Temperature and Correlations in Driven Dissipative Systems Giacomo - - PowerPoint PPT Presentation

Temperature and Correlations in Driven Dissipative Systems

Giacomo Gradenigo

Laboratoire Interdisciplinaire de Physique (LIPhy)

ENS, Lyon, 10-11-2015

Grenoble (2014-present ): E. Bertin, J.-L. Barrat, E. Ferrero Rome (2010-2012): A. Puglisi, A. Vulpiani, A. Sarracino

History

• Ph. D. : Supercooled liquids ; Trento (Italy), 2007-2009

Supervisor : P. Verrocchio. Collaborations: G. Parisi, A. Cavagna. I Giardina, T. Grigera, C. Cammarota Post-Doc : Non-equilibrium Statistical Mechanis; Rome, 2010-2012 Advisor: A. Puglisi, A. Vulpiani Collaborations: H. Touchette, R. Burioni, U. Marconi, A. Cavagna, T. Grigera,

• P. Verrocchio, A. Sarracino, D. Villamaina

Post-Doc : Glass transition; Paris, 2013-2014 Advisor: G. Biroli, S. Franz Post-Doc : Non-equilibrium Statistical Mechanics; Grenoble, 2015-present Advisor: E. Bertin, J.-L. Barrat Collaborations: E. Ferrero, A. Puglisi, G. Biroli

Interests & Reasearch topics

Equilibrium Statistical Mechanics

• Supercooled Liquids in confined geometries: effect of confinement on the

glass transition (Rome, Paris, theory, simulations).

• Effective theories for the glass transition and

Dynamical Heterogeneities in Supercooled Liquids (Trento, simulations).

Out-of-equilibrium Statistical-Mechanics

• Coarse-grained description of Granular Fluids:

Linearized Fluctuating Hydrodynamics (Rome, theory, simulations, experiments)

• Non-equilibrium fluctuations in the driven Stochastic Lorenz Gas

(1d schematic model of a granular gas): Fluctuation-Relation, Condensation of Fluctuations (Rome, Grenoble, theory, simulations)

• Anomalous diffusion in driven systems: Continuous Time Random Walks, Kinetically

Constrained Models (Rome, Grenoble, theory, simulations)

• Effective ‘’equilibrium-like’’ theories for Driven Athermal Systems (Grenoble, theory,

simulations)

• Ratchet effect (Rome, simulations)

Interests & Reasearch topics

Out-of-equilibrium Statistical-Mechanics

Coarse-grained description of Granular Fluids: Linearized Fluctuating Hydrodynamics Effective equilibrium-like theories for Driven Athermal Systems

Temperature = ? Correlations and Temperature

Dense and Dilute Granular Systems

Two-dimensional granular fluid

PART I

Amorphous packing of frictional grains (study of a model system)

PART II

PART I Temperatures and Correlations in a 2D bulk driven granular fluid

Granular Fluids: Non-Equilibrium Stationary State

Inelastic collisions

α < 1

vi vj ˆ σ v0

i

v0

j

Bulk Dissipation Bulk Driving

(energy source coupled to each particle)

∆Ecoll ∝ −([vi − vj] · ˆ σ)2(1 − α)2

Tg = 1 Nd X

i

mhv2

i i

2

Granular temperature NESS

(Non-Equilibrium Stationary State)

Vibrating plate (Time Translational Invariance, constant energy flux)

Granular Fluids: Non-Equilibrium Stationary State

Tg = 1 Nd X

i

mhv2

i i

2

Tg = Tg(φ) φ = Packing Fraction

φ ↑

Tg ↓

Inelastic collisions

α < 1

vi vj ˆ σ v0

i

v0

j

Bulk Dissipation

∆Ecoll ∝ −([vi − vj] · ˆ σ)2(1 − α)2

Vibrating plate (Time Translational Invariance, constant energy flux) Bulk Driving

(energy source coupled to each particle)

Physical meaning of the Granular Temperature = ?

Boundary Driving Vibrating vessel

Tg = 1 Nd X

i

mhv2

i i

2

Granular temperature NESS

(Non-Equilibrium Stationary State)

Study of the fluctuations within the granular gas: does really Tg plays the role of a temperature? Bulk Dissipation

∆Ecoll ∝ −([vi − vj] · ˆ σ)2(1 − α)2

(Time Translational Invariance, constant energy flux)

Physical meaning of the Granular Temperature = ?

High frequency

Tg = 1 Nd X

i

mhv2

i i

2

Granular temperature NESS

(Non-Equilibrium Stationary State)

Study of the fluctuations within the granular gas: does really Tg plays the role of a temperature? Boundary Driving

Low frequency Heat Flux

T hot

g

T cold

g

‘’Energy transport between two granular thermostats’’, C.E. Lecomte, A. Naert, J. Stat. Mech., P11004 (2014) ‘’Work exchange with a granular gas: the viewpoint of the Fluctuation Theorem’’, A. Naert, EPL 97, 20010 (2012)

Bulk Dissipation

∆Ecoll ∝ −([vi − vj] · ˆ σ)2(1 − α)2

(Time Translational Invariance, constant energy flux)

Energy injection mechanism as a ‘’thermostat’’ = ?

Tg = 1 Nd X

i

mhv2

i i

2

Granular temperature NESS

(Non-Equilibrium Stationary State)

Bulk energy injection can be modeled as a thermostat at Tb ? Which role is played by Tb-Tg? Bulk Dissipation

∆Ecoll ∝ −([vi − vj] · ˆ σ)2(1 − α)2

Vibrating plate (Time Translational Invariance, constant energy flux) Bulk Driving

(energy source coupled to each particle)

Two models for bulk driving

m ˙ vi(t) = −γbvi(t) + ξi(t) + Fcoll

i

m ˙ vi(t) = ξi(t) + Fcoll

i

TPC Van Noije, MH Ernst, E. Trizac,

• I. Pagonabarraga, PRE 59 (4), 4326

(1999)

Without viscous drag

(infinite temperature thermostat)

With viscous drag

(finite temperature thermostat)

• G. Gradenigo et al , EPL 96, 14004 (2011)
• G. Gradenigo et al , J. Stat. Mech. P08017 (2011)
• A. Puglisi et al , J. Chem. Phys. 136, 014704 (2012)

m ˙ vi(t) = ξi(t) + Fcoll

i

hξµ

i (t)ξν j (t0)i = 2 Γ δµ,νδi,jδ(t t0)

« Randomly driven granular fluids: Large-scale structure » . TPC Van Noije, MH Ernst, E. Trizac, I. Pagonabarraga, PRE 59 (4), 4326 (1999)

hv?(r)v?(r0)i ⇠ hvk(r)vk(r0)i ⇠ Γ log ✓ L |r0 r| ◆

Two-dimensional bulk driven granular fluid: Scale-free correlations

Energy Sink Energy Gain White noise

Scale-Free Correlations

Linearized Fluctuating Hydrodynamic calculations & Simulations show that

v(r) v(r0) v?(r0) vk(r0) vk(r) v⊥(r)

Coarse-grained velocity field

Two-dimensional bulk driven granular fluid: Scale-free correlations

Experiments

« Forcing and Velocity Correlations in a Vibrated Granular Monolayer »

• A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 (2002)

Smooth circulat plate (d=20cm) Steel spheres (d=1.59mm)

|Ck(r)| ∼ 1/r2

m ˙ vi(t) = ξi(t) + Fcoll

i

hξµ

i (t)ξν j (t0)i = 2 Γ δµ,νδi,jδ(t t0)

Energy Sink Energy Gain White noise

hv?(r)v?(r0)i ⇠ hvk(r)vk(r0)i ⇠ Γ log ✓ L |r0 r| ◆ Scale-Free Correlations

« Forcing and Velocity Correlations in a Vibrated Granular Monolayer »

• A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 (2002)

Rough exagonal plate (d=30cm) Steel spheres (d=3.97mm)

C⊥(r) ∼ e−r/ξ⊥ Ck(r) ∼ er/ξk

Two-dimensional bulk driven granular fluid: Finite-range correlations

m ˙ vi(t) = ξi(t) + Fcoll

i

hξµ

i (t)ξν j (t0)i = 2 Γ δµ,νδi,jδ(t t0)

Energy Sink Energy Gain White noise

hv?(r)v?(r0)i ⇠ hvk(r)vk(r0)i ⇠ Γ log ✓ L |r0 r| ◆ Scale-Free Correlations Experiments

(lattice of steel balls d=1.19 mm)

« Forcing and Velocity Correlations in a Vibrated Granular Monolayer »

• A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 (2002)

C⊥(r) ∼ e−r/ξ⊥ Ck(r) ∼ er/ξk λ0 ∼ (1 − φ)2 φ

Two-dimensional bulk driven granular fluid: Finite-range correlations

Experiments

Mean Free Path: microscopic length-scale

• f the system

/

• jj

0:6 for f?. It is surprising that this decay length has little density dependence, despite the fact that the mean free path estimated from Enskog-Boltzmann kinetic theory varies from 2:3 for 0:125 to 0:16 for 0:6. A similar

Rough exagonal plate (d=30cm) Steel spheres (d=3.97mm)

(lattice of steel balls d=1.19 mm)

Viscous drag

m ˙ vi(t) = −γbvi(t) + ξi(t) + Fcoll

i

hξµ

i (t)ξν j (t0)i = 2 Tbγb δµ,νδi,jδ(t t0)

Equilibrium Thermostat

Two-dimensional bulk driven granular fluid: The equilibrium thermostat

Inelastic Collisions Random Kicks The Limit of Elastic Collisions is well defined

Tg = Tb

‘’Distance’’ from equilibrium ∆T = Tb − Tg

α = 1

∆Ecoll ∝ −([vi − vj] · ˆ σ)2(1 − α)2

α < 1

Energy dissipated in collisions

m ˙ vi(t) = −γbvi(t) + ξi(t) + Fcoll

i

ξ⊥ = p ν/γb K0(|r0 − r|/ξ?) ∼ e|r0r|/ξ?

nhv?(r0)v?(r)i = Tgδ(2)(r0 r) + (Tb Tg)K0(|r0 r|) ξ2

? Characteristic length Finite ‘’Distance’’ from equilibrium

Two-dimensional bulk driven granular fluid: Equilibrium thermostat produces Finite-range correlations

Predictions from Linearized Fluctuating Hydrodynamic calculations

Finite extent of correlations

hξµ

i (t)ξν j (t0)i ⇠ Tbγb

• G. Gradenigo et al , EPL 96, 14004 (2011)
• G. Gradenigo et al , J. Stat. Mech. P08017 (2011)

ν = Shear viscosity

ξ⊥ = p ν/γb

Characteristic length

/

• jj

0:6 for f?. It is surprising that this decay length has little density dependence, despite the fact that the mean free path estimated from Enskog-Boltzmann kinetic theory varies from 2:3 for 0:125 to 0:16 for 0:6. A similar

ξ ξ

φlow φhigh

ξ⊥ λ0(φhigh) > ξ⊥ λ0(φlow)

Growing extent of correlations

We must look at the rescaled length!

λ0(φ)

∆T = Tb − Tg(φ)

Mean Free Path

φ ↑ Tg ↓ ∆T ↑ λ0 ↓ ξ⊥/λ0 ↑

‘’Distance’’ from equilibrium

Summary: energy injection as finite temperature bath

m ˙ vi(t) = −γbvi(t) + ξi(t) + Fcoll

i

Hot Thermostat System Cold Thermostat (inelastic collisions)

T0 = 0

Random Kicks Viscous drag

hξ2

i i ⇠ Tbγb

Finite temperature thermostat

C?,k(r) ∼ (Tb − Tg) er/ξ?,k

Inelastic collisions

Finite extent of correlations Finite ‘’distance’’ from equilibrium ∞ > Tb − Tg > 0 Tg Tb

Energy ¡Flux ¡ Energy ¡Flux ¡

• G. Gradenigo et al , EPL 96, 14004 (2011)
• G. Gradenigo et al , J. Stat. Mech. P08017 (2011)

Hot Thermostat

Tb = ∞

System Cold Thermostat (inelastic collisions)

T0 = 0

m ˙ vi(t) = ξi(t) + Fcoll

i Inelastic collisions Random kicks

hξ2

i i ⇠ Γ =

lim

γb→0 Tb→∞

Tbγb

‘’Infinite temperature’’ thermostat (elastic limit is ill-defined)

Scale-free correlations

C?,k(r) ∼ Γ log (L/r)

Infinite ‘’distance’’ from equilibrium

Tb − Tg = ∞

Tg

Energy ¡Flux ¡ Energy ¡Flux ¡

Summary: energy injection as infinite temperature bath

PART II Statistical Mechanics for a System with dry Friction

Edwards Hypothesis

Amorphous packing

• f frictional grains

Edwards Hypothesis All packings where grains occupy the same volume have the same probability

S = log X

C

δ(V ∗ − V [C)])

Microcanonical Entropy and Temperature Canonical Probability

1 TEd = ∂S ∂V

• S∗

P(C) ∼ e−V (C)/TEd

TEST OF EDWARDS ASSUMPTION: TAPPING

DYNAMIC averages THERMODYNAMIC averages

Energy injection through vertical vibration of the box (tapping)

C1 C2

Dissipation Mechanical Stability Mechanical Stability Mechanically stable configurations at fixed Volume

hOiV ∗ hOiTEd = Z−1 X

C

O(C) e−V [C]/TEd

Test Edwards Hypothesys in a model system

One-dimensional: tractable analytically Realistic: Coulomb Dry Friction Athermal: unambigous interpretation of the Effective Temperature Easily reproducible: in Numerical Simulations (done) and Experiments (not done) The model

HARMONIC CHAIN WITH DRY (Coulomb) FRICTION

Equations of motion (dynamic friction)

|(xi+1 + xi−1 − 2xi + F(t))| > µ mg

Condition to start moving (static friction)

Dynamic friction: energy dissipation External Force: energy gain

m ¨ xi = −mgµd sgn( ˙ xi) + (xi+1 + xi−1 − 2xi) + Fi(t) ˙ xi mgµd

dynamic friction coefficient static friction coefficient Harmonic ¡springs ¡

µd = 0.5 µ = 0.6

˙ xi = 0

HARMONIC CHAIN WITH DRY (Coulomb) FRICTION

Dynamic friction Gravity Harmonic ¡springs ¡

‘’Creep motion of a frictional system’’, B. Blanc, L. A. Pugnaloni, J-.C. Geminard, PRE 84, 061304 (2011) ‘’Aging of the frictional properties induced by temperature variations’’, J.-C. Geminard, E. Bertin, PRE 82, 056108 (2010) ‘’Creep motion of a granular pile induced by thermal cycling’’, T. Divoux, H. Gayvallet, J.-C. Geminard, PRL, 101, 148303 (2008)

˙ x θ mgµd sin(θ)

Temperature fluctuations: time-dependent spring stiffness Simple model for granular piles

m¨ xi = −mgµd sin(✓)sgn( ˙ xi) + (xi+1 + xi−1 − 2xi + `i+1(t) − `(t)) + mg cos(✓)

TAPPING DYNAMICS

1) External force switched on for a fixed duration t: energy injection

m ¨ xi = −mgµd sgn( ˙ xi) + (xi+1 + xi−1 − 2xi) + F ni

2) External force switched off MECHANICALLY STABLE (blocked) configuration: all particles are at rest

m ¨ xi = −mgµd sgn( ˙ xi) + (xi+1 + xi−1 − 2xi) ⇠i = xi − xi−1 − `0 e = 1 N

N

X

i=1

ξ2

i

2

Energy of the mechanically stable configurations Spring elongation

|xi+1 + xi−1 − 2xi| < µmg

Dynamics is arrested

˙ xi = 0 ∀i ρ = 1 N

N

X

i=1

ni ρ < 1

Not all the particles are pulled !

Heating Quench Heating Quench

HARMONIC CHAIN WITH DRY (Coulomb) FRICTION

After few cycles the energy of blocked configurations fluctuates around a stationary value

F > 0 F = 0

Mechanically stable configuration Mechanically stable configuration

Energy of mechanically stable configurations (N=256)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2000 4000 6000 8000 10000 12000 14000

F · τ τ

hei

τ % F fixed F % τ fixed

SPRING-SPRING CORRELATION

(IN MECHANICALLY STABLE CONFIGURATIONS) Spring-spring correlation

• 0.2

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 50

C(i-j) i-j

• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10

C(|i-j|/) |i-j|/

`(e) ∼ e e

Extent of correlation between springs grows as the energy stored by the springs

hξiξji h⇠i⇠ji ⇠ C(|i j|/`(e))

Heating Quench Heating Quench

F > 0 F = 0

Mechanically stable configuration Mechanically stable configuration

τ

• 0.2

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 50

C(i-j) i-j

• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10

C(|i-j|/) |i-j|/

• 0.2

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 50

C(i-j) i-j

• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10

C(|i-j|/) |i-j|/

• 0.2

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 50

C(i-j) i-j

• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10

C(|i-j|/) |i-j|/

fi = F fi ∼ e− (f−F )2

σ

ρ = 0.3 ρ = 0.8 fi = F ρ = 1.0

Linear increase of correlation length for all the ‘’tapping’’ dynamics we used

`(e) ∼ e

5 10 15 20 25 30 0.04 0.1 0.16 0.22

• e

` e

‘’Given a certain situation attained dynamically, physical observables are obtained by averaging over the usual equilibrium distribution at the corresponding volume, energy, etc. but restricting the sum to ‘blocked’ configurations.’’

EFFECTIVE THERMODYNAMICS ‘’Á LA EDWARDS’’

E[ξ] =

N

X

i=1

ξ2

i

2

βEd =  ∂S ∂E

• F(ξ)=1

Mechanical stability

F(ξ) = 0 ξ = {ξ1, . . . , ξN}

Springs elongations

(Barrat, Kurchan, Loreto, Sellitto, PRL, 2000)

Z Dξ e−βEdE[ξ] δ(F[ξ])

Transfer Operator Formalism (1D)

Z = Z dξ1 . . . dξN e−βEd

PN

i=1 ξ2 i 2

N

Y

i=1

Θ(µ − |ξi+1 − ξi|) ξi ξi+1 xi

EFFECTIVE THERMODYNAMICS ‘’Á LA EDWARDS’’

Z = Z dξ1 . . . dξN T(ξ1, ξ2) . . . T(ξN, ξ1) = Tr(T N)

Free-energy, energy, entropy and correlations can be calculated exactly (no explicit expressions)

T(x, y) = e−βEd x2

4 Θ(µ − |x − y|)e−βEd y2 4

SMOOTH APPROXIMATION OF THE CONSTRAINT

Z = Z dξ1 . . . dξN exp [−ξ · Aξ]

Θ(µ − |x − y|) ∼ 1 √π exp ✓ −|x − y|2 4µ2 ◆

e = 1 2 µ TEd p 2TEd + µ2

0.0001 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 10

e T

x x1/2

0.1 1 10 100 0.01 0.1 1 10

• T

x1/2

e TEd

Explicit expressions become available Transfer Operators exact result

GAUSSIAN APPROXIMATION: CORRELATION FUNCTION

0.0001 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 10

e T

x x1/2

0.1 1 10 100 0.01 0.1 1 10

• T

x1/2

e TEd

Correlations appear in the ‘’out-of-equilibrium’’ regime

0.1 1 10 100 0.01 0.1 1 10

• T

x1/2

` TEd

h⇠i⇠ji = 2µTEd p µ2 + 2TEd exp ✓ |i j| `(µ, TEd) ◆ µ2/TEd ⌧ 1 = ) ` ⇠ pTEd µ

DRIVEN ATHERMAL DYNAMICS EFFECTIVE THERMODYNAMICS m ¨ xi = Fdiss + Fel + Fext Z = Z

ξ∈blocked

Dξ e−βEdE[ξ]

Blocked configurations

`(e) ∼ e `(e) ∼ e e ∼ p TEd e ∼ √ ?

h⇠i⇠ji ⇠ exp ✓ |i j| `(e) ◆

h⇠i⇠ji ⇠ C ✓|i j| `(e) ◆

Heating Quench Heating Quench

Blocked configuration Blocked configuration

EFFECTIVE TEMPERATURE = DISSIPATED ENERGY

Driving Cycle = D.C.

0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 10 100 e ediss (points); T (lines) 0.001 0.01 0.1 0.001 0.1 10

e

ediss(points) TEd (line) ediss = − 1 N

N

X

i=1

Z

D.C.

ds vi(s)Fdiss,i(s) ediss ⌧ 1 e ∼ ediss e ∼ √ediss ediss 1

DRIVEN ¡ATHERMAL ¡ DYNAMICS ¡ EFFECTIVE ¡ THERMODYNAMICS ¡ `(e) ∼ e `(e) ∼ e e ∼ p TEd e ∼ √ediss m ¨ xi = Fdiss + Fel + Fext Z = Z

ξ∈blocked

Dξ e−βEdE[ξ]

ediss = TEd

Blocked configurations

h⇠i⇠ji ⇠ exp ✓ |i j| `(e) ◆

h⇠i⇠ji ⇠ C ✓|i j| `(e) ◆

• G. Gradenigo, E. Ferrero, E. Bertin, J.-L. Barrat, Phys. Rev. Lett. 115, 140601 (2015)

CONCLUSIONS

• PART II: Uniform Edwards measure works well for the spring-block model
• ­‑ ¡PART II: ¡Effective temperature TEd has a clear physical interpretation: energy dissipated

PERSPECTIVES

• PART II: Simulations and theory in 2D
• PART II: Dry Friction + Viscous friction. Non-uniform Edwards measure: the

same energy do not imply the same probability of configurations

• PART II: Response to external perturbations: relevance of Ted
• PART II: Correlations grow with TEd
• PART I: Solved Linear Fluctuating Hydrodynamic with viscous drag
• ­‑ ¡PART I: ¡Correlations strongly depend on the energy supply mechanism
• PART I: At finite distance from equilibrium Tb-Tg correlations decay exponentially
• PART I: Entropy production from Fluctuating Hydrodynamics: characteristic length-scale?
• PART I: Fluctuations of the entropy production in bulk driven granular gas

Thanks for your attention!

nhv⊥(k)v⊥(k)i = Tg + Tb Tg 1 + ξ2

⊥k2

Tg Tb

Fourier spectrum of velocity correlations (measured)

˙ v⊥(k, t) = −(γb + νk2)v⊥(k, t) + ξext(k, t) + ξint(k, t) hξext(k, t)ξext(k0, t0)i ⇠ νk2 Tg hξint(k, t)ξint(k0, t0)i ⇠ γbTb

Linearized fluctuating Hydrodynamics Random kicks fluctuations: External Noise Velocity fluctuations: Internal Noise

ξ⊥

• G. Gradenigo et al , EPL 96,

14004 (2011) Spheres diameter = 4mm Plate diameter = 100mm Roughness from sand blasting: [100-500] µm

‘’THERMODYNAMIC’’ ¡POTENTIALS ¡ ¡

T(x, y) ∈ L2(X x Y )

Hilbert-­‑Schmidt ¡-­‑> ¡ ¡Compact ¡ ¡

f(βEd, µ) = − 1 βEd log[λmax(βEd, µ)] e = ∂βEd(βEdf) e = λ−1

maxhλmax|∂βEdT |λmaxi

T(x, y) = e−βEd x2

4 Θ(µ − |x − y|)e−βEd y2 4

T(x, y) = T ∗(y, x)

Self-­‑Adjont ¡ ¡

T [f](x) = Z ∞

−∞

dy T(y, x)f(x)

Spectral ¡Theorem ¡

SPRING-­‑LENGHT ¡PROBABILITY ¡ DISTRIBUTIONS ¡

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

• 0.2
• 0.1

0.1 0.2

a)

1e-05 0.0001 0.001 0.01 0.1 1

• 0.2
• 0.1

0.1 0.2

b)

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

• 3
• 2
• 1

1 2 3

c)

1e-05 0.0001 0.001 0.01 0.1

• 3
• 2
• 1

1 2 3

d)

Points: ¡simulaNons ¡ Lines: ¡effecNve ¡theory ¡

P(ξ) e ∼ TEd e ∼ p TEd

PROBABILITY ¡DISTRIBUTION ¡OF ¡FORCE ¡

2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6

P(f) f

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 0.5 1 1.5 2 2.5 3 3.5 4

EffecNve ¡theory ¡

5 10 15 20 25 30 35 0.05 0.1 0.15 0.2

P(f) f

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6

P(f) f

EffecNve ¡theory ¡(lines) ¡VS ¡ simulaNons ¡(points) ¡

VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000

Edwards’ Measures for Powders and Glasses

Alain Barrat,1 Jorge Kurchan,2 Vittorio Loreto,3 and Mauro Sellitto4

1Laboratoire de Physique Théorique,* Bâtiment 210, Université de Paris-Sud, 91405 Orsay Cedex, France

Possible Test of the Thermodynamic Approach to Granular Media

David S. Dean and Alexandre Lefe `vre P H Y S I C A L R E V I E W L E T T E R S

week ending 16 MAY 2003

VOLUME 90, NUMBER 19

A statistical mechanics approach to the inherent states of granular media

Antonio Coniglioa;∗, Mario Nicodemia;b

a

Number ¡of ¡blocked ¡structures ¡in ¡fricNonal ¡ granular ¡assemblies ¡at ¡given ¡Volume ¡ Number ¡of ¡energy ¡minima ¡in ¡ models ¡of ¡glasses ¡at ¡given ¡Energy ¡

log[Nminima(E)] ∼ N log[Nblocked(V )] ∼ N

AMORPHOUS ¡PACKINGS ¡& ¡GLASSES ¡

HARMONIC ¡CHAIN ¡WITH ¡DRY ¡FRICTION ¡

• ­‑ ¡DEFINITION ¡OF ¡THE ¡MODEL ¡
• ­‑ ¡TAPPING ¡DYNAMICS ¡
• ­‑ ¡BLOCKED ¡CONFIGURATIONS ¡
• ­‑ ¡RELEVANT ¡OBSERVABLES ¡
• ­‑ ¡DEFINITION ¡OF ¡THE ¡EFFECTIVE ¡THEORY ¡
• ­‑ ¡PREDICTIONS ¡EFFECTIVE ¡THEORY ¡
• ­‑ ¡COMPARISON ¡BETWEEN ¡ ¡EFFECTIVE ¡THEORY ¡AND ¡

DRIVEN ¡ATHERMAL ¡DYNAMICS ¡

True Dynamics

Effective Thermodynamics

TEST ¡OF ¡EDWARDS ¡IN ¡ISING ¡MODEL ¡

2) ¡Quench ¡at ¡T=0: ¡only ¡spin ¡flips ¡which ¡lower ¡the ¡ energy ¡are ¡allowed ¡ 1) ¡HeaNng: ¡all ¡spins ¡are ¡flipped ¡with ¡probability ¡ ¡ HeaNng ¡ Quench ¡T=0 ¡ HeaNng ¡ Quench ¡T=0 ¡

E1 E2

TAPPING ¡DYNAMICS ¡

p p ∈ [0, 1/2[

Average ¡over ¡states ¡ colled ¡via ¡tapping ¡ dynamics ¡

‘’BLOCKED ¡CONFIGURATIONS’’ ¡

Energy ¡cannot ¡be ¡lowered ¡with ¡a ¡single ¡spin ¡flip ¡

hOiemp,E = N −1 X

i

O(Ci) δ(E Ei))

Edwards ¡measure ¡

hOiE = Z−1 X

{σ|σ∈blocked}

O(σ) e−βEdE(σ)

βEd =  ∂S ∂E

• blocked
• A. ¡Lefèvre ¡& ¡D. ¡Dean, ¡J. ¡Phys. ¡A: ¡Math. ¡Gen. ¡34 ¡(2001) ¡ ¡

TEST ¡OF ¡EDWARDS ¡IN ¡ISING ¡MODEL ¡

HeaNng ¡ Quench ¡T=0 ¡ HeaNng ¡ Quench ¡T=0 ¡

E1 E2

hOiE = Z−1 X

{σ|σ∈blocked}

O(σ) e−βEdE(σ) Z = X

σ

eβEd

P

i σiσi+1 Y

i

Θ(σi−1σi + σiσi+1)

x ≥ 0 → Θ(x) = 1 x < 0 → Θ(x) = 0

Edwards ¡measure ¡

βEd =  ∂S ∂E

• blocked

2) ¡Quench ¡at ¡T=0: ¡only ¡spin ¡flips ¡which ¡lower ¡the ¡ energy ¡are ¡allowed ¡ 1) ¡HeaNng: ¡all ¡spins ¡are ¡flipped ¡with ¡probability ¡ ¡

TAPPING ¡DYNAMICS ¡

p p ∈ [0, 1/2[

‘’BLOCKED ¡CONFIGURATIONS’’ ¡

Energy ¡cannot ¡be ¡lowered ¡with ¡a ¡single ¡spin ¡flip ¡

• A. ¡Lefèvre ¡& ¡D. ¡Dean, ¡J. ¡Phys. ¡A: ¡Math. ¡Gen. ¡34 ¡(2001) ¡ ¡

TEST ¡OF ¡EDWARDS ¡IN ¡ISING ¡MODEL ¡

HeaNng ¡ Quench ¡T=0 ¡ HeaNng ¡ Quench ¡T=0 ¡

E1 E2

2) ¡Quench ¡at ¡T=0: ¡only ¡spin ¡flips ¡which ¡lower ¡the ¡ energy ¡are ¡allowed ¡ 1) ¡HeaNng: ¡all ¡spins ¡are ¡flipped ¡with ¡probability ¡ ¡

TAPPING ¡DYNAMICS ¡

p p ∈ [0, 1/2[

‘’BLOCKED ¡CONFIGURATIONS’’ ¡

Energy ¡cannot ¡be ¡lowered ¡with ¡a ¡single ¡spin ¡flip ¡

• A. ¡Lefèvre ¡& ¡D. ¡Dean, ¡J. ¡Phys. ¡A: ¡Math. ¡Gen. ¡34 ¡(2001) ¡ ¡
• J. ¡Berg, ¡S. ¡Franz, ¡M. ¡Sellido, ¡EPJ ¡B ¡(2002) ¡ ¡
• ­‑ ¡Same ¡test ¡on ¡a ¡different ¡1D ¡spin ¡model ¡(Friedrickson-­‑Andersen) ¡
• ­‑ ¡Disagreement ¡between ¡dynamical ¡averages ¡and ¡Edwards ¡effec8ve ¡theory ¡

The ¡operator ¡(real, ¡symmetric ¡kernel) ¡has ¡an ¡orthonormal ¡basis ¡

lim

N→∞hξmξniEd =

X

b∈Sp(T )

✓ λb λmax ◆n−m

• Z ∞

−∞

dx xfb(x)fλmax (x)

• 2

T [fb](x) = λbfb(x) Z ∞

−∞

fb(x)fa(x) = δa,b

SPRING-­‑SPRING ¡CORRELATION ¡FUNCTION ¡

• 0.2

0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 30 35 40 45 50

C(i-j) i-j

Points ¡= ¡effecNve ¡thermodynamics ¡ Lines ¡= ¡tapping ¡dynamics ¡(ρ=0.3) ¡ ¡ Comparison ¡is ¡at ¡fixed ¡energy ¡

e

The ¡operator ¡(real, ¡symmetric ¡kernel) ¡has ¡an ¡orthonormal ¡basis ¡

T [fb](x) = λbfb(x) Z ∞

−∞

fb(x)fa(x) = δa,b

CORRELATION ¡FUNCTION ¡

0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 30 35

C(i-j) i-j

1e-05 0.0001 0.001 0.01 0.1 1 5 10 15 20 25 30 35 C(i-j) i-j

C(|n − m|) ∼ exp(−|n − m|/`(e)) lim

N→∞hξmξniEd =

X

b∈Sp(T )

✓ λb λmax ◆n−m

• Z ∞

−∞

dx xfb(x)fλmax (x)

• 2

0.1 1 10 100 0.01 0.1 1 10

• T

x1/2

` TEd

CORRELATION ¡FUNCTION ¡

0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 30 35

C(i-j) i-j

1e-05 0.0001 0.001 0.01 0.1 1 5 10 15 20 25 30 35 C(i-j) i-j

C(|n − m|) ∼ exp(−|n − m|/`(e))

0.1 1 10 100 0.01 0.1 1 10

• T

x1/2

` TEd

0.0001 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 10

e T

x x1/2

0.1 1 10 100 0.01 0.1 1 10

• T

x1/2

e TEd

CorrelaNons ¡appear ¡in ¡the ¡ ¡ ‘’out-­‑of-­‑equilibrium’’ ¡regime ¡

T [f](x) = Z ∞

−∞

dy T(y, x)f(x)

Z = Tr[T N]

Transfer ¡Operator ¡Formalism ¡ ¡ Z = Z dξ1 . . . dξN e−βEd

PN

i=1 ξ2 i 2

N

Y

i=1

Θ(µ − |ξi+1 − ξi|) ξi ξi+1 xi T(x, y) = e−βEd x2

4 Θ(µ − |x − y|)e−βEd y2 4

x ≥ 0 → Θ(x) = 1 x < 0 → Θ(x) = 0

EFFECTIVE THERMODYNAMICS ‘’Á LA EDWARDS’’

STATISTICAL ¡MECHANICS ¡OF ¡FRICTIONAL ¡ ATHERMAL ¡SYSTEMS ¡? ¡ ¡

Edwards ¡ ¡ Hypothesis: ¡All ¡packings ¡where ¡grains ¡

• ccupy ¡the ¡same ¡volume ¡are ¡

equiprobable ¡ ¡ AssumpNon: ¡change ¡of ¡configuraNon ¡due ¡to ¡ ‘’extensive ¡operaNons’’ ¡guarantees ¡‘’ergodicity’’ ¡

1 T = ∂S ∂E = ⇒ 1 X = ∂S ∂V

CompacNvity ¡ Grains ¡with ¡ fricNon ¡ DissipaNon ¡

e−βH

S.F ¡Edwards, ¡C.C. ¡Mounfield, ¡Physica ¡A ¡(210), ¡1994 ¡ ¡ ¡

S = log Z DC δ(V − W(C)) C = (x1, φ1, . . . , xN, φN)

Two-dimensional bulk driven granular fluid

• Inelastic binary collisions: sink of energy
• Random kicks: source of energy

m ˙ vi(t) = ξi(t) + Fcoll

i

hξµ

i (t)ξν j (t0)i = 2 Γ δµ,νδi,jδ(t t0)

v(r) v(r0) v(r) = 1 n(ε) X

i

vi Θ(ε − |ri − r|)

Coarse grained velocity field Quasi long-range order

« Randomly driven granular fluids: Large-scale structure »: TPC Van Noije, MH Ernst, E. Trizac, I. Pagonabarraga, PRE 59 (4), 4326 (1999)

vk(r) = v(r) · ˆ σr0,r

Granular Fluids: Non Equilibrium Stationary State (NESS)

Inelastic collisions

α < 1

vj = v0

j + (1 + α)

2 [(v0

i − v0 j) · ˆ

σ]ˆ σ vi = v0

i − (1 + α)

2 [(v0

i − v0 j) · ˆ

σ]ˆ σ vi vj ˆ σ v0

i

v0

j

Dissipation Injection Vibrating plate

Driven Dissipative Systems

Inelastic binary collisions

α < 1

vj = v0

j + (1 + α)

2 [(v0

i − v0 j) · ˆ

σ]ˆ σ vi = v0

i − (1 + α)

2 [(v0

i − v0 j) · ˆ

σ]ˆ σ vi vj ˆ σ v0

i

v0

j

• Energy il lost through inelastic collisions
• Energy is supplied by the vibrating vessel
• Non Equilbrium Stationary State (NESS)

Tg = 1 Nd X

i

mhv2

i i

2

Driven Dissipative Systems

Inelastic binary collisions

α < 1

vj = v0

j + (1 + α)

2 [(v0

i − v0 j) · ˆ

σ]ˆ σ vi = v0

i − (1 + α)

2 [(v0

i − v0 j) · ˆ

σ]ˆ σ

• Energy il lost through inelastic collisions
• Energy is supplied by the vibrating vessel
• Non Equilbrium Stationary State (NESS)

Tg = 1 Nd X

i

mhv2

i i

2

Tb

Energy supply mechanism: can be modeled with a thermostat?

m ˙ vi(t) = ξi(t) + Fcoll

i

hv?(r)v?(r0)i ⇠ hvk(r)vk(r0)i ⇠ Γ log ✓ L |r0 r| ◆ Theory & Simulations Experiments

« Forcing and Velocity Correlations in a Vibrated Granular Monolayer »

• A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 (2002)

Rough cexagonal plate (d=30cm) Steel spheres (d=3.97mm)

C⊥(r) ∼ e−r/ξ⊥ Ck(r) ∼ er/ξk

Two-dimensional bulk driven granular fluid: Finite-range correlations

m ˙ vi(t) = ξi(t) + Fcoll

i

hξµ

i (t)ξν j (t0)i = 2 Γ δµ,νδi,jδ(t t0)

Coarse grained field Quasi Long-range order

« Randomly driven granular fluids: Large-scale structure » TPC Van Noije, MH Ernst, E. Trizac, I. Pagonabarraga, PRE 59 (4), 4326 (1999)

vk(r) = v(r) · ˆ σr0,r v⊥(r) = v(r) − v(r) · ˆ σr0,r

hv?(r)v?(r0)i ⇠ hvk(r)vk(r0)i ⇠ Γ log ✓ L |r0 r| ◆

Two-dimensional bulk driven granular fluid: Scale-free correlations

Energy Sink Energy Gain White noise

Driven Dissipative Systems

Two-dimensional granular gas/fluid Amorphous packing

• f frictional grains

Teff Tg

Existence = ? Relation with physical

• bservables = ?

Relation with physical

• bservables = ?

Tb

Energy supply mechanism = ?

Summary: different energy injection mechanisms

m ˙ vi(t) = −γbvi(t) + ξi(t) + Fcoll

i

Viscous drag + random kicks

Two temperatures: Exponential decay of correlations:

∆T(φ) = Tb − Tg(φ) C?,k(|r − r0|) ∼ ∆T(φ) e|rr0|/ξ?,k

Correlations grows as distance from equilibrium: ∆T(φ) λ0(φ) λ0(∆T)

ξ?,k/λ0(∆T) % ∆T %

Hot Thermostat

Tb > 0 Tg > 0

System Cold Thermostat (inelastic collisions)

T0 = 0

Statistical Mechanics for Frictional Grains Packings?

|Ft| < µs |Fn|

Mechanically Stable Configurations

Amorphous packing

• f frictional grains

(1) (2)

F(12)

t

F(12)

n

F(21)

n

F(21)

t

Repeated shaking of the glass ball (above)

Energy injection through vertical vibration of the box (tapping)

C1 C2

Dissipation Mechanical Stability Mechanical Stability

Statistical Mechanics of Mechanically Stable Configurations ?

Volume Energy

P(C) ∼ e−V (C)/Teff