derivation of quantum hydrodynamic equations with fermi-dirac...

Derivation of quantumhydrodynamic equations with

Fermi-Dirac and Bose-Einsteinstatistics

Luigi Barletti (Università di Firenze)Carlo Cintolesi (Università di Trieste)

6th MMKT

Porto Ercole, june 9th 2012

A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Madelung equations

The theory of quantum fluid equations dates back to 1926,when E. Madelung discovered the hydrodynamic form ofSchrödinger equation:

∂tn + div(nu) = 0,

∂tu +12∇|u|2 +∇

(V − ~2

2∆√

n√n

)= 0,

where ψ =√

n eiS/~ and u = ∇S (and m = 1).

E. Madelung, Quantentheorie in hydrodynamischerForm. Zeitschr. f. Phys. 40, 322–326 (1926)


Bohm potential

Madelung equations look like an irrotational compressible Eulersystem with an additional term containing the Bohm potential

−~2

2∆√

n√n,

named after David Bohm, who based on it his famousinterpretation of quantum mechanics.

D. Bohm, A suggested interpretation of the quantum the-ory in terms of “hidden variables”. Physical Review 85,166–193 (1952)


Wigner functionsAn “kinetic” derivation of Madelung equations can be obtainedby using the Wigner function

w(x ,p, t) =1

(2π~)d/2

∫Rdψ

(x +

ξ

2, t)ψ

(x − ξ

2, t)

e−ip·ξ/~dξ,

and writing the equations for the moments

n(x , t) =

∫w(x ,p, t) dp, nu(x , t) =

∫p w(x ,p, t) dp.

E. Wigner, On the quantum correction for thermody-namic equilibrium. Physical Review 40, 749–759 (1932)


Mixed states

Madelung equations hold for a pure state, described by a wavefunction ψ, but for a mixed (statistical) state the system is notformally closed.

However, the derivation of quantum fluid models for collisionalsystems, necessarily requires a statistical description and,therefore, the problem arises of generalizing Madelungequations to such a situation.

The question is not merely academic, because quantum fluidmodels can be of great interest for nanoelectronics.


Quantum hydrodynamics

Indeed, a renewed interest for the subject dates back to the halfof nineties with the work of C. Gardner, who proposed aquantum hydrodynamic model based on a local quantumMaxwellian obtained from Wigner’s O(~2) corrections tothermal equilibrium.

C.L. Gardner, The quantum hydrodynamic model forsemiconductor devices. SIAM J. Appl. Math. 54(2),409–427 (1994)


Quantum entropy principle

But it is only with the work of Degond and Ringhofer that theproblem was set on a solid theoretical basis with theelaboration of the quantum version of the maximum entropyprinciple (QMEP).

P. Degond, C. Ringhofer, Quantum momenthydrodynamics and the entropy principle. J.Stat. Phys. 112(3-4), 587–628 (2003)

The QMEP has been exploited to generate several quantumfluid models of various kind.


Quantum fluid models from the QMEP (I)• P. Degond, F. Méhats, C. Ringhofer, Quantum energy-transport and

drift-diffusion models, J. Stat. Phys., 2005.

• A. Jüngel, D. Matthes, A derivation of the isothermal quantumhydrodynamic equations using entropy minimization, Z. Angew. Math.Mech., 2005.

• N. Ben Abdallah, F. Méhats, C. Negulescu. Adiabatic quantum-fluidtransport models, Commun. Math. Sci., 2006.

• A. Jüngel, D. Matthes, J. P. Milisic, Derivation of new quantumhydrodynamic equations using entropy minimization, SIAM J. Appl.Math., 2006.

• P. Degond, S. Gallego, F. Méhats. Isothermal quantum hydrodynamics:derivation, asymptotic analysis, and simulation, Multiscale Model.Simul., 2007.

• P. Degond, S. Gallego, F. Méhats, An entropic quantum drift-diffusionmodel for electron transport in resonant tunneling diodes, J. Comput.Phys., 2007.


Quantum fluid models from the QMEP (II)

• S. Brull, F. Méhats, Derivation of viscous correction terms for theisothermal quantum Euler model, Z. Angew. Math. Mech., 2010.

• L. B., F. Méhats, Quantum drift-diffusion modeling of spin transport innanostructures. J. Math. Phys., 2010.

• L. B., G. Frosali, Diffusive limit of the two-band k·p model forsemiconductors. J. Stat. Phys., 2010.

• N. Zamponi, L. B., Quantum electronic trasport in graphene: a kineticand fluid-dynamical approach. M2AS, 2011.

• N. Zamponi, Some fluid-dynamic models for quantum electron transportin graphene via entropy minimization. KRM, 2012.

• A. Jüngel. Transport Equations for Semiconductors. Springer, 2009.

• A. Jüngel. Dissipative quantum fluid models. Riv. Mat. Univ. Parma,2012.


Different statistics

Although the QMEP was originally stated for a general (convex)entropy functional, nevertheless, it has been applied only toBoltzmann entropy.

Partial exceptions:• P. Degond, S. Gallego, F. Méhats, An entropic quantum

drift-diffusion model for electron transport in resonant tunnelingdiodes, J. Comput. Phys., 2007.

• M. Trovato, L. Reggiani, Quantum maximum entropy principle fora system of identical particles. Phys. Rev. E , 2010.

• A. Jüngel, S. Krause, P. Pietra, Diffusive semiconductor momentequations using Fermi-Dirac statistics. ZAMP, 2011.

The work we are going to expose is exactly intended to fill thisgap.


The kinetic model

The starting point is the Wigner equation with BGK collisionalterm (

∂t + pm · ∇x + Θ~[V ]

)w =

g[w ]− wτ

,

where:

• w(x ,p, t), is the Wigner function of the system;

• Θ~[V ] = i~[V(x + i~

2∇p)− V

(x − i~

2∇p)]

;

• τ is a typical collisional time;

• g[w ] is the local equilibrium state given by the QMEP.


Scaling the Wigner-BGK equation

After a suitable scaling, we can write the Wigner-BGK equationas follows:

(∂t + p · ∇x + Θε[V ]) w =1α

(g[w ]− w)

• α = τt0

(hydrodynamic parameter);

• ε = ~x0p0

(semiclassical parameter).


QMEP

We assume that g[w ] is given by the QMEP:

g[w ] is the most probable state compatible with the informationwe have about it:

• g[w ] has a constant temperature T• g[w ] has the same density and current as w :

〈g[w ]〉 = n = 〈w〉, 〈pg[w ]〉 = J = 〈pw〉.

“Most probable” means that g[w ] minimizes an entropyfunctional H.Then, g[w ] is chosen as a minimizer of H among all Wignerfunctions that share with w the same moments n and J.


Entropy functional

In terms of density operators, the suitable entropy functional forthe isothermal case is the free-energy

H(%) = Tr{

T[% log %+

1λ

(1− λ%) log(1− λ%)

]+ H%

},

which also incorporates the information on particle statistics:

λ =

1, Fermi-Dirac

0, Maxwell-Boltzmann

−1, Bose-Einstein


Constrained minimization problem

The Wigner function g[w ] satisfies the QMEP if:

% = Opε(g[w ]) is a minimizer of H under the constraints

〈g[w ]〉 = n = 〈w〉, 〈pg[w ]〉 = J = 〈pw〉.

Here Opε denotes the Weyl quantization, mapping 1-1 Wignerfunctions and density operators.


The minimizer

TheoremA necessary condition for g = g[w ] to be solution of theconstrained minimization problem is that Lagrange multipliersA and B = (B1, . . . ,Bd ) exist such that

Opε(g) =(

eH(A,B)

T + λ)−1

,

whereH(A,B) = Opε

(|p−B|2

2 − A),

and A, B have to be determined as functions of n, J from themoment constraints.


Hydrodynamic limit

TheoremIn the hydrodynamic limit α→ 0, the solution wα of theWigner-BGK equation tends to the local equilibrium state g[w0],whose moments n and J satisfy the equations

∂

∂tn +

∂

∂xjJj = 0,

∂

∂tJi +

∂

∂xj

⟨pipjg[w0]

⟩+ n

∂

∂xiV = 0.


Formal closure

The unknown moment⟨pipjg[w ]

⟩can be expressed in terms of

the Lagrange multipliers as follows:

∂

∂xj

⟨pipjg[w ]

⟩=

∂

∂xj(nuiBj) +

∂

∂xiBj(nuj − Bj

)+ n

∂

∂xiA.

This provides a formal (and rather implicit) closure of thequantum hydrodynamic equations because A, B are depend onn, J through the moment constraints.


Difficulties with g[w ]

From now on, we shall simply denote by g the local equilibriumstate g[w ].

The phase-space function g is a very complicated object,involving back and forth Weyl quantization:

g = Opε−1

{exp

[Opε

(|p − B|2

2T− A

T

)]+ λ

}−1

The only hope we have to get something explicit is expanding gsemiclassically, i.e. in powers of ε.


Semiclassical expansion of g

Skipping all technical details, we find that g has the followingexpansion

g = g(0) + ε2g(2) +O(ε4)

where g(0) is the “classical” distribution

g(0) =1

e(p−B)2

2T − AT + λ

and g(2) is a complicated expression involving A, B and theirderivatives.


Semiclassical expansion of A and B

What we really need is the expansion of A, B as functions of n,J, as it results from the constraints 〈g〉 = n and 〈pg〉 = J.Using the expansion of g, it turns out that

A = A(0) + ε2A(2) +O(ε4), B = B(0) + ε2B(2) +O(ε4).

At leading order we obtain

A(0) = T φ−1d2

(n

(2πT )d/2

), B(0) = u = J/n,

where

φs(z) = −1λ

Lis (−λez) =1

Γ(s)

∫ +∞

0

t (s−1)

et−z + λdt .


Condition for invertibility

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

FD, d=3

MB, d=3

BE, d=3

BE, d=2

BE, d=1

Plots of φ d2

for somevalues of λ and d .

For λ < 0 and d ≥ 3,φ d

2ranges from 0 to

ζ( d

2

)/|λ|.

In the BE case, we have to assume n < (2πT )d2 ζ(d

2

)/|λ|.

Exceeding particles fall in the condensate phase, not describedby our model.


A and B: second-order expressions

A(2) =1

24T

[∂iuj (∂iuj − ∂jui )− 2∂i∂iA(0)

] φ0d2−2

φ0d2−1

− 124

(∂iA(0)

T

)2 φ0d2−3

φ0d2−1

B(2)i =

nd

12Tn∂j

[(∂iuj − ∂jui )φ

0d2−1

]where

∂i :=∂

∂xi

and

φ0s := φs

(A(0)

T

)= φs

(φ−1

d2

( nnd

))


Semiclassical hydrodynamic equations (I)

We can now substitute the expansions

A ≈ A(0) + ε2A(2) and B ≈ B(0) + ε2B(2)

in the quantum hydrodynamic equations

∂

∂tn +

∂

∂xj(nuj) = 0,

∂

∂t(nui) +

∂

∂xj(nuiBj) +

∂

∂xiBj(nuj − Bj

)+ n

∂

∂xi(A + V ) = 0.


Semiclassical hydrodynamic equations (II)We obtain therefore our main result:

∂

∂tn +

∂

∂xj(nuj) = 0,

∂

∂t(nui) +

∂

∂xj(nuiuj) + Tn

∂

∂xiφ−1

d2

( nnd

)+ n

∂

∂xi

(V + ε2Q(n)

)

=ε2n24T

∂

∂xi

RjkRkj φ0d2−2

2φ0d2−1

+ε2nd

12TRij

∂

∂xk

(Rjkφ

0d2−1

),

wherend = (2πT )

d2 , Rij :=

∂

∂xjui −

∂

∂xiuj


Modified Bohm potential

The term ε2Q(n) can be identified as a modified Bohmpotential:

Q(n) = − 124

2∆φ−1d2

( nnd

)φ0d2−2

φ0d2−1

+∣∣∣∇φ−1

d2

( nnd

)∣∣∣2 φ0d2−3

φ0d2−1

,

since

limλ→0

Q(n) = −16

∆√

n√n.


Limit properties of φs

From known asymptotic properties of the polylogarithms weobtain:

limλ→0

φs(z) = ez , for z ∈ R and s ∈ R

φs(z) ∼ ez , for z → −∞, λ ∈ R and s ∈ R

φs(z) ∼ zs

Γ(s + 1), for z → +∞, λ = 1 and s 6= −1,−2, . . .

φs(z) ∼ Γ(1− s)(−z)s−1, for z → 0−, λ = −1 and s < 1


The Maxwell-Boltzmann limit

In the M-B limit λ→ 0 the semiclassical equations become

∂

∂tn +

∂

∂xj(nuj) = 0,

∂

∂t(nui) +

∂

∂xj(nuiuj) + T

∂

∂xin + n

∂

∂xi

(V − ε2

6∆√

n√n

)

=ε2

12T∂

∂xj

(nRikRkj

),

(corresponding to the equations found by Jüngel-Matthes (’05) andDegond-Gallego-Méhats (’07)).


Vanishing-temperature limit

The behavior of the semiclassical hydrodynamic equations asT → 0 depends dramatically on the sign of λ.

Let us consider separately the three representative cases λ = 1(FD), λ = 0 (MB) and λ = −1 (BE).


T → 0 limit: Fermi-Dirac case

The FD case is the richest: the limit T → 0 is non-singular andyields “completely degenerate fluid” equations:

∂

∂tn +

∂

∂xj(nuj ) = 0,

∂

∂t(nui ) +

∂

∂xj(nuiuj ) +

( d2

) 2−dd Γ

( d2

) 2d

2πn∂

∂xin

2d

+ n∂

∂xi

(V − d − 2

dε2

6∆√

n√n

)=

ε2dπ

12 Γ( d

2 + 1) 2

d

[(d − 2)

dn∂

∂xi

Rjk Rkj

4n2d

+ Rij∂

∂xk

(Rjk n

d−2d

)]


T → 0 limit: Maxwell-Boltzmann case

This is the most singular case, because the limits λ→ 0 andT → 0 are somehow incompatible.

The formal limit of the semiclassical hydrodynamic equationswith BE statistics is only compatible with an irrotational fluid(R = 0) and depends on how (0,0) is approached in theparameter plane (λ,T )

Degond-Gallego-Méhats (’07) show that the fully-quantum fluidequations admit a limit, which is given by Madelung equations.


T → 0 limit: Bose-Einstein case, d ≥ 3

For λ = −1 and d ≥ 3, in the limit T → 0 the fluid is completelycondensate and, then, this case cannot be considered withinour description.


T → 0 limit: Bose-Einstein case, d = 2

In this case the limit T → 0 is only compatible with anirrotational fluid. Moreover, we have to rescale the density as

N =n

2πT.

The resulting limit equations are

∂

∂tN +

∂

∂xj(Nuj) = 0

∂

∂tui + uj

∂

∂xiuj +

∂

∂xi

(V − ε2

12∆N

)= 0


T → 0 limit: Bose-Einstein case, d = 1

In this, last, case we obtain

∂

∂tn +

∂

∂x(nu) = 0

∂

∂tu +

12∂

∂xu2 +

∂

∂x

(V − ε2

21√n∂2√n∂x2

)= 0


Conclusions

• We derived semiclassical isothermal hydrodynamicequations for Fermions or Bosons.

• The method exploits Degond and Ringhofer’s QuantumMaximum Entropy Principle and the semiclassicalexpansion of the maximizer Wigner function.

• We obtained an Euler-like system with quantumcorrections, of order ~2, involving a modified Bohmpotential and the velocity curl tensor.

• The Maxwell-Boltzmann limit and the T → 0 limit havebeen investigated.

• L.B., C. Cintolesi, Derivation of isothermal quantum fluid equations withFermi-Dirac and Bose-Einstein statistics. J. Stat. Phys. (to appear).

Thank you

derivation of quantum hydrodynamic equations with fermi-dirac...

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