. “Complex” free energy landscapegives rise to same transition, but the transition exists without multiplicity ofglassy states: KCM, plaquette models. [Jack & Garrahan, PRE ’10]

title – p.6

Thermodynamic view: RFOT

• Random First Order Transition (RFOT) theory is a theoretical frameworkconstructed over the last 30 years (Parisi, Wolynes, Götze...) using a largeset of analytical techniques.[Structural glasses and supercooled liquids, Wolynes & Lubchenko, ’12]

• Some results become exact for simple “mean-field” models, such as the

fully connected p-spin glass model: H = −∑i1···ip

Ji1···ipsi1 · · · sip .

• Recently demonstrated for hard spheres as d → ∞ [Kurchan, Zamponi et al.]

• Complex free energy landscape → sharp transitions: Onset (apparitionof metastable states), mode-coupling singularity (metastable statesdominate), 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 tohave overlap Q: [Franz & Parisi, PRL ’97]

Vq(Q) = −(T/N)

∫dr2e

−βH(r2) log

∫dr1e

−βH(r1)δ(Q−Q12)

where: Q12 = 1N

∑N

i,j=1 θ(a− |r1,i − r2,j |).

Σ(T )TK

Simple liquid

Tmct

Tonset

Q

V(Q

)

0.80.60.40.20

0.08

0.06

0.04

0.02

0

• V (Q) is a ‘large deviation’ func-

tion, mainly studied in mean-fieldRFOT limit.

• P (Q) = 〈δ(Q−Q12)〉

∼ exp[−βNV (Q)]

• Overlap fluctuations reveal evolution of multiple metastable states. Finited 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 theiroverlap Q12, record the histogram of Q12.

• (Obvious) problem: Two equilibrium configurations are typicallyuncorrelated, with mutual overlap ≪ 1 and small (nearly Gaussian)fluctuations.

T = 9, N = 256

Q

P(Q

)

0.30.20.1010−6

10−4

10−2

100

102

• Solution: Seek “large deviations” using umbrella sampling techniques.[Berthier, PRE ’13]

title – p.9

Overlap fluctuations in 3d liquid

• Idea: bias the dynamics us-

ing Wi(Q) = ki(Q−Qi)2 to ex-

plore of Q ≈ Qi.

• Reconstruct P (Q) usingreweighting techniques.

• Exponential tail below Tonset

→ phase coexistence be-tween multiple metastablestates in 3d bulk liquid.

• Static fluctuations may con-trol fluctuations and phasetransitions in trajectory space.

T = 18, 15, 13, 11, 10, 9, 7

(a) Annealed

Va(Q

)

10

5

0

T = 13, 11, 10, 8, 7

(b) Quenched

Q

Vq(Q

)

10.80.60.40.20

10

5

0

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.

εa

H = H1 +H2 − ǫaQ12

• Quenched: copy 2 is frozen.

εq

H = H1 − ǫqQ12

TK

Tonset

T

ε

Critical point

• Within RFOT: Some differences be-tween quenched and annealed cases.

• First order transition emerges from TK ,ending at a critical point near Tonset.

• Direct consequence of, but differentnature from, ideal glass transition.

title – p.11

Spin plaquette models

• Spin plaquette models are intermediate spin models between KCM andspin glass RFOT models: statics not fully trivial, localized defects and

facilitated dynamics. E.g. in d = 2 on square lattice: E = −∑�

s1s2s3s4.

• Plausible scenario for emergence of facilitated dynamics out ofinteracting Hamiltonian with glassy dynamics. [Garrahan, JPCM ’03]

• Dynamic heterogeneity similar tostandard KCM. [Jack et al., PRE ’05]

• “High-order” or “multi-point” staticcorrelations develop without finite Tphase transitions.

• For triangular plaquette model, an-nealed transition occurs [Garrahan, PRE

’14]. Quenched? -> NO (Rob).0 0.5 1 1.5 2

ε

0

0.5

1

1.5

2

T

high overlap

low overlap

critical endpoint

self-dual line

title – p.12

Numerical evidence in 3d liquid

• Investigate (T, ǫ) phase dia-gram using umbrella sampling.

• Sharp jump of the overlap be-low Tonset ≈ 10.

• Suggests coexistence regionending at a critical point.

T = 13, · · · , 7(a)

N = 256

Qǫ

10.80.60.40.200

10

20

6.43

6.33

6.17

5.98

5.09

4.17

ǫ = 0

(b)

T = 8, N = 108

Q

P(Q

)

10.80.60.40.20

10

8

6

4

2

0

• P (Q) bimodal for finite N .

• Bimodality and static suscep-tibility enhanced at larger N forT . Tc ≈ 9.8.

→ Equilibrium first-order phasetransition.

[see also Parisi & Seoane PRE ’14]title – p.13

Configurational entropy Σ(T )

• Σ = kB logN 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.

Q

P(Q

)

10.80.60.40.20

3

1.5

0

7T = 13

Σ

Harmonic spheres N = 108

βV

(Q)

1

0.8

0.6

0.4

0.2

0

• Sensible estimate:Σ = β[V (Qhigh)− V (Qlow)]

• Free energy cost to local-ize the system ‘near’ a givenconfiguration: Well-definedeven in finite d.

• Definition of ‘states’, ex-ploration of energy land-scape not needed.

[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 par-ticles: unperturbed Hamiltonian.

• Slowing down observed numerically.[Kim, Scheidler...]

TK

Tonset

TCritical point

Pinning

• Within RFOT, ideal glass transition line ex-tends up to critical point.

[Cammarota & Biroli, PNAS ’12]

• 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 alternativescenario to RFOT. [Jack & Berthier, PRE ’12]

• Crossover f⋆(T ) from competition between bulk correlations and randompinning: directly reveals growing static correlation lengthscale.

Light blue: mobile. Deep blue: frozen. Black: pinned.title – p.16

Smooth crossover

• 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, butsusceptibility remains finite as N → ∞.

• Dynamics barely slows down with f , unlike atomistic models.10

t

=

= 3

(b)

q

β = 4

β = 2.5

f

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.410

(e)

t

=

= 3

105

t

= 3

f

τ

β = 3

0

500

1000

0 0.05 0.1 0.15 0.2 0.25

title – p.17

Random pinning in 3d liquid

• Challenge: fully exploring equilibrium configuration space in thepresence of random pinning: parallel tempering. Limited (for now) to smallsystem 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 phasechange, in fully equilibrium conditions.

title – p.18

Order parameter

• We detect “glass formation” using an equilibrium, microscopic orderparameter: The global overlap Q = 〈Q12〉.

0.00 0.10 0.20 0.30 0.40c

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7ov

erla

p

T=4.8T=5.0T=5.5T=6.0T=7.0T=8.0T=13T=20

N = 64

• Gradual increase at high T to more abrupt emergence of amorphousorder at low T at well-defined c value. First-order phase transition orsmooth crossover?

title – p.19

Fluctuations: Phase coexistence

• Probability distribution function of the overlap: P (Q) = 〈δ(Q−Qαβ)〉 .

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8q

0

2

4

6

8

P(q

)

c=0c=0.063c=0.125c=0.188c=0.250c=0.281c=0.313c=0.375

T=13

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9q

0

2

4

6

8

P(q

)

c=0c=0.031c=0.078c=0.109c=0.125c=0.156c=0.188

T=4.8

N = 64

• Distributions remain nearly Gaussian at high T .

• Bimodal distributions appear at low enough T , suggestive of phasecoexistence at first-order transition, rounded by finite N effects.

title – p.20

Thermodynamic limit?

• Phase transition can only be proven using finite-size scaling techniquesto extrapolate toward N → ∞.

• Limited data support enhanced bimodality and larger susceptibility forlarger 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 fromequilibrium measurements of microscopic order parameter on both sides.

0 0.04 0.08 0.12 0.16 0.2 0.24c

0

2

4

6

8

10T

Ton

Tmct

TK from γTK from χ

glass

fluidτ=107

τ=106

τ=105τ=10

4τ=10

3

(c)

• Glass formation induced by random pinning has clear thermodynamicsignatures 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 supercooledliquids: non-Gaussian V (Q) losing convexity below ≈ Tonset.

• Adding a thermodynamic field can induce equilibrium phase transitions.

TK

Tonset

T

ε

Critical point

• Annealed coupling: first-order transitionending at simple critical point. Universal?

• Quenched coupling: first-order transitionending at random critical point. Specific toRFOT?

• Random pinning: random first order transi-tion ending at random critical point. Specificto RFOT.

• Direct probes of peculiar thermodynamic underpinnings of RFOT theory.

• A Kauzmann phase transition may exist, and its existence be decided.title – p.23