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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
Introduction Motivations
Motivations
Boltzmann-Gibbs thermodynamics
P(C) = e−βH(C)
Can’t be used in any situation!
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
Introduction Motivations
Motivations
Boltzmann-Gibbs thermodynamics
P(C) = e−βH(C)
Can’t be used in any situation!
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
Introduction Motivations
Motivations
Boltzmann-Gibbs thermodynamics
P(C) = e−βH(C)
Can’t be used in any situation!
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
Introduction Motivations
Motivations
Boltzmann-Gibbs thermodynamics
P(C) = e−βH(C)
Can’t be used in any situation!
Before reaching equilibriumtransient regime“glassy” dynamics
Non-equilibrium phenomenasystems with a current (of particles, energy)dissipative dynamics (granular media)
→ Need for a dynamical description
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 2 / 21
Introduction Outline
Outline
1 Motivationsstatics vs dynamics
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 3 / 21
Introduction Outline
Outline
1 Motivationsstatics vs dynamics
2 Models of glass formersexamplethermodynamics of historiesresultsdynamical phase coexistence
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 3 / 21
Introduction Outline
Outline
1 Motivationsstatics vs dynamics
2 Models of glass formersexamplethermodynamics of historiesresultsdynamical phase coexistence
3 Perspective
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 3 / 21
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 4 / 21
Models of glass formers Thermodynamics of histories
From configurations to histories
fluctuations of configurations
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 5 / 21
Models of glass formers Thermodynamics of histories
From configurations to histories
fluctuations of configurations
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 5 / 21
Models of glass formers Thermodynamics of histories
From configurations to histories
fluctuations of configurations
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 5 / 21
Models of glass formers Thermodynamics of histories
From configurations to histories
�����������
����� ���
� � ���
fluctuations of histories
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 5 / 21
Models of glass formers Thermodynamics of histories
From configurations to histories
������������������ ���
� !�"
fluctuations of histories
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 5 / 21
Models of glass formers Thermodynamics of histories
From configurations to histories
#�$�%�&�'�(�)*�+�, $�%
- , .�/
fluctuations of histories
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 5 / 21
Models of glass formers Example
Example
0 1 2
Independent particles
L sites n = {ni} with
{
ni = 0 inactive siteni = 1 active site
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 6 / 21
Models of glass formers Example
Example
3 4 5
Independent particles
L sites n = {ni} with
{
ni = 0 inactive siteni = 1 active site
Transition rates in each site:activation with rate W (0i → 1i) = cinactivation with rate W (1i → 0i ) = 1− c
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 6 / 21
Models of glass formers Example
Example
6 7 8
Independent particles
L sites n = {ni} with
{
ni = 0 inactive siteni = 1 active site
Transition rates in each site:activation with rate W (0i → 1i) = cinactivation with rate W (1i → 0i ) = 1− c
Equilibrium distribution: Peq(n) =∏
i
cni (1− c)1−ni
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 6 / 21
Models of glass formers Example
Example
9 : ;
Independent particles
L sites n = {ni} with
{
ni = 0 inactive siteni = 1 active site
Transition rates in each site:activation with rate W (0i → 1i) = cinactivation with rate W (1i → 0i ) = 1− c
Equilibrium distribution: Peq(n) =∏
i
cni (1− c)1−ni
Mean density of active sites: 〈n〉 =1L
∑
i
〈ni〉 = c
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 6 / 21
Models of glass formers Kinetically constrained models (KCM)
Kinetically constrained models (KCM)
Constrained dynamics: changes occur only around active sites.
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 7 / 21
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
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 7 / 21
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: 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:
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 7 / 21
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: 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
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 7 / 21
Models of glass formers Kinetically constrained models (KCM)
Trajectories
Peq(n) insufficient→ analysis of histories
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 8 / 21
Models of glass formers Kinetically constrained models (KCM)
Trajectories
Peq(n) insufficient→ analysis of histories
n(t)
t0t1 t2 t3
n(1)
n(2)
n(3)n(t)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 8 / 21
Models of glass formers Kinetically constrained models (KCM)
Trajectories
Peq(n) insufficient→ analysis of histories
n(t)
t0t1 t2 t3
n(1)
n(2)
n(3)n(t)
activity K = number of configuration changes
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 8 / 21
Models of glass formers Kinetically constrained models (KCM)
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 9 / 21
Models of glass formers Kinetically constrained models (KCM)
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 is fluctuating
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 9 / 21
Models of glass formers Kinetically constrained models (KCM)
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 is fluctuating
ρ(K ) =
∣∣∣∣∣
time-averaged density of active sitesfor histories of activity K
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 9 / 21
Models of glass formers Kinetically constrained models (KCM)
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 is fluctuating
ρ(K ) =
∣∣∣∣∣
time-averaged density of active sitesfor histories of activity K
ρ(K ) gives information on non-typical histories
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 9 / 21
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 10 / 21
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)
Thermodynamics of histories [à la Ruelle]
Ωdyn(K , t) =
∣∣∣∣∣
number of historieswith activity K between 0 et t
(grand t)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 10 / 21
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)
Thermodynamics of histories [à la Ruelle]
Ωdyn(K , t) =
∣∣∣∣∣
number of historieswith activity K between 0 et t
(grand t)
micro-canonical ensemble: technically painful
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 10 / 21
Models of glass formers Dynamical partition function
Dynamical partition function
Fluctuations (2): the canonical wayThermodynamics of configurations
Z (β, L) =∑
E
Ω(E , L) e−β E(n) = e−L f (β) (large L)
Thermodynamics of histories [à la Ruelle]
ZK(s, t) =∑
K
Ωdyn(K , t) e−s K [hist] = e−t L fK(s) (large tlarge L)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 11 / 21
Models of glass formers Dynamical partition function
Dynamical partition function
Fluctuations (2): the canonical wayThermodynamics of configurations
Z (β, L) =∑
E
Ω(E , L) e−β E(n) = e−L f (β) (large L)
Thermodynamics of histories [à la Ruelle]
ZK(s, t) =∑
K
Ωdyn(K , t) e−s K [hist] = e−t L fK(s) (large tlarge L)
(Dynamical) canonical ensembleβ conjugated to energys conjugated to activity K
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 11 / 21
Models of glass formers Dynamical partition function
Dynamical partition function
Fluctuations (2): the canonical wayThermodynamics of configurations
Z (β, L) =∑
E
Ω(E , L) e−β E(n) = e−L f (β) (large L)
Thermodynamics of histories [à la Ruelle]
ZK(s, t) =∑
K
Ωdyn(K , t) e−s K [hist] = e−t L fK(s) (large tlarge L)
Fluctuations of K
〈e−sK 〉 ∼ e−t L fK (s)∣∣∣∣∣
Dynamical free energy fK (s) ↔cumulant generating function of K .
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 11 / 21
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
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 12 / 21
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 observableinterpretation
s < 0 : more active histories (“large” K )s = 0 : equilibrium states > 0 : less active histories (“small” K )
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 12 / 21
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 observableinterpretation
s < 0 : more active histories (“large” K )s = 0 : equilibrium states > 0 : less active histories (“small” K )
Ex: time-averaged density ρ[hist] = 1Lt∫ t
0 dτ∑
i ni(τ)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 12 / 21
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 observableinterpretation
s < 0 : more active histories (“large” K )s = 0 : equilibrium states > 0 : less active histories (“small” K )
Ex: time-averaged density ρ[hist] = 1Lt∫ t
0 dτ∑
i ni(τ)
ρK (s) =〈ρ[hist] e−sK
〉
〈e−sK 〉
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 12 / 21
Models of glass formers Outline
Outline
1 Motivationsstatics vs dynamics
2 Models of glass formersexamplethermodynamics of histories
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 13 / 21
Models of glass formers Outline
Outline
1 Motivationsstatics vs dynamics
2 Models of glass formersexamplethermodynamics of historiesresultsdynamical phase coexistence
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 13 / 21
Models of glass formers Outline
Outline
1 Motivationsstatics vs dynamics
2 Models of glass formersexamplethermodynamics of historiesresultsdynamical phase coexistence
3 Perspective
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 13 / 21
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
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 14 / 21
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)
m�n_oapcqergsit�oaulv
w o\m�n_oapcqergsit�oaulv
Comparison between constrained and unconstrained dynamics
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 14 / 21
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)
L = 100L = 50
L = 200
x�y_za{c|e}g~i�zal
z\x�y_za{c|e}g~i�zal
Comparison between constrained and unconstrained dynamics
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 14 / 21
Models of glass formers Results
Dynamical phase transition: FA model (d=1)
-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
�� �\�
Comparison between constrained and unconstrained dynamics
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 15 / 21
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 16 / 21
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−sK δ(〈n〉 − Lρ
)〉∣∣∣ ∼ e−t L F(ρ,s)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 16 / 21
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−sK δ(〈n〉 − Lρ
)〉∣∣∣ ∼ e−t L F(ρ,s)
Extremalization procedure
fK(s) = minρF(ρ, s)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 16 / 21
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−sK δ(〈n〉 − Lρ
)〉∣∣∣ ∼ 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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 16 / 21
Models of glass formers Dynamical Landau free energy
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 17 / 21
Models of glass formers Dynamical Landau free energy
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)
©ª«¬ª®¯°±ª²³
«¬ª®¯�°±ª²³
Prob[0,t](ρ) ∼ e−t L F(ρ,s)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 17 / 21
Models of glass formers Dynamical Landau free energy
Dynamical Landau free-energy landscape
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 17 / 21
Models of glass formers Dynamical Landau free energy
Dynamical Landau free-energy landscape
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�úÅÄÆgÂ
ÃÇÄÂ�¶aÈÉc¶�Êg¹ÅÂ˶Ìȶ
Prob[0,t](ρ) ∼ e−t L F(ρ,s)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 17 / 21
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
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 18 / 21
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)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 19 / 21
Perspective Summary
Perspective
PerspectiveAnomalous relaxation in the FA model.Link with the (χ4 related) growing lenght.
ReferencesPRL 95 010601 (2005)J. Stat. Phys. 127 51 (2007)PRL 98 195702 (2007)JSTAT P03004 (2007)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 19 / 21
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)
ReferencesPRL 95 010601 (2005)J. Stat. Phys. 127 51 (2007)PRL 98 195702 (2007)JSTAT P03004 (2007)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 19 / 21
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
ReferencesPRL 95 010601 (2005)J. Stat. Phys. 127 51 (2007)PRL 98 195702 (2007)JSTAT P03004 (2007)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 19 / 21
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
}
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 20 / 21
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)}
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 20 / 21
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)}
Canonical description: s conjugated to K
P(C, s, t) =∑
K
e−sK P(C, K , t)
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 20 / 21
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)}
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)}
V. Lecomte (MSC-Paris 7, LPT-Orsay) Thermodynamics of histories 17/5/2007 20 / 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
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
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