chapter 1 quantum kinetic equations: an...

(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)


2Summary

1. Quantum statistical mechanics of nonequilibrium

2. Mean-Field limit

3. Quantum methods: a brief and incomplete summary

4. Hydrodynamic limits

5. Conclusion


3

1. Quantum statistical mechanics ofnonequilibrium


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


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


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


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


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)


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


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


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


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


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.


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).


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


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)


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


18

2. Mean-Field limit


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)


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)


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


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)


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


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


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) =


n(y) = |ψ(y)|2


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)


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


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


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


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) =


n(y) = ρ(y, y)


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.


32

3. Quantum methods: a brief and incompletesummary


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)


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

➟ . . .


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]


36DFT: discussion

➠ Problem: nonlinear potential not known:approximations

➟ Thomas-Fermi

➟ Kohn-Sham

➟ . . .

➠ Validity of these approximations ?


37Open systems

➠ Examples

➟ Electrons in a semiconductor

➟ Molecule in a solvant

➟ Protein in a cell

➟ . . .

➠ How to account for the environment ?


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)


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


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


41

4. Hydrodynamic limits


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’)


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]


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


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.


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


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]


48

5. Summary and conclusion


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


50Next step

➠ Derivation of quantum hydrodynamic modelsbased on the entrtopy minimization approach


1

Chapter 2

Derivation of moment models via the entropyminimization approach (classical case)

P. Degond





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


3

1. Classical description of particle systems


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 ?


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


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


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


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′


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)


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


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


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


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


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


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


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)


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


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


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 .


20Entropy minimization principle (cont)

➠ Entropy minimization:

➟ Most effective characterization of Maxwelliansfor further extensions

➠ Examples

➟ More moment constraints → Levermoremodels

➟ Quantum entropy → quantum hydro models


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


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


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


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)


25Some references for BGK

➠ Existence of weak solutions [Perthame, Pulvirenti]

➠ Numerical solutions [Dubroca, Mieussens]

➠ Generalized BGK models [Bouchut, Berthelin]


26Ref for Boltzmann: Homogeneous equation

➠ Existence and uniqueness of classical solutions[Carleman], [Arkeryd], ...

➠ Convergence to a Maxwellian as t → ∞[Desvillettes], [Wennberg], ...


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


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


29

2. The moment method and the Eulerequations


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


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


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


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


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


35Hydrodynamic scaling

Microscopic scale Macroscopic scale

η ≪ 1

➠ Rescale: x′ = εx, t′ = εt

ε(∂tfε + v · ∇xf

ε) = Q(f ε)


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


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


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


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


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


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]


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)


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


44

3. Higher order moment systems:Levermore’s approach


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}


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


47Closure problem

➠ Find a prescription for∫

fµi(v)v dv and

∫

Q(f)µi(v) dv

in terms of the moments mi


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 .


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


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


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))


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


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


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


55Hyperbolicity

➠ Hyperbolicity −→ well posedness (Godounov,Friedrichs)

➠ 6= Grad systems: not everywhere locallywell-posed


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


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


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


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


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)


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


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]


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


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].


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.


66

4. Summary, conclusion and perspectives


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


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


1

Chapter 3

Quantum hydrodynamic models derived from theentropy principle

P. Degond





2Summary

1. Quantum setting: a summary

2. QHD via entropy minimization

3. Quantum Isentropic Euler



3

1. Quantum setting: a summary


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


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


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.


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


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)


9

2. QHD via entropy minimization

Joint work with

C. RinghoferArizona State University, Tempe, USA

J. Stat. Phys. 112 (2003), pp. 587–628.


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 ?


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)


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


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


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)


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



➠ 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


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


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


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 }


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)]


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)


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)


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


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


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


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


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)


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


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


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)}


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


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


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


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


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


36Computation of W−1(α · µ) (cont)

➠ Lemma∫

p eip(x−y)


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 ∞


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)


39

3. Quantum Isentropic Euler

Joint work with

S. Gallego1 and F. Mehats2

1 MIP, Toulouse ; 2 IRMAR, Rennes

Manuscript, submitted


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


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


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’


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


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))


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


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


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


48Quantum isentropic Euler (cont)

➠ where (A, B) related with (n, nu) by the momentconditions

n(A, B) (x) =∞∑

p=1


nu(A, B) (x) =∞∑

p=1



49Computation of π

➠ To be determined: pressure tensor Π:

Π =


➠ 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)}


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}


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


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


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


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)


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


56Final expression of Π

➠ Finally

∇Π = ∇(nu ⊗ B) + (∇B)nu − n∇(A + |B|2/2)


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) =∑


nu(A, B) (x) =∑


➠ And λp(A, B), ψp(A, B): spectrum of

H(A, B) = |p|2/2 − B · p + A + |B|2/2


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)


59Free energy (cont)

➠ If V solves Poisson eq.

−∆V = n

Then, againdGdt

≤ 0

(with = for smooth solutions)


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


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


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)


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


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)


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


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


67Equivalent formulations of momentum eq. (cont)

➠ Form 4: Use continuity equation

∂tu + (∇× u) × B + ∇(u · B − |B|2/2 + V − A) = 0


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


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


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


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


72Quantum Euler for irrotational flows (cont)

➠ Advantage: only one quantity A to determinefrom the spectral problem

➠ Important special case: One-dimensional flows


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]


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


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


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


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


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


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


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


81



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.


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


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


1

Chapter 4

Diffusion models: classical case

P. Degond





2Summary

1. Drift-Diffusion model

2. Energy-Transport model


3

1. Drift-Diffusion model


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


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

➟ . . .


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


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


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


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]


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


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


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)


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


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


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 )


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)


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)


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


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


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


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


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)


23Properties of Q

➠ Mass and energy conservation

∫

Q(f)

(

1

|v|2

)

dv = 0


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


24Diffusion scaling

➠ Boltzmann eq. under diffusion scaling:

η2∂f η

∂t+ η (v · ∇xf

η −∇xV · ∇vfη) = Q(f η)


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


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



➠ 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



➠ 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)



➠ 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


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


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


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 )


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


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


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


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


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


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


1

Chapter 5

Quantum diffusion models

P. Degond





2Summary

1. Quantum energy-Transport model

2. Quantum drift-Diffusion model



3

1. Quantum energy-Transport model

P. D., F Mehats, C. Ringhofer,

J. Stat. Phys. 118 (2005), pp. 625-667


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)


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))


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



➠ 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



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)


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


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


11Properties of Q

➠ Mass and energy conservation

∫Q(w)

(1

|p|2

)dp = 0


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


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}


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


14Diffusion scaling

➠ Wigner eq. under diffusion scaling:

η2∂wη

∂t+ η(v · ∇xw

η − Θ(wη)) = Q(wη)


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

)


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


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)


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



➠ 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



➠ Suppose wη → w smoothly

Wigner-BGK eq. =⇒ Q(wη) = O(η)

=⇒ Q(w) = 0

=⇒ w = Exp(A + C|p|2/2)



➠ 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


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


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


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]



➠ 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



➠ Then

jn = −ν−1[∇Π + n∇V ] QED


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


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)


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)


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 .


31About Quantum Energy-Transport models


➟ 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]


32Quantum Drift-Diffusion model

➠ Have a nice structure (see next lecture)

➠ Hope that structure can be extended to QuantumEnergy-Transport


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


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))


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



➠ 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



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)


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


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


40Properties of Q

➠ Mass conservation∫

Q(w) dp = 0


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


41Diffusion scaling

➠ Wigner eq. under diffusion scaling:

η2∂wη

∂t+ η(v · ∇xw

η − Θ(wη)) = Q(wη)


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


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


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


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) =


with H(A, B) = |p|2/2 − B · p + A + |B|2/2


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


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


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


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


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


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)]


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)]


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.


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


55About Quantum Drift-Diffusion model


➟ 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]

(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

)


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


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


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

)


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


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


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).


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


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


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


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

chapter 1 quantum kinetic equations: an...

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