Twisting and breaking glasses:a replica approach

2014 Sept 12th　“Critical Phenomena in Random and Complex Systems” Capri

Hajime YoshinoCybermedia Center, Osaka Univ.

HY and F. Zamponi, “The shear modulus of glasses:results from the full replica symmetry breaking solution”, Phys. Rev. E 90, 022302 (2014).

See also the Poster by Corrado Raione

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

Collaborators

Francesco Zamponi (ENS, Paris)

Corrado Raione (ENS, Paris & Univ. Rome)

Financial Supports

Marc Mézard (ENS Paris)

Satoshi Okamura (Osaka Univ.)

Pierefrancesco Urbani(CEA, Saclay)

Finite temperature MD simulation of aging in a jamming system

State following approach viaFranz-Parisi potential

Shear-modulus of Hard-sphere glass in large-D limit

Replica trick to compute shear-modulusex. 3D soft-core system v(r) � r�12

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

Outline

Introduction:

Shear on the cloned liquid in the large-d limit (theory)

Aging around the jamming point (simulation)

Discussions

Glass transition

temperature

entropy

crystal

supercooled liquid

”Tg”

liquid

Tmmelting

T �

“Entropy crisis”

Ideal glass transition?

Kauzmann temperatureTK

glass

Tc

(MCT)

Supercooled liquids

Confocal scope image (E. Weeks and D. Weitz (2002))

Vibrations within cagesβ- relaxation

Structural relaxation

α-relaxation

Shear modulus: a paradox and a lesson

�Strain

stress shear modulus or “rigidity”

limN�⇥

lim��0�= lim

��0lim

N�⇥

elasticity must emerge together with plasticity

f(�) = f(0) + �� +�2

2µ + . . .

f = F/N

thermodynamicsShape of the container should not matter!

µ = 0liquidsolid

µ = 0

µ > 0

linear response,fluctuation

「水は方円の器にしたがう」水髄方円　筍子Water conforms to the shape of its container.

Intra-state and inter-state responses under shear (1RSB)

C.F. step-wise magnetic response in spin-glasses :  H. Y. and T. Rizzo, Phys. Rev. B 77, 104429 (2008).

µab = µ̂�ab�µ̂

m�µ =

m��

b=1

µab = 0

�(�)

�

µ̂

� µ̂

m�

slope

slope

�

F (�)

�s =TK

��

N

m�(T ) � T

TK� �w

�s

�w =T

��

N

H. Yoshino and M. Mézard, Phys. Rev. Lett. 105, 015504 (2010).

shear-stress (force/area)

shear-strain0

�

�

� = µ�

yield stress

stress-strain curve

�Y

“state following” : computation of the Franz-Parisi potential under shear

See the Poster by Corrado Raione

Outline

Introduction:

Shear on the cloned liquid in large-d limit (theory)

Aging around the jamming point (simulation)

DiscussionsJ. Kurchan, G. Parisi and F. Zamponi,.J. Stat. Mech. P10012 (2012).

J. Kurchan, G. Parisi, P. Urbani and F. Zamponi, J. Chem. Phys. B117, 1279 (2013).

P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi, Nature Communications 5, 3725 (2014).

P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi, arXiv:1310.2549 .

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

Basic idea of cloned liquid

m-replicas obeying the same Hamiltonian

H = H0 ��

4

�

i

�

a,b

⇥(xai � xb

i )2⇤

“Cage size”

liquid

a = 1, 2, . . . ,mxai i = 1, 2, . . . , N

solid

xa

�(x)

X1 X2

“Einstein model”

Edwards-Andeson Order Parameter（non-ergodicity order parameter)

qEA = limt��

1N

N�

i=1

�si(t)si(0)� =1N

N�

i=1

�si�2

� = �(xai � xb

i )2�

� =� � <�

��

R. Monasson, Phys. Rev. Lett. 75 2847 (1995).S. Franz and G. Parisi, J. Phys. I France 5 1401 (1995).

M. Mezard and G. Parisi, Phys. Rev. Lett. 82 747 (1999).

Repulsive contact systems

Repulsive colloids, emulsions, granular matter,..

U =�

�ij�

v(rij) rij = |ri � rj |Model potential energy

Essentially “hard-spheres” at low temperatures.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

v(r)/�

� = 2

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 �

Soft Jammed Materials – Eric R. Weeks 23

3. Other soft materials Now that I’ve introduced colloids, let’s discuss other soft systems which resemble colloids to varying degrees. 3.1 Emulsions Emulsions are similar to colloids, but rather than solid particles in a liquid, they consist of liquid droplets of one liquid, mixed into a second immiscible liquid; for example, oil droplets mixed in water. Surfactant molecules are necessary to stabilize the droplets against coalescence which is when two droplets come together and form a single droplet. A cross-section of an emulsion is shown in Fig. 3.1, and a sketch showing a droplet with the surfactants is shown in Fig. 3.2. Mayonnaise is a common example of an emulsion, made with oil droplets in water, stabilized by egg yolks as the surfactant, with extra ingredients added for taste.

Fig. 3.1. Confocal microscope image of an emulsion. The droplets (dark) are dodecane, a transparent oil. The space between the droplets is filled with a mixture of water and glycerol, designed to match the index of refraction of the dodecane droplets. The droplets are outlined with a fluorescent surfactant. The hazy green patches are free surfactant in solution, or else the tops or bottoms of other droplets. (Picture taken by ER Weeks and C Hollinger.)

Fig. 3.2. Sketch of an emulsion droplet. Not to scale: typically the surfactants are tiny molecules, whereas the droplet is micron-sized. (Sketch by C Hollinger.)

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).

v(r) = �(1� r/D)��(1� r/D)

limT�0

e�v(r)/kBT = �(r/D � 1)

r/D

Mean-field phase diagram

Kauzmann temperature

Glass close packing density (“ideal” J-point)

T

��GCP

TK(�)

�K

TMCT(�)

�MCT “jammed”

liquid glass

”unjammed ”

G. Parisi and F. Zamponi, Rev. Mod. Phys. 82, 789 (2010)L. Berthier, H. Jacquin and Z. Zamponi, Phys. Rev. Lett. 106, 135702 (2011) and Phys. Rev. E 84, 051103 (2011).

Distance to the J-point (T, ��)�� = �� �GCP

Cloned liquid theory (replica + liquid theory)

T � O(10�6)

µ � p

Experiment: rigidity of emulsionsT. G. Mason, Martin-D Lacasse, Gary Grest, Dov Levine, J Bibette, D Weitz, Physical Review E 56, 3150 (1997)

�J

“mechanical”

“entropic”

fined. However, they are well defined at larger w, and we plotGp8(w) for different radii in Fig. 6. The plateau modulus foreach emulsion rises many orders of magnitude aroundw'0.60. Emulsions comprised of smaller droplets have dis-tinctly smaller w at which the onset of the rise occurs. Athigh w, where the droplets are strongly compressed, Gp8 islarger for smaller droplets. By contrast with P, the plateaumodulus does not diverge as w approaches unity.To investigate the role of the interfacial deformation of

the droplets on the emulsion elasticity, we scale Gp8(w) by(s/R), and plot the results in Fig. 7. At high w, this scalingcollapses the data for different droplet sizes. However, at loww there are large systematic deviations from this scaling. Toreconcile these apparently different onset volume fractions,we must account for the electrostatic repulsion between theinterfaces of droplets stabilized by ionic surfactants; this al-ters the w dependences of G and P. By using weff @cf. Eq.~1!# instead of w, we account for the thin water films stabi-lizing the charges between the droplets. These thin films willmake the apparent packing size of each droplet larger. How-ever, the thickness of the film will be determined by a bal-ance between the screened electrostatic forces between drop-lets and the deformation of their interfaces. Thus the actualfilm thickness will be only weakly dependent on droplet size,but will make a relatively larger contribution for the packingof small droplets than for large droplets.The film thickness itself depends on w, but in some un-

known fashion. Thus we linearly interpolate between a maxi-mum film thickness, hmax , at low w, below rcp, where thedroplets are not deformed, and a minimum film thicknesshmin , between the facets of the nearly polyhedral droplets atwmax near w'1. Stable Newton black films of water at asimilar electrolyte concentration have been observed withhmin'50 Å @48#. This is comparable to the calculated Debyelength lD'30 Å, for 10-mM SDS solution. Thus we as-sume that hmin550 Å; this makes a larger correction for thesmaller droplets. To determine the maximum film thickness,we vary hmax until the scaled Gp8(weff) for all droplet sizescollapse onto one universal curve. We find that the filmthickness for weak compression which gives the best col-lapse is hmax5175 Å, and is the same for all droplet sizes, asshown in Fig. 8. This film thickness agrees with the mea-

sured separation between the surfaces of monodisperse fer-rofluid emulsion droplets at the same SDS concentration@49#, lending credence to its value. Near rcp, the film in-creases the volume fraction more for smaller droplets, about5% for R50.25 mm, and only 1% for R50.74 mm.The onset of a large elastic modulus now occurs near rcp,

at weff'wcrcp , as expected. We note that this value is not a

FIG. 7. The volume fraction dependence of the plateau storagemodulus Gp8(w), scaled by (s/R), for four monodisperse emul-sions having radii R50.25 mm ~d!, 0.37 mm ~n!, 0.53 mm ~j!,and 0.74 mm ~L!.

FIG. 8. The scaled plateau storage modulus Gp8/(s/R) ~smallsolid symbols!, and the scaled minimum of the loss modulusGm8 /(s/R) ~small open symbols!, as a function of weff for monodis-perse emulsions having radii R50.25 mm ~s!, 0.37 mm ~n!, 0.53mm ~h!, and 0.74 mm ~L!. The ~s! symbols are the measuredvalues of the scaled osmotic pressure P/(s/R). The maximum filmthickness has been adjusted to hmax5175 Å to give the best col-lapse of Gp8/(s/R).

FIG. 9. The frequency dependence of ~a! the storage modulus,G8(v), and ~b! the loss modulus, G9(v), for a series of effectivevolume fractions below the critical packing volume fraction wc forR'0.53 mm. The lines merely guide the eye.

3158 56T. G. MASON et al.

“mechanical”

“entropic”

pressure p

rigidity µ

measurements at room temperature

kBT/� � 10�5

Interaction between emulsions:

I. J. Jorjadze, L-L. Pontani and J. Brujic, PRL 111, 048302(2013).

� = 2v(r)/� = (1 � r/�)�

By using well-controlled emulsions consisting of droplets ofa single size @6,7#, our approach offers several advantagesover previous rheological experiments @8,9,19# which weremade using emulsions having a broad distribution of dropletsizes. Indeed, polydisperse emulsions are difficult to studybecause they contain droplets with many different Laplacepressures so that, at a fixed osmotic pressure, the large drop-lets may deform significantly while the small droplets remainessentially undeformed. Moreover, the droplet packing anddeformation cannot be easily connected to w because smalldroplets can fit into the interstices of larger packed droplets.By contrast, using monodisperse emulsions eliminates theseinherent difficulties: all the droplets have the same Laplacepressure. Moreover, the volume fraction can be simply re-lated to the packing of identical spheres, thus allowing formeaningful comparisons with theoretical predictions whichhave usually assumed that the emulsion is monodisperse andordered.The earliest calculations of P~w! and G(w) for emulsions

and foams @11–17# are based on perfectly ordered crystals ofdroplets. In such systems at a given volume fraction andapplied shear strain, all droplets are compressed equally anddeform affinely under the shear; thus all droplets have ex-actly the same shape. Describing the dependence of P andG on w then reduces to the ‘‘simpler’’ problem of solving forthe interfacial shape of a single droplet within a unit cell.Nevertheless, calculating the exact shape and area of such asingle droplet at all w.wc is a very difficult free-boundaryproblem that can only be solved analytically for simple cases@16#, or numerically @16,17#. Real emulsions, however, ex-hibit a disordered droplet structure, and a comparison of ex-perimental results to these theoretical predictions is inappro-priate. In particular, the comparison of the w dependence ofthe low-frequency plateau value of the storage modulus ofdisordered, monodisperse emulsions to the static shearmodulus predicted by these studies has demonstrated the ex-istence of significant discrepancies @18#.The origin of the elasticity of an emulsion arises from the

packing of the droplets; forces act upon each droplet due toits neighboring droplets pushing on it to withstand the os-motic pressure. However, all these forces must balance tomaintain mechanical equilibrium. Calculations of the elasticproperties of such disordered packings are complicated bythe many different droplet shapes and the necessity of main-taining mechanical equilibrium as the droplets press againstone another in differing amounts. While a general theory ofthe elasticity of disordered packings may ultimately lead to aprecise analytical description of emulsion elasticity, com-puter simulations including adequate interdroplet interactionsand accounting for the complexity associated with disordercan provide insight into the origins of the w-dependent shearmodulus. In order to understand the effects introduced bydisorder, we developed a model for compressed emulsionswhich includes a disordered structure as well as realisticdroplet deformations @10#. In this model, we formulate ananharmonic potential for the repulsion between the packeddroplets, based on numerical results obtained for individualdroplets when confined within regular cells @16#. Numericalresults for the osmotic pressure P and the static shear modu-lus G obtained from this model are in excellent agreementwith our experimental values of P and the elasticity, as can

be shown from Fig. 1. We measure the frequency dependentstorage modulus G8(w ,w), and take the low-frequency pla-teau values Gp8(w) as the static shear modulus G(w). Ourmodel of emulsions as disordered packings of repulsive ele-ments is very general, and may also be applicable to othermaterials which become elastic under an applied osmoticcompression, provided the potential between the elements isappropriately modified.The structure of this paper is as follows. In Sec. II, we

review the theoretical predictions for the osmotic pressureand shear rheology of emulsions. In Sec. III, the experimen-tal aspects of this study are described; Sec. III A describesthe emulsion preparation and the rheological measurementtechniques; Sec. III B presents the results of our measure-ments; and Sec. III C compares our experimental observa-tions to existing predictions and previous measurements. Inorder to understand the difference found between our resultsand the predictions existing for ordered arrays of droplets, inSec. IV we present the results of numerical studies based ona model that can account for disorder. In Sec. IV A, we de-scribe the details and the motivation of the model, while, inSect. IV B we present and discuss the simulation results. Abrief conclusion closes the paper.

II. THEORY

In order to understand the properties of packings of de-formable spheres, it is useful first to review the packing ofstatic, solid spheres. Their packing determines the criticalvolume fraction wc at which the onset of droplet deformationoccurs and the coordination number zc of nearest neighborstouching a given droplet. The highest volume fraction ofmonodisperse hard spheres is attained for ordered crystallinestructures, including face-centered-cubic ~fcc! and hexagonalclose packing ~hcp!. These have wc

cp5p&/6'0.74 andzccp512. By randomly varying the stacking order of theplanes, a random hexagonally close-packed ~rhcp! structurecan be made, but this does not alter either wc or zc . Otherordered packings are less dense. For example, the body-centered-cubic ~bcc! packing has wc

bcc5p)/8'0.68 andzcbcc58, while the simple cubic ~sc! packing has

FIG. 1. The scaled shear modulus and osmotic pressure as afunction of w. The computed scaled static shear modulusG/(s/R) ~1! and osmotic pressure P/(s/R) ~line!, as obtainedfrom the model presented in Sec. IV B 2, are compared with theexperimental values of Gp8(weff) ~j! and P(weff) ~s!.

56 3151OSMOTIC PRESSURE AND VISCOELASTIC SHEAR . . .

µp

M-D. Lacasse, G. S. Grest, D. Levin, T. G. Mason and D. A. Weitz, PRL 76,3448 (1996)

� > 2?

rigidity (shear-modulus) pressureWhy µ �

��� �J does not hold?

� > �GCP

� < �GCP

�GCP = 0.633353..

m� � 20.7487(�GCP � �)

yHSliq (�GCP) � 23.6238A�/m�(�GCP) � 9.72187� 10�5

ex. Soft-particle case (3-dim) Berthier-Jacquin-Zamponi (2011)

T/m� � 0.00835535(�� �GCP)

limT�0

p � 1.92178T/(�GCP � �)

limT�0

p � 0.403001(�� �GCP)

limT�0

µ̂ � 0.694315T/(�GCP � �)

limT�0

µ̂ � 0.1239496(�� �GCP)

1step RSB approximation

� = 2

limT�0

�µ̂ =1

m�

�A�

m�

�6�

�yHSliq (�GCP)3

�c1 � c2

�A�

m� + . . .

�

Berthier-Jacquin-Zamponi (2011)

Berthier-Jacquin-Zamponi (2011)

H. Yoshino, AIP Conference Proceedings 1518, 244 (2013)S. Okamura and H. Yoshino, arXiv:1306.2777 (2013).

P � T/|��|Harmonic approx. vs 1RSB approx. (2012)

�� = �J � �

“inherent structures”

shear-modulus

cage size

Brito-Wyart (2006)

µharmonic � T/|��|3/2

Ikeda-Berthier-Birol (2013)

µ � T/�

�harmonic � |��|3/2

Berthier-Jacqin-Zamponi (2011)Emulsion experiments: T. G. Mason et al (1997). Guerra-Weitz (2013)

limT�0

µ(T ) � T/|��|Yoshino (2012), Okamura-Yoshino(2013)

µ � T/�

limT�0

�(T ) � |��|

(1RSB)“meta-basins”

“unjammed” side �� < 0

Gardner transition! J. Kurchan, G. Parisi and F. Zamponi,.J. Stat. Mech. P10012 (2012).

“inherent structures”

AT=0 � 1/�

�

Durian (1995), O’Hern et. al. (2003)

“jammed” side

µharmonic ��

�� shear-modulus

cage sizeAharmonic � T/�

��Ikeda-Berthier-Birol (2013)

H ��

i,j

�

�µ

12Hµ�

ij �uµi u�

j

Harmonic approx

µ � T/A

(1RSB)“meta-basins”

limT�0

µ(T) � ��

limT�0

A(T ) � T/��

Yoshino (2012), Okamura-Yoshino(2013)

Berthier-Jacqin-Zamponi (2011)Emulsion experiments: T. G. Mason et al (1997). Guerra-Weitz (2013)

µ � T/A

�� = �J � �

Harmonic approx. vs 1RSB approx. (2012)

�� > 0 P � |��|

AT instability - Gardner’s transition

ergodicity breaking entropy crisis

jamming

J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, J. Phys. Chem. B 117, 12979 (2013).

m

1

log d

!ϕ!ϕGCP!ϕG

!ϕK!ϕ2RSBth

!ϕ1RSBth!ϕ∗

m∗

mG(!ϕ)m1RSB

th (!ϕ)m2RSB

th (!ϕ)

!ϕd

m

��

1RSB solution

mG(��)��G

Gardner’s transition

�p/� � 1/m

E. Gardner, Nucl. Phys. B 257, 747 (1985).

JRL De Almeida and D. J. Thouless, Journal of Physics A: Math. Gen. 11, 938 (1978)

Replicated liquid theory of hard spheres

x = {x1 · · ·xm} xa = ((xa)1, (xa)2, . . . , (xa)m)

Replicated Mayer function

d��

f(x, y) = �1 +m�

a=1

e��v(|(xa�ya)|)

��F =�

dx�(x)[1 � log �(x)] +12

�dxdy�(x)�(y)f(x, y)

molecule made of replicas

ua�

�

�� =2d

d�

��F (�̂, {�a})/N = 1� log � + d log m + d2 (m� 1) log(2�eD2/d2) + d

2 log det(�̂m,m)

� d2 ��F (�ab)

Order parameter

Exact expression of the free-energy as a functional of the glass order parameter

�ab =d

D2�(ua � ub)2�

J. Kurchan, G. Parisi and F. Zamponi,.J. Stat. Mech. P10012 (2012).

P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi, Nature Communications 5, 3725 (2014).

Contact potential

1+continuous Replica Symmetry Breakingprevious random first order transition(RFOT)：1 step RSBParisi’s matrix (m x m)

�̂1

�̂2

�̂1

�̂1

F�̂2

�̂2

�̂2

�̂2

�̂0 =�0

0

00

�̂1�̂2

Displacement

mm0

m1

Parisi’s equation�f(x, h)

�x= 1

2�̇(x)�

�2f(x,h)�h2 + x

��f(x,h)

�h

�2�

, f(1, h) = log ��

h�2�(1)

�.

��F�RSB = �m� 1

mdxx2 log

�x�(x)

m +� 1

x dz �(z)m

�� �� e��(m)/2

���� dh eh[1� emf(m,h)]

(note) Spinglass case : G. Parisi (1980) massless “replicon” mode, marginal stability

Hard-sphere case

eg.2 step RSB

Response in hierarchical energy landscape

χ0 = !χ2 +m1!χ1 +m0!χ0

χ1 = !χ2 +m1!χ1

χ2 = !χ2

�i =k�

j=i

mj�̃j

�O� = �dF

dh=

�

�0

w�0

�

�1��0

w�1|�0O�1 O�1 = �df�1

dh

� =d�O�dh

= �d2F

dh2

=�

�0

w�0

�

�1��0

w�1|�0

dO�1

dh+

�

�0

w�0

�

�1��0

dw�1|�0

dhO�1 +

�

�0

dw�0

dh

�

�1��0

w�1|�0O�1

= ����1 �1�0 + �Nm1� �O2�1�1 � �O�1�21 �0 + �Nm0

���O�1 �21�0 � ��O�1 �1�20

�

= ��2 + m1��1 + m0��0

Response of “cloned system” in hierarchical energy landscape　

Fml = � T

Nlog Zml Zml =

�

�0

m0/m1�

B=1

��

�1��0

e��NP

a�B f�1 (ha)

�

χ0 = !χ2 +m1!χ1 +m0!χ0

χ1 = !χ2 +m1!χ1

χ2 = !χ2

B = 1

B = 2

B = 3

m1 = 4m0 = 12

1RSB case : H. Yoshino and M. Me´zard, Phys. Rev. Lett. 105, 015504 (2010)

�ab =�2F

�ha�hb

����ha=0

= �̃2�ab + �̃1Im1ab + �̃0I

m0ab 0

0

11

11

11

11

mi

Imi =

�i =k�

j=i

mj�̃j

�ab =k�

i=0

�̃iImiab

m0 = mm1mk�1mk = 1

More generally... with k-RSB

Twist on the replicated liquid

��F (�̂, {�a})/N = 1� log � + d log m + d2 (m� 1) log(2�eD2/d2) + d

2 log det(�̂m,m)

� d2 ��

�d��2�F

��ab + �2

2 (�a � �b)2�

Replicated Mayer function (under shear)

(Compression: C(�)µ� = �µ� + ���,1�µ,1)f{�a}(x, y) = �1 +

m�

a=1

e��v(|S(�a)(xa�ya)|) S(�)µ� = �µ� + ���,1�µ,2

��F ({�a}) =�

dx�(x)[1� log �(x)] +12

�dxdy�(x)�(y)f{�a}(x, y)

ua

�1 �2...

Small strain expansion

F ({�a})/N = F ({0})/N +m�

a=1

�a�a +12

1,m�

a,b

µab�a�b + · · ·

yields shear-modulus matrix

�µab =d

2��

�

��ab

�

c( �=c)

�F��ac

� (1 � �ab)�F

��ab

�

��

b

µab = 0 “sum rule”

�µ̂(y) =1

m�(y)�(y) =

�(y)y�

� 1/m

y

dz

z2�(z) for y = x/m

!∆2

!∆2

!∆2

!∆2

!∆1

!∆1

00

00

Hierarchical RSB Hierarchical rigidity

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

µ2

µ2

µ2

µ2

µ1

µ1

µ1

µ1

µ0

µ0

1

m

m < x < 1

1 step RSB

��EA � ��d � C(��� ��d)1/2

��µab = ��µEA

��ab �

1m

�

�µ̂EA = ���1EA

1/��d1/��

µ̂EA

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. M ́ezard, PRL 105, 015504 (2010).

W. Gotze, Complex dynamics of glass-forming liquids: A mode-coupling theory, vol. 143 (Oxford University Press, USA,2009).

0

��d < �� < ��Gardner

1+continuous RSB ��Gardner < �� < ��GCP

βµEA = 1/∆EA ∝ m−κ ∝ pκ

β!µ(1) =1

mγ(1)∝ p

p ∝ 1/m → ∞

* the plateau modulus

* the "lowest" plateau modulus

κ = 1.41575..

!ϕ → !ϕ−GCP

γ(y) ∼ γ∞y−(κ−1)

E DeGiuli; E Lerner; C Brito; M Wyart, arXiv:14023834Effective medium approach + numerical simulation

�

Reference

Slave

��FFP(�) = log�

r1···rm,rm+1···rm+s

e��Pm

a=1 H[{ra}]��Pm+s

b=m+1 H[{rb},�]

= log�

r1···rm

e��Pm

a=1 H[ra](Ze(�))s

��FFP(�) = log Zm + sNVFP(�) + O(s2).

S. Franz and G. Parisi, J. Phys. I France 5 (1995) 1401.F. Krzakala and L. Zdeborova, Eur. Phys. Lett. , 90 (2010) 66002.

H[{ri}] =�

i<j

v(ri � rj)

NVFP(�) = �log Ze(�)�m � 1Zm

�

r1,...,rme��

Pma=1 H[xa]log Ze(�)

Ze(�) = log�

rm+1e��H[{rm+1

i };�]

ua

�A

cloned liquid (m-replica)

slave : “m+1”th replica

Franz-Parisi potential

lim�A�0+

limN��

“Following glassy states” under perturbation

“state following under (de)compression/shear” via Franz-Parisi potential

C. Raione, P. Urbani, H. Yoshino and F. Zamponi, submitted

Nonlinear response - yielding

See the Poster by Corrado Raione

MD simulation

�shear strain

t0 tw

� �(t) = ⇤(t� tw)⇥�

Aging

C�(t, tw) = ��(t)�(tw)�µ(t, tw) =���(t; tw)�

��

autocorrelation functionresponse function

quench t = 0

T/� = 10�3Initial configuration

Equilibrium state (liquid)

shear strain

c.f. McKenna, Narita and Lequeux, J. Rheol. 53, 489 (2009).

# of samples : 4096

N = 800, 1600� = 0.65� 0.67volume

fraction

# of particles

# of sample (initial condition/ Langevin noise)

Langevin simulation

Lee-Edwards boundary condition� = 2.5� 10�3shear-strain

temperature kBT/� = 10�5

O(t/t0) = 105time scale

Outline

Introduction:

Shear on the cloned liquid in the large-d limit (theory)

Aging around the jamming point (simulation)

Discussions

���(� + tw)�(tw)�

�τ

σ(τ ; tw)/γ

Pressure

µhamonictw

tw = 3� 102, 103, 3� 103, 5� 103, 104, 3� 104, 105

t� = 2�/�� = (� = 0.67)

� = 0.67 kBT/� = 10�5

σ(τ ; tw)/γ

β⟨σ(τ + tw)σ(tw)⟩

FDT

µharmonic

Pressure

tw

slope

slope 1

x ≃ 0.007

tw = 3� 102, 103, 3� 103, 5� 103, 104, 3� 104, 105

t� = 2�/�� = (� = 0.67)

� = 0.67 kBT/� = 10�5

N = 1600

τ

σ(τ ; tw)/γ

Pressure

µhamonictw

N = 800

inherent structures meta-basins

(1RSB)

Discussion : fluctuation within meta-basin

Thermally activated, localized “floppy modes” ?

# NOTE strong relaxation of shear-stress is possible

�xz =1V

�

i<j

rijf(rij)xij

rij

zij

rij“angular variables

E. Lerner, G. During and M. Wyart arXiv.1302.3990

�rij = |�Ri � �Rj | = 0Floppy mode:

(start) opening contact

(end) closing contact

liso � �z�1 � 1/�

�� �J

M. Wyart, PRL 109, 125502 (2012).

M. Wyart, Annales de Phys, 30 (3),1 (2005).

Response to shear of a hard-sphere glass in large-dimensional limit

Summary

2. Analysis of shear-modulus

3. State following under shear - observation of the yield process

1. Exact free-energy functional under shear

✴1+continuous RSB - (1) hierarchy of rigidities (2) anomalous scaling as ��� ��J

✴ 1RSB - jump + square-root singularity at Td

Replica can be useful for real life!