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

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Granular Mode-Coupling Theory W. Till Kranz Institute for Theoretical Physics, University Gttingen MPI Dynamics & Self-Organization, Gttingen, Germany 2013, Outline Microscopic Dynamics

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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

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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

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

Reminder: What we want

◮ A theory for the Coherent Scattering

Function

  • Edwards-Anderson Order

Parameter fq = φ(q, t → ∞)

  • Bifurcation from fq = 0 to fq > 0

◮ Hydrodynamics works well for the

fluid but is linear

  • Interactions of Modes give

Nonlinearities

log t φq(t)

η ≥ ηc

log t φq(t)

η ηc

log t φq(t)

η ≪ ηc

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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 = φ0 + φ0 φ − 1 Hydrodynamic (Free) Solution φ0(q, t) = cos(cqt) exp(−Γq2t)

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

Equation of Motion

Density Field ρ(r, t) =

i δ(r − ri(t))

Coherent Scattering Function φ(q, t) = ρq|ρq(t)

  • 0 = (∂2

t + νq∂t + Ω2 q)φ(q, t)

Speed of Sound Ωq/q Sound Damping νq Memory Kernels M(q, t) and L(q, t)

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

Equation of Motion

Density Field ρ(r, t) =

i δ(r − ri(t))

Coherent Scattering Function φ(q, t) = ρq|ρq(t)

  • 0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) Speed of Sound Ωq/q Sound Damping νq Memory Kernels M(q, t) and L(q, t)

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

Equation of Motion

Density Field ρ(r, t) =

i δ(r − ri(t))

Coherent Scattering Function φ(q, t) = ρq|ρq(t)

  • 0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) + t dτL(q, t − τ)φ(q, τ) Speed of Sound Ωq/q Sound Damping νq Memory Kernels M(q, t) and L(q, t)

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

Mode-Coupling Approximation

◮ Take into account Splitting/Merging of sound waves

MMCT[φ] = φ φ M(q, t) ≈

  • q=k+p

VqkpWqkpφ(k, t)φ(p, t) Momentum Conservation demands q = k + p Transition Rates Vqkp, Wqkp will be expressed as expectation values L(q, t) ≈ 0

◮ or at least short-lived

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

Why it helps: A schematic Model

∂2

t φ(t) + φ(t) + 4λ

t dτφ2(t − τ)∂τφ(τ) EA Order Parameter f = φ(t → ∞) follows from f 1 − f = 4λf 2

◮ Bifurcation at critical λc = 1

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

Microscopic Dynamics

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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′

n = −ǫ vn,

v′

t = vt

Energy Loss on average per collision ∆E ∝ 1 − ǫ2

v12 vn vt

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

Random Driving Force

Random Force ξi(t), gaussian distributed

◮ Average ξi = 0 ◮ Driving power PD =

  • ξ2

i

  • Stationary State as a balance between driving & dissipation
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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

∂t ∂ ∂Γ controls rate of change, ∂tA = LA.

Propagator U(t) = exp(tL), i.e., A(t) = U(t)A(0) Fun Fact L+ρq = iqjL

q

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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

∂t ∂ ∂Γ controls rate of change, ∂tA = LA.

Propagator U(t) = exp(tL), i.e., A(t) = U(t)A(0) Fun Fact L+ρq = iqjL

q

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

Our Liouville Operator

Free Streaming iL0 =

j vj · ∂ ∂rj

Collisions iT+ =

j<k(ˆ

rjk · vjk)Θ(−ˆ rjk · vjk)δ(rjk − 2a)(b+

jk − 1) ◮ Operator b+ jk implements inelastic collision rule

Driving i ˇ LD

+(t) = j ξj(t) · ∂ ∂vj

Propagator ˇ U(t) = exp+(t ˇ L+(t))

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

Our Liouville Operator

Free Streaming iL0 =

j vj · ∂ ∂rj

Collisions iT+ =

j<k(ˆ

rjk · vjk)Θ(−ˆ rjk · vjk)δ(rjk − 2a)(b+

jk − 1) ◮ Operator b+ jk implements inelastic collision rule

Driving i ˇ LD

+(t) = j ξj(t) · ∂ ∂vj

Propagator ˇ U(t) = exp+(t ˇ L+(t))

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

Averages & Averaged Quantities

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

Averages

  • A

= AΞΓ =

  • dΓf(Γ)
  • D[Ξ]A(Γ, Ξ)

◮ Average over all Trajectories of the Driving Force (for a

specific initial condition)

◮ Average over initial conditions

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

Effective Dynamics

◮ For two-point correlation functions

  • Aˇ

U(t)B =

  • A

ˇ U(t)

  • Ξ B
  • Γ = AU(t)BΓ

Averaged Dynamics LD

+ :=

  • ˇ

LD

+(t)

  • Ξ = PD
  • i

∂2 ∂v2

i

ˇ U(t) = exp+(t ˇ L+(t)) U(t) = exp(tL+)

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

More Adjoints than you’d like

Direction of Time L+, L− Quantum Scalar Product (A, B) =

  • dΓA(Γ)B∗(Γ)

Statistical Scalar Product A|B =

  • dΓf(Γ)A(Γ)B∗(Γ)

The f-Liouvillian (L±A, B) = (A, L±B) The adjoint Liouvillian

  • L†

±A|B

  • = A|L±B

Detailed Balance U(−t)A|B = A, U(t)B In Equilibrium L± = L†

± = L∓

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

More Adjoints than you’d like

Direction of Time L+, L− Quantum Scalar Product (A, B) =

  • dΓA(Γ)B∗(Γ)

Statistical Scalar Product A|B =

  • dΓf(Γ)A(Γ)B∗(Γ)

The f-Liouvillian (L±A, B) = (A, L±B) The adjoint Liouvillian

  • L†

±A|B

  • = A|L±B

Detailed Balance U(−t)A|B = A, U(t)B In Equilibrium L± = L†

± = L∓

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

More Adjoints than you’d like

Direction of Time L+, L− Quantum Scalar Product (A, B) =

  • dΓA(Γ)B∗(Γ)

Statistical Scalar Product A|B =

  • dΓf(Γ)A(Γ)B∗(Γ)

The f-Liouvillian (L±A, B) = (A, L±B) The adjoint Liouvillian

  • L†

±A|B

  • = A|L±B

Detailed Balance U(−t)A|B = A, U(t)B In Equilibrium L± = L†

± = L∓

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

Slow Variables

Conserved Quantities are Density ρq and Current Density jq =

k vkδ(r − rk) = ˆ

qjL

q + jT q

State Vector

  • ρq, jL

q

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

The Mori-Zwanzig Decomposition

Mori-Projectors P =

q

  • ρq, jL

q

ρq, jL

q

  • and Q = 1 − P

Laplace Transform ˆ g(s) = LT[g(t)] = i ∞

0 e−istg(t)dt

Propagator ˆ U(s) = (s − L+)−1 P(s − L+)−1P = [s − PL+P − PL+Q(s − QL+Q)−1QL+P]−1

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

The Mori-Zwanzig Decomposition

Mori-Projectors P =

q

  • ρq, jL

q

ρq, jL

q

  • and Q = 1 − P

Laplace Transform ˆ g(s) = LT[g(t)] = i ∞

0 e−istg(t)dt

Propagator ˆ U(s) = (s − L+)−1 P(s − L+)−1P = [s − PL+P − PL+Q(s − QL+Q)−1QL+P]−1

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

Equations of Motion

0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) + t dτL(q, t − τ)φ(q, τ)

◮ Ω2 q ∝

  • ρq|L+jL

q

jL

q|L+ρq

  • ◮ νq =
  • jL

q|L+jL q

  • ◮ M(q, t) ∝
  • jL

q|L+Q exp(tQL+Q)QL+jL q

  • ◮ L(q, t) ∝
  • ρq|L+Q exp(tQL+Q)QL+jL

q

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

Equations of Motion

0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) + t dτL(q, t − τ)φ(q, τ)

◮ Ω2 q ∝

  • ρq|L+jL

q

jL

q|L+ρq

  • ◮ νq =
  • jL

q|L+jL q

  • ◮ M(q, t) ∝
  • jL

q|L+Q exp(tQL+Q)QL+jL q

  • ◮ L(q, t) ∝
  • ρq|L+Q exp(tQL+Q)QL+jL

q

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

Equations of Motion

0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) + t dτL(q, t − τ)φ(q, τ)

◮ Ω2 q ∝

  • ρq|L+jL

q

jL

q|L+ρq

  • ◮ νq =
  • jL

q|L+jL q

  • ◮ M(q, t) ∝
  • jL

q|L+Q exp(tQL+Q)QL+jL q

  • ◮ L(q, t) ∝
  • ρq|L+Q exp(tQL+Q)QL+jL

q

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

Equations of Motion

0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) + t dτL(q, t − τ)φ(q, τ)

◮ Ω2 q ∝

  • ρq|L+jL

q

jL

q|L+ρq

  • ◮ νq =
  • jL

q|L+jL q

  • ◮ M(q, t) ∝
  • jL

q|L+Q exp(tQL+Q)QL+jL q

  • ◮ L(q, t) ∝
  • ρq|L+Q exp(tQL+Q)QL+jL

q

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

Equations of Motion

0 = (∂2

t + νq∂t + Ω2 q)φ(q, t) +

t dτM(q, t − tτ)∂τφ(q, τ) + t dτL(q, t − τ)φ(q, τ)

◮ Ω2 q ∝

  • ρq|L+jL

q

jL

q|L+ρq

  • ◮ νq =
  • jL

q|L+jL q

  • ◮ M(q, t) ∝
  • jL

q|L+Q exp(tQL+Q)QL+jL q

  • ◮ L(q, t) ∝
  • ρq|L+Q exp(tQL+Q)QL+jL

q

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

The Mode Coupling Approximation

exp(tQL+Q) ≈ |ρkρp ρkρp| exp(tQL+Q)ρkρp ρkρp| ≈ |ρkρp φ(k, t)φ(p, t) ρkρp|

◮ M(q, t) ∝

  • jL

q|L+Qρkρp

ρkρpQL+|jL

q

  • φ(k, t)φ(p, t)

◮ L(q, t) ∝ ρq|L+Qρkρp

  • ρkρpQ|L+jL

q

  • φ(k, t)φ(p, t)
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Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems

The Mode Coupling Approximation

exp(tQL+Q) ≈ |ρkρp ρkρp| exp(tQL+Q)ρkρp ρkρp| ≈ |ρkρp φ(k, t)φ(p, t) ρkρp|

◮ M(q, t) ∝

  • jL

q|L+Qρkρp

ρkρpQL+|jL

q

  • φ(k, t)φ(p, t)

◮ L(q, t) ∝ ρq|L+Qρkρp

  • ρkρpQ|L+jL

q

  • φ(k, t)φ(p, t)
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Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems

The Mode Coupling Approximation

exp(tQL+Q) ≈ |ρkρp ρkρp| exp(tQL+Q)ρkρp ρkρp| ≈ |ρkρp φ(k, t)φ(p, t) ρkρp|

◮ M(q, t) ∝

  • jL

q|L+Qρkρp

ρkρpQL+|jL

q

  • φ(k, t)φ(p, t)

◮ L(q, t) ∝ ρq|L+Qρkρp

  • ρkρpQ|L+jL

q

  • φ(k, t)φ(p, t)
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Outline Microscopic Dynamics Averages Equation of Motion The Mode-Coupling Approximation Open Problems

Simplifying Assumptions

◮ Positions and (precollisional) Velocities are uncorrelated

(Molecular Chaos)

◮ The velocity distribution factorizes ◮ The second moment of the one particle velocity pdf exists

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

Result

◮ νq = 1+ǫ 3 ωE[1 + 3j′′ 0(qd)] ◮ Ωjρ = qT ◮ Ωρj = qT

1+ǫ

2 + 1−ǫ 2 Sq

  • ◮ V 2

q = T Sq

1+ǫ

2 + 1−ǫ 2 Sq

  • M(q, t) =1 + ǫ

2 ∞ d3kSkS|q−k| × {[ˆ q · k]ck + [ˆ q · (q − k)]c|q−k|}2φ(k, t)φ(|q − k|, t) Direct Correlation Function ck := 1 − S−1

k

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

Open Problems

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

Better Distribution Function

◮ Avoid Factorization of velocities ◮ Avoid Factorization between Positions and Velocities ◮ Non Gaussian Velocity PDF not so important

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

Integration through Transients

◮ Start from Elastic Hard Spheres ◮ Switch on Inelasticity & Driving ◮ Treat as Pertubation in ITT formalism ◮ Does not work

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

Integration through Transients

◮ Start from Elastic Hard Spheres ◮ Switch on Inelasticity & Driving ◮ Treat as Pertubation in ITT formalism ◮ Does not work

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

Other Models

Easy (I think)

◮ Other Interactions ◮ (Additional) overall viscous damping

Hard (I am afraid)

◮ Nonrandom Driving Force ◮ Boundary Driving

Shearing via ITT

◮ What reference state to use? ◮ Include more Correlation Functions? ◮ Interpretation of nonlinear equations

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

Other Models

Easy (I think)

◮ Other Interactions ◮ (Additional) overall viscous damping

Hard (I am afraid)

◮ Nonrandom Driving Force ◮ Boundary Driving

Shearing via ITT

◮ What reference state to use? ◮ Include more Correlation Functions? ◮ Interpretation of nonlinear equations

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

Other Models

Easy (I think)

◮ Other Interactions ◮ (Additional) overall viscous damping

Hard (I am afraid)

◮ Nonrandom Driving Force ◮ Boundary Driving

Shearing via ITT

◮ What reference state to use? ◮ Include more Correlation Functions? ◮ Interpretation of nonlinear equations

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

Summary

◮ MCT can be generlized to NESS ◮ Predicts a Glass Transition for a values of ε ◮ Loss of Detailed Balance is clearly visible ◮ There is room for improvement

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

Thank you for your attention

◮ MCT can be generlized to NESS ◮ Predicts a Glass Transition for a values of ε ◮ Loss of Detailed Balance is clearly visible ◮ There is room for improvement