ON THE MEAN-FIELD AND CLASSICAL LIMITS OF QUANTUM MECHANICS

Transcription

1 O THE MEA-FIELD AD CLASSICAL LIMITS OF QUATUM MECHAICS FRAÇOIS GOLSE, CLÉMET MOUHOT, AD THIERRY PAUL Abstract. The main result in this paper is a new inequality bearing on solutions of the -body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of C, interaction potentials. The quantity measuring the approximation of the -body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent 2. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 3 (979), 5 23]. Our approach of this problem is based on a direct analysis of the -particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization. In memory of Louis Boutet de Monvel (94 204). Statement of the problem In nonrelativistic quantum mechanics, the dynamics of identical particles of mass m in R d is described by the linear Schrödinger equation i t Ψ = 2 xk Ψ + V (x k x l )Ψ, 2m k,l= where the unknown is Ψ Ψ(t, x,..., x ) C, the -particle wave function, while x, x 2,..., x designate the positions of the st, 2nd,..., th particle. The interaction between the kth and lth particles is given by the potential V, a realvalued measurable function defined a.e. on R d, such that () V (z) = V ( z), for a.e. z R d. Denoting the macroscopic length scale by L>0, we define a time scale T >0 such that the total interaction energy of the typical particle with the other particles is of the order of m(l/t ) 2. With the dimensionless space and time variables defined as ˆx := x/l, ˆt := t/t, the interaction potential is scaled as ˆV (ẑ) := T 2 ml 2 V (z). 99 Mathematics Subject Classification. 82C0, 35Q55 (82C05,35Q83). Key words and phrases. Schrödinger equation, Hartree equation, Liouville equation, Vlasov equation, Mean field limit, Classical limit, Monge-Kantorovich distance.

2 2 F. GOLSE, C. MOUHOT, AD T. PAUL In terms of the dimensionless parameter and the new unknown the Schrödinger equation becomes ɛ := T/mL 2 ˆΨ(ˆt, ˆx,..., ˆx ) := Ψ(t, x,..., x ), i ˆt ˆΨ = 2 ɛ ˆxk ˆΨ + ɛ k,l= ˆV (ˆx k ˆx l ) ˆΨ. In the present paper, we obtain a new estimate for solutions of this Schrödinger equation which is of particular interest in the asymptotic regime where ɛ (classical limit) and (mean field limit). Henceforth we drop all hats on rescaled quantities and consider the Cauchy problem i t Ψ ɛ, = 2 (2) ɛ xk Ψ ɛ, + V (x k x l )Ψ ɛ,, ɛ k,l= Ψ t=0 ɛ, = Ψ in ɛ,. While the discussion above applies to all types of particles, until the end of this section we restrict our attention to the case of bosons, i.e. to the case where the wave function Ψ ɛ, is a symmetric function of the space variables x,..., x. A typical example of relevant initial data for (2) is Ψ in ɛ, (x,..., x ) = (πɛ) d/4 e (xj q)2 /2ɛ e ip xj/ɛ, where p, q R d are parameters. In other words, in this example, the initial wave function is the -fold tensor product of Gaussian wave functions with width ɛ and oscillations at frequency O(/ɛ)... The mean field limit. The mean field limit is the asymptotic regime where, with ɛ > 0 fixed. Set the initial data in (2) to be Ψ in ɛ, (x,..., x ) := ψ in ɛ (x k ) with R d ψ in ɛ (x) 2 dx =, and let Ψ ɛ, be the solution of the Cauchy problem (2) which exists for all times provided that V is such that 2 ɛ xk + ɛ V (x k x l ) k,l= has a self-adjoint extension as an unbounded operator on L 2 ((R d ) ). Under various assumptions on ψ in and V, it is known that, for each t R, (R d ) Ψ ɛ, (t, x, z 2,..., z )Ψ ɛ, (t, y, z 2,..., z )dz 2... dz ψ ɛ (t, x)ψ ɛ (t, y)

3 MEA-FIELD AD CLASSICAL LIMITS 3 in some appropriate sense as, where ψ ɛ is the solution of the Hartree equation i t ψ ɛ = 2 (3) ɛ xψ ɛ + ɛ ψ ɛ(t, x) V (x z) ψ ɛ (t, z) 2 dz, R d ψ t=0 ɛ = ψɛ in. See [29, 4, 8,, 9, 0, 2, 28, 26, 27, 7] for various results in this direction, obtained under different assumptions on the regularity of the interaction potential V. Most of the physically relevant particle interactions, especially the case where V is the Coulomb potential, are covered by these results, but not always with a quantitative error estimate..2. The classical limit. The classical limit is the asymptotic regime where ɛ 0 while is kept fixed in (2) or simply ɛ 0 in (3). The formalism of the Wigner transform is perhaps the most convenient way to formulate this limit. Given Φ Φ(X) C, an element of L 2 (R n ), its Wigner transform at scale ɛ is W ɛ [Φ](X, Ξ) := (2π) n R n Φ ( X + 2 ɛy ) Φ ( X 2 ɛy ) e iξ Y dy. Assume that the initial data in (2) is a family Ψ in ɛ, such that W ɛ [Ψ in ɛ, ] F in in S ((R d R d ) ) as ɛ 0. Then, for all t R, the family of solutions Ψ ɛ, of the Cauchy problem (2) satisfies W ɛ [Ψ ɛ, (t, )] F (t,, ) in S ((R d R d ) ) as ɛ 0, where F F (t, x,..., x, ξ,..., ξ ) 0 is the solution of the following Cauchy problem for the -body Liouville equation of classical mechanics t F + ξ k xk F V (x k x l ) ξk F = 0, (4) k,l= F t=0 = F in. The classical limit of the Hartree equation (3) can be formulated similarly. Assume that the initial data in (3) is a family ψɛ in such that W ɛ [ψ in ɛ ] f in in S (R d R d ) as ɛ 0. Then, for all t R, the family of solutions ψ ɛ of the Hartree equation (3) satisfies W ɛ [ψ ɛ (t, )] f(t,, ) in S (R d R d ) as ɛ 0, where f f(t, x, ξ) 0 is the solution of the following Cauchy problem for the Vlasov equation of classical mechanics with interaction potential V : ( ) t f + ξ x f V (x z)f(t, z)dz ξ f = 0, (5) R d f t=0 = f in. See [20, 3] for results on the classical limit of quantum mechanics involving the Wigner transform.

4 4 F. GOLSE, C. MOUHOT, AD T. PAUL.3. The mean field limit in classical mechanics. There is also a notion of mean field limit in classical mechanics, which can be formulated as follows. Assume that the initial data in (4) is F in (x,..., x, ξ,..., ξ ) = f in (x k, ξ k ), where f in is a probability density on R d R d. Under various assumptions on the potential V, the solution F of (4) satisfies (R d R d ) F (t, x, x 2,..., x, ξ, ξ 2,..., ξ )dx 2 dξ 2... dx dξ f(t, x, ξ) in some appropriate sense as, where f is the solution of the Cauchy problem for the Vlasov equation (5). See [24, 6, 7] for the missing details. All these references address the problem of the mean field limit in terms of the empirical measure of the -particle system. Typically these results cover the case where V C, (R), but the case of a Coulomb, or ewtonian interaction remains open at the time of this writing. For a formulation of the same results in terms of the BBGKY hierarchy, see [5, 22]. Yet, there has been some recent progress on the case of singular potentials. In [6], the mean-field limit has been established for interaction potentials with a singularity at the origin that is weaker than that of the Coulomb potential. Another approach to the mean-field limit in the case of singular interaction involves a truncated variant of the potential with a cutoff parameter η η() > 0 assumed to vanish as the number of particles : see [6, 8, 9] for the most recent results in that direction. This cutoff parameter can be thought of as being of the order of the size of the interacting particles, as explaned in [8]. The situation described above can be summarized in the following diagram: the horizontal arrows correspond to the mean field limit, while the vertical arrows correspond to the classical limit. Schrödinger Hartree ɛ 0 ɛ 0 Liouville Vlasov However, these various limits are established by very different methods. The classical limits of either the -body Schrödinger equation or of the Hartree equation are obtained by a compactness argument and the uniqueness of the solution of the Cauchy problems (4) or (5). Error estimates for these limits require rather stringent assumptions on the regularity of the potential V and on the type of initial wave or distribution functions considered. The mean field limit in quantum mechanics (the upper horizontal arrow) comes from trace norm estimates on the infinite hierarchy of equations obtained from the BBGKY hierarchy in the large limit. The trace norm is the quantum analogue of the total variation norm on

5 MEA-FIELD AD CLASSICAL LIMITS 5 the probability measures appearing in the classical setting. In general, the total variation norm is not convenient in the context of the mean field limit, since it does not capture the distance between neighboring point particles. This suggests that the trace norm is not appropriate to obtain controls on the large (mean field) limit which remain uniform in the vanishing ɛ (classical) limit. Another notable difficulty with this problem is that the mean field limit in classical mechanics is obtained by proving the weak convergence of the -particle empirical measure in the single-particle phase space to the solution of the Vlasov equation. Since there does not seem to be any natural analogue of the notion of empirical measure for a quantum -particle system, the analogy between the quantum and the classical mean field limits is far from obvious. Our main result, stated as Theorem 2.4 below, is a new quantitative estimate for the mean field limit of quantum mechanics which is uniform in the classical limit ɛ Main result Let d be a positive integer. Henceforth we set H := L 2 (R d ), and H := H L 2 ((R d ) ) for each. We designate by L(H) the algebra of bounded linear operators on the Hilbert space H. We also denote by D(H ) the set of operators A L(H ) such that A = A 0, and trace(a) =. We are concerned with the -body Schrödinger equation written in terms of density matrices, i.e. the von eumann equation i t ρ ɛ, = 2 (6) ɛ k + V kl, ρ ɛ,, ɛ k,l= ρ t=0 ɛ, = ρ in ɛ,. where ρ ɛ, (t) D(H), while and k := I (k ) H I ( k) H, (7) (V jk ψ)(x,..., x ) := V (x k x j )ψ(x,..., x ), for each ψ H. The notation I H designates the identity on the Hilbert space H. We shall everywhere restrict our attention to symmetric density matrices, corresponding to indistinguishable particles. In other words, we assume that (8) τ σ ρ ɛ, (t)τ σ = ρ ɛ, (t) for each t R and each σ S, where τ σ is the unitary operator defined on L 2 ((R d ) ) by the formula (9) τ σ Φ(x,..., x ) := Φ(x σ (),..., x σ ()). One easily checks that the condition (0) τ σ ρ in ɛ, τ σ = ρ in ɛ,

6 6 F. GOLSE, C. MOUHOT, AD T. PAUL implies that (8) holds for each t R, since τ σ, 2 ɛ k + V kl = 0 for each σ S. ɛ k,l= On the other hand, we consider the corresponding mean field equation, i.e. the Hartree equation written in terms of the density matrix ρ ɛ (t) D(H) i t ρ ɛ = [ 2 () ɛ + ɛ ] V ρɛ, ρ ɛ, ρ t=0 ɛ = ρ in ɛ, where V ρɛ designates both the function V ρɛ (t, x) := V (x z)ρ ɛ (t, z, z)dz R d and the time-dependent multiplication operator defined on H by (V ρɛ ψ)(t, x) := V ρɛ (t, x)ψ(x). ext we formulate the mean field limit in terms of density operators. For each -particle density operator ρ D(H ), we define its first n-particle marginal density operator, denoted by ρ n for each integer n such that n, by the following conditions: { ρ n D(H n ), and trace Hn (Aρ n ) = trace H ((A I H n )ρ ) for each A L(H n ). In the mean field limit, i.e. for while ɛ > 0 is kept fixed, one expects that the sequence ρ ɛ, of first marginals of the density operators ρ ɛ, solutions of (6) converges in some topology to the solution ρ ɛ of (), provided that ρ in ɛ, approaches (ρ in ɛ ) in some appropriate sense. The difference between ρ ɛ, and ρ ɛ is measured in terms of a quantity analogous to the Monge-Kantorovich distance used in the context of optimal transport. First we define the notion of coupling between two density operators. Definition 2.. Let d be a positive integer and let H := L 2 (R d ). For each ρ, ρ D(H), let Q(ρ, ρ) be the set of R D(H 2 ) such that { traceh2 ((A I H )R) = trace H (Aρ) for each A L(H). (2) trace H2 ((I H A)R) = trace H (Aρ) ext we define two unbounded operators on H 2 L 2 (R d R d ), as follows: { (Qψ)(x, x 2 ) := (x x 2 )ψ(x, x 2 ), so that (P ψ)(x, x 2 ) := iɛ( x x2 )ψ(x, x 2 ), (3) (P P ψ)(x x 2 ) = ɛ 2 (div x div x2 )( x x2 )ψ(x, x 2 ). The quantum analogue of the Monge-Kantorovich distance with exponent 2 is defined as follows. For the definition of Monge-Kantorovich distances, also called

7 MEA-FIELD AD CLASSICAL LIMITS 7 Wasserstein distances, see formula (20) in section 3, or chapter 7 in [3], or chapter 6 in [32]. Definition 2.2. For each ρ, ρ D(H), we set MK ɛ 2(ρ, ρ) := with the following convention: inf trace H 2 ((Q Q + P P )R) /2 R Q(ρ,ρ) trace H2 ((Q Q + P P )R) := trace H2 (R /2 (Q Q + P P )R /2 ) if R /2 (Q Q + P P )R /2 is a trace-class operator, and otherwise. trace H2 ((Q Q + P P )R) := + The quantity ɛ is not a distance on D(H). In fact, for each density operator ρ on H, one has MK2(ρ, ɛ ρ) > 0 (see formula (4) below). However, MK2 ɛ can be compared with the Monge-Kantorovich distance with exponent 2 (see formula (20) below with p = 2 for a definition of this distance, and Theorem 2.3 (2) below for more information on this comparison), at least for a certain class of operators and in the limit as ɛ 0. Henceforth, we denote by P(R d ) the set of Borel probability measures on R d. For each q > 0, we define { } P q (R d ) := µ P(R d ) s.t. x q µ(dx) <. R d Theorem 2.3 (Properties of MK ɛ 2). Let d be a positive integer and let H := L 2 (R d ). For each ρ, ρ D(H) and each ɛ > 0, one has (4) MK ɛ 2(ρ, ρ) 2 2dɛ. () Let ɛ > 0 and let ρ ɛ and ρ ɛ 2 be Töplitz operators at scale ɛ on L 2 (R d ) with symbols (2πɛ) d µ and (2πɛ) d µ 2, where µ, µ 2 P 2 (R 2d ). Then MK ɛ 2(ρ ɛ, ρ ɛ 2) 2 inf trace H H((Q Q + P P ) OP T ɛ ((2πɛ) 2d µ)) µ Π(µ,µ 2) = dist MK,2 (µ, µ 2 ) 2 + 2dɛ. (2) Let ρ ɛ and ρ ɛ 2 D(H), with Husimi transforms at scale ɛ denoted respectively W ɛ [ρ ɛ ] and W ɛ [ρ ɛ 2]. Then MK ɛ 2(ρ ɛ, ρ ɛ 2) 2 dist MK,2 ( W ɛ [ρ ɛ ], W ɛ [ρ ɛ 2]) 2 2dɛ. Assume further that the Wigner transforms at scale ɛ of ρ ɛ and ρ ɛ 2, denoted respectively W ɛ [ρ ɛ ] and W ɛ [ρ ɛ 2], converge in S (R 2d ) to Wigner measures denoted respectively w and w 2 as ɛ 0. Then dist MK,2 (w, w 2 ) lim MK2(ρ ɛ ɛ, ρ ɛ 2). ɛ 0 The definitions and basic properties of Töplitz operators, Wigner and Husimi functions are recalled in Appendix B. Statement (2) in Theorem 2.3 implies in particular that the quantity MK ɛ 2 is not vanishing for all density matrices as ɛ 0 +. This property is obviously essential; otherwise, the quantity MK ɛ 2 would not be of much practical interest for controling the error in the mean field limit. Statement () in Theorem 2.3 will be used in

8 8 F. GOLSE, C. MOUHOT, AD T. PAUL choosing the initial quantum states to which our error estimate for the mean field limit will apply. The main result in this paper is the following theorem. Theorem 2.4. Let d be a positive integer. For each ɛ > 0 and each integer >, let ρ in ɛ, D(L2 ((R d ) )) satisfy (0) and let ρ in ɛ D(L 2 (R d )). Let t ρ ɛ, (t) be the solution of the quantum -body Cauchy problem (6), and let t ρ ɛ (t) be the solution of the quantum mean field Cauchy problem (). Then, for each positive integer n and all t 0, one has (5) n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)) 2 8 V e Λt 2 L + eλt Λ MKɛ 2((ρ in ɛ ), ρ in ɛ, ) 2 where Λ := Lip( V ) 2. In the particular case where ρ in ɛ, is a Töplitz operator at scale ɛ with symbol (2πɛ) d µ in ɛ, while ρin ɛ is a Töplitz operator at scale ɛ with symbol (2πɛ) d µ in ɛ, for each positive integer n and all t 0, one has (6) ( n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)) 2 2dɛ + ) dist MK,2(µ in ɛ ), µ in ɛ, ) 2 e Λt In particular, if ρ in ɛ then (7) + 8n V e Λt 2 L. Λ D(H) is a Töplitz operator at scale ɛ and ρ in ɛ, = (ρin ɛ ), n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)) 2 (2dɛ + 8 V 2L e Λt Λ ) e Λt. Observe that one cannot deduce the mean field limit of the quantum -body problem (6) from the bound on MK2(ρ ɛ ɛ (t), ρ ɛ, (t)) obtained in Theorem 2.4 in the case where ɛ > 0 is kept fixed, because of the term 2dɛ exp(λt) on the right hand side of (5). On the other hand, the mean field limit alone, i.e. for ɛ fixed, has been proved by other methods in this case (see [29, 4]). Moreover, quantitative estimates for that limit for ɛ fixed and have been obtained in [28, 26, 2]. Therefore, only the case where both and ɛ 0 remains to be treated, and the present work answers precisely this question. Indeed, the estimate in Theorem 2.4, together with the first lower bound in Theorem 2.3 (2), implies that the mean field limit, i.e. the convergence ρ n ɛ, (t) ρ ɛ (t) n for each n as is uniform as ɛ 0 and over long times intervals, in the following sense. Let c, c satisfy 0 < c < c <, and set T (c, η, ) := c ( Λ min ln ) [ ( η, ln, n(c, η, ) := min η c, c )], (where [x] designates the largest integer less than or equal to x). Then the quadratic Monge-Kantorovich distance between the Husimi transforms at scale ɛ of ρ n ɛ, (t)

9 MEA-FIELD AD CLASSICAL LIMITS 9 and ρ n ɛ (t) satisfies sup sup sup n n(c,η,) 0 t T (c,η,) 0<ɛ<η dist MK,2 ( W ɛ [ρ ɛ (t) n ], W ɛ [ρ n ɛ, (t)]) 0 in the limit as + η 0. Earlier works have discussed the mean field limit of the quantum -body problem in the small ɛ regime, more precisely, in the case where ɛ = ɛ() 0 as. The case ɛ() = /3 is of considerable importance for the mean field limit of systems of fermions, and has been investigated in [23, 30]. In the more recent reference [5], the -body problem in the fermionic case is compared to the Hartree-Fock equations, with error estimates in the Hilbert-Schmidt and trace norms see formulas (2.9)-(2.20) in Theorem 2. of [5]. Of course, convergence in either Hilbert-Schmidt or trace norm is stronger than the control (5) in terms of the quantity MK2. ɛ On the other hand, at variance with the estimate (5) above, whose right hand side grows exponentially fast in t, the bounds (2.9)-(2.20) in [5] involves a right hand side growing much faster in t specifically, of order exp(c exp(c 2 t )) for some constants c, c 2 > 0. Another difference between [5] and our work is that the inequality (5) does not postulate any dependence of ɛ in on the contrary, ɛ and are independent throughout the present paper. In [4], for each sequence ɛ ɛ() 0 as and each monokinetic solution of the Vlasov equation (5) i.e. a solution of the form f(t, x, ξ) = ρ(t, x)δ(ξ u(t, x)) Theorem. gives an asymptotic approximation rate for the convergence of the Wigner transform at scale ɛ() of ρ ɛ, to f in the sense of distributions for t [0, T ]. A priori, the time T and the convergence rate depend on the Vlasov solution f and on the sequence ɛ(). On the diagram of section.3, this result corresponds to the left vertical and bottom horizontal arrow along distinguished sequences (ɛ(), ) over time intervals which may depend on the dependence of ɛ in terms of. Another approach of the same problem can be found in [25]: it is proved that each term in the semiclassical expansion as ɛ 0 of the quantum -body problem converges as to the corresponding term in the semiclassical expansion of Hartree s equation. On the contrary, Theorem 2.4 provides a quantitative estimate of the distance between the solution of the Hartree equation and the first marginal of the solution of the quantum -body problem, for a rather general class of initial data. This estimate implies that the mean field limit, i.e. the top horizontal arrow, is uniform as ɛ 0, over arbitrary long time intervals. This estimate is the quantum analogue of the Dobrushin estimate [7] for the classical mean field limit see section 3. The new ideas used in the proof of Theorem 2.4 are (a) the use of the quantity MK ɛ 2, which behaves well with the Töplitz quantization, and can be conveniently compared with the Monge-Kantorovich distance with exponent 2 on symbols, to which it is obviously analogous; (b) an Eulerian version of Dobrushin s estimate, which avoids the traditional presentation in terms of particle trajectories as in Dobrushin s original work [7], and can therefore be easily adapted to the quantum dynamics; (c) and the adaptation of Dobrushin s estimate to the -particle Liouville equation, thereby avoiding the need of any quantum analogue of the classical notion of - particle empirical measure.

10 0 F. GOLSE, C. MOUHOT, AD T. PAUL The Eulerian version of Dobrushin s inequality (item (b) on the list above) significantly simplifies the original argument, and allows extending Dobrushin s inequality to Monge-Kantorovich distances with arbitrary finite exponents (see [2] for the original argument for the Monge-Kantorovich distance with exponent 2). Estimating directly the Monge-Kantorovich distance between the first marginal of the -particle distribution function and the solution of the mean field equation (item (c) on the list above) avoids using the fact that the -particle empirical measure is an exact solution of the mean field equation, an important feature in Dobrushin s original approach [7]. This feature is very peculiar to the mean field limit in classical Hamiltonian mechanics, and we do not know of any quantum analogue of the notion of -particle empirical measure which would exactly satisfy the mean field quantum dynamics. In other words, the mean field limit in quantum mechanics cannot be reduced to the continuous dependence of solutions of the quantum mean field equation in terms of their initial data, in some appropriate weak topology. The outline of the paper is as follows: in the next section, we present items (b)-(c) above on the mean field limit for the classical Liouville equation, leading to the Vlasov equation. The resulting estimate in Theorem 3. below improves earlier quantitative bounds of the same type obtained in [5, 22]. The proof of the uniform in ɛ estimate in Theorem 2.4 for the quantum mean field limit occupies section 5. The properties of the quantity MK ɛ 2 used in this estimate, stated in Theorem 2.3, are proved in section 4. The material on Töplitz quantization, Wigner and Husimi functions used in the proof of Theorem 2.3 is recalled in Appendix B. The Töplitz quantization is important in the analysis presented here, since it connects our main result (Theorem 2.4) with the Dobrushin proof of the mean field limit for the classical -body problem. For this reason, we dedicate our work to the memory of our friend and teacher Louis Boutet de Monvel, in recognition of his great contributions to the theory of Töplitz operators. 3. The Mean Field Limit in Classical Mechanics As a warm-up, we first discuss the mean field limit for the -body problem in classical mechanics, leading to the Vlasov equation. The approach proposed in [24, 6, 7] is based on the fact that the phase-space empirical measure of a -particle system governed by the ewton equations of classical mechanics is a weak solution of the Vlasov equation (5). The estimate of the distance between the -particle and the mean field dynamics obtained by Dobrushin [7] can be formulated in terms of propagation of chaos for the sequence of marginals of the -particle distribution, as explained in [5, 22]. The approach proposed below bears directly on the -particle distribution, i.e. the solution of the Liouville equation (4), and avoids any reference to the -particle empirical measure. Besides, the core of our argument also avoids using particle trajectories and is based on a computation formulated exclusively in terms of Eulerian coordinates. For that reason, this approach can be adapted to the quantum problem, at variance with the Dobrushin procedure [7], also used in [5, 22]. For µ, ν P(R d ), we denote by Π(µ, ν) the set of couplings of µ et ν, i.e. the set of Borel probability measures π on R d R d with first and second marginals (8) π = µ and π 2 = ν.

11 MEA-FIELD AD CLASSICAL LIMITS In other words, (9) (φ(x) + ψ(y))π(dxdy) = φ(x)µ(dx) + ψ(y)ν(dy) R d R d R d R d for each φ, ψ C b (R d ). The identity (9) can be used as a definition of the first and second marginals of π in (8). Finally, we recall the definition of the Monge- Kantorovich distance of exponent p on P p (R d ): ( ) /p (20) dist MK,p (µ, ν) := inf x y p π(dxdy). π Π(µ,ν) R d R d Theorem 3.. Assume that V Cb 2(Rd ) satisfies (). Let f in P p (R d R d ) with p. Let F be the solution of the Cauchy problem (4) for the -body Liouville equation with initial data F in := (f in ), and let f be the solution of the Cauchy problem for the Vlasov equation (5) with initial data f in. For each integer n such that n, let F(t) n := F (t, dy n+... dy dη n+... dη ) P p (R dn R dn ) be the n-th marginal of F (i.e. the marginal corresponding to the phase space distribution of the n first particles). Then n dist MK,p(f(t) n, F n (t)) p 2 p K p V p [p/2] + e Λpt L, min(p/2,) Λ p where K p := max(, p ) and Λ p := 2K p ( + 2 p Lip( V ) p ). Observe that one obtains an estimate of dist MK,p (f(t) n, F n (t)) of order O( /2 ) if p 2, corresponding to the rate predicted by the central limit theorem. Proof. The proof of Theorem 3. occupies the remaining part of the present section. 3.. The dynamics of couplings. Let π in Π((f in ), (f in ) ) satisfy (2) T σ #π in = π in, for each σ S, where T σ (x, ξ,..., x, ξ, y, η,..., y, η ) = (x σ(), ξ σ(),..., x σ(), ξ σ(), y σ(), η σ(),..., y σ(), η σ() ). It will be convenient to use the following notation (22) X := (x,..., x ), Ξ := (ξ,..., ξ ), Y := (y,..., y ), H := (η,..., η ), Let f be the solution of the Cauchy problem (5) with initial data f in, and let H ρ[f] (X (, Ξ ) := 2 ξ j 2 + V ρ[f] (x j ) ) be the mean field Hamiltonian. On the other hand, let H (Y, H ) := 2 η k 2 + V (y k y l ) k,l=

12 2 F. GOLSE, C. MOUHOT, AD T. PAUL be the microscopic (-particle) Hamiltonian. Finally, we denote by {, } the Poisson bracket on (R d R d ) defined by {φ, ψ} := ( ξj φ xj ψ ξj ψ xj φ). The following observation is the key to the Eulerian formulation of the Dobrushin type estimates, which we shall adapt to the quantum case. Lemma 3.2. Let t π (t) P 2 ((R d R d ) (R d R d ) ) be the solution of the Cauchy problem { t π + {H ρ[f(t)] (X, Ξ ) + H (Y, H ), π } 2 = 0, Then and π t=0 = π in. π (t) Π(f(t) ; F (t)), for each t R, T σ π (t) = π (t) for each t R and each σ S. Proof. Let φ φ(x, Ξ ) and ψ ψ(y, H ) Cc ((R d R d ) ) be two test functions. Then d φ(x, Ξ )π, (t, dx dξ ) dt = d φ(x, Ξ )π (t, dx dξ dy dh ) dt = {H ρ[f(t)], φ} (X, Ξ )π (t, dx dξ dy dh ) = {H ρ[f(t)], φ} (X, Ξ )π, (t, dx dξ ) since {H (Y, H ), φ(x, Ξ )} 2 = 0. The penultimate chain of equalities shows that the first marginal π, of π corresponding to the phase space variables (X, Ξ ) is a solution of the equation t π, + {H ρ[f(t)], π, } = 0. On the other hand, an elementary computation shows that the solution f of (5) satisfies t f + {H ρ[f(t)], f } = 0. Since π, and f are solutions of the same Liouville equation and π, (0) = (f in ) = f(0) we conclude from the uniqueness of the solution of the Cauchy problem for a transport equation with Lipschitz continuous coefficients that π, (t) = f(t) for all t 0.

13 MEA-FIELD AD CLASSICAL LIMITS 3 Similarly d ψ(y, H )π,2 (t, dy dh ) dt = d ψ(y, H )π (t, dx dξ dy dh ) dt = {H, ψ} (Y, H )π (t, dx dξ dy dh ) = {H, ψ} (Y, H )π,2 (t, dy dh ), since {H ρ[f(t)] (X, Ξ ), ψ(y, H )} 2 = 0. This shows that the second marginal π,2, corresponding to the phase space variables (Y, H ), is a solution to the same Liouville equation (4) as F. Since π,2 (0) = (f in ) = F (0) we conclude that π,2 (t) = F (t) for each t 0, by uniqueness of the solution of (4). Finally the time-dependent Hamiltonian satisfies Hence h f : (X, Ξ, Y, H ) H ρ[f(t)] (X, Ξ ) + H (Y, H ) h f T σ = h f for all σ S. t (π T σ #π )(t) + {h f, (π T σ #π )(t)} 2 = 0. Since (π T σ #π )(0) by (2), we conclude that π (t) = T σ #π (t) for all t 0, by uniqueness of the solution of the Cauchy problem for the Liouville equation with Hamiltonian h f The Eulerian variant of the Dobrushin estimate. Set D p (t) := ( x j y j p + ξ j η j p )π (t, dx dξ dy dh ). We recall that p and that p d/2. Then dd p = dt Hρ[f(t)] (X, Ξ ), ( x j y j p + ξ j η j p ) + H (Y, H ), ( x j y j p + ξ j η j p ) 2 2 dπ (t) dπ (t).

14 4 F. GOLSE, C. MOUHOT, AD T. PAUL First = p = Hρ[f(t)] (X, Ξ ), ( x j y j p + ξ j η j p ) { 2 ξ j 2, x j y j p } 2 + ξ j (x j y j ) x j y j p 2 p 2 {V ρ[f] (x j ), ξ j η j p } 2 V ρ[f] (x j ) (ξ j η j ) ξ j η j p 2, while = p = H (Y, H ), { 2 η j 2, x j y j p } 2 + η j (y j x j ) y j x j p 2 p 2 j, ( x j y j p + ξ j η j p ) 2 j, 2 {V (y j y k ), ξ j η j p } 2 V (y j y k ) (η j ξ j ) η j ξ j p 2. Therefore p dd p = p (ξ j η j ) (x j y j ) x j y j p 2 dπ dt ( ) (ξ j η j ) V ρ[f] (x j ) V (y j y k ) ξ j η j p 2 dπ. At this point we use Young s inequality in the form pab p a p + (p )b p max(, p )(a p + b p ) for each a, b > 0 and each p. Denoting K p := max(, p ), one has dd p dt K p + K p ( ξ j η j p + x j y j p )dπ + K p ξ j η j p dπ V ρ[f](x j ) p V (y j y k ) dπ V ρ[f](x j ) p V (y j y k ) dπ. 2K p D p + K p

15 MEA-FIELD AD CLASSICAL LIMITS 5 Let us decompose this last term as follows V ρ[f] (x j ) V (y j y k ) = V ρ[f] (x j ) + V (x j x k ) ( V (x j x k ) V (y j y k )), so that, by convexity of the function z z p on (0, ) for p, V ρ[f](x j ) p V (y j y k ) 2 p V ρ[f] (x j ) p V (x j x k ) p +2 p ( V (x j x k ) V (y j y k )). Then, by the same convexity argument as above ( V (x j x k ) V (y j y k )) Hence Since dd p dt 2 V (x j x k ) V (y j y k ) p Lip( V )p 2p Lip( V ) p (x j x k ) (y j y k ) p ( x j y j p + x k y k p ). 2K p D p + 2p K p Lip( V ) p 2 j, + 2p K p j, V ρ[f](x j ) ( x j y j p + x k y k p )dπ = 2 ( x j y j p + x k y k p )dπ l= the inequality above can be recast as dd p Λ p D p dt + 2p K p V ρ[f](x j ) with p p V (x j x k ) dπ. x l y l p dπ 2D p, p V (x j x k ) dπ, Λ p := 2K p ( + 2 p Lip( V ) p ).

16 6 F. GOLSE, C. MOUHOT, AD T. PAUL 3.3. Controling the consistency error. Let us examine the last term on the right hand side of this inequality. Since π (t) Π(f(t), F (t)) by Lemma 3.2 V ρ[f](x j ) p V (x j x k ) dπ = V ρ[f](x j ) p V (x j x k ) ρ[f] (t, dx ). Observe that this term involves only the factorized distribution f. This is the consistency error in the sense of numerical analysis. In other words, it measures by how much f fails to be an exact solution of the -body Liouville equation (4). Eventually we arrive at the inequality (23) + 2p K p V ρ[f](x j ) dd p Λ p D p dt p V (x j x k ) ρ[f] (t, dx ). The last term on the right hand side of (23) is mastered by the following inequality. Lemma 3.3. Let F be a bounded vector field on R d, and ρ be a probability density on R d. For each p > 0 and each j =,...,, one has F ρ(x j) p F (x j x k ) ρ(x m )dx m 2[p/2] + 2 (2 F min(p/2,) L )p. m= The proof of Lemma 3.3 is deferred until Appendix A. Inserting the bound in Lemma 3.3 in the right hand side of the differential inequality (23) leads to dd p dt Gronwall s inequality implies that Λ p D p + 22p K p [p/2] + min(p/2,) V p L. (24) D p (t) Dp (0)eΛpt + 2 2p K p V p L [p/2] + min(p/2,) e Λpt Λ p End of the proof of Theorem 3.. By Lemma 3.2, T σ #π (t) = π (t) for each t 0 and each permutation σ S. Thus, for each k =,...,, one has ( x k y k p + ξ k η k p )π (t, dx dξ dy dh ) = ( x y p + ξ η p )π (t, dx dξ dy dh ), We thank one of the referees who suggested using this argument instead of the error estimates in terms of Monge-Kantovich distances for the law of large numbers obtained by Fournier and Guillin [].

17 MEA-FIELD AD CLASSICAL LIMITS 7 so that D p (t) = ( x j y j p + ξ j η j p )π (t, dx dξ dy dh ) = n Denoting n ( x j y j p + ξ j η j p )π (t, dx dξ dy dh ) (25) X k := (x k,..., x ), Ξ k := (ξ k,..., ξ ) for each k =,..., we set π(t, n dx n dξ n dy n dh n ) := π (t, dx n dξ n dy n dh) n. By Fubini s theorem, for each φ C b ((R 2d ) n ), one has φ(x n, Ξ n )π(t, n dx n dξ n dy n dh n ) = φ(x n, Ξ n )π (t, dx dξ dy dh ) = φ(x n, Ξ n )f(t) (dx dξ ) = φ(x n, Ξ n )f(t) n (dx n dξ n ), while φ(y n, H n )π(t, n dx n dξ n dy n dh n ) = = = φ(y n, H n )π (t, dx dξ dy dh ) φ(y n, H n )F (t, dy dh ) φ(y n, H n )F n (t, dy n dh n ), so that (26) (27) Therefore D p (t) = n π n (t) Π(f(t) n, F n (t)) for each t 0. n ( x j y j p + ξ j η j p )π n (t, dx n dξ n dy n dh n ) n dist MK,p(f(t) n, F n (t)) p. Thus, inequality (24) implies that dist MK,p (f(t) n, F n (t)) p D p (0)eΛpt + 2 2p K p V p L [p/2] + min(p/2,) e Λpt Λ p for each t 0, each n and each π in Π((f in ), F in ) satisfying T σ π in = π in for all σ S. Finally, choose the initial coupling of the form π in := (f in ) (dx dξ )δ (X,Ξ )(Y, H ),

18 8 F. GOLSE, C. MOUHOT, AD T. PAUL i.e. Then π := D#(f in ), where D : (X, Ξ ) (X, Ξ, X, Ξ ). (28) D p (0) = since π in is supported in the diagonal ( x k y k p + ξ k η k p )π in (dx dξ dy dh ) = 0, {(X, Ξ, Y, H ) s.t. X = Y et Ξ = H }. Inserting this last piece of information in (27) leads to the inequality in the statement of Theorem 3.. Remark 3.4. In the argument above, we have not checked that f(t) n and F n(t) belong to P p ((R d ) n ) for all n and all t 0. This verification is needed on principle, because the Monge-Kantorovich distance of exponent p is defined on the set of probability measures with finite moments of order p. For each p 2, set M p (t) := ( x p + ξ p )f(t, dxdξ), t 0, where f is the solution of (5). An easy argument based on Young s inequality and the mean value inequality for V shows that M p (t) M p (0)e (p )(+2 Lip( V ))t, t 0. In particular f(t) P p (R d R d ) for all t 0 provided that f in P p (R d R d ). Therefore f(t) n P p ((R d R d ) n ) for each n, and the inequality in Theorem 3. implies that F (t) n P p(r d R d ) for all t 0 and each n =,...,, since dist MK,p (f(t) n, F (t) n ) <. 4. Proof of Theorem 2.3: Properties of MK ɛ The general lower bound (4). The lower bound (4) can be viewed as a variant of the uncertainty principle. More precisely Q Q + P P = (Q + ip ) (Q + ip ) + i(p Q Q P ) i(p Q Q P ). On the other hand, so that (P Q Q P ) = iɛ (div x (x ) + div x2 (x 2 )) = 2idɛI H2, Therefore, for each R D(H 2 ), one has Q Q + P P 2dɛI H2. trace H2 ((Q Q + P P )R) = trace H2 (R /2 (Q Q + P P )R /2 ) 2dɛ trace H2 (R) = 2dɛ.

19 MEA-FIELD AD CLASSICAL LIMITS Upper bound for Töplitz operators. The goal of this section is to prove statement () in Theorem 2.3. In the course of the proof, we shall need the following intermediate result. Everywhere in this section, the identification R 2d (q, p) q + ip C d is implicitly assumed. Lemma 4. (Construction of quantum couplings). Let ɛ > 0 and let ρ ɛ and ρ ɛ 2 be Töplitz operators at scale ɛ on L 2 (R d ) with symbols (2πɛ) d µ and (2πɛ) d µ 2, where µ, µ 2 P(R 2d ). For each coupling µ Π(µ, µ 2 ), the Töplitz operator OP T ɛ ((2πɛ) 2d µ) belongs to Q(ρ ɛ, ρ ɛ 2). Proof. Set ρ ɛ j = OPT ɛ ((2πɛ) d µ j ) for j =, 2. We recall that H = L 2 (R d ). Since µ is a probability measure on C 2d, the operator OP T ɛ ((2πɛ) 2d µ) belongs to D(H H) by (47). ext, one has OP T ɛ ((2πɛ) 2d µ) = (z, z 2 ), ɛ (z, z 2 ), ɛ µ(dz dz 2 ), C 2d where the wave function (z, z 2 ), ɛ is defined in formula (45) in Appendix B. In this section, we use the bra-ket notation, also recalled in Appendix B. Observe that so that for each A L(H). Therefore, (z, z 2 ), ɛ (z, z 2 ), ɛ = z, ɛ z, ɛ z 2, ɛ z 2, ɛ, trace H H ((A I H ) (z, z 2 ), ɛ (z, z 2 ), ɛ ) = trace H (A z, ɛ z, ɛ ) trace H ( z 2, ɛ z 2, ɛ ) = trace H (A z, ɛ z, ɛ ) z 2, ɛ z 2, ɛ = trace H (A z, ɛ z, ɛ ), trace H H ((A I H ) OP T ɛ ((2πɛ) 2d µ)) = trace H H) ((A I H ) (z, z 2 ), ɛ (z, z 2 ), ɛ )µ(dz dz 2 ) C 2d = trace H (A z, ɛ z, ɛ )µ(dz dz 2 ) C 2d = trace H (A z, ɛ z, ɛ )µ (dz ) C d Likewise, for each A L(H), = trace H (A OP T ɛ ((2πɛ) d µ ). trace H H ((I H A) OP T ɛ ((2πɛ) 2d µ) = trace H (A OP T ɛ ((2πɛ) d µ 2 ) = trace H (Aρ ɛ 2). The conclusion immediately follows. Proof of Theorem 2.3 (). For each µ Π(µ, µ 2 ), by Lemma 4. and the definition of MK ɛ 2, one has where we recall that (29) MK ɛ 2(ρ, ρ 2 ) 2 trace H H ((Q Q + P P ) OP T ɛ ((2πɛ) 2d µ), Qψ(x, x 2 ) = (x x 2 )ψ(x, x 2 ), P ψ(x, x 2 ) = iɛ( x x2 )ψ(x, x 2 ),

20 20 F. GOLSE, C. MOUHOT, AD T. PAUL so that and In other words Q Qψ(x, x 2 ) = x x 2 2 ψ(x, x 2 ), P P ψ(x, x 2 ) = ɛ 2 (div x div x2 )( x x2 )ψ(x, x 2 ). Q Q + P P = OP T ɛ ( q q p p 2 2 ) 2dɛI H2 see formula (48) in Appendix B, with f(y, y 2 ) = y y 2 2, and f = 4d. By formula (55) in Appendix B, one has (30) trace H H ((Q Q + P P ) OP T ɛ ((2πɛ) 2d µ) = ( x x ξ ξ 2 2 )µ(dx dξ dx 2 dξ 2 ) + 2dɛ. (R d ) 4 Therefore, for each µ Π(µ, µ 2 ), one has MK2(ρ ɛ, ρ 2 ) 2 ( x x ξ ξ 2 2 )µ(dx dξ dx 2 dξ 2 ) + 2dɛ. (R d ) 4 Observing that the left hand side of this inequality is independent of the coupling µ Π(µ, µ 2 ), we conclude that MK2(ρ ɛ, ρ 2 ) 2 inf ( x x ξ ξ 2 2 )µ(dx dξ dx 2 dξ 2 ) + 2dɛ π Π(µ,µ 2) (R d ) 4 = dist MK,2 (µ, µ 2 ) 2 + 2dɛ which is the sought inequality Asymptotic lower bound for MK ɛ 2. The core of the argument leading to the lower bound in Theorem 2.3 (2) combines Kantorovich duality with the convergence of Husimi functions to Wigner measures. Proof of Theorem 2.3 (2). Let a, b C b (R d ) satisfy (3) a(x, ξ ) + b(x 2, ξ 2 ) x x ξ ξ 2 2 for all x, x 2, ξ, ξ 2 R d. Hence 2 (OP T ɛ (a) I H + I H OP T ɛ (b)) = OP T ɛ (a + b) Thus, for each R ɛ Q(ρ ɛ, ρ ɛ 2), one has OP T ɛ (c) = Q Q + P P + 2dɛI H2. trace H H ((Q Q + P P )R ɛ ) trace H H ((OP T ɛ (a) I H + I H OP T ɛ (b))r ɛ ) 2dɛ = trace H (OP T ɛ (a)ρ ɛ ) + trace H (OP T ɛ (b)ρ ɛ 2) 2dɛ. Taking the inf of the left hand side as R ɛ runs through Q(ρ ɛ, ρ ɛ 2), one arrives at the inequality (32) MK ɛ 2(ρ ɛ, ρ ɛ 2) 2 trace H (OP T ɛ (a)ρ ɛ ) + trace H (OP T ɛ (b)ρ ɛ 2) 2dɛ. 2 Denoting a, b and c the functions (x, ξ, x 2, ξ 2 ) a(x, ξ ), (x, ξ, x 2, ξ 2 ) b(x 2, ξ 2 ) and (x, ξ, x 2, ξ 2 ) x x ξ ξ 2 2 respectively.

21 MEA-FIELD AD CLASSICAL LIMITS 2 ext the traces on the right hand side are expressed in terms of the Husimi functions of ρ ɛ and ρ ɛ 2 by formula (54): trace H (OP T ɛ (a)ρ ɛ ) = a(x, ξ ) W ɛ [ρ ɛ ](x, ξ )dx dξ, trace H (OP T ɛ (b)ρ ɛ 2) = b(x 2, ξ 2 ) W ɛ [ρ ɛ 2](x 2, ξ 2 )dx 2 dξ 2. Hence MK2(ρ ɛ ɛ, ρ ɛ 2) 2 + a(x, ξ ) W ɛ [ρ ɛ ](x, ξ )dx dξ b(x 2, ξ 2 ) W ɛ [ρ ɛ 2](x 2, ξ 2 )dx 2 dξ 2 2dɛ. Taking the sup of both sides of this equality over all a, b C b (R d R d ) satisfying (3) shows that ( sup a + b c a,b C b (R d R d ) a W ɛ [ρ ɛ ]dx dξ + MK ɛ 2(ρ ɛ, ρ ɛ 2) 2 b W ɛ [ρ ɛ 2]dx 2 dξ 2 ) 2dɛ = dist MK,2 ( W ɛ [ρ ɛ ], W ɛ [ρ ɛ 2]) 2 2dɛ, where the last equality follows from Kantorovich duality (Theorem in chapter of [3]). This gives the first inequality in Theorem 2.3 (2). Since the sequence of Wigner transforms of the density matrices ρ ɛ j satisfies for j =, 2, one has W ɛ [ρ ɛ j] w j in S (R d R d ) as ɛ 0 W ɛ [ρ ɛ j] w j weakly in the sense of measures on R d R d for j =, 2, by Theorem III. () in [20]. By the first inequality in Theorem 2.3 (2) already established above and Remark 6.2 in [32], lim ɛ 0 MK2(ρ ɛ ɛ, ρ ɛ 2) 2 lim dist MK,2 (W ɛ [ρ ɛ ], W ɛ [ρ ɛ 2]) 2 dist MK,2 (w, w 2 ) 2. ɛ 0 This concludes the proof of Theorem Proof of Theorem 2.4 The quantum -body Hamiltonian is H ɛ, := 2 ɛ2 j + j,k V jk, an unbounded self-adjoint operator on H := H = L 2 ((R d ) ). We denote by j the Laplacian acting on the variable x j, and by V jk the multiplication by V (x j x k ). The -body von eumann equation for the density matrix is (33) iɛ t ρ ɛ, := [H ɛ,, ρ ɛ, ], ρ ɛ, t=0 = ρ in ɛ,, with initial data ρ in ɛ, D(H ). On the other hand, for each ρ D(H), we consider the mean-field Hamiltonian H ρ ɛ := 2 ɛ2 + V ρ

22 22 F. GOLSE, C. MOUHOT, AD T. PAUL where V ρ designates the operator defined on H as the multiplication by the function x V (x x )ρ(x, x )dx =: V ρ (x). The Hartree equation for the density matrix ρ ɛ is (34) iɛ t ρ ɛ = [H ρɛ ɛ, ρ ɛ ], ρ ɛ t=0 = ρ in ɛ. By a straightforward computation, the solution ρ ɛ of the Hartree equation (34) satisfies (35) iɛ t ρ ɛ = [H ρɛ ɛ,, ρ ɛ ], ρ ɛ t=0 = (ρ in ), where H ρ ɛ, := Hρ ɛ I H + I H ρ ɛ I H I H H ρ ɛ. 5.. The quantum dynamics of couplings. Let Rɛ, in Q((ρin ɛ ), ρ in ɛ, ), and let t R ɛ, (t) D(H 2 ) be the solution of the Cauchy problem (36) iɛ t R ɛ, = [H ρɛ ɛ, I H + I H H ɛ,, R ɛ, ], R ɛ, t=0 = R in ɛ,. For each σ S, we denote by T σ the unitary operator on H 2 defined by T σ Ψ(x,..., x, y,..., y ) = Ψ(x σ (),..., x σ (), y σ (),..., y σ ()) for each Ψ H 2. We shall henceforth assume that (37) T σ Rɛ, in Tσ = Rɛ, in, for each σ S. Lemma 5.. For each t 0, one has R ɛ, (t) Q((ρ ɛ (t)), ρ ɛ, (t)). Moreover (38) T σ R ɛ, (t)t σ = R ɛ, (t) for each σ S and t 0. Proof. By definition R ɛ, (t) = U ɛ, (t/ɛ)r in ɛ, U ɛ, (t/ɛ) where d dt U ɛ, (t) = i(h ρɛ ɛ, I H + I H H ɛ, )U ɛ, (t), U ɛ, (0) = I H2. Since H ρɛ ɛ, I H + I H H ɛ, is self-adjoint, U ɛ, (t) is unitary for each t 0. Therefore R ɛ, (t) = R ɛ, (t) 0 and trace H2 (R ɛ, (t)) = trace H2 (R in ɛ, ) = for each t 0. The marginals of the density R ɛ, in the product H 2 = H H are defined by analogy with (8)-(9) in Definition 2.: { Rɛ,, D(H ) and trace H (AR ɛ,, ) = trace ((A I H2 H )R ɛ, ), (39) R ɛ,,2 D(H ) and trace H (AR ɛ,,2 ) = trace H2 ((I H A)R ɛ, ), for each A L(H ). Let A L(H ) be such that xj, A L(H ) ;

23 MEA-FIELD AD CLASSICAL LIMITS 23 then and iɛ t trace H (AR ɛ,, ) = iɛ t trace H2 ((A I H )R ɛ, ) = trace H2 ([H ρɛ ɛ, I H + I H H ɛ,, (A I H )]R ɛ, ) = trace H2 ([H ρɛ ɛ,, A] I H )R ɛ, ) = trace H ([H ρɛ ɛ,, A]R ɛ,,) = trace H (A[H ρɛ ɛ,, R ɛ,,]) R ɛ,, t=0 = R in ɛ,, = (ρ in ). By uniqueness of the solution of the Cauchy problem (35), one concludes that Similarly and R ɛ,, (t) = ρ ɛ (t), for each t 0. iɛ t trace H (AR ɛ,,2 ) = iɛ t trace H2 ((I H A)R ɛ, ) = trace H2 ([H ρɛ ɛ, I H + I H H ɛ,, (I H A)]R ɛ, ) = trace H2 ((I H ) [H ɛ,, A])R ɛ, ) = trace H ([H ɛ,, A]R ɛ,,2 ) R ɛ,,2 t=0 = R in ɛ,,2 = ρ in ɛ,. = trace H (A[H ɛ,, R ɛ,,2 ]) By uniqueness of the solution of the Cauchy problem for the von eumann equation (33), one concludes that Finally, let σ S ; R ɛ,,2 (t) = ρ ɛ, (t), for each t 0. iɛ t (T σ R ɛ, (t)tσ ) = T σ [H ρɛ ɛ, I H + I H H ɛ,, R ɛ, ]Tσ = [T σ (H ρɛ ɛ, I H + I H H ɛ, )Tσ, T σ R ɛ, Tσ ] = [τ σ H ρɛ ɛ, τ σ I H + I H τ σ H ɛ, τσ, T σ R ɛ, Tσ ] = [H ρɛ ɛ, I H + I H H ɛ,, T σ R ɛ, Tσ ] where τ σ is the unitary operator on H defined in (9), since one has obviously On the other hand τ σ H ρɛ ɛ, τ σ = H ρɛ ɛ, and τ σ H ɛ, τ σ = H ɛ,. T σ R ɛ, (0)T σ = T σ Rɛ, in (0)Tσ = Rɛ, in = R ɛ, (0), so that, by uniqueness of the solution of the Cauchy problem (36), T σ R ɛ, (t)t σ = R ɛ, (t) for each t 0 and each σ S.

24 24 F. GOLSE, C. MOUHOT, AD T. PAUL 5.2. The Dobrushin type estimate. Set D ɛ, (t) := trace (Q j Q j + Pj P j )R ɛ, (t). We compute dd ɛ, dt = ɛ trace i (Q j Q j + Pj P j )[H ρɛ ɛ, I H + I H H ɛ,, R ɛ, ] = ɛ trace i H ρɛ ɛ, I H + I H H ɛ,, = ɛ trace i H ρɛ ɛ, I H H ɛ, I H, + ɛ trace i H ɛ, I H + I H H ɛ,, (Q j Q j + Pj P j ) R ɛ, (Q j Q j + Pj P j ) R ɛ, (Q j Q j + Pj P j ) R ɛ,. Consider the second term on the right hand side of this equality. One has [ Hɛ, I H +I H H ɛ,, (Q j Q j + P j P j ) ] = [ 2 ɛ2 ( j I H + I H j ), Q j Q j ] j + [V jk I H + I H V jk, Pj P j ] j where the index j on the brackets in the right hand side indicate that the corresponding operators act on the variables x j et y j. In other words A j = I (j ) H A I ( j) H for each operator A on H. Obviously [ 2 ɛ2 ( j I H + I H j ), Q j Q j ] = 2 ɛ2 (div xj [ xj, Q j Q j ]+[ xj, Q j Q j ] xj ) 2 ɛ2 (div yj [ yj, Q j Q j ]+[ yj, Q j Q j ] yj ) = iɛ(p j Q j + Q j P j ). On the other hand [ Vjk I H + I H V jk, P j P j ] =[Vjk I H + I H V jk, P j ]P j + P j [V jk I H + I H V jk, P j ] =iɛ( V (x j x k ) V (y j y k ))P j + iɛp j ( V (x j x k ) V (y j y k ))).

25 MEA-FIELD AD CLASSICAL LIMITS 25 Therefore ɛ trace i H ɛ, I H + I H H ɛ,, trace trace 2 j,k 2 j,k By the Cauchy-Schwarz inequality while = trace (Q j Q j + Pj P j ) R ɛ, (Pj Q j + Q j P j )R ɛ, Pj ( V (x j x k ) V (y j y k ))R ɛ, ( V (x j x k ) V (y j y k ))P j R ɛ,. trace ( (P j Q j + Q j P j )R ɛ, ) trace ( (P j P j + Q j Q j )R ɛ, ), trace((p j ( V (x j x k ) V (y j y k ))+( V (x j x k ) V (y j y k ))P j )R ɛ, ) Eventually trace(p j P j R ɛ, ) + trace( V (x j x k ) V (y j y k ) 2 R ɛ, ). ɛ trace i H ɛ, I H + I H H ɛ,, (Q j Q j + Pj P j ) R ɛ, trace (Pj P j + Q j Q j )R ɛ, + trace 2 Pj P j R ɛ, j,k + trace 2 V (x j x k ) V (y j y k ) 2 R ɛ, j,k 2D ɛ, + Lip( V ) 2 trace (x j x k ) (y j y k ) 2 R ɛ, 2D ɛ, + Lip( V ) 2 trace 2 j,k 2 2 j,k ( x j y j 2 + x k y k 2 )R ɛ, ( ) = 2D ɛ, + Lip( V ) 2 4 trace x l y l 2 R ɛ, l= (2 + 4 Lip( V ) 2 )D ɛ,,

26 26 F. GOLSE, C. MOUHOT, AD T. PAUL so that dd ɛ, (2 + 4 Lip( V ) 2 )D ɛ, dt + ɛ trace i H ρɛ ɛ, I H H ɛ, I H, (Q j Q j + Pj P j ) R ɛ,. ext observe that = H ρɛ ɛ, I H H ɛ, I H, [V ρɛ (x j ), Pj P j ] k<l k<l k,l 2 k<l = (Q j Q j + Pj P j ) [V (x k x l ), (P k P k + P l P l )] ([V ρɛ (x j ), Pj ]P j + Pj [V ρɛ (x j ), P j ]) ([V (x k x l ), P k ]P k + P k [V (x k x l ), P k ]) ([V (x k x l ), P l ]P l + P l [V (x k x l ), P l ]) = ([V ρɛ (x j ), Pj ]P j + Pj [V ρɛ (x j ), P j ]) ([V (x k x l ), P k ]P k + P k [V (x k x l ), P k ]). Moreover [V ρɛ (x j ), P j ]P j + P j [V ρɛ (x j ), P j ] = iɛ( V ρɛ (x j )P j + P j V ρɛ (x j )), while [V (x k x l ), P k ] P k + P k [V (x k x l ), P k ] = iɛ( V (x k x l )P k + P k V (x k x l )).

27 MEA-FIELD AD CLASSICAL LIMITS 27 Therefore i H ρɛ ɛ, ɛ I H H ɛ, I H, + 2 = k,l = (Q j Q j + Pj P j ) ( V ρɛ (x j )P j + Pj V ρɛ (x j )) ( V (x k x l )P k + P k V (x k x l )) ( V ρɛ (x j ) P j ( ) V (x j x k ) V ρɛ (x j ) P j ) V (x j x k ). Thus ɛ trace i H ρɛ ɛ, I H H ɛ, I H, (Q j Q j + Pj P j ) R ɛ, (( ) ) = trace V ρɛ (x j ) V (x j x k ) P j R ɛ, ( ( ) ) trace Pj V ρɛ (x j ) V (x j x k ) R ɛ, trace V ρ ɛ (x j ) 2 V (x j x k ) R ɛ, + trace ( Pj ) P j R ɛ,, so that ɛ trace i H ρɛ ɛ, I H H ɛ, I H, (Q j Q j + Pj P j ) R ɛ, D ɛ, + trace V ρ ɛ (x j ) 2 V (x j x k ) R ɛ,.

28 28 F. GOLSE, C. MOUHOT, AD T. PAUL Therefore, the differential inequality satisfied by D ɛ, is recast as dd ɛ, dt (3 + 4 Lip( V ) 2 )D ɛ, + trace V ρ ɛ (x j ) V (x j x k ) 2 R ɛ, = (3 + 4 Lip( V ) 2 )D ɛ, + trace V ρ ɛ (x j ) V (x j x k ) The last inequality comes from the fact that the multiplication by V ρ ɛ (x j ) 2 V (x j x k ) acts on the variables variables x,..., x only, and that the first marginal R ɛ,, de R ɛ, is ρ ɛ by Lemma 5.. Let us finally estimate the term trace V ρ ɛ (x j ) 2 V (x j x k ) ρ ɛ. By a straightforward computation trace V ρ ɛ (x j ) = Set V (x j x )ρ ɛ (t, x, x )dx 2 ρ ɛ. V (x j x k ) V (x j x k ) f ɛ (t, x) := ρ ɛ (t, x, x), 2 l= 2 ρ ɛ ρ ɛ (t, x l, x l )dx, where ρ ɛ is the solution of the Hartree equation (34). Since ρ ɛ (t) D(H), then f ɛ (t) is a probability density on R d for each ɛ > 0 and each t 0. Applying Lemma 3.3 with p = and ρ = f ɛ (t, ) shows that with Therefore trace V ρ ɛ (x j ) V (x j x k ) Gronwall s inequality implies that 2 ρ ɛ dd ɛ, (t) ΛD ɛ, (t) + 8 dt V 2 L Λ := Lip( V ) 2. D ɛ, (t) D ɛ, (0)e Λt + 8 V 2 L e Λt Λ 8 V 2 L..

29 MEA-FIELD AD CLASSICAL LIMITS From D ɛ, to MK2(ρ ɛ ɛ (t), ρ ɛ, (t)). Let us denote by Rɛ, n the marginal density of R ɛ, corresponding to the n first particles. In other words, for each A L(H 2n ), we set where We claim that trace H2n (AR n ɛ, ) = trace H2 (S(A I H2( n) )S R ɛ, ) SΨ(x, y, x 2, y 2,..., x, y ) = Ψ(x,..., x, y,..., y ). (40) R n ɛ, Q(ρ n ɛ, ρ n ɛ, ). Indeed, for each B L(H n ), by Lemma 5., one has while trace H2n ((B I Hn )R n ɛ, ) = trace H2 (S(B I Hn I H2( n) )S R ɛ, ) = trace H2 ((B I H n ) I H )R ɛ, ) = trace H2 ((B I H n )R ɛ,, ) = trace H ((B I H n )ρ ɛ ) = trace Hn (Bρ n ɛ ), trace H2n ((I Hn B)R n ɛ, ) = trace H2 (S(I Hn B I H2( n) )S R ɛ, ) Observe further that, for each j =,...,, one has = trace H2 (I H (B I H n )R ɛ, ) = trace H2 ((B I H n )R ɛ,,2 ) = trace H ((B I H n )ρ ɛ, ) = trace Hn (Bρ n ɛ, ). trace H2 ((Q j Q j + P j P j )R ɛ, ) = trace H2 (T σ j (Q Q + P P )T σj R ɛ, ) = trace H2 ((Q Q + P P )T σj R ɛ, T σ j ) = trace H2 ((Q Q + P P )R ɛ, ), where σ j is the permutation of {,..., } exchanging and j and leaving all the other integers invariant, because of the identity (38) in Lemma 5.. Hence, for each n, each ɛ > 0 and each t 0, one has D ɛ, (t) = trace H2 ((Q j Q j + Pj P j )R ɛ, (t)) for each j =,...,, = n n trace H 2 (Q j Q j + Pj P j )R ɛ, (t) = n n trace H 2n (Q j Q j + Pj P j )Rɛ, n (t). Because of (40), this implies that D ɛ, (t) n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)),

30 30 F. GOLSE, C. MOUHOT, AD T. PAUL for each n, each ɛ > 0 and each t 0. In particular, (4) MK2(ρ ɛ ɛ (t) n, ρ n ɛ, (t)) nd ɛ, (0)e Λt + 8n V e Λt 2 L. Λ This inequality holds for each Rɛ, in condition (37). Q((ρin ɛ ), ρ in ɛ, ) satisfying the symmetry 5.4. Upper bound for D ɛ, (0). Let Q in ɛ, Q((ρin ɛ ), ρ in ɛ, ), and set R in ɛ, :=! T σ Q in ɛ, T σ S By construction, R in ɛ, D(H H ) and satisfies (37). On the other hand, for each A L(H ), one has trace H H ((I H since ρ in ɛ, A)R in ɛ, ) =! σ. trace H H ((I H σ S = trace H H! (Tσ (I H σ S = trace H H! ((I H σ S = trace H (τ! σaτ σ ρ in ɛ, ) σ S = trace H (Aτ σ ρ in! ɛ, τσ) σ S = trace H (Aρ in ɛ, ) satisfies the symmetry condition (0). By the same token trace H H ((A I H )R in ɛ, ) = trace H (A(ρ in ɛ ) ) A)T σ Q in ɛ, T σ ) A)T σ Q in ɛ, ) (τ σaτ σ ))Q in ɛ, ) for each A L(H ). Hence Rɛ, in Q((ρin ɛ ), ρ in ɛ, ). On the other hand D ɛ, (0) = trace H2 (Q j Q j + Pj P j )R in ɛ, = trace H2 (Q j Q j + Pj P j ) T σ Q in! ɛ, T σ σ S = trace H2 Tσ (Q j Q j + Pj P j ) T σ Q in ɛ,! σ S = trace H 2 (Q j Q j + Pj P j )Q in ɛ,

31 MEA-FIELD AD CLASSICAL LIMITS 3 because T σ (Q j Q j + Pj P j ) T σ = for each σ S. Inserting this in (4) shows that (Q j Q j + Pj P j ) n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)) 2 8 V e Λt 2 L Λ + eλt trace H 2 (Q j Q j + Pj P j )Q in for all Q in ɛ, Q((ρin ɛ ), ρ in ɛ, ) with n. Minimizing over Qin ɛ, ɛ, leads to n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)) 2 8 V e Λt 2 L + eλt Λ MKɛ 2((ρ in ɛ ), ρ in ɛ, ) which is precisely inequality (5). ext, in the special case where ρ in ɛ while ρ in ɛ, = OPT ɛ ((2πɛ) d µ in ɛ, (ρ in ɛ ) = OP T ɛ ((2πɛ) d (µ in ) with µin ɛ, = OP T ɛ ((2πɛ) d µ in ɛ ) with µ in ɛ P 2 (C d ), P 2((C d ) ) satisfies (0), one has ɛ ) ), and we deduce from Theorem 2.3 (2) that MK ɛ 2((ρ in ɛ ), ρ in ɛ, ) dist MK,2 ((µ in ɛ ), µ in ɛ, ) 2 + 2dɛ. Inserting this in (5) shows that n MKɛ 2(ρ ɛ (t) n, ρ n ɛ, (t)) 2 8 V e Λt 2 L Λ + e Λt ( dist MK,2((µ in ɛ ), µ in ɛ, ) 2 + 2dɛ which is (6). This concludes the proof of Theorem 2.4. Remark 5.2. Let ρ ɛ be the solution of the Cauchy problem for the Hartree equation with initial data ρ in ɛ D(H) such that trace(( x 2 ɛ 2 )ρ in ɛ ) <. Then, for each ɛ > 0 and each t 0, one has trace( x 2 ρ ɛ (t)) (trace( x 2 ρ in ɛ ) + te t (trace( ɛ 2 ρ in ɛ ) + 2 V L ) <, as a consequence of the differential inequality d dt trace ( x 2 ρ ɛ (t) ) = ( trace [ 2 ɛ, x 2 ]ρ ɛ (t) ) (( trace x 2 ɛ 2 ) ρ ɛ (t) ) and of the energy conservation ( d trace( ɛ 2 ρ ɛ (t)) + dt In particular, for each t 0 and each ɛ > 0, one has ) V (x z)ρ ɛ (t, z, z)ρ ɛ (t, x, x)dxdz = 0. trace(( x 2 ɛ 2 )ρ ɛ (t)) <. )