Finite size scaling of the dynamical free-energy in the interfacial regime of a kinetically constrained model

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Finite size scaling of the dynamical free-energy in the interfacial regime of a kinetically constrained model Thierry Bodineau 1, Vivien Lecomte 2, Cristina Toninelli

1 Finite size scaling of the dynamical free-energy in the interfacial regime of a kinetically constrained model Thierry Bodineau 1, Vivien Lecomte 2, Cristina Toninelli 2 1 DMA, ENS, Paris, France 2 LPMA, Universités Paris VI & Paris VII LPS ENS 18th April 2012 V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

2 Introduction Motivations Dynamical excitations in glass-forming liquids From: Keys et. al PRX (2011) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

3 Introduction Motivations Dynamical excitations in glass-forming liquids Can we model this simply? From: Keys et. al PRX (2011) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

4 Example 0: Introduction Motivations (in 1D for simplicity) Independent sites { ni = 0 unexcited site L sites n = {n i } with n i = 1 excited site V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

5 Example 0: Introduction Motivations (in 1D for simplicity) Independent sites L sites n = {n i } with { ni = 0 unexcited site n i = 1 excited site Transition rates in each site: excitation with rate W(0 i 1 i ) = c unexcitation with rate W(1 i 0 i ) = 1 c V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

6 Example 0: Introduction Motivations (in 1D for simplicity) Independent sites L sites n = {n i } with { ni = 0 unexcited site n i = 1 excited site Transition rates in each site: excitation with rate W(0 i 1 i ) = c unexcitation with rate W(1 i 0 i ) = 1 c Equilibrium distribution: P eq (n) = c n i (1 c) 1 n i i Mean density of excited sites: n = 1 n i = c L i Unconstrained model V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

7 Introduction Kinetically constrained models (KCM) Kinetically constrained models (KCM) Constrained dynamics: changes occur only around excited sites. Fredrickson Andersen model in 1D at least one neighbor of i must be excited to allow i to change unexcitation: excitation: 1 c 1 c c c c c V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

8 Introduction Kinetically constrained models (KCM) Kinetically constrained models (KCM) Constrained dynamics: changes occur only around excited sites. Fredrickson Andersen model in 1D at least one neighbor of i must be excited to allow i to change unexcitation: excitation: 1 c 1 c c c c c same equilibrium distribution P eq (n) with&without the constraint BUT: ageing, super-arrhenius slowing down, dynamical heterogeneity static free-energy landscape not useful need for a dynamical description V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

9 Kinetically Constrained Models Space-time bubbles of inactivity Dynamical Phase coexistence From: Merolle, Garrahan and Chandler, PNAS 102, (2005) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

10 Kinetically Constrained Models Dynamical Phase coexistence Questions Active and inactive histories having a probability of the same order Coexistence of dynamical phases? How to describe a dynamical 1 st order phase transition? Dynamical Landau free-energy landscape? V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

11 Statistics over histories Dynamical ensembles Activity of histories: order parameter Activity K = number of events = (# excitations) + (# unexcitations) (Dynamical) canonical ensemble β conjugated to energy s conjugated to activity K (statics) (dynamics) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

12 Statistics over histories Dynamical ensembles Activity of histories: order parameter Activity K = number of events = (# excitations) + (# unexcitations) (Dynamical) canonical ensemble β conjugated to energy (statics) s conjugated to activity K (dynamics) s < 0 : more active histories ( large activity K > K) s-ensemble: s = 0 : equilibrium state (equilib. activity K = K) s > 0 : less active histories ( small activity K < K) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

13 Statistics over histories Dynamical ensembles Dynamical phase transition: FA model (d=1) Density of excitations ρ(s) depending on histories. ρ(s) ρ(0) = c Ø Ý Ø Ø ÑÓÖ Ø Ú K > Kµ ¼ ÓÒ ØÖ Ò ÙÒÓÒ ØÖ Ò Ð Ø Ú K < Kµ s Comparison between constrained and unconstrained dynamics V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

14 Statistics over histories Dynamical ensembles Dynamical Landau free-energy landscape F(ρ, s) t LF(ρ,s) Prob [0,t] (ρ, s) e in the s-ensemble F(ρ, s) s > s Ô s > 0 s = s < ρ Dynamical free energy: f(s) = min F(ρ, s) ρ }{{} reached atρ=ρ(s) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

15 Statistics over histories Dynamical ensembles Scaling of the free energy in the interfacial regime Finite-size scaling of the free energy f : φ L (λ) = f ( ) λ L s = λ L t τ 0.3 φ L (λ) 0.2 x (τ) q p Active p q x + (τ) region λ c λ x 0.4 increasing L Interfacial model finite-size scaling Surface tension Σ φ L (λ) = Σ A ( λ L ) 2 3 V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

16 Scaling function Statistics over histories Dynamical ensembles Finite-size scaling of the free energy f : φ L (λ) = f ( ) λ L s = λ L λ L α( φ L (λ)+σ ) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

17 Summary and outlook Correspondence Conclusion { } finite-size scaling of dyn. free-energy f(s) { } geometrical features : of active excitations φ L (λ) }{{} finite-size φ L (λ)=f( λ L) = Σ }{{} surface tension (excitations are bubbles) ( λ α A }{{ L) } α= 2 3 (boundaries are Brownian) Questions: { the (s > 0) bubbles of excitation to How to link? the (s = 0) bubbles of inactivity How to write the correspondence in more realistic models? Link dynamical phase transition and glassy features. Reference: T. Bodineau, V. Lecomte, C. Toninelli, JSP (2012),arXiv: V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

18 Summary and outlook Correspondence Conclusion { } finite-size scaling of dyn. free-energy f(s) { } geometrical features : of active excitations φ L (λ) }{{} finite-size φ L (λ)=f( λ L) = Σ }{{} surface tension (excitations are bubbles) ( λ α A }{{ L) } α= 2 3 (boundaries are Brownian) Questions: { the (s > 0) bubbles of excitation to How to link? the (s = 0) bubbles of inactivity How to write the correspondence in more realistic models? Link dynamical phase transition and glassy features. Reference: T. Bodineau, V. Lecomte, C. Toninelli, JSP (2012),arXiv: V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April / 12

Thermodynamics of histories: application to systems with glassy dynamics

Thermodynamics of histories: application to systems with glassy dynamics Vivien Lecomte 1,2, Cécile Appert-Rolland 1, Estelle Pitard 3, Kristina van Duijvendijk 1,2, Frédéric van Wijland 1,2 Juan P. Garrahan