Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Large deviations and heterogeneities in driven or non-driven kinetically constrained models Estelle Pitard 1 CNRS, L2C, Montpellier, France MPI- Dresden- 13 July 2011 with: J.P. Garrahan (Nottingham), R.L. Jack (Bath), V. Lecomte, K. van Duijvendijk, F. van Wijland (Paris), F. Turci (Montpellier) Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Outline Dynamic transition in Kinetically Constrained Models- large deviations Phenomenology of kinetically constrained models (KCMs) Relevant order parameters for space-time trajectories: activity K Results: mean-field/ finite dimensions Driven KCMs, heterogeneities and large deviations A new dynamic phase transition for the current J Results and link with microscopic spatial heterogeneities Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Phenomenology of KCMs Spin models on a lattice / lattice gases, designed to mimick steric effects in amorphous materials: s i = 1, n i = 1: ”mobile” particle - region of low density - fast dynamics s i = − 1, n i = 0: ”blocked” particle - region of high density - slow dynamics i n i → < n > eq = c = 1 / (1 + e β ) , β = 1 / T . H = P Specific dynamical rules: Fredrickson-Andersen (FA) model in 1 dimension: a spin can flip only if at least one of its nearest neighbours is in the mobile state. ↓↑↓ ⇋ ↓↓↓ is forbidden. Mobile/blocked particles self-organize in space → dynamical correlation length ξ . How to classify time-trajectories and their activity? (F. Ritort, P. Sollich, Adv. Phys 52 , 219 (2003).) Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Relevant order parameters for space-time trajectories Ruelle formalism: from deterministic dynamical systems to continous-time Markov dynamics Observable: Activity K ( t ): number of flips between 0 and t , given a history C 0 → C 1 → .. → C t . C ′ W ( C ′ → C ) P ( C ′ , t ) − r ( C ) P ( C , t ), ∂ P Master equation: ∂ t ( C , t ) = P C ′ � = C W ( C → C ′ ) where r ( C ) = P Introduce s (analog of a temperature), conjugated to K: ˆ K e − sK P ( C , K , t ) → ∂ t ˆ P ( C , s , t ) = W K ˆ P ( C , s , t ) = P P ( C , s , t ), where W K ( s )( C , C ′ ) = e − s W ( C ′ → C ) − r ( C ) δ C , C ′ . P ( C , s , t ) = < e − sK > . C ˆ Generating function of K: Z K ( s , t ) = P For t → ∞ , Z K ( s , t ) ≃ e t ψ K ( s ) . → the large deviation function ψ K ( s ) is the largest eigenvalue of W K ( s ). Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Relevant order parameters for space-time trajectories < K > ( s , t ) t →∞ − 1 N ψ ′ Average activity: = K ( s ). Nt Analogy with the canonical ensemble: C e − β H ≃ e − Nf ( β ) , N → ∞ . space of configurations, fixed β : Z ( β ) = P space of trajectories, fixed s : C , K e − sK P ( C , K , t ) ≃ e − tf K ( s ) , t → ∞ . Z K ( s , t ) = P f K ( s ) = − ψ K ( s ): free energy for trajectories ρ K ( s ), < K > ( s , t ) : activity/chaoticity. Nt Active phase: < K > ( s , t ) / ( Nt ) > 0: s < 0. Inactive phase: < K > ( s , t ) / ( Nt ) = 0: s > 0. Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Results: Mean-Field FA W i (0 → 1) = k ′ n N , W i (1 → 0) = k n − 1 N , n = P i n i . The result is a variational principle for ψ K ( s ), involving a Landau-Ginzburg free energy F K ( ρ, s ) ( ρ : density of mobile spins): N f K ( s ) = − 1 1 N ψ K ( s ) = min F K ( ρ, s ), with ρ F K ( ρ, s ) = − 2 ρ e − s ( ρ (1 − ρ ) kk ′ ) 1 / 2 + k ′ ρ (1 − ρ ) + k ρ 2 Minima of F K ( ρ, s ) at fixed s : s > 0: inactive phase, ρ K ( s ) = 0, ψ K ( s ) / N = 0. s = 0: coexistence ρ K (0) = 0 and ρ K (0) = ρ ∗ , ψ K (0) = 0, → first order phase transition. s < 0: active phase, ρ K ( s ) > 0, ψ K ( s ) / N > 0. Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Results: Mean-Field FA F K ( ρ, s ) for different values of s : 0.4 0.3 s=-0.4 0.2 s=-0.2 s=0 free energy (FA case) s=0.2 0.1 s=0.4 0 -0.1 -0.2 -0.3 -0.4 0 0.2 0.4 0.6 0.8 1 rho Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Results: Mean-Field unconstrained model One removes the constraints: W i (0 → 1) = k ′ , W i (1 → 0) = k , for all i F K ( ρ, s ) = − 2 e − s ( ρ (1 − ρ ) kk ′ ) 1 / 2 + k ′ (1 − ρ ) + k ρ → No phase transition 1 0.8 0.6 free energy (unconstrained case) 0.4 0.2 0 -0.2 s=0.4 s=0.2 s=0 -0.4 s=-0.2 s=-0.4 -0.6 -0.8 -1 0 0.2 0.4 0.6 0.8 1 rho Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Results in finite dimensions Numerical solution using the cloning algorithm for large deviation functions (Giardina, Kurchan, Peliti 2006) . First-order phase transition for the FA model in 1d. 1 L ψ K ( s ) 0.025 0.02 L = 200 L = 100 0.015 L = 50 0.01 0.005 0 -0.005 s -0.01 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Results in finite dimensions ρ K ( s ) for the FA model in 1d. ρ K ( s ) 0.3 0.25 0.2 0.15 L = 50 L = 100 0.1 L = 200 0.05 s 0 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 “Dynamic first-order transition in kinetically constrained models of glasses”, J.P. Garrahan, R.L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, F. van Wijland, Phys. Rev. Lett. 98, 195702 (2007). “First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories”, J.P. Garrahan, R.L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, F. van Wijland, J. Phys. A 42 (2009). Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Driven KCMs, heterogeneities and large deviations 2d ASEP with kinetic constraints, (model introduced by M. Sellitto, 2008) . A particle can hop to an empty neighbouring site if it has at most 2 occupied neighbouring sites, before and after the move. Fixed density of particles ρ , periodic boundary conditions. � Driving field � E . � r ). E in one direction: p = min (1 , e For ρ > ρ c , E < E max : shear-thinning, the current J grows with E E > E max : shear-thickening, J decreases with E Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Driven KCMs, heterogeneities and large deviations Microscopic analysis: transient shear-banding at large fields, localization of the current. → very different velocity profiles for small and large driving fields. Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Large deviation functions for the activity K ( t ) and the integrated current Q ( t ): • For K , the first-order transition persists like for unforced KCMs. • For Q , there is a first-order transition only at large fields (coexistence of histories with large current and histories with no current). Absent for ASEP without constraints! • → Link between current heterogeneities and singularity in the large deviation function? Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Large deviation function for the integrated current Q ( t ): Gallavotti-Cohen symmetry. ρ=0.80 , E=2.8, L=30, clones=300 0.030 τ=700 τ=5000 τ=800 τ=10000 0.025 τ=1000 τ=50000 and 1000 clones 0.020 0.015 ψ Q 0.010 0.005 0 −1 1 −0.005 −1 0 1 s / E . 1 2 Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
Dynamic transition in KCMs- large deviations Driven KCMs, heterogeneities and large deviations Dynamical blocking walls -1 Dense domain walls play the role of kinetic traps at large fields. ρ = 0 . 82 , E = 0 ρ = 0 . 82 , E = 5 Estelle Pitard Large deviations and heterogeneities in driven or non-driven kinetically constra
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