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

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 2012 1 / 12

Introduction Motivations Dynamical excitations in glass-forming liquids From: Keys et. al PRX 1 021013 (2011) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 2 / 12

Introduction Motivations Dynamical excitations in glass-forming liquids Can we model this simply? From: Keys et. al PRX 1 021013 (2011) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 2 / 12

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 2012 3 / 12

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 2012 3 / 12

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 2012 3 / 12

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 2012 4 / 12

Kinetically Constrained Models Space-time bubbles of inactivity Dynamical Phase coexistence From: Merolle, Garrahan and Chandler, PNAS 102, 10837 (2005) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 5 / 12

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 2012 6 / 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 s conjugated to activity K (statics) (dynamics) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 7 / 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 2012 7 / 12

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 0.35 0.3 0.25 0.2 0.15 0.1 0.05-1 1 ÑÓÖ Ø Ú K > Kµ ¼ ÓÒ ØÖ Ò ÙÒÓÒ ØÖ Ò 2 3 4 Ð Ø Ú K < Kµ s Comparison between constrained and unconstrained dynamics V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 8 / 12

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) 0.2 0.15 s > s Ô 0.1 0.05-0.05 0.2 s > 0 s = 0 0.4 0.6 s < 0 0.8 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 2012 9 / 12

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 0.5 0.5 1.0 1.5 0.1 0.1 0.2 λ c λ 0.3 0 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 2012 10 / 12

Scaling function Statistics over histories Dynamical ensembles Finite-size scaling of the free energy f : φ L (λ) = f ( ) λ L s = λ L 0.5 1.0 1.5 1 2 3 4 5 6 λ 2.0 2.5 3.0 3.5 L α( φ L (λ)+σ ) V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 11 / 12

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:1111.6394 V. Lecomte (LPMA) Interfacial dynamical regime in KCM 18th April 2012 12 / 12