Transcript Spatial structures of Kinetically Constrained Models of

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
The glass transition
Vogel-Fulcher law
 K 

exp 
 T  T0 
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.
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?
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.
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.
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  )
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.


C

exp 
 
 (T  Tc ) 

C

exp

(



)
 c



  0.64
Directed (or Oriented) Percolation
Anisotropy of DP
DP is a standard continuous phase transition with
lenghts that diverge as power laws
Relationship with directed
percolation
Look at the blocked structure of the Knights model!
Blocked structures
Ergodicity*
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
*C.Toninelli, GB, DS Fisher J Stat Phys 2005
Relationship with directed
percolation
Relationship with directed
percolation
Relationship with directed
percolation
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  c  cDP
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
all the lattice.
For
 //
 //
one can typically empty
  cDP there are no blocked structure  no ergodicity breaking
c  cDP
 
This suggests a correlation length  of the order of
exp c //2 but
actually a more clever way of emptying give expc //1 [upper bound]


Systems with linear size much larger than  are unblocked
with high probability. Is this just un upper bound?No.
  exp(c
1
//


c

)  exp
 // (1 ) 
 ( c   )

Example of infinite blocked structure
constructed by putting together
elementary L by c’L 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,c’L vanishes exponentially fast as
exp(
K ' L1 than
)
for L
notlarger
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
2 exp(K '//1 )  O(1)
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.
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
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.
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.
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).
Crystal  Quasi Crystal ?
Glasses
Turbulent Crystals*
* ’Do turbulent crystals exist?’ D. Ruelle Physica V113A (1982) 619; ‘Some illformulated 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