Marginal stability in infinite dimensional Hard Spheres: the Gardner - - PowerPoint PPT Presentation

Marginal stability in infinite dimensional Hard Spheres: the Gardner transition and the fullRSB solution

Pierfrancesco Urbani

Together with: ! Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi and Francesco Zamponi

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Cargese, 28/08/2014

Nature Communications, 5, 3725 (2014)

• The simplest RFOT scheme!
• Microscopic description of marginal stability: statistics of

amorphous sphere packings.!

• Infinite dimensional Hard Spheres. First steps.!
• The 1RSB solution, phase diagram and inconsistencies.!
• Stability of the 1RSB solution. The Gardner transition in the

replicated theory.!

• The 2RSB solution: a new approximate landscape.!
• The fullRSB solution: analytical prediction for the critical

exponents at jamming.!

• The fullRSB marginal stability and the microscopic marginal

stability.

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Idea

Think about jammed packings of hard spheres as the infinite pressure limit of metastable glassy states. Apply equilibrium statistical mechanics tools developed to study glassy systems Mézard-Parisi, Parisi-Zamponi

Mean field phase diagram. Up to now the theory ha developed on the basis o hypothesis for the structur metastables states

1RSB Phase diagram

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The distribution of forces between spheres in contact displays a power law behavior at small forces The shows a power law divergence at the contact value of

Analogous to the pseudo-gap in the distribution

• f frozen fields in the Sherrington-Kirkpatrick

model.

P(f)

Wyart, Nagel, Muller

Given this and the fact that the packin marginally stable, a wonderful scaling theo

• f mean field critical exponents at jammi

can be derived (Wyart) P(f) ∼ f θ

g(r)

g(r) ∼ (r − 1)−γ

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We can hope to do practical calculations in order to give first theoretical estimates for measured physical quantities (compute the exponents!) Starting point of a systematic 1/d expansion

The properties of hard spheres at jamming seem to depend slightly on the d

(It is not the same for the glass transition where the debate has not been solved yet)

Charbonneau, Corwin, Parisi,

In the infinite dimensional limit the virial expansion takes a simple f

Mayer functi

S[ρ(x)] = Z dxρ(x) (1 − log ρ(x)) + 1 2 Z dxdyρ(x)ρ(y)f(x, y) f(x) = −

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The entropy is a function of the molecule density. Translational and rotational invariance imply This has been done by Kurchan, Parisi and Zamponi (2012) Write the saddle point equations to obtain the molecule density and them to obtain the entropy. A wonderful result is that if we take a Gaussian ansatz for the molecule density we can obtain the correct entropy. Gaussian ansatz Big simplification Monasson, Fra

Z − 2 Z x = {x1, . . . , xm} xa = X + ua ρ(x) = ρ(qab) q

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Parisi Zamponi (05-10)

packing fraction

J-line: jammed states

1RSB solution

b ϕ

p−

E

E1RS

th

b ϕ1RSB

th

b ϕd

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• 1. The jammed packings are expected to be isostatic.
• 2. Scaling of the Debye-Waller factor with the pressure

From 1RSB: Instead it was observed that The 1RSB ansatz gives hyperstatic packings

∆EA ∼ p−κ κ ∼

• 3. The distribution of forces do

show a power law behavior

∆EA ∼

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Entropic term Interaction term

where Take the 1RSB ansatz Plug it into the expression for the entropy. Find 1RSB saddle point equatio Obtain the solution. Study its stability. Expand at the third order to obtain the cubic terms and the value of the exponent parameter (MCT dynamical exponents)

(Original motivation for the work)

Obtain a stability matrix Study its eigenvalues. In particular: replicon eigenvalue

(Simple replica structure)

F[ˆ v] = lim

n→0

X

n1,...,nm:P

a na=n

n! n1! . . . nm! exp 2 4−1 2

m

X

a=1

vaa na n + 1 2

m

X

a,b=1

vab α1RSB

ab

= b α1(mδab − 1)

Mab;cd

λ = w2 w1

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The instability affects the Jamming part of the phase di

We hope to cure 1RSB inconsistencies going beyond the 1RSB s

Kurchan Parisi Urba

p−1 b ϕ

1RSB stable 1RSB unstable

b ϕd

Gardner line E

Gardner transi

1RSB uns 1RSB stable

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• Gardner (85) showed that the

equilibrium states undergo a fullRSB transition at very low temperature (below the Kauzmann point)!

• Montanari and Ricci-Tersenghi (03)

however showed that the Gardner phase affects strongly off equilibrium

• states. It may be relevant for off-

equilibrium dynamics!!

• Renewal interest from Krzakala and

Zdeborova (13) who were able to follow adiabatically glassy states showing their instability.!

• Rizzo(13): fullRSB solution.

Implications

• If in the Gardner phase

fullRSB solution we ha modes!

• The fullRSB solution is
• stable. Could the fullRSB

stability reproduce the microscopic marginal s jammed packings?

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Rizzo (13) in the ca Ising p-spin mo

Doing a perturbative 2RSB calculation just below the instability line shows that a 2RSB solution (and a fullRSB one) can be found only above a certain packing fraction

b ϕ b ϕd b ϕ1RSB

th

b ϕ2RSB

th

b ϕ∗ E E1RS

th

λ = w2 w1 > m

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Behavior at the 2RSB threshold Behavior in the high density limit

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The mean square displacement profile depends on m in a complicated way! We are interested in the behavior of the Debye-Waller factor

Reminder:

b ∆ = B B B B B B B B B B B @ b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 1 C C C C C C C C C C C A b ∆ = B B B B B B B B B B B @ b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆1 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2 b ∆2

kRSB fullRSB

∆ab → ∆(x)

∆EA = ∆(1) ∼ m?

m

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Once the expression of the fullRSB entropy is produced, one has to optimize it over the mean square displacement profile. How to solve the equations Find the variational equations. We must enforce the Parisi equations. Find the variational kRSB equations Find their continuum limit Add Lagrange multipliers to the action and then take the variations with respect to everything. At the end check that the two methods give the same eqns. Put them on a grid and discretize everything. Solve numerically the kRSB equations for successive values of k. f(1, h) = log Θ " h p 2∆(1) #

∂f(x, h) ∂x = −1 2 ˙ q(x) " ∂2f(x, h) ∂h2 f(1, h) = log(2 cosh(βh))

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y = x/m [m, 1] → [1, 1/m]

∆(y) ⇠ y−κ κ ' 1.416

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functions that appear in them.

Analogous to the distribution of fields in the SK model Roughly, the solution of the Parisi equation Asymptotic power law behavior of the mean square displacement profile

Connected with the g(r) in the gla

Solve for the exponents.

W

∆(y) ∼ y−κ b j(y, h) = − c 2y J(hyb/√γ∞)

b = 1 + c 2 b P(y, h) ⇠ 8 > < > : ycp0(hyc) for h ⇠ y−c yap1(hyb) for |h| ⇠ y−b p2(h) for h y−b

γ = a

κ = 1 + c

θ = c − a b − c γ = a b

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Whithin the fullRSB ansatz

In the Hard Spheres case

Crucial to obtain the values of the exponents (thus, also the microscopic marginal stability condition) Crucial to obtain that the fullRSB jammed packings are isostatic

1 − 2q(1) + Z 1

1

dyP(1, h)m4(1.h) = = T 2 Z 1

1

dhP(1, h)[m0(1, h)]2 ≤ T 2

l = 1 − β2(1 − 2qEA qEA = 1 N

N

X

i=1

m2

i

r = 1 N

N

X

i=1

m4

i

1 = b ϕ 2 Z 1

1

dh eh b P(y, h) ˜ f 00(y, h)2

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The value of k seems not to depend on the dimension The value extracted fr fullRSB solution seems to well the data.

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• Confirm the numerical analysis at finite pressure. !
• Try to derive the statistics of avalanches using the results on the SK

(compute the power law exponent of the avalanches size distributi

• Try to detect the Gardner point (insights from the state following

calculation)!

• Try to extend the formalism to soft spheres.!
• Try to understand why mean field theory works so well
• A Gardner transition is present in the phase diagram and affects the

jamming limit of the glass phase!

• fullRSB effects must be included to describe the statistics of packing
• The asymptotic fullRSB equations in the jamming limit give the expo

jamming!

• The microscopic marginal stability is directly linked to the fullRSB

stability!

• The infinite dimensional calculation seems to predict well the beha

finite dimensions.

Thank