Derivation of Euler’s Equations of Perfect Fluids from von Neumann’s Equation with Magnetic Field

We give a rigorous derivation of the incompressible 2D Euler equation from the von Neumann equation with an external magnetic field. The convergence is with respect to the modulated energy functional, and implies weak convergence in the sense of measures. This is the semi-classical counterpart of theorem 1.5 in (Han-Kwan and Iacobelli in Proc Am Math Soc 149(7):3045–3061, 2021). Our proof is based on a Gronwall estimate for the modulated energy functional, which in turn heavily relies on a recent functional inequality due to (Serfaty in Duke Math J 169:2887–2935, 2020).


Introduction
General background.In classical mechanics, Newton's second law of motion (also known as the N -body problem) is the following system of 2 where x k (t) ∈ R 2 (or x k (t) ∈ T 2 where T 2 is the 2−dimensional torus) and ξ k (t) ∈ R 2 are called the position and momentum respectively.From a physical perspective, the system (1.1) describes the dynamics of the positions and momenta of N identical point particles of unit mass in R 2 , interacting via an interaction potential V in the presence of fixed, constant, strong magnetic field.The parameter N should be thought of as very large, while the parameter ǫ > 0 as very small.We will be mostly concerned with the magnetic regime, although at times we may refer to the non magnetic regime as well, for which the corresponding dynamics are governed by the system On the other hand the Vlasov-Poisson system (specified for 2D) with the same strong constant magnetic field on [0, T ] × R 2 × R 2 ([0, T ] × T 2 × R 2 ) reads: (1. 3 The unknown is a time dependent probability density function f ǫ (t, x, ξ) on R 2 × R 2 (T 2 × R 2 ) and ρ ǫ is a time dependent probability density function on R 2 (T 2 ) defined by Assuming that V is chosen so that the system (1.1) is well posed, denote by Z t N := (x 1 (t), ξ 1 (t), ..., x N (t), ξ N (t)) the flow of the system (1.1) and set µ ZN (t) := 1 N N i=1 δ Z t N .Unless otherwise stated, we shall focus on the case where the position variable x is in R 2 .Denoting by P(R d × R d ) the space of probability measures on R d × R d , we have the following classical result due to Klimontovich, which enables to construct (ǫ, N )-dependent solutions to the Vlasov-Poisson (1.3) starting from the system (1.1)Proposition 1.1.For each N ≥ 1, ǫ > 0 the map t → µ ZN (t) is continuous on [0, T ] with values in P(R 2 × R 2 ) equipped with the weak topology, and is a distributional solution to equation (1.3).
As customary, we refer to the time dependent probability measure µ ZN (t) as the empirical measure or as Klimontovich solution to equation (1.3) (the latter name is justified thanks to the above proposition).In what follows, we restrict our attention to the case where V is the Green function associated to the negative Laplacian on R 2 (also known as the 2D repulsive Coulomb potential), namely Despite the fact that in this case V obviously fails to satisfy the conditions of the Cauchy-Lipschitz theorem, existence and uniqueness of solution for the system (1.1) is still ensured provided that the positions of the particles are separated initially (i.e.x in k = x in l for k = l).The Vlasov-Poisson system is also closely linked to a celebrated model from fluid dynamics, namely the incompressible 2D Euler equation in vorticity formulation, which reads u = (∇ψ) ⊥ , ω = ∆ψ, ∂ t ω + div(uω) = 0, ( where u : [0, T ] × R 2 → R 2 , ω is a scalar field on [0, T ] × R 2 and v ⊥ := (−v 2 , v 1 ) (if the domain is T 2 , it is obvious how to adjust the formulation) .Formal considerations (see e.g section 6 in [3]) suggest that equation (1.5) (on T 2 ) can be derived from equation (1.3) in the limit as ǫ → 0. In the same work [3] (especially theorem 6.1), this derivation has been made rigorous for sufficiently regular/decaying solutions of equation (1.3).The same problem has been also dealt within [9], where the authors employ compactness methods.Due to the relation between equation (1.3) and the system (1.1) provided by Klimontovich's observation, it is natural to seek a derivation of Euler from Newton's system in the presence\absence of a magnetic field.We stress that due to the singular nature of Klimontovich solutions, it is far from straightforward to extend the convergence result of [3] for Klimontovich solutions of (1.3).A recent striking functional inequality due to [19] (to be discussed in more detail in the next section) has allowed to overcome the difficulty created due to this singular behavior.This inequality (to which we refer from now on as Serfaty's inequality) allowed the same authors to derive the pressureless Euler equation from Newton's system of ODEs in the absence of a magnetic field (system (1.2)) and with monokinetic initial data.The case of non-monokinetic initial data remains a widely open problem.The derivation of the equation (1.5) from Newton's second law in the presence of a magnetic field was established in [10], again through an argument heavily relying on the above mentioned inequality.
Main contribution of the current work.In this work, we focus on the semi-classical universe.Therefore, we introduce the von Neumann equation, which is the quantum analogue of Newton's system of ODEs: The Cauchy problem for the von Neumann equation with a vector potential of the form where H N := H N,ǫ, is the quantum Hamiltonian defined by the formula and V pq is the multiplication operator corresponding to the function V (x p − x q ).As will be clarified in section 6, the operator H N can be viewed as a unbounded self-adjoint operator on L 2 (R 2N ).The Planck constant > 0 should be viewed as a very small parameter, and thus the asymptotics in the semi-classical setting are obtained as a triple limit.For technical reasons we chose to include in the Hamiltonian a quadratic confining potential of the form 1 2 N i=1 |x i | 2 .We elaborate on the reason for this choice in the next section.The unknown R N,ǫ, (t) is a symmetric density operator i.e. a bounded operator on L 2 (R 2N ) such that F N where S N is the symmetric group on N elements and where U σ is the operator defined for each In light of the previous discussion, it is natural to seek a derivation of the incompressible Euler or pressureless Euler equation from the von Neumann equation, in the presence/absence of a magnetic field respectively.In order to compare a solution of the von Neumann equation (which is an operator) with the vorticity solution of the Euler equation (which is a time dependent function on the Euclidean space), one attaches to R N,ǫ, (t) a time dependent probability density called the density of the first marginal of R N,ǫ, (t).The explicit construction of this density is recalled in the next section.Thus, deriving Euler from von Neumann reduces to proving some kind of weak convergence of the density of the first marginal to the vorticity as the parameters involved grow large or become small (according to their physical interpretation).The derivation of the pressureless Euler equation from the von Neumann equation was achieved in [7] in the limit as 1  N + → 0. One of the interesting features of the method of proof of [7] is the observation that Serfaty's inequality (which was originally applied in the context of a classical mean field limit) can be adapted to the semi-classical regime as well.This observation has also been utilized in the recent work [16], which proves a semi-classical combined mean field quasineutral limit, which is a semi-classical version of theorem 1.1 in [10].As already mentioned, the second main result of [10] is a derivation of equation (1.5) from the system (1.1), to which the authors of [10] refer to as a combined mean-field and gyrokinetic limit.Our new contribution-stated precisely in theorem 2.7 of the next section-is a derivation of the incompressible Euler equation (1.5) from the von Neumann equation (1.6), thereby complementing the above-mentioned works [7], [10], [16].Otherwise put, we prove a semi-classical combined mean field and gyrokinetic limit.The recipe for passing from classical to quantum mechanics is summarized neatly in [4]: 1. Functions on phase space are replaced by operators on the Hilbert space of square integrable functions on the underlying configuration space.
2. Integration of functions is replaced by the trace of the corresponding operators.
3. Coordinates q of the configuration space are replaced by multiplication operator q by the variable q, while momentum coordinates p are replaced by the operator p = −i ∇ ( p = −i ∇ + 1 2ǫ x ⊥ in case a magnetic field is included).
As typical in the theory of mean field limits, the argument in [10] rests upon obtaining a Gronwall estimate for a time dependent quantity which is known to control the weak convergence.This quantity is called the modulated energy.Our argument is a modification of the argument leading to a Gronwall estimate on the modulated energy in [10] , according to the above mentioned rules 1-3, which is also the central idea in the semi-classical combined mean field quasineutral limit obtained in [16].Nevertheless, it is important to point out a few points in which the present work differ from [16]: First, we insist on working on the entire plane R 2 rather than the torus T 2 , since the magnetic vector potential in question is non-periodic and so it is apriori not obvious how to even make sense of the modulated energy in the T 2 case.This in turn forces us to include a quadratic confining potential, in order to make sure that the quantum Hamiltonian can be viewed as an essentially self-adjoint operator.Another point which requires some care when working in the plane is the existence and uniqueness theory of solutions to the incompressible 2D Euler-for example, solutions with square summable velocity field cannot have a vorticity with a distinguished sign (section 3.1.3in [14])-and therefore such solutions will be inadequate for the question of interest, in which the vorticity is taken to be a probability density.Finally, the example cooked up in order to witness the initial vanishing of the modulated energy has to be adjusted to these new choices.The utility of each one of the choices we just mentioned will become clearer in the sequel.
The paper is organized as follows: In section 2 a semi-classical version of the modulated energy is introduced along with other preliminaries, and in section 3 a Gronwall estimate is established for this quantity.The calculations which lead to this estimate are more tedious in comparison to the classical setting, partially due to the fact that quantization gives rise to commutators of a differential operator with a multiplication operator-which contributes non zero terms of course.As in [7], [10], [16] this estimate heavily relies on Serfaty's inequality.Section 4 aims to explain how weak convergence is implied from this estimate-which is a simple consequence of a different (yet intimately related) inequality of Serfaty.In section 5 we construct an explicit example witnessing the asymptotic vanishing of the initial modulated energy.Finally, section 6 elaborates on the self-adjointness of the Hamiltonian and related functional analytic material.

Acknowledgement
This paper is part of the author's work towards a PhD.I would like to express my deepest graditute towards my supervisor Franҫois Golse for many fruitful discussions and for carefully reading previous versions of this manuscript and providing insightful comments and improvements.I would also like to thank Daniel Han-Kwan and Mikaela Iacobelli for a helpful correspondence, as well as to Matthew Rosenzweig for suggesting a few valuable references and offering comments which improved clarity of exposition.The author declares no conflict of interest.This work was partially supported by École Polytechnique and by the research grant "Stability analysis for nonlinear partial differential equations across multiscale applications".The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Preliminaries and Main Result
The equation which is to be derived (to which it is customary to refer to as the "target equation") is (1.5).We will work with an equivalent formulation which reads Taking the divergence of both sides of (2.1) gives We elaborate more on the objects related to the von Neumann equation (1.6).To this aim, let us fix some notations from Hilbert space theory, adapting mainly the terminology introduced in [7].Set H := L 2 (R 2 ) and for each N ≥ 1 denote H N := H ⊗N ≃ L 2 (R 2N ).As customary, L(H) stands for the normed space of bounded linear operators on H, L 1 (H) stands for the normed space of trace class operators on H and L 2 (H) stands for the normed space of Hilbert Schmidt operators on H.For each σ ∈ S N (where S N is the group of permutations on {1, ..., N }) we define the operator U σ ∈ L(H N ) by With this notation, we denote by We denote by D(H) the set of density operators on In the sequel we will take V to be as in (1.4).The unknown R N,ǫ, (t) in equation (1.6) is an element of D s (H N ).For brevity we put The Hamiltonian can be decomposed to the kinetic energy K N := K N,ǫ, and the interaction part V N := V N,ǫ which are defined by and The operator K N may be viewed as an unbounded essentially self-adjoint operator in ).This can be seen through the machinery of quadratic forms methods (see section 6 of the present work or section 2.2 in [23] for details).In section 6 we will prove the estimate ). Due to the perturbation theory developed by Kato-Rellich or Kato in [11], [12], [13] this implies that H N is essentially self-adjoint, which by Stone's theorem implies that e −itHN is a unitary operator commuting with H N -a fact which is of utter importance .We stress again that some special care is needed in the case of the 2D Coulomb potential, since originaly Kato proved that self-adjointness of the kinetic energy is invariant under perturbations which belong to L ∞ +L 2 , whereas log(|x|) obviously fails to obey this condition.The external potential 1 2 N i=1 |x i | 2 is responsible for handeling this obstcale: as clarified in section 6, it turns out that this term is helpful while establishing the self-adjointness of H N , since it compensate the divergence of log(|x|) as |x| → ∞ .This is one technical difference in comparison to the work of [10].A second related technical difference is that we confine the discussion to R 2 (rather than T 2 ), since the vector potential x ⊥ is non-periodic.Possibly, a gauge invariance principle can be applied in order to overcome this problem, but this direction remains to be pursued.
It is well known that the generalized solution to equation (1.6) is given by If R N is a density operator then both R N , √ R N are Hilbert-Schmidt operators on H N , and denote their integral kernels by k Hence ρ[R N ] is a probability density on R dN which is moreover symmetric provided R N is symmetric.We will denote by ρ N :k,ǫ, (t, •) the density associated to R N :k,ǫ, .Inspired by [10] we introduce the time dependent quantity (2. 3) The quantity E(t) is a semi-classical rescaled version of the distance introduced in [10] formula 1.10.A few remarks are in order after this definition.First, let us justify that the 2 last integrals on the second line are well defined.Since )) are obviously implied by the assumption on u stated in our main theorem (2.7) below, and explain why the product ).The convolution of the logarithm with a summable function is known to be a function of bounded mean oscillation, or in brief BMO (see 6.3, (i) in [22]).In addition, we recall that a BMO function is in the weighted space L 1 ((1+|x|) −3 ) ([22], 1.1.4).In light of this reminder we see that in order to make sense of the above integrals it suffices to require ).If we require this assumption to hold initially (t = 0), we can ensure it is propagated in time for compactly supported initial data.We elaborate on this last claim in appendix A. As for the first integral in the definition of E(t) as well as E 1 (t), we shall impose the following technical assumption which will be needed in order to justify the fact that both are well defined for all times Assumption (A).
The usefulness of assumption A will become clear while establishing conservation of energy, as stated in the following We remark that the proof of energy conservation in [7] does not consider the presence of a magnetic field.The proof of lemma (2.2) is postponed to section 6.In particular the conservation of energy includes the (non obvious at first sight) statement that both terms on the right hand side of (2.4) are separately well defined for all times t ∈ [0, T ].Assumption (A) also implies in particular which in turn makes the Riesz representation theorem available, thereby allowing to define a notion of a current, which is the quantum analog of the first moment of the solution of Vlasov-Poisson system.Denote by ∨ the anticommutator.
The current of R is the signed measure valued vector field (J 1 N :1 , J 2 N :1 ).
We now state Serfaty's remarkable functional inequality, whose considerable importance while obtaining a Gronwall estimate on the functional E(t) has already been mentioned.For Remark 2.5.After completing this work, Matthew Rosenzweig drew our attention to corollary (4.3) in [20] which provides an improved (and sharp) version of theorem 2.4, as well as to proposition (3.9) in [17] which provides a simpler proof of this improvement.
Originally, Serfaty's inequality was used in order to derive the pressureless Euler system from Newton's second order system of ODEs with monokinetic initial data.This is a classical mechanics mean field limit type result.The following lemma serves as a bridge between theorem (2.4) and the quantum settings which are relevant for us Lemma 2.6.( [7], Lemma 3.5) Set µ(t, •) := ω(t, •) + ǫU(t, •) and Then Our main theorem is ω weakly in the sense of measures.
The assumption E N,ǫ, (0) → 0 is reasonably typical, at least for ǫ, satisfying some appropriate relations, as can be seen e.g. through the example constructed in section (5).
Remark 2.9.The assumptions on (ω, u) can be realized for suitable initial data-see appendix A for more details.
The functional f(X N , µ) (and thus E 2 (t)) is not necessarily a non negative quantity.However, it turns out that it is bounded from below by a term which vanishes asymptotically.Therefore E(t) → 0 implies that the kinetic term and interaction term vanish separately: E 1 (t) → 0, E 2 (t) → 0. We have the following Proposition 2.10.

Gronwall Estimate
We present here a proof of the asymptotic vanishing of the modulated energy, as stated in the first part of theorem (2.7).In the next section we will explain how to topologize this convergence.The proof below should be regarded as a formal proof.A careful justification of the calculation below follows by the eigenfunction expansion method explained the recent work [8] and is left as an exercise, since in our view the formal calculation makes the essence of the argument more visible.In addition, in some places the dependence on the parameters ǫ, is implicit.During the last stage of the estimate we will apply Serfaty's inequality, which is valid provided the function µ is a bounded probability density.This is not necessarily true for µ = ω + ǫU, and for this reason we will need to consider Proof of theorem (2.7).
The second equality is by equation (1.6) and the third equality follows by tracing by parts ( i.e. the identity In addition We have and Hence ).With this notation we find So concluding (with Einstein's summation applied for the indices k, l) Step 2. Calculation of Ė2 (t).
We start by obtaining an evolution equation for the first marginal of ρ N,ǫ, One has V 12 ρ N :2 (t, x, y)dxdy The second term in the right hand side of equation (3.2) is d dt while the third term in the right hand side of equation (3.2) is Gathering equations (3.1) and (3.3) we conclude that (applying Einstein's summation ) Step 3. Rearrangement of terms.By equation (2.1) we have In addition, owing to claim (3.1) and noticing that u In addition, using that ∇(V ⋆ U) = ∇p we get so that Ė(t) is recast as We now apply conservation of energy.
. Proof.We compute the time derivative of the right hand side of equation (2.2).
Since the left hand side of equation (2.2) is constant in time we conclude In view of claim (3.2) Ė(t) writes By equation (3.4) we have Inserting equation (3.5) in the above yields Step 5. Gronwall inequality and conclusion.We estimate each of the J i separately: Estimating J 2 .By lemma (2.6) we can write where R(t, x, y) := U(t, y)ω(t, x) + U(t, x)ω(t, y) + ǫU(t, x)U(t, y).
We observe the following bounds Estimating J 3 , J 4 and J 5 .
5 Typicality of the Assumption E N,ǫ, (0) We explain here how to construct initial data R in N,ǫ, which witnesses the fact that the condition E(0) → 0 is nonempty.It will be convenient to use Toeplitz operators for the construction of R in N,ǫ, .We borrow some definitions and elementary facts from [5] regarding Toeplitz operators.Let z = (q, p) ∈ R d × R d .For each we consider the complex valued function on R d defined by (5.2) In addition, if ν is a probability measure then OP T (ν) is a trace class operator.We also recall the definition of the Wigner and Husimi transforms.
where j (x, y) := (x + 1 2 y, x − 1 2 y) and F 2 is the partial Fourier transform with respect to the second variable.The Husimi transform of scale is the function on R d × R d defined by the formula Remark 5.2.Denoting by G d a the centered Gaussian density on R d with covariance matrix aI we can equivalently write If µ is a finite/positive Borel probability measure on (see formula (51) in [5]).
We gather a few elementary formulas in the following Theorem 5.3.( [5], formulas (46) and ( 48) We will need the following Lemma 5.4.Let ν ∈ P(R 2 × R 2 ) have finite second moments.Then Proof.Calculation of the cross terms.Put and denote by k A (x, y) the kernel of A. It is readily checked that where the last equality is due to the formula for any polynomial g of degree ≤ m.By 3. of theorem (5.3), it follows that Calculation of the dominant terms.We utilize theorem (5.3) with Owing to formula (5.3) we find We have In addition The above inequalities evidently entail ǫtrace(( As for the quadratic term, by formulas ( 52) and (53) in [5] (or equivalently by a similar calculation to the one demonstrated in lemma (5.4)) This takes care of the kinetic part.
Step 2. The interaction part.Note that if ρ N is given as a pure tensor product ρ N = ρ ⊗N then Let ρ ǫ, (x) denote the integral kernel of OP T ((2π ) 2 ν ǫ ) evaluated at the diagonal (x, x).By a straightforward calculation Substep 2.1.J → +ǫ→0 0. We claim that J → +ǫ→0 0. Assuming that ω has compact support we have Clearly the terms in the second line are of order ǫ.The term in the first line is handled as follows.Decompose V into its negative and positive parts V = V − + V + .We have that ) for all 1 ≤ p < ∞, the same is true for the convolution V + ⋆ ω, and so since G  is an approximation to the identity we get || 1 .We split the last integral as Owing to formula (5.3), the first integral is mastered as follows By Plancheral the right hand side of equation (5.7) is where the last inequality is thanks to the elementary inequality Finally, the formula for convolution of Gaussians gives Proof.With the abbreviation which is the first inequality.The second inequality is implied from (6.3) as follows The next lemma shows that the norm associated with K N controls the norm associated with K N up to a constant Lemma 6.6.There is a constant Proof.Let us first consider the case N = 1 (with the abbreviation The second term is estimated from below as follows (applying Einstein's summation for k, l) where we used the simple observation that Π l and Π k commute.In addition, for each l, k the following relation is easily observed where we recall To conclude, equations (6.5), (6.6) and inequality (6.7) entail Integration by parts shows that so that in light of lemma (6.5)  , ǫ) .

N ǫ p<q
V − pq .The combination of theorems (6.3) and (6.4) with lemma (6.6) implies that for each 0 < a < 1 there is some b 2 = b 2 (ǫ, , N ) such that for all ϕ N ∈ C ∞ 0 (R 2N ) it holds that Therefore the combination inequalities (6.12) and (6.The following elementary observation will be used to prove lemma (2.

Definition 5 . 1 .
Let A be an unbounded operator onL 2 (R d ) with integral kernel k A ∈ S ′ (R d × R d ).The Wigner transform of scale of A is the distribution on R d × R d defined by the formula