Jamming

1
Jamming

• Andrea J. Liu
• Department of Physics Astronomy
• University of Pennsylvania
• Corey S. OHern Mechanical Engineering,
  Yale Univ.
• Leo E. Silbert Physics, Southern Ill. Univ.
• Jen M. Schwarz Physics, Syracuse Univ.
• Lincoln Chayes Mathematics, UCLA
• Sidney R. Nagel James Franck Inst., U
  Chicago
• Brought to you by NSF-DMR-0087349, DOE
  DE-FG02-03ER46087

2
Mixed Phase Transitions

• Recall random k-SAT
• Fraction of variables that are constrained obeys
• Finite-size scaling shows diverging length scale
  at rk
• Monasson, Zecchina, Kirkpatrick, Selman,
  Troyansky, Nature 400, 133 (1999).

E0, no violated clauses
Egt0, violated clauses
3
Mixed Phase Transitions

• infinite-dimensional models
• p-spin interaction spin glass Kirkpatrick,
  Thirumalai, PRL 58, 2091 (1987).
• k-core (bootstrap) Chalupa, Leath, Reich, J.
  Phys. C (1979) Pittel, Spencer, Wormald, J.Comb.
  Th. Ser. B 67, 111 (1996).
• Random k-SAT Monasson, Zecchina, Kirkpatrick,
  Selman, Troyansky, Nature 400, 133 (1999).
• - etc.
• But physicists really only care about finite
  dimensions
• Jamming transition of spheres OHern, Langer,
  Liu, Nagel, PRL 88, 075507 (2002).
• Knights models Toninelli, Biroli, Fisher, PRL 96,
  035702 (2006).
• k-core force-balance models Schwarz, Liu,
  Chayes, Europhys. Lett. 73, 560 (2006).

4
Stress Relaxation Time

• Behavior of glassforming liquids depends on how
  long you wait
• At short time scales, silly putty behaves like a
  solid
• At long time scales, silly putty behaves like a
  liquid
• Stress relaxation time t how long you need to
  wait for system to behave like liquid

5
Glass Transition

• When liquid cools, stress relaxation time
  increases
• When liquid crystallizes
• Particles order
• Stress relaxation time suddenly jumps
• When liquid is cooled through glass transition
• Particles remain disordered
• Stress relaxation time increases continuously

Picture Book of Sir John Mandevilles Travels,
ca. 1410.
6
Jamming Phase Diagram

• A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21
  (1998).

unjammed state is in equilibrium jammed state is
out of equilibrium
Problem Jamming surface is fuzzy
7
Point J

• C. S. OHern, S. A. Langer, A. J. Liu and S. R.
  Nagel, Phys. Rev. Lett. 88, 075507 (2002).
• C. S. OHern, L. E. Silbert, A. J. Liu, S. R.
  Nagel, Phys. Rev. E 68, 011306 (2003).
• Point J is special
• It is a point
• Isostatic
• Mixed first/second order zero T phase transition

soft, repulsive, finite-range spherically-symmetri
c potentials Model granular materials
8
How we study Point J

• Generate configurations near J
• e.g. Start w/ random initial positions
• Conjugate gradient energy minimization (Inherent
  structures, Stillinger Weber)
• Classify resulting configurations

Ti8
non-overlapped V0 p0
overlapped Vgt0 pgt0
or
Tf0
Tf0
9
Onset of Jamming is Onset of Overlap

• We focus on ensemble rather than individual
  configs (c.f. Torquato)
• Good ensemble is fixed f-fc, or fixed pressure

D2 D3

• Pressures for different states collapse on a
  single curve
• Shear modulus and pressure vanish at the same fc

10
How Much Does fc Vary Among States?

• Distribution of fc values narrows as system size
  grows
• Distribution approaches delta-function as N
• Essentially all configurations jam at one packing
  density
• Of course, there is a tail up to close-packed
  crystal
• J is a POINT

11
Point J is at Random Close-Packing

• Where do virtually all states jam in infinite
  system limit?
• 2d (bidisperse)
• 3d (monodisperse)
• These are values associated with random
  close-packing!

12
Point J

• Point J is special
• It is a point
• Isostatic
• Mixed first/second order zero T transition

soft, repulsive, finite-range spherically-symmetr
ic potentials
13
Number of Overlaps/Particle Z

• (2D)
• 
  (3D)

Just above fc there are Zc overlapping neighbors
per particle
Just below fc, no particles overlap
14
Isostaticity

• What is the minimum number of interparticle
  contacts needed for mechanical equilibrium?
• No friction, spherical particles, D dimensions
• Match unknowns (number of interparticle normal
  forces) to equations (force balance for
  mechanical stability)
• Number of unknowns per particleZ/2
• Number of equations per particle D
• Point J is purely geometrical!

15
Unusual Solid Properties Near Isostaticity

• L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95,
  098301 (05)
• Excess low-w modes swamp w2 Debye behavior boson
  peak
• D(w) approaches constant as f fc (M. Wyart,
  S.R. Nagel, T.A. Witten, EPL (05) )

Lowest freq mode at f-fc10-8
Density of Vibrational Modes
f- fc
16
Point J

• Point J is special
• It is a point
• Isostatic
• Mixed first/second order zero T transition

soft, repulsive, finite-range spherically-symmetr
ic potentials
17
Is there a Diverging Length Scale at J?

• For each f-fc, extract w where D(w) begins to
  drop off
• Below w , modes approach those of ordinary
  elastic solid
• We find power-law scaling
• L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95,
  098301 (2005)

w
18
Frequency Scale implies Length Scale

• The frequency w has a corresponding eigenmode
• Decompose eigenmode in plane waves
• Dominant wavevector contribution is at peak of
  fT(k,w)
• extract k
• We also expect with

19
Summary of Jamming Transition

• Mixed first-order/second-order transition
• Number of overlapping neighbors per particle
• Static shear modulus
• Diverging length scale
• And perhaps also

20
Jamming vs K-Core (Bootstrap) Percolation

• Consider lattice with coord. Zmax with sites
  indpendently occupied with probability p
• For site to be part of k-core, it must be
  occupied and have at least kd1 occupied
  neighbors
• Each of its occ. nbrs must have at least k occ.
  nbrs, etc.
• Look for percolation of the k-core

• Jammed configs at T0 are mechanically stable
• For particle to be locally stable, it must have
  at least d1 overlapping neighbors in d
  dimensions
• Each of its overlapping nbrs must have at least
  d1 overlapping nbrs, etc.
• At fgt fc all particles in load-bearing network
  have at least d1 neighbors

21
K-core Percolation on the Bethe Lattice

• K-core percolation is exactly solvable on Bethe
  lattice
• This is mean-field solution
• Let Kprobability of infinite k-connected cluster
• For kgt2 we find
• Chalupa, Leath, Reich, J. Phys. C (1979)
• Pittel, et al., J.Comb. Th. Ser. B 67, 111 (1996)

• Recall simulation results

J. M. Schwarz, A. J. Liu, L. Chayes, EPL (06)
22
K-Core Percolation in Finite Dimensions

• There appear to be at least 3 different types of
  k-core percolation transitions in finite
  dimensions
• Continuous percolation (Charybdis)
• No percolation until p1 (Scylla)
• Discontinuous percolation?
• Yes, for k-core variants
• Knights models (Toninelli, Biroli, Fisher)
• k-core with pseudo force-balance (Schwarz, Liu,
  Chayes)

23
Knights Model
Toninelli, Biroli, Fisher, PRL 96, 035702 (2006).

• Rigorous proofs that
• pclt1
• Transition is discontinuous
• Transition has diverging correlation length
• based on conjecture of anisotropic critical
  behavior in directed percolation

24
A k-Core Variant

• We introduce force-balance constraint to
  eliminate self-sustaining clusters
• Cull if klt3 or if all neighbors are on the same
  side

k3 24 possible neighbors per site Cannot have
all neighbors in upper/lower/right/left half
25
Discontinuous Transition? Yes

• The discontinuity ?c increases with system size L
• If transition were continuous, ?c would decrease
  with L

Fraction of sites in spanning cluster
26
Pclt1? Yes

• Finite-size scaling
• If pc 1, expect pc(L) 1-Ae-BL

Aizenman, Lebowitz, J. Phys. A 21, 3801 (1988)
We find
We actually have a proof now that pclt1 (Jeng,
Schwarz)
27
Diverging Correlation Length? Yes

• This value of collapses the order
  parameter data with
• For ordinary 1st-order transition,

28
Diverging Susceptibility? Yes

• How much is removed by the culling process?

29
BUT

• Exponents for k-core variants in d2 are
  different from those in mean-field!
• Mean field d2
• Why does Point J show mean-field behavior?
• Point J may have critical dimension of dc2 due
  to isostaticity (Wyart, Nagel, Witten)
• Isostaticity is a global condition not captured
  by local k-core requirement of k neighbors
• Henkes, Chakraborty, PRL 95, 198002 (2005).

30
Similarity to Other Models

• The discontinuity exponents we observe are rare
  but have been found in a few models
• Mean-field p-spin interaction spin glass
  (Kirkpatrick, Thirumalai, Wolynes)
• Mean-field dimer model (Chakraborty, et al.)
• Mean-field kinetically-constrained models
  (Fredrickson, Andersen)
• Mode-coupling approximation of glasses
  (Biroli,Bouchaud)
• These models all exhibit glassy dynamics!!
• First hint of UNIVERSALITY in jamming

31
To return to beginning.

• Recall random k-SAT
• Point J
• Hope you like jammin, too!

?-?c
0
32
Conclusions

• Point J is a special point
• Common exponents in
• different jamming models
• in mean field!
• But different in finite dimensions
• Hope you like jammin, too!
• Thanks to NSF-DMR-0087349
• DOE DE-FG02-03ER46087

33
Continuous K-Core Percolation

• Appears to be associated with self-sustaining
  clusters
• For example, k3 on triangular lattice
• pc0.69210.0005, M. C. Madeiros, C. M. Chaves,
  Physica A (1997).

Self-sustaining clusters dont exist in sphere
packings
p0.4, before culling
p0.4, after culling
p0.6, after culling
p0.65, after culling
34
No Transition Until p1

• E.g. k3 on square lattice
• There is a positive probability that there is a
  large empty square whose boundary is not
  completely occupied
• After culling process, the whole lattice will be
  empty
• Straley, van Enter J. Stat. Phys. 48, 943 (1987).
• M. Aizenmann, J. L. Lebowitz, J. Phys. A 21, 3801
  (1988).
• R. H. Schonmann, Ann. Prob. 20, 174 (1992).
• C. Toninelli, G. Biroli, D. S. Fisher, Phys. Rev.
  Lett. 92, 185504 (2004).

Voids unstable to shrinkage, not growth in sphere
packings
35
Point J and the Glass Transition

• Point J only exists for repulsive, finite-range
  potentials
• Real liquids have attractions
• Attractions serve to hold system at high enough
  density that repulsions come into play (WCA)

U
Repulsion vanishes at finite distance
r