Jamming Transitions: Glasses, Granular Media and Simple Lattice Models

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Jamming Transitions Glasses, Granular Media and
Simple Lattice Models

• Giulio Biroli
• (SPhT CEA)

Works in collaboration with Daniel S. Fisher
(Harvard University) Cristina Toninelli (LPT,
Ecole Normale Supérieure, Paris)
Jamming Percolation and Glass Transitions in
Lattice Models, cond-mat/0509661
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The glass transition
Vogel-Fulcher law
Similar phenomenology for jamming transition of
colloidal suspensions and shaken granular
media.
Eg Weeks et al. Science 0. Dauchot, G. Marty,
G. B, condmat 0507152.
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Some Fundamental Questions

• Physical mechanisms behind the Super-Arrhenius
  increase of the relaxation time.
• Does it exist the ideal glass (jamming)
  transition?
• If yes what type of transition is it? A mix of
  first and second order? Static or purely dynamic?

Is it possible to find simple short-range
finite dimensional lattice models without
quenched disorder displaying a glass (jamming)
transition?
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Kinetically Constrained Lattice Models

• KCMs have a phenomenology very similar to glass
  formers (and jamming systems) Super-Arrhenius
  behavior, non-exponential relaxation, dynamic
  heterogeneity, aging,
• Kinetic constraints mimic the cage effect (other
  justification dynamic facilitation).
• Their thermodynamics is trivial.
• On Bethe lattices they display a dynamical glass
  transition as mean field disordered systems
  (1RSB).
• In finite dimensions, in all studied models, this
  transition is wiped out by rare events.

Fredrickson Andersen Kob Andersen
Harrowell Evans Sollich Chandler Garrahan
Berthier Franz, Mulet Parisi Sherrington
Kurchan Sellitto, Biroli, Toninelli
Toninelli, Biroli Fisher.
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Knights Model and its dynamical glass transition
Kinetic Constraint the spin X can flip only if
one of the 2 couples of sites (NW, SE) is down
(empty) AND one of the 2 couples of sites (NE,SW)
is down (empty).
The rates of flip corresponds to independent
spins in a positive magnetic field.
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Dictionary spin up ? occupied site spin down
?empty site
rate 0
rate
Low T favor up spins ? high density The
equilibrium measure is the one of independent
spins (or hard core particles, ie the probability
that a site is occupied is )
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Results

• A dynamical glass transition takes place at the
  same density than site directed percolation on a
  square lattice (0.705).
• The transition is first order the
  Edwards-Anderson parameter is discontinuous
• The relaxation time scale and the dynamical
  correlation length diverge faster than any power
  law at the transition.

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Directed (or Oriented) Percolation
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Anisotropy of DP
DP is a standard continuous phase transition with
lenghts that diverge as power laws
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Relationship with directed percolation
Look at the blocked structure of the Knights
model! RK the existence of blocked structure (ie
degenerate rates) makes these KCMs very different
from the interacting particle or spin systems
studied in the past
Blocked structures Ergodicity
C.Toninelli, GB, DS Fisher J Stat Phys 2005
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Relationship with directed percolation
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Relationship with directed percolation
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Relationship with directed percolation
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Relationship with directed percolation

• If with finite probability there is an infinite
  DP cluster starting from one site then there are
  infinite blocked clusters for the Knights model ?

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• The ? at the corners might be blocked if they
  belong to blocked structures supported from the
  outside.
• This is very unlikely if the size of the square
  is much larger than
• Starting from an empty square of linear size
  one can typically empty all the lattice.
• For there are no blocked
  structure ? no ergodicity breaking?

• This suggests a correlation length of the
  order of but actually a
  more clever way of emptying give
  upper bound

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Systems with linear size much larger than
are unblocked with high probability. Is this just
un upper bound?No.
Example of infinite blocked structure constructed
by putting together elementary L by cL bricks
having DP paths connecting the smallest edges
Key point the probability of not having a DP
cluster inside a rectangular region with edges
L,cL vanishes exponentially fast as
for L not larger than the parallel DP
correlation length
Using bricks with edges of the order of the
parallel DP correlation length one can construct
a blocked structure of linear size until
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First order
The probability that the origin belongs to the
infinite blocked cluster at the transition is
strictly positive.

• The infinite blocked cluster is compact and not a
  fractal at the transition.
• The Edwards-Anderson parameter is discontinuous
  at the transition.

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Numerical results on percolation of blocked
structures
From numerics the transition is clearly first
order and with huge finite size effects in
agreement with analytical predictions
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Numerical results on the dynamics
Remarks a plateau is developing discontinously
and the relaxation timescale is increasing very
fast as expected from previous analytical results.
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Conclusion

• The Knights model has a dynamical glass
  transition at which

• The system gets jammed with a discontinuous jump
  of the order parameter.
• Time and dynamic length scales diverge faster
  than a power law similar to VFT.
• The transition is purely dynamical and the static
  correlation length is always one lattice spacing.

New type of percolation transition ? jamming
percolation
Rigorous Existence of the transition and value
of the critical density/temperature Almost
rigorous Scaling of dynamical length scales and
first order character.
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What Next?

• Study models with different dynamical rules and
  spatial dimensions
• Find a short range finite dimensional system
  without disorder with an amorphous ground state
  stable at low temperature (?Thermodynamic glass
  transition).

Glasses Turbulent Crystals
Crystal ? Quasi Crystal ??
  Do turbulent crystals exist? D. Ruelle
Physica V113A (1982) 619 Some ill-formulated
problems on regular and messy behavior in
statistical mechanics and smooth dynamics for
which I would like the advice of Yasha Sinai by
D. Ruelle