Quantum Hydrodynamics models derived from the entropy principle

Transcription

1 1 Quantum Hydrodynamics models derived from the entropy principle P. Degond (1), Ch. Ringhofer (2) (1) MIP, CNRS and Université Paul Sabatier, 118 route de Narbonne, Toulouse cedex, France (2) Dep. of Math., Arizona State University, Tempe, Arizona , USA chris/

2 Summary 2 1. Introduction 2. Review of classical hydrodynamic theories 3. Quantum setting: basics 4. Review of quantum hydrodynamic theories 5. QHD via entropy minimization 6. Quantum collision operators 7. Quantum Energy-Transport and Drift-Diffusion models 8. Summary and conclusion

3 3 1. Introduction

4 Introduction 4 Classical Quantum Microscopic Boltzmann Quantum models eq. Liouville eq. Macroscopic Fluid mech. (?) models eqs. special cases: Bohmian mech. (1 particle) perturbative ext. to many-particle

5 5 2. Review of classical hydrodynamic theories

6 Microscopic vs macroscopic models 6 Microscopic models: phase-space distribution function f(x, p, t) Boltzmann equation t f + p x f x V p f = Q(f) Macroscopic models: density n(x, t) mean velocity u(x; t) temperature T (x, t)

7 Balance equations (Euler eq.) 7 t n nu n u 2 + 3nT + x = nu nuu + nt Id (n u 2 + 5nT )u 0 n x V nu x V Euler eqs of gas dynamics. Pressure = nt : perfect gas Equation-of-State How to relate macroscopic models to microscopic eq.?

8 Moment method 8 n, q = nu, 2W = n u 2 + 3nT are moments of f: n 1 q = f p dp 2W p 2

9 Moment method 9 Natural idea: (i) multiply Boltzmann eq. by 1, p, p 2 and integrate w.r.t. p: 1 ( t f + p x f x V p f Q(f)) p dp = 0 p 2 (ii) use conservations: 1 Q(f) p p 2 dp = 0

10 Moment method (cont) 10 (iii) Get conservation eqs n 1 q + x f p t 2W p 2 p dp = 0 n x V nu x V Problem: Express fluxes in term of the conserved variables n, q, W fp i p j dp (for i j) and f p 2 p dp cannot be expressed in terms of n, q, W. conservation eqs are not closed

11 Entropy minimization (Gibbs, Levermore) 11 Closure: replace f by a solution of the entropy minimization problem: Let n, T R +, u R 3 fixed. Find min{h(f) = f(ln f 1)dp s.t. 1 n f p dp = nu p 2 n u 2 + 3nT }

12 Entropy minimization (cont) 12 Entropy minimization problem has solution: ) n p u 2 M n,u,t = exp ( (2πT ) 3/2 2T Maxwellian satisfies M n,u,t 1 p p 2 dp = n nu n u 2 + 3nT Gives the Euler eqs. of gas dynamics Levermore closures if more moments are retained

13 Quantum case 13 Idea: use the same idea for quantum models. Not so simple...

14 14 3. Quantum setting: basics

15 Density operator 15 Basic object: ρ: Hermitian, postive, trace-class operator on L 2 (R d ) s.t. Typically: Trρ = 1 ρψ = s S ρ s (ψ, φ s ) φ s for a complete orthonormal system (φ s ) s S and real numbers (ρ s ) s S such that 0 ρ s 1, ρ s = 1

16 Quantum Liouville equation 16 i t ρ = [H, ρ] + Q(ρ) H = Hamiltonian: Hψ = 2 2 ψ + V (x, t)ψ Q(ρ) unspecified: accounts for dissipation mechnisms

17 Wigner Transform 17 ρ(x, x ) integral kernel of ρ: ρψ = ρ(x, x )ψ(x ) dx W [ρ](x, p) Wigner transform of ρ: W [ρ](x, p) = ρ(x 1 2 ξ, x + 1 ξ p ξ) ei 2 dξ

18 Inverse Wigner transform (Weyl quantization) 18 Let w(x, p). ρ = W 1 (w) is the operator defined by: W 1 (w)ψ = 1 w( x + y (2π) d 2, k) ψ(y)eik(x y) dk dy w= Weyl symbol of ρ. Isometries between L 2 (Operators s.t. ρρ is trace-class) and L 2 (R 2d ): Tr{ρσ dx dp } = W [ρ](x, p)w [σ](x, p) (2π ) d

19 Wigner equation 19 Eq. for w = W [ρ]: t w + p x w + Θ [V ]w = Q(w) Θ [V ]w = i (2π) d (V (x + 2 η) V (x 2 η)) w(x, q) e iη (p q) dq dη Θ [V ]w 0 x V p w Q(w) collision operator (unspecified)

20 20 4. Review of quantum hydrodynamic theories

21 1-particle quantum hydrodynamics 21 Single state ψ Decompose i t ψ = 2 2 ψ + V (x, t)ψ ψ = ne is/ and define u = x S. Then take real and imaginary parts t n + x nu = 0 t S S 2 + V n n = 0

22 1-particle QHD (cont) 22 Take of the phase eq. t n + x nu = 0 t u + u x u = x (V + V B ) V B = 2 2 V B = Bohm potential 1 n n Pressureless Gas dynamics w. additional Bohm potential term, of order O( 2 )

23 Many-particle case 23 Procedure: Start w. statistical state, i.e. density operator Average over the probabilities ρ s = Closure problem Closure strategies: Fourier law for the heat flux [Gardner] Small temperature [Gasser, Markowich, Ringhofer] Chapman-Enskog expansion [Gardner, Ringhofer] Entropy minimization [D.,Ringhofer]

24 24 5. QHD via entropy minimization

25 Moments 25 Defined as in classical mechanics: Moments of the Wigner distribution List of monomials µ i (p) e.g. (1, p, p 2 ) µ(p) = (µ i (p)) N i=0 w(x, p) moments m[w] = (m i [w]) N i=0 m i [w] = w(x, p) µ i (p) dp e.g. m = (n, q, W )

26 Moment method Take moments of the Wigner equation: t m[w]+ x w µ p dp+ Θ[V ]w µ dp = Q(w) µ dp 26 Assume conservations Q(w) µ dp = 0 Closure problem: find an expression of the integrals at the l.h.s.

27 Entropy minimization principle 27 Entropy: H[ρ] = Tr{ρ(ln ρ 1)} ; ρ = W 1 (w) Given a set of moments m = (m i (x)) N i=0, minimize H(ρ) subject to the constraint that W [ρ](x, p) µ(p) dp = m(x) x Problem: express the moment constraints in terms of ρ

28 Moments in terms of ρ 28 Dualize the constraint: Let λ(x) = (λ i (x)) N i=0 be an arbitrary (vector) test function w(x, p) µ(p)λ(x) dx dp (2π ) d = m(x) λ(x) dx (2π ) d Tr{ρ W 1 [µ(p) λ(x)]} = m(x) λ(x) dx (2π ) d

29 Entropy minimization principle: expression 29 Given a set of (physically admissible) moments m = (m i (x)) N i=0, solve min{ H[ρ] = Tr{ρ(ln ρ 1)} Tr{ρ W 1 [µ(p) λ(x)]} = λ = (λ i (x)) N i=0 } m λ subject to: dx (2π ) d

30 Solution of the entropy problem 30 Solution is ρ α, ρ α = exp(w 1 [α(x) µ(p)]) α = (α i (x)) N i=0 is determined s.t. m[ρ α] = m w α = W [ρ α ] = Exp(α(x) µ(p)) Exp w = W [exp(w 1 (w))] (Quantum exponential) Analogy with the classical case M α = exp(α µ)

31 Quantum moment models 31 Take moments of the Wigner eq. and close with the quantum Maxwellian: t Exp(α µ) µ dp + x Exp(α µ) µ p dp + Θ[V ]Exp(α µ) µ dp = 0 Provides an evolution system for the vector function α(x, t)

32 Entropy 32 Kinetic entropy H[ρ] in terms of w = W [ρ]: dx dp H[ρ] = Tr{ρ(ln(ρ 1)} = w(ln w 1) (2π ) d with quantum log: Ln w = W [ln(w 1 (w))] Fluid entropy S(m): S(m) = H[ρ α ] = Exp(α µ)((α µ) 1) dx dp (2π ) d where α is s.t. Exp(α µ)µdp = m

33 Entropy (cont) 33 S(m) convex Define: Σ(α), Legendre dual of S Inversion of the mapping α m: δs δm = α, δσ δα = m Moment models compatible with the entropy dissipation t S(m(t)) 0 for any solution m(t) of the QHD equations

34 Quantum Hydrodynamics 34 µ = {1, p, p 2 } t n + x nu = 0 t nu + x (nuu + P) = n x V t W + x (W u + Pu + Q) = nu x V with P = pressure tensor, Q = heat flux: P = Exp(α µ)(p u)(p u)dp 2Q = Exp(α µ) p u 2 (p u)dp

35 Quantum Hydrodynamics (cont) 35 and s.t. α µ = A + B p + C p 2 Exp(α µ)µdp = (n, nu, W ) T r

36 36 6. Quantum collision operators

37 Classical Boltzmann operator 37 Q(w) = B( p p 1, Ω)(w w 1 ww 1 )dp 1 dω p + p 1 = p + p 1 p 2 + p 2 1 = p 2 + p 2 1 Ω S 2 scattering angle; B scattering cross-section Preserves mass, momentum and energy locally Kernel classical maxwellians Dissipates classical entropy Q(w) ln w dw 0

38 Quantum Boltzmann operator 38 Requirements: has the same shape as the classical one Preserves mass, momentum and energy locally Kernel quantum maxwellians Dissipate quantum entropy dx dp Q(w) Ln w (2π ) 0 d

39 Quantum Boltzmann operator (cont) 39 Q(w) = B( p p 1, Ω) (A(w) A(w) 1 A(w)A(w) 1 )dp 1 dω where A(w) is the conversion operator : A(w) = exp Ln w

40 Quantum BGK operator 40 where s.t. Q(w) = M w w M w = Exp(A + B p + C p 2 ) (M w w)µ dp = 0 for µ = (1, p, p 2 )

41 41 7. Quantum Energy-Transport and Drift-Diffusion models Collaboration w. C. Ringhofer and F. Méhats

42 Quantum Energy-Transport or Drift-Diffusion42 Rescaled Wigner equation (ε 1) ε t w + p x w + Θ [V ]w = ε 1 Q(w) Q(w) = M w w M w = Exp(A + C p 2 ) (M w w)µ dp = 0 µ = (1, p 2 ) Energy-Transport µ = 1; C = 1/2T fixed Drift-Diffusion

43 Quantum Energy-Transport (cont) 43 ε 0 (diffusion approximation) Energy transport: t Exp(A + C p 2 ) ( 1 p 2 T 2 Exp(A + C p 2 ) with T w = (p x + Θ [V ])w ) ( dp 1 p 2 ) dp = 0

44 Quantum Drift-Diffusion (cont) 44 Drift-Diffusion: t Exp(A p 2 /2T ) dp T 2 Exp(A p 2 /2T ) dp = 0 QET and QDD are consistent with quantum entropy dissipation. e.g. ET case: t S(n, W ) 0 with (n, W ) T r = Exp(A + C p 2 )(1, p 2 ) T r dp

45 2 expansions of QET and QDD 45 0 gives classical ET or DD models (diffusion models) O( 2 ) corrections to classical DD: t n x (T x n + n x (V + V B )) = 0 V B = Bohm potential: V B = n x n = 0

46 O( 2 ) corrections to classical DD or ET 46 O( 2 ) terms in quantum ET very complex model QDD up to O( 2 ) terms is consistent with quantum entropy dissipation: t S(n) 0 QET up to O( 2 ) terms is not consistent with quantum entropy dissipation.

47 47 8. Summary and conclusion

48 Summary 48 Extension of the Levermore s moment method to the quantum case Take local moments of the density operator eq. Close by a minimizer of the entropy functional leads to: Formulation of the entropy minimization problem as a global problem (local in classical mechanics) Non-local closure to the Quantum Hydrodynamics eq.

49 Summary (cont) 49 Quantum collision operators preserve mass, momentum and energy consistent w. quantum entropy decay Boltzmann or BGK type Drift-Diffusion or Energy-Transport models through-diffusion approximation of Quantum BGK Justification of Bohm potential in Quantum Drift-Diffusion Ref: D., Ringhofer, Mehats. See also [Zubarev et al]: NESOM theory

50 Perspectives 50 Verify entropy minimization problem has a solution in a reasonable sense Practical computations of model problems with QHD model 2 corrections to classical mech. Small T asymptotics Normal mode analysis of linearized model...