Granular Mode-Coupling Theory W. Till Kranz Institute for - PowerPoint PPT Presentation

Granular Mode-Coupling Theory W. Till Kranz Institute for Theoretical Physics, University Göttingen MPI Dynamics & Self-Organization, Göttingen, Germany 粒 状 の 流 れの 物 理 学 2013, 京 都

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Reminder: What we have Mean Square Displacement shows a plateau for increasing density Plateaus are a signature of caging Caging is seen as either the cause or a signature of a glass transition ◮ No Static Order Parameter is known

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Reminder: What we want η ≪ η c φ q ( t ) ◮ A theory for the Coherent Scattering Function log t η � η c • Edwards-Anderson Order Parameter f q = φ ( q , t → ∞ ) φ q ( t ) • Bifurcation from f q = 0 to f q > 0 ◮ Hydrodynamics works well for the log t fluid but is linear η ≥ η c • Interactions of Modes give Nonlinearities φ q ( t ) log t

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Interacting Sound Waves Dyson Equation for Coherent Scattering Function φ ( q , t ) = + φ 0 φ φ 0 φ − 1 Hydrodynamic (Free) Solution φ 0 ( q , t ) = cos ( cqt ) exp ( − Γ q 2 t )

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Equation of Motion Density Field ρ ( r , t ) = � i δ ( r − r i ( t )) Coherent Scattering Function φ ( q , t ) = � � ρ q | ρ q ( t ) � � 0 = ( ∂ 2 t + ν q ∂ t + Ω 2 q ) φ ( q , t ) Speed of Sound Ω q / q Sound Damping ν q Memory Kernels M ( q , t ) and L ( q , t )

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Equation of Motion Density Field ρ ( r , t ) = � i δ ( r − r i ( t )) Coherent Scattering Function φ ( q , t ) = � � ρ q | ρ q ( t ) � � � t 0 = ( ∂ 2 t + ν q ∂ t + Ω 2 q ) φ ( q , t ) + d τ M ( q , t − t τ ) ∂ τ φ ( q , τ ) 0 Speed of Sound Ω q / q Sound Damping ν q Memory Kernels M ( q , t ) and L ( q , t )

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Equation of Motion Density Field ρ ( r , t ) = � i δ ( r − r i ( t )) Coherent Scattering Function φ ( q , t ) = � � ρ q | ρ q ( t ) � � � t 0 = ( ∂ 2 t + ν q ∂ t + Ω 2 q ) φ ( q , t ) + d τ M ( q , t − t τ ) ∂ τ φ ( q , τ ) 0 � t + d τ L ( q , t − τ ) φ ( q , τ ) 0 Speed of Sound Ω q / q Sound Damping ν q Memory Kernels M ( q , t ) and L ( q , t )

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Mode-Coupling Approximation ◮ Take into account Splitting/Merging of sound waves φ M MCT [ φ ] = φ � M ( q , t ) ≈ V qkp W qkp φ ( k , t ) φ ( p , t ) q = k + p Momentum Conservation demands q = k + p Transition Rates V qkp , W qkp will be expressed as expectation values L ( q , t ) ≈ 0 ◮ or at least short-lived

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Why it helps: A schematic Model � t ∂ 2 d τφ 2 ( t − τ ) ∂ τ φ ( τ ) t φ ( t ) + φ ( t ) + 4 λ 0 EA Order Parameter f = φ ( t → ∞ ) follows from f 1 − f = 4 λ f 2 ◮ Bifurcation at critical λ c = 1

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Microscopic Dynamics

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Inelastic Hard Spheres Hard Spheres completely characterized by ◮ Mass m ◮ Radius a ◮ Coefficient of restitution ǫ ∈ [ 0 , 1 ] Collision law v t v 12 v ′ n = − ǫ v n , v ′ t = v t v n Energy Loss on average per collision ∆ E ∝ 1 − ǫ 2

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Random Driving Force Random Force ξ i ( t ) , gaussian distributed ◮ Average � ξ i � = 0 ξ 2 ◮ Driving power P D = � � i Stationary State as a balance between driving & dissipation

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems The Liouville Operator Observables A (Γ( t )) are functions of phase space Γ = ( x , p ) Liouville Operator L = ∂ Γ ∂ ∂ Γ controls rate of change, ∂ t A = L A . ∂ t Propagator U ( t ) = exp ( t L ) , i.e., A ( t ) = U ( t ) A ( 0 ) Fun Fact L + ρ q = iqj L q

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems The Liouville Operator Observables A (Γ( t )) are functions of phase space Γ = ( x , p ) Liouville Operator L = ∂ Γ ∂ ∂ Γ controls rate of change, ∂ t A = L A . ∂ t Propagator U ( t ) = exp ( t L ) , i.e., A ( t ) = U ( t ) A ( 0 ) Fun Fact L + ρ q = iqj L q

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Our Liouville Operator ∂ Free Streaming i L 0 = � j v j · ∂ r j r jk · v jk ) δ ( r jk − 2 a )( b + Collisions i T + = � j < k ( ˆ r jk · v jk )Θ( − ˆ jk − 1 ) ◮ Operator b + jk implements inelastic collision rule Driving i ˇ L D ∂ + ( t ) = � j ξ j ( t ) · ∂ v j Propagator ˇ U ( t ) = exp + ( t ˇ L + ( t ))

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Our Liouville Operator ∂ Free Streaming i L 0 = � j v j · ∂ r j r jk · v jk ) δ ( r jk − 2 a )( b + Collisions i T + = � j < k ( ˆ r jk · v jk )Θ( − ˆ jk − 1 ) ◮ Operator b + jk implements inelastic collision rule Driving i ˇ L D ∂ + ( t ) = � j ξ j ( t ) · ∂ v j Propagator ˇ U ( t ) = exp + ( t ˇ L + ( t ))

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Averages & Averaged Quantities

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Averages � � � � A � � = �� A � Ξ � Γ = d Γ f (Γ) D [Ξ] A (Γ , Ξ) ◮ Average over all Trajectories of the Driving Force (for a specific initial condition) ◮ Average over initial conditions

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Effective Dynamics ◮ For two-point correlation functions � A ˇ � ˇ � � � � U ( t ) B � � = A U ( t ) Ξ B Γ = � A U ( t ) B � Γ Averaged Dynamics ∂ 2 � � L D L D ˇ � + := + ( t ) Ξ = P D ∂ v 2 i i U ( t ) = exp + ( t ˇ ˇ L + ( t )) U ( t ) = exp ( t L + )

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems More Adjoints than you’d like Direction of Time L + , L − d Γ A (Γ) B ∗ (Γ) � Quantum Scalar Product ( A , B ) = d Γ f (Γ) A (Γ) B ∗ (Γ) � Statistical Scalar Product � A | B � = The f-Liouvillian ( L ± A , B ) = ( A , L ± B ) � � L † The adjoint Liouvillian ± A | B = � A |L ± B � Detailed Balance � U ( − t ) A | B � = � A , U ( t ) B � In Equilibrium L ± = L † ± = L ∓

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems More Adjoints than you’d like Direction of Time L + , L − d Γ A (Γ) B ∗ (Γ) � Quantum Scalar Product ( A , B ) = d Γ f (Γ) A (Γ) B ∗ (Γ) � Statistical Scalar Product � A | B � = The f-Liouvillian ( L ± A , B ) = ( A , L ± B ) � � L † The adjoint Liouvillian ± A | B = � A |L ± B � Detailed Balance � U ( − t ) A | B � = � A , U ( t ) B � In Equilibrium L ± = L † ± = L ∓

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems More Adjoints than you’d like Direction of Time L + , L − d Γ A (Γ) B ∗ (Γ) � Quantum Scalar Product ( A , B ) = d Γ f (Γ) A (Γ) B ∗ (Γ) � Statistical Scalar Product � A | B � = The f-Liouvillian ( L ± A , B ) = ( A , L ± B ) � � L † The adjoint Liouvillian ± A | B = � A |L ± B � Detailed Balance � U ( − t ) A | B � = � A , U ( t ) B � In Equilibrium L ± = L † ± = L ∓

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems Slow Variables Conserved Quantities are Density ρ q and q + j T q j L Current Density j q = � k v k δ ( r − r k ) = ˆ q � � ρ q , j L � State Vector q

Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems The Mori-Zwanzig Decomposition � and Q = 1 − P � ρ q , j L � � � ρ q , j L � Mori-Projectors P = � q q q � ∞ 0 e − ist g ( t ) dt Laplace Transform ˆ g ( s ) = LT [ g ( t )] = i Propagator ˆ U ( s ) = ( s − L + ) − 1 P ( s − L + ) − 1 P = [ s − PL + P − PL + Q ( s − QL + Q ) − 1 QL + P ] − 1