Quantum moment hydrodynamicsand the entropy principle

P. Degond (1) ∗ and C. Ringhofer (2) †

(1) MIP, UMR 5640 (CNRS-UPS-INSA)Universite Paul Sabatier

118, route de Narbonne, 31062 TOULOUSE cedex, FRANCEemail: degond@mip.ups-tlse.fr

(2) Department of Mathematics,Arizona State University,

Tempe, Arizona 85287-1804, USAemail: ringhofer@asu.edu

AbstractThis paper presents how a non-commutative version of the entropy extremalization

principle allows to construct new quantum hydrodynamic models. Our starting pointis the moment method, which consists in integrating the quantum Liouville equationwith respect to momentum p against a given vector of monomials of p. Like in theclassical case, the so-obtained moment system is not closed. Inspired from Levermore’sprocedure in the classical case [26], we propose to close the moment system by a quantum(Wigner) distribution function which minimizes the entropy subject to the constraint thatits moments are given. In contrast to the classical case, the quantum entropy is definedglobally (and not locally) as the trace of an operator. Therefore, the relation between themoments and the Lagrange multipliers of the constrained entropy minimization problembecomes non-local and the resulting moment system involves non-local operators (insteadof purely local ones in the classical case). In the present paper, we discuss some practicalaspects and consequences of this non-local feature.

Key words: Density matrix, quantum entropy, quantum moments, local quantum equilibria,quantum BGK models, quantum hydrodynamics,

AMS Subject classification: 82C10, 82C70, 82D37, 81Q05, 81S05, 81S30, 81V70,

∗supported the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. P. Degondwishes to express his gratitude to F. Mehats for very valuable remarks and comments.

†supported by NSF award Nrs. DMS0204543 and DECS0218008.

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1 Introduction

The aim of this paper is to present a new approach to quantum hydrodynamics. More precisely,starting from the quantum Liouville equation, we derive a whole hierarchy of moment modelsincluding quantum hydrodynamical models as well as higher order moment models. The so-obtained models will be referred to as ’Quantum Moment Hydrodynamics’.

The derivation of quantum hydrodynamic models from first principles has attracted consid-erable attention in the recent past. The quest for such models is driven by the growing field ofnanotechnology applications. The derivation of reliable yet computationally affordable many-particle quantum models determines the possibility of efficient industrial design and fabricationof the next generation of devices. However, few attempts have been successful in this direction.Indeed, away from the complete resolution of the Schrdinger equation (or even worse, of thequantum Liouville equation), which is computationally expensive, and the use of continuummodels with ad-hoc phenomenological closure and limited reliability, few alternatives are avail-able. The starting point for the derivation of quantum hydrodynamic models is the quantumBoltzmann equation of the form

i~∂tρ = [H, ρ] +Q(ρ), (1.1)

for the effective single particle density matrix ρ(x, y), where H is the Hamiltonian H =− ~2

2m∗ |∇x|2 + V (x), the symbol [., .] denotes the usual commutator and the operator Q modelsparticle collisions. ~ denotes the reduced Planck constant and m∗ is the particle (effective)mass. To derive fluid like models for macroscopic quantities it is convenient to consider insteadthe equivalent formulation via Wigner functions, which is of the form

∂tfw + divx(1

m∗pfw)− θ[V ]fw = Qw(fw), (1.2)

where the Wigner function fw(x, p, t) and the pseudo - differential operator θ are related to thedensity matrix ρ and the potential V via the Wigner transform

fw(x, p, t) = (2π)−d

∫Rd

ρ(x− ~2η, x+

~2η)eiη·pdη, θ[V ] =

i

~[V (x+

~2i∇p)−V (x− ~

2i∇p)] (1.3)

where ρ(x, y) is the integral kernel of ρ and d is the space dimension (an integer multiple of 3according to the number of degrees of freedom). Moment systems for the Wigner - Boltzmannequation (1.2) are equations for a given set of moments

mn(x, t) =

∫Rd

κn(p)fw(x, p, t)dp , n = 0, .., N , (1.4)

which are obtained by building the corresponding moments of (1.2). The main problem inclosing this system, i.e. expressing the highest order moments in the resulting equations interms of the lower order moments, is that the collision operator Q (or Qw in (1.2)) is notavailable in a sufficiently simple form, to be used in a Chapman - Enskog - like approach tomoment closures. Collision operators for scattering with phonons on the weak coupling limit

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have been derived and analyzed in [2], [5], [12] and quantum versions of Fokker - Planck typeoperators have been derived in [6] and [3].

First attempts towards the derivation of quantum hydrodynamic models have used BKWwaves. Writing the wave-function as ψ =

√n(x, t) exp(iS(x, t)), where n is the probability of

presence and S is the phase and inserting it into the Schrodinger equation gives rise to a setof two equations for n and n∇S which mimic the clasical density and momentum conservationequations. Compared with the classical case, the momentum equation involves an additionalterm, called the Bohm potential. This approach has been investigated in [16], [23], [20], [17].This is however a pure-state model which does not incorporate mutli-particle effects and whichcan be viewed as a zero-temperature model. Finite temperature effects have been taken intoaccount in [18] through the use of a non linear Schrodinger equation and gives rise to a mo-mentum equation with a nonlinear enthalpy relation. The equation-of-state is local (i.e. thepressure (or the enthalpy) depends locally on the density at the same point). In [22], it isargued that the Bohm potential approach is not consistent with the entropy condition. Theapproach presented in this paper brings a cure to this problem.

Other approaches [13], [11] make use of moment closures of the Wigner equation using semi-classical asymptotics (i.e. a limit ~ → 0 where ~ is the Planck constant) for thermodynamicalequilibrium states as closures. They give rise to local equations-of-state as well, which coincidewith the previous theories up to constant factors. Related to this approach are moment closuretheories of the Wigner equation using small-field asymptotics for thermodynamical equilibria[14], [15]. These give rise to nonlocal equations-of-state. However, the nonlocality enters onlythrough the potential.

The idea behind the work presented in this paper is to use a thermodynamic approach usingthe minimal amount of information about the collision operator necessary to derive momentequations. That is we assume knowledge about the corresponding moments of the collisionoperator and the existence of an entropy which is dissipated by the collisions. Our approachbears strong similarities with the theory of NESOM (for NonEquilibrium Statistical OperatorMechanics) by Zubarev and coworkers [36], [28] (see also the review paper by Luzzi [27]).However, we give a neater (and to some extent, more practical) mathematical framework which,we believe, will be useful for further developments of the theory We shall elaborate more onthe relation between our theory and NESOM in section 4, remark 4.2.

Our approach consists in taking moments of the density matrix equation (or quantumLiouville equation), and then, closing the resulting set of moment equations with an equilibriumdensity matrix. This equilibrium is found as an extremum of the entropy functional subject tothe constraints that its moments coincide with those of the density matrix we are considering.This approach is therefore similar to Levermore’s closure moment hierarchies [26] or to theextended thermodynamics approach [29] for classical systems

To be more, specific, let a set of polynomials κ(p) = (κ0(p), . . . , κN(p)) be given. Toany ρ, we associate a set of moments m[ρ] = (m0(x), . . . , mN(x)) defined by duality as therepresentation of the linear functionals λ → Tr{ρOp(λ · κ)}, where λ = (λ0(x), . . . , λN(x)),λ · κ =

∑i λi(x)κi(p) and Op means the Weyl quantization of a symbol (i.e. a function of

position x and momentum p) into an operator. This definition of moments by duality willprove more convenient in connection with the entropy minimization principle. If the chosen

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set of polynomials is equal to (1, p, |p|2/2m∗), the associated moments correspond to the usualhydrodynamic quantities, i.e. local density, momentum and energy (per particle). If a largerset is chosen, the associated moments correspond to higher order hydrodynamic quantitieslike the pressure tensor, heat flux vector, higher order heat flux tensor, etc. , according tothe terminology of extended thermodynamics [29]. For instance, the pressure tensor will beassociated with the monomials pipj where pi and pj denote the components of the momentumvector in the i-th and j-th direction respectively.

Now, we turn to the definition of an equilibrium density matrix which satisfies given momentconstraints. Such an equilibrium minimizes the entropy (defined as H(ρ) = Tr{h(ρ)} where h isa suitable convex function, like e.g. the Boltzmann entropy h(ρ) = ρ(ln(ρ)−1)). In this work, weshow that this constrained minimization problem has the solution ρm = (h′)−1(Op(µ ·κ)) whereµ = (µ0(x), . . . , µN(x)) are the Lagrange multipliers of the moment constraints {m[ρ] = mgiven}. (h′)−1 is the inverse function of the derivative of h. Functions of operators are given ameaning in the sense of functional calculus.

It is now possible to get back to the problem of deriving moment models from the quantumLiouville equation. Taking the moments of the Liouville equation (which is now a preciseconcept), we can close the chain of moment equations by using the equilibrium just defined.According to the chosen set of polynomials and the number of moments involved, we obtaina hierarchy of moments systems which we call Quantum Moment Hydrodynamics. We notethat such a closure is different from a single-state closure as it takes into account many-particleinteractions through the entropy minimization procedure. However, we should recover thesingle-state closure by a zero-temperature asymptotics (for ~ being fixed) of our moment models.The investigation of this point is left to future work. Also, the moment systems should obviouslyconstitute an approximation of the original quantum kinetic system (1.1). However, since theexpression of Q(ρ) is not known, assessing the accuracy of this approximation is extremelydelicate.

The goal of the present paper is to develop these concepts. The paper is organized asfollows: in section 2, we recall Levermore’s approach to moment closure hierarchies in theclassical case, however giving it a presentation which makes its extension to the quantumcase more natural. Then, in section 3, we extend Levermore’s approach to the quantum case,developing our definition of moment. We elaborate more on the fluid entropy and compareour theory with NESOM in section 4. Quantum moment hydrodynamics systems are derivedand studied in section 5. In particular, it is shown that quantum hydrodynamics for the usualset of moments (density, momentum and energy) differs from classical hydrodynamics by apossibly non-scalar pressure tensor and non-zero heat flux vector. These are related non locallyto the basic moments through the equilibrium density matrix. Higher order quantum momentsystems exhibit similar features. Finally, as noticed in this section, it is possible to make senseto quantum BGK relaxation systems. A conclusion is drawn in section 6. Finally, technicaldetails are deferred to two appendices (sections 7 and 8).

It should be stressed that most of the mathematical properties stated in this paper are givenonly formal proofs. Fully rigorous proofs will require a lot of mathematical developments whichare beyond the scope of the present paper. However, a practical usage of these new modelsdoes not require that all the mathematical theory is settled. Indeed, this paper should rather

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be viewed as a presentation of new models and as a programme definition for future work.The results of the present work have been announced in [9].

2 Entropy minimization principles and moment closure

hierarchies in the classical case

In this section, we mainly review the approach proposed by Levermore [26] in the classicalcase. However, we shall present the entropy minimization principle in a slightly different (butcompletely equivalent) form which will make it more suitable to an extension to the quantumcase.

In classical kinetic theory, the basic object is the particle distribution function f(x, p, t)where x ∈ Rd is the position, p ∈ Rd is the momentum, and t is the time. f is a probabilitydistribution and is therefore normalized according to∫

R2d

f(x, p) dx dp = 1 . (2.1)

In all this work, we shall assume that the total numberN of particles in the system is fixed. Thishypothesis is not essential in the classical case, but it will greatly simplify the presentation of thequantum case. The distribution function f is a solution of the so-called Boltzmann equation:

∂f

∂t+ {H, f} = Q(f) , (2.2)

where H(x, p) is the particle Hamiltonian, {H, f} is the Poisson bracket

{H, f} = ∇pH · ∇xf −∇xH · ∇pf ,

andQ(f) is a collision operator, which models the interactions of the particles among themselvesor with the surrounding medium.

On the other hand, the physics of continua considers averaged quantities which only dependon the position variable such as the mean density, momentum or energy. These quantities aredefined as moments of the distribution function f . More specifically, let κi(p), i = 0, . . . , N beN + 1 independent functions of p and denote by κ(p) = (κi(p))i=0,...,N the vector of monomials.The associated moments ki(f)(x) of f(x, p) are defined by:

ki(f)(x) =

∫Rd

f(x, p)κi(p) dp , ∀x .

The equations for k(f) = (ki(f))i=0,...,N are obtained by multiplying (2.2) by κi(p) and inte-grating with respect to p:

∂ki(f)

∂t+

∫Rd

κi(p) {H, f} dp =

∫Rd

κi(p)Q(f) dp (2.3)

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In most cases the integrals in (2.3) cannot be expressed in terms of the functions ki(f) alone.In order to reduce system (2.3) to a closed system for the functions ki, some assumptionsmust be made on the distribution function f . According to statistical physics, the most likelydistribution function (upon a certain number of realizations) is a minimum of the entropyfunctional among functions whose moments are given by the functions ki. Following [26], weuse this Ansatz and replace f in the integrals of (2.3) by this distribution function.

The minimization principle (or Gibbs minimization principle, see [4]) can be formulated asfollows. Let h be a smooth strictly convex function defined on [0,∞) and define the entropyfunctional H(f) acting on functions f(x, p) associated with h by:

H(f) =

∫R2d

h(f(x, p)) dx dp ,

Let m = (mi(x))i=0,...,N be N + 1 given functions of the position variable x. We are concernedwith the following minimization problem (Gibbs problem): find the solution fm of

H(fm) = min {H(f) | f satisfies ki(f)(x) = mi(x) , ∀x, ∀i} . (2.4)

The normalization condition (2.1) is not included in the constraints of (2.4). Rather, we assumethat the constant function is contained in κ(p), say κ0(p) = 1. Then, m0(x) is the probability ofpresence of a particle in the neighbourhood of x and is such that

∫m0(x) dx =

∫f(x, p) dx dp.

We restrict the set of moments to those satisfying∫Rd

m0(x) dx = 1 , (2.5)

so that the constraint (2.1) is satisfied, as soon as the constraints of (2.4) are satisfied.In most physics textbooks, the tradition is to use the opposite sign for the entropy (i.e.

to suppose that it is a concave function) and to write the Gibbs principle as a constrainedmaximization problem. It is the mathematicians’ usage however to use the opposite convention.Of course, the two conventions are completely equivalent.

We note that this problem can be equivalently stated as

H(fm) = min {H(f) | f satisfies Kλ(f) =

∫Rd

m · λ dx , ∀λ(x) = (λi(x))i=0,...,N} , (2.6)

where, we have introduced

Kλ(f) =

∫Rd

k(f) · λ(x) dx =

∫R2d

f(x, p)κ(p) · λ(x) dx dp , (2.7)

and, for two vectors a and b of RN+1, we write a · b =∑

i aibi. The arbitrary functionsλ = (λi(x))i=0,...,N are the Lagrange multipliers of the constraints.

According to classical optimization theory, the constrained optimization problem (2.4) canbe equivalently formulated in terms of a saddle-point problem for the Lagrangian

Lm(f, λ) = H(f)−(Kλ(f)−

∫Rd

m(x) · λ(x) dx

), (2.8)

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where the Lagrange multipliers λ = (λi(x))i=,...,N are functions of x. The saddle-point problemis formulated as follows:

H(fm) = minf

maxλLm(f, λ) = max

λmin

fLm(f, λ) , (2.9)

where now the minimum over f or the maximum over λ are unconstrained problems.Let us now consider the unconstrained problem

Lm(fλ, λ) = minfLm(f, λ) . (2.10)

Then, the necessary condition for extremality leads to

fλ(x, p) = (h′)−1(λ(x) · κ(p)) (2.11)

where h′ is the derivative of h and (h′)−1 is the inverse function of h′ (which exists since h′ isstrictly increasing). fλ is called the equilibrium distribution function and λ is called the systemof entropic variables.

Now, the solution fm of the constrained minimization problem (2.4) is an equilibrium dis-tribution function fµ where µ = µm is a solution of the unconstrained maximization problem

H(fm) = Lm(fµm , µm) = maxλLm(fλ, λ) . (2.12)

The resolution of (2.12) leads tofm = fµm , (2.13)

where µm is such thatki(fµm) = mi ,∀i = 1, . . . N , (2.14)

We note that this relation is equivalent to saying that

Kλ(fµm) =

∫Rd

m(x) · λ(x) dx , ∀λ(x) , (2.15)

This last relation is in a form which will be easily extended to the quantum case. The entropicvariables µ and the moment (or conservative) variables m are dual (in the Legendre transformsense) through the fluid entropy (see section 4 where this aspect is developed in the quantumcase).

Now, following [26], a closed set of moment equations can be derived from (2.3) by replacingf in the integrals appearing in (2.3) by the solution of the Gibbs principle associated with theconstraint that the moments are ki(f). This leads to a closed system of equations for themoments mi = ki(f) which is written as follows:

∂mi

∂t+

∫Rd

κi(p) {H, fm} dp =

∫Rd

κi(p)Q(fm) dp (2.16)

For future use, we note that system (2.16) can be written in weak form as:

∂

∂t

∫Rd

m · λ dx+

∫R2d

λ(x) · κ(p) {H, fm} dp dx =

∫R2d

λ(x) · κ(p)Q(fm) dp dx , (2.17)

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for all λ(x). We shall extend this weak form of the hydrodynamic equations to the quantumcase. The key point is to interpret the integrals in (2.17) in an operator way.

We now comment on the form (2.4) of the Gibbs minimization principle. In[26], the Gibbsprinciple is given a local form. For a function f(p), we denote by H the ’local’ entropy functional

H(f) =

∫Rd

h(f(p)) dp .

Then, for any given (N+1)-tuple of real numbers m = (mi)i=0,...,N , the local Gibbs principlereads: find f m(p) which realizes

H(f m) = min {H(f) | f(p) satisfies Ki(f) = mi , ∀i} . (2.18)

Of course, for f(x, p), we have

H(f) =

∫Rd

H(f(x, ·)) dx .

The solution of the ’local’ Gibbs minimization principle (2.18) is obviously

f m = (h′)−1(µm · κ) , (2.19)

where µm is such thatKi(f

m) = mi , ∀i . (2.20)

This defines a local mapping m ∈ RN+1 → µm ∈ RN . The solution of the global Gibbs principle(2.13) is obviously related to that of the local one (2.19) by:

µm(x) = µm(x) .

However, the global form (2.4) of Gibbs principle can be extended to the quantum case, whilethe local form cannot. Quantum mechanics is a non-local theory and requires that a non-localGibbs principle be used.

Let us review some classical cases. Suppose that the Hamiltonian is given by

H(x, p) = ε(p) + V (x, t) , ε(p) =|p|2

2m∗ ,

where ε(p) is the kinetic energy, V (x, t), the potential energy and m∗ the particle mass. Supposethat the vector κ of monomials is a d + 2-dimensional vector (d being the dimension of thephysical space) consisting of

κ(p) = (1, (pi)i=1,...,d, |p|2/(2m∗)) . (2.21)

The functions involved in κ(p) correspond to the physically conserved quantities (i.e. theprobability of presence, the d components of the momentum per particle and the energy perparticle). As often in the literature, we shall label the components of this vector according to

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κ(p) = (κ0(p), (κi(p))i=1,...,d, κd+1(p)). In thermodynamics, the associated vector of Lagrangemultipliers is usually written

λ = (µc

T,u

T,− 1

T) := (λ0, (λi)i=1,...,d, λd+1) ,

where µc ∈ R is the chemical potential, u ∈ Rd is the mean velocity and T ∈ R+ is thetemperature. The associated moment vector in turn is written

m = (n, q,W ) := (m0, (mi)i=1,...,d,md+1) ,

where n ∈ R+ is the probability of presence, q ∈ Rd is the mean momentum per particle andW ∈ R+ is the mean energy per particle. Multiplying n, q and W by the total number ofparticles N gives the number density, the fluid momentum and the fluid energy.

Let us consider specifically the Boltzmann entropy:

h(f) = kB f(ln f − 1) , (2.22)

where kB is the Boltzmann constant. In this case, the equilibrium distribution function fλ isthe classical Maxwellian:

fλ = exp{ 1

kB

(λ0 +d∑

i=1

λipi + λd+1|p|2

2m∗ )} = exp{ 1

kBT(µc +

d∑i=1

uipi −|p|2

2m∗ )} , (2.23)

Then, the mapping m→ µm relates (n, q,W ) to (µc, u, T ) in the following way:

n = h−dP (2πm∗kBT )d/2 exp

µc +m∗|u|2/2kBT

, q = m∗nu , W =1

2m∗n|u|2 +

d

2nkBT .

With this transformation, the Maxwellian takes the more familiar form

fλ =n

(2πm∗kBT )d/2exp{−|p−m∗u|2

2m∗kBT} . (2.24)

Supposing that the collision operator is mass, momentum and energy conservative, i.e.satisfies

∫Q(f)κi(p) dp = 0 for all considered κi, eqs (2.16) can be simplified and gives rise to

the usual classical hydrodynamic equations:

∂n

∂t+∇ · (nu) = 0 , (2.25)

m∗(∂nu

∂t+∇ · (nu⊗ u)) +∇(nkBT ) = −n∇V , (2.26)

∂W

∂t+∇ · (Wu) +∇ · (nkBTu) = −n∇V · u . (2.27)

Therefore, the above considered moments give rise to the balance equations for the physicallyconserved quantities (i.e. mass, momentum and energy).

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If other moments than those corresponding to the physically conserved quantities, such asthe heat flux (corresponding to κ(p) = pi|p|2/2), the pressure tensor anisotropy (i.e. κ(p) =pipj), or the higher order heat flux tensor (i.e. κ(p) = pipjpk) are considered, new sets ofhydrodynamic-like equations are obtained. This gives rise to the so-called ’moment closurehierarchies’ of Levermore [26], which are also closely related with the theory of extended ther-modynamics [29]. Then, the equilibrium distribution functions (2.11) depend on a larger di-mensional vector of Lagrange multipliers λ than the Maxwellians (2.23), which therefore appearas a special case of these equilibria. If one linearizes these equilibria about the Maxwellians(considering that the higher order moments, after convenient normalization, are small), oneobtains another moment closure method first proposed by Grad [19].

However, the following considerations should be borne in mind. First, the N -tuple of mono-mials κ(p) cannot be completely arbitrary. It has to satisfy a certain number of requirementssuch as gallilean invariance, or the fact that the moments of fλ must be defined. These con-straints are detailed in [26]. Second, the existence and uniqueness of solutions of the mini-mization problems is by far not guaranteed. Moment realizability conditions, i.e. conditionsthat guarantee that the constrained minimization problem has a solution, as well as uniquenessconditions have been studied in [24], [25], [32], [1]. When moments corresponding to physicallynon conserved quantities (i.e. such that the corresponding

∫Q(f)κi(p) dp is non zero) are con-

sidered, a particular care must be taken in the closure relation for the collision operator part ofeq. (2.3). This must be done in order to capture the correct dynamics in situations which areclose perturbations of the usual gas dynamics equations. This point is detailed in [26]. Finally,moment systems of the form (2.16) are by construction hyperbolic, which guarantees a certaindegree of well-posedness (at least in a linear sense). This point is developed in [26].

We are now going to investigate how these considerations can be extended to the quantumcase. For that purpose, we shall use the non local version of the Gibbs minimization principle.

3 Entropy minimization principles in the quantum case

We first state the entropy minimization principle in the quantum case. To be specific, weconsider a system whose objects can be described by wave-functions ψ(x) belonging to theHilbert space X = L2(Rd). In this context, the quantum equivalent of the distribution functionis the density matrix ρ which is a positive, trace-class Hermitian operator on X satisfying [31]:

Tr{ρ} = 1 . (3.1)

The normalization condition (3.1) is the quantum counterpart of (2.1). Then, the spectrum ofρ consists of a decreasing sequence of positive eigenvalues (α`)`=1,...,∞ tending to 0 as ` → ∞and the associated eigenvectors φ` constitute a Hilbert basis of X. Moreover, because of (3.1),we have

∞∑`=1

α` = 1 .

The elementary wave-function φ` represents a pure state, while the datum of a density matrixρ represents a mixed state. The eigenvalues α` represent the probability for the system to be

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in the state φ`.Any observable defined by a Hermitian operator A on X gives rise to an observation 〈A〉ρ

on the system modeled by ρ according to the formula

〈A〉ρ = Tr{ρA} ,

where ρA denotes the operator multiplication of ρ and A. Examples of such observables arethe mean position defined by the position operator X which operates through the multipli-cation by x, or the mean momentum defined by the momentum operator P = −i~∇x where~ is the reduced Planck constant and i2 = −1. More generally, we shall consider operatorsOp(a) obtained from any symbol a(x, p) of the position x and momentum p through the Weylquantization by

Op(a)φ =1

(2π~)d

∫R2d

a

(x+ y

2, p

)φ(y)ei

p(x−y)~ dp dy . (3.2)

The Weyl quantization is such that any real valued symbol gives rise to a Hermitian operator(under regularity conditions that we shall not detail see for instance [34]). The Weyl quantiza-tion (3.2) is formally related to the Wigner transform (1.3) via the formula ρ = (2π~)dOp(fw).Thus, Weyl quantization and Wigner transform are (up to a factor (2π~)d) inverse operationsone to each other and

Tr(ρOp(a)) =

∫R2d

a(x, p)fw(x, p)dxdp (3.3)

holds. Wigner functions are therefore a convenient tool to express local moments in quantummechanics. Entropy principles, however, are better expressed in terms of density matricesand operators. A class of symbol that we shall be particularly interested in are a(x, p) =λ(x) · κ(p) where λ = (λi(x))i=0,...,N is an arbitrary N+1-tuple of real valued functions andκ = (κi(p))i=0,...,N is the N+1-tuple of moment monomials.

We now choose a given N+1-tuple of real valued moment monomials κ, such that κ0(p) = 1.For any given operator ρ, we shall denote by Kλ(ρ) the expectation value of the operatorOp(λ(x) · κ(p)) (denoted Op(λ · κ) for short) i.e. for any N+1-tuple of real valued functionsλ = (λi(x))i=0,...,N :

Kλ(ρ) = Tr{ρOp(λ · κ)} =

∫R2d

λ(x) · κ(p)fw(x, p)dxdp . (3.4)

Kλ(ρ) is therefore the observation corresponding to a certain combination of the moment mono-mials κi(p), weighted by x-dependent factors λi(x). The mapping λ→ Kλ(ρ) is a linear func-tional defined on the set of real valued functions λ(x) which, by duality, defines a N+1-tupleof real valued functions m[ρ] := m(x) = (mi(x))i=0,...,N , which are functions of x, according to

Kλ(ρ) =

∫Rd

λ(x) ·m(x) dx , ∀λ(x) real valued . (3.5)

These moments are nothing but the local moments of the Wigner distribution function fw inthe usual sense:

m[ρ](x) =

∫κ(p)fw(x, p, t) ds dp .

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Therefore, (3.5) is a way to express local moments of the Wigner distribution function in termsof the density operator ρ. This formula is clearly the quantum equivalent of (2.7). Therefore,in the quantum framework, there is no local relation between ρ and its moments m, but rather,a functional one, through the duality (3.5).

For the sake of clarity, we give explicit expressions of the moments m[ρ] expressed in termsof the density matrix ρ. The proof of the following two Lemmas is just an exercise in Fouriertransforms using the definition (1.3) of the Wigner function fw and the equivalence formula(3.3), and is left to the reader. We start with the following expression of Op(λ · κ).

Lemma 3.1 Let β = (β1, . . . , βd) ∈ Nd be a multi-index (with N the set of natural integers)and denote by pβ = pβ1

1 . . . pβd

d and ∂/∂xβ = ∂/∂xβ1

1 . . . ∂/∂xβd

d . Then, for any smooth real orcomplex-valued function ν(x), we have the two following equivalent expressions of the operatorOp(pβν):

Op(pβν)φ = (−i~)|β|β1∑

γ1=0

. . .

βd∑γd=0

(β1

γ1

). . .

(βd

γd

)1

2|γ|∂ν

∂xγ

∂φ

∂xβ−γ(3.6)

Op(pβν) =1

2|β|

β1∑γ1=0

. . .

βd∑γd=0

(β1

γ1

). . .

(βd

γd

)pγνpβ−γ , (3.7)

where |β| = β1 + . . .+ βd and

(βi

γi

)is the binomial coefficient. We denote by ν the multipli-

cation operator by the function ν(x). Because ν does not commute with pi, the orders of thefactors in the right-hand side of (3.7) matters.

The local moments are given as operators acting on the integral kernel of the operator ρ by:

Lemma 3.2 Let β ∈ Nd be a multi-index and let mβ[ρ] be the moment of ρ associated withthe monomial pβ i.e. satisfying

mβ(x) =

∫Rd

pβfw(x, p)dp, Tr{ρOp(pβν(x))} =

∫Rd

mβ(x)ν(x) dx ,

for any test function ν(x). Let ρ(x, x′) be the integral kernel of the operator ρ in the positionrepresentation, i.e. the operator ρ acts on any square integrable function φ(x), x ∈ Rd accordingto

ρφ(x) =

∫Rd

ρ(x, x′)φ(x′) dx′ . (3.8)

Then, mβ is given by

mβ(x) =

(i~2

)|β| β1∑γ1=0

. . .

βd∑γd=0

(β1

γ1

). . .

(βd

γd

)(−1)|γ|

(∂

∂xγ

∂

∂x′(β−γ)ρ

)∣∣∣∣(x,x)

. (3.9)

12

This lemma shows that, as soon as the integral kernel ρ(x, x′) is smooth enough, the momentmβ is a function. Otherwise, mβ may be a singular distribution.

Now, we turn to the definition of entropy. By functional calculus, any continuous function h(and by duality, any measure) defined on the interval R+ gives rise to an operator h(ρ) definedby

h(ρ)φ =∞∑

`=1

h(α`)(φ, φ`)Xφ` ,

where (·, ·)X denotes the scalar product in X (supposed linear with respect to the left entry andantilinear with respect to the right entry). Let h be a strictly convex function on R+. Then,we define the quantum entropy H of ρ with respect to h according to

H(ρ) = Tr{h(ρ)} , (3.10)

We now formulate the quantum entropy minimization principle: let m(x) be a N+1-tupleof moment functions (defined on Rd) such that

∫m0 dx = 1. Find ρm such that

H(ρm) = min {H(ρ) | ρ satisfies Kλ(ρ) =

∫Rd

λ(x) ·m(x) dx ,

∀ real valued λ(x) = (λi(x))i=,...,N} . (3.11)

We see that this minimization problem is the quantum equivalent of problem (2.6). The con-strained optimization problem (3.11) can be rephrased as a saddle-point problem for the La-grangian

Lm(ρ, λ) = H(ρ)−(Kλ(ρ)−

∫Rd

m(x) · λ(x) dx

), (3.12)

according toH(ρm) = min

ρmax

λLm(ρ, λ) = max

λmin

ρLm(ρ, λ) , (3.13)

where now the minimum over ρ or the maximum over λ are unconstrained problems. Again,we stress the fact that λ = (λi(x))i=0,...,N is a N+1-tuple of functions of x.

To pursue the analysis, we need the following two lemmas, the proofs of which are deferredto the appendix in section 8. From now on, derivatives denoted with a δ will refer to Gateauxderivatives. The Gateaux derivative δH/δρ of a function H(ρ) (if it exists) is a linear formacting on increments δρ according to

δH

δρδρ = lim

t→0

{1

t[H(ρ+ tδρ)−H(ρ)]

}.

A necessary condition for extremality of H is that its Gateaux derivative vanishes (first orderEuler-Lagrange equation of the extremality problem).

Lemma 3.3 Let h be a strictly increasing continuously differentiable function defined on R+.Consider that H(ρ) is defined on the space of Trace-class positive self-adjoint operators ρ. ThenH is Gateaux differentiable and its Gateaux derivative δH/δρ is given by:

δH

δρδρ =

∞∑`=1

h′(α`)δρ`` = Tr{h′(ρ)δρ} , (3.14)

13

where α` are the eigenvalues of ρ, and δρ`` are the diagonal values of the perturbation operatorδρ in the basis of the eigenfunctions φ` of ρ.

Lemma 3.4 Let h be a strictly convex twice continously differentiable function on R+. Then,H is strictly convex and is twice Gateaux-differentiable, with :

δ2H

δρ2(δρ, δρ) =

∑`,r

h′(α`)− h′(αr)

α` − αr

|δρ`r|2 , (3.15)

where the quotient is understood to be h′′(α`) when α` = αr. The perturbation operator δρ isassumed Hermitian.

In fact, Nier [30] has proved infinite Frechet differentiability of h (provided h is a C∞

function) and has given proofs of formulae (3.14) and (3.15). However, his proof uses thetheory of Hellfer and Sjostrand [21] of almost analytic extensions in functional calculus. In theappendix, for the reader’s convenience, we give elementary proofs of the weaker statements 3.3and 3.4.

Let us now consider the unconstrained problem

Lm(ρλ, λ) = minρLm(ρ, λ) . (3.16)

Then, we have:

Lemma 3.5 The necessary condition for extremality for the unconstrained minimization prob-lem (3.16) is

ρλ = (h′)−1(Op(λ · κ)) (3.17)

where (h′)−1 is the inverse function of h′.

The operator ρλ is called the equilibrium density operator associated with the N+1-tuple λof entropic variables. We stress the fact that, in this theory, λ is a N+1-tuple of functions of theposition variable x and not mere constants. Formula (3.17) must be understood in the sense offunctional calculus, which is possible via the spectral theorem, since the operator Op(λ · κ) isHermitian [31]. The proof of Lemma 3.5 is deferred to the appendix, section 7.

Now, the solution ρm of the constrained minimization problem (3.11) is an equilibriumdensity operator ρλ where λ = µm is a solution of the unconstrained maximization problem

H(ρm) = Lm(ρµm , µm) = maxλLm(ρλ, λ) . (3.18)

We have:

Lemma 3.6 The solution ρm of the constrained Gibbs minimization problem (3.11) or equiv-alently, of the unconstrained maximization problem (3.18) is given by

ρm = ρµm , (3.19)

where µm is such that

Kλ(ρµm) =

∫Rd

m(x) · λ(x) dx , ∀λ(x) , (3.20)

14

By duality, relation (3.20) expresses that the moments of ρµm are m (see (3.5)). It is thequantum extension of (2.15). Again, the entropic variables µ (which are functions of x) and themoment (or conservative) variables m (which are also functions of x) are dual (in the Legendretransform sense) through the fluid entropy (which is now a functional on functions of x, seesection 4). For the proof see the appendix, section 7

4 The quantum fluid entropy

We now discuss the concept of fluid entropy in the framework of the present theory. Letm = (mi(x))i=0,...,N be a given N-tuple of fluid moments (which are functions of the positionvariable x). We define the fluid entropy S(m) by

S(m) = H(ρm) . (4.1)

For S, we can prove:

Lemma 4.1 (i) S is strictly convex.(ii) We have

δSδm

= µm , (4.2)

where δS/δm is the Gateaux derivative of S with respect to m.

Eq. (4.2) shows another aspect of the duality between µ and m, namely that they are dualthrough the entropy. Of course, this relation extends a well-known relation in the classical case.The proof can be found in the appendix, section 7.

To highlight the duality between the entropic variables µ and the moments (or conservativevariables) m, we show how relation (4.2) can be inverted by means of the Legendre dual of theentropy. Define

Σ(µ) = S(m)−∫

Rd

µ ·mdx , (4.3)

where m is such that (δS/δm)(m) = µ (or in other words, such that µm = µ, i.e. m is the setof moment of ρµ). We have the following Lemma, the proof of which is given in the appendix,section 7.

Lemma 4.2 (i) Σ is strictly concave.(ii) We have

δΣ

δµ= −m, (4.4)

The function Σ is sometimes called the Massieu-Planck function [4].

15

Remark 4.1 In equilibrium statistical mechanics (see e.g. [4]) the constraints are global i.e.are written

Tr{ρκi} = mi , ∀i = 0, . . . , N ,

where (κi)i=0,...,N are the operators corresponding to the global observables (e.g. the total

energy: κ = H). and (mi)i=0,...,N are N + 1 real numbers. Among these constraints, thefirst one, corresponding to κ0 = 1 and m0 = 1, has a special status. Indeed, it does not sayanything about the thermodynamical state of the system, but simply expresses that ρ is astatistical operator, i.e. satisfies (3.1). The associated equilibrium is written:

ρµ = (h′)−1(µ0 +N∑

i=1

µiκi) ,

where µi are now constants. If we further specialize to the Boltzmann entropy (2.22), we get:

ρµ = exp{ 1

kB

(µ0 +N∑

i=1

µiκi)} =1

Zexp{ 1

kB

(N∑

i=1

µiκi)} , (4.5)

where Z is the so-called partition function. Z can be viewed as a function of µ1, . . . , µN if ρµ

is constrained to satisfy (3.1), indeed:

Z{µ1, . . . , µN} = Tr

{exp{ 1

kB

(N∑

i=1

µiκi)}

}.

It can be shown that lnZ coincides with −Σ (up to a constant), i.e. we have:

δ lnZ

δµi

= mi , i = 1, . . . , N .

In our case however, since the µi’s are no longer constant but instead functions of x, it isimpossible to factor out the contribution of µ0 outside the exponential. This is because µ0(x)and

∑i µi(x)κi are operators which do not commute in general. So, in our framework, there

is no such object as a partition function. However, the Massieu function Σ is well-defined by(4.3) and can be used to define the moments m as functionals of the Lagrange multipliers µ.

Remark 4.2 At this point, it is appropriate to compare our approach to the NESOM theory(for NonEquilibrium Statistical Operator Mechanics), which has been pioneered by Zubarev etal (see e.g [36]) and has been later expanded by Luzzi, Vasconcellos, and coworkers. The readercan refer e.g. to [28] and [27] for reviews. This approach consists in extending formula (4.5)into a formula for a local equilibrium according to

ρµ =1

Zexp{ 1

kB

(N∑

i=1

∫µi(r, t)κi(r, t) dr)} , (4.6)

16

where κi(r, t) are local versions of the moment operators κ corresponding to the conservedquantities.

The expression∫µi(r, t)κi(r, t) dr inside the exponential bears strong similarities with our

introduction of the observable µ · κ and formula (4.6) is close to our formula (3.17) which, inthe case of the Boltzmann entropy (2.22), would reduce to

ρµ = exp{ 1

kB

(Op(µ · κ))} , (4.7)

In the cited references, the local operators κi(r, t) are not further precised. If we assume thatthey are localizations of the global moment operators κi(p) in the way outlined in the introduc-tion, i.e. something like κi(r, t) = Op(κi(p)δ(x− r)), then the expression

∑i

∫µi(r, t)κi(r, t) dr

coincides with Op(µ · κ). However, unless this is identification is made, the two approaches donot give the same results. Also, as pointed out in the previous remark, the factoring out ofpartition function in (4.6) looks somehow suspicious. However, the two approaches have clearlysimilar roots and the added value of our approach is to provide a neat mathematical frameworkto the notion of locally conserved variables by means of the operators Op(µ · κ) and the dualconcept of moments, as developed in section 3. This framework may in turn prove powerful forfuture extensions and applications of the theory.

5 Quantum moment hydrodynamics

5.1 General framework

The time evolution of the density matrix can de derived from the Schrdinger equation and isgiven by the quantum Liouville equation for the density matrix (or Von-Neumann equation):

i~∂ρ

∂t= [H, ρ] + i~Q(ρ) , (5.1)

where H is the particle Hamiltonian, [H, ρ] = Hρ − ρH is the commutator of H and ρ andQ(ρ) is an abstractly defined collision operator taking care of dissipation phenomena. To fixthe ideas, we can think of H as being the simple Hamiltonian on Rd

Hφ = − ~2

2m∗∆φ+ V φ = Op(|p|2

2m∗ + V )φ , (5.2)

where V = V (x, t) is the potential.About the collision operator Q, we specifically request the following:

(i) Q locally conserves the moments associated with κ(p), i.e.:∫Qw(fw)(x, p)κi(p) dp = 0 , i = 0, . . . , N , ∀x ∈ R3 ,

where Qw is the Wigner transform of Q,

17

(ii) Q(ρ) dissipates the quantum entropy, i.e.

Tr{Q(ρ)h′(ρ)} ≤ 0 .

An example of such an operator is the BGK operator of section 5.5 (we shall prove in [8] thatit is consistent with entropy dissipation i.e. that property (ii) holds). In [10], we also derive aBoltzmann-like collision operator for binary quantum collisions precisely on the basis of thesetwo requirements.

From (5.1), we want to derive a system of conservation equations using a similar route asthat exposed in section 2 for the classical case. First, from (5.1) we derive a system of equationsfor the moments, and then, use the equilibrium density matrix ρλ to express all the quantitiesthat cannot be directly expressed as moments. However, there is a significant difference fromthe classical case, in that the moments are not defined directly but rather, from duality throughthe linear forms Kλ(ρ). Therefore, the moment equations come naturally in duality form (orin weak form, in the sense of Partial Differential Equations theory).

Define m[ρ] to be the moments of ρ, through the duality relation (3.5). To obtain themoment equations, we compose (5.1) on the right by Op(λ · κ) and take the trace. We obtain:

∂

∂t

∫Rd

m[ρ(t)](x) · λ(x) dx = Tr

{(

[− i

~H, ρ

]+Q(ρ))Op(λ · κ)

}, ∀λ(x) . (5.3)

Eq. (5.3) is the quantum equivalent of (2.3). However, the right-hand side cannot be expressedin general in terms of the moments m[ρ(t)]. This is the closure problem.

We close eq. (5.3) by using the solution ρm = ρµm of the Gibbs minimization problem (3.11)with moments m = m[ρ(t)]. We find

∂

∂t

∫Rd

m(x, t) · λ(x) dx = Tr

{(

[− i

~H, ρµm(t)

]+Q(ρµm(t)))Op(λ · κ)

}, ∀λ(x) . (5.4)

Eq. (5.4) is a quantum moment closure system for the set of moments m, which we refer to as’Quantum Moment Hydrodynamics’. Rewriting (5.4) in terms of Wigner functions instead ofdensity matrices, one can now return to a strong formulation of the moment system. Equation(5.4) is equivalent to the weak formulation of

∂tm+

∫Rd

κ(p){∇x · (1

m∗pfmw )− θ[V ]fm

w }dp =

∫Rd

κ(p)Qw(fmw )dp, (5.5)

where fmw (x, p, t) is the closure Wigner function corresponding to ρµm(t) , i.e.

fmw (x, p, t) = (2π~)−dOp−1(ρµm(t)) = (2π)−d

∫Rd

ρµm(t)(x−~2η, x+

~2η)eiη·pdη (5.6)

holds.We now investigate several examples and consequences of this methodolgy. In all these

examples, otherwise explicitely stated, we shall restrict to the case of the Hamiltonian (5.2).

18

5.2 Quantum hydrodynamics

Our first use of the present framework is the derivation of the quantum hydrodynamic equa-tions, which are the quantum counterpart of the classical hydrodynamic system (2.25)-(2.27).For this purpose, we again consider the d+2-tuple of moment functions (2.21) i.e. κ(p) =(1, (pi)i=1,...,d, |p|2/(2m∗)). We first derive the moment equations from the Wigner functionformulation (5.5). In analogy to the classical case, change from the moment variables mi todensity n, ensemble velocity u and temperature T via the formulas

m0 = n =

∫Rd

fmw dp, mi = qi = m∗nui =

∫Rd

pifmw dp, i = 1, .., d

md+1 = W =n(m∗|u|2 + dkBT )

2=

∫Rd

|p|2

2m∗fmw dp

The moment system (5.5) is then of the form

∂tn+∇x · (nu) = 0, (b) ∂tm∗nu+∇x · (m∗uuTn+ P ) + n∇xV =< p >coll, (5.7)

(c) ∂tn(m∗|u|2 + dkBT ) +∇x · [n(m∗|u|2 + dkBT )u+ 2Pu+ 2qH ] + 2n∇xV · u =<|p|2

m∗ >coll

with the pressure tensor P and the heat flux qH given by

P =1

m∗

∫Rd

(p−m∗u)(p−m∗u)Tfmw dp, 2qH =

1

(m∗)2

∫Rd

|p−m∗u|2(p−m∗u)fmw dp . (5.8)

The closure, i.e. P and qH , has to be computed using the closure Wigner function fmw which

in turn is given by the closure operator ρµm via (5.6). Note that the quantum hydrodynamicequations coincide with the classical hydrodynamic system (2.25)-(2.27), except for the formof the pressure tensor P and the heat flux qH . (For the Maxwellian closure (2.23), P = kBnTI

and qH = 0 holds.) The terms < p >coll and < |p|2m∗ >coll on the right hand side of (5.7) denote

the corresponding moments of the collision operator evaluated at fmw and have to be computed

from the closure in terms of n, u, T in the same way as the pressure tensor and the heat flux.They vanish identically in the case when collision operator Qw conserves mass momentum andenergy.

We now turn to the computation of fmw and ρµm . Let λ be a Lagrange multiplier, i.e. a

d+2-tuple of real valued functions of x. Using the interpretation of classical thermodynamics,we can write λ = (µc

T, u

T,− 1

T) where µc(x), u(x), and T (x) will be called respectively local

chemical potential, velocity and temperature. Note that u and T will in general be differentfrom the ensemble velocity and temperature u and T in (5.7) which are defined directly fromthe moments of the Wigner function, in contrast with the classical case for which u = u andT = T holds. Then:

λ(x) · κ(p) =1

T (x)

(µc(x) + u(x) · p− |p|2

2m∗

). (5.9)

19

A straightforward computation leads to the expression of Op(λ · κ):

Op(λ · κ)φ := Op[µc, u, T ]φ

=~2

2m∗∇ · ( 1

T∇φ)− i~

1

2

(∇ ·( uTφ)

+u

T· ∇φ

)+

(µc

T+

1

4

~2

2m∗∆1

T

)φ . (5.10)

It is readily checked that Op[µc, u, T ] is a Hermitian operator.Now, as soon as T (x) ≥ 0 a.e. (which we are going to assume from now on), Op[µc, u, T ]

is an unbounded operator from below and is bounded from above. Thus, assuming that thespectrum of Op[µc, u, T ] consists of a discrete sequence of eigenvalues (a`[µc, u, T ])`=1,...,∞, wehave lim`→∞ a` = −∞. Let us denote by φ` the associated Hilbert basis of eigenfunctions. Bydefinition, the equilibrium operator ρλ = ρµc,u,T = (h′)−1(Op[µc, u, T ]) is given, for any φ ∈ Xby:

ρµc,u,T φ =∞∑

`=1

α`[µc, u, T ] (φ, φ`)X φ` , α`[µc, u, T ] = (h′)−1(a`[µc, u, T ]) . (5.11)

Like in section 2, let us denote by m = (n, q,W ) the vector of moments, with n(x) > 0 theprobability of presence (satisfying

∫n(x) dx = 1), q(x) ∈ Rd the mean momentum per particle,

and W (x) the mean energy per particle (i.e. respectively the density, momentum and energydivided by the total number of particles N ). The equilibrium operator ρn,q,W , subject to theconstraints (3.20) is a solution of the maximization problem (3.18), which, in the present case,is written:

maxµc,u,T

{∞∑

`=1

g(a`[µc, u, T ]) +

∫Rd

(µc

T(x)n(x) +

u

T(x) · q(x)− 1

T(x)W (x)

)dx

}, (5.12)

with g(s) = h◦ (h′)−1(s)−s (h′)−1(s). We note that the first term in the curly bracket of (5.12)is nothing but the Massieu-Planck potential (4.3) and that (5.12) itself is a reformulation ofrelation (4.4) as a minimization problem.

Now, if we specialize to the Boltzmann entropy (2.22), we have (h′)−1(s) = es, g(s) = −es,and so ρµc,u,T = exp{Op[µc, u, T ]} is given by (5.11) with

α`[µc, u, T ] = ea`[µc,u,T ] . (5.13)

Since ρµc,u,T is a statistical operator, it satisfies (3.1) and therefore, we must have

∞∑`=1

ea`[µc,u,T ] = 1 . (5.14)

In particular, this implies that all eigenvalues of Op[µc, u, T ] must be strictly negative or inother words, that −Op[µc, u, T ] must be an elliptic operator. Furthermore, the eigenvaluesmust tend sufficiently fast to −∞ for the series (5.14) to be convergent and have a sum equalto 1. Therefore, we must restrict the set of trial functions of the maximization problem to

20

functions which guarantee such a property to the operator. Now, the optimization problem(5.12) for the Boltzmann entropy (2.22) is written:

maxµc,u,T

{−

∞∑`=1

ea`[µc,u,T ] +

∫Rd

(µc

T(x)n(x) +

u

T(x) · q(x)− 1

T(x)W (x)

)dx

}, (5.15)

We can guess that this problem has some chances to have a unique solution. For instance,if we focus on the extremal value with respect to T (with fixed u/T and µc/T ), we see thata`[µc, u, T ] is an increasing function of T . Therefore, the first term in (5.15) is a decreasingfunction of T while the second term is an increasing one. Furthermore, as T approaches 0,a`[µc, u, T ] → −∞, exp a` → 0 and the first term tends to 0, while the second one tends to −∞(we suppose that W > 0). Conversely, if T →∞, we have a`[µc, u, T ] → 0, exp a` → 1 and thesum diverges, making the first term tend to −∞, while the second one tends to 0. Therefore,the expression to be maximized in (5.15) tends to −∞ at the boundaries of the domain ofvariation of T . Of course, this argument is not rigorous, since T is a function and some carehas to be taken in making sense to the fact that T tends to 0 or to ∞. Also, the influence ofthe other parameters u/T and µc/T has to be studied. A rigorous investigation of this problemis deferred to future work.

In classical mechanics, the system of hydrodynamics equations is hyperbolic, i.e. the matrixof the derivatives of the flux functions with respect to the state variables is diagonalizable withreal eigenvalues. These eigenvalues are the speeds of propagation of the various types of waves.Asking whether system (5.7) is hyperbolic in this sense would be meaningless. Indeed, since theflux functions are not local functions of the state variables, it would be meaningless to computethe matrix of derivatives in this way. Hyperbolicity provides a well-posedness theory, at leastlocally in time [35], [33]. The well-posedness of quantum hydrodynamics systems is an openproblem so far. The fact that an entropy is decreasing in time as the following propositionstates, should be an important ingredient in such a theory.

Proposition 5.1 Let the collision operator Qw in (1.1) dissipate the entropy, i.e. let

Tr{Q(ρ)h′(ρ)} ≤ 0, ∀ρ , (5.16)

hold. Then, any solution (n, u, T ) of system (5.7) satisfies the entropy dissipation relation:

∂

∂tS(n, q,W ) ≤ 0 . (5.17)

Proof: To prove the entropy conservation relation, we go back to the notation m = (n, q,W )and use µm = (µc/T, u/T,−1/T ) for the Lagrange multiplier of the constraint m. We write,thanks to (4.2):

d

dtS(m) =

δSδm

∂m

∂t=

∫Rd

µm∂m

∂tdx .

21

Now, using (5.4) and the cyclicity of the trace, we have

d

dtS(m) = Tr

{− i

~H[ρµm(t) ,Op(µm · κ)

]+ h′(ρµm(t))Q(ρµm(t))

}.

But, by construction, the operators Op(µm · κ) and ρµm(t) commute. Therefore, the right-handside is less than zero and the result follows.

We note that entropy conservation is not restricted to the quantum hydrodynamic model(5.7) but is valid for all quantum moment closure systems (5.4). In clasical hydrodynamics, itis a well established fact that the entropy of smooth solutions is constant in time. However,classical hydrodynamic models have discontinuous solutions (shock waves) the entropy of whichis strictly decreasing with time. It is very unlikely that quantum hydrodynamic models exhibitshock waves solutions. The meaning of the model for discontinuous solutions would even bevery unclear. For instance, what sense should we give to the operator (5.10) if the coefficientsare discontinous functions ?

Another interesting question is how the closure (5.7)-(5.8) relates to the Bohmian singlestate closure analyzed in [16] and [17]. The single state closure corresponds to the case of zerotemperature, when all particles become statistically completely independent, i.e. the Boltzmannor Fermi-Dirac distribution reduces to a δ− function. Since, when using the Boltzmann entropy,we close the moment system essentially by exp(Op(µκ)) the same limit can probably be carriedout by letting T → 0 in (5.7)(a)(b) for finite ~. Carrying out this limit is however not by nomeans simple, since it involves computing the limiting solution of the minimization problem.

5.3 Quantum moment hydrodynamics

With this term, we refer to all quantum moment closure systems (5.4) constructed from a largerbasis of monomials than that of hydrodynamics. Therefore, we consider a finite subset B of Nd

(where N is the set of natural integers) and consider multi-indices β = (β1, . . . , βd) ∈ B. Wesuppose that 0 ∈ B, ei ∈ B for all i ∈ {1, . . . , d} (where ei = (βj)j=1,...,d with βj = δij). Let b bethe cardinal of B and consider the b-tuple of moment monomials κ(p) = (pβ)β∈B. Additionally,we suppose that the space generated by κ(p) contains |p|2. In this way, it contains the hydro-dynamic monomials (1, pi, |p|2) and the associated quantum closure system (5.4) will containthe quantum hydrodynamic system (5.7) as a subcase. The vector κ may contain homogeneouspolynomials which are not monomials, like |p|4. However, for notational simplicity, we shallrestrict to κ(p) constructed from simple monomials pβ and leave this straightforward extensionto the reader.

We can now reproduce the procedure developed in the previous section. We just summarizeit now. First, let us denote by λ(x) = (λβ)β∈B a b-tuple of Lagrange multipliers. Then, theoperator Op(λ · κ) is given by:

Op(λ · κ) =∑β∈B

Op(λβ · pβ) , (5.18)

where the expression of Op(λβ · pβ) is given in Lemma 3.1.

22

From the analysis conducted in section 5.2 (at least with the Boltzmann entropy (2.22)), themaximization problem (3.18) has a non empty set of solutions only if there is a non empty set ofλ such that the operator −Op(λ·κ) is elliptic. In fact, we shall request that the set of such λ hasa non empty interior. This condition is analogous to condition (III) of Levermore’s approach[26] and is likely to lead to identical constraints on the set of monomials to be considered. Thispoint will be investigated in more detail in future work. This requirement leads to a restrictionon the possible sets B.

We now suppose that this requirement is fulfilled and that the operator Op(λ · κ) has asequence of negative eigenvalues (a`[λ])`=1,...,∞ such that a`[λ] → −∞ as ` → ∞ and thatthe associated eigenvectors φ` form a complete orthonormal basis. Let us now denote bym(x) = (mβ)β∈B a given set of moments. The maximization problem (3.18) is now stated asfollows (from now on, we shall restrict to the case of the Boltzmann entropy (2.22)):

maxλ

{−

∞∑`=1

ea`[λ] +

∫Rd

∑β∈B

λβ(x)mβ(x) dx

}. (5.19)

Let us denote by µm the value of λ which solves this maximization problem (assuming that thesolution exists and is unique) and ρm = ρµm , the associated operator

ρµm φ =∞∑

`=1

α`[µm] (φ, φ`)X φ` , α`[µ

m] = ea`[µm] . (5.20)

We shall derive the quantum moment hydrodynamics system in a similar form as for thequantum hydrodynamics system. The derivation of a form like (5.7) shall be done in a futurework. Let us consider the β-th component λβ of an arbitrary λ and denote by bβ`r[λβ] the matrixelement of the operator Op(λβp

β) in the eigenbasis φ` of ρm. As in section 5.2, we denote by

H`r the matrix element of the Hamiltonian in the same basis, and by Q`r(ρ) the matrix elementof Q(ρ). We have:

Tr{[H, ρm

]Op(λβp

β)}

=∑`,r

H`r(αr − α`)bβr`[λβ] . (5.21)

Finally, the system of quantum moment hydrodynamics equations can be written accordingto:

∂

∂t

∫Rd

mβ(x, t)λβ(x) dx = − i

~∑`,r

H`r(αr − α`)bβr`[λβ] +

∑`,r

Q`r(ρ)bβr`[λβ] , (5.22)

∀λβ(x) , ∀β ∈ B , (5.23)

This provides an evolution system for the quantities mβ, which is well adapted to a Galerkindiscretization. For this system also, provided that Q is entropy dissipative, (i.e. property (5.16)is satisfied), the entropy dissipation inequality (5.17) applies and the entropy S(m) is decreasingin time.

We point out that the solution of the maximization problem (5.20) does not always exist inthe classical case (see in particular [24], [25]). It is yet an open problem, and formidably moredifficult, to solve it in the quantum case.

23

5.4 A Galerkin discretization

The main difficulty in solving the quantum hydrodynamic model (5.7) is of course the evaluationof the pressure tensor and the heat flux vector through formulae (5.8). Indeed, this evaluationrequires the solution of the optimization problem (5.12) or (5.15). We now roughly outline thecomputational complexity (and feasibility) of this problem. The moment equations (5.3) renderthemselves to a natural Galerkin discretization in space, which preserves the entropy principle.We start by choosing a set of scalar basis functions λj(x), j = 1, .., J in space and approximatethe moment vector m = (m0, ..,mN) in (5.3) by

mn(x, t) ≈J∑

j=1

zjn(t)λj(x), n = 0, .., N . (5.24)

Equation (5.3) is then replaced by∫Rd

λj(x) ∂tm(x, t)dx = Tr{(− i

~[H, ρm] +Q(ρm))Op(λjκ)}, j = 1, .., J , (5.25)

where ρm is the solution of the constrained minimization problem

Tr(h(ρm)) = min{Tr(h(ρ)) : Tr{ρOp(λj κ)} =

∫Rd

λj mdx, j = 1, .., J} . (5.26)

Thus, we have replaced the minimization problem (3.11) by a problem with finitely manyconstraints leading, consequently to only J(N + 1) Lagrange multiplyers. Next, we replacethe density matrix ρ in (5.26) by a finite expansion. We choose orthonormal basis functionsψk(x), k = 1, .., K, and approximate the density matrix ρ by

ρm(x, y) ≈K∑

k,l=1

Rmklrkl(x, y), rkl(x, y) = ψk(x)ψl(y)

∗ ,

where Rmkl is a Hermitian matrix and the star exponent denotes complex conjugation. In order

to arrive at a finite minimization problem, we also have to replace the operator Op(λjκ) by afinite matrix, i.e.

Op(λj κ) ≈K∑

k,l=1

Γjnkl rkl(x, y), Γjn

kl =

∫Rd

∫Rd

ψk(x)∗ Op(λjκn)(x, y)ψl(y)dxdy ,

and replace the minimization problem (5.26) by

Tr(h(Rm)) = min{Tr(h(R)) : Tr{RΓjn} =

∫Rd

λjmndx, j = 1, .., J, n = 0, .., N} . (5.27)

The symbol Tr in (5.27) denotes now just the usual matrix trace and h is now to be understoodas the function of a matrix. The Galerkin equations (5.25) are then replaced by∫

Rd

λj(x)∂tmn(x, t)dx =K∑

kl=1

Γjnkl Tr{(− i

h[H, ρm] +Q(ρm))rkl}, j = 1, .., J, n = 0, .., N .

(5.28)

24

(5.28) is a system of ordinary differential equations for the expansion coefficients zjn(t) after

inserting (5.24) for the components mn of the moment vector. The system (5.27)-(5.28) nowclearly satisfies the same entropy relation as given by Proposition 5.1 on a discrete level. Thiscan be seen by multiplying (5.28) by µjn and summation over j and n, where µjn is the Lagrangemultiplier of the minimization problem (5.27), i.e. h′(Rm) =

∑jn µjnΓjn holds. Two questions

arise:

• How to choose the basis functions ψk for the density matrices and the basis functions λj

for the moments ?

• How large should K2, the number of density matrix basis functions, be compared to J ,the number of moment basis functions?

There is considerable freedom in the answer to the first question. One possibility would beto choose the ψk as the eigenfunctions of ρm itself, making the matrix Rm diagonal. Thismakes the evaluation of h(R) trivial, but has the disadvantage that we have to compute thematrices Γjn and the matrix corresponding to the Hamiltonian H anew for each time step. It isprobably preferable to choose a ’good’ basis {ψk} (say eigenfunctions of the Hamiltonian, which

are also the eigenfunctions of the equilibrium solution e−βH), compute the Γjn once and for all,and rather deal with the problem of computing the matrix function h(R) in the optimizationprocedure. An alternative would be to use the entropy variables µjn as primary variables, sinceRm = (h′)−1(

∑jn µjnΓjn) has to hold. This has the advantage of eliminating the optimization

procedure but makes the ODE system (5.28) implicit, since the term on the left hand side of(5.28) is then given by ∫

Rd

λjmndx = Tr{Γjn(h′)−1(∑j′n′

µj′n′Γj′n′)} .

The easiest way to invert this equation might again be by using an optimization procedure (i.e.a discrete version of (5.19). As to the second question: The number of degrees of freedom inthe minimization problem (5.27) is K2 (the number of elements in a K ×K hermitian matrixif we count real and imaginary parts separately). Therefore, K2 > J(N + 1) has to hold, inorder to have more variables than constraints. How much larger K2 has to be will depend onhow well we wish to approximate the minimization problem. This will have to be determinedby numerical experiment.

We conclude this section with a remark about the usefulness of Wigner functions in thiscontext. While Wigner functions are convenient for expressing local moments they are not theappropriate tool in this approach. The reason for this is the following. The reason why Wignerfunctions were useful in expressing the moment equations is that the moments of [H, ρ] can beexpressed in terms of the moments of ρ in the Wigner picture, up to a few closure terms. Thisproperty is lost here, due to taking a finite number of expansion terms for the density matrix.If we define by fkl the Wigner transform of the basis element ρkl of the density matrix, weobtain

Γjnkl = Tr{Op(λjκn)ρkl} =

∫Rd

∫Rd

λjκnfkldrdp

25

and (5.28) becomes∫Rd

λj(x)∂tmn(x, t)dx =

∫Rd

∫Rd

K∑kl=1

Γjnkl {−∇x(

1

m∗pfm)+ θ[V ]fm +Qw(fm)}fkldxdp, (5.29)

j = 1, .., J, n = 0, .., N .

Written out in more detail, the right hand side is of the form∫R4d

K∑kl=1

λj(x′)κn(p′)fkl(x

′, p′)fkl(x, p){−∇x(1

m∗pfm) + θ[V ]fm +Qw(fm)}(x, p)dxdpdx′dp′.

If nowK∑

kl=1

fkl(x′, p′)fkl(x, p) = δ(x′ − x)δ(p′ − p)

were to hold, the right hand side of (5.29) would simplify in the same way as it did in Section5.2. For a finite number K of expansion terms this can, however, not be assumed.

5.5 Quantum BGK model

Finally, we end this section about the quantum moment hierarchies by outlining the use whichcan be made of the equilibrium density matrix ρm under the moment constraint m. Indeed, adensity matrix equation including relaxation to local equilibria can be written in the spirit ofthe BGK model of rarefied gas dynamics. Such a model can be written:

∂ρ

∂t= − i

~[H, ρ]− νc(ρ− ρm[ρ(t)]) , (5.30)

where we haved denoted by m[ρ(t)] the moments associated with ρ at time t according to (3.5).Usually, we shall take the hydrodynamic moments. We shall prove in [8] that such an operatoris consistent with entropy decay. The quantity νc is a collision frequency. By construction,composing on the right by Op(λ · κ) where κ are the hydrodynamic monomials shows thatthe relaxation term ρ− ρm[ρ(t)] does not appear in the evolution of the hydrodynamic momentequations (5.3). However, the scaling νc → νc/ε, where ε� 1 is a small parameter representingthe Knudsen number (i.e. the ratio of the collision scale to the macroscopic scale), allowsto justify (at least formally) the hydrodynamic closure (5.4). In most of the literature, onlyrelaxation towards global equilibria are taken into account through equilibria ρm associated withglobal constraints (like in remark 4.1). The present approach allows to give a meaning to ’localequilibrium relaxation’ in a quantum framework. In fact, the term ’local’ is slightly misleadingin this case. It refers to the fact that moment constraints are functions of x. However, therelation between the moment constraints m and the equilibrium density matrix ρm is not alocal one, but a functional one.

A natural question is the existence of a series of perturbative models obtained through anexpansion of the solutions of (5.30) in powers of ε (after rescaling νc → νc/ε), in the spirit of

26

the Chapman-Enskog expansion of the Boltzmann equation of gas dynamics (see [26]). Thisquestion will be investigated in future work.

Another question is related to the possible existence of ’diffusion-like’ limits of model (5.30)(after a simulatneous rescaling νc → νc/ε and ∂ρ/∂t→ ε∂ρ/∂t, with ε � 1). This question isinvestigated in [8].

6 Conclusion and future work

We have developed a systematic approach to construct quantum hydrodynamical models fromappropriate closures of the quantum Liouville equation. This approach is made possible byan adequate definition of the local moment of a density operator. This concept is defined byduality through the observations of the system on a suitable class of test functions. Then,the concept of an equilibrium density matrix, a solution of the entropy minimization principlesubject to the constraints of given moments, is developed. Taking successive moments of thequantum Liouville equation (in the above defined sense), it is possible to close the chain ofequations by this equilibrium operator, giving rise to a hierarchy of models called QuantumMoment Hydrodynamic models. We have studied in more detail the quantum hydrodynamicmodel corresponding to the usual hydrodynamical moments and have shown that it is formallysimilar as the classical one apart from nonlocal expressions of the pressure tensor and heat-fluxvector.

Obviously, this paper leaves a lot of mathematical questions open: existence of solutions forthe entropy minimization problem, well-posedness of the quantum hydrodynamic equations,derivation of efficient numerical solvers, well-posedness of quantum BGK models, existence ofperturbative series expansions of solutions in the spirit of the Chapman-Enskog expansion,derivation of quantum diffusion models (like the quantum drift-diffusion), etc. Other pendingquestions are concerned with the physical restriction of the model. One can think of takinginto account non constant particle number (thus requesting the use of Fock spaces), Fermi-Dirac or Bose-Einstein statistics, spin and relativity effects together with the inclusion of themagnetic field through the Dirac equation, etc. All these questions clearly open a large field ofinvestigations for the future.

27

References

[1] P. Andries, P. Le Tallec, J. P. Perlat and B. Perthame, The Gaussian-BGK model ofBoltzmann equation with small Prandtl number, preprint.

[2] P. Argyres: Quantum kinetic equations for electrons in high electric and phonon fields,Phys. Lett. A 171 North Holland ,1992.

[3] A. Arnold, J. Lopez, P. Markowich, J. Soler: An analysis of quantum Fokker - Planckmodels: A Wigner function approach, preprint ,2002.

[4] R. Balian, From microphysics to macrophysics, Springer, 1982.

[5] J. Barker, D. Ferry: Phys. Rev. Lett. 42 ,1997.

[6] A. Caldeira, A Leggett: A path integral approach to quantum Brownian motion PhysicaA, 121:587-616 , 1983.

[7] P. Degond, Mathematical modelling of microelectronics semiconductor devices, Proceedingsof the Morningside Mathematical Center, Beijing, AMS/IP Studies in Advanced Mathe-matics, AMS Society and International Press, 2000, pp. 77–109.

[8] P. Degond, F. Mehats, C. Ringhofer, On a Quantum Energy-Transport model, in prepara-tion.

[9] P. Degond, C. Ringhofer, A note on quantum moment hydrodynamics and the entropyprinciple, C. R. Acad. Sci. Paris Ser 1 335 (2002), pp. 967–972.

[10] P. Degond, C. Ringhofer, A note on binary quantum collision operators conserving massmomentum and energy, submitted.

[11] D. Ferry, H. Grubin, Modelling of quantum transport in semiconductor devices, Solid StatePhys. 49 (1995), pp.283–448 .

[12] F. Fromlet,P. Markowich,C. Ringhofer: A Wignerfunction Approach to Phonon Scattering,VLSI Design 9 pp.339-350 ,1999.

[13] C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl.Math. 54 (1994), pp. 409–427.

[14] C. Gardner and C. Ringhofer, The smooth quantum potential for the hydrodynamic model,Phys. Rev. E 53 (1996), pp.157–167.

[15] C. Gardner and C. Ringhofer, The Chapman-Enskog Expansion and the Quantum Hydro-dynamic Model for Semiconductor Devices, VLSI Design 10 (2000), pp. 415–435.

[16] I. Gasser and A. Jungel, The quantum hydrodynamic model for semiconductors in thermalequilibrium, Z. Angew. Math. Phys. 48 (1997), pp. 45–59 .

28

[17] I. Gasser and P. A. Markowich, Quantum Hydrodynamics, Wigner Transforms and theClassical Limit, Asympt. Analysis, Vol. 14, No. 2, pp. 97-116,1997.

[18] I. Gasser, P. Markowich and C. Ringhofer, Closure conditions for classical and quantummoment hierarchies in the small temperature limit, Transp. Th. Stat. Phys. 25 (1996), pp.409–423.

[19] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), pp.331–407.

[20] H. Grubin, J. Krekovski, Quantum moment balance equations and resonant tunneling struc-tures, Solid-State Electron. bf 32 (1989), pp. 1071-1075.

[21] B. Hellfer, J. Sjostrand, Equation de Harper, Lect. Notes in Physics 345 (1989), pp. 118–197, Springer.

[22] S. Jin and X. T. Li, Multi-phase Computations of the Semiclassical Limit of the SchrodingerEquation and Related Problems: Whitham vs. Wigner, preprint, University of Wisconsin(http://www.math.wisc.edu/ jin/publications.html).

[23] A. Jungel, P. Pietra, A discretization scheme for a quasi-hydrodynamic semiconductormodel, Math. Models Meth. Appl. Sci. 7 (1997), pp. 935–955.

[24] M. Junk, Domain of definition of Levermore’s five-moment system, J. Stat. Phys. 93(1998), pp. 1143–1167.

[25] M. Junk, Maximum entropy for reduced moment problems, Math. Methods and Models inthe Applied Sciences 10 (2000), pp. 1001–1025.

[26] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996),pp. 1021–1065.

[27] R. Luzzi, untitled, electronic preprint archive, reference arXiv:cond-mat/9909160 v2 11 Sep1999.

[28] V. G. Morozov and G. Ropke, Zubarev’s method of a nonequilibrium statistical operatorand some challenges in the theory of irreversible processes, Condensed Matter Physics 1(1998), pp. 673–686.

[29] I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in NaturalPhilosophy, volume 37, Second edition. 1998.

[30] F. Nier, A variational formulation of Schrodinger-Poisson systems in dimension d ≤ 3,Comm. PDE 18 (1993), pp. 1125–1147.

[31] M. Reed and B. Simon, Methods of modern mathematical physics, vol 1, Functionalanalysis, Academic Press, 1980.

29

[32] J. Schneider, Entropic approximation in kinetic theory, preprint, Analyse non linaire ap-plique et modlisation, Universit de Toulon et du Var, BP 132, 83957 La Garde cedex,France (2001).

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[36] D. N. Zubarev, V. G. Morozov and G. Ropke, Statistical mechanics of nonequilibriumprocesses. Vol 1, basic concepts, kinetic theory, Akademie Verlag, Berlin, 1996.

30

7 Appendix: Quantum entropy minimization principle:

proofs

In this appendix, we give the proofs of the statements related with the quantum entropyminimization principle.

Proof of Lemma 3.5: The Euler-Lagrange equation for the minimization problem (3.16)is (

δLm

δρ

)∣∣∣∣(ρλ,λ)

=

(δH

δρ

)∣∣∣∣ρλ

−(δKλ

δρ

)∣∣∣∣ρλ

= 0 . (7.1)

Thanks to the linearity of Kλ(ρ) with respect to ρ, we have, for any self-adjoint, trace classoperator δρ: (

δKλ

δρ

)∣∣∣∣ρ

δρ = Tr{(δρ)Op(λ · κ)} =∑`,r

δρ`rar` ,

where we denote by a`r the matrix element of the operator Op(λ · κ) in the eigenbasis of ρ.Then, thanks to (3.14), the Euler-Lagrange equation (7.1) is written:

∞∑`=1

(h′(α`(ρ))δρ`` −

∞∑r=1

δρ`rar`

)= 0 ,

for all Hermitian, trace class operators δρ. Choosing δρ`′r′ = δ`′`δr′r + δ`′rδr′` or δρ`′r′ =i(δ`′`δr′r − δ`′rδr′`) where δ`′r′ is the Kronecker symbol, with ` 6= r, we deduce that a`r mustbe equal to zero, i.e. that ρ must be diagonal in the basis where the operator Op(λ · κ) isdiagonal. Such a basis always exist, since Op(λ · κ) is Hermitian (because λ(x) and κ(p) arereal functions) at least in the generalized sense, (i.e. in the sense of the spectral measure ifOp(λ · κ) has continuous spectrum, see [31]). Then, taking δρ`′r′ = δ`′`δr′`, we deduce that

h′(α`(ρ)) = a`` ,

i.e.α`(ρ) = (h′)−1(a``) ,

which is precisely the definition of ρ being given by (3.17).

Proof of Lemma 3.6: The Euler-Lagrange equation of the maximization problem reads(where we drop the superscript m):(

δLm

δρ

)∣∣∣∣(ρµ,µ)

(δρλ

δλ

)∣∣∣∣µ

+

(δLm

δλ

)∣∣∣∣(ρµ,µ)

= 0 . (7.2)

But, by the Euler-Lagrange equation of the minimization problem (7.1), since ρµ realizes theminimum, the first term is identically zero. Therefore, µ is characterized by(

δLm

δλ

)∣∣∣∣µ

= 0 .

31

By linearity of Lm with respect to λ, we have:(δLm

δλ

)∣∣∣∣µ

δλ = −(

Tr{ρµOp(δλ · κ)} −∫

Rd

m(x) · δλ(x) dx

)= 0 ,

for any N-tuple δλ = (δλi(x))i=1,...,N of arbitrary functions δλi(x), which is exactly relation(3.20) (with δλ = λ). It is easy to show that this extremum is indeed a maximum. Thisclassical point is left to the reader.

Proof of Lemma 4.1: (i) We use that H is strictly convex (by virtue of Lemma 3.4.Therefore, let m and m′ be two moment vectors and t ∈ [0, 1]. Then:

S(tm+ (1− t)m′) = H(ρtm+(1−t)m′)

= min {H(ρ) | Tr{ρOp(λ · κ)} =

∫Rd

λ · (tm+ (1− t)m′) dx , ∀λ(x)} .

But we have

Tr{(tρm + (1− t)ρm′)Op(λ · κ)} = tTr{ρmOp(λ · κ)}+ (1− t)Tr{ρm′

Op(λ · κ)}

= t

∫Rd

λ ·mdx+ (1− t)

∫Rd

λ ·m′ dx

=

∫Rd

λ · (tm+ (1− t)m′) dx

Therefore, tρm + (1− t)ρm′satisfies the constraints and consequently:

H(ρtm+(1−t)m′) ≤ H(tρm + (1− t)ρm′

) .

Now, using the strict convexity of H, we have:

H(ρtm+(1−t)m′) < tH(ρm) + (1− t)H(ρm′

) ,

or,S(tm+ (1− t)m′) < tS(m) + (1− t)S(m′) ,

which proves the strict convexity of S, i.e. point (i).(ii) We write:

δSδm

=δ

δm(Lm(ρµm , µm))

=δLm

δm(ρµm , µm) +

(δLm

δρ(ρµm , µm)

δρλ

δλ(µm) +

δLm

δλ(ρµm , µm)

)δµm

δm

=δLm

δm(ρµm , µm) ,

32

where the last equality follows from the fact that µm is a solution of the Euler-Lagrange equation(7.2). Therefore, for any perturbation δm = (δmi(x))i=1,...,N , we have

δSδm

δm =δLm

δm(ρµm , µm)δm =

∫Rd

µm(x) · δm(x) dx ,

which exactly means (4.2) by duality.

Proof of Lemma 4.2: (i) Σ can be defined as

Σ(µ) = minm

{S(m)−∫

Rd

µ ·mdx} ,

and a minimum of linear functions, which are concave functions, is concave.(ii) We have:

δΣ

δµδµ =

δSδm

δm

δµδµ−

∫Rd

δµ ·mdx−∫

Rd

µ · δmδµ

δµ dx .

But, with (4.2), the first term of the right-hand side cancels the last one and we have

δΣ

δµδµ = −

∫Rd

δµ ·mdx ,

which proves (4.4).

8 Appendix: properties of the quantum entropy H

The proofs of lemmas 3.3 and 3.4 can be found e.g. in [30]. Here, we give an elementary proofof these results.

Proof of Lemmas 3.3 and 3.4: Let δρ be a self-adjoint trace-class operator. By the definitionof the Gateaux differentiability, we wish to investigate if the following limit exists:

δH

δρδρ = lim

t→0

[1

t(Tr{h(ρ+ tδρ)} − Tr{h(ρ))}

].

We can choose variations δρ the expressions of which are given by either of the followingformulae in the eigenbasis φ` of ρ :

δρ`′r′ =θ

2(δ`′`δr′r + δ`′rδr′`) , (8.1)

δρ`′r′ =iθ

2(δ`′`δr′r − δ`′rδr′`) , (8.2)

where δ`′r′ is the Kronecker symbol and θ is a real number. Indeed, any variation δρ can bedecomposed into a (possibly infinite but convergent) sum of such variations. The elementary

33

δρ is still trace-class and self adjoint. It is then enough to investigate the influence of theperturbation on the space spanned by (φ`, φr). We investigate the following three cases:

(i) ` = r. Then, α`(ρ+ tδρ) = α`(ρ) + tθ and therefore,

h(α`(ρ+ tδρ)) = h(α`(ρ)) + tθ h′(α`(ρ)) +t2θ2

2h′′(α`(ρ)) + o(t2) ,

as t→ 0. Therefore, in this case,

δH

δρδρ = h′(α`(ρ))δρ`` ,

δ2H

δρ2(δρ, δρ) = h′′(α`(ρ))|δρ``|2 .

(ii) ` 6= r and α` 6= αr. Then, in the basis (φ`, φr), the matrix ρ + tδρ is written as follows(respectively in the cases (8.1) and (8.2)):

ρ+ tδρ =

(α` θt/2θt/2 αr

)or ρ+ tδρ =

(α` iθt/2

−iθt/2 αr

).

In these two cases, the eigenvalues of ρ + tδρ are the same. Suppose that α` > αr to fix theideas. Then, they are given by

α`(t) =1

2

(α` + αr +

√(α` − αr)2 + θ2t2

)= α` +

t2θ2

4(α` − αr)+ o(t2) ,

αr(t) =1

2

(α` + αr −

√(α` − αr)2 + θ2t2

)= αr −

t2θ2

4(α` − αr)+ o(t2) ,

as t→ 0. Thus,

h(α`(t)) + h(αr(t)) = h(α`) + h(αr) +t2θ2

4

h′(α`)− h′(αr)

α` − αr

+ o(t2) .

Therefore, in this case, we have:

δH

δρδρ = 0 ,

δ2H

δρ2(δρ, δρ) =

h′(α`)− h′(αr)

α` − αr

(|δρ`r|2 + |δρr`|2) .

(iii) ` 6= r and α` = αr = α. Then,

α`(t) = α+ |tθ|/2 , αr(t) = α− |tθ|/2 .

It follows that

h(α`(t)) + h(αr(t)) = 2h(α) +t2θ2

4h′′(α) + o(t2) .

Then, we deduce that:

δH

δρδρ = 0 ,

δ2H

δρ2(δρ, δρ) = h′′(α) (|δρ`r|2 + |δρr`|2) .

Collecting the results in the above three cases, leads to formulae (3.14) and (3.15).

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