Time scale separation in the East model

Paul Chleboun Joint work with:

A. Faggionato and F. Martinelli

15/07/2013

1

[Time scale separation in the low temperature East model: rigorous results, J. Stat. Mech letter (2013)] [Time scale separation and dynamic heterogeneity in the low temperature East model. arXiv:1212.2399]

Outline

• The East Model » Motivation.

» Definition.

• Relevant previous results » Relaxation time on infinite systems.

» Separation of time scales on finite systems.

• Results » Equivalence of characteristic times.

» Time scale separation (up to equilibrium length).

» Dynamical heterogeneity.

• Open problems

The East Model

• Motivation

» Glassy effects at high density (low temperature):

• Relaxation time diverges quickly (super-Arrhenius).

• Dynamical heterogeneity.

• Cooperative dynamics.

» Aim to understand the low temperature (non-equilibrium) dynamics.

• Dependence of the relaxation time on system size and temperature.

The East Model

• Graphical construction

» 1D Lattice:

» State space:

» Configurations:

» Glauber dynamics with kinetic constraint.

» Zeros facilitating. Fixed zero at the origin

[Jackle-Eisinger ‘91]

The East Model

• Graphical construction

» 1D Lattice:

» State space:

» Configurations:

» Glauber dynamics with kinetic constraint.

» Zeros facilitating. Fixed zero at the origin

The East Model

• Graphical construction

» 1D Lattice:

» State space:

» Configurations:

» Glauber dynamics with kinetic constraint.

» Zeros facilitating. Fixed zero at the origin

The East Model

• Graphical construction

» 1D Lattice:

» State space:

» Configurations:

» Glauber dynamics with kinetic constraint.

» Zeros facilitating. Fixed zero at the origin

The East Model

• Generator:

Where:

The East Model

• Stationary measure

» Constraint at site does not depend on

» reversible with respect to product Bernoulli(1-q) measure:

• Consider small q, low temperature regime

Previous Results

• Relaxation time (inverse spectral gap of )

» Finite: [Aldous, Diaconis (`02)]

» Asymptotic:

» Exponential relaxation:

» Non trivial results; the process is not monotone.

[Cancrini, Martinelli, Roberto, Toninelli , PTRF (`08)]

[Cancrini, Martinelli, Schonmann, Toninelli , JSP (`09)]

Dynamics (heuristics)

• Typical stationary inter-vacancy distance is 1/q .

• On shorter length scales, as dynamics

dominated by removing excess vacancies. » To remove a vacancy must first create vacancy to the

left (cooperative dynamics).

» Energy barrier to overcome.

» When it does over come the energy barrier happens on a much smaller timescale than it typically waits.

Dynamics (heuristics)

• Energy barrier

» Has to create at least n simultaneous zeros.

» Activation time: .

» Metastability: Actual killing is on a much shorter timescale

» Actual time reduced by an entropic factor.

Dynamics (heuristics)

[Chung, Diaconis, Graham, (`01)]

Previous Results

• Separation of time scales on finite systems.

» Universality with hierarchical coalescence process:

• ,

Dynamics on interval is approximated by a HCP.

[Faggionato, Martinelli, Roberto, Toninelli , CMP (`10)]

Previous Results

[A. Faggionato, F. Martinelli, C. Roberto, C. Toninelli (2010)]

Persistence function: Prob. has not flipped by . Initial (renewal) measure Q

Plateau behaviour

Results: Equivalence of time scales

• Relaxation time:

• Mixing time:

• Mean first passage time:

Results: Equivalence of time scales

• Relaxation time:

• Mixing time:

• Mean first passage time:

Results: Equivalence of time scales

• Relaxation time:

• Mixing time:

• Mean first passage time:

Results: Equivalence of time scales

• Relaxation time:

• Mixing time:

• Mean first passage time:

» Certainly not true on longer length scales.

Theorem 1

Theorem 2

Time Scale Separation

• On mesoscopic length scales:

with

Theorem 2

Time Scale Separation

• On mesoscopic length scales:

• Is the separation continuous?

with

No Separation at Equilibrium Length Scale

Equilibrium length scale:

• (False) conjecture [P. Sollich, M.R. Evans (2003)]

» Implies

» Motivated super-domain dynamics model.

No Separation at Equilibrium Length Scale

Theorem 3

• There is no form of time scale separation on the equilibrium length scale

Dynamical heterogeneity

• Consequences of time scale separation on mesoscopic length scales.

Theorem 4

Relaxation time bounds

• Precise bounds on relaxation time as a function of L and q.

» Key to the results on mesoscopic length scales

Theorem 5

Relaxation time bounds

• Entropy

» : Number of configurations with n zeros reachable using at most n.

» Actual reduction factor is much smaller! • Many of these configurations do not lie on the

‘saddle point’ (bottle neck).

• Estimate the true size in terms of deterministic dynamics.

• Gives rise to a lower bound on the relaxation time.

[F. Chung, P. Diaconis, R. Graham, Adv. In Appl. Math. 27 (2001)]

Open problems

• Is there continuous time scale separation on mesoscopic length scales?

• Higher dimensional ‘East processes’

Open problems

• Scaling limit on the equilibrium length scale

» Conjecture of Aldous and Diaconis (2002):

As after rescaling space by and speeding up time by the relaxation time the process converges to , such that:

1. At fixed t, is a Poisson (rate one) process on (location of zeros)

2. Each zero produces a wave of length at some rate .

3. Wave instantaneously deletes all particles it covers and replaces with a rate one Poisson process.

Summary

• Strong equivalence of characteristic time scales.

• Strong form of time scale separation and dynamical heterogeneity up to but not including the equilibrium scale.

» Precise bounds on the relaxation time, temperature and system size dependence.

• Absence of any time scale separation on the equilibrium scale.

» In this case numerical simulations were misleading, emphasizes the need for rigorous results.

Work supported by the ERC through the “Advanced Grant” PTRELSS 228032