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Thierry Paul, Mario Pulvirenti
To cite this version:
Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. DCDS-A, 2020, 39 ((4)), pp.1891-1921. �hal-01579763v5�
APPROXIMATION
2
THIERRY PAUL AND MARIO PULVIRENTI
3
Abstract. We consider theN-body quantum evolution of a particle system in the mean-field approximation. We show that thejth order marginalsFjN(t), for factorized initial dataF(0)⊗N, are explicitly expressed, moduloN−∞, out of the solutionF(t)of the corresponding non-linear mean-field equation and the solution of its linearization around F(t). The result is valid for all timest, uniformly in j=O(N12−α) for any α>0. We establish and estimate the full asymptotic expansion in integer powers of N1 ofFjN(t),j=O(√
N), whose computation at order ninvolves a finite number of operations depending onjandnbut not onN. Our results are also valid for more general models including Kac models. As a by-product we get that the rate of convergence to the mean-field limit inN1 is optimal in the sense that the first correction to the mean-field limit does not vanish.
Contents
4
1. Introduction 1
5
2. Quantum mean-field 4
6
3. Asymptotic expansion and main result 8
7
4. Proofs of Theorems 3.1 and 3.5 15
8
4.1. Recursive construction and proof of Theorem 3.1 (i)-(ii) 16
9
4.2. Estimates and proof of Theorem 3.1 (iii) 17
10
4.3. Computability and proof of Theorem 3.5 21
11
5. The Kac and “soft spheres” models 25
12
Appendix A. The asbtract model 27
13
A.1. The model 27
14
A.2. Main results similar to [26] 34
15
A.3. Asymptotic expansion 34
16
Appendix B. Derivation of the correlation hierarchy (122) 35
17
References 40
18
1. Introduction
19
The mean-field limit concerns systems of interacting (classical or quantum) particles
20
whose number diverges in a way linked with a rescaling of the interaction insuring an
21
equilibrium between interaction and kinetic energy. In the case of an additive one-body
22
kinetic energy part and a two-body interaction, and without taking in consideration
23
1
quantum statistics, this equilibrium is reached by putting in front of the interaction a
1
coupling constant proportional to the inverse of the number of particles.
2
The system is then described by isolating the evolution of one (or j) particle(s) and
3
averaging over all the other. This leads to a partial information on the system driven
4
by the so-called j-marginals. The mean-field theory ensures that the j-marginals tend,
5
as the number of particles diverges, to the j-tensor power of the solution of a non-linear
6
one-body mean-field equation (Vlasov, Hartree,...) issued from the 1-marginal on the
7
initial N-body state. This program has been achieved in many different situations,
8
and the literature concerning the mean-field approach is enormous. We refer to [30]
9
for a review and recent references.
10
As regards the fluctuations around this limit, namely the correction to be added to
11
the factorized limit in order to get better approximations of the true evolution of the
12
j-marginals, there are some results.
13
The identification of the leading order of these fluctuations with a Gaussian sto-
14
chastic process has been established in the quantum context first in [16] and in the
15
classical one in [5]. For the classical dynamics of hard spheres, the fluctuations around
16
the Boltzmann equation have been computed at leading order in [29], generalizing to
17
non-equilibrium states the results of [3]. More recently, for the quantum case, fluctu-
18
ations near the Hartree dynamics have been derived in [23] (after [22]) and in [2] also
19
for the grand canonical ensemble formalism (number of particles non fixed), using in
20
both cases the methods of second quantization (Fock space) (see also [25] for a proof
21
using the usual quantization formalism). In the case of pure states, the N-body wave
22
function is shown to be √1
N-close in L2 norm to a sum of partially factorized states
23
constructed out of the so-called Bogoliubov hierarchy. Note that these results rise a
24
problem fundamentally different from the one treated in the present paper, whose goal
25
is to compute mean-field approximation of the N-body problem with an accuracy of
26
any order in powers of N1.
27
Nevertheless the basic object of our analysis, namely the kinetic error Ej (see below
28
for the definition), is strictly related to the expectation of the fluctuations of intensive
29
observables (see e.g. [5]). However the analysis of the fluctuations problem from
30
the present point of view requires an additional analysis which goes beyond the main
31
purposes of the present paper.
32
Recently, we developed (together with S. Simonella) in [26] a method to derive mean-
1
field limits, alternative to the ones using empirical measures or direct estimates on the
2
“BBGKY-type” hierarchies (systems of coupled equations satisfied by the set of the
3
j-marginals). This method rather uses the hierarchy followed by the “kinetic errors”
4
Ej−k (defined below), already used (under the name “v-functions”) to deal with kinetic
5
limits of stochastic models [10, 7, 4, 11, 12, 6, 8, 13] and recently investigated in the
6
more singular low density limit of hard spheres [27] (note that error terms are also used
7
in [23, 22, 2, 25] for the total (pure state) wave function with a quite different point
8
of view). These quantities are, roughly speaking, the coefficient of the decomposition
9
of the j-marginal as a linear combination of the k-th tensor powers, k = 1, . . . , j, of the
10
solution of the mean-field equation issued from the 1-marginal of the initial full state.
11
We developed in [26] a strategy suitable in particular for Kac models (homogeneous
12
original one [17, 18] and non-homogeneous [9]) and quantum mean-field theory. This
13
strategy allowed us to derive the limiting factorization property of the j-marginals up
14
to, roughly speaking, j ≲ √
N. This threshold is, on the other side, the one obtained
15
by heuristic arguments as shown in [26] and rigorously in [15] for the Kac’s model. At
16
the contrary, let us recall that the quantum mean-field limit was obtained in [19] for
17
marginals of order j = o(N), for pure states initial data.
18
Here and in all this article, N denotes the number of particles of the system under
19
consideration.
20
In the present paper we provide and estimate a full asymptotic expansion in powers of
21
1
N of the difference between the evolution of j-marginals and its factorized leading order
22
form (Theorem 3.2), following a similar result for the kinetic errors Ej(t) (Theorem
23
3.1). Our results are valid for j ≤ C√
N for some explicit constant C and are valid for
24
quantum, Kac’s models and in the framework of the abstract formalism, slightly more
25
general than the one developed in [26], described in Appendix A.
26
The non-vanishing of the first correction is established, showing therefore that the
27
rate of the mean-field convergence is at most of order N1 (Corollary 3.4) : we will
28
prove that the first marginals (of order j = 1,2) are kept away from their mean-field
29
factorized limits at a distance bounded from below by CN−1, C > 0 (and not, e.g. N−2)
30
as N → ∞1
31
1This has not to be mistaken with the problem of optimality in j→ ∞versusN→ ∞for which it has been established in [19] a rate of convergence in√j
N, for pure states initial data.
Moreover, as the mean-field solution issued from the first marginal of the N body
1
symmetrical factorized initial data determines the leading order of the j-marginal, we
2
show that the additional knowledge of the linearization of the mean-field flow around
3
it gives an explicit construction of the full asymptotic expansion of the j-marginals in
4
powers of N1 uniformly in j, N satisfying j ≤ CN12−α for any C, α > 0 (Theorem 3.5).
5
Let us note the analogy with the quantum propagation of semiclassical observables,
6
driven by the classical underlying flow at leading order in the Planck constant, and
7
whose full asymptotic expansion is explicitly computable by the only knowledge of the
8
linearized flow.
9
Let us summarize in words our main result:
10
The knowledge of the mean-field flowF(t)and its linearization aroundF(t) determines
11
explicitly, modulo N−∞, uniformly for j = O(N12−α), α > 0, the j-marginals of the N-
12
body flow issued from F(0)⊗N.
13
2. Quantum mean-field
14
LetL1(L2(Rd))be the space of trace class operators onL2(Rd), with their associated
15
norms.
16
We consider the evolution of a system of N quantum particles interacting through a (real-valued) two-body, even potentialV, described for any value of the Planck constant h̵ > 0 by the Schr¨odinger equation
i̵h∂tψ = HNψ , ψ∣t=0 =ψin ∈HN ∶= L2(Rd)⊗N ,
where
HN ∶= −h̵2 2
N
∑
k=1
∆xk + 1
2N ∑
1≤k,l≤N k≠l
V(xk−xl).
We assume in this paper that V is bounded, which implies that the N-body Hamil-
17
tonian HN is self-adjoint on H2(Rd), the second Sobolev space.
18
Instead of the Schr¨odinger equation written in terms of wave functions, we shall rather consider the quantum evolution of density matrices. An N-body density matrix is an operator FN such that
0≤ FN = (FN)∗, traceHN(FN) =1.
The evolution of the density matrix FN ↦FN(t) of a N-particle system is governed
1
for any value of the Planck constant h̵ > 0 by the von Neumann equation
2
(1) ∂tFN = 1
i̵h[HN, FN],
equivalent to the Schr¨odinger equation when FN(0) is a rank one projector, modulo a
3
global phase.
4
Positivity, norm and trace are obviously preserved by (1) since HN is self-adjoint.
5
For each j = 1, . . . , N, the j-particle marginal FjN(t) of FN(t) is the unique trace class operator on Hj such that
traceHN[FN(t)(A1⊗ ⋅ ⋅ ⋅ ⊗Aj ⊗IHN−j)] = traceHj[FjN(t)(A1⊗ ⋅ ⋅ ⋅ ⊗Aj)].
for all A1, . . . , Aj bounded operators on H. Alternatively and equivalently, the FjN can
6
be defined by the partial trace ofFN on the N−j last “particles”: defining FN through
7
its integral kernel FN(x1, x′1;. . .;xN, x′N), the integral kernel of FjN is defined as (see
8
[1])
9
FjN(x1, x′1;. . .;xj, x′j) ∶= (Trj+1. . .TrNFN)(x1, x′1;. . .;xj, x′j)
∶= ∫Rd(N−j)FN(x1, x′1;. . .;xj, x′j;xj+1, xj+1;. . .;xN, xN)dxj+1⋯dxN. (2)
It will be convenient for the sequel to rewrite (1) in the following operator form
10
(3) ∂tFN = (KN +VN)FN
where KN, VN are operators on L1(L2(RN d)) defined by
11
(4) KN = 1 i̵h[−h̵2
2 ∆RdN,⋅], VN = 1 2N ∑
k,l
Vk,l with Vk,l ∶= 1
i̵h[V(xk−xl),⋅]. The self-adjointness of HN implies that
12
(5) ∥et(KN+VN)∥L1(L2(Rd))→L1(L2(Rd)) = ∥etKN∥L1(L2(RN d))→L1(L2(RN d)) = 1, t∈ R.
We will denote
13
(6) L∶= L1(L2(Rd)) so that L⊗n = L1(L2(Rnd)), n= 1, . . . , N, and, with a slight abuse of notation,
14
(7) ⎧⎪⎪
⎨⎪⎪⎩
∥⋅∥1 the trace norm on any L⊗j,
∥⋅∥ the operator norm on any L(L⊗i,L⊗j)
for i, j = 1, . . . , N (here L(L⊗i,L⊗j) is the set of bounded operators form L⊗i to L⊗j).
15
A density matrixFn ∈L⊗n is called symmetric if its integral kernelFn(x1, x′1;. . .;xn, x′n) is invariant by any permutation
(xi, x′i) ↔ (xj, x′j), i, j = 1, . . . , n.
Note that the symmetry of FN is preserved by the equation (1) due to the particular
1
form of the potential.
2
We define, for n=1, . . . , N,
3
(8) Dn = {F ∈L⊗n ∣ F > 0, ∥F∥1 =1 and F is symmetric}.
Note that FjN ∈ L⊗j (F0N = 1 ∈ L⊗0 ∶= C) and FjN > 0,∥FjN∥1 = ∥FN∥1, and obviously FjN is symmetric as FN. That is to say:
FjN ∈ Dj.
The family of j-marginals, j = 1, . . . , N, is solution of the BBGKY hierarchy of
4
equations (see [28] and also [1])
5
(9) ∂tFjN = (Kj + Tj
N)FjN + (N −j)
N Cj+1Fj+1N where:
6
(10) Kj = 1
i̵h[−h̵2
2 ∆Rjd,⋅]
7
(11) Tj = ∑
1≤i<r≤j
Ti,r with Ti,r = Vi,r and
8
(12) Cj+1Fj+1N = ∑j
i=1
Ci,j+1Fj+1N with
9
Ci,j ∶ L⊗(j+1) → L⊗j
Ci,j+1Fj+1N = Trj+1(Vi,j+1Fj+1N ) , (13)
where Trj+1 is the partial trace with respect to the (j+1)th variable, as in (2).
10
Note that, for all i≤ j = 1, . . . , N,
11
(14) ∥Tj∥ ≤j2∥V∥L∞
h̵ , and ∥Ci,j+1∥ ≤ ∥V∥L∞ h̵ .
(meant for ∥Tj∥L⊗j→L⊗j and ∥Ci,j+1∥L⊗(j+1)→L⊗j in accordance with (7)).
12
The Hartree equation is
1
(15) i̵h∂tF = [−h̵2
2 ∆+VF(x), F], F(0) ∈ D1,
where VF(x) = ∫RdV(x−y)F(y, y)dy, F(y, y′) being the integral kernel of F.
2
Note that (15) reads also
3
(16) ∂tF = K1F +Q(F, F),
with
4
(17) Q(F, F) =Tr2(V1,2(F ⊗F)).
Since V is bounded, (15) has for all time a unique solution F(t) > 0 and ∥F(t)∥1 = 1
5
(see again [28] and [1]).
6
In order to define the correlation error in an easy way, we need a bit more of notations
7
concerning the variables of integral kernels.
8
For i ≤ j = 1, . . . , N, we define the variables zi = (xi, x′i), and Zj = (z1, . . . , zj). For
9
{i1,⋯, ik} ⊂ {1,⋯, j}, we denote by Zj/{i1,⋯,ik} ∈ R2(j−k)d, the vector Zj ∶= (z1, . . . , zj)
10
after removing the components zi1, . . . zik.
11
Definition 2.1. For any j = 1, . . . , N, we define the correlation error Ej ∈ L⊗j by its
12
integral kernel
13
(18) Ej(Zj) = ∑j
k=0 ∑
1≤i1<⋅⋅⋅<ik≤j
(−1)kF(zi1). . . F(zik)Fj−kN (Zj/{i1,⋯,ik}). By convention and consistently we set
14
(19) F0N = ∥F∥1 = 1, E0 ∶= 1∈ L⊗0 ∶=C.
In [26] it was shown that (18) is inverted by the following equality:
15
(20) FjN(Zj) =∑j
k=0 ∑
1≤i1<⋅⋅⋅<ik≤j
F(zi1). . . F(zik)Ej−k(Zj/{i1,⋯,ik}), j = 0, . . . , N.
i.e. FjN is the operator of integral kernel given by (20).
16
Theorem 2.4, Theorem 2.1 and Corollary 2.2 in [26] state the following facts, among
17
others.
18
The kinetic errors Ej, j = 1, . . . , N, satisfy the system of equations
1
∂tEj = (Kj + 1
NTj)Ej +DjEj
+ Dj1Ej+1 +Dj−1Ej−1 +Dj−2Ej−2, (21)
where the operators Dj, D1j, Dj−1, Dj−2, j = 0, . . . , N, are defined at the beginning of the
2
Section 4, formulas (42)-(45).
3
We note that the operators Djα, α = 1,−1,−2 map functions of j +α variables into
4
functions of j variables.
5
Theorem 2.2 (out of Theorem 2.2. and Corollary 2.3 in [26]).
6
Let Ej(0) satisfy for some C0 > 1
7
(22) ∥Ej(0)∥1 ≤ C0j(√jN)j, j ≥ 1.
Then, for all t> 0 and all j = 1, . . . , N, one has
8
(23) ∥Ej(t)∥1 ≤ (C2eC1t∥Vh̵∥L∞)j(√jN)j, j ≥ 1.
for some C1 >0, C2 ≥ 1 explicit (see Theorem 2.2 in [26]),
9
Let us suppose moreover that the initial data for (1) is FN(0) =F(0)⊗N and F(t) is
10
the solution of (15) with initila data F(0). Then
11
(24) ∥FjN(t) −F(t)⊗j∥1 ≤ D2e
D1t∥V∥L∞
̵h j2 N,
where D2 = sup{B2,(eC0)2}, B1 = sup{B1,2C1}, B1, B2 being taken in Theorem 2.2 in
12
[26] at the value B0 = 0).
13
3. Asymptotic expansion and main result
14
15
Two questions arise naturally:
16
(1) are the estimates (23) sharp?
17
(2) Could (24) be improved with a r.h.s. of any order we wish?
18
Of course, defining FjN,n(t), n = 1, . . . , j, by its integral kernel FjN,n(Zj) =
1
∑j
k=j−n ∑
1≤i1<⋅⋅⋅<ik≤j
F(zi1). . . F(zik)Ej−k(Zj/{i1,⋯,ik}), we get by (20), (23) and (24) that,
2
for any n ≤ j, ∥FjN(t) −FjN,n∥ = O(N−(n+1)/2). However one cannot go further in
3
the approximation that is, in any case useless without the knowledge of the true
4
Ejs.
5
As we will see later on, one of our main results states that, not only estimates (23)
6
are true, but Ej(t) ∶=Nj/2Ej(t) has a full asymptotic expansion in positive powers of
7
(N1)12
8
More precisely we will show that, under the hypothesis (22) on the initial data, and
9
for all time t and all j = 1, . . . , N, there exist sequences (Ej`(t))`∈N such that
10
(25) Ej(t) ∼∑∞
`=0
Ej`(t)N−`/2
(in the sense that for all k ∈N,∥Ej(t) − ∑k
`=0Ej`(t)N−`/2∥1 = o(N−k/2)).
11
In fact part of our results will deal with coefficients Ej`(t) which will happen to have
12
a (bounded) dependence2 on N. To avoid any ambiguity with respect to this fact, we
13
precise the meaning of ∼ in (25) we consider in this paper:
14
E ∼ ∑∞
k=0ENkN−k2
⇕
∀n≥ 0,∃Nn ∈N, En, Cn > 0 such that
∀N ≥ Nn, ∣ENn∣ ≤ En and ∥E − ∑n
k=0ENkN−k2∥1 ≤CnN−k+12 .
Of course, whatever is the dependence on N of the coefficients Ek, the important point
15
is to construct an approximation of Ej(t) valid up to any order in N−12.
16
The coefficients Ej` can be determined as solutions of a partial differential equa-
17
tion which can be solved recursively. More than that, Ej`(t) turn out to be explicitly
18
computed in terms of a perturbative expansion, after the knowledge of the lineariza-
19
tion of the mean-field equation (15) around the solution of (15) with initial condition
20
F(0) = (FN(0))1 which will be discussed in detail later on.
21
2Let us remark that this situation is standard in perturbation theory, e.g. in KAM theory where the well known Arnold cut-off introduces such a dependence in the perturbation parameter.
The starting point of our analysis is the evolution equation for Ej(t), obtained by
1
the substitution Ej = N−j/2Ej in (21):
2
(26) ∂tEj = HjEj +N−12∆+jEj+1 +N−12∆−jEj−1+∆=jEj−2 where
3
(27)
⎧⎪⎪⎪⎪
⎪⎪⎪⎨⎪⎪⎪
⎪⎪⎪⎪⎩
Hj = Kj +TNj +Dj(t)
∆+j = D1j
∆−j = N D−1j
∆=j = N D−2j
the Dj′s being given by formulas (42)-(45) below. It follows that Hj,∆+j,∆−j,∆=j act on
4
functions of j, j+1, j−1, j−2 particles, namely L⊗j,L⊗j+1,L⊗j−1,L⊗j−2.
5
Inserting the expansion (25) into (26) we find for (Ejk(t))j=1,...,N,k=0,... the following
6
sequence of equations
7
(28) ∂tEjk = HjEjk+∆=jEj−2k +∆+jEj+1k−1+∆−jEj−1k−1
with the convention,
8
(29) E0k(t) =δk,0, E−1k (t) = E−2k (t) = Ej−1(t) =0 and the ones inherited from (46).
9
(28) can be solved recursively. Indeed we realize that
10
(30) ∂tEj0 = HjEj0+∆=jEj−20
can be solved by iteration in j (note that E10(t) = 0). Thus knowing Ej0, we can also
11
solve
12
(31) ∂tEj1 =HjEj1+∆=jEj−21 +∆+jEj+10 +∆−jEj−10 . by iteration in j and so on.
13
However we will see below that the computation ofEjk(t) depends actually only onEjk′′
14
k′ ≤ k, j′ ≤j+k through a number of operations depending only onj and k independent
15
of N.
16
We now introduce the two-parameter semigroup defined by
17
∂tUj(t, s) = Hj(t)Uj(t, s). (32)
Uj(s, s) = I.
The existence of Uj(t) is guaranteed by the classical theory of perturbation of semi-
1
group, Kj generating an isometric semigroup and TNj and Dj(t) being bounded. More-
2
over, let us define U(t, s) as the linearisation of the Hartree flow around F(t), namely
3
∂tU(t, s) = (K1+∆1))U(t, s), ∆1 ∶=Q(⋅, F(t)) +Q(F(t),⋅) (33)
U(s, s) = I.
We will see in Section 4.3 that Uj(t, s), when acting on symmetric states, is a per-
4
turbation of U(t, s)⊗j, and can be explicitly computed out of U(t, s) by a convergent,
5
entire, expansion in jN2∥V̵h∥. In particular, we’ll see that expansions of Uj(t, s) up to any
6
power of N1 can be explicitly obtained under the only knowledge of the linearisation of
7
the Hartree flow around F(t).
8
Using of this semigroup Uj(t, s) leads immediately to solving (28) by the family of
9
relations:
10
(34)
⎧⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎨⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎩
Ejk(t) = Uj(t,0)Ejk(0)
+ ∫s=0t Uj(t, s)(∆=jEj−2k (s) +∆+jEj+1k−1(s) +∆−jEj−1k−1(s))ds, E0k(t) = δk,0,
∆−1 (E00) ∶= −Q(F, F),
∆=2(E00) ∶= T1,2(F ⊗F) −Q(F, F) ⊗F −F ⊗Q(F, F), E−1k (t) = E−2k (t) = Ej−1(t) = 0 by convention.
We are now in position of stating the main results of the present paper.
11
Theorem 3.1. Consider for j = 0, . . . , N, k = 0, . . . , t ≥ 0 the system of recursive
12
relations (34). Then, for all t, s∈ R, the knowledge of Uj(t, s) (see Remark 3.6 below)
13
makes true the following
14
(i) Ejk(t) is explicitly determined by Ejk′′(0), j′≤ j+k, k′≤ k
15
(ii) Ejk(t) =0 if Ejk(0) =0, for all j, k, j+k odd
16
(iii) Let Ej(t) be the solution of (28) with the condition ∥Ej(0)∥ ≤ (Aj2)j/2 for some
17
A > 1. Let us take moreover Ejk(0) = δk,0Ej(0) (concerning this hypothesis, see
18
Remark 3.6 below). Then the following estimate holds true
19
(35) ∥Ej(t) − ∑2n
k=0
N−k/2Ejk(t)∥1 ≤L2n(t)N−n−12(L′2n(t)j2)j/2,
where Lk(t), L′k(t) are defined in (57) below and satisfy, as k,∣t∣ → ∞,
20
(36) logLk(t) = 3k2 (logk+ ∣t∣∥V̵h∥∞) +O(k+∣t∣∥Vh̵∥∞) and logL′k(t) = O(k+∣t∣∥Vh̵∥∞).
The proof of the theorem is given in Sections 4.1 and 4.2.
1
Note that the estimate (36) gives that Len(t) ∼ (2n)3n as n→ ∞ so that, as expected
2
in perturbation theory, the bound (35) does not provide convergence of the series
3
∑∞ k=0
N−k/2Ejk(t).
4
Let us set, for j =1, . . . , N, n= 0, . . ., Ejk(0) =δk,0Ej(0) and
5
(37) Ejn(t) = ∑2n
k=0
N−j+2kEjk(t)
and FjN,n(t) the operator of integral kernel FjN,n(t)(Zj) defined by
6
(38) FjN,n(t)(Zj) = ∑j
k=0 ∑
1≤i1<⋅⋅⋅<ik≤j
F(t)(zi1). . . F(t)(zik)Ej−kn (Zj/{i1,⋯,ik}), (that is (20) truncated at order n).
7
Theorem 3.2. Let FN(t) the solution of the quantum N body system (1) with initial
8
datum FN(0) = F⊗N, F ∈ L(L2(Rd)), F ≥ 0,TrF = 1, and F(t) the solution of the
9
Hartree equation (15) with initial datum F.
10
Then, for all n≥0 and N ≥4(e√
L′2n(t)j)2,
11
∥FjN(t) −FjN,n(t)∥1 ≤ N−n−12 2L2n(t)e
√ L′2n(t)j
√
N .
Moreover the expansion of FjN,n(t) contains only integer powers of N1.
12
Remark 3.3. The condition of factorization of the initial condition FN(0) = F⊗N,
13
equivalent to Ej(0) = δj,0, is not necessary. It can be mildly modified by taking any
14
Ej(0) satisfying (22) and the associated sequence Ej(0). We leave to the interested
15
reader the elaboration of the precise corresponding statements out of Theorem 3.1.
16
Proof. The proof is similar to the one of Corollary 2.2 in [26].
17
The fact that Ejn(t), and therefore FjN,n(t) contains only integer powers of N1 comes
18
from the fact that the factorization of FN(0) implies that Ejk(0) = δk,0δj,0 and therefore
19
Ejk(0) = Ejk(t) =0 for j+k odd.
20
Moreover
1
∥FjN(t) −FjN,n(t))∥1
≤ ∑j
k=0
( j
j −k)∥Ek−Ekn∥1 ≤ N−n−12
j
∑
k=1
(j
k)L2n(t) (L′2n(t)k2
N )
k/2
≤ N−n−12L2n(t)∑j
k=1
j(j−1). . .(j−k+1)⎛
⎝
√L′2n(t)
√N
⎞
⎠
k
kk k!
≤ N−n−12L2n(t)∑j
k=1
⎛
⎝ je√
L′2n(t)
√N
⎞
⎠
k
≤ N−n−12 2L2n(t)e
√ L′2n(t)j
√N
for N ≥ 4(e√
L′2n(t)j)2 (we used that E0(t) =E0n(t) =1 and kk!k ≤ √e2πkk ).
2
Let us remark that, under the hypothesis of Theorem 3.2, (37) gives that Ejn(t) =
3
O(N−2) for j > 2, E0n(t) = 1, E1n(t) = N−1E11(t) +O(N−2) and E2n(t) = N−1E20(t) +
4
O(N−2).
5
Therefore, keeping in FjN,1(t), given by (38), only the terms k = j −1, j −2, and
6
defining G−11 (t) = E11(t), G−12 (t) = E20(t) and G−1j (t), j > 2, by its integral kernel
7
G−1j (t)(Zj) = ∑
1≤i1<⋅⋅⋅<ij−2≤j
F(t)(zi1). . . F(t)(zij−2)E20(Zj/{i1,⋯,ij−2})
+ ∑
1≤i1<⋅⋅⋅<ij−1≤j
F(t)(zi1). . . F(t)(zij−1)E11(Zj/{i1,⋯,ij−1}), we get, by Theorem 3.2, that
8
FjN(t) −F(t)⊗j = 1
NG−1j (t) +O(N−3/2). For j =1,2, G−1j (t) ≠0 by Lemma 4.6 below.
9
For j > 2, let us pick-up a neighbourhood Ω⊂ R×R2d such that the integral kernel
∣F(t)(z)∣ ≥D >0. We get that, ∀(t, z) ∈Ω, j >2,
G−1j (t)(z, . . . , z) =F(t)(z)j−2((j−2j )E20(t)(z, z) + (j−1j )F(t)(z)E11(t)(z)),
so that G−1j (t)(z, . . . , z) = 0 would imply that F(t)(z)E11(t)(z) = −(j −2)E20(t)(z, z), incompatible with (28). Therefore, for all j =1, . . .,
∥FjN(t) −F(t)⊗j∥1 = 1
N∥G−1j (t)∥1+O(N−3/2) ≥ CN−1, C > 0, for N large enough, and we get the following by-product.
10
Corollary 3.4. The rate of convergence to the mean-field limit in N1 is optimal.
11
As we mentioned already, Uj(t, s) is given by a convergent perturbative expansion
1
out of U(t, s)⊗j where U(t, s) is the flow generated by the linearization of the Hartree
2
equation around its solution F(t).
3
More precisely, let ∆̃j = N1 Tj +Dj −∆j and, for n ∈ N, let us define the truncated
4
Dyson expansion of Uj(t, s) as
5
Ujn(t, s) = (39)
2n+1
∑
k=0 ∫stdt1...∫st2ndt2n+1U(t, t1)⊗j∆̃j(t1)U(t1, t2)⊗j∆̃j(t2). . . U(t2n, t2n+1)⊗j. For α ∈ (0,12), n ∈ N, let us define FjN,n,α(t) as the operator of integral kernel
6
FjN,n,α(t)(Zj) given by
7
(40) FjN,n,α(t)(Zj) = ∑j
k=j−[n+1/2 α ]
1≤i1<⋅⋅⋅<i∑ k≤j
F(t)(zi1). . . F(t)(zik)Ej−kn,n(Zj/{i1,⋯,ik}),
with the convention Ej−k = 0 for j −k < 0, and Ejn,n(t) ∶= N−2j
∑2n k=0
N−k2Ejk,n(t) where
8
Ejk,n(t) are the explicit solutions of the recurrence relations
9
(41) ⎧⎪⎪⎪⎪
⎨⎪⎪⎪⎪⎩
Ejk,n(t) = Ujn(t,0)Ejk,n(0)
+ ∫s=0t Ujn(t, s)(∆=jEj−2k,n(s) +∆+jEj+1k−1,n(s) +∆−jEj−1k−1,n(s))ds, Ejk(0) = δk,0Ej(0)
with the same conventions as in (34) (Ujn(t, s) is defined in (39)).
10
Obviously the solution of (41) satisfies the items (i) − (ii) of Theorem 3.1 and the
11
statements of Proposition 4.1.
12
Theorem 3.5. Let α ∈ (0,12) and C > 0. Then, under the same hypothesis as in
13
Theorem 3.2, one has, for any n∈N, t∈ R and j ≤ CN12−α,
14
∥FjN(t) −FjN,n,α(t)∥1 ≤ Mn,α,C,tN−n−12 for all N > Nn,α,C.t (Mn,α,C,t and Nn,α,C.t are given in (85)).
15
Note that the expansion ofFjN,n,α(t) contains again only integer powers of N1 and, by
16
the construction of Ujn and Proposition 4.1, its explicit computation involves a finite
17
number of operations depending only onj andn(and not inN) and the only knowledge
18
of F(t) and the solution of the Hartree equation linearized around it.
19
The proof of the theorem is given in Section 4.3.
20
Remark 3.6. [Nature of the expansion in N1 ] In the asymptotic expansion Ej(t) ∼
1
∑∞ k=[(j+1)/2]
cjk(t)N−k the coefficients cjk(t), such as each coefficient Ejk(t), depend on N
2
as well: first by the dependence in N of ∆+j = (1−Nj )Cj+1 and also by the dependence
3
in N of Uj(t, s) defined by (32). Moreover, since the condition ∥Ej(0)∥1 ≤ (Aj2)j/2
4
in Theorem 3.1 is a condition only on the size, all the results of this paper hold true
5
under any dependence of Ej(0), that is of FN(0), on N. In particular, this allows to
6
reincorporate in Ej(0) all the terms Ejk(0)N−1/2, k = 1. . ., as done in the second item
7
of Theorem 3.1.
8
4. Proofs of Theorems 3.1 and 3.5
9
Let us first recall from [26] the expression of the ingredients present in equation (21):
10
For any operator G ∈ L⊗n, n = 1, . . . , N , G(Zn) denotes its integral kernel and, for
11
any function F(Zn), n= 1, . . . , N,⋀F(Zn) is defined as the operator on L⊗n of integral
12
kernel F(Zn). Moreover J ∶= {1, . . . , j}.
13
Dj ∶ L⊗j →L⊗j Ej ↦ N −j
N ∑
i∈J
Ci,j+1(⋀F(zi)Ej(Zj+1/{i})+⋀F(zj+1)Ej(Zj)) (42)
− 1
N ∑
i≠l∈J
Ci,j+1F(zl)Ej(Zj+1/{l})
⋀
Dj1 ∶ L⊗(j+1) →L⊗j Ej+1 ↦ N −j
N Cj+1Ej+1 (43)
Dj−1 ∶ L⊗(j−1) →L⊗j Ej−1 ↦ 1
N ∑
i,r∈J
Ti,rF(zi)Ej−1(Zj/{i})
⋀
− j N ∑
i∈J
Q(F, F)(zi)Ej−1(Zj/{i})
⋀
(44)
− 1
N ∑
i≠l∈J
Ci,j+1F(zl)F(zj+1)Ej−1(Zj/{l})
⋀
− 1
N ∑
i≠l∈J
Ci,j+1⋀F(zl)F(zi)Ej−1(Zj+1/{i,l})
