Thermodynamics of histories an application to systems with … · 2016. 9. 2. · Thermodynamics of histories an application to systems with glassy dynamics Vivien Lecomte1;2, CØcile

Thermodynamics of historiesan application to systems with glassy dynamics

Vivien Lecomte1,2, Cécile Appert-Rolland1, Estelle Pitard3,

Kristina van Duijvendijk1,2, Frédéric van Wijland1,2

Juan P. Garrahan4, Robert L. Jack5,6

1Laboratoire de Physique Théorique, Université d’Orsay2Laboratoire Matière et Systèmes Complexes, Université Paris 7

3Laboratoire Colloïdes, Verres et Nanomatériaux, Université Montpellier 24School of physics and astronomy, University of Nottingham

5Rudolf Peierls Centre for Theoretical Physics, University of Oxford6Department of Chemistry, University of California

ISC – Roma – 17th May 2007V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 1 / 21

Introduction Motivations

Motivations

Boltzmann-Gibbs thermodynamics

P(C) = e−βH(C)

Before reaching equilibriumtransient regime“glassy” dynamics

Non-equilibrium phenomenasystems with a current (of particles, energy)dissipative dynamics (granular media)

V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 2 / 21


Motivations


P(C) = e−βH(C)

Can’t be used in any situation!





Motivations


P(C) = e−βH(C)

Can’t be used in any situation!



→ Need for a dynamical description


Introduction Outline

Outline

1 Motivationsstatics vs dynamics



Outline


2 Models of glass formersexamplethermodynamics of historiesresultsdynamical phase coexistence



Outline



3 Perspective


Models of glass formers

Glasses

T

E

Tm

superc

ooled

liquid

liquid

crystal

glass

Fig.1 in Ritort and Sollich, Adv. in Phys. 52 219 (2003)


Models of glass formers Thermodynamics of histories

From configurations to histories

fluctuations of configurations




��

��

� � ��

fluctuations of histories




��

� !�"





#�$�%�&�'�(�)*�+�, $�%

- , .�/



Models of glass formers Example

Example

0 1 2

Independent particles

L sites n = {ni} with

{

ni = 0 inactive siteni = 1 active site



Example

3 4 5



{


Transition rates in each site:activation with rate W (0i → 1i) = cinactivation with rate W (1i → 0i ) = 1− c



Example

6 7 8



{



Equilibrium distribution: Peq(n) =∏

i

cni (1− c)1−ni



Example

9 : ;



{



Equilibrium distribution: Peq(n) =∏

i

cni (1− c)1−ni

Mean density of active sites: 〈n〉 =1L

∑

i

〈ni〉 = c


Models of glass formers Kinetically constrained models (KCM)

Kinetically constrained models (KCM)

Constrained dynamics: changes occur only around active sites.





Fredrickson Andersen model in 1Dat least one neighbor of i must be active to allow i to change

inactivation: < = >1− c 1− c

activation: ? @ AB C D Ec c c c






inactivation: F G H1− c 1− c

activation: I J KL M N Oc c c c

same equilibrium distribution Peq(n)same mean density 〈n〉 = c

BUT:






inactivation: P Q R1− c 1− c

activation: S T UV W X Yc c c c

same equilibrium distribution Peq(n)same mean density 〈n〉 = c

BUT:AGING



Trajectories

Peq(n) insufficient→ analysis of histories



Trajectories


n(t)

t0t1 t2 t3

n(1)

n(2)

n(3)n(t)



Trajectories


n(t)

t0t1 t2 t3

n(1)

n(2)

n(3)n(t)

activity K = number of configuration changes



Statistics over histories

Classification of histories according to a time extensive parameter

activity K = number of configuration changeson average:

〈K 〉 = 2c2(1− c) L t (large t)






〈K 〉 = 2c2(1− c) L t (large t)

K is fluctuating






〈K 〉 = 2c2(1− c) L t (large t)

K is fluctuating

ρ(K ) =

∣∣∣∣∣

time-averaged density of active sitesfor histories of activity K






〈K 〉 = 2c2(1− c) L t (large t)

K is fluctuating

ρ(K ) =

∣∣∣∣∣

time-averaged density of active sitesfor histories of activity K

ρ(K ) gives information on non-typical histories


Models of glass formers Dynamical partition function

Dynamical partition function

Fluctuations (1): the micro-canonical wayThermodynamics of configurations

Ω(E , L) =

∣∣∣∣∣

number of configurationswith energy E

(large L)





Ω(E , L) =

∣∣∣∣∣


(large L)

Thermodynamics of histories [à la Ruelle]

Ωdyn(K , t) =

∣∣∣∣∣

number of historieswith activity K between 0 et t

(grand t)





Ω(E , L) =

∣∣∣∣∣


(large L)


Ωdyn(K , t) =

∣∣∣∣∣

number of historieswith activity K between 0 et t

(grand t)

micro-canonical ensemble: technically painful




Fluctuations (2): the canonical wayThermodynamics of configurations

Z (β, L) =∑

E

Ω(E , L) e−β E(n) = e−L f (β) (large L)


ZK(s, t) =∑

K

Ωdyn(K , t) e−s K [hist] = e−t L fK(s) (large tlarge L)





Z (β, L) =∑

E



ZK(s, t) =∑

K


(Dynamical) canonical ensembleβ conjugated to energys conjugated to activity K





Z (β, L) =∑

E



ZK(s, t) =∑

K


Fluctuations of K

〈e−sK 〉 ∼ e−t L fK (s)∣∣∣∣∣

Dynamical free energy fK (s) ↔cumulant generating function of K .


Models of glass formers s-state

s-state

s probes dynamical featuresaverage in the s-state of an observable O:

O(s) =〈O[hist] e−sK

〉

〈e−sK 〉

O[hist] is (for instance) an history-dependent observable



s-state



〉

〈e−sK 〉

O[hist] is (for instance) an history-dependent observableinterpretation

s < 0 : more active histories (“large” K )s = 0 : equilibrium states > 0 : less active histories (“small” K )



s-state



〉

〈e−sK 〉



Ex: time-averaged density ρ[hist] = 1Lt∫ t

0 dτ∑

i ni(τ)



s-state



〉

〈e−sK 〉



Ex: time-averaged density ρ[hist] = 1Lt∫ t

0 dτ∑

i ni(τ)

ρK (s) =〈ρ[hist] e−sK

〉

〈e−sK 〉


Models of glass formers Outline

Outline


2 Models of glass formersexamplethermodynamics of histories



Outline





Outline



3 Perspective


Models of glass formers Results

Dynamical phase transition: FA model (d=1)

-0.02 -0.01 0.01 0.02 0.03 0.04

-0.06

-0.04

-0.02

0.02

0.04

s

−fK(s)

Z\[\]�^_[a`cbedgfih�[ajlk




-0.02 -0.01 0.01 0.02 0.03 0.04

-0.06

-0.04

-0.02

0.02

0.04

s

−fK(s)

m�n_oapcqergsit�oaulv

w o\m�n_oapcqergsit�oaulv

Comparison between constrained and unconstrained dynamics




-0.02 -0.01 0.01 0.02 0.03 0.04

-0.06

-0.04

-0.02

0.02

0.04

s

−fK(s)

L = 100L = 50

L = 200

x�y_za{c|e}g~i�zal

z\x�y_za{c|e}g~i�zal





-1 1 2 3 4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

s

ρK(s)

ρK(0) = c

�g

g�g

�� \�



Models of glass formers Dynamical Landau free energy

Dynamical Landau free energy F(ρ, s)

Probability, in the s-state, to measurea time-averaged density ρ btw. 0 and t

∣∣∣∣∣∼ e−t L F(ρ,s)





∣∣∣∣∣∼ e−t L F(ρ,s)

〈

e−sK δ(〈n〉 − Lρ

)〉∣∣∣ ∼ e−t L F(ρ,s)





∣∣∣∣∣∼ e−t L F(ρ,s)

〈


)〉∣∣∣ ∼ e−t L F(ρ,s)

Extremalization procedure

fK(s) = minρF(ρ, s)





∣∣∣∣∣∼ e−t L F(ρ,s)

〈


)〉∣∣∣ ∼ e−t L F(ρ,s)

Extremalization procedure

fK(s) = minρF(ρ, s)

Minimum reached at ρ = ρK (s):

fK(s) = F(ρK (s), s)



Dynamical Landau free-energy landscape

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

ρ

F(ρ, s = 0)

¡¢£¤¥¦§¨

Prob[0,t](ρ) ∼ e−t L F(ρ,s)




0.2 0.4 0.6 0.8

-0.05

0.05

0.1

0.15

0.2

F(ρ, s)

ρ

s < 0

s = 0

s > 0

s > s ´ µ

fK(s) = minρF(ρ, s) = F(ρK(s), s)




0.2 0.4 0.6 0.8

-0.05

0.05

0.1

0.15

0.2

ρ

s = 0

F(ρ, s = 0)

¶·c¸¹º¹»¼�¹¸½¿¾s = 0 ÀÁ ¹Â�ÃºÅÄÆgÂ

ÃÇÄÂ�¶aÈÉc¶�Êg¹ÅÂË¶ÌÈ¶



Perspective Summary

Thermodynamics of histories: summary

Resultss-states ≡ tools to study dynamicsDescription of dynamical phase coexistence→ Dynamical Landau free energy F(ρ, s)←

Dynamical phase transitionCriticality in a (dynamical) “hidden” dimensionDynamical heterogeneity


Perspective Summary

Perspective

PerspectiveAnomalous relaxation in the FA model.

ReferencesPRL 95 010601 (2005)J. Stat. Phys. 127 51 (2007)PRL 98 195702 (2007)JSTAT P03004 (2007)


Perspective Summary

Perspective

PerspectiveAnomalous relaxation in the FA model.Link with the (χ4 related) growing lenght.



Perspective Summary

Perspective

PerspectiveAnomalous relaxation in the FA model.Link with the (χ4 related) growing lenght.Other glassy systems

p-spin modelsstructural glasses (Lennard-Jones)



Perspective Summary

Perspective

PerspectiveAnomalous relaxation in the FA model.Link with the (χ4 related) growing lenght.Other glassy systems

p-spin modelsstructural glasses (Lennard-Jones)

Experimental realisation of s(' 0)-states



Perspective Summary

s-modified dynamics

Markov processes:

∂t P(C, t) =∑

C′

{

W (C′ → C)P(C′, t)︸︷︷︸

gain term

−W (C → C′)P(C, t)︸︷︷︸

loss term

}


Perspective Summary

s-modified dynamics

Markov processes:

∂t P(C, t) =∑

C′

{

W (C′ → C)P(C′, t)︸︷︷︸

gain term

−W (C → C′)P(C, t)︸︷︷︸

loss term

}

More detailed dynamics for P(C, K , t):

∂t P(C, K , t) =∑

C′

{

W (C′ → C)P(C′, K−1, t)−W (C → C ′)P(C, K , t)}


Perspective Summary

s-modified dynamics

Markov processes:

∂t P(C, t) =∑

C′

{

W (C′ → C)P(C′, t)︸︷︷︸

gain term

−W (C → C′)P(C, t)︸︷︷︸

loss term

}


∂t P(C, K , t) =∑

C′

{

W (C′ → C)P(C′, K−1, t)−W (C → C ′)P(C, K , t)}

Canonical description: s conjugated to K

P(C, s, t) =∑

K

e−sK P(C, K , t)


Perspective Summary

s-modified dynamics

Markov processes:

∂t P(C, t) =∑

C′

{

W (C′ → C)P(C′, t)︸︷︷︸

gain term

−W (C → C′)P(C, t)︸︷︷︸

loss term

}


∂t P(C, K , t) =∑

C′

{

W (C′ → C)P(C′, K−1, t)−W (C → C ′)P(C, K , t)}

Canonical description: s conjugated to K

P(C, s, t) =∑

K

e−sK P(C, K , t)

s-modified dynamics [probability non-conserving]

∂t P(C, s, t) =∑

C′

{

e−sK W (C′ → C)P(C′, s, t)−W (C → C ′)P(C, s, t)}


Perspective Summary

Numerical method (with J. Tailleur)

Evaluation of large deviation functions

Z (s, t) =∑

C

P(C, s, t) =〈

e−s K〉

∼ e−t fK (s)

discrete time: Giardinà, Kurchan, Peliti [PRL 96, 120603 (2006)]

continuous time: Lecomte, Tailleur [JSTAT P03004 (2007)]

Cloning dynamics

∂tP(C, s) =∑

C′

Ws(C′ → C)P(C′, s)− rs(C)P(C, s)

︸︷︷︸

modified dynamics

+ δrs(C)P(C, s)

︸︷︷︸

cloning term

Ws(C′ → C) = e−sW (C′ → C)rs(C) =

∑

C′ 6=C Ws(C → C′)

δrs(C) = rs(C)− r(C)V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 21 / 21

Perspective Summary

Numerical method (with J. Tailleur)

Evaluation of large deviation functions

Z (s, t) =∑

C

P(C, s, t) =〈

e−s K〉

∼ e−t fK (s)

discrete time: Giardinà, Kurchan, Peliti [PRL 96, 120603 (2006)]

continuous time: Lecomte, Tailleur [JSTAT P03004 (2007)]

Cloning dynamics

∂tP(C, s) =∑

C′

Ws(C′ → C)P(C′, s)− rs(C)P(C, s)

︸︷︷︸

modified dynamics

+ δrs(C)P(C, s)

︸︷︷︸

cloning term

Ws(C′ → C) = e−sW (C′ → C)rs(C) =

∑

C′ 6=C Ws(C → C′)

δrs(C) = rs(C)− r(C)V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 21 / 21

IntroductionMotivationsOutline

Models of glass formersThermodynamics of historiesExampleKinetically constrained models (KCM)Dynamical partition functionDynamical partition functions-stateOutlineResultsDynamical Landau free energy

PerspectiveSummary

Documents

Thermodynamics of histories an application to systems with … · 2016. 9. 2. · Thermodynamics of histories an application to systems with glassy dynamics Vivien Lecomte1;2, CØcile