dynamical phase transitions and glassy...
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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
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Introduction
Out-of-equilibrium dynamics?
Dynamical phase transitions?
What are our systems of interest?
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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?
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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).
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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?
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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).
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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>
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Results: Mean-Field FA
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Results: Mean-Field FA
FK (ρ, s) for different values of s:
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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 →∞
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Results in finite dimensions
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 )
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Results in finite dimensions
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).
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Results in finite dimensions
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
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Results in finite dimensions
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).
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Driven KCMs, current heterogeneities and large deviations
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Driven KCMs, current heterogeneities and large deviations
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?
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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!
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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.
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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).
Estelle Pitard Dynamical phase transitions and glassy physics
IntroductionA dynamical phase transition for KCMs
Driven KCMs and current heterogeneities
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?
Estelle Pitard Dynamical phase transitions and glassy physics