Signatures of the full replica symmetry breaking in jamming systems - - PowerPoint PPT Presentation

Signatures of the full replica symmetry breaking in jamming systems under shear

Hajime Yoshino Cybermedia Center, Osaka University

• Aug. 11th, 2015 Japan-France Joint Seminar

"New Frontiers in Non-equilibrium Physics of Glassy Materials"

JPS Core-to-Core program 2013-2015 Non-equilibrium dynamics of soft matter and information

Collaborators Financial Supports

Synergy of Fluctuation and Structure : Quest for Universal Laws in Non-Equilibrium Systems

2013-2017 Grant-in-Aid for Scientific Research on Innovative Areas, MEXT, Japan

中山大樹 Daijyu Nakayama(Osaka Univ.) Francesco Zamponi (ENS, Paris) Corrado Raione (ENS, Paris & Sapienza-Univ. Rome) Pierefrancesco Urbani(CEA, Saclay) 岡村論 Satoshi Okamura(Osaka Univ.)

ϕ

ϕm ∼ 0.5

hexagonal lattice

Emulsions, colloids,...

Supercooled Liquid

温度効果のあるジャミング転移

エマルション (emlusion;乳濁液，乳剤)

水と油など， 混ざり合わない液体が ミセルを形成して 一方が液滴となって他方に分散している系 エマルションの圧力と剛性率の測定（室温） (大きい○=圧力, 黒シンボル=剛性率) (s: 表面張力, R:粒径)

• T. G. Mason et al. (1997)

エントロピー弾性

温度効果なしや液体では0

接触力 液滴(粒子)間の相互作用の大きさで 換算して温度T ~ 10 -5

10 µm

身近では.) マヨネーズ，木工用ボンド，など

ドデカン液滴 (in 水+グルコース)

• E. R. Weeks and
• C. Holinger(2007)

unjam jam

• 圧力と剛性率の振る舞いがほぼ同じ

→ 温度効果のない数値計算 では出てこない

• cf. C. S. O’Hern et al. (2003) 等
• E. R. Weeks,

in "Statistical Physics of Complex Fluids",

• Eds. S Maruyama & M Tokuyama

(Tohoku University Press, Sendai, Japan, 2007).

volume fraction

Model

U =

• ij

v(rij) rij = |ri − rj|

v(r) = (1 − r/D)α(1 − r/D)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

v(r)/

α = 2

r/D

kBTroom/✏ ∼ 10−5 ϕRCP ∼ 0.64

“Jammed” Glass “random” close packing

ϕg ∼ 0.58 Glass

hexagonal close packing

0.74

Colloidal crystallization

Mean-Field Picture on Glass transitions Liquid

ϕ

volume fraction

Jamming

Replica Symmetric (RS)

ϕG

Gardner transition 1step replica symmetry breaking (1RSB)

Glass

glass transition

“more” RSB λreplicon = 0

Almeida-Thouless (AT) instability much like the MF models of spin-glasses

Kurchan-Parisi-Urbani-Zamponi, (2013).

ϕd < ϕg < ϕK

• F. Zamponi’s talk

Figure 1: This figure show snaphots before/after a plastic event trigered by thermal noises. Here we used

a 2-dimensional version of the model (for the purpose of a demonstration) at volume fraction φ = 0.85 which is slightly above the jamming density φJ ∼ 0.84 (2-dim). The system is initially perturbed weakly by a shear-strain γ = 0.05 and let to relax at zero temperature by the conjugated gradient method which allows the system to relax using the harmonic modes. Then the thermal noise at (reduced) temperature T = 10−6 is switched on. The configuration of particles are represented by the circles and that of the contact forces fij = −dvij(rij)/drij are represented by bonds whose thickness is chosen to be proportional to fij. The panels a) and b) show the snapshots before/after a plastic event (which took about 104tmicro to complete). In panel c) the configuratoin of the particles before/after are overlaid : the one before the event is shown by the lighter color.

Stress relaxation process

Okamura-Yoshino, unpublished (2013)

solids under shear shear-stress (force/area) shear-strain

σ

γ

σ = µγ yield stress σY

γ

Strain

• C. Rainone, P

. Urbani, HY, F. Zamponi,

• Phys. Rev. Lett. 114, 015701 (2015).

HY, F. Zamponi, PRE90, 015701(2014).

liquid theory + replica mechanical properties

• f amorphous solids

Yoshino-Mezard, PRL 105, 015504 (2010), Yoshino, JCP 136, 214108 (2012)

Twisting replicated hardsphere liquid

Replicated Mayer function (under shear)

f{γa}(x, y) = −1 +

m

• a=1

e−βv(|S(γa)(xa−ya)|)

S(γ)µν = δµν + γδν,1δµ,2

−βF({γa}) =

• dxρ(x)[1 − log ρ(x)] + 1

2

• dxdyρ(x)ρ(y)f{γa}(x, y)

HY and F. Zamponi, Phys. Rev. E 90, 022302 (2014).

ua

γ1 γ2

...

d → ∞

−βF( ˆ ∆, {γa})/N = 1 − log ρ + d log m + d

2(m − 1) log(2πeD2/d2) + d 2 log det(ˆ

αm,m) − d

2 b

ϕ R

dλ √ 2πF

⇣ ∆ab + λ2

2 (γa − γb)2⌘

Following glassy states under shear/compression

Corrado Rainone, Pierfrancesco Urbani, Hajime Yoshino, Francesco Zamponi,

• Phys. Rev. Lett. 114, 015701 (2015)

dilatancy Yielding

βσ/d

γ

Small strain expansion

F({γa})/N = F({0})/N +

m

• a=1

σaγa + 1 2

1,m

• a,b

µabγaγb + · · ·

βµab = d 2 ϕ

• δab
• c(=c)

∂F ∂∆ac − (1 − δab) ∂F ∂∆ab

• b

µab = 0

translational invariance

• ∆2
• ∆2
• ∆2
• ∆2
• ∆1
• ∆1

Hierarchical RSB

1RSB case : HY and M. Mezard (2010), HY (2012)

µ2 µ2 µ2 µ2 µ1 µ1 µ1 µ1 µ0 µ0 1 m m < x < 1 βˆ µ(y) = 1 mγ(y) ∆(y) = γ(y) y − Z 1/m

y

dz z2 γ(z)

y = x/m

1 step RSB

• ∆EA ∼

∆d − C( ϕ − ϕd)1/2

β µab = β µEA

• δab − 1

m

• βˆ

µEA = ∆−1

EA

1/ ϕd 1/ ϕ ˆ µEA

in agreement with MCT

• G. Szamel and E. Flenner, PRL 107, 105505 (2011).
• H. Yoshino, The Journal of Chemical Physics 136, 214108 (2012).
• H. Yoshino and M. Me´zard, PRL 105, 015504 (2010).
• ϕd <

ϕ < ϕGardner

HY and F. Zamponi, Phys. Rev. E 90, 022302 (2014).

γ

1+continuous RSB

• ϕGardner <

ϕ < ϕGCP b ϕ → b ϕ−

GCP

p ∝ 1/m → ∞ γ(y) ∝ γ∞y−(κ−1) κ = 1.41575

E DeGiuli; E Lerner; C Brito; M Wyart, PNAS 111 (2014), 17054 consistent with scaling argument + effective medium computation

HY and F. Zamponi, Phys. Rev. E 90, 022302 (2014).

βµEA = 1/∆EA ∝ m−κ ∝ pκ γ

“rigidity of inherent structures”

βb µ(1) = 1 mγ(1) ∝ p

“rigidity of metabasins”

ZFC susceptibility FC susceptibility

Field Cooled/ Zero Field Cooled Susceptibilities in Spin-Glasses

χFC = β[1 − Z 1 dxQ(x)]

χZFC = β[1 − Q(1)]

Full RSB solution of the Sherrington-Kirkpatrick (SK) model (exact solution of the Edwards- Anderson spin-glass model in the limit )

d → ∞

H. Yoshino, JCP 136, 214108 (2012) (NOTE) spin-wave rigidity of spin-glass is also hierarchical reflecting RSB

• G. Kotliar, H. Sompolinsky, and A. Zippelius PRB 35, 311 (1987)

γ

Field Cooling Zero Field Cooling FC/ZFC shear response of glasses ?

Energy minimization : conjugated gradient method

T/✏ = 10−5

initial conf. obtain via MD simulation at Nakayama-Yoshino-Zamponi, in progress

shear strain µ = σ/γ

Measure the remanent shear-stress

(1) temperature quench to T=0 (working at ) (2) compression (working at T=0) ϕtarget

µ

p

FC (Temperature-quench) ZFC (Temperature-quench) ZFC (compression) FC (compression)

γ = 10−3 γ = 10−4

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.001 0.002 0.003 0.004 0.005

Simulation of densely packed soft-spheres in 3 dim.

3 dim Harmonic-sphere(binary)

N = 320, 1000

O(103) − O(104)

# of samples

d → ∞

theory

µFC ∝ p µZFC ∝ √p

Reminder:

χ2 = −βχSG δqEA = χSGh2 + . . .

Non-linear susceptibility and SG susceptibility Spinglass susceptibility

Divergence of non-linear susceptibility at spin-glass transition

m = χ0h + χ2h3 + . . .

χ2 ∝ ✓ 1 − T TSG ◆−γ

χSG = 1 N X

ij

[hSiSji2 hSii2hSji2] qEA = 1 N X

i

hSii2

Edwards-Anderson Order parameter

(T > TSG)

Non-linear shear-modulus ˆ ∆ → ˆ ∆ + δ ˆ ∆

F( ˆ ∆, {γ}) = Fentropic( ˆ ∆) + Finteraction( ˆ ∆, {γ})

fluctuation around the saddle point

cab = 1 2 X

c(∈slave),d(∈reference)

∂2Fint ∂∆ab∂∆cd

Hab,cd = ∂2F( ˆ ∆), {γ = 0} ∂∆ab∂∆cd

shear stress non-linear shear modulus

σ = µ0γ + 1 3!µ2γ3 + . . .

H. Yoshino, in progress (2015)

−βF/N = N −1 ln Z X

a<b

d∆abe−βF( ˆ

∆∗+δ ˆ ∆,{γ})

= −β ✓ F(0)/N + γ2 2 µ0 + . . . ◆ + 1 2γ4Tr(cH−1c) + . . .

βµ2 = 1 4! 1 N ∂4βF ∂γ4 = −1 2 X

λ

c(λ)2 λ → −∞

Implication of “negatively” diverging non-linear shear-modulus

Vanishing linear response regime in the Gardner’s phase see also Otsuki-Hayakawa, PRE 90, 042202 (2014)

次元ジャミング系における非線形・線形シア応答

阪大理 阪大サイバー 中山大樹 吉野 元

シア歪み に対するシア応答 の係数シアモジュラス は と定義される。粒子 ジャミング系において、これはよく知られているように の関係がある 。しか し実際上は有限のシア を用いて、有効シアモジュラス を観測すること になるので、 の に対する依存性を議論することは重要である 。シア歪み に対するシア応 答 が で展開できるとすると対称性より となる。非線形シア モジュラスは と定義される。 一方、我々はソフトポテンシャル球系で シミュレーションを行いシアモジュラスの 依存性を調べた（図 。その結果一番簡単には の関係を得た。 線形シアモジュラスと非線形シアモジュラスについて議論する。

0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.01 0.015 0.02 0.025 0.03 0.035

√γ

N=128 N=256 N=512

図 剛性率のサイズ依存性

参考文献

σ/γ γ

µ0

N → ∞

ϕ → ϕ−

G

σ γ = µ0 + 1 3!µ2γ2 + . . .

δϕ = 2.4e − 4

• D. Nakayama, H.

Yoshino, in progress (2015)

(1) rigidities of inherent structures/metabasin (2) jamming scaling as Response to shear of a hard-sphere glass in

Summary

• 2. Analysis of shear-modulus
• 1. Exact free-energy functional under shear

✴1+continuous RSB

• ϕ →

ϕJ

✴ 1RSB - jump + square-root singularity at

1) FC/ZFC under shear Nakayama-Yoshino-Zamponi, in progress 2) non-linear response (ZFC) under shear Nakayama-Yoshino, in progress

d → ∞ b ϕd

• H. Yoshino and F. Zamponi, Phys. Rev. E 90, 022302 (2014).
• 3. State following under shear/compression : jamming, melting, yielding
• C. Rainone, P. Urbani, H. Yoshino, F. Zamponi, Phys. Rev. Lett. 114, 015701 (2015).

Numerical simulations of a 3-dim soft-particle system