Temperature and Correlations in Driven Dissipative Systems Giacomo Gradenigo Laboratoire Interdisciplinaire de Physique (LIPhy) Grenoble (2014-present ): E. Bertin, J.-L. Barrat, E. Ferrero Rome (2010-2012): A. Puglisi, A. Vulpiani, A. Sarracino ENS, Lyon, 10-11-2015
History Ph. D. : Supercooled liquids ; Trento (Italy), 2007-2009 Supervisor : P. Verrocchio. Collaborations: G. Parisi, A. Cavagna. I Giardina, T. Grigera, C. Cammarota Post-Doc : Non-equilibrium Statistical Mechanis; Rome, 2010-2012 Advisor: A. Puglisi, A. Vulpiani Collaborations: H. Touchette, R. Burioni, U. Marconi, A. Cavagna, T. Grigera, P. Verrocchio, A. Sarracino, D. Villamaina Post-Doc : Glass transition; Paris, 2013-2014 Advisor: G. Biroli, S. Franz Post-Doc : Non-equilibrium Statistical Mechanics; Grenoble, 2015-present Advisor: E. Bertin, J.-L. Barrat Collaborations: E. Ferrero, A. Puglisi, G. Biroli
Interests & Reasearch topics Equilibrium Statistical Mechanics - Effective theories for the glass transition and Dynamical Heterogeneities in Supercooled Liquids ( Trento , simulations). - Supercooled Liquids in confined geometries : effect of confinement on the glass transition ( Rome , Paris , theory, simulations). Out-of-equilibrium Statistical-Mechanics - Ratchet effect ( Rome , simulations) - Coarse-grained description of Granular Fluids : Linearized Fluctuating Hydrodynamics ( Rome , theory, simulations, experiments) - Non-equilibrium fluctuations in the driven Stochastic Lorenz Gas (1d schematic model of a granular gas): Fluctuation-Relation, Condensation of Fluctuations ( Rome, Grenoble , theory, simulations) - Anomalous diffusion in driven systems: Continuous Time Random Walks, Kinetically Constrained Models ( Rome, Grenoble , theory, simulations) - Effective ‘’equilibrium-like’’ theories for Driven Athermal Systems ( Grenoble , theory, simulations)
Interests & Reasearch topics Out-of-equilibrium Statistical-Mechanics Coarse-grained description of Granular Fluids : Linearized Fluctuating Hydrodynamics Temperature = ? Correlations and Temperature Effective equilibrium-like theories for Driven Athermal Systems
Dense and Dilute Granular Systems Two-dimensional Amorphous packing of granular fluid frictional grains (study of a model system) PART I PART II
PART I Temperatures and Correlations in a 2D bulk driven granular fluid
Granular Fluids: Non-Equilibrium Stationary State (Time Translational Invariance, constant energy flux) Bulk Dissipation Bulk Driving (energy source coupled to each particle) σ ) 2 (1 − α ) 2 ∆ E coll ∝ − ([ v i − v j ] · ˆ Inelastic collisions α < 1 v i v 0 i ˆ σ Vibrating plate v 0 v j j m h v 2 i i 1 Granular NESS X T g = temperature (Non-Equilibrium Stationary State) Nd 2 i
Granular Fluids: Non-Equilibrium Stationary State (Time Translational Invariance, constant energy flux) Bulk Dissipation Bulk Driving (energy source coupled to each particle) σ ) 2 (1 − α ) 2 ∆ E coll ∝ − ([ v i − v j ] · ˆ Inelastic collisions α < 1 v i v 0 i ˆ σ Vibrating plate v 0 v j j m h v 2 i i 1 φ = Packing Fraction T g = T g ( φ ) X T g = Nd 2 T g ↓ φ ↑ i
Physical meaning of the Granular Temperature = ? (Time Translational Invariance, constant energy flux) Bulk Dissipation Boundary Driving σ ) 2 (1 − α ) 2 ∆ E coll ∝ − ([ v i − v j ] · ˆ Study of the fluctuations within the granular gas: does really T g plays the role of a temperature? Vibrating vessel m h v 2 i i 1 NESS Granular X T g = (Non-Equilibrium Stationary State) temperature Nd 2 i
Physical meaning of the Granular Temperature = ? (Time Translational Invariance, constant energy flux) Bulk Dissipation Boundary Driving σ ) 2 (1 − α ) 2 ∆ E coll ∝ − ([ v i − v j ] · ˆ Heat Flux Study of the fluctuations within the granular gas: does really T g plays the role of a temperature? ‘’Energy transport between two granular thermostats’’, C.E. Lecomte, A. Naert, J. Stat. Mech. , P11004 ( 2014 ) High frequency Low frequency ‘’Work exchange with a granular gas: the viewpoint of the T hot T cold Fluctuation Theorem’’, A. Naert, EPL 97, 20010 ( 2012 ) g g m h v 2 i i 1 NESS Granular X T g = (Non-Equilibrium Stationary State) temperature Nd 2 i
Energy injection mechanism as a ‘’thermostat’’ = ? (Time Translational Invariance, constant energy flux) Bulk Dissipation Bulk Driving (energy source coupled to each particle) σ ) 2 (1 − α ) 2 ∆ E coll ∝ − ([ v i − v j ] · ˆ Bulk energy injection can be modeled as a thermostat at T b ? Which role is played by T b -T g ? Vibrating plate m h v 2 i i 1 Granular NESS X T g = temperature (Non-Equilibrium Stationary State) Nd 2 i
Two models for bulk driving With viscous drag Without viscous drag (finite temperature thermostat) (infinite temperature thermostat) v i ( t ) = − γ b v i ( t ) + ξ i ( t ) + F coll v i ( t ) = ξ i ( t ) + F coll m ˙ m ˙ i i TPC Van Noije, MH Ernst, E. Trizac, G. Gradenigo et al , EPL 96, 14004 ( 2011 ) I. Pagonabarraga, PRE 59 (4), 4326 G. Gradenigo et al , J. Stat. Mech. P08017 ( 2011 ) ( 1999 ) A. Puglisi et al , J. Chem. Phys. 136, 014704 ( 2012 )
Two-dimensional bulk driven granular fluid: Scale-free correlations v i ( t ) = ξ i ( t ) + F coll h ξ µ m ˙ j ( t 0 ) i = 2 Γ δ µ, ν δ i,j δ ( t � t 0 ) i ( t ) ξ ν i White noise Energy Gain Energy Sink v ( r ) v ⊥ ( r ) v k ( r 0 ) Coarse-grained velocity field v k ( r ) v ? ( r 0 ) v ( r 0 ) Linearized Fluctuating Hydrodynamic calculations & Simulations show that ✓ ◆ L h v ? ( r ) v ? ( r 0 ) i ⇠ h v k ( r ) v k ( r 0 ) i ⇠ Γ log | r 0 � r | Scale-Free Correlations « Randomly driven granular fluids: Large-scale structure » . TPC Van Noije, MH Ernst, E. Trizac, I. Pagonabarraga, PRE 59 (4), 4326 ( 1999 )
Two-dimensional bulk driven granular fluid: Scale-free correlations v i ( t ) = ξ i ( t ) + F coll h ξ µ m ˙ j ( t 0 ) i = 2 Γ δ µ, ν δ i,j δ ( t � t 0 ) i ( t ) ξ ν i White noise Energy Gain Energy Sink Scale-Free ✓ ◆ L h v ? ( r ) v ? ( r 0 ) i ⇠ h v k ( r ) v k ( r 0 ) i ⇠ Γ log | r 0 � r | Correlations Experiments « Forcing and Velocity Correlations in a Vibrated Granular Monolayer » A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 ( 2002 ) Smooth circulat plate (d=20cm) Steel spheres (d=1.59mm) | C k ( r ) | ∼ 1 /r 2
Two-dimensional bulk driven granular fluid: Finite-range correlations v i ( t ) = ξ i ( t ) + F coll h ξ µ m ˙ j ( t 0 ) i = 2 Γ δ µ, ν δ i,j δ ( t � t 0 ) i ( t ) ξ ν i White noise Energy Gain Energy Sink Scale-Free ✓ ◆ L h v ? ( r ) v ? ( r 0 ) i ⇠ h v k ( r ) v k ( r 0 ) i ⇠ Γ log | r 0 � r | Correlations « Forcing and Velocity Correlations in a Vibrated Granular Monolayer » Experiments A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 ( 2002 ) Rough exagonal plate (d=30cm) (lattice of steel balls d=1.19 mm) C k ( r ) ∼ e � r/ ξ k Steel spheres (d=3.97mm) C ⊥ ( r ) ∼ e − r/ ξ ⊥
Two-dimensional bulk driven granular fluid: Finite-range correlations � � / � Mean Free Path: jj 0 : 6 � for f ? . It is surprising that this decay length has little microscopic length-scale density dependence, despite the fact that the mean free path of the system estimated from Enskog-Boltzmann kinetic theory varies λ 0 ∼ (1 − φ ) 2 from 2 : 3 � for � � 0 : 125 to 0 : 16 � for � � 0 : 6 . A similar φ « Forcing and Velocity Correlations in a Vibrated Granular Monolayer » Experiments A. Prevost, D. A. Egolf, J. S. Urbach, PRL 89 (8), 084301 ( 2002 ) Rough exagonal plate (d=30cm) (lattice of steel balls d=1.19 mm) C k ( r ) ∼ e � r/ ξ k Steel spheres (d=3.97mm) C ⊥ ( r ) ∼ e − r/ ξ ⊥
Two-dimensional bulk driven granular fluid: The equilibrium thermostat Equilibrium Thermostat v i ( t ) = − γ b v i ( t ) + ξ i ( t ) + F coll m ˙ i Random Kicks Inelastic Collisions Viscous drag h ξ µ j ( t 0 ) i = 2 T b γ b δ µ, ν δ i,j δ ( t � t 0 ) i ( t ) ξ ν σ ) 2 (1 − α ) 2 ∆ E coll ∝ − ([ v i − v j ] · ˆ Energy dissipated in collisions The Limit of Elastic Collisions is well defined T g = T b α = 1 α < 1 ‘’Distance’’ from equilibrium ∆ T = T b − T g
Two-dimensional bulk driven granular fluid: Equilibrium thermostat produces Finite-range correlations h ξ µ v i ( t ) = − γ b v i ( t ) + ξ i ( t ) + F coll j ( t 0 ) i ⇠ T b γ b i ( t ) ξ ν m ˙ i Predictions from Linearized Fluctuating Hydrodynamic calculations G. Gradenigo et al , EPL 96, 14004 ( 2011 ) G. Gradenigo et al , J. Stat. Mech. P08017 ( 2011 ) n h v ? ( r 0 ) v ? ( r ) i = T g δ (2) ( r 0 � r ) + ( T b � T g ) K 0 ( | r 0 � r | ) ξ 2 ? Finite ‘’Distance’’ from equilibrium p ν / γ b ξ ⊥ = Characteristic length K 0 ( | r 0 − r | / ξ ? ) ∼ e � | r 0 � r | / ξ ? ν = Shear viscosity Finite extent of correlations
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