## Abstract

The present work discusses the mean-field limit for the quantum *N*-body problem in the semiclassical regime. More precisely, we establish a convergence rate for the mean-field limit which is uniform as the ratio of Planck constant to the action of the typical single particle tends to zero. This convergence rate is formulated in terms of a quantum analogue of the quadratic Monge–Kantorovich or Wasserstein distance. This paper is an account of some recent collaboration with C. Mouhot, T. Paul and M. Pulvirenti.

This article is part of the themed issue ‘Hilbert’s sixth problem’.

## 1. Introduction

The Vlasov equation with *C*^{1,1} interaction potential has been derived from the *N*-body problem of classical mechanics in the large *N*, small coupling constant limit: see the works of Neunzert & Wick [1], Braun & Hepp [2] and Dobrushin [3].

On the other hand, the Hartree equation with bounded interaction potential has been derived from the *N*-body linear Schrödinger equation in the same limit: see the works of Spohn [4], Bardos *et al*. [5]. Extensions to singular interaction potentials (including the Coulomb potential) can be found in the work of Erdös & Yau [6]. Convergence rate estimates have been obtained by Rodnianski & Schlein [7], and Pickl [8] by a particularly simple argument, which seems unfortunately limited to the case of pure state initial data.

This suggests the following problem.

*Problem:* is the mean-field () limit of the quantum *N*-body problem uniform in the classical limit ()?

The first results on this problem have been obtained in the works of Graffi *et al*. and Pezzoti & Pulvirenti [9,10].

This situation is perhaps best explained by the following diagram. In this diagram, the horizontal arrows correspond to the mean-field limit—i.e. to the limit as the particle number *N* tends to infinity, while is kept fixed. The vertical arrows correspond to the classical limit of quantum dynamics—i.e. to the limit as tends to zero. The boxes refer to the governing equations corresponding to the various asymptotic regimes considered.

The first principle equation considered here is the Schrödinger equation for a system of *N* identical particles coupled by some appropriately scaled potential. The equation obtained from the *N*-body Schrödinger equation in the semiclassical regime is the *N*-body Liouville equation governing the evolution of the *N*-body distribution function in phase space. In the large *N* limit, the Hartree equation describes the mean-field limit of the *N*-body Schrödinger equation, while the Vlasov equation describes the mean-field limit of the *N*-body Liouville equation.

Our approach to this problem follows Dobrushin’s derivation [3] of the Vlasov equation from the *N*-body problem in classical mechanics (the bottom horizontal arrow). More precisely, the classical *N*-body problem is the system of ordinary differential equations
1.1Here *V* ≡*V* (*z*)∈*C*^{1,1}(**R**^{d}) is even, so that ∇*V* (*x*_{k}−*x*_{k})=0.

Dobrushin’s argument can be formulated in terms of a convergence rate involving the Monge-Kantorovich(-Rubinshtein), or Vasersthein distances, which we briefly recall. For *p*≥1, let be the set of Borel probability measures *μ* on **R**^{n} with bounded moment of order *p*, i.e.
For all , we call a *coupling* of *μ* and *ν* an element *π* of such that
for all *ϕ*,*ψ*∈*C*_{b}(**R**^{n}). The set of couplings of *μ*,*ν* is denoted by *Π*(*μ*,*ν*). Then we define the Monge–Kantorovich distance of exponent *p* by the formula
This distance metrizes the topology of weak convergence on . More precisely:
as . See ch. 7 in [11] for more details on these distances.

Let *f*≡*f*(*t*,*x*,*ξ*) be a solution of the Vlasov equation
1.2where
and where ⋆_{x} designates the convolution in the variable *x*. Elementary arguments show that, if *f*^{in} is a probability density on **R**^{d}×**R**^{d} and satisfies
then *f*(*t*,⋅,⋅) is a probability density on **R**^{d}×**R**^{d} for all *t*∈**R**, and satisfies
for all *t*∈**R**. Set
Dobrushin [3] proves that
where *L*:=Lip(∇*V*). Choose *x*_{k}(0),*ξ*_{k}(0) for *k*=1,…,*N* so that *d*_{N}(0)→0 as . (For instance, one can choose the 2*d*-tuples (*x*_{k}(0),*ξ*_{k}(0)) i.i.d. with distribution *f*|_{t=0}; this implies that *d*_{N}(0)→0 by the law of large numbers.)

In the present work, we define quantum analogues of the main elements in Dobrushin’s argument. This involves various difficulties, which are addressed as explained below.

The present paper is an account of some recent joint work in collaboration with C. Mouhot, T. Paul and M. Pulvirenti [12–15].

The results presented here belong to the more general class of rigorous derivations of simplified models describing the dynamics of large particle systems, obtained by some appropriate mathematical limiting process—as explicitly suggested in Hilbert’s sixth problem.

## 2. An Eulerian variant of Dobrushin’s argument

A first obvious difficulty is that Dobrushin’s convergence rate estimate involves transportation of statistical information along particle trajectories in a rather essential manner. Unfortunately, there is no analogue of the notion of particle trajectories in quantum mechanics. As a warm-up, we shall explain how to circumvent the need for particle trajectories in the proof of the mean-field limit in the context of classical mechanics.

First we introduce some notations for *N*-particle systems. We shall denote
where *x*_{k},*y*_{k},*ξ*_{k},*η*_{k}∈**R**^{d}. For , set

A Borel probability measure on **R**^{dN}×**R**^{dN} is said to be symmetric if it is invariant under all the transformations of the form
The set of symmetric Borel probability measures on **R**^{dN}×**R**^{dN} is denoted .

For each and *n*=1,…,*N*−1, we denote by the *n*-particle marginal of *F*_{N}, defined by the formula
Equivalently, for each *ϕ*∈*C*_{b}(**R**^{dn}×**R**^{dn}), one has

The *N*-particle Liouville equation is
2.1The unknown is *F*_{N}≡*F*_{N}(*t*,*Y*_{N},*H*_{N}), the *N*-particle distribution function, and its initial data are the symmetric Borel probability measure *F*^{in}_{N}. Observe that the characteristic curves of the Liouville equation is precisely the system of Newton’s differential equations governing the dynamics of *N*-particle systems. The method of characteristics applied to the Liouville equation implies that *F*_{N}(*t*) is the image of *F*^{in}_{N} under the section at time *t* of the flow of Newton’s system of equations (1.1). This flow is global by the Cauchy–Lipschitz theorem since *V* ∈*C*^{1,1}(**R**^{d}). With the expression for *F*_{N}(*t*) so obtained, one sees that for all *t*∈**R** if .

Our purpose in the present section is to find a bound for instead of as in Dobrushin’s original proof. We seek an argument avoiding systematically the use of particle trajectories. Besides, all the steps involved in this new proof should have clear quantum analogues.

### Theorem 2.1.

*Let* *be an even function. Let f*^{in} *be a probability density on* *such that
**and let f be the solution of the Cauchy problem for the Vlasov equation* (*1.2*) *on* *with initial data f*^{in}*. On the other hand, let* *and let F*_{N} *be the solution of the Cauchy problem for the N-particle Liouville equation* (*2.1*) *with initial data F*^{in}_{N}*. Then
**for all t*≥0, *with*

### (a) Sketch of the proof

In order to get a feeling of the argument to be adapted to the quantum setting, we briefly sketch the proof of theorem 2.1.

#### (i) A dynamics for couplings

A symmetric coupling of two symmetric *N*-particle probability measures *F*_{N} and *G*_{N} is an element of *Π*(*F*_{N},*G*_{N}) which is invariant under all transformations of the form
for . The set of symmetric couplings of *F*_{N} and *G*_{N} is denoted *Π*^{s}(*F*_{N},*G*_{N}).

### Lemma 2.2.

*Under the assumptions of theorem 2.1, let* . *Let t*↦*P*(*t*) *with values in* *be the solution of the Cauchy problem*
*Then P*(*t*)∈*Π*^{s}(*f*(*t*)^{⊗N},*F*_{N}(*t*)) *for each t*≥0.

### Proof.

Integrate both sides of the equation for *P* in (*Y*_{N},*H*_{N}) and in (*X*_{N},*Ξ*_{N}), and use the uniqueness property for the Vlasov and the Liouville equations to prove that *P*(*t*) is a coupling of *f*(*t*)^{⊗N} and *F*_{N}(*t*).

For and *P*∈*Π*((*f*(*t*)^{⊗N},*F*_{N}(*t*)) denote *P*^{σ}(*t*) the image of *P*(*t*) under the transformation
2.2Observing that *t*↦*P*(*t*)−*P*^{σ}(*t*) is a solution of the Cauchy problem above with initial condition *P*(0)−*P*^{σ}(0)=0, we use the uniqueness property for a transport equation with Lipschitz continuous coefficients to conclude that *P*(*t*)−*P*^{σ}(*t*)=0 for all *t*∈**R**. ▪

#### (ii) The functional *D*_{N}(*t*)

For each , set
where *P* is the solution of the Cauchy problem in lemma 2.2.

### Lemma 2.3.

*For all* *one has*

### Proof.

By the symmetry of *P*(*t*), one has
for all *j*=1,…,*N*. One concludes with the definition of the distance dist_{MK,2}, by setting *j*=1 in the identity above and observing that
is a coupling of *f*(*t*) and *F*^{1}_{N}(*t*). ▪

In this setting, obtaining a bound on is a problem analogous to finding moments bounds for solutions of a first-order partial differential equation.

#### (iii) Consistency versus stability

First we seek some information on the dynamics of *D*_{N}(*t*). It will be convenient to use the following notation: for *Y*_{N}=(*y*_{1},…,*y*_{N}), we set

Multiplying by (1/*N*)(|*X*_{N}−*Y*_{N}|^{2}+|*Ξ*_{N}−*H*_{N}|^{2}) each side of the equation for *P* in lemma 2.2 and integrating in all variables, one finds that
The first term on the right-hand side of this inequality comes from the obvious inequality
since *P*(*t*)≥0 and

Next
since ∇*V* is Lipschitz continuous. Hence

On the other hand, so that

### Lemma 2.4.

*Under the same assumptions as in theorem 2.1, one has*

### Proof.

One has
so that
Now, for all 1≤*j*<*k*≤*N*, one has
since
for all 1<*k*≤*N*. Hence
▪

#### (iv) Conclusion

Summarizing, we have proved that
or, equivalently,
Integrating both sides of this inequality on [0,*T*], and applying lemma 2.3, one finds that

On the other hand, let *Q*^{in}_{N} be an optimal coupling of (*f*^{in})^{⊗N} and *F*^{in}_{N}. Note that, although (*f*^{in})^{⊗N} and *F*^{in}_{N} are both symmetric, *Q*^{in}_{N} is not necessarily a symmetric coupling. However,
for all , where (*Q*^{in}_{N})^{σ} is the image of *Q*^{in}_{N} under the transformation (2.2). Hence
and this concludes the proof.

### (b) Remarks

Various remarks are in order to conclude this section on classical mechanics.

(1) The proof of theorem 2.1 is stated here in terms of the quadratic Monge–Kantorovich distance dist

_{MK,2}. However, it is clear from the proof presented above that exactly the same argument works for all*p*≥1, provided that . Dobrushin’s original argument [3] is formulated in terms of the Monge–Kantorovich distance dist_{MK,1}with exponent 1, but can be adapted to all exponents*p*≥1. (Note however that Dobrushin’s original argument is perhaps slightly more complicated in the case*p*>1, whereas the Eulerian argument presented above is essentially the same for all*p*.) However, only the case*p*=2 seems to have an obvious quantum analogue.(2) Extending the validity of theorem 2.1 to the case of a Coulomb interaction potential (leading to the well-known Vlasov–Poisson system) remains open at the time of this writing [16].

## 3. A quantum analogue of the quadratic Monge–Kantorovich distance

The quantum analogue of (Borel) probability measures on the phase space **R**^{d}×**R**^{d} is the notion of ‘density operator’ on the Hilbert space . A density operator on is a bounded operator *R* on satisfying
The set of density operators on is denoted .

By analogy with the definition of the Monge–Kantorovich distances, we next introduce the notion of *coupling between two density operators*. Let *R*_{1} and ; a coupling of *R*_{1} and *R*_{2} is an element *R* of such that
for all . The set of all couplings of *R*_{1} and *R*_{2} will be henceforth denoted .

Finally, for all , we define

Unfortunately, is not a distance on , as shown by the following lemma.

### Lemma 3.1.

*For all* *one has*

### Proof.

Set *A*_{j}:=*x*_{j}−*y*_{j} and for *j*=1,…,*d*. Then
Hence, for all

However, can be compared to the quadratic Monge–Kantorovich distance between classical probability densities associated with the quantum densities *R*_{1} and *R*_{2}. Before stating our main result in this direction, we recall two definitions.

The Wigner transform at scale of an operator (with ) is the function defined by the formula
where *r*≡*r*(*X*,*Y*) is the integral kernel of *R*. The Husimi transform at scale of is defined in terms of its Wigner transform by the formula

At variance with the Wigner transform, the Husimi transform of a density operator on is always non-negative. Indeed, one can check by an elementary computation (involving the Fourier transform of a Gaussian distribution) that
In the formulae above, we have used Dirac bra–ket notation. One recognizes in |*q*,*p*〉 the Schrödinger coherent state (see [17], §23, Problem 3). More details on the Wigner and Husimi transforms can be found in [18].

Finally, we recall the notion of Töplitz quantization. First, we recall that the family of Schrödinger coherent states recalled above provides a resolution of the identity, *viz*.
This suggests that one can associate with each Borel positive or bounded measure on the single particle phase space **R**^{d}×**R**^{d} the unbounded operator on
whose form domain is
The most fundamental properties of these Töplitz operators are
We also recall two important formulae for the Wigner and the Husimi transforms of Töplitz operators:
In particular, the second identity above suggests that, except for the unessential prefactor , Töplitz quantization can be thought of as being almost the inverse of the Husimi transform—viewing as a near identity transformation. See Appendix B of [12] for more details on the mathematical objects discussed in this section.

The main properties of the pseudo-distance are summarized in the following theorem.

### Theorem 3.2.

*For all R*_{1} *and* *one has
*

*Let* *for j*=1,2, *and let* . *Then*

In other words, although is not a distance on , it is an perturbation of the quadratic Monge–Kantorovich distance between appropriate classical densities (Töplitz symbol, or Husimi transform).

## 4. The mean-field limit in quantum mechanics

The quantum analogue of the system of Newton’s differential equations for the dynamics of *N* identical particles is the *N*-body Schrödinger equation, with unknown the *N*-body wave function *Ψ*_{N}≡*Ψ*_{N}(*t*,*x*_{1},…,*x*_{N})∈**C**:
4.1The quantum analogue of the Vlasov equation is the Hartree equation, with unknown the 1-body wave function *ψ*≡*ψ*(*t*,*x*)∈**C**:
4.2where *r*≡*r*(*t*,*X*,*Y*) is the integral kernel of *R*(*t*). In both cases, these wave functions satisfy the normalization condition

However, the analogy between quantum mechanics and classical mechanics is better understood in terms of density operators. A typical example of *N*-particle (respectively, 1-particle) density operator is |*Ψ*_{N}(*t*,⋅)〉〈*Ψ*_{N}(*t*,⋅)| (respectively, |*ψ*(*t*,⋅)〉〈*ψ*(*t*,⋅)|). The quantum analogue of the *N*-particle Liouville equation in classical mechanics is the *N*-particle Heisenberg equation
4.3where *t*↦*R*_{N}(*t*) is a curve parametrized by the time variable *t*, with values in . (The notation designates .) One easily checks that |*Ψ*_{N}(*t*,⋅)〉〈*Ψ*_{N}(*t*,⋅)| is a solution of the *N*-particle Heisenberg equation (4.3) if *Ψ*_{N}(*t*,⋅) is a solution of the *N*-particle Schrödinger equation (4.1).

Likewise, the density operator formulation of the Hartree equation is
4.4Here, *t*↦*R*(*t*) is a time-dependent element of (the set of 1-particle density operators).

A last ingredient in the setting of the mean-field limit in quantum mechanics is the notion of indistinguishable particles. Denote by *σ*↦*U*_{σ} the representation of the symmetric group defined by
The symmetry property on quantum density operators which characterizes systems of indistinguishable particles is
The set of density operators satisfying this invariance property is denoted . One can check easily that the solution of the *N*-particle Heisenberg equation *R*_{N}(*t*) belongs to for all *t*∈**R** if its initial condition belongs to .

The main result in this paper is the following convergence rate estimate for the mean-field limit of the *N*-body Heisenberg equation, which is formulated in terms of the quadratic Monge–Kantorovich distance between the Husimi transforms of the Hartree solution and of the first marginal of the *N*-particle Heisenberg solution.

### Theorem 4.1.

*Assume that the potential V is even and satisfies* *. Let* *be the solution of Hartree’s equation* (*4.4*) *with initial data* *and let* *be the solution of Heisenberg’s equation* (*4.3*) *with initial data* *. Then, for all t*≥0
*with
*

The proof of this theorem follows very closely that of theorem 2.1. We proceed by analogy between the classical and the quantum computations, using the (well-known) ‘dictionary’ summarized in the table below.

Of course, it remains to find initial data for which the quantity
is small. (The quantity above is typically of order one, since the 1/*N* prefactor is just a scaling taking into account the number of particles.) This can be done by applying the second statement in theorem 3.2, choosing initial data which are Töplitz operators.

### Theorem 4.2.

*Under the same assumptions as in theorem 4.1, assume that* *and* *are Töplitz operators of symbols* *and* *with
**Then, for each n*=1,…,*N and each t*≥0, *one has*

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## Competing interests

I declare I have no competing interests.

## Funding

No funding has been received for this article.

## Footnotes

One contribution of 14 to a theme issue ‘Hilbert's sixth problem’.

- Accepted November 26, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.