SUMMER SCHOOL - KINETIC EQUATIONS LECTURE II: CONFINED CLASSICAL TRANSPORT Shanghai, 2011.

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1 SUMMER SCHOOL - KINETIC EQUATIONS LECTURE II: CONFINED CLASSICAL TRANSPORT Shanghai, C. Ringhofer ringhofer@asu.edu, math.la.asu.edu/ chris

2 OVERVIEW Quantum and classical description of many particle systems in confined geometries and periodic media. L1 Introduction. Transport pictures (Hamilton, Schrödinger, Heisenberg, Wigner). Thermodynamics and entropy. Two different spatial scales. Induce two different time scales. Macroscopic (simpler) description on the large scale. Retain the microscopic 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.

3 CLASSICAL TRANSPORT WITH BGK OPERATOR t f + [E, f ] c = Q[f ] f = f (X, P, t): X: position P: momentum [E, f ] c : classical commutator [E, f ] c = P E X f X E P f Q[f ]: BGK collision operator, conserving an observable κ with an equilibrium M Q[f ] = 1 τ (M ρ[f ] f ) ρ: Lagrange multiplier functions κm ρ [f ] dxp = κf (X, P) dxp

4 OUTLINE Model transport complicated geometric structures (irregular pipes) 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 approximate confinement potentials. Results.

5 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 confined geometries.

6 SCHEMATIC ION CHANNEL

7 PROTEIN 1LNQ

8 MODEL HIERARCHIES FOR ION CHANNELS 1 Molecular Dynamics. (compute the background). Large computation (Roux) (O(10 7 )sec). 2 Monte Carlo. (collisions with random background, Brownian motion or Boltzmann) (Ravaioli, Saraniti) O( sec). 3 Time scale for device function: O(10 3 sec) 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.

9 GOAL Develop a model that is computationally feasible on large time scales while still incorporating more microscopic geometric information into the model. Compromise: One more free variable than standard hydrodynamics. Cross directional energy. A squeezed SHE model. Approach: Sub-band modeling. Treat transport in the confinement direction on a microcopic level. Average kinetic equations in the transport direction on long time scales.

10 CONFINED GEOMETRY Transport equation X Ω: narrow region Split: x Ω x, y Ω y ε: aspect ratio. Plates: Ω x R 2, Ω y R Pipes:Ω x R, Ω y R 2 t f + [E, f ] c = Q[f ] [E, f ] c = P E X f X E P f X = (x, y), P = (p, q), Ω = Ω x Ω y Ω y << Ω x, Ω y ε Ω x

11 SCALING DIMENSIONLESS EQUATIONS Ω y given by strong confinement in a narrow region y << x. Forces in confinement direction large compared to forces in transport direction y V >> x V. V 0 (x): gauge potential Scaling: V(x) = V 0 (x) + V 1 (εx, y) y εy, q q ε E = E xp + E yq, E yq Eyq ε E = p 2 + q 2 +V(x, y), 2m ε: aspect ratio. E xp = p 2 2m +V 0(x), E yq = q 2 2m +V 1(x, y),

12 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 with background (water). Q[f ] = S f (P, P )δ( P 2 2 P 2 2 ± ω)f (X, P ) dp γ(p)f After re-scaling P = (p, q), q q ε Q[f ] = S f (P, P )δ( p q 2 p 2 2ε 2 2 q 2 ± ω)f (X, P ) dp γ(p)f 2ε 2 Q asymptotically conserves (locally in space) the cross directional kinetic energy q 2 2

13 CONSERVATION IN THE WEAK FORMULATION Q[f ] = S f (P, P )δ( P 2 2 P 2 2 ± ω)f (X, P ) dp γ(p)f γ(p) = S f (P, P)δ( P 2 2 P 2 ± ω) dp 2 uq[f ] dpq = [u(p, q) u(p, q )]S f (P, P )δ( q2 2ε 2 q 2 2ε 2 )f (p, q ) dpqp q if u depends only on q 2 : u(p, q 2 )Q[f ] dpq = 0 A collision with the background in the confinement direction y is a rare event compared to collisions with the background in the transport direction x.

14 SCALED (APPROXIMATE) MODEL EQUATIONS t f + {E xp, f } xp = 1 ε {E yq, f } yq + 1 ε Q[f ] = 1 ε C[f ] E xp (x, p) = p V 0(x), E yq (x, y, q) = q V 1(x, y) {E xp, f } xp = x (fp) p (f x V 0 ), {E yq, f } = y (fq) q (f y V 1 ) 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 {E yq, } yq conserves E yq C conserves E yq Large Time Dynamics: (Hilbert or Chapman - Enskog) Diffusion equation with η = E yq as an additional independent variable for the density n(x, η, t).

15 LARGE TIME DYNAMICS- THE CHAPMAN - ENSKOG EXPANSION General setting for the linear case: t f + Lf = 1 ε Cf 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.)

16 PROJECTIONS Projection on the kernel M(ρ), parameterized by ρ. Pf = M(ρ), κ, Pf = κ, f CPf = CM = 0, f Split solution into κ, PCf = κ, M = κ, Cf = 0 PCf = 0, f f = f 0 + εf 1, f 0 = Pf, εf 1 = (id P)f t f 0 + PL(f 0 + εf 1 ) = 0, ε t f 1 + (id P)L(f 0 + εf 1 ) = Cf 1 Large time asymptotics ε 0 in the kinetic eaution t κ, M(ρ) + κ, L(M(ρ) + εf 1 ) = 0, (id P)LM(ρ) = Cf 1

17 SUMMARY Solve: (id P)LM(ρ) = Cf 1, κ, f 1 = 0 f 1 = 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

18 t κ, M(ρ) + κ, L(M(ρ) + εc + (id P)LM) = 0, Two cases: κ, LM 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 t ε time scale.

19 CONFINED BOLTZMANN t f + L[f ] = 1 ε C[f ] L[f ](x, y, p, q) = [E xp, f ] xp = p E xp x f x E xp p f C[f ](x, y, p, q) = [E yq, f ] yq +M(p) δ( q q 2 )f (x, y, p, q ) dp q γf

20 Conserved quantities: κ = φ(x, E yq (x, y, q)) φ(x, E yq (x, y, q))c[f ] dypq = 0, f Kernel: M[ρ] = ρ(x, E yq (x, y, q))e p 2 2 = n(x, E yq(x, y, q)) p 2 N(x, E yq (x, y, q)) e 2 N: density of states function: N(x, η) = δ(e yq (x, y, q) η) dyq ( κm = n) Projection: P[f ] = ρ(x, E yq )M(p), φ(x, E yq )P[f ] dxypq = φ(x, E yq )f dxypq

21 Diffusion equation on O( t ε 2 ) time scale t n + x F x + η F η = 0, F x = F x ( x n, η n), F η = F η ( x n, η n) η E yq : energy in the confinement direction. F x, F η : Fluxes in space and energy, given by x F x + η F η = δ(e yq η)lc + (id P)L n D dypq

22 PRACTICAL PROBLEM x F x + η F η = δ(e yq η)lc + (id P)L n D dypq The computation of the fluxes F x, F η requires the inversion of the augmented collision operator C = Q[ ] {E yq, } yq. No exact solution. Has to be done numerically. Solve a 2 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 to compute transport coefficients for a 3-D diffusion equation. Pipes: dim(y) = 2, solve a 3-D problem at any gridpoint to compute transport coefficients for a 2-D diffusion equation. Reduce computational complexity by approximating V 1 (x, y) by a harmonic potential in y.

23 COMPUTATION OF THE PSEUDO INVERSE C + [E yq, f 1 ] + M(p) δ(e yq E yq)f 1 dypq γf 1 = L[e p 2 2 n D (x, E yq] equation for the averages of f 1 over equipotential surfaces E yq = q V 1(x, y) = const. Harmonic potential approximation: V 1 (x, y) quadratic in confinement variable y with x dependent coefficients. V 1 (x, y) = 1 2 (y b(x))t G(x)(y b(x)) Equipotential surfaces become ellipsoids in R 4. f 1 written in coordinates of the energy E yq and a three dimensional angle in R 4. Invert C exactly in two of the three angular dimensions. Solve a one dimensional problem in the azimuthal angle numerically.

24 Choice of b and G For every gridpoint in the transport direction x, solve an L 2 minimization problem for the forces in the confinement direction y. Ω y y V 1 G(x)(y b(x)) 2 dy min, x Flux computation (the inversion of C) can be carried out exactly in 2 of the 3 dimensions. Reduced to an effective 1-D problem. Variable transformation: α, β [ π, p], θ [0, π] (y, q) (E(y, q), θ, α, β) C diagonal in α, β. Legendre polynomials in cos(θ).

25 y y x y y x APPROXIMATION QUALITY A toy channel Generate random charges. Compute the exact Coulomb Potential V(x, y) corresponding to these charges, and the local quadratic approximation.

26 TRAJECTORIES Molecular dynamics trajectories (x = p, p = x V) for the exact and approximate Coulomb potential and random initial conditions y y x 4

27 THE LARGE TIME DIFFUSION SYSTEM t n + x F x + η F η = 0 F x = ND x x n N Nµ x(1 + η ) n N, F η = ND η (1 + η ) n N Nµ η x n N, N, D x, D η, µ x, µ η Functionals of G(x), b(x) and of V, (computed numerically). Yields a parabolic system for n.

28 ENTROPY AND PARABOLICITY Theorem:(C. Heitzinger, CR, CMS 11) ( ) Dx µ (implies) x 0 D η µ η t e V 0n(x,η) 2 N(x,η) dxη 0

29 Protein 1LNQ, Charges for actual potential

30 NUMERICAL RESULTS (DENSITY n)

31 NUMERICAL RESULTS (FLUXES F x, F η )

32 CONCLUSION Incorporate arbitrary geometries in semiclassical transport. Evolution on time scales comparable to functional time scales of the channel. Q: Verification? Q: Quantum transport?

SUMMER SCHOOL - KINETIC EQUATIONS LECTURE I: TRANSPORT PICTURES AND DISSIPATION Shanghai, 2011.

SUMMER SCHOOL - KINETIC EQUATIONS LECTURE I: TRANSPORT PICTURES AND DISSIPATION Shanghai, 2011. C. Ringhofer ringhofer@asu.edu, math.la.asu.edu/ chris OVERVIEW Quantum and classical description of many