statistical physics methods applied to particulate matter

Statistical physics methods applied to particulatematter

Jorge Kurchan

LPS-ENS, Paris

PASI

J. Kurchan (LPS-ENS) 1 / 53

1. Statistical Mechanics:

Many particles: → a statistical approach is called for

The miracle of Liouville equation: flat measure is leftinvariant by dynamics.

No such things here, so equilibrium Stat Mech does notapply


Frictionless and un-forced

qi = pi

m

pi = −∂H∂qi

E=const

The motion is restricted to the energy shell, and a flatdistribution is conserved


If we wish to ‘stir’ or ‘shake’ the system, we need a thermostat inorder to have a stationary state

Second Principle!


Gaussian (Hoover) thermostat

E=const

The motion is restricted to the energy shell


Gaussian thermostat: deterministic

qi = pi

m

pi = −∂H∂qi

+ γ(t)pi︸︷︷︸thermostat

− fi(q)︸︷︷︸forcing

energy is conserved provided γ(t) = f•pp2 .

However, under forcing, the distribution is not flat, ergodicity isnot guaranteed!


Gaussian thermostat: noisy

qi = pi

m

pi = −∂H∂qi

+ γ(t)pi︸︷︷︸thermostat

− fi(q)︸︷︷︸forcing

− η‖i︸︷︷︸

conservativenoise

η‖i is a noise tangential to the energy surface

again, energy is conserved provided γ(t) = f•pp2 .


A chaotic system without forcing

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Forcing, thermostat

E

A A

__


A chaotic system with strong enough forcing

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Stochastic thermal bath: Langevin dynamics

qi = ∂H

∂pi= pi

m = vi

pi = −∂H∂qi− fi(q)︸︷︷︸Forcing

+ ηi(t)− γpi︸︷︷︸Thermal Bath

〈ηi(t)ηj(t′)〉 = 2γTδijδ(t− t′)

the stationary distribution is the Gibbs-Boltzmann one


Generalized Langevin equation.

q = −∂H∂q

+ η(t) +

∫ t

0dt′ R(t, t′)q′(t′)

〈η(t)η(t′)〉 = C(t, t′)

Fluctuation-dissipation:

TR(t− t′) = ∂

∂t′C(t− t′) TR(ω) = −ωC(ω)


If, for example, T = T (ω) is a growing function

we are injecting heat in the high frequencies andretrieving it in the low ones, an out of equilibrium

situation


... and of course we could inject heat at a singlefrequency (vibrating) and retrieve it in another

timescale

the important point is that a generic bath isnon-equipibrium, it does not guarantee an

equilibrium (or any known) distribution.


2. Glasses


Crystallisation

T

E

melting

crystal

liquid

E versus T .J. Kurchan (LPS-ENS) 16 / 53

Glassy solid:

T

T

E

τα

liquid

liquid

T TTk 1 2

aging

aging


(less and less) transient density profiles

ρ (x)

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t−t

density autocorrelation (t−t

w

w)

averaged over time τα

τα


{

{

ρ (x) averaged over time τα

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a

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If τα =∞ we have true states


correlations and responses

A quantity:

ρk(t) =

∫dx cos(kx)ρ(x, t)

The correlation of its fluctuations:

Ck(t, tw) = 〈ρk(t)ρk(tw)〉

The response at time t to a conjugate field hρk acting from time−∞ to tw

χ(t, tw) =δ〈ρk(t)〉δh


The α scale, in and out of equilibrium

T2

T

T

Tk 1

T3

T

τα

liquidaging

C C

T2

1 1hr 10hr

ατ

ατ (T) w(t )


{

{

ρ (x) averaged over time τα

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free energy

don’t ask

a,b,c d,e,g

landscape


Glasses in this non-ideal world age

they are continuously evolving into more compactconfigurations

or, alternatively, they are continuously nucleating stabler andstabler phases

evolution becomes slower as time passes


Aging in the correlations

T = 0:3

tC k(t)

10410310210110010�1

10.80.60.40.20Autocorrelation of density fluctuations (Lennard-Jones system, Kob-Barrat)


Aging in a response function

the stretching of a plastic bar, from an hour to four years old (Struik)


In equilibrium one should expect χ and C to be related:

C

χ


indeed:

Tχ(t, tw) = C(t, t)− C(t, tw) or T∂χ(t,t′)∂t′ = − ∂C(t,t′)

∂t′

wC(t,t )

(t,t )wχ


the fluctuation-dissipation theorem says something importantabout thermalisation:

E∗ = E + xρk +p2x2

+ω2x2

2

system

noise

response=dissipation


the fluctuation-dissipation theorem says something importantabout thermalisation:

E∗ = E + xρk +p2x2

+ω2x2

2

x = −ωx− ρk but ρk = [ρk]o︸︷︷︸bare

+

∫ t

∞dt

′ ∂χ(t, t′)

∂t′x(t

′)︸︷︷︸

back reaction

————————————

x = −ωx+ [ρk]o +∫ t∞ dt′ ∂χ(t,t

′)∂t′ x(t′)

————————————

... an oscillator in a good thermal bath of temperature Tprovided that 〈[ρk]o(t)[ρk]o(t′)〉 = −T ∂χ(t,t′)

∂t′


A concrete example for a magnetic system:

cf. Grigera Israeloff


Aging in the correlations and the response in solvable models

T2

TT

T2

ατ (T)

1

χ

T2

T

T

C

1

ατ (T)

ατ w(t )

C

1hr 10hr

ατ w(t )

1hr10hr

χ

χ

χ

C

C

1

constant temperature T

Teff


This, in turn, led to the search of effectivetemperatures in realistic models

all the temperatures in a given timescale should coincide!

FDTSelf BSelf Ak = 7:47

CorrelationSus eptibility

10.80.60.40.20

21.510.50Binary Lennard-Jones glass, simulation L. Berthier and JL Barrat


Rheology and shear thinning:the other face of aging

shear-thinning: stirring a liquid makes τα smaller, and kills aging

γ.


but surprisingly, effective temperatures are still there (for small γ)

T

χ

C

Teff

constant temperature T

ατ

C

ατ

χ

( ) ( )γ γ

.

. .


iso-τα and iso-Teff lines

Drive

Tk Tg Tc

Near Equilibrium

T


flat exploration of states (why?)

ENERGY

equilibrium inside a state

flat (metastable) distribution of states



f

Icrystal

Σ

?

1/T

gradient

gradient

1/Teff


Amazingly, one has ’laws’ for the slow out-ofequilibrium regime, quite independent of what

the system will do at infinite times

This is a new way of thinking, and it is not wellestablished yet


3. Back to dense granular matter


Granular compaction under gentle vibrations =aging

they are continuously evolving into more compactconfigurations

or, alternatively, they are continuously nucleating stabler andstabler phases

evolution becomes slower as time passes


Aging in the correlations

T = 0:3

tC k(t)

10410310210110010�1

10.80.60.40.20Autocorrelation of density fluctuations (Lennard-Jones system, Kob-Barrat)


Non-equilibrium one should NOT expect χ and C to have asimple relation

C

χ


indeed:

Tχ(t, tw)↔ C(t, t)− C(t, tw) or ∂χ(t,t′)∂t′ ↔ ∂C(t,t′)

∂t′

χ

wC(t,t )

(t,t )w


Aging in the correlations and the response in solvable models, with anon thermal bath

nothing special !

ατ w(t )

C

1hr 10hr

ατ w(t )

1hr10hr

χ χ

C

C

1

Teff

constant weak shaking

T


Rheology and shear thinning:the other face of aging

shear-thinning: stirring a granular ensemble makes τα smaller,and kills compaction

γ.


but again surprisingly, effective temperatures are still there (for

small γ)



NONEQUILIBRIUM inside a state

flat (metastable) distribution of statesENERGY



f

Icrystal

Σ

?

1/T

gradient

gradient

1/Teff


Amazingly, this is essentially Edwards’ idea, nowvalidated within solvable models

cfr. Karen’s talk.


statistical physics methods applied to particulate matter

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