A PROBLEM OF MOMENT REALIZABILITY IN QUANTUM STATISTICAL PHYSICS

This work is a generalization of the results previously obtained in [17] in a one-dimensional setting: we revisit the problem of the minimization of the quantum free energy (entropy + energy) under local constraints (moments) and prove the existence of minimizers in various configurations. While [17] addressed the 1D case on bounded domains, we treat in the present paper the multi-dimensional case as well as unbounded domains and non-linear interactions as Hartree/Hartree-Fock. Moreover, whereas [17] dealt with the first moment only, namely the charge density, we extend the results to the second moment, the current density.


Introduction
The problem of moment realizability in the quantum framework that we analyze in this paper is an essential ingredient of the recent theory developed by Degond and Ringhofer [9], see also [6,8], on the derivation of quantum hydrodynamics models from first principles. Their approach consists in transposing Levermore's [15] moment closure strategy by entropy minimization to the quantum picture. Roughly speaking, starting from the quantum Liouville equation for a density operator , they obtain an unclosed cascade of equations on moments of that is closed by a minimization of the quantum free energy. In doing so, many different models can be obtained depending on the configuration or the chosen asymptotics: Quantum Drift-Diffusion, Quantum Energy-Transport, or also Quantum Navier-Stokes, see [3,4,5,6,7,8,12,13] for more precisions.
The mathematical justification of this theory based on entropy minimization has yet to be done. The first step towards this goal is the analysis of the quantum moment problem that we started in [17] and pursue in this paper. The classical version of the moment realizability problem with applications to kinetic equations is wellknown: in the case of three moments (density, current and energy), the associated local equilibria are the classical Maxwellian and the obtained hydrodynamic model is the Euler equation ; for higher moments, the question of moment realizability was investigated in [14]. In the quantum setting, physical situations involving minimization of the free energy have already been widely addressed in the literature, particularly for the study of the stability of matter, see for instance [16,10,11] and the references therein. While the latter models involve global constraints, for instance the total number of particles in the system, the moment problem we consider here involves local constraints. In other words, focusing on the first moment only, i.e. the density n(x), we fix the local value of the density at a physical point x rather than the total number of particles. This has several consequences. First of all, the minimization problem, in particular the characterization of the minimizer, becomes considerably more difficult in that the Lagrange parameters associated to the constraints are not constant functions any longer as in the case of global constraints but functions of the position. Devising an appropriate equation for such Lagrange parameters and characterizing their regularity is a delicate task that has found partial answers in a one-dimensional setting only, see [17]. The question of the characterization for the multi-dimensional case is an open problem. The second consequence is that when prescribing local constraints, which are therefore stronger than global constraints, some additional information is added into the minimization problem. As we will see below, this allows us to show that, in some configurations, the free energy admits minimizers under local constraints, while it does not under global constraints (the free energy is not bounded from below in such a case while it is for local constraints, see [11,16]). The problem we have in mind is the minimization of a quantum free energy involving a Von Neumann entropy term (or also called Boltzmann entropy) of the form Tr( log ) for a density operator . The consequence of Theorems 2.1 and 4.3 proved in this article is the proper definition of the quantum Maxwellian used in [3,4,7,12].
The results we present in this paper generalize that of [17] in various aspects: not only we treat multi-dimensional problems, while [17] addresses the one-dimensional case only, but we also extend our previous results to unbounded domains. Besides, the theory of Degond and Ringhofer essentially considers the three first moments, namely the density, the current and the energy. We are able to treat the density and current constraints only and this is a consequence of the compactness method we are using for the proofs. There is enough compactness to tackle the first two constraints, but not enough for the last one, the energy. Moreover, non-linear systems as Hartree or Hartree-Fock systems are also considered. Our results concern the existence (and uniqueness) of minimizers, and not their characterization. As previously mentioned, the analysis of the Lagrange parameters is difficult and so far only a one-dimensional theory is available, see [17].
The paper is structured as follows: in section 2, we introduce the mathematical framework and state our main result in Theorem 2.1. For the sake of clarity of the exposition, we present here the most significant result, leaving the most general cases as extensions. Theorem 2.1 provides existence and uniqueness of minimizers in R d , d ≥ 1, for the quantum free energy with Boltzmann (or Fermi-Dirac) entropy under a local constraint of density. The proof of the theorem is carried out in section 3. The extensions of Theorem 2.1 are presented in section 4: we treat more general entropies, bounded domains, non-linear interactions as Hartree/Hartree-Fock and finally the second order constraint.

Setting of the problem and main result
As described in the introduction, for a given temperature T > 0, we will consider the problem of minimizing a free energy functional defined on density matrices by under the constraint that the density of charge n associated to is a given function n(x). Before stating our main theorem, we successively define the functional framework for density matrices, the energy functional E( ) and the entropy functional S( ).
Let us define the following space of operators on L 2 (R d ), d ∈ N: where J 1 denotes the space of trace-class operators. This space E is a Banach space endowed with the norm The energy space will be the following closed convex subspace of E: Consider now a density matrix ∈ E + , with the spectral decomposition the density of charge n associated to is defined by It can also be characterized by the weak formulation where, in the right-hand side, φ means the operator of multiplication by φ.
The kinetic energy of a density matrix reads and its entropy is defined by where β is either the Boltzmann entropy β( ) = log or the Fermi-Dirac entropy β( ) = log + (1 − ) log(1 − ); we will set More general models will be treated as extensions in Section 4, where non linear energies as well as other entropies are considered.
Let us now discuss our assumptions on the given density n(x) ≥ 0. Since Tr = n (x)dx, in order to deal with density matrices of trace one, we will assume that n(x)dx = 1. Moreover, from the definitions (2.1) and (2.3), and using the Cauchy-Schwarz inequality, one obtains ∇ √ n 2 L 2 ≤ E( ). We will thus also assume that √ n belongs to H 1 (R d ). Nevertheless, these assumptions on n are still not sufficient. Indeed, there exist density matrices of finite energy ∈ E + with entropy S( ) equal to −∞. Hence, without additional assumption on the density n, our constrained minimization problem may be ill-posed. To avoid this problem, it will be sufficient to assume that n log n belongs to L 1 (R d ). Indeed, the following crucial inequality is proved in [10]: This inequality, which can be seen as a logarithmic Sobolev inequality for systems, ensures that S( ) = i∈N * β(λ i ) is bounded from below as soon as n ρ log n belongs to L 1 (note that, as λ → 0, we have β(λ) ∼ λ log λ).
Our main result is the following theorem.
Theorem 2.1. Consider a density n(x) ≥ 0 defined a.e. on R d such that Then the following minimization problem with constraint: Theorem 2.1 is extended in section 4 to more general frameworks: other types of entropies (like C 1 ), bounded domains, non-linear interactions and the current density constraint. Let us point out that the hypothesis that n log n ∈ L 1 (R d ) is crucial for the theorem. Indeed, when the constraint is global, i.e. when only ndx is prescribed, the problem is known to be ill-posed in the sense that the functional does not admit any minimizer since it is not bounded from below [16]. It is the fact that n is prescribed locally that allows us to assume that n log n ∈ L 1 (R d ) and then to bound the free energy from below and prove the existence of minimizers.
The proof essentially relies on compactness arguments. The main difference with the method of [17] is the fact that since the problem is now posed on an unbounded domain, the Laplacian −∆ does not have a compact resolvent anymore. This compactness property of the resolvent was extensively used in [17] to prove for instance the continuity of the entropy term. Here, the absence of compactness is compensated by the fact that we prescribe n log n ∈ L 1 (R d ), and together with the logarithmic Sobolev inequality proved in (2.6) coupled to a Jensen inequality from [2], this allows us to obtain that the entropy is continuous.

Compactness of minimizing sequences
This section is devoted to the proof of our main Theorem 2.1. We denote Step 1: A is not empty. Consider the L 2 projector on √ n defined by We have This proves that the set A is not empty.
Step 2: the free energy F is bounded from below on A. The following inequality is proved in [10] after an optimization of the logarithmic Sobolev inequality (2.6) under a scaling preserving the L 2 norm: for all , we have where (λ i ) i∈N * denotes the nonincreasing sequence of eigenvalues of . Therefore, since by assumption we have n log n ∈ L 1 and n(x)dx = 1, we deduce that, for all ∈ A, we have where the constant C(n) ≥ 0 only depends on n.
Let us bound the second part of the entropy in the case ε = 1 in (2.5): the term Hence, from (3.2) and (3.3), one deduces that for all ∈ A, we have The free energy is thus bounded from below on A.

From
Step 1 and Step 2, the infimum of F on A is well-defined and is not −∞.
From now on, we consider a minimizing sequence ( k ) k∈N , i.e. a sequence satisfying Step 3: uniform bound and first convergence result. Let us prove that the minimizing sequence ( k ) k∈N is bounded in E. Since k ∈ A, we already have Moreover, from (3.5), we have Hence, the inequality (3.4) yields We have then sup Since ( k ) k∈N is a bounded sequence of E, and following for instance the arguments of [17], there exists ∈ E + such that, up to an extraction of a subsequence, k and (1 − ∆) 1/2 k (1 − ∆) 1/2 converge in the J 1 weak- * topology respectively to and This means that, for all compact operator K on L 2 (R d ) we have Step 4: satisfies the constraint. Let us prove the convergence of k in the weak J 1 topology, i.e. that for all bounded operator σ ∈ L(L 2 (R d )), To show that no loss of mass occurs at the infinity, we will use in a crucial way the fact that the density of k is a fixed L 1 function n(x). Let us introduce a truncation function χ with values in [0, 1], such that χ ≡ 1 on the centered ball of radius 1 and χ ≡ 0 outside the centered ball or radius 2. We denote χ R (x) = χ(x/R). Identifying the function χ R and the operator of multiplication by χ R , we write From Sobolev embeddings on compact domains, one deduces that, for all R > 0, the Moreover, the operators (1 − ∆) −1/2 and σ are bounded. Hence, by composition, the operator Consider now the last term in (3.9) and let us show that no mass can be lost at the infinity. Denote by σ(x, y) the integral kernel of σ and by (λ k,i , ψ k,i ) i∈N * the spectral elements of k . Notice that σ ∈ L 2 (R 3 × R 3 ). By using Cauchy-Schwarz inequalities, we get where we used that k ∈ A, i.e. that Tr( k ) = 1 and that n = n. From this last estimate and by dominated convergence, since n belongs to L 1 , one deduces that Finally, (3.9), (3.10) and (3.11) yield (3.8). This implies in particular that n = n.
To see this fact, use the characterization (2.2) of n and choose σ as the multiplication operator by the function φ in (3.8). This means that belongs to A.
Step 5: strong convergence of k . From the previous step, we know that k converges to weakly in J 1 , which implies the weak operator convergence. Moreover, since these operators are positive, we have the convergence of the norms: k J 1 = Tr( k ) = 1 = Tr( ) = J 1 . Hence, the following lemma from [19] shows that the convergence holds in the strong J 1 topology: lim Lemma 3.1 (Theorem 2.21 and addendum H of [19]). Suppose that A k → A weakly in the sense of operators and that A k We will now prove the convergence of the entropy: Tr(β( )) = lim k→+∞ Tr(β( k )). (3.13) Note that this result cannot be simply deduced by weak convergence and semicontinuity, since β is negative. Let us decompose the entropy into the sum of a singular and a regular (near 0) part: From the J 1 convergence of k , it is easy to prove the convergence of the regular part: Tr(β r ( )) = lim k→+∞ Tr(β r ( k )), (3.14) by combining two facts. First, the convergence of k to in the J 1 norm implies the convergence of the eigenvalues, see Lemma A.2 in [17]: if we denote by (λ k,i , ψ k,i ) i∈N * the nonincreasing sequence of eigenvalues and the associated eigenfunctions of k , and by (λ i , ψ i ) i∈N * the (nonincreasing) eigenvalues and eigenfunctions of , we have Since the function β r is continuous, this implies that Step 6: convergence of the entropy, part 1. In the next two steps, we prove the convergence of the singular part: where we recall that β s (λ) = λ log λ − λ. We shall use a truncation method inspired from [11]. Let us introduce two truncation functions χ and ξ with values in [0, 1], such that χ 2 + ξ 2 = 1, χ ≡ 1 on the centered ball of radius 1 and χ ≡ 0 outside the centered ball or radius 2. We denote χ R (x) = χ(x/R) and ξ R (x) = ξ(x/R) for R ≥ 1. We will use the following "Jensen inequality for traces", taken from [2]: ). Let β be a continuous and convex function defined on [0, 1] with β(0) = 0. Let ∈ E + and let X be a self-adjoint operator on L 2 (R d ) such that X 2 ≤ 1. Then we have Tr(β(X X)) ≤ Tr(Xβ( )X).

Applying this lemma yields
Tr(β s (χ R k χ R )) + Tr(β s (ξ R k ξ R )) ≤ Tr(β s ( k )). (3.20) We will pass to the limit separately in the two terms of the left-hand side. For clarity, we divide the proof of (3.13) into two steps. In this step, we treat the term Tr(β s (χ R k χ R )), R being fixed. In Step 7 we treat the other term Tr(β s (ξ R k ξ R )) and we conclude.
(3.21) and denote respectively by ( λ k,i ) i∈N * and ( λ i ) i∈N * the nonincreasing sequences of eigenvalues of k and . Since the operator of multiplication by χ R is bounded on L 2 (R d ), the strong J 1 convergence (3.12) proved in Step 5 implies that As seen above, the convergence in J 1 implies the convergence of eigenvalues, thus Hence, for all η > 0, we have as k → +∞, both sums being finite. We now claim that, for all R, Tr(β s (χ R k χ R )) = Tr(β s (χ R χ R )).

(3.26)
Let us now prove the claim (3.25). We first remark that where C is independent of k and R. Similarly, denoting by (λ i , ψ i ) i∈N * the eigenvalues and eigenfunctions of , we have Moreover, we remark that and similarly for , which yields (3.29). From (3.28) and (3.29), one deduces that ∈ E + and that n e log n e belongs to L 1 . By the logarithmic Sobolev inequality (2.6), this implies that |Tr β s ( ))| < ∞: the second part of the claim (3.25) is proved.
Let us now prove the first part of this claim, by comparing the spectrum of the operator k with the one of the harmonic oscillator H ho = −∆ + |x| 2 . Recall that the i-th eigenvalue µ i of H ho (counted with multiplicities) satisfies µ i ∼ Ci 1/d . We will use the following classical lemma proved e.g. in [17]: ∈ E + and denote by (λ i ) i≥1 the nonincreasing sequence of nonzero eigenvalues of . Let (µ i ) i≥1 be the nondecreasing sequence of eigenvalues of the quantum harmonic oscillator H ho . Then we have where ( λ k,i ) i∈N * denote the eigenvalues of k and C(R) is a constant depending on R but not on k. Let us now introduce the constant where we used that β s (λ) ∼ λ log λ near 0. We estimate: where we used a Hölder inequality. Since µ i ∼ Ci 1/d , the series i µ −2d i converges. By (3.32), this gives and the claim (3.25) is proved.
Step 7: convergence of the entropy, part 2. We now consider the second term Tr(β s (ξ R k ξ R )) in (3.20). Let k = ξ R k ξ R and = ξ R ξ R . Similarly as (3.27) and (3.28), we have and, similarly as (3.29), one has (for the right inequality, recall simply that the eigenvalues of k and belong to [0, 1]). Thus, since the left-hand side is independent of k, one deduces from n ∈ L 1 , n log n ∈ L 1 , from the definition of ξ R and from dominated convergence that lim R→+∞ sup k∈N * Tr(β s (ξ R k ξ R )) = 0 and lim R→+∞ Tr(β s (ξ R ξ R )) = 0. Let ∈ E + and let X be a self-adjoint operator on L 2 (R d ) such that X 2 ≤ 1. Then we have We use this lemma with the functions f (λ) := β s (λ) = λ log λ − λ, with the density matrices k ∈ E + or ∈ E + and with X = χ R . Recalling (3.20) (and the similar inequality for ), one gets and Tr(β s ( )) ≤ Tr(β s (χ R χ R )) ≤ Tr(β s ( )) − Tr(β s (ξ R ξ R )).
Step 8: conclusion. From (3.5), (3.7), (3.13) and Tr( k ) → Tr( ), one deduces that F ( ) ≤ inf σ∈A F (σ). Since we have proved in Step 4 that ∈ A, this shows that the infimum is realized: To conclude the proof of the theorem, it remains to remark that the strict convexity of the function β implies that → Tr(β( )) is strictly convex (see e.g. [17], Lemma 3.3). Hence the function F is also strictly convex and the minimizer is unique. The proof of Theorem 2.1 is complete.

Extensions
In this section, we give various extensions to our Theorem 2.1.

Other entropies.
We have chosen to work with the more interesting physical cases, the Boltzmann entropy or the Fermi-Dirac entropy, but one can deal with other entropies. If, instead of (2.5), we choose β as a strictly convex function, of class C 1 on [0, 1] and satisfying β(0) = 0, then one can prove that the minimization problem (2.8), with F , E and S defined by (2.9), (2.3) and (2.4), admits a unique minimizer under the following assumption on n: Note that we do not need here to assume that n log n ∈ L 1 . This case is in fact more regular than the one treated in Theorem 2.1. Indeed, the entropy is now continuous on the energy space, which was not true for β given by (2.5). In Step 5 of Section 3, we have in fact proved the following result: in the case of Neumann boundary conditions. Again, no assumption is required on the function n log n. The reason for it is that one has the following lemma: Lemma 4.2. Let β be given by (2.5), then the functional → S( ) = Tr(β( )) is continuous on E + (Ω).
This lemma is proved in the case of the dimension d = 1 in [17], but this proof can easily be extended, by an argument similar as the one that we used here in Step 6. The crucial point is that, for a density matrix in the energy space, one has (see Lemma 3.3): where (λ i ) i∈N * is the nonincreasing sequence of eigenvalues of and (µ i ) i∈N * is the nondecreasing sequence of eigenvalues of H, which satisfies the Weyl asymptotics µ i ≤ Ci 2/d .

4.3.
Other interaction terms and non linear energies. Instead of using the simple kinetic energy (2.3), one can take into account some additional terms in the energy of the density matrices, modeling interactions. In dimension d = 3 (for simplicity), consider the following energy for a density matrix, composed of four terms: The first term in (4.1) is E( ), the kinetic energy of the particles. The second term is the potential energy in a given external potential V (x). We assume that V ∈ L 3/2 (R 3 ) + L ∞ (R 3 ), for instance V (x) = m j=1 q j |x − x j | −1 models the interaction with m fixed ions. The third and the fourth terms model some non linear interactions between particles. In order to take into account the most physical cases, we consider the Hartree energy |x − y| dxdy, and the Hartree-Fock exchange energy where α and β < 0 are real-valued parameters and where (x, y) denotes the integral kernel of the operator . Let us first make a simple remark. The linear term V n dx and the Hartree term W H depend on only through its density n : since this density is prescribed in our problem (2.8), these terms will be constant ! Hence, we only have to check that they are well-defined under our assumptions. It is immediate for the case of Hartree interaction only, namely when β = 0. Indeed, the fact that √ n ∈ H 1 (R 3 ) implies by standard Sobolev embeddings that n ∈ L 1 (R 3 ) ∩ L 3 (R 3 ) ⊂ L 6 5 (R 3 ) and the Hardy-Littlewood-Sobolev inequality [18] yields .
, it is also clear that V n dx is finite from the previous regularity of n .
When the Hartree-Fock exchange term is included, we set α = −β = 1, without loss of generality. The result of Theorem 2.1 can be extended to such a case provided two facts are satisfied: W HF must be well-defined and the non-linear term W H + W HF must be lower semi-continuous. The first item follows from the Cauchy-Schwarz inequality and the simple observation that, almost everywhere on R 3 × R 3 , ( (x, y)) 2 ≤ n (x)n (y).
Recall indeed that (x, y) = i∈N * λ i ψ i (x)ψ i (y), where (λ i , ψ i ) i∈N * are the spectral elements of . This implies that |W HF | is controlled by W H which is finite. The second item is proved in [11] and uses the Fatou lemma with the fact that W H +W HF is non-negative. 4.4. Constraint on the current density. As already mentioned in the introduction, the theory of Degond and Ringhofer involves constraints on higher order moments of the density operator in addition to the density. These moments of interest are the current density and the energy density. We explain below how Theorem 2.1 can be extended to both the charge density and the current density constraints. Because of a lack of compactness, we do not tackle the energy constraint yet.
Denoting by (λ i , ψ i ) i∈N * the spectral elements of , the current density associated to is defined by This can be recast in a weak formulation as We make the following assumption: Assumption A. The functions n(x) and j(x) are given such that there exists a density operator 0 ∈ E + satisfying Tr( 0 ) = 1, n 0 = n and j 0 = j.
We already know (see Section 2) that Assumption A implies that √ n ∈ H 1 (R d ). Moreover, as consequences of Lieb-Thirring inequalities, see [1], Assumption A implies that the current density j belongs to (L q (R d )) d , with Assumption A is verified for instance if there exists u (regular enough) whose curl vanishes and such that j = nu. Indeed, since ∇ × u = 0, there exists S such that u = ∇S. Defining then Ψ = √ ne iS , a possible choice for 0 [n, j] is given by 0 [n, j] = |Ψ Ψ| . In such a context, Theorem 2.1 becomes: Theorem 4.3. Consider a charge density n(x) and a current density j(x) that verify Assumption A and such that n log n ∈ L 1 (R d ). Then the following minimization problem with constraint: min F ( ) for ∈ E + such that n = n, and j = j, where F , E and S are defined by (2.9), (2.3), (2.4), (2.5), is attained for a unique density operator.
The proof of Theorem 2.1 can easily be modified so as to include the current constraint, one only needs to verify two facts: first, that the space of admissible density operators A = { ∈ E + such that n = n, j = j} is not empty. This is a direct consequence of Assumption A. Second, that the limit of the minimizing sequence verifies the current constraint. To see this, consider a minimizing sequence ( k ) k as in Step 3 of the proof. We know from Steps 3 and 4 that k converges strongly to in J 1 , that (1 − ∆) 1/2 k (1 − ∆) 1/2 converges in the J 1 weak- * topology to (1 − ∆) 1/2 (1 − ∆) 1/2 as k → +∞ and that n = n. We have to show that j = j. For this, for all ψ ∈ (W 1,∞ (Ω)) d , denoting also by ψ the (component by component) multiplication operator by ψ, the weak formulation of the constraint reads R d j · ψ dx = −i Tr k ψ · ∇ + 1 2 ∇ · ψ .
This implies that Let us now denote by (λ k,i , ψ k,i ) i∈N * the spectral elements of k . Then since k ∈ A so that n k = n. The Lebesgue dominated convergence theorem then implies that lim R→∞ sup k∈N * |Tr ( k (ψ · ∇) (1 − χ R ))| = 0. Gathering (4.4), (4.5), (4.6) and (4.7) finally yields, when k → ∞, This means that j = j and therefore that ∈ A. The rest of the proof of Theorem 4.3 is identical to that of Theorem 2.1.