[PPT] - Commonalities between Edwards ensemble and glasses Adrian Baule PowerPoint Presentation

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Commonalities between Edwards ensemble and glasses

Adrian Baule Lin Bo Max Danisch Yuliang Jin Romain Mari Louis Portal Chaoming Song

jamlab.org

Hernan Makse City College of New York

Workshop: Physics of glassy and granular materials July 16-19, Kyoto

Wednesday, July 17, 13

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Mathematics Physics

Information theory Granular matter Glasses

3. Non-spherical shapes

Shannon (1948) Signals → High dimensional spheres Random close packing (RCP) Bernal packings (1960)

2. High-dimensional packings
1. Edwards ensemble for grains

and glass theory

Applications Random packings of hard spheres

Volume fraction Force distribution

Wednesday, July 17, 13

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3

Z(X, T) = Z exp[−W(~ x)/X] exp[−H(~ x, ~ f)/T]D~ xD ~ f

Edwards and Oakeshott, Physica A (1989)

Theoretical approach I: Statistical mechanics (Edwards’ ensemble)

Minimize volume (X=0) with constraint of force balance (T=0) and non-overlaping.

OPTIMIZATION STATISTICAL PHYS EDWARDS

instance sample packing cost function energy volume

ptimal configuration

ground state RCP at X=0 minimal cost ground state energy minimal volume

Constraint optimization problem

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Parisi and Zamponi (2010)

Theoretical approach II: Mean field theory of jammed hard-sphere (remnant of RSB solution from replica theory)

Jammed states: J-line

Approach jamming from the liquid phase.
Predict a range of RCP densities
Mean field theory based on RSB solution in the glass

phase.

Replica theory: jammed states are the infinite pressure limit of metastable hard sphere glasses

[φth, φGCP] ≈ [0.64, 0.68]

liquid state splits

max{Σ + s} s → 0

Wednesday, July 17, 13

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φGCP = 0.68 φth = 0.64 φonset φfcc = 0.74

φMRJ φc

0.64 ± 0.04 6% max{S} max{Σ + s} s → 0

φRCP = φJ

(un)Commonalities between Edwards ensemble and RT: 3d

φK = 0.62 φd = 0.58 φedw = 6/(6 + 2 √ 3) φRLP = 4/(4 + 2 √ 3)

8 densities in

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Very difficult in practice: very small range for 3d equal-size spheres

φGCP = 0.68 φth = 0.64 φonset φfcc = 0.74 φedw = 6/(6 + 2 √ 3) φRLP = 4/(4 + 2 √ 3)

φRCP φMRJ φc

0.5 0.55 0.6 0.65 0.7 0.75

φ

0.64 0.66 0.68 0.7 0.72 0.74

~

φj

Equilibrium 10

6

4*10

6

16*10

6

32*10

6

64*10

6

128*10

6

φ = φj PY EOS

crystal fluid coexistence

0.34 0.36 0.38 0.4 0.42 0.44 0.46

φ

0.46 0.47 0.48

∼

φj

10

3

10

4

10

5

10

6

10

7

φ = φj Pade EOS Glass fit

FIG. 1 (From (Skoge et al., 2006)) Evolution of the pressure during compression at rate γ in d = 3 (left) and d = 4 (right).

The density ϕ is increased at rate γ and the reduced pressure p(ϕ) = βP/ρ is measured during the process. See (Skoge et al., 2006) for details. The quantity ϕj(ϕ) =

p(ϕ)ϕ p(ϕ)−d is plotted as a function of ϕ. If the system jams at density ϕj, p → ∞ and

ϕj → ϕj. Thus the final jamming density is the point where

ϕj(ϕ) intersects the dot-dashed line ϕj = ϕ. (Left) The dotted line is the liquid (Percus-Yevick) equation of state. The curves at high γ follow the liquid branch at low density; when they leave it, the pressure increases faster and diverges at ϕj. The curves for lower γ show first a drop in the pressure, which signals

crystallization. (Right) All the curves follow the liquid equation of state (obtained from Eq.(9) of (Bishop and Whitlock, 2005))

and leave it at a density that depends on γ. In this case no crystallization is observed. For γ = 10−5 the dot-dashed line is a fit to the high-density part of the pressure (glass branch). The arrow marks the region where the pressure crosses over from the liquid to the glass branch.

3d 4d

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Z(X, T) = Z exp[−W(~ x)/X] exp[−H(~ x, ~ f)/T]D~ xD ~ f

Z(X, T) = Z exp[−W(~ x)/X]D~ x × Z exp[−H(~ f)/T]D ~ f

Song, Wang, Jin, Makse, Physica A (2010)

H =

N

X

a=1

2 6 4 @ X

b,(ab)∈E

fab ˆ nab 1 A

23

7 5

1. Full solution: Constraint optimization problem
2. Approximation: Decouple forces from geometry.
3. Edwards for volume ensemble

+ Isostaticity T=0 and X=0 optimization problem

4. Cavity method for

force ensemble

Bo, Mari, Song, Makse (2013)

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contact network second coordination shell

The Volume function is the Voronoi volume

consist of all points closer to the center of the grain than to any other

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“Easily” generalizable to other systems

equal size spheres polydisperse system ellipsoids, spherocylinders, non-convex particles, rods, sphere/ellipsoids mixtures, etc. any dimension

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“Easily” generalizable to other systems

equal size spheres polydisperse system ellipsoids, spherocylinders, non-convex particles, rods, sphere/ellipsoids mixtures, etc. any dimension

Wednesday, July 17, 13

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Analytical formula for Voronoi boundary

i

j

rij

Wi = 1 3 1 2R min

j

rij cos θij 3 ds

θij rij/ cos θij

Voronoi particle

ˆ s

Important: global minimization. Reduce to one-dimension

Wednesday, July 17, 13

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Analytical formula for Voronoi boundary

i

j

rij

Wi = 1 3 1 2R min

j

rij cos θij 3 ds

θij rij/ cos θij

Voronoi particle

ˆ s

Important: global minimization. Reduce to one-dimension

Wednesday, July 17, 13

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Analytical formula for Voronoi boundary

i

j

rij

Wi = 1 3 1 2R min

j

rij cos θij 3 ds

θij rij/ cos θij

Voronoi particle

ˆ s

Important: global minimization. Reduce to one-dimension

Wednesday, July 17, 13

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Analytical formula for Voronoi boundary

i

j

rij

Wi = 1 3 1 2R min

j

rij cos θij 3 ds

θij rij/ cos θij

Voronoi particle

ˆ s

Important: global minimization. Reduce to one-dimension

Wednesday, July 17, 13

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Average free-volume per particle

V ∗(c)

S∗(c)

Probability to find all particles outside excluded volume and surface:

w = Z ∞

1

(c3 − 1)p(c)dc = Z 1 (c3 − 1)dP>(c)

Geometrical interpretation

f cumulative dist:

c/2

Voronoi boundary

−∂P>(c) ∂c = p(c) P>(c)

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P(V ∗) = (1 − V ∗/V )N → e−ρV ∗ Mean-field approximation analogous to decorrelation principle

particles belong to bulk or in contact: Similar to car parking problem (Renyi, 1960). Probability to find a spot with in a volume V

V ∗(c)

V

Particle gas Particle gas

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Calculation of P>(c)

Particles are in contact and in the bulk:

PC(c) = e−ρsS∗(c) PB(c) = e−ρV ∗(c)

Bulk term:

ρ(w) = 1 w

Contact term:

z = geometrical coordination number

P>(c) = PB(c) × Pc(c)

mean free volume density mean free surface density

R 2R

r θ

c = r/ cos θ

V ∗(c)

S∗(c)

ˆ s

ρs(z) = 1 S∗ = √ 3 4π z

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Average Voronoi volume

Self-consistent equation: equal to zero represent the average free-volume of a single particle

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Prediction: volume fraction vs z

Aste, JSTAT 2006 X-ray tomography 300,000 grains

Equation of state agrees well with simulations and experiments

w = 2 √ 3 z

φ =

z z+2 √ 3

Theory

free volume volume function

RCP

z = 6 w = 1 √ 3

φ = 6 6 + 2 √ 3

φ = .634

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Definition of jammed state: geometric coordination z bounded by mechanical coordination Z

positions geometrical constraints effectively excludes the ordered states

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Decreasing compactivity X

Isostatic plane

Disordered Packings Forbidden zone no disordered jammed packings can exist

Edwards phase diagram for hard spheres

φRLP = 4 4 + 2 √ 3 = 0.536

φRCP =

6 6+2 √ 3

φRLP(Z) =

Z Z+2 √ 3

0.634

Wednesday, July 17, 13

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Jammed packings in high dimensions

Question: what’s the density of RCP in high dimensions? Conjecture: are disordered packings more optimal than ordered

nes?

Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound:

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Conjecture: P>(c) becomes valid in the high-dimensional limit

(I) Theoretical conjecture of g2 in high d (neglect correlations) Torquato and Stillinger, Exp. Math., 2006 (II) Factorization of P>(c) Large d 3d

Wednesday, July 17, 13

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Glass transition RT

Parisi and Zamponi, Rev. Mod. Phys. (2010)

No unified conclusion at the mean-field level (infinite d). Neither dynamics nor jamming.
Does RCP in large d have higher-order correlations missed by theory?: Test of replica th.
Edwards solution seems to corresponds to . Higher entropy state.

Edwards’ theory

Jin, Charbonneau, Meyer, Song, Zamponi, PRE (2010)

Isostatic packings (z = 2d) with unique volume fraction Isostatic packings (z = 2d) ranging volume fraction increases with dimensions

Agree with Minkowski lower bound

φth

Comparison with other theories

Wednesday, July 17, 13

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Generalizing the theory of monodisperse sphere packings

Polydisperse spheres (dimers, triangles, tetrahedra, spherocylinders, ellipsoids … ) Non-spherical objects Extra degree of freedom treated as in Onsager 1949 Distribution of radius P(r) Distribution of angles P( )

Wednesday, July 17, 13

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22

Kepler&conjecture:""
Random&sphere"packings:"

φ ≈ 0.74 φ ≈ 0.55 − 0.64

α ≈ 1.4

Donev"et"al,"Science"2004"

! Ellipsoids"pack"denser"than" spheres" ! Peak"at"aspect"ra:o" ! Spheres"appear"as"a" singular&limit&

"

Simula:on"results"on"

packings"of"ellipsoids:& RCP"

Op$mizing ¡random ¡packings ¡in ¡the ¡ space ¡of ¡object ¡shapes

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d L z < 2df

α = L d

Non-‑spherical ¡objects:

Edwards prediction

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d L z < 2df

α = L d

Non-‑spherical ¡objects:

Sta$s$cal ¡theory ¡of ¡ Voronoi ¡volume

Edwards prediction

Wednesday, July 17, 13

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d L z < 2df

α = L d

Non-‑spherical ¡objects:

Sta$s$cal ¡theory ¡of ¡ Voronoi ¡volume Evalua$ng ¡the ¡probability ¡of ¡ degenerate ¡configura$ons: ellipsoids ¡are ¡hypoconstrained

Edwards prediction

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Voronoi for non-spherical shapes

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General non-spherical shapes

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Object shape Decomposition Effective Voronoi interaction

Sphere Dimer Trimer Spherocylinder One sphere Single points Two spheres Pairs of points Three spheres Triplets Lines N spheres Ellipsoid Tetrahedron Irregular polyhedron: N faces, M vertices Two spheres Four spheres N unequal spheres Quartets of points and anti-points Pairs of points and anti-points M points and N anti-points

26

Wednesday, July 17, 13

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27

Spherocylinders.

Separa&on)lines:)

) ) ) ) )

Four)different)interac&ons:)
Line.–.Line.
Line.–.Point.
Point.–.Line.
Point.–.Point.

.

Exact.equa8on.for.each.case. . .analy8c.expressions. . .for.VB.

Wednesday, July 17, 13

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28

Calcula&on)of)coordina&on)number:) Degenerate)configura&ons)

Mechanical)equilibrium:)
3)force)equa&ons)
2)torque)equa&ons)

(torque)along)symmetry)axis) vanishes))

Linearly)independent?) Effec&ve)number)of)degrees)of)freedom) can)be)reduced!)

Zc = 2df = 10

Wednesday, July 17, 13

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29

Degenerate(configura-ons(

Mechanical(equilibrium:(
3(force(equa-ons(
2(torque(equa-ons(

(torque(along(symmetry(axis( vanishes)(

Linearly)independent?) Effec-ve(number(of(degrees(of(freedom( can(be(reduced!(

Wednesday, July 17, 13

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30

Degenerate(configura-ons(

Mechanical(equilibrium:(
3(force(equa-ons(
2(torque(equa-ons(

(torque(along(symmetry(axis( vanishes)(

Linearly)independent?) Maximal)degenerate)configura6on:(Condi-on(

f(force(balance(automa-cally(implies(torque(

balance!(

Z(α) = 2h ˜ df(α)i

Wednesday, July 17, 13

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31

z()

FCC RCP Ellipsoids theory Dimers theory Spherocylinders theory Lens-shaped particles theory

0.5 1.0 1.5 2.0 2.5 6 7 8 9 10 0.5 1.0 1.5 2.0 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74

)

a b

Dimers theory Spherocylinders theory Lens-shaped particles theory

Theory

Theory

spherocylinder

M&M candy spherocylinder spherocylinder spherocylinder dimer dimer spherocylinder spherocylinder

blate ellipsoid

spherocylinder prolate ellipsoid spherocylinder Abreu et al. 2003 Donev et al. 2004 Lu et al. 2010 Jia et al. 2007 Williams et al. 2003 Schreck et al. 2011 Faure et al. 2009 Kyrylyuk et al. 2011 Bargiel et al. 2008 Donev et al. 2004 Wouterse et al. 2009 Donev et al. 2004 Zhao et al. 2012

Simulations

Theoretical predictions

Wednesday, July 17, 13

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32

α

Dimers theory Schreck & O’Hern, 2011

1.0 1.2 1.4 1.6 1.8 2.0 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74

φ(α) d

Results'for'packing'frac2on:'dimers'

Wednesday, July 17, 13

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33

1.0 1.2 1.4 1.6 1.8 2.0 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74

α

FCC Spherocylinders theory Zhao et al, 2012 Kyrylyuk et al, 2011 Lu et al, 2010 RCP

c φ(α)

Results'for'packing'frac2on:' spherocylinders'

Wednesday, July 17, 13

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φ z(φ)

FCC RCP Dimers theory Spherocylinders theory Analytic continuation, Eq.(3) Spherical random branch Spherical ordered branch

α α

0.55 0.60 0.65 0.70 0.75 0.80 4 6 8 10 12

I. Spherical
II. Rotationally symmetric
III. Aspherical

Tetrahedra (Haji-Akbari et al 2009) Tetrahedra (Jaoshvili et al 2010) Aspherical ellipsoids (Donev et al 2004) Prolate ellipsoids (Donev et al 2004) Oblate ellipsoids (Donev et al 2004) Spherocylinders (Zhao et al 2012) Spherocylinders (Wouterse et al 2009) Spherocylinders (Williams & Philipse 2003) Dimers (Schreck & O’Hern 2011)

c

34

RCP is not singular: analytical continuation of spheres

Edwards phase diagram for many shapes

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Summary

O d e t

E

d w a r d s ! 1 . E d w a r d s e n s e m b l e t

p

r e d i c t R C P f

r

s p h e r e s . 2 . E d w a r d s e n s e m b l e f

r

n

n
s

p h e r i c a l p a r t i c l e s . 3 . E d w a r d s e n s e m b l e f

r

p a c k i n g s i n l a r g e d i m e n s i

n

s t

c
m

p a r e w i t h r e p l i c a t h e

r

y

f

h a r d s p h e r e g l a s s e s . 4 . E d w a r d s r e p l i c a t r i c k

r

c a v i t y m e t h

d

f

r

p r

p

e r a v e r a g e

v

e r q u e n c h e d d i s

r

d e r f

r

f

r

c e d i s t r i b u t i

n

f

r

a n y s y s t e m : s p h e r e s , n

n
s

p h e r e s , f r i c t i

n

a n d f r i c t i

n

l e s s , a n y d i m e n s i

n

. 5 . E x t e n d i n g M a x w e l l a r g u m e n t : C a v i t y m e t h

d

a t R S l e v e l f

r

s

l

u t i

n
n
s
l

u t i

n

t r a n s i t i

n

t

c

a l c u l a t e Z

c

f r

m

f r i c t i

n

l e s s i s

s

t a t i c g r a i n s t

f

r i c t i

n

a l g r a i n s . 6 . E d w a r d s C A V E A T : 1

5

d

n

e a t e x p e n s e

f

d r a s t i c ( y e t c

n

t r

l

l e d ) a p p r

x

i m a t i

n

s .

Wednesday, July 17, 13

SLIDE 42

φ(Z)

Z(α)

Cavity Method for Force Transmission

Z(X, T) = Z exp[−W(~ x)/X]D~ x × Z exp[−H(~ f)/T]D ~ f

Edwards volume ensemble predicts: Cavity method predicts Z: and Force Distribution:

P(f)

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~ f = −(~ f1 + ~ f2 + ~ f3) Z = 4

P(f)

Force transmission problem: back to Edwards (simplest model)

Edwards model = q-model = annealed disorder average Fix

f1 f2 f3 f

Find with constraint

Wednesday, July 17, 13

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λ2f2

P(f) = Z P(f1, λ1)P(f2, λ2)τ(λ1, λ2)δ(f − λ1f1 − λ2f2)dλ1dλ2d f1d f2

P(f) = f p e− f

p

Boltzmann equation for P(f)

f1 f2 f3 f

Boltzmann equation: assuming uncorrelated forces (MF) quenched disorder

Edwards: “Tiresomely complicated function well modelled by integrating between 0 and 1”

Fourier transform: annealed disorder : component

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F = −kT ln Z F = −kT ln Z

ln Z = lim

n→0(Zn − 1)/n

Experimentally: first find the distribution for a fixed (quenched) packing, then average over the ensemble of packings Average must be carried over a physical observable: free energy, not the partition function. quenched disorder annealed disorder Replica trick(Edwards-Anderson) Granular matter: Performed average over forces then over contact network

Annealed versus quenched disorder

Cavity Method

Wednesday, July 17, 13

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site(contact*force)* Interac/on(par/cle)*

χa({f n, f t, ˆ n a, ˆ t a}∂a)

Building the factor graph of contacts from a packing

40

a({f n, f t, ˆ n a, ˆ t a}∂a) = X

i∈∂a

~ f a

i

!

X

i∈∂a

~ r a

i × ~

f a

i

! × Y

i∈∂a

Θ(f n

i )Θ(µf n i − f t i )

Qi(f n

i , f t i )

Constraint: force balance + torque balance + repulsive + Coulomb

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Qi(f n

i , f t i )

Qa→i(f n

i , f t i )

Qb→i(f n

i , f t i )

Qc→j(f n

j , f t j)

a b c i j χa

Qa→i(f n

i , f t i ) =

1 Za→i Z dˆ ti Y

j∈∂a−i c=∂j−a

df n

j df t jdˆ

tj Qc→j(f n

j , f t j)χa({f n, f t, ˆ

n, ˆ t}∂a)

Compute marginal belief for a fix contact network

41

Qi(f n

i , f t i )

Qi(f n

i , f t i ) = 1

Zi Qa→i(f n

i , f t i ) Qb→i(f n i , f t i ),

{a, b} = ∂i

Belief propagation particle contact force

Cavity field: no average over nj

Wednesday, July 17, 13

SLIDE 48

Q(Q→) = 1 Z X

z

z R(z) Z Ω({ˆ nj})

z−1

Y

j=1

dˆ njDQ→jQ(Q→j)δ h Q→−F→

Q→j i

P(f) ∼ f θ θ = 0 Force probability over an ensemble of random graphs

42

P(f n, f t) = hQi(f n, f t)i = 1 Z Z DQa→iQ(Qa→i)Qa→i(f n, f t) 2

Degree distribution Joint distribution of contacts positions on one particle Cavity equation

10-2 10-1 100 1 2 3 4 P(f/<f>) f/<f> (A) 2D frictionless ( zc=4) exp(-2f/<f>)

10-2 10-1 100 10-6 10-4 10-2 100

=0 10-2 10-1 100 1 2 3 4 P(f/<f>) f/<f> (B) 3D frictionless ( zc=6) exp(-2f/<f>)

10-2 10-1 100 10-6 10-4 10-2 100

=0

Solved with Population Dynamics

P(f n)

Prediction:

signature of jamming

Wednesday, July 17, 13

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43

1

G(V, E), Initialize all cavity fields { }.

2

Draw an integer z with (edge-perspective) degree distribution P(z).
Then pick at random z-1fields from the population of N fields.

3

Generate a set of relative contact directions

with uniform distribution ; Particles do not overlap.

4

update the new cavity field by using the incoming fields

according to cavity equation.

5

Update all cavity fields to generate a new population. Rescale <f>=const. Run

until convergence, or until the number of iterations exceeds Tmax.

The Population Dynamics Algorithm"

Cavity Method: General Formalism Algorithm Lin Bo (CCNY) Cavity Method for Jammed Disordered Packings of Hard Particles

ψj→b(fj)

ni(1), ..., ni(z−1)

ψi→a(fi)

Wednesday, July 17, 13

SLIDE 50

 

Force probability over an ensemble of random graphs

44

10-2 10-1 100 101 1 2 3 4 P(f) f (C) exp(-1.4fn/<fn>) exp(-3.5ft/<ft>) 2D (µ=)

10-1 100 10-2 100 P(fn) fn

10-2 10-1 100 101 1 2 3 4 P(f) f (D) exp(-1.6fn/<fn>) exp(-3.8ft/<ft>) 3D (µ=)

10-1 100 10-2 100 P(fn) fn

Pµ(f n, f t)

Wednesday, July 17, 13

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45



P. Wang et al. Physica A (2010)

Cavity method

Comparison with simulations

Wednesday, July 17, 13

SLIDE 52

z > zc(µ) z < zc(µ)

46

Solution-no solution transition at Zc

3 0.5 1 1.5 2 2.5 3 3.5 4 ! (f) f

Initial fields

Q(Q→) = 1 Z X

z

z R(z) Z Ω({ˆ nj})

z−1

Y

j=1

dˆ njDQ→jQ(Q→j)δ h Q→−F→

Q→j i

50 100 150 200 1 2 3 4 Q(fn) fn 2D frictionless ( z=3.9 < zc

min)

3 6 9 1 2 3 4 Q(fn) fn Wn 2D frictionless ( z=4.5 > zc

min)

10-4 10-3 10-2 10-1 100 100 101 102 < Wn > Time Steps (A) 2D frictionless

z = 3.9
z = 4.0
z = 4.1

data discretization

Peaks= no solution Broad= solution

rder parameter:

WIDTH

Wednesday, July 17, 13

SLIDE 53

µ

47

Comparison with simulations

3 3.2 3.4 3.6 3.8 4 10-4 10-3 10-2 10-1 100 101 µ

Silbert et al. PRE (2002) Cavity method

zc(µ) zc(µ)

2D

Consistent with interpretation of as a lower bound

zc(µ)

Wednesday, July 17, 13

SLIDE 54

Definition of jammed state: isostatic condition on Z z = geometrical coordination number.

Determined by the geometry of the packing.

Z =mechanical coordination number.

Determined by force/torque balance.

Z ≤ z ≤ 2d = 6 4 = d + 1 ≤ Z ≤ 2d = 6

µ = 0 µ = ∞

z = 4 Z = 3

Wednesday, July 17, 13

SLIDE 55

Generalizing the theory of monodisperse sphere packings

Theory of monodisperse spheres

Polydisperse (binary) spheres (dimers, triangles, tetrahedrons, spherocylinders, ellipses, ellipsoids … ) Non-spherical objects Extra degree of freedom Onsager 1949 Distribution of radius P(r) Distribution of angles P( )

Wednesday, July 17, 13

SLIDE 56

Result of binary packings

Binary packings

Danisch, Jin, Makse, PRE (2010)

RCP (Z = 6)

Wednesday, July 17, 13

SLIDE 57

The partition function for hard spheres

1. The Volume Function: W (geometry)
2. Definition of jammed state:

force and torque balance Volume Ensemble + Force Ensemble Solution under different degrees of approximations

Wednesday, July 17, 13

SLIDE 58

Jammed packings in infinite dimensions

Most efficient design of signals (Information theory) Optimal packing (Sphere packing problem) Sampling theorem

Question: what’s the density of RCP in high dimensions? Conjecture: are disordered packings more optimal than ordered ones?

Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound:

Signal High-dimensional point Sloane

Wednesday, July 17, 13

SLIDE 59

Sphere packings in high dimensions

Most efficient design of signals (Information theory) Optimal packing (Sphere packing problem) Sampling theorem

Question: what’s the density of RCP in high dimensions?

Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound:

Signal High-dimensional point Sloane

Wednesday, July 17, 13

SLIDE 60

Q(Q→) = 1 Z X

z

z R(z) Z Ω({ˆ nj})

z−1

Y

j=1

dˆ njDQ→jQ(Q→j)δ h Q→−F→

Q→j i

Determination of a lower bound on average coordination number

54

3 6 9 1 2 3 4 Q(fn) fn Wn 2D frictionless ( z=4.5 > zc

min)

50 100 150 200 1 2 3 4 Q(fn) fn 2D frictionless ( z=3.9 < zc

min)

¯ zmin

c

(µ)

3 3.2 3.4 3.6 3.8 4 10-2 10-1 100 101

zc

min(µ)

µ (B)

10-1 100 10-3 10-1 101

zc

min(0)-

zc

min(µ)

zc

min(µ)-

zc

min()

µ =2/3 =0.35

10-4 10-3 10-2 10-1 100 100 101 102 < Wn > Time Steps (A) 2D frictionless

z = 3.9
z = 4.0
z = 4.1

data discretization

Wednesday, July 17, 13