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

Transcription

1 SUMMER SCHOOL - KINETIC EQUATIONS LECTURE I: TRANSPORT PICTURES AND DISSIPATION 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 Quantum transport in narrow geometries. Thin plates, narrow tubes. Sub-band modeling. L3 Semiclassical transport in narrow geometries. Free energy models. Treating complex geometries. Effective mass approximations in thin tubes. L4 Homogenization in periodic media.

3 EXAMPLE: Confinement in one direction. THIN PLATES, A SOI (SILICON OXIDE ON INSULATOR) DEVICE

4 EXAMPLE: Confinement in two directions. NARROW PIPES, AN ION CHANNEL

5 MATERIALS Webpage: Google: Christian Ringhofer courses Direct: chris/shanghai11/shanghai11.htm

6 CLASSICAL HAMILTONIAN SYSTEMS Conservation of an energy E(x, p) x : position, p : momentum f (x, p) : density of particles The Hamilton - Jacobi equations t f + [E, f ] c = 0, The classical commutator (the Poisson bracket): [E, f ] c := p E x f x E p f Measurements: A(x, p) E[A] = A(x, p)f (x, p, t) dxp

7 The classical commutator conserves any function of the energy E because of the cyclicity of the trace under the expectation. a[b, c] dxp = [a, b]c dxp t φ(e)f (x, p, t) dxp = φ(e)[e, f ] dxp = [φ(e), E]f dxp = 0 The classical energy in a vacuum: E(x, p) = 1 2m p 2 + V(x) = E kin + E pot The classical Liouville equation: t f + p m xf x V p f = 0

8 HAMILTONIAN SYSTEMS IN THE SCHRÖDINGER PICTURE 06 The Schrödinger equation for the evolution of the wave function ψ(x, t) under the Hamiltonian H. i t ψ(r, t) = Hψ = 2 2m rψ + V(r, t)ψ ψ(r, t) (the state of the system) plays the role of a (time variant) coordinate system. Observable quantities are replaced by self - adjoint operators. A: Self adjoint operator. E[A] = ψ(r, t) A[ψ](r, t) dr A self adjoint E(A) R

9 EXAMPLES Example 1: The local mass density at r = y. A[ψ](r, t) = δ(r y)ψ(r, t) n(y, t) = E(A) = ψ(y, t) 2 Example 2: The local momentum density at r = y. A[ψ](r, t) = δ(r y)i x ψ(r, t) J(y, t) = E(A) = 2Re[i ψ(y, t) y ψ(y, t)]

10 DENSITY MATRICES AND THE HEISENBERG PICTURE ρ(r, s, t): density matrix for the mixed state, (p n occupation probability of the state n) ρ(r, s, t) = n p n ψ(r, t)ψ(s, t), Expectation of observables: a(r, s): integral kernel of the operator A. A[ψ](r, t) = a(r, s)ψ(s, t) ds Expectation of A expressed via the density matrix: E[A] = Tr(a ρ) = a(r, s)ρ(s, r, t) drs The density matrix describes a probabilistic system or, alternatively, a many particle system.

11 EVOLUTION OF THE DENSITY MATRIX If all wave functions ψ n satisfy the same Schrödinger equation (identical and independent states), ρ(r, s, t) satisfies the quantum Liouville (Heisenberg) equation. i t ρ = H d ρ ρ H d =: [H d, ρ] d := [H, ρ] H d : integral kernel of the Hamiltonian H : H d (r, s) = 2 2m δ(r s) + V(r)δ(r s) H d ρ(r, s) = H d (r, q)ρ(q, s) dq [.,.]: Commutator [H, ρ] = 2 2m ( s r )ρ + (V(r) V(s))ρ

12 CONSERVATION The Heisenberg equation conserves any function of the Hamiltonian (the energy). Let φ : R R A = φ(h), a = φ(h d ) H[ψ](s) = u n (r) λ n (r)u n (s)ψ(s) ds n φ(h)[ψ](s) = n u n (r) φ(λ n (r))u n (s)ψ(s) ds (for Hilbert - Schmitt operators) [a, H d ] = 0 t E[A] = t Tr[a ρ] = 1 i Tr(a[H d, ρ]) = 1 i Tr([a, H d], ρ) = 0

13 WIGNER TRANSFORM AND WIGNER FUNCTIONS The Wigner transform: Given any linear operator A with an integral kernel ( a density matrix) a(x, y), i.e. A[ψ](x) = a(x, y)ψ(y) dy, we define the corresponding Wigner function W[a](x, ξ) = a(x 1 2 y, x y)eiy ξ dy, f (x, ξ, t) = W[ρ](x, ξ, t) Conversely, given any function g(x, ξ) we define the corresponding density matrix and operator via the inverse Wigner transform W 1 [g](r, s) = (2π) 3 g( r+s 2, ξ)eiξ (r s) dξ Self adjoint operators and density matrices correspond to real functions. a d (r, s) = a d (s, r) W[a](x, ξ) R

14 Traces of matrix products become L 2 products Tr(a b) = a(r, s)b(s, r) drs = (2π) 3 W[a](x, ξ)w[b](x, ξ) dxξ Expectations of operators: E(A) = Tr(a ρ) = W[a]W[f ](x, ξ) dxξ The Wigner transform relates the quantum mechanical to the classical picture. Observables are computed as integrals against the Wigner function in the same way as in the classical setting.

15 THE HAMILTONIAN STRUCTURE OF THE WIGNER EQUATION 16 The commutator in the Wigner picture can be conveniently expressed in terms of the operator product of two functions. Given two Wigner functions f (x, ξ), g(x, ξ), we define the operator product of f and g as the Wigner transform of the matrix product of the inverse Wigner transforms of f and g: f g = W[W 1 [f ] W 1 [g]] This can be written in pseudo differential operator notation as f g(x, ξ) = f (x+ i 2 ξ, ξ i 2 x)g(x, ξ) = g(x i 2 ξ, ξ+ i 2 x)f (x, ξ) Note: is, like the matrix product, not commutative!

16 Therefore the Wigner equation can be written in terms of the Wigner commutator as t f + [E, f ] w = 0 [E, f ] w = i (E f f E) = E(x + νi 2 ξ, ξ νi 2 x)f (x, ξ) ν=±1 For the classical energy in a vacuum E = 2 2m ξ 2 + V(x), we get [ 2 2m ξ 2 + V(x), f ] w = m ξ xf i [V(x + 1 2i ξ) V(x 1 2i ξ)]f or, written in terms of the momentum p = ξ [ 2 2m ξ 2 + V(x), f ] w = 1 m p xf i [V(x + 2i p) V(x 2i p)]f

17 THE WIGNER EQUATION t f + 1 m p xf i [V(x + 2i p) V(x 2i p)]f = 0 For 0 (formally) this reduces to the classical commutator [.,.] c.

18 THERMODYNAMICS AND ENTROPY 20 Interacting particles. information loss (entropy), system loses time reversibility. driven to a thermodynamic equilibrium. In classical systems particle collisions modeled by detailed physics. Quantum mechanical collision operators still an open problem. t f + [E, f ] = Q[f ] Q: models interactions of particles with each other or with a background.

19 BGK - TYPE OPERATORS κ(f ): quantities, conserved by the interaction. G: (entropy-) functional, dissipated by the interaction until equilibrium is reached. BGK operator: Relaxation against equilibrium. t f + [E, f ] = Q[f ], Q[f ] = 1 τ (M[f ] f ) τ: relaxation time. M[f ] maximizes G under the constraint that κ(f ) = κ(m[f ]) holds. Maximum Likelihood estimator.

20 A PROBABILISTIC DEFINITION OF ENTROPY States s 1, s 2,... assumed with probability p 1, p 2,... Experiments: s k1, s k2,.., s kn Entropy production rate I: information gained by making one additional experiment. Assumptions: I(s kn+1 ) = G(p kn+1 ) Experiments are independent. I(s 1, s 2 ) = I(s 1 ) + I(s 2 ) = G(p 1 p 2 ) : G(p) = K B ln(p), K B : Boltzmann constant. Entropy (expectation of I): G(P) = K B k p k[ln(p k ) 1], P = {p 1, p 2,...} Von Neumann entropy.

21 THE CLASSICAL CASE 24 1 κ[f ](r) == δ(x r) p f (x, p) dxdp p 2 2 G(f ) = f (ln f 1) dxdp Solution of the constrained minimization problem: M(x, p) = exp[λ 0 f (x) + λ1 f (x) p λ2 p 2 f (x) 2 ] M: Maxwellian, λ f : chemical potentials (quasi - Fermi levels) chosen such that κf = κm holds. Levermore closures.

22 QUANTUM ENTROPY Density matrix for the mixed state: ρ(x, y) = k p k ψ(x)ψ(y), p k : probability of the state k. G(P) = k p k(ln p k 1) = Tr[ρ(ln ρ id)] G is the negative physical entropy! Observable: κ: self adjoint observation operator m = Tr[κ ρ] Constrained minimization problem for the density matrix M[ρ] Tr[M[ρ](ln M[ρ] id)] = min{tr[u(ln u id)] u : Tr(κ u) = Tr(κ ρ)}

23 EXAMPLE Observe local particle density for all positions. Observation operator: κ(r, s) = δ(x r)δ(r s). Observable: m = Tr[κρ] = ρ(x, x) Compute Mas Tr[M(ln M id)] min, M(x, x) = ρ(x, x), r

24 Solution of the minimization problem G(u) min, Tr(κ u) = Tr(κ ρ) u = M[ρ] Solve constrained minimization problem using Lagrange multipliers for operators (matrices). DG(M) u = 0, u T, T := { u : Tr[κ u] = 0} T: Tangent space of the manifold given by the constraint. DG(M) T Parameterize the dual space to T by Lagrange multipliers.

25 Theorem: (Degond, CR, 02) Given a function γ(z) : R R. This induces an operator Γ(ρ) on the space of self adjoint Hilbert - Schmitt operators. ρ[ψ](r) = φ n (r)λ n φ n (s)ψ(s) ds n φ n : orthogonal eigensystem of ρ, λ n : eigenvalues. Γ(ρ)[ψ](r) = φ n (r)γ(λ n ) φ n (s)ψ(s) ds n In the same way γ (z) induces Γ (ρ). Γ(ρ)[ψ](r) = φ n (r)γ (λ n ) n φ n (s)ψ(s) ds

26 Theorem: (Degond, CR, 02) DΓ: Frechet derivative of Γ ideally: Theorem: DΓ(ρ)( ρ)] = Γ (ρ) ρ (not true) Tr[DΓ(ρ)( ρ)] = Tr[Γ (ρ) ρ]

27 APPLICATION 35 For the Von Neumann entropy, set γ(z) = z[ln(z) 1] Γ(u) = u [ln(u) 1] Tr[DΓ[M]u] = Tr[ln(M) u] = 0, u T, u : Tr(κ u) = 0 ln(m) = λκ, M = exp(λκ) with λ the Lagrange multiplier, determined by Tr(κ M) = Tr(κ ρ)

28 THE QUANTUM BGK OPERATOR Given observables κ, conserved by the particle - particle interaction. Choose the density matrix M[ρ] as M[ρ] = exp[λ ρ κ] with the Lagrange multipliers λ ρ determined by the equation Define the BGK operator Q as Solve the BGK system for the many body problem. Tr(κ M[ρ]) = Tr(κ ρ) Q[ρ] = 1 (M[ρ] ρ) τ i t ρ = [H, ρ] + i Q[ρ]

29 Theorem (Degond, Mehats, CR 05): This system dissipates the Von Neumann entropy Tr[ρ (ln(ρ) 1)] until an equilibrium ρ = M[ρ] is reached.

30 HOW TO COMPUTE THE EXPONENTIAL OF AN OPERATOR Compute M[ρ] as with λ ρ given by M = exp[ λ ρ κ], Tr(κ M) = Tr(κ ρ) The Bloch equation: Imbedding: Compute u(β) = exp( βλκ) for 0 β 1. β: inverse temperature. β u = (λ ρ κ) u, u(β = 0) = id, Tr(κ u(β = 1)) = Tr(κ ρ) M = u(β = 1)

31 APPROACH I: Solve nonlinear problem directly via Newton method Newton (Mehats 06) F(λ) = Tr(κ (u(β = 1) ρ)), F(λ) = 0 u = u(λ), u(λ + λ) u + u β u = ( λκ) u (λκ) u, u(β = 0) = 0, λ = Tr(κ (u ρ)) Tr(κ u)

32 APPROACH II Use classical approximation in the Wigner picture as pre-conditioner (Degond, CR 05): W[κ u] = W[κ] W[u] W(κ)W(u) + 2 R(u) β W[u n+1 ] = W[λκ]W[u n+1 ] W[λκ 2 R(u n )] W[κ]W[u n+1 (β = 1)] dxξ = W[κ]W[ρ] dxξ

33 BACKGROUND SCATTERING The above approach handles interaction between particles. For the applications in the next two lectures we will consider scattering with a background (i.e. phonons or charges on the walls of ion channels). The von Neumann entropy has to be replaced by a relative von Neumann entropy of the form G(ρ) = Tr[ρ (ln(ρ) + H id)] (meaning the background particles are in equilibrium.) This means for the Maxwellian M[ρ] = exp( λκ H)

34 SUMMARY The quantum mechanicl transport picture is given by i t ρ = [H, ρ] + i Q[ρ] Q[ρ] = 1 (M[ρ] ρ) τ M[ρ] = exp( λ ρ κ H), Tr(κ M) = Tr(κ ρ) in either the density matrix (Heisenberg) picture or the Wigner picture. M[ρ] has to be computed by nonlinear iteration. The two formulations (density matrix and Wigner) are equivalent. The Wigner transform allows us to relate to the classical picture by eliminating high frequency oscillations.

35 OUTLOOK For Lectures II-IV we will consider the general transport model. i t ρ = [H, ρ] + i Q[ρ], [H, ρ] = H ρ ρ H Q[ρ] = 1 (M[ρ] ρ) τ or in the Wigner picture t f + [E, f ] w = Q[ρ], [E, f ] w = E f f E Q[f ] = 1 τ (M[f ] f ) or (equivalently), in the classical case (lim for 0 t f + [E, f ] c = Q[ρ], [E, f ] c = p E x f p E x f Q[f ] = 1 τ (M[f ] f )

36 Approximate (simplified) transport picture in the presence of different length scales 8 6 G 4 y (nm) 2 S D 0 2 B x (nm)

37 Approximate (simplified) transport picture in the presence of different length scales Thin plates: quantum transport in simple geometries. GATE CONTACT SOURCE DRAIN BULK CONTACT

38 Approximate (simplified) transport picture in the presence of different length scales Thin tubes: Complex geometry given by confining potentials. (classical transport picture).