1. 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/

2. 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

3. 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?

4. 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.

5. 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.

6. 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 )

7. 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.

8. Directed (or Oriented) Percolation

9. Anisotropy of DP
DP is a standard continuous phase transition with lenghts that diverge as power laws

10. 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

11. Relationship with directed percolation

12. Relationship with directed percolation

13. Relationship with directed percolation

14. 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 

15. 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]

16. 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 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 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

17. 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.

18. 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

19. 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.

20. 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.

21. 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