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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
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