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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
1
Chapter 1
Quantum kinetic equations: an introduction
P. Degond
MIP, CNRS and Universite Paul Sabatier,
118 route de Narbonne, 31062 Toulouse cedex, France
[email protected] (see http://mip.ups-tlse.fr)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
2Summary
1. Quantum statistical mechanics of nonequilibrium
2. Mean-Field limit
3. Quantum methods: a brief and incomplete summary
4. Hydrodynamic limits
5. Conclusion
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
3
1. Quantum statistical mechanics ofnonequilibrium
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
4Wave-function
➠ State of a particle → wave-function ψ(x, t) ∈ C
➟ dPt(x) = |ψ(x, t)|2 dx = Probability of findingthe particle in dx at time t.
➟∫|ψ(x, t)|2 dx =
∫dPt(x) = 1
=⇒ ψ(·, t) ∈ L2(Rd)d dimension of base space
➠ Evolution of ψ: Schrodinger equation
i~∂tψ = Hψ
➟ ~ = Planck constant
➟ H Hamiltonian operator
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
5Hamiltonian & observables
➠ Hψ = −~2
2∆ψ + V (x, t)ψ
V (x, t) potential energy
➠ Observation of the system:
(ψ, Aψ)L2 =
∫ψ Aψ dx
A Hermitian operator (observable) on L2
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
6Example of observables
➠ Ex1. Position operator: X : ψ → xψ(x).Observation = Mean particle position:
(ψ, Xψ) =
∫x |ψ|2 dx
➠ Ex2. Momentum operator P : ψ → −i~∇ψ
(ψ, Pψ) = −∫
ψ i~∇ψ dx =
∫~k |ψ(k)|2 dk
ψ(k) = Fourier transf.
= (2π)−d/2∫
e−ik·xψ(x) dx
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
7Most general observable
➠ Any classical observable a(x, p) gives rise to aquantum observable A = Op(a) according to theWeyl quantization rule:
Op(a)ψ =1
(2π)d
∫a(
x + y
2, ~k) ψ(y)eik(x−y) dk dy
a= Weyl symbol of Op(a).
➠ Ex. 3: Classical Hamiltonian Hc = |p|2/2 + V →quantum HamiltonianOp(Hc) = H = −(~2/2)∆ + V
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
8N-particle systems
➠ ψ(x1, . . . , xN): xi coordinate of the i-th particle
➟ Classical Hamiltonian:
Hc =N∑
i=1
1
2|pi|2+
1
2
∑i6=j
φint(xi−xj)+∑
i
φext(xi)
➟ Quantum Hamiltonian:
H = −N∑
i=1
~2
2∆xi
+1
2
∑i6=j
φint(xi−xj)+∑
i
φext(xi)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
9Incompletely known states
➠ Uncertainty about the state of the system:
➟ (φs)s∈S a complete orthonormal basis of thesystem
➟ ρs lists the probability of state s:
0 ≤ ρs ≤ 1 ,∑s∈S
ρs = 1
➠ Probability of presence of the particle in theincompletely known state described by (ρs)s∈S:
P (x, t)dx =∑s∈S
ρs |φs|2 dx
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
10Density operator
➠ Incompletely known (or mixed) state (φs, ρs)s∈S
➟ Density operator ρ
ρψ =∑s∈S
ρs(ψ, φs) φs
➠ ρ is a Hermitian, positive, trace-class operator:
Trρ =∑s∈S
ρs = 1
➟ Pure state: all ρs = 0 but one ρs0= 1;
ρ = (·, φs0) φs0
= projector
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
11Evolution of ρ
➠ φs(t) solution of Shrodinger eq.
➠ uncertainty does not evolve with time:
ρs = Constant
➠ Eq. for ρ
i~∂tρ = Hρ − ρH = [H, ρ]
Quantum Liouville equation
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
12Integral kernel of ρ
➠ ρ(x, x′) integral kernel of ρ:
ρψ =
∫ρ(x, x′)ψ(x′) dx′
ρ(x, x′) =∑
s
ρsφs(x)φs(x′)
➠ Liouville eq. expressed on ρ(x, x′)
i~∂tρ = (Hx −Hx′)ρ
ρ(x′, x) = ρ(x, x′) , Trρ =
∫ρ(x, x) dx
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
13Observables and density operator
➠ Observable A. Observation of the mixed state:
〈A〉ρ =∑
s
ρs(Aφs, φs) = Tr{ρA}
➠ Example: probability of presence at x0:
P (x0) =∑s∈S
ρs |φs(x0)|2
= ρ(x0, x0) = Tr{ρ Op(δx−x0)}
Observation of the state at x = x0.
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
14Wigner transform
➠ A = Op(a):
〈Op(a)〉ρ = Tr{ρOp(a)}
=1
(2π~)d
∫W [ρ](x, p)a(x, p) dx dp
W [ρ] Wigner transform of ρ
➠ W [ρ](x0, p0) = (2π~)d〈Op(δx−x0δp−p0
)〉ρObservation of the system at (x0, p0).
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
15Wigner transform (cont)
W [ρ](x, p) =
∫ρ(x − η
2, x +
η
2) e
iη·p
~ dη
=∑
s
ρs
∫φs(x − η
2)φs(x +
η
2) e
iη·p
~ dη
➠ Note: W [ρ] real-valued but not ≥ 0W [ρ] dx dp is not a probability distributionfunction
WH [ρ] = W [ρ] ∗ G ≥ 0 , G =1
(~π)3e−(|x|2+|p|2)/~
Hussimi distribution function
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
16Wigner equation
➠ Eq. for W [ρ]:
∂tW + p · ∇xW + Θ~[V ]W = 0
Θ~[V ]W = − i
(2π)3~
∫(V (x +
~
2η) − V (x − ~
2η))
×W (x, q) eiη·(p−q) dq dη
➠ Like the classical kinetic eq. but for the fieldterm Θ~[V ]
➠ Θ~[V ]W~→0−→ −∇xV · ∇pW
➠ Note p = v (m = 1)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
17A few useful identities
∫W [ρ] W [σ]
dx dp
(2π~)d= Tr{ρ σ†}
∫a b
dx dp
(2π~)d= Tr{Op(a) Op(b)†}
W = Op−1 , Op = W−1
Weyl quantization and Wigner transformation are in-
verse operations
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
18
2. Mean-Field limit
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
19N-particle quantum system
➠ Density operator ρN on L2(R3N)
➟ Kernel ρN(x1, x′1, . . . , xN , x′
N)
➟ undistinguishability
ρN(xσ(1), x′σ(1), . . . , xσ(N), x
′σ(N)) =
ρN(x1, x′1, . . . , xN , x′
N), ∀permutation σ
➠ Liouville eq.
i~∂tρN = [HN , ρN ]
HN =N∑
i=1
1
2|pi|2 +
1
2
∑i6=j
φ(xi − xj)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
20Partial density operators
➠ Partial trace w.r.t. the N − j last variables
ρj = TrNj+1{ρN}ρj(x1, x
′1, . . . , xj, x
′j) =∫
ρN(. . . , xj+1, xj+1, . . . , xN , xN) dxj+1 . . . dxN
➠ Eq. for ρj: quantum BBGKY hierarchy
i~∂tρj = [Hj, ρj] + Qj(ρj+1)
Hj =
j∑i=1
1
2|pi|2 +
1
2
j∑i,k=1,i6=k
φ(xi − xk)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
21Quantum BBGKY hierarchy
Qj(ρj+1) = (N − j)
j∑i=1
Trj+1{[φ(xi − xj+1), ρj+1]}
➠ Eq. for ρ1
i~∂tρ1 = [H1, ρ1] + Q1(ρ2)
H1 =1
2|p1|2
Q1(ρ2) = (N − 1)Tr2{[φ(x1 − x2), ρ2]}
Q1(ρ2) = (N − 1)
∫[φ(x1 − x2) − φ(x′
1 − x2)]
×ρ2(x1, x′1, x2, x2) dx2
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
22Mean-field limit
➠ (i) Rescale φ → 1N φ and take N → ∞
➠ (ii) Propagation of chaos:
ρ2(x1, x′1, x2, x
′2) = ρ1(x1, x
′1) ρ1(x2, x
′2)
Q1(ρ2) = (1 − (1/N))
∫[φ(x1 − x2) − φ(x′
1 − x2)]
×ρ2(x1, x′1, x2, x2) dx2
≈∫
[φ(x1 − x2) − φ(x′1 − x2)]ρ
1(x2, x2) dx2
×ρ1(x1, x′1)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
23Mean-field limit (cont)
➠ i.e.
Q1(ρ2) ≈ (Vρ(x1) − Vρ(x′1))ρ
1(x1, x′1)
orQ1(ρ2) ≈ [Vρ, ρ
1]
with
Vρ(x) =
∫φ(x − y)ρ1(y, y) dy
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
24Quantum mean-field eq.
i~∂tρ = [Hmf , ρ]
Hmf =1
2|p|2 + Vρ
Vρ(x) =
∫φ(x − y)n(y) dy
n(y) = ρ(y, y)
Density operator formulation of Schrodinger mean-
field equations
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
25Schrodinger mean-field eq.
➠ Pure-state: ρ = (·, ψ)ψ is a projector where ψsatisfies Schrodinger mean-field eq.
i~∂tψ = Hmfψ
Hmf =1
2|p|2 + Vψ
Vψ(x) =
∫φ(x − y)n(y) dy
n(y) = |ψ(y)|2
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
26Mean-field limit for Fermions
➠ Fermions (such as electrons) have antisymmetricwave functions:
ψ(xσ(1), . . . , xσ(N)) = (−1)ε(σ)ψ(x1, . . . , xN)
ε(σ) = signature of the permutation σ
➠ Density matrix satisfies
ρ(x1, x′σ(1), . . . , xN , x′
σ(N)) = (−1)ε(σ)ρ(x1, x′1, . . . , xN , x′
N)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
27Mean-field closure for Fermions
➠ Hartree Mean field closure
ρ2(x1, x′1, x2, x
′2) = ρ1(x1, x
′1) ρ1(x2, x
′2)
Does not satisfy antisymmetry
➠ Instead, use
ρ2(x1, x′1, x2, x
′2) = ρ1(x1, x
′1) ρ1(x2, x
′2)−ρ1(x1, x
′2) ρ1(x2, x
′1)
’Slater determinant’ closure
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
28Exchange-correlation potential
➠ Gives
Q1(ρ2) ≈ (Vρ(x1) − Vρ(x′1))ρ
1(x1, x′1) − Q
ex(ρ1)
with
Qex
(ρ1) =
∫[φ(x1 − x2) − φ(x′
1 − x2)]ρ1(x1, x2)ρ
1(x2, x′1) dx2
Vρ(x) =
∫φ(x − y)ρ1(y, y) dy
Qex
= exchange-correlation potential
Page 29
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
29Exchange-correlation potential (cont)
➠ In short:
Qex = Tr{[φ, (ρ ⊗ ρ)ex]}2
with
(ρ ⊗ ρ)ex = ρ1(x1, x′2) ρ1(x2, x
′1)
and Tr{}2 is the trace w.r.t. the second variable
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
30Hartree-Fock mean-field model
i~∂tρ = [Hmf , ρ] − Qex(ρ)
Hmf =1
2|p|2 + Vρ
Qex = Tr{[φ, (ρ ⊗ ρ)ex]}2
Vρ(x) =
∫φ(x − y)n(y) dy
n(y) = ρ(y, y)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
31Rigorous results and comments
➠ Mean-field limit:
➟ φ smooth: [Spohn]
➟ φ = Coulomb: [Bardos, Golse, Mauser]
➟ Hartree-Fock: [Bardos, Golse, Gottlieb, Mauser]
➠ Semiclassical limit ~ → 0 of Schrodingermean-field eq.
➟ Wigner-Poisson → Vlasov-Poisson [Lions, Paul],[Markowich, Mauser]
➠ No such analogy as the BBGKY hierarchy forHard-Spheres in quantum mechanics
➟ No quantum Boltzmann eq.
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
32
3. Quantum methods: a brief and incompletesummary
Page 33
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
33’Small’ systems
➠ Ex: atoms, molecules ∼ few tens of e−
➠ Eigengvalue problem:
➟ Minimal energy (first eigenvalue)
➟ Excited states (lower spectrum)
➠ Techniques
➟ Hartree-Fock (ψ = Slater determinant)
➟ Multiconfiguration (ψ =∑
Slater det.)
➟ Born-Oppenheimer (nuclei classical)
➟ Car-Parinello (concurrent optimization)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
34Small systems: dynamics
➠ Examples
➟ Chemical reactions
➟ Surface crossings
➟ Chemical reaction control by lasers
➟ Determination of reaction intermediates
➠ Techniques
➟ Direct computation of Time-dependentSchrodinger
➟ Time-dependent Hartree-Fock
➟ . . .
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
35’Large’ systems
➠ Examples:
➟ Large molecules
➟ Crystals
➟ Molecular dynamics (change of phases)
➟ Nano-objects
➠ Density Functional Theory (DFT)
➟ Finding the minimal energy
➟ Reduces the problem to a one-particle problemin a nonlinear potential (exact)
➟ [Hohenberg], [Hohenberg-Kohn]
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
36DFT: discussion
➠ Problem: nonlinear potential not known:approximations
➟ Thomas-Fermi
➟ Kohn-Sham
➟ . . .
➠ Validity of these approximations ?
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
37Open systems
➠ Examples
➟ Electrons in a semiconductor
➟ Molecule in a solvant
➟ Protein in a cell
➟ . . .
➠ How to account for the environment ?
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
38Model for Open systems
➠ Density matrix:
ρ(x1, x′1, . . . , xN , x′
N , y1, y′1, . . . , yP , y′P )
➟ x1, . . . , xN : system under consideration
➟ y1, . . . , yP : environment variables
➠ Programme:
➟ Evolution eq. for ρ
➟ Partial trace over the y variables
➟ Closure (e.g. y at thermo equilibrium)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
39Model for Open systems (cont)
➠ Example:
➟ electron-phonon in semiconductors
➟ partial trace over phonon variables
➟ [Argyres]
➠ Problem:
➟ Leads to very complex ’collision operators’
➟ Nonlocality in space and time
➟ Very difficult to deal with numerically
➟ Validity of the closure
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
40Other route: hydrodynamic models
➠ Meso-scale:
➟ Large enough system so that a notion ofthermodynamic limit is valid
➟ Not too large s.t. quantum decoherence doesnot occur
➠ Scale separation
➟ Small scale phenomena clearly separated fromlarge scale ones
➟ small scale → local equilibrium
➟ large scale → macroscopic evolution
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
41
4. Hydrodynamic limits
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
42Difficulty w. quantum hydrodynamics
➠ 6 ∃ Boltzmann eq.
➠ What can be done:
➟ 1-particle hydrodynamics➞ Classical → pressureless gas dynamics➞ Quantum → quantum trajectories (Bohmian
mechanics)
➟ Extension of Bohmian mechanics tomany-particle: closure problem
➟ Entropy minimization principle (a la’Levermore’)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
431-particle hydrodynamics: classical case
➠ Consider the Free Transport Eq.
∂f
∂t+ v · ∇xf −∇xV · ∇vf = 0
Look for solutions of the form
f = n(x, t) δ(v − u(x, t))
➠ Then, n and u satisfy exactly Pressureless gasdynamics
∂tn + ∇x · nu = 0
∂tu + u · ∇xu = −∇xV
Non strictly hyperbolic. [Brenier], [Bouchut], [E]
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
441-particle hydrodynamics: quantum case
➠ Single state ψ
i~∂tψ = −~2
2∆ψ + V (x, t)ψ
Decompose
ψ =√
neiS/~
and define u = ∇xS. Then take real and imaginaryparts
∂tn + ∇x · nu = 0
∂tS +1
2|∇S|2 + V − ~
2
2
1√n
∆√
n = 0
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
451-particle QHD
➠ Take ∇ of the phase eq.
∂tn + ∇x · nu = 0
∂tu + u · ∇xu = −∇x(V + VB)
VB = −~2
2
1√n
∆√
n
VB = Bohm potential
➠ Pressureless Gas dynamics w. additional Bohmpotential term.
➟ If O(~2) term neglected → ClassicalHamilton-Jacobi eq.
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
46Temperature eq.
➠ Bohm potential → dispersive term: adds highfrequency oscillations
➟ Numerics delicate
➠ Question: temperature eq. ?
➟ Starting point: mixed-state (i.e. densityoperator or Wigner distribution)
➟ Average over the statistics of mixed-state
➠ Closure problem
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
47Quantum Hydrodynamic closure
➠ Classical Fourier law for the heat flux [Gardner]
➠ Small temperature asymptotics [Gasser,Markowich, Ringhofer]
➠ Chapman-Enskog expansion of phenomenologicalBGK-type collision term [Gardner, Ringhofer]
➠ Entropy minimization principle ’a la Levermore’[D. ,Ringhofer]
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
48
5. Summary and conclusion
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
49Summary
➠ Reviewed: basics of quantum statisticalmechanics of nonequilibrium systems
➟ Density operator
➟ Quantum Liouville eq.
➟ Wigner transform and Wigner eq.
➟ Mean-field limits: Hartree and Hartree-Focksystems
➠ Discussed the modeling of open systems
➠ Reviewed (briefly) previous approaches onquantum hydrodynamics
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
50Next step
➠ Derivation of quantum hydrodynamic modelsbased on the entrtopy minimization approach
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
1
Chapter 2
Derivation of moment models via the entropyminimization approach (classical case)
P. Degond
MIP, CNRS and Universite Paul Sabatier,
118 route de Narbonne, 31062 Toulouse cedex, France
[email protected] (see http://mip.ups-tlse.fr)
Page 52
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
2Summary
1. Classical description of particle systems
2. The moment method and the Euler eq.
3. Higher order moment systems: Levermore’s approach
4. Summary, conclusion and perspectives
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
3
1. Classical description of particle systems
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
4Distribution function
➠ f(x, v, t) = density in phase space (x, v)f dx dv = number of particles in dx dv
v
x
f(x, v, t)
Velocity
Position
➠ Equation satisfied by f ?
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
5Collisionless particles
➠ All particles issued from the same point (x, v) ofphase-space follow the same trajectory
X = V , V = −∇V (X , t)
v′
v
x′x
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
6Free transport equation
➠ =⇒d
dtf(X (t),V(t), t) = 0
➠ Chain rule =⇒
∂f
∂t+ v · ∇xf −∇V (x, t) · ∇vf = 0
Free transport equation
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
7Collision operator
➠ In the presence of collisions, particles would obeythe free motion equations:
➟ Rate of change of f while following theparticle motion is due to collisions
d
dt[f(X (t),V(t), t)] =
=
(
∂f
∂t+ v · ∇xf −∇V (x, t) · ∇vf
)
|(X (t),V(t),t)
= Q(f)|(X (t),V(t),t) collision operator
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
8Form of the collision operator
➠ Collision operator is
➟ local in time (collision dynamics isinstantaneous)
➟ local in space
➟ operates on v only
Q(f) = Q+(f) − Q−(f)
= Gain − Loss
OutIn
dfdt
= Q(f)v
x
v′
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
9The Boltzmann operator
➠ Models binary interactions between particles
➠ Complex form. Unnecessary for our purpose
➟ Only algebraic properties matter
➠ Boltzmann equation
∂f
∂t+ v · ∇xf −∇V (x, t) · ∇vf = Q(f)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
10Fluid variables
➠ Fluid quantities = averaged over a ’small’ volumein physical space
➠ Ex. Density n(x, t) dx = number of particles in asmall volume dx.
Mean momentum q dx =∑
i∈dx
vi
Mean energy W dx =∑
i∈dx
|vi|2/2
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
11Link w. the kinetic distribution function
➠
n
q
2W
=
∫
f
1
v
|v|2
dv
➠ n, q, W , . . . are moments of f
➟ Eqs for n, q, W , . . . are called fluid (ormacroscopic) equations
➟ To determine these equations, we need someproperties of Q
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
12Properties of Q (I): Conservations
∫
Q(f)
1
v
|v|2
dv = 0
➠ Conservation of
mass
momentum
energy
➠ 1, v, |v|2 = collisional invariant. Any collisionalinvariant is a combination of these 5 ones
Page 63
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
13Conservations (cont)
➠ Homogeneous case (∇x = 0, V = 0):
∂f
∂t= Q(f) =⇒
∂
∂t
∫
f
1
v
|v|2
dv = 0
=⇒∂
∂t
n
q
2W
dv = 0
➠ Homogeneous case =⇒ Total mass, momentumand energy are conserved
Page 64
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
14Properties of Q (II):H-theorem
➠ H-theorem∫
Q(f) ln fdv ≤ 0
➠ Define the Entropy of f :
H(f) =
∫
f(ln f − 1)dv
Note h(h) = f(ln f − 1) =⇒ h′(f) = ln f
Page 65
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
15Entropy
➠ Homogeneous situation (∇x = 0, V = 0):
∂f
∂t= Q(f) =⇒
∂H(f)
∂t=
∫
Q(f)(ln f−1)dv ≤ 0
➠ Entropy decays
➟ Rate of entropy decay = entropy dissipation
➟ Signature of irreversibility
Page 66
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
16Properties of Q (III): Equilibria
➠ Q(f) = 0 ⇐⇒ ln f is a collisional invariant
⇐⇒ ∃A, C ∈ R+, B ∈ R3s.t.
f = exp(A + B · v + C|v|2)
➠ Maxwellian: other expression
Mn,u,T =n
(2πT )3/2exp
(
−|v − u|2
2T
)
(n, u, T ) straightforwardly related w. (A, B, C)
Page 67
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
17Local vs global thermodynamic equilibrium
➠ n, u, T related w. moments n, q,W :
∫
Mn,u,T
1
v
|v|2
dv =
n
nu
n|u|2 + 3nT
➠ (n, u, T ) independent of (x, t) → Globalthermodynamic equilibrium
➠ (n, u, T ) dependent on (x, t) → Localthermodynamic equilibrium (LTE)
➟ The dynamics of (n, u, T ) → Hydrodynamicequations
Page 68
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
18Entropy decay ⇔ relaxation to Maxwellians
➠ (i) Entropy dissipation∫
Q(f) ln fdv ≤ 0 and≡ 0 iff f = Maxwellian
➠ Dynamics of the Boltzmann equation
➟ Relaxation to LTE (through entropydissipation)
➟ Slow evolution on the manifold of LTE’s
➠ Time scale separation
➟ Fast kinetic scale
➟ Slow hydrodynamic scale
Page 69
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
19Entropy minimization principle
➠ Entropy minimization subject to momentconstraints: let n, T ∈ R+, u ∈ R3 fixed.
min{H(f) =
∫
f(ln f − 1)dv s.t.
∫
f
1
v
|v|2
dv =
n
nu
n|u|2 + 3nT
}
is realized by f = Mn,u,T .
Page 70
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
20Entropy minimization principle (cont)
➠ Entropy minimization:
➟ Most effective characterization of Maxwelliansfor further extensions
➠ Examples
➟ More moment constraints → Levermoremodels
➟ Quantum entropy → quantum hydro models
Page 71
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
21BGK operator
➠ Expression of the Boltzmann operator is verycomplicated
➠ Is there a simpler operator which possesses thesame algebraic properties as the Boltzmannoperator ?
➟ Conservation of mass, momentum and energy
➟ Entropy decay
➟ Relaxation towards Maxwellian (LocalThermodynamical equilibrium)
➠ Yes: BGK operator [Bhatnagar-Gross-Krook]
➟ Plain relaxation to Maxwellians
Page 72
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
22BGK operator (cont)
Q(f) = −ν(f − Mf)
where Mf = Mn,u,T is the Maxwellian with the samemoments as f i.e. (n, u, T ) are such that
∫
(Mf − f)
1
v
|v|2
dv = 0
i.e.
n
nu
n|u|2 + 3nT
=
∫
f
1
v
|v|2
dv
Page 73
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
23Properties of BGK operator
➠ Shows the same ’algebraic’ properties as theBoltzmann operator
➠ (i) Collisional invariants:∫
Q(f)ψdv = 0,∀f ⇐⇒ ψ(v) = A+B·v+C|v|2
➠ (ii) Equilibria:
Q(f) = 0 ⇐⇒ f = Mn,u,T
Page 74
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
24Properties of BGK operator (cont)
➠ H-theorem∫
Q(f) ln fdv ≤ 0 (= 0 ⇐⇒ f = Mn,u,T )
➠ Simpler operator
➟ Theory is simpler
➟ Numerical simulations are easier
➟ Some unphysical features (Prandtl number)
Page 75
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
25Some references for BGK
➠ Existence of weak solutions [Perthame, Pulvirenti]
➠ Numerical solutions [Dubroca, Mieussens]
➠ Generalized BGK models [Bouchut, Berthelin]
Page 76
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
26Ref for Boltzmann: Homogeneous equation
➠ Existence and uniqueness of classical solutions[Carleman], [Arkeryd], ...
➠ Convergence to a Maxwellian as t → ∞[Desvillettes], [Wennberg], ...
Page 77
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
27Boltzmann: Non-homogeneous equation
➠ Difficulty: Q(f) quadratic in f
➠ ref. [DiPerna, Lions]: renormalized solutions i.e.satisfying:
(∂
∂t+ v · ∇x)β(f) = β′(f)Q(f) in D′
∀β Lipschitz, s.t. |β′(f)| ≤ C/(1 + f)
➠ Note: β′(f)Q(f) grows linearly with f
Page 78
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
28Perturbation of equilibria
➠ ref: [Ukai], [Nishida, Imai], ...
➠ M global Maxwellian (parameters (n, u, T ) areconstant indep. of x, t
➠ f = M + g, with ”g ≪ M”
➠ Decompose
Q(f) = LMg + Γ(g, g)
➠ Prove operator v · ∇xg − LMg dissipative
➠ Compensates blow-up of Γ(g, g) if g small
Page 79
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
29
2. The moment method and the Eulerequations
Page 80
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
30Moment method
➠ Natural idea: (i) multiply Boltzmann eq. by1, v, |v|2 and integrate wrt v:
∫
((∂t + v · ∇x)f − Q(f))
1
v
|v|2
dv
➠ (ii) use conservations:
∫
Q(f)
1
v
|v|2
dv = 0
Page 81
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
31Moment method (cont)
➠ (iii) Get conservation eqs
∂
∂t
n
q
2W
+ ∇x ·
∫
f
1
v
|v|2
v dv = 0
➠ Problem: Express fluxes in term of the conservedvariables n, q, W
➟∫
fvivj dv (for i 6= j) and∫
f |v|2 v dv cannotbe expressed in terms of n, q, W .
➠ conservation eqs are not closed
Page 82
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
32Fluxes
➠ Density flux:∫
fv dv = q. Define
u =q
nVelocity
➠ Momentum flux tensor:∫
fvv dv =
∫
fuu dv +
∫
f(v − u)(v − u) dv
= nuu + P
P pressure tensor, not defined in terms ofn, q, W
Page 83
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
33Fluxes (cont)
➠ Energy flux∫
f |v|2 v dv = 2(Wu + Pu + Qu)
2Q =
∫
f |v − u|2(v − u) dv
not defined in terms of n, q, W
Page 84
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
34Conservation equations
➠
∂
∂t
n
q
W
+ ∇x ·
nu
nuu + P
Wu + Pu + Q
= 0
➠ Problem: find a prescription which relates P andQ to n, u, W :
Closure problem
Page 85
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
35Hydrodynamic scaling
Microscopic scale Macroscopic scale
η ≪ 1
➠ Rescale: x′ = εx, t′ = εt
ε(∂tfε + v · ∇xf
ε) = Q(f ε)
Page 86
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
36Limit ε → 0
➠ Suppose f ε → f0 smoothly. Then
Q(f0) = 0
i.e. ∃n(x, t), u(x, t), T (x, t) s.t. f = Mn,u,T
➠
nε
nεuε
2Wε
→
n
nu
2W = n|u|2 + 3nT
Page 87
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
37Fluxes
➠
Pε =
∫
f ε(v − u)(v − u) dv −→ P = p Id
p = nT = Pressure
➠
2Qε =
∫
f ε|v − u|2(v − u) dv −→ 0
Page 88
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
38Conservation eqs as ε → 0: Euler eq.
➠
∂
∂t
n
nu
n|u|2 + 3nT
+∇x·
nu
nuu + nT Id
(n|u|2 + 5nT )u
= 0
➠ Euler eqs of gas dynamics.p = nT perfect gas Equation-of-State
Page 89
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
39Beyond Euler
➠ Problem: find order ε, ε2, . . . corrections to Eulereqs.
➠ Expand (Hilbert or Chapman-Enskog expansion):
f ε = f0 + εf1 + ε2f2 + . . .
➟ Insert in the Boltzmann eq. and solverecursively
➠ Order ε corrections → Navier-Stokes eq.
➟ Higher order corrections (Burnett,super-Burnett) unstable
Page 90
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
40Rigorous results for the hydrodynamic limit
➠ (i) Boltzmann → compressible Euler
Theorem [Caflish, CPAM 1980] n, u, T smoothsolutions of Euler on a time interval [0, t∗] (t∗ <blow-up time of regularity), with initial datan0, u0, T0.∃ε0 > 0, ∀ε < ε0, ∃f ε a solution of the Boltzmannequation with initial data Mn0,u0,T0
on [0, t∗] and
sup[0,t∗]
‖f ε(t) − Mn,u,T (t)‖ ≤ Cε
Page 91
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
41Rigorous results for the hydrodynamic limit (2)
➠ Boltzmann → incompressible Navier-Stokes
➠ Perturbation of a global Maxwellian with u = 0.
➟ Rescale velocity and time (diffusion limit)
➟ ref: [De Masi, Esposito, Lebowitz], [Bardos,Golse, Levermore], [Bardos, Ukai], [Golse,Saint-Raymond]
Page 92
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
42Why looking for new hydrodynamic systems
➠ Perturbation approach not valid when gradientsare large (i.e. ǫ not small)
➠ Higher order perturbation models (beyondNavier-Stokes) are unstable
➠ Find models which are consistent with entropydissipation (Navier-Stokes OK but not Burnett)
Page 93
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
43Higher order moment models
➠ Idea: increase the number of moments
➟ Moment system hierarchies
➟ ref. [Grad], [Muller, Ruggeri (extendedthermodynamics)], [Levermore]
➠ Try to do it consistently with the entropydissipation rule
➟ Levermore models
➟ Developped in the next section
Page 94
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
44
3. Higher order moment systems:Levermore’s approach
Page 95
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
45Moments (1)
➠ List of monomials µi(v)
µ(v) = (µi(v))Ni=0
➠ Contains hydrodynamic moments
µ0(v) = 1; µi(v) = vi, i = 1, 2, 3; µ4(v) = |v|2
➠ Example
µ(v) = {1, v, vv} Gaussian model
µ(v) = {1, v, vv, |v|2v, |v|4}
Page 96
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
46Moments (2)
➠ For a distribution function f , define:
m(f) = (mi(f))Ni=0 , mi(f) =
∫
fµi(v) dv
➠ Eq. for the i-th moment:
∂
∂tmi(f) + ∇x ·
∫
fµi(v)v dv =
∫
Q(f)µi(v) dv
➠ Note∫
Q(f)µi(v) dv 6= 0 if µi 6= hydrodynamicmonomial
Page 97
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
47Closure problem
➠ Find a prescription for∫
fµi(v)v dv and
∫
Q(f)µi(v) dv
in terms of the moments mi
Page 98
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
48Entropy minimization principle (Gibbs)
➠ Let n, T ∈ R+, u ∈ R3 fixed.
min{H(f) =
∫
f(ln f − 1)dv s.t.
∫
f
1
v
|v|2
dv =
n
nu
n|u|2 + 3nT
}
is realized by f = Mn,u,T .
Page 99
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
49Proof of Gibbs principle
➠ Euler-Lagrange eqs of the minimization problem:∃A, C ∈ R, B ∈ R3 (Lagrange multipliers) s.t.∫
(ln f − (A + B · v + C|v|2)) δf dv = 0, ∀ δf
➠ =⇒ f = exp(A + B · v + C|v|2)i.e. f = Maxwellian
Page 100
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
50Euler eqs in view of the entropy principle
➠ Euler eqs = moment system (only involvinghydrodynamical moments), closed by a solutionof the entropy minimization principle
➠ Idea [Levermore], [extended thermodynamics] Usethe same principle for higher order momentsystems
Page 101
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
51Generalized entropy minimization principle
➠ Given a set of moments m = (mi)Ni=0, solve
min{H(f) =
∫
f(ln f−1)dv s.t.
∫
fµ(v)dv = m}
➠ Solution: generalized Maxwellian:∃ vector α = (αi)
Ni=0 s.t.
f = Mα(v) = exp(α · µ(v)) = exp(N
∑
i=0
αiµi(v))
Page 102
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
52Levermore moment systems
➠ Use the generalized Maxwellian Mα as aprescription for the closure
∂
∂t
∫
Mαµ(v) dv+∇x·
∫
Mαµ(v)v dv =
∫
Q(Mα)µ(v) dv
Gives an evolution system for the parameter α
Page 103
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
53Potentials
➠ Has the form of a symmetrizable hyperbolicsystem: Define
Σ(α) =
∫
Mα dv =
∫
exp(α · µ(v)) dv
φ(α) =
∫
Mαv dv =
∫
exp(α · µ(v))v dv
Σ(α) = Massieu-Planck potential, φ = fluxpotential
∂Σ
∂α=
∫
Mαµ(v) dv ,∂φ
∂α=
∫
Mαµ(v)v dv
Page 104
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
54Symmetrized form
➠ Moment system ≡
∂
∂t
∂Σ
∂α+ ∇x ·
∂φ
∂α= r(α)
r(α) =
∫
Q(Mα)µ(v) dv
➠ or∂2Σ
∂α2
∂α
∂t+
∂2φ
∂α2· ∇xα = r(α)
∂2Σ/∂α2 =∫
Mαµ(v)µ(v) dv symmetric ≫ 0
∂2φ/∂α2 =∫
Mαµ(v)µ(v)v dv symmetric
Page 105
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
55Hyperbolicity
➠ Hyperbolicity −→ well posedness (Godounov,Friedrichs)
➠ 6= Grad systems: not everywhere locallywell-posed
Page 106
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
56Entropy
➠ S(m) = Legendre dual of Σ(α):
S(m) = α · m − Σ(α)
where α is such that
m =∂Σ
∂α(=
∫
Mαµ(v) dv)
➠ Then
α =∂S
∂m
Page 107
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
57Entropy (cont)
➠ α and m are conjugate variables.
➟ α = entropic (or intensive) variables
➟ m = conservative (or extensive) variables
➠ Link with H
S(m) =
∫
(α · µ − 1)Mα dv
=
∫
(ln Mα − 1)Mα dv = H(Mα)
Fluid entropy = Kinetic entropy evaluated atequilibrium
Page 108
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
58Levermore’s model in conservative var.
∂tm + ∇x ·∂φ
∂α
(
∂S
∂m(m)
)
= r
(
∂S
∂m(m)
)
➠ Entropy inequality
∂tS(m) + ∇x · F (m) =∂S
∂m· r
F (m) = α ·∂φ
∂α− φ(α) = Entropy flux
with α = ∂S/∂m
Page 109
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
59Entropy dissipation
∂S
∂m· r = α ·
∫
Q(Mα)µ dv
=
∫
Q(Mα) ln Mα dv ≤ 0
Thanks to H-theorem
➠ Levermore system compatible with the entropydissipation
∂tS(m) + ∇x · F (m) ≤ 0
Entropy dissipation = 0 iff Mα = standardMaxwellian Mn,u,T
Page 110
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
60Example: Gaussian closure
➠ µ(v) = {1, v, vv}.
Mα =n
(det 2πΘ)1/2exp
(
−1
2(v − u)Θ−1(v − u)
)
Θ symmetric ≫ 0 matrixα ∼ (n, u, Θ)
∂tn + ∇x · nu = 0
∂tnu + ∇x · (nuu + nΘ) = 0
∂t(nuu + nΘ) + ∇x · (nuuu + 3nΘ ∧ u) = Q(n, Θ)
Page 111
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
61Entropy in the Gaussian model
➠ Collisions
Q(n, Θ) =
∫
Q(Mα)vv dv
➠ Entropy: S = nσEntropy flux: F = nσu
σ = ln
(
n
(det 2πΘ)1/2
)
−5
2
Page 112
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
62General models: constraints
➠ If highest degree monomial of odd parity,integrals like
∫
exp(α · µ)µ dv diverge
➟ Constraint on µ: The set of α s.t. theintegrals converge has non-empty interior
➟ Highest degree monomial must have evenparity
➠ Moment realizability:
➟ characterize the set of m such that ∃α andm =
∫
exp(α · µ)µ dv
➟ ref. [Junk], [Schneider]
Page 113
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
63Example: 5 moment model (in 1D)
➠ ref. [Junk]:
➟ Moment realizability domain not convex
➟ fluid Maxwellians lie at the boundary of therealizability domain
➟ Fluxes and characteristic velocities −→ ∞when m → Maxwell.
➠ Severe drawback since collision operators relax toMaxwellians
Page 114
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
64Problems (cont)
➠ Explicit formulae for∫
exp(α · µ)µ dv and∫
exp(α · µ)µv dv not available beyond Gaussianmodel
➠ Inversion of α → m not explicit. Iterativealgorithms to solve the Legendre transform.
➠ Collision operator: r(α) =∫
Q(Mα)µ dv doesnot give the right Chapman-Enskog limit.(viscosity and heat conductivity < Navier-Stokes)
➟ Needs to correct the collision operator[Levermore, Schneider].
Page 115
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
65Practical use of Levermore’s moment models
➠ Successful applications in a selected number ofcases
➟ Gaussian model [Levemore, Morokoff]
➟ P 2 model of radiative transfer [Dubroca]
➠ Give a systematic methodology to imagine newmodels and new closures.
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
66
4. Summary, conclusion and perspectives
Page 117
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
67Summary
➠ Kinetic → fluid by the moment method
➟ closure problem
➟ Relaxation to equilibrium → Euler
➟ Correction to Euler (via the Hilbert orChapman-Enskog expansion): →Navier-Stokes or higher order models (Burnett,. . . )
➠ Transition regimes: perturbation models no morevalid when ε not small
Page 118
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
68Summary (cont) and perspectives
➠ Levermore’s attempt:
➟ closure by means of the entropy minimizationprinciple
➟ Nice features (hyperbolicity) but some flaws(moment realizability)
➠ Use the same methodology for quantumhydrodynamics
Page 119
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
1
Chapter 3
Quantum hydrodynamic models derived from theentropy principle
P. Degond
MIP, CNRS and Universite Paul Sabatier,
118 route de Narbonne, 31062 Toulouse cedex, France
[email protected] (see http://mip.ups-tlse.fr)
Page 120
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
2Summary
1. Quantum setting: a summary
2. QHD via entropy minimization
3. Quantum Isentropic Euler
4. Summary and conclusion
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
3
1. Quantum setting: a summary
Page 122
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
4Density operator
➠ Basic object: ρ: Hermitian, positive, trace-classoperator on L2(Rd) s.t.
Trρ = 1
➠ Typically:
ρψ =∑
s∈S
ρs(ψ, φs) φs
for a complete orthonormal system (φs)s∈S and real
numbers (ρs)s∈S such that 0 ≤ ρs ≤ 1,∑
ρs = 1
Page 123
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
5Quantum Liouville equation
➠
i~∂tρ = [H, ρ] + i~Q(ρ)
➠ H = Hamiltonian:
Hψ = −~2
2∆ψ + V (x, t)ψ
➠ Q(ρ) unspecified: accounts for dissipationmechnisms
Page 124
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
6Wigner Transform
➠ 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
2ξ, x +
1
2ξ) ei ξ·p
~ dξ
➠ Note: we use the momentum p instead of thevelocity v used in the classical setting. We makem = 1 so that v = p.
Page 125
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
7Inverse Wigner transform (Weyl quantization)
➠ Let w(x, p). ρ = W−1(w) = Op(w) is theoperator defined by:
W−1(w)ψ =1
(2π)d
∫w(
x + y
2, ~k) ψ(y)eik(x−y) dk dy
w= Weyl symbol of ρ.
➠ Isometries between L2 (Operators s.t. ρρ† is
trace-class) and L2(R2d):
Tr{ρσ†} =
∫W [ρ](x, p)W [σ](x, p)
dx dp
(2π~)d
Page 126
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
8Wigner equation
➠ Eq. for w = W [ρ]:
∂tw + p · ∇xw + Θ~[V ]w = Q(w)
Θ~[V ]w = − i
(2π)d~
∫(V (x +
~
2η) − V (x − ~
2η))
×w(x, q) eiη·(p−q) dq dη
➠ Θ~[V ]w~→0−→ −∇xV · ∇pw
➠ Q(w) collision operator (Wigner transf. of Q)
Page 127
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
9
2. QHD via entropy minimization
Joint work with
C. RinghoferArizona State University, Tempe, USA
J. Stat. Phys. 112 (2003), pp. 587–628.
Page 128
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
10Approach (Levermore)
➠ Take moments of a Boltzmann-like quantum eq.
i~∂tρ = [H, ρ] + i~Q(ρ)
Q(ρ) unspecified: accounts for dissipationmechnisms
➠ Close by the assumption that Q(ρ) relaxes thesystem to an equilibrium ρα defined as :
➟ an entropy minimizer
➟ constrained to have the same prescribedmoments as ρ
➠ How to define such an equilibrium ?
Page 129
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
11Moments
➠ Defined as in classical mechanics: Moments ofthe Wigner distribution
➠ List of monomials µi(p) e.g. (1, p, |p|2)
µ(p) = (µi(p))Ni=0
➠ w(x, p) → moments m[w] = (mi[w])Ni=0
mi[w] =
∫w(x, p) µi(p) dp , dp :=
dp
(2π~)d
e.g. m = (n, q,W)
Page 130
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
12Remarks
➠ Hydrodynamic moments :
n
q
2W
=
∫W [ρ]
1
p
|p|2
dp
➠ Note
mi[ρ](y) = Tr{ρW−1(µi(p)δ(x − y))}= Observation of the observable µi(p) locally atpoint y= Consistent with the quantum definition of anobservable
Page 131
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
13Moment method
➠ Take moments of the Wigner equation:
∂tm[w]+∇x·∫
w µ p dp+
∫Θ[V ]w µ dp =
∫Q(w) µ dp
➠ In general∫
Q(w) µ dp 6= 0 except for thosemoments conserved by the collision operator (e.g.mass, momentum and energy)
➠ Closure problem: find an expression of theintegrals by setting w to be a solution of theentropy minimization problem
Page 132
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
14Quantum entropy
➠ Density operator ρ
ρψ =∑
s∈S
ρs(ψ, φs)φs
for a complete orthonormal basis φs.
0 ≤ ρs ≤ 1 ,∑
s∈S
ρs = 1
➠ Entropy
H[ρ] =∑
s∈S
ρs(ln ρs − 1)
Page 133
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
15Functional calculus
➠ Let h : R → R. Then: h(ρ) defined by
h(ρ)ψ =∑
s∈S
h(ρs)(ψ, φs)φs
➠ Entropy:
H[ρ] = Tr{ρ(ln ρ − 1)}
➠ Note: we do not take into account Trρ = 1 forthe time being
Page 134
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
16Entropy minimization principle
➠ Entropy:
H[ρ] = Tr{ρ(ln ρ − 1)} ; ρ = W−1(w)
➠ Given a set of moments m = (mi(x))Ni=0,
minimize H(ρ) subject to the constraint that∫
W [ρ](x, p) µ(p) dp = m(x) ∀x
Page 135
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
17Density operator vs Wigner
➠ Problem:
➟ Entropy defined in terms of density operator
➟ Moments defined in terms of Wigner functions
➟ Non local correspondence between the tworepresentations
➠ Consequence
➟ Entropy minimization problem must be statedglobally (in space) and not locally like inclassical mechanics
➟ Requires to express the moment constraints interms of the density operator ρ
Page 136
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
18Moments in terms of ρ
➠ Dualize the constraint: Let λ(x) = (λi(x))Ni=0 be
an arbitrary (vector) test function
∫w(x, p) (µ(p) · λ(x)) dx dp =
∫m(x) · λ(x) dx
Tr{ρ W−1[µ(p) · λ(x)]} =
∫m(x) · λ(x) dx
Page 137
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
19Entropy minimization principle: expression
➠ Given a set of (physically admissible) moments
m = (mi(x))Ni=0, solve
min{ H[ρ] = Tr{ρ(ln ρ − 1)} subject to:
Tr{ρ W−1[µ(p) · λ(x)]} =
∫m · λ dx ,
∀λ = (λi(x))Ni=0 }
Page 138
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
20Entropy minimization principle: resolution
➠ Lemma: The Gateaux derivative of H is:
δH
δρδρ
def= lim
t→0
1
t(H[ρ + tδρ] − H[ρ])
= Tr{ln ρ δρ}
➠ ∃ Lagrange multipliers α(x) = (αi)Ni=0 s.t.
Tr{ln ρ δρ} = Tr{δρ W−1[µ(p) · α(x)]}
➠ ln ρ = W−1[µ(p) · α(x)]
Page 139
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
21Solution of the entropy problem
➠ Solution is ρα,
ρα = exp(W−1[α(x) · µ(p)])
α = (αi(x))Ni=0 is determined s.t. m[ρα] = m
➠ Mα = W [ρα] = Exp(α(x) · µ(p))
Exp · = W [exp(W−1(·))](Quantum exponential)
➠ Analogy with the classical case Mα = exp(α · µ)
Page 140
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
22Quantum moment models
➠ Close the moment eqs. with the quantumMaxwellian:
∂t
∫Exp(α · µ) µ dp + ∇x ·
∫Exp(α · µ) µ p dp
+
∫Θ[V ]Exp(α · µ) µ dp =
∫Q(Exp(α · µ)) µ dp
➠ Evolution system for the vector function α(x, t):Quantum Moment Model (QMM)
➠ Note: r.h.s = 0 for the hydrodynamic moments(mass, momentum and energy)
Page 141
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
23QMM via Density operator
➠ The use of density operator is often morepowerful
➠ Transform (QMM) into density operatorformalism using
Tr{ρ W−1[µ(p) · λ(x)]} =
∫m(x) · λ(x) dx
∀ vector test function λ(x) = (λi(x))Ni=0
Page 142
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
24QMM via Density operator (cont)
➠ Start from quantum Liouville eq.
∂tρ = − i
~[H, ρ] + Q(ρ)
➠ Take moments and close with equilibrium ρ = ρα
∂tTr{ραW−1(λ · µ)} = − i
~Tr{[H, ρα]W−1(λ · µ)}
+Tr{Q(ρα)W−1(λ · µ)}, ∀ test fct λ(x) = (λi(x))Ni=0
Page 143
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
25QMM via Density operator (cont)
➠ Use cyclicity of the trace:
∂t
∫m[ρα]λ dx = − i
~Tr{ρα[W−1(λ · µ),H]} +
+Tr{Q(ρα)W−1(λ · µ)}, ∀ test fct λ(x) = (λi(x))Ni=0
➠ Weak form of (QMM) using density operatorformulation
Page 144
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
26Entropy
➠ Kinetic entropy H[ρ] in terms of w = W [ρ]:
H[ρ] = Tr{ρ(ln ρ − 1)} =
∫w(Ln w − 1) dx dp
with quantum log: Ln w = W [ln(W−1(w))]
➠ Fluid entropy S(m):
S(m) = H[ρα] =
∫Exp(α · µ)((α · µ) − 1) dx dp
where α is s.t. m[α] :=∫Exp(α · µ)µdp = m
Page 145
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
27Inversion of the mapping α → m
➠ S(m) convex.
➠ S(m) =
∫α · m dx − Σ(α)
with Σ(α) Legendre dual of S:
Σ(α) =
∫Exp(α · µ)dx dp
➠ Inversion of the mapping α → m:
δS
δm= α ,
δΣ
δα= m (Gateaux derivatives)
Page 146
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
28Inversion of the mapping α → m: proof
➠ Σ(α) = Tr{ exp(W−1(α · µ)) }
➠ δ( Tr{f(ρ)} ) = Tr{ f ′(ρ) δρ }
➠ Then
δΣ = Tr{ exp(W−1(α · µ)) (W−1(δα · µ)) }
=
∫Exp(α · µ) (δα · µ)dx dp
=
∫δα · m dx
➠ δΣ/δα = m QED
Page 147
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
29Proof of inversion of α → m (cont)
➠ δS =
∫(δα · m + α · δm) dx − δΣ
➠ But, just proven that
δΣ =
∫δα · m dx
➠ Therefore
δS =
∫α · δm dx
➠ δS/δm = α QED
Page 148
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
30Entropy dissipation
➠ Moment models compatible with the entropydissipation
∂tS(m(t)) ≤ 0
for any solution m(t) of the QHD equations
➠ Proof: uses the density matrix formulation of(QMM) with choice λ = α as a test function
∂t
∫m[ρα]α dx = − i
~Tr{ρα[W−1(α · µ),H]}
+Tr{Q(ρα)W−1(α · µ)}
Page 149
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
31Entropy dissipation (cont)
➠ First term is the entropy:∫
m[ρα]α dx = Tr{ραW−1(α · µ)}
= Tr{ρα ln ρα} = Tr{ρα(ln ρα − 1)} = S(m)
➠ Second term: use cyclicity of the trace
Tr{ρα[W−1(α · µ),H]} = Tr{[ρα, ln ρα]H} = 0
➠ Q is entropy dissipative:
Tr{Q(ρα)W−1(α · µ)} = Tr{Q(ρα) ln ρα} ≤ 0
Page 150
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
32Quantum Hydrodynamic Model (QHD)
➠ µ = {1, p, |p|2}
∂tn + ∇x · nu = 0
∂tnu + ∇xΠ = −n∇xV
∂tW + ∇x · Φ = −nu · ∇xV
➠ with Π = pressure tensor, Φ = energy flux:
Π =
∫Exp(α · µ) p ⊗ p dp
2Φ =
∫Exp(α · µ) |p|2 p dp
Page 151
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
33QHD (cont)
➠ and
α · µ = A(x) + B(x) · p + C(x)|p|2
s.t.∫
Exp(α · µ)µ dp = (n, nu,W)Tr
Page 152
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
34Quantum Maxwellian
➠ Mα = Exp(α · µ) = W (exp(W−1(α · µ)))with
α · µ = A(x) + B(x) · p + C(x)|p|2
α = (A, B, C) related w. (n, nu,W) in anon-local way. Note: u 6= B/2C in general(classical = ).
➠ W−1(α · µ) is a second order differentialoperator:
W−1(α · µ)ψ = −~2∇ · (C∇ψ)
−i~(B · ∇ψ + (1/2)(∇ · B)ψ) + (A − (~2/4)∆C)ψ
Page 153
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
35Computation of W−1(α · µ)
➠ Lemma:
W−1(A) = A
W−1(B · p) = −i~(B · ∇ +1
2(∇ · B))
W−1(C|p|2) = −~2(C∆ + ∇C · ∇ +
1
4∆C)
➠ Proof
W−1(B · p) ψ =
∫B(
x + y
2) · p ψ(y)e
ip(x−y)~ dp dy
Page 154
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
36Computation of W−1(α · µ) (cont)
➠ Lemma∫
p eip(x−y)
Page 155
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
37Spectrum of W−1(α · µ)
➠ Suppose W−1(α · µ) has point spectrum only:eigenvalues as[α], eigenvectors φs[α]
W−1(α · µ) =∑
s
as(·, φs)φs
ρα = exp(W−1(α · µ)) =∑
s
eas(·, φs)φs
Trρα = 1 =⇒∑
s
eas = 1
=⇒ as < 0 and ass→∞−→ −∞
=⇒ −W−1(α · µ) elliptic operator
’ =⇒ ’ C(x) ≤ 0 + conditions at ∞
Page 156
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
38Mapping (A, B, C) → (n, u,W)
➠ Finding (A, B, C) in terms of (n, u,W) ≡minimization problem: Thanks to m = δΣ
δαand Σ
convex, this problem ⇐⇒
minα
{Σ(α) −∫
α · m dx}i.e.
minα
{∑
s
eas[α] −∫
α · m dx}
➠ Idea used in practical computations (Gallego)
Page 157
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
39
3. Quantum Isentropic Euler
Joint work with
S. Gallego1 and F. Mehats2
1 MIP, Toulouse ; 2 IRMAR, Rennes
Manuscript, submitted
Page 158
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
40Isothermal model
➠ Fixed uniform temperature T
➟ Change the entropy into the Free Energy
G(ρ) = Tr{Th(ρ) + Hρ}h(ρ) = ρ(ln ρ − 1) = Boltzmann entropy
H =|p|22
+ V = Quantum Hamiltonian
➠ Two moments are considered:
➟ Density n
➟ Momentum nu
Page 159
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
41Entropy minimization problem
➠ Find
min G(ρ) = min(Tr{Tρ(ln ρ − 1) + Hρ})subject to the moment constraints
Tr{ρφ} =
∫nφ dx
Tr{ρW−1(p · Φ)} =
∫nu · Φ dx
for all (scalar and vector) test functions φ and Φ
Page 160
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
42Solution of the entropy minimization problem
➠ Must satisfy
T ln ρ + H = A + B · p
➠ After rearrangement
ln ρ = −H(A, B)
T, H(A, B) =
|p − B|22
+ A
with
A = V − A − |B|2/2 , B = B
➠ H(A, B) = ’modified Hamiltonian’
Page 161
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
43Quantum Maxellian
➠ Density operator formulation
ρn,nu = exp(−H(A, B)
T)
➠ Quantum Maxwellian
Mn,nu = Exp(−H(A, B)
T)
➠ With (A, B) related with (n, nu) by the momentcondition
➠ T = 1 from now on
Page 162
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
44Moment reconstruction
➠ Suppose H(A, B) has discrete spectrum
➟ Eigenvalues λp(A, B), p = 1, . . . ,∞➟ Eigenfunctions ψp(A, B)
➠ Then
n(A, B) (x) =∞∑
p=1
exp(−λp(A, B)) |ψp(A, B) (x)|2
nu(A, B) (x) =∞∑
p=1
exp(−λp(A, B)) Im(~ψp (x)∇ψp (x))
Page 163
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
45Moment reconstruction: proof
➠ By construction
ρn,nu · =∞∑
p=1
exp(−λp(A, B))(·, ψp)ψp
➠ ρ diagonal in the basis (ψp)
➟ Diagonal element = exp(−λp(A, B))
➠ The multiplication operator by φ has matrixelement in this basis
φp,p′ =
∫φψpψp′ dx
Page 164
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
46Moment reconstruction: proof (cont)
➠ Trace = summing up the products of diagonalelements
Tr{ρφ} =∞∑
p=1
exp(−λp(A, B))
∫φ|ψp|2 dx
➠ Finally
n(x0) = Tr{ρδ(x − x0)}
=∞∑
p=1
exp(−λp(A, B))|ψp(x0)|2
➠ Similar computation for nu
Page 165
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
47Quantum isentropic Euler
➠ Special case of (QHD) without energy eq.
∂tn + ∇ · nu = 0
∂tnu + ∇Π = −n∇V
➠ With pressure tensor Π given by
Π =
∫Exp(−H(A, B)) p ⊗ p dp
➠ and modified Hamiltonian
H(A, B) =|p − B|2
2+ A
Page 166
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
48Quantum isentropic Euler (cont)
➠ where (A, B) related with (n, nu) by the momentconditions
n(A, B) (x) =∞∑
p=1
exp(−λp(A, B)) |ψp(A, B) (x)|2
nu(A, B) (x) =∞∑
p=1
exp(−λp(A, B)) Im(~ψp (x)∇ψp (x))
Page 167
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
49Computation of π
➠ To be determined: pressure tensor Π:
Π =
∫Exp(−H(A, B)) p ⊗ p dp
➠ Alternately∫
(∇Π) φ dx = −∫
Π∇φ dx
= −∫
Exp(−H(A, B)) (p · ∇φ)p dx dp
= −Tr{exp(−H(A, B)) W−1((p · ∇φ)p)}
Page 168
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
50Computation of moments: method
➠ Idea: use commutation with H(A, B) to reducethe degree of the p-monomial:
➠ Write (p · ∇φ)p = [H(A, B), A ] + Bwhere B is a polynomial in p of degree ≤ 1(from now on, drop W−1)
➠ Then
Tr{exp(−H(A, B)) (p · ∇φ)p} =
= Tr{exp(−H(A, B)) [H(A, B), A ]}+Tr{exp(−H(A, B)) B}
Page 169
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
51Computation of moments: method (cont)
➠ Use cyclic property of trace
Tr{exp(−H(A, B)) [H(A, B), A ]} =
= Tr{[exp(−H(A, B)) , H(A, B)]A}= 0
➠ Then
Tr{exp(−H(A, B)) (p · ∇φ)p} =
= Tr{exp(−H(A, B)) B}and the degree in p is decreased
➠ Find the convenient A and B
Page 170
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
52Commutation relations
➠ Here A = pφ → Compute [H(A, B), pφ]
➠ Lemma (commutation relations)
[φ, ψ] = 0
[p · Φ, ψ] = −i~(Φ · ∇ψ)
[p · Φ, p · Ψ] = −i~((Φ · ∇)Ψ − (Ψ · ∇)Φ) · p[|p|2/2, φ] = −i~∇φ · p[|p|2/2, pφ] = −i~(∇φ · p)p
➠ Commutation decreases the degree in p
Page 171
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
53Commutation relations: example of proof
➠ Prove [p · Φ, ψ] = −i~(Φ · ∇ψ)
➠ Lemma 1 (see above):
p · Φ = −i~ (Φ · ∇ + (∇ · Φ)/2)
➠ Use that two functions of x commute:
[(∇ · Φ), ψ] = 0
➠ Then, compute
[Φ · ∇ , ψ]f = Φ · ∇(ψf) − ψΦ · ∇f
= (Φ · ∇ψ)f
Page 172
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
54Computation of [H(A, B), pφ]
➠ H(A, B) = |p|2/2 − B · p + A + |B|2/2
➠ Then
[H(A, B), pφ] = −i~{(∇φ · p)p − (B · ∇φ)p +
+φ(∇B)p − φ∇(A + |B|2/2)}
➠ Therefore (p · ∇φ)p = [H(A, B), A ] + Bwith
A = (i/~)pφ
B = (B · ∇φ)p − φ(∇B)p + φ∇(A + |B|2/2)
Page 173
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
55Computation of Π (cont)
➠ Then
Tr{exp(−H(A, B)) (p · ∇φ)p} =
= Tr{exp(−H(A, B)) B}= Tr{exp(−H(A, B)) ((B · ∇φ)p − φ(∇B)p +
+φ∇(A + |B|2/2))}
=
∫((B · ∇φ)nu − φ(∇B)nu + nφ∇(A + |B|2/2)) dx
=
∫(−∇(nu ⊗ B) − (∇B)nu + n∇(A + |B|2/2))φ dx
= −∫
(∇Π) φ dx
Page 174
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
56Final expression of Π
➠ Finally
∇Π = ∇(nu ⊗ B) + (∇B)nu − n∇(A + |B|2/2)
Page 175
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
57Quantum Isentropic Euler
➠ Final expression:
∂tn + ∇ · nu = 0
∂tnu + ∇(nu ⊗ B) + (∇B)nu − n∇(A + |B|2/2) = −n∇V
➠ Where (A, B) are related with (n, nu) by:
n(A, B) (x) =∑
exp(−λp(A, B)) |ψp(A, B) (x)|2
nu(A, B) (x) =∑
exp(−λp(A, B)) Im(~ψp (x)∇ψp (x))
➠ And λp(A, B), ψp(A, B): spectrum of
H(A, B) = |p|2/2 − B · p + A + |B|2/2
Page 176
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
58Free energy (entropy)
➠ Fluid free energy G(n, nu):
G(n, nu) = G(ρn,nu)
= Tr{exp(−H(A, B))(−H(A, B) − 1 + H)}= Tr{exp(−H(A, B))(B · p − A − |B|2/2 − 1 + V )}
=
∫(nu · B + n(V − A − |B|2/2 − 1)) dx
➠ By construction: if V is independent of time:
dGdt
≤ 0
(with = for smooth solutions)
Page 177
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
59Free energy (cont)
➠ If V solves Poisson eq.
−∆V = n
Then, againdGdt
≤ 0
(with = for smooth solutions)
Page 178
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
60Gauge invariance
➠ Let S(x) be a smooth function. Then
exp(iS
~) H(A, B) exp(−iS
~) = H(A, B + ∇S)
➠ Proof: write
exp(iS/~)H(A, B) exp(−iS/~) − H(A, B) =
= exp(iS/~)[H(A, B), exp(−iS/~)]
and use the commutation relations
Page 179
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
61Gauge invariance (cont)
➠ Consequence 1: eigenvalues of H(A, B) andH(A, B + ∇S) are the same
➠ Consequence 2:
exp(iS
~) exp(−H(A, B)) exp(−iS
~) = exp(−H(A, B + ∇S))
The equilibrium density operators are conjugate
➠ Consequence 3: eigenvalues of exp(−H(A, B))and exp(−H(A, B + ∇S)) are the same
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
62Free energy (again)
➠ Free energy
G(n, nu) =
∫(nu · B + n(V − A − |B|2/2 − 1)) dx
Implies δGδn
= V − A − |B|2/2 = A
δGδnu
= B
➠ Legendre dual
Σ(A, B) =
∫n dx = Tr{exp(−H(A, B))} = Σ(A, B)
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
63Inversion formula
➠ Inversion formula and chain rule:
n(A, B) =δΣ
δA= −δΣ
δA
(nu)(A, B) =δΣ
δB=
δΣ
δB− B
δΣ
δA
➠ It results:
δΣ
δA= −n(A, B)
δΣ
δB= (nu)(A, B) − n(A, B) B
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
64Gauge invariance (again)
➠ eigenvalues of exp(−H(A, B)) andexp(−H(A, B + ∇S)) are the same:
Σ(A, B) = Tr{exp(−H(A, B))} =
= Tr{exp(−H(A, B + ∇S))} = Σ(A, B + ∇S)
➠ Implies
δΣ
δA(A, B + ∇S) =
δΣ
δA(A, B)
δΣ
δB(A, B + ∇S) =
δΣ
δB(A, B)
Page 183
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
65Velocity constraint
➠ Consequence 1’:
n(A, B + ∇S) = n(A, B)
(nu)(A, B + ∇S) = nu(A, B) + n(A, B)∇S
➠ Consequence 2’: ∀ test function S(x):
limt↓0
t−1(Σ(A, B + t∇S) − Σ(A, B)) = 0 =
=
∫δΣ
δB· ∇S dx =
∫(nu − nB) · ∇S dx
Meaning that ∇ · (n(u − B)) = 0
Page 184
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
66Equivalent formulations of momentum eq.
➠ Form 1 (Original formulation)
∂tnu + ∇(nu ⊗ B) + (∇B)nu − n∇(A + |B|2/2) = −n∇V
➠ Form 2: Use ∇|B|2/2 = (∇B)B
∂tnu + ∇(nu ⊗ B) + n(∇B)(u − B) + n∇(V − A) = 0
➠ Form 3: Use ∇ · (n(u − B)) = 0
∂tnu + ∇(nu ⊗ u) + n(∇× u) × (B − u) +
+n∇(V − A − |B − u|2/2) = 0
Page 185
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
67Equivalent formulations of momentum eq. (cont)
➠ Form 4: Use continuity equation
∂tu + (∇× u) × B + ∇(u · B − |B|2/2 + V − A) = 0
Page 186
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
68Irrotational flows
➠ Define the vorticity ω = ∇× u. ω satisfies
∂tω + ∇× (ω × B) = 0
Proof: take the curl of Form (4)
➠ If ω|t=0 = 0, then ω ≡ 0 for all times: irrotationalflow
➠ Irrotational flow =⇒ ∃S(x, t) s.t. u = ∇S
➠ Then:u = B = ∇S
Page 187
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
69Proof that u = B for irrotational flows
➠ Lemma: nu(A, 0) = 0
➠ Proof: nu(A, 0) =∫Exp(−H(A, 0))p dp
But H(A, 0) = |p|2/2 + A even w.r.t. p
➠ Then Exp(−H(A, 0)) even w.r.t. pNot obvious (Exp 6= exp)Prove it for powers (using Wigner)Then by series expansion, for the exponential
➠ Then nu(A, 0) = 0 by parity
Page 188
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
70u = B for irrotational flows (cont)
➠ Using the Gauge transformation
nu(A,∇S) = nu(A, 0) + n(A, 0)∇S
= 0 + n(A,∇S)∇S
➠ Shows that the solution (A, B) of the momentproblem is given by
➟ A which solves n(A, 0) = n
➟ B = ∇S = u
➠ QED if (A, B) are unique
➟ Only formal
Page 189
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
71Quantum Euler for irrotational flows
➠ Use B = u in Form (3)
∂tn + ∇ · nu = 0
∂tnu + ∇(nu ⊗ u) + n∇(V − A) = 0
∇× u = 0
➠ Where A is related with n by:
n(A) (x) =∑
exp(−λp(A, 0)) |ψp(A, 0) (x)|2
➠ (λp, ψp)(A, 0) spectrum of H(A, 0) = |p|2/2 + A
Page 190
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
72Quantum Euler for irrotational flows (cont)
➠ Advantage: only one quantity A to determinefrom the spectral problem
➠ Important special case: One-dimensional flows
Page 191
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
73Semiclassical asymptotics
➠ When ~ → 0 recover the classical isothermalEuler eqs.
➠ Retaining terms of order ~2 gives
∂tn + ∇ · nu = 0
∂tnu + ∇(nu ⊗ u) + T∇n + n∇V − ~2
6n∇(
∆√
n√n
) +
+~
2
12ω × (∇× (nω)) +
~2
24n∇(|ω|2) = 0
ω = ∇× u
➠ Already given in [Juengel, Matthes]
Page 192
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
74Semiclassical asymptotics + irrotational flows
➠ If ω = 0, system reduces to
∂tn + ∇ · nu = 0
∂tnu + ∇(nu ⊗ u) + T∇n + n∇V − ~2
6n∇(
∆√
n√n
) = 0
So-called ’Quantum Hydrodyanmic Model’
➠ Used in the literature
➟ Also in the rotational cases
➟ Heuristic derivation (only justified if T = 0)
➠ Here, the ’Quantum Hydrodyanmic Model’ isderived based on first principles
Page 193
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
75Preliminary numerical results
➠ One-dimensional model
➟ coupled with Poisson’s eq.
➟ momentum relaxation term
➠ Double barrier structure
➠ Boundary conditions
➟ Dirichlet for the wave-function (andconsequently for the density)
➟ Zero flux for the momentum
➠ Dynamics of electrons injected from the leftboundary
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(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
76Numerical results (I)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
Den
sity
Position0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
k=0
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
1
1.5
2
2.5
Position
Vel
ocity
k=0
Initial data. Left: density and potential.Right: velocity
Page 195
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
77Numerical results (II)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
Den
sity
0 0.2 0.4 0.6 0.8 10
1
2
3
4
Position0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
k=20
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2
2.5
Position
Vel
ocity
k=20
t = 0.1. Left: density and potential.Right: velocity
Page 196
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
78Numerical results (III)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
Den
sity
Position0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
k=100
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
1
1.5
2
2.5
Position
Vel
ocity
k=100
t = 0.5. Left: density and potential.Right: velocity
Page 197
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
79Numerical results (IV)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
Den
sity
Position0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
k=200
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
1
1.5
2
2.5
Position
Vel
ocity
k=200
t = 1. Left: density and potential.Right: velocity
Page 198
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
80Numerical results (V)
0 50 100 150 2000.5
1
1.5
2
2.5
3
Fre
e E
nerg
y
Time iterations
Free energy vs time
Page 199
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
81
4. Summary and conclusion
Page 200
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82Summary: quantum moment models
➠ Extension of the Levermore’s moment method tothe quantum case
➟ Take local moments of the density operator eq.
➟ Close by a minimizer of the entropy functional
➠ leads to:
➟ Formulation of the entropy minimizationproblem as a global problem (local in classicalmechanics)
➟ Non-local closure to the QuantumHydrodynamics eq.
Page 201
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
83Summary: Isothermal quantum Euler
➠ Isothermal case: entropy = free energy
➟ Analytic computation of pressure tensor
➟ System involves (n, nu) and (A, B)
➟ Related by the quantum moment problem
➠ Gauge invariance
➟ Several equivalent formulations of the model
➟ Constraint between u and B
➠ Special interest for irrotational flows
➟ Simplification: problem depends on A only
➟ One-dimensional flows
Page 202
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
84Perspectives
➠ Show entropy minimization problem has asolution in a reasonable sense
➠ Compute analytical closure for the full QHDmodel (as done for the isothermal case)
➠ Investigate Gauge invariance properties
➠ Small T asymptotics (formal for isothermal case)
➠ ~ expansion up to order ~2: see [Jungel, Matthes,
Milisic]
➠ Normal mode analysis of linearized model
Page 203
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
1
Chapter 4
Diffusion models: classical case
P. Degond
MIP, CNRS and Universite Paul Sabatier,
118 route de Narbonne, 31062 Toulouse cedex, France
[email protected] (see http://mip.ups-tlse.fr)
Page 204
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
2Summary
1. Drift-Diffusion model
2. Energy-Transport model
Page 205
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
3
1. Drift-Diffusion model
Page 206
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
4Linear Boltzmann model
➠ The simplest collisional kinetic model
∂f
∂t+ v · ∇xf −∇xV · ∇vf = Q(f)
Q(f)(v) =
∫
v′∈Rd
[W (v′ → v)f(v′) − W (v → v′)f(v)] dv′
➠ V (x, t) can be
➟ External force potential
➟ Self-consistent Mean-Field potential
➟ In all this part, considered as known
➠ W (v1 → v) ≥ 0: scattering rate
Page 207
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
5Modeling of collisions
➠ Q(f) Models collisions with the surrounding:
➟ Plasmas → electrons against ions, neutrals,. . .
➟ Semiconductors → electrons againstimpurities, phonons, . . .
➟ Nuclear reactors → neutron against fissilematerial, . . .
➟ Radiative transfer → interaction of radiationw. matter, . . .
➟ Chemiotaxis → reaction of bacteria tonutriments
➟ . . .
Page 208
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
6Detailed balance property
➠ Collision operator models relaxation tothermodynamic equilibrium w. scatteringmedium: =⇒ detailed balance property
W (v′ → v)
W (v → v′)=
M(v)
M(v′)
where M = Normalized centered Maxwellian attemperature T of the scattering medium:
T
M(v)
v
M(v) =(
12πT
)d/2e−
v2
2T
Page 209
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
7Collision operator
➠ Introduce φ(v, v′) = W (v → v′)M(v′)−1:φ symmetric φ(v, v′) = φ(v′, v) ≥ 0Then
Q(f)(v) =
∫
v′φ(v, v′)[M(v)f(v′) − M(v′)f(v)]dv′
➠ Special case: φ(v, v′) = ν = Constant
Q(f)(v) = −ν(f(v) − nM(v)) , n =
∫
f dv
BGK operator
➠ We restrict ourselves to this case for simplicity
Page 210
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
8Properties of Q
➠ Q(f)(v) = −ν(f(v) − nM(v))
➠ Conservation of particle number:∫
Q(f)dv = 0
➠ Null set of Q (equilibria) :
Q(f) = 0 ⇐⇒ ∃n ∈ R such that f = nM(v)
➠ Free energy decay :∫
Q(f)(ln f+H) dv = −
∫
ν(f−nM)(ln f−ln(nM)) dv ≤ 0
Page 211
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
9Diffusion scaling
➠ Behaviour at the macro scale −→ introduce
η =mean free path
typical macroscopic distance≪ 1
➠ change of variables (diffusion scaling):
➟ x′ = η x, t′ = η2t, F = ηF ′
η2∂f η
∂t+ η (v · ∇xf
η −∇xV · ∇vfη) = Q(f η)
➠ η → 0 describes the large scale behaviour
➟ Rigorous proof: [POUPAUD], [GOLSE-POUPAUD]
Page 212
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
10Limit η → 0: Drift-Diffusion
➠ as η → 0, f η −→ n(x, t)M(v)n(x, t) satisfies Drift-Diffusion model:
➠ Continuity equation
∂n
∂t+ ∇x · j = 0
➠ Current equation
j = −D(∇xn + nT−1∇xV )
➠ D = ν−1T = Diffusion coefficient
Page 213
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
11Limit η → 0: Sketch of proof
➠ Step 1: f η → Maxwellian nM where n = n(x, t)
➟ Chapman-Enskog expansion: definef η
1 = η−1(f η − nηM)f η
1 = O(1) as η → 0. Define f1 = limη→0 f η1
➠ Step 2: Write continuity eq.Remains valid as η → 0To be determined: Flux
➠ Step 3: Compute the flux taking the appropriatemoment of f1
Page 214
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
12Step 1: Convergence to equilibrium
➠ Suppose f η → f smoothly
Boltzmann eq. =⇒ Q(f η) = O(η)
=⇒ Q(f) = 0
=⇒ f = n(x, t)M(v)
Page 215
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
13Step 1: Chapman-Enskog expansion
➠ Write (exact): f η = nηM + ηf η1
➠ Then:1
ηQ(f η) = −νf η
1 = T f η + η∂tfη
with
T f = v · ∇xf −∇xV · ∇vf Transport operator
➠ As η → 0:
f η1 → f1 = −ν−1T f
Page 216
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
14Step 2: Continuity eq.
➠ integrate Boltzmann eq. with respect to v anduse that Q preserves particle number:
∂nη
∂t+ ∇x · j
η = 0
nη =
∫
f ηdv , jη = η−1
∫
f ηvdv =
∫
f η1 vdv
➠ nη → n and jη → j =
∫
f1vdv and:
∂n
∂t+ ∇x · j = 0
Page 217
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
15Step 3: The current eq.
➠ From f1 = −ν−1T f we compute:
f1 = −ν−1(v · ∇x −∇xV · ∇v)(nM)
= −ν−1(∇xn + nT−1∇xV ) · vM
➠ Then
j =
∫
f1vdv
= −ν−1(
∫
M(v)v ⊗ v dv)(∇xn + nT−1∇xV )
= −Tν−1(∇xn + nT−1∇xV )
Page 218
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
16About the rigorous proof
➠ Rigorous proof follows closely formal proof
➟ Convergence: weak topology enough.
➟ Error estimate f η − f = 0(η) requiresregularity estimates for the Chapman-Enskogexpansion (see e.g. [Ben Abdallah, Tayeb])
➠ Can be extended easily to the more generalcollision operator written at the beginning
➟ With suitable assumptions on the scatteringkernel W (v′ → v)
Page 219
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
17About the Drift-Diffusion model
➠ widely used by the engineers
➟ But not suitable for strongly non equilibriumphenomena
➠ Question to be investigated:
➟ Find more complex macroscopic models
➟ with a broader range of applicability
➟ using the same methodology
➠ Examples:
➟ Energy-Transport model (developed below)
➟ SHE-Fokker-Planck model (skipped)
Page 220
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
18
2. Energy-Transport model
N. Ben Abdallah, P. D., S. Genieys,
J. Stat. Phys. 84 (1996), pp. 205-231
N. Ben Abdallah, P. D.,
J. Math. Phys. 37 (1996), pp. 3306-3333
Page 221
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
19More complex BGK operator
➠ Same expression
Q(f)(v) = −ν(f − nMT (v))
with
MT (v) = (2πT )−d/2 exp(−v2/(2T ))
➠ But now T is a second free parameter s.t.
➟ (n, T ) ensure mass and energy conservation
∫
Q(f)
(
1
|v|2
)
dv = 0
Page 222
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
20Situation modeled by Q(f)
➠ Combination of an elastic and a binary collisionoperator [Ben Abdallah, D., Genieys]
➟ Semiconductors (phonon collisions treated aselastic)
➟ Plasmas (electron-ion collisions treated aselastic)
➟ . . .
➠ Energy exchanges between the particles are moreefficient than with the surrounding
➟ Possibility of a different temperature than thatof the background
Page 223
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
21Determination of (n, T )
➠ (n, T ) are given by:
∫
nMT (v)
(
1
|v|2
)
dv =
(
n
dnT
)
=
∫
f(v)
(
1
|v|2
)
dv
➠ Maxwellian can be rewritten as
nMT (v) = exp(A + C|v|2/2)
with
A = ln(n
(2πT )d/2) , C = −
1
T
➠ Note A = µ/T , µ = Chemical potential
Page 224
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
22Conservative variables vs entropic variables
➠ Energy
W =
∫
f |v|2/2 dv = dnT/2
➠ Two sets of variables
➟ Conservative variables (n,W)
➟ Entropic variables (A, C)
➟ (n,W) ←→ (A, C) is a change of variables
➟ Inversion through entropy (see below)
Page 225
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
23Properties of Q
➠ Mass and energy conservation
∫
Q(f)
(
1
|v|2
)
dv = 0
➠ Null set of Q (equilibria) :
Q(f) = 0 ⇐⇒ ∃(A, C) such that f = exp(A+C|v|2/2)
➠ Entropy decay:
∫
Q(f) ln f dv =
= −
∫
ν(f−exp(A+C|v|2/2))(ln f−(A+C|v|2/2)) dv ≤ 0
Page 226
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
24Diffusion scaling
➠ Boltzmann eq. under diffusion scaling:
η2∂f η
∂t+ η (v · ∇xf
η −∇xV · ∇vfη) = Q(f η)
Page 227
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
25Limit η → 0: Energy-Transport
➠ as η → 0, f η −→ n(x, t)MT (v)where (n, T ) satisfy the Energy-Transport model:
➠ Mass and energy conservation eqs.
∂n
∂t+ ∇x · jn = 0
∂W
∂t+ ∇x · jW + ∇xV · jn = 0
➠ With
➟ W = dnT/2 : energy
➟ jn, jW : particle and energy fluxes
Page 228
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
26Energy-Transport model (cont)
➠ Fluxes(
jn
jW
)
= −D
(
∇xA − C∇xV
∇xC
)
➠ D Diffusion matrix, symmetric, positive-definite
➠ Energy-transport model:
➟ Balance eqs. for the conservative variables
➟ Fluxes expressed in terms of the gradients ofthe entropic variables
Page 229
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
27Limit η → 0: Sketch of proof
➠ Step 1: f η → Maxwellian nMT where(n, T ) = (n, T )(x, t)
➟ Chapman-Enskog expansion: definef η
1 = η−1(f η − nηMT η)f η
1 = O(1) as η → 0. Define f1 = limη→0 f η1
➠ Step 2: Write mass and energy conservationequationsRemain valid as η → 0To be determined: Fluxes
➠ Step 3: Compute the fluxes taking theappropriate moment of f1
Page 230
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
28Step 1: Convergence to equilibrium
➠ Suppose f η → f smoothly
Boltzmann eq. =⇒ Q(f η) = O(η)
=⇒ Q(f) = 0
=⇒ f = n(x, t)MT (v) = exp(A + C|v|2/2)
Page 231
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
29Step 1: Chapman-Enskog expansion
➠ Write (exact): f η = nηMT η + ηf η1
➠ Then:1
ηQ(f η) = −νf η
1 = T f η + η∂tfη
with
T f = v · ∇xf −∇xV · ∇vf Transport operator
➠ As η → 0:
f η1 → f1 = −ν−1T f
Page 232
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
30Step 2: Mass and energy balance eqs. (finite η)
➠ Take moments of the Boltzmann eq. against 1and |v|2/2 and use that Q preserves mass andenergy:
∂nη
∂t+ ∇x · j
ηn = 0
∂Wη
∂t+ ∇x · j
ηW + ∇xV · jη
n = 0
➠ Withjηn = η−1
∫
f ηvdv =
∫
f η1 vdv
jηW = η−1
∫
f ηv |v|2/2 dv =
∫
f η1 v |v|2/2 dv
Page 233
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
31Step 2: Mass and energy balance eqs. (η → 0)
➠ As η → 0
jηn → jn =
∫
f1vdv
jηW → jW =
∫
f1v |v|2/2 dv
➠ Gives the mass and energy balance eqs in thelimit η → 0
∂n
∂t+ ∇x · jn = 0
∂W
∂t+ ∇x · jW + ∇xV · jn = 0
Page 234
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
32Step 3: Eq. for jn
➠ From f1 = −ν−1T f and
f = nMT = exp(A + C|v|2/2)
➠ Compute:
f1 = −ν−1(v · ∇x −∇xV · ∇v)(exp(A + C|v|2/2))
= −ν−1((∇xA − C∇xV ) · v(nMT ) + ∇xC · (|v|2/2) v(nMT ))
➠ Then jn =
∫
f1vdv =
= −ν−1n( (
∫
MT (v)v ⊗ v dv)(∇xA − C∇xV )
+(
∫
MT (v)v ⊗ v |v|2/2 dv)∇xC )
Page 235
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
33Step 3: Eq. for jn (cont)
➠ Denote∫
MT (v)v⊗v dv = a11T Id ,
∫
MT (v)v⊗v |v|2/2 dv = a12T2Id
a11, a12 only depend on the dimension d
➠ Then
jn = −D11(∇xA − C∇xV ) − D12∇xC
D11 = ν−1nTa11 , D12 = ν−1nT 2a12
Page 236
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
34Step 3: Eq. for jW
➠ Similar computation for jW =∫
f1v |v|2/2 dv
gives
jW = −D12(∇xA − C∇xV ) − D22∇xC
D22 = ν−1nT 3a22
➠ where a22 is defined by∫
MT (v)v ⊗ v (|v|2/2)2 dv = a22T3Id
and only depends on the dimension
Page 237
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
35Step 3: Matrix eq. for the fluxes
➠ Can be summarized in the matrix equality(
jn
jW
)
= −D
(
∇xA − C∇xV
∇xC
)
➠ With diffusion matrix
D =
(
D11 D12
D12 D22
)
➠ D symmetric positive-definite
Page 238
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
36Step 3: Positive-definiteness of D
➠ Let a, b two vectors of Rd
(
a
b
)T
D
(
a
b
)
=
∫
MT (v)|(a + b|v|2/2) · v|2 dv ≥ 0
➠ And(
a
b
)T
D
(
a
b
)
= 0 ⇐⇒ a = b = 0
Page 239
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
37About rigorous proof
➠ No rigorous proof
➟ Partial proof [Ben Abdallah, Desvillettes, Genieys] inthe case of Boltzmann + elastic operator
➠ Formal proof for more complex collision operator
➟ see [Ben Abdallah, D. Genieys]
➟ More complicated: Diffusion matrix →inversion of the linearized operator
Page 240
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
38About the Energy-Transport model
➠ increasingly used by engineers
➟ in strongly non equilibrium situations
➠ Examples:
➟ semiconductors
➟ plasmas
➟ . . .
➠ Extensions to the quantum world
➟ To be investigated in the next lectures
Page 241
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
1
Chapter 5
Quantum diffusion models
P. Degond
MIP, CNRS and Universite Paul Sabatier,
118 route de Narbonne, 31062 Toulouse cedex, France
[email protected] (see http://mip.ups-tlse.fr)
Page 242
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
2Summary
1. Quantum energy-Transport model
2. Quantum drift-Diffusion model
3. Summary and conclusion
Page 243
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
3
1. Quantum energy-Transport model
P. D., F Mehats, C. Ringhofer,
J. Stat. Phys. 118 (2005), pp. 625-667
Page 244
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
4Quantum kinetic equation
➠ Diffusion model
➟ Form of the collision operator matters
➟ 6= hydro models
➠ Need to specify Q in Liouville equation
i~ρ = [H, ρ] + i~Q(ρ)
➠ Or in the Wigner eq.
∂tw + p · ∇xw + Θ~[V ]w = Q(w)
Page 245
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
5BGK operator
➠ Classical case
➟ Relaxation to the Maxwellian
Q(f)(v) = −ν(f − exp(A + C|v|2/2))
with (A, C) such that mass and energy arepreserved
➠ Quantum case: replace the classical Maxwellianby the quantum one
Q(w)(v) = −ν(w − Exp(A + C|p|2/2))
➠ Exp w = W (exp (W−1w))
Page 246
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
6Quantum Maxwellian
➠ Wigner form. Mn,W = Exp(A + C|p|2/2)
∫Exp(A + C|p|2/2)
(1
|p|2/2
)dp =
(n
W
)
➠ Density operator form.
ρn,W = W−1(Mn,W) = exp(W−1(A + C|p|2/2))
with, ∀ test fct. φ:
Tr{ρn,W φ} =
∫nφ dx , Tr{ρn,W φ|p|2/2} =
∫Wφ dx
Page 247
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
7Entropy minimization principle
➠ Reminder: ρn,W = exp(W−1(A + C|p|2/2))satisfies the entropy minimization principle:
➟ Solve
min H[ρ] = Tr{ρ(ln ρ − 1)} subject to, ∀ test fct φ:
Tr{ρn,W φ} =
∫nφ dx , Tr{ρn,W φ|p|2/2} =
∫Wφ dx }
➠ In Wigner form
H[ρ] = Tr{ρ(ln ρ − 1)} =
∫w(Ln w − 1) dx dp
with quantum log: Ln w = W [ln(W−1(w))]
Page 248
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
8Quantum BGK operator
➠ For Wigner distribution w given,
➟ denote Mw := Mn,W s.t.n and W are the density and energy of f :
∫Mn,W
(1
|p|2/2
)dp =
∫f
(1
|p|2/2
)dp
➠ Then Quantum BGK operator is written
Q(w) = −ν(w −Mw)
➠ Density operator: call it Mρ as well
Q(ρ) = −ν(ρ −Mρ)
Page 249
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
9Situation modeled by Q(w)
➠ Similar as in the classical case but when quantumeffects need to be taken into account
➠ Energy exchanges between the particles are moreefficient than with the surrounding
➟ Possibility of a different temperature than thatof the background
Page 250
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
10Conservative variables vs entropic variables
➠ Reminder: two sets of variables
➟ Conservative variables (n,W)
➟ Entropic variables (A, C)
➟ (n,W) ←→ (A, C) is a functional change ofvariables
➟ Inversion through entropy
Page 251
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
11Properties of Q
➠ Mass and energy conservation
∫Q(w)
(1
|p|2
)dp = 0
➠ Null set of Q (equilibria) :
Q(w) = 0 ⇐⇒ ∃(A, C) such that w = Exp(A+C|p|2/2)
➠ Entropy decay:∫
Q(w)Lnw dx dp = Tr{Q(ρ) ln ρ} ≤ 0
Page 252
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
12Proof of entropy decay
➠ Proof shown in the classical case does not work(saying that Ln(w) is increasing w.r.t. w ismeaningless)
➠ Use convexity of the function Λ:
λ ∈ [0, 1] → H((1 − λ)Mρ + λρ)
➠ givesdΛ
dλ(1) ≥ Λ(1) − Λ(0)
➠ Reminder
δTr{f(ρ)} = Tr{f ′(ρ) δρ} , δH(ρ) = Tr{ln ρ δρ}
Page 253
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
13Proof of entropy decay (cont)
➠ Then
dΛ
dλ(λ) = Tr{ln((1 − λ)Mρ + λρ) (ρ −Mρ)}
➠ and
dΛ
dλ(1) = Tr{ln ρ (ρ −Mρ)} ≥ H(ρ) − H(Mρ)
➠ Entropy minimization principle
H(ρ) − H(Mρ) ≥ 0
➠ Tr{Q(ρ) ln ρ} = −νTr{ln ρ (ρ−Mρ)} ≤ 0 QED
Page 254
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
14Diffusion scaling
➠ Wigner eq. under diffusion scaling:
η2∂wη
∂t+ η(v · ∇xw
η − Θ(wη)) = Q(wη)
Page 255
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
15Limit η → 0: Quantum Energy-Transport
➠ as η → 0, wη −→ Exp(A + C|p|2/2)where (A, C) satisfy the Energy-Transport model:
➠ Mass and energy conservation eqs.
∂n
∂t+ ∇x · jn = 0
∂W∂t
+ ∇x · jW + ∇xV · jn = 0
➠ With∫Exp(A + C|p|2/2)
(1
|p|2/2
)dp =
(n
W
)
Page 256
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
16Energy-Transport model (cont)
➠ Fluxes
jn = −ν−1[∇Π + n∇V ]
jW = −ν−1[∇Q + (W Id + Π)∇V − ~2
8n∇(∆V )]
➠ with
Π(A, C) =
∫Exp(A + C|p|2/2) p ⊗ p dp
Q(A, C) =
∫Exp(A + C|p|2/2) p ⊗ p |p|2/2 dp
Page 257
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
17Structure of the model
➠ Like in the classical case:
➟ Balance eqs. for the conservative variables(n,W)
➟ Fluxes expressed in terms of the gradients ofthe entropic variables (A, C)
➠ Reminder
➟ Passage (n,W) ←→ (A, C) through entropy
➠ However, no clear symmetric positive-definitematrix structure.
➟ Symmetry is more concealed (operator-wise)
Page 258
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
18Entropy decay
➠ Entropy S(n,W) = H(Mn,W):
S(n,W) =
∫Mn,W (LnMn,W − 1) dx dp
=
∫Exp(A + C|p|2/2)(A + C|p|2/2 − 1) dx dp
=
∫(n(A − 1) + CW) dx
➠ Thend
dtS(n,W) ≤ 0
➠ Proof: similar as for hydrodynamic model
Page 259
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
19Limit η → 0: Sketch of proof
➠ Step 1: wη → Maxwellian Exp(A + C|p|2/2)where (A, C) = (A, C)(x, t)
➟ Chapman-Enskog expansion: definewη
1 = η−1(wη −Mwη)wη
1 = O(1) as η → 0. Define w1 = limη→0 wη1
➠ Step 2: Write mass and energy conservationequationsRemain valid as η → 0To be determined: Fluxes
➠ Step 3: Compute the fluxes taking theappropriate moment of w1
Page 260
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
20Step 1: Convergence to equilibrium
➠ Suppose wη → w smoothly
Wigner-BGK eq. =⇒ Q(wη) = O(η)
=⇒ Q(w) = 0
=⇒ w = Exp(A + C|p|2/2)
Page 261
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
21Step 1: Chapman-Enskog expansion
➠ Write (exact): wη = Mwη + ηwη1
➠ Then:1
ηQ(wη) = −νwη
1 = T wη + η∂twη
with
T w = v · ∇xw − Θ~[V ]w Quantum transport operator
➠ As η → 0:
wη1 → w1 = −ν−1T w
Page 262
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
22Step 2: Mass and energy balance eqs. (finite η)
➠ Take moments of the Wigner-BGK eq. against 1and |p|2/2 and use that Q preserves mass andenergy:
∂nη
∂t+ ∇x · jη
n = 0
∂Wη
∂t+ ∇x · jη
W + ∇xV · jηn = 0
➠ Withjηn = η−1
∫wηpdp =
∫wη
1pdp
jηW = η−1
∫wηp |p|2/2 dp =
∫wη
1p |p|2/2 dp
Page 263
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
23Step 2: Mass and energy balance eqs. (η → 0)
➠ As η → 0
jηn → jn =
∫w1pdp
jηW → jW =
∫w1p |p|2/2 dp
➠ Gives the mass and energy balance eqs in thelimit η → 0
∂n
∂t+ ∇x · jn = 0
∂W∂t
+ ∇x · jW + ∇xV · jn = 0
Page 264
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
24Step 3: Eq. for jn
➠ From w1 = −ν−1T w and
w = Exp(A + C|p|2/2)
➠ Compute:
w1 = −ν−1[∇x · (pExp(A + C|p|2/2))
−Θ~[V ]Exp(A + C|p|2/2)]
➠ Then jn =
∫w1pdp =
= −ν−1[∇(
∫Exp(A + C|p|2/2)p ⊗ p dp)
−∫
Θ~[V ](Exp(A + C|p|2/2)) p dp]
Page 265
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
25Step 3: Eq. for jn (cont)
➠ Lemma
∫Θ~[V ]w
1
p
|p|2/2
dp =
0
−n∇V
−nu · ∇V
➠ Lemma∫
Θ~[V ]w|p|2/2p dp = −(W Id+Π)∇V +~2
8n∇(∆V )
➠ Proof: Use definition of Θ~[V ] and simpleproperties of Fourier transform
Page 266
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
26Step 3: Eq. for jn (cont)
➠ Then
jn = −ν−1[∇Π + n∇V ] QED
Page 267
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
27Step 3: Eq. for jW
➠ Similar computation for jW =∫
w1p |p|2/2 dpgives
= −ν−1[∇(
∫Exp(A + C|p|2/2)p ⊗ p |p|2/2dp)
−∫
Θ~[V ](Exp(A+C|p|2/2)) p |p|2/2 dp]
➠ Using previous Lemma:
jW = −ν−1[∇Q + (W Id + Π)∇V − ~2
8n∇(∆V )]
QED
Page 268
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
28~ expansion
➠ Expansion of Π:
Πrs = δrs n T
+ ~2
12d n δrs(∆x ln n + 2∆x ln T + 2∇x ln n · ∇x ln T
−d+2
2|∇x ln T |2)
+~2
12n( − ∂2
rs ln n − 2∂2rs ln T − ∂r ln n ∂s ln T
−∂r ln T ∂s ln n + d+2
2∂r ln T ∂s ln T ),
With T = 2W/(dn)
Page 269
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
29~ expansion
➠ Expansion of Q:
Qrs = d+2
2δrs n T 2
+ ~2
24d n T δrs(∆x ln n + (d + 8)∆x ln T
+2(d + 4)∇x ln n · ∇x ln T + d2−4d−8
2|∇x ln T |2)
+~2
24(d + 4) n T (−∂2
rs ln n − 3∂2rs ln T
−∂r ln n ∂s ln T − ∂r ln T ∂s ln n + d2∂r ln T ∂s ln T )
With T = 2W/(dn)
Page 270
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
30~ expansion: small temperature variation
➠ |∇ ln T |/|∇ ln n| ≪ 1
Jn = −∇(
n T +~2
12dn ∆ ln n
)− n∇(V + VB[n]) ,
Jw = −∇(
d + 2
2n T 2 +
~2
24
d + 4
dn T ∆ ln n
)
−d + 4
2n T ∇VB[n] −
(d + 2
2n T +
~2
12dn ∆ ln n
)∇V
+~2
12n (∇∇ ln n)∇V +
~2
8∇∆ ln n .
Page 271
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
31About Quantum Energy-Transport models
➠ No rigorous proof
➟ existence ?
➟ convergence ?
➠ No numerical simulations (so far)
➠ In the literature
➟ quantum energy-transport models can befound
➟ But: derivation (and model itself) different
➟ e.g. extensions of the DG (Density-Gradientmodel) by [Chen & Liu, JCP 05]
Page 272
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
32Quantum Drift-Diffusion model
➠ Have a nice structure (see next lecture)
➠ Hope that structure can be extended to QuantumEnergy-Transport
Page 273
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
33
2. Quantum drift-Diffusion model
P. D., F Mehats, C. Ringhofer, J. Stat. Phys. 118 (2005), 625-667
P. D., F. M., C. R., Contemp. Math., 371 (2005), 107–131
S. Gallego, F. Mehats, SIAM Num. Anal. 43 (2005), 1828-1849
P. D., S. Gallego, F. Mehats, J. Comp. Phys., to appear
Page 274
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
34BGK operator
➠ Classical case
➟ Relaxation to the Maxwellian with fixedtemperature (T = 1 for simplicity)
Q(f)(v) = −ν(f − exp(A − |v|2/2))
with A such that mass is preservedNote: n ∼ eA up to a normalization constant
➠ Quantum case: replace the classical Maxwellianby the quantum one
Q(w)(v) = −ν(w − Exp(A − |p|2/2))
Page 275
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
35Quantum Maxwellian
➠ Wigner form. Mn = Exp(A − |p|2/2)∫
Exp(A − |p|2/2) dp = n
➠ Density operator form.
ρn = W−1(Mn) = exp(W−1(A − |p|2/2))
with, ∀ test fct. φ:
Tr{ρn φ} =
∫nφ dx
Page 276
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
36Entropy minimization principle
➠ ρn = exp(W−1(A − |p|2/2)) satisfies the Freeenergy minimization principle:
min G[ρ] = Tr{ρ(ln ρ − 1) + Hρ} subject to:
Tr{ρn φ} =
∫nφ dx , ∀ test fct φ
H = |p|2/2 + V = Hamiltonian
➠ In Wigner form
G[ρ] = Tr{ρ(ln ρ−1)+Hρ} =
∫[w(Ln w−1)+Hw] dx dp
with quantum log: Ln w = W [ln(W−1(w))]
Page 277
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
37Quantum BGK operator
➠ For Wigner distribution w given,
➟ denote Mw := Mn s.t.n is the density of f :
∫Mn dp =
∫f dp
➠ Then Quantum BGK operator is written
Q(w) = −ν(w −Mw)
➠ Density operator: call it Mρ as well
Q(ρ) = −ν(ρ −Mρ)
Page 278
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
38Situation modeled by Q(w)
➠ Similar as in the classical case but when quantumeffects need to be taken into account
➠ Energy exchanges between the particles and thesurrounding relax the temperature to thebackground temperature
Page 279
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
39Conservative variable vs entropic variable
➠ Reminder: two variables
➟ Conservative variable n
➟ Entropic variable A
➟ n ←→ A is a functional change of variables
➟ Inversion through entropy
Page 280
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
40Properties of Q
➠ Mass conservation∫
Q(w) dp = 0
➠ Null set of Q (equilibria) :
Q(w) = 0 ⇐⇒ ∃A such that w = Exp(A − |p|2/2)
➠ Free energy decay:∫
Q(w)(Lnw+H) dx dp = Tr{Q(ρ)(ln ρ+H)} ≤ 0
Proof: similar to the energy-transport case
Page 281
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
41Diffusion scaling
➠ Wigner eq. under diffusion scaling:
η2∂wη
∂t+ η(v · ∇xw
η − Θ(wη)) = Q(wη)
Page 282
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
42Limit η → 0: Quantum Drift-Diffusion
➠ as η → 0, wη −→ Exp(A − |p|2/2)where A satisfy the Energy-Transport model:
➠ Mass conservation eq.
∂n
∂t+ ∇x · jn = 0
➠ With ∫Exp(A − |p|2/2) dp = n
Page 283
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
43Drift-Diffusion model (cont)
➠ Flux
jn = −ν−1[∇Π + n∇V ]
➠ with
Π(A) =
∫Exp(A − |p|2/2) p ⊗ p dp
➠ Proof of the limit η → 0: exactly the same as inthe Energy-Transport case → omitted
Page 284
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
44Free energy decay
➠ Fluid free energy G(n) = G(Mn):
G(n) =
∫Mn,W (LnMn,W − 1 + H) dx dp
=
∫Exp(A − |p|2/2)(A − |p|2/2 − 1 + H) dx dp
=
∫n(A + V − 1) dx
➠ Then if either V independent of t or V given by
Poisson’s eq.:d
dtG(n) ≤ 0
Page 285
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
45Computation of Π
➠ To be determined: pressure tensor Π
Π(A) =
∫Exp(−H(A)) p ⊗ p dp
with modified Hamiltonian H(A) = |p|2/2 − A
➠ Π(A) = Π(−A, 0) where Π(A, B) is the pressuretensor of Isentropic Quantum Euler model
Π(A, B) =
∫Exp(−H(A, B)) p ⊗ p dp
with H(A, B) = |p|2/2 − B · p + A + |B|2/2
Page 286
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
46Computation of Π (cont)
➠ In that case, we had
∇Π(A, B) = ∇(nu ⊗ B) + (∇B)nu − n∇(A + |B|2/2)
➠ Here ∇Π(A) is deduced through B = 0 andA → −A
∇Π(A) = n∇A
Page 287
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
47Alternate expression of QDD
➠ QDD model has equivalent formulation:
∂n
∂t+ ∇x · jn = 0
jn = −ν−1(n∇(A + V ))∫Exp(A − |p|2/2) dp = n
Page 288
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
48Moment reconstruction
➠ Suppose Hamiltonian H(A) = |p|2/2 − A hasdiscrete spectrum
➟ Eigenvalues λp(A), p = 1, . . . ,∞➟ Eigenfunctions ψp(A)
➠ Then
n(A) (x) =∞∑
p=1
exp(−λp(A)) |ψp(A) (x)|2
Page 289
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
49Final expression of QDD model
➠ Continuity and current eqs.
∂n
∂t+ ∇x · jn = 0
jn = −ν−1(n∇(A + V ))
➠ n ↔ A relationship
n(A) (x) =∞∑
p=1
exp(−λp(A)) |ψp(A) (x)|2
➠ With λp(A), ψp(A) associated with modified
Hamiltonian H(A) = |p|2/2 − A
Page 290
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
50Equilibrium states
➠ Defined by jn = 0
➟ Implies A = −V
➠ Then
n (x) =∞∑
p=1
exp(−λp) |ψp (x)|2
With λp, ψp associated with the ’true’
Hamiltonian H(A) = |p|2/2 + V
➠ If n ↔ V through Poisson’s eq.Shrodinger-Poisson problem
Page 291
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
51Close to equilibrium
➠ Suppose A ≈ −V
➟ Then, replace A by −V in the moment pbm:
➠ Leads to
∂n
∂t+ ∇x · jn = 0
jn = ν−1(n∇(A + V ))
n(A) (x) =∞∑
p=1
exp(A + V − λp(−V )) |ψp(−V ) (x)|2
➠ Schrodinger-Poisson-Drift-Diffusion [Sacco et al,
Springer Lecture Notes (2004)]
Page 292
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
52~ expansion
➠ up to O(~2) terms, QDD model reads:
∂tn + ∇ · jn = 0 ,
jn = −ν−1[∇n − n∇(V + VB[n]))
VB[n] = −~2
6
1√n
∆(√
n) Bohm potential
➠ Density-Gradient model of [Ancona & Iafrate, Phys.
Rev. B (89)]
Page 293
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
53Free energy for the Density-Gradient model
➠ Free energy for the QDD model expanded up toO(~2) terms:
G2(n) =
∫
Rd
n(ln n − 1 + V + VB[n]) dx
➠ If V independent of t:
d
dtG2(n) = −
∫
Rd
1
νn|∇n+n∇(V +VB[n])|2 dx ≤ 0
➠ Similar expression if V is solved throughPoisson’s eq.
Page 294
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
54About Density-Gradient model
➠ Widely used in the literature
➟ Mathematical theory: [Ben Abdallah & Unterreiter,
ZAMP 98],
➟ Numerical methods: [Pinau, Unterreiter, SINUM
99], [Jungel, Pinau, SINUM 01]
➠ This approach
➟ Provides a derivation of DG model from firstprinciples
➟ Proves (for the first time ?) that DG modeldecreases free energy
Page 295
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
55About Quantum Drift-Diffusion model
➠ No rigorous proof
➟ existence ?
➟ convergence ?
➠ Numerical simulations
➟ The implicit semi-discretized model (coupledw. Poisson) is well-posed and has a variationalformulation [Gallego & Mehats, SIAM J. Num. Anal.
05]
Page 296
(Sum
mar
y)(C
oncl
usio
n)P
ierr
eD
egon
d-
Qua
ntum
fluid
mod
els
-C
etra
ro,s
ept2
006
56Res
onan
ttu
nnel
ing
dio
de
GaAs ( 10 21 m -3 )
GaAs ( 10 24 m -3 )
GaAs ( 10 24 m -3 )
GaAs ( 10 21 m -3 )
Al 0.3 Ga 0.7 As ( 10 21 m -3 )
Al 0.3 Ga 0.7 As ( 10 21 m -3 )
GaAs ( 10 21 m -3 )
Energy ( eV )
0 25
30
35
45
50
75
40
0.4
Pos
ition
( nm
)
Page 297
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
57Isolated diode: density
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
21
Position (nm)
Den
sity
(m
−3 )
t=0 fs
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
21
Position (nm)
Den
sity
(m
−3 )
t=10 fs
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
21
Position (nm)
Den
sity
(m
−3 )
t=1000 fs
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
21
Position (nm)
De
nsi
ty (
m−
3)
t=10000 fs
Density vs position at different times
Page 298
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
58Isolated diode: Fermi level
0 10 20 30 40 50 60 70
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Position (nm)
Ele
ctro
che
mic
al P
ote
ntia
l (V
)
t=0 fs
0 10 20 30 40 50 60 70
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Position (nm)
Ele
ctro
che
mic
al P
ote
ntia
l (V
) t=10 fs
0 10 20 30 40 50 60 70
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Position (nm)
Ele
ctro
che
mic
al P
ote
ntia
l (V
)
t=1000 fs
0 10 20 30 40 50 60 70
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Position (nm)
Ele
ctro
che
mic
al P
ote
ntia
l (V
)
t=10000 fs
Fermi level vs position at different times
Page 299
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
59Isolated diode: free energy vs time
0 1000 2000 3000 4000 5000 6000 70000.396
0.397
0.398
0.399
0.4
0.401
0.402
0.403
0.404
Time (fs)
Qua
ntum
free
ene
rgy
(eV
)
Page 300
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
60Isolated diode: comparison between models
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10−3
10−2
10−1
100
Time (fs)
||neQDD−nSP|| / ||nSP||||neQDD−nSPDD|| / ||nSPDD||
neQDD − nSP (blue) , neQDD − nSPDD (red)Relative error in L2 norm vs time
Page 301
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
61Applied bias: I − V curve
0.067me / 0.092me 0.067me / 1.5×0.092me
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
1
2
3
4
5
6
7
8
9x 10
9
Cur
rent
(Am
−2)
Voltage (V)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.5
1
1.5
2
2.5
3x 10
9
Cur
rent
(Am
−2)
Voltage (V)
1.5×0.067me / 0.092me 1.5×0.067me / 1.5×0.092me
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
9
Cur
rent
(Am
−2)
Voltage (V)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0
5
10
15x 10
8
Cur
rent
(Am
−2)
Voltage (V)
Influence of the effective mass
Page 302
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
62Density: peak to valley
0 20 40 60 800
500
1000
1500
0
5
10
15
x 1023
Time (fs)
Position (nm)
Den
sity
(m
−3 )
Density from peak (Va = 0.25V)to valley (Va = 0.31V).
Page 303
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
63Relative magnitude of the eigen-states
0 10 20 30 40 50 60 700
5
10
15x 10
23
position (nm)
Den
sity
(m
−3 )
e−λ1/(k
B T)|ψ
1|2
e−λ2/(k
B T)|ψ
2|2
e−λ3/(k
B T)|ψ
3|2
e−λ4/(k
B T)|ψ
4|2
e−λ5/(k
B T)|ψ
5|2
e−λ6/(k
B T)|ψ
6|2
n=Σ
p e−λ
p/(k
B T)|ψ
p|2
1 2
3
4
5 6 0 10 20 30 40 50 60 70
0
5
10
15x 10
23
position (nm)
Den
sity
(m
−3 )
e−λ1/(k
B T)|ψ
1|2
e−λ2/(k
B T)|ψ
2|2
e−λ3/(k
B T)|ψ
3|2
e−λ4/(k
B T)|ψ
4|2
e−λ5/(k
B T)|ψ
5|2
e−λ6/(k
B T)|ψ
6|2
n=Σ
p e−λ
p/(k
B T)|ψ
p|2
1 2
3
4
5 6
Current peak Valley
λ1 λ2 λ3 λ4 λ5 λ6 λ7
Peak 0.87 1.05 1.56 2.03 2.28 3.03 4.47
Valley 0.87 1.11 1.57 1.70 2.54 3.05 5.03
Page 304
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
64Comparison eQDD / DG
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
1
2
3
4
5
6
7
8
9x 10
9
Cu
rre
nt
(Am
−2)
Voltage (V)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.5
1
1.5
2
2.5x 10
8
Cu
rre
nt (A
m−
2)
Voltage (V)
Left: eQDD ; Right: DG
Page 305
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
65Influence of the potential
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
7
Cu
rre
nt
(Am
−2 )
Voltage (V)
0 20 40 600
0.1
0.2
0.3
0.4 eQDDDG
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
2
4
6
8
10
12x 10
9
Cur
rent
(A
m−
2 )
Voltage (V)
0 20 40 600
0.1
0.2
0.3
0.4 eQDDDG
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
1
2
3
4
5
6
7
x 107
Cur
rent
(A
m−
2 )
Voltage (V)
eQDDDG
0 20 40 600
0.1
0.2
0.3
0.4
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
2
4
6
8
10
12
14
16x 10
9
Cur
rent
(A
m−
2 )
Voltage (V)
eQDDDG
0 20 40 600
0.1
0.2
0.3
0.4
Page 306
(Summary) (Conclusion)Pierre Degond - Quantum fluid models - Cetraro, sept 2006
66Comparison between the models
0 200 400 600 800 1000 12000
1
2
3
4
5
6x 10
10
Temperature (K)
Cur
rent
(A
m−
2 )
DGCDDeQDD
➠ As T ր models are closer
➠ (DG) and (eQDD) are closer while (CDD)remains significantly away