Combined mean-field and semiclassical limits of large fermionic systems

We study the time dependent Schr\"odinger equation for large spinless fermions with the semiclassical scale $\hbar = N^{-1/3}$ in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schr\"odinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.


Introduction
In this paper, we aim to study the combined mean-field and semiclassical limit of N -fermions from timedependent Schrödinger equation to Vlasov equation. The following anti-symmetric subspace of L 2 (R 3N ) is considered for fermions, L 2 a (R 3N ) := Ψ ∈ L 2 (R 3N ) : Ψ(q π(1) , . . . , q π(N ) ) = ε(π)Ψ(q 1 , . . . , q N ) . It is known that a system of fermions initially confined in a volume of order one have kinetic energy of order N 5/3 due to the Pauli principle. Therefore, to balance the order, the scale of the interaction term should be of order N −1/3 , we refer to [6,8] for more details about this scaling. After a time rescaling of N 1/3 the Schödinger equation for N -fermions is written into By denoting the semiclassical scale = N −1/3 and multiplying both sides by 2 , one can recover the N −1 , the coupling constant for the mean field interaction. Hence one arrives at the following many body Schrödinger equation 1) where Ψ N,t ∈ L 2 a (R 3N ), Ψ N is the initial data in L 2 a (R 3N ), and V is the interacting potential. The limit from many body Schrödinger equation to the Vlasov equation has been studied extensively in the literature. Narnhofer and Sewell [34] and Spohn [46] are the first to prove this limit with the potential V assumed to be analytic and C 2 respectively.
For large N , in the mean field limit regime, the solution of many body fermionic Schrödinger equation can be approximated by the solution of the following nonlinear Hartree-Fock equation, where ω N,t is the one-particle density matrix, ̺ t (q) = N −1 ω N,t (q; q) and X N,t is a small term having the kernel X t (x, y) = N −1 V (x − y)ω N,t (x; y). In [16], for the initial data being a Slater determinant, the approximation has been proved for short time for analytic interaction potential by using BBGKY hierarchy, while [6] proved the approximation with convergence rate for arbitrary time and weakened potential in the framework of second quantization. Similar results have been extended for mixed states in [4] and for relativistic case in [7]. Recently, with the help of Fefferman-de la Llave decomposition [18,26], weaker assumptions on the interaction potential have been considered. Specifically, Coulomb potential has been considered in [38], inverse power law in [41]. Further relevant literature on the fermionic case for the meanfield limit problem of Schrödinger equation can be found in [3,20,35,36,37].
In parallel, the mean field limit for the bosonic case from many body Schrödinger system to nonlinear Hartree equation was proved in [17] for Coulomb potential. Also for Coulomb potential, the convergence with rate N 1/2 has been obtained in [40]. Later, it has been optimized to the optimal convergence rate N −1 in [11], and furthermore for stronger singular potentials in [10].
The semiclassical limit from Hartree-Fock equation to Vlasov equation has been obtained in the literature by using Wigner-Weyl transformation of the one-particle density matrix ω N,t defined by W N,t (q, p) = 2π 3ˆd y e −ip·y ω N,t x + 2 y; x − 2 y , (1. 2) which has been intensively studied in the semiclassical limit of quantum mechanics by Lions and Paul in [31]. In [5] the authors compared the inverse Wigner transform of the Vlasov solution and the solution of Hartree-Fock and get the convergence rate in the trace norm as well as Hilbert-Schmidt norm with the regular assumptions on the initial data. The works in this direction have also been extended for inverse power law potential [43], convergence rate in Schatten norm in [30], and Coulomb potential and mixed states in [42]. The convergence of relativistic Hartree dynamic to relativistic Vlasov equation has also been considered in [14]. Further convergence results from Hartree to Vlasov can be found in [1,2,21,33]. It is known that Wigner transform (1.2) is not a true probability density as it may be negative in certain phase-space. In fact, [27,32,45] concludes that the Wigner measure is non-negative if and only if the pure quantum states are Gaussian, whilst [9] state that the Wigner measure is non-negative if the state is a convex combination of coherent states. Nevertheless, it has been shown that if one convolutes the Wigner measure with a Gaussian function in phase-space, it will yield a non-negative probability measure known as Husimi measure [19,39,48]. In fact, from [19, p.21], the Husimi measure is given by where 1 k N , G = (π ) −3k exp − −1 ( k j=1 |q j | 2 + |p j | 2 ) and W (k) N,t is the Wigner transform of k-particle density matrix.
In the recent development, the convergence to Vlasov equation in the semiclassical Wasserstein pseudodistance has been proved in [23,24,25,28,29]. The semiclassical Wasserstein pseudo-distance is computed between the Husimi measure and Vlasov solution.  Figure 1: Relations of N -fermionic Schrödinger systems to other mean-field equations [22,23].
One can also show the combined limit by first taking the semiclassical limit and then the mean field limit from many particle Schrödinger to Vlasov via the Liouville equations, and the corresponding BBGKY hierarchy 1 . This has been done in [23].
Our goal, therefore, is to obtain the Vlasov equation from Schrödinger equation directly, as shown in the diagonal line of Figure 1, by taking N → ∞ and → 0 simultaneously. In order to do this, it is convenient for us to introduce the second quantization framework in our study of the quantum many-body systems. In particular, we utilize the notations in [6,8,11] where the fermionic Fock space is defined as where we denote (dx) ⊗n = dx 1 · · · dx n . The creation and annihilation operator in terms of their respective distributive forms, Due to the canonical anti-commutator relation (CAR) in the fermionic regime, we have that for all where {A, B} = AB + BA is the anti-commutator. In particular, the CAR for operator kernels hold as follow This CAR in distributive form will be frequently used in our computations. As in [6], we may write the corresponding Hamiltonian in terms of the operator valued distribution in F a by Therefore, we rewrite the Schrödinger equation in Fock space as follows, The solution to the above Cauchy problem is ψ N,t = e − i Ht ψ N , with a given initial data ψ N .
1 See Figure 1 Remark 1.1. It should be noted the states ψ N,t in our analysis stays in the N th-sector of F a due to the definition of Husimi measure which will be given later. Therefore, denoting F (n) a to be the n-th sector in F a , we say that ψ N,t ∈ F (N ) a for all t 0.
Furthermore, we use the definition of the number and kinetic energy operators as follows, respectively. We further explore the properties of the operators in (1.9) in section 2.2. Next, we shall introduce the Husimi measure. In fact, our notation follows closely with the notations in Fournais, Lewin and Solovej [19] where it deals with large fermionic particles in stationary case. The main tool in their analysis is the use of coherent state, a subtle tool that proves extremely useful in our work as well.
For any real-valued normalized function f , the coherent state is given by, 2 Similar to [12] and [19], the k-particle Husimi measure is defined as, for any 1 k N m (k) is the N -fermionic states, a(f q,p ) and a * (f q,p ) are the annihilation and creation operators respectively. Husimi measure defined in (1.11) measures how many particles, in particularly fermions, are in the k semiclassical boxes with length scaled of √ centered in its respectively phase-space pair, (q 1 , p 1 ), . . . , (q k , p k ).
In the context of this paper, we use m where the tensor products indicate Note that the function f here is a very well localized function in practice [19], therefore we may take the following assumption Assumption A1. The real-valued function f ∈ L 2 ∩W 1,∞ (R 3 ) satisfies f 2 = 1, and has compact support.
Additionally, we assume that the interaction potential to satisfy Assumption A2. V is a real-valued function such that V (−x) = V (x) and V ∈ W 2,∞ (R 3 ). 2 The function f can be any real-valued function. [19] For this paper, we set f to be compactly supported. See Assumption A1.
As is well known that in the mean field semiclassical regime, the dynamic of (1.1) can be approximated by a one particle Vlasov equation. Namely, for all q, p ∈ R 3 ∂ t m t (q, p) + p · ∇ q m t (q, p) = ∇ V * ρ t (q) · ∇ p m t (q, p), (1.13) with initial data m 0 (q, p), where m t (q, p) is the time dependent one particle probability density function, and ρ t (q) =´m t (q, p)dp. Although (1.13) is a non-linear equation, such equation would be more suitable to analyze than the increasingly large systems of Schrödinger equation. The well-posedness of the above Vlasov problem is given by Drobrushin [15] for smooth V . Now, we are ready to state the our main results.  N , the 1-particle Husimi measure of the initial data ψ N , satisfies¨d (1.14) Then, for all t 0, the k-particle Husimi measure at time t, m N,t has a weakly convergent subsequence which converges to m is a weak solution of the following infinite hierarchy in the sense of distribution, i.e. it satisfies for all k 1 that By using [47,Theorem 7.12], we have the following corollary, for t 0.
Remark 1.2. In the pioneering work by Spohn [46], he considered with p j = −i∇ j and obtained the following Vlasov hierarchy, ∂ ∂t r (N ) n (ξ 1 , η 1 , . . . , ξ n , η n , t) = n j=1 η j ∂ ∂ξ j r (N ) n (ξ 1 , η 1 , . . . , ξ n , η n , t) n+1 (ξ 1 , η 1 , . . . , ξ j , η j + k, . . . ξ n , η n , 0, −k, t), which is slightly different from Vlasov hierarchy for Husimi measure given in (1.15), or the version in (2.3) before taking the limit. The benefit of the hierarchy in (2.3) is that one observes directly the mean field and semiclassical structure in the remainder terms. The explicit formulation is helpful in getting estimates for the remainder terms in (2.3). Moreover if one can handle singular potentials (or even the Coulomb potential) for both terms separately, one expects that this new approach can be applied to obtain the limit from many body Schrördinger to Vlasov with singular potentials in the future. Since the mean field limit with singular potential has been studied with convergence rate, for example in [8], then we can utilize similar ideas to handle one of the remainder term which includes the mean field structure. In parallel, we can apply the techniques in semiclassical limit, for example in [43], to get estimates for the other remainder term.
Remark 1.3. Although the results in this article does not yield a convergent rate, the main purpose of this article is to present an alternative approach and framework, namely to rewrite the Schrödinger equation into a BBGKY type of hierarchy, and to derive estimates for the remainder terms that appear in the new hierarchy.
Remark 1.4. In Corollary 1.1, the convergence is stated in terms of 1-Wasserstein distance. For completeness, we give its definition as defined in [47] where µ and ν are probability measures and Π(µ, ν) the set of all probability measures with marginals µ and ν. The Wasserstein distance, also known as Monge-Kantorovich distance, is a distance on the set of probability measures. In fact, if we interpret the metric in L p space as the distance that measures two densities "vertically", the Wasserstein distance measures the distance between two densities "horizontally" [44].
Remark 1.5. The assumptions for initial data (1.14) and (1.16) can be realized by choosing ψ N to be the Slater-determinant. That is, for all orthonormal basis {ϕ j } ∞ j=1 , the initial data is given as Remark 1.6. Assumptions A1 and A2 are expected to be weakened to the situation that f ∈ H 1 (R 3 ), |x|f (x) ∈ L 2 (R 3 ), and V to be Coulomb potential. These will be our future projects.  N,t . In this direction, we expect to derive the rate of convergence in an appropriate distance between the Husimi measure and the solution of the Vlasov equation.
The arrangement of the paper is the following. In section 2, we give the main strategy of the proof. Followed by the reformulation of Schrödinger equation into a hierarchy of the Husimi measure, a sequence of necessary estimates on number operators, the localized number operators, and the kinetic energy operator are given, which will be contributed to do compactness argument for the Husimi measure. We leave the computation of the hierarchy to section 3.1. Furthermore, the uniform estimates for remainder terms in the hierarchy, which is another main contribution of this article, are provided in section 3.2.
2 Proof strategy through BBGKY type hierarchy for Husimi measure We first start from the many particle Schrödinger equation and derive an approximated hierarchy of time dependent Husimi measure by direct computation. Compare to the BBGKY hierarchy of Liouville equation in the classical sense, it has two families of remainder terms, which are determined by the N particle wave function from Schrödinger equation. In order to take a convergent subsequence of the k-particle Husimi measure, we derive the uniform estimates for number operator and the kinetic energy. Together with an additional estimate for localized number operator, we can show that the remainder terms are of order 1 2 −δ , for arbitrary small δ. Then the desired result will be obtained by the uniqueness of solution to the infinite hierarchy.

Reformulation: Hierarchy of time dependent Husimi measure
In this subsection, we begin by examining the dynamics of k-particle Husimi measure by using the N -body fermionic Schrödinger. The proofs of the following propositions are provided in section 3.1.
where the remainder terms R 1 and R 1 , are given by . Under the assumption in Proposition 2.1, then for 1 < k N , we have the following hierarchy where the remainder terms are denoted as

A priori estimates
In the next steps, we derive estimates in order to have compactness of each k-particle Husimi measure, as well as to prove that the remainder terms converge to zero in the sense of distribution. The estimates are derived directly from the solutions of the N -fermionic Schrödinger equation.

Properties of coherent states and Husimi measure
Here we give the properties of coherent states and Husimi measure provided in [19], which will be frequently needed in our computation. Firstly, we observe that the coherent state has a projection property, that is Lemma 2.1 (Projection of the coherent state, [19]). For every real-valued function f satisfying f 2 = 1 and the coherent states f q,p defined as in (1.10), we have that Secondly, the properties of the k-particle Husimi measure m (k) N is given as follows Lemma 2.2 (Properties of k-particle Husimi measure, [19]). Suppose for ψ N ∈ F (N ) a is normalized. Then, the following properties hold true for m (k) N : where 1 k N .
Remark 2.1. Note that as ψ N = ψ N,t , Lemma 2.2 is also valid if we replaced the stationary wavefunction ψ N , to a time-dependent ψ N,t , for t 0. Moreover, it can be obtained that for any fixed positive Following [19], we define the -weighted Fourier transformation as follows, Definition 2.1 ( -weighted Fourier transform). Let F be any real-valued function in L 2 (R 3 ). We define the -weighted Fourier transform of f to be, and its inverse transform by F −1 .
From the Definition 2.1, we have the following identity, for any G, F ∈ L 2 (R 3 ). In other words, the Dirac-delta distribution is given by

Number operator and localized number operator
In this part, we give the bounds of number operators and its corresponding localized version, both of which are used extensively in estimating the remainder terms in (2.1) and (2.3).
be the solution to Schrödinger equation in (1.1) with initial data ψ N = 1, the number operator N defined in (1.9). Then, for finite 1 k N , we have Proof. Since ψ N,t satisfies the Schrödinger equation, then for k 1, where we used the fact that H N is self-adjoint and [H N , N ] = 0. Therefore, integrating the above equation with respect to time, gives us for any 1 k N .
Remark 2.2. The expectation of the number operator is the total mass of Husimi measure. In fact, observe that Then, by (2.5) where we use Lemma 2.2 in the last equality. Moverover, if we repeat the projection above for k-times, we get where 1 k N and t 0.
More importantly, we have the following estimates for localized number operators.
such that ψ N = 1, and R be the radius of a ball such that the volume is 1. Then, for all 1 k N , we havė where χ is a characteristic function Proof. Consider first the case where k = 1. For every 1 j k, we havê where we used Lemma 2.3. Analogously, for 2 k N , where we applied Lemma 2.3 again. and it holds for every α ∈ (0, 1), s ∈ N, and x ∈ R 3 \Ω α , where C depends on the compact support and the C s norm of ϕ.
Proof. We will prove the lemma in a single-variable environment. That is, we let the momentum and space to be p = (p 1 , p 2 , p 3 ) and x = (x 1 , x 2 , x 3 ) such that x j , p j ∈ R for all j ∈ {1, 2, 3}. Then, for arbitrary x ∈ R 3 \Ω α , one of the x j s is bigger than α . Without loss of generality, we assume that |x 1 | > α and x 2 , x 3 ∈ R. Let supp ϕ ⊂ B r (0) ⊂ R 3 , we can rewrite the left hand of (2.11) into the following, we have after s times integration by parts in p 1 , where s indicates the number of time that integration by parts has been performed.

Finite moments of Husimi measure
To prove that the second moment in p of the Husimi measure is finite, we first show that the kinetic energy is bounded from above. Recall that the definition of the kinetic energy operator K, i.e., and the kinetic energy associated with ψ N is given as ψ N , Kψ N .
Lemma 2.6. Assume V ∈ W 1,∞ , then the kinetic energy is bounded in the following where C depends on ∇V ∞ .
Proof. From the Schrödinger equation, we get Note that since the commutator between kinetic and interaction term is given as Then, from (2.13), we have that .
Integrating both sides with respect to time t and we obtain the desired inequality. N,t to be the k-particle Husimi measure. Denoting the phase-space vectors q k = (q 1 , . . . , q k ) and p k = (p 1 , . . . , p k ), we have the following finite moments, where C is a constant dependent on k,˜dq 1 dp 1 (|q 1 | + |p 1 | 2 )m (1) N (q 1 , p 1 ), and ∇V ∞ . Proof. We first consider the case where k = 1. Observe that we may rewrite the kinetic energy as follows where we used the fact that To continue, we have (2.14) Since kinetic energy is real-valued, if we take the real part of (2.14), the last term in the right hand side vanishes since it is purely imaginary, yielding Note that by (2.7), we have where we recall that 3 = N −1 . Thus, taking the real part of (2.14), we have that which means, Therefore, (2.17) tells us that the second moment of the 1-particle Husimi measure in momentum space is finite if the kinetic energy is finite. Now, we turn our focus on the moment with respect to position space. From (2.1), we get ∂ t¨d q 1 dp 1 |q 1 |m Then, using intergration by parts with respect to p 1 , where R 1 is the remainder term in (2.2). Note that by Young's product inequality, we havë where we used (2.17) and Lemma 2.6 in the last inequality. Next, we want to bound the term associated with R 1 ,¨d Observer that we have, where we used (2.15), Lemma 2.2. Thus, we have that ∂ t¨d q 1 dp 1 |q 1 |m N,t (q 1 , p 1 ) (2π) 3 which gives the estimate for first moment after integrating with respect to time t.
We now consider the case of 2 k N . In this computation, we make use of the properties of k-particle Husimi measure. Namely, that the m (k) N,t is symmetric and satisfies the following equation N,t (q 1 , p 1 , . . . , q k−1 , p k−1 ).

(2.19)
Observe that for fixed 1 k N .
Then, by using the symmetricity of m where we denoted (dqdp) ⊗k−1 = dq 1 dp 1 · · · dq k−1 dp k−1 . Similar strategy is used to obtain the first moment with respect to q k . That iṡ This yields the desired conclusion.

Uniform estimates for the remainder terms
In this subsection, we give uniform estimates for the error terms that appear in (2.1) and (2.3). They are all bounded of order 1 2 −δ for arbitrary small δ > 0. The proofs of all the following propositions will be provided in section 3.2.
Proposition 2.4. Let Assumption A1 holds, then for 1 k N , we have the following bound for R k in (2.1) and (2.3). For arbitrary small δ > 0, the following estimate holds for any test function Φ ∈ C ∞ 0 (R 6k ), where C depends on D s(δ) Φ ∞ and k.
Proposition 2.5. Let Assumption A1 and A2 hold, then we have the following bound for R 1 in (2.2). For arbitrary small δ > 0, the following estimate holds for any test function Φ ∈ C ∞ 0 (R 6 ), where C depends on D s(δ) Φ ∞ .
Proposition 2.6. Suppose that Assumption A1 and A2 hold. Denote the remainders terms R k and R k as in (2.4). Then for 1 k N and arbitrary small δ > 0, the following estimates hold for any test function

22)
where C depends on D s(δ) Φ ∞ and k.

Convergence to infinite hierarchy
In this subsection, we prove that the k-particle Husimi measure m N,t has subsequence that converges weakly (as N → ∞) to a limit m (k) t in L 1 , which is a solution of the infinite hierarchy in the sense of distribution. The weak compactness of k-particle Husimi measure m    Nj,t } j∈N that converges weakly in L 1 (R 6k ) to a function (2π) 3k m (k) t , i.e. for all ϕ ∈ L ∞ (R 6k ), it holds when j → ∞ for arbitrary fixed k 1.
Proof. To apply Dunford-Pettis theorem, we need to check that it is uniformly integrable and bounded. From the previous uniform estimates that we have obtained for m where q k := (q 1 , . . . , q k ), p k := (p 1 , . . . , p k ) and C(t) is a time-dependent constant, we can check the uniform integrability. More precisely, for any ε > 0, by taking r = ε −1 (2π) 3k C(t) we have that (2.23) Furthermore, for arbitrary ε > 0, by taking δ = ε, we have that for all which means that there is no concentration for the k-particle Husimi measure.
It is shown in (2.9) that the boundedness of k-particle Husimi measure in L 1 , i.e.
Then applying directly Dunford-Pettis Theorem one obtain that k-particle Husimi measure is weakly compact in L 1 .
Proof of Theorem 1.1 and Corollary 1.1. Cantor's diagonal procedure shows that we can take the same convergent subsequence of m (k) N,t for all k 1. Then by the error estimates obtained in Propositions 2.4, 2.5, and 2.6, we can obtain that the limit satisfies the infinite hierarchy (1.15) in the sense of distribution, by directly taking the limit in the weak formulation of (2.1) and (2.3).
Observe that the estimates for the remainder terms also show that any convergent subsequence of m  Lastly, by Theorem 7.12 in [47], we would obtain the convergence in 1-Wasserstein metric.

Proof of the reformulation in section 2.1
In this subsection we supply the proofs for the reformulation of Schrödinger equation into a hierarchy of k (1 k N ) particle Husimi measure. The reformulation shares similar structure to the classical BBGKY hierarchy.
Proof of Proposition 2.1. First, observe that taking the time derivative on the Husimi measure, we have Now, focus on I 1 , we have where the last equality is just change of variable on the complex conjugate term. Then, from CAR, observe we have that where integration by parts and CAR of the operator have been used several times. Putting this back, we cancel out the the second term and get (3.1) Now, observe the following and furthermore,

(3.4)
Since the Husimi measure is actually a real-valued function, we have that N,t (q 1 , p 1 ) = Re Now, we turn our focus on II 1 , i.e., Observe that a * w a u a * x a * y a y a x =a * x a * y a y a x a * w a u + δ w=y a * x a * y a x a u − δ w=x a * x a * y a y a u + δ u=x a * w a * y a y a x − δ u=y a * w a * x a y a x .
The first term and the complex conjugate term vanishes under changes of variable, u to w and w to u. Therefore, since from assumption V (x) = V (−x), we have Now, note that mean value theorem gives and observe that since, V s(u − y) + (1 − s)(w − y) = V su + (1 − s)w − y , we can have from (3.6) the following (3.8) where we use the fact that Then we get Applying the following projection 1 (2π ) 3¨d q 2 dp 2 f q2,p2 f q2,p2 = 1, (3.11) onto a y ψ N,t , we get Putting this back into (3.10), we get the following (3.13) Therefore, we have the last term in (3.5) as Re thus we have derived the equation for m (1) N,t (q 1 , p 1 ). We have proved the reformulation from Schrödinger equation into 1-particle Husimi measure. We also observed that it contains a resemblance to the classical Vlasov equation. Next we want to prove the similar result for 2 k N .
Proof of Proposition 2.2. Now we focus on the case where 2 k N . As in the proof for the case of k = 1, we first observe that for every k ∈ N, ⊗k ψ N,t , a * x a * y a y a x a * w1 · · · a * w k a u k · · · a u1 ψ N,t =: (3.14) where the tensor product denotes (dwdu) ⊗k = dw 1 · · · dw k du 1 · · · du k . We first focus on the I 2 part of (3.14), i.e., Observe that we have where the hat indicates exclusion of that element. Putting this back into (3.15), we obtain ⊗k · ∆ uj ψ N,t , a * w1 · · · a * w k a u k · · · a uj · · · a u1 a uj ψ N,t − ∆ wj ψ N,t , a * wj a * w1 · · · a * wj · · · a * w k a u k · · · a u1 ψ N,t . (3.17) Note that, if we want to move the missing a uj or a * wj back to their original position after applying the delta function, we have for fixed j (−1) j a * w1 · · · a * w k a u k · · · a uj · · · a u1 a uj = (−1) j (−1) j−1 a * w1 · · · a * w k a u k · · · a u1 =(−1) 1 a * w1 · · · a * w k a u k · · · a u1 , (−1) j a * wj a * w1 · · · a * wj · · · a * w k a u k · · · a u1 =(−1) 1 a * w1 · · · a * w k a u k · · · a u1 .
Therefore, continuing from (3.17), we have Now, by integration by parts on (3.18) and note that the Laplacian acting on the coherent state would be similar to (3.2) and (3.3), i.e., for fixed j where 1 j k Thus, we have similar for when k = 1, the kinetic part as ∆ qj a f q k ,p k · · · a f q1,p1 ψ N,t , a f q k ,p k · · · a f q1,p1 ψ N,t .

(3.19)
Therefore it follows that Now, we turn our focus on part II 2 of (3.14), x a * y a y a x a * w1 · · · a * w k a u k · · · a u1 ψ . (3.21) For 1 k N , observe that from the CAR, we have a * w1 · · · a * w k a u k · · · a u1 a * x a * y a y a x − (−1) 8k a * x a * y a y a x a * w1 · · · a * w k a u k · · · a u1 From (3.21), we have thaẗ dxdy V (x − y) a * w1 · · · a * w k a u k · · · a u1 a * x a * y a y a x − a * x a * y a y a x a * w1 · · · a * w k a u k · · · a u1 Note that summing J 1 and J 4 , we have where the terms with V (0) cancel one another. For the remaining term, we use again CAR to obtain On the other hand, the sum of J 2 and J 2 yield By change of variable and using the fact that V (−x) = V (x), we have from (3.21) that Applying mean value theorem on the first term on right hand side, we have that ⊗k · a w k · · · a w1 a y ψ N,t , a u k · · · a u1 a y ψ N,t .

(3.24)
As in the case of k = 1, we apply the projection (3.11) onto a y ψ N,t and get further ⊗k · a w k · · · a w1 a y ψ N,t , a u k · · · a u1 1a y ψ N,t ·¨d qd p f q, p (y)ˆdv f q, p (v) a w k · · · a w1 a y ψ N,t , a u k · · · a u1 a v ψ N,t .
(3.25) Therefore, dividing both equations by 2i , we have the following equation ∆ qj a f q k ,p k · · · a f q1,p1 ψ N,t , a f q k ,p k · · · a f q1,p1 ψ N,t ·¨d qd p f q, p (y)ˆdv f q, p (v) a w k · · · a w1 a y ψ N,t , a u k · · · a u1 a v ψ N,t ⊗k · a w k · · · a w1 ψ N,t , a u k · · · a u1 ψ N,t .
(3.26) for 1 k N , p k = (p 1 , . . . , p k ) and recalling 3 = N −1 . At this point we finish the computation of the hierarchy for Husimi measure.

Proof of the uniform estimates in section 2.3
This subsection provide the proof of estimates for the error terms that appeared in the equations for m Note that in all the proofs below, we suppose, without loss of generality, that the test function Φ ∈ C ∞ 0 (R 6k ) is factorized in phase-space by family of test functions in C ∞ 0 (R 3 ) space.
Proof of Proposition 2.5 Proof. Let Φ be an arbitrary test function, then the remainder term R 1 can be written explicitly into ¨d q 1 dp 1 ∇ p1 Φ(q 1 , p 1 ) · R 1 = ¨d q 1 dp 1 ∇ p1 Φ(q 1 , p 1 ) · ¨d wdu¨dydv¨dq 2 dp 2 Then, utilizing (2.7), we may get a w a y ψ N,t , a u a y ψ N,t the appropriate terms α and s should be. By Lemma 2.5, we may bound the term i 31 , i.e., Since we assume that f is compactly supported, by Hölder inequality with respect to w and u, we have we have that where we used the change of variable √ w = w − q 1 in the last inequality. Now, since f 2 is normalized, we continue to have ˆd y a w a y ψ N,t a u a y ψ N,t On the other hand, from ii 31 we have Since f is assumed to be compactly supported, we have where we have used the fact that the test function has compact support in the q variable. Now we compare power of with the one in (3.31) and choose . Now, focus on I 3 , we use similar strategy as with II 3 .