Large deviations and heterogeneities in driven or non-driven...

Dynamic transition in KCMs- large deviationsDriven KCMs, heterogeneities and large deviations

Large deviations and heterogeneities in driven or

non-driven kinetically constrained models

Estelle Pitard

1CNRS, L2C, Montpellier, France

MPI- Dresden- 13 July 2011

with: J.P. Garrahan (Nottingham), R.L. Jack (Bath), V. Lecomte, K. vanDuijvendijk, F. van Wijland (Paris), F. Turci (Montpellier)

Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constrained


Outline

Dynamic transition in Kinetically Constrained Models- large deviations

Phenomenology of kinetically constrained models (KCMs)Relevant order parameters for space-time trajectories: activity KResults: mean-field/ finite dimensions

Driven KCMs, heterogeneities and large deviations

A new dynamic phase transition for the current JResults and link with microscopic spatial heterogeneities



Phenomenology of KCMs

Spin models on a lattice / lattice gases, designed to mimick steric effectsin amorphous materials:

si = 1, ni = 1: ”mobile” particle - region of low density - fastdynamicssi = −1, ni = 0: ”blocked” particle - region of high density - slowdynamicsH =

P

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 state.

↓↑↓⇋↓↓↓ is forbidden.

Mobile/blocked particles self-organize in space → dynamical correlationlength ξ.

How to classify time-trajectories and their activity?

(F. Ritort, P. Sollich, Adv. Phys 52, 219 (2003).)



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) =P

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

where r(C) =P

C ′ 6=C W (C → C′)

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

P̂(C , s, t) =P

K e−sKP(C , K , t) → ∂t P̂(C , s, t) = WK P̂(C , s, t),

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

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

C P̂(C , s, t) =< e−sK >.

For t → ∞, ZK (s, t) ≃ etψK (s).

→ the large deviation function ψK (s) is the largest eigenvalue of WK (s).



Relevant order parameters for space-time trajectories

Average activity: (s,t)Nt

=t→∞

− 1N

ψ′K (s).

Analogy with the canonical ensemble:

space of configurations, fixed β: Z(β) =P

C e−βH ≃ e−Nf (β),N → ∞.

space of trajectories, fixed s:ZK (s, t) =

P

C ,K e−sKP(C , K , t) ≃ e−tfK (s),t → ∞.

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

ρK (s),(s,t)

Nt: activity/chaoticity.

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) = kn−1N

, n =P

i ni .

The result is a variational principle for ψK (s), involving aLandau-Ginzburg free energy FK (ρ, s) (ρ: density of mobile spins):

1N

fK (s) = −1N

ψK (s) = minρ

FK (ρ, s), with

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

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 order

phase transition.

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



Results: Mean-Field FA

FK (ρ, s) for different values of s:

0 0.2 0.4 0.6 0.8 1rho

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

free

ene

rgy

(FA

cas

e)

s=-0.4s=-0.2s=0s=0.2s=0.4



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

0 0.2 0.4 0.6 0.8 1rho

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

free

ene

rgy

(unc

onst

rain

ed c

ase)

s=0.4s=0.2s=0s=-0.2s=-0.4



Results in finite dimensions

Numerical solution using the cloning algorithm for large deviationfunctions (Giardina, Kurchan, Peliti 2006) .

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

-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

L = 200L = 100L = 50

s

1

Lψ

K(s)



Results in finite dimensions

ρK (s) for the FA model in 1d.

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

L = 50L = 100L = 200

s

ρK(s)

“Dynamic first-order transition in kinetically constrained models ofglasses”, J.P. Garrahan, R.L. Jack, V. Lecomte, E. Pitard, K. vanDuijvendijk, F. van Wijland, Phys. Rev. Lett. 98, 195702 (2007).

“First-order dynamical phase transition in models of glasses: an approachbased 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).




2d ASEP with kinetic constraints, (model introduced by M. Sellitto, 2008).

A particle can hop to an empty neighbouring site if it has at most 2occupied neighbouring sites, before and after the move.

Fixed density of particles ρ, periodic boundary conditions.

Driving field ~E in one direction: p = min(1, e~E .~r ).

For ρ > ρc ,

E < Emax : shear-thinning, thecurrent J grows with E

E > Emax : shear-thickening, Jdecreases with E




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



Large deviation functions for the activity K (t) and the integrated current Q(t):• For K , the first-order transition persists like for unforced KCMs.• For Q, there is a first-order transition only at large fields (coexistence ofhistories with large current and histories with no current). Absent for ASEPwithout constraints!

• → Link between current heterogeneities and singularity in the large deviationfunction?



Large deviation function for the integrated current Q(t): Gallavotti-Cohensymmetry.

−1 1

ψQ

−0.005

0

0.005

0.010

0.015

0.020

0.025

0.030

s / E - 1

2

−1 0 1

ρ=0.80 , E=2.8, L=30, clones=300

τ=700

τ=800

τ=1000

τ=5000

τ=10000

τ=50000 and

1000 clones



Dynamical blocking walls -1

Dense domain walls play the role of kinetic traps at large fields.

ρ = 0.82, E = 0 ρ = 0.82, E = 5




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

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



Size effects -1

H: vertical size of the system.



Size effects -2

For E = 0,ξ(ρ) ∝ exp(exp(C/(1−ρ)))Toninelli, Biroli, Fisher, 2004.

→ Determination of thecrossover length ξ(ρ, E):dynamical correlationlength in the presence of anexternal field E?



Conclusions

Large deviation functions of generating functions in trajectories spaceprovide useful order parameters that probe active/inactive phases or largecurrent/small current phases according to the observable. s plays the roleof a ”chaoticity” temperature.

KCMs show a first-order phase transition at s = 0. In a real system, thereis coexistence between 2 different dynamical phases.

New dynamic phase transition in the case of transport: other examples?

Link between transport properties, microscopic lengths between defectsand dynamical correlation lengths?



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