Derivation of quantum

hydrodynamic equations with

Fermi-Dirac and Bose-Einstein
statistics
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 of
Schrödinger equation:

∂t n + div(nu) = 0,
√ 
~2 ∆ n

1 2
∂t u + ∇|u| + ∇ V − √ = 0,
2 2 n
√
where ψ = n eiS/~ and u = ∇S (and m = 1).

E. Madelung, Quantentheorie in hydrodynamischer

Form. Zeitschr. f. Phys. 40, 322–326 (1926)
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Bohm potential

Madelung equations look like an irrotational compressible Euler

system with an additional term containing the Bohm potential
√
~2 ∆ n
− √ ,
2 n

named after David Bohm, who based on it his famous

interpretation of quantum mechanics.

D. Bohm, A suggested interpretation of the quantum the-

ory in terms of “hidden variables”. Physical Review 85,
166–193 (1952)
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Wigner functions
An “kinetic” derivation of Madelung equations can be obtained
by using the Wigner function
Z    
1 ξ ξ
w(x, p, t) = ψ x + , t ψ x − , t e−ip·ξ/~ dξ,
(2π~)d/2 Rd 2 2

and writing the equations for the moments

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

Mixed states

Madelung equations hold for a pure state, described by a wave

function ψ, but for a mixed (statistical) state the system is not
formally closed.

However, the derivation of quantum fluid models for collisional

systems, necessarily requires a statistical description and,
therefore, the problem arises of generalizing Madelung
equations to such a situation.

The question is not merely academic, because quantum fluid

models can be of great interest for nanoelectronics.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Quantum hydrodynamics

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

C.L. Gardner, The quantum hydrodynamic model for

semiconductor devices. SIAM J. Appl. Math. 54(2),
409–427 (1994)
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Quantum entropy principle

But it is only with the work of Degond and Ringhofer that the
problem was set on a solid theoretical basis with the
elaboration of the quantum version of the maximum entropy
principle (QMEP).

P. Degond, C. Ringhofer, Quantum moment

hydrodynamics and the entropy principle. J.
Stat. Phys. 112(3-4), 587–628 (2003)

The QMEP has been exploited to generate several quantum

fluid models of various kind.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

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 quantum
hydrodynamic equations using entropy minimization, Z. Angew. Math.
Mech., 2005.
• N. Ben Abdallah, F. Méhats, C. Negulescu. Adiabatic quantum-fluid
transport models, Commun. Math. Sci., 2006.
• A. Jüngel, D. Matthes, J. P. Milisic, Derivation of new quantum
hydrodynamic 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-diffusion
model for electron transport in resonant tunneling diodes, J. Comput.
Phys., 2007.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Quantum fluid models from the QMEP (II)

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

• L. B., F. Méhats, Quantum drift-diffusion modeling of spin transport in

nanostructures. J. Math. Phys., 2010.

• L. B., G. Frosali, Diffusive limit of the two-band k·p model for

semiconductors. J. Stat. Phys., 2010.

• N. Zamponi, L. B., Quantum electronic trasport in graphene: a kinetic

and fluid-dynamical approach. M2AS, 2011.

• N. Zamponi, Some fluid-dynamic models for quantum electron transport

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

Different statistics
Although the QMEP was originally stated for a general (convex)
entropy functional, nevertheless, it has been applied only to
Boltzmann entropy.
Partial exceptions:
• P. Degond, S. Gallego, F. Méhats, An entropic quantum
drift-diffusion model for electron transport in resonant tunneling
diodes, J. Comput. Phys., 2007.

• M. Trovato, L. Reggiani, Quantum maximum entropy principle for

a system of identical particles. Phys. Rev. E , 2010.

• A. Jüngel, S. Krause, P. Pietra, Diffusive semiconductor moment

equations using Fermi-Dirac statistics. ZAMP, 2011.

The work we are going to expose is exactly intended to fill this

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

The kinetic model

The starting point is the Wigner equation with BGK collisional

term
p
g[w] − w
∂t + m · ∇x + Θ~ [V ] w = ,
τ
where:

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

• Θ~ [V ] = ~i V x + i~ i~



2 ∇p − V x − 2 ∇p ;

• τ is a typical collisional time;

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

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

Scaling the Wigner-BGK equation

After a suitable scaling, we can write the Wigner-BGK equation

as follows:

1
(∂t + p · ∇x + Θ [V ]) w = (g[w] − w)
α

• α = tτ (hydrodynamic parameter);
0

•  = x ~p (semiclassical parameter).
0 0
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

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

g[w] is the most probable state compatible with the information

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

hg[w]i = n = hwi, hpg[w]i = J = hpwi.

“Most probable” means that g[w] minimizes an entropy

functional H.
Then, g[w] is chosen as a minimizer of H among all Wigner
functions that share with w the same moments n and J.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Entropy functional

In terms of density operators, the suitable entropy functional for

the isothermal case is the free-energy
   
1
H(%) = Tr T % log % + (1 − λ%) log(1 − λ%) + H% ,
λ

which also incorporates the information on particle statistics:

1, Fermi-Dirac



λ= 0, Maxwell-Boltzmann


−1, Bose-Einstein
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Constrained minimization problem

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

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

hg[w]i = n = hwi, hpg[w]i = J = hpwi.

Here Op denotes the Weyl quantization, mapping 1-1 Wigner

functions and density operators.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

The minimizer

Theorem
A necessary condition for g = g[w] to be solution of the
constrained minimization problem is that Lagrange multipliers
A and B = (B1 , . . . , Bd ) exist such that
 H(A,B) −1
Op (g) = e T + λ ,

where  
|p−B|2
H(A, B) = Op 2 −A ,

and A, B have to be determined as functions of n, J from the

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

Hydrodynamic limit

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

∂ ∂
n+ Jj = 0,
∂t ∂xj
∂ ∂
 ∂
Ji + pi pj g[w0 ] + n V = 0.
∂t ∂xj ∂xi
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Formal closure


The unknown moment pi pj g[w] can be expressed in terms of
the Lagrange multipliers as follows:

∂
 ∂ ∂
∂
pi pj g[w] = (nui Bj ) + Bj nuj − Bj + n A.
∂xj ∂xj ∂xi ∂xi

This provides a formal (and rather implicit) closure of the

quantum hydrodynamic equations because A, B are depend on
n, J through the moment constraints.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Difficulties with g[w]

From now on, we shall simply denote by g the local equilibrium

state g[w].
The phase-space function g is a very complicated object,
involving back and forth Weyl quantization:
( " !# )−1
|p − B|2 A
g = Op −1 exp Op − +λ
2T T

The only hope we have to get something explicit is expanding g

semiclassically, i.e. in powers of .
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Semiclassical expansion of g

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

expansion
g = g (0) + 2 g (2) + O(4 )
where g (0) is the “classical” distribution

1
g (0) = (p−B)2
− TA
e 2T +λ

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

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

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 hgi = n and hpgi = J.
Using the expansion of g, it turns out that

A = A(0) + 2 A(2) + O(4 ), B = B (0) + 2 B (2) + O(4 ).

At leading order we obtain

 
(0) −1 n
A = T φd , B (0) = u = J/n,
2 (2πT )d/2

where
+∞
t (s−1)
Z
1 1
φs (z) = − Lis (−λez ) = dt.
λ Γ(s) 0 et−z +λ
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Condition for invertibility

8
FD, d=3
MB, d=3
7 BE, d=3
BE, d=2
BE, d=1
6
Plots of φ d for some
2
5 values of λ and d.
4
For λ < 0 and d ≥ 3,
3
φ d ranges from 0 to
2

2
ζ d2 /|λ|.
1

0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

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

Exceeding particles fall in the condensate phase, not described

by our model.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

A and B: second-order expressions

1 h i φ0d −2 0
1  ∂i A(0) 2 φ d2 −3
A(2) = ∂i uj (∂i uj − ∂j ui ) − 2∂i ∂i A(0) 2
−
24T φ0d −1 24 T φ0d −1
2 2

(2) nd h i
Bi = ∂j (∂i uj − ∂j ui )φ0d −1
12Tn 2

where
∂
∂i :=
∂xi
and
 A(0)    n 
φ0s := φs = φs φ−1
d
T 2 nd
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Semiclassical hydrodynamic equations (I)

We can now substitute the expansions

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

in the quantum hydrodynamic equations

∂ ∂
n+ (nuj ) = 0,
∂t ∂xj
∂ ∂ ∂
∂
(nui ) + (nui Bj ) + Bj nuj − Bj + n (A + V ) = 0.
∂t ∂xj ∂xi ∂xi
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Semiclassical hydrodynamic equations (II)

We obtain therefore our main result:

∂ ∂
n+ (nuj ) = 0,
∂t ∂xj

∂ ∂ ∂ −1  n  ∂  
(nui ) + (nui uj ) + Tn φd +n V + 2 Q(n)
∂t ∂xj ∂xi 2 nd ∂xi
0
2 n ∂ Rjk Rkj φ d2 −2 2 nd ∂  0

= + Rij Rjk φ d −1 ,
24T ∂xi 2 φ0d 12T ∂xk 2
−1 2

where
d ∂ ∂
nd = (2πT ) 2 , Rij := ui − uj
∂xj ∂xi
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Modified Bohm potential

The term 2 Q(n) can be identified as a modified Bohm

potential:

0 0
 
1  n  φ d
−2
  n  2 φ d
−3
Q(n) = − 2∆φ−1 2
+ ∇φ−1  02  ,

d 0 d
24 2 nd φd 2 n d φ d −1
−1 2 2

since
√
1∆ n
lim Q(n) = − √ .
λ→0 6 n
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Limit properties of φs

From known asymptotic properties of the polylogarithms we

obtain:

lim φs (z) = ez , for z ∈ R and s ∈ R

λ→0

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

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

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

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

The Maxwell-Boltzmann limit

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

∂ ∂
n+ (nuj ) = 0,
∂t ∂xj
√ 
2 ∆ n

∂ ∂ ∂ ∂
(nui ) + (nui uj ) + T n+n V− √
∂t ∂xj ∂xi ∂xi 6 n
2 ∂

= nRik Rkj ,
12T ∂xj

(corresponding to the equations found by Jüngel-Matthes (’05) and

Degond-Gallego-Méhats (’07)).
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Vanishing-temperature limit

The behavior of the semiclassical hydrodynamic equations as

T → 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).
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

T → 0 limit: Fermi-Dirac case

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

∂ ∂
n+ (nuj ) = 0,
∂t ∂xj

d

2−d
2
∂ ∂ 2
d
Γ d2 d ∂ 2
(nui ) + (nui uj ) + n nd
∂t ∂xj 2π ∂xi
√ 
d − 2 2 ∆ n

∂
+n V− √
∂xi d 6 n
2
 
 dπ (d − 2) ∂ Rjk Rkj ∂  d−2
=
2 n + R ij R jk n d

12 Γ d2 + 1 d d ∂xi 4n d2 ∂xk
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

T → 0 limit: Maxwell-Boltzmann case

This is the most singular case, because the limits λ → 0 and

T → 0 are somehow incompatible.

The formal limit of the semiclassical hydrodynamic equations

with BE statistics is only compatible with an irrotational fluid
(R = 0) and depends on how (0, 0) is approached in the
parameter plane (λ, T )

Degond-Gallego-Méhats (’07) show that the fully-quantum fluid

equations admit a limit, which is given by Madelung equations.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

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

For λ = −1 and d ≥ 3, in the limit T → 0 the fluid is completely

condensate and, then, this case cannot be considered within
our description.
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

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

In this case the limit T → 0 is only compatible with an

irrotational fluid. Moreover, we have to rescale the density as
n
N= .
2πT
The resulting limit equations are

∂ ∂
N+ (Nuj ) = 0
∂t ∂xj
2
 
∂ ∂ ∂
ui + uj uj + V− ∆N = 0
∂t ∂xi ∂xi 12
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

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

In this, last, case we obtain

∂ ∂
n+ (nu) = 0
∂t ∂x
√ 
2 1 ∂ 2 n

∂ 1 ∂ 2 ∂
u+ u + V− √ =0
∂t 2 ∂x ∂x 2 n ∂x 2
A short history Applying the QMEP The local equilibrium Semiclassical equations Some asymptotics Conclusions

Conclusions

• We derived semiclassical isothermal hydrodynamic

equations for Fermions or Bosons.

• The method exploits Degond and Ringhofer’s Quantum

Maximum Entropy Principle and the semiclassical
expansion of the maximizer Wigner function.

• We obtained an Euler-like system with quantum

corrections, of order ~2 , involving a modified Bohm
potential and the velocity curl tensor.

• The Maxwell-Boltzmann limit and the T → 0 limit have

been investigated.
• L.B., C. Cintolesi, Derivation of isothermal quantum fluid equations with
Fermi-Dirac and Bose-Einstein statistics. J. Stat. Phys. (to appear).
Thank you