Anomalous scaling in a kinetically constrained model in 1+1 ......Anomalous scaling in a kinetically...

Anomalous scaling in a kinetically constrained model in 1+1 dimensions

Anomalous scaling in a kinetically constrainedmodel in 1+1 dimensions

Arturo L. Zamorategui.Modelisation Stochastique

Laboratoire de Probabilites et Modeles Aleatoires.Supervisor: Vivien Lecomte

June 10, 2014

Arturo L. Zamorategui. Modelisation Stochastique Anomalous scaling in a kinetically constrained model in 1+1 dimensions


Outline

1 Introduction

2 Part 1. Statistics

3 Results

4 Part 2. Towards an effective model

5 Partial Results

6 Perspectives



Goals

1 Understand the phase coexistence of active-inactive regions at thepoint where the dynamical phase transition observed in glassysystems occurs. Very well captured by KCMs.

2 Determine an effective model of the dynamics in the phase transitionfrom the simplest possible model. Is there something simpler that aKCM ?



Introduction

A glass looks like...

2D binary mixture of hard disks withdifferent radius (Keys et al Phys. Rev.X 1, 021013, 2011).

Inactive particles vs Active particles

Start movie

Dynamic heterogeneity: inactive regions with slow dynamics and activeregions with fast dynamics.

Facilitated dynamics: mobility in a region leads to motion ofneighbouring particles. An adjacent excitation is required for both thebirth or death of an excitation.



Introduction

Activity vs Inactivity

Inactive regions surrounded by active particles.

The persistent time (tp): time a particle remains inactive before the firstmove.

The exchange time (tx): time between two consecutive moves. (Jung etal 2004).

Not a glass A glassDynamic heterogeneity 〈tp〉 = 〈tx〉 = τ 〈tx〉 � 〈tp〉

Facilitation - Particles close to excitation lines diffuse more quickly

Stokes-Einstein relations:

Dη/T = constant Satisfied Broken,

since τα ≈ 〈tp〉Dτα = constant and D ≈ δx2/〈tx〉:

τα (relaxation time) (Dτα ≈ δx2〈tp〉/〈tx〉)D (diffusion coefficient)



A KCM

A Kinetically constrained model (KCM)

Dynamic heterogeneity (bubbles of inactivity) and facilitation (excitation lines)are captured by kinetically constrained models (KCMs).

One spatial dimension (discrete) + one temporal dimension (continuous)

It is a coarse-grained model with ni = 1 if active and ni = 0 if inactive.

Dynamic is controlled by a local constraint.



A KCM

Statics vs Dynamics

Non-conservative system

Site i changes state with rate fi[(1− c)ni + c (1− ni)], where c is thedensity of active sites and fi is the kinetic constraint.

fi = 1 if constraint is satisfied and fi = 0 otherwise.

fi depends on the neighbourhood but not on ni.

Detailed balanced holds w.r.t Boltzmann distribution (Bernoulli productmeasure) with energy

∑i ni and inverse temperature β (c = 1/(1 + eβ)).

Since statics is trivial, we need a dynamical description.



A KCM

A dynamic observable: the Activity

Let C(t) be a configuration of the system at time t. We define a (discrete)observable as

A[trajectory] =K∑k=1

α(C(tk−1)→ C(tk)),

over K changes of configurations between times 0 and T .

Activity

The activity is K[trajectory] = #events, i.e. α(C(tk−1)→ C(tk)) = 1 whenconfiguration changes.



A KCM

Evolution of a KCM

The master equation for the evolution of the probability P (C, A, t) is

∂P (C, A, t)∂t

=∑C′W (C′ → C)P (C′, A− α(C′ → C), t)− r(C)P (C, A, t)

having measured a value A of the observable between 0 and t.r(C) =

∑C′W (C → C′) is the escape rate.

Continuous time: at each site there is a clock that changes the state ni after

a time ∆t given by p(∆t) = W (ni → 1− ni)e−W (ni→1−ni)∆t.



A KCM

Large deviation

To have certain dynamics one can bias a trajectory according to a parameter s,with trajectory with lower than normal activity (s > 0) or higher than normalactivity (s < 0):

P [trajectory]→ Ps[trajectory] =P [trajectory]e−sK[trajectory]

Zs,

where Zs = 〈e−sK[trajectory]〉.For large observation times:

limt→∞

lnZst→ ψK(s).

ψK(s) is a large deviation function, also known as dynamical free energy.



A KCM

First order phase transition

ψK(s) presents a singularity at s = 0:s < 0: trajectory where 〈Kt〉 is larger.

s = 0: trajectory with coexistence of active and inactive regions.

s > 0: trajectory where 〈Kt〉 is smaller.

Figure: Active-Inactive transition ( Garrahan et al (2007)).



FA model

Fredrickson-Andersen Model (FA-1f)

The transition rates in the FA model are

W (ni → 1− ni) = fi(nj)eβ(ni−1)

1 + e−β,

where fi = 1 if∑j∼i nj > 0, i.e if there is at least one active site in the

neighbourhood, otherwise fi = 0.

For an activation: W (0→ 1) = fi1

1+eβ. the term c = 1/(1 + eβ) is the

density of active sites.

Mean distance between excitations: l = 1/c



Examples FA-1f

Typical histories (with s = 0) for different densities

Histories of length L = 100 and t = 1000 for c = 0.1 and 0.6.



Statistics

Part 1



Statistics

Describe bubbles of inactivity

There is not an effective model to understand the phase coexistence ofactivity and inactivity which is generic in glasses.

Such a model must describe how bubbles of inactivity evolve in space-time .

1 Order columns ofinactivity

2 Columns → Nodes

3 Build adjacencymatrix (fromoverlap of nodes)

4 Find bubbles byidentifying onenode, itsneighbours,...



Statistics

Statistics on bubbles

Focus on one bubble...



Statistics

Geometrical Properties

Area

A

Number of Nodes

k

Temporal extension

t

Spatial extension

x

Temporal perimeter

PT

Spatial perimeter

PS



Statistics

Histograms for the Area

Varying L: Varying c:

10−1

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

Area

freque

ncy

L=50

L=100

L=150

L=200

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

Area

freque

ncy

c=0.4

c=0.5

c=0.6

Curves superimpose for different LAs c decreases, bigger bubblesappear



Statistics

Spatial Perimeter

6.6e−05 1.6e−03 4.0e−02 1.0e+00 2.5e+01 6.1e+02 1.5e+041

3

7

20

55

148

403

1097

2981

∆ t

〈PS〉

data

ζ1 =0.000

ζ2 =1.002

PS ∼ ∆t1.002 for large bubbles



Statistics

Area

6.6e−05 1.6e−03 4.0e−02 1.0e+00 2.5e+01 6.1e+02 1.5e+040

0

1

148

22026

3269017

∆ t

〈A〉

data

ζ1 =1.001

ζ2 =1.470

A ∼ ∆t1.470 for large bubbles, close to Brownian with ζ = 3/2.∗

∗S. Majumdar et al. 2005Arturo L. Zamorategui. Modelisation Stochastique Anomalous scaling in a kinetically constrained model in 1+1 dimensions


Statistics

Spatial extension

6.6e−05 1.6e−03 4.0e−02 1.0e+00 2.5e+01 6.1e+02 1.5e+041

1

2

3

4

7

12

20

33

55

90

∆ t

〈∆x〉

data

ζ1 =0.000

ζ2 =0.549

∆x ∼ ∆t0.549 for large bubbles, bigger than Brownian where ζ = 1/2



Statistics

Mean shape of bubbles

The probability that r.w. arrives to x1 at time t1 given that started at x0 attime t0 is:

Pε(x1, t1|x0, t0) =1√

2πD(t1 − t0)

[e−(x1−x0−ε)2

2D(t1−t0) − e−(x1−x0+ε)2

2D(t1−t0)

]

The mean value of x(t, t1) is:

〈x(t, t1)〉[t0,t1] =

∫∞0dxxPε(x1, t1|x, t)Pε(x, t|x0, t0)∫∞

0dxPε(x1, t1|x, t)Pε(x, t|x0, t0)



Scaling

Numerical mean shape

By scaling average curves of bubbles with duration ∆t− ε < ∆t < ∆t+ ε asF (τ) ∼ ∆t−ζf(τ∆t).

ζ = 0.5: ζ = 0.6:

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

t

x

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

t

x



(Towards an) Effective model

Part 2




FA model for small c

At low temperature (small c) the borders of the bubbles of inactivity in the FAmodel have some few active sites (left). The dynamics in the FA can bereproduced by a model of branching-coalescence (right)(Whitelan,Berthier,Garrahan 2005).

space coordinate

t

0 20 40 60 80 100 120 140 160 180

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

space coordinate

t

0 20 40 60 80 100 120 140 160 180

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

Borders of activity look like random walkers that branch, diffuse and coalesce.




Branching-Coalescing process

We can simulate the system as a branching-coalescing process if

it captures the same dynamics that the FA model

Is valid at finite time scales

Is analytically tractable




Model of branching coalescence

Initial condition with density c of active sites.

A configuration C changes after a time ∆t as p(∆t) = r(C)e−r(C)∆t where

the escape rate is

r(C) =∑i

fi(C)[(1− ni)c︸︷︷︸activation

+ni(1− c)︸︷︷︸inactivation

] = ract(C) + rinact(C)

An activation with probability: pact =ract(C)r(C) and an inactivation with

probability: pinact = rinact(C)r(C) .

A border of activity constrained to have maximum 3 active sites.




Strategy

In continuous space: two random walkers will cross an infinite number of times.

In discrete space: is reflected in small bubbles. Focus on branching/coalescingpoints of large bubbles (of duration T > 10)

10000

12000

14000

16000

18000

20000 0 50 100 150 200

t

space coordinates




Analysis

The rates are the same by time reversal symmetry. How often do we have abranching/coalescing event?

To have just branching and coalescing points we use a contraction algorithmuntil there’s no isolated point...

10000

12000

14000

16000

18000

20000 0 50 100 150 200

t

space coordinates




Analysis

We end up with a brownian net with just branching and coalescing points.

How many events occur along the borders of a bubble? The frequency ofbranching/coalescing sould not depend on size of bubbles.

10000

12000

14000

16000

18000

20000 0 50 100 150 200

t

space coordinates




Probability distribution of frequency ν

We can study the PDF of the frequency ν of branching and coalescingevents for different bubbles’ sizes.

Bubbles of duration from:

10 to 500

500 to 1000

1000 to 1500

1500 to 2000

2000 to 2500

2500 to 3000

3000 to 4000

0.00 0.02 0.04 0.06 0.08 0.100

20

40

60

80

100

Ν

PHΝL




Average frequency

Are branching/coalescencing rates independent of bubble size?

Mean frequency ν:

0 1000 2000 3000 40000.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

bubble duration HDtL

XΝ\

Average number of events:

1000 2000 3000 40000

10

20

30

40

50

bubble duration

ðev

ents




Perspectives

1 Check that the effective model from “brownian net” is realistic.

2 Convergence of the distribution of the event rates as statistics isincreased

3 Determine average frequency ν?

4 How ν and D are related to c? Scaling limits?

5 What information do we have about the phase coexistence?

6 Low temperature limits for other KCM?

7 For a biased dynamics (s < 0 or s > 0), what can be relevant fromthis simple model?




Thank you for listening!



Bibliography

Bibliography

Huang, Kerson. ”Statistical mechanics.” (1987)

Chandler et al, Annu. Rev. Phys. Chem. 2010. 61:191–217

G. H. Fredrickson and H. C. Andersen Phys. Rev. Lett.,53(13):1244–1247, 1984

T. Bodineau, V. Lecomte and C. Toninelli. J Stat Phys (2012) 147:1–17

J P Garrahan et al J. Phys. A: Math. Theor. 42 (2009) 075007

S. Majumdar et al J Stat. Phys. 119.3-4 (2005): 777-826.

Majumdar, Satya N. arXiv preprint cond-mat/0510064 (2005).

D. Chandler et al. Phys Rev X 1, 021013 (2011)



Appendix

Order parameter

Activity

K = ∆t

tobs∑t=0

L∑i=1

(ri(t+ ∆t)− ri(t))2

where L is the total number of particles considered and tobs is theamount of time the systems is observed.The mean activity is

〈K〉 = Ltobs〈(r1(t+ ∆t)− r1(t))2〉 = 2dLtobsD∆t



Appendix

Tests

For an unconstrained model, the probability of a configuration~n = (n1, n2, . . . , nL) is Punceq (~n) = cn(1− c)L−n, therefore the average valueof a site is

1

L

⟨∑i

ni

⟩unc

eq

= c .

For a constrained model where∑Li ni > 0, the probability has a

normalization factor Z Peq(~n) = cn(1−c)L−nZ

which is found to beZ = 1− (1− c)L. The average value of a site is

〈nj〉eq =c

1− (1− c)L .

We compare this value with the one obtained numerically. The bigger the

lattice, the more accurate the numerical value is.



Appendix

Other statistics

Temporal Perimeter

Area

A

Number of Nodes

k

Temporal extension

t

Spatial extension

x

Temporal perimeter

PT



Appendix

Other statistics

Spatial Perimeter

Area

A

Number of Nodes

k

Temporal extension

t

Spatial extension

x

Temporal perimeter

PT

Spatial perimeter

PS



Appendix

Other statistics

Perimeter geometry

(g)

−5 0 5 10 15 20 25 300.5

1

1.5

2

2.5

3

3.5

4

4.5

5

log(∆ t)log(PS)

data

0.00020668

1.0329

(h)

PT ∼ ∆t for all of the bubbles and PS ∼ ∆t for large bubbles.


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