dynamical phase transitions and glassy...

IntroductionA dynamical phase transition for KCMs

Driven KCMs and current heterogeneities

Dynamical phase transitions and glassy physics

Estelle Pitard

1CNRS, Laboratoire Charles Coulomb,Montpellier, France

Les Houches - 24 June 2015

with: F. Turci (Montpellier, Bristol), M. Sellitto (Napoli), L. Ciandrini, A. Parmeggiani (Montpellier),J.P. Garrahan (Nottingham), R.L. Jack (Bath), V. Lecomte, K. van Duijvendijk, F. van Wijland (Paris).

Estelle Pitard Dynamical phase transitions and glassy physics



Introduction

Out-of-equilibrium dynamics?

Dynamical phase transitions?

What are our systems of interest?




Out-of-equilibrium dynamics

Equilibrium systems: classical thermodynamics

Out-of-equilibrium

Changing the properties of the system? By changing e.g T in a canonicalsetting (= temperature quench: change of initial condition)

→ slow relaxation in a number of interesting examples

- domain growth

- competing interactions (frustration, disorder) → glasses

Driving a system with external perturbation:

- sheared complex liquids : new typical configurations with a flow

- vibrated granular (jammed) matter: non-equilibrium steady-states?




Dynamical phase transitions

Stochastic processes: master equation∂P∂t

(C , t) =∑

C ′W (C ′ → C)P(C ′, t)−∑

C ′ 6=C W (C → C ′)P(C , t)

Detailed balance: ensures that equilibrium distribution of states isstationary.

Peq(C)W (C → C ′) = Peq(C ′)W (C ′ → C)

Being far-from-equilibrium: usually stochastic processes that violatedetailed balance very strongly

A Classical example: Directed PercolationThe empty state is an absorbing phase (a trap): detailed balance is stronglyviolatedDynamical phase transition as a function of the percolation parameter p:p < pc : death of the percolating clusterp > pc : propagation of the percolating cluster




What are our systems of interest?

Systems with slow dynamics -but still detailed balance: no absorbingstate!-: interest in dynamic heterogeneity (active vs non-activetrajectories, mobile vs non-mobile particles) → search for simplemodels of glasses

The same systems with a driving force.

Our description framework: spacetime trajectories - not configurations.




What is glassy dynamics?

What is a glassy phase?

No static structural difference between fluid and glass

No thermodynamical transition, no Tc

How can one realize that a system is in a glassy state?

either drive it out-of-equilibrium or investigate its relaxationproperties→ dramatic increase in viscosity, ageing.

Importance of the dynamics and of spatio-temporal heterogeneitites(Fredrickson-Andersen 1984) → Fluctuations!

Models with

long-lived correlated spatial structuresslow, intermittent dynamics.

Our choice: Kinetically Constrained Models (KCMs).




A focus on Kinetically Constrained Models (KCMs)

Simplest models : 1d Fredrickson-Andersen (FA) model (spins) and 2dKob-Andersen (KA) model (particles).

si = 1, or ni = 1: ”mobile/active” particle, low density - fast dynamicssi = −1, or ni = 0: ”blocked/non-active” particle, high density - slowdynamicsH =

∑i ni →< n >eq= c = 1/(1 + eβ), β = 1/T .

Specific dynamical rules:

Fredrickson-Andersen (FA) model in 1 dimension: a spin can flip only if at leastone of its nearest neighbours is in the mobile/active state.

↓↑↓↓↓↓ is forbidden.

Phenomenology of kinetically constrained models (KCMs): glassybehaviour because of steric hindrance.

Mobile/blocked particles self-organize in space → glassy, slow relaxationand growing dynamical correlation lengths.

How to classify spacetime-trajectories and their activity?




A focus on Kinetically Constrained Models (KCMs)

KCMs from the point of view of space-time trajectories

Glassy behaviour because of steric hindrance/ Importance of initialconfiguration.

Relevant order parameter for space-time trajectories: activity K(t).

In the stationary state, there is a coexistence between active andinactive trajectories.

These trajectories can be probed by tuning an external -abstract-parameter s, or ”chaoticity temperature”.

Results: evidence for a dynamical phase transition in mean-field and infinite dimensions.




Relevant order parameters for space-time trajectories

Ruelle formalism: from deterministic dynamical systems to continous-timeMarkov dynamics

Observable: Activity K(t): number of flips between 0 and t, given ahistory C0 → C1 → ..→ Ct .

Master equation: ∂P∂t

(C , t) =∑

C ′W (C ′ → C)P(C ′, t)− r(C)P(C , t),where r(C) =

∑C ′ 6=C W (C → C ′)

Introduce s (analog of a temperature), conjugated to K:

P(C , s, t) =∑

K e−sKP(C ,K , t)→ new evolution equation

∂t P(C , s, t) = WK P(C , s, t),

where WK (s)(C ,C ′) = e−sW (C ′ → C)− r(C)δC ,C ′ .

Generating function of K: ZK (s, t) =∑

C P(C , s, t) =< e−sK >.

For t →∞, ZK (s, t) ' etψK (s).

→ ψK (s) is the large deviation function for the activity K .

ψK (s) is the largest eigenvalue of WK (s).




Relevant order parameters for space-time trajectories

Analogy with the canonical ensemble

Equilibrium:Configurations

Z(β,N) =∑

C e−βH

∼ e−Nf (β), N →∞.

Out-of-Equilibrium:Histories

ZK (s, t) =∑

hist,K e−sKP(hist,K , t)

∼ etψK (s), t →∞.

fK (s) = −ψK (s): free energy for trajectories

Average activity: <K>(s,t)Nt

=t→∞

− 1Nψ′K (s).

Active phase: <K>(s,t)Nt

> 0: s < 0.

Inactive phase: <K>(s,t)Nt

= 0: s > 0.




Results: Mean-Field FA

Wi (0→ 1) = k ′ nN

, Wi (1→ 0) = k n−1N

, n =∑

i ni .

Equilibrium distribution of mobile particles: Peq(n) = 1ZC nNe−βn, where

Z = (1 + ζ)N , ζ = e−β = k′

k(detailed balance).

From W one can get WK .

Symmetrization of WK : WK = Q−1WKQ, with Q(C ,C ′) = P1/2eq (C)δC ,C ′ .

WK (n, n′) = e−s(W+(n − 1)W−(n))1/2δn′,n−1 + e−s(W+(n)W−(n +1))1/2δn′,n+1 − r(n)δn,n′ , where

W+(n) = W (n→ n + 1),

W−(n) = W (n→ n − 1) and

r(n) = W+(n) + W−(n).

ψK (s): largest eigenvalue of WK can be calculated using:

ψK (s) = maxP

<P|WK |P><P|P>





One assumes a homogeneous profile of mobile particles ρ = nN

, and uses

the ansatz P(n) = eNf (ρ) in the large N limit.

The result is a variational principle for ψK (s).

Landau-Ginzburg free energy FK (ρ, s):

FK (ρ, s) = −2ρe−s(ρ(1− ρ)kk ′)1/2 + k ′ρ(1− ρ) + kρ2

and 1NfK (s) = − 1

NψK (s) = min

ρFK (ρ, s).

Minima of FK (ρ, s) at fixed s:

s > 0: inactive phase, ρK (s) = 0, ψK (s)/N = 0.

s = 0: coexistence ρK (0) = 0 and ρK (0) = ρ∗, ψK (0) = 0, → first orderphase transition.

s < 0: active phase, ρK (s) > 0, ψK (s)/N > 0.





FK (ρ, s) for different values of s:




Results: Mean-Field unconstrained model

One removes the constraints: Wi (0→ 1) = k ′, Wi (1→ 0) = k, for all i

FK (ρ, s) = −2e−s(ρ(1− ρ)kk ′)1/2 + k ′(1− ρ) + kρ

→ No phase transition




Results in finite dimensions

Numerical solution using the algorithm of Giardina, Kurchan, Peliti(discrete time Markov processes) for large deviation functions.

P(C , t) =∑

C ′W (C → C ′)P(C ′, t − 1)

solution at fixed C0:

P(C , t) =∑

C1,...,Ct−1

W (C0 → C1) . . .W (Ct−1 → C)

One looks for the large deviation function of an additive observableA = α(C0 → C1) + · · ·+ α(Ct−1 → Ct).

< e−sA >' etψα(s), t →∞





Defining Wα(s)(C → C ′) = W (C → C ′)e−sα(C→C ′),

< e−sA >=∑

C1,...,Ct

t−1∏i=0

Wα(s)(Ci → Ci+1)

but Wα(s) is not a stochastic matrix.

Introducing Y (C) =∑

C ′Wα(s)(C → C ′), and

W ′α(s)(C → C ′) = Wα(s)(C→C ′)Y (C)

, W ′α(s) is stochastic.

< e−sA >=∑

C1,...,Ct

t−1∏i=0

W ′α(s)(Ci → Ci+1)Y (Ci )





One performs the dynamics of N copies (N � 1) of the system:

each copy in configuration C is cloned with probability Y (C)stochastic evolution with W ′α(s)(C → C ′)the number of copies is sent back uniformly to N, with ratio Xt

ψα(s) = − limt→∞

1t

ln(X1 . . .Xt)

C. Giardina, J. Kurchan, L. Peliti, Phys. Rev. Lett. 96, 120603 (2006).





Numerical solution using the algorithm of Giardina, Kurchan, Peliti forlarge deviation functions.

First-order phase transition for the FA model in 1d:

derivative of large deviation function ψK (s) is discontinuous in s = 0.

Phase transition disappears when the dynamical constraint is removed.

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

L = 200L = 100L = 50

s

-0.02 -0.01 0.01 0.02 0.03 0.04

-0.06

-0.04

-0.02

0.02

s





True also for other KCMs and particle systems!Average activity: <K>(s,t)

Nt=

t→∞− 1

Nψ′K (s).

sK(s)

s

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

’rho’

FA d=1 East d=1

K(s)

s

N=50N=100N=200

N=8N=12N=24N=50N=100N=200

N=200N=100N=50

N=200N=100N=50N=24

Dynamic first-order transition in kinetically constrained models of glasses, J.P. Garrahan, R.L. Jack, V. Lecomte, E. Pitard, K. van

Duijvendijk, F. van Wijland, Phys. Rev. Lett. 98, 195702 (2007). First-order dynamical phase transition in models of glasses: an

approach based on ensembles of histories, J.P. Garrahan, R.L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, F. van Wijland, J. Phys. A

42 (2009).




Driven KCMs, heterogeneities and large deviations

2d ASEP (Asymetric Simple Exclusion Process) with kinetic constraints: amodel of driven particles at fixed density ρ on a 2d square lattice.(model introduced by M. Sellitto, 2008).

• Dynamical constraint:A particle can hop to an emptyneighbouring site if it has at most 2occupied neighbouring sites, before andafter the move• Asymmetric Exclusion Process:Driving field ~E in the horizontaldirection, limits backward steps.If hopping allowed:p = min{1, exp(E.dr)} (Metropolis).• If E = 0, this is the Kob-Andersenmodel in 2d.




Driven KCMs, heterogeneities and large deviations

A0.1=0.998

A0.2=0.982

A0.3=0.940

A0.4=0.860fit: J =A ρ(1-ρ) (1-exp(-E))

J

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

E0 1 2 3 4 5 6 7 8 9 10

ρ=0.1ρ=0.2ρ=0.3ρ=0.4

ρ=0.5ρ=0.6ρ=0.7Fit

constrained 2d ASEP1d TASEPMean Field

J ( E

= 1

0 )

0

0.1

0.2

ρ0 0.2 0.4 0.6 0.8 1.0

At low density ρ,

the current J is an increasing function of E

J is well approximated by a mean-field argument:JMF (ρ,E) = 1

4(1− e−E )ρ(1− ρ)(1− ρ3)2, but not at large densities.




Driven KCMs, current heterogeneities and large deviations

At high density ρ, the dynamical constraints induce a new transport regime.

For large densities,ρ > ρc ' 0.78,

E < Emax(ρ):”shear-thinning”, thecurrent J grows withE

E > Emax(ρ):”shear-thickening”,J decreases with E





Microscopic analysis: transient shear-banding at large fields, localization ofthe current.→ very different velocity profiles for small and large driving fields.

E=0.1, ρ=0.80 E=2.8, ρ=0.80

→ Coexistence of high current/small current regions at large fields.





Microscopic analysis: directed motion of probes can be blocked for a long timeat large E.→ very different mobilities for small and large driving fields.

Field direction

ρ=0.8 , E=0.1, τ=104

Field direction

ρ=0.8 , E=2.8, τ=104

→ Coexistence of mobile/non-mobile tracers at large field.→ Can we deduce coexistence of different dynamical phases in space-timetrajectories?




Study in trajectory space

• Large deviation functions for theactivity K(t)and the integrated current Q(t):Q(t) =

∫ t

0J(t′)dt′.

• For K , the first-order transitionexists like for non-driven KCMs.

• For Q, there is a first-ordertransition only at large fields(coexistence of histories with largecurrent and histories with no current).

• Absent for ASEP without constraints!




Spatial features: Dynamical blocking wallsTransverse domain walls of empty sites play the role of kinetic traps atlarge fields.

At small E , voids are random.

At large E , voids organize into domain walls transverse to the field.




Dynamical blocking walls

log 10

P(w

/ <w

>)

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

w / < w >0 2 4 6 8 10 12

E=0E=0.5E=1E=2E=3E=4E=5E=6




Dynamical blocking walls

Phenomenological fit of J(E) on thebasis of the effective blocking effect ofthe walls:J(E) = A(ρ)(1−e−E )(1−pblocked(ρ,E))J(E) ' A(ρ)(1− e−E )(1−α(ρ) < w >(ρ,E)).

• Large deviations and heterogeneities in a driven kinetically constrained model, F. Turci, E. Pitard, Europhys. Lett. 94, 10003 (2011).

• Driving kinetically constrained models into non-equilibrium steady states: structural and slow transport properties. F. Turci, E. Pitard,

M. Sellitto, Phys. Rev. E 86, 031112 (2012).

• Dynamical heterogeneities in a two dimensional driven glassy model: current fluctuations and finite size effects. F. Turci, E. Pitard,

Fluctuations and Noise Letters, Vol 11, Iss:3, 12420 (2012).




Conclusions and Perspectives

Large deviation functions of generating functions in trajectories spaceprovide useful order parameters that are able to identify active/inactivephases or large current/small current phases according to the observableK or Q: they quantify dynamical phase transitions.

KCMs show a first-order phase transition at s = 0.

In a physical system, this corresponds to the coexistence between 2different dynamical phases: inactive and active states induce the slowingdown of the dynamics.

Dynamic transitions and phase coexistence in realistic (Lennard-Jones)glasses → L. Hedges, R.L. Jack, J-P. Garrahan, D.C. Chandler, Science, 323, 1309 (2009).

E. Pitard, V. Lecomte, F. van Wijland, Europhys. Lett. 96 56002 (2011).

How to probe these two phases experimentally?


dynamical phase transitions and glassy...

Documents