summer school - kinetic equations lecture ii: …chris/shanghai11/l2b.pdfquantum and classical...

SUMMER SCHOOL - KINETIC EQUATIONSLECTURE II: CONFINED CLASSICAL

TRANSPORTShanghai, 2011.

C. Ringhofer

[email protected] , math.la.asu.edu/∼chris

INTRODUCTION CONFINEMENT AND SCALING LARGE TIME DYNAMICS HARMONIC APPROXIMATION RESULTS

OVERVIEW

Quantum and classical description of many particle systems inconfined geometries and periodic media.L1 Introduction. Transport pictures (Hamilton, Schrodinger,

Heisenberg, Wigner).Thermodynamics and entropy.

Two different spatial scales. Induce two different time scales.Macroscopic (simpler) description on the large scale. Retain themicroscopic transport information on the small scale.

L2 Semiclassical transport in narrow geometries.Free energy models.Treating complex geometries.Effective mass approximations in thin tubes.

L3 Quantum transport in narrow geometries.Thin plates, narrow tubes.Sub-band modeling.

L4 Homogenization in unstructured media and networks.


CLASSICAL TRANSPORT WITH BGK OPERATOR

∂tf + [E , f ]c = Q[f ]

f = f (X,P, t): X: position P: momentum

[E , f ]c: classical commutator

[E , f ]c = ∇PE · ∇Xf −∇XE · ∇Pf

Q[f ]: BGK collision operator, conserving an observable κ with anequilibriumM

Q[f ] =1τ

(Mρ[f ]− f )

ρ: Lagrange multiplier functions∫κMρ[f ] dXP =

∫κf (X,P) dXP


OUTLINE

Model transport complicated geometric structures (irregularpipes) on large time scales.

Sub-band modeling for classical transport in ion channels.

Physical principle: Strong confinement to a narrow region. ⇒Scattering processes partially conserving the energy.

Large time asymptotics⇒ diffusion equations with free energy.

Computation of transport coefficients from approximateconfinement potentials.

Results.


TRANSPORT IN NARROW PIPES

Confinement produced by repulsive charges on pipe walls.

Typical example: Ion channels (proteins in cell walls).

Transport of ions in water. Scattering with background (water).Confinement due to repulsive charges on protein walls.

Hamiltonian dynamics + collisions with a background confinedgeometries.


SCHEMATIC ION CHANNEL


PROTEIN 1LNQ


MODEL HIERARCHIES FOR ION CHANNELS

1 Molecular Dynamics. (compute the background). Largecomputation (Roux) (O(10−7)sec).

2 Monte Carlo. (collisions with random background, Brownianmotion or Boltzmann) (Ravaioli, Saraniti) O(10−5 − 10−6sec).

3 Time scale for device function: O(10−3sec)

4 Macroscopic models (Hydrodynamics, Diffusion etc.) (Jerome,Peskin) Loss of geometric information.

1-3: physical transport mechanisms.4: function of the structure on large time scales.


GOAL

Develop a model that is computationally feasible on large time scaleswhile still incorporating more microscopic geometric information intothe model.

Compromise: One more free variable than standardhydrodynamics. Cross directional energy . ’A squeezed SHEmodel’.

Approach: Sub-band modeling. Treat transport in theconfinement direction on a microcopic level. Average kineticequations in the transport direction on long time scales.


CONFINED GEOMETRY

Transport equation

∂tf + [E , f ]c = Q[f ]

[E , f ]c = ∇PE · ∇Xf −∇XE · ∇Pf

X ∈ Ω: narrow region

Split:

X = (x, y), P = (p, q), Ω = Ωx × Ωy

x ∈ Ωx, y ∈ Ωy

|Ωy| << |Ωx|, |Ωy| ≈ ε|Ωx|

ε: aspect ratio.

Plates: Ωx ⊂ R2, Ωy ⊂ RPipes:Ωx ⊂ R, Ωy ⊂ R2


SCALING→ DIMENSIONLESS EQUATIONS

Ωy given by strong confinement in a narrow region y << x.Forces in confinement direction large compared to forces intransport direction∇yV >> ∇xV .

V(x) = V0(x) + V1(εx, y)

V0(x): gauge potential

Scaling:

y→ εy, q→ qε

E = Exp + Eyq, Eyq → Eyqε

E =|p|2 + |q|2

2m+V(x, y), Exp =

|p|2

2m+V0(x), Eyq =

|q|2

2m+V1(x, y),

ε: aspect ratio.


SCALING THE COLLISIONS

Scattering with a background (water) does not conserve energy,but only mass.

Scattering exchanges a given amount ω of energy per event withbackground (water).

Q[f ] =∫

Sf (P,P′)δ( |P|2

2 −|P′|2

2 ± ω)f (X,P′) dP′ − γ(P)f

After re-scaling P = (p, q), q→ qε

Q[f ] =∫

Sf (P,P′)δ( |p|2

2 + |q|22ε2 − |p

′|22 −

|q′|22ε2 ± ω)f (X,P′) dP′ − γ(P)f

IQ asymptotically conserves (locally in space) the cross directionalkinetic energy |q|

2

2


CONSERVATION IN THE WEAK FORMULATION

Q[f ] =∫

Sf (P,P′)δ( |P|2

2 −|P′|2

2 ± ω)f (X,P′) dP′ − γ(P)f

γ(P) =

∫Sf (P′,P)δ(

|P′|2

2− |P|

2

2± ω) dP′∫

uQ[f ] dpq =

∫[u(p, q)− u(p′, q′)]Sf (P,P′)δ( |q

2|2ε2 − |q

′2|2ε2 )f (p′, q′) dpqp′q′

if u depends only on |q|2:

⇒∫

u(p, |q|2)Q[f ] dpq = 0

I A collision with the background in the confinement direction y isa rare event compared to collisions with the background in thetransport direction x.


SCALED (APPROXIMATE) MODEL EQUATIONS

∂tf + Exp, fxp = −1εEyq, fyq + 1

εQ[f ] = 1εC[f ]

Exp(x, p) =|p|2

2+ V0(x), Eyq(x, y, q) =

|q|2

2+ V1(x, y)

Exp, fxp = ∇x(fp)−∇p·(f∇xV0), Eyq, f = ∇y(fq)−∇q·(f∇yV1)

Thermodynamics (entropy maximizer) in p:

Q[f ] = M(p)∫δ( |p|

2

2 −|p′|2

2 )f (x, y, p′, q′) dp′q′ − γ(P)f

Q conserves |q|2

2 , and is local in (x, y)

The commutator Eyq, ∗yq conserves Eyq

⇒ C conserves Eyq

Large Time Dynamics: (Hilbert or Chapman - Enskog)⇒Diffusion equation with η = Eyq as an additional independentvariable for the density n(x, η, t).


LARGE TIME DYNAMICS- THE CHAPMAN - ENSKOG EXPANSION

General setting for the linear case:

∂tf + Lf =1εCf

C has a set of conserved quantities κ:〈κ, Cf 〉 = 0, ∀f

⇒ slow dynamics for the macroscopic variable 〈κ, f 〉∂t〈κ, f 〉+ 〈κ,Lf 〉 = 0

At the same time, f driven towards the kernel of C.

Parameterize the kernel manifold of C by coordinates given〈κ, f 〉. (κ and the kernel of C have to have the same dimension.)


PROJECTIONS

Projection on the kernelM(ρ), parameterized by ρ.

Pf =M(ρ), 〈κ,Pf 〉 = 〈κ, f 〉

⇒ CPf = CM = 0, ∀f

〈κ,PCf 〉 = 〈κ,M〉 = 〈κ, Cf 〉 = 0⇒ PCf = 0, ∀f

Split solution into

f = f0 + εf1, f0 = Pf , εf1 = (id − P)f

∂tf0 + PL(f0 + εf1) = 0, ε∂tf1 + (id − P)L(f0 + εf1) = Cf1

Large time asymptotics ε→ 0 in the kinetic eaution

∂t〈κ,M(ρ)〉+ 〈κ,L(M(ρ) + εf1)〉 = 0, (id − P)LM(ρ) = Cf1


SUMMARY

Solve:

(id − P)LM(ρ) = Cf1, 〈κ, f1〉 = 0⇐⇒ f1 = C+M

Compute the pseudo - inverse C+ of C.

Solve the macroscopic equation

∂t〈κ,M(ρ)〉+ 〈κ,L(M(ρ) + εC+(id − P)LM)〉 = 0,

for ρ(t).

〈κ,LM〉: convection term〈κ,LC+(id − P)LM〉: diffusion term


∂t〈κ,M(ρ)〉+ 〈κ,L(M(ρ) + εC+(id − P)LM)〉 = 0,

Two cases:

〈κ,LM〉 6= 0: Diffusive a small perturbation of convection(Navier - Stokes regime).

〈κ,LM〉 = 0: ( Diffusion)

∂t〈κ,M(ρ)〉+ ε〈κ,LC+LM〉 = 0,

Diffusion equation on tε time scale.


CONFINED BOLTZMANN

∂tf + L[f ] =1εC[f ]

L[f ](x, y, p, q) = [Exp, f ]xp = ∇pExp · ∇xf −∇xExp · ∇pf

C[f ](x, y, p, q) = −[Eyq, f ]yq+M(p)

∫δ(|q|2

2−|q

′|2

2)f (x, y, p′, q′) dp′q′−γf


Conserved quantities: κ = φ(x, Eyq(x, y, q))∫φ(x, Eyq(x, y, q))C[f ] dypq = 0, ∀f

Kernel:

M[ρ] = ρ(x, Eyq(x, y, q))e−|p|2

2 =n(x, Eyq(x, y, q))

N(x, Eyq(x, y, q))e−

|p|22

N: density of states function:

N(x, η) =

∫δ(Eyq(x, y, q)− η) dyq (⇒

∫κM = n)

Projection:P[f ] = ρ(x, Eyq)M(p),∫

φ(x, Eyq)P[f ] dxypq =

∫φ(x, Eyq)f dxypq


Diffusion equation on O( tε2 ) time scale

∂tn +∇x · Fx + ∂ηFη = 0, Fx = Fx(∇xn, ∂ηn), Fη = Fη(∇xn, ∂ηn)

η ≈ Eyq: energy in the confinement direction.

Fx,Fη : Fluxes in space and energy, given by

∇x · Fx + ∂ηFη = −∫δ(Eyq − η)LC+(id − P)L

nD

dypq


PRACTICAL PROBLEM

∇x · Fx + ∂ηFη = −∫δ(Eyq − η)LC+(id − P)L

nD

dypq

The computation of the fluxes Fx,Fη requires the inversion of theaugmented collision operator C = Q[∗]− Eyq, ∗yq.

No exact solution. Has to be done numerically. Solve a2 ∗ dim(y)− 1 dimensional problem for every grid point in (x, η)to compute the transport coefficients.

Plates: dim(y) = 1, solve a 1-D problem at any gridpoint tocompute transport coefficients for a 3-D diffusion equation.Pipes: dim(y) = 2, solve a 3-D problem at any gridpoint tocompute transport coefficients for a 2-D diffusion equation.

Reduce computational complexity by approximating V1(x, y) bya harmonic potential in y.


COMPUTATION OF THE PSEUDO INVERSE C+

−[Eyq, f1] + M(p)

∫δ(Eyq − E ′yq)f ′1 dypq− γf1 = L[e−

|p|22

nD

(x, Eyq]

⇒ equation for the averages of f1 over equipotential surfacesEyq = |q|2

2 + V1(x, y) = const.

Harmonic potential approximation:V1(x, y) quadratic in confinement variable y with x− dependentcoefficients.

V1(x, y) =12

(y− b(x))TG(x)(y− b(x))

Equipotential surfaces become ellipsoids in R4.f1 written in coordinates of the energy Eyq and a threedimensional angle in R4.Invert C exactly in two of the three angular dimensions.Solve a one dimensional problem in the azimuthal anglenumerically.


Choice of b and G

For every gridpoint in the transport direction x, solve an L2

minimization problem for the forces in the confinement direction y.∫Ωy|∇yV1 − G(x)(y− b(x))|2 dy→ min, ∀x

Flux computation (the inversion of C) can be carried out exactlyin 2 of the 3 dimensions. Reduced to an effective 1-D problem.

Variable transformation:

(y, q)↔ (E(y, q), θ, α, β)

α, β ∈ [−π, p], θ ∈ [0, π]

C diagonal in α, β. Legendre polynomials in cos(θ).


APPROXIMATION QUALITY

A toy channel

01

23

45

−4

−2

0

2

4−4

−3

−2

−1

0

1

2

3

4

xy1

y2

05

−4−3−2−101234

−4

−3

−2

−1

0

1

2

3

4

xy1

y2

Generate random charges.

Compute the exact Coulomb Potential V(x, y) corresponding tothese charges, and the local quadratic approximation.


TRAJECTORIES

Molecular dynamics trajectories (x′ = p, p′ = −∇xV) for the exactand approximate Coulomb potential and random initial conditions.

02

4

−0.6−0.4−0.200.20.40.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xy1

y2


THE LARGE TIME DIFFUSION SYSTEM

∂tn +∇x · Fx + ∂ηFη = 0

Fx = −NDx∇xnN− Nµx(1 + ∂η)

nN,

Fη = −NDη(1 + ∂η)nN− Nµη · ∇x

nN,

N,Dx,Dη, µx, µη Functionals of G(x), b(x) and of V , (computednumerically).

Yields a parabolic system for n.


ENTROPY AND PARABOLICITY

Theorem:(C. Heitzinger, CR, CMS’11)

∂t∫ eV0 n(x,η)2

N(x,η) dxη ≤ 0

(implies)(

Dx µx

Dη µη

)≤ 0


Protein 1LNQ, Charges for actual potential


NUMERICAL RESULTS (DENSITY n)


NUMERICAL RESULTS (FLUXES Fx,Fη)


CONCLUSION

Incorporate arbitrary geometries in semiclassical transport.

Evolution on time scales comparable to functional time scales ofthe channel.

Q: Verification?

Q: Quantum transport?

summer school - kinetic equations lecture ii: …chris/shanghai11/l2b.pdfquantum and classical...

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